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Young tableau
In mathematics, a Young tableau (/tæˈbloʊ, ˈtæbloʊ/; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900.[1][2] They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley.
Definitions
Note: this article uses the English convention for displaying Young diagrams and tableaux.
Diagrams
A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing order. Listing the number of boxes in each row gives a partition λ of a non-negative integer n, the total number of boxes of the diagram. The Young diagram is said to be of shape λ, and it carries the same information as that partition. Containment of one Young diagram in another defines a partial ordering on the set of all partitions, which is in fact a lattice structure, known as Young's lattice. Listing the number of boxes of a Young diagram in each column gives another partition, the conjugate or transpose partition of λ; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.
There is almost universal agreement that in labeling boxes of Young diagrams by pairs of integers, the first index selects the row of the diagram, and the second index selects the box within the row. Nevertheless, two distinct conventions exist to display these diagrams, and consequently tableaux: the first places each row below the previous one, the second stacks each row on top of the previous one. Since the former convention is mainly used by Anglophones while the latter is often preferred by Francophones, it is customary to refer to these conventions respectively as the English notation and the French notation; for instance, in his book on symmetric functions, Macdonald advises readers preferring the French convention to "read this book upside down in a mirror" (Macdonald 1979, p. 2). This nomenclature probably started out as jocular. The English notation corresponds to the one universally used for matrices, while the French notation is closer to the convention of Cartesian coordinates; however, French notation differs from that convention by placing the vertical coordinate first. The figure on the right shows, using the English notation, the Young diagram corresponding to the partition (5, 4, 1) of the number 10. The conjugate partition, measuring the column lengths, is (3, 2, 2, 2, 1).
Arm and leg length
In many applications, for example when defining Jack functions, it is convenient to define the arm length aλ(s) of a box s as the number of boxes to the right of s in the diagram λ in English notation. Similarly, the leg length lλ(s) is the number of boxes below s. The hook length of a box s is the number of boxes to the right of s or below s in English notation, including the box s itself; in other words, the hook length is aλ(s) + lλ(s) + 1.
Tableaux
A Young tableau is obtained by filling in the boxes of the Young diagram with symbols taken from some alphabet, which is usually required to be a totally ordered set. Originally that alphabet was a set of indexed variables x1, x2, x3..., but now one usually uses a set of numbers for brevity. In their original application to representations of the symmetric group, Young tableaux have n distinct entries, arbitrarily assigned to boxes of the diagram. A tableau is called standard if the entries in each row and each column are increasing. The number of distinct standard Young tableaux on n entries is given by the involution numbers
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence A000085 in the OEIS).
In other applications, it is natural to allow the same number to appear more than once (or not at all) in a tableau. A tableau is called semistandard, or column strict, if the entries weakly increase along each row and strictly increase down each column. Recording the number of times each number appears in a tableau gives a sequence known as the weight of the tableau. Thus the standard Young tableaux are precisely the semistandard tableaux of weight (1,1,...,1), which requires every integer up to n to occur exactly once.
In a standard Young tableau, the integer $k$ is a descent if $k+1$ appears in a row strictly below $k$. The sum of the descents is called the major index of the tableau.[3]
Variations
There are several variations of this definition: for example, in a row-strict tableau the entries strictly increase along the rows and weakly increase down the columns. Also, tableaux with decreasing entries have been considered, notably, in the theory of plane partitions. There are also generalizations such as domino tableaux or ribbon tableaux, in which several boxes may be grouped together before assigning entries to them.
Skew tableaux
A skew shape is a pair of partitions (λ, μ) such that the Young diagram of λ contains the Young diagram of μ; it is denoted by λ/μ. If λ = (λ1, λ2, ...) and μ = (μ1, μ2, ...), then the containment of diagrams means that μi ≤ λi for all i. The skew diagram of a skew shape λ/μ is the set-theoretic difference of the Young diagrams of λ and μ: the set of squares that belong to the diagram of λ but not to that of μ. A skew tableau of shape λ/μ is obtained by filling the squares of the corresponding skew diagram; such a tableau is semistandard if entries increase weakly along each row, and increase strictly down each column, and it is standard if moreover all numbers from 1 to the number of squares of the skew diagram occur exactly once. While the map from partitions to their Young diagrams is injective, this is not the case for the map from skew shapes to skew diagrams;[4] therefore the shape of a skew diagram cannot always be determined from the set of filled squares only. Although many properties of skew tableaux only depend on the filled squares, some operations defined on them do require explicit knowledge of λ and μ, so it is important that skew tableaux do record this information: two distinct skew tableaux may differ only in their shape, while they occupy the same set of squares, each filled with the same entries.[5] Young tableaux can be identified with skew tableaux in which μ is the empty partition (0) (the unique partition of 0).
Any skew semistandard tableau T of shape λ/μ with positive integer entries gives rise to a sequence of partitions (or Young diagrams), by starting with μ, and taking for the partition i places further in the sequence the one whose diagram is obtained from that of μ by adding all the boxes that contain a value ≤ i in T; this partition eventually becomes equal to λ. Any pair of successive shapes in such a sequence is a skew shape whose diagram contains at most one box in each column; such shapes are called horizontal strips. This sequence of partitions completely determines T, and it is in fact possible to define (skew) semistandard tableaux as such sequences, as is done by Macdonald (Macdonald 1979, p. 4). This definition incorporates the partitions λ and μ in the data comprising the skew tableau.
Overview of applications
Young tableaux have numerous applications in combinatorics, representation theory, and algebraic geometry. Various ways of counting Young tableaux have been explored and lead to the definition of and identities for Schur functions.
Many combinatorial algorithms on tableaux are known, including Schützenberger's jeu de taquin and the Robinson–Schensted–Knuth correspondence. Lascoux and Schützenberger studied an associative product on the set of all semistandard Young tableaux, giving it the structure called the plactic monoid (French: le monoïde plaxique).
In representation theory, standard Young tableaux of size k describe bases in irreducible representations of the symmetric group on k letters. The standard monomial basis in a finite-dimensional irreducible representation of the general linear group GLn are parametrized by the set of semistandard Young tableaux of a fixed shape over the alphabet {1, 2, ..., n}. This has important consequences for invariant theory, starting from the work of Hodge on the homogeneous coordinate ring of the Grassmannian and further explored by Gian-Carlo Rota with collaborators, de Concini and Procesi, and Eisenbud. The Littlewood–Richardson rule describing (among other things) the decomposition of tensor products of irreducible representations of GLn into irreducible components is formulated in terms of certain skew semistandard tableaux.
Applications to algebraic geometry center around Schubert calculus on Grassmannians and flag varieties. Certain important cohomology classes can be represented by Schubert polynomials and described in terms of Young tableaux.
Applications in representation theory
See also: Representation theory of the symmetric group
Young diagrams are in one-to-one correspondence with irreducible representations of the symmetric group over the complex numbers. They provide a convenient way of specifying the Young symmetrizers from which the irreducible representations are built. Many facts about a representation can be deduced from the corresponding diagram. Below, we describe two examples: determining the dimension of a representation and restricted representations. In both cases, we will see that some properties of a representation can be determined by using just its diagram. Young tableaux are involved in the use of the symmetric group in quantum chemistry studies of atoms, molecules and solids.[6][7]
Young diagrams also parametrize the irreducible polynomial representations of the general linear group GLn (when they have at most n nonempty rows), or the irreducible representations of the special linear group SLn (when they have at most n − 1 nonempty rows), or the irreducible complex representations of the special unitary group SUn (again when they have at most n − 1 nonempty rows). In these cases semistandard tableaux with entries up to n play a central role, rather than standard tableaux; in particular it is the number of those tableaux that determines the dimension of the representation.
Dimension of a representation
Main article: Hook length formula
Hook-lengths of the boxes for the partition 10 = 5 + 4 + 1
The dimension of the irreducible representation πλ of the symmetric group Sn corresponding to a partition λ of n is equal to the number of different standard Young tableaux that can be obtained from the diagram of the representation. This number can be calculated by the hook length formula.
A hook length hook(x) of a box x in Young diagram Y(λ) of shape λ is the number of boxes that are in the same row to the right of it plus those boxes in the same column below it, plus one (for the box itself). By the hook-length formula, the dimension of an irreducible representation is n! divided by the product of the hook lengths of all boxes in the diagram of the representation:
$\dim \pi _{\lambda }={\frac {n!}{\prod _{x\in Y(\lambda )}\operatorname {hook} (x)}}.$
The figure on the right shows hook-lengths for all boxes in the diagram of the partition 10 = 5 + 4 + 1. Thus
$\dim \pi _{\lambda }={\frac {10!}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1}}=288.$
Similarly, the dimension of the irreducible representation W(λ) of GLr corresponding to the partition λ of n (with at most r parts) is the number of semistandard Young tableaux of shape λ (containing only the entries from 1 to r), which is given by the hook-length formula:
$\dim W(\lambda )=\prod _{(i,j)\in Y(\lambda )}{\frac {r+j-i}{\operatorname {hook} (i,j)}},$
where the index i gives the row and j the column of a box.[8] For instance, for the partition (5,4,1) we get as dimension of the corresponding irreducible representation of GL7 (traversing the boxes by rows):
$\dim W(\lambda )={\frac {7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 6\cdot 7\cdot 8\cdot 9\cdot 5}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1}}=66528.$
Restricted representations
A representation of the symmetric group on n elements, Sn is also a representation of the symmetric group on n − 1 elements, Sn−1. However, an irreducible representation of Sn may not be irreducible for Sn−1. Instead, it may be a direct sum of several representations that are irreducible for Sn−1. These representations are then called the factors of the restricted representation (see also induced representation).
The question of determining this decomposition of the restricted representation of a given irreducible representation of Sn, corresponding to a partition λ of n, is answered as follows. One forms the set of all Young diagrams that can be obtained from the diagram of shape λ by removing just one box (which must be at the end both of its row and of its column); the restricted representation then decomposes as a direct sum of the irreducible representations of Sn−1 corresponding to those diagrams, each occurring exactly once in the sum.
See also
• Robinson–Schensted correspondence
• Schur–Weyl duality
Notes
1. Knuth, Donald E. (1973), The Art of Computer Programming, Vol. III: Sorting and Searching (2nd ed.), Addison-Wesley, p. 48, Such arrangements were introduced by Alfred Young in 1900.
2. Young, A. (1900), "On quantitative substitutional analysis", Proceedings of the London Mathematical Society, Series 1, 33 (1): 97–145, doi:10.1112/plms/s1-33.1.97. See in particular p. 133.
3. Stembridge, John (1989-12-01). "On the eigenvalues of representations of reflection groups and wreath products". Pacific Journal of Mathematics. Mathematical Sciences Publishers. 140 (2): 353–396. doi:10.2140/pjm.1989.140.353. ISSN 0030-8730.
4. For instance the skew diagram consisting of a single square at position (2,4) can be obtained by removing the diagram of μ = (5,3,2,1) from the one of λ = (5,4,2,1), but also in (infinitely) many other ways. In general any skew diagram whose set of non-empty rows (or of non-empty columns) is not contiguous or does not contain the first row (respectively column) will be associated to more than one skew shape.
5. A somewhat similar situation arises for matrices: the 3-by-0 matrix A must be distinguished from the 0-by-3 matrix B, since AB is a 3-by-3 (zero) matrix while BA is the 0-by-0 matrix, but both A and B have the same (empty) set of entries; for skew tableaux however such distinction is necessary even in cases where the set of entries is not empty.
6. Philip R. Bunker and Per Jensen (1998) Molecular Symmetry and Spectroscopy, 2nd ed. NRC Research Press,Ottawa pp.198-202.ISBN 9780660196282
7. R.Pauncz (1995) The Symmetric Group in Quantum Chemistry, CRC Press, Boca Raton, Florida
8. Predrag Cvitanović (2008). Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press., eq. 9.28 and appendix B.4
References
• William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997, ISBN 0-521-56724-6.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. Lecture 4
• Howard Georgi, Lie Algebras in Particle Physics, 2nd Edition - Westview
• Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp. ISBN 0-19-853530-9 MR553598
• Laurent Manivel. Symmetric Functions, Schubert Polynomials, and Degeneracy Loci. American Mathematical Society.
• Jean-Christophe Novelli, Igor Pak, Alexander V. Stoyanovskii, "A direct bijective proof of the Hook-length formula", Discrete Mathematics and Theoretical Computer Science 1 (1997), pp. 53–67.
• Bruce E. Sagan. The Symmetric Group. Springer, 2001, ISBN 0-387-95067-2
• Vinberg, E.B. (2001) [1994], "Young tableau", Encyclopedia of Mathematics, EMS Press
• Yong, Alexander (February 2007). "What is...a Young Tableau?" (PDF). Notices of the American Mathematical Society. 54 (2): 240–241. Retrieved 2008-01-16.
• Predrag Cvitanović, Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press, 2008.
External links
• Eric W. Weisstein. "Ferrers Diagram". From MathWorld—A Wolfram Web Resource.
• Eric W. Weisstein. "Young Tableau." From MathWorld—A Wolfram Web Resource.
• Semistandard tableaux entry in the FindStat database
• Standard tableaux entry in the FindStat database
| Wikipedia |
Young's convolution inequality
In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions,[1] named after William Henry Young.
Statement
Euclidean space
In real analysis, the following result is called Young's convolution inequality:[2]
Suppose $f$ is in the Lebesgue space $L^{p}(\mathbb {R} ^{d})$ and $g$ is in $L^{q}(\mathbb {R} ^{d})$ and
${\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1$
with $1\leq p,q,r\leq \infty .$ Then
$\|f*g\|_{r}\leq \|f\|_{p}\|g\|_{q}.$
Here the star denotes convolution, $L^{p}$ is Lebesgue space, and
$\|f\|_{p}={\Bigl (}\int _{\mathbb {R} ^{d}}|f(x)|^{p}\,dx{\Bigr )}^{1/p}$
denotes the usual $L^{p}$ norm.
Equivalently, if $p,q,r\geq 1$ and $ {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2$ then
$\left|\int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}}f(x)g(x-y)h(y)\,\mathrm {d} x\,\mathrm {d} y\right|\leq \left(\int _{\mathbb {R} ^{d}}\vert f\vert ^{p}\right)^{\frac {1}{p}}\left(\int _{\mathbb {R} ^{d}}\vert g\vert ^{q}\right)^{\frac {1}{q}}\left(\int _{\mathbb {R} ^{d}}\vert h\vert ^{r}\right)^{\frac {1}{r}}$
Generalizations
Young's convolution inequality has a natural generalization in which we replace $\mathbb {R} ^{d}$ by a unimodular group $G.$ If we let $\mu $ be a bi-invariant Haar measure on $G$ and we let $f,g:G\to \mathbb {R} $ or $\mathbb {C} $ be integrable functions, then we define $f*g$ by
$f*g(x)=\int _{G}f(y)g(y^{-1}x)\,\mathrm {d} \mu (y).$
Then in this case, Young's inequality states that for $f\in L^{p}(G,\mu )$ and $g\in L^{q}(G,\mu )$ and $p,q,r\in [1,\infty ]$ such that
${\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1$
we have a bound
$\lVert f*g\rVert _{r}\leq \lVert f\rVert _{p}\lVert g\rVert _{q}.$
Equivalently, if $p,q,r\geq 1$ and $ {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2$ then
$\left|\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right|\leq \left(\int _{G}\vert f\vert ^{p}\right)^{\frac {1}{p}}\left(\int _{G}\vert g\vert ^{q}\right)^{\frac {1}{q}}\left(\int _{G}\vert h\vert ^{r}\right)^{\frac {1}{r}}.$
Since $\mathbb {R} ^{d}$ is in fact a locally compact abelian group (and therefore unimodular) with the Lebesgue measure the desired Haar measure, this is in fact a generalization.
This generalization may be refined. Let $G$ and $\mu $ be as before and assume $1<p,q,r<\infty $ satisfy $ {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}+1.$ Then there exists a constant $C$ such that for any $f\in L^{p}(G,\mu )$ and any measurable function $g$ on $G$ that belongs to the weak $L^{q}$ space $L^{q,w}(G,\mu ),$ which by definition means that the following supremum
$\|g\|_{q,w}^{q}~:=~\sup _{t>0}\,t^{q}\mu (|g|>t)$
is finite, we have $f*g\in L^{r}(G,\mu )$ and[3]
$\|f*g\|_{r}~\leq ~C\,\|f\|_{p}\,\|g\|_{q,w}.$
Applications
An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the $L^{2}$ norm (that is, the Weierstrass transform does not enlarge the $L^{2}$ norm).
Proof
Proof by Hölder's inequality
Young's inequality has an elementary proof with the non-optimal constant 1.[4]
We assume that the functions $f,g,h:G\to \mathbb {R} $ are nonnegative and integrable, where $G$ is a unimodular group endowed with a bi-invariant Haar measure $\mu .$ We use the fact that $\mu (S)=\mu (S^{-1})$ for any measurable $S\subseteq G.$ Since $ p(2-{\tfrac {1}{q}}-{\tfrac {1}{r}})=q(2-{\tfrac {1}{p}}-{\tfrac {1}{r}})=r(2-{\tfrac {1}{p}}-{\tfrac {1}{q}})=1$
${\begin{aligned}&\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\\={}&\int _{G}\int _{G}\left(f(x)^{p}g(y^{-1}x)^{q}\right)^{1-{\frac {1}{r}}}\left(f(x)^{p}h(y)^{r}\right)^{1-{\frac {1}{q}}}\left(g(y^{-1}x)^{q}h(y)^{r}\right)^{1-{\frac {1}{p}}}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\end{aligned}}$
By the Hölder inequality for three functions we deduce that
${\begin{aligned}&\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\\&\leq \left(\int _{G}\int _{G}f(x)^{p}g(y^{-1}x)^{q}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{r}}}\left(\int _{G}\int _{G}f(x)^{p}h(y)^{r}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{q}}}\left(\int _{G}\int _{G}g(y^{-1}x)^{q}h(y)^{r}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{p}}}.\end{aligned}}$
The conclusion follows then by left-invariance of the Haar measure, the fact that integrals are preserved by inversion of the domain, and by Fubini's theorem.
Proof by interpolation
Young's inequality can also be proved by interpolation; see the article on Riesz–Thorin interpolation for a proof.
Sharp constant
In case $p,q>1,$ Young's inequality can be strengthened to a sharp form, via
$\|f*g\|_{r}\leq c_{p,q}\|f\|_{p}\|g\|_{q}.$
where the constant $c_{p,q}<1.$[5][6][7] When this optimal constant is achieved, the function $f$ and $g$ are multidimensional Gaussian functions.
See also
• Minkowski inequality – Inequality that established Lp spaces are normed vector spaces
Notes
1. Young, W. H. (1912), "On the multiplication of successions of Fourier constants", Proceedings of the Royal Society A, 87 (596): 331–339, doi:10.1098/rspa.1912.0086, JFM 44.0298.02, JSTOR 93120
2. Bogachev, Vladimir I. (2007), Measure Theory, vol. I, Berlin, Heidelberg, New York: Springer-Verlag, ISBN 978-3-540-34513-8, MR 2267655, Zbl 1120.28001, Theorem 3.9.4
3. Bahouri, Chemin & Danchin 2011, pp. 5–6.
4. Lieb, Elliott H.; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics (2nd ed.). Providence, R.I.: American Mathematical Society. p. 100. ISBN 978-0-8218-2783-3. OCLC 45799429.
5. Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (1): 159–182. doi:10.2307/1970980. JSTOR 1970980.
6. Brascamp, Herm Jan; Lieb, Elliott H (1976-05-01). "Best constants in Young's inequality, its converse, and its generalization to more than three functions". Advances in Mathematics. 20 (2): 151–173. doi:10.1016/0001-8708(76)90184-5.
7. Fournier, John J. F. (1977), "Sharpness in Young's inequality for convolution", Pacific Journal of Mathematics, 72 (2): 383–397, doi:10.2140/pjm.1977.72.383, MR 0461034, Zbl 0357.43002
References
• Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.
External links
• Young's Inequality for Convolutions at ProofWiki
Lp spaces
Basic concepts
• Banach & Hilbert spaces
• Lp spaces
• Measure
• Lebesgue
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L2 spaces
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$L^{\infty }$ spaces
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Maps
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Inequalities
• Babenko–Beckner
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Results
• Marcinkiewicz interpolation theorem
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For Lebesgue measure
• Isoperimetric inequality
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Applications & related
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• Sobolev space
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• Von Neumann
| Wikipedia |
Young's inequality for integral operators
In mathematical analysis, the Young's inequality for integral operators, is a bound on the $L^{p}\to L^{q}$ operator norm of an integral operator in terms of $L^{r}$ norms of the kernel itself.
Statement
Assume that $X$ and $Y$ are measurable spaces, $K:X\times Y\to \mathbb {R} $ is measurable and $q,p,r\geq 1$ are such that ${\frac {1}{q}}={\frac {1}{p}}+{\frac {1}{r}}-1$. If
$\int _{Y}|K(x,y)|^{r}\,\mathrm {d} y\leq C^{r}$ for all $x\in X$
and
$\int _{X}|K(x,y)|^{r}\,\mathrm {d} x\leq C^{r}$ for all $y\in Y$
then [1]
$\int _{X}\left|\int _{Y}K(x,y)f(y)\,\mathrm {d} y\right|^{q}\,\mathrm {d} x\leq C^{q}\left(\int _{Y}|f(y)|^{p}\,\mathrm {d} y\right)^{\frac {q}{p}}.$
Particular cases
Convolution kernel
If $X=Y=\mathbb {R} ^{d}$ and $K(x,y)=h(x-y)$, then the inequality becomes Young's convolution inequality.
See also
Young's inequality for products
Notes
1. Theorem 0.3.1 in: C. D. Sogge, Fourier integral in classical analysis, Cambridge University Press, 1993. ISBN 0-521-43464-5
| Wikipedia |
Young's inequality for products
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers.[1] The inequality is named after William Henry Young and should not be confused with Young's convolution inequality.
Young's inequality for products can be used to prove Hölder's inequality. It is also widely used to estimate the norm of nonlinear terms in PDE theory, since it allows one to estimate a product of two terms by a sum of the same terms raised to a power and scaled.
Standard version for conjugate Hölder exponents
The standard form of the inequality is the following:
Theorem — If $a\geq 0$ and $b\geq 0$ are nonnegative real numbers and if $p>1$ and $q>1$ are real numbers such that ${\frac {1}{p}}+{\frac {1}{q}}=1,$ then
$ab~\leq ~{\frac {a^{p}}{p}}+{\frac {b^{q}}{q}}.$
Equality holds if and only if $a^{p}=b^{q}.$
It can be used to prove Hölder's inequality.
Proof[2]
Since ${\tfrac {1}{p}}+{\tfrac {1}{q}}=1,$ $p-1={\tfrac {1}{q-1}}.$ A graph $y=x^{p-1}$ on the $xy$-plane is thus also a graph $x=y^{q-1}.$ From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines $x=0,x=a,y=0,y=b,$ and the fact that $y$ is always increasing for increasing $x$ and vice versa, we can see that $\int _{0}^{a}x^{p-1}\mathrm {d} x$ upper bounds the area of the rectangle below the curve (with equality when $b\geq a^{p-1}$) and $\int _{0}^{b}y^{q-1}\mathrm {d} y$ upper bounds the area of the rectangle above the curve (with equality when $b\leq a^{p-1}$). Thus, $\int _{0}^{a}x^{p-1}\mathrm {d} x+\int _{0}^{b}y^{q-1}\mathrm {d} y\geq ab,$ with equality when $b=a^{p-1}$ (or equivalently, $a^{p}=b^{q}$). Young's inequality follows from evaluating the integrals. (See below for a generalization.)
This form of Young's inequality can also be proved via Jensen's inequality.
Proof[3]
The claim is certainly true if $a=0$ or $b=0$ so henceforth assume that $a>0$ and $b>0.$ Put $t=1/p$ and $(1-t)=1/q.$ Because the logarithm function is concave,
$\ln \left(ta^{p}+(1-t)b^{q}\right)~\geq ~t\ln \left(a^{p}\right)+(1-t)\ln \left(b^{q}\right)=\ln(a)+\ln(b)=\ln(ab)$
with the equality holding if and only if $a^{p}=b^{q}.$ Young's inequality follows by exponentiating.
Young's inequality may equivalently be written as
$a^{\alpha }b^{\beta }\leq \alpha a+\beta b,\qquad \,0\leq \alpha ,\beta \leq 1,\quad \ \alpha +\beta =1.$
Where this is just the concavity of the logarithm function. Equality holds if and only if $a=b$ or $\{\alpha ,\beta \}=\{0,1\}.$
Generalizations
Theorem[4] — Suppose $a>0$ and $b>0.$ If $1<p<\infty $ and $q$ are such that ${\tfrac {1}{p}}+{\tfrac {1}{q}}=1$ then
$ab~=~\min _{0<t<\infty }\left({\frac {t^{p}a^{p}}{p}}+{\frac {t^{-q}b^{q}}{q}}\right).$
Using $t:=1$ and replacing $a$ with $a^{1/p}$ and $b$ with $b^{1/q}$ results in the inequality:
$a^{1/p}\,b^{1/q}~\leq ~{\frac {a}{p}}+{\frac {b}{q}},$
which is useful for proving Hölder's inequality.
Proof[4]
Define a real-valued function $f$ on the positive real numbers by
$f(t)~=~{\frac {t^{p}a^{p}}{p}}+{\frac {t^{-q}b^{q}}{q}}$
for every $t>0$ and then calculate its minimum.
Theorem — If $0\leq p_{i}\leq 1$ with $\sum _{i}p_{i}=1$ then
$\prod _{i}{a_{i}}^{p_{i}}~\leq ~\sum _{i}p_{i}a_{i}.$
Equality holds if and only if all the $a_{i}$s with non-zero $p_{i}$s are equal.
Elementary case
An elementary case of Young's inequality is the inequality with exponent $2,$
$ab\leq {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}},$
which also gives rise to the so-called Young's inequality with $\varepsilon $ (valid for every $\varepsilon >0$), sometimes called the Peter–Paul inequality. [5] This name refers to the fact that tighter control of the second term is achieved at the cost of losing some control of the first term – one must "rob Peter to pay Paul"
$ab~\leq ~{\frac {a^{2}}{2\varepsilon }}+{\frac {\varepsilon b^{2}}{2}}.$
Proof: Young's inequality with exponent $2$ is the special case $p=q=2.$ However, it has a more elementary proof.
Start by observing that the square of every real number is zero or positive. Therefore, for every pair of real numbers $a$ and $b$ we can write:
$0\leq (a-b)^{2}$
Work out the square of the right hand side:
$0\leq a^{2}-2ab+b^{2}$
Add $2ab$ to both sides:
$2ab\leq a^{2}+b^{2}$
Divide both sides by 2 and we have Young's inequality with exponent $2:$
$ab\leq {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}}$
Young's inequality with $\varepsilon $ follows by substituting $a'$ and $b'$ as below into Young's inequality with exponent $2:$
$a'=a/{\sqrt {\varepsilon }},\;b'={\sqrt {\varepsilon }}b.$
Matricial generalization
T. Ando proved a generalization of Young's inequality for complex matrices ordered by Loewner ordering.[6] It states that for any pair $A,B$ of complex matrices of order $n$ there exists a unitary matrix $U$ such that
$U^{*}|AB^{*}|U\preceq {\tfrac {1}{p}}|A|^{p}+{\tfrac {1}{q}}|B|^{q},$
where ${}^{*}$ denotes the conjugate transpose of the matrix and $|A|={\sqrt {A^{*}A}}.$
Standard version for increasing functions
For the standard version[7][8] of the inequality, let $f$ denote a real-valued, continuous and strictly increasing function on $[0,c]$ with $c>0$ and $f(0)=0.$ Let $f^{-1}$ denote the inverse function of $f.$ Then, for all $a\in [0,c]$ and $b\in [0,f(c)],$
$ab~\leq ~\int _{0}^{a}f(x)\,dx+\int _{0}^{b}f^{-1}(x)\,dx$
with equality if and only if $b=f(a).$
With $f(x)=x^{p-1}$ and $f^{-1}(y)=y^{q-1},$ this reduces to standard version for conjugate Hölder exponents.
For details and generalizations we refer to the paper of Mitroi & Niculescu.[9]
Generalization using Fenchel–Legendre transforms
By denoting the convex conjugate of a real function $f$ by $g,$ we obtain
$ab~\leq ~f(a)+g(b).$
This follows immediately from the definition of the convex conjugate. For a convex function $f$ this also follows from the Legendre transformation.
More generally, if $f$ is defined on a real vector space $X$ and its convex conjugate is denoted by $f^{\star }$ (and is defined on the dual space $X^{\star }$), then
$\langle u,v\rangle \leq f^{\star }(u)+f(v).$
where $\langle \cdot ,\cdot \rangle :X^{\star }\times X\to \mathbb {R} $ is the dual pairing.
Examples
The convex conjugate of $f(a)=a^{p}/p$ is $g(b)=b^{q}/q$ with $q$ such that ${\tfrac {1}{p}}+{\tfrac {1}{q}}=1,$ and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case.
The Legendre transform of $f(a)=e^{a}-1$ is $g(b)=1-b+b\ln b$, hence $ab\leq e^{a}-b+b\ln b$ for all non-negative $a$ and $b.$ This estimate is useful in large deviations theory under exponential moment conditions, because $b\ln b$ appears in the definition of relative entropy, which is the rate function in Sanov's theorem.
See also
• Convex conjugate – the ("dual") lower-semicontinuous convex function resulting from the Legendre–Fenchel transformation of a "primal" functionPages displaying wikidata descriptions as a fallback
• Integral of inverse functions – Mathematical theorem, used in calculus
• Legendre transformation – Mathematical transformation
• Young's convolution inequality
Notes
1. Young, W. H. (1912), "On classes of summable functions and their Fourier series", Proceedings of the Royal Society A, 87 (594): 225–229, Bibcode:1912RSPSA..87..225Y, doi:10.1098/rspa.1912.0076, JFM 43.1114.12, JSTOR 93236
2. Pearse, Erin. "Math 209D - Real Analysis Summer Preparatory Seminar Lecture Notes" (PDF). Retrieved 17 September 2022.
3. Bahouri, Chemin & Danchin 2011.
4. Jarchow 1981, pp. 47–55.
5. Tisdell, Chris (2013), The Peter Paul Inequality, YouTube video on Dr Chris Tisdell's YouTube channel,
6. T. Ando (1995). "Matrix Young Inequalities". In Huijsmans, C. B.; Kaashoek, M. A.; Luxemburg, W. A. J.; et al. (eds.). Operator Theory in Function Spaces and Banach Lattices. Springer. pp. 33–38. ISBN 978-3-0348-9076-2.
7. Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) [1934], Inequalities, Cambridge Mathematical Library (2nd ed.), Cambridge: Cambridge University Press, ISBN 0-521-05206-8, MR 0046395, Zbl 0047.05302, Chapter 4.8
8. Henstock, Ralph (1988), Lectures on the Theory of Integration, Series in Real Analysis Volume I, Singapore, New Jersey: World Scientific, ISBN 9971-5-0450-2, MR 0963249, Zbl 0668.28001, Theorem 2.9
9. Mitroi, F. C., & Niculescu, C. P. (2011). An extension of Young's inequality. In Abstract and Applied Analysis (Vol. 2011). Hindawi.
References
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.
External links
• Young's Inequality at PlanetMath
• Weisstein, Eric W. "Young's Inequality". MathWorld.
Measure theory
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• Absolute continuity of measures
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Other results
• Disintegration theorem
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• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
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• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
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• Spectral theory
| Wikipedia |
Young's inequality
Young's inequality may refer to:
• Young's inequality for products, bounding the product of two quantities
• Young's convolution inequality, bounding the convolution product of two functions
• Young's inequality for integral operators
See also
• William Henry Young, English mathematician (1863–1942)
• Hausdorff–Young inequality, bounding the coefficient of Fourier series
| Wikipedia |
Young's lattice
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed the representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial order on them played a key, even decisive, role. Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of Stanley (1988). It is also closely connected with the crystal bases for affine Lie algebras.
Definition
Young's lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams).
Significance
The traditional application of Young's lattice is to the description of the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. The equivalence classes of irreducible representations may be parametrized by partitions or Young diagrams, the restriction from Sn +1 to Sn is multiplicity-free, and the representation of Sn with partition p is contained in the representation of Sn +1 with partition q if and only if q covers p in Young's lattice. Iterating this procedure, one arrives at Young's semicanonical basis in the irreducible representation of Sn with partition p, which is indexed by the standard Young tableaux of shape p.
Properties
• The poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions of n have rank n. This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank.
• The poset Y is a lattice. The meet and join of two partitions are given by the intersection and the union of the corresponding Young diagrams. Because it is a lattice in which the meet and join operations are represented by intersections and unions, it is a distributive lattice.
• If a partition p covers k elements of Young's lattice for some k then it is covered by k + 1 elements. All partitions covered by p can be found by removing one of the "corners" of its Young diagram (boxes at the end both of their row and of their column). All partitions covering p can be found by adding one of the "dual corners" to its Young diagram (boxes outside the diagram that are the first such box both in their row and in their column). There is always a dual corner in the first row, and for each other dual corner there is a corner in the previous row, whence the stated property.
• If distinct partitions p and q both cover k elements of Y then k is 0 or 1, and p and q are covered by k elements. In plain language: two partitions can have at most one (third) partition covered by both (their respective diagrams then each have one box not belonging to the other), in which case there is also one (fourth) partition covering them both (whose diagram is the union of their diagrams).
• Saturated chains between ∅ and p are in a natural bijection with the standard Young tableaux of shape p: the diagrams in the chain add the boxes of the diagram of the standard Young tableau in the order of their numbering. More generally, saturated chains between q and p are in a natural bijection with the skew standard tableaux of skew shape p/q.
• The Möbius function of Young's lattice takes values 0, ±1. It is given by the formula
$\mu (q,p)={\begin{cases}(-1)^{|p|-|q|}&{\text{if the skew diagram }}p/q{\text{ is a disconnected union of squares}}\\&{\text{(no common edges);}}\\[10pt]0&{\text{otherwise}}.\end{cases}}$
Dihedral symmetry
The portion of Young's lattice lying below 1 + 1 + 1 + 1, 2 + 2 + 2, 3 + 3, and 4
Conventional diagram with partitions of the same rank at the same height
Diagram showing dihedral symmetry
Conventionally, Young's lattice is depicted in a Hasse diagram with all elements of the same rank shown at the same height above the bottom. Suter (2002) has shown that a different way of depicting some subsets of Young's lattice shows some unexpected symmetries.
The partition
$n+\cdots +3+2+1$
of the nth triangular number has a Ferrers diagram that looks like a staircase. The largest elements whose Ferrers diagrams are rectangular that lie under the staircase are these:
${\begin{array}{c}\underbrace {1+\cdots \cdots \cdots +1} _{n{\text{ terms}}}\\\underbrace {2+\cdots \cdots +2} _{n-1{\text{ terms}}}\\\underbrace {3+\cdots +3} _{n-2{\text{ terms}}}\\\vdots \\\underbrace {{}\quad n\quad {}} _{1{\text{ term}}}\end{array}}$
Partitions of this form are the only ones that have only one element immediately below them in Young's lattice. Suter showed that the set of all elements less than or equal to these particular partitions has not only the bilateral symmetry that one expects of Young's lattice, but also rotational symmetry: the rotation group of order n + 1 acts on this poset. Since this set has both bilateral symmetry and rotational symmetry, it must have dihedral symmetry: the (n + 1)st dihedral group acts faithfully on this set. The size of this set is 2n.
For example, when n = 4, then the maximal element under the "staircase" that have rectangular Ferrers diagrams are
1 + 1 + 1 + 1
2 + 2 + 2
3 + 3
4
The subset of Young's lattice lying below these partitions has both bilateral symmetry and 5-fold rotational symmetry. Hence the dihedral group D5 acts faithfully on this subset of Young's lattice.
See also
• Young–Fibonacci lattice
• Bratteli diagram
References
• Misra, Kailash C.; Miwa, Tetsuji (1990). "Crystal base for the basic representation of $U_{q}({\widehat {\mathfrak {sl}}}(n))$". Communications in Mathematical Physics. 134 (1): 79–88. Bibcode:1990CMaPh.134...79M. doi:10.1007/BF02102090. S2CID 120298905.
• Sagan, Bruce (2000). The Symmetric Group. Berlin: Springer. ISBN 0-387-95067-2.
• Stanley, Richard P. (1988). "Differential posets". Journal of the American Mathematical Society. 1 (4): 919–961. doi:10.2307/1990995. JSTOR 1990995.
• Suter, Ruedi (2002). "Young's lattice and dihedral symmetries". European Journal of Combinatorics. 23 (2): 233–238. doi:10.1006/eujc.2001.0541.
Order theory
• Topics
• Glossary
• Category
Key concepts
• Binary relation
• Boolean algebra
• Cyclic order
• Lattice
• Partial order
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Results
• Boolean prime ideal theorem
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• Knaster–Tarski theorem
• Kruskal's tree theorem
• Laver's theorem
• Mirsky's theorem
• Szpilrajn extension theorem
• Zorn's lemma
Properties & Types (list)
• Antisymmetric
• Asymmetric
• Boolean algebra
• topics
• Completeness
• Connected
• Covering
• Dense
• Directed
• (Partial) Equivalence
• Foundational
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Constructions
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Topology & Orders
• Alexandrov topology & Specialization preorder
• Ordered topological vector space
• Normal cone
• Order topology
• Order topology
• Topological vector lattice
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Related
• Antichain
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• Young's lattice
| Wikipedia |
Young measure
In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limit. Young measures have applications in the calculus of variations, especially models from material science, and the study of nonlinear partial differential equations, as well as in various optimization (or optimal control problems). They are named after Laurence Chisholm Young who invented them, already in 1937 in one dimension (curves) and later in higher dimensions in 1942.[1]
Definition
We let $\{f_{k}\}_{k=1}^{\infty }$ be a bounded sequence in $L^{p}(U,\mathbb {R} ^{m})$, where $U$ denotes an open bounded subset of $\mathbb {R} ^{n}$. Then there exists a subsequence $\{f_{k_{j}}\}_{j=1}^{\infty }\subset \{f_{k}\}_{k=1}^{\infty }$ and for almost every $x\in U$ a Borel probability measure $\nu _{x}$ on $\mathbb {R} ^{m}$ such that for each $F\in C(\mathbb {R} ^{m})$ we have $F\circ f_{k_{j}}(x){\rightharpoonup }\int _{\mathbb {R} ^{m}}F(y)d\nu _{x}(y)$ weakly in $L^{p}(U)$ if the weak limit exists (or weak star in $L^{\infty }(U)$ in case of $p=+\infty $). The measures $\nu _{x}$ are called the Young measures generated by the sequence $\{f_{k_{j}}\}_{j=1}^{\infty }$. More generally, for any Caratheodory function $f(x,A):U\times R^{m}\to R,$ the limit of $\int _{U}f(x,f_{j}(x))\ dx,$ if it exists, will be given by $\int _{U}\int _{\mathbb {R} ^{m}}f(x,A)\ d\nu _{x}(A)\ dx$.[2]
Young's original idea in the case $f\in C_{0}(U\times \mathbb {R} ^{m})$ was to for each integer $j\geq 1$ consider the uniform measure, let's say $\Gamma _{j}:=(id,f_{j})_{\sharp }L^{d}\llcorner U,$ concentrated on graph of the function $f_{j}.$ (Here, $L^{d}\llcorner U$is the restriction of the Lebesgue measure on $U.$) By taking the weak-star limit of these measures as elements of $C_{0}(U\times \mathbb {R} ^{m})^{\star },$ we have $\langle \Gamma _{j},f\rangle =\int _{U}f(x,f_{j}(x))\ dx\to \langle \Gamma ,f\rangle ,$ where $\Gamma $ is the mentioned weak limit. After a disintegration of the measure $\Gamma $ on the product space $\Omega \times \mathbb {R} ^{m},$ we get the parameterized measure $\nu _{x}$.
Example
For every asymptotically minimizing sequence $u_{n}$ of $I(u)=\int _{0}^{1}(u'(x)^{2}-1)^{2}+u'(x)^{2}dx$ subject to $u(0)=u(1)=0$ (that is, the sequence satisfies $\lim _{n\to +\infty }I(u_{n})=\inf _{u\in C^{1}([0,1])}I(u)$), and perhaps after passing to a subsequence, the sequence of derivatives $u'_{n}$ generates Young measures of the form $\nu _{x}=\alpha (x)\delta _{-1}+(1-\alpha )(x)\delta _{1}$ with $\alpha \colon [0,1]\to [0,1]$ measurable. This captures the essential features of all minimizing sequences to this problem, namely, their derivatives $u'_{k}(x)$ will tend to concentrate along the minima $\{-1,1\}$ of the integrand $(u'(x)^{2}-1)^{2}+u'(x)^{2}$.
References
1. Young, L. C. (1942). "Generalized Surfaces in the Calculus of Variations". Annals of Mathematics. 43 (1): 84–103. doi:10.2307/1968882. ISSN 0003-486X. JSTOR 1968882.
2. Pedregal, Pablo (1997). Parametrized measures and variational principles. Basel: Birkhäuser Verlag. ISBN 978-3-0348-8886-8. OCLC 812613013.
• Ball, J. M. (1989). "A version of the fundamental theorem for Young measures". In Rascle, M.; Serre, D.; Slemrod, M. (eds.). PDEs and Continuum Models of Phase Transition. Lecture Notes in Physics. Vol. 344. Berlin: Springer. pp. 207–215.
• C.Castaing, P.Raynaud de Fitte, M.Valadier (2004). Young measures on topological spaces. Dordrecht: Kluwer.{{cite book}}: CS1 maint: multiple names: authors list (link)
• L.C. Evans (1990). Weak convergence methods for nonlinear partial differential equations. Regional conference series in mathematics. American Mathematical Society.
• S. Müller (1999). Variational models for microstructure and phase transitions. Lecture Notes in Mathematics. Springer.
• P. Pedregal (1997). Parametrized Measures and Variational Principles. Basel: Birkhäuser. ISBN 978-3-0348-9815-7.
• T. Roubíček (2020). Relaxation in Optimization Theory and Variational Calculus (2nd ed.). Berlin: W. de Gruyter. ISBN 978-3-11-014542-7.
• Valadier, M. (1990). "Young measures". Methods of Nonconvex Analysis. Lecture Notes in Mathematics. Vol. 1446. Berlin: Springer. pp. 152–188.
• Young, L. C. (1937), "Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations", Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, XXX (7–9): 211–234, JFM 63.1064.01, Zbl 0019.21901, memoir presented by Stanisław Saks at the session of 16 December 1937 of the Warsaw Society of Sciences and Letters. The free PDF copy is made available by the RCIN –Digital Repository of the Scientifics Institutes.
• Young, L. C. (1969), Lectures on the Calculus of Variations and Optimal Control, Philadelphia–London–Toronto: W. B. Saunders, pp. xi+331, ISBN 9780721696409, MR 0259704, Zbl 0177.37801.
External links
• "Young measure", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• 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
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
| Wikipedia |
Young symmetrizer
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space $V^{\otimes n}$ obtained from the action of $S_{n}$ on $V^{\otimes n}$ by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young.
Definition
Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, and consider the action of $S_{n}$ given by permuting the boxes of $\lambda $. Define two permutation subgroups $P_{\lambda }$ and $Q_{\lambda }$ of Sn as follows:
$P_{\lambda }=\{g\in S_{n}:g{\text{ preserves each row of }}\lambda \}$
and
$Q_{\lambda }=\{g\in S_{n}:g{\text{ preserves each column of }}\lambda \}.$
Corresponding to these two subgroups, define two vectors in the group algebra $\mathbb {C} S_{n}$ as
$a_{\lambda }=\sum _{g\in P_{\lambda }}e_{g}$
and
$b_{\lambda }=\sum _{g\in Q_{\lambda }}\operatorname {sgn}(g)e_{g}$
where $e_{g}$ is the unit vector corresponding to g, and $\operatorname {sgn}(g)$ is the sign of the permutation. The product
$c_{\lambda }:=a_{\lambda }b_{\lambda }=\sum _{g\in P_{\lambda },h\in Q_{\lambda }}\operatorname {sgn}(h)e_{gh}$
is the Young symmetrizer corresponding to the Young tableau λ. Each Young symmetrizer corresponds to an irreducible representation of the symmetric group, and every irreducible representation can be obtained from a corresponding Young symmetrizer. (If we replace the complex numbers by more general fields the corresponding representations will not be irreducible in general.)
Construction
Let V be any vector space over the complex numbers. Consider then the tensor product vector space $V^{\otimes n}=V\otimes V\otimes \cdots \otimes V$ (n times). Let Sn act on this tensor product space by permuting the indices. One then has a natural group algebra representation $\mathbb {C} S_{n}\to \operatorname {End} (V^{\otimes n})$ on $V^{\otimes n}$ (i.e. $V^{\otimes n}$ is a right $\mathbb {C} S_{n}$ module).
Given a partition λ of n, so that $n=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{j}$, then the image of $a_{\lambda }$ is
$\operatorname {Im} (a_{\lambda }):=V^{\otimes n}a_{\lambda }\cong \operatorname {Sym} ^{\lambda _{1}}V\otimes \operatorname {Sym} ^{\lambda _{2}}V\otimes \cdots \otimes \operatorname {Sym} ^{\lambda _{j}}V.$
For instance, if $n=4$, and $\lambda =(2,2)$, with the canonical Young tableau $\{\{1,2\},\{3,4\}\}$. Then the corresponding $a_{\lambda }$ is given by
$a_{\lambda }=e_{\text{id}}+e_{(1,2)}+e_{(3,4)}+e_{(1,2)(3,4)}.$
For any product vector $v_{1,2,3,4}:=v_{1}\otimes v_{2}\otimes v_{3}\otimes v_{4}$ of $V^{\otimes 4}$ we then have
$v_{1,2,3,4}a_{\lambda }=v_{1,2,3,4}+v_{2,1,3,4}+v_{1,2,4,3}+v_{2,1,4,3}=(v_{1}\otimes v_{2}+v_{2}\otimes v_{1})\otimes (v_{3}\otimes v_{4}+v_{4}\otimes v_{3}).$
Thus the set of all $a_{\lambda }v_{1,2,3,4}$ clearly spans $\operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V$ and since the $v_{1,2,3,4}$ span $V^{\otimes 4}$ we obtain $V^{\otimes 4}a_{\lambda }=\operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V$, where we wrote informally $V^{\otimes 4}a_{\lambda }\equiv \operatorname {Im} (a_{\lambda })$.
Notice also how this construction can be reduced to the construction for $n=2$. Let $\mathbb {1} \in \operatorname {End} (V^{\otimes 2})$ be the identity operator and $S\in \operatorname {End} (V^{\otimes 2})$ the swap operator defined by $S(v\otimes w)=w\otimes v$, thus $\mathbb {1} =e_{\text{id}}$ and $S=e_{(1,2)}$. We have that
$e_{\text{id}}+e_{(1,2)}=\mathbb {1} +S$
maps into $\operatorname {Sym} ^{2}V$, more precisely
${\frac {1}{2}}(\mathbb {1} +S)$
is the projector onto $\operatorname {Sym} ^{2}V$. Then
${\frac {1}{4}}a_{\lambda }={\frac {1}{4}}(e_{\text{id}}+e_{(1,2)}+e_{(3,4)}+e_{(1,2)(3,4)})={\frac {1}{4}}(\mathbb {1} \otimes \mathbb {1} +S\otimes \mathbb {1} +\mathbb {1} \otimes S+S\otimes S)={\frac {1}{2}}(\mathbb {1} +S)\otimes {\frac {1}{2}}(\mathbb {1} +S)$
which is the projector onto $\operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V$.
The image of $b_{\lambda }$ is
$\operatorname {Im} (b_{\lambda })\cong \bigwedge ^{\mu _{1}}V\otimes \bigwedge ^{\mu _{2}}V\otimes \cdots \otimes \bigwedge ^{\mu _{k}}V$
where μ is the conjugate partition to λ. Here, $\operatorname {Sym} ^{i}V$ and $\bigwedge ^{j}V$ are the symmetric and alternating tensor product spaces.
The image $\mathbb {C} S_{n}c_{\lambda }$ of $c_{\lambda }=a_{\lambda }\cdot b_{\lambda }$ in $\mathbb {C} S_{n}$ is an irreducible representation of Sn, called a Specht module. We write
$\operatorname {Im} (c_{\lambda })=V_{\lambda }$
for the irreducible representation.
Some scalar multiple of $c_{\lambda }$ is idempotent,[1] that is $c_{\lambda }^{2}=\alpha _{\lambda }c_{\lambda }$ for some rational number $\alpha _{\lambda }\in \mathbb {Q} .$ Specifically, one finds $\alpha _{\lambda }=n!/\dim V_{\lambda }$. In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra $\mathbb {Q} S_{n}$.
Consider, for example, S3 and the partition (2,1). Then one has
$c_{(2,1)}=e_{123}+e_{213}-e_{321}-e_{312}.$
If V is a complex vector space, then the images of $c_{\lambda }$ on spaces $V^{\otimes d}$ provides essentially all the finite-dimensional irreducible representations of GL(V).
See also
• Representation theory of the symmetric group
Notes
1. See (Fulton & Harris 1991, Theorem 4.3, p. 46)
References
• William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
• Lecture 4 of Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Bruce E. Sagan. The Symmetric Group. Springer, 2001.
| Wikipedia |
YoungJu Choie
YoungJu Choie (Korean: 최영주, born June 15, 1959)[1] is a South Korean mathematician who works as a professor of mathematics at the Pohang University of Science and Technology (POSTECH). Her research interests include number theory and modular forms.[2]
YoungJu Choie
NationalitySouth Korean
Alma materTemple University
Ewha Womans University
Scientific career
FieldsMathematics
InstitutionsPohang University of Science and Technology
Doctoral advisorMarvin Knopp
Education and career
Choie graduated from Ewha Womans University in 1982,[2] and earned a doctorate in 1986 from Temple University under the supervision of Marvin Knopp.[3] After temporary positions at Ohio State University and the University of Maryland, she became an assistant professor at University of Colorado in 1989, and moved to POSTECH as a full professor in 1990. Choie became a Fellow of the American Mathematical Society in 2013.[4]
Mathematical work
Choie works on various aspects of Jacobi forms.[5][6] Together with Winfried Kohnen, she has proved upper bounds on the first sign change of Fourier coefficients of cusp forms,[7] generalizing the work of Siegel.
Selected works
• Y. Choie, Y. Park and D. Zagier, Periods of modular forms on $\Gamma _{0}(N)$ and Products of Jacobi Theta functions, Journal of the European Mathematical Society, Vol. 21, Issue 5, pp 1379–1410 (2019)
• R. Bruggeman, Y. Choie and N. Diamantis, Holomorphic automorphic forms and cohomology, Memoirs of the American Mathematical Society, 253 (2018), no. 1212, vii+167 pp. ISBN 978-1-4704-2855-6
• D. Bump and Y. Choie, “Schubert Eisenstein series”, American Journal of Mathematics Vol 136, No 6, Dec 2014, 1581-1608.
• Y. Choie and W. Kohnen, “The first sign change of Fourier coefficients of cusp forms”, American Journal of Mathematics 131 (2009), no. 2, 517-543.
• Y. Choie and D. Zagier, “Rational period functions for PSL(2, Z)”, Contemporary Mathematics, A tribute to Emil Grosswald: Number Theory and Related Analysis, 143, 89-108, 1993.
• (Book) Y. Choie and MH. Lee, "Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms”, 318 pages, Springer Monographs in Mathematics on Springer Verlag 2019 (eBook ISBN 978-3-030-29123-5)
• (Book) M. Shi, Y. Choie, A. Sharma and P. Sole, “Codes and Modular forms”, World Scientific, ISBN 978-981-121-291-8 (hardcover) | December 2019 Pages: 232
Service
Choie has been an editor of International Journal of Number Theory since 2004. In 2010–2011 she was editor-in-chief of the Bulletin of the Korean Mathematical Society.[2] She became a president of the society of Korean Women in Mathematical Sciences in 2017. Selective Public Service:
• 2020-2022: NCsoft, Non-executive director
• 2019-2020: The Presidential Advisory Council on Science and Technology, Deliberative Member, Republic of Korea.
• 2019-2020: University Councilor, Representative of Faculty at POSTECH, Pohang, Korea
• 2019-2020: Academic vice president of Korean Mathematical Society, Korea
• 2018-2021: Non-standing member of Board of Trustees, UNIST(Ulsan Institute of Science and Technology), Korea
• 2018-2020: Non-standing member of Board of Trustees, NRF(National Research Foundation), Korea
• 2018–2020, 2009-2013: Director of Pohang Mathematical Institute, POSTECH, Korea
• 2017.12-2019.11: Member, General review committee for academic research supporting the field of education, Ministry of Education
• 2017-2018: Science and Technology Innovation Board, Ministry of Science and Technology Information and communication
• 2018-2019/2020-2021 : Vice President of KOFWST/Auditor, Korea
• 2018-2019: Chief of section committee of group activities of KOFWST, Korea
• 2018: University councilor of POSTECH , Representative of Faculty
• 2017: KWMS (Korea Women in Mathematical Sciences), President, 20170101-20171231
• 2016-2017: KOFWST(Korea Federation of Women’s Sciences and Technology Academics) Member of the board of trustee
• 2016-2018: IMU Committee for Women in Mathematics (CWM) ambassadors
• 2015-2016: CRB(Chief of Research Board), National Research Foundation
• 2009-2015: Organizing Committee, 2014 Seoul ICM, Seoul, Korea
• 2013-2015: Organizing committee, ICWM, 2014, Seoul.
• 2008-2009: ICM-2014 Seoul Bitteing committee.
• 2007-2009: Head of Department of Mathematics, POSTECH, Pohang, Korea.
• 2006:WISE Mentoring Fellow, 2006-
• 2004-2007, 2012: KWMS(Korean Women in Mathematical Sciences), Board of Trustees:
Recognition
Choie has received several awards such as "The best Journal Paper Award (2002)" from the Korean Mathematical Society, "Kwon, Kyungwhan" Chaired Professor (2004) at Pohang University of Science and Technology, "The best woman Scientist of the year" award (2005) from Ministry of Science and Technology, "Amore-Pacific The best Women in Science and Technology" (2007), KOFWST (Korea Federation of Woman's Science and Technology Association) and the "2014 Distinguished research" award from Ministry of Education of Korea.[2] In 2013, Choie became one of the inaugural fellows of the American Mathematical Society.[4] Choie became the first female mathematician as member of the Korean Academy of Science and Technology in 2018.,[1] retrieved 2018-12-22. She was the first female mathematician who received (2018) the academic award of Korean Mathematical Society.
References
1. Member profile, Korean Academy of Science and Technology
2. Faculty profile, POSTECH, retrieved 2015-02-19.
3. Young-Ju Choie at the Mathematics Genealogy Project
4. "List of Fellows of the American Mathematical Society". American Mathematical Society. Retrieved 2019-10-08.
5. Choie, YoungJu; Eholzer, Wolfgang (1998-02-01). "Rankin–Cohen Operators for Jacobi and Siegel Forms". Journal of Number Theory. 68 (2): 160–177. doi:10.1006/jnth.1997.2203. ISSN 0022-314X. S2CID 17316768.
6. Choie, YoungJu (1997-05-01). "Jacobi forms and the heat operator". Mathematische Zeitschrift. 225 (1): 95–101. doi:10.1007/PL00004603. ISSN 1432-1823. S2CID 117410236.
7. Choie, YoungJu; Kohnen, Winfried (2009-03-20). "The first sign change of Fourier coefficients of cusp forms". American Journal of Mathematics. 131 (2): 517–543. doi:10.1353/ajm.0.0050. ISSN 1080-6377. S2CID 14118329.
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| Wikipedia |
Young–Deruyts development
In mathematics, the Young–Deruyts development is a method of writing invariants of an action of a group on an n-dimensional vector space V in terms of invariants depending on at most n–1 vectors (Dieudonné & Carrell 1970, 1971, p.36, 39).
References
• Dieudonné, Jean A.; Carrell, James B. (1970), "Invariant theory, old and new", Advances in Mathematics, 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525
• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102
| Wikipedia |
Young–Fibonacci lattice
In mathematics, the Young–Fibonacci graph and Young–Fibonacci lattice, named after Alfred Young and Leonardo Fibonacci, are two closely related structures involving sequences of the digits 1 and 2. Any digit sequence of this type can be assigned a rank, the sum of its digits: for instance, the rank of 11212 is 1 + 1 + 2 + 1 + 2 = 7. As was already known in ancient India, the number of sequences with a given rank is a Fibonacci number. The Young–Fibonacci lattice is an infinite modular lattice having these digit sequences as its elements, compatible with this rank structure. The Young–Fibonacci graph is the graph of this lattice, and has a vertex for each digit sequence. As the graph of a modular lattice, it is a modular graph.
The Young–Fibonacci graph and the Young–Fibonacci lattice were both initially studied in two papers by Fomin (1988) and Stanley (1988). They are named after the closely related Young's lattice and after the Fibonacci number of their elements at any given rank.
Digit sequences with a given rank
A digit sequence with rank r may be formed either by adding the digit 2 to a sequence with rank r − 2, or by adding the digit 1 to a sequence with rank r − 1. If f is the function that maps r to the number of different digit sequences of that rank, therefore, f satisfies the recurrence relation f (r) = f (r − 2) + f (r − 1) defining the Fibonacci numbers, but with slightly different initial conditions: f (0) = f (1) = 1 (there is one rank-0 string, the empty string, and one rank-1 string, consisting of the single digit 1). These initial conditions cause the sequence of values of f to be shifted by one position from the Fibonacci numbers: f (r) = Fr +1.
In the ancient Indian study of prosody, the Fibonacci numbers were used to count the number of different sequences of short and long syllables with a given total length; if the digit 1 corresponds to a short syllable, and the digit 2 corresponds to a long syllable, the rank of a digit sequence measures the total length of the corresponding sequence of syllables. See the Fibonacci number article for details.
Graphs of digit sequences
The Young–Fibonacci graph is an infinite graph, with a vertex for each string of the digits "1" and "2" (including the empty string). The neighbors of a string s are the strings formed from s by one of the following operations:
1. Insert a "1" into s, prior to the leftmost "1" (or anywhere in s if it does not already contain a "1").
2. Change the leftmost "1" of s into a "2".
3. Remove the leftmost "1" from s.
4. Change a "2" that does not have a "1" to the left of it into a "1".
It is straightforward to verify that each operation can be inverted: operations 1 and 3 are inverse to each other, as are operations 2 and 4. Therefore, the resulting graph may be considered to be undirected. However, it is usually considered to be a directed acyclic graph in which each edge connects from a vertex of lower rank to a vertex of higher rank.
As both Fomin (1988) and Stanley (1988) observe, this graph has the following properties:
• It is connected: any nonempty string may have its rank reduced by some operation, so there is a sequence of operations leading from it to the empty string, reversing which gives a directed path in the graph from the empty string to every other vertex.
• It is compatible with the rank structure: every directed path has length equal to the difference in ranks of its endpoints.
• For every two distinct nodes u and v, the number of common immediate predecessors of u and v equals the number of common immediate successors of u and v; this number is either zero or one.
• The out-degree of every vertex equals one plus its in-degree.
Fomin (1988) calls a graph with these properties a Y-graph; Stanley (1988) calls a graph with a weaker version of these properties (in which the numbers of common predecessors and common successors of any pair of nodes must be equal but may be greater than one) the graph of a differential poset.
Partial order and lattice structure
The transitive closure of the Young–Fibonacci graph is a partial order. As Stanley (1988) shows, any two vertices x and y have a unique greatest common predecessor in this order (their meet) and a unique least common successor (their join); thus, this order is a lattice, called the Young–Fibonacci lattice.
To find the meet of x and y, one may first test whether one of x and y is a predecessor of the other. A string x is a predecessor of another string y in this order exactly when the number of "2" digits remaining in y, after removing the longest common suffix of x and y, is at least as large as the number of all digits remaining in x after removing the common suffix. If x is a predecessor of y according to this test, then their meet is x, and similarly if y is a predecessor of x then their meet is y. In a second case, if neither x nor y is the predecessor of the other, but one or both of them begins with a "1" digit, the meet is unchanged if these initial digits are removed. And finally, if both x and y begin with the digit "2", the meet of x and y may be found by removing this digit from both of them, finding the meet of the resulting suffixes, and adding the "2" back to the start.
A common successor of x and y (though not necessarily the least common successor) may be found by taking a string of "2" digits with length equal to the longer of x and y. The least common successor is then the meet of the finitely many strings that are common successors of x and y and predecessors of this string of "2"s.
As Stanley (1988) further observes, the Young–Fibonacci lattice is modular. Fomin (1988) incorrectly claims that it is distributive; however, the sublattice formed by the strings {21, 22, 121, 211, 221} forms a diamond sublattice, forbidden in distributive lattices.
References
• Fomin, S. V. (1988), "Generalized Robinson–Schensted–Knuth correspondence", Journal of Mathematical Sciences, 41 (2): 979–991, doi:10.1007/BF01247093, S2CID 120902883. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR 155: 156–175, 1986.
• Stanley, Richard P. (1988), "Differential posets", Journal of the American Mathematical Society, 1 (4): 919–961, doi:10.2307/1990995, JSTOR 1990995.
| Wikipedia |
Yozo Matsushima
Yozo Matsushima (松島 与三, Matsushima Yozō, February 11, 1921 – April 9, 1983) was a Japanese mathematician.
Yozo Matsushima
Born(1921-02-11)February 11, 1921
Sakai City, Osaka Prefecture, Japan
DiedApril 9, 1983(1983-04-09) (aged 62)
Osaka, Japan
NationalityJapanese
Alma materOsaka University
Known forresearch in Lie algebras and Lie groups
AwardsAsahi Prize
Scientific career
FieldsMathematics
InstitutionsOsaka University
Nagoya University
University of Notre Dame
Doctoral advisorKenjiro Shoda
Early life
Matsushima was born on February 11, 1921, in Sakai City, Osaka Prefecture, Japan. He studied at Osaka Imperial University (later named Osaka University) and graduated with a Bachelor of Science degree in mathematics in September 1942.[1] At Osaka, he was taught by mathematicians Kenjiro Shoda. After completing his degree, he was appointed as an assistant in the Mathematical Institute of Nagoya Imperial University (later named Nagoya University).[1] These were difficult years for Japanese students and researchers because of World War II.[2]
The first paper published by Matsushima contained a proof that a conjecture of Hans Zassenhaus was false. Zassenhaus had conjectured that every semisimple Lie algebra L over a field of prime characteristic, with [L, L] = L, is the direct sum of simple ideals. Matsushima constructed a counterexample. He then developed a proof that Cartan subalgebras of a complex Lie algebra are conjugate. However, Japanese researchers were out of touch with the research done in the West, and Matsushima was unaware that French mathematician Claude Chevalley had already published a proof. When he obtained details of another paper of Chevalley through a review in Mathematical Reviews, he was able to construct the proofs for himself.[2]
Matsushima published two papers in the 1947 volume of the Proceedings of the Japan Academy (which did not appear until 1950) and three papers in the first volume of Journal of the Mathematical Society of Japan.[2]
Professorship
Matsushima became a full professor at Nagoya University in 1953. Chevalley visited Matsushima in Nagoya in 1953 and invited him to spend the following year in France. He went to France in 1954 and returned to Nagoya in December 1955. He also spent time at the University of Strasbourg. He presented some of his results to Ehresmann's seminar in Strasbourg, extending Cartan's classification of complex irreducible Lie algebras to the case of real Lie algebras.[2]
In spring 1960, Matsushima became a professor of Osaka University as successor to the chair of Shoda.[1] His research took a somewhat different direction and he wrote a series of papers on cohomology of locally symmetric spaces, collaborating with Murakami. He went to the Institute for Advanced Study in September 1962 and returned to Osaka after one year. He jointly began to organize the United States-Japan Seminar in Differential Geometry, which was held in Kyoto in June 1965. After this, he went to France and spent the academic year 1965-66 as visiting professor at the University of Grenoble. He accepted a chair at the University of Notre Dame in Notre Dame, Indiana, in September 1966.[1] He continued to collaborate with Murakami. He introduced Matsushima's formula for the Betti numbers of quotients of symmetric spaces. In 1967, he became an editor of the Journal of Differential Geometry and remained on the editorial board for the rest of his life. After 14 years at Notre Dame, he returned to Japan in 1980. A conference was organized in his honor in May 1980 before he left Notre Dame.[2]
Later life
In February 1981, a volume of papers Manifolds and Lie groups, Papers in honour of Yozo Matsushima was published by his colleagues and former students at Osaka. It also contained some papers presented to the conference held in Notre Dame in the previous May. He died on April 9, 1983, in Osaka, Japan.[2]
Honours
Matsushima received the Asahi Prize for his research on continuous groups in 1962.[2]
References
1. Murakami, Shingo (1984). "Yozô Matsushima: 1921--1983". The Osaka Journal of Mathematics. 21 (1): i–ii. Retrieved 2008-07-09.
2. O'Connor, John J.; Robertson, Edmund F., "Yozo Matsushima", MacTutor History of Mathematics Archive, University of St Andrews
• Kobayashi, Shoshichi (1984). "The mathematical work of Y. Matsushima and its development". The Osaka Journal of Mathematics. 21 (1): iii–xix. Retrieved 2008-07-10.
• Matsushima, Yozô (1992), Murakami, Shingo; Kobayashi, Shoshichi (eds.), Collected papers of Yozô Matsushima, Series in Pure Mathematics, vol. 15, River Edge, NJ: World Scientific Publishing Co. Inc., doi:10.1142/9789814360067, ISBN 9789810208141, MR 1169467
• "List of publications of Yozô Matsushima". The Osaka Journal of Mathematics. 21 (1): xx–xxii. 1984. Retrieved 2008-07-10.
• Yozo Matsushima at the Mathematics Genealogy Project
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| Wikipedia |
Yudell Luke
Yudell Leo Luke (26 June 1918 – 6 May 1983) was an American mathematician who made significant contributions to MRIGlobal, was awarded the N. T. Veatch award for Distinguished Research and Creative Activity in 1975, and appointed as Curator's Professor at the University of Missouri in 1978, a post he held until his death.
Yudell Luke
Born(1918-06-26)26 June 1918
Kansas City, Missouri, United States
Died(1983-05-06)6 May 1983
Moscow, Russia
CitizenshipAmerican
Known forSpecial functions
Hypergeometric functions
AwardsN T Veatch award for Distinguished Research and Creative Activity
Scientific career
FieldsMathematics
InstitutionsUniversity of Missouri
Luke published eight books and nearly 100 papers in a wide variety of mathematical areas, ranging from aeronautics to approximation theory. By his own estimation, Luke reviewed over 1,800 papers and books throughout his career.
Biography
Yudell Luke was born in Kansas City, Missouri, U.S. on 26 June 1918 to Jewish parents.[1] His father, David Luke, was sexton of Congregation Kerem Israel Beth Shalom. The young Luke attended the Kansas City Missouri Junior College, graduating in 1937. He read mathematics at the University of Illinois, receiving a bachelor's degree in 1939, and a Masters the following year. He then taught at the university for two years, but was called up for World War II military service in 1942.
Luke served in the United States Navy until 1946 and was stationed in Hawaii for the duration of the war. After his service, he returned to the university, where he met his future wife LaVerne (LaVerne B. (née Podolsky), 1922–2004) at the University of Illinois. They moved to Kansas City in 1946 and had four daughters, Molly, Janis, Linda, and Debra, and established the Yudell and LaVerne Luke Senior Adult Transportation Fund at the Kansas City Jewish Community Center.[1]
Soon after Luke moved to Kansas City, he was appointed to MRIGlobal (formerly Midwest Research Institute). His first position was as Head of the Mathematical Analysis Section, a position he held until his promotion to Senior Advisor for Mathematics in 1961. Luke also held posts at other universities. In 1955, he became a lecturer at the University of Missouri–Kansas City, and he also taught at the University of Kansas. After the mathematics group of MRIGlobal was disbanded in 1971, Luke was appointed professor at the University of Missouri, and in 1975, received the N T Veatch award for Distinguished Research and Creative Activity. He then became Curator's Professor at Missouri in 1978.
In 1982, an exchange programme between the University of Missouri and the University of Moscow was formed, and the following year, Luke travelled to Moscow to lecture on a series of topics as part of the programme, including special functions, asymptotic analysis and approximation theory. He died while in Russia on 6 May 1983.
Luke had a wide range of interests outside mathematics, including basketball, baseball, bridge, and cribbage. He wrote two books on the probabilities of winning at the latter.[2] He also expressed interest in opera and philosophy, and once gave a series of lectures on the history of philosophy, mainly focusing on Baruch Spinoza's ideas.
Selected bibliography
Papers
• "Rational approximations to the exponential function". Journal of the ACM, 4(1):24–29, January 1957.
Books
• "Integrals of Bessel functions". MacGraw-Hill. 1962
• "The Special Functions and Their Approximations: v. 1 (Mathematics in Science & Engineering)". Academic Press Inc. April 1969. ISBN 978-0-12-459901-7
• "The Special Functions and Their Approximations: v. 2 (Mathematics in Science & Engineering)". Academic Press Inc. 1969.
• "Cumulative Index to Mathematics of Computation 1943–1969". American Mathematical Society. December 1972. ISBN 978-0-8218-4000-9
• "Algorithms for the Computation of Mathematical Functions". Academic Press Inc. 1977. ISBN 978-0-12-459940-6
References
1. Kansas City Star (3 October 2004) Obits. Page B5.
2. O'Connor, 1998.
General
• O'Connor, John J.; Robertson, Edmund F., "Yudell Luke", MacTutor History of Mathematics Archive, University of St Andrews
• Gautschi, Walter; Wimp, Jet (1984). "In memoriam : Yudell L. Luke, 26 June 1918 – 6 May 1983". Math. Comp. 43 (168): 349–352. doi:10.1090/s0025-5718-1984-0758187-6. JSTOR 2008280.
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| Wikipedia |
Shi Yuguang
Shi Yuguang (Chinese: 史宇光; born 1969, Yinxian, Zhejiang) is a Chinese mathematician at Peking University.[1] His areas of research are geometric analysis and differential geometry.[2]
He was awarded the ICTP Ramanujan Prize in 2010, for "outstanding contributions to the geometry of complete (noncompact) Riemannian manifolds, specifically the positivity of quasi-local mass and rigidity of asymptotically hyperbolic manifolds."[3]
He earned his Ph.D. from the Chinese Academy of Sciences in 1996 under the supervision of Ding Weiyue.[4]
Technical contributions
Shi is well-known for his foundational work with Luen-Fai Tam on compact and smooth Riemannian manifolds-with-boundary whose scalar curvature is nonnegative and whose boundary is mean-convex. In particular, if the manifold has a spin structure, and if each connected component of the boundary can be isometrically embedded as a strictly convex hypersurface in Euclidean space, then the average value of the mean curvature of each boundary component is less than or equal to the average value of the mean curvature of the corresponding hypersurface in Euclidean space.
This is particularly simple in three dimensions, where every manifold has a spin structure and a result of Louis Nirenberg shows that any positively-curved Riemannian metric on the two-dimensional sphere can be isometrically embedded in three-dimensional Euclidean space in a geometrically unique way.[5] Hence Shi and Tam's result gives a striking sense in which, given a compact and smooth three-dimensional Riemannian manifold-with-boundary of nonnegative scalar curvature, whose boundary components have positive intrinsic curvature and positive mean curvature, the extrinsic geometry of the boundary components are controlled by their intrinsic geometry. More precisely, the extrinsic geometry is controlled by the extrinsic geometry of the isometric embedding uniquely determined by the intrinsic geometry.
Shi and Tam's proof adopts a method, due to Robert Bartnik, of using parabolic partial differential equations to construct noncompact Riemannian manifolds-with-boundary of nonnegative scalar curvature and prescribed boundary behavior. By combining Bartnik's construction with the given compact manifold-with-boundary, one obtains a complete Riemannian manifold which is non-differentiable along a closed and smooth hypersurface. By using Bartnik's method to relate the geometry near infinity to the geometry of the hypersurface, and by proving a positive energy theorem in which certain singularities are allowed, Shi and Tam's result follows.
From the perspective of research literature in general relativity, Shi and Tam's result is notable in proving, in certain contexts, the nonnegativity of the Brown-York quasilocal energy of J. David Brown and James W. York.[6] The ideas of Shi−Tam and Brown−York have been further developed by Mu-Tao Wang and Shing-Tung Yau, among others.
Major publication
• Yuguang Shi and Luen-Fai Tam. Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differential Geom. 62 (2002), no. 1, 79–125. doi:10.4310/jdg/1090425530
References
1. "News: Ramanujan prize awarded to Yuguang Shi". Archived from the original on 2015-02-13. Retrieved 2015-07-26.
2. http://eng.math.pku.edu.cn/en/view.php?uid=shiyg%5B%5D
3. http://www.ams.org/notices/201108/rtx110801131p.pdf
4. Shi Yuguang at the Mathematics Genealogy Project
5. Louis Nirenberg. The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6 (1953), 337–394.
6. J. David Brown and James W. York, Jr. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D (3) 47 (1993), no. 4, 1407–1419.
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Yujiro Kawamata
Yujiro Kawamata (born 1952) is a Japanese mathematician working in algebraic geometry.
Yujiro Kawamata
Nationality Japanese
Alma materUniversity of Tokyo
Known forKawamata-Viehweg vanishing theorem
Kawamata log terminal (klt) singularities
Scientific career
FieldsMathematics
InstitutionsUniversity of Tokyo
Doctoral advisorShigeru Iitaka
Career
Kawamata completed the master's course at the University of Tokyo in 1977. He was an Assistant at the University of Mannheim from 1977 to 1979 and a Miller Fellow at the University of California, Berkeley from 1981 to 1983. Kawamata is now a professor at the University of Tokyo. He won the Mathematical Society of Japan Autumn award (1988) and the Japan Academy of Sciences award (1990) for his work in algebraic geometry.
Research
Kawamata was involved in the development of the minimal model program in the 1980s. The program aims to show that every algebraic variety is birational to one of an especially simple type: either a minimal model or a Fano fiber space. The Kawamata-Viehweg vanishing theorem, strengthening the Kodaira vanishing theorem, is a method. Building on that, Kawamata proved the basepoint-free theorem. The cone theorem and contraction theorem, central results in the theory, are the result of a joint effort by Kawamata, Kollár, Mori, Reid, and Shokurov.[1]
After Mori proved the existence of minimal models in dimension 3 in 1988, Kawamata and Miyaoka clarified the structure of minimal models by proving the abundance conjecture in dimension 3.[2] Kawamata used analytic methods in Hodge theory to prove the Iitaka conjecture over a base of dimension 1.[3]
More recently, a series of papers by Kawamata related the derived category of coherent sheaves on an algebraic variety to geometric properties in the spirit of minimal model theory.[4]
Notes
1. Y. Kawamata, K. Matsuda, and K. Matsuki. Introduction to the minimal model program. Algebraic Geometry, Sendai 1985. North-Holland (1987), 283-360.
2. Y. Kawamata. Abundance theorem for minimal threefolds. Invent. Math. 108 (1992), 229-246.
3. Y. Kawamata. Kodaira dimension of algebraic fiber spaces over curves. Invent. Math. 66 (1982), 57-71.
4. Y. Kawamata. D-equivalence and K-equivalence. J. Diff. Geom. 61 (2002), 147-171.
References
• Kawamata, Yujiro (1982), "Kodaira dimension of algebraic fiber spaces over curves", Inventiones Mathematicae, 66: 57–71, Bibcode:1982InMat..66...57K, doi:10.1007/BF01404756, MR 0652646, S2CID 123007245
• Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji (1987), "Introduction to the minimal model program", Algebraic Geometry, Sendai 1985, Advanced Studies in Pure Mathematics, vol. 10, North-Holland, pp. 283–360, ISBN 0-444-70313-6, MR 0946243
• Kawamata, Yujiro (1992), "Abundance theorem for minimal threefolds", Inventiones Mathematicae, 108: 229–246, Bibcode:1992InMat.108..229K, doi:10.1007/BF02100604, MR 1161091, S2CID 121956975
• Kawamata, Yujiro (2002), "D-equivalence and K-equivalence", Journal of Differential Geometry, 61: 147–171, arXiv:math/0205287, Bibcode:2002math......5287K, doi:10.4310/jdg/1090351323, MR 1949787, S2CID 8778816
• Kawamata, Yujiro (2014), Kōjigen daisū tayōtairon / 高次元代数多様体論 (Higher Dimensional Algebraic Varieties), Iwanami Shoten, ISBN 978-4000075985
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Yukio Matsumoto
Yukio Matsumoto (松本 幸夫, Matsumoto Yukio; * 1944) is a japanese mathematician, who worked mostly in the field of geometric topology and low-dimensional topology. He was a former professor for mathematics at the university of Tokyo.[1]
He received his Ph.D in 1973 from the university of Tokyo and his supervisor was Ichiro Tamura.[2]
In 1984 he won the Iyanaga Prize of the Mathematical Society of Japan.[1][3][4]
Selected publications
Solo
• Matsumoto, Yukio (2001). An Introduction to Morse Theory. Translations of Mathematical Monographs. ISBN 978-0821810224.
Joint
• Kojima, Sadayoshi; Matsumoto, Yukio; Saito, Kyōji; Seppälä, Mika (1995). Topology and Teichmüller spaces. World Scientific Publishing. doi:10.1142/3122.
References
1. "Yukio Matsumoto". researchmap.jp. researchmap. Retrieved 2022-11-08.
2. Yukio Matsumoto at the Mathematics Genealogy Project
3. "List of Spring and Autumn Prizes Winners". www.mathsoc.jp.
4. https://www.gakushuin.ac.jp/univ/new/pamphlet-english.pdf
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Goodman and Kruskal's gamma
In statistics, Goodman and Kruskal's gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities. It measures the strength of association of the cross tabulated data when both variables are measured at the ordinal level. It makes no adjustment for either table size or ties. Values range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.
This statistic (which is distinct from Goodman and Kruskal's lambda) is named after Leo Goodman and William Kruskal, who proposed it in a series of papers from 1954 to 1972.[1][2][3][4]
Definition
The estimate of gamma, G, depends on two quantities:
• Ns, the number of pairs of cases ranked in the same order on both variables (number of concordant pairs),
• Nd, the number of pairs of cases ranked in reversed order on both variables (number of reversed pairs),
where "ties" (cases where either of the two variables in the pair are equal) are dropped. Then
$G={\frac {N_{s}-N_{d}}{N_{s}+N_{d}}}\ .$
This statistic can be regarded as the maximum likelihood estimator for the theoretical quantity $\gamma $, where
$\gamma ={\frac {P_{s}-P_{d}}{P_{s}+P_{d}}}\ ,$
and where Ps and Pd are the probabilities that a randomly selected pair of observations will place in the same or opposite order respectively, when ranked by both variables.
Critical values for the gamma statistic are sometimes found by using an approximation, whereby a transformed value, t of the statistic is referred to Student t distribution, where
$t\approx G{\sqrt {\frac {N_{s}+N_{d}}{n(1-G^{2})}}}\ ,$
and where n is the number of observations (not the number of pairs):
$n\neq N_{s}+N_{d}.\,$
Yule's Q
A special case of Goodman and Kruskal's gamma is Yule's Q, also known as the Yule coefficient of association,[5] which is specific to 2×2 matrices. Consider the following contingency table of events, where each value is a count of an event's frequency:
YesNoTotals
Positive aba+b
Negative cdc+d
Totals a+cb+dn
Yule's Q is given by:
$Q={\frac {ad-bc}{ad+bc}}\ .$
Although computed in the same fashion as Goodman and Kruskal's gamma, it has a slightly broader interpretation because the distinction between nominal and ordinal scales becomes a matter of arbitrary labeling for dichotomous distinctions. Thus, whether Q is positive or negative depends merely on which pairings the analyst considers to be concordant, but is otherwise symmetric.
Q varies from −1 to +1. −1 reflects total negative association, +1 reflects perfect positive association and 0 reflects no association at all. The sign depends on which pairings the analyst initially considered to be concordant, but this choice does not affect the magnitude.
In term of the odds ratio OR, Yule's Q is given by
$Q={\frac {{OR}-1}{{OR}+1}}\ .$
and so Yule's Q and Yule's Y are related by
$Q={\frac {2Y}{1+Y^{2}}}\ ,$
$Y={\frac {1-{\sqrt {1-Q^{2}}}}{Q}}\ .$
See also
• Kendall tau rank correlation coefficient
• Goodman and Kruskal's lambda
• Yule's Y, also known as the coefficient of colligation
References
1. Goodman, Leo A.; Kruskal, William H. (1954). "Measures of Association for Cross Classifications". Journal of the American Statistical Association. 49 (268): 732–764. doi:10.2307/2281536. JSTOR 2281536.
2. Goodman, Leo A.; Kruskal, William H. (1959). "Measures of Association for Cross Classifications. II: Further Discussion and References". Journal of the American Statistical Association. 54 (285): 123–163. doi:10.1080/01621459.1959.10501503. JSTOR 2282143.
3. Goodman, Leo A.; Kruskal, William H. (1963). "Measures of Association for Cross Classifications III: Approximate Sampling Theory". Journal of the American Statistical Association. 58 (302): 310–364. doi:10.1080/01621459.1963.10500850. JSTOR 2283271.
4. Goodman, Leo A.; Kruskal, William H. (1972). "Measures of Association for Cross Classifications, IV: Simplification of Asymptotic Variances". Journal of the American Statistical Association. 67 (338): 415–421. doi:10.1080/01621459.1972.10482401. JSTOR 2284396.
5. Yule, G U. (1912). "On the methods of measuring association between two attributes". Journal of the Royal Statistical Society. 49 (6): 579–652. doi:10.2307/2340126. JSTOR 2340126.
Further reading
• Sheskin, D.J. (2007) The Handbook of Parametric and Nonparametric Statistical Procedures. Chapman & Hall/CRC, ISBN 9781584888147
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| Wikipedia |
Coefficient of colligation
In statistics, Yule's Y, also known as the coefficient of colligation, is a measure of association between two binary variables. The measure was developed by George Udny Yule in 1912,[1][2] and should not be confused with Yule's coefficient for measuring skewness based on quartiles.
Formula
For a 2×2 table for binary variables U and V with frequencies or proportions
V = 0V = 1
U = 0ab
U = 1cd
Yule's Y is given by
$Y={\frac {{\sqrt {ad}}-{\sqrt {bc}}}{{\sqrt {ad}}+{\sqrt {bc}}}}.$
Yule's Y is closely related to the odds ratio OR = ad/(bc) as is seen in following formula:
$Y={\frac {{\sqrt {OR}}-1}{{\sqrt {OR}}+1}}$
Yule's Y varies from −1 to +1. −1 reflects total negative correlation, +1 reflects perfect positive association while 0 reflects no association at all. These correspond to the values for the more common Pearson correlation.
Yule's Y is also related to the similar Yule's Q, which can also be expressed in terms of the odds ratio. Q and Y are related by:
$Q={\frac {2Y}{1+Y^{2}}}\ ,$
$Y={\frac {1-{\sqrt {1-Q^{2}}}}{Q}}\ .$
Interpretation
Yule's Y gives the fraction of perfect association in per unum (multiplied by 100 it represents this fraction in a more familiar percentage). Indeed, the formula transforms the original 2×2 table in a crosswise symmetric table wherein b = c = 1 and a = d = √OR.
For a crosswise symmetric table with frequencies or proportions a = d and b = c it is very easy to see that it can be split up in two tables. In such tables association can be measured in a perfectly clear way by dividing (a – b) by (a + b). In transformed tables b has to be substituted by 1 and a by √OR. The transformed table has the same degree of association (the same OR) as the original not-crosswise symmetric table. Therefore, the association in asymmetric tables can be measured by Yule's Y, interpreting it in just the same way as with symmetric tables. Of course, Yule's Y and (a − b)/(a + b) give the same result in crosswise symmetric tables, presenting the association as a fraction in both cases.
Yule's Y measures association in a substantial, intuitively understandable way and therefore it is the measure of preference to measure association.
Examples
The following crosswise symmetric table
V = 0V = 1
U = 04010
U = 11040
can be split up into two tables:
V = 0V = 1
U = 01010
U = 11010
and
V = 0V = 1
U = 0300
U = 1030
It is obvious that the degree of association equals 0.6 per unum (60%).
The following asymmetric table can be transformed in a table with an equal degree of association (the odds ratios of both tables are equal).
V = 0V = 1
U = 031
U = 139
Here follows the transformed table:
V = 0V = 1
U = 031
U = 113
The odds ratios of both tables are equal to 9. Y = (3 − 1)/(3 + 1) = 0.5 (50%)
References
1. Yule, G. Udny (1912). "On the Methods of Measuring Association Between Two Attributes". Journal of the Royal Statistical Society. 75 (6): 579–652. doi:10.2307/2340126. JSTOR 2340126.
2. Michel G. Soete. A new theory on the measurement of association between two binary variables in medical sciences: association can be expressed in a fraction (per unum, percentage, pro mille....) of perfect association (2013), e-article, BoekBoek.be
| Wikipedia |
Simpson's paradox
Simpson's paradox is a phenomenon in probability and statistics in which a trend appears in several groups of data but disappears or reverses when the groups are combined. This result is often encountered in social-science and medical-science statistics,[1][2][3] and is particularly problematic when frequency data are unduly given causal interpretations.[4] The paradox can be resolved when confounding variables and causal relations are appropriately addressed in the statistical modeling[4][5] (e.g., through cluster analysis[6]).
Simpson's paradox has been used to illustrate the kind of misleading results that the misuse of statistics can generate.[7][8]
Edward H. Simpson first described this phenomenon in a technical paper in 1951,[9] but the statisticians Karl Pearson (in 1899[10]) and Udny Yule (in 1903[11]) had mentioned similar effects earlier. The name Simpson's paradox was introduced by Colin R. Blyth in 1972.[12] It is also referred to as Simpson's reversal, the Yule–Simpson effect, the amalgamation paradox, or the reversal paradox.[13]
Mathematician Jordan Ellenberg argues that Simpson's paradox is misnamed as "there's no contradiction involved, just two different ways to think about the same data" and suggests that its lesson "isn't really to tell us which viewpoint to take but to insist that we keep both the parts and the whole in mind at once."[14]
Examples
UC Berkeley gender bias
One of the best-known examples of Simpson's paradox comes from a study of gender bias among graduate school admissions to University of California, Berkeley. The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance.[15][16]
All Men Women
Applicants Admitted Applicants Admitted Applicants Admitted
Total 12,763 41% 8,442 44% 4,321 35%
However, when taking into account the information about departments being applied to, the different rejection percentages reveal the different difficulty of getting into the department, and at the same time it showed that women tended to apply to more competitive departments with lower rates of admission, even among qualified applicants (such as in the English department), whereas men tended to apply to less competitive departments with higher rates of admission (such as in the engineering department). The pooled and corrected data showed a "small but statistically significant bias in favor of women".[16]
The data from the six largest departments are listed below:
Department All Men Women
Applicants Admitted Applicants Admitted Applicants Admitted
A 933 64% 825 62% 108 82%
B 585 63% 560 63% 25 68%
C 918 35% 325 37% 593 34%
D 792 34% 417 33% 375 35%
E 584 25% 191 28% 393 24%
F 714 6% 373 6% 341 7%
Total 4526 39% 2691 45% 1835 30%
Legend:
greater percentage of successful applicants than the other gender
greater number of applicants than the other gender
bold - the two 'most applied for' departments for each gender
The entire data showed total of 4 out of 85 departments to be significantly biased against women, while 6 to be significantly biased against men (not all present in the 'six largest departments' table above). Notably, the numbers of biased departments were not the basis for the conclusion, but rather it was the gender admissions pooled across all departments, while weighing by each department's rejection rate across all of its applicants.[16]
Kidney stone treatment
Another example comes from a real-life medical study[17] comparing the success rates of two treatments for kidney stones.[18] The table below shows the success rates (the term success rate here actually means the success proportion) and numbers of treatments for treatments involving both small and large kidney stones, where Treatment A includes open surgical procedures and Treatment B includes closed surgical procedures. The numbers in parentheses indicate the number of success cases over the total size of the group.
Treatment
Stone size
Treatment A Treatment B
Small stones Group 1
93% (81/87)
Group 2
87% (234/270)
Large stones Group 3
73% (192/263)
Group 4
69% (55/80)
Both 78% (273/350)83% (289/350)
The paradoxical conclusion is that treatment A is more effective when used on small stones, and also when used on large stones, yet treatment B appears to be more effective when considering both sizes at the same time. In this example, the "lurking" variable (or confounding variable) causing the paradox is the size of the stones, which was not previously known to researchers to be important until its effects were included.
Which treatment is considered better is determined by which success ratio (successes/total) is larger. The reversal of the inequality between the two ratios when considering the combined data, which creates Simpson's paradox, happens because two effects occur together:
1. The sizes of the groups, which are combined when the lurking variable is ignored, are very different. Doctors tend to give cases with large stones the better treatment A, and the cases with small stones the inferior treatment B. Therefore, the totals are dominated by groups 3 and 2, and not by the two much smaller groups 1 and 4.
2. The lurking variable, stone size, has a large effect on the ratios; i.e., the success rate is more strongly influenced by the severity of the case than by the choice of treatment. Therefore, the group of patients with large stones using treatment A (group 3) does worse than the group with small stones, even if the latter used the inferior treatment B (group 2).
Based on these effects, the paradoxical result is seen to arise because the effect of the size of the stones overwhelms the benefits of the better treatment (A). In short, the less effective treatment B appeared to be more effective because it was applied more frequently to the small stones cases, which were easier to treat.[18]
Batting averages
A common example of Simpson's paradox involves the batting averages of players in professional baseball. It is possible for one player to have a higher batting average than another player each year for a number of years, but to have a lower batting average across all of those years. This phenomenon can occur when there are large differences in the number of at bats between the years. Mathematician Ken Ross demonstrated this using the batting average of two baseball players, Derek Jeter and David Justice, during the years 1995 and 1996:[19][20]
Year
Batter
1995 1996 Combined
Derek Jeter 12/48 .250 183/582 .314 195/630 .310
David Justice 104/411 .253 45/140 .321 149/551 .270
In both 1995 and 1996, Justice had a higher batting average (in bold type) than Jeter did. However, when the two baseball seasons are combined, Jeter shows a higher batting average than Justice. According to Ross, this phenomenon would be observed about once per year among the possible pairs of players.[19]
Vector interpretation
Simpson's paradox can also be illustrated using a 2-dimensional vector space.[21] A success rate of $ {\frac {p}{q}}$ (i.e., successes/attempts) can be represented by a vector ${\vec {A}}=(q,p)$, with a slope of $ {\frac {p}{q}}$. A steeper vector then represents a greater success rate. If two rates $ {\frac {p_{1}}{q_{1}}}$ and $ {\frac {p_{2}}{q_{2}}}$ are combined, as in the examples given above, the result can be represented by the sum of the vectors $(q_{1},p_{1})$ and $(q_{2},p_{2})$, which according to the parallelogram rule is the vector $(q_{1}+q_{2},p_{1}+p_{2})$, with slope $ {\frac {p_{1}+p_{2}}{q_{1}+q_{2}}}$.
Simpson's paradox says that even if a vector ${\vec {L}}_{1}$ (in orange in figure) has a smaller slope than another vector ${\vec {B}}_{1}$ (in blue), and ${\vec {L}}_{2}$ has a smaller slope than ${\vec {B}}_{2}$, the sum of the two vectors ${\vec {L}}_{1}+{\vec {L}}_{2}$ can potentially still have a larger slope than the sum of the two vectors ${\vec {B}}_{1}+{\vec {B}}_{2}$, as shown in the example. For this to occur one of the orange vectors must have a greater slope than one of the blue vectors (here ${\vec {L}}_{2}$ and ${\vec {B}}_{1}$), and these will generally be longer than the alternatively subscripted vectors – thereby dominating the overall comparison.
Correlation between variables
Simpson's reversal can also arise in correlations, in which two variables appear to have (say) a positive correlation towards one another, when in fact they have a negative correlation, the reversal having been brought about by a "lurking" confounder. Berman et al.[22] give an example from economics, where a dataset suggests overall demand is positively correlated with price (that is, higher prices lead to more demand), in contradiction of expectation. Analysis reveals time to be the confounding variable: plotting both price and demand against time reveals the expected negative correlation over various periods, which then reverses to become positive if the influence of time is ignored by simply plotting demand against price.
Psychology
Psychological interest in Simpson's paradox seeks to explain why people deem sign reversal to be impossible at first, offended by the idea that an action preferred both under one condition and under its negation should be rejected when the condition is unknown. The question is where people get this strong intuition from, and how it is encoded in the mind.
Simpson's paradox demonstrates that this intuition cannot be derived from either classical logic or probability calculus alone, and thus led philosophers to speculate that it is supported by an innate causal logic that guides people in reasoning about actions and their consequences.[4] Savage's sure-thing principle[12] is an example of what such logic may entail. A qualified version of Savage's sure thing principle can indeed be derived from Pearl's do-calculus[4] and reads: "An action A that increases the probability of an event B in each subpopulation Ci of C must also increase the probability of B in the population as a whole, provided that the action does not change the distribution of the subpopulations." This suggests that knowledge about actions and consequences is stored in a form resembling Causal Bayesian Networks.
Probability
A paper by Pavlides and Perlman presents a proof, due to Hadjicostas, that in a random 2 × 2 × 2 table with uniform distribution, Simpson's paradox will occur with a probability of exactly 1⁄60.[23] A study by Kock suggests that the probability that Simpson's paradox would occur at random in path models (i.e., models generated by path analysis) with two predictors and one criterion variable is approximately 12.8 percent; slightly higher than 1 occurrence per 8 path models.[24]
Simpson's second paradox
A second, less well-known paradox was also discussed in Simpson's 1951 paper. It can occur when the "sensible interpretation" is not necessarily found in the separated data, like in the Kidney Stone example, but can instead reside in the combined data. Whether the partitioned or combined form of the data should be used hinges on the process giving rise to the data, meaning the correct interpretation of the data cannot always be determined by simply observing the tables.[25]
Judea Pearl has shown that, in order for the partitioned data to represent the correct causal relationships between any two variables, $X$ and $Y$, the partitioning variables must satisfy a graphical condition called "back-door criterion":[26][27]
1. They must block all spurious paths between $X$ and $Y$
2. No variable can be affected by $X$
This criterion provides an algorithmic solution to Simpson's second paradox, and explains why the correct interpretation cannot be determined by data alone; two different graphs, both compatible with the data, may dictate two different back-door criteria.
When the back-door criterion is satisfied by a set Z of covariates, the adjustment formula (see Confounding) gives the correct causal effect of X on Y. If no such set exists, Pearl's do-calculus can be invoked to discover other ways of estimating the causal effect.[4][28] The completeness of do-calculus [29][28] can be viewed as offering a complete resolution of the Simpson's paradox.
Criticism
One criticism is that the paradox is not really a paradox at all, but rather a failure to properly account for confounding variables or to consider causal relationships between variables.[30]
Another criticism of the apparent Simpson's paradox is that it may be a result of the specific way that data is stratified or grouped. The phenomenon may disappear or even reverse if the data is stratified differently or if different confounding variables are considered. Simpson's example actually highlighted a phenomenon called noncollapsibility,[31] which occurs when subgroups with high proportions do not make simple averages when combined. This suggests that the paradox may not be a universal phenomenon, but rather a specific instance of a more general statistical issue.
Critics of the apparent Simpson's paradox also argue that the focus on the paradox may distract from more important statistical issues, such as the need for careful consideration of confounding variables and causal relationships when interpreting data.[32]
Despite these criticisms, the apparent Simpson's paradox remains a popular and intriguing topic in statistics and data analysis. It continues to be studied and debated by researchers and practitioners in a wide range of fields, and it serves as a valuable reminder of the importance of careful statistical analysis and the potential pitfalls of simplistic interpretations of data.
See also
• Aliasing – Signal processing effect
• Anscombe's quartet – Four data sets with the same descriptive statistics, yet very different distributions
• Berkson's paradox – Tendency to misinterpret statistical experiments involving conditional probabilities
• Cherry picking – Fallacy of incomplete evidence
• Condorcet paradox – Situation in social choice theory where collective preferences are cyclic
• Ecological fallacy – Logical fallacy that occurs when group characteristics are applied to individuals
• Gerrymandering – Form of political manipulation
• Low birth-weight paradox – Statistical quirk of babies' birth weights
• Modifiable areal unit problem – Source of statistical bias
• Prosecutor's fallacy – Error in thinking which involves under-valuing base rate informationPages displaying short descriptions of redirect targets
• Will Rogers phenomenon – phenomenon in which moving an element from one set to another set raises the average values of both setsPages displaying wikidata descriptions as a fallback
• Spurious correlation
• Omitted-variable bias
References
1. Clifford H. Wagner (February 1982). "Simpson's Paradox in Real Life". The American Statistician. 36 (1): 46–48. doi:10.2307/2684093. JSTOR 2684093.
2. Holt, G. B. (2016). Potential Simpson's paradox in multicenter study of intraperitoneal chemotherapy for ovarian cancer. Journal of Clinical Oncology, 34(9), 1016–1016.
3. Franks, Alexander; Airoldi, Edoardo; Slavov, Nikolai (2017). "Post-transcriptional regulation across human tissues". PLOS Computational Biology. 13 (5): e1005535. arXiv:1506.00219. Bibcode:2017PLSCB..13E5535F. doi:10.1371/journal.pcbi.1005535. ISSN 1553-7358. PMC 5440056. PMID 28481885.
4. Judea Pearl. Causality: Models, Reasoning, and Inference, Cambridge University Press (2000, 2nd edition 2009). ISBN 0-521-77362-8.
5. Kock, N., & Gaskins, L. (2016). Simpson's paradox, moderation and the emergence of quadratic relationships in path models: An information systems illustration. International Journal of Applied Nonlinear Science, 2(3), 200–234.
6. Rogier A. Kievit, Willem E. Frankenhuis, Lourens J. Waldorp and Denny Borsboom, Simpson's paradox in psychological science: a practical guide https://doi.org/10.3389/fpsyg.2013.00513
7. Robert L. Wardrop (February 1995). "Simpson's Paradox and the Hot Hand in Basketball". The American Statistician, 49 (1): pp. 24–28.
8. Alan Agresti (2002). "Categorical Data Analysis" (Second edition). John Wiley and Sons ISBN 0-471-36093-7
9. Simpson, Edward H. (1951). "The Interpretation of Interaction in Contingency Tables". Journal of the Royal Statistical Society, Series B. 13: 238–241.
10. Pearson, Karl; Lee, Alice; Bramley-Moore, Lesley (1899). "Genetic (reproductive) selection: Inheritance of fertility in man, and of fecundity in thoroughbred racehorses". Philosophical Transactions of the Royal Society A. 192: 257–330. doi:10.1098/rsta.1899.0006.
11. G. U. Yule (1903). "Notes on the Theory of Association of Attributes in Statistics". Biometrika. 2 (2): 121–134. doi:10.1093/biomet/2.2.121.
12. Colin R. Blyth (June 1972). "On Simpson's Paradox and the Sure-Thing Principle". Journal of the American Statistical Association. 67 (338): 364–366. doi:10.2307/2284382. JSTOR 2284382.
13. I. J. Good, Y. Mittal (June 1987). "The Amalgamation and Geometry of Two-by-Two Contingency Tables". The Annals of Statistics. 15 (2): 694–711. doi:10.1214/aos/1176350369. ISSN 0090-5364. JSTOR 2241334.
14. Ellenberg, Jordan (May 25, 2021). Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy and Everything Else. New York: Penguin Press. p. 228. ISBN 978-1-9848-7905-9. OCLC 1226171979.
15. David Freedman, Robert Pisani, and Roger Purves (2007), Statistics (4th edition), W. W. Norton. ISBN 0-393-92972-8.
16. P.J. Bickel, E.A. Hammel and J.W. O'Connell (1975). "Sex Bias in Graduate Admissions: Data From Berkeley" (PDF). Science. 187 (4175): 398–404. Bibcode:1975Sci...187..398B. doi:10.1126/science.187.4175.398. PMID 17835295. S2CID 15278703. Archived (PDF) from the original on 2016-06-04.
17. C. R. Charig; D. R. Webb; S. R. Payne; J. E. Wickham (29 March 1986). "Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy". Br Med J (Clin Res Ed). 292 (6524): 879–882. doi:10.1136/bmj.292.6524.879. PMC 1339981. PMID 3083922.
18. Steven A. Julious; Mark A. Mullee (3 December 1994). "Confounding and Simpson's paradox". BMJ. 309 (6967): 1480–1481. doi:10.1136/bmj.309.6967.1480. PMC 2541623. PMID 7804052.
19. Ken Ross. "A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (Paperback)" Pi Press, 2004. ISBN 0-13-147990-3. 12–13
20. Statistics available from Baseball-Reference.com: Data for Derek Jeter; Data for David Justice.
21. Kocik Jerzy (2001). "Proofs without Words: Simpson's Paradox" (PDF). Mathematics Magazine. 74 (5): 399. doi:10.2307/2691038. JSTOR 2691038. Archived (PDF) from the original on 2010-06-12.
22. Berman, S. DalleMule, L. Greene, M., Lucker, J. (2012), "Simpson's Paradox: A Cautionary Tale in Advanced Analytics Archived 2020-05-10 at the Wayback Machine", Significance.
23. Marios G. Pavlides & Michael D. Perlman (August 2009). "How Likely is Simpson's Paradox?". The American Statistician. 63 (3): 226–233. doi:10.1198/tast.2009.09007. S2CID 17481510.
24. Kock, N. (2015). How likely is Simpson's paradox in path models? International Journal of e-Collaboration, 11(1), 1–7.
25. Norton, H. James; Divine, George (August 2015). "Simpson's paradox ... and how to avoid it". Significance. 12 (4): 40–43. doi:10.1111/j.1740-9713.2015.00844.x.
26. Pearl, Judea (2014). "Understanding Simpson's Paradox". The American Statistician. 68 (1): 8–13. doi:10.2139/ssrn.2343788. S2CID 2626833.
27. Pearl, Judea (1993). "Graphical Models, Causality, and Intervention". Statistical Science. 8 (3): 266–269. doi:10.1214/ss/1177010894.
28. Pearl, J.; Mackenzie, D. (2018). The Book of Why: The New Science of Cause and Effect. New York, NY: Basic Books.
29. Shpitser, I.; Pearl, J. (2006). Dechter, R.; Richardson, T.S. (eds.). "Identification of Conditional Interventional Distributions". Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence. Corvallis, OR: AUAI Press: 437–444.
30. Blyth, Colin R. (June 1972). "On Simpson's Paradox and the Sure-Thing Principle". Journal of the American Statistical Association. 67 (338): 364–366. doi:10.1080/01621459.1972.10482387. ISSN 0162-1459.
31. Greenland, Sander (2021-11-01). "Noncollapsibility, confounding, and sparse-data bias. Part 2: What should researchers make of persistent controversies about the odds ratio?". Journal of Clinical Epidemiology. 139: 264–268. doi:10.1016/j.jclinepi.2021.06.004. ISSN 0895-4356. PMID 34119647.
32. Hernán, Miguel A.; Clayton, David; Keiding, Niels (June 2011). "The Simpson's paradox unraveled". International Journal of Epidemiology. 40 (3): 780–785. doi:10.1093/ije/dyr041. ISSN 1464-3685. PMC 3147074. PMID 21454324.
Bibliography
• Leila Schneps and Coralie Colmez, Math on trial. How numbers get used and abused in the courtroom, Basic Books, 2013. ISBN 978-0-465-03292-1. (Sixth chapter: "Math error number 6: Simpson's paradox. The Berkeley sex bias case: discrimination detection").
External links
Wikimedia Commons has media related to Simpson's paradox.
• Simpson's Paradox at the Stanford Encyclopedia of Philosophy, by Jan Sprenger and Naftali Weinberger.
• How statistics can be misleading – Mark Liddell – TED-Ed video and lesson.
• Pearl, Judea, "Understanding Simpson’s Paradox" (PDF)
• Simpson's Paradox, a short article by Alexander Bogomolny on the vector interpretation of Simpson's paradox
• The Wall Street Journal column "The Numbers Guy" for December 2, 2009 dealt with recent instances of Simpson's paradox in the news. Notably a Simpson's paradox in the comparison of unemployment rates of the 2009 recession with the 1983 recession.
• At the Plate, a Statistical Puzzler: Understanding Simpson's Paradox by Arthur Smith, August 20, 2010
• Simpson's Paradox, a video by Henry Reich of MinutePhysics
| Wikipedia |
Yule–Simon distribution
In probability and statistics, the Yule–Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon. Simon originally called it the Yule distribution.[1]
Yule–Simon
Probability mass function
Yule–Simon PMF on a log-log scale. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Yule–Simon CMF. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters $\rho >0\,$ shape (real)
Support $k\in \{1,2,\dotsc \}$
PMF $\rho \operatorname {B} (k,\rho +1)$
CDF $1-k\operatorname {B} (k,\rho +1)$
Mean ${\frac {\rho }{\rho -1}}$ for $\rho >1$
Mode $1$
Variance ${\frac {\rho ^{2}}{(\rho -1)^{2}(\rho -2)}}$ for $\rho >2$
Skewness ${\frac {(\rho +1)^{2}{\sqrt {\rho -2}}}{(\rho -3)\rho }}\,$ for $\rho >3$
Ex. kurtosis $\rho +3+{\frac {11\rho ^{3}-49\rho -22}{(\rho -4)(\rho -3)\rho }}$ for $\rho >4$
MGF does not exist
CF ${\frac {\rho }{\rho +1}}{}_{2}F_{1}(1,1;\rho +2;e^{i\,t})e^{i\,t}$
The probability mass function (pmf) of the Yule–Simon (ρ) distribution is
$f(k;\rho )=\rho \operatorname {B} (k,\rho +1),$
for integer $k\geq 1$ and real $\rho >0$, where $\operatorname {B} $ is the beta function. Equivalently the pmf can be written in terms of the rising factorial as
$f(k;\rho )={\frac {\rho \Gamma (\rho +1)}{(k+\rho )^{\underline {\rho +1}}}},$
where $\Gamma $ is the gamma function. Thus, if $\rho $ is an integer,
$f(k;\rho )={\frac {\rho \,\rho !\,(k-1)!}{(k+\rho )!}}.$ !\,(k-1)!}{(k+\rho )!}}.}
The parameter $\rho $ can be estimated using a fixed point algorithm.[2]
The probability mass function f has the property that for sufficiently large k we have
$f(k;\rho )\approx {\frac {\rho \Gamma (\rho +1)}{k^{\rho +1}}}\propto {\frac {1}{k^{\rho +1}}}.$
This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: $f(k;\rho )$ can be used to model, for example, the relative frequency of the $k$th most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a (typically small) power of $k$.
Occurrence
The Yule–Simon distribution arose originally as the limiting distribution of a particular model studied by Udny Yule in 1925 to analyze the growth in the number of species per genus in some higher taxa of biotic organisms.[3] The Yule model makes use of two related Yule processes, where a Yule process is defined as a continuous time birth process which starts with one or more individuals. Yule proved that when time goes to infinity, the limit distribution of the number of species in a genus selected uniformly at random has a specific form and exhibits a power-law behavior in its tail. Thirty years later, the Nobel laureate Herbert A. Simon proposed a time-discrete preferential attachment model to describe the appearance of new words in a large piece of a text. Interestingly enough, the limit distribution of the number of occurrences of each word, when the number of words diverges, coincides with that of the number of species belonging to the randomly chosen genus in the Yule model, for a specific choice of the parameters. This fact explains the designation Yule–Simon distribution that is commonly assigned to that limit distribution. In the context of random graphs, the Barabási–Albert model also exhibits an asymptotic degree distribution that equals the Yule–Simon distribution in correspondence of a specific choice of the parameters and still presents power-law characteristics for more general choices of the parameters. The same happens also for other preferential attachment random graph models.[4]
The preferential attachment process can also be studied as an urn process in which balls are added to a growing number of urns, each ball being allocated to an urn with probability linear in the number (of balls) the urn already contains.
The distribution also arises as a compound distribution, in which the parameter of a geometric distribution is treated as a function of random variable having an exponential distribution. Specifically, assume that $W$ follows an exponential distribution with scale $1/\rho $ or rate $\rho $:
$W\sim \operatorname {Exponential} (\rho ),$
with density
$h(w;\rho )=\rho \exp(-\rho w).$
Then a Yule–Simon distributed variable K has the following geometric distribution conditional on W:
$K\sim \operatorname {Geometric} (\exp(-W)).$
The pmf of a geometric distribution is
$g(k;p)=p(1-p)^{k-1}$
for $k\in \{1,2,\dotsc \}$. The Yule–Simon pmf is then the following exponential-geometric compound distribution:
$f(k;\rho )=\int _{0}^{\infty }g(k;\exp(-w))h(w;\rho )\,dw.$
The maximum likelihood estimator for the parameter $\rho $ given the observations $k_{1},k_{2},k_{3},\dots ,k_{N}$ is the solution to the fixed point equation
$\rho ^{(t+1)}={\frac {N+a-1}{b+\sum _{i=1}^{N}\sum _{j=1}^{k_{i}}{\frac {1}{\rho ^{(t)}+j}}}},$
where $b=0,a=1$ are the rate and shape parameters of the gamma distribution prior on $\rho $.
This algorithm is derived by Garcia[2] by directly optimizing the likelihood. Roberts and Roberts[5]
generalize the algorithm to Bayesian settings with the compound geometric formulation described above. Additionally, Roberts and Roberts[5] are able to use the Expectation Maximisation (EM) framework to show convergence of the fixed point algorithm. Moreover, Roberts and Roberts[5] derive the sub-linearity of the convergence rate for the fixed point algorithm. Additionally, they use the EM formulation to give 2 alternate derivations of the standard error of the estimator from the fixed point equation. The variance of the $\lambda $ estimator is
$\operatorname {Var} ({\hat {\lambda }})={\frac {1}{{\frac {N}{{\hat {\lambda }}^{2}}}-\sum _{i=1}^{N}\sum _{j=1}^{k_{i}}{\frac {1}{({\hat {\lambda }}+j)^{2}}}}},$
the standard error is the square root of the quantity of this estimate divided by N.
Generalizations
The two-parameter generalization of the original Yule distribution replaces the beta function with an incomplete beta function. The probability mass function of the generalized Yule–Simon(ρ, α) distribution is defined as
$f(k;\rho ,\alpha )={\frac {\rho }{1-\alpha ^{\rho }}}\;\mathrm {B} _{1-\alpha }(k,\rho +1),\,$
with $0\leq \alpha <1$. For $\alpha =0$ the ordinary Yule–Simon(ρ) distribution is obtained as a special case. The use of the incomplete beta function has the effect of introducing an exponential cutoff in the upper tail.
See also
• Zeta distribution
• Scale-free network
• Beta negative binomial distribution
Bibliography
• Colin Rose and Murray D. Smith, Mathematical Statistics with Mathematica. New York: Springer, 2002, ISBN 0-387-95234-9. (See page 107, where it is called the "Yule distribution".)
References
1. Simon, H. A. (1955). "On a class of skew distribution functions". Biometrika. 42 (3–4): 425–440. doi:10.1093/biomet/42.3-4.425.
2. Garcia Garcia, Juan Manuel (2011). "A fixed-point algorithm to estimate the Yule-Simon distribution parameter". Applied Mathematics and Computation. 217 (21): 8560–8566. doi:10.1016/j.amc.2011.03.092.
3. Yule, G. U. (1924). "A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S". Philosophical Transactions of the Royal Society B. 213 (402–410): 21–87. doi:10.1098/rstb.1925.0002.
4. Pachon, Angelica; Polito, Federico; Sacerdote, Laura (2015). "Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models". Journal of Statistical Physics. 162 (6): 1608–1638. arXiv:1503.06150. doi:10.1007/s10955-016-1462-7. S2CID 119168040.
5. Roberts, Lucas; Roberts, Denisa (2017). "An Expectation Maximization Framework for Preferential Attachment Models". arXiv:1710.08511 [stat.CO].
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
| Wikipedia |
Yuli Rudyak
Yuli B. Rudyak is a professor of mathematics at the University of Florida in Gainesville, Florida. He obtained his doctorate from Moscow State University under the supervision of M. M. Postnikov. His main research interests are geometry and topology and symplectic topology.
Books
• Rudyak, Yu. (1998), On Thom spectra, orientability, and cobordism. With a foreword by Haynes Miller, Springer Monographs in Mathematics, Berlin: Springer-Verlag
Reviewer Donald M. Davis (mathematician) for MathSciNet wrote: "This book provides an excellent and thorough treatment of various topics related to cobordism. It should become an indispensable tool for advanced graduate students and workers in algebraic topology."[1]
The book listed 118 cites at Google Scholar in 2011.[2]
Personal life
Rudyak is the father of Marina Rudyak, who is an Assistant Professor of Chinese Studies at the University of Heidelberg.
Notes
1. MR1627486
2. Google Scholar search results
External links
• Webpage at UFL
| Wikipedia |
Yulia Gel
Yulia R. Gel is a professor in the Department of Mathematical Sciences at the University of Texas at Dallas[1] and an adjunct professor in the Department of Statistics and Actuarial Science of the University of Waterloo.[2]
Yulia Gel
Education
• Saint Petersburg State University
• University of Washington
Known forTopological Data Analysis
Scientific career
FieldsStatistics
Institutions
• University of Texas at Dallas
• University of Waterloo
Academic advisorsVladimir N. Fomin
Early life and education
Gel earned her doctorate in mathematics at Saint Petersburg State University in Russia, under the supervision of Vladimir N. Fomin.[3] After postdoctoral research at the University of Washington, she joined the Waterloo faculty in 2004, and moved to Dallas in 2014.[4]
Research and career
Prior to joining the University of Texas at Dallas, Yulia Gel served as an Assistant/Associate Professor with tenure in the Department of Statistics and Actuarial Sciences at the University of Waterloo, Canada, from 2004 to 2014. She has also held visiting positions at prominent institutions such as NASA Jet Propulsion Lab (Caltech), the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK), Johns Hopkins University, University of California at Berkeley, and George Washington University.
Yulia Gel has a diverse range of research interests that span statistical foundations of data science, machine learning, topological and geometric methods in statistics, and topological data analysis. Her work focuses on graph mining, inference for random graphs and complex networks, uncertainty quantification in network analysis, data depth on networks, time series analysis, spatio-temporal processes, and climate informatics. She is particularly interested in the application of statistical and data science techniques to domains such as healthcare predictive analytics and climate informatics.
Awards and honors
In 2014 Yulia was elected as a Fellow of the American Statistical Association" for theoretical contributions to nonparametric aspects of spatiotemporal processes; for promoting the application of modern statistical methodologies in law, public policy, and the environmental sciences; and for championing the advancement of women and other under-represented groups in the mathematical and physical sciences."[5]
References
1. Mathematics faculty and research, UT Dallas, retrieved 2016-07-12.
2. "Yulia Gel". Statistics and Actuarial Science. University of Waterloo. 23 February 2015. Retrieved 8 December 2017.
3. Yulia Gel at the Mathematics Genealogy Project
4. Golbeck, Amanda L.; Olkin, Ingram; Gel, Yulia R., eds. (2015), "About the Editors", Leadership and Women in Statistics, CRC Press, ISBN 9781482236453.
5. ASA Honors 63 New Fellows (PDF), American Statistical Association, June 11, 2014, retrieved 2016-07-11.
External links
• Google scholar profile
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Yuliya Mishura
Yuliya Stepanivna Mishura (Ukrainian: Юлія Степанівна Мішура) is a Ukrainian mathematician specializing in probability theory and mathematical finance. She is a professor at the Taras Shevchenko National University of Kyiv.[1]
Education and career
Mishura earned a Ph.D. in 1978 from the Taras Shevchenko National University of Kyiv with a dissertation on Limit Theorems for Functionals from Stochastic Fields supervised by Dmitrii Sergeevich Silvestrov. She earned a Dr. Sci. from the National Academy of Sciences of Ukraine in 1990 with a dissertation Martingale Methods in the Theory of Stochastic Fields.[1][2]
She became an assistant professor in the Faculty of Mechanics and Mathematics at National Taras Shevchenko University of Kyiv in 1976. She has been a full professor since 1991, and head of the Department of Probability, Statistics and Actuarial Mathematics since 2003.[1]
With Kęstutis Kubilius, she is the founding co-editor-in-chief of the journal Modern Stochastics: Theory and Applications.[3] She is the editor-in-chief of the journal Theory of Probability and Mathematical Statistics.
Books
Mishura is the author of many monographs and textbooks.[1] They include:
• Discrete-Time Approximations and Limit Theorems In Applications to Financial Markets (with Kostiantyn Ralchenko, De Gruyter Series in Probability and Stochastics, 2021)
• Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations (with G. Kulinich, S. Kushnirenko, Vol.9 Bocconi & Springer Series, Mathematics, Statistics, Finance and Economics, 2020)
• Fractional Brownian Motion. Approximations and Projections (with Oksana Banna, Kostiantyn Ralchenko, Sergiy Shklyar, Wiley-ISTE, 2019)
• Stochastic Analysis of Mixed Fractional Gaussian Processes (ISTE Press, 2018)[4]
• Theory and Statistical Applications of Stochastic Processes (with Georgiy Shevchenko, ISTE Press and John Wiley & Sons, 2017)
• Parameter Estimation in Fractional Diffusion Models (with Kęstutis Kubilius and Kostiantyn Ralchenko, Bocconi University Press and Springer, 2017)[5]
• Ruin Probabilities: Smoothness, Bounds, Supermartingale Approach (with Olena Ragulina, ISTE Press, 2016)
• Financial Mathematics: Optimization in Insurance and Finance Set (ISTE Press, 2016)[6]
• Theory of Stochastic Processes: With Applications to Financial Mathematics And Risk Theory (with Gusak, Kukush, Kulik, and Pilipenko, Problem Books in Mathematics, Springer, 2010)[7]
• Stochastic Calculus for Fractional Brownian Motion and Related Processes (Lecture Notes in Mathematics 1929, Springer, 2008)[8]
References
1. "Yuliya Mishura", Employee profile, Taras Shevchenko National University of Kyiv, retrieved 2020-03-29
2. Yuliya Mishura at the Mathematics Genealogy Project
3. "Editors-in-chief", Modern Stochastics: Theory and Applications, retrieved 2020-03-29
4. Review of Stochastic Analysis of Mixed Fractional Gaussian Processes:
• Aurzada, Frank, Mathematical Reviews, MR 3793191{{citation}}: CS1 maint: untitled periodical (link)
5. Reviews of Parameter Estimation in Fractional Diffusion Models:
• Kolnogorov, Alex V., zbMATH, Zbl 1388.60006{{citation}}: CS1 maint: untitled periodical (link)
• Lu, Fei, Mathematical Reviews, MR 3752152{{citation}}: CS1 maint: untitled periodical (link)
6. Review of Financial Mathematics:
• Vives, Josep, zbMATH, Zbl 1371.91001{{citation}}: CS1 maint: untitled periodical (link)
7. Reviews of Theory of Stochastic Processes:
• Hein, Claudia, zbMATH, Zbl 1189.60001{{citation}}: CS1 maint: untitled periodical (link)
• Hand, David J. (December 2010), International Statistical Review, 78 (3): 461, doi:10.1111/j.1751-5823.2010.00122_15.x, JSTOR 27919877{{citation}}: CS1 maint: untitled periodical (link)
• Castellacci, Giuseppe (2011), Mathematical Reviews, MR 2572942{{citation}}: CS1 maint: untitled periodical (link)
• Myers, Donald E. (August 2011), Technometrics, 53 (3): 324–325, JSTOR 23210411{{citation}}: CS1 maint: untitled periodical (link)
8. Reviews of Stochastic Calculus for Fractional Brownian Motion and Related Processes:
• Gapeev, Pavel, zbMATH, Zbl 1138.60006{{citation}}: CS1 maint: untitled periodical (link)
• Nourdin, Ivan (2008), Mathematical Reviews, MR 2378138{{citation}}: CS1 maint: untitled periodical (link)
External links
• Yuliya Mishura publications indexed by Google Scholar
Authority control
International
• VIAF
Academics
• DBLP
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• IdRef
| Wikipedia |
Yunqing Tang
Yunqing Tang is a mathematician specialising in number theory and arithmetic geometry and an Assistant Professor at University of California, Berkeley. She was awarded the SASTRA Ramanujan Prize in 2022 for "having established, by herself and in collaboration, a number of striking results on some central problems in arithmetic geometry and number theory".[1][2]
Yunqing Tang was born in China and secured a BSc degree from Beijing University in 2011 and then moved to Harvard University for higher education from where she graduated with a Ph D degree in 2016 under the guidance of Mark Kisin. She was associated with Princeton University in several capacities. First she was with the IAS Princeton during 2016-2017, then as an instructor from July 2017 to Jan 2020 and then as an assistant professor from July 2021 to June 2022, In between, she worked as a researcher at CNRS from February 2020 to June 2021. She is with University of California, Berkeley since July 2022.[3] [4]
Work
The citation for SASTRA Ramnujan Prize summarizes Yunqing Tang's contributions to mathematics thus:[2]
"The prize notes that her works display a remarkable combination of sophisticated techniques, in which the arithmetic and geometry of modular curves and of Shimura varieties play a central role, and have strong links with the discoveries of Srinivasa Ramanujan in the area of modular equations. ... she established a new special case of the Ogus conjecture concerning cycles in de Rham cohomology of abelian varieties. She has shown that any abelian surface with real multiplication has infinitely many primes with split reduction. She resolved of the long-standing unbounded coefficient conjecture of Atkin and Swinnertin-Dyer that algebraic functions which are not invariant under any congruence subgroup of SL2(Z), must have unbounded denominators. The study of algebraic functions that are related to the moduli of elliptic integrals, stems from Ramanujan’s own investigations and the plethora of beautiful modular identities that he discovered."
Awards and recognition
The awards and recognition conferred on Yunqing Tang include:[4]
• SASTRA Ramanujan Prize, 2022.
• AWM Dissertation Prize, awarded for outstanding Ph.D dissertations by female students in the US, 2016.
• New World Mathematics Award, Gold Medal for Ph.D thesis awarded for outstanding Chinese mathematics students worldwide, 2016.
• Merit Research Fellowship, Graduate School of Arts and Sciences, Harvard University, 2015 – 2016.
References
1. The Hindu Bubeau (3 October 2022). "SASTRA Ramanujan Prize for 2022 goes to Yunqing Tang". The Hindu. Retrieved 8 November 2022.
2. "YUNQING TANG TO RECEIVE 2022 SASTRA RAMANUJAN PRIZE" (PDF). University of California, Berkeley. Retrieved 8 November 2022.
3. "Yunqing Tang". Princeton University. Retrieved 8 November 2022.
4. "Curriculun Vitae of Yunqing Tang" (PDF). Princeton University. Retrieved 8 November 2022.
| Wikipedia |
Yupana
A yupana (from Quechua: yupay 'count')[1] is a counting board used to perform arithmetic operations, dating back to the time of the Incas. Very little documentation exists concerning its precise physical form or how it was used.
See also: Mathematics of the Incas
Inca Empire
Inca society
• Education
• Religion
• Mythology
• Architecture
• Engineering
• Roads
• Army
• Agriculture
• Ayllu
• Cuisine
Inca history
• Kingdom of Cusco
• Inca Empire
• History of Cusco
• Chimor–Inca War
• Invasion of Chile
• Civil War
• Spanish conquest
• Neo-Inca State
Types
The term yupana refers to two distinct classes of objects:
• Table Yupana (or archaeological yupana): a system of geometric boxes of different sizes and materials. The first example of this type was found in 1869 in the Ecuadorian province of Azuay and prompted searches for more of these objects. All examples of the archaeological yupana vary greatly from each other.[2] Some archaeological yupanas found in Manchán (an archaeological site in Casma) and Huacones-Vilcahuasi (in Cañete) were embedded into the floor.
• Poma de Ayala Yupana: a picture on page 360 of El primer nueva corónica y buen gobierno, written by the Amerindian chronicler Felipe Guaman Poma de Ayala shows a 5x4 chessboard (shown right).[3] The chessboard, though resembling a table yupana, differs from this style in most notably in each of its rectangular trays have the same dimensions, while table yupanas have trays of other polygonal shapes of differing sizes.
Although very different from each other, most scholars who have dealt with table yupanas have extended reasoning and theories to the Poma de Ayala yupana and vice versa, perhaps in an attempt to find a unifying thread or a common method of creation. For example, the Nueva coronica (New Chronicle) discovered in 1916 in the library of Copenhagen contained evidence that a portion of the studies on the Poma de Ayala yupana were based on previous studies and theories regarding table yupanas.[2]
History
Several chroniclers of the Indies described, in brief, this Incan abacus and its operation.
Felipe Guaman Poma de Ayala
The first was Guaman Poma de Ayala around the year 1615 who wrote:
... They count using tables, numbered in increments one hundred thousand to ten thousand, one hundred to ten, and onward until they arrive at one. They keep records of everything that happens in this realm: holidays and Sundays, months and years. The accountants and treasurers of the kingdom are found in every city, town, or indigenous village...
— [3]
In addition to providing this brief description, Poma de Ayala drew a picture of the yupana: a board of five rows and four columns with each cell holding a series of black and white circles.
José de Acosta
Predating Pomo de Ayala's writings, in 1596 The Jesuit father José de Acosta wrote:
... Well, seeing another group which uses kernels of corn is an enchanting thing, as a very embarrassing account, which he will have a very good accountant do by pen and ink, to see how each contribution fits with so many people, taking so much from over there and adding so much from here, with another hundred small pieces, these Indians will take their kernels and put one here, three there, eight I don't know where; they will move a kernel from here, they will barter three from there, and, in fact, they leave with their account done punctually without missing a mark, and much more they know how to put into account and account for what each can pay or give, that we will know how to give to each of them as ascertained by pen and ink. If this is not ingenuity and these men are beasts, let whoever wishes to judge it so judge it, for what I judge to be true is that in what they apply they give us great advantages.
— [4]
Juan de Velasco
In 1841, Father Juan de Velasco wrote:
... these teachers were using something like a series of trays made of wood, stone, or clay, with different separations, in which they put stones of different shapes, colors and angularities...
— [5]
Table yupana
Various table yupana have been found across Ecuador and Peru.
The Chordeleg Yupana
The earliest known example of a table yupana was found in 1869 in Chordeleg, Azuay Province, Ecuador. A rectangular table (33x27 cm) of wood consisting of 17 compartments, 14 of which are square, 2 are rectangular, and one of which is octagonal. Two edges of the table contain other square compartments (12x12 cm) raised and arranged side by side, upon which two square platforms (7x7 cm), are superimposed. These structures are called "towers". The table's compartments are symmetrical with respect to the diagonal of the rectangular compartments. The four sides of the board are also engraved with images of human heads and a crocodile.[2] As a result of this discovery, Charles Wiener a systematic study of these objects in 1877. Wiener concluded that the table yupanas served to calculate the taxes that farmers paid to the Incan empire.[6]
The Caraz Yupana
Found at Caraz between 1878 and 1879, this table yupana differs from that of Chordeleg as the material of construction is the stone and the central octagonal compartment is replaced with a rectangular one; towers also have three shelves instead of two.[2]
The Callejón de Huaylas Yupana
A series of table yupanas much different from the first, was described by Erland Nordenskiöld in 1931. These yupana, made of stone, boast a series of rectangular and square compartments. The tower has two rectangular compartments. The compartments are arranged symmetrically with respect to the axis of the smaller side of the table.[2]
The Triangular Yupana
These yupana, made of stone, have 18 triangular compartments. On one side there is a rectangular tower with one level and three triangular compartments. In the central part there are four square compartments.[2]
The Chan Chan Yupana
Identical to the yupana of Chordeleg, both for the material and the arrangement of the compartments, this table yupana was found in the Chan Chan archaeological complex in Peru in 1967.[2]
The Cárhua de la Bahía Yupana
Discovered in the Peruvian province of Pisco, these are two table yupana in clay and bone. The first is rectangular (47x32 cm), has 22 square (5x5 cm) and three rectangular (16x18 cm) compartments, and has no towers. The second yupana is rectangular (32x23 cm) and has 22 square compartments, two L-shaped compartments and three rectangular compartments in the center. The compartments are arranged symmetrically with respect to the axis of the longer side.[2]
• Fig. A - Structure of a “Chordeleg” table-yupana. Colouring to differentiate the compartments.
• Fig. B - Identication of a stereotyped color
• Fig. C - Really existing tocapu catalogued by Victoria de la Jara
• Fig. D - Other tocapu pattern, possible stylization of the previous one
• Fig. E - Tocapu called “llave inca”, Inca key
The Huancarcuchu Yupana
Discovered in Northern Ecuador by Max Uhle in 1922, this yupana is made of stone and its compartments are drawn onto the surface of the tablet. It has the shape of a pyramid consisting of 10 overlapping rectangles: four on the first level, three on the second, two in the third and one in the fourth. This yupana is the one that is closest to the picture by Poma de Ayala in Nueva Coronica, while having a line fewer and being partially drawn.[2]
The Florio Yupana
C. Florio presents a study [7] which does not identify a yupana in these archaeological findings, but an object whose name has been forgotten and remains unknown. Instead, this object is used to connect to the tocapu (an ideogram already used by pre-Incas civilizations) called “llave inca” (i.e. Inca key) to the yanantin-masintin philosophy. The scholar justifies this based on from the lack of objective evidence that recognizes this object as a yupana, a belief that consolidated over years without repetition or demonstration of this hypothesis, and with the crossing of data from the Miccinelli Documents and the tocapu catalogued by Victoria de la Jara.
Supposing to color the different compartments of the table yupana (fig. A), C. Florio identifies a drawing (fig. B) very similar to an existing tocapu (fig. C) catalogued by Victoria de la Jara. In addition, in the tocapu reported in figure D, also catalogued by V. de la Jara, Florio identifies a stylization of tocapu C and the departure point for creating the tocapu “llave Inca” (Inca key). She finds the relation between the table yupana and the Inca key also similar in their connection with the concept of duality: the table yupana structure is clearly dual and Blas Valera in “Exsul Immeritus Blas Valera populo suo” (one of the two Miccinelli Documents) describes the "Inca key" tocapu as representing the concept of the “opposite forces” and the “number 2”, both strictly linked to the concept of duality.[8]
According to C. Florio, the real yupana used by the Incas is that of Guáman Poma, but with more columns and rows. The Poma de Ayala yupana would have represented just the part of the yupana useful for carrying out a specific calculation, which Florio identifies to be multiplication (see below).
Theories Based On the Poma de Ayala Yupana
Henry Wassen
In 1931, Henry Wassén studied the Poma de Ayala yupana, proposing for the first time a possible representation of the numbers on the board and the operations of addition and multiplication. He interpreted the white circles as gaps carved into yupana into which the seeds described by chroniclers would be inserted: so the white circles correspond to empty gaps, while the blacks circles correspond to the same gaps filled with a black seed.[2]
The numbering system at the base of the yupana was positional notation in base 10 (in line with the writings of the chroniclers of the Indies).
The representation of the numbers then followed a vertical progression such that the numbers 1-9 were positioned in the first row from the bottom, the second row contained the tens, the third contained the hundreds, and so on.
Wassen proposed a progression of values of the seeds that depends on their position in the table: 1, 5, 15, 30, respectively, depending on which seeds occupy a gap in the first, second, third and fourth columns (see the table below). Only a maximum of five seeds could be included in a box belonging to the first column, so that the maximum value of that box was 5, multiplied by the power of the corresponding row. These seeds could be replaced with one seed of the next column, useful during arithmetic operations. According to the theory of Wassen, therefore, the operations of sum and product were carried out horizontally.
This theory received a lot of criticism due to the high complexity of the calculations and was therefore considered inadequate and soon abandoned.
The following table shows the number 13457 as it would appear on Wassen's yupana:
Wassen's Yupana
Powers\Values151530
104 •◦◦◦◦ ◦◦◦ ◦◦ ◦
103 •••◦◦ ◦◦◦ ◦◦ ◦
102 ••••◦ ◦◦◦ ◦◦ ◦
101 ◦◦◦◦◦ •◦◦ ◦◦ ◦
100 ••◦◦◦ •◦◦ ◦◦ ◦
Representation of 13457
This first interpretation of the Poma de Ayala yupana was the starting point for the theories developed by subsequent authors, into the modern writing. No researcher moved away from the positional numbering system until 2008.
Emilio Mendizabal
Emilio Mendizabal was the first to propose in 1976 that the Inca a representation based on the progression 1, 2, 3, 5 in addition to the decimal representation.[9] In the same publication, Mendizabal pointed out that the series of numbers 1, 2, 3, and 5, appear in Poma de Ayala's drawing, and are part of the Fibonacci sequence, and stressed the importance of the "magic" that the number 5 contained for civilizations of Northern Peru, similar in significance to the number 8 for the civilizations of Southern Peru.[2]
Radicati di Primeglio
In 1979, Carlos Radicati di Primeglio emphasized the difference of table yupana from that of Poma de Ayala, describing the state-of-the-art research and advanced theories so far. He also proposed the algorithms for calculating the four basic arithmetic operations for the Poma de Ayala yupana, according to a new interpretation for which it was possible to have up to nine seeds in each box with a vertical progression of powers of ten.[2] Radicati associated each gap with a value of 1.
The following table shows the number 13457 as it would appear on Radicati's yupana:
Radicati's Yupana
Powers\Values1111
104 •◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
103 •••◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
102 ••••◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
101 •••••
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
100 •••••
••◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
Representation of 13457
William Burns Glynn
In 1981, the English textile engineer William Burns Glynn proposed a positional base 10 solution for the yupana of Poma de Ayala.[10]
Glynn, as Radicati, adopted Wassen's idea of full and empty gaps, as well as a vertical progression of the powers of ten, but proposed an architecture that allowed yupana users to greatly simplify the arithmetic operations themselves.
The horizontal progression of the values of the seeds in its representation is 1, 1, 1 for the first three columns, such that in each row is possible to deposit a maximum of ten seeds (5 + 3 + 2 seeds). Ten seeds in any row corresponds to a single seed in the line above it.
The last column in Glynn's yupana is dedicated to the "memory", a place that can hold up to ten seeds before they are moved to the upper line. According to the author, this is very useful during arithmetic operations in order to reduce the possibility of error.
Glynn's solution has been adopted in various teaching projects all over the world, and even today some of its variants are used in some schools of South America.[11][12]
The following table shows the number 13457 as it would appear on Glynn's yupana:
Glynn's Yupana
Potenze\Valori111Memoria
104 •◦◦◦◦ ◦◦◦ ◦◦ ◦
103 •••◦◦ ◦◦◦ ◦◦ ◦
102 ••••◦ ◦◦◦ ◦◦ ◦
101 ••••• ◦◦◦ ◦◦ ◦
100 ••••• ••◦ ◦◦ ◦
Nicolino de Pasquale
In 2001, the Italian engineer Nicolino de Pasquale proposed a positional solution in base 40 of the Poma de Ayala yupana, taking the representation theory of Fibonacci already proposed by Emilio Mendizabal and developing it for the four operations.
De Pasquale's yupana also adopts a vertical progression to represent numbers by powers of 40. The representation of the numbers is based on the fact that the sum of the values of the circles in each row is 39, if each circle takes the value 5 in the first column, 3 in the second column, 2 in the third and 1 in the fourth one; it is thus possible to represent 39 numbers, united to neutral element ( zero or "no seeds" in the table); this forms the basis of 40 symbols necessary for the numbering system.[13]
The following table shows one of the possible representations of the number 13457 in De Pasquale's yupana:
De Pasquale's Yupana
Powers\Values5321
404 ◦◦◦◦◦ ◦◦◦ ◦◦ ◦
403 ◦◦◦◦◦ ◦◦◦ ◦◦ ◦
402 •◦◦◦◦ ◦◦◦ •◦ •
401 ••◦◦◦ ••◦ ◦◦ ◦
400 ••◦◦◦ •◦◦ •• ◦
After its publication, De Pasquale's theory sparked great controversy among researchers who fell into two primary groups: a group supporting the base 10 theory and another supporting the base 40 theory. The Spanish chronicles written of the conquest of the Americas indicated that the Incas used a decimal system and since 2003 the base 10 theory has been proposed as the basis for calculating both with the abacus and the quipu[14]
De Pasquale has recently proposed the use of yupana as astronomical calendar running in mixed base 36/40[15] and provided his own interpretation of the Quechua word huno, translating it as "0.1".[16] This interpretation diverges from all chroniclers of the Indies, especially Domingo de Santo Tomas[1] who in 1560 translated huno into chunga guaranga (ten thousand).
Cinzia Florio
In 2008 Cinzia Florio proposed an alternative and revolutionary approach compared to all the theories proposed so far. Florio's newer theory deviated from the positional numbering system and adopted additive, or sign-value notation.[17]
Relying exclusively on Poma de Ayala's design, Florio explained the arrangement of white and black circles and interpreted the use of the yupana as a board for computing multiplications, in which the multiplicand is represented in the right column, the multiplier in the two central columns, and the product in the left column, illustrated in the following table:
Florio's Yupana
ProductMultiplierMultiplierMultiplicand
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
The theory differs from all the previous in several aspects: first, the white and black circles would not be gaps that could be filled with a seed, but rather different colors of seeds, representing respectively tens and ones (this according to the chronicler Juan de Velasco).[5]
Secondly, the multiplicand is entered in the first column respecting the sign-value notation: so, the seeds can be entered in any order and the number is given by the sum of the values of these seeds.
The multiplier is represented as the sum of two factors, since the procedure for obtaining the product is based on the distributive property of multiplication over addition.
According to Florio, the multiplication table drawn by Poma de Ayala with provision of the seeds represented the calculation: 32 x 5, where the multiplier 5 is decomposed into 3 + 2. The sequence of numbers 1,2,3,5 would be causal, contingent to the calculation done and unrelated to the Fibonacci series.
Florio's Yupana
ProductMultiplicatorMultiplicatorMultiplicand
3X2X
◦◦◦•• ◦◦• •• ◦
◦◦◦◦• ◦◦• ◦◦ •
••••• ◦◦◦ ◦• ◦
◦◦◦◦• ◦◦• ◦• ◦
◦◦◦•• ••• ◦◦ •
151(160)966432
Key: ◦ = 10; • = 1; The operation represented is: 32 x 5 = 32 x (2 + 3) = (32 x 2) + (32 x 3) = 64 + 96 = 160
The numbers represented in the columns are, from left to right:
• 32 (the multiplicand),
• 64 = 32 x 2 and 32 x 3 = 96 (which together constitute the multiplicand, multiplied by the two factors in which the multiplier has been broken down)
• 151 (the product)
The final number in this computation (which is incorrect) is the basis for all possible criticisms of this interpretation, since 160, not 151, is the sum of 96 and 64. Florio notes, however, that the mistake could have been on the part of Poma de Ayala in the original drawing, in designing a space as being occupied by a black circle instead of a white one. In this case, changing just one black circle into a white one in the final column gives us the number 160, the correct product.
Poma de Ayala's yupana cannot represent every multiplicand either, it is necessary to extend the yupana vertically (adding rows) to represent numbers whose sum of digits exceeds 5. The case is the same for the multipliers: to represent all the numbers is necessary to extend the number of columns. Apart from the supposed erroneous calculation (or erroneous representation by the designer), this is the only theory that identifies in Poma de Ayala's yupana a mathematical and consistent message (multiplication) and not a series of random numbers as in other interpretations.
Andrés Chirinos (2010)
In October 2010, Peruvian researcher Andrés Chirinos with the support of the Spanish Agency for International Development Cooperation (ACEID), revised older drawings and descriptions chronicled by Poma de Ayala, and finally deciphered the use of the yupana: a table with eleven holes which Chirinos calls a "Pre-Columbian Calculator", capable of adding, subtracting, dividing, and multiplying, making him hopeful of applying this information to the investigation of how quipus were used and functioned.[18]
See also
• Quipu
• Inca Empire
• Numbering System
References
1. Santo Tomas, "Lexicon o Vocabulario de la lengua general del Peru", 1560
2. Radicati di Primeglio, "Il sistema contabile degli Inca: Yupana e Quipu", 1979
3. Guaman Poma de Ayala, "Primer Nueva Coronica y Buen Gobierno", 1615
4. José de Acosta - Historia Natural y Moral de las Indias - Libro VI cap XVIII (De los memoriales y cuentas que usaron los Indios del Perú)
5. Juan Velasco - “Historia del Reino de Quito” - 1841 44, Tomo II, 7
6. Wong Torres, Zelma (2014-03-16). "Origen de la Contabilidad a Traves del Tiempo". Quipukamayoc. 11 (21): 105. doi:10.15381/quipu.v11i21.5496. ISSN 1609-8196.
7. C. Florio, "Recovering memory - The Inca Key as Yanantin"
8. piruanorum., Laurencich Minelli, Laura, ed. lit. Valera, Blas. Exsul Immeritus Blas valera Populu Suo. Cumis, Joan Antonio. Historia et rudimenta linguae (2009). Exsul Immeritus Blas Valera populo suo e Historia et rudimenta linguae piruanorum : indios, gesuiti e spagnoli in due documenti segreti sul Perù del XVII secolo. Cooperativa Libraria Universitaria Editrice Bologna. OCLC 912444132.{{cite book}}: CS1 maint: multiple names: authors list (link)
9. Emilio., Mendizábal Losack (1976). La pasión racionalista andina. [Universidad Nacional Mayor de San Marcos]. OCLC 10567025.
10. William Burns Glynn, "Calculation table of the Incas", Bol. Lima No. 11, 1981, 1-15.
11. Mora & Valero "La Yupana come strumento pedagogico alle elementari"
12. Fiorentino, "La yupana elettronica: uno strumento per la didattica interculturale della matematica"
13. N. De Pasquale "Il volo del condor", Pescara Informa, 2001
14. Lorenzi, Incan counting system as easy as 1,2,3,5 (2004)
15. N. De Pasquale, "The Saved Kingdom"
16. N. De Pasquale, "Decimal Guaman Poma"
17. C. Florio, "Incontri e disincontri nella individuazione di una relazione matematica nella yupana in Guaman Poma de Ayala", Salerno, 14-15 maggio e 10-12 Dicembre 2008 - Oédipus Editore, 2009
18. Vega, Beatriz (2010-11-20). "Lo Relativo en la Matemática. El Caso de la Proporcionalidad en el 3° Ciclo de la EGB". Yupana (5): 41–52. doi:10.14409/yu.v1i5.260. ISSN 2362-5562.
External links
• Gilsdorf - Ethnomathematics of the Inkas
• Heliane Seline - Mathematics through cultures
• O'Connor & Robertson - Mathematics of the Incas
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• (in Spanish) Poma de Ayala - El Primer Nueva Coronica y Buen Gobierno
• (in Spanish) José De Acosta - Historia Natural y Moral de las Indias
• (in Spanish) Velasco - Historia del reyno de Quito del America del Sur
Theory by Wassen and table-Yupana
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• (in Italian) Fiorentino - La yupana elettronica: uno strumento per la didattica interculturale della matematica
Theory by De Pasquale
• (in Italian) Università Bocconi di Milano - La Matematica nelle civiltà pre-colombiane
• (in English)Incan counting system as easy as 1,2,3,5 - by Rossella Lorenzi
• (in Italian) Notizie sulla numerazione Inca e sulla yupana
• (in Italian) Un italiano scopre l'enigma della matematica inca
• (in Italian) Il Sole 24 Ore Domenica 10 Novembre 2002 – N. 308 – Pagina 35 - di Antonio Aimi - SCIENZA E FILOSOFIA Matematica precolombiana Scoperto il metodo di calcolo degli Inca
• (in Italian) L'unione Sarda - I numeri della natura nella scacchiera degli Inca - di Andrea Mameli
• (in English) "Guaman Poma Game, by N. De Pasquale, D. D'Ottavio
Theory by C. Florio
• (in Italian) Florio - Incontri e disincontri nella individuazione di una relazione matematica nella yupana in Guaman Poma de Ayala
• (in Spanish) Florio - Encuentros y Desencuentros en la identificación de unarelación matemática en la yupana de Guaman Poma de Ayala
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Yuri A. Kuznetsov
Yuri A. Kuznetsov is a Russian-American mathematician currently the M. D. Anderson Chair Professor of Mathematics at University of Houston and Editor-in-Chief of Journal of Numerical Mathematics.[1][2]
References
1. "Yuri A. Kuznetsov". uh.edu. Retrieved May 10, 2017.
2. "Yuri Kuznetsov". ras.ru. Retrieved May 10, 2017.
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Yuri Burago
Yuri Dmitrievich Burago (Russian: Ю́рий Дми́триевич Бура́го; born 1936) is a Russian mathematician. He works in differential and convex geometry.
Yuri Burago
Yuri D. Burago at Oberwolfach in 2006. Photo courtesy MFO.
Born1936
NationalityRussian
Alma materSt. Petersburg State University
AwardsLeroy P. Steele Prize (2014)[1]
Scientific career
FieldsMathematics
InstitutionsSt. Petersburg State University
Doctoral advisorVictor Zalgaller
Aleksandr Aleksandrov
Doctoral studentsSergei Ivanov
Grigori Perelman
Education and career
Burago studied at Leningrad University, where he obtained his Ph.D. and Habilitation degrees. His advisors were Victor Zalgaller and Aleksandr Aleksandrov.
Yuri is a creator (with his students Perelman and Petrunin, and M. Gromov) of what is known now as Alexandrov Geometry. Also brought geometric inequalities to the state of art.
Burago is the head of the Laboratory of Geometry and Topology that is part of the St. Petersburg Department of Steklov Institute of Mathematics.[2] He took part in a report for the United States Civilian Research and Development Foundation for the Independent States of the former Soviet Union.[3]
Works
• Burago, Dmitri; Yuri Burago; Sergei Ivanov (2001-06-12) [1984]. A Course in Metric Geometry. American Mathematical Society (publisher). pp. 417. ISBN 978-0-8218-2129-9.
• Burago, Yuri; Zalgaller, Victor (February 1988) [1980]. Geometric Inequalities. Transl. from Russian by A.B. Sossinsky. Springer Verlag. ISBN 3-540-13615-0.[4]
His other books and papers include:
• Geometry III: Theory of Surfaces (1992)[4]
• Potential Theory and Function Theory for Irregular Regions (1969)[4]
• Isoperimetric inequalities in the theory of surfaces of bounded external curvature (1970)[4]
Students
He has advised Grigori Perelman, who solved the Poincaré conjecture, one of the seven Millennium Prize Problems. Burago was an advisor to Perelman during the latter's post-graduate research at St. Petersburg Department of Steklov Institute of Mathematics.
Footnotes
1. "The Leroy P Steele Prize of the AMS". n.d. Retrieved 5 January 2023.
2. Laboratory of Geometry and Topology
3. U.S. Civilian Research and Development Foundation for the Independent States of the former Soviet Union Archived December 7, 2006, at the Wayback Machine. 2001 Program Report.
4. Bibliography
External links
• Burago's page on the site of Steklov Mathematical Institute
• Yuri Burago at the Mathematics Genealogy Project
• Yuri Dmitrievich Burago in the Oberwolfach Photo Collection
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Yuri Luchko
Yuri Luchko is a German professor of mathematics at the Berlin University of Applied Sciences and Technology. His 90 works were peer-reviewed and appeared in such journals as the Fractional Calculus and Applied Analysis and Journal of Mathematical Analysis and Applications, among others.[1]
References
1. "Yuri Luchko". Google Scholar. Retrieved December 11, 2013.
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Yuri Petunin
Yuri Ivanovich Petunin (Russian: Юрий Иванович Петунин) was a Soviet and Ukrainian mathematician. Petunin was born in the city of Michurinsk (USSR) on September 30, 1937. After graduating from the Tambov State Pedagogical Institute he began his studies at Voronezh State University under the supervision of S.G Krein. He completed his postgraduate studies in 1962, and in 1968 he received his Doctor of Science Degree, the highest scientific degree awarded in the Soviet Union. In 1970 he joined the faculty of the computational mathematics department at Kyiv State University.
Yuri Ivanovich Petunin
Born(1937-09-30)September 30, 1937
Michurinsk, Tambov Oblast, USSR
DiedJune 1, 2011(2011-06-01) (aged 73)
Kyiv, Ukraine
NationalitySoviet Union
Ukraine
Known forFunctional analysis, Mathematical statistics, Biology
Scientific career
FieldsMathematician
InstitutionsKyiv State University
Doctoral advisorSelim Krein
Yuri Petunin is highly regarded for his results in functional analysis. He developed the theory of Scales in Banach spaces,[1] the theory of characteristics of linear manifolds in conjugate Banach spaces,[2] and with S.G. Krein and E.M. Semenov contributed to the theory of interpolation of linear operators.[3] He solved Banach's problem of norming subspaces in conjugate Banach spaces[2] as well as a problem posted by Calderón and Lions concerning interpolation in factor spaces.[3]
In addition to his work in functional analysis, Professor Petunin made significant contributions to pattern recognition and mathematical statistics. He also worked on developing differential diagnostics for oncological disease.[4] The Vysochanskij–Petunin inequality that bears his name formally justifies the so-called 3-sigma rule for unimodal distributions, a rule that has been broadly used in statistics since the time of Gauss. In the area of pattern recognition he developed a theory of linear discriminant rules where he investigated the problems of linear separability of any number of sets in n-dimensional space.[5]
In his later years Yuri Petunin returned to the area of functional analysis where he had begun his scientific research. Together with his colleagues at the department of computational mathematics, he successfully worked toward a solution of Hilbert's 20th problem.[6]
See also
• Vysochanskii–Petunin inequality
References
1. S G Krein and Yu I Petunin, Scales of Banach spaces, 1966 Russ. Math. Surv. 21, 85–129
2. Yu. I. Petunin and A. N. Plichko, The Theory of the Characteristics of Subspaces and Its Applications [in Russian], Vishcha Shkola, Kyiv (1980)
3. S.G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of linear operators, Providence, R.I. : American Mathematical Society, 1982. vii, 375 p.,ISBN 0821845047
4. R.I. Andrushkiw, N.V. Boroday, D.A. Klyushin, Yu.I. Petunin. Computer-aided cytogenetic method of cancer diagnosis. — New York: Nova Publishers, 2007.
5. Yu. I. Petunin and G.A Shuldeshov, The theory of linear discriminant functions I,II, Cybernetics (Russian) no1,2, pp. 34–44, 31–35, 1981.
6. D.A. Klyushin, S.I. Lyasko, D.A. Nomirovskii, Yu.I. Petunin, Vladimir Semenov, Generalized Solutions of Operator Equations and Extreme Elements (Springer Optimization and Its Applications, Vol. 55)
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Gradshteyn and Ryzhik
Gradshteyn and Ryzhik (GR) is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is Table of Integrals, Series, and Products.
Table of Integrals, Series, and Products
Gradshteyn and Ryzhik, seventh English edition, 2007
AuthorRyzhik, Gradshteyn, Geronimus, Tseytlin et al.
CountryRussia
LanguageRussian, German, Polish, English, Japanese, Chinese
GenreMath
PublisherAcademic Press
Publication date
1943, 2015
Since its first publication in 1943, it was considerably expanded and it soon became a "classic" and highly regarded reference for mathematicians, scientists and engineers. After the deaths of the original authors, the work was maintained and further expanded by other editors.
At some stage a German and English dual-language translation became available, followed by Polish, English-only and Japanese versions. After several further editions, the Russian and German-English versions went out of print and have not been updated after the fall of the Iron Curtain, but the English version is still being actively maintained and refined by new editors, and it has recently been retranslated back into Russian as well.
Overview
One of the valuable characteristics of Gradshteyn and Ryzhik compared to similar compilations is that most listed integrals are referenced. The literature list contains 92 main entries and 140 additional entries (in the eighth English edition). The integrals are classified by numbers, which haven't changed from the fourth Russian up to the seventh English edition (the numbering in older editions as well as in the eighth English edition is not fully compatible). The book does not only contain the integrals, but also lists additional properties and related special functions. It also includes tables for integral transforms. Another advantage of Gradshteyn and Ryzhik compared to computer algebra systems is the fact that all special functions and constants used in the evaluation of the integrals are listed in a registry as well, thereby allowing reverse lookup of integrals based on special functions or constants.
On the downsides, Gradshteyn and Ryzhik has become known to contain a relatively high number of typographical errors even in newer editions, which has repeatedly led to the publication of extensive errata lists. Earlier English editions were also criticized for their poor translation of mathematical terms[1][2][3] and mediocre print quality.[1][2][4][5]
History
The work was originally compiled by the Russian mathematicians Iosif Moiseevich Ryzhik (Russian: Иосиф Моисеевич Рыжик, German: Jossif Moissejewitsch Ryschik)[6][nb 1] and Izrail Solomonovich Gradshteyn (Russian: Израиль Соломонович Градштейн, German: Israil Solomonowitsch Gradstein).[6][nb 2] While some contents were original, significant portions were collected from other previously existing integral tables like David Bierens de Haan's Nouvelles tables d'intégrales définies (1867),[6][7] Václav Jan Láska's Sammlung von Formeln der reinen und angewandten Mathematik (1888–1894)[6][8] or Edwin Plimpton Adams' and Richard Lionel Hippisley's Smithsonian Mathematical Formulae and Tables of Elliptic Functions (1922).[6][9]
The first edition, which contained about 5 000 formulas,[10][11][nb 3] was authored by Ryzhik,[nb 1] who had already published a book on special functions in 1936[6][12] and died during World War II around 1941.[6] Not announcing this fact, his compilation was published posthumously[6][nb 1] in 1943, followed by a second corrected edition in his name in 1948.[nb 4]
The third edition (1951) was worked on by Gradshteyn,[13] who also introduced the chapter numbering system in decimal notation. Gradshteyn planned considerable expansion for the fourth edition, a work he could not finish due to his own death.[6][nb 2] Therefore, the fourth (1962/1963) and fifth (1971) editions were continued by Yuri Veniaminovich Geronimus (Russian: Юрий Вениаминович Геронимус, German: Juri Weniaminowitsch Geronimus)[6][nb 5] and Michail Yulyevich Tseytlin (Russian: Михаил Ю́льевич Цейтлин, German: Michael Juljewitsch Zeitlin).[nb 6] The fourth edition contained about 12 000 formulas already.[14][nb 3]
Based on the third Russian edition, the first German-English edition with 5 400 formulas[15][nb 3] was published in 1957 by the East-German Deutscher Verlag der Wissenschaften (DVW) with German translations by Christa[nb 7] and Lothar Berg[nb 8] and the English texts by Martin Strauss.[nb 9] In Zentralblatt für Mathematik Karl Prachar wrote:[16]
"Die sehr reichhaltigen Tafeln wurden nur aus dem Russischen ins Deutsche und Englische übersetzt."
(Translation: The very comprehensive tables were only translated into German and English language.)
In 1963, it was followed by the second edition, a reprint edition with a four-page inlet listing corrections compiled by Eldon Robert Hansen.
Derived from the 1963 edition, but considerably expanded and split into two volumes, the third German-English edition by Ludwig Boll[nb 10] was finally published by MIR Moscow in 1981 (with permission of DVW and distributed through Verlag Harri Deutsch in the Western world); it incorporated the material of the fifth Russian edition (1971) as well.[nb 11]
Pending this third German-English edition an English-only edition by Alan Jeffrey[nb 12] was published in 1965. Lacking a clear designation by itself it was variously known as first, third or fourth English edition, as it was based on the then-current fourth Russian edition. The formulas were photographically reproduced and the text translated. This still held true for the expanded fourth English edition in 1980, which added chapters 10 to 17.[17]
Both of these editions saw multiple print runs each incorporating newly found corrections. Starting with the third printing, updated table entries were marked by adding a small superscript number to the entry number indicating the corresponding print run ("3" etc.), a convention carried over into later editions by continuing to increase the superscript number as kind of a revision number (no longer directly corresponding with actual print runs).
The fifth edition (1994), which contained close to 20 000 formulas,[18][nb 3] was electronically reset[3] in preparation for a CD-ROM issue of the fifth edition (1996) and in anticipation of further editions. Since the sixth edition (2000), now corresponding with superscript number "10", Daniel Zwillinger[nb 13] started contributing as well. The last edition being edited by Jeffrey before his death[nb 12] was the seventh English edition published in 2007 (with superscript number "11").[19] This edition has been retranslated back into Russian as "seventh Russian edition" in 2011.[20][nb 11]
For the eighth edition (2014/2015, with superscript number "12") Zwillinger took over the role of the editor. He was assisted by Victor Hugo Moll.[21][nb 14] In order to make room for additional information without increasing the size of the book significantly, the former chapters 11 (on algebraic inequalities), chapters 13 to 16 (on matrices and related results, determinants, norms, ordinary differential equations) and chapter 18 (on z-transforms) worth about 50 pages in total were removed and some chapters renumbered (12 to 11, 17 to 12). This edition contains more than 10 000 entries.[21][nb 3]
Related projects
In 1995, Alan Jeffrey published his Handbook of Mathematical Formulas and Integrals.[22] It was partially based on the fifth English edition of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products and meant as an companion, but written to be more accessible for students and practitioners.[22] It went through four editions up to 2008.[22][23][24][25] The fourth edition also took advantage of changes incorporated into the seventh English edition of Gradshteyn and Ryzhik.[25]
Inspired by a 1988 paper in which Ilan Vardi proved several integrals in Gradshteyn and Ryzhik,[26] Victor Hugo Moll with George Boros started a project to prove all integrals listed in Gradshteyn and Ryzhik and add additional commentary and references.[27] In the foreword of the book Irresistible Integrals (2004), they wrote:[28]
It took a short time to realize that this task was monumental.
Nevertheless, the efforts have meanwhile resulted in about 900 entries from Gradshteyn and Ryzhik discussed in a series of more than 30 articles[29][30][31] of which papers 1 to 28[lower-alpha 1] have been published in issues 14 to 26 of Scientia, Universidad Técnica Federico Santa María (UTFSM), between 2007 and 2015[60] and compiled into a two-volume book series Special Integrals of Gradshteyn and Ryzhik: the Proofs (2014–2015) already.[61][62]
Editions
Russian editions
• Рыжик, И. М. (1943). Таблицы интегралов, сумм, рядов и произведений [Tables of integrals, sums, series and products] (in Russian) (1 ed.). Moscow: Gosudarstvennoe Izdatel'stvo Tehniko-Teoretičeskoj Literatury (Государственное издательство технико-теоретической литературы) (GITTL / ГИТТЛ) (Tehteoretizdat / Техтеоретиздат). LCCN ltf89006085. 400 pages.[10][63]
• Рыжик, И. М. (1948). Таблицы интегралов, сумм, рядов и произведений (in Russian) (2 ed.). Moscow: Gosudarstvennoe Izdatel'stvo Tehniko-Teoretičeskoj Literatury (Государственное издательство технико-теоретической литературы) (GITTL / ГИТТЛ) (Tehteoretizdat / Техтеоретиздат). 400 pages.[11]
• Рыжик, И. М.; Градштейн, И. С. (1951). Таблицы интегралов, сумм, рядов и произведений (in Russian) (3 ed.). Moscow: Gosudarstvennoe Izdatel'stvo Tehniko-Teoretičeskoj Literatury (Государственное издательство технико-теоретической литературы) (GITTL / ГИТТЛ) (Tehteoretizdat / Техтеоретиздат). LCCN 52034158. 464 pages (+ errata inlet).[63][64][65]
• Градштейн, И. С.; Рыжик, И. М. (1963) [1962]. Геронимус, Ю. В.; Цейтлин, М. Ю́. (eds.). Tablitsy Integralov, Summ, Riadov I Proizvedenii Таблицы интегралов, сумм, рядов и произведений (in Russian) (4 ed.). Moscow: Gosudarstvennoe Izdatel'stvo Fiziko-Matematicheskoy Literatury (Государственное издательство физико-математической литературы) (Fizmatgiz / Физматгиз). LCCN 63027211. 1100 pages (+ 8 page errata inlet in later print runs).[14] Errata:[66]
• Градштейн, И. С.; Рыжик, И. М. (1971). Геронимус, Ю. В.; Цейтлин, М. Ю́. (eds.). Таблицы интегралов, сумм, рядов и произведений (in Russian) (5 ed.). Nauka (Наука). LCCN 78876185. Dark brown fake-leather, 1108 pages.[nb 11]
• Градштейн, И. С.; Рыжик, И. М.; Геронимус, Ю. В.; Цейтлин, М. Ю́. (2011). Jeffrey, Alan; Zwillinger, Daniel (eds.). Таблицы интегралов, ряда и продуктов [Table of Integrals, Series, and Products] (in Russian). Translated by Maximov, Vasily Vasilyevich [in Russian] (7 ed.). Saint-Petersburg, Russia: BHV (БХВ-Петербург). ISBN 978-5-9775-0360-0. GR:11. l+1182 pages.[20][nb 11]
German editions
• Ryshik, Jossif Moissejewitsch; Gradstein, Israil Solomonowitsch (1957). Tafeln / Tables: Summen-, Produkt- und Integral-Tafeln / Tables of Series, Products, and Integrals (in German and English). Translated by Berg, Christa; Berg, Lothar; Strauss, Martin. Foreword by Schröder, Kurt Erich (1 ed.). Berlin, Germany: Deutscher Verlag der Wissenschaften. LCCN 58028629. DNB-IDN 454242255, Lizenz-Nr. 206, 435/2/57. Retrieved 2016-02-21. Gray or green linen, xxiii+438 pages.[15][16] Errata:[65][67][68][69][70][71][72]
• Ryshik, Jossif Moissejewitsch; Gradstein, Israil Solomonowitsch (1963). Tafeln / Tables: Summen-, Produkt- und Integral-Tafeln / Tables of Series, Products, and Integrals (in German and English). Translated by Berg, Christa; Berg, Lothar; Strauss, Martin. Foreword by Schröder, Kurt Erich (2nd corrected ed.). Berlin, Germany: VEB Deutscher Verlag der Wissenschaften (DVW). LCCN 63025905. DNB-IDN 579497747, Lizenz-Nr. 206, 435/93/63. Retrieved 2016-02-21. (Printed by VEB Druckerei "Thomas Müntzer", Bad Langensalza. Distributed in the USA by Plenum Press, Inc., New York.) Green linen, xxiii+438 pages + 4 page errata inlet. Errata:[70]
• Gradstein, Israil Solomonowitsch; Ryshik, Jossif Moissejewitsch (1981). Geronimus, Juri Weneaminowitsch; Zeitlin, Michael Juljewitsch (eds.). Tafeln / Tables: Summen-, Produkt- und Integraltafeln / Tables of Series, Products, and Integrals (in German and English). Translated by Boll, Ludwig (3 ed.). Berlin / Thun / Frankfurt am Main / Moscow: Verlag Harri Deutsch / Verlag MIR Moscow. ISBN 3-87144-350-6. LCCN 82202345. DNB-IDN 551448512, 881086274, 881086282. Gray linen with gilded embossing by A. W. Schipow, 2 volumes, 677+3 & 504 pages.[73][74]
Polish edition
• Ryżyk, I. M.; Gradsztejn, I. S. (1964). Tablice całek, sum, szeregów i iloczynów (in Polish). Translated by Malesiński, Roman (1 ed.). Warsaw, Poland: Państwowe Wydawnictwo Naukowe (PWN). OCLC 749996828. VIAF 309184374. Retrieved 2016-02-16. Light grayish cover, 464 pages.
English editions
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (February 1966) [1965]. Jeffrey, Alan (ed.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (3 ed.). Academic Press. ISBN 0-12-294750-9. LCCN 65029097. Retrieved 2016-02-21. Black cloth hardcover with gilt titles, white dust jacket, xiv+1086 pages.[1] Errata:[1][72][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (1980). Jeffrey, Alan (ed.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (4th corrected and enlarged ed.). Academic Press, Inc. ISBN 978-0-12-294760-5. GR:4,5,6,7. Retrieved 2016-02-21. xlvi+1160 pages.[2][17] Errata:[2][17][93][94][95][96][97][98][99][100][101]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (January 1994). Jeffrey, Alan (ed.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (5 ed.). Academic Press, Inc. MR 1243179. Retrieved 2016-02-21. Blue hardcover with green or blue rectangular and gilt titles, xlvii+1204 pages.[3][18][4][5] (A CD-ROM version with ISBN 0-12-294756-8 / ISBN 978-0-12-294756-8 and LCCN 96-801532 was prepared by Lightbinders, Inc. in July 1996.[102][103][4][5]) Errata:[3][104][105][106][107][4][5][108]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (July 2000). Jeffrey, Alan; Zwillinger, Daniel (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (6 ed.). Academic Press, Inc. ISBN 978-0-12-294757-5. MR 1773820. GR:10. Retrieved 2016-02-21. Red cover, xlvii+1163 pages.[109] (A reprint edition "积分, 级数和乘积表" by World Books Press became available in China under ISBN 7-5062-6546-X / ISBN 978-7-5062-6546-1 in April 2004.) Errata:[32][41][45][109][110][111][112][113]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (February 2007). Jeffrey, Alan; Zwillinger, Daniel (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (7 ed.). Academic Press, Inc. ISBN 978-0-12-373637-6. MR 2360010. GR:11. Retrieved 2016-02-21. xlviii+1171 pages, with CD-ROM.[19][114] (A reprint edition "积分, 级数和乘积表" by Beijing World Publishing Corporation (世界图书出版公司北京公司 / WPCBJ) became available in China under ISBN 7-5062-8235-6 / ISBN 978-7-5062-8235-2 in May 2007.) Errata:[41][45][47][51][53][57][59][115]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. GR:12. Retrieved 2016-02-21. xlvi+1133 pages.[21] Errata:[116][30][117]
Japanese edition
• Градштейн (Guradoshu グラドシュ), И. С.; Рыжик (Rijiku リジク), И. М. (December 1983). Sūgaku daikōshikishū 数学大公式集 [Large mathematics collection] (in Japanese). Translated by Otsuki, Yoshihiko (大槻 義彦) [in Japanese] (1 ed.). Tokyo, Japan: Maruzen (丸善). ISBN 978-4-621-02796-7. NCID BN00561932. JPNO JP84018271. Retrieved 2016-04-06. xv+1085 pages.
See also
• Prudnikov, Brychkov and Marichev (PBM)
• Bronshtein and Semendyayev (BS)
• Abramowitz and Stegun (AS)
• NIST Handbook of Mathematical Functions (DLMF)
• Jahnke and Emde (JE)
• Magnus and Oberhettinger (MO)
Notes
1. [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]
1. Iosif Moiseevich Ryzhik (Иосиф Моисеевич Рыжик)[nb 15] (1918?–1941?). VIAF 15286520. GND 107340518, 1087809320. (NB. Some sources identify him as a sergeant (сержантом) born in 1918, originally from Vitebsk (Витебска), who was drafted into the army in 1939 from Chkalovsk (Чкаловска), Orenburg (Оренбург), and got missing in December 1941. However, since a birth year 1918 would have made him a very young author (18), this could also have been a namesake. In the foreword of the first edition of the book, Ryzhik thanked three mathematicians of the Moscow Mathematical Society for their suggestions and advice: Vyacheslav Vassilievich Stepanov (Вячеслав Васильевич Степанов), Aleksei Ivanovich Markushevich (Алексей Иванович Маркушевич), and Ilya Nikolaevich Bronshtein (Илья Николаевич Бронштейн), suggesting that he must have been in some way associated with this group.)
2. Izrail Solomonovich Gradshteyn (Израиль Соломонович Градштейн) (1899, Odessa – 1958, Moscow). VIAF 20405466, VIAF 310677818, VIAF 270418384. ISNI 0000000116049405. GND 11526194X.
3. Following the sources, this article distinguishes between the documented number of formulas and the number of entries.
4. The fact that Ryzhik's death was not announced before the third edition of the book in 1951 might indicate that his status was unclear for a number of years, or, in the case of the first edition, that typesetting had already started, but actual production of the book had to be delayed and was then finalized in his absence as a consequence of the war.
5. Yuri Veniaminovich Geronimus (Юрий Венеаминович Геронимус) (1923–2013), GND 131451812.
6. Michail Yulyevich Tseytlin (Михаил Ю́льевич Цейтлин), also as M. Yu. Ceitlin, Michael Juljewitsch Zeitlin, Michael Juljewitsch Zeitlein, Michael Juljewitsch Tseitlin, Mikhail Juljewitsch Tseitlin (?–).
7. Christa Berg née Jahncke (?–), GND 122341597 (this entry contains an incorrect birth year and some incorrectly associated books).
8. Lothar Berg (1930-07-28 to 2015-07-27), GND 117708054.
9. Martin D. H. Strauss also as Martin D. H. Strauß (1907-03-18 Pillau, Baltijsk, Ostpreußen – 1978-05-17, East-Berlin, GDR), GND 139569200, German physicist and philosopher.
10. Ludwig Boll (1911-12-10 Gaulsheim, Germany – 1984-12-02), GND 1068090308, German mathematician.
11. The seventh Russian edition (2011) was named after the seventh English edition (2007), of which it was a retranslation. There was no sixth genuinely Russian edition. The English series of editions was originally (1965) based on the fourth Russian edition (1962/1963). It is unknown if any changes for the fifth Russian edition (1971) or the third German-English edition (1981), which did incorporate material from the fifth Russian edition, were reflected in any of the English editions in between (and thereby in the seventh Russian edition as well).
12. Alan Jeffrey (1929-07-16 to 2010-06-06), GND 113118120.
13. Daniel "Dan" Ian Zwillinger (1957–), GND 172475694.
14. Victor Hugo Moll (1956–), GND 173099572.
References
1. Shanks, Daniel (October 1966). "Reviews and Descriptions of Tables and Books 85: Table of Integrals, Series, and Products by I. S. Gradshteyn, I. M. Ryzhik" (PDF). Mathematics of Computation (review). 20 (96): 616–617. doi:10.2307/2003554. JSTOR 2003554. RMT:85. Retrieved 2016-03-04.
2. Luke, Yudell Leo (January 1981). "Reviews and Descriptions of Tables and Books 5: I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980" (PDF). Mathematics of Computation (review). 36 (153): 310–314. doi:10.2307/2007757. JSTOR 2007757. MSC:7.95,7.100. Retrieved 2016-03-04.
3. Kölbig, Kurt Siegfried (January 1995). "Reviews and Descriptions of Tables and Books 1: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Alan Jeffrey, ed.), Academic Press, Boston, 1994" (PDF). Mathematics of Computation. 64 (209): 439–441. doi:10.2307/2153347. JSTOR 2153347. MSC:00A22,33-00,44-00,65-00. Retrieved 2016-03-03.
4. Wimp, Jet (April 1997). "Tables of Integrals, Series and Products By I. S. Gradshteyn and I. M. Ryzhik, edited by Alan Jeffrey". American Mathematical Monthly.
5. Wimp, Jet (October 1997) [1997-09-30]. Koepf, Wolfram (ed.). "2. Tables of Integrals, Series and Products By I. S. Gradshteyn and I. M. Ryzhik, edited by Alan Jeffrey" (PDF). Books and Journals: Reviews. Orthogonal Polynomials and Special Functions. Vol. 8, no. 1. SIAM Activity Group on Orthogonal Polynomials and Special Functions. pp. 13–16. Archived (PDF) from the original on 2022-01-20. Retrieved 2022-01-23.
6. Wolfram, Stephen (2005-10-08). "The History and Future of Special Functions". Wolfram Technology Conference, Festschrift for Oleg Marichev, in honor of his 60th birthday (speech, blog post). Champaign, IL, USA: Stephen Wolfram, LLC. The story behind Gradshteyn-Ryzhik. Archived from the original on 2016-04-07. Retrieved 2016-04-06. […] In 1936 Iosif Moiseevich Ryzhik had a book entitled Special Functions published by the United Moscow-Leningrad Scientific-Technical Publisher. Ryzhik died in 1941, either during the siege of Leningrad, or fighting on the Russian front. In 1943, a table of formulas was published under Ryzhik's name by the Governmental Moscow-Leningrad Technical-Theoretical Publisher. The only thing the book seems to say about its origins is that it's responding to the shortage of books of formulas. It says that some integrals marked in it are original, but the others mostly come from three books—a French one from 1858, a German one from 1894, and an American one from 1922. It explains that effort went into the ordering of the integrals, and that some are simplified by using a new special function s equal to Gamma[x+y-1]/(Gamma[x]Gamma[y]). It then thanks three fairly prominent mathematicians from Moscow University. That's basically all we know about Ryzhik. […] Israil Solomonovitch Gradshteyn was born in 1899 in Odessa, and became a professor of mathematics at Moscow State University. But in 1948, he was fired as part of the Soviet attack on Jewish academics. To make money, he wanted to write a book. And so he decided to build on Ryzhik's tables. Apparently he never met Ryzhik. But he created a new edition, and by the third edition, the book was known as Gradshteyn-Ryzhik. […] Gradshteyn died of natural causes in Moscow in 1958. Though somehow there developed an urban legend that one of the authors of Gradshteyn-Ryzhik had been shot as a piece of anti-Semitism on the grounds that an error in their tables had caused an airplane crash. […] Meanwhile, starting around 1953, Yurii Geronimus, who had met Gradshteyn at Moscow State University, began helping with the editing of the tables, and actually added the appendices on special functions. Later on, several more people were involved. And when the tables were published in the West, there were arguments about royalties. But Geronimus [in 2005 was] still alive and well and living in Jerusalem, and Oleg phoned him […]
7. Bierens de Haan, David (1867). Nouvelles tables d'intégrales définies [New tables of definite integrals] (in French) (1 ed.). Leiden, Netherlands: P. Engels. Retrieved 2016-04-17. (NB. This book had a precursor in 1858 named Tables d'intégrales définies (published by C. G. van der Post in Amsterdam) with supplement Supplément aux tables d'intégrales définies in ca. 1864. Cited as BI (БХ) in GR.)
8. Láska, Václav Jan (1888–1894). Sammlung von Formeln der reinen und angewandten Mathematik [Compilation of formulae of pure and applied mathematics] (in German). Vol. 1–3 (1 ed.). Braunschweig, Germany: Friedrich Vieweg und Sohn. OCLC 24624148. Retrieved 2016-04-17. (NB. The book writes the author's name as Wenzel Láska. Cited as LA (Ла) in GR.)
9. Adams, Edwin Plimpton; Hippisley, Richard Lionel (1922). Greenhill, Alfred George (ed.). Smithsonian Mathematical Formulae and Tables of Elliptic Functions. Smithsonian Miscellaneous Collections. Vol. 74 (1 ed.). Washington D.C., USA: Smithsonian Institution. Retrieved 2016-04-17. (NB. Cited as AD (А) in GR.)
10. Archibald, Raymond Clare (October 1945). "Recent Mathematical Tables 219: I. M. Ryzhik, Tablitsy Integralov, Summ, Riadov i Proizvedeniĭ, Leningrad, OGIZ, 1943" (PDF). Mathematical Tables and Other Aids to Computation (MTAG). American Mathematical Society. 1 (12): 442–443. RMT:219. Retrieved 2016-03-04.
11. Hahn, Wolfgang (1950-07-05). "Rydzik, I. M.: Tabellen für Integrale, Summen, Reihen und Produkte. 2. Aufl. Moskau, Leningrad: Staatsverlag für techn.-theor. Lit., 1948". Zentralblatt für Mathematik (review) (in German). Berlin, Germany. 34 (1/3): 70. Zbl 0034.07001. Retrieved 2016-02-16.
12. Stepanov, Vyacheslav Vassilievich (1936). Preface. Специальные функции: Собрание формул и вспомогательные таблицы [Special functions: A collection of formulas and an auxiliary table]. By Ryzhik, Iosif Moiseevich (in Russian) (1 ed.). Moscow / Leningrad: Объединенное научно-техническое издательство, ONTI. Гострансизд-во. Глав. ред. общетехнич. лит-ры и номографии. Archived from the original on 2016-04-09. Retrieved 2016-04-09. (160 pages.)
13. Rosenfeld (Розенфельд), Boris Abramowitsch (Борис Абрамович) [in German] (2003). "Math.ru" Об Исааке Моисеевиче Ягломе [About Isaac Moiseevich Yaglom]. Мат. просвещение (Mat. enlightenment) (in Russian): 25–28. Archived from the original on 2022-01-12. Retrieved 2022-01-12 – via math.ru. […] во время антисемитской кампании, известной как «борьба с космополитизмом», был уволен вместе с И.М. Гельфандом и И. С. Градштейном […] [during the antisemitic campaign known as the "fight against cosmopolitanism", he was fired along with I. M. Gelfand and I. S. Gradstein.]
14. Bruins, Evert Marie [in German] (1964-01-02). "Gradšteĭn, I. S., und I. M. Ryžik: Integral-, Summen-, Reihen- und Produkttafeln. (Tablicy integralov, summ, rjadov i proizvedeniĭ) 4. Aufl. überarb. unter Mitwirkung von Ju. V. Geronimus und M. Ju. Ceĭtlin. Moskau: Staatsverlag für physikalisch-mathematische Literatur, 1962". Zentralblatt für Mathematik (list) (in German). 103 (1): 38. Zbl 0103.03801. Retrieved 2016-02-16.
15. Wrench, Jr., John William (October 1960). "Reviews and Descriptions of Tables and Books 69: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln: Tables of Series, Products, and Integrals, VEB Deutscher Verlag der Wissenschaften, Berlin" (PDF). Mathematics of Computation. 14 (72): 381–382. doi:10.2307/2003905. JSTOR 2003905. RMT:69. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
16. Prachar, Karl (1959-09-15). "Ryshik, I. M. und I. S. Gradstein: Summen-, Produkt- und Integraltafeln / Tables of series, products and integrals. Berlin: VEB Deutscher Verlag der Wissenschaften, 1957". Zentralblatt für Mathematik (review) (in German). 80 (2): 337–338. Zbl 0080.33703. Archived from the original on 2016-02-17. Retrieved 2016-02-12.
17. Papp, Frank J. "Gradshteyn, I. S.; Ryzhik, I. M.: Tables of integrals, series, and products. Corr. and enl. ed. by Alan Jeffrey. Incorporating the 4th ed. by Yu. V. Geronimus and M. Yu. Tseytlin (M. Yu. Tsejtlin). Transl. from the Russian – New York – London – Toronto. Volumes 1, 2. German and English Transl". Zentralblatt für Mathematik und ihre Grenzgebiete (review). 521: 193. MR 0582453. Zbl 0521.33001. Retrieved 2016-02-16.
18. "Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Transl. from the Russian by Scripta Technica, Inc. 5th ed. Boston, MA: Academic Press, Inc. (1994)". Zentralblatt MATH. 1994. ISBN 9780122947551. Zbl 0918.65002. Retrieved 2016-02-16.
19. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (February 2007). Jeffrey, Alan; Zwillinger, Daniel (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica (7 ed.). Academic Press, Inc. ISBN 978-0-12-373637-6. MR 2360010. GR:11. Retrieved 2016-02-21.
20. Таблицы интегралов, ряда и продуктов [Table of Integrals, Series, and Products] (PDF) (in Russian) (7 ed.). BHV (БХВ-Петербург). 2011. ISBN 978-5-9775-0360-0. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
21. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. GR:12. Retrieved 2016-02-21.
22. Jeffrey, Alan (1995-01-01). Handbook of Mathematical Formulas and Integrals (1 ed.). Academic Press, Inc. ISBN 978-0-12-382580-3.
23. Jeffrey, Alan (2000-08-01). Handbook of Mathematical Formulas and Integrals (2 ed.). Academic Press, Inc. ISBN 978-0-12-382251-2.
24. Jeffrey, Alan (2004). Handbook of Mathematical Formulas and Integrals (3 ed.). Academic Press, Inc. (published 2003-11-21). ISBN 978-0-12-382256-7. Archived from the original on 2022-01-01. Retrieved 2016-03-01.
25. Jeffrey, Alan; Dai, Hui-Hui (2008-02-01). Handbook of Mathematical Formulas and Integrals (4 ed.). Academic Press, Inc. ISBN 978-0-12-374288-9. Retrieved 2016-03-01. (NB. Contents of companion CD-ROM: )
26. Vardi, Ilan (April 1988). "Integrals: An Introduction to Analytic Number Theory" (PDF). American Mathematical Monthly. 95 (4): 308–315. doi:10.2307/2323562. JSTOR 2323562. Archived (PDF) from the original on 2016-03-15. Retrieved 2016-03-14.
27. Moll, Victor Hugo (April 2010) [2009-08-30]. "Seized Opportunities" (PDF). Notices of the American Mathematical Society. 57 (4): 476–484. Archived (PDF) from the original on 2016-04-08. Retrieved 2016-04-08.
28. Boros, George; Moll, Victor Hugo (2006) [September 2004]. Irresistible Integrals. Symbolics, Analysis and Experiments in the Evaluation of Integrals (reprinted 1st ed.). Cambridge University Press (CUP). p. xi. ISBN 978-0-521-79186-1. Retrieved 2016-02-22. (NB. This edition contains many typographical errors.)
29. Moll, Victor Hugo; Vignat, Christophe. "The integrals in Gradshteyn and Ryzhik. Part 29: Chebyshev polynomials" (PDF). Scientia. Series A: Mathematical Sciences. Archived from the original on 2016-03-13. Retrieved 2016-03-13.{{cite journal}}: CS1 maint: unfit URL (link) (NB. This paper discusses 19 GR entries: 1.14.2.1, 1.320, 2.18.1.9, 3.753.2, 3.771.8, 6.611, 7.341.1, 7.341.2, 7.342, 7.343.1, 7.344.1, 7.344.2, 7.346, 7.348, 7.349, 7.355.1, 7.355.2, 8.411.1, 8.921. )
30. Amdeberhan, Tewodros; Dixit, Atul; Guan, Xiao; Jiu, Lin; Kuznetsov, Alexey; Moll, Victor Hugo; Vignat, Christophe. "The integrals in Gradshteyn and Ryzhik. Part 30: Trigonometric functions" (PDF). Scientia. Series A: Mathematical Sciences. Archived from the original on 2016-03-13. Retrieved 2016-03-13.{{cite journal}}: CS1 maint: unfit URL (link) (NB. This paper discusses 51 GR entries: 1.320.1, 1.320.3, 1.320.5, 1.320.7, 2.01.5, 2.01.6, 2.01.7, 2.01.8, 2.01.9, 2.01.10, 2.01.11, 2.01.12, 2.01.13, 2.01.14, 2.513.1, 2.513.2, 2.513.3, 2.513.4, 3.231.5, 3.274.2, 3.541.8, 3.611.3, 3.621.3, 3.621.4, 3.624.6, 3.631.16, 3.661.3, 3.661.4, 3.675.1, 3.675.2, 3.688.1, 3.721.1, 3.747.7, 3.761.4, 4.381.1, 4.381.2, 4.381.3, 4.381.4, 4.521.1, 6.671.7, 6.671.8, 7.244.1, 7.244.2, 7.531.1, 7.531.2, 8.230.1, 8.230.2, 8.361.7, 8.370, 8.910.2, 8.911.1. It also contains 1 errata for GR entry 3.541.8. )
31. Gonzalez, Ivan; Kohl, Karen T.; Moll, Victor Hugo (2014-06-13) [2014-03-19]. "Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets" (PDF). Scientia. Series A: Mathematical Sciences (published 2014). 25: 65–84. Retrieved 2016-03-13. (NB. This paper is also incorporated into volume II.)
32. Moll, Victor Hugo (2006-11-06) [2006-07-21]. "The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 14: 1–6. Archived from the original (PDF) on 2017-02-02. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 10 GR entries: 3.419.2, 3.419.3, 3.419.4, 3.419.5, 3.419.6, 4.232.3, 4.261.4, 4.262.3, 4.263.1, 4.264.3. )
33. Moll, Victor Hugo (2006-11-06) [2006-06-27]. "The integrals in Gradshteyn and Ryzhik. Part 2: Elementary logarithmic integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 14: 7–15. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 5 GR entries: 4.231.1, 4.231.5, 4.231.6, 4.232.1, 4.232.2. )
34. Moll, Victor Hugo (2007-01-16) [2006-12-27]. "The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of logarithms and exponentials" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 15: 31–36. arXiv:0705.0175. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 8 GR entries: 4.331.1, 4.335.1, 4.335.3, 4.352.1, 4.352.2, 4.352.3, 4.352.4, 4.353.2. )
35. Moll, Victor Hugo (2007-01-16) [2006-12-27]. "The integrals in Gradshteyn and Ryzhik. Part 4: The gamma function" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 15: 37–46. arXiv:0705.0179. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 44 GR entries: 3.324.2, 3.326.1, 3.326.2, 3.328, 3.351.3, 3.371, 3.381.4, 3.382.2, 3.434.1, 3.434.2, 3.461.2, 3.461.3, 3.462.9, 3.471.3, 3.478.1, 3.478.2, 3.481.1, 3.481.2, 4.215.1, 4.215.2, 4.215.3, 4.215.4, 4.229.1, 4.229.3, 4.229.4, 4.269.3, 4.272.5, 4.272.6, 4.272.7, 4.325.8, 4.325.11, 4.325.12,, 4.331.1 4.333, 4.335.1, 4.335.3, 4.355.1, 4.355.3, 4.355.4, 4.358.2, 4.358.3, 4.358.4, 4.369.1, 4.369.2. )
36. Amdeberhan, Tewodros; Medina, Luis A.; Moll, Victor Hugo (2007-01-16) [2006-12-27]. "The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 15: 47–60. arXiv:0705.2379. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 10 GR entries: 3.621.1, 3.621.3, 3.621.4, 3.761.11, 3.764.1, 3.764.2, 3.821.3, 3.821.14, 3.822.1, 3.822.2. )
37. Moll, Victor Hugo (2007-10-31) [2007-08-31]. "The integrals in Gradshteyn and Ryzhik. Part 6: The beta function" (PDF). Scientia. Series A: Mathematical Sciences (published 2008). 16: 9–24. arXiv:0707.2121. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 65 GR entries: 3.191.1, 3.191.2, 3.191.3, 3.192.1, 3.192.2, 3.193, 3.194.3, 3.194.4, 3.194.6, 3.194.7, 3.196.2, 3.196.3, 3.196.4, 3.196.5, 3.216.1, 3.216.2, 3.217, 3.218, 3.221.1, 3.221.2, 3.222.2, 3.223.1, 3.223.2, 3.223.3, 3.224, 3.225.1, 3.225.2, 3.225.3, 3.226.1, 3.226.2, 3.241.2, 3.241.4, 3.241.5, 3.248.1, 3.248.2, 3.248.3, 3.249.1, 3.249.2, 3.249.5, 3.249.7, 3.249.8, 3.251.1, 3.251.2, 3.251.3, 3.251.4, 3.251.5, 3.251.6, 3.251.8, 3.251.9, 3.251.10, 3.251.11, 3.267.1, 3.267.2, 3.267.3, 3.311.3, 3.311.9, 3.312.1, 3.313.1, 3.313.2, 3.314, 3.457.3, 4.251.1, 4.273, 4.275.1, 4.321.1. )
38. Amdeberhan, Tewodros; Moll, Victor Hugo (2007-10-31) [2007-08-21]. "The integrals in Gradshteyn and Ryzhik. Part 7: Elementary examples" (PDF). Scientia. Series A: Mathematical Sciences (published 2008). 16: 25–39. arXiv:0707.2122. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 30 GR entries: 2.321.1, 2.321.2, 2.322.1, 2.322.2, 2.322.3, 2.322.4, 3.195, 3.248.4, 3.248.6, 3.249.1, 3.249.6, 3.252.1, 3.252.2, 3.252.3, 3.268.1, 3.310, 3.311.1, 3.351.1, 3.351.2, 3.351.7, 3.351.8, 3.351.9, 3.353.4, 3.411.19, 3.411.20, 3.471.1, 3.622.3, 3.622.4, 4.212.7, 4.222.1. )
39. Moll, Victor Hugo; Rosenberg, Jason; Straub, Armin; Whitworth, Pat (2007-10-31) [2007-08-31]. "The integrals in Gradshteyn and Ryzhik. Part 8: Combinations of powers, exponentials and logarithms" (PDF). Scientia. Series A: Mathematical Sciences (published 2008). 16: 41–50. arXiv:0707.2123. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 7 GR entries: 3.351.1, 4.331.1, 4.351.1, 4.351.2, 4.353.3, 4.362.1, 8.350.1. )
40. Amdeberhan, Tewodros; Moll, Victor Hugo; Rosenberg, Jason; Straub, Armin; Whitworth, Pat (2008-11-18) [2007-11-29]. "The integrals in Gradshteyn and Ryzhik. Part 9: Combinations of logarithms, rational and trigonometric functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2009). 17: 27–44. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 45 GR entries: 2.148.4, 3.747.7, 4.223.1, 4.223.2, 4.224.1, 4.224.2, 4.224.3, 4.224.4, 4.224.5, 4.224.6, 4.225.1, 4.225.2, 4.227.1, 4.227.2, 4.227.3, 4.227.9, 4.227.10, 4.227.11, 4.227.13, 4.227.14, 4.227.15, 4.231.1, 4.231.2, 4.231.3, 4.231.4, 4.231.8, 4.231.9, 4.231.11, 4.231.12, 4.231.13, 4.231.14, 4.231.15, 4.231.19, 4.231.20, 4.233.1, 4.261.8, 4.291.1, 4.291.2, 4.295.5, 4.295.6, 4.295.11, 4.521.1, 4.531.1, 8.366.3, 8.380.3. )
41. Medina, Luis A.; Moll, Victor Hugo (2008-11-18) [2007-11-29]. "The integrals in Gradshteyn and Ryzhik. Part 10: The digamma function" (PDF). Scientia. Series A: Mathematical Sciences (published 2009). 17: 45–66. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 76 GR entries: 3.219, 3.231.1, 3.231.3, 3.231.5, 3.231.6, 3.233, 3.234.1, 3.235, 3.244.2, 3.244.3, 3.265, 3.268.2, 3.269.1, 3.269.3, 3.311.5, 3.311.6, 3.311.7, 3.311.8, 3.311.10, 3.311.11, 3.311.12, 3.312.2, 3.316, 3.317.1, 3.317.2, 3.427.1, 3.427.2, 3.429, 3.434.2, 3.435.3, 3.435.4, 3.442.3, 3.457.1, 3.463, 3.467, 3.469.2, 3.469.3, 3.471.14, 3.475.1, 3.475.2, 3.475.3, 3.476.1, 3.476.2, 4.241.1, 4.241.2, 4.241.3, 4.241.4, 4.241.5, 4.241.7, 4.241.8, 4.241.9, 4.241.10, 4.241.11, 4.243, 4.244.1, 4.244.2, 4.244.3, 4.245.1, 4.245.2, 4.246, 4.247.1, 4.247.2, 4.251.4, 4.253.1, 4.254.1, 4.254.6, 4.256, 4.271.15, 4.275.2, 4.281.1, 4.281.4, 4.281.5, 4.293.8, 4.293.13, 4.331.1, 8.371.2. )
42. Boyadzhiev, Khristo N.; Medina, Luis A.; Moll, Victor Hugo (2009-03-16) [2008-07-02]. "The integrals in Gradshteyn and Ryzhik. Part 11: The incomplete beta function" (PDF). Scientia. Series A: Mathematical Sciences (published 2009). 18: 61–75. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 52 GR entries: 3.222.1, 3.231.2, 3.231.4, 3.241.1, 3.244.1, 3.249.4, 3.251.7, 3.269.2, 3.311.2, 3.311.13, 3.522.4, 3.541.6, 3.541.7, 3.541.8, 3.622.2, 3.623.2, 3.623.3, 3.624.1, 3.635.1, 3.651.1, 3.651.2, 3.656.1, 3.981.3, 4.231.1, 4.231.6, 4.231.11, 4.231.12, 4.231.14, 4.231.19, 4.231.20, 4.234.1, 4.234.2, 4.251.3, 4.254.4, 4.261.2, 4.261.6, 4.261.11, 4.262.1, 4.262.4, 4.263.2, 4.264.1, 4.265, 4.266.1, 4.271.1, 4.271.16, 8.361.7, 8.365.4, 8.366.3, 8.366.11, 8.366.12, 8.366.13, 8.370. )
43. Moll, Victor Hugo; Posey, Ronald A. (2009-03-16) [2008-07-02]. "The integrals in Gradshteyn and Ryzhik. Part 12: Some logarithmic integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2009). 18: 77–84. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 6 GR entries: 4.231.1, 4.233.1, 4.233.2, 4.233.3, 4.233.4, 4.261.8. )
44. Moll, Victor Hugo (2010-10-10) [2009-07-07]. "The integrals in Gradshteyn and Ryzhik. Part 13: Trigonometric forms of the beta function" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 19: 91–96. arXiv:1004.2439. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 21 GR entries: 3.621.1, 3.621.2, 3.621.3, 3.621.4, 3.621.5, 3.621.6, 3.621.7, 3.622.1, 3.623.1, 3.624.2, 3.624.3, 3.624.4, 3.624.5, 3.625.1, 3.625.2, 3.625.3, 3.625.4, 3.626.1, 3.626.2, 3.627, 3.628. )
45. Amdeberhan, Tewodros; Moll, Victor Hugo (2010-10-10) [2009-07-07]. "The integrals in Gradshteyn and Ryzhik. Part 14: An elementary evaluation of entry 3.411.5" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 19: 97–103. arXiv:1004.2440. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 1 GR entry: 3.411.5. )
46. Albano, Matthew; Amdeberhan, Tewodros; Beyerstedt, Erin; Moll, Victor Hugo (2010-07-18) [2010-04-20]. "The integrals in Gradshteyn and Ryzhik. Part 15: Frullani integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 19: 113–119. arXiv:1005.2940. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 12 GR entries: 3.232, 3.329, 3.412.1, 3.434.2, 3.436, 3.476.1, 3.484, 4.267.8, 4.297.7, 4.319.3, 4.324.2, 4.536.2. )
47. Boettner, Stefan Thomas; Moll, Victor Hugo (2010-07-21) [2010-03-22]. "The integrals in Gradshteyn and Ryzhik. Part 16: Complete elliptic integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 20: 45–59. arXiv:1005.2941. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 36 GR entries: 1.421.1, 3.166.16, 3.166.18, 3.721.1, 3.821.7, 3.841.1, 3.841.2, 3.841.3, 3.841.4, 3.842.3, 3.842.4, 3.844.1, 3.844.2, 3.844.3, 3.844.4, 3.844.5, 3.844.6, 3.844.7, 3.844.8, 3.846.1, 3.846.2, 3.846.3, 3.846.4, 3.846.5, 3.846.6, 3.846.7, 3.846.8, 4.242.1, 4.395.1, 4.414.1, 4.432.1, 4.522.4, 4.522.5, 4.522.6, 4.522.7, 8.129.1. )
48. Amdeberhan, Tewodros; Boyadzhiev, Khristo N.; Moll, Victor Hugo (2010-07-21) [2010-03-22]. "The integrals in Gradshteyn and Ryzhik. Part 17: The Riemann zeta function" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 20: 61–71. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 36 GR entries: 3.333.1, 3.333.2, 3.411.1, 3.411.2, 3.411.3, 3.411.4, 3.411.6, 3.411.7, 3.411.8, 3.411.9, 3.411.10, 3.411.11, 3.411.12, 3.411.13, 3.411.14, 3.411.15, 3.411.17, 3.411.18, 3.411.21, 3.411.22, 3.411.25, 3.411.26, 4.231.1, 4.231.2, 4.261.12, 4.261.13, 4.262.1, 4.262.2, 4.262.5, 4.262.6, 4.264.1, 4.264.2, 4.266.1, 4.266.2, 4.271.1, 4.271.2. )
49. Koutschan, Christoph; Moll, Victor Hugo (2010-08-21) [2010-04-26]. "The integrals in Gradshteyn and Ryzhik. Part 18: Some automatic proofs" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 20: 93–111. Archived (PDF) from the original on 2016-03-14. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 7 GR entries: 2.148.3, 3.953, 4.535.1, 6.512.1, 7.322, 7.349, 7.512.5. )
50. Albano, Matthew; Amdeberhan, Tewodros; Beyerstedt, Erin; Moll, Victor Hugo (2011-04-13) [2010-12-23]. "The integrals in Gradshteyn and Ryzhik. Part 19: The error function" (PDF). Scientia. Series A: Mathematical Sciences (published 2011). 21: 25–42. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 42 GR entries: 3.321.1, 3.321.2, 3.321.3, 3.321.4, 3.321.5, 3.321.6, 3.321.7, 3.322.1, 3.322.2, 3.323.1, 3.323.2, 3.325, 3.361.1, 3.361.2, 3.361.3, 3.362.1, 3.362.2, 3.363.1, 3.363.2, 3.369, 3.382.4, 3.461.5, 3.462.5, 3.462.6, 3.462.7, 3.462.8, 3.464, 3.466.1, 3.466.2, 3.471.15, 3.471.16, 3.472.1, 6.281.1, 6.282.1, 6.283.1, 6.283.2, 6.294.1, 8.258.1, 8.258.2, 8.258.3, 8.258.4, 8.258.5. )
51. Kohl, Karen T.; Moll, Victor Hugo (2011-04-13) [2010-12-23]. "The integrals in Gradshteyn and Ryzhik. Part 20: Hypergeometric functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2011). 21: 43–54. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 26 GR entries: 3.194.1, 3.194.2, 3.194.5, 3.196.1, 3.197.1, 3.197.2, 3.197.3, 3.197.4, 3.197.5, 3.197.6, 3.197.7, 3.197.8, 3.197.9, 3.197.10, 3.197.11, 3.197.12, 3.198, 3.199, 3.227.1, 3.254.1, 3.254.2, 3.259.2, 3.312.3, 3.389.1, 9.121.4, 9.131.1. )
52. Boyadzhiev, Khristo N.; Moll, Victor Hugo (2012-10-20) [2012-05-15]. "The integrals in Gradshteyn and Ryzhik. Part 21: Hyperbolic functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 22: 109–127. Archived (PDF) from the original on 2016-03-14. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 75 GR entries: 2.423.9, 3.231.2, 3.231.5, 3.265, 3.511.1, 3.511.2, 3.511.4, 3.511.5, 3.511.8, 3.512.1, 3.512.2, 3.521.1, 3.521.2, 3.522.3, 3.522.6, 3.522.8, 3.522.10, 3.523.2, 3.523.3, 3.523.4, 3.523.5, 3.523.6, 3.523.7, 3.523.8, 3.523.9, 3.523.10, 3.523.11, 3.523.12, 3.524.2, 3.524.4, 3.524.6, 3.524.8, 3.524.9, 3.524.10, 3.524.11, 3.524.12, 3.524.13, 3.524.14, 3.524.15, 3.524.16, 3.524.17, 3.524.18, 3.524.19, 3.524.20, 3.524.21, 3.524.22, 3.524.23, 3.525.1, 3.525.2, 3.525.3, 3.525.4, 3.525.5, 3.525.6, 3.525.7, 3.525.8, 3.527.1, 3.527.2, 3.527.3, 3.527.4, 3.527.5, 3.527.6, 3.527.7, 3.527.8, 3.527.9, 3.527.10, 3.527.11, 3.527.12, 3.527.13, 3.527.14, 3.527.15, 3.527.16, 3.543.2, 4.113.3, 4.231.12, 8.365.9. )
53. Glasser, Larry; Kohl, Karen T.; Koutschan, Christoph; Moll, Victor Hugo; Straub, Armin (2012-10-20) [2012-05-15]. "The integrals in Gradshteyn and Ryzhik. Part 22: Bessel-K functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 22: 129–151. Archived (PDF) from the original on 2016-03-14. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 36 GR entries: 3.323.3, 3.324.1, 3.337.1, 3.364.3, 3.365.2, 3.366.2, 3.372, 3.383.3, 3.383.8, 3.387.1, 3.387.3, 3.387.6, 3.388.2, 3.389.4, 3.391, 3.395.1, 3.461.6, 3.461.7, 3.461.8, 3.461.9, 3.462.20, 3.462.21, 3.462.22, 3.462.23, 3.462.24, 3.462.25, 3.469.1, 3.471.4, 3.471.8, 3.471.9, 3.471.12, 3.478.4, 3.479.1, 3.547.2, 3.547.4, 8.432.6. )
54. Medina, Luis A.; Moll, Victor Hugo (2012-06-25) [2012-02-01]. "The integrals in Gradshteyn and Ryzhik. Part 23: Combination of logarithms and rational functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 23: 1–18. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 54 GR entries: 3.417.1, 3.417.2, 4.212.7, 4.224.5, 4.224.6, 4.225.1, 4.225.2, 4.231.1, 4.231.2, 4.231.8, 4.231.9, 4.231.10, 4.231.11, 4.231.16, 4.231.17, 4.231.18, 4.233.1, 4.234.3, 4.234.6, 4.234.7, 4.234.8, 4.262.7, 4.262.8, 4.262.9, 4.291.1, 4.291.2, 4.291.3, 4.291.4, 4.291.5, 4.291.6, 4.291.7, 4.291.8, 4.291.9, 4.291.10, 4.291.11, 4.291.12, 4.291.13, 4.291.14, 4.291.15, 4.291.16, 4.291.17, 4.291.18, 4.291.19, 4.291.20, 4.291.21, 4.291.22, 4.291.23, 4.291.24, 4.291.25, 4.291.26, 4.291.27, 4.291.28, 4.291.29, 4.291.30. )
55. McInturff, Kim; Moll, Victor Hugo (2012-07-28) [2012-02-01]. "The integrals in Gradshteyn and Ryzhik. Part 24: Polylogarithm functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 23: 45–51. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 10 GR entries: 3.418.1, 3.514.1, 3.531.1, 3.531.2, 3.531.3, 3.531.4, 3.531.5, 3.531.6, 3.531.7, 9.513.1. )
56. Moll, Victor Hugo (2012-07-28) [2012-02-01]. "The integrals in Gradshteyn and Ryzhik. Part 25: Evaluation by series" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 23: 53–65. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 18 GR entries: 3.194.8, 3.311.4, 3.342, 3.451.1, 3.451.2, 3.466.3, 3.747.7, 4.221.1, 4.221.2, 4.221.3, 4.251.5, 4.251.6, 4.269.1, 4.269.2, 8.339.1, 8.339.2, 8.365.4, 8.366.3. )
57. Boyadzhiev, Khristo N.; Moll, Victor Hugo (2015-01-30) [2014-09-19]. "The integrals in Gradshteyn and Ryzhik. Part 26: The exponential integral" (PDF). Scientia. Series A: Mathematical Sciences (published 2015). 26: 19–30. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 41 GR entries: 3.327, 3.351.4, 3.351.5, 3.351.6, 3.352.1, 3.352.2, 3.352.3, 3.352.4, 3.352.5, 3.352.6, 3.353.1, 3.353.2, 3.353.3, 3.353.4, 3.353.5, 3.354.1, 3.354.2, 3.354.3, 3.354.4, 3.355.1, 3.355.2, 3.355.3, 3.355.4, 3.356.1, 3.356.2, 3.356.3, 3.356.4, 3.357.1, 3.357.2, 3.357.3, 3.357.4, 3.357.5, 3.357.6, 3.358.1, 3.358.2, 3.358.3, 3.358.4, 4.211.1, 4.211.2, 4.212.1, 4.212.2. )
58. Medina, Luis A.; Moll, Victor Hugo (2015-01-30) [2014-09-16]. "The integrals in Gradshteyn and Ryzhik. Part 27: More logarithmic examples" (PDF). Scientia. Series A: Mathematical Sciences (published 2015). 26: 31–47. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 37 GR entries: 3.194.3, 3.231.1, 3.231.5, 3.231.6, 3.241.3, 3.621.3, 3.621.5, 4.224.2, 4.231.1, 4.231.2, 4.231.8, 4.233.5, 4.234.4, 4.234.5, 4.235.1, 4.235.2, 4.235.3, 4.235.4, 4.241.5, 4.241.6, 4.241.7, 4.241.11, 4.251.1, 4.251.2, 4.252.1, 4.252.2, 4.252.3, 4.252.4, 4.254.2, 4.254.3, 4.255.2, 4.255.3, 4.257.1, 4.261.9, 4.261.15, 4.261.16, 4.297.8. )
59. Dixit, Atul; Moll, Victor Hugo (2015-02-01) [2014-09-30]. "The integrals in Gradshteyn and Ryzhik. Part 28: The confluent hypergeometric function and Whittaker functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2015). 26: 49–61. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 17 GR entries: 7.612.1, 7.612.2, 7.621.1, 7.621.2, 7.621.3, 7.621.4, 7.621.5, 7.621.6, 7.621.7, 7.621.8, 7.621.9, 7.621.10, 7.621.11, 7.621.12, 8.334.2, 8.703, 9.211.4. )
60. Moll, Victor Hugo (2012). "Index of the papers in Revista Scientia with formulas from GR". Retrieved 2016-02-17.
61. Moll, Victor Hugo (2014-10-01). Special Integrals of Gradshteyn and Ryzhik: the Proofs. Monographs and Research Notes in Mathematics. Vol. I (1 ed.). Chapman and Hall/CRC Press/Taylor & Francis Group, LLC (published 2014-11-12). ISBN 978-1-4822-5651-2. Retrieved 2016-02-12. (NB. This volume compiles Scientia papers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 from issues 14 to 19.)
62. Moll, Victor Hugo (2015-08-24). Special Integrals of Gradshteyn and Ryzhik: the Proofs. Monographs and Research Notes in Mathematics. Vol. II (1 ed.). Chapman and Hall/CRC Press/Taylor & Francis Group, LLC (published 2015-10-27). ISBN 978-1-4822-5653-6. Retrieved 2016-02-12. (NB. This volume compiles 14 Scientia articles from issues 20, 21, 22, 23, 25 and 26 including papers 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 and one unnumbered paper.)
63. "Ryžik, I. M.: Tafeln von Integralen, Summen, Reihen und Produkten. Moskau-Leningrad: Staatsverlag für technisch-theoretische Literatur, 1943". Zentralblatt für Mathematik (list) (in German). 60 (1): 123. 1957-04-01. Zbl 0060.12305. Retrieved 2016-02-16.
64. Prachar, Karl (1952-09-01). "Ryžik, I. M. und I. S. Gradštejn: Tafeln von Integralen, Summen, Reihen und Produkten. 3. umgearb. Aufl. Moskau-Leningrad: Staatsverlag für technisch-theoretische Literatur, 1951". Zentralblatt für Mathematik (review) (in German). 44 (1/5): 133. Zbl 0044.13303. Retrieved 2016-02-16.
65. Wrench, Jr., John William (October 1960). "Table Errata 293: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln: Tables of Series, Products, and Integrals, Deutscher Verlag der Wissenschaften, Berlin, 1957" (PDF). Mathematics of Computation. 14 (72): 401–403. JSTOR 2003934. MTE:293. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
66. Градштейн, И. С.; Рыжик, И. М. (1971). "Errata in 4th edition". In Геронимус, Ю. В.; Цейтлин, М. Ю́. (eds.). Таблицы интегралов, сумм, рядов и произведений (in Russian) (5 ed.). Nauka (Наука). pp. 1101–1108. (NB. The 8-page errata list in later print runs of the fourth Russian edition affected 189 table entries.)
67. Ryshik-Gradstein: Summen-, Produkt- und Integral-Tafeln: Berichtigungen zur 1. Auflage (in German). Berlin, Germany: VEB Deutscher Verlag der Wissenschaften. 1962. MR 0175273. (NB. This brochure was available free of charge from the publisher on request.)
68. "Ryshik-Gradstein: Tafeln Summen Produkte Integrale: Berichtigungen zur 1. Auflage". L'Enseignement Mathématique. Bulletin Bibliographique: Livres Nouveaux (in French and German). 9: 5. 1963. Retrieved 2016-03-04. Die Berichtigung wird den Interessenten auf Anfrage kostenlos durch den Verlag geliefert.
69. Rodman, Richard B. (January 1963). "Table Errata 326: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln, Deutscher Verlag der Wissenschaften, Berlin, 1957" (PDF). Mathematics of Computation. 17 (81): 100–103. JSTOR 2003754. MTE:326. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
70. Schmieg, Glenn M. (July 1966). "Errata 392: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln: Tables of Series, Products, and Integrals, VEB Deutscher Verlag der Wissenschaften, Berlin, 1957" (PDF). Mathematics of Computation. 20 (95): 468–471. doi:10.2307/2003630. JSTOR 2003630. MTE:392. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
71. Filliben, James J. [at Wikidata] (January 1970). "Table Errata 456: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln, VEB Deutscher Verlag der Wissenschaften, Berlin, 1957" (PDF). Mathematics of Computation. 24 (109): 239–242. JSTOR 2004912. MTE:456. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
72. MacKinnon, Robert F. (January 1972). "Table Errata 486: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 26 (117): 305–307. doi:10.1090/s0025-5718-1972-0415970-6. JSTOR 2004754. MR 0415970. MTE:486. Retrieved 2016-03-04.
73. "Gradshtejn, I. S.; Ryzhik, I. M.: Summen-, Produkt- und Integraltafeln. Band 1, 2. Deutsch und englisch. Übers. aus dem Russischen auf der Basis der 5. russ. Aufl., überarb. von J. Geronimus und M. Zeitlin. Tables of series, products, and integrals. Volumes 1, 2. German and English Transl". Zentralblatt für Mathematik und ihre Grenzgebiete (list) (in German). 448: 395. Zbl 0448.65002. Retrieved 2016-02-16.
74. "Gradshtejn, I. S.; Ryzhik, I. M.: Summen-, Produkt- und Integraltafeln. Band 1, 2. Deutsch und englisch. Übers. aus dem Russischen auf der Basis der 5. russ. Aufl., überarb. von J. Geronimus und M. Zeitlin. Tables of series, products, and integrals. Volumes 1, 2. German and English Transl". Zentralblatt für Mathematik und ihre Grenzgebiete (list) (in German). 456: 355. Zbl 0456.65001. Retrieved 2016-02-16.
75. Fettis, Henry E. (April 1967). "Table Errata 408: I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series and Products, Fourth Edition, Academic Press, New York, 1965" (PDF). Mathematics of Computation. 21 (98): 293–295. JSTOR 2004214. MTE:408. Retrieved 2016-03-04.
76. Blake, J. R.; Wood, Van E.; Glasser, M. Lawrence; Fettis, Henry E.; Hansen, Eldon Robert; Patrick, Merrell R. (October 1968). "Table Errata 428: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, fourth edition, prepared by Yu. V. Geronimus & M. Yu. Tseytlin, Academic Press, New York, 1965" (PDF). Mathematics of Computation. 22 (104): 903–907. JSTOR 2004606. MR 0239122. MTE:428. Retrieved 2016-03-04. (NB. See 1972 corrigenda by Fettis and 1979 corrigenda by Anderson.)
77. Corrington, Murlan S.; Fettis, Henry E. (April 1969). "Table Errata 437: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 23 (106): 467–472. JSTOR 2004471. MR 0415966. Retrieved 2016-03-04.
78. Bradley, Lee C. (October 1969). "Table Errata 446: I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series and Products, 4th edition, Academic Press, New York, 1965" (PDF). Mathematics of Computation. 23 (108): 891–892, s15–s17. doi:10.1090/s0025-5718-1969-0415968-8. JSTOR 2004993. MR 0415968. MTE:446. Retrieved 2016-03-04.
79. Young, A. T. (January 1970). "Table Errata 451: A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Tables of Integral Transforms, McGraw-Hill Book Co., New York, 1954" (PDF). Mathematics of Computation. 24 (109): 239–242. doi:10.2307/2004614. JSTOR 2004912. MR 0257656. MTE:451. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16. (NB. The error also affects entry 3.252.10 on page 297 in GR.)
80. Muhlhausen, Carl W.; Konowalow, Daniel D. (January 1971). "Table Errata 473: I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 25 (113): 199–201. JSTOR 2005156. MR 0415969. MTE:473. Retrieved 2016-03-04.
81. Nash, John C. (April 1972). "Table Errata 492: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 26 (118): 597–599. JSTOR 2005198. MR 0415971. MTE:492. Retrieved 2016-03-04.
82. Fettis, Henry E. (April 1972). "Corrigendum: MTE 428, Math. Comp., v.22, 1968, pp. 903–907" (PDF). Mathematics of Computation. 26 (118): 601. doi:10.2307/2005199. JSTOR 2005199. MR 0415973. Retrieved 2016-03-04. (NB. This corrigenda applies to MTE 428.)
83. Ojo, Akin; Sadiku, J. (April 1973). "Table Errata 503: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 27 (122): 451–452. doi:10.1090/s0025-5718-1973-0415972-0. JSTOR 2005649. MR 0415972. MTE:503. Retrieved 2016-03-04. (NB. See 1982 corrigenda by Fettis.)
84. Fettis, Henry E. (April 1982). "Corrigenda: Ojo, Akin; Sadiku, J. (1973). Table Errata 503: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 38 (158): 657. doi:10.1090/S0025-5718-1982-0645681-X. JSTOR 2007312. MR 0645681. Retrieved 2016-03-14. (NB. This corrigenda applies to MTE 503.)
85. Scherzinger, Ann (October 1976). "Table Errata 528: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 30 (136): 899. doi:10.1090/s0025-5718-1976-0408192-x. JSTOR 2005420. MR 0408192. MTE:528. Retrieved 2016-03-04.
86. Carr, Alistair R. (April 1977). "Table Errata 534: I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 31 (138): 614–616. doi:10.1090/s0025-5718-1977-0428676-9. JSTOR 2006446. MR 0428676. MTE:534. Retrieved 2016-03-04.
87. Robinson, Neville I. (January 1978). "Table Errata 550: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 32 (141): 317–320. JSTOR 2006287. MR 0478539. MTE:550. Retrieved 2016-03-04.
88. Fettis, Henry E. (January 1979). "Table Errata 557: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 33 (145): 430–431. JSTOR 2006060. MTE:557. Retrieved 2016-03-04.
89. Anderson, Michael (January 1979). "Corrigenda: p. 906 of MTE 428. I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 33 (145): 432–433. doi:10.2307/2006061. JSTOR 2006061. Retrieved 2016-03-04. (NB. This corrigenda applies to MTE 428.)
90. McGregor, John Ross (April 1979). "Table Errata 564: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 33 (146): 845–846. JSTOR 2006322. MTE:564. Retrieved 2016-03-04.
91. Birtwistle, David T. (October 1979). "Table Errata 565: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 33 (148): 1377. JSTOR 2006476. MTE:565. Retrieved 2016-03-04.
92. Gallas, Jason A. Carlson (October 1980). "Table Errata 572: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 35 (152): 1444. doi:10.1090/S0025-5718-1980-0583522-8. JSTOR 2006418. MR 0583522. MTE:572. Retrieved 2016-03-04.
93. Fettis, Henry E. (January 1981). "Table Errata 577: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 36 (153): 317–318. doi:10.1090/S0025-5718-1981-0595074-8. JSTOR 2007758. MR 0595074. MTE:577. Retrieved 2016-03-04.
94. Fettis, Henry E. (January 1981). "Table Errata 582: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 36 (153): 315–320. JSTOR 2007758. MR 0595074. MTE:582. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-04. (NB. See 1982 corrigenda by Fettis.)
95. Fettis, Henry E. (January 1982). "Corrigenda: Fettis, Henry E. (1981). Table Errata 582: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 38 (157): 337. doi:10.1090/S0025-5718-1982-0637313-1. JSTOR 2007492. MR 0637313. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16. (NB. This corrigenda applies to MTE 582.)
96. van Haeringen, Hendrik; Kok, Lambrecht P. [at Wikidata] (October 1982). "Table Errata 589: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and enlarged edition, Academic Press, New York, First printing, 1980" (PDF). Mathematics of Computation. 39 (160): 747–757. doi:10.1090/S0025-5718-1982-0669666-2. JSTOR 2007357. MR 0669666. MTE:589. Retrieved 2016-02-22.
97. van Haeringen, Hendrik; Kok, Lambrecht P. [at Wikidata]. "I. S. Gradshteyn & I. M. Ryzhik, Tables of integrals, series, and products. Math. comput. 39, 747–757 (1982)". Zentralblatt für Mathematik und ihre Grenzgebiete (review). 521: 193. Zbl 0521.33002. Retrieved 2016-02-16.
98. Fettis, Henry E.; Deutsch, Emeric; Krupnikov, Ernst D. (October 1983). "Table Errata 601: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and enlarged edition, Academic Press, New York, First Printing, 1980" (PDF). Mathematics of Computation. 41 (164): 780–783. doi:10.1090/S0025-5718-1983-0717727-2. JSTOR 2007718. MR 0717727. MTE:601. Retrieved 2016-03-04.
99. Solt, György (October 1986). "Table Errata 607: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition prepared by A. Jeffrey, Academic Press, New York, 1980" (PDF). Mathematics of Computation. 47 (176): 767–768. doi:10.1090/S0025-5718-1986-0856719-2. JSTOR 2008202. MR 0856719. MTE:607. Retrieved 2016-03-03.
100. "Errors in the Integral Tables of Gradshteyn and Ryzhik with Correct Results from Mathematica". Mathematica Information Centre / Wolfram Library Archive: Technical Notes. Champaign, IL, USA: Wolfram Research, Inc. 2004 [2003]. Archived from the original on 2003-04-25. Retrieved 2016-02-16.
101. "Errors in the Integral Tables of Gradshteyn and Ryzhik with Correct Results from Mathematica" (Technical note). Champaign, IL, USA: Wolfram Research, Inc. 2004 [2003]. Archived from the original on 2004-06-19. Retrieved 2016-03-04.
102. "Table of integrals, series, and products. Ed. by Alan Jeffrey. CD-ROM version 1.0 for PC, MAC, and UNIX computers. 5th ed. (English) San Diego, CA: Academic Press (1996)". Zentralblatt MATH. 1996. ISBN 9780122947568. Zbl 0918.65001. Retrieved 2016-02-16.
103. Rosenblum, Marvin (October 1996). Koepf, Wolfram (ed.). "4. Table of Integrals, Series, and Products, CD-ROM Version 1.0 Edited by Alan Jeffrey" (PDF). Books and Journals: Review. Orthogonal Polynomials and Special Functions. Vol. 7, no. 1. SIAM Activity Group on Orthogonal Polynomials and Special Functions. pp. 11–12. Archived (PDF) from the original on 2022-01-20. Retrieved 2022-01-23.
104. Kölbig, Kurt Siegfried (June 1996) [1995]. "Corrigenda: I. S. Gradshteyn & I. M. Ryzhik; Table of Integrals, Series, and Products, Fifth edition, Academic Press, Boston" (PDF). CERN Computing and Networks Division. CN/95/15. Retrieved 2016-02-12.
105. Kölbig, Kurt Siegfried (January 1995). "Table errata 617: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Alan Jeffrey, ed.), Academic Press, Boston, 1994" (PDF). Mathematics of Computation. 64 (209): 449–460. doi:10.1090/S0025-5718-1995-1270626-0. JSTOR 2153354. MR 1270626. MTE:617. Retrieved 2016-03-03.
106. Lambert, Adeline (January 1997). "Table Errata 628". Mathematics of Computation. 66 (217): 463. JSTOR 2153671. MR 1388890. MTE:628.
107. Fikioris, George (October 1998). "Table Errata 634". Mathematics of Computation. 67 (224): 1753–1754. JSTOR 2584882. MR 1625064. MTE:634. MSC:00A22,33-00,65-00.
108. Ruderman, Dan L. (2001-01-22). "Errors in Gradshteyn and Ryzhik, 5th ed". Archived from the original on 2007-02-18. 3.381.3, 3.411.6, 3.721.3, 3.761.2, 3.761.9, 3.897.1, 6.561.13, 8.350.2
109. Rangarajan, Sarukkai Krishnamachari. "Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. 6th ed. San Diego, CA: Academic Press". Zentralblatt MATH. ISBN 9780122947575. Zbl 0981.65001. Retrieved 2016-02-16.
110. "Table Errata 636". Mathematics of Computation. 71 (239): 1335–1336. July 2002. JSTOR 2698918. MTE:636.
111. "Table Errata 637". Mathematics of Computation. 71 (239): 1335–1336. July 2002. JSTOR 2698918. MTE:637.
112. Zwillinger, Daniel; Jeffrey, Alan (2005-11-10). "Errata for Tables of Integrals, Series, and Products, 6th edition by I. S. Gradshteyn and M. Ryzhik edited by Alan Jeffrey and Daniel Zwillinger, Academic Press, Orlando, Florida, 2000, ISBN 0-12-294757-6" (PDF). Archived (PDF) from the original on 2016-03-08. Retrieved 2016-03-08. (NB. This list of 64 pages has 398 entries. According to Daniel Zwillinger it is incomplete.)
113. De Vos, Alexis (2020-11-09) [2009-03-19]. "Alexis De Vos". Universiteit Gent, Belgium. Archived from the original on 2021-06-13. Retrieved 2022-01-12. […] Finally, he is the proud discoverer of an error in equation 3.454.1 of the Gradshteyn and Ryzhik "Tables of integrals, series, and products". See errata for 6th edition by Alan Jeffrey and Daniel Zwillinger, pages 1 and 19. The error is now corrected in the 7th edition page 363 (with acknowledgement in page xxvi). […]
114. "Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. With one CD-ROM (Windows, Macintosh and UNIX). 7th ed. Amsterdam: Elsevier/Academic Press (ISBN 978-0-12-373637-6)". Zentralblatt MATH. ISBN 9780123736376. Zbl 1208.65001. Retrieved 2016-02-16.
115. Zwillinger, Daniel; Jeffrey, Alan (2008-04-11). "Errata for Tables of Integrals, Series, and Products (7th edition) by I. S. Gradshteyn and M. Ryzhik edited by Alan Jeffrey and Daniel Zwillinger, Academic Press, Orlando, Florida, 2007, ISBN 0-12-373637-4" (PDF). Archived (PDF) from the original on 2016-03-08. Retrieved 2016-03-08. (NB. This list of 7 pages has 42 entries. According to Daniel Zwillinger it is incomplete.)
116. Veestraeten, Dirk (2015-07-24) [2015-06-21]. Written at Amsterdam, Netherlands. "Some remarks, generalizations and misprints in the integrals in Gradshteyn and Ryzhik". Scientia. Series A: Mathematical Sciences. Valparaıso, Chile: Universidad Tecnica Federico Santa Marıa. 26: 115–131. ISSN 0716-8446. S2CID 124124467. Archived (PDF) from the original on 2021-12-26. Retrieved 2021-12-26. (18 pages)
117. Zwillinger, Daniel; Moll, Victor Hugo (2021-04-23) [2014-10-06]. "Errata for Tables of Integrals, Series, and Products (8th edition) by I. S. Gradshteyn and M. Ryzhik edited by Daniel Zwillinger and Victor Moll, Academic Press, 2014, ISBN 0-12-384933-0" (PDF) (6 ed.). Archived (PDF) from the original on 2021-04-25. Retrieved 2021-12-26. (NB. This list of 33 pages has 191 entries.)
External links
• Zwillinger, Daniel. "Gradshteyn and Ryzhik: Table of Integrals, Series, and Products (Home Page)". Archived from the original on 2016-03-08. Retrieved 2016-03-08.
• Moll, Victor Hugo. "List with the formulas and proofs in GR". Archived from the original on 2010-01-09. Retrieved 2016-03-08.
• "SCIENTIA, Series A: Mathematical Sciences – Official Journal of the Universidad Técnica Federico Santa María". Universidad Técnica Federico Santa María (UTFSM). Retrieved 2016-03-08.
• Blackley, Jonathan "Seamus" (2021-06-12) [2021-06-11]. "One of my most cherished things, a tool so useful and powerful that I honesty have come to look at it as a friend: Table of Integrals, Series, and Products". twitter. Archived from the original on 2021-09-11. Retrieved 2022-01-12.
| Wikipedia |
Yuri Zhuravlyov (mathematician)
Yuri Ivanovich Zhuravlyov (Russian: Юрий Иванович Журавлёв; 14 January 1935 – 14 January 2022) was a Soviet and Russian mathematician specializing in the algebraic theory of algorithms. His research in applied mathematics and computer science was foundational for a number of specialties within discrete mathematics, pattern recognition, and predictive analysis. Zhuravlyov was a full member of the Russian Academy of Sciences and the chairman of its "Applied Mathematics and Informatics" section. He was also the editor-in-chief of the international journal Pattern Recognition and Image Analysis.
Yuri Zhuravlyov
Born
Yuri Ivanovich Zhuravlyov
(1935-01-14)14 January 1935
Voronezh, RSFSR, Soviet Union
Died14 January 2022(2022-01-14) (aged 87)
Moscow, Russia
NationalityRussian
EducationFull Member RAS (1992)
Alma materMoscow State University
Scientific career
FieldsMathematics
InstitutionsDorodnitsyn Computing Centre, Moscow State University
Biography
Zhuravlyov was born on 14 January 1935 in Voronezh in the former Soviet Union. In 1952, after finishing high school, he applied and was accepted into the Mathematics Department at Moscow State University. Under the direction of Alexey Lyapunov, he completed his first serious work on the minimization of partially defined boolean functions. The work was published in 1955 and awarded first prize at the All-Soviet student research competition.
In 1957, Zhuravlyov completed his master's thesis on a solution to the problem of finding words in a finite set with consideration for its construction. In 1959, he completed his doctoral work which involved a proof for lack of local unsolvability for constructing the minimal disjunctive normal form.
In 1959, he moved to Novosibirsk, where he pursued government-sponsored research and taught algebra and mathematical logic at the Novosibirsk University. In 1966, he began research into pattern recognition. His first serious work in this field related to the identification of deposits in the area of gold mining. He then developed a model of algorithms for calculating estimates that became foundational for numerous subsequent research and works in the field.
In 1969, Zhuravlyov moved to Moscow to head the Pattern Recognition Lab at the Central Soviet Computing Center. In 1970, he also joined the faculty of the Moscow Institute of Physics and Technology as a full professor.
Throughout the 1970s and 1980s, Zhuravlyov published a series of seminal works in applied mathematics and informatics. In 1991, he founded the journal Pattern Recognition and Image Analysis. In 1992, he was invited to join the Russian Academy of Sciences. In 1997, he became a professor at Moscow State University.
Zhuravlyov died in Moscow on 14 January 2022, on his 87th birthday.[1]
References
1. Юрий Иванович Журавлев (14.01.1935 – 14.01.2022) (in Russian)
External links
• Maik journal page
• Mathematics Genealogy Project
• Springer journal page
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• ISNI
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• Germany
• Latvia
• Poland
Academics
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| Wikipedia |
Yurii Egorov
Yurii (or Yuri) Vladimirovich Egorov (Юрий Владимирович Егоров, born 14 July 1938 in Moscow, died October 2018 in Toulouse) was a Russian-Soviet mathematician who specialized in differential equations.
Biography
In 1960 he completed his undergraduate studies at the Mechanics and Mathematics Faculty of Moscow State University (MSU). In 1963 from MSU he received his Ph.D. with the thesis "Некоторые задачи теории оптимального управления в бесконечномерных пространствах" ("Some Problems of Optimal Control Theory in Infinite-Dimensional Spaces"). In 1970 from MSU he received his Russian doctorate of sciences (Doctor Nauk) with thesis: "О локальных свойствах псевдодифференциальных операторов главного типа" ("Local Properties of Pseudodifferential Operators of Principal Type"). He was employed at MSU from 1961 to 1992, and he was a full professor in the Department of Differential Equations of the Mechanics and Mathematics Faculty there from 1973 to 1992. Since 1992 he has been a professor of mathematics at Paul Sabatier University (Toulouse III).
Egorov's research deals with differential equations and applications in mathematical physics, spectral theory, and optimal control theory. In 1970 he was an Invited Speaker of the ICM in Nice.[1]
Awards
• 1981 — Lomonosov Memorial Prize (established in 1944) — for his series of publications on "Субэллиптические операторы и их применения к исследованию краевых задач" (Subelliptic operators and their applications to the study of boundary value problems)
• 1988 — USSR State Prize (with several co-authors) — for their series of publications (1958–1985) on "Исследования краевых задач для дифференциальных операторов и их приложения в математической физике" (Research on boundary value problems and their applications in mathematical physics)
• 1998 — Petrovsky Award (jointly with V. A. Kondratiev) for their series of publications on "Исследование спектра эллиптических операторов" (The study of the spectra of elliptic operators)
Selected publications
Articles
• "The canonical transformations of pseudodifferential operators." Uspekhi Matematicheskikh Nauk 24, no. 5 (1969): 235–236.
• "On the solubility of differential equations with simple characteristics." Russian Mathematical Surveys 26, no. 2 (1971): 113.
• with Mikhail Aleksandrovich Shubin: "Linear partial differential equations. Foundations of the classical theory." Itogi Nauki i Tekhniki. Seriya" Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya" 30 (1988): 5–255.
• "A contribution to the theory of generalized functions." Russian Mathematical Surveys 45, no. 5 (1990): 1.
• with Vladimir Aleksandrovich Kondrat'ev and Olga Arsen'evna Oleynik: "Asymptotic behaviour of the solutions of non-linear elliptic and parabolic systems in tube domains." Sbornik: Mathematics 189, no. 3 (1998): 359–382.
• Victor A. Galaktionov, Vladimir A. Kondratiev, and Stanislav I. Pohozaev: "On the necessary conditions of global existence to a quasilinear inequality in the half-space." Comptes Rendus de l'Académie des Sciences-Series I-Mathematics 330, no. 2 (2000): 93–98.
Books
• with Vladimir A. Kondratiev: On spectral theory of elliptic operators. Operator theory, advances and applications ; vol. 89. Basel; Boston: Birkhäuser Verlag. 1996. ISBN 9783764353902; x+328 pages{{cite book}}: CS1 maint: postscript (link)
• with Bert-Wolfgang Schulze: Pseudo-differential operators, singularities, applications. Operator theory, advances and applications ; vol. 93. Basel; Boston: Birkhäuser Verlag. 1997. ISBN 9783764354848; xiii+349 pages{{cite book}}: CS1 maint: postscript (link)
References
1. Egorov, Yu V. "On the local solvability of pseudodifferential equations." In Actes du Congrès International des Mathématiciens, Tome 2, pp. 717–722. 1970.
Authority control
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Other
• IdRef
| Wikipedia |
Yurii Nesterov
Yurii Nesterov is a Russian mathematician, an internationally recognized expert in convex optimization, especially in the development of efficient algorithms and numerical optimization analysis. He is currently a professor at the University of Louvain (UCLouvain).
Yurii Nesterov
2005 in Oberwolfach
Born (1956-01-25) January 25, 1956
Moscow, USSR
CitizenshipBelgium
Alma materMoscow State University (1977)
Awards
• Dantzig Prize, 2000
• John von Neumann Theory Prize, 2009
• EURO Gold Medal, 2016
Scientific career
Fields
• Convex optimization,
• Semidefinite programming,
• Nonlinear programming,
• Numerical analysis,
• Applied mathematics
Institutions
• UCLouvain
• National Research University
• Central Economic Mathematical Institute
Doctoral advisorBoris Polyak
Biography
In 1977, Yurii Nesterov graduated in applied mathematics at Moscow State University. From 1977 to 1992 he was a researcher at the Central Economic Mathematical Institute of the Russian Academy of Sciences. Since 1993, he has been working at UCLouvain, specifically in the Department of Mathematical Engineering from the Louvain School of Engineering, Center for Operations Research and Econometrics.
In 2000, Nesterov received the Dantzig Prize.[1]
In 2009, Nesterov won the John von Neumann Theory Prize.[2]
In 2016, Nesterov received the EURO Gold Medal.[3]
Academic work
Nesterov is most famous for his work in convex optimization, including his 2004 book, considered a canonical reference on the subject.[4] His main novel contribution is an accelerated version of gradient descent that converges considerably faster than ordinary gradient descent (commonly referred as Nesterov momentum, Nesterov Acceleration or Nesterov accelerated gradient, in short — NAG).[5][6][7][8][9] This method, sometimes called "FISTA", was further developed by Beck & Teboulle in their 2009 paper "A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems".[10]
His work with Arkadi Nemirovski in their 1994 book[11] is the first to point out that the interior point method can solve convex optimization problems, and the first to make a systematic study of semidefinite programming (SDP). Also in this book, they introduced the self-concordant functions which are useful in the analysis of Newton's method.[12]
References
1. "The George B. Dantzig Prize". 2000. Retrieved December 12, 2014.
2. "John Von Neumann Theory Prize". 2009. Retrieved June 4, 2014.
3. "EURO Gold Medal". 2016. Retrieved August 20, 2016.
4. Nesterov, Yurii (2004). Introductory lectures on convex optimization : A basic course. Kluwer Academic Publishers. CiteSeerX 10.1.1.693.855. ISBN 978-1402075537.
5. Nesterov, Y (1983). "A method for unconstrained convex minimization problem with the rate of convergence $O(1/k^{2})$". Doklady AN USSR. 269: 543–547.
6. Walkington, Noel J. (2023). "Nesterov's Method for Convex Optimization". SIAM Review. 65 (2): 539–562. doi:10.1137/21M1390037. ISSN 0036-1445.
7. Bubeck, Sebastien (April 1, 2013). "ORF523: Nesterov's Accelerated Gradient Descent". Retrieved June 4, 2014.
8. Bubeck, Sebastien (March 6, 2014). "Nesterov's Accelerated Gradient Descent for Smooth and Strongly Convex Optimization". Retrieved June 4, 2014.
9. "The zen of gradient descent". blog.mrtz.org. Retrieved 2023-05-13.
10. Beck, Amir; Teboulle, Marc (2009-01-01). "A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems". SIAM Journal on Imaging Sciences. 2 (1): 183–202. doi:10.1137/080716542.
11. Nesterov, Yurii; Arkadii, Nemirovskii (1995). Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics. ISBN 978-0898715156.
12. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (PDF). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
External links
• Official website
John von Neumann Theory Prize
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Yurii Reshetnyak
Yurii Grigorievich Reshetnyak (Russian: Ю́рий Григо́рьевич Решетня́к, 26 September 1929 – 17 December 2021) was a Soviet and Russian mathematician and academician.[1]
Yurii Grigorievich Reshetnyak
Юрий Григорьевич Решетняк
Born(1929-09-26)26 September 1929
Leningrad,
RSFSR, USSR
Died17 December 2021(2021-12-17) (aged 92)
Novosibirsk, Russia
CitizenshipUSSR, Russia
Alma materLeningrad State University
Scientific career
FieldsMathematics
Doctoral advisorA. D. Aleksandrov
He worked in geometry and the theory of functions of a real variable. He was known for his work in the Reshetnyak gluing theorem. Reshetnyak received the 2000 Lobachevsky Prize from the Russian Academy of Sciences.[2]
Reshetnyak died on 17 December 2021, at the age of 92.[3]
Selected publications
• Space mappings with bounded distortion. Translations of Mathematical Monographs. Vol. 73. Providence, RI: American Mathematical Society. 1989. ISBN 0-8218-4526-8; 362 pp.{{cite book}}: CS1 maint: postscript (link)[4]
• with A. D. Aleksandrov: General theory of irregular curves [translated from the Russian by L. Ya. Yuzina]. Dordrecht & Boston: Kluwer Academic Publishers. 1989. ISBN 9027728119; x+288 pp.{{cite book}}: CS1 maint: postscript (link)
References
1. Решетняк Юрий Григорьевич
2. Lobachecvsky Prize, Russian Academy of Sciences. Accessed January 13, 2014
3. В Новосибирске скончался известный математик Юрий Решетняк (in Russian)
4. Vuorinen, Matti (1991). "Review: Space mappings with bounded distortion by Yu. G. Reshetnyak" (PDF). Bull. Amer. Math. Soc. (N.S.). 24 (2): 408–415. doi:10.1090/s0273-0979-1991-16051-9.
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Yuri Manin
Yuri Ivanovich Manin (Russian: Ю́рий Ива́нович Ма́нин; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.
Yuri Manin
Manin in 2006
Born
Yuri Ivanovich Manin
(1937-02-16)16 February 1937
Simferopol, Crimean ASSR, Russian SFSR, Soviet Union
Died7 January 2023(2023-01-07) (aged 85)
NationalityRussian
Alma mater
• Moscow State University
• Steklov Mathematics Institute (PhD)
Known forManin conjecture
Manin matrix
Manin obstruction
Manin triple
Manin–Drinfeld theorem
Manin–Mumford conjecture
ADHM construction
Gauss–Manin connection
Cartier–Manin operator
CH-quasigroup
Modular symbol
Quantum simulator
Awards
• Nemmers Prize in Mathematics (1994)
• Schock Prize (1999)
• Cantor Medal (2002)
• Bolyai Prize (2010)
• King Faisal International Prize (2002)
Scientific career
FieldsMathematics
Institutions
• Max-Planck-Institut für Mathematik
• Northwestern University
Doctoral advisorIgor Shafarevich
Doctoral students
• Alexander Beilinson
• Vladimir Berkovich
• Ivan Cherednik
• Mariusz Wodzicki
• Vladimir Drinfeld
• Ha Huy Khoai
• Vasilli Iskovskikh
• Mikhail Kapranov
• Victor Kolyvagin
• Alexander L. Rosenberg
• Vyacheslav Shokurov
• Alexei Skorobogatov
• Yuri Tschinkel
Life and career
Manin was born on 16 February 1937 in Simferopol, Crimean ASSR, Soviet Union.[1]
He received a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He became a professor at the Max-Planck-Institut für Mathematik in Bonn, where he was director from 1992 to 2005 and then director emeritus.[2][1] He was also a professor emeritus at Northwestern University.[3]
He had over the years more than 40 doctoral students, including Vladimir Berkovich, Mariusz Wodzicki, Alexander Beilinson, Ivan Cherednik, Alexei Skorobogatov, Vladimir Drinfeld, Mikhail Kapranov, Vyacheslav Shokurov, Ralph Kaufmann, Arend Bayer, Victor Kolyvagin and Hà Huy Khoái.[4]
Manin died on 7 January 2023.[1]
Research
Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations. The Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties.[5][6]
He developed the Manin obstruction, indicating the role of the Brauer group in accounting for obstructions to the Hasse principle via Grothendieck's theory of global Azumaya algebras, setting off a generation of further work.[7][8]
Manin pioneered the field of arithmetic topology (along with John Tate, David Mumford, Michael Artin, and Barry Mazur).[9] He also formulated the Manin conjecture, which predicts the asymptotic behaviour of the number of rational points of bounded height on algebraic varieties.[10]
In mathematical physics, Manin wrote on Yang–Mills theory, quantum information, and mirror symmetry.[11][12] He was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable.[13]
He wrote a book on cubic surfaces and cubic forms, showing how to apply both classical and contemporary methods of algebraic geometry, as well as nonassociative algebra.[14]
Awards
He was awarded the Brouwer Medal in 1987, the first Nemmers Prize in Mathematics in 1994, the Schock Prize of the Royal Swedish Academy of Sciences in 1999, the Cantor Medal of the German Mathematical Society in 2002, the King Faisal International Prize in 2002, and the Bolyai Prize of the Hungarian Academy of Sciences in 2010.[1]
In 1990, he became a foreign member of the Royal Netherlands Academy of Arts and Sciences.[15] He was a member of eight other academies of science and was also an honorary member of the London Mathematical Society.[1]
Selected works
• Mathematics as metaphor – selected essays. American Mathematical Society. 2009.
• Rational points of algebraic curves over function fields. {{cite book}}: |work= ignored (help)
• Manin, Yu I. (1965). "Algebraic topology of algebraic varieties". Russian Mathematical Surveys. 20 (6): 183–192. Bibcode:1965RuMaS..20..183M. doi:10.1070/RM1965v020n06ABEH001192. S2CID 250895773.
• Frobenius manifolds, quantum cohomology, and moduli spaces. American Mathematical Society. 1999.[16]
• Quantum groups and non commutative geometry. Montreal: Centre de Recherches Mathématiques. 1988.
• Topics in non-commutative geometry. Princeton University Press. 1991. ISBN 9780691635781.[17]
• Gauge field theory and complex geometry. Grundlehren der mathematischen Wissenschaften. Springer. 1988.[18]
• Cubic forms - algebra, geometry, arithmetics. North Holland. 1986.
• A course in mathematical logic. Springer. 1977.,[19] second expanded edition with new chapters by the author and Boris Zilber, Springer 2010.
• Computable and Uncomputable. Moscow. 1980.{{cite book}}: CS1 maint: location missing publisher (link)[13]
• Mathematics and physics. Birkhäuser. 1981.
• Manin, Yu. I. (1984). "New dimensions in geometry". Arbeitstagung. Lectures Notes in Mathematics. Vol. 1111. Bonn: Springer. pp. 59–101. doi:10.1007/BFb0084585. ISBN 978-3-540-15195-1.
• Manin, Yuri; Kostrikin, Alexei I. (1989). Linear algebra and geometry. London, England: Gordon and Breach. doi:10.1201/9781466593480. ISBN 9780429073816. S2CID 124713118.
• Manin, Yuri; Gelfand, Sergei (1994). Homological algebra. Encyclopedia of Mathematical Sciences. Springer.
• Manin, Yuri; Gelfand, Sergei Gelfand (1996). Methods of Homological algebra. Springer Monographs in Mathematics. Springer. doi:10.1007/978-3-662-12492-5. ISBN 978-3-642-07813-2.
• Manin, Yuri; Kobzarev, Igor (1989). Elementary Particles: mathematics, physics and philosophy. Dordrecht: Kluwer.
• Manin, Yuri; Panchishkin, Alexei A. (1995). Introduction to Number theory. Springer.
• Manin, Yuri I. (2000). "Moduli, Motives, Mirrors". European Congress of Mathematics. Progress in Mathematics. Barcelona. pp. 53–73. doi:10.1007/978-3-0348-8268-2_4. hdl:21.11116/0000-0004-357E-4. ISBN 978-3-0348-9497-5.{{cite book}}: CS1 maint: location missing publisher (link)
• Classical computing, quantum computing and Shor´s factoring algorithm (PDF). Bourbaki Seminar. 1999.{{cite book}}: CS1 maint: location missing publisher (link)
• Rademacher, Hans; Toeplitz, Otto (2002). Von Zahlen und Figuren [From Numbers and Figures] (in German). doi:10.1007/978-3-662-36239-6. ISBN 978-3-662-35411-7.
• Manin, Yuri; Marcolli, Matilde (2002). "Holography principle and arithmetic of algebraic curves". Advances in Theoretical and Mathematical Physics. Max-Planck-Institut für Mathematik, Bonn: International Press. 5 (3): 617–650. doi:10.4310/ATMP.2001.v5.n3.a6. S2CID 25731842.
• Manin, Yu. I. (December 1991). "Three-dimensional hyperbolic geometry as ∞-adic Arakelov geometry". Inventiones Mathematicae. 104 (1): 223–243. Bibcode:1991InMat.104..223M. doi:10.1007/BF01245074. S2CID 121350567.
• Mathematik, Kunst und Zivilisation [Mathematics, Art and Civilisation]. Die weltweit besten mathematischen Artikel im 21. Jahrhundert. Vol. 3. e-enterprise. 2014. ISBN 978-3-945059-15-9.
See also
• Arithmetic topology
• Noncommutative residue
References
1. "Max Planck Institute for Mathematics in Bonn Mourns Death of Yuri Manin". Max Planck Institute for Mathematics. 8 January 2023. Retrieved 8 January 2023.
2. "Yuri Manin | Max Planck Institute for Mathematics". www.mpim-bonn.mpg.de. Retrieved 6 August 2018.
3. "Emeriti Faculty: Department of Mathematics – Northwestern University". math.northwestern.edu. Retrieved 6 August 2018.
4. Yuri Manin at the Mathematics Genealogy Project
5. Manin, Ju. I. (1958), "Algebraic curves over fields with differentiation", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya (in Russian), 22: 737–756, MR 0103889 English translation in Manin, Ju. I. (1964) [1958], "Algebraic curves over fields with differentiation", American Mathematical Society translations: 22 papers on algebra, number theory and differential geometry, vol. 37, Providence, R.I.: American Mathematical Society, pp. 59–78, ISBN 978-0-8218-1737-7, MR 0103889
6. "Gauss-Manin connection", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
7. Serge Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 250–258. ISBN 3-540-61223-8. Zbl 0869.11051.
8. Alexei N. Skorobogatov (1999). Appendix A by S. Siksek: 4-descent. "Beyond the Manin obstruction". Inventiones Mathematicae. 135 (2): 399–424. arXiv:alg-geom/9711006. Bibcode:1999InMat.135..399S. doi:10.1007/s002220050291. S2CID 14285244. Zbl 0951.14013.
9. Morishita, Masanori (2012). "Introduction". Knots and Primes. Universitext. London: Springer. pp. 1–7. doi:10.1007/978-1-4471-2158-9_1. ISBN 978-1-4471-2157-2.
10. Franke, J.; Manin, Y. I.; Tschinkel, Y. (1989). "Rational points of bounded height on Fano varieties". Inventiones Mathematicae. 95 (2): 421–435. Bibcode:1989InMat..95..421F. doi:10.1007/bf01393904. MR 0974910. S2CID 121044839. Zbl 0674.14012.
11. Atiyah, Michael; Drinfeld, Vladimir; Hitchin, Nigel; Manin, Yuri (1978). "Construction of instantons". Physics Letters A. 65 (3): 185–187. Bibcode:1978PhLA...65..185A. doi:10.1016/0375-9601(78)90141-X.
12. Devchand, Chandrashekar; Ogievetsky, Victor I. (1996). "Integrability of N=3 super Yang-Mills equations". Topics in statistical and theoretical physics. Amer. Math. Soc. Transl. Ser. 2. Vol. 177. Providence, RI: American Mathematical Society. pp. 51–58.
13. Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 10 May 2013. Retrieved 4 March 2013.
14. Manin: Cubic forms – algebra, geometry, arithmetics, North Holland 1986
15. "Y.I. Manin". Royal Netherlands Academy of Arts and Sciences. Retrieved 19 July 2015.
16. Getzler, Ezra (2001). "Review: Frobenius manifolds, quantum cohomology, and moduli spaces by Yuri I. Manin". Bull. Amer. Math. Soc. (N.S.). 38 (1): 101–108. doi:10.1090/S0273-0979-00-00888-0.
17. Penkov, Ivan (1993). "Review: Topics in non-commutative geometry by Yuri I. Manin". Bull. Amer. Math. Soc. (N.S.). 29 (1): 106–111. doi:10.1090/S0273-0979-1993-00391-4.
18. LeBrun, Claude (1989). "Review: Gauge field theory and complex geometry by Yuri I. Manin; trans. by N. Koblitz and J. R. King". Bull. Amer. Math. Soc. (N.S.). 21 (1): 192–196. doi:10.1090/S0273-0979-1989-15816-3.
19. Shoenfield, J. R. (1979). "Review: A course in mathematical logic by Yu. I Manin" (PDF). Bull. Amer. Math. Soc. (N.S.). 1 (3): 539–541. doi:10.1090/s0273-0979-1979-14613-5.
Further reading
• Némethi, A. (April 2011). "Yuri Ivanovich Manin" (PDF). Acta Mathematica Hungarica. 133 (1–2): 1–13. doi:10.1007/s10474-011-0151-x.
• Jean-Paul Pier (1 January 2000). Development of Mathematics 1950–2000. Springer Science & Business Media. p. 1116. ISBN 978-3-7643-6280-5.
External links
Wikiquote has quotations related to Yuri Manin.
• Manin's page at Max-Planck-Institut für Mathematik website
• Good Proofs are Proofs that Make us Wiser, interview by Martin Aigner and Vasco A. Schmidt
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Mathematics
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Yuriy Drozd
Yuriy Drozd (Ukrainian: Юрій Анатолійович Дрозд; born October 15, 1944) is a Ukrainian mathematician working primarily in algebra. He is a Corresponding Member of the National Academy of Sciences of Ukraine and head of the Department of Algebra and Topology at the Institute of Mathematics of the National Academy of Sciences of Ukraine.
Yuriy Drozd
Born (1944-10-15) 15 October 1944
Kyiv, Ukrainian SSR
NationalityUkrainian
Alma materTaras Shevchenko National University of Kyiv,
Steklov Institute of Mathematics
AwardsState Prize of Ukraine in Science and Technology
Scientific career
Fieldsmathematics, algebra, representation theory, algebraic geometry
InstitutionsInstitute of Mathematics of NAS of Ukraine
Doctoral advisorIgor Shafarevich
Doctoral studentsVolodymyr Mazorchuk
Biography
Yiriy Drozd graduated from Kyiv University in 1966, pursuing a postgraduate degree at the Institute of Mathematics of the National Academy of Sciences of Ukraine in 1969. His PhD dissertation On Some Questions of the Theory of Integral Representations (1970) was supervised by Igor Shafarevich.
From 1969 to 2006 Drozd worked at the Faculty of Mechanics and Mathematics at Kyiv University (at first as lecturer, then as associate professor and full professor). From 1980 to 1998 he headed the Department of Algebra and Mathematical Logic. Since 2006 he has been the head of the Department of Algebra and Topology (until 2014 - the Department of Algebra) of the Institute of Mathematics of the National Academy of Sciences of Ukraine.
His doctoral students include Volodymyr Mazorchuk.
References
• Mathematics Genealogy Project.
• Institute of Mathematics of the National Academy of Sciences of Ukraine.
• Personal site.
• Oberwolfach Photo Collection.
External links
• Yuriy Drozd, Introduction to Algebraic Geometry (course lecture notes, University of Kaiserslautern).
• Yuriy Drozd, Vector Bundles over Projective Curves.
• Yuriy Drozd, General Properties of Surface Singularities.
• Drozd, Yuriy; Kirichenko, Vladimir (1994). Finite-Dimensional Algebras. Springer. ISBN 978-3-642-76244-4.
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Yuri Aleksandrovich Brychkov
Yury Aleksandrovich Brychkov (Russian: Юрий Александрович Брычков; born 29 February 1944 in Moscow, Russia) is a Russian mathematician.
Yury Aleksandrovich Brychkov
Юрий Александрович Брычков
Born (1944-02-29) 29 February 1944
Moscow, USSR
Alma materMoscow State University
Known forTables of Series, Special Functions, Integral Transforms
Scientific career
FieldsMathematics
Doctoral advisorYuri Mikhailovich Shirokov
He graduated from Moscow State University in 1966 and worked on quantum field theory at the Steklov Mathematical Institute of the Russian Academy of Sciences, under the supervision of Yuri Mikhailovich Shirokov. He received his PhD in 1971 and he has been with the Dorodnicyn Computing Centre of the Russian Academy of Sciences since 1969.
Yu. A. Brychkov has worked on various topics of pure mathematics, and he has made contributions to the fields of special functions and integral transforms. He has also worked on the computer implementation of special functions at the University of Waterloo,[1] Maplesoft, and Wolfram Research.[2] He is a founding editor of the Journal of Integral Transforms and Special Functions,[3] and has authored a number of handbooks, including the five volume Integrals and Series (Gordon and Breach Science Publishers, 1986–1992).[4]
Works
• Brychkov, Yu. A.; Prudnikov, A. P. (1989). Integral transformations of generalized functions. New York-London: Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 2-88124-705-9. (342 pages)
• Brychkov, Yu. A.; Marichev, O. I.; Prudnikov, A. P. (1989). Tables of indefinite integrals. New York-London: Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 2-88124-710-5. (192 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. Integraly i ryady Интегралы и ряды [Integrals and series] (in Russian). Vol. Set 1-3 (1 ed.). Nauka (Наука). Moscow. 1981−1986.
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. Integrals and series. Vol. Set 1-5. Gordon & Breach Science Publishers / CRC Press. New York-London. 1986−1992.
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1986). Integrals and Series. Vol. 1: Elementary Functions. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 978-2-88124-089-8. OCLC 916363878. (798 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1986). Integrals and Series. Vol. 2: Special functions. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 978-2-88124-090-4. OCLC 50653126. (750 pages.)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1990). Integrals and Series. Vol. 3: More special functions. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 978-2-88124-682-1. OCLC 916363880. (800 pages.)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1992). Integrals and Series. Vol. 4: Direct Laplace Transforms. Gordon & Breach Science Publishers / CRC Press. New York-London. OCLC 63722509. (Second printing: 1998.) (xviii+618 pages.)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1992). Integrals and Series. Vol. 5: Inverse Laplace Transforms. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 2-88124838-1. OCLC 489706146. (xx+595 pages.)
• Brychkov, Yu. A.; Glaeske, H.-Ju.; Prudnikov, A. P.; Vu Kim, Tuan (1992). Multidimensional integral transformations. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 2-88124-839-X. (386 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (2003). Integraly i ryady Интегралы и ряды [Integrals and series] (in Russian). Vol. Set 1-3 (2nd revised ed.). Fizmatlit (Физматлит). ISBN 978-5-9221-0322-0.
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Elementary functions] (in Russian). Vol. 1: Elementarnye funktsii (Элементарные функции) (2nd revised ed.). Fizmatlit (Физматлит). Moscow. ISBN 978-5-9221-0323-7. OCLC 937142305. (630 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Special functions] (in Russian). Vol. 2: Spetsialnye funktsii (Специальные функции) (2nd revised ed.). Fizmatlit (Физматлит). Moscow. ISBN 978-5-9221-0324-4. (663 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Special functions. Further chapters.] (in Russian). Vol. 3: Spetsialnye funktsii. Dopolnitelnye glavy (2nd revised ed.). Fizmatlit (Физматлит). Moscow. ISBN 978-5-9221-0325-1. (710 pages)
• Brychkov, Yu. A. (2008). Handbook of special functions. Derivatives, integrals, series and other formulas. CRC Press. Boca Raton. ISBN 978-1-58488-956-4. (xx+680 pages)
• Brychkov, Yu. A.; Marichev, O. I.; Savischenko., N. V. (2018). Handbook of Mellin Transforms. CRC Press. Boca Raton. ISBN 978-1-13835-335-0. (xx+587 pages)
References
1. "University of Waterloo: Telephone Directory, September 2002" (PDF). Information Systems & Technology, University of Waterloo. Retrieved 2018-05-10.
2. "Wolfram Blog: Author Index: Yury Brychkov". Wolfram. Retrieved 2018-05-10.
3. "Integral Transforms and Special Functions: Editorial Board". Integral Transforms and Special Functions. 26 (12): ebi. 2015. doi:10.1080/10652469.2015.1088624.
4. Integrals and series / A.P. Prudnikov, Yu. A. Brychkov, O.I. Marichev; translated from the Russian by N.M. Queen. 1986. ISBN 9782881240973. Retrieved 2018-05-10. {{cite book}}: |website= ignored (help)
External links
• New Derivatives of the Bessel Functions Have Been Discovered
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Yuri Gurevich
Yuri Gurevich, Professor Emeritus at the University of Michigan, is an American computer scientist and mathematician and the inventor of abstract state machines.
Gurevich was born and educated in the Soviet Union.[1] He taught mathematics there and then in Israel before moving to the United States in 1982. The best-known work of his Soviet period is on the classical decision problem.[2] In Israel, Gurevich worked with Saharon Shelah on monadic second-order theories.[3] The Forgetful Determinacy Theorem of Gurevich–Harrington is of that period as well.[4]
From 1982 to 1998, Gurevich taught computer science at the University of Michigan, where he started to work on various aspects of computational complexity theory[5] including average case complexity.[6] He became one of the founders of the emerging field of finite model theory.[7]
Most importantly, he became interested in the problem of what an algorithm is. This led him to the theory of abstract state machines (ASMs). The ASM Thesis says that, behaviorally, every algorithm is an ASM.[8] A few convincing axioms enabled derivation of the sequential ASM thesis[9] and the Church–Turing thesis.[10] The ASM thesis has also been proven for some other classes of algorithms.[11][12]
From 1998 to 2018, Gurevich was with Microsoft Research where he founded a group on Foundations of Software Engineering. The group built Spec Explorer based on the theory of abstract state machines. The tool was adopted by the Windows team; a modified version of the tool helped Microsoft meet the European Union demands for high-level executable specifications. Later, Gurevich worked with different Microsoft groups on various efficiency, safety, and security issues,[13] including access control,[14] differential compression,[15] and privacy.[16]
Since 1988, Gurevich has managed the column on Logic in Computer Science in the Bulletin of the European Association for Theoretical Computer Science.[17] Since 2013 Gurevich has worked primarily on quantum computing,[18] while continuing research in his traditional areas.
Gurevich is a 2020 AAAS Fellow,[19] a 1997 ACM Fellow,[20] a 1995 Guggenheim Fellow,[21] an inaugural fellow of the European Association for Theoretical Computer Science,[22] a member of Academia Europaea, and Dr. Honoris Causa of Hasselt University in Belgium and of Ural State University in Russia.
References
1. Blass, Andreas; Dershowitz, Nachum; Reisig, Wolfgang (2010), Blass, Andreas; Dershowitz, Nachum; Reisig, Wolfgang (eds.), "Yuri, Logic, and Computer Science", Fields of Logic and Computation, Berlin, Heidelberg: Springer Berlin Heidelberg, vol. 6300, pp. 1–48, doi:10.1007/978-3-642-15025-8_1, ISBN 978-3-642-15024-1, retrieved 2023-07-05
2. E. Börger, E. Grädel, and Y. Gurevich. The Classical Decision Problem. Springer, 1997.
3. Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman (eds.), Model-Theoretic Logics, Springer, 1985, 479-506.
4. Y. Gurevich and L. Harrington. Automata, Trees, and Games. STOC '82: Proceedings of the Fourteenth annual ACM Symposium on Theory of Computing, 1982, 60-65.
5. Y. Gurevich and S. Shelah. Expected computation time for Hamiltonian Path Problem. SIAM Journal on Computing 16:3, 1987, 486-502.
6. Y. Gurevich. Average case completeness. Journal of Computer and System Sciences 42:3, 1991, 346-398.
7. Y. Gurevich. Toward logic tailored for computational complexity. In M Richter et al. (eds.), Computation and Proof Theory. Springer Lecture Notes in Mathematics 1104, 1984, 175-216.
8. Y. Gurevich. Evolving Algebra 1993: Lipari Guide. In E. Börger (ed.), Specification and Validation Methods, Oxford University Press, 1995, 9–36. https://arxiv.org/abs/1808.06255
9. Y. Gurevich. Sequential Abstract State Machines capture sequential algorithms. ACM Transactions on Computational Logic 1(1), 2000.
10. N. Dershowitz and Y. Gurevich. A natural axiomatization of computability and proof of Church’s Thesis. Bulletin of Symbolic Logic 14:3, 2008, 299-350.
11. A. Blass and Y. Gurevich. Abstract State Machines Capture Parallel Algorithms. ACM Transactions on Computational Logic 4(4), 2003, 578–651, and 9(3), 2008, article 19.
12. A. Blass, Y. Gurevich, D. Rosenzweig, and B. Rossman. Interactive Small-Step Algorithms II: Abstract State Machines and the Characterization Theorem. Logical Methods in Computer Science 3(4), 2007, paper 4.
13. "Google Patents".
14. A. Blass, Y. Gurevich, M. Moskal, and I. Neeman. Evidential authorization. In S. Nanz (ed), The Future of Software Engineering, Springer 2011, 77–99.
15. N. Bjørner, A. Blass, and Y. Gurevich. Content-dependent chunking for differential compression: The local maximum approach. Journal of Computer Systems Science 76(3-4), 2010, 154-203.
16. Y. Gurevich, E. Hudis, and J.M. Wing. Inverse privacy. Communications of the ACM 59(7), 2016, 38-42.
17. https://eatcs.org/index.php/eatcs-bulletin
18. A. Bocharov, Y. Gurevich, and K.M. Svore. Efficient decomposition of single-qubit gates into V basis circuits. Physical Review A 88:1, 2013.
19. AAAS Fellows, retrieved on Jan 11, 2021.
20. ACM Fellows, Association for Computing Machinery. Accessed February 16, 2010.
21. Fellows List, Archived June 22, 2011, at the Wayback Machine John Simon Guggenheim Memorial Foundation. Accessed February 16, 2010.
22. "EATCS names 2014 fellows", Milestones: Computer Science Awards, Appointments, Communications of the ACM, 58 (1): 24, January 2015, doi:10.1145/2686734, S2CID 11485095
External links
• Gurevich's Homepage
• Yuri Gurevich, Mathematics Genealogy Project
Wikimedia Commons has media related to Yuri Gurevich.
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Yutaka Taniyama
Yutaka Taniyama (谷山 豊, Taniyama Yutaka[1], 12 November 1927 – 17 November 1958) was a Japanese mathematician known for the Taniyama–Shimura conjecture.
Yutaka Taniyama
谷山 豊
Born(1927-11-12)12 November 1927
Kisai near Tokyo, Japan
Died17 November 1958(1958-11-17) (aged 31)
Tokyo, Japan
Alma materUniversity of Tokyo
Known forContributions in Algebraic Number Theory, Taniyama–Shimura conjecture
Scientific career
FieldsMathematics
InstitutionsUniversity of Tokyo
Contribution
Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and refined case of this conjecture for elliptic curves over rationals is called the Taniyama–Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro Shimura. The names Taniyama, Shimura and Weil have all been attached to this conjecture, but the idea is essentially due to Taniyama.
“Taniyama's interests were in algebraic number theory and his fame is mainly due to two problems posed by him at the symposium on Algebraic Number Theory held in Tokyo and Nikko in 1955. His meeting with André Weil at this symposium was to have a major influence on Taniyama's work. These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field. This conjecture proved to be a major component in the proof of Fermat's Last Theorem by Andrew Wiles.”[2]
In 1986 Ken Ribet proved that if the Taniyama–Shimura conjecture held, then so would Fermat's Last Theorem, which inspired Andrew Wiles to work for a number of years in secrecy on it, and to prove enough of it to prove Fermat's Last Theorem. Owing to the pioneering contribution of Wiles and the efforts of a number of mathematicians, the Taniyama–Shimura conjecture was finally proven in 1999. The original Taniyama conjecture for elliptic curves over arbitrary number fields remains open.
In an episode of Nova (American TV program) on the proof of Fermat's Last Theorem, reflecting on Taniyama's work, Goro Shimura stated:
Taniyama was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction and so eventually he got right answers. I tried to imitate him, but I found out that it is very difficult to make good mistakes.[3] [4]
Depression and death
In 1958, Taniyama worked for University of Tokyo as an assistant (joshu), was engaged, and was offered a position at the Institute for Advanced Study in Princeton, New Jersey. On 17 November 1958, Taniyama committed suicide. He left a note explaining how far he had progressed with his teaching duties, and apologizing to his colleagues for the trouble he was causing them. His suicide note read:
Until yesterday I had no definite intention of killing myself. But more than a few must have noticed that lately I have been tired both physically and mentally. As to the cause of my suicide, I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future. There may be someone to whom my suicide will be troubling or a blow to a certain degree. I sincerely hope that this incident will cast no dark shadow over the future of that person. At any rate, I cannot deny that this is a kind of betrayal, but please excuse it as my last act in my own way, as I have been doing my own way all my life.
Although his note is mostly enigmatic it does mention tiredness and a loss of confidence in his future. Taniyama's ideas had been criticized as unsubstantiated and his behavior had occasionally been deemed peculiar. Goro Shimura mentioned that he suffered from depression. Taniyama also mentioned in the note his concern that some might be harmed by his suicide and his hope that the act would not cast "a dark shadow over that person."
About a month later, Misako Suzuki, the woman whom he was planning to marry, also committed suicide by carbon monoxide poisoning, leaving a note reading: "We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him."
After Taniyama's death, Goro Shimura stated that:
He was always kind to his colleagues, especially to his juniors, and he genuinely cared about their welfare. He was the moral support of many of those who came into mathematical contact with him, including of course myself. Probably he was never conscious of this role he was playing. But I feel his noble generosity in this respect even more strongly now than when he was alive. And yet nobody was able to give him any support when he desperately needed it. Reflecting on this, I am overwhelmed by the bitterest grief.
See also
• Taniyama group
Notes
1. Taniyama's given name 豊 was intended to be read as Toyo, but was frequently misread as the more common form Yutaka, which he eventually adopted as his own name.
2. Yutaka Taniyama biography, University of St Andrews, Scotland: https://www-history.mcs.st-and.ac.uk/Biographies/Taniyama.html
3. "The Proof". Nova. Season 25. Episode 4. 28 October 1997. 14:21 minutes in. PBS. Transcript of episode.
4. "Fermat's Last Theorem". Horizon. 1995. 12:08 minutes in. BBC.
Publications
• Shimura, Goro; Taniyama, Yutaka (1961), Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Tokyo: The Mathematical Society of Japan, MR 0125113 This book is hard to find, but an expanded version was later published as Shimura, Goro (1997). Abelian Varieties with Complex Multiplication and Modular Functions (Hardcover ed.). Princeton University Press. ISBN 978-0-691-01656-6.
References
• Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections", The Bulletin of the London Mathematical Society, 21 (2): 186–196, doi:10.1112/blms/21.2.186, ISSN 0024-6093, MR 0976064
• Singh, Simon (hardcover, 1998). Fermat's Enigma. Bantam Books. ISBN 0-8027-1331-9 (previously published under the title Fermat's Last Theorem).
• Weil, André, "Y. Taniyama", Sugaku-no Ayumi, 6 (4): 21–22, Reprinted in Weil's collected works, volume II
External links
• O'Connor, John J.; Robertson, Edmund F., "Yutaka Taniyama", MacTutor History of Mathematics Archive, University of St Andrews
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Yutaka Yamamoto (mathematician)
Yutaka Yamamoto (山本 裕, Yamamoto Yutaka, born 29 March 1950), is a Japanese mathematician working in systems theory, control theory, and signal processing.
References
• YYfest 2010 Symposium on Systems, Control, and Signal Processing In honor of Yutaka Yamamoto on the occasion of his 60th birthday Kyoto University. 29–31 March 2010
• Jan C. Willems; Shinji Hara; Yoshito Ohta; Hisaya Fujioka, eds. (2010). Perspectives in Mathematical System Theory, Control, and Signal Processing: A Festschrift in Honor of Yutaka Yamamoto on the Occasion of his 60th Birthday. Springer Science & Business Media. ISBN 978-3-540-93917-7.
External links
• Personal website at Kyoto University
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Yuval Flicker
Yuval Zvi Flicker (Hebrew: יוּבַל צְבִי פְלִיקֶר; born 1955 in Israel) is an American mathematician. His primary research interests include automorphic representations.[1]
Yuval Flicker
Born (1955-01-03) 3 January 1955
Kfar Saba, Israel
NationalityIsrael, United States
Alma materUniversity of Cambridge
Hebrew University of Jerusalem
Tel Aviv University
AwardsAlexander von Humboldt Fellow, Fulbright Award, Lady Davis Fellow, Simons Foundation Fellow, NUS Senior Fellow
Scientific career
FieldsMathematics
InstitutionsOhio State University
Ariel University
Doctoral advisorAlan Baker
He received his PhD degree from the University of Cambridge in 1978. His thesis advisor was Alan Baker, in the area of transcendental number theory.[1][2]
He taught at Princeton University, Columbia University, Harvard University and Ohio State University, where he now has the title of Faculty Emeritus.[3] He also worked with David Kazhdan[4] and Pierre Deligne.[1][5]
Education
Born 1955 in Kfar-Saba, raised in Ramat-Gan, Flicker studied Mathematics and Philosophy at Tel-Aviv University gaining a BA in 1973, then he studied Mathematics at the Hebrew University gaining an MA in 1974. After that he studied Part III of the Mathematical Tripos at DPMMS, Cambridge University in 1974-75, where he was awarded his PhD under the supervision of Fields Medalist Alan Baker in 1978. His dissertation was "Linear forms on Abelian Varieties over Local Fields". He was a Post Doctoral scholar at the Institute for Advanced Study Princeton 1978-79, at Columbia University 1979-81, at Princeton University 1981-85, and at Harvard University 1985-87. He worked as a member of the Mathematics Department at the Ohio State University from 1987 to 2015.
Research
Flicker's research interests include Automorphic and Admissible Representations, Automorphic forms over function fields, Arithmetic Geometry, Lifting of Representations, Hecke-Iwahori algebras, p-adic automorphic forms, Galois Cohomology, Local-Global Principles, Motives, Algebraic Groups, Covering Groups, Shimura Varieties. He coauthored works with David Kazhdan,[4] Pierre Deligne,[5] his students[6] and other scholars.[7] He acknowledges influence of Joseph Bernstein[8] and of Vladimir Drinfeld.[9] He is the author of several books.
Dissemination
Flicker visited and lectured at the Universities of Mannheim, Bielefeld, Münster, Essen, Köln, HU Berlin supported by a Humboldt Stiftung, DAAD and SFB; at MPIM in Bonn; at University of Tokyo; at TIFR Bombay (and later TIFR Mumbai); at University of Santiago, Chile; at University of Buenos Aires supported by a Fulbright award; at the Chinese Academy of Sciences; at National University of Singapore supported by an NUS Senior Fellowship; at the Hebrew University of Jerusalem supported by a Lady Davis Fellowship and Schonbrunn Professorship, and Simons Fellowship; at IMPA Rio de Janeiro; at Erzincan University supported by TÜBİTAK.
Flicker endorsed An Open Letter to Richard Riley, United States Secretary of Education.
Books
Yuval Flicker is the author of a number of books including:
• Arthur's Invariant Trace Formula and Comparison of Inner Forms (2016)[10]
• Drinfeld Moduli Schemes and Automorphic Forms (2013)[11]
• Automorphic Representations of Low Rank Groups (2006)[12]
• Automorphic Forms and Shimura Varieties of PGSp(2) (2005)[13]
• Matching of Orbital Integrals on GL(4) and GSp(2) (1999)[14]
External links
• Home Page at Ohio State
• Math Department Ohio State
• Personal Home Page
References
1. "Yuval Flicker OSU CV" (PDF).
2. Yuval Zvi Flicker at the Mathematics Genealogy Project.
3. "Yuval Flicker". Ohio State University. Retrieved 22 October 2021.
4. "Metaplectic correspondence". Publications Mathématiques de l'IHÉS.
5. "Counting local systems with principal unipotent local monodromy". Annals of Mathematics.
6. "Twister Character of a Small Representations of PGL(4)" (PDF). Moscow Mathematical Journal.
7. "Grothendieck's Theorem on Non-Abelian H2 and Local-Global Principles" (PDF). Journal of the American Mathematical Society.
8. "K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Part 2". Proceedings of Symposia in Pure Mathematics.
9. "Eisenstein Series and the Trace Formula for GL(2) over a Function Field" (PDF). Documenta Mathematica.
10. Birkhäuser Basel, ISBN 978-3-319-31593-5.
11. Springer-Verlag New York, ISBN 978-1-4614-5888-3.
12. World Scientific, ISBN 978-981-256-803-8.
13. World Scientific, ISBN 978-981-256-403-0.
14. Memoirs of the American Mathematical Society 655, AMS, ISBN 978-0-8218-0959-4.
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Yves André
Yves André (born December 11, 1959) is a French mathematician, specializing in arithmetic geometry.
Yves André
André at Oberwolfach, 2007
Born (1959-12-11) December 11, 1959
NationalityFrench
Alma materPierre and Marie Curie University
AwardsPrix Paul Doistau–Émile Blutet (2011)
Member of the Academia Europaea (2015)
Scientific career
Doctoral advisorDaniel Bertrand
Biography
André received his doctorate in 1984 from Pierre and Marie Curie University (Paris VI) with thesis advisor Daniel Bertrand and thesis Structure de Hodge, équations différentielles p-adiques, et indépendance algébrique de périodes d'intégrales abéliennes.[1] He became at CNRS in 1985 a Researcher, in 2000 a Research Director 2nd Class, and in 2009 a Research Director 1st Class (at École Normale Supérieure and Institut de mathématiques de Jussieu – Paris Rive Gauche).[2]
Research
In 1989, he formulated the one-dimensional-subvariety case of what is now known as the André-Oort conjecture on special subvarieties of Shimura varieties.[3] Only partial results have been proven so far; by André himself and by Jonathan Pila in 2009. In 2016, André used Scholze's method of perfectoid spaces to prove Melvin Hochster's direct summand conjecture that any finite extension of a regular commutative ring splits as a module.[4][5]
Awards
In 2011, André received the Prix Paul Doistau–Émile Blutet of the Académie des Sciences. In 2015, he was elected as a Member of the Academia Europaea. He was an invited speaker at the 2018 International Congress of Mathematicians in Rio de Janeiro and gave a talk titled Perfectoid spaces and the homological conjectures.[6]
Selected publications
• André, Yves (1989). G-Functions and Geometry A Publication of the Max-Planck-Institut für Mathematik, Bonn. Wiesbaden. ISBN 978-3-663-14108-2. OCLC 860266118.{{cite book}}: CS1 maint: location missing publisher (link)
• André, Yves (18 October 2022). "Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part". Compositio Mathematica (in French). 82 (1): 1–24. ISSN 1570-5846. Retrieved 22 November 2022.
• Andre, Yves (1996). "On the Shafarevich and Tate conjectures for hyperkähler varieties". Mathematische Annalen. Springer Science and Business Media LLC. 305 (1): 205–248. doi:10.1007/bf01444219. ISSN 0025-5831. S2CID 122949797.
• André, Yves; Baldassarri, F. (2001). De Rham cohomology of differential modules on algebraic varieties. Basel, Switzerland: Birkhäuser. ISBN 978-3-0348-8336-8. OCLC 679321692.
• Period mappings and differential equations. From C to Cp: Tohoku-Hokkaido Lectures in Arithmetic Geometry, Tokyo, Memoirs Mathematical Society of Japan 2003 (with appendix by F. Kato, N. Tsuzuki)
• "Une introduction aux motifs (Motifs purs, motifs mixtes, périodes)". Société Mathématique de France (in French). Retrieved 22 November 2022.
• André, Yves (2009). "Galois theory, motives and transcendental numbers". Renormalization and Galois Theories. IRMA Lectures in Mathematics and Theoretical Physics. Vol. 15. Zuerich, Switzerland: European Mathematical Society Publishing House. pp. 165–177. doi:10.4171/073-1/4. ISBN 978-3-03719-073-9. S2CID 16880343.
• André, Yves (7 December 2017). "La conjecture du facteur direct". Publications mathématiques de l'IHÉS (in French). Springer Science and Business Media LLC. 127 (1): 71–93. arXiv:1609.00345. doi:10.1007/s10240-017-0097-9. ISSN 0073-8301. S2CID 254170253.
References
1. Yves André at the Mathematics Genealogy Project
2. "Yves André". Academia Europaea.
3. "G-functions and geometry", Vieweg 1989
4. André, Yves (2016). "La conjecture du facteur direct". arXiv:1609.00345 [math.AG].
5. Bhatt, Bhargav (2016). "On the direct summand conjecture and its derived variant". arXiv:1608.08882 [math.AG]..
6. André, Yves (2018). "Perfectoid spaces and the homological conjectures". arXiv:1801.10006 [math.AC].
External links
• "Yves André - Grothendieck et les équations différentielles". YouTube. 4 April 2016.
• "Yves André - Direct summand conjecture and perfectoid Abhyankar lemma: an overview". YouTube. 11 November 2016.
• "Yves André: What is... a motivic Galois group". YouTube. 18 January 2018.
• "Yves André: Periods of relative 1 motives". YouTube. 18 January 2018.
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Yves Colin de Verdière
Yves Colin de Verdière is a French mathematician.
Yves Colin de Verdière
NationalityFrench
Alma materParis Diderot University
Known forColin de Verdière graph invariant
AwardsPrize Ampère
Fellow of the United States National Academy of Sciences
Émile Picard Medal
Scientific career
FieldsMathematics
InstitutionsJoseph Fourier University
Doctoral advisorMarcel Berger
Life
He studied at the École Normale Supérieure in Paris in the late 1960s, obtained his Ph.D. in 1973, and then spent the bulk of his working life as faculty at Joseph Fourier University in Grenoble. He retired in December 2005.
Work
Colin de Verdière is known for work in spectral theory, in particular on the semiclassical limit of quantum mechanics (including quantum chaos); in graph theory where he introduced a new graph invariant, the Colin de Verdière graph invariant; and on a variety of other subjects within Riemannian geometry and number theory.
Honors and awards
His contributions have been recognized by several awards: senior member of the Institut Universitaire de France from 1991 to 2001; Prize Ampère of the French Academy of Sciences in 1999; Fellow of the American Academy of Arts and Sciences in 2004; Émile Picard Medal of the French Academy of Sciences in 2018. He was an invited speaker at the International Congress of Mathematicians, held in Berkeley, California in 1986.
External links
• Yves Colin de Verdière at the Mathematics Genealogy Project
• "Colin de Verdière's home page".
• "Conference in honour of his retirement". Archived from the original on 2006-05-06.
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Yves Meyer
Yves F. Meyer (French: [mɛjɛʁ]; born 19 July 1939) is a French mathematician. He is among the progenitors of wavelet theory, having proposed the Meyer wavelet. Meyer was awarded the Abel Prize in 2017.
Yves Meyer
Yves Meyer giving a lecture in 2016.
Born (1939-07-19) 19 July 1939
Paris, France
NationalityFrench
EducationÉcole Normale Supérieure
University of Strasbourg
Known forWavelet theory
AwardsSalem Prize
Carl Friedrich Gauss Prize
Abel Prize
Princess of Asturias Award
Scientific career
FieldsMathematics
ThesisIdéaux Fermés de L1 dans Lesquels une Suite Approche l'Identité (1966)
Doctoral advisorJean-Pierre Kahane
Doctoral students
• Pascal Auscher
• Aline Bonami
• Albert Cohen
Biography
Born in Paris in a Jewish family, Yves Meyer studied at the Lycée Carnot in Tunis;[1] he won the French General Student Competition (Concours Général) in Mathematics, and was placed first in the entrance examination for the École Normale Supérieure in 1957.[2] He obtained his Ph.D. in 1966, under the supervision of Jean-Pierre Kahane.[3][4] The Mexican historian Jean Meyer is his cousin.
Yves Meyer taught at the Prytanée national militaire during his military service (1960–1963), then was a teaching assistant at the Université de Strasbourg (1963–1966), a professor at Université Paris-Sud (1966–1980), a professor at École Polytechnique (1980–1986), a professor at Université Paris-Dauphine (1985–1995), a senior researcher at the Centre national de la recherche scientifique (CNRS) (1995–1999), an invited professor at the Conservatoire National des Arts et Métiers (2000), a professor at École Normale Supérieure de Cachan (1999–2003), and has been a professor emeritus at Ecole Normale Supérieure de Cachan since 2004.
He was awarded the 2010 Gauss Prize for fundamental contributions to number theory, operator theory and harmonic analysis, and his pivotal role in the development of wavelets and multiresolution analysis.[3] He also received the 2017 Abel Prize "for his pivotal role in the development of the mathematical theory of wavelets."[5]
Publications
• Meyer, Yves (1970). Nombres de Pisot, nombres de Salem, et analyse harmonique (in French). Berlin New York: Springer-Verlag. ISBN 978-3-540-36243-2. OCLC 295014081.
• Algebraic numbers and harmonic analysis. Burlington: Elsevier Science. 1972. ISBN 978-0-08-095412-7. OCLC 761646828.
• Meyer, Yves (1990). Ondelettes et opérateurs (in French). Paris: Hermann. ISBN 978-2-7056-6125-0. OCLC 945745937.
• Meyer, Yves (22 April 1993). Wavelets and Operators. D. H. Salinger. Cambridge University Press. doi:10.1017/cbo9780511623820. ISBN 978-0-521-42000-6.[6]
Awards and recognitions
• He is a member of the Académie des Sciences since 1993.[7]
• Meyer was an Invited Speaker at the ICM in 1970 in Nice, in 1983 in Warsaw,[8] and in 1990 in Kyoto.[9]
• In 2010, Yves Meyer was awarded the Carl Friedrich Gauss Prize.[3]
• In 2012 he became a fellow of the American Mathematical Society.[10]
• In 2017 he was awarded the Abel Prize for his pivotal role in developing the mathematical theory of wavelets.[11]
• In 2020 he received the Princess of Asturias Award for Technical and Scientific Research.[12]
See also
• Wavelet
• Alex Grossmann
• Meyer wavelet
• Compressed sensing
• Harmonious set
• JPEG 2000
• Meyer set
• Ingrid Daubechies
• Jean Morlet
References
1. "Home". lyceecarnotdetunis.com.
2. Société de Mathématiques Appliquées et Industrielles : Yves Meyer.
3. "Carl Friedrich Gauss Prize – Yves Meyer". International Congress of Mathematicians 2010, Hyderabad, India. Archived from the original on 23 September 2010.
4. Yves F. Meyer at the Mathematics Genealogy Project
5. "2017: Yves Meyer". www.abelprize.no. Retrieved 22 July 2022.{{cite web}}: CS1 maint: url-status (link)
6. Chui, Charles K. (1996). "Review: Wavelets and operators, by Yves Meyer; A friendly guide to wavelets, by Gerald Kaiser". Bull. Amer. Math. Soc. (N.S.). 33 (1): 131–134. doi:10.1090/s0273-0979-96-00635-0.
7. Académie des Sciences : Yves Meyer. Archived 9 August 2011 at the Wayback Machine
8. Meyer, Yves. "Intégrales singulières, opérateurs multilinéaires, analyse complexe et équations aux dérivées partielles." Proc. Intern. Cong. Math (1983): 1001–1010.
9. Meyer, Yves F. "Wavelets and applications." Proc. Intern. Cong. Math (1990): 1619–1626.
10. List of Fellows of the American Mathematical Society, retrieved 4 February 2013.
11. "Abel Prize 2017: Yves Meyer wins 'maths Nobel' for work on wavelets". The Guardian. 21 March 2017.
12. "Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès, Princess of Asturias Award for Technical and Scientific Research 2020". Princess of Asturias Foundation. Retrieved 23 June 2020.
External links
• Société Mathématiques de France : Lecture by Yves Meyer (2009)
• Yves Meyer at the Mathematics Genealogy Project
• Gauss prize 2010
Abel Prize laureates
• 2003 Jean-Pierre Serre
• 2004 Michael Atiyah
• Isadore Singer
• 2005 Peter Lax
• 2006 Lennart Carleson
• 2007 S. R. Srinivasa Varadhan
• 2008 John G. Thompson
• Jacques Tits
• 2009 Mikhail Gromov
• 2010 John Tate
• 2011 John Milnor
• 2012 Endre Szemerédi
• 2013 Pierre Deligne
• 2014 Yakov Sinai
• 2015 John Forbes Nash Jr.
• Louis Nirenberg
• 2016 Andrew Wiles
• 2017 Yves Meyer
• 2018 Robert Langlands
• 2019 Karen Uhlenbeck
• 2020 Hillel Furstenberg
• Grigory Margulis
• 2021 László Lovász
• Avi Wigderson
• 2022 Dennis Sullivan
• 2023 Luis Caffarelli
Laureates of the Prince or Princess of Asturias Award for Technical and Scientific Research
Prince of Asturias Award for Technical and Scientific Research
1980s
• 1981: Alberto Sols
• 1982: Manuel Ballester
• 1983: Luis Antonio Santaló Sors
• 1984: Antonio Garcia-Bellido
• 1985: David Vázquez Martínez and Emilio Rosenblueth
• 1986: Antonio González González
• 1987: Jacinto Convit and Pablo Rudomín
• 1988: Manuel Cardona and Marcos Moshinsky
• 1989: Guido Münch
1990s
• 1990: Santiago Grisolía and Salvador Moncada
• 1991: Francisco Bolívar Zapata
• 1992: Federico García Moliner
• 1993: Amable Liñán
• 1994: Manuel Patarroyo
• 1995: Manuel Losada Villasante and Instituto Nacional de Biodiversidad of Costa Rica
• 1996: Valentín Fuster
• 1997: Atapuerca research team
• 1998: Emilio Méndez Pérez and Pedro Miguel Echenique Landiríbar
• 1999: Ricardo Miledi and Enrique Moreno González
2000s
• 2000: Robert Gallo and Luc Montagnier
• 2001: Craig Venter, John Sulston, Francis Collins, Hamilton Smith and Jean Weissenbach
• 2002: Lawrence Roberts, Robert E. Kahn, Vinton Cerf and Tim Berners-Lee
• 2003: Jane Goodall
• 2004: Judah Folkman, Tony Hunter, Joan Massagué, Bert Vogelstein and Robert Weinberg
• 2005: Antonio Damasio
• 2006: Juan Ignacio Cirac
• 2007: Peter Lawrence and Ginés Morata
• 2008: Sumio Iijima, Shuji Nakamura, Robert Langer, George M. Whitesides and Tobin Marks
• 2009: Martin Cooper and Raymond Tomlinson
2010s
• 2010: David Julius, Baruch Minke and Linda Watkins
• 2011: Joseph Altman, Arturo Álvarez-Buylla and Giacomo Rizzolatti
• 2012: Gregory Winter and Richard A. Lerner
• 2013: Peter Higgs, François Englert and European Organization for Nuclear Research CERN
• 2014: Avelino Corma Canós, Mark E. Davis and Galen D. Stucky
Princess of Asturias Award for Technical and Scientific Research
2010s
• 2015: Emmanuelle Charpentier and Jennifer Doudna
• 2016: Hugh Herr
• 2017: Rainer Weiss, Kip S. Thorne, Barry C. Barish and the LIGO Scientific Collaboration
• 2018: Svante Pääbo
• 2019: Joanne Chory and Sandra Myrna Díaz
2020s
• 2020: Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès
• 2021: Katalin Karikó, Drew Weissman, Philip Felgner, Uğur Şahin, Özlem Türeci, Derrick Rossi and Sarah Gilbert
• 2022: Geoffrey Hinton, Yann LeCun, Yoshua Bengio and Demis Hassabis
• 2023: Jeffrey I. Gordon, Everett Peter Greenberg and Bonnie Bassler
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| Wikipedia |
Yvette Amice
Yvette Amice (June 4, 1936 – July 4, 1993) was a French mathematician whose research concerned number theory and p-adic analysis.[1] She was president of the Société mathématique de France.[1]
Education
Amice studied mathematics at the École normale supérieure de jeunes filles in Sèvres, beginnining in 1956 and earning her agrégation in 1959.[1] She became an assistant at the Faculté des sciences de Paris until 1964, when she completed a state doctorate under the supervision of Charles Pisot. Her dissertation was Interpolation p-adique [p-adic interpolation].[1][2]
Career
On completing her doctorate, she became maître de conférences at the University of Poitiers and then, in 1966, professor at the University of Bordeaux. She returned to Poitiers in 1968 but then in 1970 became one of the founding professors of Paris Diderot University, where she was vice president from 1978 to 1981.
In 1975 she became president of the Société mathématique de France.[1]
Textbook
Amice was the author of a textbook on the p-adic number system, Les nombres p-adiques (Presses Universitaires de France, 1975).[3]
References
1. Barsky, Daniel; Kahane, Jean-Pierre (1994), "Yvette Amice (1936–1993)" (PDF), Gazette des Mathématiciens (61): 83–87, MR 1289341.
2. Yvette Amice at the Mathematics Genealogy Project
3. Review of Les nombres p-adiques by W. Bartenwerfer, MR0447195 (in German).
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| Wikipedia |
Yvette Kosmann-Schwarzbach
Yvette Kosmann-Schwarzbach (born 30 April 1941)[1] is a French mathematician and professor.
Yvette Kosmann-Schwarzbach
Born (1941-04-30) 30 April 1941
NationalityFrench
Alma materUniversity of Paris
Known forKosmann lift
Scientific career
FieldsMathematics
InstitutionsÉcole polytechnique
University of Lille
ThesisDérivées de Lie des spineurs (1970)
Doctoral advisorAndré Lichnerowicz
Websitehttps://www.cmls.polytechnique.fr/perso/kosmann/
Education and career
Kosmann-Schwarzbach obtained her doctoral degree in 1970 at the University of Paris under supervision of André Lichnerowicz on a dissertation titled Dérivées de Lie des spineurs (Lie derivatives of spinors).[2]
She worked at Lille University of Science and Technology, and since 1993 at the École polytechnique.
Research
Kosmann-Schwarzbach is the author of over fifty articles on differential geometry, algebra and mathematical physics, of two books on Lie groups and on the Noether theorem, as well as the co-editor of several books concerning the theory of integrable systems. The Kosmann lift in differential geometry is named after her.[3][4]
Works
• Groups and Symmetries: From Finite Groups to Lie Groups. Translated by Stephanie Frank Singer. Springer 2010, ISBN 978-0387788654.[5]
• The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. Translated by Bertram Schwarzbach. Springer 2011, ISBN 978-0387878676.[6]
References
1. Birth date from Library of Congress and French National Library, retrieved 2019-10-13
2. Yvette Kosmann-Schwarzbach at the Mathematics Genealogy Project
3. Fatibene, L.; Ferraris, M.; Francaviglia, M.; Godina, M. (28 August – 1 September 1995). Janyska, J.; Kolář, I.; Slovák, J. (eds.). "A geometric definition of Lie derivative for Spinor Fields". Proceedings of the 6th International Conference on Differential Geometry and Applications. Brno, Czech Republic: Masaryk University: 549–558.
4. Godina M. and Matteucci P. (2003), Reductive G-structures and Lie derivatives, Journal of Geometry and Physics, 47, pp. 66–86
5. Reviews of Groups and Symmetries: Aloysius Helminck (2011), MR2553682; Thomas R. Hagedorn (2010), MAA Reviews; Ilka Agricola, Zbl 1132.20001; Eugene Kryachko, Zbl 1201.20001.
6. Reviews of The Noether Theorems: Jeremy Gray (2008), Historia Mathematica, doi:10.1016/j.hm.2007.06.002; Narciso Román-Roy (2012), MR2761345; Michael Berg (2011), MAA Reviews; Teodora-Liliana Rădulescu, Zbl 1128.01024; Reinhard Siegmund-Schultze, Zbl 1216.01011.
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Yvonne Dold-Samplonius
Yvonne Dold-Samplonius (20 May 1937 – 16 June 2014) was a Dutch mathematician and historian who specialized in the history of Islamic mathematics during the Middle age. She was particularly interested in the mathematical methods used by Islamic architects and builders of the Middle Ages for measurements of volumes and measurements of religious buildings or in the design of muqarnas.
Biography
Born on 20 May 1937 in Haarlem, Yvonne Samplonius obtained her degree in mathematics and Arabic from the University of Amsterdam (Doktoratsexamen) in 1966.[1] Yvonne Dold-Samplonius married in 1965 the German mathematician Albrecht Dold. She studied from 1966 to 1967 at Harvard University under the direction of Professor John E. Murdoch.[1] She obtained in 1977 a PhD for her analysis of the treatise Kitāb al-mafrādāt li Aqāţun (Book of Assumptions of Aqātun) under the supervision of Prof. Evert Marie Bruins and Prof. Juan Vernet.[2][3]
She came into contact with the work of the Persian mathematician, physicist and astronomer Abū Sahl al-Qūhī, who worked in Baghdad in the 10th century and worked on the geometrical forms of buildings.[4] Through his work, she became interested in the geometrical calculations that helped building many domes of palaces and mosques, called muqarnas, in the Arab world and Persia.[5][6][7] She wrote articles on the Islamic mathematicians Jamshīd al-Kāshī and Abu-Abdullah Muhammad ibn Īsa Māhānī in the Dictionary of the Middle Ages and in the Dictionary of Scientific Biography.[8][9]
In her last years her interest shifted to mathematics in Islamic architecture from an historic point of view.[7][10] Since 1995, she has been an associate member of the Interdisciplinary Center for Scientific Computing (IWR) of the University of Heidelberg, with whom she has published several videos on Islamic geometrical art.[11][12] In 1985, she is visiting professor at the University of Siena. In 2000, she organized with Joseph Dauben the conference "2000 Years of Transmission of Mathematical Ideas".[13] In 2002, she became a Corresponding Member of the International Academy of the History of Sciences and was elected effective member in 2007. She was made honorary citizen of Kashan in Iran in 2000.[11]
Publications
• Yvonne Dold-Samplonius, Dissertation: Book of Assumptions by Aqatun (Kitab al-Mafrudat li-Aqatun), Amsterdam 1977.
• Yvonne Dold-Samplonius : Practical Arabic Mathematics: Measuring the Muqarnas by al-Kashi, Centaurus 35, 193–242, (1992/3).
• Yvonne Dold-Samplonius : How al-Kashi Measures the Muqarnas: A Second Look, M. Folkerts (Ed.), Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich, Wolfenbütteler Mittelalter-Studien Vol. 10, 56 – 90, Wiesbaden, (1996).
• Yvonne Dold-Samplonius : Calculation of Arches and Domes in 15th Century Samarkand, Nexus Network Journal, Vol. 2(3), (2000).
• Yvonne Dold-Samplonius : Calculating Surface Areas and Volumes in Islamic Architecture, The Enterprise of Science in Islam, New Perspectives, Eds. Jan P. Hogendijk et Abdelhamid I. Sabra, MIT Press, Cambridge Mass. pp. 235–265, (2003).
• Yvonne Dold-Samplonius, Silvia L. Harmsen : The Muqarnas Plate Found at Takht-i Sulaiman, A New Interpretation, Muqarnas Vol. 22, Leiden, pp. 85–94, (2005).
Videos
• Yvonne Dold-Samplonius, Christoph Kindl, Norbert Quien : Qubba for al-Kashi, Video, Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, American Mathematical Society, (1996). Qubba for al-Kashi on YouTube
• Yvonne Dold-Samplonius, Silvia L. Harmsen, Susanne Krömker, Michael Winckler : Magic of Muqarnas, Video, Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, (2005).
References
1. de Waard, Peter (2014). "Yvonne Dold-Samplonius 1937–2014" (PDF). Obituary University of Heidelberg (in Dutch).
2. Dold-Samplonius, Yvonne (1977). Book of Assumptions by Aquatun: Text-critical Edition.
3. Dold-Samplonius, Yvonne (1978). "Some remarks on the Book of Assumptions by Aqatun" (PDF). Journal for the History of Arabic Science. 2 (2): 255–263.
4. Dold-Samplonius, Yvonne (2008) [1970–80]. "Al-Qūhī (or Al-Kūhī), Abū Sahl Wayjan Ibn Rustam". Complete Dictionary of Scientific Biography. Encyclopedia.com.
5. Dold-Samplonius, Yvonne (1992-10-01). "Practical Arabic Mathematics: Measuring the Muqarnas by al-K¯ash¯i" (PDF). Centaurus. 35 (3): 193–242. Bibcode:1992Cent...35..193D. doi:10.1111/j.1600-0498.1992.tb00699.x. ISSN 1600-0498.
6. Dold-Samplonius, Yvonne; Hermelink, Heinrich (1970). "Al-Jayyānī, Abū'Abd Allāh Muḥammad Ibn Mu'ādh". Complete Dictionary of Scientific Biography. Encyclopedia.com.
7. Dold-Samplonius, Yvonne; Harmsen, Silvia L. (2005). "The Muqarnas Plate Found at Takht-I Sulayman: A New Interpretation". Muqarnas. 22: 85–94. doi:10.1163/22118993_02201005. JSTOR 25482424.
8. Dold-Samplonius, Yvonne. "Al-Kāshī | Muslim astronomer and mathematician". Encyclopedia Britannica. Retrieved 2018-02-02.
9. Dold-Samplonius, Yvonne (2008) [1970-80]. "Al-Māhānī, Abū 'Abd Allāh Muḥammad Ibn 'Īsā". Complete Dictionary of Scientific Biography. Encyclopedia.com.
10. Dold-Samplonius, Yvonne (2003). "Calculating Surface Areas and Volumes in Islamic Architecture". In Hogendijk, J. P.; Sabra, A. I. (eds.). The Enterprise of Science in Islam: New Perspectives. MIT Press. pp. 235–265. ISBN 978-0-262-19482-2.
11. IWR - History of Islamic Mathematics (2014). "Curriculum Vitae – Dr. Yvonne Dold-Samplonius". www.iwr.uni-heidelberg.de. Retrieved 2018-02-04.
12. Yvonne, Dold-Samplonius; Silvia, Harmsen; Susanne, Krömker; Michael J., Winckler (2005). "Magic of Muqarnas" (in German). doi:10.11588/heidok.00017446. {{cite journal}}: Cite journal requires |journal= (help)
13. Dold-Samplonius, Yvonne; Dauben, Joseph W., eds. (2002). From China to Paris: 2000 Years Transmission of Mathematical Ideas. Franz Steiner Verlag. ISBN 978-3-515-08223-5.
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| Wikipedia |
Z* theorem
In mathematics, George Glauberman's Z* theorem is stated as follows:
Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inverse image in G of the center of G/O(G).
This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer–Suzuki theorem to deal with some small cases).
Details
The original paper Glauberman (1966) gave several criteria for an element to lie outside Z*(G). Its theorem 4 states:
For an element t in T, it is necessary and sufficient for t to lie outside Z*(G) that there is some g in G and abelian subgroup U of T satisfying the following properties:
1. g normalizes both U and the centralizer CT(U), that is g is contained in N = NG(U) ∩ NG(CT(U))
2. t is contained in U and tg ≠ gt
3. U is generated by the N-conjugates of t
4. the exponent of U is equal to the order of t
Moreover g may be chosen to have prime power order if t is in the center of T, and g may be chosen in T otherwise.
A simple corollary is that an element t in T is not in Z*(G) if and only if there is some s ≠ t such that s and t commute and s and t are G-conjugate.
A generalization to odd primes was recorded in Guralnick & Robinson (1993): if t is an element of prime order p and the commutator [t, g] has order coprime to p for all g, then t is central modulo the p′-core. This was also generalized to odd primes and to compact Lie groups in Mislin & Thévenaz (1991), which also contains several useful results in the finite case.
Henke & Semeraro (2015) have also studied an extension of the Z* theorem to pairs of groups (G, H) with H a normal subgroup of G.
Works cited
• Dade, Everett C. (1971), "Character theory pertaining to finite simple groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969, Boston, MA: Academic Press, pp. 249–327, ISBN 978-0-12-563850-0, MR 0360785 gives a detailed proof of the Brauer–Suzuki theorem.
• Glauberman, George (1966), "Central elements in core-free groups", Journal of Algebra, 4 (3): 403–420, doi:10.1016/0021-8693(66)90030-5, ISSN 0021-8693, MR 0202822, Zbl 0145.02802
• Guralnick, Robert M.; Robinson, Geoffrey R. (1993), "On extensions of the Baer-Suzuki theorem", Israel Journal of Mathematics, 82 (1): 281–297, doi:10.1007/BF02808114, ISSN 0021-2172, MR 1239051, Zbl 0794.20029
• Henke, Ellen; Semeraro, Jason (1 October 2015). "Centralizers of normal subgroups and the Z*-theorem". Journal of Algebra. 439: 511–514. arXiv:1411.1932. doi:10.1016/j.jalgebra.2015.06.027.
• Mislin, Guido; Thévenaz, Jacques (1991), "The Z*-theorem for compact Lie groups", Mathematische Annalen, 291 (1): 103–111, doi:10.1007/BF01445193, ISSN 0025-5831, MR 1125010
| Wikipedia |
Fisher's z-distribution
Fisher's z-distribution is the statistical distribution of half the logarithm of an F-distribution variate:
$z={\frac {1}{2}}\log F$
"z-distribution" redirects here. For the distribution related to z-scores, see Normal distribution § Standard normal distribution.
Fisher's z
Probability density function
Parameters $d_{1}>0,\ d_{2}>0$ deg. of freedom
Support $x\in (-\infty ;+\infty )\!$ ;+\infty )\!}
PDF ${\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{\left(d_{1}+d_{2}\right)/2}}}\!$
Mode $0$
It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto.[1] Nowadays one usually uses the F-distribution instead.
The probability density function and cumulative distribution function can be found by using the F-distribution at the value of $x'=e^{2x}\,$. However, the mean and variance do not follow the same transformation.
The probability density function is[2][3]
$f(x;d_{1},d_{2})={\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{(d_{1}+d_{2})/2}}},$
where B is the beta function.
When the degrees of freedom becomes large ($d_{1},d_{2}\rightarrow \infty $), the distribution approaches normality with mean[2]
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \bar{x} = \frac 1 2 \left( \frac 1 {d_2} - \frac 1 {d_1} \right)}
and variance
$\sigma _{x}^{2}={\frac {1}{2}}\left({\frac {1}{d_{1}}}+{\frac {1}{d_{2}}}\right).$
Related distribution
• If $X\sim \operatorname {FisherZ} (n,m)$ then $e^{2X}\sim \operatorname {F} (n,m)\,$ (F-distribution)
• If $X\sim \operatorname {F} (n,m)$ then ${\tfrac {\log X}{2}}\sim \operatorname {FisherZ} (n,m)$
References
1. Fisher, R. A. (1924). "On a Distribution Yielding the Error Functions of Several Well Known Statistics" (PDF). Proceedings of the International Congress of Mathematics, Toronto. 2: 805–813. Archived from the original (PDF) on April 12, 2011.
2. Leo A. Aroian (December 1941). "A study of R. A. Fisher's z distribution and the related F distribution". The Annals of Mathematical Statistics. 12 (4): 429–448. doi:10.1214/aoms/1177731681. JSTOR 2235955.
3. Charles Ernest Weatherburn (1961). A first course in mathematical statistics.
External links
• MathWorld entry
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
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| Wikipedia |
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.[1][2]
It can be considered as a discrete-time equivalent of the Laplace transform (s-domain).[3] This similarity is explored in the theory of time-scale calculus.
Whereas the continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, the discrete-time Fourier transform is evaluated over the unit circle of the z-domain. What is roughly the s-domain's left half-plane, is now the inside of the complex unit circle; what is the z-domain's outside of the unit circle, roughly corresponds to the right half-plane of the s-domain.
One of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or numerical approximation. Such methods tend not to be accurate except in the vicinity of the complex unity, i.e. at low frequencies.
History
The basic idea now known as the Z-transform was known to Laplace, and it was re-introduced in 1947 by W. Hurewicz[4][5] and others as a way to treat sampled-data control systems used with radar. It gives a tractable way to solve linear, constant-coefficient difference equations. It was later dubbed "the z-transform" by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952.[6][7]
The modified or advanced Z-transform was later developed and popularized by E. I. Jury.[8][9]
The idea contained within the Z-transform is also known in mathematical literature as the method of generating functions which can be traced back as early as 1730 when it was introduced by de Moivre in conjunction with probability theory.[10] From a mathematical view the Z-transform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.
Definition
The Z-transform can be defined as either a one-sided or two-sided transform. (Just like we have the one-sided Laplace transform and the two-sided Laplace transform.) [11]
Bilateral Z-transform
The bilateral or two-sided Z-transform of a discrete-time signal $x[n]$ is the formal power series $X(z)$ defined as
$X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=-\infty }^{\infty }x[n]z^{-n}$
(Eq.1)
where $n$ is an integer and $z$ is, in general, a complex number:
$z=Ae^{j\phi }=A\cdot (\cos {\phi }+j\sin {\phi })$
where $A$ is the magnitude of $z$, $j$ is the imaginary unit, and $\phi $ is the complex argument (also referred to as angle or phase) in radians.
Unilateral Z-transform
Alternatively, in cases where $x[n]$ is defined only for $n\geq 0$, the single-sided or unilateral Z-transform is defined as
$X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=0}^{\infty }x[n]z^{-n}.$
(Eq.2)
In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system.
An important example of the unilateral Z-transform is the probability-generating function, where the component $x[n]$ is the probability that a discrete random variable takes the value $n$, and the function $X(z)$ is usually written as $X(s)$ in terms of $s=z^{-1}$. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.
Inverse Z-transform
The inverse Z-transform is
$x[n]={\mathcal {Z}}^{-1}\{X(z)\}={\frac {1}{2\pi j}}\oint _{C}X(z)z^{n-1}dz$
(Eq.3)
where C is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal (see Example 2), this means the path C must encircle all of the poles of $X(z)$.
A special case of this contour integral occurs when C is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when $X(z)$ is stable, that is, when all the poles are inside the unit circle. With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or Fourier series, of the periodic values of the Z-transform around the unit circle:
$x[n]={\frac {1}{2\pi }}\int _{-\pi }^{+\pi }X(e^{j\omega })e^{j\omega n}d\omega .$
(Eq.4)
The Z-transform with a finite range of n and a finite number of uniformly spaced z values can be computed efficiently via Bluestein's FFT algorithm. The discrete-time Fourier transform (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.
Region of convergence
The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges.
$\mathrm {ROC} =\left\{z:\left|\sum _{n=-\infty }^{\infty }x[n]z^{-n}\right|<\infty \right\}$
Example 1 (no ROC)
Let $x[n]=0.5^{n}\ $. Expanding x[n] on the interval (−∞, ∞) it becomes
$x[n]=\left\{\dots ,0.5^{-3},0.5^{-2},0.5^{-1},1,0.5,0.5^{2},0.5^{3},\dots \right\}=\left\{\dots ,2^{3},2^{2},2,1,0.5,0.5^{2},0.5^{3},\dots \right\}.$
Looking at the sum
$\sum _{n=-\infty }^{\infty }x[n]z^{-n}\to \infty .$
Therefore, there are no values of z that satisfy this condition.
Example 2 (causal ROC)
Let $x[n]=0.5^{n}u[n]\ $ (where u is the Heaviside step function). Expanding x[n] on the interval (−∞, ∞) it becomes
$x[n]=\left\{\dots ,0,0,0,1,0.5,0.5^{2},0.5^{3},\dots \right\}.$
Looking at the sum
$\sum _{n=-\infty }^{\infty }x[n]z^{-n}=\sum _{n=0}^{\infty }0.5^{n}z^{-n}=\sum _{n=0}^{\infty }\left({\frac {0.5}{z}}\right)^{n}={\frac {1}{1-0.5z^{-1}}}.$
The last equality arises from the infinite geometric series and the equality only holds if |0.5z−1| < 1, which can be rewritten in terms of z as |z| > 0.5. Thus, the ROC is |z| > 0.5. In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".
Example 3 (anti causal ROC)
Let $x[n]=-(0.5)^{n}u[-n-1]\ $ (where u is the Heaviside step function). Expanding x[n] on the interval (−∞, ∞) it becomes
$x[n]=\left\{\dots ,-(0.5)^{-3},-(0.5)^{-2},-(0.5)^{-1},0,0,0,0,\dots \right\}.$
Looking at the sum
$\sum _{n=-\infty }^{\infty }x[n]z^{-n}=-\sum _{n=-\infty }^{-1}0.5^{n}z^{-n}=-\sum _{m=1}^{\infty }\left({\frac {z}{0.5}}\right)^{m}=-{\frac {0.5^{-1}z}{1-0.5^{-1}z}}=-{\frac {1}{0.5z^{-1}-1}}={\frac {1}{1-0.5z^{-1}}}.$
Using the infinite geometric series, again, the equality only holds if |0.5−1z| < 1 which can be rewritten in terms of z as |z| < 0.5. Thus, the ROC is |z| < 0.5. In this case the ROC is a disc centered at the origin and of radius 0.5.
What differentiates this example from the previous example is only the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
Examples conclusion
Examples 2 & 3 clearly show that the Z-transform X(z) of x[n] is unique when and only when specifying the ROC. Creating the pole–zero plot for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will never contain poles.
In example 2, the causal system yields an ROC that includes |z| = ∞ while the anticausal system in example 3 yields an ROC that includes |z| = 0.
In systems with multiple poles it is possible to have a ROC that includes neither |z| = ∞ nor |z| = 0. The ROC creates a circular band. For example,
$x[n]=0.5^{n}u[n]-0.75^{n}u[-n-1]$
has poles at 0.5 and 0.75. The ROC will be 0.5 < |z| < 0.75, which includes neither the origin nor infinity. Such a system is called a mixed-causality system as it contains a causal term (0.5)nu[n] and an anticausal term −(0.75)nu[−n−1].
The stability of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., |z| = 1) then the system is stable. In the above systems the causal system (Example 2) is stable because |z| > 0.5 contains the unit circle.
Let us assume we are provided a Z-transform of a system without a ROC (i.e., an ambiguous x[n]). We can determine a unique x[n] provided we desire the following:
• Stability
• Causality
For stability the ROC must contain the unit circle. If we need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. If we need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. If we need both stability and causality, all the poles of the system function must be inside the unit circle.
The unique x[n] can then be found.
Properties
Properties of the z-transform
Property
Time domain Z-domain Proof ROC
Definition of Z-transform $x[n]$ $X(z)$ $X(z)={\mathcal {Z}}\{x[n]\}$ (definition of the z-transform)
$x[n]={\mathcal {Z}}^{-1}\{X(z)\}$ (definition of the inverse z-transform)
$r_{2}<|z|<r_{1}$
Linearity $a_{1}x_{1}[n]+a_{2}x_{2}[n]$ $a_{1}X_{1}(z)+a_{2}X_{2}(z)$ ${\begin{aligned}X(z)&=\sum _{n=-\infty }^{\infty }(a_{1}x_{1}(n)+a_{2}x_{2}(n))z^{-n}\\&=a_{1}\sum _{n=-\infty }^{\infty }x_{1}(n)z^{-n}+a_{2}\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}\\&=a_{1}X_{1}(z)+a_{2}X_{2}(z)\end{aligned}}$ Contains ROC1 ∩ ROC2
Time expansion $x_{K}[n]={\begin{cases}x[r],&n=Kr\\0,&n\notin K\mathbb {Z} \end{cases}}$
with $K\mathbb {Z} :=\{Kr:r\in \mathbb {Z} \}$ :=\{Kr:r\in \mathbb {Z} \}}
$X(z^{K})$ ${\begin{aligned}X_{K}(z)&=\sum _{n=-\infty }^{\infty }x_{K}(n)z^{-n}\\&=\sum _{r=-\infty }^{\infty }x(r)z^{-rK}\\&=\sum _{r=-\infty }^{\infty }x(r)(z^{K})^{-r}\\&=X(z^{K})\end{aligned}}$ $R^{\frac {1}{K}}$
Decimation $x[Kn]$ ${\frac {1}{K}}\sum _{p=0}^{K-1}X\left(z^{\tfrac {1}{K}}\cdot e^{-i{\tfrac {2\pi }{K}}p}\right)$ ohio-state.edu or ee.ic.ac.uk
Time delay $x[n-k]$
with $k>0$ and $x:x[n]=0\ \forall n<0$
$z^{-k}X(z)$ ${\begin{aligned}Z\{x[n-k]\}&=\sum _{n=0}^{\infty }x[n-k]z^{-n}\\&=\sum _{j=-k}^{\infty }x[j]z^{-(j+k)}&&j=n-k\\&=\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}\\&=z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}\\&=z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}&&x[\beta ]=0,\beta <0\\&=z^{-k}X(z)\end{aligned}}$ ROC, except z = 0 if k > 0 and z = ∞ if k < 0
Time advance $x[n+k]$
with $k>0$
Bilateral Z-transform:
$z^{k}X(z)$
Unilateral Z-transform:[12]
$z^{k}X(z)-z^{k}\sum _{n=0}^{k-1}x[n]z^{-n}$
First difference backward $x[n]-x[n-1]$
with $x[n]=0$ for $n<0$
$(1-z^{-1})X(z)$ Contains the intersection of ROC of X1(z) and z ≠ 0
First difference forward $x[n+1]-x[n]$ $(z-1)X(z)-zx[0]$
Time reversal $x[-n]$ $X(z^{-1})$ ${\begin{aligned}{\mathcal {Z}}\{x(-n)\}&=\sum _{n=-\infty }^{\infty }x(-n)z^{-n}\\&=\sum _{m=-\infty }^{\infty }x(m)z^{m}\\&=\sum _{m=-\infty }^{\infty }x(m){(z^{-1})}^{-m}\\&=X(z^{-1})\\\end{aligned}}$ ${\tfrac {1}{r_{1}}}<|z|<{\tfrac {1}{r_{2}}}$
Scaling in the z-domain $a^{n}x[n]$ $X(a^{-1}z)$ ${\begin{aligned}{\mathcal {Z}}\left\{a^{n}x[n]\right\}&=\sum _{n=-\infty }^{\infty }a^{n}x(n)z^{-n}\\&=\sum _{n=-\infty }^{\infty }x(n)(a^{-1}z)^{-n}\\&=X(a^{-1}z)\end{aligned}}$ $|a|r_{2}<|z|<|a|r_{1}$
Complex conjugation $x^{*}[n]$ $X^{*}(z^{*})$ ${\begin{aligned}{\mathcal {Z}}\{x^{*}(n)\}&=\sum _{n=-\infty }^{\infty }x^{*}(n)z^{-n}\\&=\sum _{n=-\infty }^{\infty }\left[x(n)(z^{*})^{-n}\right]^{*}\\&=\left[\sum _{n=-\infty }^{\infty }x(n)(z^{*})^{-n}\right]^{*}\\&=X^{*}(z^{*})\end{aligned}}$
Real part $\operatorname {Re} \{x[n]\}$ ${\tfrac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]$
Imaginary part $\operatorname {Im} \{x[n]\}$ ${\tfrac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]$
Differentiation in the z-domain $nx[n]$ $-z{\frac {dX(z)}{dz}}$ ${\begin{aligned}{\mathcal {Z}}\{nx(n)\}&=\sum _{n=-\infty }^{\infty }nx(n)z^{-n}\\&=z\sum _{n=-\infty }^{\infty }nx(n)z^{-n-1}\\&=-z\sum _{n=-\infty }^{\infty }x(n)(-nz^{-n-1})\\&=-z\sum _{n=-\infty }^{\infty }x(n){\frac {d}{dz}}(z^{-n})\\&=-z{\frac {dX(z)}{dz}}\end{aligned}}$ ROC, if $X(z)$ is rational;
ROC possibly excluding the boundary, if $X(z)$ is irrational[13]
Convolution $x_{1}[n]*x_{2}[n]$ $X_{1}(z)X_{2}(z)$ ${\begin{aligned}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\}&={\mathcal {Z}}\left\{\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\right\}\\&=\sum _{n=-\infty }^{\infty }\left[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\right]z^{-n}\\&=\sum _{l=-\infty }^{\infty }x_{1}(l)\left[\sum _{n=-\infty }^{\infty }x_{2}(n-l)z^{-n}\right]\\&=\left[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}\right]\!\!\left[\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}\right]\\&=X_{1}(z)X_{2}(z)\end{aligned}}$ Contains ROC1 ∩ ROC2
Cross-correlation $r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]$ $R_{x_{1},x_{2}}(z)=X_{1}^{*}({\tfrac {1}{z^{*}}})X_{2}(z)$ Contains the intersection of ROC of $X_{1}({\tfrac {1}{z^{*}}})$ and $X_{2}(z)$
Accumulation $\sum _{k=-\infty }^{n}x[k]$ ${\frac {1}{1-z^{-1}}}X(z)$ ${\begin{aligned}\sum _{n=-\infty }^{\infty }\sum _{k=-\infty }^{n}x[k]z^{-n}&=\sum _{n=-\infty }^{\infty }(x[n]+\cdots +x[-\infty ])z^{-n}\\&=X(z)\left(1+z^{-1}+z^{-2}+\cdots \right)\\&=X(z)\sum _{j=0}^{\infty }z^{-j}\\&=X(z){\frac {1}{1-z^{-1}}}\end{aligned}}$
Multiplication $x_{1}[n]x_{2}[n]$ ${\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}({\tfrac {z}{v}})v^{-1}\mathrm {d} v$ -
Parseval's theorem
$\sum _{n=-\infty }^{\infty }x_{1}[n]x_{2}^{*}[n]\quad =\quad {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}^{*}({\tfrac {1}{v^{*}}})v^{-1}\mathrm {d} v$
Initial value theorem: If x[n] is causal, then
$x[0]=\lim _{z\to \infty }X(z).$
Final value theorem: If the poles of (z − 1)X(z) are inside the unit circle, then
$x[\infty ]=\lim _{z\to 1}(z-1)X(z).$
Table of common Z-transform pairs
Here:
$u:n\mapsto u[n]={\begin{cases}1,&n\geq 0\\0,&n<0\end{cases}}$
is the unit (or Heaviside) step function and
$\delta :n\mapsto \delta [n]={\begin{cases}1,&n=0\\0,&n\neq 0\end{cases}}$
is the discrete-time unit impulse function (cf Dirac delta function which is a continuous-time version). The two functions are chosen together so that the unit step function is the accumulation (running total) of the unit impulse function.
Signal, $x[n]$Z-transform, $X(z)$ROC
1$\delta [n]$1all z
2$\delta [n-n_{0}]$$z^{-n_{0}}$$z\neq 0$
3$u[n]\,$${\frac {1}{1-z^{-1}}}$$|z|>1$
4$-u[-n-1]$${\frac {1}{1-z^{-1}}}$$|z|<1$
5$nu[n]$${\frac {z^{-1}}{(1-z^{-1})^{2}}}$$|z|>1$
6$-nu[-n-1]\,$${\frac {z^{-1}}{(1-z^{-1})^{2}}}$$|z|<1$
7$n^{2}u[n]$${\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}$$|z|>1\,$
8$-n^{2}u[-n-1]\,$${\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}$$|z|<1\,$
9$n^{3}u[n]$${\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}$$|z|>1\,$
10$-n^{3}u[-n-1]$${\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}$$|z|<1\,$
11$a^{n}u[n]$${\frac {1}{1-az^{-1}}}$$|z|>|a|$
12$-a^{n}u[-n-1]$${\frac {1}{1-az^{-1}}}$$|z|<|a|$
13$na^{n}u[n]$${\frac {az^{-1}}{(1-az^{-1})^{2}}}$$|z|>|a|$
14$-na^{n}u[-n-1]$${\frac {az^{-1}}{(1-az^{-1})^{2}}}$$|z|<|a|$
15$n^{2}a^{n}u[n]$${\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}$$|z|>|a|$
16$-n^{2}a^{n}u[-n-1]$${\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}$$|z|<|a|$
17$\left({\begin{array}{c}n+m-1\\m-1\end{array}}\right)a^{n}u[n]$${\frac {1}{(1-az^{-1})^{m}}}$, for positive integer $m$[13]$|z|>|a|$
18$(-1)^{m}\left({\begin{array}{c}-n-1\\m-1\end{array}}\right)a^{n}u[-n-m]$${\frac {1}{(1-az^{-1})^{m}}}$, for positive integer $m$[13]$|z|<|a|$
19$\cos(\omega _{0}n)u[n]$${\frac {1-z^{-1}\cos(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}$$|z|>1$
20$\sin(\omega _{0}n)u[n]$${\frac {z^{-1}\sin(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}$$|z|>1$
21$a^{n}\cos(\omega _{0}n)u[n]$${\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}$$|z|>|a|$
22$a^{n}\sin(\omega _{0}n)u[n]$${\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}$$|z|>|a|$
Relationship to Fourier series and Fourier transform
Further information: Discrete-time Fourier transform § Relationship to the Z-transform
For values of $z$ in the region $|z|=1$, known as the unit circle, we can express the transform as a function of a single, real variable, ω, by defining $z=e^{j\omega }$. And the bi-lateral transform reduces to a Fourier series:
$\sum _{n=-\infty }^{\infty }x[n]\ z^{-n}=\sum _{n=-\infty }^{\infty }x[n]\ e^{-j\omega n},$
(Eq.4)
which is also known as the discrete-time Fourier transform (DTFT) of the $x[n]$ sequence. This 2π-periodic function is the periodic summation of a Fourier transform, which makes it a widely used analysis tool. To understand this, let $X(f)$ be the Fourier transform of any function, $x(t)$, whose samples at some interval, T, equal the x[n] sequence. Then the DTFT of the x[n] sequence can be written as follows.
$\underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x[n]}\ e^{-j2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X(f-k/T).$
(Eq.5)
When T has units of seconds, $\scriptstyle f$ has units of hertz. Comparison of the two series reveals that $\omega =2\pi fT$ is a normalized frequency with unit of radian per sample. The value ω = 2π corresponds to $ f={\frac {1}{T}}$. And now, with the substitution $ f={\frac {\omega }{2\pi T}},$ Eq.4 can be expressed in terms of the Fourier transform, X(•):
$\sum _{n=-\infty }^{\infty }x[n]\ e^{-j\omega n}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }\underbrace {X\left({\tfrac {\omega }{2\pi T}}-{\tfrac {k}{T}}\right)} _{X\left({\frac {\omega -2\pi k}{2\pi T}}\right)}.$
(Eq.6)
As parameter T changes, the individual terms of Eq.5 move farther apart or closer together along the f-axis. In Eq.6 however, the centers remain 2π apart, while their widths expand or contract. When sequence x(nT) represents the impulse response of an LTI system, these functions are also known as its frequency response. When the $x(nT)$ sequence is periodic, its DTFT is divergent at one or more harmonic frequencies, and zero at all other frequencies. This is often represented by the use of amplitude-variant Dirac delta functions at the harmonic frequencies. Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler discrete Fourier transform (DFT). (See Discrete-time Fourier transform § Periodic data.)
Relationship to Laplace transform
Further information: Laplace transform § Z-transform
Bilinear transform
The bilinear transform can be used to convert continuous-time filters (represented in the Laplace domain) into discrete-time filters (represented in the Z-domain), and vice versa. The following substitution is used:
$s={\frac {2}{T}}{\frac {(z-1)}{(z+1)}}$
to convert some function $H(s)$ in the Laplace domain to a function $H(z)$ in the Z-domain (Tustin transformation), or
$z=e^{sT}\approx {\frac {1+sT/2}{1-sT/2}}$
from the Z-domain to the Laplace domain. Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire $j\omega $ axis of the s-plane onto the unit circle in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the $j\omega $ axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the $j\omega $ axis is in the region of convergence of the Laplace transform.
Starred transform
Given a one-sided Z-transform, X(z), of a time-sampled function, the corresponding starred transform produces a Laplace transform and restores the dependence on sampling parameter, T:
${\bigg .}X^{*}(s)=X(z){\bigg |}_z=e^{sT}}$
The inverse Laplace transform is a mathematical abstraction known as an impulse-sampled function.
Linear constant-coefficient difference equation
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the autoregressive moving-average equation.
$\sum _{p=0}^{N}y[n-p]\alpha _{p}=\sum _{q=0}^{M}x[n-q]\beta _{q}$
Both sides of the above equation can be divided by α0, if it is not zero, normalizing α0 = 1 and the LCCD equation can be written
$y[n]=\sum _{q=0}^{M}x[n-q]\beta _{q}-\sum _{p=1}^{N}y[n-p]\alpha _{p}.$
This form of the LCCD equation is favorable to make it more explicit that the "current" output y[n] is a function of past outputs y[n − p], current input x[n], and previous inputs x[n − q].
Transfer function
Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields
$Y(z)\sum _{p=0}^{N}z^{-p}\alpha _{p}=X(z)\sum _{q=0}^{M}z^{-q}\beta _{q}$
and rearranging results in
$H(z)={\frac {Y(z)}{X(z)}}={\frac {\sum _{q=0}^{M}z^{-q}\beta _{q}}{\sum _{p=0}^{N}z^{-p}\alpha _{p}}}={\frac {\beta _{0}+z^{-1}\beta _{1}+z^{-2}\beta _{2}+\cdots +z^{-M}\beta _{M}}{\alpha _{0}+z^{-1}\alpha _{1}+z^{-2}\alpha _{2}+\cdots +z^{-N}\alpha _{N}}}.$
Zeros and poles
From the fundamental theorem of algebra the numerator has M roots (corresponding to zeros of H) and the denominator has N roots (corresponding to poles). Rewriting the transfer function in terms of zeros and poles
$H(z)={\frac {(1-q_{1}z^{-1})(1-q_{2}z^{-1})\cdots (1-q_{M}z^{-1})}{(1-p_{1}z^{-1})(1-p_{2}z^{-1})\cdots (1-p_{N}z^{-1})}},$
where qk is the kth zero and pk is the kth pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the pole–zero plot.
In addition, there may also exist zeros and poles at z = 0 and z = ∞. If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.
By factoring the denominator, partial fraction decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the impulse response and the linear constant coefficient difference equation of the system.
Output response
If such a system H(z) is driven by a signal X(z) then the output is Y(z) = H(z)X(z). By performing partial fraction decomposition on Y(z) and then taking the inverse Z-transform the output y[n] can be found. In practice, it is often useful to fractionally decompose $\textstyle {\frac {Y(z)}{z}}$ before multiplying that quantity by z to generate a form of Y(z) which has terms with easily computable inverse Z-transforms.
See also
• Advanced Z-transform
• Bilinear transform
• Difference equation (recurrence relation)
• Discrete convolution
• Discrete-time Fourier transform
• Finite impulse response
• Formal power series
• Generating function
• Generating function transformation
• Laplace transform
• Laurent series
• Least-squares spectral analysis
• Probability-generating function
• Star transform
• Zak transform
• Zeta function regularization
References
1. Mandal, Jyotsna Kumar (2020). "Z-Transform-Based Reversible Encoding". Reversible Steganography and Authentication via Transform Encoding. Studies in Computational Intelligence. Vol. 901. Singapore: Springer Singapore. pp. 157–195. doi:10.1007/978-981-15-4397-5_7. ISBN 978-981-15-4396-8. ISSN 1860-949X. S2CID 226413693. Z is a complex variable. Z-transform converts the discrete spatial domain signal into complex frequency domain representation. Z-transform is derived from the Laplace transform.
2. Lynn, Paul A. (1986). "The Laplace Transform and the z-transform". Electronic Signals and Systems. London: Macmillan Education UK. pp. 225–272. doi:10.1007/978-1-349-18461-3_6. ISBN 978-0-333-39164-8. Laplace Transform and the z-transform are closely related to the Fourier Transform. z-transform is especially suitable for dealing with discrete signals and systems. It offers a more compact and convenient notation than the discrete-time Fourier Transform.
3. Palani, S. (2021-08-26). "The z-Transform Analysis of Discrete Time Signals and Systems". Signals and Systems. Cham: Springer International Publishing. pp. 921–1055. doi:10.1007/978-3-030-75742-7_9. ISBN 978-3-030-75741-0. S2CID 238692483. z-transform is the discrete counterpart of Laplace transform. z-transform converts difference equations of discrete time systems to algebraic equations which simplifies the discrete time system analysis. Laplace transform and z-transform are common except that Laplace transform deals with continuous time signals and systems.
4. E. R. Kanasewich (1981). Time Sequence Analysis in Geophysics. University of Alberta. pp. 186, 249. ISBN 978-0-88864-074-1.
5. E. R. Kanasewich (1981). Time sequence analysis in geophysics (3rd ed.). University of Alberta. pp. 185–186. ISBN 978-0-88864-074-1.
6. Ragazzini, J. R.; Zadeh, L. A. (1952). "The analysis of sampled-data systems". Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry. 71 (5): 225–234. doi:10.1109/TAI.1952.6371274. S2CID 51674188.
7. Cornelius T. Leondes (1996). Digital control systems implementation and computational techniques. Academic Press. p. 123. ISBN 978-0-12-012779-5.
8. Eliahu Ibrahim Jury (1958). Sampled-Data Control Systems. John Wiley & Sons.
9. Eliahu Ibrahim Jury (1973). Theory and Application of the Z-Transform Method. Krieger Pub Co. ISBN 0-88275-122-0.
10. Eliahu Ibrahim Jury (1964). Theory and Application of the Z-Transform Method. John Wiley & Sons. p. 1.
11. Jackson, Leland B. (1996). "The z Transform". Digital Filters and Signal Processing. Boston, MA: Springer US. pp. 29–54. doi:10.1007/978-1-4757-2458-5_3. ISBN 978-1-4419-5153-3. z transform is to discrete-time systems what the Laplace transform is to continuous-time systems. z is a complex variable. This is sometimes referred to as the two-sided z transform, with the one-sided z transform being the same except for a summation from n = 0 to infinity. The primary use of the one sided transform ... is for causal sequences, in which case the two transforms are the same anyway. We will not, therefore, make this distinction and will refer to ... as simply the z transform of x(n).
12. Bolzern, Paolo; Scattolini, Riccardo; Schiavoni, Nicola (2015). Fondamenti di Controlli Automatici (in Italian). MC Graw Hill Education. ISBN 978-88-386-6882-1.
13. A. R. Forouzan (2016). "Region of convergence of derivative of Z transform". Electronics Letters. 52 (8): 617–619. Bibcode:2016ElL....52..617F. doi:10.1049/el.2016.0189. S2CID 124802942.
Further reading
• Refaat El Attar, Lecture notes on Z-Transform, Lulu Press, Morrisville NC, 2005. ISBN 1-4116-1979-X.
• Ogata, Katsuhiko, Discrete Time Control Systems 2nd Ed, Prentice-Hall Inc, 1995, 1987. ISBN 0-13-034281-5.
• Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal Processing, 2nd Edition, Prentice Hall Signal Processing Series. ISBN 0-13-754920-2.
External links
• "Z-transform", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Numerical inversion of the Z-transform
• Z-Transform table of some common Laplace transforms
• Mathworld's entry on the Z-transform
• Z-Transform threads in Comp.DSP
• A graphic of the relationship between Laplace transform s-plane to Z-plane of the Z transform
• A video-based explanation of the Z-Transform for engineers
• What is the z-Transform?
Digital signal processing
Theory
• Detection theory
• Discrete signal
• Estimation theory
• Nyquist–Shannon sampling theorem
Sub-fields
• Audio signal processing
• Digital image processing
• Speech processing
• Statistical signal processing
Techniques
• Z-transform
• Advanced z-transform
• Matched Z-transform method
• Bilinear transform
• Constant-Q transform
• Discrete cosine transform (DCT)
• Discrete Fourier transform (DFT)
• Discrete-time Fourier transform (DTFT)
• Impulse invariance
• Integral transform
• Laplace transform
• Post's inversion formula
• Starred transform
• Zak transform
Sampling
• Aliasing
• Anti-aliasing filter
• Downsampling
• Nyquist rate / frequency
• Oversampling
• Quantization
• Sampling rate
• Undersampling
• Upsampling
Authority control: National
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| Wikipedia |
Z-factor
The Z-factor is a measure of statistical effect size. It has been proposed for use in high-throughput screening (where it is also known as Z-prime[1]), and commonly written as Z' to judge whether the response in a particular assay is large enough to warrant further attention.
Background
In high-throughput screens, experimenters often compare a large number (hundreds of thousands to tens of millions) of single measurements of unknown samples to positive and negative control samples. The particular choice of experimental conditions and measurements is called an assay. Large screens are expensive in time and resources. Therefore, prior to starting a large screen, smaller test (or pilot) screens are used to assess the quality of an assay, in an attempt to predict if it would be useful in a high-throughput setting. The Z-factor is an attempt to quantify the suitability of a particular assay for use in a full-scale, high-throughput screen.
Definition
The Z-factor is defined in terms of four parameters: the means ($\mu $) and standard deviations ($\sigma $) of both the positive (p) and negative (n) controls ($\mu _{p}$, $\sigma _{p}$, and $\mu _{n}$, $\sigma _{n}$). Given these values, the Z-factor is defined as:
${\text{Z-factor}}=1-{3(\sigma _{p}+\sigma _{n}) \over |\mu _{p}-\mu _{n}|}$
In practice, the Z-factor is estimated from the sample means and sample standard deviations
${\text{Estimated Z-factor}}=1-{3({\hat {\sigma }}_{p}+{\hat {\sigma }}_{n}) \over |{\hat {\mu }}_{p}-{\hat {\mu }}_{n}|}$
Interpretation
The following interpretations for the Z-factor are taken from:[2]
Z-factorInterpretation
1.0Ideal. Z-factors can never exceed 1.
between 0.5 and 1.0An excellent assay. Note that if $\sigma _{p}=\sigma _{n}$, 0.5 is equivalent to a separation of 12 standard deviations between $\mu _{p}$ and $\mu _{n}$.
between 0 and 0.5A marginal assay.
less than 0There is too much overlap between the positive and negative controls for the assay to be useful.
Note that by the standards of many types of experiments, a zero Z-factor would suggest a large effect size, rather than a borderline useless result as suggested above. For example, if σp=σn=1, then μp=6 and μn=0 gives a zero Z-factor. But for normally-distributed data with these parameters, the probability that the positive control value would be less than the negative control value is less than 1 in 105. Extreme conservatism is used in high throughput screening due to the large number of tests performed.
Limitations
The constant factor 3 in the definition of the Z-factor is motivated by the normal distribution, for which more than 99% of values occur within 3 standard deviations of the mean. If the data follow a strongly non-normal distribution, the reference points (e.g. the meaning of a negative value) may be misleading. Another issue is that the usual estimates of the mean and standard deviation are not robust; accordingly many users in the high-throughput screening community prefer the "Robust Z-prime" which substitutes the median for the mean and the median absolute deviation for the standard deviation.[3] Extreme values (outliers) in either the positive or negative controls can adversely affect the Z-factor, potentially leading to an apparently unfavorable Z-factor even when the assay would perform well in actual screening .[4] In addition, the application of the single Z-factor-based criterion to two or more positive controls with different strengths in the same assay will lead to misleading results .[5] The absolute sign in the Z-factor makes it inconvenient to derive the statistical inference of Z-factor mathematically [6] . A recently proposed statistical parameter, strictly standardized mean difference (SSMD), can address these issues[5] [6] [7] . One estimate of SSMD is robust to outliers.
See also
• high-throughput screening
• SSMD
• Z-score or Standard score
References
1. "Orbitrap LC-MS - US". thermofisher.com.
2. Zhang, JH; Chung, TDY; Oldenburg, KR (1999). "A simple statistical parameter for use in evaluation and validation of high throughput screening assays". Journal of Biomolecular Screening. 4 (2): 67–73. doi:10.1177/108705719900400206. PMID 10838414. S2CID 36577200.
3. Birmingham, Amanda; et al. (August 2009). "Statistical Methods for Analysis of High-Throughput RNA Interference Screens". Nat Methods. 6 (8): 569–575. doi:10.1038/nmeth.1351. PMC 2789971. PMID 19644458.
4. Sui Y, Wu Z (2007). "Alternative Statistical Parameter for High-Throughput Screening Assay Quality Assessment". Journal of Biomolecular Screening. 12 (2): 229–34. doi:10.1177/1087057106296498. PMID 17218666.
5. Zhang XHD, Espeseth AS, Johnson E, Chin J, Gates A, Mitnaul L, Marine SD, Tian J, Stec EM, Kunapuli P, Holder DJ, Heyse JF, Stulovici B, Ferrer M (2008). "Integrating experimental and analytic approaches to improve data quality in genome-wide RNAi screens". Journal of Biomolecular Screening. 13 (5): 378–89. doi:10.1177/1087057108317145. PMID 18480473. S2CID 22679273.
6. Zhang, XHD (2007). "A pair of new statistical parameters for quality control in RNA interference high-throughput screening assays". Genomics. 89 (4): 552–61. doi:10.1016/j.ygeno.2006.12.014. PMID 17276655.
7. Zhang, XHD (2008). "Novel analytic criteria and effective plate designs for quality control in genome-wide RNAi screens". Journal of Biomolecular Screening. 13 (5): 363–77. doi:10.1177/1087057108317062. PMID 18567841. S2CID 12688742.
Further reading
• Kraybill, B. (2005) "Quantitative Assay Evaluation and Optimization" (unpublished note)
• Zhang XHD (2011) "Optimal High-Throughput Screening: Practical Experimental Design and Data Analysis for Genome-scale RNAi Research, Cambridge University Press"
| Wikipedia |
Z function
In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half. It is also called the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function. It can be defined in terms of the Riemann–Siegel theta function and the Riemann zeta function by
$Z(t)=e^{i\theta (t)}\zeta \left({\frac {1}{2}}+it\right).$
It follows from the functional equation of the Riemann zeta function that the Z function is real for real values of t. It is an even function, and real analytic for real values. It follows from the fact that the Riemann-Siegel theta function and the Riemann zeta function are both holomorphic in the critical strip, where the imaginary part of t is between −1/2 and 1/2, that the Z function is holomorphic in the critical strip also. Moreover, the real zeros of Z(t) are precisely the zeros of the zeta function along the critical line, and complex zeros in the Z function critical strip correspond to zeros off the critical line of the Riemann zeta function in its critical strip.
The Riemann–Siegel formula
Calculation of the value of Z(t) for real t, and hence of the zeta function along the critical line, is greatly expedited by the Riemann–Siegel formula. This formula tells us
$Z(t)=2\sum _{n^{2}<t/2\pi }n^{-1/2}\cos(\theta (t)-t\log n)+R(t),$
where the error term R(t) has a complex asymptotic expression in terms of the function
$\Psi (z)={\frac {\cos 2\pi (z^{2}-z-1/16)}{\cos 2\pi z}}$
and its derivatives. If $u=\left({\frac {t}{2\pi }}\right)^{1/4}$, $N=\lfloor u^{2}\rfloor $ and $p=u^{2}-N$ then
$R(t)\sim (-1)^{N-1}\left(\Psi (p)u^{-1}-{\frac {1}{96\pi ^{2}}}\Psi ^{(3)}(p)u^{-3}+\cdots \right)$
where the ellipsis indicates we may continue on to higher and increasingly complex terms.
Other efficient series for Z(t) are known, in particular several using the incomplete gamma function. If
$Q(a,z)={\frac {\Gamma (a,z)}{\Gamma (a)}}={\frac {1}{\Gamma (a)}}\int _{z}^{\infty }u^{a-1}e^{-u}\,du$
then an especially nice example is
$Z(t)=2\Re \left(e^{i\theta (t)}\left(\sum _{n=1}^{\infty }Q\left({\frac {s}{2}},\pi in^{2}\right)-{\frac {\pi ^{s/2}e^{\pi is/4}}{s\Gamma \left({\frac {s}{2}}\right)}}\right)\right)$
Behavior of the Z function
From the critical line theorem, it follows that the density of the real zeros of the Z function is
${\frac {c}{2\pi }}\log {\frac {t}{2\pi }}$
for some constant c > 2/5. Hence, the number of zeros in an interval of a given size slowly increases. If the Riemann hypothesis is true, all of the zeros in the critical strip are real zeros, and the constant c is one. It is also postulated that all of these zeros are simple zeros.
An Omega theorem
Because of the zeros of the Z function, it exhibits oscillatory behavior. It also slowly grows both on average and in peak value. For instance, we have, even without the Riemann hypothesis, the Omega theorem that
$Z(t)=\Omega \left(\exp \left({\frac {3}{4}}{\sqrt {\frac {\log t}{\log \log t}}}\right)\right),$
where the notation means that $Z(t)$ divided by the function within the Ω does not tend to zero with increasing t.
Average growth
The average growth of the Z function has also been much studied. We can find the root mean square (abbreviated RMS) average from
${\frac {1}{T}}\int _{0}^{T}Z(t)^{2}dt\sim \log T$
or
${\frac {1}{T}}\int _{T}^{2T}Z(t)^{2}dt\sim \log T$
which tell us that the RMS size of Z(t) grows as ${\sqrt {\log t}}$.
This estimate can be improved to
${\frac {1}{T}}\int _{0}^{T}Z(t)^{2}dt=\log T+(2\gamma -2\log(2\pi )-1)+O(T^{-15/22})$
If we increase the exponent, we get an average value which depends more on the peak values of Z. For fourth powers, we have
${\frac {1}{T}}\int _{0}^{T}Z(t)^{4}dt\sim {\frac {1}{2\pi ^{2}}}(\log T)^{4}$
from which we may conclude that the fourth root of the mean fourth power grows as ${\frac {1}{2^{1/4}{\sqrt {\pi }}}}\log t.$
The Lindelöf hypothesis
Main article: Lindelöf hypothesis
Higher even powers have been much studied, but less is known about the corresponding average value. It is conjectured, and follows from the Riemann hypothesis, that
${\frac {1}{T}}\int _{0}^{T}Z(t)^{2k}\,dt=o(T^{\varepsilon })$
for every positive ε. Here the little "o" notation means that the left hand side divided by the right hand side does converge to zero; in other words little o is the negation of Ω. This conjecture is called the Lindelöf hypothesis, and is weaker than the Riemann hypothesis. It is normally stated in an important equivalent form, which is
$Z(t)=o(t^{\varepsilon });$
in either form it tells us the rate of growth of the peak values cannot be too high. The best known bound on this rate of growth is not strong, telling us that any $\epsilon >{\frac {89}{570}}\approx 0.156$ is suitable. It would be astonishing to find that the Z function grew anywhere close to as fast as this. Littlewood proved that on the Riemann hypothesis,
$Z(t)=o\left(\exp \left({\frac {10\log t}{\log \log t}}\right)\right),$
and this seems far more likely.
References
• Edwards, H.M. (1974). Riemann's zeta function. Pure and Applied Mathematics. Vol. 58. New York-London: Academic Press. ISBN 0-12-232750-0. Zbl 0315.10035.
• Ivić, Aleksandar (2013). The theory of Hardy's Z-function. Cambridge Tracts in Mathematics. Vol. 196. Cambridge: Cambridge University Press. ISBN 978-1-107-02883-8. Zbl 1269.11075.
• Paris, R. B.; Kaminski, D. (2001). Asymptotics and Mellin-Barnes Integrals. Encyclopedia of Mathematics and Its Applications. Vol. 85. Cambridge: Cambridge University Press. ISBN 0-521-79001-8. Zbl 0983.41019.
• Ramachandra, K. (February 1996). Lectures on the mean-value and Omega-theorems for the Riemann Zeta-function. Lectures on Mathematics and Physics. Mathematics. Tata Institute of Fundamental Research. Vol. 85. Berlin: Springer-Verlag. ISBN 3-540-58437-4. Zbl 0845.11003.
• Titchmarsh, E. C. (1986) [1951]. Heath-Brown, D.R. (ed.). The Theory of the Riemann Zeta-Function (second revised ed.). Oxford University Press.
External links
• Weisstein, Eric W. "Riemann–Siegel Functions". MathWorld.
• Wolfram Research – Riemann-Siegel function Z (includes function plotting and evaluation)
| Wikipedia |
Z-matrix (mathematics)
In mathematics, the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form:
$Z=(z_{ij});\quad z_{ij}\leq 0,\quad i\neq j.$
Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term quasinegative matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made.
The Jacobian of a competitive dynamical system is a Z-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is J, then (−J) is a Z-matrix.
Related classes are L-matrices, M-matrices, P-matrices, Hurwitz matrices and Metzler matrices. L-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices and P-matrices are nonsingular M-matrices.
In the context of quantum complexity theory, these are referred to as stoquastic operators.[1]
See also
• Hurwitz matrix
• M-matrix
• Metzler matrix
• P-matrix
References
1. Bravyi, Sergey; DiVincenzo, David P.; Oliveira, Roberto I.; Terhal, Barbara M. (2006). "The Complexity of Stoquastic Local Hamiltonian Problems". arXiv:quant-ph/0606140.
• Huan T.; Cheng G.; Cheng X. (1 April 2006). "Modified SOR-type iterative method for Z-matrices". Applied Mathematics and Computation. 175 (1): 258–268. doi:10.1016/j.amc.2005.07.050.
• Saad, Y. (1996). Iterative methods for sparse linear systems (2nd ed.). Philadelphia, PA.: Society for Industrial and Applied Mathematics. p. 28. ISBN 0-534-94776-X.
• Berman, Abraham; Plemmons, Robert J. (2014). Nonnegative Matrices in the Mathematical Sciences. Academic Press. ISBN 9781483260860.
Matrix classes
Explicitly constrained entries
• Alternant
• Anti-diagonal
• Anti-Hermitian
• Anti-symmetric
• Arrowhead
• Band
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• Bisymmetric
• Block-diagonal
• Block
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• Conference
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Constant
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With specific applications
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Used in statistics
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Related terms
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• Mathematics portal
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| Wikipedia |
Mahler's 3/2 problem
In mathematics, Mahler's 3/2 problem concerns the existence of "Z-numbers".
A Z-number is a real number x such that the fractional parts of
$x\left({\frac {3}{2}}\right)^{n}$
are less than 1/2 for all positive integers n. Kurt Mahler conjectured in 1968 that there are no Z-numbers.
More generally, for a real number α, define Ω(α) as
$\Omega (\alpha )=\inf _{\theta >0}\left({\limsup _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace -\liminf _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace }\right).$
Mahler's conjecture would thus imply that Ω(3/2) exceeds 1/2. Flatto, Lagarias, and Pollington showed[1] that
$\Omega \left({\frac {p}{q}}\right)>{\frac {1}{p}}$
for rational p/q > 1 in lowest terms.
References
1. Flatto, Leopold; Lagarias, Jeffrey C.; Pollington, Andrew D. (1995). "On the range of fractional parts of ζ { (p/q)n }". Acta Arithmetica. LXX (2): 125–147. doi:10.4064/aa-70-2-125-147. ISSN 0065-1036. Zbl 0821.11038.
• Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. ISBN 0-8218-3387-1. Zbl 1033.11006.
| Wikipedia |
Integer points in convex polyhedra
The study of integer points in convex polyhedra[1] is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear equations with nonnegative coefficients have" or "how many solutions does an integer linear program have". Counting integer points in polyhedra or other questions about them arise in representation theory, commutative algebra, algebraic geometry, statistics, and computer science.[2]
The set of integer points, or, more generally, the set of points of an affine lattice, in a polyhedron is called Z-polyhedron,[3] from the mathematical notation $\mathbb {Z} $ or Z for the set of integer numbers.[4]
Properties
For a lattice Λ, Minkowski's theorem relates the number d(Λ) (the volume of a fundamental parallelepiped of the lattice) and the volume of a given symmetric convex set S to the number of lattice points contained in S.
The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d(Λ) as well.
Applications
Loop optimization
In certain approaches to loop optimization, the set of the executions of the loop body is viewed as the set of integer points in a polyhedron defined by loop constraints.
See also
• Convex lattice polytope
• Pick's theorem
References and notes
1. In some contexts convex polyhedra are called simply "polyhedra".
2. Integer points in polyhedra. Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, ACM--SIAM Joint Summer Research Conference, 2006
3. The term "Z-polyhedron" is also used as a synonym to convex lattice polytope, the convex hull of finitely many points in an affine lattice.
4. "Computations on Iterated Spaces" in: The Compiler Design Handbook: Optimizations and Machine Code Generation, CRC Press 2007, 2nd edition, ISBN 1-4200-4382-X, p.15-7
Further reading
• Barvinok, Alexander; Beck, Matthias; Haase, Christian; Reznick, Bruce; Welker, Volkmar (2005), Integer Points In Polyhedra: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held in Snowbird, UT, July 13–17, 2003, Contemporary Mathematics, vol. 374, Providence, RI: American Mathematical Society, doi:10.1090/conm/374, ISBN 0-8218-3459-2, MR 2134757
• Barvinok, Alexander (2008), Integer Points In Polyhedra, Zurich Lectures in Advanced Mathematics, Zürich: European Mathematical Society, doi:10.4171/052, ISBN 978-3-03719-052-4, MR 2455889
• Beck, Matthias; Haase, Christian; Reznick, Bruce; Vergne, Michèle; Welker, Volkmar; Yoshida, Ruriko (2008), Integer Points In Polyhedra: Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics (PDF), Contemporary Mathematics, vol. 452, Providence, RI: American Mathematical Society, doi:10.1090/conm/452, ISBN 978-0-8218-4173-0, MR 2416261
| Wikipedia |
Z-test
A Z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-tests test the mean of a distribution. For each significance level in the confidence interval, the Z-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's t-test whose critical values are defined by the sample size (through the corresponding degrees of freedom). Both the Z-test and Student's t-test have similarities in that they both help determine the significance of a set of data. However, the z-test is rarely used in practice because the population deviation is difficult to determine.
Applicability
Because of the central limit theorem, many test statistics are approximately normally distributed for large samples. Therefore, many statistical tests can be conveniently performed as approximate Z-tests if the sample size is large or the population variance is known. If the population variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n < 30), the Student's t-test may be more appropriate (in some cases, n < 50, as described below).
Procedure
How to perform a Z test when T is a statistic that is approximately normally distributed under the null hypothesis is as follows:
First, estimate the expected value μ of T under the null hypothesis, and obtain an estimate s of the standard deviation of T.
Second, determine the properties of T : one tailed or two tailed.
For Null hypothesis H0: μ≥μ0 vs alternative hypothesis H1: μ<μ0 , it is lower/left-tailed (one tailed).
For Null hypothesis H0: μ≤μ0 vs alternative hypothesis H1: μ>μ0 , it is upper/right-tailed (one tailed).
For Null hypothesis H0: μ=μ0 vs alternative hypothesis H1: μ≠μ0 , it is two-tailed.
Third, calculate the standard score:
$Z={\frac {({\bar {X}}-\mu _{0})}{\sigma }},$
which one-tailed and two-tailed p-values can be calculated as Φ(Z)(for lower/left-tailed tests), Φ(−Z) (for upper/right-tailed tests) and 2Φ(−|Z|) (for two-tailed tests) where Φ is the standard normal cumulative distribution function.
Use in location testing
1. The term "Z-test" is often used to refer specifically to the one-sample location test comparing the mean of a set of measurements to a given constant when the sample variance is known. For example, if the observed data X1, ..., Xn are (i) independent, (ii) have a common mean μ, and (iii) have a common variance σ2, then the sample average X has mean μ and variance ${\frac {\sigma ^{2}}{n}}$.
2. The null hypothesis is that the mean value of X is a given number μ0. We can use X as a test-statistic, rejecting the null hypothesis if X − μ0 is large.
3. To calculate the standardized statistic $Z={\frac {({\bar {X}}-\mu _{0})}{s}}$, we need to either know or have an approximate value for σ2, from which we can calculate $s^{2}={\frac {\sigma ^{2}}{n}}$ . In some applications, σ2 is known, but this is uncommon.
4. If the sample size is moderate or large, we can substitute the sample variance for σ2, giving a plug-in test. The resulting test will not be an exact Z-test since the uncertainty in the sample variance is not accounted for—however, it will be a good approximation unless the sample size is small.
5. A t-test can be used to account for the uncertainty in the sample variance when the data are exactly normal.
6. Difference between Z-test and t-test: Z-test is used when sample size is large (n>50), or the population variance is known. t-test is used when sample size is small (n<50) and population variance is unknown.
7. There is no universal constant at which the sample size is generally considered large enough to justify use of the plug-in test. Typical rules of thumb: the sample size should be 50 observations or more.
8. For large sample sizes, the t-test procedure gives almost identical p-values as the Z-test procedure.
9. Other location tests that can be performed as Z-tests are the two-sample location test and the paired difference test.
Conditions
For the Z-test to be applicable, certain conditions must be met.
• Nuisance parameters should be known, or estimated with high accuracy (an example of a nuisance parameter would be the standard deviation in a one-sample location test). Z-tests focus on a single parameter, and treat all other unknown parameters as being fixed at their true values. In practice, due to Slutsky's theorem, "plugging in" consistent estimates of nuisance parameters can be justified. However if the sample size is not large enough for these estimates to be reasonably accurate, the Z-test may not perform well.
• The test statistic should follow a normal distribution. Generally, one appeals to the central limit theorem to justify assuming that a test statistic varies normally. There is a great deal of statistical research on the question of when a test statistic varies approximately normally. If the variation of the test statistic is strongly non-normal, a Z-test should not be used.
If estimates of nuisance parameters are plugged in as discussed above, it is important to use estimates appropriate for the way the data were sampled. In the special case of Z-tests for the one or two sample location problem, the usual sample standard deviation is only appropriate if the data were collected as an independent sample.
In some situations, it is possible to devise a test that properly accounts for the variation in plug-in estimates of nuisance parameters. In the case of one and two sample location problems, a t-test does this.
Example
Suppose that in a particular geographic region, the mean and standard deviation of scores on a reading test are 100 points, and 12 points, respectively. Our interest is in the scores of 55 students in a particular school who received a mean score of 96. We can ask whether this mean score is significantly lower than the regional mean—that is, are the students in this school comparable to a simple random sample of 55 students from the region as a whole, or are their scores surprisingly low?
First calculate the standard error of the mean:
$\mathrm {SE} ={\frac {\sigma }{\sqrt {n}}}={\frac {12}{\sqrt {55}}}={\frac {12}{7.42}}=1.62$
where ${\sigma }$ is the population standard deviation.
Next calculate the z-score, which is the distance from the sample mean to the population mean in units of the standard error:
$z={\frac {M-\mu }{\mathrm {SE} }}={\frac {96-100}{1.62}}=-2.47$
In this example, we treat the population mean and variance as known, which would be appropriate if all students in the region were tested. When population parameters are unknown, a Student's t-test should be conducted instead.
The classroom mean score is 96, which is −2.47 standard error units from the population mean of 100. Looking up the z-score in a table of the standard normal distribution cumulative probability, we find that the probability of observing a standard normal value below −2.47 is approximately 0.5 − 0.4932 = 0.0068. This is the one-sided p-value for the null hypothesis that the 55 students are comparable to a simple random sample from the population of all test-takers. The two-sided p-value is approximately 0.014 (twice the one-sided p-value).
Another way of stating things is that with probability 1 − 0.014 = 0.986, a simple random sample of 55 students would have a mean test score within 4 units of the population mean. We could also say that with 98.6% confidence we reject the null hypothesis that the 55 test takers are comparable to a simple random sample from the population of test-takers.
The Z-test tells us that the 55 students of interest have an unusually low mean test score compared to most simple random samples of similar size from the population of test-takers. A deficiency of this analysis is that it does not consider whether the effect size of 4 points is meaningful. If instead of a classroom, we considered a subregion containing 900 students whose mean score was 99, nearly the same z-score and p-value would be observed. This shows that if the sample size is large enough, very small differences from the null value can be highly statistically significant. See statistical hypothesis testing for further discussion of this issue.
Z-tests other than location tests
Location tests are the most familiar Z-tests. Another class of Z-tests arises in maximum likelihood estimation of the parameters in a parametric statistical model. Maximum likelihood estimates are approximately normal under certain conditions, and their asymptotic variance can be calculated in terms of the Fisher information. The maximum likelihood estimate divided by its standard error can be used as a test statistic for the null hypothesis that the population value of the parameter equals zero. More generally, if ${\hat {\theta }}$ is the maximum likelihood estimate of a parameter θ, and θ0 is the value of θ under the null hypothesis,
${\frac {{\hat {\theta }}-\theta _{0}}{{\rm {SE}}({\hat {\theta }})}}$
can be used as a Z-test statistic.
When using a Z-test for maximum likelihood estimates, it is important to be aware that the normal approximation may be poor if the sample size is not sufficiently large. Although there is no simple, universal rule stating how large the sample size must be to use a Z-test, simulation can give a good idea as to whether a Z-test is appropriate in a given situation.
Z-tests are employed whenever it can be argued that a test statistic follows a normal distribution under the null hypothesis of interest. Many non-parametric test statistics, such as U statistics, are approximately normal for large enough sample sizes, and hence are often performed as Z-tests.
See also
• Normal distribution
• Standard normal table
• Standard score
• Student's t-test
References
• Sprinthall, R. C. (2011). Basic Statistical Analysis (9th ed.). Pearson Education. ISBN 978-0-205-05217-2.
• Casella, G., Berger, R. L. (2002). Statistical Inference. Duxbury Press. ISBN 0-534-24312-6.
• Douglas C.Montgomery, George C.Runger.(2014). Applied Statistics And Probability For Engineers.(6th ed.). John Wiley & Sons, inc. ISBN 9781118539712, 9781118645062.
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| Wikipedia |
G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules.
The term G-module is also used for the more general notion of an R-module on which G acts linearly (i.e. as a group of R-module automorphisms).
Definition and basics
Let $G$ be a group. A left $G$-module consists of[1] an abelian group $M$ together with a left group action $\rho :G\times M\to M$ such that
g·(a1 + a2) = g·a1 + g·a2
where g·a denotes ρ(g,a). A right G-module is defined similarly. Given a left G-module M, it can be turned into a right G-module by defining a·g = g−1·a.
A function f : M → N is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if f is both a group homomorphism and G-equivariant.
The collection of left (respectively right) G-modules and their morphisms form an abelian category G-Mod (resp. Mod-G). The category G-Mod (resp. Mod-G) can be identified with the category of left (resp. right) ZG-modules, i.e. with the modules over the group ring Z[G].
A submodule of a G-module M is a subgroup A ⊆ M that is stable under the action of G, i.e. g·a ∈ A for all g ∈ G and a ∈ A. Given a submodule A of M, the quotient module M/A is the quotient group with action g·(m + A) = g·m + A.
Examples
• Given a group G, the abelian group Z is a G-module with the trivial action g·a = a.
• Let M be the set of binary quadratic forms f(x, y) = ax2 + 2bxy + cy2 with a, b, c integers, and let G = SL(2, Z) (the 2×2 special linear group over Z). Define
$(g\cdot f)(x,y)=f((x,y)g^{t})=f\left((x,y)\cdot {\begin{bmatrix}\alpha &\gamma \\\beta &\delta \end{bmatrix}}\right)=f(\alpha x+\beta y,\gamma x+\delta y),$
where
$g={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}$
and (x, y)g is matrix multiplication. Then M is a G-module studied by Gauss.[2] Indeed, we have
$g(h(f(x,y)))=gf((x,y)h^{t})=f((x,y)h^{t}g^{t})=f((x,y)(gh)^{t})=(gh)f(x,y).$
• If V is a representation of G over a field K, then V is a G-module (it is an abelian group under addition).
Topological groups
If G is a topological group and M is an abelian topological group, then a topological G-module is a G-module where the action map G×M → M is continuous (where the product topology is taken on G×M).[3]
In other words, a topological G-module is an abelian topological group M together with a continuous map G×M → M satisfying the usual relations g(a + a′) = ga + ga′, (gg′)a = g(g′a), and 1a = a.
Notes
1. Curtis, Charles W.; Reiner, Irving (1962), Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons (Reedition 2006 by AMS Bookstore), ISBN 978-0-470-18975-7.
2. Kim, Myung-Hwan (1999), Integral Quadratic Forms and Lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea, American Mathematical Soc.
3. D. Wigner (1973). "Algebraic cohomology of topological groups". Trans. Amer. Math. Soc. 178: 83–93. doi:10.1090/s0002-9947-1973-0338132-7.
References
• Chapter 6 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
| Wikipedia |
ZJ theorem
In mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then Op′(G)Z(J(S)) is a normal subgroup of G, for any Sylow p-subgroup S.
Notation and definitions
• J(S) is the Thompson subgroup of a p-group S: the subgroup generated by the abelian subgroups of maximal order.
• Z(H) means the center of a group H.
• Op′ is the maximal normal subgroup of G of order coprime to p, the p′-core
• Op is the maximal normal p-subgroup of G, the p-core.
• Op′,p(G) is the maximal normal p-nilpotent subgroup of G, the p′,p-core, part of the upper p-series.
• For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-stable if whenever P is a p-subgroup of G such that POp′(G) is normal in G, and [P,x,x] = 1, then the image of x in NG(P)/CG(P) is contained in a normal p-subgroup of NG(P)/CG(P).
• For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-constrained if the centralizer CG(P) is contained in Op′,p(G) whenever P is a Sylow p-subgroup of Op′,p(G).
References
• Glauberman, George (1968), "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, doi:10.4153/cjm-1968-107-2, ISSN 0008-414X, MR 0230807
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
• Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra, 13 (2): 149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR 0245683
| Wikipedia |
NL (complexity)
In computational complexity theory, NL (Nondeterministic Logarithmic-space) is the complexity class containing decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space.
Unsolved problem in computer science:
${\mathsf {L{\overset {?}{=}}NL}}$
(more unsolved problems in computer science)
NL is a generalization of L, the class for logspace problems on a deterministic Turing machine. Since any deterministic Turing machine is also a nondeterministic Turing machine, we have that L is contained in NL.
NL can be formally defined in terms of the computational resource nondeterministic space (or NSPACE) as NL = NSPACE(log n).
Important results in complexity theory allow us to relate this complexity class with other classes, telling us about the relative power of the resources involved. Results in the field of algorithms, on the other hand, tell us which problems can be solved with this resource. Like much of complexity theory, many important questions about NL are still open (see Unsolved problems in computer science).
Occasionally NL is referred to as RL due to its probabilistic definition below; however, this name is more frequently used to refer to randomized logarithmic space, which is not known to equal NL.
NL-complete problems
Several problems are known to be NL-complete under log-space reductions, including ST-connectivity and 2-satisfiability. ST-connectivity asks, for nodes S and T in a directed graph, whether T is reachable from S. 2-satisfiability asks, given a propositional formula of which each clause is the disjunction of two literals, if there is a variable assignment that makes the formula true. An example instance, where $\neg $ indicates not, might be:
$(x_{1}\vee \neg x_{3})\wedge (\neg x_{2}\vee x_{3})\wedge (\neg x_{1}\vee \neg x_{2})$
Containments
It is known that NL is contained in P, since there is a polynomial-time algorithm for 2-satisfiability, but it is not known whether NL = P or whether L = NL. It is known that NL = co-NL, where co-NL is the class of languages whose complements are in NL. This result (the Immerman–Szelepcsényi theorem) was independently discovered by Neil Immerman and Róbert Szelepcsényi in 1987; they received the 1995 Gödel Prize for this work.
In circuit complexity, NL can be placed within the NC hierarchy. In Papadimitriou 1994, Theorem 16.1, we have:
${\mathsf {NC_{1}\subseteq L\subseteq NL\subseteq NC_{2}}}$.
More precisely, NL is contained in AC1. It is known that NL is equal to ZPL, the class of problems solvable by randomized algorithms in logarithmic space and unbounded time, with no error. It is not, however, known or believed to be equal to RLP or ZPLP, the polynomial-time restrictions of RL and ZPL, which some authors refer to as RL and ZPL.
We can relate NL to deterministic space using Savitch's theorem, which tells us that any nondeterministic algorithm can be simulated by a deterministic machine in at most quadratically more space. From Savitch's theorem, we have directly that:
${\mathsf {NL\subseteq SPACE}}(\log ^{2}n)\ \ \ \ {\text{equivalently, }}{\mathsf {NL\subseteq L}}^{2}.$
This was the strongest deterministic-space inclusion known in 1994 (Papadimitriou 1994 Problem 16.4.10, "Symmetric space"). Since larger space classes are not affected by quadratic increases, the nondeterministic and deterministic classes are known to be equal, so that for example we have PSPACE = NPSPACE.
Alternative definitions
Probabilistic definition
Suppose C is the complexity class of decision problems solvable in logarithmithic space with probabilistic Turing machines that never accept incorrectly but are allowed to reject incorrectly less than 1/3 of the time; this is called one-sided error. The constant 1/3 is arbitrary; any x with 0 ≤ x < 1/2 would suffice.
It turns out that C = NL. Notice that C, unlike its deterministic counterpart L, is not limited to polynomial time, because although it has a polynomial number of configurations it can use randomness to escape an infinite loop. If we do limit it to polynomial time, we get the class RL, which is contained in but not known or believed to equal NL.
There is a simple algorithm that establishes that C = NL. Clearly C is contained in NL, since:
• If the string is not in the language, both reject along all computation paths.
• If the string is in the language, an NL algorithm accepts along at least one computation path and a C algorithm accepts along at least two-thirds of its computation paths.
To show that NL is contained in C, we simply take an NL algorithm and choose a random computation path of length n, and execute this 2n times. Because no computation path exceeds length n, and because there are 2n computation paths in all, we have a good chance of hitting the accepting one (bounded below by a constant).
The only problem is that we don't have room in log space for a binary counter that goes up to 2n. To get around this we replace it with a randomized counter, which simply flips n coins and stops and rejects if they all land on heads. Since this event has probability 2−n, we expect to take 2n steps on average before stopping. It only needs to keep a running total of the number of heads in a row it sees, which it can count in log space.
Because of the Immerman–Szelepcsényi theorem, according to which NL is closed under complements, the one-sided error in these probabilistic computations can be replaced by zero-sided error. That is, these problems can be solved by probabilistic Turing machines that use logarithmic space and never make errors. The corresponding complexity class that also requires the machine to use only polynomial time is called ZPLP.
Thus, when we only look at space alone, it seems that randomization and nondeterminism are equally powerful.
Certificate definition
NL can equivalently be characterised by certificates, analogous to classes such as NP. Consider a deterministic logarithmic-space bounded Turing machine that has an additional read-only read-once input tape. A language is in NL if and only if such a Turing machine accepts any word in the language for an appropriate choice of certificate in its additional input tape, and rejects any word not in the language regardless of the certificate.[1]
Cem Say and Abuzer Yakaryılmaz have proven that the deterministic logarithmic-space Turing machine in the statement above can be replaced by a bounded-error probabilistic constant-space Turing machine that is allowed to use only a constant number of random bits.[2]
Descriptive complexity
There is a simple logical characterization of NL: it contains precisely those languages expressible in first-order logic with an added transitive closure operator.
Closure properties
The class NL is closed under the operations complementation, union, and therefore intersection, concatenation, and Kleene star.
Notes
1. Arora, Sanjeev; Barak, Boaz (2009). "Definition 4.19". Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978-0-521-42426-4.
2. A. C. Cem Say, Abuzer Yakaryılmaz, "Finite state verifiers with constant randomness," Logical Methods in Computer Science, Vol. 10(3:6)2014, pp. 1-17.
References
• Complexity Zoo: NL
• Papadimitriou, C. (1994). "Chapter 16: Logarithmic Space". Computational Complexity. Addison-Wesley. ISBN 0-201-53082-1.
• Michael Sipser (27 June 1997). "Sections 8.4–8.6: The Classes L and NL, NL-completeness, NL equals coNL". Introduction to the Theory of Computation. PWS Publishing. pp. 294–302. ISBN 0-534-94728-X.
• Introduction to Complexity Theory: Lecture 7. Oded Goldreich. Proposition 6.1. Our C is what Goldreich calls badRSPACE(log n).
Important complexity classes
Considered feasible
• DLOGTIME
• AC0
• ACC0
• TC0
• L
• SL
• RL
• NL
• NL-complete
• NC
• SC
• CC
• P
• P-complete
• ZPP
• RP
• BPP
• BQP
• APX
• FP
Suspected infeasible
• UP
• NP
• NP-complete
• NP-hard
• co-NP
• co-NP-complete
• AM
• QMA
• PH
• ⊕P
• PP
• #P
• #P-complete
• IP
• PSPACE
• PSPACE-complete
Considered infeasible
• EXPTIME
• NEXPTIME
• EXPSPACE
• 2-EXPTIME
• ELEMENTARY
• PR
• R
• RE
• ALL
Class hierarchies
• Polynomial hierarchy
• Exponential hierarchy
• Grzegorczyk hierarchy
• Arithmetical hierarchy
• Boolean hierarchy
Families of classes
• DTIME
• NTIME
• DSPACE
• NSPACE
• Probabilistically checkable proof
• Interactive proof system
List of complexity classes
| Wikipedia |
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.[1] In terms of set-builder notation, that is
$A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}.$[2][3]
"Cartesian square" redirects here. For Cartesian squares in category theory, see Cartesian square (category theory).
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).[4]
One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
The Cartesian product is named after René Descartes,[5] whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
Set-theoretic definition
A rigorous definition of the Cartesian product requires a domain to be specified in the set-builder notation. In this case the domain would have to contain the Cartesian product itself. For defining the Cartesian product of the sets $A$ and $B$, one such domain is the set ${\mathcal {P}}({\mathcal {P}}(A\cup B))$ where ${\mathcal {P}}$ denotes the power set. Then the Cartesian product of the sets $A$ and $B$ would be defined as[6]
$A\times B=\{x\in {\mathcal {P}}({\mathcal {P}}(A\cup B))\mid \exists a\in A\exists b\in B:x=(a,b)\}.$
Examples
A deck of cards
An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits {♠, ♥, ♦, ♣} form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), …, (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}.
Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), …, (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.
These two sets are distinct, even disjoint, but there is a natural bijection between them, under which (3, ♣) corresponds to (♣, 3) and so on.
A two-dimensional coordinate system
The main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way, and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates. Usually, such a pair's first and second components are called its x and y coordinates, respectively (see picture). The set of all such pairs (i.e., the Cartesian product ℝ×ℝ, with ℝ denoting the real numbers) is thus assigned to the set of all points in the plane.
Most common implementation (set theory)
Main article: Implementation of mathematics in set theory
A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, Kuratowski's definition, is $(x,y)=\{\{x\},\{x,y\}\}$. Under this definition, $(x,y)$ is an element of ${\mathcal {P}}({\mathcal {P}}(X\cup Y))$, and $X\times Y$ is a subset of that set, where ${\mathcal {P}}$ represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, and specification. Since functions are usually defined as a special case of relations, and relations are usually defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is necessarily prior to most other definitions.
Non-commutativity and non-associativity
Let A, B, C, and D be sets.
The Cartesian product A × B is not commutative,
$A\times B\neq B\times A,$[4]
because the ordered pairs are reversed unless at least one of the following conditions is satisfied:[7]
• A is equal to B, or
• A or B is the empty set.
For example:
A = {1,2}; B = {3,4}
A × B = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
B × A = {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}
A = B = {1,2}
A × B = B × A = {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}
A = {1,2}; B = ∅
A × B = {1,2} × ∅ = ∅
B × A = ∅ × {1,2} = ∅
Strictly speaking, the Cartesian product is not associative (unless one of the involved sets is empty).
$(A\times B)\times C\neq A\times (B\times C)$
If for example A = {1}, then (A × A) × A = {((1, 1), 1)} ≠ {(1, (1, 1))} = A × (A × A).
Intersections, unions, and subsets
See also: List of set identities and relations
Example sets
A = {y ∈ ℝ : 1 ≤ y ≤ 4}, B = {x ∈ ℝ : 2 ≤ x ≤ 5},
and C = {x ∈ ℝ : 4 ≤ x ≤ 7}, demonstrating
A × (B∩C) = (A×B) ∩ (A×C),
A × (B∪C) = (A×B) ∪ (A×C), and
A × (B \ C) = (A×B) \ (A×C)
Example sets
A = {x ∈ ℝ : 2 ≤ x ≤ 5}, B = {x ∈ ℝ : 3 ≤ x ≤ 7},
C = {y ∈ ℝ : 1 ≤ y ≤ 3}, D = {y ∈ ℝ : 2 ≤ y ≤ 4}, demonstrating
(A∩B) × (C∩D) = (A×C) ∩ (B×D).
(A∪B) × (C∪D) ≠ (A×C) ∪ (B×D) can be seen from the same example.
The Cartesian product satisfies the following property with respect to intersections (see middle picture).
$(A\cap B)\times (C\cap D)=(A\times C)\cap (B\times D)$
In most cases, the above statement is not true if we replace intersection with union (see rightmost picture).
$(A\cup B)\times (C\cup D)\neq (A\times C)\cup (B\times D)$
In fact, we have that:
$(A\times C)\cup (B\times D)=[(A\setminus B)\times C]\cup [(A\cap B)\times (C\cup D)]\cup [(B\setminus A)\times D]$
For the set difference, we also have the following identity:
$(A\times C)\setminus (B\times D)=[A\times (C\setminus D)]\cup [(A\setminus B)\times C]$
Here are some rules demonstrating distributivity with other operators (see leftmost picture):[7]
${\begin{aligned}A\times (B\cap C)&=(A\times B)\cap (A\times C),\\A\times (B\cup C)&=(A\times B)\cup (A\times C),\\A\times (B\setminus C)&=(A\times B)\setminus (A\times C),\end{aligned}}$
$(A\times B)^{\complement }=\left(A^{\complement }\times B^{\complement }\right)\cup \left(A^{\complement }\times B\right)\cup \left(A\times B^{\complement }\right)\!,$
where $A^{\complement }$ denotes the absolute complement of A.
Other properties related with subsets are:
${\text{if }}A\subseteq B{\text{, then }}A\times C\subseteq B\times C;$
${\text{if both }}A,B\neq \emptyset {\text{, then }}A\times B\subseteq C\times D\!\iff \!A\subseteq C{\text{ and }}B\subseteq D.$[8]
Cardinality
See also: Cardinal arithmetic
The cardinality of a set is the number of elements of the set. For example, defining two sets: A = {a, b} and B = {5, 6}. Both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements:
A × B = {(a,5), (a,6), (b,5), (b,6)}.
where each element of A is paired with each element of B, and where each pair makes up one element of the output set. The number of values in each element of the resulting set is equal to the number of sets whose Cartesian product is being taken; 2 in this case. The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is,
|A × B| = |A| · |B|.[4]
In this case, |A × B| = 4
Similarly
|A × B × C| = |A| · |B| · |C|
and so on.
The set A × B is infinite if either A or B is infinite, and the other set is not the empty set.[9]
Cartesian products of several sets
n-ary Cartesian product
The Cartesian product can be generalized to the n-ary Cartesian product over n sets X1, ..., Xn as the set
$X_{1}\times \cdots \times X_{n}=\{(x_{1},\ldots ,x_{n})\mid x_{i}\in X_{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}$
of n-tuples. If tuples are defined as nested ordered pairs, it can be identified with (X1 × ⋯ × Xn−1) × Xn. If a tuple is defined as a function on {1, 2, …, n} that takes its value at i to be the ith element of the tuple, then the Cartesian product X1×⋯×Xn is the set of functions
$\{x:\{1,\ldots ,n\}\to X_{1}\cup \cdots \cup X_{n}\ |\ x(i)\in X_{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.$
n-ary Cartesian power
The Cartesian square of a set X is the Cartesian product X2 = X × X. An example is the 2-dimensional plane R2 = R × R where R is the set of real numbers:[1] R2 is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).
The n-ary Cartesian power of a set X, denoted $X^{n}$, can be defined as
$X^{n}=\underbrace {X\times X\times \cdots \times X} _{n}=\{(x_{1},\ldots ,x_{n})\ |\ x_{i}\in X\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.$
An example of this is R3 = R × R × R, with R again the set of real numbers,[1] and more generally Rn.
The n-ary Cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a special case, the 0-ary Cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
Infinite Cartesian products
It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If I is any index set, and $\{X_{i}\}_{i\in I}$ is a family of sets indexed by I, then the Cartesian product of the sets in $\{X_{i}\}_{i\in I}$ is defined to be
$\prod _{i\in I}X_{i}=\left\{\left.f:I\to \bigcup _{i\in I}X_{i}\ \right|\ \forall i\in I.\ f(i)\in X_{i}\right\},$
that is, the set of all functions defined on the index set I such that the value of the function at a particular index i is an element of Xi. Even if each of the Xi is nonempty, the Cartesian product may be empty if the axiom of choice, which is equivalent to the statement that every such product is nonempty, is not assumed. $\prod _{i\in I}X_{i}$ may also be denoted ${\mathsf {X}}$${}_{i\in I}X_{i}$.[10]
For each j in I, the function
$\pi _{j}:\prod _{i\in I}X_{i}\to X_{j},$
defined by $\pi _{j}(f)=f(j)$ is called the jth projection map.
Cartesian power is a Cartesian product where all the factors Xi are the same set X. In this case,
$\prod _{i\in I}X_{i}=\prod _{i\in I}X$
is the set of all functions from I to X, and is frequently denoted XI. This case is important in the study of cardinal exponentiation. An important special case is when the index set is $\mathbb {N} $, the natural numbers: this Cartesian product is the set of all infinite sequences with the ith term in its corresponding set Xi. For example, each element of
$\prod _{n=1}^{\infty }\mathbb {R} =\mathbb {R} \times \mathbb {R} \times \cdots $
can be visualized as a vector with countably infinite real number components. This set is frequently denoted $\mathbb {R} ^{\omega }$, or $\mathbb {R} ^{\mathbb {N} }$.
Other forms
Abbreviated form
If several sets are being multiplied together (e.g., X1, X2, X3, …), then some authors[11] choose to abbreviate the Cartesian product as simply ×Xi.
Cartesian product of functions
If f is a function from X to A and g is a function from Y to B, then their Cartesian product f × g is a function from X × Y to A × B with
$(f\times g)(x,y)=(f(x),g(y)).$
This can be extended to tuples and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.
Cylinder
Let $A$ be a set and $B\subseteq A$. Then the cylinder of $B$ with respect to $A$ is the Cartesian product $B\times A$ of $B$ and $A$.
Normally, $A$ is considered to be the universe of the context and is left away. For example, if $B$ is a subset of the natural numbers $\mathbb {N} $, then the cylinder of $B$ is $B\times \mathbb {N} $.
Definitions outside set theory
Category theory
Although the Cartesian product is traditionally applied to sets, category theory provides a more general interpretation of the product of mathematical structures. This is distinct from, although related to, the notion of a Cartesian square in category theory, which is a generalization of the fiber product.
Exponentiation is the right adjoint of the Cartesian product; thus any category with a Cartesian product (and a final object) is a Cartesian closed category.
Graph theory
In graph theory, the Cartesian product of two graphs G and H is the graph denoted by G × H, whose vertex set is the (ordinary) Cartesian product V(G) × V(H) and such that two vertices (u,v) and (u′,v′) are adjacent in G × H, if and only if u = u′ and v is adjacent with v′ in H, or v = v′ and u is adjacent with u′ in G. The Cartesian product of graphs is not a product in the sense of category theory. Instead, the categorical product is known as the tensor product of graphs.
See also
• Binary relation
• Concatenation of sets of strings
• Coproduct
• Cross product
• Direct product of groups
• Empty product
• Euclidean space
• Exponential object
• Finitary relation
• Join (SQL) § Cross join
• Orders on the Cartesian product of totally ordered sets
• Axiom of power set (to prove the existence of the Cartesian product)
• Product (category theory)
• Product topology
• Product type
• Ultraproduct
References
1. Weisstein, Eric W. "Cartesian Product". mathworld.wolfram.com. Retrieved September 5, 2020.
2. Warner, S. (1990). Modern Algebra. Dover Publications. p. 6.
3. Nykamp, Duane. "Cartesian product definition". Math Insight. Retrieved September 5, 2020.
4. "Cartesian Product". web.mnstate.edu. Archived from the original on July 18, 2020. Retrieved September 5, 2020.
5. "Cartesian". Merriam-Webster.com. 2009. Retrieved December 1, 2009.
6. Corry, S. "A Sketch of the Rudiments of Set Theory" (PDF). Retrieved May 5, 2023.
7. Singh, S. (August 27, 2009). Cartesian product. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/
8. Cartesian Product of Subsets. (February 15, 2011). ProofWiki. Retrieved 05:06, August 1, 2011 from https://proofwiki.org/w/index.php?title=Cartesian_Product_of_Subsets&oldid=45868
9. Peter S. (1998). A Crash Course in the Mathematics of Infinite Sets. St. John's Review, 44(2), 35–59. Retrieved August 1, 2011, from http://www.mathpath.org/concepts/infinity.htm
10. F. R. Drake, Set Theory: An Introduction to Large Cardinals, p.24. Studies in Logic and the Foundations of Mathematics, vol. 76 (1978). ISBN 0-7204-2200-0.
11. Osborne, M., and Rubinstein, A., 1994. A Course in Game Theory. MIT Press.
External links
• Cartesian Product at ProvenMath
• "Direct product", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• How to find the Cartesian Product, Education Portal Academy
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| Wikipedia |
Zadeh's rule
In mathematical optimization, Zadeh's rule (also known as the least-entered rule) is an algorithmic refinement of the simplex method for linear optimization.
The rule was proposed around 1980 by Norman Zadeh (son of Lotfi A. Zadeh), and has entered the folklore of convex optimization since then.[1]
Zadeh offered a reward of $1,000 to anyone who can show that the rule admits polynomially many iterations or to prove that there is a family of linear programs on which the pivoting rule requires subexponentially many iterations to find the optimum.[2]
Algorithm
Zadeh's rule belongs to the family of history-based improvement rules which, during a run of the simplex algorithm, retain supplementary data in addition to the current basis of the linear program.
In particular, the rule chooses among all improving variables one which has entered the basis least often, intuitively ensuring that variables that might yield a substantive improvement in the long run but only a small improvement in a single step will be applied after a linear number of steps.
The supplementary data structure in Zadeh's algorithm can therefore be modeled as an occurrence record, mapping all variables to natural numbers, monitoring how often a particular variable has entered the basis. In every iteration, the algorithm then selects an improving variable that is minimal with respect to the retained occurrence record.
Note that the rule does not explicitly specify which particular improving variable should enter the basis in case of a tie.
Superpolynomial lower bound
Zadeh's rule has been shown to have at least super-polynomial time complexity in the worse-case by constructing a family of Markov decision processes on which the policy iteration algorithm requires a super-polynomial number of steps.[3][4]
Running the simplex algorithm with Zadeh's rule on the induced linear program then yields a super-polynomial lower bound.
The result was presented at the "Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?" IPAM workshop in 2011 by Oliver Friedmann.[5] Zadeh, although not working in academia anymore at that time, attended the Workshop and honored his original proposal.[6]
Exponential lower bound
Friedmann's original result has since been strengthened by the construction of an exponential instance for Zadeh's rule.[7]
Notes
1. Zadeh, Norman (1980). "What is the worst case behaviour of the simplex algorithm?". Technical Report, Department of Operations Research, Stanford.
2. Ziegler, Günter (2004). "Typical and extremal linear programs". The Sharpest Cut (MPS-Siam Series on Optimization: 217–230. doi:10.1137/1.9780898718805.ch14. ISBN 978-0-89871-552-1.
3. Friedmann, Oliver (2011). "A subexponential lower bound for Zadeh's pivoting rule for solving linear programs and games". Proceedings of the 15th International Conference on Integer Programming and Combinatorial Optimization (IPCO). pp. 192–206.
4. Disser, Y.; Hopp, A.V. (2019). "On Friedmann's Subexponential Lower Bound for Zadeh's Pivot Rule". Proceedings of the 20th Conference on Integer Programming and Combinatorial Optimization (IPCO). pp. 168–180.
5. "Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?".
6. "Günter Ziegler: 1000$ from Beverly Hills for a Math Problem. (IPAM remote blogging.)". 20 January 2011.
7. Disser, Yann; Friedmann, Oliver; Hopp, Alexander V. (2022). "An exponential lower bound for Zadeh's pivot rule". Mathematical Programming.
| Wikipedia |
Sara Zahedi
Sara Zahedi (born 1981 in Tehran)[1] is an Iranian-Swedish mathematician who works in computational fluid dynamics and holds an associate professorship in numerical analysis at the Royal Institute of Technology (KTH) in Sweden. She is one of ten winners and the only female winner of the European Mathematical Society Prize for 2016 "for her outstanding research regarding the development and analysis of numerical algorithms for partial differential equations with a focus on applications to problems with dynamically changing geometry". The topic of Zahedi's EMS Prize lecture was her recent research on the CutFEM method of solving fluid dynamics problems with changing boundary geometry, such as arise when simulating the dynamics of systems of two immiscible liquids.[2] This method combines level set methods to represent the domain boundaries as cuts through an underlying uniform grid, together with numerical simulation techniques that can adapt to the complex geometries of grid cells cut by these boundaries.[3]
When Zahedi was ten years old, with her father having been killed by the regime after the Iranian Revolution, her mother sent her on her own as a refugee to Sweden, and only rejoined her some years later.[1][4] She was drawn to mathematics in part because she understood mathematics better than the Swedish language,[4] and to fluid mechanics because of its real-world applications.[1] She earned a master's degree from KTH in 2006, and a doctorate in 2011;[2] her dissertation, Numerical Methods for Fluid Interface Problems, was supervised by Gunilla Kreiss.[5] After postdoctoral studies at Uppsala University, she returned to KTH as an assistant professor in 2014.[2]
Selected publications
• Olsson, Elin; Kreiss, Gunilla; Zahedi, Sara (2007), "A conservative level set method for two phase flow. II", Journal of Computational Physics, 225 (1): 785–807, Bibcode:2007JCoPh.225..785O, doi:10.1016/j.jcp.2006.12.027, MR 2346700.
• Hansbo, Peter; Larson, Mats G.; Zahedi, Sara (2014), "A cut finite element method for a Stokes interface problem", Applied Numerical Mathematics, 85: 90–114, arXiv:1205.5684, doi:10.1016/j.apnum.2014.06.009, MR 3239219, S2CID 119167194.
References
1. Timon, Ágata (July 20, 2016), "Me gusta resolver problemas matemáticos del mundo real: Sara Zahedi, del Royal Institute of Technology (Suecia), ganadora del premio EMS a matemáticos jóvenes", El Mundo (in Spanish).
2. Prize laureates, 7th Eur. Congress of Mathematics, July 18–22, 2016, retrieved 2016-07-26.
3. Burman, Erik; Claus, Susanne; Hansbo, Peter; Larson, Mats G.; Massing, André (December 2014), "CutFEM: Discretizing geometry and partial differential equations" (PDF), International Journal for Numerical Methods in Engineering, 104 (7): 472–501, Bibcode:2015IJNME.104..472B, doi:10.1002/nme.4823.
4. "A refugee's story: 'Math was a language I understood'", Don't call me a prodigy: the rising stars of European mathematics, Deutsche Welle, July 18, 2016, retrieved 2016-07-26.
5. Numerical Methods for Fluid Interface Problems, dissertations.se, retrieved 2016-07-26.
External links
• Home page
• Sara Zahedi publications indexed by Google Scholar
Mathematics in Iran
Mathematicians
Before
20th Century
• Abu al-Wafa' Buzjani
• Jamshīd al-Kāshī (al-Kashi's theorem)
• Omar Khayyam (Khayyam-Pascal's triangle, Khayyam-Saccheri quadrilateral, Khayyam's Solution of Cubic Equations)
• Al-Mahani
• Muhammad Baqir Yazdi
• Nizam al-Din al-Nisapuri
• Al-Nayrizi
• Kushyar Gilani
• Ayn al-Quzat Hamadani
• Al-Isfahani
• Al-Isfizari
• Al-Khwarizmi (Al-jabr)
• Najm al-Din al-Qazwini al-Katibi
• Nasir al-Din al-Tusi
• Al-Biruni
Modern
• Maryam Mirzakhani
• Caucher Birkar
• Sara Zahedi
• Farideh Firoozbakht (Firoozbakht's conjecture)
• S. L. Hakimi (Havel–Hakimi algorithm)
• Siamak Yassemi
• Freydoon Shahidi (Langlands–Shahidi method)
• Hamid Naderi Yeganeh
• Esmail Babolian
• Ramin Takloo-Bighash
• Lotfi A. Zadeh (Fuzzy mathematics, Fuzzy set, Fuzzy logic)
• Ebadollah S. Mahmoodian
• Reza Sarhangi (The Bridges Organization)
• Siavash Shahshahani
• Gholamhossein Mosaheb
• Amin Shokrollahi
• Reza Sadeghi
• Mohammad Mehdi Zahedi
• Mohsen Hashtroodi
• Hossein Zakeri
• Amir Ali Ahmadi
Prize Recipients
Fields Medal
• Maryam Mirzakhani (2014)
• Caucher Birkar (2018)
EMS Prize
• Sara Zahedi (2016)
Satter Prize
• Maryam Mirzakhani (2013)
Organizations
• Iranian Mathematical Society
Institutions
• Institute for Research in Fundamental Sciences
Authority control
International
• VIAF
National
• Germany
Academics
• DBLP
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• zbMATH
| Wikipedia |
Zahorski theorem
In mathematics, Zahorski's theorem is a theorem of real analysis. It states that a necessary and sufficient condition for a subset of the real line to be the set of points of non-differentiability of a continuous real-valued function, is that it be the union of a Gδ set and a ${G_{\delta }}_{\sigma }$ set of zero measure.
This result was proved by Zygmunt Zahorski in 1939 and first published in 1941.
References
• Zahorski, Zygmunt (1941), "Punktmengen, in welchen eine stetige Funktion nicht differenzierbar ist", Rec. Math. (Mat. Sbornik), Nouvelle Série (in Russian and German), 9 (51): 487–510, MR 0004869.
• Zahorski, Zygmunt (1946), "Sur l'ensemble des points de non-dérivabilité d'une fonction continue" (French translation of 1941 Russian paper), Bulletin de la Société Mathématique de France, 74: 147–178, doi:10.24033/bsmf.1381, MR 0022592.
| Wikipedia |
Porous set
In mathematics, a porous set is a concept in the study of metric spaces. Like the concepts of meagre and measure zero sets, a porous set can be considered "sparse" or "lacking bulk"; however, porous sets are not equivalent to either meagre sets or measure zero sets, as shown below.
Definition
Let (X, d) be a complete metric space and let E be a subset of X. Let B(x, r) denote the closed ball in (X, d) with centre x ∈ X and radius r > 0. E is said to be porous if there exist constants 0 < α < 1 and r0 > 0 such that, for every 0 < r ≤ r0 and every x ∈ X, there is some point y ∈ X with
$B(y,\alpha r)\subseteq B(x,r)\setminus E.$
A subset of X is called σ-porous if it is a countable union of porous subsets of X.
Properties
• Any porous set is nowhere dense. Hence, all σ-porous sets are meagre sets (or of the first category).
• If X is a finite-dimensional Euclidean space Rn, then porous subsets are sets of Lebesgue measure zero.
• However, there does exist a non-σ-porous subset P of Rn which is of the first category and of Lebesgue measure zero. This is known as Zajíček's theorem.
• The relationship between porosity and being nowhere dense can be illustrated as follows: if E is nowhere dense, then for x ∈ X and r > 0, there is a point y ∈ X and s > 0 such that
$B(y,s)\subseteq B(x,r)\setminus E.$
However, if E is also porous, then it is possible to take s = αr (at least for small enough r), where 0 < α < 1 is a constant that depends only on E.
References
• Reich, Simeon; Zaslavski, Alexander J. (2002). "Two convergence results for continuous descent methods". Electronic Journal of Differential Equations. 2002 (24): 1–11. ISSN 1072-6691.
• Zajíček, L. (1987–1988). "Porosity and σ-porosity". Real Anal. Exchange. 13 (2): 314–350. doi:10.2307/44151885. ISSN 0147-1937. JSTOR 44151885. MR943561
| Wikipedia |
Zakai equation
In filtering theory the Zakai equation is a linear stochastic partial differential equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear stochastic partial differential equation for the normalized density of the hidden state. In principle either approach allows one to estimate a quantity function (the state of a dynamical system) from noisy measurements, even when the system is non-linear (thus generalizing the earlier results of Wiener and Kalman for linear systems and solving a central problem in estimation theory). The application of this approach to a specific engineering situation may be problematic however, as these equations are quite complex.[1][2] The Zakai equation is a bilinear stochastic partial differential equation. It was named after Moshe Zakai.[3]
Overview
Assume the state of the system evolves according to
$dx=f(x,t)dt+dw$
and a noisy measurement of the system state is available:
$dz=h(x,t)dt+dv$
where $w,v$ are independent Wiener processes. Then the unnormalized conditional probability density $p(x,t)$ of the state at time t is given by the Zakai equation:
$dp=L(p)dt+ph^{T}dz$
where the operator $L(p)=-\sum {\frac {\partial (f_{i}p)}{\partial x_{i}}}+{\frac {1}{2}}\sum {\frac {\partial ^{2}p}{\partial x_{i}\partial x_{j}}}$
As previously mentioned, $p$ is an unnormalized density and thus does not necessarily integrate to 1. After solving for $p$, integration and normalization can be done if desired (an extra step not required in the Kushner approach).
Note that if the last term on the right hand side is omitted (by choosing h identically zero), the result is a nonstochastic PDE: the familiar Fokker–Planck equation, which describes the evolution of the state when no measurement information is available.
See also
• Kushner equation
• Kalman filter
• Wiener filter
References
1. Sritharan, S. S. (1994). "Nonlinear filtering of stochastic Navier–Stokes equations". In Funaki, T.; Woyczynski, W. A. (eds.). Nonlinear Stochastic PDEs: Burgers Turbulence and Hydrodynamic Limit (PDF). Springer-Verlag. pp. 247–260. ISBN 0-387-94624-1.
2. Hobbs, S. L.; Sritharan, S. S. (1996). "Nonlinear filtering theory for stochastic reaction–diffusion equations". In Gretsky, N.; Goldstein, J.; Uhl, J. J. (eds.). Probability and Modern Analysis (PDF). Marcel Dekker. pp. 219–234.
3. Zakai, M. (1969). "On the optimal filtering of diffusion processes". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 11 (3): 230–243. doi:10.1007/BF00536382. MR 0242552. S2CID 119763576. Zbl 0164.19201.
Further reading
• Grigelionis, B.; Mikulevičius, R. (1983). "Stochastic evolution equations and densities of the conditional distributions". Theory and Application of Random Fields. Berlin: Springer. pp. 49–88. doi:10.1007/BFb0044682.
• Schuss, Zeev (2012). "Nonlinear Filtering and Smoothing of Diffusions". Nonlinear Filtering and Optimal Phase Tracking. Boston: Springer. pp. 85–106. doi:10.1007/978-1-4614-0487-3_3. ISBN 978-1-4614-0486-6.
| Wikipedia |
Knizhnik–Zamolodchikov equations
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the N-point functions of affine primary fields and can be derived using either the formalism of Lie algebras or that of vertex algebras.
The structure of the genus-zero part of the conformal field theory is encoded in the monodromy properties of these equations. In particular, the braiding and fusion of the primary fields (or their associated representations) can be deduced from the properties of the four-point functions, for which the equations reduce to a single matrix-valued first-order complex ordinary differential equation of Fuchsian type.
Originally the Russian physicists Vadim Knizhnik and Alexander Zamolodchikov derived the equations for the SU(2) Wess–Zumino–Witten model using the classical formulas of Gauss for the connection coefficients of the hypergeometric differential equation.
Definition
Let ${\hat {\mathfrak {g}}}_{k}$ denote the affine Lie algebra with level k and dual Coxeter number h. Let v be a vector from a zero mode representation of ${\hat {\mathfrak {g}}}_{k}$ and $\Phi (v,z)$ the primary field associated with it. Let $t^{a}$ be a basis of the underlying Lie algebra ${\mathfrak {g}}$, $t_{i}^{a}$ their representation on the primary field $\Phi (v_{i},z)$ and η the Killing form. Then for $i,j=1,2,\ldots ,N$ the Knizhnik–Zamolodchikov equations read
$\left((k+h)\partial _{z_{i}}+\sum _{j\neq i}{\frac {\sum _{a,b}\eta _{ab}t_{i}^{a}\otimes t_{j}^{b}}{z_{i}-z_{j}}}\right)\left\langle \Phi (v_{N},z_{N})\dots \Phi (v_{1},z_{1})\right\rangle =0.$
Informal derivation
The Knizhnik–Zamolodchikov equations result from the Sugawara construction of the Virasoro algebra from the affine Lie algebra. More specifically, they result from applying the identity
$L_{-1}={\frac {1}{2(k+h)}}\sum _{k\in \mathbf {Z} }\sum _{a,b}\eta _{ab}J_{-k}^{a}J_{k-1}^{b}$
to the affine primary field $\Phi (v_{i},z_{i})$ in a correlation function of affine primary fields. In this context, only the terms $k=0,1$ are non-vanishing. The action of $J_{-1}^{a}$ can then be rewritten using global Ward identities,
$\left(\left(J_{-1}^{a}\right)_{i}+\sum _{j\neq i}{\frac {t_{j}^{a}}{z_{i}-z_{j}}}\right)\left\langle \Phi (v_{N},z_{N})\dots \Phi (v_{1},z_{1})\right\rangle =0,$
and $L_{-1}$ can be identified with the infinitesimal translation operator ${\frac {\partial }{\partial z}}$.
Mathematical formulation
Since the treatment in Tsuchiya & Kanie (1988), the Knizhnik–Zamolodchikov equation has been formulated mathematically in the language of vertex algebras due to Borcherds (1986) and Frenkel, Lepowsky & Meurman (1988). This approach was popularized amongst theoretical physicists by Goddard (1988) harvtxt error: no target: CITEREFGoddard1988 (help) and amongst mathematicians by Kac (1996) harvtxt error: no target: CITEREFKac1996 (help).
The vacuum representation H0 of an affine Kac–Moody algebra at a fixed level can be encoded in a vertex algebra. The derivation d acts as the energy operator L0 on H0, which can be written as a direct sum of the non-negative integer eigenspaces of L0, the zero energy space being generated by the vacuum vector Ω. The eigenvalue of an eigenvector of L0 is called its energy. For every state a in L there is a vertex operator V(a,z) which creates a from the vacuum vector Ω, in the sense that
$V(a,0)\Omega =a.$
The vertex operators of energy 1 correspond to the generators of the affine algebra
$X(z)=\sum X(n)z^{-n-1}$
where X ranges over the elements of the underlying finite-dimensional simple complex Lie algebra ${\mathfrak {g}}$.
There is an energy 2 eigenvector L−2Ω which give the generators Ln of the Virasoro algebra associated to the Kac–Moody algebra by the Segal–Sugawara construction
$T(z)=\sum L_{n}z^{-n-2}.$
If a has energy α, then the corresponding vertex operator has the form
$V(a,z)=\sum V(a,n)z^{-n-\alpha }.$
The vertex operators satisfy
${\begin{aligned}{\frac {d}{dz}}V(a,z)&=\left[L_{-1},V(a,z)\right]=V\left(L_{-1}a,z\right)\\\left[L_{0},V(a,z)\right]&=\left(z^{-1}{\frac {d}{dz}}+\alpha \right)V(a,z)\end{aligned}}$
as well as the locality and associativity relations
$V(a,z)V(b,w)=V(b,w)V(a,z)=V(V(a,z-w)b,w).$
These last two relations are understood as analytic continuations: the inner products with finite energy vectors of the three expressions define the same polynomials in z±1, w±1 and (z − w)−1 in the domains |z| < |w|, |z| > |w| and |z – w| < |w|. All the structural relations of the Kac–Moody and Virasoro algebra can be recovered from these relations, including the Segal–Sugawara construction.
Every other integral representation Hi at the same level becomes a module for the vertex algebra, in the sense that for each a there is a vertex operator Vi(a, z) on Hi such that
$V_{i}(a,z)V_{i}(b,w)=V_{i}(b,w)V_{i}(a,z)=V_{i}(V(a,z-w)b,w).$
The most general vertex operators at a given level are intertwining operators Φ(v, z) between representations Hi and Hj where v lies in Hk. These operators can also be written as
$\Phi (v,z)=\sum \Phi (v,n)z^{-n-\delta }$
but δ can now be rational numbers. Again these intertwining operators are characterized by properties
$V_{j}(a,z)\Phi (v,w)=\Phi (v,w)V_{i}(a,w)=\Phi \left(V_{k}(a,z-w)v,w\right)$
and relations with L0 and L−1 similar to those above.
When v is in the lowest energy subspace for L0 on Hk, an irreducible representation of ${\mathfrak {g}}$, the operator Φ(v, w) is called a primary field of charge k.
Given a chain of n primary fields starting and ending at H0, their correlation or n-point function is defined by
$\left\langle \Phi (v_{1},z_{1})\Phi (v_{2},z_{2})\cdots \Phi (v_{n},z_{n})\right\rangle =\left(\Phi \left(v_{1},z_{1}\right)\Phi \left(v_{2},z_{2}\right)\cdots \Phi \left(v_{n},z_{n}\right)\Omega ,\Omega \right).$
In the physics literature the vi are often suppressed and the primary field written Φi(zi), with the understanding that it is labelled by the corresponding irreducible representation of ${\mathfrak {g}}$.
Vertex algebra derivation
If (Xs) is an orthonormal basis of ${\mathfrak {g}}$ for the Killing form, the Knizhnik–Zamolodchikov equations may be deduced by integrating the correlation function
$\sum _{s}\left\langle X_{s}(w)X_{s}(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\rangle (w-z)^{-1}$
first in the w variable around a small circle centred at z; by Cauchy's theorem the result can be expressed as sum of integrals around n small circles centred at the zj's:
${1 \over 2}(k+h)\left\langle T(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\rangle =-\sum _{j,s}\left\langle X_{s}(z)\Phi (v_{1},z_{1})\cdots \Phi (X_{s}v_{j},z_{j})\cdots \Phi (v_{n},z_{n})\right\rangle (z-z_{j})^{-1}.$
Integrating both sides in the z variable about a small circle centred on zi yields the ith Knizhnik–Zamolodchikov equation.
Lie algebra derivation
It is also possible to deduce the Knizhnik–Zamodchikov equations without explicit use of vertex algebras. The termΦ(vi, zi) may be replaced in the correlation function by its commutator with Lr where r = 0, ±1. The result can be expressed in terms of the derivative with respect to zi. On the other hand, Lr is also given by the Segal–Sugawara formula:
${\begin{aligned}L_{0}&=(k+h)^{-1}\sum _{s}\left[{\frac {1}{2}}X_{s}(0)^{2}+\sum _{m>0}X_{s}(-m)X_{s}(m)\right]\\L_{\pm 1}&=(k+h)^{-1}\sum _{s}\sum _{m\geq 0}X_{s}(-m\pm 1)X_{s}(m)\end{aligned}}$
After substituting these formulas for Lr, the resulting expressions can be simplified using the commutator formulas
$[X(m),\Phi (a,n)]=\Phi (Xa,m+n).$
Original derivation
The original proof of Knizhnik & Zamolodchikov (1984), reproduced in Tsuchiya & Kanie (1988), uses a combination of both of the above methods. First note that for X in ${\mathfrak {g}}$
$\left\langle X(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\rangle =\sum _{j}\left\langle \Phi (v_{1},z_{1})\cdots \Phi (Xv_{j},z_{j})\cdots \Phi (v_{n},z_{n})\right\rangle (z-z_{j})^{-1}.$
Hence
$\sum _{s}\langle X_{s}(z)\Phi (z_{1},v_{1})\cdots \Phi (X_{s}v_{i},z_{i})\cdots \Phi (v_{n},z_{n})\rangle =\sum _{j}\sum _{s}\langle \cdots \Phi (X_{s}v_{j},z_{j})\cdots \Phi (X_{s}v_{i},z_{i})\cdots \rangle (z-z_{j})^{-1}.$
On the other hand,
$\sum _{s}X_{s}(z)\Phi \left(X_{s}v_{i},z_{i}\right)=(z-z_{i})^{-1}\Phi \left(\sum _{s}X_{s}^{2}v_{i},z_{i}\right)+(k+g){\partial \over \partial z_{i}}\Phi (v_{i},z_{i})+O(z-z_{i})$
so that
$(k+g){\frac {\partial }{\partial z_{i}}}\Phi (v_{i},z_{i})=\lim _{z\to z_{i}}\left[\sum _{s}X_{s}(z)\Phi \left(X_{s}v_{i},z_{i}\right)-(z-z_{i})^{-1}\Phi \left(\sum _{s}X_{s}^{2}v_{i},z_{i}\right)\right].$
The result follows by using this limit in the previous equality.
Monodromy representation of KZ equation
In conformal field theory along the above definition the n-point correlation function of the primary field satisfies KZ equation. In particular, for ${\mathfrak {sl}}_{2}$ and non negative integers k there are $k+1$ primary fields $\Phi _{j}(z_{j})$ 's corresponding to spin j representation ($j=0,1/2,1,3/2,\ldots ,k/2$). The correlation function $\Psi (z_{1},\dots ,z_{n})$ of the primary fields $\Phi _{j}(z_{j})$ 's for the representation $(\rho ,V_{i})$ takes values in the tensor product $V_{1}\otimes \cdots \otimes V_{n}$ and its KZ equation is
$(k+2){\frac {\partial }{\partial z_{i}}}\Psi =\sum _{i,j\neq i}{\frac {\Omega _{ij}}{z_{i}-z_{j}}}\Psi $,
where $\Omega _{ij}=\sum _{a}\rho _{i}(J^{a})\otimes \rho _{j}(J_{a})$ as the above informal derivation.
This n-point correlation function can be analytically continued as multi-valued holomorphic function to the domain $X_{n}\subset \mathbb {C} ^{n}$ with $z_{i}\neq z_{j}$ for $i\neq j$. Due to this analytic continuation, the holonomy of the KZ equation can be described by the braid group $B_{n}$ introduced by Emil Artin.[1] In general, A complex semi-simple Lie algebra ${\mathfrak {g}}$ and its representations $(\rho ,V_{i})$ give the linear representation of braid group
$\theta \colon B_{n}\rightarrow V_{1}\otimes \cdots \otimes V_{n}$
as the holonomy of KZ equation. Oppositely, a KZ equation gives the linear representation of braid groups as its holonomy.
The action on $V_{1}\otimes \dots \otimes V_{n}$ by the analytic continuation of KZ equation is called monodromy representation of KZ equation. In particular, if all $V_{i}$ 's have spin 1/2 representation then the linear representation obtained from KZ equation agrees with the representation constructed from operator algebra theory by Vaughan Jones. It is known that the monodromy representation of KZ equation with a general semi-simple Lie algebra agrees with the linear representation of braid group given by R-matrix of the corresponding quantum group.
Applications
• Representation theory of affine Lie algebra and quantum groups
• Braid groups
• Topology of hyperplane complements
• Knot theory and 3-folds
See also
• Quantum KZ equations
References
1. Kohno 2002
• Baik, Jinho; Deift, Percy; Johansson, Kurt (June 1999), "On the distribution of the length of the longest increasing subsequence of random permutations" (PDF), J. Amer. Math. Soc., 12 (4): 1119–78, doi:10.1090/S0894-0347-99-00307-0, S2CID 11355968
• Knizhnik, V.G.; Zamolodchikov, A.B. (1984), "Current Algebra and Wess–Zumino Model in Two-Dimensions", Nucl. Phys. B, 247 (1): 83–103, Bibcode:1984NuPhB.247...83K, doi:10.1016/0550-3213(84)90374-2
• Tsuchiya, A.; Kanie, Y. (1988), Vertex operators in conformal field theory on P(1) and monodromy representations of braid group, Adv. Stud. Pure Math., vol. 16, pp. 297–372 (Erratum in volume 19, pp. 675–682.)
• Borcherds, Richard (1986), "Vertex algebras, Kac–Moody algebras, and the Monster", Proc. Natl. Acad. Sci. USA, 83 (10): 3068–3071, Bibcode:1986PNAS...83.3068B, doi:10.1073/pnas.83.10.3068, PMC 323452, PMID 16593694
• Frenkel, Igor; Lepowsky, James; Meurman, Arne (1988), Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, ISBN 0-12-267065-5
• Goddard, Peter (1989), "Meromorphic conformal field theory", in Kac, Victor G. (ed.), Infinite Dimensional Lie Algebras And Groups, Advanced Series In Mathematical Physics, vol. 7, World Scientific, pp. 556–587, ISBN 978-981-4663-17-5
• Kac, Victor (1998), Vertex algebras for beginners, University Lecture Series, vol. 10, American Mathematical Society, ISBN 0-8218-0643-2
• Etingof, Pavel I.; Frenkel, Igor; Kirillov, Alexander A. (1998), Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations, Mathematical Surveys and Monographs, vol. 58, American Mathematical Society, ISBN 0821804960
• Frenkel, Edward; Ben-Zvi, David (2001), Vertex algebras and Algebraic Curves, Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, ISBN 0-8218-2894-0
• Kohno, Toshitake (2002), Conformal Field Theory and Topology, Translation of Mathematical Monographs, vol. 210, American Mathematical Society, ISBN 978-0821821305
| Wikipedia |
Zappa–Szép product
In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It is named after Guido Zappa (1940) and Jenő Szép (1950) although it was independently studied by others including B.H. Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904).[1]
Internal Zappa–Szép products
Let G be a group with identity element e, and let H and K be subgroups of G. The following statements are equivalent:
• G = HK and H ∩ K = {e}
• For each g in G, there exists a unique h in H and a unique k in K such that g = hk.
If either (and hence both) of these statements hold, then G is said to be an internal Zappa–Szép product of H and K.
Examples
Let G = GL(n,C), the general linear group of invertible n × n matrices over the complex numbers. For each matrix A in G, the QR decomposition asserts that there exists a unique unitary matrix Q and a unique upper triangular matrix R with positive real entries on the main diagonal such that A = QR. Thus G is a Zappa–Szép product of the unitary group U(n) and the group (say) K of upper triangular matrices with positive diagonal entries.
One of the most important examples of this is Philip Hall's 1937 theorem on the existence of Sylow systems for soluble groups. This shows that every soluble group is a Zappa–Szép product of a Hall p'-subgroup and a Sylow p-subgroup, and in fact that the group is a (multiple factor) Zappa–Szép product of a certain set of representatives of its Sylow subgroups.
In 1935, George Miller showed that any non-regular transitive permutation group with a regular subgroup is a Zappa–Szép product of the regular subgroup and a point stabilizer. He gives PSL(2,11) and the alternating group of degree 5 as examples, and of course every alternating group of prime degree is an example. This same paper gives a number of examples of groups which cannot be realized as Zappa–Szép products of proper subgroups, such as the quaternion group and the alternating group of degree 6.
External Zappa–Szép products
As with the direct and semidirect products, there is an external version of the Zappa–Szép product for groups which are not known a priori to be subgroups of a given group. To motivate this, let G = HK be an internal Zappa–Szép product of subgroups H and K of the group G. For each k in K and each h in H, there exist α(k, h) in H and β(k, h) in K such that kh = α(k, h) β(k, h). This defines mappings α : K × H → H and β : K × H → K which turn out to have the following properties:
• α(e, h) = h and β(k, e) = k for all h in H and k in K.
• α(k1k2, h) = α(k1, α(k2, h))
• β(k, h1h2) = β(β(k, h1), h2)
• α(k, h1h2) = α(k, h1) α(β(k, h1), h2)
• β(k1k2, h) = β(k1, α(k2, h)) β(k2, h)
for all h1, h2 in H, k1, k2 in K. From these, it follows that
• For each k in K, the mapping h ↦ α(k, h) is a bijection of H.
• For each h in H, the mapping k ↦ β(k, h) is a bijection of K.
(Indeed, suppose α(k, h1) = α(k, h2). Then h1 = α(k−1k, h1) = α(k−1, α(k, h1)) = α(k−1, α(k, h2)) = h2. This establishes injectivity, and for surjectivity, use h = α(k, α(k−1, h)).)
More concisely, the first three properties above assert the mapping α : K × H → H is a left action of K on (the underlying set of) H and that β : K × H → K is a right action of H on (the underlying set of) K. If we denote the left action by h → kh and the right action by k → kh, then the last two properties amount to k(h1h2) = kh1 kh1h2 and (k1k2)h = k1k2h k2h.
Turning this around, suppose H and K are groups (and let e denote each group's identity element) and suppose there exist mappings α : K × H → H and β : K × H → K satisfying the properties above. On the cartesian product H × K, define a multiplication and an inversion mapping by, respectively,
• (h1, k1) (h2, k2) = (h1 α(k1, h2), β(k1, h2) k2)
• (h, k)−1 = (α(k−1, h−1), β(k−1, h−1))
Then H × K is a group called the external Zappa–Szép product of the groups H and K. The subsets H × {e} and {e} × K are subgroups isomorphic to H and K, respectively, and H × K is, in fact, an internal Zappa–Szép product of H × {e} and {e} × K.
Relation to semidirect and direct products
Let G = HK be an internal Zappa–Szép product of subgroups H and K. If H is normal in G, then the mappings α and β are given by, respectively, α(k,h) = k h k− 1 and β(k, h) = k. This is easy to see because $(h_{1}k_{1})(h_{2}k_{2})=(h_{1}k_{1}h_{2}k_{1}^{-1})(k_{1}k_{2})$ and $h_{1}k_{1}h_{2}k_{1}^{-1}\in H$ since by normality of $H$, $k_{1}h_{2}k_{1}^{-1}\in H$. In this case, G is an internal semidirect product of H and K.
If, in addition, K is normal in G, then α(k,h) = h. In this case, G is an internal direct product of H and K.
References
1. Martin W. Liebeck; Cheryl E. Praeger; Jan Saxl (2010). Regular Subgroups of Primitive Permutation Groups. American Mathematical Soc. pp. 1–2. ISBN 978-0-8218-4654-4.
• Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, MR 0224703, OCLC 527050, Kap. VI, §4.
• Michor, P. W. (1989), "Knit products of graded Lie algebras and groups", Proceedings of the Winter School on Geometry and Physics, Srni, Suppl. Rendiconti Circolo Matematico di Palermo, Ser. II, 22: 171–175, arXiv:math/9204220, Bibcode:1992math......4220M.
• Miller, G. A. (1935), "Groups which are the products of two permutable proper subgroups", Proceedings of the National Academy of Sciences, 21 (7): 469–472, Bibcode:1935PNAS...21..469M, doi:10.1073/pnas.21.7.469, PMC 1076628, PMID 16588002
• Szép, J. (1950), "On the structure of groups which can be represented as the product of two subgroups", Acta Sci. Math. Szeged, 12: 57–61.
• Takeuchi, M. (1981), "Matched pairs of groups and bismash products of Hopf algebras", Comm. Algebra, 9 (8): 841–882, doi:10.1080/00927878108822621.
• Zappa, G. (1940), "Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili traloro", Atti Secondo Congresso Un. Mat. Ital., Bologna{{citation}}: CS1 maint: location missing publisher (link); Edizioni Cremonense, Rome, (1942) 119–125.
• Agore, A.L.; Chirvasitu, A.; Ion, B.; Militaru, G. (2007), Factorization problems for finite groups, arXiv:math/0703471, Bibcode:2007math......3471A, doi:10.1007/s10468-009-9145-6, S2CID 18024087.
• Brin, M. G. (2005). "On the Zappa-Szép Product". Communications in Algebra. 33 (2): 393–424. arXiv:math/0406044. doi:10.1081/AGB-200047404. S2CID 15169734.
| Wikipedia |
Turán's brick factory problem
Unsolved problem in mathematics:
Can any complete bipartite graph be drawn with fewer crossings than the number given by Zarankiewicz?
(more unsolved problems in mathematics)
In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a brick factory during World War II.[1]
A drawing method found by Kazimierz Zarankiewicz has been conjectured to give the correct answer for every complete bipartite graph, and the statement that this is true has come to be known as the Zarankiewicz crossing number conjecture. The conjecture remains open, with only some special cases solved.[2]
Origin and formulation
During World War II, Hungarian mathematician Pál Turán was forced to work in a brick factory, pushing wagon loads of bricks from kilns to storage sites. The factory had tracks from each kiln to each storage site, and the wagons were harder to push at the points where tracks crossed each other. Turán was inspired by this situation to ask how the factory might be redesigned to minimize the number of crossings between these tracks.[1]
Mathematically, this problem can be formalized as asking for a graph drawing of a complete bipartite graph, whose vertices represent kilns and storage sites, and whose edges represent the tracks from each kiln to each storage site. The graph should be drawn in the plane with each vertex as a point, each edge as a curve connecting its two endpoints, and no vertex placed on an edge that it is not incident to. A crossing is counted whenever two edges that are disjoint in the graph have a nonempty intersection in the plane. The question is then, what is the minimum number of crossings in such a drawing?[2][3]
Turán's formulation of this problem is often recognized as one of the first studies of the crossing numbers of graphs.[4] (Another independent formulation of the same concept occurred in sociology, in methods for drawing sociograms,[5] and a much older puzzle, the three utilities problem, can be seen as a special case of the brick factory problem with three kilns and three storage facilities.[6]) Crossing numbers have since gained greater importance, as a central object of study in graph drawing[7] and as an important tool in VLSI design[8] and discrete geometry.[9]
Upper bound
Both Zarankiewicz and Kazimierz Urbanik saw Turán speak about the brick factory problem in different talks in Poland in 1952,[3] and independently published attempted solutions of the problem, with equivalent formulas for the number of crossings.[10][11] As both of them showed, it is always possible to draw the complete bipartite graph Km,n (a graph with m vertices on one side, n vertices on the other side, and mn edges connecting the two sides) with a number of crossings equal to
$\operatorname {cr} (K_{m,n})\leq {\biggl \lfloor }{\frac {n}{2}}{\biggr \rfloor }{\biggl \lfloor }{\frac {n-1}{2}}{\biggr \rfloor }{\biggl \lfloor }{\frac {m}{2}}{\biggr \rfloor }{\biggl \lfloor }{\frac {m-1}{2}}{\biggr \rfloor }.$
The construction is simple: place m vertices on the x-axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points to the left and right of the y-axis. Similarly, place n vertices on the y-axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points above and below the x-axis. Then, connect every point on the x-axis by a straight line segment to every point on the y-axis.[3]
However, their proofs that this formula is optimal, that is, that there can be no drawings with fewer crossings, were erroneous. The gap was not discovered until eleven years after publication, nearly simultaneously by Gerhard Ringel and Paul Kainen.[12] Nevertheless, it is conjectured that Zarankiewicz's and Urbanik's formula is optimal. This has come to be known as the Zarankiewicz crossing number conjecture. Although some special cases of it are known to be true, the general case remains open.[2]
Lower bounds
Since Km,n and Kn,m are isomorphic, it is enough to consider the case where m ≤ n. In addition, for m ≤ 2 Zarankiewicz's construction gives no crossings, which of course cannot be bested. So the only nontrivial cases are those for which m and n are both ≥ 3.
Zarankiewicz's attempted proof of the conjecture, although invalid for the general case of Km,n, works for the case m = 3. It has since been extended to other small values of m, and the Zarankiewicz conjecture is known to be true for the complete bipartite graphs Km,n with m ≤ 6.[13] The conjecture is also known to be true for K7,7, K7,8, and K7,9.[14] If a counterexample exists, that is, a graph Km,n requiring fewer crossings than the Zarankiewicz bound, then in the smallest counterexample both m and n must be odd.[13]
For each fixed choice of m, the truth of the conjecture for all Km,n can be verified by testing only a finite number of choices of n.[15] More generally, it has been proven that every complete bipartite graph requires a number of crossings that is (for sufficiently large graphs) at least 83% of the number given by the Zarankiewicz bound. Closing the gap between this lower bound and the upper bound remains an open problem.[16]
Rectilinear crossing numbers
If edges are required to be drawn as straight line segments, rather than arbitrary curves, then some graphs need more crossings than they would when drawn with curved edges. However, the upper bound established by Zarankiewicz for the crossing numbers of complete bipartite graphs can be achieved using only straight edges. Therefore, if the Zarankiewicz conjecture is correct, then the complete bipartite graphs have rectilinear crossing numbers equal to their crossing numbers.[17]
References
1. Turán, P. (1977), "A note of welcome", Journal of Graph Theory, 1: 7–9, doi:10.1002/jgt.3190010105.
2. Pach, János; Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, vol. 152, American Mathematical Society, pp. 126–127.
3. Beineke, Lowell; Wilson, Robin (2010), "The early history of the brick factory problem", The Mathematical Intelligencer, 32 (2): 41–48, doi:10.1007/s00283-009-9120-4, MR 2657999, S2CID 122588849.
4. Foulds, L. R. (1992), Graph Theory Applications, Universitext, Springer, p. 71, ISBN 9781461209331.
5. Bronfenbrenner, Urie (1944), "The graphic presentation of sociometric data", Sociometry, 7 (3): 283–289, doi:10.2307/2785096, JSTOR 2785096, The arrangement of subjects on the diagram, while haphazard in part, is determined largely by trial and error with the aim of minimizing the number of intersecting lines.
6. Bóna, Miklós (2011), A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, World Scientific, pp. 275–277, ISBN 9789814335232. Bóna introduces the puzzle (in the form of three houses to be connected to three wells) on p. 275, and writes on p. 277 that it "is equivalent to the problem of drawing K3,3 on a plane surface without crossings".
7. Schaefer, Marcus (2014), "The graph crossing number and its variants: a survey", The Electronic Journal of Combinatorics: #DS21
8. Leighton, T. (1983), Complexity Issues in VLSI, Foundations of Computing Series, Cambridge, MA: MIT Press
9. Székely, L. A. (1997), "Crossing numbers and hard Erdős problems in discrete geometry", Combinatorics, Probability and Computing, 6 (3): 353–358, doi:10.1017/S0963548397002976, MR 1464571, S2CID 36602807
10. Zarankiewicz, K. (1954), "On a problem of P. Turan concerning graphs", Fundamenta Mathematicae, 41: 137–145, doi:10.4064/fm-41-1-137-145, MR 0063641.
11. Urbaník, K. (1955), "Solution du problème posé par P. Turán", Colloq. Math., 3: 200–201. As cited by Székely, László A. (2001) [1994], "Zarankiewicz crossing number conjecture", Encyclopedia of Mathematics, EMS Press
12. Guy, Richard K. (1969), "The decline and fall of Zarankiewicz's theorem", Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968), Academic Press, New York, pp. 63–69, MR 0253931.
13. Kleitman, Daniel J. (1970), "The crossing number of K5,n", Journal of Combinatorial Theory, 9 (4): 315–323, doi:10.1016/s0021-9800(70)80087-4, MR 0280403.
14. Woodall, D. R. (1993), "Cyclic-order graphs and Zarankiewicz's crossing-number conjecture", Journal of Graph Theory, 17 (6): 657–671, doi:10.1002/jgt.3190170602, MR 1244681.
15. Christian, Robin; Richter, R. Bruce; Salazar, Gelasio (2013), "Zarankiewicz's conjecture is finite for each fixed m", Journal of Combinatorial Theory, Series B, 103 (2): 237–247, doi:10.1016/j.jctb.2012.11.001, MR 3018068.
16. de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, R. B.; Salazar, G. (2006), "Improved bounds for the crossing numbers of Km,n and Kn", SIAM Journal on Discrete Mathematics, 20 (1): 189–202, arXiv:math/0404142, doi:10.1137/S0895480104442741, MR 2257255, S2CID 1509054.
17. Kainen, Paul C. (1968), "On a problem of P. Erdős", Journal of Combinatorial Theory, 5 (4): 374–377, doi:10.1016/s0021-9800(68)80013-4, MR 0231744
External links
• Weisstein, Eric W., "Zarankiewicz's Conjecture", MathWorld
| Wikipedia |
Zarhin trick
In mathematics, the Zarhin trick is a method for eliminating the polarization of abelian varieties A by observing that the abelian variety A4 × Â4 is principally polarized. The method was introduced by Zarhin (1974) in his proof of the Tate conjecture over global fields of positive characteristic.
References
• Zarhin, Ju. G. (1974), "A remark on endomorphisms of abelian varieties over function fields of finite characteristic", Mathematics of the USSR-Izvestiya, 8 (3): 477–480, doi:10.1070/IM1974v008n03ABEH002115, ISSN 0373-2436, MR 0354689
| Wikipedia |
Oscar Zariski
Oscar Zariski (April 24, 1899 – July 4, 1986) was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century.
Oscar Zariski
Oscar Zariski (1899–1986)
Born
Russian: О́скар Зари́сский
(1899-04-24)April 24, 1899
Kobrin, Russian Empire
DiedJuly 4, 1986(1986-07-04) (aged 87)
Brookline, Massachusetts, United States
NationalityAmerican
Alma materUniversity of Kyiv
University of Rome
Known forContributions to algebraic geometry
AwardsCole Prize in Algebra (1944)
National Medal of Science (1965)
Wolf Prize (1981)
Steele Prize (1981)
Scientific career
FieldsMathematics
InstitutionsJohns Hopkins University
University of Illinois
Harvard University
Doctoral advisorGuido Castelnuovo
Doctoral studentsS. S. Abhyankar
Michael Artin
Iacopo Barsotti
Irvin Cohen
Daniel Gorenstein
Robin Hartshorne
Heisuke Hironaka
Steven Kleiman
Joseph Lipman
David Mumford
Maxwell Rosenlicht
Pierre Samuel
Abraham Seidenberg
Education
Zariski was born Oscher (also transliterated as Ascher or Osher) Zaritsky to a Jewish family (his parents were Bezalel Zaritsky and Hanna Tennenbaum) and in 1918 studied at the University of Kyiv. He left Kyiv in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi.
Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski.
Johns Hopkins University years
Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz. He had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school. The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed. It is still an important reference.
It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry. He addressed the question of rigour by recourse to commutative algebra. The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points. A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety. The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use also valuation theory to describe the phenomena such as blowing up (balloon-style, rather than explosively).
Harvard University years
After spending a year 1946–1947 at the University of Illinois at Urbana–Champaign, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969. In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests. The two sets of foundations weren't reconciled at that point.
At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman—thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation. Zariski himself worked on equisingularity theory. Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.
Zariski proposed the first example of a Zariski surface in 1958.
Views
Zariski was a Jewish atheist.[1]
Awards and recognition
Zariski was elected to the United States National Academy of Sciences in 1944,[2] the American Academy of Arts and Sciences in 1948,[3] and the American Philosophical Society in 1951.[4] Zariski was awarded the Steele Prize in 1981, and in the same year the Wolf Prize in Mathematics with Lars Ahlfors. He wrote also Commutative Algebra in two volumes, with Pierre Samuel. His papers have been published by MIT Press, in four volumes. In 1997 a conference was held in his honor in Obergurgl, Austria.[5][6]
Publications
• Zariski, Oscar (2004) [1935], Abhyankar, Shreeram S.; Lipman, Joseph; Mumford, David (eds.), Algebraic surfaces, Classics in mathematics (second supplemented ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-58658-6, MR 0469915[7]
• Zariski, Oscar (1958), Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, vol. 4, The Mathematical Society of Japan, Tokyo, MR 0097403
• Zariski, Oscar (1969) [1958], Cohn, James (ed.), An introduction to the theory of algebraic surfaces, Lecture notes in mathematics, vol. 83, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0082246, ISBN 978-3-540-04602-8, MR 0263819
• Zariski, Oscar; Samuel, Pierre (1975) [1958], Commutative algebra I, Graduate Texts in Mathematics, vol. 28, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90089-6, MR 0090581[8]
• Zariski, Oscar; Samuel, Pierre (1975) [1960], Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876[9]
• Zariski, Oscar (2006) [1973], Kmety, François; Merle, Michel; Lichtin, Ben (eds.), The moduli problem for plane branches, University Lecture Series, vol. 39, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2983-7, MR 0414561(original title): Le problème des modules pour les branches planes{{citation}}: CS1 maint: postscript (link)[10]
• Zariski, Oscar (1972), Collected papers. Vol. I: Foundations of algebraic geometry and resolution of singularities, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-08049-1, MR 0505100
• Zariski, Oscar (1973), Collected papers. Vol. II: Holomorphic functions and linear systems, Mathematicians of Our Time, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-01038-2, MR 0505100
• Zariski, Oscar (1978), Artin, Michael; Mazur, Barry (eds.), Collected papers. Volume III. Topology of curves and surfaces, and special topics in the theory of algebraic varieties, Mathematicians of Our Time, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-24021-5, MR 0505104
• Zariski, Oscar (1979), Lipman, Joseph; Teissier, Bernard (eds.), Collected papers. Vol. IV. Equisingularity on algebraic varieties, Mathematicians of Our Time, vol. 16, MIT Press, ISBN 978-0-262-08049-1, MR 0545653
See also
• Zariski ring
• Zariski tangent space
• Zariski surface
• Zariski topology
• Zariski–Riemann surface
• Zariski space (disambiguation)
• Zariski's lemma
• Zariski's main theorem
Notes
1. Carol Parikh (2008). The Unreal Life of Oscar Zariski. Springer. p. 5. ISBN 9780387094298. And yet it did, even though since moving into the boarding house he had become an atheist and most of his friends, including his best friend, were Russians.
2. "Oscar Zariski". www.nasonline.org. Retrieved 2023-02-16.
3. "Oscar Zariski". American Academy of Arts & Sciences. 9 February 2023. Retrieved 2023-02-16.
4. "APS Member History". search.amphilsoc.org. Retrieved 2023-02-16.
5. Herwig Hauser; Joseph Lipman; Frans Oort; Adolfo Quirós (14 February 2000). Resolution of Singularities: A research textbook in tribute to Oscar Zariski Based on the courses given at the Working Week in Obergurgl, Austria, September 7–14, 1997. Springer Science & Business Media. ISBN 978-3-7643-6178-5.
6. Bogomolov, Fedor; Tschinkel, Yuri (2001). "Book Review: Alterations and resolution of singularities". Bulletin of the American Mathematical Society. 39 (1): 95–101. doi:10.1090/S0273-0979-01-00922-3. ISSN 0273-0979.
7. Lefschetz, Solomon (1936). "Review: Algebraic Surfaces, by Oscar Zariski" (PDF). Bulletin of the American Mathematical Society. 42 (1, Part 2): 13–14. doi:10.1090/s0002-9904-1936-06238-5.
8. Herstein, I. N. (1959). "Review: Commutative algebra, Vol. 1, by Oscar Zariski and Pierre Samuel" (PDF). Bull. Amer. Math. Soc. 6 (1): 26–30. doi:10.1090/S0002-9904-1959-10267-6.
9. Auslander, M. (1962). "Review: Commutative algebra, Vol. II, by O. Zariski and P. Samuel" (PDF). Bull. Amer. Math. Soc. 68 (1): 12–13. doi:10.1090/s0002-9904-1962-10674-0.
10. Washburn, Sherwood (1988). "Review: Le problème des modules pour les branches planes, by Oscar Zariski, with an appendix by Bernard Teissier" (PDF). Bull. Amer. Math. Soc. (N.S.). 18 (2): 209–214. doi:10.1090/s0273-0979-1988-15651-0.
References
• Blass, Piotr (2013), "The influence of Oscar Zariski on algebraic geometry" (PDF), Notices of the American Mathematical Society
• Mumford, David (1986), "Oscar Zariski: 1899–1986" (PDF), Notices of the American Mathematical Society, 33 (6): 891–894, ISSN 0002-9920, MR 0860889
• Parikh, Carol (2009) [1991], The Unreal Life of Oscar Zariski, Springer, ISBN 9780387094304, MR 1086628
• Gouvêa, Fernando Q. (1 January 2009). "Review of The Unreal Life of Oscar Zariski by Carol Parikh". MAA Reviews, Mathematical Association of America, maa.org.
External links
Wikiquote has quotations related to Oscar Zariski.
• O'Connor, John J.; Robertson, Edmund F., "Oscar Zariski", MacTutor History of Mathematics Archive, University of St Andrews
• Oscar Zariski at the Mathematics Genealogy Project
• Biography from United States Naval Academy.
Laureates of the Wolf Prize in Mathematics
1970s
• Israel Gelfand / Carl L. Siegel (1978)
• Jean Leray / André Weil (1979)
1980s
• Henri Cartan / Andrey Kolmogorov (1980)
• Lars Ahlfors / Oscar Zariski (1981)
• Hassler Whitney / Mark Krein (1982)
• Shiing-Shen Chern / Paul Erdős (1983/84)
• Kunihiko Kodaira / Hans Lewy (1984/85)
• Samuel Eilenberg / Atle Selberg (1986)
• Kiyosi Itô / Peter Lax (1987)
• Friedrich Hirzebruch / Lars Hörmander (1988)
• Alberto Calderón / John Milnor (1989)
1990s
• Ennio de Giorgi / Ilya Piatetski-Shapiro (1990)
• Lennart Carleson / John G. Thompson (1992)
• Mikhail Gromov / Jacques Tits (1993)
• Jürgen Moser (1994/95)
• Robert Langlands / Andrew Wiles (1995/96)
• Joseph Keller / Yakov G. Sinai (1996/97)
• László Lovász / Elias M. Stein (1999)
2000s
• Raoul Bott / Jean-Pierre Serre (2000)
• Vladimir Arnold / Saharon Shelah (2001)
• Mikio Sato / John Tate (2002/03)
• Grigory Margulis / Sergei Novikov (2005)
• Stephen Smale / Hillel Furstenberg (2006/07)
• Pierre Deligne / Phillip A. Griffiths / David B. Mumford (2008)
2010s
• Dennis Sullivan / Shing-Tung Yau (2010)
• Michael Aschbacher / Luis Caffarelli (2012)
• George Mostow / Michael Artin (2013)
• Peter Sarnak (2014)
• James G. Arthur (2015)
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| Wikipedia |
Zariski's finiteness theorem
In algebra, Zariski's finiteness theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case.[1] Precisely, it states:
Given a normal domain A, finitely generated as an algebra over a field k, if L is a subfield of the field of fractions of A containing k such that $\operatorname {tr.deg} _{k}(L)\leq 2$, then the k-subalgebra $L\cap A$ is finitely generated.
References
1. http://aix1.uottawa.ca/~ddaigle/articles/H14survey.pdf
• Zariski, O. (1954). "Interprétations algébrico-géométriques du quatorzième problème de Hilbert". Bull. Sci. Math. (2). 78: 155–168.
| Wikipedia |
Zariski's lemma
In algebra, Zariski's lemma, proved by Oscar Zariski (1947), states that, if a field K is finitely generated as an associative algebra over another field k, then K is a finite field extension of k (that is, it is also finitely generated as a vector space).
An important application of the lemma is a proof of the weak form of Hilbert's Nullstellensatz:[1] if I is a proper ideal of $k[t_{1},...,t_{n}]$ (k algebraically closed field), then I has a zero; i.e., there is a point x in $k^{n}$ such that $f(x)=0$ for all f in I. (Proof: replacing I by a maximal ideal ${\mathfrak {m}}$, we can assume $I={\mathfrak {m}}$ is maximal. Let $A=k[t_{1},...,t_{n}]$ and $\phi :A\to A/{\mathfrak {m}}$ be the natural surjection. By the lemma $A/{\mathfrak {m}}$ is a finite extension. Since k is algebraically closed that extension must be k. Then for any $f\in {\mathfrak {m}}$,
$f(\phi (t_{1}),\cdots ,\phi (t_{n}))=\phi (f(t_{1},\cdots ,t_{n}))=0$;
that is to say, $x=(\phi (t_{1}),\cdots ,\phi (t_{n}))$ is a zero of ${\mathfrak {m}}$.)
The lemma may also be understood from the following perspective. In general, a ring R is a Jacobson ring if and only if every finitely generated R-algebra that is a field is finite over R.[2] Thus, the lemma follows from the fact that a field is a Jacobson ring.
Proofs
Two direct proofs, one of which is due to Zariski, are given in Atiyah–MacDonald.[3][4] For Zariski's original proof, see the original paper.[5] Another direct proof in the language of Jacobson rings is given below. The lemma is also a consequence of the Noether normalization lemma. Indeed, by the normalization lemma, K is a finite module over the polynomial ring $k[x_{1},\ldots ,x_{d}]$ where $x_{1},\ldots ,x_{d}$ are elements of K that are algebraically independent over k. But since K has Krull dimension zero and since an integral ring extension (e.g., a finite ring extension) preserves Krull dimensions, the polynomial ring must have dimension zero; i.e., $d=0$.
The following characterization of a Jacobson ring contains Zariski's lemma as a special case. Recall that a ring is a Jacobson ring if every prime ideal is an intersection of maximal ideals. (When A is a field, A is a Jacobson ring and the theorem below is precisely Zariski's lemma.)
Theorem — [2] Let A be a ring. Then the following are equivalent.
1. A is a Jacobson ring.
2. Every finitely generated A-algebra B that is a field is finite over A.
Proof: 2. $\Rightarrow $ 1.: Let ${\mathfrak {p}}$ be a prime ideal of A and set $B=A/{\mathfrak {p}}$. We need to show the Jacobson radical of B is zero. For that end, let f be a nonzero element of B. Let ${\mathfrak {m}}$ be a maximal ideal of the localization $B[f^{-1}]$. Then $B[f^{-1}]/{\mathfrak {m}}$ is a field that is a finitely generated A-algebra and so is finite over A by assumption; thus it is finite over $B=A/{\mathfrak {p}}$ and so is finite over the subring $B/{\mathfrak {q}}$ where ${\mathfrak {q}}={\mathfrak {m}}\cap B$. By integrality, ${\mathfrak {q}}$ is a maximal ideal not containing f.
1. $\Rightarrow $ 2.: Since a factor ring of a Jacobson ring is Jacobson, we can assume B contains A as a subring. Then the assertion is a consequence of the next algebraic fact:
(*) Let $B\supset A$ be integral domains such that B is finitely generated as A-algebra. Then there exists a nonzero a in A such that every ring homomorphism $\phi :A\to K$, K an algebraically closed field, with $\phi (a)\neq 0$ extends to ${\widetilde {\phi }}:B\to K$.
Indeed, choose a maximal ideal ${\mathfrak {m}}$ of A not containing a. Writing K for some algebraic closure of $A/{\mathfrak {m}}$, the canonical map $\phi :A\to A/{\mathfrak {m}}\hookrightarrow K$ extends to ${\widetilde {\phi }}:B\to K$. Since B is a field, ${\widetilde {\phi }}$ is injective and so B is algebraic (thus finite algebraic) over $A/{\mathfrak {m}}$. We now prove (*). If B contains an element that is transcendental over A, then it contains a polynomial ring over A to which φ extends (without a requirement on a) and so we can assume B is algebraic over A (by Zorn's lemma, say). Let $x_{1},\dots ,x_{r}$ be the generators of B as A-algebra. Then each $x_{i}$ satisfies the relation
$a_{i0}x_{i}^{n}+a_{i1}x_{i}^{n-1}+\dots +a_{in}=0,\,\,a_{ij}\in A$
where n depends on i and $a_{i0}\neq 0$. Set $a=a_{10}a_{20}\dots a_{r0}$. Then $B[a^{-1}]$ is integral over $A[a^{-1}]$. Now given $\phi :A\to K$, we first extend it to ${\widetilde {\phi }}:A[a^{-1}]\to K$ by setting ${\widetilde {\phi }}(a^{-1})=\phi (a)^{-1}$. Next, let ${\mathfrak {m}}=\operatorname {ker} {\widetilde {\phi }}$. By integrality, ${\mathfrak {m}}={\mathfrak {n}}\cap A[a^{-1}]$ for some maximal ideal ${\mathfrak {n}}$ of $B[a^{-1}]$. Then ${\widetilde {\phi }}:A[a^{-1}]\to A[a^{-1}]/{\mathfrak {m}}\to K$ extends to $B[a^{-1}]\to B[a^{-1}]/{\mathfrak {n}}\to K$. Restrict the last map to B to finish the proof. $\square $
Notes
1. Milne 2017, Theorem 2.12.
2. Atiyah & MacDonald 1969, Ch 5. Exercise 25.
3. Atiyah & MacDonald 1969, Ch 5. Exercise 18.
4. Atiyah & MacDonald 1969, Proposition 7.9.
5. Zariski 1947, pp. 362–368.
Sources
• Atiyah, Michael; MacDonald, Ian G. (1969). Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics. Addison–Wesley. ISBN 0-201-40751-5.
• Milne, James (19 March 2017). "Algebraic Geometry". Retrieved 1 February 2022.
• Zariski, Oscar (April 1947). "A new proof of Hilbert's Nullstellensatz". Bulletin of the American Mathematical Society. 53 (4): 362–368. doi:10.1090/s0002-9904-1947-08801-7. MR 0020075.
| Wikipedia |
Correspondence (algebraic geometry)
In algebraic geometry, a correspondence between algebraic varieties V and W is a subset R of V×W, that is closed in the Zariski topology. In set theory, a subset of a Cartesian product of two sets is called a binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations. There are some important examples, even when V and W are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves.
However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on intersection theory,[1] uses the definition above. In literature, however, a correspondence from a variety X to a variety Y is often taken to be a subset Z of X×Y such that Z is finite and surjective over each component of X. Note the asymmetry in this latter definition; which talks about a correspondence from X to Y rather than a correspondence between X and Y. The typical example of the latter kind of correspondence is the graph of a function f:X→Y. Correspondences also play an important role in the construction of motives (cf. presheaf with transfers).[2]
See also
• Adequate equivalence relation
References
1. Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
2. Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284
| Wikipedia |
Zariski geometry
In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular.
Definition
A Zariski geometry consists of a set X and a topological structure on each of the sets
X, X2, X3, ...
satisfying certain axioms.
(N) Each of the Xn is a Noetherian topological space, of dimension at most n.
Some standard terminology for Noetherian spaces will now be assumed.
(A) In each Xn, the subsets defined by equality in an n-tuple are closed. The mappings
Xm → Xn
defined by projecting out certain coordinates and setting others as constants are all continuous.
(B) For a projection
p: Xm → Xn
and an irreducible closed subset Y of Xm, p(Y) lies between its closure Z and Z \ Z′ where Z′ is a proper closed subset of Z. (This is quantifier elimination, at an abstract level.)
(C) X is irreducible.
(D) There is a uniform bound on the number of elements of a fiber in a projection of any closed set in Xm, other than the cases where the fiber is X.
(E) A closed irreducible subset of Xm, of dimension r, when intersected with a diagonal subset in which s coordinates are set equal, has all components of dimension at least r − s + 1.
The further condition required is called very ample (cf. very ample line bundle). It is assumed there is an irreducible closed subset P of some Xm, and an irreducible closed subset Q of P× X2, with the following properties:
(I) Given pairs (x, y), (x′, y′) in X2, for some t in P, the set of (t, u, v) in Q includes (t, x, y) but not (t, x′, y′)
(J) For t outside a proper closed subset of P, the set of (x, y) in X2, (t, x, y) in Q is an irreducible closed set of dimension 1.
(K) For all pairs (x, y), (x′, y′) in X2, selected from outside a proper closed subset, there is some t in P such that the set of (t, u, v) in Q includes (t, x, y) and (t, x′, y′).
Geometrically this says there are enough curves to separate points (I), and to connect points (K); and that such curves can be taken from a single parametric family.
Then Hrushovski and Zilber prove that under these conditions there is an algebraically closed field K, and a non-singular algebraic curve C, such that its Zariski geometry of powers and their Zariski topology is isomorphic to the given one. In short, the geometry can be algebraized.
References
• Hrushovski, Ehud; Zilber, Boris (1996). "Zariski Geometries" (PDF). Journal of the American Mathematical Society. 9 (1): 1–56. doi:10.1090/S0894-0347-96-00180-4.
| Wikipedia |
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in real or complex analysis; in particular, it is not Hausdorff.[1] This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space.
The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces.
The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety.[1] In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology.
The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz, that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal. Another basic idea of Grothendieck's scheme theory is to consider as points, not only the usual points corresponding to maximal ideals, but also all (irreducible) algebraic varieties, which correspond to prime ideals. Thus the Zariski topology on the set of prime ideals (spectrum) of a commutative ring is the topology such that a set of prime ideals is closed if and only if it is the set of all prime ideals that contain a fixed ideal.
Zariski topology of varieties
In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, which were introduced by Grothendieck around 1960), the Zariski topology is defined on algebraic varieties.[2] The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed, algebraically closed field k (in classical algebraic geometry, k is usually the field of complex numbers).
Affine varieties
First, we define the topology on the affine space $\mathbb {A} ^{n},$ formed by the n-tuples of elements of k. The topology is defined by specifying its closed sets, rather than its open sets, and these are taken simply to be all the algebraic sets in $\mathbb {A} ^{n}.$ That is, the closed sets are those of the form
$V(S)=\{x\in \mathbb {A} ^{n}\mid f(x)=0,\forall f\in S\}$
where S is any set of polynomials in n variables over k. It is a straightforward verification to show that:
• V(S) = V((S)), where (S) is the ideal generated by the elements of S;
• For any two ideals of polynomials I, J, we have
1. $V(I)\cup V(J)\,=\,V(IJ);$
2. $V(I)\cap V(J)\,=\,V(I+J).$
It follows that finite unions and arbitrary intersections of the sets V(S) are also of this form, so that these sets form the closed sets of a topology (equivalently, their complements, denoted D(S) and called principal open sets, form the topology itself). This is the Zariski topology on $\mathbb {A} ^{n}.$
If X is an affine algebraic set (irreducible or not) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some $\mathbb {A} ^{n}.$ Equivalently, it can be checked that:
• The elements of the affine coordinate ring
$A(X)\,=\,k[x_{1},\dots ,x_{n}]/I(X)$
act as functions on X just as the elements of $k[x_{1},\dots ,x_{n}]$ act as functions on $\mathbb {A} ^{n}$; here, I(X) is the ideal of all polynomials vanishing on X.
• For any set of polynomials S, let T be the set of their images in A(X). Then the subset of X
$V'(T)=\{x\in X\mid f(x)=0,\forall f\in T\}$
(these notations are not standard) is equal to the intersection with X of V(S).
This establishes that the above equation, clearly a generalization of the definition of the closed sets in $\mathbb {A} ^{n}$ above, defines the Zariski topology on any affine variety.
Projective varieties
Recall that n-dimensional projective space $\mathbb {P} ^{n}$ is defined to be the set of equivalence classes of non-zero points in $\mathbb {A} ^{n+1}$ by identifying two points that differ by a scalar multiple in k. The elements of the polynomial ring $k[x_{0},\dots ,x_{n}]$ are not functions on $\mathbb {P} ^{n}$ because any point has many representatives that yield different values in a polynomial; however, for homogeneous polynomials the condition of having zero or nonzero value on any given projective point is well-defined since the scalar multiple factors out of the polynomial. Therefore, if S is any set of homogeneous polynomials we may reasonably speak of
$V(S)=\{x\in \mathbb {P} ^{n}\mid f(x)=0,\forall f\in S\}.$
The same facts as above may be established for these sets, except that the word "ideal" must be replaced by the phrase "homogeneous ideal", so that the V(S), for sets S of homogeneous polynomials, define a topology on $\mathbb {P} ^{n}.$ As above the complements of these sets are denoted D(S), or, if confusion is likely to result, D′(S).
The projective Zariski topology is defined for projective algebraic sets just as the affine one is defined for affine algebraic sets, by taking the subspace topology. Similarly, it may be shown that this topology is defined intrinsically by sets of elements of the projective coordinate ring, by the same formula as above.
Properties
An important property of Zariski topologies is that they have a base consisting of simple elements, namely the D(f) for individual polynomials (or for projective varieties, homogeneous polynomials) f. That these form a basis follows from the formula for the intersection of two Zariski-closed sets given above (apply it repeatedly to the principal ideals generated by the generators of (S)). The open sets in this base are called distinguished or basic open sets. The importance of this property results in particular from its use in the definition of an affine scheme.
By Hilbert's basis theorem and some elementary properties of Noetherian rings, every affine or projective coordinate ring is Noetherian. As a consequence, affine or projective spaces with the Zariski topology are Noetherian topological spaces, which implies that any closed subset of these spaces is compact.
However, except for finite algebraic sets, no algebraic set is ever a Hausdorff space. In the old topological literature "compact" was taken to include the Hausdorff property, and this convention is still honored in algebraic geometry; therefore compactness in the modern sense is called "quasicompactness" in algebraic geometry. However, since every point (a1, ..., an) is the zero set of the polynomials x1 - a1, ..., xn - an, points are closed and so every variety satisfies the T1 axiom.
Every regular map of varieties is continuous in the Zariski topology. In fact, the Zariski topology is the weakest topology (with the fewest open sets) in which this is true and in which points are closed. This is easily verified by noting that the Zariski-closed sets are simply the intersections of the inverse images of 0 by the polynomial functions, considered as regular maps into $\mathbb {A} ^{1}.$
Spectrum of a ring
In modern algebraic geometry, an algebraic variety is often represented by its associated scheme, which is a topological space (equipped with additional structures) that is locally homeomorphic to the spectrum of a ring.[3] The spectrum of a commutative ring A, denoted Spec A, is the set of the prime ideals of A, equipped with the Zariski topology, for which the closed sets are the sets
$V(I)=\{P\in \operatorname {Spec} A\mid P\supset I\}$
where I is an ideal.
To see the connection with the classical picture, note that for any set S of polynomials (over an algebraically closed field), it follows from Hilbert's Nullstellensatz that the points of V(S) (in the old sense) are exactly the tuples (a1, ..., an) such that the ideal generated by the polynomials x1 − a1, ..., xn − an contains S; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form. Thus, V(S) is "the same as" the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.
Another way, perhaps more similar to the original, to interpret the modern definition is to realize that the elements of A can actually be thought of as functions on the prime ideals of A; namely, as functions on Spec A. Simply, any prime ideal P has a corresponding residue field, which is the field of fractions of the quotient A/P, and any element of A has a reflection in this residue field. Furthermore, the elements that are actually in P are precisely those whose reflection vanishes at P. So if we think of the map, associated to any element a of A:
$e_{a}\colon {\bigl (}P\in \operatorname {Spec} A{\bigr )}\mapsto \left({\frac {a\;{\bmod {P}}}{1}}\in \operatorname {Frac} (A/P)\right)$
("evaluation of a"), which assigns to each point its reflection in the residue field there, as a function on Spec A (whose values, admittedly, lie in different fields at different points), then we have
$e_{a}(P)=0\Leftrightarrow P\in V(a)$
More generally, V(I) for any ideal I is the common set on which all the "functions" in I vanish, which is formally similar to the classical definition. In fact, they agree in the sense that when A is the ring of polynomials over some algebraically closed field k, the maximal ideals of A are (as discussed in the previous paragraph) identified with n-tuples of elements of k, their residue fields are just k, and the "evaluation" maps are actually evaluation of polynomials at the corresponding n-tuples. Since as shown above, the classical definition is essentially the modern definition with only maximal ideals considered, this shows that the interpretation of the modern definition as "zero sets of functions" agrees with the classical definition where they both make sense.
Just as Spec replaces affine varieties, the Proj construction replaces projective varieties in modern algebraic geometry. Just as in the classical case, to move from the affine to the projective definition we need only replace "ideal" by "homogeneous ideal", though there is a complication involving the "irrelevant maximal ideal," which is discussed in the cited article.
Examples
• Spec k, the spectrum of a field k is the topological space with one element.
• Spec ℤ, the spectrum of the integers has a closed point for every prime number p corresponding to the maximal ideal (p) ⊂ ℤ, and one non-closed generic point (i.e., whose closure is the whole space) corresponding to the zero ideal (0). So the closed subsets of Spec ℤ are precisely the whole space and the finite unions of closed points.
• Spec k[t], the spectrum of the polynomial ring over a field k: such a polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k[t]. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal, and the set of the closed points is homeomorphic with the affine line k equipped with its Zariski topology. Because of this homeomorphism, some authors call affine line the spectrum of k[t]. If k is not algebraically closed, for example the field of the real numbers, the picture becomes more complicated because of the existence of non-linear irreducible polynomials. In this case, the spectrum consists of one closed point for each monic irreducible polynomial, and a generic point corresponding to the zero ideal. For example, the spectrum of ℝ[t] consists of the closed points (x − a), for a in ℝ, the closed points (x2 + px + q) where p, q are in ℝ and with negative discriminant p2 − 4q < 0, and finally a generic point (0). For any field, the closed subsets of Spec k[t] are finite unions of closed points, and the whole space. (This results from the fact that k[t] is a principal ideal domain, and, in a principal ideal domain, the prime ideals that contain an ideal are the prime factors of the prime factorization of a generator of the ideal).
Further properties
The most dramatic change in the topology from the classical picture to the new is that points are no longer necessarily closed; by expanding the definition, Grothendieck introduced generic points, which are the points with maximal closure, that is the minimal prime ideals. The closed points correspond to maximal ideals of A. However, the spectrum and projective spectrum are still T0 spaces: given two points P, Q, which are prime ideals of A, at least one of them, say P, does not contain the other. Then D(Q) contains P but, of course, not Q.
Just as in classical algebraic geometry, any spectrum or projective spectrum is (quasi)compact, and if the ring in question is Noetherian then the space is a Noetherian space. However, these facts are counterintuitive: we do not normally expect open sets, other than connected components, to be compact, and for affine varieties (for example, Euclidean space) we do not even expect the space itself to be compact. This is one instance of the geometric unsuitability of the Zariski topology. Grothendieck solved this problem by defining the notion of properness of a scheme (actually, of a morphism of schemes), which recovers the intuitive idea of compactness: Proj is proper, but Spec is not.
See also
• Spectral space
Citations
1. Hulek 2003, p. 19, 1.1.1..
2. Mumford 1999.
3. Dummit & Foote 2004.
References
• Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3 ed.). Wiley. pp. 71–72. ISBN 9780471433347.
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052
• Hulek, Klaus (2003). Elementary Algebraic Geometry. AMS. ISBN 978-0-8218-2952-3.
• Mumford, David (1999) [1967]. The Red Book of Varieties and Schemes. Lecture Notes in Mathematics. Vol. 1358 (expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.). Berlin, New York: Springer-Verlag. doi:10.1007/b62130. ISBN 978-3-540-63293-1. MR 1748380.
• Todd Rowland. "Zariski Topology". MathWorld.
| Wikipedia |
Zariski ring
In commutative algebra, a Zariski ring is a commutative Noetherian topological ring A whose topology is defined by an ideal ${\mathfrak {a}}$ contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced by Oscar Zariski (1946) under the name "semi-local ring" which now means something different, and named "Zariski rings" by Pierre Samuel (1953). Examples of Zariski rings are noetherian local rings with the topology induced by the maximal ideal, and ${\mathfrak {a}}$-adic completions of Noetherian rings.
Let A be a Noetherian topological ring with the topology defined by an ideal ${\mathfrak {a}}$. Then the following are equivalent.
• A is a Zariski ring.
• The completion ${\widehat {A}}$ is faithfully flat over A (in general, it is only flat over A).
• Every maximal ideal is closed.
References
• Atiyah, Michael F.; Macdonald, Ian G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802
• Samuel, Pierre (1953), Algèbre locale, Mémor. Sci. Math., vol. 123, Paris: Gauthier-Villars, MR 0054995
• Zariski, Oscar (1946), "Generalized semi-local rings", Summa Brasil. Math., 1 (8): 169–195, MR 0022835
• Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876
| Wikipedia |
Zariski surface
In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface. In particular, all Zariski surfaces are unirational. They were named by Piotr Blass in 1977 after Oscar Zariski who used them in 1958 to give examples of unirational surfaces in characteristic p > 0 that are not rational. (In characteristic 0 by contrast, Castelnuovo's theorem implies that all unirational surfaces are rational.)
For spaces of valuations, see Zariski–Riemann surface.
Zariski surfaces are birational to surfaces in affine 3-space A3 defined by irreducible polynomials of the form
$z^{p}=f(x,y).\ $
The following problem was posed by Oscar Zariski in 1971: Let S be a Zariski surface with vanishing geometric genus. Is S necessarily a rational surface? For p = 2 and for p = 3 the answer to the above problem is negative as shown in 1977 by Piotr Blass in his University of Michigan Ph.D. thesis and by William E. Lang in his Harvard Ph.D. thesis in 1978. Kentaro Mitsui (2014) announced further examples giving a negative answer to Zariski's question in every characteristic p>0 . His method however is non constructive at the moment and we do not have explicit equations for p>3.
See also
• List of algebraic surfaces
References
• Blass, Piotr; Lang, Jeffrey (1987), Zariski surfaces and differential equations in characteristic p>0, Monographs and Textbooks in Pure and Applied Mathematics, vol. 106, New York: Marcel Dekker Inc., ISBN 978-0-8247-7637-4, MR 0879599
• Mitsui, Kentaro (2014), "On a question of Zariski on Zariski surfaces", Math. Z., 276 (1–2): 237–242, doi:10.1007/s00209-013-1195-0, MR 3150201
• Zariski, Oscar (1958), "On Castelnuovo's criterion of rationality pa=P2=0 of an algebraic surface", Illinois Journal of Mathematics, 2: 303–315, ISSN 0019-2082, MR 0099990
| Wikipedia |
Zariski tangent space
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.
Motivation
For example, suppose given a plane curve C defined by a polynomial equation
F(X,Y) = 0
and take P to be the origin (0,0). Erasing terms of higher order than 1 would produce a 'linearised' equation reading
L(X,Y) = 0
in which all terms XaYb have been discarded if a + b > 1.
We have two cases: L may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to C at (0,0) is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take P as a general point on C; it is better to say 'affine space' and then note that P is a natural origin, rather than insist directly that it is a vector space.)
It is easy to see that over the real field we can obtain L in terms of the first partial derivatives of F. When those both are 0 at P, we have a singular point (double point, cusp or something more complicated). The general definition is that singular points of C are the cases when the tangent space has dimension 2.
Definition
The cotangent space of a local ring R, with maximal ideal ${\mathfrak {m}}$ is defined to be
${\mathfrak {m}}/{\mathfrak {m}}^{2}$
where ${\mathfrak {m}}$2 is given by the product of ideals. It is a vector space over the residue field k:= R/${\mathfrak {m}}$. Its dual (as a k-vector space) is called tangent space of R.[1]
This definition is a generalization of the above example to higher dimensions: suppose given an affine algebraic variety V and a point v of V. Morally, modding out ${\mathfrak {m}}$2 corresponds to dropping the non-linear terms from the equations defining V inside some affine space, therefore giving a system of linear equations that define the tangent space.
The tangent space $T_{P}(X)$ and cotangent space $T_{P}^{*}(X)$ to a scheme X at a point P is the (co)tangent space of ${\mathcal {O}}_{X,P}$. Due to the functoriality of Spec, the natural quotient map $f:R\rightarrow R/I$ induces a homomorphism $g:{\mathcal {O}}_{X,f^{-1}(P)}\rightarrow {\mathcal {O}}_{Y,P}$ for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed $T_{P}(Y)$ in $T_{f^{-1}P}(X)$.[2] Since morphisms of fields are injective, the surjection of the residue fields induced by g is an isomorphism. Then a morphism k of the cotangent spaces is induced by g, given by
${\mathfrak {m}}_{P}/{\mathfrak {m}}_{P}^{2}$
$\cong ({\mathfrak {m}}_{f^{-1}P}/I)/(({\mathfrak {m}}_{f^{-1}P}^{2}+I)/I)$
$\cong {\mathfrak {m}}_{f^{-1}P}/({\mathfrak {m}}_{f^{-1}P}^{2}+I)$
$\cong ({\mathfrak {m}}_{f^{-1}P}/{\mathfrak {m}}_{f^{-1}P}^{2})/\mathrm {Ker} (k).$
Since this is a surjection, the transpose $k^{*}:T_{P}(Y)\rightarrow T_{f^{-1}P}(X)$ is an injection.
(One often defines the tangent and cotangent spaces for a manifold in the analogous manner.)
Analytic functions
If V is a subvariety of an n-dimensional vector space, defined by an ideal I, then R = Fn / I, where Fn is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at x is
mn / (I+mn2),
where mn is the maximal ideal consisting of those functions in Fn vanishing at x.
In the planar example above, I = (F(X,Y)), and I+m2 = (L(X,Y))+m2.
Properties
If R is a Noetherian local ring, the dimension of the tangent space is at least the dimension of R:
dim m/m2 ≧ dim R
R is called regular if equality holds. In a more geometric parlance, when R is the local ring of a variety V at a point v, one also says that v is a regular point. Otherwise it is called a singular point.
The tangent space has an interpretation in terms of K[t]/(t2), the dual numbers for K; in the parlance of schemes, morphisms from Spec K[t]/(t2) to a scheme X over K correspond to a choice of a rational point x ∈ X(k) and an element of the tangent space at x.[3] Therefore, one also talks about tangent vectors. See also: tangent space to a functor.
In general, the dimension of the Zariski tangent space can be extremely large. For example, let $C^{1}(\mathbf {R} )$ be the ring of continuously differentiable real-valued functions on $\mathbf {R} $. Define $R=C_{0}^{1}(\mathbf {R} )$ to be the ring of germs of such functions at the origin. Then R is a local ring, and its maximal ideal m consists of all germs which vanish at the origin. The functions $x^{\alpha }$ for $\alpha \in (1,2)$ define linearly independent vectors in the Zariski cotangent space $m/m^{2}$, so the dimension of $m/m^{2}$ is at least the ${\mathfrak {c}}$, the cardinality of the continuum. The dimension of the Zariski tangent space $(m/m^{2})^{*}$ is therefore at least $2^{\mathfrak {c}}$. On the other hand, the ring of germs of smooth functions at a point in an n-manifold has an n-dimensional Zariski cotangent space.[lower-alpha 1]
See also
• Tangent cone
• Jet (mathematics)
Notes
1. https://mathoverflow.net/questions/44705/cardinalities-larger-than-the-continuum-in-areas-besides-set-theory/44733#44733
Citations
1. Eisenbud & Harris 1998, I.2.2, pg. 26.
2. Smoothness and the Zariski Tangent Space, James McKernan, 18.726 Spring 2011 Lecture 5
3. Hartshorne 1977, Exercise II 2.8.
Sources
• Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5 – via Internet Archive.
• Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. New York: Springer-Verlag. ISBN 978-0-387-90244-9. MR 0463157.
• Zariski, Oscar (1947). "The concept of a simple point of an abstract algebraic variety". Transactions of the American Mathematical Society. 62: 1–52. doi:10.1090/S0002-9947-1947-0021694-1. MR 0021694. Zbl 0031.26101.
External links
• Zariski tangent space. V.I. Danilov (originator), Encyclopedia of Mathematics.
| Wikipedia |
Zariski's main theorem
In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski (1943), is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational.
Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called Zariski's main theorem are as follows:
• The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original form of his main theorem.
• A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset.
• The total transform of a normal point under a proper birational morphism is connected.
• A closely related theorem of Grothendieck describes the structure of quasi-finite morphisms of schemes, which implies Zariski's original main theorem.
• Several results in commutative algebra that imply the geometric form of Zariski's main theorem.
• A normal local ring is unibranch, which is a variation of the statement that the transform of a normal point is connected.
• The local ring of a normal point of a variety is analytically normal. This is a strong form of the statement that it is unibranch.
The name "Zariski's main theorem" comes from the fact that Zariski labelled it as the "MAIN THEOREM" in Zariski (1943).
Zariski's main theorem for birational morphisms
Let f be a birational mapping of algebraic varieties V and W. Recall that f is defined by a closed subvariety $\Gamma \subset V\times W$ (a "graph" of f) such that the projection on the first factor $p_{1}$ induces an isomorphism between an open $U\subset V$ and $p_{1}^{-1}(U)$, and such that $p_{2}\circ p_{1}^{-1}$ is an isomorphism on U too. The complement of U in V is called a fundamental variety or indeterminacy locus, and the image of a subset of V under $p_{2}\circ p_{1}^{-1}$ is called a total transform of it.
The original statement of the theorem in (Zariski 1943, p. 522) reads:
MAIN THEOREM: If W is an irreducible fundamental variety on V of a birational correspondence T between V and V′ and if T has no fundamental elements on V′ then — under the assumption that V is locally normal at W — each irreducible component of the transform T[W] is of higher dimension than W.
Here T is essentially a morphism from V′ to V that is birational, W is a subvariety of the set where the inverse of T is not defined whose local ring is normal, and the transform T[W] means the inverse image of W under the morphism from V′ to V.
Here are some variants of this theorem stated using more recent terminology. Hartshorne (1977, Corollary III.11.4) calls the following connectedness statement "Zariski's Main theorem":
If f:X→Y is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of Y is connected.
The following consequence of it (Theorem V.5.2,loc.cit.) also goes under this name:
If f:X→Y is a birational transformation of projective varieties with Y normal, then the total transform of a fundamental point of f is connected and of dimension at least 1.
Examples
• Suppose that V is a smooth variety of dimension greater than 1 and V′ is given by blowing up a point W on V. Then V is normal at W, and the component of the transform of W is a projective space, which has dimension greater than W as predicted by Zariski's original form of his main theorem.
• In the previous example the transform of W was irreducible. It is easy to find examples where the total transform is reducible by blowing up other points on the transform. For example, if V′ is given by blowing up a point W on V and then blowing up another point on this transform, the total transform of W has 2 irreducible components meeting at a point. As predicted by Hartshorne's form of the main theorem, the total transform is connected and of dimension at least 1.
• For an example where W is not normal and the conclusion of the main theorem fails, take V′ to be a smooth variety, and take V to be given by identifying two distinct points on V′, and take W to be the image of these two points. Then W is not normal, and the transform of W consists of two points, which is not connected and does not have positive dimension.
Zariski's main theorem for quasifinite morphisms
In EGA III, Grothendieck calls the following statement which does not involve connectedness a "Main theorem" of Zariski Grothendieck (1961, Théorème 4.4.3):
If f:X→Y is a quasi-projective morphism of Noetherian schemes then the set of points that are isolated in their fiber is open in X. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over Y.
In EGA IV, Grothendieck observed that the last statement could be deduced from a more general theorem about the structure of quasi-finite morphisms, and the latter is often referred to as the "Zariski's main theorem in the form of Grothendieck". It is well known that open immersions and finite morphisms are quasi-finite. Grothendieck proved that under the hypothesis of separatedness all quasi-finite morphisms are compositions of such Grothendieck (1966, Théorème 8.12.6):
if Y is a quasi-compact separated scheme and $f:X\to Y$ is a separated, quasi-finite, finitely presented morphism then there is a factorization into $X\to Z\to Y$, where the first map is an open immersion and the second one is finite.
The relation between this theorem about quasi-finite morphisms and Théorème 4.4.3 of EGA III quoted above is that if f:X→Y is a projective morphism of varieties, then the set of points that are isolated in their fiber is quasifinite over Y. Then structure theorem for quasi-finite morphisms applies and yields the desired result.
Zariski's main theorem for commutative rings
Zariski (1949) reformulated his main theorem in terms of commutative algebra as a statement about local rings. Grothendieck (1961, Théorème 4.4.7) generalized Zariski's formulation as follows:
If B is an algebra of finite type over a local Noetherian ring A, and n is a maximal ideal of B which is minimal among ideals of B whose inverse image in A is the maximal ideal m of A, then there is a finite A-algebra A′ with a maximal ideal m′ (whose inverse image in A is m) such that the localization Bn is isomorphic to the A-algebra A′m′.
If in addition A and B are integral and have the same field of fractions, and A is integrally closed, then this theorem implies that A and B are equal. This is essentially Zariski's formulation of his main theorem in terms of commutative rings.
Zariski's main theorem: topological form
A topological version of Zariski's main theorem says that if x is a (closed) point of a normal complex variety it is unibranch; in other words there are arbitrarily small neighborhoods U of x such that the set of non-singular points of U is connected (Mumford 1999, III.9).
The property of being normal is stronger than the property of being unibranch: for example, a cusp of a plane curve is unibranch but not normal.
Zariski's main theorem: power series form
A formal power series version of Zariski's main theorem says that if x is a normal point of a variety then it is analytically normal; in other words the completion of the local ring at x is a normal integral domain (Mumford 1999, III.9).
See also
• Deligne's connectedness theorem
• Fulton–Hansen connectedness theorem
• Grothendieck's connectedness theorem
• Stein factorization
• Theorem on formal functions
References
• Danilov, V.I. (2001) [1994], "Zariski theorem", Encyclopedia of Mathematics, EMS Press
• Grothendieck, Alexandre (1961), Eléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie, Publications Mathématiques de l'IHÉS, vol. 11, pp. 5–167
• Grothendieck, Alexandre (1966), Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie, Publications Mathématiques de l'IHÉS, vol. 28, pp. 43–48
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Mumford, David (1999) [1988], The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358 (expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.), Berlin, New York: Springer-Verlag, doi:10.1007/b62130, ISBN 978-3-540-63293-1, MR 1748380
• Peskine, Christian (1966), "Une généralisation du main theorem de Zariski", Bull. Sci. Math. (2), 90: 119–127
• Raynaud, Michel (1970), Anneaux locaux henséliens, Lecture Notes in Mathematics, vol. 169, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069571, ISBN 978-3-540-05283-8, MR 0277519
• Zariski, Oscar (1943), "Foundations of a general theory of birational correspondences.", Trans. Amer. Math. Soc., 53 (3): 490–542, doi:10.2307/1990215, JSTOR 1990215, MR 0008468
• Zariski, Oscar (1949), "A simple analytical proof of a fundamental property of birational transformations.", Proc. Natl. Acad. Sci. U.S.A., 35 (1): 62–66, Bibcode:1949PNAS...35...62Z, doi:10.1073/pnas.35.1.62, JSTOR 88284, MR 0028056, PMC 1062959, PMID 16588856
External links
• Is there an intuitive reason for Zariski's main theorem?
| Wikipedia |
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.
| Wikipedia |
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
| Wikipedia |
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
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| Wikipedia |
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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| Wikipedia |
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.
| Wikipedia |
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.
| Wikipedia |
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
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| Wikipedia |
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
| Wikipedia |
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
| Wikipedia |
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
| Wikipedia |
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
• Who was Who
• Dictionary of National Biography
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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| Wikipedia |
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.
| Wikipedia |
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
| Wikipedia |
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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| Wikipedia |
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
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• Determinacy
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• Axiom schema
• replacement
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Set types
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Set theorists
• Paul Bernays
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• Abraham Fraenkel
• Kurt Gödel
• Thomas Jech
• John von Neumann
• Willard Quine
• Bertrand Russell
• Thoralf Skolem
• Ernst Zermelo
| Wikipedia |
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.
| Wikipedia |
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)
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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.
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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.
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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).
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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
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• 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.
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| Wikipedia |
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.
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| Wikipedia |
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.
Set theory
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