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Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group $H_{2}(G,\mathbb {Z} )$ of a group G. It was introduced by Issai Schur (1904) in his work on projective representations.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
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Finite groups
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• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
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• Discrete groups
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• Integers ($\mathbb {Z} $)
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Modular groups
• PSL(2, $\mathbb {Z} $)
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Infinite dimensional Lie group
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Algebraic groups
• Linear algebraic group
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• Elliptic curve
Examples and properties
The Schur multiplier $\operatorname {M} (G)$ of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, then the order of $\operatorname {M} (G)$ is not divisible by p. In particular, if all Sylow p-subgroups of G are cyclic, then $\operatorname {M} (G)$ is trivial.
For instance, the Schur multiplier of the nonabelian group of order 6 is the trivial group since every Sylow subgroup is cyclic. The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups has order 2.
The Schur multipliers of the finite simple groups are given at the list of finite simple groups. The covering groups of the alternating and symmetric groups are of considerable recent interest.
Relation to projective representations
Schur's original motivation for studying the multiplier was to classify projective representations of a group, and the modern formulation of his definition is the second cohomology group $H^{2}(G,\mathbb {C} ^{\times })$. A projective representation is much like a group representation except that instead of a homomorphism into the general linear group $\operatorname {GL} (n,\mathbb {C} )$, one takes a homomorphism into the projective general linear group $\operatorname {PGL} (n,\mathbb {C} )$. In other words, a projective representation is a representation modulo the center.
Schur (1904, 1907) showed that every finite group G has associated to it at least one finite group C, called a Schur cover, with the property that every projective representation of G can be lifted to an ordinary representation of C. The Schur cover is also known as a covering group or Darstellungsgruppe. The Schur covers of the finite simple groups are known, and each is an example of a quasisimple group. The Schur cover of a perfect group is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up to isoclinism.
Relation to central extensions
The study of such covering groups led naturally to the study of central and stem extensions.
A central extension of a group G is an extension
$1\to K\to C\to G\to 1$
where $K\leq Z(C)$ is a subgroup of the center of C.
A stem extension of a group G is an extension
$1\to K\to C\to G\to 1$
where $K\leq Z(C)\cap C'$ is a subgroup of the intersection of the center of C and the derived subgroup of C; this is more restrictive than central.[1]
If the group G is finite and one considers only stem extensions, then there is a largest size for such a group C, and for every C of that size the subgroup K is isomorphic to the Schur multiplier of G. If the finite group G is moreover perfect, then C is unique up to isomorphism and is itself perfect. Such C are often called universal perfect central extensions of G, or covering group (as it is a discrete analog of the universal covering space in topology). If the finite group G is not perfect, then its Schur covering groups (all such C of maximal order) are only isoclinic.
It is also called more briefly a universal central extension, but note that there is no largest central extension, as the direct product of G and an abelian group form a central extension of G of arbitrary size.
Stem extensions have the nice property that any lift of a generating set of G is a generating set of C. If the group G is presented in terms of a free group F on a set of generators, and a normal subgroup R generated by a set of relations on the generators, so that $G\cong F/R$, then the covering group itself can be presented in terms of F but with a smaller normal subgroup S, that is, $C\cong F/S$. Since the relations of G specify elements of K when considered as part of C, one must have $S\leq [F,R]$.
In fact if G is perfect, this is all that is needed: C ≅ [F,F]/[F,R] and M(G) ≅ K ≅ R/[F,R]. Because of this simplicity, expositions such as (Aschbacher 2000, §33) handle the perfect case first. The general case for the Schur multiplier is similar but ensures the extension is a stem extension by restricting to the derived subgroup of F: M(G) ≅ (R ∩ [F, F])/[F, R]. These are all slightly later results of Schur, who also gave a number of useful criteria for calculating them more explicitly.
Relation to efficient presentations
In combinatorial group theory, a group often originates from a presentation. One important theme in this area of mathematics is to study presentations with as few relations as possible, such as one relator groups like Baumslag–Solitar groups. These groups are infinite groups with two generators and one relation, and an old result of Schreier shows that in any presentation with more generators than relations, the resulting group is infinite. The borderline case is thus quite interesting: finite groups with the same number of generators as relations are said to have a deficiency zero. For a group to have deficiency zero, the group must have a trivial Schur multiplier because the minimum number of generators of the Schur multiplier is always less than or equal to the difference between the number of relations and the number of generators, which is the negative deficiency. An efficient group is one where the Schur multiplier requires this number of generators.[2]
A fairly recent topic of research is to find efficient presentations for all finite simple groups with trivial Schur multipliers. Such presentations are in some sense nice because they are usually short, but they are difficult to find and to work with because they are ill-suited to standard methods such as coset enumeration.
Relation to topology
In topology, groups can often be described as finitely presented groups and a fundamental question is to calculate their integral homology $H_{n}(G,\mathbb {Z} )$. In particular, the second homology plays a special role and this led Heinz Hopf to find an effective method for calculating it. The method in (Hopf 1942) is also known as Hopf's integral homology formula and is identical to Schur's formula for the Schur multiplier of a finite group:
$H_{2}(G,\mathbb {Z} )\cong (R\cap [F,F])/[F,R]$
where $G\cong F/R$ and F is a free group. The same formula also holds when G is a perfect group.[3]
The recognition that these formulas were the same led Samuel Eilenberg and Saunders Mac Lane to the creation of cohomology of groups. In general,
$H_{2}(G,\mathbb {Z} )\cong {\bigl (}H^{2}(G,\mathbb {C} ^{\times }){\bigr )}^{*}$
where the star denotes the algebraic dual group. Moreover, when G is finite, there is an unnatural isomorphism
${\bigl (}H^{2}(G,\mathbb {C} ^{\times }){\bigr )}^{*}\cong H^{2}(G,\mathbb {C} ^{\times }).$
The Hopf formula for $H_{2}(G)$ has been generalised to higher dimensions. For one approach and references see the paper by Everaert, Gran and Van der Linden listed below.
A perfect group is one whose first integral homology vanishes. A superperfect group is one whose first two integral homology groups vanish. The Schur covers of finite perfect groups are superperfect. An acyclic group is a group all of whose reduced integral homology vanishes.
Applications
The second algebraic K-group K2(R) of a commutative ring R can be identified with the second homology group H2(E(R), Z) of the group E(R) of (infinite) elementary matrices with entries in R.[4]
See also
• Quasisimple group
The references from Clair Miller give another view of the Schur Multiplier as the kernel of a morphism κ: G ∧ G → G induced by the commutator map.
Notes
1. Rotman 1994, p. 553
2. Johnson & Robertson 1979, pp. 275–289
3. Rosenberg 1994, Theorems 4.1.3, 4.1.19
4. Rosenberg 1994, Corollary 4.2.10
References
• Aschbacher, Michael (2000), Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10 (2nd ed.), Cambridge University Press, ISBN 978-0-521-78145-9, MR 1777008, Zbl 0997.20001
• Hopf, Heinz (1942), "Fundamentalgruppe und zweite Bettische Gruppe", Commentarii Mathematici Helvetici, 14: 257–309, doi:10.1007/BF02565622, ISSN 0010-2571, MR 0006510, Zbl 0027.09503
• Johnson, David Lawrence; Robertson, Edmund Frederick (1979), "Finite groups of deficiency zero", in Wall, C.T.C. (ed.), Homological Group Theory, London Mathematical Society Lecture Note Series, vol. 36, Cambridge University Press, ISBN 978-0-521-22729-2, Zbl 0423.20029
• Kuzmin, Leonid Viktorovich (2001) [1994], "Schur multiplicator", Encyclopedia of Mathematics, EMS Press
• Rosenberg, Jonathan (1994), Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, ISBN 978-0-387-94248-3, MR 1282290, Zbl 0801.19001 Errata
• Rotman, Joseph J. (1994), An introduction to the theory of groups, Springer-Verlag, ISBN 978-0-387-94285-8
• Schur, Issai (1904), "Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen.", Journal für die reine und angewandte Mathematik (in German), 127: 20–50, ISSN 0075-4102, JFM 35.0155.01
• Schur, Issai (1907), "Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen.", Journal für die reine und angewandte Mathematik (in German), 1907 (132): 85–137, doi:10.1515/crll.1907.132.85, ISSN 0075-4102, JFM 38.0174.02
• Van der Kallen, Wilberd (1984), "Review: F. Rudolf Beyl and Jürgen Tappe, Group extensions, representations, and the Schur multiplicator", Bulletin of the American Mathematical Society, 10 (2): 330–3, doi:10.1090/s0273-0979-1984-15273-x
• Wiegold, James (1982), "The Schur multiplier: an elementary approach", Groups–St. Andrews 1981 (St. Andrews, 1981), London Math. Soc. Lecture Note Ser., vol. 71, Cambridge University Press, pp. 137–154, MR 0679156, Zbl 0502.20003
• Miller, Clair (1952), "The second homology of a group", Proceedings of the American Mathematical Society, 3 (4): 588–595, doi:10.1090/s0002-9939-1952-0049191-5, Zbl 0047.25703
• Dennis, R.K. (1976), In search of new "Homology" functors having a close relationship to K-theory, Cornell University
• Brown, R.; Johnson, D.L.; Robertson, E.F. (1987), "Some computations of non-abelian tensor products of groups", Journal of Algebra, 111: 177–202, doi:10.1016/0021-8693(87)90248-1, Zbl 0626.20038
• Ellis, G.J.; Leonard, F. (1995), "Computing Schur multipliers and tensor products of finite groups", Proceedings of the Royal Irish Academy, 95A (2): 137–147, ISSN 0035-8975, JSTOR 20490165, Zbl 0863.20010
• Ellis, Graham J. (1998), "The Schur multiplier of a pair of groups", Applied Categorical Structures, 6 (3): 355–371, doi:10.1023/A:1008652316165, Zbl 0948.20026
• Eick, Bettina; Nickel, Werner (2008), "Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group", Journal of Algebra, 320 (2): 927–944, doi:10.1016/j.jalgebra.2008.02.041, Zbl 1163.20022
• Everaert, Tomas; Gran, Marino; Van der Linden, Tim (2008), "Higher Hopf formulae for homology via Galois theory", Advances in Mathematics, 217 (5): 2231–67, arXiv:math/0701815, doi:10.1016/j.aim.2007.11.001, Zbl 1140.18012
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Wikipedia
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Schur class
In mathematics, the Schur class consists of the Schur functions: the holomorphic functions from the open unit disk to the closed unit disk. These functions were studied by Schur (1918).
The Schur parameters γj of a Schur function f0 are defined recursively by
$\gamma _{j}=f_{j}(0)$
$zf_{j+1}={\frac {f_{j}(z)-\gamma _{j}}{1-{\overline {\gamma _{j}}}f_{j}(z)}}.$
The Schur parameters γj all have absolute value at most 1.
This gives a continued fraction expansion of the Schur function f0 by repeatedly using the fact that
$f_{j}(z)=\gamma _{j}+{\frac {1-|\gamma _{j}|^{2}}{{\overline {\gamma _{j}}}+{\frac {1}{zf_{j+1}(z)}}}}$
which gives
$f_{0}(z)=\gamma _{0}+{\frac {1-|\gamma _{0}|^{2}}{{\overline {\gamma _{0}}}+{\frac {1}{z\gamma _{1}+{\frac {z(1-|\gamma _{1}|^{2})}{{\overline {\gamma _{1}}}+{\frac {1}{z\gamma _{2}+\cdots }}}}}}}}.$
See also
• Szegő polynomial
References
• Schur, I. (1918), "Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. I, II", J. Reine Angew. Math. (in German), Berlin: Walter de Gruyter, 147: 205–232, doi:10.1515/crll.1917.147.205, JFM 46.0475.01
• Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 1. Classical theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3446-6, MR 2105088
• Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 2. Spectral theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3675-0, MR 2105089
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Wikipedia
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Schur product theorem
In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur[1] (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in Journal für die reine und angewandte Mathematik.[2][3])
We remark that the converse of the theorem holds in the following sense. If $M$ is a symmetric matrix and the Hadamard product $M\circ N$ is positive definite for all positive definite matrices $N$, then $M$ itself is positive definite.
Proof
Proof using the trace formula
For any matrices $M$ and $N$, the Hadamard product $M\circ N$ considered as a bilinear form acts on vectors $a,b$ as
$a^{*}(M\circ N)b=\operatorname {tr} \left(M^{\textsf {T}}\operatorname {diag} \left(a^{*}\right)N\operatorname {diag} (b)\right)$
where $\operatorname {tr} $ is the matrix trace and $\operatorname {diag} (a)$ is the diagonal matrix having as diagonal entries the elements of $a$.
Suppose $M$ and $N$ are positive definite, and so Hermitian. We can consider their square-roots $M^{\frac {1}{2}}$ and $N^{\frac {1}{2}}$, which are also Hermitian, and write
$\operatorname {tr} \left(M^{\textsf {T}}\operatorname {diag} \left(a^{*}\right)N\operatorname {diag} (b)\right)=\operatorname {tr} \left({\overline {M}}^{\frac {1}{2}}{\overline {M}}^{\frac {1}{2}}\operatorname {diag} \left(a^{*}\right)N^{\frac {1}{2}}N^{\frac {1}{2}}\operatorname {diag} (b)\right)=\operatorname {tr} \left({\overline {M}}^{\frac {1}{2}}\operatorname {diag} \left(a^{*}\right)N^{\frac {1}{2}}N^{\frac {1}{2}}\operatorname {diag} (b){\overline {M}}^{\frac {1}{2}}\right)$
Then, for $a=b$, this is written as $\operatorname {tr} \left(A^{*}A\right)$ for $A=N^{\frac {1}{2}}\operatorname {diag} (a){\overline {M}}^{\frac {1}{2}}$ and thus is strictly positive for $A\neq 0$, which occurs if and only if $a\neq 0$. This shows that $(M\circ N)$ is a positive definite matrix.
Case of M = N
Let $X$ be an $n$-dimensional centered Gaussian random variable with covariance $\langle X_{i}X_{j}\rangle =M_{ij}$. Then the covariance matrix of $X_{i}^{2}$ and $X_{j}^{2}$ is
$\operatorname {Cov} \left(X_{i}^{2},X_{j}^{2}\right)=\left\langle X_{i}^{2}X_{j}^{2}\right\rangle -\left\langle X_{i}^{2}\right\rangle \left\langle X_{j}^{2}\right\rangle $
Using Wick's theorem to develop $\left\langle X_{i}^{2}X_{j}^{2}\right\rangle =2\left\langle X_{i}X_{j}\right\rangle ^{2}+\left\langle X_{i}^{2}\right\rangle \left\langle X_{j}^{2}\right\rangle $ we have
$\operatorname {Cov} \left(X_{i}^{2},X_{j}^{2}\right)=2\left\langle X_{i}X_{j}\right\rangle ^{2}=2M_{ij}^{2}$
Since a covariance matrix is positive definite, this proves that the matrix with elements $M_{ij}^{2}$ is a positive definite matrix.
General case
Let $X$ and $Y$ be $n$-dimensional centered Gaussian random variables with covariances $\left\langle X_{i}X_{j}\right\rangle =M_{ij}$, $\left\langle Y_{i}Y_{j}\right\rangle =N_{ij}$ and independent from each other so that we have
$\left\langle X_{i}Y_{j}\right\rangle =0$ for any $i,j$
Then the covariance matrix of $X_{i}Y_{i}$ and $X_{j}Y_{j}$ is
$\operatorname {Cov} \left(X_{i}Y_{i},X_{j}Y_{j}\right)=\left\langle X_{i}Y_{i}X_{j}Y_{j}\right\rangle -\left\langle X_{i}Y_{i}\right\rangle \left\langle X_{j}Y_{j}\right\rangle $
Using Wick's theorem to develop
$\left\langle X_{i}Y_{i}X_{j}Y_{j}\right\rangle =\left\langle X_{i}X_{j}\right\rangle \left\langle Y_{i}Y_{j}\right\rangle +\left\langle X_{i}Y_{i}\right\rangle \left\langle X_{j}Y_{j}\right\rangle +\left\langle X_{i}Y_{j}\right\rangle \left\langle X_{j}Y_{i}\right\rangle $
and also using the independence of $X$ and $Y$, we have
$\operatorname {Cov} \left(X_{i}Y_{i},X_{j}Y_{j}\right)=\left\langle X_{i}X_{j}\right\rangle \left\langle Y_{i}Y_{j}\right\rangle =M_{ij}N_{ij}$
Since a covariance matrix is positive definite, this proves that the matrix with elements $M_{ij}N_{ij}$ is a positive definite matrix.
Proof of positive semidefiniteness
Let $M=\sum \mu _{i}m_{i}m_{i}^{\textsf {T}}$ and $N=\sum \nu _{i}n_{i}n_{i}^{\textsf {T}}$. Then
$M\circ N=\sum _{ij}\mu _{i}\nu _{j}\left(m_{i}m_{i}^{\textsf {T}}\right)\circ \left(n_{j}n_{j}^{\textsf {T}}\right)=\sum _{ij}\mu _{i}\nu _{j}\left(m_{i}\circ n_{j}\right)\left(m_{i}\circ n_{j}\right)^{\textsf {T}}$
Each $\left(m_{i}\circ n_{j}\right)\left(m_{i}\circ n_{j}\right)^{\textsf {T}}$ is positive semidefinite (but, except in the 1-dimensional case, not positive definite, since they are rank 1 matrices). Also, $\mu _{i}\nu _{j}>0$ thus the sum $M\circ N$ is also positive semidefinite.
Proof of definiteness
To show that the result is positive definite requires even further proof. We shall show that for any vector $a\neq 0$, we have $a^{\textsf {T}}(M\circ N)a>0$. Continuing as above, each $a^{\textsf {T}}\left(m_{i}\circ n_{j}\right)\left(m_{i}\circ n_{j}\right)^{\textsf {T}}a\geq 0$, so it remains to show that there exist $i$ and $j$ for which corresponding term above is nonzero. For this we observe that
$a^{\textsf {T}}(m_{i}\circ n_{j})(m_{i}\circ n_{j})^{\textsf {T}}a=\left(\sum _{k}m_{i,k}n_{j,k}a_{k}\right)^{2}$
Since $N$ is positive definite, there is a $j$ for which $n_{j}\circ a\neq 0$ (since otherwise $n_{j}^{\textsf {T}}a=\sum _{k}(n_{j}\circ a)_{k}=0$ for all $j$), and likewise since $M$ is positive definite there exists an $i$ for which $\sum _{k}m_{i,k}(n_{j}\circ a)_{k}=m_{i}^{\textsf {T}}(n_{j}\circ a)\neq 0.$ However, this last sum is just $\sum _{k}m_{i,k}n_{j,k}a_{k}$. Thus its square is positive. This completes the proof.
References
1. Schur, J. (1911). "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen". Journal für die reine und angewandte Mathematik. 1911 (140): 1–28. doi:10.1515/crll.1911.140.1. S2CID 120411177.
2. Zhang, Fuzhen, ed. (2005). The Schur Complement and Its Applications. Numerical Methods and Algorithms. Vol. 4. doi:10.1007/b105056. ISBN 0-387-24271-6., page 9, Ch. 0.6 Publication under J. Schur
3. Ledermann, W. (1983). "Issai Schur and His School in Berlin". Bulletin of the London Mathematical Society. 15 (2): 97–106. doi:10.1112/blms/15.2.97.
External links
• Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen at EUDML
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Wikipedia
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Schur–Horn theorem
In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are Kostant's convexity theorem, Atiyah–Guillemin–Sternberg convexity theorem, Kirwan convexity theorem.
Statement
Schur–Horn theorem — Theorem. Let $d_{1},\dots ,d_{N}$ and $\lambda _{1},\dots ,\lambda _{N}$ be two sequences of real numbers arranged in a non-increasing order. There is a Hermitian matrix with diagonal values $d_{1},\dots ,d_{N}$ (in this order, starting with $d_{1}$ at the top-left) and eigenvalues $\lambda _{1},\dots ,\lambda _{N}$ if and only if
$\sum _{i=1}^{n}d_{i}\leq \sum _{i=1}^{n}\lambda _{i}\qquad n=1,\dots ,N$
and
$\sum _{i=1}^{N}d_{i}=\sum _{i=1}^{N}\lambda _{i}.$
The inequalities above may alternatively be written:
${\begin{alignedat}{7}d_{1}&\;\leq \;&&\lambda _{1}\\[0.3ex]d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}\\[0.3ex]\vdots &\;\leq &&\vdots \\[0.3ex]d_{N-1}+\cdots +d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}\\[0.3ex]d_{N}+d_{N-1}+\cdots +d_{2}+d_{1}&\;=&&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}+\lambda _{N}.\\[0.3ex]\end{alignedat}}$
The Schur–Horn theorem may thus be restated more succinctly and in plain English:
Schur–Horn theorem: Given any non-increasing real sequences of desired diagonal elements $d_{1}\geq \cdots \geq d_{N}$ and desired eigenvalues $\lambda _{1}\geq \cdots \geq \lambda _{N},$ there exists a Hermitian matrix with these eigenvalues and diagonal elements if and only if these two sequences have the same sum and for every possible integer $n,$ the sum of the first $n$ desired diagonal elements never exceeds the sum of the first $n$ desired eigenvalues.
Reformation allowing unordered diagonals and eigenvalues
Although this theorem requires that $d_{1}\geq \cdots \geq d_{N}$ and $\lambda _{1}\geq \cdots \geq \lambda _{N}$ be non-increasing, it is possible to reformulate this theorem without these assumptions.
We start with the assumption $\lambda _{1}\geq \cdots \geq \lambda _{N}.$ The left hand side of the theorem's characterization (that is, "there exists a Hermitian matrix with these eigenvalues and diagonal elements") depends on the order of the desired diagonal elements $d_{1},\dots ,d_{N}$ (because changing their order would change the Hermitian matrix whose existence is in question) but it does not depend on the order of the desired eigenvalues $\lambda _{1},\dots ,\lambda _{N}.$
On the right hand right hand side of the characterization, only the values of $\lambda _{1}+\cdots +\lambda _{n}$ depend on the assumption $\lambda _{1}\geq \cdots \geq \lambda _{N}.$ Notice that this assumption means that the expression $\lambda _{1}+\cdots +\lambda _{n}$ is just notation for the sum of the $n$ largest desired eigenvalues. Replacing the expression $\lambda _{1}+\cdots +\lambda _{n}$ with this written equivalent makes the assumption $\lambda _{1}\geq \cdots \geq \lambda _{N}$ completely unnecessary:
Schur–Horn theorem: Given any $N$ desired real eigenvalues and a non-increasing real sequence of desired diagonal elements $d_{1}\geq \cdots \geq d_{N},$ there exists a Hermitian matrix with these eigenvalues and diagonal elements if and only if these two sequences have the same sum and for every possible integer $n,$ the sum of the first $n$ desired diagonal elements never exceeds the sum of the $n$ largest desired eigenvalues.
Permutation polytope generated by a vector
The permutation polytope generated by ${\tilde {x}}=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n}$ denoted by ${\mathcal {K}}_{\tilde {x}}$ is defined as the convex hull of the set $\{(x_{\pi (1)},x_{\pi (2)},\ldots ,x_{\pi (n)})\in \mathbb {R} ^{n}:\pi \in S_{n}\}.$ Here $S_{n}$ denotes the symmetric group on $\{1,2,\ldots ,n\}.$ In other words, the permutation polytope generated by $(x_{1},\dots ,x_{n})$ is the convex hull of the set of all points in $\mathbb {R} ^{n}$ that can be obtained by rearranging the coordinates of $(x_{1},\dots ,x_{n}).$ The permutation polytope of $(1,1,2),$ for instance, is the convex hull of the set $\{(1,1,2),(1,2,1),(2,1,1)\},$ which in this case is the solid (filled) triangle whose vertices are the three points in this set. Notice, in particular, that rearranging the coordinates of $(x_{1},\dots ,x_{n})$ does not change the resulting permutation polytope; in other words, if a point ${\tilde {y}}$ can be obtained from ${\tilde {x}}=(x_{1},\dots ,x_{n})$ by rearranging its coordinates, then ${\mathcal {K}}_{\tilde {y}}={\mathcal {K}}_{\tilde {x}}.$
The following lemma characterizes the permutation polytope of a vector in $\mathbb {R} ^{n}.$
Lemma[1][2] — If $x_{1}\geq \cdots \geq x_{n},$ and $y_{1}\geq \cdots \geq y_{n},$ have the same sum $x_{1}+\cdots +x_{n}=y_{1}+\cdots +y_{n},$ then the following statements are equivalent:
1. ${\tilde {y}}:=(y_{1},\cdots ,y_{n})\in {\mathcal {K}}_{\tilde {x}}.$
2. $y_{1}\leq x_{1},$ and $y_{1}+y_{2}\leq x_{1}+x_{2},$ and $\ldots ,$ and $y_{1}+y_{2}+\cdots +y_{n-1}\leq x_{1}+x_{2}+\cdots +x_{n-1}$
3. There exist a sequence of points ${\tilde {x}}_{1},\dots ,{\tilde {x}}_{n}$ in ${\mathcal {K}}_{\tilde {x}},$ starting with ${\tilde {x}}_{1}={\tilde {x}}$ and ending with ${\tilde {x}}_{n}={\tilde {y}}$ such that ${\tilde {x}}_{k+1}=t{\tilde {x}}_{k}+(1-t)\tau ({\tilde {x_{k}}})$ for each $k$ in $\{1,2,\ldots ,n-1\},$ some transposition $\tau $ in $S_{n},$ and some $t$ in $[0,1],$ depending on $k.$
Reformulation of Schur–Horn theorem
In view of the equivalence of (i) and (ii) in the lemma mentioned above, one may reformulate the theorem in the following manner.
Theorem. Let $d_{1},\dots ,d_{N}$ and $\lambda _{1},\dots ,\lambda _{N}$ be real numbers. There is a Hermitian matrix with diagonal entries $d_{1},\dots ,d_{N}$ and eigenvalues $\lambda _{1},\dots ,\lambda _{N}$ if and only if the vector $(d_{1},\ldots ,d_{n})$ is in the permutation polytope generated by $(\lambda _{1},\ldots ,\lambda _{n}).$
Note that in this formulation, one does not need to impose any ordering on the entries of the vectors $d_{1},\dots ,d_{N}$ and $\lambda _{1},\dots ,\lambda _{N}.$
Proof of the Schur–Horn theorem
Let $A=(a_{jk})$ be a $n\times n$ Hermitian matrix with eigenvalues $\{\lambda _{i}\}_{i=1}^{n},$ counted with multiplicity. Denote the diagonal of $A$ by ${\tilde {a}},$ thought of as a vector in $\mathbb {R} ^{n},$ and the vector $(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})$ by ${\tilde {\lambda }}.$ Let $\Lambda $ be the diagonal matrix having $\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}$ on its diagonal.
($\Rightarrow $) $A$ may be written in the form $U\Lambda U^{-1},$ where $U$ is a unitary matrix. Then
$a_{ii}=\sum _{j=1}^{n}\lambda _{j}|u_{ij}|^{2},\;i=1,2,\ldots ,n.$
Let $S=(s_{ij})$ be the matrix defined by $s_{ij}=|u_{ij}|^{2}.$ Since $U$ is a unitary matrix, $S$ is a doubly stochastic matrix and we have ${\tilde {a}}=S{\tilde {\lambda }}.$ By the Birkhoff–von Neumann theorem, $S$ can be written as a convex combination of permutation matrices. Thus ${\tilde {a}}$ is in the permutation polytope generated by ${\tilde {\lambda }}.$ This proves Schur's theorem.
($\Leftarrow $) If ${\tilde {a}}$ occurs as the diagonal of a Hermitian matrix with eigenvalues $\{\lambda _{i}\}_{i=1}^{n},$ then $t{\tilde {a}}+(1-t)\tau ({\tilde {a}})$ also occurs as the diagonal of some Hermitian matrix with the same set of eigenvalues, for any transposition $\tau $ in $S_{n}.$ One may prove that in the following manner.
Let $\xi $ be a complex number of modulus $1$ such that ${\overline {\xi a_{jk}}}=-\xi a_{jk}$ and $U$ be a unitary matrix with $\xi {\sqrt {t}},{\sqrt {t}}$ in the $j,j$ and $k,k$ entries, respectively, $-{\sqrt {1-t}},\xi {\sqrt {1-t}}$ at the $j,k$ and $k,j$ entries, respectively, $1$ at all diagonal entries other than $j,j$ and $k,k,$ and $0$ at all other entries. Then $UAU^{-1}$ has $ta_{jj}+(1-t)a_{kk}$ at the $j,j$ entry, $(1-t)a_{jj}+ta_{kk}$ at the $k,k$ entry, and $a_{ll}$ at the $l,l$ entry where $l\neq j,k.$ Let $\tau $ be the transposition of $\{1,2,\dots ,n\}$ that interchanges $j$ and $k.$
Then the diagonal of $UAU^{-1}$ is $t{\tilde {a}}+(1-t)\tau ({\tilde {a}}).$
$\Lambda $ is a Hermitian matrix with eigenvalues $\{\lambda _{i}\}_{i=1}^{n}.$ Using the equivalence of (i) and (iii) in the lemma mentioned above, we see that any vector in the permutation polytope generated by $(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}),$ occurs as the diagonal of a Hermitian matrix with the prescribed eigenvalues. This proves Horn's theorem.
Symplectic geometry perspective
The Schur–Horn theorem may be viewed as a corollary of the Atiyah–Guillemin–Sternberg convexity theorem in the following manner. Let ${\mathcal {U}}(n)$ denote the group of $n\times n$ unitary matrices. Its Lie algebra, denoted by ${\mathfrak {u}}(n),$ is the set of skew-Hermitian matrices. One may identify the dual space ${\mathfrak {u}}(n)^{*}$ with the set of Hermitian matrices ${\mathcal {H}}(n)$ via the linear isomorphism $\Psi :{\mathcal {H}}(n)\rightarrow {\mathfrak {u}}(n)^{*}$ :{\mathcal {H}}(n)\rightarrow {\mathfrak {u}}(n)^{*}} defined by $\Psi (A)(B)=\mathrm {tr} (iAB)$ for $A\in {\mathcal {H}}(n),B\in {\mathfrak {u}}(n).$ The unitary group ${\mathcal {U}}(n)$ acts on ${\mathcal {H}}(n)$ by conjugation and acts on ${\mathfrak {u}}(n)^{*}$ by the coadjoint action. Under these actions, $\Psi $ is an ${\mathcal {U}}(n)$-equivariant map i.e. for every $U\in {\mathcal {U}}(n)$ the following diagram commutes,
Let ${\tilde {\lambda }}=(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})\in \mathbb {R} ^{n}$ and $\Lambda \in {\mathcal {H}}(n)$ denote the diagonal matrix with entries given by ${\tilde {\lambda }}.$ Let ${\mathcal {O}}_{\tilde {\lambda }}$ denote the orbit of $\Lambda $ under the ${\mathcal {U}}(n)$-action i.e. conjugation. Under the ${\mathcal {U}}(n)$-equivariant isomorphism $\Psi ,$ the symplectic structure on the corresponding coadjoint orbit may be brought onto ${\mathcal {O}}_{\tilde {\lambda }}.$ Thus ${\mathcal {O}}_{\tilde {\lambda }}$ is a Hamiltonian ${\mathcal {U}}(n)$-manifold.
Let $\mathbb {T} $ denote the Cartan subgroup of ${\mathcal {U}}(n)$ which consists of diagonal complex matrices with diagonal entries of modulus $1.$ The Lie algebra ${\mathfrak {t}}$ of $\mathbb {T} $ consists of diagonal skew-Hermitian matrices and the dual space ${\mathfrak {t}}^{*}$ consists of diagonal Hermitian matrices, under the isomorphism $\Psi .$ In other words, ${\mathfrak {t}}$ consists of diagonal matrices with purely imaginary entries and ${\mathfrak {t}}^{*}$ consists of diagonal matrices with real entries. The inclusion map ${\mathfrak {t}}\hookrightarrow {\mathfrak {u}}(n)$ induces a map $\Phi :{\mathcal {H}}(n)\cong {\mathfrak {u}}(n)^{*}\rightarrow {\mathfrak {t}}^{*},$ :{\mathcal {H}}(n)\cong {\mathfrak {u}}(n)^{*}\rightarrow {\mathfrak {t}}^{*},} which projects a matrix $A$ to the diagonal matrix with the same diagonal entries as $A.$ The set ${\mathcal {O}}_{\tilde {\lambda }}$ is a Hamiltonian $\mathbb {T} $-manifold, and the restriction of $\Phi $ to this set is a moment map for this action.
By the Atiyah–Guillemin–Sternberg theorem, $\Phi ({\mathcal {O}}_{\tilde {\lambda }})$ is a convex polytope. A matrix $A\in {\mathcal {H}}(n)$ is fixed under conjugation by every element of $\mathbb {T} $ if and only if $A$ is diagonal. The only diagonal matrices in ${\mathcal {O}}_{\tilde {\lambda }}$ are the ones with diagonal entries $\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}$ in some order. Thus, these matrices generate the convex polytope $\Phi ({\mathcal {O}}_{\tilde {\lambda }}).$ This is exactly the statement of the Schur–Horn theorem.
Notes
1. Kadison, R. V., Lemma 5, The Pythagorean Theorem: I. The finite case, Proc. Natl. Acad. Sci. USA, vol. 99 no. 7 (2002):4178–4184 (electronic)
2. Kadison, R. V.; Pedersen, G. K., Lemma 13, Means and Convex Combinations of Unitary Operators, Math. Scand. 57 (1985),249–266
References
• Schur, Issai, Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsber. Berl. Math. Ges. 22 (1923), 9–20.
• Horn, Alfred, Doubly stochastic matrices and the diagonal of a rotation matrix, American Journal of Mathematics 76 (1954), 620–630.
• Kadison, R. V.; Pedersen, G. K., Means and Convex Combinations of Unitary Operators, Math. Scand. 57 (1985),249–266.
• Kadison, R. V., The Pythagorean Theorem: I. The finite case, Proc. Natl. Acad. Sci. USA, vol. 99 no. 7 (2002):4178–4184 (electronic)
External links
• MathWorld
• Terry Tao: 254A, Notes 3a: Eigenvalues and sums of Hermitian matrices
• Sheela Devadas, Peter J. Haine, Keaton Stubis: The Schur-Horn Theorem
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Schur–Weyl duality
Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. It is named after two pioneers of representation theory of Lie groups, Issai Schur, who discovered the phenomenon, and Hermann Weyl, who popularized it in his books on quantum mechanics and classical groups as a way of classifying representations of unitary and general linear groups.
Schur–Weyl duality can be proven using the double centralizer theorem.[1]
Description
Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. Consider the tensor space
$\mathbb {C} ^{n}\otimes \mathbb {C} ^{n}\otimes \cdots \otimes \mathbb {C} ^{n}$ with k factors.
The symmetric group Sk on k letters acts on this space (on the left) by permuting the factors,
$\sigma (v_{1}\otimes v_{2}\otimes \cdots \otimes v_{k})=v_{\sigma ^{-1}(1)}\otimes v_{\sigma ^{-1}(2)}\otimes \cdots \otimes v_{\sigma ^{-1}(k)}.$
The general linear group GLn of invertible n×n matrices acts on it by the simultaneous matrix multiplication,
$g(v_{1}\otimes v_{2}\otimes \cdots \otimes v_{k})=gv_{1}\otimes gv_{2}\otimes \cdots \otimes gv_{k},\quad g\in {\text{GL}}_{n}.$
These two actions commute, and in its concrete form, the Schur–Weyl duality asserts that under the joint action of the groups Sk and GLn, the tensor space decomposes into a direct sum of tensor products of irreducible modules (for these two groups) that actually determine each other,
$\mathbb {C} ^{n}\otimes \mathbb {C} ^{n}\otimes \cdots \otimes \mathbb {C} ^{n}=\bigoplus _{D}\pi _{k}^{D}\otimes \rho _{n}^{D}.$
The summands are indexed by the Young diagrams D with k boxes and at most n rows, and representations $\pi _{k}^{D}$ of Sk with different D are mutually non-isomorphic, and the same is true for representations $\rho _{n}^{D}$ of GLn.
The abstract form of the Schur–Weyl duality asserts that two algebras of operators on the tensor space generated by the actions of GLn and Sk are the full mutual centralizers in the algebra of the endomorphisms $\mathrm {End} _{\mathbb {C} }(\mathbb {C} ^{n}\otimes \mathbb {C} ^{n}\otimes \cdots \otimes \mathbb {C} ^{n}).$
Example
Suppose that k = 2 and n is greater than one. Then the Schur–Weyl duality is the statement that the space of two-tensors decomposes into symmetric and antisymmetric parts, each of which is an irreducible module for GLn:
$\mathbb {C} ^{n}\otimes \mathbb {C} ^{n}=S^{2}\mathbb {C} ^{n}\oplus \Lambda ^{2}\mathbb {C} ^{n}.$
The symmetric group S2 consists of two elements and has two irreducible representations, the trivial representation and the sign representation. The trivial representation of S2 gives rise to the symmetric tensors, which are invariant (i.e. do not change) under the permutation of the factors, and the sign representation corresponds to the skew-symmetric tensors, which flip the sign.
Proof
First consider the following setup:
• G a finite group,
• $A=\mathbb {C} [G]$ the group algebra of G,
• $U$ a finite-dimensional right A-module, and
• $B=\operatorname {End} _{G}(U)$, which acts on U from the left and commutes with the right action of G (or of A). In other words, $B$ is the centralizer of $A$ in the endomorphism ring $\operatorname {End} (U)$.
The proof uses two algebraic lemmas.
Lemma 1 — [2] If $W$ is a simple left A-module, then $U\otimes _{A}W$ is a simple left B-module.
Proof: Since U is semisimple by Maschke's theorem, there is a decomposition $U=\bigoplus _{i}U_{i}^{\oplus m_{i}}$ into simple A-modules. Then $U\otimes _{A}W=\bigoplus _{i}(U_{i}\otimes _{A}W)^{\oplus m_{i}}$. Since A is the left regular representation of G, each simple G-module appears in A and we have that $U_{i}\otimes _{A}W=\mathbb {C} $ (respectively zero) if and only if $U_{i},W$ correspond to the same simple factor of A (respectively otherwise). Hence, we have: $U\otimes _{A}W=(U_{i_{0}}\otimes _{A}W)^{\oplus m_{i_{0}}}=\mathbb {C} ^{\oplus m_{i_{0}}}.$ Now, it is easy to see that each nonzero vector in $\mathbb {C} ^{\oplus m_{i_{0}}}$ generates the whole space as a B-module and so $U\otimes _{A}W$ is simple. (In general, a nonzero module is simple if and only if each of its nonzero cyclic submodule coincides with the module.) $\square $
Lemma 2 — [3] When $U=V^{\otimes d}$ and G is the symmetric group ${\mathfrak {S}}_{d}$, a subspace of $U$ is a B-submodule if and only if it is invariant under $\operatorname {GL} (V)$; in other words, a B-submodule is the same as a $\operatorname {GL} (V)$-submodule.
Proof: Let $W=\operatorname {End} (V)$. The $W\hookrightarrow \operatorname {End} (U),w\mapsto w^{d}=d!w\otimes \cdots \otimes w$. Also, the image of W spans the subspace of symmetric tensors $\operatorname {Sym} ^{d}(W)$. Since $B=\operatorname {Sym} ^{d}(W)$, the image of $W$ spans $B$. Since $\operatorname {GL} (V)$ is dense in W either in the Euclidean topology or in the Zariski topology, the assertion follows. $\square $
The Schur–Weyl duality now follows. We take $G={\mathfrak {S}}_{d}$ to be the symmetric group and $U=V^{\otimes d}$ the d-th tensor power of a finite-dimensional complex vector space V.
Let $V^{\lambda }$ denote the irreducible ${\mathfrak {S}}_{d}$-representation corresponding to a partition $\lambda $ and $m_{\lambda }=\dim V^{\lambda }$. Then by Lemma 1
$S^{\lambda }(V):=V^{\otimes d}\otimes _{{\mathfrak {S}}_{d}}V^{\lambda }$
is irreducible as a $\operatorname {GL} (V)$-module. Moreover, when $A=\bigoplus _{\lambda }(V^{\lambda })^{\oplus m_{\lambda }}$ is the left semisimple decomposition, we have:[4]
$V^{\otimes d}=V^{\otimes d}\otimes _{A}A=\bigoplus _{\lambda }(V^{\otimes d}\otimes _{{\mathfrak {S}}_{d}}V^{\lambda })^{\oplus m_{\lambda }}$,
which is the semisimple decomposition as a $\operatorname {GL} (V)$-module.
Generalizations
The Brauer algebra plays the role of the symmetric group in the generalization of the Schur-Weyl duality to the orthogonal and symplectic groups.
More generally, the partition algebra and its subalgebras give rise to a number of generalizations of the Schur-Weyl duality.
See also
• Partition algebra
Notes
1. Etingof, Pavel; Golberg, Oleg; Hensel, Sebastian; Liu, Tiankai; Schwendner, Alex; Vaintrob, Dmitry; Yudovina, Elena (2011), Introduction to representation theory. With historical interludes by Slava Gerovitch, Zbl 1242.20001, Theorem 5.18.4
2. Fulton & Harris, Lemma 6.22. harvnb error: no target: CITEREFFultonHarris (help)
3. Fulton & Harris, Lemma 6.23. harvnb error: no target: CITEREFFultonHarris (help)
4. Fulton & Harris, Theorem 6.3. (2), (4) harvnb error: no target: CITEREFFultonHarris (help)
References
• 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.
• Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. The Schur lectures (1992) (Tel Aviv), 1–182, Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995. MR1321638
• Issai Schur, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen. Dissertation. Berlin. 76 S (1901) JMF 32.0165.04
• Issai Schur, Über die rationalen Darstellungen der allgemeinen linearen Gruppe. Sitzungsberichte Akad. Berlin 1927, 58–75 (1927) JMF 53.0108.05
• Sengupta, Ambar N. (2012). "Chapter 10: Character Duality". Representing Finite Groups, A Semimsimple Introduction. Springer. ISBN 978-1-4614-1232-8. OCLC 875741967.
• Hermann Weyl, The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J., 1939. xii+302 pp. MR0000255
External links
• How to constructively/combinatorially prove Schur-Weyl duality?
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Wikipedia
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Schwarz–Christoffel mapping
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction. They were introduced independently by Elwin Christoffel in 1867 and Hermann Schwarz in 1869.
Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces, hyperbolic art, and fluid dynamics.
Definition
Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping f from the upper half-plane
$\{\zeta \in \mathbb {C} :\operatorname {Im} \zeta >0\}$ :\operatorname {Im} \zeta >0\}}
to the interior of the polygon. The function f maps the real axis to the edges of the polygon. If the polygon has interior angles $\alpha ,\beta ,\gamma ,\ldots $, then this mapping is given by
$f(\zeta )=\int ^{\zeta }{\frac {K}{(w-a)^{1-(\alpha /\pi )}(w-b)^{1-(\beta /\pi )}(w-c)^{1-(\gamma /\pi )}\cdots }}\,\mathrm {d} w$
where $K$ is a constant, and $a<b<c<\cdots $ are the values, along the real axis of the $\zeta $ plane, of points corresponding to the vertices of the polygon in the $z$ plane. A transformation of this form is called a Schwarz–Christoffel mapping.
The integral can be simplified by mapping the point at infinity of the $\zeta $ plane to one of the vertices of the $z$ plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant $K$. Conventionally, the point at infinity would be mapped to the vertex with angle $\alpha $.
In practice, to find a mapping to a specific polygon one needs to find the $a<b<c<\cdots $ values which generate the correct polygon side lengths. This requires solving a set of nonlinear equations, and in most cases can only be done numerically.[1]
Example
Consider a semi-infinite strip in the z plane. This may be regarded as a limiting form of a triangle with vertices P = 0, Q = π i, and R (with R real), as R tends to infinity. Now α = 0 and β = γ = π⁄2 in the limit. Suppose we are looking for the mapping f with f(−1) = Q, f(1) = P, and f(∞) = R. Then f is given by
$f(\zeta )=\int ^{\zeta }{\frac {K}{(w-1)^{1/2}(w+1)^{1/2}}}\,\mathrm {d} w.\,$
Evaluation of this integral yields
$z=f(\zeta )=C+K\operatorname {arcosh} \zeta ,$
where C is a (complex) constant of integration. Requiring that f(−1) = Q and f(1) = P gives C = 0 and K = 1. Hence the Schwarz–Christoffel mapping is given by
$z=\operatorname {arcosh} \zeta .$
This transformation is sketched below.
Other simple mappings
Triangle
A mapping to a plane triangle with interior angles $\pi a,\,\pi b$ and $\pi (1-a-b)$ is given by
$z=f(\zeta )=\int ^{\zeta }{\frac {dw}{(w-1)^{1-a}(w+1)^{1-b}}},$
which can be expressed in terms of hypergeometric functions.
Square
The upper half-plane is mapped to the square by
$z=f(\zeta )=\int ^{\zeta }{\frac {\mathrm {d} w}{\sqrt {w(1-w^{2})}}}={\sqrt {2}}\,F\left({\sqrt {\zeta +1}};{\sqrt {2}}/2\right),$
where F is the incomplete elliptic integral of the first kind.
General triangle
The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map.
See also
• The Schwarzian derivative appears in the theory of Schwarz–Christoffel mappings.
References
1. Driscoll, Toby. "Schwarz-Christoffel mapping". www.math.udel.edu. Retrieved 2021-05-17.{{cite web}}: CS1 maint: url-status (link)
• Christoffel, Elwin Bruno (1867). "Sul problema delle temperature stazionarie e la rappresentazione di una data superficie" [On the problem of stationary temperatures and the representation of a given surface]. Annali di Matematica Pura ed Applicata (in Italian). 1: 89–103. doi:10.1007/BF02419161. S2CID 121089696.
• Driscoll, Tobin A.; Trefethen, Lloyd N. (2002). Schwarz–Christoffel Mapping. Cambridge University Press. doi:10.1017/CBO9780511546808. ISBN 9780521807265.
• Schwarz, Hermann Amandus (1869). "Ueber einige Abbildungsaufgaben" [About some mapping problems]. Crelle's Journal (in German). 1869 (70): 105–120. doi:10.1515/crll.1869.70.105. S2CID 121291546.
• Forsyth, Andrew Russell (1918) [1st ed. 1893]. Theory of Functions of a Complex Variable. Cambridge. §§267–270, pp. 665–677.
• Nehari, Zeev (1982) [1952], Conformal mapping, New York: Dover Publications, ISBN 978-0-486-61137-2, MR 0045823
• The Conformal Hyperbolic Square and Its Ilk Chamberlain Fong, Bridges Finland Conference Proceedings, 2016
Further reading
An analogue of SC mapping that works also for multiply-connected is presented in: Case, James (2008), "Breakthrough in Conformal Mapping" (PDF), SIAM News, 41 (1).
External links
• "Schwarz–Christoffel transformation". PlanetMath.
• Schwarz–Christoffel toolbox (software for MATLAB)
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Wikipedia
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Schwartz–Zippel lemma
In mathematics, the Schwartz–Zippel lemma (also called the DeMillo–Lipton–Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determining whether a given multivariate polynomial is the 0-polynomial (or identically equal to 0). It was discovered independently by Jack Schwartz,[1] Richard Zippel,[2] and Richard DeMillo and Richard J. Lipton, although DeMillo and Lipton's version was shown a year prior to Schwartz and Zippel's result.[3] The finite field version of this bound was proved by Øystein Ore in 1922.[4]
Statement and proof of the lemma
Theorem 1 (Schwartz, Zippel). Let
$P\in F[x_{1},x_{2},\ldots ,x_{n}]$
be a non-zero polynomial of total degree d ≥ 0 over a field F. Let S be a finite subset of F and let r1, r2, ..., rn be selected at random independently and uniformly from S. Then
$\Pr[P(r_{1},r_{2},\ldots ,r_{n})=0]\leq {\frac {d}{|S|}}.$
Equivalently, the Lemma states that for any finite subset S of F, if Z(P) is the zero set of P, then
$|Z(P)\cap S^{n}|\leq d\cdot |S|^{n-1}.$
Proof. The proof is by mathematical induction on n. For n = 1, as was mentioned before, P can have at most d roots. This gives us the base case. Now, assume that the theorem holds for all polynomials in n − 1 variables. We can then consider P to be a polynomial in x1 by writing it as
$P(x_{1},\dots ,x_{n})=\sum _{i=0}^{d}x_{1}^{i}P_{i}(x_{2},\dots ,x_{n}).$
Since P is not identically 0, there is some i such that $P_{i}$ is not identically 0. Take the largest such i. Then $\deg P_{i}\leq d-i$, since the degree of $x_{1}^{i}P_{i}$ is at most d.
Now we randomly pick $r_{2},\dots ,r_{n}$ from S. By the induction hypothesis, $\Pr[P_{i}(r_{2},\ldots ,r_{n})=0]\leq {\frac {d-i}{|S|}}.$
If $P_{i}(r_{2},\ldots ,r_{n})\neq 0$, then $P(x_{1},r_{2},\ldots ,r_{n})$ is of degree i (and thus not identically zero) so
$\Pr[P(r_{1},r_{2},\ldots ,r_{n})=0|P_{i}(r_{2},\ldots ,r_{n})\neq 0]\leq {\frac {i}{|S|}}.$
If we denote the event $P(r_{1},r_{2},\ldots ,r_{n})=0$ by A, the event $P_{i}(r_{2},\ldots ,r_{n})=0$ by B, and the complement of B by $B^{c}$, we have
${\begin{aligned}\Pr[A]&=\Pr[A\cap B]+\Pr[A\cap B^{c}]\\&=\Pr[B]\Pr[A|B]+\Pr[B^{c}]\Pr[A|B^{c}]\\&\leq \Pr[B]+\Pr[A|B^{c}]\\&\leq {\frac {d-i}{|S|}}+{\frac {i}{|S|}}={\frac {d}{|S|}}\end{aligned}}$
Applications
The importance of the Schwartz–Zippel Theorem and Testing Polynomial Identities follows from algorithms which are obtained to problems that can be reduced to the problem of polynomial identity testing.
Zero testing
For example, is
$(x_{1}+3x_{2}-x_{3})(3x_{1}+x_{4}-1)\cdots (x_{7}-x_{2})\equiv 0\ ?$ ?}
To solve this, we can multiply it out and check that all the coefficients are 0. However, this takes exponential time. In general, a polynomial can be algebraically represented by an arithmetic formula or circuit.
Comparison of two polynomials
Given a pair of polynomials $p_{1}(x)$ and $p_{2}(x)$, is
$p_{1}(x)\equiv p_{2}(x)$?
This problem can be solved by reducing it to the problem of polynomial identity testing. It is equivalent to checking if
$[p_{1}(x)-p_{2}(x)]\equiv 0.$
Hence if we can determine that
$p(x)\equiv 0,$
where
$p(x)=p_{1}(x)\;-\;p_{2}(x),$
then we can determine whether the two polynomials are equivalent.
Comparison of polynomials has applications for branching programs (also called binary decision diagrams). A read-once branching program can be represented by a multilinear polynomial which computes (over any field) on {0,1}-inputs the same Boolean function as the branching program, and two branching programs compute the same function if and only if the corresponding polynomials are equal. Thus, identity of Boolean functions computed by read-once branching programs can be reduced to polynomial identity testing.
Comparison of two polynomials (and therefore testing polynomial identities) also has applications in 2D-compression, where the problem of finding the equality of two 2D-texts A and B is reduced to the problem of comparing equality of two polynomials $p_{A}(x,y)$ and $p_{B}(x,y)$.
Primality testing
Given $n\in \mathbb {N} $, is $n$ a prime number?
A simple randomized algorithm developed by Manindra Agrawal and Somenath Biswas can determine probabilistically whether $n$ is prime and uses polynomial identity testing to do so.
They propose that all prime numbers n (and only prime numbers) satisfy the following polynomial identity:
$(1+z)^{n}=1+z^{n}({\mbox{mod}}\;n).$
This is a consequence of the Frobenius endomorphism.
Let
${\mathcal {P}}_{n}(z)=(1+z)^{n}-1-z^{n}.$
Then ${\mathcal {P}}_{n}(z)=0\;({\mbox{mod}}\;n)$ iff n is prime. The proof can be found in [4]. However, since this polynomial has degree $n$, where $n$ may or may not be a prime, the Schwartz–Zippel method would not work. Agrawal and Biswas use a more sophisticated technique, which divides ${\mathcal {P}}_{n}$ by a random monic polynomial of small degree.
Prime numbers are used in a number of applications such as hash table sizing, pseudorandom number generators and in key generation for cryptography. Therefore, finding very large prime numbers (on the order of (at least) $10^{350}\approx 2^{1024}$) becomes very important and efficient primality testing algorithms are required.
Perfect matching
Let $G=(V,E)$ be a graph of n vertices where n is even. Does G contain a perfect matching?
Theorem 2 (Tutte 1947): A Tutte matrix determinant is not a 0-polynomial if and only if there exists a perfect matching.
A subset D of E is called a matching if each vertex in V is incident with at most one edge in D. A matching is perfect if each vertex in V has exactly one edge that is incident to it in D. Create a Tutte matrix A in the following way:
$A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1{\mathit {n}}}\\a_{21}&a_{22}&\cdots &a_{2{\mathit {n}}}\\\vdots &\vdots &\ddots &\vdots \\a_{{\mathit {n}}1}&a_{{\mathit {n}}2}&\ldots &a_{\mathit {nn}}\end{bmatrix}}$
where
$a_{ij}={\begin{cases}x_{ij}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i<j\\-x_{ji}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i>j\\0\;\;\;\;{\mbox{otherwise}}.\end{cases}}$
The Tutte matrix determinant (in the variables xij, $i<j$ ) is then defined as the determinant of this skew-symmetric matrix which coincides with the square of the pfaffian of the matrix A and is non-zero (as polynomial) if and only if a perfect matching exists. One can then use polynomial identity testing to find whether G contains a perfect matching. There exists a deterministic black-box algorithm for graphs with polynomially bounded permanents (Grigoriev & Karpinski 1987).[5]
In the special case of a balanced bipartite graph on $n=m+m$ vertices this matrix takes the form of a block matrix
$A={\begin{pmatrix}0&X\\-X^{t}&0\end{pmatrix}}$
if the first m rows (resp. columns) are indexed with the first subset of the bipartition and the last m rows with the complementary subset. In this case the pfaffian coincides with the usual determinant of the m × m matrix X (up to sign). Here X is the Edmonds matrix.
Determinant of a matrix with polynomial entries
Let
$p(x_{1},x_{2},\ldots ,x_{n})$
be the determinant of the polynomial matrix.
Currently, there is no known sub-exponential time algorithm that can solve this problem deterministically. However, there are randomized polynomial algorithms whose analysis requires a bound on the probability that a non-zero polynomial will have roots at randomly selected test points.
Notes
1. Schwartz 1980.
2. Zippel 1979.
3. DeMillo & Lipton 1978.
4. Ö. Ore, Über höhere Kongruenzen. Norsk Mat. Forenings Skrifter Ser. I (1922), no. 7, 15 pages.
5. Grigoriev & Karpinski 1987.
References
• Agrawal, Manindra; Biswas, Somenath (2003-02-21). "Primality and Identity Testing via Chinese Remaindering". Journal of the ACM. 50 (4): 429–443. doi:10.1145/792538.792540. S2CID 13145079. Retrieved 2008-06-15.
• Berman, Piotr; Karpinski, Marek; Larmore, Lawrence L.; Plandowski, Wojciech; Rytter, Wojciech (2002). "On the Complexity of Pattern Matching for Highly Compressed Two-Dimensional Texts" (ps). Journal of Computer and System Sciences. 65 (2): 332–350. doi:10.1006/jcss.2002.1852. Retrieved 2008-06-15.
• Grigoriev, Dima; Karpinski, Marek (1987). "The matching problem for bipartite graphs with polynomially bounded permanents is in NC". Proceedings of the 28th Annual Symposium on Foundations of Computer Science (FOCS 1987), Los Angeles, California, USA, 27-29 October 1987. IEEE Computer Society. pp. 166–172. doi:10.1109/SFCS.1987.56. ISBN 978-0-8186-0807-0. S2CID 14476361.
• Moshkovitz, Dana (2010). An Alternative Proof of The Schwartz-Zippel Lemma. ECCC TR10-096
• DeMillo, Richard A.; Lipton, Richard J. (1978). "A probabilistic remark on algebraic program testing". Information Processing Letters. 7 (4): 193–195. doi:10.1016/0020-0190(78)90067-4.
• Rudich, Steven (2004). AMS (ed.). Computational Complexity Theory. IAS/Park City Mathematics Series. Vol. 10. ISBN 978-0-8218-2872-4.
• Schwartz, Jacob T. (October 1980). "Fast probabilistic algorithms for verification of polynomial identities" (PDF). Journal of the ACM. 27 (4): 701–717. CiteSeerX 10.1.1.391.1254. doi:10.1145/322217.322225. S2CID 8314102. Retrieved 2008-06-15.
• Tutte, W.T. (April 1947). "The factorization of linear graphs". J. London Math. Soc. 22 (2): 107–111. doi:10.1112/jlms/s1-22.2.107. hdl:10338.dmlcz/128215.
• Zippel, Richard (1979). "Probabilistic algorithms for sparse polynomials". In Ng, Edward W. (ed.). Symbolic and Algebraic Computation, EUROSAM '79, An International Symposiumon Symbolic and Algebraic Computation, Marseille, France, June 1979, Proceedings. Lecture Notes in Computer Science. Vol. 72. Springer. pp. 216–226. doi:10.1007/3-540-09519-5_73. ISBN 978-3-540-09519-4.
• Zippel, Richard (February 1989). "An Explicit Separation of Relativised Random Polynomial Time and Relativised Deterministic Polynomial Time" (ps). Retrieved 2008-06-15.
• Zippel, Richard (1993). Springer (ed.). Effective Polynomial Computation. The Springer International Series in Engineering and Computer Science. Vol. 241. ISBN 978-0-7923-9375-7.
External links
• The Curious History of the Schwartz–Zippel Lemma, by Richard J. Lipton
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Wikipedia
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Schwartz space
In mathematics, Schwartz space ${\mathcal {S}}$ is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space ${\mathcal {S}}^{*}$ of ${\mathcal {S}}$, that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function.
For the Schwartz space of a semisimple Lie group, see Harish-Chandra's Schwartz space. For the Schwartz space of a locally compact abelian group, see Schwartz–Bruhat function.
Schwartz space is named after French mathematician Laurent Schwartz.
Definition
Let $\mathbb {N} $ be the set of non-negative integers, and for any $n\in \mathbb {N} $, let $\mathbb {N} ^{n}:=\underbrace {\mathbb {N} \times \dots \times \mathbb {N} } _{n{\text{ times}}}$ be the n-fold Cartesian product. The Schwartz space or space of rapidly decreasing functions on $\mathbb {R} ^{n}$ is the function space
$S\left(\mathbb {R} ^{n},\mathbb {C} \right):=\left\{f\in C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )\mid \forall \alpha ,\beta \in \mathbb {N} ^{n},\|f\|_{\alpha ,\beta }<\infty \right\},$
where $C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )$ is the function space of smooth functions from $\mathbb {R} ^{n}$ into $\mathbb {C} $, and
$\|f\|_{\alpha ,\beta }:=\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }(D^{\beta }f)(x)\right|.$
Here, $\sup $ denotes the supremum, and we used multi-index notation, i.e. $x^{\alpha }:=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}}$ and $D^{\beta }:=\partial _{1}^{\beta _{1}}\partial _{2}^{\beta _{2}}\ldots \partial _{n}^{\beta _{n}}$.
To put common language to this definition, one could consider a rapidly decreasing function as essentially a function f(x) such that f(x), f ′(x), f ′′(x), ... all exist everywhere on R and go to zero as x→ ±∞ faster than any reciprocal power of x. In particular, S(Rn, C) is a subspace of the function space C∞(Rn, C) of smooth functions from Rn into C.
Examples of functions in the Schwartz space
• If α is a multi-index, and a is a positive real number, then
$x^{\alpha }e^{-a|x|^{2}}\in {\mathcal {S}}(\mathbf {R} ^{n}).$
• Any smooth function f with compact support is in S(Rn). This is clear since any derivative of f is continuous and supported in the support of f, so (xαDβ) f has a maximum in Rn by the extreme value theorem.
• Because the Schwartz space is a vector space, any polynomial $\phi (x^{\alpha })$ can by multiplied by a factor $e^{-ax^{2}}$ for $a>0$ a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials inside a Schwartz space.
Properties
Analytic properties
• From Leibniz's rule, it follows that 𝒮(Rn) is also closed under pointwise multiplication:
If f, g ∈ 𝒮(Rn) then the product fg ∈ 𝒮(Rn).
• The Fourier transform is a linear isomorphism F:𝒮(Rn) → 𝒮(Rn).
• If f ∈ 𝒮(R) then f is uniformly continuous on R.
• 𝒮(Rn) is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers.
• Both 𝒮(Rn) and its strong dual space are also:
1. complete Hausdorff locally convex spaces,
2. nuclear Montel spaces,
It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology,[1]
1. Ultrabornological spaces,
2. reflexive barrelled Mackey spaces.
Relation of Schwartz spaces with other topological vector spaces
• If 1 ≤ p ≤ ∞, then 𝒮(Rn) ⊂ Lp(Rn).
• If 1 ≤ p < ∞, then 𝒮(Rn) is dense in Lp(Rn).
• The space of all bump functions, C∞
c
(Rn)
, is included in 𝒮(Rn).
See also
• Bump function
• Schwartz–Bruhat function
• Nuclear space
References
1. Trèves 2006, pp. 351–359.
Sources
• Hörmander, L. (1990). The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis) (2nd ed.). Berlin: Springer-Verlag. ISBN 3-540-52343-X.
• Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press. ISBN 0-12-585050-6.
• Stein, Elias M.; Shakarchi, Rami (2003). Fourier Analysis: An Introduction (Princeton Lectures in Analysis I). Princeton: Princeton University Press. ISBN 0-691-11384-X.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
This article incorporates material from Space of rapidly decreasing functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
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Wikipedia
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Schwartz kernel theorem
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space ${\mathcal {D}}$ of test functions. The space ${\mathcal {D}}$ itself consists of smooth functions of compact support.
Statement of the theorem
Let $X$ and $Y$ be open sets in $\mathbb {R} ^{n}$. Every distribution $k\in {\mathcal {D}}'(X\times Y)$ defines a continuous linear map $K\colon {\mathcal {D}}(Y)\to {\mathcal {D}}'(X)$ such that
$\left\langle k,u\otimes v\right\rangle =\left\langle Kv,u\right\rangle $
(1)
for every $u\in {\mathcal {D}}(X),v\in {\mathcal {D}}(Y)$. Conversely, for every such continuous linear map $K$ there exists one and only one distribution $k\in {\mathcal {D}}'(X\times Y)$ such that (1) holds. The distribution $k$ is the kernel of the map $K$.
Note
Given a distribution $k\in {\mathcal {D}}'(X\times Y)$ one can always write the linear map K informally as
$Kv=\int _{Y}k(\cdot ,y)v(y)dy$
so that
$\langle Kv,u\rangle =\int _{X}\int _{Y}k(x,y)v(y)u(x)dydx$.
Integral kernels
The traditional kernel functions $K(x,y)$ of two variables of the theory of integral operators having been expanded in scope to include their generalized function analogues, which are allowed to be more singular in a serious way, a large class of operators from ${\mathcal {D}}$ to its dual space ${\mathcal {D}}'$ of distributions can be constructed. The point of the theorem is to assert that the extended class of operators can be characterised abstractly, as containing all operators subject to a minimum continuity condition. A bilinear form on ${\mathcal {D}}$ arises by pairing the image distribution with a test function.
A simple example is that the natural embedding of the test function space ${\mathcal {D}}$ into ${\mathcal {D}}'$ - sending every test function $f$ into the corresponding distribution $[f]$ - corresponds to the delta distribution
$\delta (x-y)$
concentrated at the diagonal of the underlined Euclidean space, in terms of the Dirac delta function $\delta $. While this is at most an observation, it shows how the distribution theory adds to the scope. Integral operators are not so 'singular'; another way to put it is that for $K$ a continuous kernel, only compact operators are created on a space such as the continuous functions on $[0,1]$. The operator $I$ is far from compact, and its kernel is intuitively speaking approximated by functions on $[0,1]\times [0,1]$ with a spike along the diagonal $x=y$ and vanishing elsewhere.
This result implies that the formation of distributions has a major property of 'closure' within the traditional domain of functional analysis. It was interpreted (comment of Jean Dieudonné) as a strong verification of the suitability of the Schwartz theory of distributions to mathematical analysis more widely seen. In his Éléments d'analyse volume 7, p. 3 he notes that the theorem includes differential operators on the same footing as integral operators, and concludes that it is perhaps the most important modern result of functional analysis. He goes on immediately to qualify that statement, saying that the setting is too 'vast' for differential operators, because of the property of monotonicity with respect to the support of a function, which is evident for differentiation. Even monotonicity with respect to singular support is not characteristic of the general case; its consideration leads in the direction of the contemporary theory of pseudo-differential operators.
Smooth manifolds
Dieudonné proves a version of the Schwartz result valid for smooth manifolds, and additional supporting results, in sections 23.9 to 23.12 of that book.
Generalization to nuclear spaces
Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in Grothendieck 1955. We have the following generalization of the theorem.
Schwartz kernel theorem:[1] Suppose that X is nuclear, Y is locally convex, and v is a continuous bilinear form on $X\times Y$. Then v originates from a space of the form $X_{A^{\prime }}^{\prime }{\widehat {\otimes }}_{\epsilon }Y_{B^{\prime }}^{\prime }$ where $A^{\prime }$ and $B^{\prime }$ are suitable equicontinuous subsets of $X^{\prime }$ and $Y^{\prime }$. Equivalently, v is of the form,
$v(x,y)=\sum _{i=1}^{\infty }\lambda _{i}\left\langle x,x_{i}^{\prime }\right\rangle \left\langle y,y_{i}^{\prime }\right\rangle $ for all $(x,y)\in X\times Y$
where $\left(\lambda _{i}\right)\in l^{1}$ and each of $\{x_{1}^{\prime },x_{2}^{\prime },\ldots \}$ and $\{y_{1}^{\prime },y_{2}^{\prime },\ldots \}$ are equicontinuous. Furthermore, these sequences can be taken to be null sequences (i.e. converging to 0) in $X_{A^{\prime }}^{\prime }$ and $Y_{B^{\prime }}^{\prime }$, respectively.
See also
• Fredholm kernel – type of a kernel on a Banach spacePages displaying wikidata descriptions as a fallback
• Injective tensor product
• Nuclear operator
• Nuclear space – A generalization of finite dimensional Euclidean spaces different from Hilbert spaces
• Projective tensor product – tensor product defined on two topological vector spacesPages displaying wikidata descriptions as a fallback
• Rigged Hilbert space – Construction linking the study of "bound" and continuous eigenvalues in functional analysis
• Trace class – Compact operator for which a finite trace can be defined
References
1. Schaefer & Wolff 1999, p. 172.
Bibliography
• Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
• Hörmander, L. (1983). The analysis of linear partial differential operators I. Grundl. Math. Wissenschaft. Vol. 256. Springer. doi:10.1007/978-3-642-96750-4. ISBN 3-540-12104-8. MR 0717035..
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.
External links
• G. L. Litvinov (2001) [1994], "Nuclear bilinear form", Encyclopedia of Mathematics, EMS Press
Topological tensor products and nuclear spaces
Basic concepts
• Auxiliary normed spaces
• Nuclear space
• Tensor product
• Topological tensor product
• of Hilbert spaces
Topologies
• Inductive tensor product
• Injective tensor product
• Projective tensor product
Operators/Maps
• Fredholm determinant
• Fredholm kernel
• Hilbert–Schmidt operator
• Hypocontinuity
• Integral
• Nuclear
• between Banach spaces
• Trace class
Theorems
• Grothendieck trace theorem
• Schwartz kernel theorem
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
|
Wikipedia
|
Schwarz's list
In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by Hermann Schwarz (1873, p. 323) when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the cases in which the hypergeometric equation has a finite monodromy group, or equivalently has two independent solutions that are algebraic functions. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of a cyclic group), and was first derived by Schwarz by methods of complex analytic geometry. Correspondingly the statement is not directly in terms of the parameters specifying the hypergeometric equation, but in terms of quantities used to describe certain spherical triangles.
The wider importance of the table, for general second-order differential equations in the complex plane, was shown by Felix Klein, who proved a result to the effect that cases of finite monodromy for such equations and regular singularities could be attributed to changes of variable (complex analytic mappings of the Riemann sphere to itself) that reduce the equation to hypergeometric form. In fact more is true: Schwarz's list underlies all second-order equations with regular singularities on compact Riemann surfaces having finite monodromy, by a pullback from the hypergeometric equation on the Riemann sphere by a complex analytic mapping, of degree computable from the equation's data.[1][2]
Number $\lambda $ $\mu $ $\nu $ area/$\pi $ polyhedron
1 1/2 1/2 p/n (≤ 1/2) p/n Dihedral
2 1/2 1/3 1/3 1/6 Tetrahedral
3 2/3 1/3 1/3 2/6 Tetrahedral
4 1/2 1/3 1/4 1/12 Cube/octahedron
5 2/3 1/4 1/4 2/12 Cube/octahedron
6 1/2 1/3 1/5 1/30 Icosahedron/Dodecahedron
7 2/5 1/3 1/3 2/30 Icosahedron/Dodecahedron
8 2/3 1/5 1/5 2/30 Icosahedron/Dodecahedron
9 1/2 2/5 1/5 3/30 Icosahedron/Dodecahedron
10 3/5 1/3 1/5 4/30 Icosahedron/Dodecahedron
11 2/5 2/5 2/5 6/30 Icosahedron/Dodecahedron
12 2/3 1/3 1/5 6/30 Icosahedron/Dodecahedron
13 4/5 1/5 1/5 6/30 Icosahedron/Dodecahedron
14 1/2 2/5 1/3 7/30 Icosahedron/Dodecahedron
15 3/5 2/5 1/3 10/30 Icosahedron/Dodecahedron
The numbers $\lambda ,\mu ,\nu $ are (up to permutations, sign changes and addition of $(\ell ,m,n)\in \mathbb {Z} ^{3}$ with $\ell +m+n$ even) the differences $1-c,c-a-b,b-a$ of the exponents of the hypergeometric differential equation at the three singular points $0,1,\infty $. They are rational numbers if and only if $a,b$ and $c$ are, a point that matters in arithmetic rather than geometric approaches to the theory.
Further work
An extension of Schwarz's results was given by T. Kimura, who dealt with cases where the identity component of the differential Galois group of the hypergeometric equation is a solvable group.[3][4] A general result connecting the differential Galois group G and the monodromy group Γ states that G is the Zariski closure of Γ — this theorem is attributed in the book of Matsuda to Michio Kuga. By general differential Galois theory, the resulting Kimura-Schwarz table classifies cases of integrability of the equation by algebraic functions and quadratures.
Another relevant list is that of K. Takeuchi, who classified the (hyperbolic) triangle groups that are arithmetic groups (85 examples).[5]
Émile Picard sought to extend the work of Schwarz in complex geometry, by means of a generalized hypergeometric function, to construct cases of equations where the monodromy was a discrete group in the projective unitary group PU(1, n). Pierre Deligne and George Mostow used his ideas to construct lattices in the projective unitary group. This work recovers in the classical case the finiteness of Takeuchi's list, and by means of a characterisation of the lattices they construct that are arithmetic groups, provided new examples of non-arithmetic lattices in PU(1, n).[6]
Baldassari applied the Klein universality, to discuss algebraic solutions of the Lamé equation by means of the Schwarz list.[7]
Other hypergeometric functions which can be expressed algebraically, like those on Schwarz's list, arise in theoretical physics in the context of $T{\overline {T}}$ deformations of two-dimensional gauge theories.[8]
See also
• Schwarz triangle
Notes
1. A modern treatment is in Baldassarri, F.; Dwork, B. (1979). "On second order linear differential equations with algebraic solutions". American Journal of Mathematics. 101 (1): 42–76. doi:10.2307/2373938. JSTOR 2373938. MR 0527825..
2. Baldassari, F. (1986–1987). "Towards a Schwarz list for Lamé's differential operators via division points on elliptic curves". Groupe de travail d'analyse ultramétrique. 14: Exposé no. 22, 17 pp.; see pp. 5-6.
3. Kimura, Tosihusa (1969–1970). "On Riemann's equations which are solvable by quadratures" (PDF). Funkcialaj Ekvacioj. 12: 269–281. MR 0277789.
4. Morales-Ruiz, Juan J.; Ramis, Jean Pierre (2001). "A note on the non-integrability of some Hamiltonian systems with a homogeneous potential". Methods and Applications of Analysis. 8 (1): 113–120. doi:10.4310/MAA.2001.v8.n1.a5. MR 1867496.; see p. 116 for the formulation.
5. Takeuchi, Kisao (1977). "Arithmetic triangle groups". Journal of the Mathematical Society of Japan. 29 (1): 91–106. doi:10.2969/jmsj/02910091. MR 0429744.
6. Deligne, Pierre; Mostow, G. D. (1986). "Monodromy of hypergeometric functions and non-lattice integral monodromy" (PDF). Publications mathématiques de l'IHÉS. 63: 5–89. doi:10.1007/BF02831622. S2CID 121385846.
7. F. Baldassarri, On algebraic solutions of Lamé’s differential equation, J. Differential Equations 41 (1) (1981) 44–58. Correction in Algebraic Solutions of the Lamé Equation, Revisited (PDF), by Robert S. Maier.
8. Brennan, T. Daniel; Ferko, Christian; Sethi, Savdeep (2020). "A non-abelian analogue of DBI from TT". SciPost Physics. 8 (4): 052. arXiv:1912.12389. doi:10.21468/SciPostPhys.8.4.052. S2CID 209515455.
References
• Matsuda, Michihiko (1985). Lectures on algebraic solutions of hypergeometric differential equations (PDF). Lectures in Mathematics. Vol. 15. Tokyo: Kinokuniya Company Ltd. MR1104881.
• Schwarz, H. A. (1873). "Ueber diejenigen Fälle in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt". Journal für die reine und angewandte Mathematik. 75: 292–335. ISSN 0075-4102.
External links
• Towards a nonlinear Schwarz's list (PDF)
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Wikipedia
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Schwartz topological vector space
In functional analysis and related areas of mathematics, Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets. These spaces were introduced by Alexander Grothendieck.
Definition
A Hausdorff locally convex space X with continuous dual $X^{\prime }$, X is called a Schwartz space if it satisfies any of the following equivalent conditions:[1]
1. For every closed convex balanced neighborhood U of the origin in X, there exists a neighborhood V of 0 in X such that for all real r > 0, V can be covered by finitely many translates of rU.
2. Every bounded subset of X is totally bounded and for every closed convex balanced neighborhood U of the origin in X, there exists a neighborhood V of 0 in X such that for all real r > 0, there exists a bounded subset B of X such that V ⊆ B + rU.
Properties
Every quasi-complete Schwartz space is a semi-Montel space. Every Fréchet Schwartz space is a Montel space.[2]
The strong dual space of a complete Schwartz space is an ultrabornological space.
Examples and sufficient conditions
• Vector subspace of Schwartz spaces are Schwartz spaces.
• The quotient of a Schwartz space by a closed vector subspace is again a Schwartz space.
• The Cartesian product of any family of Schwartz spaces is again a Schwartz space.
• The weak topology induced on a vector space by a family of linear maps valued in Schwartz spaces is a Schwartz space if the weak topology is Hausdorff.
• The locally convex strict inductive limit of any countable sequence of Schwartz spaces (with each space TVS-embedded in the next space) is again a Schwartz space.
Counter-examples
Every infinite-dimensional normed space is not a Schwartz space.[2]
There exist Fréchet spaces that are not Schwartz spaces and there exist Schwartz spaces that are not Montel spaces.[2]
See also
• Auxiliary normed space
• Montel space – Barrelled space where closed and bounded subsets are compact
References
1. Khaleelulla 1982, p. 32.
2. Khaleelulla 1982, pp. 32–63.
Bibliography
• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). doi:10.5802/aif.16. MR 0042609.
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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Schwartz–Bruhat function
In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the space of Schwartz–Bruhat functions.
Definitions
• On a real vector space $\mathbb {R} ^{n}$, the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space ${\mathcal {S}}(\mathbb {R} ^{n})$.
• On a torus, the Schwartz–Bruhat functions are the smooth functions.
• On a sum of copies of the integers, the Schwartz–Bruhat functions are the rapidly decreasing functions.
• On an elementary group (i.e., an abelian locally compact group that is a product of copies of the reals, the integers, the circle group, and finite groups), the Schwartz–Bruhat functions are the smooth functions all of whose derivatives are rapidly decreasing.[1]
• On a general locally compact abelian group $G$, let $A$ be a compactly generated subgroup, and $B$ a compact subgroup of $A$ such that $A/B$ is elementary. Then the pullback of a Schwartz–Bruhat function on $A/B$ is a Schwartz–Bruhat function on $G$, and all Schwartz–Bruhat functions on $G$ are obtained like this for suitable $A$ and $B$. (The space of Schwartz–Bruhat functions on $G$ is endowed with the inductive limit topology.)
• On a non-archimedean local field $K$, a Schwartz–Bruhat function is a locally constant function of compact support.
• In particular, on the ring of adeles $\mathbb {A} _{K}$ over a global field $K$, the Schwartz–Bruhat functions $f$ are finite linear combinations of the products $\prod _{v}f_{v}$ over each place $v$ of $K$, where each $f_{v}$ is a Schwartz–Bruhat function on a local field $K_{v}$ and $f_{v}=\mathbf {1} _{{\mathcal {O}}_{v}}$ is the characteristic function on the ring of integers ${\mathcal {O}}_{v}$ for all but finitely many $v$. (For the archimedean places of $K$, the $f_{v}$ are just the usual Schwartz functions on $\mathbb {R} ^{n}$, while for the non-archimedean places the $f_{v}$ are the Schwartz–Bruhat functions of non-archimedean local fields.)
• The space of Schwartz–Bruhat functions on the adeles $\mathbb {A} _{K}$ is defined to be the restricted tensor product[2] $\bigotimes _{v}'{\mathcal {S}}(K_{v}):=\varinjlim _{E}\left(\bigotimes _{v\in E}{\mathcal {S}}(K_{v})\right)$ of Schwartz–Bruhat spaces ${\mathcal {S}}(K_{v})$ of local fields, where $E$ is a finite set of places of $K$. The elements of this space are of the form $f=\otimes _{v}f_{v}$, where $f_{v}\in {\mathcal {S}}(K_{v})$ for all $v$ and $f_{v}|_{{\mathcal {O}}_{v}}=1$ for all but finitely many $v$. For each $x=(x_{v})_{v}\in \mathbb {A} _{K}$ we can write $f(x)=\prod _{v}f_{v}(x_{v})$, which is finite and thus is well defined.[3]
Examples
• Every Schwartz–Bruhat function $f\in {\mathcal {S}}(\mathbb {Q} _{p})$ can be written as $f=\sum _{i=1}^{n}c_{i}\mathbf {1} _{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}$, where each $a_{i}\in \mathbb {Q} _{p}$, $k_{i}\in \mathbb {Z} $, and $c_{i}\in \mathbb {C} $.[4] This can be seen by observing that $\mathbb {Q} _{p}$ being a local field implies that $f$ by definition has compact support, i.e., $\operatorname {supp} (f)$ has a finite subcover. Since every open set in $\mathbb {Q} _{p}$ can be expressed as a disjoint union of open balls of the form $a+p^{k}\mathbb {Z} _{p}$ (for some $a\in \mathbb {Q} _{p}$ and $k\in \mathbb {Z} $) we have
$\operatorname {supp} (f)=\coprod _{i=1}^{n}(a_{i}+p^{k_{i}}\mathbb {Z} _{p})$. The function $f$ must also be locally constant, so $f|_{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}=c_{i}\mathbf {1} _{a_{i}+p^{k_{i}}\mathbb {Z} _{p}}$ for some $c_{i}\in \mathbb {C} $. (As for $f$ evaluated at zero, $f(0)\mathbf {1} _{\mathbb {Z} _{p}}$ is always included as a term.)
• On the rational adeles $\mathbb {A} _{\mathbb {Q} }$ all functions in the Schwartz–Bruhat space ${\mathcal {S}}(\mathbb {A} _{\mathbb {Q} })$ are finite linear combinations of $\prod _{p\leq \infty }f_{p}=f_{\infty }\times \prod _{p<\infty }f_{p}$ over all rational primes $p$, where $f_{\infty }\in {\mathcal {S}}(\mathbb {R} )$, $f_{p}\in {\mathcal {S}}(\mathbb {Q} _{p})$, and $f_{p}=\mathbf {1} _{\mathbb {Z} _{p}}$ for all but finitely many $p$. The sets $\mathbb {Q} _{p}$ and $\mathbb {Z} _{p}$ are the field of p-adic numbers and ring of p-adic integers respectively.
Properties
The Fourier transform of a Schwartz–Bruhat function on a locally compact abelian group is a Schwartz–Bruhat function on the Pontryagin dual group. Consequently, the Fourier transform takes tempered distributions on such a group to tempered distributions on the dual group. Given the (additive) Haar measure on $\mathbb {A} _{K}$ the Schwartz–Bruhat space ${\mathcal {S}}(\mathbb {A} _{K})$ is dense in the space $L^{2}(\mathbb {A} _{K},dx).$
Applications
In algebraic number theory, the Schwartz–Bruhat functions on the adeles can be used to give an adelic version of the Poisson summation formula from analysis, i.e., for every $f\in {\mathcal {S}}(\mathbb {A} _{K})$ one has $\sum _{x\in K}f(ax)={\frac {1}{|a|}}\sum _{x\in K}{\hat {f}}(a^{-1}x)$, where $a\in \mathbb {A} _{K}^{\times }$. John Tate developed this formula in his doctoral thesis to prove a more general version of the functional equation for the Riemann zeta function. This involves giving the zeta function of a number field an integral representation in which the integral of a Schwartz–Bruhat function, chosen as a test function, is twisted by a certain character and is integrated over $\mathbb {A} _{K}^{\times }$ with respect to the multiplicative Haar measure of this group. This allows one to apply analytic methods to study zeta functions through these zeta integrals.[5]
References
1. Osborne, M. Scott (1975). "On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups". Journal of Functional Analysis. 19: 40–49. doi:10.1016/0022-1236(75)90005-1.
2. Bump, p.300
3. Ramakrishnan, Valenza, p.260
4. Deitmar, p.134
5. Tate, John T. (1950), "Fourier analysis in number fields, and Hecke's zeta-functions", Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-0-9502734-2-6, MR 0217026
• Osborne, M. Scott (1975). "On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups". Journal of Functional Analysis. 19: 40–49. doi:10.1016/0022-1236(75)90005-1.
• Gelfand, I. M.; et al. (1990). Representation Theory and Automorphic Functions. Boston: Academic Press. ISBN 0-12-279506-7.
• Bump, Daniel (1998). Automorphic Forms and Representations. Cambridge: Cambridge University Press. ISBN 978-0521658188.
• Deitmar, Anton (2012). Automorphic Forms. Berlin: Springer-Verlag London. ISBN 978-1-4471-4434-2. ISSN 0172-5939.
• Ramakrishnan, V.; Valenza, R. J. (1999). Fourier Analysis on Number Fields. New York: Springer-Verlag. ISBN 978-0387984360.
• Tate, John T. (1950), "Fourier analysis in number fields, and Hecke's zeta-functions", Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-0-9502734-2-6, MR 0217026
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Wikipedia
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Schwarz–Ahlfors–Pick theorem
In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model.
The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane H to itself, will not increase the Poincaré distance between points. The unit disk U with the Poincaré metric has negative Gaussian curvature −1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit disk to other negatively curved surfaces:
Theorem (Schwarz–Ahlfors–Pick). Let U be the unit disk with Poincaré metric $\rho $; let S be a Riemann surface endowed with a Hermitian metric $\sigma $ whose Gaussian curvature is ≤ −1; let $f:U\rightarrow S$ be a holomorphic function. Then
$\sigma (f(z_{1}),f(z_{2}))\leq \rho (z_{1},z_{2})$
for all $z_{1},z_{2}\in U.$
A generalization of this theorem was proved by Shing-Tung Yau in 1973.[1]
References
1. Osserman, Robert (September 1999). "From Schwarz to Pick to Ahlfors and Beyond" (PDF). Notices of the AMS. 46 (8): 868–873.
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Wikipedia
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Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.
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Statement
Let $\mathbf {D} =\{z:|z|<1\}$ be the open unit disk in the complex plane $\mathbb {C} $ centered at the origin, and let $f:\mathbf {D} \rightarrow \mathbb {C} $ be a holomorphic map such that $f(0)=0$ and $|f(z)|\leq 1$ on $\mathbf {D} $.
Then $|f(z)|\leq |z|$ for all $z\in \mathbf {D} $, and $|f'(0)|\leq 1$.
Moreover, if $|f(z)|=|z|$ for some non-zero $z$ or $|f'(0)|=1$, then $f(z)=az$ for some $a\in \mathbb {C} $ with $|a|=1$.[1]
Proof
The proof is a straightforward application of the maximum modulus principle on the function
$g(z)={\begin{cases}{\frac {f(z)}{z}}\,&{\mbox{if }}z\neq 0\\f'(0)&{\mbox{if }}z=0,\end{cases}}$
which is holomorphic on the whole of $D$, including at the origin (because $f$ is differentiable at the origin and fixes zero). Now if $D_{r}=\{z:|z|\leq r\}$ denotes the closed disk of radius $r$ centered at the origin, then the maximum modulus principle implies that, for $r<1$, given any $z\in D_{r}$, there exists $z_{r}$ on the boundary of $D_{r}$ such that
$|g(z)|\leq |g(z_{r})|={\frac {|f(z_{r})|}{|z_{r}|}}\leq {\frac {1}{r}}.$
As $r\rightarrow 1$ we get $|g(z)|\leq 1$.
Moreover, suppose that $|f(z)|=|z|$ for some non-zero $z\in D$, or $|f'(0)|=1$. Then, $|g(z)|=1$ at some point of $D$. So by the maximum modulus principle, $g(z)$ is equal to a constant $a$ such that $|a|=1$. Therefore, $f(z)=az$, as desired.
Schwarz–Pick theorem
A variant of the Schwarz lemma, known as the Schwarz–Pick theorem (after Georg Pick), characterizes the analytic automorphisms of the unit disc, i.e. bijective holomorphic mappings of the unit disc to itself:
Let $f:\mathbf {D} \to \mathbf {D} $ be holomorphic. Then, for all $z_{1},z_{2}\in \mathbf {D} $,
$\left|{\frac {f(z_{1})-f(z_{2})}{1-{\overline {f(z_{1})}}f(z_{2})}}\right|\leq \left|{\frac {z_{1}-z_{2}}{1-{\overline {z_{1}}}z_{2}}}\right|$
and, for all $z\in \mathbf {D} $,
${\frac {\left|f'(z)\right|}{1-\left|f(z)\right|^{2}}}\leq {\frac {1}{1-\left|z\right|^{2}}}.$
The expression
$d(z_{1},z_{2})=\tanh ^{-1}\left|{\frac {z_{1}-z_{2}}{1-{\overline {z_{1}}}z_{2}}}\right|$
is the distance of the points $z_{1}$, $z_{2}$ in the Poincaré metric, i.e. the metric in the Poincaré disc model for hyperbolic geometry in dimension two. The Schwarz–Pick theorem then essentially states that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric. If equality holds throughout in one of the two inequalities above (which is equivalent to saying that the holomorphic map preserves the distance in the Poincaré metric), then $f$ must be an analytic automorphism of the unit disc, given by a Möbius transformation mapping the unit disc to itself.
An analogous statement on the upper half-plane $\mathbf {H} $ can be made as follows:
Let $f:\mathbf {H} \to \mathbf {H} $ be holomorphic. Then, for all $z_{1},z_{2}\in \mathbf {H} $,
$\left|{\frac {f(z_{1})-f(z_{2})}{{\overline {f(z_{1})}}-f(z_{2})}}\right|\leq {\frac {\left|z_{1}-z_{2}\right|}{\left|{\overline {z_{1}}}-z_{2}\right|}}.$
This is an easy consequence of the Schwarz–Pick theorem mentioned above: One just needs to remember that the Cayley transform $W(z)=(z-i)/(z+i)$ maps the upper half-plane $\mathbf {H} $ conformally onto the unit disc $\mathbf {D} $. Then, the map $W\circ f\circ W^{-1}$ is a holomorphic map from $\mathbf {D} $ onto $\mathbf {D} $. Using the Schwarz–Pick theorem on this map, and finally simplifying the results by using the formula for $W$, we get the desired result. Also, for all $z\in \mathbf {H} $,
${\frac {\left|f'(z)\right|}{{\text{Im}}(f(z))}}\leq {\frac {1}{{\text{Im}}(z)}}.$
If equality holds for either the one or the other expressions, then $f$ must be a Möbius transformation with real coefficients. That is, if equality holds, then
$f(z)={\frac {az+b}{cz+d}}$
with $a,b,c,d\in \mathbb {R} $ and $ad-bc>0$.
Proof of Schwarz–Pick theorem
The proof of the Schwarz–Pick theorem follows from Schwarz's lemma and the fact that a Möbius transformation of the form
${\frac {z-z_{0}}{{\overline {z_{0}}}z-1}},\qquad |z_{0}|<1,$
maps the unit circle to itself. Fix $z_{1}$ and define the Möbius transformations
$M(z)={\frac {z_{1}-z}{1-{\overline {z_{1}}}z}},\qquad \varphi (z)={\frac {f(z_{1})-z}{1-{\overline {f(z_{1})}}z}}.$
Since $M(z_{1})=0$ and the Möbius transformation is invertible, the composition $\varphi (f(M^{-1}(z)))$ maps $0$ to $0$ and the unit disk is mapped into itself. Thus we can apply Schwarz's lemma, which is to say
$\left|\varphi \left(f(M^{-1}(z))\right)\right|=\left|{\frac {f(z_{1})-f(M^{-1}(z))}{1-{\overline {f(z_{1})}}f(M^{-1}(z))}}\right|\leq |z|.$
Now calling $z_{2}=M^{-1}(z)$ (which will still be in the unit disk) yields the desired conclusion
$\left|{\frac {f(z_{1})-f(z_{2})}{1-{\overline {f(z_{1})}}f(z_{2})}}\right|\leq \left|{\frac {z_{1}-z_{2}}{1-{\overline {z_{1}}}z_{2}}}\right|.$
To prove the second part of the theorem, we rearrange the left-hand side into the difference quotient and let $z_{2}$ tend to $z_{1}$.
Further generalizations and related results
The Schwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds.
De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of $f$ at $0$ in case $f$ is injective; that is, univalent.
The Koebe 1/4 theorem provides a related estimate in the case that $f$ is univalent.
See also
• Nevanlinna–Pick interpolation
References
1. Theorem 5.34 in Rodriguez, Jane P. Gilman, Irwin Kra, Rubi E. (2007). Complex analysis : in the spirit of Lipman Bers ([Online] ed.). New York: Springer. p. 95. ISBN 978-0-387-74714-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See Section 2.3)
• S. Dineen (1989). The Schwarz Lemma. Oxford. ISBN 0-19-853571-6.
This article incorporates material from Schwarz lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia
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Bayesian information criterion
In statistics, the Bayesian information criterion (BIC) or Schwarz information criterion (also SIC, SBC, SBIC) is a criterion for model selection among a finite set of models; models with lower BIC are generally preferred. It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion (AIC).
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When fitting models, it is possible to increase the maximum likelihood by adding parameters, but doing so may result in overfitting. Both BIC and AIC attempt to resolve this problem by introducing a penalty term for the number of parameters in the model; the penalty term is larger in BIC than in AIC for sample sizes greater than 7.[1]
The BIC was developed by Gideon E. Schwarz and published in a 1978 paper,[2] where he gave a Bayesian argument for adopting it.
Definition
The BIC is formally defined as[3][lower-alpha 1]
$\mathrm {BIC} =k\ln(n)-2\ln({\widehat {L}}).\ $
where
• ${\hat {L}}$ = the maximized value of the likelihood function of the model $M$, i.e. ${\hat {L}}=p(x\mid {\widehat {\theta }},M)$, where ${\widehat {\theta }}$ are the parameter values that maximize the likelihood function;
• $x$ = the observed data;
• $n$ = the number of data points in $x$, the number of observations, or equivalently, the sample size;
• $k$ = the number of parameters estimated by the model. For example, in multiple linear regression, the estimated parameters are the intercept, the $q$ slope parameters, and the constant variance of the errors; thus, $k=q+2$.
Derivation
Konishi and Kitagawa[5]: 217 derive the BIC to approximate the distribution of the data, integrating out the parameters using Laplace's method, starting with the following model evidence:
$p(x\mid M)=\int p(x\mid \theta ,M)\pi (\theta \mid M)\,d\theta $
where $\pi (\theta \mid M)$ is the prior for $\theta $ under model $M$.
The log-likelihood, $\ln(p(x|\theta ,M))$, is then expanded to a second order Taylor series about the MLE, ${\widehat {\theta }}$, assuming it is twice differentiable as follows:
$\ln(p(x\mid \theta ,M))=\ln({\widehat {L}})-{\frac {n}{2}}(\theta -{\widehat {\theta }})^{\operatorname {T} }{\mathcal {I}}(\theta )(\theta -{\widehat {\theta }})+R(x,\theta ),$
where ${\mathcal {I}}(\theta )$ is the average observed information per observation, and $R(x,\theta )$ denotes the residual term. To the extent that $R(x,\theta )$ is negligible and $\pi (\theta \mid M)$ is relatively linear near ${\widehat {\theta }}$, we can integrate out $\theta $ to get the following:
$p(x\mid M)\approx {\hat {L}}{\left({\frac {2\pi }{n}}\right)}^{\frac {k}{2}}|{\mathcal {I}}({\widehat {\theta }})|^{-{\frac {1}{2}}}\pi ({\widehat {\theta }})$
As $n$ increases, we can ignore $|{\mathcal {I}}({\widehat {\theta }})|$ and $\pi ({\widehat {\theta }})$ as they are $O(1)$. Thus,
$p(x\mid M)=\exp \left(\ln {\widehat {L}}-{\frac {k}{2}}\ln(n)+O(1)\right)=\exp \left(-{\frac {\mathrm {BIC} }{2}}+O(1)\right),$
where BIC is defined as above, and ${\widehat {L}}$ either (a) is the Bayesian posterior mode or (b) uses the MLE and the prior $\pi (\theta \mid M)$ has nonzero slope at the MLE. Then the posterior
$p(M\mid x)\propto p(x\mid M)p(M)\approx \exp \left(-{\frac {\mathrm {BIC} }{2}}\right)p(M)$
Usage
When picking from several models, ones with lower BIC values are generally preferred. The BIC is an increasing function of the error variance $\sigma _{e}^{2}$ and an increasing function of k. That is, unexplained variation in the dependent variable and the number of explanatory variables increase the value of BIC. However, a lower BIC does not necessarily indicate one model is better than another. Because it involves approximations, the BIC is merely a heuristic. In particular, differences in BIC should never be treated like transformed Bayes factors.
It is important to keep in mind that the BIC can be used to compare estimated models only when the numerical values of the dependent variable[lower-alpha 2] are identical for all models being compared. The models being compared need not be nested, unlike the case when models are being compared using an F-test or a likelihood ratio test.
Properties
• The BIC generally penalizes free parameters more strongly than the Akaike information criterion, though it depends on the size of n and relative magnitude of n and k.
• It is independent of the prior.
• It can measure the efficiency of the parameterized model in terms of predicting the data.
• It penalizes the complexity of the model where complexity refers to the number of parameters in the model.
• It is approximately equal to the minimum description length criterion but with negative sign.
• It can be used to choose the number of clusters according to the intrinsic complexity present in a particular dataset.
• It is closely related to other penalized likelihood criteria such as Deviance information criterion and the Akaike information criterion.
Limitations
The BIC suffers from two main limitations[6]
1. the above approximation is only valid for sample size $n$ much larger than the number $k$ of parameters in the model.
2. the BIC cannot handle complex collections of models as in the variable selection (or feature selection) problem in high-dimension.[6]
Gaussian special case
Under the assumption that the model errors or disturbances are independent and identically distributed according to a normal distribution and the boundary condition that the derivative of the log likelihood with respect to the true variance is zero, this becomes (up to an additive constant, which depends only on n and not on the model):[7]
$\mathrm {BIC} =n\ln({\widehat {\sigma _{e}^{2}}})+k\ln(n)\ $
where ${\widehat {\sigma _{e}^{2}}}$ is the error variance. The error variance in this case is defined as
${\widehat {\sigma _{e}^{2}}}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\widehat {x_{i}}})^{2}.$
which is a biased estimator for the true variance.
In terms of the residual sum of squares (RSS) the BIC is
$\mathrm {BIC} =n\ln(RSS/n)+k\ln(n)\ $
When testing multiple linear models against a saturated model, the BIC can be rewritten in terms of the deviance $\chi ^{2}$ as:[8]
$\mathrm {BIC} =\chi ^{2}+k\ln(n)$
where $k$ is the number of model parameters in the test.
See also
• Akaike information criterion
• Bayes factor
• Bayesian model comparison
• Deviance information criterion
• Hannan–Quinn information criterion
• Jensen–Shannon divergence
• Kullback–Leibler divergence
• Minimum message length
Notes
1. The AIC, AICc and BIC defined by Claeskens and Hjort[4] are the negatives of those defined in this article and in most other standard references.
2. A dependent variable is also called a response variable or an outcome variable. See Regression analysis.
References
1. See the review paper: Stoica, P.; Selen, Y. (2004), "Model-order selection: a review of information criterion rules", IEEE Signal Processing Magazine (July): 36–47, doi:10.1109/MSP.2004.1311138, S2CID 17338979.
2. Schwarz, Gideon E. (1978), "Estimating the dimension of a model", Annals of Statistics, 6 (2): 461–464, doi:10.1214/aos/1176344136, MR 0468014.
3. Wit, Ernst; Edwin van den Heuvel; Jan-Willem Romeyn (2012). "'All models are wrong...': an introduction to model uncertainty" (PDF). Statistica Neerlandica. 66 (3): 217–236. doi:10.1111/j.1467-9574.2012.00530.x. S2CID 7793470.
4. Claeskens, G.; Hjort, N. L. (2008), Model Selection and Model Averaging, Cambridge University Press
5. Konishi, Sadanori; Kitagawa, Genshiro (2008). Information criteria and statistical modeling. Springer. ISBN 978-0-387-71886-6.
6. Giraud, C. (2015). Introduction to high-dimensional statistics. Chapman & Hall/CRC. ISBN 9781482237948.
7. Priestley, M.B. (1981). Spectral Analysis and Time Series. Academic Press. ISBN 978-0-12-564922-3. (p. 375).
8. Kass, Robert E.; Raftery, Adrian E. (1995), "Bayes Factors", Journal of the American Statistical Association, 90 (430): 773–795, doi:10.2307/2291091, ISSN 0162-1459, JSTOR 2291091.
Further reading
• Bhat, H. S.; Kumar, N (2010). "On the derivation of the Bayesian Information Criterion" (PDF). Archived from the original (PDF) on 28 March 2012. {{cite journal}}: Cite journal requires |journal= (help)
• Findley, D. F. (1991). "Counterexamples to parsimony and BIC". Annals of the Institute of Statistical Mathematics. 43 (3): 505–514. doi:10.1007/BF00053369. S2CID 58910242.
• Kass, R. E.; Wasserman, L. (1995). "A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion". Journal of the American Statistical Association. 90 (431): 928–934. doi:10.2307/2291327. JSTOR 2291327.
• Liddle, A. R. (2007). "Information criteria for astrophysical model selection". Monthly Notices of the Royal Astronomical Society. 377 (1): L74–L78. arXiv:astro-ph/0701113. Bibcode:2007MNRAS.377L..74L. doi:10.1111/j.1745-3933.2007.00306.x. S2CID 2884450.
• McQuarrie, A. D. R.; Tsai, C.-L. (1998). Regression and Time Series Model Selection. World Scientific.
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Schwarz alternating method
In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869–1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved. This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907. It furnished a scheme for uniformizing the union of two regions knowing how to uniformize each of them separately, provided their intersection was topologically a disk or an annulus. From 1870 onwards Carl Neumann also contributed to this theory.
In the 1950s Schwarz's method was generalized in the theory of partial differential equations to an iterative method for finding the solution of an elliptic boundary value problem on a domain which is the union of two overlapping subdomains. It involves solving the boundary value problem on each of the two subdomains in turn, taking always the last values of the approximate solution as the next boundary conditions. It is used in numerical analysis, under the name multiplicative Schwarz method (in opposition to additive Schwarz method) as a domain decomposition method.
History
It was first formulated by H. A. Schwarz[1] and served as a theoretical tool: its convergence for general second order elliptic partial differential equations was first proved much later, in 1951, by Solomon Mikhlin.[2]
The algorithm
The original problem considered by Schwarz was a Dirichlet problem (with the Laplace's equation) on a domain consisting of a circle and a partially overlapping square. To solve the Dirichlet problem on one of the two subdomains (the square or the circle), the value of the solution must be known on the border: since a part of the border is contained in the other subdomain, the Dirichlet problem must be solved jointly on the two subdomains. An iterative algorithm is introduced:
1. Make a first guess of the solution on the circle's boundary part that is contained in the square
2. Solve the Dirichlet problem on the circle
3. Use the solution in (2) to approximate the solution on the square's boundary
4. Solve the Dirichlet problem on the square
5. Use the solution in (4) to approximate the solution on the circle's boundary, then go to step (2).
At convergence, the solution on the overlap is the same when computed on the square or on the circle.
Optimized Schwarz methods
The convergence speed depends on the size of the overlap between the subdomains, and on the transmission conditions (boundary conditions used in the interface between the subdomains). It is possible to increase the convergence speed of the Schwarz methods by choosing adapted transmission conditions: theses methods are then called Optimized Schwarz methods.[3]
See also
• Uniformization theorem
• Schwarzian derivative
• Schwarz triangle map
• Schwarz reflection principle
• Additive Schwarz method
Notes
1. See his paper (Schwarz 1870b)
2. See the paper (Mikhlin 1951): a comprehensive exposition was given by the same author in later books
3. Gander, Martin J.; Halpern, Laurence; Nataf, Frédéric (2001), "Optimized Schwarz Methods", 12th International Conference on Domain Decomposition Methods (PDF)
References
Original papers
• Schwarz, H.A. (1869), "Über einige Abbildungsaufgaben", J. Reine Angew. Math., 1869 (70): 105–120, doi:10.1515/crll.1869.70.105, S2CID 121291546
• Schwarz, H.A. (1870a), "Über die Integration der partiellen Differentialgleichung ∂2u/∂x2 + ∂2u/∂y2 = 0 unter vorgeschriebenen Grenz- und Unstetigkeitbedingungen", Monatsberichte der Königlichen Akademie der Wissenschaft zu Berlin: 767–795
• Schwarz, H. A. (1870b), "Über einen Grenzübergang durch alternierendes Verfahren", Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15: 272–286, JFM 02.0214.02
• Neumann, Carl (1870), "Zur Theorie des Potentiales", Math. Ann., 2 (3): 514, doi:10.1007/bf01448242, S2CID 122015888
• Neumann, Carl (1877), Untersuchungen über das logarithmische und Newton'sche Potential, Teubner
• Neumann, Carl (1884), Vorlesungen über Riemann's Theorie der abelschen Integrale (2nd ed.), Teubner
Conformal mapping and harmonic functions
• Nevanlinna, Rolf (1939), "Über das alternierende Verfahren von Schwarz", J. Reine Angew. Math., 1939 (180): 121–128, doi:10.1515/crll.1939.180.121, S2CID 199546268
• Nevanlinna, Rolf (1939), "Bemerkungen zum alternierenden Verfahren", Monatshefte für Mathematik und Physik, 48: 500–508, doi:10.1007/bf01696203, S2CID 123260734
• Nevanlinna, Rolf (1953), Uniformisierung, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. 64, Springer
• Sario, Leo (1953), "Alternating method on arbitrary Riemann surfaces", Pacific J. Math., 3 (3): 631–645, doi:10.2140/pjm.1953.3.631
• Morgenstern, Dietrich (1956), "Begründung des alternierenden Verfahrens durch Orthogonalprojektion", Z. Angew. Math. Mech., 36 (7–8): 255–256, Bibcode:1956ZaMM...36..255M, doi:10.1002/zamm.19560360711, hdl:10338.dmlcz/100409
• Cohn, Harvey (1980), Conformal mapping on Riemann surfaces, Dover, pp. 242–262, ISBN 0-486-64025-6, Chapter 12, Alternating Procedures
• Garnett, John B.; Marshall, Donald E. (2005), Harmonic Measure, Cambridge University Press, ISBN 1139443097
• Freitag, Eberhard (2011), Complex analysis. 2. Riemann surfaces, several complex variables, abelian functions, higher modular functions, Springer, ISBN 978-3-642-20553-8
• de Saint-Gervais, Henri Paul (2016), Uniformization of Riemann Surfaces: revisiting a hundred-year-old theorem, Heritage of European Mathematics, translated by Robert G. Burns, European Mathematical Society, doi:10.4171/145, ISBN 978-3-03719-145-3, translation of French text
• Chorlay, Renaud (2007), L'émergence du couple local-global dans les théories géométriques, de Bernhard Riemann à la théorie des faisceaux (PDF), pp. 123–134 (cited in de Saint-Gervais)
• Bottazzini, Umberto; Gray, Jeremy (2013), Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, ISBN 978-1461457251
PDEs and numerical analysis
• Mikhlin, S.G. (1951), "On the Schwarz algorithm", Doklady Akademii Nauk SSSR, n. Ser. (in Russian), 77: 569–571, MR 0041329, Zbl 0054.04204
External links
• Solomentsev, E.D. (2001) [1994], "Schwarz alternating method", Encyclopedia of Mathematics, EMS Press
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Wikipedia
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Schwarz function
The Schwarz function of a curve in the complex plane is an analytic function which maps the points of the curve to their complex conjugates. It can be used to generalize the Schwarz reflection principle to reflection across arbitrary analytic curves, not just across the real axis.
Not to be confused with Schwartz function or Schwarz triangle function.
The Schwarz function exists for analytic curves. More precisely, for every non-singular, analytic Jordan arc $\Gamma $ in the complex plane, there is an open neighborhood $\Omega $ of $\Gamma $ and a unique analytic function $S$ on $\Omega $ such that $S(z)={\overline {z}}$ for every $z\in \Gamma $.[1]
The "Schwarz function" was named by Philip J. Davis and Henry O. Pollak (1958) in honor of Hermann Schwarz,[2][3] who introduced the Schwarz reflection principle for analytic curves in 1870.[4] However, the Schwarz function does not explicitly appear in Schwarz's works.[5]
Examples
The unit circle is described by the equation $|z|^{2}=1$, or ${\overline {z}}=1/z$. Thus, the Schwarz function of the unit circle is $S(z)=1/z$.
A more complicated example is an ellipse defined by $(x/a)^{2}+(y/b)^{2}=1$. The Schwarz function can be found by substituting $\textstyle x={\frac {z+{\overline {z}}}{2}}$ and $\textstyle y={\frac {z-{\overline {z}}}{2i}}$ and solving for ${\overline {z}}$. The result is:[6]
$S(z)={\frac {1}{a^{2}-b^{2}}}\left((a^{2}+b^{2})z-2ab{\sqrt {z^{2}+b^{2}-a^{2}}}\right)$.
This is analytic on the complex plane minus a branch cut along the line segment between the foci $\pm {\sqrt {a^{2}-b^{2}}}$.
References
1. Shapiro 1992, p. 3
2. Davis, Phillip; Pollak, Henry (January 1958). "On the Analytic Continuation of Mapping Functions" (PDF). Transactions of the American Mathematical Society. 87 (1): 198–225. doi:10.2307/1993097. JSTOR 1993097.
3. Needham 1997, p. 255
4. Schwarz, H.A. (1870). "Ueber die Integration der paritellen Differentialgleichung $\textstyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0$ unter vorgeschriebenen Grenz- und Unstetigkeitsbedingungen". Monatsberichte der Königlichen Preussische Akademie des Wissenschaften zu Berlin: 767–795. Reprinted in: Schwarz, H.A. (1890). Gesammelte Mathematische Abhandlungen. Vol. II. pp. 144–171.
5. Shapiro 1992, p. 2
6. Needham 1997, p. 256
• Davis, Philip J. (1974). The Schwarz function and its applications. Carus Monographs 17. Mathematical Association of America. ISBN 978-0-883-85017-6. OCLC 912405492.
• Needham, Tristan (1997). Visual Complex Analysis. Clarendon Press. ISBN 978-0-19-853447-1.
• Shapiro, Harold S. (1992-03-18). The Schwarz Function and Its Generalization to Higher Dimensions. John Wiley & Sons. ISBN 978-0-471-57127-8. OCLC 924755133.
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Wikipedia
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Schwarz integral formula
In complex analysis, a branch of mathematics, the Schwarz integral formula, named after Hermann Schwarz, allows one to recover a holomorphic function, up to an imaginary constant, from the boundary values of its real part.
Unit disc
Let f be a function holomorphic on the closed unit disc {z ∈ C | |z| ≤ 1}. Then
$f(z)={\frac {1}{2\pi i}}\oint _{|\zeta |=1}{\frac {\zeta +z}{\zeta -z}}\operatorname {Re} (f(\zeta ))\,{\frac {d\zeta }{\zeta }}+i\operatorname {Im} (f(0))$
for all |z| < 1.
Upper half-plane
Let f be a function holomorphic on the closed upper half-plane {z ∈ C | Im(z) ≥ 0} such that, for some α > 0, |zα f(z)| is bounded on the closed upper half-plane. Then
$f(z)={\frac {1}{\pi i}}\int _{-\infty }^{\infty }{\frac {u(\zeta ,0)}{\zeta -z}}\,d\zeta ={\frac {1}{\pi i}}\int _{-\infty }^{\infty }{\frac {\operatorname {Re} (f)(\zeta +0i)}{\zeta -z}}\,d\zeta $
for all Im(z) > 0.
Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral; this is because the additional decay condition makes the conditions for this formula more stringent.
Corollary of Poisson integral formula
The formula follows from Poisson integral formula applied to u:[1][2]
$u(z)={\frac {1}{2\pi }}\int _{0}^{2\pi }u(e^{i\psi })\operatorname {Re} {e^{i\psi }+z \over e^{i\psi }-z}\,d\psi \qquad {\text{for }}|z|<1.$
By means of conformal maps, the formula can be generalized to any simply connected open set.
Notes and references
1. Lectures on Entire Functions, p. 9, at Google Books
2. The derivation without an appeal to the Poisson formula can be found at: https://planetmath.org/schwarzandpoissonformulas Archived 2021-12-24 at the Wayback Machine
• Ahlfors, Lars V. (1979), Complex Analysis, Third Edition, McGraw-Hill, ISBN 0-07-085008-9
• Remmert, Reinhold (1990), Theory of Complex Functions, Second Edition, Springer, ISBN 0-387-97195-5
• Saff, E. B., and A. D. Snider (1993), Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Second Edition, Prentice Hall, ISBN 0-13-327461-6
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Wikipedia
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Schwarz lantern
In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. It is formed by stacked rings of isosceles triangles, arranged within each ring in the same pattern as an antiprism. The resulting shape can be folded from paper, and is named after mathematician Hermann Schwarz and for its resemblance to a cylindrical paper lantern.[1] It is also known as Schwarz's boot,[2] Schwarz's polyhedron,[3] or the Chinese lantern.[4]
As Schwarz showed, for the surface area of a polyhedron to converge to the surface area of a curved surface, it is not sufficient to simply increase the number of rings and the number of isosceles triangles per ring. Depending on the relation of the number of rings to the number of triangles per ring, the area of the lantern can converge to the area of the cylinder, to a limit arbitrarily larger than the area of the cylinder, or to infinity—in other words, the area can diverge. The Schwarz lantern demonstrates that sampling a curved surface by close-together points and connecting them by small triangles is inadequate to ensure an accurate approximation of area, in contrast to the accurate approximation of arc length by inscribed polygonal chains.
The phenomenon that closely sampled points can lead to inaccurate approximations of area has been called the Schwarz paradox.[5][6] The Schwarz lantern is an instructive example in calculus and highlights the need for care when choosing a triangulation for applications in computer graphics and the finite element method.
History and motivation
Archimedes approximated the circumference of circles by the lengths of inscribed or circumscribed regular polygons.[7][8] More generally, the length of any smooth or rectifiable curve can be defined as the supremum of the lengths of polygonal chains inscribed in them.[1] However, for this to work correctly, the vertices of the polygonal chains must lie on the given curve, rather than merely near it. Otherwise, in a counterexample sometimes known as the staircase paradox, polygonal chains of vertical and horizontal line segments of total length $2$ can lie arbitrarily close to a diagonal line segment of length ${\sqrt {2}}$, converging in distance to the diagonal segment but not converging to the same length. The Schwarz lantern provides a counterexample for surface area rather than length,[9] and shows that for area, requiring vertices to lie on the approximated surface is not enough to ensure an accurate approximation.[1]
German mathematician Hermann Schwarz (1843–1921) devised his construction in the late 19th century[lower-alpha 1] as a counterexample to the erroneous definition in J. A. Serret's 1868 book Cours de calcul differentiel et integral,[12] which incorrectly states that:
Soit une portion de surface courbe terminée par un contour $C$; nous nommerons aire de cette surface la limite $S$ vers laquelle tend l'aire d'une surface polyédrale inscrite formée de faces triangulaires et terminee par un contour polygonal $\Gamma $ ayant pour limite le contour $C$.
Il faut démontrer que la limite $S$ existe et qu'elle est indépendante de la loi suivant laquelle décroissent les faces de la surface polyedrale inscrite.
Let a portion of curved surface be bounded by a contour $C$; we will define the area of this surface to be the limit $S$ tended towards by the area of an inscribed polyhedral surface formed from triangular faces and bounded by a polygonal contour $\Gamma $ whose limit is the contour $C$.
It must be shown that the limit $S$ exists and that it is independent of the law according to which the faces of the inscribed polyhedral surface shrink.
Independently of Schwarz, Giuseppe Peano found the same counterexample.[10] At the time, Peano was a student of Angelo Genocchi, who, from communication with Schwarz, already knew about the difficulty of defining surface area. Genocchi informed Charles Hermite, who had been using Serret's erroneous definition in his course. Hermite asked Schwarz for details, revised his course, and published the example in the second edition of his lecture notes (1883).[11] The original note from Schwarz to Hermite was not published until the second edition of Schwarz's collected works in 1890.[13][14]
An instructive example of the value of careful definitions in calculus,[5] the Schwarz lantern also highlights the need for care in choosing a triangulation for applications in computer graphics and for the finite element method for scientific and engineering simulations.[6][15] In computer graphics, scenes are often described by triangulated surfaces, and accurate rendering of the illumination of those surfaces depends on the direction of the surface normals. A poor choice of triangulation, as in the Schwarz lantern, can produce an accordion-like surface whose normals are far from the normals of the approximated surface, and the closely-spaced sharp folds of this surface can also cause problems with aliasing.[6]
The failure of Schwarz lanterns to converge to the cylinder's area only happens when they include highly obtuse triangles, with angles close to 180°. In restricted classes of Schwarz lanterns using angles bounded away from 180°, the area converges to the same area as the cylinder as the number of triangles grows to infinity. The finite element method, in its most basic form, approximates a smooth function (often, the solution to a physical simulation problem in science or engineering) by a piecewise-linear function on a triangulation. The Schwarz lantern's example shows that, even for simple functions such as the height of a cylinder above a plane through its axis, and even when the function values are calculated accurately at the triangulation vertices, a triangulation with angles close to 180° can produce highly inaccurate simulation results. This motivates mesh generation methods for which all angles are bounded away from 180°, such as nonobtuse meshes.[15]
Construction
The discrete polyhedral approximation considered by Schwarz can be described by two parameters: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): m , the number of rings of triangles in the Schwarz lantern; and $n$, half of the number of triangles per ring.[16][lower-alpha 2] For a single ring ($m=1$), the resulting surface consists of the triangular faces of an antiprism of order $n$. For larger values of $m$, the Schwarz lantern is formed by stacking $m$ of these antiprisms.[6] To construct a Schwarz lantern that approximates a given right circular cylinder, the cylinder is sliced by parallel planes into $m$ congruent cylindrical rings. These rings have $m+1$ circular boundaries—two at the ends of the given cylinder, and $m-1$ more where it was sliced. In each circle, $n$ vertices of the Schwarz lantern are spaced equally, forming a regular polygon. These polygons are rotated by an angle of $\pi /n$ from one circle to the next, so that each edge from a regular polygon and the nearest vertex on the next circle form the base and apex of an isosceles triangle. These triangles meet edge-to-edge to form the Schwarz lantern, a polyhedral surface that is topologically equivalent to the cylinder.[16]
Origami crease pattern for a Schwarz lantern with $m=16$ and $n=4$
Detail of a boot from the painting Saint Florian (1473) by Francesco del Cossa, showing Yoshimura buckling
Ignoring top and bottom vertices, each vertex touches two apex angles and four base angles of congruent isosceles triangles, just as it would in a tessellation of the plane by triangles of the same shape. As a consequence, the Schwarz lantern can be folded from a flat piece of paper, with this tessellation as its crease pattern.[18] This crease pattern has been called the Yoshimura pattern,[19] after the work of Y. Yoshimura on the Yoshimura buckling pattern of cylindrical surfaces under axial compression, which can be similar in shape to the Schwarz lantern.[20]
Area
The area of the Schwarz lantern, for any cylinder and any particular choice of the parameters $m$ and $n$, can be calculated by a straightforward application of trigonometry. A cylinder of radius $r$ and length $\ell $ has area $2\pi r\ell $. For a Schwarz lantern with parameters $m$ and $n$, each band is a shorter cylinder of length $\ell /m$, approximated by $2n$ isosceles triangles. The length of the base of each triangle can be found from the formula for the edge length of a regular $n$-gon, namely[16]
$2r\sin {\frac {\pi }{n}}.$
The height $h$ of each triangle can be found by applying the Pythagorean theorem to a right triangle formed by the apex of the triangle, the midpoint of the base, and the midpoint of the arc of the circle bounded by the endpoints of the base. The two sides of this right triangle are the length $\ell /m$ of the cylindrical band, and the sagitta of the arc,[lower-alpha 3] giving the formula[16]
$h^{2}=\left({\frac {\ell }{m}}\right)^{2}+\left(r\left(1-\cos {\frac {\pi }{n}}\right)\right)^{2}.$
Combining the formula for the area of each triangle from its base and height, and the total number $2mn$ of the triangles, gives the Schwarz lantern a total area of[16]
$A(m,n)=2mn\left(r\sin {\frac {\pi }{n}}\right){\sqrt {\left({\frac {\ell }{m}}\right)^{2}+r^{2}\left(1-\cos {\frac {\pi }{n}}\right)^{2}}}.$
Limits
The Schwarz lanterns, for large values of both parameters, converge uniformly to the cylinder that they approximate.[21] However, because there are two free parameters $m$ and $n$, the limiting area of the Schwarz lantern, as both $m$ and $n$ become arbitrarily large, can be evaluated in different orders, with different results. If $m$ is fixed while $n$ grows, and the resulting limit is then evaluated for arbitrarily large choices of $m$, one obtains[16]
$\lim _{m\to \infty }\lim _{n\to \infty }A(m,n)=2\pi r\ell ,$
the correct area for the cylinder. In this case, the inner limit already converges to the same value, and the outer limit is superfluous. Geometrically, substituting each cylindrical band by a band of very sharp isosceles triangles accurately approximates its area.[16]
On the other hand, reversing the ordering of the limits gives[16]
$\lim _{n\to \infty }\lim _{m\to \infty }A(m,n)=\infty .$
In this case, for a fixed choice of $n$, as $m$ grows and the length $\ell /m$ of each cylindrical band becomes arbitrarily small, each corresponding band of isosceles triangles becomes nearly planar. Each triangle approaches the triangle formed by two consecutive edges of a regular $2n$-gon, and the area of the whole band of triangles approaches $2n$ times the area of one of these planar triangles, a finite number. However, the number $m$ of these bands grows arbitrarily large; because the lantern's area grows in approximate proportion to $m$, it also becomes arbitrarily large.[16]
It is also possible to fix a functional relation between $m$ and $n$, and to examine the limit as both parameters grow large simultaneously, maintaining this relation. Different choices of this relation can lead to either of the two behaviors described above, convergence to the correct area or divergence to infinity. For instance, setting $m=cn$ (for an arbitrary constant $c$) and taking the limit for large $n$ leads to convergence to the correct area, while setting $m=cn^{3}$ leads to divergence. A third type of limiting behavior is obtained by setting $m=cn^{2}$. For this choice,
$\lim _{n\to \infty }A(cn^{2},n)=2\pi r{\sqrt {\ell ^{2}+{\frac {r^{2}\pi ^{4}c^{2}}{4}}}}.$
In this case, the area of the Schwarz lantern, parameterized in this way, converges, but to a larger value than the area of the cylinder. Any desired larger area can be obtained by making an appropriate choice of the constant $c$.[16]
See also
• Kaleidocycle, a chain of tetrahedra linked edge-to-edge like a degenerate Schwarz lantern with $n=2$
• Runge's phenomenon, another example of failure of convergence
Notes
1. Gandon & Perrin (2009) place the timing more precisely as the early 1890s,[10] but this is contradicted by Hermite's use of this example in 1883. Kennedy (1980) dates Schwarz's communication to Genocchi on this topic to 1880, and Peano's rediscovery to 1882.[11]
2. Other sources may use different parameterizations; for instance, Dubrovsky (1991) uses $k$ instead of $m$ to denote the number of cylinders.[17]
3. The sagitta of a circular arc is the distance from the midpoint of the arc to the midpoint of its chord.
References
1. Makarov, Boris; Podkorytov, Anatolii (2013). "Section 8.2.4". Real analysis: measures, integrals and applications. Universitext. Berlin: Springer-Verlag. pp. 415–416. doi:10.1007/978-1-4471-5122-7. ISBN 978-1-4471-5121-0. MR 3089088.
2. Bernshtein, D. (March–April 1991). "Toy store: Latin triangles and fashionable footwear" (PDF). Quantum: The Magazine of Math and Science. Vol. 1, no. 4. p. 64.
3. Wells, David (1991). "Schwarz's polyhedron". The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 225–226. ISBN 978-0-14-011813-1.
4. Berger, Marcel (1987). Geometry I. Universitext. Berlin: Springer-Verlag. pp. 263–264. doi:10.1007/978-3-540-93815-6. ISBN 978-3-540-11658-5. MR 2724360.
5. Atneosen, Gail H. (March 1972). "The Schwarz paradox: An interesting problem for the first-year calculus student". The Mathematics Teacher. 65 (3): 281–284. doi:10.5951/MT.65.3.0281. JSTOR 27958821.
6. Glassner, A. (1997). "The perils of problematic parameterization". IEEE Computer Graphics and Applications. 17 (5): 78–83. doi:10.1109/38.610212.
7. Traub, Gilbert (1984). The Development of the Mathematical Analysis of Curve Length from Archimedes to Lebesgue (Doctoral dissertation). New York University. p. 470. MR 2633321. ProQuest 303305072.
8. Brodie, Scott E. (1980). "Archimedes' axioms for arc-length and area". Mathematics Magazine. 53 (1): 36–39. doi:10.1080/0025570X.1980.11976824. JSTOR 2690029. MR 0560018.
9. Ogilvy, C. Stanley (1962). "Note to page 7". Tomorrow's Math: Unsolved Problems for the Amateur. Oxford University Press. pp. 155–161.
10. Gandon, Sébastien; Perrin, Yvette (2009). "Le problème de la définition de l'aire d'une surface gauche: Peano et Lebesgue" (PDF). Archive for History of Exact Sciences (in French). 63 (6): 665–704. doi:10.1007/s00407-009-0051-4. JSTOR 41134329. MR 2550748. S2CID 121535260.
11. Kennedy, Hubert C. (1980). Peano: Life and works of Giuseppe Peano. Studies in the History of Modern Science. Vol. 4. Dordrecht & Boston: D. Reidel Publishing Co. pp. 9–10. ISBN 90-277-1067-8. MR 0580947.
12. Serret, J. A. (1868). Cours de calcul différentiel et intégral, Tome second: Calcul intégral (in French). Paris: Gauthier-Villars. p. 296.
13. Schwarz, H. A. (1890). "Sur une définition erronée de l'aire d'une surface courbe". Gesammelte Mathematische Abhandlungen von H. A. Schwarz (in French). Verlag von Julius Springer. pp. 309–311.
14. Archibald, Thomas (2002). "Charles Hermite and German mathematics in France". In Parshall, Karen Hunger; Rice, Adrian C. (eds.). Mathematics unbound: the evolution of an international mathematical research community, 1800–1945. Papers from the International Symposium held at the University of Virginia, Charlottesville, VA, May 27–29, 1999. History of Mathematics. Vol. 23. Providence, Rhode Island: American Mathematical Society. pp. 123–137. MR 1907173. See footnote 60, p. 135.
15. Bern, M.; Mitchell, S.; Ruppert, J. (1995). "Linear-size nonobtuse triangulation of polygons". Discrete & Computational Geometry. 14 (4): 411–428. doi:10.1007/BF02570715. MR 1360945. S2CID 120526239.
16. Zames, Frieda (September 1977). "Surface area and the cylinder area paradox". The Two-Year College Mathematics Journal. 8 (4): 207–211. doi:10.2307/3026930. JSTOR 3026930.
17. Dubrovsky, Vladimir (March–April 1991). "In search of a definition of surface area" (PDF). Quantum: The Magazine of Math and Science. Vol. 1, no. 4. pp. 6–9.
18. Lamb, Evelyn (30 November 2013). "Counterexamples in origami". Roots of unity. Scientific American.
19. Miura, Koryo; Tachi, Tomohiro (2010). "Synthesis of rigid-foldable cylindrical polyhedra" (PDF). Symmetry: Art and Science, 8th Congress and Exhibition of ISIS. Gmünd.
20. Yoshimura, Yoshimaru (July 1955). On the mechanism of buckling of a circular cylindrical shell under axial compression. Technical Memorandum 1390. National Advisory Committee for Aeronautics.
21. Polthier, Konrad (2005). "Computational aspects of discrete minimal surfaces" (PDF). In Hoffman, David (ed.). Global theory of minimal surfaces: Proceedings of the Clay Mathematical Institute Summer School held in Berkeley, CA, June 25 – July 27, 2001. Clay Mathematics Proceedings. Vol. 2. Providence, Rhode Island: American Mathematical Society. pp. 65–111. doi:10.1016/j.cagd.2005.06.010. MR 2167256.
External links
• Bogomolny, Alexander. "The Schwarz Lantern Explained". Cut-the-knot.
• Marx, Jean-Pierre (17 September 2016). "The Schwarz Lantern". Math Counterexamples.
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Wikipedia
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Schwarz minimal surface
In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz.
In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces.[1][2] They were later named by Alan Schoen in his seminal report that described the gyroid and other triply periodic minimal surfaces.[3]
The surfaces were generated using symmetry arguments: given a solution to Plateau's problem for a polygon, reflections of the surface across the boundary lines also produce valid minimal surfaces that can be continuously joined to the original solution. If a minimal surface meets a plane at right angles, then the mirror image in the plane can also be joined to the surface. Hence given a suitable initial polygon inscribed in a unit cell periodic surfaces can be constructed.[4]
The Schwarz surfaces have topological genus 3, the minimal genus of triply periodic minimal surfaces.[5]
They have been considered as models for periodic nanostructures in block copolymers, electrostatic equipotential surfaces in crystals,[6] and hypothetical negatively curved graphite phases.[7]
Schwarz P ("Primitive")
Schoen named this surface 'primitive' because it has two intertwined congruent labyrinths, each with the shape of an inflated tubular version of the simple cubic lattice. While the standard P surface has cubic symmetry the unit cell can be any rectangular box, producing a family of minimal surfaces with the same topology.[8]
It can be approximated by the implicit surface
$\cos(x)+\cos(y)+\cos(z)=0\ $.[9]
The P surface has been considered for prototyping tissue scaffolds with a high surface-to-volume ratio and porosity.[10]
Schwarz D ("Diamond")
Schoen named this surface 'diamond' because it has two intertwined congruent labyrinths, each having the shape of an inflated tubular version of the diamond bond structure. It is sometimes called the F surface in the literature.
It can be approximated by the implicit surface
$\sin(x)\sin(y)\sin(z)+\sin(x)\cos(y)\cos(z)+\cos(x)\sin(y)\cos(z)+\cos(x)\cos(y)\sin(z)=0.\ $
An exact expression exists in terms of elliptic integrals, based on the Weierstrass representation.[11]
Schwarz H ("Hexagonal")
The H surface is similar to a catenoid with a triangular boundary, allowing it to tile space.
Schwarz CLP ("Crossed layers of parallels")
Illustrations
• http://www.susqu.edu/brakke/evolver/examples/periodic/periodic.html
• http://www.indiana.edu/~minimal/archive/Triply/genus3.html
• http://www.thphys.uni-heidelberg.de/~biophys/index.php?lang=e&n1=research_tpms
• https://web.archive.org/web/20160225062057/http://homepages.ulb.ac.be/~morahman/gallery/schwartz.html
• http://virtualmathmuseum.org/Surface/gallery_m.html
References
1. H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.
2. E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimalflächen", Akad. Abhandlungen, Helsingfors, 1883.
3. Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)
4. Hermann Karcher, Konrad Polthier, "Construction of Triply Periodic Minimal Surfaces", Phil. Trans. R. Soc. Lond. A 16 September 1996 vol. 354 no. 1715 2077–2104
5. "Alan Schoen geometry".
6. Mackay, Alan L. (April 1985). "Periodic minimal surfaces". Nature. 314 (6012): 604–606. Bibcode:1985Natur.314..604M. doi:10.1038/314604a0. S2CID 4267918.
7. Terrones, H.; Mackay, A. L. (December 1994). "Negatively curved graphite and triply periodic minimal surfaces". Journal of Mathematical Chemistry. 15 (1): 183–195. doi:10.1007/BF01277558. S2CID 123561096.
8. W. H. Meeks. The theory of triply-periodic minimal surfaces. Indiana University Math. Journal, 39 (3):877-936, 1990.
9. "Triply Periodic Level Surfaces". Archived from the original on 2019-02-12. Retrieved 2019-02-10.
10. Jaemin Shin, Sungki Kim, Darae Jeong, Hyun Geun Lee, Dongsun Lee, Joong Yeon Lim, and Junseok Kim, Finite Element Analysis of Schwarz P Surface Pore Geometries for Tissue-Engineered Scaffolds, Mathematical Problems in Engineering, Volume 2012, Article ID 694194, doi:10.1155/2012/694194
11. Paul J.F. Gandy, Djurdje Cvijović, Alan L. Mackay, Jacek Klinowski, Exact computation of the triply periodic D (`diamond') minimal surface, Chemical Physics Letters, Volume 314, Issues 5–6, 10 December 1999, Pages 543–551
Minimal surfaces
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Wikipedia
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Schwarz triangle function
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a Schwarz triangle, although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a Möbius triangle, the inverse of the Schwarz triangle function is a single-valued automorphic function for that triangle's triangle group. More specifically, it is a modular function.
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Formula
Let πα, πβ, and πγ be the interior angles at the vertices of the triangle in radians. Each of α, β, and γ may take values between 0 and 1 inclusive. Following Nehari,[1] these angles are in clockwise order, with the vertex having angle πα at the origin and the vertex having angle πγ lying on the real line. The Schwarz triangle function can be given in terms of hypergeometric functions as:
$s(\alpha ,\beta ,\gamma ;z)=z^{\alpha }{\frac {_{2}F_{1}\left(a',b';c';z\right)}{_{2}F_{1}\left(a,b;c;z\right)}}$
where
a = (1−α−β−γ)/2,
b = (1−α+β−γ)/2,
c = 1−α,
a′ = a − c + 1 = (1+α−β−γ)/2,
b′ = b − c + 1 = (1+α+β−γ)/2, and
c′ = 2 − c = 1 + α.
This function maps the upper half-plane to a spherical triangle if α + β + γ > 1, or a hyperbolic triangle if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: a = 0, $_{2}F_{1}\left(a,b;c;z\right)=1$, and the formula reduces to that given by the Schwarz–Christoffel transformation.
Derivation
Through the theory of complex ordinary differential equations with regular singular points and the Schwarzian derivative, the triangle function can be expressed as the quotient of two solutions of a hypergeometric differential equation with real coefficients and singular points at 0, 1 and ∞. By the Schwarz reflection principle, the reflection group induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two-dimensional representation corresponds to the monodromy of the ordinary differential equation and induces a group of Möbius transformations on quotients of hypergeometric functions.[2]
Singular points
This mapping has regular singular points at z = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,[3]
${\begin{aligned}s(0)&=0,\\[4mu]s(1)&={\frac {\Gamma (1-a')\Gamma (1-b')\Gamma (c')}{\Gamma (1-a)\Gamma (1-b)\Gamma (c)}},\\[8mu]s(\infty )&=\exp \left(i\pi \alpha \right){\frac {\Gamma (1-a')\Gamma (b)\Gamma (c')}{\Gamma (1-a)\Gamma (b')\Gamma (c)}},\end{aligned}}$
where $ \Gamma (x)$ is the gamma function.
Near each singular point, the function may be approximated as
${\begin{aligned}s_{0}(z)&=z^{\alpha }(1+O(z)),\\[6mu]s_{1}(z)&=(1-z)^{\gamma }(1+O(1-z)),\\[6mu]s_{\infty }(z)&=z^{\beta }(1+O(1/z)),\end{aligned}}$
where $O(x)$ is big O notation.
Inverse
When α, β, and γ are rational, the triangle is a Schwarz triangle. When each of α, β, and γ are either the reciprocal of an integer or zero, the triangle is a Möbius triangle, i.e. a non-overlapping Schwarz triangle. For a Möbius triangle, the inverse is a modular function.
In the spherical case, that modular function is a rational function. For Euclidean triangles, the inverse can be expressed using elliptical functions.[4]
Ideal triangles
When α = 0 the triangle is degenerate, lying entirely on the real line. If either of β or γ are non-zero, the angles can be permuted so that the positive value is α, but that is not an option for an ideal triangle having all angles zero.
Instead, a mapping to an ideal triangle with vertices at 0, 1, and ∞ is given by in terms of the complete elliptic integral of the first kind:
$i{\frac {K(1-z)}{K(z)}}$.
This expression is the inverse of the modular lambda function.[5]
Extensions
The Schwarz–Christoffel transformation gives the mapping from the upper half-plane to any Euclidean polygon.
The methodology used to derive the Schwarz triangle function earlier can be applied more generally to arc-edged polygons. However, for an n-sided polygon, the solution has n-3 additional parameters, which are difficult to determine in practice.[6] See Schwarzian derivative § Conformal mapping of circular arc polygons for more details.
Applications
L. P. Lee used Schwarz triangle functions to derive conformal map projections onto polyhedral surfaces.[4]
References
1. Nehari 1975, p. 309.
2. Nehari 1975, pp. 198–208.
3. Nehari 1975, pp. 315−316.
4. Lee, Laurence (1976). Conformal Projections based on Elliptic Functions. Cartographica Monographs. Vol. 16. University of Toronto Press. ISBN 9780919870161. Chapters also published in The Canadian Cartographer. 13 (1). 1976.
5. Nehari 1975, pp. 316–318.
6. Nehari 1975, p. 202.
Sources
• Ahlfors, Lars V. (1979). Complex analysis: an introduction to the theory of analytic functions of one complex variable (3 ed.). New York: McGraw-Hill. ISBN 0-07-000657-1. OCLC 4036464.
• Carathéodory, Constantin (1954). Theory of functions of a complex variable. Vol. 2. Translated by F. Steinhardt. Chelsea. OCLC 926250115.
• Hille, Einar (1997). Ordinary differential equations in the complex domain. Mineola, N.Y.: Dover Publications. ISBN 0-486-69620-0. OCLC 36225146.
• Nehari, Zeev (1975). Conformal mapping. New York: Dover Publications. ISBN 0-486-61137-X. OCLC 1504503.
• Sansone, Giovanni; Gerretsen, Johan (1969). Lectures on the theory of functions of a complex variable. II: Geometric theory. Wolters-Noordhoff. OCLC 245996162.
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Wikipedia
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Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in Schwarz (1873).
These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group.
A Schwarz triangle is represented by three rational numbers (p q r), each representing the angle at a vertex. The value n⁄d means the vertex angle is d⁄n of the half-circle. "2" means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a non-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is a three-parameter family of Möbius triangles, and no exceptional objects.
Solution space
A fundamental domain triangle (p q r), with vertex angles π⁄p, π⁄q, and π⁄r, can exist in different spaces depending on the value of the sum of the reciprocals of these integers:
${\begin{aligned}{\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}&>1{\text{ : Sphere}}\\[8pt]{\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}&=1{\text{ : Euclidean plane}}\\[8pt]{\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}&<1{\text{ : Hyperbolic plane}}\end{aligned}}$ : Sphere}}\\[8pt]{\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}&=1{\text{ : Euclidean plane}}\\[8pt]{\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}&<1{\text{ : Hyperbolic plane}}\end{aligned}}}
This is simply a way of saying that in Euclidean space the interior angles of a triangle sum to π, while on a sphere they sum to an angle greater than π, and on hyperbolic space they sum to less.
Graphical representation
A Schwarz triangle is represented graphically by a triangular graph. Each node represents an edge (mirror) of the Schwarz triangle. Each edge is labeled by a rational value corresponding to the reflection order, being π/vertex angle.
Schwarz triangle (p q r) on sphere
Schwarz triangle graph
Order-2 edges represent perpendicular mirrors that can be ignored in this diagram. The Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden.
A Coxeter group can be used for a simpler notation, as (p q r) for cyclic graphs, and (p q 2) = [p,q] for (right triangles), and (p 2 2) = [p]×[].
A list of Schwarz triangles
Möbius triangles for the sphere
(2 2 2) or [2,2]
(3 2 2) or [3,2]
...
(3 3 2) or [3,3]
(4 3 2) or [4,3]
(5 3 2) or [5,3]
Schwarz triangles with whole numbers, also called Möbius triangles, include one 1-parameter family and three exceptional cases:
1. [p,2] or (p 2 2) – Dihedral symmetry,
2. [3,3] or (3 3 2) – Tetrahedral symmetry,
3. [4,3] or (4 3 2) – Octahedral symmetry,
4. [5,3] or (5 3 2) – Icosahedral symmetry,
Schwarz triangles for the sphere by density
The Schwarz triangles (p q r), grouped by density:
Density Dihedral Tetrahedral Octahedral Icosahedral
d(2 2 n/d)
1(2 3 3)(2 3 4)(2 3 5)
2(3/2 3 3)(3/2 4 4)(3/2 5 5), (5/2 3 3)
3(2 3/2 3)(2 5/2 5)
4(3 4/3 4)(3 5/3 5)
5(2 3/2 3/2)(2 3/2 4)
6(3/2 3/2 3/2)(5/2 5/2 5/2), (3/2 3 5), (5/4 5 5)
7(2 3 4/3)(2 3 5/2)
8(3/2 5/2 5)
9(2 5/3 5)
10(3 5/3 5/2), (3 5/4 5)
11(2 3/2 4/3)(2 3/2 5)
13(2 3 5/3)
14(3/2 4/3 4/3)(3/2 5/2 5/2), (3 3 5/4)
16(3 5/4 5/2)
17(2 3/2 5/2)
18(3/2 3 5/3), (5/3 5/3 5/2)
19(2 3 5/4)
21(2 5/4 5/2)
22(3/2 3/2 5/2)
23(2 3/2 5/3)
26(3/2 5/3 5/3)
27(2 5/4 5/3)
29(2 3/2 5/4)
32(3/2 5/4 5/3)
34(3/2 3/2 5/4)
38(3/2 5/4 5/4)
42(5/4 5/4 5/4)
Triangles for the Euclidean plane
(3 3 3)
(4 4 2)
(6 3 2)
Density 1:
1. (3 3 3) – 60-60-60 (equilateral),
2. (4 4 2) – 45-45-90 (isosceles right),
3. (6 3 2) – 30-60-90,
Density 2:
1. (6 6 3/2) - 120-30-30 triangle
Density ∞:
1. (4 4/3 ∞)
2. (3 3/2 ∞)
3. (6 6/5 ∞)
Triangles for the hyperbolic plane
(7 3 2)
(8 3 2)
(5 4 2)
(4 3 3)
(4 4 3)
(∞ ∞ ∞)
Fundamental domains of (p q r) triangles
Density 1:
• (2 3 7), (2 3 8), (2 3 9) ... (2 3 ∞)
• (2 4 5), (2 4 6), (2 4 7) ... (2 4 ∞)
• (2 5 5), (2 5 6), (2 5 7) ... (2 5 ∞)
• (2 6 6), (2 6 7), (2 6 8) ... (2 6 ∞)
• (3 3 4), (3 3 5), (3 3 6) ... (3 3 ∞)
• (3 4 4), (3 4 5), (3 4 6) ... (3 4 ∞)
• (3 5 5), (3 5 6), (3 5 7) ... (3 5 ∞)
• (3 6 6), (3 6 7), (3 6 8) ... (3 6 ∞)
• ...
• (∞ ∞ ∞)
Density 2:
• (3/2 7 7), (3/2 8 8), (3/2 9 9) ... (3/2 ∞ ∞)
• (5/2 4 4), (5/2 5 5), (5/2 6 6) ... (5/2 ∞ ∞)
• (7/2 3 3), (7/2 4 4), (7/2 5 5) ... (7/2 ∞ ∞)
• (9/2 3 3), (9/2 4 4), (9/2 5 5) ... (9/2 ∞ ∞)
• ...
Density 3:
• (2 7/2 7), (2 9/2 9), (2 11/2 11) ...
Density 4:
• (7/3 3 7), (8/3 3 8), (3 10/3 10), (3 11/3 11) ...
Density 6:
• (7/4 7 7), (9/4 9 9), (11/4 11 11) ...
• (7/2 7/2 7/2), (9/2 9/2 9/2), ...
Density 10:
• (3 7/2 7)
The (2 3 7) Schwarz triangle is the smallest hyperbolic Schwarz triangle, and as such is of particular interest. Its triangle group (or more precisely the index 2 von Dyck group of orientation-preserving isometries) is the (2,3,7) triangle group, which is the universal group for all Hurwitz groups – maximal groups of isometries of Riemann surfaces. All Hurwitz groups are quotients of the (2,3,7) triangle group, and all Hurwitz surfaces are tiled by the (2,3,7) Schwarz triangle. The smallest Hurwitz group is the simple group of order 168, the second smallest non-abelian simple group, which is isomorphic to PSL(2,7), and the associated Hurwitz surface (of genus 3) is the Klein quartic.
The (2 3 8) triangle tiles the Bolza surface, a highly symmetric (but not Hurwitz) surface of genus 2.
The triangles with one noninteger angle, listed above, were first classified by Anthony W. Knapp in.[1] A list of triangles with multiple noninteger angles is given in.[2]
Tessellation by Schwarz triangles
In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For triangles without "cusps"—angles equal to zero or equivalently vertices on the real axis—the elementary approach of Carathéodory (1954) will be followed. For triangles with one or two cusps, elementary arguments of Evans (1973), simplifying the approach of Hecke (1935), will be used: in the case of a Schwarz triangle with one angle zero and another a right angle, the orientation-preserving subgroup of the reflection group of the triangle is a Hecke group. For an ideal triangle in which all angles are zero, so that all vertices lie on the real axis, the existence of the tessellation will be established by relating it to the Farey series described in Hardy & Wright (2008) and Series (2015). In this case the tessellation can be considered as that associated with three touching circles on the Riemann sphere, a limiting case of configurations associated with three disjoint non-nested circles and their reflection groups, the so-called "Schottky groups", described in detail in Mumford, Series & Wright (2015). Alternatively—by dividing the ideal triangle into six triangles with angles 0, π/2 and π/3—the tessellation by ideal triangles can be understood in terms of tessellations by triangles with one or two cusps.
Triangles without cusps
Suppose that the hyperbolic triangle Δ has angles π/a, π/b and π/c with a, b, c integers greater than 1. The hyperbolic area of Δ equals π – π/a – π/b – π/c, so that
${\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}<1.$
The construction of a tessellation will first be carried out for the case when a, b and c are greater than 2.[3]
The original triangle Δ gives a convex polygon P1 with 3 vertices. At each of the three vertices the triangle can be successively reflected through edges emanating from the vertices to produce 2m copies of the triangle where the angle at the vertex is π/m. The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point. The union of these new triangles together with the original triangle form a connected shape P2. It is made up of triangles which only intersect in edges or vertices, forms a convex polygon with all angles less than or equal to π and each side being the edge of a reflected triangle. In the case when an angle of Δ equals π/3, a vertex of P2 will have an interior angle of π, but this does not affect the convexity of P2. Even in this degenerate case when an angle of π arises, the two collinear edge are still considered as distinct for the purposes of the construction.
The construction of P2 can be understood more clearly by noting that some triangles or tiles are added twice, the three which have a side in common with the original triangle. The rest have only a vertex in common. A more systematic way of performing the tiling is first to add a tile to each side (the reflection of the triangle in that edge) and then fill in the gaps at each vertex. This results in a total of 3 + (2a – 3) + (2b - 3) + (2c - 3) = 2(a + b + c) - 6 new triangles. The new vertices are of two types. Those which are vertices of the triangles attached to sides of the original triangle, which are connected to 2 vertices of Δ. Each of these lie in three new triangles which intersect at that vertex. The remainder are connected to a unique vertex of Δ and belong to two new triangles which have a common edge. Thus there are 3 + (2a – 4) + (2b - 4) + (2c - 4) = 2(a + b + c) - 9 new vertices. By construction there is no overlapping. To see that P2 is convex, it suffices to see that the angle between sides meeting at a new vertex make an angle less than or equal to π. But the new vertices lies in two or three new triangles, which meet at that vertex, so the angle at that vertex is no greater than 2π/3 or π, as required.
This process can be repeated for P2 to get P3 by first adding tiles to each edge of P2 and then filling in the tiles round each vertex of P2. Then the process can be repeated from P3, to get P4 and so on, successively producing Pn from Pn – 1. It can be checked inductively that these are all convex polygons, with non-overlapping tiles. Indeed, as in the first step of the process there are two types of tile in building Pn from Pn – 1, those attached to an edge of Pn – 1 and those attached to a single vertex. Similarly there are two types of vertex, one in which two new tiles meet and those in which three tiles meet. So provided that no tiles overlap, the previous argument shows that angles at vertices are no greater than π and hence that Pn is a convex polygon.[lower-alpha 1]
It therefore has to be verified that in constructing Pn from Pn − 1:[4]
(a) the new triangles do not overlap with Pn − 1 except as already described;
(b) the new triangles do not overlap with each other except as already described;
(c) the geodesic from any point in Δ to a vertex of the polygon Pn – 1 makes an angle ≤ 2π/3 with each of the edges of the polygon at that vertex.
To prove (a), note that by convexity, the polygon Pn − 1 is the intersection of the convex half-spaces defined by the full circular arcs defining its boundary. Thus at a given vertex of Pn − 1 there are two such circular arcs defining two sectors: one sector contains the interior of Pn − 1, the other contains the interiors of the new triangles added around the given vertex. This can be visualized by using a Möbius transformation to map the upper half plane to the unit disk and the vertex to the origin; the interior of the polygon and each of the new triangles lie in different sectors of the unit disk. Thus (a) is proved.
Before proving (c) and (b), a Möbius transformation can be applied to map the upper half plane to the unit disk and a fixed point in the interior of Δ to the origin.
The proof of (c) proceeds by induction. Note that the radius joining the origin to a vertex of the polygon Pn − 1 makes an angle of less than 2π/3 with each of the edges of the polygon at that vertex if exactly two triangles of Pn − 1 meet at the vertex, since each has an angle less than or equal to π/3 at that vertex. To check this is true when three triangles of Pn − 1 meet at the vertex, C say, suppose that the middle triangle has its base on a side AB of Pn − 2. By induction the radii OA and OB makes angles of less than or equal to 2π/3 with the edge AB. In this case the region in the sector between the radii OA and OB outside the edge AB is convex as the intersection of three convex regions. By induction the angles at A and B are greater than or equal to π/3. Thus the geodesics to C from A and B start off in the region; by convexity, the triangle ABC lies wholly inside the region. The quadrilateral OACB has all its angles less than π (since OAB is a geodesic triangle), so is convex. Hence the radius OC lies inside the angle of the triangle ABC near C. Thus the angles between OC and the two edges of Pn – 1 meeting at C are less than or equal to π/3 + π/3 = 2π/3, as claimed.
To prove (b), it must be checked how new triangles in Pn intersect.
First consider the tiles added to the edges of Pn – 1. Adopting similar notation to (c), let AB be the base of the tile and C the third vertex. Then the radii OA and OB make angles of less than or equal to 2π/3 with the edge AB and the reasoning in the proof of (c) applies to prove that the triangle ABC lies within the sector defined by the radii OA and OB. This is true for each edge of Pn – 1. Since the interiors of sectors defined by distinct edges are disjoint, new triangles of this type only intersect as claimed.
Next consider the additional tiles added for each vertex of Pn – 1. Taking the vertex to be A, three are two edges AB1 and AB2 of Pn – 1 that meet at A. Let C1 and C2 be the extra vertices of the tiles added to these edges. Now the additional tiles added at A lie in the sector defined by radii OB1 and OB2. The polygon with vertices C2 O, C1, and then the vertices of the additional tiles has all its internal angles less than π and hence is convex. It is therefore wholly contained in the sector defined by the radii OC1 and OC2. Since the interiors of these sectors are all disjoint, this implies all the claims about how the added tiles intersect.
Finally it remains to prove that the tiling formed by the union of the triangles covers the whole of the upper half plane. Any point z covered by the tiling lies in a polygon Pn and hence a polygon Pn +1 . It therefore lies in a copy of the original triangle Δ as well as a copy of P2 entirely contained in Pn +1 . The hyperbolic distance between Δ and the exterior of P2 is equal to r > 0. Thus the hyperbolic distance between z and points not covered by the tiling is at least r. Since this applies to all points in the tiling, the set covered by the tiling is closed. On the other hand, the tiling is open since it coincides with the union of the interiors of the polygons Pn. By connectivity, the tessellation must cover the whole of the upper half plane.
To see how to handle the case when an angle of Δ is a right angle, note that the inequality
${\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}<1$.
implies that if one of the angles is a right angle, say a = 2, then both b and c are greater than 2 and one of them, b say, must be greater than 3. In this case, reflecting the triangle across the side AB gives an isosceles hyperbolic triangle with angles π/c, π/c and 2π/b. If 2π/b ≤ π/3, i.e. b is greater than 5, then all the angles of the doubled triangle are less than or equal to π/3. In that case the construction of the tessellation above through increasing convex polygons adapts word for word to this case except that around the vertex with angle 2π/b, only b—and not 2b—copies of the triangle are required to tile a neighborhood of the vertex. This is possible because the doubled triangle is isosceles. The tessellation for the doubled triangle yields that for the original triangle on cutting all the larger triangles in half.[5]
It remains to treat the case when b equals 4 or 5. If b = 4, then c ≥ 5: in this case if c ≥ 6, then b and c can be switched and the argument above applies, leaving the case b = 4 and c = 5. If b = 5, then c ≥ 4. The case c ≥ 6 can be handled by swapping b and c, so that the only extra case is b = 5 and c = 5. This last isosceles triangle is the doubled version of the first exceptional triangle, so only that triangle Δ1—with angles π/2, π/4 and π/5 and hyperbolic area π/20—needs to be considered (see below). Carathéodory (1954) handles this case by a general method which works for all right angled triangles for which the two other angles are less than or equal to π/4. The previous method for constructing P2, P3, ... is modified by adding an extra triangle each time an angle 3π/2 arises at a vertex. The same reasoning applies to prove there is no overlapping and that the tiling covers the hyperbolic upper half plane.[5]
On the other hand, the given configuration gives rise to an arithmetic triangle group. These were first studied in Fricke & Klein (1897). and have given rise to an extensive literature. In 1977 Takeuchi obtained a complete classification of arithmetic triangle groups (there are only finitely many) and determined when two of them are commensurable. The particular example is related to Bring's curve and the arithmetic theory implies that the triangle group for Δ1 contains the triangle group for the triangle Δ2 with angles π/4, π/4 and π/5 as a non-normal subgroup of index 6.[6]
Doubling the triangles Δ1 and Δ2, this implies that there should be a relation between 6 triangles Δ3 with angles π/2, π/5 and π/5 and hyperbolic area π/10 and a triangle Δ4 with angles π/5, π/5 and π/10 and hyperbolic area 3π/5. Threlfall (1932) established such a relation directly by completely elementary geometric means, without reference to the arithmetic theory: indeed as illustrated in the fifth figure below, the quadrilateral obtained by reflecting across a side of a triangle of type Δ4 can be tiled by 12 triangles of type Δ3. The tessellation by triangles of the type Δ4 can be handled by the main method in this section; this therefore proves the existence of the tessellation by triangles of type Δ3 and Δ1.[7]
• Tessellation by triangles with angles π/2, π/5 and π/5
• Tessellation obtained by coalescing two triangles
• Tiling with pentagons formed from 10 (2,5,5) triangles
• Adjusting to tiling by triangles with angles π/5, π/10, π/10
• Tiling 2 (5,10,10) triangles with 12 (2,5,5) triangles
Triangles with one or two cusps
In the case of a Schwarz triangle with one or two cusps, the process of tiling becomes simpler; but it is easier to use a different method going back to Hecke to prove that these exhaust the hyperbolic upper half plane.
In the case of one cusp and non-zero angles π/a, π/b with a, b integers greater than one, the tiling can be envisaged in the unit disk with the vertex having angle π/a at the origin. The tiling starts by adding 2a – 1 copies of the triangle at the origin by successive reflections. This results in a polygon P1 with 2a cusps and between each two 2a vertices each with an angle π/b. The polygon is therefore convex. For each non-ideal vertex of P1, the unique triangle with that vertex can be similar reflected around that vertex, thus adding 2b – 1 new triangles, 2b – 1 new ideal points and 2 b – 1 new vertices with angle π/a. The resulting polygon P2 is thus made up of 2a(2b – 1) cusps and the same number of vertices each with an angle of π/a, so is convex. The process can be continued in this way to obtain convex polygons P3, P4, and so on. The polygon Pn will have vertices having angles alternating between 0 and π/a for n even and between 0 and π/b for n odd. By construction the triangles only overlap at edges or vertices, so form a tiling.[8]
• Tessellation by triangle with angles 0, π/3, π/5
• Tessellation by triangle with angles 0, π/5, π/2
• Tessellation by triangle with angles 0, 0, π/5
The case where the triangle has two cusps and one non-zero angle π/a can be reduced to the case of one cusp by observing that the trinale is the double of a triangle with one cusp and non-zero angles π/a and π/b with b = 2. The tiling then proceeds as before.[9]
To prove that these give tessellations, it is more convenient to work in the upper half plane. Both cases can be treated simultaneously, since the case of two cusps is obtained by doubling a triangle with one cusp and non-zero angles π/a and π/2. So consider the geodesic triangle in the upper half plane with angles 0, π/a, π/b with a, b integers greater than one. The interior of such a triangle can be realised as the region X in the upper half plane lying outside the unit disk |z| ≤ 1 and between two lines parallel to the imaginary axis through points u and v on the unit circle. Let Γ be the triangle group generated by the three reflections in the sides of the triangle.
To prove that the successive reflections of the triangle cover the upper half plane, it suffices to show that for any z in the upper half plane there is a g in Γ such that g(z) lies in X. This follows by an argument of Evans (1973), simplified from the theory of Hecke groups. Let λ = Re a and μ = Re b so that, without loss of generality, λ < 0 ≤ μ. The three reflections in the sides are given by
$R_{1}(z)={\frac {1}{\overline {z}}},\ R_{2}(z)=-{\overline {z}}+\lambda ,\ R_{3}(z)=-{\overline {z}}+\mu .$
Thus T = R3∘R2 is translation by μ − λ. It follows that for any z1 in the upper half plane, there is an element g1 in the subgroup Γ1 of Γ generated by T such that w1 = g1(z1) satisfies λ ≤ Re w1 ≤ μ, i.e. this strip is a fundamental domain for the translation group Γ1. If |w1| ≥ 1, then w1 lies in X and the result is proved. Otherwise let z2 = R1(w1) and find g2 Γ1 such that w2 = g2(z2) satisfies λ ≤ Re w2 ≤ μ. If |w2| ≥ 1 then the result is proved. Continuing in this way, either some wn satisfies |wn| ≥ 1, in which case the result is proved; or |wn| < 1 for all n. Now since gn + 1 lies in Γ1 and |wn| < 1,
$\operatorname {Im} g_{n+1}(z_{n+1})=\operatorname {Im} z_{n+1}=\operatorname {Im} {\frac {w_{n}}{|w_{n}|{}^{2}}}={\frac {\operatorname {Im} w_{n}}{|w_{n}|{}^{2}}}.$
In particular
$\operatorname {Im} w_{n+1}\geq \operatorname {Im} w_{n}$
and
${\frac {\operatorname {Im} w_{n+1}}{\operatorname {Im} w_{n}}}=|w_{n}|^{-2}\geq 1.$
Thus, from the inequality above, the points (wn) lies in the compact set |z| ≤ 1, λ ≤ Re z ≤ μ and Im z ≥ Im w 1. It follows that |wn| tends to 1; for if not, then there would be an r < 1 such that |wm| ≤ r for inifitely many m and then the last equation above would imply that Im wn tends to infinity, a contradiction.
Let w be a limit point of the wn, so that |w| = 1. Thus w lies on the arc of the unit circle between u and v. If w ≠ u, v, then R1 wn would lie in X for n sufficiently large, contrary to assumption. Hence w =u or v. Hence for n sufficiently large wn lies close to u or v and therefore must lie in one of the reflections of the triangle about the vertex u or v, since these fill out neighborhoods of u and v. Thus there is an element g in Γ such that g(wn) lies in X. Since by construction wn is in the Γ-orbit of z1, it follows that there is a point in this orbit lying in X, as required.[10]
Ideal triangles
The tessellation for an ideal triangle with all its vertices on the unit circle and all its angles 0 can be considered as a special case of the tessellation for a triangle with one cusp and two now zero angles π/3 and π/2. Indeed, the ideal triangle is made of six copies one-cusped triangle obtained by reflecting the smaller triangle about the vertex with angle π/3.
• Tessellation for triangle with angles 0, π/3 and π/2
• Tessellation for ideal triangle
• Line drawing of tessellation by ideal triangles
Each step of the tiling, however, is uniquely determined by the positions of the new cusps on the circle, or equivalently the real axis; and these points can be understood directly in terms of Farey series following Series (2015), Hatcher (2013, pp. 20–32) and Hardy & Wright (2008, pp. 23–31). This starts from the basic step that generates the tessellation, the reflection of an ideal triangle in one of its sides. Reflection corresponds to the process of inversion in projective geometry and taking the projective harmonic conjugate, which can be defined in terms of the cross ratio. In fact if p, q, r, s are distinct points in the Riemann sphere, then there is a unique complex Möbius transformation g sending p, q and s to 0, ∞ and 1 respectively. The cross ratio (p, q; r, s) is defined to be g(r) and is given by the formula
$(p,q;r,s)={\frac {(p-r)(q-s)}{(p-s)(q-r)}}.$
By definition it is invariant under Möbius transformations. If a, b lie on the real axis, the harmonic conjugate of c with respect to a and b is defined to be the unique real number d such that (a, b; c, d) = −1. So for example if a = 1 and b = –1, the conjugate of r is 1/r. In general Möbius invariance can be used to obtain an explicit formula for d in terms of a, b and c. Indeed, translating the centre t = (a + b)/2 of the circle with diameter having endpoints a and b to 0, d – t is the harmonic conjugate of c – t with respect to a - t and b – t. The radius of the circle is ρ = (b – a)/2 so (d - t)/ρ is the harmonic conjugate of (c – t)/ρ with respect to 1 and -1. Thus
${\frac {d-t}{\rho }}={\frac {\rho }{c-t}}$
so that
$d={\frac {\rho ^{2}}{r-t}}+t={\frac {(c-a)b+(c-b)a}{(c-a)+(c-b)}}.$
It will now be shown that there is a parametrisation of such ideal triangles given by rationals in reduced form
$a={\frac {p_{1}}{q_{1}}},\ b={\frac {p_{1}+p_{2}}{q_{1}+q_{2}}},\ c={\frac {p_{2}}{q_{2}}}$
with a and c satisfying the "neighbour condition" p2q1 − q2p1 = 1.
The middle term b is called the Farey sum or mediant of the outer terms and written
$b=a\oplus c.$
The formula for the reflected triangle gives
$d={\frac {p_{1}+2p_{2}}{q_{1}+2q_{2}}}=a\oplus b.$
Similarly the reflected triangle in the second semicircle gives a new vertex b ⊕ c. It is immediately verified that a and b satisfy the neighbour condition, as do b and c.
Now this procedure can be used to keep track of the triangles obtained by successively reflecting the basic triangle Δ with vertices 0, 1 and ∞. It suffices to consider the strip with 0 ≤ Re z ≤ 1, since the same picture is reproduced in parallel strips by applying reflections in the lines Re z = 0 and 1. The ideal triangle with vertices 0, 1, ∞ reflects in the semicircle with base [0,1] into the triangle with vertices a = 0, b = 1/2, c = 1. Thus a = 0/1 and c = 1/1 are neighbours and b = a ⊕ c. The semicircle is split up into two smaller semicircles with bases [a,b] and [b,c]. Each of these intervals splits up into two intervals by the same process, resulting in 4 intervals. Continuing in this way, results into subdivisions into 8, 16, 32 intervals, and so on. At the nth stage, there are 2n adjacent intervals with 2n + 1 endpoints. The construction above shows that successive endpoints satisfy the neighbour condition so that new endpoints resulting from reflection are given by the Farey sum formula.
To prove that the tiling covers the whole hyperbolic plane, it suffices to show that every rational in [0,1] eventually occurs as an endpoint. There are several ways to see this. One of the most elementary methods is described in Graham, Knuth & Patashnik (1994) in their development—without the use of continued fractions—of the theory of the Stern-Brocot tree, which codifies the new rational endpoints that appear at the nth stage. They give a direct proof that every rational appears. Indeed, starting with {0/1,1/1}, successive endpoints are introduced at level n+1 by adding Farey sums or mediants (p+r)/(q+s) between all consecutive terms p/q, r/s at the nth level (as described above). Let x = a/b be a rational lying between 0 and 1 with a and b coprime. Suppose that at some level x is sandwiched between successive terms p/q < x < r/s. These inequalities force aq – bp ≥ 1 and br – as ≥ 1 and hence, since rp – qs = 1,
$a+b=(r+s)(ap-bq)+(p+q)(br-as)\geq p+q+r+s.$
This puts an upper bound on the sum of the numerators and denominators. On the other hand, the mediant (p+r)/(q+s) can be introduced and either equals x, in which case the rational x appears at this level; or the mediant provides a new interval containing x with strictly larger numerator-and-denominator sum. The process must therefore terminate after at most a + b steps, thus proving that x appears.[11]
A second approach relies on the modular group G = SL(2,Z).[12] The Euclidean algorithm implies that this group is generated by the matrices
$S={\begin{pmatrix}0&1\\-1&0\end{pmatrix}},\,\,\,T={\begin{pmatrix}1&1\\0&1\end{pmatrix}}.$
In fact let H be the subgroup of G generated by S and T. Let
$g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$
be an element of SL(2,Z). Thus ad − cb = 1, so that a and c are coprime. Let
$v={\begin{pmatrix}a\\c\end{pmatrix}},\,\,\,u={\begin{pmatrix}1\\0\end{pmatrix}}.$
Applying S if necessary, it can be assumed that |a| > |c| (equality is not possible by coprimeness). We write a = mc + r with 0 ≤ r ≤ |c|. But then
$T^{-m}{\begin{pmatrix}a\\c\end{pmatrix}}={\begin{pmatrix}r\\c\end{pmatrix}}.$
This process can be continued until one of the entries is 0, in which case the other is necessarily ±1. Applying a power of S if necessary, it follows that v = h u for some h in H. Hence
$h^{-1}g={\begin{pmatrix}1&p\\0&q\end{pmatrix}}$
with p, q integers. Clearly p = 1, so that h−1g = Tq. Thus g = h Tq lies in H as required.
To prove that all rationals in [0,1] occur, it suffices to show that G carries Δ onto triangles in the tessellation. This follows by first noting that S and T carry Δ on to such a triangle: indeed as Möbius transformations, S(z) = –1/z and T(z) = z + 1, so these give reflections of Δ in two of its sides. But then S and T conjugate the reflections in the sides of Δ into reflections in the sides of SΔ and TΔ, which lie in Γ. Thus G normalizes Γ. Since triangles in the tessellation are exactly those of the form gΔ with g in Γ, it follows that S and T, and hence all elements of G, permute triangles in the tessellation. Since every rational is of the form g(0) for g in G, every rational in [0,1] is the vertex of a triangle in the tessellation.
The reflection group and tessellation for an ideal triangle can also be regarded as a limiting case of the Schottky group for three disjoint unnested circles on the Riemann sphere. Again this group is generated by hyperbolic reflections in the three circles. In both cases the three circles have a common circle which cuts them orthogonally. Using a Möbius transformation, it may be assumed to be the unit circle or equivalently the real axis in the upper half plane.[13]
Approach of Siegel
In this subsection the approach of Carl Ludwig Siegel to the tessellation theorem for triangles is outlined. Siegel's less elementary approach does not use convexity, instead relying on the theory of Riemann surfaces, covering spaces and a version of the monodromy theorem for coverings. It has been generalized to give proofs of the more general Poincaré polygon theorem. (Note that the special case of tiling by regular n-gons with interior angles 2π/n is an immediate consequence of the tessellation by Schwarz triangles with angles π/n, π/n and π/2.)[14][15]
Let Γ be the free product Z2 ∗ Z2 ∗ Z2. If Δ = ABC is a Schwarz triangle with angles π/a, π/b and π/c, where a, b, c ≥ 2, then there is a natural map of Γ onto the group generated by reflections in the sides of Δ. Elements of Γ are described by a product of the three generators where no two adjacent generators are equal. At the vertices A, B and C the product of reflections in the sides meeting at the vertex define rotations by angles 2π/a, 2π/b and 2π/c; Let gA, gB and gC be the corresponding products of generators of Γ = Z2 ∗ Z2 ∗ Z2. Let Γ0 be the normal subgroup of index 2 of Γ, consisting of elements that are the product of an even number of generators; and let Γ1 be the normal subgroup of Γ generated by (gA)a, (gB)b and (gC)c. These act trivially on Δ. Let Γ = Γ/Γ1 and Γ0 = Γ0/Γ1.
The disjoint union of copies of Δ indexed by elements of Γ with edge identifications has the natural structure of a Riemann surface Σ. At an interior point of a triangle there is an obvious chart. As a point of the interior of an edge the chart is obtained by reflecting the triangle across the edge. At a vertex of a triangle with interior angle π/n, the chart is obtained from the 2n copies of the triangle obtained by reflecting it successively around that vertex. The group Γ acts by deck transformations of Σ, with elements in Γ0 acting as holomorphic mappings and elements not in Γ0 acting as antiholomorphic mappings.
There is a natural map P of Σ into the hyperbolic plane. The interior of the triangle with label g in Γ is taken onto g(Δ), edges are taken to edges and vertices to vertices. It is also easy to verify that a neighbourhood of an interior point of an edge is taken into a neighbourhood of the image; and similarly for vertices. Thus P is locally a homeomorphism and so takes open sets to open sets. The image P(Σ), i.e. the union of the translates g(Δ), is therefore an open subset of the upper half plane. On the other hand, this set is also closed. Indeed, if a point is sufficiently close to Δ it must be in a translate of Δ. Indeed, a neighbourhood of each vertex is filled out the reflections of Δ and if a point lies outside these three neighbourhoods but is still close to Δ it must lie on the three reflections of Δ in its sides. Thus there is δ > 0 such that if z lies within a distance less than δ from Δ, then z lies in a Γ-translate of Δ. Since the hyperbolic distance is Γ-invariant, it follows that if z lies within a distance less than δ from Γ(Δ) it actually lies in Γ(Δ), so this union is closed. By connectivity it follows that P(Σ) is the whole upper half plane.
On the other hand, P is a local homeomorphism, so a covering map. Since the upper half plane is simply connected, it follows that P is one-one and hence the translates of Δ tessellate the upper half plane. This is a consequence of the following version of the monodromy theorem for coverings of Riemann surfaces: if Q is a covering map between Riemann surfaces Σ1 and Σ2, then any path in Σ2 can be lifted to a path in Σ1 and any two homotopic paths with the same end points lift to homotopic paths with the same end points; an immediate corollary is that if Σ2 is simply connected, Q must be a homeomorphism.[16] To apply this, let Σ1 = Σ, let Σ2 be the upper half plane and let Q = P. By the corollary of the monodromy theorem, P must be one-one.
It also follows that g(Δ) = Δ if and only if g lies in Γ1, so that the homomorphism of Γ0 into the Möbius group is faithful.
Hyperbolic reflection groups
See also: Uniform tilings in hyperbolic plane
The tessellation of the Schwarz triangles can be viewed as a generalization of the theory of infinite Coxeter groups, following the theory of hyperbolic reflection groups developed algebraically by Jacques Tits[17] and geometrically by Ernest Vinberg.[18] In the case of the Lobachevsky or hyperbolic plane, the ideas originate in the nineteenth-century work of Henri Poincaré and Walther von Dyck. As Joseph Lehner has pointed out in Mathematical Reviews, however, rigorous proofs that reflections of a Schwarz triangle generate a tessellation have often been incomplete, his own 1964 book "Discontinuous Groups and Automorphic Forms", being one example.[19][20] Carathéodory's elementary treatment in his 1950 textbook "Funktiontheorie", translated into English in 1954, and Siegel's 1954 account using the monodromy principle are rigorous proofs. The approach using Coxeter groups will be summarised here, within the general framework of classification of hyperbolic reflection groups.[21]
Let r, s and t be symbols and let a, b, c ≥ 2 be integers, possibly ∞, with
${1 \over a}+{1 \over b}+{1 \over c}<1.$
Define Γ to be the group with presentation having generators r, s and t that are all involutions and satisfy (st)a = 1, (tr)b = 1 and (rs)c = 1. If one of the integers is infinite, then the product has infinite order. The generators r, s and t are called the simple reflections.
Set A = cos π / a if a ≥ 2 is finite and cosh x with x > 0 otherwise; similarly set B = cos π / b or cosh y and C = cos π / c or cosh z.[22] Let er, es and et be a basis for a 3-dimensional real vector space V with symmetric bilinear form Λ such that Λ(es,et) = − A, Λ(et,er) = − B and Λ(er,es) = − C, with the three diagonal entries equal to one. The symmetric bilinear form Λ is non-degenerate with signature (2,1). Define ρ(v) = v − 2 Λ(v,er) er, σ(v) = v − 2 Λ(v,es) es and τ(v) = v − 2 Λ(v,et) et.
Theorem (geometric representation). The operators ρ, σ and τ are involutions on V, with respective eigenvectors er, es and et with simple eigenvalue −1. The products of the operators have orders corresponding to the presentation above (so στ has order a, etc). The operators ρ, σ and τ induce a representation of Γ on V which preserves Λ.
The bilinear form Λ for the basis has matrix
$M={\begin{pmatrix}1&-C&-B\\-C&1&-A\\-B&-A&1\\\end{pmatrix}},$
so has determinant 1−A2−B2−C2−2ABC. If c = 2, say, then the eigenvalues of the matrix are 1 and 1 ± (A2+B2)½. The condition a−1 + b−1 < ½ immediately forces A2+B2 > 1, so that Λ must have signature (2,1). So in general a, b, c ≥ 3. Clearly the case where all are equal to 3 is impossible. But then the determinant of the matrix is negative while its trace is positive. As a result two eigenvalues are positive and one negative, i.e. Λ has signature (2,1). Manifestly ρ, σ and τ are involutions, preserving Λ with the given −1 eigenvectors.
To check the order of the products like στ, it suffices to note that:
1. the reflections σ and τ generate a finite or infinite dihedral group;
2. the 2-dimensional linear span U of es and et is invariant under σ and τ, with the restriction of Λ positive-definite;
3. W, the orthogonal complement of U, is negative-definite on Λ, and σ and τ act trivially on W.
(1) is clear since if γ = στ generates a normal subgroup with σγσ−1 = γ−1. For (2), U is invariant by definition and the matrix is positive-definite since 0 < cos π / a < 1. Since Λ has signature (2,1), a non-zero vector w in W must satisfy Λ(w,w) < 0. By definition, σ has eigenvalues 1 and –1 on U, so w must be fixed by σ. Similarly w must be fixed by τ, so that (3) is proved. Finally in (1)
$\sigma ({\mathbf {e} }_{s})=-{\mathbf {e} }_{s},\,\,\,\,\,\sigma ({\mathbf {e} }_{t})=2\cos(\pi /a){\mathbf {e} }_{s}+{\mathbf {e} }_{t},\,\,\,\,\,\tau ({\mathbf {e} }_{s})=2\cos(\pi /a){\mathbf {e} }_{s}+{\mathbf {e} }_{t},\,\,\,\,\,\tau ({\mathbf {e} }_{t})=-{\mathbf {e} }_{t},$
so that, if a is finite, the eigenvalues of στ are -1, ς and ς−1, where ς = exp 2πi / a; and if a is infinite, the eigenvalues are -1, X and X−1, where X = exp 2x. Moreover a straightforward induction argument shows that if θ = π / a then[23]
$(\sigma \tau )^{m}({\mathbf {e} }_{s})=[\sin(2m+1)\theta /\sin \theta ]{\mathbf {e} }_{s}+[\sin 2m\theta /\sin \theta ]{\mathbf {e} }_{t},$
$\tau (\sigma \tau )^{m}({\mathbf {e} }_{s})=[\sin(2m+1)\theta /\sin \theta ]{\mathbf {e} }_{s}+[\sin(2m+2)\theta /\sin \theta ]{\mathbf {e} }_{t}$
and if x > 0 then
$(\sigma \tau )^{m}({\mathbf {e} }_{s})=[\sinh(2m+1)x/\sinh x]{\mathbf {e} }_{s}+[\sinh 2mx/\sinh x]{\mathbf {e} }_{t},$
$\tau (\sigma \tau )^{m}({\mathbf {e} }_{s})=[\sinh(2m+1)x/\sinh x]{\mathbf {e} }_{s}+[\sinh(2m+2)x/\sinh x]{\mathbf {e} }_{t}.$[lower-alpha 2]
Let Γa be the dihedral subgroup of Γ generated by s and t, with analogous definitions for Γb and Γc. Similarly define Γr to be the cyclic subgroup of Γ given by the 2-group {1,r}, with analogous definitions for Γs and Γt. From the properties of the geometric representation, all six of these groups act faithfully on V. In particular Γa can be identified with the group generated by σ and τ; as above it decomposes explicitly as a direct sum of the 2-dimensional irreducible subspace U and the 1-dimensional subspace W with a trivial action. Thus there is a unique vector w = er + λ es + μ et in W satisfying σ(w) = w and τ(w) = w. Explicitly λ = (C + AB)/(1 – A2) and μ = (B + AC)/(1 – A2).
Remark on representations of dihedral groups. It is well known that, for finite-dimensional real inner product spaces, two orthogonal involutions S and T can be decomposed as an orthogonal direct sum of 2-dimensional or 1-dimensional invariant spaces; for example, this can be deduced from the observation of Paul Halmos and others, that the positive self-adjoint operator (S – T)2 commutes with both S and T. In the case above, however, where the bilinear form Λ is no longer a positive definite inner product, different ad hoc reasoning has to be given.
Theorem (Tits). The geometric representation of the Coxeter group is faithful.
This result was first proved by Tits in the early 1960s and first published in the text of Bourbaki (1968) with its numerous exercises. In the text, the fundamental chamber was introduced by an inductive argument; exercise 8 in §4 of Chapter V was expanded by Vinay Deodhar to develop a theory of positive and negative roots and thus shorten the original argument of Tits.[24]
Let X be the convex cone of sums κer + λes + μet with real non-negative coefficients, not all of them zero. For g in the group Γ, define ℓ(g), the word length or length, to be the minimum number of reflections from r, s and t required to write g as an ordered composition of simple reflections. Define a positive root to be a vector ger, ges or ger lying in X, with g in Γ.[lower-alpha 3]
It is routine to check from the definitions that[25]
• if |ℓ(gq) – ℓ(g)| = 1 for a simple reflection q and, if g ≠ 1, there is always a simple reflection q such that ℓ(g) = ℓ(gq) + 1;
• for g and h in Γ, ℓ(gh) ≤ ℓ(g) + ℓ(h).
Proposition. If g is in Γ and ℓ(gq) = ℓ(g) ± 1 for a simple reflection q, then geq lies in ±X, and is therefore a positive or negative root, according to the sign.
Replacing g by gq, only the positive sign needs to be considered. The assertion will be proved by induction on ℓ(g) = m, it being trivial for m = 0. Assume that ℓ(gs) = ℓ(g) + 1. If ℓ(g) = m > 0, without less of generality it may be assumed that the minimal expression for g ends with ...t. Since s and t generate the dihedral group Γa, g can be written as a product g = hk, where k = (st)n or t(st)n and h has a minimal expression that ends with ...r, but never with s or t. This implies that ℓ(hs) = ℓ(h) + 1 and ℓ(ht) = ℓ(h) + 1. Since ℓ(h) < m, the induction hypothesis shows that both hes and het lie in X. It therefore suffices to show that kes has the form λes + μet with λ, μ ≥ 0, not both 0. But that has already been verified in the formulas above.[25]
Corollary (proof of Tits' theorem). The geometric representation is faithful.
It suffices to show that if g fixes er, es and et, then g = 1. Considering a minimal expression for g ≠ 1, the conditions ℓ(gq) = ℓ(g) + 1 clearly cannot be simultaneously satisfied by the three simple reflections q.
Note that, as a consequence of Tits' theorem, the generators g = st, h = tr and k = rs satisfy ga = 1, hb = 1 and kc = 1 with ghk = 1. This gives a presentation of the orientation-preserving index 2 normal subgroup Γ1 of Γ. The presentation corresponds to the fundamental domain obtained by reflecting two sides of the geodesic triangle to form a geodesic parallelogram (a special case of Poincaré's polygon theorem).[26]
Further consequences. The roots are the disjoint union of the positive roots and the negative roots. The simple reflection q permutes every positive root other than eq. For g in Γ, ℓ(g) is the number of positive roots made negative by g.
Fundamental domain and Tits cone.[27]
Let G be the 3-dimensional closed Lie subgroup of GL(V) preserving Λ. As V can be identified with a 3-dimensional Lorentzian or Minkowski space with signature (2,1), the group G is isomorphic to the Lorentz group O(2,1) and therefore SL±(2,R) / {±I}.[lower-alpha 4] Choosing e to be a positive root vector in X, the stabilizer of e is a maximal compact subgroup K of G isomorphic to O(2). The homogeneous space X = G / K is a symmetric space of constant negative curvature, which can be identified with the 2-dimensional hyperboloid or Lobachevsky plane ${\mathfrak {H}}^{2}$. The discrete group Γ acts discontinuously on G / K: the quotient space Γ \ G / K is compact if a, b and c are all finite, and of finite area otherwise. Results about the Tits fundamental chamber have a natural interpretation in terms of the corresponding Schwarz triangle, which translate directly into the properties of the tessellation of the geodesic triangle through the hyperbolic reflection group Γ. The passage from Coxeter groups to tessellation can first be found in the exercises of §4 of Chapter V of Bourbaki (1968), due to Tits, and in Iwahori (1966); currently numerous other equivalent treatments are available, not always directly phrased in terms of symmetric spaces.
Approach of Maskit, de Rham and Beardon
Maskit (1971) gave a general proof of Poincaré's polygon theorem in hyperbolic space; a similar proof was given in de Rham (1971). Specializing to the hyperbolic plane and Schwarz triangles, this can be used to give a modern approach for establishing that the existence of Schwarz triangle tessellations, as described in Beardon (1983) and Maskit (1988). The Swiss mathematicians de la Harpe (1990) harvtxt error: no target: CITEREFde_la_Harpe1990 (help) and Haefliger have provided an introductory account, taking geometric group theory as their starting point.[28]
See also
• List of uniform polyhedra by Schwarz triangle
• Wythoff symbol
• Wythoff construction
• Uniform polyhedron
• Nonconvex uniform polyhedron
• Density (polytope)
• Goursat tetrahedron
• Regular hyperbolic tiling
• Uniform tilings in hyperbolic plane
Notes
1. As in the case of P2, if an angle of Δ equals π/3, vertices where the interior angle is π stay marked as vertices and colinear edges are not coallesced.
2. In the limit as x tends to 0, (στ)m(es) = (2m+1)es + 2met and τ(στ)m(es) = (2m+1)es + (2m+2)et.
3. Here Γ is regarded as acting on V through the geometric representation.
4. SL±(2,R) is the subgroup of GL(2,R) with determinant ±1.
References
1. A. W. Knapp, Doubly generated Fuchsian groups, Michigan Mathematical Journal 15 (1968), no. 3, 289–304
2. Klimenko and Sakuma, Two-generator discrete subgroups of Isom( H 2 ) containing orientation-reversing elements, Geometriae Dedicata October 1998, Volume 72, Issue 3, pp 247-282
3. Carathéodory 1954, pp. 177–181
4. Carathéodory 1954, pp. 178−180
5. Carathéodory 1954, pp. 181–182
6. See:
• Takeuchi 1977a
• Takeuchi 1977b
• Weber 2005
7. See:
• Threlfall 1932, pp. 20–22, Figure 9
• Weber 2005
8. Carathéodory 1954, p. 183
9. Carathéodory 1954, p. 184
10. See:
• Evans 1973, pp. 108−109
• Berndt & Knopp 2008, pp. 16−17
11. Graham, Knuth & Patashnik 1994, p. 118
12. Series 2015
13. See:
• McMullen 1998
• Mumford, Series & Wright 2015
14. Siegel 1971, pp. 85–87
15. For proofs of Poincaré's polygon theorem, see
• Maskit 1971
• de Rham 1971
• Beardon 1983, pp. 242–249
• Iversen 1992, pp. 200–208
• Epstein & Petronio 1994
• Berger 2010, pp. 616–617
16. Beardon 1984, pp. 106–107, 110–111
17. See:
• Tits 2013
• Bourbaki 1981 harvnb error: no target: CITEREFBourbaki1981 (help)
• Humphreys 1990
18. See:
• Vinberg 1971
• Vinberg 1985
• Shvartsman & Vinberg harvnb error: no target: CITEREFShvartsmanVinberg (help)
19. Lehner 1964 harvnb error: no target: CITEREFLehner1964 (help)
20. Maskit 1971
21. See:
• Brown 1989
• Humphreys 1990
• Abremko & Brown 2007 harvnb error: no target: CITEREFAbremkoBrown2007 (help)
• Davis 2008
22. Heckman 2017. sfn error: no target: CITEREFHeckman2017 (help)
23. Howlett 1996
24. See:
• Tits 2013
• Bourbaki 1968
• Steinberg 1968
• Hiller 1980 harvnb error: no target: CITEREFHiller1980 (help)
• Deodhar 1982, Deodhar 1986
• Humphreys 1990
• Howlett 1996
• Heckman 2017 harvnb error: no target: CITEREFHeckman2017 (help)
25. See:
• Humphreys 1990
• Howlett 1996
• Heckman 2017 harvnb error: no target: CITEREFHeckman2017 (help)
26. See:
• Magnus, Karrass & Solitar 1976
• Magnus 1974
• Iversen 1992
• Ellis 2019
27. See:
• Tits 2013
• Bourbaki 1968
• Maxwell 1982
• Abramenko & Brown 2007
• Davis 2008
• Heckman 2017 harvnb error: no target: CITEREFHeckman2017 (help)
28. See:
• Milnor 1975
• Beardon 1983, pp. 242–249
• Iversen 1992, pp. 200–208
• Bridson & Haefliger 1999
• Berger 2010, pp. 616–617
• Coxeter, H.S.M. (1973), Regular Polytopes (Third ed.), Dover Publications, ISBN 0-486-61480-8, Table 3: Schwarz's Triangles
• Schwarz, H. A. (1873), "Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt", Journal für die reine und angewandte Mathematik, 1873 (75): 292–335, doi:10.1515/crll.1873.75.292, ISSN 0075-4102, S2CID 121698536 (Note that Coxeter references this as "Zur Theorie der hypergeometrischen Reihe", which is the short title used in the journal page headers).
• Wenninger, Magnus J. (1979), "An introduction to the notion of polyhedral density", Spherical models, CUP Archive, pp. 132–134, ISBN 978-0-521-22279-2
• Abramenko, Peter; Brown, Kenneth S. (2007). Buildings: Theory and Applications. Graduate Texts in Mathematics. Vol. 248. Springer-Verlag. ISBN 978-0-387-78834-0. MR 2439729.
• Beardon, Alan F. (1983), "Poincaré's Theorem", The Geometry of Discrete Groups, Graduate Texts in Mathematics, vol. 91, Springer, pp. 242–252, ISBN 0-387-90788-2
• Beardon, Alan F. (1984), "A primer on Riemann surfaces", London Mathematical Society Lecture Note Series, Cambridge University Press, 78, ISBN 0521271045
• Berger, Marcel (2010), Geometry revealed. A Jacob's ladder to modern higher geometry, translated by Lester Senechal, Springer, ISBN 978-3-540-70996-1
• Berndt, Bruce C.; Knopp, Marvin I. (2008), Hecke's theory of modular forms and Dirichlet series, Monographs in Number Theory, vol. 5, World Scientific, ISBN 978-981-270-635-5
• Bourbaki, Nicolas (1968). "Chapitre IV : Groupes de Coxeter et systèmes de Tits • Chapitre V : Groupes engendrés par des réflexions". Groupes et algèbres de Lie. Éléments de mathématique (in French). Paris: Hermann. pp. 1–56, 57–141. MR 0240238. Reprinted by Masson in 1981 as ISBN 2-225-76076-4.
• Bridson, Martin R.; Haefliger, André (1999). "I. Basic material on SL2(R), discrete subgroups, and the upper half-plane". Metric spaces of non-positive curvature (PDF). Grundlehren der mathematischen Wissenschaften. Vol. 319. Springer-Verlag. ISBN 3-540-64324-9. MR 1744486.
• Brown, Kenneth S. (1989). Buildings. Springer-Verlag. ISBN 0-387-96876-8. MR 0969123.
• Carathéodory, Constantin (1954), Theory of functions of a complex variable, vol. 2, translated by F. Steinhardt, Chelsea
• Davis, Michael W. (2008), "Appendix D. The Geometric Representation", The geometry and topology of Coxeter groups, London Mathematical Society Monographs, vol. 32, Princeton University Press, pp. 439–447, ISBN 978-0-691-13138-2
• de la Harpe, Pierre (1991). "An invitation to Coxeter groups". Group theory from a geometrical viewpoint (Trieste, 1990). World Scientific. pp. 193–253. MR 1170367.
• Deodhar, Vinay V. (1982). "On the root system of a Coxeter group". Comm. Algebra. 10 (6): 611–630. doi:10.1080/00927878208822738. MR 0647210.
• Deodhar, Vinay V. (1986). "Some characterizations of Coxeter groups". Enseign. Math. 32: 111–120. MR 0850554.
• de Rham, G. (1971). "Sur les polygones générateurs de groupes fuchsiens". Enseign. Math. (in French). 17: 49–61.
• Ellis, Graham (2019). "Triangle groups". An Invitation to Computational Homotopy. Oxford University Press. pp. 441–444. ISBN 978-0-19-883298-0. MR 3971587.
• Epstein, David B. A.; Petronio, Carlo (1994). "An exposition of Poincaré's polyhedron theorem". Enseign. Math. 40: 113–170. MR 1279064.
• Evans, Ronald (1973), "A fundamental region for Hecke's modular group", Journal of Number Theory, 5 (2): 108–115, Bibcode:1973JNT.....5..108E, doi:10.1016/0022-314x(73)90063-2
• Fricke, Robert; Klein, Felix (1897), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen (in German), B. G. Teubner, ISBN 978-1-4297-0551-6, JFM 28.0334.01
• Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994), Concrete mathematics (2nd ed.), Addison-Wesley, pp. 116–118, ISBN 0-201-55802-5
• Hardy, G. H.; Wright, E. M. (2008), An introduction to the theory of numbers (6th ed.), Oxford University Press, ISBN 978-0-19-921986-5
• Hatcher, Allen (2013). "1. The Farey Diagram". Topology of Numbers (PDF). Cornell University. Retrieved 25 February 2022.
• Hecke, E. (1935), "Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung", Mathematische Annalen (in German), 112: 664–699, doi:10.1007/bf01565437
• Heckman, Gert J. (2018). "Coxeter Groups" (PDF). Radboud University Nijmegen. Retrieved 3 March 2022.
• Hiller, Howard (1982). Geometry of Coxeter groups. Research Notes in Mathematics. Vol. 54. Pitman. ISBN 0-273-08517-4. MR 0649068.
• Howlett, Robert (1996). "Introduction to Coxeter Groups". Australian National University Geometric Group Theory Workshop. Sydney.{{cite book}}: CS1 maint: location missing publisher (link)
• Humphreys, James E. (1990). Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics. Vol. 29. Cambridge University Press. ISBN 0-521-37510-X. MR 1066460.
• Iversen, Birger (1992), Hyperbolic geometry, London Mathematical Society Student Texts, vol. 25, Cambridge University Press, ISBN 0-521-43508-0
• Iwahori, Nagayoshi (1966). "On discrete reflection groups on symmetric Riemannian manifolds". Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965). Tokyo: Nippon Hyoronsha. pp. 57–62. MR 0217741.
• Magnus, Wilhelm (1974), Noneuclidean tesselations and their groups, Pure and Applied Mathematics, vol. 61, Academic Press
• Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (1976). Combinatorial group theory: Presentations of groups in terms of generators and relations (Second revised ed.). Dover Books. MR 0207802.
• Maskit, Bernard (1971), "On Poincaré's theorem for fundamental polygons", Advances in Mathematics, 7 (3): 219–230, doi:10.1016/s0001-8708(71)80003-8
• Maskit, Bernard (1988). "Poincaré's theorem". Kleinian groups. Grundlehren der mathematischen Wissenschaften. Vol. 287. Springer-Verlag. ISBN 3-540-17746-9. MR 0959135.
• Maxwell, George (1982). "Sphere packings and hyperbolic reflection groups". J. Algebra. 79: 78–97. doi:10.1016/0021-8693(82)90318-0. MR 0679972.
• McMullen, Curtis T. (1998), "Hausdorff dimension and conformal dynamics. III. Computation of dimension", American Journal of Mathematics, 120: 691–721, doi:10.1353/ajm.1998.0031, S2CID 15928775
• Milnor, John (1975). "On the 3-dimensional Brieskorn manifolds M(p,q,r)". Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox). Ann. of Math. Studies. Vol. 84. Princeton University Press. pp. 175–225. MR 0418127.
• Mumford, David; Series, Caroline; Wright, David (2015), Indra's pearls. The vision of Felix Klein, Cambridge University Press, ISBN 978-1-107-56474-9
• Series, Caroline (2015), Continued fractions and hyperbolic geometry, Loughborough LMS Summer School (PDF), retrieved 15 February 2017
• Siegel, C. L. (1971), Topics in complex function theory, vol. II. Automorphic functions and abelian integrals, translated by A. Shenitzer; M. Tretkoff, Wiley-Interscience, pp. 85–87, ISBN 0-471-60843-2
• Steinberg, Robert (1968). Endomorphisms of linear algebraic groups. Memoirs of the American Mathematical Society. Vol. 80. American Mathematical Society. MR 0230728.
• Takeuchi, Kisao (1977a), "Arithmetic triangle groups", Journal of the Mathematical Society of Japan, 29: 91–106, doi:10.2969/jmsj/02910091
• Takeuchi, Kisao (1977b), "Commensurability classes of arithmetic triangle groups", Journal of the Faculty of Science, the University of Tokyo, Section IA, Mathematics, 24: 201–212
• Threlfall, W. (1932), "Gruppenbilder" (PDF), Abhandlungen der Mathematisch-physischen Klasse der Sachsischen Akademie der Wissenschaften, Hirzel, 41: 1–59
• Tits, Jacques (2013). "Groupes et géométries de Coxeter". In F. Buekenhout; B. M. Mühlherr; J-P. Tignol; H. Van Maldeghem (eds.). Œuvres/Collected works, Volume I. Heritage of European Mathematics (in French). Zürich: European Mathematical Society. pp. 803–817. ISBN 978-3-03719-126-2. This manuscript was the foundational text for the theory of Coxeter groups, used for preparing Chapter IV of Bourbaki's Groupes et Algèbres de Lie; it was first published in 2001.
• Vinberg, Ernest B. (1971). Translated by P. Flor. "Discrete linear groups generated by reflections". Mathematics of the USSR-Izvestiya. 5 (5): 1083–1119. Bibcode:1971IzMat...5.1083V. doi:10.1070/IM1971v005n05ABEH001203. MR 0302779.
• Vinberg, Ernest B. (1985). Translated by J. Wiegold. "Hyperbolic reflection groups". Russian Mathematical Surveys. London Mathematical Society. 40 (1): 31–75. Bibcode:1985RuMaS..40...31V. doi:10.1070/RM1985v040n01ABEH003527. S2CID 250912767.
• Vinberg, Ernest B.; Shvartsman, O. V. (1993). "Discrete groups of motions of spaces of constant curvature". Geometry II: Spaces of Constant Curvature. Encyclopaedia Math. Sci. Vol. 29. Springer-Verlag. pp. 139–248. ISBN 3-540-52000-7. MR 1254933.
• Weber, Matthias (2005), "Kepler's small stellated dodecahedron as a Riemann surface", Pacific Journal of Mathematics, 220: 167–182, doi:10.2140/pjm.2005.220.167
External links
• Weisstein, Eric W. "Schwarz triangle". MathWorld.
• Klitzing, Richard. "3D The general Schwarz triangle (p q r) and the generalized incidence matrices of the corresponding polyhedra".
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Wikipedia
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Schwarzian derivative
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces. It is named after the German mathematician Hermann Schwarz.
Definition
The Schwarzian derivative of a holomorphic function f of one complex variable z is defined by
$(Sf)(z)=\left({\frac {f''(z)}{f'(z)}}\right)'-{\frac {1}{2}}\left({\frac {f''(z)}{f'(z)}}\right)^{2}={\frac {f'''(z)}{f'(z)}}-{\frac {3}{2}}\left({\frac {f''(z)}{f'(z)}}\right)^{2}.$
The same formula also defines the Schwarzian derivative of a C3 function of one real variable. The alternative notation
$\{f,z\}=(Sf)(z)$
is frequently used.
Properties
The Schwarzian derivative of any Möbius transformation
$g(z)={\frac {az+b}{cz+d}}$
is zero. Conversely, the Möbius transformations are the only functions with this property. Thus, the Schwarzian derivative precisely measures the degree to which a function fails to be a Möbius transformation.[1]
If g is a Möbius transformation, then the composition g o f has the same Schwarzian derivative as f; and on the other hand, the Schwarzian derivative of f o g is given by the chain rule
$(S(f\circ g))(z)=(Sf)(g(z))\cdot g'(z)^{2}.$
More generally, for any sufficiently differentiable functions f and g
$S(f\circ g)=\left((Sf)\circ g\right)\cdot (g')^{2}+Sg.$
When f and g are smooth real-valued functions, this implies that all iterations of a function with negative (or positive) Schwarzian will remain negative (resp. positive), a fact of use in the study of one-dimensional dynamics.[2]
Introducing the function of two complex variables[3]
$F(z,w)=\log \left({\frac {f(z)-f(w)}{z-w}}\right),$
its second mixed partial derivative is given by
${\frac {\partial ^{2}F(z,w)}{\partial z\,\partial w}}={f^{\prime }(z)f^{\prime }(w) \over (f(z)-f(w))^{2}}-{1 \over (z-w)^{2}},$
and the Schwarzian derivative is given by the formula:
$(Sf)(w)=\left.6\cdot {\partial ^{2}F(z,w) \over \partial z\,\partial w}\right\vert _{z=w}.$
The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has
$(Sw)(v)=-\left({\frac {dw}{dv}}\right)^{2}(Sv)(w)$
or more explicitly, $Sf+(f')^{2}((Sf^{-1})\circ f)=0$. This follows from the chain rule above.
Geometric interpretation
William Thurston interprets the Schwarzian derivative as a measure of how much a conformal map deviates from a Möbius transformation.[1] Let $f$ be a conformal mapping in a neighborhood of $z_{0}\in \mathbb {C} $. Then there exists a unique Möbius transformation $M$ such that $M,f$ has the same 0, 1, 2-th order derivatives at $z_{0}$.
Now $(M^{-1}\circ f)(z-z_{0})=z_{0}+(z-z_{0})+{\frac {1}{6}}a(z-z_{0})^{3}+\cdots $. To explicitly solve for $a$, it suffices to solve the case of $z_{0}=0$. Let $M^{-1}(z)={\frac {Az+B}{Cz+1}}$, and solve for $A,B,C$ that would make the first three coefficients of $M^{-1}\circ f$ to equal to 0, 1, 0. Plugging it into the fourth coefficient, we get $a=(Sf)(z_{0})$.
After a translation, rotation, and scaling of the complex plane, we have $(M^{-1}\circ f)(z)=z+z^{3}+O(z^{4})$ in a neighborhood of zero. Up to the third order, then, this function maps the circle of radius $r$ to the curve defined by $(r\cos \theta +r^{3}\cos 3\theta ,r\sin \theta +r^{3}\sin 3\theta )$, where $\theta \in [0,2\pi ]$. This curve is, up to the fourth order, an ellipse with semiaxes $r+r^{3},r-r^{3}$:
${\frac {(r\cos \theta +r^{3}\cos 3\theta )^{2}}{(r+r^{3})^{2}}}+{\frac {(r\sin \theta +r^{3}\sin 3\theta )^{2}}{(r-r^{3})^{2}}}={\frac {1+8r^{4}\sin ^{2}(2\theta )+O(r^{6})}{(1-r^{4})^{2}}}\rightarrow 1+8r^{4}\sin ^{2}(2\theta )+O(r^{6})$
as $r\rightarrow 0$Since Möbius transformations always map circles to circles or lines, the amount of elliptic-ness measures the deviation of $f$ from a Möbius transform.
Differential equation
The Schwarzian derivative has a fundamental relation with a second-order linear ordinary differential equation in the complex plane.[4] Let $f_{1}(z)$ and $f_{2}(z)$ be two linearly independent holomorphic solutions of
${\frac {d^{2}f}{dz^{2}}}+Q(z)f(z)=0.$
Then the ratio $g(z)=f_{1}(z)/f_{2}(z)$ satisfies
$(Sg)(z)=2Q(z)$
over the domain on which $f_{1}(z)$ and $f_{2}(z)$ are defined, and $f_{2}(z)\neq 0.$ The converse is also true: if such a g exists, and it is holomorphic on a simply connected domain, then two solutions $f_{1}$ and $f_{2}$ can be found, and furthermore, these are unique up to a common scale factor.
When a linear second-order ordinary differential equation can be brought into the above form, the resulting Q is sometimes called the Q-value of the equation.
Note that the Gaussian hypergeometric differential equation can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way.
Conditions for univalence
If f is a holomorphic function on the unit disc, D, then W. Kraus (1932) and Nehari (1949) proved that a necessary condition for f to be univalent is[5]
$|S(f)|\leq 6(1-|z|^{2})^{-2}.$
Conversely if f(z) is a holomorphic function on D satisfying
$|S(f)(z)|\leq 2(1-|z|^{2})^{-2},$
then Nehari proved that f is univalent.[6]
In particular a sufficient condition for univalence is[7]
$|S(f)|\leq 2.$
Conformal mapping of circular arc polygons
The Schwarzian derivative and associated second-order ordinary differential equation can be used to determine the Riemann mapping between the upper half-plane or unit circle and any bounded polygon in the complex plane, the edges of which are circular arcs or straight lines. For polygons with straight edges, this reduces to the Schwarz–Christoffel mapping, which can be derived directly without using the Schwarzian derivative. The accessory parameters that arise as constants of integration are related to the eigenvalues of the second-order differential equation. Already in 1890 Felix Klein had studied the case of quadrilaterals in terms of the Lamé differential equation.[8][9][10]
Let Δ be a circular arc polygon with angles πα1, ..., παn in clockwise order. Let f : H → Δ be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points a1, ..., an on the real axis. Then p(x) = S(f)(x) is real-valued for x real and not one of the points. By the Schwarz reflection principle p(x) extends to a rational function on the complex plane with a double pole at ai:
$p(z)=\sum _{i=1}^{n}{\frac {(1-\alpha _{i}^{2})}{2(z-a_{i})^{2}}}+{\frac {\beta _{i}}{z-a_{i}}}.$
The real numbers βi are called accessory parameters. They are subject to three linear constraints:
$\sum \beta _{i}=0$
$\sum 2a_{i}\beta _{i}+\left(1-\alpha _{i}^{2}\right)=0$
$\sum a_{i}^{2}\beta _{i}+a_{i}\left(1-\alpha _{i}^{2}\right)=0$
which correspond to the vanishing of the coefficients of $z^{-1},z^{-2}$ and $z^{-3}$ in the expansion of p(z) around z = ∞. The mapping f(z) can then be written as
$f(z)={u_{1}(z) \over u_{2}(z)},$
where $u_{1}(z)$ and $u_{2}(z)$ are linearly independent holomorphic solutions of the linear second-order ordinary differential equation
$u^{\prime \prime }(z)+{\tfrac {1}{2}}p(z)u(z)=0.$
There are n−3 linearly independent accessory parameters, which can be difficult to determine in practise.
For a triangle, when n = 3, there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and f(z) is the Schwarz triangle function, which can be written in terms of hypergeometric functions.
For a quadrilateral the accessory parameters depend on one independent variable λ. Writing U(z) = q(z)u(z) for a suitable choice of q(z), the ordinary differential equation takes the form
$a(z)U^{\prime \prime }(z)+b(z)U^{\prime }(z)+(c(z)+\lambda )U(z)=0.$
Thus $q(z)u_{i}(z)$ are eigenfunctions of a Sturm–Liouville equation on the interval $[a_{i},a_{i+1}]$. By the Sturm separation theorem, the non-vanishing of $u_{2}(z)$ forces λ to be the lowest eigenvalue.
Complex structure on Teichmüller space
Universal Teichmüller space is defined to be the space of real analytic quasiconformal mappings of the unit disc D, or equivalently the upper half-plane H, onto itself, with two mappings considered to be equivalent if on the boundary one is obtained from the other by composition with a Möbius transformation. Identifying D with the lower hemisphere of the Riemann sphere, any quasiconformal self-map f of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere ${\tilde {f}}$ onto itself. In fact ${\tilde {f}}$ is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation
${\frac {\partial F}{\partial {\bar {z}}}}=\mu (z){\frac {\partial F}{\partial z}},$
where μ is the bounded measurable function defined by
$\mu (z)={\frac {\partial f}{\partial {\bar {z}}}}{\bigg /}{\frac {\partial f}{\partial z}}$
on the lower hemisphere, extended to 0 on the upper hemisphere.
Identifying the upper hemisphere with D, Lipman Bers used the Schwarzian derivative to define a mapping
$g=S({\tilde {f}}),$
which embeds universal Teichmüller space into an open subset U of the space of bounded holomorphic functions g on D with the uniform norm. Frederick Gehring showed in 1977 that U is the interior of the closed subset of Schwarzian derivatives of univalent functions.[11][12][13]
For a compact Riemann surface S of genus greater than 1, its universal covering space is the unit disc D on which its fundamental group Γ acts by Möbius transformations. The Teichmüller space of S can be identified with the subspace of the universal Teichmüller space invariant under Γ. The holomorphic functions g have the property that
$g(z)\,dz^{2}$
is invariant under Γ, so determine quadratic differentials on S. In this way, the Teichmüller space of S is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on S.
Diffeomorphism group of the circle
Crossed homomorphisms
The transformation property
$S(f\circ g)=\left(S(f)\circ g\right)\cdot (g')^{2}+S(g).$
allows the Schwarzian derivative to be interpreted as a continuous 1-cocycle or crossed homomorphism of the diffeomorphism group of the circle with coefficients in the module of densities of degree 2 on the circle.[14] Let Fλ(S1) be the space of tensor densities of degree λ on S1. The group of orientation-preserving diffeomorphisms of S1, Diff(S1), acts on Fλ(S1) via pushforwards. If f is an element of Diff(S1) then consider the mapping
$f\to S(f^{-1}).$
In the language of group cohomology the chain-like rule above says that this mapping is a 1-cocycle on Diff(S1) with coefficients in F2(S1). In fact
$H^{1}({\text{Diff}}(\mathbf {S} ^{1});F_{2}(\mathbf {S} ^{1}))=\mathbf {R} $
and the 1-cocycle generating the cohomology is f → S(f−1). The computation of 1-cohomology is a particular case of the more general result
$H^{1}({\text{Diff}}(\mathbf {S} ^{1});F_{\lambda }(\mathbf {S} ^{1}))=\mathbf {R} \,\,\mathrm {for} \,\,\lambda =0,1,2\,\,\mathrm {and} \,\,(0)\,\,\mathrm {otherwise.} $
Note that if G is a group and M a G-module, then the identity defining a crossed homomorphism c of G into M can be expressed in terms of standard homomorphisms of groups: it is encoded in a homomorphism 𝜙 of G into the semidirect product $M\rtimes G$ such that the composition of 𝜙 with the projection $M\rtimes G$ onto G is the identity map; the correspondence is by the map C(g) = (c(g), g). The crossed homomorphisms form a vector space and containing as a subspace the coboundary crossed homomorphisms b(g) = g ⋅ m − m for m in M. A simple averaging argument shows that, if K is a compact group and V a topological vector space on which K acts continuously, then the higher cohomology groups vanish Hm(K, V) = (0) for m > 0. In particular for 1-cocycles χ with
$\chi (xy)=\chi (x)+x\cdot \chi (y),$
averaging over y, using left invariant of the Haar measure on K gives
$\chi (x)=m-x\cdot m,$
with
$m=\int _{K}\chi (y)\,dy.$
Thus by averaging it may be assumed that c satisfies the normalisation condition c(x) = 0 for x in Rot(S1). Note that if any element x in G satisfies c(x) = 0 then C(x) = (0,x). But then, since C is a homomorphism, C(xgx−1) = C(x)C(g)C(x)−1, so that c satisfies the equivariance condition c(xgx−1) = x ⋅ c(g). Thus it may be assumed that the cocycle satisfies these normalisation conditions for Rot(S1). The Schwarzian derivative in fact vanishes whenever x is a Möbius transformation corresponding to SU(1,1). The other two 1-cycles discussed below vanish only on Rot(S1) (λ = 0, 1).
There is an infinitesimal version of this result giving a 1-cocycle for Vect(S1), the Lie algebra of smooth vector fields, and hence for the Witt algebra, the subalgebra of trigonometric polynomial vector fields. Indeed, when G is a Lie group and the action of G on M is smooth, there is a Lie algebraic version of crossed homomorphism obtained by taking the corresponding homomorphisms of the Lie algebras (the derivatives of the homomorphisms at the identity). This also makes sense for Diff(S1) and leads to the 1-cocycle
$s\left(f\,{d \over d\theta }\right)={d^{3}f \over d\theta ^{3}}\,(d\theta )^{2}$
which satisfies the identity
$s([X,Y])=X\cdot s(Y)-Y\cdot s(X).$
In the Lie algebra case, the coboundary maps have the form b(X) = X ⋅ m for m in M. In both cases the 1-cohomology is defined as the space of crossed homomorphisms modulo coboundaries. The natural correspondence between group homomorphisms and Lie algebra homomorphisms leads to the "van Est inclusion map"
$H^{1}(\operatorname {Diff} (\mathbf {S} ^{1});F_{\lambda }(\mathbf {S} ^{1}))\hookrightarrow H^{1}(\operatorname {Vect} (\mathbf {S} ^{1});F_{\lambda }(\mathbf {S} ^{1})),$
In this way the calculation can be reduced to that of Lie algebra cohomology. By continuity this reduces to the computation of crossed homomorphisms 𝜙 of the Witt algebra into Fλ(S1). The normalisations conditions on the group crossed homomorphism imply the following additional conditions for 𝜙:
$\varphi (\operatorname {Ad} (x)X)=x\cdot \varphi (X),\,\,\varphi (d/d\theta )=0$
for x in Rot(S1).
Following the conventions of Kac & Raina (1987), a basis of the Witt algebra is given by
$d_{n}=ie^{in\theta }\,{d \over d\theta }$
so that [dm,dn] = (m – n) dm + n. A basis for the complexification of Fλ(S1) is given by
$v_{n}=e^{in\theta }\,(d\theta )^{\lambda },$
so that
$d_{m}\cdot v_{n}=-(n+\lambda m)v_{n+m},\,\,g_{\zeta }\cdot v_{n}=\zeta ^{n}v_{n},$
for gζ in Rot(S1) = T. This forces 𝜙(dn) = an ⋅ vn for suitable coefficients an. The crossed homomorphism condition 𝜙([X,Y]) = X𝜙(Y) – Y𝜙(X) gives a recurrence relation for the an:
$(m-n)a_{m+n}=(m+\lambda n)a_{m}-(n+\lambda m)a_{n}.$
The condition 𝜙(d/dθ) = 0, implies that a0 = 0. From this condition and the recurrence relation, it follows that up to scalar multiples, this has a unique non-zero solution when λ equals 0, 1 or 2 and only the zero solution otherwise. The solution for λ = 1 corresponds to the group 1-cocycle $\varphi _{1}(f)=f^{\prime \prime }/f^{\prime }\,d\theta $. The solution for λ = 0 corresponds to the group 1-cocycle 𝜙0(f) = log f' . The corresponding Lie algebra 1-cocycles for λ = 0, 1, 2 are given up to a scalar multiple by
$\varphi _{\lambda }\left(F{d \over d\theta }\right)={d^{\lambda +1}F \over d\theta ^{\lambda +1}}\,(d\theta )^{\lambda }.$
Central extensions
The crossed homomorphisms in turn give rise to the central extension of Diff(S1) and of its Lie algebra Vect(S1), the so-called Virasoro algebra.
Coadjoint action
The group Diff(S1) and its central extension also appear naturally in the context of Teichmüller theory and string theory.[15] In fact the homeomorphisms of S1 induced by quasiconformal self-maps of D are precisely the quasisymmetric homeomorphisms of S1; these are exactly homeomorphisms which do not send four points with cross ratio 1/2 to points with cross ratio near 1 or 0. Taking boundary values, universal Teichmüller can be identified with the quotient of the group of quasisymmetric homeomorphisms QS(S1) by the subgroup of Möbius transformations Moeb(S1). (It can also be realized naturally as the space of quasicircles in C.) Since
$\operatorname {Moeb} (\mathbf {S} ^{1})\subset \operatorname {Diff} (\mathbf {S} ^{1})\subset {\text{QS}}(\mathbf {S} ^{1})$
the homogeneous space Diff(S1)/Moeb(S1) is naturally a subspace of universal Teichmüller space. It is also naturally a complex manifold and this and other natural geometric structures are compatible with those on Teichmüller space. The dual of the Lie algebra of Diff(S1) can be identified with the space of Hill's operators on S1
${d^{2} \over d\theta ^{2}}+q(\theta ),$
and the coadjoint action of Diff(S1) invokes the Schwarzian derivative. The inverse of the diffeomorphism f sends the Hill's operator to
${d^{2} \over d\theta ^{2}}+f^{\prime }(\theta )^{2}\,q\circ f(\theta )+{\tfrac {1}{2}}S(f)(\theta ).$
Pseudogroups and connections
The Schwarzian derivative and the other 1-cocycle defined on Diff(S1) can be extended to biholomorphic between open sets in the complex plane. In this case the local description leads to the theory of analytic pseudogroups, formalizing the theory of infinite-dimensional groups and Lie algebras first studied by Élie Cartan in the 1910s. This is related to affine and projective structures on Riemann surfaces as well as the theory of Schwarzian or projective connections, discussed by Gunning, Schiffer and Hawley.
A holomorphic pseudogroup Γ on C consists of a collection of biholomorphisms f between open sets U and V in C which contains the identity maps for each open U, which is closed under restricting to opens, which is closed under composition (when possible), which is closed under taking inverses and such that if a biholomorphisms is locally in Γ, then it too is in Γ. The pseudogroup is said to be transitive if, given z and w in C, there is a biholomorphism f in Γ such that f(z) = w. A particular case of transitive pseudogroups are those which are flat, i.e. contain all complex translations Tb(z) = z + b. Let G be the group, under composition, of formal power series transformations F(z) = a1z + a2z2 + .... with a1 ≠ 0. A holomorphic pseudogroup Γ defines a subgroup A of G, namely the subgroup defined by the Taylor series expansion about 0 (or "jet") of elements f of Γ with f(0) = 0. Conversely if Γ is flat it is uniquely determined by A: a biholomorphism f on U is contained in Γ in if and only if the power series of T–f(a) ∘ f ∘ Ta lies in A for every a in U: in other words the formal power series for f at a is given by an element of A with z replaced by z − a; or more briefly all the jets of f lie in A.[16]
The group G has a natural homomorphisms onto the group Gk of k-jets obtained by taking the truncated power series taken up to the term zk. This group acts faithfully on the space of polynomials of degree k (truncating terms of order higher than k). Truncations similarly define homomorphisms of Gk onto Gk − 1; the kernel consists of maps f with f(z) = z + bzk, so is Abelian. Thus the group Gk is solvable, a fact also clear from the fact that it is in triangular form for the basis of monomials.
A flat pseudogroup Γ is said to be "defined by differential equations" if there is a finite integer k such that homomorphism of A into $''G''<sub>''k''</sub>$ is faithful and the image is a closed subgroup. The smallest such k is said to be the order of Γ. There is a complete classification of all subgroups A that arise in this way which satisfy the additional assumptions that the image of A in Gk is a complex subgroup and that G1 equals C*: this implies that the pseudogroup also contains the scaling transformations Sa(z) = az for a ≠ 0, i.e. contains A contains every polynomial az with a ≠ 0.
The only possibilities in this case are that k = 1 and A = {az: a ≠ 0}; or that k = 2 and A = {az/(1−bz) : a ≠ 0}. The former is the pseudogroup defined by affine subgroup of the complex Möbius group (the az + b transformations fixing ∞); the latter is the pseudogroup defined by the whole complex Möbius group.
This classification can easily be reduced to a Lie algebraic problem since the formal Lie algebra ${\mathfrak {g}}$ of G consists of formal vector fields F(z) d/dz with F a formal power series. It contains the polynomial vectors fields with basis dn = zn+1 d/dz (n ≥ 0), which is a subalgebra of the Witt algebra. The Lie brackets are given by [dm,dn] = (n − m)dm+n. Again these act on the space of polynomials of degree ≤ k by differentiation—it can be identified with C[[z]]/(zk+1)—and the images of d0, ..., dk – 1 give a basis of the Lie algebra of Gk. Note that Ad(Sa) dn= a–n dn. Let ${\mathfrak {a}}$ denote the Lie algebra of A: it is isomorphic to a subalgebra of the Lie algebra of Gk. It contains d0 and is invariant under Ad(Sa). Since ${\mathfrak {a}}$ is a Lie subalgebra of the Witt algebra, the only possibility is that it has basis d0 or basis d0, dn for some n ≥ 1. There are corresponding group elements of the form f(z)= z + bzn+1 + .... Composing this with translations yields T–f(ε) ∘ f ∘ T ε(z) = cz + dz2 + ... with c, d ≠ 0. Unless n = 2, this contradicts the form of subgroup A; so n = 2.[17]
The Schwarzian derivative is related to the pseudogroup for the complex Möbius group. In fact if f is a biholomorphism defined on V then 𝜙2(f) = S(f) is a quadratic differential on V. If g is a bihomolorphism defined on U and g(V) ⊆ U, S(f ∘ g) and S(g) are quadratic differentials on U; moreover S(f) is a quadratic differential on V, so that g∗S(f) is also a quadratic differential on U. The identity
$S(f\circ g)=g_{*}S(f)+S(g)$
is thus the analogue of a 1-cocycle for the pseudogroup of biholomorphisms with coefficients in holomorphic quadratic differentials. Similarly $\varphi _{0}(f)=\log f^{\prime }$ and $\varphi _{1}(f)=f^{\prime \prime }/f^{\prime }$ are 1-cocycles for the same pseudogroup with values in holomorphic functions and holomorphic differentials. In general 1-cocycle can be defined for holomorphic differentials of any order so that
$\varphi (f\circ g)=g_{*}\varphi (f)+\varphi (g).$
Applying the above identity to inclusion maps j, it follows that 𝜙(j) = 0; and hence that if f1 is the restriction of f2, so that f2 ∘ j = f1, then 𝜙(f1) = 𝜙 (f2). On the other hand, taking the local holomororphic flow defined by holomorphic vector fields—the exponential of the vector fields—the holomorphic pseudogroup of local biholomorphisms is generated by holomorphic vector fields. If the 1-cocycle 𝜙 satisfies suitable continuity or analyticity conditions, it induces a 1-cocycle of holomorphic vector fields, also compatible with restriction. Accordingly, it defines a 1-cocycle on holomorphic vector fields on C:[18]
$\varphi ([X,Y])=X\varphi (Y)-Y\varphi (X).$
Restricting to the Lie algebra of polynomial vector fields with basis dn = zn+1 d/dz (n ≥ −1), these can be determined using the same methods of Lie algebra cohomology (as in the previous section on crossed homomorphisms). There the calculation was for the whole Witt algebra acting on densities of order k, whereas here it is just for a subalgebra acting on holomorphic (or polynomial) differentials of order k. Again, assuming that 𝜙 vanishes on rotations of C, there are non-zero 1-cocycles, unique up to scalar multiples. only for differentials of degree 0, 1 and 2 given by the same derivative formula
$\varphi _{k}\left(p(z){d \over dz}\right)=p^{(k+1)}(z)\,(dz)^{k},$
where p(z) is a polynomial.
The 1-cocycles define the three pseudogroups by 𝜙k(f) = 0: this gives the scaling group (k = 0); the affine group (k = 1); and the whole complex Möbius group (k = 2). So these 1-cocycles are the special ordinary differential equations defining the pseudogroup. More significantly they can be used to define corresponding affine or projective structures and connections on Riemann surfaces. If Γ is a pseudogroup of smooth mappings on Rn, a topological space M is said to have a Γ-structure if it has a collection of charts f that are homeomorphisms from open sets Vi in M to open sets Ui in Rn such that, for every non-empty intersection, the natural map from fi (Ui ∩ Uj) to fj (Ui ∩ Uj) lies in Γ. This defines the structure of a smooth n-manifold if Γ consists of local diffeomorphims and a Riemann surface if n = 2—so that R2 ≡ C—and Γ consists of biholomorphisms. If Γ is the affine pseudogroup, M is said to have an affine structure; and if Γ is the Möbius pseudogroup, M is said to have a projective structure. Thus a genus one surface given as C/Λ for some lattice Λ ⊂ C has an affine structure; and a genus p > 1 surface given as the quotient of the upper half plane or unit disk by a Fuchsian group has a projective structure.[19]
Gunning in 1966 describes how this process can be reversed: for genus p > 1, the existence of a projective connection, defined using the Schwarzian derivative 𝜙2 and proved using standard results on cohomology, can be used to identify the universal covering surface with the upper half plane or unit disk (a similar result holds for genus 1, using affine connections and 𝜙1).[19]
See also
• Riccati equation
Notes
1. Thurston, William P. "Zippers and univalent functions." The Bieberbach conjecture (West Lafayette, Ind., 1985) 21 (1986): 185-197.
2. Weisstein, Eric W. "Schwarzian Derivative." From MathWorld—A Wolfram Web Resource.
3. Schiffer 1966
4. Hille 1976, pp. 374–401
5. Lehto 1987, p. 60
6. Duren 1983
7. Lehto 1987, p. 90
8. Nehari 1952
9. von Koppenfels & Stallmann 1959
10. Klein 1922
11. Ahlfors 1966
12. Lehto 1987
13. Imayoshi & Taniguchi 1992
14. Ovsienko & Tabachnikov 2005, pp. 21–22
15. Pekonen 1995
16. Sternberg 1983, pp. 421–424
17. Gunning 1978
18. Libermann harvnb error: no target: CITEREFLibermann (help)
19. Gunning 1966
References
• Ahlfors, Lars (1966), Lectures on quasiconformal mappings, Van Nostrand, pp. 117–146, Chapter 6, "Teichmüller Spaces"
• Duren, Peter L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, pp. 258–265, ISBN 978-0-387-90795-6]
• Guieu, Laurent; Roger, Claude (2007), L'algèbre et le groupe de Virasoro, Montreal: CRM, ISBN 978-2-921120-44-9
• Gunning, R. C. (1966), Lectures on Riemann surfaces, Princeton Mathematical Notes, Princeton University Press
• Gunning, R. C. (1978), On uniformization of complex manifolds: the role of connections, Mathematical Notes, vol. 22, Princeton University Press, ISBN 978-0-691-08176-2
• Hille, Einar (1976), Ordinary differential equations in the complex domain, Dover, pp. 374–401, ISBN 978-0-486-69620-1, Chapter 10, "The Schwarzian".
• Imayoshi, Y.; Taniguchi, M. (1992), An introduction to Teichmüller spaces, Springer-Verlag, ISBN 978-4-431-70088-3
• Kac, V. G.; Raina, A. K. (1987), Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, World Scientific, ISBN 978-9971-50-395-6
• von Koppenfels, W.; Stallmann, F. (1959), Praxis der konformen Abbildung, Die Grundlehren der mathematischen Wissenschaften, vol. 100, Springer-Verlag, pp. 114–141, Section 12, "Mapping of polygons with circular arcs".
• Klein, Felix (1922), Collected works, vol. 2, Springer-Verlag, pp. 540–549, "On the theory of generalized Lamé functions".
• Lehto, Otto (1987), Univalent functions and Teichmüller spaces, Springer-Verlag, pp. 50–59, 111–118, 196–205, ISBN 978-0-387-96310-5
• Libermann, Paulette (1959), "Pseudogroupes infinitésimaux attachés aux pseudogroupes de Lie", Bull. Soc. Math. France, 87: 409–425, doi:10.24033/bsmf.1536
• 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), Conformal mapping, Dover, pp. 189–226, ISBN 978-0-486-61137-2
• Ovsienko, V.; Tabachnikov, S. (2005), Projective Differential Geometry Old and New, Cambridge University Press, ISBN 978-0-521-83186-4
• Ovsienko, Valentin; Tabachnikov, Sergei (2009), "What Is . . . the Schwarzian Derivative?" (PDF), AMS Notices, 56 (1): 34–36
• Pekonen, Osmo (1995), "Universal Teichmüller space in geometry and physics", J. Geom. Phys., 15 (3): 227–251, arXiv:hep-th/9310045, Bibcode:1995JGP....15..227P, doi:10.1016/0393-0440(94)00007-Q, S2CID 119598450
• Schiffer, Menahem (1966), "Half-Order Differentials on Riemann Surfaces", SIAM Journal on Applied Mathematics, 14 (4): 922–934, doi:10.1137/0114073, JSTOR 2946143, S2CID 120194068
• Segal, Graeme (1981), "Unitary representations of some infinite-dimensional groups", Comm. Math. Phys., 80 (3): 301–342, Bibcode:1981CMaPh..80..301S, doi:10.1007/bf01208274, S2CID 121367853
• Sternberg, Shlomo (1983), Lectures on differential geometry (Second ed.), Chelsea Publishing, ISBN 978-0-8284-0316-0
• Takhtajan, Leon A.; Teo, Lee-Peng (2006), Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc., vol. 183
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Wikipedia
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Schwarz reflection principle
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane. In notation, if $F(z)$ is a function that satisfies the above requirements, then its extension to the rest of the complex plane is given by the formula,
$F({\bar {z}})={\overline {F(z)}}.$
This article is about the reflection principle in complex analysis. For reflection principles of set theory, see Reflection principle.
That is, we make the definition that agrees along the real axis.
The result proved by Hermann Schwarz is as follows. Suppose that F is a continuous function on the closed upper half plane $\left\{z\in \mathbb {C} \mid \operatorname {Im} (z)\geq 0\right\}$, holomorphic on the upper half plane $\left\{z\in \mathbb {C} \mid \operatorname {Im} (z)>0\right\}$, which takes real values on the real axis. Then the extension formula given above is an analytic continuation to the whole complex plane.[1]
In practice it would be better to have a theorem that allows F certain singularities, for example F a meromorphic function. To understand such extensions, one needs a proof method that can be weakened. In fact Morera's theorem is well adapted to proving such statements. Contour integrals involving the extension of F clearly split into two, using part of the real axis. So, given that the principle is rather easy to prove in the special case from Morera's theorem, understanding the proof is enough to generate other results.
The principle also adapts to apply to harmonic functions.
See also
• Kelvin transform
• Method of image charges
• Schwarz function
References
1. Cartan, Henri. Elementary theory of analytic functions of one or several variables. p. 75.
External links
• "Riemann-Schwarz principle", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Todd Rowland. "Schwarz reflection principle". MathWorld.
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Wikipedia
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Conic constant
In geometry, the conic constant (or Schwarzschild constant,[1] after Karl Schwarzschild) is a quantity describing conic sections, and is represented by the letter K. The constant is given by
$K=-e^{2},$
where e is the eccentricity of the conic section.
The equation for a conic section with apex at the origin and tangent to the y axis is
$y^{2}-2Rx+(K+1)x^{2}=0$
alternately
$x={\dfrac {y^{2}}{R+{\sqrt {R^{2}-(K+1)y^{2}}}}}$
where R is the radius of curvature at x = 0.
This formulation is used in geometric optics to specify oblate elliptical (K > 0), spherical (K = 0), prolate elliptical (0 > K > −1), parabolic (K = −1), and hyperbolic (K < −1) lens and mirror surfaces. When the paraxial approximation is valid, the optical surface can be treated as a spherical surface with the same radius.
Some non-optical design references use the letter p as the conic constant. In these cases, p = K + 1.
References
1. Rakich, Andrew (2005-08-18). Sasian, Jose M; Koshel, R. John; Juergens, Richard C (eds.). "The 100th birthday of the conic constant and Schwarzschild's revolutionary papers in optics". Novel Optical Systems Design and Optimization VIII. International Society for Optics and Photonics. 5875: 587501. Bibcode:2005SPIE.5875....1R. doi:10.1117/12.635041. S2CID 119718303.
• Smith, Warren J. (2008). Modern Optical Engineering, 4th ed. McGraw-Hill Professional. pp. 512–515. ISBN 978-0-07-147687-4.
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Wikipedia
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Schwarzschild geodesics
In general relativity, Schwarzschild geodesics describe the motion of test particles in the gravitational field of a central fixed mass $ M,$ that is, motion in the Schwarzschild metric. Schwarzschild geodesics have been pivotal in the validation of Einstein's theory of general relativity. For example, they provide accurate predictions of the anomalous precession of the planets in the Solar System and of the deflection of light by gravity.
General relativity
$G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$
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Schwarzschild geodesics pertain only to the motion of particles of masses so small they contribute little to the gravitational field. However, they are highly accurate in many astrophysical scenarios provided that $ m$ is many-fold smaller than the central mass $ M$, e.g., for planets orbiting their sun. Schwarzschild geodesics are also a good approximation to the relative motion of two bodies of arbitrary mass, provided that the Schwarzschild mass $ M$ is set equal to the sum of the two individual masses $ m_{1}$ and $ m_{2}$. This is important in predicting the motion of binary stars in general relativity.
Historical context
The Schwarzschild metric is named in honour of its discoverer Karl Schwarzschild, who found the solution in 1915, only about a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution.
In 1931, Yusuke Hagihara published a paper showing that the trajectory of a test particle in the Schwarzschild metric can be expressed in terms of elliptic functions.[1]
Samuil Kaplan in 1949 has shown that there is a minimum radius for the circular orbit to be stable in Schwarzschild metric.[2]
Schwarzschild metric
An exact solution to the Einstein field equations is the Schwarzschild metric, which corresponds to the external gravitational field of an uncharged, non-rotating, spherically symmetric body of mass $ M$. The Schwarzschild solution can be written as[3]
$c^{2}{d\tau }^{2}=\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}dt^{2}-{\frac {dr^{2}}{1-{\frac {r_{\text{s}}}{r}}}}-r^{2}d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\varphi ^{2}$
where
$\tau $, in the case of a test particle of small positive mass, is the proper time (time measured by a clock moving with the particle) in seconds,
$c$ is the speed of light in meters per second,
$t$ is, for $r>r_{\text{s}}$, the time coordinate (time measured by a stationary clock at infinity) in seconds,
$r$ is, for $r>r_{\text{s}}$, the radial coordinate (circumference of a circle centered at the star divided by $2\pi $) in meters,
$\theta $ is the colatitude (angle from North) in radians,
$\varphi $ is the longitude in radians, and
$r_{\text{s}}$ is the Schwarzschild radius of the massive body (in meters), which is related to its mass $ M$ by
$r_{\text{s}}={\frac {2GM}{c^{2}}},$
where $ G$ is the gravitational constant. The classical Newtonian theory of gravity is recovered in the limit as the ratio $ {\frac {r_{\text{s}}}{r}}$ goes to zero. In that limit, the metric returns to that defined by special relativity.
In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius $ r_{\text{s}}$ of the Earth is roughly 9 mm (3⁄8 inch); at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The Schwarzschild radius of the Sun is much larger, roughly 2953 meters, but at its surface, the ratio $ {\frac {r_{\text{s}}}{r}}$ is roughly 4 parts in a million. A white dwarf star is much denser, but even here the ratio at its surface is roughly 250 parts in a million. The ratio only becomes large close to ultra-dense objects such as neutron stars (where the ratio is roughly 50%) and black holes.
Orbits of test particles
We may simplify the problem by using symmetry to eliminate one variable from consideration. Since the Schwarzschild metric is symmetrical about $ \theta ={\frac {\pi }{2}}$, any geodesic that begins moving in that plane will remain in that plane indefinitely (the plane is totally geodesic). Therefore, we orient the coordinate system so that the orbit of the particle lies in that plane, and fix the $ \theta $ coordinate to be $ {\frac {\pi }{2}}$ so that the metric (of this plane) simplifies to
$c^{2}d\tau ^{2}=\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}dt^{2}-{\frac {dr^{2}}{1-{\frac {r_{\text{s}}}{r}}}}-r^{2}d\varphi ^{2}.$
Two constants of motion (values that do not change over proper time $\tau $) can be identified (cf. the derivation given below). One is the total energy $ E$:
$\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}={\frac {E}{mc^{2}}}.$
and the other is the specific angular momentum:
$h={\frac {L}{\mu }}=r^{2}{\frac {d\varphi }{d\tau }},$
where $ L$ is the total angular momentum of the two bodies, and $ \mu $ is the reduced mass. When $ M\gg m$, the reduced mass is approximately equal to $ m$. Sometimes it is assumed that $ m=\mu $. In the case of the planet Mercury this simplification introduces an error more than twice as large as the relativistic effect. When discussing geodesics, $ m$ can be considered fictitious, and what matters are the constants $ {\frac {E}{m}}$ and $ h$. In order to cover all possible geodesics, we need to consider cases in which $ {\frac {E}{m}}$ is infinite (giving trajectories of photons) or imaginary (for tachyonic geodesics). For the photonic case, we also need to specify a number corresponding to the ratio of the two constants, namely $ {\frac {mh}{E}}$, which may be zero or a non-zero real number.
Substituting these constants into the definition of the Schwarzschild metric
$c^{2}=\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}\left({\frac {dt}{d\tau }}\right)^{2}-{\frac {1}{1-{\frac {r_{\text{s}}}{r}}}}\left({\frac {dr}{d\tau }}\right)^{2}-r^{2}\left({\frac {d\varphi }{d\tau }}\right)^{2},$
yields an equation of motion for the radius as a function of the proper time $ \tau $:
$\left({\frac {dr}{d\tau }}\right)^{2}={\frac {E^{2}}{m^{2}c^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left(c^{2}+{\frac {h^{2}}{r^{2}}}\right).$
The formal solution to this is
$\tau =\int {\frac {dr}{\pm {\sqrt {{\frac {E^{2}}{m^{2}c^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left(c^{2}+{\frac {h^{2}}{r^{2}}}\right)}}}}.$
Note that the square root will be imaginary for tachyonic geodesics.
Using the relation higher up between $ {\frac {dt}{d\tau }}$ and $ E$, we can also write
$t=\int {\frac {dr}{\pm c\left(1-{\frac {r_{\text{s}}}{r}}\right){\sqrt {1-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left(c^{2}+{\frac {h^{2}}{r^{2}}}\right){\frac {m^{2}c^{2}}{E^{2}}}}}}}.$
Since asymptotically the integrand is inversely proportional to $ r-r_{\text{s}}$, this shows that in the $ r,\theta ,\varphi ,t$ frame of reference if $ r$ approaches $ r_{\text{s}}$ it does so exponentially without ever reaching it. However, as a function of $ \tau $, $ r$ does reach $ r_{\text{s}}$.
The above solutions are valid while the integrand is finite, but a total solution may involve two or an infinity of pieces, each described by the integral but with alternating signs for the square root.
When $ E=mc^{2}$ and $ h=0$, we can solve for $ t$ and $ \tau $ explicitly:
${\begin{aligned}t&={\text{constant}}\pm {\frac {r_{\text{s}}}{c}}\left({\frac {2}{3}}\left({\frac {r}{r_{\text{s}}}}\right)^{\frac {3}{2}}+2{\sqrt {\frac {r}{r_{\text{s}}}}}+\ln {\frac {\left|{\sqrt {\frac {r}{r_{\text{s}}}}}-1\right|}{{\sqrt {\frac {r}{r_{\text{s}}}}}+1}}\right)\\\tau &={\text{constant}}\pm {\frac {2}{3}}{\frac {r_{\text{s}}}{c}}\left({\frac {r}{r_{\text{s}}}}\right)^{\frac {3}{2}}\end{aligned}}$
and for photonic geodesics ($ m=0$) with zero angular momentum
${\begin{aligned}t&={\text{constant}}\pm {\frac {1}{c}}\left(r+r_{\text{s}}\ln \left|{\frac {r}{r_{\text{s}}}}-1\right|\right)\\\tau &={\text{constant}}.\end{aligned}}$
(Although the proper time is trivial in the photonic case, one can define an affine parameter $ \lambda $, and then the solution to the geodesic equation is $ r=c_{1}\lambda +c_{2}$.)
Another solvable case is that in which $ E=0$ and $ t$ and $ \varphi $ are constant. In the volume where $ r<r_{\text{s}}$ this gives for the proper time
$\tau ={\text{constant}}\pm {\frac {r_{\text{s}}}{c}}\left(\arcsin {\sqrt {\frac {r}{r_{\text{s}}}}}-{\sqrt {{\frac {r}{r_{\text{s}}}}\left(1-{\frac {r}{r_{\text{s}}}}\right)}}\right).$
This is close to solutions with $ {\frac {E^{2}}{m^{2}}}$ small and positive. Outside of $ r_{\text{s}}$ the $ E=0$ solution is tachyonic and the "proper time" is space-like:
$\tau ={\text{constant}}\pm i{\frac {r_{\text{s}}}{c}}\left(\ln \left({\sqrt {\frac {r}{r_{\text{s}}}}}+{\sqrt {{\frac {r}{r_{\text{s}}}}-1}}\right)+{\sqrt {{\frac {r}{r_{\text{s}}}}\left({\frac {r}{r_{\text{s}}}}-1\right)}}\right).$
This is close to other tachyonic solutions with $ {\frac {E^{2}}{m^{2}}}$ small and negative. The constant $ t$ tachyonic geodesic outside $ r_{\text{s}}$ is not continued by a constant $ t$ geodesic inside $ r_{\text{s}}$, but rather continues into a "parallel exterior region" (see Kruskal–Szekeres coordinates). Other tachyonic solutions can enter a black hole and re-exit into the parallel exterior region. The constant $ t$ solution inside the event horizon ($ r_{\text{s}}$) is continued by a constant $ t$ solution in a white hole.
When the angular momentum is not zero we can replace the dependence on proper time by a dependence on the angle $ \varphi $ using the definition of $ h$
$\left({\frac {dr}{d\varphi }}\right)^{2}=\left({\frac {dr}{d\tau }}\right)^{2}\left({\frac {d\tau }{d\varphi }}\right)^{2}=\left({\frac {dr}{d\tau }}\right)^{2}\left({\frac {r^{2}}{h}}\right)^{2},$
which yields the equation for the orbit
$\left({\frac {dr}{d\varphi }}\right)^{2}={\frac {r^{4}}{b^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left({\frac {r^{4}}{a^{2}}}+r^{2}\right)$
where, for brevity, two length-scales, $ a$ and $ b$, have been defined by
${\begin{aligned}a&={\frac {h}{c}},\\b&={\frac {cL}{E}}={\frac {hmc}{E}}.\end{aligned}}$
Note that in the tachyonic case, $ a$ will be imaginary and $ b$ real or infinite.
The same equation can also be derived using a Lagrangian approach[4] or the Hamilton–Jacobi equation[5] (see below). The solution of the orbit equation is
$\varphi =\int {\frac {dr}{\pm r^{2}{\sqrt {{\frac {1}{b^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left({\frac {1}{a^{2}}}+{\frac {1}{r^{2}}}\right)}}}}.$
This can be expressed in terms of the Weierstrass elliptic function $ \wp $.[6]
Local and delayed velocities
Unlike in classical mechanics, in Schwarzschild coordinates $ {\frac {{\rm {d}}r}{{\rm {d}}\tau }}$ and $ r\ {\frac {{\rm {d}}\varphi }{{\rm {d}}\tau }}$ are not the radial $ v_{\parallel }$ and transverse $ v_{\perp }$ components of the local velocity $ v$ (relative to a stationary observer), instead they give the components for the celerity which are related to $ v$ by
${\frac {{\rm {d}}r}{{\rm {d}}\tau }}=v_{\parallel }{\sqrt {1-{\frac {r_{\text{s}}}{r}}}}\ \gamma $
for the radial and
${\frac {{\rm {d}}\varphi }{{\rm {d}}\tau }}={\frac {v_{\perp }}{r}}\ \gamma $
for the transverse component of motion, with $ v^{2}=v_{\parallel }^{2}+v_{\perp }^{2}$. The coordinate bookkeeper far away from the scene observes the shapiro-delayed velocity $ {\hat {v}}$, which is given by the relation
${\hat {v}}_{\perp }=v_{\perp }{\sqrt {1-{\frac {r_{\text{s}}}{r}}}}$ and ${\hat {v}}_{\parallel }=v_{\parallel }\left(1-{\frac {r_{\text{s}}}{r}}\right)$.
The time dilation factor between the bookkeeper and the moving test-particle can also be put into the form
${\frac {{\rm {d}}\tau }{{\rm {d}}t}}={\frac {\sqrt {1-{\frac {r_{\text{s}}}{r}}}}{\gamma }}$
where the numerator is the gravitational, and the denominator is the kinematic component of the time dilation. For a particle falling in from infinity the left factor equals the right factor, since the in-falling velocity $ v$ matches the escape velocity $ c{\sqrt {\frac {r_{\text{s}}}{r}}}$ in this case.
The two constants angular momentum $ L$ and total energy $ E$ of a test-particle with mass $ m$ are in terms of $ v$
$L=m\ v_{\perp }\ r\ \gamma $
and
$E=mc^{2}\ {\sqrt {1-{\frac {r_{\text{s}}}{r}}}}\ \gamma $
where
$E=E_{\rm {rest}}+E_{\rm {kin}}+E_{\rm {pot}}$
and
$E_{\rm {rest}}=mc^{2}\ ,\ \ E_{\rm {kin}}=(\gamma -1)mc^{2}\ ,\ \ E_{\rm {pot}}=\left({\sqrt {1-{\frac {r_{\text{s}}}{r}}}}-1\right)\ \gamma \ mc^{2}$
For massive testparticles $ \gamma $ is the Lorentz factor $ \gamma =1/{\sqrt {1-v^{2}/c^{2}}}$ and $ \tau $ is the proper time, while for massless particles like photons $ \gamma $ is set to $ 1$ and $ \tau $ takes the role of an affine parameter. If the particle is massless $ E_{\rm {rest}}$ is replaced with $ E_{\rm {kin}}$ and $ mc^{2}$ with $ hf$, where $ h$ is the Planck constant and $ f$ the locally observed frequency.
Exact solution using elliptic functions
The fundamental equation of the orbit is easier to solve[note 1] if it is expressed in terms of the inverse radius $ u={\frac {1}{r}}$
$\left({\frac {du}{d\varphi }}\right)^{2}={\frac {1}{b^{2}}}-\left(1-ur_{\text{s}}\right)\left({\frac {1}{a^{2}}}+u^{2}\right)$
The right-hand side of this equation is a cubic polynomial, which has three roots, denoted here as $ u_{1}$, $ u_{2}$, and $ u_{3}$
$\left({\frac {du}{d\varphi }}\right)^{2}=r_{\text{s}}\left(u-u_{1}\right)\left(u-u_{2}\right)\left(u-u_{3}\right)$
The sum of the three roots equals the coefficient of the $ u^{2}$ term
$u_{1}+u_{2}+u_{3}={\frac {1}{r_{\text{s}}}}$
A cubic polynomial with real coefficients can either have three real roots, or one real root and two complex conjugate roots. If all three roots are real numbers, the roots are labeled so that $ u_{1}<u_{2}<u_{3}$. If instead there is only one real root, then that is denoted as $ u_{3}$; the complex conjugate roots are labeled $ u_{1}$ and $ u_{2}$. Using Descartes' rule of signs, there can be at most one negative root; $ u_{1}$ is negative if and only if $ b<a$. As discussed below, the roots are useful in determining the types of possible orbits.
Given this labeling of the roots, the solution of the fundamental orbital equation is
$u=u_{1}+\left(u_{2}-u_{1}\right)\,\mathrm {sn} ^{2}\left({\frac {1}{2}}\varphi {\sqrt {r_{\text{s}}\left(u_{3}-u_{1}\right)}}+\delta \right)$
where $ \mathrm {sn} $ represents the sinus amplitudinus function (one of the Jacobi elliptic functions) and $ \delta $ is a constant of integration reflecting the initial position. The elliptic modulus $ k$ of this elliptic function is given by the formula
$k={\sqrt {\frac {u_{2}-u_{1}}{u_{3}-u_{1}}}}$
Newtonian limit
To recover the Newtonian solution for the planetary orbits, one takes the limit as the Schwarzschild radius $ r_{\text{s}}$ goes to zero. In this case, the third root $ u_{3}$ becomes roughly $ {\frac {1}{r_{\text{s}}}}$, and much larger than $ u_{1}$ or $ u_{2}$. Therefore, the modulus $ k$ tends to zero; in that limit, $ \mathrm {sn} $ becomes the trigonometric sine function
$u=u_{1}+\left(u_{2}-u_{1}\right)\,\sin ^{2}\left({\frac {1}{2}}\varphi +\delta \right)$
Consistent with Newton's solutions for planetary motions, this formula describes a focal conic of eccentricity $ e$
$e={\frac {u_{2}-u_{1}}{u_{2}+u_{1}}}$
If $ u_{1}$ is a positive real number, then the orbit is an ellipse where $ u_{1}$ and $ u_{2}$ represent the distances of furthest and closest approach, respectively. If $ u_{1}$ is zero or a negative real number, the orbit is a parabola or a hyperbola, respectively. In these latter two cases, $ u_{2}$ represents the distance of closest approach; since the orbit goes to infinity ($ u=0$), there is no distance of furthest approach.
Roots and overview of possible orbits
A root represents a point of the orbit where the derivative vanishes, i.e., where $ {\frac {du}{d\phi }}=0$. At such a turning point, $ u$ reaches a maximum, a minimum, or an inflection point, depending on the value of the second derivative, which is given by the formula
${\frac {d^{2}u}{d\varphi ^{2}}}={\frac {r_{\text{s}}}{2}}\left[\left(u-u_{2}\right)\left(u-u_{3}\right)+\left(u-u_{1}\right)\left(u-u_{3}\right)+\left(u-u_{1}\right)\left(u-u_{2}\right)\right]$
If all three roots are distinct real numbers, the second derivative is positive, negative, and positive at u1, u2, and u3, respectively. It follows that a graph of u versus φ may either oscillate between u1 and u2, or it may move away from u3 towards infinity (which corresponds to r going to zero). If u1 is negative, only part of an "oscillation" will actually occur. This corresponds to the particle coming from infinity, getting near the central mass, and then moving away again toward infinity, like the hyperbolic trajectory in the classical solution.
If the particle has just the right amount of energy for its angular momentum, u2 and u3 will merge. There are three solutions in this case. The orbit may spiral in to $ r={\frac {1}{u_{2}}}={\frac {1}{u_{3}}}$, approaching that radius as (asymptotically) a decreasing exponential in φ, $ \tau $, or $ t$. Or one can have a circular orbit at that radius. Or one can have an orbit that spirals down from that radius to the central point. The radius in question is called the inner radius and is between $ {\frac {3}{2}}$ and 3 times rs. A circular orbit also results when $ u_{2}$ is equal to $ u_{1}$, and this is called the outer radius. These different types of orbits are discussed below.
If the particle comes at the central mass with sufficient energy and sufficiently low angular momentum then only $ u_{1}$ will be real. This corresponds to the particle falling into a black hole. The orbit spirals in with a finite change in φ.
Precession of orbits
The function sn and its square sn2 have periods of 4K and 2K, respectively, where K is defined by the equation[note 2]
$K=\int _{0}^{1}{\frac {dy}{\sqrt {\left(1-y^{2}\right)\left(1-k^{2}y^{2}\right)}}}$
Therefore, the change in φ over one oscillation of $ u$ (or, equivalently, one oscillation of $ r$) equals[7]
$\Delta \varphi ={\frac {4K}{\sqrt {r_{\text{s}}\left(u_{3}-u_{1}\right)}}}$
In the classical limit, u3 approaches $ {\frac {1}{r_{\text{s}}}}$ and is much larger than $ u_{1}$ or $ u_{2}$. Hence, $ k^{2}$ is approximately
$k^{2}={\frac {u_{2}-u_{1}}{u_{3}-u_{1}}}\approx r_{\text{s}}\left(u_{2}-u_{1}\right)\ll 1$
For the same reasons, the denominator of Δφ is approximately
${\frac {1}{\sqrt {r_{\text{s}}\left(u_{3}-u_{1}\right)}}}={\frac {1}{\sqrt {1-r_{\text{s}}\left(2u_{1}+u_{2}\right)}}}\approx 1+{\frac {1}{2}}r_{\text{s}}\left(2u_{1}+u_{2}\right)$
Since the modulus $ k$ is close to zero, the period K can be expanded in powers of $ k$; to lowest order, this expansion yields
$K\approx \int _{0}^{1}{\frac {dy}{\sqrt {1-y^{2}}}}\left(1+{\frac {1}{2}}k^{2}y^{2}\right)={\frac {\pi }{2}}\left(1+{\frac {k^{2}}{4}}\right)$
Substituting these approximations into the formula for Δφ yields a formula for angular advance per radial oscillation
$\delta \varphi =\Delta \varphi -2\pi \approx {\frac {3}{2}}\pi r_{\text{s}}\left(u_{1}+u_{2}\right)$
For an elliptical orbit, $ u_{1}$ and $ u_{2}$ represent the inverses of the longest and shortest distances, respectively. These can be expressed in terms of the ellipse's semi-major axis $ A$ and its orbital eccentricity $ e$,
${\begin{aligned}r_{\text{max}}&={\frac {1}{u_{1}}}=A(1+e)\\r_{\text{min}}&={\frac {1}{u_{2}}}=A(1-e)\end{aligned}}$
giving
$u_{1}+u_{2}={\frac {2}{A\left(1-e^{2}\right)}}$
Substituting the definition of $ r_{\text{s}}$ gives the final equation
$\delta \varphi \approx {\frac {6\pi GM}{c^{2}A\left(1-e^{2}\right)}}$
Bending of light by gravity
In the limit as the particle mass m goes to zero (or, equivalently if the light is heading directly toward the central mass, as the length-scale a goes to infinity), the equation for the orbit becomes
$\varphi =\int {\frac {dr}{r^{2}{\sqrt {{\frac {1}{b^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {1}{r^{2}}}}}}}$
Expanding in powers of $ {\frac {r_{\text{s}}}{r}}$, the leading order term in this formula gives the approximate angular deflection δφ for a massless particle coming in from infinity and going back out to infinity:
$\delta \varphi \approx {\frac {2r_{\text{s}}}{b}}={\frac {4GM}{c^{2}b}}.$
Here, $ b$ is the impact parameter, somewhat greater than the distance of closest approach, $ r_{3}$:[8]
$b=r_{3}{\sqrt {\frac {r_{3}}{r_{3}-r_{\text{s}}}}}$
Although this formula is approximate, it is accurate for most measurements of gravitational lensing, due to the smallness of the ratio $ {\frac {r_{\text{s}}}{r}}$. For light grazing the surface of the sun, the approximate angular deflection is roughly 1.75 arcseconds, roughly one millionth part of a circle.
Relation to Newtonian physics
Effective radial potential energy
The equation of motion for the particle derived above
$\left({\frac {dr}{d\tau }}\right)^{2}={\frac {E^{2}}{m^{2}c^{2}}}-c^{2}+{\frac {r_{\text{s}}c^{2}}{r}}-{\frac {L^{2}}{m\mu r^{2}}}+{\frac {r_{\text{s}}L^{2}}{m\mu r^{3}}}$
can be rewritten using the definition of the Schwarzschild radius rs as
${\frac {1}{2}}m\left({\frac {dr}{d\tau }}\right)^{2}=\left[{\frac {E^{2}}{2mc^{2}}}-{\frac {1}{2}}mc^{2}\right]+{\frac {GMm}{r}}-{\frac {L^{2}}{2\mu r^{2}}}+{\frac {G(M+m)L^{2}}{c^{2}\mu r^{3}}},$
which is equivalent to a particle moving in a one-dimensional effective potential
$V(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2\mu r^{2}}}-{\frac {G(M+m)L^{2}}{c^{2}\mu r^{3}}}$
The first two terms are well-known classical energies, the first being the attractive Newtonian gravitational potential energy and the second corresponding to the repulsive "centrifugal" potential energy; however, the third term is an attractive energy unique to general relativity. As shown below and elsewhere, this inverse-cubic energy causes elliptical orbits to precess gradually by an angle δφ per revolution
$\delta \varphi \approx {\frac {6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}}$
where $ A$ is the semi-major axis and $ e$ is the eccentricity.
The third term is attractive and dominates at small $ r$ values, giving a critical inner radius rinner at which a particle is drawn inexorably inwards to $ r=0$; this inner radius is a function of the particle's angular momentum per unit mass or, equivalently, the $ a$ length-scale defined above.
Circular orbits and their stability
The effective potential $ V$ can be re-written in terms of the length $ a={\frac {h}{c}}$.
$V(r)={\frac {\mu c^{2}}{2}}\left[-{\frac {r_{\text{s}}}{r}}+{\frac {a^{2}}{r^{2}}}-{\frac {r_{\text{s}}a^{2}}{r^{3}}}\right]$
Circular orbits are possible when the effective force is zero
$F=-{\frac {dV}{dr}}=-{\frac {\mu c^{2}}{2r^{4}}}\left[r_{\text{s}}r^{2}-2a^{2}r+3r_{\text{s}}a^{2}\right]=0$
i.e., when the two attractive forces — Newtonian gravity (first term) and the attraction unique to general relativity (third term) — are exactly balanced by the repulsive centrifugal force (second term). There are two radii at which this balancing can occur, denoted here as rinner and router
${\begin{aligned}r_{\text{outer}}&={\frac {a^{2}}{r_{\text{s}}}}\left(1+{\sqrt {1-{\frac {3r_{\text{s}}^{2}}{a^{2}}}}}\right)\\[3pt]r_{\text{inner}}&={\frac {a^{2}}{r_{\text{s}}}}\left(1-{\sqrt {1-{\frac {3r_{\text{s}}^{2}}{a^{2}}}}}\right)={\frac {3a^{2}}{r_{\text{outer}}}}\end{aligned}}$
which are obtained using the quadratic formula. The inner radius rinner is unstable, because the attractive third force strengthens much faster than the other two forces when r becomes small; if the particle slips slightly inwards from rinner (where all three forces are in balance), the third force dominates the other two and draws the particle inexorably inwards to r = 0. At the outer radius, however, the circular orbits are stable; the third term is less important and the system behaves more like the non-relativistic Kepler problem.
When $ a$ is much greater than $ r_{\text{s}}$ (the classical case), these formulae become approximately
${\begin{aligned}r_{\text{outer}}&\approx {\frac {2a^{2}}{r_{\text{s}}}}\\[3pt]r_{\text{inner}}&\approx {\frac {3}{2}}r_{\text{s}}\end{aligned}}$
Substituting the definitions of $ a$ and rs into router yields the classical formula for a particle of mass $ m$ orbiting a body of mass $ M$.
$r_{\mathrm {outer} }^{3}={\frac {G(M+m)}{\omega _{\varphi }^{2}}}$
where ωφ is the orbital angular speed of the particle. This formula is obtained in non-relativistic mechanics by setting the centrifugal force equal to the Newtonian gravitational force:
${\frac {GMm}{r^{2}}}=\mu \omega _{\varphi }^{2}r$
Where $ \mu $ is the reduced mass.
In our notation, the classical orbital angular speed equals
$\omega _{\varphi }^{2}\approx {\frac {GM}{r_{\mathrm {outer} }^{3}}}=\left({\frac {r_{\text{s}}c^{2}}{2r_{\mathrm {outer} }^{3}}}\right)=\left({\frac {r_{\text{s}}c^{2}}{2}}\right)\left({\frac {r_{\text{s}}^{3}}{8a^{6}}}\right)={\frac {c^{2}r_{\text{s}}^{4}}{16a^{6}}}$
At the other extreme, when a2 approaches 3rs2 from above, the two radii converge to a single value
$r_{\mathrm {outer} }\approx r_{\mathrm {inner} }\approx 3r_{\text{s}}$
The quadratic solutions above ensure that router is always greater than 3rs, whereas rinner lies between 3⁄2 rs and 3rs. Circular orbits smaller than 3⁄2 rs are not possible. For massless particles, a goes to infinity, implying that there is a circular orbit for photons at rinner = 3⁄2 rs. The sphere of this radius is sometimes known as the photon sphere.
Precession of elliptical orbits
The orbital precession rate may be derived using this radial effective potential V. A small radial deviation from a circular orbit of radius router will oscillate stably with an angular frequency
$\omega _{r}^{2}={\frac {1}{m}}\left[{\frac {d^{2}V}{dr^{2}}}\right]_{r=r_{\mathrm {outer} }}$
which equals
$\omega _{r}^{2}=\left({\frac {c^{2}r_{\text{s}}}{2r_{\mathrm {outer} }^{4}}}\right)\left(r_{\mathrm {outer} }-r_{\mathrm {inner} }\right)=\omega _{\varphi }^{2}{\sqrt {1-{\frac {3r_{\text{s}}^{2}}{a^{2}}}}}$
Taking the square root of both sides and performing a Taylor series expansion yields
$\omega _{r}=\omega _{\varphi }\left[1-{\frac {3r_{\text{s}}^{2}}{4a^{2}}}+{\mathcal {O}}\left({\frac {r_{\text{s}}^{4}}{a^{4}}}\right)\right]$
Multiplying by the period T of one revolution gives the precession of the orbit per revolution
$\delta \varphi =T\left(\omega _{\varphi }-\omega _{r}\right)\approx 2\pi \left({\frac {3r_{\text{s}}^{2}}{4a^{2}}}\right)={\frac {3\pi m^{2}c^{2}}{2L^{2}}}r_{\text{s}}^{2}$
where we have used ωφT = 2п and the definition of the length-scale a. Substituting the definition of the Schwarzschild radius rs gives
$\delta \varphi \approx {\frac {3\pi m^{2}c^{2}}{2L^{2}}}\left({\frac {4G^{2}M^{2}}{c^{4}}}\right)={\frac {6\pi G^{2}M^{2}m^{2}}{c^{2}L^{2}}}$
This may be simplified using the elliptical orbit's semiaxis A and eccentricity e related by the formula
${\frac {h^{2}}{G(M+m)}}=A\left(1-e^{2}\right)$
to give the precession angle
$\delta \varphi \approx {\frac {6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}}$
Mathematical derivations of the orbital equation
Christoffel symbols
The non-vanishing Christoffel symbols for the Schwarzschild-metric are:[9]
${\begin{aligned}\Gamma _{rt}^{t}=-\Gamma _{rr}^{r}&={\frac {r_{\text{s}}}{2r(r-r_{\text{s}})}}\\[3pt]\Gamma _{tt}^{r}&=-{\frac {r_{\text{s}}(r-r_{\text{s}})}{2r^{3}}}\\[3pt]\Gamma _{\phi \phi }^{r}&=(r_{\text{s}}-r)\sin ^{2}(\theta )\\[3pt]\Gamma _{\theta \theta }^{r}&=r_{\text{s}}-r\\[3pt]\Gamma _{r\theta }^{\theta }=\Gamma _{r\phi }^{\phi }&={\frac {1}{r}}\\[3pt]\Gamma _{\phi \phi }^{\theta }&=-\sin(\theta )\cos(\theta )\\[3pt]\Gamma _{\theta \phi }^{\phi }&=\cot(\theta )\end{aligned}}$
Geodesic equation
According to Einstein's theory of general relativity, particles of negligible mass travel along geodesics in the space-time. In flat space-time, far from a source of gravity, these geodesics correspond to straight lines; however, they may deviate from straight lines when the space-time is curved. The equation for the geodesic lines is[10]
${\frac {d^{2}x^{\lambda }}{dq^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{dq}}{\frac {dx^{\nu }}{dq}}=0$
where Γ represents the Christoffel symbol and the variable $ q$ parametrizes the particle's path through space-time, its so-called world line. The Christoffel symbol depends only on the metric tensor $ g_{\mu \nu }$, or rather on how it changes with position. The variable $ q$ is a constant multiple of the proper time $ \tau $ for timelike orbits (which are traveled by massive particles), and is usually taken to be equal to it. For lightlike (or null) orbits (which are traveled by massless particles such as the photon), the proper time is zero and, strictly speaking, cannot be used as the variable $ q$. Nevertheless, lightlike orbits can be derived as the ultrarelativistic limit of timelike orbits, that is, the limit as the particle mass m goes to zero while holding its total energy fixed.
Therefore, to solve for the motion of a particle, the most straightforward way is to solve the geodesic equation, an approach adopted by Einstein[11] and others.[12] The Schwarzschild metric may be written as
$c^{2}d\tau ^{2}=w(r)c^{2}dt^{2}-v(r)dr^{2}-r^{2}d\theta ^{2}-r^{2}\sin ^{2}\theta d\phi ^{2}\,$
where the two functions $ w(r)=1-{\frac {r_{\text{s}}}{r}}$and its reciprocal $ v(r)={\frac {1}{w(r)}}$are defined for brevity. From this metric, the Christoffel symbols $ \Gamma _{\mu \nu }^{\lambda }$may be calculated, and the results substituted into the geodesic equations
${\begin{aligned}0&={\frac {d^{2}\theta }{dq^{2}}}+{\frac {2}{r}}{\frac {d\theta }{dq}}{\frac {dr}{dq}}-\sin \theta \cos \theta \left({\frac {d\phi }{dq}}\right)^{2}\\[3pt]0&={\frac {d^{2}\phi }{dq^{2}}}+{\frac {2}{r}}{\frac {d\phi }{dq}}{\frac {dr}{dq}}+2\cot \theta {\frac {d\phi }{dq}}{\frac {d\theta }{dq}}\\[3pt]0&={\frac {d^{2}t}{dq^{2}}}+{\frac {1}{w}}{\frac {dw}{dr}}{\frac {dt}{dq}}{\frac {dr}{dq}}\\[3pt]0&={\frac {d^{2}r}{dq^{2}}}-{\frac {1}{v}}{\frac {dv}{dr}}\left({\frac {dr}{dq}}\right)^{2}-{\frac {r}{v}}\left({\frac {d\theta }{dq}}\right)^{2}-{\frac {r\sin ^{2}\theta }{v}}\left({\frac {d\phi }{dq}}\right)^{2}+{\frac {c^{2}}{2v}}{\frac {dw}{dr}}\left({\frac {dt}{dq}}\right)^{2}\end{aligned}}$
It may be verified that $ \theta ={\frac {\pi }{2}}$ is a valid solution by substitution into the first of these four equations. By symmetry, the orbit must be planar, and we are free to arrange the coordinate frame so that the equatorial plane is the plane of the orbit. This $ \theta $ solution simplifies the second and fourth equations.
To solve the second and third equations, it suffices to divide them by $ {\frac {d\phi }{dq}}$ and $ {\frac {dt}{dq}}$, respectively.
${\begin{aligned}0&={\frac {d}{dq}}\left[\ln {\frac {d\phi }{dq}}+\ln r^{2}\right]\\[3pt]0&={\frac {d}{dq}}\left[\ln {\frac {dt}{dq}}+\ln w\right],\end{aligned}}$
which yields two constants of motion.
Lagrangian approach
Because test particles follow geodesics in a fixed metric, the orbits of those particles may be determined using the calculus of variations, also called the Lagrangian approach.[13] Geodesics in space-time are defined as curves for which small local variations in their coordinates (while holding their endpoints events fixed) make no significant change in their overall length s. This may be expressed mathematically using the calculus of variations
$0=\delta s=\delta \int ds=\delta \int {\sqrt {g_{\mu \nu }{\frac {dx^{\mu }}{d\tau }}{\frac {dx^{\nu }}{d\tau }}}}d\tau =\delta \int {\sqrt {2T}}d\tau $
where τ is the proper time, s = cτ is the arc-length in space-time and T is defined as
$2T=c^{2}=\left({\frac {ds}{d\tau }}\right)^{2}=g_{\mu \nu }{\frac {dx^{\mu }}{d\tau }}{\frac {dx^{\nu }}{d\tau }}=\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}\left({\frac {dt}{d\tau }}\right)^{2}-{\frac {1}{1-{\frac {r_{\text{s}}}{r}}}}\left({\frac {dr}{d\tau }}\right)^{2}-r^{2}\left({\frac {d\varphi }{d\tau }}\right)^{2}$
in analogy with kinetic energy. If the derivative with respect to proper time is represented by a dot for brevity
${\dot {x}}^{\mu }={\frac {dx^{\mu }}{d\tau }}$
T may be written as
$2T=c^{2}=\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}\left({\dot {t}}\right)^{2}-{\frac {1}{1-{\frac {r_{\text{s}}}{r}}}}\left({\dot {r}}\right)^{2}-r^{2}\left({\dot {\varphi }}\right)^{2}$
Constant factors (such as c or the square root of two) don't affect the answer to the variational problem; therefore, taking the variation inside the integral yields Hamilton's principle
$0=\delta \int {\sqrt {2T}}d\tau =\int {\frac {\delta T}{\sqrt {2T}}}d\tau ={\frac {1}{c}}\delta \int Td\tau .$
The solution of the variational problem is given by Lagrange's equations
${\frac {d}{d\tau }}\left({\frac {\partial T}{\partial {\dot {x}}^{\sigma }}}\right)={\frac {\partial T}{\partial x^{\sigma }}}.$
When applied to t and φ, these equations reveal two constants of motion
${\begin{aligned}{\frac {d}{d\tau }}\left[r^{2}{\frac {d\varphi }{d\tau }}\right]&=0,\\{\frac {d}{d\tau }}\left[\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}\right]&=0,\end{aligned}}$
which may be expressed in terms of two constant length-scales, $ a$ and $ b$
${\begin{aligned}r^{2}{\frac {d\varphi }{d\tau }}&=ac,\\\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}&={\frac {a}{b}}.\end{aligned}}$
As shown above, substitution of these equations into the definition of the Schwarzschild metric yields the equation for the orbit.
Hamiltonian approach
A Lagrangian solution can be recast into an equivalent Hamiltonian form.[14] In this case, the Hamiltonian $H$ is given by
$2H=c^{2}={\frac {p_{t}^{2}}{c^{2}\left(1-{\frac {r_{\text{s}}}{r}}\right)}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)p_{r}^{2}-{\frac {p_{\theta }^{2}}{r^{2}}}-{\frac {p_{\varphi }^{2}}{r^{2}\sin ^{2}\theta }}$
Once again, the orbit may be restricted to $ \theta ={\frac {\pi }{2}}$by symmetry. Since $ t$ and $ \varphi $ do not appear in the Hamiltonian, their conjugate momenta are constant; they may be expressed in terms of the speed of light $ c$ and two constant length-scales $ a$ and $ b$
${\begin{aligned}p_{\varphi }&=-ac\\p_{\theta }&=0\\p_{t}&={\frac {ac^{2}}{b}}\end{aligned}}$
The derivatives with respect to proper time are given by
${\begin{aligned}{\frac {dr}{d\tau }}&={\frac {\partial H}{\partial p_{r}}}=-\left(1-{\frac {r_{\text{s}}}{r}}\right)p_{r}\\{\frac {d\varphi }{d\tau }}&={\frac {\partial H}{\partial p_{\varphi }}}={\frac {-p_{\varphi }}{r^{2}}}={\frac {ac}{r^{2}}}\\{\frac {dt}{d\tau }}&={\frac {\partial H}{\partial p_{t}}}={\frac {p_{t}}{c^{2}\left(1-{\frac {r_{\text{s}}}{r}}\right)}}={\frac {a}{b\left(1-{\frac {r_{\text{s}}}{r}}\right)}}\end{aligned}}$
Dividing the first equation by the second yields the orbital equation
${\frac {dr}{d\varphi }}=-{\frac {r^{2}}{ac}}\left(1-{\frac {r_{\text{s}}}{r}}\right)p_{r}$
The radial momentum pr can be expressed in terms of r using the constancy of the Hamiltonian $ H={\frac {c^{2}}{2}}$; this yields the fundamental orbital equation
$\left({\frac {dr}{d\varphi }}\right)^{2}={\frac {r^{4}}{b^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left({\frac {r^{4}}{a^{2}}}+r^{2}\right)$
Hamilton–Jacobi approach
The orbital equation can be derived from the Hamilton–Jacobi equation.[15] The advantage of this approach is that it equates the motion of the particle with the propagation of a wave, and leads neatly into the derivation of the deflection of light by gravity in general relativity, through Fermat's principle. The basic idea is that, due to gravitational slowing of time, parts of a wave-front closer to a gravitating mass move more slowly than those further away, thus bending the direction of the wave-front's propagation.
Using general covariance, the Hamilton–Jacobi equation for a single particle of unit mass can be expressed in arbitrary coordinates as
$g^{\mu \nu }{\frac {\partial S}{\partial x^{\mu }}}{\frac {\partial S}{\partial x^{\nu }}}=c^{2}.$
This is equivalent to the Hamiltonian formulation above, with the partial derivatives of the action taking the place of the generalized momenta. Using the Schwarzschild metric gμν, this equation becomes
${\frac {1}{c^{2}\left(1-{\frac {r_{\text{s}}}{r}}\right)}}\left({\frac {\partial S}{\partial t}}\right)^{2}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left({\frac {\partial S}{\partial r}}\right)^{2}-{\frac {1}{r^{2}}}\left({\frac {\partial S}{\partial \varphi }}\right)^{2}=c^{2}$
where we again orient the spherical coordinate system with the plane of the orbit. The time t and azimuthal angle φ are cyclic coordinates, so that the solution for Hamilton's principal function S can be written
$S=-p_{t}t+p_{\varphi }\varphi +S_{r}(r)\,$
where $p_{t}$ and $p_{\varphi }$ are the constant generalized momenta. The Hamilton–Jacobi equation gives an integral solution for the radial part $S_{r}(r)$
$S_{r}(r)=\int ^{r}{\frac {dr}{1-{\frac {r_{\text{s}}}{r}}}}{\sqrt {{\frac {p_{t}^{2}}{c^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left(c^{2}+{\frac {p_{\varphi }^{2}}{r^{2}}}\right)}}.$
Taking the derivative of Hamilton's principal function S with respect to the conserved momentum pφ yields
${\frac {\partial S}{\partial p_{\varphi }}}=\varphi +{\frac {\partial S_{r}}{\partial p_{\varphi }}}=\mathrm {constant} $
which equals
$\varphi -\int ^{r}{\frac {p_{\varphi }dr}{r^{2}{\sqrt {{\frac {p_{t}^{2}}{c^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left(c^{2}+{\frac {p_{\varphi }^{2}}{r^{2}}}\right)}}}}=\mathrm {constant} $
Taking an infinitesimal variation in φ and r yields the fundamental orbital equation
$\left({\frac {dr}{d\varphi }}\right)^{2}={\frac {r^{4}}{b^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left({\frac {r^{4}}{a^{2}}}+r^{2}\right).$
where the conserved length-scales a and b are defined by the conserved momenta by the equations
${\begin{aligned}{\frac {\partial S}{\partial \varphi }}=p_{\varphi }&=-ac\\{\frac {\partial S}{\partial t}}=p_{t}&={\frac {ac^{2}}{b}}\end{aligned}}$
Hamilton's principle
The action integral for a particle affected only by gravity is
$S=\int {-mc^{2}d\tau }=-mc\int {c{\frac {d\tau }{dq}}dq}=-mc\int {{\sqrt {-g_{\mu \nu }{\frac {dx^{\mu }}{dq}}{\frac {dx^{\nu }}{dq}}}}dq}$
where $ \tau $ is the proper time and $ q$ is any smooth parameterization of the particle's world line. If one applies the calculus of variations to this, one again gets the equations for a geodesic. To simplify the calculations, one first takes the variation of the square of the integrand. For the metric and coordinates of this case and assuming that the particle is moving in the equatorial plane $ \theta ={\frac {\pi }{2}}$, that square is
$\left(c{\frac {d\tau }{dq}}\right)^{2}=-g_{\mu \nu }{\frac {dx^{\mu }}{dq}}{\frac {dx^{\nu }}{dq}}=\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}\left({\frac {dt}{dq}}\right)^{2}-{\frac {1}{1-{\frac {r_{\text{s}}}{r}}}}\left({\frac {dr}{dq}}\right)^{2}-r^{2}\left({\frac {d\varphi }{dq}}\right)^{2}\,.$
Taking variation of this gives
$\delta \left(c{\frac {d\tau }{dq}}\right)^{2}=2c^{2}{\frac {d\tau }{dq}}\delta {\frac {d\tau }{dq}}=\delta \left[\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}\left({\frac {dt}{dq}}\right)^{2}-{\frac {1}{1-{\frac {r_{\text{s}}}{r}}}}\left({\frac {dr}{dq}}\right)^{2}-r^{2}\left({\frac {d\varphi }{dq}}\right)^{2}\right]\,.$
Motion in longitude
Vary with respect to longitude $ \varphi $ only to get
$2c^{2}{\frac {d\tau }{dq}}\delta {\frac {d\tau }{dq}}=-2r^{2}{\frac {d\varphi }{dq}}\delta {\frac {d\varphi }{dq}}\,.$
Divide by $ 2c{\frac {d\tau }{dq}}$ to get the variation of the integrand itself
$c\,\delta {\frac {d\tau }{dq}}=-{\frac {r^{2}}{c}}{\frac {d\varphi }{d\tau }}\delta {\frac {d\varphi }{dq}}=-{\frac {r^{2}}{c}}{\frac {d\varphi }{d\tau }}{\frac {d\delta \varphi }{dq}}\,.$
Thus
$0=\delta \int {c{\frac {d\tau }{dq}}dq}=\int {c\delta {\frac {d\tau }{dq}}dq}=\int {-{\frac {r^{2}}{c}}{\frac {d\varphi }{d\tau }}{\frac {d\delta \varphi }{dq}}dq}\,.$
Integrating by parts gives
$0=-{\frac {r^{2}}{c}}{\frac {d\varphi }{d\tau }}\delta \varphi -\int {{\frac {d}{dq}}\left[-{\frac {r^{2}}{c}}{\frac {d\varphi }{d\tau }}\right]\delta \varphi dq}\,.$
The variation of the longitude is assumed to be zero at the end points, so the first term disappears. The integral can be made nonzero by a perverse choice of $ \delta \varphi $ unless the other factor inside is zero everywhere. So the equation of motion is
${\frac {d}{dq}}\left[-{\frac {r^{2}}{c}}{\frac {d\varphi }{d\tau }}\right]=0\,.$
Motion in time
Vary with respect to time $ t$ only to get
$2c^{2}{\frac {d\tau }{dq}}\delta {\frac {d\tau }{dq}}=2\left(1-{\frac {r_{\text{s}}}{r}}\right)c^{2}{\frac {dt}{dq}}\delta {\frac {dt}{dq}}\,.$
Divide by $ 2c{\frac {d\tau }{dq}}$ to get the variation of the integrand itself
$c\delta {\frac {d\tau }{dq}}=c\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}\delta {\frac {dt}{dq}}=c\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}{\frac {d\delta t}{dq}}\,.$
Thus
$0=\delta \int {c{\frac {d\tau }{dq}}dq}=\int {c\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}{\frac {d\delta t}{dq}}dq}\,.$
Integrating by parts gives
$0=c\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}\delta t-\int {{\frac {d}{dq}}\left[c\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}\right]\delta tdq}\,.$
So the equation of motion is
${\frac {d}{dq}}\left[c\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}\right]=0\,.$
Conserved momenta
Integrate these equations of motion to determine the constants of integration getting
${\begin{aligned}L=p_{\phi }&=mr^{2}{\frac {d\varphi }{d\tau }}\,,\\E=-p_{t}&=mc^{2}\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {dt}{d\tau }}\,.\end{aligned}}$
These two equations for the constants of motion $ L$ (angular momentum) and $ E$ (energy) can be combined to form one equation that is true even for photons and other massless particles for which the proper time along a geodesic is zero.
${\frac {d\varphi }{dt}}=\left(1-{\frac {r_{\text{s}}}{r}}\right){\frac {L\,c^{2}}{E\,r^{2}}}\,.$
Radial motion
Substituting
${\frac {d\varphi }{d\tau }}={\frac {L}{m\,r^{2}}}\,$
and
${\frac {dt}{d\tau }}={\frac {E}{\left(1-{\frac {r_{\text{s}}}{r}}\right)m\,c^{2}}}\,$
into the metric equation (and using $ \theta ={\frac {\pi }{2}}$) gives
$c^{2}={\frac {1}{1-{\frac {r_{\text{s}}}{r}}}}\,{\frac {E^{2}}{m^{2}c^{2}}}-{\frac {1}{1-{\frac {r_{\text{s}}}{r}}}}\left({\frac {dr}{d\tau }}\right)^{2}-{\frac {1}{r^{2}}}\,{\frac {L^{2}}{m^{2}}}\,,$
from which one can derive
${\left({\frac {dr}{d\tau }}\right)}^{2}={\frac {E^{2}}{m^{2}c^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left(c^{2}+{\frac {L^{2}}{m^{2}r^{2}}}\right)\,,$
which is the equation of motion for $ r$. The dependence of $ r$ on $ \varphi $ can be found by dividing this by
${\left({\frac {d\varphi }{d\tau }}\right)}^{2}={\frac {L^{2}}{m^{2}r^{4}}}$
to get
${\left({\frac {dr}{d\varphi }}\right)}^{2}={\frac {E^{2}r^{4}}{L^{2}c^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left({\frac {m^{2}c^{2}r^{4}}{L^{2}}}+r^{2}\right)\,$
which is true even for particles without mass. If length scales are defined by
$a={\frac {L}{m\,c}}$
and
$b={\frac {L\,c}{E}}\,,$
then the dependence of $ r$ on $ \varphi $ simplifies to
${\left({\frac {dr}{d\varphi }}\right)}^{2}={\frac {r^{4}}{b^{2}}}-\left(1-{\frac {r_{\text{s}}}{r}}\right)\left({\frac {r^{4}}{a^{2}}}+r^{2}\right)\,.$
See also
• Classical central-force problem
• Frame fields in general relativity
• Kepler problem
• Two-body problem in general relativity
Notes
1. This substitution of $ u$ for $ r$ is also common in classical central-force problems, since it also renders those equations easier to solve. For further information, please see the article on the classical central-force problem.
2. In the mathematical literature, K is known as the complete elliptic integral of the first kind; for further information, please see the article on elliptic integrals.
References
1. Kozai, Yoshihide (1998). "Development of Celestial Mechanics in Japan". Planet. Space Sci. 46 (8): 1031–36. Bibcode:1998P&SS...46.1031K. doi:10.1016/s0032-0633(98)00033-6.
2. Kaplan, Samuil (1949). "On Circular Orbits in Einstein's Theory of Gravitation". J. Exp. Theor. Phys. 19 (10): 951–952. arXiv:2201.07971. Bibcode:1949ZhETF..19..951K.
3. Landau and Lifshitz, pp. 299–301.
4. Whittaker 1937.
5. Landau and Lifshitz (1975), pp. 306–309.
6. Gibbons, G. W.; Vyska, M. (February 29, 2012). "The application of Weierstrass elliptic functions to Schwarzschild null geodesics". Classical and Quantum Gravity. 29 (6): 065016. arXiv:1110.6508. Bibcode:2012CQGra..29f5016G. doi:10.1088/0264-9381/29/6/065016. S2CID 119675906.
7. Synge, pp. 294–295.
8. arXiv.org: gr-qc/9907034v1.
9. Sean Carroll: Lecture Notes on General Relativity, Chapter 7, Eq. 7.33
10. Weinberg, p. 122.
11. Einstein, pp. 95–96.
12. Weinberg, pp. 185–188; Wald, pp. 138–139.
13. Synge, pp. 290–292; Adler, Bazin, and Schiffer, pp. 179–182; Whittaker, pp. 390–393; Pauli, p. 167.
14. Lanczos, pp. 331–338.
15. Landau and Lifshitz, pp. 306–307; Misner, Thorne, and Wheeler, pp. 636–679.
Bibliography
• Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einstein'schen Theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1, 189–196.
• scan of the original paper
• text of the original paper, in Wikisource
• translation by Antoci and Loinger
• a commentary on the paper, giving a simpler derivation
• Schwarzschild, K. (1916). Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1, 424-?.
• Flamm, L (1916). "Beiträge zur Einstein'schen Gravitationstheorie". Physikalische Zeitschrift. 17: 448–?.
• Adler, R; Bazin M; Schiffer M (1965). Introduction to General Relativity. New York: McGraw-Hill Book Company. pp. 177–193. ISBN 978-0-07-000420-7.
• Einstein, A (1956). The Meaning of Relativity (5th ed.). Princeton, New Jersey: Princeton University Press. pp. 92–97. ISBN 978-0-691-02352-6.
• Hagihara, Y (1931). "Theory of the relativistic trajectories in a gravitational field of Schwarzschild". Japanese Journal of Astronomy and Geophysics. 8: 67–176. Bibcode:1931AOTok..31...67H. ISSN 0368-346X.
• Lanczos, C (1986). The Variational Principles of Mechanics (4th ed.). New York: Dover Publications. pp. 330–338. ISBN 978-0-486-65067-8.
• Landau, LD; Lifshitz, EM (1975). The Classical Theory of Fields. Course of Theoretical Physics. Vol. 2 (revised 4th English ed.). New York: Pergamon Press. pp. 299–309. ISBN 978-0-08-018176-9.
• Misner, CW; Thorne, K & Wheeler, JA (1973). Gravitation. San Francisco: W. H. Freeman. pp. Chapter 25 (pp. 636–687), §33.5 (pp. 897–901), and §40.5 (pp. 1110–1116). ISBN 978-0-7167-0344-0. (See Gravitation (book).)
• Pais, A. (1982). Subtle is the Lord: The Science and the Life of Albert Einstein. Oxford University Press. pp. 253–256. ISBN 0-19-520438-7.
• Pauli, W (1958). Theory of Relativity. Translated by G. Field. New York: Dover Publications. pp. 40–41, 166–169. ISBN 978-0-486-64152-2.
• Rindler, W (1977). Essential Relativity: Special, General, and Cosmological (revised 2nd ed.). New York: Springer Verlag. pp. 143–149. ISBN 978-0-387-10090-6.
• Roseveare, N. T (1982). Mercury's perihelion, from Leverrier to Einstein. Oxford: University Press. ISBN 0-19-858174-2.
• Synge, JL (1960). Relativity: The General Theory. Amsterdam: North-Holland Publishing. pp. 289–298. ISBN 978-0-7204-0066-3.
• Wald, RM (1984). General Relativity. Chicago: The University of Chicago Press. pp. 136–146. ISBN 978-0-226-87032-8.
• Walter, S. (2007). "Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910". In Renn, J. (ed.). The Genesis of General Relativity. Vol. 3. Berlin: Springer. pp. 193–252.
• Weinberg, S (1972). Gravitation and Cosmology. New York: John Wiley and Sons. pp. 185–201. ISBN 978-0-471-92567-5.
• Whittaker, ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies (4th ed.). New York: Dover Publications. pp. 389–393. ISBN 978-1-114-28944-4.
External links
• Excerpt from Reflections on Relativity by Kevin Brown.
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Wikipedia
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Schwinger parametrization
Schwinger parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops.
Using the well-known observation that
${\frac {1}{A^{n}}}={\frac {1}{(n-1)!}}\int _{0}^{\infty }du\,u^{n-1}e^{-uA},$
Julian Schwinger noticed that one may simplify the integral:
$\int {\frac {dp}{A(p)^{n}}}={\frac {1}{\Gamma (n)}}\int dp\int _{0}^{\infty }du\,u^{n-1}e^{-uA(p)}={\frac {1}{\Gamma (n)}}\int _{0}^{\infty }du\,u^{n-1}\int dp\,e^{-uA(p)},$
for Re(n)>0.
Another version of Schwinger parametrization is:
${\frac {i}{A+i\epsilon }}=\int _{0}^{\infty }du\,e^{iu(A+i\epsilon )},$
which is convergent as long as $\epsilon >0$ and $A\in \mathbb {R} $.[1] It is easy to generalize this identity to n denominators.
See also
• Feynman parametrization
References
1. Schwartz, M. D. (2014). "33". Quantum Field Theory and the Standard Model (9 ed.). Cambridge University Press. p. 705. ISBN 9781107034730.
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Wikipedia
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Currying
In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function $f$ that takes three arguments creates a nested unary function $g$, so that the code
${\text{let }}x=f(a,b,c)$
gives $x$ the same value as the code
${\begin{aligned}{\text{let }}h=g(a)\\{\text{let }}i=h(b)\\{\text{let }}x=i(c),\end{aligned}}$
or called in sequence,
${\text{let }}x=g(a)(b)(c).$
In a more mathematical language, a function that takes two arguments, one from $X$ and one from $Y$, and produces outputs in $Z,$ by currying is translated into a function that takes a single argument from $X$ and produces as outputs functions from $Y$ to $Z.$ This is a natural one-to-one correspondence between these two types of functions, so that the sets together with functions between them form a Cartesian closed category. The currying of a function with more than two arguments can then be defined by induction. Currying is related to, but not the same as, partial application.
Currying is useful in both practical and theoretical settings. In functional programming languages, and many others, it provides a way of automatically managing how arguments are passed to functions and exceptions. In theoretical computer science, it provides a way to study functions with multiple arguments in simpler theoretical models which provide only one argument. The most general setting for the strict notion of currying and uncurrying is in the closed monoidal categories, which underpins a vast generalization of the Curry–Howard correspondence of proofs and programs to a correspondence with many other structures, including quantum mechanics, cobordisms and string theory.[1] It was introduced by Gottlob Frege,[2][3] developed by Moses Schönfinkel,[3][4][5][6] and further developed by Haskell Curry.[7][8]
Uncurrying is the dual transformation to currying, and can be seen as a form of defunctionalization. It takes a function $f$ whose return value is another function $g$, and yields a new function $f'$ that takes as parameters the arguments for both $f$ and $g$, and returns, as a result, the application of $f$ and subsequently, $g$, to those arguments. The process can be iterated.
Motivation
Currying provides a way for working with functions that take multiple arguments, and using them in frameworks where functions might take only one argument. For example, some analytical techniques can only be applied to functions with a single argument. Practical functions frequently take more arguments than this. Frege showed that it was sufficient to provide solutions for the single argument case, as it was possible to transform a function with multiple arguments into a chain of single-argument functions instead. This transformation is the process now known as currying.[9] All "ordinary" functions that might typically be encountered in mathematical analysis or in computer programming can be curried. However, there are categories in which currying is not possible; the most general categories which allow currying are the closed monoidal categories.
Some programming languages almost always use curried functions to achieve multiple arguments; notable examples are ML and Haskell, where in both cases all functions have exactly one argument. This property is inherited from lambda calculus, where multi-argument functions are usually represented in curried form.
Currying is related to, but not the same as partial application. In practice, the programming technique of closures can be used to perform partial application and a kind of currying, by hiding arguments in an environment that travels with the curried function.
Illustration
Suppose we have a function $f:\mathbb {R} \times \mathbb {R} \to \mathbb {R} $ which takes two real number ($\mathbb {R} $) arguments and outputs real numbers, and it is defined by $f(x,y)=x+y^{2}$. Currying translates this into a function $h$ which takes a single real argument and outputs functions from $\mathbb {R} $ to $\mathbb {R} $. In symbols, $h:\mathbb {R} \to \mathbb {R} ^{\mathbb {R} }$, where $\mathbb {R} ^{\mathbb {R} }$denotes the set of all functions that take a single real argument and produce real outputs. For every real number $x$, define the function $h_{x}:\mathbb {R} \to \mathbb {R} $ by $h_{x}(y)=x+y^{2}$, and then define the function $h:\mathbb {R} \to \mathbb {R} ^{\mathbb {R} }$ by $h(x)=h_{x}$. So for instance, $h(2)$ is the function that sends its real argument $y$ to the output $2+y^{2}$, or $h(2)(y)=h_{2}(y)=2+y^{2}$. We see that in general
$h(x)(y)=x+y^{2}=f(x,y)$
so that the original function $f$ and its currying $h$ convey exactly the same information. In this situation, we also write
${\text{curry}}(f)=h.$
This also works for functions with more than two arguments. If $f$ were a function of three arguments $f(x,y,z)$, its currying $h$ would have the property
$f(x,y,z)=h(x)(y)(z).$
History
The "Curry" in "Currying" is a reference to logician Haskell Curry, who used the concept extensively, but Moses Schönfinkel had the idea six years before Curry.[10] The alternative name "Schönfinkelisation" has been proposed.[11] In the mathematical context, the principle can be traced back to work in 1893 by Frege.[2][3]
The originator of the word "currying" is not clear. David Turner says the word was coined by Christopher Strachey in his 1967 lecture notes Fundamental Concepts in Programming Languages,[12] but although the concept is mentioned, the word "currying" does not appear in the notes.[4] John C. Reynolds defined "currying" in a 1972 paper, but did not claim to have coined the term.[5]
Definition
Currying is most easily understood by starting with an informal definition, which can then be molded to fit many different domains. First, there is some notation to be established. The notation $X\to Y$ denotes all functions from $X$ to $Y$. If $f$ is such a function, we write $f\colon X\to Y$. Let $X\times Y$ denote the ordered pairs of the elements of $X$ and $Y$ respectively, that is, the Cartesian product of $X$ and $Y$. Here, $X$ and $Y$ may be sets, or they may be types, or they may be other kinds of objects, as explored below.
Given a function
$f\colon (X\times Y)\to Z$,
currying constructs a new function
$h\colon X\to (Y\to Z)$.
That is, $h$ takes an argument from $X$ and returns a function that maps $Y$ to $Z$. It is defined by
$h(x)(y)=f(x,y)$
for $x$ from $X$ and $y$ from $Y$. We then also write
${\text{curry}}(f)=h.$
Uncurrying is the reverse transformation, and is most easily understood in terms of its right adjoint, the function $\operatorname {apply} .$
Set theory
In set theory, the notation $Y^{X}$ is used to denote the set of functions from the set $X$ to the set $Y$. Currying is the natural bijection between the set $A^{B\times C}$ of functions from $B\times C$ to $A$, and the set $(A^{C})^{B}$ of functions from $B$ to the set of functions from $C$ to $A$. In symbols:
$A^{B\times C}\cong (A^{C})^{B}$
Indeed, it is this natural bijection that justifies the exponential notation for the set of functions. As is the case in all instances of currying, the formula above describes an adjoint pair of functors: for every fixed set $C$, the functor $B\mapsto B\times C$ is left adjoint to the functor $A\mapsto A^{C}$.
In the category of sets, the object $Y^{X}$ is called the exponential object.
Function spaces
In the theory of function spaces, such as in functional analysis or homotopy theory, one is commonly interested in continuous functions between topological spaces. One writes ${\text{Hom}}(X,Y)$ (the Hom functor) for the set of all functions from $X$ to $Y$, and uses the notation $Y^{X}$ to denote the subset of continuous functions. Here, ${\text{curry}}$ is the bijection
${\text{curry}}:{\text{Hom}}(X\times Y,Z)\to {\text{Hom}}(X,{\text{Hom}}(Y,Z)),$
while uncurrying is the inverse map. If the set $Y^{X}$ of continuous functions from $X$ to $Y$ is given the compact-open topology, and if the space $Y$ is locally compact Hausdorff, then
${\text{curry}}:Z^{X\times Y}\to (Z^{Y})^{X}$
is a homeomorphism. This is also the case when $X$, $Y$ and $Y^{X}$ are compactly generated,[13]: chapter 5 [14] although there are more cases.[15][16]
One useful corollary is that a function is continuous if and only if its curried form is continuous. Another important result is that the application map, usually called "evaluation" in this context, is continuous (note that eval is a strictly different concept in computer science.) That is,
${\begin{aligned}&&{\text{eval}}:Y^{X}\times X\to Y\\&&(f,x)\mapsto f(x)\end{aligned}}$
is continuous when $Y^{X}$ is compact-open and $Y$ locally compact Hausdorff.[17] These two results are central for establishing the continuity of homotopy, i.e. when $X$ is the unit interval $I$, so that $Z^{I\times Y}\cong (Z^{Y})^{I}$ can the thought of as either a homotopy of two functions from $Y$ to $Z$, or, equivalently, a single (continuous) path in $Z^{Y}$.
Algebraic topology
In algebraic topology, currying serves as an example of Eckmann–Hilton duality, and, as such, plays an important role in a variety of different settings. For example, loop space is adjoint to reduced suspensions; this is commonly written as
$[\Sigma X,Z]\approxeq [X,\Omega Z]$
where $[A,B]$ is the set of homotopy classes of maps $A\rightarrow B$, and $\Sigma A$ is the suspension of A, and $\Omega A$ is the loop space of A. In essence, the suspension $\Sigma X$ can be seen as the cartesian product of $X$ with the unit interval, modulo an equivalence relation to turn the interval into a loop. The curried form then maps the space $X$ to the space of functions from loops into $Z$, that is, from $X$ into $\Omega Z$.[17] Then ${\text{curry}}$ is the adjoint functor that maps suspensions to loop spaces, and uncurrying is the dual.[17]
The duality between the mapping cone and the mapping fiber (cofibration and fibration)[13]: chapters 6,7 can be understood as a form of currying, which in turn leads to the duality of the long exact and coexact Puppe sequences.
In homological algebra, the relationship between currying and uncurrying is known as tensor-hom adjunction. Here, an interesting twist arises: the Hom functor and the tensor product functor might not lift to an exact sequence; this leads to the definition of the Ext functor and the Tor functor.
Domain theory
In order theory, that is, the theory of lattices of partially ordered sets, ${\text{curry}}$ is a continuous function when the lattice is given the Scott topology.[18] Scott-continuous functions were first investigated in the attempt to provide a semantics for lambda calculus (as ordinary set theory is inadequate to do this). More generally, Scott-continuous functions are now studied in domain theory, which encompasses the study of denotational semantics of computer algorithms. Note that the Scott topology is quite different than many common topologies one might encounter in the category of topological spaces; the Scott topology is typically finer, and is not sober.
The notion of continuity makes its appearance in homotopy type theory, where, roughly speaking, two computer programs can be considered to be homotopic, i.e. compute the same results, if they can be "continuously" refactored from one to the other.
Lambda calculi
In theoretical computer science, currying provides a way to study functions with multiple arguments in very simple theoretical models, such as the lambda calculus, in which functions only take a single argument. Consider a function $f(x,y)$ taking two arguments, and having the type $(X\times Y)\to Z$, which should be understood to mean that x must have the type $X$, y must have the type $Y$, and the function itself returns the type $Z$. The curried form of f is defined as
${\text{curry}}(f)=\lambda x.(\lambda y.(f(x,y)))$
where $\lambda $ is the abstractor of lambda calculus. Since curry takes, as input, functions with the type $(X\times Y)\to Z$, one concludes that the type of curry itself is
${\text{curry}}:((X\times Y)\to Z)\to (X\to (Y\to Z))$
The → operator is often considered right-associative, so the curried function type $X\to (Y\to Z)$ is often written as $X\to Y\to Z$. Conversely, function application is considered to be left-associative, so that $f(x,y)$ is equivalent to
$(({\text{curry}}(f)\;x)\;y)={\text{curry}}(f)\;x\;y$.
That is, the parenthesis are not required to disambiguate the order of the application.
Curried functions may be used in any programming language that supports closures; however, uncurried functions are generally preferred for efficiency reasons, since the overhead of partial application and closure creation can then be avoided for most function calls.
Type theory
In type theory, the general idea of a type system in computer science is formalized into a specific algebra of types. For example, when writing $f\colon X\to Y$, the intent is that $X$ and $Y$ are types, while the arrow $\to $ is a type constructor, specifically, the function type or arrow type. Similarly, the Cartesian product $X\times Y$ of types is constructed by the product type constructor $\times $.
The type-theoretical approach is expressed in programming languages such as ML and the languages derived from and inspired by it: CaML, Haskell and F#.
The type-theoretical approach provides a natural complement to the language of category theory, as discussed below. This is because categories, and specifically, monoidal categories, have an internal language, with simply-typed lambda calculus being the most prominent example of such a language. It is important in this context, because it can be built from a single type constructor, the arrow type. Currying then endows the language with a natural product type. The correspondence between objects in categories and types then allows programming languages to be re-interpreted as logics (via Curry–Howard correspondence), and as other types of mathematical systems, as explored further, below.
Logic
Under the Curry–Howard correspondence, the existence of currying and uncurrying is equivalent to the logical theorem $((A\land B)\to C)\Leftrightarrow (A\to (B\to C))$, as tuples (product type) corresponds to conjunction in logic, and function type corresponds to implication.
The exponential object $Q^{P}$ in the category of Heyting algebras is normally written as material implication $P\to Q$. Distributive Heyting algebras are Boolean algebras, and the exponential object has the explicit form $\neg P\lor Q$, thus making it clear that the exponential object really is material implication.[19]
Category theory
The above notions of currying and uncurrying find their most general, abstract statement in category theory. Currying is a universal property of an exponential object, and gives rise to an adjunction in cartesian closed categories. That is, there is a natural isomorphism between the morphisms from a binary product $f\colon (X\times Y)\to Z$ and the morphisms to an exponential object $g\colon X\to Z^{Y}$.
This generalizes to a broader result in closed monoidal categories: Currying is the statement that the tensor product and the internal Hom are adjoint functors; that is, for every object $B$ there is a natural isomorphism:
$\mathrm {Hom} (A\otimes B,C)\cong \mathrm {Hom} (A,B\Rightarrow C).$
Here, Hom denotes the (external) Hom-functor of all morphisms in the category, while $B\Rightarrow C$ denotes the internal hom functor in the closed monoidal category. For the category of sets, the two are the same. When the product is the cartesian product, then the internal hom $B\Rightarrow C$ becomes the exponential object $C^{B}$.
Currying can break down in one of two ways. One is if a category is not closed, and thus lacks an internal hom functor (possibly because there is more than one choice for such a functor). Another way is if it is not monoidal, and thus lacks a product (that is, lacks a way of writing down pairs of objects). Categories that do have both products and internal homs are exactly the closed monoidal categories.
The setting of cartesian closed categories is sufficient for the discussion of classical logic; the more general setting of closed monoidal categories is suitable for quantum computation.[20]
The difference between these two is that the product for cartesian categories (such as the category of sets, complete partial orders or Heyting algebras) is just the Cartesian product; it is interpreted as an ordered pair of items (or a list). Simply typed lambda calculus is the internal language of cartesian closed categories; and it is for this reason that pairs and lists are the primary types in the type theory of LISP, Scheme and many functional programming languages.
By contrast, the product for monoidal categories (such as Hilbert space and the vector spaces of functional analysis) is the tensor product. The internal language of such categories is linear logic, a form of quantum logic; the corresponding type system is the linear type system. Such categories are suitable for describing entangled quantum states, and, more generally, allow a vast generalization of the Curry–Howard correspondence to quantum mechanics, to cobordisms in algebraic topology, and to string theory.[1] The linear type system, and linear logic are useful for describing synchronization primitives, such as mutual exclusion locks, and the operation of vending machines.
Contrast with partial function application
Currying and partial function application are often conflated.[21] One of the significant differences between the two is that a call to a partially applied function returns the result right away, not another function down the currying chain; this distinction can be illustrated clearly for functions whose arity is greater than two.[22]
Given a function of type $f\colon (X\times Y\times Z)\to N$, currying produces ${\text{curry}}(f)\colon X\to (Y\to (Z\to N))$. That is, while an evaluation of the first function might be represented as $f(1,2,3)$, evaluation of the curried function would be represented as $f_{\text{curried}}(1)(2)(3)$, applying each argument in turn to a single-argument function returned by the previous invocation. Note that after calling $f_{\text{curried}}(1)$, we are left with a function that takes a single argument and returns another function, not a function that takes two arguments.
In contrast, partial function application refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given the definition of $f$ above, we might fix (or 'bind') the first argument, producing a function of type ${\text{partial}}(f)\colon (Y\times Z)\to N$. Evaluation of this function might be represented as $f_{\text{partial}}(2,3)$. Note that the result of partial function application in this case is a function that takes two arguments.
Intuitively, partial function application says "if you fix the first argument of the function, you get a function of the remaining arguments". For example, if function div stands for the division operation x/y, then div with the parameter x fixed at 1 (i.e., div 1) is another function: the same as the function inv that returns the multiplicative inverse of its argument, defined by inv(y) = 1/y.
The practical motivation for partial application is that very often the functions obtained by supplying some but not all of the arguments to a function are useful; for example, many languages have a function or operator similar to plus_one. Partial application makes it easy to define these functions, for example by creating a function that represents the addition operator with 1 bound as its first argument.
Partial application can be seen as evaluating a curried function at a fixed point, e.g. given $f\colon (X\times Y\times Z)\to N$ and $a\in X$ then ${\text{curry}}({\text{partial}}(f)_{a})(y)(z)={\text{curry}}(f)(a)(y)(z)$ or simply ${\text{partial}}(f)_{a}={\text{curry}}_{1}(f)(a)$ where ${\text{curry}}_{1}$ curries f's first parameter.
Thus, partial application is reduced to a curried function at a fixed point. Further, a curried function at a fixed point is (trivially), a partial application. For further evidence, note that, given any function $f(x,y)$, a function $g(y,x)$ may be defined such that $g(y,x)=f(x,y)$. Thus, any partial application may be reduced to a single curry operation. As such, curry is more suitably defined as an operation which, in many theoretical cases, is often applied recursively, but which is theoretically indistinguishable (when considered as an operation) from a partial application.
So, a partial application can be defined as the objective result of a single application of the curry operator on some ordering of the inputs of some function.
See also
• Tensor-hom adjunction
• Lazy evaluation
• Closure (computer science)
• S m
n
theorem
• Closed monoidal category
Notes
1. John C. Baez and Mike Stay, "Physics, Topology, Logic and Computation: A Rosetta Stone", (2009) ArXiv 0903.0340 in New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95-174.
2. Gottlob Frege, Grundgesetze der Arithmetik I, Jena: Verlag Hermann Pohle, 1893, §36.
3. Willard Van Orman Quine, introduction to Moses Schönfinkel's "Bausteine der mathematischen Logik", pp. 355–357, esp. 355. Translated by Stefan Bauer-Mengelberg as "On the building blocks of mathematical logic" in Jean van Heijenoort (1967), A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, pp. 355–66.
4. Strachey, Christopher (2000). "Fundamental Concepts in Programming Languages". Higher-Order and Symbolic Computation. 13: 21. CiteSeerX 10.1.1.332.3161. doi:10.1023/A:1010000313106. S2CID 14124601. There is a device originated by Schönfinkel, for reducing operators with several operands to the successive application of single operand operators. (Reprinted lecture notes from 1967.)
5. Reynolds, John C. (1972). "Definitional interpreters for higher-order programming languages". Proceedings of the ACM annual conference on - ACM '72. pp. 717–740. doi:10.1145/800194.805852. S2CID 163294. In the last line we have used a trick called Currying (after the logician H. Curry) to solve the problem of introducing a binary operation into a language where all functions must accept a single argument. (The referee comments that although "Currying" is tastier, "Schönfinkeling" might be more accurate.) {{cite book}}: |journal= ignored (help)
6. Kenneth Slonneger and Barry L. Kurtz. Formal Syntax and Semantics of Programming Languages. 1995. p. 144.
7. Henk Barendregt, Erik Barendsen, "Introduction to Lambda Calculus", March 2000, page 8.
8. Curry, Haskell; Feys, Robert (1958). Combinatory logic. Vol. I (2 ed.). Amsterdam, Netherlands: North-Holland Publishing Company.
9. Graham Hutton. "Frequently Asked Questions for comp.lang.functional: Currying". nott.ac.uk.
10. Curry, Haskell B. (1980). "Some Philosophical Aspects of Combinatory Logic". Studies in Logic and the Foundations of Mathematics. 101: 85–101. doi:10.1016/S0049-237X(08)71254-0. ISBN 9780444853455. Some contemporary logicians call this way of looking at a function "currying", because I made extensive use of it; but Schönfinkel had the idea some 6 years before I did.
11. I. Heim and A. Kratzer (1998). Semantics in Generative Grammar. Blackwell.
12. Turner, David (1 Jun 1997). "Programming language, Currying, or Schonfinkeling?". computer-programming-forum.com. Retrieved 3 March 2022.
13. J.P. May, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics ISBN 0-226-51183-9
14. Compactly generated topological space at the nLab
15. P. I. Booth and J. Tillotson, "Monoidal closed, Cartesian closed and convenient categories of topological spaces", Pacific Journal of Mathematics, 88 (1980) pp.33-53.
16. Convenient category of topological spaces at the nLab
17. Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for proof.)
18. Barendregt, H.P. (1984). The Lambda Calculus. North-Holland. ISBN 978-0-444-87508-2. (See theorems 1.2.13, 1.2.14)
19. Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic (1992) Springer ISBN 0-387-97710-4 (See Chapter 1, pp.48-57)
20. Samson Abramsky and Bob Coecke, "A Categorical Semantics for Quantum Protocols."
21. "The Uncarved Blog: Partial Function Application is not Currying". uncarved.com. Archived from the original on 2016-10-23.
22. "Functional Programming in 5 Minutes". Slides. 15 May 2013.
References
• Schönfinkel, Moses (1924). "Über die Bausteine der mathematischen Logik". Math. Ann. 92 (3–4): 305–316. doi:10.1007/BF01448013. S2CID 118507515.
• Heim, Irene; Kratzer, Angelika (1998). Semantics in a Generative Grammar. Malden: Blackwall Publishers. ISBN 0-631-19712-5.
External links
Look up currying in Wiktionary, the free dictionary.
• Currying Schonfinkelling at the Portland Pattern Repository
• Currying != Generalized Partial Application! - post at Lambda-the-Ultimate.org
Software design patterns
Gang of Four
patterns
Creational
• Abstract factory
• Builder
• Factory method
• Prototype
• Singleton
Structural
• Adapter
• Bridge
• Composite
• Decorator
• Facade
• Flyweight
• Proxy
Behavioral
• Chain of responsibility
• Command
• Interpreter
• Iterator
• Mediator
• Memento
• Observer
• State
• Strategy
• Template method
• Visitor
Concurrency
patterns
• Active object
• Balking
• Binding properties
• Double-checked locking
• Event-based asynchronous
• Guarded suspension
• Join
• Lock
• Monitor
• Proactor
• Reactor
• Read write lock
• Scheduler
• Scheduled-task pattern
• Thread pool
• Thread-local storage
Architectural
patterns
• Front controller
• Interceptor
• MVC
• ADR
• ECS
• n-tier
• Specification
• Publish–subscribe
• Naked objects
• Service locator
• Active record
• Identity map
• Data access object
• Data transfer object
• Inversion of control
• Model 2
• Broker
Other
patterns
• Blackboard
• Business delegate
• Composite entity
• Dependency injection
• Intercepting filter
• Lazy loading
• Mock object
• Null object
• Object pool
• Servant
• Twin
• Type tunnel
• Method chaining
• Delegation
Books
• Design Patterns
• Enterprise Integration Patterns
People
• Christopher Alexander
• Erich Gamma
• Ralph Johnson
• John Vlissides
• Grady Booch
• Kent Beck
• Ward Cunningham
• Martin Fowler
• Robert Martin
• Jim Coplien
• Douglas Schmidt
• Linda Rising
Communities
• The Hillside Group
• The Portland Pattern Repository
See also
• Anti-pattern
• Architectural pattern
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Wikipedia
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Schoenflies problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.
Original formulation
The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane.
An alternative statement is that if $C\subset \mathbb {R} ^{2}$ is a simple closed curve, then there is a homeomorphism $f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}$ such that $f(C)$ is the unit circle in the plane. Elementary proofs can be found in Newman (1939), Cairns (1951), Moise (1977) and Thomassen (1992). The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case of a continuous curve is then deduced by approximating by polygons. The theorem is also an immediate consequence of Carathéodory's extension theorem for conformal mappings, as discussed in Pommerenke (1992, p. 25).
If the curve is smooth then the homeomorphism can be chosen to be a diffeomorphism. Proofs in this case rely on techniques from differential topology. Although direct proofs are possible (starting for example from the polygonal case), existence of the diffeomorphism can also be deduced by using the smooth Riemann mapping theorem for the interior and exterior of the curve in combination with the Alexander trick for diffeomorphisms of the circle and a result on smooth isotopy from differential topology.[1]
Such a theorem is valid only in two dimensions. In three dimensions there are counterexamples such as Alexander's horned sphere. Although they separate space into two regions, those regions are so twisted and knotted that they are not homeomorphic to the inside and outside of a normal sphere.
Proofs of the Jordan–Schoenflies theorem
For smooth or polygonal curves, the Jordan curve theorem can be proved in a straightforward way. Indeed, the curve has a tubular neighbourhood, defined in the smooth case by the field of unit normal vectors to the curve or in the polygonal case by points at a distance of less than ε from the curve. In a neighbourhood of a differentiable point on the curve, there is a coordinate change in which the curve becomes the diameter of an open disk. Taking a point not on the curve, a straight line aimed at the curve starting at the point will eventually meet the tubular neighborhood; the path can be continued next to the curve until it meets the disk. It will meet it on one side or the other. This proves that the complement of the curve has at most two connected components. On the other hand, using the Cauchy integral formula for the winding number, it can be seen that the winding number is constant on connected components of the complement of the curve, is zero near infinity and increases by 1 when crossing the curve. Hence the curve separates the plane into exactly two components, its "interior" and its "exterior", the latter being unbounded. The same argument works for a piecewise differentiable Jordan curve.[2]
Polygonal curve
Given a simple closed polygonal curve in the plane, the piecewise linear Jordan–Schoenflies theorem states that there is a piecewise linear homeomorphism of the plane, with compact support, carrying the polygon onto a triangle and taking the interior and exterior of one onto the interior and exterior of the other.[3]
The interior of the polygon can be triangulated by small triangles, so that the edges of the polygon form edges of some of the small triangles. Piecewise linear homeomorphisms can be made up from special homeomorphisms obtained by removing a diamond from the plane and taking a piecewise affine map, fixing the edges of the diamond, but moving one diagonal into a V shape. Compositions of homeomorphisms of this kind give rise to piecewise linear homeomorphisms of compact support; they fix the outside of a polygon and act in an affine way on a triangulation of the interior. A simple inductive argument shows that it is always possible to remove a free triangle—one for which the intersection with the boundary is a connected set made up of one or two edges—leaving a simple closed Jordan polygon. The special homeomorphisms described above or their inverses provide piecewise linear homeomorphisms which carry the interior of the larger polygon onto the polygon with the free triangle removed. Iterating this process it follows that there is a piecewise linear homeomorphism of compact support carrying the original polygon onto a triangle.[4]
Because the homeomorphism is obtained by composing finite many homeomorphisms of the plane of compact support, it follows that the piecewise linear homeomorphism in the statement of the piecewise linear Jordan-Schoenflies theorem has compact support.
As a corollary, it follows that any homeomorphism between simple closed polygonal curves extends to a homeomorphism between their interiors.[5] For each polygon there is a homeomorphism of a given triangle onto the closure of their interior. The three homeomorphisms yield a single homeomorphism of the boundary of the triangle. By the Alexander trick this homeomorphism can be extended to a homeomorphism of closure of interior of the triangle. Reversing this process this homeomorphism yields a homeomorphism between the closures of the interiors of the polygonal curves.
Continuous curve
The Jordan-Schoenflies theorem for continuous curves can be proved using Carathéodory's theorem on conformal mapping. It states that the Riemann mapping between the interior of a simple Jordan curve and the open unit disk extends continuously to a homeomorphism between their closures, mapping the Jordan curve homeomorphically onto the unit circle.[6] To prove the theorem, Carathéodory's theorem can be applied to the two regions on the Riemann sphere defined by the Jordan curve. This will result in homeomorphisms between their closures and the closed disks |z| ≤ 1 and |z| ≥ 1. The homeomorphisms from the Jordan curve to the circle will differ by a homeomorphism of the circle which can be extended to the unit disk (or its complement) by the Alexander trick. Composition with this homeomorphism will yield a pair of homeomorphisms which match on the Jordan curve and therefore define a homeomorphism of the Riemann sphere carrying the Jordan curve onto the unit circle.
The continuous case can also be deduced from the polygonal case by approximating the continuous curve by a polygon.[7] The Jordan curve theorem is first deduced by this method. The Jordan curve is given by a continuous function on the unit circle. It and the inverse function from its image back to the unit circle are uniformly continuous. So dividing the circle up into small enough intervals, there are points on the curve such that the line segments joining adjacent points lie close to the curve, say by ε. Together these line segments form a polygonal curve. If it has self-intersections, these must also create polygonal loops. Erasing these loops, results in a polygonal curve without self-intersections which still lies close to the curve; some of its vertices might not lie on the curve, but they all lie within a neighbourhood of the curve. The polygonal curve divides the plane into two regions, one bounded region U and one unbounded region V. Both U and V ∪ ∞ are continuous images of the closed unit disk. Since the original curve is contained within a small neighbourhood of the polygonal curve, the union of the images of slightly smaller concentric open disks entirely misses the original curve and their union excludes a small neighbourhood of the curve. One of the images is a bounded open set consisting of points around which the curve has winding number one; the other is an unbounded open set consisting of points of winding number zero. Repeating for a sequence of values of ε tending to 0, leads to a union of open path-connected bounded sets of points of winding number one and a union of open path-connected unbounded sets of winding number zero. By construction these two disjoint open path-connected sets fill out the complement of the curve in the plane.[8]
Given the Jordan curve theorem, the Jordan-Schoenflies theorem can be proved as follows.[9]
• The first step is to show that a dense set of points on the curve are accessible from the inside of the curve, i.e. they are at the end of a line segment lying entirely in the interior of the curve. In fact, a given point on the curve is arbitrarily close to some point in the interior and there is a smallest closed disk about that point which intersects the curve only on its boundary; those boundary points are close to the original point on the curve and by construction are accessible.
• The second step is to prove that given finitely many accessible points Ai on the curve connected to line segments AiBi in its interior, there are disjoint polygonal curves in the interior with vertices on each of the line segments such that their distance to the original curve is arbitrarily small. This requires tessellations of the plane by uniformly small tiles such that if two tiles meet they have a side or a segment of a side in common: examples are the standard hexagonal tessellation; or the standard brickwork tiling by rectangles or squares with common or stretch bonds. It suffices to construct a polygonal path so that its distance to the Jordan curve is arbitrarily small. Orient the tessellation such no side of a tiles is parallel to any AiBi. The size of the tiles can be taken arbitrarily small. Take the union of all the closed tiles containing at least one point of the Jordan curve. Its boundary is made up of disjoint polygonal curves. If the size of the tiles is sufficiently small, the endpoints Bi will lie in the interior of exactly one of the polygonal boundary curves. Its distance to the Jordan curve is less than twice the diameter of the tiles, so is arbitrarily small.
• The third step is to prove that any homeomorphism f between the curve and a given triangle can be extended to a homeomorphism between the closures of their interiors. In fact take a sequence ε1, ε2, ε3, ... decreasing to zero. Choose finitely many points Ai on the Jordan curve Γ with successive points less than ε1 apart. Make the construction of the second step with tiles of diameter less than ε1 and take Ci to be the points on the polygonal curve Γ1 intersecting AiBi. Take the points f(Ai) on the triangle. Fix an origin in the triangle Δ and scale the triangle to get a smaller one Δ1 at a distance less than ε1 from the original triangle. Let Di be the points at the intersection of the radius through f(Ai) and the smaller triangle. There is a piecewise linear homeomorphism F1 of the polygonal curve onto the smaller triangle carrying Ci onto Di. By the Jordan-Schoenflies theorem it extends to a homeomorphism F1 between the closure of their interiors. Now carry out the same process for ε2 with a new set of points on the Jordan curve. This will produce a second polygonal path Γ2 between Γ1 and Γ. There is likewise a second triangle Δ2 between Δ1 and Δ. The line segments for the accessible points on Γ divide the polygonal region between Γ2 and Γ1 into a union of polygonal regions; similarly for radii for the corresponding points on Δ divides the region between Δ2 and Δ1 into a union of polygonal regions. The homeomorphism F1 can be extended to homeomorphisms between the different polygons, agreeing on common edges (closed intervals on line segments or radii). By the polygonal Jordan-Schoenflies theorem, each of these homeomorphisms extends to the interior of the polygon. Together they yield a homeomorphism F2 of the closure of the interior of Γ2 onto the closure of the interior of Δ2; F2 extends F1. Continuing in this way produces polygonal curves Γn and triangles Δn with a homomeomorphism Fn between the closures of their interiors; Fn extends Fn – 1. The regions inside the Γn increase to the region inside Γ; and the triangles Δn increase to Δ. The homeomorphisms Fn patch together to give a homeomorphism F from the interior of Γ onto the interior of Δ. By construction it has limit f on the boundary curves Γ and Δ. Hence F is the required homeomorphism.
• The fourth step is to prove that any homeomorphism between Jordan curves can be extended to a homeomorphism between the closures of their interiors. By the result of the third step, it is sufficient to show that any homeomorphism of the boundary of a triangle extends to a homeomorphism of the closure of its interior. This is a consequence of the Alexander trick. (The Alexander trick also establishes a homeomorphism between the solid triangle and the closed disk: the homeomorphism is just the natural radial extension of the projection of the triangle onto its circumcircle with respect to its circumcentre.)
• The final step is to prove that given two Jordan curves there is a homeomorphism of the plane of compact support carrying one curve onto the other. In fact each Jordan curve lies inside the same large circle and in the interior of each large circle there are radii joining two diagonally opposite points to the curve. Each configuration divide the plane into the exterior of the large circle, the interior of the Jordan curve and the region between the two into two bounded regions bounded by Jordan curves (formed of two radii, a semicircle, and one of the halves of the Jordan curve). Take the identity homeomorphism of the large circle; piecewise linear homeomorphisms between the two pairs of radii; and a homeomorphism between the two pairs of halves of the Jordan curves given by a linear reparametrization. The 4 homeomorphisms patch together on the boundary arcs to yield a homeomorphism of the plane given by the identity off the large circle and carrying one Jordan curve onto the other.
Smooth curve
Proofs in the smooth case depend on finding a diffeomorphism between the interior/exterior of the curve and the closed unit disk (or its complement in the extended plane). This can be solved for example by using the smooth Riemann mapping theorem, for which a number of direct methods are available, for example through the Dirichlet problem on the curve or Bergman kernels.[10] (Such diffeomorphisms will be holomorphic on the interior and exterior of the curve; more general diffeomorphisms can be constructed more easily using vector fields and flows.) Regarding the smooth curve as lying inside the extended plane or 2-sphere, these analytic methods produce smooth maps up to the boundary between the closure of the interior/exterior of the smooth curve and those of the unit circle. The two identifications of the smooth curve and the unit circle will differ by a diffeomorphism of the unit circle. On the other hand, a diffeomorphism f of the unit circle can be extended to a diffeomorphism F of the unit disk by the Alexander extension:
$\displaystyle {F(re^{i\theta })=r\exp[i\psi (r)g(\theta )+i(1-\psi (r))\theta ],}$
where ψ is a smooth function with values in [0,1], equal to 0 near 0 and 1 near 1, and f(eiθ) = eig(θ), with g(θ + 2π) = g(θ) + 2π. Composing one of the diffeomorphisms with the Alexander extension allows the two diffeomorphisms to be patched together to give a homeomorphism of the 2-sphere which restricts to a diffeomorphism on the closed unit disk and the closures of its complement which it carries onto the interior and exterior of the original smooth curve. By the isotopy theorem in differential topology,[11] the homeomorphism can be adjusted to a diffeomorphism on the whole 2-sphere without changing it on the unit circle. This diffeomorphism then provides the smooth solution to the Schoenflies problem.
The Jordan-Schoenflies theorem can be deduced using differential topology. In fact it is an immediate consequence of the classification up to diffeomorphism of smooth oriented 2-manifolds with boundary, as described in Hirsch (1994). Indeed, the smooth curve divides the 2-sphere into two parts. By the classification each is diffeomorphic to the unit disk and—taking into account the isotopy theorem—they are glued together by a diffeomorphism of the boundary. By the Alexander trick, such a diffeomorphism extends to the disk itself. Thus there is a diffeomorphism of the 2-sphere carrying the smooth curve onto the unit circle.
On the other hand, the diffeomorphism can also be constructed directly using the Jordan-Schoenflies theorem for polygons and elementary methods from differential topology, namely flows defined by vector fields.[12] When the Jordan curve is smooth (parametrized by arc length) the unit normal vectors give a non-vanishing vector field X0 in a tubular neighbourhood U0 of the curve. Take a polygonal curve in the interior of the curve close to the boundary and transverse to the curve (at the vertices the vector field should be strictly within the angle formed by the edges). By the piecewise linear Jordan–Schoenflies theorem, there is a piecewise linear homeomorphism, affine on an appropriate triangulation of the interior of the polygon, taking the polygon onto a triangle. Take an interior point P in one of the small triangles of the triangulation. It corresponds to a point Q in the image triangle. There is a radial vector field on the image triangle, formed of straight lines pointing towards Q. This gives a series of lines in the small triangles making up the polygon. Each defines a vector field Xi on a neighbourhood Ui of the closure of the triangle. Each vector field is transverse to the sides, provided that Q is chosen in "general position" so that it is not collinear with any of the finitely many edges in the triangulation. Translating if necessary, it can be assumed that P and Q are at the origin 0. On the triangle containing P the vector field can be taken to be the standard radial vector field. Similarly the same procedure can be applied to the outside of the smooth curve, after applying Möbius transformation to map it into the finite part of the plane and ∞ to 0. In this case the neighbourhoods Ui of the triangles have negative indices. Take the vector fields Xi with a negative sign, pointing away from the point at infinity. Together U0 and the Ui's with i ≠ 0 form an open cover of the 2-sphere. Take a smooth partition of unity ψi subordinate to the cover Ui and set
$\displaystyle {X=\sum \psi _{i}\cdot X_{i}.}$
X is a smooth vector field on the two sphere vanishing only at 0 and ∞. It has index 1 at 0 and -1 at ∞. Near 0 the vector field equals the radial vector field pointing towards 0. If αt is the smooth flow defined by X, the point 0 is an attracting point and ∞ a repelling point. As t tends to +∞, the flow send points to 0; while as t tends to –∞ points are sent to ∞. Replacing X by f⋅X with f a smooth positive function, changes the parametrization of the integral curves of X, but not the integral curves themselves. For an appropriate choice of f equal to 1 outside a small annulus near 0, the integral curves starting at points of the smooth curve will all reach smaller circle bounding the annulus at the same time s. The diffeomorphism αs therefore carries the smooth curve onto this small circle. A scaling transformation, fixing 0 and ∞, then carries the small circle onto the unit circle. Composing these diffeomorphisms gives a diffeomorphism carrying the smooth curve onto the unit circle.
Generalizations
There does exist a higher-dimensional generalization due to Morton Brown (1960) and independently Barry Mazur (1959) with Morse (1960), which is also called the generalized Schoenflies theorem. It states that, if an (n − 1)-dimensional sphere S is embedded into the n-dimensional sphere Sn in a locally flat way (that is, the embedding extends to that of a thickened sphere), then the pair (Sn, S) is homeomorphic to the pair (Sn, Sn−1), where Sn−1 is the equator of the n-sphere. Brown and Mazur received the Veblen Prize for their contributions. Both the Brown and Mazur proofs are considered "elementary" and use inductive arguments.
The Schoenflies problem can be posed in categories other than the topologically locally flat category, i.e. does a smoothly (piecewise-linearly) embedded (n − 1)-sphere in the n-sphere bound a smooth (piecewise-linear) n-ball? For n = 4, the problem is still open for both categories. See Mazur manifold. For n ≥ 5 the question in the smooth category has an affirmative answer, and follows from the h-cobordism theorem.
Notes
1. See:
• Hirsch 1994
• Shastri 2011
• Napier & Ramachandran 2011
• Taylor 2011
• Kerzman 1977
• Bell & Krantz 1987
• Bell 1992
2. Katok & Climenhaga 2008
3. See:
• Moise 1977
• Bing 1983
4. Moise 1977, pp. 26–29
5. Bing 1983, p. 29
6. See:
• Carathéodory 1913
• Goluzin 1969, p. 44
• Pommerenke 1975
7. See:
• Moise 1977
• Bing 1983
8. See:
• Bing 1983
• Katok & Climenhaga 2008, Lecture 36
9. Bing & 1983, pp. 29–32
10. See:
• Napier & Ramachandran 2011
• Taylor 2011
• Kerzman 1977
• Bell & Krantz 1987
• Bell 1992
11. See:
• Hirsch 1994, p. 182, Theorem 1.9
• Shastri 2011, p. 173, Theorem 6.4.3
12. See:
• Smale 1961
• Milnor 1965
• Hirsch 1994
• Shastri 2011
• Matsumoto 2002
• Nicolaescu 2011
References
• Bell, Steven R.; Krantz, Steven G. (1987), "Smoothness to the boundary of conformal maps", Rocky Mountain Journal of Mathematics, 17: 23–40, doi:10.1216/rmj-1987-17-1-23
• Bell, Steven R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 978-0-8493-8270-3
• Bing, R. H. (1983), The Geometric Topology of 3-Manifolds, Colloquium Publications -, vol. 40, American Mathematical Society, ISBN 978-0-8218-1040-8
• Brown, Morton (1960), "A proof of the generalized Schoenflies theorem", Bulletin of the American Mathematical Society, 66 (2): 74–76, CiteSeerX 10.1.1.228.5491, doi:10.1090/S0002-9904-1960-10400-4, MR 0117695
• Cairns, Stewart S. (1951), "An Elementary Proof of the Jordan-Schoenflies Theorem", Proceedings of the American Mathematical Society, 2 (6): 860–867, doi:10.1090/S0002-9939-1951-0046635-9, MR 0046635
• Carathéodory, Constantin (1913), "Zur Ränderzuordnung bei konformer Abbildung", Göttingen Nachrichten: 509–518
• do Carmo, Manfredo P. (1976), Differential geometry of curves and surfaces, Prentice-Hall, ISBN 978-0-13-212589-5
• Goluzin, Gennadiĭ M. (1969), Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, vol. 26, American Mathematical Society
• Hirsch, Morris (1994), Differential topology (2nd ed.), Springer
• Katok, Anatole B.; Climenhaga, Vaughn (2008), Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them, Student Mathematical Library, vol. 46, American Mathematical Society, ISBN 978-0-8218-4679-7
• Kerzman, Norberto (1977), A Monge-Ampére equation in complex analysis, Proc. Symp. Pure Math., vol. XXX, Providence, RI: American Mathematical Society, MR 0454082
• Matsumoto, Yukio (2002), An introduction to Morse theory, Translations of Mathematical Monographs, vol. 208, American Mathematical Society, ISBN 978-0821810224
• Mazur, Barry (1959), "On embeddings of spheres", Bulletin of the American Mathematical Society, 65 (2): 59–65, doi:10.1090/S0002-9904-1959-10274-3, MR 0117693
• Milnor, John (1965), Lectures on the h-cobordism theorem, Princeton University Press
• Moise, Edwin E. (1977), Geometric topology in dimensions 2 and 3, Graduate texts in mathematics, vol. 47, New York-Heidelberg: Springer-Verlag, doi:10.1007/978-1-4612-9906-6, ISBN 978-0-387-90220-3, MR 0488059
• Morse, Marston (1960), "A reduction of the Schoenflies extension problem", Bulletin of the American Mathematical Society, 66 (2): 113–115, doi:10.1090/S0002-9904-1960-10420-X, MR 0117694
• Napier, Terrence; Ramachandran, Mohan (2011), An Introduction to Riemann Surfaces, Springer, ISBN 978-0-8176-4692-9
• Newman, Maxwell Herman Alexander (1939), Elements of the topology of plane sets of points, Cambridge University Press
• Nicolaescu, Liviu (2011), An invitation to Morse theory (2nd ed.), Springer, ISBN 9781461411048
• Pommerenke, Christian (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht
• Pommerenke, Christian (1992), Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften, vol. 299, Springer, ISBN 978-3540547518
• Schoenflies, A. (1906), "Beitrage zur Theorie der Punktmengen III", Mathematische Annalen, 62 (2): 286–328, doi:10.1007/bf01449982, S2CID 123992220
• Shastri, Anant R. (2011), Elements of differential topology, CRC Press, ISBN 9781439831601
• Smale, Stephen (1961), "On gradient dynamical systems", Annals of Mathematics, 74 (1): 199–206, doi:10.2307/1970311, JSTOR 1970311
• Taylor, Michael E. (2011), Partial differential equations I. Basic theory, Applied Mathematical Sciences, vol. 115 (Second ed.), Springer, ISBN 978-1-4419-7054-1
• Thomassen, Carsten (1992), "The Jordan-Schoenflies Theorem and the Classification of Surfaces", American Mathematical Monthly, 99 (2): 116–130, doi:10.2307/2324180, JSTOR 2324180
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Wikipedia
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Splitting circle method
In mathematics, the splitting circle method is a numerical algorithm for the numerical factorization of a polynomial and, ultimately, for finding its complex roots. It was introduced by Arnold Schönhage in his 1982 paper The fundamental theorem of algebra in terms of computational complexity (Technical report, Mathematisches Institut der Universität Tübingen). A revised algorithm was presented by Victor Pan in 1998. An implementation was provided by Xavier Gourdon in 1996 for the Magma and PARI/GP computer algebra systems.
General description
The fundamental idea of the splitting circle method is to use methods of complex analysis, more precisely the residue theorem, to construct factors of polynomials. With those methods it is possible to construct a factor of a given polynomial $p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots +p_{0}$ for any region of the complex plane with a piecewise smooth boundary. Most of those factors will be trivial, that is constant polynomials. Only regions that contain roots of p(x) result in nontrivial factors that have exactly those roots of p(x) as their own roots, preserving multiplicity.
In the numerical realization of this method one uses disks D(c,r) (center c, radius r) in the complex plane as regions. The boundary circle of a disk splits the set of roots of p(x) in two parts, hence the name of the method. To a given disk one computes approximate factors following the analytical theory and refines them using Newton's method. To avoid numerical instability one has to demand that all roots are well separated from the boundary circle of the disk. So to obtain a good splitting circle it should be embedded in a root free annulus A(c,r,R) (center c, inner radius r, outer radius R) with a large relative width R/r.
Repeating this process for the factors found, one finally arrives at an approximative factorization of the polynomial at a required precision. The factors are either linear polynomials representing well isolated zeros or higher order polynomials representing clusters of zeros.
Details of the analytical construction
Newton's identities are a bijective relation between the elementary symmetric polynomials of a tuple of complex numbers and its sums of powers. Therefore, it is possible to compute the coefficients of a polynomial
$p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots +p_{0}=(x-z_{1})\cdots (x-z_{n})$
(or of a factor of it) from the sums of powers of its zeros
$t_{m}=z_{1}^{m}+\cdots +z_{n}^{m}$, $m=0,1,\dots ,n$
by solving the triangular system that is obtained by comparing the powers of u in the following identity of formal power series
$a_{n-1}+2\,a_{n-2}\,u+\cdots +(n-1)\,a_{1}\,u^{n-2}+n\,a_{0}\,u^{n-1}$
$=-(1+a_{n-1}\,u+\cdots +a_{1}\,u^{n-1}+a_{0}\,u^{n})\cdot (t_{1}+t_{2}\,u+t_{3}\,u^{2}+\dots +t_{n}\,u^{n-1}+\cdots ).$
If $G\subset \mathbb {C} $ is a domain with piecewise smooth boundary C and if the zeros of p(x) are pairwise distinct and not on the boundary C, then from the residue theorem of residual calculus one gets
${\frac {1}{2\pi \,i}}\oint _{C}{\frac {p'(z)}{p(z)}}z^{m}\,dz=\sum _{z\in G:\,p(z)=0}{\frac {p'(z)z^{m}}{p'(z)}}=\sum _{z\in G:\,p(z)=0}z^{m}.$
The identity of the left to the right side of this equation also holds for zeros with multiplicities. By using the Newton identities one is able to compute from those sums of powers the factor
$f(x):=\prod _{z\in G:\,p(z)=0}(x-z)$
of p(x) corresponding to the zeros of p(x) inside G. By polynomial division one also obtains the second factor g(x) in p(x) = f(x)g(x).
The commonly used regions are circles in the complex plane. Each circle gives raise to a split of the polynomial p(x) in factors f(x) and g(x). Repeating this procedure on the factors using different circles yields finer and finer factorizations. This recursion stops after a finite number of proper splits with all factors being nontrivial powers of linear polynomials.
The challenge now consists in the conversion of this analytical procedure into a numerical algorithm with good running time. The integration is approximated by a finite sum of a numerical integration method, making use of the fast Fourier transform for the evaluation of the polynomials p(x) and p'(x). The polynomial f(x) that results will only be an approximate factor. To ensure that its zeros are close to the zeros of p inside G and only to those, one must demand that all zeros of p are far away from the boundary C of the region G.
Basic numerical observation
(Schönhage 1982) Let $p\in \mathbb {C} [X]$ be a polynomial of degree n which has k zeros inside the circle of radius 1/2 and the remaining n-k zeros outside the circle of radius 2. With N=O(k) large enough, the approximation of the contour integrals using N points results in an approximation $f_{0}$ of the factor f with error
$\|f-f_{0}\|\leq 2^{2k-N}\,nk\,100/98$,
where the norm of a polynomial is the sum of the moduli of its coefficients.
Since the zeros of a polynomial are continuous in its coefficients, one can make the zeros of $f_{0}$ as close as wanted to the zeros of f by choosing N large enough. However, one can improve this approximation faster using a Newton method. Division of p with remainder yields an approximation $g_{0}$ of the remaining factor g. Now
$p-f_{0}g_{0}=(f-f_{0})g_{0}+(g-g_{0})f_{0}+(f-f_{0})(g-g_{0})$,
so discarding the last second order term one has to solve $p-f_{0}g_{0}=f_{0}\Delta g+g_{0}\Delta f$ using any variant of the extended Euclidean algorithm to obtain the incremented approximations $f_{1}=f_{0}+\Delta f$ and $g_{1}=g_{0}+\Delta g$. This is repeated until the increments are zero relative to the chosen precision.
Graeffe iteration
The crucial step in this method is to find an annulus of relative width 4 in the complex plane that contains no zeros of p and contains approximately as many zeros of p inside as outside of it. Any annulus of this characteristic can be transformed, by translation and scaling of the polynomial, into the annulus between the radii 1/2 and 2 around the origin. But, not every polynomial admits such a splitting annulus.
To remedy this situation, the Graeffe iteration is applied. It computes a sequence of polynomials
$p_{0}=p,\qquad p_{j+1}(x)=(-1)^{\deg p}p_{j}({\sqrt {x}})\,p_{j}(-{\sqrt {x}}),$
where the roots of $p_{j}(x)$ are the $2^{j}$-th dyadic powers of the roots of the initial polynomial p. By splitting $p_{j}(x)=e(x^{2})+x\,o(x^{2})$ into even and odd parts, the succeeding polynomial is obtained by purely arithmetic operations as $p_{j+1}(x)=(-1)^{\deg p}(e(x)^{2}-x\,o(x)^{2})$. The ratios of the absolute moduli of the roots increase by the same power $2^{j}$ and thus tend to infinity. Choosing j large enough one finally finds a splitting annulus of relative width 4 around the origin.
The approximate factorization of $p_{j}(x)\approx f_{j}(x)\,g_{j}(x)$ is now to be lifted back to the original polynomial. To this end an alternation of Newton steps and Padé approximations is used. It is easy to check that
${\frac {p_{j-1}(x)}{g_{j}(x^{2})}}\approx {\frac {f_{j-1}(x)}{g_{j-1}(-x)}}$
holds. The polynomials on the left side are known in step j, the polynomials on the right side can be obtained as Padé approximants of the corresponding degrees for the power series expansion of the fraction on the left side.
Finding a good circle
Making use of the Graeffe iteration and any known estimate for the absolute value of the largest root one can find estimates R of this absolute value of any precision. Now one computes estimates for the largest and smallest distances $R_{j}>r_{j}>0$ of any root of p(x) to any of the five center points 0, 2R, −2R, 2Ri, −2Ri and selects the one with the largest ratio $R_{j}/r_{j}$ between the two. By this construction it can be guaranteed that $R_{j}/r_{j}>e^{0{.}3}\approx 1.35$ for at least one center. For such a center there has to be a root-free annulus of relative width $\textstyle e^{0{.}3/n}\approx 1+{\frac {0{.}3}{n}}$. After $\textstyle 3+\log _{2}(n)$ Graeffe iterations, the corresponding annulus of the iterated polynomial has a relative width greater than 11 > 4, as required for the initial splitting described above (see Schönhage (1982)). After $\textstyle 4+\log _{2}(n)+\log _{2}(2+\log _{2}(n))$ Graeffe iterations, the corresponding annulus has a relative width greater than $\textstyle 2^{13{.}8}\cdot n^{6{.}9}>(64\cdot n^{3})^{2}$, allowing a much simplified initial splitting (see Malajovich/Zubelli (1997))
To locate the best root-free annulus one uses a consequence of the Rouché theorem: For k = 1, ..., n − 1 the polynomial equation
$\,0=\sum _{j\neq k}|p_{j}|u^{j}-|p_{k}|u^{k},$
u > 0, has, by Descartes' rule of signs zero or two positive roots $u_{k}<v_{k}$. In the latter case, there are exactly k roots inside the (closed) disk $D(0,u_{k})$ and $A(0,u_{k},v_{k})$ is a root-free (open) annulus.
References
• Schönhage, Arnold (1982): The fundamental theorem of algebra in terms of computational complexity. Preliminary Report, Math. Inst. Univ. Tübingen (1982), 49 pages. (ps.gz)
• Gourdon, Xavier (1996). Combinatoire, Algorithmique et Geometrie des Polynomes. Paris: Ecole Polytechnique.
• V. Y. Pan (1996). "Optimal and nearly optimal algorithms for approximating polynomial zeros". Comput. Math. Appl. 31 (12): 97–138. doi:10.1016/0898-1221(96)00080-6.
• V. Y. Pan (1997). "Solving a polynomial equation: Some history and recent progresses". SIAM Review. 39 (2): 187–220. doi:10.1137/S0036144595288554.
• Gregorio Malajovich and Jorge P. Zubelli (1997). "A fast and stable algorithm for splitting polynomials". Computers & Mathematics with Applications. 33. No 3 (2): 1–23. doi:10.1016/S0898-1221(96)00233-7.
• Pan, Victor (1998). Algorithm for Approximating Complex Polynomial Zeros
• Pan, Victor (2002). Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding
• Magma documentation. Real and Complex Fields: Element Operations.
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Schönhardt polyhedron
In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices. It is named after German mathematician Erich Schönhardt, who described it in 1928.[1] The same polyhedra have also been studied in connection with Cauchy's rigidity theorem as an example where polyhedra with two different shapes have faces of the same shapes.
Construction
One way of constructing the Schönhardt polyhedron starts with a triangular prism, with two parallel equilateral triangles as its faces. One of the triangles is rotated around the centerline of the prism, breaking the square faces of the prism into pairs of triangles. If each of these pairs is chosen to be non-convex, the Schönhardt polyhedron is the result.[2]
Properties
The Schönhardt polyhedron has six vertices, twelve edges, and eight triangular faces. The six vertices of the Schönhardt polyhedron can be used to form fifteen unordered pairs of vertices. Twelve of these fifteen pairs form edges of the polyhedron: there are six edges in the two equilateral triangle faces, and six edges connecting the two triangles. The remaining three edges form diagonals of the polyhedron, but lie entirely outside the polyhedron.[3]
The convex hull of the Schönhardt polyhedron is another polyhedron with the same six vertices, and a different set of twelve edges and eight triangular faces; like the Schönhardt polyhedron, it is combinatorially equivalent to a regular octahedron. The symmetric difference of the Schönhardt polyhedron consists of three tetrahedra, each lying between one of the concave dihedral edges of the Schönhardt polyhedron and one of the exterior diagonals. Thus, the Schönhardt polyhedron can be formed by removing these three tetrahedra from a convex (but irregular) octahedron.[4]
Impossibility of triangulation
It is impossible to partition the Schönhardt polyhedron into tetrahedra whose vertices are vertices of the polyhedron. More strongly, there is no tetrahedron that lies entirely inside the Schönhardt polyhedron and has vertices of the polyhedron as its four vertices. This follows from the following two properties of the Schönhardt polyhedron:[3]
• Every triangle formed by its edges is one of its faces. Therefore, because it is not a tetrahedron itself, every tetrahedron formed by four of its vertices must have an edge that it does not share with the Schönhardt polyhedron.[3]
• Every diagonal that connects two of its vertices but is not an edge of the Schönhardt polyhedron lies outside the polyhedron. Therefore, every tetrahedron that uses such a diagonal as one of its edges must also lie in part outside the Schönhardt polyhedron.[3]
Jumping polyhedron
In connection with the theory of flexible polyhedra, instances of the Schönhardt polyhedron form a "jumping polyhedron": a polyhedron that has two different rigid states, both having the same face shapes and the same orientation (convex or concave) of each edge. A model whose surface is made of a stiff but somewhat deformable material, such as cardstock, can be made to "jump" between the two shapes. A solid model could not change shape in this way. Neither could a model made of a more rigid material like glass: although it could exist in either of the two shapes, it would be unable to deform sufficiently to move between them. This stands in contrast to Cauchy's rigidity theorem, according to which, for each convex polyhedron, there is no other polyhedron having the same face shapes and edge orientations.[5]
Related constructions
It was shown by Rambau (2005) that the Schönhardt polyhedron can be generalized to other polyhedra, combinatorially equivalent to antiprisms, that cannot be triangulated. These polyhedra are formed by connecting regular k-gons in two parallel planes, twisted with respect to each other, in such a way that k of the 2k edges that connect the two k-gons have concave dihedrals. For sufficiently small twisting angles, the result has no triangulation.[4][6] Another polyhedron that cannot be triangulated is Jessen's icosahedron, combinatorially equivalent to a regular icosahedron.[2]
In a different direction, Bagemihl (1948) constructed a polyhedron that shares with the Schönhardt polyhedron the property that it has no internal diagonals. The tetrahedron and the Császár polyhedron have no diagonals at all: every pair of vertices in these polyhedra forms an edge.[3] It remains an open question whether there are any other polyhedra (with manifold boundary) without diagonals,[7] although there exist non-manifold surfaces with no diagonals and any number of vertices greater than five.[8]
Applications
Ruppert & Seidel (1992) used Schönhardt's polyhedron as the basis for a proof that it is NP-complete to determine whether a non-convex polyhedron can be triangulated. The proof uses many copies of the Schönhardt polyhedron, with its top face removed, as gadgets within a larger polyhedron. Any triangulation of the overall polyhedron must include a tetrahedron connecting the bottom face of each gadget to a vertex in the rest of the polyhedron that can see this bottom face. The complex pattern of obstructions between tetrahedra of this type can be used to simulate Boolean logic components in a reduction from the Boolean satisfiability problem.[4][10]
References
1. Schönhardt, E. (1928), "Über die Zerlegung von Dreieckspolyedern in Tetraeder", Mathematische Annalen, 98: 309–312, doi:10.1007/BF01451597
2. Bezdek, Andras; Carrigan, Braxton (2016), "On nontriangulable polyhedra", Beiträge zur Algebra und Geometrie, 57 (1): 51–66, doi:10.1007/s13366-015-0248-4, MR 3457762, S2CID 118484882
3. Bagemihl, F. (1948), "On indecomposable polyhedra", American Mathematical Monthly, 55 (7): 411–413, doi:10.2307/2306130, JSTOR 2306130
4. De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco (2010), "Example 3.6.1: Schönhardt's polyhedron", Triangulations: Structures for algorithms and applications, Algorithms and Computation in Mathematics, vol. 25, Berlin: Springer-Verlag, pp. 133–134, doi:10.1007/978-3-642-12971-1, ISBN 978-3-642-12970-4, MR 2743368
5. Grünbaum, Branko (1975), Lectures on lost mathematics (PDF), pp. 41–42
6. Rambau, J. (2005), "On a generalization of Schönhardt's polyhedron" (PDF), in Goodman, Jacob E.; Pach, János; Welzl, Emo (eds.), Combinatorial and Computational Geometry, MSRI Publications, vol. 52, Cambridge: Cambridge University Press, pp. 501–516
7. Ziegler, Günter M. (2008), "Polyhedral surfaces of high genus", in Bobenko, A. I.; Schröder, P.; Sullivan, J. M.; et al. (eds.), Discrete Differential Geometry, Oberwolfach Seminars, vol. 38, Springer-Verlag, pp. 191–213, arXiv:math/0412093, doi:10.1007/978-3-7643-8621-4_10, ISBN 978-3-7643-8620-7, math.MG/0412093
8. Szabó, Sándor (1984), "Polyhedra without diagonals", Periodica Mathematica Hungarica, 15 (1): 41–49, doi:10.1007/BF02109370; Szabó, Sándor (2009), "Polyhedra without diagonals II", Periodica Mathematica Hungarica, 58 (2): 181–187, doi:10.1007/s10998-009-10181-x
9. Tobie, Roger (2012), "What Can You Learn From a Hole in the Ground?", Proceeding of the Natural Philosophy Alliance, 19th Annual Conference, 25-28 July, 2012, Albquerque, New Mexico, vol. 9, Lulu Press, p. 628, ISBN 978-1-105-95509-9
10. Ruppert, J.; Seidel, R. (1992), "On the difficulty of triangulating three-dimensional nonconvex polyhedra", Discrete & Computational Geometry, 7: 227–253, doi:10.1007/BF02187840
External links
• Three Untetrahedralizable Objects, D. Eppstein. Includes a rotatable 3d model of the Schönhardt polyhedron.
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Schützenberger group
In abstract algebra, in semigroup theory, a Schützenberger group is a certain group associated with a Green H-class of a semigroup.[1] The Schützenberger groups associated with different H-classes are different. However, the groups associated with two different H-classes contained in the same D-class of a semigroup are isomorphic. Moreover, if the H-class itself were a group, the Schützenberger group of the H-class would be isomorphic to the H-class. In fact, there are two Schützenberger groups associated with a given H-class and each is antiisomorphic to the other.
The Schützenberger group was discovered by Marcel-Paul Schützenberger in 1957[2][3] and the terminology was coined by A. H. Clifford.[4]
The Schützenberger group
Let S be a semigroup and let S1 be the semigroup obtained by adjoining an identity element 1 to S (if S already has an identity element, then S1 = S). Green's H-relation in S is defined as follows: If a and b are in S then
a H b ⇔ there are u, v, x, y in S1 such that ua = ax = b and vb = by = a.
For a in S, the set of all b' s in S such that a H b is the Green H-class of S containing a, denoted by Ha.
Let H be an H-class of the semigroup S. Let T(H) be the set of all elements t in S1 such that Ht is a subset of H itself. Each t in T(H) defines a transformation, denoted by γt, of H by mapping h in H to ht in H. The set of all these transformations of H, denoted by Γ(H), is a group under composition of mappings (taking functions as right operators). The group Γ(H) is the Schützenberger group associated with the H-class H.
Examples
If H is a maximal subgroup of a monoid M, then H is an H-class, and it is naturally isomorphic to its own Schützenberger group.
In general, one has that the cardinality of H and its Schützenberger group coincide for any H-class H.
Applications
It is known that a monoid with finitely many left and right ideals is finitely presented (or just finitely generated) if and only if all of its Schützenberger groups are finitely presented (respectively, finitely generated). Similarly such a monoid is residually finite if and only if all of its Schützenberger groups are residually finite.
References
1. "The Schützenberger Group of an H-class in the Semigroup of Binary Relations by Robert L. Brandon, Darel W. Hardy, George Markowsky, Missouri University of Science and Technology, 1972-12-01".
2. Marcel-Paul Schützenberger (1957). "D-representation des demi-groupes". C. R. Acad. Sci. Paris. 244: 1994–1996. (MR 19, 249)
3. Clifford, Alfred Hoblitzelle; Preston, Gordon Bamford (1961). The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No. 7. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-0272-4. MR 0132791. (pp. 63–66)
4. Wilf, Herbert; et al. (August 29, 1996). "Marcel-Paul Schützenberger (1920–1996)". The Electronic Journal of Combinatorics. Retrieved 2015-12-30.
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Jeu de taquin
In the mathematical field of combinatorics, jeu de taquin is a construction due to Marcel-Paul Schützenberger (1977) which defines an equivalence relation on the set of skew standard Young tableaux. A jeu de taquin slide is a transformation where the numbers in a tableau are moved around in a way similar to how the pieces in the fifteen puzzle move. Two tableaux are jeu de taquin equivalent if one can be transformed into the other via a sequence of such slides.
"Jeu de taquin" (literally "teasing game") is the French name for the fifteen puzzle.
Definition of a jeu de taquin slide
Given a skew standard Young tableau T of skew shape $\lambda /\mu $, pick an adjacent empty cell c that can be added to the skew diagram $\lambda \setminus \mu $; what this means is that c must share at least one edge with some cell in T, and $\{c\}\cup \lambda \setminus \mu $ must also be a skew diagram. There are two kinds of slide, depending on whether c lies to the upper left of T or to the lower right. Suppose to begin with that c lies to the upper left. Slide the number from its neighbouring cell into c; if c has neighbours both to its right and below, then pick the smallest of these two numbers, favoring the one below. (This rule is designed so that the tableau property of having increasing rows and columns will be preserved.) If the cell that just has been emptied has no neighbour to its right or below, then the slide is completed. Otherwise, slide a number into that cell according to the same rule as before, and continue in this way until the slide is completed. After this transformation, the resulting tableau (with the now-empty cell removed) is still a skew (or possibly straight) standard Young tableau.
The other kind of slide, when c lies to the lower right of T, just goes in the opposite direction. In this case, one slides numbers into an empty cell from the neighbour to its left or above, picking the larger number whenever there is a choice. The two types of slides are mutual inverses – a slide of one kind can be undone using a slide of the other kind.
The two slides described above are referred to as slides into the cell c. The first kind of slide (when c lies to the upper left of T) is said to be an inward slide; the second kind is referred to as an outward slide.
The word "slide" is synonymous to the French word "glissement", which is occasionally also used in English literature.
Subtleties
Jeu-de-taquin slides change not only the relative order of the entries of a tableau, but also its shape. In the definition given above, the result of a jeu-de-taquin slide is given as a skew diagram along with a skew standard tableau having it as shape. Often, it is better to work with skew shapes rather than skew diagrams. (Recall that every skew shape $\lambda /\mu $ gives rise to a skew diagram $\lambda \setminus \mu $, but this is not an injective correspondence because, e. g., the distinct skew shapes $(2,1)/(2)$ and $(1,1)/(1)$ yield the same skew diagram.) For this reason, it is useful to modify the above definition of a jeu-de-taquin slide in such a way that, when given a skew shape along with a skew standard tableau and an addable cell as an input, it yields a well-defined skew shape along with a skew standard tableau at its output. This is done as follows: An inward slide of a skew tableau T of skew shape $\lambda /\mu $ into a cell c is defined as above when c is a corner of $\mu $ (that is, when $\mu \setminus \left\{c\right\}$ is a Young diagram), and the resulting skew shape is set to be $(\lambda \setminus \left\{d\right\})/(\mu \setminus \left\{c\right\})$ where d is the empty cell at the end of the sliding procedure. An outward slide of a skew tableau T of skew shape $\lambda /\mu $ into a cell c is defined as above when c is a cocorner of $\lambda $ (that is, when $\lambda \cup \left\{c\right\}$ is a Young diagram), and the resulting skew shape is set to be $(\lambda \cup \left\{c\right\})/(\mu \cup \left\{d\right\})$ where d is the empty cell at the end of the sliding procedure.
Generalization to skew semistandard tableaux
Jeu de taquin slides generalize to skew semistandard (as opposed to skew standard) tableaux and retain most of their properties in that generality. The only change that has to be made to the definition of an inward slide, in order for it to generalize, is a rule on how to proceed when the (temporarily) empty cell has neighbours below and to its right, and these neighbours are filled with equal numbers. In this situation, the neighbour below (not the one to the right) has to be slid into the empty cell. A similar change is needed in the definition of an outward slide (where one has to choose the neighbor above). These changes may seem arbitrary, but they actually make the "only reasonable choices"—meaning the only choices that keep the columns of the tableau (disregarding the empty cell) strictly increasing (as opposed to just weakly increasing).
Rectification
Given a skew standard or skew semistandard tableau T, one can iteratively apply inward slides to T until the tableau becomes straight-shape (which means no more inward slides are possible). This can generally be done in many different ways (one can freely choose into which cell to slide first), but the resulting straight-shape tableau is known to be the same for all possible choices. This tableau is called the rectification of T.
Jeu-de-taquin equivalence
Two skew semistandard tableaux T and S are said to be jeu-de-taquin equivalent if one can transform one of them into the other using a sequence (possibly empty) of slides (both inward and outward slides being allowed). Equivalently, two skew semistandard tableaux T and S are jeu-de-taquin equivalent if and only if they have the same rectification.
Reading words and Knuth equivalence
There are various ways to associate a word (in the sense of combinatorics, i. e., a finite sequence of elements of an alphabet—here the set of positive integers) to every Young tableau. We choose the one apparently most popular: We associate to every Young tableau T the word obtained by concatenating the rows of T from the bottom row to the top row. (Each row of T is seen as a word simply by reading its entries from left to right, and we draw Young tableaux in English notation so that the longest row of a straight-shape tableau appears at the top.) This word will be referred to as the reading word, or briefly as the word, of T.
It can then be shown that two skew semistandard tableaux T and S are jeu-de-taquin equivalent if and only if the reading words of T and S are Knuth equivalent. As a consequence, the rectification of a skew semistandard tableau T can also be obtained as the insertion tableau of the reading word of T under the Robinson-Schensted correspondence.
The Schützenberger involution
Jeu de taquin can be used to define an operation on standard Young tableaux of any given shape, which turns out to be an involution, although this is not obvious from the definition. One starts by emptying the square in the top-left corner, turning the tableau into a skew tableau with one less square. Now apply a jeu de taquin slide to turn that skew tableau into a straight one, which will free one square on the outside border. Then fill this square with the negative of the value that was originally removed at the top-left corner; this negated value is considered part of a new tableau rather than of the original tableau, and its position will not change in the sequel. Now as long as the original tableau has some entries left, repeat the operation of removing the entry x of the top-left corner, performing a jeu de taquin slide on what is left of the original tableau, and placing the value −x into the square so freed. When all entries of the original tableau have been handled, their negated values are arranged in such a way that rows and columns are increasing. Finally one can add an appropriate constant to all entries to obtain a Young tableau with positive entries.
Applications
Jeu de taquin is closely connected to such topics as the Robinson–Schensted–Knuth correspondence, the Littlewood–Richardson rule, and Knuth equivalence.
References
• Désarménien, J. (2001) [1994], "Jeu de taquin", Encyclopedia of Mathematics, EMS Press
• Sagan, Bruce E. (2001), The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics 203 (2nd ed.), New York: Springer, ISBN 0-387-95067-2
• Fulton, William (1997), Young Tableaux, London Mathematical Society Student Texts 35 (1st ed.), Melbourne: Cambridge University Press, ISBN 0-521-56144-2
• Haiman, M. D. (1992). "Dual equivalence with applications, including a conjecture of Proctor". Discrete Mathematics. 99: 79–113. doi:10.1016/0012-365X(92)90368-P.
• Schützenberger, Marcel-Paul (1977), "La correspondance de Robinson", in Foata, Dominique (ed.), Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Lecture Notes in Math., vol. 579, Berlin: Springer, pp. 59–113, doi:10.1007/BFb0090012, ISBN 978-3-540-08143-2
• Stanley, Richard P. (1999), Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics 62, vol. 2, Cambridge University Press, ISBN 0-521-56069-1
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Science and Math Institute (Tacoma, Washington)
Tacoma Science and Math Institute (also known as SAMi), is a public high school in the Tacoma Public Schools district. It is located in Metro Parks Tacoma in Tacoma, Washington. The school offers an integrated inquiry-based curriculum for students in grades 9-12 that combines the arts, science, math, and environmental and marine studies. It operates in partnership with local organizations, including the Point Defiance Zoo and Aquarium as well as local universities.[2] SAMi also operates in partnership with other local schools, including its sister schools Tacoma School of the Arts (SOTA) and Industrial Design Engineering and Art (iDEA).
Science and Math institute
Address
5502 Five Mile Drive
Tacoma
,
WA
98407
Coordinates47°18′11″N 122°30′46″W
Information
TypeHigh School
Established2009
PrincipalJon Ketler, Liz Minks, Joni Hall
Faculty26.90 (FTE)[1]
Enrollment537 (2018-19)[1]
Student to teacher ratio19.96[1]
Websitesami.tacomaschools.org
History and facilities
SAMi was started in the Fall of 2009, when they opened their doors to the first freshman class (of 2013). The school expanded by adding one grade per year and as of 2012 offered all four high school grades (freshman, sophomore, junior and senior) in its program. In 2012-2013 the school began accepting Grade 8 students for the graduating class of 2017.[2]
The school began with nine portable classrooms located near the Pt. Defiance boat trailer parking lot, in front of the parks maintenance and greenhouse facility, to the south of the Ferry Dock, the Tacoma Yacht Club, and Anthony's Diner. In the summer of 2015, the portable classrooms were moved to Pt. Defiance's Camp Six, the former site of a logging museum. The old location of SAMi has since been dug out during Tacoma's Destination Point Defiance renovation process.[3]
Since its inception the school has been seeking to expand the effective area of the campus into the park and use buildings and structures in the park as classrooms. In 2009-2010 the school was granted access to the parks greenhouses for a plant biology class, which continued until the demolishing of the greenhouses in 2016. In the 2010–2011 school year regular access was granted to the Point Defiance Zoo & Aquarium, the Pagoda and the historical Lodge.[4] It also holds its Outdoor Education class in the areas around the Fort Nisqually Living History Museum.[5]
SAMi was restricted from using the pagoda after an arsonist set it on fire on April 15, 2011.[6] The building sustained significant damage and the arsonist is now in custody.[7] The restored and renovated pagoda is still used by SAMi for classes.
In the February 2013 bond election, voters approved funding for a new permanent facility for the school adjacent to the Point Defiance Zoo and Aquarium. Construction was expected to be completed by 2015, but the building was not opened until October 2017.[8] This new building, the Environmental Learning Center (ELC), is designed to function as an extra 8 classrooms and a learning space for preschoolers. During the summer, the Point Defiance Zoo will use it as an outreach space.
The primary school buildings now consist of portables at the old Camp 6, the Environmental Learning Center (ELC), Pagoda, and Point Defiance Zoo & Aquarium classrooms.[9] The whole of Point Defiance Park is considered the schools campus, according to the school's staff. However, student roaming (usually only allowed during the student's lunch break) is prohibited south of Pearl Street area. This is loosely enforced.
Mission and goals
"The mission of the Science and Math Institute is to provide a creative path of learning that emphasizes individual expression and growth through science and mathematics as central elements in academic achievement and life-long endeavors."[2]
SAMi intends to keep expanding its campus, including allowing students access to nearby downtown Ruston.
Academics
Many different types of classes are offered at the Science and Math Institute. Aside from the required high school classes,[10] SAMI offers science and art oriented electives. Students also are enrolled in classes earlier than they would be at most other high schools in the district and other areas.
SAMi uses a system called "Pathways" that assists students with selecting the best classes that will benefit their future, whether it be a career or simply college endeavors. Presently the two pathways are 'Natural Sciences' and 'Physical Sciences'
Students also can enroll in two classes per day at SOTA and/or iDEA by splitting the academic day between SAMi, SOTA and iDEA.
Current Leadership
SAMi's leaderships is split between 'downtown' and on campus. Two co-directors (vice-principles) run day to day operations while reporting to the coalition schools (SAMI, SOTA, IDEA) central admin (principal's) who are based out of downtown.
Coalition School Central admin
2016- 21: Jon Ketler & Kristen Tinder
2021- 23: Jon Ketler & Zack Varnell
Recent Co-directors (Vice Principles) at SAMI
2016 - 17: Liz Minks & Ralph Harrison
2017 - 19: Liz Minks & Joni Hall
2019 - 21: Liz Minks & Anne Tsuneshi
2021 - 23: Liz Minks & Joni Hall[11]
Enrollment
Students from the high school venture out to local middle schools and present their school to all of the middle school students in a massive presentation. Potential applicants are told details of the school and how they can be a part of it.
The student selection process is simple. An interested person submits their application with their best school work under the graduation requirement categories and then are interviewed by a SAMi student and a staff member. If a student completes the application and takes part in the interview, their name is put into lottery system which will produce the next school years names, depending on how many new students SAMi takes that year.
SAMI at times has been critiqued for only accepting students into their 9th grade, generally not accepting transfers during students 10th, 11th, or 12th grades years from in or out of district.
References
1. "Tacoma Science and Math Institute". National Center for Education Statistics. Retrieved May 29, 2020.
2. "What is SAMI?" (PDF). SAMI. Retrieved 24 October 2014.
3. Cafazzo, Debbie (July 30, 2015). "Logs to logarithms: SAMI gets interim home at former Camp 6 site". The News Tribune. Retrieved September 12, 2021.
4. "Lodge at Point Defiance Park". Metro Parts Tacoma > Lodge at Point Defiance Park. Archived from the original on 3 April 2012. Retrieved 24 October 2014.
5. "Metro Parks Tacoma > Fort Nisqually Living History Museum". Metro Parks Tacoma. Archived from the original on 2 May 2012. Retrieved 24 October 2014.
6. "Police: Point Defiance Pagoda Fire May Be Connected To Break-In". www.kirotv.com. 2011-04-16. Retrieved 2012-03-11.
7. "Teen arrested for Pt. Defiance Park fire - KCPQ". Q13fox.com. 2011-04-25. Retrieved 2012-03-11.
8. "Construction Planning" (PDF). Tacoma.k12. Retrieved 24 October 2014.
9. "Class Locations - Science and Math Institute". Tacoma SAMI. September 12, 2021. Retrieved September 12, 2021.{{cite web}}: CS1 maint: url-status (link)
10. "Earn High School Credit". OSPI. Retrieved 24 October 2014.
11. "Staff List - Science and Math Institute". Tacoma SAMi. September 12, 2021. Retrieved September 12, 2021.{{cite web}}: CS1 maint: url-status (link)
External links
• Official website
• School District school webpage
• OSPI School Report Card 2011-2012
• Point Defiance Zoo & Aquarium
• Innovation Schools Share Insights at AWB Legislative Day, Olympia Business Watch, 2012-01
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Wikipedia
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Computational science
Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science that uses advanced computing capabilities to understand and solve complex physical problems. This includes
• Algorithms (numerical and non-numerical): mathematical models, computational models, and computer simulations developed to solve sciences (e.g, physical, biological, and social), engineering, and humanities problems
• Computer hardware that develops and optimizes the advanced system hardware, firmware, networking, and data management components needed to solve computationally demanding problems
• The computing infrastructure that supports both the science and engineering problem solving and the developmental computer and information science
In practical use, it is typically the application of computer simulation and other forms of computation from numerical analysis and theoretical computer science to solve problems in various scientific disciplines. The field is different from theory and laboratory experiments, which are the traditional forms of science and engineering. The scientific computing approach is to gain understanding through the analysis of mathematical models implemented on computers. Scientists and engineers develop computer programs and application software that model systems being studied and run these programs with various sets of input parameters. The essence of computational science is the application of numerical algorithms[1] and computational mathematics. In some cases, these models require massive amounts of calculations (usually floating-point) and are often executed on supercomputers or distributed computing platforms.
The computational scientist
The term computational scientist is used to describe someone skilled in scientific computing. Such a person is usually a scientist, an engineer, or an applied mathematician who applies high-performance computing in different ways to advance the state-of-the-art in their respective applied disciplines in physics, chemistry, or engineering.
Computational science is now commonly considered a third mode of science , complementing and adding to experimentation/observation and theory (see image).[2] Here, one defines a system as a potential source of data,[3] an experiment as a process of extracting data from a system by exerting it through its inputs[4] and a model (M) for a system (S) and an experiment (E) as anything to which E can be applied in order to answer questions about S.[5] A computational scientist should be capable of:
• recognizing complex problems
• adequately conceptualizing the system containing these problems
• designing a framework of algorithms suitable for studying this system: the simulation
• choosing a suitable computing infrastructure (parallel computing/grid computing/supercomputers)
• hereby, maximizing the computational power of the simulation
• assessing to what level the output of the simulation resembles the systems: the model is validated
• adjusting the conceptualization of the system accordingly
• repeat the cycle until a suitable level of validation is obtained: the computational scientist trusts that the simulation generates adequately realistic results for the system under the studied conditions
Substantial effort in computational sciences has been devoted to developing algorithms, efficient implementation in programming languages, and validating computational results. A collection of problems and solutions in computational science can be found in Steeb, Hardy, Hardy, and Stoop (2004).[6]
Philosophers of science addressed the question to what degree computational science qualifies as science, among them Humphreys[7] and Gelfert.[8] They address the general question of epistemology: how does gain insight from such computational science approaches? Tolk[9] uses these insights to show the epistemological constraints of computer-based simulation research. As computational science uses mathematical models representing the underlying theory in executable form, in essence, they apply modeling (theory building) and simulation (implementation and execution). While simulation and computational science are our most sophisticated way to express our knowledge and understanding, they also come with all constraints and limits already known for computational solutions.
Applications of computational science
Problem domains for computational science/scientific computing include:
Predictive computational science
Predictive computational science is a scientific discipline concerned with the formulation, calibration, numerical solution, and validation of mathematical models designed to predict specific aspects of physical events, given initial and boundary conditions, and a set of characterizing parameters and associated uncertainties.[10] In typical cases, the predictive statement is formulated in terms of probabilities. For example, given a mechanical component and a periodic loading condition, "the probability is (say) 90% that the number of cycles at failure (Nf) will be in the interval N1<Nf<N2".[11]
Urban complex systems
In 2018, over half the world's population lives in cities.[12] By 2050, the United Nations estimates, 68% of the world's population will be urban.[12] This urban growth is focused in the urban populations of developing countries where city dwellers will more than double, increasing from 2.5 billion in 2009 to almost 5.2 billion in 2050. Cities are massively complex systems created by humans, made up of humans, and governed by humans. Trying to predict, understand and somehow shape the development of cities in the future requires complex thinking and computational models and simulations to help mitigate challenges and possible disasters. The focus of research in urban complex systems is, through modeling and simulation, to build a greater understanding of city dynamics and help prepare for the coming urbanization.
Computational finance
Main article: Computational finance
In financial markets, huge volumes of interdependent assets are traded by a large number of interacting market participants in different locations and time zones. Their behavior is of unprecedented complexity and the characterization and measurement of the risk inherent to this highly diverse set of instruments is typically based on complicated mathematical and computational models. Solving these models exactly in closed form, even at a single instrument level, is typically not possible, and therefore we have to look for efficient numerical algorithms. This has become even more urgent and complex recently, as the credit crisis has clearly demonstrated the role of cascading effects going from single instruments through portfolios of single institutions to even the interconnected trading network. Understanding this requires a multi-scale and holistic approach where interdependent risk factors such as market, credit, and liquidity risk are modeled simultaneously and at different interconnected scales.
Computational biology
Exciting new developments in biotechnology are now revolutionizing biology and biomedical research. Examples of these techniques are high-throughput sequencing, high-throughput quantitative PCR, intra-cellular imaging, in-situ hybridization of gene expression, three-dimensional imaging techniques like Light Sheet Fluorescence Microscopy, and Optical Projection (micro)-Computer Tomography. Given the massive amounts of complicated data that is generated by these techniques, their meaningful interpretation, and even their storage, form major challenges calling for new approaches. Going beyond current bioinformatics approaches, computational biology needs to develop new methods to discover meaningful patterns in these large data sets. Model-based reconstruction of gene networks can be used to organize the gene expression data in a systematic way and to guide future data collection. A major challenge here is to understand how gene regulation is controlling fundamental biological processes like biomineralization and embryogenesis. The sub-processes like gene regulation, organic molecules interacting with the mineral deposition process, cellular processes, physiology, and other processes at the tissue and environmental levels are linked. Rather than being directed by a central control mechanism, biomineralization and embryogenesis can be viewed as an emergent behavior resulting from a complex system in which several sub-processes on very different temporal and spatial scales (ranging from nanometer and nanoseconds to meters and years) are connected into a multi-scale system. One of the few available options to understand such systems is by developing a multi-scale model of the system.
Complex systems theory
Using information theory, non-equilibrium dynamics, and explicit simulations, computational systems theory tries to uncover the true nature of complex adaptive systems.
Computational science and engineering
Computational science and engineering (CSE) is a relatively new discipline that deals with the development and application of computational models and simulations, often coupled with high-performance computing, to solve complex physical problems arising in engineering analysis and design (computational engineering) as well as natural phenomena (computational science). CSE has been described as the "third mode of discovery" (next to theory and experimentation).[13] In many fields, computer simulation is integral and therefore essential to business and research. Computer simulation provides the capability to enter fields that are either inaccessible to traditional experimentation or where carrying out traditional empirical inquiries is prohibitively expensive. CSE should neither be confused with pure computer science, nor with computer engineering, although a wide domain in the former is used in CSE (e.g., certain algorithms, data structures, parallel programming, high-performance computing), and some problems in the latter can be modeled and solved with CSE methods (as an application area).
Methods and algorithms
Algorithms and mathematical methods used in computational science are varied. Commonly applied methods include:
• Computer algebra,[14][15][16][17] including symbolic computation in fields such as statistics, equation solving, algebra, calculus, geometry, linear algebra, tensor analysis (multilinear algebra), optimization
• Numerical analysis,[18][19][20][21] including Computing derivatives by finite differences
• Application of Taylor series as convergent and asymptotic series
• Computing derivatives by Automatic differentiation (AD)
• Finite element method for solving PDEs[22][23]
• High order difference approximations via Taylor series and Richardson extrapolation
• Methods of integration[24] on a uniform mesh: rectangle rule (also called midpoint rule), trapezoid rule, Simpson's rule
• Runge–Kutta methods for solving ordinary differential equations
• Newton's method[25]
• Discrete Fourier transform
• Monte Carlo methods[26][27]
• Numerical linear algebra,[28][29][30] including decompositions and eigenvalue algorithms
• Linear programming[31][32]
• Branch and cut
• Branch and bound
• Molecular dynamics, Car–Parrinello molecular dynamics
• Space mapping
• Time stepping methods for dynamical systems
Historically and today, Fortran remains popular for most applications of scientific computing.[33][34] Other programming languages and computer algebra systems commonly used for the more mathematical aspects of scientific computing applications include GNU Octave, Haskell,[33] Julia,[33] Maple,[34] Mathematica,[35][36][37][38][39] MATLAB,[40][41][42] Python (with third-party SciPy library[43][44][45]), Perl (with third-party PDL library), R,[46] Scilab,[47][48] and TK Solver. The more computationally intensive aspects of scientific computing will often use some variation of C or Fortran and optimized algebra libraries such as BLAS or LAPACK. In addition, parallel computing is heavily used in scientific computing to find solutions of large problems in a reasonable amount of time. In this framework, the problem is either divided over many cores on a single CPU node (such as with OpenMP), divided over many CPU nodes networked together (such as with MPI), or is run on one or more GPUs (typically using either CUDA or OpenCL).
Computational science application programs often model real-world changing conditions, such as weather, airflow around a plane, automobile body distortions in a crash, the motion of stars in a galaxy, an explosive device, etc. Such programs might create a 'logical mesh' in computer memory where each item corresponds to an area in space and contains information about that space relevant to the model. For example, in weather models, each item might be a square kilometer; with land elevation, current wind direction, humidity, temperature, pressure, etc. The program would calculate the likely next state based on the current state, in simulated time steps, solving differential equations that describe how the system operates, and then repeat the process to calculate the next state.
Conferences and journals
In 2001, the International Conference on Computational Science (ICCS) was first organized. Since then, it has been organized yearly. ICCS is an A-rank conference in the CORE ranking.[49]
The Journal of Computational Science published its first issue in May 2010.[50][51][52] The Journal of Open Research Software was launched in 2012.[53] The ReScience C initiative, which is dedicated to replicating computational results, was started on GitHub in 2015.[54]
Education
At some institutions, a specialization in scientific computation can be earned as a "minor" within another program (which may be at varying levels). However, there are increasingly many bachelor's, master's, and doctoral programs in computational science. The joint degree program master program computational science at the University of Amsterdam and the Vrije Universiteit in computational science was first offered in 2004. In this program, students:
• learn to build computational models from real-life observations;
• develop skills in turning these models into computational structures and in performing large-scale simulations;
• learn theories that will give a firm basis for the analysis of complex systems;
• learn to analyze the results of simulations in a virtual laboratory using advanced numerical algorithms.
ETH Zurich offers a bachelor's and master's degree in Computational Science and Engineering. The degree equips students with the ability to understand scientific problem and apply numerical methods to solve such problems. The directions of specializations include Physics, Chemistry, Biology and other Scientific and Engineering disciplines.
George Mason University was one of the early pioneers first offering a multidisciplinary doctorate Ph.D. program in Computational Sciences and Informatics in 1992 that focused on a number of specialty areas, including bioinformatics, computational chemistry, earth systems, and global changes, computational mathematics, computational physics, space sciences, and computational statistics.
The School of Computational and Integrative Sciences, Jawaharlal Nehru University (erstwhile School of Information Technology[55]) also offers a vibrant master's science program for computational science with two specialties: Computational Biology and Complex Systems.[56]
Subfields
• Bioinformatics
• Car–Parrinello molecular dynamics
• Cheminformatics
• Chemometrics
• Computational archaeology
• Computational astrophysics
• Computational biology
• Computational chemistry
• Computational materials science
• Computational economics
• Computational electromagnetics
• Computational engineering
• Computational finance
• Computational fluid dynamics
• Computational forensics
• Computational geophysics
• Computational history
• Computational informatics
• Computational intelligence
• Computational law
• Computational linguistics
• Computational mathematics
• Computational mechanics
• Computational neuroscience
• Computational particle physics
• Computational physics
• Computational sociology
• Computational statistics
• Computational sustainability
• Computer algebra
• Computer simulation
• Financial modeling
• Geographic information science
• Geographic information system (GIS)
• High-performance computing
• Machine learning
• Network analysis
• Neuroinformatics
• Numerical linear algebra
• Numerical weather prediction
• Pattern recognition
• Scientific visualization
• Simulation
See also
• Computational science and engineering
• Modeling and simulation
• Comparison of computer algebra systems
• Differentiable programming
• List of molecular modeling software
• List of numerical analysis software
• List of statistical packages
• Timeline of scientific computing
• Simulated reality
• Extensions for Scientific Computation (XSC)
References
1. Nonweiler T. R., 1986. Computational Mathematics: An Introduction to Numerical Approximation, John Wiley and Sons
2. Graduate Education for Computational Science and Engineering.Siam.org, Society for Industrial and Applied Mathematics (SIAM) website; accessed Feb 2013.
3. Siegler, Bernard (1976). Theory of Modeling and Simulation.
4. Cellier, François (1990). Continuous System Modelling.
5. Minski, Marvin (1965). Models,Minds, Machines.
6. Steeb W.-H., Hardy Y., Hardy A. and Stoop R., 2004. Problems and Solutions in Scientific Computing with C++ and Java Simulations, World Scientific Publishing. ISBN 981-256-112-9
7. Humphreys, Paul. Extending ourselves: Computational science, empiricism, and scientific method. Oxford University Press, 2004.
8. Gelfert, Axel. 2016. How to do science with models: A philosophical primer. Cham: Springer.
9. Tolk, Andreas. "Learning Something Right from Models That Are Wrong: Epistemology of Simulation." In Concepts and Methodologies for Modeling and Simulation, edited by L. Yilmaz, pp. 87-106, Cham: Springer International Publishing, 2015.
10. Oden, J.T., Babuška, I. and Faghihi, D., 2017. Predictive computational science: Computer predictions in the presence of uncertainty. Encyclopedia of Computational Mechanics. Second Edition, pp. 1-26.
11. Szabó B, Actis R and Rusk D. Validation of notch sensitivity factors. Journal of Verification, Validation and Uncertainty Quantification. 4 011004, 2019
12. "68% of the world population projected to live in urban areas by 2050, says UN | UN DESA | United Nations Department of Economic and Social Affairs". www.un.org. Retrieved 2021-12-31.
13. "Computational Science and Engineering Program: Graduate Student Handbook" (PDF). cseprograms.gatech.edu. September 2009. Archived from the original (PDF) on 2014-10-14. Retrieved 2017-08-26.
14. Von Zur Gathen, J., & Gerhard, J. (2013). Modern computer algebra. Cambridge University Press.
15. Geddes, K. O., Czapor, S. R., & Labahn, G. (1992). Algorithms for computer algebra. Springer Science & Business Media.
16. Albrecht, R. (2012). Computer algebra: symbolic and algebraic computation (Vol. 4). Springer Science & Business Media.
17. Mignotte, M. (2012). Mathematics for computer algebra. Springer Science & Business Media.
18. Stoer, J., & Bulirsch, R. (2013). Introduction to numerical analysis. Springer Science & Business Media.
19. Conte, S. D., & De Boor, C. (2017). Elementary numerical analysis: an algorithmic approach. Society for Industrial and Applied Mathematics.
20. Greenspan, D. (2018). Numerical Analysis. CRC Press.
21. Linz, P. (2019). Theoretical numerical analysis. Courier Dover Publications.
22. Brenner, S., & Scott, R. (2007). The mathematical theory of finite element methods (Vol. 15). Springer Science & Business Media.
23. Oden, J. T., & Reddy, J. N. (2012). An introduction to the mathematical theory of finite elements. Courier Corporation.
24. Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation.
25. Peter Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Second printed edition. Series Computational Mathematics 35, Springer (2006)
26. Hammersley, J. (2013). Monte carlo methods. Springer Science & Business Media.
27. Kalos, M. H., & Whitlock, P. A. (2009). Monte carlo methods. John Wiley & Sons.
28. Demmel, J. W. (1997). Applied numerical linear algebra. SIAM.
29. Ciarlet, P. G., Miara, B., & Thomas, J. M. (1989). Introduction to numerical linear algebra and optimization. Cambridge University Press.
30. Trefethen, Lloyd; Bau III, David (1997). Numerical Linear Algebra (1st ed.). Philadelphia: SIAM.
31. Vanderbei, R. J. (2015). Linear programming. Heidelberg: Springer.
32. Gass, S. I. (2003). Linear programming: methods and applications. Courier Corporation.
33. Phillips, Lee (2014-05-07). "Scientific computing's future: Can any coding language top a 1950s behemoth?". Ars Technica. Retrieved 2016-03-08.
34. Landau, Rubin (2014-05-07). "A First Course in Scientific Computing" (PDF). Princeton University. Retrieved 2016-03-08.
35. Mathematica 6 Scientific Computing World, May 2007
36. Maeder, R. E. (1991). Programming in mathematica. Addison-Wesley Longman Publishing Co., Inc.
37. Stephen Wolfram. (1999). The MATHEMATICA® book, version 4. Cambridge University Press.
38. Shaw, W. T., & Tigg, J. (1993). Applied Mathematica: getting started, getting it done. Addison-Wesley Longman Publishing Co., Inc.
39. Marasco, A., & Romano, A. (2001). Scientific Computing with Mathematica: Mathematical Problems for Ordinary Differential Equations; with a CD-ROM. Springer Science & Business Media.
40. Quarteroni, A., Saleri, F., & Gervasio, P. (2006). Scientific computing with MATLAB and Octave. Berlin: Springer.
41. Gander, W., & Hrebicek, J. (Eds.). (2011). Solving problems in scientific computing using Maple and Matlab®. Springer Science & Business Media.
42. Barnes, B., & Fulford, G. R. (2011). Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB. Chapman and Hall/CRC.
43. Jones, E., Oliphant, T., & Peterson, P. (2001). SciPy: Open source scientific tools for Python.
44. Bressert, E. (2012). SciPy and NumPy: an overview for developers. " O'Reilly Media, Inc.".
45. Blanco-Silva, F. J. (2013). Learning SciPy for numerical and scientific computing. Packt Publishing Ltd.
46. Ihaka, R., & Gentleman, R. (1996). R: a language for data analysis and graphics. Journal of computational and graphical statistics, 5(3), 299-314.
47. Bunks, C., Chancelier, J. P., Delebecque, F., Goursat, M., Nikoukhah, R., & Steer, S. (2012). Engineering and scientific computing with Scilab. Springer Science & Business Media.
48. Thanki, R. M., & Kothari, A. M. (2019). Digital image processing using SCILAB. Springer International Publishing.
49. "ICCS - International Conference on Computational Science". Retrieved 2022-01-21.
50. Sloot, Peter; Coveney, Peter; Dongarra, Jack (2010). "Redirecting". Journal of Computational Science. 1 (1): 3–4. doi:10.1016/j.jocs.2010.04.003.
51. Seidel, Edward; Wing, Jeannette M. (2010). "Redirecting". Journal of Computational Science. 1 (1): 1–2. doi:10.1016/j.jocs.2010.04.004. S2CID 211478325.
52. Sloot, Peter M.A. (2010). "Computational science: A kaleidoscopic view into science". Journal of Computational Science. 1 (4): 189. doi:10.1016/j.jocs.2010.11.001.
53. "Announcing the Journal of Open Research Software - a software metajournal". software.ac.uk. Retrieved 2021-12-31.
54. Rougier, Nicolas P.; Hinsen, Konrad; Alexandre, Frédéric; Arildsen, Thomas; Barba, Lorena A.; Benureau, Fabien C.Y.; Brown, C. Titus; Buyl, Pierre de; Caglayan, Ozan; Davison, Andrew P.; Delsuc, Marc-André; Detorakis, Georgios; Diem, Alexandra K.; Drix, Damien; Enel, Pierre; Girard, Benoît; Guest, Olivia; Hall, Matt G.; Henriques, Rafael N.; Hinaut, Xavier; Jaron, Kamil S.; Khamassi, Mehdi; Klein, Almar; Manninen, Tiina; Marchesi, Pietro; McGlinn, Daniel; Metzner, Christoph; Petchey, Owen; Plesser, Hans Ekkehard; Poisot, Timothée; Ram, Karthik; Ram, Yoav; Roesch, Etienne; Rossant, Cyrille; Rostami, Vahid; Shifman, Aaron; Stachelek, Joseph; Stimberg, Marcel; Stollmeier, Frank; Vaggi, Federico; Viejo, Guillaume; Vitay, Julien; Vostinar, Anya E.; Yurchak, Roman; Zito, Tiziano (December 2017). "Sustainable computational science: the ReScience initiative". PeerJ Comput Sci. 3. e142. arXiv:1707.04393. Bibcode:2017arXiv170704393R. doi:10.7717/peerj-cs.142. PMC 8530091. PMID 34722870. S2CID 7392801.
55. "SCIS | Welcome to Jawaharlal Nehru University". www.jnu.ac.in. Archived from the original on 2013-03-10.
56. "SCIS: Program of Study | Welcome to Jawaharlal Nehru University". www.jnu.ac.in. Archived from the original on 7 February 2020. Retrieved 31 December 2021.
Additional sources
• E. Gallopoulos and A. Sameh, "CSE: Content and Product". IEEE Computational Science and Engineering Magazine, 4(2):39–43 (1997)
• G. Hager and G. Wellein, Introduction to High Performance Computing for Scientists and Engineers, Chapman and Hall (2010)
• A.K. Hartmann, Practical Guide to Computer Simulations, World Scientific (2009)
• Journal Computational Methods in Science and Technology (open access), Polish Academy of Sciences
• Journal Computational Science and Discovery, Institute of Physics
• R.H. Landau, C.C. Bordeianu, and M. Jose Paez, A Survey of Computational Physics: Introductory Computational Science, Princeton University Press (2008)
External links
Wikiversity has learning resources about Scientific computing
Wikimedia Commons has media related to Computational science.
• Journal of Computational Science
• The Journal of Open Research Software
• The National Center for Computational Science at Oak Ridge National Laboratory
Computational science
Biology
• Anatomy
• Biological systems
• Genomics
• Neuroscience
• Phylogenetics
Chemistry
• Electronic structure
• Molecular mechanics
Physics
• Astrophysics
• Electromagnetics
• Fluid dynamics
• Mechanics
• Particle physics
Linguistics
• Semantics
• Lexicology
Social science
• Politics
• Sociology
• Economics
Other
• Finance
• Materials science
• Mathematics
Differentiable computing
General
• Differentiable programming
• Information geometry
• Statistical manifold
• Automatic differentiation
• Neuromorphic engineering
• Pattern recognition
• Tensor calculus
• Computational learning theory
• Inductive bias
Concepts
• Gradient descent
• SGD
• Clustering
• Regression
• Overfitting
• Hallucination
• Adversary
• Attention
• Convolution
• Loss functions
• Backpropagation
• Normalization (Batchnorm)
• Activation
• Softmax
• Sigmoid
• Rectifier
• Regularization
• Datasets
• Augmentation
• Diffusion
• Autoregression
Applications
• Machine learning
• In-context learning
• Artificial neural network
• Deep learning
• Scientific computing
• Artificial Intelligence
• Language model
• Large language model
Hardware
• IPU
• TPU
• VPU
• Memristor
• SpiNNaker
Software libraries
• TensorFlow
• PyTorch
• Keras
• Theano
• JAX
• Flux.jl
Implementations
Audio–visual
• AlexNet
• WaveNet
• Human image synthesis
• HWR
• OCR
• Speech synthesis
• Speech recognition
• Facial recognition
• AlphaFold
• DALL-E
• Midjourney
• Stable Diffusion
Verbal
• Word2vec
• Seq2seq
• BERT
• LaMDA
• Bard
• NMT
• Project Debater
• IBM Watson
• GPT-2
• GPT-3
• ChatGPT
• GPT-4
• GPT-J
• Chinchilla AI
• PaLM
• BLOOM
• LLaMA
Decisional
• AlphaGo
• AlphaZero
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Wikipedia
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Quantification (science)
In mathematics and empirical science, quantification (or quantitation) is the act of counting and measuring that maps human sense observations and experiences into quantities. Quantification in this sense is fundamental to the scientific method.
For formal and natural language features to indicate quantity, see Quantifier (logic) and Quantifier (linguistics).
Natural science
Some measure of the undisputed general importance of quantification in the natural sciences can be gleaned from the following comments:
• "these are mere facts, but they are quantitative facts and the basis of science."[1]
• It seems to be held as universally true that "the foundation of quantification is measurement."[2]
• There is little doubt that "quantification provided a basis for the objectivity of science."[3]
• In ancient times, "musicians and artists ... rejected quantification, but merchants, by definition, quantified their affairs, in order to survive, made them visible on parchment and paper."[4]
• Any reasonable "comparison between Aristotle and Galileo shows clearly that there can be no unique lawfulness discovered without detailed quantification."[5]
• Even today, "universities use imperfect instruments called 'exams' to indirectly quantify something they call knowledge."[6]
This meaning of quantification comes under the heading of pragmatics.
In some instances in the natural sciences a seemingly intangible concept may be quantified by creating a scale—for example, a pain scale in medical research, or a discomfort scale at the intersection of meteorology and human physiology such as the heat index measuring the combined perceived effect of heat and humidity, or the wind chill factor measuring the combined perceived effects of cold and wind.
Social sciences
In the social sciences, quantification is an integral part of economics and psychology. Both disciplines gather data – economics by empirical observation and psychology by experimentation – and both use statistical techniques such as regression analysis to draw conclusions from it.
In some instances a seemingly intangible property may be quantified by asking subjects to rate something on a scale—for example, a happiness scale or a quality-of-life scale—or by the construction of a scale by the researcher, as with the index of economic freedom. In other cases, an unobservable variable may be quantified by replacing it with a proxy variable with which it is highly correlated—for example, per capita gross domestic product is often used as a proxy for standard of living or quality of life.
Frequently in the use of regression, the presence or absence of a trait is quantified by employing a dummy variable, which takes on the value 1 in the presence of the trait or the value 0 in the absence of the trait.
Quantitative linguistics is an area of linguistics that relies on quantification. For example,[7] indices of grammaticalization of morphemes, such as phonological shortness, dependence on surroundings, and fusion with the verb, have been developed and found to be significantly correlated across languages with stage of evolution of function of the morpheme.
Hard versus soft science
The ease of quantification is one of the features used to distinguish hard and soft sciences from each other. Scientists often consider hard sciences to be more scientific or rigorous, but this is disputed by social scientists who maintain that appropriate rigor includes the qualitative evaluation of the broader contexts of qualitative data. In some social sciences such as sociology, quantitative data are difficult to obtain, either because laboratory conditions are not present or because the issues involved are conceptual but not directly quantifiable. Thus in these cases qualitative methods are preferred.
See also
• Calibration
• Internal standard
• Isotope dilution
• Physical quantity
• Quantitative analysis (chemistry)
• Standard addition
References
1. Cattell, James McKeen; and Farrand, Livingston (1896) "Physical and mental measurements of the students of Columbia University", The Psychological Review, Vol. 3, No. 6 (1896), pp. 618–648; p. 648 quoted in James McKeen Cattell (1860–1944) Psychologist, Publisher, and Editor.
2. Wilks, Samuel Stanley (1961) "Some Aspects of Quantification in Science", Isis, Vol. 52, No. 2 (1961), pp. 135–142; p. 135
3. Hong, Sungook (2004) "History of Science: Building Circuits of Trust", Science, Vol. 305, No. 5690 (10 September 2004), pp. 1569–1570
4. Crosby, Alfred W. (1996) The Measure of Reality: Quantification and Western Society, Cambridge University Press, 1996, p. 201
5. Langs, Robert J. (1987) "Psychoanalysis as an Aristotelian Science—Pathways to Copernicus and a Modern-Day Approach", Contemporary Psychoanalysis, Vol. 23 (1987), pp. 555–576
6. Lynch, Aaron (1999) "Misleading Mix of Religion and Science," Journal of Memetics: Evolutionary Models of Information Transmission, Vol. 3, No. 1 (1999)
7. Bybee, Joan; Perkins, Revere; and Pagliuca, William. (1994) The Evolution of Grammar, Univ. of Chicago Press: ch. 4.
Further reading
Look up quantification in Wiktionary, the free dictionary.
• Crosby, Alfred W. (1996) The Measure of Reality: Quantification and Western Society, 1250–1600. Cambridge University Press.
• Wiese, Heike, 2003. Numbers, Language, and the Human Mind. Cambridge University Press. ISBN 0-521-83182-2.
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Scipione del Ferro
Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician who first discovered a method to solve the depressed cubic equation.
Scipione del Ferro
Born6 February 1465
Bologna
Died5 November 1526(1526-11-05) (aged 61)
Bologna
NationalityItalian
Alma materUniversity of Bologna
Known forSolving the depressed cubic equation
Scientific career
FieldsMathematics
InstitutionsUniversity of Bologna
Life
Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filippa Ferro. His father, Floriano, worked in the paper industry, which owed its existence to the invention of the press in the 1450s and which probably allowed Scipione to access various works during the early stages of his life. He married and had a daughter, who was named Filippa after his mother.
He likely studied at the University of Bologna, where he was appointed a lecturer there in Arithmetic and Geometry in 1496. During his last years, he also undertook commercial work.
Diffusion of his work
There are no surviving scripts from del Ferro. This is in large part due to his resistance to communicating his works. Instead of publishing his ideas, he would only show them to a small, select group of friends and students.
It is suspected that this is due to the practice of mathematicians at the time of publicly challenging one another. When a mathematician accepted another's challenge, each mathematician needed to solve the other's problems. The loser in a challenge often lost funding or his university position. Del Ferro was fearful of being challenged and likely kept his greatest work secret so that he could use it to defend himself in the event of a challenge.
Despite this secrecy, he had a notebook where he recorded all his important discoveries. After he died in 1526, his son-in-law Annibale della Nave inherited this notebook, who was also a mathematician and married to del Ferro's daughter, Filippa. Nave was a former student of del Ferro's, and he replaced del Ferro at the University of Bologna after his death.
In 1543, Gerolamo Cardano and Lodovico Ferrari (one of Cardano's students) travelled to Bologna to meet Nave and learn about his late father-in-law's notebook, where the solution to the depressed cubic equation appeared.
The solution of the cubic equation
Mathematicians from del Ferro's time knew that the general cubic equation could be simplified to one of two cases called the depressed cubic equation, for positive numbers $p$,$q$,$x$:
$x^{3}+px=q,\,$
$x^{3}=px+q.\,$
The term in $x^{2}$ can always be removed by letting $x=x'+a$ for an appropriate constant $a$.
While it is not known today with certainty what method del Ferro used, it is thought that he used the fact that $x={\sqrt {a+{\sqrt {b}}}}+{\sqrt {a-{\sqrt {b}}}}$ solves the equation $x^{2}=(2{\sqrt {a^{2}-b}})x^{0}+2a$ to conjecture that $x={\sqrt[{3}]{a+{\sqrt {b}}}}+{\sqrt[{3}]{a-{\sqrt {b}}}}$ solves $x^{3}=(3{\sqrt[{3}]{a^{2}-b}})x+2a$. This turns out to be true.
Then with the appropriate substitution of parameters, one can derive a solution to the depressed cubic:
${\sqrt[{3}]{{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}+{\sqrt[{3}]{{\frac {q}{2}}-{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}.$
There are conjectures about whether del Ferro worked on a solution to the cubic equation as a result of Luca Pacioli's short tenure at the University of Bologna in 1501–1502. Pacioli had previously declared in Summa de arithmetica that he believed a solution to the equation to be impossible, fueling wide interest in the mathematical community.
It is unknown whether Scipione del Ferro solved both cases or not. However, in 1925, manuscripts were discovered by Bortolotti which contained del Ferro's method and made Bortolotti suspect that del Ferro had solved both cases.
Cardano, in his book Ars Magna (published in 1545) states that it was del Ferro who was the first to solve the cubic equation and that the solution he gives is del Ferro's method.
Other contributions
Del Ferro also made other important contributions to the rationalization of fractions with denominators containing sums of cube roots.
He also investigated geometry problems with a compass set at a fixed angle, but little is known about his work in this area.
References
• O'Connor, John; Robertson, Edmund (1999). "MacTutor History of Mathematics". University of St. Andrews.
• Notable Mathematicians, Online Edition. Gale Group.
• Cardano, Gerolamo (1545). Ars Magna.
• Masotti, Arnaldo. Dictionary of Scientific Biography. pp. 595–597.
• Merino, Orlando (2006). A short history of complex numbers.
• García Venturini, Alejandro. Matemáticos Que Hicieron Historia.
• Stewart, Ian (2004). Galois Theory, Third Edition. Chapman & Hall/CRC Mathematics.
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Irene Sciriha
Irene Sciriha Aquilina is a Maltese mathematician specializing in spectral graph theory and chemical graph theory.[1] A particular topic of her research has been the singular graphs, graphs whose adjacency matrix is a singular matrix, and the nut graphs, singular graphs all of whose nontrivial induced subgraphs are non-singular.[2] She is a professor of mathematics at the University of Malta.[3]
Professor
Irene Sciriha Aquilina
Born
Valletta, Malta
Alma materRoyal University of Malta (BSc)
University of Reading (PhD)
Scientific career
FieldsMathematics
InstitutionsUniversity of Malta
ThesisOn some aspects of graph spectra (1998)
Doctoral advisorsAnthony Hilton
Stanley Fiorini
Education and career
Sciriha studied mathematics at the University of Malta, earning bachelor's and master's degrees[3] as the only woman studying mathematics or physics there at that time.[2] She completed a PhD in 1998 at the University of Reading in England. Her dissertation, On some aspects of graph spectra, was jointly supervised by Anthony Hilton and Stanley Fiorini.[4]
She began teaching at the University of Malta in 1971.[3] She was convenor of European Women in Mathematics from 2000 to 2001.[1]
Recognition
Sciriha is a Fellow of the Institute of Combinatorics and its Applications.[3] One of her students, chemist Martha Borg, won the Turner Prize at the University of Sheffield for a doctoral dissertation co-advised by Sciriha and Patrick W. Fowler.[5]
References
1. Irene Sciriha, European Women in Mathematics, retrieved 2021-05-27; History, European Women in Mathematics, retrieved 2021-05-27
2. "Woman Scientist of the Month: Irene Sciriha Aquilina", European Platform of Women Scientists, 3 February 2020, retrieved 2021-05-27
3. "Prof. Irene Sciriha Aquilina", Staff profiles, University of Malta, retrieved 2021-05-27
4. Irene Sciriha at the Mathematics Genealogy Project
5. "Dr Martha Borg awarded the UK Turner Prize for an outstanding thesis", Newspoint, University of Malta, retrieved 2021-05-27; Jones, Becky Catrin (2021), "A promising early career researcher", Think Magazine, University of Malta, vol. 34, pp. 60–63
External links
• Irene Sciriha publications indexed by Google Scholar
• Home page
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Hilbert's third problem
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Carl Friedrich Gauss,[1] David Hilbert conjectured that this is not always possible. This was confirmed within the year by his student Max Dehn, who proved that the answer in general is "no" by producing a counterexample.[2]
The answer for the analogous question about polygons in 2 dimensions is "yes" and had been known for a long time; this is the Wallace–Bolyai–Gerwien theorem.
Unknown to Hilbert and Dehn, Hilbert's third problem was also proposed independently by Władysław Kretkowski for a math contest of 1882 by the Academy of Arts and Sciences of Kraków, and was solved by Ludwik Antoni Birkenmajer with a different method than Dehn's. Birkenmajer did not publish the result, and the original manuscript containing his solution was rediscovered years later.[3]
History and motivation
The formula for the volume of a pyramid,
${\frac {{\text{base area}}\times {\text{height}}}{3}},$
had been known to Euclid, but all proofs of it involve some form of limiting process or calculus, notably the method of exhaustion or, in more modern form, Cavalieri's principle. Similar formulas in plane geometry can be proven with more elementary means. Gauss regretted this defect in two of his letters to Christian Ludwig Gerling, who proved that two symmetric tetrahedra are equidecomposable.[3]
Gauss' letters were the motivation for Hilbert: is it possible to prove the equality of volume using elementary "cut-and-glue" methods? Because if not, then an elementary proof of Euclid's result is also impossible.
Dehn's answer
Dehn's proof is an instance in which abstract algebra is used to prove an impossibility result in geometry. Other examples are doubling the cube and trisecting the angle.
Two polyhedra are called scissors-congruent if the first can be cut into finitely many polyhedral pieces that can be reassembled to yield the second. Any two scissors-congruent polyhedra have the same volume. Hilbert asks about the converse.
For every polyhedron $P$, Dehn defines a value, now known as the Dehn invariant $\operatorname {D} (P)$, with the property that, if $P$ is cut into polyhedral pieces $P_{1},P_{2},\dots P_{n}$, then
$\operatorname {D} (P)=\operatorname {D} (P_{1})+\operatorname {D} (P_{2})+\cdots +\operatorname {D} (P_{n}).$
In particular, if two polyhedra are scissors-congruent, then they have the same Dehn invariant. He then shows that every cube has Dehn invariant zero while every regular tetrahedron has non-zero Dehn invariant. Therefore, these two shapes cannot be scissors-congruent.
A polyhedron's invariant is defined based on the lengths of its edges and the angles between its faces. If a polyhedron is cut into two, some edges are cut into two, and the corresponding contributions to the Dehn invariants should therefore be additive in the edge lengths. Similarly, if a polyhedron is cut along an edge, the corresponding angle is cut into two. Cutting a polyhedron typically also introduces new edges and angles; their contributions must cancel out. The angles introduced when a cut passes through a face add to $\pi $, and the angles introduced around an edge interior to the polyhedron add to $2\pi $. Therefore, the Dehn invariant is defined in such a way that integer multiples of angles of $\pi $ give a net contribution of zero.
All of the above requirements can be met by defining $\operatorname {D} (P)$ as an element of the tensor product of the real numbers $\mathbb {R} $ (representing lengths of edges) and the quotient space $\mathbb {R} /(\mathbb {Q} \pi )$ (representing angles, with all rational multiples of $\pi $ replaced by zero). For some purposes, this definition can be made using the tensor product of modules over $\mathbb {Z} $ (or equivalently of abelian groups), while other aspects of this topic make use of a vector space structure on the invariants, obtained by considering the two factors $\mathbb {R} $ and $\mathbb {R} /(\mathbb {Q} \pi )$ to be vector spaces over $\mathbb {Q} $ and taking the tensor product of vector spaces over $\mathbb {Q} $. This choice of structure in the definition does not make a difference in whether two Dehn invariants, defined in either way, are equal or unequal.
For any edge $e$ of a polyhedron $P$, let $\ell (e)$ be its length and let $\theta (e)$ denote the dihedral angle of the two faces of $P$ that meet at $e$, measured in radians and considered modulo rational multiples of $\pi $. The Dehn invariant is then defined as
$\operatorname {D} (P)=\sum _{e}\ell (e)\otimes \theta (e)$
where the sum is taken over all edges $e$ of the polyhedron $P$. It is a valuation.
Further information
In light of Dehn's theorem above, one might ask "which polyhedra are scissors-congruent"? Sydler (1965) showed that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant.[4] Børge Jessen later extended Sydler's results to four dimensions. In 1990, Dupont and Sah provided a simpler proof of Sydler's result by reinterpreting it as a theorem about the homology of certain classical groups.[5]
Debrunner showed in 1980 that the Dehn invariant of any polyhedron with which all of three-dimensional space can be tiled periodically is zero.[6]
Unsolved problem in mathematics:
In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?
(more unsolved problems in mathematics)
Jessen also posed the question of whether the analogue of Jessen's results remained true for spherical geometry and hyperbolic geometry. In these geometries, Dehn's method continues to work, and shows that when two polyhedra are scissors-congruent, their Dehn invariants are equal. However, it remains an open problem whether pairs of polyhedra with the same volume and the same Dehn invariant, in these geometries, are always scissors-congruent.[7]
Original question
Hilbert's original question was more complicated: given any two tetrahedra T1 and T2 with equal base area and equal height (and therefore equal volume), is it always possible to find a finite number of tetrahedra, so that when these tetrahedra are glued in some way to T1 and also glued to T2, the resulting polyhedra are scissors-congruent?
Dehn's invariant can be used to yield a negative answer also to this stronger question.
See also
• Hill tetrahedron
• Onorato Nicoletti
References
1. Carl Friedrich Gauss: Werke, vol. 8, pp. 241 and 244
2. Dehn, Max (1901). "Ueber den Rauminhalt". Mathematische Annalen. 55 (3): 465–478. doi:10.1007/BF01448001. S2CID 120068465.
3. Ciesielska, Danuta; Ciesielski, Krzysztof (2018-05-29). "Equidecomposability of Polyhedra: A Solution of Hilbert's Third Problem in Kraków before ICM 1900". The Mathematical Intelligencer. 40 (2): 55–63. doi:10.1007/s00283-017-9748-4. ISSN 0343-6993.
4. Sydler, J.-P. (1965). "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions". Comment. Math. Helv. 40: 43–80. doi:10.1007/bf02564364. S2CID 123317371.
5. Dupont, Johan; Sah, Chih-Han (1990). "Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences". Acta Math. 164 (1–2): 1–27. doi:10.1007/BF02392750.
6. Debrunner, Hans E. (1980). "Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln". Arch. Math. 35 (6): 583–587. doi:10.1007/BF01235384. S2CID 121301319.
7. Dupont, Johan L. (2001), Scissors congruences, group homology and characteristic classes, Nankai Tracts in Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, p. 6, doi:10.1142/9789812810335, ISBN 978-981-02-4507-8, MR 1832859, archived from the original on 2016-04-29.
Further reading
• Benko, D. (2007). "A New Approach to Hilbert's Third Problem". The American Mathematical Monthly. 114 (8): 665–676. doi:10.1080/00029890.2007.11920458. S2CID 7213930.
• Schwartz, Rich (2010). "The Dehn–Sydler Theorem Explained" (PDF).
• Koji, Shiga; Toshikazu Sunada (2005). A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra. American Mathematical Society.
External links
• Proof of Dehn's Theorem at Everything2
• Weisstein, Eric W. "Dehn Invariant". MathWorld.
• Dehn Invariant at Everything2
• Hazewinkel, M. (2001) [1994], "Dehn invariant", Encyclopedia of Mathematics, EMS Press
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Schnyder's theorem
In graph theory, Schnyder's theorem is a characterization of planar graphs in terms of the order dimension of their incidence posets. It is named after Walter Schnyder, who published its proof in 1989.
The incidence poset P(G) of an undirected graph G with vertex set V and edge set E is the partially ordered set of height 2 that has V ∪ E as its elements. In this partial order, there is an order relation x < y when x is a vertex, y is an edge, and x is one of the two endpoints of y.
The order dimension of a partial order is the smallest number of total orderings whose intersection is the given partial order; such a set of orderings is called a realizer of the partial order. Schnyder's theorem states that a graph G is planar if and only if the order dimension of P(G) is at most three.
Extensions
This theorem has been generalized by Brightwell and Trotter (1993, 1997) to a tight bound on the dimension of the height-three partially ordered sets formed analogously from the vertices, edges and faces of a convex polyhedron, or more generally of an embedded planar graph: in both cases, the order dimension of the poset is at most four. However, this result cannot be generalized to higher-dimensional convex polytopes, as there exist four-dimensional polytopes whose face lattices have unbounded order dimension.
Even more generally, for abstract simplicial complexes, the order dimension of the face poset of the complex is at most 1 + d, where d is the minimum dimension of a Euclidean space in which the complex has a geometric realization (Ossona de Mendez 1999, 2002).
Other graphs
As Schnyder observes, the incidence poset of a graph G has order dimension two if and only if the graph is a path or a subgraph of a path. For, in when an incidence poset has order dimension is two, its only possible realizer consists of two total orders that (when restricted to the graph's vertices) are the reverse of each other. Any other two orders would have an intersection that includes an order relation between two vertices, which is not allowed for incidence posets. For these two orders on the vertices, an edge between consecutive vertices can be included in the ordering by placing it immediately following the later of the two edge endpoints, but no other edges can be included.
If a graph can be colored with four colors, then its incidence poset has order dimension at most four (Schnyder 1989).
The incidence poset of a complete graph on n vertices has order dimension $\Theta (\log \log n)$ (Spencer 1971).
References
• Brightwell, G.; Trotter, W. T. (1993), "The order dimension of convex polytopes", SIAM Journal on Discrete Mathematics, 6 (2): 230–245, doi:10.1137/0406018, MR 1215230.
• Brightwell, G.; Trotter, W. T. (1997), "The order dimension of planar maps", SIAM Journal on Discrete Mathematics, 10 (4): 515–528, CiteSeerX 10.1.1.127.1016, doi:10.1137/S0895480192238561, MR 1477654.
• Ossona de Mendez, P. (1999), "Geometric realization of simplicial complexes", in Kratochvil, J. (ed.), Proc. Int. Symp. Graph Drawing (GD 1999), Lecture Notes in Computer Science, vol. 1731, Springer-Verlag, pp. 323–332, doi:10.1007/3-540-46648-7_33, MR 1856785.
• Ossona de Mendez, P. (2002), "Realization of posets" (PDF), Journal of Graph Algorithms and Applications, 6 (1): 149–153, doi:10.7155/jgaa.00048, MR 1898206.
• Schnyder, W. (1989), "Planar graphs and poset dimension", Order, 5 (4): 323–343, doi:10.1007/BF00353652, MR 1010382, S2CID 122785359.
• Spencer, J. (1971), "Minimal scrambling sets of simple orders", Acta Mathematica Academiae Scientiarum Hungaricae, 22 (3–4): 349–353, doi:10.1007/bf01896428, MR 0292722, S2CID 123142998.
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Freyd cover
In the mathematical discipline of category theory, the Freyd cover or scone category is a construction that yields a set-like construction out of a given category. The only requirement is that the original category has a terminal object. The scone category inherits almost any categorical construct the original category has. Scones can be used to generally describe proofs that use logical relations.
Definition
Formally, the scone of a category C with a terminal object 1 is the comma category $1_{\text{Set}}\downarrow \operatorname {Hom} _{C}(1,-)$.[1]
References
1. Freyd cover at the nLab
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Physics
Physics is the natural science of matter, involving the study of matter,[lower-alpha 1] its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.[2] Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves.[lower-alpha 2][3][4][5] A scientist who specializes in the field of physics is called a physicist.
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Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest.[6] Over much of the past two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the Scientific Revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right.[lower-alpha 3] Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences[3] and suggest new avenues of research in these and other academic disciplines such as mathematics and philosophy.
Advances in physics often enable new technologies. For example, advances in the understanding of electromagnetism, solid-state physics, and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons;[3] advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.
History
The word "physics" originates from Ancient Greek: φυσική (ἐπιστήμη), romanized: physikḗ (epistḗmē), meaning "knowledge of nature".[8][9][10]
Ancient astronomy
Astronomy is one of the oldest natural sciences. Early civilizations dating back before 3000 BCE, such as the Sumerians, ancient Egyptians, and the Indus Valley Civilisation, had a predictive knowledge and a basic awareness of the motions of the Sun, Moon, and stars. The stars and planets, believed to represent gods, were often worshipped. While the explanations for the observed positions of the stars were often unscientific and lacking in evidence, these early observations laid the foundation for later astronomy, as the stars were found to traverse great circles across the sky,[6] which could not explain the positions of the planets.
According to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy.[11] Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies,[12] while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey; later Greek astronomers provided names, which are still used today, for most constellations visible from the Northern Hemisphere.[13]
Natural philosophy
Natural philosophy has its origins in Greece during the Archaic period (650 BCE – 480 BCE), when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause.[14] They proposed ideas verified by reason and observation, and many of their hypotheses proved successful in experiment;[15] for example, atomism was found to be correct approximately 2000 years after it was proposed by Leucippus and his pupil Democritus.[16]
Medieval European and Islamic
The Western Roman Empire fell in the fifth century, and this resulted in a decline in intellectual pursuits in the western part of Europe. By contrast, the Eastern Roman Empire (also known as the Byzantine Empire) resisted the attacks from the barbarians, and continued to advance various fields of learning, including physics.[17]
In the sixth century, Isidore of Miletus created an important compilation of Archimedes' works that are copied in the Archimedes Palimpsest.
In sixth-century Europe John Philoponus, a Byzantine scholar, questioned Aristotle's teaching of physics and noted its flaws. He introduced the theory of impetus. Aristotle's physics was not scrutinized until Philoponus appeared; unlike Aristotle, who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics Philoponus wrote:
But this is completely erroneous, and our view may be corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a very small one. And so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other[19]
Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries later,[20] during the Scientific Revolution. Galileo cited Philoponus substantially in his works when arguing that Aristotelian physics was flawed.[21][22] In the 1300s Jean Buridan, a teacher in the faculty of arts at the University of Paris, developed the concept of impetus. It was a step toward the modern ideas of inertia and momentum.[23]
Islamic scholarship inherited Aristotelian physics from the Greeks and during the Islamic Golden Age developed it further, especially placing emphasis on observation and a priori reasoning, developing early forms of the scientific method.
Although Aristotle's principles of physics was criticized, it is important to identify the evidence he based his views off of. When thinking of the history of science and math, it is notable to acknowledge the contributions made by older scientists. Aristotle's science was the backbone of the science we learn in schools today. Aristotle published many biological works including The Parts of Animals, in which he discusses both biological science and natural science as well. It is also integral to mention the role Aristotle had in the progression of physics and metaphysics and how his beliefs and findings are still being taught in science classes to this day. The explanations that Aristotle gives for his findings are also very simple. When thinking of the elements, Aristotle believed that each element (earth, fire, water, air) had its own natural place. Meaning that because of the density of these elements, they will revert back to their own specific place in the atmosphere.[24] So, because of their weights, fire would be at the very top, air right underneath fire, then water, then lastly earth. He also stated that when a small amount of one element enters the natural place of another, the less abundant element will automatically go into its own natural place. For example, if there is a fire on the ground, if you pay attention, the flames go straight up into the air as an attempt to go back into its natural place where it belongs. Aristotle called his metaphysics "first philosophy" and characterized it as the study of "being as being".[25] Aristotle defined the paradigm of motion as a being or entity encompassing different areas in the same body.[25] Meaning that if a person is at a certain location (A) they can move to a new location (B) and still take up the same amount of space. This is involved with Aristotle's belief that motion is a continuum. In terms of matter, Aristotle believed that the change in category (ex. place) and quality (ex. color) of an object is defined as "alteration". But, a change in substance is a change in matter. This is also very close to our idea of matter today.
He also devised his own laws of motion that include 1) heavier objects will fall faster, the speed being proportional to the weight and 2) the speed of the object that is falling depends inversely on the density object it is falling through (ex. density of air).[26] He also stated that, when it comes to violent motion (motion of an object when a force is applied to it by a second object) that the speed that object moves, will only be as fast or strong as the measure of force applied to it.[26] This is also seen in the rules of velocity and force that is taught in physics classes today. These rules are not necessarily what we see in our physics today but, they are very similar. It is evident that these rules were the backbone for other scientists to come revise and edit his beliefs.
The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics (also known as Kitāb al-Manāẓir), written by Ibn al-Haytham, in which he presented the alternative to the ancient Greek idea about vision. In his Treatise on Light as well as in his Kitāb al-Manāẓir, he presented a study of the phenomenon of the camera obscura (his thousand-year-old version of the pinhole camera) and delved further into the way the eye itself works.Using the knowledge of previous scholars, he was able to begin to explain how light enters the eye. He asserted that the light ray is focused, but the actual explanation of how light projected to the back of the eye had to wait until 1604. His Treatise on Light explained the camera obscura, hundreds of years before the modern development of photography.[27]
The seven-volume Book of Optics (Kitab al-Manathir) hugely influenced thinking across disciplines from the theory of visual perception to the nature of perspective in medieval art, in both the East and the West, for more than 600 years. Many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to Johannes Kepler.
The translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build devices that replicated those Ibn al-Haytham had built and understand the way vision works.
Classical
Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics.[28]
Major developments in this period include the replacement of the geocentric model of the Solar System with the heliocentric Copernican model, the laws governing the motion of planetary bodies (determined by Kepler between 1609 and 1619), Galileo's pioneering work on telescopes and observational astronomy in the 16th and 17th Centuries, and Isaac Newton's discovery and unification of the laws of motion and universal gravitation (that would come to bear his name).[29] Newton also developed calculus,[lower-alpha 4] the mathematical study of continuous change, which provided new mathematical methods for solving physical problems.[30]
The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from research efforts during the Industrial Revolution as energy needs increased.[31] The laws comprising classical physics remain very widely used for objects on everyday scales travelling at non-relativistic speeds, since they provide a very close approximation in such situations, and theories such as quantum mechanics and the theory of relativity simplify to their classical equivalents at such scales. Inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century.
Modern
Modern physics began in the early 20th century with the work of Max Planck in quantum theory and Albert Einstein's theory of relativity. Both of these theories came about due to inaccuracies in classical mechanics in certain situations. Classical mechanics predicted that the speed of light depends on the motion of the observer, which could not be resolved with the constant speed predicted by Maxwell's equations of electromagnetism. This discrepancy was corrected by Einstein's theory of special relativity, which replaced classical mechanics for fast-moving bodies and allowed for a constant speed of light.[32] Black-body radiation provided another problem for classical physics, which was corrected when Planck proposed that the excitation of material oscillators is possible only in discrete steps proportional to their frequency. This, along with the photoelectric effect and a complete theory predicting discrete energy levels of electron orbitals, led to the theory of quantum mechanics improving on classical physics at very small scales.[33]
Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger and Paul Dirac.[33] From this early work, and work in related fields, the Standard Model of particle physics was derived.[34] Following the discovery of a particle with properties consistent with the Higgs boson at CERN in 2012,[35] all fundamental particles predicted by the standard model, and no others, appear to exist; however, physics beyond the Standard Model, with theories such as supersymmetry, is an active area of research.[36] Areas of mathematics in general are important to this field, such as the study of probabilities and groups.
Philosophy
In many ways, physics stems from ancient Greek philosophy. From Thales' first attempt to characterize matter, to Democritus' deduction that matter ought to reduce to an invariant state the Ptolemaic astronomy of a crystalline firmament, and Aristotle's book Physics (an early book on physics, which attempted to analyze and define motion from a philosophical point of view), various Greek philosophers advanced their own theories of nature. Physics was known as natural philosophy until the late 18th century.[lower-alpha 5]
By the 19th century, physics was realized as a discipline distinct from philosophy and the other sciences. Physics, as with the rest of science, relies on philosophy of science and its "scientific method" to advance our knowledge of the physical world.[38] The scientific method employs a priori reasoning as well as a posteriori reasoning and the use of Bayesian inference to measure the validity of a given theory.[39]
The development of physics has answered many questions of early philosophers but has also raised new questions. Study of the philosophical issues surrounding physics, the philosophy of physics, involves issues such as the nature of space and time, determinism, and metaphysical outlooks such as empiricism, naturalism and realism.[40]
Many physicists have written about the philosophical implications of their work, for instance Laplace, who championed causal determinism,[41] and Erwin Schrödinger, who wrote on quantum mechanics.[42][43] The mathematical physicist Roger Penrose has been called a Platonist by Stephen Hawking,[44] a view Penrose discusses in his book, The Road to Reality.[45] Hawking referred to himself as an "unashamed reductionist" and took issue with Penrose's views.[46]
Core theories
Physics deals with a wide variety of systems, although certain theories are used by all physicists. Each of these theories was experimentally tested numerous times and found to be an adequate approximation of nature. For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at a speed much less than the speed of light. These theories continue to be areas of active research today. Chaos theory, a remarkable aspect of classical mechanics, was discovered in the 20th century, three centuries after the original formulation of classical mechanics by Newton (1642–1727).
These central theories are important tools for research into more specialized topics, and any physicist, regardless of their specialization, is expected to be literate in them. These include classical mechanics, quantum mechanics, thermodynamics and statistical mechanics, electromagnetism, and special relativity.
Classical
Classical physics includes the traditional branches and topics that were recognized and well-developed before the beginning of the 20th century—classical mechanics, acoustics, optics, thermodynamics, and electromagnetism. Classical mechanics is concerned with bodies acted on by forces and bodies in motion and may be divided into statics (study of the forces on a body or bodies not subject to an acceleration), kinematics (study of motion without regard to its causes), and dynamics (study of motion and the forces that affect it); mechanics may also be divided into solid mechanics and fluid mechanics (known together as continuum mechanics), the latter include such branches as hydrostatics, hydrodynamics, aerodynamics, and pneumatics. Acoustics is the study of how sound is produced, controlled, transmitted and received.[47] Important modern branches of acoustics include ultrasonics, the study of sound waves of very high frequency beyond the range of human hearing; bioacoustics, the physics of animal calls and hearing,[48] and electroacoustics, the manipulation of audible sound waves using electronics.[49]
Optics, the study of light, is concerned not only with visible light but also with infrared and ultraviolet radiation, which exhibit all of the phenomena of visible light except visibility, e.g., reflection, refraction, interference, diffraction, dispersion, and polarization of light. Heat is a form of energy, the internal energy possessed by the particles of which a substance is composed; thermodynamics deals with the relationships between heat and other forms of energy. Electricity and magnetism have been studied as a single branch of physics since the intimate connection between them was discovered in the early 19th century; an electric current gives rise to a magnetic field, and a changing magnetic field induces an electric current. Electrostatics deals with electric charges at rest, electrodynamics with moving charges, and magnetostatics with magnetic poles at rest.
Modern
Modern physics
${\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi _{n}(t)\rangle $
$G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$
Schrödinger and Einstein field equations
Founders
• Max Planck
• Albert Einstein
• Niels Bohr
• Max Born
• Werner Heisenberg
• Erwin Schrödinger
• Pascual Jordan
• Wolfgang Pauli
• Paul Dirac
• Ernest Rutherford
• Louis de Broglie
• Satyendra Nath Bose
Concepts
• Topology
• Space
• Time
• Energy
• Matter
• Work
• Randomness
• Information
• Entropy
• Mind
• Light
• Particle
• Wave
Branches
• Applied
• Experimental
• Theoretical
• Mathematical
• Philosophy of physics
• Quantum mechanics
• Quantum field theory
• Quantum information
• Quantum computation
• Electromagnetism
• Weak interaction
• Electroweak interaction
• Strong interaction
• Atomic
• Particle
• Nuclear
• Atomic, molecular, and optical
• Condensed matter
• Statistical
• Complex systems
• Non-linear dynamics
• Biophysics
• Neurophysics
• Plasma physics
• Special relativity
• General relativity
• Astrophysics
• Cosmology
• Theories of gravitation
• Quantum gravity
• Theory of everything
Scientists
• Witten
• Röntgen
• Becquerel
• Lorentz
• Planck
• Curie
• Wien
• Skłodowska-Curie
• Sommerfeld
• Rutherford
• Soddy
• Onnes
• Einstein
• Wilczek
• Born
• Weyl
• Bohr
• Kramers
• Schrödinger
• de Broglie
• Laue
• Bose
• Compton
• Pauli
• Walton
• Fermi
• van der Waals
• Heisenberg
• Dyson
• Zeeman
• Moseley
• Hilbert
• Gödel
• Jordan
• Dirac
• Wigner
• Hawking
• P. W. Anderson
• Lemaître
• Thomson
• Poincaré
• Wheeler
• Penrose
• Millikan
• Nambu
• von Neumann
• Higgs
• Hahn
• Feynman
• Yang
• Lee
• Lenard
• Salam
• 't Hooft
• Veltman
• Bell
• Gell-Mann
• J. J. Thomson
• Raman
• Bragg
• Bardeen
• Shockley
• Chadwick
• Lawrence
• Zeilinger
• Goudsmit
• Uhlenbeck
Categories
• Modern physics
Classical physics is generally concerned with matter and energy on the normal scale of observation, while much of modern physics is concerned with the behavior of matter and energy under extreme conditions or on a very large or very small scale. For example, atomic and nuclear physics study matter on the smallest scale at which chemical elements can be identified. The physics of elementary particles is on an even smaller scale since it is concerned with the most basic units of matter; this branch of physics is also known as high-energy physics because of the extremely high energies necessary to produce many types of particles in particle accelerators. On this scale, ordinary, commonsensical notions of space, time, matter, and energy are no longer valid.[50]
The two chief theories of modern physics present a different picture of the concepts of space, time, and matter from that presented by classical physics. Classical mechanics approximates nature as continuous, while quantum theory is concerned with the discrete nature of many phenomena at the atomic and subatomic level and with the complementary aspects of particles and waves in the description of such phenomena. The theory of relativity is concerned with the description of phenomena that take place in a frame of reference that is in motion with respect to an observer; the special theory of relativity is concerned with motion in the absence of gravitational fields and the general theory of relativity with motion and its connection with gravitation. Both quantum theory and the theory of relativity find applications in many areas of modern physics.[51]
Fundamental concepts in modern physics
• Causality
• Covariance
• Action
• Physical field
• Symmetry
• Physical interaction
• Statistical ensemble
• Quantum
• Wave
• Particle
Difference
While physics itself aims to discover universal laws, its theories lie in explicit domains of applicability.
Loosely speaking, the laws of classical physics accurately describe systems whose important length scales are greater than the atomic scale and whose motions are much slower than the speed of light. Outside of this domain, observations do not match predictions provided by classical mechanics. Einstein contributed the framework of special relativity, which replaced notions of absolute time and space with spacetime and allowed an accurate description of systems whose components have speeds approaching the speed of light. Planck, Schrödinger, and others introduced quantum mechanics, a probabilistic notion of particles and interactions that allowed an accurate description of atomic and subatomic scales. Later, quantum field theory unified quantum mechanics and special relativity. General relativity allowed for a dynamical, curved spacetime, with which highly massive systems and the large-scale structure of the universe can be well-described. General relativity has not yet been unified with the other fundamental descriptions; several candidate theories of quantum gravity are being developed.
Relation to other fields
Prerequisites
Mathematics provides a compact and exact language used to describe the order in nature. This was noted and advocated by Pythagoras,[52] Plato,[53] Galileo,[54] and Newton. Some theorists, like Hilary Putnam and Penelope Maddy, hold that logical truths, and therefore mathematical reasoning, depend on the empirical world. This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world, which may explain the peculiar relation between these fields.
Physics uses mathematics[55] to organise and formulate experimental results. From those results, precise or estimated solutions are obtained, or quantitative results, from which new predictions can be made and experimentally confirmed or negated. The results from physics experiments are numerical data, with their units of measure and estimates of the errors in the measurements. Technologies based on mathematics, like computation have made computational physics an active area of research.
Ontology is a prerequisite for physics, but not for mathematics. It means physics is ultimately concerned with descriptions of the real world, while mathematics is concerned with abstract patterns, even beyond the real world. Thus physics statements are synthetic, while mathematical statements are analytic. Mathematics contains hypotheses, while physics contains theories. Mathematics statements have to be only logically true, while predictions of physics statements must match observed and experimental data.
The distinction is clear-cut, but not always obvious. For example, mathematical physics is the application of mathematics in physics. Its methods are mathematical, but its subject is physical.[56] The problems in this field start with a "mathematical model of a physical situation" (system) and a "mathematical description of a physical law" that will be applied to that system. Every mathematical statement used for solving has a hard-to-find physical meaning. The final mathematical solution has an easier-to-find meaning, because it is what the solver is looking for.
Pure physics is a branch of fundamental science (also called basic science). Physics is also called "the fundamental science" because all branches of natural science like chemistry, astronomy, geology, and biology are constrained by laws of physics.[57] Similarly, chemistry is often called the central science because of its role in linking the physical sciences. For example, chemistry studies properties, structures, and reactions of matter (chemistry's focus on the molecular and atomic scale distinguishes it from physics). Structures are formed because particles exert electrical forces on each other, properties include physical characteristics of given substances, and reactions are bound by laws of physics, like conservation of energy, mass, and charge. Physics is applied in industries like engineering and medicine.
Application and influence
Applied physics is a general term for physics research, which is intended for a particular use. An applied physics curriculum usually contains a few classes in an applied discipline, like geology or electrical engineering. It usually differs from engineering in that an applied physicist may not be designing something in particular, but rather is using physics or conducting physics research with the aim of developing new technologies or solving a problem.
The approach is similar to that of applied mathematics. Applied physicists use physics in scientific research. For instance, people working on accelerator physics might seek to build better particle detectors for research in theoretical physics.
Physics is used heavily in engineering. For example, statics, a subfield of mechanics, is used in the building of bridges and other static structures. The understanding and use of acoustics results in sound control and better concert halls; similarly, the use of optics creates better optical devices. An understanding of physics makes for more realistic flight simulators, video games, and movies, and is often critical in forensic investigations.
With the standard consensus that the laws of physics are universal and do not change with time, physics can be used to study things that would ordinarily be mired in uncertainty. For example, in the study of the origin of the earth, one can reasonably model earth's mass, temperature, and rate of rotation, as a function of time allowing one to extrapolate forward or backward in time and so predict future or prior events. It also allows for simulations in engineering that drastically speed up the development of a new technology.
But there is also considerable interdisciplinarity, so many other important fields are influenced by physics (e.g., the fields of econophysics and sociophysics).
Research
Scientific method
Physicists use the scientific method to test the validity of a physical theory. By using a methodical approach to compare the implications of a theory with the conclusions drawn from its related experiments and observations, physicists are better able to test the validity of a theory in a logical, unbiased, and repeatable way. To that end, experiments are performed and observations are made in order to determine the validity or invalidity of the theory.[58]
A scientific law is a concise verbal or mathematical statement of a relation that expresses a fundamental principle of some theory, such as Newton's law of universal gravitation.[59]
Theory and experiment
Main articles: Theoretical physics and Experimental physics
Theorists seek to develop mathematical models that both agree with existing experiments and successfully predict future experimental results, while experimentalists devise and perform experiments to test theoretical predictions and explore new phenomena. Although theory and experiment are developed separately, they strongly affect and depend upon each other. Progress in physics frequently comes about when experimental results defy explanation by existing theories, prompting intense focus on applicable modelling, and when new theories generate experimentally testable predictions, which inspire the development of new experiments (and often related equipment).[60]
Physicists who work at the interplay of theory and experiment are called phenomenologists, who study complex phenomena observed in experiment and work to relate them to a fundamental theory.[61]
Theoretical physics has historically taken inspiration from philosophy; electromagnetism was unified this way.[lower-alpha 6] Beyond the known universe, the field of theoretical physics also deals with hypothetical issues,[lower-alpha 7] such as parallel universes, a multiverse, and higher dimensions. Theorists invoke these ideas in hopes of solving particular problems with existing theories; they then explore the consequences of these ideas and work toward making testable predictions.
Experimental physics expands, and is expanded by, engineering and technology. Experimental physicists who are involved in basic research design and perform experiments with equipment such as particle accelerators and lasers, whereas those involved in applied research often work in industry, developing technologies such as magnetic resonance imaging (MRI) and transistors. Feynman has noted that experimentalists may seek areas that have not been explored well by theorists.[62]
Scope and aims
Physics covers a wide range of phenomena, from elementary particles (such as quarks, neutrinos, and electrons) to the largest superclusters of galaxies. Included in these phenomena are the most basic objects composing all other things. Therefore, physics is sometimes called the "fundamental science".[57] Physics aims to describe the various phenomena that occur in nature in terms of simpler phenomena. Thus, physics aims to both connect the things observable to humans to root causes, and then connect these causes together.
For example, the ancient Chinese observed that certain rocks (lodestone and magnetite) were attracted to one another by an invisible force. This effect was later called magnetism, which was first rigorously studied in the 17th century. But even before the Chinese discovered magnetism, the ancient Greeks knew of other objects such as amber, that when rubbed with fur would cause a similar invisible attraction between the two.[63] This was also first studied rigorously in the 17th century and came to be called electricity. Thus, physics had come to understand two observations of nature in terms of some root cause (electricity and magnetism). However, further work in the 19th century revealed that these two forces were just two different aspects of one force—electromagnetism. This process of "unifying" forces continues today, and electromagnetism and the weak nuclear force are now considered to be two aspects of the electroweak interaction. Physics hopes to find an ultimate reason (theory of everything) for why nature is as it is (see section Current research below for more information).[64]
Research fields
Contemporary research in physics can be broadly divided into nuclear and particle physics; condensed matter physics; atomic, molecular, and optical physics; astrophysics; and applied physics. Some physics departments also support physics education research and physics outreach.[65]
Since the 20th century, the individual fields of physics have become increasingly specialised, and today most physicists work in a single field for their entire careers. "Universalists" such as Einstein (1879–1955) and Lev Landau (1908–1968), who worked in multiple fields of physics, are now very rare.[lower-alpha 8]
The major fields of physics, along with their subfields and the theories and concepts they employ, are shown in the following table.
FieldSubfieldsMajor theoriesConcepts
Nuclear and particle physics Nuclear physics, Nuclear astrophysics, Particle physics, Astroparticle physics, Particle physics phenomenology Standard Model, Quantum field theory, Quantum electrodynamics, Quantum chromodynamics, Electroweak theory, Effective field theory, Lattice field theory, Gauge theory, Supersymmetry, Grand Unified Theory, Superstring theory, M-theory, AdS/CFT correspondence Fundamental interaction (gravitational, electromagnetic, weak, strong), Elementary particle, Spin, Antimatter, Spontaneous symmetry breaking, Neutrino oscillation, Seesaw mechanism, Brane, String, Quantum gravity, Theory of everything, Vacuum energy
Atomic, molecular, and optical physics Atomic physics, Molecular physics, Atomic and molecular astrophysics, Chemical physics, Optics, Photonics Quantum optics, Quantum chemistry, Quantum information science Photon, Atom, Molecule, Diffraction, Electromagnetic radiation, Laser, Polarization (waves), Spectral line, Casimir effect
Condensed matter physics Solid-state physics, High-pressure physics, Low-temperature physics, Surface physics, Nanoscale and mesoscopic physics, Polymer physics BCS theory, Bloch's theorem, Density functional theory, Fermi gas, Fermi liquid theory, Many-body theory, Statistical mechanics Phases (gas, liquid, solid), Bose–Einstein condensate, Electrical conduction, Phonon, Magnetism, Self-organization, Semiconductor, superconductor, superfluidity, Spin,
Astrophysics Astronomy, Astrometry, Cosmology, Gravitation physics, High-energy astrophysics, Planetary astrophysics, Plasma physics, Solar physics, Space physics, Stellar astrophysics Big Bang, Cosmic inflation, General relativity, Newton's law of universal gravitation, Lambda-CDM model, Magnetohydrodynamics Black hole, Cosmic background radiation, Cosmic string, Cosmos, Dark energy, Dark matter, Galaxy, Gravity, Gravitational radiation, Gravitational singularity, Planet, Solar System, Star, Supernova, Universe
Applied physics Accelerator physics, Acoustics, Agrophysics, Atmospheric physics, Biophysics, Chemical physics, Communication physics, Econophysics, Engineering physics, Fluid dynamics, Geophysics, Laser physics, Materials physics, Medical physics, Nanotechnology, Optics, Optoelectronics, Photonics, Photovoltaics, Physical chemistry, Physical oceanography, Physics of computation, Plasma physics, Solid-state devices, Quantum chemistry, Quantum electronics, Quantum information science, Vehicle dynamics
Nuclear and particle
Particle physics is the study of the elementary constituents of matter and energy and the interactions between them.[66] In addition, particle physicists design and develop the high-energy accelerators,[67] detectors,[68] and computer programs[69] necessary for this research. The field is also called "high-energy physics" because many elementary particles do not occur naturally but are created only during high-energy collisions of other particles.[70]
Currently, the interactions of elementary particles and fields are described by the Standard Model.[71] The model accounts for the 12 known particles of matter (quarks and leptons) that interact via the strong, weak, and electromagnetic fundamental forces.[71] Dynamics are described in terms of matter particles exchanging gauge bosons (gluons, W and Z bosons, and photons, respectively).[72] The Standard Model also predicts a particle known as the Higgs boson.[71] In July 2012 CERN, the European laboratory for particle physics, announced the detection of a particle consistent with the Higgs boson,[73] an integral part of the Higgs mechanism.
Nuclear physics is the field of physics that studies the constituents and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those in nuclear medicine and magnetic resonance imaging, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology.
Atomic, molecular, and optical
Atomic, molecular, and optical physics (AMO) is the study of matter—matter and light—matter interactions on the scale of single atoms and molecules. The three areas are grouped together because of their interrelationships, the similarity of methods used, and the commonality of their relevant energy scales. All three areas include both classical, semi-classical and quantum treatments; they can treat their subject from a microscopic view (in contrast to a macroscopic view).
Atomic physics studies the electron shells of atoms. Current research focuses on activities in quantum control, cooling and trapping of atoms and ions,[74][75][76] low-temperature collision dynamics and the effects of electron correlation on structure and dynamics. Atomic physics is influenced by the nucleus (see hyperfine splitting), but intra-nuclear phenomena such as fission and fusion are considered part of nuclear physics.
Molecular physics focuses on multi-atomic structures and their internal and external interactions with matter and light. Optical physics is distinct from optics in that it tends to focus not on the control of classical light fields by macroscopic objects but on the fundamental properties of optical fields and their interactions with matter in the microscopic realm.
Condensed matter
Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter.[77][78] In particular, it is concerned with the "condensed" phases that appear whenever the number of particles in a system is extremely large and the interactions between them are strong.[79]
The most familiar examples of condensed phases are solids and liquids, which arise from the bonding by way of the electromagnetic force between atoms.[80] More exotic condensed phases include the superfluid[81] and the Bose–Einstein condensate[82] found in certain atomic systems at very low temperature, the superconducting phase exhibited by conduction electrons in certain materials,[83] and the ferromagnetic and antiferromagnetic phases of spins on atomic lattices.[84]
Condensed matter physics is the largest field of contemporary physics. Historically, condensed matter physics grew out of solid-state physics, which is now considered one of its main subfields.[85] The term condensed matter physics was apparently coined by Philip Anderson when he renamed his research group—previously solid-state theory—in 1967.[86] In 1978, the Division of Solid State Physics of the American Physical Society was renamed as the Division of Condensed Matter Physics.[85] Condensed matter physics has a large overlap with chemistry, materials science, nanotechnology and engineering.[79]
Astrophysics
Astrophysics and astronomy are the application of the theories and methods of physics to the study of stellar structure, stellar evolution, the origin of the Solar System, and related problems of cosmology. Because astrophysics is a broad subject, astrophysicists typically apply many disciplines of physics, including mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.[87]
The discovery by Karl Jansky in 1931 that radio signals were emitted by celestial bodies initiated the science of radio astronomy. Most recently, the frontiers of astronomy have been expanded by space exploration. Perturbations and interference from the earth's atmosphere make space-based observations necessary for infrared, ultraviolet, gamma-ray, and X-ray astronomy.
Physical cosmology is the study of the formation and evolution of the universe on its largest scales. Albert Einstein's theory of relativity plays a central role in all modern cosmological theories. In the early 20th century, Hubble's discovery that the universe is expanding, as shown by the Hubble diagram, prompted rival explanations known as the steady state universe and the Big Bang.
The Big Bang was confirmed by the success of Big Bang nucleosynthesis and the discovery of the cosmic microwave background in 1964. The Big Bang model rests on two theoretical pillars: Albert Einstein's general relativity and the cosmological principle. Cosmologists have recently established the ΛCDM model of the evolution of the universe, which includes cosmic inflation, dark energy, and dark matter.
Numerous possibilities and discoveries are anticipated to emerge from new data from the Fermi Gamma-ray Space Telescope over the upcoming decade and vastly revise or clarify existing models of the universe.[88][89] In particular, the potential for a tremendous discovery surrounding dark matter is possible over the next several years.[90] Fermi will search for evidence that dark matter is composed of weakly interacting massive particles, complementing similar experiments with the Large Hadron Collider and other underground detectors.
IBEX is already yielding new astrophysical discoveries: "No one knows what is creating the ENA (energetic neutral atoms) ribbon" along the termination shock of the solar wind, "but everyone agrees that it means the textbook picture of the heliosphere—in which the Solar System's enveloping pocket filled with the solar wind's charged particles is plowing through the onrushing 'galactic wind' of the interstellar medium in the shape of a comet—is wrong."[91]
Current research
Research in physics is continually progressing on a large number of fronts.
In condensed matter physics, an important unsolved theoretical problem is that of high-temperature superconductivity.[92] Many condensed matter experiments are aiming to fabricate workable spintronics and quantum computers.[79][93]
In particle physics, the first pieces of experimental evidence for physics beyond the Standard Model have begun to appear. Foremost among these are indications that neutrinos have non-zero mass. These experimental results appear to have solved the long-standing solar neutrino problem, and the physics of massive neutrinos remains an area of active theoretical and experimental research. The Large Hadron Collider has already found the Higgs boson, but future research aims to prove or disprove the supersymmetry, which extends the Standard Model of particle physics. Research on the nature of the major mysteries of dark matter and dark energy is also currently ongoing.[94]
Although much progress has been made in high-energy, quantum, and astronomical physics, many everyday phenomena involving complexity,[95] chaos,[96] or turbulence[97] are still poorly understood. Complex problems that seem like they could be solved by a clever application of dynamics and mechanics remain unsolved; examples include the formation of sandpiles, nodes in trickling water, the shape of water droplets, mechanisms of surface tension catastrophes, and self-sorting in shaken heterogeneous collections.[lower-alpha 9][98]
These complex phenomena have received growing attention since the 1970s for several reasons, including the availability of modern mathematical methods and computers, which enabled complex systems to be modeled in new ways. Complex physics has become part of increasingly interdisciplinary research, as exemplified by the study of turbulence in aerodynamics and the observation of pattern formation in biological systems. In the 1932 Annual Review of Fluid Mechanics, Horace Lamb said:[99]
I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.
Education
This section is an excerpt from Physics education.[edit]
Physics education or physics teaching refers to the education methods currently used to teach physics. The occupation is called physics educator or physics teacher. Physics education research refers to an area of pedagogical research that seeks to improve those methods. Historically, physics has been taught at the high school and college level primarily by the lecture method together with laboratory exercises aimed at verifying concepts taught in the lectures. These concepts are better understood when lectures are accompanied with demonstration, hand-on experiments, and questions that require students to ponder what will happen in an experiment and why. Students who participate in active learning for example with hands-on experiments learn through self-discovery. By trial and error they learn to change their preconceptions about phenomena in physics and discover the underlying concepts. Physics education is part of the broader area of science education.
Career
This section is an excerpt from Physicist.[edit]
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.[100][101] Physicists generally are interested in the root or ultimate causes of phenomena, and usually frame their understanding in mathematical terms. They work across a wide range of research fields, spanning all length scales: from sub-atomic and particle physics, through biological physics, to cosmological length scales encompassing the universe as a whole. The field generally includes two types of physicists: experimental physicists who specialize in the observation of natural phenomena and the development and analysis of experiments, and theoretical physicists who specialize in mathematical modeling of physical systems to rationalize, explain and predict natural phenomena.[100]
Physicists can apply their knowledge towards solving practical problems or to developing new technologies (also known as applied physics or engineering physics).[102][103][104]
See also
• Earth science – Fields of natural science related to Earth
• Neurophysics – branch of biophysics dealing with the development and use of physical methods to gain information about the nervous systemPages displaying wikidata descriptions as a fallback
• Psychophysics – Branch of knowledge relating physical stimuli and psychological perception
• Quantum physics – Description of physical properties at the atomic and subatomic scalePages displaying short descriptions of redirect targets
• Relationship between mathematics and physics – Study of how mathematics and physics relate to each other
• Science tourism – Travel to notable science locations
Lists
• List of important publications in physics
• List of physicists
• Lists of physics equations
Notes
1. At the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept.[1]
2. The term "universe" is defined as everything that physically exists: the entirety of space and time, all forms of matter, energy and momentum, and the physical laws and constants that govern them. However, the term "universe" may also be used in slightly different contextual senses, denoting concepts such as the cosmos or the philosophical world.
3. Francis Bacon's 1620 Novum Organum was critical in the development of scientific method.[7]
4. Calculus was independently developed at around the same time by Gottfried Wilhelm Leibniz; while Leibniz was the first to publish his work and develop much of the notation used for calculus today, Newton was the first to develop calculus and apply it to physical problems. See also Leibniz–Newton calculus controversy
5. Noll notes that some universities still use this title.[37]
6. See, for example, the influence of Kant and Ritter on Ørsted.
7. Concepts which are denoted hypothetical can change with time. For example, the atom of nineteenth-century physics was denigrated by some, including Ernst Mach's critique of Ludwig Boltzmann's formulation of statistical mechanics. By the end of World War II, the atom was no longer deemed hypothetical.
8. Yet, universalism is encouraged in the culture of physics. For example, the World Wide Web, which was innovated at CERN by Tim Berners-Lee, was created in service to the computer infrastructure of CERN, and was/is intended for use by physicists worldwide. The same might be said for arXiv.org
9. See the work of Ilya Prigogine, on 'systems far from equilibrium', and others.
References
1. Feynman, Leighton & Sands 1963, p. I-2 "If, in some cataclysm, all [] scientific knowledge were to be destroyed [save] one sentence [...] what statement would contain the most information in the fewest words? I believe it is [...] that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another ..."
2. Maxwell 1878, p. 9 "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events."
3. Young & Freedman 2014, p. 1 "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."
4. Young & Freedman 2014, p. 2 "Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena."
5. Holzner 2006, p. 7 "Physics is the study of your world and the world and universe around you."
6. Krupp 2003
7. Cajori 1917, pp. 48–49
8. "physics". Online Etymology Dictionary. Archived from the original on 24 December 2016. Retrieved 1 November 2016.
9. "physic". Online Etymology Dictionary. Archived from the original on 24 December 2016. Retrieved 1 November 2016.
10. φύσις, φυσική, ἐπιστήμη. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project
11. Aaboe 1991
12. Clagett 1995
13. Thurston 1994
14. Singer 2008, p. 35
15. Lloyd 1970, pp. 108–109
16. Gill, N.S. "Atomism – Pre-Socratic Philosophy of Atomism". About Education. Archived from the original on 10 July 2014. Retrieved 1 April 2014.
17. Lindberg 1992, p. 363.
18. Smith 2001, Book I [6.85], [6.86], p. 379; Book II, [3.80], p. 453.
19. "John Philoponus, Commentary on Aristotle's Physics". Archived from the original on 11 January 2016. Retrieved 15 April 2018.
20. Galileo (1638). Two New Sciences. in order to better understand just how conclusive Aristotle's demonstration is, we may, in my opinion, deny both of his assumptions. And as to the first, I greatly doubt that Aristotle ever tested by experiment whether it be true that two stones, one weighing ten times as much as the other, if allowed to fall, at the same instant, from a height of, say, 100 cubits, would so differ in speed that when the heavier had reached the ground, the other would not have fallen more than 10 cubits.
Simp. – His language would seem to indicate that he had tried the experiment, because he says: We see the heavier; now the word see shows that he had made the experiment.
Sagr. – But I, Simplicio, who have made the test can assure[107] you that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound, provided both are dropped from a height of 200 cubits.
21. Lindberg 1992, p. 162.
22. "John Philoponus". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2018. Archived from the original on 22 April 2018. Retrieved 11 April 2018.
23. "John Buridan". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2018. Archived from the original on 22 April 2018. Retrieved 11 April 2018.
24. tbcaldwe (14 October 2012). "Natural Philosophy: Aristotle | Physics 139". Retrieved 17 December 2022.
25. "Aristotle – Physics and metaphysics". www.britannica.com. Retrieved 17 December 2022.
26. "Aristotle". galileoandeinstein.phys.virginia.edu. Retrieved 17 December 2022.
27. Howard & Rogers 1995, pp. 6–7
28. Ben-Chaim 2004
29. Guicciardini 1999
30. Allen 1997
31. "The Industrial Revolution". Schoolscience.org, Institute of Physics. Archived from the original on 7 April 2014. Retrieved 1 April 2014.
32. O'Connor & Robertson 1996a
33. O'Connor & Robertson 1996b
34. "The Standard Model". DONUT. Fermilab. 29 June 2001. Archived from the original on 31 May 2014. Retrieved 1 April 2014.
35. Cho 2012
36. Womersley, J. (February 2005). "Beyond the Standard Model" (PDF). Symmetry. Vol. 2, no. 1. pp. 22–25. Archived (PDF) from the original on 24 September 2015.
37. Noll, Walter (23 June 2006). "On the Past and Future of Natural Philosophy" (PDF). Journal of Elasticity. 84 (1): 1–11. doi:10.1007/s10659-006-9068-y. S2CID 121957320. Archived (PDF) from the original on 18 April 2016.
38. Rosenberg 2006, Chapter 1
39. Godfrey-Smith 2003, Chapter 14: "Bayesianism and Modern Theories of Evidence"
40. Godfrey-Smith 2003, Chapter 15: "Empiricism, Naturalism, and Scientific Realism?"
41. Laplace 1951
42. Schrödinger 1983
43. Schrödinger 1995
44. Hawking & Penrose 1996, p. 4 "I think that Roger is a Platonist at heart but he must answer for himself."
45. Penrose 2004
46. Penrose et al. 1997
47. "acoustics". Encyclopædia Britannica. Archived from the original on 18 June 2013. Retrieved 14 June 2013.
48. "Bioacoustics – the International Journal of Animal Sound and its Recording". Taylor & Francis. Archived from the original on 5 September 2012. Retrieved 31 July 2012.
49. "Acoustics and You (A Career in Acoustics?)". Acoustical Society of America. Archived from the original on 4 September 2015. Retrieved 21 May 2013.
50. Tipler & Llewellyn 2003, pp. 269, 477, 561
51. Tipler & Llewellyn 2003, pp. 1–4, 115, 185–187
52. Dijksterhuis 1986
53. Mastin 2010 harvnb error: no target: CITEREFMastin2010 (help) "Although usually remembered today as a philosopher, Plato was also one of ancient Greece's most important patrons of mathematics. Inspired by Pythagoras, he founded his Academy in Athens in 387 BC, where he stressed mathematics as a way of understanding more about reality. In particular, he was convinced that geometry was the key to unlocking the secrets of the universe. The sign above the Academy entrance read: 'Let no-one ignorant of geometry enter here.'"
54. Toraldo Di Francia 1976, p. 10 'Philosophy is written in that great book which ever lies before our eyes. I mean the universe, but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures, without whose help it is humanly impossible to comprehend a single word of it, and without which one wanders in vain through a dark labyrinth.' – Galileo (1623), The Assayer"
55. "Applications of Mathematics to the Sciences". 25 January 2000. Archived from the original on 10 May 2015. Retrieved 30 January 2012.
56. "Journal of Mathematical Physics". Archived from the original on 18 August 2014. Retrieved 31 March 2014. [Journal of Mathematical Physics] purpose is the publication of papers in mathematical physics—that is, the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.
57. The Feynman Lectures on Physics Vol. I Ch. 3: The Relation of Physics to Other Sciences; see also reductionism and special sciences
58. Ellis, G.; Silk, J. (16 December 2014). "Scientific method: Defend the integrity of physics". Nature. 516 (7531): 321–323. Bibcode:2014Natur.516..321E. doi:10.1038/516321a. PMID 25519115.
59. Honderich 1995, pp. 474–476
60. "Has theoretical physics moved too far away from experiments? Is the field entering a crisis and, if so, what should we do about it?". Perimeter Institute for Theoretical Physics. June 2015. Archived from the original on 21 April 2016.
61. "Phenomenology". Max Planck Institute for Physics. Archived from the original on 7 March 2016. Retrieved 22 October 2016.
62. Feynman 1965, p. 157 "In fact experimenters have a certain individual character. They ... very often do their experiments in a region in which people know the theorist has not made any guesses."
63. Stewart, J. (2001). Intermediate Electromagnetic Theory. World Scientific. p. 50. ISBN 978-981-02-4471-2.
64. Weinberg, S. (1993). Dreams of a Final Theory: The Search for the Fundamental Laws of Nature. Hutchinson Radius. ISBN 978-0-09-177395-3.
65. Redish, E. "Science and Physics Education Homepages". University of Maryland Physics Education Research Group. Archived from the original on 28 July 2016.
66. "Division of Particles & Fields". American Physical Society. Archived from the original on 29 August 2016. Retrieved 18 October 2012.
67. Halpern 2010
68. Grupen 1999
69. Walsh 2012
70. "High Energy Particle Physics Group". Institute of Physics. Archived from the original on 29 May 2019. Retrieved 18 October 2012.
71. Oerter 2006
72. Gribbin, Gribbin & Gribbin 1998
73. "CERN experiments observe particle consistent with long-sought Higgs boson". CERN. 4 July 2012. Archived from the original on 14 November 2012. Retrieved 18 October 2012.
74. "Atomic, Molecular, and Optical Physics". MIT Department of Physics. Archived from the original on 27 February 2014. Retrieved 21 February 2014.
75. "Korea University, Physics AMO Group". Archived from the original on 1 March 2014. Retrieved 21 February 2014.
76. "Aarhus Universitet, AMO Group". Archived from the original on 7 March 2014. Retrieved 21 February 2014.
77. Taylor & Heinonen 2002
78. Girvin, Steven M.; Yang, Kun (28 February 2019). Modern Condensed Matter Physics. Cambridge University Press. ISBN 978-1-108-57347-4. Archived from the original on 25 February 2021. Retrieved 23 August 2020.
79. Cohen 2008
80. Moore 2011, pp. 255–258
81. Leggett 1999
82. Levy 2001
83. Stajic, Coontz & Osborne 2011
84. Mattis 2006
85. "History of Condensed Matter Physics". American Physical Society. Archived from the original on 12 September 2011. Retrieved 31 March 2014.
86. "Philip Anderson". Princeton University, Department of Physics. Archived from the original on 8 October 2011. Retrieved 15 October 2012.
87. "BS in Astrophysics". University of Hawaii at Manoa. Archived from the original on 4 April 2016. Retrieved 14 October 2016.
88. "NASA – Q&A on the GLAST Mission". Nasa: Fermi Gamma-ray Space Telescope. NASA. 28 August 2008. Archived from the original on 25 April 2009. Retrieved 29 April 2009.
89. See also Nasa – Fermi Science Archived 3 April 2010 at the Wayback Machine and NASA – Scientists Predict Major Discoveries for GLAST Archived 2 March 2009 at the Wayback Machine.
90. "Dark Matter". NASA. 28 August 2008. Archived from the original on 13 January 2012. Retrieved 30 January 2012.
91. Kerr 2009
92. Leggett, A.J. (2006). "What DO we know about high Tc?" (PDF). Nature Physics. 2 (3): 134–136. Bibcode:2006NatPh...2..134L. doi:10.1038/nphys254. S2CID 122055331. Archived from the original (PDF) on 10 June 2010.
93. Wolf, S.A.; Chtchelkanova, A.Y.; Treger, D.M. (2006). "Spintronics – A retrospective and perspective" (PDF). IBM Journal of Research and Development. 50: 101–110. doi:10.1147/rd.501.0101. S2CID 41178069. Archived from the original (PDF) on 24 September 2020.
94. Gibney, E. (2015). "LHC 2.0: A new view of the Universe". Nature. 519 (7542): 142–143. Bibcode:2015Natur.519..142G. doi:10.1038/519142a. PMID 25762263.
95. National Research Council & Committee on Technology for Future Naval Forces 1997, p. 161
96. Kellert 1993, p. 32
97. Eames, I.; Flor, J.B. (2011). "New developments in understanding interfacial processes in turbulent flows". Philosophical Transactions of the Royal Society A. 369 (1937): 702–705. Bibcode:2011RSPTA.369..702E. doi:10.1098/rsta.2010.0332. PMID 21242127. Richard Feynman said that 'Turbulence is the most important unsolved problem of classical physics'
98. National Research Council (2007). "What happens far from equilibrium and why". Condensed-Matter and Materials Physics: the science of the world around us. pp. 91–110. doi:10.17226/11967. ISBN 978-0-309-10969-7. Archived from the original on 4 November 2016.
– Jaeger, Heinrich M.; Liu, Andrea J. (2010). "Far-From-Equilibrium Physics: An Overview". arXiv:1009.4874 [cond-mat.soft].
99. Goldstein 1969
100. Rosen, Joe (2009). Encyclopedia of Physics. Infobase Publishing. p. 247.
101. "physicist". Merriam-Webster Dictionary. "a scientist who studies or is a specialist in physics"
102. "Industrial Physicists: Primarily specializing in Physics" (PDF). American Institute for Physics. October 2016.
103. "Industrial Physicists: Primarily specializing in Engineering" (PDF). American Institute for Physics. October 2016.
104. "Industrial Physicists: Primarily specializing outside of STEM sectors" (PDF). American Institute for Physics. October 2016.
Sources
• Aaboe, A. (1991). "Mesopotamian Mathematics, Astronomy, and Astrology". The Cambridge Ancient History. Vol. III (2nd ed.). Cambridge University Press. ISBN 978-0-521-22717-9.
• Abazov, V.; et al. (DØ Collaboration) (12 June 2007). "Direct observation of the strange 'b' baryon $\Xi _{b}^{-}$". Physical Review Letters. 99 (5): 052001. arXiv:0706.1690v2. Bibcode:2007PhRvL..99e2001A. doi:10.1103/PhysRevLett.99.052001. PMID 17930744. S2CID 11568965.
• Allen, D. (10 April 1997). "Calculus". Texas A&M University. Retrieved 1 April 2014.
• Ben-Chaim, M. (2004). Experimental Philosophy and the Birth of Empirical Science: Boyle, Locke and Newton. Aldershot: Ashgate Publishing. ISBN 978-0-7546-4091-2. OCLC 53887772.
• Cajori, Florian (1917). A History of Physics in Its Elementary Branches: Including the Evolution of Physical Laboratories. Macmillan.
• Cho, A. (13 July 2012). "Higgs Boson Makes Its Debut After Decades-Long Search". Science. 337 (6091): 141–143. Bibcode:2012Sci...337..141C. doi:10.1126/science.337.6091.141. PMID 22798574.
• Clagett, M. (1995). Ancient Egyptian Science. Vol. 2. Philadelphia: American Philosophical Society.
• Cohen, M.L. (2008). "Fifty Years of Condensed Matter Physics". Physical Review Letters. 101 (5): 25001–25006. Bibcode:2008PhRvL.101y0001C. doi:10.1103/PhysRevLett.101.250001. PMID 19113681.
• Dijksterhuis, E.J. (1986). The mechanization of the world picture: Pythagoras to Newton. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-08403-9. Archived from the original on 5 August 2011.
• Feynman, R.P.; Leighton, R.B.; Sands, M. (1963). The Feynman Lectures on Physics. Vol. 1. ISBN 978-0-201-02116-5.
• Feynman, R.P. (1965). The Character of Physical Law. ISBN 978-0-262-56003-0.
• Godfrey-Smith, P. (2003). Theory and Reality: An Introduction to the Philosophy of Science. ISBN 978-0-226-30063-4.
• Goldstein, S. (1969). "Fluid Mechanics in the First Half of this Century". Annual Review of Fluid Mechanics. 1 (1): 1–28. Bibcode:1969AnRFM...1....1G. doi:10.1146/annurev.fl.01.010169.000245.
• Gribbin, J.R.; Gribbin, M.; Gribbin, J. (1998). Q is for Quantum: An Encyclopedia of Particle Physics. Free Press. Bibcode:1999qqep.book.....G. ISBN 978-0-684-85578-3.
• Grupen, Klaus (10 July 1999). "Instrumentation in Elementary Particle Physics: VIII ICFA School". AIP Conference Proceedings. 536: 3–34. arXiv:physics/9906063. Bibcode:2000AIPC..536....3G. doi:10.1063/1.1361756. S2CID 119476972.
• Guicciardini, N. (1999). Reading the Principia: The Debate on Newton's Methods for Natural Philosophy from 1687 to 1736. New York: Cambridge University Press. ISBN 978-0521640664.
• Halpern, P. (2010). Collider: The Search for the World's Smallest Particles. John Wiley & Sons. ISBN 978-0-470-64391-4.
• Hawking, S.; Penrose, R. (1996). The Nature of Space and Time. ISBN 978-0-691-05084-3.
• Holzner, S. (2006). Physics for Dummies. John Wiley & Sons. Bibcode:2005pfd..book.....H. ISBN 978-0-470-61841-7. Physics is the study of your world and the world and universe around you.
• Honderich, T., ed. (1995). The Oxford Companion to Philosophy (1 ed.). Oxford: Oxford University Press. pp. 474–476. ISBN 978-0-19-866132-0.
• Howard, Ian; Rogers, Brian (1995). Binocular Vision and Stereopsis. Oxford University Press. ISBN 978-0-19-508476-4.
• Kellert, S.H. (1993). In the Wake of Chaos: Unpredictable Order in Dynamical Systems. University of Chicago Press. ISBN 978-0-226-42976-2.
• Kerr, R.A. (16 October 2009). "Tying Up the Solar System With a Ribbon of Charged Particles". Science. 326 (5951): 350–351. doi:10.1126/science.326_350a. PMID 19833930.
• Krupp, E.C. (2003). Echoes of the Ancient Skies: The Astronomy of Lost Civilizations. Dover Publications. ISBN 978-0-486-42882-6. Retrieved 31 March 2014.
• Laplace, P.S. (1951). A Philosophical Essay on Probabilities. Translated from the 6th French edition by Truscott, F.W. and Emory, F.L. New York: Dover Publications.
• Leggett, A.J. (1999). "Superfluidity". Reviews of Modern Physics. 71 (2): S318–S323. Bibcode:1999RvMPS..71..318L. doi:10.1103/RevModPhys.71.S318.
• Levy, Barbara G. (December 2001). "Cornell, Ketterle, and Wieman Share Nobel Prize for Bose-Einstein Condensates". Physics Today. 54 (12): 14. Bibcode:2001PhT....54l..14L. doi:10.1063/1.1445529.
• Lindberg, David (1992). The Beginnings of Western Science. University of Chicago Press.
• Lloyd, G.E.R. (1970). Early Greek Science: Thales to Aristotle. London; New York: Chatto and Windus; W. W. Norton & Company. ISBN 978-0-393-00583-7.
• Mattis, D.C. (2006). The Theory of Magnetism Made Simple. World Scientific. ISBN 978-981-238-579-6.
• Maxwell, J.C. (1878). Matter and Motion. D. Van Nostrand. ISBN 978-0-486-66895-6. matter and motion.
• Moore, J.T. (2011). Chemistry For Dummies (2 ed.). John Wiley & Sons. ISBN 978-1-118-00730-3.
• National Research Council; Committee on Technology for Future Naval Forces (1997). Technology for the United States Navy and Marine Corps, 2000–2035 Becoming a 21st-Century Force: Volume 9: Modeling and Simulation. Washington, DC: The National Academies Press. ISBN 978-0-309-05928-2.
• O'Connor, J.J.; Robertson, E.F. (February 1996a). "Special Relativity". MacTutor History of Mathematics archive. University of St Andrews. Retrieved 1 April 2014.
• O'Connor, J.J.; Robertson, E.F. (May 1996b). "A History of Quantum Mechanics". MacTutor History of Mathematics archive. University of St Andrews. Retrieved 1 April 2014.
• Oerter, R. (2006). The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics. Pi Press. ISBN 978-0-13-236678-6.
• Penrose, R.; Shimony, A.; Cartwright, N.; Hawking, S. (1997). The Large, the Small and the Human Mind. Cambridge University Press. ISBN 978-0-521-78572-3.
• Penrose, R. (2004). The Road to Reality. ISBN 978-0-679-45443-4.
• Rosenberg, Alex (2006). Philosophy of Science. Routledge. ISBN 978-0-415-34317-6.
• Schrödinger, E. (1983). My View of the World. Ox Bow Press. ISBN 978-0-918024-30-5.
• Schrödinger, E. (1995). The Interpretation of Quantum Mechanics. Ox Bow Press. ISBN 978-1-881987-09-3.
• Singer, C. (2008). A Short History of Science to the 19th Century. Streeter Press.
• Smith, A. Mark (2001). Alhacen's Theory of Visual Perception: A Critical Edition, with English Translation and Commentary, of the First Three Books of Alhacen's De Aspectibus, the Medieval Latin Version of Ibn al-Haytham's Kitāb al-Manāẓir, 2 vols. Transactions of the American Philosophical Society. Vol. 91. Philadelphia: American Philosophical Society. ISBN 978-0-87169-914-5. OCLC 47168716.
• Smith, A. Mark (2001a). "Alhacen's Theory of Visual Perception: A Critical Edition, with English Translation and Commentary, of the First Three Books of Alhacen's "De aspectibus", the Medieval Latin Version of Ibn al-Haytham's "Kitāb al-Manāẓir": Volume One". Transactions of the American Philosophical Society. 91 (4): i–clxxxi, 1–337. doi:10.2307/3657358. JSTOR 3657358.
• Smith, A. Mark (2001b). "Alhacen's Theory of Visual Perception: A Critical Edition, with English Translation and Commentary, of the First Three Books of Alhacen's "De aspectibus", the Medieval Latin Version of Ibn al-Haytham's "Kitāb al-Manāẓir": Volume Two". Transactions of the American Philosophical Society. 91 (5): 339–819. doi:10.2307/3657357. JSTOR 3657357.
• Stajic, Jelena; Coontz, R.; Osborne, I. (8 April 2011). "Happy 100th, Superconductivity!". Science. 332 (6026): 189. Bibcode:2011Sci...332..189S. doi:10.1126/science.332.6026.189. PMID 21474747.
• Taylor, P.L.; Heinonen, O. (2002). A Quantum Approach to Condensed Matter Physics. Cambridge University Press. ISBN 978-0-521-77827-5.
• Thurston, H. (1994). Early Astronomy. Springer.
• Tipler, Paul; Llewellyn, Ralph (2003). Modern Physics. W. H. Freeman. ISBN 978-0-7167-4345-3.
• Toraldo Di Francia, G. (1976). The Investigation of the Physical World. ISBN 978-0-521-29925-1.
• Walsh, K.M. (1 June 2012). "Plotting the Future for Computing in High-Energy and Nuclear Physics". Brookhaven National Laboratory. Archived from the original on 29 July 2016. Retrieved 18 October 2012.
• Young, H.D.; Freedman, R.A. (2014). Sears and Zemansky's University Physics with Modern Physics Technology Update (13th ed.). Pearson Education. ISBN 978-1-292-02063-1.
External links
• Physics at Quanta Magazine
• Usenet Physics FAQ – FAQ compiled by sci.physics and other physics newsgroups
• Website of the Nobel Prize in physics – Award for outstanding contributions to the subject
• World of Physics – Online encyclopedic dictionary of physics
• Nature Physics – Academic journal
• Physics – Online magazine by the American Physical Society
• Physics/Publications at Curlie – Directory of physics related media
• The Vega Science Trust – Science videos, including physics
• HyperPhysics website – Physics and astronomy mind-map from Georgia State University
• Physics at MIT OpenCourseWare – Online course material from Massachusetts Institute of Technology
• The Feynman Lectures on Physics
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Wikipedia
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Scorer's function
In mathematics, the Scorer's functions are special functions studied by Scorer (1950) and denoted Gi(x) and Hi(x).
Hi(x) and -Gi(x) solve the equation
$y''(x)-x\ y(x)={\frac {1}{\pi }}$
and are given by
$\mathrm {Gi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\sin \left({\frac {t^{3}}{3}}+xt\right)\,dt,$
$\mathrm {Hi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\exp \left(-{\frac {t^{3}}{3}}+xt\right)\,dt.$
The Scorer's functions can also be defined in terms of Airy functions:
${\begin{aligned}\mathrm {Gi} (x)&{}=\mathrm {Bi} (x)\int _{x}^{\infty }\mathrm {Ai} (t)\,dt+\mathrm {Ai} (x)\int _{0}^{x}\mathrm {Bi} (t)\,dt,\\\mathrm {Hi} (x)&{}=\mathrm {Bi} (x)\int _{-\infty }^{x}\mathrm {Ai} (t)\,dt-\mathrm {Ai} (x)\int _{-\infty }^{x}\mathrm {Bi} (t)\,dt.\end{aligned}}$
• Plot of the Scorer function Gi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the derivative of the Scorer function Hi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the derivative of the Scorer function Gi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the Scorer function Hi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
References
• Olver, F. W. J. (2010), "Scorer functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Scorer, R. S. (1950), "Numerical evaluation of integrals of the form $I=\int _{x_{1}}^{x_{2}}f(x)e^{i\phi (x)}dx$ and the tabulation of the function ${\rm {Gi}}(z)={\frac {1}{\pi }}\int _{0}^{\infty }{\rm {sin}}\left(uz+{\frac {1}{3}}u^{3}\right)du$", The Quarterly Journal of Mechanics and Applied Mathematics, 3: 107–112, doi:10.1093/qjmam/3.1.107, ISSN 0033-5614, MR 0037604
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Wikipedia
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Histogram
A histogram is an approximate representation of the distribution of numerical data. The term was first introduced by Karl Pearson.[1] To construct a histogram, the first step is to "bin" (or "bucket") the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The bins (intervals) must be adjacent and are often (but not required to be) of equal size.[2]
Histogram
One of the Seven Basic Tools of Quality
First described byKarl Pearson
PurposeTo roughly assess the probability distribution of a given variable by depicting the frequencies of observations occurring in certain ranges of values.
If the bins are of equal size, a bar is drawn over the bin with height proportional to the frequency—the number of cases in each bin. A histogram may also be normalized to display "relative" frequencies showing the proportion of cases that fall into each of several categories, with the sum of the heights equaling 1.
However, bins need not be of equal width; in that case, the erected rectangle is defined to have its area proportional to the frequency of cases in the bin.[3] The vertical axis is then not the frequency but frequency density—the number of cases per unit of the variable on the horizontal axis. Examples of variable bin width are displayed on Census bureau data below.
As the adjacent bins leave no gaps, the rectangles of a histogram touch each other to indicate that the original variable is continuous.[4]
Histograms give a rough sense of the density of the underlying distribution of the data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the x-axis are all 1, then a histogram is identical to a relative frequency plot.
The histogram is one of the seven basic tools of quality control.[5]
Histograms are sometimes confused with bar charts. A histogram is used for continuous data, where the bins represent ranges of data, while a bar chart is a plot of categorical variables. Some authors recommend that bar charts have gaps between the rectangles to clarify the distinction.[6][7]
A bar graph and a histogram are two common types of graphical representations of data. While they may look similar, there are some key differences between the two that are important to understand.
A bar graph is a chart that uses bars to represent the frequency or quantity of different categories of data. The bars can be either vertical or horizontal, and they are typically arranged either horizontally or vertically to make it easy to compare the different categories. Bar graphs are useful for displaying data that can be divided into discrete categories, such as the number of students in different grade levels at a school.
A histogram, on the other hand, is a graph that shows the distribution of numerical data. It is a type of bar chart that shows the frequency or number of observations within different numerical ranges, called bins. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The histogram provides a visual representation of the distribution of the data, showing the number of observations that fall within each bin. This can be useful for identifying patterns and trends in the data, and for making comparisons between different datasets.[8]
Examples
This is the data for the histogram to the right, using 500 items:
Bin/IntervalCount/Frequency
−3.5 to −2.519
−2.5 to −1.5132
−1.5 to −0.51109
−0.5 to 0.49180
0.5 to 1.49132
1.5 to 2.4934
2.5 to 3.494
The words used to describe the patterns in a histogram are: "symmetric", "skewed left" or "right", "unimodal", "bimodal" or "multimodal".
• Symmetric, unimodal
• Skewed right
• Skewed left
• Bimodal
• Multimodal
• Symmetric
It is a good idea to plot the data using several different bin widths to learn more about it. Here is an example on tips given in a restaurant.
• Tips using a $1 bin width, skewed right, unimodal
• Tips using a 10c bin width, still skewed right, multimodal with modes at $ and 50c amounts, indicates rounding, also some outliers
The U.S. Census Bureau found that there were 124 million people who work outside of their homes.[9] Using their data on the time occupied by travel to work, the table below shows the absolute number of people who responded with travel times "at least 30 but less than 35 minutes" is higher than the numbers for the categories above and below it. This is likely due to people rounding their reported journey time. The problem of reporting values as somewhat arbitrarily rounded numbers is a common phenomenon when collecting data from people.
Data by absolute numbers
IntervalWidthQuantityQuantity/width
054180836
55136872737
105186183723
155196343926
205179813596
25571901438
305163693273
3553212642
4054122824
45159200613
60306461215
9060343557
This histogram shows the number of cases per unit interval as the height of each block, so that the area of each block is equal to the number of people in the survey who fall into its category. The area under the curve represents the total number of cases (124 million). This type of histogram shows absolute numbers, with Q in thousands.
Data by proportion
IntervalWidthQuantity (Q)Q/total/width
0541800.0067
55136870.0221
105186180.0300
155196340.0316
205179810.0290
25571900.0116
305163690.0264
35532120.0052
40541220.0066
451592000.0049
603064610.0017
906034350.0005
This histogram differs from the first only in the vertical scale. The area of each block is the fraction of the total that each category represents, and the total area of all the bars is equal to 1 (the fraction meaning "all"). The curve displayed is a simple density estimate. This version shows proportions, and is also known as a unit area histogram.
In other words, a histogram represents a frequency distribution by means of rectangles whose widths represent class intervals and whose areas are proportional to the corresponding frequencies: the height of each is the average frequency density for the interval. The intervals are placed together in order to show that the data represented by the histogram, while exclusive, is also contiguous. (E.g., in a histogram it is possible to have two connecting intervals of 10.5–20.5 and 20.5–33.5, but not two connecting intervals of 10.5–20.5 and 22.5–32.5. Empty intervals are represented as empty and not skipped.)[10]
Mathematical definitions
The data used to construct a histogram are generated via a function mi that counts the number of observations that fall into each of the disjoint categories (known as bins). Thus, if we let n be the total number of observations and k be the total number of bins, the histogram data mi meet the following conditions:
$n=\sum _{i=1}^{k}{m_{i}}.$
A histogram can be thought of as a simplistic kernel density estimation, which uses a kernel to smooth frequencies over the bins. This yields a smoother probability density function, which will in general more accurately reflect distribution of the underlying variable. The density estimate could be plotted as an alternative to the histogram, and is usually drawn as a curve rather than a set of boxes. Histograms are nevertheless preferred in applications, when their statistical properties need to be modeled. The correlated variation of a kernel density estimate is very difficult to describe mathematically, while it is simple for a histogram where each bin varies independently.
An alternative to kernel density estimation is the average shifted histogram,[11] which is fast to compute and gives a smooth curve estimate of the density without using kernels.
Cumulative histogram
A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. That is, the cumulative histogram Mi of a histogram mj is defined as:
$M_{i}=\sum _{j=1}^{i}{m_{j}}.$
Number of bins and width
There is no "best" number of bins, and different bin sizes can reveal different features of the data. Grouping data is at least as old as Graunt's work in the 17th century, but no systematic guidelines were given[12] until Sturges' work in 1926.[13]
Using wider bins where the density of the underlying data points is low reduces noise due to sampling randomness; using narrower bins where the density is high (so the signal drowns the noise) gives greater precision to the density estimation. Thus varying the bin-width within a histogram can be beneficial. Nonetheless, equal-width bins are widely used.
Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. Depending on the actual data distribution and the goals of the analysis, different bin widths may be appropriate, so experimentation is usually needed to determine an appropriate width. There are, however, various useful guidelines and rules of thumb.[14]
The number of bins k can be assigned directly or can be calculated from a suggested bin width h as:
$k=\left\lceil {\frac {\max x-\min x}{h}}\right\rceil .$
The braces indicate the ceiling function.
Square-root choice
$k=\lceil {\sqrt {n}}\rceil \,$
which takes the square root of the number of data points in the sample (used by Excel's Analysis Toolpak histograms and many other) and rounds to the next integer.[15]
Sturges' formula
Sturges' formula[13] is derived from a binomial distribution and implicitly assumes an approximately normal distribution.
$k=\lceil \log _{2}n\rceil +1,\,$
Sturges' formula implicitly bases bin sizes on the range of the data, and can perform poorly if n < 30, because the number of bins will be small—less than seven—and unlikely to show trends in the data well. On the other extreme, Sturges' formula may overestimate bin width for very large datasets, resulting in oversmoothed histograms.[16] It may also perform poorly if the data are not normally distributed.
When compared to Scott's rule and the Terrell-Scott rule, two other widely accepted formulas for histogram bins, the output of Sturges' formula is closest when n ≈ 100.[16]
Rice rule
$k=\lceil 2{\sqrt[{3}]{n}}\rceil ,$
The Rice Rule [17] is presented as a simple alternative to Sturges' rule.
Doane's formula
Doane's formula[18] is a modification of Sturges' formula which attempts to improve its performance with non-normal data.
$k=1+\log _{2}(n)+\log _{2}\left(1+{\frac {|g_{1}|}{\sigma _{g_{1}}}}\right)$
where $g_{1}$ is the estimated 3rd-moment-skewness of the distribution and
$\sigma _{g_{1}}={\sqrt {\frac {6(n-2)}{(n+1)(n+3)}}}$
Scott's normal reference rule
Bin width $h$ is given by
$h={\frac {3.49{\hat {\sigma }}}{\sqrt[{3}]{n}}},$
where ${\hat {\sigma }}$ is the sample standard deviation. Scott's normal reference rule[19] is optimal for random samples of normally distributed data, in the sense that it minimizes the integrated mean squared error of the density estimate.[12]
Freedman–Diaconis' choice
The Freedman–Diaconis rule gives bin width $h$ as:[20][12]
$h=2{\frac {\operatorname {IQR} (x)}{\sqrt[{3}]{n}}},$
which is based on the interquartile range, denoted by IQR. It replaces 3.5σ of Scott's rule with 2 IQR, which is less sensitive than the standard deviation to outliers in data.
Minimizing cross-validation estimated squared error
This approach of minimizing integrated mean squared error from Scott's rule can be generalized beyond normal distributions, by using leave-one out cross validation:[21][22]
${\underset {h}{\operatorname {arg\,min} }}{\hat {J}}(h)={\underset {h}{\operatorname {arg\,min} }}\left({\frac {2}{(n-1)h}}-{\frac {n+1}{n^{2}(n-1)h}}\sum _{k}N_{k}^{2}\right)$
Here, $N_{k}$ is the number of datapoints in the kth bin, and choosing the value of h that minimizes J will minimize integrated mean squared error.
Shimazaki and Shinomoto's choice
The choice is based on minimization of an estimated L2 risk function[23]
${\underset {h}{\operatorname {arg\,min} }}{\frac {2{\bar {m}}-v}{h^{2}}}$
where $\textstyle {\bar {m}}$ and $\textstyle v$ are mean and biased variance of a histogram with bin-width $\textstyle h$, $\textstyle {\bar {m}}={\frac {1}{k}}\sum _{i=1}^{k}m_{i}$ and $\textstyle v={\frac {1}{k}}\sum _{i=1}^{k}(m_{i}-{\bar {m}})^{2}$.
Variable bin widths
Rather than choosing evenly spaced bins, for some applications it is preferable to vary the bin width. This avoids bins with low counts. A common case is to choose equiprobable bins, where the number of samples in each bin is expected to be approximately equal. The bins may be chosen according to some known distribution or may be chosen based on the data so that each bin has $\approx n/k$ samples. When plotting the histogram, the frequency density is used for the dependent axis. While all bins have approximately equal area, the heights of the histogram approximate the density distribution.
For equiprobable bins, the following rule for the number of bins is suggested:[24]
$k=2n^{2/5}$
This choice of bins is motivated by maximizing the power of a Pearson chi-squared test testing whether the bins do contain equal numbers of samples. More specifically, for a given confidence interval $\alpha $ it is recommended to choose between 1/2 and 1 times the following equation:[25]
$k=4\left({\frac {2n^{2}}{\Phi ^{-1}(\alpha )}}\right)^{\frac {1}{5}}$
Where $\Phi ^{-1}$ is the probit function. Following this rule for $\alpha =0.05$ would give between $1.88n^{2/5}$ and $3.77n^{2/5}$; the coefficient of 2 is chosen as an easy-to-remember value from this broad optimum.
Remark
A good reason why the number of bins should be proportional to ${\sqrt[{3}]{n}}$ is the following: suppose that the data are obtained as $n$ independent realizations of a bounded probability distribution with smooth density. Then the histogram remains equally "rugged" as $n$ tends to infinity. If $s$ is the "width" of the distribution (e. g., the standard deviation or the inter-quartile range), then the number of units in a bin (the frequency) is of order $nh/s$ and the relative standard error is of order ${\sqrt {s/(nh)}}$. Compared to the next bin, the relative change of the frequency is of order $h/s$ provided that the derivative of the density is non-zero. These two are of the same order if $h$ is of order $s/{\sqrt[{3}]{n}}$, so that $k$ is of order ${\sqrt[{3}]{n}}$. This simple cubic root choice can also be applied to bins with non-constant widths.
Applications
• In hydrology the histogram and estimated density function of rainfall and river discharge data, analysed with a probability distribution, are used to gain insight in their behaviour and frequency of occurrence.[27] An example is shown in the blue figure.
• In many Digital image processing programs there is an histogram tool, which show you the distribution of the contrast / brightness of the pixels.
See also
Wikimedia Commons has media related to Histograms.
• Data and information visualization
• Data binning
• Density estimation
• Kernel density estimation, a smoother but more complex method of density estimation
• Entropy estimation
• Freedman–Diaconis rule
• Image histogram
• Pareto chart
• Seven basic tools of quality
• V-optimal histograms
References
1. Pearson, K. (1895). "Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 186: 343–414. Bibcode:1895RSPTA.186..343P. doi:10.1098/rsta.1895.0010.
2. Howitt, D.; Cramer, D. (2008). Introduction to Statistics in Psychology (Fourth ed.). Prentice Hall. ISBN 978-0-13-205161-3.
3. Freedman, D.; Pisani, R.; Purves, R. (1998). Statistics (Third ed.). W. W. Norton. ISBN 978-0-393-97083-8.
4. Charles Stangor (2011) "Research Methods For The Behavioral Sciences". Wadsworth, Cengage Learning. ISBN 9780840031976.
5. Nancy R. Tague (2004). "Seven Basic Quality Tools". The Quality Toolbox. Milwaukee, Wisconsin: American Society for Quality. p. 15. Retrieved 2010-02-05.
6. Naomi, Robbins. "A Histogram is NOT a Bar Chart". Forbes. Retrieved 31 July 2018.
7. M. Eileen Magnello (December 2006). "Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician". The New Zealand Journal for the History and Philosophy of Science and Technology. 1 volume. OCLC 682200824.
8. "Histogram maker". histogram maker.
9. US 2000 census.
10. Dean, S., & Illowsky, B. (2009, February 19). Descriptive Statistics: Histogram. Retrieved from the Connexions Web site: http://cnx.org/content/m16298/1.11/
11. David W. Scott (December 2009). "Averaged shifted histogram". Wiley Interdisciplinary Reviews: Computational Statistics. 2:2 (2): 160–164. doi:10.1002/wics.54. S2CID 122986682.
12. Scott, David W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. New York: John Wiley.
13. Sturges, H. A. (1926). "The choice of a class interval". Journal of the American Statistical Association. 21 (153): 65–66. doi:10.1080/01621459.1926.10502161. JSTOR 2965501.
14. e.g. § 5.6 "Density Estimation", W. N. Venables and B. D. Ripley, Modern Applied Statistics with S (2002), Springer, 4th edition. ISBN 0-387-95457-0.
15. "EXCEL Univariate: Histogram".
16. Scott, David W. (2009). "Sturges' rule". WIREs Computational Statistics. 1 (3): 303–306. doi:10.1002/wics.35. S2CID 197483064.
17. Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University (chapter 2 "Graphing Distributions", section "Histograms")
18. Doane DP (1976) Aesthetic frequency classification. American Statistician, 30: 181–183
19. Scott, David W. (1979). "On optimal and data-based histograms". Biometrika. 66 (3): 605–610. doi:10.1093/biomet/66.3.605.
20. Freedman, David; Diaconis, P. (1981). "On the histogram as a density estimator: L2 theory" (PDF). Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 57 (4): 453–476. CiteSeerX 10.1.1.650.2473. doi:10.1007/BF01025868. S2CID 14437088.
21. Wasserman, Larry (2004). All of Statistics. New York: Springer. p. 310. ISBN 978-1-4419-2322-6.
22. Stone, Charles J. (1984). "An asymptotically optimal histogram selection rule" (PDF). Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer.
23. Shimazaki, H.; Shinomoto, S. (2007). "A method for selecting the bin size of a time histogram". Neural Computation. 19 (6): 1503–1527. CiteSeerX 10.1.1.304.6404. doi:10.1162/neco.2007.19.6.1503. PMID 17444758. S2CID 7781236.
24. Jack Prins; Don McCormack; Di Michelson; Karen Horrell. "Chi-square goodness-of-fit test". NIST/SEMATECH e-Handbook of Statistical Methods. NIST/SEMATECH. p. 7.2.1.1. Retrieved 29 March 2019.
25. Moore, David (1986). "3". In D'Agostino, Ralph; Stephens, Michael (eds.). Goodness-of-Fit Techniques. New York, NY, USA: Marcel Dekker Inc. p. 70. ISBN 0-8247-7487-6.
26. A calculator for probability distributions and density functions
27. An illustration of histograms and probability density functions
Further reading
• Lancaster, H.O. An Introduction to Medical Statistics. John Wiley and Sons. 1974. ISBN 0-471-51250-8
External links
Wikimedia Commons has media related to Histogram.
Look up histogram in Wiktionary, the free dictionary.
• Exploring Histograms, an essay by Aran Lunzer and Amelia McNamara
• Journey To Work and Place Of Work (location of census document cited in example)
• Smooth histogram for signals and images from a few samples
• Histograms: Construction, Analysis and Understanding with external links and an application to particle Physics.
• A Method for Selecting the Bin Size of a Histogram
• Histograms: Theory and Practice, some great illustrations of some of the Bin Width concepts derived above.
• Histograms the Right Way
• Interactive histogram generator
• Matlab function to plot nice histograms
• Dynamic Histogram in MS Excel
• Histogram construction and manipulation using Java applets, and charts on SOCR
• Toolbox for constructing the best histograms
Seven basic tools of quality
• Cause-and-effect diagram
• Check sheet
• Control chart
• Histogram
• Pareto chart
• Scatter diagram
• Stratification
Quality (business)
Statistics
• Outline
• Index
Descriptive statistics
Continuous data
Center
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• Median
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Dispersion
• Average absolute deviation
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• Interquartile range
• Percentile
• Range
• Standard deviation
• Variance
Shape
• Central limit theorem
• Moments
• Kurtosis
• L-moments
• Skewness
Count data
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Summary tables
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Dependence
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Graphics
• Bar chart
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Data collection
Study design
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Interval estimation
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Testing hypotheses
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Specific tests
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Rank statistics
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Bayesian inference
• Bayesian probability
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Correlation
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Regression analysis
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Linear regression
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Generalized linear model
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Partition of variance
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Categorical / Multivariate / Time-series / Survival analysis
Categorical
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Multivariate
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General
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Specific tests
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Time domain
• Autocorrelation (ACF)
• partial (PACF)
• Cross-correlation (XCF)
• ARMA model
• ARIMA model (Box–Jenkins)
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Frequency domain
• Spectral density estimation
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• Wavelet
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Survival
Survival function
• Kaplan–Meier estimator (product limit)
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Hazard function
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Test
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Applications
Biostatistics
• Bioinformatics
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• Medical statistics
Engineering statistics
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Social statistics
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Neighborhood semantics
Neighborhood semantics, also known as Scott–Montague semantics, is a formal semantics for modal logics. It is a generalization, developed independently by Dana Scott and Richard Montague, of the more widely known relational semantics for modal logic. Whereas a relational frame $\langle W,R\rangle $ consists of a set W of worlds (or states) and an accessibility relation R intended to indicate which worlds are alternatives to (or, accessible from) others, a neighborhood frame $\langle W,N\rangle $ still has a set W of worlds, but has instead of an accessibility relation a neighborhood function
$N:W\to 2^{2^{W}}$
that assigns to each element of W a set of subsets of W. Intuitively, each family of subsets assigned to a world are the propositions necessary at that world, where 'proposition' is defined as a subset of W (i.e. the set of worlds at which the proposition is true). Specifically, if M is a model on the frame, then
$M,w\models \square \varphi \Longleftrightarrow (\varphi )^{M}\in N(w),$
where
$(\varphi )^{M}=\{u\in W\mid M,u\models \varphi \}$
is the truth set of $\varphi $.
Neighborhood semantics is used for the classical modal logics that are strictly weaker than the normal modal logic K.
Correspondence between relational and neighborhood models
To every relational model M = (W, R, V) there corresponds an equivalent (in the sense of having pointwise-identical modal theories) neighborhood model M' = (W, N, V) defined by
$N(w)=\{(\varphi )^{M}\mid M,w\models \Box \varphi \}.$
The fact that the converse fails gives a precise sense to the remark that neighborhood models are a generalization of relational ones. Another (perhaps more natural) generalization of relational structures are general frames.
References
• Chellas, B.F. Modal Logic. Cambridge University Press, 1980.
• Montague, R. "Universal Grammar", Theoria 36, 373–98, 1970.
• Scott, D. "Advice on modal logic", in Philosophical Problems in Logic, ed. Karel Lambert. Reidel, 1970.
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Scott continuity
In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of the supremum of D, i.e. $\sqcup f[D]=f(\sqcup D)$, where $\sqcup $ is the directed join.[1] When $Q$ is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets.[2]
A subset O of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty intersection with O. The Scott-open subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology.[1]
The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.[3]
Scott-continuous functions show up in the study of models for lambda calculi[3] and the denotational semantics of computer programs.
Properties
A Scott-continuous function is always monotonic.
A subset of a directed complete partial order is closed with respect to the Scott topology induced by the partial order if and only if it is a lower set and closed under suprema of directed subsets.[4]
A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T0 separation axiom).[4] However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial.[4] The Scott-open sets form a complete lattice when ordered by inclusion.[5]
For any Kolmogorov space, the topology induces an order relation on that space, the specialization order: x ≤ y if and only if every open neighbourhood of x is also an open neighbourhood of y. The order relation of a dcpo D can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology. However, a dcpo equipped with the Scott topology need not be sober: the specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology.[4]
Examples
The open sets in a given topological space when ordered by inclusion form a lattice on which the Scott topology can be defined. A subset X of a topological space T is compact with respect to the topology on T (in the sense that every open cover of X contains a finite subcover of X) if and only if the set of open neighbourhoods of X is open with respect to the Scott topology.[5]
For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.[6]
Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic.[7]
See also
• Alexandrov topology
• Upper topology
Footnotes
1. Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 978-0-521-36062-3.
2. Scott topology at the nLab
3. Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill (ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.
4. Abramsky, S.; Jung, A. (1994). "Domain theory" (PDF). In Abramsky, S.; Gabbay, D.M.; Maibaum, T.S.E. (eds.). Handbook of Logic in Computer Science. Vol. III. Oxford University Press. ISBN 978-0-19-853762-5.
5. Bauer, Andrej & Taylor, Paul (2009). "The Dedekind Reals in Abstract Stone Duality". Mathematical Structures in Computer Science. 19 (4): 757–838. CiteSeerX 10.1.1.424.6069. doi:10.1017/S0960129509007695. S2CID 6774320. Retrieved October 8, 2010.
6. Barendregt, H.P. (1984). The Lambda Calculus. North-Holland. ISBN 978-0-444-87508-2. (See theorems 1.2.13, 1.2.14)
7. N. Belnap (1975) "How Computers Should Think", pages 30 to 56 in Contemporary Aspects of Philosophy, Gilbert Ryle editor, Oriel Press ISBN 0-85362-161-6
References
• "Scott Topology". PlanetMath.
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Scott A. Mitchell
Scott Alan Mitchell is a researcher of applied mathematics in the Center for Computing Research at Sandia National Laboratories.
Scott Alan Mitchell
Scott A. Mitchell with whiteboard and bicycle
CitizenshipUnited States
Alma materUniversity of Wisconsin–Madison
Cornell University
Known forMesh generation
Scientific career
FieldsApplied Mathematics
Computational Geometry
Computer Graphics
InstitutionsSandia National Laboratories
Doctoral advisorStephen A. Vavasis
Websitewww.sandia.gov/~samitch/
Background
Mitchell received a B.S in Applied Math, Engineering & Physics from the University of Wisconsin-Madison (1988), and an M.S. (1991) and Ph.D. (1993) in Applied Math from Cornell University. He worked the summer of 1991 at Xerox PARC (now PARC). Since 1992 he has been at Sandia National Laboratories in the Center for Computing Research, with several different roles. He researched theoretical computational geometry meshing from 1992—1993. He contributed to applied meshing in the CUBIT project: R&D 1993—2000, project leader 2000—2002, R&D 2015—. He managed Sandia's Optimization and Uncertainty Estimation department, and had programmatic roles on the Laboratory Directed Research and Development program and NNSA's ASC program from 2002-2007. He researched informatics and applying persistent homology from 2008-2011. Since 2011 he researches mesh generation and sampling.
He served on the committee of the Meshing Roundtable and International Symposium on Computational Geometry SoCG conferences. He served as a guest editor for the journal CAD. As an adjunct professor, he taught a small graduate course on computational geometry at the University of New Mexico. He is a member of ACM and SIAM.
Research
He published[1][2][3] algorithms in the areas of mesh generation, reconstruction and sampling, for the contexts of computational geometry, simulation, computer graphics and uncertainty quantification. His main contributions have been geometric algorithms with provable correctness and output quality guarantees. His PhD thesis was the first tetrahedral meshing algorithm with guarantees on both the number of elements and their shape. He is also well known for a series of papers on whisker weaving and other algorithms for hexahedral mesh generation using the dual spatial twist continuum. He used optimization for mesh generation, specifically interval assignment, deciding the right number of edges locally so the model can be meshed globally. Since 2011 he contributed sampling algorithms for computer graphics and uncertainty quantification, and algorithms for mesh generation (including duality) and surface reconstruction.
References
1. Scott A. Mitchell publications indexed by Google Scholar
2. Scott A. Mitchell at DBLP Bibliography Server
3. Scott A. Mitchell author profile page at the ACM Digital Library
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Scott Sheffield
Scott Sheffield (born October 20, 1973) is a professor of mathematics at the Massachusetts Institute of Technology.[1] His primary research field is theoretical probability.
Scott Sheffield
Born (1973-10-20) October 20, 1973
NationalityAmerican
Alma materStanford University
AwardsLoève Prize (2011)
Rollo Davidson Prize (2006)
Clay Research Award (2017)
Scientific career
FieldsMathematician
InstitutionsMIT
ThesisRandom surfaces: large deviations and gradient Gibbs measure classifications (2003)
Doctoral advisorAmir Dembo
Doctoral studentsEwain Gwynne, Nina Holden, Xin Sun
Websitemath.mit.edu/~sheffield/
Research
Much of Sheffield's work examines conformal invariant objects which arise in the study of two-dimensional statistical physics models. He studies the Schramm-Loewner evolution SLE(κ) and its relations to a variety of other random objects. For example, he proved that SLE describes the interface between two Liouville quantum gravity surfaces that have been conformally welded together.[2] In joint work with Oded Schramm, he showed that contour lines of the Gaussian free field are related to SLE(4).[3][4] With Jason Miller, he developed the theory of Gaussian free field flow lines, which include SLE(κ) for all values of κ, as well as many variants of SLE.[5]
Sheffield and Bertrand Duplantier proved the Knizhnik-Polyakov-Zamolodchikov (KPZ) relation for fractal scaling dimensions in Liouville quantum gravity.[6] Sheffield also defined the conformal loop ensembles, which serve as scaling limits of the collection of all interfaces in various statistical physics models.[7] In joint work with Wendelin Werner, he described the conformal loop ensembles as the outer boundaries of clusters of Brownian loops.[8]
In addition to these contributions, Sheffield has also proved results regarding internal diffusion limited aggregation, dimers, game theory, partial differential equations, and Lipschitz extension theory.[9]
Teaching
Since 2011, Sheffield has taught 18.600 (formerly 18.440), the introductory probability course at MIT.[10]
Education and career
Sheffield graduated from Harvard University in 1998 with an A.B. and A.M. in mathematics. In 2003, he received his Ph.D. in mathematics from Stanford University. Before becoming a professor at MIT, Sheffield held postdoctoral positions at Microsoft Research, the University of California at Berkeley, and the Institute for Advanced Study. He was also an associate professor at New York University.[11]
Awards
Scott Sheffield received the Loève Prize, the Presidential Early Career Award for Scientists and Engineers, the Sloan Research Fellowship, and the Rollo Davidson Prize. He was also an invited speaker at the 2010 meeting of the International Congress of Mathematicians and a plenary speaker in 2022. In 2017 he received the Clay Research Award jointly with Jason Miller.[12] He was elected to the American Academy of Arts and Sciences in 2021.[13] In 2023 he received the Leonard Eisenbud Prize for Mathematics and Physics of the AMS jointly with Jason Miller.
Books
• Scott Sheffield (2005), Random Surfaces, American Mathematical Society[14]
References
1. "Scott Sheffield". MIT.edu. Retrieved December 12, 2021.
2. Sheffield, Scott (2010). "Conformal weldings of random surfaces: SLE and the quantum gravity zipper". arXiv:1012.4797. {{cite journal}}: Cite journal requires |journal= (help)
3. Schramm, Oded; Sheffield, Scott (2006). "Contour lines of the two-dimensional discrete Gaussian free field". arXiv:math/0605337. Bibcode:2006math......5337S. {{cite journal}}: Cite journal requires |journal= (help)
4. Schramm, Oded; Sheffield, Scott (2010). "A contour line of the continuum Gaussian free field". arXiv:1008.2447. {{cite journal}}: Cite journal requires |journal= (help)
5. Miller, Jason; Sheffield, Scott (2012). "Imaginary Geometry I: Interacting SLEs". arXiv:1201.1496. {{cite journal}}: Cite journal requires |journal= (help)
6. Duplantier, Bertrand; Sheffield, Scott (2008). "Liouville Quantum Gravity and KPZ". arXiv:0808.1560. {{cite journal}}: Cite journal requires |journal= (help)
7. Sheffield, Scott (2006). "Exploration trees and conformal loop ensembles". arXiv:math/0609167. Bibcode:2006math......9167S. {{cite journal}}: Cite journal requires |journal= (help)
8. Sheffield, Scott; Werner, Wendelin (2010). "Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction". arXiv:1006.2374. {{cite journal}}: Cite journal requires |journal= (help)
9. 2011 Loève Prize announcement, http://www.stat.berkeley.edu/users/aldous/Research/sheffield.pdf
10. http://math.mit.edu/~sheffield/teach.html
11. "CV". MIT. Retrieved August 23, 2013.
12. Clay Research Award 2017
13. "New Members Elected in 2021". American Academy of Arts and Sciences. Retrieved April 22, 2021.
14. Sheffield, Scott (2005). Random surfaces. Société mathématique de France. p. vi+175. ISBN 2856291872. Retrieved September 21, 2019.
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Scott core theorem
In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott, (Scott 1973). The precise statement is as follows:
Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable.
A simplified proof is given in (Rubinstein & Swarup 1990), and a stronger uniqueness statement is proven in (Harris & Scott 1996).
References
• Harris, Luke; Scott, G. Peter (1996), "The uniqueness of compact cores for 3-manifolds", Pacific Journal of Mathematics, 172 (1): 139–150, doi:10.2140/pjm.1996.172.139, ISSN 0030-8730, MR 1379290
• Rubinstein, J. H.; Swarup, G. A. (1990), "On Scott's core theorem", The Bulletin of the London Mathematical Society, 22 (5): 495–498, doi:10.1112/blms/22.5.495, MR 1082023
• Scott, G. Peter (1973), "Compact submanifolds of 3-manifolds", Journal of the London Mathematical Society, Second Series, 7 (2): 246–250, doi:10.1112/jlms/s2-7.2.246, MR 0326737
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Scott's trick
In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy.
The method relies on the axiom of regularity but not on the axiom of choice. It can be used to define representatives for ordinal numbers in ZF, Zermelo–Fraenkel set theory without the axiom of choice (Forster 2003:182). The method was introduced by Dana Scott (1955).
Beyond the problem of defining set representatives for ordinal numbers, Scott's trick can be used to obtain representatives for cardinal numbers and more generally for isomorphism types, for example, order types of linearly ordered sets (Jech 2003:65). It is credited to be indispensable (even in the presence of the axiom of choice) when taking ultrapowers of proper classes in model theory. (Kanamori 1994:47)
Application to cardinalities
The use of Scott's trick for cardinal numbers shows how the method is typically employed. The initial definition of a cardinal number is an equivalence class of sets, where two sets are equivalent if there is a bijection between them. The difficulty is that almost every equivalence class of this relation is a proper class, and so the equivalence classes themselves cannot be directly manipulated in set theories, such as Zermelo–Fraenkel set theory, that only deal with sets. It is often desirable in the context of set theory to have sets that are representatives for the equivalence classes. These sets are then taken to "be" cardinal numbers, by definition.
In Zermelo–Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice is not assumed, for some cardinal numbers it may not be possible to find such an ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representatives.
Scott's trick assigns representatives differently, using the fact that for every set $A$ there is a least rank $V_{\alpha }$ in the cumulative hierarchy when some set of the same cardinality as $A$ appears. Thus one may define the representative of the cardinal number of $A$ to be the set of all sets of rank $V_{\alpha }$ that have the same cardinality as $A$. This definition assigns a representative to every cardinal number even when not every set can be well-ordered (an assumption equivalent to the axiom of choice). It can be carried out in Zermelo–Fraenkel set theory, without using the axiom of choice, but making essential use of the axiom of regularity.
Scott's trick in general
Let $\sim $ be an equivalence relation of sets. Let $a$ be a set and $[a]$ its equivalence class with respect to $\sim $. If $V\cap [a]$ is non-empty, we can define a set, which represents $[a]$, even if $[a]$ is a proper class. Namely, there exists a least ordinal $\alpha $, such that $V_{\alpha }\cap [a]$ is non-empty. This intersection is a set, so we can take it as the representative of $[a]$. We didn't use regularity for this construction.
The axiom of regularity is equivalent to $a\in V$ for all sets $a$ (see Regularity, the cumulative hierarchy and types). So in particular, if we assume the axiom of regularity, then $V\cap [a]$ will be non-empty for all sets $a$ and equivalence relations $\sim $, since $a\in V\cap [a]$. To summarize: given the axiom of regularity, we can find representatives of every equivalence class, for any equivalence relation.
References
• Thomas Forster (2003), Logic, Induction and Sets, Cambridge University Press. ISBN 0-521-53361-9
• Thomas Jech, Set Theory, 3rd millennium (revised) ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2
• Akihiro Kanamori: The Higher Infinite. Large Cardinals in Set Theory from their Beginnings., Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp.
• Scott, Dana (1955), "Definitions by abstraction in axiomatic set theory" (PDF), Bulletin of the American Mathematical Society, 61 (5): 442, doi:10.1090/S0002-9904-1955-09941-5
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Scottish Café
The Scottish Café (Polish: Kawiarnia Szkocka) was a café in Lwów, Poland (now Lviv, Ukraine) where, in the 1930s and 1940s, mathematicians from the Lwów School of Mathematics collaboratively discussed research problems, particularly in functional analysis and topology.
Stanisław Ulam recounts that the tables of the café had marble tops, so they could write in pencil, directly on the table, during their discussions. To keep the results from being lost, and after becoming annoyed with their writing directly on the table tops, Stefan Banach's wife provided the mathematicians with a large notebook, which was used for writing the problems and answers and eventually became known as the Scottish Book. The book—a collection of solved, unsolved, and even probably unsolvable problems—could be borrowed by any of the guests of the café. Solving any of the problems was rewarded with prizes, with the most difficult and challenging problems having expensive prizes (during the Great Depression and on the eve of World War II), such as a bottle of fine brandy.[1]
For problem 153, which was later recognized as being closely related to Stefan Banach's "basis problem", Stanisław Mazur offered the prize of a live goose. This problem was solved only in 1972 by Per Enflo, who was presented with the live goose in a ceremony that was broadcast throughout Poland.[2]
The café building now houses the Szkocka Restaurant & Bar (named for the original Scottish Café) and the Atlas Deluxe hotel at the street address of 27 Taras Shevchenko Prospekt.
See also
The following mathematicians were associated with the Lwów School of Mathematics or contributed to The Scottish Book:
• Stefan Banach
• Karol Borsuk
• Mark Kac
• Stefan Kaczmarz
• Bronisław Knaster
• Kazimierz Kuratowski
• Stanisław Saks
• Juliusz Schauder
• Hugo Steinhaus
• Stanisław Ulam
• Gus Ward
References
1. Mauldin, ed.
2. Mauldin, ed.; Kaluza.
• Kałuża, Roman (1996). Ann Kostant and Wojbor Woyczyński (ed.). Through a reporter's eyes: The life of Stefan Banach. Birkhäuser. ISBN 0-8176-3772-9. MR 1392949.
• R. Daniel Mauldin, ed. (1981). The Scottish Book: Mathematics from the Scottish Café (Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979). Boston, Mass.: Birkhäuser. pp. xiii+268 pp. (2 plates). ISBN 3-7643-3045-7. MR 0666400.
External links
• Scottish book
• Scottish book Web page at Home Page of Stefan Banach at Adam Mickiewicz University in Poznań website
• Manuscript of Scottish book (PDF)
• Typescript of English version of Scottish book (PDF)
• Kawiarnia Szkocka at the MacTutor archive
• "The Life of Stefan Banach" review by Sheldon Axler
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Scott–Curry theorem
In mathematical logic, the Scott–Curry theorem is a result in lambda calculus stating that if two non-empty sets of lambda terms A and B are closed under beta-convertibility then they are recursively inseparable.[1]
Explanation
A set A of lambda terms is closed under beta-convertibility if for any lambda terms X and Y, if $X\in A$ and X is β-equivalent to Y then $Y\in A$. Two sets A and B of natural numbers are recursively separable if there exists a computable function $f:\mathbb {N} \rightarrow \{0,1\}$ such that $f(a)=0$ if $a\in A$ and $f(b)=1$ if $b\in B$. Two sets of lambda terms are recursively separable if their corresponding sets under a Gödel numbering are recursively separable, and recursively inseparable otherwise.
The Scott–Curry theorem applies equally to sets of terms in combinatory logic with weak equality. It has parallels to Rice's theorem in computability theorem, which states that all non-trivial semantic properties of programs are undecidable.
The theorem has the immediate consequence that it is an undecidable problem to determine if two lambda terms are β-equivalent.
Proof
The proof is adapted from Barendregt in The Lambda Calculus.[2]
Let A and B be closed under beta-convertibility and let a and b be lambda term representations of elements from A and B respectively. Suppose for a contradiction that f is a lambda term representing a computable function such that $fx=0$ if $x\in A$ and $fx=1$ if $x\in B$ (where equality is β-equality). Then define $G\equiv \lambda x.{\text{if}}\ ({\text{zero?}}\ (fx))ab$. Here, ${\text{zero?}}$ is true if its argument is zero and false otherwise, and ${\text{if}}$ is the identity so that ${\text{if}}\ bxy$ is equal to x if b is true and y if b is false.
Then $x\in C\implies Gx=a$ and similarly, $x\notin C\implies Gx=b$. By the Second Recursion Theorem, there is a term X which is equal to f applied to the Church numeral of its Gödel numbering, X'. Then $X\in C$ implies that $X=G(X')=b$ so in fact $X\notin C$. The reverse assumption $X\notin C$ gives $X=G(X')=a$ so $X\in C$. Either way we arise at a contradiction, and so f cannot be a function which separates A and B. Hence A and B are recursively inseparable.
History
Dana Scott first proved the theorem in 1963. The theorem, in a slightly less general form, was independently proven by Haskell Curry.[3] It was published in Curry's 1969 paper "The undecidability of λK-conversion".[4]
References
1. Hindley, J.R.; Seldin, J.P. (1986). Introduction to Combinators and (lambda) Calculus. Cambridge Monographs on Mathematical Physics. Cambridge University Press. ISBN 9780521268967. LCCN lc85029908.
2. Barendregt, H.P. (1985). The Lambda Calculus: Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics. Vol. 103 (3rd ed.). Elsevier Science. ISBN 0444875085.
3. Gabbay, D.M.; Woods, J. (2009). Logic from Russell to Church. Handbook of the History of Logic. Elsevier Science. ISBN 9780080885476.
4. Curry, Haskell B. (1969). "The undecidability of λK-conversion". Journal of Symbolic Logic. January 1969: 10–14.
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Scott–Potter set theory
An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott and the philosopher George Boolos.
Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers, Peano arithmetic and the other usual number systems, and the theory of relations.
ZU etc.
Preliminaries
This section and the next follow Part I of Potter (2004) closely. The background logic is first-order logic with identity. The ontology includes urelements as well as sets, which makes it clear that there can be sets of entities defined by first-order theories not based on sets. The urelements are not essential in that other mathematical structures can be defined as sets, and it is permissible for the set of urelements to be empty.
Some terminology peculiar to Potter's set theory:
• ι is a definite description operator and binds a variable. (In Potter's notation the iota symbol is inverted.)
• The predicate U holds for all urelements (non-collections).
• ιxΦ(x) exists iff (∃!x)Φ(x). (Potter uses Φ and other upper-case Greek letters to represent formulas.)
• {x : Φ(x)} is an abbreviation for ιy(not U(y) and (∀x)(x ∈ y ⇔ Φ(x))).
• a is a collection if {x : x∈a} exists. (All sets are collections, but not all collections are sets.)
• The accumulation of a, acc(a), is the set {x : x is an urelement or ∃b∈a (x∈b or x⊂b)}.
• If ∀v∈V(v = acc(V∩v)) then V is a history.
• A level is the accumulation of a history.
• An initial level has no other levels as members.
• A limit level is a level that is neither the initial level nor the level above any other level.
• A set is a subcollection of some level.
• The birthday of set a, denoted V(a), is the lowest level V such that a⊂V.
Axioms
The following three axioms define the theory ZU.
Creation: ∀V∃V' (V∈V' ).
Remark: There is no highest level, hence there are infinitely many levels. This axiom establishes the ontology of levels.
Separation: An axiom schema. For any first-order formula Φ(x) with (bound) variables ranging over the level V, the collection {x∈V : Φ(x)} is also a set. (See Axiom schema of separation.)
Remark: Given the levels established by Creation, this schema establishes the existence of sets and how to form them. It tells us that a level is a set, and all subsets, definable via first-order logic, of levels are also sets. This schema can be seen as an extension of the background logic.
Infinity: There exists at least one limit level. (See Axiom of infinity.)
Remark: Among the sets Separation allows, at least one is infinite. This axiom is primarily mathematical, as there is no need for the actual infinite in other human contexts, the human sensory order being necessarily finite. For mathematical purposes, the axiom "There exists an inductive set" would suffice.
Further existence premises
The following statements, while in the nature of axioms, are not axioms of ZU. Instead, they assert the existence of sets satisfying a stated condition. As such, they are "existence premises," meaning the following. Let X denote any statement below. Any theorem whose proof requires X is then formulated conditionally as "If X holds, then..." Potter defines several systems using existence premises, including the following two:
• ZfU =df ZU + Ordinals;
• ZFU =df Separation + Reflection.
Ordinals: For each (infinite) ordinal α, there exists a corresponding level Vα.
Remark: In words, "There exists a level corresponding to each infinite ordinal." Ordinals makes possible the conventional Von Neumann definition of ordinal numbers.
Let τ(x) be a first-order term.
Replacement: An axiom schema. For any collection a, ∀x∈a[τ(x) is a set] → {τ(x) : x∈a} is a set.
Remark: If the term τ(x) is a function (call it f(x)), and if the domain of f is a set, then the range of f is also a set.
Reflection: Let Φ denote a first-order formula in which any number of free variables are present. Let Φ(V) denote Φ with these free variables all quantified, with the quantified variables restricted to the level V.
Then ∃V[Φ→Φ(V)] is an axiom.
Remark: This schema asserts the existence of a "partial" universe, namely the level V, in which all properties Φ holding when the quantified variables range over all levels, also hold when these variables range over V only. Reflection turns Creation, Infinity, Ordinals, and Replacement into theorems (Potter 2004: §13.3).
Let A and a denote sequences of nonempty sets, each indexed by n.
Countable Choice: Given any sequence A, there exists a sequence a such that:
∀n∈ω[an∈An].
Remark. Countable Choice enables proving that any set must be one of finite or infinite.
Let B and C denote sets, and let n index the members of B, each denoted Bn.
Choice: Let the members of B be disjoint nonempty sets. Then:
∃C∀n[C∩Bn is a singleton].
Discussion
The von Neumann universe implements the "iterative conception of set" by stratifying the universe of sets into a series of "levels," with the sets at a given level being the members of the sets making up the next higher level. Hence the levels form a nested and well-ordered sequence, and would form a hierarchy if set membership were transitive. The resulting iterative conception steers clear, in a well-motivated way, of the well-known paradoxes of Russell, Burali-Forti, and Cantor. These paradoxes all result from the unrestricted use of the principle of comprehension that naive set theory allows. Collections such as "the class of all sets" or "the class of all ordinals" include sets from all levels of the hierarchy. Given the iterative conception, such collections cannot form sets at any given level of the hierarchy and thus cannot be sets at all. The iterative conception has gradually become more accepted over time, despite an imperfect understanding of its historical origins.
Boolos's (1989) axiomatic treatment of the iterative conception is his set theory S, a two sorted first order theory involving sets and levels.
Scott's theory
Scott (1974) did not mention the "iterative conception of set," instead proposing his theory as a natural outgrowth of the simple theory of types. Nevertheless, Scott's theory can be seen as an axiomatization of the iterative conception and the associated iterative hierarchy.
Scott began with an axiom he declined to name: the atomic formula x∈y implies that y is a set. In symbols:
∀x,y∃a[x∈y→y=a].
His axiom of Extensionality and axiom schema of Comprehension (Separation) are strictly analogous to their ZF counterparts and so do not mention levels. He then invoked two axioms that do mention levels:
• Accumulation. A given level "accumulates" all members and subsets of all earlier levels. See the above definition of accumulation.
• Restriction. All collections belong to some level.
Restriction also implies the existence of at least one level and assures that all sets are well-founded.
Scott's final axiom, the Reflection schema, is identical to the above existence premise bearing the same name, and likewise does duty for ZF's Infinity and Replacement. Scott's system has the same strength as ZF.
Potter's theory
Potter (1990, 2004) introduced the idiosyncratic terminology described earlier in this entry, and discarded or replaced all of Scott's axioms except Reflection; the result is ZU. ZU, like ZF, cannot be finitely axiomatized. ZU differs from ZFC in that it:
• Includes no axiom of extensionality because the usual extensionality principle follows from the definition of collection and an easy lemma.
• Admits nonwellfounded collections. However Potter (2004) never invokes such collections, and all sets (collections which are contained in a level) are wellfounded. No theorem in Potter would be overturned if an axiom stating that all collections are sets were added to ZU.
• Includes no equivalents of Choice or the axiom schema of Replacement.
Hence ZU is closer to the Zermelo set theory of 1908, namely ZFC minus Choice, Replacement, and Foundation. It is stronger than this theory, however, since cardinals and ordinals can be defined, despite the absence of Choice, using Scott's trick and the existence of levels, and no such definition is possible in Zermelo set theory. Thus in ZU, an equivalence class of:
• Equinumerous sets from a common level is a cardinal number;
• Isomorphic well-orderings, also from a common level, is an ordinal number.
Similarly the natural numbers are not defined as a particular set within the iterative hierarchy, but as models of a "pure" Dedekind algebra. "Dedekind algebra" is Potter's name for a set closed under a unary injective operation, successor, whose domain contains a unique element, zero, absent from its range. Because the theory of Dedekind algebras is categorical (all models are isomorphic), any such algebra can proxy for the natural numbers.
Although Potter (2004) devotes an entire appendix to proper classes, the strength and merits of Scott–Potter set theory relative to the well-known rivals to ZFC that admit proper classes, namely NBG and Morse–Kelley set theory, have yet to be explored.
Scott–Potter set theory resembles NFU in that the latter is a recently (Jensen 1967) devised axiomatic set theory admitting both urelements and sets that are not well-founded. But the urelements of NFU, unlike those of ZU, play an essential role; they and the resulting restrictions on Extensionality make possible a proof of NFU's consistency relative to Peano arithmetic. But nothing is known about the strength of NFU relative to Creation+Separation, NFU+Infinity relative to ZU, and of NFU+Infinity+Countable Choice relative to ZU + Countable Choice.
Unlike nearly all writing on set theory in recent decades, Potter (2004) mentions mereological fusions. His collections are also synonymous with the "virtual sets" of Willard Quine and Richard Milton Martin: entities arising from the free use of the principle of comprehension that can never be admitted to the universe of discourse.
See also
• Foundation of mathematics
• Hierarchy (mathematics)
• List of set theory topics
• Philosophy of mathematics
• S (Boolos 1989)
• Von Neumann universe
• Zermelo set theory
• ZFC
References
• George Boolos, 1971, "The iterative conception of set," Journal of Philosophy 68: 215–31. Reprinted in Boolos 1999. Logic, Logic, and Logic. Harvard Univ. Press: 13-29.
• --------, 1989, "Iteration Again," Philosophical Topics 42: 5-21. Reprinted in Boolos 1999. Logic, Logic, and Logic. Harvard Univ. Press: 88-104.
• Potter, Michael, 1990. Sets: An Introduction. Oxford Univ. Press.
• ------, 2004. Set Theory and its Philosophy. Oxford Univ. Press.
• Dana Scott, 1974, "Axiomatizing set theory" in Jech, Thomas, J., ed., Axiomatic Set Theory II, Proceedings of Symposia in Pure Mathematics 13. American Mathematical Society: 207–14.
External links
Review of Potter(1990):
• McGee, Vann, "" "Journal of Symbolic Logic 1993":1077-1078
Reviews of Potter (2004):
• Bays, Timothy, 2005, "Review," Notre Dame Philosophical Reviews.
• Uzquiano, Gabriel, 2005, "Review," Philosophia Mathematica 13: 308-46.
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Scutoid
A scutoid is a particular type of geometric solid between two parallel surfaces. The boundary of each of the surfaces (and of all the other parallel surfaces between them) either is a polygon or resembles a polygon, but is not necessarily planar, and the vertices of the two end polygons are joined by either a curve or a Y-shaped connection on at least one of the edges, but not necessarily all of the edges. Scutoids present at least one vertex between these two planes. Scutoids are not necessarily convex, and lateral faces are not necessarily planar, so several scutoids can pack together to fill all the space between the two parallel surfaces. They may be more generally described as a mix between a frustum and a prismatoid.[1][2]
Naming
The object was first described by Gómez-Gálvez et al. in a paper entitled Scutoids are a geometrical solution to three-dimensional packing of epithelia, and published in July 2018.[1] Officially, the name scutoid was coined because of its resemblance to the shape of the scutum and scutellum in some insects, such as beetles in the subfamily Cetoniinae.[1] Unofficially, Clara Grima has stated that while working on the project, the shape was temporarily called an Escu-toid as a joke after the biology group leader Luis M. Escudero.[3][4] Since his last name, "Escudero", means "squire" (from Latin scutarius = shield-bearer), the temporary name was modified slightly to become "scutoid".
Appearance in nature
The shape, however odd, is a building block of multicellular organisms; complex life might never have emerged on Earth without it.
— Alan Burdick, We Are All Scutoids: A Brand New Shape, Explained[2]
Epithelial cells adopt the "scutoidal shape" under certain circumstances.[1] In epithelia, cells can 3D-pack as scutoids, facilitating tissue curvature. This is fundamental to the shaping of the organs during development.[1][5][6]
"Scutoid is a prismatoid to which one extra mid-level vertex has been added. This extra vertex forces some of the "faces" of the resulting object to curve. This means that Scutoids are not polyhedra, because not all of their faces are planar. ... For the computational biologists who created/discovered the Scutoid, the key property of the shape is that it can combine with itself and other geometric objects like frustums to create 3D packings of epithelial cells."
- Laura Taalman[7][8]
Cells in the developing lung epithelium have been found to have more complex shapes than the term "scutoid", inspired by the simple scutellum of beetles, suggests.[9] When "scutoids" exhibit multiple Y-shaped connections or vertices along their axis, they have therefore been called "punakoids" instead,[10] as their shape is more reminiscent of the Pancake Rocks in Punakaiki, New Zealand.
Potential uses
The scutoid explains how epithelial cells (the cells that line and protect organs such as the skin) efficiently pack in three dimensions.[1] As epithelial tissue bends or grows, the cells have to take on new shapes to pack together using the least amount of energy possible, and until the scutoid's discovery, it was assumed that epithelial cells packed in mostly frustums, as well as other prism-like shapes.[3] Now, with the knowledge of how epithelial cells pack, it opens up many new possibilities in terms of artificial organs. The scutoid may be applied to making better artificial organs, allowing for things like effective organ replacements, recognizing if a person's cells are packing correctly or not, and ways to fix that problem.[3]
References
1. Gómez-Gálvez, Pedro; Vicente-Munuera, Pablo; Tagua, Antonio; Forja, Cristina; Castro, Ana M.; Letrán, Marta; Valencia-Expósito, Andrea; Grima, Clara; Bermúdez-Gallardo, Marina (27 July 2018). "Scutoids are a geometrical solution to three-dimensional packing of epithelia". Nature Communications. 9 (1): 2960. Bibcode:2018NatCo...9.2960G. doi:10.1038/s41467-018-05376-1. ISSN 2041-1723. PMC 6063940. PMID 30054479.
2. Burdick, Alan (30 July 2018). "We Are All Scutoids: A Brand-New Shape, Explained". The New Yorker. Retrieved 3 August 2018.
3. standupmaths (3 August 2018). "THE SCUTOID: did scientists discover a new shape?". Retrieved 3 August 2018 – via YouTube.
4. Supplementary Movie from Gómez-Gálvez et al. 2018, Electronic supplementary material
5. Boddy, Jessica. "The 'Scutoid' Is Geometry's Newest Shape, and It Could Be All Over Your Body". Gizmodo. Retrieved 29 July 2018.
6. "Scientists have discovered a brand-new three-dimensional shape". Newsweek. 27 July 2018. Retrieved 29 July 2018.
7. Taalman, Laura [@mathgrrl] (28 July 2018). "Have you read the @Nature article introducing the new mathematical shape called the SCUTOID? This cutting-edge science is now 3D printable: www.thingiverse.com/thing:3024272" (Tweet). Retrieved 3 August 2018 – via Twitter.
8. Taalman, Laura. "Pair of Packable Scutoids by mathgrrl on Shapeways". Shapeways.com. Retrieved 3 August 2018.
9. Gómez, Harold F.; Dumond, Mathilde; Hodel, Leonie; Vetter, Roman; Iber, Dagmar (5 October 2021). "3D cell neighbour dynamics in growing pseudostratified epithelia". eLife. 10: e68135. doi:10.7554/eLife.68135. PMC 8570695. PMID 34609280.
10. Iber, Dagmar; Vetter, Roman (12 May 2022). "3D Organisation of Cells in Pseudostratified Epithelia". Frontiers in Physics. 10: 898160. Bibcode:2022FrP....10.8160I. doi:10.3389/fphy.2022.898160.
External links
• THE SCUTOID: did scientists discover a new shape? - Matt Parker at Youtube
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
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Search problem
In the mathematics of computational complexity theory, computability theory, and decision theory, a search problem is a type of computational problem represented by a binary relation. Intuitively, the problem consists in finding structure "y" in object "x". An algorithm is said to solve the problem if at least one corresponding structure exists, and then one occurrence of this structure is made output; otherwise, the algorithm stops with an appropriate output ("not found" or any message of the like).
Every search problem also has a corresponding decision problem, namely
$L(R)=\{x\mid \exists yR(x,y)\}.\,$
This definition may be generalized to n-ary relations using any suitable encoding which allows multiple strings to be compressed into one string (for instance by listing them consecutively with a delimiter).
More formally, a relation R can be viewed as a search problem, and a Turing machine which calculates R is also said to solve it. More formally, if R is a binary relation such that field(R) ⊆ Γ+ and T is a Turing machine, then T calculates R if:
• If x is such that there is some y such that R(x, y) then T accepts x with output z such that R(x, z) (there may be multiple y, and T need only find one of them)
• If x is such that there is no y such that R(x, y) then T rejects x
(Note that the graph of a partial function is a binary relation, and if T calculates a partial function then there is at most one possible output.)
Such problems occur very frequently in graph theory and combinatorial optimization, for example, where searching for structures such as particular matchings, optional cliques, particular stable sets, etc. are subjects of interest.
Definition
A search problem is often characterized by:[1]
• A set of states
• A start state
• A goal state or goal test: a boolean function which tells us whether a given state is a goal state
• A successor function: a mapping from a state to a set of new states
Objective
Find a solution when not given an algorithm to solve a problem, but only a specification of what a solution looks like.[1]
Search method
• Generic search algorithm: given a graph, start nodes, and goal nodes, incrementally explore paths from the start nodes.
• Maintain a frontier of paths from the start node that have been explored.
• As search proceeds, the frontier expands into the unexplored nodes until a goal node is encountered.
• The way in which the frontier is expanded defines the search strategy.[1]
Input: a graph,
a set of start nodes,
Boolean procedure goal(n) that tests if n is a goal node.
frontier := {s : s is a start node};
while frontier is not empty:
select and remove path <n0, ..., nk> from frontier;
if goal(nk)
return <n0, ..., nk>;
for every neighbor n of nk
add <n0, ..., nk, n> to frontier;
end while
See also
• Unbounded search operator
• Decision problem
• Optimization problem
• Counting problem (complexity)
• Function problem
• Search games
References
1. Leyton-Brown, Kevin. "Graph Search" (PDF). ubc. Retrieved 7 February 2013.
This article incorporates material from search problem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Seashell surface
In mathematics, a seashell surface is a surface made by a circle which spirals up the z-axis while decreasing its own radius and distance from the z-axis. Not all seashell surfaces describe actual seashells found in nature.
Parametrization
The following is a parameterization of one seashell surface:
${\begin{aligned}x&{}={\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\cos(2v)(1+\cos u)+\cos 2v\\\\y&{}={\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\sin(2v)(1+\cos u)+\sin 2v\\\\z&{}={\frac {10v}{2\pi }}+{\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\sin(u)+15\end{aligned}}$
where $0\leq u<2\pi $ and $-2\pi \leq v<2\pi $\\
Various authors have suggested different models for the shape of shell. David M. Raup proposed a model where there is one magnification for the x-y plane, and another for the x-z plane. Chris Illert[1] proposed a model where the magnification is scalar, and the same for any sense or direction with an equation like
${\vec {F}}\left({\theta ,\varphi }\right)=e^{\alpha \varphi }\left({\begin{array}{*{20}c}{\cos \left(\varphi \right),}&{-\sin(\varphi ),}&{\rm {0}}\\{\sin(\varphi ),}&{\cos \left(\varphi \right),}&0\\{0,}&{\rm {0,}}&1\\\end{array}}\right){\vec {F}}\left({\theta ,0}\right)$
which starts with an initial generating curve ${\vec {F}}\left({\theta ,0}\right)$ and applies a rotation and exponential magnification.
See also
• Helix
• Seashell
• Spiral
References
1. Dr Chris Illert was awarded his Ph.D. on 26 September 2013 at the University of Western Sydney http://www.uws.edu.au/__data/assets/image/0004/547060/2013_ICS_Graduates.jpg.
• Weisstein, Eric W. "Seashell". MathWorld.
• C. Illert (Feb. 1983), "the mathematics of Gnomonic seashells", Mathematical Biosciences 63(1): 21-56.
• C. Illert (1987), "Part 1, seashell geometry", Il Nuovo Cimento 9D(7): 702-813.
• C. Illert (1989), "Part 2, tubular 3D seashell surfaces", Il Nuovo Cimento 11D(5): 761-780.
• C. Illert (Oct 1990),"Nipponites mirabilis, a challenge to seashell theory?", Il Nuovo Cimento 12D(10): 1405-1421.
• C. Illert (Dec 1990), "elastic conoidal spires", Il Nuovo Cimento 12D(12): 1611-1632.
• C. Illert & C. Pickover (May 1992), "generating irregularly oscillating fossil seashells", IEE Computer Graphics & Applications 12(3):18-22.
• C. Illert (July 1995), "Australian supercomputer graphics exhibition", IEEE Computer Graphics & Applications 15(4):89-91.
• C. Illert (Editor 1995), "Proceedings of the First International Conchology Conference, 2-7 Jan 1995, Tweed Shire, Australia", publ. by Hadronic Press, Florida USA. 219 pages.
• C. Illert & R. Santilli (1995), "Foundations of Theoretical Conchology", publ. by Hadronic Press, Florida USA. 183 pages plus coloured plates.
• Deborah R. Fowler, Hans Meinhardt, and Przemyslaw Prusinkiewicz. Modeling seashells. Proceedings of SIGGRAPH '92 (Chicago, Illinois, July 26–31, 1992), In Computer Graphics, 26, 2, (July 1992), ACM SIGGRAPH, New York, pp. 379–387.
• Callum Galbraith, Przemyslaw Prusinkiewicz, and Brian Wyvill. Modeling a Murex cabritii sea shell with a structured implicit surface modeler. The Visual Computer vol. 18, pp. 70–80. http://algorithmicbotany.org/papers/murex.tvc2002.html
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Seasonal subseries plot
Seasonal subseries plots are a graphical tool to visualize and detect seasonality in a time series.[1] Seasonal subseries plots involves the extraction of the seasons from a time series into a subseries. Based on a selected periodicity, it is an alternative plot that emphasizes the seasonal patterns are where the data for each season are collected together in separate mini time plots.[2]
Seasonal subseries plots enables the underlying seasonal pattern to be seen clearly, and also shows the changes in seasonality over time. Especially, it allows to detect changes between different seasons, changes within a particular season over time.
However, this plot is only useful if the period of the seasonality is already known. In many cases, this will in fact be known. For example, monthly data typically has a period of 12. If the period is not known, an autocorrelation plot or spectral plot can be used to determine it. If there is a large number of observations, then a box plot may be preferable.
Definition
Seasonal sub-series plots are formed by
• Vertical axis: response variable
• Horizontal axis: time of year; for example, with monthly data, all the January values are plotted (in chronological order), then all the February values, and so on.
The horizontal line displays the mean value for each month over the time series.
The analyst must specify the length of the seasonal pattern before generating this plot. In most cases, the analyst will know this from the context of the problem and data collection.
Importance
It is important to know when analyzing a time series if there is a significant seasonality effect. The seasonal subseries plot is an excellent tool for determining if there is a seasonal pattern. The seasonal subseries plot can provide answers to the following questions:
• Do the data exhibit a seasonal pattern?
• What is the nature of the seasonality?
• Is there a within-group pattern (e.g., do January and July exhibit similar patterns)?
• Are there any outliers once seasonality has been accounted for?
• Is the seasonality changing over time?[3]
Practically, seasonal subseries plots are often inspected as a preliminary screening tool. They allow visual inferences to be drawn from data prior to modelling and forecasting.
Related techniques
• Autocorrelation plot
• Box plot
• Recurrence plot
• Run sequence plot
• Time series
• Seasonal plot
Software
Seasonal subseries plots can be implemented in the R programming language using function monthplot().
Example
The following R code results in the above seasonal deviation plot of antidiabetic drug sales;
> monthplot(a10, ylab= "$ million" , xlab= "Month", xaxt= "n", main= "Seasonal deviation plot: antidiabetic drug sales")
> axis(1, at=1:12, labels=month.abb, cex=0.8)
References
1. "Seasonal Subseries Plot". NIST/SEMATECH e-Handbook of Statistical Methods. National Institute of Standards and Technology. Retrieved 12 May 2015.
2. 2.1 Graphics | OTexts. Retrieved 2016-05-12. {{cite book}}: |website= ignored (help)
3. Chapter 2 Time series graphics | Forecasting: Principles and Practice (2nd ed).
Further reading
• Cleveland, William (1993). Visualizing Data. Hobart Press. ISBN 9780963488404.
• Maindonald, Braun (2010). Data Analysis and Graphics Using R: An Example-Based Approach. 3rd ed. Cambridge, UK: Cambridge University Press.
• Hyndman, R.J.; Koehler, A.B. (2006). "Another look at measures of forecast accuracy". International Journal of Forecasting. 22 (4): 679–688. doi:10.1016/j.ijforecast.2006.03.001.
External links
• Seasonal Subseries Plot
This article incorporates public domain material from the National Institute of Standards and Technology.
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Sebastian Finsterwalder
Sebastian Finsterwalder (4 October 1862 – 4 December 1951) was a German mathematician and glaciologist. Acknowledged as the "father of glacier photogrammetry";[1][2] he pioneered the use of repeat photography as a temporal surveying instrument in measurement of the geology and structure of the Alps and their glacier flows.[3] The measurement techniques he developed and the data he produced are still in use to discover evidence for climate change.[4][5][lower-alpha 1][7][8]
Sebastian Finsterwalder
Born( 1862-10-04)4 October 1862
Rosenheim, Kingdom of Bavaria
Died4 December 1951(1951-12-04) (aged 89)
Munich, Germany
Alma materUniversity of Tübingen
Known for
• Aerodynamics
• Photogrammetry
• Finsterwaldersche fields method
• Froude–Finsterwalder equation
Spouse
Franziska Mallepell
(m. 1892)
Children
• Richard Finsterwalder (1899-1963), Professor at the Technical University in Hanover and Munich,
• Ulrich Finsterwalder (1897-1988), a civil engineer.
AwardsHelmert commemorative medallion for excellence by the German Association of Surveying
Scientific career
FieldsMathematics, geometry, surveying, topography, aerodynamics and geology
InstitutionsTechnical University of Munich
Doctoral advisorAlexander von Brill
Doctoral studentsHans Jörg Stetter
Life
Sebastian Finsterwalder was born 4 October 1862 in Rosenheim, son of Johann Nepomuk Finsterwalder, a master baker from Antdorf near Weilheim, Upper Bavaria, and Anna Amman of Rosenheim.[9] He died 4 December 1951 in Munich[10]).[11] He was a Bavarian mathematician and surveyor.[12] In 1892 he married Franziska Mallepell (d. 1953) from Brixen, South Tyrol. Their two sons worked in similar fields; Richard Finsterwalder (1899-1963), Professor at the Technical University in Hanover and Munich, and Ulrich Finsterwalder (1897-1988), a civil engineer.
A keen mountaineer, Finsterwalder became interested, through the influence of his friend E. Richter, in alpine fossils as indicators of the geology and structure of the Alps and their glaciers. His desire for accurate, but also less costly, motion measurements on glaciers led him to glaciological applications of photogrammetry in geodesy.[13]
In 1886, aged 24, he received his doctorate from the University of Tübingen, under the guidance of the algebraic geometer Alexander von Brill. Finsterwalder observed that Rudolf Sturm's analysis of the "homography problem" (1869) can be used to solve the problem of 3D-reconstruction using point matches in two images; which is the mathematical foundation of photogrammetry.
Finsterwalder pioneered geodetic surveys in the high mountains. At the age of 27 years he conducted a first glacier mapping project at Vernagtferner in the Ötztal Alps, Austria.
Research and applications of photogrammetry
Following the 1878 work of Italian engineer Pio Paganini of the Istituto Geografico Militare[lower-alpha 2] and others,[lower-alpha 3] Finstenwalder advanced methods for reconstruction and measurements of three-dimensional objects from photographic images.
He was appointed professor at the Technical University of Munich in 1891, succeeding his teacher, A. Voss, at the Department of Analytical Geometry, Differential and Integral Calculus (remaining at the university for forty years until 1931). The next year, he married, and completed the first recording of the Bavarian glacier in Wettersteingebirge and the Berchtesgaden Alps.
He applied the technique of plane table photogrammetry in addition to a conventional geodetic survey, assisted by the novel lightweight, accurate phototheodolite that he had developed for high-mountain applications. The device was based on the prototype phototheodolite developed by Albrecht Meydenbauer (1834-1921) for architectural applications. From 1890 Finsterwalder also employed aerial photography,[16] reconstituting the topography of the area of Gars am Inn in 1899 from a pair of balloon photographs using mathematical calculations of many points in the images.[17]
In 1897 Finsterwalder addressed the German Mathematical Society, and he described some of the results of projective geometry he was applying to photogrammetry.[18] His theory of large triangle meshes became known as the "Finsterwaldersche fields method" (1915). His analytical approach was laborious however, prompting development of analogue instrumentation with stereo measurement permitting faster optical/mechanical reconstruction of the photographic data arrays to determine object points.[19] This was assisted by new technology; Carl Pulfrich's stereocomparator (1901) and Eduard Ritter von Orel's stereoautograph (1907), both instruments built by the company Carl Zeiss.[20]
In 1911 he took over the chair of descriptive geometry, turning down offers of appointment from Vienna, Berlin and Potsdam.
Aerodynamics
Felix Klein commissioned Finsterwalder while the latter was professor of mathematics at the Munich polytechnic, to write on aerodynamics for his Enzyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (EMW) (tr. 'Encyclopedia of mathematical sciences including their applications'). The article, which he submitted in August 1902, more than a year before the Wrights achieved powered flight is thus prescient in its insights into the mathematics behind this new field of engineering. Finsterwalder also worked with Martin Kutta (1867-1944) at the Institute in Munich to devise formulas relating to the lift on an aerofoil in terms of the circulation round it. Kutta's habilitation thesis, completed in the same year, 1902, with which Finsterwalder assisted, contains the Kutta-Joukowski theorem giving the lift on an aerofoil.
Glacier flow in the Ötztal Alps
In 1922 Finsterwalder mapped the topography of the Ötztal Alps[lower-alpha 4] focusing on two glaciers, i.e. Gepatschferner and Weißseeferner, using stereophotogrammetry.[21] During this work he discovered Ölgruben rock glacier and the rock glacier north of Krummgampenspitze.[22] In 1923 and 1924 Finsterwalder measured a flow velocity profile across Ölgruben rock glacier.[23][24] Because of Finsterwalder's efforts, Ölgruben rock glacier became the subject of a notably extended, longitudinal study of flow velocity with high value in climate research,[25] with repeat surveys undertaken by Wolfgang Pillewizer in 1938, 1939, and 1953 using photogrammetry,[26] and which is still ongoing, employing modern satellite-based positioning techniques.[27] His son Richard assisted in the mapping project in the Ötztal Alps and went on to advance his father's studies.
Other contributions
Under his leadership the Bavarian International Commission for Geodesy undertook precise gravity measurements with relative gravimeters throughout Bavaria.
Honours
• 1965 Finsterwalder High School in his birthplace Rosenheim was named after him.
• Finsterwalder Glacier is named after him.
• 1915 President of the German Mathematical Society.
• 1943 awarded Helmert commemorative medallion for excellence by the German Association of Surveying.
• 1938 Asteroid 1482 (Sebastiana) was named after him.
Publications
• Finsterwalder, S. (1890) "Die Photogrammetrie in den italienischen Hochalpen," Mittheilungen des Deutschen und Österreichischen Alpenvereins, vol. 16, nº 1, 1890, pp. 6–9
• Finsterwalder, S., Muret, E., (1901). Les variations périodiques des glaciers. VIme Rapport, 1900. Extrait des Archives des Sciences physiques et naturelles 106/4 (12), 118– 131.
• Finsterwalder, S., Muret, E., (1902). Les variations périodiques des glaciers. VIIme Rapport, 1901. Extrait des Archives des Sciences physiques et naturelles 107/4 (14), 282– 302.
• Finsterwalder, S., Muret, E., (1903). Les variations périodiques des glaciers. VIIIme Rapport, 1902. Extrait des Archives des Sciences physiques et naturelles 108/4 (15), 661– 677.
• Finsterwalder, S., (1928) Geleitworte zur Karte des Gepatschferners. Zeitschrift für Gletscherkunde, 16, 20–41.
See also
• Plane table
• Rephotography
Notes
1. Terrestrial (ground-based or close-range) photogrammetry was one of the first successful methods for detecting and quantifying surface changes in rock glaciers. Flow velocity was a typical parameter derived from this. The 2D or even 3D kinematics of the rock glacier surface is needed for rheological models. In recent years, active rock glaciers have also become the focus of climate change research. Climate warming influences flow/creep velocity of rock glaciers, which can thus be seen as indicators of environmental change in mountainous regions. Melting of the subsurface ice causes surface lowering, which in the worst case may lead to active landslide activity and even a total collapse of the rock glacier surface.[6]
2. Finsterwalder remarked in 1890 that in Italy, thousands of square kilometres of alpine territory had already been photographically surveyed—with hardly anyone taking notice in Germany. What astonished Finsterwalder most was the skill with which the topographers of the I.G.M. [Istituto Topografico Militare] transformed the photos into maps. Anyone with an interest in mapmaking "will absorb himself with greatest pleasure into the many details of this map and will never stop to admire the accuracy and fidelity with which everything is overheard from nature."[14]
3. Photogrammetry – the art of making measurements using images – is the task of determining an object or its dimensions using photographs. Preliminary work on this problem was done by Lambert in what he referred to as "inverting the perspective" and by Beautemps-Beaupre (1791-1793). In surveying these methods were first tested by A. Laussedat (1852-59). Starting in 1855 I. Porro began developing instruments for photogrammetry. A. Meydenbauer brought architectural photogrammetry to high level. W. Jordan17 and C. Koppe approached the problem from the standpoint of geodesy, and G. Hauck approached it from a theoretical point of view. Photogrammetry was practiced on a large scale in Italy by L. P. Paganini since 1880 and in Canada by E. Deville since 1889. S. Finsterwalder has been doing aerial photogrammetry from balloons since 1890. C. Pulfrich has been using stereoscopy since 1890. A. Laussedat has collected material on the history of photographic methods and equipment."[15]
4. These Alps are the location in which 'Ötzi the Iceman' was found; a well-preserved natural mummy of a man who lived about 3,300 BCE.
References
1. Brunner, K., 2006. Karten dokumentieren den Rückzug der Gletscher seit 1850. In: K. Kriz, W. Cartwright, A. Pucher and M. Kinberger (eds), Kartographie als Kommunikationsmedium. Wiener Schriften zur Geographie und Kartographie, 17, Institut für Geographie und Regionalforschung, Universität Wien, pp. 191–200.
2. Rinner, K. and Burkhardt, R. (eds), 1972. Gletscherphotogrammetrie. In: Handbuch der Vermessungskunde. Photogrammetrie, Band III a/2, in German, J.B. Metzlersche Verlagsbuchhandlung, Stuttgart, pp. 1428–1470.
3. Konecny, G. (2014). Geoinformation: remote sensing, photogrammetry and geographic information systems. CRC Press.
4. Kaiser, T. (2014). Implications of changing climate on Zugspitze glaciers in southern Germany. 12th issue• August 2014.
5. Bavarian glaciers in climate change - a status report. Bavarian State Ministry for Environment and Health, Munich, 2012, p 21
6. Kaufmann, V. (2012) 'The evolution of rock glacier monitoring using terrestrial photogrammetry: the example of Äußeres Hochebenkar rock glacier (Austria)' Austrian Journal of Earth Sciences Volume 105/2 Vienna 2012 pp. 63–7
7. Keutterling, A. Thomas, A. (2006) Monitoring glacier elevation and volume changes with digital photogrammetry and GIS at Gepatschferner glacier, Austria International Journal of Remote Sensing Vol. 27, Iss. 19, 2006
8. Finsterwalder, S., (1928) Geleitworte zur Karte des Gepatschferners. Zeitschrift für Gletscherkunde, 16, 20-41.
9. (in German) Führung durch „Rosenheim wird Stadt“ Archived 23 April 2021 at the Wayback Machine
10. (in German) Finsterwalder, Sebastian Mathematiker, * 4.10.1862 Rosenheim/Inn, † 4.12.1951 München. (katholisch) Deutsche Biographie
11. Walther HOFMANN: Sebastian Finsterwalder, in: Neue Deutsche Biographie Bd. 5, S. 166-167. [Walther Hofmann Sebastian Finsterwalder, in: New German Biography Vol 5, pp. 166–167. ]
12. Robert SAUER / Max KNEISSL: Sebastian Finsterwalder, in: Jahrbuch der Bayerischen Akademie der Wissenschaften für 1952, S. 200-204. [Robert SAUER / Max KNEISSL Sebastian Finsterwalder, in: Yearbook of the Bavarian Academy of Sciences, 1952, pp. 200–204.]
13. Albertz, J. (2010). 100 Years German Society for Photogrammetry, Remote Sensing, and Geoinformation. Deutsche Gesellschaft für Photogrammetrie, Fernerkundung und Geoinformatione.V., ISBN 978-3-00-031038-6, 144 pp.
14. Albertz, J., 2010. 100 Years German Society for Photogrammetry, Remote Sensing, and Geoinformation. Deutsche Gesellschaft für Photogrammetrie, Fernerkundung und Geoinformatione.V., ISBN 978-3-00-031038-6, 144 pp.
15. Finsterwalder, S. (1906) Photogrammetrie. In: Encyklopcidie der Mathematischen Wissenschaften mit Einschluft ihrer Anwendungen. Band VI, Teil1, Geodcisie und Geophysik. Leipzig: B.G. Teubner 1906-1925. pp. 98–116.]
16. Kneissl, M. (1942) Sebastian Finsterwalder zum 80. Geburtstag. Bildmessung und Luftbildwesen. 11, 53-64.
17. Finsterwalder, S.: Eine Grundaufgabe der Photogrammetrie und ihre Anwendung auf Ballonaufnahmen. Abh. Bayer. Akad. Wiss., 2. Abt. 22, 225-260 (1903).
18. Finsterwalder, S. (1897) Die geometrischen Grundlagen der Photogrammetrie. Jahresber deutsch Math-Verein. 6 (2), 1-41
19. Konecny, G. (2002) Geoinformation: Remote Sensing, Photogrammetry and Geographic Information Systems. CRC Press. p. 9
20. Finsterwalder was doctoral advisor to Heinrich Erfle (1884–1923) a German optician who spent most of his career at Carl Zeiss.
21. Finsterwalder, S., (1928) Geleitworte zur Karte des Gepatschferners. Zeitschrift für Gletscherkunde, 16, 20-41.
22. (in German) Östliche Krummgampenspitze 3090 m 10135 ft.
23. Finsterwalder, S., (1928) Geleitworte zur Karte des Gepatschferners. Zeitschrift für Gletscherkunde, 16, 20-41
24. Pillewizer, W. (1957). Untersuchungen an Blockströmen der Ötztaler Alpen. In: E. Fels (ed), Geomorphologische Abhandlungen: Otto Maull zum 70. Geburtstage gewidmet. Abhandlungen des Geographischen Instituts der Freien Universität Berlin, 5, pp. 37–50.
25. Fischer, Andrea (2013) 'Long-term glacier monitoring at the LTER test sites Hintereisferner, Kesselwandferner and Jamtalferner and other glaciers in Tyrol: a source of ancillary information for biological succession studies'. In Plant Ecology & Diversity Volume 6, Issue 3-4, December 2013, pages 537-547 Published online: 27 Sep 2013
26. Pillewizer, W. (1957). Untersuchungen an Blockströmen der Ötztaler Alpen. In: E. Fels (ed), Geomorphologische Abhandlungen: Otto Maull zum 70. Geburtstage gewidmet. Abhandlungen des Geographischen Instituts der Freien Universität Berlin, 5, pp. 37–50: see Figure 2
27. Hausmann, H, Krainer, K., Brückl, E. and Mostler, W.(2007). Creep of Two Alpine Rock Glaciers – Observation and Modelling (Ötztal- and Stubai Alps, Austria). In: V. Kaufmann & W. Sulzer (eds), Proceedings of the 9th International Symposium on High Mountain Remote Sensing Cartography. Grazer Schriften der Geographie und Raumforschung, 43, Institute of Geography and Regional Science, University of Graz, 145-150.
Literature and links
• Seligman, G. (1949) Research on Glacier Flow. An Historical Outline. Geografiska Annaler, Vol. 31, Glaciers and Climate: Geophysical and Geomorphological Essays, Wiley / Swedish Society for Anthropology and Geography pp. 228–238
• Kaufmann, V. (2012) The evolution of rock glacier monitoring using terrestrial photogrammetry: the example of Äußeres Hochebenkar rock glacier (Austria) Austrian Journal of Earth Sciences Volume 105/2 Vienna 2012 pp. 63–77
• Leather Charles Steger : Astronomical and Physical Geodesy Volume 5 of the "Manual of Surveying" (ed. Jordan Eggert Kneissl, publishing JBMetzler, Stuttgart in 1969.
• Walther Welsch et al. evaluation of geodetic monitoring measurements. Manual of Engineering Geodesy (ed. M.Möser, H.Schlemmer et al.), Wichmann-Verlag Heidelberg, 2000.
• G. Clauß, in: Zs. f. Vermessungswesen, 1932, S. 721-26 ( P );
• R. Rehlen, H. Heß u. M. Lagally, in: Zs. f. Gletscherkde. 20, 1932, S. IX-XXI ( P )
• O. v. Gruber, in: S. F. z. 75. Geburtstag, Festschr. d. Dt. Ges. f. Photogrammetrie, 1937;
• M. Kneißl, S. F. z. 80. Geburtstag, in: Bildmessung u. Luftbildwesen 17, 1942, S. 53-64 ( vollst. W- Verz., 123 Nr. )
• ders., in: Zs. f. Vermessungswesen 77, 1952, S. 1-3 ( P )
• Richard Finsterwalder, in: SB d. Bayer. Ak. d. Wiss., 1953, S. 257;
• ders., in: Geist u. Gestalt, Biogr. Btrr. z. Gesch. d. Bayer. Ak. d. Wiss...II, 1959, S. 65-69 ( L )
• G. Faber, ebd., S. 34 f. ( P ebd. III, S. 183);
• Pogg. IV-VII a. – Slg. math. Modelle v. F. im Math. Inst. d. TH München.
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Wikipedia
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Secant method
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method. However, the secant method predates Newton's method by over 3000 years.[1]
The method
For finding a zero of a function f, the secant method is defined by the recurrence relation.
$x_{n}=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}={\frac {x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}}.$
As can be seen from this formula, two initial values x0 and x1 are required. Ideally, they should be chosen close to the desired zero.
Derivation of the method
Starting with initial values x0 and x1, we construct a line through the points (x0, f(x0)) and (x1, f(x1)), as shown in the picture above. In slope–intercept form, the equation of this line is
$y={\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{0})+f(x_{0}).$
The root of this linear function, that is the value of x such that y = 0 is
$x=x_{1}-f(x_{1}){\frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}}.$
We then use this new value of x as x2 and repeat the process, using x1 and x2 instead of x0 and x1. We continue this process, solving for x3, x4, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between xn and xn−1):
${\begin{aligned}x_{2}&=x_{1}-f(x_{1}){\frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}},\\[6pt]x_{3}&=x_{2}-f(x_{2}){\frac {x_{2}-x_{1}}{f(x_{2})-f(x_{1})}},\\[6pt]&\,\,\,\vdots \\[6pt]x_{n}&=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}.\end{aligned}}$
Convergence
The iterates $x_{n}$ of the secant method converge to a root of $f$ if the initial values $x_{0}$ and $x_{1}$ are sufficiently close to the root. The order of convergence is $\varphi $, where
$\varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618$
is the golden ratio. In particular, the convergence is super linear, but not quite quadratic.
This result only holds under some technical conditions, namely that $f$ be twice continuously differentiable and the root in question be simple (i.e., with multiplicity 1).
If the initial values are not close enough to the root, then there is no guarantee that the secant method converges. There is no general definition of "close enough", but the criterion has to do with how "wiggly" the function is on the interval $[x_{0},x_{1}]$. For example, if $f$ is differentiable on that interval and there is a point where $f'=0$ on the interval, then the algorithm may not converge.
Comparison with other root-finding methods
The secant method does not require that the root remain bracketed, like the bisection method does, and hence it does not always converge. The false position method (or regula falsi) uses the same formula as the secant method. However, it does not apply the formula on $x_{n-1}$ and $x_{n-2}$, like the secant method, but on $x_{n-1}$ and on the last iterate $x_{k}$ such that $f(x_{k})$ and $f(x_{n-1})$ have a different sign. This means that the false position method always converges; however, only with a linear order of convergence. Bracketing with a super-linear order of convergence as the secant method can be attained with improvements to the false position method (see Regula falsi § Improvements in regula falsi) such as the ITP method or Illinois method.
The recurrence formula of the secant method can be derived from the formula for Newton's method
$x_{n}=x_{n-1}-{\frac {f(x_{n-1})}{f'(x_{n-1})}}$
by using the finite-difference approximation, for a small $\epsilon $:
$f'(x_{n-1})\approx {\frac {f(x_{n-1})-f(x_{n-2})}{x_{n-1}-x_{n-2}}}\approx {\frac {f(x_{n-1}+{\frac {\epsilon }{2}})-f(x_{n-1}-{\frac {\epsilon }{2}})}{\epsilon }}$
The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a quasi-Newton method.
If we compare Newton's method with the secant method, we see that Newton's method converges faster (order 2 against φ ≈ 1.6). However, Newton's method requires the evaluation of both $f$ and its derivative $f'$ at every step, while the secant method only requires the evaluation of $f$. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating $f$ takes as much time as evaluating its derivative and we neglect all other costs, we can do two steps of the secant method (decreasing the logarithm of the error by a factor φ2 ≈ 2.6) for the same cost as one step of Newton's method (decreasing the logarithm of the error by a factor 2), so the secant method is faster. If, however, we consider parallel processing for the evaluation of the derivative, Newton's method proves its worth, being faster in time, though still spending more steps.
Generalization
Broyden's method is a generalization of the secant method to more than one dimension.
The following graph shows the function f in red and the last secant line in bold blue. In the graph, the x intercept of the secant line seems to be a good approximation of the root of f.
Computational example
Below, the secant method is implemented in the Python programming language.
It is then applied to find a root of the function f(x) = x2 − 612 with initial points $x_{0}=10$ and $x_{1}=30$
def secant_method(f, x0, x1, iterations):
"""Return the root calculated using the secant method."""
for i in range(iterations):
x2 = x1 - f(x1) * (x1 - x0) / float(f(x1) - f(x0))
x0, x1 = x1, x2
# Apply a stopping criterion here (see below)
return x2
def f_example(x):
return x ** 2 - 612
root = secant_method(f_example, 10, 30, 5)
print(f"Root: {root}") # Root: 24.738633748750722
It is very important to have a good stopping criterion above, otherwise, due to limited numerical precision of floating point numbers, the algorithm can return inaccurate results if running for too many iterations. For example, the loop above can stop when one of these is reached first: abs(x0 - x1) < tol, or abs(x0/x1-1) < tol, or abs(f(x1)) < tol. [2]
Notes
1. Papakonstantinou, Joanna; Tapia, Richard (2013). "Origin and evolution of the secant method in one dimension". American Mathematical Monthly. 120 (6): 500–518. doi:10.4169/amer.math.monthly.120.06.500. JSTOR 10.4169/amer.math.monthly.120.06.500. S2CID 17645996 – via JSTOR.
2. "MATLAB TUTORIAL for the First Course. Part 1.3: Secant Methods".
See also
• False position method
References
• Avriel, Mordecai (1976). Nonlinear Programming: Analysis and Methods. Prentice Hall. pp. 220–221. ISBN 0-13-623603-0.
• Allen, Myron B.; Isaacson, Eli L. (1998). Numerical analysis for applied science. John Wiley & Sons. pp. 188–195. ISBN 978-0-471-55266-6.
External links
• Secant Method Notes, PPT, Mathcad, Maple, Mathematica, Matlab at Holistic Numerical Methods Institute
• Weisstein, Eric W. "Secant Method". MathWorld.
Root-finding algorithms
Bracketing (no derivative)
• Bisection method
• Regula falsi
• ITP Method
Newton
• Newton's method
Quasi-Newton
• Muller's method
• Secant method
Hybrid methods
• Brent's method
• Ridders' method
Polynomial methods
• Bairstow's method
• Jenkins–Traub method
• Laguerre's method
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Wikipedia
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Secant variety
In algebraic geometry, the secant variety $\operatorname {Sect} (V)$, or the variety of chords, of a projective variety $V\subset \mathbb {P} ^{r}$ is the Zariski closure of the union of all secant lines (chords) to V in $\mathbb {P} ^{r}$:[1]
$\operatorname {Sect} (V)=\bigcup _{x,y\in V}{\overline {xy}}$
(for $x=y$, the line ${\overline {xy}}$ is the tangent line.) It is also the image under the projection $p_{3}:(\mathbb {P} ^{r})^{3}\to \mathbb {P} ^{r}$ of the closure Z of the incidence variety
$\{(x,y,r)|x\wedge y\wedge r=0\}$.
Note that Z has dimension $2\dim V+1$ and so $\operatorname {Sect} (V)$ has dimension at most $2\dim V+1$.
More generally, the $k^{th}$ secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on $V$. It may be denoted by $\Sigma _{k}$. The above secant variety is the first secant variety. Unless $\Sigma _{k}=\mathbb {P} ^{r}$, it is always singular along $\Sigma _{k-1}$, but may have other singular points.
If $V$ has dimension d, the dimension of $\Sigma _{k}$ is at most $kd+d+k$. A useful tool for computing the dimension of a secant variety is Terracini's lemma.
Examples
A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space $\mathbb {P} ^{3}$ as follows.[2] Let $C\subset \mathbb {P} ^{r}$ be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if $r>3$, then there is a point p on $\mathbb {P} ^{r}$ that is not on S and so we have the projection $\pi _{p}$ from p to a hyperplane H, which gives the embedding $\pi _{p}:C\hookrightarrow H\simeq \mathbb {P} ^{r-1}$. Now repeat.
If $S\subset \mathbb {P} ^{5}$ is a surface that does not lie in a hyperplane and if $\operatorname {Sect} (S)\neq \mathbb {P} ^{5}$, then S is a Veronese surface.[3]
References
1. Griffiths & Harris 1994, pg. 173
2. Griffiths & Harris 1994, pg. 215
3. Griffiths & Harris 1994, pg. 179
• Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., ISBN 978-1107602724
• Griffiths, P.; Harris, J. (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 617. ISBN 0-471-05059-8.
• Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3
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Wikipedia
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Secant line
In geometry, a secant is a line that intersects a curve at a minimum of two distinct points.[1] The word secant comes from the Latin word secare, meaning to cut.[2] In the case of a circle, a secant intersects the circle at exactly two points. A chord is the line segment determined by the two points, that is, the interval on the secant whose ends are the two points.[3]
For the secant trigonometric function, see Secant (trigonometry).
Circles
Further information: Circle § Chord
A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a secant line, at one point a tangent line and at no points an exterior line. A chord is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord.
In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in his treatment, are usually proved.
For example, Theorem (Elementary Circular Continuity):[4] If ${\mathcal {C}}$ is a circle and $\ell $ a line that contains a point A that is inside ${\mathcal {C}}$ and a point B that is outside of ${\mathcal {C}}$ then $\ell $ is a secant line for ${\mathcal {C}}$.
In some situations phrasing results in terms of secant lines instead of chords can help to unify statements. As an example of this consider the result:[5]
If two secant lines contain chords AB and CD in a circle and intersect at a point P that is not on the circle, then the line segment lengths satisfy AP⋅PB = CP⋅PD.
If the point P lies inside the circle this is Euclid III.35, but if the point is outside the circle the result is not contained in the Elements. However, Robert Simson following Christopher Clavius demonstrated this result, sometimes called the intersecting secants theorem, in their commentaries on Euclid.[6]
Curves
For curves more complicated than simple circles, the possibility that a line that intersects a curve in more than two distinct points arises. Some authors define a secant line to a curve as a line that intersects the curve in two distinct points. This definition leaves open the possibility that the line may have other points of intersection with the curve. When phrased this way the definitions of a secant line for circles and curves are identical and the possibility of additional points of intersection just does not occur for a circle.
Secants and tangents
Secants may be used to approximate the tangent line to a curve, at some point P, if it exists. Define a secant to a curve by two points, P and Q, with P fixed and Q variable. As Q approaches P along the curve, if the slope of the secant approaches a limit value, then that limit defines the slope of the tangent line at P.[1] The secant lines PQ are the approximations to the tangent line. In calculus, this idea is the geometric definition of the derivative.
A tangent line to a curve at a point P may be a secant line to that curve if it intersects the curve in at least one point other than P. Another way to look at this is to realize that being a tangent line at a point P is a local property, depending only on the curve in the immediate neighborhood of P, while being a secant line is a global property since the entire domain of the function producing the curve needs to be examined.
Sets and n-secants
The concept of a secant line can be applied in a more general setting than Euclidean space. Let K be a finite set of k points in some geometric setting. A line will be called an n-secant of K if it contains exactly n points of K.[7] For example, if K is a set of 50 points arranged on a circle in the Euclidean plane, a line joining two of them would be a 2-secant (or bisecant) and a line passing through only one of them would be a 1-secant (or unisecant). A unisecant in this example need not be a tangent line to the circle.
This terminology is often used in incidence geometry and discrete geometry. For instance, the Sylvester–Gallai theorem of incidence geometry states that if n points of Euclidean geometry are not collinear then there must exist a 2-secant of them. And the original orchard-planting problem of discrete geometry asks for a bound on the number of 3-secants of a finite set of points.
Finiteness of the set of points is not essential in this definition, as long as each line can intersect the set in only a finite number of points.
See also
• Elliptic curve, a curve for which every secant has a third point of intersection, from which most of a group law may be defined
• Mean value theorem, that every secant of the graph of a smooth function has a parallel tangent line
• Quadrisecant, a line that intersects four points of a curve (usually a space curve)
• Secant plane, the three-dimensional equivalent of a secant line
• Secant variety, the union of secant lines and tangent lines to a given projective variety
References
1. Protter, Murray H.; Protter, Philip E. (1988), Calculus with Analytic Geometry, Jones & Bartlett Learning, p. 62, ISBN 9780867200935.
2. Redgrove, Herbert Stanley (1913), Experimental Mensuration: An Elementary Test-book of Inductive Geometry, Van Nostrand, p. 167.
3. Gullberg, Jan (1997), Mathematics: From the Birth of Numbers, W. W. Norton & Company, p. 387, ISBN 9780393040029.
4. Venema, Gerard A. (2006), Foundations of Geometry, Pearson/Prentice-Hall, p. 229, ISBN 978-0-13-143700-5
5. Jacobs, Harold R. (1974), Geometry, W. H. Freeman & Co., p. 482, ISBN 0-7167-0456-0
6. Heath, Thomas L. (1956), The thirteen books of Euclid's Elements (Vol. 2), Dover, p. 73
7. Hirschfeld, J. W. P. (1979), Projective Geometries over Finite Fields, Oxford University Press, p. 70, ISBN 0-19-853526-0
External links
• Weisstein, Eric W. "Secant line". MathWorld.
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Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space $T$ is second-countable if there exists some countable collection ${\mathcal {U}}=\{U_{i}\}_{i=1}^{\infty }$ of open subsets of $T$ such that any open subset of $T$ can be written as a union of elements of some subfamily of ${\mathcal {U}}$. A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.
Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rn) with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.
Properties
Second-countability is a stronger notion than first-countability. A space is first-countable if each point has a countable local base. Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space. However any uncountable discrete space is first-countable but not second-countable.
Second-countability implies certain other topological properties. Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover). The reverse implications do not hold. For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.[1] Therefore, the lower limit topology on the real line is not metrizable.
In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactness are all equivalent properties.
Urysohn's metrization theorem states that every second-countable, Hausdorff regular space is metrizable. It follows that every such space is completely normal as well as paracompact. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.
Other properties
• A continuous, open image of a second-countable space is second-countable.
• Every subspace of a second-countable space is second-countable.
• Quotients of second-countable spaces need not be second-countable; however, open quotients always are.
• Any countable product of a second-countable space is second-countable, although uncountable products need not be.
• The topology of a second-countable T1 space has cardinality less than or equal to c (the cardinality of the continuum).
• Any base for a second-countable space has a countable subfamily which is still a base.
• Every collection of disjoint open sets in a second-countable space is countable.
Examples and counterexamples
• Consider the disjoint countable union $X=[0,1]\cup [2,3]\cup [4,5]\cup \dots \cup [2k,2k+1]\cup \dotsb $. Define an equivalence relation and a quotient topology by identifying the left ends of the intervals - that is, identify 0 ~ 2 ~ 4 ~ … ~ 2k and so on. X is second-countable, as a countable union of second-countable spaces. However, X/~ is not first-countable at the coset of the identified points and hence also not second-countable.
• The above space is not homeomorphic to the same set of equivalence classes endowed with the obvious metric: i.e. regular Euclidean distance for two points in the same interval, and the sum of the distances to the left hand point for points not in the same interval -- yielding a strictly coarser topology than the above space. It is a separable metric space (consider the set of rational points), and hence is second-countable.
• The long line is not second-countable, but is first-countable.
Notes
1. Willard, theorem 16.11, p. 112
References
• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4
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Second derivative
In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation:
$\mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}},$
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where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change. The last expression ${\tfrac {d^{2}{\boldsymbol {x}}}{dt^{2}}}$ is the second derivative of position (x) with respect to time.
On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.
Second derivative power rule
The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows:
${\frac {d^{2}}{dx^{2}}}\left[x^{n}\right]={\frac {d}{dx}}{\frac {d}{dx}}\left[x^{n}\right]={\frac {d}{dx}}\left[nx^{n-1}\right]=n{\frac {d}{dx}}\left[x^{n-1}\right]=n(n-1)x^{n-2}.$
Notation
Further information: Notation for differentiation
The second derivative of a function $f(x)$ is usually denoted $f''(x)$.[1][2] That is:
$f''=\left(f'\right)'$
When using Leibniz's notation for derivatives, the second derivative of a dependent variable y with respect to an independent variable x is written
${\frac {d^{2}y}{dx^{2}}}.$
This notation is derived from the following formula:
${\frac {d^{2}y}{dx^{2}}}\,=\,{\frac {d}{dx}}\left({\frac {dy}{dx}}\right).$
Example
Given the function
$f(x)=x^{3},$
the derivative of f is the function
$f^{\prime }(x)=3x^{2}.$
The second derivative of f is the derivative of $f^{\prime }$, namely
$f^{\prime \prime }(x)=6x.$
Relation to the graph
Concavity
The second derivative of a function f can be used to determine the concavity of the graph of f.[2] A function whose second derivative is positive will be concave up (also referred to as convex), meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (also simply called concave), and its tangent lines will lie above the graph of the function.
Inflection points
Main article: Inflection point
If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.
Second derivative test
Main article: Second derivative test
The relation between the second derivative and the graph can be used to test whether a stationary point for a function (i.e., a point where $f'(x)=0$) is a local maximum or a local minimum. Specifically,
• If $f^{\prime \prime }(x)<0$, then $f$ has a local maximum at $x$.
• If $f^{\prime \prime }(x)>0$, then $f$ has a local minimum at $x$.
• If $f^{\prime \prime }(x)=0$, the second derivative test says nothing about the point $x$, a possible inflection point.
The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration. Clearly, the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time, the velocity will become negative and the vehicle will reverse. The same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration.
Limit
It is possible to write a single limit for the second derivative:
$f''(x)=\lim _{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.$
The limit is called the second symmetric derivative.[3][4] Note that the second symmetric derivative may exist even when the (usual) second derivative does not.
The expression on the right can be written as a difference quotient of difference quotients:
${\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}.$
This limit can be viewed as a continuous version of the second difference for sequences.
However, the existence of the above limit does not mean that the function $f$ has a second derivative. The limit above just gives a possibility for calculating the second derivative—but does not provide a definition. A counterexample is the sign function $\operatorname {sgn}(x)$, which is defined as:
$\operatorname {sgn}(x)={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}$
The sign function is not continuous at zero, and therefore the second derivative for $x=0$ does not exist. But the above limit exists for $x=0$:
${\begin{aligned}\lim _{h\to 0}{\frac {\operatorname {sgn}(0+h)-2\operatorname {sgn}(0)+\operatorname {sgn}(0-h)}{h^{2}}}&=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)-2\cdot 0+\operatorname {sgn}(-h)}{h^{2}}}\\&=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)+(-\operatorname {sgn}(h))}{h^{2}}}=\lim _{h\to 0}{\frac {0}{h^{2}}}=0.\end{aligned}}$
Quadratic approximation
Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function f. This is the quadratic function whose first and second derivatives are the same as those of f at a given point. The formula for the best quadratic approximation to a function f around the point x = a is
$f(x)\approx f(a)+f'(a)(x-a)+{\tfrac {1}{2}}f''(a)(x-a)^{2}.$
This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a.
Eigenvalues and eigenvectors of the second derivative
For many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors of the second derivative can be obtained. For example, assuming $x\in [0,L]$ and homogeneous Dirichlet boundary conditions (i.e., $v(0)=v(L)=0$), the eigenvalues are $\lambda _{j}=-{\tfrac {j^{2}\pi ^{2}}{L^{2}}}$ and the corresponding eigenvectors (also called eigenfunctions) are $v_{j}(x)={\sqrt {\tfrac {2}{L}}}\sin \left({\tfrac {j\pi x}{L}}\right)$. Here, $v''_{j}(x)=\lambda _{j}v_{j}(x),\,j=1,\ldots ,\infty .$
For other well-known cases, see Eigenvalues and eigenvectors of the second derivative.
Generalization to higher dimensions
The Hessian
Main article: Hessian matrix
The second derivative generalizes to higher dimensions through the notion of second partial derivatives. For a function f: R3 → R, these include the three second-order partials
${\frac {\partial ^{2}f}{\partial x^{2}}},\;{\frac {\partial ^{2}f}{\partial y^{2}}},{\text{ and }}{\frac {\partial ^{2}f}{\partial z^{2}}}$
and the mixed partials
${\frac {\partial ^{2}f}{\partial x\,\partial y}},\;{\frac {\partial ^{2}f}{\partial x\,\partial z}},{\text{ and }}{\frac {\partial ^{2}f}{\partial y\,\partial z}}.$
If the function's image and domain both have a potential, then these fit together into a symmetric matrix known as the Hessian. The eigenvalues of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the second partial derivative test.)
The Laplacian
Main article: Laplace operator
Another common generalization of the second derivative is the Laplacian. This is the differential operator $\nabla ^{2}$ (or $\Delta $) defined by
$\nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.$
The Laplacian of a function is equal to the divergence of the gradient, and the trace of the Hessian matrix.
See also
• Chirpyness, second derivative of instantaneous phase
• Finite difference, used to approximate second derivative
• Second partial derivative test
• Symmetry of second derivatives
References
1. "Content - The second derivative". amsi.org.au. Retrieved 2020-09-16.
2. "Second Derivatives". Math24. Retrieved 2020-09-16.
3. A. Zygmund (2002). Trigonometric Series. Cambridge University Press. pp. 22–23. ISBN 978-0-521-89053-3.
4. Thomson, Brian S. (1994). Symmetric Properties of Real Functions. Marcel Dekker. p. 1. ISBN 0-8247-9230-0.
Further reading
Print
• Anton, Howard; Bivens, Irl; Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single and Multivariable (8th ed.), New York: Wiley, ISBN 978-0-471-47244-5
• Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1
• Apostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications, vol. 1 (2nd ed.), Wiley, ISBN 978-0-471-00007-5
• Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN 978-0-03-029558-4
• Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company, ISBN 978-0-618-60624-5
• Spivak, Michael (September 1994), Calculus (3rd ed.), Publish or Perish, ISBN 978-0-914098-89-8
• Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7
• Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), New York: St. Martin's Press, ISBN 978-0-312-18548-0
Online books
• Crowell, Benjamin (2003), Calculus
• Garrett, Paul (2004), Notes on First-Year Calculus
• Hussain, Faraz (2006), Understanding Calculus
• Keisler, H. Jerome (2000), Elementary Calculus: An Approach Using Infinitesimals
• Mauch, Sean (2004), Unabridged Version of Sean's Applied Math Book, archived from the original on 2006-04-15
• Sloughter, Dan (2000), Difference Equations to Differential Equations
• Strang, Gilbert (1991), Calculus
• Stroyan, Keith D. (1997), A Brief Introduction to Infinitesimal Calculus, archived from the original on 2005-09-11
• Wikibooks, Calculus
External links
• Discrete Second Derivative from Unevenly Spaced Points
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Second-order propositional logic
A second-order propositional logic is a propositional logic extended with quantification over propositions. A special case are the logics that allow second-order Boolean propositions, where quantifiers may range either just over the Boolean truth values, or over the Boolean-valued truth functions.
The most widely known formalism is the intuitionistic logic with impredicative quantification, System F. Parigot (1997) showed how this calculus can be extended to admit classical logic.
See also
• Boolean satisfiability problem
• Second-order arithmetic
• Second-order logic
• Type theory
References
• Parigot, Michel (Dec 1997). "Proofs of strong normalisation for second order classical natural deduction". Journal of Symbolic Logic (published 12 March 2014). 62 (4): 1461–1479. doi:10.2307/2275652. ISSN 0022-4812. JSTOR 2275652.
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Feigenbaum constants
In mathematics, specifically bifurcation theory, the Feigenbaum constants /ˈfaɪɡənˌbaʊm/[1] are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum.
History
Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate. Feigenbaum made this discovery in 1975,[2][3] and he officially published it in 1978.[4]
The first constant
The first Feigenbaum constant δ is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map
$x_{i+1}=f(x_{i}),$
where f(x) is a function parameterized by the bifurcation parameter a.
It is given by the limit[5]
$\delta =\lim _{n\to \infty }{\frac {a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}}=4.669\,201\,609\,\ldots ,$
where an are discrete values of a at the nth period doubling.
Names
• Feigenbaum constant
• Feigenbaum bifurcation velocity
• delta
Value
• 30 decimal places : δ = 4.669201609102990671853203820466…
• (sequence A006890 in the OEIS)
• A simple rational approximation is: 621/133, which is correct to 5 significant values (when rounding). For more precision use 1228/263, which is correct to 7 significant values.
• Is approximately equal to 10(1/π − 1), with an error of 0.0047%
Non-linear maps
To see how this number arises, consider the real one-parameter map
$f(x)=a-x^{2}.$
Here a is the bifurcation parameter, x is the variable. The values of a for which the period doubles (e.g. the largest value for a with no period-2 orbit, or the largest a with no period-4 orbit), are a1, a2 etc. These are tabulated below:[6]
n Period Bifurcation parameter (an) Ratio an−1 − an−2/an − an−1
1 2 0.75 —
2 4 1.25 —
3 8 1.3680989 4.2337
4 16 1.3940462 4.5515
5 32 1.3996312 4.6458
6 64 1.4008286 4.6639
7 128 1.4010853 4.6682
8 256 1.4011402 4.6689
The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the logistic map
$f(x)=ax(1-x)$
with real parameter a and variable x. Tabulating the bifurcation values again:[7]
n Period Bifurcation parameter (an) Ratio an−1 − an−2/an − an−1
1 2 3 —
2 4 3.4494897 —
3 8 3.5440903 4.7514
4 16 3.5644073 4.6562
5 32 3.5687594 4.6683
6 64 3.5696916 4.6686
7 128 3.5698913 4.6692
8 256 3.5699340 4.6694
Fractals
In the case of the Mandelbrot set for complex quadratic polynomial
$f(z)=z^{2}+c$
the Feigenbaum constant is the limiting ratio between the diameters of successive circles on the real axis in the complex plane (see animation on the right).
n Period = 2n Bifurcation parameter (cn) Ratio $={\dfrac {c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}}$
1 2 −0.75 —
2 4 −1.25 —
3 8 −1.3680989 4.2337
4 16 −1.3940462 4.5515
5 32 −1.3996312 4.6458
6 64 −1.4008287 4.6639
7 128 −1.4010853 4.6682
8 256 −1.4011402 4.6689
9 512 −1.401151982029
10 1024 −1.401154502237
∞ −1.4011551890…
Bifurcation parameter is a root point of period-2n component. This series converges to the Feigenbaum point c = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.
Other maps also reproduce this ratio; in this sense the Feigenbaum constant in bifurcation theory is analogous to π in geometry and e in calculus.
The second constant
The second Feigenbaum constant or Feigenbaum's alpha constant (sequence A006891 in the OEIS),
$\alpha =2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218...,$
is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold). A negative sign is applied to α when the ratio between the lower subtine and the width of the tine is measured.[8]
These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).[8]
A simple rational approximation is 13/11 × 17/11 × 37/27 = 8177/3267.
Other values
The period-3 window in the logistic map also has a period-doubling route to chaos, and it has its own two Feigenbaum constants. $\delta =55.26,\alpha =9.277$ (Appendix F.2 [9]).
Properties
Both numbers are believed to be transcendental, although they have not been proven to be so.[10] In fact, there is no known proof that either constant is even irrational.
The first proof of the universality of the Feigenbaum constants was carried out by Oscar Lanford—with computer-assistance—in 1982[11] (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987[12]). Over the years, non-numerical methods were discovered for different parts of the proof, aiding Mikhail Lyubich in producing the first complete non-numerical proof.[13]
See also
• Bifurcation diagram
• Bifurcation theory
• Cascading failure
• Feigenbaum function
• List of chaotic maps
Notes
1. The Feigenbaum Constant (4.669) - Numberphile, retrieved 7 February 2023
2. Feigenbaum, M. J. (1976). "Universality in complex discrete dynamics" (PDF). Los Alamos Theoretical Division Annual Report 1975–1976.
3. Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996). Chaos: An Introduction to Dynamical Systems. Springer. ISBN 0-387-94677-2.
4. Feigenbaum, Mitchell J. (1978). "Quantitative universality for a class of nonlinear transformations". Journal of Statistical Physics. 19 (1): 25–52. Bibcode:1978JSP....19...25F. doi:10.1007/BF01020332. S2CID 124498882.
5. Jordan, D. W.; Smith, P. (2007). Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th ed.). Oxford University Press. ISBN 978-0-19-920825-8.
6. Alligood, p. 503.
7. Alligood, p. 504.
8. Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos. Studies in Nonlinearity. Perseus Books. ISBN 978-0-7382-0453-6.
9. Hilborn, Robert C. (2000). Chaos and nonlinear dynamics: an introduction for scientists and engineers (2nd ed.). Oxford: Oxford University Press. ISBN 0-19-850723-2. OCLC 44737300.
10. Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
11. Lanford III, Oscar (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc. 6 (3): 427–434. doi:10.1090/S0273-0979-1982-15008-X.
12. Eckmann, J. P.; Wittwer, P. (1987). "A complete proof of the Feigenbaum conjectures". Journal of Statistical Physics. 46 (3–4): 455. Bibcode:1987JSP....46..455E. doi:10.1007/BF01013368. S2CID 121353606.
13. Lyubich, Mikhail (1999). "Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness Conjecture". Annals of Mathematics. 149 (2): 319–420. arXiv:math/9903201. Bibcode:1999math......3201L. doi:10.2307/120968. JSTOR 120968. S2CID 119594350.
References
• Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences Springer, 1996, ISBN 978-0-38794-677-1
• Briggs, Keith (July 1991). "A Precise Calculation of the Feigenbaum Constants" (PDF). Mathematics of Computation. 57 (195): 435–439. Bibcode:1991MaCom..57..435B. doi:10.1090/S0025-5718-1991-1079009-6.
• Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
• Broadhurst, David (22 March 1999). "Feigenbaum constants to 1018 decimal places".
• Weisstein, Eric W. "Feigenbaum Constant". MathWorld.
External links
• Feigenbaum Constant – from Wolfram MathWorld
• OEIS sequence A006890 (Decimal expansion of Feigenbaum bifurcation velocity)
OEIS sequence A006891 (Decimal expansion of Feigenbaum reduction parameter)
OEIS sequence A195102 (Decimal expansion of the parameter for the biquadratic solution of the Feigenbaum-Cvitanovic equation)
• Feigenbaum constant – PlanetMath
• Moriarty, Philip; Bowley, Roger (2009). "δ – Feigenbaum Constant". Sixty Symbols. Brady Haran for the University of Nottingham.
• Thurlby, Judi (2021). Rigorous calculations of renormalisation fixed points and attractors (PhD). U. Portsmouth.
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Wikipedia
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Isomorphism theorems
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
History
The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether.
Three years later, B.L. van der Waerden published his influential Moderne Algebra, the first abstract algebra textbook that took the groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, and van der Waerden himself on ideals as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.
Groups
We first present the isomorphism theorems of the groups.
Note on numbers and names
Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.
Comparison of the names of the group isomorphism theorems
Comment Author Theorem A Theorem B Theorem C
No "third" theorem Jacobson[1] Fundamental theorem of homomorphisms (Second isomorphism theorem) "often called the first isomorphism theorem"
van der Waerden,[2] Durbin[4] Fundamental theorem of homomorphisms First isomorphism theorem Second isomorphism theorem
Knapp[5] (No name) Second isomorphism theorem First isomorphism theorem
Grillet[6] Homomorphism theorem Second isomorphism theorem First isomorphism theorem
Three numbered theorems (Other convention per Grillet) First isomorphism theorem Third isomorphism theorem Second isomorphism theorem
Rotman[7] First isomorphism theorem Second isomorphism theorem Third isomorphism theorem
Fraleigh[8] (No name) Second isomorphism theorem Third isomorphism theorem
Dummit & Foote[9] First isomorphism theorem Second or Diamond isomorphism theorem Third isomorphism theorem
No numbering Milne[10] Homomorphism theorem Isomorphism theorem Correspondence theorem
Scott[11] Homomorphism theorem Isomorphism theorem Freshman theorem
It is less common to include the Theorem D, usually known as the lattice theorem or the correspondence theorem, as one of isomorphism theorems, but when included, it is the last one.
Theorem A (groups)
See also: Fundamental theorem on homomorphisms
Let G and H be groups, and let f : G → H be a homomorphism. Then:
1. The kernel of f is a normal subgroup of G,
2. The image of f is a subgroup of H, and
3. The image of f is isomorphic to the quotient group G / ker(f).
In particular, if f is surjective then H is isomorphic to G / ker(f).
This theorem is usually called the first isomorphism theorem.
Theorem B (groups)
Let $G$ be a group. Let $S$ be a subgroup of $G$, and let $N$ be a normal subgroup of $G$. Then the following hold:
1. The product $SN$ is a subgroup of $G$,
2. The subgroup $N$ is a normal subgroup of $SN$,
3. The intersection $S\cap N$ is a normal subgroup of $S$, and
4. The quotient groups $(SN)/N$ and $S/(S\cap N)$ are isomorphic.
Technically, it is not necessary for $N$ to be a normal subgroup, as long as $S$ is a subgroup of the normalizer of $N$ in $G$. In this case, $N$ is not a normal subgroup of $G$, but $N$ is still a normal subgroup of the product $SN$.
This theorem is sometimes called the second isomorphism theorem,[10] diamond theorem[12] or the parallelogram theorem.[13]
An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting $G=\operatorname {GL} _{2}(\mathbb {C} )$, the group of invertible 2 × 2 complex matrices, $S=\operatorname {SL} _{2}(\mathbb {C} )$, the subgroup of determinant 1 matrices, and $N$ the normal subgroup of scalar matrices $\mathbb {C} ^{\times }\!I=\left\{\left({\begin{smallmatrix}a&0\\0&a\end{smallmatrix}}\right):a\in \mathbb {C} ^{\times }\right\}$, we have $S\cap N=\{\pm I\}$, where $I$ is the identity matrix, and $SN=\operatorname {GL} _{2}(\mathbb {C} )$. Then the second isomorphism theorem states that:
$\operatorname {PGL} _{2}(\mathbb {C} ):=\operatorname {GL} _{2}\left(\mathbb {C} )/(\mathbb {C} ^{\times }\!I\right)\cong \operatorname {SL} _{2}(\mathbb {C} )/\{\pm I\}=:\operatorname {PSL} _{2}(\mathbb {C} )$
Theorem C (groups)
Let $G$ be a group, and $N$ a normal subgroup of $G$. Then
1. If $K$ is a subgroup of $G$ such that $N\subseteq K\subseteq G$, then $G/N$ has a subgroup isomorphic to $K/N$.
2. Every subgroup of $G/N$ is of the form $K/N$ for some subgroup $K$ of $G$ such that $N\subseteq K\subseteq G$.
3. If $K$ is a normal subgroup of $G$ such that $N\subseteq K\subseteq G$, then $G/N$ has a normal subgroup isomorphic to $K/N$.
4. Every normal subgroup of $G/N$ is of the form $K/N$ for some normal subgroup $K$ of $G$ such that $N\subseteq K\subseteq G$.
5. If $K$ is a normal subgroup of $G$ such that $N\subseteq K\subseteq G$, then the quotient group $(G/N)/(K/N)$ is isomorphic to $G/K$.
The last statement is sometimes referred to as the third isomorphism theorem. The first four statements are often subsumed under Theorem D below, and referred to as the lattice theorem, correspondence theorem, or fourth isomorphism theorem.
Theorem D (groups)
Main article: Lattice theorem
Let $G$ be a group, and $N$ a normal subgroup of $G$. The canonical projection homomorphism $G\rightarrow G/N$ defines a bijective correspondence between the set of subgroups of $G$ containing $N$ and the set of (all) subgroups of $G/N$. Under this correspondence normal subgroups correspond to normal subgroups.
This theorem is sometimes called the correspondence theorem, the lattice theorem, and the fourth isomorphism theorem.
The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.[14]
Discussion
The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism $f:G\rightarrow H$. The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into $\iota \circ \pi $, where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object $\ker f$ and a monomorphism $\kappa :\ker f\rightarrow G$ :\ker f\rightarrow G} (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from $\ker f$ to $H$ and $G/\ker f$.
If the sequence is right split (i.e., there is a morphism σ that maps $G/\operatorname {ker} f$ to a π-preimage of itself), then G is the semidirect product of the normal subgroup $\operatorname {im} \kappa $ and the subgroup $\operatorname {im} \sigma $. If it is left split (i.e., there exists some $\rho :G\rightarrow \operatorname {ker} f$ such that $\rho \circ \kappa =\operatorname {id} _{{\text{ker}}f}$), then it must also be right split, and $\operatorname {im} \kappa \times \operatorname {im} \sigma $ is a direct product decomposition of G. In general, the existence of a right split does not imply the existence of a left split; but in an abelian category (such as that of abelian groups), left splits and right splits are equivalent by the splitting lemma, and a right split is sufficient to produce a direct sum decomposition $\operatorname {im} \kappa \oplus \operatorname {im} \sigma $. In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence $0\rightarrow G/\operatorname {ker} f\rightarrow H\rightarrow \operatorname {coker} f\rightarrow 0$.
In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups of G, while the intersection S ∩ N is the meet.
The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects.
Rings
The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.
Theorem A (rings)
Let $R$ and $S$ be rings, and let $\varphi :R\rightarrow S$ be a ring homomorphism. Then:
1. The kernel of $\varphi $ is an ideal of $R$,
2. The image of $\varphi $ is a subring of $S$, and
3. The image of $\varphi $ is isomorphic to the quotient ring $R/\ker \varphi $.
In particular, if $\varphi $ is surjective then $S$ is isomorphic to $R/\ker \varphi $.[15]
Theorem B (rings)
Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then:
1. The sum S + I = {s + i | s ∈ S, i ∈ I } is a subring of R,
2. The intersection S ∩ I is an ideal of S, and
3. The quotient rings (S + I) / I and S / (S ∩ I) are isomorphic.
Theorem C (rings)
Let R be a ring, and I an ideal of R. Then
1. If $A$ is a subring of $R$ such that $I\subseteq A\subseteq R$, then $A/I$ is a subring of $R/I$.
2. Every subring of $R/I$ is of the form $A/I$ for some subring $A$ of $R$ such that $I\subseteq A\subseteq R$.
3. If $J$ is an ideal of $R$ such that $I\subseteq J\subseteq R$, then $J/I$ is an ideal of $R/I$.
4. Every ideal of $R/I$ is of the form $J/I$ for some ideal $J$ of $R$ such that $I\subseteq J\subseteq R$.
5. If $J$ is an ideal of $R$ such that $I\subseteq J\subseteq R$, then the quotient ring $(R/I)/(J/I)$ is isomorphic to $R/J$.
Theorem D (rings)
Let $I$ be an ideal of $R$. The correspondence $A\leftrightarrow A/I$ is an inclusion-preserving bijection between the set of subrings $A$ of $R$ that contain $I$ and the set of subrings of $R/I$. Furthermore, $A$ (a subring containing $I$) is an ideal of $R$ if and only if $A/I$ is an ideal of $R/I$.[16]
Modules
The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces (modules over a field) and abelian groups (modules over $\mathbb {Z} $) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem.
In the following, "module" will mean "R-module" for some fixed ring R.
Theorem A (modules)
Let M and N be modules, and let φ : M → N be a module homomorphism. Then:
1. The kernel of φ is a submodule of M,
2. The image of φ is a submodule of N, and
3. The image of φ is isomorphic to the quotient module M / ker(φ).
In particular, if φ is surjective then N is isomorphic to M / ker(φ).
Theorem B (modules)
Let M be a module, and let S and T be submodules of M. Then:
1. The sum S + T = {s + t | s ∈ S, t ∈ T} is a submodule of M,
2. The intersection S ∩ T is a submodule of M, and
3. The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic.
Theorem C (modules)
Let M be a module, T a submodule of M.
1. If $S$ is a submodule of $M$ such that $T\subseteq S\subseteq M$, then $S/T$ is a submodule of $M/T$.
2. Every submodule of $M/T$ is of the form $S/T$ for some submodule $S$ of $M$ such that $T\subseteq S\subseteq M$.
3. If $S$ is a submodule of $M$ such that $T\subseteq S\subseteq M$, then the quotient module $(M/T)/(S/T)$ is isomorphic to $M/S$.
Theorem D (modules)
Let $M$ be a module, $N$ a submodule of $M$. There is a bijection between the submodules of $M$ that contain $N$ and the submodules of $M/N$. The correspondence is given by $A\leftrightarrow A/N$ for all $A\supseteq N$. This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between the lattice of submodules of $M/N$ and the lattice of submodules of $M$ that contain $N$).[17]
Universal algebra
To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations.
A congruence on an algebra $A$ is an equivalence relation $\Phi \subseteq A\times A$ that forms a subalgebra of $A\times A$ considered as an algebra with componentwise operations. One can make the set of equivalence classes $A/\Phi $ into an algebra of the same type by defining the operations via representatives; this will be well-defined since $\Phi $ is a subalgebra of $A\times A$. The resulting structure is the quotient algebra.
Theorem A (universal algebra)
Let $f:A\rightarrow B$ be an algebra homomorphism. Then the image of $f$ is a subalgebra of $B$, the relation given by $\Phi :f(x)=f(y)$ (i.e. the kernel of $f$) is a congruence on $A$, and the algebras $A/\Phi $ and $\operatorname {im} f$ are isomorphic. (Note that in the case of a group, $f(x)=f(y)$ iff $f(xy^{-1})=1$, so one recovers the notion of kernel used in group theory in this case.)
Theorem B (universal algebra)
Given an algebra $A$, a subalgebra $B$ of $A$, and a congruence $\Phi $ on $A$, let $\Phi _{B}=\Phi \cap (B\times B)$ be the trace of $\Phi $ in $B$ and $[B]^{\Phi }=\{K\in A/\Phi :K\cap B\neq \emptyset \}$ the collection of equivalence classes that intersect $B$. Then
1. $\Phi _{B}$ is a congruence on $B$,
2. $\ [B]^{\Phi }$ is a subalgebra of $A/\Phi $, and
3. the algebra $[B]^{\Phi }$ is isomorphic to the algebra $B/\Phi _{B}$.
Theorem C (universal algebra)
Let $A$ be an algebra and $\Phi ,\Psi $ two congruence relations on $A$ such that $\Psi \subseteq \Phi $. Then $\Phi /\Psi =\{([a']_{\Psi },[a'']_{\Psi }):(a',a'')\in \Phi \}=[\ ]_{\Psi }\circ \Phi \circ [\ ]_{\Psi }^{-1}$ is a congruence on $A/\Psi $, and $A/\Phi $ is isomorphic to $(A/\Psi )/(\Phi /\Psi ).$
Theorem D (universal algebra)
Let $A$ be an algebra and denote $\operatorname {Con} A$ the set of all congruences on $A$. The set $\operatorname {Con} A$ is a complete lattice ordered by inclusion.[18] If $\Phi \in \operatorname {Con} A$ is a congruence and we denote by $\left[\Phi ,A\times A\right]\subseteq \operatorname {Con} A$ the set of all congruences that contain $\Phi $ (i.e. $\left[\Phi ,A\times A\right]$ is a principal filter in $\operatorname {Con} A$, moreover it is a sublattice), then the map $\alpha :\left[\Phi ,A\times A\right]\to \operatorname {Con} (A/\Phi ),\Psi \mapsto \Psi /\Phi $ :\left[\Phi ,A\times A\right]\to \operatorname {Con} (A/\Phi ),\Psi \mapsto \Psi /\Phi } is a lattice isomorphism.[19][20]
Note
1. Jacobson (2009), sec 1.10
2. van der Waerden, Algebra (1994).
3. Durbin (2009), sec. 54
4. [the names are] essentially the same as [van der Waerden 1994][3]
5. Knapp (2016), sec IV 2
6. Grillet (2007), sec. I 5
7. Rotman (2003), sec. 2.6
8. Fraleigh (2003), Chap. 34
9. Dummit, David Steven (2004). Abstract algebra. Richard M. Foote (Third ed.). Hoboken, NJ. pp. 97–98. ISBN 0-471-43334-9. OCLC 52559229.{{cite book}}: CS1 maint: location missing publisher (link)
10. Milne (2013), Chap. 1, sec. Theorems concerning homomorphisms
11. Scott (1964), secs 2.2 and 2.3
12. I. Martin Isaacs (1994). Algebra: A Graduate Course. American Mathematical Soc. p. 33. ISBN 978-0-8218-4799-2.
13. Paul Moritz Cohn (2000). Classic Algebra. Wiley. p. 245. ISBN 978-0-471-87731-8.
14. Wilson, Robert A. (2009). The Finite Simple Groups. Graduate Texts in Mathematics 251. Springer-Verlag London. p. 7. doi:10.1007/978-1-84800-988-2. ISBN 978-1-4471-2527-3.
15. Moy, Samuel (2022). "An Introduction to the Theory of Field Extensions" (PDF). UChicago Department of Math. Retrieved Dec 20, 2022.
16. Dummit, David S.; Foote, Richard M. (2004). Abstract algebra. Hoboken, NJ: Wiley. p. 246. ISBN 978-0-471-43334-7.
17. Dummit and Foote (2004), p. 349
18. Burris and Sankappanavar (2012), p. 37
19. Burris and Sankappanavar (2012), p. 49
20. Sun, William. "Is there a general form of the correspondence theorem?". Mathematics StackExchange. Retrieved 20 July 2019.
References
• Noether, Emmy, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Mathematische Annalen 96 (1927) pp. 26–61
• McLarty, Colin, "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the rise of functors". The Architecture of Modern Mathematics: Essays in history and philosophy (edited by Jeremy Gray and José Ferreirós), Oxford University Press (2006) pp. 211–35.
• Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 9780486471891
• Cohn, Paul M., Universal algebra, Chapter II.3 p. 57
• Milne, James S. (2013), Group Theory, 3.13
• van der Waerden, B. I. (1994), Algebra, vol. 1 (9 ed.), Springer-Verlag
• Dummit, David S.; Foote, Richard M. (2004). Abstract algebra. Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7.
• Burris, Stanley; Sankappanavar, H. P. (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9.
• Scott, W. R. (1964), Group Theory, Prentice Hall
• Durbin, John R. (2009). Modern Algebra: An Introduction (6 ed.). Wiley. ISBN 978-0-470-38443-5.
• Knapp, Anthony W. (2016), Basic Algebra (Digital second ed.)
• Grillet, Pierre Antoine (2007), Abstract Algebra (2 ed.), Springer
• Rotman, Joseph J. (2003), Advanced Modern Algebra (2 ed.), Prentice Hall, ISBN 0130878685
• Hungerford, Thomas W. (1980), Algebra (Graduate Texts in Mathematics, 73), Springer, ISBN 0387905189
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Wikipedia
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Wheat and chessboard problem
The wheat and chessboard problem (sometimes expressed in terms of rice grains) is a mathematical problem expressed in textual form as:
If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish?
The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares, then the sum of grains on all 64 squares is: 1 + 2 + 4 + 8 + ... and so forth for the 64 squares. The total number of grains can be shown to be 264−1 or 18,446,744,073,709,551,615 (eighteen quintillion, four hundred forty-six quadrillion, seven hundred forty-four trillion, seventy-three billion, seven hundred nine million, five hundred fifty-one thousand, six hundred and fifteen, over 1.4 trillion metric tons), which is over 2,000 times the annual world production of wheat.[1]
This exercise can be used to demonstrate how quickly exponential sequences grow, as well as to introduce exponents, zero power, capital-sigma notation and geometric series. Updated for modern times using pennies and a hypothetical question such as "Would you rather have a million dollars or a penny on day one, doubled every day until day 30?", the formula has been used to explain compound interest. (Doubling would yield over one billion seventy three million pennies, or over 10 million dollars: 230−1=1,073,741,823).[2][3]
Origins
The problem appears in different stories about the invention of chess. One of them includes the geometric progression problem. The story is first known to have been recorded in 1256 by Ibn Khallikan.[4] Another version has the inventor of chess (in some tellings Sessa, an ancient Indian Minister) request his ruler give him wheat according to the wheat and chessboard problem. The ruler laughs it off as a meager prize for a brilliant invention, only to have court treasurers report the unexpectedly huge number of wheat grains would outstrip the ruler's resources. Versions differ as to whether the inventor becomes a high-ranking advisor or is executed.[5]
Macdonnell also investigates the earlier development of the theme.[6]
[According to al-Masudi's early history of India], shatranj, or chess was invented under an Indian king, who expressed his preference for this game over backgammon. [...] The Indians, he adds, also calculated an arithmetical progression with the squares of the chessboard. [...] The early fondness of the Indians for enormous calculations is well known to students of their mathematics, and is exemplified in the writings of the great astronomer Āryabaṭha (born 476 A.D.). [...] An additional argument for the Indian origin of this calculation is supplied by the Arabic name for the square of the chessboard, (بيت, "beit"), 'house'. [...] For this has doubtless a historical connection with its Indian designation koṣṭhāgāra, 'store-house', 'granary' [...].
Solutions
The simple, brute-force solution is just to manually double and add each step of the series:
$T_{64}$ = 1 + 2 + 4 + ..... + 9,223,372,036,854,775,808 = 18,446,744,073,709,551,615
where $T_{64}$ is the total number of grains.
The series may be expressed using exponents:
$T_{64}=2^{0}+2^{1}+2^{2}+\cdots +2^{63}$
and, represented with capital-sigma notation as:
$\sum _{i=0}^{63}2^{i}.\,$
It can also be solved much more easily using:
$T_{64}=2^{64}-1.\,$
A proof of which is:
$s=2^{0}+2^{1}+2^{2}+\cdots +2^{63}.$
Multiply each side by 2:
$2s=2^{1}+2^{2}+2^{3}+\cdots +2^{63}+2^{64}.$
Subtract original series from each side:
${\begin{aligned}2s-s&=\qquad \quad {\cancel {2^{1}}}+{\cancel {2^{2}}}+\cdots +{\cancel {2^{63}}}+2^{64}\\&\quad -2^{0}-{\cancel {2^{1}}}-{\cancel {2^{2}}}-\cdots -{\cancel {2^{63}}}\\&=2^{64}-2^{0}\\\therefore s&=2^{64}-1.\end{aligned}}$
The solution above is a particular case of the sum of a geometric series, given by
$a+ar+ar^{2}+ar^{3}+\cdots +ar^{n-1}=\sum _{k=0}^{n-1}ar^{k}=a\,{\frac {1-r^{n}}{1-r}},$
where $a$ is the first term of the series, $r$ is the common ratio and $n$ is the number of terms.
In this problem $a=1$, $r=2$ and $n=64$.
The exercise of working through this problem may be used to explain and demonstrate exponents and the quick growth of exponential and geometric sequences. It can also be used to illustrate sigma notation. When expressed as exponents, the geometric series is: 20 + 21 + 22 + 23 + ... and so forth, up to 263. The base of each exponentiation, "2", expresses the doubling at each square, while the exponents represent the position of each square (0 for the first square, 1 for the second, and so on.).
The number of grains is the 64th Mersenne number.
Second half of the chessboard
In technology strategy, the "second half of the chessboard" is a phrase, coined by Ray Kurzweil,[7] in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy. While the number of grains on the first half of the chessboard is large, the amount on the second half is vastly (232 > 4 billion times) larger.
The number of grains of wheat on the first half of the chessboard is 1 + 2 + 4 + 8 + ... + 2,147,483,648, for a total of 4,294,967,295 (232 − 1) grains, or about 279 tonnes of wheat (assuming 65 mg as the mass of one grain of wheat).[8]
The number of grains of wheat on the second half of the chessboard is 232 + 233 + 234 + ... + 263, for a total of 264 − 232 grains. This is equal to the square of the number of grains on the first half of the board, plus itself. The first square of the second half alone contains one more grain than the entire first half. On the 64th square of the chessboard alone, there would be 263 = 9,223,372,036,854,775,808 grains, more than two billion times as many as on the first half of the chessboard.
On the entire chessboard there would be 264 − 1 = 18,446,744,073,709,551,615 grains of wheat, weighing about 1,199,000,000,000 metric tons. This is over 1,600 times the global production of wheat (729 million metric tons in 2014 and 780.8 million tonnes in 2019).[9]
Use
Carl Sagan titled the second chapter of his final book The Persian Chessboard and wrote, referring to bacteria, that "Exponentials can't go on forever, because they will gobble up everything."[10] Similarly, The Limits to Growth uses the story to present suggested consequences of exponential growth: "Exponential growth never can go on very long in a finite space with finite resources."[11]
Donald Knuth used to pay a finder's fee reward check for coding errors found in his TeX and Metafont programs. It followed the audacious scheme inspired by the wheat and chessboard problem.[12]
See also
• Legend of the Ambalappuzha Paal Payasam
• Malthusian growth model
• Moore's law
• Orders of magnitude (data)
• Technology strategy
• The Limits to Growth
References
1. In the period 2020–21 this was an estimated 772.64 million metric tonnes, "Global Wheat Production Statistics since 1990". Retrieved 2022-05-25.{{cite web}}: CS1 maint: url-status (link)
2. "A Penny Doubled Every Day for 30 Days = $10.7M" – via www.bloomberg.com.
3. "Doubling Pennies". Mathforum.org. Retrieved 2017-08-09.
4. Clifford A. Pickover (2009), The Math Book: From Pythagoras to the 57th Dimension, New York : Sterling. ISBN 9781402757969. p. 102
5. Tahan, Malba (1993). The Man Who Counted: A Collection of Mathematical Adventures. New York: W.W. Norton & Co. pp. 113–115. ISBN 0393309347. Retrieved 2015-04-05.
6. Macdonell, A. A. (1898). "The Origin and Early History of Chess". Journal of the Royal Asiatic Society of Great Britain & Ireland. 30 (1): 117–141. doi:10.1017/S0035869X00146246. S2CID 163963500.
7. Kurzweil, Ray (1999). The Age of Spiritual Machines: When Computers Exceed Human Intelligence. New York: Penguin. p. 37. ISBN 0-670-88217-8. Retrieved 2015-04-06.
8. "Encyclopedia Britannica: Grain, unit of weight". 29 April 2004. Retrieved 2 March 2017.
9. "FAOSTAT". faostat3.fao.org. Retrieved 2 March 2017.
10. Sagan, Carl (1997). Billions and Billions: Thoughts On Life And Death At the Brink Of The Millennium. New York: Ballantine Books. p. 17. ISBN 0-345-37918-7.
11. Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III (1972). The Limits to Growth, p. 21, at Google Books. New York: University Books. ISBN 0-87663-165-0. Retrieved 2015-04-05.
12. Weisstein, Eric W. "Wheat and Chessboard Problem". MathWorld.
External links
Look up wheat and chessboard problem in Wiktionary, the free dictionary.
• Weisstein, Eric W. "Wheat and Chessboard Problem". MathWorld.
• Salt and chessboard problem - A variation on the wheat and chessboard problem with measurements of each square.
• Learning materials related to Math Adventures/Wheat and the Chessboard at Wikiversity
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Second Hardy–Littlewood conjecture
In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the first Hardy–Littlewood conjecture, the second Hardy–Littlewood conjecture was proposed by G. H. Hardy and John Edensor Littlewood in 1923.[1]
Second Hardy–Littlewood conjecture
Plot of $\pi (x)+\pi (y)-\pi (x+y)$ for $x,y\leq 200$
FieldNumber theory
Conjectured byG. H. Hardy
John Edensor Littlewood
Conjectured in1923
Open problemyes
Statement
The conjecture states that
$\pi (x+y)\leq \pi (x)+\pi (y)$
for integers x, y ≥ 2, where π(z) denotes the prime-counting function, giving the number of prime numbers up to and including z.
Connection to the first Hardy–Littlewood conjecture
The statement of the second Hardy–Littlewood conjecture is equivalent to the statement that the number of primes from x + 1 to x + y is always less than or equal to the number of primes from 1 to y. This was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime k-tuples, and the first violation is expected to likely occur for very large values of x.[2][3] For example, an admissible k-tuple (or prime constellation) of 447 primes can be found in an interval of y = 3159 integers, while π(3159) = 446. If the first Hardy–Littlewood conjecture holds, then the first such k-tuple is expected for x greater than 1.5 × 10174 but less than 2.2 × 101198.[4]
References
1. Hardy, G. H.; Littlewood, J. E. (1923). "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes". Acta Math. 44 (44): 1–70. doi:10.1007/BF02403921..
2. Hensley, Douglas; Richards, Ian. "Primes in intervals". Acta Arith. 25 (1973/74): 375–391. doi:10.4064/aa-25-4-375-391. MR 0396440.
3. Richards, Ian (1974). "On the Incompatibility of Two Conjectures Concerning Primes". Bull. Amer. Math. Soc. 80: 419–438. doi:10.1090/S0002-9904-1974-13434-8.
4. "447-tuple calculations". Retrieved 2008-08-12.
External links
• Engelsma, Thomas J. "k-tuple Permissible Patterns". Retrieved 2008-08-12.
• Oliveira e Silva, Tomás. "Admissible prime constellations". Retrieved 2008-08-12.
Prime number conjectures
• Hardy–Littlewood
• 1st
• 2nd
• Agoh–Giuga
• Andrica's
• Artin's
• Bateman–Horn
• Brocard's
• Bunyakovsky
• Chinese hypothesis
• Cramér's
• Dickson's
• Elliott–Halberstam
• Firoozbakht's
• Gilbreath's
• Grimm's
• Landau's problems
• Goldbach's
• weak
• Legendre's
• Twin prime
• Legendre's constant
• Lemoine's
• Mersenne
• Oppermann's
• Polignac's
• Pólya
• Schinzel's hypothesis H
• Waring's prime number
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Wikipedia
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Meagre set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms.
The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre.
Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis.
Definitions
Throughout, $X$ will be a topological space.
The definition of meagre set uses the notion of a nowhere dense subset of $X,$ that is, a subset of $X$ whose closure has empty interior. See the corresponding article for more details.
A subset of $X$ is called meagre in $X,$ a meagre subset of $X,$ or of the first category in $X$ if it is a countable union of nowhere dense subsets of $X$.[1] Otherwise, the subset is called nonmeagre in $X,$ a nonmeagre subset of $X,$ or of the second category in $X.$[1] The qualifier "in $X$" can be omitted if the ambient space is fixed and understood from context.
A topological space is called meagre (respectively, nonmeagre) if it is a meagre (respectively, nonmeagre) subset of itself.
A subset $A$ of $X$ is called comeagre in $X,$ or residual in $X,$ if its complement $X\setminus A$ is meagre in $X$. (This use of the prefix "co" is consistent with its use in other terms such as "cofinite".) A subset is comeagre in $X$ if and only if it is equal to a countable intersection of sets, each of whose interior is dense in $X.$
Remarks on terminology
The notions of nonmeagre and comeagre should not be confused. If the space $X$ is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets. If the space $X$ is nonmeager, no set is at the same time meagre and comeager, every comeagre set is nonmeagre, and there can be nonmeagre sets that are not comeagre, that is, with nonmeagre complement. See the Examples section below.
As an additional point of terminology, if a subset $A$ of a topological space $X$ is given the subspace topology induced from $X$, one can talk about it being a meagre space, namely being a meagre subset of itself (when considered as a topological space in its own right). In this case $A$ can also be called a meagre subspace of $X$, meaning a meagre space when given the subspace topology. Importantly, this is not the same as being meagre in the whole space $X$. (See the Properties and Examples sections below for the relationship between the two.) Similarly, a nonmeagre subspace will be a set that is nonmeagre in itself, which is not the same as being nonmeagre in the whole space. Be aware however that in the context of topological vector spaces some authors may use the phrase "meagre/nonmeagre subspace" to mean a vector subspace that is a meagre/nonmeagre set relative to the whole space.[2]
The terms first category and second category were the original ones used by René Baire in his thesis of 1899.[3] The meagre terminology was introduced by Bourbaki in 1948.[4][5]
Examples
The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space.
In the nonmeagre space $X=[0,1]\cup ([2,3]\cap \mathbb {Q} )$ the set $[2,3]\cap \mathbb {Q} $ is meagre. The set $[0,1]$ is nonmeagre and comeagre.
In the nonmeagre space $X=[0,2]$ the set $[0,1]$ is nonmeagre. But it is not comeagre, as its complement $(1,2]$ is also nonmeagre.
A countable T1 space without isolated point is meagre. So it is also meagre in any space that contains it as a subspace. For example, $\mathbb {Q} $ is both a meagre subspace of $\mathbb {R} $ (that is, meagre in itself with the subspace topology induced from $\mathbb {R} $) and a meagre subset of $\mathbb {R} .$
The Cantor set is nowhere dense in $\mathbb {R} $ and hence meagre in $\mathbb {R} .$ But it is nonmeagre in itself, since it is a complete metric space.
The set $([0,1]\cap \mathbb {Q} )\cup \{2\}$ is not nowhere dense in $\mathbb {R} $, but it is meagre in $\mathbb {R} $. It is nonmeagre in itself (since as a subspace it contains an isolated point).
The line $\mathbb {R} \times \{0\}$ is meagre in the plane $\mathbb {R} ^{2}.$ But it is a nonmeagre subspace, that is, it is nonmeagre in itself.
The set $S=(\mathbb {Q} \times \mathbb {Q} )\cup (\mathbb {R} \times \{0\})$ is a meagre subset of $\mathbb {R} ^{2}$ even though its meagre subset $\mathbb {R} \times \{0\}$ is a nonmeagre subspace (that is, $\mathbb {R} $ is not a meagre topological space).[6] A countable Hausdorff space without isolated points is meagre, whereas any topological space that contains an isolated point is nonmeagre.[6] Because the rational numbers are countable, they are meagre as a subset of the reals and as a space—that is, they do not form a Baire space.
Any topological space that contains an isolated point is nonmeagre[6] (because no set containing the isolated point can be nowhere dense). In particular, every nonempty discrete space is nonmeagre.
There is a subset $H$ of the real numbers $\mathbb {R} $ that splits every nonempty open set into two nonmeagre sets. That is, for every nonempty open set $U\subseteq \mathbb {R} $, the sets $U\cap H$ and $U\setminus H$ are both nonmeagre.
In the space $C([0,1])$ of continuous real-valued functions on $[0,1]$ with the topology of uniform convergence, the set $A$ of continuous real-valued functions on $[0,1]$ that have a derivative at some point is meagre.[7][8] Since $C([0,1])$ is a complete metric space, it is nonmeagre. So the complement of $A$, which consists of the continuous real-valued nowhere differentiable functions on $[0,1],$ is comeagre and nonmeagre. In particular that set is not empty. This is one way to show the existence of continuous nowhere differentiable functions.
Characterizations and sufficient conditions
Every nonempty Baire space is nonmeagre. In particular, by the Baire category theorem every nonempty complete metric space and every nonempty locally compact Hausdorff space is nonmeagre.
Every Baire space is nonmeagre but there exist nonmeagre spaces that are not Baire spaces.[6] Since complete (pseudo)metric spaces as well as Hausdorff locally compact spaces are Baire spaces, they are also nonmeagre spaces.[6]
Any subset of a meagre set is a meagre set, as is the union of countably many meagre sets.[9] If $h:X\to X$ is a homeomorphism then a subset $S\subseteq X$ is meagre if and only if $h(S)$ is meagre.[9]
Every nowhere dense subset is a meagre set.[9] Consequently, any closed subset of $X$ whose interior in $X$ is empty is of the first category of $X$ (that is, it is a meager subset of $X$).
The Banach category theorem[10] states that any space $X,$ the union of any countable family of open sets of the first category is of the first category.
All subsets and all countable unions of meagre sets are meagre. Thus the meagre subsets of a fixed space form a σ-ideal of subsets, a suitable notion of negligible set. Dually, all supersets and all countable intersections comeagre sets are comeagre. Every superset of a nonmeagre set is nonmeagre.
Suppose $A\subseteq Y\subseteq X,$ where $Y$ has the subspace topology induced from $X.$ The set $A$ may be meagre in $X$ without being meagre in $Y.$ However the following results hold:[5]
• If $A$ is meagre in $Y,$ then $A$ is meagre in $X.$
• If $Y$ is open in $X,$ then $A$ is meagre in $Y$ if and only if $A$ is meagre in $X.$
• If $Y$ is dense in $X,$ then $A$ is meagre in $Y$ if and only if $A$ is meagre in $X.$
And correspondingly for nonmeagre sets:
• If $A$ is nonmeagre in $X,$ then $A$ is nonmeagre in $Y.$
• If $Y$ is open in $X,$ then $A$ is nonmeagre in $Y$ if and only if $A$ is nonmeagre in $X.$
• If $Y$ is dense in $X,$ then $A$ is nonmeagre in $Y$ if and only if $A$ is nonmeagre in $X.$
In particular, every subset of $X$ that is meagre in itself is meagre in $X.$ Every subset of $X$ that is nonmeagre in $X$ is nonmeagre in itself. And for an open set or a dense set in $X,$ being meagre in $X$ is equivalent to being meagre in itself, and similarly for the nonmeagre property.
A topological space $X$ is nonmeagre if and only if every countable intersection of dense open sets in $X$ is nonempty.[11]
Any superset of a comeagre set is comeagre, as is the intersection of countably many comeagre sets (because countable union of countable sets is countable).
Properties
A nonmeagre locally convex topological vector space is a barreled space.[6]
Every nowhere dense subset of $X$ is meagre. Consequently, any closed subset with empty interior is meagre. Thus a closed subset of $X$ that is of the second category in $X$ must have non-empty interior in $X$[12] (because otherwise it would be nowhere dense and thus of the first category).
If $B\subseteq X$ is of the second category in $X$ and if $S_{1},S_{2},\ldots $ are subsets of $X$ such that $B\subseteq S_{1}\cup S_{2}\cup \cdots $ then at least one $S_{n}$ is of the second category in $X.$
Meagre subsets and Lebesgue measure
There exist nowhere dense subsets (which are thus meagre subsets) that have positive Lebesgue measure.[6]
A meagre set in $\mathbb {R} $ need not have Lebesgue measure zero, and can even have full measure. For example, in the interval $[0,1]$ fat Cantor sets, like the Smith–Volterra–Cantor set, are closed nowhere dense and they can be constructed with a measure arbitrarily close to $1.$ The union of a countable number of such sets with measure approaching $1$ gives a meagre subset of $[0,1]$ with measure $1.$[13]
Dually, there can be nonmeagre sets with measure zero. The complement of any meagre set of measure $1$ in $[0,1]$ (for example the one in the previous paragraph) has measure $0$ and is comeagre in $[0,1],$ and hence nonmeagre in $[0,1]$ since $[0,1]$ is a Baire space.
Here is another example of a nonmeagre set in $\mathbb {R} $ with measure $0$:
$\bigcap _{m=1}^{\infty }\bigcup _{n=1}^{\infty }\left(r_{n}-\left({\tfrac {1}{2}}\right)^{n+m},r_{n}+\left({\tfrac {1}{2}}\right)^{n+m}\right)$
where $r_{1},r_{2},\ldots $ is a sequence that enumerates the rational numbers.
Relation to Borel hierarchy
Just as a nowhere dense subset need not be closed, but is always contained in a closed nowhere dense subset (viz, its closure), a meagre set need not be an $F_{\sigma }$ set (countable union of closed sets), but is always contained in an $F_{\sigma }$ set made from nowhere dense sets (by taking the closure of each set).
Dually, just as the complement of a nowhere dense set need not be open, but has a dense interior (contains a dense open set), a comeagre set need not be a $G_{\delta }$ set (countable intersection of open sets), but contains a dense $G_{\delta }$ set formed from dense open sets.
Banach–Mazur game
Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game. Let $Y$ be a topological space, ${\mathcal {W}}$ be a family of subsets of $Y$ that have nonempty interiors such that every nonempty open set has a subset belonging to ${\mathcal {W}},$ and $X$ be any subset of $Y.$ Then there is a Banach–Mazur game $MZ(X,Y,{\mathcal {W}}).$ In the Banach–Mazur game, two players, $P$ and $Q,$ alternately choose successively smaller elements of ${\mathcal {W}}$ to produce a sequence $W_{1}\supseteq W_{2}\supseteq W_{3}\supseteq \cdots .$ Player $P$ wins if the intersection of this sequence contains a point in $X$; otherwise, player $Q$ wins.
Theorem — For any ${\mathcal {W}}$ meeting the above criteria, player $Q$ has a winning strategy if and only if $X$ is meagre.
Erdos-Sierpinski duality
Many arguments about meagre sets also apply to null sets, i.e. sets of Lebesgue measure 0. The Erdos-Sierpinski duality theorem states that if the continuum hypothesis holds, there is an involution from reals to reals where the image of a null set of reals is a meagre set, and vice versa.[14] In fact, the image of a set of reals under the map is null if and only if the original set was meagre, and vice versa.[15]
See also
• Barrelled space – Type of topological vector space
• Generic property – Property holding for typical examples, for analogs to residual
• Negligible set – Mathematical set regarded as insignificant, for analogs to meagre
• Property of Baire – Difference of an open set by a meager set
Notes
1. Narici & Beckenstein 2011, p. 389.
2. Schaefer, Helmut H. (1966). "Topological Vector Spaces". Macmillan.
3. Baire, René (1899). "Sur les fonctions de variables réelles". Annali di Mat. Pura ed Appl. 3: 1–123., page 65
4. Oxtoby, J. (1961). "Cartesian products of Baire spaces" (PDF). Fundamenta Mathematicae. 49 (2): 157–166. doi:10.4064/fm-49-2-157-166."Following Bourbaki [...], a topological space is called a Baire space if ..."
5. Bourbaki 1989, p. 192.
6. Narici & Beckenstein 2011, pp. 371–423.
7. Banach, S. (1931). "Über die Baire'sche Kategorie gewisser Funktionenmengen". Studia Math. 3 (1): 174–179. doi:10.4064/sm-3-1-174-179.
8. Willard 2004, Theorem 25.5.
9. Rudin 1991, p. 43.
10. Oxtoby 1980, p. 62.
11. Willard 2004, Theorem 25.2.
12. Rudin 1991, pp. 42–43.
13. "Is there a measure zero set which isn't meagre?". MathOverflow.
14. Quintanilla, M. (2022). "The real numbers in inner models of set theory". arXiv:2206.10754. (p.25)
15. S. Saito, The Erdos-Sierpinski Duality Theorem, notes. Accessed 18 January 2023.
Bibliography
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063.
• Oxtoby, John C. (1980). "The Banach Category Theorem". Measure and Category (Second ed.). New York: Springer. pp. 62–65. ISBN 0-387-90508-1.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
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Wikipedia
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First class constraint
A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishing of all the constraints). To calculate the first class constraint, one assumes that there are no second class constraints, or that they have been calculated previously, and their Dirac brackets generated.[1]
Main article: Dirac bracket
First and second class constraints were introduced by Dirac (1950, p.136, 1964, p.17) as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate.[2][3]
The terminology of first and second class constraints is confusingly similar to that of primary and secondary constraints, reflecting the manner in which these are generated. These divisions are independent: both first and second class constraints can be either primary or secondary, so this gives altogether four different classes of constraints.
Poisson brackets
Consider a Poisson manifold M with a smooth Hamiltonian over it (for field theories, M would be infinite-dimensional).
Suppose we have some constraints
$f_{i}(x)=0,$
for n smooth functions
$\{f_{i}\}_{i=1}^{n}$
These will only be defined chartwise in general. Suppose that everywhere on the constrained set, the n derivatives of the n functions are all linearly independent and also that the Poisson brackets
$\{f_{i},f_{j}\}$
and
$\{f_{i},H\}$
all vanish on the constrained subspace.
This means we can write
$\{f_{i},f_{j}\}=\sum _{k}c_{ij}^{k}f_{k}$
for some smooth functions $c_{ij}^{k}$ −−there is a theorem showing this; and
$\{f_{i},H\}=\sum _{j}v_{i}^{j}f_{j}$
for some smooth functions $v_{i}^{j}$.
This can be done globally, using a partition of unity. Then, we say we have an irreducible first-class constraint (irreducible here is in a different sense from that used in representation theory).
Geometric theory
For a more elegant way, suppose given a vector bundle over ${\mathcal {M}}$, with $n$-dimensional fiber $V$. Equip this vector bundle with a connection. Suppose too we have a smooth section f of this bundle.
Then the covariant derivative of f with respect to the connection is a smooth linear map $\nabla f$ from the tangent bundle $T{\mathcal {M}}$ to $V$, which preserves the base point. Assume this linear map is right invertible (i.e. there exists a linear map $g$ such that $(\Delta f)g$ is the identity map) for all the fibers at the zeros of f. Then, according to the implicit function theorem, the subspace of zeros of f is a submanifold.
The ordinary Poisson bracket is only defined over $C^{\infty }(M)$, the space of smooth functions over M. However, using the connection, we can extend it to the space of smooth sections of f if we work with the algebra bundle with the graded algebra of V-tensors as fibers.
Assume also that under this Poisson bracket, $\{f,f\}=0$ (note that it's not true that $\{g,g\}=0$ in general for this "extended Poisson bracket" anymore) and $\{f,H\}=0$ on the submanifold of zeros of f (If these brackets also happen to be zero everywhere, then we say the constraints close off shell). It turns out the right invertibility condition and the commutativity of flows conditions are independent of the choice of connection. So, we can drop the connection provided we are working solely with the restricted subspace.
Intuitive meaning
What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other on the constrained subspace; or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constraint flows all bring the point to another point on the constrained subspace.
Since we wish to restrict ourselves to the constrained subspace only, this suggests that the Hamiltonian, or any other physical observable, should only be defined on that subspace. Equivalently, we can look at the equivalence class of smooth functions over the symplectic manifold, which agree on the constrained subspace (the quotient algebra by the ideal generated by the f 's, in other words).
The catch is, the Hamiltonian flows on the constrained subspace depend on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this.
Look at the orbits of the constrained subspace under the action of the symplectic flows generated by the f 's. This gives a local foliation of the subspace because it satisfies integrability conditions (Frobenius theorem). It turns out if we start with two different points on a same orbit on the constrained subspace and evolve both of them under two different Hamiltonians, respectively, which agree on the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows will always lie in the same orbit at equal times. It also turns out if we have two smooth functions A1 and B1, which are constant over orbits at least on the constrained subspace (i.e. physical observables) (i.e. {A1,f}={B1,f}=0 over the constrained subspace)and another two A2 and B2, which are also constant over orbits such that A1 and B1 agrees with A2 and B2 respectively over the restrained subspace, then their Poisson brackets {A1, B1} and {A2, B2} are also constant over orbits and agree over the constrained subspace.
In general, one cannot rule out "ergodic" flows (which basically means that an orbit is dense in some open set), or "subergodic" flows (which an orbit dense in some submanifold of dimension greater than the orbit's dimension). We can't have self-intersecting orbits.
For most "practical" applications of first-class constraints, we do not see such complications: the quotient space of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a differentiable manifold, which can be turned into a symplectic manifold by projecting the symplectic form of M onto it (this can be shown to be well defined). In light of the observation about physical observables mentioned earlier, we can work with this more "physical" smaller symplectic manifold, but with 2n fewer dimensions.
In general, the quotient space is a bit difficult to work with when doing concrete calculations (not to mention nonlocal when working with diffeomorphism constraints), so what is usually done instead is something similar. Note that the restricted submanifold is a bundle (but not a fiber bundle in general) over the quotient manifold. So, instead of working with the quotient manifold, we can work with a section of the bundle instead. This is called gauge fixing.
The major problem is this bundle might not have a global section in general. This is where the "problem" of global anomalies comes in, for example. A global anomaly is different from the Gribov ambiguity, which is when a gauge fixing doesn't work to fix a gauge uniquely, in a global anomaly, there is no consistent definition of the gauge field. A global anomaly is a barrier to defining a quantum gauge theory discovered by Witten in 1980.
What have been described are irreducible first-class constraints. Another complication is that Δf might not be right invertible on subspaces of the restricted submanifold of codimension 1 or greater (which violates the stronger assumption stated earlier in this article). This happens, for example in the cotetrad formulation of general relativity, at the subspace of configurations where the cotetrad field and the connection form happen to be zero over some open subset of space. Here, the constraints are the diffeomorphism constraints.
One way to get around this is this: For reducible constraints, we relax the condition on the right invertibility of Δf into this one: Any smooth function that vanishes at the zeros of f is the fiberwise contraction of f with (a non-unique) smooth section of a ${\bar {V}}$-vector bundle where ${\bar {V}}$ is the dual vector space to the constraint vector space V. This is called the regularity condition.
Constrained Hamiltonian dynamics from a Lagrangian gauge theory
First of all, we will assume the action is the integral of a local Lagrangian that only depends up to the first derivative of the fields. The analysis of more general cases, while possible is more complicated. When going over to the Hamiltonian formalism, we find there are constraints. Recall that in the action formalism, there are on shell and off shell configurations. The constraints that hold off shell are called primary constraints while those that only hold on shell are called secondary constraints.
Examples
Consider the dynamics of a single point particle of mass m with no internal degrees of freedom moving in a pseudo-Riemannian spacetime manifold S with metric g. Assume also that the parameter τ describing the trajectory of the particle is arbitrary (i.e. we insist upon reparametrization invariance). Then, its symplectic space is the cotangent bundle T*S with the canonical symplectic form ω.
If we coordinatize T * S by its position x in the base manifold S and its position within the cotangent space p, then we have a constraint
f = m2 −g(x)−1(p,p) = 0 .
The Hamiltonian H is, surprisingly enough, H = 0. In light of the observation that the Hamiltonian is only defined up to the equivalence class of smooth functions agreeing on the constrained subspace, we can use a new Hamiltonian H '= f instead. Then, we have the interesting case where the Hamiltonian is the same as a constraint! See Hamiltonian constraint for more details.
Consider now the case of a Yang–Mills theory for a real simple Lie algebra L (with a negative definite Killing form η) minimally coupled to a real scalar field σ, which transforms as an orthogonal representation ρ with the underlying vector space V under L in ( d − 1) + 1 Minkowski spacetime. For l in L, we write
ρ(l)[σ]
as
l[σ]
for simplicity. Let A be the L-valued connection form of the theory. Note that the A here differs from the A used by physicists by a factor of i and g. This agrees with the mathematician's convention.
The action S is given by
$S[\mathbf {A} ,\sigma ]=\int d^{d}x{\frac {1}{4g^{2}}}\eta ((\mathbf {g} ^{-1}\otimes \mathbf {g} ^{-1})(\mathbf {F} ,\mathbf {F} ))+{\frac {1}{2}}\alpha (\mathbf {g} ^{-1}(D\sigma ,D\sigma ))$
where g is the Minkowski metric, F is the curvature form
$d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} $
(no is or gs!) where the second term is a formal shorthand for pretending the Lie bracket is a commutator, D is the covariant derivative
Dσ = dσ − A[σ]
and α is the orthogonal form for ρ.
What is the Hamiltonian version of this model? Well, first, we have to split A noncovariantly into a time component φ and a spatial part A→. Then, the resulting symplectic space has the conjugate variables σ, πσ (taking values in the underlying vector space of ${\bar {\rho }}$, the dual rep of ρ), A→, π→A, φ and πφ. For each spatial point, we have the constraints, πφ=0 and the Gaussian constraint
${\vec {D}}\cdot {\vec {\pi }}_{A}-\rho '(\pi _{\sigma },\sigma )=0$
where since ρ is an intertwiner
$\rho :L\otimes V\rightarrow V$,
ρ ' is the dualized intertwiner
$\rho ':{\bar {V}}\otimes V\rightarrow L$
( L is self-dual via η). The Hamiltonian,
$H_{f}=\int d^{d-1}x{\frac {1}{2}}\alpha ^{-1}(\pi _{\sigma },\pi _{\sigma })+{\frac {1}{2}}\alpha ({\vec {D}}\sigma \cdot {\vec {D}}\sigma )-{\frac {g^{2}}{2}}\eta ({\vec {\pi }}_{A},{\vec {\pi }}_{A})-{\frac {1}{2g^{2}}}\eta (\mathbf {B} \cdot \mathbf {B} )-\eta (\pi _{\phi },f)-<\pi _{\sigma },\phi [\sigma ]>-\eta (\phi ,{\vec {D}}\cdot {\vec {\pi }}_{A}).$
The last two terms are a linear combination of the Gaussian constraints and we have a whole family of (gauge equivalent)Hamiltonians parametrized by f. In fact, since the last three terms vanish for the constrained states, we may drop them.
Second class constraints
In a constrained Hamiltonian system, a dynamical quantity is second class if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a second class constraint.
See Dirac brackets for diverse illustrations.
An example: a particle confined to a sphere
Before going on to the general theory, consider a specific example step by step to motivate the general analysis.
Start with the action describing a Newtonian particle of mass m constrained to a spherical surface of radius R within a uniform gravitational field g. When one works in Lagrangian mechanics, there are several ways to implement a constraint: one can switch to generalized coordinates that manifestly solve the constraint, or one can use a Lagrange multiplier while retaining the redundant coordinates so constrained.
In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice). For pedagogical reasons, instead, consider the problem in (redundant) Cartesian coordinates, with a Lagrange multiplier term enforcing the constraint.
The action is given by
$S=\int dtL=\int dt\left[{\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-mgz+{\frac {\lambda }{2}}(x^{2}+y^{2}+z^{2}-R^{2})\right]$
where the last term is the Lagrange multiplier term enforcing the constraint.
Of course, as indicated, we could have just used different, non-redundant, spherical coordinates and written it as
$S=\int dt\left[{\frac {mR^{2}}{2}}({\dot {\theta }}^{2}+\sin ^{2}(\theta ){\dot {\phi }}^{2})+mgR\cos(\theta )\right]$
instead, without extra constraints; but we are considering the former coordinatization to illustrate constraints.
The conjugate momenta are given by
$p_{x}=m{\dot {x}}$, $p_{y}=m{\dot {y}}$, $p_{z}=m{\dot {z}}$, $p_{\lambda }=0$ .
Note that we can't determine •λ from the momenta.
The Hamiltonian is given by
$H={\vec {p}}\cdot {\dot {\vec {r}}}+p_{\lambda }{\dot {\lambda }}-L={\frac {p^{2}}{2m}}+p_{\lambda }{\dot {\lambda }}+mgz-{\frac {\lambda }{2}}(r^{2}-R^{2})$.
We cannot eliminate •λ at this stage yet. We are here treating •λ as a shorthand for a function of the symplectic space which we have yet to determine and not as an independent variable. For notational consistency, define u1 = •λ from now on. The above Hamiltonian with the pλ term is the "naive Hamiltonian". Note that since, on-shell, the constraint must be satisfied, one cannot distinguish, on-shell, between the naive Hamiltonian and the above Hamiltonian with the undetermined coefficient, •λ = u1.
We have the primary constraint
pλ=0.
We require, on the grounds of consistency, that the Poisson bracket of all the constraints with the Hamiltonian vanish at the constrained subspace. In other words, the constraints must not evolve in time if they are going to be identically zero along the equations of motion.
From this consistency condition, we immediately get the secondary constraint
${\begin{aligned}0&=\{H,p_{\lambda }\}_{\text{PB}}\\&=\sum _{i}{\frac {\partial H}{\partial q_{i}}}{\frac {\partial p_{\lambda }}{\partial p_{i}}}-{\frac {\partial H}{\partial p_{i}}}{\frac {\partial p_{\lambda }}{\partial q_{i}}}\\&={\frac {\partial H}{\partial \lambda }}\\&={\frac {1}{2}}(r^{2}-R^{2})\\&\Downarrow \\0&=r^{2}-R^{2}\end{aligned}}$
This constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient u2, enlarging the Hamiltonian to
$H={\frac {p^{2}}{2m}}+mgz-{\frac {\lambda }{2}}(r^{2}-R^{2})+u_{1}p_{\lambda }+u_{2}(r^{2}-R^{2})~.$
Similarly, from this secondary constraint, we find the tertiary constraint
${\begin{aligned}0&=\{H,r^{2}-R^{2}\}_{PB}\\&=\{H,x^{2}\}_{PB}+\{H,y^{2}\}_{PB}+\{H,z^{2}\}_{PB}\\&={\frac {\partial H}{\partial p_{x}}}2x+{\frac {\partial H}{\partial p_{y}}}2y+{\frac {\partial H}{\partial p_{z}}}2z\\&={\frac {2}{m}}(p_{x}x+p_{y}y+p_{z}z)\\&\Downarrow \\0&={\vec {p}}\cdot {\vec {r}}\end{aligned}}$
Again, one should add this constraint into the Hamiltonian, since, on-shell, no one can tell the difference. Therefore, so far, the Hamiltonian looks like
$H={\frac {p^{2}}{2m}}+mgz-{\frac {\lambda }{2}}(r^{2}-R^{2})+u_{1}p_{\lambda }+u_{2}(r^{2}-R^{2})+u_{3}{\vec {p}}\cdot {\vec {r}}~,$
where u1, u2, and u3 are still completely undetermined.
Note that, frequently, all constraints that are found from consistency conditions are referred to as secondary constraints and secondary, tertiary, quaternary, etc., constraints are not distinguished.
We keep turning the crank, demanding this new constraint have vanishing Poisson bracket
$0=\{{\vec {p}}\cdot {\vec {r}},\,H\}_{PB}={\frac {p^{2}}{m}}-mgz+\lambda r^{2}-2u_{2}r^{2}.$
We might despair and think that there is no end to this, but because one of the new Lagrange multipliers has shown up, this is not a new constraint, but a condition that fixes the Lagrange multiplier:
$u_{2}={\frac {\lambda }{2}}+{\frac {1}{r^{2}}}\left({\frac {p^{2}}{2m}}-{\frac {1}{2}}mgz\right).$
Plugging this into our Hamiltonian gives us (after a little algebra)
$H={\frac {p^{2}}{2m}}(2-{\frac {R^{2}}{r^{2}}})+{\frac {1}{2}}mgz(1+{\frac {R^{2}}{r^{2}}})+u_{1}p_{\lambda }+u_{3}{\vec {p}}\cdot {\vec {r}}$
Now that there are new terms in the Hamiltonian, one should go back and check the consistency conditions for the primary and secondary constraints. The secondary constraint's consistency condition gives
${\frac {2}{m}}{\vec {r}}\cdot {\vec {p}}+2u_{3}r^{2}=0.$
Again, this is not a new constraint; it only determines that
$u_{3}=-{\frac {{\vec {r}}\cdot {\vec {p}}}{mr^{2}}}~.$
At this point there are no more constraints or consistency conditions to check!
Putting it all together,
$H=\left(2-{\frac {R^{2}}{r^{2}}}\right){\frac {p^{2}}{2m}}+{\frac {1}{2}}\left(1+{\frac {R^{2}}{r^{2}}}\right)mgz-{\frac {({\vec {r}}\cdot {\vec {p}})^{2}}{mr^{2}}}+u_{1}p_{\lambda }$.
When finding the equations of motion, one should use the above Hamiltonian, and as long as one is careful to never use constraints before taking derivatives in the Poisson bracket then one gets the correct equations of motion. That is, the equations of motion are given by
${\dot {\vec {r}}}=\{{\vec {r}},\,H\}_{PB},\quad {\dot {\vec {p}}}=\{{\vec {p}},\,H\}_{PB},\quad {\dot {\lambda }}=\{\lambda ,\,H\}_{PB},\quad {\dot {p}}_{\lambda }=\{p_{\lambda },H\}_{PB}.$
Before analyzing the Hamiltonian, consider the three constraints,
$\varphi _{1}=p_{\lambda },\quad \varphi _{2}=r^{2}-R^{2},\quad \varphi _{3}={\vec {p}}\cdot {\vec {r}}.$
Note the nontrivial Poisson bracket structure of the constraints. In particular,
$\{\varphi _{2},\varphi _{3}\}=2r^{2}\neq 0.$
The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but even on-shell it is nonzero. Therefore, φ2 and φ3 are second class constraints while φ1 is a first class constraint. Note that these constraints satisfy the regularity condition.
Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace. However, Dirac noticed that we can turn the underlying differential manifold of the symplectic space into a Poisson manifold using his eponymous modified bracket, called the Dirac bracket, such that this Dirac bracket of any (smooth) function with any of the second class constraints always vanishes.
Effectively, these brackets (illustrated for this spherical surface in the Dirac bracket article) project the system back onto the constraints surface. If one then wished to canonically quantize this system, then one need promote the canonical Dirac brackets,[4] not the canonical Poisson brackets to commutation relations.
Examination of the above Hamiltonian shows a number of interesting things happening. One thing to note is that, on-shell when the constraints are satisfied, the extended Hamiltonian is identical to the naive Hamiltonian, as required. Also, note that λ dropped out of the extended Hamiltonian. Since φ1 is a first class primary constraint, it should be interpreted as a generator of a gauge transformation. The gauge freedom is the freedom to choose λ, which has ceased to have any effect on the particle's dynamics. Therefore, that λ dropped out of the Hamiltonian, that u1 is undetermined, and that φ1 = pλ is first class, are all closely interrelated.
Note that it would be more natural not to start with a Lagrangian with a Lagrange multiplier, but instead take r² − R² as a primary constraint and proceed through the formalism: The result would the elimination of the extraneous λ dynamical quantity. However, the example is more edifying in its current form.
See also: Dirac bracket
Example: Proca action
Another example we will use is the Proca action. The fields are $A^{\mu }=({\vec {A}},\phi )$ and the action is
$S=\int d^{d}xdt\left[{\frac {1}{2}}E^{2}-{\frac {1}{4}}B_{ij}B_{ij}-{\frac {m^{2}}{2}}A^{2}+{\frac {m^{2}}{2}}\phi ^{2}\right]$
where
${\vec {E}}\equiv -\nabla \phi -{\dot {\vec {A}}}$
and
$B_{ij}\equiv {\frac {\partial A_{j}}{\partial x_{i}}}-{\frac {\partial A_{i}}{\partial x_{j}}}$.
$({\vec {A}},-{\vec {E}})$ and $(\phi ,\pi )$ are canonical variables. The second class constraints are
$\pi \approx 0$
and
$\nabla \cdot {\vec {E}}+m^{2}\phi \approx 0$.
The Hamiltonian is given by
$H=\int d^{d}x\left[{\frac {1}{2}}E^{2}+{\frac {1}{4}}B_{ij}B_{ij}-\pi \nabla \cdot {\vec {A}}+{\vec {E}}\cdot \nabla \phi +{\frac {m^{2}}{2}}A^{2}-{\frac {m^{2}}{2}}\phi ^{2}\right]$.
See also
• Dirac bracket
• Holonomic constraint
• Analysis of flows
References
1. Ingemar Bengtsson, Stockholm University. "Constrained Hamiltonian Systems" (PDF). Stockholm University. Retrieved 29 May 2018. We start from a Lagrangian L ( q, ̇ q ), derive the canonical momenta, postulate the naive Poisso n brackets, and compute the Hamiltonian. For simplicity, one assumes that no second class constraints occur, or if they do, that they have been dealt with already and the naive brackets replaced with Dirac brackets. There remain a set of constraints [...]
2. Dirac, Paul A. M. (1950), "Generalized Hamiltonian dynamics", Canadian Journal of Mathematics, 2: 129–148, doi:10.4153/CJM-1950-012-1, ISSN 0008-414X, MR 0043724, S2CID 119748805
3. Dirac, Paul A. M. (1964), Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, vol. 2, Belfer Graduate School of Science, New York, ISBN 9780486417134, MR 2220894. Unabridged reprint of original, Dover Publications, New York, NY, 2001.
4. Corrigan, E.; Zachos, C. K. (1979). "Non-local charges for the supersymmetric σ-model". Physics Letters B. 88 (3–4): 273. Bibcode:1979PhLB...88..273C. doi:10.1016/0370-2693(79)90465-9.
Further reading
• Falck, N. K.; Hirshfeld, A. C. (1983). "Dirac-bracket quantisation of a constrained nonlinear system: The rigid rotator". European Journal of Physics. 4 (1): 5–9. Bibcode:1983EJPh....4....5F. doi:10.1088/0143-0807/4/1/003. S2CID 250845310.
• Homma, T.; Inamoto, T.; Miyazaki, T. (1990). "Schrödinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space". Physical Review D. 42 (6): 2049–2056. Bibcode:1990PhRvD..42.2049H. doi:10.1103/PhysRevD.42.2049. PMID 10013054.
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Second covariant derivative
In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields.
See also: Exterior covariant derivative
Definition
Formally, given a (pseudo)-Riemannian manifold (M, g) associated with a vector bundle E → M, let ∇ denote the Levi-Civita connection given by the metric g, and denote by Γ(E) the space of the smooth sections of the total space E. Denote by T*M the cotangent bundle of M. Then the second covariant derivative can be defined as the composition of the two ∇s as follows: [1]
$\Gamma (E){\stackrel {\nabla }{\longrightarrow }}\Gamma (T^{*}M\otimes E){\stackrel {\nabla }{\longrightarrow }}\Gamma (T^{*}M\otimes T^{*}M\otimes E).$
For example, given vector fields u, v, w, a second covariant derivative can be written as
$(\nabla _{u,v}^{2}w)^{a}=u^{c}v^{b}\nabla _{c}\nabla _{b}w^{a}$
by using abstract index notation. It is also straightforward to verify that
$(\nabla _{u}\nabla _{v}w)^{a}=u^{c}\nabla _{c}v^{b}\nabla _{b}w^{a}=u^{c}v^{b}\nabla _{c}\nabla _{b}w^{a}+(u^{c}\nabla _{c}v^{b})\nabla _{b}w^{a}=(\nabla _{u,v}^{2}w)^{a}+(\nabla _{\nabla _{u}v}w)^{a}.$
Thus
$\nabla _{u,v}^{2}w=\nabla _{u}\nabla _{v}w-\nabla _{\nabla _{u}v}w.$
When the torsion tensor is zero, so that $[u,v]=\nabla _{u}v-\nabla _{v}u$, we may use this fact to write Riemann curvature tensor as [2]
$R(u,v)w=\nabla _{u,v}^{2}w-\nabla _{v,u}^{2}w.$
Similarly, one may also obtain the second covariant derivative of a function f as
$\nabla _{u,v}^{2}f=u^{c}v^{b}\nabla _{c}\nabla _{b}f=\nabla _{u}\nabla _{v}f-\nabla _{\nabla _{u}v}f.$
Again, for the torsion-free Levi-Civita connection, and for any vector fields u and v, when we feed the function f into both sides of
$\nabla _{u}v-\nabla _{v}u=[u,v]$
we find
$(\nabla _{u}v-\nabla _{v}u)(f)=[u,v](f)=u(v(f))-v(u(f)).$.
This can be rewritten as
$\nabla _{\nabla _{u}v}f-\nabla _{\nabla _{v}u}f=\nabla _{u}\nabla _{v}f-\nabla _{v}\nabla _{u}f,$
so we have
$\nabla _{u,v}^{2}f=\nabla _{v,u}^{2}f.$
That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives.
Notes
1. Parker, Thomas H. "Geometry Primer" (PDF). Retrieved 2 January 2015., pp. 7
2. Jean Gallier and Dan Guralnik. "Chapter 13: Curvature in Riemannian Manifolds" (PDF). Retrieved 2 January 2015.
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Deviation of a local ring
In commutative algebra, the deviations of a local ring R are certain invariants εi(R) that measure how far the ring is from being regular.
Definition
The deviations εn of a local ring R with residue field k are non-negative integers defined in terms of its Poincaré series P(t) by
$P(t)=\sum _{n\geq 0}t^{n}\operatorname {Tor} _{n}^{R}(k,k)=\prod _{n\geq 0}{\frac {(1+t^{2n+1})^{\varepsilon _{2n}}}{(1-t^{2n+2})^{\varepsilon _{2n+1}}}}.$
The zeroth deviation ε0 is the embedding dimension of R (the dimension of its tangent space). The first deviation ε1 vanishes exactly when the ring R is a regular local ring, in which case all the higher deviations also vanish. The second deviation ε2 vanishes exactly when the ring R is a complete intersection ring, in which case all the higher deviations vanish.
References
• Gulliksen, T. H. (1971), "A homological characterization of local complete intersections", Compositio Mathematica, 23: 251–255, ISSN 0010-437X, MR 0301008
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Plane (mathematics)
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.
When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space.
Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional or planar space.[1]
Euclidean plane
This section is an excerpt from Euclidean plane.[edit]
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E2. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement.
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.
The set $\mathbb {R} ^{2}$ of the pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane, since every Euclidean plane is isomorphic to it.
Embedding in three-dimensional space
This section is an excerpt from Euclidean planes in three-dimensional space.[edit]
In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space $\mathbb {R} ^{3}$. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.
While a pair of real numbers $\mathbb {R} ^{2}$ suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space $\mathbb {R} ^{3}$.
Elliptic plane
This section is an excerpt from Elliptic geometry § Elliptic plane.[edit]
The elliptic plane is the real projective plane provided with a metric. Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere tangent to it. With O the center of the hemisphere, a point P in σ determines a line OP intersecting the hemisphere, and any line L ⊂ σ determines a plane OL which intersects the hemisphere in half of a great circle. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line of σ corresponds to this plane; instead a line at infinity is appended to σ. As any line in this extension of σ corresponds to a plane through O, and since any pair of such planes intersects in a line through O, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed.[2]
Given P and Q in σ, the elliptic distance between them is the measure of the angle POQ, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition of distance".[3]: 82 This venture into abstraction in geometry was followed by Felix Klein and Bernhard Riemann leading to non-Euclidean geometry and Riemannian geometry.
Projective plane
This section is an excerpt from Projective plane.[edit]
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point.
Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane.[4] This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane.
A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces. Such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes.
Further generalizations
In addition to its familiar geometric structure, with isomorphisms that are isometries with respect to the usual inner product, the plane may be viewed at various other levels of abstraction. Each level of abstraction corresponds to a specific category.
At one extreme, all geometrical and metric concepts may be dropped to leave the topological plane, which may be thought of as an idealized homotopically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent the open disc, is the basic topological neighborhood used to construct surfaces (or 2-manifolds) classified in low-dimensional topology. Isomorphisms of the topological plane are all continuous bijections. The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem.
The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but collinearity and ratios of distances on any line are preserved.
Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability.
In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to the complex plane and the major area of complex analysis. The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation.
In the same way as in the real case, the plane may also be viewed as the simplest, one-dimensional (over the complex numbers) complex manifold, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are all conformal bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a multiplication by a complex number and a translation.
In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere tangent to the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point. This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature.
Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane. The latter possibility finds an application in the theory of special relativity in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is a timelike hypersurface in three-dimensional Minkowski space.)
Topological and differential geometric notions
The one-point compactification of the plane is homeomorphic to a sphere (see stereographic projection); the open disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the (compact) sphere. The result of this compactification is a manifold referred to as the Riemann sphere or the complex projective line. The projection from the Euclidean plane to a sphere without a point is a diffeomorphism and even a conformal map.
The plane itself is homeomorphic (and diffeomorphic) to an open disk. For the hyperbolic plane such diffeomorphism is conformal, but for the Euclidean plane it is not.
See also
• Affine plane
• Hyperbolic geometry
References
1. Janich, P.; Zook, D. (1992). Euclid's Heritage. Is Space Three-Dimensional?. The Western Ontario Series in Philosophy of Science. Springer Netherlands. p. 50. ISBN 978-0-7923-2025-8. Retrieved 2023-03-11.
2. H. S. M. Coxeter (1965) Introduction to Geometry, page 92
3. Cayley, Arthur (1859), "A sixth memoir upon quantics", Philosophical Transactions of the Royal Society of London, 149: 61–90, doi:10.1098/rstl.1859.0004, ISSN 0080-4614, JSTOR 108690
4. The phrases "projective plane", "extended affine plane" and "extended Euclidean plane" may be distinguished according to whether the line at infinity is regarded as special (in the so-called "projective" plane it is not, in the "extended" planes it is) and to whether Euclidean metric is regarded as meaningful (in the projective and affine planes it is not). Similarly for projective or extended spaces of other dimensions.
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Eccentricity (mathematics)
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.
For the eccentricity of a vertex in a graph, see Eccentricity (graph theory).
One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular:
• The eccentricity of a circle is $0.$
• The eccentricity of an ellipse which is not a circle is between $0$ and $1.$
• The eccentricity of a parabola is $1.$
• The eccentricity of a hyperbola is greater than $1.$
• The eccentricity of a pair of lines is $\infty $
Two conic sections with the same eccentricity are similar.
Definitions
Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as e.
The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis vertical, the eccentricity is[1]
$e={\frac {\sin \beta }{\sin \alpha }},\ \ 0<\alpha <90^{\circ },\ 0\leq \beta \leq 90^{\circ }\ ,$
where β is the angle between the plane and the horizontal and α is the angle between the cone's slant generator and the horizontal. For $\beta =0$ the plane section is a circle, for $\beta =\alpha $ a parabola. (The plane must not meet the vertex of the cone.)
The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a: that is, $e={\frac {c}{a}}$ (lacking a center, the linear eccentricity for parabolas is not defined). It is worth to note that a parabola can be treated as an ellipse or a hyperbola, but with one focal point at infinity.
Alternative names
The eccentricity is sometimes called the first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called the numerical eccentricity.
In the case of ellipses and hyperbolas the linear eccentricity is sometimes called the half-focal separation.
Notation
Three notational conventions are in common use:
1. e for the eccentricity and c for the linear eccentricity.
2. ε for the eccentricity and e for the linear eccentricity.
3. e or ϵ< for the eccentricity and f for the linear eccentricity (mnemonic for half-focal separation).
This article uses the first notation.
Values
Conic sectionEquationEccentricity (e)Linear eccentricity (c)
Circle $x^{2}+y^{2}=r^{2}$ $0$ $0$
Ellipse ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$ or ${\frac {y^{2}}{a^{2}}}+{\frac {x^{2}}{b^{2}}}=1$ where $a>b$ ${\sqrt {1-{\frac {b^{2}}{a^{2}}}}}$ ${\sqrt {a^{2}-b^{2}}}$
Parabola $x^{2}=4ay$ $1$ undefined ($\infty $)
Hyperbola ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$ or ${\frac {y^{2}}{a^{2}}}-{\frac {x^{2}}{b^{2}}}=1$ ${\sqrt {1+{\frac {b^{2}}{a^{2}}}}}$ ${\sqrt {a^{2}+b^{2}}}$
Here, for the ellipse and the hyperbola, a is the length of the semi-major axis and b is the length of the semi-minor axis.
When the conic section is given in the general quadratic form
$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,$
the following formula gives the eccentricity e if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse, and not an imaginary ellipse:[2]
$e={\sqrt {\frac {2{\sqrt {(A-C)^{2}+B^{2}}}}{\eta (A+C)+{\sqrt {(A-C)^{2}+B^{2}}}}}}$
where $\eta =1$ if the determinant of the 3×3 matrix
${\begin{bmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{bmatrix}}$
is negative or $\eta =-1$ if that determinant is positive.
Ellipses
The eccentricity of an ellipse is strictly less than 1. When circles (which have eccentricity 0) are counted as ellipses, the eccentricity of an ellipse is greater than or equal to 0; if circles are given a special category and are excluded from the category of ellipses, then the eccentricity of an ellipse is strictly greater than 0.
For any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis. In the coordinate system with origin at the ellipse's center and x-axis aligned with the major axis, points on the ellipse satisfy the equation
${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,$
with foci at coordinates $(\pm c,0)$ for $ c={\sqrt {a^{2}-b^{2}}}.$
We define a number of related additional concepts (only for ellipses):
NameSymbolin terms of a and bin terms of e
First eccentricity $e$ ${\sqrt {1-{\frac {b^{2}}{a^{2}}}}}$ $e$
Second eccentricity $e'$ ${\sqrt {{\frac {a^{2}}{b^{2}}}-1}}$ ${\frac {e}{\sqrt {1-e^{2}}}}$
Third eccentricity $e''={\sqrt {m}}$ ${\frac {\sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}+b^{2}}}}$ ${\frac {e}{\sqrt {2-e^{2}}}}$
Angular eccentricity $\alpha $ $\cos ^{-1}\left({\frac {b}{a}}\right)$ $\sin ^{-1}e$
Other formulae for the eccentricity of an ellipse
The eccentricity of an ellipse is, most simply, the ratio of the linear eccentricity c (distance between the center of the ellipse and each focus) to the length of the semimajor axis a.
$e={\frac {c}{a}}.$
The eccentricity is also the ratio of the semimajor axis a to the distance d from the center to the directrix:
$e={\frac {a}{d}}.$
The eccentricity can be expressed in terms of the flattening f (defined as $f=1-b/a$ for semimajor axis a and semiminor axis b):
$e={\sqrt {1-(1-f)^{2}}}={\sqrt {f(2-f)}}.$
(Flattening may be denoted by g in some subject areas if f is linear eccentricity.)
Define the maximum and minimum radii $r_{\text{max}}$ and $r_{\text{min}}$ as the maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). Then with semimajor axis a, the eccentricity is given by
$e={\frac {r_{\text{max}}-r_{\text{min}}}{r_{\text{max}}+r_{\text{min}}}}={\frac {r_{\text{max}}-r_{\text{min}}}{2a}},$
which is the distance between the foci divided by the length of the major axis.
Hyperbolas
The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular hyperbola is ${\sqrt {2}}$.
Quadrics
The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane). But: conic sections may occur on surfaces of higher order, too (see image).
Celestial mechanics
In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the apocenter distance is close to the pericenter distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipses, in Keplerian, i.e., $1/r$ potentials.
Analogous classifications
A number of classifications in mathematics use derived terminology from the classification of conic sections by eccentricity:
• Classification of elements of SL2(R) as elliptic, parabolic, and hyperbolic – and similarly for classification of elements of PSL2(R), the real Möbius transformations.
• Classification of discrete distributions by variance-to-mean ratio; see cumulants of some discrete probability distributions for details.
• Classification of partial differential equations is by analogy with the conic sections classification; see elliptic, parabolic and hyperbolic partial differential equations.[3]
See also
• Kepler orbits
• Eccentricity vector
• Orbital eccentricity
• Roundness (object)
• Conic constant
References
1. Thomas, George B.; Finney, Ross L. (1979), Calculus and Analytic Geometry (fifth ed.), Addison-Wesley, p. 434. ISBN 0-201-07540-7
2. Ayoub, Ayoub B., "The eccentricity of a conic section", The College Mathematics Journal 34(2), March 2003, 116-121.
3. "Classification of Linear PDEs in Two Independent Variables". Retrieved 2 July 2013.
External links
Wikimedia Commons has media related to Eccentricity.
• MathWorld: Eccentricity
Gravitational orbits
Types
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Maneuvers
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Second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by $\mathrm {I\!I} $ (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.
Surface in R3
Motivation
The second fundamental form of a parametric surface S in R3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:
$z=L{\frac {x^{2}}{2}}+Mxy+N{\frac {y^{2}}{2}}+{\text{higher order terms}}\,,$
and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form
$L\,dx^{2}+2M\,dx\,dy+N\,dy^{2}\,.$
For a smooth point P on S, one can choose the coordinate system so that the plane z = 0 is tangent to S at P, and define the second fundamental form in the same way.
Classical notation
The second fundamental form of a general parametric surface is defined as follows. Let r = r(u,v) be a regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u and v by ru and rv. Regularity of the parametrization means that ru and rv are linearly independent for any (u,v) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product ru × rv is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:
$\mathbf {n} ={\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}\,.$
The second fundamental form is usually written as
$\mathrm {I\!I} =L\,du^{2}+2M\,du\,dv+N\,dv^{2}\,,$
its matrix in the basis {ru, rv} of the tangent plane is
${\begin{bmatrix}L&M\\M&N\end{bmatrix}}\,.$
The coefficients L, M, N at a given point in the parametric uv-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed with the aid of the dot product as follows:
$L=\mathbf {r} _{uu}\cdot \mathbf {n} \,,\quad M=\mathbf {r} _{uv}\cdot \mathbf {n} \,,\quad N=\mathbf {r} _{vv}\cdot \mathbf {n} \,.$
For a signed distance field of Hessian H, the second fundamental form coefficients can be computed as follows:
$L=-\mathbf {r} _{u}\cdot \mathbf {H} \cdot \mathbf {r} _{u}\,,\quad M=-\mathbf {r} _{u}\cdot \mathbf {H} \cdot \mathbf {r} _{v}\,,\quad N=-\mathbf {r} _{v}\cdot \mathbf {H} \cdot \mathbf {r} _{v}\,.$
Physicist's notation
The second fundamental form of a general parametric surface S is defined as follows.
Let r = r(u1,u2) be a regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to uα by rα, α = 1, 2. Regularity of the parametrization means that r1 and r2 are linearly independent for any (u1,u2) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product r1 × r2 is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:
$\mathbf {n} ={\frac {\mathbf {r} _{1}\times \mathbf {r} _{2}}{|\mathbf {r} _{1}\times \mathbf {r} _{2}|}}\,.$
The second fundamental form is usually written as
$\mathrm {I\!I} =b_{\alpha \beta }\,du^{\alpha }\,du^{\beta }\,.$
The equation above uses the Einstein summation convention.
The coefficients bαβ at a given point in the parametric u1u2-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed in terms of the normal vector n as follows:
$b_{\alpha \beta }=r_{\,\alpha \beta }^{\ \ \,\gamma }n_{\gamma }\,.$
Hypersurface in a Riemannian manifold
In Euclidean space, the second fundamental form is given by
$\mathrm {I\!I} (v,w)=-\langle d\nu (v),w\rangle \nu $
where $\nu $ is the Gauss map, and $d\nu $ the differential of $\nu $ regarded as a vector-valued differential form, and the brackets denote the metric tensor of Euclidean space.
More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the shape operator (denoted by S) of a hypersurface,
$\mathrm {I} \!\mathrm {I} (v,w)=\langle S(v),w\rangle n=-\langle \nabla _{v}n,w\rangle n=\langle n,\nabla _{v}w\rangle n\,,$
where ∇vw denotes the covariant derivative of the ambient manifold and n a field of normal vectors on the hypersurface. (If the affine connection is torsion-free, then the second fundamental form is symmetric.)
The sign of the second fundamental form depends on the choice of direction of n (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of orientation of the surface).
Generalization to arbitrary codimension
The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values in the normal bundle and it can be defined by
$\mathrm {I\!I} (v,w)=(\nabla _{v}w)^{\bot }\,,$
where $(\nabla _{v}w)^{\bot }$ denotes the orthogonal projection of covariant derivative $\nabla _{v}w$ onto the normal bundle.
In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:
$\langle R(u,v)w,z\rangle =\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle .$
This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium.
For general Riemannian manifolds one has to add the curvature of ambient space; if N is a manifold embedded in a Riemannian manifold (M,g) then the curvature tensor RN of N with induced metric can be expressed using the second fundamental form and RM, the curvature tensor of M:
$\langle R_{N}(u,v)w,z\rangle =\langle R_{M}(u,v)w,z\rangle +\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle \,.$
See also
• First fundamental form
• Gaussian curvature
• Gauss–Codazzi equations
• Shape operator
• Third fundamental form
• Tautological one-form
References
• Guggenheimer, Heinrich (1977). "Chapter 10. Surfaces". Differential Geometry. Dover. ISBN 0-486-63433-7.
• Kobayashi, Shoshichi & Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 2 (New ed.). Wiley-Interscience. ISBN 0-471-15732-5.
• Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume 3). Publish or Perish. ISBN 0-914098-72-1.
External links
• Steven Verpoort (2008) Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects from Katholieke Universiteit Leuven.
Various notions of curvature defined in differential geometry
Differential geometry
of curves
• Curvature
• Torsion of a curve
• Frenet–Serret formulas
• Radius of curvature (applications)
• Affine curvature
• Total curvature
• Total absolute curvature
Differential geometry
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• Principal curvatures
• Gaussian curvature
• Mean curvature
• Darboux frame
• Gauss–Codazzi equations
• First fundamental form
• Second fundamental form
• Third fundamental form
Riemannian geometry
• Curvature of Riemannian manifolds
• Riemann curvature tensor
• Ricci curvature
• Scalar curvature
• Sectional curvature
Curvature of connections
• Curvature form
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Second-generation wavelet transform
In signal processing, the second-generation wavelet transform (SGWT) is a wavelet transform where the filters (or even the represented wavelets) are not designed explicitly, but the transform consists of the application of the Lifting scheme. Actually, the sequence of lifting steps could be converted to a regular discrete wavelet transform, but this is unnecessary because both design and application is made via the lifting scheme. This means that they are not designed in the frequency domain, as they are usually in the classical (so to speak first generation) transforms such as the DWT and CWT). The idea of moving away from the Fourier domain was introduced independently by David Donoho and Harten in the early 1990s.
Calculating transform
The input signal $f$ is split into odd $\gamma _{1}$ and even $\lambda _{1}$ samples using shifting and downsampling. The detail coefficients $\gamma _{2}$ are then interpolated using the values of $\gamma _{1}$ and the prediction operator on the even values:
$\gamma _{2}=\gamma _{1}-P(\lambda _{1})\,$
The next stage (known as the updating operator) alters the approximation coefficients using the detailed ones:
$\lambda _{2}=\lambda _{1}+U(\gamma _{2})\,$
The functions prediction operator $P$ and updating operator $U$ effectively define the wavelet used for decomposition. For certain wavelets the lifting steps (interpolating and updating) are repeated several times before the result is produced.
The idea can be expanded (as used in the DWT) to create a filter bank with a number of levels. The variable tree used in wavelet packet decomposition can also be used.
Advantages
The SGWT has a number of advantages over the classical wavelet transform in that it is quicker to compute (by a factor of 2) and it can be used to generate a multiresolution analysis that does not fit a uniform grid. Using a priori information the grid can be designed to allow the best analysis of the signal to be made. The transform can be modified locally while preserving invertibility; it can even adapt to some extent to the transformed signal.
References
• Wim Sweldens: Second-Generation Wavelets: Theory and Application
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Wikipedia
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Class formation
In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory.
Definitions
A formation is a topological group G together with a topological G-module A on which G acts continuously.
A layer E/F of a formation is a pair of open subgroups E, F of G such that F is a finite index subgroup of E. It is called a normal layer if F is a normal subgroup of E, and a cyclic layer if in addition the quotient group is cyclic. If E is a subgroup of G, then AE is defined to be the elements of A fixed by E. We write
Hn(E/F)
for the Tate cohomology group Hn(E/F, AF) whenever E/F is a normal layer. (Some authors think of E and F as fixed fields rather than subgroup of G, so write F/E instead of E/F.) In applications, G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure.
A class formation is a formation such that for every normal layer E/F
H1(E/F) is trivial, and
H2(E/F) is cyclic of order |E/F|.
In practice, these cyclic groups come provided with canonical generators uE/F ∈ H2(E/F), called fundamental classes, that are compatible with each other in the sense that the restriction (of cohomology classes) of a fundamental class is another fundamental class. Often the fundamental classes are considered to be part of the structure of a class formation.
A formation that satisfies just the condition H1(E/F)=1 is sometimes called a field formation. For example, if G is any finite group acting on a field L and A=L×, then this is a field formation by Hilbert's theorem 90.
Examples
The most important examples of class formations (arranged roughly in order of difficulty) are as follows:
• Archimedean local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by complex conjugation.
• Finite fields: The module A is the integers (with trivial G-action), and G is the absolute Galois group of a finite field, which is isomorphic to the profinite completion of the integers.
• Local class field theory of characteristic p>0: The module A is the separable algebraic closure of the field of formal Laurent series over a finite field, and G is the Galois group.
• Non-archimedean local class field theory of characteristic 0: The module A is the algebraic closure of a field of p-adic numbers, and G is the Galois group.
• Global class field theory of characteristic p>0: The module A is the union of the groups of idele classes of separable finite extensions of some function field over a finite field, and G is the Galois group.
• Global class field theory of characteristic 0: The module A is the union of the groups of idele classes of algebraic number fields, and G is the Galois group of the rational numbers (or some algebraic number field) acting on A.
It is easy to verify the class formation property for the finite field case and the archimedean local field case, but the remaining cases are more difficult. Most of the hard work of class field theory consists of proving that these are indeed class formations. This is done in several steps, as described in the sections below.
The first inequality
The first inequality of class field theory states that
|H0(E/F)| ≥ |E/F|
for cyclic layers E/F. It is usually proved using properties of the Herbrand quotient, in the more precise form
|H0(E/F)| = |E/F|×|H1(E/F)|.
It is fairly straightforward to prove, because the Herbrand quotient is easy to work out, as it is multiplicative on short exact sequences, and is 1 for finite modules.
Before about 1950, the first inequality was known as the second inequality, and vice versa.
The second inequality
The second inequality of class field theory states that
|H0(E/F)| ≤ |E/F|
for all normal layers E/F.
For local fields, this inequality follows easily from Hilbert's theorem 90 together with the first inequality and some basic properties of group cohomology.
The second inequality was first proved for global fields by Weber using properties of the L series of number fields, as follows. Suppose that the layer E/F corresponds to an extension k⊂K of global fields. By studying the Dedekind zeta function of K one shows that the degree 1 primes of K have Dirichlet density given by the order of the pole at s=1, which is 1 (When K is the rationals, this is essentially Euler's proof that there are infinitely many primes using the pole at s=1 of the Riemann zeta function.) As each prime in k that is a norm is the product of deg(K/k)= |E/F| distinct degree 1 primes of K, this shows that the set of primes of k that are norms has density 1/|E/F|. On the other hand, by studying Dirichlet L-series of characters of the group H0(E/F), one shows that the Dirichlet density of primes of k representing the trivial element of this group has density 1/|H0(E/F)|. (This part of the proof is a generalization of Dirichlet's proof that there are infinitely many primes in arithmetic progressions.) But a prime represents a trivial element of the group H0(E/F) if it is equal to a norm modulo principal ideals, so this set is at least as dense as the set of primes that are norms. So
1/|H0(E/F)| ≥ 1/|E/F|
which is the second inequality.
In 1940 Chevalley found a purely algebraic proof of the second inequality, but it is longer and harder than Weber's original proof. Before about 1950, the second inequality was known as the first inequality; the name was changed because Chevalley's algebraic proof of it uses the first inequality.
Takagi defined a class field to be one where equality holds in the second inequality. By the Artin isomorphism below, H0(E/F) is isomorphic to the abelianization of E/F, so equality in the second inequality holds exactly for abelian extensions, and class fields are the same as abelian extensions.
The first and second inequalities can be combined as follows. For cyclic layers, the two inequalities together prove that
H1(E/F)|E/F| = H0(E/F) ≤ |E/F|
so
H0(E/F) = |E/F|
and
H1(E/F) = 1.
Now a basic theorem about cohomology groups shows that since H1(E/F) = 1 for all cyclic layers, we have
H1(E/F) = 1
for all normal layers (so in particular the formation is a field formation). This proof that H1(E/F) is always trivial is rather roundabout; no "direct" proof of it (whatever this means) for global fields is known. (For local fields the vanishing of H1(E/F) is just Hilbert's theorem 90.)
For cyclic group, H0 is the same as H2, so H2(E/F) = |E/F| for all cyclic layers. Another theorem of group cohomology shows that since H1(E/F) = 1 for all normal layers and H2(E/F) ≤ |E/F| for all cyclic layers, we have
H2(E/F)≤ |E/F|
for all normal layers. (In fact, equality holds for all normal layers, but this takes more work; see the next section.)
The Brauer group
The Brauer groups H2(E/*) of a class formation are defined to be the direct limit of the groups H2(E/F) as F runs over all open subgroups of E. An easy consequence of the vanishing of H1 for all layers is that the groups H2(E/F) are all subgroups of the Brauer group. In local class field theory the Brauer groups are the same as Brauer groups of fields, but in global class field theory the Brauer group of the formation is not the Brauer group of the corresponding global field (though they are related).
The next step is to prove that H2(E/F) is cyclic of order exactly |E/F|; the previous section shows that it has at most this order, so it is sufficient to find some element of order |E/F| in H2(E/F).
The proof for arbitrary extensions uses a homomorphism from the group G onto the profinite completion of the integers with kernel G∞, or in other words a compatible sequence of homomorphisms of G onto the cyclic groups of order n for all n, with kernels Gn. These homomorphisms are constructed using cyclic cyclotomic extensions of fields; for finite fields they are given by the algebraic closure, for non-archimedean local fields they are given by the maximal unramified extensions, and for global fields they are slightly more complicated. As these extensions are given explicitly one can check that they have the property that H2(G/Gn) is cyclic of order n, with a canonical generator. It follows from this that for any layer E, the group H2(E/E∩G∞) is canonically isomorphic to Q/Z. This idea of using roots of unity was introduced by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem.
For general layers E,F there is an exact sequence
$0\rightarrow H^{2}(E/F)\cap H^{2}(E/E\cap G_{\infty })\rightarrow H^{2}(E/E\cap G_{\infty })\rightarrow H^{2}(F/F\cap G_{\infty })$
The last two groups in this sequence can both be identified with Q/Z and the map between them is then multiplication by |E/F|. So the first group is canonically isomorphic to Z/nZ. As H2(E/F) has order at most Z/nZ is must be equal to Z/nZ (and in particular is contained in the middle group)).
This shows that the second cohomology group H2(E/F) of any layer is cyclic of order |E/F|, which completes the verification of the axioms of a class formation. With a little more care in the proofs, we get a canonical generator of H2(E/F), called the fundamental class.
It follows from this that the Brauer group H2(E/*) is (canonically) isomorphic to the group Q/Z, except in the case of the archimedean local fields R and C when it has order 2 or 1.
Tate's theorem and the Artin map
Tate's theorem in group cohomology is as follows. Suppose that A is a module over a finite group G and a is an element of H2(G,A), such that for every subgroup E of G
• H1(E,A) is trivial, and
• H2(E,A) is generated by Res(a) which has order E.
Then cup product with a is an isomorphism
• Hn(G,Z) → Hn+2(G,A).
If we apply the case n=−2 of Tate's theorem to a class formation, we find that there is an isomorphism
• H−2(E/F,Z) → H0(E/F,AF)
for any normal layer E/F. The group H−2(E/F,Z) is just the abelianization of E/F, and the group H0(E/F,AF) is AE modulo the group of norms of AF. In other words, we have an explicit description of the abelianization of the Galois group E/F in terms of AE.
Taking the inverse of this isomorphism gives a homomorphism
AE → abelianization of E/F,
and taking the limit over all open subgroups F gives a homomorphism
AE → abelianization of E,
called the Artin map. The Artin map is not necessarily surjective, but has dense image. By the existence theorem below its kernel is the connected component of AE (for class field theory), which is trivial for class field theory of non-archimedean local fields and for function fields, but is non-trivial for archimedean local fields and number fields.
The Takagi existence theorem
The main remaining theorem of class field theory is the Takagi existence theorem, which states that every finite index closed subgroup of the idele class group is the group of norms corresponding to some abelian extension. The classical way to prove this is to construct some extensions with small groups of norms, by first adding in many roots of unity, and then taking Kummer extensions and Artin–Schreier extensions. These extensions may be non-abelian (though they are extensions of abelian groups by abelian groups); however, this does not really matter, as the norm group of a non-abelian Galois extension is the same as that of its maximal abelian extension (this can be shown using what we already know about class fields). This gives enough (abelian) extensions to show that there is an abelian extension corresponding to any finite index subgroup of the idele class group.
A consequence is that the kernel of the Artin map is the connected component of the identity of the idele class group, so that the abelianization of the Galois group of F is the profinite completion of the idele class group.
For local class field theory, it is also possible to construct abelian extensions more explicitly using Lubin–Tate formal group laws. For global fields, the abelian extensions can be constructed explicitly in some cases: for example, the abelian extensions of the rationals can be constructed using roots of unity, and the abelian extensions of quadratic imaginary fields can be constructed using elliptic functions, but finding an analog of this for arbitrary global fields is an unsolved problem.
Weil group
Main article: Weil group
This is not a Weyl group and has no connection with the Weil–Châtelet group or the Mordell–Weil group
The Weil group of a class formation with fundamental classes uE/F ∈ H2(E/F, AF) is a kind of modified Galois group, introduced by Weil (1951) and used in various formulations of class field theory, and in particular in the Langlands program.
If E/F is a normal layer, then the Weil group U of E/F is the extension
1 → AF → U → E/F → 1
corresponding to the fundamental class uE/F in H2(E/F, AF). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers G/F, for F an open subgroup of G.
The reciprocity map of the class formation (G, A) induces an isomorphism from AG to the abelianization of the Weil group.
See also
• Abelian extension
• Artin L-function
• Artin reciprocity
• Class field theory
• Complex multiplication
• Galois cohomology
• Hasse norm theorem
• Herbrand quotient
• Hilbert class field
• Kronecker–Weber theorem
• Local class field theory
• Takagi existence theorem
• Tate cohomology group
References
• Artin, Emil; Tate, John (2009) [1952], Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, MR 0223335
• Kawada, Yukiyosi (1971), "Class formations", 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Providence, R.I.: American Mathematical Society, pp. 96–114
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 0554237, esp. chapter XI: Class formations
• Tate, J. (1979), "Number theoretic background", Automorphic forms, representations, and L-functions Part 2, Proc. Sympos. Pure Math., vol. XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 978-0-8218-1435-2
• Weil, André (1951), "Sur la theorie du corps de classes", Journal of the Mathematical Society of Japan, 3: 1–35, doi:10.2969/jmsj/00310001, ISSN 0025-5645, MR 0044569, reprinted in volume I of his collected papers, ISBN 0-387-90330-5
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Second-order cone programming
A second-order cone program (SOCP) is a convex optimization problem of the form
minimize $\ f^{T}x\ $
subject to
$\lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i},\quad i=1,\dots ,m$
$Fx=g\ $
where the problem parameters are $f\in \mathbb {R} ^{n},\ A_{i}\in \mathbb {R} ^{{n_{i}}\times n},\ b_{i}\in \mathbb {R} ^{n_{i}},\ c_{i}\in \mathbb {R} ^{n},\ d_{i}\in \mathbb {R} ,\ F\in \mathbb {R} ^{p\times n}$, and $g\in \mathbb {R} ^{p}$. $x\in \mathbb {R} ^{n}$ is the optimization variable. $\lVert x\rVert _{2}$ is the Euclidean norm and $^{T}$ indicates transpose.[1] The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function $(Ax+b,c^{T}x+d)$ to lie in the second-order cone in $\mathbb {R} ^{n_{i}+1}$.[1]
SOCPs can be solved by interior point methods[2] and in general, can be solved more efficiently than semidefinite programming (SDP) problems.[3] Some engineering applications of SOCP include filter design, antenna array weight design, truss design, and grasping force optimization in robotics.[4] Applications in quantitative finance include portfolio optimization; some market impact constraints, because they are not linear, cannot be solved by quadratic programming but can be formulated as SOCP problems.[5][6][7]
Second-order cone
The standard or unit second-order cone of dimension $n+1$ is defined as
${\mathcal {C}}_{n+1}=\left\{{\begin{bmatrix}x\\t\end{bmatrix}}{\Bigg |}x\in \mathbb {R} ^{n},t\in \mathbb {R} ,||x||_{2}\leq t\right\}$.
The second-order cone is also known by quadratic cone, ice-cream cone, or Lorentz cone. The second-order cone in $\mathbb {R} ^{3}$ is $\left\{(x,y,z){\Big |}{\sqrt {x^{2}+y^{2}}}\leq z\right\}$.
The set of points satisfying a second-order cone constraint is the inverse image of the unit second-order cone under an affine mapping:
$\lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i}\Leftrightarrow {\begin{bmatrix}A_{i}\\c_{i}^{T}\end{bmatrix}}x+{\begin{bmatrix}b_{i}\\d_{i}\end{bmatrix}}\in {\mathcal {C}}_{n_{i}+1}$
and hence is convex.
The second-order cone can be embedded in the cone of the positive semidefinite matrices since
$||x||\leq t\Leftrightarrow {\begin{bmatrix}tI&x\\x^{T}&t\end{bmatrix}}\succcurlyeq 0,$
i.e., a second-order cone constraint is equivalent to a linear matrix inequality (Here $M\succcurlyeq 0$ means $M$ is semidefinite matrix). Similarly, we also have,
$\lVert A_{i}x+b_{i}\rVert _{2}\leq c_{i}^{T}x+d_{i}\Leftrightarrow {\begin{bmatrix}(c_{i}^{T}x+d_{i})I&A_{i}x+b_{i}\\(A_{i}x+b_{i})^{T}&c_{i}^{T}x+d_{i}\end{bmatrix}}\succcurlyeq 0$.
Relation with other optimization problems
When $A_{i}=0$ for $i=1,\dots ,m$, the SOCP reduces to a linear program. When $c_{i}=0$ for $i=1,\dots ,m$, the SOCP is equivalent to a convex quadratically constrained linear program.
Convex quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating the objective function as a constraint.[4] Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semidefinite program.[4] The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation.[3] In fact, while any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP,[8] it is known that there exist convex semialgebraic sets that are not representable by SDPs, that is, there exist convex semialgebraic sets that can not be written as a feasible region of a SDP.[9]
Examples
Quadratic constraint
Consider a convex quadratic constraint of the form
$x^{T}Ax+b^{T}x+c\leq 0.$
This is equivalent to the SOCP constraint
$\lVert A^{1/2}x+{\frac {1}{2}}A^{-1/2}b\rVert \leq \left({\frac {1}{4}}b^{T}A^{-1}b-c\right)^{\frac {1}{2}}$
Stochastic linear programming
Consider a stochastic linear program in inequality form
minimize $\ c^{T}x\ $
subject to
$\mathbb {P} (a_{i}^{T}x\leq b_{i})\geq p,\quad i=1,\dots ,m$
where the parameters $a_{i}\ $ are independent Gaussian random vectors with mean ${\bar {a}}_{i}$ and covariance $\Sigma _{i}\ $ and $p\geq 0.5$. This problem can be expressed as the SOCP
minimize $\ c^{T}x\ $
subject to
${\bar {a}}_{i}^{T}x+\Phi ^{-1}(p)\lVert \Sigma _{i}^{1/2}x\rVert _{2}\leq b_{i},\quad i=1,\dots ,m$
where $\Phi ^{-1}(\cdot )\ $ is the inverse normal cumulative distribution function.[1]
Stochastic second-order cone programming
We refer to second-order cone programs as deterministic second-order cone programs since data defining them are deterministic. Stochastic second-order cone programs are a class of optimization problems that are defined to handle uncertainty in data defining deterministic second-order cone programs.[10]
Solvers and scripting (programming) languages
Name License Brief info
AMPLcommercialAn algebraic modeling language with SOCP support
Artelys Knitrocommercial
CPLEXcommercial
FICO Xpresscommercial
Gurobi Optimizercommercial
MATLABcommercialThe coneprog function solves SOCP problems[11] using an interior-point algorithm[12]
MOSEKcommercialparallel interior-point algorithm
NAG Numerical LibrarycommercialGeneral purpose numerical library with SOCP solver
References
1. Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (PDF). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved July 15, 2019.
2. Potra, lorian A.; Wright, Stephen J. (1 December 2000). "Interior-point methods". Journal of Computational and Applied Mathematics. 124 (1–2): 281–302. Bibcode:2000JCoAM.124..281P. doi:10.1016/S0377-0427(00)00433-7.
3. Fawzi, Hamza (2019). "On representing the positive semidefinite cone using the second-order cone". Mathematical Programming. 175 (1–2): 109–118. arXiv:1610.04901. doi:10.1007/s10107-018-1233-0. ISSN 0025-5610. S2CID 119324071.
4. Lobo, Miguel Sousa; Vandenberghe, Lieven; Boyd, Stephen; Lebret, Hervé (1998). "Applications of second-order cone programming". Linear Algebra and Its Applications. 284 (1–3): 193–228. doi:10.1016/S0024-3795(98)10032-0.
5. "Solving SOCP" (PDF).
6. "portfolio optimization" (PDF).
7. Li, Haksun (16 January 2022). Numerical Methods Using Java: For Data Science, Analysis, and Engineering. APress. pp. Chapter 10. ISBN 978-1484267967.
8. Scheiderer, Claus (2020-04-08). "Second-order cone representation for convex subsets of the plane". arXiv:2004.04196 [math.OC].
9. Scheiderer, Claus (2018). "Spectrahedral Shadows". SIAM Journal on Applied Algebra and Geometry. 2 (1): 26–44. doi:10.1137/17M1118981. ISSN 2470-6566.
10. Alzalg, Baha M. (2012-10-01). "Stochastic second-order cone programming: Applications models". Applied Mathematical Modelling. 36 (10): 5122–5134. doi:10.1016/j.apm.2011.12.053. ISSN 0307-904X.
11. "Second-order cone programming solver - MATLAB coneprog". MathWorks. 2021-03-01. Retrieved 2021-07-15.
12. "Second-Order Cone Programming Algorithm - MATLAB & Simulink". MathWorks. 2021-03-01. Retrieved 2021-07-15.
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Second-order predicate
In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument.[1] Compare higher-order predicate.
The idea of second order predication was introduced by the German mathematician and philosopher Frege. It is based on his idea that a predicate such as "is a philosopher" designates a concept, rather than an object.[2] Sometimes a concept can itself be the subject of a proposition, such as in "There are no Bosnian philosophers". In this case, we are not saying anything of any Bosnian philosophers, but of the concept "is a Bosnian philosopher" that it is not satisfied. Thus the predicate "is not satisfied" attributes something to the concept "is a Bosnian philosopher", and is thus a second-level predicate.
This idea is the basis of Frege's theory of number.[3]
References
1. Yaqub, Aladdin M. (2013), An Introduction to Logical Theory, Broadview Press, p. 288, ISBN 9781551119939.
2. Oppy, Graham (2007), Ontological Arguments and Belief in God, Cambridge University Press, p. 145, ISBN 9780521039000.
3. Kremer, Michael (1985), "Frege's theory of number and the distinction between function and object", Philosophical Studies, 47 (3): 313–323, doi:10.1007/BF00355206, MR 0788101.
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Second partial derivative test
In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.
The Hessian approximates the function at a critical point with a second-degree polynomial.
Functions of two variables
Suppose that f(x, y) is a differentiable real function of two variables whose second partial derivatives exist and are continuous. The Hessian matrix H of f is the 2 × 2 matrix of partial derivatives of f:
$H(x,y)={\begin{bmatrix}f_{xx}(x,y)&f_{xy}(x,y)\\f_{yx}(x,y)&f_{yy}(x,y)\end{bmatrix}}.$
Define D(x, y) to be the determinant
$D(x,y)=\det(H(x,y))=f_{xx}(x,y)f_{yy}(x,y)-\left(f_{xy}(x,y)\right)^{2}$
of H. Finally, suppose that (a, b) is a critical point of f, that is, that fx(a, b) = fy(a, b) = 0. Then the second partial derivative test asserts the following:[1]
1. If D(a, b) > 0 and fxx(a, b) > 0 then (a, b) is a local minimum of f.
2. If D(a, b) > 0 and fxx(a, b) < 0 then (a, b) is a local maximum of f.
3. If D(a, b) < 0 then (a, b) is a saddle point of f.
4. If D(a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive).
Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that fxx fyy − fxy2 is positive at (x, y) implies that fxx and fyy have the same sign there. Therefore, the second condition, that fxx be greater (or less) than zero, could equivalently be that fyy or tr(H) = fxx + fyy be greater (or less) than zero at that point.
A condition implicit in the statement of the test is that if $f_{xx}=0$ or $f_{yy}=0$, it must be the case that $D(a,b)\leq 0,$ and therefore only cases 3 or 4 are possible.
Functions of many variables
For a function f of three or more variables, there is a generalization of the rule above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point. The following test can be applied at any critical point a for which the Hessian matrix is invertible:
1. If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a.
2. If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a.
3. If the Hessian has both positive and negative eigenvalues then a is a saddle point for f (and in fact this is true even if a is degenerate).
In those cases not listed above, the test is inconclusive.[2]
For functions of three or more variables, the determinant of the Hessian does not provide enough information to classify the critical point, because the number of jointly sufficient second-order conditions is equal to the number of variables, and the sign condition on the determinant of the Hessian is only one of the conditions. Note that in the one-variable case, the Hessian condition simply gives the usual second derivative test.
In the two variable case, $D(a,b)$ and $f_{xx}(a,b)$ are the principal minors of the Hessian. The first two conditions listed above on the signs of these minors are the conditions for the positive or negative definiteness of the Hessian. For the general case of an arbitrary number n of variables, there are n sign conditions on the n principal minors of the Hessian matrix that together are equivalent to positive or negative definiteness of the Hessian (Sylvester's criterion): for a local minimum, all the principal minors need to be positive, while for a local maximum, the minors with an odd number of rows and columns need to be negative and the minors with an even number of rows and columns need to be positive. See Hessian matrix#Bordered Hessian for a discussion that generalizes these rules to the case of equality-constrained optimization.
Examples
To find and classify the critical points of the function
$z=f(x,y)=(x+y)(xy+xy^{2})$,
we first set the partial derivatives
${\frac {\partial z}{\partial x}}=y(2x+y)(y+1)$ and ${\frac {\partial z}{\partial y}}=x\left(3y^{2}+2y(x+1)+x\right)$
equal to zero and solve the resulting equations simultaneously to find the four critical points
$(0,0),(0,-1),(1,-1)$ and $\left({\frac {3}{8}},-{\frac {3}{4}}\right)$.
In order to classify the critical points, we examine the value of the determinant D(x, y) of the Hessian of f at each of the four critical points. We have
${\begin{aligned}D(a,b)&=f_{xx}(a,b)f_{yy}(a,b)-\left(f_{xy}(a,b)\right)^{2}\\&=2b(b+1)\cdot 2a(a+3b+1)-(2a+2b+4ab+3b^{2})^{2}.\end{aligned}}$
Now we plug in all the different critical values we found to label them; we have
$D(0,0)=0;~~D(0,-1)=-1;~~D(1,-1)=-1;~~D\left({\frac {3}{8}},-{\frac {3}{4}}\right)={\frac {27}{128}}.$
Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at $\left({\frac {3}{8}},-{\frac {3}{4}}\right)$ since $f_{xx}=-{\frac {3}{8}}<0$. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f.)
Notes
1. Stewart 2004 harvnb error: no target: CITEREFStewart2004 (help), p. 803.
2. Kurt Endl/Wolfgang Luh: Analysis II. Aula-Verlag 1972, 7th edition 1989, ISBN 3-89104-455-0, pp. 248-258 (German)
References
• James Stewart (2005). Multivariable Calculus: Concepts & Contexts. Brooks/Cole. ISBN 0-534-41004-9.
External links
• Relative Minimums and Maximums - Paul's Online Math Notes - Calc III Notes (Lamar University)
• Weisstein, Eric W. "Second Derivative Test". MathWorld.
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First-player and second-player win
In combinatorial game theory, a two-player deterministic perfect information turn-based game is a first-player-win if with perfect play the first player to move can always force a win. Similarly, a game is second-player-win if with perfect play the second player to move can always force a win. With perfect play, if neither side can force a win, the game is a draw.
Some games with relatively small game trees have been proven to be first or second-player wins. For example, the game of nim with the classic 3–4–5 starting position is a first-player-win game. However, Nim with the 1-3-5-7 starting position is a second-player-win. The classic game of Connect Four has been mathematically proven to be first-player-win.
With perfect play, checkers has been determined to be a draw; neither player can force a win.[1] Another example of a game which leads to a draw with perfect play is tic-tac-toe, and this includes play from any opening move.
Significant theory has been completed in the effort to solve chess. It has been speculated that there may be first-move advantage which can be detected when the game is played imperfectly (such as with all humans and all current chess engines). However, with perfect play, it remains unsolved as to whether the game is a first-player win (White), a second player win (Black), or a forced draw.[2][3][4]
See also
• Solved game
• Strategy-stealing argument
• Zugzwang
• Determinacy
• Combinatorial game theory
• First-move advantage in chess
References
1. Schaeffer, J.; Burch, N.; Bjornsson, Y.; Kishimoto, A.; Muller, M.; Lake, R.; Lu, P.; Sutphen, S. (2007). "Checkers Is Solved". Science. 317 (5844): 1518–1522. Bibcode:2007Sci...317.1518S. doi:10.1126/science.1144079. PMID 17641166. S2CID 10274228. Retrieved 2008-11-24.
2. J.W.H.M. Uiterwijk, H.J. van den Herik. "The Advantage of the Initiative". (August 1999).
3. Shannon, C. (March 1950). "Programming a Computer for Playing Chess" (PDF). Philosophical Magazine. 7. 41 (314). Archived from the original (PDF) on 2010-07-06. Retrieved 2008-06-27.
4. Victor Allis (1994). "PhD thesis: Searching for Solutions in Games and Artificial Intelligence" (PDF). Department of Computer Science. University of Limburg. Retrieved 2012-07-14.
Topics in game theory
Definitions
• Congestion game
• Cooperative game
• Determinacy
• Escalation of commitment
• Extensive-form game
• First-player and second-player win
• Game complexity
• Graphical game
• Hierarchy of beliefs
• Information set
• Normal-form game
• Preference
• Sequential game
• Simultaneous game
• Simultaneous action selection
• Solved game
• Succinct game
Equilibrium
concepts
• Bayesian Nash equilibrium
• Berge equilibrium
• Core
• Correlated equilibrium
• Epsilon-equilibrium
• Evolutionarily stable strategy
• Gibbs equilibrium
• Mertens-stable equilibrium
• Markov perfect equilibrium
• Nash equilibrium
• Pareto efficiency
• Perfect Bayesian equilibrium
• Proper equilibrium
• Quantal response equilibrium
• Quasi-perfect equilibrium
• Risk dominance
• Satisfaction equilibrium
• Self-confirming equilibrium
• Sequential equilibrium
• Shapley value
• Strong Nash equilibrium
• Subgame perfection
• Trembling hand
Strategies
• Backward induction
• Bid shading
• Collusion
• Forward induction
• Grim trigger
• Markov strategy
• Dominant strategies
• Pure strategy
• Mixed strategy
• Strategy-stealing argument
• Tit for tat
Classes
of games
• Bargaining problem
• Cheap talk
• Global game
• Intransitive game
• Mean-field game
• Mechanism design
• n-player game
• Perfect information
• Large Poisson game
• Potential game
• Repeated game
• Screening game
• Signaling game
• Stackelberg competition
• Strictly determined game
• Stochastic game
• Symmetric game
• Zero-sum game
Games
• Go
• Chess
• Infinite chess
• Checkers
• Tic-tac-toe
• Prisoner's dilemma
• Gift-exchange game
• Optional prisoner's dilemma
• Traveler's dilemma
• Coordination game
• Chicken
• Centipede game
• Lewis signaling game
• Volunteer's dilemma
• Dollar auction
• Battle of the sexes
• Stag hunt
• Matching pennies
• Ultimatum game
• Rock paper scissors
• Pirate game
• Dictator game
• Public goods game
• Blotto game
• War of attrition
• El Farol Bar problem
• Fair division
• Fair cake-cutting
• Cournot game
• Deadlock
• Diner's dilemma
• Guess 2/3 of the average
• Kuhn poker
• Nash bargaining game
• Induction puzzles
• Trust game
• Princess and monster game
• Rendezvous problem
Theorems
• Arrow's impossibility theorem
• Aumann's agreement theorem
• Folk theorem
• Minimax theorem
• Nash's theorem
• Negamax theorem
• Purification theorem
• Revelation principle
• Sprague–Grundy theorem
• Zermelo's theorem
Key
figures
• Albert W. Tucker
• Amos Tversky
• Antoine Augustin Cournot
• Ariel Rubinstein
• Claude Shannon
• Daniel Kahneman
• David K. Levine
• David M. Kreps
• Donald B. Gillies
• Drew Fudenberg
• Eric Maskin
• Harold W. Kuhn
• Herbert Simon
• Hervé Moulin
• John Conway
• Jean Tirole
• Jean-François Mertens
• Jennifer Tour Chayes
• John Harsanyi
• John Maynard Smith
• John Nash
• John von Neumann
• Kenneth Arrow
• Kenneth Binmore
• Leonid Hurwicz
• Lloyd Shapley
• Melvin Dresher
• Merrill M. Flood
• Olga Bondareva
• Oskar Morgenstern
• Paul Milgrom
• Peyton Young
• Reinhard Selten
• Robert Axelrod
• Robert Aumann
• Robert B. Wilson
• Roger Myerson
• Samuel Bowles
• Suzanne Scotchmer
• Thomas Schelling
• William Vickrey
Miscellaneous
• All-pay auction
• Alpha–beta pruning
• Bertrand paradox
• Bounded rationality
• Combinatorial game theory
• Confrontation analysis
• Coopetition
• Evolutionary game theory
• First-move advantage in chess
• Game Description Language
• Game mechanics
• Glossary of game theory
• List of game theorists
• List of games in game theory
• No-win situation
• Solving chess
• Topological game
• Tragedy of the commons
• Tyranny of small decisions
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Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x2. The adjective which corresponds to squaring is quadratic.
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1)2 = x2 + 2x + 1.
One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers x), the square of x is the same as the square of its additive inverse −x. That is, the square function satisfies the identity x2 = (−x)2. This can also be expressed by saying that the square function is an even function.
In real numbers
The squaring operation defines a real function called the square function or the squaring function. Its domain is the whole real line, and its image is the set of nonnegative real numbers.
The square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a monotonic function on the interval [0, +∞). On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on (−∞,0]. Hence, zero is the (global) minimum of the square function. The square x2 of a number x is less than x (that is x2 < x) if and only if 0 < x < 1, that is, if x belongs to the open interval (0,1). This implies that the square of an integer is never less than the original number x.
Every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of only one number, itself. For this reason, it is possible to define the square root function, which associates with a non-negative real number the non-negative number whose square is the original number.
No square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. The lack of real square roots for the negative numbers can be used to expand the real number system to the complex numbers, by postulating the imaginary unit i, which is one of the square roots of −1.
The property "every non-negative real number is a square" has been generalized to the notion of a real closed field, which is an ordered field such that every non-negative element is a square and every polynomial of odd degree has a root. The real closed fields cannot be distinguished from the field of real numbers by their algebraic properties: every property of the real numbers, which may be expressed in first-order logic (that is expressed by a formula in which the variables that are quantified by ∀ or ∃ represent elements, not sets), is true for every real closed field, and conversely every property of the first-order logic, which is true for a specific real closed field is also true for the real numbers.
In geometry
There are several major uses of the square function in geometry.
The name of the square function shows its importance in the definition of the area: it comes from the fact that the area of a square with sides of length l is equal to l2. The area depends quadratically on the size: the area of a shape n times larger is n2 times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a sphere is proportional to the square of its radius, a fact that is manifested physically by the inverse-square law describing how the strength of physical forces such as gravity varies according to distance.
The square function is related to distance through the Pythagorean theorem and its generalization, the parallelogram law. Euclidean distance is not a smooth function: the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance (denoted d2 or r2), which has a paraboloid as its graph, is a smooth and analytic function.
The dot product of a Euclidean vector with itself is equal to the square of its length: v⋅v = v2. This is further generalised to quadratic forms in linear spaces via the inner product. The inertia tensor in mechanics is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size (length).
There are infinitely many Pythagorean triples, sets of three positive integers such that the sum of the squares of the first two equals the square of the third. Each of these triples gives the integer sides of a right triangle.
In abstract algebra and number theory
The square function is defined in any field or ring. An element in the image of this function is called a square, and the inverse images of a square are called square roots.
The notion of squaring is particularly important in the finite fields Z/pZ formed by the numbers modulo an odd prime number p. A non-zero element of this field is called a quadratic residue if it is a square in Z/pZ, and otherwise, it is called a quadratic non-residue. Zero, while a square, is not considered to be a quadratic residue. Every finite field of this type has exactly (p − 1)/2 quadratic residues and exactly (p − 1)/2 quadratic non-residues. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.
More generally, in rings, the square function may have different properties that are sometimes used to classify rings.
Zero may be the square of some non-zero elements. A commutative ring such that the square of a non zero element is never zero is called a reduced ring. More generally, in a commutative ring, a radical ideal is an ideal I such that $x^{2}\in I$ implies $x\in I$. Both notions are important in algebraic geometry, because of Hilbert's Nullstellensatz.
An element of a ring that is equal to its own square is called an idempotent. In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in integral domains. However, the ring of the integers modulo n has 2k idempotents, where k is the number of distinct prime factors of n. A commutative ring in which every element is equal to its square (every element is idempotent) is called a Boolean ring; an example from computer science is the ring whose elements are binary numbers, with bitwise AND as the multiplication operation and bitwise XOR as the addition operation.
In a totally ordered ring, x2 ≥ 0 for any x. Moreover, x2 = 0 if and only if x = 0.
In a supercommutative algebra where 2 is invertible, the square of any odd element equals zero.
If A is a commutative semigroup, then one has
$\forall x,y\in A\quad (xy)^{2}=xyxy=xxyy=x^{2}y^{2}.$
In the language of quadratic forms, this equality says that the square function is a "form permitting composition". In fact, the square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling. The doubling method was formalized by A. A. Albert who started with the real number field $\mathbb {R} $ and the square function, doubling it to obtain the complex number field with quadratic form x2 + y2, and then doubling again to obtain quaternions. The doubling procedure is called the Cayley–Dickson construction, and has been generalized to form algebras of dimension 2n over a field F with involution.
The square function z2 is the "norm" of the composition algebra $\mathbb {C} $, where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras.
In complex numbers
See also: Exponentiation § Powers of complex numbers
On complex numbers, the square function $z\to z^{2}$ is a twofold cover in the sense that each non-zero complex number has exactly two square roots.
The square of the absolute value of a complex number is called its absolute square, squared modulus, squared magnitude, or squared norm.[1] It is the product of the complex number with its complex conjugate, and equals the sum of the squares of the real and imaginary parts of the complex number.
The absolute square of a complex number is always a nonnegative real number, that is zero if and only if the complex number is zero. It is easier to compute than the absolute value (no square root), and is a smooth real-valued function. Because of these two properties, the absolute square is often preferred to the absolute value for explicit computations and when methods of mathematical analysis are involved (for example optimization or integration).
Other uses
Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also in physics where many units are defined using squares and inverse squares: see below.
Least squares is the standard method used with overdetermined systems.
Squaring is used in statistics and probability theory in determining the standard deviation of a set of values, or a random variable. The deviation of each value xi from the mean ${\overline {x}}$ of the set is defined as the difference $x_{i}-{\overline {x}}$. These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the variance, and its square root is the standard deviation.
See also
• Exponentiation by squaring
• Polynomial SOS, the representation of a non-negative polynomial as the sum of squares of polynomials
• Hilbert's seventeenth problem, for the representation of positive polynomials as a sum of squares of rational functions
• Square-free polynomial
• Cube (algebra)
• Metric tensor
• Quadratic equation
• Polynomial ring
• Sums of squares (disambiguation page with various relevant links)
Related identities
Algebraic (need a commutative ring)
• Difference of two squares
• Brahmagupta–Fibonacci identity, related to complex numbers in the sense discussed above
• Euler's four-square identity, related to quaternions in the same way
• Degen's eight-square identity, related to octonions in the same way
• Lagrange's identity
Other
• Pythagorean trigonometric identity
• Parseval's identity
Related physical quantities
• acceleration, length per square time
• cross section (physics), an area-dimensioned quantity
• coupling constant (has square charge in the denominator, and may be expressed with square distance in the numerator)
• kinetic energy (quadratic dependence on velocity)
• specific energy, a (square velocity)-dimensioned quantity
Footnotes
1. Weisstein, Eric W. "Absolute Square". mathworld.wolfram.com.
Further reading
• Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1, ISBN 0-8218-4402-4
• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
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Strong law of small numbers
In mathematics, the "strong law of small numbers" is the humorous law that proclaims, in the words of Richard K. Guy (1988):[1]
There aren't enough small numbers to meet the many demands made of them.
In other words, any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few. Earlier (1980) this "law" was reported by Martin Gardner.[2] Guy's subsequent 1988 paper of the same title gives numerous examples in support of this thesis. (This paper earned him the MAA Lester R. Ford Award.)
Second strong law of small numbers
Guy also formulated a second strong law of small numbers:
When two numbers look equal, it ain't necessarily so![3]
Guy explains this latter law by the way of examples: he cites numerous sequences for which observing the first few members may lead to a wrong guess about the generating formula or law for the sequence. Many of the examples are the observations of other mathematicians.[3]
One example Guy gives is the conjecture that $2^{p}-1$ is prime—in fact, a Mersenne prime—when $p$ is prime; but this conjecture, while true for $p$ = 2, 3, 5 and 7, fails for $p$ = 11 (and for many other values).
Another relates to the prime number race: primes congruent to 3 modulo 4 appear to be more numerous than those congruent to 1; however this is false, and first ceases being true at 26861.
A geometric example concerns Moser's circle problem (pictured), which appears to have the solution of $2^{n-1}$ for $n$ points, but this pattern breaks at and above $n=6$.
See also
• Insensitivity to sample size
• Law of large numbers (unrelated, but the origin of the name)
• Mathematical coincidence
• Pigeonhole principle
• Representativeness heuristic
Notes
1. Guy, Richard K. (1988). "The strong law of small numbers" (PDF). The American Mathematical Monthly. 95 (8): 697–712. doi:10.2307/2322249. JSTOR 2322249.
2. Gardner, Martin (December 1980). "Patterns in primes are a clue to the strong law of small numbers". Mathematical Games. Scientific American. 243 (6): 18–28. JSTOR 24966473.
3. Guy, Richard K. (1990). "The second strong law of small numbers". Mathematics Magazine. 63 (1): 3–20. doi:10.2307/2691503. JSTOR 2691503.
External links
• Caldwell, Chris. "Law of small numbers". The Prime Glossary.
• Weisstein, Eric W. "Strong Law of Small Numbers". MathWorld.
• Carnahan, Scott (2007-10-27). "Small finite sets". Secret Blogging Seminar, notes on a talk by Jean-Pierre Serre on properties of small finite sets.{{cite web}}: CS1 maint: postscript (link)
• Amos Tversky; Daniel Kahneman (August 1971). "Belief in the law of small numbers". Psychological Bulletin. 76 (2): 105–110. CiteSeerX 10.1.1.592.3838. doi:10.1037/h0031322. people have erroneous intuitions about the laws of chance. In particular, they regard a sample randomly drawn from a population as highly representative, I.e., similar to the population in all essential characteristics.
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Secondary calculus and cohomological physics
In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.
Secondary calculus
Secondary calculus acts on the space of solutions of a system of partial differential equations (usually non-linear equations). When the number of independent variables is zero, i.e. the equations are algebraic ones, secondary calculus reduces to classical differential calculus.
All objects in secondary calculus are cohomology classes of differential complexes growing on diffieties. The latter are, in the framework of secondary calculus, the analog of smooth manifolds.
Cohomological physics
Cohomological physics was born with Gauss's theorem, describing the electric charge contained inside a given surface in terms of the flux of the electric field through the surface itself. Flux is the integral of a differential form and, consequently, a de Rham cohomology class. It is not by chance that formulas of this kind, such as the well known Stokes formula, though being a natural part of classical differential calculus, have entered in modern mathematics from physics.
Classical analogues
All the constructions in classical differential calculus have an analog in secondary calculus. For instance, higher symmetries of a system of partial differential equations are the analog of vector fields on differentiable manifolds. The Euler operator, which associates to each variational problem the corresponding Euler–Lagrange equation, is the analog of the classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order. More generally, the analog of differential forms in secondary calculus are the elements of the first term of the so-called C-spectral sequence, and so on.
The simplest diffieties are infinite prolongations of partial differential equations, which are subvarieties of infinite jet spaces. The latter are infinite dimensional varieties that can not be studied by means of standard functional analysis. On the contrary, the most natural language in which to study these objects is differential calculus over commutative algebras. Therefore, the latter must be regarded as a fundamental tool of secondary calculus. On the other hand, differential calculus over commutative algebras gives the possibility to develop algebraic geometry as if it were differential geometry.
Theoretical physics
Recent developments of particle physics, based on quantum field theories and its generalizations, have led to understand the deep cohomological nature of the quantities describing both classical and quantum fields. The turning point was the discovery of the famous BRST transformation. For instance, it was understood that observables in field theory are classes in horizontal de Rham cohomology which are invariant under the corresponding gauge group and so on. This current in modern theoretical physics is actually growing and it is called Cohomological Physics.
It is relevant that secondary calculus and cohomological physics, which developed for twenty years independently from each other, arrived at the same results. Their confluence took place at the international conference Secondary Calculus and Cohomological Physics (Moscow, August 24–30, 1997).
Prospects
A large number of modern mathematical theories harmoniously converges in the framework of secondary calculus, for instance: commutative algebra and algebraic geometry, homological algebra and differential topology, Lie group and Lie algebra theory, differential geometry, etc.
See also
• Differential calculus over commutative algebras – part of commutative algebraPages displaying wikidata descriptions as a fallback
• Spectrum of a ring – Set of a ring's prime ideals
References
• I. S. Krasil'shchik, Calculus over Commutative Algebras: a concise user's guide, Acta Appl. Math. 49 (1997) 235—248; DIPS-01/98
• I. S. Krasil'shchik, A. M. Verbovetsky, Homological Methods in Equations of Mathematical Physics, Open Ed. and Sciences, Opava (Czech Rep.), 1998; DIPS-07/98.
• I. S. Krasil'shchik, A. M. Vinogradov (eds.), Symmetries and conservation laws for differential equations of mathematical physics, Translations of Math. Monographs 182, Amer. Math. Soc., 1999.
• J. Nestruev, Smooth Manifolds and Observables, Graduate Texts in Mathematics 220, Springer, 2002, doi:10.1007/978-3-030-45650-4.
• A. M. Vinogradov, The C-spectral sequence, Lagrangian formalism, and conservation laws I. The linear theory, J. Math. Anal. Appl. 100 (1984) 1—40; Diffiety Inst. Library.
• A. M. Vinogradov, The C-spectral sequence, Lagrangian formalism, and conservation laws II. The nonlinear theory, J. Math. Anal. Appl. 100 (1984) 41—129; Diffiety Inst. Library.
• A. M. Vinogradov, From symmetries of partial differential equations towards secondary (`quantized') calculus, J. Geom. Phys. 14 (1994) 146—194; Diffiety Inst. Library.
• A. M. Vinogradov, Introduction to Secondary Calculus, Proc. Conf. Secondary Calculus and Cohomology Physics (M. Henneaux, I. S. Krasil'shchik, and A. M. Vinogradov, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, Rhode Island, 1998; DIPS-05/98.
• A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus, Translations of Math. Monographs 204, Amer. Math. Soc., 2001.
External links
• The Diffiety Institute
• Diffiety School
Manifolds (Glossary)
Basic concepts
• Topological manifold
• Atlas
• Differentiable/Smooth manifold
• Differential structure
• Smooth atlas
• Submanifold
• Riemannian manifold
• Smooth map
• Submersion
• Pushforward
• Tangent space
• Differential form
• Vector field
Main results (list)
• Atiyah–Singer index
• Darboux's
• De Rham's
• Frobenius
• Generalized Stokes
• Hopf–Rinow
• Noether's
• Sard's
• Whitney embedding
Maps
• Curve
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• in Lie theory
• Foliation
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Types of
manifolds
• Closed
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• (Almost) Contact
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• G-structure
• Hadamard
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• Lie algebra
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• (Pseudo−, Sub−) Riemannian
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• Tame
Tensors
Vectors
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• bundle
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Covectors
• Closed/Exact
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• bundle
• De Rham cohomology
• Differential form
• Vector-valued
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• flow
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• density
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Bundles
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Connections
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Related
• Classification of manifolds
• Gauge theory
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Generalizations
• Banach manifold
• Diffeology
• Diffiety
• Fréchet manifold
• K-theory
• Orbifold
• Secondary calculus
• over commutative algebras
• Sheaf
• Stratifold
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Wikipedia
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Secondary measure
In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.
Introduction
Under certain assumptions that we will specify further, it is possible to obtain the existence of a secondary measure and even to express it.
For example, if one works in the Hilbert space L2([0, 1], R, ρ)
$\forall x\in [0,1],\qquad \mu (x)={\frac {\rho (x)}{{\frac {\varphi ^{2}(x)}{4}}+\pi ^{2}\rho ^{2}(x)}}$
with
$\varphi (x)=\lim _{\varepsilon \to 0^{+}}2\int _{0}^{1}{\frac {(x-t)\rho (t)}{(x-t)^{2}+\varepsilon ^{2}}}\,dt$
in the general case, or:
$\varphi (x)=2\rho (x){\text{ln}}\left({\frac {x}{1-x}}\right)-2\int _{0}^{1}{\frac {\rho (t)-\rho (x)}{t-x}}\,dt$
when ρ satisfies a Lipschitz condition.
This application φ is called the reducer of ρ.
More generally, μ et ρ are linked by their Stieltjes transformation with the following formula:
$S_{\mu }(z)=z-c_{1}-{\frac {1}{S_{\rho }(z)}}$
in which c1 is the moment of order 1 of the measure ρ.
These secondary measures, and the theory around them, lead to some surprising results, and make it possible to find in an elegant way quite a few traditional formulas of analysis, mainly around the Euler Gamma function, Riemann Zeta function, and Euler's constant.
They also allowed the clarification of integrals and series with a tremendous effectiveness, though it is a priori difficult.
Finally they make it possible to solve integral equations of the form
$f(x)=\int _{0}^{1}{\frac {g(t)-g(x)}{t-x}}\rho (t)\,dt$
where g is the unknown function, and lead to theorems of convergence towards the Chebyshev and Dirac measures.
The broad outlines of the theory
Let ρ be a measure of positive density on an interval I and admitting moments of any order. We can build a family {Pn} of orthogonal polynomials for the inner product induced by ρ. Let us call {Qn} the sequence of the secondary polynomials associated with the family P. Under certain conditions there is a measure for which the family Q is orthogonal. This measure, which we can clarify from ρ is called a secondary measure associated initial measure ρ.
When ρ is a probability density function, a sufficient condition so that μ, while admitting moments of any order can be a secondary measure associated with ρ is that its Stieltjes Transformation is given by an equality of the type:
$S_{\mu }(z)=a\left(z-c_{1}-{\frac {1}{S_{\rho }(z)}}\right),$
a is an arbitrary constant and c1 indicating the moment of order 1 of ρ.
For a = 1 we obtain the measure known as secondary, remarkable since for n ≥ 1 the norm of the polynomial Pn for ρ coincides exactly with the norm of the secondary polynomial associated Qn when using the measure μ.
In this paramount case, and if the space generated by the orthogonal polynomials is dense in L2(I, R, ρ), the operator Tρ defined by
$f(x)\mapsto \int _{I}{\frac {f(t)-f(x)}{t-x}}\rho (t)dt$
creating the secondary polynomials can be furthered to a linear map connecting space L2(I, R, ρ) to L2(I, R, μ) and becomes isometric if limited to the hyperplane Hρ of the orthogonal functions with P0 = 1.
For unspecified functions square integrable for ρ we obtain the more general formula of covariance:
$\langle f/g\rangle _{\rho }-\langle f/1\rangle _{\rho }\times \langle g/1\rangle _{\rho }=\langle T_{\rho }(f)/T_{\rho }(g)\rangle _{\mu }.$
The theory continues by introducing the concept of reducible measure, meaning that the quotient ρ/μ is element of L2(I, R, μ). The following results are then established:
The reducer φ of ρ is an antecedent of ρ/μ for the operator Tρ. (In fact the only antecedent which belongs to Hρ).
For any function square integrable for ρ, there is an equality known as the reducing formula:
$\langle f/\varphi \rangle _{\rho }=\langle T_{\rho }(f)/1\rangle _{\rho }$.
The operator
$f\mapsto \varphi \times f-T_{\rho }(f)$
defined on the polynomials is prolonged in an isometry Sρ linking the closure of the space of these polynomials in L2(I, R, ρ2μ−1) to the hyperplane Hρ provided with the norm induced by ρ.
Under certain restrictive conditions the operator Sρ acts like the adjoint of Tρ for the inner product induced by ρ.
Finally the two operators are also connected, provided the images in question are defined, by the fundamental formula of composition:
$T_{\rho }\circ S_{\rho }\left(f\right)={\frac {\rho }{\mu }}\times (f).$
Case of the Lebesgue measure and some other examples
The Lebesgue measure on the standard interval [0, 1] is obtained by taking the constant density ρ(x) = 1.
The associated orthogonal polynomials are called Legendre polynomials and can be clarified by
$P_{n}(x)={\frac {d^{n}}{dx^{n}}}\left(x^{n}(1-x)^{n}\right).$
The norm of Pn is worth
${\frac {n!}{\sqrt {2n+1}}}.$
The recurrence relation in three terms is written:
$2(2n+1)XP_{n}(X)=-P_{n+1}(X)+(2n+1)P_{n}(X)-n^{2}P_{n-1}(X).$
The reducer of this measure of Lebesgue is given by
$\varphi (x)=2\ln \left({\frac {x}{1-x}}\right).$
The associated secondary measure is then clarified as
$\mu (x)={\frac {1}{\ln ^{2}\left({\frac {x}{1-x}}\right)+\pi ^{2}}}$.
If we normalize the polynomials of Legendre, the coefficients of Fourier of the reducer φ related to this orthonormal system are null for an even index and are given by
$C_{n}(\varphi )=-{\frac {4{\sqrt {2n+1}}}{n(n+1)}}$
for an odd index n.
The Laguerre polynomials are linked to the density ρ(x) = e−x on the interval I = [0, ∞). They are clarified by
$L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}(x^{n}e^{-x})=\sum _{k=0}^{n}{\binom {n}{k}}(-1)^{k}{\frac {x^{k}}{k!}}$
and are normalized.
The reducer associated is defined by
$\varphi (x)=2\left(\ln(x)-\int _{0}^{\infty }e^{-t}\ln |x-t|dt\right).$
The coefficients of Fourier of the reducer φ related to the Laguerre polynomials are given by
$C_{n}(\varphi )=-{\frac {1}{n}}\sum _{k=0}^{n-1}{\frac {1}{\binom {n-1}{k}}}.$
This coefficient Cn(φ) is no other than the opposite of the sum of the elements of the line of index n in the table of the harmonic triangular numbers of Leibniz.
The Hermite polynomials are linked to the Gaussian density
$\rho (x)={\frac {e^{-{\frac {x^{2}}{2}}}}{\sqrt {2\pi }}}$
on I = R.
They are clarified by
$H_{n}(x)={\frac {1}{\sqrt {n!}}}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-{\frac {x^{2}}{2}}}\right)$
and are normalized.
The reducer associated is defined by
$\varphi (x)=-{\frac {2}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }te^{-{\frac {t^{2}}{2}}}\ln |x-t|\,dt.$
The coefficients of Fourier of the reducer φ related to the system of Hermite polynomials are null for an even index and are given by
$C_{n}(\varphi )=(-1)^{\frac {n+1}{2}}{\frac {\left({\frac {n-1}{2}}\right)!}{\sqrt {n!}}}$
for an odd index n.
The Chebyshev measure of the second form. This is defined by the density
$\rho (x)={\frac {8}{\pi }}{\sqrt {x(1-x)}}$
on the interval [0, 1].
It is the only one which coincides with its secondary measure normalised on this standard interval. Under certain conditions it occurs as the limit of the sequence of normalized secondary measures of a given density.
Examples of non-reducible measures
Jacobi measure on (0, 1) of density
$\rho (x)={\frac {2}{\pi }}{\sqrt {\frac {1-x}{x}}}.$
Chebyshev measure on (−1, 1) of the first form of density
$\rho (x)={\frac {1}{\pi {\sqrt {1-x^{2}}}}}.$
Sequence of secondary measures
The secondary measure μ associated with a probability density function ρ has its moment of order 0 given by the formula
$d_{0}=c_{2}-c_{1}^{2},$
where c1 and c2 indicating the respective moments of order 1 and 2 of ρ.
To be able to iterate the process then, one 'normalizes' μ while defining ρ1 = μ/d0 which becomes in its turn a density of probability called naturally the normalised secondary measure associated with ρ.
We can then create from ρ1 a secondary normalised measure ρ2, then defining ρ3 from ρ2 and so on. We can therefore see a sequence of successive secondary measures, created from ρ0 = ρ, is such that ρn+1 that is the secondary normalised measure deduced from ρn
It is possible to clarify the density ρn by using the orthogonal polynomials Pn for ρ, the secondary polynomials Qn and the reducer associated φ. That gives the formula
$\rho _{n}(x)={\frac {1}{d_{0}^{n-1}}}{\frac {\rho (x)}{\left(P_{n-1}(x){\frac {\varphi (x)}{2}}-Q_{n-1}(x)\right)^{2}+\pi ^{2}\rho ^{2}(x)P_{n-1}^{2}(x)}}.$
The coefficient $d_{0}^{n-1}$ is easily obtained starting from the leading coefficients of the polynomials Pn−1 and Pn. We can also clarify the reducer φn associated with ρn, as well as the orthogonal polynomials corresponding to ρn.
A very beautiful result relates the evolution of these densities when the index tends towards the infinite and the support of the measure is the standard interval [0, 1].
Let
$xP_{n}(x)=t_{n}P_{n+1}(x)+s_{n}P_{n}(x)+t_{n-1}P_{n-1}(x)$
be the classic recurrence relation in three terms. If
$\lim _{n\mapsto \infty }t_{n}={\tfrac {1}{4}},\quad \lim _{n\mapsto \infty }s_{n}={\tfrac {1}{2}},$
then the sequence {ρn} converges completely towards the Chebyshev density of the second form
$\rho _{tch}(x)={\frac {8}{\pi }}{\sqrt {x(1-x)}}$.
These conditions about limits are checked by a very broad class of traditional densities. A derivation of the sequence of secondary measures and convergence can be found in [1]
Equinormal measures
One calls two measures thus leading to the same normalised secondary density. It is remarkable that the elements of a given class and having the same moment of order 1 are connected by a homotopy. More precisely, if the density function ρ has its moment of order 1 equal to c1, then these densities equinormal with ρ are given by a formula of the type:
$\rho _{t}(x)={\frac {t\rho (x)}{\left({\tfrac {1}{2}}(t-1)(x-c_{1})\varphi (x)-t\right)^{2}+\pi ^{2}\rho ^{2}(x)(t-1)^{2}(x-c_{1})^{2}}},$
t describing an interval containing ]0, 1].
If μ is the secondary measure of ρ, that of ρt will be tμ.
The reducer of ρt is
$\varphi _{t}(x)={\frac {2(x-c_{1})-tG(x)}{\left((x-c_{1})-t{\tfrac {1}{2}}G(x)\right)^{2}+t^{2}\pi ^{2}\mu ^{2}(x)}}$
by noting G(x) the reducer of μ.
Orthogonal polynomials for the measure ρt are clarified from n = 1 by the formula
$P_{n}^{t}(x)={\frac {tP_{n}(x)+(1-t)(x-c_{1})Q_{n}(x)}{\sqrt {t}}}$
with Qn secondary polynomial associated with Pn.
It is remarkable also that, within the meaning of distributions, the limit when t tends towards 0 per higher value of ρt is the Dirac measure concentrated at c1.
For example, the equinormal densities with the Chebyshev measure of the second form are defined by:
$\rho _{t}(x)={\frac {2t{\sqrt {1-x^{2}}}}{\pi \left[t^{2}+4(1-t)x^{2}\right]}},$
with t describing ]0, 2]. The value t = 2 gives the Chebyshev measure of the first form.
A few beautiful applications
In the formulas below G is Catalan's constant, γ is the Euler's constant, β2n is the Bernoulli number of order 2n, H2n+1 is the harmonic number of order 2n+1 and Ei is the Exponential integral function.
${\frac {1}{\ln(p)}}={\frac {1}{p-1}}+\int _{0}^{\infty }{\frac {1}{(x+p)(\ln ^{2}(x)+\pi ^{2})}}dx\qquad \qquad \forall p>1$
$\gamma =\int _{0}^{\infty }{\frac {\ln(1+{\frac {1}{x}})}{\ln ^{2}(x)+\pi ^{2}}}dx$
$\gamma ={\frac {1}{2}}+\int _{0}^{\infty }{\frac {\overline {(x+1)\cos(\pi x)}}{x+1}}dx$
The notation $x\mapsto {\overline {(x+1)\cos(\pi x)}}$ indicating the 2 periodic function coinciding with $x\mapsto (x+1)\cos(\pi x)$ on (−1, 1).
$\gamma ={\frac {1}{2}}+\sum _{k=1}^{n}{\frac {\beta _{2k}}{2k}}-{\frac {\beta _{2n}}{\zeta (2n)}}\int _{1}^{\infty }\lfloor t\rfloor \cos(2\pi t)t^{-2n-1}dt$
$\beta _{k}={\frac {(-1)^{k}k!}{\pi }}{\text{Im}}\left(\int _{-\infty }^{\infty }{\frac {e^{x}}{(1+e^{x})(x-i\pi )^{k}}}dx\right)$
$\int _{0}^{1}\ln ^{2n}\left({\frac {x}{1-x}}\right)\,dx=(-1)^{n+1}(2^{2n}-2)\beta _{2n}\pi ^{2n}$
$\int _{0}^{1}\cdots \int _{0}^{1}\left(\sum _{k=1}^{2n}{\frac {\ln(t_{k})}{\prod _{i\neq k}(t_{k}-t_{i})}}\right)\,dt_{1}\cdots dt_{2n}={\tfrac {1}{2}}(-1)^{n+1}(2\pi )^{2n}\beta _{2n}$
$\int _{0}^{\infty }{\frac {e^{-\alpha x}}{\Gamma (x+1)}}dx=e^{e^{-\alpha }}-1+\int _{0}^{\infty }{\frac {1-e^{-x}}{(\ln(x)+\alpha )^{2}+\pi ^{2}}}{\frac {dx}{x}}\qquad \qquad \forall \alpha \in \mathbf {R} $
$\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sum _{k=0}^{n-1}{\frac {1}{\binom {n-1}{k}}}\right)^{2}={\tfrac {4}{9}}\pi ^{2}=\int _{0}^{\infty }4\left(\mathrm {Ei} (1,-x)+i\pi \right)^{2}e^{-3x}\,dx.$
${\frac {23}{15}}-\ln(2)=\sum _{n=0}^{\infty }{\frac {1575}{2(n+1)(2n+1)(4n-3)(4n-1)(4n+1)(4n+5)(4n+7)(4n+9)}}$
$G=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4^{k+1}}}\left({\frac {1}{(4k+3)^{2}}}+{\frac {2}{(4k+2)^{2}}}+{\frac {2}{(4k+1)^{2}}}\right)+{\frac {\pi }{8}}\ln(2)$
$G={\frac {\pi }{8}}\ln(2)+\sum _{n=0}^{\infty }(-1)^{n}{\frac {H_{2n+1}}{2n+1}}.$
If the measure ρ is reducible and let φ be the associated reducer, one has the equality
$\int _{I}\varphi ^{2}(x)\rho (x)\,dx={\frac {4\pi ^{2}}{3}}\int _{I}\rho ^{3}(x)\,dx.$
If the measure ρ is reducible with μ the associated reducer, then if f is square integrable for μ, and if g is square integrable for ρ and is orthogonal with P0 = 1 one has equivalence:
$f(x)=\int _{I}{\frac {g(t)-g(x)}{t-x}}\rho (t)dt\Leftrightarrow g(x)=(x-c_{1})f(x)-T_{\mu }(f(x))={\frac {\varphi (x)\mu (x)}{\rho (x)}}f(x)-T_{\rho }\left({\frac {\mu (x)}{\rho (x)}}f(x)\right)$
c1 indicates the moment of order 1 of ρ and Tρ the operator
$g(x)\mapsto \int _{I}{\frac {g(t)-g(x)}{t-x}}\rho (t)\,dt.$
In addition, the sequence of secondary measures has applications in Quantum Mechanics. The sequence gives rise to the so-called sequence of residual spectral densities for specialized Pauli-Fierz Hamiltonians. This also provides a physical interpretation for the sequence of secondary measures. [1]
See also
• Orthogonal polynomials
• Probability
References
1. Mappings of open quantum systems onto chain representations and Markovian embeddings, M. P. Woods, R. Groux, A. W. Chin, S. F. Huelga, M. B. Plenio. https://arxiv.org/abs/1111.5262
External links
• personal page of Roland Groux about the theory of secondary measures
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Secondary polynomials
In mathematics, the secondary polynomials $\{q_{n}(x)\}$ associated with a sequence $\{p_{n}(x)\}$ of polynomials orthogonal with respect to a density $\rho (x)$ are defined by
$q_{n}(x)=\int _{\mathbb {R} }\!{\frac {p_{n}(t)-p_{n}(x)}{t-x}}\rho (t)\,dt.$
To see that the functions $q_{n}(x)$ are indeed polynomials, consider the simple example of $p_{0}(x)=x^{3}.$ Then,
${\begin{aligned}q_{0}(x)&{}=\int _{\mathbb {R} }\!{\frac {t^{3}-x^{3}}{t-x}}\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!{\frac {(t-x)(t^{2}+tx+x^{2})}{t-x}}\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!(t^{2}+tx+x^{2})\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!t^{2}\rho (t)\,dt+x\int _{\mathbb {R} }\!t\rho (t)\,dt+x^{2}\int _{\mathbb {R} }\!\rho (t)\,dt\end{aligned}}$
which is a polynomial $x$ provided that the three integrals in $t$ (the moments of the density $\rho $) are convergent.
See also
• Secondary measure
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Section conjecture
In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism $\pi _{1}(X)\to \operatorname {Gal} (k)$, where $X$ is a complete smooth curve of genus at least 2 over a field $k$ that is finitely generated over $\mathbb {Q} $, in terms of decomposition groups of rational points of $X$. The conjecture was introduced by Alexander Grothendieck (1997) in a 1983 letter to Gerd Faltings.
References
• Grothendieck, Alexander (1997), "Brief an G. Faltings", in Schneps, Leila; Lochak, Pierre (eds.), Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge University Press, pp. 49–58, ISBN 978-0-521-59642-8, MR 1483108
External links
• "Why is the section conjecture important?". mathoverflow.net.
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Section formula
In coordinate geometry, Section formula is used to find the ratio in which a line segment is divided by a point internally or externally.[1] It is used to find out the centroid, incenter and excenters of a triangle. In physics, it is used to find the center of mass of systems, equilibrium points, etc.[2][3][4][5]
Internal Divisions
If point P (lying on AB) divides the line segment AB joining the points $\mathrm {A} (x_{1},y_{1})$ and $\mathrm {B} (x_{2},y_{2})$ in the ratio m:n, then
$P=\left({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}}\right)$[6]
The ratio m:n can also be written as $m/n:1$, or $k:1$, where $k=m/n$. So, the coordinates of point $P$ dividing the line segment joining the points $\mathrm {A} (x_{1},y_{1})$ and $\mathrm {B} (x_{2},y_{2})$ are:
$\left({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}}\right)$
$=\left({\frac {{\frac {m}{n}}x_{2}+x_{1}}{{\frac {m}{n}}+1}},{\frac {{\frac {m}{n}}y_{2}+y_{1}}{{\frac {m}{n}}+1}}\right)$
$=\left({\frac {kx_{2}+x_{1}}{k+1}},{\frac {ky_{2}+y_{1}}{k+1}}\right)$[4][5]
Similarly, the ratio can also be written as $k:k-1$, and the coordinates of P are $((1-k)x_{1}+kx_{2},(1-k)y_{1}+ky_{2})$.[1]
Proof
Triangles $PAQ\sim BPC$.
${\begin{aligned}{\frac {AP}{BP}}={\frac {AQ}{CP}}={\frac {PQ}{BC}}\\{\frac {m}{n}}={\frac {x-x_{1}}{x_{2}-x}}={\frac {y-y_{1}}{y_{2}-y}}\\mx_{2}-mx=nx-nx_{1},my_{2}-my=ny-ny_{1}\\mx+nx=mx_{2}+nx_{1},my+ny=my_{2}+ny_{1}\\(m+n)x=mx_{2}+nx_{1},(m+n)y=my_{2}+ny_{1}\\x={\frac {mx_{2}+nx_{1}}{m+n}},y={\frac {my_{2}+ny_{1}}{m+n}}\\\end{aligned}}$
External Divisions
If a point P (lying on the extension of AB) divides AB in the ratio m:n then
$P=\left({\dfrac {mx_{2}-nx_{1}}{m-n}},{\dfrac {my_{2}-ny_{1}}{m-n}}\right)$[6]
Midpoint formula
Main article: Midpoint
The midpoint of a line segment divides it internally in the ratio $ 1:1$. Applying the Section formula for internal division:[4][5]
$P=\left({\dfrac {x_{1}+x_{2}}{2}},{\dfrac {y_{1}+y_{2}}{2}}\right)$
Derivation
$P=\left({\dfrac {mx_{2}+nx_{1}}{m+n}},{\dfrac {my_{2}+ny_{1}}{m+n}}\right)$
$=\left({\frac {1\cdot x_{1}+1\cdot x_{2}}{1+1}},{\frac {1\cdot y_{1}+1\cdot y_{2}}{1+1}}\right)$
$=\left({\dfrac {x_{1}+x_{2}}{2}},{\dfrac {y_{1}+y_{2}}{2}}\right)$
Centroid
The centroid of a triangle is the intersection of the medians and divides each median in the ratio $ 2:1$. Let the vertices of the triangle be $A(x_{1},y_{1})$, $ B(x_{2},y_{2})$ and $ C(x_{3},y_{3})$. So, a median from point A will intersect BC at $ \left({\frac {x_{2}+x_{3}}{2}},{\frac {y_{2}+y_{3}}{2}}\right)$. Using the section formula, the centroid becomes:
$\left({\frac {x_{1}+x_{2}+x_{3}}{3}},{\frac {y_{1}+y_{2}+y_{3}}{3}}\right)$
In 3-Dimensions
Let A and B be two points with Cartesian coordinates (x1, y1, z1) and (x2, y2, z2) and P be a point on the line through A and B. If $AP:PB=m:n$. Then the section formulae give the coordinates of P as
$\left({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}},{\frac {mz_{2}+nz_{1}}{m+n}}\right)$[7]
If, instead, P is a point on the line such that $AP:PB=k:1-k$, its coordinates are $((1-k)x_{1}+kx_{2},(1-k)y_{1}+ky_{2},(1-k)z_{1}+kz_{2})$.[7]
In vectors
The position vector of a point P dividing the line segment joining the points A and B whose position vectors are ${\vec {a}}$ and ${\vec {b}}$
1. in the ratio $m:n$ internally, is given by ${\frac {n{\vec {a}}+m{\vec {b}}}{m+n}}$[8][1]
2. in the ratio $m:n$ externally, is given by ${\frac {m{\vec {b}}-n{\vec {a}}}{m-n}}$[8]
See also
• Cross-section Formula
• Distance Formula
• Midpoint Formula
References
1. Clapham, Christopher; Nicholson, James (2014-09-18), "section formulae", The Concise Oxford Dictionary of Mathematics, Oxford University Press, doi:10.1093/acref/9780199679591.001.0001, ISBN 978-0-19-967959-1, retrieved 2020-10-30
2. "Section Formula | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-10-16.
3. https://ncert.nic.in/ncerts/l/jemh107.pdf
4. Aggarwal, R.S. Secondary School Mathematics for Class 10. Bharti Bhawan Publishers & Distributors (1 January 2020). ISBN 978-9388704519.
5. Sharma, R.D. Mathematics for Class 10. Dhanpat Rai Publication (1 January 2020). ISBN 978-8194192640.
6. Loney, S L. The Elements of Coordinate Geometry (Part-1).
7. Clapham, Christopher; Nicholson, James (2014-09-18), "section formulae", The Concise Oxford Dictionary of Mathematics, Oxford University Press, doi:10.1093/acref/9780199679591.001.0001, ISBN 978-0-19-967959-1, retrieved 2020-10-30
8. https://ncert.nic.in/ncerts/l/leep210.pdf
External links
• section-formula by GeoGebra
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Section (fiber bundle)
In the mathematical field of topology, a section (or cross section)[1] of a fiber bundle $E$ is a continuous right inverse of the projection function $\pi $. In other words, if $E$ is a fiber bundle over a base space, $B$:
$\pi \colon E\to B$
then a section of that fiber bundle is a continuous map,
$\sigma \colon B\to E$
such that
$\pi (\sigma (x))=x$ for all $x\in B$.
A section is an abstract characterization of what it means to be a graph. The graph of a function $g\colon B\to Y$ can be identified with a function taking its values in the Cartesian product $E=B\times Y$, of $B$ and $Y$:
$\sigma \colon B\to E,\quad \sigma (x)=(x,g(x))\in E.$
Let $\pi \colon E\to B$ be the projection onto the first factor: $\pi (x,y)=x$. Then a graph is any function $\sigma $ for which $\pi (\sigma (x))=x$.
The language of fibre bundles allows this notion of a section to be generalized to the case when $E$ is not necessarily a Cartesian product. If $\pi \colon E\to B$ is a fibre bundle, then a section is a choice of point $\sigma (x)$ in each of the fibres. The condition $\pi (\sigma (x))=x$ simply means that the section at a point $x$ must lie over $x$. (See image.)
For example, when $E$ is a vector bundle a section of $E$ is an element of the vector space $E_{x}$ lying over each point $x\in B$. In particular, a vector field on a smooth manifold $M$ is a choice of tangent vector at each point of $M$: this is a section of the tangent bundle of $M$. Likewise, a 1-form on $M$ is a section of the cotangent bundle.
Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space $B$ is a smooth manifold $M$, and $E$ is assumed to be a smooth fiber bundle over $M$ (i.e., $E$ is a smooth manifold and $\pi \colon E\to M$ is a smooth map). In this case, one considers the space of smooth sections of $E$ over an open set $U$, denoted $C^{\infty }(U,E)$. It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., $C^{k}$ sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces).
Local and global sections
Fiber bundles do not in general have such global sections (consider, for example, the fiber bundle over $S^{1}$ with fiber $F=\mathbb {R} \setminus \{0\}$ obtained by taking the Möbius bundle and removing the zero section), so it is also useful to define sections only locally. A local section of a fiber bundle is a continuous map $s\colon U\to E$ where $U$ is an open set in $B$ and $\pi (s(x))=x$ for all $x$ in $U$. If $(U,\varphi )$ is a local trivialization of $E$, where $\varphi $ is a homeomorphism from $\pi ^{-1}(U)$ to $U\times F$ (where $F$ is the fiber), then local sections always exist over $U$ in bijective correspondence with continuous maps from $U$ to $F$. The (local) sections form a sheaf over $B$ called the sheaf of sections of $E$.
The space of continuous sections of a fiber bundle $E$ over $U$ is sometimes denoted $C(U,E)$, while the space of global sections of $E$ is often denoted $\Gamma (E)$ or $\Gamma (B,E)$.
Extending to global sections
Sections are studied in homotopy theory and algebraic topology, where one of the main goals is to account for the existence or non-existence of global sections. An obstruction denies the existence of global sections since the space is too "twisted". More precisely, obstructions "obstruct" the possibility of extending a local section to a global section due to the space's "twistedness". Obstructions are indicated by particular characteristic classes, which are cohomological classes. For example, a principal bundle has a global section if and only if it is trivial. On the other hand, a vector bundle always has a global section, namely the zero section. However, it only admits a nowhere vanishing section if its Euler class is zero.
Generalizations
Obstructions to extending local sections may be generalized in the following manner: take a topological space and form a category whose objects are open subsets, and morphisms are inclusions. Thus we use a category to generalize a topological space. We generalize the notion of a "local section" using sheaves of abelian groups, which assigns to each object an abelian group (analogous to local sections).
There is an important distinction here: intuitively, local sections are like "vector fields" on an open subset of a topological space. So at each point, an element of a fixed vector space is assigned. However, sheaves can "continuously change" the vector space (or more generally abelian group).
This entire process is really the global section functor, which assigns to each sheaf its global section. Then sheaf cohomology enables us to consider a similar extension problem while "continuously varying" the abelian group. The theory of characteristic classes generalizes the idea of obstructions to our extensions.
See also
• Fibration
• Gauge theory
• Principal bundle
• Pullback bundle
• Vector bundle
Notes
1. Husemöller, Dale (1994), Fibre Bundles, Springer Verlag, p. 12, ISBN 0-387-94087-1
References
• Norman Steenrod, The Topology of Fibre Bundles, Princeton University Press (1951). ISBN 0-691-00548-6.
• David Bleecker, Gauge Theory and Variational Principles, Addison-Wesley publishing, Reading, Mass (1981). ISBN 0-201-10096-7.
• Husemöller, Dale (1994), Fibre Bundles, Springer Verlag, ISBN 0-387-94087-1
External links
• Fiber Bundle, PlanetMath
• Weisstein, Eric W. "Fiber Bundle". MathWorld.
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Sectorial operator
In mathematics, more precisely in operator theory, a sectorial operator is a linear operator on a Banach space, whose spectrum in an open sector in the complex plane and whose resolvent is uniformly bounded from above outside any larger sector. Such operators might be unbounded.
Sectorial operators have applications in the theory of elliptic and parabolic partial differential equations.
Sectorial operator
Let $(X,\|\cdot \|)$ be a Banach space. Let $A$ be a (not necessarily bounded) linear operator on $X$ and $\sigma (A)$ its spectrum.
For the angle $0<\omega \leq \pi $, we define the open sector
$\Sigma _{\omega }:=\{z\in \mathbb {C} \setminus \{0\}:|\operatorname {arg} z|<\omega \}$,
and set $\Sigma _{0}:=(0,\infty )$ if $\omega =0$.
Now, fix an angle $\omega \in [0,\pi )$.
The operator $A$ is called sectorial with angle $\omega $ if[1]
$\sigma (A)\subset {\overline {\Sigma _{\omega }}}$
and if
$\sup \limits _{\lambda \in \mathbb {C} \setminus {\overline {\Sigma _{\psi }}}}|\lambda |\|(\lambda -A)^{-1}\|<\infty $.
for every larger angle $\psi \in (\omega ,\pi )$. The set of sectorial operators with angle $\omega $ is denoted by $\operatorname {Sect} (\omega )$.
Remarks
• If $\omega \neq 0$, then $\Sigma _{\omega }$ is open and symmetric over the positive real axis with angular aperture $2\omega $.
Bibliography
• Markus Haase (2010), Birkhäuser Basel (ed.), The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169, doi:10.1007/3-7643-7698-8, ISBN 978-3-7643-7697-0
• Atsushi Yagi (2010), Springer, Berlin, Heidelberg (ed.), "Sectorial Operators", Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics, pp. 55–116, doi:10.1007/978-3-642-04631-5_2, ISBN 978-3-642-04630-8{{citation}}: CS1 maint: multiple names: editors list (link)
• Markus Haase (2003), Universität Ulm (ed.), The Functional Calculus for Sectorial Operators and Similarity Methods
References
1. Haase, Markus (2006). The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications. p. 19. doi:10.1007/3-7643-7698-8. ISBN 978-3-7643-7697-0.
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Secular variation
The secular variation of a time series is its long-term, non-periodic variation (see decomposition of time series). Whether a variation is perceived as secular or not depends on the available timescale: a variation that is secular over a timescale of centuries may be a segment of what is, over a timescale of millions of years, a periodic variation. Natural quantities often have both periodic and secular variations. Secular variation is sometimes called secular trend or secular drift when the emphasis is on a linear long-term trend.
The term is used wherever time series are applicable in history, economics, operations research, biological anthropology, and astronomy (particularly celestial mechanics) such as VSOP (planets).
Etymology
The word secular, from the Latin root saecularis ("of an age, occurring once in an age"),[1] has two basic meanings: I. Of or pertaining to the world (from which secularity is derived), and II. Of or belonging to an age or long period. The latter use appeared in the 18th century in the sense of "living or lasting for an age or ages". In the 19th century terms like secular acceleration and secular variation appeared in astronomy, and similar language was used in economics by 1895.[2]
Astronomy
In astronomy, secular variations are distinguished from periodic phenomena. In particular, astronomical ephemerides use secular to label the longest duration or non-oscillatory perturbations in the motion of planets, contrasted with periodic perturbations which exhibit repetition over the course of a given time frame. In this context it is referred to as secular motion. Solar System ephemerides are essential for the navigation of spacecraft and for all kinds of space observations of the planets, their natural satellites, stars and galaxies.
Most of the known perturbations to motion in stable, regular, and well-determined dynamical systems tend to be periodic at some level, but in many-body systems, chaotic dynamics result in some effects which are unidirectional (for example, planetary migration).
Solar System
Secular phenomena create variations in the orbits of the Moon and the planets. The solar emission spectrum and the solar wind follow secular trends due to migration through the galactic plane. Consensus has determined these to have been among the smallest of factors to influence climate and extinction during human evolution, dwarfed by complex solar cycles and magnetic cycles.
Moon
The secular acceleration of the Moon depends on tidal forces. It was discovered early but it was some time before it was correctly explained.[3]
Earth
Depending on the time frame, perturbations can appear secular even if they are actually periodic. An example of this is the precession of the Earth's axis considered over the time frame of a few hundred or thousand years. When viewed in this timeframe the so-called "precession of the equinoxes" can appear to be a secular phenomenon since the axial precession takes 25,771.5 years. Thus monitoring it over a much smaller timeframe appears to simply result in a "drift" of the position of the equinox in the plane of the ecliptic of approximately one degree per 71.6 years,[4] influencing the Milankovitch cycles.[5]
Planets
Secular variation also refers to long-term trends in the orbits of all of the planets. Several attempts have from time to time been undertaken to analyze and predict such gravitational deviations for planets, observing ordinary satellite orbits. Others are referred to as post-keplerian effects.
Variations Séculaires des Orbites Planétaires (VSOP) is a modern numerical model[6] that tries to address the problem.
Market trends
Market trends are classified as secular, primary and secondary for long, medium and short time frames.[7] Some traders identify market trends using technical analysis.
Geomagnetic secular variation
Geomagnetic secular variation refers to some changes in the Earth's magnetic field. The field has variations on timescales from milliseconds to millions of years – its rapid ones mostly come from currents in the ionosphere and magnetosphere. The secular variations are those over periods of a year or more, reflecting changes in the Earth's core. Phenomena associated with these include geomagnetic jerk, westward drift and geomagnetic reversals.[8]
Biological anthropology
A secular trend, widely tapered off and in some places ended, in which case a discrete developmental shift, has been found to apply across the continents in the average age of onset of puberty (menarche/first menstruation and beginning of breast development) of girls from the 1940s to 2010s: beginning roughly 4 months earlier per decade. This is largely believed to be caused by nutritional changes in children over time.[9][10][11][12][13]
References
1. "secular (adj.)". Etymology Online.
2. "secular, adj. and n.". Oxford English Dictionary.
3. Jyri B. Kolesnik; Revision of the tidal acceleration of the Moon and the tidal deceleration of the Earth's rotation from historical optical observations of planets, in ISBN 2-901057-45-4 (2001) pp. 231 - 234.
4. Lowrie, William (2004). Fundamentals of Geophysics. Cambridge University Press. ISBN 978-0-521-46164-1.
5. Jurij B. Kolesnik; A new approach to interpretation of the non-precessional equinox motion, in Journées 2000 - systèmes de référence spatio-temporels. J2000, a fundamental epoch for origins of reference systems and astronomical models, Paris, Septembre 2000, edited by N. Capitaine, Observatoire de Paris (2001), pp. 119 – 120. ISBN 2-901057-45-4
6. Bretagnon, P. (1982). "Théorie du mouvement de l'ensemble des planètes. Solution VSOP82". Astronomy & Astrophysics. 114: 278–288. Bibcode:1982A&A...114..278B.
7. Edwards, R.; McGee, J.; Bessetti, W. H. C. (2007). Technical Analysis of Stock Trends. CRC Press. p. 17. ISBN 978-0-8493-3772-7.
8. Merrill, Ronald T.; McElhinny, Michael W.; McFadden, Phillip L. (1996). The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle. International Geophysics Series. Vol. 63. Academic Press. ISBN 9780124912458.
9. Okasha, M; McCarron, P; McEwen, J; Smith, GD (2001). "Age at menarche: secular trends and association with adult anthropometric measures". Annals of Human Biology. 28 (1): 68–78. doi:10.1080/03014460150201896. PMID 11201332.
10. Wattigney, WA; Srinivasan, SR; Chen, W; Greenlund, KJ; Berenson, GS (1999). "Secular trend of earlier onset of menarche with increasing obesity in black and white girls: the Bogalusa Heart Study". Ethnicity & Disease. 9 (2): 181–189. PMID 10421080.
11. Prentice, S; Fulford, AJ; Jarjou, LM; Goldberg, GR; Prentice, A (2010). "Evidence for a downward secular trend in age of menarche in a rural Gambian population". Annals of Human Biology. 37 (5): 717–721. doi:10.3109/03014461003727606. PMC 3575631. PMID 20465526.
12. Biro, Frank; Galvez, MP; Greenspan, LC; Succop, PA; Vangeepuram, N (Sep 2010), "Pubertal assessment method and baseline characteristics in a mixed longitudinal study of girls", Pediatrics, 126 (3): e583–90, doi:10.1542/peds.2009-3079, PMC 4460992, PMID 20696727
13. Euling, Susan; Herman-Giddens, Marcia; Lee, Peter; Selevan, Sherry (February 2008). "Examination of US Puberty-Timing Data from 1940 to 1994 for Secular Trends: Panel Findings". Pediatrics. 121 (3).
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Flood fill
Flood fill, also called seed fill, is a flooding algorithm that determines and alters the area connected to a given node in a multi-dimensional array with some matching attribute. It is used in the "bucket" fill tool of paint programs to fill connected, similarly-colored areas with a different color, and in games such as Go and Minesweeper for determining which pieces are cleared. A variant called boundary fill uses the same algorithms but is defined as the area connected to a given node that does not have a particular attribute.[1]
Note that flood filling is not suitable for drawing filled polygons, as it will miss some pixels in more acute corners. [2] Instead, see Even-odd rule and Nonzero-rule.
The algorithm parameters
The traditional flood-fill algorithm takes three parameters: a start node, a target color, and a replacement color. The algorithm looks for all nodes in the array that are connected to the start node by a path of the target color and changes them to the replacement color. For a boundary-fill, in place of the target color, a border color would be supplied.
In order to generalize the algorithm in the common way, the following descriptions will instead have two routines available. [3] One called Inside which returns true for unfilled points that, by their color, would be inside the filled area, and one called Set which fills a pixel/node. Any node that has Set called on it must then no longer be Inside.
Depending on whether we consider nodes touching at the corners connected or not, we have two variations: eight-way and four-way respectively.
Stack-based recursive implementation (four-way)
The earliest-known, implicitly stack-based, recursive, four-way flood-fill implementation goes as follows:[4][5][6][7]
Flood-fill (node):
1. If node is not Inside return.
2. Set the node
3. Perform Flood-fill one step to the south of node.
4. Perform Flood-fill one step to the north of node
5. Perform Flood-fill one step to the west of node
6. Perform Flood-fill one step to the east of node
7. Return.
Though easy to understand, the implementation of the algorithm used above is impractical in languages and environments where stack space is severely constrained (e.g. Microcontrollers).
Moving the recursion into a data structure
Four-way flood fill using a queue for storage
Four-way flood fill using a stack for storage
Moving the recursion into a data structure (either a stack or a queue) prevents a stack overflow. It is similar to the simple recursive solution, except that instead of making recursive calls, it pushes the nodes onto a stack or queue for consumption, with the choice of data structure affecting the proliferation pattern:
Flood-fill (node):
1. Set Q to the empty queue or stack.
2. Add node to the end of Q.
3. While Q is not empty:
4. Set n equal to the first element of Q.
5. Remove first element from Q.
6. If n is Inside:
Set the n
Add the node to the west of n to the end of Q.
Add the node to the east of n to the end of Q.
Add the node to the north of n to the end of Q.
Add the node to the south of n to the end of Q.
7. Continue looping until Q is exhausted.
8. Return.
Further potential optimizations
• Check and set each node's pixel color before adding it to the stack/queue, reducing stack/queue size.
• Use a loop for the east/west directions, queuing pixels above/below as you go (making it similar to the span filling algorithms, below).
• Interleave two or more copies of the code with extra stacks/queues, to allow out-of-order processors more opportunity to parallelize.
• Use multiple threads (ideally with slightly different visiting orders, so they don't stay in the same area).[6]
Advantages
• Very simple algorithm - easy to make bug-free. [6]
Disadvantages
• Uses a lot of memory, particularly when using a stack. [6]
• Tests most filled pixels a total of four times.
• Not suitable for pattern filling, as it requires pixel test results to change.
• Access pattern is not cache-friendly, for the queuing variant.
• Cannot easily optimize for multi-pixel words or bitplanes. [2]
Span filling
It's possible to optimize things further by working primarily with spans, a row with constant y. The first published complete example works on the following basic principle. [1]
1. Starting with a seed point, fill left and right. Keep track of the leftmost filled point lx and rightmost filled point rx. This defines the span.
2. Scan from lx to rx above and below the seed point, searching for new seed points to continue with.
As an optimisation, the scan algorithm does not need restart from every seed point, but only those at the start of the next span. Using a stack explores spans depth first, whilst a queue explores spans breadth first.
This algorithm is the most popular, for both citations and implementations , despite testing most filled pixels three times in total.
fn fill(x, y):
if not Inside(x, y) then return
let s = new empty stack or queue
Add (x, y) to s
while s is not empty:
Remove an (x, y) from s
let lx = x
while Inside(lx - 1, y):
Set(lx - 1, y)
lx = lx - 1
while Inside(x, y):
Set(x, y)
x = x + 1
scan(lx, x - 1, y + 1, s)
scan(lx, x - 1, y - 1, s)
fn scan(lx, rx, y, s):
let span_added = false
for x in lx .. rx:
if not Inside(x, y):
span_added = false
else if not span_added:
Add (x, y) to s
span_added = true
Over time, the following optimizations were realized:
• When a new scan would be entirely within a grandparent span, it would certainly only find filled pixels, and so wouldn't need queueing. [1][6][3]
• Further, when a new scan overlaps a grandparent span, only the overhangs (U-turns and W-turns) need to be scanned. [3]
• It's possible to fill while scanning for seeds [6]
The final, combined-scan-and-fill span filler was then published in 1990. In pseudo-code form: [8]
fn fill(x, y):
if not Inside(x, y) then return
let s = new empty queue or stack
Add (x, x, y, 1) to s
Add (x, x, y - 1, -1) to s
while s is not empty:
Remove an (x1, x2, y, dy) from s
let x = x1
if Inside(x, y):
while Inside(x - 1, y):
Set(x - 1, y)
x = x - 1
if x < x1:
Add (x, x1 - 1, y - dy, -dy) to s
while x1 <= x2:
while Inside(x1, y):
Set(x1, y)
x1 = x1 + 1
if x1 > x:
Add (x, x1 - 1, y + dy, dy) to s
if x1 - 1 > x2:
Add (x2 + 1, x1 - 1, y - dy, -dy) to s
x1 = x1 + 1
while x1 < x2 and not Inside(x1, y):
x1 = x1 + 1
x = x1
Advantages
• 2–8x faster than the pixel-recursive algorithm.
• Access pattern is cache and bitplane-friendly. [6]
• Can draw a horizontal line rather than setting individual pixels. [6]
Disadvantages
• Still visits pixels it has already filled. (For the popular algorithm, 3 scans of most pixels. For the final one, only doing extra scans of pixels where there are holes in the filled area.) [3]
• Not suitable for pattern filling, as it requires pixel test results to change.
Adding pattern filling support
Two common ways to make the span and pixel-based algorithms support pattern filling are either to use a unique color as a plain fill and then replace that with a pattern or to keep track (in a 2d boolean array or as regions) of which pixels have been visited, using it to indicate pixels are no longer fillable. Inside must then return false for such visited pixels. [3]
Graph-theoretic filling
Some theorists applied explicit graph theory to the problem, treating spans of pixels, or aggregates of such, as nodes and studying their connectivity. The first published graph theory algorithm worked similarly to the span filling, above, but had a way to detect when it would duplicate filling of spans. [9] Unfortunately, it had bugs that made it not complete some fills. [10] A corrected algorithm was later published with a similar basis in graph theory; however, it alters the image as it goes along, to temporarily block off potential loops, complicating the programmatic interface.[10] A later published algorithm depended on the boundary being distinct from everything else in the image and so isn't suitable for most uses; [11][3] it also requires an extra bit per pixel for bookkeeping. [3]
Advantages
• Suitable for pattern filling, directly, as it never retests filled pixels. [9] [10] [11]
• Double the speed of the original span algorithm, for uncomplicated fills. [3]
• Access pattern is cache and bitplane-friendly. [6][3]
Disadvantages
• Regularly, a span has to be compared to every other 'front' in the queue, which significantly slows down complicated fills. [3]
• Switching back and forth between graph theoretic and pixel domains complicates understanding.
• The code is fairly complicated, increasing the chances of bugs.
Walk-based filling (Fixed-memory method)
A method exists that uses essentially no memory for four-connected regions by pretending to be a painter trying to paint the region without painting themselves into a corner. This is also a method for solving mazes. The four pixels making the primary boundary are examined to see what action should be taken. The painter could find themselves in one of several conditions:
1. All four boundary pixels are filled.
2. Three of the boundary pixels are filled.
3. Two of the boundary pixels are filled.
4. One boundary pixel is filled.
5. Zero boundary pixels are filled.
Where a path or boundary is to be followed, the right-hand rule is used. The painter follows the region by placing their right-hand on the wall (the boundary of the region) and progressing around the edge of the region without removing their hand.
For case #1, the painter paints (fills) the pixel the painter is standing upon and stops the algorithm.
For case #2, a path leading out of the area exists. Paint the pixel the painter is standing upon and move in the direction of the open path.
For case #3, the two boundary pixels define a path which, if we painted the current pixel, may block us from ever getting back to the other side of the path. We need a "mark" to define where we are and which direction we are heading to see if we ever get back to exactly the same pixel. If we already created such a "mark", then we preserve our previous mark and move to the next pixel following the right-hand rule.
A mark is used for the first 2-pixel boundary that is encountered to remember where the passage started and in what direction the painter was moving. If the mark is encountered again and the painter is traveling in the same direction, then the painter knows that it is safe to paint the square with the mark and to continue in the same direction. This is because (through some unknown path) the pixels on the other side of the mark can be reached and painted in the future. The mark is removed for future use.
If the painter encounters the mark but is going in a different direction, then some sort of loop has occurred, which caused the painter to return to the mark. This loop must be eliminated. The mark is picked up, and the painter then proceeds in the direction indicated previously by the mark using a left-hand rule for the boundary (similar to the right-hand rule but using the painter's left hand). This continues until an intersection is found (with three or more open boundary pixels). Still using the left-hand rule the painter now searches for a simple passage (made by two boundary pixels). Upon finding this two-pixel boundary path, that pixel is painted. This breaks the loop and allows the algorithm to continue.
For case #4, we need to check the opposite 8-connected corners to see whether they are filled or not. If either or both are filled, then this creates a many-path intersection and cannot be filled. If both are empty, then the current pixel can be painted and the painter can move following the right-hand rule.
The algorithm trades time for memory. For simple shapes it is very efficient. However, if the shape is complex with many features, the algorithm spends a large amount of time tracing the edges of the region trying to ensure that all can be painted.
This algorithm was first available commercially in 1981 on a Vicom Image Processing system manufactured by Vicom Systems, Inc. A walking algorithm was published in 1994.[12] The classic recursive flood fill algorithm was available on the Vicom system as well.
Pseudocode
This is a pseudocode implementation of an optimal fixed-memory flood-fill algorithm written in structured English:
The variables
• cur, mark, and mark2 each hold either pixel coordinates or a null value
• NOTE: when mark is set to null, do not erase its previous coordinate value. Keep those coordinates available to be recalled if necessary.
• cur-dir, mark-dir, and mark2-dir each hold a direction (left, right, up, or down)
• backtrack and findloop each hold boolean values
• count is an integer
The algorithm
NOTE: All directions (front, back, left, right) are relative to cur-dir
set cur to starting pixel
set cur-dir to default direction
clear mark and mark2 (set values to null)
set backtrack and findloop to false
while front-pixel is empty do
move forward
end while
jump to START
MAIN LOOP:
move forward
if right-pixel is inside then
if backtrack is true and findloop is false and either front-pixel or left-pixel is inside then
set findloop to true
end if
turn right
PAINT:
move forward
end if
START:
set count to number of non-diagonally adjacent pixels filled (front/back/left/right ONLY)
if count is not 4 then
do
turn right
while front-pixel is inside
do
turn left
while front-pixel is not inside
end if
switch count
case 1
if backtrack is true then
set findloop to true
else if findloop is true then
if mark is null then
restore mark
end if
else if front-left-pixel and back-left-pixel are both inside then
clear mark
set cur
jump to PAINT
end if
end case
case 2
if back-pixel is not inside then
if front-left-pixel is inside then
clear mark
set cur
jump to PAINT
end if
else if mark is not set then
set mark to cur
set mark-dir to cur-dir
clear mark2
set findloop and backtrack to false
else
if mark2 is not set then
if cur is at mark then
if cur-dir is the same as mark-dir then
clear mark
turn around
set cur
jump to PAINT
else
set backtrack to true
set findloop to false
set cur-dir to mark-dir
end if
else if findloop is true then
set mark2 to cur
set mark2-dir to cur-dir
end if
else
if cur is at mark then
set cur to mark2
set cur-dir to mark2-dir
clear mark and mark2
set backtrack to false
turn around
set cur
jump to PAINT
else if cur at mark2 then
set mark to cur
set cur-dir and mark-dir to mark2-dir
clear mark2
end if
end if
end if
end case
case 3
clear mark
set cur
jump to PAINT
end case
case 4
set cur
done
end case
end switch
end MAIN LOOP
Advantages
• Constant memory usage.
Disadvantages
• Access pattern is not cache or bitplane-friendly.
• Can spend a lot of time walking around loops before closing them.
Vector implementations
Version 0.46 of Inkscape includes a bucket fill tool, giving output similar to ordinary bitmap operations and indeed using one: the canvas is rendered, a flood fill operation is performed on the selected area and the result is then traced back to a path. It uses the concept of a boundary condition.
See also
• Breadth-first search
• Depth-first search
• Graph traversal
• Connected-component labeling
• Dijkstra's algorithm
• Watershed (image processing)
External links
• Sample implementations for recursive and non-recursive, classic and scanline flood fill, by Lode Vandevenne.
• Sample Java implementation using Q non-recursive.
References
1. Smith, Alvy Ray (1979). Tint Fill. SIGGRAPH '79: Proceedings of the 6th annual conference on Computer graphics and interactive techniques. pp. 276–283. doi:10.1145/800249.807456.
2. Ackland, Bryan D; Weste, Neil H (1981). The edge flag algorithm — A fill method for raster scan displays. IEEE Transactions on Computers (Volume: C-30, Issue: 1). pp. 41–48. doi:10.1109/TC.1981.6312155.
3. Fishkin, Kenneth P; Barsky, Brian A (1985). An Analysis and Algorithm for Filling Propagation. Computer-Generated Images: The State of the Art Proceedings of Graphics Interface ’85. pp. 56–76. doi:10.1007/978-4-431-68033-8_6.
4. Newman, William M; Sproull, Robert Fletcher (1979). Principles of Interactive Computer Graphics (2nd ed.). McGraw-Hill. p. 253. ISBN 978-0-07-046338-7.
5. Pavlidis, Theo (1982). Algorithms for Graphics and Image Processing. Springer-Verlag. p. 181. ISBN 978-3-642-93210-6.
6. Levoy, Marc (1982). Area Flooding Algorithms. SIGGRAPH 1981 Two-Dimensional Computer Animation course notes.
7. Foley, J D; van Dam, A; Feiner, S K; Hughes, S K (1990). Computer Graphics: Principles and Practice (2nd ed.). Addison–Wesley. pp. 979–982. ISBN 978-0-201-84840-3.
8. Heckbert, Paul S (1990). "IV.10: A Seed Fill Algorithm". In Glassner, Andrew S (ed.). Graphics Gems. Academic Press. pp. 275–277. ISBN 0122861663.
9. Lieberman, Henry (1978). How to Color in a Coloring Book. SIGGRAPH '78: Proceedings of the 5th annual conference on Computer graphics and interactive techniques. pp. 111–116. doi:10.1145/800248.807380.
10. Shani, Uri (1980). Filling regions in binary raster images: A graph-theoretic approach. SIGGRAPH '80: Proceedings of the 7th annual conference on Computer graphics and interactive techniques. pp. 321–327. doi:10.1145/800250.807511.
11. Pavlidis, Theo (1981). Contour Filling in Raster Graphics. SIGGRAPH '81: Proceedings of the 8th annual conference on Computer graphics and interactive techniques. pp. 29–36. doi:10.1145/800224.806786.
12. Henrich, Dominik (1994). Space-efficient region filling in raster graphics. The Visual Computer. pp. 205–215. doi:10.1007/BF01901287.
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Wikipedia
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Initial condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value,[1]: pp. 160 is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time.
The initial condition of a vibrating string
Evolution from the initial condition
In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons.
Linear system
Discrete time
A linear matrix difference equation of the homogeneous (having no constant term) form $X_{t+1}=AX_{t}$ has closed form solution $X_{t}=A^{t}X_{0}$ predicated on the vector $X_{0}$ of initial conditions on the individual variables that are stacked into the vector; $X_{0}$ is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being the number of time lags in the system. The initial conditions in this linear system do not affect the qualitative nature of the future behavior of the state variable X; that behavior is stable or unstable based on the eigenvalues of the matrix A but not based on the initial conditions.
Alternatively, a dynamic process in a single variable x having multiple time lags is
$x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.$
Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. Again the initial conditions do not affect the qualitative nature of the variable's long-term evolution. The solution of this equation is found by using its characteristic equation $\lambda ^{k}-a_{1}\lambda ^{k-1}-a_{2}\lambda ^{k-2}-\cdots -a_{k-1}\lambda -a_{k}=0$ to obtain the latter's k solutions, which are the characteristic values $\lambda _{1},\dots ,\lambda _{k},$ for use in the solution equation
$x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.$
Here the constants $c_{1},\dots ,c_{k}$ are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition $x_{t}$ Is known.
Continuous time
A differential equation system of the first order with n variables stacked in a vector X is
${\frac {dX}{dt}}=AX.$
Its behavior through time can be traced with a closed form solution conditional on an initial condition vector $X_{0}$. The number of required initial pieces of information is the dimension n of the system times the order k = 1 of the system, or n. The initial conditions do not affect the qualitative behavior (stable or unstable) of the system.
A single kth order linear equation in a single variable x is
${\frac {d^{k}x}{dt^{k}}}+a_{k-1}{\frac {d^{k-1}x}{dt^{k-1}}}+\cdots +a_{1}{\frac {dx}{dt}}+a_{0}x=0.$
Here the number of initial conditions necessary for obtaining a closed form solution is the dimension n = 1 times the order k, or simply k. In this case the k initial pieces of information will typically not be different values of the variable x at different points in time, but rather the values of x and its first k – 1 derivatives, all at some point in time such as time zero. The initial conditions do not affect the qualitative nature of the system's behavior. The characteristic equation of this dynamic equation is $\lambda ^{k}+a_{k-1}\lambda ^{k-1}+\cdots +a_{1}\lambda +a_{0}=0,$ whose solutions are the characteristic values $\lambda _{1},\dots ,\lambda _{k};$ these are used in the solution equation
$x(t)=c_{1}e^{\lambda _{1}t}+\cdots +c_{k}e^{\lambda _{k}t}.$
This equation and its first k – 1 derivatives form a system of k equations that can be solved for the k parameters $c_{1},\dots ,c_{k},$ given the known initial conditions on x and its k – 1 derivatives' values at some time t.
Nonlinear systems
Another initial condition
Evolution of this initial condition for an example PDE
Nonlinear systems can exhibit a substantially richer variety of behavior than linear systems can. In particular, the initial conditions can affect whether the system diverges to infinity or whether it converges to one or another attractor of the system. Each attractor, a (possibly disconnected) region of values that some dynamic paths approach but never leave, has a (possibly disconnected) basin of attraction such that state variables with initial conditions in that basin (and nowhere else) will evolve toward that attractor. Even nearby initial conditions could be in basins of attraction of different attractors (see for example Newton's method#Basins of attraction).
Moreover, in those nonlinear systems showing chaotic behavior, the evolution of the variables exhibits sensitive dependence on initial conditions: the iterated values of any two very nearby points on the same strange attractor, while each remaining on the attractor, will diverge from each other over time. Thus even on a single attractor the precise values of the initial conditions make a substantial difference for the future positions of the iterates. This feature makes accurate simulation of future values difficult, and impossible over long horizons, because stating the initial conditions with exact precision is seldom possible and because rounding error is inevitable after even only a few iterations from an exact initial condition.
Empirical laws and initial conditions
Every empirical law has the disquieting quality that one does not know its limitations. We have seen that there are regularities in the events in the world around us which can be formulated in terms of mathematical concepts with an uncanny accuracy. There are, on the other hand, aspects of the world concerning which we do not believe in the existence of any accurate regularities. We call these initial conditions.[2]
See also
• Boundary condition
• Initialization vector, in cryptography
References
1. Baumol, William J. (1970). Economic Dynamics: An Introduction (3rd ed.). London: Collier-Macmillan. ISBN 0-02-306660-1.
2. Wigner, Eugene P. (1960). "The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. Archived from the original (PDF) on February 12, 2021.
External links
• Quotations related to Initial condition at Wikiquote
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Wikipedia
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Seema Nanda
Seema Nanda is an Indian mathematician. In her research she applies mathematics to study problems in biology, engineering and finance. Her research interests are primarily in solving real world problems using mathematics and computations.[1]
Education
Her education in mathematics was at the Courant Institute of Mathematical Sciences of New York University, where she obtained her Ph.D in 1998. Her PhD thesis was in the area of probability theory and was supervised by Charles M. Newman.[2]
Career
Prior to her current position as a faculty at the Tata Institute of Fundamental Research in Bangalore, she held cross-disciplinary academic positions at the University of Tennessee and at Harvey Mudd College. She switched careers from working in the corporate world to academic research in 2004. Before returning to academia she worked as a quantitative analyst for an investment bank in New York City. She is also interested in encouraging the youth of India to understand science and mathematics. In 2012 she founded an NGO called Leora Trust which aims to empower girls through education.
Awards
She was a recipient of the Bella Manel prize (given to a promising female student of mathematics at NYU) in 1996.
References
1. Home page Archived 2014-10-21 at the Wayback Machine, Tata Institute, retrieved 2015-02-17.
2. Seema Nanda at the Mathematics Genealogy Project
External links
• Leora Trust
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
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Wikipedia
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Seesaw theorem
In algebraic geometry, the seesaw theorem, or seesaw principle, says roughly that a limit of trivial line bundles over complete varieties is a trivial line bundle. It was introduced by André Weil in a course at the University of Chicago in 1954–1955, and is related to Severi's theory of correspondences.
The seesaw theorem is proved using proper base change. It can be used to prove the theorem of the cube.
Statement
Lang (1959, p.241) originally stated the seesaw principle in terms of divisors. It is now more common to state it in terms of line bundles as follows (Mumford 2008, Corollary 6, section 5). Suppose L is a line bundle over X×T, where X is a complete variety and T is an algebraic set. Then the set of points t of T such that L is trivial on X×t is closed. Moreover if this set is the whole of T then L is the pullback of a line bundle on T. Mumford (2008, section 10) also gave a more precise version, showing that there is a largest closed subscheme of T such that L is the pullback of a line bundle on the subscheme.
References
• Lang, Serge (1959), Abelian varieties, Interscience Tracts in Pure and Applied Mathematics, vol. 7, New York: Interscience Publishers, Inc., MR 0106225
• Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290
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0
0 (zero) is a number representing an empty quantity. As a number, 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and other algebraic structures.
← −1 0 1 →
−1 0 1 2 3 4 5 6 7 8 9 →
• List of numbers
• Integers
← 0 10 20 30 40 50 60 70 80 90 →
Cardinal0, zero, "oh" (/oʊ/), nought, naught, nil
OrdinalZeroth, noughth, 0th
Binary02
Ternary03
Senary06
Octal08
Duodecimal012
Hexadecimal016
Arabic, Kurdish, Persian, Sindhi, Urdu٠
Hindu Numerals०
Chinese零, 〇
Burmese၀
Khmer០
Thai๐
Bangla০
In place-value notation such as decimal, 0 also serves as a numerical digit to indicate that that position's power of 10 is not multiplied by anything or added to the resulting number. This concept appears to have been difficult to discover.
Common names for the number 0 in English are zero, nought, naught (/nɔːt/), nil. In contexts where at least one adjacent digit distinguishes it from the letter O, the number is sometimes pronounced as oh or o (/oʊ/). Informal or slang terms for 0 include zilch and zip. Historically, ought, aught (/ɔːt/), and cipher have also been used.
Etymology
Main articles: Names for the number 0 and Names for the number 0 in English
The word zero came into the English language via French zéro from the Italian zero, a contraction of the Venetian zevero form of Italian zefiro via ṣafira or ṣifr.[1] In pre-Islamic time the word ṣifr (Arabic صفر) had the meaning "empty".[2] Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India.[2] The first known English use of zero was in 1598.[3]
The Italian mathematician Fibonacci (c. 1170 – c. 1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term zephyrum. This became zefiro in Italian, and was then contracted to zero in Venetian. The Italian word zefiro was already in existence (meaning "west wind" from Latin and Greek Zephyrus) and may have influenced the spelling when transcribing Arabic ṣifr.[4]
Modern usage
Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "nothing" and "none" are often used. Sometimes, the word "nought" or "naught" is used.
It is often called "oh" in the context of reading out a string of digits, such as telephone numbers, street addresses, credit card numbers, military time, or years (e.g. the area code 201 would be pronounced "two oh one"; a year such as 1907 is often pronounced "nineteen oh seven"). The presence of other digits, indicating that the string contains only numbers, avoids confusion with the letter O. For this reason, systems that include strings with both letters and numbers (e.g. Canadian postal codes) may exclude the use of the letter O.
Slang words for zero include "zip", "zilch", "nada", and "scratch".[5]
"Nil" is used for many sports in British English. Several sports have specific words for a score of zero, such as "love" in tennis – from French l'œuf, "the egg" – and "duck" in cricket, a shortening of "duck's egg"; "goose egg" is another general slang term used for zero.[5]
History
Ancient Near East
nfr
heart with trachea
beautiful, pleasant, good
Ancient Egyptian numerals were of base 10.[6] They used hieroglyphs for the digits and were not positional. By 1770 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this line.[7]
By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated base 60 positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. In a tablet unearthed at Kish (dating to as early as 700 BC), the scribe Bêl-bân-aplu used three hooks as a placeholder in the same Babylonian system.[8] By 300 BC, a punctuation symbol (two slanted wedges) was co-opted to serve as this placeholder.[9][10]
The Babylonian placeholder was not a true zero because it was not used alone, nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60) looked the same, because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.
Pre-Columbian Americas
The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its vigesimal (base-20) positional numeral system. Many different glyphs, including the partial quatrefoil were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC.[lower-alpha 1]
Since the eight earliest Long Count dates appear outside the Maya homeland,[11] it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.[12] Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates.
Although zero became an integral part of Maya numerals, with a different, empty tortoise-like "shell shape" used for many depictions of the "zero" numeral, it is assumed not to have influenced Old World numeral systems.
Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.
Classical antiquity
The ancient Greeks had no symbol for zero (μηδέν, pronounced 'midén'), and did not use a digit placeholder for it.[13] According to mathematician Charles Seife, the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in astronomy after 500 BC, representing it with the lowercase Greek letter ό (όμικρον) or omicron.[14] However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into Greek numerals.[14] Greeks seemed to have a philosophical opposition to using zero as a number.[14] Other scholars give the Greek partial adoption of the Babylonian zero a later date, with the scientist Andreas Nieder giving a date of after 400 BC and the mathematician Robert Kaplan dating it after the conquests of Alexander.[15][16]
Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", leading to philosophical and, by the medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.[17]
By AD 150, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (—°)[18][19] in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest.[20] This Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World.[21] Ptolemy used it many times in his Almagest (VI.8) for the magnitude of solar and lunar eclipses. It represented the value of both digits and minutes of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was the angular diameter of the Sun. Minutes of immersion was tabulated from 0′0″ to 31′20″ to 0′0″, where 0′0″ used the symbol as a placeholder in two positions of his sexagesimal positional numeral system,[lower-alpha 2] while the combination meant a zero angle. Minutes of immersion was also a continuous function 1/12 31′20″ √d(24−d) (a triangular pulse with convex sides), where d was the digit function and 31′20″ was the sum of the radii of the Sun's and Moon's discs.[22] Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none.
The earliest use of zero in the calculation of the Julian Easter occurred before AD 311, at the first entry in a table of epacts as preserved in an Ethiopic document for the years 311 to 369, using a Ge'ez word for "none" (English translation is "0" elsewhere) alongside Ge'ez numerals (based on Greek numerals), which was translated from an equivalent table published by the Church of Alexandria in Medieval Greek.[23] This use was repeated in 525 in an equivalent table, that was translated via the Latin nulla or "none" by Dionysius Exiguus, alongside Roman numerals.[24] When division produced zero as a remainder, nihil, meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals by Bede—or his colleagues—around AD 725.[25]
China
The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, and Japanese records dated from the 18th century, describe how the 4th century BC Chinese counting rods system enabled one to perform decimal calculations. As noted in Xiahou Yang's Suanjing (425–468 AD) that states that to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places,[27] According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero".[26] The counting rod system is considered a positional notation system.[28]
In AD 690, Empress Wǔ promulgated Zetian characters, one of which was "〇"; originally meaning 'star', it subsequently came to represent zero.
Zero was not treated as a number at that time, but as a "vacant position".[29] Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol for zero.[30] Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen in The Nine Chapters on the Mathematical Art.[31]
India
Pingala (c. 3rd/2nd century BC[32]), a Sanskrit prosody scholar,[33] used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to Morse code.[34] Pingala used the Sanskrit word śūnya explicitly to refer to zero.[32]
The concept of zero as a written digit in the decimal place value notation was developed in India.[35] A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetic for merchants.[36] In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from AD 224–383, AD 680–779, and AD 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together.[37][38][39]
The Lokavibhāga, a Jain text on cosmology surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 (Saka era 380), uses a decimal place-value system, including a zero. In this text, śūnya ("void, empty") is also used to refer to zero.[40]
The Aryabhatiya (c. 500), states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding".[41][42][43]
Rules governing the use of zero appeared in Brahmagupta's Brahmasputha Siddhanta (7th century), which states the sum of zero with itself as zero, and incorrectly division by zero as:[44][45]
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Epigraphy
A black dot is used as a decimal placeholder in the Bakhshali manuscript, portions of which date from AD 224–993.[46]
There are numerous copper plate inscriptions, with the same small o in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.[8]
A stone tablet found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of "605" in Khmer numerals (a set of numeral glyphs for the Hindu–Arabic numeral system). The number is the year of the inscription in the Saka era, corresponding to a date of AD 683.[47]
The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior, in India, dated 876.[48][49]
Transmission to Islamic culture
See also: History of the Hindu–Arabic numeral system
The Arabic-language inheritance of science was largely Greek,[50] followed by Hindu influences.[51] In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others.
In AD 813, astronomical tables were prepared by a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals;[51] and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero.[52] This book was later translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started to acquire a meaning of any arithmetic based on decimals.[51]
Muhammad ibn Ahmad al-Khwarizmi, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ṣifr.[53]
Transmission to Europe
The Hindu–Arabic numeral system (base 10) reached Western Europe in the 11th century, via Al-Andalus, through Spanish Muslims, the Moors, together with knowledge of classical astronomy and instruments like the astrolabe; Gerbert of Aurillac is credited with reintroducing the lost teachings into Catholic Europe. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician Fibonacci or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating:
After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus (Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written.[54][55][56]
Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called algorismus after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco, about 1235 and was one of the earliest scientific books to be printed in 1488. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe.
Mathematics
See also: Null (mathematics)
0 is the integer immediately preceding 1. Zero is an even number[57] because it is divisible by 2 with no remainder. 0 is neither positive nor negative,[58] or both positive and negative; cf. Signed zero.[59] Many definitions[60] include 0 as a natural number, in which case it is the only natural number that is not positive. Zero is a number which quantifies a count or an amount of null size. In most cultures, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted.
As a value or a number, zero is not the same as the digit zero, used in numeral systems with positional notation. Successive positions of digits have higher weights, so the digit zero is used inside a numeral to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system (e.g., the number 02). In some instances, a leading zero may be used to distinguish a number.
Elementary algebra
The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number (as well as an algebraic number and a complex number).
The number 0 is neither positive nor negative, and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors, and cannot be composite because it cannot be expressed as a product of prime numbers (as 0 must always be one of the factors).[61] Zero is, however, even (i.e., a multiple of 2, as well as being a multiple of any other integer, rational, or real number).
The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated.
• Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition.
• Subtraction: x − 0 = x and 0 − x = −x.
• Multiplication: x · 0 = 0 · x = 0.
• Division: 0/x = 0, for nonzero x. But x/0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule.
• Exponentiation: x0 = x/x = 1, except that the case x = 0 is considered undefined in some contexts. For all positive real x, 0x = 0.
The expression 0/0, which may be obtained in an attempt to determine the limit of an expression of the form f(x)/g(x) as a result of applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not mean that the limit sought is necessarily undefined; rather, it means that the limit of f(x)/g(x), if it exists, must be found by another method, such as l'Hôpital's rule.
The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.
Other branches of mathematics
• In set theory, 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is defined to be the empty set. When this is done, the empty set is the von Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.
• Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set.
• In propositional logic, 0 may be used to denote the truth value false.
• In abstract algebra, 0 is commonly used to denote a zero element, which is a neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined).
• In lattice theory, 0 may denote the bottom element of a bounded lattice.
• In category theory, 0 is sometimes used to denote an initial object of a category.
• In recursion theory, 0 can be used to denote the Turing degree of the partial computable functions.
Related mathematical terms
• A zero of a function f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros, these are called the roots of the function. This is related to zeros of a holomorphic function.
• The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. The zero function is the only function that is both even and odd. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map.
• Several branches of mathematics have zero elements, which generalize either the property 0 + x = x, or the property 0 · x = 0, or both.
Physics
The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an absolute temperature (as measured in kelvins), zero is the lowest possible value (negative temperatures are defined, but negative-temperature systems are not actually colder). This is in contrast to for example temperatures on the Celsius scale, where zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.
Chemistry
Zero has been proposed as the atomic number of the theoretical element tetraneutron. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in its own right. This would create an element with no protons and no charge on its nucleus.
As early as 1926, Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.
Computer science
The most common practice throughout human history has been to start counting at one, and this is the practice in early classic computer programming languages such as Fortran and COBOL. However, in the late 1950s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in C, so that for an array of n items the sequence of array indices runs from 0 to n−1.
There can be confusion between 0- and 1-based indexing; for example, Java's JDBC indexes parameters from 1 although Java itself uses 0-based indexing.[62]
In databases, it is possible for a field not to have a value. It is then said to have a null value.[63] For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to three-valued logic. No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null result.[64]
A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types).
In mathematics, −0 and +0 is equivalent to 0; both −0 and +0 represent exactly the same number, i.e., there is no "positive zero" or "negative zero" distinct from zero. However, in some computer hardware signed number representations, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as signed zero, with the latter form sometimes called negative zero. These representations include the signed magnitude and one's complement binary integer representations (but not the two's complement binary form used in most modern computers), and most floating-point number representations (such as IEEE 754 and IBM S/390 floating-point formats).
In binary, 0 represents the value for "off", which means no electricity flow.[65]
Zero is the value of false in many programming languages.
The Unix epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1970.[66][67][68]
The Classic Mac OS epoch and Palm OS epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1904.[69]
Many APIs and operating systems that require applications to return an integer value as an exit status typically use zero to indicate success and non-zero values to indicate specific error or warning conditions.
Programmers often use a slashed zero to avoid confusion with the letter "O".[70]
Other fields
• In comparative zoology and cognitive science, recognition that some animals display awareness of the concept of zero leads to the conclusion that the capability for numerical abstraction arose early in the evolution of species.[71]
• In telephony, pressing 0 is often used for dialling out of a company network or to a different city or region, and 00 is used for dialling abroad. In some countries, dialling 0 places a call for operator assistance.
• DVDs that can be played in any region are sometimes referred to as being "region 0"
• Roulette wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run).
• In Formula One, if the reigning World Champion no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the reigning champion won the title with. This happened in 1993 and 1994, with Damon Hill driving car 0, due to the reigning World Champion (Nigel Mansell and Alain Prost respectively) not competing in the championship.
• On the U.S. Interstate Highway System, in most states exits are numbered based on the nearest milepost from the highway's western or southern terminus within that state. Several that are less than half a mile (800 m) from state boundaries in that direction are numbered as Exit 0.
• In numismatics, €0 notes have been designed for commemorative purposes.[72] The Zero rupee note has been circulated in India as a means of protesting political corruption.
Symbols and representations
Main article: Symbols for zero
The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0.[73] Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.[73]
A slashed zero ($0\!\!\!{/}$) can be used to distinguish the number from the letter (mostly used in computing, navigation and in the military). The digit 0 with a dot in the center seems to have originated as an option on IBM 3270 displays and has continued with some modern computer typefaces such as Andalé Mono, and in some airline reservation systems. One variation uses a short vertical bar instead of the dot. Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). A further distinction is made in falsification-hindering typeface as used on German car number plates by slitting open the digit 0 on the upper right side. In some systems either the letter O or the numeral 0, or both, are excluded from use, to avoid confusion.
Year label
In the BC calendar era, the year 1 BC is the first year before AD 1; there is not a year zero. By contrast, in astronomical year numbering, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so forth.[74]
See also
• Brahmagupta
• Aryabhata
• Division by zero
• Grammatical number
• Gwalior Fort
• Mathematical constant
• Number theory
• Peano axioms
• Signed zero
Notes
1. No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.
2. Each place in Ptolemy's sexagesimal system was written in Greek numerals from 0 to 59, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20.
References
1. See:
• Douglas Harper (2011), Zero Archived 3 July 2017 at the Wayback Machine, Etymology Dictionary, Quote="figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", c. 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher", translation of Sanskrit sunya-m "empty place, desert, naught";
• Menninger, Karl (1992). Number words and number symbols: a cultural history of numbers. Courier Dover Publications. pp. 399–404. ISBN 978-0-486-27096-8. Archived from the original on 25 April 2016. Retrieved 5 January 2016.;
• "zero, n." OED Online. Oxford University Press. December 2011. Archived from the original on 7 March 2012. Retrieved 4 March 2012. French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr
2. See:
• Smithsonian Institution, Oriental Elements of Culture in the Occident, p. 518, at Google Books, Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives, Quote="Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya.";
• Jan Gullberg (1997), Mathematics: From the Birth of Numbers, W.W. Norton & Co., ISBN 978-0-393-04002-9, p. 26, Quote = Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.;
• Robert Logan (2010), The Poetry of Physics and the Physics of Poetry, World Scientific, ISBN 978-981-4295-92-5, p. 38, Quote = The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr.
3. Zero Archived 6 December 2017 at the Wayback Machine, Merriam Webster online Dictionary
4. Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 978-0-471-39340-5.
5. 'Aught' synonyms Archived 23 August 2014 at the Wayback Machine, Thesaurus.com – Retrieved April 2013.
6. "Egyptian numerals". mathshistory.st-andrews.ac.uk. Archived from the original on 15 November 2019. Retrieved 21 December 2019.
7. Joseph, George Gheverghese (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third ed.). Princeton UP. p. 86. ISBN 978-0-691-13526-7.
8. Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
9. "Zero". Maths History. Archived from the original on 21 September 2021. Retrieved 2021-09-07.
10. "Babylonian mathematics: View as single page". www.open.edu. Archived from the original on 7 September 2021. Retrieved 2021-09-07.
11. Diehl, p. 186
12. Mortaigne, Véronique (28 November 2014). "The golden age of Mayan civilisation – exhibition review". The Guardian. Archived from the original on 28 November 2014. Retrieved 10 October 2015.
13. Wallin, Nils-Bertil (19 November 2002). "The History of Zero". YaleGlobal online. The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Archived from the original on 25 August 2016. Retrieved 1 September 2016.
14. Seife, Charles (1 September 2000). Zero: The Biography of a Dangerous Idea. Penguin. p. 39. ISBN 978-0-14-029647-1. OCLC 1005913932. Archived from the original on 30 April 2022. Retrieved 30 April 2022.
15. Nieder, Andreas (19 November 2019). A Brain for Numbers: The Biology of the Number Instinct. MIT Press. p. 286. ISBN 978-0-262-35432-5. Archived from the original on 30 April 2022. Retrieved 30 April 2022.
16. Kaplan, Robert (28 October 1999). The Nothing that Is: A Natural History of Zero. Oxford University Press. p. 17. ISBN 978-0-19-802945-8. OCLC 466431211. Archived from the original on 24 June 2014. Retrieved 30 April 2022.
17. Huggett, Nick (2019). "Zeno's Paradoxes". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2019 ed.). Metaphysics Research Lab, Stanford University. Archived from the original on 10 January 2021. Retrieved 2020-08-09.
18. Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2 ed.). Dover Publications. pp. 13–14, plate 2. ISBN 978-0-486-22332-2.
19. Mercier, Raymond. "Consideration of the Greek symbol 'zero'" (PDF). Home of Kairos. Archived (PDF) from the original on 5 November 2020. Retrieved 28 March 2020.
20. Ptolemy (1998) [1984, c.150]. Ptolemy's Almagest. Translated by Toomer, G. J. Princeton University Press. pp. 306–307. ISBN 0-691-00260-6.
21. O'Connor, J J; Robertson, E F. "A history of Zero". MacTutor History of Mathematics. Archived from the original on 7 April 2020. Retrieved 28 March 2020.
22. Pedersen, Olaf (2010) [1974]. A Survey of the Almagest. Springer. pp. 232–235. ISBN 978-0-387-84825-9.
23. Neugebauer, Otto (2016) [1979]. Ethiopic Astronomy and Computus (Red Sea Press ed.). Red Sea Press. pp. 25, 53, 93, 183, Plate I. ISBN 978-1-56902-440-9.. The pages in this edition have numbers six less than the same pages in the original edition.
24. Deckers, Michael (2003) [525]. "Cyclus Decemnovennalis Dionysii – Nineteen Year Cycle of Dionysius". Archived from the original on 15 January 2019.
25. C. W. Jones, ed., Opera Didascalica, vol. 123C in Corpus Christianorum, Series Latina.
26. Hodgkin, Luke (2005). A History of Mathematics : From Mesopotamia to Modernity: From Mesopotamia to Modernity. Oxford University Press. p. 85. ISBN 978-0-19-152383-0.
27. O'Connor, J.J. (Jan 2004). "Chinese numerals". Mac Tutor. School of Mathematics and Statistics University of St Andrews, Scotland. Archived from the original on 14 June 2020. Retrieved 14 June 2020.
28. Crossley, Lun. 1999, p. 12 "the ancient Chinese system is a place notation system"
29. Kang-Shen Shen; John N. Crossley; Anthony W.C. Lun; Hui Liu (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford UP. p. 35. ISBN 978-0-19-853936-0. zero was regarded as a number in India ... whereas the Chinese employed a vacant position
30. "Mathematics in the Near and Far East" (PDF). grmath4.phpnet.us. p. 262. Archived (PDF) from the original on 4 November 2013. Retrieved 7 June 2012.
31. Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. pp. 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
32. Plofker, Kim (2009). Mathematics in India. Princeton University Press. pp. 54–56. ISBN 978-0-691-12067-6. In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero
33. Vaman Shivaram Apte (1970). Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India. Motilal Banarsidass. pp. 648–649. ISBN 978-81-208-0045-8. Archived from the original on 22 April 2017. Retrieved 21 April 2017.
34. "Math for Poets and Drummers" (PDF). people.sju.edu. Archived from the original (PDF) on 22 January 2019. Retrieved 20 December 2015.
35. Bourbaki, Nicolas Elements of the History of Mathematics (1998), p. 46
36. Weiss, Ittay (20 September 2017). "Nothing matters: How India's invention of zero helped create modern mathematics". The Conversation. Archived from the original on 12 July 2018. Retrieved 12 July 2018.
37. Devlin, Hannah (13 September 2017). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Archived from the original on 20 November 2017. Retrieved 14 September 2017.
38. Revell, Timothy (14 September 2017). "History of zero pushed back 500 years by ancient Indian text". New Scientist. Archived from the original on 25 October 2017. Retrieved 25 October 2017.
39. "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 14 September 2017. Archived from the original on 14 September 2017. Retrieved 25 October 2017.
40. Ifrah, Georges (2000), p. 416.
41. Aryabhatiya of Aryabhata, translated by Walter Eugene Clark.
42. O'Connor, Robertson, J.J., E.F. "Aryabhata the Elder". School of Mathematics and Statistics University of St Andrews, Scotland. Archived from the original on 11 July 2015. Retrieved 26 May 2013.{{cite web}}: CS1 maint: multiple names: authors list (link)
43. William L. Hosch, ed. (15 August 2010). The Britannica Guide to Numbers and Measurement (Math Explained). The Rosen Publishing Group. pp. 97–98. ISBN 978-1-61530-108-9. Archived from the original on 4 August 2016. Retrieved 26 September 2016.
44. Algebra with Arithmetic of Brahmagupta and Bhaskara, translated to English by Henry Thomas Colebrooke (1817) London
45. Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. New York: Oxford University Press. pp. 68–75. ISBN 978-0-19-514237-2.
46. "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. 13 September 2017. Archived from the original on 20 November 2017. Retrieved 14 September 2017.
47. Cœdès, George, "A propos de l'origine des chiffres arabes," Bulletin of the School of Oriental Studies, University of London, Vol. 6, No. 2, 1931, pp. 323–328. Diller, Anthony, "New Zeros and Old Khmer," The Mon-Khmer Studies Journal, Vol. 25, 1996, pp. 125–132.
48. Casselman, Bill. "All for Nought". ams.org. University of British Columbia), American Mathematical Society. Archived from the original on 6 December 2015. Retrieved 20 December 2015.
49. Ifrah, Georges (2000), p. 400.
50. Pannekoek, A. (1961). A History of Astronomy. George Allen & Unwin. p. 165.
51. Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith: Constantine to Dante – A.D. 325–1300, Simon & Schuster, ISBN 978-0-9650007-5-8, p. 241, "The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the Siddhantas – Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables."
52. Brezina, Corona (2006). Al-Khwarizmi: The Inventor of Algebra. The Rosen Publishing Group. ISBN 978-1-4042-0513-0. Archived from the original on 29 February 2020. Retrieved 26 September 2016.
53. Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith, Simon & Schuster, ISBN 978-0-9650007-5-8, p. 241, "In 976, Muhammad ibn Ahmad, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems called ṣifr, "empty" whence our cipher."
54. Sigler, L., Fibonacci's Liber Abaci. English translation, Springer, 2003.
55. Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly 11/1 (February 1973), pp. 99–104.
56. Hansen, Alice (2008-06-09). Primary Mathematics: Extending Knowledge in Practice. SAGE. ISBN 978-0-85725-233-3. Archived from the original on 7 March 2021. Retrieved 7 November 2020.
57. Lemma B.2.2, The integer 0 is even and is not odd, in Penner, Robert C. (1999). Discrete Mathematics: Proof Techniques and Mathematical Structures. World Scientific. p. 34. ISBN 978-981-02-4088-2.
58. W., Weisstein, Eric. "Zero". mathworld.wolfram.com. Archived from the original on 1 June 2013. Retrieved 4 April 2018.{{cite web}}: CS1 maint: multiple names: authors list (link)
59. Weil, Andre (2012-12-06). Number Theory for Beginners. Springer Science & Business Media. ISBN 978-1-4612-9957-8. Archived from the original on 14 June 2021. Retrieved 6 April 2021.
60. Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). The historical roots of elementary mathematics. Courier Dover Publications. pp. 254–255. ISBN 978-0-486-13968-5. Archived from the original on 23 June 2016. Retrieved 5 January 2016., Extract of pp. 254–255 Archived 10 May 2016 at the Wayback Machine
61. Reid, Constance (1992). From zero to infinity: what makes numbers interesting (4th ed.). Mathematical Association of America. p. 23. ISBN 978-0-88385-505-8. zero neither prime nor composite
62. "ResultSet (Java Platform SE 8 )". docs.oracle.com. Archived from the original on 9 May 2022. Retrieved 2022-05-09.
63. Wu, X.; Ichikawa, T.; Cercone, N. (25 October 1996). Knowledge-Base Assisted Database Retrieval Systems. World Scientific. ISBN 978-981-4501-75-0. Archived from the original on 31 March 2022. Retrieved 7 November 2020.
64. "Null values and the nullable type". IBM. 12 December 2018. Archived from the original on 23 November 2021. Retrieved 23 November 2021. In regard to services, sending a null value as an argument in a remote service call means that no data is sent. Because the receiving parameter is nullable, the receiving function creates a new, uninitialized value for the missing data then passes it to the requested service function.
65. Chris Woodford 2006, p. 9. sfn error: no target: CITEREFChris_Woodford2006 (help)
66. Paul DuBois. "MySQL Cookbook: Solutions for Database Developers and Administrators" Archived 24 February 2017 at the Wayback Machine 2014. p. 204.
67. Arnold Robbins; Nelson Beebe. "Classic Shell Scripting" Archived 24 February 2017 at the Wayback Machine. 2005. p. 274
68. Iztok Fajfar. "Start Programming Using HTML, CSS, and JavaScript" Archived 24 February 2017 at the Wayback Machine. 2015. p. 160.
69. Darren R. Hayes. "A Practical Guide to Computer Forensics Investigations" Archived 24 February 2017 at the Wayback Machine. 2014. p. 399
70. "Font Survey: 42 of the Best Monospaced Programming Fonts". codeproject.com. 18 August 2010. Archived from the original on 24 January 2012. Retrieved 22 July 2021.
71. Cepelewicz, Jordana Animals Count and Use Zero. How Far Does Their Number Sense Go? Archived 18 August 2021 at the Wayback Machine, Quanta, August 9, 2021
72. ""Zero euro" banknote creator Richard FAILLE strikes again!". 25 June 2017. Archived from the original on 25 April 2023. Retrieved 25 April 2023.
73. Bemer, R. W. (1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh". Communications of the ACM. 10 (8): 513–518. doi:10.1145/363534.363563. S2CID 294510.
74. Steel, Duncan (2000). Marking time: the epic quest to invent the perfect calendar. John Wiley & Sons. p. 113. ISBN 978-0-471-29827-4. In the B.C./A.D. scheme there is no year zero. After 31 December 1 BC came 1 January AD 1. ... If you object to that no-year-zero scheme, then don't use it: use the astronomer's counting scheme, with negative year numbers.
Bibliography
• Aczel, Amir D. (2015). Finding Zero. New York: Palgrave Macmillan. ISBN 978-1-137-27984-2.
• Asimov, Isaac (1978). "Nothing Counts". Asimov on Numbers. New York: Pocket Books. ISBN 978-0-671-82134-0. OCLC 1105483009.
• Barrow, John D. (2001). The Book of Nothing. Vintage. ISBN 0-09-928845-1.
• Woodford, Chris (2006). Digital Technology. Evans Brothers. ISBN 978-0-237-52725-9. Archived from the original on 17 August 2019. Retrieved 24 March 2016.
Historical studies
• Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3-540-64767-8.
• Diehl, Richard A. (2004). The Olmecs: America's First Civilization. London: Thames & Hudson.
• Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 0-471-39340-1.
• Kaplan, Robert (2000). The Nothing That Is: A Natural History of Zero. Oxford University Press.
• Seife, Charles (2000). Zero: The Biography of a Dangerous Idea. Penguin USA. ISBN 0-14-029647-6.
External links
Look up zero in Wiktionary, the free dictionary.
• Searching for the World's First Zero
• A History of Zero
• Zero Saga
• The History of Algebra
• Edsger W. Dijkstra: Why numbering should start at zero, EWD831 (PDF of a handwritten manuscript)
• Zero on In Our Time at the BBC
• Weisstein, Eric W. "0". MathWorld.
• Texts on Wikisource:
• "Zero". Encyclopædia Britannica (11th ed.). 1911.
• "Zero". Encyclopedia Americana. 1920.
Integers
0s
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700s
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800s
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900s
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≥1000
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Wikipedia
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Segal category
In mathematics, a Segal category is a model of an infinity category introduced by Hirschowitz & Simpson (1998), based on work of Graeme Segal in 1974.
References
• Hirschowitz, André; Simpson, Carlos (1998). "Descente pour les n-champs" (in French). arXiv:math/9807049.
• Joyal, A. (2008), The theory of quasi-categories and its applications, lectures at CRM Barcelona (PDF), pp. 164–169, archived from the original (PDF) on 2011-07-06
External links
• Segal category at the nLab
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Wikipedia
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Segal's conjecture
Segal's Burnside ring conjecture, or, more briefly, the Segal conjecture, is a theorem in homotopy theory, a branch of mathematics. The theorem relates the Burnside ring of a finite group G to the stable cohomotopy of the classifying space BG. The conjecture was made in the mid 1970s by Graeme Segal and proved in 1984 by Gunnar Carlsson. As of 2016, this statement is still commonly referred to as the Segal conjecture, even though it now has the status of a theorem.
Statement of the theorem
The Segal conjecture has several different formulations, not all of which are equivalent. Here is a weak form: there exists, for every finite group G, an isomorphism
$\varprojlim \pi _{S}^{0}\left(BG_{+}^{(k)}\right)\to {\widehat {A}}(G).$
Here, lim denotes the inverse limit, πS* denotes the stable cohomotopy ring, B denotes the classifying space, the superscript k denotes the k-skeleton, and the subscript + denotes the addition of a disjoint basepoint. On the right-hand side, the hat denotes the completion of the Burnside ring with respect to its augmentation ideal.
The Burnside ring
The Burnside ring of a finite group G is constructed from the category of finite G-sets as a Grothendieck group. More precisely, let M(G) be the commutative monoid of isomorphism classes of finite G-sets, with addition the disjoint union of G-sets and identity element the empty set (which is a G-set in a unique way). Then A(G), the Grothendieck group of M(G), is an abelian group. It is in fact a free abelian group with basis elements represented by the G-sets G/H, where H varies over the subgroups of G. (Note that H is not assumed here to be a normal subgroup of G, for while G/H is not a group in this case, it is still a G-set.) The ring structure on A(G) is induced by the direct product of G-sets; the multiplicative identity is the (isomorphism class of any) one-point set, which becomes a G-set in a unique way.
The Burnside ring is the analogue of the representation ring in the category of finite sets, as opposed to the category of finite-dimensional vector spaces over a field (see motivation below). It has proven to be an important tool in the representation theory of finite groups.
The classifying space
Main article: Classifying space
For any topological group G admitting the structure of a CW-complex, one may consider the category of principal G-bundles. One can define a functor from the category of CW-complexes to the category of sets by assigning to each CW-complex X the set of principal G-bundles on X. This functor descends to a functor on the homotopy category of CW-complexes, and it is natural to ask whether the functor so obtained is representable. The answer is affirmative, and the representing object is called the classifying space of the group G and typically denoted BG. If we restrict our attention to the homotopy category of CW-complexes, then BG is unique. Any CW-complex that is homotopy equivalent to BG is called a model for BG.
For example, if G is the group of order 2, then a model for BG is infinite-dimensional real projective space. It can be shown that if G is finite, then any CW-complex modelling BG has cells of arbitrarily large dimension. On the other hand, if G = Z, the integers, then the classifying space BG is homotopy equivalent to the circle S1.
Motivation and interpretation
The content of the theorem becomes somewhat clearer if it is placed in its historical context. In the theory of representations of finite groups, one can form an object $R[G]$ called the representation ring of $G$ in a way entirely analogous to the construction of the Burnside ring outlined above. The stable cohomotopy is in a sense the natural analog to complex K-theory, which is denoted $KU^{*}$. Segal was inspired to make his conjecture after Michael Atiyah proved the existence of an isomorphism
$KU^{0}(BG)\to {\widehat {R}}[G]$
which is a special case of the Atiyah–Segal completion theorem.
References
• Adams, J. Frank (1980). "Graeme Segal's Burnside ring conjecture". Topology Symposium, Siegen 1979. Lecture Notes in Mathematics. Vol. 788. Berlin: Springer. pp. 378–395. MR 0585670.
• Carlsson, Gunnar (1984). "Equivariant stable homotopy and Segal's Burnside ring conjecture". Annals of Mathematics. 120 (2): 189–224. doi:10.2307/2006940. JSTOR 2006940. MR 0763905.
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Segal space
In mathematics, a Segal space is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More precisely, a simplicial set, considered as a simplicial discrete space, satisfies the Segal conditions iff it is the nerve of a category. The condition for Segal spaces is a homotopical version of this.
Complete Segal spaces were introduced by Rezk (2001) as models for (∞, 1)-categories.
References
• Rezk, Charles (2001), "A model for the homotopy theory of homotopy theory", Transactions of the American Mathematical Society, 353 (3): 973–1007, doi:10.1090/S0002-9947-00-02653-2, ISSN 0002-9947, MR 1804411
External links
• Segal space at the nLab
• Complete Segal space at the nLab
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Segal–Bargmann space
In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions F in n complex variables satisfying the square-integrability condition:
$\|F\|^{2}:=\pi ^{-n}\int _{\mathbb {C} ^{n}}|F(z)|^{2}\exp(-|z|^{2})\,dz<\infty ,$
where here dz denotes the 2n-dimensional Lebesgue measure on $\mathbb {C} ^{n}.$ It is a Hilbert space with respect to the associated inner product:
$\langle F\mid G\rangle =\pi ^{-n}\int _{\mathbb {C} ^{n}}{\overline {F(z)}}G(z)\exp(-|z|^{2})\,dz.$
The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see Bargmann (1961) and Segal (1963). Basic information about the material in this section may be found in Folland (1989) and Hall (2000) . Segal worked from the beginning in the infinite-dimensional setting; see Baez, Segal & Zhou (1992) and Section 10 of Hall (2000) for more information on this aspect of the subject.
Properties
A basic property of this space is that pointwise evaluation is continuous, meaning that for each $a\in \mathbb {C} ^{n},$ there is a constant C such that
$|F(a)|<C\|F\|.$
It then follows from the Riesz representation theorem that there exists a unique Fa in the Segal–Bargmann space such that
$F(a)=\langle F_{a}\mid F\rangle .$
The function Fa may be computed explicitly as
$F_{a}(z)=\exp({\overline {a}}\cdot z)$
where, explicitly,
${\overline {a}}\cdot z=\sum _{j=1}^{n}{\overline {a_{j}}}z_{j}.$
The function Fa is called the coherent state (applied in mathematical physics) with parameter a, and the function
$\kappa (a,z):={\overline {F_{a}(z)}}$
is known as the reproducing kernel for the Segal–Bargmann space. Note that
$F(a)=\langle F_{a}\mid F\rangle =\pi ^{-n}\int _{\mathbb {C} ^{n}}\kappa (a,z)F(z)\exp(-|z|^{2})\,dz,$
meaning that integration against the reproducing kernel simply gives back (i.e., reproduces) the function F, provided, of course that F is an element of the space (and in particular is holomorphic).
Note that
$\|F_{a}\|^{2}=\langle F_{a}\mid F_{a}\rangle =F_{a}(a)=\exp(|a|^{2}).$
It follows from the Cauchy–Schwarz inequality that elements of the Segal–Bargmann space satisfy the pointwise bounds
$|F(a)|\leq \|F_{a}\|\|F\|=\exp(|a|^{2}/2)\|F\|.$
Quantum mechanical interpretation
One may interpret a unit vector in the Segal–Bargmann space as the wave function for a quantum particle moving in $\mathbb {R} ^{n}.$ In this view, $\mathbb {C} ^{n}$ plays the role of the classical phase space, whereas $\mathbb {R} ^{n}$ is the configuration space. The restriction that F be holomorphic is essential to this interpretation; if F were an arbitrary square-integrable function, it could be localized into an arbitrarily small region of the phase space, which would go against the uncertainty principle. Since, however, F is required to be holomorphic, it satisfies the pointwise bounds described above, which provides a limit on how concentrated F can be in any region of phase space.
Given a unit vector F in the Segal–Bargmann space, the quantity
$\pi ^{-n}|F(z)|^{2}\exp(-|z|^{2})$
may be interpreted as a sort of phase space probability density for the particle. Since the above quantity is manifestly non-negative, it cannot coincide with the Wigner function of the particle, which usually has some negative values. In fact, the above density coincides with the Husimi function of the particle, which is obtained from the Wigner function by smearing with a Gaussian. This connection will be made more precise below, after we introduce the Segal–Bargmann transform.
The canonical commutation relations
One may introduce annihilation operators $a_{j}$ and creation operators $a_{j}^{*}$ on the Segal–Bargmann space by setting
$a_{j}=\partial /\partial z_{j}$
and
$a_{j}^{*}=z_{j}$
These operators satisfy the same relations as the usual creation and annihilation operators, namely, the $a_{j}$ and $a_{j}^{*}$ commute among themselves and
$\left[a_{j},a_{k}^{*}\right]=\delta _{j,k}$
Furthermore, the adjoint of $a_{j}$ with respect to the Segal–Bargmann inner product is $a_{j}^{*}.$ (This is suggested by the notation, but not at all obvious from the formulas for $a_{j}$ and $a_{j}^{*}$!) Indeed, Bargmann was led to introduce the particular form of the inner product on the Segal–Bargmann space precisely so that the creation and annihilation operators would be adjoints of each other.
We may now construct self-adjoint "position" and "momentum" operators Aj and Bj by the formulas:
$A_{j}=(a_{j}+a_{j}^{*})/{\sqrt {2}}$
$B_{j}=(a_{j}-a_{j}^{*})/(i{\sqrt {2}})$
These operators satisfy the ordinary canonical commutation relations, and it can be shown that that they act irreducibly on the Segal–Bargmann space; see Section 14.4 of Hall (2013).
The Segal–Bargmann transform
Since the operators Aj and Bj from the previous section satisfy the Weyl relations and act irreducibly on the Segal–Bargmann space, the Stone–von Neumann theorem applies. Thus, there is a unitary map B from the position Hilbert space $L^{2}(\mathbb {R} ^{n})$ to the Segal–Bargmann space that intertwines these operators with the usual position and momentum operators.
The map B may be computed explicitly as a modified double Weierstrass transform,
$(Bf)(z)=\int _{\mathbb {R} ^{n}}\exp[-(z\cdot z-2{\sqrt {2}}z\cdot x+x\cdot x)/2]f(x)\,dx,$
where dx is the n-dimensional Lebesgue measure on $\mathbb {R} ^{n}$ and where z is in $\mathbb {C} ^{n}.$ See Bargmann (1961) and Section 14.4 of Hall (2013). One can also describe (Bf)(z) as the inner product of f with an appropriately normalized coherent state with parameter z, where, now, we express the coherent states in the position representation instead of in the Segal–Bargmann space.
We may now be more precise about the connection between the Segal–Bargmann space and the Husimi function of a particle. If f is a unit vector in $L^{2}(\mathbb {R} ^{n}),$ then we may form a probability density on $\mathbb {C} ^{n}$ as
$\pi ^{-n}|(Bf)(z)|^{2}\exp(-|z|^{2})~.$
The claim is then that the above density is the Husimi function of f, which may be obtained from the Wigner function of f by convolving with a double Gaussian (the Weierstrass transform). This fact is easily verified by using the formula for Bf along with the standard formula for the Husimi function in terms of coherent states.
Since B is unitary, its Hermitian adjoint is its inverse. Recalling that the measure on $\mathbb {C} ^{n}$ is $e^{-|z|^{2}}\,dz$, we thus obtain one inversion formula for B as
$f(x)=\int _{\mathbb {C} ^{n}}\exp[-({\overline {z}}\cdot {\overline {z}}-2{\sqrt {2}}{\overline {z}}\cdot x+x\cdot x)/2](Bf)(z)e^{-|z|^{2}}\,dz.$
Since, however, Bf is a holomorphic function, there can be many integrals involving Bf that give the same value. (Think of the Cauchy integral formula.) Thus, there can be many different inversion formulas for the Segal–Bargmann transform B.
Another useful inversion formula is[1]
$f(x)=C\exp(-|x|^{2}/2)\int _{\mathbb {R} ^{n}}(Bf)(x+iy)\exp(-|y|^{2}/2)\,dy,$
where
$C=\pi ^{-n/4}(2\pi )^{-n/2}.$
This inversion formula may be understood as saying that the position "wave function" f may be obtained from the phase-space "wave function" Bf by integrating out the momentum variables. This is to be contrasted to the Wigner function, where the position probability density is obtained from the phase space (quasi-)probability density by integrating out the momentum variables.
Generalizations
There are various generalizations of the Segal–Bargmann space and transform. In one of these,[2][3] the role of the configuration space $\mathbb {R} ^{n}$ is played by the group manifold of a compact Lie group, such as SU(N). The role of the phase space $\mathbb {C} ^{n}$ is then played by the complexification of the compact Lie group, such as $\operatorname {SL} (N,\mathbb {C} )$ in the case of SU(N). The various Gaussians appearing in the ordinary Segal–Bargmann space and transform are replaced by heat kernels. This generalized Segal–Bargmann transform could be applied, for example, to the rotational degrees of freedom of a rigid body, where the configuration space is the compact Lie groups SO(3).
This generalized Segal–Bargmann transform gives rise to a system of coherent states, known as heat kernel coherent states. These have been used widely in the literature on loop quantum gravity.
See also
• Theta representation
• Hardy space
References
1. B.C. Hall, "The range of the heat operator", in The Ubiquitous Heat Kernel, edited by Jay Jorgensen and Lynne H. Walling, AMS 2006, pp. 203–231
2. B.C. Hall, "The Segal–Bargmann 'coherent state' transform for compact Lie groups", Journal of Functional Analysis 122 (1994), 103–151
3. B.C. Hall, "The inverse Segal–Bargmann transform for compact Lie groups", Journal of Functional Analysis 143 (1997), 98–116
Sources
• Bargmann, V. (1961), "On a Hilbert space of analytic functions and an associated integral transform", Communications on Pure and Applied Mathematics, 14 (3): 187, doi:10.1002/cpa.3160140303, hdl:10338.dmlcz/143587
• Segal, I. E. (1963), "Mathematical problems of relativistic physics", in Kac, M. (ed.), Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. II, Lectures in Applied Mathematics, American Mathematical Society, Chap. VI, LCCN 62-21480
• Folland, G. (1989), Harmonic Analysis in Phase Space, Princeton University Press, ISBN 978-0691085289
• Baez, J.; Segal, I. E.; Zhou, Z. (1992), Introduction to Algebraic and Constructive Quantum Field Theory, Princeton University Press, ISBN 978-0691605128
• Hall, B. C (2000), "Holomorphic methods in analysis and mathematical physics", in Pérez-Esteva, S.; Villegas-Blas, C. (eds.), First Summer School in Analysis and Mathematical Physics: Quantization, the Segal-Bargmann Transform and Semiclassical Analysis, Contemporary Mathematics, vol. 260, AMS, pp. 1–59, ISBN 978-0-8218-2115-2
• Hall, B. C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer Verlag, doi:10.1007/978-1-4614-7116-5, ISBN 978-1-4614-7115-8, S2CID 117837329
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