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Simplicial commutative ring
In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If A is a simplicial commutative ring, then it can be shown that $\pi _{0}A$ is a ring and $\pi _{i}A$ are modules over that ring (in fact, $\pi _{*}A$ is a graded ring over $\pi _{0}A$.)
A topology-counterpart of this notion is a commutative ring spectrum.
Examples
• The ring of polynomial differential forms on simplexes.
Graded ring structure
Let A be a simplicial commutative ring. Then the ring structure of A gives $\pi _{*}A=\oplus _{i\geq 0}\pi _{i}A$ the structure of a graded-commutative graded ring as follows.
By the Dold–Kan correspondence, $\pi _{*}A$ is the homology of the chain complex corresponding to A; in particular, it is a graded abelian group. Next, to multiply two elements, writing $S^{1}$ for the simplicial circle, let $x:(S^{1})^{\wedge i}\to A,\,\,y:(S^{1})^{\wedge j}\to A$ be two maps. Then the composition
$(S^{1})^{\wedge i}\times (S^{1})^{\wedge j}\to A\times A\to A$,
the second map the multiplication of A, induces $(S^{1})^{\wedge i}\wedge (S^{1})^{\wedge j}\to A$. This in turn gives an element in $\pi _{i+j}A$. We have thus defined the graded multiplication $\pi _{i}A\times \pi _{j}A\to \pi _{i+j}A$. It is associative because the smash product is. It is graded-commutative (i.e., $xy=(-1)^{|x||y|}yx$) since the involution $S^{1}\wedge S^{1}\to S^{1}\wedge S^{1}$ introduces a minus sign.
If M is a simplicial module over A (that is, M is a simplicial abelian group with an action of A), then the similar argument shows that $\pi _{*}M$ has the structure of a graded module over $\pi _{*}A$ (cf. Module spectrum).
Spec
By definition, the category of affine derived schemes is the opposite category of the category of simplicial commutative rings; an object corresponding to A will be denoted by $\operatorname {Spec} A$.
See also
• E_n-ring
References
• What is a simplicial commutative ring from the point of view of homotopy theory?
• What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?
• Reference request - CDGA vs. sAlg in char. 0
• A. Mathew, Simplicial commutative rings, I.
• B. Toën, Simplicial presheaves and derived algebraic geometry
• P. Goerss and K. Schemmerhorn, Model categories and simplicial methods
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Simplicial complex recognition problem
The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether it is homeomorphic to another fixed simplicial complex. The problem is undecidable for complexes of dimension 5 or more.[1][2]: 9–11
Background
An abstract simplicial complex (ASC) is family of sets that is closed under taking subsets (the subset of a set in the family is also a set in the family). Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each set with k elements in the ASC is mapped to a (k-1)-dimensional simplex in the GSC. Thus, an ASC provides a finite representation of a geometric object. Given an ASC, one can ask several questions regarding the topology of the GSC it represents.
Homeomorphism problem
The homeomorphism problem is: given two finite simplicial complexes representing smooth manifolds, decide if they are homeomorphic.
• If the complexes are of dimension at most 3, then the problem is decidable. This follows from the proof of the geometrization conjecture.
• For every d ≥ 4, the homeomorphism problem for d-dimensional simplicial complexes is undecidable.[3]
The same is true if "homeomorphic" is replaced with "piecewise-linear homeomorphic".
Recognition problem
The recognition problem is a sub-problem of the homeomorphism problem, in which one simplicial complex is given as a fixed parameter. Given another simplicial complex as an input, the problem is to decide whether it is homeomorphic to the given fixed complex.
• The recognition problem is decidable for the 3-dimensional sphere $S^{3}$.[4] That is, there is an algorithm that can decide whether any given simplicial complex is homeomorphic to the boundary of a 4-dimensional ball.
• The recognition problem is undecidable for the d-dimensional sphere $S^{d}$ for any d ≥ 5. The proof is by reduction to the word problem for groups. From this, it can be proved that the recognition problem is undecidable for any fixed compact d-dimensional manifold with d ≥ 5.
• As of 2014, it is open whether the recognition problem is decidable for the 4-dimensional sphere $S^{4}$.[2]: 11
Manifold problem
The manifold problem is: given a finite simplicial complex, is it homeomorphic to a manifold? The problem is undecidable; the proof is by reduction from the word problem for groups.[2]: 11
References
1. Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, ISBN 9780387979700.
2. Poonen, Bjorn (2014-10-25). "Undecidable problems: a sampler". arXiv:1204.0299 [math.LO].
3. "A. Markov, "The insolubility of the problem of homeomorphy", Dokl. Akad. Nauk SSSR, 121:2 (1958), 218–220". www.mathnet.ru. Retrieved 2022-11-27.
4. Matveev, Sergei (2003), Matveev, Sergei (ed.), "Algorithmic Recognition of S3", Algorithmic Topology and Classification of 3-Manifolds, Algorithms and Computation in Mathematics, Berlin, Heidelberg: Springer, vol. 9, pp. 193–214, doi:10.1007/978-3-662-05102-3_5, ISBN 978-3-662-05102-3, retrieved 2022-11-27
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Simplicial group
In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The Dold–Kan correspondence says that a simplicial abelian group may be identified with a chain complex. In fact it can be shown that any simplicial abelian group $A$ is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces, $\prod _{i\geq 0}K(\pi _{i}A,i).$[1]
A commutative monoid in the category of simplicial abelian groups is a simplicial commutative ring.
Eckmann (1945) discusses a simplicial analogue of the fact that a cohomology class on a Kähler manifold has a unique harmonic representative and deduces Kirchhoff's circuit laws from these observations.
References
1. Paul Goerss and Rick Jardine (1999, Ch 3. Proposition 2.20)
• Eckmann, Beno (1945), "Harmonische Funktionen und Randwertaufgaben in einem Komplex", Commentarii Mathematici Helvetici, 17: 240–255, doi:10.1007/BF02566245, MR 0013318
• Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.
• Charles Weibel, An introduction to homological algebra
External links
• simplicial group at the nLab
• What is a simplicial commutative ring from the point of view of homotopy theory?
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Simplicial homology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0).
Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead).[1]: sec.5.3.2
Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space.[2]: sec.8.6 As a result, it gives a computable way to distinguish one space from another.
Definitions
Orientations
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v0,...,vk), with the rule that two orderings define the same orientation if and only if they differ by an even permutation. Thus every simplex has exactly two orientations, and switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1-simplex amounts to choosing one of the two possible directions, and choosing an orientation of a 2-simplex amounts to choosing what "counterclockwise" should mean.
Chains
Let S be a simplicial complex. A simplicial k-chain is a finite formal sum
$\sum _{i=1}^{N}c_{i}\sigma _{i},\,$
where each ci is an integer and σi is an oriented k-simplex. In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation. For example,
$(v_{0},v_{1})=-(v_{1},v_{0}).$
The group of k-chains on S is written Ck. This is a free abelian group which has a basis in one-to-one correspondence with the set of k-simplices in S. To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices.
Boundaries and cycles
Let σ = (v0,...,vk) be an oriented k-simplex, viewed as a basis element of Ck. The boundary operator
$\partial _{k}:C_{k}\rightarrow C_{k-1}$
is the homomorphism defined by:
$\partial _{k}(\sigma )=\sum _{i=0}^{k}(-1)^{i}(v_{0},\dots ,{\widehat {v_{i}}},\dots ,v_{k}),$
where the oriented simplex
$(v_{0},\dots ,{\widehat {v_{i}}},\dots ,v_{k})$
is the ith face of σ, obtained by deleting its ith vertex.
In Ck, elements of the subgroup
$Z_{k}:=\ker \partial _{k}$
are referred to as cycles, and the subgroup
$B_{k}:=\operatorname {im} \partial _{k+1}$
is said to consist of boundaries.
Boundaries of boundaries
Because $(-1)^{i+j-1}(v_{0},\dots ,{\widehat {v_{i}}},\dots ,{\widehat {\widehat {v_{j}}}},\dots ,v_{k})=-(-1)^{i+j}(v_{0},\dots ,{\widehat {\widehat {v_{i}}}},\dots ,{\widehat {v_{j}}},\dots ,v_{k})$, where ${\widehat {\widehat {v_{x}}}}$ is the second face removed, $\partial ^{2}=0$. In geometric terms, this says that the boundary of anything has no boundary. Equivalently, the abelian groups
$(C_{k},\partial _{k})$
form a chain complex. Another equivalent statement is that Bk is contained in Zk.
As an example, consider a tetrahedron with vertices oriented as w,x,y,z. By definition, its boundary is given by: xyz - wyz + wxz - wxy. The boundary of the boundary is given by: (yz-xz+xy)-(yz-wz+wy)+(xz-wz+wx)-(xy-wy+wx) = 0.
Homology groups
The kth homology group Hk of S is defined to be the quotient abelian group
$H_{k}(S)=Z_{k}/B_{k}\,.$
It follows that the homology group Hk(S) is nonzero exactly when there are k-cycles on S which are not boundaries. In a sense, this means that there are k-dimensional holes in the complex. For example, consider the complex S obtained by gluing two triangles (with no interior) along one edge, shown in the image. The edges of each triangle can be oriented so as to form a cycle. These two cycles are by construction not boundaries (since every 2-chain is zero). One can compute that the homology group H1(S) is isomorphic to Z2, with a basis given by the two cycles mentioned. This makes precise the informal idea that S has two "1-dimensional holes".
Holes can be of different dimensions. The rank of the kth homology group, the number
$\beta _{k}=\operatorname {rank} (H_{k}(S))\,$
is called the kth Betti number of S. It gives a measure of the number of k-dimensional holes in S.
Example
Homology groups of a triangle
Let S be a triangle (without its interior), viewed as a simplicial complex. Thus S has three vertices, which we call v0, v1, v2, and three edges, which are 1-dimensional simplices. To compute the homology groups of S, we start by describing the chain groups Ck:
• C0 is isomorphic to Z3 with basis (v0), (v1), (v2),
• C1 is isomorphic to Z3 with a basis given by the oriented 1-simplices (v0, v1), (v0, v2), and (v1, v2).
• C2 is the trivial group, since there is no simplex like $(v_{0},v_{1},v_{2})$ because the triangle has been supposed without its interior. So are the chain groups in other dimensions.
The boundary homomorphism ∂: C1 → C0 is given by:
$\partial (v_{0},v_{1})=(v_{1})-(v_{0})$
$\partial (v_{0},v_{2})=(v_{2})-(v_{0})$
$\partial (v_{1},v_{2})=(v_{2})-(v_{1})$
Since C−1 = 0, every 0-chain is a cycle (i.e. Z0 = C0); moreover, the group B0 of the 0-boundaries is generated by the three elements on the right of these equations, creating a two-dimensional subgroup of C0. So the 0th homology group H0(S) = Z0/B0 is isomorphic to Z, with a basis given (for example) by the image of the 0-cycle (v0). Indeed, all three vertices become equal in the quotient group; this expresses the fact that S is connected.
Next, the group of 1-cycles is the kernel of the homomorphism ∂ above, which is isomorphic to Z, with a basis given (for example) by (v0,v1) − (v0,v2) + (v1,v2). (A picture reveals that this 1-cycle goes around the triangle in one of the two possible directions.) Since C2 = 0, the group of 1-boundaries is zero, and so the 1st homology group H1(S) is isomorphic to Z/0 ≅ Z. This makes precise the idea that the triangle has one 1-dimensional hole.
Next, since by definition there are no 2-cycles, C2 = 0 (the trivial group). Therefore the 2nd homology group H2(S) is zero. The same is true for Hi(S) for all i not equal to 0 or 1. Therefore, the homological connectivity of the triangle is 0 (it is the largest k for which the reduced homology groups up to k are trivial).
Homology groups of higher-dimensional simplices
Let S be a tetrahedron (without its interior), viewed as a simplicial complex. Thus S has four 0-dimensional vertices, six 1-dimensional edges, and four 2-dimensional faces. The construction of the homology groups of a tetrahedron is described in detail here.[3] It turns out that H0(S) is isomorphic to Z, H2(S) is isomorphic to Z too, and all other groups are trivial.Therefore, the homological connectivity of the tetrahedron is 0.
If the tetrahedron contains its interior, then H2(S) is trivial too.
In general, if S is a d-dimensional simplex, the following holds:
• If S is considered without its interior, then H0(S) = Z and Hd−1(S) = Z and all other homologies are trivial;
• If S is considered with its interior, then H0(S) = Z and all other homologies are trivial.
Simplicial maps
Main article: Simplicial map
Let S and T be simplicial complexes. A simplicial map f from S to T is a function from the vertex set of S to the vertex set of T such that the image of each simplex in S (viewed as a set of vertices) is a simplex in T. A simplicial map f: S → T determines a homomorphism of homology groups Hk(S) → Hk(T) for each integer k. This is the homomorphism associated to a chain map from the chain complex of S to the chain complex of T. Explicitly, this chain map is given on k-chains by
$f((v_{0},\ldots ,v_{k}))=(f(v_{0}),\ldots ,f(v_{k}))$
if f(v0), ..., f(vk) are all distinct, and otherwise f((v0, ..., vk)) = 0.
This construction makes simplicial homology a functor from simplicial complexes to abelian groups. This is essential to applications of the theory, including the Brouwer fixed point theorem and the topological invariance of simplicial homology.
Related homologies
Singular homology is a related theory that is better adapted to theory rather than computation. Singular homology is defined for all topological spaces and depends only on the topology, not any triangulation; and it agrees with simplicial homology for spaces which can be triangulated.[4]: thm.2.27 Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as image analysis, medical imaging, and data analysis in general.
Another related theory is Cellular homology.
Applications
A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.) in which one wishes to find a topological feature. Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex. However, the data points have to first be triangulated, meaning one replaces the data with a simplicial complex approximation. Computation of persistent homology[5] involves analysis of homology at different resolutions, registering homology classes (holes) that persist as the resolution is changed. Such features can be used to detect structures of molecules, tumors in X-rays, and cluster structures in complex data.
More generally, simplicial homology plays a central role in topological data analysis, a technique in the field of data mining.
Implementations
• A MATLAB toolbox for computing persistent homology, Plex (Vin de Silva, Gunnar Carlsson), is available at this site.
• Stand-alone implementations in C++ are available as part of the Perseus, Dionysus and PHAT software projects.
• For Python, there are libraries such as scikit-tda, Persim, giotto-tda and GUDHI, the latter aimed at generating topological features for machine learning. These can be found at the PyPI repository.
See also
• Simplicial homotopy
References
1. Prasolov, V. V. (2006), Elements of combinatorial and differential topology, American Mathematical Society, ISBN 0-8218-3809-1, MR 2233951
2. Armstrong, M. A. (1983), Basic topology, Springer-Verlag, ISBN 0-387-90839-0, MR 0705632
3. Wildberger, Norman J. (2012). "More homology computations". Archived from the original on 2021-12-22.
4. Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 0-521-79540-0, MR 1867354
5. Edelsbrunner, H.; Letscher, D.; Zomorodian, A. (2002). "Topological Persistence and Simplification". Discrete & Computational Geometry. 28: 511–533. doi:10.1007/s00454-002-2885-2.
Robins, V. (Summer 1999). "Towards computing homology from finite approximations" (PDF). Topology Proceedings. 24: 503–532.
External links
• Topological methods in scientific computing
• Computational homology (also cubical homology)
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Simplicial homotopy
In algebraic topology, a simplicial homotopy[1]pg 23 is an analog of a homotopy between topological spaces for simplicial sets. If
$f,g:X\to Y$
are maps between simplicial sets, a simplicial homotopy from f to g is a map
$h:X\times \Delta ^{1}\to Y$
such that the diagram (see ) formed by f, g and h commute; the key is to use the diagram that results in $f(x)=h(x,0)$ and $g(x)=h(x,1)$ for all x in X.
See also
• Kan complex
• Dold–Kan correspondence (under which a chain homotopy corresponds to a simplicial homotopy)
• Simplicial homology
References
1. Goerss, Paul G.; Jardin, John F. (2009). Simplicial Homotopy Theory. Birkhäuser Basel. ISBN 978-3-0346-0188-7. OCLC 837507571.
External links
• http://ncatlab.org/nlab/show/simplicial+homotopy
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Link (simplicial complex)
The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local structure of the complex at the vertex.
Link of a vertex
Given an abstract simplicial complex X and $ v$ a vertex in $ V(X)$, its link $ \operatorname {Lk} (v,X)$ is a set containing every face $ \tau \in X$ such that $ v\not \in \tau $ and $ \tau \cup \{v\}$ is a face of X.
• In the special case in which X is a 1-dimensional complex (that is: a graph), $ \operatorname {Lk} (v,X)$ contains all vertices $ u\neq v$ such that $ \{u,v\}$ is an edge in the graph; that is, $ \operatorname {Lk} (v,X)={\mathcal {N}}(v)=$the neighborhood of $ v$ in the graph.
Given a geometric simplicial complex X and $ v\in V(X)$, its link $ \operatorname {Lk} (v,X)$ is a set containing every face $ \tau \in X$ such that $ v\not \in \tau $ and there is a simplex in $ X$ that has $ v$ as a vertex and $ \tau $ as a face.[1]: 3 Equivalently, the join $ v\star \tau $ is a face in $ X$.[2]: 20
• As an example, suppose v is the top vertex of the tetrahedron at the left. Then the link of v is the triangle at the base of the tetrahedron. This is because, for each edge of that triangle, the join of v with the edge is a triangle (one of the three triangles at the sides of the tetrahedron); and the join of v with the triangle itself is the entire tetrahedron.
An alternative definition is: the link of a vertex $ v\in V(X)$ is the graph Lk(v, X) constructed as follows. The vertices of Lk(v, X) are the edges of X incident to v. Two such edges are adjacent in Lk(v, X) iff they are incident to a common 2-cell at v.
• The graph Lk(v, X) is often given the topology of a ball of small radius centred at v; it is an analog to a sphere centered at a point.[3]
Link of a face
The definition of a link can be extended from a single vertex to any face.
Given an abstract simplicial complex X and any face $ \sigma $ of X, its link $ \operatorname {Lk} (\sigma ,X)$ is a set containing every face $ \tau \in X$ such that $ \sigma ,\tau $ are disjoint and $ \tau \cup \sigma $ is a face of X: $ \operatorname {Lk} (\sigma ,X):=\{\tau \in X:~\tau \cap \sigma =\emptyset ,~\tau \cup \sigma \in X\}$.
Given a geometric simplicial complex X and any face $ \sigma \in X$, its link $ \operatorname {Lk} (\sigma ,X)$ is a set containing every face $ \tau \in X$ such that $ \sigma ,\tau $ are disjoint and there is a simplex in $ X$ that has both $ v$ and $ \tau $ as faces.[1]: 3
Examples
The link of a vertex of a tetrahedron is a triangle – the three vertices of the link corresponds to the three edges incident to the vertex, and the three edges of the link correspond to the faces incident to the vertex. In this example, the link can be visualized by cutting off the vertex with a plane; formally, intersecting the tetrahedron with a plane near the vertex – the resulting cross-section is the link.
Another example is illustrated below. There is a two-dimensional simplicial complex. At the left, a vertex is marked in yellow. At the right, the link of that vertex is marked in green.
• A vertex and its link.
Properties
• For any simplicial complex X, every link $ \operatorname {Lk} (\sigma ,X)$ is downward-closed, and therefore it is a simplicial complex too; it is a sub-complex of X.
• Because X is simplicial, there is a set isomorphism between $ \operatorname {Lk} (\sigma ,X)$ and the set $X_{\sigma }:=\{\rho \in X{\text{ such that }}\sigma \subseteq \rho \}$: every $ \tau \in \operatorname {Lk} (\sigma ,X)$ corresponds to $ \tau \cup \sigma $, which is in $X_{\sigma }$.
Link and star
A concept closely related to the link is the star.
Given an abstract simplicial complex X and any face $ \sigma \in X$,$ V(X)$, its star $ \operatorname {St} (\sigma ,X)$ is a set containing every face $ \tau \in X$ such that $ \tau \cup \sigma $ is a face of X. In the special case in which X is a 1-dimensional complex (that is: a graph), $ \operatorname {St} (v,X)$ contains all edges $ \{u,v\}$ for all vertices $ u$ that are neighbors of $ v$. That is, it is a graph-theoretic star centered at $ u$.
Given a geometric simplicial complex X and any face $ \sigma \in X$, its star $ \operatorname {St} (\sigma ,X)$ is a set containing every face $ \tau \in X$ such that there is a simplex in $ X$ having both $ \sigma $ and $ \tau $ as faces: $ \operatorname {St} (\sigma ,X):=\{\tau \in X:\exists \rho \in X:\tau ,\sigma {\text{ are faces of }}\rho \}$. In other words, it is the closure of the set $ \{\rho \in X:\sigma {\text{ is a face of }}\rho \}$ -- the set of simplices having $ \sigma $ as a face.
So the link is a subset of the star. The star and link are related as follows:
• For any $ \sigma \in X$, $ \operatorname {Lk} (\sigma ,X)=\{\tau \in \operatorname {St} (\sigma ,X):\tau \cap \sigma =\emptyset \}$. [1]: 3
• For any $ v\in V(X)$, $ \operatorname {St} (v,X)=v\star \operatorname {Lk} (v,X)$, that is, the star of $ v$ is the cone of its link at $ v$.[2]: 20
An example is illustrated below. There is a two-dimensional simplicial complex. At the left, a vertex is marked in yellow. At the right, the star of that vertex is marked in green.
• A vertex and its star.
See also
• Vertex figure - a geometric concept similar to the simplicial link.
References
1. Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15
2. C. P. Rourke and B. J. Sanderson (1972). Introduction to Piecewise-Linear Topology. doi:10.1007/978-3-642-81735-9. ISBN 978-3-540-11102-3.
3. Bridson, Martin; Haefliger, André (1999), Metric spaces of non-positive curvature, Springer, ISBN 3-540-64324-9
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Simplicial localization
In category theory, a branch of mathematics, the simplicial localization of a category C with respect to a class W of morphisms of C is a simplicial category LC whose $\pi _{0}$ is the localization $C[W^{-1}]$ of C with respect to W; that is, $\pi _{0}LC(x,y)=C[W^{-1}](x,y)$ for any objects x, y in C. The notion is due to Dwyer and Kan.
References
• W. G. Dwyer and Dan Kan, Simplicial localizations of categories Archived 2014-03-24 at the Wayback Machine
• http://math.mit.edu/~mdono/_Juvitop.pdf Archived 2013-11-05 at the Wayback Machine
External links
• simplicial localization at the nLab
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Simplicial manifold
In physics, the term simplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study of Regge calculus. These objects combine attributes of a simplex with those of a manifold. There is no standard usage of this term in mathematics, and so the concept can refer to a triangulation in topology, or a piecewise linear manifold, or one of several different functors from either the category of sets or the category of simplicial sets to the category of manifolds.
Not to be confused with Symplectic manifold.
A manifold made out of simplices
A simplicial manifold is a simplicial complex for which the geometric realization is homeomorphic to a topological manifold. This is essentially the concept of a triangulation in topology. This can mean simply that a neighborhood of each vertex (i.e. the set of simplices that contain that point as a vertex) is homeomorphic to a n-dimensional ball.
A simplicial object built from manifolds
A simplicial manifold is also a simplicial object in the category of manifolds. This is a special case of a simplicial space in which, for each n, the space of n-simplices is a manifold.
For example, if G is a Lie group, then the simplicial nerve of G has the manifold $G^{n}$ as its space of n-simplices. More generally, G can be a Lie groupoid.
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Simplicial map
A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex.[1] Simplicial maps can be used to approximate continuous functions between topological spaces that can be triangulated; this is formalized by the simplicial approximation theorem.
A simplicial isomorphism is a bijective simplicial map such that both it and its inverse are simplicial.
Definitions
A simplicial map is defined in slightly different ways in different contexts.
Abstract simplicial complexes
Let K and L be two abstract simplicial complexes (ASC). A simplicial map of K into L is a function from the vertices of K to the vertices of L, $f:V(K)\to V(L)$, that maps every simplex in K to a simplex in L. That is, for any $\sigma \in K$, $f(\sigma )\in L$.[2]: 14, Def.1.5.2 As an example, let K be ASC containing the sets {1,2},{2,3},{3,1} and their subsets, and let L be the ASC containing the set {4,5,6} and its subsets. Define a mapping f by: f(1)=f(2)=4, f(3)=5. Then f is a simplicial mapping, since f({1,2})={4} which is a simplex in L, f({2,3})=f({3,1})={4,5} which is also a simplex in L, etc.
If $f$ is not bijective, it may map k-dimensional simplices in K to l-dimensional simplices in L, for any l ≤ k. In the above example, f maps the one-dimensional simplex {1,2} to the zero-dimensional simplex {4}.
If $f$ is bijective, and its inverse $f^{-1}$ is a simplicial map of L into K, then $f$ is called a simplicial isomorphism. Isomorphic simplicial complexes are essentially "the same", up ro a renaming of the vertices. The existence of an isomorphism between L and K is usually denoted by $K\cong L$.[2]: 14 The function f defined above is not an isomorphism since it is not bijective. If we modify the definition to f(1)=4, f(2)=5, f(3)=6, then f is bijective but it is still not an isomorphism, since $f^{-1}$ is not simplicial: $f^{-1}(\{4,5,6\})=\{1,2,3\}$, which is not a simplex in K. If we modify L by removing {4,5,6}, that is, L is the ASC containing only the sets {4,5},{5,6},{6,4} and their subsets, then f is an isomorphism.
Geometric simplicial complexes
Let K and L be two geometric simplicial complexes (GSC). A simplicial map of K into L is a function $f:K\to L$ such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex $\sigma \in K$, $\operatorname {conv} (f(V(\sigma )))\in L$. Note that this implies that vertices of K are mapped to vertices of L. [1]
Equivalently, one can define a simplicial map as a function from the underlying space of K (the union of simplices in K) to the underlying space of L, $f:|K|\to |L|$, that maps every simplex in K linearly to a simplex in L. That is, for any simplex $\sigma \in K$, $f(\sigma )\in L$, and in addition, $f\vert _{\sigma }$ (the restriction of $f$ to $\sigma $) is a linear function.[3]: 16 [4]: 3 Every simplicial map is continuous.
Simplicial maps are determined by their effects on vertices. In particular, there are a finite number of simplicial maps between two given finite simplicial complexes.
A simplicial map between two ASCs induces a simplicial map between their geometric realizations (their underlying polyhedra) using barycentric coordinates. This can be defined precisely.[2]: 15, Def.1.5.3 Let K, L be to ASCs, and let $f:V(K)\to V(L)$ be a simplicial map. The affine extension of $f$ is a mapping $|f|:|K|\to |L|$ defined as follows. For any point $x\in |K|$, let $\sigma $ be its support (the unique simplex containing x in its interior), and denote the vertices of $\sigma $ by $v_{0},\ldots ,v_{k}$. The point $x$ has a unique representation as a convex combination of the vertices, $x=\sum _{i=0}^{k}a_{i}v_{i}$ with $a_{i}\geq 0$ and $\sum _{i=0}^{k}a_{i}=1$ (the $a_{i}$ are the barycentric coordinates of $x$). We define $|f|(x):=\sum _{i=0}^{k}a_{i}f(v_{i})$. This |f| is a simplicial map of |K| into |L|; it is a continuous function. If f is injective, then |f| is injective; if f is an isomorphism between K and L, then |f| is a homeomorphism between |K| and |L|.[2]: 15, Prop.1.5.4
Simplicial approximation
Let $f\colon |K|\to |L|$ be a continuous map between the underlying polyhedra of simplicial complexes and let us write ${\text{st}}(v)$ for the star of a vertex. A simplicial map $f_{\triangle }\colon K\to L$ such that $f({\text{st}}(v))\subseteq {\text{st}}(f_{\triangle }(v))$, is called a simplicial approximation to $f$.
A simplicial approximation is homotopic to the map it approximates. See simplicial approximation theorem for more details.
Piecewise-linear maps
Let K and L be two GSCs. A function $f:|K|\to |L|$ is called piecewise-linear (PL) if there exist a subdivision K' of K, and a subdivision L' of L, such that $f:|K'|\to |L'|$ is a simplicial map of K' into L'. Every simplicial map is PL, but the opposite is not true. For example, suppose |K| and |L| are two triangles, and let $f:|K|\to |L|$ be a non-linear function that maps the leftmost half of |K| linearly into the leftmost half of |L|, and maps the rightmost half of |K| linearly into the rightmostt half of |L|. Then f is PL, since it is a simplicial map between a subdivision of |K| into two triangles and a subdivision of |L| into two triangles. This notion is an adaptation of the general notion of a piecewise-linear function to simplicial complexes.
A PL homeomorphism between two polyhedra |K| and |L| is a PL mapping such that the simplicial mapping between the subdivisions, $f:|K'|\to |L'|$, is a homeomorphism.
References
1. Munkres, James R. (1995). Elements of Algebraic Topology. Westview Press. ISBN 978-0-201-62728-2.
2. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
3. Colin P. Rourke and Brian J. Sanderson (1982). Introduction to Piecewise-Linear Topology. New York: Springer-Verlag. doi:10.1007/978-3-642-81735-9. ISBN 978-3-540-11102-3.
4. Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15
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Simplicial polytope
In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only triangular faces[1] and corresponds via Steinitz's theorem to a maximal planar graph.
Not to be confused with Simple polytope.
They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons.
Examples
Simplicial polyhedra include:
• Bipyramids
• Gyroelongated dipyramids
• Deltahedra (equilateral triangles)
• Platonic
• tetrahedron, octahedron, icosahedron
• Johnson solids:
• triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, gyroelongated square dipyramid
• Catalan solids:
• triakis tetrahedron, triakis octahedron, tetrakis hexahedron, disdyakis dodecahedron, triakis icosahedron, pentakis dodecahedron, disdyakis triacontahedron
Simplicial tilings:
• Regular:
• triangular tiling
• Laves tilings:
• tetrakis square tiling, triakis triangular tiling, kisrhombille tiling
Simplicial 4-polytopes include:
• convex regular 4-polytope
• 4-simplex, 16-cell, 600-cell
• Dual convex uniform honeycombs:
• Disphenoid tetrahedral honeycomb
• Dual of cantitruncated cubic honeycomb
• Dual of omnitruncated cubic honeycomb
• Dual of cantitruncated alternated cubic honeycomb
Simplicial higher polytope families:
• simplex
• cross-polytope (Orthoplex)
See also
• Simplicial complex
• Delaunay triangulation
Notes
1. Polyhedra, Peter R. Cromwell, 1997. (p.341)
References
• Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. ISBN 0-521-66405-5.
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Simplicial presheaf
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s.[1] Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.[2]
Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf $\operatorname {Hom} (-,U)$. Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf).
Example: Let G be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf $BG$. For example, one might set $B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} $. These types of examples appear in K-theory.
If $f:X\to Y$ is a local weak equivalence of simplicial presheaves, then the induced map $\mathbb {Z} f:\mathbb {Z} X\to \mathbb {Z} Y$ is also a local weak equivalence.
Homotopy sheaves of a simplicial presheaf
Let F be a simplicial presheaf on a site. The homotopy sheaves $\pi _{*}F$ of F is defined as follows. For any $f:X\to Y$ in the site and a 0-simplex s in F(X), set $(\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))$ and $(\pi _{i}^{\text{pr}}(F,s))(f)=\pi _{i}(F(Y),f^{*}(s))$. We then set $\pi _{i}F$ to be the sheaf associated with the pre-sheaf $\pi _{i}^{\text{pr}}F$.
Model structures
The category of simplicial presheaves on a site admits many different model structures.
Some of them are obtained by viewing simplicial presheaves as functors
$S^{op}\to \Delta ^{op}Sets$
The category of such functors is endowed with (at least) three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations in the first are maps
${\mathcal {F}}\to {\mathcal {G}}$
such that
${\mathcal {F}}(U)\to {\mathcal {G}}(U)$
is a weak equivalence / fibration of simplicial sets, for all U in the site S. The injective model structure is similar, but with weak equivalences and cofibrations instead.
Stack
Main article: Stack (mathematics)
A simplicial presheaf F on a site is called a stack if, for any X and any hypercovering H →X, the canonical map
$F(X)\to \operatorname {holim} F(H_{n})$
is a weak equivalence as simplicial sets, where the right is the homotopy limit of
$[n]=\{0,1,\dots ,n\}\mapsto F(H_{n})$.
Any sheaf F on the site can be considered as a stack by viewing $F(X)$ as a constant simplicial set; this way, the category of sheaves on the site is included as a subcategory to the homotopy category of simplicial presheaves on the site. The inclusion functor has a left adjoint and that is exactly $F\mapsto \pi _{0}F$.
If A is a sheaf of abelian group (on the same site), then we define $K(A,1)$ by doing classifying space construction levelwise (the notion comes from the obstruction theory) and set $K(A,i)=K(K(A,i-1),1)$. One can show (by induction): for any X in the site,
$\operatorname {H} ^{i}(X;A)=[X,K(A,i)]$
where the left denotes a sheaf cohomology and the right the homotopy class of maps.
See also
• cubical set
• N-group (category theory)
Notes
1. http://ncatlab.org/nlab/files/ToenStacksNAC.pdf
2. Jardine 2007, §1
Further reading
• Konrad Voelkel, Model structures on simplicial presheaves
References
• Jardine, J.F. (2004). "Generalised sheaf cohomology theories". In Greenlees, J. P. C. (ed.). Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, 9--20 September 2002. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 131. Dordrecht: Kluwer Academic. pp. 29–68. ISBN 1-4020-1833-9. Zbl 1063.55004.
• Jardine, J.F. (2007). "Simplicial presheaves" (PDF).
• B. Toën, Simplicial presheaves and derived algebraic geometry
External links
• J.F. Jardine's homepage
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Simplicial space
In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological spaces.[1]
References
1. Baues, Hans Joachim (1995), "Homotopy types", in James, I. M. (ed.), Handbook of Algebraic Topology, Amsterdam: North-Holland, pp. 1–72, doi:10.1016/B978-044481779-2/50002-X, ISBN 9780444817792, MR 1361886. See in particular p. 8.
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Simplicial volume
In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a certain measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes.
Given a closed and oriented manifold, one defines the simplicial norm by minimizing the sum of the absolute values of the coefficients over all singular chains homologous to a given cycle. The simplicial volume is the simplicial norm of the fundamental class.[1][2]
It is named after Mikhail Gromov, who introduced it in 1982. With William Thurston, he proved that the simplicial volume of a finite volume hyperbolic manifold is proportional to the hyperbolic volume.[1]
The simplicial volume is equal to twice the Thurston norm[3]
Thurston also used the simplicial volume to prove that hyperbolic volume decreases under hyperbolic Dehn surgery.[4]
References
1. Benedetti, Riccardo; Petronio, Carlo (1992), Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, p. 105, doi:10.1007/978-3-642-58158-8, ISBN 3-540-55534-X, MR 1219310.
2. Ratcliffe, John G. (2006), Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149 (2nd ed.), Berlin: Springer, p. 555, doi:10.1007/978-1-4757-4013-4, ISBN 978-0387-33197-3, MR 2249478, S2CID 123040867.
3. Gabai, David (January 1983). "Foliations and the topology of 3-manifolds". Journal of Differential Geometry. 18 (3): 445–503. doi:10.4310/jdg/1214437784. ISSN 0022-040X.
4. Benedetti & Petronio (1992), pp. 196ff.
• Michael Gromov. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.
External links
• Simplicial volume at the Manifold Atlas.
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Simplicially enriched category
In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies to simplicial objects in Cat (the category of small categories). Simplicially enriched categories can, however, be identified with simplicial objects in Cat whose object part is constant, or more precisely, whose all face and degeneracy maps are bijective on objects. Simplicially enriched categories can model (∞, 1)-categories, but the dictionary has to be carefully built. Namely many notions, limits for example, are different from the limits in the sense of enriched category theory.
References
• Goerss, Paul; Jardine, John (2009), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Birkhäuser Basel, ISBN 978-3-7643-6064-1, MR 1711612
• Lurie, Jacob (2009), Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, arXiv:math.CT/0608040, ISBN 978-0-691-14049-0, MR 2522659
External links
• Simplicially enriched category at the nLab
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
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Morass (set theory)
In axiomatic set theory, a mathematical discipline, a morass is an infinite combinatorial structure, used to create "large" structures from a "small" number of "small" approximations. They were invented by Ronald Jensen for his proof that cardinal transfer theorems hold under the axiom of constructibility. A far less complex but equivalent variant known as a simplified morass was introduced by Velleman, and the term morass is now often used to mean these simpler structures.
Overview
Whilst it is possible to define so-called gap-n morasses for n > 1, they are so complex that focus is usually restricted to the gap-1 case, except for specific applications. The "gap" is essentially the cardinal difference between the size of the "small approximations" used and the size of the ultimate structure.
A (gap-1) morass on an uncountable regular cardinal κ (also called a (κ,1)-morass) consists of a tree of height κ + 1, with the top level having κ+-many nodes. The nodes are taken to be ordinals, and functions π between these ordinals are associated to the edges in the tree order. It is required that the ordinal structure of the top level nodes be "built up" as the direct limit of the ordinals in the branch to that node by the maps π, so the lower level nodes can be thought of as approximations to the (larger) top level node. A long list of further axioms is imposed to have this happen in a particularly "nice" way.[1][2]
Variants and equivalents
Velleman[2] and Shelah and Stanley[3] independently developed forcing axioms equivalent to the existence of morasses, to facilitate their use by non-experts. Going further, Velleman[4] showed that the existence of morasses is equivalent to simplified morasses, which are vastly simpler structures. However, the only known construction of a simplified morass in Gödel's constructible universe is by means of morasses, so the original notion retains interest.
Other variants on morasses, generally with added structure, have also appeared over the years. These include universal morasses,[5] whereby every subset of κ is built up through the branches of the morass, mangroves,[6] which are morasses stratified into levels (mangals) at which every branch must have a node, and quagmires.[7]
Simplified morass
Velleman [8] defined gap-1 simplified morasses which are much simpler than gap-1 morasses, and showed that the existence of gap-1 morasses is equivalent to the existence of gap-1 simplified morasses.
Roughly speaking: a (κ,1)-simplified morass M = < φ→, F⇒ > contains a sequence φ→ = < φβ : β ≤ κ > of ordinals such that φβ < κ for β < κ and φκ = κ+, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of monotone mappings from φα to φβ for α < β ≤ κ with specific (easy but important) conditions.
Velleman's clear definition can be found in,[9] where he also constructed (ω0,1) simplified morasses in ZFC. In [10] he gave similar simple definitions for gap-2 simplified morasses, and in [11] he constructed (ω0,2) simplified morasses in ZFC.
Higher gap simplified morasses for any n ≥ 1 were defined by Morgan [12] and Szalkai,.[13][14]
Roughly speaking: a (κ,n + 1)-simplified morass (of Szalkai) M = < M→, F⇒ > contains a sequence M→ = < Mβ : β ≤ κ > of (< κ,n)-simplified morass-like structures for β < κ and Mκ a (κ+,n) -simplified morass, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of mappings from Mα to Mβ for α < β ≤ κ with specific conditions.
References
1. K. Devlin. Constructibility. Springer, Berlin, 1984.
2. Velleman, Daniel J. (1982). "Morasses, diamond, and forcing". Ann. Math. Logic. 23: 199–281. doi:10.1016/0003-4843(82)90005-5. Zbl 0521.03034.
3. Shelah, S.; Stanley, L. (1982). "S-forcing, I: A "black box" theorem for morasses, with applications: Super-Souslin trees and generalizing Martin's axiom". Israel Journal of Mathematics. 43: 185–224. doi:10.1007/BF02761942.
4. Velleman, Dan (1984). "Simplified morasses". Journal of Symbolic Logic. 49 (1): 257–271. doi:10.2307/2274108. Zbl 0575.03035.
5. K. Devlin. Aspects of Constructibility, Lecture Notes in Mathematics 354, Springer, Berlin, 1973.
6. Brooke-Taylor, A.; Friedman, S. (2009). "Large cardinals and gap-1 morasses". Annals of Pure and Applied Logic. 159 (1–2): 71–99. arXiv:0801.1912. doi:10.1016/j.apal.2008.10.007. Zbl 1165.03033.
7. Kanamori, Akihiro (1983). "Morasses in combinatorial set theory". In Mathias, A.R.D. (ed.). Surveys in set theory. London Mathematical Society Lecture Note Series. Vol. 87. Cambridge: Cambridge University Press. pp. 167–196. ISBN 0-521-27733-7. Zbl 0525.03036.
8. D. Velleman. Simplified Morasses, Journal of Symbolic Logic 49, No. 1 (1984), pp 257–271.
9. D. Velleman. Simplified Morasses, Journal of Symbolic Logic 49, No. 1 (1984), pp 257–271.
10. D. Velleman. Simplified Gap-2 Morasses, Annals of Pure and Applied Logic 34, (1987), pp 171–208.
11. D. Velleman. Gap-2 Morasses of Height ω0, Journal of Symbolic Logic 52, (1987), pp 928–938.
12. Ch. Morgan. The Equivalence of Morasses and Simplified Morasses in the Finite Gap Case, PhD.Thesis, Merton College, UK, 1989.
13. I. Szalkai. Higher Gap Simplified Morasses and Combinatorial Applications, PhD-Thesis (in Hungarian), ELTE, Budapest, 1991. English abstract: http://math.uni-pannon.hu/~szalkai/Szalkai-1991d-MorassAbst-.pdf
14. I. Szalkai. An Inductive Definition of Higher Gap Simplified Morasses, Publicationes Mathematicae Debrecen 58 (2001), pp 605–634. http://math.uni-pannon.hu/~szalkai/Szalkai-2001a-IndMorass.pdf
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Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected[1]) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.
Definition and equivalent formulations
A topological space $X$ is called simply connected if it is path-connected and any loop in $X$ defined by $f:S^{1}\to X$ can be contracted to a point: there exists a continuous map $F:D^{2}\to X$ such that $F$ restricted to $S^{1}$ is $f.$ Here, $S^{1}$ and $D^{2}$ denotes the unit circle and closed unit disk in the Euclidean plane respectively.
An equivalent formulation is this: $X$ is simply connected if and only if it is path-connected, and whenever $p:[0,1]\to X$ and $q:[0,1]\to X$ are two paths (that is, continuous maps) with the same start and endpoint ($p(0)=q(0)$ and $p(1)=q(1)$), then $p$ can be continuously deformed into $q$ while keeping both endpoints fixed. Explicitly, there exists a homotopy $F:[0,1]\times [0,1]\to X$ such that $F(x,0)=p(x)$ and $F(x,1)=q(x).$
A topological space $X$ is simply connected if and only if $X$ is path-connected and the fundamental group of $X$ at each point is trivial, i.e. consists only of the identity element. Similarly, $X$ is simply connected if and only if for all points $x,y\in X,$ the set of morphisms $\operatorname {Hom} _{\Pi (X)}(x,y)$ in the fundamental groupoid of $X$ has only one element.[2]
In complex analysis: an open subset $X\subseteq \mathbb {C} $ is simply connected if and only if both $X$ and its complement in the Riemann sphere are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes a nice example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. It might also be worth pointing out that a relaxation of the requirement that $X$ be connected leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has a connected extended complement exactly when each of its connected components are simply connected.
Informal discussion
Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected.
The definition rules out only handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of any dimension, is called contractibility.
Examples
• The Euclidean plane $\mathbb {R} ^{2}$ is simply connected, but $\mathbb {R} ^{2}$ minus the origin $(0,0)$ is not. If $n>2,$ then both $\mathbb {R} ^{n}$ and $\mathbb {R} ^{n}$ minus the origin are simply connected.
• Analogously: the n-dimensional sphere $S^{n}$ is simply connected if and only if $n\geq 2.$
• Every convex subset of $\mathbb {R} ^{n}$ is simply connected.
• A torus, the (elliptic) cylinder, the Möbius strip, the projective plane and the Klein bottle are not simply connected.
• Every topological vector space is simply connected; this includes Banach spaces and Hilbert spaces.
• For $n\geq 2,$ the special orthogonal group $\operatorname {SO} (n,\mathbb {R} )$ is not simply connected and the special unitary group $\operatorname {SU} (n)$ is simply connected.
• The one-point compactification of $\mathbb {R} $ is not simply connected (even though $\mathbb {R} $ is simply connected).
• The long line $L$ is simply connected, but its compactification, the extended long line $L^{*}$ is not (since it is not even path connected).
Properties
A surface (two-dimensional topological manifold) is simply connected if and only if it is connected and its genus (the number of handles of the surface) is 0.
A universal cover of any (suitable) space $X$ is a simply connected space which maps to $X$ via a covering map.
If $X$ and $Y$ are homotopy equivalent and $X$ is simply connected, then so is $Y.$
The image of a simply connected set under a continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is $\mathbb {C} \setminus \{0\},$ which is not simply connected.
The notion of simple connectedness is important in complex analysis because of the following facts:
• The Cauchy's integral theorem states that if $U$ is a simply connected open subset of the complex plane $\mathbb {C} ,$ and $f:U\to \mathbb {C} $ is a holomorphic function, then $f$ has an antiderivative $F$ on $U,$ and the value of every line integral in $U$ with integrand $f$ depends only on the end points $u$ and $v$ of the path, and can be computed as $F(v)-F(u).$ The integral thus does not depend on the particular path connecting $u$ and $v,$
• The Riemann mapping theorem states that any non-empty open simply connected subset of $\mathbb {C} $ (except for $\mathbb {C} $ itself) is conformally equivalent to the unit disk.
The notion of simple connectedness is also a crucial condition in the Poincaré conjecture.
See also
• Fundamental group – Mathematical group of the homotopy classes of loops in a topological space
• Deformation retract – Continuous, position-preserving mapping from a topological space into a subspacePages displaying short descriptions of redirect targets
• n-connected space
• Path-connected – Topological space that is connectedPages displaying short descriptions of redirect targets
• Unicoherent space
References
1. "n-connected space in nLab". ncatlab.org. Retrieved 2017-09-17.
2. Ronald, Brown (June 2006). Topology and Groupoids. Academic Search Complete. North Charleston: CreateSpace. ISBN 1419627228. OCLC 712629429.
• Spanier, Edwin (December 1994). Algebraic Topology. Springer. ISBN 0-387-94426-5.
• Conway, John (1986). Functions of One Complex Variable I. Springer. ISBN 0-387-90328-3.
• Bourbaki, Nicolas (2005). Lie Groups and Lie Algebras. Springer. ISBN 3-540-43405-4.
• Gamelin, Theodore (January 2001). Complex Analysis. Springer. ISBN 0-387-95069-9.
• Joshi, Kapli (August 1983). Introduction to General Topology. New Age Publishers. ISBN 0-85226-444-5.
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Simply typed lambda calculus
The simply typed lambda calculus ($\lambda ^{\to }$), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor ($\to $) that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus.[1]
The term simple type is also used to refer extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The simple types, except for full recursion, are still considered simple because the Church encodings of such structures can be done using only $\to $ and suitable type variables, while polymorphism and dependency cannot.
Syntax
In this article, the symbols $\sigma $ and $\tau $ are used to range over types. Informally, the function type $\sigma \to \tau $ refers to the type of functions that, given an input of type $\sigma $, produce an output of type $\tau $. By convention, $\to $ associates to the right: $\sigma \to \tau \to \rho $ is read as $\sigma \to (\tau \to \rho )$.
To define the types, a set of base types, $B$, must first be defined. These are sometimes called atomic types or type constants. With this fixed, the syntax of types is:
$\tau ::=\tau \to \tau \mid T\quad \mathrm {where} \quad T\in B$ ::=\tau \to \tau \mid T\quad \mathrm {where} \quad T\in B} .
For example, $B=\{a,b\}$, generates an infinite set of types starting with $a,b,$$a\to a,$$a\to b,b\to b,$$b\to a,$$a\to (a\to a),\ldots ,$$(b\to a)\to (a\to b),\ldots $
A set of term constants is also fixed for the base types. For example, it might assumed that a base type nat, and the term constants could be the natural numbers. In the original presentation, Church used only two base types: $o$ for "the type of propositions" and $\iota $ for "the type of individuals". The type $o$ has no term constants, whereas $\iota $ has one term constant. Frequently the calculus with only one base type, usually $o$, is considered.
The syntax of the simply typed lambda calculus is essentially that of the lambda calculus itself. The term $x{\mathbin {:}}\tau $ denotes that the variable $x$ is of type $\tau $. The term syntax, in BNF, is then:
$e::=x\mid \lambda x{\mathbin {:}}\tau .e\mid e\,e\mid c$
where $c$ is a term constant.
That is, variable reference, abstractions, application, and constant. A variable reference $x$ is bound if it is inside of an abstraction binding $x$. A term is closed if there are no unbound variables.
In comparison, the syntax of untyped lambda calculus has no such typing or term constants:
$e::=x\mid \lambda x.e\mid e\,e$
Whereas in typed lambda calculus every abstraction (i.e. function) must specify the type of its argument.
Typing rules
To define the set of well-typed lambda terms of a given type, one defines a typing relation between terms and types. First, one introduces typing contexts or typing environments $\Gamma ,\Delta ,\dots $, which are sets of typing assumptions. A typing assumption has the form $x{\mathbin {:}}\sigma $, meaning $x$ has type $\sigma $.
The typing relation $\Gamma \vdash e{\mathbin {:}}\sigma $ indicates that $e$ is a term of type $\sigma $ in context $\Gamma $. In this case $e$ is said to be well-typed (having type $\sigma $). Instances of the typing relation are called typing judgements. The validity of a typing judgement is shown by providing a typing derivation, constructed using typing rules (wherein the premises above the line allow us to derive the conclusion below the line). Simply-typed lambda calculus uses these rules:
${\frac {x{\mathbin {:}}\sigma \in \Gamma }{\Gamma \vdash x{\mathbin {:}}\sigma }}$ (1) ${\frac {c{\text{ is a constant of type }}T}{\Gamma \vdash c{\mathbin {:}}T}}$ (2)
${\frac {\Gamma ,x{\mathbin {:}}\sigma \vdash e{\mathbin {:}}\tau }{\Gamma \vdash (\lambda x{\mathbin {:}}\sigma .~e){\mathbin {:}}(\sigma \to \tau )}}$ (3) ${\frac {\Gamma \vdash e_{1}{\mathbin {:}}\sigma \to \tau \quad \Gamma \vdash e_{2}{\mathbin {:}}\sigma }{\Gamma \vdash e_{1}~e_{2}{\mathbin {:}}\tau }}$ (4)
In words,
1. If $x$ has type $\sigma $ in the context, then $x$ has type $\sigma $.
2. Term constants have the appropriate base types.
3. If, in a certain context with $x$ having type $\sigma $, $e$ has type $\tau $, then, in the same context without $x$, $\lambda x{\mathbin {:}}\sigma .~e$ has type $\sigma \to \tau $.
4. If, in a certain context, $e_{1}$ has type $\sigma \to \tau $, and $e_{2}$ has type $\sigma $, then $e_{1}~e_{2}$ has type $\tau $.
Examples of closed terms, i.e. terms typable in the empty context, are:
• For every type $\tau $, a term $\lambda x{\mathbin {:}}\tau .x{\mathbin {:}}\tau \to \tau $ (identity function/I-combinator),
• For types $\sigma ,\tau $, a term $\lambda x{\mathbin {:}}\sigma .\lambda y{\mathbin {:}}\tau .x{\mathbin {:}}\sigma \to \tau \to \sigma $ (the K-combinator), and
• For types $\tau ,\tau ',\tau ''$, a term $\lambda x{\mathbin {:}}\tau \to \tau '\to \tau ''.\lambda y{\mathbin {:}}\tau \to \tau '.\lambda z{\mathbin {:}}\tau .xz(yz):(\tau \to \tau '\to \tau '')\to (\tau \to \tau ')\to \tau \to \tau ''$ (the S-combinator).
These are the typed lambda calculus representations of the basic combinators of combinatory logic.
Each type $\tau $ is assigned an order, a number $o(\tau )$. For base types, $o(T)=0$; for function types, $o(\sigma \to \tau )={\mbox{max}}(o(\sigma )+1,o(\tau ))$. That is, the order of a type measures the depth of the most left-nested arrow. Hence:
$o(\iota \to \iota \to \iota )=1$
$o((\iota \to \iota )\to \iota )=2$
Semantics
Intrinsic vs. extrinsic interpretations
Broadly speaking, there are two different ways of assigning meaning to the simply typed lambda calculus, as to typed languages more generally, variously called intrinsic vs. extrinsic, ontological vs. semantical, or Church-style vs. Curry-style.[1][3][4] An intrinsic semantics only assigns meaning to well-typed terms, or more precisely, assigns meaning directly to typing derivations. This has the effect that terms differing only by type annotations can nonetheless be assigned different meanings. For example, the identity term $\lambda x{\mathbin {:}}{\mathtt {int}}.~x$ on integers and the identity term $\lambda x{\mathbin {:}}{\mathtt {bool}}.~x$ on booleans may mean different things. (The classic intended interpretations are the identity function on integers and the identity function on boolean values.) In contrast, an extrinsic semantics assigns meaning to terms regardless of typing, as they would be interpreted in an untyped language. In this view, $\lambda x{\mathbin {:}}{\mathtt {int}}.~x$ and $\lambda x{\mathbin {:}}{\mathtt {bool}}.~x$ mean the same thing (i.e., the same thing as $\lambda x.~x$).
The distinction between intrinsic and extrinsic semantics is sometimes associated with the presence or absence of annotations on lambda abstractions, but strictly speaking this usage is imprecise. It is possible to define an extrinsic semantics on annotated terms simply by ignoring the types (i.e., through type erasure), as it is possible to give an intrinsic semantics on unannotated terms when the types can be deduced from context (i.e., through type inference). The essential difference between intrinsic and extrinsic approaches is just whether the typing rules are viewed as defining the language, or as a formalism for verifying properties of a more primitive underlying language. Most of the different semantic interpretations discussed below can be seen through either an intrinsic or extrinsic perspective.
Equational theory
The simply typed lambda calculus has the same equational theory of βη-equivalence as untyped lambda calculus, but subject to type restrictions. The equation for beta reduction
$(\lambda x{\mathbin {:}}\sigma .~t)\,u=_{\beta }t[x:=u]$
holds in context $\Gamma $ whenever $\Gamma ,x{\mathbin {:}}\sigma \vdash t{\mathbin {:}}\tau $ and $\Gamma \vdash u{\mathbin {:}}\sigma $, while the equation for eta reduction
$\lambda x{\mathbin {:}}\sigma .~t\,x=_{\eta }t$
holds whenever $\Gamma \vdash t\!:\sigma \to \tau $ and $x$ does not appear free in $t$.
Operational semantics
Likewise, the operational semantics of simply typed lambda calculus can be fixed as for the untyped lambda calculus, using call by name, call by value, or other evaluation strategies. As for any typed language, type safety is a fundamental property of all of these evaluation strategies. Additionally, the strong normalization property described below implies that any evaluation strategy will terminate on all simply typed terms.
Categorical semantics
The simply typed lambda calculus (with $\beta \eta $-equivalence) is the internal language of Cartesian closed categories (CCCs), as was first observed by Joachim Lambek.[5] Given any specific CCC, the basic types of the corresponding lambda calculus are just the objects, and the terms are the morphisms. Conversely, every simply typed lambda calculus gives a CCC whose objects are the types, and morphisms are equivalence classes of terms.
To make the correspondence clear, a type constructor for the Cartesian product is typically added to the above. To preserve the categoricity of the Cartesian product, one adds typing rules for pairing, projection, and a unit term. Given two terms $s{\mathbin {:}}\sigma $ and $t{\mathbin {:}}\tau $, the term $(s,t)$ has type $\sigma \times \tau $. Likewise, if one has a term $u{\mathbin {:}}\tau _{1}\times \tau _{2}$, then there are terms $\pi _{1}(u){\mathbin {:}}\tau _{1}$ and $\pi _{2}(u){\mathbin {:}}\tau _{2}$ where the $\pi _{i}$ correspond to the projections of the Cartesian product. The unit term, of type 1, is written as $()$ and vocalized as 'nil', is the final object. The equational theory is extended likewise, so that one has
$\pi _{1}(s{\mathbin {:}}\sigma ,t{\mathbin {:}}\tau )=s{\mathbin {:}}\sigma $
$\pi _{2}(s{\mathbin {:}}\sigma ,t{\mathbin {:}}\tau )=t{\mathbin {:}}\tau $
$(\pi _{1}(u{\mathbin {:}}\sigma \times \tau ),\pi _{2}(u{\mathbin {:}}\sigma \times \tau ))=u{\mathbin {:}}\sigma \times \tau $
$t{\mathbin {:}}1=()$
This last is read as "if t has type 1, then it reduces to nil".
The above can then be turned into a category by taking the types as the objects. The morphisms $\sigma \to \tau $ are equivalence classes of pairs $(x{\mathbin {:}}\sigma ,t{\mathbin {:}}\tau )$ where x is a variable (of type $\sigma $) and t is a term (of type $\tau $), having no free variables in it, except for (optionally) x. Closure is obtained from currying and application, as usual.
More precisely, there exist functors between the category of Cartesian closed categories, and the category of simply-typed lambda theories.
It is common to extend this case to closed symmetric monoidal categories by using a linear type system. The reason for this is that the CCC is a special case of the closed symmetric monoidal category, which is typically taken to be the category of sets. This is fine for laying the foundations of set theory, but the more general topos seems to provide a superior foundation.
Proof-theoretic semantics
The simply typed lambda calculus is closely related to the implicational fragment of propositional intuitionistic logic, i.e., minimal logic, via the Curry–Howard isomorphism: terms correspond precisely to proofs in natural deduction, and inhabited types are exactly the tautologies of minimal logic.
Alternative syntaxes
The presentation given above is not the only way of defining the syntax of the simply typed lambda calculus. One alternative is to remove type annotations entirely (so that the syntax is identical to the untyped lambda calculus), while ensuring that terms are well-typed via Hindley–Milner type inference. The inference algorithm is terminating, sound, and complete: whenever a term is typable, the algorithm computes its type. More precisely, it computes the term's principal type, since often an unannotated term (such as $\lambda x.~x$) may have more than one type (${\mathtt {int}}\to {\mathtt {int}}$, ${\mathtt {bool}}\to {\mathtt {bool}}$, etc., which are all instances of the principal type $\alpha \to \alpha $).
Another alternative presentation of simply typed lambda calculus is based on bidirectional type checking, which requires more type annotations than Hindley–Milner inference but is easier to describe. The type system is divided into two judgments, representing both checking and synthesis, written $\Gamma \vdash e\Leftarrow \tau $ and $\Gamma \vdash e\Rightarrow \tau $ respectively. Operationally, the three components $\Gamma $, $e$, and $\tau $ are all inputs to the checking judgment $\Gamma \vdash e\Leftarrow \tau $, whereas the synthesis judgment $\Gamma \vdash e\Rightarrow \tau $ only takes $\Gamma $ and $e$ as inputs, producing the type $\tau $ as output. These judgments are derived via the following rules:
${\frac {x{\mathbin {:}}\sigma \in \Gamma }{\Gamma \vdash x\Rightarrow \sigma }}$ [1] ${\frac {c{\text{ is a constant of type }}T}{\Gamma \vdash c\Rightarrow T}}$ [2]
${\frac {\Gamma ,x{\mathbin {:}}\sigma \vdash e\Leftarrow \tau }{\Gamma \vdash \lambda x.~e\Leftarrow \sigma \to \tau }}$ [3] ${\frac {\Gamma \vdash e_{1}\Rightarrow \sigma \to \tau \quad \Gamma \vdash e_{2}\Leftarrow \sigma }{\Gamma \vdash e_{1}~e_{2}\Rightarrow \tau }}$ [4]
${\frac {\Gamma \vdash e\Rightarrow \tau }{\Gamma \vdash e\Leftarrow \tau }}$ [5] ${\frac {\Gamma \vdash e\Leftarrow \tau }{\Gamma \vdash (e{\mathbin {:}}\tau )\Rightarrow \tau }}$ [6]
Observe that rules [1]–[4] are nearly identical to rules (1)–(4) above, except for the careful choice of checking or synthesis judgments. These choices can be explained like so:
1. If $x{\mathbin {:}}\sigma $ is in the context, we can synthesize type $\sigma $ for $x$.
2. The types of term constants are fixed and can be synthesized.
3. To check that $\lambda x.~e$ has type $\sigma \to \tau $ in some context, we extend the context with $x{\mathbin {:}}\sigma $ and check that $e$ has type $\tau $.
4. If $e_{1}$ synthesizes type $\sigma \to \tau $ (in some context), and $e_{2}$ checks against type $\sigma $ (in the same context), then $e_{1}~e_{2}$ synthesizes type $\tau $.
Observe that the rules for synthesis are read top-to-bottom, whereas the rules for checking are read bottom-to-top. Note in particular that we do not need any annotation on the lambda abstraction in rule [3], because the type of the bound variable can be deduced from the type at which we check the function. Finally, we explain rules [5] and [6] as follows:
1. To check that $e$ has type $\tau $, it suffices to synthesize type $\tau $.
2. If $e$ checks against type $\tau $, then the explicitly annotated term $(e{\mathbin {:}}\tau )$ synthesizes $\tau $.
Because of these last two rules coercing between synthesis and checking, it is easy to see that any well-typed but unannotated term can be checked in the bidirectional system, so long as we insert "enough" type annotations. And in fact, annotations are needed only at β-redexes.
General observations
Given the standard semantics, the simply typed lambda calculus is strongly normalizing: that is, well-typed terms always reduce to a value, i.e., a $\lambda $ abstraction. This is because recursion is not allowed by the typing rules: it is impossible to find types for fixed-point combinators and the looping term $\Omega =(\lambda x.~x~x)(\lambda x.~x~x)$. Recursion can be added to the language by either having a special operator ${\mathtt {fix}}_{\alpha }$of type $(\alpha \to \alpha )\to \alpha $ or adding general recursive types, though both eliminate strong normalization.
Since it is strongly normalising, it is decidable whether or not a simply typed lambda calculus program halts: in fact, it always halts. We can therefore conclude that the language is not Turing complete.
Important results
• Tait showed in 1967 that $\beta $-reduction is strongly normalizing.[6] As a corollary $\beta \eta $-equivalence is decidable. Statman showed in 1979 that the normalisation problem is not elementary recursive,[7] a proof which was later simplified by Mairson.[8] The problem is known to be in the set ${\mathcal {E}}^{4}$ of the Grzegorczyk hierarchy.[9] A purely semantic normalisation proof (see normalisation by evaluation) was given by Berger and Schwichtenberg in 1991.[10]
• The unification problem for $\beta \eta $-equivalence is undecidable. Huet showed in 1973 that 3rd order unification is undecidable[11] and this was improved upon by Baxter in 1978[12] then by Goldfarb in 1981[13] by showing that 2nd order unification is already undecidable. A proof that higher order matching (unification where only one term contains existential variables) is decidable was announced by Colin Stirling in 2006, and a full proof was published in 2009.[14]
• We can encode natural numbers by terms of the type $(o\to o)\to (o\to o)$ (Church numerals). Schwichtenberg showed in 1975 that in $\lambda ^{\to }$ exactly the extended polynomials are representable as functions over Church numerals;[15] these are roughly the polynomials closed up under a conditional operator.
• A full model of $\lambda ^{\to }$ is given by interpreting base types as sets and function types by the set-theoretic function space. Friedman showed in 1975 that this interpretation is complete for $\beta \eta $-equivalence, if the base types are interpreted by infinite sets.[16] Statman showed in 1983 that $\beta \eta $-equivalence is the maximal equivalence which is typically ambiguous, i.e. closed under type substitutions (Statman's Typical Ambiguity Theorem).[17] A corollary of this is that the finite model property holds, i.e. finite sets are sufficient to distinguish terms which are not identified by $\beta \eta $-equivalence.
• Plotkin introduced logical relations in 1973 to characterize the elements of a model which are definable by lambda terms.[18] In 1993 Jung and Tiuryn showed that a general form of logical relation (Kripke logical relations with varying arity) exactly characterizes lambda definability.[19] Plotkin and Statman conjectured that it is decidable whether a given element of a model generated from finite sets is definable by a lambda term (Plotkin–Statman conjecture). The conjecture was shown to be false by Loader in 2001.[20]
Notes
1. Church, Alonzo (June 1940). "A formulation of the simple theory of types" (PDF). Journal of Symbolic Logic. 5 (2): 56–68. doi:10.2307/2266170. JSTOR 2266170. S2CID 15889861. Archived from the original (PDF) on 12 January 2019.
2. Pfenning, Frank. "Church and Curry: Combining Intrinsic and Extrinsic Typing" (PDF): 1. Retrieved 26 February 2022. {{cite journal}}: Cite journal requires |journal= (help)
3. Curry, Haskell B (1934-09-20). "Functionality in Combinatory Logic". Proceedings of the National Academy of Sciences of the United States of America. 20 (11): 584–90. Bibcode:1934PNAS...20..584C. doi:10.1073/pnas.20.11.584. ISSN 0027-8424. PMC 1076489. PMID 16577644. (presents an extrinsically typed combinatory logic, later adapted by others to the lambda calculus)[2]
4. Reynolds, John (1998). Theories of Programming Languages. Cambridge, England: Cambridge University Press. pp. 327, 334. ISBN 9780521594141.
5. Lambek, J. (1986). "Cartesian closed categories and typed λ-calculi". Combinators and Functional Programming Languages. Lecture Notes in Computer Science. Vol. 242. Springer. pp. 136–175. doi:10.1007/3-540-17184-3_44. ISBN 978-3-540-47253-7.
6. Tait, W. W. (August 1967). "Intensional interpretations of functionals of finite type I". The Journal of Symbolic Logic. 32 (2): 198–212. doi:10.2307/2271658. ISSN 0022-4812. JSTOR 2271658. S2CID 9569863.
7. Statman, Richard (1 July 1979). "The typed λ-calculus is not elementary recursive". Theoretical Computer Science. 9 (1): 73–81. doi:10.1016/0304-3975(79)90007-0. ISSN 0304-3975.
8. Mairson, Harry G. (14 September 1992). "A simple proof of a theorem of Statman". Theoretical Computer Science. 103 (2): 387–394. doi:10.1016/0304-3975(92)90020-G. ISSN 0304-3975.
9. Statman, Richard (July 1979). "The typed λ-calculus is not elementary recursive". Theoretical Computer Science. 9 (1): 73–81. doi:10.1016/0304-3975(79)90007-0. ISSN 0304-3975.
10. Berger, U.; Schwichtenberg, H. (July 1991). "An inverse of the evaluation functional for typed lambda -calculus". [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science. pp. 203–211. doi:10.1109/LICS.1991.151645. ISBN 0-8186-2230-X. S2CID 40441974.
11. Huet, Gerard P. (1 April 1973). "The undecidability of unification in third order logic". Information and Control. 22 (3): 257–267. doi:10.1016/S0019-9958(73)90301-X. ISSN 0019-9958.
12. Baxter, Lewis D. (1 August 1978). "The undecidability of the third order dyadic unification problem". Information and Control. 38 (2): 170–178. doi:10.1016/S0019-9958(78)90172-9. ISSN 0019-9958.
13. Goldfarb, Warren D. (1 January 1981). "The undecidability of the second-order unification problem". Theoretical Computer Science. 13 (2): 225–230. doi:10.1016/0304-3975(81)90040-2. ISSN 0304-3975.
14. Stirling, Colin (22 July 2009). "Decidability of higher-order matching". Logical Methods in Computer Science. 5 (3): 1–52. arXiv:0907.3804. doi:10.2168/LMCS-5(3:2)2009. S2CID 1478837.
15. Schwichtenberg, Helmut (1 September 1975). "Definierbare Funktionen imλ-Kalkül mit Typen". Archiv für mathematische Logik und Grundlagenforschung (in German). 17 (3): 113–114. doi:10.1007/BF02276799. ISSN 1432-0665. S2CID 11598130.
16. Friedman, Harvey (1975). "Equality between functionals". Logic Colloquium. Lecture Notes in Mathematics. Springer. 453: 22–37. doi:10.1007/BFb0064870. ISBN 978-3-540-07155-6.
17. Statman, R. (1 December 1983). "$\lambda $-definable functionals and $\beta \eta $ conversion". Archiv für mathematische Logik und Grundlagenforschung. 23 (1): 21–26. doi:10.1007/BF02023009. ISSN 1432-0665. S2CID 33920306.
18. Plotkin, G.D. (1973). Lambda-definability and logical relations (PDF) (Technical report). Edinburgh University. Retrieved 30 September 2022.
19. Jung, Achim; Tiuryn, Jerzy (1993). "A new characterization of lambda definability". Typed Lambda Calculi and Applications. Lecture Notes in Computer Science. Springer. 664: 245–257. doi:10.1007/BFb0037110. ISBN 3-540-56517-5.
20. Loader, Ralph (2001). "The Undecidability of λ-Definability". Logic, Meaning and Computation: Essays in Memory of Alonzo Church. Springer Netherlands: 331–342. doi:10.1007/978-94-010-0526-5_15. ISBN 978-94-010-3891-1.
References
• H. Barendregt, Lambda Calculi with Types, Handbook of Logic in Computer Science, Volume II, Oxford University Press, 1993. ISBN 0-19-853761-1.
External links
• Loader, Ralph (February 1998). "Notes on Simply Typed Lambda Calculus".
• "Church's Type Theory" entry in the Stanford Encyclopedia of Philosophy
Alonzo Church
Notable ideas
• Lambda calculus
• Simply typed lambda calculus
• Church–Turing thesis
• Church encoding
• Frege–Church ontology
• Church–Rosser theorem
Students
• Alan Turing
• C. Anthony Anderson
• Peter Andrews
• George Alfred Barnard
• William Boone
• Martin Davis
• William Easton
• Alfred Foster
• Leon Henkin
• John George Kemeny
• Stephen Cole Kleene
• Simon B. Kochen
• Maurice L'Abbé
• Isaac Malitz
• Gary R. Mar
• Michael O. Rabin
• Nicholas Rescher
• Hartley Rogers, Jr
• J. Barkley Rosser
• Dana Scott
• Norman Shapiro
• Raymond Smullyan
Institutions
• Princeton University
• University of California, Los Angeles
Family
• Alonzo Church (college president)
• A. C. Croom
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Wikipedia
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Simply connected at infinity
In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for any compact subset C of X, there is a compact set D in X containing C so that the induced map
$\pi _{1}(X-D)\to \pi _{1}(X-C)$
is the zero map. Intuitively, this is the property that loops far away from a small subspace of X can be collapsed, no matter how bad the small subspace is.
The Whitehead manifold is an example of a 3-manifold that is contractible but not simply connected at infinity. Since this property is invariant under homeomorphism, this proves that the Whitehead manifold is not homeomorphic to R3.
However, it is a theorem of John R. Stallings[1] that for $n\geq 5$, a contractible n-manifold is homeomorphic to Rn precisely when it is simply connected at infinity.
References
1. "Theory : Chapter 10" (PDF). Math.rutgers.edu. Retrieved 2015-03-08.
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Wikipedia
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Nonabelian Hodge correspondence
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.
The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem which defines a correspondence between stable vector bundles and unitary representations of the fundamental group of a compact Riemann surface. In fact the Narasimhan–Seshadri theorem may be obtained as a special case of the nonabelian Hodge correspondence by setting the Higgs field to zero.
History
It was proven by M. S. Narasimhan and C. S. Seshadri in 1965 that stable vector bundles on a compact Riemann surface correspond to irreducible projective unitary representations of the fundamental group.[1] This theorem was phrased in a new light in the work of Simon Donaldson in 1983, who showed that stable vector bundles correspond to Yang–Mills connections, whose holonomy gives the representations of the fundamental group of Narasimhan and Seshadri.[2] The Narasimhan–Seshadri theorem was generalised from the case of compact Riemann surfaces to the setting of compact Kähler manifolds by Donaldson in the case of algebraic surfaces, and in general by Karen Uhlenbeck and Shing-Tung Yau.[3][4] This correspondence between stable vector bundles and Hermitian Yang–Mills connections is known as the Kobayashi–Hitchin correspondence.
The Narasimhan–Seshadri theorem concerns unitary representations of the fundamental group. Nigel Hitchin introduced a notion of a Higgs bundle as an algebraic object which should correspond to complex representations of the fundamental group (in fact the terminology "Higgs bundle" was introduced by Carlos Simpson after the work of Hitchin). The first instance of the nonabelian Hodge theorem was proven by Hitchin, who considered the case of rank two Higgs bundles over a compact Riemann surface.[5] Hitchin showed that a polystable Higgs bundle corresponds to a solution of Hitchin's equations, a system of differential equations obtained as a dimensional reduction of the Yang–Mills equations to dimension two. It was shown by Donaldson in this case that solutions to Hitchin's equations are in correspondence with representations of the fundamental group.[6]
The results of Hitchin and Donaldson for Higgs bundles of rank two on a compact Riemann surface were vastly generalised by Carlos Simpson and Kevin Corlette. The statement that polystable Higgs bundles correspond to solutions of Hitchin's equations was proven by Simpson.[7][8] The correspondence between solutions of Hitchin's equations and representations of the fundamental group was shown by Corlette.[9]
Definitions
In this section we recall the objects of interest in the nonabelian Hodge theorem.[7][8]
Higgs bundles
Main article: Higgs bundle
A Higgs bundle over a compact Kähler manifold $(X,\omega )$ is a pair $(E,\Phi )$ where $E\to X$ is a holomorphic vector bundle and $\Phi :E\to E\otimes {\boldsymbol {\Omega }}^{1}$ is an $\operatorname {End} (E)$-valued holomorphic $(1,0)$-form on $X$, called the Higgs field. Additionally, the Higgs field must satisfy $\Phi \wedge \Phi =0$.
A Higgs bundle is (semi-)stable if, for every proper, non-zero coherent subsheaf ${\mathcal {F}}\subset E$ which is preserved by the Higgs field, so that $\Phi ({\mathcal {F}})\subset {\mathcal {F}}\otimes {\boldsymbol {\Omega }}^{1}$, one has
${\frac {\deg({\mathcal {F}})}{\operatorname {rank} ({\mathcal {F}})}}<{\frac {\deg(E)}{\operatorname {rank} (E)}}\quad {\text{(resp. }}\leq {\text{)}}.$
This rational number is called the slope, denoted $\mu (E)$, and the above definition mirrors that of a stable vector bundle. A Higgs bundle is polystable if it is a direct sum of stable Higgs bundles of the same slope, and is therefore semi-stable.
Hermitian Yang–Mills connections and Hitchin's equations
See also: Hermitian Yang–Mills connection
The generalisation of Hitchin's equation to higher dimension can be phrased as an analog of the Hermitian Yang–Mills equations for a certain connection constructed out of the pair $(E,\Phi )$. A Hermitian metric $h$ on a Higgs bundle $(E,\Phi )$ gives rise to a Chern connection $\nabla _{A}$ and curvature $F_{A}$. The condition that $\Phi $ is holomorphic can be phrased as ${\bar {\partial }}_{A}\Phi =0$. Hitchin's equations, on a compact Riemann surface, state that
${\begin{cases}&F_{A}+[\Phi ,\Phi ^{*}]=\lambda \operatorname {Id} _{E}\\&{\bar {\partial }}_{A}\Phi =0\end{cases}}$
for a constant $\lambda =-2\pi i\mu (E)$. In higher dimensions these equations generalise as follows. Define a connection $D$ on $E$ by $D=\nabla _{A}+\Phi +\Phi ^{*}$. This connection is said to be a Hermitian Yang–Mills connection (and the metric a Hermitian Yang–Mills metric) if
$\Lambda _{\omega }F_{D}=\lambda \operatorname {Id} _{E}.$
This reduces to Hitchin's equations for a compact Riemann surface. Note that the connection $D$ is not a Hermitian Yang–Mills connection in the usual sense, as it is not unitary, and the above condition is a non-unitary analogue of the normal HYM condition.
Representations of the fundamental group and harmonic metrics
A representation of the fundamental group $\rho \colon \pi _{1}(X)\to \operatorname {GL} (r,\mathbb {C} )$ gives rise to a vector bundle with flat connection as follows. The universal cover ${\hat {X}}$ of $X$ is a principal bundle over $X$ with structure group $\pi _{1}(X)$. Thus there is an associated bundle to ${\hat {X}}$ given by
$E={\hat {X}}\times _{\rho }\mathbb {C} ^{r}.$
This vector bundle comes naturally equipped with a flat connection $D$. If $h$ is a Hermitian metric on $E$, define an operator $D_{h}''$ as follows. Decompose $D=\partial +{\bar {\partial }}$ into operators of type $(1,0)$ and $(0,1)$, respectively. Let $A'$ be the unique operator of type $(1,0)$ such that the $(1,0)$-connection $A'+{\bar {\partial }}$ preserves the metric $h$. Define $\Phi =(\partial -A')/2$, and set $D_{h}''={\bar {\partial }}+\Phi $. Define the pseudocurvature of $h$ to be $G_{h}=(D_{h}'')^{2}$.
The metric $h$ is said to be harmonic if
$\Lambda _{\omega }G_{h}=0.$
Notice that the condition $G_{h}=0$ is equivalent to the three conditions ${\bar {\partial }}^{2}=0,{\bar {\partial }}\Phi =0,\Phi \wedge \Phi =0$, so if $G_{h}=0$ then the pair $(E,\Phi )$ defines a Higgs bundle with holomorphic structure on $E$ given by the Dolbeault operator ${\bar {\partial }}$.
It is a result of Corlette that if $h$ is harmonic, then it automatically satisfies $G_{h}=0$ and so gives rise to a Higgs bundle.[9]
Moduli spaces
Main article: Moduli space
To each of the three concepts: Higgs bundles, flat connections, and representations of the fundamental group, one can define a moduli space. This requires a notion of isomorphism between these objects. In the following, fix a smooth complex vector bundle $E$. Every Higgs bundle will be considered to have the underlying smooth vector bundle $E$.
• (Higgs bundles) The group of complex gauge transformations ${\mathcal {G}}^{\mathbb {C} }$ acts on the set ${\mathcal {H}}$ of Higgs bundles by the formula $g\cdot (E,\Phi )=(g\cdot E,g\Phi g^{-1})$. If ${\mathcal {H}}^{ss}$ and ${\mathcal {H}}^{s}$ denote the subsets of semistable and stable Higgs bundles, respectively, then one obtains moduli spaces
$M_{Dol}^{ss}:={\mathcal {H}}^{ss}//{\mathcal {G}}^{\mathcal {C}},\qquad M_{Dol}^{s}:={\mathcal {H}}^{s}/{\mathcal {G}}^{\mathcal {C}}$
where these quotients are taken in the sense of geometric invariant theory, so orbits whose closures intersect are identified in the moduli space. These moduli spaces are called the Dolbeault moduli spaces. Notice that by setting $\Phi =0$, one obtains as subsets the moduli spaces of semi-stable and stable holomorphic vector bundles $N_{Dol}^{ss}\subset M_{Dol}^{ss}$ and $N_{Dol}^{s}\subset M_{Dol}^{s}$. It is also true that if one defines the moduli space $M_{Dol}^{ps}$ of polystable Higgs bundles then this space is isomorphic to the space of semi-stable Higgs bundles, as every gauge orbit of semi-stable Higgs bundles contains in its closure a unique orbit of polystable Higgs bundles.
• (Flat connections) The group complex gauge transformations also acts on the set ${\mathcal {A}}$ of flat connections $\nabla $ on the smooth vector bundle $E$. Define the moduli spaces
$M_{dR}:={\mathcal {A}}//{\mathcal {G}}^{\mathcal {C}},\qquad M_{dR}^{*}:={\mathcal {A}}^{*}/{\mathcal {G}}^{\mathcal {C}},$
where ${\mathcal {A}}^{*}$ denotes the subset consisting of irreducible flat connections $\nabla $ which do not split as a direct sum $\nabla =\nabla _{1}\oplus \nabla _{2}$ on some splitting $E=E_{1}\oplus E_{2}$ of the smooth vector bundle $E$. These moduli spaces are called the de Rham moduli spaces.
• (Representations) The set of representations $\operatorname {Hom} (\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))$ of the fundamental group of $X$ is acted on by the general linear group by conjugation of representations. Denote by the superscripts $+$ and $*$ the subsets consisting of semisimple representations and irreducible representations respectively. Then define moduli spaces
$M_{B}^{+}=\operatorname {Hom} ^{+}(\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))//G,\qquad M_{B}^{*}=\operatorname {Hom} ^{*}(\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))/G$
of semisimple and irreducible representations, respectively. These quotients are taken in the sense of geometric invariant theory, where two orbits are identified if their closures intersect. These moduli spaces are called the Betti moduli spaces.
Statement
The nonabelian Hodge theorem can be split into two parts. The first part was proved by Donaldson in the case of rank two Higgs bundles over a compact Riemann surface, and in general by Corlette.[6][9] In general the nonabelian Hodge theorem holds for a smooth complex projective variety $X$, but some parts of the correspondence hold in more generality for compact Kähler manifolds.
Nonabelian Hodge theorem (part 1) — A representation $\rho :\pi _{1}(X)\to \operatorname {GL} (r,\mathbb {C} )$ :\pi _{1}(X)\to \operatorname {GL} (r,\mathbb {C} )} of the fundamental group is semisimple if and only if the flat vector bundle $E={\hat {X}}\times _{\rho }\mathbb {C} ^{r}$ admits a harmonic metric. Furthermore the representation is irreducible if and only if the flat vector bundle is irreducible.
The second part of the theorem was proven by Hitchin in the case of rank two Higgs bundles on a compact Riemann surface, and in general by Simpson.[5][7][8]
Nonabelian Hodge theorem (part 2) — A Higgs bundle $(E,\Phi )$ has a Hermitian Yang–Mills metric if and only if it is polystable. This metric is a harmonic metric, and therefore arises from a semisimple representation of the fundamental group, if and only if the Chern classes $c_{1}(E)$ and $c_{2}(E)$ vanish. Furthermore, a Higgs bundle is stable if and only if it admits an irreducible Hermitian Yang–Mills connection, and therefore comes from an irreducible representation of the fundamental group.
Combined together, the correspondence can be phrased as follows:
Nonabelian Hodge theorem — A Higgs bundle (which is topologically trivial) arises from a semisimple representation of the fundamental group if and only if it is polystable. Furthermore it arises from an irreducible representation if and only if it is stable.
In terms of moduli spaces
The nonabelian Hodge correspondence not only gives a bijection of sets, but homeomorphisms of moduli spaces. Indeed, if two Higgs bundles are isomorphic, in the sense that they can be related by a gauge transformation and therefore correspond to the same point in the Dolbeault moduli space, then the associated representations will also be isomorphic, and give the same point in the Betti moduli space. In terms of the moduli spaces the nonabelian Hodge theorem can be phrased as follows.
Nonabelian Hodge theorem (moduli space version) — There are homeomorphisms $M_{Dol}^{ss}\cong M_{dR}\cong M_{B}^{+}$ of moduli spaces which restrict to homeomorphisms $M_{Dol}^{s}\cong M_{dR}^{*}\cong M_{B}^{*}$.
In general these moduli spaces will be not just topological spaces, but have some additional structure. For example, the Dolbeault moduli space and Betti moduli space $M_{Dol}^{ss},M_{B}^{+}$ are naturally complex algebraic varieties, and where it is smooth, the de Rham moduli space $M_{dR}$ is a Riemannian manifold. On the common locus where these moduli spaces are smooth, the map $M_{dR}\to M_{B}^{+}$ is a diffeomorphism, and since $M_{B}^{+}$ is a complex manifold on the smooth locus, $M_{dR}$ obtains a compatible Riemannian and complex structure, and is therefore a Kähler manifold.
Similarly, on the smooth locus, the map $M_{B}^{+}\to M_{Dol}^{ss}$ is a diffeomorphism. However, even though the Dolbeault and Betti moduli spaces both have natural complex structures, these are not isomorphic. In fact, if they are denoted $I,J$ (for the associated integrable almost complex structures) then $IJ=-JI$. In particular if one defines a third almost complex structure by $K=IJ$ then $I^{2}=J^{2}=K^{2}=IJK=-\operatorname {Id} $. If one combines these three complex structures with the Riemannian metric coming from $M_{dR}$, then on the smooth locus the moduli spaces become a Hyperkähler manifold.
Relation to Hitchin–Kobayashi correspondence and unitary representations
See also: Kobayashi–Hitchin correspondence
If one sets the Higgs field $\Phi $ to zero, then a Higgs bundle is simply a holomorphic vector bundle. This gives an inclusion $N_{Dol}^{ss}\subset M_{Dol}^{ss}$ of the moduli space of semi-stable holomorphic vector bundles into the moduli space of Higgs bundles. The Hitchin–Kobayashi correspondence gives a correspondence between holomorphic vector bundles and Hermitian Yang–Mills connections over compact Kähler manifolds, and can therefore be seen as a special case of the nonabelian Hodge correspondence.
When the underlying vector bundle is topologically trivial, the holonomy of a Hermitian Yang–Mills connection will give rise to a unitary representation of the fundamental group, $\rho :\pi _{1}(X)\to \operatorname {U} (r)$ :\pi _{1}(X)\to \operatorname {U} (r)} . The subset of the Betti moduli space corresponding to the unitary representations, denoted $N_{B}^{+}$, will get mapped isomorphically onto the moduli space of semi-stable vector bundles $N_{Dol}^{ss}$.
Examples
Rank one Higgs bundles on compact Riemann surfaces
The special case where the rank of the underlying vector bundle is one gives rise to a simpler correspondence.[10] Firstly, every line bundle is stable, as there are no proper non-zero subsheaves. In this case, a Higgs bundle consists of a pair $(L,\Phi )$ of a holomorphic line bundle and a holomorphic $(1,0)$-form, since the endomorphism of a line bundle are trivial. In particular, the Higgs field is uncoupled from the holomorphic line bundle, so the moduli space $M_{Dol}$ will split as a product, and the one-form automatically satisfies the condition $\Phi \wedge \Phi =0$. The gauge group of a line bundle is commutative, and so acts trivially on the Higgs field $\Phi $ by conjugation. Thus the moduli space can be identified as a product
$M_{Dol}=\operatorname {Jac} (X)\times H^{0}(X,{\boldsymbol {\Omega }}^{1})$
of the Jacobian variety of $X$, classifying all holomorphic line bundles up to isomorphism, and the vector space $H^{0}(X,{\boldsymbol {\Omega }}^{1})$ of holomorphic $(1,0)$-forms.
In the case of rank one Higgs bundles on compact Riemann surfaces, one obtains a further description of the moduli space. The fundamental group of a compact Riemann surface, a surface group, is given by
$\pi _{1}(X)=\langle a_{1},\dots ,a_{g},b_{1},\dots ,b_{g}\mid [a_{1},b_{1}]\cdots [a_{g},b_{g}]=e\rangle $
where $g$ is the genus of the Riemann surface. The representations of $\pi _{1}(X)$ into the general linear group $\operatorname {GL} (1,\mathbb {C} )=\mathbb {C} ^{*}$ are therefore given by $2g$-tuples of non-zero complex numbers:
$\operatorname {Hom} (\pi _{1}(X),\mathbb {C} ^{*})=(\mathbb {C} ^{*})^{2g}.$
Since $\mathbb {C} ^{*}$ is abelian, the conjugation on this space is trivial, and the Betti moduli space is $M_{B}=(\mathbb {C} ^{*})^{2g}$. On the other hand, by Serre duality, the space of holomorphic $(1,0)$-forms is dual to the sheaf cohomology $H^{1}(X,{\mathcal {O}}_{X})$. The Jacobian variety is an Abelian variety given by the quotient
$\operatorname {Jac} (X)={\frac {H^{1}(X,{\mathcal {O}}_{X})}{H^{1}(X,\mathbb {Z} )}},$
so has tangent spaces given by the vector space $H^{1}(X,{\mathcal {O}}_{X})$, and cotangent bundle
$T^{*}\operatorname {Jac} (X)=\operatorname {Jac} (X)\times H^{1}(X,{\mathcal {O}}_{X})^{*}=\operatorname {Jac} (X)\times H^{0}(X,{\boldsymbol {\Omega }}^{1})=M_{Dol}.$
That is, the Dolbeault moduli space, the moduli space of holomorphic Higgs line bundles, is simply the cotangent bundle to the Jacobian, the moduli space of holomorphic line bundles. The nonabelian Hodge correspondence therefore gives a diffeomorphism
$T^{*}\operatorname {Jac} (X)\cong (\mathbb {C} ^{*})^{2g}$
which is not a biholomorphism. One can check that the natural complex structures on these two spaces are different, and satisfy the relation $IJ=-JI$, giving a hyperkähler structure on the cotangent bundle to the Jacobian.
Generalizations
It is possible to define the notion of a principal $G$-Higgs bundle for a complex reductive algebraic group $G$, a version of Higgs bundles in the category of principal bundles. There is a notion of a stable principal bundle, and one can define a stable principal $G$-Higgs bundle. A version of the nonabelian Hodge theorem holds for these objects, relating principal $G$-Higgs bundles to representations of the fundamental group into $G$.[7][8][11]
Nonabelian Hodge theory
The correspondence between Higgs bundles and representations of the fundamental group can be phrased as a kind of nonabelian Hodge theorem, which is to say, an analogy of the Hodge decomposition of a Kähler manifold, but with coefficients in the nonabelian group $\operatorname {GL} (n,\mathbb {C} )$ instead of the abelian group $\mathbb {C} $. The exposition here follows the discussion by Oscar Garcia-Prada in the appendix to Wells' Differential Analysis on Complex Manifolds.[12]
Hodge decomposition
The Hodge decomposition of a compact Kähler manifold decomposes the complex de Rham cohomology into the finer Dolbeault cohomology:
$H_{dR}^{k}(X,\mathbb {C} )=\bigoplus _{p+q=k}H_{Dol}^{p,q}(X).$
At degree one this gives a direct sum
$H^{1}(X,\mathbb {C} )=H^{0,1}(X)\oplus H^{1,0}(X)\cong H^{1}(X,{\mathcal {O}}_{X})\oplus H^{0}(X,{\boldsymbol {\Omega }}^{1})$
where we have applied the Dolbeault theorem to phrase the Dolbeault cohomology in terms of sheaf cohomology of the sheaf of holomorphic $(1,0)$-forms ${\boldsymbol {\Omega }}^{1},$ and the structure sheaf ${\mathcal {O}}_{X}$ of holomorphic functions on $X$.
Nonabelian cohomology
When constructing sheaf cohomology, the coefficient sheaf ${\mathcal {F}}$ is always a sheaf of abelian groups. This is because for an abelian group, every subgroup is normal, so the quotient group
${\check {H}}^{k}(X,{\mathcal {F}})=Z^{k}(X,{\mathcal {F}})/B^{k}(X,{\mathcal {F}})$
of sheaf cocycles by sheaf coboundaries is always well-defined. When the sheaf ${\mathcal {F}}$ is not abelian, these quotients are not necessarily well-defined, and so sheaf cohomology theories do not exist, except in the following special cases:
• $k=0$: The 0th sheaf cohomology group is always the space of global sections of the sheaf ${\mathcal {F}}$, so is always well-defined even if ${\mathcal {F}}$ is nonabelian.
• $k=1$: The 1st sheaf cohomology set is well-defined for a nonabelian sheaf ${\mathcal {F}}$, but it is not itself a quotient group.
• $k=2$: In some special cases, an analogue of the second degree sheaf cohomology can be defined for nonabelian sheaves using the theory of gerbes.
A key example of nonabelian cohomology occurs when the coefficient sheaf is ${\mathcal {GL}}(r,\mathbb {C} )$, the sheaf of holomorphic functions into the complex general linear group. In this case it is a well-known fact from Čech cohomology that the cohomology set
${\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))$
is in one-to-one correspondence with the set of holomorphic vector bundles of rank $r$ on $X$, up to isomorphism. Notice that there is a distinguished holomorphic vector bundle of rank $r$, the trivial vector bundle, so this is actually a cohomology pointed set. In the special case $r=1$ the general linear group is the abelian group $\mathbb {C} ^{*}$ of non-zero complex numbers with respect to multiplication. In this case one obtains the group of holomorphic line bundles up to isomorphism, otherwise known as the Picard group.
Nonabelian Hodge theorem
The first cohomology group $H^{1}(X,\mathbb {C} )$ is isomorphic to the group of homomorphisms from the fundamental group $\pi _{1}(X)$ to $\mathbb {C} $. This can be understood, for example, by applying the Hurewicz theorem. Thus the regular Hodge decomposition mentioned above may be phrased as
$\operatorname {Hom} (\pi _{1}(X),\mathbb {C} )\cong H^{1}(X,{\mathcal {O}}_{X})\oplus H^{0}(X,{\boldsymbol {\Omega }}^{1}).$
The nonabelian Hodge correspondence gives an analogy of this statement of the Hodge theorem for nonabelian cohomology, as follows. A Higgs bundle consists of a pair $(E,\Phi )$ where $E$ is a holomorphic vector bundle, and $\Phi \in H^{0}(X,\operatorname {End} (E)\otimes {\boldsymbol {\Omega }}^{1})$ is a holomorphic, endomorphism-valued $(1,0)$-form. The holomorphic vector bundle $E$ may be identified with an element of ${\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))$ as mentioned above. Thus a Higgs bundle may be thought of as an element of the direct product
$(E,\Phi )\in {\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))\oplus H^{0}(X,\operatorname {End} (E)\otimes {\boldsymbol {\Omega }}^{1}).$
The nonabelian Hodge correspondence gives an isomorphism from the moduli space of $\operatorname {GL} (r,\mathbb {C} )$-representations of the fundamental group $\pi _{1}(X)$ to the moduli space of Higgs bundles, which could therefore be written as an isomorphism
$\operatorname {Rep} (\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))\cong {\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))\oplus H^{0}(X,\operatorname {End} (E)\otimes {\boldsymbol {\Omega }}^{1}).$
This can be seen as an analogy of the regular Hodge decomposition above. The moduli space of representations $\operatorname {Rep} (\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))$ plays the role of the first cohomology of $X$ with nonabelian coefficients, the cohomology set ${\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))$ plays the role of the space $H^{1}(X,{\mathcal {O}}_{X})$, and the group $H^{0}(X,\operatorname {End} (E)\otimes {\boldsymbol {\Omega }}^{1})$ plays the role of the holomorphic (1,0)-forms $H^{0}(X,{\boldsymbol {\Omega }}^{1})$.
The isomorphism here is written $\cong $, but this is not an actual isomorphism of sets, as the moduli space of Higgs bundles is not literally given by the direct sum above, as this is only an analogy.
Hodge structure
See also: Hodge structure
The moduli space $M_{Dol}^{ss}$ of semi-stable Higgs bundles has a natural action of the multiplicative group $\mathbb {C} ^{*}$, given by scaling the Higgs field: $\lambda \cdot (E,\Phi )=(E,\lambda \Phi )$ for $\lambda \in \mathbb {C} ^{*}$. For abelian cohomology, such a $\mathbb {C} ^{*}$ action gives rise to a Hodge structure, which is a generalisation of the Hodge decomposition of the cohomology of a compact Kähler manifold. One way of understanding the nonabelian Hodge theorem is to use the $\mathbb {C} ^{*}$ action on the moduli space $M_{B}^{+}$ to obtain a Hodge filtration. This can lead to new topological invariants of the underlying manifold $X$. For example, one obtains restrictions on which groups may appear as the fundamental groups of compact Kähler manifolds in this way.[7]
References
1. Narasimhan, M. S.; Seshadri, C. S. (1965). "Stable and unitary vector bundles on a compact Riemann surface". Annals of Mathematics. 82 (3): 540–567. doi:10.2307/1970710. JSTOR 1970710. MR 0184252.
2. Donaldson, Simon K. (1983), "A new proof of a theorem of Narasimhan and Seshadri", Journal of Differential Geometry, 18 (2): 269–277, doi:10.4310/jdg/1214437664, MR 0710055
3. Donaldson, Simon K. (1985). "Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundle". Proceedings of the London Mathematical Society. 3. 50 (1): 1–26. doi:10.1112/plms/s3-50.1.1. MR 0765366.
4. Uhlenbeck, Karen; Yau, Shing-Tung (1986), "On the existence of Hermitian–Yang–Mills connections in stable vector bundles", Communications on Pure and Applied Mathematics, 39: S257–S293, doi:10.1002/cpa.3160390714, ISSN 0010-3640, MR 0861491
5. Hitchin, Nigel J. (1987). "The self-duality equations on a Riemann surface". Proceedings of the London Mathematical Society. 55 (1): 59–126. doi:10.1112/plms/s3-55.1.59. MR 0887284.
6. Donaldson, Simon K. (1987). "Twisted harmonic maps and the self-duality equations". Proceedings of the London Mathematical Society. 55 (1): 127–131. doi:10.1112/plms/s3-55.1.127. MR 0887285.
7. Simpson, Carlos T. (1991), "Nonabelian Hodge theory", Proceedings of the International Congress of Mathematicians (Kyoto, 1990) (PDF), vol. 1, Tokyo: Math. Soc. Japan, pp. 747–756, MR 1159261
8. Simpson, Carlos T. (1992). "Higgs bundles and local systems". Publications Mathématiques de l'IHÉS. 75: 5–95. doi:10.1007/BF02699491. MR 1179076. S2CID 56417181.
9. Corlette, Kevin (1988). "Flat G-bundles with canonical metrics". Journal of Differential Geometry. 28 (3): 361–382. doi:10.4310/jdg/1214442469. MR 0965220.
10. Goldman, William M.; Xia, Eugene Z. (2008). "Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces". Memoirs of the American Mathematical Society. 193 (904): viii+69 pp. arXiv:math/0402429. doi:10.1090/memo/0904. ISSN 0065-9266. MR 2400111. S2CID 2865489.
11. Anchouche, Boudjemaa; Biswas, Indranil (2001). "Einstein–Hermitian connections on polystable principal bundles over a compact Kähler manifold" (PDF). American Journal of Mathematics. 123 (2): 207–228. doi:10.1353/ajm.2001.0007. MR 1828221. S2CID 122182133.
12. Wells, Raymond O., Jr. (1980). Differential analysis on complex manifolds. Graduate Texts in Mathematics. Vol. 65 (2nd ed.). New York-Berlin: Springer-Verlag. ISBN 0-387-90419-0. MR 0608414.{{cite book}}: CS1 maint: multiple names: authors list (link)
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Wikipedia
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Sims conjecture
In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims.[1] He conjectured that if $G$ is a primitive permutation group on a finite set $S$ and $G_{\alpha }$ denotes the stabilizer of the point $\alpha $ in $S$, then there exists an integer-valued function $f$ such that $f(d)\geq |G_{\alpha }|$ for $d$ the length of any orbit of $G_{\alpha }$ in the set $S\setminus \{\alpha \}$.
The conjecture was proven by Peter Cameron, Cheryl Praeger, Jan Saxl, and Gary Seitz using the classification of finite simple groups, in particular the fact that only finitely many isomorphism types of sporadic groups exist.
The theorem reads precisely as follows.[2]
Theorem — There exists a function $f:\mathbb {N} \to \mathbb {N} $ such that whenever $G$ is a primitive permutation group and $h>1$ is the length of a non-trivial orbit of a point stabilizer $H$ in $G$, then the order of $H$ is at most $f(h)$.
Thus, in a primitive permutation group with "large" stabilizers, these stabilizers cannot have any small orbit. A consequence of their proof is that there exist only finitely many connected distance-transitive graphs having degree greater than 2.[3][4][5]
References
1. Sims, Charles C. (1967). "Graphs and finite permutation groups". Mathematische Zeitschrift. 95 (1): 76–86. doi:10.1007/BF01117534. S2CID 186227555.
2. Pyber, László; Tracey, Gareth (2021). "Some simplifications in the proof of the Sims conjecture". arXiv:2102.06670 [math.GR].
3. Cameron, Peter J.; Praeger, Cheryl E.; Saxl, Jan; Seitz, Gary M. (1983). "On the Sims conjecture and distance transitive graphs". Bulletin of the London Mathematical Society. 15 (5): 499–506. doi:10.1112/blms/15.5.499.
4. Cameron, Peter J. (1982). "There are only finitely many distance-transitive graphs of given valency greater than two". Combinatorica. 2 (1): 9–13. doi:10.1007/BF02579277. S2CID 6483108.
5. Isaacs, I. Martin (2011). Finite Group Theory. American Mathematical Society. ISBN 9780821843444. OCLC 935038216.
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Wikipedia
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Robert Simson
Robert Simson (14 October 1687 – 1 October 1768) was a Scottish mathematician and professor of mathematics at the University of Glasgow. The Simson line is named after him.[1]
Robert Simson
Robert Simson
Born(1687-10-14)14 October 1687
Died1 October 1768(1768-10-01) (aged 80)
His college residence at Glasgow
Resting placeRamshorn Cemetery
NationalityBritish
Alma materUniversity of Glasgow
Scientific career
FieldsMathematics, especially geometry
InstitutionsUniversity of Glasgow
Christ's Hospital
Notable studentsMaclaurin
Matthew Stewart
William Trail
James Williamson (mathematician)
InfluencesEdmond Halley
Biography
Robert Simson was born on 14 October 1687,[2][3] probably the eldest of the seventeen children, all male, of John Simson, a Glasgow merchant, and Agnes, daughter of Patrick Simson, minister of Renfrew; only six of them reached adulthood.[4]
Simson matriculated at the University of Glasgow in 1701, intending to enter the Church. He followed the course in the faculty of arts (Latin, Greek, logic, natural philosophy) and then concentrated on studying theology and Semitic languages.[2] Mathematics was not taught at the university, but by reading Sinclair's Tuyrocinia Mathematica in Novem Tractatus and then Euclid’s Elements Simson soon became deeply interested in mathematics and especially geometry. His efforts impressed the university Senate to such an extent that they offered him the chair of mathematics, to replace the recently-dismissed Sinclair. As he had had no formal training in the subject, Simson turned down the offer but agreed to take up the post a year later, during which time he would increase his knowledge of mathematics.
After a failed attempt to go to Oxford, Simson spent his year in London at Christ's Hospital. During this time he made valuable contacts with several prominent mathematicians, including John Caswell, James Jurin (secretary of the Royal Society), Humphrey Ditton and, most importantly, Edmond Halley.[2]
Simson was admitted professor of mathematics at Glasgow, aged 23, on 20 November 1711, where his first task was to design a two-year course in mathematics, some of which he taught himself; his lectures included geometry, of course, and algebra, logarithms and optics. Among his students were Maclaurin, Matthew Stewart, and William Trail. He resigned the post in 1761, and was succeeded by another of his pupils Rev Prof James Williamson FRSE (1725-1795).[5]
During his time at Glasgow Simson noted in 1753[6] that, as the Fibonacci numbers increased in magnitude, the ratio between adjacent numbers approached the golden ratio, whose value is
$\varphi ={\frac {1+{\sqrt {5}}}{2}}=$1.618033988749....
As for the man himself, “Simson appears to have been tall and of good stature. In spite of his great scholarship he was a modest, unassuming man who was very cautious in promoting his own work. He enjoyed good company and presided over the weekly meetings of a dining club that he had instituted … He had a special interest in botany, in which he was an acknowledged expert”.[7]
Robert Simson did not marry. He died, aged 80, in his college residence at Glasgow on 1 October 1768, and was interred in the Blackfriars Burying Ground (now known as Ramshorn Cemetery), where, in the south wall, is placed to his memory a plain marble tablet, with a highly and justly complimentary inscription”.[8] Simson's library, including some of his own works, was bequeathed to the university on his death. It consists of about 850 printed books, mainly early mathematical and astronomical texts.[9]
Subscriptions towards the erection of a monument to Dr Simson were collected in 1865, with the Senate of the College of Glasgow, the (thirteenth) Earl of Eglinton and Winton, and the Earl Stanhope each donating £10; and John Carrick Moore – the first cousin twice removed of Robert Simson – giving £15.[10] The memorial, designed by Frederick Thomas Pilkington, is “a large octagonal monument with carved Egyptian details, topped with a ball finial”.[11] It is situated on a hilltop in West Kilbride cemetery.
Works
Simson's contributions to mathematical knowledge took the form of critical editions and commentaries on the works of the ancient geometers.[7] The first of his published writings is a paper in the Philosophical Transactions (1723, vol. xl. p. 330) on Euclid's Porisms.
Then followed Sectionum conicarum libri V. (Edinburgh, 1735), a second edition of which, with additions, appeared in 1750. The first three books of this treatise were translated into English and, several times, printed as The Elements of the Conic Sections. In 1749, was published Apollonii Pergaei locorum planorum libri II., a restoration of Apollonius's lost treatise, founded on the lemmas given in the seventh book of Pappus's Mathematical Collection.
In 1756, appeared, both in Latin and in English, the first edition of his Euclid's Elements. This work, which contained only the first six and the eleventh and twelfth books, and to which, in its English version, he added the Data in 1762, was for long the standard text of Euclid in England.
After Simson's death, restorations of Apollonius's treatise De section determinata and of Euclid's treatise De Porismatibus were printed for private circulation in 1776, at the expense of Earl Stanhope, in a volume with the title Roberti Simson opera quaedam reliqua. The volume contains also dissertations on Logarithms and on the Limits of Quantities and Ratios, and a few problems illustrating the ancient geometrical analysis.[12]
References
1. Robert Simson. University of Glasgow (multi-tab page)
2. Sneddon, Ian N. "Simon, Robert". Encyclopedia.com. Retrieved 27 April 2023.
3. Tweddle gives the date as 18 October in "Simson, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. (Subscription or UK public library membership required.)
4. Simpson, Helen A (1927). Early Records of Simpson Families in Scotland, North Ireland, and Eastern United States (PDF). Philadelphia: J. B. Lippincott Company I. Retrieved 29 April 2023.
5. "Mathematics in Glasgow University in 1883".
6. Rankin, R A (August 1995). "Robert Simson". MacTutor. Retrieved 1 May 2023.
7. "Simson, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. (Subscription or UK public library membership required.)
8. "Sketch of the life of Dr Robert Simson, of Glasgow". The Ardrossan and Saltcoats Herald. 4 February 1865. p. 3.
9. "Archives and Special Collections: Simson Collection". University of Glasgow. Retrieved 30 April 2023.
10. "Simson Monument". The Ardrossan and Saltcoats Herald. 4 October 1865.
11. "Robert Simson's Memorial". Ayrshire & Arran. Retrieved 30 April 2023.
12. Simson, Robert; Clow, James. Opera quaedam reliqua etc. Glasguae: Andreas Foulis. Retrieved 1 May 2023.
Further reading
• William Trail (1812) Life and Writings of Robert Simson at Google Books
• Charles Hutton (1815) Mathematical and Philosophical Dictionary, volume II, p. 395-398 (online copy, p. 395, at Google Books)
External links
Wikimedia Commons has media related to Robert Simson.
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Simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. For large numbers of local optima, SA can find the global optima.[1] It is often used when the search space is discrete (for example the traveling salesman problem, the boolean satisfiability problem, protein structure prediction, and job-shop scheduling). For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to exact algorithms such as gradient descent or branch and bound.
The name of the algorithm comes from annealing in metallurgy, a technique involving heating and controlled cooling of a material to alter its physical properties. Both are attributes of the material that depend on their thermodynamic free energy. Heating and cooling the material affects both the temperature and the thermodynamic free energy or Gibbs energy. Simulated annealing can be used for very hard computational optimization problems where exact algorithms fail; even though it usually achieves an approximate solution to the global minimum, it could be enough for many practical problems.
The problems solved by SA are currently formulated by an objective function of many variables, subject to several mathematical constraints. In practice, the constraint can be penalized as part of the objective function.
Similar techniques have been independently introduced on several occasions, including Pincus (1970),[2] Khachaturyan et al (1979,[3] 1981[4]), Kirkpatrick, Gelatt and Vecchi (1983), and Cerny (1985).[5] In 1983, this approach was used by Kirkpatrick, Gelatt Jr., Vecchi,[6] for a solution of the traveling salesman problem. They also proposed its current name, simulated annealing.
This notion of slow cooling implemented in the simulated annealing algorithm is interpreted as a slow decrease in the probability of accepting worse solutions as the solution space is explored. Accepting worse solutions allows for a more extensive search for the global optimal solution. In general, simulated annealing algorithms work as follows. The temperature progressively decreases from an initial positive value to zero. At each time step, the algorithm randomly selects a solution close to the current one, measures its quality, and moves to it according to the temperature-dependent probabilities of selecting better or worse solutions, which during the search respectively remain at 1 (or positive) and decrease toward zero.
The simulation can be performed either by a solution of kinetic equations for probability density functions,[7][8] or by using a stochastic sampling method.[6][9] The method is an adaptation of the Metropolis–Hastings algorithm, a Monte Carlo method to generate sample states of a thermodynamic system, published by N. Metropolis et al. in 1953.[10]
Overview
The state s of some physical systems, and the function E(s) to be minimized, is analogous to the internal energy of the system in that state. The goal is to bring the system, from an arbitrary initial state, to a state with the minimum possible energy.
The basic iteration
At each step, the simulated annealing heuristic considers some neighboring state s* of the current state s, and probabilistically decides between moving the system to state s* or staying in-state s. These probabilities ultimately lead the system to move to states of lower energy. Typically this step is repeated until the system reaches a state that is good enough for the application, or until a given computation budget has been exhausted.
The neighbours of a state
Optimization of a solution involves evaluating the neighbours of a state of the problem, which are new states produced through conservatively altering a given state. For example, in the travelling salesman problem each state is typically defined as a permutation of the cities to be visited, and the neighbors of any state are the set of permutations produced by swapping any two of these cities. The well-defined way in which the states are altered to produce neighboring states is called a "move", and different moves give different sets of neighboring states. These moves usually result in minimal alterations of the last state, in an attempt to progressively improve the solution through iteratively improving its parts (such as the city connections in the traveling salesman problem).
Simple heuristics like hill climbing, which move by finding better neighbour after better neighbour and stop when they have reached a solution which has no neighbours that are better solutions, cannot guarantee to lead to any of the existing better solutions – their outcome may easily be just a local optimum, while the actual best solution would be a global optimum that could be different. Metaheuristics use the neighbours of a solution as a way to explore the solution space, and although they prefer better neighbours, they also accept worse neighbours in order to avoid getting stuck in local optima; they can find the global optimum if run for a long enough amount of time.
Acceptance probabilities
The probability of making the transition from the current state $s$ to a candidate new state $s_{\mathrm {new} }$ is specified by an acceptance probability function $P(e,e_{\mathrm {new} },T)$, that depends on the energies $e=E(s)$ and $e_{\mathrm {new} }=E(s_{\mathrm {new} })$ of the two states, and on a global time-varying parameter $T$ called the temperature. States with a smaller energy are better than those with a greater energy. The probability function $P$ must be positive even when $e_{\mathrm {new} }$ is greater than $e$. This feature prevents the method from becoming stuck at a local minimum that is worse than the global one.
When $T$ tends to zero, the probability $P(e,e_{\mathrm {new} },T)$ must tend to zero if $e_{\mathrm {new} }>e$ and to a positive value otherwise. For sufficiently small values of $T$, the system will then increasingly favor moves that go "downhill" (i.e., to lower energy values), and avoid those that go "uphill." With $T=0$ the procedure reduces to the greedy algorithm, which makes only the downhill transitions.
In the original description of simulated annealing, the probability $P(e,e_{\mathrm {new} },T)$ was equal to 1 when $e_{\mathrm {new} }<e$—i.e., the procedure always moved downhill when it found a way to do so, irrespective of the temperature. Many descriptions and implementations of simulated annealing still take this condition as part of the method's definition. However, this condition is not essential for the method to work.
The $P$ function is usually chosen so that the probability of accepting a move decreases when the difference $e_{\mathrm {new} }-e$ increases—that is, small uphill moves are more likely than large ones. However, this requirement is not strictly necessary, provided that the above requirements are met.
Given these properties, the temperature $T$ plays a crucial role in controlling the evolution of the state $s$ of the system with regard to its sensitivity to the variations of system energies. To be precise, for a large $T$, the evolution of $s$ is sensitive to coarser energy variations, while it is sensitive to finer energy variations when $T$ is small.
The annealing schedule
Fast
Slow
Example illustrating the effect of cooling schedule on the performance of simulated annealing. The problem is to rearrange the pixels of an image so as to minimize a certain potential energy function, which causes similar colors to attract at short range and repel at a slightly larger distance. The elementary moves swap two adjacent pixels. These images were obtained with a fast cooling schedule (left) and a slow cooling schedule (right), producing results similar to amorphous and crystalline solids, respectively.
The name and inspiration of the algorithm demand an interesting feature related to the temperature variation to be embedded in the operational characteristics of the algorithm. This necessitates a gradual reduction of the temperature as the simulation proceeds. The algorithm starts initially with $T$ set to a high value (or infinity), and then it is decreased at each step following some annealing schedule—which may be specified by the user, but must end with $T=0$ towards the end of the allotted time budget. In this way, the system is expected to wander initially towards a broad region of the search space containing good solutions, ignoring small features of the energy function; then drift towards low-energy regions that become narrower and narrower, and finally move downhill according to the steepest descent heuristic.
For any given finite problem, the probability that the simulated annealing algorithm terminates with a global optimal solution approaches 1 as the annealing schedule is extended.[11] This theoretical result, however, is not particularly helpful, since the time required to ensure a significant probability of success will usually exceed the time required for a complete search of the solution space.[12]
Pseudocode
The following pseudocode presents the simulated annealing heuristic as described above. It starts from a state s0 and continues until a maximum of kmax steps have been taken. In the process, the call neighbour(s) should generate a randomly chosen neighbour of a given state s; the call random(0, 1) should pick and return a value in the range [0, 1], uniformly at random. The annealing schedule is defined by the call temperature(r), which should yield the temperature to use, given the fraction r of the time budget that has been expended so far.
• Let s = s0
• For k = 0 through kmax (exclusive):
• T ← temperature( 1 - (k+1)/kmax )
• Pick a random neighbour, snew ← neighbour(s)
• If P(E(s), E(snew), T) ≥ random(0, 1):
• s ← snew
• Output: the final state s
Selecting the parameters
In order to apply the simulated annealing method to a specific problem, one must specify the following parameters: the state space, the energy (goal) function E(), the candidate generator procedure neighbour(), the acceptance probability function P(), and the annealing schedule temperature() AND initial temperature init_temp. These choices can have a significant impact on the method's effectiveness. Unfortunately, there are no choices of these parameters that will be good for all problems, and there is no general way to find the best choices for a given problem. The following sections give some general guidelines.
Sufficiently near neighbour
Simulated annealing may be modeled as a random walk on a search graph, whose vertices are all possible states, and whose edges are the candidate moves. An essential requirement for the neighbour() function is that it must provide a sufficiently short path on this graph from the initial state to any state which may be the global optimum – the diameter of the search graph must be small. In the traveling salesman example above, for instance, the search space for n = 20 cities has n! = 2,432,902,008,176,640,000 (2.4 quintillion) states; yet the number of neighbors of each vertex is $\sum _{k=1}^{n-1}k={\frac {n(n-1)}{2}}=190$ edges (coming from n choose 2), and the diameter of the graph is $n-1$.
Transition probabilities
To investigate the behavior of simulated annealing on a particular problem, it can be useful to consider the transition probabilities that result from the various design choices made in the implementation of the algorithm. For each edge $(s,s')$ of the search graph, the transition probability is defined as the probability that the simulated annealing algorithm will move to state $s'$ when its current state is $s$. This probability depends on the current temperature as specified by temperature(), on the order in which the candidate moves are generated by the neighbour() function, and on the acceptance probability function P(). (Note that the transition probability is not simply $P(e,e',T)$, because the candidates are tested serially.)
Acceptance probabilities
The specification of neighbour(), P(), and temperature() is partially redundant. In practice, it's common to use the same acceptance function P() for many problems, and adjust the other two functions according to the specific problem.
In the formulation of the method by Kirkpatrick et al., the acceptance probability function $P(e,e',T)$ was defined as 1 if $e'<e$, and $\exp(-(e'-e)/T)$ otherwise. This formula was superficially justified by analogy with the transitions of a physical system; it corresponds to the Metropolis–Hastings algorithm, in the case where T=1 and the proposal distribution of Metropolis–Hastings is symmetric. However, this acceptance probability is often used for simulated annealing even when the neighbour() function, which is analogous to the proposal distribution in Metropolis–Hastings, is not symmetric, or not probabilistic at all. As a result, the transition probabilities of the simulated annealing algorithm do not correspond to the transitions of the analogous physical system, and the long-term distribution of states at a constant temperature $T$ need not bear any resemblance to the thermodynamic equilibrium distribution over states of that physical system, at any temperature. Nevertheless, most descriptions of simulated annealing assume the original acceptance function, which is probably hard-coded in many implementations of SA.
In 1990, Moscato and Fontanari,[13] and independently Dueck and Scheuer,[14] proposed that a deterministic update (i.e. one that is not based on the probabilistic acceptance rule) could speed-up the optimization process without impacting on the final quality. Moscato and Fontanari conclude from observing the analogous of the "specific heat" curve of the "threshold updating" annealing originating from their study that "the stochasticity of the Metropolis updating in the simulated annealing algorithm does not play a major role in the search of near-optimal minima". Instead, they proposed that "the smoothening of the cost function landscape at high temperature and the gradual definition of the minima during the cooling process are the fundamental ingredients for the success of simulated annealing." The method subsequently popularized under the denomination of "threshold accepting" due to Dueck and Scheuer's denomination. In 2001, Franz, Hoffmann and Salamon showed that the deterministic update strategy is indeed the optimal one within the large class of algorithms that simulate a random walk on the cost/energy landscape.[15]
Efficient candidate generation
When choosing the candidate generator neighbour(), one must consider that after a few iterations of the simulated annealing algorithm, the current state is expected to have much lower energy than a random state. Therefore, as a general rule, one should skew the generator towards candidate moves where the energy of the destination state $s'$ is likely to be similar to that of the current state. This heuristic (which is the main principle of the Metropolis–Hastings algorithm) tends to exclude "very good" candidate moves as well as "very bad" ones; however, the former are usually much less common than the latter, so the heuristic is generally quite effective.
In the traveling salesman problem above, for example, swapping two consecutive cities in a low-energy tour is expected to have a modest effect on its energy (length); whereas swapping two arbitrary cities is far more likely to increase its length than to decrease it. Thus, the consecutive-swap neighbour generator is expected to perform better than the arbitrary-swap one, even though the latter could provide a somewhat shorter path to the optimum (with $n-1$ swaps, instead of $n(n-1)/2$).
A more precise statement of the heuristic is that one should try first candidate states $s'$ for which $P(E(s),E(s'),T)$ is large. For the "standard" acceptance function $P$ above, it means that $E(s')-E(s)$ is on the order of $T$ or less. Thus, in the traveling salesman example above, one could use a neighbour() function that swaps two random cities, where the probability of choosing a city-pair vanishes as their distance increases beyond $T$.
Barrier avoidance
When choosing the candidate generator neighbour() one must also try to reduce the number of "deep" local minima—states (or sets of connected states) that have much lower energy than all its neighbouring states. Such "closed catchment basins" of the energy function may trap the simulated annealing algorithm with high probability (roughly proportional to the number of states in the basin) and for a very long time (roughly exponential on the energy difference between the surrounding states and the bottom of the basin).
As a rule, it is impossible to design a candidate generator that will satisfy this goal and also prioritize candidates with similar energy. On the other hand, one can often vastly improve the efficiency of simulated annealing by relatively simple changes to the generator. In the traveling salesman problem, for instance, it is not hard to exhibit two tours $A$, $B$, with nearly equal lengths, such that (1) $A$ is optimal, (2) every sequence of city-pair swaps that converts $A$ to $B$ goes through tours that are much longer than both, and (3) $A$ can be transformed into $B$ by flipping (reversing the order of) a set of consecutive cities. In this example, $A$ and $B$ lie in different "deep basins" if the generator performs only random pair-swaps; but they will be in the same basin if the generator performs random segment-flips.
Cooling schedule
The physical analogy that is used to justify simulated annealing assumes that the cooling rate is low enough for the probability distribution of the current state to be near thermodynamic equilibrium at all times. Unfortunately, the relaxation time—the time one must wait for the equilibrium to be restored after a change in temperature—strongly depends on the "topography" of the energy function and on the current temperature. In the simulated annealing algorithm, the relaxation time also depends on the candidate generator, in a very complicated way. Note that all these parameters are usually provided as black box functions to the simulated annealing algorithm. Therefore, the ideal cooling rate cannot be determined beforehand, and should be empirically adjusted for each problem. Adaptive simulated annealing algorithms address this problem by connecting the cooling schedule to the search progress. Other adaptive approach as Thermodynamic Simulated Annealing,[16] automatically adjusts the temperature at each step based on the energy difference between the two states, according to the laws of thermodynamics.
Restarts
Sometimes it is better to move back to a solution that was significantly better rather than always moving from the current state. This process is called restarting of simulated annealing. To do this we set s and e to sbest and ebest and perhaps restart the annealing schedule. The decision to restart could be based on several criteria. Notable among these include restarting based on a fixed number of steps, based on whether the current energy is too high compared to the best energy obtained so far, restarting randomly, etc.
Related methods
• Interacting Metropolis–Hasting algorithms (a.k.a. sequential Monte Carlo[17]) combines simulated annealing moves with an acceptance-rejection of the best fitted individuals equipped with an interacting recycling mechanism.
• Quantum annealing uses "quantum fluctuations" instead of thermal fluctuations to get through high but thin barriers in the target function.
• Stochastic tunneling attempts to overcome the increasing difficulty simulated annealing runs have in escaping from local minima as the temperature decreases, by 'tunneling' through barriers.
• Tabu search normally moves to neighbouring states of lower energy, but will take uphill moves when it finds itself stuck in a local minimum; and avoids cycles by keeping a "taboo list" of solutions already seen.
• Dual-phase evolution is a family of algorithms and processes (to which simulated annealing belongs) that mediate between local and global search by exploiting phase changes in the search space.
• Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.
• Genetic algorithms maintain a pool of solutions rather than just one. New candidate solutions are generated not only by "mutation" (as in SA), but also by "recombination" of two solutions from the pool. Probabilistic criteria, similar to those used in SA, are used to select the candidates for mutation or combination, and for discarding excess solutions from the pool.
• Memetic algorithms search for solutions by employing a set of agents that both cooperate and compete in the process; sometimes the agents' strategies involve simulated annealing procedures for obtaining high quality solutions before recombining them.[18] Annealing has also been suggested as a mechanism for increasing the diversity of the search.[19]
• Graduated optimization digressively "smooths" the target function while optimizing.
• Ant colony optimization (ACO) uses many ants (or agents) to traverse the solution space and find locally productive areas.
• The cross-entropy method (CE) generates candidates solutions via a parameterized probability distribution. The parameters are updated via cross-entropy minimization, so as to generate better samples in the next iteration.
• Harmony search mimics musicians in improvisation process where each musician plays a note for finding a best harmony all together.
• Stochastic optimization is an umbrella set of methods that includes simulated annealing and numerous other approaches.
• Particle swarm optimization is an algorithm modelled on swarm intelligence that finds a solution to an optimization problem in a search space, or model and predict social behavior in the presence of objectives.
• The runner-root algorithm (RRA) is a meta-heuristic optimization algorithm for solving unimodal and multimodal problems inspired by the runners and roots of plants in nature.
• Intelligent water drops algorithm (IWD) which mimics the behavior of natural water drops to solve optimization problems
• Parallel tempering is a simulation of model copies at different temperatures (or Hamiltonians) to overcome the potential barriers.
• Multi-objective simulated annealing algorithms have been used in multi-objective optimization.[20]
See also
• Adaptive simulated annealing
• Automatic label placement
• Combinatorial optimization
• Dual-phase evolution
• Graph cuts in computer vision
• Intelligent water drops algorithm
• Markov chain
• Molecular dynamics
• Multidisciplinary optimization
• Particle swarm optimization
• Place and route
• Quantum annealing
• Traveling salesman problem
References
1. "What is Simulated Annealing?". www.cs.cmu.edu. Retrieved 2023-05-13.
2. Pincus, Martin (Nov–Dec 1970). "A Monte-Carlo Method for the Approximate Solution of Certain Types of Constrained Optimization Problems". Journal of the Operations Research Society of America. 18 (6): 967–1235. doi:10.1287/opre.18.6.1225.
3. Khachaturyan, A.: Semenovskaya, S.: Vainshtein B., Armen (1979). "Statistical-Thermodynamic Approach to Determination of Structure Amplitude Phases". Soviet Physics, Crystallography. 24 (5): 519–524.{{cite journal}}: CS1 maint: multiple names: authors list (link)
4. Khachaturyan, A.; Semenovskaya, S.; Vainshtein, B. (1981). "The Thermodynamic Approach to the Structure Analysis of Crystals". Acta Crystallographica. A37 (5): 742–754. Bibcode:1981AcCrA..37..742K. doi:10.1107/S0567739481001630.{{cite journal}}: CS1 maint: multiple names: authors list (link)
5. Laarhoven, P. J. M. van (Peter J. M.) (1987). Simulated annealing : theory and applications. Aarts, E. H. L. (Emile H. L.). Dordrecht: D. Reidel. ISBN 90-277-2513-6. OCLC 15548651.
6. Kirkpatrick, S.; Gelatt Jr, C. D.; Vecchi, M. P. (1983). "Optimization by Simulated Annealing". Science. 220 (4598): 671–680. Bibcode:1983Sci...220..671K. CiteSeerX 10.1.1.123.7607. doi:10.1126/science.220.4598.671. JSTOR 1690046. PMID 17813860. S2CID 205939.
7. Khachaturyan, A.; Semenovskaya, S.; Vainshtein, B. (1979). "Statistical-Thermodynamic Approach to Determination of Structure Amplitude Phases". Sov.Phys. Crystallography. 24 (5): 519–524.
8. Khachaturyan, A.; Semenovskaya, S.; Vainshtein, B. (1981). "The Thermodynamic Approach to the Structure Analysis of Crystals". Acta Crystallographica. 37 (A37): 742–754. Bibcode:1981AcCrA..37..742K. doi:10.1107/S0567739481001630.
9. Černý, V. (1985). "Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm". Journal of Optimization Theory and Applications. 45: 41–51. doi:10.1007/BF00940812. S2CID 122729427.
10. Metropolis, Nicholas; Rosenbluth, Arianna W.; Rosenbluth, Marshall N.; Teller, Augusta H.; Teller, Edward (1953). "Equation of State Calculations by Fast Computing Machines". The Journal of Chemical Physics. 21 (6): 1087. Bibcode:1953JChPh..21.1087M. doi:10.1063/1.1699114. OSTI 4390578. S2CID 1046577.
11. Granville, V.; Krivanek, M.; Rasson, J.-P. (1994). "Simulated annealing: A proof of convergence". IEEE Transactions on Pattern Analysis and Machine Intelligence. 16 (6): 652–656. doi:10.1109/34.295910.
12. Nolte, Andreas; Schrader, Rainer (1997), "A Note on the Finite Time Behaviour of Simulated Annealing", Operations Research Proceedings 1996, Berlin, Heidelberg: Springer Berlin Heidelberg, vol. 1996, pp. 175–180, doi:10.1007/978-3-642-60744-8_32, ISBN 978-3-540-62630-5, retrieved 2023-02-06
13. Moscato, P.; Fontanari, J.F. (1990), "Stochastic versus deterministic update in simulated annealing", Physics Letters A, 146 (4): 204–208, Bibcode:1990PhLA..146..204M, doi:10.1016/0375-9601(90)90166-L
14. Dueck, G.; Scheuer, T. (1990), "Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing", Journal of Computational Physics, 90 (1): 161–175, Bibcode:1990JCoPh..90..161D, doi:10.1016/0021-9991(90)90201-B, ISSN 0021-9991
15. Franz, A.; Hoffmann, K.H.; Salamon, P (2001), "Best optimal strategy for finding ground states", Physical Review Letters, 86 (3): 5219–5222, doi:10.1103/PhysRevLett.86.5219, PMID 11384462
16. De Vicente, Juan; Lanchares, Juan; Hermida, Román (2003). "Placement by thermodynamic simulated annealing". Physics Letters A. 317 (5–6): 415–423. Bibcode:2003PhLA..317..415D. doi:10.1016/j.physleta.2003.08.070.
17. Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay (2006). "Sequential Monte Carlo samplers". Journal of the Royal Statistical Society, Series B. 68 (3): 411–436. arXiv:cond-mat/0212648. doi:10.1111/j.1467-9868.2006.00553.x. S2CID 12074789.
18. Moscato, Pablo (June 1993). "An introduction to population approaches for optimization and hierarchical objective functions: A discussion on the role of tabu search". Annals of Operations Research. 41 (2): 85–121. doi:10.1007/BF02022564. S2CID 35382644.
19. Moscato, P. (1989). "On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms". Caltech Concurrent Computation Program (report 826).
20. Deb, Bandyopadhyay (June 2008). "A Simulated Annealing-Based Multiobjective Optimization Algorithm: AMOSA". IEEE Transactions on Evolutionary Computation. 12 (3): 269–283. doi:10.1109/TEVC.2007.900837. S2CID 12107321.
Further reading
• A. Das and B. K. Chakrabarti (Eds.), Quantum Annealing and Related Optimization Methods, Lecture Note in Physics, Vol. 679, Springer, Heidelberg (2005)
• Weinberger, E. (1990). "Correlated and uncorrelated fitness landscapes and how to tell the difference". Biological Cybernetics. 63 (5): 325–336. doi:10.1007/BF00202749. S2CID 851736.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 10.12. Simulated Annealing Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
• Strobl, M.A.R.; Barker, D. (2016). "On simulated annealing phase transitions in phylogeny reconstruction". Molecular Phylogenetics and Evolution. 101: 46–55. doi:10.1016/j.ympev.2016.05.001. PMC 4912009. PMID 27150349.
• V.Vassilev, A.Prahova: "The Use of Simulated Annealing in the Control of Flexible Manufacturing Systems", International Journal INFORMATION THEORIES & APPLICATIONS, VOLUME 6/1999
External links
• Simulated Annealing A Javascript app that allows you to experiment with simulated annealing. Source code included.
• "General Simulated Annealing Algorithm" An open-source MATLAB program for general simulated annealing exercises.
• Self-Guided Lesson on Simulated Annealing A Wikiversity project.
• Google in superposition of using, not using quantum computer Ars Technica discusses the possibility that the D-Wave computer being used by Google may, in fact, be an efficient simulated annealing co-processor.
• A Simulated Annealing-Based Multiobjective Optimization Algorithm: AMOSA.
Major subfields of optimization
• Convex programming
• Fractional programming
• Integer programming
• Quadratic programming
• Nonlinear programming
• Stochastic programming
• Robust optimization
• Combinatorial optimization
• Infinite-dimensional optimization
• Metaheuristics
• Constraint satisfaction
• Multiobjective optimization
• Simulated annealing
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Wikipedia
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Simultaneous embedding
Simultaneous embedding is a technique in graph drawing and information visualization for visualizing two or more different graphs on the same or overlapping sets of labeled vertices, while avoiding crossings within both graphs. Crossings between an edge of one graph and an edge of the other graph are allowed.[1]
If edges are allowed to be drawn as polylines or curves, then any planar graph may be drawn without crossing with its vertices in arbitrary positions in the plane, where the same vertex placement provides a simultaneous embedding.[1]
There are two restricted models: simultaneous geometric embedding, where each graph must be drawn planarly with line segments representing its edges rather than more complex curves, restricting the two given graphs to subclasses of the planar graphs, and simultaneous embedding with fixed edges, where curves or bends are allowed in the edges, but any edge in both graphs must be represented by the same curve in both drawings.[1] In the unrestricted model, any two planar graphs can have a simultaneous embedding.
Definition
Simultaneous embedding is a technique in graph drawing and information visualization for visualizing two or more different graphs on the same or overlapping sets of labeled vertices, while avoiding crossings within both graphs. Crossings between an edge of one graph and an edge of the other graph are allowed; it is only crossings between two edges of the same graph that are disallowed.[1]
If edges are allowed to be drawn as polylines or curves, then any planar graph may be drawn without crossings with its vertices in arbitrary positions in the plane. Using the same vertex placement for two graphs provides a simultaneous embedding of the two graphs. Research has concentrated on finding drawings with few bends, or with few crossings between edges from the two graphs.[1]
There are two restricted models: simultaneous geometric embedding and simultaneous embedding with fixed edges, where curves or bends are allowed in the edges, but any edge present in both graphs must be represented by the same curve in both drawings. When a simultaneous geometric embedding exists, it automatically is also a simultaneous embedding with fixed edges.[1]
For simultaneous embedding problems on more than two graphs, it is standard to assume that all pairs of input graphs have the same intersection as each other; that is, the edge and vertex sets of the graphs form a sunflower. This constraint is known as sunflower intersection.[1]
Simultaneous embedding is closely related to thickness, the minimum number of planar subgraphs that can cover all of the edges of a given graph, and geometric thickness, the minimum number of edge colors needed in a straight-line drawing of a given graph with no crossing between same-colored edges. In particular, the thickness of a given graph is two, if the graph's edges can be partitioned into two subgraphs that have a simultaneous embedding, and the geometric thickness is two, if the edges can be partitioned into two subgraphs with simultaneous geometric embedding.[2]
Geometric
In simultaneous geometric embedding each graph must be drawn as a planar graph with line segments representing its edges rather than more complex curves, restricting the two given graphs to subclasses of the planar graphs. Many results on simultaneous geometric embedding are based on the idea that the Cartesian coordinates of the two given graphs' vertices can be derived from properties of the two graphs. One of the most basic results of this type is the fact that any two path graphs on the same vertex set always have a simultaneous embedding. To find such an embedding, one can use the position of a vertex in the first path as its x-coordinate, and the position of the same vertex in the second path as its y-coordinate. In this way, the first path will be drawn as an x-monotone polyline, a type of curve that is automatically non-self-crossing, and the second path will similarly be drawn as a y-monotone polyline.
This type of drawing places the vertices in an integer lattice of dimensions linear in the graph sizes. Similarly defined layouts also work, with larger but still linear grid sizes, when both graphs are caterpillars or when both are cycle graphs. A simultaneous embedding in a grid of linear dimensions is also possible for any number of graphs that are all stars. Other pairs of graph types that always admit a simultaneous embedding, but that might need larger grid sizes, include a wheel graph and a cycle graph, a tree and a matching, or a pair of graphs both of which have maximum degree two. However, pairs of planar graphs and a matching, or of a Angelini, Geyer, Neuwirth and Kaufmann showed that a tree and a path exist, that have no simultaneous geometric embedding.[3][4]
Testing whether two graphs admit a simultaneous geometric embedding is NP-hard.[1][5] More precisely, it is complete for the existential theory of the reals. The proof of this result also implies that for some pairs of graphs that have simultaneous geometric embeddings, the smallest grid on which they can be drawn has doubly exponential size.[6] [2] When a simultaneous geometric embedding exists, it automatically is also a simultaneous embedding with fixed edges.[1]
Fixed edges
In simultaneous embedding with fixed edges, curves or bends are allowed in the edges, but any edge present in both graphs must be represented by the same curve in both drawings.[1] The classification of different types of input as always having an embedding or as sometimes not being possible depends not only on the two types of graphs to be drawn, but also on the structure of their intersection. For instance, it is always possible to find such an embedding when both of the two given graphs are outerplanar graphs and their intersection is a linear forest, with at most one bend per edge and with vertex coordinates and bend points all belonging to a grid of polynomial area. However, there exist other pairs of outerplanar graphs with more complex intersections that have no such embedding. It is also possible to find a simultaneous embedding with fixed edges for any pair of a planar graph and a tree.[7][8][9]
Unsolved problem in mathematics:
Can a simultaneous embedding with fixed edges for two given graphs be found in polynomial time?
(more unsolved problems in mathematics)
It is an open question whether the existence of a simultaneous embedding with fixed edges for two given graphs can be tested in polynomial time. However, for three or more graphs, the problem is NP-complete. When simultaneous embeddings with fixed edges do exist, they can be found in polynomial time for pairs of outerplanar graphs, and for Biconnected graphs, i.e. pairs of graphs whose intersection is biconnected.[1][10][11][12]
Unrestricted
Any two planar graphs can have a simultaneous embedding. This may be done in a grid of polynomial area, with at most two bends per edge. Any two subhamiltonian graphs have a simultaneous embedding with at most one bend per edge.[1][8][13]
References
1. Bläsius, Thomas; Kobourov, Stephen G.; Rutter, Ignaz (2013), "Simultaneous embedding of planar graphs", in Tamassia, Roberto (ed.), Handbook of Graph Drawing and Visualization, CRC Press, pp. 349–383, ISBN 9781420010268
2. Duncan, Christian; Eppstein, David; Kobourov, Stephen G. (2004), "The geometric thickness of low degree graphs", Proc. 20th ACM Symposium on Computational Geometry, ACM, pp. 340–346, arXiv:cs.CG/0312056, doi:10.1145/997817.997868, S2CID 7595249.
3. Brass, Peter; Cenek, Eowyn; Duncan, Christian A.; Efrat, Alon; Erten, Cesim; Ismailescu, Dan P.; Kobourov, Stephen G.; Lubiw, Anna; Mitchell, Joseph S. B. (2007), "On simultaneous planar graph embeddings", Computational Geometry Theory & Applications, 36 (2): 117–130, doi:10.1016/j.comgeo.2006.05.006, MR 2278011.
4. Cabello, Sergio; van Kreveld, Marc; Liotta, Giuseppe; Meijer, Henk; Speckmann, Bettina; Verbeek, Kevin (2011), "Geometric simultaneous embeddings of a graph and a matching", Journal of Graph Algorithms and Applications, 15 (1): 79–96, CiteSeerX 10.1.1.487.4749, doi:10.7155/jgaa.00218, MR 2776002.
5. Estrella-Balderrama, Alejandro; Gassner, Elisabeth; Jünger, Michael; Percan, Merijam; Schaefer, Marcus; Schulz, Michael (2008), "Simultaneous geometric graph embeddings", Graph Drawing: 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers, Lecture Notes in Computer Science, vol. 4875, Berlin: Springer, pp. 280–290, doi:10.1007/978-3-540-77537-9_28, MR 2427826.
6. Cardinal, Jean; Kusters, Vincent (2015), "The complexity of simultaneous geometric graph embedding", Journal of Graph Algorithms and Applications, 19 (1): 259–272, doi:10.7155/jgaa.00356, MR 3344782, S2CID 12662906.
7. Bläsius, Kobourov & Rutter (2013), Figure 11.5.
8. Di Giacomo, Emilio; Liotta, Giuseppe (2007), "Simultaneous embedding of outerplanar graphs, paths, and cycles", International Journal of Computational Geometry & Applications, 17 (2): 139–160, doi:10.1142/S0218195907002276, MR 2309902.
9. Frati, Fabrizio (2007), "Embedding graphs simultaneously with fixed edges", Graph Drawing: 14th International Symposium, GD 2006, Karlsruhe, Germany, September 18–20, 2006, Revised Papers, Lecture Notes in Computer Science, vol. 4372, Berlin: Springer, pp. 108–113, doi:10.1007/978-3-540-70904-6_12, MR 2393910.
10. Fowler, J. Joseph; Jünger, Michael; Kobourov, Stephen G.; Schulz, Michael (2011), "Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges", Computational Geometry Theory & Applications, 44 (8): 385–398, doi:10.1016/j.comgeo.2011.02.002, MR 2805957.
11. Gassner, Elisabeth; Jünger, Michael; Percan, Merijam; Schaefer, Marcus; Schulz, Michael (2006), "Simultaneous graph embeddings with fixed edges", Graph-Theoretic Concepts in Computer Science: 32nd International Workshop, WG 2006, Bergen, Norway, June 22-24, 2006, Revised Papers (PDF), Lecture Notes in Computer Science, vol. 4271, Berlin: Springer, pp. 325–335, doi:10.1007/11917496_29, MR 2290741.
12. Haeupler, Bernhard; Jampani, Krishnam Raju; Lubiw, Anna (2013), "Testing simultaneous planarity when the common graph is 2-connected", Journal of Graph Algorithms and Applications, 17 (3): 147–171, arXiv:1009.4517, doi:10.7155/jgaa.00289, MR 3043207.
13. Di Giacomo, Emilio; Liotta, Giuseppe (2005), "A note on simultaneous embedding of planar graphs", 21st European Workshop on Computational Geometry (PDF), Eindhoven University of Technology.
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Wikipedia
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System of equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a:
• System of linear equations,
• System of nonlinear equations,
• System of bilinear equations,
• System of polynomial equations,
• System of differential equations, or a
• System of difference equations
See also
• Simultaneous equations model, a statistical model in the form of simultaneous linear equations
• Elementary algebra, for elementary methods
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Wikipedia
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Simultaneous perturbation stochastic approximation
Simultaneous perturbation stochastic approximation (SPSA) is an algorithmic method for optimizing systems with multiple unknown parameters. It is a type of stochastic approximation algorithm. As an optimization method, it is appropriately suited to large-scale population models, adaptive modeling, simulation optimization, and atmospheric modeling. Many examples are presented at the SPSA website http://www.jhuapl.edu/SPSA. A comprehensive book on the subject is Bhatnagar et al. (2013). An early paper on the subject is Spall (1987) and the foundational paper providing the key theory and justification is Spall (1992).
SPSA is a descent method capable of finding global minima, sharing this property with other methods as simulated annealing. Its main feature is the gradient approximation that requires only two measurements of the objective function, regardless of the dimension of the optimization problem. Recall that we want to find the optimal control $u^{*}$ with loss function $J(u)$:
$u^{*}=\arg \min _{u\in U}J(u).$
Both Finite Differences Stochastic Approximation (FDSA) and SPSA use the same iterative process:
$u_{n+1}=u_{n}-a_{n}{\hat {g}}_{n}(u_{n}),$
where $u_{n}=((u_{n})_{1},(u_{n})_{2},\ldots ,(u_{n})_{p})^{T}$ represents the $n^{th}$ iterate, ${\hat {g}}_{n}(u_{n})$ is the estimate of the gradient of the objective function $g(u)={\frac {\partial }{\partial u}}J(u)$ evaluated at ${u_{n}}$, and $\{a_{n}\}$ is a positive number sequence converging to 0. If $u_{n}$ is a p-dimensional vector, the $i^{th}$ component of the symmetric finite difference gradient estimator is:
FD: $({\hat {g_{n}}}(u_{n}))_{i}={\frac {J(u_{n}+c_{n}e_{i})-J(u_{n}-c_{n}e_{i})}{2c_{n}}},$
1 ≤i ≤p, where $e_{i}$ is the unit vector with a 1 in the $i^{th}$ place, and $c_{n}$is a small positive number that decreases with n. With this method, 2p evaluations of J for each $g_{n}$ are needed. When p is large, this estimator loses efficiency.
Let now $\Delta _{n}$ be a random perturbation vector. The $i^{th}$ component of the stochastic perturbation gradient estimator is:
SP: $({\hat {g_{n}}}(u_{n}))_{i}={\frac {J(u_{n}+c_{n}\Delta _{n})-J(u_{n}-c_{n}\Delta _{n})}{2c_{n}(\Delta _{n})_{i}}}.$
Remark that FD perturbs only one direction at a time, while the SP estimator disturbs all directions at the same time (the numerator is identical in all p components). The number of loss function measurements needed in the SPSA method for each $g_{n}$ is always 2, independent of the dimension p. Thus, SPSA uses p times fewer function evaluations than FDSA, which makes it a lot more efficient.
Simple experiments with p=2 showed that SPSA converges in the same number of iterations as FDSA. The latter follows approximately the steepest descent direction, behaving like the gradient method. On the other hand, SPSA, with the random search direction, does not follow exactly the gradient path. In average though, it tracks it nearly because the gradient approximation is an almost unbiased estimator of the gradient, as shown in the following lemma.
Convergence lemma
Denote by
$b_{n}=E[{\hat {g}}_{n}|u_{n}]-\nabla J(u_{n})$
the bias in the estimator ${\hat {g}}_{n}$. Assume that $\{(\Delta _{n})_{i}\}$ are all mutually independent with zero-mean, bounded second moments, and $E(|(\Delta _{n})_{i}|^{-1})$ uniformly bounded. Then $b_{n}$→0 w.p. 1.
Sketch of the proof
The main idea is to use conditioning on $\Delta _{n}$ to express $E[({\hat {g}}_{n})_{i}]$ and then to use a second order Taylor expansion of $J(u_{n}+c_{n}\Delta _{n})_{i}$ and $J(u_{n}-c_{n}\Delta _{n})_{i}$. After algebraic manipulations using the zero mean and the independence of $\{(\Delta _{n})_{i}\}$, we get
$E[({\hat {g}}_{n})_{i}]=(g_{n})_{i}+O(c_{n}^{2})$
The result follows from the hypothesis that $c_{n}$→0.
Next we resume some of the hypotheses under which $u_{t}$ converges in probability to the set of global minima of $J(u)$. The efficiency of the method depends on the shape of $J(u)$, the values of the parameters $a_{n}$ and $c_{n}$ and the distribution of the perturbation terms $\Delta _{ni}$. First, the algorithm parameters must satisfy the following conditions:
• $a_{n}$ >0, $a_{n}$→0 when n→∝ and $\sum _{n=1}^{\infty }a_{n}=\infty $. A good choice would be $a_{n}={\frac {a}{n}};$ a>0;
• $c_{n}={\frac {c}{n^{\gamma }}}$, where c>0, $\gamma \in \left[{\frac {1}{6}},{\frac {1}{2}}\right]$;
• $\sum _{n=1}^{\infty }({\frac {a_{n}}{c_{n}}})^{2}<\infty $
• $\Delta _{ni}$ must be mutually independent zero-mean random variables, symmetrically distributed about zero, with $\Delta _{ni}<a_{1}<\infty $. The inverse first and second moments of the $\Delta _{ni}$ must be finite.
A good choice for $\Delta _{ni}$ is the Rademacher distribution, i.e. Bernoulli +-1 with probability 0.5. Other choices are possible too, but note that the uniform and normal distributions cannot be used because they do not satisfy the finite inverse moment conditions.
The loss function J(u) must be thrice continuously differentiable and the individual elements of the third derivative must be bounded: $|J^{(3)}(u)|<a_{3}<\infty $. Also, $|J(u)|\rightarrow \infty $ as $u\rightarrow \infty $.
In addition, $\nabla J$ must be Lipschitz continuous, bounded and the ODE ${\dot {u}}=g(u)$ must have a unique solution for each initial condition. Under these conditions and a few others, $u_{k}$ converges in probability to the set of global minima of J(u) (see Maryak and Chin, 2008).
It has been shown that differentiability is not required: continuity and convexity are sufficient for convergence.[1]
Extension to second-order (Newton) methods
It is known that a stochastic version of the standard (deterministic) Newton-Raphson algorithm (a “second-order” method) provides an asymptotically optimal or near-optimal form of stochastic approximation. SPSA can also be used to efficiently estimate the Hessian matrix of the loss function based on either noisy loss measurements or noisy gradient measurements (stochastic gradients). As with the basic SPSA method, only a small fixed number of loss measurements or gradient measurements are needed at each iteration, regardless of the problem dimension p. See the brief discussion in Stochastic gradient descent.
References
• Bhatnagar, S., Prasad, H. L., and Prashanth, L. A. (2013), Stochastic Recursive Algorithms for Optimization: Simultaneous Perturbation Methods, Springer .
• Hirokami, T., Maeda, Y., Tsukada, H. (2006) "Parameter estimation using simultaneous perturbation stochastic approximation", Electrical Engineering in Japan, 154 (2), 30–3
• Maryak, J.L., and Chin, D.C. (2008), "Global Random Optimization by Simultaneous Perturbation Stochastic Approximation," IEEE Transactions on Automatic Control, vol. 53, pp. 780-783.
• Spall, J. C. (1987), “A Stochastic Approximation Technique for Generating Maximum Likelihood Parameter Estimates,” Proceedings of the American Control Conference, Minneapolis, MN, June 1987, pp. 1161–1167.
• Spall, J. C. (1992), “Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation,” IEEE Transactions on Automatic Control, vol. 37(3), pp. 332–341.
• Spall, J.C. (1998). "Overview of the Simultaneous Perturbation Method for Efficient Optimization" 2. Johns Hopkins APL Technical Digest, 19(4), 482–492.
• Spall, J.C. (2003) Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control, Wiley. ISBN 0-471-33052-3 (Chapter 7)
1. He, Ying; Fu, Michael C.; Steven I., Marcus (August 2003). "Convergence of simultaneous perturbation stochastic approximation for nondifferentiable optimization". IEEE Transactions on Automatic Control. 48 (8): 1459–1463. doi:10.1109/TAC.2003.815008. Retrieved March 6, 2022.
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Wikipedia
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Sine and cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle $\theta $, the sine and cosine functions are denoted simply as $\sin \theta $ and $\cos \theta $.[1]
"Sine" and "Cosine" redirect here. For other uses, see Sine (disambiguation) and Cosine (disambiguation). "Sine" is not to be confused with Sign, Sign (mathematics) or the sign function.
Sine and cosine
General information
General definition${\begin{aligned}&\sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[8pt]\end{aligned}}$
Fields of applicationTrigonometry, Fourier series, etc.
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
More generally, the definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period.
Notation
Main article: Trigonometric functions § Notation
Sine and cosine are written using functional notation with the abbreviations sin and cos.
Often, if the argument is simple enough, the function value will be written without parentheses, as sin θ rather than as sin(θ).
Each of sine and cosine is a function of an angle, which is usually expressed in terms of radians or degrees. Except where explicitly stated otherwise, this article assumes that the angle is measured in radians.
Definitions
Right-angled triangle definitions
To define the sine and cosine of an acute angle α, start with a right triangle that contains an angle of measure α; in the accompanying figure, angle α in triangle ABC is the angle of interest. The three sides of the triangle are named as follows:
• The opposite side is the side opposite to the angle of interest, in this case side a.
• The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
• The adjacent side is the remaining side, in this case side b. It forms a side of (and is adjacent to) both the angle of interest (angle A) and the right angle.
Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side, divided by the length of the hypotenuse:[2]
$\sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\qquad \cos(\alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}$
The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides.[2]
As stated, the values $\sin(\alpha )$ and $\cos(\alpha )$ appear to depend on the choice of right triangle containing an angle of measure α. However, this is not the case: all such triangles are similar, and so the ratios are the same for each of them.
Unit circle definitions
In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.
Let a line through the origin intersect the unit circle, making an angle of θ with the positive half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos(θ) and sin(θ), respectively. This definition is consistent with the right-angled triangle definition of sine and cosine when $0<\theta <{\frac {\pi }{2}}$: because the length of the hypotenuse of the unit circle is always 1, $\sin(\theta )={\frac {\text{opposite}}{\text{hypotenuse}}}={\frac {\text{opposite}}{1}}={\text{opposite}}$. The length of the opposite side of the triangle is simply the y-coordinate. A similar argument can be made for the cosine function to show that $\cos(\theta )={\frac {\text{adjacent}}{\text{hypotenuse}}}$ when $0<\theta <{\frac {\pi }{2}}$, even under the new definition using the unit circle. tan(θ) is then defined as ${\frac {\sin(\theta )}{\cos(\theta )}}$, or, equivalently, as the slope of the line segment.
Using the unit circle definition has the advantage that the angle can be extended to any real argument. This can also be achieved by requiring certain symmetries, and that sine be a periodic function.
Complex exponential function definitions
Main article: Euler's formula
The exponential function $e^{z}$ is defined on the entire domain of the complex numbers. The definition of sine and cosine can be extended to all complex numbers via
$\sin z={\frac {e^{iz}-e^{-iz}}{2i}}$
$\cos z={\frac {e^{iz}+e^{-iz}}{2}}$
These can be reversed to give Euler's formula
$e^{iz}=\cos z+i\sin z$
$e^{-iz}=\cos z-i\sin z$
When plotted on the complex plane, the function $e^{ix}$ for real values of $x$ traces out the unit circle in the complex plane.
When $x$ is a real number, sine and cosine simplify to the imaginary and real parts of $e^{ix}$ or $e^{-ix}$, as:
$\sin x=\operatorname {Im} (e^{ix})=-\operatorname {Im} (e^{-ix})$
$\cos x=\operatorname {Re} (e^{ix})=\operatorname {Re} (e^{-ix})$
When $z=x+iy$ for real values $x$ and $y$, sine and cosine can be expressed in terms of real sines, cosines, and hyperbolic functions as
${\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}$
Differential equation definition
$(\cos \theta ,\sin \theta )$ is the solution $(x(\theta ),y(\theta ))$ to the two-dimensional system of differential equations $y'(\theta )=x(\theta )$ and $x'(\theta )=-y(\theta )$ with the initial conditions $y(0)=0$ and $x(0)=1$. One could interpret the unit circle in the above definitions as defining the phase space trajectory of the differential equation with the given initial conditions.
The animation above shows how the sine function (in red) is graphed from the y-coordinate (red dot) of a point on the unit circle (in green), at an angle of θ. The cosine (in blue) is the x-coordinate. It can be interpreted as a phase space trajectory of the system of differential equations $y'(\theta )=x(\theta )$ and $x'(\theta )=-y(\theta )$ starting from the initial conditions $y(0)=0$ and $x(0)=1$.
Series definitions
The successive derivatives of sine, evaluated at zero, can be used to determine its Taylor series. Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine, and that the derivative of cosine is the negative of sine. This means the successive derivatives of sin(x) are cos(x), -sin(x), -cos(x), sin(x), continuing to repeat those four functions. The (4n+k)-th derivative, evaluated at the point 0:
$\sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}$
where the superscript represents repeated differentiation. This implies the following Taylor series expansion at x = 0. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x (where x is the angle in radians):[3]
${\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\end{aligned}}$
Taking the derivative of each term gives the Taylor series for cosine:
${\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}$
Continued fraction definitions
The sine function can also be represented as a generalized continued fraction:
$\sin(x)={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}.$
$\cos(x)={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}.$
The continued fraction representations can be derived from Euler's continued fraction formula and express the real number values, both rational and irrational, of the sine and cosine functions.
Identities
Main article: List of trigonometric identities
Exact identities (using radians):
These apply for all values of $\theta $.
$\sin(\theta )=\cos \left({\frac {\pi }{2}}-\theta \right)=\cos \left(\theta -{\frac {\pi }{2}}\right)$
$\cos(\theta )=\sin \left({\frac {\pi }{2}}-\theta \right)=\sin \left(\theta +{\frac {\pi }{2}}\right)$
Reciprocals
The reciprocal of sine is cosecant, i.e., the reciprocal of $\sin {\theta }$ is $\csc {\theta }$. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the length of the hypotenuse to that of the adjacent side.
$\csc {\theta }={\frac {1}{\sin {\theta }}}={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}$
$\sec {\theta }={\frac {1}{\cos {\theta }}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}$
Inverses
The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin−1). The inverse function of cosine is arccosine (arccos, acos, or cos−1). (The superscript of −1 in sin−1 and cos−1 denotes the inverse of a function, not exponentiation.) As sine and cosine are not injective, their inverses are not exact inverse functions, but partial inverse functions. For example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0 etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain, the expression arcsin(x) will evaluate only to a single value, called its principal value. The standard range of principal values for arcsin is from −π/2 to π/2 and the standard range for arccos is from 0 to π.
$\theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right).$
where (for some integer k):
${\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&y=-\arccos(x)+2\pi k\end{aligned}}$
By definition, arcsin and arccos satisfy the equations:
$\sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x$
and
${\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq \theta \leq \pi \end{aligned}}$
Pythagorean trigonometric identity
The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:[1]
$\cos ^{2}(\theta )+\sin ^{2}(\theta )=1$
where sin2(x) means (sin(x))2.
Double angle formulas
Sine and cosine satisfy the following double angle formulas:
$\sin(2\theta )=2\sin(\theta )\cos(\theta )$
$\cos(2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )=2\cos ^{2}(\theta )-1=1-2\sin ^{2}(\theta )$
The cosine double angle formula implies that sin2 and cos2 are, themselves, shifted and scaled sine waves. Specifically,[4]
$\sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}$
The graph shows both the sine function and the sine squared function, with the sine in blue and sine squared in red. Both graphs have the same shape, but with different ranges of values, and different periods. Sine squared has only positive values, but twice the number of periods.
Derivative and integrals
See also: List of integrals of trigonometric functions and Differentiation of trigonometric functions
The derivatives of sine and cosine are:
${\frac {d}{dx}}\sin(x)=\cos(x)\qquad {\frac {d}{dx}}\cos(x)=-\sin(x)$
and their antiderivatives are:
$\int \sin(x)\,dx=-\cos(x)+C$
$\int \cos(x)\,dx=\sin(x)+C$
where C denotes the constant of integration.[1]
Properties relating to the quadrants
The table below displays many of the key properties of the sine function (sign, monotonicity, convexity), arranged by the quadrant of the argument. For arguments outside those in the table, one may compute the corresponding information by using the periodicity $\sin(\alpha +2\pi )=\sin(\alpha )$ of the sine function.
Quadrant Angle Sine Cosine
Degrees Radians Sign Monotony Convexity Sign Monotony Convexity
1st quadrant, I $0^{\circ }<x<90^{\circ }$ $0<x<{\frac {\pi }{2}}$ $+$ increasing concave $+$ decreasing concave
2nd quadrant, II $90^{\circ }<x<180^{\circ }$ ${\frac {\pi }{2}}<x<\pi $ $+$ decreasing concave $-$ decreasing convex
3rd quadrant, III $180^{\circ }<x<270^{\circ }$ $\pi <x<{\frac {3\pi }{2}}$ $-$ decreasing convex $-$ increasing convex
4th quadrant, IV $270^{\circ }<x<360^{\circ }$ ${\frac {3\pi }{2}}<x<2\pi $ $-$ increasing convex $+$ increasing concave
The following table gives basic information at the boundary of the quadrants.
Degrees Radians $\sin(x)$ $\cos(x)$
Value Point type Value Point type
$0^{\circ }$ $0$ $0$ Root, inflection $1$ Maximum
$90^{\circ }$ ${\frac {\pi }{2}}$ $1$ Maximum $0$ Root, inflection
$180^{\circ }$ $\pi $ $0$ Root, inflection $-1$ Minimum
$270^{\circ }$ ${\frac {3\pi }{2}}$ $-1$ Minimum $0$ Root, inflection
Fixed points
Main article: Dottie number
Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is $\sin(0)=0$. The only real fixed point of the cosine function is called the Dottie number. That is, the Dottie number is the unique real root of the equation $\cos(x)=x.$ The decimal expansion of the Dottie number is $0.739085\ldots $.[5]
Arc length
The arc length of the sine curve between $0$ and $t$ is
$\int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} (t,1/{\sqrt {2}}),$
where $\operatorname {E} (\varphi ,k)$ is the incomplete elliptic integral of the second kind with modulus $k$. It cannot be expressed using elementary functions.
The arc length for a full period is[6]
$L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\frac {2\pi }{\varpi }}+2\varpi =7.640395578\ldots $
where $\Gamma $ is the gamma function and $\varpi $ is the lemniscate constant.[6][7]
Laws
Main articles: Law of sines and Law of cosines
The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:
${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}.$
This is equivalent to the equality of the first three expressions below:
${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,$
where R is the triangle's circumradius.
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
The law of cosines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:
$a^{2}+b^{2}-2ab\cos(C)=c^{2}$
In the case where $C=\pi /2$, $\cos(C)=0$ and this becomes the Pythagorean theorem: for a right triangle, $a^{2}+b^{2}=c^{2},$ where c is the hypotenuse.
Special values
For certain integral numbers x of degrees, the values of sin(x) and cos(x) are particularly simple and can be expressed without nested square roots. A table of these angles is given below. For more complex angle expressions see Exact trigonometric values § Common angles.
Angle, x sin(x) cos(x)
Degrees Radians Gradians Turns Exact Decimal Exact Decimal
0° 0 0g 0 0 0 1 1
15° 1/12π 16+2/3g 1/24 ${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ 0.2588 ${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ 0.9659
30° 1/6π 33+1/3g 1/12 1/2 0.5 ${\frac {\sqrt {3}}{2}}$ 0.8660
45° 1/4π 50g 1/8 ${\frac {\sqrt {2}}{2}}$ 0.7071 ${\frac {\sqrt {2}}{2}}$ 0.7071
60° 1/3π 66+2/3g 1/6 ${\frac {\sqrt {3}}{2}}$ 0.8660 1/2 0.5
75° 5/12π 83+1/3g 5/24 ${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ 0.9659 ${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ 0.2588
90° 1/2π 100g 1/4 1 1 0 0
90 degree increments:
x in degrees 0°90°180°270°360°
x in radians 0π/2π3π/22π
x in gons 0100g200g300g400g
x in turns 01/41/23/41
sin x 010−10
cos x 10−101
Relationship to complex numbers
Main article: Trigonometric functions § Relationship to exponential function (Euler's formula)
Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates (r, φ):
$z=r(\cos(\varphi )+i\sin(\varphi ))$
The real and imaginary parts are:
$\operatorname {Re} (z)=r\cos(\varphi )$
$\operatorname {Im} (z)=r\sin(\varphi )$
where r and φ represent the magnitude and angle of the complex number z.
For any real number θ, Euler's formula says that:
$e^{i\theta }=\cos(\theta )+i\sin(\theta )$
Therefore, if the polar coordinates of z are (r, φ), $z=re^{i\varphi }.$
Complex arguments
Applying the series definition of the sine and cosine to a complex argument, z, gives:
${\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=-i\sinh \left(iz\right)\\\cos(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{-iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}$
where sinh and cosh are the hyperbolic sine and cosine. These are entire functions.
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:
${\begin{aligned}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)-\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)-i\sin(x)\sinh(y)\\\end{aligned}}$
Partial fraction and product expansions of complex sine
Using the partial fraction expansion technique in complex analysis, one can find that the infinite series
$\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}$
both converge and are equal to $ {\frac {\pi }{\sin(\pi z)}}$. Similarly, one can show that
${\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.$
Using product expansion technique, one can derive
$\sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).$
Alternatively, the infinite product for the sine can be proved using complex Fourier series.
Proof of the infinite product for the sine
Using complex Fourier series, the function $\cos(zx)$ can be decomposed as
$\cos(zx)={\frac {z\sin(\pi z)}{\pi }}\displaystyle \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}\,e^{inx}}{z^{2}-n^{2}}},\,z\in \mathbb {C} \setminus \mathbb {Z} ,\,x\in [-\pi ,\pi ].$
Setting $x=\pi $ yields
$\cos(\pi z)={\frac {z\sin(\pi z)}{\pi }}\displaystyle \sum _{n=-\infty }^{\infty }{\frac {1}{z^{2}-n^{2}}}={\frac {z\sin(\pi z)}{\pi }}\left({\frac {1}{z^{2}}}+2\displaystyle \sum _{n=1}^{\infty }{\frac {1}{z^{2}-n^{2}}}\right).$
Therefore, we get
$\pi \cot(\pi z)={\frac {1}{z}}+2\displaystyle \sum _{n=1}^{\infty }{\frac {z}{z^{2}-n^{2}}}.$
The function $\pi \cot(\pi z)$ is the derivative of $\ln(\sin(\pi z))+C_{0}$. Furthermore, if $ {\frac {df}{dz}}={\frac {z}{z^{2}-n^{2}}}$, then the function $f$ such that the emerged series converges on some open and connected subset of $\mathbb {C} $ is $ f={\frac {1}{2}}\ln \left(1-{\frac {z^{2}}{n^{2}}}\right)+C_{1}$, which can be proved using the Weierstrass M-test. The interchange of the sum and derivative is justified by uniform convergence. It follows that
$\ln(\sin(\pi z))=\ln(z)+\displaystyle \sum _{n=1}^{\infty }\ln \left(1-{\frac {z^{2}}{n^{2}}}\right)+C.$
Exponentiating gives
$\sin(\pi z)=ze^{C}\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).$
Since $ \lim _{z\to 0}{\frac {\sin(\pi z)}{z}}=\pi $ and $ \lim _{z\to 0}\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)=1$, we have $e^{C}=\pi $. Hence
$\sin(\pi z)=\pi z\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)$
for some open and connected subset of $\mathbb {C} $. Let $ a_{n}(z)=-{\frac {z^{2}}{n^{2}}}$. Since $ \sum _{n=1}^{\infty }|a_{n}(z)|$ converges uniformly on any closed disk, $ \prod _{n=1}^{\infty }(1+a_{n}(z))$ converges uniformly on any closed disk as well.[8] It follows that the infinite product is holomorphic on $\mathbb {C} $. By the identity theorem, the infinite product for the sine is valid for all $z\in \mathbb {C} $, which completes the proof. $\blacksquare $
Usage of complex sine
sin(z) is found in the functional equation for the Gamma function,
$\Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},$
which in turn is found in the functional equation for the Riemann zeta-function,
$\zeta (s)=2(2\pi )^{s-1}\Gamma (1-s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1-s).$
As a holomorphic function, sin z is a 2D solution of Laplace's equation:
$\Delta u(x_{1},x_{2})=0.$
The complex sine function is also related to the level curves of pendulums.[9]
Complex graphs
Sine function in the complex plane
real component imaginary component magnitude
Arcsine function in the complex plane
real component imaginary component magnitude
History
Main article: History of trigonometry
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).[10]
The sine and cosine functions can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period (Aryabhatiya and Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.[11]
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.[12] With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant.[12] Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents.[13][14] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.[14]
The first published use of the abbreviations sin, cos, and tan is by the 16th-century French mathematician Albert Girard; these were further promulgated by Euler (see below). The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.[15] Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722).[16] Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.[11]
Etymology
Main article: History of trigonometry § Etymology
Etymologically, the word sine derives from the Sanskrit word jyā 'bow-string'[17][18] or more specifically its synonym jīvá (both adopted from Ancient Greek χορδή 'string'[19]), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see jyā, koti-jyā and utkrama-jyā). This was transliterated in Arabic as jība, which is however meaningless in that language and abbreviated jb (جب). Since Arabic is written without short vowels, jb was interpreted as the homograph jaib, jayb (جيب), which means 'bosom', 'pocket', 'fold'. When the Arabic texts of Al-Battani and al-Khwārizmī were translated into Medieval Latin in the 12th century by Gerard of Cremona, he used the Latin equivalent sinus (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a toga over the breast').[11][20][21] Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.[22][23] The English form sine was introduced in the 1590s.[24]
The word cosine derives from an abbreviation of the Latin complementi sinus 'sine of the complementary angle' as cosinus in Edmund Gunter's Canon triangulorum (1620), which also includes a similar definition of cotangens.[25][26][27]
Software implementations
There is no standard algorithm for calculating sine and cosine. IEEE 754, the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.[28]
Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. sin(1022).
A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or linearly interpolate between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.
The CORDIC algorithm is commonly used in scientific calculators.
The sine and cosine functions, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, they are typically abbreviated to sin and cos.
Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.
In programming languages, sin and cos are typically either a built-in function or found within the language's standard math library.
For example, the C standard library defines sine functions within math.h: sin(double), sinf(float), and sinl(long double). The parameter of each is a floating point value, specifying the angle in radians. Each function returns the same data type as it accepts. Many other trigonometric functions are also defined in math.h, such as for cosine, arc sine, and hyperbolic sine (sinh).
Similarly, Python defines math.sin(x) and math.cos(x) within the built-in math module. Complex sine and cosine functions are also available within the cmath module, e.g. cmath.sin(z). CPython's math functions call the C math library, and use a double-precision floating-point format.
Turns based implementations
Some software libraries provide implementations of sine and cosine using the input angle in half-turns, a half-turn being an angle of 180 degrees or $\pi $ radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.[29][30] In MATLAB, OpenCL, R, Julia, CUDA, and ARM, these function are called sinpi and cospi.[29][31][30][32][33][34] For example, sinpi(x) would evaluate to $\sin(\pi x),$ where x is expressed in half-turns, and consequently the final input to the function, πx can be interpreted in radians by sin.
The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing $2\pi $, $\pi $, and $ {\frac {\pi }{2}}$ in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.
Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo $ {\frac {\pi }{2}}$ involves inaccuracies in representing $ {\frac {\pi }{2}}$.
For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution.[35] If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to $ {\frac {\pi }{2048}}$ would be incurred.
See also
• Āryabhaṭa's sine table
• Bhaskara I's sine approximation formula
• Discrete sine transform
• Euler's formula
• Generalized trigonometry
• Hyperbolic function
• Dixon elliptic functions
• Lemniscate elliptic functions
• Law of sines
• List of periodic functions
• List of trigonometric identities
• Madhava series
• Madhava's sine table
• Optical sine theorem
• Polar sine—a generalization to vertex angles
• Proofs of trigonometric identities
• Sinc function
• Sine and cosine transforms
• Sine integral
• Sine quadrant
• Sine wave
• Sine–Gordon equation
• Sinusoidal model
• SOH-CAH-TOA
• Trigonometric functions
• Trigonometric integral
Citations
1. Weisstein, Eric W. "Sine". mathworld.wolfram.com. Retrieved 2020-08-29.
2. "Sine, Cosine, Tangent". www.mathsisfun.com. Retrieved 2020-08-29.
3. See Ahlfors, pages 43–44.
4. "Sine-squared function". Retrieved August 9, 2019.
5. "OEIS A003957". oeis.org. Retrieved 2019-05-26.
6. "A105419 - Oeis".
7. Adlaj, Semjon (2012). "An Eloquent Formula for the Perimeter of an Ellipse" (PDF). American Mathematical Society. p. 1097.
8. Rudin, Walter (1987). Real and Complex Analysis (Third ed.). McGraw-Hill Book Company. ISBN 0-07-100276-6. p. 299, Theorem 15.4
9. "Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?". math.stackexchange.com. Retrieved 2019-08-12.
10. Brendan, T. (February 1965). "How Ptolemy constructed trigonometry tables". The Mathematics Teacher. 58 (2): 141–149 – via JSTOR.
11. Merzbach, Uta C.; Boyer, Carl B. (2011), A History of Mathematics (3rd ed.), John Wiley & Sons: It was Robert of Chester's translation from the Arabic that resulted in our word "sine". The Hindus had given the name jiva to the half-chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language there is also the word jaib meaning "bay" or "inlet". When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence, he used the word sinus, the Latin word for "bay" or "inlet".
12. Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. p. 74. Archived from the original on 2013-10-19. Retrieved 2010-07-13.
13. Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer Science+Business Media. ISBN 978-1-4020-0260-1.
14. "trigonometry". Encyclopedia Britannica.
15. Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer. ISBN 9783540647676.
16. "Why the sine has a simple derivative Archived 2011-07-20 at the Wayback Machine", in Historical Notes for Calculus Teachers Archived 2011-07-20 at the Wayback Machine by V. Frederick Rickey Archived 2011-07-20 at the Wayback Machine
17. "How the Trig Functions Got their Names". Ask Dr. Math. Drexel University. Retrieved 2 March 2010.
18. J J O'Connor and E F Robertson (June 1996). "The trigonometric functions". Retrieved 2 March 2010.
19. See Plofker, Mathematics in India, Princeton University Press, 2009, p. 257
See "Clark University". Archived from the original on 15 June 2008.
See Maor (1998), chapter 3, regarding the etymology.
20. Eli Maor (1998), Trigonometric Delights, Princeton: Princeton University Press, p. 35-36.
21. Victor J. Katz (2008), A History of Mathematics, Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. "A History of Mathematics" (PDF). Archived (PDF) from the original on 2015-04-14. Retrieved 2015-04-09.: The English word “sine” comes from a series of mistranslations of the Sanskrit jyā-ardha (chord-half). Āryabhaṭa frequently abbreviated this term to jyā or its synonym jīvá. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabic word jiba. But since Arabic is written without vowels, later writers interpreted the consonants jb as jaib, which means bosom or breast. In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word sinus, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf.
22. Smith, D.E. (1958) [1925], History of Mathematics, vol. I, Dover, p. 202, ISBN 0-486-20429-4
23. Various sources credit the first use of sinus to either
• Plato Tiburtinus's 1116 translation of the Astronomy of Al-Battani
• Gerard of Cremona's translation of the Algebra of al-Khwārizmī
• Robert of Chester's 1145 translation of the tables of al-Khwārizmī
See Merlet, A Note on the History of the Trigonometric Functions in Ceccarelli (ed.), International Symposium on History of Machines and Mechanisms, Springer, 2004
See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.
See Katx, Victor (July 2008). A history of mathematics (3rd ed.). Boston: Pearson. p. 210 (sidebar). ISBN 978-0321387004.
24. The anglicized form is first recorded in 1593 in Thomas Fale's Horologiographia, the Art of Dialling.
25. Gunter, Edmund (1620). Canon triangulorum.
26. Roegel, Denis, ed. (6 December 2010). "A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938. Archived from the original on 28 July 2017. Retrieved 28 July 2017.
27. "cosine".
28. Zimmermann, Paul (2006), "Can we trust floating-point numbers?", Grand Challenges of Informatics (PDF), p. 14/31, archived (PDF) from the original on 2011-07-16, retrieved 2010-09-11
29. "MATLAB Documentation sinpi
30. "R Documentation sinpi
31. "OpenCL Documentation sinpi
32. "Julia Documentation sinpi
33. "CUDA Documentation sinpi
34. "ARM Documentation sinpi
35. "ALLEGRO Angle Sensor Datasheet
References
• Traupman, Ph.D., John C. (1966), The New College Latin & English Dictionary, Toronto: Bantam, ISBN 0-553-27619-0
• Webster's Seventh New Collegiate Dictionary, Springfield: G. & C. Merriam Company, 1969
External links
Look up sine in Wiktionary, the free dictionary.
• Media related to Sine function at Wikimedia Commons
Look up sine and cosine in Wiktionary, the free dictionary.
Trigonometric and hyperbolic functions
Groups
• Trigonometric
• Sine and cosine
• Inverse trigonometric
• Hyperbolic
• Inverse hyperbolic
Other
• Versine
• Exsecant
• Jyā, koti-jyā and utkrama-jyā
• atan2
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Wikipedia
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Scientific calculator
A scientific calculator is an electronic calculator, either desktop or handheld, designed to perform calculations using basic (addition, subtraction, multiplication, division) and complex (trigonometric, hyperbolic, etc.) mathematical operations and functions. They have completely replaced slide rules and are used in both educational and professional settings.
Left: Texas Instruments TI-30X IIS calculator with a two-tier LCD. The upper dot-matrix area can display input formulae and symbols.
Right: The TI-84 Plus—A typical graphing calculator by Texas Instruments
In some areas of study scientific calculators have been replaced by graphing calculators and financial calculators which have the capabilities of a scientific calculator along with the capability to graph input data and functions.
Functions
When electronic calculators were originally marketed they normally had only four or five capabilities (addition, subtraction, multiplication, division and square root). Modern scientific calculators generally have many more capabilities than the original four or five function calculator, and the capabilities differ between manufacturers and models.
The capabilities of a modern scientific calculator include:
• Scientific notation
• Floating-point decimal arithmetic
• Logarithmic functions, using both base 10 and base e
• Trigonometric functions (some including hyperbolic trigonometry)
• Exponential functions and roots beyond the square root
• Quick access to constants such as pi and e
In addition, high-end scientific calculators generally include some or all of the following:
• Cursor controls to edit equations and view previous calculations (some calculators such as the LCD-8310, badge engineered under both Olympia and United Office keep the number of the previous result on-screen for convenience while the new calculation is being entered.[1])
• Hexadecimal, binary, and octal calculations, including basic Boolean mathematics
• Complex numbers
• Fractions calculations
• Statistics and probability calculations
• Programmability — see Programmable calculator
• Equation solving
• Matrix calculations
• Calculus
• Letters that can be used for spelling words or including variables into an equation
• Conversion of units
• Physical constants
While most scientific calculators have traditionally used a single-line display similar to traditional pocket calculators, many of them have more digits (10 to 12), sometimes with extra digits for the floating-point exponent. A few have multi-line displays, with some models from Hewlett-Packard, Texas Instruments (both US manufacturers), Casio, Sharp, and Canon (all three Japanese makers) using dot matrix displays similar to those found on graphing calculators.
Uses
Scientific calculators are used widely in situations that require quick access to certain mathematical functions, especially those that were once looked up in mathematical tables, such as trigonometric functions or logarithms. They are also used for calculations of very large or very small numbers, as in some aspects of astronomy, physics, and chemistry.
They are very often required for math classes from the junior high school level through college, and are generally either permitted or required on many standardized tests covering math and science subjects; as a result, many are sold into educational markets to cover this demand, and some high-end models include features making it easier to translate a problem on a textbook page into calculator input, e.g. by providing a method to enter an entire problem in as it is written on the page using simple formatting tools.
History
The first scientific calculator that included all of the basic ideas above was the programmable Hewlett-Packard HP-9100A,[2] released in 1968, though the Wang LOCI-2 and the Mathatronics Mathatron[3] had some features later identified with scientific calculator designs. The HP-9100 series was built entirely from discrete transistor logic with no integrated circuits, and was one of the first uses of the CORDIC algorithm for trigonometric computation in a personal computing device, as well as the first calculator based on reverse Polish notation (RPN) entry. HP became closely identified with RPN calculators from then on, and even today some of their high-end calculators (particularly the long-lived HP-12C financial calculator and the HP-48 series of graphing calculators) still offer RPN as their default input mode due to having garnered a very large following.
The HP-35, introduced on February 1, 1972, was Hewlett-Packard's first pocket calculator and the world's first handheld scientific calculator.[4] Like some of HP's desktop calculators it used RPN. Introduced at US$395, the HP-35 was available from 1972 to 1975.
Texas Instruments (TI), after the production of several units with scientific notation, introduced a handheld scientific calculator on January 15, 1974, in the form of the SR-50.[5] TI continues to be a major player in the calculator market, with their long-running TI-30 series being one of the most widely used scientific calculators in classrooms.
Casio, Canon, and Sharp have also been major players, with Casio's FX series (beginning with the Casio FX-1 in 1972[6]) being a very common brand, used particularly in schools. Casio is also a major player in the graphing calculator market, and was the first company to produce one (Casio fx-7000G).
See also
• Formula calculator
• Calculator input methods
• Software calculators
References
1. "Nostalgia & Fun With Calculators". Homo Ludditus. 10 February 2019.
2. HP-9100A/B at hpmuseum.org
3. "across the editor's desk: COMPUTING AND DATA PROCESSING NEWSLETTER - THE MATHATRON" (PDF). Computers and Automation. XIII (3): 43. Mar 1964. Retrieved 2020-09-05.
4. HP-35 Scientific Calculator Awarded IEEE Milestone
5. SR-50 page at datamath.org
6. Casio FX-1 Desktop Scientific Calculator
Wikimedia Commons has media related to Scientific calculators.
Computer sizes
Classes of computers
Microcomputer,
personal
computer
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Wikipedia
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Trigonometric integral
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
For simple integrals of trigonometric functions, see List of integrals of trigonometric functions.
Sine integral
The different sine integral definitions are
$\operatorname {Si} (x)=\int _{0}^{x}{\frac {\sin t}{t}}\,dt$
$\operatorname {si} (x)=-\int _{x}^{\infty }{\frac {\sin t}{t}}\,dt~.$
Note that the integrand ${\frac {\sin(t)}{t}}$ is the sinc function, and also the zeroth spherical Bessel function. Since sinc is an even entire function (holomorphic over the entire complex plane), Si is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints.
By definition, Si(x) is the antiderivative of sin x / x whose value is zero at x = 0, and si(x) is the antiderivative whose value is zero at x = ∞. Their difference is given by the Dirichlet integral,
$\operatorname {Si} (x)-\operatorname {si} (x)=\int _{0}^{\infty }{\frac {\sin t}{t}}\,dt={\frac {\pi }{2}}\quad {\text{ or }}\quad \operatorname {Si} (x)={\frac {\pi }{2}}+\operatorname {si} (x)~.$
In signal processing, the oscillations of the sine integral cause overshoot and ringing artifacts when using the sinc filter, and frequency domain ringing if using a truncated sinc filter as a low-pass filter.
Related is the Gibbs phenomenon: If the sine integral is considered as the convolution of the sinc function with the heaviside step function, this corresponds to truncating the Fourier series, which is the cause of the Gibbs phenomenon.
Cosine integral
The different cosine integral definitions are
$\operatorname {Cin} (x)=\int _{0}^{x}{\frac {1-\cos t}{t}}\,dt~,$
$\operatorname {Ci} (x)=-\int _{x}^{\infty }{\frac {\cos t}{t}}\,dt=\gamma +\ln x-\int _{0}^{x}{\frac {1-\cos t}{t}}\,dt\qquad ~{\text{ for }}~\left|\operatorname {Arg} (x)\right|<\pi ~,$
where γ ≈ 0.57721566 ... is the Euler–Mascheroni constant. Some texts use ci instead of Ci.
Ci(x) is the antiderivative of cos x / x (which vanishes as $x\to \infty $). The two definitions are related by
$\operatorname {Ci} (x)=\gamma +\ln x-\operatorname {Cin} (x)~.$
Cin is an even, entire function. For that reason, some texts treat Cin as the primary function, and derive Ci in terms of Cin.
Hyperbolic sine integral
The hyperbolic sine integral is defined as
$\operatorname {Shi} (x)=\int _{0}^{x}{\frac {\sinh(t)}{t}}\,dt.$
It is related to the ordinary sine integral by
$\operatorname {Si} (ix)=i\operatorname {Shi} (x).$
Hyperbolic cosine integral
The hyperbolic cosine integral is
$\operatorname {Chi} (x)=\gamma +\ln x+\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt\qquad ~{\text{ for }}~\left|\operatorname {Arg} (x)\right|<\pi ~,$
where $\gamma $ is the Euler–Mascheroni constant.
It has the series expansion
$\operatorname {Chi} (x)=\gamma +\ln(x)+{\frac {x^{2}}{4}}+{\frac {x^{4}}{96}}+{\frac {x^{6}}{4320}}+{\frac {x^{8}}{322560}}+{\frac {x^{10}}{36288000}}+O(x^{12}).$
Auxiliary functions
Trigonometric integrals can be understood in terms of the so-called "auxiliary functions"
${\begin{array}{rcl}f(x)&\equiv &\int _{0}^{\infty }{\frac {\sin(t)}{t+x}}\,dt&=&\int _{0}^{\infty }{\frac {e^{-xt}}{t^{2}+1}}\,dt&=&\operatorname {Ci} (x)\sin(x)+\left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\cos(x)~,\\g(x)&\equiv &\int _{0}^{\infty }{\frac {\cos(t)}{t+x}}\,dt&=&\int _{0}^{\infty }{\frac {te^{-xt}}{t^{2}+1}}\,dt&=&-\operatorname {Ci} (x)\cos(x)+\left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\sin(x)~.\end{array}}$
Using these functions, the trigonometric integrals may be re-expressed as (cf. Abramowitz & Stegun, p. 232)
${\begin{array}{rcl}{\frac {\pi }{2}}-\operatorname {Si} (x)=-\operatorname {si} (x)&=&f(x)\cos(x)+g(x)\sin(x)~,\qquad {\text{ and }}\\\operatorname {Ci} (x)&=&f(x)\sin(x)-g(x)\cos(x)~.\\\end{array}}$
Nielsen's spiral
The spiral formed by parametric plot of si , ci is known as Nielsen's spiral.
$x(t)=a\times \operatorname {ci} (t)$
$y(t)=a\times \operatorname {si} (t)$
The spiral is closely related to the Fresnel integrals and the Euler spiral. Nielsen's spiral has applications in vision processing, road and track construction and other areas.[1]
Expansion
Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument.
Asymptotic series (for large argument)
$\operatorname {Si} (x)\sim {\frac {\pi }{2}}-{\frac {\cos x}{x}}\left(1-{\frac {2!}{x^{2}}}+{\frac {4!}{x^{4}}}-{\frac {6!}{x^{6}}}\cdots \right)-{\frac {\sin x}{x}}\left({\frac {1}{x}}-{\frac {3!}{x^{3}}}+{\frac {5!}{x^{5}}}-{\frac {7!}{x^{7}}}\cdots \right)$
$\operatorname {Ci} (x)\sim {\frac {\sin x}{x}}\left(1-{\frac {2!}{x^{2}}}+{\frac {4!}{x^{4}}}-{\frac {6!}{x^{6}}}\cdots \right)-{\frac {\cos x}{x}}\left({\frac {1}{x}}-{\frac {3!}{x^{3}}}+{\frac {5!}{x^{5}}}-{\frac {7!}{x^{7}}}\cdots \right)~.$
These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at ℜ(x) ≫ 1.
Convergent series
$\operatorname {Si} (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}}=x-{\frac {x^{3}}{3!\cdot 3}}+{\frac {x^{5}}{5!\cdot 5}}-{\frac {x^{7}}{7!\cdot 7}}\pm \cdots $
$\operatorname {Ci} (x)=\gamma +\ln x+\sum _{n=1}^{\infty }{\frac {(-1)^{n}x^{2n}}{2n(2n)!}}=\gamma +\ln x-{\frac {x^{2}}{2!\cdot 2}}+{\frac {x^{4}}{4!\cdot 4}}\mp \cdots $
These series are convergent at any complex x, although for |x| ≫ 1, the series will converge slowly initially, requiring many terms for high precision.
Derivation of series expansion
From the Maclaurin series expansion of sine:
$\sin \,x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+{\frac {x^{9}}{9!}}-{\frac {x^{11}}{11!}}+\cdots $
${\frac {\sin \,x}{x}}=1-{\frac {x^{2}}{3!}}+{\frac {x^{4}}{5!}}-{\frac {x^{6}}{7!}}+{\frac {x^{8}}{9!}}-{\frac {x^{10}}{11!}}+\cdots $
$\therefore \int {\frac {\sin \,x}{x}}dx=x-{\frac {x^{3}}{3!\cdot 3}}+{\frac {x^{5}}{5!\cdot 5}}-{\frac {x^{7}}{7!\cdot 7}}+{\frac {x^{9}}{9!\cdot 9}}-{\frac {x^{11}}{11!\cdot 11}}+\cdots $
Relation with the exponential integral of imaginary argument
The function
$\operatorname {E} _{1}(z)=\int _{1}^{\infty }{\frac {\exp(-zt)}{t}}\,dt\qquad ~{\text{ for }}~\Re (z)\geq 0$
is called the exponential integral. It is closely related to Si and Ci,
$\operatorname {E} _{1}(ix)=i\left(-{\frac {\pi }{2}}+\operatorname {Si} (x)\right)-\operatorname {Ci} (x)=i\operatorname {si} (x)-\operatorname {ci} (x)\qquad ~{\text{ for }}~x>0~.$
As each respective function is analytic except for the cut at negative values of the argument, the area of validity of the relation should be extended to (Outside this range, additional terms which are integer factors of π appear in the expression.)
Cases of imaginary argument of the generalized integro-exponential function are
$\int _{1}^{\infty }\cos(ax){\frac {\ln x}{x}}\,dx=-{\frac {\pi ^{2}}{24}}+\gamma \left({\frac {\gamma }{2}}+\ln a\right)+{\frac {\ln ^{2}a}{2}}+\sum _{n\geq 1}{\frac {(-a^{2})^{n}}{(2n)!(2n)^{2}}}~,$
which is the real part of
$\int _{1}^{\infty }e^{iax}{\frac {\ln x}{x}}\,dx=-{\frac {\pi ^{2}}{24}}+\gamma \left({\frac {\gamma }{2}}+\ln a\right)+{\frac {\ln ^{2}a}{2}}-{\frac {\pi }{2}}i\left(\gamma +\ln a\right)+\sum _{n\geq 1}{\frac {(ia)^{n}}{n!n^{2}}}~.$
Similarly
$\int _{1}^{\infty }e^{iax}{\frac {\ln x}{x^{2}}}\,dx=1+ia\left[-{\frac {\pi ^{2}}{24}}+\gamma \left({\frac {\gamma }{2}}+\ln a-1\right)+{\frac {\ln ^{2}a}{2}}-\ln a+1\right]+{\frac {\pi a}{2}}{\Bigl (}\gamma +\ln a-1{\Bigr )}+\sum _{n\geq 1}{\frac {(ia)^{n+1}}{(n+1)!n^{2}}}~.$
Efficient evaluation
Padé approximants of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. The following formulae, given by Rowe et al. (2015),[2] are accurate to better than 10−16 for 0 ≤ x ≤ 4,
${\begin{array}{rcl}\operatorname {Si} (x)&\approx &x\cdot \left({\frac {\begin{array}{l}1-4.54393409816329991\cdot 10^{-2}\cdot x^{2}+1.15457225751016682\cdot 10^{-3}\cdot x^{4}-1.41018536821330254\cdot 10^{-5}\cdot x^{6}\\~~~+9.43280809438713025\cdot 10^{-8}\cdot x^{8}-3.53201978997168357\cdot 10^{-10}\cdot x^{10}+7.08240282274875911\cdot 10^{-13}\cdot x^{12}\\~~~-6.05338212010422477\cdot 10^{-16}\cdot x^{14}\end{array}}{\begin{array}{l}1+1.01162145739225565\cdot 10^{-2}\cdot x^{2}+4.99175116169755106\cdot 10^{-5}\cdot x^{4}+1.55654986308745614\cdot 10^{-7}\cdot x^{6}\\~~~+3.28067571055789734\cdot 10^{-10}\cdot x^{8}+4.5049097575386581\cdot 10^{-13}\cdot x^{10}+3.21107051193712168\cdot 10^{-16}\cdot x^{12}\end{array}}}\right)\\&~&\\\operatorname {Ci} (x)&\approx &\gamma +\ln(x)+\\&&x^{2}\cdot \left({\frac {\begin{array}{l}-0.25+7.51851524438898291\cdot 10^{-3}\cdot x^{2}-1.27528342240267686\cdot 10^{-4}\cdot x^{4}+1.05297363846239184\cdot 10^{-6}\cdot x^{6}\\~~~-4.68889508144848019\cdot 10^{-9}\cdot x^{8}+1.06480802891189243\cdot 10^{-11}\cdot x^{10}-9.93728488857585407\cdot 10^{-15}\cdot x^{12}\\\end{array}}{\begin{array}{l}1+1.1592605689110735\cdot 10^{-2}\cdot x^{2}+6.72126800814254432\cdot 10^{-5}\cdot x^{4}+2.55533277086129636\cdot 10^{-7}\cdot x^{6}\\~~~+6.97071295760958946\cdot 10^{-10}\cdot x^{8}+1.38536352772778619\cdot 10^{-12}\cdot x^{10}+1.89106054713059759\cdot 10^{-15}\cdot x^{12}\\~~~+1.39759616731376855\cdot 10^{-18}\cdot x^{14}\\\end{array}}}\right)\end{array}}$
The integrals may be evaluated indirectly via auxiliary functions $f(x)$ and $g(x)$, which are defined by
$\operatorname {Si} (x)={\frac {\pi }{2}}-f(x)\cos(x)-g(x)\sin(x)$
$\operatorname {Ci} (x)=f(x)\sin(x)-g(x)\cos(x)$
or equivalently
$f(x)\equiv \left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\cos(x)+\operatorname {Ci} (x)\sin(x)$
$g(x)\equiv \left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\sin(x)-\operatorname {Ci} (x)\cos(x)$
For $x\geq 4$ the Padé rational functions given below approximate $f(x)$ and $g(x)$ with error less than 10−16:[2]
${\begin{array}{rcl}f(x)&\approx &{\dfrac {1}{x}}\cdot \left({\frac {\begin{array}{l}1+7.44437068161936700618\cdot 10^{2}\cdot x^{-2}+1.96396372895146869801\cdot 10^{5}\cdot x^{-4}+2.37750310125431834034\cdot 10^{7}\cdot x^{-6}\\~~~+1.43073403821274636888\cdot 10^{9}\cdot x^{-8}+4.33736238870432522765\cdot 10^{10}\cdot x^{-10}+6.40533830574022022911\cdot 10^{11}\cdot x^{-12}\\~~~+4.20968180571076940208\cdot 10^{12}\cdot x^{-14}+1.00795182980368574617\cdot 10^{13}\cdot x^{-16}+4.94816688199951963482\cdot 10^{12}\cdot x^{-18}\\~~~-4.94701168645415959931\cdot 10^{11}\cdot x^{-20}\end{array}}{\begin{array}{l}1+7.46437068161927678031\cdot 10^{2}\cdot x^{-2}+1.97865247031583951450\cdot 10^{5}\cdot x^{-4}+2.41535670165126845144\cdot 10^{7}\cdot x^{-6}\\~~~+1.47478952192985464958\cdot 10^{9}\cdot x^{-8}+4.58595115847765779830\cdot 10^{10}\cdot x^{-10}+7.08501308149515401563\cdot 10^{11}\cdot x^{-12}\\~~~+5.06084464593475076774\cdot 10^{12}\cdot x^{-14}+1.43468549171581016479\cdot 10^{13}\cdot x^{-16}+1.11535493509914254097\cdot 10^{13}\cdot x^{-18}\end{array}}}\right)\\&&\\g(x)&\approx &{\dfrac {1}{x^{2}}}\cdot \left({\frac {\begin{array}{l}1+8.1359520115168615\cdot 10^{2}\cdot x^{-2}+2.35239181626478200\cdot 10^{5}\cdot x^{-4}+3.12557570795778731\cdot 10^{7}\cdot x^{-6}\\~~~+2.06297595146763354\cdot 10^{9}\cdot x^{-8}+6.83052205423625007\cdot 10^{10}\cdot x^{-10}+1.09049528450362786\cdot 10^{12}\cdot x^{-12}\\~~~+7.57664583257834349\cdot 10^{12}\cdot x^{-14}+1.81004487464664575\cdot 10^{13}\cdot x^{-16}+6.43291613143049485\cdot 10^{12}\cdot x^{-18}\\~~~-1.36517137670871689\cdot 10^{12}\cdot x^{-20}\end{array}}{\begin{array}{l}1+8.19595201151451564\cdot 10^{2}\cdot x^{-2}+2.40036752835578777\cdot 10^{5}\cdot x^{-4}+3.26026661647090822\cdot 10^{7}\cdot x^{-6}\\~~~+2.23355543278099360\cdot 10^{9}\cdot x^{-8}+7.87465017341829930\cdot 10^{10}\cdot x^{-10}+1.39866710696414565\cdot 10^{12}\cdot x^{-12}\\~~~+1.17164723371736605\cdot 10^{13}\cdot x^{-14}+4.01839087307656620\cdot 10^{13}\cdot x^{-16}+3.99653257887490811\cdot 10^{13}\cdot x^{-18}\end{array}}}\right)\\\end{array}}$
See also
• Logarithmic integral
• Tanc function
• Tanhc function
• Sinhc function
• Coshc function
References
1. Gray (1993). Modern Differential Geometry of Curves and Surfaces. Boca Raton. p. 119.{{cite book}}: CS1 maint: location missing publisher (link)
2. Rowe, B.; et al. (2015). "GALSIM: The modular galaxy image simulation toolkit". Astronomy and Computing. 10: 121. arXiv:1407.7676. Bibcode:2015A&C....10..121R. doi:10.1016/j.ascom.2015.02.002. S2CID 62709903.
• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 5". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 231. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Further reading
• Mathar, R.J. (2009). "Numerical evaluation of the oscillatory integral over exp(iπx)·x1/x between 1 and ∞". Appendix B. arXiv:0912.3844 [math.CA].
• Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). "Section 6.8.2 – Cosine and Sine Integrals". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
• Sloughter, Dan. "Sine Integral Taylor series proof" (PDF). Difference Equations to Differential Equations.
• Temme, N.M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", 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.
External links
• http://mathworld.wolfram.com/SineIntegral.html
• "Integral sine", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Integral cosine", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Nonelementary integrals
• Elliptic integral
• Error function
• Exponential integral
• Fresnel integral
• Logarithmic integral function
• Trigonometric integral
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Wikipedia
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Sina Greenwood
Sina Ruth Greenwood is a New Zealand mathematician whose interests include continuum theory, discrete dynamical systems, inverse limits, set-valued analysis, and Volterra spaces.[1][2] She is an associate professor of mathematics and Associate Dean Pacific in the faculty of science at the University of Auckland.[1]
Education and career
Greenwood's parents emigrated from Samoa to Whanganui in New Zealand, shortly before Greenwood was born; they moved from there to Auckland when she was a child. She earned a bachelor's degree at the University of Auckland, and after some time in Australia became a secondary school teacher in Auckland.[2]
Returning to the University of Auckland for graduate study in mathematics, she earned a master's degree and then completed her PhD in 1999, under the joint supervision of David Gauld and David W. Mcintyre. Her dissertation was Nonmetrisable Manifolds.[2][3][4] She and three other students who finished their doctorates at the same time became the first topologists to earn a doctorate at Auckland.[2]
After postdoctoral research, funded by a New Zealand Science and Technology Post-Doctoral Fellowship, she obtained a permanent position at the University of Auckland as a lecturer in 2004,[2][5] later becoming an associate professor.[2] Beyond mathematics, her work at the university has also included advocating for the interests of Pasifika and Māori students.[2][5]
Recognition
Greenwood is a Fellow of the New Zealand Mathematical Society,[6] elected in 2018.[2]
References
1. "Dr Sina Ruth Greenwood", University directory, University of Auckland, retrieved 2022-04-07
2. Gauld, David (December 2019), "Profile: Sina Greenwood" (PDF), NZMS Newsletter, New Zealand Mathematical Society (137): 20–21, retrieved 2022-04-07
3. Sina Greenwood at the Mathematics Genealogy Project
4. Greenwood, Sina (1999). Nonmetrisable Manifolds (Doctoral thesis). ResearchSpace@Auckland, University of Auckland. hdl:2292/697.
5. "New Colleagues", NZMS Newsletter (91), August 2004
6. NZMS Accreditation, New Zealand Mathematical Society, retrieved 2022-04-07
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Sinai–Ruelle–Bowen measure
In the mathematical discipline of ergodic theory, a Sinai–Ruelle–Bowen (SRB) measure is an invariant measure that behaves similarly to, but is not an ergodic measure. In order to be ergodic, the time average would need to be equal the space average for almost all initial states $x\in X$, with $X$ being the phase space.[1] For an SRB measure $\mu $, it suffices that the ergodicity condition be valid for initial states in a set $B(\mu )$ of positive Lebesgue measure.[2]
The initial ideas pertaining to SRB measures were introduced by Yakov Sinai, David Ruelle and Rufus Bowen in the less general area of Anosov diffeomorphisms and axiom A attractors.[3][4][5]
Definition
Let $T:X\rightarrow X$ be a map. Then a measure $\mu $ defined on $X$ is an SRB measure if there exist $U\subset X$ of positive Lebesgue measure, and $V\subset U$ with same Lebesgue measure, such that:[2][6]
$\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{i=0}^{n}\varphi (T^{i}x)=\int _{U}\varphi \,d\mu $
for every $x\in V$ and every continuous function $\varphi :U\rightarrow \mathbb {R} $.
One can see the SRB measure $\mu $ as one that satisfies the conclusions of Birkhoff's ergodic theorem on a smaller set contained in $X$.
Existence of SRB measures
The following theorem establishes sufficient conditions for the existence of SRB measures. It considers the case of Axiom A attractors, which is simpler, but it has been extended times to more general scenarios.[7]
Theorem 1:[7] Let $T:X\rightarrow X$ be a $C^{2}$ diffeomorphism with an Axiom A attractor ${\mathcal {A}}\subset X$. Assume that this attractor is irreducible, that is, it is not the union of two other sets that are also invariant under $T$. Then there is a unique Borelian measure $\mu $, with $\mu (X)=1$,[lower-alpha 1] characterized by the following equivalent statements:
1. $\mu $ is an SRB measure;
2. $\mu $ has absolutely continuous measures conditioned on the unstable manifold and submanifolds thereof;
3. $h(T)=\int \log \left|\det(DT)|_{E^{u}}\right|\,d\mu $, where $h$ is the Kolmogorov–Sinai entropy, $E^{u}$ is the unstable manifold and $D$ is the differential operator.
Also, in these conditions $\left(T,X,{\mathcal {B}}(X),\mu \right)$ is a measure-preserving dynamical system.
It has also been proved that the above are equivalent to stating that $\mu $ equals the zero-noise limit stationary distribution of a Markov chain with states $T^{i}(x)$.[8] That is, consider that to each point $x\in X$ is associated a transition probability $P_{\varepsilon }(\cdot \mid x)$ with noise level $\varepsilon $ that measures the amount of uncertainty of the next state, in a way such that:
$\lim _{\varepsilon \rightarrow 0}P_{\varepsilon }(\cdot \mid x)=\delta _{Tx}(\cdot ),$
where $\delta $ is the Dirac measure. The zero-noise limit is the stationary distribution of this Markov chain when the noise level approaches zero. The importance of this is that it states mathematically that the SRB measure is a "good" approximation to practical cases where small amounts of noise exist,[8] though nothing can be said about the amount of noise that is tolerable.
See also
• Quasi-invariant measure
• Krylov–Bogolyubov theorem
• Gibbs measure
Notes
1. If it does not integrate to one, there will be infinite such measures, each being equal to the other except for a multiplicative constant.
References
1. Walters, Peter (2000). An Introduction to Ergodic Theory. Springer.
2. Bonatti, C.; Viana, M. (2000). "SRB measures for partially hyperbolic systems whose central direction is mostly contracting". Israel Journal of Mathematics. 115 (1): 157–193. doi:10.1007/BF02810585. S2CID 10139213.
3. Bowen, Robert Edward (1975). "Ergodic theory of axiom A diffeomorphisms". Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics. Vol. 470. Springer. pp. 63–76. doi:10.1007/978-3-540-77695-6_4.
4. Ruelle, David (1976). "A measure associated with axiom A attractors". American Journal of Mathematics. 98 (3): 619–654. doi:10.2307/2373810. JSTOR 2373810.
5. Sinai, Yakov G. (1972). "Gibbs measures in ergodic theory". Russian Mathematical Surveys. 27 (4): 21–69. doi:10.1070/RM1972v027n04ABEH001383.
6. Metzger, R. J. (2000). "Sinai–Ruelle–Bowen measures for contracting Lorenz maps and flows". Annales de l'Institut Henri Poincaré C. 17 (2): 247–276. Bibcode:2000AIHPC..17..247M. doi:10.1016/S0294-1449(00)00111-6.
7. Young, L. S. (2002). "What are SRB measures, and which dynamical systems have them?". Journal of Statistical Physics. 108 (5–6): 733–754. doi:10.1023/A:1019762724717. S2CID 14403405.
8. Cowieson, W.; Young, L. S. (2005). "SRB measures as zero-noise limits". Ergodic Theory and Dynamical Systems. 25 (4): 1115–1138. doi:10.1017/S0143385704000604. S2CID 15640353.
Chaos theory
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Wikipedia
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Sinc numerical methods
In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques[1] for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by
$C(f,h)(x)=\sum _{k=-\infty }^{\infty }f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)$
where the step size h>0 and where the sinc function is defined by
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/":): \textrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x}
Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers.
The truncated Sinc expansion of f is defined by the following series:
$C_{M,N}(f,h)(x)=\displaystyle \sum _{k=-M}^{N}f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)$ .
Sinc numerical methods cover
• function approximation,
• approximation of derivatives,
• approximate definite and indefinite integration,
• approximate solution of initial and boundary value ordinary differential equation (ODE) problems,
• approximation and inversion of Fourier and Laplace transforms,
• approximation of Hilbert transforms,
• approximation of definite and indefinite convolution,
• approximate solution of partial differential equations,
• approximate solution of integral equations,
• construction of conformal maps.
Indeed, Sinc are ubiquitous for approximating every operation of calculus
In the standard setup of the sinc numerical methods, the errors (in big O notation) are known to be $O\left(e^{-c{\sqrt {n}}}\right)$ with some c>0, where n is the number of nodes or bases used in the methods. However, Sugihara[2] has recently found that the errors in the Sinc numerical methods based on double exponential transformation are $O\left(e^{-{\frac {kn}{\ln n}}}\right)$ with some k>0, in a setup that is also meaningful both theoretically and practically and are found to be best possible in a certain mathematical sense.
Reading
• Stenger, Frank (2011). Handbook of Sinc Numerical Methods. Boca Raton, Florida: CRC Press. ISBN 9781439821596.
• Lund, John; Bowers, Kenneth (1992). Sinc Methods for Quadrature and Differential Equations. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). ISBN 9780898712988.
References
1. Stenger, F. (2000). "Summary of sinc numerical methods". Journal of Computational and Applied Mathematics. 121: 379–420. doi:10.1016/S0377-0427(00)00348-4.
2. Sugihara, M.; Matsuo, T. (2004). "Recent developments of the Sinc numerical methods". Journal of Computational and Applied Mathematics. 164–165: 673. doi:10.1016/j.cam.2003.09.016.
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Wikipedia
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Shannon wavelet
In functional analysis, the Shannon wavelet (or sinc wavelets) is a decomposition that is defined by signal analysis by ideal bandpass filters. Shannon wavelet may be either of real or complex type.
Shannon wavelet is not well-localized(noncompact) in the time domain,but its Fourier transform is band-limited(compact support). Hence Shannon wavelet has poor time localization but has good frequency localization. These characteristics are in stark contrast to those of the Haar wavelet. The Haar and sinc systems are Fourier duals of each other.
Definition
Sinc funcition is the starting point for the definition of the shannon wavelet.
Scaling function
First, we define the scaling function to be the sinc function.
$\phi ^{\text{(Sha)}}(t):={\frac {\sin \pi t}{\pi t}}=\operatorname {sinc} (t).$
And define the dilated and translated intances to be
$\phi _{k}^{n}(t):=2^{n/2}\phi ^{\text{(Sha)}}(2^{n}t-k)$
where the parameter $n,k$ means the dilation and the translation for the wavelet respectively.
Then we can derive the Fourier transform of the scaling function:
$\Phi ^{\text{(Sha)}}(\omega )={\frac {1}{2\pi }}\Pi ({\frac {\omega }{2\pi }})={\begin{cases}{\frac {1}{2\pi }},&{\mbox{if }}{|\omega |\leq \pi },\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}$ where the (normalised) gate function is defined by
$\Pi (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 1/2},\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}$ Also for the dilated and translated instances of scaling function: $\Phi _{k}^{n}(\omega )={\frac {2^{-n/2}}{2\pi }}e^{-i\omega (k+1)/2^{n}}\Pi ({\frac {\omega }{2^{n+1}\pi }})$
Mother wavelet
Use $\Phi ^{\text{(Sha)}}$ and multiresolution approximation we can derive the Fourier transform of the Mother wavelet:
$\Psi ^{\text{(Sha)}}(\omega )={\frac {1}{2\pi }}e^{-i\omega }{\bigg (}\Pi ({\frac {\omega }{\pi }}-{\frac {3}{2}})+\Pi ({\frac {\omega }{\pi }}+{\frac {3}{2}}){\bigg )}$
And the dilated and translated instances:
$\Psi _{k}^{n}(\omega )={\frac {2^{-n/2}}{2\pi }}e^{-i\omega (k+1)/2^{n}}{\bigg (}\Pi ({\frac {\omega }{2^{n}\pi }}-{\frac {3}{2}})+\Pi ({\frac {\omega }{2^{n}\pi }}+{\frac {3}{2}}){\bigg )}$
Then the shannon mother wavelet function and the family of dilated and translated instances can be obtained by the inverse Fourier transform:
$\psi ^{\text{(Sha)}}(t)={\frac {\sin \pi (t-(1/2))-\sin 2\pi (t-(1/2))}{\pi (t-1/2)}}=\operatorname {sinc} {\bigg (}t-{\frac {1}{2}}{\bigg )}-2\operatorname {sinc} {\bigg (}2(t-{\frac {1}{2}}){\bigg )}$
$\psi _{k}^{n}(t)=2^{n/2}\psi ^{\text{(Sha)}}(2^{n}t-k)$
Property of mother wavelet and scaling function
• Mother wavelets are orthonormal, namely,
$<\psi _{k}^{n}(t),\psi _{h}^{m}(t)>=\delta ^{nm}\delta _{hk}={\begin{cases}1,&{\text{if }}h=k{\text{ and }}n=m\\0,&{\text{otherwise}}\end{cases}}$
• The translated instances of scaling function at level $n=0$ are orthogonal
$<\phi _{k}^{0}(t),\phi _{h}^{0}(t)>=\delta ^{kh}$
• The translated instances of scaling function at level $n=0$ are orthogonal to the mother wavelets
$<\phi _{k}^{0}(t),\psi _{h}^{m}(t)>=0$
• Shannon wavelets has an infinite number of vanishing moments.
Reconstruction of a Function by Shannon Wavelets
Suppose $f(x)\in L_{2}(\mathbb {R} )$ such that $\operatorname {supp} \operatorname {FT} \{f\}\subset [-\pi ,\pi ]$ and for any dilation and the translation parameter $n,k$,
${\Bigg |}\int _{-\infty }^{\infty }f(t)\phi _{k}^{0}(t)dt{\Bigg |}<\infty $, ${\Bigg |}\int _{-\infty }^{\infty }f(t)\psi _{k}^{n}(t)dt{\Bigg |}<\infty $
Then
$f(t)=\sum _{k=\infty }^{\infty }\alpha _{k}\phi _{k}^{0}(t)$ is uniformly convergent, where $\alpha _{k}=f(k)$
Real Shannon wavelet
The Fourier transform of the Shannon mother wavelet is given by:
$\Psi ^{(\operatorname {Sha} )}(w)=\prod \left({\frac {w-3\pi /2}{\pi }}\right)+\prod \left({\frac {w+3\pi /2}{\pi }}\right).$
where the (normalised) gate function is defined by
$\prod (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 1/2},\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}$
The analytical expression of the real Shannon wavelet can be found by taking the inverse Fourier transform:
$\psi ^{(\operatorname {Sha} )}(t)=\operatorname {sinc} \left({\frac {t}{2}}\right)\cdot \cos \left({\frac {3\pi t}{2}}\right)$
or alternatively as
$\psi ^{(\operatorname {Sha} )}(t)=2\cdot \operatorname {sinc} (2t)-\operatorname {sinc} (t),$
where
$\operatorname {sinc} (t):={\frac {\sin {\pi t}}{\pi t}}$
is the usual sinc function that appears in Shannon sampling theorem.
This wavelet belongs to the $C^{\infty }$-class of differentiability, but it decreases slowly at infinity and has no bounded support, since band-limited signals cannot be time-limited.
The scaling function for the Shannon MRA (or Sinc-MRA) is given by the sample function:
$\phi ^{(Sha)}(t)={\frac {\sin \pi t}{\pi t}}=\operatorname {sinc} (t).$
Complex Shannon wavelet
In the case of complex continuous wavelet, the Shannon wavelet is defined by
$\psi ^{(CSha)}(t)=\operatorname {sinc} (t)\cdot e^{-2\pi it}$,
References
• S.G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999, ISBN 0-12-466606-X
• C.S. Burrus, R.A. Gopinath, H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, Prentice-Hall, 1988, ISBN 0-13-489600-9.
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Wikipedia
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Sindoni Tower
The Sindoni Tower (also known as Filippo Sindoni Tower or the Torre Sindoni in Spanish) is a building located in the Venezuelan city of Maracay. It is the highest tower in the city. It has an estimated height of 125 meters, and about 32 floors, making it the eighth tallest building in Venezuela, as well as being one of the newest buildings in height. It was opened in 1999. The tower is mostly used as a commercial office space.[1]
Sindoni Tower
General information
StatusCompleted
TypeOffice
LocationMaracay, Venezuela
Completed1999
OwnerGrupo de Empresas Sindoni C. A.
Height
Roof125 m (410 ft)
Technical details
Floor count32
See also
• List of tallest buildings in South America
• List of tallest buildings in Venezuela
References
1. "Torre Sindoni - The Skyscraper Center". www.skyscrapercenter.com. Retrieved 2022-08-09.
10°14′55″N 67°35′26″W
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Wikipedia
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Law of sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
${\frac {a}{\sin {\alpha }}}\,=\,{\frac {b}{\sin {\beta }}}\,=\,{\frac {c}{\sin {\gamma }}}\,=\,2R,$
Law of Sines
Figure 1, With circumcircle
Figure 2, Without circumcircle
Two triangles labelled with the components of the law of sines. α, β and γ are the angles associated with the vertices at capital A, B, and C, respectively. Lower-case a, b, and c are the lengths of the sides opposite them. (a is opposite α, etc.)
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals;
${\frac {\sin {\alpha }}{a}}\,=\,{\frac {\sin {\beta }}{b}}\,=\,{\frac {\sin {\gamma }}{c}}.$
The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ambiguous case) and the technique gives two possible values for the enclosed angle.
The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines.
The law of sines can be generalized to higher dimensions on surfaces with constant curvature.[1]
History
H.J.J. Wilson's book Eastern Science[2] states that the 7th century Indian mathematician Brahmagupta describes what we now know as the law of sines in his astronomical treatise Brāhmasphuṭasiddhānta. In his partial translation of this work, Colebrooke[3] translates Brahmagupta's statement of the sine rule as: The product of the two sides of a triangle, divided by twice the perpendicular, is the central line; and the double of this is the diameter of the central line.
According to Ubiratàn D'Ambrosio and Helaine Selin, the spherical law of sines was discovered in the 10th century. It is variously attributed to Abu-Mahmud Khojandi, Abu al-Wafa' Buzjani, Nasir al-Din al-Tusi and Abu Nasr Mansur.[4]
Ibn Muʿādh al-Jayyānī's The book of unknown arcs of a sphere in the 11th century contains the spherical law of sines.[5] The plane law of sines was later stated in the 13th century by Nasīr al-Dīn al-Tūsī. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles, and provided proofs for this law.[6]
According to Glen Van Brummelen, "The Law of Sines is really Regiomontanus's foundation for his solutions of right-angled triangles in Book IV, and these solutions are in turn the bases for his solutions of general triangles."[7] Regiomontanus was a 15th-century German mathematician.
Proof
The area of any triangle can be written as one half of its base times its height. Selecting one side of the triangle as the base, the height of the triangle relative to that base is computed as the length of another side times the sine of the angle between the chosen side and the base. Thus depending on the selection of the base, the area T of the triangle can be written as any of:
$T={\frac {1}{2}}b\left(c\sin {\alpha }\right)={\frac {1}{2}}c\left(a\sin {\beta }\right)={\frac {1}{2}}a\left(b\sin {\gamma }\right).$
Multiplying these by 2/abc gives
${\frac {2T}{abc}}={\frac {\sin {\alpha }}{a}}={\frac {\sin {\beta }}{b}}={\frac {\sin {\gamma }}{c}}\,.$
The ambiguous case of triangle solution
When using the law of sines to find a side of a triangle, an ambiguous case occurs when two separate triangles can be constructed from the data provided (i.e., there are two different possible solutions to the triangle). In the case shown below they are triangles ABC and ABC′.
Given a general triangle, the following conditions would need to be fulfilled for the case to be ambiguous:
• The only information known about the triangle is the angle α and the sides a and c.
• The angle α is acute (i.e., α < 90°).
• The side a is shorter than the side c (i.e., a < c).
• The side a is longer than the altitude h from angle β, where h = c sin α (i.e., a > h).
If all the above conditions are true, then each of angles β and β′ produces a valid triangle, meaning that both of the following are true:
${\gamma }'=\arcsin {\frac {c\sin {\alpha }}{a}}\quad {\text{or}}\quad {\gamma }=\pi -\arcsin {\frac {c\sin {\alpha }}{a}}.$
From there we can find the corresponding β and b or β′ and b′ if required, where b is the side bounded by vertices A and C and b′ is bounded by A and C′.
Examples
The following are examples of how to solve a problem using the law of sines.
Example 1
Given: side a = 20, side c = 24, and angle γ = 40°. Angle α is desired.
Using the law of sines, we conclude that
${\frac {\sin \alpha }{20}}={\frac {\sin(40^{\circ })}{24}}.$
$\alpha =\arcsin \left({\frac {20\sin(40^{\circ })}{24}}\right)\approx 32.39^{\circ }.$
Note that the potential solution α = 147.61° is excluded because that would necessarily give α + β + γ > 180°.
Example 2
If the lengths of two sides of the triangle a and b are equal to x, the third side has length c, and the angles opposite the sides of lengths a, b, and c are α, β, and γ respectively then
${\begin{aligned}&\alpha =\beta ={\frac {180^{\circ }-\gamma }{2}}=90^{\circ }-{\frac {\gamma }{2}}\\[6pt]&\sin \alpha =\sin \beta =\sin \left(90^{\circ }-{\frac {\gamma }{2}}\right)=\cos \left({\frac {\gamma }{2}}\right)\\[6pt]&{\frac {c}{\sin \gamma }}={\frac {a}{\sin \alpha }}={\frac {x}{\cos \left({\frac {\gamma }{2}}\right)}}\\[6pt]&{\frac {c\cos \left({\frac {\gamma }{2}}\right)}{\sin \gamma }}=x\end{aligned}}$
Relation to the circumcircle
In the identity
${\frac {a}{\sin {\alpha }}}={\frac {b}{\sin {\beta }}}={\frac {c}{\sin {\gamma }}},$
the common value of the three fractions is actually the diameter of the triangle's circumcircle. This result dates back to Ptolemy.[8][9]
Proof
As shown in the figure, let there be a circle with inscribed $\triangle ABC$ and another inscribed $\triangle ADB$ that passes through the circle's center O. The $\angle AOD$ has a central angle of $180^{\circ }$ and thus $\angle ABD=90^{\circ }$, by Thales's theorem. Since $\triangle ABD$ is a right triangle,
$\sin {\delta }={\frac {\text{opposite}}{\text{hypotenuse}}}={\frac {c}{2R}},$
where $ R={\frac {d}{2}}$ is the radius of the circumscribing circle of the triangle.[9] Angles ${\gamma }$ and ${\delta }$ have the same central angle thus they are the same, by the inscribed angle theorem: ${\gamma }={\delta }$. Therefore,
$\sin {\delta }=\sin {\gamma }={\frac {c}{2R}}.$
Rearranging yields
$2R={\frac {c}{\sin {\gamma }}}.$
Repeating the process of creating $\triangle ADB$ with other points gives
${\frac {a}{\sin {\alpha }}}={\frac {b}{\sin {\beta }}}={\frac {c}{\sin {\gamma }}}=2R.$
Relationship to the area of the triangle
The area of a triangle is given by $ T={\frac {1}{2}}ab\sin \theta $, where $\theta $ is the angle enclosed by the sides of lengths a and b. Substituting the sine law into this equation gives
$T={\frac {1}{2}}ab\cdot {\frac {c}{2R}}.$
Taking $R$ as the circumscribing radius,[10]
$T={\frac {abc}{4R}}.$
It can also be shown that this equality implies
${\begin{aligned}{\frac {abc}{2T}}&={\frac {abc}{2{\sqrt {s(s-a)(s-b)(s-c)}}}}\\[6pt]&={\frac {2abc}{\sqrt {{(a^{2}+b^{2}+c^{2})}^{2}-2(a^{4}+b^{4}+c^{4})}}},\end{aligned}}$
where T is the area of the triangle and s is the semiperimeter $ s={\frac {1}{2}}\left(a+b+c\right).$
The second equality above readily simplifies to Heron's formula for the area.
The sine rule can also be used in deriving the following formula for the triangle's area: denoting the semi-sum of the angles' sines as $ S={\frac {1}{2}}\left(\sin A+\sin B+\sin C\right)$, we have[11]
$T=4R^{2}{\sqrt {S\left(S-\sin A\right)\left(S-\sin B\right)\left(S-\sin C\right)}}$
where $R$ is the radius of the circumcircle: $2R={\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}$.
The spherical law of sines
The spherical law of sines deals with triangles on a sphere, whose sides are arcs of great circles.
Suppose the radius of the sphere is 1. Let a, b, and c be the lengths of the great-arcs that are the sides of the triangle. Because it is a unit sphere, a, b, and c are the angles at the center of the sphere subtended by those arcs, in radians. Let A, B, and C be the angles opposite those respective sides. These are dihedral angles between the planes of the three great circles.
Then the spherical law of sines says:
${\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}.$
Vector proof
Consider a unit sphere with three unit vectors OA, OB and OC drawn from the origin to the vertices of the triangle. Thus the angles α, β, and γ are the angles a, b, and c, respectively. The arc BC subtends an angle of magnitude a at the centre. Introduce a Cartesian basis with OA along the z-axis and OB in the xz-plane making an angle c with the z-axis. The vector OC projects to ON in the xy-plane and the angle between ON and the x-axis is A. Therefore, the three vectors have components:
$\mathbf {OA} ={\begin{pmatrix}0\\0\\1\end{pmatrix}},\quad \mathbf {OB} ={\begin{pmatrix}\sin c\\0\\\cos c\end{pmatrix}},\quad \mathbf {OC} ={\begin{pmatrix}\sin b\cos A\\\sin b\sin A\\\cos b\end{pmatrix}}.$
The scalar triple product, OA ⋅ (OB × OC) is the volume of the parallelepiped formed by the position vectors of the vertices of the spherical triangle OA, OB and OC. This volume is invariant to the specific coordinate system used to represent OA, OB and OC. The value of the scalar triple product OA ⋅ (OB × OC) is the 3 × 3 determinant with OA, OB and OC as its rows. With the z-axis along OA the square of this determinant is
${\begin{aligned}{\bigl (}\mathbf {OA} \cdot (\mathbf {OB} \times \mathbf {OC} ){\bigr )}^{2}&=\left(\det {\begin{pmatrix}\mathbf {OA} &\mathbf {OB} &\mathbf {OC} \end{pmatrix}}\right)^{2}\\[4pt]&={\begin{vmatrix}0&0&1\\\sin c&0&\cos c\\\sin b\cos A&\sin b\sin A&\cos b\end{vmatrix}}^{2}=\left(\sin b\sin c\sin A\right)^{2}.\end{aligned}}$
Repeating this calculation with the z-axis along OB gives (sin c sin a sin B)2, while with the z-axis along OC it is (sin a sin b sin C)2. Equating these expressions and dividing throughout by (sin a sin b sin c)2 gives
${\frac {\sin ^{2}A}{\sin ^{2}a}}={\frac {\sin ^{2}B}{\sin ^{2}b}}={\frac {\sin ^{2}C}{\sin ^{2}c}}={\frac {V^{2}}{\sin ^{2}(a)\sin ^{2}(b)\sin ^{2}(c)}},$
where V is the volume of the parallelepiped formed by the position vector of the vertices of the spherical triangle. Consequently, the result follows.
It is easy to see how for small spherical triangles, when the radius of the sphere is much greater than the sides of the triangle, this formula becomes the planar formula at the limit, since
$\lim _{a\to 0}{\frac {\sin a}{a}}=1$
and the same for sin b and sin c.
Geometric proof
Consider a unit sphere with:
$OA=OB=OC=1$
Construct point $D$ and point $E$ such that $\angle ADO=\angle AEO=90^{\circ }$
Construct point $A'$ such that $\angle A'DO=\angle A'EO=90^{\circ }$
It can therefore be seen that $\angle ADA'=B$ and $\angle AEA'=C$
Notice that $A'$ is the projection of $A$ on plane $OBC$. Therefore $\angle AA'D=\angle AA'E=90^{\circ }$
By basic trigonometry, we have:
${\begin{aligned}AD&=\sin c\\AE&=\sin b\end{aligned}}$
But $AA'=AD\sin B=AE\sin C$
Combining them we have:
${\begin{aligned}\sin c\sin B&=\sin b\sin C\\\Rightarrow {\frac {\sin B}{\sin b}}&={\frac {\sin C}{\sin c}}\end{aligned}}$
By applying similar reasoning, we obtain the spherical law of sine:
${\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}$
See also: Spherical trigonometry, Spherical law of cosines, and Half-side formula
Other proofs
A purely algebraic proof can be constructed from the spherical law of cosines. From the identity $\sin ^{2}A=1-\cos ^{2}A$ and the explicit expression for $\cos A$ from the spherical law of cosines
${\begin{aligned}\sin ^{2}\!A&=1-\left({\frac {\cos a-\cos b\,\cos c}{\sin b\,\sin c}}\right)^{2}\\&={\frac {\left(1-\cos ^{2}\!b\right)\left(1-\cos ^{2}\!c\right)-\left(\cos a-\cos b\,\cos c\right)^{2}}{\sin ^{2}\!b\,\sin ^{2}\!c}}\\[8pt]{\frac {\sin A}{\sin a}}&={\frac {\left[1-\cos ^{2}\!a-\cos ^{2}\!b-\cos ^{2}\!c+2\cos a\cos b\cos c\right]^{1/2}}{\sin a\sin b\sin c}}.\end{aligned}}$
Since the right hand side is invariant under a cyclic permutation of $a,\;b,\;c$ the spherical sine rule follows immediately.
The figure used in the Geometric proof above is used by and also provided in Banerjee[12] (see Figure 3 in this paper) to derive the sine law using elementary linear algebra and projection matrices.
Hyperbolic case
In hyperbolic geometry when the curvature is −1, the law of sines becomes
${\frac {\sin A}{\sinh a}}={\frac {\sin B}{\sinh b}}={\frac {\sin C}{\sinh c}}\,.$
In the special case when B is a right angle, one gets
$\sin C={\frac {\sinh c}{\sinh b}}$
which is the analog of the formula in Euclidean geometry expressing the sine of an angle as the opposite side divided by the hypotenuse.
See also: Hyperbolic triangle
The case of surfaces of constant curvature
Define a generalized sine function, depending also on a real parameter K:
$\sin _{K}x=x-{\frac {Kx^{3}}{3!}}+{\frac {K^{2}x^{5}}{5!}}-{\frac {K^{3}x^{7}}{7!}}+\cdots .$
The law of sines in constant curvature K reads as[1]
${\frac {\sin A}{\sin _{K}a}}={\frac {\sin B}{\sin _{K}b}}={\frac {\sin C}{\sin _{K}c}}\,.$
By substituting K = 0, K = 1, and K = −1, one obtains respectively the Euclidean, spherical, and hyperbolic cases of the law of sines described above.
Let pK(r) indicate the circumference of a circle of radius r in a space of constant curvature K. Then pK(r) = 2π sinK r. Therefore, the law of sines can also be expressed as:
${\frac {\sin A}{p_{K}(a)}}={\frac {\sin B}{p_{K}(b)}}={\frac {\sin C}{p_{K}(c)}}\,.$
This formulation was discovered by János Bolyai.[13]
Higher dimensions
A tetrahedron has four triangular facets. The absolute value of the polar sine (psin) of the normal vectors to the three facets that share a vertex of the tetrahedron, divided by the area of the fourth facet will not depend upon the choice of the vertex:
${\begin{aligned}&{\frac {\left|\operatorname {psin} (\mathbf {b} ,\mathbf {c} ,\mathbf {d} )\right|}{\mathrm {Area} _{a}}}={\frac {\left|\operatorname {psin} (\mathbf {a} ,\mathbf {c} ,\mathbf {d} )\right|}{\mathrm {Area} _{b}}}={\frac {\left|\operatorname {psin} (\mathbf {a} ,\mathbf {b} ,\mathbf {d} )\right|}{\mathrm {Area} _{c}}}={\frac {\left|\operatorname {psin} (\mathbf {a} ,\mathbf {b} ,\mathbf {c} )\right|}{\mathrm {Area} _{d}}}\\[4pt]={}&{\frac {(3\operatorname {Volume} _{\mathrm {tetrahedron} })^{2}}{2!~\mathrm {Area} _{a}\mathrm {Area} _{b}\mathrm {Area} _{c}\mathrm {Area} _{d}}}\,.\end{aligned}}$
More generally, for an n-dimensional simplex (i.e., triangle (n = 2), tetrahedron (n = 3), pentatope (n = 4), etc.) in n-dimensional Euclidean space, the absolute value of the polar sine of the normal vectors of the facets that meet at a vertex, divided by the hyperarea of the facet opposite the vertex is independent of the choice of the vertex. Writing V for the hypervolume of the n-dimensional simplex and P for the product of the hyperareas of its (n − 1)-dimensional facets, the common ratio is
${\frac {\left|\operatorname {psin} (\mathbf {b} ,\ldots ,\mathbf {z} )\right|}{\mathrm {Area} _{a}}}=\cdots ={\frac {\left|\operatorname {psin} (\mathbf {a} ,\ldots ,\mathbf {y} )\right|}{\mathrm {Area} _{z}}}={\frac {(nV)^{n-1}}{(n-1)!P}}.$
See also
• Gersonides
• Half-side formula – for solving spherical triangles
• Law of cosines
• Law of tangents
• Law of cotangents
• Mollweide's formula – for checking solutions of triangles
• Solution of triangles
• Surveying
References
1. "Generalized law of sines". mathworld.
2. Wilson, H.J.J., Eastern Science, John Murray Publishers, 1952, p46.
3. Colebrooke, Henry Thomas, Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara, London John Murray, 1817, pp. 299-300, URL: https://archive.org/details/algebrawitharith00brahuoft/page/298/mode/2up
4. Sesiano just lists al-Wafa as a contributor. Sesiano, Jacques (2000) "Islamic mathematics" pp. 137–157, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2
5. O'Connor, John J.; Robertson, Edmund F., "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", MacTutor History of Mathematics Archive, University of St Andrews
6. Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9.
7. Glen Van Brummelen (2009). "The mathematics of the heavens and the earth: the early history of trigonometry". Princeton University Press. p.259. ISBN 0-691-12973-8
8. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1–3, 1967
9. "Law of Sines". www.pballew.net. Retrieved 2018-09-18.
10. Mr. T's Math Videos (2015-06-10), Area of a Triangle and Radius of its Circumscribed Circle, archived from the original on 2021-12-11, retrieved 2018-09-18
11. Mitchell, Douglas W., "A Heron-type area formula in terms of sines," Mathematical Gazette 93, March 2009, 108–109.
12. Banerjee, Sudipto (2004), "Revisiting Spherical Trigonometry with Orthogonal Projectors", The College Mathematics Journal, Mathematical Association of America, 35 (5): 375–381, doi:10.1080/07468342.2004.11922099, S2CID 122277398Text online {{citation}}: External link in |postscript= (help)CS1 maint: postscript (link)
13. Katok, Svetlana (1992). Fuchsian groups. Chicago: University of Chicago Press. p. 22. ISBN 0-226-42583-5.
External links
Wikimedia Commons has media related to Law of sines.
• "Sine theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• The Law of Sines at cut-the-knot
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Singapore math
Singapore math (or Singapore maths in British English[1]) is a teaching method based on the national mathematics curriculum used for first through sixth grade in Singaporean schools.[2][3] The term was coined in the United States[4] to describe an approach originally developed in Singapore to teach students to learn and master fewer mathematical concepts at greater detail as well as having them learn these concepts using a three-step learning process: concrete, pictorial, and abstract.[2][3] In the concrete step, students engage in hands-on learning experiences using physical objects which can be everyday items such as paper clips, toy blocks or math manipulates such as counting bears, link cubes and fraction discs.[5] This is followed by drawing pictorial representations of mathematical concepts. Students then solve mathematical problems in an abstract way by using numbers and symbols.[6]
The development of Singapore math began in the 1980s when Singapore's Ministry of Education developed its own mathematics textbooks that focused on problem solving and developing thinking skills.[3][7] Outside Singapore, these textbooks were adopted by several schools in the United States and in other countries such as Canada, Israel, the Netherlands, Indonesia, Chile, Jordan, India, Pakistan, Thailand, Malaysia, Japan, South Korea, the Philippines and the United Kingdom.[1][8][9][10] Early adopters of these textbooks in the U.S. included parents interested in homeschooling as well as a limited number of schools.[3] These textbooks became more popular since the release of scores from international education surveys such as Trends in International Mathematics and Science Study (TIMSS) and Programme for International Student Assessment (PISA), which showed Singapore at the top three of the world since 1995.[11][12] U.S. editions of these textbooks have since been adopted by a large number of school districts as well as charter and private schools.[3]
History
Before the development of its own mathematics textbooks in the 1980s, Singapore imported its mathematics textbooks from other countries.[13] In 1981, the Curriculum Development Institute of Singapore (CDIS) (currently the Curriculum Planning and Development Division) began to develop its own mathematics textbooks and curriculum. The CDIS developed and distributed a textbook series for elementary schools in Singapore called Primary Mathematics, which was first published in 1982 and subsequently revised in 1992 to emphasize problem solving.[14][15] In the late 1990s, the country's Ministry of Education opened the elementary school textbook market to private companies, and Marshall Cavendish, a local and private publisher of educational materials, began to publish and market the Primary Mathematics textbooks.[1][15][16]
Following Singapore's curricular and instructional initiatives, dramatic improvements in math proficiency among Singaporean students on international assessments were observed.[1] TIMSS, an international assessment for math and science among fourth and eighth graders, ranked Singapore's fourth and eighth grade students first in mathematics four times (1995, 1999, 2003, and 2015) among participating nations.[11][14][12] Likewise, the Organisation for Economic Co-operation and Development (OECD)'s Programme for International Student Assessment (PISA), a worldwide study of 15-year-old school students' scholastic performance in mathematics, science, and reading, has ranked Singaporean students first in 2015,[17] and second after Shanghai, China in 2009 and 2012.[18][19]
Since the TIMSS publication of Singapore's high ranking in mathematics, professional mathematicians in the U.S. took a closer look at Singapore mathematics textbooks such as Primary Mathematics.[11] The term Singapore math was originally coined in the U.S. to describe the teaching approach based on these textbooks.[4] In 2005, the American Institutes for Research (AIR) published a study, which concluded that U.S. schools could benefit from adopting these textbooks.[11] The textbooks were already distributed in the U.S. by Singapore Math, Inc., a private venture based in Oregon.[14] Early users of these textbooks in the U.S. included parents interested in homeschooling as well as a limited number of schools.[3] They became more popular since the release of the TIMSS scores showing Singapore's top ranking.[11] As of 2004, U.S. versions of Singapore mathematics textbooks were adopted in over 200 U.S. schools.[3][8] Schools and counties that had adopted these textbooks reported improvements in their students' performance.[8][11][16][20] Singapore math textbooks were also used in schools from other countries such as Canada, Israel, and the United Kingdom.[1][8][9]
Features
Covers fewer topics in greater depth
Compared to a traditional U.S. math curriculum, Singapore math focuses on fewer topics but covers them in greater detail.[3] Each semester-level Singapore math textbook builds upon prior knowledge and skills, with students mastering them before moving on to the next grade. Students, therefore, need not re-learn these skills at the next grade level.[2] By the end of sixth grade, Singapore math students have mastered multiplication and division of fractions and can solve difficult multi-step word problems.[21]
In the U.S., it was found that Singapore math emphasizes the essential math skills recommended in the 2006 Focal Points publication by the National Council of Teachers of Mathematics (NCTM), the 2008 final report by the National Mathematics Advisory Panel, and the proposed Common Core State Standards, though it generally progresses to topics at an earlier grade level compared to U.S. standards.[22][23]
Three-step learning process
Main article: Arithmetic
Singapore math teaches students mathematical concepts in a three-step learning process: concrete, pictorial, and abstract.[3] This learning process was based on the work of an American psychologist, Jerome Bruner. In the 1960s, Bruner found that people learn in three stages by first handling real objects before transitioning to pictures and then to symbols.[24] The Singapore government later adapted this approach to their math curriculum in the 1980s.
The first of the three steps is concrete, wherein students learn while handling objects such as chips, dice, or paper clips.[5] Students learn to count these objects (e.g., paper clips) by physically lining them up in a row. They then learn basic arithmetic operations such as addition or subtraction by physically adding or removing the objects from each row.[24]
Students then transition to the pictorial step by drawing diagrams called "bar-models" to represent specific quantities of an object.[11][24] This involves drawing a rectangular bar to represent a specific quantity. For instance, if a short bar represents five paper clips, a bar that is twice as long would represent ten. By visualizing the difference between the two bars, students learn to solve problems of addition by adding one bar to the other, which will, in this instance, produce an answer of fifteen paper clips. They can use this method to solve other mathematical problems involving subtraction, multiplication, and division.[11][21] Bar modeling is far more efficient than the "guess-and-check" approach, in which students simply guess combinations of numbers until they stumble onto the solution.[11]
Once students have learned to solve mathematical problems using bar modeling, they begin to solve mathematical problems with exclusively abstract tools: numbers and symbols.
Bar modeling
Bar modeling is a pictorial method used to solve word problems in arithmetic.[21][25] These bar models can come in multiple forms such as a whole-part or a comparison model.
With the whole-part model, students would draw a rectangular bar to represent a "whole" larger quantity, which can be subdivided into two or more "parts." A student could be exposed to a word problem involving addition such as:
If John has 70 apples and Jane has 30 apples, how many apples do they both have?
The solution to this problem could be solved by drawing one bar and dividing it into two parts, with the longer part as 70 and the shorter part as 30. By visualizing these two parts, students would simply solve the above word problem by adding both parts together to build a whole bar of 100. Conversely, a student could use whole-part model to solve a subtraction problem such as 100 - 70, by having the longer part be 70 and the whole bar be 100. They would then solve the problem by inferring the shorter part to be 30.
The whole-part model can also be used to solve problems involving multiplication or division.[26] A multiplication problem could be presented as follows:
How much money would Jane have if she saved $30 each week for 4 weeks in a row?
The student could solve this multiplication problem by drawing one bar to represent the unknown answer, and subdivide that bar into four equal parts, with each part representing $30. Based on the drawn model, the student could then visualize this problem as providing a solution of $120.
Unlike the whole-part model, a comparison model involves comparing two bars of unequal lengths.[21][25] It can be used to solve a subtraction problem such as the following:
John needs to walk 100 miles to reach his home. So far, he has walked 70 miles. How many miles does he have left to walk home?
By using the comparison model, the student would draw one long bar to represent 100 and another shorter bar to represent 70. By comparing these two bars, students could then solve for the difference between the two numbers, which in this case is 30 miles. Like the whole-part model, the comparison model can also be used to solve word problems involving addition, multiplication, and division.
See also
• Common Core State Standards Initiative
• Mathematics education
• Programme for International Student Assessment
• Trends in International Mathematics and Science Study
References
1. The Independent (July 2, 2009). "Box clever: Singapore's magic formula for maths success". The Independent.
2. Brown, Laura L. "What's Singapore Math?". PBS. Retrieved September 19, 2013.
3. Hu, Winnie (September 30, 2010). "Making Math Lessons as Easy as 1, Pause, 2, Pause ..." The New York Times. New York, NY.
4. Jackson, Bill (July 26, 2011). "Going Beyond Singapore Math: Resisting Quick Fixes" (PDF). Singapore Math Source. Retrieved July 19, 2014.
5. Knake, Lindsay (December 2011). "Saginaw Township elementary schools implement hands-on Singapore math program". MLive. Grand Rapids, MI.
6. Jackson, Bill (October 10, 2012). "My view: America's students can benefit from Singapore math". CNN. Atlanta, GA.
7. Wright, Gerard (May 12, 2008). "Mathematics Mighty Ducks". The Age. Australia.
8. Prystay, Cris (December 13, 2004). "As math skills slip, U.S. schools seek answers from Asia". The Wall Street Journal.
9. Wong, Khoon Yoong; Lee, Ngan Hoe (February 19, 2009). "Singapore education and mathematics curriculum". In Wong Koon Yoong; Lee Peng Yee; Berinderjeet Kaur; Foong Pui Yee; Ng Swee Fong (eds.). Mathematics Education: The Singapore Journey. Vol. 2. Singapore: World Scientific Publishing. pp. 13–47. ISBN 978-981-283-375-4.
10. "Mathemagis: Introducing Singapore Math in the Philippines". SmartParenting.com.ph. 2012-04-12. Retrieved 2019-09-27.
11. Garelick, Barry (Fall 2006). "Miracle math: A successful program from Singapore tests the limits of school reform in the suburbs". Educational Next. 6.
12. Gurney-Read, Josie (November 29, 2016). "Revealed: World pupil rankings in science and maths - TIMSS results in full". The Daily Telegraph.
13. Lee, Peng Yee (September 12, 2008). "Sixty years of mathematics syllabi and textbooks in Singapore". In Usiskin, Zalman; Willmore, Edwin (eds.). Mathematics Curriculum in Pacific Rim Countries—China, Japan, Korea, and Singapore Proceedings of a Conference. Information Age Publishing. pp. 85–92. ISBN 978-1-59311-953-9.
14. Garelick, Barry (2006). "A tale of two countries and one school district". Nonpartisan Education Review. 6 (8). Archived from the original on 2013-09-21. Retrieved 2013-09-20.
15. Fang, Yanping; Lee, Christine Kim-Eng; Haron, Sharifah Thalha Bte Syed (February 19, 2009). "Lesson study in mathematics: Three cases in Singapore". In Wong Koon Yoong; Lee Peng Yee; Berinderjeet Kaur; Foong Pui Yee; Ng Swee Fong (eds.). Mathematics Education: The Singapore Journey. Vol. 2. Singapore: World Scientific Publishing. pp. 104–129. ISBN 978-981-283-375-4.
16. Landsberg, Mitchell (March 9, 2008). "In L.A., Singapore math has added value". Los Angeles Times. Los Angeles, CA.
17. Coughlan, Sean (December 6, 2016). "Pisa tests: Singapore top in global education rankings". BBC.
18. Dillon, sam (December 7, 2010). "Top test scores from shanghai stun educators". New York Times. New York, NY.
19. The Economist (December 7, 2013). "Finn-ished". The Economist.
20. Moroney, Kyle (December 2, 2013). "How Common Core standards are affecting elementary, middle school math classes". MLive.
21. Hoven, John; Garelick, Barry (November 2007). "Singapore Math: Using the bar model approach, Singapore textbooks enable students to solve difficult math problems--and learn how to think symbolically" (PDF). Educational Leadership. 65: 28–21. Archived from the original (PDF) on 2013-10-19. Retrieved 2013-09-20.
22. National Mathematics Advisory Panel (March 2008). "Foundations for Success: The Final Report of the National Mathematics Advisory Panel" (PDF). U.S. Department of Education. Retrieved December 13, 2013.
23. Garland, Sarah (October 16, 2013). "How Does Common Core Compare?". Huffington Post.
24. BBC (December 2, 2013). "Can the Singapore method help your children learn maths?". BBC.
25. Frank Schaffer Publications (June 2009). "Introduction to Singapore Math". 70 Must-Know Word Problems, Grade 7 (Singapore Math) (workbook ed.). Frank Schaffer Publications. pp. 3–8. ISBN 978-0-7682-4016-0.
26. Jackson, Bill. "Singapore math bar model strategy" (PDF). The Daily Riff. Retrieved December 16, 2013.
External links
• BBC (December 2, 2013). "Can the Singapore method help your children learn maths?". BBC.
• Jackson, B. "Singapore Math Demystified!". The Daily Riff. Retrieved September 20, 2013.
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Single-entry single-exit
In mathematics graph theory, a single-entry single-exit (SESE) region in a given graph is an ordered edge pair.
For example, with the ordered edge pair, (a, b) of distinct control-flow edges a and b where:
1. a dominates b
2. b postdominates a
3. Every cycle containing a also contains b and vice versa.
where a node x is said to dominate node y in a directed graph if every path from start to y includes x. A node x is said to postdominate a node y if every path from y to end includes x.
So, a and b refer to the entry and exit edge, respectively.
• The first condition ensures that every path from start into the region passes through the region’s entry edge, a.
• The second condition ensures that every path from inside the region to end passes through the region’s exit edge, b.
• The first two conditions are necessary but not enough to characterize SESE regions: since backedges do not alter the dominance or postdominance relationships, the first two conditions alone do not prohibit backedges entering or exiting the region.
• The third condition encodes two constraints: every path from inside the region to a point 'above' a passed through b, and every path from a point 'below' b to a point inside the region passes through a.[1]
References
1. The program structure tree: computing control regions in linear time
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Matrix unit
In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1.[1][2] The matrix unit with a 1 in the ith row and jth column is denoted as $E_{ij}$. For example, the 3 by 3 matrix unit with i = 1 and j = 2 is
$E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}$
Not to be confused with unit matrix, unitary matrix, or invertible matrix.
A vector unit is a standard unit vector.
A single-entry matrix generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1.
Properties
The set of m by n matrix units is a basis of the space of m by n matrices.[2]
The product of two matrix units of the same square shape $n\times n$ satisfies the relation
$E_{ij}E_{kl}=\delta _{jk}E_{il},$
where $\delta _{jk}$ is the Kronecker delta.[2]
The group of scalar n-by-n matrices over a ring R is the centralizer of the subset of n-by-n matrix units in the set of n-by-n matrices over R.[2]
The matrix norm (induced by the same two vector norms) of a matrix unit is equal to 1.
When multiplied by another matrix, it isolates a specific row or column in arbitrary position. For example, for any 3-by-3 matrix A:[3]
$E_{23}A=\left[{\begin{matrix}0&0&0\\a_{31}&a_{32}&a_{33}\\0&0&0\end{matrix}}\right].$
$AE_{23}=\left[{\begin{matrix}0&0&a_{12}\\0&0&a_{22}\\0&0&a_{32}\end{matrix}}\right].$
References
1. Artin, Michael. Algebra. Prentice Hall. p. 9.
2. Lam, Tsit-Yuen (1999). "Chapter 17: Matrix Rings". Lectures on Modules and Rings. Graduate Texts in Mathematics. Vol. 189. Springer Science+Business Media. pp. 461–479.
3. Marcel Blattner (2009). "B-Rank: A top N Recommendation Algorithm". arXiv:0908.2741 [physics.data-an].
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Asymptotic gain model
The asymptotic gain model[1][2] (also known as the Rosenstark method[3]) is a representation of the gain of negative feedback amplifiers given by the asymptotic gain relation:
$G=G_{\infty }\left({\frac {T}{T+1}}\right)+G_{0}\left({\frac {1}{T+1}}\right)\ ,$
where $T$ is the return ratio with the input source disabled (equal to the negative of the loop gain in the case of a single-loop system composed of unilateral blocks), G∞ is the asymptotic gain and G0 is the direct transmission term. This form for the gain can provide intuitive insight into the circuit and often is easier to derive than a direct attack on the gain.
Figure 1 shows a block diagram that leads to the asymptotic gain expression. The asymptotic gain relation also can be expressed as a signal flow graph. See Figure 2. The asymptotic gain model is a special case of the extra element theorem.
As follows directly from limiting cases of the gain expression, the asymptotic gain G∞ is simply the gain of the system when the return ratio approaches infinity:
$G_{\infty }=G\ {\Big |}_{T\rightarrow \infty }\ ,$
while the direct transmission term G0 is the gain of the system when the return ratio is zero:
$G_{0}=G\ {\Big |}_{T\rightarrow 0}\ .$
Advantages
• This model is useful because it completely characterizes feedback amplifiers, including loading effects and the bilateral properties of amplifiers and feedback networks.
• Often feedback amplifiers are designed such that the return ratio T is much greater than unity. In this case, and assuming the direct transmission term G0 is small (as it often is), the gain G of the system is approximately equal to the asymptotic gain G∞.
• The asymptotic gain is (usually) only a function of passive elements in a circuit, and can often be found by inspection.
• The feedback topology (series-series, series-shunt, etc.) need not be identified beforehand as the analysis is the same in all cases.
Implementation
Direct application of the model involves these steps:
1. Select a dependent source in the circuit.
2. Find the return ratio for that source.
3. Find the gain G∞ directly from the circuit by replacing the circuit with one corresponding to T = ∞.
4. Find the gain G0 directly from the circuit by replacing the circuit with one corresponding to T = 0.
5. Substitute the values for T, G∞ and G0 into the asymptotic gain formula.
These steps can be implemented directly in SPICE using the small-signal circuit of hand analysis. In this approach the dependent sources of the devices are readily accessed. In contrast, for experimental measurements using real devices or SPICE simulations using numerically generated device models with inaccessible dependent sources, evaluating the return ratio requires special methods.
Connection with classical feedback theory
Classical feedback theory neglects feedforward (G0). If feedforward is dropped, the gain from the asymptotic gain model becomes
$G=G_{\infty }{\frac {T}{1+T}}={\frac {G_{\infty }T}{1+{\frac {1}{G_{\infty }}}G_{\infty }T}}\ ,$
while in classical feedback theory, in terms of the open loop gain A, the gain with feedback (closed loop gain) is:
$A_{\mathrm {FB} }={\frac {A}{1+{\beta }_{\mathrm {FB} }A}}\ .$
Comparison of the two expressions indicates the feedback factor βFB is:
$\beta _{\mathrm {FB} }={\frac {1}{G_{\infty }}}\ ,$
while the open-loop gain is:
$A=G_{\infty }\ T\ .$
If the accuracy is adequate (usually it is), these formulas suggest an alternative evaluation of T: evaluate the open-loop gain and G∞ and use these expressions to find T. Often these two evaluations are easier than evaluation of T directly.
Examples
The steps in deriving the gain using the asymptotic gain formula are outlined below for two negative feedback amplifiers. The single transistor example shows how the method works in principle for a transconductance amplifier, while the second two-transistor example shows the approach to more complex cases using a current amplifier.
Single-stage transistor amplifier
Consider the simple FET feedback amplifier in Figure 3. The aim is to find the low-frequency, open-circuit, transresistance gain of this circuit G = vout / iin using the asymptotic gain model.
The small-signal equivalent circuit is shown in Figure 4, where the transistor is replaced by its hybrid-pi model.
Return ratio
It is most straightforward to begin by finding the return ratio T, because G0 and G∞ are defined as limiting forms of the gain as T tends to either zero or infinity. To take these limits, it is necessary to know what parameters T depends upon. There is only one dependent source in this circuit, so as a starting point the return ratio related to this source is determined as outlined in the article on return ratio.
The return ratio is found using Figure 5. In Figure 5, the input current source is set to zero, By cutting the dependent source out of the output side of the circuit, and short-circuiting its terminals, the output side of the circuit is isolated from the input and the feedback loop is broken. A test current it replaces the dependent source. Then the return current generated in the dependent source by the test current is found. The return ratio is then T = −ir / it. Using this method, and noticing that RD is in parallel with rO, T is determined as:
$T=g_{\mathrm {m} }\left(R_{\mathrm {D} }\ ||r_{\mathrm {O} }\right)\approx g_{\mathrm {m} }R_{\mathrm {D} }\ ,$
where the approximation is accurate in the common case where rO >> RD. With this relationship it is clear that the limits T → 0, or ∞ are realized if we let transconductance gm → 0, or ∞.[5]
Asymptotic gain
Finding the asymptotic gain G∞ provides insight, and usually can be done by inspection. To find G∞ we let gm → ∞ and find the resulting gain. The drain current, iD = gm vGS, must be finite. Hence, as gm approaches infinity, vGS also must approach zero. As the source is grounded, vGS = 0 implies vG = 0 as well.[6] With vG = 0 and the fact that all the input current flows through Rf (as the FET has an infinite input impedance), the output voltage is simply −iin Rf. Hence
$G_{\infty }={\frac {v_{\mathrm {out} }}{i_{\mathrm {in} }}}=-R_{\mathrm {f} }\ .$
Alternatively G∞ is the gain found by replacing the transistor by an ideal amplifier with infinite gain - a nullor.[7]
Direct feedthrough
To find the direct feedthrough $G_{0}$ we simply let gm → 0 and compute the resulting gain. The currents through Rf and the parallel combination of RD || rO must therefore be the same and equal to iin. The output voltage is therefore iin (RD || rO).
Hence
$G_{0}={\frac {v_{out}}{i_{in}}}=R_{D}\|r_{O}\approx R_{D}\ ,$
where the approximation is accurate in the common case where rO >> RD.
Overall gain
The overall transresistance gain of this amplifier is therefore:
$G={\frac {v_{out}}{i_{in}}}=-R_{f}{\frac {g_{m}R_{D}}{1+g_{m}R_{D}}}+R_{D}{\frac {1}{1+g_{m}R_{D}}}={\frac {R_{D}\left(1-g_{m}R_{f}\right)}{1+g_{m}R_{D}}}\ .$
Examining this equation, it appears to be advantageous to make RD large in order make the overall gain approach the asymptotic gain, which makes the gain insensitive to amplifier parameters (gm and RD). In addition, a large first term reduces the importance of the direct feedthrough factor, which degrades the amplifier. One way to increase RD is to replace this resistor by an active load, for example, a current mirror.
Two-stage transistor amplifier
Figure 6 shows a two-transistor amplifier with a feedback resistor Rf. This amplifier is often referred to as a shunt-series feedback amplifier, and analyzed on the basis that resistor R2 is in series with the output and samples output current, while Rf is in shunt (parallel) with the input and subtracts from the input current. See the article on negative feedback amplifier and references by Meyer or Sedra.[8][9] That is, the amplifier uses current feedback. It frequently is ambiguous just what type of feedback is involved in an amplifier, and the asymptotic gain approach has the advantage/disadvantage that it works whether or not you understand the circuit.
Figure 6 indicates the output node, but does not indicate the choice of output variable. In what follows, the output variable is selected as the short-circuit current of the amplifier, that is, the collector current of the output transistor. Other choices for output are discussed later.
To implement the asymptotic gain model, the dependent source associated with either transistor can be used. Here the first transistor is chosen.
Return ratio
The circuit to determine the return ratio is shown in the top panel of Figure 7. Labels show the currents in the various branches as found using a combination of Ohm's law and Kirchhoff's laws. Resistor R1 = RB // rπ1 and R3 = RC2 // RL. KVL from the ground of R1 to the ground of R2 provides:
$i_{\mathrm {B} }=-v_{\pi }{\frac {1+R_{2}/R_{1}+R_{\mathrm {f} }/R_{1}}{(\beta +1)R_{2}}}\ .$
KVL provides the collector voltage at the top of RC as
$v_{\mathrm {C} }=v_{\pi }\left(1+{\frac {R_{\mathrm {f} }}{R_{1}}}\right)-i_{\mathrm {B} }r_{\pi 2}\ .$
Finally, KCL at this collector provides
$i_{\mathrm {T} }=i_{\mathrm {B} }-{\frac {v_{\mathrm {C} }}{R_{\mathrm {C} }}}\ .$
Substituting the first equation into the second and the second into the third, the return ratio is found as
$T=-{\frac {i_{\mathrm {R} }}{i_{\mathrm {T} }}}=-g_{\mathrm {m} }{\frac {v_{\pi }}{i_{\mathrm {T} }}}$
$={\frac {g_{\mathrm {m} }R_{\mathrm {C} }}{\left(1+{\frac {R_{\mathrm {f} }}{R_{1}}}\right)\left(1+{\frac {R_{\mathrm {C} }+r_{\pi 2}}{(\beta +1)R_{2}}}\right)+{\frac {R_{\mathrm {C} }+r_{\pi 2}}{(\beta +1)R_{1}}}}}\ .$
Gain G0 with T = 0
The circuit to determine G0 is shown in the center panel of Figure 7. In Figure 7, the output variable is the output current βiB (the short-circuit load current), which leads to the short-circuit current gain of the amplifier, namely βiB / iS:
$G_{0}={\frac {\beta i_{B}}{i_{S}}}\ .$
Using Ohm's law, the voltage at the top of R1 is found as
$(i_{S}-i_{R})R_{1}=i_{R}R_{f}+v_{E}\ \ ,$
or, rearranging terms,
$i_{S}=i_{R}\left(1+{\frac {R_{f}}{R_{1}}}\right)+{\frac {v_{E}}{R_{1}}}\ .$
Using KCL at the top of R2:
$i_{R}={\frac {v_{E}}{R_{2}}}+(\beta +1)i_{B}\ .$
Emitter voltage vE already is known in terms of iB from the diagram of Figure 7. Substituting the second equation in the first, iB is determined in terms of iS alone, and G0 becomes:
$G_{0}={\frac {\beta }{(\beta +1)\left(1+{\frac {R_{f}}{R_{1}}}\right)+(r_{\pi 2}+R_{C})\left[{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\left(1+{\frac {R_{f}}{R_{1}}}\right)\right]}}$
Gain G0 represents feedforward through the feedback network, and commonly is negligible.
Gain G∞ with T → ∞
The circuit to determine G∞ is shown in the bottom panel of Figure 7. The introduction of the ideal op amp (a nullor) in this circuit is explained as follows. When T → ∞, the gain of the amplifier goes to infinity as well, and in such a case the differential voltage driving the amplifier (the voltage across the input transistor rπ1) is driven to zero and (according to Ohm's law when there is no voltage) it draws no input current. On the other hand, the output current and output voltage are whatever the circuit demands. This behavior is like a nullor, so a nullor can be introduced to represent the infinite gain transistor.
The current gain is read directly off the schematic:
$G_{\infty }={\frac {\beta i_{B}}{i_{S}}}=\left({\frac {\beta }{\beta +1}}\right)\left(1+{\frac {R_{f}}{R_{2}}}\right)\ .$
Comparison with classical feedback theory
Using the classical model, the feed-forward is neglected and the feedback factor βFB is (assuming transistor β >> 1):
$\beta _{FB}={\frac {1}{G_{\infty }}}\approx {\frac {1}{(1+{\frac {R_{f}}{R_{2}}})}}={\frac {R_{2}}{(R_{f}+R_{2})}}\ ,$
and the open-loop gain A is:
$A=G_{\infty }T\approx {\frac {\left(1+{\frac {R_{f}}{R_{2}}}\right)g_{m}R_{C}}{\left(1+{\frac {R_{f}}{R_{1}}}\right)\left(1+{\frac {R_{C}+r_{\pi 2}}{(\beta +1)R_{2}}}\right)+{\frac {R_{C}+r_{\pi 2}}{(\beta +1)R_{1}}}}}\ .$
Overall gain
The above expressions can be substituted into the asymptotic gain model equation to find the overall gain G. The resulting gain is the current gain of the amplifier with a short-circuit load.
Gain using alternative output variables
In the amplifier of Figure 6, RL and RC2 are in parallel. To obtain the transresistance gain, say Aρ, that is, the gain using voltage as output variable, the short-circuit current gain G is multiplied by RC2 // RL in accordance with Ohm's law:
$A_{\rho }=G\left(R_{\mathrm {C2} }//R_{\mathrm {L} }\right)\ .$
The open-circuit voltage gain is found from Aρ by setting RL → ∞.
To obtain the current gain when load current iL in load resistor RL is the output variable, say Ai, the formula for current division is used: iL = iout × RC2 / ( RC2 + RL ) and the short-circuit current gain G is multiplied by this loading factor:
$A_{i}=G\left({\frac {R_{C2}}{R_{C2}+R_{L}}}\right)\ .$
Of course, the short-circuit current gain is recovered by setting RL = 0 Ω.
References and notes
1. Middlebrook, RD: Design-oriented analysis of feedback amplifiers; Proc. of National Electronics Conference, Vol. XX, Oct. 1964, pp. 1–4
2. Rosenstark, Sol (1986). Feedback amplifier principles. NY: Collier Macmillan. p. 15. ISBN 0-02-947810-3.
3. Palumbo, Gaetano & Salvatore Pennisi (2002). Feedback amplifiers: theory and design. Boston/Dordrecht/London: Kluwer Academic. pp. §3.3 pp. 69–72. ISBN 0-7923-7643-9.
4. Paul R. Gray, Hurst P J Lewis S H & Meyer RG (2001). Analysis and design of analog integrated circuits (Fourth ed.). New York: Wiley. Figure 8.42 p. 604. ISBN 0-471-32168-0.
5. Although changing RD // rO also could force the return ratio limits, these resistor values affect other aspects of the circuit as well. It is the control parameter of the dependent source that must be varied because it affects only the dependent source.
6. Because the input voltage vGS approaches zero as the return ratio gets larger, the amplifier input impedance also tends to zero, which means in turn (because of current division) that the amplifier works best if the input signal is a current. If a Norton source is used, rather than an ideal current source, the formal equations derived for T will be the same as for a Thévenin voltage source. Note that in the case of input current, G∞ is a transresistance gain.
7. Verhoeven CJ, van Staveren A, Monna GL, Kouwenhoven MH, Yildiz E (2003). Structured electronic design: negative-feedback amplifiers. Boston/Dordrecht/London: Kluwer Academic. pp. §2.3 – §2.5 pp. 34–40. ISBN 1-4020-7590-1.
8. P R Gray; P J Hurst; S H Lewis & R G Meyer (2001). Analysis and Design of Analog Integrated Circuits (Fourth ed.). New York: Wiley. pp. 586–587. ISBN 0-471-32168-0.
9. A. S. Sedra & K.C. Smith (2004). Microelectronic Circuits (Fifth ed.). New York: Oxford. Example 8.4, pp. 825–829 and PSpice simulation pp. 855–859. ISBN 0-19-514251-9.
See also
• Blackman's theorem
• Extra element theorem
• Mason's gain formula
• Feedback amplifiers
• Return ratio
• Signal-flow graph
External links
• Lecture notes on the asymptotic gain model
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Wikipedia
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Multivalued function
In mathematics, a multivalued function, also called multifunction and many-valued function, is a set-valued function with continuity properties that allow considering it locally as an ordinary function.
Not to be confused with Multivariate function.
Multivalued functions arise commonly in applications of the implicit function theorem, since this theorem can be viewed as asserting the existence of a multivalued function. In particular, the inverse function of a differentiable function is a multivalued function, and is single-valued only when the original function is monotonic. For example, the complex logarithm is a multivalued function, as the inverse of the exponential function. It cannot be considered as an ordinary function, since, when one follows one value of the logarithm along a circle centered at 0, one gets another value than the starting one after a complete turn. This phenomenon is called monodromy.
Another common way for defining a multivalued function is analytic continuation, which generates commonly some monodromy: analytic continuation along a closed curve may generate a final value that differs from the starting value.
Multivalued functions arise also as solutions of differential equations, where the different values are parametrized by initial conditions.
Motivation
The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function $f(z)$ in some neighbourhood of a point $z=a$. This is the case for functions defined by the implicit function theorem or by a Taylor series around $z=a$. In such a situation, one may extend the domain of the single-valued function $f(z)$ along curves in the complex plane starting at $a$. In doing so, one finds that the value of the extended function at a point $z=b$ depends on the chosen curve from $a$ to $b$; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.
For example, let $f(z)={\sqrt {z}}\,$ be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of $z=1$ in the complex plane, and then further along curves starting at $z=1$, so that the values along a given curve vary continuously from ${\sqrt {1}}=1$. Extending to negative real numbers, one gets two opposite values for the square root—for example ±i for –1—depending on whether the domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for nth roots, logarithms, and inverse trigonometric functions.
To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the principal value, producing a single-valued function on the whole plane which is discontinuous along certain boundary curves. Alternatively, dealing with the multivalued function allows having something that is everywhere continuous, at the cost of possible value changes when one follows a closed path (monodromy). These problems are resolved in the theory of Riemann surfaces: to consider a multivalued function $f(z)$ as an ordinary function without discarding any values, one multiplies the domain into a many-layered covering space, a manifold which is the Riemann surface associated to $f(z)$.
Examples
• Every real number greater than zero has two real square roots, so that square root may be considered a multivalued function. For example, we may write ${\sqrt {4}}=\pm 2=\{2,-2\}$; although zero has only one square root, ${\sqrt {0}}=\{0\}$.
• Each nonzero complex number has two square roots, three cube roots, and in general n nth roots. The only nth root of 0 is 0.
• The complex logarithm function is multiple-valued. The values assumed by $\log(a+bi)$ for real numbers $a$ and $b$ are $\log {\sqrt {a^{2}+b^{2}}}+i\arg(a+bi)+2\pi ni$ for all integers $n$.
• Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have
$\tan \left({\tfrac {\pi }{4}}\right)=\tan \left({\tfrac {5\pi }{4}}\right)=\tan \left({\tfrac {-3\pi }{4}}\right)=\tan \left({\tfrac {(2n+1)\pi }{4}}\right)=\cdots =1.$
As a consequence, arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan x to −π/2 < x < π/2 – a domain over which tan x is monotonically increasing. Thus, the range of arctan(x) becomes −π/2 < y < π/2. These values from a restricted domain are called principal values.
• The antiderivative can be considered as a multivalued function. The antiderivative of a function is the set of functions whose derivative is that function. The constant of integration follows from the fact that the derivative of a constant function is 0.
• Inverse hyperbolic functions over the complex domain are multiple-valued because hyperbolic functions are periodic along the imaginary axis. Over the reals, they are single-valued, except for arcosh and arsech.
These are all examples of multivalued functions that come about from non-injective functions. Since the original functions do not preserve all the information of their inputs, they are not reversible. Often, the restriction of a multivalued function is a partial inverse of the original function.
Branch points
Main article: Branch point
Multivalued functions of a complex variable have branch points. For example, for the nth root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units i and −i are branch points. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range. A suitable interval may be found through use of a branch cut, a kind of curve that connects pairs of branch points, thus reducing the multilayered Riemann surface of the function to a single layer. As in the case with real functions, the restricted range may be called the principal branch of the function.
Applications
In physics, multivalued functions play an increasingly important role. They form the mathematical basis for Dirac's magnetic monopoles, for the theory of defects in crystals and the resulting plasticity of materials, for vortices in superfluids and superconductors, and for phase transitions in these systems, for instance melting and quark confinement. They are the origin of gauge field structures in many branches of physics.
Further reading
• H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, World Scientific (Singapore, 2008) (also available online)
• H. Kleinert, Gauge Fields in Condensed Matter, Vol. I: Superflow and Vortex Lines, 1–742, Vol. II: Stresses and Defects, 743–1456, World Scientific, Singapore, 1989 (also available online: Vol. I and Vol. II)
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Wikipedia
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Single-parameter utility
In mechanism design, an agent is said to have single-parameter utility if his valuation of the possible outcomes can be represented by a single number. For example, in an auction for a single item, the utilities of all agents are single-parametric, since they can be represented by their monetary evaluation of the item. In contrast, in a combinatorial auction for two or more related items, the utilities are usually not single-parametric, since they are usually represented by their evaluations to all possible bundles of items.
Notation
There is a set $X$ of possible outcomes.
There are $n$ agents which have different valuations for each outcome.
In general, each agent can assign a different and unrelated value to every outcome in $X$.
In the special case of single-parameter utility, each agent $i$ has a publicly known outcome proper subset $W_{i}\subset X$ which are the "winning outcomes" for agent $i$ (e.g., in a single-item auction, $W_{i}$ contains the outcome in which agent $i$ wins the item).
For every agent, there is a number $v_{i}$ which represents the "winning-value" of $i$. The agent's valuation of the outcomes in $X$ can take one of two values:[1]: 228
• $v_{i}$ for each outcome in $W_{i}$;
• 0 for each outcome in $X\setminus W_{i}$.
The vector of the winning-values of all agents is denoted by $v$.
For every agent $i$, the vector of all winning-values of the other agents is denoted by $v_{-i}$. So $v\equiv (v_{i},v_{-i})$.
A social choice function is a function that takes as input the value-vector $v$ and returns an outcome $x\in X$. It is denoted by ${\text{Outcome}}(v)$ or ${\text{Outcome}}(v_{i},v_{-i})$.
Monotonicity
The weak monotonicity property has a special form in single-parameter domains. A social choice function is weakly-monotonic if for every agent $i$ and every $v_{i},v_{i}',v_{-i}$, if:
${\text{Outcome}}(v_{i},v_{-i})\in W_{i}$ and
$v'_{i}\geq v_{i}>0$ then:
${\text{Outcome}}(v'_{i},v_{-i})\in W_{i}$
I.e, if agent $i$ wins by declaring a certain value, then he can also win by declaring a higher value (when the declarations of the other agents are the same).
The monotonicity property can be generalized to randomized mechanisms, which return a probability-distribution over the space $X$.[1]: 334 The WMON property implies that for every agent $i$ and every $v_{i},v_{i}',v_{-i}$, the function:
$\Pr[{\text{Outcome}}(v_{i},v_{-i})\in W_{i}]$
is a weakly-increasing function of $v_{i}$.
Critical value
For every weakly-monotone social-choice function, for every agent $i$ and for every vector $v_{-i}$, there is a critical value $c_{i}(v_{-i})$, such that agent $i$ wins if-and-only-if his bid is at least $c_{i}(v_{-i})$.
For example, in a second-price auction, the critical value for agent $i$ is the highest bid among the other agents.
In single-parameter environments, deterministic truthful mechanisms have a very specific format.[1]: 334 Any deterministic truthful mechanism is fully specified by the set of functions c. Agent $i$ wins if and only if his bid is at least $c_{i}(v_{-i})$, and in that case, he pays exactly $c_{i}(v_{-i})$.
Deterministic implementation
It is known that, in any domain, weak monotonicity is a necessary condition for implementability. I.e, a social-choice function can be implemented by a truthful mechanism, only if it is weakly-monotone.
In a single-parameter domain, weak monotonicity is also a sufficient condition for implementability. I.e, for every weakly-monotonic social-choice function, there is a deterministic truthful mechanism that implements it. This means that it is possible to implement various non-linear social-choice functions, e.g. maximizing the sum-of-squares of values or the min-max value.
The mechanism should work in the following way:[1]: 229
• Ask the agents to reveal their valuations, $v$.
• Select the outcome based on the social-choice function: $x={\text{Outcome}}[v]$.
• Every winning agent (every agent $i$ such that $x\in W_{i}$) pays a price equal to the critical value: ${\text{Price}}_{i}(x,v_{-i})=-c_{i}(v_{-i})$.
• Every losing agent (every agent $i$ such that $x\notin W_{i}$) pays nothing: ${\text{Price}}_{i}(x,v_{-i})=0$.
This mechanism is truthful, because the net utility of each agent is:
• $v_{i}-c_{i}(v_{-i})$ if he wins;
• 0 if he loses.
Hence, the agent prefers to win if $v_{i}>c_{-i}$ and to lose if $v_{i}<c_{-i}$, which is exactly what happens when he tells the truth.
Randomized implementation
A randomized mechanism is a probability-distribution on deterministic mechanisms. A randomized mechanism is called truthful-in-expectation if truth-telling gives the agent a largest expected value.
In a randomized mechanism, every agent $i$ has a probability of winning, defined as:
$w_{i}(v_{i},v_{-i}):=\Pr[{\text{Outcome}}(v_{i},v_{-i})\in W_{i}]$
and an expected payment, defined as:
$\mathbb {E} [{\text{Payment}}_{i}(v_{i},v_{-i})]$
In a single-parameter domain, a randomized mechanism is truthful-in-expectation if-and-only if:[1]: 232
• The probability of winning, $w_{i}(v_{i},v_{-i})$, is a weakly-increasing function of $v_{i}$;
• The expected payment of an agent is:
$\mathbb {E} [{\text{Payment}}_{i}(v_{i},v_{-i})]=v_{i}\cdot w_{i}(v_{i},v_{-i})-\int _{0}^{v_{i}}w_{i}(t,v_{-i})dt$
Note that in a deterministic mechanism, $w_{i}(v_{i},v_{-i})$ is either 0 or 1, the first condition reduces to weak-monotonicity of the Outcome function and the second condition reduces to charging each agent his critical value.
Single-parameter vs. multi-parameter domains
When the utilities are not single-parametric (e.g. in combinatorial auctions), the mechanism design problem is much more complicated. The VCG mechanism is one of the only mechanisms that works for such general valuations.
See also
• Single peaked preferences
References
1. Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.
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Wikipedia
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Singleton (mathematics)
In mathematics, a singleton, also known as a unit set[1] or one-point set, is a set with exactly one element. For example, the set $\{0\}$ is a singleton whose single element is $0$.
Part of a series on statistics
Probability theory
• Probability
• Axioms
• Determinism
• System
• Indeterminism
• Randomness
• Probability space
• Sample space
• Event
• Collectively exhaustive events
• Elementary event
• Mutual exclusivity
• Outcome
• Singleton
• Experiment
• Bernoulli trial
• Probability distribution
• Bernoulli distribution
• Binomial distribution
• Normal distribution
• Probability measure
• Random variable
• Bernoulli process
• Continuous or discrete
• Expected value
• Markov chain
• Observed value
• Random walk
• Stochastic process
• Complementary event
• Joint probability
• Marginal probability
• Conditional probability
• Independence
• Conditional independence
• Law of total probability
• Law of large numbers
• Bayes' theorem
• Boole's inequality
• Venn diagram
• Tree diagram
For a sequence with one member, see 1-tuple.
Properties
Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains,[1] thus 1 and {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as $\{\{1,2,3\}\}$ is a singleton as it contains a single element (which itself is a set, however, not a singleton).
A set is a singleton if and only if its cardinality is 1. In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton $\{0\}.$
In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set A, the axiom applied to A and A asserts the existence of $\{A,A\},$ which is the same as the singleton $\{A\}$ (since it contains A, and no other set, as an element).
If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the single element of S. Thus every singleton is a terminal object in the category of sets.
A singleton has the property that every function from it to any arbitrary set is injective. The only non-singleton set with this property is the empty set.
Every singleton set is an ultra prefilter. If $X$ is a set and $x\in X$ then the upward of $\{x\}$ in $X,$ which is the set $\{S\subseteq X:x\in S\},$ is a principal ultrafilter on $X.$[2] Moreover, every principal ultrafilter on $X$ is necessarily of this form.[2] The ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called free ultrafilters). Every net valued in a singleton subset $X$ of is an ultranet in $X.$
The Bell number integer sequence counts the number of partitions of a set (OEIS: A000110), if singletons are excluded then the numbers are smaller (OEIS: A000296).
In category theory
Structures built on singletons often serve as terminal objects or zero objects of various categories:
• The statement above shows that the singleton sets are precisely the terminal objects in the category Set of sets. No other sets are terminal.
• Any singleton admits a unique topological space structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category.
• Any singleton admits a unique group structure (the unique element serving as identity element). These singleton groups are zero objects in the category of groups and group homomorphisms. No other groups are terminal in that category.
Definition by indicator functions
Let S be a class defined by an indicator function
$b:X\to \{0,1\}.$
Then S is called a singleton if and only if there is some $y\in X$ such that for all $x\in X,$
$b(x)=(x=y).$
Definition in Principia Mathematica
The following definition was introduced by Whitehead and Russell[3]
$\iota $‘$x={\hat {y}}(y=x)$ Df.
The symbol $\iota $‘$x$ denotes the singleton $\{x\}$ and ${\hat {y}}(y=x)$ denotes the class of objects identical with $x$ aka $\{y:y=x\}$. This occurs as a definition in the introduction, which, in places, simplifies the argument in the main text, where it occurs as proposition 51.01 (p.357 ibid.). The proposition is subsequently used to define the cardinal number 1 as
$1={\hat {\alpha }}((\exists x)\alpha =\iota $‘$x)$ Df.
That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)
See also
• Class (set theory) – Collection of sets in mathematics that can be defined based on a property of its members
• Uniqueness quantification – Logical property of being the one and only object satisfying a condition
References
1. Stoll, Robert (1961). Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. pp. 5–6.
2. Dolecki & Mynard 2016, pp. 27–54.
3. Whitehead, Alfred North; Bertrand Russell (1910). Principia Mathematica. Vol. I. p. 37.
• Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
Set theory
Overview
• Set (mathematics)
Axioms
• Adjunction
• Choice
• countable
• dependent
• global
• Constructibility (V=L)
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Wikipedia
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Singular boundary method
In numerical analysis, the singular boundary method (SBM) belongs to a family of meshless boundary collocation techniques which include the method of fundamental solutions (MFS),[1][2][3] boundary knot method (BKM),[4] regularized meshless method (RMM),[5] boundary particle method (BPM),[6] modified MFS,[7] and so on. This family of strong-form collocation methods is designed to avoid singular numerical integration and mesh generation in the traditional boundary element method (BEM) in the numerical solution of boundary value problems with boundary nodes, in which a fundamental solution of the governing equation is explicitly known.
The salient feature of the SBM is to overcome the fictitious boundary in the method of fundamental solution, while keeping all merits of the latter. The method offers several advantages over the classical domain or boundary discretization methods, among which are:
• meshless. The method requires neither domain nor boundary meshing but boundary-only discretization points;
• integration-free. The numerical integration of singular or nearly singular kernels could be otherwise troublesome, expensive, and complicated, as in the case, for example, the boundary element method;
• boundary-only discretization for homogeneous problems. The SBM shares all the advantages of the BEM over domain discretization methods such as the finite element or finite difference methods;
• to overcome the perplexing fictitious boundary in the method of fundamental solutions (see Figs. 1 and 2), thanks to the introduction of the concept of the origin intensity factor, which isolates the singularity of the fundamental solutions.
The SBM provides a significant and promising alternative to popular boundary-type methods such as the BEM and MFS, in particular, for infinite domain, wave, thin-walled structures, and inverse problems.
History of the singular boundary method
The methodology of the SBM was firstly proposed by Chen and his collaborators in 2009.[8][9] The basic idea is to introduce a concept of the origin intensity factor to isolate the singularity of the fundamental solutions so that the source points can be placed directly on the real boundary. In comparison, the method of fundamental solutions requires a fictitious boundary for placing the source points to avoid the singularity of fundamental solution. The SBM has since been successfully applied to a variety of physical problems, such as potential problems,[10][11] infinite domain problem,[12] Helmholtz problem,[13] and plane elasticity problem.[14]
There are the two techniques to evaluate the origin intensity factor. The first approach is to place a cluster of sample nodes inside the problem domain and to calculate the algebraic equations. The strategy leads to extra computational costs and makes the method is not as efficient as expected compared to the MFS. The second approach[15][16] is to employ a regularization technique to cancel the singularities of the fundamental solution and its derivatives. Consequently, the origin intensity factors can be determined directly without using any sample nodes. This scheme makes the method more stable, accurate, efficient, and extends its applicability.
Recent developments
Boundary layer effect problems
Like all the other boundary-type numerical methods, also it is observed that the SBM encounters a dramatic drop of solution accuracy at the region nearby boundary. Unlike singularity at origin, the fundamental solution at near-boundary regions remains finite. However, instead of being a flat function, the interpolation function develops a sharp peak as the field point approaches the boundary. Consequently, the kernels become “nearly singular” and can not accurately be calculated. This is similar to the so-called boundary layer effect encountered in the BEM-based methods.
A nonlinear transformation, based on the sinh function, can be employed to remove or damp out the rapid variations of the nearly singular kernels.[17] As a result, the troublesome boundary layer effect in the SBM has been successfully remedied. The implementation of this transformation is straightforward and can easily be embedded in existing SBM programs. For the test problems studied, very promising results are obtained even when the distance between the field point and the boundary is as small as 1×10−10.
Large-scale problems
Like the MFS and BEM, the SBM will produce dense coefficient matrices, whose operation count and the memory requirements for matrix equation buildup are of the order of O(N2) which is computationally too expensive to simulate large-scale problems.
The fast multipole method (FMM) can reduce both CPU time and memory requirement from O(N2) to O(N) or O(NlogN). With the help of FMM, the SBM can be fully capable of solving a large scale problem of several million unknowns on a desktop. This fast algorithm dramatically expands the applicable territory of the SBM to far greater problems than were previously possible.
See also
• Meshfree methods
• Radial basis function
• Trefftz method
References
1. method of fundamental solutions (MFS)
2. Golberg MA, Chen CS, Ganesh M, "Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions", Eng Anal Bound Elem 2000;24(7–8): 539–47.
3. Fairweather G, Karageorghis A, "The method of fundamental solutions for elliptic boundary value problems", Adv Comput Math 1998;9(1): 69–95.
4. Chen W, Tanaka M, "A meshless, integration-free, and boundary-only RBF technique Archived 2016-03-04 at the Wayback Machine", Comput Math Appl 2002;43(3–5): 379–91.
5. D.L. Young, K.H. Chen, C.W. Lee, "Novel meshless method for solving the potential problems with arbitrary domain", J Comput Phys 2005;209(1): 290–321.
6. boundary particle method (BPM)
7. Sarler B, "Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions", Eng Anal Bound Elem 2009;33(12): 1374–82.
8. Chen W, "Singular boundary method: A novel, simple, meshfree, boundary collocation numerical method", Chin J Solid Mech 2009;30(6): 592–9.
9. Chen W, Wang FZ, "A method of fundamental solutions without fictitious boundary Archived 2015-06-06 at the Wayback Machine", Eng Anal Bound Elem 2010;34(5): 530–32.
10. Wei X, Chen W, Fu ZJ, "Solving inhomogeneous problems by singular boundary method", J Mar SCI Tech 2012; 20(5).
11. Chen W, Fu ZJ, Wei X, "Potential Problems by Singular Boundary Method Satisfying Moment Condition", Comput Model Eng Sci 2009;54(1): 65–85.
12. Chen W, Fu Z, "A novel numerical method for infinite domain potential problems", Chin Sci Bull 2010;55(16): 1598–603.
13. Fu ZJ, Chen W, "A novel boundary meshless method for radiation and scattering problems", Advances in Boundary Element Techniques XI, Proceedings of the 11th international Conference, 12–14 July 2010, 83–90, Published by EC Ltd, United Kingdom (ISBN 978-0-9547783-7-8)
14. Gu Y, Chen W, Zhang CZ., "Singular boundary method for solving plane strain elastostatic problems", Int J Solids Struct 2011;48(18): 2549–56.
15. Chen W, Gu Y, "Recent advances on singular boundary method", Joint International Workshop on Trefftz Method VI and Method of Fundamental Solution II, Taiwan 2011.
16. Gu Y, Chen, W, "Improved singular boundary method for three dimensional potential problems", Chinese Journal of Theoretical and Applied Mechanics, 2012, 44(2): 351-360 (in Chinese)
17. Gu Y, Chen W, Zhang J, "Investigation on near-boundary solutions by singular boundary method", Eng Anal Bound Elem 2012;36(8): 117–82.
External links
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Wikipedia
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Singular cardinals hypothesis
In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal.
According to Mitchell (1992), the singular cardinals hypothesis is:
If κ is any singular strong limit cardinal, then 2κ = κ+.
Here, κ+ denotes the successor cardinal of κ.
Since SCH is a consequence of GCH, which is known to be consistent with ZFC, SCH is consistent with ZFC. The negation of SCH has also been shown to be consistent with ZFC, if one assumes the existence of a sufficiently large cardinal number. In fact, by results of Moti Gitik, ZFC + ¬SCH is equiconsistent with ZFC + the existence of a measurable cardinal κ of Mitchell order κ++.
Another form of the SCH is the following statement:
2cf(κ) < κ implies κcf(κ) = κ+,
where cf denotes the cofinality function. Note that κcf(κ)= 2κ for all singular strong limit cardinals κ. The second formulation of SCH is strictly stronger than the first version, since the first one only mentions strong limits. From a model in which the first version of SCH fails at ℵω and GCH holds above ℵω+2, we can construct a model in which the first version of SCH holds but the second version of SCH fails, by adding ℵω Cohen subsets to ℵn for some n.
Jack Silver proved that if κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Silver's original proof used generic ultrapowers. The following important fact follows from Silver's theorem: if the singular cardinals hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals. In particular, then, if $\kappa $ is the least counterexample to the singular cardinals hypothesis, then $\mathrm {cf} (\kappa )=\mathrm {\omega } $.
The negation of the singular cardinals hypothesis is intimately related to violating the GCH at a measurable cardinal. A well-known result of Dana Scott is that if the GCH holds below a measurable cardinal $\kappa $ on a set of measure one—i.e., there is normal $\kappa $-complete ultrafilter D on ${\mathcal {P}}(\kappa )$ such that $\{\alpha <\kappa \mid 2^{\alpha }=\alpha ^{+}\}\in D$, then $2^{\kappa }=\kappa ^{+}$. Starting with $\kappa $ a supercompact cardinal, Silver was able to produce a model of set theory in which $\kappa $ is measurable and in which $2^{\kappa }>\kappa ^{+}$. Then, by applying Prikry forcing to the measurable $\kappa $, one gets a model of set theory in which $\kappa $ is a strong limit cardinal of countable cofinality and in which $2^{\kappa }>\kappa ^{+}$—a violation of the SCH. Gitik, building on work of Woodin, was able to replace the supercompact in Silver's proof with measurable of Mitchell order $\kappa ^{++}$. That established an upper bound for the consistency strength of the failure of the SCH. Gitik again, using results of inner model theory, was able to show that a measurable cardinal of Mitchell order $\kappa ^{++}$ is also the lower bound for the consistency strength of the failure of SCH.
A wide variety of propositions imply SCH. As was noted above, GCH implies SCH. On the other hand, the proper forcing axiom, which implies $2^{\aleph _{0}}=\aleph _{2}$ and hence is incompatible with GCH also implies SCH. Solovay showed that large cardinals almost imply SCH—in particular, if $\kappa $ is strongly compact cardinal, then the SCH holds above $\kappa $. On the other hand, the non-existence of (inner models for) various large cardinals (below a measurable cardinal of Mitchell order $\kappa ^{++}$) also imply SCH.
References
• Thomas Jech: Properties of the gimel function and a classification of singular cardinals, Fundamenta Mathematicae 81 (1974): 57–64.
• William J. Mitchell, "On the singular cardinal hypothesis," Trans. Amer. Math. Soc., volume 329 (2): pp. 507–530, 1992.
• Jason Aubrey, The Singular Cardinals Problem (PDF), VIGRE expository report, Department of Mathematics, University of Michigan.
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Wikipedia
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Singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups $H_{n}(X).$ Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory simplicial homology).
In brief, singular homology is constructed by taking maps of the standard n-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation – mapping each n-dimensional simplex to its (n−1)-dimensional boundary – induces the singular chain complex. The singular homology is then the homology of the chain complex. The resulting homology groups are the same for all homotopy equivalent spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology is expressible as a functor from the category of topological spaces to the category of graded abelian groups.
Singular simplices
A singular n-simplex in a topological space X is a continuous function (also called a map) $\sigma $ from the standard n-simplex $\Delta ^{n}$ to X, written $\sigma :\Delta ^{n}\to X.$ :\Delta ^{n}\to X.} This map need not be injective, and there can be non-equivalent singular simplices with the same image in X.
The boundary of $\sigma ,$ denoted as $\partial _{n}\sigma ,$ is defined to be the formal sum of the singular (n − 1)-simplices represented by the restriction of $\sigma $ to the faces of the standard n-simplex, with an alternating sign to take orientation into account. (A formal sum is an element of the free abelian group on the simplices. The basis for the group is the infinite set of all possible singular simplices. The group operation is "addition" and the sum of simplex a with simplex b is usually simply designated a + b, but a + a = 2a and so on. Every simplex a has a negative −a.) Thus, if we designate $\sigma $ by its vertices
$[p_{0},p_{1},\ldots ,p_{n}]=[\sigma (e_{0}),\sigma (e_{1}),\ldots ,\sigma (e_{n})]$
corresponding to the vertices $e_{k}$ of the standard n-simplex $\Delta ^{n}$ (which of course does not fully specify the singular simplex produced by $\sigma $), then
$\partial _{n}\sigma =\partial _{n}[p_{0},p_{1},\ldots ,p_{n}]=\sum _{k=0}^{n}(-1)^{k}[p_{0},\ldots ,p_{k-1},p_{k+1},\ldots ,p_{n}]=\sum _{k=0}^{n}(-1)^{k}\sigma \mid _{e_{0},\ldots ,e_{k-1},e_{k+1},\ldots ,e_{n}}$
is a formal sum of the faces of the simplex image designated in a specific way.[1] (That is, a particular face has to be the restriction of $\sigma $ to a face of $\Delta ^{n}$ which depends on the order that its vertices are listed.) Thus, for example, the boundary of $\sigma =[p_{0},p_{1}]$ (a curve going from $p_{0}$ to $p_{1}$) is the formal sum (or "formal difference") $[p_{1}]-[p_{0}]$.
Singular chain complex
The usual construction of singular homology proceeds by defining formal sums of simplices, which may be understood to be elements of a free abelian group, and then showing that we can define a certain group, the homology group of the topological space, involving the boundary operator.
Consider first the set of all possible singular n-simplices $\sigma _{n}(X)$ on a topological space X. This set may be used as the basis of a free abelian group, so that each singular n-simplex is a generator of the group. This set of generators is of course usually infinite, frequently uncountable, as there are many ways of mapping a simplex into a typical topological space. The free abelian group generated by this basis is commonly denoted as $C_{n}(X)$. Elements of $C_{n}(X)$ are called singular n-chains; they are formal sums of singular simplices with integer coefficients.
The boundary $\partial $ is readily extended to act on singular n-chains. The extension, called the boundary operator, written as
$\partial _{n}:C_{n}\to C_{n-1},$
is a homomorphism of groups. The boundary operator, together with the $C_{n}$, form a chain complex of abelian groups, called the singular complex. It is often denoted as $(C_{\bullet }(X),\partial _{\bullet })$ or more simply $C_{\bullet }(X)$.
The kernel of the boundary operator is $Z_{n}(X)=\ker(\partial _{n})$, and is called the group of singular n-cycles. The image of the boundary operator is $B_{n}(X)=\operatorname {im} (\partial _{n+1})$, and is called the group of singular n-boundaries.
It can also be shown that $\partial _{n}\circ \partial _{n+1}=0$, implying $B_{n}(X)\subseteq Z_{n}(X)$. The $n$-th homology group of $X$ is then defined as the factor group
$H_{n}(X)=Z_{n}(X)/B_{n}(X).$
The elements of $H_{n}(X)$ are called homology classes.[2]
Homotopy invariance
If X and Y are two topological spaces with the same homotopy type (i.e. are homotopy equivalent), then
$H_{n}(X)\cong H_{n}(Y)\,$
for all n ≥ 0. This means homology groups are homotopy invariants, and therefore topological invariants.
In particular, if X is a connected contractible space, then all its homology groups are 0, except $H_{0}(X)\cong \mathbb {Z} $.
A proof for the homotopy invariance of singular homology groups can be sketched as follows. A continuous map f: X → Y induces a homomorphism
$f_{\sharp }:C_{n}(X)\rightarrow C_{n}(Y).$
It can be verified immediately that
$\partial f_{\sharp }=f_{\sharp }\partial ,$
i.e. f# is a chain map, which descends to homomorphisms on homology
$f_{*}:H_{n}(X)\rightarrow H_{n}(Y).$
We now show that if f and g are homotopically equivalent, then f* = g*. From this follows that if f is a homotopy equivalence, then f* is an isomorphism.
Let F : X × [0, 1] → Y be a homotopy that takes f to g. On the level of chains, define a homomorphism
$P:C_{n}(X)\rightarrow C_{n+1}(Y)$
that, geometrically speaking, takes a basis element σ: Δn → X of Cn(X) to the "prism" P(σ): Δn × I → Y. The boundary of P(σ) can be expressed as
$\partial P(\sigma )=f_{\sharp }(\sigma )-g_{\sharp }(\sigma )-P(\partial \sigma ).$
So if α in Cn(X) is an n-cycle, then f#(α ) and g#(α) differ by a boundary:
$f_{\sharp }(\alpha )-g_{\sharp }(\alpha )=\partial P(\alpha ),$
i.e. they are homologous. This proves the claim.[3]
Homology groups of common spaces
The table below shows the k-th homology groups $H_{k}(X)$ of n-dimensional real projective spaces RPn, complex projective spaces, CPn, a point, spheres Sn($n\geq 1$), and a 3-torus T3 with integer coefficients.
Space Homotopy type
RPn[4] $\mathbf {Z} $ k = 0 and k = n odd
$\mathbf {Z} /2\mathbf {Z} $ k odd, 0 < k < n
0 otherwise
CPn[5] $\mathbf {Z} $ k = 0,2,4,...,2n
0 otherwise
point[6] $\mathbf {Z} $ k = 0
0 otherwise
Sn $\mathbf {Z} $ k = 0,n
0 otherwise
T3[7] $\mathbf {Z} $ k = 0,3
$\mathbf {Z} $3 k = 1,2
0 otherwise
Functoriality
The construction above can be defined for any topological space, and is preserved by the action of continuous maps. This generality implies that singular homology theory can be recast in the language of category theory. In particular, the homology group can be understood to be a functor from the category of topological spaces Top to the category of abelian groups Ab.
Consider first that $X\mapsto C_{n}(X)$ is a map from topological spaces to free abelian groups. This suggests that $C_{n}(X)$ might be taken to be a functor, provided one can understand its action on the morphisms of Top. Now, the morphisms of Top are continuous functions, so if $f:X\to Y$ is a continuous map of topological spaces, it can be extended to a homomorphism of groups
$f_{*}:C_{n}(X)\to C_{n}(Y)\,$
by defining
$f_{*}\left(\sum _{i}a_{i}\sigma _{i}\right)=\sum _{i}a_{i}(f\circ \sigma _{i})$
where $\sigma _{i}:\Delta ^{n}\to X$ is a singular simplex, and $\sum _{i}a_{i}\sigma _{i}\,$ is a singular n-chain, that is, an element of $C_{n}(X)$. This shows that $C_{n}$ is a functor
$C_{n}:\mathbf {Top} \to \mathbf {Ab} $
from the category of topological spaces to the category of abelian groups.
The boundary operator commutes with continuous maps, so that $\partial _{n}f_{*}=f_{*}\partial _{n}$. This allows the entire chain complex to be treated as a functor. In particular, this shows that the map $X\mapsto H_{n}(X)$ is a functor
$H_{n}:\mathbf {Top} \to \mathbf {Ab} $
from the category of topological spaces to the category of abelian groups. By the homotopy axiom, one has that $H_{n}$ is also a functor, called the homology functor, acting on hTop, the quotient homotopy category:
$H_{n}:\mathbf {hTop} \to \mathbf {Ab} .$
This distinguishes singular homology from other homology theories, wherein $H_{n}$ is still a functor, but is not necessarily defined on all of Top. In some sense, singular homology is the "largest" homology theory, in that every homology theory on a subcategory of Top agrees with singular homology on that subcategory. On the other hand, the singular homology does not have the cleanest categorical properties; such a cleanup motivates the development of other homology theories such as cellular homology.
More generally, the homology functor is defined axiomatically, as a functor on an abelian category, or, alternately, as a functor on chain complexes, satisfying axioms that require a boundary morphism that turns short exact sequences into long exact sequences. In the case of singular homology, the homology functor may be factored into two pieces, a topological piece and an algebraic piece. The topological piece is given by
$C_{\bullet }:\mathbf {Top} \to \mathbf {Comp} $
which maps topological spaces as $X\mapsto (C_{\bullet }(X),\partial _{\bullet })$ and continuous functions as $f\mapsto f_{*}$. Here, then, $C_{\bullet }$ is understood to be the singular chain functor, which maps topological spaces to the category of chain complexes Comp (or Kom). The category of chain complexes has chain complexes as its objects, and chain maps as its morphisms.
The second, algebraic part is the homology functor
$H_{n}:\mathbf {Comp} \to \mathbf {Ab} $
which maps
$C_{\bullet }\mapsto H_{n}(C_{\bullet })=Z_{n}(C_{\bullet })/B_{n}(C_{\bullet })$
and takes chain maps to maps of abelian groups. It is this homology functor that may be defined axiomatically, so that it stands on its own as a functor on the category of chain complexes.
Homotopy maps re-enter the picture by defining homotopically equivalent chain maps. Thus, one may define the quotient category hComp or K, the homotopy category of chain complexes.
Coefficients in R
Given any unital ring R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module. That is, rather than performing the above constructions from the starting point of free abelian groups, one instead uses free R-modules in their place. All of the constructions go through with little or no change. The result of this is
$H_{n}(X;R)\ $
which is now an R-module. Of course, it is usually not a free module. The usual homology group is regained by noting that
$H_{n}(X;\mathbb {Z} )=H_{n}(X)$
when one takes the ring to be the ring of integers. The notation Hn(X; R) should not be confused with the nearly identical notation Hn(X, A), which denotes the relative homology (below).
The universal coefficient theorem provides a mechanism to calculate the homology with R coefficients in terms of homology with usual integer coefficients using the short exact sequence
$0\to H_{n}(X;\mathbb {Z} )\otimes R\to H_{n}(X;R)\to Tor_{1}(H_{n-1}(X;\mathbb {Z} ),R)\to 0.$
where Tor is the Tor functor.[8] Of note, if R is torsion-free, then Tor_1(G, R) = 0 for any G, so the above short exact sequence reduces to an isomorphism between $H_{n}(X;\mathbb {Z} )\otimes R$ and $H_{n}(X;R).$
Relative homology
Main article: Relative homology
For a subspace $A\subset X$, the relative homology Hn(X, A) is understood to be the homology of the quotient of the chain complexes, that is,
$H_{n}(X,A)=H_{n}(C_{\bullet }(X)/C_{\bullet }(A))$
where the quotient of chain complexes is given by the short exact sequence
$0\to C_{\bullet }(A)\to C_{\bullet }(X)\to C_{\bullet }(X)/C_{\bullet }(A)\to 0.$[9]
Reduced homology
The reduced homology of a space X, annotated as ${\tilde {H}}_{n}(X)$ is a minor modification to the usual homology which simplifies expressions of some relationships and fulfils the intuiton that all homology groups of a point should be zero.
For the usual homology defined on a chain complex:
$\dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\partial _{0}}{\longrightarrow \,}}0$
To define the reduced homology, we augment the chain complex with an additional $\mathbb {Z} $ between $C_{0}$ and zero:
$\dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\epsilon }{\longrightarrow \,}}\mathbb {Z} \to 0$
where $\epsilon \left(\sum _{i}n_{i}\sigma _{i}\right)=\sum _{i}n_{i}$. This can be justified by interpreting the empty set as "(-1)-simplex", which means that $C_{-1}\simeq \mathbb {Z} $.
The reduced homology groups are now defined by ${\tilde {H}}_{n}(X)=\ker(\partial _{n})/\mathrm {im} (\partial _{n+1})$ for positive n and ${\tilde {H}}_{0}(X)=\ker(\epsilon )/\mathrm {im} (\partial _{1})$. [10]
For n > 0, $H_{n}(X)={\tilde {H}}_{n}(X)$, while for n = 0, $H_{0}(X)={\tilde {H}}_{0}(X)\oplus \mathbb {Z} .$
Cohomology
Main article: Cohomology
By dualizing the homology chain complex (i.e. applying the functor Hom(-, R), R being any ring) we obtain a cochain complex with coboundary map $\delta $. The cohomology groups of X are defined as the homology groups of this complex; in a quip, "cohomology is the homology of the co [the dual complex]".
The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a differential graded algebra as follows:
• the graded set of groups form a graded R-module;
• this can be given the structure of a graded R-algebra using the cup product;
• the Bockstein homomorphism β gives a differential.
There are additional cohomology operations, and the cohomology algebra has addition structure mod p (as before, the mod p cohomology is the cohomology of the mod p cochain complex, not the mod p reduction of the cohomology), notably the Steenrod algebra structure.
Betti homology and cohomology
Since the number of homology theories has become large (see Category:Homology theory), the terms Betti homology and Betti cohomology are sometimes applied (particularly by authors writing on algebraic geometry) to the singular theory, as giving rise to the Betti numbers of the most familiar spaces such as simplicial complexes and closed manifolds.
Extraordinary homology
Main article: Extraordinary homology theory
If one defines a homology theory axiomatically (via the Eilenberg–Steenrod axioms), and then relaxes one of the axioms (the dimension axiom), one obtains a generalized theory, called an extraordinary homology theory. These originally arose in the form of extraordinary cohomology theories, namely K-theory and cobordism theory. In this context, singular homology is referred to as ordinary homology.
See also
• Derived category
• Excision theorem
• Hurewicz theorem
• Simplicial homology
• Cellular homology
References
1. Hatcher, 105
2. Hatcher, 108
3. Theorem 2.10. Hatcher, 111
4. Hatcher, 144
5. Hatcher, 140
6. Hatcher, 110
7. Hatcher, 142-143
8. Hatcher, 264
9. Hatcher, 115
10. Hatcher, 110
• Allen Hatcher, Algebraic topology. Cambridge University Press, ISBN 0-521-79160-X and ISBN 0-521-79540-0
• J.P. May, A Concise Course in Algebraic Topology, Chicago University Press ISBN 0-226-51183-9
• Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, ISBN 0-387-96678-1
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Wikipedia
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Singularity (mathematics)
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.[1][2][3]
For example, the function
$f(x)={\frac {1}{x}}$
has a singularity at $x=0$, where the value of the function is not defined, as involving a division by zero. The absolute value function $g(x)=|x|$ also has a singularity at $x=0$, since it is not differentiable there.[4]
The algebraic curve defined by $\left\{(x,y):y^{3}-x^{2}=0\right\}$ in the $(x,y)$ coordinate system has a singularity (called a cusp) at $(0,0)$. For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.
Real analysis
In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not).
To describe the way these two types of limits are being used, suppose that $f(x)$ is a function of a real argument $x$, and for any value of its argument, say $c$, then the left-handed limit, $f(c^{-})$, and the right-handed limit, $f(c^{+})$, are defined by:
$f(c^{-})=\lim _{x\to c}f(x)$, constrained by $x<c$ and
$f(c^{+})=\lim _{x\to c}f(x)$, constrained by $x>c$.
The value $f(c^{-})$ is the value that the function $f(x)$ tends towards as the value $x$ approaches $c$ from below, and the value $f(c^{+})$ is the value that the function $f(x)$ tends towards as the value $x$ approaches $c$ from above, regardless of the actual value the function has at the point where $x=c$ .
There are some functions for which these limits do not exist at all. For example, the function
$g(x)=\sin \left({\frac {1}{x}}\right)$
does not tend towards anything as $x$ approaches $c=0$. The limits in this case are not infinite, but rather undefined: there is no value that $g(x)$ settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.
The possible cases at a given value $c$ for the argument are as follows.
• A point of continuity is a value of $c$ for which $f(c^{-})=f(c)=f(c^{+})$, as one expects for a smooth function. All the values must be finite. If $c$ is not a point of continuity, then a discontinuity occurs at $c$.
• A type I discontinuity occurs when both $f(c^{-})$ and $f(c^{+})$ exist and are finite, but at least one of the following three conditions also applies:
• $f(c^{-})\neq f(c^{+})$;
• $f(x)$ is not defined for the case of $x=c$; or
• $f(c)$ has a defined value, which, however, does not match the value of the two limits.
Type I discontinuities can be further distinguished as being one of the following subtypes:
• A jump discontinuity occurs when $f(c^{-})\neq f(c^{+})$, regardless of whether $f(c)$ is defined, and regardless of its value if it is defined.
• A removable discontinuity occurs when $f(c^{-})=f(c^{+})$, also regardless of whether $f(c)$ is defined, and regardless of its value if it is defined (but which does not match that of the two limits).
• A type II discontinuity occurs when either $f(c^{-})$ or $f(c^{+})$ does not exist (possibly both). This has two subtypes, which are usually not considered separately:
• An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote.
• An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits $f(c^{-})$ or $f(c^{+})$ does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if valid answers are extended to include $\pm \infty $.
In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.
Coordinate singularities
A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an n-vector representation).
Complex analysis
In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities, and the branch points.
Isolated singularities
Suppose that $\ f\ $ is a function that is complex differentiable in the complement of a point $\ a\ $ in an open subset $\ U\ $ of the complex numbers $\ \mathbb {C} ~.$ Then:
• The point $\ a\ $ is a removable singularity of $\ f\ $ if there exists a holomorphic function $\ g\ $ defined on all of $\ U\ $ such that $\ f(z)=g(z)\ $ for all $\ z\ $ in $\ U\smallsetminus \{a\}~.$ The function $\ g\ $ is a continuous replacement for the function $\ f~.$[5]
• The point $\ a\ $ is a pole or non-essential singularity of $\ f\ $ if there exists a holomorphic function $\ g\ $ defined on $\ U\ $ with $\ g(a)\ $ nonzero, and a natural number $\ n\ $ such that $\ f(z)={\frac {g(z)}{\ (z-a)^{n}\ }}\ $ for all $\ z\ $ in $\ U\smallsetminus \{a\}~.$ The least such number $\ n\ $ is called the order of the pole. The derivative at a non-essential singularity itself has a non-essential singularity, with $\ n\ $ increased by 1 (except if $\ n\ $ is 0 so that the singularity is removable).
• The point $\ a\ $ is an essential singularity of $\ f\ $ if it is neither a removable singularity nor a pole. The point $\ a\ $ is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.[1]
Nonisolated singularities
Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types:
• Cluster points: limit points of isolated singularities. If they are all poles, despite admitting Laurent series expansions on each of them, then no such expansion is possible at its limit.
• Natural boundaries: any non-isolated set (e.g. a curve) on which functions cannot be analytically continued around (or outside them if they are closed curves in the Riemann sphere).
Branch points
Branch points are generally the result of a multi-valued function, such as $\ {\sqrt {z\ }}\ $ or $\ \log(z)\ ,$ which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, even though it must connect two different branch points (such as $\ z=0\ $ and $\ z=\infty \ $ for $\ \log(z)\ $) which are fixed in place.
Finite-time singularity
A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and Partial Differential Equations – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form $x^{-\alpha },$ of which the simplest is hyperbolic growth, where the exponent is (negative) 1: $x^{-1}.$ More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses $(t_{0}-t)^{-\alpha }$ (using t for time, reversing direction to $-t$ so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time $t_{0}$).
An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the Painlevé paradox (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the Euler's Disk toy).
Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation" (simplistic models yield infinite human population in finite time).
Algebraic geometry and commutative algebra
In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation y2 − x3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the x-axis is a "double tangent."
For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.
An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring.
See also
• Catastrophe theory
• Defined and undefined
• Degeneracy (mathematics)
• Division by zero
• Hyperbolic growth
• Pathological (mathematics)
• Singular solution
• Removable singularity
References
1. "Singularities, Zeros, and Poles". mathfaculty.fullerton.edu. Retrieved 2019-12-12.
2. "Singularity | complex functions". Encyclopedia Britannica. Retrieved 2019-12-12.
3. "Singularity (mathematics)". TheFreeDictionary.com. Retrieved 2019-12-12.
4. Berresford, Geoffrey C.; Rockett, Andrew M. (2015). Applied Calculus. Cengage Learning. p. 151. ISBN 978-1-305-46505-3.
5. Weisstein, Eric W. "Singularity". mathworld.wolfram.com. Retrieved 2019-12-12.
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Singular function
In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties:
• f is continuous on [a, b]. (**)
• there exists a set N of measure 0 such that for all x outside of N the derivative f ′(x) exists and is zero, that is, the derivative of f vanishes almost everywhere.
• f is non-constant on [a, b].
A standard example of a singular function is the Cantor function, which is sometimes called the devil's staircase (a term also used for singular functions in general). There are, however, other functions that have been given that name. One is defined in terms of the circle map.
If f(x) = 0 for all x ≤ a and f(x) = 1 for all x ≥ b, then the function can be taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor an absolutely continuous random variable (since the probability density is zero everywhere it exists).
Singular functions occur, for instance, as sequences of spatially modulated phases or structures in solids and magnets, described in a prototypical fashion by the Frenkel–Kontorova model and by the ANNNI model, as well as in some dynamical systems. Most famously, perhaps, they lie at the center of the fractional quantum Hall effect.
When referring to functions with a singularity
When discussing mathematical analysis in general, or more specifically real analysis or complex analysis or differential equations, it is common for a function which contains a mathematical singularity to be referred to as a 'singular function'. This is especially true when referring to functions which diverge to infinity at a point or on a boundary. For example, one might say, "1/x becomes singular at the origin, so 1/x is a singular function."
Advanced techniques for working with functions that contain singularities have been developed in the subject called distributional or generalized function analysis. A weak derivative is defined that allows singular functions to be used in partial differential equations, etc.
See also
• Absolute continuity
• Mathematical singularity
• Generalized function
• Distribution
• Minkowski's question-mark function
References
(**) This condition depends on the references [1]
1. "Singular function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Lebesgue, H. (1955–1961), Theory of functions of a real variable, F. Ungar
• Halmos, P.R. (1950), Measure theory, v. Nostrand
• Royden, H.L (1988), Real Analysis, Prentice-Hall, Englewood Cliffs, New Jersey
• Lebesgue, H. (1928), Leçons sur l'intégration et la récherche des fonctions primitives, Gauthier-Villars
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Wikipedia
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Singular integral
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator
$T(f)(x)=\int K(x,y)f(y)\,dy,$
whose kernel function K : Rn×Rn → R is singular along the diagonal x = y. Specifically, the singularity is such that |K(x, y)| is of size |x − y|−n asymptotically as |x − y| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp(Rn).
The Hilbert transform
Main article: Hilbert transform
The archetypal singular integral operator is the Hilbert transform H. It is given by convolution against the kernel K(x) = 1/(πx) for x in R. More precisely,
$H(f)(x)={\frac {1}{\pi }}\lim _{\varepsilon \to 0}\int _{|x-y|>\varepsilon }{\frac {1}{x-y}}f(y)\,dy.$
The most straightforward higher dimension analogues of these are the Riesz transforms, which replace K(x) = 1/x with
$K_{i}(x)={\frac {x_{i}}{|x|^{n+1}}}$
where i = 1, ..., n and $x_{i}$ is the i-th component of x in Rn. All of these operators are bounded on Lp and satisfy weak-type (1, 1) estimates.[1]
Singular integrals of convolution type
Main article: Singular integral operators of convolution type
A singular integral of convolution type is an operator T defined by convolution with a kernel K that is locally integrable on Rn\{0}, in the sense that
$T(f)(x)=\lim _{\varepsilon \to 0}\int _{|y-x|>\varepsilon }K(x-y)f(y)\,dy.$
(1)
Suppose that the kernel satisfies:
1. The size condition on the Fourier transform of K
${\hat {K}}\in L^{\infty }(\mathbf {R} ^{n})$
2. The smoothness condition: for some C > 0,
$\sup _{y\neq 0}\int _{|x|>2|y|}|K(x-y)-K(x)|\,dx\leq C.$
Then it can be shown that T is bounded on Lp(Rn) and satisfies a weak-type (1, 1) estimate.
Property 1. is needed to ensure that convolution (1) with the tempered distribution p.v. K given by the principal value integral
$\operatorname {p.v.} \,\,K[\phi ]=\lim _{\epsilon \to 0^{+}}\int _{|x|>\epsilon }\phi (x)K(x)\,dx$
is a well-defined Fourier multiplier on L2. Neither of the properties 1. or 2. is necessarily easy to verify, and a variety of sufficient conditions exist. Typically in applications, one also has a cancellation condition
$\int _{R_{1}<|x|<R_{2}}K(x)\,dx=0,\ \forall R_{1},R_{2}>0$
which is quite easy to check. It is automatic, for instance, if K is an odd function. If, in addition, one assumes 2. and the following size condition
$\sup _{R>0}\int _{R<|x|<2R}|K(x)|\,dx\leq C,$
then it can be shown that 1. follows.
The smoothness condition 2. is also often difficult to check in principle, the following sufficient condition of a kernel K can be used:
• $K\in C^{1}(\mathbf {R} ^{n}\setminus \{0\})$
• $|\nabla K(x)|\leq {\frac {C}{|x|^{n+1}}}$
Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.[2]
Singular integrals of non-convolution type
These are even more general operators. However, since our assumptions are so weak, it is not necessarily the case that these operators are bounded on Lp.
Calderón–Zygmund kernels
A function K : Rn×Rn → R is said to be a Calderón–Zygmund kernel if it satisfies the following conditions for some constants C > 0 and δ > 0.[2]
1. $|K(x,y)|\leq {\frac {C}{|x-y|^{n}}}$
2. $|K(x,y)-K(x',y)|\leq {\frac {C|x-x'|^{\delta }}{{\bigl (}|x-y|+|x'-y|{\bigr )}^{n+\delta }}}{\text{ whenever }}|x-x'|\leq {\frac {1}{2}}\max {\bigl (}|x-y|,|x'-y|{\bigr )}$
3. $|K(x,y)-K(x,y')|\leq {\frac {C|y-y'|^{\delta }}{{\bigl (}|x-y|+|x-y'|{\bigr )}^{n+\delta }}}{\text{ whenever }}|y-y'|\leq {\frac {1}{2}}\max {\bigl (}|x-y'|,|x-y|{\bigr )}$
Singular integrals of non-convolution type
T is said to be a singular integral operator of non-convolution type associated to the Calderón–Zygmund kernel K if
$\int g(x)T(f)(x)\,dx=\iint g(x)K(x,y)f(y)\,dy\,dx,$
whenever f and g are smooth and have disjoint support.[2] Such operators need not be bounded on Lp
Calderón–Zygmund operators
A singular integral of non-convolution type T associated to a Calderón–Zygmund kernel K is called a Calderón–Zygmund operator when it is bounded on L2, that is, there is a C > 0 such that
$\|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}},$
for all smooth compactly supported ƒ.
It can be proved that such operators are, in fact, also bounded on all Lp with 1 < p < ∞.
The T(b) theorem
The T(b) theorem provides sufficient conditions for a singular integral operator to be a Calderón–Zygmund operator, that is for a singular integral operator associated to a Calderón–Zygmund kernel to be bounded on L2. In order to state the result we must first define some terms.
A normalised bump is a smooth function φ on Rn supported in a ball of radius 1 and centred at the origin such that |∂α φ(x)| ≤ 1, for all multi-indices |α| ≤ n + 2. Denote by τx(φ)(y) = φ(y − x) and φr(x) = r−nφ(x/r) for all x in Rn and r > 0. An operator is said to be weakly bounded if there is a constant C such that
$\left|\int T{\bigl (}\tau ^{x}(\varphi _{r}){\bigr )}(y)\tau ^{x}(\psi _{r})(y)\,dy\right|\leq Cr^{-n}$
for all normalised bumps φ and ψ. A function is said to be accretive if there is a constant c > 0 such that Re(b)(x) ≥ c for all x in R. Denote by Mb the operator given by multiplication by a function b.
The T(b) theorem states that a singular integral operator T associated to a Calderón–Zygmund kernel is bounded on L2 if it satisfies all of the following three conditions for some bounded accretive functions b1 and b2:[3]
1. $M_{b_{2}}TM_{b_{1}}$ is weakly bounded;
2. $T(b_{1})$ is in BMO;
3. $T^{t}(b_{2}),$ is in BMO, where Tt is the transpose operator of T.
See also
• Singular integral operators on closed curves
Notes
1. Stein, Elias (1993). "Harmonic Analysis". Princeton University Press.
2. Grafakos, Loukas (2004), "7", Classical and Modern Fourier Analysis, New Jersey: Pearson Education, Inc.
3. David; Semmes; Journé (1985). "Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation" (in French). Vol. 1. Revista Matemática Iberoamericana. pp. 1–56.
References
• Calderon, A. P.; Zygmund, A. (1952), "On the existence of certain singular integrals", Acta Mathematica, 88 (1): 85–139, doi:10.1007/BF02392130, ISSN 0001-5962, MR 0052553, Zbl 0047.10201.
• Calderon, A. P.; Zygmund, A. (1956), "On singular integrals", American Journal of Mathematics, The Johns Hopkins University Press, 78 (2): 289–309, doi:10.2307/2372517, ISSN 0002-9327, JSTOR 2372517, MR 0084633, Zbl 0072.11501.
• Coifman, Ronald; Meyer, Yves (1997), Wavelets: Calderón-Zygmund and multilinear operators, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, pp. xx+315, ISBN 0-521-42001-6, MR 1456993, Zbl 0916.42023.
• Mikhlin, Solomon G. (1948), "Singular integral equations", UMN, 3 (25): 29–112, MR 0027429 (in Russian).
• Mikhlin, Solomon G. (1965), Multidimensional singular integrals and integral equations, International Series of Monographs in Pure and Applied Mathematics, vol. 83, Oxford–London–Edinburgh–New York City–Paris–Frankfurt: Pergamon Press, pp. XII+255, MR 0185399, Zbl 0129.07701.
• Mikhlin, Solomon G.; Prössdorf, Siegfried (1986), Singular Integral Operators, Berlin–Heidelberg–New York City: Springer Verlag, p. 528, ISBN 0-387-15967-3, MR 0867687, Zbl 0612.47024, (European edition: ISBN 3-540-15967-3).
• Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, Princeton, NJ: Princeton University Press, pp. XIV+287, ISBN 0-691-08079-8, MR 0290095, Zbl 0207.13501
External links
• Stein, Elias M. (October 1998). "Singular Integrals: The Roles of Calderón and Zygmund" (PDF). Notices of the American Mathematical Society. 45 (9): 1130–1140.
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Wikipedia
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Integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign.[1] In mathematical notation, integral equations may thus be expressed as being of the form:
$f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});I^{1}(u),I^{2}(u),I^{3}(u),...,I^{m}(u))=0$
For equations of integer unknowns, see Diophantine equation.
where $I^{i}(u)$ is an integral operator acting on u.[1] Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.[1] A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:
$f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});D^{1}(u),D^{2}(u),D^{3}(u),...,D^{m}(u))=0$
where $D^{i}(u)$ may be viewed as a differential operator of order i.[1] Due to this close connection between differential and integral equations, one can often convert between the two.[1] For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation.[1] In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form.[2] See also, for example, Green's function and Fredholm theory.
Classification and overview
Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations.[1] These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation.[1] These comments are made concrete through the following definitions and examples:
Linearity
Linear: An integral equation is linear if the unknown function u(x) and its integrals appear linear in the equation.[1] Hence, an example of a linear equation would be:[1]
$u(x)=f(x)+\int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u(t)dt$
As a note on naming convention: i) u(x) is called the unknown function, ii) f(x) is called a known function, iii) K(x,t) is a function of two variables and often called the Kernel function, and iv) λ is an unknown factor or parameter, which plays the same role as the eigenvalue in linear algebra.[1] Nonlinear: An integral equation is nonlinear if the unknown function u(x) or any of its integrals appear nonlinear in the equation.[1] Hence, examples of nonlinear equations would be the equation above if we replaced u(t) with $u^{2}(x),\,\,cos(u(x)),\,{\text{or }}\,e^{u(x)}$, such as:
$u(x)=f(x)+\int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u^{2}(t)dt$
Certain kinds of nonlinear integral equations have specific names.[3] A selection of such equations are:[3]
• Nonlinear Volterra integral equations of the second kind which have the general form: $u(x)=f(x)+\lambda \int _{a}^{x}K(x,t)\,F(x,t,u(t))\,dt,$ where F is a known function.[3]
• Nonlinear Fredholm integral equations of the second kind which have the general form: $f(x)=F(x,\int _{a}^{b}K(x,y,f(x),f(y))\,dy)$.[3]
• A special type of nonlinear Fredholm integral equations of the second kind is given by the form: $f(x)=g(x)+\int _{a}^{b}K(x,y,f(x),f(y))\,dy$, which has the two special subclasses:[3]
• Urysohn equation: $f(x)=g(x)+\int _{a}^{b}k(x,y,f(y))\,dy$.[3]
• Hammerstein equation: $f(x)=g(x)+\int _{a}^{b}k(x,y)\,G(y,f(y))\,dy$.[3]
More information on the Hammerstein equation and different versions of the Hammerstein equation can be found in the Hammerstein section below.
Location of the unknown equation
First kind: An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign.[3] An example would be: $f(x)=\int _{a}^{b}K(x,t)\,u(t)\,dt$.[3]
Second kind: An integral equation is called an integral equation of the second kind if the unknown function also appears outside the integral.[3]
Third kind: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form:[3]
$g(t)u(t)+\lambda \int _{a}^{b}K(t,x)u(x)\,dx=f(t)$
where g(t) vanishes at least once in the interval [a,b][4][5] or where g(t) vanishes at a finite number of points in (a,b).[6]
Limits of Integration
Fredholm: An integral equation is called a Fredholm integral equation if both of the limits of integration in all integrals are fixed and constant.[1] An example would be that the integral is taken over a fixed subset of $\mathbb {R} ^{n}$.[3] Hence, the following two examples are Fredholm equations:[1]
• Fredholm equation of the first type: $f(x)=\int _{a}^{b}K(x,t)\,u(t)\,dt$.
• Fredholm equation of the second type: $u(x)=f(x)+\lambda \int _{a}^{b}K(x,t)\,u(t)\,dt.$
Note that we can express integral equations such as those above also using integral operator notation.[7] For example, we can define the Fredholm integral operator as:
$({\mathcal {F}}y)(t):=\int _{t_{0}}^{T}K(t,s)\,y(s)\,ds.$
Hence, the above Fredholm equation of the second kind may be written compactly as:[7]
$y(t)=g(t)+\lambda ({\mathcal {F}}y)(t).$
Volterra: An integral equation is called a Volterra integral equation if at least one of the limits of integration is a variable.[1] Hence, the integral is taken over a domain varying with the variable of integration.[3] Examples of Volterra equations would be:[1]
• Volterrra integral equation of the first kind: $f(x)=\int _{a}^{x}K(x,t)\,u(t)\,dt$
• Volterrra integral equation of the second kind: $u(x)=f(x)+\lambda \int _{a}^{x}K(x,t)\,u(t)\,dt.$
As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator ${\mathcal {V}}:C(I)\to C(I)$, as follows:[3]
$({\mathcal {V}}\phi )(t):=\int _{t_{0}}^{t}K(t,s)\,\phi (s)\,ds$
where $t\in I=[t_{0},T]$ and K(t,s) is called the kernel and must be continuous on the interval $D:=\{(t,s):0\leq s\leq t\leq T\leq \infty \}$.[3] Hence, the Volterra integral equation of the first kind may be written as:[3]
$({\mathcal {V}}y)(t)=g(t)$
with $g(0)=0$. In addition, a linear Volterra integral equation of the second kind for an unknown function $y(t)$ and a given continuous function $g(t)$ on the interval $I$ where $t\in I$:
$y(t)=g(t)+({\mathcal {V}}y)(t).$
Volterra-Fredholm: In higher dimensions, integral equations such as Fredholm-Volterra integral equations (VFIE) exist.[3] A VFIE has the form:
$u(t,x)=g(t,x)+({\mathcal {T}}u)(t,x)$
with $x\in \Omega $ and $\Omega $ being a closed bounded region in $\mathbb {R} ^{d}$ with piecewise smooth boundary.[3] The Fredholm-Volterrra Integral Operator ${\mathcal {T}}:C(I\times \Omega )\to C(I\times \Omega )$ is defined as:[3]
$({\mathcal {T}}u)(t,x):=\int _{0}^{t}\int _{\Omega }K(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds.$
Note that while throughout this article, the bounds of the integral are usually written as intervals, this need not be the case.[7] In general, integral equations don't always need to be defined over an interval $[a,b]=I$, but could also be defined over a curve or surface.[7]
Homogeneity
Homogenous: An integral equation is called homogeneous if the known function $f$ is identically zero.[1]
Inhomogenous: An integral equation is called inhomogeneous if the known function $f$ is nonzero.[1]
Regularity
Regular: An integral equation is called regular if the integrals used are all proper integrals.[7]
Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is an improper integral.[7] This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated.[1]
Examples include:[1]
$F(\lambda )=\int _{-\infty }^{\infty }e^{-i\lambda x}u(x)\,dx$
$L[u(x)]=\int _{0}^{\infty }e^{-\lambda x}u(x)\,dx$
These two integral equations are the Fourier transform and the Laplace transform of u(x), respectively, with both being Fredholm equations of the first kind with kernel $K(x,t)=e^{-i\lambda x}$ and $K(x,t)=e^{-\lambda x}$, respectively.[1] Another example of a singular integral equation in which the kernel becomes unbounded is:[1]
$x^{2}=\int _{0}^{x}{\frac {1}{\sqrt {x-t}}}\,u(t)\,dt.$
This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation:[7]
$g(x)=\int _{a}^{x}{\frac {f(y)}{\sqrt {x-y}}}\,dy$
Strongly singular: An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.[7]
Integro-differential equations
An Integro-differential equation, as the name suggests, combines differential and integral operators into one equation.[1] There are many version including the Volterra integro-differential equation and delay type equations as defined below.[3] For example, using the Volterra operator as defined above, the Volterra integro-differential equation may be written as:[3]
$y'(t)=f(t,y(t))+(V_{\alpha }y)(t)$
For delay problems, we can define the delay integral operator $({\mathcal {W}}_{\theta ,\alpha }y)$ as:[3]
$({\mathcal {W}}_{\theta ,\alpha }y)(t):=\int _{\theta (t)}^{t}(t-s)^{-\alpha }\cdot k_{2}(t,s,y(s),y'(s))\,ds$
where the delay integro-differential equation may be expressed as:[3]
$y'(t)=f(t,y(t),y(\theta (t)))+({\mathcal {W}}_{\theta ,\alpha }y)(t).$
Volterra integral equations
Uniqueness and existence theorems in 1D
The solution to a linear Volterra integral equation of the first kind, given by the equation:
$({\mathcal {V}}y)(t)=g(t)$
can be described by the following uniqueness and existence theorem.[3] Recall that the Volterra integral operator ${\mathcal {V}}:C(I)\to C(I)$, can be defined as follows:[3]
$({\mathcal {V}}\phi )(t):=\int _{t_{0}}^{t}K(t,s)\,\phi (s)\,ds$
where $t\in I=[t_{0},T]$ and K(t,s) is called the kernel and must be continuous on the interval $D:=\{(t,s):0\leq s\leq t\leq T\leq \infty \}$.[3]
Theorem — Assume that $K$ satisfies $K\in C(D),\,\partial K/\partial t\in C(D)$ and $\vert K(t,t)\vert \geq k_{0}>0$ for some $t\in I.$ Then for any $g\in C^{1}(I)$ with $g(0)=0$ the integral equation above has a unique solution in $y\in C(I)$.
The solution to a linear Volterra integral equation of the second kind, given by the equation:[3]
$y(t)=g(t)+({\mathcal {V}}y)(t)$
can be described by the following uniqueness and existence theorem.[3]
Theorem — Let $K\in C(D)$ and let $R$ denote the resolvent Kernel associated with $K$. Then, for any $g\in C(I)$ , the second-kind Volterra integral equation has a unique solution $y\in C(I)$ and this solution is given by: $y(t)=g(t)+\int _{0}^{t}R(t,s)\,g(s)\,ds$.
Volterra integral equations in ℝ2
A Volterra Integral equation of the second kind can be expressed as follows:[3]
$u(t,x)=g(t,x)+\int _{0}^{x}\int _{0}^{y}K(x,\xi ,y,\eta )\,u(\xi ,\eta )\,d\eta \,d\xi $
where $(x,y)\in \Omega :=[0,X]\times [0,Y]$ :=[0,X]\times [0,Y]} , $g\in C(\Omega )$, $K\in C(D_{2})$ and $D_{2}:=\{(x,\xi ,y,\eta ):0\leq \xi \leq x\leq X,0\leq \eta \leq y\leq Y\}$.[3] This integral equation has a unique solution $u\in C(\Omega )$ given by:[3]
$u(t,x)=g(t,x)+\int _{0}^{x}\int _{0}^{y}R(x,\xi ,y,\eta )\,g(\xi ,\eta )\,d\eta \,d\xi $
where $R$ is the resolvent kernel of K.[3]
Uniqueness and existence theorems of Fredhom-Volterra equations
As defined above, a VFIE has the form:
$u(t,x)=g(t,x)+({\mathcal {T}}u)(t,x)$
with $x\in \Omega $ and $\Omega $ being a closed bounded region in $\mathbb {R} ^{d}$ with piecewise smooth boundary.[3] The Fredholm-Volterrra Integral Operator ${\mathcal {T}}:C(I\times \Omega )\to C(I\times \Omega )$ is defined as:[3]
$({\mathcal {T}}u)(t,x):=\int _{0}^{t}\int _{\Omega }K(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds.$
In the case where the Kernel K may be written as $K(t,s,x,\xi )=k(t-s)H(x,\xi )$, K is called the positive memory kernel.[3] With this in mind, we can now introduce the following theorem:[3]
Theorem — If the linear VFIE given by: $u(t,x)=g(t,x)+\int _{0}^{t}\int _{\Omega }K(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds$ with $(t,x)\in I\times \Omega $ satisfies the following conditions:
• $g\in C(I\times \Omega )$, and
• $K\in C(D\times \Omega ^{2})$ where $D:=\{(t,s):0\leq s\leq t\leq T\}$ and $\Omega ^{2}=\Omega \times \Omega $
Then the VFIE has a unique solution $u\in C(I\times \Omega )$ given by $u(t,x)=g(t,x)+\int _{0}^{t}\int _{\Omega }R(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds$ where $R\in C(D\times \Omega ^{2})$ is called the Resolvent Kernel and is given by the limit of the Neumann series for the Kernel $K$ and solves the resolvent equations: $R(t,s,x,\xi )=K(t,s,x,\xi )+\int _{0}^{t}\int _{\Omega }K(t,v,x,z)R(v,s,z,\xi )\,dz\,dv=K(t,s,x,\xi )+\int _{0}^{t}\int _{\Omega }R(t,v,x,z)K(v,s,z,\xi )\,dz\,dv$
Special Volterra equations
A special type of Volterra equation which is used in various applications is defined as follows:[3]
$y(t)=g(t)+(V_{\alpha }y)(t)$
where $t\in I=[t_{0},T]$, the function g(t) is continuous on the interval $I$, and the Volterra integral operator $(V_{\alpha }t)$ is given by:
$(V_{\alpha }t)(t):=\int _{t_{0}}^{t}(t-s)^{-\alpha }\cdot k(t,s,y(s))\,ds$
with $(0\leq \alpha <1)$.[3]
Converting IVP to integral equations
In the following section, we give an example of how to convert an initial value problem (IVP) into an integral equation. There are multiple motivations for doing so, among them being that integral equations can often be more readily solvable and are more suitable for proving existence and uniqueness theorems.[7]
The following example was provided by Wazwaz on pages 1 and 2 in his book.[1] We examine the IVP given by the equation:
$u'(t)=2tu(t),\,\,\,\,\,\,\,x\geq 0$
and the initial condition:
$u(0)=1$
If we integrate both sides of the equation, we get:
$\int _{0}^{x}u'(t)dt=\int _{0}^{x}2tu(t)dt$
and by the fundamental theorem of calculus, we obtain:
$u(x)-u(0)=\int _{0}^{x}2tu(t)dt$
Rearranging the equation above, we get the integral equation:
$u(x)=1+\int _{0}^{x}2tu(t)dt$
which is a Volterra integral equation of the form:
$u(x)=f(x)+\int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u(t)dt$
where K(x,t) is called the kernel and equal to 2t, and f(x)=1.[1]
Power series solution for integral equations
See also: Liouville–Neumann series
In many cases, if the Kernel of the integral equation is of the form K(xt) and the Mellin transform of K(t) exists, we can find the solution of the integral equation
$g(s)=s\int _{0}^{\infty }K(st)\,f(t)\,dt$
in the form of a power series
$f(t)=\sum _{n=0}^{\infty }{\frac {a_{n}}{M(n+1)}}t^{n}$
where
$g(s)=\sum _{n=0}^{\infty }a_{n}s^{-n},\qquad M(n+1)=\int _{0}^{\infty }K(t)\,t^{n}\,dt$
are the Z-transform of the function g(s), and M(n + 1) is the Mellin transform of the Kernel.
Numerical solution
It is worth noting that integral equations often do not have an analytical solution, and must be solved numerically. An example of this is evaluating the electric-field integral equation (EFIE) or magnetic-field integral equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem.
One method to solve numerically requires discretizing variables and replacing integral by a quadrature rule
$\sum _{j=1}^{n}w_{j}K\left(s_{i},t_{j}\right)u(t_{j})=f(s_{i}),\qquad i=0,1,\dots ,n.$
Then we have a system with n equations and n variables. By solving it we get the value of the n variables
$u(t_{0}),u(t_{1}),\dots ,u(t_{n}).$
Integral equations as a generalization of eigenvalue equations
Further information: Fredholm theory
Certain homogeneous linear integral equations can be viewed as the continuum limit of eigenvalue equations. Using index notation, an eigenvalue equation can be written as
$\sum _{j}M_{i,j}v_{j}=\lambda v_{i}$
where M = [Mi,j] is a matrix, v is one of its eigenvectors, and λ is the associated eigenvalue.
Taking the continuum limit, i.e., replacing the discrete indices i and j with continuous variables x and y, yields
$\int K(x,y)\,\varphi (y)\,dy=\lambda \,\varphi (x),$
where the sum over j has been replaced by an integral over y and the matrix M and the vector v have been replaced by the kernel K(x, y) and the eigenfunction φ(y). (The limits on the integral are fixed, analogously to the limits on the sum over j.) This gives a linear homogeneous Fredholm equation of the second type.
In general, K(x, y) can be a distribution, rather than a function in the strict sense. If the distribution K has support only at the point x = y, then the integral equation reduces to a differential eigenfunction equation.
In general, Volterra and Fredholm integral equations can arise from a single differential equation, depending on which sort of conditions are applied at the boundary of the domain of its solution.
Wiener–Hopf integral equations
Main article: Wiener–Hopf method
$y(t)=\lambda x(t)+\int _{0}^{\infty }k(t-s)\,x(s)\,ds,\qquad 0\leq t<\infty .$
Originally, such equations were studied in connection with problems in radiative transfer, and more recently, they have been related to the solution of boundary integral equations for planar problems in which the boundary is only piecewise smooth.
Hammerstein equations
A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form:[3]
$g(t)=\int _{0}^{t}K(t,s)\,G(s,y(s))\,ds.$
Under certain regularity conditions, the equation is equivalent to the implicit Volterra integral equation of the second-kind:[3]
$G(t,y(t))=g_{1}(t)-\int _{0}^{t}K_{1}(t,s)\,G(s,y(s))\,ds$
where:
$g_{1}(t):={\frac {g'(t)}{K(t,t)}}\,\,\,\,\,\,\,{\text{and}}\,\,\,\,\,\,\,K_{1}(t,s):=-{\frac {1}{K(t,t)}}{\frac {\partial K(t,s)}{\partial t}}.$
The equation may however also be expressed in operator form which motivates the definition of the following operator called the nonlinear Volterra-Hammerstein operator:[3]
$({\mathcal {H}}y)(t):=\int _{0}^{t}K(t,s)\,G(s,y(s))\,ds$
Here $G:I\times \mathbb {R} \to \mathbb {R} $ is a smooth function while the kernel K may be continuous, i.e. bounded, or weakly singular.[3] The corresponding second-kind Volterra integral equation called the Volterra-Hammerstein Integral Equation of the second kind, or simply Hammerstein equation for short, can be expressed as:[3]
$y(t)=g(t)+({\mathcal {H}}y)(t)$
In certain applications, the nonlinearity of the function G may be treated as being only semi-linear in the form of:[3]
$G(s,y)=y+H(s,y)$
In this case, we the following semi-linear Volterra integral equation:[3]
$y(t)=g(t)+({\mathcal {H}}y)(t)=g(t)+\int _{0}^{t}K(t,s)[y(s)+H(s,y(s))]\,ds$
In this form, we can state an existence and uniqueness theorem for the semi-linear Hammerstein integral equation.[3]
Theorem — Suppose that the semi-linear Hammerstein equation has a unique solution $y\in C(I)$ and $H:I\times \mathbb {R} \to \mathbb {R} $ be a Lipschitz continuous function. Then the solution of this eqution may be written in the form: $y(t)=y_{l}(t)+\int _{0}^{t}R(t,s)\,H(s,y(s))\,ds$ where $y_{l}(t)$ denotes the unique solution of the linear part of the equation above and is given by: $y_{l}(t)=g(t)+\int _{0}^{t}R(t,s)\,g(s)\,ds$ with $R(t,s)$ denoting the resolvent kernel.
We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, ${\mathcal {N}}$ defined as follows:[3]
$({\mathcal {N}}\phi )(t):=G(t,\phi (t))$
More about this can be found on page 75 of this book.[3]
Applications
Integral equations are important in many applications. Problems in which integral equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations.
• Actuarial science (ruin theory[8])
• Computational electromagnetics
• Boundary element method
• Inverse problems
• Marchenko equation (inverse scattering transform)
• Options pricing under jump-diffusion[9]
• Radiative transfer
• Viscoelasticity
• Fluid mechanics
See also
• Differential equation
• Integro-differential equation
• Ruin theory
• Volterra integral equation
Bibliography
• Agarwal, Ravi P., and Donal O'Regan. Integral and Integrodifferential Equations: Theory, Method and Applications. Gordon and Breach Science Publishers, 2000.[10]
• Brunner, Hermann. Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, 2004.[3]
• Burton, T. A. Volterra Integral and Differential Equations. Elsevier, 2005.[11]
• Chapter 7 It Mod 02-14-05 - Ira A. Fulton College of Engineering. https://www.et.byu.edu/~vps/ET502WWW/NOTES/CH7m.pdf.[12]
• Corduneanu, C. Integral Equations and Applications. Cambridge University Press, 2008.[13]
• Hackbusch, Wolfgang. Integral Equations Theory and Numerical Treatment. Birkhäuser, 1995.[7]
• Hochstadt, Harry. Integral Equations. Wiley-Interscience/John Wiley & Sons, 1989.[14]
• "Integral Equation." From Wolfram MathWorld, https://mathworld.wolfram.com/IntegralEquation.html.[15]
• "Integral Equation." Integral Equation - Encyclopedia of Mathematics, https://encyclopediaofmath.org/wiki/Integral_equation.[16]
• Jerri, Abdul J. Introduction to Integral Equations with Applications. Sampling Publishing, 2007.[17]
• Pipkin, A. C. A Course on Integral Equations. Springer-Verlag, 1991.[18]
• Polëiìanin A. D., and Alexander V. Manzhirov. Handbook of Integral Equations. Chapman & Hall/CRC, 2008.[19]
• Wazwaz, Abdul-Majid. A First Course in Integral Equations. World Scientific, 2015.[1]
References
1. Wazwaz, Abdul-Majid (2005). A First Course in Integral Equation. World Scientific.
2. admin (2022-09-10). "Maxwell's Equations: Derivation in Integral and Differential form". Ox Science. Retrieved 2022-12-10.
3. Brunner, Hermann (2004). Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press.
4. Bart, G. R.; Warnock, R. L. (November 1973). "Linear Integral Equations of the Third Kind". SIAM Journal on Mathematical Analysis. 4 (4): 609–622. doi:10.1137/0504053. ISSN 0036-1410.
5. Shulaia, D. (2017-12-01). "Integral equations of the third kind for the case of piecewise monotone coefficients". Transactions of A. Razmadze Mathematical Institute. 171 (3): 396–410. doi:10.1016/j.trmi.2017.05.002. ISSN 2346-8092.
6. Sukavanam, N. (1984-05-01). "A Fredholm-type theory for third-kind linear integral equations". Journal of Mathematical Analysis and Applications. 100 (2): 478–485. doi:10.1016/0022-247X(84)90096-9. ISSN 0022-247X.
7. Hackbusch, Wolfgang (1995). Integral Equations Theory and Numerical Treatment. Birkhauser.
8. "Lecture Notes on Risk Theory" (PDF). 2010.
9. Sachs, E. W.; Strauss, A. K. (2008-11-01). "Efficient solution of a partial integro-differential equation in finance". Applied Numerical Mathematics. 58 (11): 1687–1703. doi:10.1016/j.apnum.2007.11.002. ISSN 0168-9274.
10. Donal., Agarwal, Ravi P. O'Regan (2000). Integral and integrodifferential equations : theory, method and applications. Gordon and Breach Science Publishers. ISBN 90-5699-221-X. OCLC 44617552.{{cite book}}: CS1 maint: multiple names: authors list (link)
11. Burton, T.A. (2005). Volterra Integral and Differential Equations. Elsevier.
12. "Chapter 7 It Mod 02-14-05 - Ira A. Fulton College of Engineering" (PDF).
13. Corduneanu, C. (2008). Integral Equations and Applications. Cambridge University Press.
14. Hochstadt, Harry (1989). Integral Equations. Wiley-Interscience/John Wiley & Sons.
15. "Integral Equation".
16. "Integral equation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2022-11-14.
17. Jerri, Abdul J. Introduction to integral equations with applications. ISBN 0-9673301-1-4. OCLC 852490911.
18. Pipkin, A.C. (1991). A Course on Integral Equations. Springer-Verlag.
19. Polëiìanin, A.D. (2008). Handbook of Integral Equation. Chapman & Hall/CRC.
Further reading
• Kendall E. Atkinson The Numerical Solution of Integral Equations of the Second Kind. Cambridge Monographs on Applied and Computational Mathematics, 1997.
• George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.
• Harry Bateman (1910) History and Present State of the Theory of Integral Equations, Report of the British Association.
• Andrei D. Polyanin and Alexander V. Manzhirov Handbook of Integral Equations. CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4.
• E. T. Whittaker and G. N. Watson. A Course of Modern Analysis Cambridge Mathematical Library.
• M. Krasnov, A. Kiselev, G. Makarenko, Problems and Exercises in Integral Equations, Mir Publishers, Moscow, 1971
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Chapter 19. Integral Equations and Inverse Theory". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
External links
• Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
• Integral Equations: Index at EqWorld: The World of Mathematical Equations.
• "Integral equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Integral Equations (MIT OpenCourseWare)
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Singular integral operators of convolution type
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L2 is evident because the Fourier transform converts them into multiplication operators. Continuity on Lp spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on Lp spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.
L2 theory
Hilbert transform on the circle
See also: Harmonic conjugate
The theory for L2 functions is particularly simple on the circle.[1][2] If f ∈ L2(T), then it has a Fourier series expansion
$f(\theta )=\sum _{n\in \mathbf {Z} }a_{n}e^{in\theta }.$
Hardy space H2(T) consists of the functions for which the negative coefficients vanish, an = 0 for n < 0. These are precisely the square-integrable functions that arise as boundary values of holomorphic functions in the open unit disk. Indeed, f is the boundary value of the function
$F(z)=\sum _{n\geq 0}a_{n}z^{n},$
in the sense that the functions
$f_{r}(\theta )=F(re^{i\theta }),$
defined by the restriction of F to the concentric circles |z| = r, satisfy
$\|f_{r}-f\|_{2}\rightarrow 0.$
The orthogonal projection P of L2(T) onto H2(T) is called the Szegő projection. It is a bounded operator on L2(T) with operator norm 1. By Cauchy's theorem
$F(z)={1 \over 2\pi i}\int _{|\zeta |=1}{\frac {f(\zeta )}{\zeta -z}}\,d\zeta ={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\theta ) \over 1-e^{-i\theta }z}\,d\theta .$
Thus
$F(re^{i\varphi })={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\varphi -\theta ) \over 1-re^{i\theta }}\,d\theta .$
When r = 1, the integrand on the right-hand side has a singularity at θ = 0. The truncated Hilbert transform is defined by
$H_{\varepsilon }f(\varphi )={i \over \pi }\int _{\varepsilon \leq |\theta |\leq \pi }{f(\varphi -\theta ) \over 1-e^{i\theta }}\,d\theta ={1 \over \pi }\int _{|\zeta -e^{i\varphi }|\geq \delta }{f(\zeta ) \over \zeta -e^{i\varphi }}\,d\zeta ,$
where δ = |1 – eiε|. Since it is defined as convolution with a bounded function, it is a bounded operator on L2(T). Now
$H_{\varepsilon }{1}={i \over \pi }\int _{\varepsilon }^{\pi }2\Re (1-e^{i\theta })^{-1}\,d\theta ={i \over \pi }\int _{\varepsilon }^{\pi }1\,d\theta =i-{i\varepsilon \over \pi }.$
If f is a polynomial in z then
$H_{\varepsilon }f(z)-{i(1-\varepsilon ) \over \pi }f(z)={1 \over \pi i}\int _{|\zeta -z|\geq \delta }{f(\zeta )-f(z) \over \zeta -z}\,d\zeta .$
By Cauchy's theorem the right-hand side tends to 0 uniformly as ε, and hence δ, tends to 0. So
$H_{\varepsilon }f\rightarrow if$
uniformly for polynomials. On the other hand, if u(z) = z it is immediate that
${\overline {H_{\varepsilon }f}}=-u^{-1}H_{\varepsilon }(u{\overline {f}}).$
Thus if f is a polynomial in z−1 without constant term
$H_{\varepsilon }f\rightarrow -if$ uniformly.
Define the Hilbert transform on the circle by
$H=i(2P-I).$
Thus if f is a trigonometric polynomial
$H_{\varepsilon }f\rightarrow Hf$ uniformly.
It follows that if f is any L2 function
$H_{\varepsilon }f\rightarrow Hf$ in the L2 norm.
This is an immediate consequence of the result for trigonometric polynomials once it is established that the operators Hε are uniformly bounded in operator norm. But on [–π,π]
$(1-e^{i\theta })^{-1}=[(1-e^{i\theta })^{-1}-i\theta ^{-1}]+i\theta ^{-1}.$
The first term is bounded on the whole of [–π,π], so it suffices to show that the convolution operators Sε defined by
$S_{\varepsilon }f(\varphi )=\int _{\varepsilon \leq |\theta |\leq \pi }f(\varphi -\theta )\theta ^{-1}\,d\theta $
are uniformly bounded. With respect to the orthonormal basis einθ convolution operators are diagonal and their operator norms are given by taking the supremum of the moduli of the Fourier coefficients. Direct computation shows that these all have the form
${\frac {1}{\pi }}\left|\int _{a}^{b}{\sin t \over t}\,dt\right|$
with 0 < a < b. These integrals are well known to be uniformly bounded.
It also follows that, for a continuous function f on the circle, Hεf converges uniformly to Hf, so in particular pointwise. The pointwise limit is a Cauchy principal value, written
$Hf=\mathrm {P.V.} \,{1 \over \pi }\int {f(\zeta ) \over \zeta -e^{i\varphi }}\,d\zeta .$
If f is just in L2 then Hεf converges to Hf pointwise almost everywhere. In fact define the Poisson operators on L2 functions by
$T_{r}\left(\sum a_{n}e^{in\theta }\right)=\sum r^{|n|}a_{n}e^{in\theta },$
for r < 1. Since these operators are diagonal, it is easy to see that Trf tends to f in L2 as r increases to 1. Moreover, as Lebesgue proved, Trf also tends pointwise to f at each Lebesgue point of f. On the other hand, it is also known that TrHf − H1 − r f tends to zero at each Lebesgue point of f. Hence H1 – r f tends pointwise to f on the common Lebesgue points of f and Hf and therefore almost everywhere.[3][4][5]
Results of this kind on pointwise convergence are proved more generally below for Lp functions using the Poisson operators and the Hardy–Littlewood maximal function of f.
The Hilbert transform has a natural compatibility with orientation-preserving diffeomorphisms of the circle.[6] Thus if H is a diffeomorphism of the circle with
$H(e^{i\theta })=e^{ih(\theta )},\,\,\,h(\theta +2\pi )=h(\theta )+2\pi ,$
then the operators
$H_{\varepsilon }^{h}f(e^{i\varphi })={\frac {1}{\pi }}\int _{|e^{ih(\theta )}-e^{ih(\varphi )}|\geq \varepsilon }{\frac {f(e^{i\theta })}{e^{i\theta }-e^{i\varphi }}}e^{i\theta }\,d\theta ,$
are uniformly bounded and tend in the strong operator topology to H. Moreover, if Vf(z) = f(H(z)), then VHV−1 − H is an operator with smooth kernel, so a Hilbert–Schmidt operator.
In fact if G is the inverse of H with corresponding function g(θ), then
$(VH_{\varepsilon }^{h}V^{-1}-H_{\varepsilon })f(e^{i\varphi })={1 \over \pi }\int _{|e^{i\theta }-e^{i\varphi }|\geq \varepsilon }\left[{g^{\prime }(\theta )e^{ig(\theta )} \over e^{ig(\theta )}-e^{ig(\varphi )}}-{e^{i\theta } \over e^{i\theta }-e^{i\varphi }}\right]\,f(e^{i\theta })\,d\theta .$
Since the kernel on the right hand side is smooth on T × T, it follows that the operators on the right hand side are uniformly bounded and hence so too are the operators Hεh. To see that they tend strongly to H, it suffices to check this on trigonometric polynomials. In that case
$H_{\varepsilon }^{h}f(\zeta )={1 \over \pi i}\int _{|H(z)-H(\zeta )|\geq \varepsilon }{\frac {f(z)}{z-\zeta }}dz={1 \over \pi i}\int _{|H(z)-H(\zeta )|\geq \varepsilon }{f(z)-f(\zeta ) \over z-\zeta }\,dz+{\frac {f(\zeta )}{\pi i}}\int _{|H(z)-H(\zeta )|\geq \varepsilon }{dz \over z-\zeta }.$
In the first integral the integrand is a trigonometric polynomial in z and ζ and so the integral is a trigonometric polynomial in ζ. It tends in L2 to the trigonometric polynomial
${1 \over \pi i}\int {f(z)-f(\zeta ) \over z-\zeta }\,dz.$
The integral in the second term can be calculated by the principle of the argument. It tends in L2 to the constant function 1, so that
$\lim _{\varepsilon \to 0}H_{\varepsilon }^{h}f(\zeta )=f(\zeta )+{1 \over \pi i}\int {f(z)-f(\zeta ) \over z-\zeta }\,dz,$
where the limit is in L2. On the other hand, the right hand side is independent of the diffeomorphism. Since for the identity diffeomorphism, the left hand side equals Hf, it too equals Hf (this can also be checked directly if f is a trigonometric polynomial). Finally, letting ε → 0,
$(VHV^{-1}-H)f(e^{i\varphi })={\frac {1}{\pi }}\int \left[{g^{\prime }(\theta )e^{ig(\theta )} \over e^{ig(\theta )}-e^{ig(\varphi )}}-{e^{i\theta } \over e^{i\theta }-e^{i\varphi }}\right]\,f(e^{i\theta })\,d\theta .$
The direct method of evaluating Fourier coefficients to prove the uniform boundedness of the operator Hε does not generalize directly to Lp spaces with 1 < p < ∞. Instead a direct comparison of Hεf with the Poisson integral of the Hilbert transform is used classically to prove this. If f has Fourier series
$f(e^{i\theta })=\sum _{n\in \mathbf {Z} }a_{n}e^{in\theta },$
its Poisson integral is defined by
$P_{r}f(e^{i\theta })=\sum _{n\in \mathbf {Z} }a_{n}r^{|n|}e^{in\theta }={1 \over 2\pi }\int _{0}^{2\pi }{(1-r^{2})f(e^{i\theta }) \over 1-2r\cos \theta +r^{2}}\,d\theta =K_{r}\star f(e^{i\theta }),$
where the Poisson kernel Kr is given by
$K_{r}(e^{i\theta })=\sum _{n\in \mathbf {Z} }r^{|n|}e^{in\theta }={1-r^{2} \over 1-2r\cos \theta +r^{2}}.$
In f is in Lp(T) then the operators Pr satisfy
$\|P_{r}f-f\|_{p}\rightarrow 0.$
In fact the Kr are positive so
$\|K_{r}\|_{1}={1 \over 2\pi }\int _{0}^{2\pi }K_{r}(e^{i\theta })\,d\theta =1.$
Thus the operators Pr have operator norm bounded by 1 on Lp. The convergence statement above follows by continuity from the result for trigonometric polynomials, where it is an immediate consequence of the formula for the Fourier coefficients of Kr.
The uniform boundedness of the operator norm of Hε follows because HPr − H1−r is given as convolution by the function ψr, where[7]
${\begin{aligned}\psi _{r}(e^{i\theta })&=1+{\frac {1-r}{1+r}}\cot \left({\tfrac {\theta }{2}}\right)K_{r}(e^{i\theta })\\&\leq 1+{\frac {1-r}{1+r}}\cot \left({\tfrac {1-r}{2}}\right)K_{r}(e^{i\theta })\end{aligned}}$
for 1 − r ≤ |θ| ≤ π, and, for |θ| < 1 − r,
$\psi _{r}(e^{i\theta })=1+{2r\sin \theta \over 1-2r\cos \theta +r^{2}}.$
These estimates show that the L1 norms ∫ |ψr| are uniformly bounded. Since H is a bounded operator, it follows that the operators Hε are uniformly bounded in operator norm on L2(T). The same argument can be used on Lp(T) once it is known that the Hilbert transform H is bounded in operator norm on Lp(T).
Hilbert transform on the real line
See also: Hilbert transform
As in the case of the circle, the theory for L2 functions is particularly easy to develop. In fact, as observed by Rosenblum and Devinatz, the two Hilbert transforms can be related using the Cayley transform.[8]
The Hilbert transform HR on L2(R) is defined by
${\widehat {H_{\mathbf {R} }f}}=\left(i\chi _{[0,\infty )}-i\chi _{(-\infty ,0]}\right){\widehat {f}},$
where the Fourier transform is given by
${\widehat {f}}(t)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-itx}\,dx.$
Define the Hardy space H2(R) to be the closed subspace of L2(R) consisting of functions for which the Fourier transform vanishes on the negative part of the real axis. Its orthogonal complement is given by functions for which the Fourier transform vanishes on the positive part of the real axis. It is the complex conjugate of H2(R). If PR is the orthogonal projection onto H2(R), then
$H_{\mathbf {R} }=i(2P_{\mathbf {R} }-I).$
The Cayley transform
$C(x)={x-i \over x+i}$
carries the extended real line onto the circle, sending the point at ∞ to 1, and the upper halfplane onto the unit disk.
Define the unitary operator from L2(T) onto L2(R) by
$Uf(x)=\pi ^{-1/2}(x+i)^{-1}f(C(x)).$
This operator carries the Hardy space of the circle H2(T) onto H2(R). In fact for |w| < 1, the linear span of the functions
$f_{w}(z)={\frac {1}{1-wz}}$
is dense in H2(T). Moreover,
$Uf_{w}(x)={\frac {1}{\sqrt {\pi }}}{\frac {1}{(1-w)(x-{\overline {z}})}}$
where
$z=C^{-1}({\overline {w}}).$
On the other hand, for z ∈ H, the linear span of the functions
$g_{z}(t)=e^{itz}\chi _{[0,\infty )}(t)$
is dense in L2((0,∞)). By the Fourier inversion formula, they are the Fourier transforms of
$h_{z}(x)={\widehat {g_{z}}}(-x)={i \over {\sqrt {2\pi }}}(x+z)^{-1},$
so the linear span of these functions is dense in H2(R). Since U carries the fw's onto multiples of the hz's, it follows that U carries H2(T) onto H2(R). Thus
$UH_{\mathbf {T} }U^{*}=H_{\mathbf {R} }.$
In Nikolski (1986), part of the L2 theory on the real line and the upper halfplane is developed by transferring the results from the circle and the unit disk. The natural replacements for concentric circles in the disk are lines parallel to the real axis in H. Under the Cayley transform, these correspond to circles in the disk that are tangent to the unit circle at the point one. The behaviour of functions in H2(T) on these circles is part of the theory of Carleson measures. However, the theory of singular integrals can be developed more easily by working directly on R.
H2(R) consists exactly of L2 functions f that arise of boundary values of holomorphic functions on H in the following sense:[9] f is in H2 provided that there is a holomorphic function F(z) on H such that the functions fy(x) = f(x + iy) for y > 0 are in L2 and fy tends to f in L2 as y → 0. In this case F is necessarily unique and given by Cauchy's integral formula:
$F(z)={1 \over 2\pi i}\int _{-\infty }^{\infty }{f(s) \over s-z}\,ds.$
In fact, identifying H2 with L2(0,∞) via the Fourier transform, for y > 0 multiplication by e−yt on L2(0,∞) induces a contraction semigroup Vy on H2. Hence for f in L2
${1 \over 2\pi i}\int _{-\infty }^{\infty }{f(s) \over s-z}\,ds={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(s){\widehat {g_{z}}}(s)\,ds={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f}}(s)g_{z}(s)\,ds=V_{y}Pf(x).$
If f is in H2, F(z) is holomorphic for Im z > 0, since the family of L2 functions gz depends holomorphically on z. Moreover, fy = Vyf tends to f in H2 since this is true for the Fourier transforms. Conversely if such an F exists, by Cauchy's integral theorem and the above identity applied to fy
$f_{y+t}=V_{t}Pf_{y}$
for t > 0. Letting t tend to 0, it follows that Pfy = fy, so that fy lies in H2. But then so too does the limit f. Since
$V_{t}f_{y}=f_{y+t}=V_{y}f_{t},$
uniqueness of F follows from
$f_{t}=\lim _{y\to 0}f_{y+t}=\lim _{y\to 0}V_{t}f_{y}=V_{t}f.$
For f in L2, the truncated Hilbert transforms are defined by
${\begin{aligned}H_{\varepsilon ,R}f(x)&={1 \over \pi }\int _{\varepsilon \leq |y-x|\leq R}{f(y) \over x-y}\,dy={1 \over \pi }\int _{\varepsilon \leq |y|\leq R}{f(x-y) \over y}\,dy\\H_{\varepsilon }f(x)&={1 \over \pi }\int _{|y-x|\geq \varepsilon }{f(y) \over x-y}\,dy={1 \over \pi }\int _{|y|\geq \varepsilon }{f(x-y) \over y}\,dy.\end{aligned}}$
The operators Hε,R are convolutions by bounded functions of compact support, so their operator norms are given by the uniform norm of their Fourier transforms. As before the absolute values have the form
${1 \over {\sqrt {2\pi }}}\left|\int _{a}^{b}{2\sin t \over t}\,dt\right|.$
with 0 < a < b, so the operators Hε,R are uniformly bounded in operator norm. Since Hε,Rf tends to Hεf in L2 for f with compact support, and hence for arbitrary f, the operators Hε are also uniformly bounded in operator norm.
To prove that Hε f tends to Hf as ε tends to zero, it suffices to check this on a dense set of functions. On the other hand,
${\overline {H_{\varepsilon }f}}=-H_{\varepsilon }({\overline {f}}),$
so it suffices to prove that Hεf tends to if for a dense set of functions in H2(R), for example the Fourier transforms of smooth functions g with compact support in (0,∞). But the Fourier transform f extends to an entire function F on C, which is bounded on Im(z) ≥ 0. The same is true of the derivatives of g. Up to a scalar these correspond to multiplying F(z) by powers of z. Thus F satisfies a Paley-Wiener estimate for Im(z) ≥ 0:[10]
$|F^{(m)}(z)|\leq K_{N,m}(1+|z|)^{-N}$
for any m, N ≥ 0. In particular, the integral defining Hεf(x) can be computed by taking a standard semicircle contour centered on x. It consists of a large semicircle with radius R and a small circle radius ε with the two portions of the real axis between them. By Cauchy's theorem, the integral round the contour is zero. The integral round the large contour tends to zero by the Paley-Wiener estimate. The integral on the real axis is the limit sought. It is therefore given as minus the limit on the small semicircular contour. But this is the limit of
${1 \over \pi }\int _{\Gamma }{F(z) \over z-x}\,dz.$
Where Γ is the small semicircular contour, oriented anticlockwise. By the usual techniques of contour integration, this limit equals if(x).[11] In this case, it is easy to check that the convergence is dominated in L2 since
$H_{\varepsilon }f(x)={\frac {1}{\pi }}\int _{|y-x|\geq \varepsilon }{\frac {f(y)-f(x)}{y-x}}\,dy={\frac {1}{\pi }}\int _{|y-x|\geq \varepsilon }\int _{0}^{1}f^{\prime }(x+t(y-x))\,dt\,dy$
so that convergence is dominated by
$G(x)={\frac {1}{2\pi }}\int _{0}^{1}\int _{-\infty }^{\infty }|f^{\prime }(x+ty)|\,dy$
which is in L2 by the Paley-Wiener estimate.
It follows that for f on L2(R)
$H_{\varepsilon }f\rightarrow Hf.$
This can also be deduced directly because, after passing to Fourier transforms, Hε and H become multiplication operators by uniformly bounded functions. The multipliers for Hε tend pointwise almost everywhere to the multiplier for H, so the statement above follows from the dominated convergence theorem applied to the Fourier transforms.
As for the Hilbert transform on the circle, Hεf tends to Hf pointwise almost everywhere if f is an L2 function. In fact, define the Poisson operators on L2 functions by
$T_{y}f(x)=\int _{-\infty }^{\infty }P_{y}(x-t)f(t)\,dt,$
where the Poisson kernel is given by
$P_{y}(x)={\frac {y}{\pi (x^{2}+y^{2})}}.$
for y > 0. Its Fourier transform is
${\widehat {P_{y}}}(t)=e^{-y|t|},$
from which it is easy to see that Tyf tends to f in L2 as y increases to 0. Moreover, as Lebesgue proved, Tyf also tends pointwise to f at each Lebesgue point of f. On the other hand, it is also known that TyHf – Hyf tends to zero at each Lebesgue point of f. Hence Hεf tends pointwise to f on the common Lebesgue points of f and Hf and therefore almost everywhere.[12][13] The absolute values of the functions Tyf − f and TyHf – Hyf can be bounded pointwise by multiples of the maximal function of f.[14]
As for the Hilbert transform on the circle, the uniform boundedness of the operator norms of Hε follows from that of the Tε if H is known to be bounded, since HTε − Hε is the convolution operator by the function
$g_{\varepsilon }(x)={\begin{cases}{\frac {x}{\pi (x^{2}+\varepsilon ^{2})}}&|x|\leq \varepsilon \\{\frac {x}{\pi (x^{2}+\varepsilon ^{2})}}-{\frac {1}{\pi x}}&|x|>\varepsilon \end{cases}}$
The L1 norms of these functions are uniformly bounded.
Riesz transforms in the complex plane
Main article: Riesz transform
The complex Riesz transforms R and R* in the complex plane are the unitary operators on L2(C) defined as multiplication by z/|z| and its conjugate on the Fourier transform of an L2 function f:
${\widehat {Rf}}(z)={{\overline {z}} \over |z|}{\widehat {f}}(z),\,\,\,{\widehat {R^{*}f}}(z)={z \over |z|}{\widehat {f}}(z).$
Identifying C with R2, R and R* are given by
$R=-iR_{1}+R_{2},\,\,\,R^{*}=-iR_{1}-R_{2},$
where R1 and R2 are the Riesz transforms on R2 defined below.
On L2(C), the operator R and its integer powers are unitary. They can also be expressed as singular integral operators:[15]
${R^{k}f(w)=\lim _{\varepsilon \to 0}\int _{|z-w|\geq \varepsilon }M_{k}(w-z)f(z)\,dx\,dy,}$
where
$M_{k}(z)={k \over 2\pi i^{k}}{z^{k} \over |z|^{k+2}}\,\,\,\,(k\geq 1),\,\,\,\,M_{-k}(z)={\overline {M_{k}(z)}}.$
Defining the truncated higher Riesz transforms as
${R_{\varepsilon }^{(k)}f(w)=\int _{|z-w|\geq \varepsilon }M_{k}(w-z)f(z)\,dx\,dy,}$
these operators can be shown to be uniformly bounded in operator norm. For odd powers this can be deduced by the method of rotation of Calderón and Zygmund, described below.[16] If the operators are known to be bounded in operator norm it can also be deduced using the Poisson operators.[17]
The Poisson operators Ts on R2 are defined for s > 0 by
${T_{s}f(x)={1 \over 2\pi }\int _{\mathbf {R} ^{2}}{sf(x) \over (|x-t|^{2}+s^{2})^{3/2}}\,dt.}$
They are given by convolution with the functions
${P_{s}(x)={s \over 2\pi (|x|^{2}+s^{2})^{3/2}}.}$
Ps is the Fourier transform of the function e− s|x|, so under the Fourier transform they correspond to multiplication by these functions and form a contraction semigroup on L2(R2). Since Py is positive and integrable with integral 1, the operators Ts also define a contraction semigroup on each Lp space with 1 < p < ∞.
The higher Riesz transforms of the Poisson kernel can be computed:
${R^{k}P_{s}(z)={k \over 2\pi i^{k}}{z^{k} \over (|z|^{2}+s^{2})^{k/2+1}}}$
for k ≥ 1 and the complex conjugate for − k. Indeed, the right hand side is a harmonic function F(x,y,s) of three variable and for such functions[18]
${T_{s_{1}}F(x,y,s_{2})=F(x,y,s_{1}+s_{2}).}$
As before the operators
${T_{\varepsilon }R^{k}-R_{\varepsilon }^{(k)}}$
are given by convolution with integrable functions and have uniformly bounded operator norms. Since the Riesz transforms are unitary on L2(C), the uniform boundedness of the truncated Riesz transforms implies that they converge in the strong operator topology to the corresponding Riesz transforms.
The uniform boundedness of the difference between the transform and the truncated transform can also be seen for odd k using the Calderón-Zygmund method of rotation.[19][20] The group T acts by rotation on functions on C via
${U_{\theta }f(z)=f(e^{i\theta }z).}$
This defines a unitary representation on L2(C) and the unitary operators Rθ commute with the Fourier transform. If A is a bounded operator on L2(R) then it defines a bounded operator A(1) on L2(C) simply by making A act on the first coordinate. With the identification L2(R2) = L2(R) ⊗ L2(R), A(1) = A ⊗ I. If φ is a continuous function on the circle then a new operator can be defined by
${B={1 \over 2\pi }\int _{0}^{2\pi }\varphi (\theta )U_{\theta }A^{(1)}U_{\theta }^{*}\,d\theta .}$
This definition is understood in the sense that
${(Bf,g)={1 \over 2\pi }\int _{0}^{2\pi }\varphi (\theta )(U_{\theta }A^{(1)}U_{\theta }^{*}f,g)\,d\theta }$
for any f, g in L2(C). It follows that
${\|B\|\leq {1 \over 2\pi }\int _{0}^{2\pi }|\varphi (\theta )|\cdot \|A\|\,d\theta .}$
Taking A to be the Hilbert transform H on L2(R) or its truncation Hε, it follows that
${\begin{aligned}R&={1 \over 2\pi }\int _{0}^{2\pi }e^{-i\theta }U_{\theta }H^{(1)}U_{\theta }^{*}\,d\theta ,\\R_{\varepsilon }&={1 \over 2\pi }\int _{0}^{2\pi }e^{-i\theta }U_{\theta }H_{\varepsilon }^{(1)}U_{\theta }^{*}\,d\theta .\end{aligned}}$
Taking adjoints gives a similar formula for R* and its truncation. This gives a second way to verify estimates of the norms of R, R* and their truncations. It has the advantage of being applicable also for Lp spaces.
The Poisson operators can also be used to show that the truncated higher Riesz transforms of a function tend to the higher Riesz transform at the common Lebesgue points of the function and its transform. Indeed, (RkTε − R(k)ε)f → 0 at each Lebesgue point of f; while (Rk − RkTε)f → 0 at each Lebesgue point of Rkf.[21]
Beurling transform in the complex plane
See also: Beltrami equation
Since
${{\overline {z}} \over z}=\left({{\overline {z}} \over |z|}\right)^{2},$
the Beurling transform T on L2 is the unitary operator equal to R2. This relation has been used classically in Vekua (1962) and Ahlfors (1966) to establish the continuity properties of T on Lp spaces. The results on the Riesz transform and its powers show that T is the limit in the strong operator topology of the truncated operators
$T_{\varepsilon }f(w)=-{\frac {1}{\pi }}\iint _{|z-w|\geq \varepsilon }{\frac {f(z)}{(w-z)^{2}}}dxdy.$
Accordingly, Tf can be written as a Cauchy principal value integral:
$Tf(w)=-{\frac {1}{\pi }}P.V.\iint {\frac {f(z)}{(w-z)^{2}}}dxdy=-{\frac {1}{\pi }}\lim _{\varepsilon \to 0}\iint _{|z-w|\geq \varepsilon }{\frac {f(z)}{(w-z)^{2}}}dx\,dy.$
From the description of T and T* on Fourier transforms, it follows that if f is smooth of compact support
${\begin{aligned}T(\partial _{z}f)&=\partial _{z}T(f),\\T(\partial _{\overline {z}}f)&=\partial _{\overline {z}}T(f).\end{aligned}}$
Like the Hilbert transform in one dimension, the Beurling transform has a compatibility with conformal changes of coordinate. Let Ω be a bounded region in C with smooth boundary ∂Ω and let φ be a univalent holomorphic map of the unit disk D onto Ω extending to a smooth diffeomorphism of the circle onto ∂Ω. If χΩ is the characteristic function of Ω, the operator can χΩTχΩ defines an operator T(Ω) on L2(Ω). Through the conformal map φ, it induces an operator, also denoted T(Ω), on L2(D) which can be compared with T(D). The same is true of the truncations Tε(Ω) and Tε(D).
Let Uε be the disk |z − w| < ε and Vε the region |φ(z) − φ(w)| < ε. On L2(D)
${\begin{aligned}T_{\varepsilon }(\Omega )f(w)&=-{\frac {1}{\pi }}\iint _{D\backslash V_{\varepsilon }}\left[{\varphi ^{\prime }(w)\varphi ^{\prime }(z) \over (\varphi (z)-\varphi (w))^{2}}f(z)\right]dx\,dy,\\T_{\varepsilon }(D)f(w)&=-{1 \over \pi }\iint _{D\backslash U_{\varepsilon }}{f(z) \over (z-w)^{2}}\,dx\,dy,\end{aligned}}$
and the operator norms of these truncated operators are uniformly bounded. On the other hand, if
$T_{\varepsilon }^{\prime }(D)f(w)=-{1 \over \pi }\iint _{D\backslash V_{\varepsilon }}{\frac {f(z)}{(z-w)^{2}}}dx\,dy,$
then the difference between this operator and Tε(Ω) is a truncated operator with smooth kernel K(w,z):
$K(w,z)=-{1 \over \pi }\left[{\varphi '(w)\varphi '(z) \over (\varphi (z)-\varphi (w))^{2}}-{1 \over (z-w)^{2}}\right].$
So the operators T′ε(D) must also have uniformly bounded operator norms. To see that their difference tends to 0 in the strong operator topology, it is enough to check this for f smooth of compact support in D. By Green's theorem[22]
$\left(T_{\varepsilon }(D)-T_{\varepsilon }^{\prime }(D)\right)f(w)={\frac {1}{\pi }}\iint _{U_{\varepsilon }}{\partial _{z}f(z) \over z-w}dx\,dy-{1 \over \pi }\iint _{V_{\varepsilon }}{\partial _{z}f(z) \over z-w}dx\,dy+{1 \over 2\pi i}\int _{\partial U_{\varepsilon }}{\frac {f(z)}{z-w}}d{\overline {z}}-{\frac {1}{2\pi i}}\int _{\partial V_{\varepsilon }}{f(z) \over z-w}\,d{\overline {z}}.$
All four terms on the right hand side tend to 0. Hence the difference T(Ω) − T(D) is the Hilbert–Schmidt operator with kernel K.
For pointwise convergence there is simple argument due to Mateu & Verdera (2006) showing that the truncated integrals converge to Tf precisely at its Lebesgue points, that is almost everywhere.[23] In fact T has the following symmetry property for f, g ∈ L2(C)
$\iint (Tf)g=-{1 \over \pi }\lim \int _{|z-w|\geq \varepsilon }{\frac {f(w)g(z)}{(w-z)^{2}}}=\iint f(Tg).$
On the other hand, if χ is the characteristic function of the disk D(z,ε) with centre z and radius ε, then
$T\chi (w)=-\varepsilon ^{2}{\frac {1-\chi (w)}{(w-z)^{2}}}.$
Hence
$T_{\varepsilon }(f)(z)={1 \over \pi \varepsilon ^{2}}\iint f(T\chi )={1 \over \pi \varepsilon ^{2}}\iint (Tf)\chi =\mathbf {Av} _{D(z,\varepsilon )}\,Tf.$
By the Lebesgue differentiation theorem, the right-hand side converges to Tf at the Lebesgue points of Tf.
Riesz transforms in higher dimensions
Main article: Riesz transform
For f in the Schwartz space of Rn, the jth Riesz transform is defined by
$R_{j}f(x)=c_{n}\lim _{\varepsilon \to 0}\int _{|y|\geq \varepsilon }f(x-y){y_{j} \over |y|^{n+1}}dy={\frac {c_{n}}{n-1}}\int \partial _{j}f(x-y){1 \over |y|^{n-1}}dy,$
where
$c_{n}=\Gamma \left({\tfrac {n+1}{2}}\right)\pi ^{-{\frac {n+1}{2}}}.$
Under the Fourier transform:
${\widehat {R_{j}f}}(t)={it_{j} \over |t|}{\widehat {f}}(t).$
Thus Rj corresponds to the operator ∂jΔ−1/2, where Δ = −∂12 − ⋯ −∂n2 denotes the Laplacian on Rn. By definition Rj is a bounded and skew-adjoint operator for the L2 norm and
$R_{1}^{2}+\cdots +R_{n}^{2}=-I.$
The corresponding truncated operators
$R_{j,\varepsilon }f(x)=c_{n}\int _{|y|\geq \varepsilon }f(x-y){y_{j} \over |y|^{n+1}}dy$
are uniformly bounded in the operator norm. This can either be proved directly or can be established by the Calderón−Zygmund method of rotations for the group SO(n).[24] This expresses the operators Rj and their truncations in terms of the Hilbert transforms in one dimension and its truncations. In fact if G = SO(n) with normalised Haar measure and H(1) is the Hilbert transform in the first coordinate, then
${\begin{aligned}R_{j}&=\int _{G}\varphi (g)gH^{(1)}g^{-1}\,dg,\\R_{j,\varepsilon }&=\int _{G}\varphi (g)gH_{\varepsilon }^{(1)}g^{-1}\,dg,\\R_{j,\varepsilon ,R}&=\int _{G}\varphi (g)gH_{\varepsilon ,R}^{(1)}g^{-1}\,dg.\end{aligned}}$
where φ(g) is the (1,j) matrix coefficient of g.
In particular for f ∈ L2, Rj,εf → Rjf in L2. Moreover, Rj,εf tends to Rj almost everywhere. This can be proved exactly as for the Hilbert transform by using the Poisson operators defined on L2(Rn) when Rn is regarded as the boundary of a halfspace in Rn+1. Alternatively it can be proved directly from the result for the Hilbert transform on R using the expression of Rj as an integral over G.[25][26]
The Poisson operators Ty on Rn are defined for y > 0 by[27]
$T_{y}f(x)=c_{n}\int _{\mathbf {R} ^{n}}{\frac {yf(x)}{\left(|x-t|^{2}+y^{2}\right)^{\frac {n+1}{2}}}}dt.$
They are given by convolution with the functions
$P_{y}(x)=c_{n}{\frac {y}{\left(|x|^{2}+y^{2}\right)^{\frac {n+1}{2}}}}.$
Py is the Fourier transform of the function e−y|x|, so under the Fourier transform they correspond to multiplication by these functions and form a contraction semigroup on L2(Rn). Since Py is positive and integrable with integral 1, the operators Ty also define a contraction semigroup on each Lp space with 1 < p < ∞.
The Riesz transforms of the Poisson kernel can be computed
$R_{j}P_{\varepsilon }(x)=c_{n}{\frac {x_{j}}{\left(|x|^{2}+\varepsilon ^{2}\right)^{\frac {n+1}{2}}}}.$
The operator RjTε is given by convolution with this function. It can be checked directly that the operators RjTε − Rj,ε are given by convolution with functions uniformly bounded in L1 norm. The operator norm of the difference is therefore uniformly bounded. We have (RjTε − Rj,ε)f → 0 at each Lebesgue point of f; while (Rj − RjTε)f → 0 at each Lebesgue point of Rjf. So Rj,εf → Rjf on the common Lebesgue points of f and Rjf.
Lp theory
Elementary proofs of M. Riesz theorem
The theorem of Marcel Riesz asserts that singular integral operators that are continuous for the L2 norm are also continuous in the Lp norm for 1 < p < ∞ and that the operator norms vary continuously with p.
Bochner's proof for Hilbert transform on the circle[28]
Once it is established that the operator norms of the Hilbert transform on Lp(T) are bounded for even integers, it follows from the Riesz–Thorin interpolation theorem and duality that they are bounded for all p with 1 < p < ∞ and that the norms vary continuously with p. Moreover, the arguments with the Poisson integral can be applied to show that the truncated Hilbert transforms Hε are uniformly bounded in operator norm and converge in the strong operator topology to H.
It is enough to prove the bound for real trigonometric polynomials without constant term:
$f\left(e^{i\theta }\right)=\sum _{m=1}^{N}a_{m}e^{im\theta }+a_{-m}e^{-im\theta },\qquad a_{-m}={\overline {a_{m}}}.$
Since f + iHf is a polynomial in eiθ without constant term
${\frac {1}{2\pi }}\int _{0}^{2\pi }(f+iHf)^{2n}\,d\theta =0.$
Hence, taking the real part and using Hölder's inequality:
$\|Hf\|_{2n}^{2n}\leq \sum _{k=0}^{n-1}{2n \choose 2k}\left|\left((Hf)^{2k},f^{2n-2k}\right)\right|\leq \sum _{k=0}^{n-1}{2n \choose 2k}\|Hf\|_{2n}^{2k}\cdot \|f\|_{2n}^{2n-2k}.$
So the M. Riesz theorem follows by induction for p an even integer and hence for all p with 1 < p < ∞.
Cotlar's proof for Hilbert transform on the line[29]
Once it is established that the operator norms of the Hilbert transform on Lp(R) are bounded when p is a power of 2, it follows from the Riesz–Thorin interpolation theorem and duality that they are bounded for all p with 1 < p < ∞ and that the norms vary continuously with p. Moreover, the arguments with the Poisson integral can be applied to show that the truncated Hilbert transforms Hε are uniformly bounded in operator norm and converge in the strong operator topology to H.
It is enough to prove the bound when f is a Schwartz function. In that case the following identity of Cotlar holds:
$(Hf)^{2}=f^{2}+2H(fH(f)).$
In fact, write f = f+ + f− according to the ±i eigenspaces of H. Since f ± iHf extend to holomorphic functions in the upper and lower half plane, so too do their squares. Hence
$f^{2}-(Hf)^{2}=\left(f_{+}+f_{-}\right)^{2}+\left(f_{+}-f_{-}\right)^{2}=2\left(f_{+}^{2}+f_{-}^{2}\right)=-2iH\left(f_{+}^{2}-f_{-}^{2}\right)=-2H(f(Hf)).$
(Cotlar's identity can also be verified directly by taking Fourier transforms.)
Hence, assuming the M. Riesz theorem for p = 2n,
$\|Hf\|_{2^{n+1}}^{2}=\left\|(Hf)^{2}\right\|_{2^{n}}\leq \left\|f^{2}\right\|_{2^{n}}+2\|H(fH(f))\|_{2^{n}}\leq \|f\|_{2^{n+1}}^{2}+2\|H\|_{2^{n}}\|f\|_{2^{n+1}}\|Hf\|_{2^{n+1}}.$
Since
$R^{2}>1+2\|H\|_{2^{n}}R$
for R sufficiently large, the M. Riesz theorem must also hold for p = 2n+1.
Exactly the same method works for the Hilbert transform on the circle.[30] The same identity of Cotlar is easily verified on trigonometric polynomials f by writing them as the sum of the terms with non-negative and negative exponents, i.e. the ±i eigenfunctions of H. The Lp bounds can therefore be established when p is a power of 2 and follow in general by interpolation and duality.
Calderón–Zygmund method of rotation
The method of rotation for Riesz transforms and their truncations applies equally well on Lp spaces for 1 < p < ∞. Thus these operators can be expressed in terms of the Hilbert transform on R and its truncations. The integration of the functions Φ from the group T or SO(n) into the space of operators on Lp is taken in the weak sense:
$\left(\int _{G}\Phi (x)\,dx\,f,g\right)=\int _{G}(\Phi (x)f,g)\,dx$
where f lies in Lp and g lies in the dual space Lq with 1/p + 1/q. It follows that Riesz transforms are bounded on Lp and that the differences with their truncations are also uniformly bounded. The continuity of the Lp norms of a fixed Riesz transform is a consequence of the Riesz–Thorin interpolation theorem.
Pointwise convergence
The proofs of pointwise convergence for Hilbert and Riesz transforms rely on the Lebesgue differentiation theorem, which can be proved using the Hardy-Littlewood maximal function.[31] The techniques for the simplest and best-known case, namely the Hilbert transform on the circle, are a prototype for all the other transforms. This case is explained in detail here.
Let f be in Lp(T) for p > 1. The Lebesgue differentiation theorem states that
${A(\varepsilon )={1 \over 2\varepsilon }\int _{x-\varepsilon }^{x+\varepsilon }|f(t)-f(x)|\,dt\to 0}$
for almost all x in T.[32][33][34] The points at which this holds are called the Lebesgue points of f. Using this theorem it follows that if f is an integrable function on the circle, the Poisson integral Trf tends pointwise to f at each Lebesgue point of f. In fact, for x fixed, A(ε) is a continuous function on [0,π]. Continuity at 0 follows because x is a Lebesgue point and elsewhere because, if h is an integrable function, the integral of |h| on intervals of decreasing length tends to 0 by Hölder's inequality.
Letting r = 1 − ε, the difference can be estimated by two integrals:
$2\pi |T_{r}f(x)-f(x)|=\int _{0}^{2\pi }|(f(x-y)-f(x))P_{r}(y)|\,dy\leq \int _{|y|\leq \varepsilon }+\int _{|y|\geq \varepsilon }.$
The Poisson kernel has two important properties for ε small
${\begin{aligned}\sup _{y\in [-\varepsilon ,\varepsilon ]}|P_{1-\varepsilon }(y)|&\leq \varepsilon ^{-1}.\\\sup _{y\notin (-\varepsilon ,\varepsilon )}|P_{1-\varepsilon }(y)|&\to 0.\end{aligned}}$
The first integral is bounded by A(ε) by the first inequality so tends to zero as ε goes to 0; the second integral tends to 0 by the second inequality.
The same reasoning can be used to show that T1 − εHf – Hεf tends to zero at each Lebesgue point of f.[35] In fact the operator T1 − εHf has kernel Qr + i, where the conjugate Poisson kernel Qr is defined by
${Q_{r}(\theta )={2r\sin \theta \over 1-2r\cos \theta +r^{2}}.}$
Hence
${2\pi |T_{1-\varepsilon }Hf(x)-H_{\varepsilon }f(x)|\leq \int _{|y|\leq \varepsilon }|f(x-y)-f(x)|\cdot |Q_{r}(y)|\,dy+\int _{|y|\geq \varepsilon }|f(x-y)-f(x)|\cdot |Q_{1}(y)-Q_{r}(y)|\,dy.}$
The conjugate Poisson kernel has two important properties for ε small
${\begin{aligned}\sup _{y\in [-\varepsilon ,\varepsilon ]}|Q_{1-\varepsilon }(y)|&\leq \varepsilon ^{-1}.\\\sup _{y\notin (-\varepsilon ,\varepsilon )}|Q_{1}(y)-Q_{1-\varepsilon }(y)|&\to 0.\end{aligned}}$
Exactly the same reasoning as before shows that the two integrals tend to 0 as ε → 0.
Combining these two limit formulas it follows that Hεf tends pointwise to Hf on the common Lebesgue points of f and Hf and therefore almost everywhere.[36][37][38]
Maximal functions
See also: Hardy–Littlewood maximal function
Much of the Lp theory has been developed using maximal functions and maximal transforms. This approach has the advantage that it also extends to L1 spaces in an appropriate "weak" sense and gives refined estimates in Lp spaces for p > 1. These finer estimates form an important part of the techniques involved in Lennart Carleson's solution in 1966 of Lusin's conjecture that the Fourier series of L2 functions converge almost everywhere.[39] In the more rudimentary forms of this approach, the L2 theory is given less precedence: instead there is more emphasis on the L1 theory, in particular its measure-theoretic and probabilistic aspects; results for other Lp spaces are deduced by a form of interpolation between L1 and L∞ spaces. The approach is described in numerous textbooks, including the classics Zygmund (1977) and Katznelson (1968). Katznelson's account is followed here for the particular case of the Hilbert transform of functions in L1(T), the case not covered by the development above. F. Riesz's proof of convexity, originally established by Hardy, is established directly without resorting to Riesz−Thorin interpolation.[40][41]
If f is an L1 function on the circle its maximal function is defined by[42]
${f^{*}(t)=\sup _{0<h\leq \pi }{1 \over 2h}\int _{t-h}^{t+h}|f(s)|\,ds.}$
f* is finite almost everywhere and is of weak L1 type. In fact for λ > 0 if
${E_{f}(\lambda )=\{x:\,|f(x)|>\lambda \},\,\,f_{\lambda }=\chi _{E(\lambda )}f,}$
then[43]
$m(E_{f^{*}}(\lambda ))\leq {8 \over \lambda }\int _{E_{f}(\lambda )}|f|\leq {8\|f\|_{1} \over \lambda },$
where m denotes Lebesgue measure.
The Hardy−Littlewood inequality above leads to a proof that almost every point x of T is a Lebesgue point of an integrable function f, so that
$\lim _{h\to 0}{\frac {\int _{x-h}^{x+h}|f(t)-f(x)|\,dt}{2h}}\to 0.$
In fact, let
$\omega (f)(x)=\limsup _{h\to 0}{\frac {\int _{x-h}^{x+h}|f(t)-f(x)|\,dt}{2h}}\leq f^{*}(x)+|f(x)|.$
If g is continuous, then the ω(g) =0, so that ω(f − g) = ω(f). On the other hand, f can be approximated arbitrarily closely in L1 by continuous g. Then, using Chebychev's inequality,
$m\{x:\,\omega (f)(x)>\lambda \}=m\{x:\,\omega (f-g)(x)>\lambda \}\leq m\{x:\,(f-g)^{*}(x)>\lambda \}+m\{x:\,|f(x)-g(x)|>\lambda \}\leq C\lambda ^{-1}\|f-g\|_{1}.$
The right-hand side can be made arbitrarily small, so that ω(f) = 0 almost everywhere.
The Poisson integrals of an L1 function f satisfy[44]
${|T_{r}f|\leq f^{*}.}$
It follows that Tr f tends to f pointwise almost everywhere. In fact let
${\Omega (f)=\limsup _{r\to 1}|T_{r}f-f|.}$
If g is continuous, then the difference tends to zero everywhere, so Ω(f − g) = Ω(f). On the other hand, f can be approximated arbitrarily closely in L1 by continuous g. Then, using Chebychev's inequality,
$m\{x:\,\Omega (f)(x)>\lambda \}=m\{x:\,\Omega (f-g)(x)>\lambda \}\leq m\{x:\,(f-g)^{*}(x)>\lambda \}+m\{x:\,|f(x)-g(x)|>\lambda \}\leq C\lambda ^{-1}\|f-g\|_{1}.$
The right-hand side can be made arbitrarily small, so that Ω(f) = 0 almost everywhere. A more refined argument shows that convergence occurs at each Lebesgue point of f.
If f is integrable the conjugate Poisson integrals are defined and given by convolution by the kernel Qr. This defines Hf inside |z| < 1. To show that Hf has a radial limit for almost all angles,[45] consider
${F(z)=\exp(-f(z)-iHf(z)),}$
where f(z) denotes the extension of f by Poisson integral. F is holomorphic in the unit disk with |F(z)| ≤ 1. The restriction of F to a countable family of concentric circles gives a sequence of functions in L∞(T) which has a weak g limit in L∞(T) with Poisson integral F. By the L2 results, g is the radial limit for almost all angles of F. It follows that Hf(z) has a radial limit almost everywhere. This is taken as the definition of Hf on T, so that TrH f tends pointwise to H almost everywhere. The function Hf is of weak L1 type.[46]
The inequality used above to prove pointwise convergence for Lp function with 1 < p < ∞ make sense for L1 functions by invoking the maximal function. The inequality becomes
${|H_{\varepsilon }f-T_{1-\varepsilon }Hf|\leq 4f^{*}.}$
Let
${\omega (f)=\limsup _{\varepsilon \to 0}|H_{\varepsilon }f-T_{1-\varepsilon }Hf|.}$
If g is smooth, then the difference tends to zero everywhere, so ω(f − g) = ω(f). On the other hand, f can be approximated arbitrarily closely in L1 by smooth g. Then
$m\{x:\,\omega (f)(x)>\lambda \}=m\{x:\,\omega (f-g)(x)>\lambda \}\leq m\{x:\,4(f-g)^{*}(x)>\lambda \}\leq C\lambda ^{-1}\|f-g\|_{1}.$
The right hand side can be made arbitrarily small, so that ω(f) = 0 almost everywhere. Thus the difference for f tends to zero almost everywhere. A more refined argument can be given[47] to show that, as in case of Lp, the difference tends to zero at all Lebesgue points of f. In combination with the result for the conjugate Poisson integral, it follows that, if f is in L1(T), then Hεf converges to Hf almost everywhere, a theorem originally proved by Privalov in 1919.
General theory
Calderón & Zygmund (1952) introduced general techniques for studying singular integral operators of convolution type. In Fourier transform the operators are given by multiplication operators. These will yield bounded operators on L2 if the corresponding multiplier function is bounded. To prove boundedness on Lp spaces, Calderón and Zygmund introduced a method of decomposing L1 functions, generalising the rising sun lemma of F. Riesz. This method showed that the operator defined a continuous operator from L1 to the space of functions of weak L1. The Marcinkiewicz interpolation theorem and duality then implies that the singular integral operator is bounded on all Lp for 1 < p < ∞. A simple version of this theory is described below for operators on R. As de Leeuw (1965) showed, results on R can be deduced from corresponding results for T by restricting the multiplier to the integers, or equivalently periodizing the kernel of the operator. Corresponding results for the circle were originally established by Marcinkiewicz in 1939. These results generalize to Rn and Tn. They provide an alternative method for showing that the Riesz transforms, the higher Riesz transforms and in particular the Beurling transform define bounded operators on Lp spaces.[48]
Calderón-Zygmund decomposition
See also: Rising sun lemma and Calderón–Zygmund lemma
Let f be a non-negative integrable or continuous function on [a,b]. Let I = (a,b). For any open subinterval J of [a,b], let fJ denote the average of |f| over J. Let α be a positive constant greater than fI. Divide I into two equal intervals (omitting the midpoint). One of these intervals must satisfy fJ < α since their sum is 2fI so less than 2α. Otherwise the interval will satisfy α ≤ fJ < 2α. Discard such intervals and repeat the halving process with the remaining interval, discarding intervals using the same criterion. This can be continued indefinitely. The discarded intervals are disjoint and their union is an open set Ω. For points x in the complement, they lie in a nested set of intervals with lengths decreasing to 0 and on each of which the average of f is bounded by α. If f is continuous these averages tend to |f(x)|. If f is only integrable this is only true almost everywhere, for it is true at the Lebesgue points of f by the Lebesgue differentiation theorem. Thus f satisfies |f(x)| ≤ α almost everywhere on Ωc, the complement of Ω. Let Jn be the set of discarded intervals and define the "good" function g by
${g(x)=\chi _{J_{n}}(f)\,\,\,(x\in J_{n}),\,\,\,\,\,g(x)=f(x)\,\,\,(x\in \Omega ^{c}).}$
By construction |g(x)| ≤ 2α almost everywhere and
${\|g\|_{1}\leq \|f\|_{1}.}$
Combining these two inequalities gives
${\|g\|_{p}^{p}\leq (2\alpha )^{p-1}\|f\|_{1}.}$
Define the "bad" function b by b = f − g. Thus b is 0 off Ω and equal to f minus its average on Jn. So the average of b on Jn is zero and
${\|b\|_{1}\leq 2\|f\|_{1}.}$
Moreover, since |b| ≥ α on Ω
${m(\Omega )\leq \alpha ^{-1}\|f\|_{1}.}$
The decomposition
$\displaystyle {f(x)=g(x)+b(x)}$
is called the Calderón–Zygmund decomposition.[49]
Multiplier theorem
See also: Multiplier (Fourier analysis) and Marcinkiewicz interpolation theorem
Let K(x) be a kernel defined on R\{0} such that
$W(f)=\lim _{\varepsilon \to 0}\int _{|x|\geq \varepsilon }K(x)f(x)\,dx$
exists as a tempered distribution for f a Schwartz function. Suppose that the Fourier transform of T is bounded, so that convolution by W defines a bounded operator T on L2(R). Then if K satisfies Hörmander's condition
$A=\sup _{y\neq 0}\int _{|x|\geq 2|y|}|K(x-y)-K(x)|\,dx<\infty ,$
then T defines a bounded operator on Lp for 1 < p < ∞ and a continuous operator from L1 into functions of weak type L1.[50]
In fact by the Marcinkiewicz interpolation argument and duality, it suffices to check that if f is smooth of compact support then
$m\{x:\,|Tf(x)|\geq 2\lambda \}\leq (2A+4\|T\|)\cdot \lambda ^{-1}\|f\|_{1}.$
Take a Calderón−Zygmund decomposition of f as above
$f(x)=g(x)+b(x)$
with intervals Jn and with α = λμ, where μ > 0. Then
$m\{x:\,|Tf(x)|\geq 2\lambda \}\leq m\{x:\,|Tg(x)|\geq \lambda \}+m\{x:\,|Tb(x)|\geq \lambda \}.$
The term for g can be estimated using Chebychev's inequality:
$m\{x:\,|Tg(x)|\geq 2\lambda \}\leq \lambda ^{-2}\|Tg\|_{2}^{2}\leq \lambda ^{-2}\|T\|^{2}\|g\|_{2}^{2}\leq 2\lambda ^{-1}\mu \|T\|^{2}\|f\|_{1}.$
If J* is defined to be the interval with the same centre as J but twice the length, the term for b can be broken up into two parts:
$m\{x:\,|Tb(x)|\geq \lambda \}\leq m\{x:\,x\notin \cup J_{n}^{*},\,\,\,|Tb(x)|\geq \lambda \}+m(\cup J_{n}^{*}).$
The second term is easy to estimate:
$m(\cup J_{n}^{*})\leq \sum m(J_{n}^{*})=2\sum m(J_{n})\leq 2\lambda ^{-1}\mu ^{-1}\|f\|_{1}.$
To estimate the first term note that
$b=\sum b_{n},\qquad b_{n}=(f-\mathbf {Av} _{J_{n}}(f))\chi _{J_{n}}.$
Thus by Chebychev's inequality:
$m\{x:\,x\notin \cup J_{m}^{*},\,\,\,|Tb(x)|\geq \lambda \}\leq \lambda ^{-1}\int _{(\cup J_{m}^{*})^{c}}|Tb(x)|\,dx\leq \lambda ^{-1}\sum _{n}\int _{(J_{n}^{*})^{c}}|Tb_{n}(x)|\,dx.$
By construction the integral of bn over Jn is zero. Thus, if yn is the midpoint of Jn, then by Hörmander's condition:
$\int _{(J_{n}^{*})^{c}}|Tb_{n}(x)|\,dx=\int _{(J_{n}^{*})^{c}}\left|\int _{J_{n}}(K(x-y)-K(x-y_{n}))b_{n}(y)\,dy\right|\,dx\leq \int _{J_{n}}|b_{n}(y)|\int _{(J_{n}^{*})^{c}}|K(x-y)-K(x-y_{n})|\,dxdy\leq A\|b_{n}\|_{1}.$
Hence
$m\left\{x:\,x\notin \cup J_{m}^{*},|Tb(x)|\geq \lambda \right\}\leq \lambda ^{-1}A\|b\|_{1}\leq 2A\lambda ^{-1}\|f\|_{1}.$
Combining the three estimates gives
$m\{x:\,|Tf(x)|\geq \lambda \}\leq \left(2\mu \|T\|^{2}+2\mu ^{-1}+2A\right)\lambda ^{-1}\|f\|_{1}.$
The constant is minimized by taking
$\mu =\|T\|^{-1}.$
The Markinciewicz interpolation argument extends the bounds to any Lp with 1 < p < 2 as follows.[51] Given a > 0, write
$f=f_{a}+f^{a},$
where fa = f if |f| < a and 0 otherwise and fa = f if |f| ≥ a and 0 otherwise. Then by Chebychev's inequality and the weak type L1 inequality above
$m\{x:\,|Tf(x)|>a\}\leq m\left\{x:\,|Tf_{a}(x)|>{\tfrac {a}{2}}\right\}+m\left\{x:\,|Tf^{a}(x)|>{\tfrac {a}{2}}\right\}\leq 4a^{-2}\|T\|^{2}\|f_{a}\|_{2}^{2}+Ca^{-1}\|f^{a}\|_{1}.$
Hence
${\begin{aligned}\|Tf\|_{p}^{p}&=p\int _{0}^{\infty }a^{p-1}m\{x:\,|Tf(x)|>a\}\,da\\&\leq p\int _{0}^{\infty }a^{p-1}\left(4a^{-2}\|T\|^{2}\|f_{a}\|_{2}^{2}+Ca^{-1}\|f^{a}\|_{1}\right)da\\&=4\|T\|^{2}\iint _{|f(x)|<a}|f(x)|^{2}a^{p-3}\,dx\,da+2C\iint _{|f(x)|\geq a}|f(x)|a^{p-2}\,dx\,da\\&\leq \left(4\|T\|^{2}(2-p)^{-1}+C(p-1)^{-1}\right)\int |f|^{p}\\&=C_{p}\|f\|_{p}^{p}.\end{aligned}}$
By duality
$\|Tf\|_{q}\leq C_{p}\|f\|_{q}.$
Continuity of the norms can be shown by a more refined argument[52] or follows from the Riesz–Thorin interpolation theorem.
Notes
1. Torchinsky 2004, pp. 65–66
2. Bell 1992, pp. 14–15
3. Krantz 1999
4. Torchinsky 1986 harvnb error: no target: CITEREFTorchinsky1986 (help)
5. Stein & Rami 2005, pp. 112–114 harvnb error: no target: CITEREFSteinRami2005 (help)
6. See:
• Mikhlin & Prössdorf 1986
• Segal 1981
• Pressley & Segal 1986
7. Garnett 2007, p. 102
8. See:
• Devinatz 1967
• Rosenblum & Rovnyak 1997
• Rosenblum & Rovnyak 1994
• Nikolski 1986
9. Stein & Shakarchi 2005, pp. 213–221
10. Hörmander 1990
11. Titchmarsh, 1939 & 102–105 harvnb error: no target: CITEREFTitchmarsh1939102–105 (help)
12. See:
• Krantz 1999
• Torchinsky 1986 harvnb error: no target: CITEREFTorchinsky1986 (help)
• Duoandikoetxea 2001, pp. 49–51
13. Stein & Shakarchi 2005, pp. 112–114
14. Stein & Weiss 1971
15. Astala, Ivaniecz & Martin 2009, pp. 101–102 harvnb error: no target: CITEREFAstalaIvanieczMartin2009 (help)
16. Grafakos 2005 harvnb error: no target: CITEREFGrafakos2005 (help)
17. Stein & Weiss 1971
18. Stein & Weiss 1971, p. 51
19. Grafakos 2008
20. Stein & Weiss 1971, pp. 222–223
21. Stein & Weiss 1971
22. Astala, Iwaniecz & Martin 2009, pp. 93–95 harvnb error: no target: CITEREFAstalaIwanieczMartin2009 (help)
23. Astala, Iwaniecz & Martin 2009, pp. 97–98 harvnb error: no target: CITEREFAstalaIwanieczMartin2009 (help)
24. Grafokos 2008, pp. 272–274 harvnb error: no target: CITEREFGrafokos2008 (help)
25. Grafakos 2008
26. Stein & Weiss 1971, pp. 222–223, 236–237
27. Stein & Weiss 1971
28. Grafakos 2005, p. 215−216 harvnb error: no target: CITEREFGrafakos2005 (help)
29. Grafakos 2005, p. 255−257 harvnb error: no target: CITEREFGrafakos2005 (help)
30. Gohberg & Krupnik 1992, pp. 19–20
31. See:
• Stein & Weiss 1971, pp. 12–13
• Torchinsky 2004
32. Torchinsky 2005, pp. 41–42 harvnb error: no target: CITEREFTorchinsky2005 (help)
33. Katznelson 1968, pp. 10–21
34. Stein, Shakarchi & 112-114 harvnb error: no target: CITEREFSteinShakarchi112-114 (help)
35. Garnett 2007, pp. 102–103
36. Krantz 1999
37. Torchinsky 1986 harvnb error: no target: CITEREFTorchinsky1986 (help)
38. Stein & Shakarchi 2005, pp. 112–114
39. Arias de Reyna 2002
40. Duren 1970, pp. 8–10, 14
41. See also:
• Torchinsky 2005 harvnb error: no target: CITEREFTorchinsky2005 (help)
• Grafakos 2008
• Krantz 1999
42. Krantz 1999, p. 71
43. Katznelson 1968, pp. 74–75
44. Katznelson 1968, p. 76
45. Katznelson 1968, p. 64
46. Katznelson 1968, p. 66
47. Katznelson 2004, pp. 78–79 harvnb error: no target: CITEREFKatznelson2004 (help)
48. See:
• Hörmander 1990
• Torchinsky 2005 harvnb error: no target: CITEREFTorchinsky2005 (help)
• Grafakos 2008
• Stein 1970
• Stein & Weiss 1971, pp. 257–267
49. Torchinsky 2005, pp. 74–76, 84–85 harvnb error: no target: CITEREFTorchinsky2005 (help)
50. Grafakos 2008, pp. 290–293
51. Hörmander 1990, p. 245
52. Torchinsky 2005, pp. 87–91 harvnb error: no target: CITEREFTorchinsky2005 (help)
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• Stein, Elías M.; Weiss, Guido L. (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, ISBN 069108078X
• Stein, Elias M.; Shakarchi, Rami (2005), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton Lectures in Analysis, vol. 3, Princeton University Press, ISBN 0691113866
• Titchmarsh, E. C. (1939), The Theory of Functions (2nd ed.), Oxford University Press, ISBN 0198533497
• Torchinsky, Alberto (2004), Real-Variable Methods in Harmonic Analysis, Dover, ISBN 0-486-43508-3
• Vekua, I. N. (1962), Generalized analytic functions, Pergamon Press
• Zygmund, Antoni (1977), Trigonometric Series. Vol. I, II (2nd ed.), Cambridge University Press, ISBN 0-521-07477-0
• Zygmund, Antoni (1971), Intégrales singulières, Lecture Notes in Mathematics, vol. 204, Springer-Verlag
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Wikipedia
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Singular integral operators on closed curves
In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that of the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transform satisfies the jump relations of Plemelj and Sokhotski, which express the original function as the difference between the boundary values of holomorphic functions on the region and its complement. Singular integral operators have been studied on various classes of functions, including Hölder spaces, Lp spaces and Sobolev spaces. In the case of L2 spaces—the case treated in detail below—other operators associated with the closed curve, such as the Szegő projection onto Hardy space and the Neumann–Poincaré operator, can be expressed in terms of the Cauchy transform and its adjoint.
Operators on the unit circle
Main article: Singular integral operators of convolution type
If f is in L2(T), then it has a Fourier series expansion[1][2]
$\displaystyle {f(\theta )=\sum _{n\in {\mathbf {Z} }}a_{n}e^{in\theta }.}$
Hardy space H2(T) consists of the functions for which the negative coefficients vanish, an = 0 for n < 0. These are precisely the square-integrable functions that arise as boundary values of holomorphic functions in the unit disk |z| < 1. Indeed, f is the boundary value of the function
$\displaystyle {F(z)=\sum _{n\geq 0}a_{n}z^{n},}$
in the sense that the functions
$\displaystyle {f_{r}(\theta )=F(re^{i\theta })},$
defined by the restriction of F to the concentric circles |z| = r, satisfy
$\displaystyle {\|f_{r}-f\|_{2}\rightarrow 0.}$
The orthogonal projection P of L2(T) onto H2(T) is called the Szegő projection. It is a bounded operator on L2(T) with operator norm 1.
By Cauchy's theorem
$\displaystyle {F(z)={1 \over 2\pi i}\int _{|\zeta |=1}{f(\zeta ) \over \zeta -z}\,d\zeta ={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\theta ) \over 1-e^{-i\theta }z}\,d\theta .}$
Thus
$\displaystyle {F(re^{i\varphi })={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\varphi -\theta ) \over 1-re^{i\theta }}\,d\theta .}$
When r equals 1, the integrand on the right hand side has a singularity at θ = 0. The truncated Hilbert transform is defined by
$\displaystyle {H^{\varepsilon }f(\varphi )={i \over \pi }\int _{\varepsilon \leq |\theta |\leq \pi }{f(\varphi -\theta ) \over 1-e^{i\theta }}\,d\theta ={1 \over \pi }\int _{|\zeta -e^{i\varphi }|\geq \delta }{f(\zeta ) \over \zeta -e^{i\varphi }}\,d\zeta ,}$
where δ = |1 – eiε|. Since it is defined as convolution with a bounded function, it is a bounded operator on L2(T). Now
$\displaystyle {H^{\varepsilon }{1}={i \over \pi }\int _{\varepsilon }^{\pi }2\Re (1-e^{i\theta })^{-1}\,d\theta ={i \over \pi }\int _{\varepsilon }^{\pi }1\,d\theta =i-{i\varepsilon \over \pi }.}$
If f is a polynomial in z then
$\displaystyle {H^{\varepsilon }f(z)-{i(1-\varepsilon ) \over \pi }f(z)={1 \over \pi i}\int _{|\zeta -z|\geq \delta }{f(\zeta )-f(z) \over \zeta -z}\,d\zeta .}$
By Cauchy's theorem the right hand side tends to 0 uniformly as ε, and hence δ, tends to 0. So
$\displaystyle {H^{\varepsilon }f\rightarrow if}$
uniformly for polynomials. On the other hand, if u(z) = z it is immediate that
$\displaystyle {{\overline {H^{\varepsilon }f}}=-u^{-1}H^{\varepsilon }(u{\overline {f}}).}$
Thus if f is a polynomial in z−1 without constant term
$\displaystyle {H^{\varepsilon }f\rightarrow -if}$ uniformly.
Define the Hilbert transform on the circle by
$\displaystyle {H=i(2P-I).}$
Thus if f is a trigonometric polynomial
$\displaystyle {H^{\varepsilon }f\rightarrow Hf}$ uniformly.
It follows that if f is any L2 function
$\displaystyle {H^{\varepsilon }f\rightarrow Hf}$ in the L2 norm.
This is a consequence of the result for trigonometric polynomials since the Hε are uniformly bounded in operator norm: indeed their Fourier coefficients are uniformly bounded.
It also follows that, for a continuous function f on the circle, Hεf converges uniformly to Hf, so in particular pointwise. The pointwise limit is a Cauchy principal value, written
$\displaystyle {Hf=\mathrm {P.V.} \,{1 \over \pi }\int {f(\zeta ) \over \zeta -e^{i\varphi }}\,d\zeta .}$
The Hilbert transform has a natural compatibility with orientation-preserving diffeomorphisms of the circle.[3] Thus if H is a diffeomorphism of the circle with
$\displaystyle {H(e^{i\theta })=e^{ih(\theta )},\,\,\,h(\theta +2\pi )=h(\theta )+2\pi ,}$
then the operators
$\displaystyle {H_{h}^{\varepsilon }f(e^{i\varphi })={1 \over \pi }\int _{|e^{ih(\theta )}-e^{ih(\varphi )}|\geq \varepsilon }{f(e^{i\theta }) \over e^{i\theta }-e^{i\varphi }}\,e^{i\theta }\,d\theta ,}$
are uniformly bounded and tend in the strong operator topology to H. Moreover, if Vf(z) = f(H(z)), then VHV−1 – H is an operator with smooth kernel, so a Hilbert–Schmidt operator.
Hardy spaces
See also: Hardy space
The Hardy space on the unit circle can be generalized to any multiply connected bounded domain Ω with smooth boundary ∂Ω. The Hardy space H2(∂Ω) can be defined in a number of equivalent ways. The simplest way to define it is as the closure in L2(∂Ω) of the space of holomorphic functions on Ω which extend continuously to smooth functions on the closure of Ω. As Walsh proved, in a result that was a precursor of Mergelyan's theorem, any holomorphic function on Ω that extends continuously to the closure can be approximated in the uniform norm by a rational function with poles in the complementary region Ωc. If Ω is simply connected, then the rational function can be taken to be a polynomial. There is a counterpart of this theorem on the boundary, the Hartogs–Rosenthal theorem, which states that any continuous function ∂Ω can be approximated in the uniform norm by rational functions with poles in the complement of ∂Ω. It follows that for a simply connected domain when ∂Ω is a simple closed curve, H2(∂Ω) is just the closure of the polynomials; in general it is the closure of the space of rational functions with poles lying off ∂Ω.[4]
On the unit circle an L2 function f with Fourier series expansion
$\displaystyle {f(e^{i\theta })=\sum a_{n}e^{in\theta }}$
has a unique extension to a harmonic function in the unit disk given by the Poisson integral
$\displaystyle {f(re^{i\theta })=P_{r}f(e^{i\theta })=\sum a_{n}r^{|n|}e^{in\theta }.}$
In particular
$\displaystyle {\|P_{r}f\|^{2}=\sum |a_{n}|^{2}r^{2|n|},}$
so that the norms increase to the value at r = 1, the norm of f. A similar in the complement of the unit disk where the harmonic extension is given by
$\displaystyle {F_{R}(e^{i\theta })=F(Re^{i\theta })=\sum a_{n}R^{-|n|}a_{n}e^{in\theta }.}$
In this case the norms increase from the value at R = ∞ to the norm of f, the value at R = 1.
A similar result holds for a harmonic function f on a simply connected region with smooth boundary provided the L2 norms are taken over the level curves in a tubular neighbourhood of the boundary.[5] Using vector notation v(t) = (x(t), y(t)) to parametrize the boundary curve by arc length, the following classical formulas hold:
$\displaystyle {{\dot {\mathbf {v} }}\cdot {\dot {\mathbf {v} }}=1,\,\,\,{\ddot {\mathbf {v} }}\cdot {\dot {\mathbf {v} }}=0.}$
Thus the unit tangent vector t(t) at t and oriented normal vector n(t) are given by
$\displaystyle {\mathbf {t} ={\dot {\mathbf {v} }},\,\,\,\,\,\,\mathbf {n} =(-{\dot {y}},{\dot {x}}).}$
The constant relating the acceleration vector to the normal vector is the curvature of the curve:
$\displaystyle {{\ddot {\mathbf {v} }}=\kappa (t)\,\mathbf {n} (t),\,\,\,\,\,\kappa (t)={\ddot {\mathbf {v} }}\cdot \mathbf {n} ={\ddot {y}}{\dot {x}}-{\ddot {x}}{\dot {y}}.}$
There are two further formulas of Frenet:
$\displaystyle {{\dot {\mathbf {n} }}=-\kappa \mathbf {t} ,\,\,\,{\ddot {\mathbf {n} }}={\dot {\kappa }}\mathbf {n} -\kappa ^{2}\mathbf {t} .}$
A tubular neighbourhood of the boundary is given by
$\displaystyle {\mathbf {v} _{s}(t)=\mathbf {v} (t)+s\mathbf {n} (t),}$
so that the level curves ∂Ωs with s constant bound domains Ωs. Moreover[6]
$\displaystyle {{\dot {\mathbf {v} }}_{s}(t)=(1-s\kappa )\mathbf {t} ,\,\,\,\,\partial _{s}|{\dot {\mathbf {v} }}_{s}|=-\kappa .}$
Hence differentiating the integral means with respect to s, the derivative in the direction of the inward pointing normal, gives
$\displaystyle {\partial _{s}\int _{\partial \Omega _{s}}|f|^{2}=-\int _{\partial \Omega _{s}}(\partial _{n}f{\overline {f}}+f{\overline {\partial _{n}f}})-\int _{\partial \Omega _{s}}\kappa (1-\kappa s)^{-1}|f|^{2}=-2\iint _{\Omega _{s}}|\nabla f|^{2}-\int _{\partial \Omega _{s}}\kappa (1-\kappa s)^{-1}|f|^{2},}$
using Green's theorem. Thus for s small
$\displaystyle {\partial _{s}\|f|_{\partial \Omega _{s}}\|^{2}\leq M\|f|_{\partial \Omega _{s}}\|^{2},}$
for some constant M independent of f. This implies that
$\displaystyle {\partial _{s}e^{-Ms}\|f|_{\partial \Omega _{s}}\|^{2}\leq 0,}$
so that, on integrating this inequality, the norms are bounded near the boundary:
$\displaystyle {\|f|_{\partial \Omega _{s}}\|\leq e^{Ms/2}\|f|_{\partial \Omega }\|.}$
This inequality shows that a function in the L2 Hardy space H2(Ω) leads, via the Cauchy integral operator C, to a holomorphic function on Ω satisfying the classical condition that the integral means
$\displaystyle {\int _{\partial \Omega _{s}}|f|^{2}}$
are bounded. Furthermore, the restrictions fs of f to ∂Ωs, which can be naturally identified with ∂Ω, tend in L2 to the original function in Hardy space.[7] In fact H2(Ω) has been defined as the closure in L2(Ω) of rational functions (which can be taken to be polynomials if Ω is simply connected). Any rational function with poles only in Ωc can be recovered inside Ω from its boundary value g by Cauchy's integral formula
$\displaystyle {Cg(a)={1 \over 2\pi i}\int _{\partial \Omega }{g(z) \over z-a}\,dz.}$
The estimates above show that the functions Cg|∂Ωs depend continuously on Cg|∂Ω. Moreover, in this case the functions tend uniformly to the boundary value and hence also in L2, using the natural identification of the spaces L2(∂Ωs) with L2(∂Ω). Since Ch can be defined for any L2 function as a holomorphic function on Ω since h is integrable on ∂Ω. Since h is a limit in L2 of rational functions g, the same results hold for h and Ch, with the same inequalities for the integral means. Equally well h is the limit in L2(∂Ω) of the functions Ch|∂Ωs.
The estimates above for the integral means near the boundary show that Cf lies in L2(Ω) and that its L2 norm can be bounded in terms of that of f. Since Cf is also holomorphic, it lies in the Bergman space A2(Ω) of Ω. Thus the Cauchy integral operator C defines a natural mapping from the Hardy space of the boundary into the Bergman space of the interior.[8]
The Hardy space H2(Ω) has a natural partner, namely the closure in L2(∂Ω) of boundary values of rational functions vanishing at ∞ with poles only in Ω. Denoting this subspace by H2+(∂Ω) to distinguish it from the original Hardy space, which will also denoted by H2−(∂Ω), the same reasoning as above can be applied. When applied to a function h in H2+(∂Ω), the Cauchy integral operator defines a holomorphic function F in Ωc vanishing at ∞ such that near the boundary the restriction of F to the level curves, each identified with the boundary, tend in L2 to h. Unlike the case of the circle, H2−(∂Ω) and H2+(∂Ω) are not orthogonal spaces. By the Hartogs−Rosenthal theorem, their sum is dense in L2(∂Ω). As shown below, these are the ±i eigenspaces of the Hilbert transform on ∂Ω, so their sum is in fact direct and the whole of L2(∂Ω).
Hilbert transform on a closed curve
For a bounded simply connected domain Ω in the complex plane with smooth boundary ∂Ω, the theory of the Hilbert transform can be deduced by direct comparison with the Hilbert transform for the unit circle.[9]
To define the Hilbert transform H∂Ω on L2(∂Ω), take ∂Ω to be parametrized by arclength and thus a function z(t). The Hilbert transform is defined to be the limit in the strong operator topology of the truncated operators H∂Ωε defined by
$\displaystyle {H_{\partial \Omega }^{\varepsilon }f(s)={1 \over \pi i}\int _{|s-t|\geq \varepsilon }\,\,\,\,{f(t) \over z(t)-z(s)}\,{\dot {z}}(t)\,dt.}$
To make the comparison it will be convenient to apply a scaling transformation in C so that the length of ∂Ω is 2π. (This only changes the operators above by a fixed positive factor.) There is then a canonical unitary isomorphism of L2(∂Ω) onto L2(T), so the two spaces can be identified. The truncated operators H∂Ωε can be compared directly with the truncated Hilbert transform Hε:
$\displaystyle {H_{\partial \Omega }^{\varepsilon }g(s)-H^{\varepsilon }g(s)={1 \over \pi i}\int _{|t-s|\geq \varepsilon }K(s,t)\cdot g(t)\,dt,}$
where
$\displaystyle {K(u,v)={{\dot {z}}(t) \over z(t)-z(s)}-{ie^{it} \over e^{it}-e^{is}}=\partial _{t}\log \left({z(t)-z(s) \over e^{it}-e^{is}}\right).}$
The kernel K is thus smooth on T × T, so the difference above tends in the strong topology to the Hilbert–Schmidt operator defined by the kernel. It follows that the truncated operators H∂Ωε are uniformly bounded in norm and have a limit in the strong operator topology denoted H∂Ω and called the Hilbert transform on ∂Ω.
$\displaystyle {H_{\partial \Omega }g=\lim _{\varepsilon \rightarrow 0}H_{\partial \Omega }^{\varepsilon }g.}$
Letting ε tend to 0 above yields
$\displaystyle {H_{\partial \Omega }g(s)-Hg(s)={1 \over \pi i}\int _{0}^{2\pi }K(s,t)\cdot g(t)\,dt.}$
Since H is skew-adjoint and H∂Ω differs from H by a Hilbert–Schmidt operator with smooth kernel, it follows that H∂Ω + H∂Ω* is a Hilbert-Schmidt operator with smooth kernel. The kernel can also be computed explicitly using the truncated Hilbert transforms for ∂Ω:
$\displaystyle {C(s,t)={1 \over \pi i}\left({{\dot {z}}(t) \over z(t)-z(s)}-{{\overline {{\dot {z}}(s)}} \over {\overline {z(t)}}-{\overline {z(s)}}}\right),}$
and it can be verified directly that this is a smooth function on T × T.[10]
Plemelj–Sokhotski relation
See also: Sokhotski–Plemelj theorem
Let C− and C+ be the Cauchy integral operators for Ω and Ωc. Then
$\displaystyle {C_{\pm }\circ H=\pm iC_{\pm }.}$
Since the operators C−, C+ and H are bounded, it suffices to check this on rational functions F with poles off ∂Ω and vanishing at ∞ by the Hartogs–Rosenthal theorem. The rational function can be written as a sum of functions F = F− + F+ where F− has poles only in Ωc and F+ has poles only in Let f, f± be the restrictions of f, f± to ∂Ω. By Cauchy's integral formula
$\displaystyle {C_{\pm }f_{\pm }=F_{\pm },\,\,\,C_{\pm }f_{\mp }=0.}$
On the other hand, it is straightforward to check that[11]
$\displaystyle {Hf_{\pm }=\pm if_{\pm }.}$
Indeed, by Cauchy's theorem, since F− is holomorphic in Ω,
$\displaystyle {\int _{\partial \Omega ,\,\,|z-w|\geq \varepsilon }{f_{-}(z) \over z-w}\,dz=-\int _{|z-w|=\varepsilon ,\,\,z\in \Omega }{F_{-}(z) \over z-w}\,dz.}$
As ε tends to 0, the latter integral tends to πi f−(w) by the residue calculus. A similar argument applies to f+, taking the circular contour on the right inside Ωc.[12]
By continuity it follows that H acts as multiplication by i on H2− and as multiplication by −i on H2+. Since these spaces are closed and their sum dense, it follows that
$\displaystyle {H^{2}=-I.}$
Moreover, H2− and H2+ must be the ±i eigenspaces of H, so their sum is the whole of L2(∂Ω). The Plemelj–Sokhotski relation for f in L2(∂Ω) is the relation
$\displaystyle {-iHf=C_{-}f|_{\partial \Omega }-C_{+}f|_{\partial \Omega }.}$
It has been verified for f in the Hardy spaces H2±(∂Ω), so is true also for their sum. The Cauchy idempotent E is defined by
$\displaystyle {H=i(2E-I).}$
The range of E is thus H2−(∂Ω) and that of I − E is H2+(∂Ω). From the above[13]
$\displaystyle {Ef=C_{-}f|_{\partial \Omega },\,\,\,\,(I-E)f=C_{+}f|_{\partial \Omega }.}$
Operators on a closed curve
See also: Szegő kernel and Neumann–Poincaré operator
Two other operators defined on a closed curve ∂Ω can be expressed in terms of the Hilbert and Cauchy transforms H and E. [14]
The Szegő projection P is defined to be the orthogonal projection onto Hardy space H2(∂Ω). Since E is an idempotent with range H2(∂Ω), P is given by the Kerzman–Stein formula:
$\displaystyle {P=E(I+E-E^{*})^{-1}.}$
Indeed, since E − E* is skew-adjoint its spectrum is purely imaginary, so the operator I + E − E* is invertible.[15] It is immediate that
$\displaystyle {EP=P,\,\,\,PE=E.}$
Hence PE* = P. So
$\displaystyle {P(I+E-E^{*})=P+E-P=E.}$
Since the operator H + H* is a Hilbert–Schmidt operator with smooth kernel, the same is true for E − E*.[16]
Moreover, if J is the conjugate-linear operator of complex conjugation and U the operator of multiplication by the unit tangent vector:
$\displaystyle {Jf(t)={\overline {f(t)}},\,\,\,Uf(t)={\dot {z}}(t)\cdot f(t).}$
then the formula for the truncated Hilbert transform on ∂Ω immediately yields the following identity for adjoints
$\displaystyle {(H^{\varepsilon })^{*}=JUH^{\varepsilon }U^{*}J.}$
Letting ε tend to 0, it follows that
$\displaystyle {H^{*}=JUHU^{*}J}$
and hence
$\displaystyle {I-E^{*}=JUEU^{*}J.}$
The comparison with the Hilbert transform for the circle shows that commutators of H and E with diffeomorphisms of the circle are Hilbert–Schmidt operators. Similar their commutators with the multiplication operator corresponding to a smooth function f on the circle is also Hilbert–Schmidt operators. Up to a constant the kernel of the commutator with H is given by the smooth function
$\displaystyle {A(s,t)={f(s)-f(t) \over z(s)-z(t)}.}$
The Neumann–Poincaré operator T is defined on real functions f as
$\displaystyle {Tf(w)={1 \over 2\pi }\int _{\partial \Omega }\partial _{n}(\log |z-w|)f(z)={1 \over 2}\Re (Hf)(w).}$
Writing h = f + ig,[17]
$\displaystyle {2Th=\Re (Hf)+i\Re (Hg)={1 \over 2}(Hf+JHf+iHg+iJHg)={1 \over 2}(H+JHJ)h}$
so that
$\displaystyle {T={1 \over 4}(H+JHJ)={1 \over 4}(H+UH^{*}U^{*}),}$
a Hilbert–Schmidt operator.
Classical definition of Hardy space
The classical definition of Hardy space is as the space of holomorphic functions F on Ω for which the functions Fs = F|∂Ωs have bounded norm in L2(∂Ω). An argument based on the Carathéodory kernel theorem shows that this condition is satisfied whenever there is a family of Jordan curves in Ω, eventually containing any compact subset in their interior, on which the integral means of F are bounded.[18]
To prove that the classical definition of Hardy space gives the space H2(∂Ω), take F as above. Some subsequence hn = Fsn converges weakly in L2(∂Ω) to h say. It follows that Ch = F in Ω. In fact, if Cn is the Cauchy integral operator corresponding to Ωsn, then[19]
$\displaystyle {Ch(a)-F(a)=Ch(a)-C_{n}h_{n}(a)=C(h-h_{n})(a)+[(C-C_{n})h_{n}](a).}$
Since the first term on the right hand side is defined by pairing h − hn with a fixed L2 function, it tends to zero. If zn(t) is the complex number corresponding to vsn, then
$\displaystyle {[(C-C_{n})h_{n}](a)={1 \over 2\pi i}\int h_{n}(t)\left({{\dot {z}}(t) \over z(t)-a}-{{\dot {z}}_{n}(t) \over z_{n}(t)-a}\right)\,dt.}$
This integral tends to zero because the L2 norms of hn are uniformly bounded while the bracketed expression in the integrand tends to 0 uniformly and hence in L2.
Thus F = Ch. On the other hand, if E is the Cauchy idempotent with range H2(∂Ω), then C ∘ E = C. Hence F =Ch = C (Eh). As already shown Fs tends to Ch in L2(∂Ω). But a subsequence tends weakly to h. Hence Ch = h and therefore the two definitions are equivalent.[20]
Generalizations
The theory for multiply connected bounded domains with smooth boundary follows easily from the simply connected case.[21] There are analogues of the operators H, E and P. On a given component of the boundary, the singular contributions to H and E come from the singular integral on that boundary component, so the technical parts of the theory are direct consequences of the simply connected case.
Singular integral operators on spaces of Hölder continuous functions are discussed in Gakhov (1990). Their action on Lp and Sobolev spaces is discussed in Mikhlin & Prössdorf (1986).
Notes
1. Torchinsky 2004, pp. 65–66
2. Bell 1992, pp. 14–15
3. See:
• Mikhlin & Prössdorf 1986
• Segal 1981
• Pressley & Segal 1986
4. See:
• Bell 1992
• Gamelin 2005
• Duren 1970
• Conway 1995
• Conway 2000
5. Bell 1992, pp. 19–20
6. Bell 1992, pp. 19–22
7. Bell 1992, pp. 16–21
8. Bell 1992, p. 22
9. See:
• Gohberg & Krupnik 1992
• Gakhov 1990
10. Bell 1992, pp. 15–16
11. See:
• Gakhov 1990
• Bell 1992
• Goluzin 1969
• Gohberg & Krupnik 1992
12. Titchmarsh 1939
13. Bell 1992
14. See:
• Kerzman & Stein 1978
• Bell 1992
• Shapiro 1992
15. Shapiro 1992, p. 65
16. Bell 1992
17. Shapiro 1992, pp. 66–67
18. Duren 1970, p. 168
19. Bell 1992, pp. 17–18
20. Bell 1992, pp. 19–20
21. See:
• Gakhov 1990
• Mikhlin & Prössdorf 1986
References
• Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X
• Bell, S. R. (2016), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics (2nd ed.), CRC Press, ISBN 9781498727211
• Conway, John B. (1995), Functions of one complex variable II, Graduate texts in mathematics, vol. 159, Springer, p. 197, ISBN 0387944605
• Conway, John B. (2000), A course in operator theory, Graduate Studies in Mathematics, vol. 21, American Mathematical Society, pp. 175–176, ISBN 0821820656
• David, Guy (1984), "Opérateurs intégraux singuliers sur certaines courbes du plan complexe", Ann. Sci. École Norm. Sup., 17: 157–189, doi:10.24033/asens.1469
• Duren, Peter L. (1970), Theory of Hp spaces, Pure and Applied Mathematics, vol. 38, Academic Press
• Gakhov, F. D. (1990), Boundary value problems. Reprint of the 1966 translation, Dover Publications, ISBN 0-486-66275-6
• Gamelin, Theodore W. (2005), Uniform algebras (2nd ed.), American Mathematical Society, pp. 46–47, ISBN 0821840495
• Garnett, J. B. (2007), Bounded analytic functions, Graduate Texts in Mathematics, vol. 236, Springer, ISBN 978-0-387-33621-3
• Gohberg, Israel; Krupnik, Naum (1992), One-dimensional linear singular integral equations. I. Introduction, Operator Theory: Advances and Applications, vol. 53, Birkhäuser, ISBN 3-7643-2584-4
• Goluzin, G. M. (1969), Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, vol. 26, American Mathematical Society
• Katznelson, Yitzhak (2004), An Introduction to Harmonic Analysis, Cambridge University Press, ISBN 978-0-521-54359-0
• Kerzman, N.; Stein, E. M. (1978), "The Cauchy kernel, the Szegö kernel, and the Riemann mapping function", Math. Ann., 236: 85–93, doi:10.1007/bf01420257, S2CID 121336615
• Muskhelishvili, N. I. (1992), Singular integral equations. Boundary problems of function theory and their application to mathematical physics, Dover, ISBN 0-486-66893-2
• Mikhlin, Solomon G.; Prössdorf, Siegfried (1986), Singular integral operators, Springer-Verlag, ISBN 3-540-15967-3
• Pressley, Andrew; Segal, Graeme (1986), Loop groups, Oxford University Press, ISBN 0-19-853535-X
• Segal, Graeme (1981), "Unitary representations of some infinite-dimensional groups", Comm. Math. Phys., 80 (3): 301–342, doi:10.1007/bf01208274, S2CID 121367853
• Shapiro, H. S. (1992), The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9, Wiley-Interscience, ISBN 0-471-57127-X
• Titchmarsh, E. C. (1939), The Theory of Functions (2nd ed.), Oxford University Press, ISBN 0-19-853349-7
• Torchinsky, Alberto (2004), Real-Variable Methods in Harmonic Analysis, Dover, ISBN 0-486-43508-3
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Wikipedia
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Singular measure
In mathematics, two positive (or signed or complex) measures $\mu $ and $\nu $ defined on a measurable space $(\Omega ,\Sigma )$ are called singular if there exist two disjoint measurable sets $A,B\in \Sigma $ whose union is $\Omega $ such that $\mu $ is zero on all measurable subsets of $B$ while $\nu $ is zero on all measurable subsets of $A.$ This is denoted by $\mu \perp \nu .$
A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples.
Examples on Rn
As a particular case, a measure defined on the Euclidean space $\mathbb {R} ^{n}$ is called singular, if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure.
Example. A discrete measure.
The Heaviside step function on the real line,
$H(x)\ {\stackrel {\mathrm {def} }{=}}{\begin{cases}0,&x<0;\\1,&x\geq 0;\end{cases}}$
has the Dirac delta distribution $\delta _{0}$ as its distributional derivative. This is a measure on the real line, a "point mass" at $0.$ However, the Dirac measure $\delta _{0}$ is not absolutely continuous with respect to Lebesgue measure $\lambda ,$ nor is $\lambda $ absolutely continuous with respect to $\delta _{0}:$ $\lambda (\{0\})=0$ but $\delta _{0}(\{0\})=1;$ if $U$ is any open set not containing 0, then $\lambda (U)>0$ but $\delta _{0}(U)=0.$
Example. A singular continuous measure.
The Cantor distribution has a cumulative distribution function that is continuous but not absolutely continuous, and indeed its absolutely continuous part is zero: it is singular continuous.
Example. A singular continuous measure on $\mathbb {R} ^{2}.$
The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
See also
• Absolute continuity (measure theory) – Form of continuity for functionsPages displaying short descriptions of redirect targets
• Lebesgue's decomposition theorem
• Singular distribution – distribution concentrated on a set of measure zeroPages displaying wikidata descriptions as a fallback
References
• Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. ISBN 1-58488-347-2.
• J Taylor, An Introduction to Measure and Probability, Springer, 1996. ISBN 0-387-94830-9.
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
This article incorporates material from singular measure on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia
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Complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers.[1] Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
This article is about a topic in the theory of elliptic curves. For information about multiplication of complex numbers, see complex numbers.
It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.[2]
There is also the higher-dimensional complex multiplication theory of abelian varieties A having enough endomorphisms in a certain precise sense, roughly that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules.
Example of the imaginary quadratic field extension
Consider an imaginary quadratic field $ K=\mathbb {Q} \left({\sqrt {-d}}\right),\,d\in \mathbb {Z} ,d>0$. An elliptic function $f$ is said to have complex multiplication if there is an algebraic relation between $f(z)$ and $f(\lambda z)$ for all $\lambda $ in $K$.
Conversely, Kronecker conjectured – in what became known as the Kronecker Jugendtraum – that every abelian extension of $K$ could be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication. To this day this remains one of the few cases of Hilbert's twelfth problem which has actually been solved.
An example of an elliptic curve with complex multiplication is
$\mathbb {C} /(\theta \mathbb {Z} [i])$
where Z[i] is the Gaussian integer ring, and θ is any non-zero complex number. Any such complex torus has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as
$Y^{2}=4X^{3}-aX$
for some $a\in \mathbb {C} $, which demonstrably has two conjugate order-4 automorphisms sending
$Y\to \pm iY,\quad X\to -X$
in line with the action of i on the Weierstrass elliptic functions.
More generally, consider the lattice Λ, an additive group in the complex plane, generated by $\omega _{1},\omega _{2}$. Then we define the Weierstrass function of the variable $z$ in $\mathbb {C} $ as follows:
$\wp (z;\Lambda )=\wp (z;\omega _{1},\omega _{2})={\frac {1}{z^{2}}}+\sum _{(m,n)\neq (0,0)}\left\{{\frac {1}{(z+m\omega _{1}+n\omega _{2})^{2}}}-{\frac {1}{\left(m\omega _{1}+n\omega _{2}\right)^{2}}}\right\},$
and
$g_{2}=60\sum _{(m,n)\neq (0,0)}(m\omega _{1}+n\omega _{2})^{-4}$
$g_{3}=140\sum _{(m,n)\neq (0,0)}(m\omega _{1}+n\omega _{2})^{-6}.$
Let $\wp '$ be the derivative of $\wp $. Then we obtain an isomorphism of complex Lie groups:
$w\mapsto (\wp (w):\wp '(w):1)\in \mathbb {P} ^{2}(\mathbb {C} )$
from the complex torus group $\mathbb {C} /\Lambda $ to the projective elliptic curve defined in homogeneous coordinates by
$E=\left\{(x:y:z)\in \mathbb {C} ^{3}\mid y^{2}z=4x^{3}-g_{2}xz^{2}-g_{3}z^{3}\right\}$
and where the point at infinity, the zero element of the group law of the elliptic curve, is by convention taken to be $(0:1:0)$. If the lattice defining the elliptic curve is actually preserved under multiplication by (possibly a proper subring of) the ring of integers ${\mathfrak {o}}_{K}$ of $K$, then the ring of analytic automorphisms of $E=\mathbb {C} /\Lambda $ turns out to be isomorphic to this (sub)ring.
If we rewrite $\tau =\omega _{1}/\omega _{2}$ where $\operatorname {Im} \tau >0$ and $\Delta (\Lambda )=g_{2}(\Lambda )^{3}-27g_{3}(\Lambda )^{2}$, then
$j(\tau )=j(E)=j(\Lambda )=2^{6}3^{3}g_{2}(\Lambda )^{3}/\Delta (\Lambda )\ .$
This means that the j-invariant of $E$ is an algebraic number – lying in $K$ – if $E$ has complex multiplication.
Abstract theory of endomorphisms
The ring of endomorphisms of an elliptic curve can be of one of three forms: the integers Z; an order in an imaginary quadratic number field; or an order in a definite quaternion algebra over Q.[3]
When the field of definition is a finite field, there are always non-trivial endomorphisms of an elliptic curve, coming from the Frobenius map, so every such curve has complex multiplication (and the terminology is not often applied). But when the base field is a number field, complex multiplication is the exception. It is known that, in a general sense, the case of complex multiplication is the hardest to resolve for the Hodge conjecture.
Kronecker and abelian extensions
Kronecker first postulated that the values of elliptic functions at torsion points should be enough to generate all abelian extensions for imaginary quadratic fields, an idea that went back to Eisenstein in some cases, and even to Gauss. This became known as the Kronecker Jugendtraum; and was certainly what had prompted Hilbert's remark above, since it makes explicit class field theory in the way the roots of unity do for abelian extensions of the rational number field, via Shimura's reciprocity law.
Indeed, let K be an imaginary quadratic field with class field H. Let E be an elliptic curve with complex multiplication by the integers of K, defined over H. Then the maximal abelian extension of K is generated by the x-coordinates of the points of finite order on some Weierstrass model for E over H.[4]
Many generalisations have been sought of Kronecker's ideas; they do however lie somewhat obliquely to the main thrust of the Langlands philosophy, and there is no definitive statement currently known.
Sample consequence
It is no accident that
$e^{\pi {\sqrt {163}}}=262537412640768743.99999999999925007\dots \,$
or equivalently,
$e^{\pi {\sqrt {163}}}=640320^{3}+743.99999999999925007\dots \,$
is so close to an integer. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of modular forms, and the fact that
$\mathbf {Z} \left[{\frac {1+{\sqrt {-163}}}{2}}\right]$
is a unique factorization domain.
Here $(1+{\sqrt {-163}})/2$ satisfies α2 = α − 41. In general, S[α] denotes the set of all polynomial expressions in α with coefficients in S, which is the smallest ring containing α and S. Because α satisfies this quadratic equation, the required polynomials can be limited to degree one.
Alternatively,
$e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+743.99999999999925007\dots \,$
an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner numbers.
Singular moduli
The points of the upper half-plane τ which correspond to the period ratios of elliptic curves over the complex numbers with complex multiplication are precisely the imaginary quadratic numbers.[5] The corresponding modular invariants j(τ) are the singular moduli, coming from an older terminology in which "singular" referred to the property of having non-trivial endomorphisms rather than referring to a singular curve.[6]
The modular function j(τ) is algebraic on imaginary quadratic numbers τ:[7] these are the only algebraic numbers in the upper half-plane for which j is algebraic.[8]
If Λ is a lattice with period ratio τ then we write j(Λ) for j(τ). If further Λ is an ideal a in the ring of integers OK of a quadratic imaginary field K then we write j(a) for the corresponding singular modulus. The values j(a) are then real algebraic integers, and generate the Hilbert class field H of K: the field extension degree [H:K] = h is the class number of K and the H/K is a Galois extension with Galois group isomorphic to the ideal class group of K. The class group acts on the values j(a) by [b] : j(a) → j(ab).
In particular, if K has class number one, then j(a) = j(O) is a rational integer: for example, j(Z[i]) = j(i) = 1728.
See also
• Algebraic Hecke character
• Heegner point
• Hilbert's twelfth problem
• Lubin–Tate formal group, local fields
• Drinfeld shtuka, global function field case
• Wiles's proof of Fermat's Last Theorem
Citations
1. Silverman 2009, p. 69, Remark 4.3.
2. Reid, Constance (1996), Hilbert, Springer, p. 200, ISBN 978-0-387-94674-0
3. Silverman 1986, p. 102.
4. Serre 1967, p. 295.
5. Silverman 1986, p. 339.
6. Silverman 1994, p. 104.
7. Serre 1967, p. 293.
8. Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 56. ISBN 0-521-20461-5. Zbl 0297.10013.
References
• Borel, A.; Chowla, S.; Herz, C. S.; Iwasawa, K.; Serre, J.-P. Seminar on complex multiplication. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957–58. Lecture Notes in Mathematics, No. 21 Springer-Verlag, Berlin-New York, 1966
• Husemöller, Dale H. (1987). Elliptic curves. Graduate Texts in Mathematics. Vol. 111. With an appendix by Ruth Lawrence. Springer-Verlag. ISBN 0-387-96371-5. Zbl 0605.14032.
• Lang, Serge (1983). Complex multiplication. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 255. New York: Springer-Verlag. ISBN 0-387-90786-6. Zbl 0536.14029.
• Serre, J.-P. (1967). "XIII. Complex multiplication". In Cassels, J.W.S.; Fröhlich, Albrecht (eds.). Algebraic Number Theory. Academic Press. pp. 292–296.
• Shimura, Goro (1971). Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan. Vol. 11. Tokyo: Iwanami Shoten. Zbl 0221.10029.
• Shimura, Goro (1998). Abelian varieties with complex multiplication and modular functions. Princeton Mathematical Series. Vol. 46. Princeton, NJ: Princeton University Press. ISBN 0-691-01656-9. Zbl 0908.11023.
• Silverman, Joseph H. (1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 106. Springer-Verlag. ISBN 0-387-96203-4. Zbl 0585.14026.
• Silverman, Joseph H. (2009). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 106 (2nd ed.). Springer Science. doi:10.1007/978-0-387-09494-6. ISBN 978-0-387-09493-9.
• Silverman, Joseph H. (1994). Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 151. Springer-Verlag. ISBN 0-387-94328-5. Zbl 0911.14015.
External links
• Complex multiplication from PlanetMath.org
• Examples of elliptic curves with complex multiplication from PlanetMath.org
• Ribet, Kenneth A. (October 1995). "Galois Representations and Modular Forms". Bulletin of the American Mathematical Society. 32 (4): 375–402. arXiv:math/9503219. CiteSeerX 10.1.1.125.6114. doi:10.1090/s0273-0979-1995-00616-6. S2CID 16786407.
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Wikipedia
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Cofinality
In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.
Not to be confused with cofiniteness.
This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.
Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net.
Examples
• The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subset).
• In particular, the cofinality of any nonzero finite ordinal, or indeed any finite directed set, is 1, since such sets have a greatest element.
• Every cofinal subset of a partially ordered set must contain all maximal elements of that set. Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.
• In particular, let $A$ be a set of size $n,$ and consider the set of subsets of $A$ containing no more than $m$ elements. This is partially ordered under inclusion and the subsets with $m$ elements are maximal. Thus the cofinality of this poset is $n$ choose $m.$
• A subset of the natural numbers $\mathbb {N} $ is cofinal in $\mathbb {N} $ if and only if it is infinite, and therefore the cofinality of $\aleph _{0}$ is $\aleph _{0}.$ Thus $\aleph _{0}$ is a regular cardinal.
• The cofinality of the real numbers with their usual ordering is $\aleph _{0},$ since $\mathbb {N} $ is cofinal in $\mathbb {R} .$ The usual ordering of $\mathbb {R} $ is not order isomorphic to $c,$ the cardinality of the real numbers, which has cofinality strictly greater than $\aleph _{0}.$ This demonstrates that the cofinality depends on the order; different orders on the same set may have different cofinality.
Properties
If $A$ admits a totally ordered cofinal subset, then we can find a subset $B$ that is well-ordered and cofinal in $A.$ Any subset of $B$ is also well-ordered. Two cofinal subsets of $B$ with minimal cardinality (that is, their cardinality is the cofinality of $B$) need not be order isomorphic (for example if $B=\omega +\omega ,$ then both $\omega +\omega $ and $\{\omega +n:n<\omega \}$ viewed as subsets of $B$ have the countable cardinality of the cofinality of $B$ but are not order isomorphic.) But cofinal subsets of $B$ with minimal order type will be order isomorphic.
Cofinality of ordinals and other well-ordered sets
The cofinality of an ordinal $\alpha $ is the smallest ordinal $\delta $ that is the order type of a cofinal subset of $\alpha .$ The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.
Thus for a limit ordinal $\alpha ,$ there exists a $\delta $-indexed strictly increasing sequence with limit $\alpha .$ For example, the cofinality of $\omega ^{2}$ is $\omega ,$ because the sequence $\omega \cdot m$ (where $m$ ranges over the natural numbers) tends to $\omega ^{2};$ but, more generally, any countable limit ordinal has cofinality $\omega .$ An uncountable limit ordinal may have either cofinality $\omega $ as does $\omega _{\omega }$ or an uncountable cofinality.
The cofinality of 0 is 0. The cofinality of any successor ordinal is 1. The cofinality of any nonzero limit ordinal is an infinite regular cardinal.
Regular and singular ordinals
Main article: Regular cardinal
A regular ordinal is an ordinal that is equal to its cofinality. A singular ordinal is any ordinal that is not regular.
Every regular ordinal is the initial ordinal of a cardinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular. Assuming the axiom of choice, $\omega _{\alpha +1}$ is regular for each $\alpha .$ In this case, the ordinals $0,1,\omega ,\omega _{1},$ and $\omega _{2}$ are regular, whereas $2,3,\omega _{\omega },$ and $\omega _{\omega \cdot 2}$ are initial ordinals that are not regular.
The cofinality of any ordinal $\alpha $ is a regular ordinal, that is, the cofinality of the cofinality of $\alpha $ is the same as the cofinality of $\alpha .$ So the cofinality operation is idempotent.
Cofinality of cardinals
If $\kappa $ is an infinite cardinal number, then $\operatorname {cf} (\kappa )$ is the least cardinal such that there is an unbounded function from $\operatorname {cf} (\kappa )$ to $\kappa ;$ ;} $\operatorname {cf} (\kappa )$ is also the cardinality of the smallest set of strictly smaller cardinals whose sum is $\kappa ;$ ;} more precisely
$\mathrm {cf} (\kappa )=\min \left\{|I|\ :\ \kappa =\sum _{i\in I}\lambda _{i}\ \land \ {\text{ for all such }}i\,\lambda _{i}<\kappa \right\}$ :\ \kappa =\sum _{i\in I}\lambda _{i}\ \land \ {\text{ for all such }}i\,\lambda _{i}<\kappa \right\}}
That the set above is nonempty comes from the fact that
$\kappa =\bigcup _{i\in \kappa }\{i\}$
that is, the disjoint union of $\kappa $ singleton sets. This implies immediately that $\operatorname {cf} (\kappa )\leq \kappa .$ The cofinality of any totally ordered set is regular, so $\operatorname {cf} (\kappa )=\operatorname {cf} (\operatorname {cf} (\kappa )).$
Using König's theorem, one can prove $\kappa <\kappa ^{\operatorname {cf} (\kappa )}$ and $\kappa <\operatorname {cf} \left(2^{\kappa }\right)$ for any infinite cardinal $\kappa .$
The last inequality implies that the cofinality of the cardinality of the continuum must be uncountable. On the other hand,
$\aleph _{\omega }=\bigcup _{n<\omega }\aleph _{n}.$
The ordinal number ω being the first infinite ordinal, so that the cofinality of $\aleph _{\omega }$ is card(ω) = $\aleph _{0}.$ (In particular, $\aleph _{\omega }$ is singular.) Therefore,
$2^{\aleph _{0}}\neq \aleph _{\omega }.$
(Compare to the continuum hypothesis, which states $2^{\aleph _{0}}=\aleph _{1}.$)
Generalizing this argument, one can prove that for a limit ordinal $\delta $
$\mathrm {cf} (\aleph _{\delta })=\mathrm {cf} (\delta ).$
On the other hand, if the axiom of choice holds, then for a successor or zero ordinal $\delta $
$\mathrm {cf} (\aleph _{\delta })=\aleph _{\delta }.$
See also
• Club set – Set theory concept
• Initial ordinal – mathematical conceptPages displaying wikidata descriptions as a fallback
References
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
Order theory
• Topics
• Glossary
• Category
Key concepts
• Binary relation
• Boolean algebra
• Cyclic order
• Lattice
• Partial order
• Preorder
• Total order
• Weak ordering
Results
• Boolean prime ideal theorem
• Cantor–Bernstein theorem
• Cantor's isomorphism theorem
• Dilworth's theorem
• Dushnik–Miller theorem
• Hausdorff maximal principle
• Knaster–Tarski theorem
• Kruskal's tree theorem
• Laver's theorem
• Mirsky's theorem
• Szpilrajn extension theorem
• Zorn's lemma
Properties & Types (list)
• Antisymmetric
• Asymmetric
• Boolean algebra
• topics
• Completeness
• Connected
• Covering
• Dense
• Directed
• (Partial) Equivalence
• Foundational
• Heyting algebra
• Homogeneous
• Idempotent
• Lattice
• Bounded
• Complemented
• Complete
• Distributive
• Join and meet
• Reflexive
• Partial order
• Chain-complete
• Graded
• Eulerian
• Strict
• Prefix order
• Preorder
• Total
• Semilattice
• Semiorder
• Symmetric
• Total
• Tolerance
• Transitive
• Well-founded
• Well-quasi-ordering (Better)
• (Pre) Well-order
Constructions
• Composition
• Converse/Transpose
• Lexicographic order
• Linear extension
• Product order
• Reflexive closure
• Series-parallel partial order
• Star product
• Symmetric closure
• Transitive closure
Topology & Orders
• Alexandrov topology & Specialization preorder
• Ordered topological vector space
• Normal cone
• Order topology
• Order topology
• Topological vector lattice
• Banach
• Fréchet
• Locally convex
• Normed
Related
• Antichain
• Cofinal
• Cofinality
• Comparability
• Graph
• Duality
• Filter
• Hasse diagram
• Ideal
• Net
• Subnet
• Order morphism
• Embedding
• Isomorphism
• Order type
• Ordered field
• Ordered vector space
• Partially ordered
• Positive cone
• Riesz space
• Upper set
• Young's lattice
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Wikipedia
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Singular distribution
In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero.
Not to be confused with Singular distribution (differential geometry).
Other names
These distributions are sometimes called singular continuous distributions, since their cumulative distribution functions are singular and continuous.
Properties
Such distributions are not absolutely continuous with respect to Lebesgue measure.
A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function, since the Lebesgue integral of any such function would be zero.
In general, distributions can be described as a discrete distribution (with a probability mass function), an absolutely continuous distribution (with a probability density), a singular distribution (with neither), or can be decomposed into a mixture of these.
Example
An example is the Cantor distribution; its cumulative distribution function is a devil's staircase. Less curious examples appear in higher dimensions. For example, the upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
See also
• Singular measure
• Lebesgue's decomposition theorem
External links
• Singular distribution in the Encyclopedia of Mathematics
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
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Wikipedia
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Singular solution
A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions.
In some cases, the term singular solution is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve. A singular solution in this stronger sense is often given as tangent to every solution from a family of solutions. By tangent we mean that there is a point x where ys(x) = yc(x) and y's(x) = y'c(x) where yc is a solution in a family of solutions parameterized by c. This means that the singular solution is the envelope of the family of solutions.
Usually, singular solutions appear in differential equations when there is a need to divide in a term that might be equal to zero. Therefore, when one is solving a differential equation and using division one must check what happens if the term is equal to zero, and whether it leads to a singular solution. The Picard–Lindelöf theorem, which gives sufficient conditions for unique solutions to exist, can be used to rule out the existence of singular solutions. Other theorems, such as the Peano existence theorem, give sufficient conditions for solutions to exist without necessarily being unique, which can allow for the existence of singular solutions.
A divergent solution
Consider the homogeneous linear ordinary differential equation
$xy'(x)+2y(x)=0,\,\!$
where primes denote derivatives with respect to x. The general solution to this equation is
$y(x)=Cx^{-2}.\,\!$
For a given $C$, this solution is smooth except at $x=0$ where the solution is divergent. Furthermore, for a given $x\not =0$, this is the unique solution going through $(x,y(x))$.
Failure of uniqueness
Consider the differential equation
$y'(x)^{2}=4y(x).\,\!$
A one-parameter family of solutions to this equation is given by
$y_{c}(x)=(x-c)^{2}.\,\!$
Another solution is given by
$y_{s}(x)=0.\,\!$
Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. By considering the two sets of solutions above, one can see that the solution fails to be unique when $y=0$. (It can be shown that for $y>0$ if a single branch of the square root is chosen, then there is a local solution which is unique using the Picard–Lindelöf theorem.) Thus, the solutions above are all singular solutions, in the sense that solution fails to be unique in a neighbourhood of one or more points. (Commonly, we say "uniqueness fails" at these points.) For the first set of solutions, uniqueness fails at one point, $x=c$, and for the second solution, uniqueness fails at every value of $x$. Thus, the solution $y_{s}$ is a singular solution in the stronger sense that uniqueness fails at every value of x. However, it is not a singular function since it and all its derivatives are continuous.
In this example, the solution $y_{s}(x)=0$ is the envelope of the family of solutions $y_{c}(x)=(x-c)^{2}$. The solution $y_{s}$ is tangent to every curve $y_{c}(x)$ at the point $(c,0)$.
The failure of uniqueness can be used to construct more solutions. These can be found by taking two constant $c_{1}<c_{2}$ and defining a solution $y(x)$ to be $(x-c_{1})^{2}$ when $x<c_{1}$, to be $0$ when $c_{1}\leq x\leq c_{2}$, and to be $(x-c_{2})^{2}$ when $x>c_{2}$. Direct calculation shows that this is a solution of the differential equation at every point, including $x=c_{1}$ and $x=c_{2}$. Uniqueness fails for these solutions on the interval $c_{1}\leq x\leq c_{2}$, and the solutions are singular, in the sense that the second derivative fails to exist, at $x=c_{1}$ and $x=c_{2}$.
Further example of failure of uniqueness
The previous example might give the erroneous impression that failure of uniqueness is directly related to $y(x)=0$. Failure of uniqueness can also be seen in the following example of a Clairaut's equation:
$y(x)=x\cdot y'+(y')^{2}\,\!$
We write y' = p and then
$y(x)=x\cdot p+(p)^{2}.$
Now, we shall take the differential according to x:
$p=y'=p+xp'+2pp'$
which by simple algebra yields
$0=(2p+x)p'.$
This condition is solved if 2p+x=0 or if p′=0.
If p' = 0 it means that y' = p = c = constant, and the general solution of this new equation is:
$y_{c}(x)=c\cdot x+c^{2}$
where c is determined by the initial value.
If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives
$y_{s}(x)=-{\tfrac {1}{2}}x^{2}+(-{\tfrac {1}{2}}x)^{2}=-{\tfrac {1}{4}}x^{2}.$
Now we shall check when these solutions are singular solutions. If two solutions intersect each other, that is, they both go through the same point (x,y), then there is a failure of uniqueness for a first-order ordinary differential equation. Thus, there will be a failure of uniqueness if a solution of the first form intersects the second solution.
The condition of intersection is : ys(x) = yc(x). We solve
$c\cdot x+c^{2}=y_{c}(x)=y_{s}(x)=-{\tfrac {1}{4}}x^{2}$
to find the intersection point, which is $(-2c,-c^{2})$.
We can verify that the curves are tangent at this point y's(x) = y'c(x). We calculate the derivatives:
$y_{c}'(-2c)=c\,\!$
$y_{s}'(-2c)=-{\tfrac {1}{2}}x|_{x=-2c}=c.\,\!$
Hence,
$y_{s}(x)=-{\tfrac {1}{4}}\cdot x^{2}\,\!$
is tangent to every member of the one-parameter family of solutions
$y_{c}(x)=c\cdot x+c^{2}\,\!$
of this Clairaut equation:
$y(x)=x\cdot y'+(y')^{2}.\,\!$
See also
• Chandrasekhar equation
• Chrystal's equation
• Caustic (mathematics)
• Envelope (mathematics)
• Initial value problem
• Picard–Lindelöf theorem
Bibliography
• Rozov, N.Kh. (2001) [1994], "Singular solution", Encyclopedia of Mathematics, EMS Press
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Wikipedia
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Singular trace
In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling.
American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of trace class operators.[1][2] Therefore, in distinction to the finite-dimensional case, in infinite dimensions the canonical operator trace is not the unique trace up to scaling. The operator trace is the continuous extension of the matrix trace from finite rank operators to all trace class operators, and the term singular derives from the fact that a singular trace vanishes where the matrix trace is supported, analogous to a singular measure vanishing where Lebesgue measure is supported.
Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes.[3][4] In heuristic terms, a singular trace corresponds to a way of summing numbers a1, a2, a3, ... that is completely orthogonal or 'singular' with respect to the usual sum a1 + a2 + a3 + ... . This allows mathematicians to sum sequences like the harmonic sequence (and operators with similar spectral behaviour) that are divergent for the usual sum. In similar terms a (noncommutative) measure theory or probability theory can be built for distributions like the Cauchy distribution (and operators with similar spectral behaviour) that do not have finite expectation in the usual sense.
Origin
By 1950 French mathematician Jacques Dixmier, a founder of the semifinite theory of von Neumann algebras,[5] thought that a trace on the bounded operators of a separable Hilbert space would automatically be normal up to some trivial counterexamples.[6]: 217 Over the course of 15 years Dixmier, aided by a suggestion of Nachman Aronszajn and inequalities proved by Joseph Hersch, developed an example of a non-trivial yet non-normal trace on weak trace-class operators,[7] disproving his earlier view. Singular traces based on Dixmier's construction are called Dixmier traces.
Independently and by different methods, German mathematician Albrecht Pietsch (de) investigated traces on ideals of operators on Banach spaces.[8] In 1987 Nigel Kalton answered a question of Pietsch by showing that the operator trace is not the unique trace on quasi-normed proper subideals of the trace-class operators on a Hilbert space.[9] József Varga independently studied a similar question.[10] To solve the question of uniqueness of the trace on the full ideal of trace-class operators, Kalton developed a spectral condition for the commutator subspace of trace class operators following on from results of Gary Weiss.[1] A consequence of the results of Weiss and the spectral condition of Kalton was the existence of non-trivial singular traces on trace class operators .[2][6]: 185
Also independently, and from a different direction, Mariusz Wodzicki investigated the noncommutative residue, a trace on classical pseudo-differential operators on a compact manifold that vanishes on trace class pseudo-differential operators of order less than the negative of the dimension of the manifold.[11]
Definition
A trace φ on a two-sided ideal J of the bounded linear operators B(H) on a separable Hilbert space H is a linear functional φ:J → $\mathbb {C} $ such that φ(AB) = φ(BA) for all operators A from J and B from B(H). That is, a trace is a linear functional on J that vanishes on the commutator subspace Com(J) of J.
A trace φ is singular if φ(A) = 0 for every A from the subideal of finite rank operators F(H) within J.
Existence and characterisation
Singular traces are characterised by the spectral Calkin correspondence between two-sided ideals of bounded operators on Hilbert space and rearrangement invariant sequence spaces. Using the spectral characterisation of the commutator subspace due to Ken Dykema, Tadeusz Figiel, Gary Weiss and Mariusz Wodzicki,[12] to every trace φ on a two-sided ideal J there is a unique symmetric functional f on the corresponding Calkin sequence space j such that
$\varphi (A)={\rm {f}}(\mu (A))$
(1)
for every positive operator A belonging to J.[6] Here μ: J+ → j+ is the map from a positive operator to its singular values. A singular trace φ corresponds to a symmetric functional f on the sequence space j that vanishes on c00, the sequences with a finite number of non-zero terms.
The characterisation parallels the construction of the usual operator trace where
${\rm {Tr}}(A)=\sum _{n=0}^{\infty }\mu (n,A)=\sum \mu (A)$
for A a positive trace class operator. The trace class operators and the sequence space of summable sequences are in Calkin correspondence. (The sum Σ is a symmetric functional on the space of summable sequences.)
Existence
A non-zero trace φ exists on a two-sided ideal J of operators on a separable Hilbert space if the co-dimension of its commutator subspace is not zero. There are ideals that admit infinitely many linearly independent non-zero singular traces. For example, the commutator subspace of the ideal of weak trace-class operators contains the ideal of trace class operators and every positive operator in the commutator subspace of the weak trace class is trace class.[12] Consequently, every trace on the weak trace class ideal is singular and the co-dimension of the weak trace class ideal commutator subspace is infinite.[6]: 191 Not all of the singular traces on the weak trace class ideal are Dixmier traces.[6]: 316
Lidskii formulation
The trace of a square matrix is the sum of its eigenvalues. Lidskii's formula extends this result to functional analysis and states that the trace of a trace class operator A is given by the sum of its eigenvalues,[13]
${\rm {Tr}}(A)=\sum _{n=0}^{\infty }\lambda (n,A)=\sum (\lambda (A)).$
The characterisation (1) of a trace φ on positive operators of a two-ideal J as a symmetric functional applied to singular values can be improved to the statement that the trace φ on any operator in J is given by the same symmetric functional applied to eigenvalue sequences, provided that the eigenvalues of all operators in J belong to the Calkin sequence space j.[14] In particular, if a bounded operator A belongs to J whenever there is a bounded operator B in J such that
$\prod _{k=0}^{n}\mu (k,A)\leq \prod _{k=0}^{n}\mu (k,B)$
(2)
for every natural number n, then for each trace φ on J there is a unique symmetric functional f on the Calkin space j with
$\varphi (A)={\rm {f}}(\lambda (A))$
(3)
where λ(A) is the sequence of eigenvalues of an operator A in J rearranged so that the absolute value of the eigenvalues is decreasing. If A is quasi-nilpotent then λ(A) is the zero sequence. Most two-sided ideals satisfy the property (2), including all Banach ideals and quasi-Banach ideals.
Equation (3) is the precise statement that singular traces measure asymptotic spectral behaviour of operators.
Fredholm formulation
The trace of a square matrix is the sum of its diagonal elements. In functional analysis the corresponding formula for trace class operators is
${\rm {Tr}}(A)=\sum _{n=0}^{\infty }\langle Ae_{n},e_{n}\rangle =\sum (\{\langle Ae_{n},e_{n}\rangle \}_{n=0}^{\infty })$
where { en }n=0∞ is an arbitrary orthonormal basis of the separable Hilbert space H. Singular traces do not have an equivalent formulation for arbitrary bases. Only when φ(A)=0 will an operator A generally satisfy
$\varphi (A)={\rm {f}}(\{\langle Ae_{n},e_{n}\rangle \}_{n=0}^{\infty })$
for a singular trace φ and an arbitrary orthonormal basis { en }n=0∞ .[6]: 242
The diagonal formulation is often used instead of the Lidskii formulation to calculate the trace of products, since eigenvalues of products are hard to determine. For example, in quantum statistical mechanics the expectation of an observable S is calculated against a fixed trace-class energy density operator T by the formula
$\langle S\rangle ={\rm {Tr}}(ST)=\sum _{n=0}^{\infty }\langle Se_{n},e_{n}\rangle \lambda (n,T)=v_{T}(\{\langle Se_{n},e_{n}\rangle \}_{n=0}^{\infty })$
where vT belongs to (l∞)* ≅ l1. The expectation is calculated from the expectation values ⟨Sen, en⟩ and the probability ⟨Pn⟩ = λ(n,T) of the system being in the bound quantum state en. Here Pn is the projection operator onto the one-dimensional subspace spanned by the energy eigenstate en. The eigenvalues of the product, λ(n,ST), have no equivalent interpretation.
There are results for singular traces of products.[15] For a product ST where S is bounded and T is selfadjoint and belongs to a two-sided ideal J then
$\varphi (ST)={\rm {f}}(\{\langle Se_{n},e_{n}\rangle \lambda (n,T)\}_{n=0}^{\infty })=v_{\varphi ,T}(\{\langle Se_{n},e_{n}\rangle \}_{n=0}^{\infty })$
for any trace φ on J. The orthonormal basis { en }n=0∞ must be ordered so that Ten = μ(n,T)en, n=0,1,2... . When φ is singular and φ(T)=1 then vφ,T is a linear functional on l∞ that extends the limit at infinity on the convergent sequences c. The expectation ⟨S⟩ = φ(ST) in this case has the property that ⟨Pn⟩= 0 for each n, or that there is no probability of being in a bound quantum state. That
$\langle S\rangle ={\text{``limit at infinity''}}\langle Se_{n},e_{n}\rangle $
has led to a link between singular traces, the correspondence principle, and classical limits,.[6]: ch 12
Use in noncommutative geometry
See also: Dixmier trace and Noncommutative residue
The first application of singular traces was the noncommutative residue, a trace on classical pseudo-differential operators on a compact manifold that vanishes on trace class pseudo-differential operators of order less than the negative of the dimension of the manifold, introduced Mariusz Wodzicki and Victor Guillemin independently .[11][16] Alain Connes characterised the noncommutative residue within noncommutative geometry, Connes' generalisation of differential geometry, using Dixmier traces.[3]
An expectation involving a singular trace and non-trace class density is used in noncommutative geometry,
$\int S={\rm {Tr}}_{\omega }(S|D|^{-d}).$
(4)
Here S is a bounded linear operator on the Hilbert space L2(X) of square-integrable functions on a d-dimensional closed manifold X, Trω is a Dixmier trace on the weak trace class ideal, and the density |D|−d in the weak trace class ideal is the dth power of the 'line element' |D|−1 where D is a Dirac type operator suitably normalised so that Trω(|D|−d)=1.
The expectation (4) is an extension of the Lebesgue integral on the commutative algebra of essentially bounded functions acting by multiplication on L2(X) to the full noncommutative algebra of bounded operators on L2(X).[15] That is,
$\int M_{f}=\int _{X}f(x)\,dx.$
where dx is the volume form on X, f is an essentially bounded function, and Mf is the bounded operator Mf h(x) = (fh)(x) for any square-integrable function h in L2(X). Simultaneously, the expectation (4) is the limit at infinity of the quantum expectations S → ⟨Sen,en⟩ defined by the eigenvectors of the Laplacian on X. More precisely, for many bounded operators on L2(X), included all zero-order classical pseudo-differential operators and operators of the form Mf where f is an essentially bounded function, the sequence ⟨Sen, en⟩ logarithmically converges and[6]: 384
$\int S=\lim _{n\to \infty }{\frac {\sum _{k=0}^{n}{\frac {1}{1+k}}\langle Se_{k},e_{k}\rangle }{\sum _{k=0}^{n}{\frac {1}{1+k}}}}$
These properties are linked to the spectrum of Dirac type operators and not to Dixmier traces; they still hold if the Dixmier trace in (4) is replaced by any trace on weak trace class operators.[15]
Examples
Suppose H is a separable infinite-dimensional Hilbert space.
Ideals without traces
• Bounded operators. Paul Halmos showed in 1954 that every bounded operator on a separable infinite-dimensional Hilbert space is the sum of two commutators.[17] That is, Com(B(H)) = B(H) and the co-dimension of the commutator subspace of B(H) is zero. The bounded linear operators admit no everywhere defined traces. The qualification is relevant; as a von Neumann algebra B(H) admits semifinite (strong-densely defined) traces.
Modern examination of the commutator subspace involves checking its spectral characterisation. The following ideals have no traces since the Cesàro means of positive sequences from the Calkin corresponding sequence space belong back in the sequence space, indicating that the ideal and its commutator subspace are equal.
• Compact operators. The commutator subspace Com(K(H)) = K(H) where K(H) denotes the compact linear operators. The ideal of compact operators admits no traces.
• Schatten p-ideals. The commutator subspace Com(Lp) = Lp, p > 1, where Lp denotes the Schatten p-ideal,
$L_{p}=\{A\in K(H):\left(\sum _{n=0}^{\infty }\mu (n,A)^{p}\right)^{\frac {1}{p}}<\infty \},$
and μ(A) denotes the sequence of singular values of a compact operator A. The Schatten ideals for p > 1 admit no traces.
• Lorentz p-ideals or weak-Lp ideals. The commutator subspace Com(Lp,∞) = Lp,∞, p > 1, where
$L_{p,\infty }=\{A\in K(H):\mu (n,A)=O(n^{-{\frac {1}{p}}})\}$
is the weak-Lp ideal. The weak-Lp ideals, p > 1, admit no traces. The weak-Lp ideals are equal to the Lorentz ideals (below) with concave function ψ(n)=n1−1/p.
Ideals with traces
• Finite rank operators. It is checked from the spectral condition that the kernel of the operator trace Tr and the commutator subspace of the finite rank operators are equal, ker Tr = Com(F(H)). It follows that the commutator subspace Com(F(H)) has co-dimension 1 in F(H). Up to scaling Tr is the unique trace on F(H).
• Trace class operators. The trace class operators L1 have Com(L1) strictly contained in ker Tr. The co-dimension of the commutator subspace is therefore greater than one, and is shown to be infinite.[18] Whilst Tr is, up to scaling, the unique continuous trace on L1 for the norm ||A||1 = Tr(|A|), the ideal of trace class operators admits infinitely many linearly independent and non-trivial singular traces.
• Weak trace class operators. Since Com(L1,∞)+ = (L1)+ the co-dimension of the commutator subspace of the weak-L1 ideal is infinite. Every trace on weak trace class operators vanishes on trace class operators, and hence is singular. The weak trace class operators form the smallest ideal where every trace on the ideal must be singular.[18] Dixmier traces provide an explicit construction of traces on the weak trace class operators.
${\rm {Tr}}_{\omega }(A)=\omega \left(\left\{{\frac {1}{\log(1+n)}}\sum _{k=0}^{n}\lambda (k,A)\right\}_{n=0}^{\infty }\right),\quad A\in L_{1,\infty }.$
This formula is valid for every weak trace class operator A and involves the eigenvalues ordered in decreasing absolute value. Also ω can be any extension to l∞ of the ordinary limit, it does not need to be dilation invariant as in Dixmier's original formulation. Not all of the singular traces on the weak trace class ideal are Dixmier traces.[6]: 316
• k-tensor weak trace class ideals. The weak-Lp ideals, p > 1, admit no traces as explained above. They are not the right setting for higher order factorisations of the traces on the weak trace class ideal L1,∞. For a natural number k ≥ 1 the ideals
$E_{\otimes k}=\{A\in K(H):\mu (n,A)=O(\log ^{k-1}(n)/n)\}$
form the appropriate setting. They have commutator subspaces of infinite co-dimension that form a chain such that E⊗k-1 ⊂ Com(E⊗k) (with the convention that E0 = L1). Dixmier traces on E⊗k have the form
${\rm {Tr}}_{\omega }^{k}(A)=\omega \left(\left\{{\frac {1}{\log ^{k}(1+n)}}\sum _{j=0}^{n}\lambda (j,A)\right\}_{n=0}^{\infty }\right),\quad A\in E_{\otimes k}.$
• Lorentz ψ-ideals. The natural setting for Dixmier traces is on a Lorentz ψ-ideal for a concave increasing function ψ : [0,∞) → [0,∞),
$L_{\psi }=\{A\in K(H):{\frac {1}{\psi (1+n)}}\sum _{j=0}^{n}\mu (n,A)<\infty \}.$
There are some ω that extend the ordinary limit to l∞ such that
${\rm {Tr}}_{\omega }^{\psi }(A)=\omega \left(\left\{{\frac {1}{\psi (1+n)}}\sum _{j=0}^{n}\lambda (j,A)\right\}_{n=0}^{\infty }\right),\quad A\in L_{\psi }$
is a singular trace if and only if[6]: 225
$\liminf _{n\to \infty }{\frac {\psi (2n)}{\psi (n)}}=1.$
The principal ideal generated by any compact operator A with μ(A)=ψ' is called the 'small ideal' inside Lψ. The k-tensor weak trace class ideal is the small ideal inside the Lorentz ideal with ψ=logk.
• Fully symmetric ideals generalise Lorentz ideals. Dixmier traces form all the fully symmetric traces on a Lorentz ideal up to scaling, and form a weak* dense subset of the fully symmetric traces on a general fully symmetric ideal. It is known the fully symmetric traces are a strict subset of the positive traces on a fully symmetric ideal.[6]: 109 Therefore, Dixmier traces are not the full set of positive traces on Lorentz ideals.
Notes
1. Weiss, Gary (1980). "Commutators of Hilbert-Schmidt Operators, II". Integral Equations and Operator Theory. 3 (4): 574–600. doi:10.1007/BF01702316.
2. N. J. Kalton (1989). "Trace-class operators and commutators" (PDF). Journal of Functional Analysis. 86: 41–74. doi:10.1016/0022-1236(89)90064-5. Archived from the original (PDF) on 2017-08-10. Retrieved 2013-07-28.
3. Connes, Alain (1988). "The action functional in noncommutative geometry" (PDF). Communications in Mathematical Physics. 117 (4): 673–683. Bibcode:1988CMaPh.117..673C. doi:10.1007/bf01218391. S2CID 14261310.
4. A. Connes (1995). Noncommutative Geometry (PDF). New York: Academic Press. ISBN 978-0-08-057175-1.
5. J. Dixmier (1957). Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann. Paris: Gauthier-Villars.,
6. S. Lord, F. A. Sukochev. D. Zanin (2012). Singular traces: theory and applications. Berlin: De Gruyter. doi:10.1515/9783110262551. ISBN 978-3-11-026255-1.
7. J. Dixmier (1966). "Existence de traces non normales". Comptes Rendus de l'Académie des Sciences, Série A et B. 262: A1107–A1108.
8. A. Pietsch (1981). "Operator ideals with a trace". Mathematische Nachrichten. 100: 61–91. doi:10.1002/mana.19811000105.
9. N. J. Kalton (1987). "Unusual traces on operator ideals" (PDF). Mathematische Nachrichten. 134: 119–130. doi:10.1002/mana.19871340108.
10. J. V. Varga (1989). "Traces on irregular ideals" (PDF). Proceedings of the American Mathematical Society. 107 (3): 715–723. doi:10.1090/s0002-9939-1989-0984818-8.
11. M. Wodzicki (1984). "Local invariants of spectral asymmetry". Inventiones Mathematicae. 75: 143–177. Bibcode:1984InMat..75..143W. doi:10.1007/bf01403095.
12. K. Dykema; T. Figiel; G. Weiss; M. Wodzicki (2004). "Commutator structure of operator ideals" (PDF). Advances in Mathematics. 185: 1–79. doi:10.1016/s0001-8708(03)00141-5.
13. V. B. Lidskii (1959). "Conditions for completeness of a system of root subspaces for non-selfadjoint operators with discrete spectrum". Tr. Mosk. Mat. Obs. 8: 83–120.
14. N. J. Kalton; S. Lord; D. Potapov; F. Sukochev (2013). "Traces of compact operators and the noncommutative residue" (PDF). Advances in Mathematics. 235: 1–55. arXiv:1210.3423. doi:10.1016/j.aim.2012.11.007.
15. V. Guillemin (1985). "A new proof of Weyl's formula on the asymptotic distribution of eigenvalues". Advances in Mathematics. 55 (2): 131–160. doi:10.1016/0001-8708(85)90018-0.
16. P. Halmos (1954). "Commutators of operators. II". American Journal of Mathematics. 76 (1): 191–198. doi:10.2307/2372409. JSTOR 2372409.
17. V. Kaftal; G. Weiss (2002). "Traces, ideals, and arithmetic means". Proceedings of the National Academy of Sciences. 99 (11): 7356–7360. Bibcode:2002PNAS...99.7356K. doi:10.1073/pnas.112074699. PMC 124235. PMID 12032287.
References
• S. Lord, F. A. Sukochev. D. Zanin (2012). Singular traces: theory and applications. Berlin: De Gruyter. doi:10.1515/9783110262551. ISBN 978-3-11-026255-1.
• B. Simon (2005). Trace ideals and their applications. Providence, RI: Amer. Math. Soc. ISBN 978-0-82-183581-4.
• A. Pietsch (1981). "Operator ideals with a trace". Mathematische Nachrichten. 100: 61–91. doi:10.1002/mana.19811000105.
• A. Pietsch (1987). Eigenvalues and s-numbers. Cambridge, UK: Cambridge University Press. ISBN 978-0-52-132532-5.
• S. Albeverio; D. Guido; A. Ponosov; S. Scarlatti (1996). "Singular traces and compact operators" (PDF). Journal of Functional Analysis. 137 (2): 281–302. doi:10.1006/jfan.1996.0047.
• M. Wodzicki (2002). "Vestigia investiganda" (PDF). Moscow Mathematical Journal. 2 (4): 769–798. doi:10.17323/1609-4514-2002-2-4-769-798.
• A. Connes (1994). Noncommutative geometry. Boston, MA: Academic Press. ISBN 978-0-12-185860-5.
See also
• Dixmier trace
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Wikipedia
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Singular value
In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator $T:X\rightarrow Y$ acting between Hilbert spaces $X$ and $Y$, are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator $T^{*}T$ (where $T^{*}$ denotes the adjoint of $T$).
The singular values are non-negative real numbers, usually listed in decreasing order (σ1(T), σ2(T), …). The largest singular value σ1(T) is equal to the operator norm of T (see Min-max theorem).
If T acts on Euclidean space $\mathbb {R} ^{n}$, there is a simple geometric interpretation for the singular values: Consider the image by $T$ of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of $T$ (the figure provides an example in $\mathbb {R} ^{2}$).
The singular values are the absolute values of the eigenvalues of a normal matrix A, because the spectral theorem can be applied to obtain unitary diagonalization of $A$ as $A=U\Lambda U^{*}$. Therefore, $ {\sqrt {A^{*}A}}={\sqrt {U\Lambda ^{*}\Lambda U^{*}}}=U\left|\Lambda \right|U^{*}$.
Most norms on Hilbert space operators studied are defined using s-numbers. For example, the Ky Fan-k-norm is the sum of first k singular values, the trace norm is the sum of all singular values, and the Schatten norm is the pth root of the sum of the pth powers of the singular values. Note that each norm is defined only on a special class of operators, hence s-numbers are useful in classifying different operators.
In the finite-dimensional case, a matrix can always be decomposed in the form $\mathbf {U\Sigma V^{*}} $, where $\mathbf {U} $ and $\mathbf {V^{*}} $ are unitary matrices and $\mathbf {\Sigma } $ is a rectangular diagonal matrix with the singular values lying on the diagonal. This is the singular value decomposition.
Basic properties
For $A\in \mathbb {C} ^{m\times n}$, and $i=1,2,\ldots ,\min\{m,n\}$.
Min-max theorem for singular values. Here $U:\dim(U)=i$ is a subspace of $\mathbb {C} ^{n}$ of dimension $i$.
${\begin{aligned}\sigma _{i}(A)&=\min _{\dim(U)=n-i+1}\max _{\underset {\|x\|_{2}=1}{x\in U}}\left\|Ax\right\|_{2}.\\\sigma _{i}(A)&=\max _{\dim(U)=i}\min _{\underset {\|x\|_{2}=1}{x\in U}}\left\|Ax\right\|_{2}.\end{aligned}}$
Matrix transpose and conjugate do not alter singular values.
$\sigma _{i}(A)=\sigma _{i}\left(A^{\textsf {T}}\right)=\sigma _{i}\left(A^{*}\right).$
For any unitary $U\in \mathbb {C} ^{m\times m},V\in \mathbb {C} ^{n\times n}.$
$\sigma _{i}(A)=\sigma _{i}(UAV).$
Relation to eigenvalues:
$\sigma _{i}^{2}(A)=\lambda _{i}\left(AA^{*}\right)=\lambda _{i}\left(A^{*}A\right).$
Relation to trace:
$\sum _{i=1}^{n}\sigma _{i}^{2}={\text{tr}}\ A^{\ast }A$.
If $A^{\top }A$ is full rank, the product of singular values is ${\sqrt {\det A^{\top }A}}$.
If $AA^{\top }$ is full rank, the product of singular values is ${\sqrt {\det AA^{\top }}}$.
If $A$ is full rank, the product of singular values is $|\det A|$.
Inequalities about singular values
See also.[1]
Singular values of sub-matrices
For $A\in \mathbb {C} ^{m\times n}.$
1. Let $B$ denote $A$ with one of its rows or columns deleted. Then
$\sigma _{i+1}(A)\leq \sigma _{i}(B)\leq \sigma _{i}(A)$
2. Let $B$ denote $A$ with one of its rows and columns deleted. Then
$\sigma _{i+2}(A)\leq \sigma _{i}(B)\leq \sigma _{i}(A)$
3. Let $B$ denote an $(m-k)\times (n-l)$ submatrix of $A$. Then
$\sigma _{i+k+l}(A)\leq \sigma _{i}(B)\leq \sigma _{i}(A)$
Singular values of A + B
For $A,B\in \mathbb {C} ^{m\times n}$
1. $\sum _{i=1}^{k}\sigma _{i}(A+B)\leq \sum _{i=1}^{k}(\sigma _{i}(A)+\sigma _{i}(B)),\quad k=\min\{m,n\}$
2. $\sigma _{i+j-1}(A+B)\leq \sigma _{i}(A)+\sigma _{j}(B).\quad i,j\in \mathbb {N} ,\ i+j-1\leq \min\{m,n\}$
Singular values of AB
For $A,B\in \mathbb {C} ^{n\times n}$
1. ${\begin{aligned}\prod _{i=n}^{i=n-k+1}\sigma _{i}(A)\sigma _{i}(B)&\leq \prod _{i=n}^{i=n-k+1}\sigma _{i}(AB)\\\prod _{i=1}^{k}\sigma _{i}(AB)&\leq \prod _{i=1}^{k}\sigma _{i}(A)\sigma _{i}(B),\\\sum _{i=1}^{k}\sigma _{i}^{p}(AB)&\leq \sum _{i=1}^{k}\sigma _{i}^{p}(A)\sigma _{i}^{p}(B),\end{aligned}}$
2. $\sigma _{n}(A)\sigma _{i}(B)\leq \sigma _{i}(AB)\leq \sigma _{1}(A)\sigma _{i}(B)\quad i=1,2,\ldots ,n.$
For $A,B\in \mathbb {C} ^{m\times n}$[2]
$2\sigma _{i}(AB^{*})\leq \sigma _{i}\left(A^{*}A+B^{*}B\right),\quad i=1,2,\ldots ,n.$
Singular values and eigenvalues
For $A\in \mathbb {C} ^{n\times n}$.
1. See[3]
$\lambda _{i}\left(A+A^{*}\right)\leq 2\sigma _{i}(A),\quad i=1,2,\ldots ,n.$
2. Assume $\left|\lambda _{1}(A)\right|\geq \cdots \geq \left|\lambda _{n}(A)\right|$. Then for $k=1,2,\ldots ,n$:
1. Weyl's theorem
$\prod _{i=1}^{k}\left|\lambda _{i}(A)\right|\leq \prod _{i=1}^{k}\sigma _{i}(A).$
2. For $p>0$.
$\sum _{i=1}^{k}\left|\lambda _{i}^{p}(A)\right|\leq \sum _{i=1}^{k}\sigma _{i}^{p}(A).$
History
This concept was introduced by Erhard Schmidt in 1907. Schmidt called singular values "eigenvalues" at that time. The name "singular value" was first quoted by Smithies in 1937. In 1957, Allahverdiev proved the following characterization of the nth s-number:[4]
$s_{n}(T)=\inf {\big \{}\,\|T-L\|:L{\text{ is an operator of finite rank }}<n\,{\big \}}.$
This formulation made it possible to extend the notion of s-numbers to operators in Banach space.
See also
• Condition number
• Cauchy interlacing theorem or Poincaré separation theorem
• Schur–Horn theorem
• Singular value decomposition
References
1. R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991. Chap. 3
2. X. Zhan. Matrix Inequalities. Springer-Verlag, Berlin, Heidelberg, 2002. p.28
3. R. Bhatia. Matrix Analysis. Springer-Verlag, New York, 1997. Prop. III.5.1
4. I. C. Gohberg and M. G. Krein. Introduction to the Theory of Linear Non-selfadjoint Operators. American Mathematical Society, Providence, R.I.,1969. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18.
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Wikipedia
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Singularity function
Singularity functions are a class of discontinuous functions that contain singularities, i.e. they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory.[1][2][3] The functions are notated with brackets, as $\langle x-a\rangle ^{n}$ where n is an integer. The "$\langle \rangle $" are often referred to as singularity brackets . The functions are defined as:
n $\langle x-a\rangle ^{n}$
$<0$ ${\frac {d^{|n+1|}}{dx^{|n+1|}}}\delta (x-a)\,$
-2 ${\frac {d}{dx}}\delta (x-a)\,$
-1 $\delta (x-a)\,$
0 $H(x-a)\,$
1 $(x-a)H(x-a)\,$
2 $(x-a)^{2}H(x-a)$
$\geq 0$ $(x-a)^{n}H(x-a)$
where: δ(x) is the Dirac delta function, also called the unit impulse. The first derivative of δ(x) is also called the unit doublet. The function $H(x)$ is the Heaviside step function: H(x) = 0 for x < 0 and H(x) = 1 for x > 0. The value of H(0) will depend upon the particular convention chosen for the Heaviside step function. Note that this will only be an issue for n = 0 since the functions contain a multiplicative factor of x − a for n > 0. $\langle x-a\rangle ^{1}$ is also called the Ramp function.
Integration
Integrating $\langle x-a\rangle ^{n}$ can be done in a convenient way in which the constant of integration is automatically included so the result will be 0 at x = a.
$\int \langle x-a\rangle ^{n}dx={\begin{cases}\langle x-a\rangle ^{n+1},&n<0\\{\frac {\langle x-a\rangle ^{n+1}}{n+1}},&n\geq 0\end{cases}}$
Example beam calculation
The deflection of a simply supported beam as shown in the diagram, with constant cross-section and elastic modulus, can be found using Euler–Bernoulli beam theory. Here we are using the sign convention of downwards forces and sagging bending moments being positive.
Load distribution:
$w=-3{\text{ N}}\langle x-0\rangle ^{-}1\ +\ 6{\text{ Nm}}^{-1}\langle x-2{\text{ m}}\rangle ^{0}\ -\ 9{\text{ N}}\langle x-4{\text{ m}}\rangle ^{-}1\ -\ 6{\text{ Nm}}^{-1}\langle x-4{\text{ m}}\rangle ^{0}\ $
Shear force:
$S=\int w\,dx$
$S=-3{\text{ N}}\langle x-0\rangle ^{0}\ +\ 6{\text{ Nm}}^{-1}\langle x-2{\text{ m}}\rangle ^{1}\ -\ 9{\text{ N}}\langle x-4{\text{ m}}\rangle ^{0}\ -\ 6{\text{ Nm}}^{-1}\langle x-4{\text{ m}}\rangle ^{1}\,$
Bending moment:
$M=-\int S\,dx$
$M=3{\text{ N}}\langle x-0\rangle ^{1}\ -\ 3{\text{ Nm}}^{-1}\langle x-2{\text{ m}}\rangle ^{2}\ +\ 9{\text{ N}}\langle x-4{\text{ m}}\rangle ^{1}\ +\ 3{\text{ Nm}}^{-1}\langle x-4{\text{ m}}\rangle ^{2}\,$
Slope:
$u'={\frac {1}{EI}}\int M\,dx$
Because the slope is not zero at x = 0, a constant of integration, c, is added
$u'={\frac {1}{EI}}\left({\frac {3}{2}}{\text{ N}}\langle x-0\rangle ^{2}\ -\ 1{\text{ Nm}}^{-1}\langle x-2{\text{ m}}\rangle ^{3}\ +\ {\frac {9}{2}}{\text{ N}}\langle x-4{\text{ m}}\rangle ^{2}\ +\ 1{\text{ Nm}}^{-1}\langle x-4{\text{ m}}\rangle ^{3}\ +\ c\right)\,$
Deflection:
$u=\int u'\,dx$
$u={\frac {1}{EI}}\left({\frac {1}{2}}{\text{ N}}\langle x-0\rangle ^{3}\ -\ {\frac {1}{4}}{\text{ Nm}}^{-1}\langle x-2{\text{ m}}\rangle ^{4}\ +\ {\frac {3}{2}}{\text{ N}}\langle x-4{\text{ m}}\rangle ^{3}\ +\ {\frac {1}{4}}{\text{ Nm}}^{-1}\langle x-4{\text{ m}}\rangle ^{4}\ +\ cx\right)\,$
The boundary condition u = 0 at x = 4 m allows us to solve for c = −7 Nm2
See also
• Macaulay brackets
• Macaulay's method
References
1. Zemanian, A. H. (1965), Distribution Theory and Transform Analysis, McGraw-Hill Book Company
2. Hoskins, R. F. (1979), Generalised Functions, Halsted Press
3. Lighthill, M.J. (1958), Fourier Analysis and Generalized Functions, Cambridge University Press
External links
• Singularity Functions (Tim Lahey)
• Singularity functions (J. Lubliner, Department of Civil and Environmental Engineering)
• Beams: Deformation by Singularity Functions (Dr. Ibrahim A. Assakkaf)
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Wikipedia
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Singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string. Perhaps the string will also touch itself without crossing, like an underlined "U". This is another kind of singularity. Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the "U" away from the "underline".
This article is about the mathematical discipline. For other geometric uses, see Singular point of a curve. For other mathematical uses, see Singularity (mathematics). For non-mathematical uses, see Singularity (disambiguation).
Vladimir Arnold defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the properties undergo sudden change under a small variation of the parameters. These situations are called perestroika (Russian: перестройка), bifurcations or catastrophes. Classifying the types of changes and characterizing sets of parameters which give rise to these changes are some of the main mathematical goals. Singularities can occur in a wide range of mathematical objects, from matrices depending on parameters to wavefronts.[1]
How singularities may arise
In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.
Other ways in which singularities occur is by degeneration of manifold structure. The presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired "corners" in a process of folding up, resembling the creasing of a table napkin.
Singularities in algebraic geometry
Algebraic curve singularities
Historically, singularities were first noticed in the study of algebraic curves. The double point at (0, 0) of the curve
$y^{2}=x^{2}+x^{3}$
and the cusp there of
$y^{2}=x^{3}\ $
are qualitatively different, as is seen just by sketching. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong. It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves.
It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
The general position of singularities in algebraic geometry
Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system. One can say that the extrinsic meaning of a singular point isn't in question; it is just that in intrinsic terms the coordinates in the ambient space don't straightforwardly translate the geometry of the algebraic variety at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka's fundamental theorem on resolution of singularities (in birational geometry in characteristic 0). This means that the simple process of "lifting" a piece of string off itself, by the "obvious" use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general collapse (through multiple processes). This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is).
The smooth theory and catastrophes
At about the same time as Hironaka's work, the catastrophe theory of René Thom was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where the level set develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials. To compensate, only the stable phenomena are considered. One can argue that in nature, anything destroyed by tiny changes is not going to be observed; the visible is the stable. Whitney had shown that in low numbers of variables the stable structure of critical points is very restricted, in local terms. Thom built on this, and his own earlier work, to create a catastrophe theory supposed to account for discontinuous change in nature.
Arnold's view
While Thom was an eminent mathematician, the subsequent fashionable nature of elementary catastrophe theory as propagated by Christopher Zeeman caused a reaction, in particular on the part of Vladimir Arnold.[2] He may have been largely responsible for applying the term singularity theory to the area including the input from algebraic geometry, as well as that flowing from the work of Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of equivalence relations on singular points, and germs. Technically this involves group actions of Lie groups on spaces of jets; in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives. Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics.
Duality
An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality is also disallowed. A major advance was the introduction of intersection cohomology, which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of perverse sheaf in homological algebra.
Other possible meanings
The theory mentioned above does not directly relate to the concept of mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable singularity. The monodromy theory of differential equations, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, monodromy studies the way a covering map can degenerate, while singularity theory studies the way a manifold can degenerate; and these fields are linked.
See also
• Tangent
• Zariski tangent space
• General position
• Contact (mathematics)
• Singular solution
• Stratification (mathematics)
• Intersection homology
• Mixed Hodge structure
• Whitney umbrella
• Round function
• Victor Goryunov
Notes
1. Arnold, V. I. (2000). "Singularity Theory". www.newton.ac.uk. Isaac Newton Institute for Mathematical Sciences. Retrieved 31 May 2016.
2. Arnold 1992
References
• V.I. Arnold (1992). Catastrophe Theory. Springer-Verlag. ISBN 978-3540548119.
• E. Brieskorn; H. Knörrer (1986). Plane Algebraic Curves. Birkhauser-Verlag. ISBN 978-3764317690.
• R. Abraham and J. Marsden (1987). Foundations of Mechanics, Second Edition. Benjamin/Cummings Publishing Company.
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
• Diffeomorphism
• Local
• Geodesic
• Exponential map
• in Lie theory
• Foliation
• Immersion
• Integral curve
• Lie derivative
• Section
• Submersion
Types of
manifolds
• Closed
• (Almost) Complex
• (Almost) Contact
• Fibered
• Finsler
• Flat
• G-structure
• Hadamard
• Hermitian
• Hyperbolic
• Kähler
• Kenmotsu
• Lie group
• Lie algebra
• Manifold with boundary
• Oriented
• Parallelizable
• Poisson
• Prime
• Quaternionic
• Hypercomplex
• (Pseudo−, Sub−) Riemannian
• Rizza
• (Almost) Symplectic
• Tame
Tensors
Vectors
• Distribution
• Lie bracket
• Pushforward
• Tangent space
• bundle
• Torsion
• Vector field
• Vector flow
Covectors
• Closed/Exact
• Covariant derivative
• Cotangent space
• bundle
• De Rham cohomology
• Differential form
• Vector-valued
• Exterior derivative
• Interior product
• Pullback
• Ricci curvature
• flow
• Riemann curvature tensor
• Tensor field
• density
• Volume form
• Wedge product
Bundles
• Adjoint
• Affine
• Associated
• Cotangent
• Dual
• Fiber
• (Co) Fibration
• Jet
• Lie algebra
• (Stable) Normal
• Principal
• Spinor
• Subbundle
• Tangent
• Tensor
• Vector
Connections
• Affine
• Cartan
• Ehresmann
• Form
• Generalized
• Koszul
• Levi-Civita
• Principal
• Vector
• Parallel transport
Related
• Classification of manifolds
• Gauge theory
• History
• Morse theory
• Moving frame
• Singularity theory
Generalizations
• Banach manifold
• Diffeology
• Diffiety
• Fréchet manifold
• K-theory
• Orbifold
• Secondary calculus
• over commutative algebras
• Sheaf
• Stratifold
• Supermanifold
• Stratified space
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Wikipedia
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Sinhc function
In mathematics, the sinhc function appears frequently in papers about optical scattering,[1] Heisenberg spacetime[2] and hyperbolic geometry.[3] For $z\neq 0$, it is defined as[4][5]
$\operatorname {sinhc} (z)={\frac {\sinh(z)}{z}}$
The sinhc function is the hyperbolic analogue of the sinc function, defined by $\sin x/x$. It is a solution of the following differential equation:
$w(z)z-2\,{\frac {d}{dz}}w(z)-z{\frac {d^{2}}{dz^{2}}}w(z)=0$
Properties
The first-order derivative is given by
${\frac {\cosh(z)}{z}}-{\frac {\sinh(z)}{z^{2}}}$
The Taylor series expansion is
$\sum _{i=0}^{\infty }{\frac {z^{2i}}{(2i+1)!}}.$
The Padé approximant is
$\operatorname {sinhc} \left(z\right)=\left(1+{\frac {53272705}{360869676}}\,{z}^{2}+{\frac {38518909}{7217393520}}\,{z}^{4}+{\frac {269197963}{3940696861920}}\,{z}^{6}+{\frac {4585922449}{15605159573203200}}\,{z}^{8}\right)\left(1-{\frac {2290747}{120289892}}\,{z}^{2}+{\frac {1281433}{7217393520}}\,{z}^{4}-{\frac {560401}{562956694560}}\,{z}^{6}+{\frac {1029037}{346781323848960}}\,{z}^{8}\right)^{-1}$
In terms of other special functions
• $\operatorname {sinhc} (z)={\frac {{\rm {KummerM}}(1,\,2,\,2\,z)}{e^{z}}}$, where ${\rm {KummerM}}(a,b,z)$ is Kummer's confluent hypergeometric function.
• $\operatorname {sinhc} (z)={\frac {\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {z}}\right)}{e^{z}}}$, where ${\rm {HeunB}}(q,\alpha ,\gamma ,\delta ,\epsilon ,z)$ is the biconfluent Heun function.
• $\operatorname {sinhc} (z)=1/2\,{\frac {{\rm {WhittakerM}}(0,\,1/2,\,2\,z)}{z}}$, where ${\rm {WhittakerM}}(a,b,z)$ is a Whittaker function.
Gallery
See also
• Tanc function
• Tanhc function
• Sinhc integral
• Coshc function
References
1. den Outer, P. N.; Lagendijk, Ad; Nieuwenhuizen, Th. M. (1993-06-01). "Location of objects in multiple-scattering media". Journal of the Optical Society of America A. 10 (6): 1209. Bibcode:1993JOSAA..10.1209D. doi:10.1364/JOSAA.10.001209. ISSN 1084-7529.
2. Körpinar, Talat (2014). "New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime". International Journal of Theoretical Physics. 53 (9): 3208–3218. Bibcode:2014IJTP...53.3208K. doi:10.1007/s10773-014-2118-5. ISSN 0020-7748. S2CID 121715858.
3. Nilgün Sönmez, A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry, International Mathematical Forum, 4, 2009, no. 38, 1877–1881
4. ten Thije Boonkkamp, J. H. M.; van Dijk, J.; Liu, L.; Peerenboom, K. S. C. (2012). "Extension of the Complete Flux Scheme to Systems of Conservation Laws". Journal of Scientific Computing. 53 (3): 552–568. doi:10.1007/s10915-012-9588-5. ISSN 0885-7474. S2CID 8455136.
5. Weisstein, Eric W. "Sinhc Function". mathworld.wolfram.com. Retrieved 2022-11-17.
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Wikipedia
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Sine-Gordon equation
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation for a function $\varphi $ dependent on two variables typically denoted $x$ and $t$, involving the wave operator and the sine of $\varphi $.
It was originally introduced by Edmond Bour (1862) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space.[1] The equation was rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model.[2]
This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions,[3] and is an example of an integrable PDE. Among well-known integrable PDEs, the sine-Gordon equation is the only relativistic system due to its Lorentz invariance.
Origin of the equation in differential geometry
There are two equivalent forms of the sine-Gordon equation. In the (real) space-time coordinates, denoted $(x,t)$, the equation reads:[4]
$\varphi _{tt}-\varphi _{xx}+\sin \varphi =0,$
where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (u, v), akin to asymptotic coordinates where
$u={\frac {x+t}{2}},\quad v={\frac {x-t}{2}},$
the equation takes the form[5]
$\varphi _{uv}=\sin \varphi .$
This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of surfaces of constant Gaussian curvature K = −1, also called pseudospherical surfaces. There is a distinguished coordinate system for such a surface in which the coordinate mesh u = constant, v = constant is given by the asymptotic lines parameterized with respect to the arc length. The first fundamental form of the surface in these coordinates has a special form
$ds^{2}=du^{2}+2\cos \varphi \,du\,dv+dv^{2},$
where $\varphi $ expresses the angle between the asymptotic lines, and for the second fundamental form, $L=N=0,M=\sin \varphi $. Then the Gauss–Codazzi equation expressing a compatibility condition between the first and second fundamental forms results in the sine-Gordon equation.
This analysis shows that any pseudospherical surface gives rise to a solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily singular due to the Hilbert embedding theorem. In the simplest case, the pseudosphere, also known as the tractroid, corresponds to a static one-soliton, but the tractroid has a singular cusp at its equator.
Conversely, one can start with a solution to the sine-Gordon equation to obtain a pseudosphere uniquely up to rigid transformations. There is a theorem, sometimes called the fundamental theorem of surfaces, that if a pair of matrix-valued bilinear forms satisfy the Gauss–Codazzi equations, then they are the first and second fundamental forms of an embedded surface in 3-dimensional space. Solutions to the sine-Gordon equation can be used to construct such matrices by using the forms obtained above.
New solutions from old
The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Bianchi and Bäcklund led to the discovery of Bäcklund transformations. Another transformation of pseudospherical surfaces is the Lie transform introduced by Sophus Lie in 1879, which corresponds to Lorentz boosts for solutions of the sine-Gordon equation.[6]
There are also some more straightforward ways to construct new solutions but which do not give new surfaces. Since the sine-Gordon equation is odd, the negative of any solution is another solution. However this does not give a new surface, as the sign-change comes down to a choice of direction for the normal to the surface. New solutions can be found by translating the solution: if $\varphi $ is a solution, then so is $\varphi +2n\pi $ for $n$ an integer.
Naming
The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics:[4]
$\varphi _{tt}-\varphi _{xx}+\varphi =0.$
The sine-Gordon equation is the Euler–Lagrange equation of the field whose Lagrangian density is given by
${\mathcal {L}}_{\text{SG}}(\varphi )={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-1+\cos \varphi .$
Using the Taylor series expansion of the cosine in the Lagrangian,
$\cos(\varphi )=\sum _{n=0}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}},$
it can be rewritten as the Klein–Gordon Lagrangian plus higher-order terms:
${\begin{aligned}{\mathcal {L}}_{\text{SG}}(\varphi )&={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-{\frac {\varphi ^{2}}{2}}+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}\\&={\mathcal {L}}_{\text{KG}}(\varphi )+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n}}{(2n)!}}.\end{aligned}}$
Soliton solutions
An interesting feature of the sine-Gordon equation is the existence of soliton and multisoliton solutions.
1-soliton solutions
The sine-Gordon equation has the following 1-soliton solutions:
$\varphi _{\text{soliton}}(x,t):=4\arctan \left(e^{m\gamma (x-vt)+\delta }\right),$
where
$\gamma ^{2}={\frac {1}{1-v^{2}}},$
and the slightly more general form of the equation is assumed:
$\varphi _{tt}-\varphi _{xx}+m^{2}\sin \varphi =0.$
The 1-soliton solution for which we have chosen the positive root for $\gamma $ is called a kink and represents a twist in the variable $\varphi $ which takes the system from one constant solution $\varphi =0$ to an adjacent constant solution $\varphi =2\pi $. The states $\varphi \cong 2\pi n$ are known as vacuum states, as they are constant solutions of zero energy. The 1-soliton solution in which we take the negative root for $\gamma $ is called an antikink. The form of the 1-soliton solutions can be obtained through application of a Bäcklund transform to the trivial (vacuum) solution and the integration of the resulting first-order differentials:
$\varphi '_{u}=\varphi _{u}+2\beta \sin {\frac {\varphi '+\varphi }{2}},$
$\varphi '_{v}=-\varphi _{v}+{\frac {2}{\beta }}\sin {\frac {\varphi '-\varphi }{2}}{\text{ with }}\varphi =\varphi _{0}=0$
for all time.
The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model introduced by Julio Rubinstein in 1970.[7] Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge $\theta _{\text{K}}=-1$. The alternative counterclockwise (right-handed) twist with topological charge $\theta _{\text{AK}}=+1$ will be an antikink.
2-soliton solutions
Multi-soliton solutions can be obtained through continued application of the Bäcklund transform to the 1-soliton solution, as prescribed by a Bianchi lattice relating the transformed results.[9] The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a phase shift. Since the colliding solitons recover their velocity and shape, such an interaction is called an elastic collision.
The kink-kink solution is given by
$\varphi _{K/K}(x,t)=4\arctan \left({\frac {v^{2}\sinh {\frac {x}{\sqrt {1-v^{2}}}}}{{\sqrt {v^{2}-1}}\cosh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)$
while the kink-antikink solution is given by
$\varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2}}}}}{\sinh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)$
Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a breather. There are known three types of breathers: standing breather, traveling large-amplitude breather, and traveling small-amplitude breather.[10]
The standing breather solution is given by
$\varphi (x,t)=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right).$
3-soliton solutions
3-soliton collisions between a traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather, the shift of the breather $\Delta _{\text{B}}$ is given by
$\Delta _{\text{B}}={\frac {2\operatorname {artanh} {\sqrt {(1-\omega ^{2})(1-v_{\text{K}}^{2})}}}{\sqrt {1-\omega ^{2}}}},$
where $v_{\text{K}}$ is the velocity of the kink, and $\omega $ is the breather's frequency.[10] If the old position of the standing breather is $x_{0}$, after the collision the new position will be $x_{0}+\Delta _{\text{B}}$.
Bäcklund transformation
See also: Bäcklund transform
Suppose that $\varphi $ is a solution of the sine-Gordon equation
$\varphi _{uv}=\sin \varphi .\,$
Then the system
${\begin{aligned}\psi _{u}&=\varphi _{u}+2a\sin {\Bigl (}{\frac {\psi +\varphi }{2}}{\Bigr )}\\\psi _{v}&=-\varphi _{v}+{\frac {2}{a}}\sin {\Bigl (}{\frac {\psi -\varphi }{2}}{\Bigr )}\end{aligned}}\,\!$
where a is an arbitrary parameter, is solvable for a function $\psi $ which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform, as both $\varphi $ and $\psi $ are solutions to the same equation, that is, the sine-Gordon equation.
By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.
For example, if $\varphi $ is the trivial solution $\varphi \equiv 0$, then $\psi $ is the one-soliton solution with $a$ related to the boost applied to the soliton.
Topological charge and energy
The topological charge or winding number of a solution $\varphi $ is
$N={\frac {1}{2\pi }}\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi }}\left[\varphi (x=\infty ,t)-\varphi (x=-\infty ,t)\right].$
The energy of a solution $\varphi $ is
$E=\int _{\mathbb {R} }\left\{{\frac {1}{2}}(\varphi _{t}^{2}+\varphi _{x}^{2})+[1+\cos \varphi ]\right\}$
(which is the Hamiltonian for the Lagrangian for the sine-Gordon model).
The topological charge is conserved if the energy is finite. The topological charge does not determine the solution, even up to Lorentz boosts. Both the trivial solution and the soliton-antisoliton pair solution have $N=0$.
Zero-curvature formulation
The sine-Gordon equation is equivalent to the curvature of a particular ${\mathfrak {su}}(2)$-connection on $\mathbb {R} ^{2}$ being equal to zero.[11]
Explicitly, with coordinates $(u,v)$ on $\mathbb {R} ^{2}$, the connection components $A_{\mu }$ are given by
$A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2}}\varphi _{u}\\{\frac {i}{2}}\varphi _{u}&-i\lambda \end{pmatrix}}={\frac {1}{2}}\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},$
$A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda }}\cos \varphi &-{\frac {1}{4\lambda }}\sin \varphi \\{\frac {1}{4\lambda }}\sin \varphi &{\frac {i}{4\lambda }}\cos \varphi \end{pmatrix}}=-{\frac {1}{4\lambda }}i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda }}i\cos \varphi \sigma _{3},$
where the $\sigma _{i}$ are the Pauli matrices. Then the zero-curvature equation
$\partial _{v}A_{u}-\partial _{u}A_{v}+[A_{u},A_{v}]=0$
is equivalent to the sine-Gordon equation $\varphi _{uv}=\sin \varphi $. The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined $F_{\mu \nu }=[\partial _{\mu }-A_{\mu },\partial _{\nu }-A_{\nu }]$.
The pair of matrices $A_{u}$ and $A_{v}$ are also known as a Lax pair for the sine-Gordon equation, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.
Related equations
The sinh-Gordon equation is given by[12]
$\varphi _{xx}-\varphi _{tt}=\sinh \varphi .$
This is the Euler–Lagrange equation of the Lagrangian
${\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})-\cosh \varphi .$
Another closely related equation is the elliptic sine-Gordon equation or Euclidean sine-Gordon equation, given by
$\varphi _{xx}+\varphi _{yy}=\sin \varphi ,$
where $\varphi $ is now a function of the variables x and y. This is no longer a soliton equation, but it has many similar properties, as it is related to the sine-Gordon equation by the analytic continuation (or Wick rotation) y = it.
The elliptic sinh-Gordon equation may be defined in a similar way.
Another similar equation comes from the Euler–Lagrange equation for Liouville field theory
$\varphi _{xx}-\varphi _{tt}=2e^{2\varphi }.$
A generalization is given by Toda field theory.[13] More precisely, Liouville field theory is the Toda field theory for the finite Kac–Moody algebra ${\mathfrak {sl}}_{2}$, while sin(h)-Gordon is the Toda field theory for the affine Kac–Moody algebra ${\hat {\mathfrak {sl}}}_{2}$.
Infinite volume and on a half line
One can also consider the sine-Gordon model on a circle,[14] on a line segment, or on a half line.[15] It is possible to find boundary conditions which preserve the integrability of the model.[15] On a half line the spectrum contains boundary bound states in addition to the solitons and breathers.[15]
Quantum sine-Gordon model
In quantum field theory the sine-Gordon model contains a parameter that can be identified with the Planck constant. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of breathers.[16][17][18] The number of the breathers depends on the value of the parameter. Multiparticle production cancels on mass shell.
Semi-classical quantization of the sine-Gordon model was done by Ludwig Faddeev and Vladimir Korepin.[19] The exact quantum scattering matrix was discovered by Alexander Zamolodchikov.[20] This model is S-dual to the Thirring model, as discovered by Coleman. [21] This is sometimes known as the Coleman correspondence and serves as an example of boson-fermion correspondence in the interacting case. This article also showed that the constants appearing in the model behave nicely under renormalization: there are three parameters $\alpha _{0},\beta $ and $\gamma _{0}$. Coleman showed $\alpha _{0}$ receives only a multiplicative correction, $\gamma _{0}$ receives only an additive correction, and $\beta $ is not renormalized. Further, for a critical, non-zero value $\beta ={\sqrt {4\pi }}$, the theory is in fact dual to a free massive Dirac field theory.
The quantum sine-Gordon equation should be modified so the exponentials become vertex operators
${\mathcal {L}}_{QsG}={\frac {1}{2}}\partial _{\mu }\varphi \partial ^{\mu }\varphi +{\frac {1}{2}}m_{0}\varphi ^{2}-\alpha (V_{\beta }+V_{-\beta })$
with $V_{\beta }=:e^{i\beta \varphi }:$, where the semi-colons denote normal ordering. A possible mass term is included.
Regimes of renormalizability
For different values of the parameter $\beta ^{2}$, the renormalizability properties of the sine-Gordon theory change.[22] The identification of these regimes is attributed to Jürg Fröhlich.
The finite regime is $\beta ^{2}<4\pi $, where no counterterms are needed to render the theory well-posed. The super-renormalizable regime is $4\pi <\beta ^{2}<8\pi $, where a finite number of counterterms are needed to render the theory well-posed. More counterterms are needed for each threshold ${\frac {n}{n+1}}8\pi $ passed.[23] For $\beta ^{2}>8\pi $, the theory becomes ill-defined (Coleman 1975). The boundary values are $\beta ^{2}=4\pi $ and $\beta ^{2}=8\pi $, which are respectively the free fermion point, as the theory is dual to a free fermion via the Coleman correspondence, and the self-dual point, where the vertex operators form an affine sl2 subalgebra, and the theory becomes strictly renormalizable (renormalizable, but not super-renormalizable).
Stochastic sine-Gordon model
The stochastic or dynamical sine-Gordon model has been studied by Martin Hairer and Hao Shen [24] allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.
The equation is
$\partial _{t}u={\frac {1}{2}}\Delta u+c\sin(\beta u+\theta )+\xi ,$
where $c,\beta ,\theta $ are real-valued constants, and $\xi $ is space-time white noise. The space dimension is fixed to 2. In the proof of existence of solutions, the thresholds $\beta ^{2}={\frac {n}{n+1}}8\pi $ again play a role in determining convergence of certain terms.
Supersymmetric sine-Gordon model
A supersymmetric extension of the sine-Gordon model also exists.[25] Integrability preserving boundary conditions for this extension can be found as well.[25]
Physical applications
The sine-Gordon model arises as the continuum limit of the Frenkel–Kontorova model which models crystal dislocations.
Dynamics in long Josephson junctions are well-described by the sine-Gordon equations, and conversely provide a useful experimental system for studying the sine-Gordon model.[26]
The sine-Gordon model is in the same universality class as the effective action for a Coulomb gas of vortices and anti-vortices in the continuous classical XY model, which is a model of magnetism.[27][28] The Kosterlitz–Thouless transition for vortices can therefore be derived from a renormalization group analysis of the sine-Gordon field theory.[29][30]
The sine-Gordon equation also arises as the formal continuum limit of a different model of magnetism, the quantum Heisenberg model, in particular the XXZ model.[31]
See also
• Josephson effect
• Fluxon
• Shape waves
References
1. Bour, Edmond (1862). "Theorie de la deformation des surfaces". Journal de l'École impériale polytechnique. 22 (39): 1–148. OCLC 55567842.
2. Frenkel J, Kontorova T (1939). "On the theory of plastic deformation and twinning". Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya. 1: 137–149.
3. Hirota, Ryogo (November 1972). "Exact Solution of the Sine-Gordon Equation for Multiple Collisions of Solitons". Journal of the Physical Society of Japan. 33 (5): 1459–1463. Bibcode:1972JPSJ...33.1459H. doi:10.1143/JPSJ.33.1459.
4. Rajaraman, R. (1989). Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. North-Holland Personal Library. Vol. 15. North-Holland. pp. 34–45. ISBN 978-0-444-87047-6.
5. Polyanin, Andrei D.; Valentin F. Zaitsev (2004). Handbook of Nonlinear Partial Differential Equations. Chapman & Hall/CRC Press. pp. 470–492. ISBN 978-1-58488-355-5.
6. Terng, C. L., & Uhlenbeck, K. (2000). "Geometry of solitons" (PDF). Notices of the AMS. 47 (1): 17–25.{{cite journal}}: CS1 maint: multiple names: authors list (link)
7. Rubinstein, Julio (1970). "Sine-Gordon equation". Journal of Mathematical Physics. 11 (1): 258–266. Bibcode:1970JMP....11..258R. doi:10.1063/1.1665057.
8. Georgiev D. D., Papaioanou S. N., Glazebrook J. F. (2004). "Neuronic system inside neurons: molecular biology and biophysics of neuronal microtubules". Biomedical Reviews. 15: 67–75. doi:10.14748/bmr.v15.103.{{cite journal}}: CS1 maint: uses authors parameter (link)
9. Rogers, C.; W. K. Schief (2002). Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. New York: Cambridge University Press. ISBN 978-0-521-01288-1.
10. Miroshnichenko A. E., Vasiliev A. A., Dmitriev S. V. Solitons and Soliton Collisions.
11. Dunajski, Maciej (2010). Solitons, instantons, and twistors. Oxford: Oxford University Press. p. 49. ISBN 978-0-19-857063-9.
12. Polyanin, Andrei D.; Zaitsev, Valentin F. (16 December 2011). Handbook of Nonlinear Partial Differential Equations (Second ed.). Boca Raton: CRC Press. p. 485. ISBN 978-1-4200-8723-9.
13. Yuanxi, Xie; Tang, Jiashi (February 2006). "A unified method for solving sinh-Gordon–type equations". Il Nuovo Cimento B. 121 (2): 115–121. Bibcode:2006NCimB.121..115X. doi:10.1393/ncb/i2005-10164-6.
14. McKean, H. P. (1981). "The sine-Gordon and sinh-Gordon equations on the circle". Communications on Pure and Applied Mathematics. 34 (2): 197–257. doi:10.1002/cpa.3160340204.
15. Bowcock, Peter; Tzamtzis, Georgios (2007). "The complex sine-Gordon model on a half line". Journal of High Energy Physics. 2007 (3): 047. arXiv:hep-th/0203139. Bibcode:2007JHEP...03..047B. doi:10.1088/1126-6708/2007/03/047. S2CID 119501952.
16. Korepin, V. E. (1979). "Direct calculation of the S matrix in the massive Thirring model". Theoretical and Mathematical Physics. 41 (2): 953–967. Bibcode:1979TMP....41..953K. doi:10.1007/bf01028501. S2CID 121527379.
17. Takada, Satoshi; Misawa, Susumu (1981). "The Quantum Sine-Gordon Model and the Fermi-Bose Relation". Progress of Theoretical Physics. 66 (1): 101–117. Bibcode:1981PThPh..66..101T. doi:10.1143/ptp.66.101.
18. Bogoliubov, N. M.; Korepin, V. E.; Izergin, A. G. (1985). "Structure of the vacuum in the quantum sine-Gordon model". Physics Letters B. 159 (4): 345–347. Bibcode:1985PhLB..159..345B. doi:10.1016/0370-2693(85)90264-3.
19. Faddeev, L. D.; Korepin, V. E. (1978). "Quantum theory of solitons". Physics Reports. 42 (1): 1–87. Bibcode:1978PhR....42....1F. doi:10.1016/0370-1573(78)90058-3.
20. Zamolodchikov, Alexander B.; Zamolodchikov, Alexey B. (1978). "Relativistic factorized S-matrix in two dimensions having O(N) isotopic symmetry". Nuclear Physics B. 133 (3): 525–535. Bibcode:1978NuPhB.133..525Z. doi:10.1016/0550-3213(78)90239-0.
21. Coleman, Sidney (15 April 1975). "Quantum sine-Gordon equation as the massive Thirring model". Physical Review D. 11 (8): 2088–2097. Bibcode:1975PhRvD..11.2088C. doi:10.1103/PhysRevD.11.2088. Retrieved 27 January 2023.
22. Fröb, Markus B.; Cadamuro, Daniela (2022). "Local operators in the Sine-Gordon model: $\partial_μϕ\, \partial_νϕ$ and the stress tensor". arXiv:2205.09223 [math-ph].
23. Chandra, Ajay; Hairer, Martin; Shen, Hao (2018). "The dynamical sine-Gordon model in the full subcritical regime". arXiv:1808.02594 [math.PR].
24. Hairer, Martin; Shen, Hao (February 2016). "The Dynamical Sine-Gordon Model". Communications in Mathematical Physics. 341 (3): 933–989. arXiv:1409.5724. Bibcode:2016CMaPh.341..933H. doi:10.1007/s00220-015-2525-3. S2CID 253750515. Retrieved 14 May 2023.
25. Inami, Takeo; Odake, Satoru; Zhang, Yao-Zhong (1995). "Supersymmetric extension of the sine-Gordon theory with integrable boundary interactions". Physics Letters B. 359 (1): 118–124. arXiv:hep-th/9506157. Bibcode:1995PhLB..359..118I. doi:10.1016/0370-2693(95)01072-X. S2CID 18230581.
26. Mazo, Juan J.; Ustinov, Alexey V. (2014). "The sine-Gordon Equation in Josephson-Junction Arrays". The sine-Gordon Model and its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics. Springer International Publishing. pp. 155–175. ISBN 978-3-319-06722-3. Retrieved 22 August 2023.
27. José, Jorge (15 November 1976). "Sine-Gordon theory and the classical two-dimensional x − y model". Physical Review D. 14 (10): 2826–2829. Bibcode:1976PhRvD..14.2826J. doi:10.1103/PhysRevD.14.2826.
28. Fröhlich, Jürg (October 1976). "Classical and quantum statistical mechanics in one and two dimensions: Two-component Yukawa — and Coulomb systems". Communications in Mathematical Physics. 47 (3): 233–268. Bibcode:1976CMaPh..47..233F. doi:10.1007/BF01609843. S2CID 120798940.
29. Ohta, T.; Kawasaki, K. (1 August 1978). "Renormalization Group Theory of the Interfacial Roughening Transition". Progress of Theoretical Physics. 60 (2): 365–379. Bibcode:1978PThPh..60..365O. doi:10.1143/PTP.60.365.
30. Kogut, John B. (1 October 1979). "An introduction to lattice gauge theory and spin systems". Reviews of Modern Physics. 51 (4): 659–713. Bibcode:1979RvMP...51..659K. doi:10.1103/RevModPhys.51.659.
31. Faddeev, L. D. (1996). "How Algebraic Bethe Ansatz works for integrable model". arXiv:hep-th/9605187.
External links
• sine-Gordon equation at EqWorld: The World of Mathematical Equations.
• Sinh-Gordon Equation at EqWorld: The World of Mathematical Equations.
• sine-Gordon equation Archived 2012-03-16 at the Wayback Machine at NEQwiki, the nonlinear equations encyclopedia.
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See also: Template:Quantum mechanics topics
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Sinistral and dextral
Sinistral and dextral, in some scientific fields, are the two types of chirality ("handedness") or relative direction. The terms are derived from the Latin words for "left" (sinister) and "right" (dexter). Other disciplines use different terms (such as dextro- and laevo-rotary in chemistry, or clockwise and anticlockwise in physics) or simply use left and right (as in anatomy).
Relative direction and chirality are distinct concepts. Relative direction is from the point of view of the observer; a completely symmetric object has a left side and a right side, from the observer's point of view, if the top and bottom and direction of observation are defined. Chirality, however, is observer-independent: no matter how one looks at a right-hand screw thread, it remains different from a left-hand screw thread. Therefore, a symmetric object has sinistral and dextral directions arbitrarily defined by the position of the observer, while an asymmetric object that shows chirality may have sinistral and dextral directions defined by characteristics of the object, regardless of the position of the observer.
Biology
Gastropods
Because the coiled shells of gastropods are asymmetric, they possess a quality called chirality–the "handedness" of an asymmetric structure.
Over 90%[1] of gastropod species have shells in which the direction of the coil is dextral (right-handed). A small minority of species and genera have shells in which the coils are almost always sinistral (left-handed). A very few species show an even mixture of dextral and sinistral individuals (for example, Amphidromus perversus).[2]
Flatfish
The most obvious characteristic of flatfish, other than their flatness, is their asymmetric morphology: both eyes are on the same side of the head in the adult fish. In some families of flatfish, the eyes are always on the right side of the body (dextral or right-eyed flatfish), and in others, they are always on the left (sinistral or left-eyed flatfish). Primitive spiny turbots include equal numbers of right- and left-sided individuals, and are generally more symmetric than other families.[3]
Geology
In geology, the terms sinistral and dextral refer to the horizontal component of movement of blocks on either side of a fault or the sense of movement within a shear zone. These are terms of relative direction, as the movement of the blocks is described relative to each other when viewed from above. Movement is sinistral (left-handed) if the block on the other side of the fault moves to the left, or if straddling the fault the left side moves toward the observer. Movement is dextral (right-handed) if the block on the other side of the fault moves to the right, or if straddling the fault the right side moves toward the observer.[4]
See also
• Dexter and sinister, as used in heraldry
• Helicity (disambiguation)
• Jeremy (snail)
• Laterality
• Left and right (disambiguation)
• Symmetry
References
1. Schilthuizen M. & Davison A. (2005). "The convoluted evolution of snail chirality". Naturwissenschaften 92(11): 504–515. doi:10.1007/s00114-005-0045-2.
2. Amphidromus perversus (Linnaeus, 1758).
3. Chapleau, Francois & Amaoka, Kunio (1998). Paxton, J.R. & Eschmeyer, W.N. (eds.). Encyclopedia of Fishes. San Diego: Academic Press. xxx. ISBN 0-12-547665-5.
4. Park, R.G. (2004). Foundation of Structural Geology (3 ed.). Routledge. p. 11. ISBN 978-0-7487-5802-9.
External links
Look up sinistral or dextral in Wiktionary, the free dictionary.
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Bubble sort
Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the input list element by element, comparing the current element with the one after it, swapping their values if needed. These passes through the list are repeated until no swaps had to be performed during a pass, meaning that the list has become fully sorted. The algorithm, which is a comparison sort, is named for the way the larger elements "bubble" up to the top of the list.
Bubble sort
Static visualization of bubble sort[1]
ClassSorting algorithm
Data structureArray
Worst-case performance$O(n^{2})$ comparisons, $O(n^{2})$ swaps
Best-case performance$O(n)$ comparisons, $O(1)$ swaps
Average performance$O(n^{2})$ comparisons, $O(n^{2})$ swaps
Worst-case space complexity$O(n)$ total, $O(1)$ auxiliary
This simple algorithm performs poorly in real world use and is used primarily as an educational tool. More efficient algorithms such as quicksort, timsort, or merge sort are used by the sorting libraries built into popular programming languages such as Python and Java. However, if parallel processing is allowed, bubble sort sorts in O(n) time, making it considerably faster than parallel implementations of insertion sort or selection sort which do not parallelize as effectively. [2][3]
History
The earliest description of the Bubble sort algorithm was in a 1956 paper by mathematician and actuary Edward Harry Friend,[4] Sorting on electronic computer systems,[5] published in the third issue of the third volume of the Journal of the Association of Computing Machinery (ACM) , as a "Sorting exchange algorithm". Friend described the fundamentals of the algorithm, and, although initially his paper went unnoticed, some years later, it was rediscovered by many computer scientists, including Kenneth E. Iverson who coined its current name.
Analysis
Performance
Bubble sort has a worst-case and average complexity of $O(n^{2})$, where $n$ is the number of items being sorted. Most practical sorting algorithms have substantially better worst-case or average complexity, often $O(n\log n)$. Even other $O(n^{2})$ sorting algorithms, such as insertion sort, generally run faster than bubble sort, and are no more complex. For this reason, bubble sort is rarely used in practice.
Like insertion sort, bubble sort is adaptive, giving it an advantage over algorithms like quicksort. This means that it may outperform those algorithms in cases where the list is already mostly sorted (having a small number of inversions), despite the fact that it has worse average-case time complexity. For example, bubble sort is $O(n)$ on a list that is already sorted, while quicksort would still perform its entire $O(n\log n)$ sorting process.
While any sorting algorithm can be made $O(n)$ on a presorted list simply by checking the list before the algorithm runs, improved performance on almost-sorted lists is harder to replicate.
Rabbits and turtles
The distance and direction that elements must move during the sort determine bubble sort's performance because elements move in different directions at different speeds. An element that must move toward the end of the list can move quickly because it can take part in successive swaps. For example, the largest element in the list will win every swap, so it moves to its sorted position on the first pass even if it starts near the beginning. On the other hand, an element that must move toward the beginning of the list cannot move faster than one step per pass, so elements move toward the beginning very slowly. If the smallest element is at the end of the list, it will take $n-1$ passes to move it to the beginning. This has led to these types of elements being named rabbits and turtles, respectively, after the characters in Aesop's fable of The Tortoise and the Hare.
Various efforts have been made to eliminate turtles to improve upon the speed of bubble sort. Cocktail sort is a bi-directional bubble sort that goes from beginning to end, and then reverses itself, going end to beginning. It can move turtles fairly well, but it retains $O(n^{2})$ worst-case complexity. Comb sort compares elements separated by large gaps, and can move turtles extremely quickly before proceeding to smaller and smaller gaps to smooth out the list. Its average speed is comparable to faster algorithms like quicksort.
Step-by-step example
Take an array of numbers "5 1 4 2 8", and sort the array from lowest number to greatest number using bubble sort. In each step, elements written in bold are being compared. Three passes will be required;
First Pass
( 5 1 4 2 8 ) → ( 1 5 4 2 8 ), Here, algorithm compares the first two elements, and swaps since 5 > 1.
( 1 5 4 2 8 ) → ( 1 4 5 2 8 ), Swap since 5 > 4
( 1 4 5 2 8 ) → ( 1 4 2 5 8 ), Swap since 5 > 2
( 1 4 2 5 8 ) → ( 1 4 2 5 8 ), Now, since these elements are already in order (8 > 5), algorithm does not swap them.
Second Pass
( 1 4 2 5 8 ) → ( 1 4 2 5 8 )
( 1 4 2 5 8 ) → ( 1 2 4 5 8 ), Swap since 4 > 2
( 1 2 4 5 8 ) → ( 1 2 4 5 8 )
( 1 2 4 5 8 ) → ( 1 2 4 5 8 )
Now, the array is already sorted, but the algorithm does not know if it is completed. The algorithm needs one additional whole pass without any swap to know it is sorted.
Third Pass
( 1 2 4 5 8 ) → ( 1 2 4 5 8 )
( 1 2 4 5 8 ) → ( 1 2 4 5 8 )
( 1 2 4 5 8 ) → ( 1 2 4 5 8 )
( 1 2 4 5 8 ) → ( 1 2 4 5 8 )
Implementation
Pseudocode implementation
In pseudocode the algorithm can be expressed as (0-based array):
procedure bubbleSort(A : list of sortable items)
n := length(A)
repeat
swapped := false
for i := 1 to n-1 inclusive do
{ if this pair is out of order }
if A[i-1] > A[i] then
{ swap them and remember something changed }
swap(A[i-1], A[i])
swapped := true
end if
end for
until not swapped
end procedure
Optimizing bubble sort
The bubble sort algorithm can be optimized by observing that the n-th pass finds the n-th largest element and puts it into its final place. So, the inner loop can avoid looking at the last n − 1 items when running for the n-th time:
procedure bubbleSort(A : list of sortable items)
n := length(A)
repeat
swapped := false
for i := 1 to n - 1 inclusive do
if A[i - 1] > A[i] then
swap(A[i - 1], A[i])
swapped := true
end if
end for
n := n - 1
until not swapped
end procedure
More generally, it can happen that more than one element is placed in their final position on a single pass. In particular, after every pass, all elements after the last swap are sorted, and do not need to be checked again. This allows to skip over many elements, resulting in about a worst case 50% improvement in comparison count (though no improvement in swap counts), and adds very little complexity because the new code subsumes the "swapped" variable:
To accomplish this in pseudocode, the following can be written:
procedure bubbleSort(A : list of sortable items)
n := length(A)
repeat
newn := 0
for i := 1 to n - 1 inclusive do
if A[i - 1] > A[i] then
swap(A[i - 1], A[i])
newn := i
end if
end for
n := newn
until n ≤ 1
end procedure
Alternate modifications, such as the cocktail shaker sort attempt to improve on the bubble sort performance while keeping the same idea of repeatedly comparing and swapping adjacent items.
Use
Although bubble sort is one of the simplest sorting algorithms to understand and implement, its O(n2) complexity means that its efficiency decreases dramatically on lists of more than a small number of elements. Even among simple O(n2) sorting algorithms, algorithms like insertion sort are usually considerably more efficient.
Due to its simplicity, bubble sort is often used to introduce the concept of an algorithm, or a sorting algorithm, to introductory computer science students. However, some researchers such as Owen Astrachan have gone to great lengths to disparage bubble sort and its continued popularity in computer science education, recommending that it no longer even be taught.[6]
The Jargon File, which famously calls bogosort "the archetypical [sic] perversely awful algorithm", also calls bubble sort "the generic bad algorithm".[7] Donald Knuth, in The Art of Computer Programming, concluded that "the bubble sort seems to have nothing to recommend it, except a catchy name and the fact that it leads to some interesting theoretical problems", some of which he then discusses.[8]
Bubble sort is asymptotically equivalent in running time to insertion sort in the worst case, but the two algorithms differ greatly in the number of swaps necessary. Experimental results such as those of Astrachan have also shown that insertion sort performs considerably better even on random lists. For these reasons many modern algorithm textbooks avoid using the bubble sort algorithm in favor of insertion sort.
Bubble sort also interacts poorly with modern CPU hardware. It produces at least twice as many writes as insertion sort, twice as many cache misses, and asymptotically more branch mispredictions. Experiments by Astrachan sorting strings in Java show bubble sort to be roughly one-fifth as fast as an insertion sort and 70% as fast as a selection sort.[6]
In computer graphics bubble sort is popular for its capability to detect a very small error (like swap of just two elements) in almost-sorted arrays and fix it with just linear complexity (2n). For example, it is used in a polygon filling algorithm, where bounding lines are sorted by their x coordinate at a specific scan line (a line parallel to the x axis) and with incrementing y their order changes (two elements are swapped) only at intersections of two lines. Bubble sort is a stable sort algorithm, like insertion sort.
Variations
• Odd–even sort is a parallel version of bubble sort, for message passing systems.
• Passes can be from right to left, rather than left to right. This is more efficient for lists with unsorted items added to the end.
• Cocktail shaker sort alternates leftwards and rightwards passes.
Debate over name
Bubble sort has been occasionally referred to as a "sinking sort".[9]
For example, Donald Knuth describes the insertion of values at or towards their desired location as letting "[the value] settle to its proper level", and that "this method of sorting has sometimes been called the sifting or sinking technique.[10]
This debate is perpetuated by the ease with which one may consider this algorithm from two different but equally valid perspectives:
1. The larger values might be regarded as heavier and therefore be seen to progressively sink to the bottom of the list
2. The smaller values might be regarded as lighter and therefore be seen to progressively bubble up to the top of the list.
In popular culture
In 2007, former Google CEO Eric Schmidt asked then-presidential candidate Barack Obama during an interview about the best way to sort one million integers; Obama paused for a moment and replied: "I think the bubble sort would be the wrong way to go."[11][12]
Notes
1. Cortesi, Aldo (27 April 2007). "Visualising Sorting Algorithms". Retrieved 16 March 2017.
2. "[JDK-6804124] (coll) Replace "modified mergesort" in java.util.Arrays.sort with timsort - Java Bug System". bugs.openjdk.java.net. Retrieved 2020-01-11.
3. Peters, Tim (2002-07-20). "[Python-Dev] Sorting". Retrieved 2020-01-11.
4. "EDWARD FRIEND Obituary (2019) - Washington, DC - The Washington Post". Legacy.com.
5. Friend, Edward H. (1956). "Sorting on Electronic Computer Systems". Journal of the ACM. 3 (3): 134–168. doi:10.1145/320831.320833. S2CID 16071355.
6. Astrachan, Owen (2003). "Bubble sort: an archaeological algorithmic analysis" (PDF). ACM SIGCSE Bulletin. 35 (1): 1–5. doi:10.1145/792548.611918. ISSN 0097-8418.
7. "jargon, node: bogo-sort". www.jargon.net.
8. Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Second Edition. Addison-Wesley, 1998. ISBN 0-201-89685-0. Pages 106–110 of section 5.2.2: Sorting by Exchanging. "[A]lthough the techniques used in the calculations [to analyze the bubble sort] are instructive, the results are disappointing since they tell us that the bubble sort isn't really very good at all. Compared to straight insertion […], bubble sorting requires a more complicated program and takes about twice as long!" (Quote from the first edition, 1973.)
9. Black, Paul E. (24 August 2009). "bubble sort". Dictionary of Algorithms and Data Structures. National Institute of Standards and Technology. Retrieved 1 October 2014.
10. Knuth, Donald (1997). The Art of Computer Programming: Volume 3: Searching and Sorting. p. 80. ISBN 0201896850.
11. Lai Stirland, Sarah (2007-11-14). "Obama Passes His Google Interview". Wired. Retrieved 2020-10-27.
12. Barack Obama, Eric Schmidt (Nov 14, 2007). Barack Obama | Candidates at Google (Video) (YouTube). Mountain View, CA 94043 The Googleplex: Talks at Google. Event occurs at 23:20. Archived from the original on September 7, 2019. Retrieved Sep 18, 2019.{{cite AV media}}: CS1 maint: location (link)
References
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Problem 2-2, pg.40.
• Sorting in the Presence of Branch Prediction and Caches
• Fundamentals of Data Structures by Ellis Horowitz, Sartaj Sahni and Susan Anderson-Freed ISBN 81-7371-605-6
• Owen Astrachan. Bubble Sort: An Archaeological Algorithmic Analysis
• Computer Integrated Manufacturing by Spasic PhD, Srdic MSc, Open Source, 1987.
External links
The Wikibook Algorithm implementation has a page on the topic of: Bubble sort
Wikimedia Commons has media related to Bubble sort.
Wikiversity has learning resources about Bubble sort
• Martin, David R. (2007). "Animated Sorting Algorithms: Bubble Sort". Archived from the original on 2015-03-03. – graphical demonstration
• "Lafore's Bubble Sort". (Java applet animation)
• OEIS sequence A008302 (Table (statistics) of the number of permutations of [n] that need k pair-swaps during the sorting)
Sorting algorithms
Theory
• Computational complexity theory
• Big O notation
• Total order
• Lists
• Inplacement
• Stability
• Comparison sort
• Adaptive sort
• Sorting network
• Integer sorting
• X + Y sorting
• Transdichotomous model
• Quantum sort
Exchange sorts
• Bubble sort
• Cocktail shaker sort
• Odd–even sort
• Comb sort
• Gnome sort
• Proportion extend sort
• Quicksort
Selection sorts
• Selection sort
• Heapsort
• Smoothsort
• Cartesian tree sort
• Tournament sort
• Cycle sort
• Weak-heap sort
Insertion sorts
• Insertion sort
• Shellsort
• Splaysort
• Tree sort
• Library sort
• Patience sorting
Merge sorts
• Merge sort
• Cascade merge sort
• Oscillating merge sort
• Polyphase merge sort
Distribution sorts
• American flag sort
• Bead sort
• Bucket sort
• Burstsort
• Counting sort
• Interpolation sort
• Pigeonhole sort
• Proxmap sort
• Radix sort
• Flashsort
Concurrent sorts
• Bitonic sorter
• Batcher odd–even mergesort
• Pairwise sorting network
• Samplesort
Hybrid sorts
• Block merge sort
• Kirkpatrick–Reisch sort
• Timsort
• Introsort
• Spreadsort
• Merge-insertion sort
Other
• Topological sorting
• Pre-topological order
• Pancake sorting
• Spaghetti sort
Impractical sorts
• Stooge sort
• Slowsort
• Bogosort
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Wikipedia
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Sinkhorn's theorem
Sinkhorn's theorem states that every square matrix with positive entries can be written in a certain standard form.
Theorem
If A is an n × n matrix with strictly positive elements, then there exist diagonal matrices D1 and D2 with strictly positive diagonal elements such that D1AD2 is doubly stochastic. The matrices D1 and D2 are unique modulo multiplying the first matrix by a positive number and dividing the second one by the same number.[1] [2]
Sinkhorn–Knopp algorithm
A simple iterative method to approach the double stochastic matrix is to alternately rescale all rows and all columns of A to sum to 1. Sinkhorn and Knopp presented this algorithm and analyzed its convergence.[3] This is essentially the same as the Iterative proportional fitting algorithm, well known in survey statistics.
Analogues and extensions
The following analogue for unitary matrices is also true: for every unitary matrix U there exist two diagonal unitary matrices L and R such that LUR has each of its columns and rows summing to 1.[4]
The following extension to maps between matrices is also true (see Theorem 5[5] and also Theorem 4.7[6]): given a Kraus operator that represents the quantum operation Φ mapping a density matrix into another,
$S\mapsto \Phi (S)=\sum _{i}B_{i}SB_{i}^{*},$
that is trace preserving,
$\sum _{i}B_{i}^{*}B_{i}=I,$
and, in addition, whose range is in the interior of the positive definite cone (strict positivity), there exist scalings xj, for j in {0,1}, that are positive definite so that the rescaled Kraus operator
$S\mapsto x_{1}\Phi (x_{0}^{-1}Sx_{0}^{-1})x_{1}=\sum _{i}(x_{1}B_{i}x_{0}^{-1})S(x_{1}B_{i}x_{0}^{-1})^{*}$
is doubly stochastic. In other words, it is such that both,
$x_{1}\Phi (x_{0}^{-1}Ix_{0}^{-1})x_{1}=I,$
as well as for the adjoint,
$x_{0}^{-1}\Phi ^{*}(x_{1}Ix_{1})x_{0}^{-1}=I,$
where I denotes the identity operator.
Applications
In the 2010s Sinkhorn's theorem came to be used to find solutions of entropy-regularised optimal transport problems.[7] This has been of interest in machine learning because such "Sinkhorn distances" can be used to evaluate the difference between data distributions and permutations.[8][9][10] This improves the training of machine learning algorithms, in situations where maximum likelihood training may not be the best method.
References
1. Sinkhorn, Richard. (1964). "A relationship between arbitrary positive matrices and doubly stochastic matrices." Ann. Math. Statist. 35, 876–879. doi:10.1214/aoms/1177703591
2. Marshall, A.W., & Olkin, I. (1967). "Scaling of matrices to achieve specified row and column sums." Numerische Mathematik. 12(1), 83–90. doi:10.1007/BF02170999
3. Sinkhorn, Richard, & Knopp, Paul. (1967). "Concerning nonnegative matrices and doubly stochastic matrices". Pacific J. Math. 21, 343–348.
4. Idel, Martin; Wolf, Michael M. (2015). "Sinkhorn normal form for unitary matrices". Linear Algebra and Its Applications. 471: 76–84. arXiv:1408.5728. doi:10.1016/j.laa.2014.12.031. S2CID 119175915.
5. Georgiou, Tryphon; Pavon, Michele (2015). "Positive contraction mappings for classical and quantum Schrödinger systems". Journal of Mathematical Physics. 56 (3): 033301–1–24. arXiv:1405.6650. Bibcode:2015JMP....56c3301G. doi:10.1063/1.4915289. S2CID 119707158.
6. Gurvits, Leonid (2004). "Classical complexity and quantum entanglement". Journal of Computational Science. 69 (3): 448–484. doi:10.1016/j.jcss.2004.06.003.
7. Cuturi, Marco (2013). "Sinkhorn distances: Lightspeed computation of optimal transport". Advances in neural information processing systems. pp. 2292–2300.
8. Mensch, Arthur; Blondel, Mathieu; Peyre, Gabriel (2019). "Geometric losses for distributional learning". Proc ICML 2019. arXiv:1905.06005.
9. Mena, Gonzalo; Belanger, David; Munoz, Gonzalo; Snoek, Jasper (2017). "Sinkhorn networks: Using optimal transport techniques to learn permutations". NIPS Workshop in Optimal Transport and Machine Learning.
10. Kogkalidis, Konstantinos; Moortgat, Michael; Moot, Richard (2020). "Neural Proof Nets". Proceedings of the 24th Conference on Computational Natural Language Learning.
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Wikipedia
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Trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions[1][2]) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions.
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Notation
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example sin(x). Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression $\sin x+y$ would typically be interpreted to mean $\sin(x)+y,$ so parentheses are required to express $\sin(x+y).$
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example $\sin ^{2}x$ and $\sin ^{2}(x)$ denote $\sin(x)\cdot \sin(x),$ not $\sin(\sin x).$ This differs from the (historically later) general functional notation in which $f^{2}(x)=(f\circ f)(x)=f(f(x)).$
However, the exponent ${-1}$ is commonly used to denote the inverse function, not the reciprocal. For example $\sin ^{-1}x$ and $\sin ^{-1}(x)$ denote the inverse trigonometric function alternatively written $\arcsin x\colon $ The equation $\theta =\sin ^{-1}x$ implies $\sin \theta =x,$ not $\theta \cdot \sin x=1.$ In this case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than ${-1}$ are not in common use.
Right-angled triangle definitions
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ. Thus these six ratios define six functions of θ, which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ, and adjacent represents the side between the angle θ and the right angle.[3][4]
sine
$\sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}$
cosecant
$\csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}$
cosine
$\cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}$
secant
$\sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}$
tangent
$\tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}$
cotangent
$\cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}$
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π/2 radians. Therefore $\sin(\theta )$ and $\cos(90^{\circ }-\theta )$ represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
Summary of relationships between trigonometric functions[5]
Function Description Relationship
using radians using degrees
sine opposite/hypotenuse $\sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}$ $\sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}$
cosine adjacent/hypotenuse $\cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,$ $\cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,$
tangent opposite/adjacent $\tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}$ $\tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}$
cotangent adjacent/opposite $\cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}$ $\cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}$
secant hypotenuse/adjacent $\sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}$ $\sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}$
cosecant hypotenuse/opposite $\csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}$ $\csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}$
Radians versus degrees
In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient. One common unit is degrees, in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics).
However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function, via power series,[6] or as solutions to differential equations given particular initial values[7] (see below), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle given in radians.[6] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions.[8] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2π (≈ 6.28) rad. For real number x, the notations sin x, cos x, etc. refer to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/π)°, so that, for example, sin π = sin 180° when we take x = π. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π/180 ≈ 0.0175.
Unit-circle definitions
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and $ {\frac {\pi }{2}}$ radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let ${\mathcal {L}}$ be the ray obtained by rotating by an angle θ the positive half of the x-axis (counterclockwise rotation for $\theta >0,$ and clockwise rotation for $\theta <0$). This ray intersects the unit circle at the point $\mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).$ The ray ${\mathcal {L}},$ extended to a line if necessary, intersects the line of equation $x=1$ at point $\mathrm {B} =(1,y_{\mathrm {B} }),$ and the line of equation $y=1$ at point $\mathrm {C} =(x_{\mathrm {C} },1).$ The tangent line to the unit circle at the point A, is perpendicular to ${\mathcal {L}},$ and intersects the y- and x-axes at points $\mathrm {D} =(0,y_{\mathrm {D} })$ and $\mathrm {E} =(x_{\mathrm {E} },0).$ The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A. That is,
$\cos \theta =x_{\mathrm {A} }\quad $ and $\quad \sin \theta =y_{\mathrm {A} }.$[10]
In the range $0\leq \theta \leq \pi /2$, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse. And since the equation $x^{2}+y^{2}=1$ holds for all points $\mathrm {P} =(x,y)$ on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity.
$\cos ^{2}\theta +\sin ^{2}\theta =1.$
The other trigonometric functions can be found along the unit circle as
$\tan \theta =y_{\mathrm {B} }\quad $ and $\quad \cot \theta =x_{\mathrm {C} },$
$\csc \theta \ =y_{\mathrm {D} }\quad $ and $\quad \sec \theta =x_{\mathrm {E} }.$
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
$\tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.$
Since a rotation of an angle of $\pm 2\pi $ does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of $2\pi $. Thus trigonometric functions are periodic functions with period $2\pi $. That is, the equalities
$\sin \theta =\sin \left(\theta +2k\pi \right)\quad $ and $\quad \cos \theta =\cos \left(\theta +2k\pi \right)$
hold for any angle θ and any integer k. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that $2\pi $ is the smallest value for which they are periodic (i.e., $2\pi $ is the fundamental period of these functions). However, after a rotation by an angle $\pi $, the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of $\pi $. That is, the equalities
$\tan \theta =\tan(\theta +k\pi )\quad $ and $\quad \cot \theta =\cot(\theta +k\pi )$
hold for any angle θ and any integer k.
Algebraic values
The algebraic expressions for the most important angles are as follows:
$\sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0$ (zero angle)
$\sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}$
$\sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}$
$\sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}$
$\sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1$ (right angle)
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.[11]
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
• For an angle which, measured in degrees, is a multiple of three, the exact trigonometric values of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by ruler and compass.
• For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.
• For an angle which, expressed in degrees, is a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of nth roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.
• For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem, proved in 1966.
Simple algebraic values
Main article: Exact trigonometric values § Common angles
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
Angle, θ, in $\sin(\theta )$ $\cos(\theta )$ $\tan(\theta )$
radians degrees
$0$ $0^{\circ }$ $0$ $1$ $0$
${\frac {\pi }{12}}$ $15^{\circ }$ ${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ ${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ $2-{\sqrt {3}}$
${\frac {\pi }{6}}$ $30^{\circ }$ ${\frac {1}{2}}$ ${\frac {\sqrt {3}}{2}}$ ${\frac {\sqrt {3}}{3}}$
${\frac {\pi }{4}}$ $45^{\circ }$ ${\frac {\sqrt {2}}{2}}$ ${\frac {\sqrt {2}}{2}}$ $1$
${\frac {\pi }{3}}$ $60^{\circ }$ ${\frac {\sqrt {3}}{2}}$ ${\frac {1}{2}}$ ${\sqrt {3}}$
${\frac {5\pi }{12}}$ $75^{\circ }$ ${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ ${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ $2+{\sqrt {3}}$
${\frac {\pi }{2}}$ $90^{\circ }$ $1$ $0$ Undefined
In calculus
The modern trend in mathematics is to build geometry from calculus rather than the converse. Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus.
Trigonometric functions are differentiable and analytic at every point where they are defined; that is, everywhere for the sine and the cosine, and, for the tangent, everywhere except at π/2 + kπ for every integer k.
The trigonometric function are periodic functions, and their primitive period is 2π for the sine and the cosine, and π for the tangent, which is increasing in each open interval (π/2 + kπ, π/2 + (k + 1)π). At each end point of these intervals, the tangent function has a vertical asymptote.
In calculus, there are two equivalent definitions of trigonometric functions, either using power series or differential equations. These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.
Definition by differential equations
Sine and cosine can be defined as the unique solution to the initial value problem:
${\frac {d}{dx}}\sin x=\cos x,\ {\frac {d}{dx}}\cos x=-\sin x,\ \sin(0)=0,\ \cos(0)=1.$
Differentiating again, $ {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x$ and $ {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x$, so both sine and cosine are solutions of the ordinary differential equation
$y''+y=0.$
Applying the quotient rule to the tangent $\tan x=\sin x/\cos x$, we derive
${\frac {d}{dx}}\tan x={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}=1+\tan ^{2}x=\sec ^{2}x.$
Power series expansion
Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions[12]
${\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}.\end{aligned}}$
The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form $ (2k+1){\frac {\pi }{2}}$ for the tangent and the secant, or $k\pi $ for the cotangent and the cosecant, where k is an arbitrary integer.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.[13]
More precisely, defining
Un, the nth up/down number,
Bn, the nth Bernoulli number, and
En, is the nth Euler number,
one has the following series expansions:[14]
${\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}}{(2n+1)!}}x^{2n+1}\\[8mu]&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}$
${\begin{aligned}\csc x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}$
${\begin{aligned}\sec x&=\sum _{n=0}^{\infty }{\frac {U_{2n}}{(2n)!}}x^{2n}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\\[5mu]&=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}$
${\begin{aligned}\cot x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}$
Continued fraction expansion
The following expansions are valid in the whole complex plane:
$\sin x={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}$
$\cos x={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}$
$\tan x={\cfrac {x}{1-{\cfrac {x^{2}}{3-{\cfrac {x^{2}}{5-{\cfrac {x^{2}}{7-\ddots }}}}}}}}={\cfrac {1}{{\cfrac {1}{x}}-{\cfrac {1}{{\cfrac {3}{x}}-{\cfrac {1}{{\cfrac {5}{x}}-{\cfrac {1}{{\cfrac {7}{x}}-\ddots }}}}}}}}$
The last one was used in the historically first proof that π is irrational.[15]
Partial fraction expansion
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:[16]
$\pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.$
This identity can be proved with the Herglotz trick.[17] Combining the (–n)th with the nth term lead to absolutely convergent series:
$\pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}.$
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
$\pi \csc \pi x=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{x+n}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}},$
$\pi ^{2}\csc ^{2}\pi x=\sum _{n=-\infty }^{\infty }{\frac {1}{(x+n)^{2}}},$
$\pi \sec \pi x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n+1)}{(n+{\tfrac {1}{2}})^{2}-x^{2}}},$
$\pi \tan \pi x=2x\sum _{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.$
Infinite product expansion
The following infinite product for the sine is of great importance in complex analysis:
$\sin z=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .$
For the proof of this expansion, see Sine. From this, it can be deduced that
$\cos z=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{(n-1/2)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .$
Relationship to exponential function (Euler's formula)
Euler's formula relates sine and cosine to the exponential function:
$e^{ix}=\cos x+i\sin x.$
This formula is commonly considered for real values of x, but it remains true for all complex values.
Proof: Let $f_{1}(x)=\cos x+i\sin x,$ and $f_{2}(x)=e^{ix}.$ One has $df_{j}(x)/dx=if_{j}(x)$ for j = 1, 2. The quotient rule implies thus that $d/dx\,(f_{1}(x)/f_{2}(x))=0$. Therefore, $f_{1}(x)/f_{2}(x)$ is a constant function, which equals 1, as $f_{1}(0)=f_{2}(0)=1.$ This proves the formula.
One has
${\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}$
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
${\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}$
When x is real, this may be rewritten as
$\cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).$
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity $e^{a+b}=e^{a}e^{b}$ for simplifying the result.
Definitions using functional equations
One can also define the trigonometric functions using various functional equations.
For example,[18] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula
$\cos(x-y)=\cos x\cos y+\sin x\sin y\,$
and the added condition
$0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.$
In the complex plane
The sine and cosine of a complex number $z=x+iy$ can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:
${\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}$
By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of $z$ becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
Trigonometric functions in the complex plane
$\sin z\,$
$\cos z\,$
$\tan z\,$
$\cot z\,$
$\sec z\,$
$\csc z\,$
Basic identities
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π/2], see Proofs of trigonometric identities). For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
Parity
The cosine and the secant are even functions; the other trigonometric functions are odd functions. That is:
${\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}$
Periods
All trigonometric functions are periodic functions of period 2π. This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k, one has
${\begin{array}{lrl}\sin(x+&2k\pi )&=\sin x\\\cos(x+&2k\pi )&=\cos x\\\tan(x+&k\pi )&=\tan x\\\cot(x+&k\pi )&=\cot x\\\csc(x+&2k\pi )&=\csc x\\\sec(x+&2k\pi )&=\sec x.\end{array}}$
Pythagorean identity
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
$\sin ^{2}x+\cos ^{2}x=1$.
Dividing through by either $\cos ^{2}x$ or $\sin ^{2}x$ gives
$\tan ^{2}x+1=\sec ^{2}x$
and
$1+\cot ^{2}x=\csc ^{2}x$.
Sum and difference formulas
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.
Sum
${\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\[5mu]\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\[5mu]\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}$
Difference
${\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\[5mu]\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\[5mu]\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}$
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
${\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\[5mu]\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\[5mu]\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}$
These identities can be used to derive the product-to-sum identities.
By setting $t=\tan {\tfrac {1}{2}}\theta ,$ all trigonometric functions of $\theta $ can be expressed as rational fractions of $t$:
${\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\[5mu]\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\[5mu]\tan \theta &={\frac {2t}{1-t^{2}}}.\end{aligned}}$
Together with
$d\theta ={\frac {2}{1+t^{2}}}\,dt,$
this is the tangent half-angle substitution, which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.
Derivatives and antiderivatives
The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration.
$f(x)$$f'(x)$$ \int f(x)\,dx$
$\sin x$$\cos x$$-\cos x+C$
$\cos x$$-\sin x$$\sin x+C$
$\tan x$$\sec ^{2}x$$\ln \left|\sec x\right|+C$
$\csc x$$-\csc x\cot x$$\ln \left|\csc x-\cot x\right|+C$
$\sec x$$\sec x\tan x$$\ln \left|\sec x+\tan x\right|+C$
$\cot x$$-\csc ^{2}x$$\ln \left|\sin x\right|+C$
Note: For $0<x<\pi $ the integral of $\csc x$ can also be written as $-\operatorname {arsinh} (\cot x),$ and for the integral of $\sec x$ for $-\pi /2<x<\pi /2$ as $\operatorname {arsinh} (\tan x),$ where $\operatorname {arsinh} $ is the inverse hyperbolic sine.
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
${\begin{aligned}{\frac {d\cos x}{dx}}&={\frac {d}{dx}}\sin(\pi /2-x)=-\cos(\pi /2-x)=-\sin x\,,\\{\frac {d\csc x}{dx}}&={\frac {d}{dx}}\sec(\pi /2-x)=-\sec(\pi /2-x)\tan(\pi /2-x)=-\csc x\cot x\,,\\{\frac {d\cot x}{dx}}&={\frac {d}{dx}}\tan(\pi /2-x)=-\sec ^{2}(\pi /2-x)=-\csc ^{2}x\,.\end{aligned}}$
Inverse functions
Main article: Inverse trigonometric functions
The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
FunctionDefinitionDomainSet of principal values
$y=\arcsin x$$\sin y=x$$-1\leq x\leq 1$$ -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}$
$y=\arccos x$$\cos y=x$$-1\leq x\leq 1$$ 0\leq y\leq \pi $
$y=\arctan x$$\tan y=x$$-\infty <x<\infty $$ -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}$
$y=\operatorname {arccot} x$$\cot y=x$$-\infty <x<\infty $$ 0<y<\pi $
$y=\operatorname {arcsec} x$$\sec y=x$$x<-1{\text{ or }}x>1$$ 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}$
$y=\operatorname {arccsc} x$$\csc y=x$$x<-1{\text{ or }}x>1$$ -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0$
The notations sin−1, cos−1, etc. are often used for arcsin and arccos, etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "arcsecond".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms.
Applications
Main article: Uses of trigonometry
Angles and sides of a triangle
In this section A, B, C denote the three (interior) angles of a triangle, and a, b, c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
Law of sines
Main article: Law of sines
The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:
${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},$
where Δ is the area of the triangle, or, equivalently,
${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,$
where R is the triangle's circumradius.
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
Law of cosines
Main article: Law of cosines
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:
$c^{2}=a^{2}+b^{2}-2ab\cos C,$
or equivalently,
$\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.$
In this formula the angle at C is opposite to the side c. This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem.
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
Law of tangents
Main article: Law of tangents
The law of tangents says that:
${\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}$.
Law of cotangents
Main article: Law of cotangents
If s is the triangle's semiperimeter, (a + b + c)/2, and r is the radius of the triangle's incircle, then rs is the triangle's area. Therefore Heron's formula implies that:
$r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}$.
The law of cotangents says that:[19]
$\cot {\frac {A}{2}}={\frac {s-a}{r}}$
It follows that
${\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.$
Periodic functions
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.
Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.[20]
Under rather general conditions, a periodic function f (x) can be expressed as a sum of sine waves or cosine waves in a Fourier series.[21] Denoting the sine or cosine basis functions by φk, the expansion of the periodic function f (t) takes the form:
$f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).$
For example, the square wave can be written as the Fourier series
$f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.$
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
History
Main article: History of trigonometry
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) can be traced back to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.[22] (See Aryabhata's sine table.)
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.[23] With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant.[23] Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Circa 830, Habash al-Hasib al-Marwazi discovered the cotangent, and produced tables of tangents and cotangents.[24][25] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.[25] The trigonometric functions were later studied by mathematicians including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.
Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series.[26] (See Madhava series and Madhava's sine table.)
The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.[27]
The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).[28]
The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie.[29]
In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x.[30] Though introduced as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series. He presented "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).[22]
A few functions were common historically, but are now seldom used, such as the chord, the versine (which appeared in the earliest tables[22]), the coversine, the haversine,[31] the exsecant and the excosecant. The list of trigonometric identities shows more relations between these functions.
• crd(θ) = 2 sin(θ/2)
• versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)
• coversin(θ) = 1 − sin(θ) = versin(π/2 − θ)
• haversin(θ) = 1/2versin(θ) = sin2(θ/2)
• exsec(θ) = sec(θ) − 1
• excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1
Etymology
Main article: History of trigonometry § Etymology
The word sine derives[32] from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin.[33] The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".[34]
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle.[35]
The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.[36][37]
See also
• All Students Take Calculus – a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane
• Bhaskara I's sine approximation formula
• Small-angle approximation
• Differentiation of trigonometric functions
• Generalized trigonometry
• Generating trigonometric tables
• Hyperbolic function
• List of integrals of trigonometric functions
• List of periodic functions
• List of trigonometric identities
• Polar sine – a generalization to vertex angles
• Proofs of trigonometric identities
• Versine – for several less used trigonometric functions and unit circle diagrams of all functions
• Chord (geometry)#In trigonometry
Notes
1. Klein, Christian Felix (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). Vol. 1 (3rd ed.). Berlin: J. Springer.
2. Klein, Christian Felix (2004) [1932]. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company. ISBN 978-0-48643480-3. Archived from the original on 2018-02-15. Retrieved 2017-08-13.
3. Protter & Morrey (1970, pp. APP-2, APP-3)
4. "Sine, Cosine, Tangent". www.mathsisfun.com. Retrieved 2020-08-29.
5. Protter & Morrey (1970, p. APP-7)
6. Rudin, Walter, 1921–2010. Principles of mathematical analysis (Third ed.). New York. ISBN 0-07-054235-X. OCLC 1502474.{{cite book}}: CS1 maint: multiple names: authors list (link)
7. Diamond, Harvey (2014). "Defining Exponential and Trigonometric Functions Using Differential Equations". Mathematics Magazine. 87 (1): 37–42. doi:10.4169/math.mag.87.1.37. ISSN 0025-570X. S2CID 126217060.
8. Spivak, Michael (1967). "15". Calculus. Addison-Wesley. pp. 256–257. LCCN 67-20770.
9. Heng, H. H.; Cheng, Khoo; Talbert, J. F. (2001). Additional Mathematics. Pearson Education South Asia. ISBN 978-981-235-211-8.
10. Bityutskov, V.I. (2011-02-07). "Trigonometric Functions". Encyclopedia of Mathematics. Archived from the original on 2017-12-29. Retrieved 2017-12-29.
11. Larson, Ron (2013). Trigonometry (9th ed.). Cengage Learning. p. 153. ISBN 978-1-285-60718-4. Archived from the original on 2018-02-15. Extract of page 153 Archived 15 February 2018 at the Wayback Machine
12. See Ahlfors, pp. 43–44.
13. Stanley, Enumerative Combinatorics, Vol I., p. 149
14. Abramowitz; Weisstein.
15. Lambert, Johann Heinrich (2004) [1768], "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", in Berggren, Lennart; Borwein, Jonathan M.; Borwein, Peter B. (eds.), Pi, a source book (3rd ed.), New York: Springer-Verlag, pp. 129–140, ISBN 0-387-20571-3
16. Aigner, Martin; Ziegler, Günter M. (2000). Proofs from THE BOOK (Second ed.). Springer-Verlag. p. 149. ISBN 978-3-642-00855-9. Archived from the original on 2014-03-08.
17. Remmert, Reinhold (1991). Theory of complex functions. Springer. p. 327. ISBN 978-0-387-97195-7. Archived from the original on 2015-03-20. Extract of page 327 Archived 20 March 2015 at the Wayback Machine
18. Kannappan, Palaniappan (2009). Functional Equations and Inequalities with Applications. Springer. ISBN 978-0387894911.
19. The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, pp. 529–530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.
20. Farlow, Stanley J. (1993). Partial differential equations for scientists and engineers (Reprint of Wiley 1982 ed.). Courier Dover Publications. p. 82. ISBN 978-0-486-67620-3. Archived from the original on 2015-03-20.
21. See for example, Folland, Gerald B. (2009). "Convergence and completeness". Fourier Analysis and its Applications (Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 77ff. ISBN 978-0-8218-4790-9. Archived from the original on 2015-03-19.
22. Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7, p. 210.
23. Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. p. 74. Archived from the original on 2013-10-19. Retrieved 2010-07-13.
24. Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer Science+Business Media. ISBN 978-1-4020-0260-1.
25. "trigonometry". Encyclopedia Britannica.
26. O'Connor, J. J.; Robertson, E. F. "Madhava of Sangamagrama". School of Mathematics and Statistics University of St Andrews, Scotland. Archived from the original on 2006-05-14. Retrieved 2007-09-08.
27. Van Brummelen, Glen (2018). "The end of an error: Bianchini, Regiomontanus, and the tabulation of stellar coordinates". Archive for History of Exact Sciences. 72 (5): 547–563. doi:10.1007/s00407-018-0214-2. JSTOR 45211959. S2CID 240294796.
28. "Fincke biography". Archived from the original on 2017-01-07. Retrieved 2017-03-15.
29. O'Connor, John J.; Robertson, Edmund F., "Trigonometric functions", MacTutor History of Mathematics Archive, University of St Andrews
30. Bourbaki, Nicolás (1994). Elements of the History of Mathematics. Springer. ISBN 9783540647676.
31. Nielsen (1966, pp. xxiii–xxiv)
32. The anglicized form is first recorded in 1593 in Thomas Fale's Horologiographia, the Art of Dialling.
33. Various sources credit the first use of sinus to either
• Plato Tiburtinus's 1116 translation of the Astronomy of Al-Battani
• Gerard of Cremona's translation of the Algebra of al-Khwārizmī
• Robert of Chester's 1145 translation of the tables of al-Khwārizmī
See Merlet, A Note on the History of the Trigonometric Functions in Ceccarelli (ed.), International Symposium on History of Machines and Mechanisms, Springer, 2004
See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.
See Katx, Victor (July 2008). A history of mathematics (3rd ed.). Boston: Pearson. p. 210 (sidebar). ISBN 978-0321387004.
34. See Plofker, Mathematics in India, Princeton University Press, 2009, p. 257
See "Clark University". Archived from the original on 2008-06-15.
See Maor (1998), chapter 3, regarding the etymology.
35. Oxford English Dictionary
36. Gunter, Edmund (1620). Canon triangulorum.
37. Roegel, Denis, ed. (2010-12-06). "A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938. Archived from the original on 2017-07-28. Retrieved 2017-07-28.
References
• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Lars Ahlfors, Complex Analysis: an introduction to the theory of analytic functions of one complex variable, second edition, McGraw-Hill Book Company, New York, 1966.
• Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.
• Cajori, Florian (1929). "§2.2.1. Trigonometric Notations". A History of Mathematical Notations. Vol. 2. Open Court. pp. 142–179 (¶511–537).
• Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (1991).
• Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
• Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," IEEE Trans. Computers 45 (3), 328–339 (1996).
• Maor, Eli, Trigonometric Delights, Princeton Univ. Press. (1998). Reprint edition (2002): ISBN 0-691-09541-8.
• Needham, Tristan, "Preface"" to Visual Complex Analysis. Oxford University Press, (1999). ISBN 0-19-853446-9.
• Nielsen, Kaj L. (1966), Logarithmic and Trigonometric Tables to Five Places (2nd ed.), New York: Barnes & Noble, LCCN 61-9103
• O'Connor, J. J., and E. F. Robertson, "Trigonometric functions", MacTutor History of Mathematics archive. (1996).
• O'Connor, J. J., and E. F. Robertson, "Madhava of Sangamagramma", MacTutor History of Mathematics archive. (2000).
• Pearce, Ian G., "Madhava of Sangamagramma" Archived 2006-05-05 at the Wayback Machine, MacTutor History of Mathematics archive. (2002).
• Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042
• Weisstein, Eric W., "Tangent" from MathWorld, accessed 21 January 2006.
External links
Wikibooks has a book on the topic of: Trigonometry
• "Trigonometric functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Visionlearning Module on Wave Mathematics
• GonioLab Visualization of the unit circle, trigonometric and hyperbolic functions
• q-Sine Article about the q-analog of sin at MathWorld
• q-Cosine Article about the q-analog of cos at MathWorld
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Wikipedia
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Sinus totus
In trigonometry, the sinus totus (Latin for "total sine") was historically the radius of the base circle used to construct a sine table; that is, the maximum possible value of the sine.
Letting the notation $\operatorname {Sin} \theta $ stand for the historical sine, and $\sin \theta $ stand for the modern sine function,
$\operatorname {Sin} \theta =R\sin \theta ,$
where $R$ is the sinus totus, $R=\operatorname {Sin} 90^{\circ }.$
References
• Gupta, Radha Charan (1977). "Indian values of the sinus totus". Indian Journal of History of Science Calcutta. 13 (2): 125–143. Presented at the 15th International Congress of History of Science, Edinburgh, 1977. Reprinted in Ramasubramanian, K., ed. (2019). Gaṇitānanda: Selected Works of Radha Charan Gupta on History of Mathematics. Springer. pp. 397–415. doi:10.1007/978-981-13-1229-8_39.
• Roegel, Denis (2021). A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries (Report). LORIA (Université de Lorraine, CNRS, INRIA). hal-03330572.
• Van Brummelen, Glen (2021). "1. European Trigonometry Comes of Age". The Doctrine of Triangles: A History of Modern Trigonometry. Princeton University Press. pp. 5, 11, 13, 33. doi:10.1515/9780691219875-002.
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Wikipedia
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Versine
The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Aryabhatia,[1] Section I) trigonometric tables. The versine of an angle is 1 minus its cosine.
For the archaic unit of mass, see Versine in WP:de.
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
Look up versine or versed sine in Wiktionary, the free dictionary.
There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation.
Overview
The versine[3][4][5][6][7] or versed sine[8][9][10][11][12] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin, sinver,[13][14] vers, ver[15] or siv.[16][17] In Latin, it is known as the sinus versus (flipped sine), versinus, versus, or sagitta (arrow).[18]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to
$\operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}$
There are several related functions corresponding to the versine:
• The versed cosine,[19][nb 1] or vercosine, abbreviated vercosin, vercos, or vcs.
• The coversed sine or coversine[20] (in Latin, cosinus versus or coversinus), abbreviated coversin,[21] covers,[22][23][24] cosiv, or cvs[25]
• The coversed cosine[26] or covercosine, abbreviated covercosin, covercos, or cvc
In full analogy to the above-mentioned four functions another set of four "half-value" functions exists as well:
• The haversed sine[27] or haversine (Latin semiversus),[28][29] abbreviated haversin, semiversin, semiversinus, havers, hav,[30][31] hvs,[nb 2] sem, or hv,[32] most famous from the haversine formula used historically in navigation
• The haversed cosine[33] or havercosine, abbreviated havercosin, havercos, hac or hvc
• The hacoversed sine, hacoversine,[21] or cohaversine, abbreviated hacoversin, semicoversin, hacovers, hacov[34] or hcv
• The hacoversed cosine,[35] hacovercosine, or cohavercosine, abbreviated hacovercosin, hacovercos or hcc
History and applications
Versine and coversine
The ordinary sine function (see note on etymology) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine (sinus versus).[37] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle:
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos(θ) (equal to the length of line OC) and versin(θ) (equal to the length of line CD) is the radius OD (with length 1). Illustrated this way, the sine is vertical (rectus, literally "straight") while the versine is horizontal (versus, literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow,[18][36] from the Arabic usage sahem[38] of the same meaning. This itself comes from the Indian word 'sara' (arrow) that was commonly used to refer to "utkrama-jya". If the arc ADB of the double-angle Δ = 2θ is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph).[36]
In 1821, Cauchy used the terms sinus versus (siv) for the versine and cosinus versus (cosiv) for the coversine.[16][17][nb 1]
Historically, the versed sine was considered one of the most important trigonometric functions.[12][37][38]
As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient.[12] Even with a calculator or computer, round-off errors make it advisable to use the sin2 formula for small θ.
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle (θ = 0, 2π, …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half-chord) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°).[37]
The versine appears as an intermediate step in the application of the half-angle formula sin2(θ/2) = 1/2versin(θ), derived by Ptolemy, that was used to construct such tables.
Haversine
The haversine, in particular, was important in navigation because it appears in the haversine formula, which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the earth's radius vs. sphere) given angular positions (e.g., longitude and latitude). One could also use sin2(θ/2) directly, but having a table of the haversine removed the need to compute squares and square roots.[12]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801.[14][39]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords".[40][41][18]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers.) was coined[42] by James Inman[14][43][44] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the earth using spherical trigonometry for applications in navigation.[3][42] Inman also used the terms nat. versine and nat. vers. for versines.[3]
Other high-regarded tables of haversines were those of Richard Farley in 1856[40][45] and John Caulfield Hannyngton in 1876.[40][46]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995[47][48] or in a more compact method for sight reduction since 2014.[32]
Modern uses
Whilst the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2π) of a versine or, more commonly, a haversine (or havercosine) waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann, Hann–Poisson and Tukey windows), because it smoothly (continuous in value and slope) "turns on" from zero to one (for haversine) and back to zero.[nb 2] In these applications, it is named Hann function or raised-cosine filter. Likewise, the havercosine is used in raised-cosine distributions in probability theory and statistics.
In the form of sin2(θ) the haversine of the double-angle Δ describes the relation between spreads and angles in rational trigonometry, a proposed reformulation of metrical planar and solid geometries by Norman John Wildberger since 2005.[49]
Mathematical identities
Definitions
${\textrm {versin}}(\theta ):=2\sin ^{2}\!\left({\frac {\theta }{2}}\right)=1-\cos(\theta )\,$[4]
${\textrm {coversin}}(\theta ):={\textrm {versin}}\!\left({\frac {\pi }{2}}-\theta \right)=1-\sin(\theta )\,$[4]
${\textrm {vercosin}}(\theta ):=2\cos ^{2}\!\left({\frac {\theta }{2}}\right)=1+\cos(\theta )\,$[19]
${\textrm {covercosin}}(\theta ):={\textrm {vercosin}}\!\left({\frac {\pi }{2}}-\theta \right)=1+\sin(\theta )\,$[26]
${\textrm {haversin}}(\theta ):={\frac {{\textrm {versin}}(\theta )}{2}}=\sin ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1-\cos(\theta )}{2}}\,$[4]
${\textrm {hacoversin}}(\theta ):={\frac {{\textrm {coversin}}(\theta )}{2}}={\frac {1-\sin(\theta )}{2}}\,$[21]
${\textrm {havercosin}}(\theta ):={\frac {{\textrm {vercosin}}(\theta )}{2}}=\cos ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1+\cos(\theta )}{2}}\,$[33]
${\textrm {hacovercosin}}(\theta ):={\frac {{\textrm {covercosin}}(\theta )}{2}}={\frac {1+\sin(\theta )}{2}}\,$[35]
Circular rotations
The functions are circular rotations of each other.
${\begin{aligned}\mathrm {versin} (\theta )&=\mathrm {coversin} \left(\theta +{\frac {\pi }{2}}\right)=\mathrm {vercosin} \left(\theta +\pi \right)=\mathrm {covercosin} \left(\theta +{\frac {3\pi }{2}}\right)\\\mathrm {haversin} (\theta )&=\mathrm {hacoversin} \left(\theta +{\frac {\pi }{2}}\right)=\mathrm {havercosin} \left(\theta +\pi \right)=\mathrm {hacovercosin} \left(\theta +{\frac {3\pi }{2}}\right)\end{aligned}}$
Derivatives and integrals
${\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {versin} (x)=\sin {x}$[50] $\int \mathrm {versin} (x)\,\mathrm {d} x=x-\sin {x}+C$[4][50]
${\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {vercosin} (x)=-\sin {x}$ $\int \mathrm {vercosin} (x)\,\mathrm {d} x=x+\sin {x}+C$
${\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {coversin} (x)=-\cos {x}$[20] $\int \mathrm {coversin} (x)\,\mathrm {d} x=x+\cos {x}+C$[20]
${\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {covercosin} (x)=\cos {x}$ $\int \mathrm {covercosin} (x)\,\mathrm {d} x=x-\cos {x}+C$
${\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {haversin} (x)={\frac {\sin {x}}{2}}$[27] $\int \mathrm {haversin} (x)\,\mathrm {d} x={\frac {x-\sin {x}}{2}}+C$[27]
${\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {havercosin} (x)={\frac {-\sin {x}}{2}}$ $\int \mathrm {havercosin} (x)\,\mathrm {d} x={\frac {x+\sin {x}}{2}}+C$
${\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {hacoversin} (x)={\frac {-\cos {x}}{2}}$ $\int \mathrm {hacoversin} (x)\,\mathrm {d} x={\frac {x+\cos {x}}{2}}+C$
${\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {hacovercosin} (x)={\frac {\cos {x}}{2}}$ $\int \mathrm {hacovercosin} (x)\,\mathrm {d} x={\frac {x-\cos {x}}{2}}+C$
Inverse functions
Inverse functions like arcversine[34] (arcversin, arcvers,[8][34] avers,[51][52] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine[34] (arccoversin, arccovers,[8][34] acovers,[51][52] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav,[34] haversin−1,[53] invhav,[34][54][55][56] ahav,[34][51][52] ahvs, ahv, hav−1[57][58]), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
$\operatorname {arcversin} (y)=\arccos \left(1-y\right)\,$[34][51][52]
$\operatorname {arcvercos} (y)=\arccos \left(y-1\right)\,$
$\operatorname {arccoversin} (y)=\arcsin \left(1-y\right)\,$[34][51][52]
$\operatorname {arccovercos} (y)=\arcsin \left(y-1\right)\,$
$\operatorname {archaversin} (y)=2\arcsin \left({\sqrt {y}}\right)=\arccos \left(1-2y\right)\,$[34][51][52][53][54][55][57][58]
$\operatorname {archavercos} (y)=2\arccos \left({\sqrt {y}}\right)=\arccos \left(2y-1\right)$
$\operatorname {archacoversin} (y)=\arcsin \left(1-2y\right)\,$
$\operatorname {archacovercos} (y)=\arcsin \left(2y-1\right)\,$
Other properties
These functions can be extended into the complex plane.[50][20][27]
Maclaurin series:[27]
${\begin{aligned}\operatorname {versin} (z)&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{(2k)!}}\\\operatorname {haversin} (z)&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{2(2k)!}}\end{aligned}}$
$\lim _{\theta \to 0}{\frac {\operatorname {versin} (\theta )}{\theta }}=0$[8]
${\begin{aligned}{\frac {\operatorname {versin} (\theta )+\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}-{\frac {\operatorname {exsec} (\theta )+\operatorname {excsc} (\theta )}{\operatorname {exsec} (\theta )-\operatorname {excsc} (\theta )}}&={\frac {2\operatorname {versin} (\theta )\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}\\[3pt][\operatorname {versin} (\theta )+\operatorname {exsec} (\theta )]\,[\operatorname {coversin} (\theta )+\operatorname {excsc} (\theta )]&=\sin(\theta )\cos(\theta )\end{aligned}}$[8]
Approximations
When the versine v is small in comparison to the radius r, it may be approximated from the half-chord length L (the distance AC shown above) by the formula[59]
$v\approx {\frac {L^{2}}{2r}}.$
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s (AD in the figure above) by the formula
$s\approx L+{\frac {v^{2}}{r}}$
This formula was known to the Chinese mathematician Shen Kuo, and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing.[60]
A more accurate approximation used in engineering[61] is
$v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}$
Arbitrary curves and chords
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8v/L2 goes to the instantaneous curvature. This usage is especially common in rail transport, where it describes measurements of the straightness of the rail tracks[62] and it is the basis of the Hallade method for rail surveying.
The term sagitta (often abbreviated sag) is used similarly in optics, for describing the surfaces of lenses and mirrors.
See also
• Trigonometric identities
• Exsecant and excosecant
• Versiera (Witch of Agnesi)
• Exponential minus 1
• Natural logarithm plus 1
Notes
1. Some English sources confuse the versed cosine with the coversed sine. Historically (f.e. in Cauchy, 1821), the sinus versus (versine) was defined as siv(θ) = 1−cos(θ), the cosinus versus (what is now also known as coversine) as cosiv(θ) = 1−sin(θ), and the vercosine as vcsθ = 1+cos(θ). However, in their 2009 English translation of Cauchy's work, Bradley and Sandifer associate the cosinus versus (and cosiv) with the versed cosine (what is now also known as vercosine) rather than the coversed sine. Similarly, in their 1968/2000 work, Korn and Korn associate the covers(θ) function with the versed cosine instead of the coversed sine.
2. The abbreviation hvs sometimes used for the haversine function in signal processing and filtering is also sometimes used for the unrelated Heaviside step function.
References
1. The Āryabhaṭīya by Āryabhaṭa
2. Haslett, Charles (September 1855). Hackley, Charles W. (ed.). The Mechanic's, Machinist's, Engineer's Practical Book of Reference: Containing tables and formulæ for use in superficial and solid mensuration; strength and weight of materials; mechanics; machinery; hydraulics, hydrodynamics; marine engines, chemistry; and miscellaneous recipes. Adapted to and for the use of all classes of practical mechanics. Together with the Engineer's Field Book: Containing formulæ for the various of running and changing lines, locating side tracks and switches, &c., &c. Tables of radii and their logarithms, natural and logarithmic versed sines and external secants, natural sines and tangents to every degree and minute of the quadrant, and logarithms from the natural numbers from 1 to 10,000. New York, USA: James G. Gregory, successor of W. A. Townsend & Co. (Stringer & Townsend). Retrieved 2017-08-13. […] Still there would be much labor of computation which may be saved by the use of tables of external secants and versed sines, which have been employed with great success recently by the Engineers on the Ohio and Mississippi Railroad, and which, with the formulas and rules necessary for their application to the laying down of curves, drawn up by Mr. Haslett, one of the Engineers of that Road, are now for the first time given to the public. […] In presenting this work to the public, the Author claims for it the adaptation of a new principle in trigonometrical analysis of the formulas generally used in field calculations. Experience has shown, that versed sines and external secants as frequently enter into calculations on curves as sines and tangents; and by their use, as illustrated in the examples given in this work, it is believed that many of the rules in general use are much simplified, and many calculations concerning curves and running lines made less intricate, and results obtained with more accuracy and far less trouble, than by any methods laid down in works of this kind. The examples given have all been suggested by actual practice, and will explain themselves. […] As a book for practical use in field work, it is confidently believed that this is more direct in the application of rules and facility of calculation than any work now in use. In addition to the tables generally found in books of this kind, the author has prepared, with great labor, a Table of Natural and Logarithmic Versed Sines and External Secants, calculated to degrees, for every minute; also, a Table of Radii and their Logarithms, from 1° to 60°. […]{{cite book}}: CS1 maint: url-status (link) 1856 edition
3. Inman, James (1835) [1821]. Navigation and Nautical Astronomy: For the Use of British Seamen (3 ed.). London, UK: W. Woodward, C. & J. Rivington. Retrieved 2015-11-09. (Fourth edition: .)
4. Zucker, Ruth (1983) [June 1964]. "Chapter 4.3.147: Elementary Transcendental Functions - Circular functions". In Abramowitz, Milton; Stegun, Irene Ann (eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 78. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
5. Tapson, Frank (2004). "Background Notes on Measures: Angles". 1.4. Cleave Books. Archived from the original on 2007-02-09. Retrieved 2015-11-12.
6. Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) [1987]. "32.13. The Cosine cos(x) and Sine sin(x) functions - Cognate functions". An Atlas of Functions: with Equator, the Atlas Function Calculator (2 ed.). Springer Science+Business Media, LLC. p. 322. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48806-6. LCCN 2008937525.
7. Beebe, Nelson H. F. (2017-08-22). "Chapter 11.1. Sine and cosine properties". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA: Springer International Publishing AG. p. 301. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721.
8. Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Review Exercises [100] Secondary Trigonometric Functions". Written at Ann Arbor, Michigan, USA. Trigonometry. Vol. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. pp. 125–127. Retrieved 2017-08-12.{{cite book}}: CS1 maint: url-status (link)
9. Boyer, Carl Benjamin (1969) [1959]. "5: Commentary on the Paper of E. J. Dijksterhuis (The Origins of Classical Mechanics from Aristotle to Newton)". In Clagett, Marshall (ed.). Critical Problems in the History of Science (3 ed.). Madison, Milwaukee, and London: University of Wisconsin Press, Ltd. pp. 185–190. ISBN 0-299-01874-1. LCCN 59-5304. 9780299018740. Retrieved 2015-11-16.
10. Swanson, Todd; Andersen, Janet; Keeley, Robert (1999). "5 (Trigonometric Functions)" (PDF). Precalculus: A Study of Functions and Their Applications. Harcourt Brace & Company. p. 344. Archived (PDF) from the original on 2003-06-17. Retrieved 2015-11-12.
11. Korn, Grandino Arthur; Korn, Theresa M. (2000) [1961]. "Appendix B: B9. Plane and Spherical Trigonometry: Formulas Expressed in Terms of the Haversine Function". Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review (3 ed.). Mineola, New York, USA: Dover Publications, Inc. pp. 892–893. ISBN 978-0-486-41147-7. (See errata.)
12. Calvert, James B. (2007-09-14) [2004-01-10]. "Trigonometry". Archived from the original on 2007-10-02. Retrieved 2015-11-08.
13. Edler von Braunmühl, Anton (1903). Vorlesungen über Geschichte der Trigonometrie - Von der Erfindung der Logarithmen bis auf die Gegenwart [Lectures on history of trigonometry - from the invention of logarithms up to the present] (in German). Vol. 2. Leipzig, Germany: B. G. Teubner. p. 231. Retrieved 2015-12-09.
14. Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. Vol. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago, USA: Open court publishing company. p. 172. ISBN 978-1-60206-714-1. 1602067147. Retrieved 2015-11-11. The haversine first appears in the tables of logarithmic versines of José de Mendoza y Rios (Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of James Inman (1821). See J. D. White in Nautical Magazine (February and July 1926). (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
15. Shaneyfelt, Ted V. "德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?". Hilo, Hawaii: University of Hawaii. Archived from the original on 2015-09-19. Retrieved 2015-11-08.
16. Cauchy, Augustin-Louis (1821). "Analyse Algébrique". Cours d'Analyse de l'Ecole royale polytechnique (in French). Vol. 1. L'Imprimerie Royale, Debure frères, Libraires du Roi et de la Bibliothèque du Roi. access-date=2015-11-07--> (reissued by Cambridge University Press, 2009; ISBN 978-1-108-00208-0)
17. Bradley, Robert E.; Sandifer, Charles Edward (2010-01-14) [2009]. Buchwald, J. Z. (ed.). Cauchy's Cours d'analyse: An Annotated Translation. pp. 10, 285. doi:10.1007/978-1-4419-0549-9. ISBN 978-1-4419-0548-2. LCCN 2009932254. 1441905499, 978-1-4419-0549-9. Retrieved 2015-11-09. {{cite book}}: |work= ignored (help) (See errata.)
18. van Brummelen, Glen Robert (2013). Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. ISBN 9780691148922. 0691148929. Retrieved 2015-11-10.
19. Weisstein, Eric Wolfgang. "Vercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-24. Retrieved 2015-11-06.
20. Weisstein, Eric Wolfgang. "Coversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-11-27. Retrieved 2015-11-06.
21. Weisstein, Eric Wolfgang. "Hacoversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-29. Retrieved 2015-11-06.
22. Ludlow, Henry Hunt; Bass, Edgar Wales (1891). Elements of Trigonometry with Logarithmic and Other Tables (3 ed.). Boston, USA: John Wiley & Sons. p. 33. Retrieved 2015-12-08.
23. Wentworth, George Albert (1903) [1887]. Plane Trigonometry (2 ed.). Boston, USA: Ginn and Company. p. 5.
24. Kenyon, Alfred Monroe; Ingold, Louis (1913). Trigonometry. New York, USA: The Macmillan Company. pp. 8–9. Retrieved 2015-12-08.
25. Anderegg, Frederick; Roe, Edward Drake (1896). Trigonometry: For Schools and Colleges. Boston, USA: Ginn and Company. p. 10. Retrieved 2015-12-08.
26. Weisstein, Eric Wolfgang. "Covercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-28. Retrieved 2015-11-06.
27. Weisstein, Eric Wolfgang. "Haversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-03-10. Retrieved 2015-11-06.
28. Fulst, Otto (1972). "17, 18". In Lütjen, Johannes; Stein, Walter; Zwiebler, Gerhard (eds.). Nautische Tafeln (in German) (24 ed.). Bremen, Germany: Arthur Geist Verlag.
29. Sauer, Frank (2015) [2004]. "Semiversus-Verfahren: Logarithmische Berechnung der Höhe" (in German). Hotheim am Taunus, Germany: Astrosail. Archived from the original on 2013-09-17. Retrieved 2015-11-12.
30. Rider, Paul Reece; Davis, Alfred (1923). Plane Trigonometry. New York, USA: D. Van Nostrand Company. p. 42. Retrieved 2015-12-08.
31. "Haversine". Wolfram Language & System: Documentation Center. 7.0. 2008. Archived from the original on 2014-09-01. Retrieved 2015-11-06.
32. Rudzinski, Greg (July 2015). Ix, Hanno. "Ultra compact sight reduction". Ocean Navigator. Portland, ME, USA: Navigator Publishing LLC (227): 42–43. ISSN 0886-0149. Retrieved 2015-11-07.
33. Weisstein, Eric Wolfgang. "Havercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-29. Retrieved 2015-11-06.
34. van Vlijmen, Oscar (2005-12-28) [2003]. "Goniology". Eenheden, constanten en conversies. Archived from the original on 2009-10-28. Retrieved 2015-11-28.
35. Weisstein, Eric Wolfgang. "Hacovercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-29. Retrieved 2015-11-06.
36. "sagitta". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
37. Boyer, Carl Benjamin; Merzbach, Uta C. (1991-03-06) [1968]. A History of Mathematics (2 ed.). New York, USA: John Wiley & Sons. ISBN 978-0471543978. 0471543977. Retrieved 2019-08-10.
38. Miller, Jeff (2007-09-10). "Earliest Known Uses of Some of the Words of Mathematics (V)". New Port Richey, Florida, USA. Archived from the original on 2015-09-05. Retrieved 2015-11-10.
39. de Mendoza y Ríos, Joseph (1795). Memoria sobre algunos métodos nuevos de calcular la longitud por las distancias lunares: y aplicación de su teórica á la solucion de otros problemas de navegacion (in Spanish). Madrid, Spain: Imprenta Real.
40. Archibald, Raymond Clare (1945). "Recent Mathematical Tables : 197[C, D].—Natural and Logarithmic Haversines..." Mathematical Tables and Other Aids to Computation. 1 (11): 421–422. doi:10.1090/S0025-5718-45-99080-6.
41. Andrew, James (1805). Astronomical and Nautical Tables with Precepts for finding the Latitude and Longitude of Places. Vol. T. XIII. London. pp. 29–148. (A 7-place haversine table from 0° to 120° in intervals of 10".)
42. "haversine". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989.
43. White, J. D. (February 1926). "(unknown title)". Nautical Magazine. (NB. According to Cajori, 1929, this journal has a discussion on the origin of haversines.)
44. White, J. D. (July 1926). "(unknown title)". Nautical Magazine. (NB. According to Cajori, 1929, this journal has a discussion on the origin of haversines.)
45. Farley, Richard (1856). Natural Versed Sines from 0 to 125°, and Logarithmic Versed Sines from 0 to 135°. London.{{cite book}}: CS1 maint: location missing publisher (link) (A haversine table from 0° to 125°/135°.)
46. Hannyngton, John Caulfield (1876). Haversines, Natural and Logarithmic, used in Computing Lunar Distances for the Nautical Almanac. London.{{cite book}}: CS1 maint: location missing publisher (link) (A 7-place haversine table from 0° to 180°, log. haversines at intervals of 15", nat. haversines at intervals of 10".)
47. Stark, Bruce D. (1997) [1995]. Stark Tables for Clearing the Lunar Distance and Finding Universal Time by Sextant Observation Including a Convenient Way to Sharpen Celestial Navigation Skills While On Land (2 ed.). Starpath Publications. ISBN 978-0914025214. 091402521X. Retrieved 2015-12-02. (NB. Contains a table of Gaussian logarithms lg(1+10−x).)
48. Kalivoda, Jan (2003-07-30). "Bruce Stark - Tables for Clearing the Lunar Distance and Finding G.M.T. by Sextant Observation (1995, 1997)" (Review). Prague, Czech Republic. Archived from the original on 2004-01-12. Retrieved 2015-12-02.
49. Wildberger, Norman John (2005). Divine Proportions: Rational Trigonometry to Universal Geometry (1 ed.). Australia: Wild Egg Pty Ltd. ISBN 0-9757492-0-X. Retrieved 2015-12-01.
50. Weisstein, Eric Wolfgang. "Versine". MathWorld. Wolfram Research, Inc. Archived from the original on 2010-03-31. Retrieved 2015-11-05.
51. Simpson, David G. (2001-11-08). "AUXTRIG" (Fortran 90 source code). Greenbelt, Maryland, USA: NASA Goddard Space Flight Center. Archived from the original on 2008-06-16. Retrieved 2015-10-26.
52. van den Doel, Kees (2010-01-25). "jass.utils Class Fmath". JASS - Java Audio Synthesis System. 1.25. Archived from the original on 2007-09-02. Retrieved 2015-10-26.
53. mf344 (2014-07-04). "Lost but lovely: The haversine". Plus magazine. maths.org. Archived from the original on 2014-07-18. Retrieved 2015-11-05.
54. Skvarc, Jure (1999-03-01). "identify.py: An asteroid_server client which identifies measurements in MPC format". Fitsblink (Python source code). Archived from the original on 2008-11-20. Retrieved 2015-11-28.
55. Skvarc, Jure (2014-10-27). "astrotrig.py: Astronomical trigonometry related functions" (Python source code). Ljubljana, Slovenia: Telescope Vega, University of Ljubljana. Archived from the original on 2015-11-28. Retrieved 2015-11-28.
56. Ballew, Pat (2007-02-08) [2003]. "Versine". Math Words, page 4. Versine. Archived from the original on 2007-02-08. Retrieved 2015-11-28.
57. Weisstein, Eric Wolfgang. "Inverse Haversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2008-06-08. Retrieved 2015-10-05.
58. "InverseHaversine". Wolfram Language & System: Documentation Center. 7.0. 2008. Retrieved 2015-11-05.
59. Woodward, Ernest (December 1978). Geometry - Plane, Solid & Analytic Problem Solver. Problem Solvers Solution Guides. Research & Education Association (REA). p. 359. ISBN 978-0-87891-510-1.
60. Needham, Noel Joseph Terence Montgomery (1959). Science and Civilisation in China: Mathematics and the Sciences of the Heavens and the Earth. Vol. 3. Cambridge University Press. p. 39. ISBN 9780521058018.
61. Boardman, Harry (1930). Table For Use in Computing Arcs, Chords and Versines. Chicago Bridge and Iron Company. p. 32.
62. Nair, P. N. Bhaskaran (1972). "Track measurement systems—concepts and techniques". Rail International. International Railway Congress Association, International Union of Railways. 3 (3): 159–166. ISSN 0020-8442. OCLC 751627806.
Further reading
• Hawking, Stephen William, ed. (2002). On the Shoulders of Giants: The Great Works of Physics and Astronomy. Philadelphia, USA: Running Press. ISBN 0-7624-1698-X. LCCN 2002100441. Retrieved 2017-07-31.{{cite book}}: CS1 maint: url-status (link)
External links
• Pegg, Jr., Ed. "Sagitta, Apothem, and Chord". The Wolfram Demonstrations Project.
• Trigonometric Functions at GeoGebra.org
Trigonometric and hyperbolic functions
Groups
• Trigonometric
• Sine and cosine
• Inverse trigonometric
• Hyperbolic
• Inverse hyperbolic
Other
• Versine
• Exsecant
• Jyā, koti-jyā and utkrama-jyā
• atan2
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Wikipedia
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Sinusoidal spiral
In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates
$r^{n}=a^{n}\cos(n\theta )\,$
where a is a nonzero constant and n is a rational number other than 0. With a rotation about the origin, this can also be written
$r^{n}=a^{n}\sin(n\theta ).\,$
The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including:
• Rectangular hyperbola (n = −2)
• Line (n = −1)
• Parabola (n = −1/2)
• Tschirnhausen cubic (n = −1/3)
• Cayley's sextet (n = 1/3)
• Cardioid (n = 1/2)
• Circle (n = 1)
• Lemniscate of Bernoulli (n = 2)
The curves were first studied by Colin Maclaurin.
Equations
Differentiating
$r^{n}=a^{n}\cos(n\theta )\,$
and eliminating a produces a differential equation for r and θ:
${\frac {dr}{d\theta }}\cos n\theta +r\sin n\theta =0$.
Then
$\left({\frac {dr}{ds}},\ r{\frac {d\theta }{ds}}\right)\cos n\theta {\frac {ds}{d\theta }}=\left(-r\sin n\theta ,\ r\cos n\theta \right)=r\left(-\sin n\theta ,\ \cos n\theta \right)$
which implies that the polar tangential angle is
$\psi =n\theta \pm \pi /2$
and so the tangential angle is
$\varphi =(n+1)\theta \pm \pi /2$.
(The sign here is positive if r and cos nθ have the same sign and negative otherwise.)
The unit tangent vector,
$\left({\frac {dr}{ds}},\ r{\frac {d\theta }{ds}}\right)$,
has length one, so comparing the magnitude of the vectors on each side of the above equation gives
${\frac {ds}{d\theta }}=r\cos ^{-1}n\theta =a\cos ^{-1+{\tfrac {1}{n}}}n\theta $.
In particular, the length of a single loop when $n>0$ is:
$a\int _{-{\tfrac {\pi }{2n}}}^{\tfrac {\pi }{2n}}\cos ^{-1+{\tfrac {1}{n}}}n\theta \ d\theta $
The curvature is given by
${\frac {d\varphi }{ds}}=(n+1){\frac {d\theta }{ds}}={\frac {n+1}{a}}\cos ^{1-{\tfrac {1}{n}}}n\theta $.
Properties
The inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve's value of n. For example, the inverse of the lemniscate of Bernoulli is a rectangular hyperbola.
The isoptic, pedal and negative pedal of a sinusoidal spiral are different sinusoidal spirals.
One path of a particle moving according to a central force proportional to a power of r is a sinusoidal spiral.
When n is an integer, and n points are arranged regularly on a circle of radius a, then the set of points so that the geometric mean of the distances from the point to the n points is a sinusoidal spiral. In this case the sinusoidal spiral is a polynomial lemniscate.
Wikimedia Commons has media related to Sinusoidal spiral.
References
• Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Spiral" p. 213–214
• "Sinusoidal spiral" at www.2dcurves.com
• "Sinusoidal Spirals" at The MacTutor History of Mathematics
• Weisstein, Eric W. "Sinusoidal Spiral". MathWorld.
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Wikipedia
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Siobhán Vernon
Siobhán Vernon (née O'Shea) was the first Irish-born woman to get a PhD in pure mathematics in Ireland, in 1964.[1]
Siobhán Vernon
Born
Siobhán O'Shea
(1932-02-22)22 February 1932
Macroom, County Cork
Died18 September 2002(2002-09-18) (aged 70)
NationalityIrish
EducationUniversity College, Cork
OccupationMathematician
SpousePeter Vernon
Parent(s)Joseph J O'Shea and M O'Shea
Early life and education
Siobhán O'Shea was born in Macroom, County Cork, in 1932 and was the daughter of Joseph J. O'Shea and his wife M. O'Shea.[2]
Her post-primary education was at the Convent of Mercy in Macroom, but she also attended the De La Salle, a secondary school for boys, for the higher level mathematics classes required for the Irish Leaving Certificate examination. She entered University College, Cork in 1949 and was awarded a college scholarship in 1950, based on the results of her first year examinations. She graduated in 1952 with a first class honours B.Sc. in Mathematics and Mathematical Physics. She went on to complete her M.Sc. in Mathematics and Mathematics Statistics, awarded in 1954.[3]
Career
Siobhán Vernon worked as a demonstrator in the Department of Mathematics at University College, Cork while she completed her M.Sc. and was then appointed Senior Demonstrator.[2] Encouraged by Dr Patrick Brendan Kennedy, Siobhán began to publish research in 1956 and was appointed to the full-time post of Assistant in 1957. Continuing her research career, she spent a year as a visiting lecturer in Royal Holloway College, University of London, in 1962-63.[3]
Returning to University College, Cork, she submitted her published papers for the award of PhD, which was awarded in 1964 by the National University of Ireland. She was appointed lecturer in 1965.[3]
Following her marriage to geologist Peter Vernon, Siobhán reduced her teaching to half-time, as they raised their four children. She later returned to full time teaching, retiring in 1988.
Awards
In 1995 she was honoured with a Catherine McAuley award as a distinguished past pupil by the Convent of Mercy in Macroom.[3]
Later life and death
Her final publication came after her retirement, when she contributed a chapter on Paddy Kennedy in Creators of Mathematics: The Irish Connection.[4] She died on 18 September 2002.[5]
References
1. "Siobhán Vernon". Atlas of Irish Mathematics & Mathematicians. Retrieved 12 March 2017.
2. O'Connor, John J.; Robertson, Edmund F., "Siobhán Vernon", MacTutor History of Mathematics Archive, University of St Andrews
3. "Appreciation Dr Siobhan Vernon". The Irish Times. 31 December 2002. Retrieved 12 March 2017.
4. Long, Maebh (2007). "Irish Mathematicians" (PDF). Kaleidoscope. 1.1: 3.
5. Barry, Patrick D. (Winter 2002). "Dr Siobhán Vernon Obituary" (PDF). The College Courier.
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
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Wikipedia
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Sion's minimax theorem
In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem, named after Maurice Sion.
It states:
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex subset of a linear topological space. If $f$ is a real-valued function on $X\times Y$ with
$f(x,\cdot )$ upper semicontinuous and quasi-concave on $Y$, $\forall x\in X$, and
$f(\cdot ,y)$ lower semicontinuous and quasi-convex on $X$, $\forall y\in Y$
then,
$\min _{x\in X}\sup _{y\in Y}f(x,y)=\sup _{y\in Y}\min _{x\in X}f(x,y).$
See also
• Parthasarathy's theorem
• Saddle point
References
• Sion, Maurice (1958). "On general minimax theorems". Pacific Journal of Mathematics. 8 (1): 171–176. doi:10.2140/pjm.1958.8.171. MR 0097026. Zbl 0081.11502.
• Komiya, Hidetoshi (1988). "Elementary proof for Sion's minimax theorem". Kodai Mathematical Journal. 11 (1): 5–7. doi:10.2996/kmj/1138038812. MR 0930413. Zbl 0646.49004.
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Wikipedia
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Michael Sipser
Michael Fredric Sipser (born September 17, 1954) is an American theoretical computer scientist who has made early contributions to computational complexity theory. He is a professor of applied mathematics and was the Dean of Science at the Massachusetts Institute of Technology.
Michael Sipser
Born
Michael Fredric Sipser
(1954-09-17) September 17, 1954
Brooklyn, New York
NationalityAmerican
Alma mater
• Cornell University
• University of California at Berkeley
Awards
• Fellow, American Academy of Arts and Sciences
• Fellow, American Mathematical Society
• Fellow, Association for Computing Machinery
Scientific career
Fields
• Mathematics
• Computer Science
InstitutionsMIT
ThesisNondeterminism and the Size of Two-Way Finite Automata (1980)
Doctoral advisorManuel Blum
Doctoral students
• Lance Fortnow
• Sofya Raskhodnikova
• Leonard Schulman
• Daniel Spielman
• Andrew Sutherland
• Yiqun Lisa Yin
Websitemath.mit.edu/~sipser/
Biography
Sipser was born and raised in Brooklyn, New York and moved to Oswego, New York when he was 12 years old. He earned his BA in mathematics from Cornell University in 1974 and his PhD in engineering from the University of California at Berkeley in 1980 under the direction of Manuel Blum.[1][2]
He joined MIT's Laboratory for Computer Science as a research associate in 1979 and then was a Research Staff Member at IBM Research in San Jose. In 1980, he joined the MIT faculty. He spent the 1985-1986 academic year on the faculty of the University of California at Berkeley and then returned to MIT. From 2004 until 2014, he served as head of the MIT Mathematics department. He was appointed Interim Dean of the MIT School of Science in 2013 and Dean in 2014.[3] He served as Dean until 2020, when he was followed by Nergis Mavalvala.[4] He is a fellow of the American Academy of Arts and Sciences.[5] In 2015 he was elected as a fellow of the American Mathematical Society "for contributions to complexity theory and for leadership and service to the mathematical community."[6] He was elected as an ACM Fellow in 2017.[7]
Scientific career
Sipser specializes in algorithms and complexity theory, specifically efficient error correcting codes, interactive proof systems, randomness, quantum computation, and establishing the inherent computational difficulty of problems. He introduced the method of probabilistic restriction for proving super-polynomial lower bounds on circuit complexity in a paper joint with Merrick Furst and James B. Saxe.[8] Their result was later improved to be an exponential lower bound by Andrew Yao and Johan Håstad.[9]
In an early derandomization theorem, Sipser showed that BPP is contained in the polynomial hierarchy,[10] subsequently improved by Peter Gács and Clemens Lautemann to form what is now known as the Sipser-Gács-Lautemann theorem. Sipser also established a connection between expander graphs and derandomization.[11] He and his PhD student Daniel Spielman introduced expander codes, an application of expander graphs.[12] With fellow graduate student David Lichtenstein, Sipser proved that Go is PSPACE hard.[13]
In quantum computation theory, he introduced the adiabatic algorithm jointly with Edward Farhi, Jeffrey Goldstone, and Samuel Gutmann.[14]
Sipser has long been interested in the P versus NP problem. In 1975, he wagered an ounce of gold with Leonard Adleman that the problem would be solved with a proof that P≠NP by the end of the 20th century. Sipser sent Adleman an American Gold Eagle coin in 2000 because the problem remained (and remains) unsolved.[15]
Notable books
Sipser is the author of Introduction to the Theory of Computation,[16] a textbook for theoretical computer science.
Personal life
Sipser lives in Cambridge, Massachusetts with his wife, Ina, and has two children: a daughter, Rachel, who graduated from New York University, and a younger son, Aaron, who graduated from MIT.[1]
References
1. Trafton, Anne, "Michael Sipser named dean of the School of Science: Sipser has served as interim dean since Marc Kastner’s departure", MIT News Office, June 5, 2014
2. Michael Sipser at the Mathematics Genealogy Project
3. MIT Mathematics | People Directory Archived 2008-12-18 at the Wayback Machine
4. "School of Science | MIT History". Retrieved 2020-08-25.
5. "Membership". American Academy of Arts and Sciences. Retrieved 23 September 2014.
6. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2015-11-16.
7. ACM Recognizes 2017 Fellows for Making Transformative Contributions and Advancing Technology in the Digital Age, Association for Computing Machinery, December 11, 2017, retrieved 2017-11-13
8. Furst, Merrick; Saxe, James B.; Sipser, Michael (1984). "Parity, circuits, and the polynomial-time hierarchy". Mathematical Systems Theory. 17 (1): 13–27. doi:10.1007/BF01744431. MR 0738749. S2CID 14677270.
9. "Research Vignette: Hard Problems All The Way Up | Simons Institute for the Theory of Computing". simons.berkeley.edu. 30 July 2015. Retrieved 2015-09-17.
10. Sipser, Michael (1983). "A complexity theoretic approach to randomness". Proceedings of the 15th ACM Symposium on Theory of Computing.
11. Sipser, Michael (1986). "Expanders, Randomness, or Time versus Space". Proceedings of the Conference on Structure in Complexity. Lecture Notes in Computer Science. 223: 325–329. doi:10.1007/3-540-16486-3_108. ISBN 978-3-540-16486-9.
12. Sipser, Michael; Spielman, Daniel (1996). "Expander Codes" (PDF). IEEE Transactions on Information Theory. 42 (6): 1710–1722. doi:10.1109/18.556667.
13. Lichtenstein, David; Sipser, Michael (1980-04-01). "GO Is Polynomial-Space Hard". J. ACM. 27 (2): 393–401. doi:10.1145/322186.322201. ISSN 0004-5411. S2CID 29498352.
14. Farhi, Edward; Goldstone, Jeffrey; Gutmann, Sam; Sipser, Michael (2000-01-28). "Quantum Computation by Adiabatic Evolution". arXiv:quant-ph/0001106.
15. Pavlus, John (2012-01-01). "Machines of the Infinite". Scientific American. 307 (3): 66–71. Bibcode:2012SciAm.307c..66P. doi:10.1038/scientificamerican0912-66. PMID 22928263.
16. Sipser, Michael (2012-06-27). Introduction to the Theory of Computation (3 ed.). Cengage Learning. ISBN 978-1133187790.
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Sipser–Lautemann theorem
In computational complexity theory, the Sipser–Lautemann theorem or Sipser–Gács–Lautemann theorem states that bounded-error probabilistic polynomial (BPP) time is contained in the polynomial time hierarchy, and more specifically Σ2 ∩ Π2.
In 1983, Michael Sipser showed that BPP is contained in the polynomial time hierarchy.[1] Péter Gács showed that BPP is actually contained in Σ2 ∩ Π2. Clemens Lautemann contributed by giving a simple proof of BPP’s membership in Σ2 ∩ Π2, also in 1983.[2] It is conjectured that in fact BPP=P, which is a much stronger statement than the Sipser–Lautemann theorem.
Proof
Here we present the Lautemann's proof.[2] Without loss of generality, a machine M ∈ BPP with error ≤ 2−|x| can be chosen. (All BPP problems can be amplified to reduce the error probability exponentially.) The basic idea of the proof is to define a Σ2 sentence that is equivalent to stating that x is in the language, L, defined by M by using a set of transforms of the random variable inputs.
Since the output of M depends on random input, as well as the input x, it is useful to define which random strings produce the correct output as A(x) = {r | M(x,r) accepts}. The key to the proof is to note that when x ∈ L, A(x) is very large and when x ∉ L, A(x) is very small. By using bitwise parity, ⊕, a set of transforms can be defined as A(x) ⊕ t={r ⊕ t | r ∈ A(x)}. The first main lemma of the proof shows that the union of a small finite number of these transforms will contain the entire space of random input strings. Using this fact, a Σ2 sentence and a Π2 sentence can be generated that is true if and only if x ∈ L (see conclusion).
Lemma 1
The general idea of lemma one is to prove that if A(x) covers a large part of the random space $R=\{1,0\}^{|r|}$ then there exists a small set of translations that will cover the entire random space. In more mathematical language:
If ${\frac {|A(x)|}{|R|}}\geq 1-{\frac {1}{2^{|x|}}}$, then $\exists t_{1},t_{2},\ldots ,t_{|r|}$, where $t_{i}\in \{1,0\}^{|r|}$ such that $\bigcup _{i}A(x)\oplus t_{i}=R.$
Proof. Randomly pick t1, t2, ..., t|r|. Let $S=\bigcup _{i}A(x)\oplus t_{i}$ (the union of all transforms of A(x)).
So, for all r in R,
$\Pr[r\notin S]=\Pr[r\notin A(x)\oplus t_{1}]\cdot \Pr[r\notin A(x)\oplus t_{2}]\cdots \Pr[r\notin A(x)\oplus t_{|r|}]\leq {\frac {1}{2^{|x|\cdot |r|}}}.$
The probability that there will exist at least one element in R not in S is
$\Pr {\Bigl [}\bigvee _{i}(r_{i}\notin S){\Bigr ]}\leq \sum _{i}{\frac {1}{2^{|x|\cdot |r|}}}={\frac {2^{|r|}}{2^{|x|\cdot |r|}}}<1.$
Therefore
$\Pr[S=R]\geq 1-{\frac {2^{|r|}}{2^{|x|\cdot |r|}}}>0.$
Thus there is a selection for each $t_{1},t_{2},\ldots ,t_{|r|}$ such that
$\bigcup _{i}A(x)\oplus t_{i}=R.$
Lemma 2
The previous lemma shows that A(x) can cover every possible point in the space using a small set of translations. Complementary to this, for x ∉ L only a small fraction of the space is covered by $S=\bigcup _{i}A(x)\oplus t_{i}$. We have:
${\frac {|S|}{|R|}}\leq |r|\cdot {\frac {|A(x)|}{|R|}}\leq |r|\cdot 2^{-|x|}<1$
because $|r|$ is polynomial in $|x|$.
Conclusion
The lemmas show that language membership of a language in BPP can be expressed as a Σ2 expression, as follows.
$x\in L\iff \exists t_{1},t_{2},\dots ,t_{|r|}\,\forall r\in R\bigvee _{1\leq i\leq |r|}(M(x,r\oplus t_{i}){\text{ accepts}}).$
That is, x is in language L if and only if there exist $|r|$ binary vectors, where for all random bit vectors r, TM M accepts at least one random vector ⊕ ti.
The above expression is in Σ2 in that it is first existentially then universally quantified. Therefore BPP ⊆ Σ2. Because BPP is closed under complement, this proves BPP ⊆ Σ2 ∩ Π2.
Stronger version
The theorem can be strengthened to ${\mathsf {BPP}}\subseteq {\mathsf {MA}}\subseteq {\mathsf {S}}_{2}^{P}\subseteq \Sigma _{2}\cap \Pi _{2}$ (see MA, SP
2
).[3][4]
References
1. Sipser, Michael (1983). "A complexity theoretic approach to randomness". Proceedings of the 15th ACM Symposium on Theory of Computing. ACM Press: 330–335. CiteSeerX 10.1.1.472.8218.
2. Lautemann, Clemens (1983). "BPP and the polynomial hierarchy". Inf. Proc. Lett. 17. 17 (4): 215–217. doi:10.1016/0020-0190(83)90044-3.
3. Canetti, Ran (1996). "More on BPP and the polynomial-time hierarchy". Information Processing Letters. 57 (5): 237–241. doi:10.1016/0020-0190(96)00016-6.
4. Russell, Alexander; Sundaram, Ravi (1998). "Symmetric alternation captures BPP". Computational Complexity. 7 (2): 152–162. CiteSeerX 10.1.1.219.3028. doi:10.1007/s000370050007. ISSN 1016-3328. S2CID 15331219.
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Andrew Searle Hart
Sir Andrew Searle Hart (1811–1890) was an Anglo-Irish mathematician and Vice-Provost of Trinity College Dublin (TCD).
Sir Andrew Searle Hart
Born(1811-03-14)14 March 1811
Limerick, Ireland
Died13 April 1890(1890-04-13) (aged 79)
Kilderry House, Donegal, Ireland
OccupationMathematician and Vice-Provost of Trinity College, Dublin
Early life and background
He was the youngest son of the Rev. George Vaughan Hart of Glenalla, County Donegal, and his wife Maria Murray, daughter of the Very Rev. John Hume, dean of Derry, and was born at Limerick on (1811-03-14)14 March 1811. His grandfather, Lieutenant John Hart, a younger son of the family, was killed in action at the Battle of the Monongahela.
His father took possession of the Glenalla and Carrablagh estates from the Murrays, to whom his wife was related.
He was a descendant of Henry Hart, who came to Ireland with the army of Elizabeth I. Another relation, Sir Eustace Hart, married Lady Mary de Vere, a daughter of John de Vere, 16th Earl of Oxford and a sister of the 17th Earl of Oxford,[1] who is a proposed alternative to the authorship of the works by William Shakespeare.
His mother, Maria Murray Hume, was from the same family as the philosopher David Hume. Sir Andrew's first cousin once removed was James Deacon Hume, the 18th century economist and civil servant.[2] On the Murray side, Hart was a direct descendant of the Murrays of Cockpool and of Sir William Murray, who married Isabel Randolph, a sister of Thomas Randolph, 1st Earl of Moray and a niece of Robert the Bruce.[3][4][5]
He was educated at Foyle College and by a private tutor before entering TCD in 1828, where he became the class-fellow and intimate friend of Isaac Butt, with whom he always preserved a warm friendship although they differed in politics. Hart graduated BA 1833, proceeded MA 1839, and LL.B. and LL.D. 1840. He was elected a fellow on 15 June 1835.[6] From 1827 to 1832 he was the Donegall Lecturer in Mathematics at TCD.
Academic career
Hart obtained much reputation as a mathematician, and published useful treatises on hydrostatics and mechanics. Between 1849 and 1861 he contributed valuable papers to the Cambridge and Dublin Mathematical Journal, to the 'Proceedings of the Irish Academy,' and to the Quarterly Journal of Mathematics, chiefly on the subject of geodesic lines and on curves.
Hart also served as Professor of Real and Personal Property in King's Inns, Dublin.[7]
Hart's Theorem
Hart's most important contribution was contained in his paper Extension of Terquem's theorem respecting the circle which bisects three sides of a triangle (1861).[7] Hart wrote this paper after carrying out an investigation suggested by William Rowan Hamilton in a letter to Hart.[7] In addition, Hart corresponded with George Salmon on the same topic.[7] This paper contains the result which became known as Hart's Theorem, which is a generalisation of Feuerbach's Theorem. Hart's Theorem states:
Taking any three of the eight circles which touch three others, a circle can be described to touch these three, and to touch a fourth circle of the eight touching circles.[7][8]
In Principles of geometry (1925), Henry Baker sums up Hart's Theorem as follows:
Given three lines in a plane, there are four circles touching them; these circles, we know, are all touched by another circle, the nine-points circle (Feuerbach's theorem; see Vol. II). In other words, given three lines, we can add to them a circle such that the four, these lines and the circle, are all touched by four other circles. In the present chapter we show how, given any three circles in a plane, we can add to them another circle, which we call the Hart circle, such that the four circles are all touched by four other circles (Hart, 'Quart. J. of Math.', IV (1861), p. 260). The three original circles are in fact touched by eight other circles, as we shall prove. There are fourteen ways of choosing, from these eight, four circles which touch another circle. In six of these ways, the four circles chosen have a common orthogonal circle; and the four circles consisting of the original circles, and their Hart circle, have also a common orthogonal circle. We have shown that circles in a plane may be regarded as projections of plane sections of a quadric. We prove the results enunciated as theorems for such plane sections. This appears greatly to increase the interest and clearness of the matter.[7]
Hart was co-opted as a Senior Fellow of TCD on 10 July 1858.[6]
In February 1873, Hart made up part of the delegation sent to London on behalf of TCD to lobby members of parliament to vote against the Irish University Bill[7]
He was elected Vice-Provost of TCD in 1876,[6] and at this time undertook many of the duties of the then provost, Humphrey Lloyd, that ill health had permitted him from carrying out.[7]
Personal life
He married in 1840 Frances, daughter of Sir Henry McDougall, Q.C., of Dublin; she died in 1876. Two sons, George Vaughan Hart (1841-1912), a barrister, and Henry Chichester Hart (1847-1908), a botanist and explorer, of Carrablagh House, Donegal, survived him.[6] The youngest son, William Hume Hart (1852-1887) predeceased Sir Andrew.
Sir Andrew Hart took an active interest in the affairs of the Irish Church, and was for many years a member of the general synod and representative church body.
On 25 January 1886 he was knighted at Dublin Castle by the lord-lieutenant, Lord Carnarvon, "in recognition of his academic rank and attainments."[6]
Sir Andrew Hart died suddenly at the house of his brother-in-law and cousin (his sister had married her cousin[9]), George Vaughan Hart, of Kilderry, County Donegal, on (1890-04-13)13 April 1890.[6]
Photograph
• Sir Andrew Hart with his eight grandchildren, Dublin, 1888
Left to right: Norah Searle Hart (1879-1965), Adelaide (Ada) Hart (1876-197?), George Vaughan (Vaughan) Hart (1877-1928), Hilda Chichester Hart (1882-1967), Tristram Beresford Hart (1884-1963), Ruth Hart (1886-1977), Sir Andrew Searle Hart, Ethel Hart (1875-1964). All the minors shown except Ada were children of George Vaughan Hart. Ada was the daughter of William Hume Hart.
Ancestry
Ancestors of Andrew Searle Hart
16. Captain Henry Hart (grandson of Captain Henry Hart)
8. Colonel George Hart
17. Anne Beresford (daughter of Sir Tristram Beresford, 1st Baronet and Sarah Sackville)
4. John Hart
18. George Vaughan of Buncrana Castle
9. Marianna Vaughan
2. Rev. George Vaughan Ledwich Hart
20. John Barnard
10. Nathaniel Barnard (brother of Rt. Rev William Barnard, the Bishop of Derry)
21. Isabella ?
5. Catherine Barnard
1. Sir Andrew Searle Hart
24. Rev. William Hume (scion of Home of Blackadder)
12. Dr James Hume (brother of John Hume (bishop))
25. Jane Robertson
6. Very Rev. John Hume
26. Nathaniel Rokeby
13. Elizabeth Rokeby
27. Elizabeth ?
3. Maria Murray Hume
28. John Murray (son of George Murray of Broughton)
14. Captain James Murray
29. Bathia Freeman
7. Jane Murray (sister of Brig-Gen. John Murray)
15. Nikola Anna Johnstone
Publications
1. 'An Elementary Treatise on Mechanics,' 1844; 2nd edit. 1847.
2. 'On the Form of Geodesic Lines through the Umbilic of an Ellipsoid, Proceedings of the Royal Irish Academy 4 (1847-1850), 274,' 1849.
3. 'Geometrical demonstration of some properties of geodesic lines, Cambridge and Dublin Mathematical Journal 4 (1849), 80–84.'
4. 'On geodesic lines traced on a surface of the second degree, Cambridge and Dublin Mathematical Journal 4 (1849), 192–194.'
5. 'An Elementary Treatise on Hydrostatics and Hydrodynamics,' 1846; another edit. 1850.
6. 'An account of some transformations of curves, Cambridge and Dublin Mathematical Journal 8 (1853), 47–50.'
7. 'On the porism of the in-and-circumscribed triangle, Quarterly Journal of Pure and Applied Mathematics 2 (1858), 143.'
8. 'Extension of Terquem's theorem respecting the circle which bisects three sides of a triangle, Quarterly Journal of Pure and Applied Mathematics 4 (1861), 260–261.'
9. 'On Nine-Point Contact of Cubic Curves, The Transactions of the Royal Irish Academy 25, Science (1875), 559–565.'
10. 'On the Intersections of Plane Curves of the Third Order, The Transactions of the Royal Irish Academy 26, Science (1879), 449–452.'
11. 'On Twisted Quartics, Hermathena 5 (10) (1884), 164–170.'
12. 'On the Linear Relations between the Nine Points of Intersection of a System of Plane Cubic Curves, Hermathena 6 (13) (1887), 286–289.'
References
1. Charles Mosley, editor. Burke's Peerage, Baronetage & Knightage, 107th edition, 3 volumes (Wilmington, Delaware, U.S.A.: Burke's Peerage (Genealogical Books) Ltd, 2003), volume 2, page 2348.
2. "Maria Murray Hume". Archived from the original on 12 March 2017. Retrieved 11 March 2017.
3. Bain, Joseph, FSA (Scot)., The Edwards in Scotland, 1296 – 1377, Edinburgh, 1901:61 & 66
4. Weis, Fredk., Lewis, et al., The Magna Charta Sureties 1215, 5th edition, Baltimore, 2002: 50
5. Richardson, Douglas, Plantagenet Ancestry, Baltimore, Md., 2004: 682
6. Boase 1891.
7. Andrew Searle Hart (1811-1890)
8. Hart Circle
9. Jane Maria Hart, ThePeerage.com
Attribution
This article incorporates text from a publication now in the public domain: Boase, George Clement (1891). "Hart, Andrew Searle". In Stephen, Leslie; Lee, Sidney (eds.). Dictionary of National Biography. Vol. 25. London: Smith, Elder & Co. pp. 56–57.
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Christopher Wren
Sir Christopher Wren FRS (/rɛn/;[2] 30 October 1632 [O.S. 20 October] – 8 March 1723 [O.S. 25 February])[3][4] was an English architect, astronomer, mathematician and physicist who was one of the most highly acclaimed architects in the history of England.[4] Known for his work in the English Baroque style,[4] he was accorded responsibility for rebuilding 52 churches in the City of London after the Great Fire in 1666, including what is regarded as his masterpiece, St Paul's Cathedral, on Ludgate Hill, completed in 1710.[5]
Sir
Christopher Wren
FRS
Wren in a portrait by Godfrey Kneller (1711)
Born30 October 1632 [O.S. 20 October]
East Knoyle, Wiltshire, England
Died8 March 1723 [O.S. 25 February]
(aged 90)[1]
St James's, London, England
NationalityEnglish (later British)
Alma materWadham College, Oxford
Known forDesigner of 54 London churches, including St Paul's Cathedral, as well as many notable secular buildings in London after the Great Fire
Spouses
Faith Coghill
(m. 1669; died 1675)
Jane Fitzwilliam
(m. 1677; died 1680)
Children4
Parent(s)Christopher Wren the Elder
Mary Cox
Scientific career
FieldsArchitecture, physics, astronomy and mathematics
InstitutionsAll Souls' College, Oxford
Academic advisorsWilliam Oughtred
Surveyor of the King's Works
In office
1669–1718
Preceded byJohn Denham
Succeeded byWilliam Benson
3rd President of the Royal Society
In office
1680–1682
Preceded byJoseph Williamson
Succeeded byJohn Hoskyns
Member of the English Parliament
1701–1702Weymouth and Melcombe Regis
6 March – 17 May 1690
11 January – 14 May 1689
New Windsor
1685–1687Plympton Erle
The principal creative responsibility for a number of the churches is now more commonly attributed to others in his office, especially Nicholas Hawksmoor. Other notable buildings by Wren include the Royal Hospital Chelsea, the Old Royal Naval College, Greenwich, and the south front of Hampton Court Palace.
Educated in Latin and Aristotelian physics at the University of Oxford, Wren was a founder of the Royal Society and served as its president from 1680 to 1682.[4] His scientific work was highly regarded by Isaac Newton and Blaise Pascal.
Life and works
Wren was born in East Knoyle in Wiltshire, the only surviving son of Christopher Wren the Elder (1589–1658) and Mary Cox, the only child of the Wiltshire squire Robert Cox from Fonthill Bishop. Christopher Sr. was, at that time, the rector of East Knoyle and, later, Dean of Windsor. It was while they were living at East Knoyle that all their children were born; Mary, Catherine and Susan were all born by 1628 but then several children who were born, died within a few weeks of their birth. Their son Christopher was born in 1632 then, two years later, another daughter named Elizabeth was born. Mary must have died shortly after the birth of Elizabeth, although there does not appear to be any surviving record of the date. Through Mary Cox, however, the family became well off financially for, as the only heir, she had inherited her father's estate.[6]
As a child Wren "seem'd consumptive".[7] Although a sickly child, he would survive into robust old age. He was first taught at home by a private tutor and his father. After his father's royal appointment as Dean of Windsor in March 1635, his family spent part of each year there, but little is known about Wren's life at Windsor. He spent his first eight years at East Knoyle and was educated by the Rev. William Shepherd, a local clergyman.[6]
Little is known of Wren's schooling thereafter, during dangerous times when his father's Royal associations would have required the family to keep a very low profile from the ruling Parliamentary authorities. It was a tough time in his life, but one which would go on to have a significant impact upon his later works. The story that he was at Westminster School between 1641 and 1646 is substantiated only by Parentalia, the biography compiled by his son, a fourth Christopher, which places him there "for some short time" before going up to Oxford (in 1650); however, it is entirely consistent with headmaster Doctor Busby's well-documented practice of educating the sons of impoverished Royalists and Puritans alike, irrespective of current politics or his own position.[8]
Some of Wren's youthful exercises preserved or recorded (though few are datable) showed that he received a thorough grounding in Latin and also learned to draw. According to Parentalia, he was "initiated" in the principles of mathematics by William Holder, who married Wren's elder sister Susan (or Susanna) in 1643. His drawing was put to academic use in providing many of the anatomical drawings for the anatomy textbook of the brain, Cerebri Anatome (1664), published by Thomas Willis, which coined the term "neurology".[9] During this time period, Wren became interested in the design and construction of mechanical instruments. It was probably through Holder that Wren met Sir Charles Scarburgh whom Wren assisted in his anatomical studies. Another sister Anne Brunsell, married a clergyman and is buried in Stretham.[10]
On 25 June 1650, Wren entered Wadham College, Oxford, where he studied Latin and the works of Aristotle. It is anachronistic to imagine that he received scientific training in the modern sense. However, Wren became closely associated with John Wilkins, the Warden of Wadham. The Wilkins circle was a group whose activities led to the formation of the Royal Society, comprising a number of distinguished mathematicians, creative workers and experimental philosophers. This connection probably influenced Wren's studies of science and mathematics at Oxford. He graduated B.A. in 1651, and two years later received M.A.[11]
1653–1664
After receiving his M.A. in 1653, Wren was elected a fellow of All Souls' College in the same year and began an active period of research and experiment in Oxford.[12] Among these were a number of physiological experiments on dogs, including one now recognized as the first injection of fluids into the bloodstream of a live animal under laboratory conditions. At Oxford he became part of the group around John Wilkins, he was key to the correspondence network known as the Invisible College,[13] within All Souls arms the arms of Wren's Robert Boyle appear in the colonnade of the Great Quadrangle, opposite the arms of the Hill family of Shropshire, close by a sundial designed by Boyle's friend Wren.[14]
His days as a fellow of All Souls ended when Wren was appointed Professor of Astronomy at Gresham College, London, in 1657.[15][16] He was there provided with a set of rooms and a stipend and required to give weekly lectures in both Latin and English.[16] Wren took up this new work with enthusiasm. He continued to meet the men with whom he had frequent discussions in Oxford. They attended his London lectures and in 1660, initiated formal weekly meetings. It was from these meetings that the Royal Society, England's premier scientific body, was to develop. He undoubtedly played a major role in the early life of what would become the Royal Society; his great breadth of expertise in so many different subjects helped in the exchange of ideas between the various scientists. In fact, the report on one of these meetings reads:
Memorandum November 28, 1660. These persons following according to the usual custom of most of them, met together at Gresham College to hear Mr Wren's lecture, viz. The Lord Brouncker, Mr Boyle, Mr Bruce, Sir Robert Moray, Sir Paule Neile, Dr Wilkins, Dr Goddard, Dr Petty, Mr Ball, Mr Rooke, Mr Wren, Mr Hill. And after the lecture was ended they did according to the usual manner, withdraw for mutual converse.[17]
In 1662, they proposed a society "for the promotion of Physico-Mathematicall Experimental Learning". This body received its Royal Charter from Charles II and "The Royal Society of London for Improving Natural Knowledge" was formed. In addition to being a founder member of the Society, Wren was president of the Royal Society from 1680 to 1682.[4]
In 1661, Wren was elected Savilian Professor of Astronomy at Oxford, and in 1669 he was appointed Surveyor of Works to Charles II. From 1661 until 1668 Wren's life was based in Oxford, although his attendance at meetings of the Royal Society meant that he had to make periodic trips to London.[15]
The main sources for Wren's scientific achievements are the records of the Royal Society. His scientific works ranged from astronomy, optics, the problem of finding longitude at sea, cosmology, mechanics, microscopy, surveying, medicine and meteorology. He observed, measured, dissected, built models and employed, invented and improved a variety of instruments.[18]
1665–1723
It was probably around this time that Sir Christopher Wren was drawn into redesigning a battered St Paul's Cathedral. Making a trip to Paris in 1665, Wren studied architecture, which had reached a climax of creativity, and perused the drawings of Bernini, the great Italian sculptor and architect, who himself was visiting Paris at the time. Returning from Paris, he made his first design for St Paul's. A week later, however, the Great Fire destroyed two-thirds of the city. Wren submitted his plans for rebuilding the city to King Charles II, although they were never adopted. With his appointment as King's Surveyor of Works in 1669, he had a presence in the general process of rebuilding the city, but was not directly involved with the rebuilding of houses or companies' halls. Wren was personally responsible for the rebuilding of 51 churches; however, it is not necessarily true to say that each of them represented his own fully developed design.
Wren was knighted on 14 November 1673.[19] This honour was bestowed on him after his resignation from the Savilian chair in Oxford, by which time he had already begun to make his mark as an architect, both in services to the Crown and in playing an important part in rebuilding London after the Great Fire.
Additionally, he was sufficiently active in public affairs to be returned as Member of Parliament on four occasions.[20] Wren first stood for Parliament in a by-election in 1667 for the Cambridge University constituency, losing by six votes to Sir Charles Wheler.[21] He was unsuccessful again in a by-election for the Oxford University constituency in 1674, losing to Thomas Thynne.[22] At his third attempt Wren was successful, and he sat for Plympton Erle during the Loyal Parliament of 1685 to 1687.[23] Wren was returned for New Windsor on 11 January 1689 in the general election, but his election was declared void on 14 May 1689.[24] He was elected again for New Windsor on 6 March 1690, but this election was declared void on 17 May 1690.[25] Over a decade later he was elected unopposed for Weymouth and Melcombe Regis at the November 1701 general election. He retired at the general election the following year.[26]
Wren's career was well established by 1669, and it may have been his appointment as Surveyor of the King's Works early that year that persuaded him that he could finally afford to marry. In 1669, the 37-year-old Wren married his childhood neighbour, the 33-year-old Faith Coghill, daughter of Sir John Coghill of Bletchingdon. Little is known of Faith, but a love letter from Wren survives, which reads, in part:
I have sent your Watch at last & envy the felicity of it, that it should be soe near your side & soe often enjoy your Eye. ... .but have a care for it, for I have put such a spell into it; that every Beating of the Balance will tell you 'tis the Pulse of my Heart, which labors as much to serve you and more trewly than the Watch; for the Watch I beleeve will sometimes lie, and sometimes be idle & unwilling ... but as for me you may be confident I shall never ...[27]
This brief marriage produced two children: Gilbert, born October 1672, who suffered from convulsions and died at about 18 months old, and Christopher, born February 1675. The younger Christopher was trained by his father to be an architect. It was this Christopher that supervised the topping out ceremony of St Paul's in 1710 and wrote the famous Parentalia, or, Memoirs of the family of the Wrens. Faith Wren died of smallpox on 3 September 1675. She was buried in the chancel of St Martin-in-the-Fields beside the infant Gilbert. A few days later Wren's mother-in-law, Lady Coghill, arrived to take the infant Christopher back with her to Oxfordshire to raise.
In 1677, 17 months after the death of his first wife, Wren remarried, this time to Jane Fitzwilliam, daughter of William FitzWilliam, 2nd Baron FitzWilliam,[28] and his wife Jane Perry, the daughter of a prosperous London merchant.
She was a mystery to Wren's friends and companions. Robert Hooke, who often saw Wren two or three times every week, had, as he recorded in his diary, never even heard of her, and was not to meet her till six weeks after the marriage.[29] As with the first marriage, this too produced two children: a daughter Jane (1677–1702); and a son William, "Poor Billy" born June 1679, who was developmentally delayed.
Like the first, this second marriage was also brief. Jane Wren died of tuberculosis in September 1680. She was buried alongside Faith and Gilbert in the chancel of St Martin-in-the-Fields. Wren was never to marry again; he lived to be over 90 years old and of those years was married only nine.
Bletchingdon was the home of Wren's brother-in-law William Holder, who was rector of the local church. Holder had been a Fellow of Pembroke College, Oxford. An intellectual of considerable ability, he is said to have been the figure who introduced Wren to arithmetic and geometry.[30]
Wren's later life was not without criticisms and attacks on his competence and his taste. In 1712, the Letter Concerning Design of Anthony Ashley Cooper, third Earl of Shaftesbury, circulated in manuscript. Proposing a new British style of architecture, Shaftesbury censured Wren's cathedral, his taste and his long-standing control of royal works. Although Wren was appointed to the Fifty New Churches Commission in 1711, he was left only with nominal charge of a board of works when the surveyorship started in 1715. On 26 April 1718, on the pretext of failing powers, he was dismissed in favour of William Benson.[31]
In 1713, he bought the manor of Wroxall, Warwickshire, from the Burgoyne family, to which his son Christopher retired in 1716 after losing his post as Clerk of Works.[32] Several of Wren's descendants would be buried there in the Church of St Leonard.
Death
The Wren family estate was at The Old Court House in the area of Hampton Court. He had been given a lease on the property by Queen Anne in lieu of salary arrears for building St Paul's.[33] For convenience Wren also leased a house on St James's Street in London. According to a 19th-century legend, he would often go to London to pay unofficial visits to St Paul's, to check on the progress of "my greatest work". On one of these trips to London, at the age of ninety, he caught a cold and on 25 February 1723 a servant who tried to awaken Wren from his nap found that he had died in his sleep.[34]
Wren was laid to rest on 5 March 1723. His body was placed in the southeast corner of the crypt of St Paul's. There is a memorial to him in the crypt at St Paul's Cathedral.[35] beside those of his daughter Jane, his sister Susan Holder, and her husband William.[36] The plain stone plaque was written by Wren's eldest son and heir, Christopher Wren the Younger[37] The inscription, which is also inscribed in a circle of black marble on the main floor beneath the centre of the dome, reads:
SUBTUS CONDITUR HUIUS ECCLESIÆ ET VRBIS CONDITOR CHRISTOPHORUS WREN, QUI VIXIT ANNOS ULTRA NONAGINTA, NON SIBI SED BONO PUBLICO. LECTOR SI MONUMENTUM REQUIRIS CIRCUMSPICE Obijt XXV Feb: An°: MDCCXXIII Æt: XCI.
which translates from Latin as:[38]
Here in its foundations lies the architect of this church and city, Christopher Wren, who lived beyond ninety years, not for his own profit but for the public good. Reader, if you seek his monument – look around you. Died 25 Feb. 1723, age 91.
His obituary was published in the Post Boy No. 5244 London 2 March 1723:[39]
Sir Christopher Wren who died on Monday last in the 91st year of his age, was the only son of Dr. Chr. Wren, Dean of Windsor & Wolverhampton, Registar of the Garter, younger brother of Dr. Mathew (sic) Wren Ld Bp of Ely, a branch of the ancient family of Wrens of Binchester in the Bishoprick [sic] of Durham
1653. Elected from Wadham into fellowship of All Souls
1657. Professor of Astronomy Gresham College London
1660. Savilian Professor. Oxford
After 1666. Surveyor General for Rebuilding the Cathedral Church of St.Paul and the Parochial Churches & all other Public Buildings which he lived to finish
1669. Surveyor General till April 26. 1718
1680. President of the Royal Society
1698. Surveyor General & Sub Commissioner for Repairs to Westminster Abbey by Act of Parliament, continued till death.
His body is to be deposited in the Great Vault under the Dome of the Cathedral of St. Paul.
"The Curious and Entire Libraries of Sir Christopher Wren", and of his son, were auctioned by Langford and Cock at Mr Cock's in Covent Garden on 24–27 October 1748.[40]
Scientific career
One of Wren's friends, Robert Hooke, scientist and architect and a fellow Westminster Schoolboy, said of him "Since the time of Archimedes there scarce ever met in one man in so great perfection such a mechanical hand and so philosophical mind."
When a fellow of All Souls, Wren constructed a transparent beehive for scientific observation; he began observing the moon, which was to lead to the invention of micrometers for the telescope. According to Parentalia (pp. 210–211), his solid model of the moon attracted the attention of the King who commanded Wren to perfect it and present it to him.
He contrived an artificial Eye, truly and dioptrically made (as large as a Tennis-Ball) representing the Picture as Nature makes it: The Cornea, and Crystalline were Glass, the other Humours, Water.
— Parentalia, p. 209
He experimented on terrestrial magnetism and had taken part in medical experiments while at Wadham College, performing the first successful injection of a substance into the bloodstream (of a dog). In Gresham College, he did experiments involving determining longitude through magnetic variation and through lunar observation to help with navigation, and helped construct a 35-foot (11 m) telescope with Sir Paul Neile. Wren also studied and improved the microscope and telescope at this time. He had also been making observations of the planet Saturn from around 1652 with the aim of explaining its appearance. His hypothesis was written up in De corpore saturni but before the work was published, Huygens presented his theory of the rings of Saturn. Immediately Wren recognised this as a better hypothesis than his own and De corpore saturni was never published. In addition, he constructed an exquisitely detailed lunar model and presented it to the king. In 1658, he found the length of an arc of the cycloid using an exhaustion proof based on dissections to reduce the problem to summing segments of chords of a circle which are in geometric progression.
A year into Wren's appointment as a Savilian Professor in Oxford, the Royal Society was created and Wren became an active member. As Savilian Professor, Wren studied mechanics thoroughly, especially elastic collisions and pendulum motions. He also directed his far-ranging intelligence to the study of meteorology: in 1662, he invented the tipping bucket rain gauge and, in 1663, designed a "weather-clock" that would record temperature, humidity, rainfall and barometric pressure. A working weather clock based on Wren's design was completed by Robert Hooke in 1679.[41]
In addition, Wren experimented on muscle functionality, hypothesizing that the swelling and shrinking of muscles might proceed from a fermentative motion arising from the mixture of two heterogeneous fluids. Although this is incorrect, it was at least founded upon observation and may mark a new outlook on medicine: specialisation.
Another topic to which Wren contributed was optics. He published a description of an engine to create perspective drawings and he discussed the grinding of conical lenses and mirrors. Out of this work came another of Wren's important mathematical results, namely that the hyperboloid of revolution is a ruled surface. These results were published in 1669.[42] In subsequent years, Wren continued with his work with the Royal Society, although after the 1680s his scientific interests seem to have waned: no doubt his architectural and official duties absorbed more time.
It was a problem posed by Wren that serves as an ultimate source to the conception of Newton's Principia Mathematica Philosophiae Naturalis. Robert Hooke had theorised that planets, moving in vacuo, describe orbits around the Sun because of a rectilinear inertial motion by the tangent and an accelerated motion towards the Sun. Wren's challenge to Halley and Hooke, for the reward of a book worth thirty shillings, was to provide, within the context of Hooke's hypothesis, a mathematical theory linking Kepler's laws with a specific force law. Halley took the problem to Newton for advice, prompting the latter to write a nine-page answer, De motu corporum in gyrum, which was later to be expanded into the Principia.[43]
Mentioned above are only a few of Wren's scientific works. He also studied other areas, ranging from agriculture, ballistics, water and freezing, light and refraction, to name only a few. Thomas Birch's History of the Royal Society (1756–57) is one of the most important sources of our knowledge not only of the origins of the Society, but also the day-to-day running of the Society. It is in these records that most of Wren's known scientific works are recorded.
Architectural career
Wren was a prominent man of science at the height of the Scientific Revolution. The Scientific Revolution seemed to promise a merger of the science of mechanics and the art of building. In Galileo Galilei's Two New Sciences the first science is not dynamics, for which the book is now better known, but rather the strength of materials, which Galileo had recognized 30 years earlier as a “science that is very necessary in making machines and buildings of all kinds.” In 1624 Henry Wotton, the British ambassador to Venice, published a book on architecture in which he analyzed in a rudimentary way the structure of a stone arch. Moreover, in the 17th century, it was people who would now be called scientists who were awarded the commissions to design and build monumental structures. In Turin, Guarino Guarini, a mathematician, devised the plans for such celebrated buildings as the Royal Church of Saint Lawrence, the Chapel of the Holy Shroud and the Palazzo Carignano. In Paris, Claude Perrault, a physician and an anatomist, designed the façade of the Louvre and the observatory of the Académie Française. In London, it was Wren and Hooke who collaborated as chief architect and city surveyor after the city was devastated by the Great Fire of 1666.
In 1661, just months after taking his post at Oxford, Wren was invited by Charles II to oversee the construction of new harbour defences at Tangier—then-newly under British control. Wren ultimately excused himself from the King's offer. Letters dated to the end of 1661 note that in addition to the Tangier project, Charles II had also sought Wren for consultation regarding repairs to Old St Paul's Cathedral, the reconstruction of which would ultimately be the architect's magnum opus. Speaking of Wren's vocational transition from academic to architect-engineer, biographer Adrian Tinniswood writes "the use of mathematicians in military fortification was not unusual... Perhaps Wren also had experience of the business of fortification, more than we know."[15]
Early architectural work
Pembroke Chapel
Sheldonian Theatre
Emmanuel College Chapel
Wren's first known foray into architecture came after his uncle, Matthew Wren, Bishop of Ely, offered to finance a new chapel for Pembroke College, Cambridge. Matthew commissioned his nephew for the design, finding the architecturally inexperienced Christopher to be both ideologically sympathetic and stylistically deferential. Wren produced his design in the Winter of 1662 or 1663 and the chapel was completed in 1665.
Wren's second, similarly collegiate work followed soon after, when he was commissioned to design Oxford's "New Theatre," financed by Gilbert Sheldon.[44] His design for the structure was met with lukewarm to negative reception, with even Wren's defenders admitting the young architect to have not yet been "capable of handling a large architectural composition with assurance".[15] Adrian Tinniswood credits the building's flaws to "Sheldon's refusal to pay for an elaborate exterior, Wren's inability to find an adequate external expression for a building which was wholly conditioned by the functionality of its interior space and, ...his refusal to bend the knee to classical authority in the way that our experience of eighteenth-century architecture has conditioned us to believe is right."[15] Prior to the theatre's 1669 completion, Wren had received further commissions for the Garden Quadrangle at Trinity College, Oxford, and the chapel of Emmanuel College, Cambridge.[15]
Wren left for Paris in July 1665 on his first and only trip abroad. In France, the architect encountered an architectural milieu more closely linked to the ideals of the Italian Renaissance. Wren also met Gian Lorenzo Bernini, who was "widely acknowledged by contemporaries as the greatest artist of the century". Though Bernini's concrete influence on Wren's designs was transmitted via published plans and engravings, the encounter surely impacted the budding architect and his vocational trajectory.[15]
St Paul's Cathedral
St Paul's Cathedral in London has always been the highlight of Wren's reputation. His association with it spans his whole architectural career, including the 36 years between the start of the new building and the declaration by parliament of its completion in 1711.Letters document Wren's involvement in St Paul as early as 1661, when he was consulted by Charles II regarding repairs to the medieval structure.[15] In the spring of 1666, he made his first design for a dome for St Paul's. It was accepted in principle on 27 August 1666. One week later, however, the Great Fire of London reduced two-thirds of the City to a smoking desert and old St Paul's to ruin. Wren was most likely at Oxford at the time, but the news, so fantastically relevant to his future, drew him at once to London. Between 5 and 11 September, he ascertained the precise area of devastation, worked out a plan for rebuilding the City and submitted it to Charles II. Others also submitted plans. However, no new plan proceeded any further than the paper on which it was drawn. A Rebuilding of London Act which provided rebuilding of some essential buildings was passed in 1666. In 1669, the King's Surveyor of Works died and Wren was promptly installed.
The development of Wren's design for St Paul's Cathedral
Greek Cross Design (1673)
The Warrant Design (1674)
The cathedral as built
It was not until 1670 that the pace of rebuilding started accelerating. A second rebuilding act was passed that year, raising the tax on coal and thus providing a source of funds for rebuilding of churches destroyed within the City of London. Wren presented his initial "First Model" for St Paul's. This plan was accepted, and demolition of the old cathedral began. By 1672, however, this design seemed too modest, and Wren met his critics by producing a design of spectacular grandeur. This modified design, called "Great Model", was accepted by the King and the construction started in November 1673. However, this design failed to satisfy the chapter and clerical opinion generally; moreover, it had an economic drawback. Wren was confined to a "cathedral form" desired by the clergy. In 1674 he produced the rather meagre Classical-Gothic compromise known as the Warrant Design. However, this design, called so from the royal warrant of 14 May 1675 attached to the drawings, is not the design upon which work had begun a few weeks before.
St Paul's Cathedral
West front
Dome
Nave
The cathedral that Wren started to build bears only a slight resemblance to the Warrant Design. In 1697, the first service was held in the cathedral when Wren was 65. There was still, however, no dome. Finally, in 1711 the cathedral was declared complete, and Wren was paid the half of his salary that, in the hope of accelerating progress, Parliament had withheld for 14 years since 1697. The cathedral had been built for 36 years under his direction, and the only disappointment he had about his masterpiece was the dome: against his wishes, the commission engaged Thornhill to paint the inner dome in false perspective and finally authorised a balustrade around the roof line. This diluted the hard edge Wren had intended for his cathedral, and elicited the apt parthian comment that "ladies think nothing well without an edging".[45]
Later career
During the 1670s, Wren received significant secular commissions. Among many of his notable designs at this time, the monument (1671–76)[46] commemorating the Great Fire also involved Robert Hooke, but Wren was in control of the final design, the Royal Observatory (1675–76),[46] and the Wren Library at Trinity College, Cambridge (1676–84)[46] were the most important ones.
In 1682, Wren advised that the original statues of the King's Beasts on St George's Chapel, Windsor be removed. The pinnacles were left bare until 1925, when replica statues were installed.[47]
By historical accident, all Wren's large-scale secular commissions dated from after the 1680s. At the age of 50 his personal development, as was that of English architecture, was ready for monumental but humane architecture, in which the scales of individual parts relate both to the whole and to the people who used them. The first large project Wren designed, the Chelsea Hospital (1682–92),[46] does not entirely satisfy the eye in this respect, but met its brief with distinction and such success that even in the 21st century it fulfils its original function. The reconstruction of the stateroom at Windsor Castle was notable for the integration of architecture, sculpture and painting. This commission was in the hand of Hugh May, who died in February 1684, before the construction finished; Wren assumed his post and finalised the works.
Between 1683 and 1685 he was much occupied in designing the King's House, Winchester, where Charles II had hoped to spend his declining years, but which was never completed. When Wren promised that it would be complete within a year the King, who was conscious of his mortality, replied that " a year is a great time in my life".
After the death of Charles II in 1685, Wren's attention was directed mainly to Whitehall (1685–87).[46] The new king, James II, required a new chapel and also ordered a new gallery, council chamber and a riverside apartment for the Queen. Later, when James II was removed from the throne, Wren took on architectural projects such as Kensington Palace (1689–96)[46] and Hampton Court (1689–1700).[46]
The erection of the present Windsor Guildhall was begun in 1687, under the direction of Sir Thomas Fitz (or Fiddes) but there is a story that on his death in 1689, the task was taken over by Sir Christopher Wren. It was completed at a cost of £2687 – 1s – 6d. The new building was supported around its perimeter by stone columns, providing a covered area beneath as a venue for corn markets.
The story is widely told that the borough Council demanded that Wren should insert additional columns within the covered area, in order to support the weight of the heavy building above; Wren, however, was adamant that these were not necessary. Eventually, the council insisted and, in due course, the extra supporting columns were built, but Wren made them slightly short, so that they do not quite touch the ceiling, hence proving his claim that they were not necessary. However, there is little evidence that Wren was ever involved in the design or construction of the Guildhall. It is now believed that the story grew out of Wren's connections with Windsor and that his son, also called Christopher Wren, who served as a Member of Parliament for Windsor, commissioned the statue of Prince George of Denmark in 1713 on the south end of the building and his name was engraved underneath. The pillars were probably moved into the corn market from the east side of the building when an extension was added in 1829.[48] The gaps at the top of the pillars are now filled with tiles smaller than the capitals.
Wren did not pursue his work on architectural design as actively as he had before the 1690s, although he still played important roles in a number of royal commissions. In 1696 he was appointed Surveyor of Greenwich Naval Hospital,[46] and in 1698 he was appointed Surveyor of Westminster Abbey.[49] He resigned from the former role in 1716 but held the latter until his death, approving with a wavering signature[50] Burlington's revisions of Wren's own earlier designs for the great Archway of Westminster School.
Freemasonry
Since at least the 18th century, the Lodge of Antiquity No. 2, one of the four founding Masonic Lodges of the Premier Grand Lodge of England in 1717, has claimed Christopher Wren to have been its Master at the Goose and Gridiron at St. Paul's churchyard.[51] Whilst he was rebuilding the cathedral he is said to have been "adopted" on 18 May 1691 (that is, accepted as a sort of honorary member or patron, rather than an operative). Their 18th-century maul with its 1827 inscription claiming that it was used by Wren for the foundation stone of St. Paul's, belonging to the Lodge and on display in the Library and Museum of Freemasonry in London, corroborates the story. James Anderson made the claims in his widely circulated Constitutions while many of Wren's friends were still alive, but he made many highly creative claims as to the history or legends of Freemasonry. There is also a clear possibility of confusion between the operative workmen's lodges which might naturally have welcomed the boss, and the "speculative" or gentlemen's lodges which became highly fashionable just after Wren's death. By the standards of his time, a gentleman like Wren would not generally join an artisan body; however the workmen of St Paul's cathedral would naturally have sought the patronage or "interest" of their employer, and within Wren's lifetime there was a predominantly gentlemen's Lodge at the Rummer and Grapes, a mile upriver at Westminster (where Wren had been to School).
In 1788, the Lodge of Antiquity thought they were buying a portrait of Wren which now dominates Lodge Room 10, in the same building as the Museum; but it is now identified with William Talman, not Wren. Nevertheless, this recorded event and many old records attest to the fact that Antiquity thought that Wren had been its Master, at a time when it still held its minute books for the relevant years (which were lost by Preston at some date after 1778).
The evidence of whether Wren was a speculative freemason is the subject of the Prestonian Lecture[52] of 2011, which concludes on the evidence of two obituaries and Aubrey's memoirs, with supporting materials, that he did indeed attend the closed meeting in 1691, probably of the Lodge of Antiquity, but that there is nothing to suggest that he was ever a Grand Officer as claimed by Anderson.
Achievement and legacy
Christopher Wren appeared on the reverse of the first British £50 banknote (Series D) issued in modern times. The notes were printed between 1981 and 1994, and were in circulation until 1996.[53]
Greenwich Hospital, designed largely by Wren, is a designated World Heritage Site
In 1997, UNESCO inscribed Wren's Greenwich Hospital on the World Heritage list, citing the complex's "outstanding architectural and artistic achievements".[54]
Bibliography
• Wren, Christopher; Ames, Joseph; Wren, Stephen (1750). Parentalia, or, Memoirs of the family of the Wrens.
See also
• List of works by Christopher Wren
• List of Christopher Wren churches in London
• Thomas Gilbert, one of Wren's apprentices and adaptant of his architectural style
• Gresham Professor of Astronomy
• List of presidents of the Royal Society
Wren appears, or is mentioned in several Restoration-era novels or movies.
• The novel Hawksmoor by Peter Ackroyd, which features a fictionalised Christopher Wren
• He also features as an important secondary character in Rosalind Laker's (Barbara Ovstedal) novel Circle of Pearls.
• He is mentioned in the 2004 film The Libertine, starring Johnny Depp, Rosamund Pike and John Malkovich.
• For the character created by Agatha Christie, see the play The Mousetrap
References
Citations
1. From the 12th century to 1752, the legal year in England began on 25 March Old Style. Wren died in 1722 O.S. according to the pre-1752 calendar (see Paul Welberry Kent, Allan Chapman, eds., Robert Hooke and the English Renaissance, Gracewing Publishing, 2005, p. 47).
2. Wells, John C. (2008), Longman Pronunciation Dictionary (3rd ed.), Longman, p. 908, ISBN 9781405881180
3. Here both Old Style and New Style dates are given, with "Old Style" meaning: according to the Julian calendar but with the year starting on 1 January. Dates elsewhere in this article are Old Style in the same way, except where both styles are given. Using New Style dates for Wren's birth and death, even though he lived in England in the Old Style era, avoids confusion about his age at death.
4. "Sir Christopher Wren | English architect". Encyclopedia Britannica. Retrieved 31 August 2018.
5. "Sir Christopher Wren (1632–1723)". Retrieved 31 August 2018.
6. "Question: Wren's connection with Wiltshire". Wiltshire and Swindon History Centre. Wiltshire Council. 17 May 2003. Retrieved 14 May 2023.
7. Wren, Ames & Wren 1750
8. "Sir Christopher Wren". www.encyclopedia.com. Retrieved 31 August 2018.
9. "Five depictions of the brain – The Psychologist". bps.org.uk. Retrieved 31 March 2017.
10. Pevsner, Nikolaus (1970). The buildings of England: Cambridgeshire (2nd ed.). Penguin Books. p. 462. ISBN 0-14-071010-8.
11. Downes, Kerry (2007). Christopher Wren. New York: Oxford University Press. ISBN 9780199215249. OCLC 83977472.
12. Bolton, Glorney (1956). Sir Christopher Wren. Hutchinson. p. 37.
13. Higgitt, Rebekah (20 October 2014). "Google Doodle forgets to celebrate Christopher Wren the man of science". The Guardian. ISSN 0261-3077. Retrieved 21 February 2023.
14. History of Science Museum Oxford University. "The Virtual Oxford Science Walk".
15. Tinniswood, Adrian (2002). His Invention So Fertile: A Life of Christopher Wren. Pimlico. pp. 115–129. ISBN 978-0-7126-7364-8.
16. Rabbitts, Paul (2019). Sir Christopher Wren. Bloomsbury Publishing. p. 13. ISBN 978-1-78442-323-0.
17. "Sir Christopher Wren". The MacTutor History of Mathematics archive. Retrieved 30 September 2006.
18. Windsor, Alan (March 1984). "John Soane: The Making of an Architect Pierre de La Ruffinière Du Prey". Journal of the Society of Architectural Historians. 43 (1): 84–85. doi:10.2307/989987. JSTOR 989987.
19. Meridew, John (1848). A Catalogue of Engraved Portraits of Nobility, Gentry, Clergymen and Others, Born, Resident In, Or Connected with the County of Warwick: Alphabetically Arranged, with Names of the Painters and Engravers, ... to which are Added Numerous Biographical Notices, ... p. 77.
20. "Sir Christopher Wren, 1632–1723". The History of Parliament. Retrieved 15 September 2016.
21. "Cambridge University, 1660–1690". The History of Parliament. Retrieved 15 September 2016.
22. "Oxford University, 1660–1690". The History of Parliament. Retrieved 15 September 2016.
23. "Plympton Erle, 1660–1690". The History of Parliament. Retrieved 15 September 2016.
24. "New Windsor, 1660–1690". The History of Parliament. Retrieved 15 September 2016.
25. "New Windsor, 1690–1715". The History of Parliament. Retrieved 15 September 2016.
26. "Weymouth and Melcolme Regis, 1690–1715". The History of Parliament. Retrieved 15 September 2016.
27. Tinniswood 2001, p. 184 (Some time earlier, Faith had dropped her wristwatch into a pool of water. It had been sent to Wren in London for it to be repaired. This letter was part of a package.)
28. "Christopher Wren - Biography". Maths History. Retrieved 7 October 2022.
29. Tinniswood 2001, p. 239
30. Davies, C.S.L. (2008). "The Youth and Education of Christopher Wren". The English Historical Review. 123 (501): 300–327. doi:10.1093/ehr/cen008. ISSN 0013-8266. JSTOR 20108454.
31. Downes, Kerry (2004). "Wren, Sir Christopher (1632–1723), architect, mathematician, and astronomer". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/30019. Retrieved 16 June 2019. (Subscription or UK public library membership required.)
32. "Parishes: Wroxall". British History Online. Retrieved 10 September 2018.
33. Buchanan, Clare (11 April 2013). "Sir Christopher Wren's magnificent home up for sale". Richmond and Twickenham Times. London. Archived from the original on 12 November 2013. Retrieved 7 July 2013.
34. Tinniswood 2001, p. 366
35. "Memorials of St Paul's Cathedral" Sinclair, W. p. 469: London; Chapman & Hall, Ltd; 1909.
36. "Discover the Crypt – St Paul's Cathedral, London, UK". stpauls.co.uk. Retrieved 6 September 2009.
37. Elmes 1852, p. 411
38. Masters, Tom; Fallon, Steve; Maric, Vesna (2008). London. Lonely Planet Publications. p. 111. ISBN 978-1-74104-712-7.
39. Bolton, Arthur T.; Hendry, H. Duncan, eds. (1941). The Wren Society Volume XVIII. Oxford University Press. p. 181.
40. Cock, Christopher (1748). A Catalogue of the Curious and Entire Libraries of Sir Christopher Wren, Knt. and Christopher Wren, Esq. his son, etc. London: Christopher Cock.
41. Multhauf, Robert P. (1961). "The Introduction of Self-Registering Meteorological Instruments". United States National Museum Bulletin.
42. Wren, Christophoro (1669). "Generatio corporis cylindroidis hyperbolici, elaborandis lentibus hyperbolicis accommodati, auth. Christophoro Wren L L D. Et Regiorum Ædificiorum Præfecto, nec non-Soc. Regiæ Sodali". Philosophical Transactions of the Royal Society of London. 4 (48): 961–962. Bibcode:1669RSPT....4..961W. doi:10.1098/rstl.1669.0018.
43. Grattan-Guinness, Ivor, ed.; Landmark Writings in Western Mathematics, 1st ed., 2005, pp. 64–65
44. Geraghty, Anthony (2002). "Wren's Preliminary Design for the Sheldonian Theatre". Architectural History. 45: 275–288. doi:10.2307/1568785. ISSN 0066-622X. JSTOR 1568785.
45. Bolton and Hendry, eds., The Wren Society, 20 vols.
46. Downes 1988, p. 131
47. London, H. Stanford (1953). The Queen's Beasts. Newman Neame. p. 15.
48. Marson, Pamela; Mitchell, Brigitte (2015). Windsor Guildhall: History and Tour. Friends of the Windsor & Royal Borough Museum. p. 7. ISBN 9780-9010-3309-3.
49. Jardine 2003, p. 440
50. Westminster Abbey Muniments
51. "Manifesto of 1778 issued by The Lodge of Antiquity, formerly The Old Lodge of St Paul, to preserve the Ancient Landmarks of Freemasonry, Brotherly Love, Relief and Truth" (PDF). Lodgeroomus.net. Archived from the original (PDF) on 21 July 2012.
52. Campbell 2011
53. Dutton, Roy (2009). Financial Meltdown. Infodial. p. 233. ISBN 978-0-9556554-3-2.
54. Centre, UNESCO World Heritage. "Maritime Greenwich". UNESCO World Heritage Centre. Retrieved 10 July 2021.
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• Danzer, Gerald A.; Klor De Alva, J. Jorge; Krieger, Larry S. (2003). The Americans. Rand McNally. ISBN 978-0-618-37719-0.
• Downes, Kerry (1988). The Architecture of Wren (second ed.). Redhedge. ISBN 978-0-9513877-0-2.
• Elmes, James (1852). Sir Christopher Wren and his times. Chapman & Hall.
• Escott, John (1996). London. Oxford University Press. ISBN 978-0-19-422801-5.
• Hart, Vaughan (1995). St Paul's Cathedral: Sir Christopher Wren. Phaedon. ISBN 978-0-7148-2998-2.
• Hart, Vaughan (2020) Christopher Wren: In Search of Eastern Antiquity. Yale University Press. ISBN 9781913107079
• Hart, Vaughan, ‘London's Standard: Christopher Wren and the Heraldry of the Monument’, in RES: Journal of Anthropology and Aesthetics, vol.73/74, Autumn 2020, pp. 325–39
• Jardine, Lisa (2003). On a Grander Scale: The Outstanding Career of Sir Christopher Wren. HarperCollins. ISBN 978-0-00-710775-9. paperback ISBN 0-00-710776-5
• Tinniswood, Adrian (2001). His Invention So Fertile: A Life of Christopher Wren. Oxford University Press. ISBN 978-0-19-514989-0.
• Ward, J. (1740). The lives of the professors of Gresham College. John Moore in Bartholomew lane.
External links
Wikimedia Commons has media related to Christopher Wren.
Wikiquote has quotations related to Christopher Wren.
• "Wren, Christopher (1632-1723)" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
• Middleton, John Henry (1911). "Wren, Sir Christopher" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 28 (11th ed.). Cambridge University Press. pp. 843–844.
• 'Scientists and Craftsmen in Sir Christopher Wren's London', lecture by Professor Allan Chapman, Gresham College, 23 April 2008 (available in text, audio and video formats).
• Life and times of Sir Christopher Wren on a Freemasonry website
• View interiors of Wren Churches in 360 degrees
Presidents of the Royal Society
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18th century
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19th century
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20th century
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21st century
• Lord May (2000)
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• Sir Paul Nurse (2010)
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• Adrian Smith (2020)
Savilian Professors
Chairs established by Sir Henry Savile
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Sir Cumference
Sir Cumference is a series of children's educational books about math by Cindy Neuschwander and Wayne Geehan.
The books have been studied for their use in mathematics education.[1]
Characters
Most of the characters of the book are named after math terms, such as Sir Cumference (circumference).
Sir Cumference
Sir Cumference is a knight in the kingdom of Camelot. He has a wife called Lady Di of Ameter and a son named Radius.
Di of Ameter
Di of Ameter is the wife of Sir Cumference. In the first book, she came up with all the different shapes of the table (parallelogram, square, etc.) and in Sir Cumference and the Dragon of Pi, she stayed with Sir Cumference when he turned into a dragon.
Radius
Radius is the son of Di of Ameter and Sir Cumference. He has a friend named Vertex in Sir Cumference and the Sword in the Cone, and plays an important role in both Sir Cumference and the Dragon of Pi and The Sword in the Cone first by turning his father to a dragon and back, and later assisting Vertex in becoming King. He is the focus of Sir Cumference and the Great Knight of Angleland, in which he becomes a knight after rescuing King Lell and his pair of dragons.
Vertex
Vertex is the best friend of Radius. He appears on the first page of Sir Cumference and the Sword in the Cone. He is quoted saying, "I've found out why King Arthur called us all here!" Sir Cumference and Radius agree Vertex should be the heir to the throne.
Series
Currently, there are 11 books in the series:
1. Sir Cumference and the First Round Table (1997)
2. Sir Cumference and the Dragon of Pi (1999)
3. Sir Cumference and the Great Knight of Angleland (2001)
4. Sir Cumference and the Sword in the Cone (2003)
5. Sir Cumference and the Isle of Immeter (2006)
6. Sir Cumference and All the Kings Tens (2009)
7. Sir Cumference and the Viking's Map (2012)
8. Sir Cumference and the Off-the-Charts Dessert (2013)
9. Sir Cumference and the Roundabout Battle (2015)
10. Sir Cumference and the Fracton Faire (2017)
11. Sir Cumference Gets Decima's Point (2020)
References
1. Long, Betty B.; Crocker, Deborah A. (2000). "Adventures with Sir Cumference: Standard Shapes and Nonstandard Units". Teaching Children Mathematics. 7 (4): 242–245. doi:10.5951/TCM.7.4.0242. ISSN 1073-5836. JSTOR 41197576.
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Sir Edmund Whittaker Memorial Prize
The Sir Edmund Whittaker Memorial Prize is awarded every four years by the Edinburgh Mathematical Society to an outstanding young mathematician having a specified connection with Scotland. It is named after Sir Edmund Whittaker.
History
After the death of Sir Edmund Whittaker in 1956, his son John Macnaghten Whittaker donated on behalf of the Whittaker Family the sum of £500 to the Edinburgh Mathematical Society to establish a prize for mathematical work in memory of his father. As of 2009, the award money remains £500.[1]
Winners
• 2017 Arend Bayer (University of Edinburgh)
• 2013 Stuart White (University of Glasgow)
• 2009 Agata Smoktunowicz (University of Edinburgh)[2]
• 2005 Tom Bridgeland (University of Sheffield)
• 2001 Michael McQuillan and J A Sherratt
• 1997 Alan D Rendall (Max-Planck-Institut für Gravitationsphysik)
• 1993 Mitchell A. Berger and Alan W. Reid[3]
• 1989 A A Lacey and Michael Röckner
• 1985 John Mackintosh Howie
• 1981 John M. Ball (University of Oxford)
• 1977 Gavin Brown and C A Stuart
• 1973 A M Davie (University of Edinburgh)
• 1970 Derek J S Robinson (University of Illinois)
• 1965 John Bryce McLeod (University of Oxford)
• 1961 A G Mackie and Andrew H. Wallace
See also
• List of mathematics awards
References
1. "Whittaker rules 2013". Edinburgh Mathematical Society. Archived from the original on 5 December 2014.
2. "Whittaker Prize : EMS". Edinburgh Mathematical Society. Archived from the original on 5 December 2014. Retrieved 30 November 2014.
3. "Sir Edmund Whittaker Memorial Prize". Proceedings of the Edinburgh Mathematical Society. 37 (2): 359–360. June 1994. doi:10.1017/S0013091500006131. ISSN 1464-3839.
External links
• O'Connor, John J.; Robertson, Edmund F., "EMS Whittaker Prize", MacTutor History of Mathematics archive, University of St Andrews
• O'Connor, John J.; Robertson, Edmund F., "Winners of the EMS Whittaker Prize", MacTutor History of Mathematics archive, University of St Andrews
• "Edinburgh Mathematical Society". Edinburgh Mathematical Society. Archived from the original on 14 July 2001. Retrieved 15 November 2020.
• "Prizes". Edinburgh Mathematical Society. Retrieved 15 November 2020.
Sir Edmund Taylor Whittaker FRS FRSE LLD ScD
Fields
• Mathematics
• Astronomy
• Mathematical physics
• History of science
Notable works
• A Course of Modern Analysis (1902)
• Analytical Dynamics of Particles and Rigid Bodies (1904)
• A History of the Theories of Aether and Electricity, from the age of Descartes to the Close of the Nineteenth Century (1910)
• A History of the Theories of Aether and Electricity, the Classic Theories (1951)
• A History of the Theories of Aether and Electricity, the Modern Theories (1900-1926) (1953)
Eponym of
• Whittaker function
• Whittaker model
• Whittaker–Nyquist–Kotelnikov–Shannon sampling theorem
• Whittaker–Shannon interpolation formula
• Sir Edmund Whittaker Memorial Prize
Notable research
• Rapidity
• Special functions
• Electromagnetism
• General relativity
• Harmonic functions
• Automorphic functions
• Confluent hypergeometric functions
• Numerical analysis
Notable family members
• John Macnaghten Whittaker (son)
• Edward Copson (son-in-law)
Notable disputes
• Lorentz-Poincaré-Einstein controversy
• Fictitious Problems in Mathematics
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Wikipedia
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James Lighthill
Sir Michael James Lighthill FRS FRAeS[1] (23 January 1924 – 17 July 1998) was a British applied mathematician, known for his pioneering work in the field of aeroacoustics[2][3][4][5][6] and for writing the Lighthill report on artificial intelligence.
Sir
James Lighthill
FRS FRAeS
Michael James Lighthill
Born(1924-01-23)23 January 1924
Paris, France
Died17 July 1998(1998-07-17) (aged 74)
Sark, Channel Islands
NationalityBritish
Alma materCambridge University
Known forLighthill report
Lighthill's equation
Lighthill's eighth power law
Lighthill mechanism
Aeroacoustics
Fluid dynamics
AwardsTimoshenko Medal (1963)
Royal Medal (1964)
Elliott Cresson Medal (1975)
Naylor Prize and Lectureship (1977)
IMA Gold Medal (1982)
Otto Laporte Award (1984)
Copley Medal (1998)
Scientific career
FieldsMathematics,
Acoustics
InstitutionsVictoria University of Manchester
University College London
Cambridge University
Imperial College London
Doctoral studentsGerald B. Whitham
Biography
James Lighthill was born to Ernest Balzar Lichtenberg and Marjorie Holmes: an Alsatian mining engineer who changed his name to Lighthill in 1917, and the daughter of an engineer. The family lived in Paris until 1927, when the father retired and returned to live in England. As a young man, James Lighthill was known as Michael Lighthill.[7]
Lighthill was educated at Winchester College, and graduated with a BA from Trinity College, Cambridge in 1943.[8] He specialised in fluid dynamics, and worked at the National Physical Laboratory at Trinity. Between 1946 and 1959 he was Beyer Professor of Applied Mathematics at the University of Manchester. Lighthill then moved from Manchester to become director of the Royal Aircraft Establishment at Farnborough. There he worked on the development of television and communications satellites, and on the development of crewed spacecraft. This latter work was used in the development of the Concorde supersonic airliner.
In 1955, together with G. B. Whitham, Lighthill set out the first comprehensive theory of kinematic waves[9][10] (an application of the method of characteristics), with a multitude of applications, prime among them fluid flow and traffic flow.
Lighthill's early work included two dimensional aerofoil theory, and supersonic flow around solids of revolution. In addition to the dynamics of gas at high speeds he studied shock and blast waves and introduced the squirmer model. He is credited with founding the subject of aeroacoustics, a subject vital to the reduction of noise in jet engines. Lighthill's eighth power law states that the acoustic power radiated by a jet engine is proportional to the eighth power of the jet speed.[11] He also founded non-linear acoustics, and showed that the same non-linear differential equations could model both flood waves in rivers and traffic flow in highways.
In 1958, Lightill was elected to the American Academy of Arts and Sciences.[12]
In 1964 he became the Royal Society's resident professor at Imperial College London, before returning to Trinity College, Cambridge, five years later as Lucasian Professor of Mathematics, a chair he held until 1979, when he was succeeded by Stephen Hawking. Lighthill then became Provost of University College London (UCL) – a post he held until 1989.
Lighthill founded the Institute of Mathematics and its Applications (IMA) in 1964, alongside Professor Sir Bryan Thwaites. In 1968, he was awarded an Honorary Degree (Doctor of Science) by the University of Bath.[13] In 1972 he was invited to deliver the MacMillan Memorial Lecture to the Institution of Engineers and Shipbuilders in Scotland. He chose the subject "Aquatic Animal Locomotion".[14]
Lighthill was elected to the American Philosophical Society in 1970.[15]
In the early 1970s, partly in reaction to significant internal discord within that field, the Science Research Council (SRC), as it was then known, asked Lighthill to compile a review of academic research in Artificial Intelligence. Lighthill's report, which was published in 1973 and became known as the "Lighthill report," was highly critical of basic research in foundational areas such as robotics and language processing, and "formed the basis for the decision by the British government to end support for AI research in all but two universities",[16] starting what is sometimes referred to as the "AI winter".
In 1976, Lighthill was elected to the United States National Academy of Sciences.[17]
In 1982, Lighthill and Alan B. Tayler were jointly awarded the first ever Gold Medal of the Institute of Mathematics and its Applications in recognition of their "outstanding contributions to mathematics and its applications over a period of years".[18] In 1983 Lighthill was awarded the Ludwig Prandtl Ring from the Deutsche Gesellschaft für Luft- und Raumfahrt (German Society for Aeronautics and Astronautics) for "outstanding contribution in the field of aerospace engineering".
His hobby was open-water swimming. He died in the water in 1998 when the mitral valve in his heart ruptured while he was swimming round the island of Sark, a feat which he had accomplished many times before.[19]
Publications
• Lighthill, M. J. (1952). "On sound generated aerodynamically. I. General theory". Proceedings of the Royal Society A. 211 (1107): 564–587. Bibcode:1952RSPSA.211..564L. doi:10.1098/rspa.1952.0060. S2CID 124316233.
• Lighthill, M. J. (1954). "On sound generated aerodynamically. II. Turbulence as a source of sound". Proceedings of the Royal Society A. 222 (1148): 1–32. Bibcode:1954RSPSA.222....1L. doi:10.1098/rspa.1954.0049. S2CID 123268161.
• Lighthill, M. J. (1958). Introduction to Fourier Analysis. Cambridge Monographs on Mechanics. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-09128-2.
• Lighthill, M. J. (1958). Introduction to Fourier analysis and generalised functions. New York: Cambridge University Press. ISBN 978-0-521-05556-7.[20]
• Lighthill, M. J. (1960). Higher approximations in aerodynamics theory. Princeton University Press. ISBN 978-0-691-07976-9.
• Lighthill, M. J. (1986). An informal introduction to theoretical fluid mechanics. Oxford: Clarendon Press. ISBN 978-0-19-853630-7.
• Lighthill, M. J. (1987). Mathematical Biofluiddynamics. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial Mathematics. ISBN 978-0-89871-014-4.
• Lighthill, M. J. (2001). Waves in fluids. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-01045-0.
• Lighthill, M. J. (1997). Hussaini, M. Yousuff (ed.). Collected papers of Sir James Lighthill. Oxford: Oxford University Press. ISBN 978-0-19-509222-6.
See also
• James Lighthill House
External links
• Lighthill Papers at University College London
References
1. Pedley, Tim J. (2001). "Sir (Michael) James Lighthill. 23 January 1924 – 17 July 1998: Elected F.R.S. 1953". Biographical Memoirs of Fellows of the Royal Society. 47: 333–356. doi:10.1098/rsbm.2001.0019.
2. O'Connor, John J.; Robertson, Edmund F., "James Lighthill", MacTutor History of Mathematics Archive, University of St Andrews
3. James Lighthill at the Mathematics Genealogy Project
4. "The Oxford Dictionary of National Biography". Oxford Dictionary of National Biography (online ed.). Oxford University Press. 2004. doi:10.1093/ref:odnb/68885. (Subscription or UK public library membership required.)
5. "Engines of Ingenuity No. 2250: Sir Michael James Lighthill by John H. Lienhard". Retrieved 28 July 2011.
6. Pedley, T. J. (2001). "James Lighthill and his contributions to fluid mechanics". Annual Review of Fluid Mechanics. 33: 1–41. Bibcode:2001AnRFM..33....1P. doi:10.1146/annurev.fluid.33.1.1.
7. "Michael James Lighthill". MacTutor History of Mathematics Archive. Retrieved 3 November 2020.
8. "Michael James Lighthill". University of St Andrews. Retrieved 25 August 2015.
9. Lighthill, M. J.; Whitham, G. B. (1955). "On Kinematic Waves. I. Flood Movement in Long Rivers". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 229 (1178): 281. Bibcode:1955RSPSA.229..281L. CiteSeerX 10.1.1.205.4573. doi:10.1098/rspa.1955.0088. S2CID 18301080.
10. Lighthill, M. J.; Whitham, G. B. (1955). "On Kinematic Waves. II. A Theory of Traffic Flow on Long Crowded Roads". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 229 (1178): 317. Bibcode:1955RSPSA.229..317L. doi:10.1098/rspa.1955.0089. S2CID 15210652.
11. Crighton, David (March 1999). "Obituary: James Lighthill". Physics Today. 52 (3): 104–106. Bibcode:1999PhT....52c.104C. doi:10.1063/1.882537.
12. "Michael James Lighthill". American Academy of Arts & Sciences. Retrieved 9 September 2022.
13. "Corporate Information". Archived from the original on 25 May 2016. Retrieved 27 February 2012.
14. "Hugh Miller Macmillan". Macmillan Memorial Lectures. Institution of Engineers and Shipbuilders in Scotland. Archived from the original on 4 October 2018. Retrieved 29 January 2019.
15. "APS Member History". search.amphilsoc.org. Retrieved 9 September 2022.
16. Russell, Stuart J.; Norvig, Peter (2003), Artificial Intelligence: A Modern Approach (2nd ed.), Upper Saddle River, New Jersey: Prentice Hall, ISBN 0-13-790395-2
17. "M. James Lighthill". www.nasonline.org. Retrieved 9 September 2022.
18. "IMA Gold Medal". Retrieved 16 May 2018. Institute of Mathematics and its Applications
19. Crighton, D., 1999, J. Fluid Mech., vol. 386, pp. 1–3
20. Lees, Milton (1959). "Review: Introduction to Fourier analysis and generalised functions, by M. J. Lighthill". Bull. Amer. Math. Soc. 65 (4): 248–249. doi:10.1090/S0002-9904-1959-10325-6.
Further reading
• Smith, Peter K.; Jordan, Dominic William (2007). Nonlinear ordinary differential equations: an introduction for scientists and engineers. Oxford [Oxfordshire]: Oxford University Press. ISBN 978-0-19-920825-8.
Lucasian Professors of Mathematics
• Isaac Barrow (1664)
• Isaac Newton (1669)
• William Whiston (1702)
• Nicholas Saunderson (1711)
• John Colson (1739)
• Edward Waring (1760)
• Isaac Milner (1798)
• Robert Woodhouse (1820)
• Thomas Turton (1822)
• George Biddell Airy (1826)
• Charles Babbage (1828)
• Joshua King (1839)
• George Stokes (1849)
• Joseph Larmor (1903)
• Paul Dirac (1932)
• James Lighthill (1969)
• Stephen Hawking (1979)
• Michael Green (2009)
• Michael Cates (2015)
Copley Medallists (1951–2000)
• David Keilin (1951)
• Paul Dirac (1952)
• Albert Kluyver (1953)
• E. T. Whittaker (1954)
• Ronald Fisher (1955)
• Patrick Blackett (1956)
• Howard Florey (1957)
• John Edensor Littlewood (1958)
• Macfarlane Burnet (1959)
• Harold Jeffreys (1960)
• Hans Krebs (1961)
• Cyril Norman Hinshelwood (1962)
• Paul Fildes (1963)
• Sydney Chapman (1964)
• Alan Hodgkin (1965)
• Lawrence Bragg (1966)
• Bernard Katz (1967)
• Tadeusz Reichstein (1968)
• Peter Medawar (1969)
• Alexander R. Todd (1970)
• Norman Pirie (1971)
• Nevill Francis Mott (1972)
• Andrew Huxley (1973)
• W. V. D. Hodge (1974)
• Francis Crick (1975)
• Dorothy Hodgkin (1976)
• Frederick Sanger (1977)
• Robert Burns Woodward (1978)
• Max Perutz (1979)
• Derek Barton (1980)
• Peter D. Mitchell (1981)
• John Cornforth (1982)
• Rodney Robert Porter (1983)
• Subrahmanyan Chandrasekhar (1984)
• Aaron Klug (1985)
• Rudolf Peierls (1986)
• Robin Hill (1987)
• Michael Atiyah (1988)
• César Milstein (1989)
• Abdus Salam (1990)
• Sydney Brenner (1991)
• George Porter (1992)
• James D. Watson (1993)
• Frederick Charles Frank (1994)
• Frank Fenner (1995)
• Alan Cottrell (1996)
• Hugh Huxley (1997)
• James Lighthill (1998)
• John Maynard Smith (1999)
• Alan Battersby (2000)
John von Neumann Lecturers
• Lars Ahlfors (1960)
• Mark Kac (1961)
• Jean Leray (1962)
• Stanislaw Ulam (1963)
• Solomon Lefschetz (1964)
• Freeman Dyson (1965)
• Eugene Wigner (1966)
• Chia-Chiao Lin (1967)
• Peter Lax (1968)
• George F. Carrier (1969)
• James H. Wilkinson (1970)
• Paul Samuelson (1971)
• Jule Charney (1974)
• James Lighthill (1975)
• René Thom (1976)
• Kenneth Arrow (1977)
• Peter Henrici (1978)
• Kurt O. Friedrichs (1979)
• Keith Stewartson (1980)
• Garrett Birkhoff (1981)
• David Slepian (1982)
• Joseph B. Keller (1983)
• Jürgen Moser (1984)
• John W. Tukey (1985)
• Jacques-Louis Lions (1986)
• Richard M. Karp (1987)
• Germund Dahlquist (1988)
• Stephen Smale (1989)
• Andrew Majda (1990)
• R. Tyrrell Rockafellar (1992)
• Martin D. Kruskal (1994)
• Carl de Boor (1996)
• William Kahan (1997)
• Olga Ladyzhenskaya (1998)
• Charles S. Peskin (1999)
• Persi Diaconis (2000)
• David Donoho (2001)
• Eric Lander (2002)
• Heinz-Otto Kreiss (2003)
• Alan C. Newell (2004)
• Jerrold E. Marsden (2005)
• George C. Papanicolaou (2006)
• Nancy Kopell (2007)
• David Gottlieb (2008)
• Franco Brezzi (2009)
• Bernd Sturmfels (2010)
• Ingrid Daubechies (2011)
• John M. Ball (2012)
• Stanley Osher (2013)
• Leslie Greengard (2014)
• Jennifer Tour Chayes (2015)
• Donald Knuth (2016)
• Bernard J. Matkowsky (2017)
• Charles F. Van Loan (2018)
• Margaret H. Wright (2019)
• Nick Trefethen (2020)
• Chi-Wang Shu (2021)
• Leah Keshet (2022)
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Michael Atiyah
Sir Michael Francis Atiyah OM FRS FRSE FMedSci FAA FREng[4] (/əˈtiːə/; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry.[5] His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004.
Sir
Michael Atiyah
OM FRS FRSE FMedSci FAA FREng
Michael Atiyah in 2007
Born
Michael Francis Atiyah
(1929-04-22)22 April 1929
Hampstead, London, England
Died11 January 2019(2019-01-11) (aged 89)
Edinburgh, Scotland
Education
• Trinity College, Cambridge (BA, PhD)
Known forAtiyah algebroid
Atiyah conjecture
Atiyah conjecture on configurations
Atiyah flop
Atiyah–Bott formula
Atiyah–Bott fixed-point theorem
Atiyah–Floer conjecture
Atiyah–Hirzebruch spectral sequence
Atiyah–Jones conjecture
Atiyah–Hitchin–Singer theorem
Atiyah–Singer index theorem
Atiyah–Segal completion theorem
ADHM construction
Fredholm module
Eta invariant
K-theory
KR-theory
Pin group
Toric manifold
Awards
• Berwick Prize (1961)
• Fields Medal (1966)
• Royal Medal (1968)
• De Morgan Medal (1980)
• Copley Medal (1988)
• Abel Prize (2004)
Scientific career
FieldsMathematics
Institutions
• University of Oxford
• Institute for Advanced Study
• University of Leicester
• University of Edinburgh
• University of Cambridge
ThesisSome Applications of Topological Methods in Algebraic Geometry (1955)
Doctoral advisorW. V. D. Hodge[1][2]
Doctoral students
• Simon Donaldson
• K. David Elworthy
• Nigel Hitchin[3]
• Lisa Jeffrey
• Frances Kirwan
• Peter Kronheimer
• Ruth Lawrence
• George Lusztig
• Ian R. Porteous
• Graeme Segal
• David O. Tall[2]
Other notable studentsEdward Witten
Early life and education
Atiyah was born on 22 April 1929 in Hampstead, London, England, the son of Jean (née Levens) and Edward Atiyah.[6] His mother was Scottish and his father was a Lebanese Orthodox Christian. He had two brothers, Patrick (deceased) and Joe, and a sister, Selma (deceased).[7] Atiyah went to primary school at the Diocesan school in Khartoum, Sudan (1934–1941), and to secondary school at Victoria College in Cairo and Alexandria (1941–1945); the school was also attended by European nobility displaced by the Second World War and some future leaders of Arab nations.[8] He returned to England and Manchester Grammar School for his HSC studies (1945–1947) and did his national service with the Royal Electrical and Mechanical Engineers (1947–1949). His undergraduate and postgraduate studies took place at Trinity College, Cambridge (1949–1955).[9] He was a doctoral student of William V. D. Hodge[2] and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry.[1][2]
Atiyah was a member of the British Humanist Association.[10]
During his time at Cambridge, he was president of The Archimedeans.[11]
Career and research
Atiyah spent the academic year 1955–1956 at the Institute for Advanced Study, Princeton, then returned to Cambridge University, where he was a research fellow and assistant lecturer (1957–1958), then a university lecturer and tutorial fellow at Pembroke College, Cambridge (1958–1961). In 1961, he moved to the University of Oxford, where he was a reader and professorial fellow at St Catherine's College (1961–1963).[9] He became Savilian Professor of Geometry and a professorial fellow of New College, Oxford, from 1963 to 1969. He took up a three-year professorship at the Institute for Advanced Study in Princeton after which he returned to Oxford as a Royal Society Research Professor and professorial fellow of St Catherine's College. He was president of the London Mathematical Society from 1974 to 1976.[9]
I started out by changing local currency into foreign currency everywhere I travelled as a child and ended up making money. That's when my father realised that I would be a mathematician some day.
Michael Atiyah[12]
Atiyah was president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002.[13] He also contributed to the foundation of the InterAcademy Panel on International Issues, the Association of European Academies (ALLEA), and the European Mathematical Society (EMS).[14]
Within the United Kingdom, he was involved in the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and was its first director (1990–1996). He was President of the Royal Society (1990–1995), Master of Trinity College, Cambridge (1990–1997),[13] Chancellor of the University of Leicester (1995–2005),[13] and president of the Royal Society of Edinburgh (2005–2008).[15] From 1997 until his death in 2019 he was an honorary professor in the University of Edinburgh. He was a Trustee of the James Clerk Maxwell Foundation.[16]
Atiyah's mathematical collaborators included Raoul Bott, Friedrich Hirzebruch[17] and Isadore Singer, and his students included Graeme Segal, Nigel Hitchin, Simon Donaldson, and Edward Witten.[18] Together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is used in counting the number of independent solutions to differential equations. Some of his more recent work was inspired by theoretical physics, in particular instantons and monopoles, which are responsible for some corrections in quantum field theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004.
Collaborations
Atiyah collaborated with many mathematicians. His three main collaborations were with Raoul Bott on the Atiyah–Bott fixed-point theorem and many other topics, with Isadore M. Singer on the Atiyah–Singer index theorem, and with Friedrich Hirzebruch on topological K-theory,[19] all of whom he met at the Institute for Advanced Study in Princeton in 1955.[20] His other collaborators included; J. Frank Adams (Hopf invariant problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins problem), Howard Donnelly (L-functions), Vladimir G. Drinfeld (instantons), Johan L. Dupont (singularities of vector fields), Lars Gårding (hyperbolic differential equations), Nigel J. Hitchin (monopoles), William V. D. Hodge (Integrals of the second kind), Michael Hopkins (K-theory), Lisa Jeffrey (topological Lagrangians), John D. S. Jones (Yang–Mills theory), Juan Maldacena (M-theory), Yuri I. Manin (instantons), Nick S. Manton (Skyrmions), Vijay K. Patodi (spectral asymmetry), A. N. Pressley (convexity), Elmer Rees (vector bundles), Wilfried Schmid (discrete series representations), Graeme Segal (equivariant K-theory), Alexander Shapiro[21] (Clifford algebras), L. Smith (homotopy groups of spheres), Paul Sutcliffe (polyhedra), David O. Tall (lambda rings), John A. Todd (Stiefel manifolds), Cumrun Vafa (M-theory), Richard S. Ward (instantons) and Edward Witten (M-theory, topological quantum field theories).[22]
His later research on gauge field theories, particularly Yang–Mills theory, stimulated important interactions between geometry and physics, most notably in the work of Edward Witten.[23]
If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner.
Michael Atiyah[24]
Atiyah's students included Peter Braam 1987, Simon Donaldson 1983, K. David Elworthy 1967, Howard Fegan 1977, Eric Grunwald 1977, Nigel Hitchin 1972, Lisa Jeffrey 1991, Frances Kirwan 1984, Peter Kronheimer 1986, Ruth Lawrence 1989, George Lusztig 1971, Jack Morava 1968, Michael Murray 1983, Peter Newstead 1966, Ian R. Porteous 1961, John Roe 1985, Brian Sanderson 1963, Rolph Schwarzenberger 1960, Graeme Segal 1967, David Tall 1966, and Graham White 1982.[2]
Other contemporary mathematicians who influenced Atiyah include Roger Penrose, Lars Hörmander, Alain Connes and Jean-Michel Bismut.[25] Atiyah said that the mathematician he most admired was Hermann Weyl,[26] and that his favourite mathematicians from before the 20th century were Bernhard Riemann and William Rowan Hamilton.[27]
The seven volumes of Atiyah's collected papers include most of his work, except for his commutative algebra textbook;[28] the first five volumes are divided thematically and the sixth and seventh arranged by date.
Algebraic geometry (1952–1958)
Main article: Algebraic geometry
Atiyah's early papers on algebraic geometry (and some general papers) are reprinted in the first volume of his collected works.[29]
As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on twisted cubics.[30] He started research under W. V. D. Hodge and won the Smith's prize for 1954 for a sheaf-theoretic approach to ruled surfaces,[31] which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology.[32] His PhD thesis with Hodge was on a sheaf-theoretic approach to Solomon Lefschetz's theory of integrals of the second kind on algebraic varieties, and resulted in an invitation to visit the Institute for Advanced Study in Princeton for a year.[33] While in Princeton he classified vector bundles on an elliptic curve (extending Alexander Grothendieck's classification of vector bundles on a genus 0 curve), by showing that any vector bundle is a sum of (essentially unique) indecomposable vector bundles,[34] and then showing that the space of indecomposable vector bundles of given degree and positive dimension can be identified with the elliptic curve.[35] He also studied double points on surfaces,[36] giving the first example of a flop, a special birational transformation of 3-folds that was later heavily used in Shigefumi Mori's work on minimal models for 3-folds.[37] Atiyah's flop can also be used to show that the universal marked family of K3 surfaces is not Hausdorff.[38]
K-theory (1959–1974)
Main article: K-theory
Atiyah's works on K-theory, including his book on K-theory[39] are reprinted in volume 2 of his collected works.[40]
The simplest nontrivial example of a vector bundle is the Möbius band (pictured on the right): a strip of paper with a twist in it, which represents a rank 1 vector bundle over a circle (the circle in question being the centerline of the Möbius band). K-theory is a tool for working with higher-dimensional analogues of this example, or in other words for describing higher-dimensional twistings: elements of the K-group of a space are represented by vector bundles over it, so the Möbius band represents an element of the K-group of a circle.[41]
Topological K-theory was discovered by Atiyah and Friedrich Hirzebruch[42] who were inspired by Grothendieck's proof of the Grothendieck–Riemann–Roch theorem and Bott's work on the periodicity theorem. This paper only discussed the zeroth K-group; they shortly after extended it to K-groups of all degrees,[43] giving the first (nontrivial) example of a generalized cohomology theory.
Several results showed that the newly introduced K-theory was in some ways more powerful than ordinary cohomology theory. Atiyah and Todd[44] used K-theory to improve the lower bounds found using ordinary cohomology by Borel and Serre for the James number, describing when a map from a complex Stiefel manifold to a sphere has a cross section. (Adams and Grant-Walker later showed that the bound found by Atiyah and Todd was best possible.) Atiyah and Hirzebruch[45] used K-theory to explain some relations between Steenrod operations and Todd classes that Hirzebruch had noticed a few years before. The original solution of the Hopf invariant one problem operations by J. F. Adams was very long and complicated, using secondary cohomology operations. Atiyah showed how primary operations in K-theory could be used to give a short solution taking only a few lines, and in joint work with Adams[46] also proved analogues of the result at odd primes.
The Atiyah–Hirzebruch spectral sequence relates the ordinary cohomology of a space to its generalized cohomology theory.[43] (Atiyah and Hirzebruch used the case of K-theory, but their method works for all cohomology theories).
Atiyah showed[47] that for a finite group G, the K theory of its classifying space, BG, is isomorphic to the completion of its character ring:
$K(BG)\cong R(G)^{\wedge }.$
The same year[48] they proved the result for G any compact connected Lie group. Although soon the result could be extended to all compact Lie groups by incorporating results from Graeme Segal's thesis,[49] that extension was complicated. However a simpler and more general proof was produced by introducing equivariant K-theory, i.e. equivalence classes of G-vector bundles over a compact G-space X.[50] It was shown that under suitable conditions the completion of the equivariant K theory of X is isomorphic to the ordinary K-theory of a space, $X_{G}$, which fibred over BG with fibre X:
$K_{G}(X)^{\wedge }\cong K(X_{G}).$
The original result then followed as a corollary by taking X to be a point: the left hand side reduced to the completion of R(G) and the right to K(BG). See Atiyah–Segal completion theorem for more details.
He defined new generalized homology and cohomology theories called bordism and cobordism, and pointed out that many of the deep results on cobordism of manifolds found by René Thom, C. T. C. Wall, and others could be naturally reinterpreted as statements about these cohomology theories.[51] Some of these cohomology theories, in particular complex cobordism, turned out to be some of the most powerful cohomology theories known.
"Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine."
Michael Atiyah[52]
He introduced[53] the J-group J(X) of a finite complex X, defined as the group of stable fiber homotopy equivalence classes of sphere bundles; this was later studied in detail by J. F. Adams in a series of papers, leading to the Adams conjecture.
With Hirzebruch he extended the Grothendieck–Riemann–Roch theorem to complex analytic embeddings,[53] and in a related paper[54] they showed that the Hodge conjecture for integral cohomology is false. The Hodge conjecture for rational cohomology is, as of 2008, a major unsolved problem.[55]
The Bott periodicity theorem was a central theme in Atiyah's work on K-theory, and he repeatedly returned to it, reworking the proof several times to understand it better. With Bott he worked out an elementary proof,[56] and gave another version of it in his book.[57] With Bott and Shapiro he analysed the relation of Bott periodicity to the periodicity of Clifford algebras;[58] although this paper did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood. He found a proof of several generalizations using elliptic operators;[59] this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem.[60]
Index theory (1963–1984)
Main article: Atiyah–Singer index theorem
Atiyah's work on index theory is reprinted in volumes 3 and 4 of his collected works.[61][62]
The index of a differential operator is closely related to the number of independent solutions (more precisely, it is the differences of the numbers of independent solutions of the differential operator and its adjoint). There are many hard and fundamental problems in mathematics that can easily be reduced to the problem of finding the number of independent solutions of some differential operator, so if one has some means of finding the index of a differential operator these problems can often be solved. This is what the Atiyah–Singer index theorem does: it gives a formula for the index of certain differential operators, in terms of topological invariants that look quite complicated but are in practice usually straightforward to calculate.
Several deep theorems, such as the Hirzebruch–Riemann–Roch theorem, are special cases of the Atiyah–Singer index theorem. In fact the index theorem gave a more powerful result, because its proof applied to all compact complex manifolds, while Hirzebruch's proof only worked for projective manifolds. There were also many new applications: a typical one is calculating the dimensions of the moduli spaces of instantons. The index theorem can also be run "in reverse": the index is obviously an integer, so the formula for it must also give an integer, which sometimes gives subtle integrality conditions on invariants of manifolds. A typical example of this is Rochlin's theorem, which follows from the index theorem.
The most useful piece of advice I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is both simple and non-trivial.
Michael Atiyah[63]
The index problem for elliptic differential operators was posed in 1959 by Gel'fand.[64] He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Hirzebruch and Borel had proved the integrality of the  genus of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Singer in 1961).
The first announcement of the Atiyah–Singer theorem was their 1963 paper.[65] The proof sketched in this announcement was inspired by Hirzebruch's proof of the Hirzebruch–Riemann–Roch theorem and was never published by them, though it is described in the book by Palais.[66] Their first published proof[67] was more similar to Grothendieck's proof of the Grothendieck–Riemann–Roch theorem, replacing the cobordism theory of the first proof with K-theory, and they used this approach to give proofs of various generalizations in a sequence of papers from 1968 to 1971.
Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space Y. In this case the index is an element of the K theory of Y, rather than an integer.[68] If the operators in the family are real, then the index lies in the real K theory of Y. This gives a little extra information, as the map from the real K theory of Y to the complex K-theory is not always injective.[69]
With Bott, Atiyah found an analogue of the Lefschetz fixed-point formula for elliptic operators, giving the Lefschetz number of an endomorphism of an elliptic complex in terms of a sum over the fixed points of the endomorphism.[70] As special cases their formula included the Weyl character formula, and several new results about elliptic curves with complex multiplication, some of which were initially disbelieved by experts.[71] Atiyah and Segal combined this fixed point theorem with the index theorem as follows. If there is a compact group action of a group G on the compact manifold X, commuting with the elliptic operator, then one can replace ordinary K-theory in the index theorem with equivariant K-theory. For trivial groups G this gives the index theorem, and for a finite group G acting with isolated fixed points it gives the Atiyah–Bott fixed point theorem. In general it gives the index as a sum over fixed point submanifolds of the group G.[72]
Atiyah[73] solved a problem asked independently by Hörmander and Gel'fand, about whether complex powers of analytic functions define distributions. Atiyah used Hironaka's resolution of singularities to answer this affirmatively. An ingenious and elementary solution was found at about the same time by J. Bernstein, and discussed by Atiyah.[74]
As an application of the equivariant index theorem, Atiyah and Hirzebruch showed that manifolds with effective circle actions have vanishing Â-genus.[75] (Lichnerowicz showed that if a manifold has a metric of positive scalar curvature then the Â-genus vanishes.)
With Elmer Rees, Atiyah studied the problem of the relation between topological and holomorphic vector bundles on projective space. They solved the simplest unknown case, by showing that all rank 2 vector bundles over projective 3-space have a holomorphic structure.[76] Horrocks had previously found some non-trivial examples of such vector bundles, which were later used by Atiyah in his study of instantons on the 4-sphere.
Atiyah, Bott and Vijay K. Patodi[77] gave a new proof of the index theorem using the heat equation.
If the manifold is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index. These conditions can be local (like demanding that the sections in the domain vanish at the boundary) or more complicated global conditions (like requiring that the sections in the domain solve some differential equation). The local case was worked out by Atiyah and Bott, but they showed that many interesting operators (e.g., the signature operator) do not admit local boundary conditions. To handle these operators, Atiyah, Patodi and Singer introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder, and also introduced the Atiyah–Patodi–Singer eta invariant. This resulted in a series of papers on spectral asymmetry,[78] which were later unexpectedly used in theoretical physics, in particular in Witten's work on anomalies.
The fundamental solutions of linear hyperbolic partial differential equations often have Petrovsky lacunas: regions where they vanish identically. These were studied in 1945 by I. G. Petrovsky, who found topological conditions describing which regions were lacunas. In collaboration with Bott and Lars Gårding, Atiyah wrote three papers updating and generalizing Petrovsky's work.[79]
Atiyah[80] showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient. The kernel of the elliptic operator is in general infinite-dimensional in this case, but it is possible to get a finite index using the dimension of a module over a von Neumann algebra; this index is in general real rather than integer valued. This version is called the L2 index theorem, and was used by Atiyah and Schmid[81] to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's discrete series representations of semisimple Lie groups. In the course of this work they found a more elementary proof of Harish-Chandra's fundamental theorem on the local integrability of characters of Lie groups.[82]
With H. Donnelly and I. Singer, he extended Hirzebruch's formula (relating the signature defect at cusps of Hilbert modular surfaces to values of L-functions) from real quadratic fields to all totally real fields.[83]
Gauge theory (1977–1985)
Main article: Gauge theory (mathematics)
Many of his papers on gauge theory and related topics are reprinted in volume 5 of his collected works.[84] A common theme of these papers is the study of moduli spaces of solutions to certain non-linear partial differential equations, in particular the equations for instantons and monopoles. This often involves finding a subtle correspondence between solutions of two seemingly quite different equations. An early example of this which Atiyah used repeatedly is the Penrose transform, which can sometimes convert solutions of a non-linear equation over some real manifold into solutions of some linear holomorphic equations over a different complex manifold.
In a series of papers with several authors, Atiyah classified all instantons on 4-dimensional Euclidean space. It is more convenient to classify instantons on a sphere as this is compact, and this is essentially equivalent to classifying instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant. With Hitchin and Singer[85] he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principal bundle over a compact 4-dimensional Riemannian manifold (the Atiyah–Hitchin–Singer theorem). For example, the dimension of the space of SU2 instantons of rank k>0 is 8k−3. To do this they used the Atiyah–Singer index theorem to calculate the dimension of the tangent space of the moduli space at a point; the tangent space is essentially the space of solutions of an elliptic differential operator, given by the linearization of the non-linear Yang–Mills equations. These moduli spaces were later used by Donaldson to construct his invariants of 4-manifolds. Atiyah and Ward used the Penrose correspondence to reduce the classification of all instantons on the 4-sphere to a problem in algebraic geometry.[86] With Hitchin he used ideas of Horrocks to solve this problem, giving the ADHM construction of all instantons on a sphere; Manin and Drinfeld found the same construction at the same time, leading to a joint paper by all four authors.[87] Atiyah reformulated this construction using quaternions and wrote up a leisurely account of this classification of instantons on Euclidean space as a book.[88]
The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics.
Michael Atiyah[89]
Atiyah's work on instanton moduli spaces was used in Donaldson's work on Donaldson theory. Donaldson showed that the moduli space of (degree 1) instantons over a compact simply connected 4-manifold with positive definite intersection form can be compactified to give a cobordism between the manifold and a sum of copies of complex projective space. He deduced from this that the intersection form must be a sum of one-dimensional ones, which led to several spectacular applications to smooth 4-manifolds, such as the existence of non-equivalent smooth structures on 4-dimensional Euclidean space. Donaldson went on to use the other moduli spaces studied by Atiyah to define Donaldson invariants, which revolutionized the study of smooth 4-manifolds, and showed that they were more subtle than smooth manifolds in any other dimension, and also quite different from topological 4-manifolds. Atiyah described some of these results in a survey talk.[90]
Green's functions for linear partial differential equations can often be found by using the Fourier transform to convert this into an algebraic problem. Atiyah used a non-linear version of this idea.[91] He used the Penrose transform to convert the Green's function for the conformally invariant Laplacian into a complex analytic object, which turned out to be essentially the diagonal embedding of the Penrose twistor space into its square. This allowed him to find an explicit formula for the conformally invariant Green's function on a 4-manifold.
In his paper with Jones,[92] he studied the topology of the moduli space of SU(2) instantons over a 4-sphere. They showed that the natural map from this moduli space to the space of all connections induces epimorphisms of homology groups in a certain range of dimensions, and suggested that it might induce isomorphisms of homology groups in the same range of dimensions. This became known as the Atiyah–Jones conjecture, and was later proved by several mathematicians.[93]
Harder and M. S. Narasimhan described the cohomology of the moduli spaces of stable vector bundles over Riemann surfaces by counting the number of points of the moduli spaces over finite fields, and then using the Weil conjectures to recover the cohomology over the complex numbers.[94] Atiyah and R. Bott used Morse theory and the Yang–Mills equations over a Riemann surface to reproduce and extending the results of Harder and Narasimhan.[95]
An old result due to Schur and Horn states that the set of possible diagonal vectors of an Hermitian matrix with given eigenvalues is the convex hull of all the permutations of the eigenvalues. Atiyah proved a generalization of this that applies to all compact symplectic manifolds acted on by a torus, showing that the image of the manifold under the moment map is a convex polyhedron,[96] and with Pressley gave a related generalization to infinite-dimensional loop groups.[97]
Duistermaat and Heckman found a striking formula, saying that the push-forward of the Liouville measure of a moment map for a torus action is given exactly by the stationary phase approximation (which is in general just an asymptotic expansion rather than exact). Atiyah and Bott[98] showed that this could be deduced from a more general formula in equivariant cohomology, which was a consequence of well-known localization theorems. Atiyah showed[99] that the moment map was closely related to geometric invariant theory, and this idea was later developed much further by his student F. Kirwan. Witten shortly after applied the Duistermaat–Heckman formula to loop spaces and showed that this formally gave the Atiyah–Singer index theorem for the Dirac operator; this idea was lectured on by Atiyah.[100]
With Hitchin he worked on magnetic monopoles, and studied their scattering using an idea of Nick Manton.[101] His book[102] with Hitchin gives a detailed description of their work on magnetic monopoles. The main theme of the book is a study of a moduli space of magnetic monopoles; this has a natural Riemannian metric, and a key point is that this metric is complete and hyperkähler. The metric is then used to study the scattering of two monopoles, using a suggestion of N. Manton that the geodesic flow on the moduli space is the low energy approximation to the scattering. For example, they show that a head-on collision between two monopoles results in 90-degree scattering, with the direction of scattering depending on the relative phases of the two monopoles. He also studied monopoles on hyperbolic space.[103]
Atiyah showed[104] that instantons in 4 dimensions can be identified with instantons in 2 dimensions, which are much easier to handle. There is of course a catch: in going from 4 to 2 dimensions the structure group of the gauge theory changes from a finite-dimensional group to an infinite-dimensional loop group. This gives another example where the moduli spaces of solutions of two apparently unrelated nonlinear partial differential equations turn out to be essentially the same.
Atiyah and Singer found that anomalies in quantum field theory could be interpreted in terms of index theory of the Dirac operator;[105] this idea later became widely used by physicists.
Later work (1986–2019)
Many of the papers in the 6th volume[106] of his collected works are surveys, obituaries, and general talks. Atiyah continued to publish subsequently, including several surveys, a popular book,[107] and another paper with Segal on twisted K-theory.
One paper[108] is a detailed study of the Dedekind eta function from the point of view of topology and the index theorem.
Several of his papers from around this time study the connections between quantum field theory, knots, and Donaldson theory. He introduced the concept of a topological quantum field theory, inspired by Witten's work and Segal's definition of a conformal field theory.[109] His book “The Geometry and Physics of Knots” [110] describes the new knot invariants found by Vaughan Jones and Edward Witten in terms of topological quantum field theories, and his paper with L. Jeffrey[111] explains Witten's Lagrangian giving the Donaldson invariants.
He studied skyrmions with Nick Manton,[112] finding a relation with magnetic monopoles and instantons, and giving a conjecture for the structure of the moduli space of two skyrmions as a certain subquotient of complex projective 3-space.
Several papers[113] were inspired by a question of Jonathan Robbins (called the Berry–Robbins problem), who asked if there is a map from the configuration space of n points in 3-space to the flag manifold of the unitary group. Atiyah gave an affirmative answer to this question, but felt his solution was too computational and studied a conjecture that would give a more natural solution. He also related the question to Nahm's equation, and introduced the Atiyah conjecture on configurations.
But for most practical purposes, you just use the classical groups. The exceptional Lie groups are just there to show you that the theory is a bit bigger; it is pretty rare that they ever turn up.
Michael Atiyah[114]
With Juan Maldacena and Cumrun Vafa,[115] and E. Witten[116] he described the dynamics of M-theory on manifolds with G2 holonomy. These papers seem to be the first time that Atiyah worked on exceptional Lie groups.
In his papers with M. Hopkins[117] and G. Segal[118] he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in theoretical physics.
In October 2016, he claimed[119] a short proof of the non-existence of complex structures on the 6-sphere. His proof, like many predecessors, is considered flawed by the mathematical community, even after the proof was rewritten in a revised form.[120][121]
At the 2018 Heidelberg Laureate Forum, he claimed to have solved the Riemann hypothesis, Hilbert's eighth problem, by contradiction using the fine-structure constant. Again, the proof did not hold up and the hypothesis remains one of the six unsolved Millennium Prize Problems in mathematics, as of 2023.[122][123]
Bibliography
Books
This subsection lists all books written by Atiyah; it omits a few books that he edited.
• Atiyah, Michael F.; Macdonald, Ian G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802. A classic textbook covering standard commutative algebra.
• Atiyah, Michael F. (1970), Vector fields on manifolds, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200, Cologne: Westdeutscher Verlag, MR 0263102. Reprinted as (Atiyah 1988b, item 50).
• Atiyah, Michael F. (1974), Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Berlin, New York: Springer-Verlag, MR 0482866. Reprinted as (Atiyah 1988c, item 78).
• Atiyah, Michael F. (1979), Geometry of Yang–Mills fields, Scuola Normale Superiore Pisa, Pisa, MR 0554924. Reprinted as (Atiyah 1988e, item 99).
• Atiyah, Michael F.; Hitchin, Nigel (1988), The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, doi:10.1515/9781400859306, ISBN 978-0-691-08480-0, MR 0934202. Reprinted as (Atiyah 2004, item 126).
• Atiyah, Michael F. (1988a), Collected works. Vol. 1 Early papers: general papers, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853275-0, MR 0951892.
• Atiyah, Michael F. (1988b), Collected works. Vol. 2 K-theory, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853276-7, MR 0951892.
• Atiyah, Michael F. (1988c), Collected works. Vol. 3 Index theory: 1, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853277-4, MR 0951892.
• Atiyah, Michael F. (1988d), Collected works. Vol. 4 Index theory: 2, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853278-1, MR 0951892.
• Atiyah, Michael F. (1988e), Collected works. Vol. 5 Gauge theories, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853279-8, MR 0951892.
• Atiyah, Michael F. (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, ISBN 978-0-201-09394-0, MR 1043170. First edition (1967) reprinted as (Atiyah 1988b, item 45).
• Atiyah, Michael F. (1990), The geometry and physics of knots, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, doi:10.1017/CBO9780511623868, ISBN 978-0-521-39521-2, MR 1078014. Reprinted as (Atiyah 2004, item 136).
• Atiyah, Michael F. (2004), Collected works. Vol. 6, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853099-2, MR 2160826.
• Atiyah, Michael F. (2007), Siamo tutti matematici (Italian: We are all mathematicians), Roma: Di Renzo Editore, p. 96, ISBN 978-88-8323-157-5
• Atiyah, Michael (2014), Collected works. Vol. 7. 2002-2013, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-968926-2, MR 3223085.
• Atiyah, Michael F.; Iagolnitzer, Daniel; Chong, Chitat (2015), Fields Medallists' Lectures (3rd Edition), World Scientific, doi:10.1142/9652, ISBN 978-981-4696-18-0.
Selected papers
• Atiyah, Michael F. (1961), "Characters and cohomology of finite groups", Inst. Hautes Études Sci. Publ. Math., 9: 23–64, doi:10.1007/BF02698718, S2CID 54764252. Reprinted in (Atiyah 1988b, paper 29).
• Atiyah, Michael F.; Hirzebruch, Friedrich (1961), Vector bundles and homogeneous spaces, Proceedings of Symposia in Pure Mathematics, vol. 3, pp. 7–38, doi:10.1090/pspum/003/0139181, ISBN 9780821814031. Reprinted in (Atiyah 1988b, paper 28).
• Atiyah, Michael F.; Segal, Graeme B. (1969), "Equivariant K-Theory and Completion", Journal of Differential Geometry, 3 (1–2): 1–18, doi:10.4310/jdg/1214428815. Reprinted in (Atiyah 1988b, paper 49).
• Atiyah, Michael F. (1976), "Elliptic operators, discrete groups and von Neumann algebras", Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974), Asterisque, vol. 32–33, Soc. Math. France, Paris, pp. 43–72, MR 0420729. Reprinted in (Atiyah 1988d, paper 89). Formulation of the Atiyah "Conjecture" on the rationality of the L2-Betti numbers.
• Atiyah, Michael F.; Singer, Isadore M. (1963), "The Index of Elliptic Operators on Compact Manifolds", Bull. Amer. Math. Soc., 69 (3): 322–433, doi:10.1090/S0002-9904-1963-10957-X. An announcement of the index theorem. Reprinted in (Atiyah 1988c, paper 56).
• Atiyah, Michael F.; Singer, Isadore M. (1968a), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715. This gives a proof using K-theory instead of cohomology. Reprinted in (Atiyah 1988c, paper 64).
• Atiyah, Michael F.; Segal, Graeme B. (1968), "The Index of Elliptic Operators: II", Annals of Mathematics, Second Series, 87 (3): 531–545, doi:10.2307/1970716, JSTOR 1970716. This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K-theory. Reprinted in (Atiyah 1988c, paper 65).
• Atiyah, Michael F.; Singer, Isadore M. (1968b), "The Index of Elliptic Operators III", Annals of Mathematics, Second Series, 87 (3): 546–604, doi:10.2307/1970717, JSTOR 1970717. This paper shows how to convert from the K-theory version to a version using cohomology. Reprinted in (Atiyah 1988c, paper 66).
• Atiyah, Michael F.; Singer, Isadore M. (1971), "The Index of Elliptic Operators IV", Annals of Mathematics, Second Series, 93 (1): 119–138, doi:10.2307/1970756, JSTOR 1970756 This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family. Reprinted in (Atiyah 1988c, paper 67).
• Atiyah, Michael F.; Singer, Isadore M. (1971), "The Index of Elliptic Operators V", Annals of Mathematics, Second Series, 93 (1): 139–149, doi:10.2307/1970757, JSTOR 1970757. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information. Reprinted in (Atiyah 1988c, paper 68).
• Atiyah, Michael F.; Bott, Raoul (1966), "A Lefschetz Fixed Point Formula for Elliptic Differential Operators", Bull. Am. Math. Soc., 72 (2): 245–50, doi:10.1090/S0002-9904-1966-11483-0. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex. Reprinted in (Atiyah 1988c, paper 61).
• Atiyah, Michael F.; Bott, Raoul (1967), "A Lefschetz Fixed Point Formula for Elliptic Complexes: I", Annals of Mathematics, Second Series, 86 (2): 374–407, doi:10.2307/1970694, JSTOR 1970694 (reprinted in (Atiyah 1988c, paper 61))and Atiyah, Michael F.; Bott, Raoul (1968), "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications", Annals of Mathematics, Second Series, 88 (3): 451–491, doi:10.2307/1970721, JSTOR 1970721. Reprinted in (Atiyah 1988c, paper 62). These give the proofs and some applications of the results announced in the previous paper.
• Atiyah, Michael F.; Bott, Raoul; Patodi, Vijay K. (1973), "On the heat equation and the index theorem" (PDF), Invent. Math., 19 (4): 279–330, Bibcode:1973InMat..19..279A, doi:10.1007/BF01425417, MR 0650828, S2CID 115700319; Atiyah, Michael F.; Bott, R.; Patodi, V. K. (1975), "Errata", Invent. Math., 28 (3): 277–280, Bibcode:1975InMat..28..277A, doi:10.1007/BF01425562, MR 0650829 Reprinted in (Atiyah 1988d, paper 79, 79a).
• Atiyah, Michael F.; Schmid, Wilfried (1977), "A geometric construction of the discrete series for semisimple Lie groups", Invent. Math., 42: 1–62, Bibcode:1977InMat..42....1A, doi:10.1007/BF01389783, MR 0463358, S2CID 189831012; Atiyah, Michael F.; Schmid, Wilfried (1979), "Erratum", Invent. Math., 54 (2): 189–192, Bibcode:1979InMat..54..189A, doi:10.1007/BF01408936, MR 0550183. Reprinted in (Atiyah 1988d, paper 90).
• Atiyah, Michael (2010), Edinburgh Lectures on Geometry, Analysis and Physics, arXiv:1009.4827v1, Bibcode:2010arXiv1009.4827A
Awards and honours
In 1966, when he was thirty-seven years old, he was awarded the Fields Medal,[124] for his work in developing K-theory, a generalized Lefschetz fixed-point theorem and the Atiyah–Singer theorem, for which he also won the Abel Prize jointly with Isadore Singer in 2004.[125] Among other prizes he has received are the Royal Medal of the Royal Society in 1968,[126] the De Morgan Medal of the London Mathematical Society in 1980, the Antonio Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987,[127] the Copley Medal of the Royal Society in 1988,[128] the Benjamin Franklin Medal for Distinguished Achievement in the Sciences of the American Philosophical Society in 1993,[129] the Jawaharlal Nehru Birth Centenary Medal of the Indian National Science Academy in 1993,[130] the President's Medal from the Institute of Physics in 2008,[131] the Grande Médaille of the French Academy of Sciences in 2010[132] and the Grand Officier of the French Légion d'honneur in 2011.[133]
He was elected a foreign member of the National Academy of Sciences, the American Academy of Arts and Sciences (1969),[134] the Académie des Sciences, the Akademie Leopoldina, the Royal Swedish Academy, the Royal Irish Academy, the Royal Society of Edinburgh, the American Philosophical Society, the Indian National Science Academy, the Chinese Academy of Science, the Australian Academy of Science, the Russian Academy of Science, the Ukrainian Academy of Science, the Georgian Academy of Science, the Venezuela Academy of Science, the Norwegian Academy of Science and Letters, the Royal Spanish Academy of Science, the Accademia dei Lincei and the Moscow Mathematical Society.[9][13] In 2012, he became a fellow of the American Mathematical Society.[135] He was also appointed as a Honorary Fellow[4] of the Royal Academy of Engineering[4] in 1993.
Atiyah was awarded honorary degrees by the universities of Birmingham, Bonn, Chicago, Cambridge, Dublin, Durham, Edinburgh, Essex, Ghent, Helsinki, Lebanon, Leicester, London, Mexico, Montreal, Oxford, Reading, Salamanca, St. Andrews, Sussex, Wales, Warwick, the American University of Beirut, Brown University, Charles University in Prague, Harvard University, Heriot–Watt University, Hong Kong (Chinese University), Keele University, Queen's University (Canada), The Open University, University of Waterloo, Wilfrid Laurier University, Technical University of Catalonia, and UMIST.[9][13][136][137]
Atiyah was made a Knight Bachelor in 1983[9] and made a member of the Order of Merit in 1992.[13]
The Michael Atiyah building[138] at the University of Leicester and the Michael Atiyah Chair in Mathematical Sciences[139] at the American University of Beirut were named after him.
Personal life
Atiyah married Lily Brown on 30 July 1955, with whom he had three sons, John, David and Robin. Atiyah's eldest son John died on 24 June 2002 while on a walking holiday in the Pyrenees with his wife Maj-Lis.
Lily Atiyah died on 13 March 2018 at the age of 90[5][7][9] while Sir Michael Atiyah died less than a year later on 11 January 2019, aged 89.[140][141]
See also
• List of presidents of the Royal Society
References
1. Atiyah, Michael Francis (1955). Some applications of topological methods in algebraic geometry. repository.cam.ac.uk (PhD thesis). University of Cambridge. Archived from the original on 18 November 2017. Retrieved 17 November 2017.
2. Michael Atiyah at the Mathematics Genealogy Project
3. Hitchin, Nigel J. (1972). Differentiable manifolds : the space of harmonic spinors. bodleian.ox.ac.uk (DPhil thesis). University of Oxford. OCLC 500473357. EThOS uk.bl.ethos.459281.
4. "List of Fellows". Archived from the original on 8 June 2016. Retrieved 28 October 2014.
5. O'Connor, John J.; Robertson, Edmund F., "Michael Atiyah", MacTutor History of Mathematics Archive, University of St Andrews
6. "ATIYAH, Sir Michael (Francis)". Who's Who. Vol. 2014 (online edition via Oxford University Press ed.). A & C Black. (Subscription or UK public library membership required.)
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9. Atiyah 1988a, p. xi
10. "Distinguished mathematician and supporter of Humanism."
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13. Atiyah 2004, p. ix
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15. Royal Society of Edinburgh announcement, archived from the original on 20 November 2008, retrieved 14 August 2008
16. "James Clerk Maxwell Foundation Annual Report and Summary Accounts" (PDF). 2019.
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18. "Edward Witten – Adventures in physics and math (Kyoto Prize lecture 2014)" (PDF).
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21. Alexander Shapiro at the Mathematics Genealogy Project
22. Atiyah 2004, pp. xi–xxv
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24. Atiyah 1988a, paper 12, p. 233
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32. Atiyah 1988a, p. 1
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34. Atiyah 1988a, paper 5
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36. Atiyah 1988a, paper 8
37. Matsuki 2002.
38. Barth et al. 2004
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42. Atiyah 1988b, paper 24
43. Atiyah 1988b, paper 28
44. Atiyah 1988b, paper 26
45. Atiyah 1988a, papers 30,31
46. Atiyah 1988b, paper 42
47. Atiyah 1961
48. Atiyah & Hirzebruch 1961
49. Segal 1968
50. Atiyah & Segal 1969
51. Atiyah 1988b, paper 34
52. Atiyah 2004, paper 160, p. 7
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61. Atiyah 1988c
62. Atiyah 1988d
63. Atiyah 1988a, paper 17, p. 76
64. Gel'fand 1960
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66. Palais 1965
67. Atiyah & Singer 1968a
68. Atiyah 1988c, paper 67
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71. Atiyah 1988c, p. 3
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73. Atiyah 1988c, paper 73
74. Atiyah 1988a, paper 15
75. Atiyah 1988c, paper 74
76. Atiyah 1988c, paper 76
77. Atiyah, Bott & Patodi 1973
78. Atiyah 1988d, papers 80–83
79. Atiyah 1988d, papers 84, 85, 86
80. Atiyah 1976
81. Atiyah & Schmid 1977
82. Atiyah 1988d, paper 91
83. Atiyah 1988d, papers 92, 93
84. Atiyah 1988e.
85. Atiyah 1988e, papers 94, 97
86. Atiyah 1988e, paper 95
87. Atiyah 1988e, paper 96
88. Atiyah 1988e, paper 99
89. Atiyah 1988a, paper 19, p. 13
90. Atiyah 1988e, paper 112
91. Atiyah 1988e, paper 101
92. Atiyah 1988e, paper 102
93. Boyer et al. 1993
94. Harder & Narasimhan 1975
95. Atiyah 1988e, papers 104–105
96. Atiyah 1988e, paper 106
97. Atiyah 1988e, paper 108
98. Atiyah 1988e, paper 109
99. Atiyah 1988e, paper 110
100. Atiyah 1988e, paper 124
101. Atiyah 1988e, papers 115, 116
102. Atiyah & Hitchin 1988
103. Atiyah 1988e, paper 118
104. Atiyah 1988e, paper 117
105. Atiyah 1988e, papers 119, 120, 121
106. Michael Atiyah 2004
107. Atiyah 2007
108. Atiyah 2004, paper 127
109. Atiyah 2004, paper 132
110. Atiyah 1990
111. Atiyah 2004, paper 139
112. Atiyah 2004, papers 141, 142
113. Atiyah 2004, papers 163, 164, 165, 166, 167, 168
114. Atiyah 1988a, paper 19, p. 19
115. Atiyah 2004, paper 169
116. Atiyah 2004, paper 170
117. Atiyah 2004, paper 172
118. Atiyah 2004, paper 173
119. Atiyah, Michael (2016). "The Non-Existent Complex 6-Sphere". arXiv:1610.09366 [math.DG].
120. What is the current understanding regarding complex structures on the 6-sphere? (MathOverflow), retrieved 24 September 2018
121. Atiyah's May 2018 paper on the 6-sphere (MathOverflow), retrieved 24 September 2018
122. "Skepticism surrounds renowned mathematician's attempted proof of 160-year-old hypothesis". Science | AAAS. 24 September 2018. Archived from the original on 26 September 2018. Retrieved 26 September 2018.
123. "Riemann hypothesis likely remains unsolved despite claimed proof". Archived from the original on 24 September 2018. Retrieved 24 September 2018.
124. Fields medal citation: Cartan, Henri (1968), "L'oeuvre de Michael F. Atiyah", Proceedings of International Conference of Mathematicians (Moscow, 1966), Izdatyel'stvo Mir, Moscow, pp. 9–14
125. "2004: Sir Michael Francis Atiyah and Isadore M. Singer". www.abelprize.no. Retrieved 22 August 2022.{{cite web}}: CS1 maint: url-status (link)
126. Royal archive winners 1989–1950, archived from the original on 9 June 2008, retrieved 14 August 2008
127. Sir Michael Atiyah FRS, Newton institute, archived from the original on 31 May 2008, retrieved 14 August 2008
128. Copley archive winners 1989–1900, archived from the original on 9 June 2008, retrieved 14 August 2008
129. "Benjamin Franklin Medal for Distinguished Achievement in the Sciences Recipients". American Philosophical Society. Archived from the original on 24 September 2012. Retrieved 27 November 2011.
130. Jawaharlal Nehru Birth Centenary Medal, archived from the original on 10 July 2012, retrieved 14 August 2008
131. 2008 President's medal, retrieved 14 August 2008
132. La Grande Medaille, archived from the original on 1 August 2010, retrieved 25 January 2011
133. Legion d'honneur, archived from the original on 24 September 2011, retrieved 11 September 2011
134. "Book of Members, 1780-2010: Chapter A" (PDF). American Academy of Arts and Sciences. Archived (PDF) from the original on 10 May 2011. Retrieved 27 April 2011.
135. List of Fellows of the American Mathematical Society Archived 5 August 2013 at the Wayback Machine, retrieved 3 November 2012.
136. "Heriot-Watt University Edinburgh: Honorary Graduates". www1.hw.ac.uk. Archived from the original on 18 April 2016. Retrieved 4 April 2016.
137. Honorary Doctorates, Charles University in Prague, retrieved 4 May 2018
138. The Michael Atiyah building, archived from the original on 9 February 2009, retrieved 14 August 2008
139. American University of Beirut establishes the Michael Atiyah Chair in Mathematical Sciences, archived from the original on 3 April 2008, retrieved 14 August 2008
140. "Michael Atiyah 1929-2019". University of Oxford Mathematical Institute. 11 January 2019. Archived from the original on 11 January 2019. Retrieved 11 January 2019.
141. "A tribute to former President of the Royal Society Sir Michael Atiyah OM FRS (1929 - 2019)". The Royal Society. 11 January 2019. Archived from the original on 11 January 2019. Retrieved 11 January 2019.
Sources
• Boyer, Charles P.; Hurtubise, J. C.; Mann, B. M.; Milgram, R. J. (1993), "The topology of instanton moduli spaces. I. The Atiyah–Jones conjecture", Annals of Mathematics, Second Series, 137 (3): 561–609, doi:10.2307/2946532, ISSN 0003-486X, JSTOR 2946532, MR 1217348
• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Berlin: Springer, p. 334, ISBN 978-3-540-00832-3
• Gel'fand, Israel M. (1960), "On elliptic equations", Russ. Math. Surv., 15 (3): 113–123, Bibcode:1960RuMaS..15..113G, doi:10.1070/rm1960v015n03ABEH004094. Reprinted in volume 1 of his collected works, p. 65–75, ISBN 0-387-13619-3. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data.
• Harder, G.; Narasimhan, M. S. (1975), "On the cohomology groups of moduli spaces of vector bundles on curves", Mathematische Annalen, 212 (3): 215–248, doi:10.1007/BF01357141, ISSN 0025-5831, MR 0364254, S2CID 117851906, archived from the original on 5 March 2016, retrieved 30 September 2013
• Matsuki, Kenji (2002), Introduction to the Mori program, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-5602-9, ISBN 978-0-387-98465-0, MR 1875410
• Palais, Richard S. (1965), Seminar on the Atiyah–Singer Index Theorem, Annals of Mathematics Studies, vol. 57, S.l.: Princeton Univ Press, ISBN 978-0-691-08031-4. This describes the original proof of the index theorem. (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)
• Segal, Graeme B. (1968), "The representation ring of a compact Lie group", Inst. Hautes Études Sci. Publ. Math., 34: 113–128, doi:10.1007/BF02684592, S2CID 55847918.
• Yau, Shing-Tung; Chan, Raymond H., eds. (1999), "Sir Michael Atiyah: a great mathematician of the twentieth century", Asian J. Math., International Press, 3 (1): 1–332, ISBN 978-1-57146-080-6, MR 1701915, archived from the original on 8 August 2008.
• Yau, Shing-Tung, ed. (2005), The Founders of Index Theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer, International Press, p. 358, ISBN 978-1-57146-120-9, archived from the original on 7 February 2006.
External links
Wikiquote has quotations related to Michael Atiyah
• Michael Atiyah tells his life story at Web of Stories
• The celebrations of Michael Atiyah's 80th birthday in Edinburgh, 20-24 April 2009
• Mathematical descendants of Michael Atiyah
• "Sir Michael Atiyah on math, physics and fun", superstringtheory.com, Official Superstring theory web site], retrieved 14 August 2008
• Atiyah, Michael, Beauty in Mathematics (video, 3m14s), retrieved 14 August 2008
• Atiyah, Michael, The nature of space (Online lecture), retrieved 14 August 2008
• Batra, Amba (8 November 2003), Maths guru with Einstein's dream prefers chalk to mouse. (Interview with Atiyah.), Delhi newsline, archived from the original on 8 February 2009, retrieved 14 August 2008
• Michael Atiyah at the Mathematics Genealogy Project
• Halim, Hala (1998), "Michael Atiyah:Euclid and Victoria", Al-Ahram Weekly On-line, no. 391, archived from the original on 16 August 2004, retrieved 26 August 2008
• Meek, James (21 April 2004), "Interview with Michael Atiyah", The Guardian, London, retrieved 14 August 2008
• Sir Michael Atiyah FRS, Isaac Newton Institute, retrieved 14 August 2008
• "Atiyah and Singer receive 2004 Abel prize" (PDF), Notices of the American Mathematical Society, 51 (6): 650–651, 2006, retrieved 14 August 2008
• Raussen, Martin; Skau, Christian (24 May 2004), Interview with Michael Atiyah and Isadore Singer, retrieved 14 August 2008
• Photos of Michael Francis Atiyah, Oberwolfach photo collection, retrieved 14 August 2008
• Wade, Mike (21 April 2009), Maths and the bomb: Sir Michael Atiyah at 80, London: Timesonline, retrieved 12 May 2010
• List of works of Michael Atiyah from Celebratio Mathematica
• Connes, Alain; Kouneiher, Joseph (2019). "Sir Michael Atiyah, a Knight Mathematician : A tribute to Michael Atiyah, an inspiration and a friend". Notices of the American Mathematical Society. 66 (10): 1660–1685. arXiv:1910.07851. Bibcode:2019arXiv191007851C. doi:10.1090/noti1981. S2CID 204743755.
• Portraits of Michael Atiyah at the National Portrait Gallery, London
Abel Prize laureates
• 2003 Jean-Pierre Serre
• 2004 Michael Atiyah
• Isadore Singer
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• John Edensor Littlewood (1958)
• Macfarlane Burnet (1959)
• Harold Jeffreys (1960)
• Hans Krebs (1961)
• Cyril Norman Hinshelwood (1962)
• Paul Fildes (1963)
• Sydney Chapman (1964)
• Alan Hodgkin (1965)
• Lawrence Bragg (1966)
• Bernard Katz (1967)
• Tadeusz Reichstein (1968)
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• Norman Pirie (1971)
• Nevill Francis Mott (1972)
• Andrew Huxley (1973)
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• George Porter (1992)
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• Frederick Charles Frank (1994)
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• Alan Battersby (2000)
De Morgan Medallists
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• A. G. Greenhill (1902)
• H. F. Baker (1905)
• J. W. L. Glaisher (1908)
• Horace Lamb (1911)
• J. Larmor (1914)
• W. H. Young (1917)
• E. W. Hobson (1920)
• P. A. MacMahon (1923)
• A. E. H. Love (1926)
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• Bertrand Russell (1932)
• E. T. Whittaker (1935)
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• Louis Mordell (1941)
• Sydney Chapman (1944)
• George Neville Watson (1947)
• A. S. Besicovitch (1950)
• E. C. Titchmarsh (1953)
• G. I. Taylor (1956)
• W. V. D. Hodge (1959)
• Max Newman (1962)
• Philip Hall (1965)
• Mary Cartwright (1968)
• Kurt Mahler (1971)
• Graham Higman (1974)
• C. Ambrose Rogers (1977)
• Michael Atiyah (1980)
• K. F. Roth (1983)
• J. W. S. Cassels (1986)
• D. G. Kendall (1989)
• Albrecht Fröhlich (1992)
• W. K. Hayman (1995)
• R. A. Rankin (1998)
• J. A. Green (2001)
• Roger Penrose (2004)
• Bryan John Birch (2007)
• Keith William Morton (2010)
• John Griggs Thompson (2013)
• Timothy Gowers (2016)
• Andrew Wiles (2019)
Masters of Trinity College, Cambridge
• John Redman
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Richard V. Southwell
Sir Richard Vynne Southwell, FRS[2] (2 July 1888 – 9 December 1970) was a British mathematician who specialised in applied mechanics as an engineering science academic.[4][5]
Richard Southwell
Born
Richard Vynne Southwell
2 July 1888
Norwich
Died9 December 1970(1970-12-09) (aged 82)
NationalityBritish
Alma materTrinity College, Cambridge[1]
AwardsTimoshenko Medal (1959)
Elliott Cresson Medal (1964)
Fellow of the Royal Society[2]
Scientific career
FieldsMathematics
InstitutionsUniversity of Oxford
Imperial College London
Doctoral studentsLeslie Fox
Olgierd Zienkiewicz[3]
Education and career
Richard Southwell was educated at Norwich School and Trinity College, Cambridge, where in 1912 he achieved first class degree results in both the mathematical and mechanical science tripos.[6] In 1914, he became a Fellow of Trinity, and a lecturer in Mechanical Sciences.
Southwell was in the Royal Naval Air Service during World War I. After World War I, he was head of the Aerodynamics and Structures Divisions at the Royal Aircraft Establishment, Farnborough.
In 1920, he moved to the National Physical Laboratory. He then returned to Trinity College in 1925 as Fellow and Mathematics Lecturer in 1925. Next, in 1929, he moved to Oxford University as Professor of Engineering Science and Fellow of Brasenose College. Here, he developed a research group, including Derman Christopherson, with whom he worked on his relaxation method. He became a member of a number of UK governmental technical committees, including for the Air Ministry, at the time when the R100 and R101 airships were being conceived.
Southwell was Rector at Imperial College, London from 1942 until his retirement in 1948.[4][7][8] He continued his research at Imperial College. He was also involved in the opening a new student residence, Selkirk Hall.
Scientific contribution
As a scientist, Southwell developed relaxation methods for solving partial differential equations in engineering and theoretical physics during the 1930 and the 1940s. The equations had first to be discretised by the finite difference methods. Then, the values of the function of the grids would have to be iteratively adjusted so that the discretised equation would be satisfied. At the time, digital computers did not exist, and the computations had to be done by hand. Southwell developed various techniques to speed up the calculations. For instance, in 1935, he used multiple grids for that purpose, a technique which would later be elaborated into the multigrid method.[9]
Honours
Southwell received the following honours and recognition for his achievements:
• Professor at University of Oxford (1925)
• Fellow of the Royal Society (1925)[2]
• Worcester Reed Warner medal, ASME (1941)
• Member of the National Academy of Sciences (1943)
• Timoshenko Medal (1959)
• Elliott Cresson Medal (1964)
Southwell was also honoured with a knighthood.
Publications
• Stress Calculation in Frameworks by the method of relaxation of constraints Proc. Roy. Soc. A 151, 56 (1935); Proc. Roy. Soc. A 153, 41 (1935).
• Relaxation methods in engineering science : a treatise on approximate computation (Oxford Univ. Press – 1940)
• An Introduction to the Theory of Elasticity for Engineers and Physicists, 2nd ed. London: (Oxford University Press, 1941)
• Relaxation Methods in Theoretical Physics, a continuation of the treatise, Relaxation methods in engineering science (Oxford University Press – 1946)
References
1. SOUTHWELL, Sir Richard Vynne, Who Was Who, A & C Black, 1920–2015; online edn, Oxford University Press, 2014
2. Christopherson, D. G. (1972). "Richard Vynne Southwell 1888-1970". Biographical Memoirs of Fellows of the Royal Society. 18: 549–565. doi:10.1098/rsbm.1972.0020. S2CID 122050415.
3. Richard V. Southwell at the Mathematics Genealogy Project
4. Sir Richard Southwell, MA, LLD, FRS: Rector 1942–48, Imperial College, London, UK.
5. "Annual Report". 1975.
6. A. G. Pugsley, ‘Southwell, Sir Richard Vynne (1888–1970)’, rev. H. C. G. Matthew, Oxford Dictionary of National Biography, Oxford University Press, 2004; online edn, Jan 2011, accessed 2 November 2013
7. Hannah Guy, The history of Imperial College London, 1907–2007, Imperial College Press, 2007. Page 749.
8. "Imperial College of Science and Technology: Sir Richard Southwell, F.R.S". Nature. 162 (4105): 16. 1948. Bibcode:1948Natur.162Q..16.. doi:10.1038/162016a0.
9. A. O. Demuren, Application of Multigrid Methods to solve Navier-Stokes Equations, NASA Technical Memorandum no. 102359 (1989).
External links
• Centenary of Imperial college with a short biography of R. V. Southwell
• Biography in Portuguese (Brasil)
• Early Numerical Linear Algebra in the UK
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Roger Penrose
Sir Roger Penrose OM FRS HonFInstP (born 8 August 1931)[1] is a British mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics.[2] He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fellow of Wadham College, Oxford, and an honorary fellow of St John's College, Cambridge, and University College London.[3][4][5]
Sir
Roger Penrose
OM FRS HonFInstP
Penrose in 2011
Born (1931-08-08) 8 August 1931
Colchester, England, UK
Alma mater
• University College London (BSc)
• St John's College, Cambridge (MSc, PhD)
Known for
List of contributions
• Moore–Penrose inverse
• Twistor theory
• Spin network
• Abstract index notation
• Black hole bomb
• Geometry of spacetime
• Cosmic censorship
• Illumination problem
• Weyl curvature hypothesis
• Penrose inequalities
• Penrose interpretation of quantum mechanics
• Diósi–Penrose model
• Newman–Penrose formalism
• GHP formalism
• Penrose diagram
• Penrose inequality
• Penrose process
• Penrose tiling
• Penrose triangle
• Penrose stairs
• Penrose–Hawking singularity theorems
• Penrose graphical notation
• Penrose transform
• Penrose–Terrell effect
• pp-wave spacetime
• Schrödinger–Newton equations
• Orch-OR/Penrose–Lucas argument
• FELIX experiment
• Trapped surface
• Andromeda paradox
• Conformal cyclic cosmology
Spouses
Joan Isabel Wedge
(m. 1959, divorced)
Vanessa Thomas
(m. 1988)
[1]
Children4
RelativesLionel Penrose (father), Roland Penrose (uncle), Jonathan Penrose (brother), Oliver Penrose (brother), Shirley Hodgson (sister), Antony Penrose (cousin)
Awards
List of awards
• Adams Prize (1966)
• Heineman Prize (1971)
• Fellow of the Royal Society (1972)
• Eddington Medal (1975)
• Royal Medal (1985)
• Wolf Prize (1988)
• Dirac Medal (1989)
• Albert Einstein Medal (1990)
• Naylor Prize and Lectureship (1991)
• Knight Bachelor (1994)
• James Scott Prize Lectureship (1997–2000)
• Karl Schwarzschild Medal (2000)
• De Morgan Medal (2004)
• Dalton Medal (2005)
• Copley Medal (2008)
• Fonseca Prize (2011)
• Nobel Prize in Physics (2020)
Scientific career
FieldsMathematical physics, tessellations
Institutions
• Cornell University
• Bedford College, London
• Princeton University
• Rice University
• Syracuse University
• King's College, London
• Birkbeck, University of London
• Wadham College, Oxford
• Polish Academy of Sciences
ThesisTensor Methods in Algebraic Geometry (1958)
Doctoral advisorJohn A. Todd
Other academic advisorsW. V. D. Hodge
Doctoral students
• Andrew Hodges
• Lane Hughston
• Richard Jozsa
• Claude LeBrun
• John McNamara
• Tristan Needham
• Tim Poston
• Asghar Qadir
• Richard S. Ward
Influenced
• Michael Atiyah
• Stuart Hameroff
Penrose has contributed to the mathematical physics of general relativity and cosmology. He has received several prizes and awards, including the 1988 Wolf Prize in Physics, which he shared with Stephen Hawking for the Penrose–Hawking singularity theorems,[6] and the 2020 Nobel Prize in Physics "for the discovery that black hole formation is a robust prediction of the general theory of relativity".[7][8][9][10][lower-alpha 1] He is regarded as one of the greatest living physicists, mathematicians and scientists, and is particularly noted for the breadth and depth of his work in both natural and formal sciences.[11][12][13][14][15][16][17][18][19]
Early life and education
Born in Colchester, Essex, Roger Penrose is a son of medical doctor Margaret (Leathes) and psychiatrist and geneticist Lionel Penrose.[lower-alpha 2] His paternal grandparents were J. Doyle Penrose, an Irish-born artist, and The Hon. Elizabeth Josephine, daughter of Alexander Peckover, 1st Baron Peckover; his maternal grandparents were physiologist John Beresford Leathes and Russian Jewish[20] Sonia Marie Natanson.[21][22][23] His uncle was artist Roland Penrose, whose son with photographer Lee Miller is Antony Penrose.[24][25] Penrose is the brother of physicist Oliver Penrose, of geneticist Shirley Hodgson, and of chess Grandmaster Jonathan Penrose.[26][27] Their stepfather was the mathematician and computer scientist Max Newman.
Penrose spent World War II as a child in Canada where his father worked in London, Ontario.[28] Penrose studied at University College School.[1] He attended University College London and attained a first class degree in mathematics[26] from University of London in 1952.
In 1955, while a student, Penrose reintroduced the E. H. Moore generalised matrix inverse, also known as the Moore–Penrose inverse,[29] after it had been reinvented by Arne Bjerhammar in 1951.[30] Having started research under the professor of geometry and astronomy, Sir W. V. D. Hodge, Penrose finished his PhD at St John's College, Cambridge, in 1958, with a thesis on tensor methods in algebraic geometry[31] supervised by algebraist and geometer John A. Todd.[32] He devised and popularised the Penrose triangle in the 1950s in collaboration with his father, describing it as "impossibility in its purest form", and exchanged material with the artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it.[33][34] Escher's Waterfall, and Ascending and Descending were in turn inspired by Penrose.[35]
As reviewer Manjit Kumar puts it:
As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of Escher's work. Soon he was trying to conjure up impossible figures of his own and discovered the tribar – a triangle that looks like a real, solid three-dimensional object, but isn't. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that simultaneously loops up and down. An article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces.[36]
Research and career
Penrose spent the academic year 1956–57 as an assistant lecturer at Bedford College, London and was then a research fellow at St John's College, Cambridge. During that three-year post, he married Joan Isabel Wedge, in 1959. Before the fellowship ended Penrose won a NATO Research Fellowship for 1959–61, first at Princeton and then at Syracuse University. Returning to the University of London, Penrose spent two years, 1961–63, as a researcher at King's College, London, before returning to the United States to spend the year 1963–64 as a visiting associate professor at the University of Texas at Austin.[37] He later held visiting positions at Yeshiva, Princeton, and Cornell during 1966–67 and 1969.
In 1964, while a reader at Birkbeck College, London, (and having had his attention drawn from pure mathematics to astrophysics by the cosmologist Dennis Sciama, then at Cambridge)[26] in the words of Kip Thorne of Caltech, "Roger Penrose revolutionised the mathematical tools that we use to analyse the properties of spacetime".[38][39] Until then, work on the curved geometry of general relativity had been confined to configurations with sufficiently high symmetry for Einstein's equations to be solvable explicitly, and there was doubt about whether such cases were typical. One approach to this issue was by the use of perturbation theory, as developed under the leadership of John Archibald Wheeler at Princeton.[40] The other, and more radically innovative, approach initiated by Penrose was to overlook the detailed geometrical structure of spacetime and instead concentrate attention just on the topology of the space, or at most its conformal structure, since it is the latter – as determined by the lay of the lightcones – that determines the trajectories of lightlike geodesics, and hence their causal relationships. The importance of Penrose's epoch-making paper "Gravitational Collapse and Space-Time Singularities"[41] was not its only result, summarised roughly as that if an object such as a dying star implodes beyond a certain point, then nothing can prevent the gravitational field getting so strong as to form some kind of singularity. It also showed a way to obtain similarly general conclusions in other contexts, notably that of the cosmological Big Bang, which he dealt with in collaboration with Dennis Sciama's most famous student, Stephen Hawking.[42][43][44] The Penrose–Hawking singularity theorems were inspired by Amal Kumar Raychaudhuri's Raychaudhuri equation.
It was in the local context of gravitational collapse that the contribution of Penrose was most decisive, starting with his 1969 cosmic censorship conjecture,[45] to the effect that any ensuing singularities would be confined within a well-behaved event horizon surrounding a hidden space-time region for which Wheeler coined the term black hole, leaving a visible exterior region with strong but finite curvature, from which some of the gravitational energy may be extractable by what is known as the Penrose process, while accretion of surrounding matter may release further energy that can account for astrophysical phenomena such as quasars.[46][47][48]
Following up his "weak cosmic censorship hypothesis", Penrose went on, in 1979, to formulate a stronger version called the "strong censorship hypothesis". Together with the Belinski–Khalatnikov–Lifshitz conjecture and issues of nonlinear stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. Also from 1979, dates Penrose's influential Weyl curvature hypothesis on the initial conditions of the observable part of the universe and the origin of the second law of thermodynamics.[49] Penrose and James Terrell independently realised that objects travelling near the speed of light will appear to undergo a peculiar skewing or rotation. This effect has come to be called the Terrell rotation or Penrose–Terrell rotation.[50][51]
In 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature (2,2).[52][53]
Penrose is well known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane nonperiodically, and are the first tilings to exhibit fivefold rotational symmetry. In 1984, such patterns were observed in the arrangement of atoms in quasicrystals.[54] Another noteworthy contribution is his 1971 invention of spin networks, which later came to form the geometry of spacetime in loop quantum gravity.[55] He was influential in popularizing what are commonly known as Penrose diagrams (causal diagrams).[56]
In 1983, Penrose was invited to teach at Rice University in Houston, by the then provost Bill Gordon. He worked there from 1983 to 1987.[57] His doctoral students have included, among others, Andrew Hodges,[58] Lane Hughston, Richard Jozsa, Claude LeBrun, John McNamara, Tristan Needham, Tim Poston,[59] Asghar Qadir, and Richard S. Ward.
In 2004, Penrose released The Road to Reality: A Complete Guide to the Laws of the Universe, a 1,099-page comprehensive guide to the Laws of Physics that includes an explanation of his own theory. The Penrose Interpretation predicts the relationship between quantum mechanics and general relativity, and proposes that a quantum state remains in superposition until the difference of space-time curvature attains a significant level.[60][61]
Penrose is the Francis and Helen Pentz Distinguished Visiting Professor of Physics and Mathematics at Pennsylvania State University.[62]
An earlier universe
In 2010, Penrose reported possible evidence, based on concentric circles found in Wilkinson Microwave Anisotropy Probe data of the cosmic microwave background sky, of an earlier universe existing before the Big Bang of our own present universe.[63] He mentions this evidence in the epilogue of his 2010 book Cycles of Time,[64] a book in which he presents his reasons, to do with Einstein's field equations, the Weyl curvature C, and the Weyl curvature hypothesis (WCH), that the transition at the Big Bang could have been smooth enough for a previous universe to survive it.[65][66] He made several conjectures about C and the WCH, some of which were subsequently proved by others, and he also popularized his conformal cyclic cosmology (CCC) theory.[67] In this theory, Penrose postulates that at the end of the universe all matter is eventually contained within black holes which subsequently evaporate via Hawking radiation. At this point, everything contained within the universe consists of photons which "experience" neither time nor space. There is essentially no difference between an infinitely large universe consisting only of photons and an infinitely small universe consisting only of photons. Therefore, a singularity for a Big Bang and an infinitely expanded universe are equivalent. [68]
In simple terms, Penrose believes that the singularity in Einstein's field equation at the Big Bang is only an apparent singularity, similar to the well-known apparent singularity at the event horizon of a black hole.[46] The latter singularity can be removed by a change of coordinate system, and Penrose proposes a different change of coordinate system that will remove the singularity at the big bang.[69] One implication of this is that the major events at the Big Bang can be understood without unifying general relativity and quantum mechanics, and therefore we are not necessarily constrained by the Wheeler–DeWitt equation, which disrupts time.[70][71] Alternatively, one can use the Einstein–Maxwell–Dirac equations.[72]
Consciousness
Penrose has written books on the connection between fundamental physics and human (or animal) consciousness. In The Emperor's New Mind (1989), he argues that known laws of physics are inadequate to explain the phenomenon of consciousness.[73] Penrose proposes the characteristics this new physics may have and specifies the requirements for a bridge between classical and quantum mechanics (what he calls correct quantum gravity).[74] Penrose uses a variant of Turing's halting theorem to demonstrate that a system can be deterministic without being algorithmic. (For example, imagine a system with only two states, ON and OFF. If the system's state is ON when a given Turing machine halts and OFF when the Turing machine does not halt, then the system's state is completely determined by the machine; nevertheless, there is no algorithmic way to determine whether the Turing machine stops.)[75][76]
Penrose believes that such deterministic yet non-algorithmic processes may come into play in the quantum mechanical wave function reduction, and may be harnessed by the brain. He argues that computers today are unable to have intelligence because they are algorithmically deterministic systems. He argues against the viewpoint that the rational processes of the mind are completely algorithmic and can thus be duplicated by a sufficiently complex computer.[77] This contrasts with supporters of strong artificial intelligence, who contend that thought can be simulated algorithmically. He bases this on claims that consciousness transcends formal logic because factors such as the insolubility of the halting problem and Gödel's incompleteness theorem prevent an algorithmically based system of logic from reproducing such traits of human intelligence as mathematical insight.[77][78] These claims were originally espoused by the philosopher John Lucas of Merton College, Oxford.[79] G. Hirase has paraphrased Penrose' argument and reinforced it.[80]
The Penrose–Lucas argument about the implications of Gödel's incompleteness theorem for computational theories of human intelligence has been criticised by mathematicians, computer scientists and philosophers. Many experts in these fields assert that Penrose's argument fails, though different authors may choose different aspects of the argument to attack.[81] Marvin Minsky, a leading proponent of artificial intelligence, was particularly critical, stating that Penrose "tries to show, in chapter after chapter, that human thought cannot be based on any known scientific principle." Minsky's position is exactly the opposite – he believed that humans are, in fact, machines, whose functioning, although complex, is fully explainable by current physics. Minsky maintained that "one can carry that quest [for scientific explanation] too far by only seeking new basic principles instead of attacking the real detail. This is what I see in Penrose's quest for a new basic principle of physics that will account for consciousness."[82]
Penrose responded to criticism of The Emperor's New Mind with his follow-up 1994 book Shadows of the Mind, and in 1997 with The Large, the Small and the Human Mind. In those works, he also combined his observations with those of anesthesiologist Stuart Hameroff.[83]
Penrose and Hameroff have argued that consciousness is the result of quantum gravity effects in microtubules, which they dubbed Orch-OR (orchestrated objective reduction). Max Tegmark, in a paper in Physical Review E,[84] calculated that the time scale of neuron firing and excitations in microtubules is slower than the decoherence time by a factor of at least 10,000,000,000. The reception of the paper is summed up by this statement in Tegmark's support: "Physicists outside the fray, such as IBM's John A. Smolin, say the calculations confirm what they had suspected all along. 'We're not working with a brain that's near absolute zero. It's reasonably unlikely that the brain evolved quantum behavior'".[85] Tegmark's paper has been widely cited by critics of the Penrose–Hameroff position.
In their reply to Tegmark's paper, also published in Physical Review E, the physicists Scott Hagan, Jack Tuszyński and Hameroff[86][87] claimed that Tegmark did not address the Orch-OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than the much smaller separations stipulated for Orch-OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark's, but still well short of the 25 ms required if the quantum processing in the theory was to be linked to the 40 Hz gamma synchrony, as Orch-OR suggested. To bridge this gap, the group made a series of proposals.[86] They supposed that the interiors of neurons could alternate between liquid and gel states. In the gel state, it was further hypothesized that the water electrical dipoles are oriented in the same direction, along the outer edge of the microtubule tubulin subunits.[86] Hameroff et al. proposed that this ordered water could screen any quantum coherence within the tubulin of the microtubules from the environment of the rest of the brain. Each tubulin also has a tail extending out from the microtubules, which is negatively charged, and therefore attracts positively charged ions. It is suggested that this could provide further screening. Further to this, there was a suggestion that the microtubules could be pumped into a coherent state by biochemical energy.[88]
Finally, he suggested that the configuration of the microtubule lattice might be suitable for quantum error correction, a means of holding together quantum coherence in the face of environmental interaction.[88]
Hameroff, in a lecture in part of a Google Tech talks series exploring quantum biology, gave an overview of current research in the area, and responded to subsequent criticisms of the Orch-OR model.[89] In addition to this, a 2011 paper by Roger Penrose and Stuart Hameroff published in the Journal of Cosmology gives an updated model of their Orch-OR theory, in light of criticisms, and discusses the place of consciousness within the universe.[90]
Phillip Tetlow, although himself supportive of Penrose's views, acknowledges that Penrose's ideas about the human thought process are at present a minority view in scientific circles, citing Minsky's criticisms and quoting science journalist Charles Seife's description of Penrose as "one of a handful of scientists" who believe that the nature of consciousness suggests a quantum process.[85]
In January 2014, Hameroff and Penrose ventured that a discovery of quantum vibrations in microtubules by Anirban Bandyopadhyay of the National Institute for Materials Science in Japan[91] supports the hypothesis of Orch-OR theory.[92] A reviewed and updated version of the theory was published along with critical commentary and debate in the March 2014 issue of Physics of Life Reviews.[93]
Publications
His popular publications include:
• The Emperor's New Mind: Concerning Computers, Minds, and The Laws of Physics (1989)[94]
• Shadows of the Mind: A Search for the Missing Science of Consciousness (1994)[95]
• The Road to Reality: A Complete Guide to the Laws of the Universe (2004)[96]
• Cycles of Time: An Extraordinary New View of the Universe (2010)[97]
• Fashion, Faith, and Fantasy in the New Physics of the Universe (2016)[98]
His co-authored publications include:
• The Nature of Space and Time (with Stephen Hawking) (1996)[99]
• The Large, the Small and the Human Mind (with Abner Shimony, Nancy Cartwright, and Stephen Hawking) (1997)[100]
• White Mars: The Mind Set Free (with Brian Aldiss) (1999)[101]
His academic books include:
• Techniques of Differential Topology in Relativity (1972, ISBN 0-89871-005-7)
• Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (with Wolfgang Rindler, 1987) ISBN 0-521-33707-0 (paperback)
• Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (with Wolfgang Rindler, 1988) (reprint), ISBN 0-521-34786-6 (paperback)
His forewords to other books include:
• Foreword to “The Map and the Territory: Exploring the foundations of science, thought and reality” by Shyam Wuppuluri and Francisco Antonio Doria. Published by Springer in "The Frontiers Collection", 2018.[102]
• Foreword to Beating the Odds: The Life and Times of E. A. Milne, written by Meg Weston Smith. Published by World Scientific Publishing Co in June 2013.[103]
• Foreword to "A Computable Universe" by Hector Zenil. Published by World Scientific Publishing Co in December 2012.[104]
• Foreword to Quantum Aspects of Life by Derek Abbott, Paul C. W. Davies, and Arun K. Pati. Published by Imperial College Press in 2008.[105]
• Foreword to Fearful Symmetry by Anthony Zee's. Published by Princeton University Press in 2007.[106]
Awards and honours
Penrose has been awarded many prizes for his contributions to science. In 1971, he was awarded the Dannie Heineman Prize for Astrophysics. He was elected a Fellow of the Royal Society (FRS) in 1972. In 1975, Stephen Hawking and Penrose were jointly awarded the Eddington Medal of the Royal Astronomical Society. In 1985, he was awarded the Royal Society Royal Medal. Along with Stephen Hawking, he was awarded the prestigious Wolf Foundation Prize for Physics in 1988.
In 1989, Penrose was awarded the Dirac Medal and Prize of the British Institute of Physics. He was also made an Honorary Fellow of the Institute of Physics (HonFInstP).[107]
In 1990, Penrose was awarded the Albert Einstein Medal for outstanding work related to the work of Albert Einstein by the Albert Einstein Society. In 1991, he was awarded the Naylor Prize of the London Mathematical Society. From 1992 to 1995, he served as President of the International Society on General Relativity and Gravitation. In 1994, Penrose was knighted for services to science.[108] In the same year, he was also awarded an Honorary Degree (Doctor of Science) by the University of Bath,[109] and became a member of Polish Academy of Sciences. In 1998, he was elected Foreign Associate of the United States National Academy of Sciences.[110] In 2000, he was appointed a Member of the Order of Merit (OM).[111]
In 2004, he was awarded the De Morgan Medal for his wide and original contributions to mathematical physics.[112] To quote the citation from the London Mathematical Society:
His deep work on General Relativity has been a major factor in our understanding of black holes. His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics. His tilings of the plane underlie the newly discovered quasi-crystals.[113]
In 2005, Penrose was awarded an honorary doctorate by Warsaw University and Katholieke Universiteit Leuven (Belgium), and in 2006 by the University of York. In 2006, he also won the Dirac Medal given by the University of New South Wales. In 2008, Penrose was awarded the Copley Medal. He is also a Distinguished Supporter of Humanists UK and one of the patrons of the Oxford University Scientific Society.
He was elected to the American Philosophical Society in 2011.[114] The same year, he was also awarded the Fonseca Prize by the University of Santiago de Compostela.
In 2012, Penrose was awarded the Richard R. Ernst Medal by ETH Zürich for his contributions to science and strengthening the connection between science and society. In 2015 Penrose was awarded an honorary doctorate by CINVESTAV-IPN (Mexico).[115]
In 2017, he was awarded the Commandino Medal at the Urbino University for his contributions to the history of science.
In 2020, Penrose was awarded one half of the Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity, a half-share also going to Reinhard Genzel and Andrea Ghez for the discovery of a supermassive compact object at the centre of our galaxy.[9]
Personal life
Penrose married Vanessa Thomas, director of Academic Development at Cokethorpe School and former head of mathematics at Abingdon School,[116][117] with whom he has one son (Maxwell Sebastian Penrose).[118][116] He has three sons from a previous marriage to American Joan Isabel Penrose (née Wedge), whom he married in 1959.[119][120]
Religious views
During an interview with BBC Radio 4 on 25 September 2010, Penrose stated, "I'm not a believer myself. I don't believe in established religions of any kind."[121] He regards himself as an agnostic.[122] In the 1991 film A Brief History of Time, he also said, "I think I would say that the universe has a purpose, it's not somehow just there by chance … some people, I think, take the view that the universe is just there and it runs along—it's a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don't think that's a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it."[123]
Penrose is a patron of Humanists UK.[124]
See also
• List of things named after Roger Penrose
References
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Notes
1. The 2020 Nobel Prize was also awarded jointly to Reinhard Genzel and Andrea Ghez for their work on black holes.
2. Penrose and his father shared mathematical concepts with Dutch graphic artist M. C. Escher which were incorporated into a lot of pieces, including Waterfall, which is based on the 'Penrose triangle', and Up and Down.
External links
Wikiquote has quotations related to Roger Penrose.
Wikimedia Commons has media related to Roger Penrose.
• Awake in the Universe – Penrose debates how creativity, the most elusive of faculties, has helped us unlock the country of the mind and the mysteries of the cosmos with Bonnie Greer.
• Works by or about Roger Penrose at Internet Archive
• Dangerous Knowledge on YouTube – Penrose was one of the principal interviewees in a BBC documentary about the mathematics of infinity directed by David Malone
• Penrose's new theory "Aeons Before the Big Bang?":
• Original 2005 lecture: "Before the Big Bang? A new perspective on the Weyl curvature hypothesis" Archived 7 August 2009 at the Wayback Machine (Isaac Newton Institute for Mathematical Sciences, Cambridge, 11 November 2005).
• Original publication: "Before the Big Bang: an outrageous new perspective and its implications for particle physics". Proceedings of EPAC 2006. Edinburgh. 2759–2762 (cf. also Hill, C.D. & Nurowski, P. (2007) "On Penrose's 'Before the Big Bang' ideas". Ithaca)
• Revised 2009 lecture: "Aeons Before the Big Bang?" (Georgia Institute of Technology, Center for Relativistic Astrophysics)
• BBC interview on the new theory on YouTube
• Roger Penrose on The Forum
• Penrose on sidestepping reason on YouTube
• Hilary Putnam's review of Penrose's 'Shadows of the Mind' claiming that Penrose's use of Godel's Incompleteness Theorem is fallacious Archived 28 November 2007 at the Wayback Machine
• Beyond the Doubting of a Shadow: A Reply to Commentaries on Shadows of the Mind at the Wayback Machine (archived 18 June 2008)
• Penrose Tiling found in Islamic Architecture
• Two theories for the formation of quasicrystals resembling Penrose tilings
• Tegmark, Max (2000). "The importance of quantum decoherence in brain processes". Physical Review E. 61 (4): 4194–4206. arXiv:quant-ph/9907009. Bibcode:2000PhRvE..61.4194T. doi:10.1103/physreve.61.4194. PMID 11088215. S2CID 17140058.
• "Biological feasibility of quantum states in the brain" – (a disputation of Tegmark's result by Hagan, Hameroff, and Tuszyński)
• Tegmarks's rejoinder to Hagan et al.
• "Toilet Paper Plagiarism" at the Wayback Machine (archived 12 March 2005) – D. Trull about Penrose's lawsuit concerning the use of his Penrose tilings on toilet paper
• Roger Penrose: A Knight on the tiles (Plus Magazine)
• Penrose's Gifford Lecture biography
• Quantum-Mind
• Audio: Roger Penrose in conversation on the BBC World Service discussion show
• Roger Penrose speaking about Hawking's new book on Premier Christian Radio
• "The Cyclic Universe – A conversation with Roger Penrose", Ideas Roadshow, 2013
• Forbidden crystal symmetry in mathematics and architecture, filmed event at the Royal Institution, October 2013
• Oxford Mathematics Interviews: "Extra Time: Professor Sir Roger Penrose in conversation with Andrew Hodges." These two films explore the development of Sir Roger Penrose's thought over more than 60 years, ending with his most recent theories and predictions. 51 min and 42 min. (Mathematical Institute)
• BBC Radio 4 – The Life Scientific – Roger Penrose on Black Holes – 22 November 2016 Sir Roger Penrose talks to Jim Al-Khalili about his trailblazing work on how black holes form, the problems with quantum physics and his portrayal in films about Stephen Hawking.
• The Penrose Institute Website
• A chess problem holds the key to human consciousness?, Chessbase
• Roger Penrose on Nobelprize.org
Roger Penrose
Books
• The Emperor's New Mind (1989)
• Shadows of the Mind (1994)
• The Road to Reality (2004)
• Cycles of Time (2010)
• Fashion, Faith, and Fantasy in the New Physics of the Universe (2016)
Coauthored books
• The Nature of Space and Time (with Stephen Hawking) (1996)
• The Large, the Small and the Human Mind (with Abner Shimony, Nancy Cartwright and Stephen Hawking) (1997)
• White Mars or, The Mind Set Free (with Brian W. Aldiss) (1999)
Academic works
• Techniques of Differential Topology in Relativity (1972)
• Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (with Wolfgang Rindler) (1987)
• Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (with Wolfgang Rindler) (1988)
Concepts
• Twistor theory
• Spin network
• Abstract index notation
• Black hole bomb
• Geometry of spacetime
• Cosmic censorship
• Weyl curvature hypothesis
• Penrose inequalities
• Penrose interpretation of quantum mechanics
• Moore–Penrose inverse
• Newman–Penrose formalism
• Penrose diagram
• Penrose–Hawking singularity theorems
• Penrose inequality
• Penrose process
• Penrose tiling
• Penrose triangle
• Penrose stairs
• Penrose graphical notation
• Penrose transform
• Penrose–Terrell effect
• Orchestrated objective reduction/Penrose–Lucas argument
• FELIX experiment
• Trapped surface
• Andromeda paradox
• Conformal cyclic cosmology
Related
• Lionel Penrose (father)
• Oliver Penrose (brother)
• Jonathan Penrose (brother)
• Shirley Hodgson (sister)
• John Beresford Leathes (grandfather)
• Illumination problem
• Quantum mind
Articles related to Roger Penrose
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Laureates of the Wolf Prize in Physics
1970s
• Chien-Shiung Wu (1978)
• George Uhlenbeck / Giuseppe Occhialini (1979)
1980s
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• Freeman Dyson / Gerardus 't Hooft / Victor Weisskopf (1981)
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• Conyers Herring / Philippe Nozières (1984–85)
• Mitchell Feigenbaum / Albert J. Libchaber (1986)
• Herbert Friedman / Bruno Rossi / Riccardo Giacconi (1987)
• Roger Penrose / Stephen Hawking (1988)
1990s
• Pierre-Gilles de Gennes / David J. Thouless (1990)
• Maurice Goldhaber / Valentine Telegdi (1991)
• Joseph H. Taylor Jr. (1992)
• Benoît Mandelbrot (1993)
• Vitaly Ginzburg / Yoichiro Nambu (1994–95)
• John Wheeler (1996–97)
• Yakir Aharonov / Michael Berry (1998)
• Dan Shechtman (1999)
2000s
• Raymond Davis Jr. / Masatoshi Koshiba (2000)
• Bertrand Halperin / Anthony Leggett (2002–03)
• Robert Brout / François Englert / Peter Higgs (2004)
• Daniel Kleppner (2005)
• Albert Fert / Peter Grünberg (2006–07)
2010s
• John F. Clauser / Alain Aspect / Anton Zeilinger (2010)
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• Jacob Bekenstein (2012)
• Peter Zoller / Juan Ignacio Cirac (2013)
• James D. Bjorken / Robert P. Kirshner (2015)
• Yoseph Imry (2016)
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• Charles H. Bennett / Gilles Brassard (2018)
2020s
• Rafi Bistritzer / Pablo Jarillo-Herrero / Allan H. MacDonald (2020)
• Giorgio Parisi (2021)
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• Graham Higman (1974)
• C. Ambrose Rogers (1977)
• Michael Atiyah (1980)
• K. F. Roth (1983)
• J. W. S. Cassels (1986)
• D. G. Kendall (1989)
• Albrecht Fröhlich (1992)
• W. K. Hayman (1995)
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• J. A. Green (2001)
• Roger Penrose (2004)
• Bryan John Birch (2007)
• Keith William Morton (2010)
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• Andrew Wiles (2019)
Relativity
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Fundamental
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General
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• Introduction
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• 1916
• 1917: Barkla
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1926–1950
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• 1934
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1976–2000
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• 1997: Chu / Cohen-Tannoudji / Phillips
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• 1999: 't Hooft / Veltman
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2001–
present
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• 2022: Aspect / Clauser / Zeilinger
2020 Nobel Prize laureates
Chemistry
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Sister Beiter conjecture
In mathematics, the Sister Beiter conjecture is a conjecture about the size of coefficients of ternary cyclotomic polynomials (i.e. where the index is the product of three prime numbers). It is named after Marion Beiter, a Catholic nun who first proposed it in 1968.[1]
Background
For $n\in \mathbb {N} _{>0}$ the maximal coefficient (in absolute value) of the cyclotomic polynomial $\Phi _{n}(x)$ is denoted by $A(n)$.
Let $3\leq p\leq q\leq r$ be three prime numbers. In this case the cyclotomic polynomial $\Phi _{pqr}(x)$ is called ternary. In 1895, A. S. Bang[2] proved that $A(pqr)\leq p-1$. This implies the existence of $M(p):=\max \limits _{p\leq q\leq r{\text{ prime}}}A(pqr)$ such that $1\leq M(p)\leq p-1$.
Statement
Sister Beiter conjectured[1] in 1968 that $M(p)\leq {\frac {p+1}{2}}$. This was later disproved, but a corrected Sister Beiter conjecture was put forward as $M(p)\leq {\frac {2}{3}}p$.
Status
A preprint[3] from 2023 explains the history in detail and claims to prove this corrected conjecture. Explicitly it claims to prove
$M(p)\leq {\frac {2}{3}}p{\text{ and }}\lim \limits _{p\rightarrow \infty }{\frac {M(p)}{p}}={\frac {2}{3}}.$
References
1. Beiter, Marion (April 1968). "Magnitude of the Coefficients of the Cyclotomic Polynomial $F_{pqr}(x)$". The American Mathematical Monthly. 75 (4): 370–372. doi:10.2307/2313416. JSTOR 2313416.
2. Bang, A.S. (1895). "Om Lingingen $\Phi _{n}(x)=0$". Tidsskr. Math. 6: 6–12.
3. Juran, Branko; Moree, Pieter; Riekert, Adrian; Schmitz, David; Völlmecke, Julian (2023). "A proof of the corrected Sister Beiter cyclotomic coefficient conjecture inspired by Zhao and Zhang". arXiv:2304.09250 [math.NT].
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Mary Celine Fasenmyer
Mary Celine Fasenmyer, RSM (October 4, 1906, Crown, Pennsylvania – December 27, 1996, Erie, Pennsylvania) was an American mathematician and Catholic religious sister. She is most noted for her work on hypergeometric functions and linear algebra.[1]
Biography
Fasenmyer grew up in Pennsylvania's oil country, and displayed mathematical talent in high school. For ten years after her graduation she taught and studied at Mercyhurst College in Erie, where she joined the Sisters of Mercy. She pursued her mathematical studies in Pittsburgh and the University of Michigan, obtaining her doctorate in 1946 under the direction of Earl Rainville, with a dissertation entitled Some Generalized Hypergeometric Polynomials.[2]
After earning her Ph.D., Fasenmyer published two papers which expanded on her doctorate work. These would be further elaborated by Doron Zeilberger and Herbert Wilf into "WZ theory", which allowed computerized proof of many combinatorial identities. After this, she returned to Mercyhurst to teach and did not engage in further research.
Fasenmyer died in 1996.
"Sister Celine's" method
Fasenmyer is most remembered for the method that bears her name, first described in her Ph.D. thesis concerning recurrence relations in hypergeometric series.[1] The thesis demonstrated a purely algorithmic method to find recurrence relations satisfied by sums of terms of a hypergeometric polynomial, and requires only the series expansions of the polynomial. The beauty of her method is that it lends itself readily to computer automation. The work of Wilf and Zeilberger generalized the algorithm and established its correctness.
The hypergeometric polynomials she studied are called Sister Celine's polynomials.
References
1. Rosen, KH and Michaels, JG (2000) Handbook of Discrete and Combinatorial Mathematics, CRC Press.
2. Murray, MAM (2001) Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America, MIT Press.
Publications
• Fasenmyer, Mary Celine (1947), "Some generalized hypergeometric polynomials", Bulletin of the American Mathematical Society, (1945 University of Michigan PhD thesis), 53 (8): 806–812, doi:10.1090/S0002-9904-1947-08893-5, ISSN 0002-9904, MR 0022276, Zbl 0032.15402
• Fasenmyer, Mary Celine (1949), "A note on pure recurrence relations", The American Mathematical Monthly, 56 (1): 14–17, doi:10.1080/00029890.1949.11990232, ISSN 0002-9890, JSTOR 2305810, MR 0030044, Zbl 0032.41002
External links
• O'Connor, John J.; Robertson, Edmund F., "Mary Celine Fasenmyer", MacTutor History of Mathematics Archive, University of St Andrews
• "Sister Celine's Methods, Theorems, and Demonstrations" (PDF). Archived from the original (PDF) on 2010-06-01. (251 KB)
• Weisstein, Eric W. "Sister Celine's Method". MathWorld.
• Marko Petkovsek, Herbert Wilf and Doron Zeilberger (1996). A=B. AK Peters. pp. 57–58. ISBN 1-56881-063-6.
• Herbert Wilf and Lily Yen talk to Sister Celine (1993)
• Mary Celine Fasenmyer at the Mathematics Genealogy Project
• "Sister Mary Celine Fasenmyer", Biographies of Women Mathematicians, Agnes Scott College
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Mary Domitilla Thuener
Mary Domitilla Thuener OSB (Eleanor Margaret Thuener, 1880–1977) was a nun and mathematician who served as the first head of Villa Madonna College.[1]
Early life and education
Thuener was born on October 25, 1880. Her father was an immigrant from Germany who married an American; they had seven children but only three survived. Eleanor, the oldest, was born in Allegheny, Pennsylvania. She completed her studies at St. Mary’s Academy in Monroe, Michigan in 1905, took orders as a Benedictine nun, and entered the St. Waldburg convent in Covington, Kentucky, taking the name Mary Domitilla. There she came to work as a teacher in two local Catholic schools.[1]
By taking evening classes at St. Xavier College, Thuener completed a bachelor's degree in 1920. She completed a master's degree in 1923, in the women's college associated with the Catholic University of America.[1]
In 1929, she left her work at Thomas More College for additional study at the Catholic University of America, completing a PhD in 1932. Her dissertation, supervised by Aubrey E. Landry, was On the Number and Reality of the Self-Symmetric Quadrilaterals In-and-Circumscribed to the Triangular-Symmetric Rational Quartic.
Leadership
In 1921, the Benedictines of Covington founded Villa Madonna College, later to become Thomas More College. Thuener became its first dean,[1][2] and also taught mathematics there. In 1929 she left to earn her doctorate. She then returned to Villa Madonna as a mathematics and physics instructor.[1]
She served as prioress of St. Waldburg's beginning in 1943.[1][2]
Thuener died on September 29, 1977.
References
1. Green, Judy; LaDuke, Jeanne (2008), Pioneering Women in American Mathematics — The Pre-1940 PhD's, History of Mathematics, vol. 34 (1st ed.), American Mathematical Society, The London Mathematical Society, ISBN 978-0-8218-4376-5 Biography on pp. 599–600 of the Supplementary Material at AMS
2. Harmeling, Sister Deborah; Kremer, Deborah Kohl (2012), Benedictine Sisters of St. Walburg Monastery, Images of America, Arcadia Publishing, pp. 23, 67, ISBN 9780738590622
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Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.
Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry.
There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety, this is completely accurate—it is possible to recover a sober space from its associated site. However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies. Conversely, there are Grothendieck topologies that do not come from topological spaces.
The term "Grothendieck topology" has changed in meaning. In Artin (1962) it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning. Giraud (1964) modified the definition to use sieves rather than covers. Much of the time this does not make much difference, as each Grothendieck pretopology determines a unique Grothendieck topology, though quite different pretopologies can give the same topology.
Overview
Main article: History of topos theory
André Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they define. His conjectures postulated that there should be a cohomology theory of algebraic varieties that gives number-theoretic information about their defining equations. This cohomology theory was known as the "Weil cohomology", but using the tools he had available, Weil was unable to construct it.
In the early 1960s, Alexander Grothendieck introduced étale maps into algebraic geometry as algebraic analogues of local analytic isomorphisms in analytic geometry. He used étale coverings to define an algebraic analogue of the fundamental group of a topological space. Soon Jean-Pierre Serre noticed that some properties of étale coverings mimicked those of open immersions, and that consequently it was possible to make constructions that imitated the cohomology functor $H^{1}$. Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory that he suspected would be the Weil cohomology. To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use étale coverings instead. Grothendieck also saw how to phrase the definition of covering abstractly; this is where the definition of a Grothendieck topology comes from.
Definition
Motivation
The classical definition of a sheaf begins with a topological space $X$. A sheaf associates information to the open sets of $X$. This information can be phrased abstractly by letting $O(X)$ be the category whose objects are the open subsets $U$ of $X$ and whose morphisms are the inclusion maps $V\rightarrow U$ of open sets $U$ and $V$ of $X$. We will call such maps open immersions, just as in the context of schemes. Then a presheaf on $X$ is a contravariant functor from $O(X)$ to the category of sets, and a sheaf is a presheaf that satisfies the gluing axiom (here including the separation axiom). The gluing axiom is phrased in terms of pointwise covering, i.e., $\{U_{i}\}$ covers $U$ if and only if $\bigcup _{i}U_{i}=U$. In this definition, $U_{i}$ is an open subset of $X$. Grothendieck topologies replace each $U_{i}$ with an entire family of open subsets; in this example, $U_{i}$ is replaced by the family of all open immersions $V_{ij}\to U_{i}$. Such a collection is called a sieve. Pointwise covering is replaced by the notion of a covering family; in the above example, the set of all $\{V_{ij}\to U_{i}\}_{j}$ as $i$ varies is a covering family of $U$. Sieves and covering families can be axiomatized, and once this is done open sets and pointwise covering can be replaced by other notions that describe other properties of the space $X$.
Sieves
In a Grothendieck topology, the notion of a collection of open subsets of U stable under inclusion is replaced by the notion of a sieve. If c is any given object in C, a sieve on c is a subfunctor of the functor Hom(−, c); (this is the Yoneda embedding applied to c). In the case of O(X), a sieve S on an open set U selects a collection of open subsets of U that is stable under inclusion. More precisely, consider that for any open subset V of U, S(V) will be a subset of Hom(V, U), which has only one element, the open immersion V → U. Then V will be considered "selected" by S if and only if S(V) is nonempty. If W is a subset of V, then there is a morphism S(V) → S(W) given by composition with the inclusion W → V. If S(V) is non-empty, it follows that S(W) is also non-empty.
If S is a sieve on X, and f: Y → X is a morphism, then left composition by f gives a sieve on Y called the pullback of S along f, denoted by f$^{\ast }$S. It is defined as the fibered product S ×Hom(−, X) Hom(−, Y) together with its natural embedding in Hom(−, Y). More concretely, for each object Z of C, f$^{\ast }$S(Z) = { g: Z → Y | fg $\in $S(Z) }, and f$^{\ast }$S inherits its action on morphisms by being a subfunctor of Hom(−, Y). In the classical example, the pullback of a collection {Vi} of subsets of U along an inclusion W → U is the collection {Vi∩W}.
Grothendieck topology
A Grothendieck topology J on a category C is a collection, for each object c of C, of distinguished sieves on c, denoted by J(c) and called covering sieves of c. This selection will be subject to certain axioms, stated below. Continuing the previous example, a sieve S on an open set U in O(X) will be a covering sieve if and only if the union of all the open sets V for which S(V) is nonempty equals U; in other words, if and only if S gives us a collection of open sets that cover U in the classical sense.
Axioms
The conditions we impose on a Grothendieck topology are:
• (T 1) (Base change) If S is a covering sieve on X, and f: Y → X is a morphism, then the pullback f$\ast $S is a covering sieve on Y.
• (T 2) (Local character) Let S be a covering sieve on X, and let T be any sieve on X. Suppose that for each object Y of C and each arrow f: Y → X in S(Y), the pullback sieve f$\ast $T is a covering sieve on Y. Then T is a covering sieve on X.
• (T 3) (Identity) Hom(−, X) is a covering sieve on X for any object X in C.
The base change axiom corresponds to the idea that if {Ui} covers U, then {Ui ∩ V} should cover U ∩ V. The local character axiom corresponds to the idea that if {Ui} covers U and {Vij}j $\in $Ji covers Ui for each i, then the collection {Vij} for all i and j should cover U. Lastly, the identity axiom corresponds to the idea that any set is covered by itself via the identity map.
Grothendieck pretopologies
In fact, it is possible to put these axioms in another form where their geometric character is more apparent, assuming that the underlying category C contains certain fibered products. In this case, instead of specifying sieves, we can specify that certain collections of maps with a common codomain should cover their codomain. These collections are called covering families. If the collection of all covering families satisfies certain axioms, then we say that they form a Grothendieck pretopology. These axioms are:
• (PT 0) (Existence of fibered products) For all objects X of C, and for all morphisms X0 → X that appear in some covering family of X, and for all morphisms Y → X, the fibered product X0 ×X Y exists.
• (PT 1) (Stability under base change) For all objects X of C, all morphisms Y → X, and all covering families {Xα → X}, the family {Xα ×X Y → Y} is a covering family.
• (PT 2) (Local character) If {Xα → X} is a covering family, and if for all α, {Xβα → Xα} is a covering family, then the family of composites {Xβα → Xα → X} is a covering family.
• (PT 3) (Isomorphisms) If f: Y → X is an isomorphism, then {f} is a covering family.
For any pretopology, the collection of all sieves that contain a covering family from the pretopology is always a Grothendieck topology.
For categories with fibered products, there is a converse. Given a collection of arrows {Xα → X}, we construct a sieve S by letting S(Y) be the set of all morphisms Y → X that factor through some arrow Xα → X. This is called the sieve generated by {Xα → X}. Now choose a topology. Say that {Xα → X} is a covering family if and only if the sieve that it generates is a covering sieve for the given topology. It is easy to check that this defines a pretopology.
(PT 3) is sometimes replaced by a weaker axiom:
• (PT 3') (Identity) If 1X : X → X is the identity arrow, then {1X} is a covering family.
(PT 3) implies (PT 3'), but not conversely. However, suppose that we have a collection of covering families that satisfies (PT 0) through (PT 2) and (PT 3'), but not (PT 3). These families generate a pretopology. The topology generated by the original collection of covering families is then the same as the topology generated by the pretopology, because the sieve generated by an isomorphism Y → X is Hom(−, X). Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent.
Sites and sheaves
See also: Topos
Let C be a category and let J be a Grothendieck topology on C. The pair (C, J) is called a site.
A presheaf on a category is a contravariant functor from C to the category of all sets. Note that for this definition C is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology. Consequently, we define a sheaf on a site to be a presheaf F such that for all objects X and all covering sieves S on X, the natural map Hom(Hom(−, X), F) → Hom(S, F), induced by the inclusion of S into Hom(−, X), is a bijection. Halfway in between a presheaf and a sheaf is the notion of a separated presheaf, where the natural map above is required to be only an injection, not a bijection, for all sieves S. A morphism of presheaves or of sheaves is a natural transformation of functors. The category of all sheaves on C is the topos defined by the site (C, J).
Using the Yoneda lemma, it is possible to show that a presheaf on the category O(X) is a sheaf on the topology defined above if and only if it is a sheaf in the classical sense.
Sheaves on a pretopology have a particularly simple description: For each covering family {Xα → X}, the diagram
$F(X)\rightarrow \prod _{\alpha \in A}F(X_{\alpha }){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\prod _{\alpha ,\beta \in A}F(X_{\alpha }\times _{X}X_{\beta })$
must be an equalizer. For a separated presheaf, the first arrow need only be injective.
Similarly, one can define presheaves and sheaves of abelian groups, rings, modules, and so on. One can require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring, module, etc.) object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent.
Examples of sites
The discrete and indiscrete topologies
Let C be any category. To define the discrete topology, we declare all sieves to be covering sieves. If C has all fibered products, this is equivalent to declaring all families to be covering families. To define the indiscrete topology, also known as the coarse or chaotic topology,[1] we declare only the sieves of the form Hom(−, X) to be covering sieves. The indiscrete topology is generated by the pretopology that has only isomorphisms for covering families. A sheaf on the indiscrete site is the same thing as a presheaf.
The canonical topology
Let C be any category. The Yoneda embedding gives a functor Hom(−, X) for each object X of C. The canonical topology is the biggest (finest) topology such that every representable presheaf, i.e. presheaf of the form Hom(−, X), is a sheaf. A covering sieve or covering family for this site is said to be strictly universally epimorphic because it consists of the legs of a colimit cone (under the full diagram on the domains of its constituent morphisms) and these colimits are stable under pullbacks along morphisms in C. A topology that is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is called subcanonical. Subcanonical sites are exactly the sites for which every presheaf of the form Hom(−, X) is a sheaf. Most sites encountered in practice are subcanonical.
Small site associated to a topological space
We repeat the example that we began with above. Let X be a topological space. We defined O(X) to be the category whose objects are the open sets of X and whose morphisms are inclusions of open sets. Note that for an open set U and a sieve S on U, the set S(V) contains either zero or one element for every open set V. The covering sieves on an object U of O(X) are those sieves S satisfying the following condition:
• If W is the union of all the sets V such that S(V) is non-empty, then W = U.
This notion of cover matches the usual notion in point-set topology.
This topology can also naturally be expressed as a pretopology. We say that a family of inclusions {Vα $\subseteq $ U} is a covering family if and only if the union $\cup $Vα equals U. This site is called the small site associated to a topological space X.
Big site associated to a topological space
Let Spc be the category of all topological spaces. Given any family of functions {uα : Vα → X}, we say that it is a surjective family or that the morphisms uα are jointly surjective if $\cup $ uα(Vα) equals X. We define a pretopology on Spc by taking the covering families to be surjective families all of whose members are open immersions. Let S be a sieve on Spc. S is a covering sieve for this topology if and only if:
• For all Y and every morphism f : Y → X in S(Y), there exists a V and a g : V → X such that g is an open immersion, g is in S(V), and f factors through g.
• If W is the union of all the sets f(Y), where f : Y → X is in S(Y), then W = X.
Fix a topological space X. Consider the comma category Spc/X of topological spaces with a fixed continuous map to X. The topology on Spc induces a topology on Spc/X. The covering sieves and covering families are almost exactly the same; the only difference is that now all the maps involved commute with the fixed maps to X. This is the big site associated to a topological space X . Notice that Spc is the big site associated to the one point space. This site was first considered by Jean Giraud.
The big and small sites of a manifold
Let M be a manifold. M has a category of open sets O(M) because it is a topological space, and it gets a topology as in the above example. For two open sets U and V of M, the fiber product U ×M V is the open set U ∩ V, which is still in O(M). This means that the topology on O(M) is defined by a pretopology, the same pretopology as before.
Let Mfd be the category of all manifolds and continuous maps. (Or smooth manifolds and smooth maps, or real analytic manifolds and analytic maps, etc.) Mfd is a subcategory of Spc, and open immersions are continuous (or smooth, or analytic, etc.), so Mfd inherits a topology from Spc. This lets us construct the big site of the manifold M as the site Mfd/M. We can also define this topology using the same pretopology we used above. Notice that to satisfy (PT 0), we need to check that for any continuous map of manifolds X → Y and any open subset U of Y, the fibered product U ×Y X is in Mfd/M. This is just the statement that the preimage of an open set is open. Notice, however, that not all fibered products exist in Mfd because the preimage of a smooth map at a critical value need not be a manifold.
Topologies on the category of schemes
See also: List of topologies on the category of schemes
The category of schemes, denoted Sch, has a tremendous number of useful topologies. A complete understanding of some questions may require examining a scheme using several different topologies. All of these topologies have associated small and big sites. The big site is formed by taking the entire category of schemes and their morphisms, together with the covering sieves specified by the topology. The small site over a given scheme is formed by only taking the objects and morphisms that are part of a cover of the given scheme.
The most elementary of these is the Zariski topology. Let X be a scheme. X has an underlying topological space, and this topological space determines a Grothendieck topology. The Zariski topology on Sch is generated by the pretopology whose covering families are jointly surjective families of scheme-theoretic open immersions. The covering sieves S for Zar are characterized by the following two properties:
• For all Y and every morphism f : Y → X in S(Y), there exists a V and a g : V → X such that g is an open immersion, g is in S(V), and f factors through g.
• If W is the union of all the sets f(Y), where f : Y → X is in S(Y), then W = X.
Despite their outward similarities, the topology on Zar is not the restriction of the topology on Spc! This is because there are morphisms of schemes that are topologically open immersions but that are not scheme-theoretic open immersions. For example, let A be a non-reduced ring and let N be its ideal of nilpotents. The quotient map A → A/N induces a map Spec A/N → Spec A, which is the identity on underlying topological spaces. To be a scheme-theoretic open immersion it must also induce an isomorphism on structure sheaves, which this map does not do. In fact, this map is a closed immersion.
The étale topology is finer than the Zariski topology. It was the first Grothendieck topology to be closely studied. Its covering families are jointly surjective families of étale morphisms. It is finer than the Nisnevich topology, but neither finer nor coarser than the cdh and l′ topologies.
There are two flat topologies, the fppf topology and the fpqc topology. fppf stands for fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat, of finite presentation, and is quasi-finite. fpqc stands for fidèlement plate et quasi-compacte, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat. In both categories, a covering family is defined to be a family that is a cover on Zariski open subsets.[2] In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover.[3] These topologies are closely related to descent. The fpqc topology is finer than all the topologies mentioned above, and it is very close to the canonical topology.
Grothendieck introduced crystalline cohomology to study the p-torsion part of the cohomology of characteristic p varieties. In the crystalline topology, which is the basis of this theory, the underlying category has objects given by infinitesimal thickenings together with divided power structures. Crystalline sites are examples of sites with no final object.
Continuous and cocontinuous functors
There are two natural types of functors between sites. They are given by functors that are compatible with the topology in a certain sense.
Continuous functors
If (C, J) and (D, K) are sites and u : C → D is a functor, then u is continuous if for every sheaf F on D with respect to the topology K, the presheaf Fu is a sheaf with respect to the topology J. Continuous functors induce functors between the corresponding topoi by sending a sheaf F to Fu. These functors are called pushforwards. If ${\tilde {C}}$ and ${\tilde {D}}$ denote the topoi associated to C and D, then the pushforward functor is $u_{s}:{\tilde {D}}\to {\tilde {C}}$.
us admits a left adjoint us called the pullback. us need not preserve limits, even finite limits.
In the same way, u sends a sieve on an object X of C to a sieve on the object uX of D. A continuous functor sends covering sieves to covering sieves. If J is the topology defined by a pretopology, and if u commutes with fibered products, then u is continuous if and only if it sends covering sieves to covering sieves and if and only if it sends covering families to covering families. In general, it is not sufficient for u to send covering sieves to covering sieves (see SGA IV 3, Exemple 1.9.3).
Cocontinuous functors
Again, let (C, J) and (D, K) be sites and v : C → D be a functor. If X is an object of C and R is a sieve on vX, then R can be pulled back to a sieve S as follows: A morphism f : Z → X is in S if and only if v(f) : vZ → vX is in R. This defines a sieve. v is cocontinuous if and only if for every object X of C and every covering sieve R of vX, the pullback S of R is a covering sieve on X.
Composition with v sends a presheaf F on D to a presheaf Fv on C, but if v is cocontinuous, this need not send sheaves to sheaves. However, this functor on presheaf categories, usually denoted ${\hat {v}}^{*}$, admits a right adjoint ${\hat {v}}_{*}$. Then v is cocontinuous if and only if ${\hat {v}}_{*}$ sends sheaves to sheaves, that is, if and only if it restricts to a functor $v_{*}:{\tilde {C}}\to {\tilde {D}}$. In this case, the composite of ${\hat {v}}^{*}$ with the associated sheaf functor is a left adjoint of v* denoted v*. Furthermore, v* preserves finite limits, so the adjoint functors v* and v* determine a geometric morphism of topoi ${\tilde {C}}\to {\tilde {D}}$.
Morphisms of sites
A continuous functor u : C → D is a morphism of sites D → C (not C → D) if us preserves finite limits. In this case, us and us determine a geometric morphism of topoi ${\tilde {C}}\to {\tilde {D}}$. The reasoning behind the convention that a continuous functor C → D is said to determine a morphism of sites in the opposite direction is that this agrees with the intuition coming from the case of topological spaces. A continuous map of topological spaces X → Y determines a continuous functor O(Y) → O(X). Since the original map on topological spaces is said to send X to Y, the morphism of sites is said to as well.
A particular case of this happens when a continuous functor admits a left adjoint. Suppose that u : C → D and v : D → C are functors with u right adjoint to v. Then u is continuous if and only if v is cocontinuous, and when this happens, us is naturally isomorphic to v* and us is naturally isomorphic to v*. In particular, u is a morphism of sites.
See also
• Fibered category
• Lawvere–Tierney topology
Notes
1. SGA IV, II 1.1.4.
2. SGA III1, IV 6.3.
3. SGA III1, IV 6.3, Proposition 6.3.1(v).
References
• Artin, Michael (1962). Grothendieck topologies. Notes on a Seminar Spring 1962. Department of Mathematics, Harvard University. OCLC 680377057. Zbl 0208.48701.
• Demazure, Michel; Grothendieck, Alexandre, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie — 1962–64 — Schémas en groupes — (SGA 3) vol. 1. Lecture notes in mathematics (in French). Vol. 151. Springer. pp. xv+564. Zbl 0212.52810.
• Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie — 1963–64 — Théorie des topos et cohomologie étale des schémas — (SGA 4) vol. 1. Lecture notes in mathematics (in French). Vol. 269. Springer. xix+525. doi:10.1007/BFb0081551. ISBN 978-3-540-37549-4.
• Giraud, Jean (1964), "Analysis situs", Séminaire Bourbaki, 1962/63. Fasc. 3, Paris: Secrétariat mathématique, MR 0193122
• Shatz, Stephen S. (1972). Profinite groups, arithmetic, and geometry. Annals of Mathematics Studies. Vol. 67. Princeton University Press. ISBN 0-691-08017-8. MR 0347778. Zbl 0236.12002.
• Nisnevich, Yevsey A. (2012) [1989]. "The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory". In Jardine, J. F.; Snaith, V. P. (eds.). Algebraic K-theory: connections with geometry and topology. Proceedings of the NATO Advanced Study Institute held in Lake Louise, Alberta, December 7–11, 1987. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences. Vol. 279. Springer. pp. 241–342. doi:10.1007/978-94-009-2399-7_11. ISBN 978-94-009-2399-7. Zbl 0715.14009.
External links
• The birthday of Grothendieck topologies
• The birthday of Grothendieck topologies (non-archived version)
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Siu's semicontinuity theorem
In complex analysis, the Siu semicontinuity theorem implies that the Lelong number of a closed positive current on a complex manifold is semicontinuous. More precisely, the points where the Lelong number is at least some constant form a complex subvariety. This was conjectured by Harvey & King (1972) and proved by Siu (1973, 1974). Demailly (1987) generalized Siu's theorem to more general versions of the Lelong number.
References
• Demailly, Jean-Pierre (1987), "Nombres de Lelong généralisés, théorèmes d'intégralité et d'analyticité", Acta Mathematica, 159 (3): 153–169, doi:10.1007/BF02392558, ISSN 0001-5962, MR 0908144
• Harvey, F. Reese; King, James R. (1972), "On the structure of positive currents", Inventiones Mathematicae, 15: 47–52, doi:10.1007/BF01418641, ISSN 0020-9910, MR 0296348
• Siu, Yum-Tong (1973), "Analyticity of sets associated to Lelong numbers and the extension of meromorphic maps", Bulletin of the American Mathematical Society, 79 (6): 1200–1205, doi:10.1090/S0002-9904-1973-13378-6, ISSN 0002-9904, MR 0330505
• Siu, Yum-Tong (1974), "Analyticity of sets associated to Lelong numbers and the extension of closed positive currents", Inventiones Mathematicae, 27 (1–2): 53–156, doi:10.1007/BF01389965, ISSN 0020-9910, MR 0352516
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Wikipedia
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Sivaguru S. Sritharan
Sivaguru S. Sritharan (also known as S. S. Sritharan) is an American aerodynamicist and mathematician.[1]
Professor
Sivaguru S. Sritharan
Dr. Sivaguru S. Sritharan
Sritharan served in civilian universities such as University of Southern California and University of Wyoming as faculty member and head of the department and also in the Department of Defense (U. S. Navy and U. S. Air Force) in various capacities ranging from scientist to leadership roles, and also held visiting positions at several international institutions.[1]
He served as the Vice Chancellor at the Ramaiah University of Applied Sciences in Bengaluru, India.[1]
Education
Sritharan had his high schooling at Jaffna Central College. He then joined at University of Sri Lanka (Peradeniya) and obtained a BSc (Honors) degree in mechanical engineering. He obtained a Master of Science degree in aeronautics and astronautics from University of Washington and a master's degree and Ph.D. in applied mathematics from University of Arizona.[2][1]
Career
Sritharan served as the first Provost and Vice Chancellor of the Air Force Institute of Technology at Dayton, Ohio and as the Dean of the Graduate School of Engineering and Applied Sciences at the Naval Postgraduate School, Monterey, California.[1]
He was a Professor and Head of the Department of Mathematics at University of Wyoming and Head of the Science and Technology Branch at the Naval Information Warfare Systems Command in San Diego.[1]
Contributions
Sritharan is known for his research contributions in rigorous mathematical theory, optimal control and stochastic analysis of fluid mechanics and magneto-hydrodynamics.[3][4]
His notable contributions include:
1. Developing dynamic programming method for the equations of fluid dynamics. This subject is closely related to reinforcement learning in the language of machine learning.[5]
2. First complete proof of the Pontryagin’s Maximum Principle for fluid dynamic equations with state constraints, as a joint work with UCLA mathematician Hector. O. Fattorini.[6]
3. Developing robust (H-infinity) control theory for fluid dynamics as a joint work with Romanian mathematician Viorel P. Barbu.[7]
4. First successful rigorous theory establishing a direct stochastic analogy to the famous Jacques-Louis Lions and G. Prodi (1959) on existence and uniqueness theorem for the two dimensional Navier-Stokes equation as a joint work with J. L. Menaldi utilizing a subtle local monotonicity property.[8]
5. Proving Large Deviation Principle for stochastic Navier-Stokes equation as a joint work with P. Sundar to estimate the probability of rare events.[9]
Bibliography
• Sritharan, S.S. (2019), Invariant Manifold Theory for Hydrodynamic Transition, Courier Dover Publications, ISBN 9780486828282
• Sritharan, S.S. (1998), Optimal Control of Viscous Flow, SIAM, ISBN 9780898714067
References
1. "Vice Chancellor". Ramaiah University of Applied Sciences. Retrieved July 19, 2020.
2. "SIVAGURU S. SRITHARAN". ContactOut. Retrieved July 19, 2020.
3. Sritharan, S.S. (2019), Invariant Manifold Theory for Hydrodynamic Transition, Courier Dover Publications, ISBN 9780486828282
4. Sritharan, S.S. (1998), Optimal Control of Viscous Flow, SIAM, ISBN 9780898714067
5. Sritharan, S.S. (1991), ""Dynamic Programming of the Navier-Stokes Equations," in Systems and Control Letters, Vol. 16, No. 4, pp. 299-307", Systems & Control Letters, Elsevier, 16 (4): 299–307, doi:10.1016/0167-6911(91)90020-F, retrieved July 20, 2020
6. Fattorini, H. O.; Sritharan, S.S. (1994), ""Necessary and Sufficient Conditions for Optimal Controls in Viscous Flow," Proceedings of the Royal Society of Edinburgh, Series A, Vol. 124A, pp. 211-251", Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Proceedings of the Royal Society, 124 (2): 211–251, doi:10.1017/S0308210500028444, S2CID 18018847, retrieved July 20, 2020
7. Barbu, V.; Sritharan, S.S. (1998), "H-infinity-control theory of fluid dynamics," Proceedings of The Royal Society of London, Series A, pp. 3009-3033, Vol. 356, No. 1979, November 1998 (PDF), Proceedings of the Royal Society, retrieved July 20, 2020
8. Menaldi, J. L.; Sritharan, S.S. (2002), "Stochastic 2-D Navier-Stokes equation," Applied Mathematics and Optimization, 46, 2002, pp. 31-53, Wayne State University, retrieved July 20, 2020
9. Sundar, P.; Sritharan, S.S. (2006), "Large Deviations for Two-dimensional Stochastic Navier-Stokes Equations", Stochastic Processes, Theory and Applications, Vol. 116, Issue 11, (2006), 1636-1659 (PDF), Elsevier, retrieved July 20, 2020
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6-sphere coordinates
In mathematics, 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the 3D Cartesian coordinates across the unit 2-sphere $x^{2}+y^{2}+z^{2}=1$. They are so named because the loci where one coordinate is constant form spheres tangent to the origin from one of six sides (depending on which coordinate is held constant and whether its value is positive or negative). They have nothing whatsoever to do with the 6-sphere, which is an object of considerable interest in its own right.
The three coordinates are
$u={\frac {x}{x^{2}+y^{2}+z^{2}}},\quad v={\frac {y}{x^{2}+y^{2}+z^{2}}},\quad w={\frac {z}{x^{2}+y^{2}+z^{2}}}.$
Since inversion is its own inverse, the equations for x, y, and z in terms of u, v, and w are similar:
$x={\frac {u}{u^{2}+v^{2}+w^{2}}},\quad y={\frac {v}{u^{2}+v^{2}+w^{2}}},\quad z={\frac {w}{u^{2}+v^{2}+w^{2}}}.$
This coordinate system is $R$-separable for the 3-variable Laplace equation.
See also
• Multiplicative inverse (for 1-dimensional version)
• Y-Δ transform (unrelated, but similar formula for comparison)
• 6-sphere
References
• Moon, P. and Spencer, D. E. 6-sphere Coordinates. Fig. 4.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 122–123, 1988.
• Weisstein, Eric W. "6-sphere coordinates". MathWorld.
• Six-Sphere Coordinates by Michael Schreiber, the Wolfram Demonstrations Project.
Orthogonal coordinate systems
Two dimensional
• Cartesian
• Polar (Log-polar)
• Parabolic
• Bipolar
• Elliptic
Three dimensional
• Cartesian
• Cylindrical
• Spherical
• Parabolic
• Paraboloidal
• Oblate spheroidal
• Prolate spheroidal
• Ellipsoidal
• Elliptic cylindrical
• Toroidal
• Bispherical
• Bipolar cylindrical
• Conical
• 6-sphere
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Hexafoil
The hexafoil is a design with six-fold dihedral symmetry composed from six vesica piscis lenses arranged radially around a central point, often shown enclosed in a circumference of another six lenses. It is also sometimes known as a "daisy wheel".[1] A second, quite different, design is also sometimes referred to by this name; see alternate symbol.
The design is found as a rosette ornament in artwork dating back to at least the Late Bronze Age.[2]
Construction
The seven overlapping circles grid forms a triangular lattice, seen here with hexagonal rings of 1, 7, 19, 37, 61, 91 circles.[3]
The pattern figure can be drawn by pen and compass, by creating seven interlinking circles of the same diameter touching the previous circle's center. The second circle is centered at any point on the first circle. All following circles are centered on the intersection of two other circles.
The design is sometimes expanded into a regular overlapping circles grid. Bartfeld (2005) describes the construction: "This design consists of circles having a 1-[inch] radius, with each point of intersection serving as a new center. The design can be expanded ad infinitum depending upon the number of times the odd-numbered points are marked off."[4]
Usage
The hexafoil has been very widely used throughout European folk art for a very long period of time. It is attested from at least the beginning of the Late Bronze Age,[2] represented, for example, on ornamental golden disks found in Shaft Grave III at Mycenae (16th century BC).[5] It is also found in some Cantabrian stelae, dated to the Iron Age, as well as Norwegian bronze kettles from the same period[6]
The six-petal rosette is common in 17th to 20th century folk art throughout Europe.
In Portugal, it is common to find it in medieval churches and cathedrals, as the engraved signature of a mason; but also as decoration and symbol of protection on the chimneys of old houses in Alentejo (at times together with the lauburu, or with the pentagram).
In Galicia (Spain) hexafoils are found since the Iron Age in torc terminals and decoration, and is still used in folk art.[7]
It can also be found in the Pyrenees (Cantabria, Navarre, Aragon).[8]
In the United Kingdom the hexafoil is commonly found on churches, but also in barns and private buildings, as well as on cross slabs.[6] The use of the hexafoil as a folk magic symbol was brought from the United Kingdom to Australia by settlers, where six leaf designs with concentric circles have been found in homes and occasionally in public buildings to serve as a sign of protection.[9][10]
The hexafoil was also widely used on gravestones in Colonial America, especially popular in parts of Connecticut, Massachusetts, and Pennsylvania. The design was commonly used from the later 17th century until the early 19th century.
The design is also known as "Sun of the Alps" (Sole delle Alpi) in Italy from its widespread use in alpine folk art.[11] It resembles a pattern often found in that area on buildings.[12] It is used in the coat of arms of Lecco Province. It has also been used as the emblem of Padanian nationalism in northern Italy since the 1990s. In 2001, Editoriale Nord, the publishing company of La Padania, registered the green-on-white design as a trademark.[13]
In Norway it can mostly be found on wooden objects, such as beer bowls, clothes smoothing boards, milk butts, wooden chests, beds, and so on, but it can also be found on the doors of buildings. In Norwegian it's sometimes known as "Olavsrose" (rose of Olaf), although that name is used for another symbol as well.[6]
In Lithuania the hexafoil was found on wooden beer bowls, on spindles, but also on other wooden objects. It is known as "little sun" (saulute) in Lithuanian.[6]
In the Tatra mountains, southeastern Poland and western Ukraine, the mark was commonly carved on roof beams inside peasant huts. In Ukraine it was known as "the symbol of Perun" (Peruna znak) and "the thunder mark" (gromovoi znak).[14]
In the Russian North the hexafoil was carved near the outside roof of peasant houses to protect them against lightning. The symbol was known as the thunder sign (gromovoi znak) or the thunder wheel (gromovoe koleso), and was associated with the thunder god Perun.[15]
Gallery
• Khachkar with hexafoils, swastikas and sauwastikas in Sanahin, Armenia
• Ornamented gold disks from Shaft Grave III at Mycenae (16th century BC), (Archaeological Museum, Istanbul)
• Nepali sicle (hasiya) with its carrier (khurpeto) carved with the Aryan Star/Flower of Life symbol
• Cup with Flower of Life motif from Idalion, Cyprus, 8th-7th century BCE (Museum of Louvre, Paris)
• Flower of Life on a mosaic from Apaša (Ephesos)
• Floor decoration from the palace of King Ashurbanipal, Northern Iraq, 645 BCE (Museum of Louvre, Paris)
• Mosaic floor from a bathhouse in Herod's Palace, Jerusalem, 1st century BCE
• Roman-era mosaic, Domus dell'Ortaglia, Brescia, 2nd century CE
• Cantabrian stele at the Monastery of Iranzu, Navarre
• Selection of carvings from the hillfort of Santa Trega, Galicia (La Tène period, c. 1st century BC)
• Detail of an Imperial Guardian Lion at the Gate of Supreme Harmony, Forbidden City, Beijing, China
• Facade of the medieval church in Galdo degli Alburni, Province of Salerno
• Hexafoils featured prominently on a Colonial New England gravestone carved by Obadiah Wheeler in Franklin Connecticut.
• Facade of the church in San Domenico, Lucera, Province of Foggia (ca. 1300)
• ceiling beam displayed at the Rural Architecture Museum of Sanok (Poland) dated to 1681.
• Masonry in casa Federici, Erbanno, Val Camonica
• Wall painting on the facade of farmhouse Runcata in St. Ulrich in Gröden, Ortisei, Val Gardena, South Tyrol
• Green-on-white "Sun of the Alps" as used by the Lega Nord and in Padanian nationalism
• Perun's sign as used within Slavic Native Faith
Origin
The origin and meaning of the symbol are not known, but many researchers have independently suggested that it is of religious origin,[16] and very likely served as a protective symbol.[17][1][15] There are two main theories for its meaning and origin.
Solar symbol
Peralta Labrador (1989) cites a proposal according to which the design in the La Tène (Celtic) period was a solar symbol associated with the god Taranis.[18] Other researchers have also described it as a solar symbol,[19][1] but no reasoning for this has been given. However, the Lithuanian ("little sun") and Italian ("sun of the Alps") names do suggest a solar origin.
Thunder wheel
Garshol (2021) suggests that the rosette is actually a wheel with spokes, and that it originally signified the Proto-Indo-European thunder god Perkwunos, later becoming associated with his various incarnations, such as Perun, Tarḫunz, Taranis, Thor and Jupiter. The Russian and Ukrainian names of the symbol, as well as other more involved arguments, are given as rationale.[6]
Alternate symbol
The name hexafoil is sometimes also used to refer to a different geometric design that is used as a traditional element of Gothic architecture,[20] created by overlapping six circular arcs to form a flower-like image.[21][22] The hexafoil design is modeled after the six petal lily, for its symbolism of purity and relation to the Trinity.[23] The hexafoil form is created from a series of compound units, and exists as a more complex variation of the same extruded figure.[24] Other forms similar to the hexafoil include the trefoil, quatrefoil, and cinquefoil.[25]
The other hexafoil design is implemented in various Gothic buildings constructed in the 12th through 16th century. The traditional design is used in cloisters, triforiums and stained glass windows of famous buildings such as Notre-Dame, Salisbury Cathedral, and Regensburg Cathedral.[26] Stone cut-out hexafoils are displayed in a plate tracery style in the Salisbury Cathedral, creating a pattern along the triforium.[27]
It can also be see as a framing design in Bible moralisée.[28] They are often rendered in red, blue, gold or vibrant orange and surround biblical scenes in the bible.[28][29] The hexafoil style of framing was often used in conjunction with architectural framing to provide the text with more depth, creativity, invention, and volume.[28] Old Testament illustrations were surrounded by hexafoil frames while moralization depictions favored architectural frames.[29]
See also
Wikimedia Commons has media related to Flower of Life.
• Overlapping circles grid
• Sudarshana Chakra
• Triquetra
• Triskelion
• Foil (architecture)
References
1. Easton, Timothy (2016). "Apotropaic symbols and other measures for protecting buildings against misfortune". In Hutton, Ronald (ed.). Physical Evidence for Ritual Acts, Sorcery, and Witchcraft in Christian Britain. Palgrave Historical Studies in Witchcraft and Magic. pp. 39–67. ISBN 978-1-137-44482-0.
2. Høyrup, J. (2000). "Geometrical Patterns in the Pre-classical Greek Area. Prospecting the Borderland between Decoration, Art, and Structural Inquiry" (PDF). Revue d'histoire des mathématiques. 6 (1): 5–58.
3. Islamic Art and Geometric Design: Activities for Learning
4. Bartfeld, Martha (2005). How to Create Sacred Geometry Mandalas. Santa Fe, NM: Mandalart Creations. p. 35., citing Drunvalo Melchizedek, The Ancient Secret of the Flower of Life (1999). The attribution of the term "Flower of Life" to Melchizedek (1999) is also found in Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, Inc. (published May 14, 2002), pp. 43 and 873–874, ISBN 1-57955-008-8 and in Weisstein, Eric W. (12 December 2002), CRC Concise Encyclopedia of Mathematics, Second Edition, CRC Press (published 2002), p. 1079, ISBN 1420035223.
5. Excavated by Heinrich Schliemann in 1876. Schliemann, Mykenae (1878), pp. 165–172.
6. Garshol, Lars Marius (2021). "Olav's Rose, Perun's Mark, Taranis's Wheel". Peregrinations. 7 (4): 121–151.
7. Romero, Bieito (2019). Simboloxía Máxica en Galicia. A Coruña: Baía Edicións. pp. 70–95. ISBN 978-84-9995-329-8.
8. Ariel Golan, Prehistoric Religion: Mythology, Symbolism, (2003), p. 54.
9. Mysterious hexafoil markings in Australian homes point to hidden magical past, Nicole Dyer and Damien Larkins, ABC News Online, 2017-02-22
10. Evans, Ian J. "Defence Against the Devil: Apotropaic Marks in Australia". {{cite journal}}: Cite journal requires |journal= (help)
11. "Il significato del simbolo del Sole delle Alpi" (in Italian). Lega Nord. Archived from the original on January 12, 2014. Retrieved December 1, 2014.
12. Ivano Dorboló (June 6, 2010). "The church of S.Egidio and the Sun of the Alps symbol". Storia di Confine – Valli di Natisone. Retrieved November 9, 2015.
13. Ufficio Italiano Brevetti e Marchi, registrazione del simbolo del Sole delle Alpi. According to Rosanna Sapori of Radio Padania Libera, the trademark as of August 2010 was owned by Silvio Berlusconi, who would have obtained it in exchange for the bailout of the bankrupt Credieuronord bank «Vi racconto perché Bossi è prigioniero di Berlusconi» Il Riformista 28 August 2010.
14. Areta Kovalska (October 10, 2018). "A Protection Symbol for the Home: The Six-Petal Rosette on the Crossbeams of Galicia". Forgotten Galicia. Retrieved 2021-12-27.
15. Ivanits, Linda J. (1989). Russian Folk Belief. M. E. Sharpe. p. 17.
16. Simonett, Christoph (1965). Die Bauernhäuser des Kantons Graubünden. Verlag Schweizerische Gesellschaft für Volkskunde. p. 224-226.
17. Weiser-Aall, Lily (1947). "Magiske tegn på norske trekar". By og Bygd. 5: 127.
18. Eduardo Peralta Labrador, Las estelas discoideas de Cantabria in: Estelas discoideas de la Peninsula Iberica (1989), pp 425–466, citing the opinion of José María Blázquez Martínez.
19. Østmoe Kostveit, Åsta (1997). Kors i kake, skurd i tre (in Norwegian). Landbruksforlaget. p. 58.
20. Hartop, Christopher; Norton, Jonathan (2008), Geometry and the silversmith: the Domcha Collection, John Adamson, ISBN 9780952432289, The trefoil, quatrefoil, hexafoil and octofoil, essential elements of Gothic architecture, all figure in medieval silver.
21. Passmore, Augustine C. (1904), Handbook of Technical Terms Used in Architecture and Building and Their Allied Trades and Subjects, Scott, Greenwood, and Company, p. 178, A geometrical figure used in tracery; it is composed of six lobes or parts of circles joining each other.
22. Rugoff, Milton (1976), The Britannica encyclopedia of American art: a special educational supplement to the Encyclopædia Britannica, Encyclopædia Britannica Educational Corp., p. 636, A geometrical figure with six lobes, used as the form of a silver platter or a wooden decorative panel.
23. Laxton, William (1856). The Civil Engineer and Architect's Journal. Published for the proprietor.
24. Griffith, William Pettit (1845). The Natural System of Architecture, as Opposed to the Artificial System of the Present Day. Gilbert and Rivington.
25. Chiffriller, Joe (2002). "Tips & Tricks to Gothic Geometry" (PDF). New York Carver – via PBworks.
26. "The World's Best Photos of hexafoil - Flickr Hive Mind". hiveminer.com. Retrieved 2018-11-12.
27. "Salisbury Cathedral". Khan Academy. Retrieved 2018-11-17.
28. Husband, Timothy B. (2008). The Art of Illumination: The Limbourg Brothers and the Belles Heures of Jean de France, Duc de Berry. Metropolitan Museum of Art, Yale University Press. p. 282. ISBN 9781588392947. hexafoil in architecture.
29. "Microfilms and Fascimilies Database // Medieval Institute Library // University of Notre Dame". medieval-microfilms-and-facsimiles.library.nd.edu. Archived from the original on 2019-07-28. Retrieved 2018-11-12.
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Wikipedia
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Ice-type model
In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice.[1] Variants have been proposed as models of certain ferroelectric[2] and antiferroelectric[3] crystals.
In 1967, Elliott H. Lieb found the exact solution to a two-dimensional ice model known as "square ice".[4] The exact solution in three dimensions is only known for a special "frozen" state.[5]
Description
An ice-type model is a lattice model defined on a lattice of coordination number 4. That is, each vertex of the lattice is connected by an edge to four "nearest neighbours". A state of the model consists of an arrow on each edge of the lattice, such that the number of arrows pointing inwards at each vertex is 2. This restriction on the arrow configurations is known as the ice rule. In graph theoretic terms, the states are Eulerian orientations of an underlying 4-regular undirected graph. The partition function also counts the number of nowhere-zero 3-flows.[6]
For two-dimensional models, the lattice is taken to be the square lattice. For more realistic models, one can use a three-dimensional lattice appropriate to the material being considered; for example, the hexagonal ice lattice is used to analyse ice.
At any vertex, there are six configurations of the arrows which satisfy the ice rule (justifying the name "six-vertex model"). The valid configurations for the (two-dimensional) square lattice are the following:
The energy of a state is understood to be a function of the configurations at each vertex. For square lattices, one assumes that the total energy $E$ is given by
$E=n_{1}\epsilon _{1}+n_{2}\epsilon _{2}+\ldots +n_{6}\epsilon _{6},$
for some constants $\epsilon _{1},\ldots ,\epsilon _{6}$, where $n_{i}$ here denotes the number of vertices with the $i$th configuration from the above figure. The value $\epsilon _{i}$ is the energy associated with vertex configuration number $i$.
One aims to calculate the partition function $Z$ of an ice-type model, which is given by the formula
$Z=\sum \exp(-E/k_{\rm {B}}T),$
where the sum is taken over all states of the model, $E$ is the energy of the state, $k_{\rm {B}}$ is the Boltzmann constant, and $T$ is the system's temperature.
Typically, one is interested in the thermodynamic limit in which the number $N$ of vertices approaches infinity. In that case, one instead evaluates the free energy per vertex $f$ in the limit as $N\to \infty $, where $f$ is given by
$f=-k_{\rm {B}}TN^{-1}\log Z.$
Equivalently, one evaluates the partition function per vertex $W$ in the thermodynamic limit, where
$W=Z^{1/N}.$
The values $f$ and $W$ are related by
$f=-k_{\rm {B}}T\log W.$
Physical justification
Several real crystals with hydrogen bonds satisfy the ice model, including ice[1] and potassium dihydrogen phosphate KH
2
PO
4
[2] (KDP). Indeed, such crystals motivated the study of ice-type models.
In ice, each oxygen atom is connected by a bond to four other oxygens, and each bond contains one hydrogen atom between the terminal oxygens. The hydrogen occupies one of two symmetrically located positions, neither of which is in the middle of the bond. Pauling argued[1] that the allowed configuration of hydrogen atoms is such that there are always exactly two hydrogens close to each oxygen, thus making the local environment imitate that of a water molecule, H
2
O. Thus, if we take the oxygen atoms as the lattice vertices and the hydrogen bonds as the lattice edges, and if we draw an arrow on a bond which points to the side of the bond on which the hydrogen atom sits, then ice satisfies the ice model. Similar reasoning applies to show that KDP also satisfies the ice model.
In recent years, ice-type models have been explored as descriptions of pyrochlore spin ice[7] and artificial spin ice systems,[8][9] in which geometrical frustration in the interactions between bistable magnetic moments ("spins") leads to "ice-rule" spin configurations being favoured. Recently such analogies have been extended to explore the circumstances under which spin-ice systems may be accurately described by the Rys F-model.[10][11][12][13]
Specific choices of vertex energies
On the square lattice, the energies $\epsilon _{1},\ldots ,\epsilon _{6}$ associated with vertex configurations 1-6 determine the relative probabilities of states, and thus can influence the macroscopic behaviour of the system. The following are common choices for these vertex energies.
The ice model
When modeling ice, one takes $\epsilon _{1}=\epsilon _{2}=\ldots =\epsilon _{6}=0$, as all permissible vertex configurations are understood to be equally likely. In this case, the partition function $Z$ equals the total number of valid states. This model is known as the ice model (as opposed to an ice-type model).
The KDP model of a ferroelectric
Slater[2] argued that KDP could be represented by an ice-type model with energies
$\epsilon _{1}=\epsilon _{2}=0,\epsilon _{3}=\epsilon _{4}=\epsilon _{5}=\epsilon _{6}>0$
For this model (called the KDP model), the most likely state (the least-energy state) has all horizontal arrows pointing in the same direction, and likewise for all vertical arrows. Such a state is a ferroelectric state, in which all hydrogen atoms have a preference for one fixed side of their bonds.
Rys F model of an antiferroelectric
The Rys $F$ model[3] is obtained by setting
$\epsilon _{1}=\epsilon _{2}=\epsilon _{3}=\epsilon _{4}>0,\epsilon _{5}=\epsilon _{6}=0.$
The least-energy state for this model is dominated by vertex configurations 5 and 6. For such a state, adjacent horizontal bonds necessarily have arrows in opposite directions and similarly for vertical bonds, so this state is an antiferroelectric state.
The zero field assumption
If there is no ambient electric field, then the total energy of a state should remain unchanged under a charge reversal, i.e. under flipping all arrows. Thus one may assume without loss of generality that
$\epsilon _{1}=\epsilon _{2},\quad \epsilon _{3}=\epsilon _{4},\quad \epsilon _{5}=\epsilon _{6}$
This assumption is known as the zero field assumption, and holds for the ice model, the KDP model, and the Rys F model.
History
The ice rule was introduced by Linus Pauling in 1935 to account for the residual entropy of ice that had been measured by William F. Giauque and J. W. Stout.[14] The residual entropy, $S$, of ice is given by the formula
$S=k_{\rm {B}}\log Z=k_{\rm {B}}\,N\,\log W,$
where $k_{\rm {B}}$ is the Boltzmann constant, $N$ is the number of oxygen atoms in the piece of ice, which is always taken to be large (the thermodynamic limit) and $Z=W^{N}$ is the number of configurations of the hydrogen atoms according to Pauling's ice rule. Without the ice rule we would have $W=4$ since the number of hydrogen atoms is $2N$ and each hydrogen has two possible locations. Pauling estimated that the ice rule reduces this to $W=1.5$, a number that would agree extremely well with the Giauque-Stout measurement of $S$. It can be said that Pauling's calculation of $S$ for ice is one of the simplest, yet most accurate applications of statistical mechanics to real substances ever made. The question that remained was whether, given the model, Pauling's calculation of $W$, which was very approximate, would be sustained by a rigorous calculation. This became a significant problem in combinatorics.
Both the three-dimensional and two-dimensional models were computed numerically by John F. Nagle in 1966[15] who found that $W=1.50685\pm 0.00015$ in three-dimensions and $W=1.540\pm 0.001$ in two-dimensions. Both are amazingly close to Pauling's rough calculation, 1.5.
In 1967, Lieb found the exact solution of three two-dimensional ice-type models: the ice model,[4] the Rys $F$ model,[16] and the KDP model.[17] The solution for the ice model gave the exact value of $W$ in two-dimensions as
$W_{2D}=\left({\frac {4}{3}}\right)^{3/2}=1.5396007....$
which is known as Lieb's square ice constant.
Later in 1967, Bill Sutherland generalised Lieb's solution of the three specific ice-type models to a general exact solution for square-lattice ice-type models satisfying the zero field assumption.[18]
Still later in 1967, C. P. Yang[19] generalised Sutherland's solution to an exact solution for square-lattice ice-type models in a horizontal electric field.
In 1969, John Nagle derived the exact solution for a three-dimensional version of the KDP model, for a specific range of temperatures.[5] For such temperatures, the model is "frozen" in the sense that (in the thermodynamic limit) the energy per vertex and entropy per vertex are both zero. This is the only known exact solution for a three-dimensional ice-type model.
Relation to eight-vertex model
The eight-vertex model, which has also been exactly solved, is a generalisation of the (square-lattice) six-vertex model: to recover the six-vertex model from the eight-vertex model, set the energies for vertex configurations 7 and 8 to infinity. Six-vertex models have been solved in some cases for which the eight-vertex model has not; for example, Nagle's solution for the three-dimensional KDP model[5] and Yang's solution of the six-vertex model in a horizontal field.[19]
Boundary conditions
This ice model provide an important 'counterexample' in statistical mechanics: the bulk free energy in the thermodynamic limit depends on boundary conditions.[20] The model was analytically solved for periodic boundary conditions, anti-periodic, ferromagnetic and domain wall boundary conditions. The six vertex model with domain wall boundary conditions on a square lattice has specific significance in combinatorics, it helps to enumerate alternating sign matrices. In this case the partition function can be represented as a determinant of a matrix (whose dimension is equal to the size of the lattice), but in other cases the enumeration of $W$ does not come out in such a simple closed form.
Clearly, the largest $W$ is given by free boundary conditions (no constraint at all on the configurations on the boundary), but the same $W$ occurs, in the thermodynamic limit, for periodic boundary conditions,[21] as used originally to derive $W_{2D}$.
3-colorings of a lattice
The number of states of an ice type model on the internal edges of a finite simply connected union of squares of a lattice is equal to one third of the number of ways to 3-color the squares, with no two adjacent squares having the same color. This correspondence between states is due to Andrew Lenard and is given as follows. If a square has color i = 0, 1, or 2, then the arrow on the edge to an adjacent square goes left or right (according to an observer in the square) depending on whether the color in the adjacent square is i+1 or i−1 mod 3. There are 3 possible ways to color a fixed initial square, and once this initial color is chosen this gives a 1:1 correspondence between colorings and arrangements of arrows satisfying the ice-type condition.
See also
• Eight-vertex model
Notes
1. Pauling, L. (1935). "The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement". Journal of the American Chemical Society. 57 (12): 2680–2684. doi:10.1021/ja01315a102.
2. Slater, J. C. (1941). "Theory of the Transition in KH2PO4". Journal of Chemical Physics. 9 (1): 16–33. Bibcode:1941JChPh...9...16S. doi:10.1063/1.1750821.
3. Rys, F. (1963). "Über ein zweidimensionales klassisches Konfigurationsmodell". Helvetica Physica Acta. 36: 537.
4. Lieb, E. H. (1967). "Residual Entropy of Square Ice". Physical Review. 162 (1): 162–172. Bibcode:1967PhRv..162..162L. doi:10.1103/PhysRev.162.162.
5. Nagle, J. F. (1969). "Proof of the first order phase transition in the Slater KDP model". Communications in Mathematical Physics. 13 (1): 62–67. Bibcode:1969CMaPh..13...62N. doi:10.1007/BF01645270. S2CID 122432926.
6. Mihail, M.; Winkler, P. (1992). "On the Number of Eularian Orientations of a Graph". SODA '92 Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics. pp. 138–145. ISBN 978-0-89791-466-6.
7. Bramwell, Steven T; Harris, Mark J (2020-09-02). "The history of spin ice". Journal of Physics: Condensed Matter. 32 (37): 374010. Bibcode:2020JPCM...32K4010B. doi:10.1088/1361-648X/ab8423. ISSN 0953-8984. PMID 32554893.
8. Wang, R. F.; Nisoli, C.; Freitas, R. S.; Li, J.; McConville, W.; Cooley, B. J.; Lund, M. S.; Samarth, N.; Leighton, C.; Crespi, V. H.; Schiffer, P. (January 2006). "Artificial 'spin ice' in a geometrically frustrated lattice of nanoscale ferromagnetic islands". Nature. 439 (7074): 303–306. arXiv:cond-mat/0601429. Bibcode:2006Natur.439..303W. doi:10.1038/nature04447. ISSN 1476-4687. PMID 16421565. S2CID 1462022.
9. Perrin, Yann; Canals, Benjamin; Rougemaille, Nicolas (December 2016). "Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice". Nature. 540 (7633): 410–413. arXiv:1610.01316. Bibcode:2016Natur.540..410P. doi:10.1038/nature20155. ISSN 1476-4687. PMID 27894124. S2CID 4409371.
10. Jaubert, L. D. C.; Lin, T.; Opel, T. S.; Holdsworth, P. C. W.; Gingras, M. J. P. (2017-05-19). "Spin ice Thin Film: Surface Ordering, Emergent Square ice, and Strain Effects". Physical Review Letters. 118 (20): 207206. arXiv:1608.08635. Bibcode:2017PhRvL.118t7206J. doi:10.1103/PhysRevLett.118.207206. ISSN 0031-9007. PMID 28581768. S2CID 118688211.
11. Arroo, Daan M.; Bramwell, Steven T. (2020-12-22). "Experimental measures of topological sector fluctuations in the F-model". Physical Review B. 102 (21): 214427. Bibcode:2020PhRvB.102u4427A. doi:10.1103/PhysRevB.102.214427. ISSN 2469-9950. S2CID 222290448.
12. Nisoli, Cristiano (2020-11-01). "Topological order of the Rys F-model and its breakdown in realistic square spin ice: Topological sectors of Faraday loops". Europhysics Letters. 132 (4): 47005. arXiv:2004.02107. Bibcode:2020EL....13247005N. doi:10.1209/0295-5075/132/47005. ISSN 0295-5075. S2CID 221891692.
13. Schánilec, V.; Brunn, O.; Horáček, M.; Krátký, S.; Meluzín, P.; Šikola, T.; Canals, B.; Rougemaille, N. (2022-07-07). "Approaching the Topological Low-Energy Physics of the F Model in a Two-Dimensional Magnetic Lattice". Physical Review Letters. 129 (2): 027202. Bibcode:2022PhRvL.129b7202S. doi:10.1103/PhysRevLett.129.027202. ISSN 0031-9007. PMID 35867462. S2CID 250378329.
14. Giauque, W. F.; Stout, Stout (1936). "The entropy of water and third law of thermodynamics. The heat capacity of ice from 15 to 273K". Journal of the American Chemical Society. 58 (7): 1144–1150. Bibcode:1936JAChS..58.1144G. doi:10.1021/ja01298a023.
15. Nagle, J. F. (1966). "Lattice Statistics of Hydrogen Bonded Crystals. I. The Residual Entropy of Ice". Journal of Mathematical Physics. 7 (8): 1484–1491. Bibcode:1966JMP.....7.1484N. doi:10.1063/1.1705058.
16. Lieb, E. H. (1967). "Exact Solution of the Problem of the Entropy of Two-Dimensional Ice". Physical Review Letters. 18 (17): 692–694. Bibcode:1967PhRvL..18..692L. doi:10.1103/PhysRevLett.18.692.
17. Lieb, E. H. (1967). "Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric". Physical Review Letters. 19 (3): 108–110. Bibcode:1967PhRvL..19..108L. doi:10.1103/PhysRevLett.19.108.
18. Sutherland, B. (1967). "Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals". Physical Review Letters. 19 (3): 103–104. Bibcode:1967PhRvL..19..103S. doi:10.1103/PhysRevLett.19.103.
19. Yang, C. P. (1967). "Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals". Physical Review Letters. 19 (3): 586–588. Bibcode:1967PhRvL..19..586Y. doi:10.1103/PhysRevLett.19.586.
20. Korepin, V.; Zinn-Justin, P. (2000). "Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions". Journal of Physics A. 33 (40): 7053–7066. arXiv:cond-mat/0004250. Bibcode:2000JPhA...33.7053K. doi:10.1088/0305-4470/33/40/304. S2CID 2143060.
21. Brascamp, H. J.; Kunz, H.; Wu, F. Y. (1973). "Some rigorous results for the vertex model in statistical mechanics". Journal of Mathematical Physics. 14 (12): 1927–1932. Bibcode:1973JMP....14.1927B. doi:10.1063/1.1666271.
Further reading
• Lieb, E.H.; Wu, F.Y. (1972), "Two Dimensional Ferroelectric Models", in C. Domb; M. S. Green (eds.), Phase Transitions and Critical Phenomena, vol. 1, New York: Academic Press, pp. 331–490
• Baxter, Rodney J. (1982), Exactly solved models in statistical mechanics (PDF), London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-083180-7, MR 0690578, archived from the original (PDF) on 2021-04-14, retrieved 2012-08-12
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Wikipedia
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6
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.[1]
← 5 6 7 →
−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 →
Cardinalsix
Ordinal6th
(sixth)
Numeral systemsenary
Factorization2 × 3
Divisors1, 2, 3, 6
Greek numeralϚ´
Roman numeralVI, vi, ↅ
Greek prefixhexa-/hex-
Latin prefixsexa-/sex-
Binary1102
Ternary203
Senary106
Octal68
Duodecimal612
Hexadecimal616
Greekστ (or ΣΤ or ς)
Arabic, Kurdish, Sindhi, Urdu٦
Persian۶
Amharic፮
Bengali৬
Chinese numeral六，陸
Devanāgarī६
Gujarati૬
Hebrewו
Khmer៦
Thai๖
Telugu౬
Tamil௬
Saraiki٦
Malayalam൬
In mathematics
Six is the smallest positive integer which is neither a square number nor a prime number. It is the second smallest composite number after four, equal to the sum and the product of its three proper divisors (1, 2 and 3).[1] As such, 6 is the only number that is both the sum and product of three consecutive positive numbers. It is the smallest perfect number, which are numbers that are equal to their aliquot sum, or sum of their proper divisors.[1][2] It is also the largest of the four all-Harshad numbers (1, 2, 4, and 6).[3]
6 is a pronic number and the only semiprime to be.[4] It is the first discrete biprime (2 × 3)[5] which makes it the first member of the (2 × q) discrete biprime family, where q is a higher prime. All primes above 3 are of the form 6n ± 1 for n ≥ 1.
As a perfect number:
• 6 is related to the Mersenne prime 3, since 21(22 – 1) = 6. (The next perfect number is 28.)
• 6 is the only even perfect number that is not the sum of successive odd cubes.[6]
• 6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, 25.
Six is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist; sixty (10 × 6) and ninety (15 × 6) are the next two.[7]
All integers $n$ that are multiples of 6 are pseudoperfect (all multiples of a pseudoperfect number are pseudoperfect). Six is also the smallest Granville number, or ${\mathcal {S}}$-perfect number.[8]
Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler".[9] Six is a congruent number.[10]
6 is the second primary pseudoperfect number,[11] and harmonic divisor number.[12] It is also the second superior highly composite number,[13] and the last to also be a primorial.
There are six different ways in which 100 can be expressed as the sum of two prime numbers: (3 + 97), (11 + 89), (17 + 83), (29 + 71), (41 + 59) and (47 + 53).
There is not a prime $p$ such that the multiplicative order of 2 modulo $p$ is 6, that is, $ord_{p}(2)=6$ By Zsigmondy's theorem, if $n$ is a natural number that is not 1 or 6, then there is a prime $p$ such that $ord_{p}(2)=n$. See A112927 for such $p$.
The ring of integer of the sixth cyclotomic field Q(ζ6) , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where $\omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}$.
The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.[14]
There are six basic trigonometric functions: sin, cos, sec, csc, tan, and cot.[15]
The smallest non-abelian group is the symmetric group $\mathrm {S_{3}} $ which has 3! = 6 elements.[1]
Six is a triangular number[16] and so is its square (36). It is the first octahedral number, preceding 19.[17]
A six-sided polygon is a hexagon,[1] one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of a power of 2 (namely 21) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon with a compass and straightedge alone.
Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem.[18] The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.
There is only one non-trivial magic hexagon: it is of order-3 and made of nineteen cells, with a magic constant of 38. All rows and columns in a 6 × 6 magic square collectively generate a magic sum of 666 (which is doubly triangular). On the other hand, Graeco-Latin squares with order 6 do not exist; if $n$ is a natural number that is not 2 or 6, then there is a Graeco-Latin square of order $n$.[19]
The cube is one of five Platonic solids, with a total of six squares as faces. It is the only regular polyhedron that can generate a uniform honeycomb on its own, which is also self-dual. The cuboctahedron, which is an Archimedean solid that is one of two quasiregular polyhedra, has eight triangles and six squares as faces. Inside, its vertex arrangement can be interpreted as three hexagons that intersect to form an equatorial hexagonal hemi-face, by-which the cuboctahedron is dissected into triangular cupolas. This solid is also the only polyhedron with radial equilateral symmetry, where its edges and long radii are of equal length; its one of only four polytopes with this property — the others are the hexagon, the tesseract (as the four-dimensional analogue of the cube), and the 24-cell. Only six polygons are faces of non-prismatic uniform polyhedra such as the Platonic solids or the Archimedean solids: the triangle, the square, the pentagon, the hexagon, the octagon, and the decagon. If self-dual images of the tetrahedron are considered distinct, then there are a total of six regular polyhedra that are formed by three different Weyl groups in the third dimension (based on tetrahedral, octahedral and icosahedral symmetries).
How closely the shape of an object resembles that of a perfect sphere is called its sphericity, calculated by:[20]
$\Psi ={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}={\frac {A_{s}}{A_{p}}},$
where $A_{s}$ is the surface area of the sphere, $V_{p}$ the volume of the object, and $A_{p}$ the surface area of the object.
In four dimensions, there are a total of six convex regular polytopes: the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell.
$\mathrm {S_{6}} $, with 720 = 6! elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4, the four-dimensional 5-cell, and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number $n$ for which there is a construction of $n$ isomorphic objects on an $n$-set $A$, invariant under all permutations of $A$, but not naturally in one-to-one correspondence with the elements of $A$. This can also be expressed category theoretically: consider the category whose objects are the $n$ element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for $n=6$.
In the classification of finite simple groups, twenty of twenty-six sporadic groups in the happy family are part of three families of groups which divide the order of the friendly giant, the largest sporadic group: five first generation Mathieu groups, seven second generation subquotients of the Leech lattice, and eight third generation subgroups of the friendly giant. The remaining six sporadic groups do not divide the order of the friendly giant, which are termed the pariahs (Ly, O'N, Ru, J4, J3, and J1).[21]
List of basic calculations
Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 50 100 1000
6 × x 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 150 300 600 6000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
6 ÷ x 6 3 2 1.5 1.2 1 0.857142 0.75 0.6 0.6 0.54 0.5 0.461538 0.428571 0.4
x ÷ 6 0.16 0.3 0.5 0.6 0.83 1 1.16 1.3 1.5 1.6 1.83 2 2.16 2.3 2.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
6x 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336 13060694016
x6 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 2985984 4826809
Greek and Latin word parts
Hexa
Hexa is classical Greek for "six".[1] Thus:
• "Hexadecimal" combines hexa- with the Latinate decimal to name a number base of 16[22]
• A hexagon is a regular polygon with six sides[23]
• L'Hexagone is a French nickname for the continental part of Metropolitan France for its resemblance to a regular hexagon
• A hexahedron is a polyhedron with six faces, with a cube being a special case[24]
• Hexameter is a poetic form consisting of six feet per line
• A "hex nut" is a nut with six sides, and a hex bolt has a six-sided head
• The prefix "hexa-" also occurs in the systematic name of many chemical compounds, such as hexane which has 6 carbon atoms (C6H14).
The prefix sex-
Sex- is a Latin prefix meaning "six".[1] Thus:
• Senary is the ordinal adjective meaning "sixth"[25]
• People with sexdactyly have six fingers on each hand
• The measuring instrument called a sextant got its name because its shape forms one-sixth of a whole circle
• A group of six musicians is called a sextet
• Six babies delivered in one birth are sextuplets
• Sexy prime pairs – Prime pairs differing by six are sexy, because sex is the Latin word for six.[26][27]
The SI prefix for 10006 is exa- (E), and for its reciprocal atto- (a).
Evolution of the Hindu-Arabic digit
The evolution of our modern digit 6 appears rather simple when compared with the other digits. The modern 6 can be traced back to the Brahmi numerals of India, which are first known from the Edicts of Ashoka c. 250 BCE.[28][29][30][31] It was written in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.[32]
On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a "b" is not practical.
Just as in most modern typefaces, in typefaces with text figures the character for the digit 6 usually has an ascender, as, for example, in .[33]
This digit resembles an inverted 9. To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels.
In music
In artists
• Les Six ("The Six" in English) was a group consisting of the French composers Georges Auric, Louis Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc and Germaine Tailleferre in the 1920s[34]
• Bands with the number six in their name include Six Organs of Admittance,[35] 6 O'Clock Saints, Electric Six,[36] Eve 6, Los Xey (sei is Basque for "six"), Out On Blue Six, Six In Six, Sixpence None the Richer,[37] Slant 6,[38] Vanity 6, and You Me At Six[39]
• #6 is the pseudonym of American musician Shawn Crahan, when performing with the band Slipknot
In instruments
• A standard guitar has six strings[40]
• Most woodwind instruments have six basic holes or keys (e.g., bassoon, clarinet, pennywhistle, saxophone); these holes or keys are usually not given numbers or letters in the fingering charts
In music theory
• There are six whole tones in an octave.[41]
• There are six semitones in a tritone.[42]
In works
• "Six geese a-laying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas".[43]
• Divided in six arias, Hexachordum Apollinis is generally regarded as one of the pinnacles of Johann Pachelbel's oeuvre.[44]
• The theme of the sixth album by Dream Theater, Six Degrees of Inner Turbulence, was the number six: the album has six songs, and the sixth song — that is, the complete second disc — explores the stories of six individuals suffering from various mental illnesses.[45]
• Aristotle gave six elements of tragedy, the first of which is Mythos.[46]
In religion
See also: 666
• In Judaism:
• Six points on a Star of David[47]
• Six orders of the Mishnah[48]
• Six symbolic foods placed on the Passover Seder Plate[49]
• God took six days to create the world in the Old Testament Book of Genesis;[50] humankind was created on day 6. In the City of God, Augustine of Hippo suggested (book 11, chapter 30) that God's creation of the world took six days because 6 is a perfect number.[51]
• The Jewish holiday of Shavuot starts on the sixth day of the Hebrew month of Sivan[52]
• Seraphs have six wings.[53]
• In Islam:
• There are Six articles of belief[54]
• Fasting six days of Shawwal, together with the month of Ramadan, is equivalent to fasting the whole year[55]
• In Hindu theology, a trasarenu is the combination of six celestial paramānus (atoms).
• In Taoism:
• Six Lines of a Hexagram
• Six Ministries of Huang Di[56]
In science
Astronomy
• Messier object M6, a magnitude 4.5 open cluster in the constellation Scorpius, also known as the Butterfly Cluster[57]
• The New General Catalogue object NGC 6, a spiral galaxy in the constellation Andromeda
• The Roman numeral VI:
• Stands for subdwarfs in the Yerkes spectral classification scheme
• (Usually) stands for the sixth-discovered satellite of a planet or minor planet (e.g. Jupiter VI)
• 6 Hebe
Biology
• The cells of a beehive are six-sided.[58]
• Insects have six legs.[59]
• Six kingdoms in the taxonomic rank below domain (biology); Animalia, Plantae, Fungi, Protista, Archaea/Archaeabacteria, and Bacteria/Eubacteria. See Kingdom (biology).[60]
• The six elements most common in biomolecules are called the CHNOPS elements; the letters stand for the chemical abbreviations of carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur. See CHON.[61]
Chemistry
• A benzene molecule has a ring of six carbon atoms.[62]
• 6 is the atomic number of carbon.[63]
• The sixfold symmetry of snowflakes arises from the hexagonal crystal structure of ordinary ice.[64]
• A hexamer is an oligomer made of six subunits.
Medicine
• There are six tastes in traditional Indian medicine (Ayurveda): sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body.[65]
• Phase 6 is one of six pandemic influenza phases.[66]
Physics
• In the Standard Model of particle physics, there are six types of quarks and six types of leptons.[67]
• In statistical mechanics, the six-vertex model has six possible configurations of arrows at each vertex[68]
• There are six colors in the RGB color wheel: (primary) red, blue, green, (secondary) cyan, magenta, and yellow. (See Tertiary color)[69]
• In three-dimensional Euclidean space, there are six unknown support reactions for a statically determinate structure: one force in each of the three dimensions, and one moment through each of three possible orthogonal planes.
In sports
• The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit.[70] They are the oldest remaining teams in the league, though not necessarily the first six; they comprised the entire league from 1942 to 1967.
• Number of players:
• In association football (soccer), the number of substitutes combined by both teams, that are allowed in the game.
• In box lacrosse, the number of players per team, including the goaltender, that are on the floor at any one time, excluding penalty situations.[71]
• In ice hockey, the number of players per team, including the goaltender, that are on the ice at any one time during regulation play, excluding penalty situations. (Some leagues reduce the number of players on the ice during overtime.)[72]
• In volleyball:
• Six players from each team on each side play against each other.[73]
• Standard rules only allow six total substitutions per team per set. (Substitutions involving the libero, a defensive specialist who can only play in the back row, are not counted against this limit.)
• Six-man football is a variant of American or Canadian football, played by smaller schools with insufficient enrollment to field the traditional 11-man (American) or 12-man (Canadian) squad.[74]
• Scoring:
• In both American and Canadian football, 6 points are awarded for a touchdown.[75]
• In Australian rules football, 6 points are awarded for a goal, scored when a kicked ball passes between the defending team's two inner goalposts without having been touched by another player.
• In cricket, six runs are scored for the batting team when the ball is hit to the boundary or the ground beyond it without having touched the ground in the field.
• In basketball, the ball used for women's full-court competitions is designated "size 6".[76]
• In most rugby league competitions (but not the Super League, which uses static squad numbering), the jersey number 6 is worn by the starting five-eighth (Southern Hemisphere term) or stand-off (Northern Hemisphere term).
• In rugby union, the starting blindside flanker wears jersey number 6. (Some teams use "left" and "right" flankers instead of "openside" and "blindside", with 6 being worn by the starting left flanker.)[77]
In technology
• On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry Pearl it is the key for J and K, and on the BlackBerry 8700 series and Curve 8900 with full keyboard, it is the key for F
• The "6-meter band" in amateur radio includes the frequencies from 50 to 54 MHz
• 6 is the resin identification code used in recycling to identify polystyrene[78]
In calendars
• In the ancient Roman calendar, Sextilis was the sixth month. After the Julian reform, June became the sixth month and Sextilis was renamed August[79]
• Sextidi was the sixth day of the décade in the French Revolutionary calendar[80]
In the arts and entertainment
Games
• The number of sides on a cube, hence the highest number on a standard die[81]
• The six-sided tiles on a hex grid are used in many tabletop and board games.
• The highest number on one end of a standard domino
Comics and cartoons
• The Super 6, a 1966 animated cartoon series featuring six different super-powered heroes.[82]
Literature
• The Power of Six is a book written by Pittacus Lore, and the second in the Lorien Legacies series.[83]
• Number 6 is a character in the book series Lorien Legacies
TV
• Number Six (Tricia Helfer), is a family of fictional characters from the reimagined science fiction television series, Battlestar Galactica
• Number 6, the main protagonist in The Prisoner played by Patrick McGoohan, and portrayed by Jim Caviezel in the remake.
• Six is a character in the television series Blossom played by Jenna von Oÿ.[84]
• Six is the nickname of Kal Varrik, a central character in the television series Dark Matter, played by Roger Cross.[85]
• Six is a History channel series that chronicles the operations and daily lives of SEAL Team Six.[86]
• Six Feet Under, an HBO series that ran from 2005 to 2011.[87]
Movies
• Number 6 (Teresa Palmer) is a character in the movie I Am Number Four (2011).[88]
• The 6th Day (2000), starring Arnold Schwarzenegger.[89]
• The Sixth Sense (1999), written and directed by M. Night Shyamalan and starring Haley Joel Osment and Bruce Willis.[90]
• Girl 6 (1996), directed by Spike Lee.[91]
Musicals
• Six is a modern retelling of the lives of the six wives of Henry VIII presented as a pop concert.[92]
Anthropology
• The name of the smallest group of Cub Scouts and Guiding's equivalent Brownies, traditionally consisting of six people and is led by a "sixer".
• A coffin is traditionally buried six feet under the ground; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead[93]
• There are said to be no more than six degrees of separation between any two people on Earth.[94]
• In Western astrology, Virgo is the 6th astrological sign of the Zodiac[95]
• The Six Dynasties form part of Chinese history[96]
• Six is a lucky number in Chinese culture.[97]
• The Birmingham Six were a British miscarriage of justice, held in prison for 16 years.[98]
• "Six" is used as an informal slang term for the British Secret Intelligence Service, MI6.[99]
In other fields
• Six pack is a common form of packaging for six bottles or cans of drink (especially beer), and by extension, other assemblages of six items.[100]
• In Pythagorean numerology (a pseudoscience), the number 6 is the digit of balance, harmony and organization of the home and family
• The fundamental flight instruments lumped together on a cockpit display are often called the Basic Six or six-pack.
• The number of dots in a braille cell.[101]
• See also Six degrees (disambiguation).
• Extrasensory perception is sometimes called the "sixth sense".[102]
• Six Flags is an American company running amusement parks and theme parks in the U.S., Canada, and Mexico.[103]
• In the U.S. Army "Six" as part of a radio call sign is used by the commanding officer of a unit, while subordinate platoon leaders usually go by "One".[104] (For a similar example see also: Rainbow Six.)
See also
• List of highways numbered 6
References
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2. Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 11. ISBN 978-1-84800-000-1.
3. Weisstein, Eric W. "Harshad Number". mathworld.wolfram.com. Retrieved 2020-08-03.
4. "Sloane's A002378: Pronic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2020-11-30.
5. Sloane, N. J. A. (ed.). "Sequence A001358 (Semiprimes (or biprimes): products of two primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-03.
6. David Wells, The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books (1987): 67
7. Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
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12. Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
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32. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.66
33. Negru, John (1988). Computer Typesetting. Van Nostrand Reinhold. p. 59. ISBN 978-0-442-26696-7. slight ascenders that rise above the cap height ( in 4 and 6 )
34. Auric, Georges; Durey, Louis; Honegger, Arthur; Milhaud, Darius; Poulenc, Francis; Tailleferre, Germaine (2014-08-20). Caramel Mou and Other Great Piano Works of "Les Six": Pieces by Auric, Durey, Honegger, Milhaud, Poulenc and Tailleferre (in French). Courier Corporation. ISBN 978-0-486-49340-4.
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37. "Sixpence None The Richer". GRAMMY.com. 2020-05-19. Retrieved 2020-08-04.
38. "Slant 6 | Biography & History". AllMusic. Retrieved 2020-08-04.
39. "You Me at Six | Biography & History". AllMusic. Retrieved 2020-08-04.
40. "Definition of GUITAR". www.merriam-webster.com. Retrieved 2020-08-04.
41. D'Amante, Elvo (1994-01-01). Music Fundamentals: Pitch Structures and Rhythmic Design. Scarecrow Press. p. 194. ISBN 978-1-4616-6985-2. The division of an octave into six equal parts is referred to as the whole-tone scale
42. Horsley, Charles Edward (1876). A Text Book of Harmony: For the Use of Schools and Students. Sampson Low, Marston, Searle, & Rivington. p. 4. Like the Tritone , it contains six semitones
43. Tribble, Mimi (2004). 300 Ways to Make the Best Christmas Ever!: Decorations, Carols, Crafts & Recipes for Every Kind of Christmas Tradition. Sterling Publishing Company, Inc. p. 145. ISBN 978-1-4027-1685-0. Six geese a-laying
44. Staines, Joe (2010-05-17). The Rough Guide to Classical Music. Penguin. p. 393. ISBN 978-1-4053-8321-9. ...the six arias with variations collected under the title Hexachordum Apollinis (1699)...
45. Hegarty, Paul; Halliwell, Martin (2011-06-23). Beyond and Before: Progressive Rock since the 1960s. Bloomsbury Publishing USA. p. 169. ISBN 978-1-4411-1480-8. Six Degrees of Inner Turbulence
46. Curran, Angela (2015-10-05). Routledge Philosophy Guidebook to Aristotle and the Poetics. Routledge. p. 133. ISBN 978-1-317-67706-2. THE SIX QUALITATIVE ELEMENTS OF TRAGEDY
47. Plaut, W. Gunther (1991). The Magen David: How the Six-pointed Star Became an Emblem for the Jewish People. B'nai B'rith Books. ISBN 978-0-910250-16-0. How the Six-pointed Star Became an Emblem for the Jewish People
48. Lauterbach, Jacob Zallel (1916). Midrash and Mishnah: A Study in the Early History of the Halakah. Bloch. p. 9. Six orders of Mishnah
49. Rosen, Ceil; Rosen, Moishe (2006-05-01). Christ in the Passover. Moody Publishers. p. 79. ISBN 978-1-57567-480-3. Six symbolic foods
50. Repcheck, Jack (2008-12-15). The Man Who Found Time: James Hutton And The Discovery Of Earth's Antiquity. Basic Books. ISBN 978-0-7867-4399-5. it actually took only six days to create the earth
51. "CHURCH FATHERS: City of God, Book XI (St. Augustine)". www.newadvent.org. Retrieved 2020-08-04. These works are recorded to have been completed in six days (the same day being six times repeated), because six is a perfect number
52. Grossman, Grace Cohen; Ahlborn, Richard E.; Institution, Smithsonian (1997). Judaica at the Smithsonian: Cultural Politics as Cultural Model. Smithsonian Institution Press. p. 228. Shavuot falls on the sixth day of the Hebrew month of Sivan
53. Robertson, William Archibald Scott (1880). The crypt of Canterbury cathedral; its architecture, its history, and its frescoes. Mitchell & Hughes. p. 91. ...but seraphs , with six wings
54. Shapera, Paul M. (2009-08-15). Iran's Religious Leaders. The Rosen Publishing Group, Inc. p. 10. ISBN 978-1-4358-5283-9. Islam has six articles of faith
55. Algül, Hüseyin (2005). The Blessed Days and Nights of the Islamic Year. Tughra Books. p. 65. ISBN 978-1-932099-93-5. ...it was blessed to fast for six days in the month of Shawwal...
56. Bary, William Theodore De; DeBary, William T.; Chan, Wing-tsit; Lufrano, Richard; Ching, Julia; Johnson, David; Liu, Kwang-Ching; Mungello, David (1999). Sources of Chinese Tradition. Columbia University Press. ISBN 978-0-231-11270-3. ...and the Six Ministries were made...
57. Rhoads, Samuel E. (1996). The Sky Tonight: A Guided Tour of the Stars Over Hawai'i. Bishop Museum Press. ISBN 978-0-930897-93-2. Three Messier objects are visible in this part of the sky : M6 , M7 and M8 .
58. Sedgwick, Marcus (2011-07-05). White Crow. Roaring Brook Press. p. 145. ISBN 978-1-4299-7634-3. The cells of honeycombs are six-sided because a hexagon is the most material-efficient tessellation
59. Parker, Steve (2005). Ant Lions, Wasps & Other Insects. Capstone. p. 16. ISBN 978-0-7565-1250-7. Insects have six legs each...
60. Pendarvis, Murray P.; Crawley, John L. (2019-02-01). Exploring Biology in the Laboratory: Core Concepts. Morton Publishing Company. p. 10. ISBN 978-1-61731-899-3. ...presently at least six kingdoms are recognized;
61. Mader, Sylvia S. (2004). Biology. McGraw-Hill. p. 20. ISBN 978-0-07-291934-9. The acronym CHNOPS helps us remember these six elements
62. DESS, Fritz Dufour, MBA (2018-09-19). The Realities of Reality - Part II: Making Sense of Why Modern Science Advances (Volume 1). Fritz Dufour. p. 100. The benzene molecule has its six carbon atoms in a ring{{cite book}}: CS1 maint: multiple names: authors list (link)
63. Starr, Cecie; Evers, Christine (2012-05-10). Biology Today and Tomorrow without Physiology. Cengage Learning. p. 25. ISBN 978-1-133-36536-5. For example, the atomic number of carbon is 6,
64. Webb, Stephen; Webb, Professor of Australian Studies Stephen (2004-05-25). Out of this World: Colliding Universes, Branes, Strings, and Other Wild Ideas of Modern Physics. Springer Science & Business Media. p. 16. ISBN 978-0-387-02930-6. snowflake, with its familiar sixfold rotational symmetry
65. Woo, Teri Moser; Robinson, Marylou V. (2015-08-03). Pharmacotherapeutics For Advanced Practice Nurse Prescribers. F.A. Davis. p. 145. ISBN 978-0-8036-4581-3. Ayurvedic herbology is based on the tridoshic theory that there exist six basic tastes
66. Pandemic Influenza Preparedness and Response Guidance for Healthcare Workers and Healthcare Employers. OSHA, U.S. Department of Labor. 2007. p. 8. The WHO Plan describes six phases of increasing public health risk associated with the emergence of a new influenza
67. Sanghera, Paul (2011-03-08). Quantum Physics for Scientists and Technologists: Fundamental Principles and Applications for Biologists, Chemists, Computer Scientists, and Nanotechnologists. John Wiley & Sons. p. 64. ISBN 978-0-470-92269-9. ...there are six types of quarks and six types of leptons.
68. Jimbo, M.; Jimbo, Michio; Miwa, Tetsuji; Tsuchiya, Akihiro (1989). Integrable Systems in Quantum Field Theory and Statistical Mechanics. Academic Press. p. 588. ISBN 978-0-12-385342-4. Allowed configurations in the six-vertex model and their statistical weights
69. Sloan, Robin James Stuart (2015-05-07). Virtual Character Design for Games and Interactive Media. CRC Press. p. 34. ISBN 978-1-4665-9820-1. placing six primaries around the wheel in the following order: red, yellow, green, cyan, blue, magenta.
70. Bamford, Tab (2016-10-15). 100 Things Blackhawks Fans Should Know & Do Before They Die. Triumph Books. ISBN 978-1-63319-638-4. the Original Six
71. Stillwell, Jim L. (1987). Making and Using Creative Play Equipment. Human Kinetics Publishers. p. 36. ISBN 978-0-87322-084-2. Indoor Lacrosse . This is played with six players per team
72. Williams, Heather (2019). Hockey: A Guide for Players and Fans. Capstone. p. 16. ISBN 978-1-5435-7458-6. There are six players per team on the ice at one time.
73. Sports, The National Alliance For Youth (2009-05-11). Coaching Volleyball For Dummies. John Wiley & Sons. p. 48. ISBN 978-0-470-53398-7. In a regulation volleyball match with six players on each side of the court,
74. "sixmanfootball.com". www.sixmanfootball.com. Retrieved 2020-08-06.
75. "How Football Teams Can Score Points in Game Play". dummies. Retrieved 2020-08-06.
76. "Basketball Sizes Chart: What Size Ball Should a Player Use?". Basketball For Coaches. 2018-09-12. Retrieved 2020-08-06.
77. "Rugby flanker (#6 & #7): A position specific guide". Atrox Rugby. 2019-02-01. Retrieved 2020-08-06.
78. Stevens, E. S. (2002). Green Plastics: An Introduction to the New Science of Biodegradable Plastics. Princeton University Press. p. 45. ISBN 978-0-691-04967-0.
79. Bunson, Matthew (2014-05-14). Encyclopedia of the Roman Empire. Infobase Publishing. p. 90. ISBN 978-1-4381-1027-1. Augustus was also originally called Sextilis, the sixth month.
80. Nicolas, Sir Nicholas Harris (1833). The Chronology of History: Containing Tables, Calculations and Statements, Indispensable for Ascertaining the Dates of Historical Events and of Public and Private Documents from the Earliest Period to the Present Time. Longham, Rees, Orme, Brown, Green, & Longman and John Taylor. p. 172. SEXTIDI , or " Jour de la Révolution , "
81. Schumer, Peter D. (2004-02-11). Mathematical Journeys. John Wiley & Sons. p. 88. ISBN 978-0-471-22066-4. Roll two dice, a standard six-sided die numbered 1 through 6
82. DataBase, The Big Cartoon. "The Super 6 Episode Guide -DePatie-Freleng Ent". Big Cartoon DataBase (BCDB). Retrieved 2020-08-06.
83. "'The Power of Six' trailer". EW.com. Retrieved 2020-08-06.
84. Terrace, Vincent (1993). Television Character and Story Facts: Over 110,000 Details from 1,008 Shows, 1945-1992. McFarland & Company. p. 54. ISBN 978-0-89950-891-7. Every Monday Blossom and Six ( who also...
85. "Kal Varrick - Dark Matter". Universe Guide. Archived from the original on September 15, 2020. Retrieved 2020-08-06.
86. "Six: A Review". 2017-03-23. Retrieved 2020-08-06.
87. "The 100 Greatest TV Shows Of All Time". Empire. 2019-10-16. Retrieved 2020-08-06.
88. "Character profile for Number 6 from I Am Number Four (Lorien Legacies, #1) (page 1)". www.goodreads.com. Retrieved 2020-08-06.
89. The 6th Day (2000) - Roger Spottiswoode | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 2020-08-06
90. The Sixth Sense (1999) - M. Night Shyamalan | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 2020-08-06
91. Lee, Spike (1996-03-22), Girl 6 (Comedy, Drama), Theresa Randle, Isaiah Washington, Spike Lee, Jenifer Lewis, Fox Searchlight Pictures, 40 Acres & A Mule Filmworks, retrieved 2021-05-05
92. "Six the Musical". www.sixthemusical.com. Retrieved 2020-08-06.
93. Rimes, Wendy (2016-04-01). "The Reason Why The Dead Are Buried Six Feet Below The Ground". Elite Readers. Retrieved 2020-08-06.
94. "Six Degrees of Peggy Bacon". www.aaa.si.edu. Retrieved 2020-08-06.
95. "Virgo | constellation and astrological sign". Encyclopedia Britannica. Retrieved 2020-08-06.
96. Wilkinson, Endymion Porter; Wilkinson, Scholar and Diplomat (Eu Ambassador to China 1994-2001) Endymion (2000). Chinese History: A Manual. Harvard Univ Asia Center. p. 11. ISBN 978-0-674-00249-4.
97. "6 is a lucky number in Chinese culture - Google Search". www.google.com. Retrieved 2020-08-06. Two of the luckiest numbers in China are '6' and '8'.
98. Peirce, Gareth (2011-03-12). "The Birmingham Six: Have we learned from our disgraceful past?". The Guardian. ISSN 0261-3077. Retrieved 2020-08-06.
99. Smith, Michael (2011-10-31). Six: The Real James Bonds 1909-1939. Biteback Publishing. ISBN 978-1-84954-264-7.
100. "Are Plastic Six-Pack Rings Still Ensnaring Wildlife?". Environment. 2018-09-19. Retrieved 2020-08-06.
101. "What Is Braille? | American Foundation for the Blind". www.afb.org. Retrieved 2020-08-06.
102. Walker, Kathryn; Innes, Brian (2009). Mysteries of the Mind. Crabtree Publishing Company. p. 5. ISBN 978-0-7787-4149-7. this is sometimes called a " sixth sense "
103. "Six Flags Reopens With Enhanced Safety Protocols | IAAPA". www.iaapa.org. Retrieved 2020-08-06.
104. Mason, Robert (1983). Chickenhawk. London: Corgi Books. p. 141. ISBN 978-0-552-12419-5.
• The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66–68
• A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links ISBN 0-521-42385-6
• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 67 - 69
External links
Look up six in Wiktionary, the free dictionary.
• The Number 6
• The Positive Integer 6
• Prime curiosities: 6
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Six circles theorem
In geometry, the six circles theorem relates to a chain of six circles together with a triangle, such that each circle is tangent to two sides of the triangle and also to the preceding circle in the chain. The chain closes, in the sense that the sixth circle is always tangent to the first circle.[1][2] It is assumed in this construction that all circles lie within the triangle, and all points of tangency lie on the sides of the triangle. If the problem is generalized to allow circles that may not be within the triangle, and points of tangency on the lines extending the sides of the triangle, then the sequence of circles eventually reaches a periodic sequence of six circles, but may take arbitrarily many steps to reach this periodicity.[3]
The name may also refer to Miquel's six circles theorem, the result that if five circles have four triple points of intersection then the remaining four points of intersection lie on a sixth circle.
References
1. Evelyn, C. J. A.; Money-Coutts, G. B.; Tyrrell, John Alfred (1974). The Seven Circles Theorem and Other New Theorems. London: Stacey International. pp. 49–58. ISBN 978-0-9503304-0-2.
2. Wells, David (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 231. ISBN 0-14-011813-6.
3. Ivanov, Dennis; Tabachnikov, Serge (2016). "The six circles theorem revisited". American Mathematical Monthly. 123 (7): 689–698. arXiv:1312.5260. doi:10.4169/amer.math.monthly.123.7.689. MR 3539854. S2CID 17597937.
External links
• Weisstein, Eric W. "Six Circles Theorem". MathWorld.
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Magnitude (mathematics)
In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs.
In physics, magnitude can be defined as quantity or distance.
History
The Greeks distinguished between several types of magnitude,[1] including:
• Positive fractions
• Line segments (ordered by length)
• Plane figures (ordered by area)
• Solids (ordered by volume)
• Angles (ordered by angular magnitude)
They proved that the first two could not be the same, or even isomorphic systems of magnitude.[2] They did not consider negative magnitudes to be meaningful, and magnitude is still primarily used in contexts in which zero is either the smallest size or less than all possible sizes.
Numbers
Main article: Absolute value
The magnitude of any number $x$ is usually called its absolute value or modulus, denoted by $|x|$.[3]
Real numbers
The absolute value of a real number r is defined by:[4]
$\left|r\right|=r,{\text{ if }}r{\text{ ≥ }}0$
$\left|r\right|=-r,{\text{ if }}r<0.$
Absolute value may also be thought of as the number's distance from zero on the real number line. For example, the absolute value of both 70 and −70 is 70.
Complex numbers
A complex number z may be viewed as the position of a point P in a 2-dimensional space, called the complex plane. The absolute value (or modulus) of z may be thought of as the distance of P from the origin of that space. The formula for the absolute value of z = a + bi is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space:[5]
$\left|z\right|={\sqrt {a^{2}+b^{2}}}$
where the real numbers a and b are the real part and the imaginary part of z, respectively. For instance, the modulus of −3 + 4i is ${\sqrt {(-3)^{2}+4^{2}}}=5$. Alternatively, the magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, ${\bar {z}}$, where for any complex number $z=a+bi$, its complex conjugate is ${\bar {z}}=a-bi$.
$\left|z\right|={\sqrt {z{\bar {z}}}}={\sqrt {(a+bi)(a-bi)}}={\sqrt {a^{2}-abi+abi-b^{2}i^{2}}}={\sqrt {a^{2}+b^{2}}}$
(where $i^{2}=-1$).
Vector spaces
Euclidean vector space
Main article: Euclidean norm
A Euclidean vector represents the position of a point P in a Euclidean space. Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real numbers (the Cartesian coordinates of P): x = [x1, x2, ..., xn]. Its magnitude or length, denoted by $\|x\|$,[6] is most commonly defined as its Euclidean norm (or Euclidean length):[7]
$\|\mathbf {x} \|={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.$
For instance, in a 3-dimensional space, the magnitude of [3, 4, 12] is 13 because ${\sqrt {3^{2}+4^{2}+12^{2}}}={\sqrt {169}}=13.$ This is equivalent to the square root of the dot product of the vector with itself:
$\|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}.$
The Euclidean norm of a vector is just a special case of Euclidean distance: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector x:
1. $\left\|\mathbf {x} \right\|,$
2. $\left|\mathbf {x} \right|.$
A disadvantage of the second notation is that it can also be used to denote the absolute value of scalars and the determinants of matrices, which introduces an element of ambiguity.
Normed vector spaces
Main article: Normed vector space
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude.
A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space.[8] The norm of a vector v in a normed vector space can be considered to be the magnitude of v.
Pseudo-Euclidean space
In a pseudo-Euclidean space, the magnitude of a vector is the value of the quadratic form for that vector.
Logarithmic magnitudes
When comparing magnitudes, a logarithmic scale is often used. Examples include the loudness of a sound (measured in decibels), the brightness of a star, and the Richter scale of earthquake intensity. Logarithmic magnitudes can be negative. In the natural sciences, a logarithmic magnitude is typically referred to as a level.
Order of magnitude
Main article: Order of magnitude
Orders of magnitude denote differences in numeric quantities, usually measurements, by a factor of 10—that is, a difference of one digit in the location of the decimal point.
Other mathematical measures
This section is an excerpt from Measure (mathematics).[edit]
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.
The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.
See also
• Number sense
• Vector notation
• Set size
References
1. Heath, Thomas Smd. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.
2. Bloch, Ethan D. (2011), The Real Numbers and Real Analysis, Springer, p. 52, ISBN 9780387721774, The idea of incommensurable pairs of lengths of line segments was discovered in ancient Greece.
3. "Magnitude Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-08-23.
4. Mendelson, Elliott (2008). Schaum's Outline of Beginning Calculus. McGraw-Hill Professional. p. 2. ISBN 978-0-07-148754-2.
5. Ahlfors, Lars V. (1953). Complex Analysis. Tokyo: McGraw Hill Kogakusha.
6. Nykamp, Duane. "Magnitude of a vector definition". Math Insight. Retrieved August 23, 2020.
7. Howard Anton; Chris Rorres (12 April 2010). Elementary Linear Algebra: Applications Version. John Wiley & Sons. ISBN 978-0-470-43205-1.
8. Golan, Johnathan S. (January 2007), The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed.), Springer, ISBN 978-1-4020-5494-5
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Size
Size in general is the magnitude or dimensions of a thing. More specifically, geometrical size (or spatial size) can refer to linear dimensions (length, width, height, diameter, perimeter), area, or volume. Size can also be measured in terms of mass, especially when assuming a density range.
"Physical dimension" redirects here. For the dimension of a physical quantity, see Dimension (physics).
In mathematical terms, "size is a concept abstracted from the process of measuring by comparing a longer to a shorter".[1] Size is determined by the process of comparing or measuring objects, which results in the determination of the magnitude of a quantity, such as length or mass, relative to a unit of measurement. Such a magnitude is usually expressed as a numerical value of units on a previously established spatial scale, such as meters or inches.
The sizes with which humans tend to be most familiar are body dimensions (measures of anthropometry), which include measures such as human height and human body weight. These measures can, in the aggregate, allow the generation of commercially useful distributions of products that accommodate expected body sizes,[2] as with the creation of clothing sizes and shoe sizes, and with the standardization of door frame dimensions, ceiling heights, and bed sizes. The human experience of size can lead to a psychological tendency towards size bias,[3] wherein the relative importance or perceived complexity of organisms and other objects is judged based on their size relative to humans, and particularly whether this size makes them easy to observe without aid.
Human perception
Humans most frequently perceive the size of objects through visual cues.[4] One common means of perceiving size is to compare the size of a newly observed object with the size of a familiar object whose size is already known. Binocular vision gives humans the capacity for depth perception, which can be used to judge which of several objects is closer, and by how much, which allows for some estimation of the size of the more distant object relative to the closer object. This also allows for the estimation of the size of large objects based on comparison of closer and farther parts of the same object. The perception of size can be distorted by manipulating these cues, for example through the creation of forced perspective.
Some measures of size may also be determined by sound. Visually impaired humans often use echolocation to determine features of their surroundings, such as the size of spaces and objects. However, even humans who lack this ability can tell if a space that they are unable to see is large or small from hearing sounds echo in the space. Size can also be determined by touch, which is a process of haptic perception.
The sizes of objects that can not readily be measured merely by sensory input may be evaluated with other kinds of measuring instruments. For example, objects too small to be seen with the naked eye may be measured when viewed through a microscope, while objects too large to fit within the field of vision may be measured using a telescope, or through extrapolation from known reference points. However, even very advanced measuring devices may still present a limited field of view.
Terminology
Objects being described by their relative size are often described as being comparatively big and little, or large and small, although "big and little tend to carry affective and evaluative connotations, whereas large and small tend to refer only to the size of a thing".[5] A wide range of other terms exist to describe things by their relative size, with small things being described for example as tiny, miniature, or minuscule, and large things being described as, for example, huge, gigantic, or enormous. Objects are also typically described as tall or short specifically relative to their vertical height, and as long or short specifically relative to their length along other directions.
Although the size of an object may be reflected in its mass or its weight, each of these is a different concept. In scientific contexts, mass refers loosely to the amount of "matter" in an object (though "matter" may be difficult to define), whereas weight refers to the force experienced by an object due to gravity.[6] An object with a mass of 1.0 kilogram will weigh approximately 9.81 newtons (newton is the unit of force, while kilogram is the unit of mass) on the surface of the Earth (its mass multiplied by the gravitational field strength). Its weight will be less on Mars (where gravity is weaker), more on Saturn, and negligible in space when far from any significant source of gravity, but it will always have the same mass. Two objects of equal size, however, may have very different mass and weight, depending on the composition and density of the objects. By contrast, if two objects are known to have roughly the same composition, then some information about the size of one can be determined by measuring the size of the other, and determining the difference in weight between the two. For example, if two blocks of wood are equally dense, and it is known that one weighs ten kilograms and the other weighs twenty kilograms, and that the ten kilogram block has a volume of one cubic foot, then it can be deduced that the twenty kilogram block has a volume of two cubic feet.
Conceptualization and generalization
The concept of size is often applied to ideas that have no physical reality. In mathematics, magnitude is the size of a mathematical object, which is an abstract object with no concrete existence. Magnitude is a property by which the object can be compared as larger or smaller than other objects of the same kind. More formally, an object's magnitude is an ordering (or ranking) of the class of objects to which it belongs. There are various other mathematical concepts of size for sets, such as:
• measure (mathematics), a systematic way to assign to each suitable subset a number
• cardinality (equal if there is a bijection), of a set is a measure of the "number of elements of the set"
• for well-ordered sets: ordinal number (equal if there is an order-isomorphism)
In statistics (hypothesis testing), the "size" of the test refers to the rate of false positives, denoted by α. In astronomy, the magnitude of brightness or intensity of a star is measured on a logarithmic scale. Such a scale is also used to measure the intensity of an earthquake, and this intensity is often referred to as the "size" of the event.[7] In computing, file size is a measure of the size of a computer file, typically measured in bytes. The actual amount of disk space consumed by the file depends on the file system. The maximum file size a file system supports depends on the number of bits reserved to store size information and the total size of the file system in terms of its capacity to store bits of information.
In physics, the Planck length, denoted ℓP, is a unit of length, equal to 1.616199(97)×10−35 metres. It is a unit in the system of Planck units, developed by physicist Max Planck. The Planck length is defined in terms of three fundamental physical constants: the speed of light, the Planck constant, and the Newtonian constant of gravitation. In contrast, the largest observable thing is the observable universe. The comoving distance – the distance as would be measured at a specific time, including the present – between Earth and the edge of the observable universe is 46 billion light-years (14×10^9 pc), making the diameter of the observable universe about 91 billion light-years (28×10^9 pc).
In poetry, fiction, and other literature, size is occasionally assigned to characteristics that do not have measurable dimensions, such as the metaphorical reference to the size of a person's heart as a shorthand for describing their typical degree of kindness or generosity. With respect to physical size, the concept of resizing is occasionally presented in fairy tales, fantasy, and science fiction, placing humans in a different context within their natural environment by depicting them as having physically been made exceptionally large or exceptionally small through some fantastic means.
See also
• Dimensional instruments
• Orders of magnitude (length)
References
1. C. Smoryński, History of Mathematics: A Supplement (2008), p. 76.
2. Thomas T. Samaras, Human Body Size and the Laws of Scaling (2007), p. 3.
3. "The notion that bacteria are primitive, unsophisticated organisms stems from what I would call size chauvinism". Matthews, Clifford (1995). Cosmic beginnings and human ends : where science and religion meet. Chicago and LaSalle, Ill: Open Court. p. 208. ISBN 978-0-8126-9270-9. OCLC 31435749.
4. Bennett L. Schwartz, John H. Krantz, Sensation and Perception (2015), Chapter 7: "Depth and Size Perception", p. 169-199.
5. John R. Taylor, The Mental Corpus: How Language is Represented in the Mind (2012), p. 108.
6. de Silva, G.M.S. (2002), Basic Metrology for ISO 9000 Certification, Butterworth-Heinemann
7. See, e.g., Robert A. Meyers, Extreme Environmental Events: Complexity in Forecasting and Early Warning (2010), p. 364, stating "[t]he corner frequency scales with the size of the earthquake measured by the seismic moment".
External links
Look up biggity, resize, resized, resizing, or size in Wiktionary, the free dictionary.
• Media related to Size at Wikimedia Commons
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Size functor
Given a size pair $(M,f)\ $ where $M\ $ is a manifold of dimension $n\ $ and $f\ $ is an arbitrary real continuous function defined on it, the $i$-th size functor,[1] with $i=0,\ldots ,n\ $, denoted by $F_{i}\ $, is the functor in $Fun(\mathrm {Rord} ,\mathrm {Ab} )\ $, where $\mathrm {Rord} \ $ is the category of ordered real numbers, and $\mathrm {Ab} \ $ is the category of Abelian groups, defined in the following way. For $x\leq y\ $, setting $M_{x}=\{p\in M:f(p)\leq x\}\ $, $M_{y}=\{p\in M:f(p)\leq y\}\ $, $j_{xy}\ $ equal to the inclusion from $M_{x}\ $ into $M_{y}\ $, and $k_{xy}\ $ equal to the morphism in $\mathrm {Rord} \ $ from $x\ $ to $y\ $,
• for each $x\in \mathbb {R} \ $, $F_{i}(x)=H_{i}(M_{x});\ $
• $F_{i}(k_{xy})=H_{i}(j_{xy}).\ $
In other words, the size functor studies the process of the birth and death of homology classes as the lower level set changes. When $M\ $ is smooth and compact and $f\ $ is a Morse function, the functor $F_{0}\ $ can be described by oriented trees, called $H_{0}\ $ − trees.
The concept of size functor was introduced as an extension to homology theory and category theory of the idea of size function. The main motivation for introducing the size functor originated by the observation that the size function $\ell _{(M,f)}(x,y)\ $ can be seen as the rank of the image of $H_{0}(j_{xy}):H_{0}(M_{x})\rightarrow H_{0}(M_{y})$.
The concept of size functor is strictly related to the concept of persistent homology group,[2] studied in persistent homology. It is worth to point out that the $i\ $-th persistent homology group coincides with the image of the homomorphism $F_{i}(k_{xy})=H_{i}(j_{xy}):H_{i}(M_{x})\rightarrow H_{i}(M_{y})$.
See also
• Size theory
• Size function
• Size homotopy group
• Size pair
References
1. Cagliari, Francesca; Ferri, Massimo; Pozzi, Paola (2001). "Size functions from a categorical viewpoint". Acta Applicandae Mathematicae. 67 (3): 225–235. doi:10.1023/A:1011923819754.
2. Edelsbrunner, Herbert; Letscher, David; Zomorodian, Afra (2002). "Topological Persistence and Simplification". Discrete & Computational Geometry. 28 (4): 511–533. doi:10.1007/s00454-002-2885-2.
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Size homotopy group
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair $(M,\varphi )$ is given, where $M$ is a closed manifold of class $C^{0}\ $ and $\varphi :M\to \mathbb {R} ^{k}$ is a continuous function. Consider the lexicographical order $\preceq $ on $\mathbb {R} ^{k}$ defined by setting $(x_{1},\ldots ,x_{k})\preceq (y_{1},\ldots ,y_{k})\ $ if and only if $x_{1}\leq y_{1},\ldots ,x_{k}\leq y_{k}$. For every $Y\in \mathbb {R} ^{k}$ set $M_{Y}=\{Z\in \mathbb {R} ^{k}:Z\preceq Y\}$.
Assume that $P\in M_{X}\ $ and $X\preceq Y\ $. If $\alpha \ $, $\beta \ $ are two paths from $P\ $ to $P\ $ and a homotopy from $\alpha \ $ to $\beta \ $, based at $P\ $, exists in the topological space $M_{Y}\ $, then we write $\alpha \approx _{Y}\beta \ $. The first size homotopy group of the size pair $(M,\varphi )\ $ computed at $(X,Y)\ $ is defined to be the quotient set of the set of all paths from $P\ $ to $P\ $ in $M_{X}\ $ with respect to the equivalence relation $\approx _{Y}\ $, endowed with the operation induced by the usual composition of based loops.[1]
In other words, the first size homotopy group of the size pair $(M,\varphi )\ $ computed at $(X,Y)\ $ and $P\ $ is the image $h_{XY}(\pi _{1}(M_{X},P))\ $ of the first homotopy group $\pi _{1}(M_{X},P)\ $ with base point $P\ $ of the topological space $M_{X}\ $, when $h_{XY}\ $ is the homomorphism induced by the inclusion of $M_{X}\ $ in $M_{Y}\ $.
The $n$-th size homotopy group is obtained by substituting the loops based at $P\ $ with the continuous functions $\alpha :S^{n}\to M\ $ taking a fixed point of $S^{n}\ $ to $P\ $, as happens when higher homotopy groups are defined.
See also
• Size function
• Size functor
• Size pair
• Natural pseudodistance
References
1. Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society – Simon Stevin, 6:455–464, 1999.
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Size theory
In mathematics, size theory studies the properties of topological spaces endowed with $\mathbb {R} ^{k}$-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance between size pairs. A survey of size theory can be found in .[1]
History and applications
The beginning of size theory is rooted in the concept of size function, introduced by Frosini.[2] Size functions have been initially used as a mathematical tool for shape comparison in computer vision and pattern recognition.[3][4][5][6][7][8][9][10]
An extension of the concept of size function to algebraic topology was made in the 1999 Frosini and Mulazzani paper [11] where size homotopy groups were introduced, together with the natural pseudodistance for $\mathbb {R} ^{k}$-valued functions. An extension to homology theory (the size functor) was introduced in 2001.[12] The size homotopy group and the size functor are strictly related to the concept of persistent homology group [13] studied in persistent homology. It is worth to point out that the size function is the rank of the $0$-th persistent homology group, while the relation between the persistent homology group and the size homotopy group is analogous to the one existing between homology groups and homotopy groups.
In size theory, size functions and size homotopy groups are seen as tools to compute lower bounds for the natural pseudodistance. Actually, the following link exists between the values taken by the size functions $\ell _{(N,\psi )}({\bar {x}},{\bar {y}})$, $\ell _{(M,\varphi )}({\tilde {x}},{\tilde {y}})$ and the natural pseudodistance $d((M,\varphi ),(N,\psi ))$ between the size pairs $(M,\varphi ),\ (N,\psi )$ ,[14][15]
${\text{If }}\ell _{(N,\psi )}({\bar {x}},{\bar {y}})>\ell _{(M,\varphi )}({\tilde {x}},{\tilde {y}}){\text{ then }}d((M,\varphi ),(N,\psi ))\geq \min\{{\tilde {x}}-{\bar {x}},{\bar {y}}-{\tilde {y}}\}.$
An analogous result holds for size homotopy group.[11]
The attempt to generalize size theory and the concept of natural pseudodistance to norms that are different from the supremum norm has led to the study of other reparametrization invariant norms.[16]
See also
• Size function
• Natural pseudodistance
• Size functor
• Size homotopy group
• Size pair
• Matching distance
References
1. Silvia Biasotti, Leila De Floriani, Bianca Falcidieno, Patrizio Frosini, Daniela Giorgi, Claudia Landi, Laura Papaleo, Michela Spagnuolo, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys, vol. 40 (2008), n. 4, 12:1–12:87.
2. Patrizio Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bulletin of the Australian Mathematical Society, 42(3):407–416, 1990.
3. Alessandro Verri, Claudio Uras, Patrizio Frosini and Massimo Ferri, On the use of size functions for shape analysis, Biological Cybernetics, 70:99–107, 1993.
4. Patrizio Frosini and Claudia Landi, Size functions and morphological transformations, Acta Applicandae Mathematicae, 49(1):85–104, 1997.
5. Alessandro Verri and Claudio Uras, Metric-topological approach to shape representation and recognition, Image Vision Comput., 14:189–207, 1996.
6. Alessandro Verri and Claudio Uras, Computing size functions from edge maps, Internat. J. Comput. Vision, 23(2):169–183, 1997.
7. Françoise Dibos, Patrizio Frosini and Denis Pasquignon, The use of size functions for comparison of shapes through differential invariants, Journal of Mathematical Imaging and Vision, 21(2):107–118, 2004.
8. Michele d'Amico, Patrizio Frosini and Claudia Landi, Using matching distance in Size Theory: a survey, International Journal of Imaging Systems and Technology, 16(5):154–161, 2006.
9. Andrea Cerri, Massimo Ferri, Daniela Giorgi: Retrieval of trademark images by means of size functions Graphical Models 68:451–471, 2006.
10. Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, Bianca Falcidieno: Size functions for comparing 3D models. Pattern Recognition 41:2855–2873, 2008.
11. Patrizio Frosini and Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society – Simon Stevin, 6:455–464 1999.
12. Francesca Cagliari, Massimo Ferri and Paola Pozzi, Size functions from a categorical viewpoint, Acta Applicandae Mathematicae, 67(3):225–235, 2001.
13. Herbert Edelsbrunner, David Letscher and Afra Zomorodian, Topological Persistence and Simplification, Discrete and Computational Geometry, 28(4):511–533, 2002.
14. Patrizio Frosini and Claudia Landi, Size Theory as a Topological Tool for Computer Vision, Pattern Recognition And Image Analysis, 9(4):596–603, 1999.
15. Pietro Donatini and Patrizio Frosini, Lower bounds for natural pseudodistances via size functions, Archives of Inequalities and Applications, 2(1):1–12, 2004.
16. Patrizio Frosini, Claudia Landi: Reparametrization invariant norms. Transactions of the American Mathematical Society 361:407–452, 2009.
Topology
Fields
• General (point-set)
• Algebraic
• Combinatorial
• Continuum
• Differential
• Geometric
• low-dimensional
• Homology
• cohomology
• Set-theoretic
• Digital
Key concepts
• Open set / Closed set
• Interior
• Continuity
• Space
• compact
• Connected
• Hausdorff
• metric
• uniform
• Homotopy
• homotopy group
• fundamental group
• Simplicial complex
• CW complex
• Polyhedral complex
• Manifold
• Bundle (mathematics)
• Second-countable space
• Cobordism
Metrics and properties
• Euler characteristic
• Betti number
• Winding number
• Chern number
• Orientability
Key results
• Banach fixed-point theorem
• De Rham cohomology
• Invariance of domain
• Poincaré conjecture
• Tychonoff's theorem
• Urysohn's lemma
• Category
• Mathematics portal
• Wikibook
• Wikiversity
• Topics
• general
• algebraic
• geometric
• Publications
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Skein relation
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots. In general, the converse does not hold.
Skein relations are often used to give a simple definition of knot polynomials. A skein relation gives a linear relation between the values of a knot polynomial on a collection of three links which differ from each other only in a small region. For some knot polynomials, such as the Conway, Alexander, and Jones polynomials, the relevant skein relations are sufficient to calculate the polynomial recursively.
Definition
A skein relationship requires three link diagrams that are identical except at one crossing. The three diagrams must exhibit the three possibilities that could occur for the two line segments at that crossing, one of the lines could pass under, the same line could be over or the two lines might not cross at all. Link diagrams must be considered because a single skein change can alter a diagram from representing a knot to one representing a link and vice versa. Depending on the knot polynomial in question, the links (or tangles) appearing in a skein relation may be oriented or unoriented.
The three diagrams are labelled as follows. Turn the three link diagram so the directions at the crossing in question are both roughly northward. One diagram will have northwest over northeast, it is labelled L−. Another will have northeast over northwest, it's L+. The remaining diagram is lacking that crossing and is labelled L0.
(The labelling is independent of direction insofar as it remains the same if all directions are reversed. Thus polynomials on undirected knots are unambiguously defined by this method. However, the directions on links are a vital detail to retain as one recurses through a polynomial calculation.)
It is also sensible to think in a generative sense, by taking an existing link diagram and "patching" it to make the other two—just so long as the patches are applied with compatible directions.
To recursively define a knot (link) polynomial, a function F is fixed and for any triple of diagrams and their polynomials labelled as above,
$F{\Big (}L_{-},L_{0},L_{+}{\Big )}=0$
or more pedantically
$F{\Big (}L_{-}(x),L_{0}(x),L_{+}(x),x{\Big )}=0$ for all $x$
(Finding an F which produces polynomials independent of the sequences of crossings used in a recursion is no trivial exercise.)
More formally, a skein relation can be thought of as defining the kernel of a quotient map from the planar algebra of tangles. Such a map corresponds to a knot polynomial if all closed diagrams are taken to some (polynomial) multiple of the image of the empty diagram.
Example
Sometime in the early 1960s, Conway showed how to compute the Alexander polynomial using skein relations. As it is recursive, it is not quite so direct as Alexander's original matrix method; on the other hand, parts of the work done for one knot will apply to others. In particular, the network of diagrams is the same for all skein-related polynomials.
Let function P from link diagrams to Laurent series in ${\sqrt {x}}$ be such that $P({\rm {unknot}})=1$ and a triple of skein-relation diagrams $(L_{-},L_{0},L_{+})$ satisfies the equation
$P(L_{-})=(x^{-1/2}-x^{1/2})P(L_{0})+P(L_{+})$
Then P maps a knot to one of its Alexander polynomials.
In this example, we calculate the Alexander polynomial of the cinquefoil knot (), the alternating knot with five crossings in its minimal diagram. At each stage we exhibit a relationship involving a more complex link and two simpler diagrams. Note that the more complex link is on the right in each step below except the last. For convenience, let A = x−1/2−x1/2.
To begin, we create two new diagrams by patching one of the cinquefoil's crossings (highlighted in yellow) so
P() = A × P() + P()
The first diagram is actually a trefoil; the second diagram is two unknots with four crossings. Patching the latter
P() = A × P() + P()
gives, again, a trefoil, and two unknots with two crossings (the Hopf link ). Patching the trefoil
P() = A × P() + P()
gives the unknot and, again, the Hopf link. Patching the Hopf link
P() = A × P() + P()
gives a link with 0 crossings (unlink) and an unknot. The unlink takes a bit of sneakiness:
P() = A × P() + P()
Computations
We now have enough relations to compute the polynomials of all the links we've encountered, and can use the above equations in reverse order to work up to the cinquefoil knot itself. The calculation is described in the table below, where ? denotes the unknown quantity we are solving for in each relation:
knot name diagrams P (diagram)
skein equation ? P in full
unknot defined as 1 x→1
unlink 1=A?+1 0 x→0
Hopf link 0=A1+? -A x→x1/2-x−1/2
trefoil 1=A(-A)+? 1+A2 x→x−1-1+x
4 crossing link -A=A(1+A2)+? -A(2+A2) x→-x−3/2+x−1/2-x1/2+x3/2
cinquefoil 1+A2=A(-A(2+A2))+? 1+3A2+A4 x→x−2-x−1+1-x+x2
Thus the Alexander polynomial for a cinquefoil is P(x) = x−2 -x−1 +1 -x +x2.
Sources
• American Mathematical Society, Knots and Their Polynomials, Feature Column.
• Weisstein, Eric W. "Skein Relationship". MathWorld.
• Morton, Hugh R.; Lukac, Sascha G. (2003), "HOMFLY polynomial of decorated Hopf link", Journal of Knot Theory and Its Ramifications, 12: 395–416, arXiv:math.GT/0108011, doi:10.1142/s0218216503002536.
Knot theory (knots and links)
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Wikipedia
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Steinitz's theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.[1]
This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions.[2] It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes.[3] Additionally, it has been applied in graph drawing, as a way to construct three-dimensional visualizations of abstract graphs.[4] Branko Grünbaum has called this theorem "the most important and deepest known result on 3-polytopes."[5]
The theorem appears in a 1922 publication of Ernst Steinitz,[6] after whom it is named. It can be proven by mathematical induction (as Steinitz did), by finding the minimum-energy state of a two-dimensional spring system and lifting the result into three dimensions, or by using the circle packing theorem. Several extensions of the theorem are known, in which the polyhedron that realizes a given graph has additional constraints; for instance, every polyhedral graph is the graph of a convex polyhedron with integer coordinates, or the graph of a convex polyhedron all of whose edges are tangent to a common midsphere.
Definitions and statement of the theorem
An undirected graph is a system of vertices and edges, each edge connecting two of the vertices. As is common in graph theory, for the purposes of Steinitz's theorem these graphs are restricted to being finite (the vertices and edges are finite sets) and simple (no two edges connect the same two vertices, and no edge connects a vertex to itself). From any polyhedron one can form a graph, by letting the vertices of the graph correspond to the vertices of the polyhedron and by connecting any two graph vertices by an edge whenever the corresponding two polyhedron vertices are the endpoints of an edge of the polyhedron. This graph is known as the skeleton of the polyhedron.[7]
A graph is planar if it can be drawn with its vertices as points in the Euclidean plane, and its edges as curves that connect these points, such that no two edge curves cross each other and such that the point representing a vertex lies on the curve representing an edge only when the vertex is an endpoint of the edge. By Fáry's theorem, every planar drawing can be straightened so that the curves representing the edges are line segments. A graph is 3-connected if it has more than three vertices and, after the removal of any two of its vertices, any other pair of vertices remain connected by a path. Steinitz's theorem states that these two conditions are both necessary and sufficient to characterize the skeletons of three-dimensional convex polyhedra: a given graph $G$ is the graph of a convex three-dimensional polyhedron, if and only if $G$ is planar and 3-vertex-connected.[5][8]
Proofs
One direction of Steinitz's theorem (the easier direction to prove) states that the graph of every convex polyhedron is planar and 3-connected. As shown in the illustration, planarity can be shown by using a Schlegel diagram: if one places a light source near one face of the polyhedron, and a plane on the other side, the shadows of the polyhedron edges will form a planar graph, embedded in such a way that the edges are straight line segments. The 3-connectivity of a polyhedral graph is a special case of Balinski's theorem that the graph of any $k$-dimensional convex polytope is $k$-connected. The connectivity of the graph of a polytope, after removing any $k-1$ of its vertices, can be proven by choosing one more vertex $v$, finding a linear function that is zero on the resulting set of $k$ vertices, and following the paths generated by the simplex method to connect every vertex to one of two extreme vertices of the linear function, with the chosen vertex $v$ connected to both.[9]
The other, more difficult, direction of Steinitz's theorem states that every planar 3-connected graph is the graph of a convex polyhedron. There are three standard approaches for this part: proofs by induction, lifting two-dimensional Tutte embeddings into three dimensions using the Maxwell–Cremona correspondence, and methods using the circle packing theorem to generate a canonical polyhedron.
Induction
Although Steinitz's original proof was not expressed in terms of graph theory, it can be rewritten in those terms, and involves finding a sequence of Δ-Y and Y-Δ transforms that reduce any 3-connected planar graph to $K_{4}$, the graph of the tetrahedron. A Y-Δ transform removes a degree-three vertex from a graph, adding edges between all of its former neighbors if those edges did not already exist; the reverse transformation, a Δ-Y transform, removes the edges of a triangle from a graph and replaces them by a new degree-three vertex adjacent to the same three vertices. Once such a sequence is found, it can be reversed and converted into geometric operations that build up the desired polyhedron step by step starting from a tetrahedron. Each Y-Δ transform in the reversed sequence can be performed geometrically by slicing off a degree-three vertex from a polyhedron. A Δ-Y transform in the reversed sequence can be performed geometrically by removing a triangular face from a polyhedron and extending its neighboring faces until the point where they meet, but only when that triple intersection point of the three neighboring faces is on the far side of the removed face from the polyhedron. When the triple intersection point is not on the far side of this face, a projective transformation of the polyhedron suffices to move it to the correct side. Therefore, by induction on the number of Δ-Y and Y-Δ transforms needed to reduce a given graph to $K_{4}$, every polyhedral graph can be realized as a polyhedron.[5]
A later work by Epifanov strengthened Steinitz's proof that every polyhedral graph can be reduced to $K_{4}$ by Δ-Y and Y-Δ transforms. Epifanov proved that if two vertices are specified in a planar graph, then the graph can be reduced to a single edge between those terminals by combining Δ-Y and Y-Δ transforms with series–parallel reductions.[10] Epifanov's proof was complicated and non-constructive, but it was simplified by Truemper using methods based on graph minors. Truemper observed that every grid graph is reducible by Δ-Y and Y-Δ transforms in this way, that this reducibility is preserved by graph minors, and that every planar graph is a minor of a grid graph.[11] This idea can be used to replace Steinitz's lemma that a reduction sequence exists. After this replacement, the rest of the proof can be carried out using induction in the same way as Steinitz's original proof.[8] For these proofs, carried out using any of the ways of finding sequences of Δ-Y and Y-Δ transforms, there exist polyhedral graphs that require a nonlinear number of steps. More precisely, every planar graph can be reduced using a number of steps at most proportional to $n^{3/2}$, and infinitely many graphs require a number of steps at least proportional to $n^{3/2}$, where $n$ is the number of vertices in the graph.[12][13]
An alternative form of induction proof is based on removing edges (and compressing out the degree-two vertices that might be left after this removal) or contracting edges and forming a minor of the given planar graph. Any polyhedral graph can be reduced to $K_{4}$ by a linear number of these operations, and again the operations can be reversed and the reversed operations performed geometrically, giving a polyhedral realization of the graph. However, while it is simpler to prove that a reduction sequence exists for this type of argument, and the reduction sequences are shorter, the geometric steps needed to reverse the sequence are more complicated.[14]
Lifting
Equilibrium stress on the graph of a cube
A frustum lifting the stressed drawing (with the same 2d positions) into 3d
If a graph is drawn in the plane with straight line edges, then an equilibrium stress is defined as an assignment of nonzero real numbers (weights) to the edges, with the property that each vertex is in the position given by the weighted average of its neighbors. According to the Maxwell–Cremona correspondence, an equilibrium stress can be lifted to a piecewise linear continuous three-dimensional surface such that the edges forming the boundaries between the flat parts of the surface project to the given drawing. The weight and length of each edge determines the difference in slopes of the surface on either side of the edge, and the condition that each vertex is in equilibrium with its neighbors is equivalent to the condition that these slope differences cause the surface to meet up with itself correctly in the neighborhood of the vertex. Positive weights translate to convex dihedral angles between two faces of the piecewise linear surface, and negative weights translate to concave dihedral angles. Conversely, every continuous piecewise-linear surface comes from an equilibrium stress in this way. If a finite planar graph is drawn and given an equilibrium stress in such a way that all interior edges of the drawing have positive weights, and all exterior edges have negative weights, then by translating this stress into a three-dimensional surface in this way, and then replacing the flat surface representing the exterior of the graph by its complement in the same plane, one obtains a convex polyhedron, with the additional property that its perpendicular projection onto the plane has no crossings.[15][16]
The Maxwell–Cremona correspondence has been used to obtain polyhedral realizations of polyhedral graphs by combining it with a planar graph drawing method of W. T. Tutte, the Tutte embedding. Tutte's method begins by fixing one face of a polyhedral graph into convex position in the plane. This face will become the outer face of a drawing of a graph. The method continues by setting up a system of linear equations in the vertex coordinates, according to which each remaining vertex should be placed at the average of its neighbors. Then as Tutte showed, this system of equations will have a unique solution in which each face of the graph is drawn as a convex polygon.[17] Intuitively, this solution describes the pattern that would be obtained by replacing the interior edges of the graph by ideal springs and letting them settle to their minimum-energy state.[18] The result is almost an equilibrium stress: if one assigns weight one to each interior edge, then each interior vertex of the drawing is in equilibrium. However, it is not always possible to assign negative numbers to the exterior edges so that they, too, are in equilibrium. Such an assignment is always possible when the outer face is a triangle, and so this method can be used to realize any polyhedral graph that has a triangular face. If a polyhedral graph does not contain a triangular face, its dual graph does contain a triangle and is also polyhedral, so one can realize the dual in this way and then realize the original graph as the polar polyhedron of the dual realization.[4][19] An alternative method for realizing polyhedra using liftings avoids duality by choosing any face with at most five vertices as the outer face. Every polyhedral graph has such a face, and by choosing the fixed shape of this face more carefully, the Tutte embedding of the rest of the graph can be lifted.[20]
Circle packing
According to one variant of the circle packing theorem, for every polyhedral graph, there exists a system of circles in the plane or on any sphere, representing the vertices and faces of the graph, so that:
• each two adjacent vertices of the graph are represented by tangent circles,
• each two adjacent faces of the graph are represented by tangent circle,
• each pair of a vertex and a face that it touches are represented by circles that cross at a right angle, and
• all other pairs of circles are separated from each other.[21]
The same system of circles forms a representation of the dual graph by swapping the roles of circles that represent vertices, and circles that represent faces. From any such representation on a sphere, embedded into three-dimensional Euclidean space, one can form a convex polyhedron that is combinatorially equivalent to the given graph, as an intersection of half-spaces whose boundaries pass through the face circles. From each vertex of this polyhedron, the horizon on the sphere, seen from that vertex, is the circle that represents it. This horizon property determines the three-dimensional position of each vertex, and the polyhedron can be equivalently defined as the convex hull of the vertices, positioned in this way. The sphere becomes the midsphere of the realization: each edge of the polyhedron is tangent to the sphere, at a point where two tangent vertex circles cross two tangent face circles.[22]
Realizations with additional properties
Integer coordinates
It is possible to prove a stronger form of Steinitz's theorem, that any polyhedral graph can be realized by a convex polyhedron whose coordinates are integers.[23] For instance, Steinitz's original induction-based proof can be strengthened in this way. However, the integers that would result from Steinitz's construction are doubly exponential in the number of vertices of the given polyhedral graph. Writing down numbers of this magnitude in binary notation would require an exponential number of bits.[19] Geometrically, this means that some features of the polyhedron may have size doubly exponentially larger than others, making the realizations derived from this method problematic for applications in graph drawing.[4]
Subsequent researchers have found lifting-based realization algorithms that use only a linear number of bits per vertex.[20][24] It is also possible to relax the requirement that the coordinates be integers, and assign coordinates in such a way that the $x$-coordinates of the vertices are distinct integers in the range from 0 to $2n-4$ and the other two coordinates are real numbers in the unit interval, so that each edge has length at least one while the overall polyhedron has linear volume.[25][26] Some polyhedral graphs are known to be realizable on grids of only polynomial size; in particular this is true for the pyramids (realizations of wheel graphs), prisms (realizations of prism graphs), and stacked polyhedra (realizations of Apollonian networks).[27]
Another way of stating the existence of integer realizations is that every three-dimensional convex polyhedron has a combinatorially equivalent integer polyhedron.[23] For instance, the regular dodecahedron is not itself an integer polyhedron, because of its regular pentagon faces, but it can be realized as an equivalent integer pyritohedron.[20] This is not always possible in higher dimensions, where there exist polytopes (such as the ones constructed from the Perles configuration) that have no integer equivalent.[28]
Equal slopes
A Halin graph is a special case of a polyhedral graph, formed from a planar-embedded tree (with no degree-two vertices) by connecting the leaves of the tree into a cycle. For Halin graphs, one can choose polyhedral realizations of a special type: the outer cycle forms a horizontal convex base face, and every other face lies directly above the base face (as in the polyhedra realized through lifting), with all of these upper faces having the same slope. Polyhedral surfaces with equal-slope faces over any base polygon (not necessarily convex) can be constructed from the polygon's straight skeleton, and an equivalent way of describing this realization is that the two-dimensional projection of the tree onto the base face forms its straight skeleton. The proof of this result uses induction: any rooted tree may reduced to a smaller tree by removing the leaves from an internal node whose children are all leaves, the Halin graph formed from the smaller tree has a realization by the induction hypothesis, and it is possible to modify this realization in order to add any number of leaf children to the tree node whose children were removed.[29]
Specifying the shape of a face
In any polyhedron that represents a given polyhedral graph $G$, the faces of $G$ are exactly the cycles in $G$ that do not separate $G$ into two components: that is, removing a facial cycle from $G$ leaves the rest of $G$ as a connected subgraph. Such cycles are called peripheral cycles. Thus, the combinatorial structure of the faces (but not their geometric shapes) is uniquely determined from the graph structure. Another strengthening of Steinitz's theorem, by Barnette and Grünbaum, states that for any polyhedral graph, any face of the graph, and any convex polygon representing that face, it is possible to find a polyhedral realization of the whole graph that has the specified shape for the designated face. This is related to a theorem of Tutte, that any polyhedral graph can be drawn in the plane with all faces convex and any specified shape for its outer face. However, the planar graph drawings produced by Tutte's method do not necessarily lift to convex polyhedra. Instead, Barnette and Grünbaum prove this result using an inductive method.[30] It is also always possible, given a polyhedral graph $G$ and an arbitrary cycle $C$ in $G$, to find a realization for which $C$ forms the silhouette of the realization under parallel projection.[31]
Tangent spheres
The realization of polyhedra using the circle packing theorem provides another strengthening of Steinitz's theorem: every 3-connected planar graph may be represented as a convex polyhedron in such a way that all of its edges are tangent to the same unit sphere, the midsphere of the polyhedron.[22] By performing a carefully chosen Möbius transformation of a circle packing before transforming it into a polyhedron, it is possible to find a polyhedral realization that realizes all the symmetries of the underlying graph, in the sense that every graph automorphism is a symmetry of the polyhedral realization.[32][33] More generally, if $G$ is a polyhedral graph and $K$ is any smooth three-dimensional convex body, it is possible to find a polyhedral representation of $G$ in which all edges are tangent to $K$.[34]
Circle packing methods can also be used to characterize the graphs of polyhedra that have a circumsphere through all their vertices, or an insphere tangent to all of their faces. (The polyhedra with a circumsphere are also significant in hyperbolic geometry as the ideal polyhedra.) In both cases, the existence of a sphere is equivalent to the solvability of a system of linear inequalities on positive real variables associated with each edge of the graph. In the case of the insphere, these variables must sum to exactly one on each face cycle of the graph, and to more than one on each non-face cycle. Dually, for the circumsphere, the variables must sum to one at each vertex, and more than one across each cut with two or more vertices on each side of the cut. Although there may be exponentially many linear inequalities to satisfy, a solution (if one exists) can be found in polynomial time using the ellipsoid method. The values of the variables from a solution determine the angles between pairs of circles in a circle packing whose corresponding polyhedron has the desired relation to its sphere.[35][36]
Related results
In any dimension higher than three, the algorithmic Steinitz problem consists of determining whether a given lattice is the face lattice of a convex polytope. It is unlikely to have polynomial time complexity, as it is NP-hard and more strongly complete for the existential theory of the reals, even for four-dimensional polytopes, by Richter-Gebert's universality theorem.[38] Here, the existential theory of the reals is a class of computational problems that can be formulated in terms of finding real variables that satisfy a given system of polynomial equations and inequalities. For the algorithmic Steinitz problem, the variables of such a problem can be the vertex coordinates of a polytope, and the equations and inequalities can be used to specify the flatness of each face in the given face lattice and the convexity of each angle between faces. Completeness means that every other problem in this class can be transformed into an equivalent instance of the algorithmic Steinitz problem, in polynomial time. The existence of such a transformation implies that, if the algorithmic Steinitz problem has a polynomial time solution, then so does every problem in the existential theory of the reals, and every problem in NP.[39] However, because a given graph may correspond to more than one face lattice, it is difficult to extend this completeness result to the problem of recognizing the graphs of 4-polytopes. Determining the computational complexity of this graph recognition problem remains open.[40]
Researchers have also found graph-theoretic characterizations of the graphs of certain special classes of three-dimensional non-convex polyhedra[37][41] and four-dimensional convex polytopes.[40][42][43] However, in both cases, the general problem remains unsolved. Indeed, even the problem of determining which complete graphs are the graphs of non-convex polyhedra (other than $K_{4}$ for the tetrahedron and $K_{7}$ for the Császár polyhedron) remains unsolved.[44]
Eberhard's theorem partially characterizes the multisets of polygons that can be combined to form the faces of a convex polyhedron. It can be proven by forming a 3-connected planar graph with the given set of polygon faces, and then applying Steinitz's theorem to find a polyhedral realization of that graph.[3]
László Lovász has shown a correspondence between polyhedral representations of graphs and matrices realizing the Colin de Verdière graph invariants of the same graphs. The Colin de Verdière invariant is the maximum corank of a weighted adjacency matrix of the graph, under some additional conditions that are irrelevant for polyhedral graphs. These are square symmetric matrices indexed by the vertices, with the weight of vertex $i$ in the diagonal coefficient $M_{i,i}$ and with the weight of edge $i,j$ in the off-diagonal coefficients $M_{i,j}$ and $M_{j,i}$. When vertices $i$ and $j$ are not adjacent, the coefficient $M_{i,j}$ is required to be zero. This invariant is at most three if and only if the graph is a planar graph. As Lovász shows, when the graph is polyhedral, a representation of it as a polyhedron can be obtained by finding a weighted adjacency matrix of corank three, finding three vectors forming a basis for its nullspace, using the coefficients of these vectors as coordinates for the vertices of a polyhedron, and scaling these vertices appropriately.[45]
History
The history of Steinitz's theorem is described by Grünbaum (2007),[46] who notes its first appearance in a cryptic form in a publication of Ernst Steinitz, originally written in 1916.[6] Steinitz provided more details in later lecture notes, published after his 1928 death. Although modern treatments of Steinitz's theorem state it as a graph-theoretic characterization of polyhedra, Steinitz did not use the language of graphs.[46] The graph-theoretic formulation of the theorem was introduced in the early 1960s by Branko Grünbaum and Theodore Motzkin, with its proof also converted to graph theory in Grünbaum's 1967 text Convex Polytopes.[46] The work of Epifanov on Δ-Y and Y-Δ transforms, strengthening Steinitz's proof, was motivated by other problems than the characterization of polyhedra. Truemper (1989) credits Grünbaum with observing the relevance of this work for Steinitz's theorem.[11]
The Maxwell–Cremona correspondence between stress diagrams and polyhedral liftings was developed in a series of papers by James Clerk Maxwell from 1864 to 1870, based on earlier work of Pierre Varignon, William Rankine, and others, and was popularized in the late 19th century by Luigi Cremona.[47] The observation that this correspondence can be used with the Tutte embedding to prove Steinitz's theorem comes from Eades & Garvan (1995);[4] see also Richter-Gebert (1996).[38]
The circle packing theorem was proved by Paul Koebe in 1936[48][49] and (independently) by E. M. Andreev in 1970;[49][50] it was popularized in the mid-1980s by William Thurston, who (despite citing Koebe and Andreev) is often credited as one of its discoverers.[49] Andreev's version of the theorem was already formulated as a Steinitz-like characterization for certain polyhedra in hyperbolic space,[50] and the use of circle packing to realize polyhedra with midspheres comes from the work of Thurston.[51] The problem of characterizing polyhedra with inscribed or circumscribed spheres, eventually solved using a method based on circle packing realizations, goes back to unpublished work of René Descartes circa 1630[52] and to Jakob Steiner in 1832;[35][53] the first examples of polyhedra that have no realization with a circumsphere or insphere were given by Steinitz in 1928.[35][54]
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7. More technically, this graph is the 1-skeleton; see Grünbaum (2003), p. 138, and Ziegler (1995), p. 64.
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15. Maxwell, J. Clerk (1864), "On reciprocal figures and diagrams of forces", Philosophical Magazine, 4th Series, 27 (182): 250–261, doi:10.1080/14786446408643663
16. Whiteley, Walter (1982), "Motions and stresses of projected polyhedra", Structural Topology, 7: 13–38, hdl:2099/989, MR 0721947
17. Tutte, W. T. (1963), "How to draw a graph", Proceedings of the London Mathematical Society, 13: 743–767, doi:10.1112/plms/s3-13.1.743, MR 0158387
18. Brandes, Ulrik (2001), "Drawing on physical analogies", in Kaufmann, Michael; Wagner, Dorothea (eds.), Drawing Graphs: Methods and Models, Lecture Notes in Computer Science, vol. 2025, Berlin: Springer, pp. 71–86, CiteSeerX 10.1.1.9.5023, doi:10.1007/3-540-44969-8_4, MR 1880146
19. Onn, Shmuel; Sturmfels, Bernd (1994), "A quantitative Steinitz' theorem", Beiträge zur Algebra und Geometrie, 35 (1): 125–129, MR 1287206
20. Ribó Mor, Ares; Rote, Günter; Schulz, André (2011), "Small grid embeddings of 3-polytopes", Discrete & Computational Geometry, 45 (1): 65–87, arXiv:0908.0488, doi:10.1007/s00454-010-9301-0, MR 2765520, S2CID 10141034
21. Brightwell, Graham R.; Scheinerman, Edward R. (1993), "Representations of planar graphs", SIAM Journal on Discrete Mathematics, 6 (2): 214–229, doi:10.1137/0406017, MR 1215229
22. Ziegler, Günter M. (2007), "Convex polytopes: extremal constructions and f-vector shapes. Section 1.3: Steinitz's theorem via circle packings", in Miller, Ezra; Reiner, Victor; Sturmfels, Bernd (eds.), Geometric Combinatorics, IAS/Park City Mathematics Series, vol. 13, American Mathematical Society, pp. 628–642, ISBN 978-0-8218-3736-8
23. Grünbaum (2003), theorem 13.2.3, p. 244, states this in an equivalent form where the coordinates are rational numbers.
24. Buchin, Kevin; Schulz, André (2010), "On the number of spanning trees a planar graph can have", in de Berg, Mark; Meyer, Ulrich (eds.), Algorithms - ESA 2010, 18th Annual European Symposium, Liverpool, UK, September 6-8, 2010, Proceedings, Part I, Lecture Notes in Computer Science, vol. 6346, Springer, pp. 110–121, CiteSeerX 10.1.1.746.942, doi:10.1007/978-3-642-15775-2_10, ISBN 978-3-642-15774-5, MR 2762847, S2CID 42211547
25. Chrobak, Marek; Goodrich, Michael T.; Tamassia, Roberto (1996), "Convex drawings of graphs in two and three dimensions", Proceedings of the 12th ACM Symposium on Computational Geometry (SoCG '96), ACM, pp. 319–328, doi:10.1145/237218.237401, S2CID 1015103
26. Schulz, André (2011), "Drawing 3-polytopes with good vertex resolution", Journal of Graph Algorithms and Applications, 15 (1): 33–52, doi:10.7155/jgaa.00216, MR 2776000
27. Demaine, Erik D.; Schulz, André (2017), "Embedding stacked polytopes on a polynomial-size grid", Discrete & Computational Geometry, 57 (4): 782–809, arXiv:1403.7980, doi:10.1007/s00454-017-9887-6, MR 3639604, S2CID 104867
28. Grünbaum (2003), p. 96a.
29. Aichholzer, Oswin; Cheng, Howard; Devadoss, Satyan L.; Hackl, Thomas; Huber, Stefan; Li, Brian; Risteski, Andrej (2012), "What makes a Tree a Straight Skeleton?" (PDF), Proceedings of the 24th Canadian Conference on Computational Geometry (CCCG'12)
30. Barnette, David W.; Grünbaum, Branko (1970), "Preassigning the shape of a face", Pacific Journal of Mathematics, 32 (2): 299–306, doi:10.2140/pjm.1970.32.299, MR 0259744
31. Barnette, David W. (1970), "Projections of 3-polytopes", Israel Journal of Mathematics, 8 (3): 304–308, doi:10.1007/BF02771563, MR 0262923, S2CID 120791830
32. Hart, George W. (1997), "Calculating canonical polyhedra", Mathematica in Education and Research, 6 (3): 5–10
33. Bern, Marshall W.; Eppstein, David (2001), "Optimal Möbius transformations for information visualization and meshing", in Dehne, Frank K. H. A.; Sack, Jörg-Rüdiger; Tamassia, Roberto (eds.), Algorithms and Data Structures, 7th International Workshop, WADS 2001, Providence, RI, USA, August 8-10, 2001, Proceedings, Lecture Notes in Computer Science, vol. 2125, Springer, pp. 14–25, arXiv:cs/0101006, doi:10.1007/3-540-44634-6_3, S2CID 3266233
34. Schramm, Oded (1992), "How to cage an egg", Inventiones Mathematicae, 107 (3): 543–560, Bibcode:1992InMat.107..543S, doi:10.1007/BF01231901, MR 1150601, S2CID 189830473
35. Rivin, Igor (1996), "A characterization of ideal polyhedra in hyperbolic 3-space", Annals of Mathematics, Second Series, 143 (1): 51–70, doi:10.2307/2118652, JSTOR 2118652, MR 1370757
36. Dillencourt, Michael B.; Smith, Warren D. (1996), "Graph-theoretical conditions for inscribability and Delaunay realizability", Discrete Mathematics, 161 (1–3): 63–77, doi:10.1016/0012-365X(95)00276-3, MR 1420521, S2CID 16382428
37. Eppstein, David; Mumford, Elena (2014), "Steinitz theorems for simple orthogonal polyhedra", Journal of Computational Geometry, 5 (1): 179–244, doi:10.20382/jocg.v5i1a10, MR 3259910, S2CID 8531578
38. Richter-Gebert, Jürgen (1996), Realization Spaces of Polytopes, Lecture Notes in Mathematics, vol. 1643, Springer-Verlag, CiteSeerX 10.1.1.2.3495, doi:10.1007/BFb0093761, ISBN 978-3-540-62084-6, MR 1482230
39. Schaefer, Marcus (2013), "Realizability of graphs and linkages", in Pach, János (ed.), Thirty Essays on Geometric Graph Theory, New York: Springer, pp. 461–482, doi:10.1007/978-1-4614-0110-0_24, MR 3205168
40. Eppstein, David (2020), "Treetopes and their graphs", Discrete & Computational Geometry, 64 (2): 259–289, arXiv:1510.03152, doi:10.1007/s00454-020-00177-0, MR 4131546, S2CID 213885326
41. Hong, Seok-Hee; Nagamochi, Hiroshi (2011), "Extending Steinitz's theorem to upward star-shaped polyhedra and spherical polyhedra", Algorithmica, 61 (4): 1022–1076, doi:10.1007/s00453-011-9570-x, MR 2852056, S2CID 12622357
42. Blind, Roswitha; Mani-Levitska, Peter (1987), "Puzzles and polytope isomorphisms", Aequationes Mathematicae, 34 (2–3): 287–297, doi:10.1007/BF01830678, MR 0921106, S2CID 120222616
43. Kalai, Gil (1988), "A simple way to tell a simple polytope from its graph", Journal of Combinatorial Theory, Series A, 49 (2): 381–383, doi:10.1016/0097-3165(88)90064-7, MR 0964396
44. Ziegler, Günter M. (2008), "Polyhedral surfaces of high genus", Discrete Differential Geometry, Oberwolfach Seminars, vol. 38, Springer, pp. 191–213, arXiv:math/0412093, doi:10.1007/978-3-7643-8621-4_10, ISBN 978-3-7643-8620-7, MR 2405667, S2CID 15911143
45. Lovász, László (2001), "Steinitz representations of polyhedra and the Colin de Verdière number", Journal of Combinatorial Theory, Series B, 82 (2): 223–236, doi:10.1006/jctb.2000.2027, MR 1842113
46. Grünbaum, Branko (2007), "Graphs of polyhedra; polyhedra as graphs", Discrete Mathematics, 307 (3–5): 445–463, doi:10.1016/j.disc.2005.09.037, hdl:1773/2276, MR 2287486
47. Erickson, Jeff; Lin, Patrick (2020), "A toroidal Maxwell–Cremona-Delaunay correspondence", in Cabello, Sergio; Chen, Danny Z. (eds.), 36th International Symposium on Computational Geometry (SoCG 2020), Leibniz International Proceedings in Informatics (LIPIcs), vol. 164, Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum für Informatik, pp. 40:1–40:17, arXiv:2003.10057, doi:10.4230/LIPIcs.SoCG.2020.40, ISBN 978-3-95977-143-6, S2CID 209514295
48. Koebe, Paul (1936), "Kontaktprobleme der Konformen Abbildung", Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig: Mathematisch-Physische Klasse (in German), 88: 141–164
49. Stephenson, Kenneth (2003), "Circle packing: a mathematical tale" (PDF), Notices of the American Mathematical Society, 50 (11): 1376–1388, CiteSeerX 10.1.1.101.5592, MR 2011604
50. Andreev, E. M. (1970), "Convex polyhedra in Lobačevskiĭ spaces", Matematicheskii Sbornik, 81 (123): 445–478, Bibcode:1970SbMat..10..413A, doi:10.1070/SM1970v010n03ABEH001677, MR 0259734
51. Schramm, Oded (1991), "Existence and uniqueness of packings with specified combinatorics", Israel Journal of Mathematics, 73 (3): 321–341, doi:10.1007/BF02773845, MR 1135221, S2CID 121855202; see discussion following Corollary 3.8, p. 329
52. Federico, Pasquale Joseph (1982), Descartes on Polyhedra: A Study of the "De solidorum elementis", Sources in the History of Mathematics and Physical Sciences, vol. 4, Springer, p. 52
53. Steiner, Jakob (1832), "Question 77", Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander (in German), Berlin: G. Fincke, p. 316
54. Steinitz, Ernst (1928), "Über isoperimetrische Probleme bei konvexen Polyedern", Journal für die Reine und Angewandte Mathematik (in German), 1928 (159): 133–143, doi:10.1515/crll.1928.159.133, MR 1581158, S2CID 199546274
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Wikipedia
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Skeleton (category theory)
In mathematics, a skeleton of a category is a subcategory that, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the "smallest" equivalent category, which captures all "categorical properties" of the original. In fact, two categories are equivalent if and only if they have isomorphic skeletons. A category is called skeletal if isomorphic objects are necessarily identical.
Definition
A skeleton of a category C is an equivalent category D in which no two distinct objects are isomorphic. It is generally considered to be a subcategory. In detail, a skeleton of C is a category D such that:
• D is a subcategory of C: every object of D is an object of C
$\mathrm {Ob} (D)\subseteq \mathrm {Ob} (C)$
for every pair of objects d1 and d2 of D, the morphisms in D are morphisms in C, i.e.
$\mathrm {Hom} _{D}(d_{1},d_{2})\subseteq \mathrm {Hom} _{C}(d_{1},d_{2})$
and the identities and compositions in D are the restrictions of those in C.
• The inclusion of D in C is full, meaning that for every pair of objects d1 and d2 of D we strengthen the above subset relation to an equality:
$\mathrm {Hom} _{D}(d_{1},d_{2})=\mathrm {Hom} _{C}(d_{1},d_{2})$
• The inclusion of D in C is essentially surjective: Every C-object is isomorphic to some D-object.
• D is skeletal: No two distinct D-objects are isomorphic.
Existence and uniqueness
It is a basic fact that every small category has a skeleton; more generally, every accessible category has a skeleton. (This is equivalent to the axiom of choice.) Also, although a category may have many distinct skeletons, any two skeletons are isomorphic as categories, so up to isomorphism of categories, the skeleton of a category is unique.
The importance of skeletons comes from the fact that they are (up to isomorphism of categories), canonical representatives of the equivalence classes of categories under the equivalence relation of equivalence of categories. This follows from the fact that any skeleton of a category C is equivalent to C, and that two categories are equivalent if and only if they have isomorphic skeletons.
Examples
• The category Set of all sets has the subcategory of all cardinal numbers as a skeleton.
• The category K-Vect of all vector spaces over a fixed field $K$ has the subcategory consisting of all powers $K^{(\alpha )}$, where α is any cardinal number, as a skeleton; for any finite m and n, the maps $K^{m}\to K^{n}$ are exactly the n × m matrices with entries in K.
• FinSet, the category of all finite sets has FinOrd, the category of all finite ordinal numbers, as a skeleton.
• The category of all well-ordered sets has the subcategory of all ordinal numbers as a skeleton.
• A preorder, i.e. a small category such that for every pair of objects $A,B$, the set ${\mbox{Hom}}(A,B)$ either has one element or is empty, has a partially ordered set as a skeleton.
See also
• Glossary of category theory
• Thin category
References
• Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). Abstract and Concrete Categories. Originally published by John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
• Robert Goldblatt (1984). Topoi, the Categorial Analysis of Logic (Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications.
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Wikipedia
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Skellam distribution
The Skellam distribution is the discrete probability distribution of the difference $N_{1}-N_{2}$ of two statistically independent random variables $N_{1}$ and $N_{2},$ each Poisson-distributed with respective expected values $\mu _{1}$ and $\mu _{2}$. It is useful in describing the statistics of the difference of two images with simple photon noise, as well as describing the point spread distribution in sports where all scored points are equal, such as baseball, hockey and soccer.
Skellam
Probability mass function
Examples of the probability mass function for the Skellam distribution. The horizontal axis is the index k. (The function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters $\mu _{1}\geq 0,~~\mu _{2}\geq 0$
Support $k\in \{\ldots ,-2,-1,0,1,2,\ldots \}$
PMF $e^{-(\mu _{1}\!+\!\mu _{2})}\left({\frac {\mu _{1}}{\mu _{2}}}\right)^{k/2}\!\!I_{k}(2{\sqrt {\mu _{1}\mu _{2}}})$
Mean $\mu _{1}-\mu _{2}\,$
Median N/A
Variance $\mu _{1}+\mu _{2}\,$
Skewness ${\frac {\mu _{1}-\mu _{2}}{(\mu _{1}+\mu _{2})^{3/2}}}$
Ex. kurtosis ${\frac {1}{\mu _{1}+\mu _{2}}}$
MGF $e^{-(\mu _{1}+\mu _{2})+\mu _{1}e^{t}+\mu _{2}e^{-t}}$
CF $e^{-(\mu _{1}+\mu _{2})+\mu _{1}e^{it}+\mu _{2}e^{-it}}$
The distribution is also applicable to a special case of the difference of dependent Poisson random variables, but just the obvious case where the two variables have a common additive random contribution which is cancelled by the differencing: see Karlis & Ntzoufras (2003) for details and an application.
The probability mass function for the Skellam distribution for a difference $K=N_{1}-N_{2}$ between two independent Poisson-distributed random variables with means $\mu _{1}$ and $\mu _{2}$ is given by:
$p(k;\mu _{1},\mu _{2})=\Pr\{K=k\}=e^{-(\mu _{1}+\mu _{2})}\left({\mu _{1} \over \mu _{2}}\right)^{k/2}I_{k}(2{\sqrt {\mu _{1}\mu _{2}}})$
where Ik(z) is the modified Bessel function of the first kind. Since k is an integer we have that Ik(z)=I|k|(z).
Derivation
The probability mass function of a Poisson-distributed random variable with mean μ is given by
$p(k;\mu )={\mu ^{k} \over k!}e^{-\mu }.\,$
for $k\geq 0$ (and zero otherwise). The Skellam probability mass function for the difference of two independent counts $K=N_{1}-N_{2}$ is the convolution of two Poisson distributions: (Skellam, 1946)
${\begin{aligned}p(k;\mu _{1},\mu _{2})&=\sum _{n=-\infty }^{\infty }p(k+n;\mu _{1})p(n;\mu _{2})\\&=e^{-(\mu _{1}+\mu _{2})}\sum _{n=\max(0,-k)}^{\infty }{{\mu _{1}^{k+n}\mu _{2}^{n}} \over {n!(k+n)!}}\end{aligned}}$
Since the Poisson distribution is zero for negative values of the count $(p(N<0;\mu )=0)$, the second sum is only taken for those terms where $n\geq 0$ and $n+k\geq 0$. It can be shown that the above sum implies that
${\frac {p(k;\mu _{1},\mu _{2})}{p(-k;\mu _{1},\mu _{2})}}=\left({\frac {\mu _{1}}{\mu _{2}}}\right)^{k}$
so that:
$p(k;\mu _{1},\mu _{2})=e^{-(\mu _{1}+\mu _{2})}\left({\mu _{1} \over \mu _{2}}\right)^{k/2}I_{|k|}(2{\sqrt {\mu _{1}\mu _{2}}})$
where I k(z) is the modified Bessel function of the first kind. The special case for $\mu _{1}=\mu _{2}(=\mu )$ is given by Irwin (1937):
$p\left(k;\mu ,\mu \right)=e^{-2\mu }I_{|k|}(2\mu ).$
Using the limiting values of the modified Bessel function for small arguments, we can recover the Poisson distribution as a special case of the Skellam distribution for $\mu _{2}=0$.
Properties
As it is a discrete probability function, the Skellam probability mass function is normalized:
$\sum _{k=-\infty }^{\infty }p(k;\mu _{1},\mu _{2})=1.$
We know that the probability generating function (pgf) for a Poisson distribution is:
$G\left(t;\mu \right)=e^{\mu (t-1)}.$
It follows that the pgf, $G(t;\mu _{1},\mu _{2})$, for a Skellam probability mass function will be:
${\begin{aligned}G(t;\mu _{1},\mu _{2})&=\sum _{k=-\infty }^{\infty }p(k;\mu _{1},\mu _{2})t^{k}\\[4pt]&=G\left(t;\mu _{1}\right)G\left(1/t;\mu _{2}\right)\\[4pt]&=e^{-(\mu _{1}+\mu _{2})+\mu _{1}t+\mu _{2}/t}.\end{aligned}}$
Notice that the form of the probability-generating function implies that the distribution of the sums or the differences of any number of independent Skellam-distributed variables are again Skellam-distributed. It is sometimes claimed that any linear combination of two Skellam distributed variables are again Skellam-distributed, but this is clearly not true since any multiplier other than $\pm 1$ would change the support of the distribution and alter the pattern of moments in a way that no Skellam distribution can satisfy.
The moment-generating function is given by:
$M\left(t;\mu _{1},\mu _{2}\right)=G(e^{t};\mu _{1},\mu _{2})=\sum _{k=0}^{\infty }{t^{k} \over k!}\,m_{k}$
which yields the raw moments mk . Define:
$\Delta \ {\stackrel {\mathrm {def} }{=}}\ \mu _{1}-\mu _{2}\,$
$\mu \ {\stackrel {\mathrm {def} }{=}}\ (\mu _{1}+\mu _{2})/2.\,$
Then the raw moments mk are
$m_{1}=\left.\Delta \right.\,$
$m_{2}=\left.2\mu +\Delta ^{2}\right.\,$
$m_{3}=\left.\Delta (1+6\mu +\Delta ^{2})\right.\,$
The central moments M k are
$M_{2}=\left.2\mu \right.,\,$
$M_{3}=\left.\Delta \right.,\,$
$M_{4}=\left.2\mu +12\mu ^{2}\right..\,$
The mean, variance, skewness, and kurtosis excess are respectively:
${\begin{aligned}\operatorname {E} (n)&=\Delta ,\\[4pt]\sigma ^{2}&=2\mu ,\\[4pt]\gamma _{1}&=\Delta /(2\mu )^{3/2},\\[4pt]\gamma _{2}&=1/2.\end{aligned}}$
The cumulant-generating function is given by:
$K(t;\mu _{1},\mu _{2})\ {\stackrel {\mathrm {def} }{=}}\ \ln(M(t;\mu _{1},\mu _{2}))=\sum _{k=0}^{\infty }{t^{k} \over k!}\,\kappa _{k}$
which yields the cumulants:
$\kappa _{2k}=\left.2\mu \right.$
$\kappa _{2k+1}=\left.\Delta \right..$
For the special case when μ1 = μ2, an asymptotic expansion of the modified Bessel function of the first kind yields for large μ:
$p(k;\mu ,\mu )\sim {1 \over {\sqrt {4\pi \mu }}}\left[1+\sum _{n=1}^{\infty }(-1)^{n}{\{4k^{2}-1^{2}\}\{4k^{2}-3^{2}\}\cdots \{4k^{2}-(2n-1)^{2}\} \over n!\,2^{3n}\,(2\mu )^{n}}\right].$
(Abramowitz & Stegun 1972, p. 377). Also, for this special case, when k is also large, and of order of the square root of 2μ, the distribution tends to a normal distribution:
$p(k;\mu ,\mu )\sim {e^{-k^{2}/4\mu } \over {\sqrt {4\pi \mu }}}.$
These special results can easily be extended to the more general case of different means.
Bounds on weight above zero
If $X\sim \operatorname {Skellam} (\mu _{1},\mu _{2})$, with $\mu _{1}<\mu _{2}$, then
${\frac {\exp(-({\sqrt {\mu _{1}}}-{\sqrt {\mu _{2}}})^{2})}{(\mu _{1}+\mu _{2})^{2}}}-{\frac {e^{-(\mu _{1}+\mu _{2})}}{2{\sqrt {\mu _{1}\mu _{2}}}}}-{\frac {e^{-(\mu _{1}+\mu _{2})}}{4\mu _{1}\mu _{2}}}\leq \Pr\{X\geq 0\}\leq \exp(-({\sqrt {\mu _{1}}}-{\sqrt {\mu _{2}}})^{2})$
Details can be found in Poisson distribution#Poisson races
References
• Abramowitz, Milton; Stegun, Irene A., eds. (June 1965). Handbook of mathematical functions with formulas, graphs, and mathematical tables (Unabridged and unaltered republ. [der Ausg.] 1964, 5. Dover printing ed.). Dover Publications. pp. 374–378. ISBN 0486612724. Retrieved 27 September 2012.
• Irwin, J. O. (1937) "The frequency distribution of the difference between two independent variates following the same Poisson distribution." Journal of the Royal Statistical Society: Series A, 100 (3), 415–416. JSTOR 2980526
• Karlis, D. and Ntzoufras, I. (2003) "Analysis of sports data using bivariate Poisson models". Journal of the Royal Statistical Society, Series D, 52 (3), 381–393. doi:10.1111/1467-9884.00366
• Karlis D. and Ntzoufras I. (2006). Bayesian analysis of the differences of count data. Statistics in Medicine, 25, 1885–1905.
• Skellam, J. G. (1946) "The frequency distribution of the difference between two Poisson variates belonging to different populations". Journal of the Royal Statistical Society, Series A, 109 (3), 296. JSTOR 2981372
See also
• Ratio distribution for (truncated) Poisson distributions
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
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