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Wikipedia:Moncef Ben Salem#0	Moncef Ben Salem (Tunisian Arabic: المنصف بن سالم; February 1, 1953 – March 24, 2015) was a Tunisian politician and university professor. He served as the Minister of Higher Education and Scientific Research under Prime Minister Hamadi Jebali. == Biography == === Early life === Moncef Ben Salem was born on 1 April 1953. He received a BA degree in Mathematics and Physics 1972 and a master's degree in Maths in 1974. He received PhDs in Maths and Physics from the University of Toulouse and Supméca in Paris. He was a member of the Union of Arab Mathematicians and Physicians from 1980 to 1987. He co-founded the Ecole Nationale d'Ingénieurs de Sfax in 1983 === Political activism and career === He was a member of the Ennahda Movement. He was critical of Presidents Habib Bourguiba and Zine El Abidine Ben Ali, calling Bourguiba a "zionist". As a result of his political activism under Ben Ali, he was jailed for eighteen months from 1987 to 1989, and from 1990 to 1993. In 1987, his passport was revoked, and he was forbidden from leaving Tunisia or travelling inside the country for twenty years, as were his children. From 1993 to 2011, he was forced to live under surveillance. He was also forbidden to work as a university professor. During those years, he was supported by the Canadian Committee of Scientists and Scholars and the American Mathematical Society. He later taught at the University of Maryland in the US, at the French National Centre for Scientific Research in France, Italy, Germany, Belgium, and at the University of Sfax in Tunisia. === Minister === On 20 December 2011, after former President Ben Ali was deposed, he joined the Jebali Cabinet as Minister of Higher Education. === Death === He died on March 24, 2015, leaving a widow and 4 children. == References ==
Wikipedia:Monika Ludwig#0	Monika Ludwig (born 1966 in Cologne) is an Austrian mathematician, University Professor of Convex and Discrete Geometry at the Vienna University of Technology. == Academic career == Ludwig earned a Dipl.-Ing. degree from the Vienna University of Technology in 1990, and a doctorate in 1994 under the supervision of Peter M. Gruber. She remained at the same university as an assistant and associate professor from 1994 until 2007, when she moved to the Polytechnic Institute of New York University. She returned to the Vienna University of Technology as a full professor in 2010. == Awards and honors == Ludwig won the Edmund and Rosa Hlawka Prize of the Austrian Academy of Sciences, given to an outstanding Austrian researcher in geometry of numbers or numerical analysis under the age of 30, in 1998. She won the Prize of the Austrian Mathematical Society in 2004. She became a corresponding member of the Austrian Academy of Sciences in 2011, and a fellow of the American Mathematical Society in 2012. She became a full member of the Austrian Academy of Sciences in 2013. == Notable publications == Ludwig, Monika; Reitzner, Matthias (15 October 1999), "A characterization of affine surface area", Advances in Mathematics, 147 (1): 138–172, doi:10.1006/aima.1999.1832 Ludwig, Monika (25 December 2002), "Projection bodies and valuations", Advances in Mathematics, 172 (2): 158–168, doi:10.1016/S0001-8708(02)00021-X Ludwig, Monika (15 July 2003), "Ellipsoids and matrix-valued valuations", Duke Mathematical Journal, 119 (1): 159–188, CiteSeerX 10.1.1.207.4952, doi:10.1215/S0012-7094-03-11915-8 Ludwig, Monika (2005), "Minkowski valuations", Transactions of the American Mathematical Society, 357 (10): 4191–4213, doi:10.1090/S0002-9947-04-03666-9 Haberl, Christoph; Ludwig, Monika (2006), "A characterization of L p {\displaystyle L_{p}} intersection bodies", International Mathematics Research Notices, 2006: 10548, doi:10.1155/IMRN/2006/10548 Ludwig, Monika (2006), "Intersection bodies and valuations", American Journal of Mathematics, 128 (6): 1409–1428, CiteSeerX 10.1.1.60.8035, doi:10.1353/ajm.2006.0046, JSTOR 40068039, S2CID 11303696 Ludwig, Monika (20 August 2010), "General affine surface areas", Advances in Mathematics, 224 (6): 2346–2360, arXiv:0908.2191, doi:10.1016/j.aim.2010.02.004 Ludwig, Monika; Reitzner, Matthias (2010), "A classification of S L ( n ) {\displaystyle SL(n)} invariant valuations", Annals of Mathematics, Second series, 172 (2): 1219–1267, doi:10.4007/annals.2010.172.1219, JSTOR 29764637 == References == == External links == Monika Ludwig publications indexed by Google Scholar
Wikipedia:Monique Dauge#0	Monique Dauge (born 1956) is a French mathematician and numerical analyst specializing in partial differential equations, spectral theory, and applications to scientific computing. She is an emeritus senior researcher at the French National Centre for Scientific Research (CNRS), associated with the University of Rennes 1. == Education and career == Dauge was born on 6 October 1956 in Nantes, and earned a diploma and agrégation in 1978 at the University of Nantes. In 1980 she defended a doctoral thesis at Nantes, Etude de l’opérateur de Stokes dans un polygone : régularité, singularités et théorème d’indice, and in 1986 she completed her habilitation there with the habilitation thesis Régularités et singularités des solutions de problèmes aux limites elliptiques sur des domaines singuliers de type à coins, supervised by Lai The Pham. Meanwhile, she became junior researcher for the CNRS in 1980, and researcher in 1984, both associated with the University of Nantes. In 1996 she became a director of research for the CNRS, and moved to the University of Rennes. She retired as an emeritus senior researcher in 2021. == Selected publications == Dauge is the author of Elliptic boundary value problems on corner domains: Smoothness and asymptotics of solutions (Lecture Notes in Mathematics 1341, Springer, 1988). She is the coauthor of Spectral methods for axisymmetric domains: Numerical algorithms and tests (with Christine Bernardi and Yvon Maday, with contributions from Mejdi Azaïez, Gauthier-Villars, 1999). Her many research publications include the highly cited paper "Vector potentials in three‐dimensional non‐smooth domains" (with Chérif Amrouche, Christine Bernardi, and Vivette Girault, Mathematical Methods in the Applied Sciences, 1998). == References == == External links == Home page Monique Dauge publications indexed by Google Scholar
Wikipedia:Monique Laurent#0	Monique Laurent (born 1960) is a French computer scientist and mathematician who is an expert in mathematical optimization. She is a researcher at the Centrum Wiskunde & Informatica in Amsterdam where she is also a member of the Management Team. Laurent also holds a part-time position as a professor of econometrics and operations research at Tilburg University. == Education and career == Laurent earned a doctorate from Paris Diderot University in 1986, under the supervision of Michel Deza. She worked at CNRS from 1988 to 1997, when she moved to CWI. She took a second position at Tilburg in 2009. == Book == With Deza, Laurent is the author of the book Geometry of Cuts and Metrics (Algorithms and Combinatorics 15, Springer, 1997). == Awards and honors == She was an invited speaker at the International Congress of Mathematicians in 2014. She was elected as a fellow of the Society for Industrial and Applied Mathematics in 2017, "for contributions to discrete and polynomial optimization and revealing interactions between them". She has been a member of the Royal Netherlands Academy of Arts and Sciences since 2018. In 2024, she was awarded a Gauss Lecture by the German Mathematical Society. == References == == External links == Home page at CWI Google scholar profile Media related to Monique Laurent (mathematician) at Wikimedia Commons
Wikipedia:Monique Teillaud#0	Monique Teillaud is a French researcher in computational geometry at the French Institute for Research in Computer Science and Automation (INRIA) in Nancy, France. She moved to Nancy in 2014 from a different INRIA center in Sophia Antipolis, where she was one of the developers of CGAL, a software library of computational geometry algorithms. Teillaud graduated from the École Normale Supérieure de Jeunes Filles in 1985, she then got a position at École nationale supérieure d'informatique pour l'industrie et l'entreprise before moving to Inria in 1989. She completed her Ph.D. in 1991 at Paris-Sud University under the supervision of Jean-Daniel Boissonnat. She was the 2008 program chair of the Symposium on Computational Geometry. She is also the author or editor of two books in computational geometry: Towards Dynamic Randomized Algorithms in Computational Geometry (Lecture Notes in Computer Science 758, Springer, 1993) Effective Computational Geometry for Curves and Surfaces (edited with Boissonat, Springer, 2007) == References == == External links == Monique Teillaud publications indexed by Google Scholar
Wikipedia:Monk's formula#0	In mathematics, Monk's formula, found by Monk (1959), is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold. Write tij for the transposition (i j), and si = ti,i+1. Then 𝔖sr = x1 + ⋯ + xr, and Monk's formula states that for a permutation w, S s r S w = ∑ i ≤ r < j ℓ ( w t i j ) = ℓ ( w ) + 1 S w t i j , {\displaystyle {\mathfrak {S}}_{s_{r}}{\mathfrak {S}}_{w}=\sum _{{i\leq r<j} \atop {\ell (wt_{ij})=\ell (w)+1}}{\mathfrak {S}}_{wt_{ij}},} where ℓ ( w ) {\displaystyle \ell (w)} is the length of w. The pairs (i, j) appearing in the sum are exactly those such that i ≤ r < j, wi < wj, and there is no i < k < j with wi < wk < wj; each wtij is a cover of w in Bruhat order. == References == Monk, D. (1959), "The geometry of flag manifolds", Proceedings of the London Mathematical Society, Third Series, 9 (2): 253–286, CiteSeerX 10.1.1.1033.7188, doi:10.1112/plms/s3-9.2.253, ISSN 0024-6115, MR 0106911
Wikipedia:Monodromy#0	In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of monodromy comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a monodromy group: a group of transformations acting on the data that encodes what happens as we "run round" in one dimension. Lack of monodromy is sometimes called polydromy. == Definition == Let X {\displaystyle X} be a connected and locally connected based topological space with base point x {\displaystyle x} , and let p : X ~ → X {\displaystyle p:{\tilde {X}}\to X} be a covering with fiber F = p − 1 ( x ) {\displaystyle F=p^{-1}(x)} . For a loop γ : [ 0 , 1 ] → X {\displaystyle \gamma :[0,1]\to X} based at x {\displaystyle x} , denote a lift under the covering map, starting at a point x ~ ∈ F {\displaystyle {\tilde {x}}\in F} , by γ ~ {\displaystyle {\tilde {\gamma }}} . Finally, we denote by x ~ ⋅ γ {\displaystyle {\tilde {x}}\cdot \gamma } the endpoint γ ~ ( 1 ) {\displaystyle {\tilde {\gamma }}(1)} , which is generally different from x ~ {\displaystyle {\tilde {x}}} . There are theorems which state that this construction gives a well-defined group action of the fundamental group π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} on F {\displaystyle F} , and that the stabilizer of x ~ {\displaystyle {\tilde {x}}} is exactly p ∗ ( π 1 ( X ~ , x ~ ) ) {\displaystyle p_{}\left(\pi _{1}\left({\tilde {X}},{\tilde {x}}\right)\right)} , that is, an element [ γ ] {\displaystyle [\gamma ]} fixes a point in F {\displaystyle F} if and only if it is represented by the image of a loop in X ~ {\displaystyle {\tilde {X}}} based at x ~ {\displaystyle {\tilde {x}}} . This action is called the monodromy action and the corresponding homomorphism π 1 ( X , x ) → Aut ⁡ ( H ∗ ( F x ) ) {\displaystyle \pi _{1}(X,x)\to \operatorname {Aut} (H_{}(F_{x}))} into the automorphism group on F {\displaystyle F} is the algebraic monodromy. The image of this homomorphism is the monodromy group. There is another map π 1 ( X , x ) → Diff ⁡ ( F x ) / Is ⁡ ( F x ) {\displaystyle \pi _{1}(X,x)\to \operatorname {Diff} (F_{x})/\operatorname {Is} (F_{x})} whose image is called the topological monodromy group. == Example == These ideas were first made explicit in complex analysis. In the process of analytic continuation, a function that is an analytic function F ( z ) {\displaystyle F(z)} in some open subset E {\displaystyle E} of the punctured complex plane C ∖ { 0 } {\displaystyle \mathbb {C} \backslash \{0\}} may be continued back into E {\displaystyle E} , but with different values. For example, take F ( z ) = log ⁡ ( z ) E = { z ∈ C ∣ Re ⁡ ( z ) > 0 } . {\displaystyle {\begin{aligned}F(z)&=\log(z)\\E&=\{z\in \mathbb {C} \mid \operatorname {Re} (z)>0\}.\end{aligned}}} Then analytic continuation anti-clockwise round the circle \| z \| = 1 {\displaystyle \|z\|=1} will result in the return not to F ( z ) {\displaystyle F(z)} but to F ( z ) + 2 π i . {\displaystyle F(z)+2\pi i.} In this case the monodromy group is the infinite cyclic group, and the covering space is the universal cover of the punctured complex plane. This cover can be visualized as the helicoid with parametric equations ( x , y , z ) = ( ρ cos ⁡ θ , ρ sin ⁡ θ , θ ) {\displaystyle (x,y,z)=(\rho \cos \theta ,\rho \sin \theta ,\theta )} restricted to ρ > 0 {\displaystyle \rho >0} . The covering map is a vertical projection, in a sense collapsing the spiral in the obvious way to get a punctured plane. == Differential equations in the complex domain == One important application is to differential equations, where a single solution may give further linearly independent solutions by analytic continuation. Linear differential equations defined in an open, connected set S {\displaystyle S} in the complex plane have a monodromy group, which (more precisely) is a linear representation of the fundamental group of S {\displaystyle S} , summarising all the analytic continuations round loops within S {\displaystyle S} . The inverse problem, of constructing the equation (with regular singularities), given a representation, is a Riemann–Hilbert problem. For a regular (and in particular Fuchsian) linear system one usually chooses as generators of the monodromy group the operators M j {\displaystyle M_{j}} corresponding to loops each of which circumvents just one of the poles of the system counterclockwise. If the indices j {\displaystyle j} are chosen in such a way that they increase from 1 {\displaystyle 1} to p + 1 {\displaystyle p+1} when one circumvents the base point clockwise, then the only relation between the generators is the equality M 1 ⋯ M p + 1 = id {\displaystyle M_{1}\cdots M_{p+1}=\operatorname {id} } . The Deligne–Simpson problem is the following realisation problem: For which tuples of conjugacy classes in GL ⁡ ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} do there exist irreducible tuples of matrices M j {\displaystyle M_{j}} from these classes satisfying the above relation? The problem has been formulated by Pierre Deligne and Carlos Simpson was the first to obtain results towards its resolution. An additive version of the problem about residua of Fuchsian systems has been formulated and explored by Vladimir Kostov. The problem has been considered by other authors for matrix groups other than GL ⁡ ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} as well. == Topological and geometric aspects == In the case of a covering map, we look at it as a special case of a fibration, and use the homotopy lifting property to "follow" paths on the base space X {\displaystyle X} (we assume it path-connected for simplicity) as they are lifted up into the cover C {\displaystyle C} . If we follow round a loop based at x {\displaystyle x} in X {\displaystyle X} , which we lift to start at c {\displaystyle c} above x {\displaystyle x} , we'll end at some c ∗ {\displaystyle c^{}} again above x {\displaystyle x} ; it is quite possible that c ≠ c ∗ {\displaystyle c\neq c^{}} , and to code this one considers the action of the fundamental group π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} as a permutation group on the set of all c {\displaystyle c} , as a monodromy group in this context. In differential geometry, an analogous role is played by parallel transport. In a principal bundle B {\displaystyle B} over a smooth manifold M {\displaystyle M} , a connection allows "horizontal" movement from fibers above m {\displaystyle m} in M {\displaystyle M} to adjacent ones. The effect when applied to loops based at m {\displaystyle m} is to define a holonomy group of translations of the fiber at m {\displaystyle m} ; if the structure group of B {\displaystyle B} is G {\displaystyle G} , it is a subgroup of G {\displaystyle G} that measures the deviation of B {\displaystyle B} from the product bundle M × G {\displaystyle M\times G} . === Monodromy groupoid and foliations === Analogous to the fundamental groupoid it is possible to get rid of the choice of a base point and to define a monodromy groupoid. Here we consider (homotopy classes of) lifts of paths in the base space X {\displaystyle X} of a fibration p : X ~ → X {\displaystyle p:{\tilde {X}}\to X} . The result has the structure of a groupoid over the base space X {\displaystyle X} . The advantage is that we can drop the condition of connectedness of X {\displaystyle X} . Moreover the construction can also be generalized to foliations: Consider ( M , F ) {\displaystyle (M,{\mathcal {F}})} a (possibly singular) foliation of M {\displaystyle M} . Then for every path in a leaf of F {\displaystyle {\mathcal {F}}} we can consider its induced diffeomorphism on local transversal sections through the endpoints. Within a simply connected chart this diffeomorphism becomes unique and especially canonical between different transversal sections if we go over to the germ of the diffeomorphism around the endpoints. In this way it also becomes independent of the path (between fixed endpoints) within a simply connected chart and is therefore invariant under homotopy. == Definition via Galois theory == Let F ( x ) {\displaystyle \mathbb {F} (x)} denote the field of the rational functions in the variable x {\displaystyle x} over the field F {\displaystyle \mathbb {F} } , which is the field of fractions of the polynomial ring F [ x ] {\displaystyle \mathbb {F} [x]} . An element y = f ( x ) {\displaystyle y=f(x)} of F ( x ) {\displaystyle \mathbb {F} (x)} determines a finite field extension [ F ( x ) : F ( y ) ] {\displaystyle [\mathbb {F} (x):\mathbb {F} (y)]} . This extension is generally not Galois but has Galois closure L ( f ) {\displaystyle L(f)} . The associated Galois group of the extension [ L ( f ) : F ( y ) ] {\displaystyle [L(f):\mathbb {F} (y)]} is called the monodromy group of f {\displaystyle f} . In the case of F = C {\displaystyle \mathbb {F} =\mathbb {C} } Riemann surface theory enters and allows for the geometric interpretation given above. In the case that the extension [ C ( x ) : C ( y ) ] {\displaystyle [\mathbb {C} (x):\mathbb {C} (y)]} is already Galois, the associated monodromy group is sometimes called a group of deck transformations. This has connections with the Galois theory of covering spaces leading to the Riemann existence theorem. == See also == Braid group Monodromy matrix Monodromy theorem Mapping class group (of a punctured disk) == Notes == == References == V. I. Danilov (2001) [1994], "Monodromy", Encyclopedia of Mathematics, EMS Press "Group-groupoids and monodromy groupoids", O. Mucuk, B. Kılıçarslan, T. ¸Sahan, N. Alemdar, Topology and its Applications 158 (2011) 2034–2042 doi:10.1016/j.topol.2011.06.048 R. Brown Topology and Groupoids (2006). P.J. Higgins, "Categories and groupoids", van Nostrand (1971) TAC Reprint H. Żołądek, "The Monodromy Group", Birkhäuser Basel 2006; doi: 10.1007/3-7643-7536-1
Wikipedia:Monogenic function#0	A monogenic function is a complex function with a single finite derivative. More precisely, a function f ( z ) {\displaystyle f(z)} defined on A ⊆ C {\displaystyle A\subseteq \mathbb {C} } is called monogenic at ζ ∈ A {\displaystyle \zeta \in A} , if f ′ ( ζ ) {\displaystyle f'(\zeta )} exists and is finite, with: f ′ ( ζ ) = lim z → ζ f ( z ) − f ( ζ ) z − ζ {\displaystyle f'(\zeta )=\lim _{z\to \zeta }{\frac {f(z)-f(\zeta )}{z-\zeta }}} Alternatively, it can be defined as the above limit having the same value for all paths. Functions can either have a single derivative (monogenic) or infinitely many derivatives (polygenic), with no intermediate cases. Furthermore, a function f ( x ) {\displaystyle f(x)} which is monogenic ∀ ζ ∈ B {\displaystyle \forall \zeta \in B} , is said to be monogenic on B {\displaystyle B} , and if B {\displaystyle B} is a domain of C {\displaystyle \mathbb {C} } , then it is analytic as well (The notion of domains can also be generalized in a manner such that functions which are monogenic over non-connected subsets of C {\displaystyle \mathbb {C} } , can show a weakened form of analyticity) The term monogenic was coined by Cauchy. == References ==
Wikipedia:Monoidal category action#0	In mathematics, a monoidal category (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗ : C × C → C {\displaystyle \otimes :\mathbf {C} \times \mathbf {C} \to \mathbf {C} } that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, for which monoidal categories can be considered an abstraction, is a system of data types closed under a type constructor that takes two types and builds an aggregate type. The types serve as the objects, and ⊗ is the aggregate constructor. The associativity up to isomorphism is then a way of expressing that different ways of aggregating the same data—such as ( ( a , b ) , c ) {\displaystyle ((a,b),c)} and ( a , ( b , c ) ) {\displaystyle (a,(b,c))} —store the same information even though the aggregate values need not be the same. The aggregate type may be analogous to the operation of addition (type sum) or of multiplication (type product). For type product, the identity object is the unit ( ) {\displaystyle ()} , so there is only one inhabitant of the type, and that is why a product with it is always isomorphic to the other operand. For type sum, the identity object is the void type, which stores no information, and it is impossible to address an inhabitant. The concept of monoidal category does not presume that values of such aggregate types can be taken apart; on the contrary, it provides a framework that unifies classical and quantum information theory. In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the category. They are also used in the definition of an enriched category. Monoidal categories have numerous applications outside of category theory proper. They are used to define models for the multiplicative fragment of intuitionistic linear logic. They also form the mathematical foundation for the topological order in condensed matter physics. Braided monoidal categories have applications in quantum information, quantum field theory, and string theory. == Formal definition == A monoidal category is a category C {\displaystyle \mathbf {C} } equipped with a monoidal structure. A monoidal structure consists of the following: a bifunctor ⊗ : C × C → C {\displaystyle \otimes \colon \mathbf {C} \times \mathbf {C} \to \mathbf {C} } called the monoidal product, or tensor product, an object I {\displaystyle I} called the monoidal unit, unit object, or identity object, three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation: is associative: there is a natural (in each of three arguments A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} ) isomorphism α {\displaystyle \alpha } , called associator, with components α A , B , C : A ⊗ ( B ⊗ C ) ≅ ( A ⊗ B ) ⊗ C {\displaystyle \alpha _{A,B,C}\colon A\otimes (B\otimes C)\cong (A\otimes B)\otimes C} , has I {\displaystyle I} as left and right identity: there are two natural isomorphisms λ {\displaystyle \lambda } and ρ {\displaystyle \rho } , respectively called left and right unitor, with components λ A : I ⊗ A ≅ A {\displaystyle \lambda _{A}\colon I\otimes A\cong A} and ρ A : A ⊗ I ≅ A {\displaystyle \rho _{A}\colon A\otimes I\cong A} . Note that a good way to remember how λ {\displaystyle \lambda } and ρ {\displaystyle \rho } act is by alliteration; Lambda, λ {\displaystyle \lambda } , cancels the identity on the left, while Rho, ρ {\displaystyle \rho } , cancels the identity on the right. The coherence conditions for these natural transformations are: for all A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} and D {\displaystyle D} in C {\displaystyle \mathbf {C} } , the pentagon diagram commutes; for all A {\displaystyle A} and B {\displaystyle B} in C {\displaystyle \mathbf {C} } , the triangle diagram commutes. A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal category. == Examples == Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category. For example: Set, the category of sets with the Cartesian product, any particular one-element set serving as the unit. Cat, the category of small categories with the product category, where the category with one object and only its identity map is the unit. Dually, any category with finite coproducts is monoidal with the coproduct as the monoidal product and the initial object as the unit. Such a monoidal category is called cocartesian monoidal R-Mod, the category of modules over a commutative ring R, is a monoidal category with the tensor product of modules ⊗R serving as the monoidal product and the ring R (thought of as a module over itself) serving as the unit. As special cases one has: K-Vect, the category of vector spaces over a field K, with the one-dimensional vector space K serving as the unit. Ab, the category of abelian groups, with the group of integers Z serving as the unit. For any commutative ring R, the category of R-algebras is monoidal with the tensor product of algebras as the product and R as the unit. The category of pointed spaces (restricted to compactly generated spaces for example) is monoidal with the smash product serving as the product and the pointed 0-sphere (a two-point discrete space) serving as the unit. The category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit. Just like for any category E, the full subcategory spanned by any given object is a monoid, it is the case that for any 2-category E, and any object C in Ob(E), the full 2-subcategory of E spanned by {C} is a monoidal category. In the case E = Cat, we get the endofunctors example above. Bounded-above meet semilattices are strict symmetric monoidal categories: the product is meet and the identity is the top element. Any ordinary monoid ( M , ⋅ , 1 ) {\displaystyle (M,\cdot ,1)} is a small monoidal category with object set M {\displaystyle M} , only identities for morphisms, ⋅ {\displaystyle \cdot } as tensor product and 1 {\displaystyle 1} as its identity object. Conversely, the set of isomorphism classes (if such a thing makes sense) of a monoidal category is a monoid w.r.t. the tensor product. Any commutative monoid ( M , ⋅ , 1 ) {\displaystyle (M,\cdot ,1)} can be realized as a monoidal category with a single object. Recall that a category with a single object is the same thing as an ordinary monoid. By an Eckmann-Hilton argument, adding another monoidal product on M {\displaystyle M} requires the product to be commutative. == Properties and associated notions == It follows from the three defining coherence conditions that a large class of diagrams (i.e. diagrams whose morphisms are built using α {\displaystyle \alpha } , λ {\displaystyle \lambda } , ρ {\displaystyle \rho } , identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated that all such diagrams commute. There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid from abstract algebra. Ordinary monoids are precisely the monoid objects in the cartesian monoidal category Set. Further, any (small) strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product). Monoidal functors are the functors between monoidal categories that preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product. Every monoidal category can be seen as the category B(∗, ∗) of a bicategory B with only one object, denoted ∗. The concept of a category C enriched in a monoidal category M replaces the notion of a set of morphisms between pairs of objects in C with the notion of an M-object of morphisms between every two objects in C. === Free strict monoidal category === For every category C, the free strict monoidal category Σ(C) can be constructed as follows: its objects are lists (finite sequences) A1, ..., An of objects of C; there are arrows between two objects A1, ..., Am and B1, ..., Bn only if m = n, and then the arrows are lists (finite sequences) of arrows f1: A1 → B1, ..., fn: An → Bn of C; the tensor product of two objects A1, ..., An and B1, ..., Bm is the concatenation A1, ..., An, B1, ..., Bm of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists. The identity object is the empty list. This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad on Cat. == Specializations == If, in a monoidal category, A ⊗ B {\displaystyle A\otimes B} and B ⊗ A {\displaystyle B\otimes A} are naturally isomorphic in a manner compatible with the coherence conditions, we speak of a braided monoidal category. If, moreover, this natural isomorphism is its own inverse, we have a symmetric monoidal category. A closed monoidal category is a monoidal category where the functor X ↦ X ⊗ A {\displaystyle X\mapsto X\otimes A} has a right adjoint, which is called the "internal Hom-functor" X ↦ H o m C ( A , X ) {\displaystyle X\mapsto \mathrm {Hom} _{\mathbf {C} }(A,X)} . Examples include cartesian closed categories such as Set, the category of sets, and compact closed categories such as FdVect, the category of finite-dimensional vector spaces. Autonomous categories (or compact closed categories or rigid categories) are monoidal categories in which duals with nice properties exist; they abstract the idea of FdVect. Dagger symmetric monoidal categories, equipped with an extra dagger functor, abstracting the idea of FdHilb, finite-dimensional Hilbert spaces. These include the dagger compact categories. Tannakian categories are monoidal categories enriched over a field, which are very similar to representation categories of linear algebraic groups. === Preordered monoids === A preordered monoid is a monoidal category in which for every two objects c , c ′ ∈ O b ( C ) {\displaystyle c,c'\in \mathrm {Ob} (\mathbf {C} )} , there exists at most one morphism c → c ′ {\displaystyle c\to c'} in C. In the context of preorders, a morphism c → c ′ {\displaystyle c\to c'} is sometimes notated c ≤ c ′ {\displaystyle c\leq c'} . The reflexivity and transitivity properties of an order, defined in the traditional sense, are incorporated into the categorical structure by the identity morphism and the composition formula in C, respectively. If c ≤ c ′ {\displaystyle c\leq c'} and c ′ ≤ c {\displaystyle c'\leq c} , then the objects c , c ′ {\displaystyle c,c'} are isomorphic which is notated c ≅ c ′ {\displaystyle c\cong c'} . Introducing a monoidal structure to the preorder C involves constructing an object I ∈ C {\displaystyle I\in \mathbf {C} } , called the monoidal unit, and a functor C × C → C {\displaystyle \mathbf {C} \times \mathbf {C} \to \mathbf {C} } , denoted by " ⋅ {\displaystyle \;\cdot \;} ", called the monoidal multiplication. I {\displaystyle I} and ⋅ {\displaystyle \cdot } must be unital and associative, up to isomorphism, meaning: ( c 1 ⋅ c 2 ) ⋅ c 3 ≅ c 1 ⋅ ( c 2 ⋅ c 3 ) {\displaystyle (c_{1}\cdot c_{2})\cdot c_{3}\cong c_{1}\cdot (c_{2}\cdot c_{3})} and I ⋅ c ≅ c ≅ c ⋅ I {\displaystyle I\cdot c\cong c\cong c\cdot I} . As · is a functor, if c 1 → c 1 ′ {\displaystyle c_{1}\to c_{1}'} and c 2 → c 2 ′ {\displaystyle c_{2}\to c_{2}'} then ( c 1 ⋅ c 2 ) → ( c 1 ′ ⋅ c 2 ′ ) {\displaystyle (c_{1}\cdot c_{2})\to (c_{1}'\cdot c_{2}')} . The other coherence conditions of monoidal categories are fulfilled through the preorder structure as every diagram commutes in a preorder. The natural numbers are an example of a monoidal preorder: having both a monoid structure (using + and 0) and a preorder structure (using ≤) forms a monoidal preorder as m ≤ n {\displaystyle m\leq n} and m ′ ≤ n ′ {\displaystyle m'\leq n'} implies m + m ′ ≤ n + n ′ {\displaystyle m+m'\leq n+n'} . The free monoid on some generating set produces a monoidal preorder, producing the semi-Thue system. == See also == Skeleton (category theory) Spherical category Monoidal category action == References == == External links == Media related to Monoidal category at Wikimedia Commons
Wikipedia:Monomial basis#0	In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial). == One indeterminate == The polynomial ring K[x] of univariate polynomials over a field K is a K-vector space, which has 1 , x , x 2 , x 3 , … {\displaystyle 1,x,x^{2},x^{3},\ldots } as an (infinite) basis. More generally, if K is a ring then K[x] is a free module which has the same basis. The polynomials of degree at most d form also a vector space (or a free module in the case of a ring of coefficients), which has { 1 , x , x 2 , … , x d − 1 , x d } {\displaystyle \{1,x,x^{2},\ldots ,x^{d-1},x^{d}\}} as a basis. The canonical form of a polynomial is its expression on this basis: a 0 + a 1 x + a 2 x 2 + ⋯ + a d x d , {\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{d}x^{d},} or, using the shorter sigma notation: ∑ i = 0 d a i x i . {\displaystyle \sum _{i=0}^{d}a_{i}x^{i}.} The monomial basis is naturally totally ordered, either by increasing degrees 1 < x < x 2 < ⋯ , {\displaystyle 1<x<x^{2}<\cdots ,} or by decreasing degrees 1 > x > x 2 > ⋯ . {\displaystyle 1>x>x^{2}>\cdots .} == Several indeterminates == In the case of several indeterminates x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} a monomial is a product x 1 d 1 x 2 d 2 ⋯ x n d n , {\displaystyle x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{n}^{d_{n}},} where the d i {\displaystyle d_{i}} are non-negative integers. As x i 0 = 1 , {\displaystyle x_{i}^{0}=1,} an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular 1 = x 1 0 x 2 0 ⋯ x n 0 {\displaystyle 1=x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0}} is a monomial. Similar to the case of univariate polynomials, the polynomials in x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} form a vector space (if the coefficients belong to a field) or a free module (if the coefficients belong to a ring), which has the set of all monomials as a basis, called the monomial basis. The homogeneous polynomials of degree d {\displaystyle d} form a subspace which has the monomials of degree d = d 1 + ⋯ + d n {\displaystyle d=d_{1}+\cdots +d_{n}} as a basis. The dimension of this subspace is the number of monomials of degree d {\displaystyle d} , which is ( d + n − 1 d ) = n ( n + 1 ) ⋯ ( n + d − 1 ) d ! , {\displaystyle {\binom {d+n-1}{d}}={\frac {n(n+1)\cdots (n+d-1)}{d!}},} where ( d + n − 1 d ) {\textstyle {\binom {d+n-1}{d}}} is a binomial coefficient. The polynomials of degree at most d {\displaystyle d} form also a subspace, which has the monomials of degree at most d {\displaystyle d} as a basis. The number of these monomials is the dimension of this subspace, equal to ( d + n d ) = ( d + n n ) = ( d + 1 ) ⋯ ( d + n ) n ! . {\displaystyle {\binom {d+n}{d}}={\binom {d+n}{n}}={\frac {(d+1)\cdots (d+n)}{n!}}.} In contrast to the univariate case, there is no natural total order of the monomial basis in the multivariate case. For problems which require choosing a total order, such as Gröbner basis computations, one generally chooses an admissible monomial order – that is, a total order on the set of monomials such that m < n ⟺ m q < n q {\displaystyle m<n\iff mq<nq} and 1 ≤ m {\displaystyle 1\leq m} for every monomial m , n , q . {\displaystyle m,n,q.} == See also == Horner's method Polynomial sequence Newton polynomial Lagrange polynomial Legendre polynomial Bernstein form Chebyshev form
Wikipedia:Monomial representation#0	In the mathematical fields of representation theory and group theory, a linear representation ρ {\displaystyle \rho } (rho) of a group G {\displaystyle G} is a monomial representation if there is a finite-index subgroup H {\displaystyle H} and a one-dimensional linear representation σ {\displaystyle \sigma } of H {\displaystyle H} , such that ρ {\displaystyle \rho } is equivalent to the induced representation I n d H G σ {\displaystyle \mathrm {Ind} _{H}^{G_{\sigma }}} . Alternatively, one may define it as a representation whose image is in the monomial matrices. Here for example G {\displaystyle G} and H {\displaystyle H} may be finite groups, so that induced representation has a classical sense. The monomial representation is only a little more complicated than the permutation representation of G {\displaystyle G} on the cosets of H {\displaystyle H} . It is necessary only to keep track of scalars coming from σ {\displaystyle \sigma } applied to elements of H {\displaystyle H} . == Definition == To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple ( V , X , ( V x ) x ∈ X ) {\displaystyle (V,X,(V_{x})_{x\in X})} where V {\displaystyle V} is a finite-dimensional complex vector space, X {\displaystyle X} is a finite set and ( V x ) x ∈ X {\displaystyle (V_{x})_{x\in X}} is a family of one-dimensional subspaces of V {\displaystyle V} such that V = ⊕ x ∈ X V x {\displaystyle V=\oplus _{x\in X}V_{x}} . Now Let G {\displaystyle G} be a group, the monomial representation of G {\displaystyle G} on V {\displaystyle V} is a group homomorphism ρ : G → G L ( V ) {\displaystyle \rho :G\to \mathrm {GL} (V)} such that for every element g ∈ G {\displaystyle g\in G} , ρ ( g ) {\displaystyle \rho (g)} permutes the V x {\displaystyle V_{x}} 's, this means that ρ {\displaystyle \rho } induces an action by permutation of G {\displaystyle G} on X {\displaystyle X} . == References == "Monomial representation", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Karpilovsky, Gregory (1985). Projective Representations of Finite Groups. M. Dekker. ISBN 978-0-8247-7313-7.
Wikipedia:Monster vertex algebra#0	The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by applying the Goddard–Thorn theorem of string theory to construct the monster Lie algebra, an infinite-dimensional generalized Kac–Moody algebra acted on by the monster. The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex algebra products. It can be constructed as conformal field theory describing 24 free bosons compactified on the torus induced by the Leech lattice and orbifolded by the two-element reflection group. == References == Borcherds, Richard (1986), "Vertex algebras, Kac-Moody algebras, and the Monster", Proceedings of the National Academy of Sciences of the United States of America, 83 (10): 3068–3071, Bibcode:1986PNAS...83.3068B, doi:10.1073/pnas.83.10.3068, PMC 323452, PMID 16593694 Meurman, Arne; Frenkel, Igor; Lepowsky, James (1988), Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Boston, MA: Academic Press, pp. liv+508 pp, ISBN 978-0-12-267065-7
Wikipedia:More Maths Grads#0	More Maths Grads was a three-year project run from 2007 to 2010 by a consortium of British mathematics organisations which aimed to increase the supply of mathematical sciences graduates in England and to widen participation within the mathematical sciences from groups of learners who have not previously been well represented in higher education. == History == The project was launched to address a perceived problem with numbers of students studying mathematics at university - that higher education participation had increased since 2001 but numbers studying mathematical sciences remained almost constant, and had particular focus on encouraging participation from groups of learners who were not well represented in higher education. The project was initially called The Increasing the Supply of Mathematical Science Graduates programme before later being renamed More Maths Grads. Funding of £3.3M was provided by the Higher Education Funding Council for England under the 'Strategically Important Subjects' initiative. More Maths Grads was led by the Maths, Stats & OR Network on behalf of a consortium which also included the Institute of Mathematics and its Applications, the London Mathematical Society, the Royal Statistical Society, and HoDoMS, the Heads of Departments of Mathematical Sciences. The project concentrated its activity on three regions: West Midlands, Yorkshire & Humberside and London. It worked in collaboration with Coventry University, University of Leeds, Queen Mary, University of London and Sheffield Hallam University. It was overseen by a steering committee chaired by Duncan Lawson. The project was managed first by Helen Orr and later by Makhan Singh. == Work areas == The More Maths Grads project ran four strands of activity: Careers theme, producing information about career opportunities with mathematics; Student theme, focused on enrichment activities; Teaching theme, professional development for teachers; HE Curriculum theme, research about the current higher education mathematical sciences curriculum. The HE Curriculum theme was concerned with curriculum content and also issues around student experience and teaching practice == Legacy == The conclusion of More Maths Grads after three years was marked by a Parliamentary Reception 'Where will maths take you?' on 27 January 2010, hosted by Charles Clarke MP, who claimed the project had made "an impact in improving standards of mathematics education and increasing the number continuing to study mathematics". At the event, project manager Makhan Singh claimed the project had "touched the lives of tens of thousands school students, plus many more members of the wider public" and highlighted the resources and good practice generated by the project, which included the Maths in a Box resource. The project was followed by the National HE STEM Programme, which built on its work. == References == == External links == Project web site
Wikipedia:Moritz Allé#0	Moritz Allé (1837–1913) was an Austrian astronomer and mathematician, one of the teachers of Nikola Tesla. == Scientific career == Allé studied mathematics at the University of Vienna. After his university graduation, Allé startet his professional career as an assistant at the Vienna Observatory in 1856. He was appointed Adjunkt (senior position) at the observatory in Kraków in 1859. In 1860 he completed his PhD at the University of Kiel. In 1862 he was appointed Adjunkt at the observatory in Prague. It was there where he completed his habilitation in mathematics in 1863. In 1867, Allé was appointed professor of mathematics at the Joanneum in Graz and was elected as its rector in 1875/76. In June 1882 Allé was appointed professor of mathematics at the Deutsche Technische Hochschule in Prague. There, he was also dean for five periods and different faculties but declined serving as rector after his election in June 1887 for health reasons. In 1896 he succeeded Josef Kolbe at the chair of mathematics at the Polytechnic Institute in Vienna, where he served as rector as well in 1900/01. Upon his retirement in 1906, Allé was awarded the imperial title of a Hofrat. Allé was known as an excellent and sympathetic teacher and thus much appreciated by his students. In his autobiography, Nikola Tesla describes him as the most brilliant lecturer to whom I ever listened. == Private life == Moritz Allé passed away on April 6, 1913 and was buried on April 8 in Baden. == Scientific contributions == The early scientific work of Allé led to a number of publications in astronomy where he contributed on orbit determination of planets and comets. He later gave up this field of research in favour of mathematics. == Publications == Allé, M.:Die Opposition der Calliope im Jahre 1857. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 21, 1856, pp. 379–381. Allé, M.:Die Bahn der Laetitia.Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 24, 1857, pp. 159–161. Allé, M.:Die Bahn der Leda. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 32, 1858, pp. 258f. Allé, M.:Die Bahn der Nemausa. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 38, 1859, pp. 749–757. Allé, M.:De methodis variis perturbationes speciales dictas computandi. Kiel, 1860.Disertace na univerzitě v Kielu. Allé, M.:Die Bahn der Leda.Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 43, 1861, pp. 585f. Allé, M.:Über die Entwicklung von Funktionen in Reihen, die nach einer besonderen Gattung algebraischer Ausdrücke fortschreiten. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 52, 1865, pp. 453–478. Allé, M.:Ein Beitrag zur Theorie der Funktionen von drei Veränderlichen.Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 72, 1875, pp. 289–310. Allé, M.:Über die Bewegungsgleichungen eines Systems von Punkten. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 73, 1876, pp. 25–46. Allé, M.: Zur Theorie des Gaussschen Krümmungsmasses. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 74, 1877, pp. 9–38. Allé, M.:Beiträge zur Theorie des Doppelverhältnisses und zur Raumkollineation. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 85, 1881, pp. 1021–1034. Allé, M.:Über die Ableitung der Gleichungen der drehenden Bewegung eines starren Körpers nach der Grassmannschen Analyse. Mitteilungen der Mathematischen Gesellschaft zu Prag, 1892, pp. 64–68. Allé, M.:Ein Beitrag zur Theorie der Evoluten. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 113, 1904, pp. 53–70. Allé, M.: Über infinitesimale Transformation. Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien, 113, 1904, pp. 681–720. == References == == External links == Allé bibliography, Lenka Lomtatidze, 2003. Moritz Allé at the Mathematics Genealogy Project
Wikipedia:Morphism of algebraic varieties#0	In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the concepts of rational and birational maps are widely used as well; they are partial functions that are defined locally by rational fractions instead of polynomials. An algebraic variety has naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces. == Definition == If X and Y are closed subvarieties of A n {\displaystyle \mathbb {A} ^{n}} and A m {\displaystyle \mathbb {A} ^{m}} (so they are affine varieties), then a regular map f : X → Y {\displaystyle f\colon X\to Y} is the restriction of a polynomial map A n → A m {\displaystyle \mathbb {A} ^{n}\to \mathbb {A} ^{m}} . Explicitly, it has the form: f = ( f 1 , … , f m ) {\displaystyle f=(f_{1},\dots ,f_{m})} where the f i {\displaystyle f_{i}} s are in the coordinate ring of X: k [ X ] = k [ x 1 , … , x n ] / I , {\displaystyle k[X]=k[x_{1},\dots ,x_{n}]/I,} where I is the ideal defining X (note: two polynomials f and g define the same function on X if and only if f − g is in I). The image f(X) lies in Y, and hence satisfies the defining equations of Y. That is, a regular map f : X → Y {\displaystyle f:X\to Y} is the same as the restriction of a polynomial map whose components satisfy the defining equations of Y {\displaystyle Y} . More generally, a map f : X→Y between two varieties is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of f(x) such that f(U) ⊂ V and the restricted function f : U→V is regular as a function on some affine charts of U and V. Then f is called regular, if it is regular at all points of X. Note: It is not immediately obvious that the two definitions coincide: if X and Y are affine varieties, then a map f : X→Y is regular in the first sense if and only if it is so in the second sense. Also, it is not immediately clear whether regularity depends on a choice of affine charts (it does not.) This kind of a consistency issue, however, disappears if one adopts the formal definition. Formally, an (abstract) algebraic variety is defined to be a particular kind of a locally ringed space. When this definition is used, a morphism of varieties is just a morphism of locally ringed spaces. The composition of regular maps is again regular; thus, algebraic varieties form the category of algebraic varieties where the morphisms are the regular maps. Regular maps between affine varieties correspond contravariantly in one-to-one to algebra homomorphisms between the coordinate rings: if f : X→Y is a morphism of affine varieties, then it defines the algebra homomorphism f # : k [ Y ] → k [ X ] , g ↦ g ∘ f {\displaystyle f^{\#}:k[Y]\to k[X],\,g\mapsto g\circ f} where k [ X ] , k [ Y ] {\displaystyle k[X],k[Y]} are the coordinate rings of X and Y; it is well-defined since g ∘ f = g ( f 1 , … , f m ) {\displaystyle g\circ f=g(f_{1},\dots ,f_{m})} is a polynomial in elements of k [ X ] {\displaystyle k[X]} . Conversely, if ϕ : k [ Y ] → k [ X ] {\displaystyle \phi :k[Y]\to k[X]} is an algebra homomorphism, then it induces the morphism ϕ a : X → Y {\displaystyle \phi ^{a}:X\to Y} given by: writing k [ Y ] = k [ y 1 , … , y m ] / J , {\displaystyle k[Y]=k[y_{1},\dots ,y_{m}]/J,} ϕ a = ( ϕ ( y 1 ¯ ) , … , ϕ ( y m ¯ ) ) {\displaystyle \phi ^{a}=(\phi ({\overline {y_{1}}}),\dots ,\phi ({\overline {y_{m}}}))} where y ¯ i {\displaystyle {\overline {y}}_{i}} are the images of y i {\displaystyle y_{i}} 's. Note ϕ a # = ϕ {\displaystyle {\phi ^{a}}^{\#}=\phi } as well as f # a = f . {\displaystyle {f^{\#}}^{a}=f.} In particular, f is an isomorphism of affine varieties if and only if f# is an isomorphism of the coordinate rings. For example, if X is a closed subvariety of an affine variety Y and f is the inclusion, then f# is the restriction of regular functions on Y to X. See #Examples below for more examples. == Regular functions == In the particular case that Y {\displaystyle Y} equals A 1 {\displaystyle \mathbb {A} ^{1}} the regular maps f : X → A 1 {\displaystyle f:X\rightarrow \mathbb {A} ^{1}} are called regular functions, and are algebraic analogs of smooth functions studied in differential geometry. The ring of regular functions (that is the coordinate ring or more abstractly the ring of global sections of the structure sheaf) is a fundamental object in affine algebraic geometry. The only regular function on a projective variety is constant (this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis). A scalar function f : X → A 1 {\displaystyle f:X\rightarrow \mathbb {A} ^{1}} is regular at a point x {\displaystyle x} if, in some open affine neighborhood of x {\displaystyle x} , it is a rational function that is regular at x {\displaystyle x} ; i.e., there are regular functions g {\displaystyle g} , h {\displaystyle h} near x {\displaystyle x} such that f = g / h {\displaystyle f=g/h} and h {\displaystyle h} does not vanish at x {\displaystyle x} . Caution: the condition is for some pair (g, h) not for all pairs (g, h); see Examples. If X is a quasi-projective variety; i.e., an open subvariety of a projective variety, then the function field k(X) is the same as that of the closure X ¯ {\displaystyle {\overline {X}}} of X and thus a rational function on X is of the form g/h for some homogeneous elements g, h of the same degree in the homogeneous coordinate ring k [ X ¯ ] {\displaystyle k[{\overline {X}}]} of X ¯ {\displaystyle {\overline {X}}} (cf. Projective variety#Variety structure). Then a rational function f on X is regular at a point x if and only if there are some homogeneous elements g, h of the same degree in k [ X ¯ ] {\displaystyle k[{\overline {X}}]} such that f = g/h and h does not vanish at x. This characterization is sometimes taken as the definition of a regular function. == Comparison with a morphism of schemes == If X = Spec ⁡ A {\displaystyle X=\operatorname {Spec} A} and Y = Spec ⁡ B {\displaystyle Y=\operatorname {Spec} B} are affine schemes, then each ring homomorphism ϕ : B → A {\displaystyle \phi :B\rightarrow A} determines a morphism ϕ a : X → Y , p ↦ ϕ − 1 ( p ) {\displaystyle \phi ^{a}:X\to Y,\,{\mathfrak {p}}\mapsto \phi ^{-1}({\mathfrak {p}})} by taking the pre-images of prime ideals. All morphisms between affine schemes are of this type and gluing such morphisms gives a morphism of schemes in general. Now, if X, Y are affine varieties; i.e., A, B are integral domains that are finitely generated algebras over an algebraically closed field k, then, working with only the closed points, the above coincides with the definition given at #Definition. (Proof: If f : X → Y is a morphism, then writing ϕ = f # {\displaystyle \phi =f^{\#}} , we need to show m f ( x ) = ϕ − 1 ( m x ) {\displaystyle {\mathfrak {m}}_{f(x)}=\phi ^{-1}({\mathfrak {m}}_{x})} where m x , m f ( x ) {\displaystyle {\mathfrak {m}}_{x},{\mathfrak {m}}_{f(x)}} are the maximal ideals corresponding to the points x and f(x); i.e., m x = { g ∈ k [ X ] ∣ g ( x ) = 0 } {\displaystyle {\mathfrak {m}}_{x}=\{g\in k[X]\mid g(x)=0\}} . This is immediate.) This fact means that the category of affine varieties can be identified with a full subcategory of affine schemes over k. Since morphisms of varieties are obtained by gluing morphisms of affine varieties in the same way morphisms of schemes are obtained by gluing morphisms of affine schemes, it follows that the category of varieties is a full subcategory of the category of schemes over k. For more details, see [1]. == Examples == The regular functions on A n {\displaystyle \mathbb {A} ^{n}} are exactly the polynomials in n {\displaystyle n} variables and the regular functions on P n {\displaystyle \mathbb {P} ^{n}} are exactly the constants. Let X {\displaystyle X} be the affine curve y = x 2 {\displaystyle y=x^{2}} . Then f : X → A 1 , ( x , y ) ↦ x {\displaystyle f:X\to \mathbf {A} ^{1},\,(x,y)\mapsto x} is a morphism; it is bijective with the inverse g ( x ) = ( x , x 2 ) {\displaystyle g(x)=(x,x^{2})} . Since g {\displaystyle g} is also a morphism, f {\displaystyle f} is an isomorphism of varieties. Let X {\displaystyle X} be the affine curve y 2 = x 3 + x 2 {\displaystyle y^{2}=x^{3}+x^{2}} . Then f : A 1 → X , t ↦ ( t 2 − 1 , t 3 − t ) {\displaystyle f:\mathbf {A} ^{1}\to X,\,t\mapsto (t^{2}-1,t^{3}-t)} is a morphism. It corresponds to the ring homomorphism f # : k [ X ] → k [ t ] , g ↦ g ( t 2 − 1 , t 3 − t ) , {\displaystyle f^{\#}:k[X]\to k[t],\,g\mapsto g(t^{2}-1,t^{3}-t),} which is seen to be injective (since f is surjective). Continuing the preceding example, let U = A1 − {1}. Since U is the complement of the hyperplane t = 1, U is affine. The restriction f : U → X {\displaystyle f:U\to X} is bijective. But the corresponding ring homomorphism is the inclusion k [ X ] = k [ t 2 − 1 , t 3 − t ] ↪ k [ t , ( t − 1 ) − 1 ] {\displaystyle k[X]=k[t^{2}-1,t^{3}-t]\hookrightarrow k[t,(t-1)^{-1}]} , which is not an isomorphism and so the restriction f \|U is not an isomorphism. Let X be the affine curve x2 + y2 = 1 and let f ( x , y ) = 1 − y x . {\displaystyle f(x,y)={1-y \over x}.} Then f is a rational function on X. It is regular at (0, 1) despite the expression since, as a rational function on X, f can also be written as f ( x , y ) = x 1 + y {\displaystyle f(x,y)={x \over 1+y}} . Let X = A2 − (0, 0). Then X is an algebraic variety since it is an open subset of a variety. If f is a regular function on X, then f is regular on D A 2 ( x ) = A 2 − { x = 0 } {\displaystyle D_{\mathbf {A} ^{2}}(x)=\mathbf {A} ^{2}-\{x=0\}} and so is in k [ D A 2 ( x ) ] = k [ A 2 ] [ x − 1 ] = k [ x , x − 1 , y ] {\displaystyle k[D_{\mathbf {A} ^{2}}(x)]=k[\mathbf {A} ^{2}][x^{-1}]=k[x,x^{-1},y]} . Similarly, it is in k [ x , y , y − 1 ] {\displaystyle k[x,y,y^{-1}]} . Thus, we can write: f = g x n = h y m {\displaystyle f={g \over x^{n}}={h \over y^{m}}} where g, h are polynomials in k[x, y]. But this implies g is divisible by xn and so f is in fact a polynomial. Hence, the ring of regular functions on X is just k[x, y]. (This also shows that X cannot be affine since if it were, X is determined by its coordinate ring and thus X = A2.) Suppose P 1 = A 1 ∪ { ∞ } {\displaystyle \mathbf {P} ^{1}=\mathbf {A} ^{1}\cup \{\infty \}} by identifying the points (x : 1) with the points x on A1 and ∞ = (1 : 0). There is an automorphism σ of P1 given by σ(x : y) = (y : x); in particular, σ exchanges 0 and ∞. If f is a rational function on P1, then σ # ( f ) = f ( 1 / z ) {\displaystyle \sigma ^{\#}(f)=f(1/z)} and f is regular at ∞ if and only if f(1/z) is regular at zero. Taking the function field k(V) of an irreducible algebraic curve V, the functions F in the function field may all be realised as morphisms from V to the projective line over k. (cf. #Properties) The image will either be a single point, or the whole projective line (this is a consequence of the completeness of projective varieties). That is, unless F is actually constant, we have to attribute to F the value ∞ at some points of V. For any algebraic varieties X, Y, the projection p : X × Y → X , ( x , y ) ↦ x {\displaystyle p:X\times Y\to X,\,(x,y)\mapsto x} is a morphism of varieties. If X and Y are affine, then the corresponding ring homomorphism is p # : k [ X ] → k [ X × Y ] = k [ X ] ⊗ k k [ Y ] , f ↦ f ⊗ 1 {\displaystyle p^{\#}:k[X]\to k[X\times Y]=k[X]\otimes _{k}k[Y],\,f\mapsto f\otimes 1} where ( f ⊗ 1 ) ( x , y ) = f ( p ( x , y ) ) = f ( x ) {\displaystyle (f\otimes 1)(x,y)=f(p(x,y))=f(x)} . == Properties == A morphism between varieties is continuous with respect to Zariski topologies on the source and the target. The image of a morphism of varieties need not be open nor closed (for example, the image of A 2 → A 2 , ( x , y ) ↦ ( x , x y ) {\displaystyle \mathbf {A} ^{2}\to \mathbf {A} ^{2},\,(x,y)\mapsto (x,xy)} is neither open nor closed). However, one can still say: if f is a morphism between varieties, then the image of f contains an open dense subset of its closure (cf. constructible set). A morphism f:X→Y of algebraic varieties is said to be dominant if it has dense image. For such an f, if V is a nonempty open affine subset of Y, then there is a nonempty open affine subset U of X such that f(U) ⊂ V and then f # : k [ V ] → k [ U ] {\displaystyle f^{\#}:k[V]\to k[U]} is injective. Thus, the dominant map f induces an injection on the level of function fields: k ( Y ) = lim → ⁡ k [ V ] ↪ k ( X ) , g ↦ g ∘ f {\displaystyle k(Y)=\varinjlim k[V]\hookrightarrow k(X),\,g\mapsto g\circ f} where the direct limit runs over all nonempty open affine subsets of Y. (More abstractly, this is the induced map from the residue field of the generic point of Y to that of X.) Conversely, every inclusion of fields k ( Y ) ↪ k ( X ) {\displaystyle k(Y)\hookrightarrow k(X)} is induced by a dominant rational map from X to Y. Hence, the above construction determines a contravariant-equivalence between the category of algebraic varieties over a field k and dominant rational maps between them and the category of finitely generated field extension of k. If X is a smooth complete curve (for example, P1) and if f is a rational map from X to a projective space Pm, then f is a regular map X → Pm. In particular, when X is a smooth complete curve, any rational function on X may be viewed as a morphism X → P1 and, conversely, such a morphism as a rational function on X. On a normal variety (in particular, a smooth variety), a rational function is regular if and only if it has no poles of codimension one. This is an algebraic analog of Hartogs' extension theorem. There is also a relative version of this fact; see [2]. A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism (a counterexample is given by a Frobenius morphism t ↦ t p {\displaystyle t\mapsto t^{p}} .) On the other hand, if f is bijective birational and the target space of f is a normal variety, then f is biregular. (cf. Zariski's main theorem.) A regular map between complex algebraic varieties is a holomorphic map. (There is actually a slight technical difference: a regular map is a meromorphic map whose singular points are removable, but the distinction is usually ignored in practice.) In particular, a regular map into the complex numbers is just a usual holomorphic function (complex-analytic function). == Morphisms to a projective space == Let f : X → P m {\displaystyle f:X\to \mathbf {P} ^{m}} be a morphism from a projective variety to a projective space. Let x be a point of X. Then some i-th homogeneous coordinate of f(x) is nonzero; say, i = 0 for simplicity. Then, by continuity, there is an open affine neighborhood U of x such that f : U → P m − { y 0 = 0 } {\displaystyle f:U\to \mathbf {P} ^{m}-\{y_{0}=0\}} is a morphism, where yi are the homogeneous coordinates. Note the target space is the affine space Am through the identification ( a 0 : ⋯ : a m ) = ( 1 : a 1 / a 0 : ⋯ : a m / a 0 ) ∼ ( a 1 / a 0 , … , a m / a 0 ) {\displaystyle (a_{0}:\dots :a_{m})=(1:a_{1}/a_{0}:\dots :a_{m}/a_{0})\sim (a_{1}/a_{0},\dots ,a_{m}/a_{0})} . Thus, by definition, the restriction f \|U is given by f \| U ( x ) = ( g 1 ( x ) , … , g m ( x ) ) {\displaystyle f\|_{U}(x)=(g_{1}(x),\dots ,g_{m}(x))} where gi's are regular functions on U. Since X is projective, each gi is a fraction of homogeneous elements of the same degree in the homogeneous coordinate ring k[X] of X. We can arrange the fractions so that they all have the same homogeneous denominator say f0. Then we can write gi = fi/f0 for some homogeneous elements fi's in k[X]. Hence, going back to the homogeneous coordinates, f ( x ) = ( f 0 ( x ) : f 1 ( x ) : ⋯ : f m ( x ) ) {\displaystyle f(x)=(f_{0}(x):f_{1}(x):\dots :f_{m}(x))} for all x in U and by continuity for all x in X as long as the fi's do not vanish at x simultaneously. If they vanish simultaneously at a point x of X, then, by the above procedure, one can pick a different set of fi's that do not vanish at x simultaneously (see Note at the end of the section.) In fact, the above description is valid for any quasi-projective variety X, an open subvariety of a projective variety X ¯ {\displaystyle {\overline {X}}} ; the difference being that fi's are in the homogeneous coordinate ring of X ¯ {\displaystyle {\overline {X}}} . Note: The above does not say a morphism from a projective variety to a projective space is given by a single set of polynomials (unlike the affine case). For example, let X be the conic y 2 = x z {\displaystyle y^{2}=xz} in P2. Then two maps ( x : y : z ) ↦ ( x : y ) {\displaystyle (x:y:z)\mapsto (x:y)} and ( x : y : z ) ↦ ( y : z ) {\displaystyle (x:y:z)\mapsto (y:z)} agree on the open subset { ( x : y : z ) ∈ X ∣ x ≠ 0 , z ≠ 0 } {\displaystyle \{(x:y:z)\in X\mid x\neq 0,z\neq 0\}} of X (since ( x : y ) = ( x y : y 2 ) = ( x y : x z ) = ( y : z ) {\displaystyle (x:y)=(xy:y^{2})=(xy:xz)=(y:z)} ) and so defines a morphism f : X → P 1 {\displaystyle f:X\to \mathbf {P} ^{1}} . == Fibers of a morphism == The important fact is: In Mumford's red book, the theorem is proved by means of Noether's normalization lemma. For an algebraic approach where the generic freeness plays a main role and the notion of "universally catenary ring" is a key in the proof, see Eisenbud, Ch. 14 of "Commutative algebra with a view toward algebraic geometry." In fact, the proof there shows that if f is flat, then the dimension equality in 2. of the theorem holds in general (not just generically). == Degree of a finite morphism == Let f: X → Y be a finite surjective morphism between algebraic varieties over a field k. Then, by definition, the degree of f is the degree of the finite field extension of the function field k(X) over fk(Y). By generic freeness, there is some nonempty open subset U in Y such that the restriction of the structure sheaf OX to f−1(U) is free as OY\|U-module. The degree of f is then also the rank of this free module. If f is étale and if X, Y are complete, then for any coherent sheaf F on Y, writing χ for the Euler characteristic, χ ( f ∗ F ) = deg ⁡ ( f ) χ ( F ) . {\displaystyle \chi (f^{}F)=\deg(f)\chi (F).} (The Riemann–Hurwitz formula for a ramified covering shows the "étale" here cannot be omitted.) In general, if f is a finite surjective morphism, if X, Y are complete and F a coherent sheaf on Y, then from the Leray spectral sequence H p ⁡ ( Y , R q f ∗ f ∗ F ) ⇒ H p + q ⁡ ( X , f ∗ F ) {\displaystyle \operatorname {H} ^{p}(Y,R^{q}f_{}f^{}F)\Rightarrow \operatorname {H} ^{p+q}(X,f^{}F)} , one gets: χ ( f ∗ F ) = ∑ q = 0 ∞ ( − 1 ) q χ ( R q f ∗ f ∗ F ) . {\displaystyle \chi (f^{}F)=\sum _{q=0}^{\infty }(-1)^{q}\chi (R^{q}f_{}f^{}F).} In particular, if F is a tensor power L ⊗ n {\displaystyle L^{\otimes n}} of a line bundle, then R q f ∗ ( f ∗ F ) = R q f ∗ O X ⊗ L ⊗ n {\displaystyle R^{q}f_{}(f^{}F)=R^{q}f_{}{\mathcal {O}}_{X}\otimes L^{\otimes n}} and since the support of R q f ∗ O X {\displaystyle R^{q}f_{}{\mathcal {O}}_{X}} has positive codimension if q is positive, comparing the leading terms, one has: deg ⁡ ( f ∗ L ) = deg ⁡ ( f ) deg ⁡ ( L ) {\displaystyle \operatorname {deg} (f^{}L)=\operatorname {deg} (f)\operatorname {deg} (L)} (since the generic rank of f ∗ O X {\displaystyle f_{}{\mathcal {O}}_{X}} is the degree of f.) If f is étale and k is algebraically closed, then each geometric fiber f−1(y) consists exactly of deg(f) points. == See also == Algebraic function Smooth morphism Étale morphisms – The algebraic analogue of local diffeomorphisms. Resolution of singularities contraction morphism == Notes == == Citations == == References ==
Wikipedia:Morrie's law#0	Morrie's law is a special trigonometric identity. Its name is due to the physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name because he learned it during his childhood from a boy with the name Morrie Jacobs and afterwards remembered it for all of his life. == Identity and generalisation == cos ⁡ ( 20 ∘ ) ⋅ cos ⁡ ( 40 ∘ ) ⋅ cos ⁡ ( 80 ∘ ) = 1 8 . {\displaystyle \cos(20^{\circ })\cdot \cos(40^{\circ })\cdot \cos(80^{\circ })={\frac {1}{8}}.} It is a special case of the more general identity 2 n ⋅ ∏ k = 0 n − 1 cos ⁡ ( 2 k α ) = sin ⁡ ( 2 n α ) sin ⁡ ( α ) {\displaystyle 2^{n}\cdot \prod _{k=0}^{n-1}\cos(2^{k}\alpha )={\frac {\sin(2^{n}\alpha )}{\sin(\alpha )}}} with n = 3 and α = 20° and the fact that sin ⁡ ( 160 ∘ ) sin ⁡ ( 20 ∘ ) = sin ⁡ ( 180 ∘ − 20 ∘ ) sin ⁡ ( 20 ∘ ) = 1 , {\displaystyle {\frac {\sin(160^{\circ })}{\sin(20^{\circ })}}={\frac {\sin(180^{\circ }-20^{\circ })}{\sin(20^{\circ })}}=1,} since sin ⁡ ( 180 ∘ − x ) = sin ⁡ ( x ) . {\displaystyle \sin(180^{\circ }-x)=\sin(x).} == Similar identities == A similar identity for the sine function also holds: sin ⁡ ( 20 ∘ ) ⋅ sin ⁡ ( 40 ∘ ) ⋅ sin ⁡ ( 80 ∘ ) = 3 8 . {\displaystyle \sin(20^{\circ })\cdot \sin(40^{\circ })\cdot \sin(80^{\circ })={\frac {\sqrt {3}}{8}}.} Moreover, dividing the second identity by the first, the following identity is evident: tan ⁡ ( 20 ∘ ) ⋅ tan ⁡ ( 40 ∘ ) ⋅ tan ⁡ ( 80 ∘ ) = 3 = tan ⁡ ( 60 ∘ ) . {\displaystyle \tan(20^{\circ })\cdot \tan(40^{\circ })\cdot \tan(80^{\circ })={\sqrt {3}}=\tan(60^{\circ }).} == Proof == === Geometric proof of Morrie's law === Consider a regular nonagon A B C D E F G H I {\displaystyle ABCDEFGHI} with side length 1 {\displaystyle 1} and let M {\displaystyle M} be the midpoint of A B {\displaystyle AB} , L {\displaystyle L} the midpoint B F {\displaystyle BF} and J {\displaystyle J} the midpoint of B D {\displaystyle BD} . The inner angles of the nonagon equal 140 ∘ {\displaystyle 140^{\circ }} and furthermore γ = ∠ F B M = 80 ∘ {\displaystyle \gamma =\angle FBM=80^{\circ }} , β = ∠ D B F = 40 ∘ {\displaystyle \beta =\angle DBF=40^{\circ }} and α = ∠ C B D = 20 ∘ {\displaystyle \alpha =\angle CBD=20^{\circ }} (see graphic). Applying the cosinus definition in the right angle triangles △ B F M {\displaystyle \triangle BFM} , △ B D L {\displaystyle \triangle BDL} and △ B C J {\displaystyle \triangle BCJ} then yields the proof for Morrie's law: 1 = \| A B \| = 2 ⋅ \| M B \| = 2 ⋅ \| B F \| ⋅ cos ⁡ ( γ ) = 2 2 \| B L \| cos ⁡ ( γ ) = 2 2 ⋅ \| B D \| ⋅ cos ⁡ ( γ ) ⋅ cos ⁡ ( β ) = 2 3 ⋅ \| B J \| ⋅ cos ⁡ ( γ ) ⋅ cos ⁡ ( β ) = 2 3 ⋅ \| B C \| ⋅ cos ⁡ ( γ ) ⋅ cos ⁡ ( β ) ⋅ cos ⁡ ( α ) = 2 3 ⋅ 1 ⋅ cos ⁡ ( γ ) ⋅ cos ⁡ ( β ) ⋅ cos ⁡ ( α ) = 8 ⋅ cos ⁡ ( 80 ∘ ) ⋅ cos ⁡ ( 40 ∘ ) ⋅ cos ⁡ ( 20 ∘ ) {\displaystyle {\begin{aligned}1&=\|AB\|\\&=2\cdot \|MB\|\\&=2\cdot \|BF\|\cdot \cos(\gamma )\\&=2^{2}\|BL\|\cos(\gamma )\\&=2^{2}\cdot \|BD\|\cdot \cos(\gamma )\cdot \cos(\beta )\\&=2^{3}\cdot \|BJ\|\cdot \cos(\gamma )\cdot \cos(\beta )\\&=2^{3}\cdot \|BC\|\cdot \cos(\gamma )\cdot \cos(\beta )\cdot \cos(\alpha )\\&=2^{3}\cdot 1\cdot \cos(\gamma )\cdot \cos(\beta )\cdot \cos(\alpha )\\&=8\cdot \cos(80^{\circ })\cdot \cos(40^{\circ })\cdot \cos(20^{\circ })\end{aligned}}} === Algebraic proof of the generalised identity === Recall the double angle formula for the sine function sin ⁡ ( 2 α ) = 2 sin ⁡ ( α ) cos ⁡ ( α ) . {\displaystyle \sin(2\alpha )=2\sin(\alpha )\cos(\alpha ).} Solve for cos ⁡ ( α ) {\displaystyle \cos(\alpha )} cos ⁡ ( α ) = sin ⁡ ( 2 α ) 2 sin ⁡ ( α ) . {\displaystyle \cos(\alpha )={\frac {\sin(2\alpha )}{2\sin(\alpha )}}.} It follows that: cos ⁡ ( 2 α ) = sin ⁡ ( 4 α ) 2 sin ⁡ ( 2 α ) cos ⁡ ( 4 α ) = sin ⁡ ( 8 α ) 2 sin ⁡ ( 4 α ) ⋮ cos ⁡ ( 2 n − 1 α ) = sin ⁡ ( 2 n α ) 2 sin ⁡ ( 2 n − 1 α ) . {\displaystyle {\begin{aligned}\cos(2\alpha )&={\frac {\sin(4\alpha )}{2\sin(2\alpha )}}\\[6pt]\cos(4\alpha )&={\frac {\sin(8\alpha )}{2\sin(4\alpha )}}\\&\,\,\,\vdots \\\cos \left(2^{n-1}\alpha \right)&={\frac {\sin \left(2^{n}\alpha \right)}{2\sin \left(2^{n-1}\alpha \right)}}.\end{aligned}}} Multiplying all of these expressions together yields: cos ⁡ ( α ) cos ⁡ ( 2 α ) cos ⁡ ( 4 α ) ⋯ cos ⁡ ( 2 n − 1 α ) = sin ⁡ ( 2 α ) 2 sin ⁡ ( α ) ⋅ sin ⁡ ( 4 α ) 2 sin ⁡ ( 2 α ) ⋅ sin ⁡ ( 8 α ) 2 sin ⁡ ( 4 α ) ⋯ sin ⁡ ( 2 n α ) 2 sin ⁡ ( 2 n − 1 α ) . {\displaystyle \cos(\alpha )\cos(2\alpha )\cos(4\alpha )\cdots \cos \left(2^{n-1}\alpha \right)={\frac {\sin(2\alpha )}{2\sin(\alpha )}}\cdot {\frac {\sin(4\alpha )}{2\sin(2\alpha )}}\cdot {\frac {\sin(8\alpha )}{2\sin(4\alpha )}}\cdots {\frac {\sin \left(2^{n}\alpha \right)}{2\sin \left(2^{n-1}\alpha \right)}}.} The intermediate numerators and denominators cancel leaving only the first denominator, a power of 2 and the final numerator. Note that there are n terms in both sides of the expression. Thus, ∏ k = 0 n − 1 cos ⁡ ( 2 k α ) = sin ⁡ ( 2 n α ) 2 n sin ⁡ ( α ) , {\displaystyle \prod _{k=0}^{n-1}\cos \left(2^{k}\alpha \right)={\frac {\sin \left(2^{n}\alpha \right)}{2^{n}\sin(\alpha )}},} which is equivalent to the generalization of Morrie's law. == See also == Viète's formula, same identity taking α = 2 − n x {\displaystyle \alpha =2^{-n}x} on Morrie's law List of trigonometric identities == References == == Further reading == Glen Van Brummelen: Trigonometry: A Very Short Introduction. Oxford University Press, 2020, ISBN 9780192545466, pp. 79–83 Ernest C. Anderson: Morrie's Law and Experimental Mathematics. In: Journal of recreational mathematics, 1998 == External links == Weisstein, Eric W. "Morrie's Law". MathWorld.
Wikipedia:Morris Birkbeck Pell#0	Morris Birkbeck Pell (31 March 1827 – 7 May 1879) was an American-Australian mathematician, professor, lawyer and actuary. He became the inaugural Professor of Mathematics and Natural Philosophy at the University of Sydney in 1852, and continued in the role until ill health enforced his retirement in 1877. He was for many years a member of the University Senate, and councillor and secretary of the Royal Society of New South Wales. == Early life == Pell's mother Eliza Birkbeck (1797-1880) was a daughter of Morris Birkbeck (1764-1825), the English agricultural innovator, social reformer and antislavery campaigner. In 1817-18 Birkbeck, with George Flower, had founded a utopian colony, the English Settlement, in the Illinois Territory of the United States, and Birkbeck laid out the new town there of Albion, Illinois. A widower since 1804, Birkbeck had brought his seven children with him to America, and it was there that his daughter Eliza met and married Gilbert Titus Pell (1796-1860), who came from a prominent family of New York politicians. Gilbert Pell was descended from Sir John Pell (1643-1702), Lord of Pelham Manor, New York—who was the son of English mathematician Dr. John Pell (1611-1685), and nephew and heir of early American pioneer and settler Thomas Pell. Gilbert Pell served as a representative in the Illinois legislature, and in the 1850s was appointed United States envoy to Mexico. Morris Pell was born of this union in the new settlement of Albion in 1827, their third child and only son. In 1835 the family separated and Mrs Pell took her children first to Poughkeepsie, New York, then to Plymouth, England, in 1841, where Morris attended the New Grammar School. In 1849 he graduated as Senior Wrangler in mathematics at Cambridge University—a position once regarded as "the greatest intellectual achievement attainable in Britain." == Career == In 1852, aged 24, Pell was chosen from twenty-six candidates to become the first Professor of Mathematics and Natural Philosophy at the newly opened University of Sydney, in the British colony of New South Wales, Australia. With his new wife Jane Juliana (née Rusden), his mother and two sisters he sailed from England to Australia on the Asiatic and became one of the university's three foundation professors. Professor Pell gave the first lecture in Mathematics on 13 October 1852, two days after the university's inauguration, to all 24 students of the university. One of them, William Windeyer, later to become Chancellor of the university, wrote in his diary: "Went to a lecture at 10 with Mr Pell, who amused as well as instructed, I think I shall like him ...". In 1854, in evidence to a New South Wales Legislative Council select committee on education, Pell advocated the opening of a secular grammar school. In 1859 he testified to the New South Wales Legislative Assembly select committees on the Sydney Grammar School and the University of Sydney, regarding the composition of the University Senate, the adverse effect of clergy on enrolments, the value of liberal studies in the education of businessmen and squatters, and the beneficial effect of the university on secondary education. His evidence resulted in ex-officio membership of the University Senate for professors. He was a member of the Senate from 1861 to 1877 and after resignation was re-elected to the senate in 1878 by members of convocation. Pell was a member of the Australian Philosophical Society from 1856 and served on its council in 1858. Subsequently, Queen Victoria granted Royal Assent to the Society and it was renamed the Royal Society of New South Wales. Pell was a member and its secretary from 1867, and a member of its council from 1869. For many years almost crippled by an injury to his spine, Pell resigned in mid-1877 as professor of mathematics at Sydney University, on a pension of £412 10s. == Personal life == On 7 May 1879, aged 52, he died of "progressive paralysis" (see Motor neuron disease) and was buried in the Balmain Cemetery in Sydney. He was survived by his estranged wife Julia (née Rusden), five sons and three daughters. == References == == External links == Morris Birkbeck Pell at the Australian Dictionary of Biography Online
Wikipedia:Moscow Mathematical Papyrus#0	The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geometry, and algebra. Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, where it remains today. Based on the palaeography and orthography of the hieratic text, the text was most likely written down in the 13th Dynasty and based on older material probably dating to the Twelfth Dynasty of Egypt, roughly 1850 BC. Approximately 5.5 m (18 ft) long and varying between 3.8 and 7.6 cm (1.5 and 3 in) wide, its format was divided by the Soviet Orientalist Vasily Vasilievich Struve in 1930 into 25 problems with solutions. It is a well-known mathematical papyrus, usually referenced together with the Rhind Mathematical Papyrus. The Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two. == Exercises contained in the Moscow Papyrus == The problems in the Moscow Papyrus follow no particular order, and the solutions of the problems provide much less detail than those in the Rhind Mathematical Papyrus. The papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively. The remaining problems are more common in nature. === Ship's part problems === Problems 2 and 3 are ship's part problems. One of the problems calculates the length of a ship's rudder and the other computes the length of a ship's mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long. === Aha problems === Aha problems involve finding unknown quantities (referred to as aha, "stack") if the sum of the quantity and part(s) of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance, problem 19 asks one to calculate a quantity taken 1+1⁄2 times and added to 4 to make 10. In other words, in modern mathematical notation one is asked to solve 3 2 x + 4 = 10 {\displaystyle {\frac {3}{2}}x+4=10} . === Pefsu problems === Most of the problems are pefsu problems (see: Egyptian algebra): 10 of the 25 problems. A pefsu measures the strength of the beer made from a hekat of grain pefsu = number loaves of bread (or jugs of beer) number of heqats of grain {\displaystyle {\mbox{pefsu}}={\frac {\mbox{number loaves of bread (or jugs of beer)}}{\mbox{number of heqats of grain}}}} A higher pefsu number means weaker bread or beer. The pefsu number is mentioned in many offering lists. For example, problem 8 translates as: (1) Example of calculating 100 loaves of bread of pefsu 20 (2) If someone says to you: "You have 100 loaves of bread of pefsu 20 (3) to be exchanged for beer of pefsu 4 (4) like 1/2 1/4 malt-date beer" (5) First calculate the grain required for the 100 loaves of the bread of pefsu 20 (6) The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer (7) The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain. (8) Calculate 1/2 of 5 heqat, the result will be 2 1/2 (9) Take this 2 1/2 four times (10) The result is 10. Then you say to him: (11) "Behold! The beer quantity is found to be correct." === Baku problems === Problems 11 and 23 are Baku problems. These calculate the output of workers. Problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to? Problem 23 finds the output of a shoemaker given that he has to cut and decorate sandals. === Geometry problems === Seven of the twenty-five problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere (problem 10) and finding the volume of a frustum (a truncated pyramid). == Two geometry problems == === Problem 10 === The tenth problem of the Moscow Mathematical Papyrus asks for a calculation of the surface area of a hemisphere (Struve, Gillings) or possibly the area of a semi-cylinder (Peet). Below we assume that the problem refers to the area of a hemisphere. The text of problem 10 runs like this: "Example of calculating a basket. You are given a basket with a mouth of 4 1/2. What is its surface? Take 1/9 of 9 (since) the basket is half an egg-shell. You get 1. Calculate the remainder which is 8. Calculate 1/9 of 8. You get 2/3 + 1/6 + 1/18. Find the remainder of this 8 after subtracting 2/3 + 1/6 + 1/18. You get 7 + 1/9. Multiply 7 + 1/9 by 4 + 1/2. You get 32. Behold this is its area. You have found it correctly." The solution amounts to computing the area as Area = ( ( ( 2 × diameter ) × 8 9 ) × 8 9 ) × diameter = 128 81 ( diameter ) 2 {\displaystyle {\text{Area}}=(((2\times {\text{diameter}})\times {\frac {8}{9}})\times {\frac {8}{9}})\times {\text{diameter}}={\frac {128}{81}}({\text{diameter}})^{2}} The formula calculates for the area of a hemisphere, where the scribe of the Moscow Papyrus used 256 81 ≈ 3.16049 {\displaystyle {\frac {256}{81}}\approx 3.16049} to approximate π. === Problem 14: Volume of frustum of square pyramid === The fourteenth problem of the Moscow Mathematical calculates the volume of a frustum. Problem 14 states that a pyramid has been truncated in such a way that the top area is a square of length 2 units, the bottom a square of length 4 units, and the height 6 units, as shown. The volume is found to be 56 cubic units, which is correct. The solution to the problem indicates that the Egyptians knew the correct formula for obtaining the volume of a truncated pyramid: V = 1 3 h ( a 2 + a b + b 2 ) {\displaystyle V={\frac {1}{3}}h(a^{2}+ab+b^{2})} where a and b are the base and top side lengths of the truncated pyramid and h is the height. Researchers have speculated how the Egyptians might have arrived at the formula for the volume of a frustum but the derivation of this formula is not given in the papyrus. == Summary == Richard J. Gillings gave a cursory summary of the Papyrus' contents. Numbers with overlines denote the unit fraction having that number as denominator, e.g. 4 ¯ = 1 4 {\displaystyle {\bar {4}}={\frac {1}{4}}} ; unit fractions were common objects of study in ancient Egyptian mathematics. == Other papyri == Other mathematical texts from Ancient Egypt include: Berlin Papyrus 6619 Egyptian Mathematical Leather Roll Lahun Mathematical Papyri Rhind Mathematical Papyrus General papyri: Papyrus Harris I Rollin Papyrus For the 2/n tables see: RMP 2/n table == See also == List of ancient Egyptian papyri == Notes == == References == === Full text of the Moscow Mathematical Papyrus === Struve, Vasilij Vasil'evič, and Boris Turaev. 1930. Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer === Other references === Allen, Don. April 2001. The Moscow Papyrus and Summary of Egyptian Mathematics. Imhausen, A., Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden 2003. Mathpages.com. The Prismoidal Formula. O'Connor and Robertson, 2000. Mathematics in Egyptian Papyri. Truman State University, Math and Computer Science Division. Mathematics and the Liberal Arts: Ancient Egypt and The Moscow Mathematical Papyrus. Williams, Scott W. Mathematicians of the African Diaspora, containing a page on Egyptian Mathematics Papyri. Zahrt, Kim R. W. Thoughts on Ancient Egyptian Mathematics Archived 2011-09-27 at the Wayback Machine.
Wikipedia:Mosely snowflake#0	The Mosely snowflake (after Jeannine Mosely) is a Sierpiński–Menger type of fractal obtained in two variants either by the operation opposite to creating the Sierpiński-Menger snowflake or Cantor dust i.e. not by leaving but by removing eight of the smaller 1/3-scaled corner cubes and the central one from each cube left from the previous recursion (lighter) or by removing only corner cubes (heavier). In one dimension this operation (i.e. the recursive removal of two side line segments) is trivial and converges only to single point. It resembles the original water snowflake of snow. By construction the Hausdorff dimension of the lighter snowflake is d H = log 3 ⁡ ( 27 − 9 ) = ln ⁡ 18 / ln ⁡ 3 ≈ 2.630929 {\displaystyle d_{H}=\log _{3}(27-9)=\ln 18/\ln 3\approx 2.630929} and the heavier d H = log 3 ⁡ ( 27 − 8 ) = ln ⁡ 19 / ln ⁡ 3 ≈ 2.680143 {\displaystyle d_{H}=\log _{3}(27-8)=\ln 19/\ln 3\approx 2.680143} . == See also == Menger sponge == References == Wertheim, Margaret; Mosely, Jeannine (2011), The Mosely snowflake sponge: construction guide, California : USC Libraries{{citation}}: CS1 maint: publisher location (link).
Wikipedia:Moses Lemans#0	Moses Lemans (November 5, 1785, Naarden, Netherlands – October 17, 1832, Amsterdam, Netherlands) was a Dutch-Jewish Hebraist and mathematician, and a leader of the Haskalah movement in Holland. He was a founder of the Jewish Mathematicians' Association, Mathesis Artium Genetrix, and published a number of works on Hebrew grammar and mathematics. Born in Naarden, Lemans was educated by his father and (in mathematics) by Judah Littwack. He helped found Hanokh la na'ar al pi darkho, a society for reform in Jewish education, for which he published a number of Hebrew textbooks. In 1818 he was appointed head of the first school for needy Jews in Amsterdam, and in 1828 teacher of mathematics in the Amsterdam gymnasium. == Works == In 1808 he published Ma'amar Imrah Ẓerufah (Article on Pure Speech), in which he advocated for the abandonment of Ashkenazi pronunciation of Hebrew in favour of the Sephardi one, and some years later a Hebrew grammar, Rudimenta (1820). In collaboration with Samuel Israel Mulder he published a Hebrew-Dutch dictionary in 1829-1831. The most notable of his Hebrew poems is an epic on the Belgian Revolution. Lemans was also involved in efforts to propagate among Jews of the Netherlands a knowledge of the Dutch language, by translating prayer-books into Dutch. == References == This article incorporates text from a publication now in the public domain: Isidore Singer and E. Slijper (1901–1906). "Lemans, Moses". In Singer, Isidore; et al. (eds.). The Jewish Encyclopedia. New York: Funk & Wagnalls. == Further reading == A. Dellavilla (1852), Allon Muẓẓav. Michman-Melkman (1967), Leshonenu la-Am, 18, 76–90, 120–35. Teisjure l'Ange (1833), Algemeene Konst-en-Letterbode, ii., Nos. 37, 38. Ulman (1836), Jaarboeken voor de Israëliten in Nederland, 2 (1836), 297–312.
Wikipedia:Moshe Goldberg#0	Moshe Goldberg (Hebrew: משה גולדברג‎; born 1945) is an Israeli mathematician. He is a professor emeritus of mathematics at the Technion – Israel Institute of Technology. == Early life == Moshe Goldberg was born and raised in Tel Aviv. His parents, Gad and Rachel Raya Goldberg, immigrated from Poland and Lithuania to Palestine shortly after Hitler became Germany's chancellor in 1933. After completing his undergraduate studies, Goldberg served in the Israel Defense Forces for three years. Released at the rank of captain, he resumed his studies, earning his Ph.D. from Tel Aviv University in 1973 under the supervision of Saul Abarbanel. == Academic career == After a postdoctoral position at the University of California, Los Angeles (UCLA), Goldberg joined the Technion – Israel Institute of Technology in 1979, and in due course became the Ruth and Samuel Jaffe Professor of Mathematics. Goldberg began his scientific career in computational fluid dynamics. He then turned to other topics, including numerical analysis of hyperbolic and parabolic partial differential systems, linear and Multilinear algebra, matrix and operator theory, functional analysis, and various types of algebras. He held visiting positions at California Institute of Technology (Caltech), UCLA, University of California Santa Barbara, and Université Paris Dauphine (Paris 9). Goldberg published over 80 research papers. In 2013 he retired as professor emeritus. == References ==
Wikipedia:Moshe Meiselman#0	Moshe Meiselman is an American-born Orthodox rabbi and rosh yeshiva (dean) of Yeshiva Toras Moshe in Jerusalem, which he established in 1982. He also founded and served as principal of Yeshiva University of Los Angeles (YULA) from 1977 to 1982. He is a descendant of the Lithuanian Jewish Soloveitchik rabbinic dynasty. == Early life and education == Rabbi Meiselman was born to Harry Meiselman, a dental surgeon, and Shulamit Soloveitchik, a teacher and Jewish school principal who attended New York University and Radcliffe College. On his father's side, he is a descendent of the rebbe (hereditary leader of a hasidic dynasty) Baruch of Kossov. On his mother's side, he is a descendant of the Soloveitchik rabbinic dynasty. His maternal grandfather was Rabbi Moshe Soloveichik and his maternal great-grandfather was Rabbi Chaim Soloveitchik, known as Reb Chaim Brisker. His mother, Shulamit, authored the book The Soloveitchik Heritage: A Daughter's Memoir (1995). Rabbi Meiselman was a nephew of Rabbi Dr. Joseph B. Soloveitchik, rosh yeshiva of R.I.E.T.S., with whom, according to Rabbi Meiselman, he had study sessions on a near daily basis from the time he was 18 until he was 29 years old. Rabbi Meiselman graduated from high school at the Boston Latin School and then went on to attend Harvard College (which all three of Soloveitchik's children and his American grandchildren attended) and the Massachusetts Institute of Technology where he earned his doctorate in mathematics in 1967 with the thesis "The Operation Ring for Connective K-Theory". == Career == Rabbi Meiselman began his career teaching mathematics at City University of New York. After his marriage in 1971, he became a maggid shiur at Beis Medrash L'Torah in Skokie. Afterward, he taught at Yeshivas Brisk (Brisk Rabbinical College) in Chicago, headed for a time by his uncle, Rabbi Ahron Soloveichik. In 1977 he moved to the West Coast and founded the Yeshiva University of Los Angeles (YULA), opening separate high school programs for boys and girls, a yeshivah gedolah, and a kolel. He also served as a posek (arbiter of Jewish law) for the local community. In 1982, having built up enrollment to nearly 400 male and female students in YULA's various divisions, Rabbi Meiselman moved to Israel to open a yeshiva for American students, together with co-rosh yeshiva Rabbi Doniel Lehrfield (Rabbi Lehrfield and several other faculty members subsequently left to start another yeshiva, Bais Yisroel). He named the new school Toras Moshe after his grandfather, Moshe. He selected Rabbis Michel Shurkin and Moshe Twersky, both close students of Rabbi Soloveitchik, to head the teaching staff. In 2011, Meiselman reported about his yeshiva that "We have 96 boys in the beis medrash and 44 in the kollel, and almost all of our kollel yungerleit are home-grown". Rabbi Meiselman is one of several grandchildren of Rabbi Moshe Soloveichik who have established yeshivas in Israel, perhaps the most famous being Rabbi Aharon Lichtenstein, son-in-law of Rabbi Joseph Soloveitchik, who established Yeshivat Har Etzion in the late 1960s. Yeshivat Reshit, a popular yeshiva in Israel for American students in Beit Shemesh, was established by the Rabbis Marcus, also descendants of Rabbi Moshe Soloveitchik. Rabbi Meiselman is the author of several books and numerous magazine articles. His Jewish Woman in Jewish Law (1978) sparked much discussion among authors and feminists for his traditional Jewish response to feminism. Additionally, Rabbi Meiselman has authored Tiferes Tzvi, a commentary to the Rambam, as well as numerous articles on Talmudic study and thought in Hebrew. == Philosophies and controversies == === Torah, Chazal and Science === Rabbi Meiselman's 2013 book, Torah, Chazal and Science, promotes the theory that all unqualified scientific statements of the Talmudic sages are sourced from the word of God, who can not be wrong, and are therefore immutable: "All of Chazal’s (the Talmudic sages') definitive statements are to be taken as absolute fact [even] outside the realm of halakhah (Jewish law)". The flip side of this thesis, and another major theme of the book, is that modern science is transitory and unreliable compared to the divine wisdom of the sages. His book has been criticized by Rabbi Dr. Natan Slifkin and others. === The Holocaust === Following the opinion of some Haredi thinkers in the area of Holocaust theology, Meiselman has argued that the Holocaust was the result of Jewish cultural assimilation in Western Europe in the early twentieth century. He writes that "the turning away from the status of an am ha-nivhar, a chosen people, and the frightening rush toward assimilation were, according to the rules that govern Jewish destiny, the real causes for the Holocaust." === The State of Israel === Rabbi Meiselman subscribes to Haredi views regarding the State of Israel and the Israel Defence Forces. He has stated that it is forbidden for a yeshiva student to join the Israeli army, and has criticized the Nachal Haredi, stating in an interview that Nachal Haredi has "not been successful in maintaining commitment to Torah." In 2013, Rabbi Meiselman sat on the dais at a rally in NY against conscription of yeshiva students into the Israeli army. Both Satmar Rebbes were involved in the planning of, and also sat at the dais at, this rally. === Modern Orthodoxy === In commenting on Modern Orthodox innovations with regard to women, Rabbi Meiselman has stated that "when it comes to the rabbis and the people who are at the forefront of pushing for these changes so that they can 'update' Orthodoxy to conform with today’s 'progressive' cultural norm ... [the] common denominator between nearly all of them is that they are largely ignorant of halacha and devoid of serious Torah scholarship. If your knowledge of Torah and halacha are limited, then you are not limited by halacha. One is never confined by things that one doesn't know and never learned!" Some of Yeshiva Toras Moshe's faculty members dissuade students from enrolling at Yeshiva University when they leave Toras Moshe, while others are less opposed. == Personal == Meiselman is married to Rivkah Leah Eichenstein. == Bibliography == === Books === Torah, Chazal and Science. Israel Bookshop Publications. 2013. ISBN 978-1-60091-243-6. Jewish Woman in Jewish Law. Ktav Publishing House. 1978. ISBN 0-87068-329-2. Meiselman, Moshe (1974). Esheth Hayil in Perspective: The Role of Women in Judaism. === Selected articles === "Commitment: Reviewed by Moshe Meiselman" (PDF). Jewish Action. Orthodox Union: 89–94. 2005. "The Rav, Feminism and Public Policy: An Insider's Overview". Tradition. 33 (1): 5–30. The Incomparable Gaon of Vilna. Orthodox Union. 1997. == References == == External links == "Book Review: 'Torah, Chazal and Science'", San Diego Jewish World, March 27, 2014
Wikipedia:Moshe Shaked#0	Moshe Shaked (February 21, 1945 - October 28, 2014) was an American mathematician and statistician. He was a student at the Hebrew University of Jerusalem and at the University of Rochester, where he completed his Ph.D. in 1975, under Albert W. Marshall. Shaked held various positions at the University of New Mexico, the University of British Columbia, and at Indiana University. He became a full professor of mathematics at the University of Arizona in 1986, and he was a Professor Emeritus at Arizona since 2013. Moshe Shaked was a Fellow of the Institute of Mathematical Statistics. Shaked was a leading figure in stochastic order and distribution theory. He published widely in applied probability and statistics. He became most celebrated internationally for his collection of influential papers on stochastic order and multivariate dependence. Shaked’s contribution also includes pioneering studies on stochastic convexity and on multivariate phase-type distributions, with important applications in reliability modelling and queueing analysis. He made significant contributions to multivariate aging notions and multivariate life distributions, as well as to accelerated life tests (inference, non-parametric approach and goodness of fit). Moshe Shaked was born in Jerusalem to a Polish immigrant family. He was a knowledgeable ancient-coin enthusiast and a passionate museum-goer. His son, Tal Shaked, is an American chess grandmaster. == External links == Moshe Shaked at the Mathematics Genealogy Project Moshe Shaked - Google Scholar Citations Moshe Shaked, 1945–2014, IMS Bulletin Stochastic Orders in Reliability and Risk In Honor of Professor Moshe Shaked, Haijun Li and Xiaohu Li (Eds.), 2013, Springer Stochastic Orders, Moshe Shaked and George Shanthikumar, 2007, Springer
Wikipedia:Moshe Zakai#0	Moshe Zakai (Hebrew: משה זכאי; December 22, 1926 – November 27, 2015) was a Distinguished Professor at the Technion, Israel in electrical engineering, member of the Israel Academy of Sciences and Humanities and Rothschild Prize winner. == Biography == Moshe Zakai was born in Sokółka, Poland, to his parents Rachel and Eliezer Zakheim with whom he immigrated to Israel in 1936. He got the BSc degree in electrical engineering from the Technion – Israel Institute of Technology in 1951. He joined the scientific department of the Defense Minister of Israel, where he was assigned to research and development of radar systems. From 1956 to 1958, he did graduate work at the University of Illinois on an Israeli Government Fellowship, and was awarded the PhD in electrical engineering. He then returned to the scientific department as head of the communication research group. In 1965, he joined the faculty of the Technion as an associate professor. In 1969, he was promoted to the rank of professor and in 1970, he was appointed the holder of the Fondiller Chair in Telecommunication. He was appointed distinguished professor in 1985. From 1970 until 1973, he served as the dean of the faculty of Electrical Engineering, and from 1976 to 1978 he served as vice president of academic affairs. He retired in 1998 as distinguished professor emeritus. Moshe Zakai was married to Shulamit (Mita) Briskman, they have 3 children and 12 grandchildren. == Major awards == 1973 Fellow of the Institute of Electrical and Electronics Engineers (IEEE) 1988 Fellow of the Institute of Mathematical Statistics 1989 Foreign member of the US National Academy of Engineering 1993 Member of the Israel Academy of Sciences and Humanities 1993 The IEEE Control Systems Award 1994 The Rothschild Prize in Engineering == Research == === Background === Zakai's main research concentrated on the study of the theory of stochastic processes and its application to information and control problems; namely, problems of noise in communication radar and control systems. The basic class of random processes which represent the noise in such systems are known as "white noise" or the "Wiener process" where the white noise is "something like a derivative" of the Wiener process. Since these processes vary quickly with time, the classical differential and integral calculus is not applicable to such processes. In the 1940s Kiyoshi Itō developed a stochastic calculus (the Ito calculus) for such random processes. === The relation between classical and Ito calculi === From the results of Ito it became clear, back in the 1950s, that if a sequence of smooth functions which present the input to a physical system converge to something like a Brownian motion, then the sequence of outputs of the system do not converge in the classical sense. Several papers written by Eugene Wong and Zakai clarified the relation between the two approaches. This opened up the way to the application of the Ito calculus to problems in physics and engineering. These results are often referred to as Wong-Zakai corrections or theorems. === Nonlinear filtering === The solution to the problem of the optimal filtering of a wide class of linear dynamical system is known as the Kalman filter. This led to the same problem for nonlinear dynamical systems. The results for this case were highly complicated and were initially studied by Stratonovich in 1959 - 1960 and later by Kushner in 1964, leading to the Kushner-Stratonovich equation, a non-linear stochastic partial differential equation (SPDE) for the conditional probability density representing the optimal filter. Around 1967, Zakai derived a considerably simpler SPDE for an unnormalized version of the optimal filter density. It is known as the Zakai equation, and it has the great advantage of being a linear SPDE. The Zakai equation has been the starting point for further research work in this field. === Comparing practical solutions with the optimal solution === In many cases the optimal design of communication or radar operating under noise is too complicated to be practical, while practical solutions are known. In such cases it is extremely important to know how close the practical solution is to the theoretically optimal one. === Extension of the Ito calculus to the two-parameter processes === White noise and Brownian motion (the Wiener process) are functions of a single parameter, namely time. For problems such as rough surfaces it is necessary to extend the Ito calculus to two parameter Brownian sheets. Several papers which he wrote jointly with Wong extend the Ito integral to a "two-parameter" time. They also showed that every functional of the Brownian sheet can be represented as an extended integral. === The Malliavin calculus and its application === In addition to the Ito calculus, Paul Malliavin developed in the 1970s a "stochastic calculus of variations", now known as the Malliavin calculus. It turned out that in this setup it is possible to define a stochastic integral which will include the Ito integral. The papers of Zakai with David Nualart, Ali Süleyman Üstünel and Zeitouni promoted the understanding and applicability of the Malliavin calculus. The monograph of Üstünel and Zakai deals with the application of the Malliavin calculus to derive relations between the Wiener process and other processes which are in some sense "similar" to the probability law of the Wiener process. In the last decade he extended to transformations which are in some sense a "rotation" of the Wiener process and with Ustunel extended to some general cases results of information theory which were known for simpler spaces. === Further information === On his life and research, see pages xi–xiv of the volume in honor of Zakai's 65 birthday. For the list of publications until 1990, see pages xv–xx. For publications between 1990 and 2000, see [17]. For later publications search for M Zakai in arXiv. == References ==
Wikipedia:Moshé Machover#0	Moshé Machover (Hebrew: משה מחובר; born 1936) is a mathematician, philosopher, and socialist activist, noted for his writings against Zionism. Born to a Jewish family in Tel Aviv, then part of the British Mandate of Palestine, Machover moved to Britain in 1968 where he became a naturalised citizen. He was a founder of Matzpen, the Israeli Socialist Organisation, in 1962. == Career == Machover has written extensively on the conflict in the Middle East. In 1961, while still members of the Israeli Communist Party, Machover and Akiva Orr, under the pseudonym 'A Israeli', wrote the anti-Zionist analysis of the Arab-Israeli conflict Shalom, Shalom ve'ein Shalom (Hebrew: שלום, שלום, ואין שלום; "Peace, Peace, and there is no Peace"). The intention of the book was to explain, from publicly available sources, why in 1956 "Ben-Gurion preferred to invade Egypt, alongside France and Britain, rather than to make peace with Egypt". In the course of writing the book, "It became clear to us that the roots of the Israeli–Arab conflict lay, not in the conflict between Israel and the Arab states, but rather in the conflict between Zionist colonialism and the Palestinians over the land of Palestine and its independence". This thesis was an implicit challenge to the line of the Israeli Communist Party, which considered Israel's alliance with the U.S. to be a matter of political choice, not deriving from the colonial nature of the state. When Machover and Orr followed this by criticising the party's adherence to the Soviet line, and called for the publication of the party's history, they were expelled. Machover and Orr, together with others expelled at the same time, then established Matzpen. Together with Jabra Nicola (pen name A. Said), Machover developed the position, adopted by Matzpen, that the solution to the Israeli Palestinian problem is in a struggle to defeat Zionism and its allies – imperialism and Arab Reaction – and "rally to itself a wider struggle for the political and social liberation of the Middle East as a whole". The struggle for Palestinian liberation can succeed only when the Palestinian and Israeli masses enter "a joint struggle with the revolutionary forces in the Arab world". Machover was a lecturer in mathematics at the Hebrew University of Jerusalem from 1960 to 1964, and again from 1966 to 1968; during 1964–66, he was a visiting lecturer at the University of Bristol in England. In 1968, Machover moved permanently to London where he became Reader in Mathematical Logic at King's College London until 1995. He joined Kings from the Chelsea College of Science and Technology's Department of History and Philosophy of Science which was merged into the Philosophy Department in 1993. Since 1995, he has been Professor of Philosophy at the University of London. In London, together with Orr and Shimon Tzabar, Machover established the Israeli Revolutionary Action Committee Abroad. In 1971, Machover, Orr and Haim Hanegbi published an article in the New Left Review on "The Class Nature of Israeli Society". This article, which has been frequently republished, is included together with several more of Machover's early writings on the Middle East in the collection The Other Israel: the radical case against Zionism. In 1975, Machover was one of the founders of Khamsin, the "journal of revolutionary socialists of the Middle East". Many of his articles from Khamsin are included in the collection Forbidden Agendas. In 2012, Haymarket Press published a collection of Machover's essays from 1966 to 2010, under the title Israelis and Palestinians Conflict and Resolution. == October 2017 expulsion and readmission into the Labour Party == Machover is a Labour Party member of the Hampstead and Kilburn Constituency Labour Party. In October 2017, he was expelled from the Labour Party on suspicion that of being associated with the Communist Party of Great Britain (Provisional Central Committee), in contravention of party rules. The expulsion came when an article he wrote for the CPGB (PCC)'s newspaper, the Weekly Worker was being investigated as, according to the party's head of disputes, it "appears to meet the International Holocaust Remembrance Alliance definition of antisemitism". The Guardian subsequently published a letter of protest undersigned by 139 Labour Party members, including Sir Geoffrey Bindman, dismissing the insinuation of anti-Semitism as "personally offensive and politically dangerous". His expulsion was rescinded on 30 October. == Personal life == Machover's son Daniel Machover is a solicitor in London, specialising in human and civil rights cases. == Technical publications == Lectures on Non-Standard Analysis (with J. Hirschfeld, 1969) A Course in Mathematical Logic (with J. L. Bell, 1977) Laws of Chaos: A Probabilistic Approach to Political Economy (with E. Farjoun, London, 1983) Set Theory, Logic and their Limitations (1996) The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes (with D. Felsenthal, 1998). == Selected books == Peace, Peace and there is no Peace (with Akiva Orr) (Hebrew), (English) The Other Israel: the radical case against Zionism Israelis and Palestinians: Conflict and Resolution, Haymarket Books, Paperback, February 2012 == References == == External links == Articles in Weekly Worker (newspaper of the Communist Party of Great Britain (Provisional Central Committee) Moshé Machover Archive at Marxists Internet Archive Articles on the University of Cork Palestine Solidarity Campaign website Israelis and Palestinians: Conflict and Resolution, Barry Amiel and Norman Melburn Trust Annual Lecture, 30. November 2006
Wikipedia:Motivic zeta function#0	In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.} Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta function Arithmetic zeta function Artin–Mazur zeta function of a dynamical system Barnes zeta function or double zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta function of a number field Duursma zeta function of error-correcting codes Epstein zeta function of a quadratic form Goss zeta function of a function field Hasse–Weil zeta function of a variety Height zeta function of a variety Hurwitz zeta function, a generalization of the Riemann zeta function Igusa zeta function Ihara zeta function of a graph L-function, a "twisted" zeta function Lefschetz zeta function of a morphism Lerch zeta function, a generalization of the Riemann zeta function Local zeta function of a characteristic-p variety Matsumoto zeta function Minakshisundaram–Pleijel zeta function of a Laplacian Motivic zeta function of a motive Multiple zeta function, or Mordell–Tornheim zeta function of several variables p-adic zeta function of a p-adic number Prime zeta function, like the Riemann zeta function, but only summed over primes Riemann zeta function, the archetypal example Ruelle zeta function Selberg zeta function of a Riemann surface Shimizu L-function Shintani zeta function Subgroup zeta function Witten zeta function of a Lie group Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special case for the poset of integer divisibility is related as a formal Dirichlet series to the Riemann zeta function. Zeta function of an operator or spectral zeta function == See also == Other functions called zeta functions, but not analogous to the Riemann zeta function Jacobi zeta function Weierstrass zeta function Topics related to zeta functions Artin conjecture Birch and Swinnerton-Dyer conjecture Riemann hypothesis and the generalized Riemann hypothesis. Selberg class S Explicit formulae for L-functions Trace formula == External links == A directory of all known zeta functions
Wikipedia:Motoko Kotani#0	Motoko Kotani (Japanese: 小谷元子, born 1960) is a Japanese applied mathematician, specializing in discrete geometric analysis and crystallography, and an academic administrator. She is the executive vice president for research for Tohoku University, the former executive director of Riken, the former president of the Mathematical Society of Japan, and the president-elect of the International Science Council. == Education == Kotani graduated from the University of Tokyo in 1983. She went to Tokyo Metropolitan University for graduate study, earning a master's degree in 1985 and completing her doctorate in 1990. == Academic career == She was a lecturer at Toho University from 1990 to 1997, and an associate professor there from 1997 to 1999, with a term as a postdoctoral researcher in Germany at the Max Planck Institute for Mathematics from 1993 to 1994. In 1999, she moved to the Mathematics Institute of Tohoku University. She was named as a distinguished professor there in 2008 and as director of the WPI Research Center, Advanced Institute for Materials Research in 2012. She became executive director of Riken from 2017 to 2020, while continuing to hold a position as a researcher at Tohoku. In 2020 she was named executive vice president for research. == Service == Kotani was president of the Mathematical Society of Japan from 2015 to 2016. She became president-elect of the International Science Council in 2021, for a three-year term beginning in 2024. In 2022 the Japanese Ministry of Foreign Affairs named her as Science and Technology Co-Advisor to the Minister for Foreign Affairs. == Recognition == Kotani was the 2005 winner of the Saruhashi Prize. == References == == External links == English-language home page at Tohoku University Voices of Women at IHES: Motoko Kotani
Wikipedia:Motor variable#0	In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. Motor variable is used here in place of split-complex variable for euphony and tradition. For example, f ( z ) = u ( z ) + j v ( z ) , z = x + j y , x , y ∈ R , j 2 = + 1 , u ( z ) , v ( z ) ∈ R . {\displaystyle f(z)=u(z)+j\ v(z),\ z=x+jy,\ x,y\in R,\quad j^{2}=+1,\quad u(z),v(z)\in R.} Functions of a motor variable provide a context to extend real analysis and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with motor variables, and more generally in hypercomplex analysis. == Elementary functions == Let D = { z = x + j y : x , y ∈ R } {\displaystyle \{z=x+jy:x,y\in R\}} , the split-complex plane. The following exemplar functions f have domain and range in D: The action of a hyperbolic versor u = exp ⁡ ( a j ) = cosh ⁡ a + j sinh ⁡ a {\displaystyle u=\exp(aj)=\cosh a+j\sinh a} is combined with translation to produce the affine transformation f ( z ) = u z + c {\displaystyle f(z)=uz+c\ } . When c = 0, the function is equivalent to a squeeze mapping. The squaring function has no analogy in ordinary complex arithmetic. Let f ( z ) = z 2 {\displaystyle f(z)=z^{2}\ } and note that f ( − 1 ) = f ( j ) = f ( − j ) = 1. {\displaystyle f(-1)=f(j)=f(-j)=1.\ } The result is that the four quadrants are mapped into one, the identity component: U 1 = { z ∈ D :∣ y ∣< x } {\displaystyle U_{1}=\{z\in D:\mid y\mid <x\}} , and there are four square roots for elements of this component but no square roots for elements of the other three components. Note that z z ∗ = 1 {\displaystyle zz^{}=1\ } forms the unit hyperbola x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1} . Thus, the reciprocation f ( z ) = 1 / z = z ∗ / ∣ z ∣ 2 where ∣ z ∣ 2 = z z ∗ {\displaystyle f(z)=1/z=z^{}/\mid z\mid ^{2}{\text{where}}\mid z\mid ^{2}=zz^{*}} involves the hyperbola as curve of reference as opposed to the circle in C. == Linear fractional transformations == Using the concept of a projective line over a ring, the projective line P(D) is formed. The construction uses homogeneous coordinates with split-complex number components. The projective line P(D) is transformed by linear fractional transformations: [ z : 1 ] ( a c b d ) = [ a z + b : c z + d ] , {\displaystyle [z:1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=[az+b:cz+d],} sometimes written f ( z ) = a z + b c z + d , {\displaystyle f(z)={\frac {az+b}{cz+d}},} provided cz + d is a unit in D. Elementary linear fractional transformations include hyperbolic rotations ( u 0 0 1 ) , {\displaystyle {\begin{pmatrix}u&0\\0&1\end{pmatrix}},} translations ( 1 0 t 1 ) , {\displaystyle {\begin{pmatrix}1&0\\t&1\end{pmatrix}},} and the inversion ( 0 1 1 0 ) . {\displaystyle {\begin{pmatrix}0&1\\1&0\end{pmatrix}}.} Each of these has an inverse, and compositions fill out a group of linear fractional transformations. The motor variable is characterized by hyperbolic angle in its polar coordinates, and this angle is preserved by motor variable linear fractional transformations just as circular angle is preserved by the Möbius transformations of the ordinary complex plane. Transformations preserving angles are called conformal, so linear fractional transformations are conformal maps. Transformations bounding regions can be compared: For example, on the ordinary complex plane, the Cayley transform carries the upper half-plane to the unit disk, thus bounding it. A mapping of the identity component U1 of D into a rectangle provides a comparable bounding action: f ( z ) = 1 z + 1 / 2 , f : U 1 → T {\displaystyle f(z)={\frac {1}{z+1/2}},\quad f:U_{1}\to T} where T = {z = x + jy : \|y\| < x < 1 or \|y\| < 2 – x when 1 ≤ x <2}. To realize the linear fractional transformations as bijections on the projective line a compactification of D is used. See the section given below. == Exp, log, and square root == The exponential function carries the whole plane D into U1: e x = ∑ n = 0 ∞ x n n ! = ∑ n = 0 ∞ x 2 n ( 2 n ) ! + ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! = cosh ⁡ x + sinh ⁡ x {\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}+\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}=\cosh x+\sinh x} . Thus when x = bj, then ex is a hyperbolic versor. For the general motor variable z = a + bj, one has e z = e a ( cosh ⁡ b + j sinh ⁡ b ) {\displaystyle e^{z}=e^{a}(\cosh b+j\ \sinh b)\ } . In the theory of functions of a motor variable special attention should be called to the square root and logarithm functions. In particular, the plane of split-complex numbers consists of four connected components { U 1 , − U 1 , j U 1 , − j U 1 } , {\displaystyle \{U_{1},-U_{1},jU_{1},-jU_{1}\},} and the set of singular points that have no inverse: the diagonals z = x ± x j, x ∈ R. The identity component, namely {z : x > \|y\| } = U1, is the range of the squaring function and the exponential. Thus it is the domain of the square root and logarithm functions. The other three quadrants do not belong in the domain because square root and logarithm are defined as one-to-one inverses of the squaring function and the exponential function. Graphic description of the logarithm of D is given by Motter & Rosa in their article "Hyperbolic Calculus" (1998). == D-holomorphic functions == The Cauchy–Riemann equations that characterize holomorphic functions on a domain in the complex plane have an analogue for functions of a motor variable. An approach to D-holomorphic functions using a Wirtinger derivative was given by Motter & Rossa: The function f = u + j v is called D-holomorphic when 0 = ( ∂ ∂ x − j ∂ ∂ y ) ( u + j v ) = u x − j 2 v y + j ( v x − u y ) . {\displaystyle 0\ =\ \left({\partial \over \partial x}-j{\partial \over \partial y}\right)(u+jv)=\ u_{x}-j^{2}v_{y}+j(v_{x}-u_{y}).} By considering real and imaginary components, a D-holomorphic function satisfies u x = v y , v x = u y . {\displaystyle u_{x}=v_{y},\quad v_{x}=u_{y}.} These equations were published in 1893 by Georg Scheffers, so they have been called Scheffers' conditions. The comparable approach in harmonic function theory can be viewed in a text by Peter Duren. It is apparent that the components u and v of a D-holomorphic function f satisfy the wave equation, associated with D'Alembert, whereas components of C-holomorphic functions satisfy Laplace's equation. == La Plata lessons == At the National University of La Plata in 1935, J.C. Vignaux, an expert in convergence of infinite series, contributed four articles on the motor variable to the university's annual periodical. He is the sole author of the introductory one, and consulted with his department head A. Durañona y Vedia on the others. In "Sobre las series de numeros complejos hiperbolicos" he says (p. 123): This system of hyperbolic complex numbers [motor variables] is the direct sum of two fields isomorphic to the field of real numbers; this property permits explication of the theory of series and of functions of the hyperbolic complex variable through the use of properties of the field of real numbers. He then proceeds, for example, to generalize theorems due to Cauchy, Abel, Mertens, and Hardy to the domain of the motor variable. In the primary article, cited below, he considers D-holomorphic functions, and the satisfaction of d’Alembert's equation by their components. He calls a rectangle with sides parallel to the diagonals y = x and y = − x, an isotropic rectangle since its sides are on isotropic lines. He concludes his abstract with these words: Isotropic rectangles play a fundamental role in this theory since they form the domains of existence for holomorphic functions, domains of convergence of power series, and domains of convergence of functional series. Vignaux completed his series with a six-page note on the approximation of D-holomorphic functions in a unit isotropic rectangle by Bernstein polynomials. While there are some typographical errors as well as a couple of technical stumbles in this series, Vignaux succeeded in laying out the main lines of the theory that lies between real and ordinary complex analysis. The text is especially impressive as an instructive document for students and teachers due to its exemplary development from elements. Furthermore, the entire excursion is rooted in "its relation to Émile Borel’s geometry" so as to underwrite its motivation. == Bireal variable == In 1892 Corrado Segre recalled the tessarine algebra as bicomplex numbers. Naturally the subalgebra of real tessarines arose and came to be called the bireal numbers. In 1946 U. Bencivenga published an essay on the dual numbers and the split-complex numbers where he used the term bireal number. He also described some of the function theory of the bireal variable. The essay was studied at University of British Columbia in 1949 when Geoffrey Fox wrote his master's thesis "Elementary function theory of a hypercomplex variable and the theory of conformal mapping in the hyperbolic plane". On page 46 Fox reports "Bencivenga has shown that a function of a bireal variable maps the hyperbolic plane into itself in such a manner that, at those points for which the derivative of a function exists and does not vanish, hyperbolic angles are preserved in the mapping". G. Fox proceeds to provide the polar decomposition of a bireal variable and discusses hyperbolic orthogonality. Starting from a different definition he proves on page 57 Theorem 3.42 : Two vectors are mutually orthogonal if and only if their unit vectors are mutually reflections of one another in one or another of the diagonal lines through 0. Fox focuses on "bilinear transformations" w = α z + β γ z + δ {\displaystyle w={\frac {\alpha z+\beta }{\gamma z+\delta }}} , where α , β , γ , δ {\displaystyle \alpha ,\beta ,\gamma ,\delta } are bireal constants. To cope with singularity he augments the plane with a single point at infinity (page 73). Among his novel contributions to function theory is the concept of an interlocked system. Fox shows that for a bireal k satisfying (a − b)2 < \|k\| < (a + b)2 the hyperbolas \|z\| = a2 and \|z − k\| = b2 do not intersect (form an interlocked system). He then shows that this property is preserved by bilinear transformations of a bireal variable. == Compactification == The multiplicative inverse function is so important that extreme measures are taken to include it in the mappings of differential geometry. For instance, the complex plane is rolled up to the Riemann sphere for ordinary complex arithmetic. For split-complex arithmetic a hyperboloid is used instead of a sphere: H = { ( x , y , z ) : z 2 + x 2 − y 2 = 1 } . {\displaystyle H=\{(x,y,z):z^{2}+x^{2}-y^{2}=1\}.} As with the Riemann sphere, the method is stereographic projection from P = (0, 0, 1) through t = (x, y, 0) to the hyperboloid. The line L = Pt is parametrized by s in L = { ( s x , s y , 1 − s ) : s ∈ R } {\displaystyle L=\{(sx,sy,1-s):s\in R\}} so that it passes P when s is zero and t when s is one. From H ∩ L it follows that ( 1 − s ) 2 + ( s x ) 2 − ( s y ) 2 = 1 , so that s = 2 1 + x 2 − y 2 . {\displaystyle (1-s)^{2}+(sx)^{2}-(sy)^{2}=1,{\text{ so that}}\quad s={\frac {2}{1+x^{2}-y^{2}}}.} If t is on the null cone, then s = 2 and (2x, ±2x, – 1) is on H, the opposite points (2x, ±2x, 1) make up the light cone at infinity that is the image of the null cone under inversion. Note that for t with y 2 > 1 + x 2 , {\displaystyle y^{2}>1+x^{2},} s is negative. The implication is that the back-ray through P to t provides the point on H. These points t are above and below the hyperbola conjugate to the unit hyperbola. The compactification must be completed in P3R with homogeneous coordinates (w, x, y, z) where w = 1 specifies the affine space (x, y, z) used so far. Hyperboloid H is absorbed into the projective conic { ( w , x , y , z ) ∈ P 3 R : z 2 + x 2 = y 2 + w 2 } , {\displaystyle \{(w,x,y,z)\in P^{3}R:z^{2}+x^{2}=y^{2}+w^{2}\},} which is a compact space. Walter Benz performed the compactification by using a mapping due to Hans Beck. Isaak Yaglom illustrated a two-step compactification as above, but with the split-complex plane tangent to the hyperboloid. In 2015 Emanuello & Nolder performed the compactification by first embedding the motor plane into a torus, and then making it projective by identifying antipodal points. == References == Francesco Catoni, Dino Boccaletti, & Roberto Cannata (2008) Mathematics of Minkowski Space-Time, Birkhäuser Verlag, Basel. Chapter 7: Functions of a hyperbolic variable. Shahram Dehdasht + seven others (2021) "Conformal Hyperbolic Optics", Physical Review Research 3,033281 doi:10.1103/PhysRevResearch.3.033281
Wikipedia:Motzkin–Taussky theorem#0	The Motzkin–Taussky theorem is a result from operator and matrix theory about the representation of a sum of two bounded, linear operators (resp. matrices). The theorem was proven by Theodore Motzkin and Olga Taussky-Todd. The theorem is used in perturbation theory, where e.g. operators of the form T + x T 1 {\displaystyle T+xT_{1}} are examined. == Statement == Let X {\displaystyle X} be a finite-dimensional complex vector space. Furthermore, let A , B ∈ B ( X ) {\displaystyle A,B\in B(X)} be such that all linear combinations T = α A + β B {\displaystyle T=\alpha A+\beta B} are diagonalizable for all α , β ∈ C {\displaystyle \alpha ,\beta \in \mathbb {C} } . Then all eigenvalues of T {\displaystyle T} are of the form λ T = α λ A + β λ B {\displaystyle \lambda _{T}=\alpha \lambda _{A}+\beta \lambda _{B}} (i.e. they are linear in α {\displaystyle \alpha } und β {\displaystyle \beta } ) and λ A , λ B {\displaystyle \lambda _{A},\lambda _{B}} are independent of the choice of α , β {\displaystyle \alpha ,\beta } . Here λ A {\displaystyle \lambda _{A}} stands for an eigenvalue of A {\displaystyle A} . === Comments === Motzkin and Taussky call the above property of the linearity of the eigenvalues in α , β {\displaystyle \alpha ,\beta } property L. == Bibliography == Kato, Tosio (1995). Perturbation Theory for Linear Operators. Classics in Mathematics. Vol. 132 (2 ed.). Berlin, Heidelberg: Springer. p. 86. doi:10.1007/978-3-642-66282-9. ISBN 978-3-540-58661-6. Friedland, Shmuel (1981). "A generalization of the Motzkin-Taussky theorem". Linear Algebra and Its Applications. 36: 103–109. doi:10.1016/0024-3795(81)90223-8. == Notes ==
Wikipedia:Mountain pass theorem#0	The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points. == Statement == The assumptions of the theorem are: I {\displaystyle I} is a functional from a Hilbert space H to the reals, I ∈ C 1 ( H , R ) {\displaystyle I\in C^{1}(H,\mathbb {R} )} and I ′ {\displaystyle I'} is Lipschitz continuous on bounded subsets of H, I {\displaystyle I} satisfies the Palais–Smale compactness condition, I [ 0 ] = 0 {\displaystyle I[0]=0} , there exist positive constants r and a such that I [ u ] ≥ a {\displaystyle I[u]\geq a} if ‖ u ‖ = r {\displaystyle \Vert u\Vert =r} , and there exists v ∈ H {\displaystyle v\in H} with ‖ v ‖ > r {\displaystyle \Vert v\Vert >r} such that I [ v ] ≤ 0 {\displaystyle I[v]\leq 0} . If we define: Γ = { g ∈ C ( [ 0 , 1 ] ; H ) \| g ( 0 ) = 0 , g ( 1 ) = v } {\displaystyle \Gamma =\{\mathbf {g} \in C([0,1];H)\,\vert \,\mathbf {g} (0)=0,\mathbf {g} (1)=v\}} and: c = inf g ∈ Γ max 0 ≤ t ≤ 1 I [ g ( t ) ] , {\displaystyle c=\inf _{\mathbf {g} \in \Gamma }\max _{0\leq t\leq 1}I[\mathbf {g} (t)],} then the conclusion of the theorem is that c is a critical value of I. == Visualization == The intuition behind the theorem is in the name "mountain pass." Consider I as describing elevation. Then we know two low spots in the landscape: the origin because I [ 0 ] = 0 {\displaystyle I[0]=0} , and a far-off spot v where I [ v ] ≤ 0 {\displaystyle I[v]\leq 0} . In between the two lies a range of mountains (at ‖ u ‖ = r {\displaystyle \Vert u\Vert =r} ) where the elevation is high (higher than a>0). In order to travel along a path g from the origin to v, we must pass over the mountains—that is, we must go up and then down. Since I is somewhat smooth, there must be a critical point somewhere in between. (Think along the lines of the mean-value theorem.) The mountain pass lies along the path that passes at the lowest elevation through the mountains. Note that this mountain pass is almost always a saddle point. For a proof, see section 8.5 of Evans. == Weaker formulation == Let X {\displaystyle X} be Banach space. The assumptions of the theorem are: Φ ∈ C ( X , R ) {\displaystyle \Phi \in C(X,\mathbf {R} )} and have a Gateaux derivative Φ ′ : X → X ∗ {\displaystyle \Phi '\colon X\to X^{}} which is continuous when X {\displaystyle X} and X ∗ {\displaystyle X^{}} are endowed with strong topology and weak* topology respectively. There exists r > 0 {\displaystyle r>0} such that one can find certain ‖ x ′ ‖ > r {\displaystyle \\|x'\\|>r} with max ( Φ ( 0 ) , Φ ( x ′ ) ) < inf ‖ x ‖ = r Φ ( x ) =: m ( r ) {\displaystyle \max \,(\Phi (0),\Phi (x'))<\inf \limits _{\\|x\\|=r}\Phi (x)=:m(r)} . Φ {\displaystyle \Phi } satisfies weak Palais–Smale condition on { x ∈ X ∣ m ( r ) ≤ Φ ( x ) } {\displaystyle \{x\in X\mid m(r)\leq \Phi (x)\}} . In this case there is a critical point x ¯ ∈ X {\displaystyle {\overline {x}}\in X} of Φ {\displaystyle \Phi } satisfying m ( r ) ≤ Φ ( x ¯ ) {\displaystyle m(r)\leq \Phi ({\overline {x}})} . Moreover, if we define Γ = { c ∈ C ( [ 0 , 1 ] , X ) ∣ c ( 0 ) = 0 , c ( 1 ) = x ′ } {\displaystyle \Gamma =\{c\in C([0,1],X)\mid c\,(0)=0,\,c\,(1)=x'\}} then Φ ( x ¯ ) = inf c ∈ Γ max 0 ≤ t ≤ 1 Φ ( c ( t ) ) . {\displaystyle \Phi ({\overline {x}})=\inf _{c\,\in \,\Gamma }\max _{0\leq t\leq 1}\Phi (c\,(t)).} For a proof, see section 5.5 of Aubin and Ekeland. == References == == Further reading == Aubin, Jean-Pierre; Ekeland, Ivar (2006). Applied Nonlinear Analysis. Dover Books. ISBN 0-486-45324-3. Bisgard, James (2015). "Mountain Passes and Saddle Points". SIAM Review. 57 (2): 275–292. doi:10.1137/140963510. Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2. Jabri, Youssef (2003). The Mountain Pass Theorem, Variants, Generalizations and Some Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press. ISBN 0-521-82721-3. Mawhin, Jean; Willem, Michel (1989). "The Mountain Pass Theorem and Periodic Solutions of Superlinear Convex Autonomous Hamiltonian Systems". Critical Point Theory and Hamiltonian Systems. New York: Springer-Verlag. pp. 92–97. ISBN 0-387-96908-X. McOwen, Robert C. (1996). "Mountain Passes and Saddle Points". Partial Differential Equations: Methods and Applications. Upper Saddle River, NJ: Prentice Hall. pp. 206–208. ISBN 0-13-121880-8.
Wikipedia:Muhammad Habibar Rahman#0	Muhammad Habibur Rahman (1 January 1923 - 15 April 1971) was a Bengali intellectual who was killed in the Bangladesh Liberation war and is considered a martyr in Bangladesh. == Early life == Rahman was born in Baliadhar, Noakhali District, East Bengal, British India on 1 January 1923. He finished his SSC from Dattapara High School in 1938 and HSC from Calcutta Islamia College in 1940. He finished his undergraduate studies in mathematics from Presidency College in Kolkata. He completed his master's degree in mathematics from the Aligarh University. == Career == He joined Dhaka College as a professor of mathematics in 1946. In 1951 he received government funding to study in Cambridge University in the United Kingdom. He graduated from Cambridge in 1953 after finishing the Tripos in mathematics. He worked in Presidency College in Kolkata before joining Rajshahi University in 1954. He joined as a professor of mathematics and by in 1958 had been promoted to reader. In 1962 he pursued higher studies in applied mathematics in the United States. From 1964 to 1966 he served as the chairman of the Department of Mathematics at Rajshahi University. From 1967 to 1970 he served as the provost of Ameer Ali Hall of Rajshahi University after which returned to being the chairman of the Department of Mathematics. He was a member of the Dhaka Rationalist club. == Death == The Pakistan Army on 15 April 1971 captured him from his home in front of his family and he never came back, is presumed to be dead. Rajshahi University named Shaheed Habibur Rahman Hall after him. The dorm has a bust of him in its entrance. He was also awarded with "Ekushey Padak" (Lit: TwentyFirst Award) second highest civilian award in Bangladesh. == References ==
Wikipedia:Muhammad Rafique (mathematician)#0	Muhammad Rafique (2 January 1940 — 16 June 1996) was a Pakistani mathematician and professor of mathematics at the Punjab University. He was a versatile scholar who authored textbooks on computer language and special relativity. He was the co-author of textbook Group Theory for High Energy Physicists, which was eventually published years after his death in 2016. == Biographical overview == Rafique was born in Lahore, Punjab in British India on 2 January 1940 in Lahore, Punjab in India and was educated at the Punjab University where he graduated with BA with first-class honours in Mathematics in 1960. He served in the Faculty of Mathematics at the Punjab University where he graduated with MA in Mathematics in 1964, and earned a scholarship to study mathematics in the United Kingdom.: 256 He attended the University of North Wales where he graduated with a PhD in Mathematics in 1967.: 256 Upon returning to Pakistan, he joined the Punjab University and taught there until 1971 when he joined the International Center for Theoretical Physics in Italy as a post-doctoral scholar.: xi From 1972 to 1977, Rafique worked at the Institute of Nuclear Science and Technology where he contributed his work on fast neutron calculations for atomic weapons which built his interests in the theory of relativity and nuclear physics.: 74–75 From 1977 to 1982, he served on the faculty of University of Tripoli in Libya, and served as the Head of Department of Mathematics at the Punjab University from 1983 until 1992 when he went to teach mathematics at the King Fahd University in Saudi Arabia. His tenure at the King Fahd University was short-lived. He died due to cardiac arrest in June 1996. Although a mathematician, Rafique was a prolific author on physics, was writing a college text on group theory's applications on high energy physics with Mohammad Saleem at the time of his death in 1992. The college book was eventually published in 2015-2016 by British publisher Taylor & Francis. === Textbooks === Saleem, Mohammad; Rafique, Muhammad (2016). Group Theory for High Energy Physicists. Taylor & Francis. ISBN 978-1-4665-1064-7. Retrieved 27 April 2020. Rafique, Muhammad (1993). FORTRAN Programming on a Personal Computer. Lahore, Punj. Pakistan: Punjab University Press. Rafique, Muhammad (1983). Differential Equations for Scientists and Engineers. Lahore, Punj. Pakistan: Punjab University Press. Rafique, Muhammad (1985). Practical Geometry. Lahore, Punj. Pakistan: Punjab University Press. Saleem, Mohammad; Rafique, Muhammad (1992). Special Relativity: Applications to Particle Physics and the classical theory of fields. Ellis Horwood. ISBN 978-0-13-827106-0. Retrieved 27 April 2020. == See also == Definite integral == References == == External links == Punjab University
Wikipedia:Muhammad Raziuddin Siddiqui#0	Muhammad Raziuddin Siddiqui, FPAS, NI, HI, SI (Urdu: محمد رضی الدین صدیقی, [rəzɪ.ʊd̪ːiːn sɪˈd̪ːiːqi]; 8 January 1908 – 8 January 1998), also known as Dr. Razi, was a Pakistani theoretical physicist and a mathematician who played a role in Pakistan's education system, and Pakistan's indigenous development of nuclear weapons. An educationist and a scientist, Siddiqui established educational research institutes and universities in his country. During the 1940s in Europe, he contributed in mathematical physics and worked on general relativity and the theory of relativity, nuclear energy, and quantum gravity. He was one of the notable students of Albert Einstein. He had been the vice-chancellor of four Pakistani universities, and the first vice-chancellor of Quaid-e-Azam University and served as the Emeritus professor of Physics there until his death in 1998. == Biography == === Life and education === Raziuddin Siddiqui was born on 8 January 1908 in Hyderabad- Deccan, India to Mohammed Muzaffer uddin Siddiqui and Baratunnisa Begum. His family consisted of one elder brother, Mohammed Zakiuddin Siddiqui and two sisters, Abida Begum and Sajida Begum, he was the youngest in the family. He attended the newly established Osmania University. After passing the Rashidia Exams in 1918, Siddiqui completed his matriculation from Osmania University in 1921, and earned a Bachelor of Arts (BA) in mathematics, with distinction, in 1925. === Siddiqui in Europe === Siddiqui was then awarded a scholarship from the Government of the State of Hyderabad to pursue higher studies in United Kingdom where he completed his MA in mathematics, under Paul Dirac from the University of Cambridge in 1928. Then, he proceeded further to work for his PhD at the University of Leipzig in Germany (Weimar Republic). He studied mathematics and quantum mechanics under Albert Einstein. He completed his PhD in theoretical physics, writing a brief research thesis on the Theory of relativity and the nuclear binding energy. He did his post doctoral work at the University of Paris, France. == Research in theoretical physics == In Europe, while Siddique was working on his post-doctoral research at the Paris University, he had the opportunity to meet with the members of "The Paris Group" where he had led the discussions on unsolved problems in physics and in mathematics. During his stay in Great Britain, he studied Quantum mechanics and published scientific papers at the Cavendish Laboratory. === Return to India === In 1931, Siddiqui then returned to Hyderabad, British Indian Empire, and joined Osmania University there as an associate professor of mathematics. During 1948–49, he served as vice-chancellor of Osmania, appointed by the governor of Andhra Pradesh. === Move to Pakistan === After the Partition of India led to the independence of Pakistan in 1947, at the request of the Government of Pakistan, Siddiqui migrated to Karachi, Pakistan in 1950, along with some of his family. His brother Zakiuddin and one of his sisters, Sajida Begum, remained in Hyderabad, India with their families and parents. His father Muzaffer uddin Siddiqui died while his visit to Raziuddin Siddiqui in Pakistan later in his years. In Karachi, Siddiqui joined the Karachi University's teaching faculty and taught as professor of applied mathematics there. In 1953, he was simultaneously appointed to the post of vice-chancellor of the University of Sindh and the University of Peshawar. Siddiqui founded the first mathematical society in Pakistan in 1952 by the name of "All Pakistan Mathematics Association", and remained its president until 1972. In 1956, Siddiqui helped establish Nuclear power in Pakistan and its expansion in the country by first joining the newly established Pakistan Atomic Energy Commission (PAEC) and then establishing the first science directorate on mathematical physics. In 1964, he moved to Islamabad, where he joined PAEC. There he began his academic research in theoretical physics. In 1965, with the establishment of Quaid-e-Azam University (QAU), Siddiqui was appointed as its first vice-chancellor by the then foreign minister Zulfikar Ali Bhutto. He was one of the first professors of Physics at Quaid-e-Azam University where he also served as the chairman of the Physics Department. He continued his tenure until 1972, when he rejoined PAEC at the request of Prime Minister Bhutto. During the 1960s, he helped convince President of Pakistan Ayub Khan to make a proposed university a research institution. He, at first, established the "Institute of Physics" at the QAU, and invited Professor Riazuddin to be its first director, and the dean of the faculty. Then, Riazuddin, with the help of his mentor, Dr. Abdus Salam, convinced the then PAEC chairman Dr. Ishrat Hussain Usmani to send all the theoreticians to the Institute of Physics to form a physics group. This established the "Theoretical Physics Group" (TPG), which later designed nuclear weapons for Pakistan. With the establishment of TPG, Siddiqui began to work with Abdus Salam, and on his advice began research in Theoretical Physics at PAEC. In 1970, he established the Mathematical Physics Group (MPG) at PAEC, where he led academic research in advanced mathematics. He also delegated mathematicians to PAEC to specialise in their fields at the MPG Division of PAEC. === Pakistani nuclear weapons program === After the Indo-Pakistani War of 1971, Siddiqui joined the Pakistan Atomic Energy Commission (PAEC) at the request of Prime Minister Zulfikar Ali Bhutto. Siddiqui was the first full-time Technical Member of PAEC and was responsible for preparation of its charter. During the 1970s, Siddiqui worked on problems in theoretical physics with Pakistani theoretical physicists in the nuclear weapons programme. Previously, he had worked in Europe, including carrying out nuclear research in the British nuclear weapon program, and the French atomic program. At PAEC, he became a mentor to some of the country's academic scientists. At PAEC, he was the director of the Mathematical Physics Group (MPG) and was tasked with performing mathematical calculations involved in nuclear fission and supercomputing. While both MPG and Theoretical Physics Group (TPG) had reported directly to Abdus Salam, Siddiqui co-ordinated each meeting with the scientists of TPG and mathematicians of the MPG. At PAEC, he directed the mathematical research directly involving the theory of general relativity, and helped establish the quantum computers laboratories at PAEC. Since theoretical physics plays a major role in identifying the parameters of nuclear physics, Siddiqui started the work on special relativity's complex applications, the 'relativity of simultaneity'. His Mathematical Physics Group undertook the research and performed calculations on the 'relativity of simultaneity' during the process of weapon detonation, where multiple explosive energy rays are bound to release in the same isolate and close medium at the same time interval. === Post-war === After his work at PAEC, Siddiqui again joined Quaid-e-Azam University's Physics Faculty. As professor of physics, he continued his research at the Institute of Physics, QAU. He helped develop the higher education sector, and placed mainframe policies in the institution. == Death and legacy == Siddiqui remained in Islamabad, and had associated himself with Quaid-e-Azam University. In 1990, he was made Professor Emeritus of Physics and Mathematics there. He died on 8 January 1998, at the age of 90. Siddiqui's biography was written by scientists who had worked with him. In 1960, due to his efforts to expand education, he was awarded the third-highest civilian award of Pakistan, Sitara-i-Imtiaz, from the then-President of Pakistan, Field Marshal Ayub Khan. In 1981, he was awarded the second highest civilian award, Hilal-i-Imtiaz, from President General Muhammad Zia-ul-Haq due to his efforts in Pakistan's atomic program, and for popularising science in Pakistan. In May 1998, the Government of Pakistan awarded him the highest civilian award, the Nishan-i-Imtiaz, posthumously by Prime Minister Nawaz Sharif when Pakistan conducted its first successful nuclear tests, 'Chagai-I'. == Family == His eldest daughter, Dr. Shirin Tahir-Kheli, is a former special assistant to the president of the United States of America, and Senior Adviser for women's empowerment. == Civil awards == Sitara-i-Imtiaz (1960) Hilal-i-Imtiaz (1981) Nishan-e-Imtiaz (1998) Gold Medal, Pakistan Academy of Sciences (1950) Gold Medal, Pakistan Mathematical Society (1980) Gold Medallion, Pakistan Physical Society (1953) Doctorate of Science Honoris Causa, Osmania University (1938) == Books == Quantum Mechanics and its Physics Dastan-e-Riazi (The Tale of Mathematics) Izafiat Tasawur-e-Zaman-o-Makaan Experiences in science and education by M. Raziuddin Siddiqui, published in 1977. Establishing a new university in a developing country: Policies and procedures by M. Raziuddin Siddiqui, published in 1990. == See also == Abdus Salam Salimuzzaman Siddiqui Quaid-i-Azam University Nuclear weapon == References == === Bibliography === == External links == Muhammad Raziuddin Siddiqui Dr. Raziuddin Siddiqui Memorial Library Ias.ac.in Iiit.ac.in, Iqbal Ka Tasawwuf-e-Zaman-o-MakaN at Digital Library of India
Wikipedia:Muhammad al-Rudani#0	Muhammad al-Rudani (Arabic: محمد بن سليمان الروداني) (c. 1627 – 1683) was a Moroccan polymath who was active as an astronomer, grammarian, jurist, logician, mathematician and poet. == Biography == Al-Rudani was born in c. 1627 in Taroudant. He was of Shilha origin. After studying in his hometown at the Great Mosque of Taroudant and its Madrasa, he continued his studies in the Zaouia Nasiriyya under Mohammed ibn Nasir for four years, the Zaouia of Dila, in Marrakesh and in Fez. His teachers in Morocco were: the theologian Isa al-Sugtani (d.1651), the chronologist Muhammad ibn Said al-Marghiti (d. 1679), and the grammarian Muhammad al-Murabit al-Dilai' (d. 1678). Afterwards, he left to study in the Islamic east. Thus, in the early 1650s, he stayed in Algiers, where he studied under the logician Said ibn Ibrahim Qaddura. In Egypt and the Levant, he studied under Ali al-Ajhuri, Shihab al-Din al-Khafaji, Shihab al-Din al-Qaliyubi, Muhammad ibn Ahmad al-Shubri, al-Shaikh Sultan, Khayr al-Din al-Ramli, Muhammad al-Naqib ibn Hamza al-Hasani and Ibn Balban. He got an ijazah from all of these scholars. He died in Damascus on October 31, 1683. == Works == Some of his works are: Silat al-khalaf bi-mawsūl al-salaf, an extensive record of the various chains of certifications he had received Bahjat al-tullāb fı̄ l-'amal bil-asturlāb, a treatise on the astrolabe Maqāsid al-'awālı̄ bi-qalā'id al-la'ālı̄, a didactic poem on ilm al-mı̄qāt (chronology) with a prose commentary al-Nāfi'a 'alā l-āla al-jāmi'a completed in 1661, a treatise describing the spherical astrolabe that he had constructed. Jam' al-fawā'id min Jāmi' al-usūl wa Majma' al-zawā'id, a compilation of hadith == References == == Sources == Ayduz, Salim (2007). "Rudānī: Abū ʿAbdallāh Muḥammad ibn Sulaymān (Muḥammad) al-Fāsī ibn Ṭāhir al-Rudānī al-Sūsī al-Mālikī [al-Maghribī]". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. p. 990. ISBN 978-0-387-31022-0. Berque, Jacques (1955). "La littérature marocaine et l'Orient au XVIIe siècle". Arabica. 2 (3): 295–312. doi:10.1163/157005855X00031. ISSN 0570-5398. JSTOR 4054895. Hajji, Mohamed (2001). "al-Rudani, Muhammad ibn Sulayman". In Toufiq, Ahmed; Hajji, Mohamed (eds.). Ma'lamat al-Maghrib (Encyclopedia of Morocco) (in Arabic). Vol. 13. al-Jamī'a al-Maghribiyya li-l-Ta'līf wa-l-Tarjama wa-l-Nashr. El-Rouayheb, Khaled (2015). Islamic Intellectual History in the Seventeenth Century. Cambridge University Press. ISBN 978-1-107-04296-4.
Wikipedia:Muhammad ibn Muhammad al-Fulani al-Kishnawi#0	Muhammad ibn Muhammad al-AlFulani al-Katsinawi was a prominent 18th century Fulani scholar, mathematician, astrologer, Arabic grammarian and jurist from Katsina, present-day Northern Nigeria. Al-Kishnawi studied at the Gobarau Minaret in Katsina before leaving for Cairo, Egypt in 1732, where he published in Arabic a work titled, "A Treatise on the Magical Use of the Letters of the Alphabet" which is a mathematical scholarly manuscript of procedures for constructing magic squares up to the order 11. == Life == Muhammad al-Kishnawi was a Fulani scholar born in Dan Rako in Katsina (in modern-day Katsina state). Dan Rako was known for its association with the Wangara traders from Mali, who had established a presence in the area. The town was later sacked by Muhammad Bello and it no longer exists. He was born into a Muslim family and studied the religion and its holy book, the Quran. Among his teachers were Muhamamd al-Wali al-Burnawi, a famous scholar from Kanem-Bornu, Muhammad Fudi, the father of Usman dan Fodio, and Muhammad al-Bindu "Booro Binndi", another famous scholar from Kanem-Bornu. He became famous in Hausaland and Bornu and attracted many students.: 15 : 255–257 Sometime before 1730, he left Katsina to embark on a pilgrimage to the Hijaz. He writes:When the Deliverer of Destiny and Sempiternal Will delivered me, and the Usher of Divine Mercy ushered me to visit His good Prophet, upon him be the best of prayers and most devoted salutations, and to perform the pilgrimage of His holy sanctified House, I stayed there for some time and grew through these prayers... [and] spent of my duty to thank Him, the Gracious for variegating an areborerum for me, a small utterance [shaʿrat lisānan] indeed for that greatest of graces [niʿam] that He has bestowed upon me….: 249 The journey to Mecca was arduous, and it was common for West African pilgrims to take breaks in Cairo before continuing their journey. This was a practice observed by notable figures like Mansa Musa, the famous Malian king, during his pilgrimage in the 14th century. Following a similar route, al-Kishnawi also stopped in Cairo before proceeding to Mecca and eventually settling in Medina. During his time in the Hijaz, al-Kishnawi had the opportunity to meet and learn from scholars from various parts of the Islamic world. Around the years 1733-1734, he relocated to Cairo, where he found accommodation near Al-Azhar University. He dedicated himself to writing, and during his first four years in Cairo, he completed several notable works, including Al-Durr al-manẓūm, Bahjat al-āfāq, Bulūgh al-arab, and Durar al-yawāqī. Al-Kishnawi became famous in Egypt, later becoming the teacher of Hassan al-Jabarti, the father of the renowned Egyptian historian Abd al-Rahman al-Jabarti. Abd al-Rahman writes that his father “learned the art of numerical and literal magic squares and the art of fractions” from al-Kishnawi. In 1741, Al-Kishnawi died at the age of 42 in the home of Hassan al-Jabarti in Cairo. He was buried in the Hall of Scholars in Cairo. == Notable works == Many of his works are now at the al-Azhar Library in Cairo. Some are preserved in Dar al-kutub, and some archives in Morocco, Nigeria and London.: 15 As words of encouragement to the readers he writes:Do not give up, for that is ignorance and not according to the rules of this art ... Like the lover, you cannot hope to achieve success without infinite perseverance.Some of his notable works are: Bulūgh al-arab min kalām al-ʿarab: a work on Arabic grammar dated to around 1736-7. Bughyat al-mawālī fī tarjamat Muḥammad al-Wālī: a biography of Muhamamd al-Wali al-Burnawi (one of his teachers). Manḥ al-quddū: a didactic poem on logic drawn from the Mukhtaṣar of al-Sanūsī. Izālat al-‘ubū ‘an wajh minaḥ al-quddūs: a commentary on the Mukhtaṣar of al-Sanūsī. A commentary on Kitāb al-durr wa’l-tiryāq fī ‘ilm al-awfāq by Abd al-Rahman al-Jurjanī on the science of letters and the great names of God, completed on 6 September 1734. Three treatises on Durar al-yawāqīt fī ‘ilm al-ḥurūf wa’l asmā’. Mughnī al-mawāfī ‘an jamī‘ al-khawāf: a numerological work on the magic square completed on 29 January 1733. Al-Durr al-manẓūm wa khulāṣat al-sirr al-maktūm fī ‘ilm al-ṭalāsim wa’l-nujūm: his famous commentary on the three domains of the "secret sciences", completed on 20 December 1733.: 264–265 : 141 == References ==
Wikipedia:Multibrot set#0	In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set. z ↦ z d + c . {\displaystyle z\mapsto z^{d}+c.\,} where d ≥ 2. The exponent d may be further generalized to negative and fractional values. == Examples == Sources: The case of d = 2 {\displaystyle d=2\,} is the classic Mandelbrot set from which the name is derived. The sets for other values of d also show fractal images when they are plotted on the complex plane. Each of the examples of various powers d shown below is plotted to the same scale. Values of c belonging to the set are black. Values of c that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, depending on the number of recursions that caused a value to exceed a fixed magnitude in the Escape Time algorithm. === Positive powers === The example d = 2 is the original Mandelbrot set. The examples for d > 2 are often called multibrot sets. These sets include the origin and have fractal perimeters, with (d − 1)-fold rotational symmetry. === Negative powers === When d is negative the set appears to surround but does not include the origin, However this is just an artifact of the fixed maximum radius allowed by the Escape Time algorithm, and is not a limit of the sets that actually have a shape in the middle with an no hole (You can see this by using the Lyapunov exponent [No hole because the origin diverges to undefined not infinity because the origin {0 or 0+0i} taken to a negative power becomes undefined]). There is interesting complex behaviour in the contours between the set and the origin, in a star-shaped area with (1 − d)-fold rotational symmetry. The sets appear to have a circular perimeter, however this is an artifact of the fixed maximum radius allowed by the Escape Time algorithm, and is not a limit of the sets that actually extend in all directions to infinity. === Fractional powers === === Rendering along the exponent === An alternative method is to render the exponent along the vertical axis. This requires either fixing the real or the imaginary value, and rendering the remaining value along the horizontal axis. The resulting set rises vertically from the origin in a narrow column to infinity. Magnification reveals increasing complexity. The first prominent bump or spike is seen at an exponent of 2, the location of the traditional Mandelbrot set at its cross-section. The third image here renders on a plane that is fixed at a 45-degree angle between the real and imaginary axes. == Rendering images == All the above images are rendered using an Escape Time algorithm that identifies points outside the set in a simple way. Much greater fractal detail is revealed by plotting the Lyapunov exponent, as shown by the example below. The Lyapunov exponent is the error growth-rate of a given sequence. First calculate the iteration sequence with N iterations, then calculate the exponent as λ = lim N → ∞ 1 N ln ⁡ \| z \| {\displaystyle \lambda =\lim _{N\to \infty }{\frac {1}{N}}\ln \|\mathbf {z} \|} and if the exponent is negative the sequence is stable. The white pixels in the picture are the parameters c for which the exponent is positive aka unstable. The colours show the periods of the cycles which the orbits are attracted to. All points colored dark-blue (outside) are attracted by a fixed point, all points in the middle (lighter blue) are attracted by a period 2 cycle and so on. === Pseudocode === ESCAPE TIME ALGORITHM for each pixel on the screen do x = x0 = x co-ordinate of pixel y = y0 = y co-ordinate of pixel iteration := 0 max_iteration := 1000 while (xx + yy ≤ (22) and iteration < max_iteration do / INSERT CODE(S)FOR Z^d FROM TABLE BELOW */ iteration := iteration + 1 if iteration = max_iteration then colour := black else colour := iteration plot(x0, y0, colour) The complex value z has coordinates (x,y) on the complex plane and is raised to various powers inside the iteration loop by codes shown in this table. Powers not shown in the table can be obtained by concatenating the codes shown. == References ==
Wikipedia:Multifractal system#0	A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. Multifractal systems are common in nature. They include the length of coastlines, mountain topography, fully developed turbulence, real-world scenes, heartbeat dynamics, human gait and activity, human brain activity, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in fluid dynamics to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more. The origin of multifractality in sequential (time series) data has been attributed to mathematical convergence effects related to the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie exponential dispersion models, as well as the geometric Tweedie models. The first convergence effect yields monofractal sequences, and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences. Multifractal analysis is used to investigate datasets, often in conjunction with other methods of fractal and lacunarity analysis. The technique entails distorting datasets extracted from patterns to generate multifractal spectra that illustrate how scaling varies over the dataset. Multifractal analysis has been used to decipher the generating rules and functionalities of complex networks. Multifractal analysis techniques have been applied in a variety of practical situations, such as predicting earthquakes and interpreting medical images. == Definition == In a multifractal system s {\displaystyle s} , the behavior around any point is described by a local power law: s ( x → + a → ) − s ( x → ) ∼ a h ( x → ) . {\displaystyle s({\vec {x}}+{\vec {a}})-s({\vec {x}})\sim a^{h({\vec {x}})}.} The exponent h ( x → ) {\displaystyle h({\vec {x}})} is called the singularity exponent, as it describes the local degree of singularity or regularity around the point x → {\displaystyle {\vec {x}}} . The ensemble formed by all the points that share the same singularity exponent is called the singularity manifold of exponent h, and is a fractal set of fractal dimension D ( h ) : {\displaystyle D(h):} the singularity spectrum. The curve D ( h ) {\displaystyle D(h)} versus h {\displaystyle h} is called the singularity spectrum and fully describes the statistical distribution of the variable s {\displaystyle s} . In practice, the multifractal behaviour of a physical system X {\displaystyle X} is not directly characterized by its singularity spectrum D ( h ) {\displaystyle D(h)} . Rather, data analysis gives access to the multiscaling exponents ζ ( q ) , q ∈ R {\displaystyle \zeta (q),\ q\in {\mathbb {R} }} . Indeed, multifractal signals generally obey a scale invariance property that yields power-law behaviours for multiresolution quantities, depending on their scale a {\displaystyle a} . Depending on the object under study, these multiresolution quantities, denoted by T X ( a ) {\displaystyle T_{X}(a)} , can be local averages in boxes of size a {\displaystyle a} , gradients over distance a {\displaystyle a} , wavelet coefficients at scale a {\displaystyle a} , etc. For multifractal objects, one usually observes a global power-law scaling of the form: ⟨ T X ( a ) q ⟩ ∼ a ζ ( q ) {\displaystyle \langle T_{X}(a)^{q}\rangle \sim a^{\zeta (q)}\ } at least in some range of scales and for some range of orders q {\displaystyle q} . When such behaviour is observed, one talks of scale invariance, self-similarity, or multiscaling. == Estimation == Using so-called multifractal formalism, it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum D ( h ) {\displaystyle D(h)} and the multi-scaling exponents ζ ( q ) {\displaystyle \zeta (q)} through a Legendre transform. While the determination of D ( h ) {\displaystyle D(h)} calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the ζ ( q ) {\displaystyle \zeta (q)} relies on the use of statistical averages and linear regressions in log-log diagrams. Once the ζ ( q ) {\displaystyle \zeta (q)} are known, one can deduce an estimate of D ( h ) , {\displaystyle D(h),} thanks to a simple Legendre transform. Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. The ζ ( q ) {\displaystyle \zeta (q)} are statistically interpreted, as they characterize the evolution of the distributions of the T X ( a ) {\displaystyle T_{X}(a)} as a {\displaystyle a} goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from Gaussian models. Modelling as a multiplicative cascade also leads to estimation of multifractal properties. This methods works reasonably well, even for relatively small datasets. A maximum likely fit of a multiplicative cascade to the dataset not only estimates the complete spectrum but also gives reasonable estimates of the errors. == Estimating multifractal scaling from box counting == Multifractal spectra can be determined from box counting on digital images. First, a box counting scan is done to determine how the pixels are distributed; then, this "mass distribution" becomes the basis for a series of calculations. The chief idea is that for multifractals, the probability P {\displaystyle P} of a number of pixels m {\displaystyle m} , appearing in a box i {\displaystyle i} , varies as box size ϵ {\displaystyle \epsilon } , to some exponent α {\displaystyle \alpha } , which changes over the image, as in Eq.0.0 (NB: For monofractals, in contrast, the exponent does not change meaningfully over the set). P {\displaystyle P} is calculated from the box-counting pixel distribution as in Eq.2.0. ϵ {\displaystyle \epsilon } = an arbitrary scale (box size in box counting) at which the set is examined i {\displaystyle i} = the index for each box laid over the set for an ϵ {\displaystyle \epsilon } m [ i , ϵ ] {\displaystyle m_{[i,\epsilon ]}} = the number of pixels or mass in any box, i {\displaystyle i} , at size ϵ {\displaystyle \epsilon } N ϵ {\displaystyle N_{\epsilon }} = the total boxes that contained more than 0 pixels, for each ϵ {\displaystyle \epsilon } P {\displaystyle P} is used to observe how the pixel distribution behaves when distorted in certain ways as in Eq.3.0 and Eq.3.1: Q {\displaystyle Q} = an arbitrary range of values to use as exponents for distorting the data set When Q = 1 {\displaystyle Q=1} , Eq.3.0 equals 1, the usual sum of all probabilities, and when Q = 0 {\displaystyle Q=0} , every term is equal to 1, so the sum is equal to the number of boxes counted, N ϵ {\displaystyle N_{\epsilon }} . These distorting equations are further used to address how the set behaves when scaled or resolved or cut up into a series of ϵ {\displaystyle \epsilon } -sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following: An important feature of Eq.3.0 is that it can also be seen to vary according to scale raised to the exponent τ {\displaystyle \tau } in Eq.4.0: Thus, a series of values for τ ( Q ) {\displaystyle \tau _{(Q)}} can be found from the slopes of the regression line for the log of Eq.3.0 versus the log of ϵ {\displaystyle \epsilon } for each Q {\displaystyle Q} , based on Eq.4.1: For the generalized dimension: α ( Q ) {\displaystyle \alpha _{(Q)}} is estimated as the slope of the regression line for log A ϵ {\displaystyle \epsilon } ,Q versus log ϵ {\displaystyle \epsilon } where: Then f ( α ( Q ) ) {\displaystyle f_{\left(\alpha _{(Q)}\right)}} is found from Eq.5.3. The mean τ ( Q ) {\displaystyle \tau _{(Q)}} is estimated as the slope of the log-log regression line for τ ( Q ) [ ϵ ] {\displaystyle \tau _{{(Q)}_{[\epsilon ]}}} versus ϵ {\displaystyle \epsilon } , where: In practice, the probability distribution depends on how the dataset is sampled, so optimizing algorithms have been developed to ensure adequate sampling. == Applications == Multifractal analysis has been successfully used in many fields, including physical, information, and biological sciences. For example, the quantification of residual crack patterns on the surface of reinforced concrete shear walls. === Dataset distortion analysis === Multifractal analysis has been used in several scientific fields to characterize various types of datasets. In essence, multifractal analysis applies a distorting factor to datasets extracted from patterns, to compare how the data behave at each distortion. This is done using graphs known as multifractal spectra, analogous to viewing the dataset through a "distorting lens", as shown in the illustration. Several types of multifractal spectra are used in practise. ==== DQ vs Q ==== One practical multifractal spectrum is the graph of DQ vs Q, where DQ is the generalized dimension for a dataset and Q is an arbitrary set of exponents. The expression generalized dimension thus refers to a set of dimensions for a dataset (detailed calculations for determining the generalized dimension using box counting are described below). ==== Dimensional ordering ==== The general pattern of the graph of DQ vs Q can be used to assess the scaling in a pattern. The graph is generally decreasing, sigmoidal around Q=0, where D(Q=0) ≥ D(Q=1) ≥ D(Q=2). As illustrated in the figure, variation in this graphical spectrum can help distinguish patterns. The image shows D(Q) spectra from a multifractal analysis of binary images of non-, mono-, and multi-fractal sets. As is the case in the sample images, non- and mono-fractals tend to have flatter D(Q) spectra than multifractals. The generalized dimension also gives important specific information. D(Q=0) is equal to the capacity dimension, which—in the analysis shown in the figures here—is the box counting dimension. D(Q=1) is equal to the information dimension, and D(Q=2) to the correlation dimension. This relates to the "multi" in multifractal, where multifractals have multiple dimensions in the D(Q) versus Q spectra, but monofractals stay rather flat in that area. ==== f(α) versus α ==== Another useful multifractal spectrum is the graph of f ( α ) {\displaystyle f(\alpha )} versus α {\displaystyle \alpha } (see calculations). These graphs generally rise to a maximum that approximates the fractal dimension at Q=0, and then fall. Like DQ versus Q spectra, they also show typical patterns useful for comparing non-, mono-, and multi-fractal patterns. In particular, for these spectra, non- and mono-fractals converge on certain values, whereas the spectra from multifractal patterns typically form humps over a broader area. === Generalized dimensions of species abundance distributions in space === One application of Dq versus Q in ecology is characterizing the distribution of species. Traditionally the relative species abundances is calculated for an area without taking into account the locations of the individuals. An equivalent representation of relative species abundances are species ranks, used to generate a surface called the species-rank surface, which can be analyzed using generalized dimensions to detect different ecological mechanisms like the ones observed in the neutral theory of biodiversity, metacommunity dynamics, or niche theory. == See also == de Rham curve – Continuous fractal curve obtained as the image of Cantor space Fractional Brownian motion – Probability theory concept Detrended fluctuation analysis – Method to detect power-law scaling in time series Tweedie distributions – Family of probability distributionsPages displaying short descriptions of redirect targets Markov switching multifractal – model of asset returnsPages displaying wikidata descriptions as a fallback == References == == Further reading == Falconer, Kenneth J. (2014). "17. Multifractal measures". Fractal geometry: mathematical foundations and applications (3. ed., 1. publ ed.). Chichester: Wiley. ISBN 978-1-119-94239-9. Barabási, A.- L.; Stanley, H. E., eds. (1995), "Multi-affine surfaces", Fractal Concepts in Surface Growth, Cambridge: Cambridge University Press, pp. 262–268, doi:10.1017/CBO9780511599798.026, ISBN 978-0-521-48318-6, retrieved 2024-06-05 G, Evertsz C. J.; Mandelbrot, Benoît B. (1992). "Multifractal measures" (PDF). Chaos and Fractals New Frontiers of Science: 922–953. Archived from the original (PDF) on 2023-07-13. Mandelbrot, Benoît B. (1997). Fractals and scaling in finance: discontinuity, concentration, risk. Selecta. New York, NY Berlin Heidelberg: Springer. ISBN 978-0-387-98363-9. Harte, David (2001-06-26). Multifractals. Chapman and Hall/CRC. doi:10.1201/9781420036008. ISBN 978-0-429-12366-5. Stanley H.E., Meakin P. (1988). "Multifractal phenomena in physics and chemistry" (Review). Nature. 335 (6189): 405–9. Bibcode:1988Natur.335..405S. doi:10.1038/335405a0. S2CID 4318433. Arneodo, Alain; Audit, Benjamin; Kestener, Pierre; Roux, Stephane (2008). "Wavelet-based multifractal analysis". Scholarpedia. 3 (3): 4103. Bibcode:2008SchpJ...3.4103A. doi:10.4249/scholarpedia.4103. ISSN 1941-6016. == External links == Veneziano, Daniele; Essiam, Albert K. (June 1, 2003). "Flow through porous media with multifractal hydraulic conductivity". Water Resources Research. 39 (6): 1166. Bibcode:2003WRR....39.1166V. doi:10.1029/2001WR001018. ISSN 1944-7973. Movies of visualizations of multifractals
Wikipedia:Multilinear form#0	In abstract algebra and multilinear algebra, a multilinear form on a vector space V {\displaystyle V} over a field K {\displaystyle K} is a map f : V k → K {\displaystyle f\colon V^{k}\to K} that is separately K {\displaystyle K} -linear in each of its k {\displaystyle k} arguments. More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces. A multilinear k {\displaystyle k} -form on V {\displaystyle V} over R {\displaystyle \mathbb {R} } is called a (covariant) k {\displaystyle {\boldsymbol {k}}} -tensor, and the vector space of such forms is usually denoted T k ( V ) {\displaystyle {\mathcal {T}}^{k}(V)} or L k ( V ) {\displaystyle {\mathcal {L}}^{k}(V)} . == Tensor product == Given a k {\displaystyle k} -tensor f ∈ T k ( V ) {\displaystyle f\in {\mathcal {T}}^{k}(V)} and an ℓ {\displaystyle \ell } -tensor g ∈ T ℓ ( V ) {\displaystyle g\in {\mathcal {T}}^{\ell }(V)} , a product f ⊗ g ∈ T k + ℓ ( V ) {\displaystyle f\otimes g\in {\mathcal {T}}^{k+\ell }(V)} , known as the tensor product, can be defined by the property ( f ⊗ g ) ( v 1 , … , v k , v k + 1 , … , v k + ℓ ) = f ( v 1 , … , v k ) g ( v k + 1 , … , v k + ℓ ) , {\displaystyle (f\otimes g)(v_{1},\ldots ,v_{k},v_{k+1},\ldots ,v_{k+\ell })=f(v_{1},\ldots ,v_{k})g(v_{k+1},\ldots ,v_{k+\ell }),} for all v 1 , … , v k + ℓ ∈ V {\displaystyle v_{1},\ldots ,v_{k+\ell }\in V} . The tensor product of multilinear forms is not commutative; however it is bilinear and associative: f ⊗ ( a g 1 + b g 2 ) = a ( f ⊗ g 1 ) + b ( f ⊗ g 2 ) {\displaystyle f\otimes (ag_{1}+bg_{2})=a(f\otimes g_{1})+b(f\otimes g_{2})} , ( a f 1 + b f 2 ) ⊗ g = a ( f 1 ⊗ g ) + b ( f 2 ⊗ g ) , {\displaystyle (af_{1}+bf_{2})\otimes g=a(f_{1}\otimes g)+b(f_{2}\otimes g),} and ( f ⊗ g ) ⊗ h = f ⊗ ( g ⊗ h ) . {\displaystyle (f\otimes g)\otimes h=f\otimes (g\otimes h).} If ( v 1 , … , v n ) {\displaystyle (v_{1},\ldots ,v_{n})} forms a basis for an n {\displaystyle n} -dimensional vector space V {\displaystyle V} and ( ϕ 1 , … , ϕ n ) {\displaystyle (\phi ^{1},\ldots ,\phi ^{n})} is the corresponding dual basis for the dual space V ∗ = T 1 ( V ) {\displaystyle V^{}={\mathcal {T}}^{1}(V)} , then the products ϕ i 1 ⊗ ⋯ ⊗ ϕ i k {\displaystyle \phi ^{i_{1}}\otimes \cdots \otimes \phi ^{i_{k}}} , with 1 ≤ i 1 , … , i k ≤ n {\displaystyle 1\leq i_{1},\ldots ,i_{k}\leq n} form a basis for T k ( V ) {\displaystyle {\mathcal {T}}^{k}(V)} . Consequently, T k ( V ) {\displaystyle {\mathcal {T}}^{k}(V)} has dimension n k {\displaystyle n^{k}} . == Examples == === Bilinear forms === If k = 2 {\displaystyle k=2} , f : V × V → K {\displaystyle f:V\times V\to K} is referred to as a bilinear form. A familiar and important example of a (symmetric) bilinear form is the standard inner product (dot product) of vectors. === Alternating multilinear forms === An important class of multilinear forms are the alternating multilinear forms, which have the additional property that f ( x σ ( 1 ) , … , x σ ( k ) ) = sgn ⁡ ( σ ) f ( x 1 , … , x k ) , {\displaystyle f(x_{\sigma (1)},\ldots ,x_{\sigma (k)})=\operatorname {sgn}(\sigma )f(x_{1},\ldots ,x_{k}),} where σ : N k → N k {\displaystyle \sigma :\mathbf {N} _{k}\to \mathbf {N} _{k}} is a permutation and sgn ⁡ ( σ ) {\displaystyle \operatorname {sgn}(\sigma )} denotes its sign (+1 if even, –1 if odd). As a consequence, alternating multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e., σ ( p ) = q , σ ( q ) = p {\displaystyle \sigma (p)=q,\sigma (q)=p} and σ ( i ) = i , 1 ≤ i ≤ k , i ≠ p , q {\displaystyle \sigma (i)=i,1\leq i\leq k,i\neq p,q} ): f ( x 1 , … , x p , … , x q , … , x k ) = − f ( x 1 , … , x q , … , x p , … , x k ) . {\displaystyle f(x_{1},\ldots ,x_{p},\ldots ,x_{q},\ldots ,x_{k})=-f(x_{1},\ldots ,x_{q},\ldots ,x_{p},\ldots ,x_{k}).} With the additional hypothesis that the characteristic of the field K {\displaystyle K} is not 2, setting x p = x q = x {\displaystyle x_{p}=x_{q}=x} implies as a corollary that f ( x 1 , … , x , … , x , … , x k ) = 0 {\displaystyle f(x_{1},\ldots ,x,\ldots ,x,\ldots ,x_{k})=0} ; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors use this last condition as the defining property of alternating forms. This definition implies the property given at the beginning of the section, but as noted above, the converse implication holds only when char ⁡ ( K ) ≠ 2 {\displaystyle \operatorname {char} (K)\neq 2} . An alternating multilinear k {\displaystyle k} -form on V {\displaystyle V} over R {\displaystyle \mathbb {R} } is called a multicovector of degree k {\displaystyle {\boldsymbol {k}}} or k {\displaystyle {\boldsymbol {k}}} -covector, and the vector space of such alternating forms, a subspace of T k ( V ) {\displaystyle {\mathcal {T}}^{k}(V)} , is generally denoted A k ( V ) {\displaystyle {\mathcal {A}}^{k}(V)} , or, using the notation for the isomorphic kth exterior power of V ∗ {\displaystyle V^{}} (the dual space of V {\displaystyle V} ), ⋀ k V ∗ {\textstyle \bigwedge ^{k}V^{}} . Note that linear functionals (multilinear 1-forms over R {\displaystyle \mathbb {R} } ) are trivially alternating, so that A 1 ( V ) = T 1 ( V ) = V ∗ {\displaystyle {\mathcal {A}}^{1}(V)={\mathcal {T}}^{1}(V)=V^{}} , while, by convention, 0-forms are defined to be scalars: A 0 ( V ) = T 0 ( V ) = R {\displaystyle {\mathcal {A}}^{0}(V)={\mathcal {T}}^{0}(V)=\mathbb {R} } . The determinant on n × n {\displaystyle n\times n} matrices, viewed as an n {\displaystyle n} argument function of the column vectors, is an important example of an alternating multilinear form. ==== Exterior product ==== The tensor product of alternating multilinear forms is, in general, no longer alternating. However, by summing over all permutations of the tensor product, taking into account the parity of each term, the exterior product ( ∧ {\displaystyle \wedge } , also known as the wedge product) of multicovectors can be defined, so that if f ∈ A k ( V ) {\displaystyle f\in {\mathcal {A}}^{k}(V)} and g ∈ A ℓ ( V ) {\displaystyle g\in {\mathcal {A}}^{\ell }(V)} , then f ∧ g ∈ A k + ℓ ( V ) {\displaystyle f\wedge g\in {\mathcal {A}}^{k+\ell }(V)} : ( f ∧ g ) ( v 1 , … , v k + ℓ ) = 1 k ! ℓ ! ∑ σ ∈ S k + ℓ ( sgn ⁡ ( σ ) ) f ( v σ ( 1 ) , … , v σ ( k ) ) g ( v σ ( k + 1 ) , … , v σ ( k + ℓ ) ) , {\displaystyle (f\wedge g)(v_{1},\ldots ,v_{k+\ell })={\frac {1}{k!\ell !}}\sum _{\sigma \in S_{k+\ell }}(\operatorname {sgn}(\sigma ))f(v_{\sigma (1)},\ldots ,v_{\sigma (k)})g(v_{\sigma (k+1)},\ldots ,v_{\sigma (k+\ell )}),} where the sum is taken over the set of all permutations over k + ℓ {\displaystyle k+\ell } elements, S k + ℓ {\displaystyle S_{k+\ell }} . The exterior product is bilinear, associative, and graded-alternating: if f ∈ A k ( V ) {\displaystyle f\in {\mathcal {A}}^{k}(V)} and g ∈ A ℓ ( V ) {\displaystyle g\in {\mathcal {A}}^{\ell }(V)} then f ∧ g = ( − 1 ) k ℓ g ∧ f {\displaystyle f\wedge g=(-1)^{k\ell }g\wedge f} . Given a basis ( v 1 , … , v n ) {\displaystyle (v_{1},\ldots ,v_{n})} for V {\displaystyle V} and dual basis ( ϕ 1 , … , ϕ n ) {\displaystyle (\phi ^{1},\ldots ,\phi ^{n})} for V ∗ = A 1 ( V ) {\displaystyle V^{}={\mathcal {A}}^{1}(V)} , the exterior products ϕ i 1 ∧ ⋯ ∧ ϕ i k {\displaystyle \phi ^{i_{1}}\wedge \cdots \wedge \phi ^{i_{k}}} , with 1 ≤ i 1 < ⋯ < i k ≤ n {\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n} form a basis for A k ( V ) {\displaystyle {\mathcal {A}}^{k}(V)} . Hence, the dimension of A k ( V ) {\displaystyle {\mathcal {A}}^{k}(V)} for n-dimensional V {\displaystyle V} is ( n k ) = n ! ( n − k ) ! k ! {\textstyle {\tbinom {n}{k}}={\frac {n!}{(n-k)!\,k!}}} . === Differential forms === Differential forms are mathematical objects constructed via tangent spaces and multilinear forms that behave, in many ways, like differentials in the classical sense. Though conceptually and computationally useful, differentials are founded on ill-defined notions of infinitesimal quantities developed early in the history of calculus. Differential forms provide a mathematically rigorous and precise framework to modernize this long-standing idea. Differential forms are especially useful in multivariable calculus (analysis) and differential geometry because they possess transformation properties that allow them be integrated on curves, surfaces, and their higher-dimensional analogues (differentiable manifolds). One far-reaching application is the modern statement of Stokes' theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions. The synopsis below is primarily based on Spivak (1965) and Tu (2011). ==== Definition of differential k-forms and construction of 1-forms ==== To define differential forms on open subsets U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} , we first need the notion of the tangent space of R n {\displaystyle \mathbb {R} ^{n}} at p {\displaystyle p} , usually denoted T p R n {\displaystyle T_{p}\mathbb {R} ^{n}} or R p n {\displaystyle \mathbb {R} _{p}^{n}} . The vector space R p n {\displaystyle \mathbb {R} _{p}^{n}} can be defined most conveniently as the set of elements v p {\displaystyle v_{p}} ( v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} , with p ∈ R n {\displaystyle p\in \mathbb {R} ^{n}} fixed) with vector addition and scalar multiplication defined by v p + w p := ( v + w ) p {\displaystyle v_{p}+w_{p}:=(v+w)_{p}} and a ⋅ ( v p ) := ( a ⋅ v ) p {\displaystyle a\cdot (v_{p}):=(a\cdot v)_{p}} , respectively. Moreover, if ( e 1 , … , e n ) {\displaystyle (e_{1},\ldots ,e_{n})} is the standard basis for R n {\displaystyle \mathbb {R} ^{n}} , then ( ( e 1 ) p , … , ( e n ) p ) {\displaystyle ((e_{1})_{p},\ldots ,(e_{n})_{p})} is the analogous standard basis for R p n {\displaystyle \mathbb {R} _{p}^{n}} . In other words, each tangent space R p n {\displaystyle \mathbb {R} _{p}^{n}} can simply be regarded as a copy of R n {\displaystyle \mathbb {R} ^{n}} (a set of tangent vectors) based at the point p {\displaystyle p} . The collection (disjoint union) of tangent spaces of R n {\displaystyle \mathbb {R} ^{n}} at all p ∈ R n {\displaystyle p\in \mathbb {R} ^{n}} is known as the tangent bundle of R n {\displaystyle \mathbb {R} ^{n}} and is usually denoted T R n := ⋃ p ∈ R n R p n {\textstyle T\mathbb {R} ^{n}:=\bigcup _{p\in \mathbb {R} ^{n}}\mathbb {R} _{p}^{n}} . While the definition given here provides a simple description of the tangent space of R n {\displaystyle \mathbb {R} ^{n}} , there are other, more sophisticated constructions that are better suited for defining the tangent spaces of smooth manifolds in general (see the article on tangent spaces for details). A differential k {\displaystyle {\boldsymbol {k}}} -form on U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} is defined as a function ω {\displaystyle \omega } that assigns to every p ∈ U {\displaystyle p\in U} a k {\displaystyle k} -covector on the tangent space of R n {\displaystyle \mathbb {R} ^{n}} at p {\displaystyle p} , usually denoted ω p := ω ( p ) ∈ A k ( R p n ) {\displaystyle \omega _{p}:=\omega (p)\in {\mathcal {A}}^{k}(\mathbb {R} _{p}^{n})} . In brief, a differential k {\displaystyle k} -form is a k {\displaystyle k} -covector field. The space of k {\displaystyle k} -forms on U {\displaystyle U} is usually denoted Ω k ( U ) {\displaystyle \Omega ^{k}(U)} ; thus if ω {\displaystyle \omega } is a differential k {\displaystyle k} -form, we write ω ∈ Ω k ( U ) {\displaystyle \omega \in \Omega ^{k}(U)} . By convention, a continuous function on U {\displaystyle U} is a differential 0-form: f ∈ C 0 ( U ) = Ω 0 ( U ) {\displaystyle f\in C^{0}(U)=\Omega ^{0}(U)} . We first construct differential 1-forms from 0-forms and deduce some of their basic properties. To simplify the discussion below, we will only consider smooth differential forms constructed from smooth ( C ∞ {\displaystyle C^{\infty }} ) functions. Let f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a smooth function. We define the 1-form d f {\displaystyle df} on U {\displaystyle U} for p ∈ U {\displaystyle p\in U} and v p ∈ R p n {\displaystyle v_{p}\in \mathbb {R} _{p}^{n}} by ( d f ) p ( v p ) := D f \| p ( v ) {\displaystyle (df)_{p}(v_{p}):=Df\|_{p}(v)} , where D f \| p : R n → R {\displaystyle Df\|_{p}:\mathbb {R} ^{n}\to \mathbb {R} } is the total derivative of f {\displaystyle f} at p {\displaystyle p} . (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps (also known as coordinate functions) π i : R n → R {\displaystyle \pi ^{i}:\mathbb {R} ^{n}\to \mathbb {R} } , defined by x ↦ x i {\displaystyle x\mapsto x^{i}} , where x i {\displaystyle x^{i}} is the ith standard coordinate of x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} . The 1-forms d π i {\displaystyle d\pi ^{i}} are known as the basic 1-forms; they are conventionally denoted d x i {\displaystyle dx^{i}} . If the standard coordinates of v p ∈ R p n {\displaystyle v_{p}\in \mathbb {R} _{p}^{n}} are ( v 1 , … , v n ) {\displaystyle (v^{1},\ldots ,v^{n})} , then application of the definition of d f {\displaystyle df} yields d x p i ( v p ) = v i {\displaystyle dx_{p}^{i}(v_{p})=v^{i}} , so that d x p i ( ( e j ) p ) = δ j i {\displaystyle dx_{p}^{i}((e_{j})_{p})=\delta _{j}^{i}} , where δ j i {\displaystyle \delta _{j}^{i}} is the Kronecker delta. Thus, as the dual of the standard basis for R p n {\displaystyle \mathbb {R} _{p}^{n}} , ( d x p 1 , … , d x p n ) {\displaystyle (dx_{p}^{1},\ldots ,dx_{p}^{n})} forms a basis for A 1 ( R p n ) = ( R p n ) ∗ {\displaystyle {\mathcal {A}}^{1}(\mathbb {R} _{p}^{n})=(\mathbb {R} _{p}^{n})^{}} . As a consequence, if ω {\displaystyle \omega } is a 1-form on U {\displaystyle U} , then ω {\displaystyle \omega } can be written as ∑ a i d x i {\textstyle \sum a_{i}\,dx^{i}} for smooth functions a i : U → R {\displaystyle a_{i}:U\to \mathbb {R} } . Furthermore, we can derive an expression for d f {\displaystyle df} that coincides with the classical expression for a total differential: d f = ∑ i = 1 n D i f d x i = ∂ f ∂ x 1 d x 1 + ⋯ + ∂ f ∂ x n d x n . {\displaystyle df=\sum _{i=1}^{n}D_{i}f\;dx^{i}={\partial f \over \partial x^{1}}\,dx^{1}+\cdots +{\partial f \over \partial x^{n}}\,dx^{n}.} [Comments on notation: In this article, we follow the convention from tensor calculus and differential geometry in which multivectors and multicovectors are written with lower and upper indices, respectively. Since differential forms are multicovector fields, upper indices are employed to index them. The opposite rule applies to the components of multivectors and multicovectors, which instead are written with upper and lower indices, respectively. For instance, we represent the standard coordinates of vector v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} as ( v 1 , … , v n ) {\displaystyle (v^{1},\ldots ,v^{n})} , so that v = ∑ i = 1 n v i e i {\textstyle v=\sum _{i=1}^{n}v^{i}e_{i}} in terms of the standard basis ( e 1 , … , e n ) {\displaystyle (e_{1},\ldots ,e_{n})} . In addition, superscripts appearing in the denominator of an expression (as in ∂ f ∂ x i {\textstyle {\frac {\partial f}{\partial x^{i}}}} ) are treated as lower indices in this convention. When indices are applied and interpreted in this manner, the number of upper indices minus the number of lower indices in each term of an expression is conserved, both within the sum and across an equal sign, a feature that serves as a useful mnemonic device and helps pinpoint errors made during manual computation.] ==== Basic operations on differential k-forms ==== The exterior product ( ∧ {\displaystyle \wedge } ) and exterior derivative ( d {\displaystyle d} ) are two fundamental operations on differential forms. The exterior product of a k {\displaystyle k} -form and an ℓ {\displaystyle \ell } -form is a ( k + ℓ ) {\displaystyle (k+\ell )} -form, while the exterior derivative of a k {\displaystyle k} -form is a ( k + 1 ) {\displaystyle (k+1)} -form. Thus, both operations generate differential forms of higher degree from those of lower degree. The exterior product ∧ : Ω k ( U ) × Ω ℓ ( U ) → Ω k + ℓ ( U ) {\displaystyle \wedge :\Omega ^{k}(U)\times \Omega ^{\ell }(U)\to \Omega ^{k+\ell }(U)} of differential forms is a special case of the exterior product of multicovectors in general (see above). As is true in general for the exterior product, the exterior product of differential forms is bilinear, associative, and is graded-alternating. More concretely, if ω = a i 1 … i k d x i 1 ∧ ⋯ ∧ d x i k {\displaystyle \omega =a_{i_{1}\ldots i_{k}}\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}} and η = a j 1 … i ℓ d x j 1 ∧ ⋯ ∧ d x j ℓ {\displaystyle \eta =a_{j_{1}\ldots i_{\ell }}dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\ell }}} , then ω ∧ η = a i 1 … i k a j 1 … j ℓ d x i 1 ∧ ⋯ ∧ d x i k ∧ d x j 1 ∧ ⋯ ∧ d x j ℓ . {\displaystyle \omega \wedge \eta =a_{i_{1}\ldots i_{k}}a_{j_{1}\ldots j_{\ell }}\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}\wedge dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\ell }}.} Furthermore, for any set of indices { α 1 … , α m } {\displaystyle \{\alpha _{1}\ldots ,\alpha _{m}\}} , d x α 1 ∧ ⋯ ∧ d x α p ∧ ⋯ ∧ d x α q ∧ ⋯ ∧ d x α m = − d x α 1 ∧ ⋯ ∧ d x α q ∧ ⋯ ∧ d x α p ∧ ⋯ ∧ d x α m . {\displaystyle dx^{\alpha _{1}}\wedge \cdots \wedge dx^{\alpha _{p}}\wedge \cdots \wedge dx^{\alpha _{q}}\wedge \cdots \wedge dx^{\alpha _{m}}=-dx^{\alpha _{1}}\wedge \cdots \wedge dx^{\alpha _{q}}\wedge \cdots \wedge dx^{\alpha _{p}}\wedge \cdots \wedge dx^{\alpha _{m}}.} If I = { i 1 , … , i k } {\displaystyle I=\{i_{1},\ldots ,i_{k}\}} , J = { j 1 , … , j ℓ } {\displaystyle J=\{j_{1},\ldots ,j_{\ell }\}} , and I ∩ J = ∅ {\displaystyle I\cap J=\varnothing } , then the indices of ω ∧ η {\displaystyle \omega \wedge \eta } can be arranged in ascending order by a (finite) sequence of such swaps. Since d x α ∧ d x α = 0 {\displaystyle dx^{\alpha }\wedge dx^{\alpha }=0} , I ∩ J ≠ ∅ {\displaystyle I\cap J\neq \varnothing } implies that ω ∧ η = 0 {\displaystyle \omega \wedge \eta =0} . Finally, as a consequence of bilinearity, if ω {\displaystyle \omega } and η {\displaystyle \eta } are the sums of several terms, their exterior product obeys distributivity with respect to each of these terms. The collection of the exterior products of basic 1-forms { d x i 1 ∧ ⋯ ∧ d x i k ∣ 1 ≤ i 1 < ⋯ < i k ≤ n } {\displaystyle \{dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}\mid 1\leq i_{1}<\cdots <i_{k}\leq n\}} constitutes a basis for the space of differential k-forms. Thus, any ω ∈ Ω k ( U ) {\displaystyle \omega \in \Omega ^{k}(U)} can be written in the form ω = ∑ i 1 < ⋯ < i k a i 1 … i k d x i 1 ∧ ⋯ ∧ d x i k , ( ∗ ) {\displaystyle \omega =\sum _{i_{1}<\cdots <i_{k}}a_{i_{1}\ldots i_{k}}\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}},\qquad ()} where a i 1 … i k : U → R {\displaystyle a_{i_{1}\ldots i_{k}}:U\to \mathbb {R} } are smooth functions. With each set of indices { i 1 , … , i k } {\displaystyle \{i_{1},\ldots ,i_{k}\}} placed in ascending order, () is said to be the standard presentation of ω {\displaystyle \omega } . In the previous section, the 1-form d f {\displaystyle df} was defined by taking the exterior derivative of the 0-form (continuous function) f {\displaystyle f} . We now extend this by defining the exterior derivative operator d : Ω k ( U ) → Ω k + 1 ( U ) {\displaystyle d:\Omega ^{k}(U)\to \Omega ^{k+1}(U)} for k ≥ 1 {\displaystyle k\geq 1} . If the standard presentation of k {\displaystyle k} -form ω {\displaystyle \omega } is given by (), the ( k + 1 ) {\displaystyle (k+1)} -form d ω {\displaystyle d\omega } is defined by d ω := ∑ i 1 < … < i k d a i 1 … i k ∧ d x i 1 ∧ ⋯ ∧ d x i k . {\displaystyle d\omega :=\sum _{i_{1}<\ldots <i_{k}}da_{i_{1}\ldots i_{k}}\wedge dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}.} A property of d {\displaystyle d} that holds for all smooth forms is that the second exterior derivative of any ω {\displaystyle \omega } vanishes identically: d 2 ω = d ( d ω ) ≡ 0 {\displaystyle d^{2}\omega =d(d\omega )\equiv 0} . This can be established directly from the definition of d {\displaystyle d} and the equality of mixed second-order partial derivatives of C 2 {\displaystyle C^{2}} functions (see the article on closed and exact forms for details). ==== Integration of differential forms and Stokes' theorem for chains ==== To integrate a differential form over a parameterized domain, we first need to introduce the notion of the pullback of a differential form. Roughly speaking, when a differential form is integrated, applying the pullback transforms it in a way that correctly accounts for a change-of-coordinates. Given a differentiable function f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} and k {\displaystyle k} -form η ∈ Ω k ( R m ) {\displaystyle \eta \in \Omega ^{k}(\mathbb {R} ^{m})} , we call f ∗ η ∈ Ω k ( R n ) {\displaystyle f^{}\eta \in \Omega ^{k}(\mathbb {R} ^{n})} the pullback of η {\displaystyle \eta } by f {\displaystyle f} and define it as the k {\displaystyle k} -form such that ( f ∗ η ) p ( v 1 p , … , v k p ) := η f ( p ) ( f ∗ ( v 1 p ) , … , f ∗ ( v k p ) ) , {\displaystyle (f^{}\eta )_{p}(v_{1p},\ldots ,v_{kp}):=\eta _{f(p)}(f_{}(v_{1p}),\ldots ,f_{}(v_{kp})),} for v 1 p , … , v k p ∈ R p n {\displaystyle v_{1p},\ldots ,v_{kp}\in \mathbb {R} _{p}^{n}} , where f ∗ : R p n → R f ( p ) m {\displaystyle f_{}:\mathbb {R} _{p}^{n}\to \mathbb {R} _{f(p)}^{m}} is the map v p ↦ ( D f \| p ( v ) ) f ( p ) {\displaystyle v_{p}\mapsto (Df\|_{p}(v))_{f(p)}} . If ω = f d x 1 ∧ ⋯ ∧ d x n {\displaystyle \omega =f\,dx^{1}\wedge \cdots \wedge dx^{n}} is an n {\displaystyle n} -form on R n {\displaystyle \mathbb {R} ^{n}} (i.e., ω ∈ Ω n ( R n ) {\displaystyle \omega \in \Omega ^{n}(\mathbb {R} ^{n})} ), we define its integral over the unit n {\displaystyle n} -cell as the iterated Riemann integral of f {\displaystyle f} : ∫ [ 0 , 1 ] n ω = ∫ [ 0 , 1 ] n f d x 1 ∧ ⋯ ∧ d x n := ∫ 0 1 ⋯ ∫ 0 1 f d x 1 ⋯ d x n . {\displaystyle \int _{[0,1]^{n}}\omega =\int _{[0,1]^{n}}f\,dx^{1}\wedge \cdots \wedge dx^{n}:=\int _{0}^{1}\cdots \int _{0}^{1}f\,dx^{1}\cdots dx^{n}.} Next, we consider a domain of integration parameterized by a differentiable function c : [ 0 , 1 ] n → A ⊂ R m {\displaystyle c:[0,1]^{n}\to A\subset \mathbb {R} ^{m}} , known as an n-cube. To define the integral of ω ∈ Ω n ( A ) {\displaystyle \omega \in \Omega ^{n}(A)} over c {\displaystyle c} , we "pull back" from A {\displaystyle A} to the unit n-cell: ∫ c ω := ∫ [ 0 , 1 ] n c ∗ ω . {\displaystyle \int _{c}\omega :=\int _{[0,1]^{n}}c^{*}\omega .} To integrate over more general domains, we define an n {\displaystyle {\boldsymbol {n}}} -chain C = ∑ i n i c i {\textstyle C=\sum _{i}n_{i}c_{i}} as the formal sum of n {\displaystyle n} -cubes and set ∫ C ω := ∑ i n i ∫ c i ω . {\displaystyle \int _{C}\omega :=\sum _{i}n_{i}\int _{c_{i}}\omega .} An appropriate definition of the ( n − 1 ) {\displaystyle (n-1)} -chain ∂ C {\displaystyle \partial C} , known as the boundary of C {\displaystyle C} , allows us to state the celebrated Stokes' theorem (Stokes–Cartan theorem) for chains in a subset of R m {\displaystyle \mathbb {R} ^{m}} : If ω {\displaystyle \omega } is a smooth ( n − 1 ) {\displaystyle (n-1)} -form on an open set A ⊂ R m {\displaystyle A\subset \mathbb {R} ^{m}} and C {\displaystyle C} is a smooth n {\displaystyle n} -chain in A {\displaystyle A} , then ∫ C d ω = ∫ ∂ C ω {\displaystyle \int _{C}d\omega =\int _{\partial C}\omega } .Using more sophisticated machinery (e.g., germs and derivations), the tangent space T p M {\displaystyle T_{p}M} of any smooth manifold M {\displaystyle M} (not necessarily embedded in R m {\displaystyle \mathbb {R} ^{m}} ) can be defined. Analogously, a differential form ω ∈ Ω k ( M ) {\displaystyle \omega \in \Omega ^{k}(M)} on a general smooth manifold is a map ω : p ∈ M ↦ ω p ∈ A k ( T p M ) {\displaystyle \omega :p\in M\mapsto \omega _{p}\in {\mathcal {A}}^{k}(T_{p}M)} . Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary and even certain "rough" domains (see the article on Stokes' theorem for details). == See also == Bilinear map Exterior algebra Homogeneous polynomial Linear form Multilinear map == References ==
Wikipedia:Multiple rule-based problems#0	Multiple rule-based problems are problems containing various conflicting rules and restrictions. Such problems typically have an "optimal" solution, found by striking a balance between the various restrictions, without directly defying any of the aforementioned restrictions. Solutions to such problems can either require complex, non-linear thinking processes, or can instead require mathematics-based solutions in which an optimal solution is found by setting the various restrictions as equations, and finding an appropriate maximum value when all equations are added. These problems may thus require more working information as compared to causal relationship problem solving or single rule-based problem solving. The multiple rule-based problem solving is more likely to increase cognitive load than are the other two types of problem solving. == References ==
Wikipedia:Multiple-scale analysis#0	In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove (unwanted) secular terms. The latter puts constraints on the approximate solution, which are called solvability conditions. Mathematics research from about the 1980s proposes that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see center manifold and slow manifold). == Example: undamped Duffing equation == === Differential equation and energy conservation === As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation: d 2 y d t 2 + y + ε y 3 = 0 , {\displaystyle {\frac {d^{2}y}{dt^{2}}}+y+\varepsilon y^{3}=0,} y ( 0 ) = 1 , d y d t ( 0 ) = 0 , {\displaystyle y(0)=1,\qquad {\frac {dy}{dt}}(0)=0,} which is a second-order ordinary differential equation describing a nonlinear oscillator. A solution y(t) is sought for small values of the (positive) nonlinearity parameter 0 < ε ≪ 1. The undamped Duffing equation is known to be a Hamiltonian system: d p d t = − ∂ H ∂ q , d q d t = + ∂ H ∂ p , with H = 1 2 p 2 + 1 2 q 2 + 1 4 ε q 4 , {\displaystyle {\frac {dp}{dt}}=-{\frac {\partial H}{\partial q}},\qquad {\frac {dq}{dt}}=+{\frac {\partial H}{\partial p}},\quad {\text{ with }}\quad H={\tfrac {1}{2}}p^{2}+{\tfrac {1}{2}}q^{2}+{\tfrac {1}{4}}\varepsilon q^{4},} with q = y(t) and p = dy/dt. Consequently, the Hamiltonian H(p, q) is a conserved quantity, a constant, equal to H0 = ⁠1/2⁠ + ⁠1/4⁠ ε for the given initial conditions. This implies that both q and p have to be bounded: \| q \| ≤ 1 + 1 2 ε and \| p \| ≤ 1 + 1 2 ε for all t . {\displaystyle \left\|q\right\|\leq {\sqrt {1+{\tfrac {1}{2}}\varepsilon }}\quad {\text{ and }}\quad \left\|p\right\|\leq {\sqrt {1+{\tfrac {1}{2}}\varepsilon }}\qquad {\text{ for all }}t.} The bound on q is found by equating H with p = 0 to H0: 1 2 q 2 + 1 4 ε q 4 = 1 2 + 1 4 ε {\displaystyle {\tfrac {1}{2}}q^{2}+{\tfrac {1}{4}}\varepsilon q^{4}={\tfrac {1}{2}}+{\tfrac {1}{4}}\varepsilon } , and then dropping the q4 term. This is indeed an upper bound on \|q\|, though keeping the q4 term gives a smaller bound with a more complicated formula. === Straightforward perturbation-series solution === A regular perturbation-series approach to the problem proceeds by writing y ( t ) = y 0 ( t ) + ε y 1 ( t ) + O ( ε 2 ) {\textstyle y(t)=y_{0}(t)+\varepsilon y_{1}(t)+{\mathcal {O}}(\varepsilon ^{2})} and substituting this into the undamped Duffing equation. Matching powers of ε {\textstyle \varepsilon } gives the system of equations d 2 y 0 d t 2 + y 0 = 0 , d 2 y 1 d t 2 + y 1 = − y 0 3 . {\displaystyle {\begin{aligned}{\frac {d^{2}y_{0}}{dt^{2}}}+y_{0}&=0,\\{\frac {d^{2}y_{1}}{dt^{2}}}+y_{1}&=-y_{0}^{3}.\end{aligned}}} Solving these subject to the initial conditions yields y ( t ) = cos ⁡ ( t ) + ε [ 1 32 cos ⁡ ( 3 t ) − 1 32 cos ⁡ ( t ) − 3 8 t sin ⁡ ( t ) ⏟ secular ] + O ( ε 2 ) . {\displaystyle y(t)=\cos(t)+\varepsilon \left[{\tfrac {1}{32}}\cos(3t)-{\tfrac {1}{32}}\cos(t)-\underbrace {{\tfrac {3}{8}}\,t\,\sin(t)} _{\text{secular}}\right]+{\mathcal {O}}(\varepsilon ^{2}).} Note that the last term between the square braces is secular: it grows without bound for large \|t\|. In particular, for t = O ( ε − 1 ) {\displaystyle t=O(\varepsilon ^{-1})} this term is O(1) and has the same order of magnitude as the leading-order term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution. === Method of multiple scales === To construct a solution that is valid beyond t = O ( ϵ − 1 ) {\displaystyle t=O(\epsilon ^{-1})} , the method of multiple-scale analysis is used. Introduce the slow scale t1: t 1 = ε t {\displaystyle t_{1}=\varepsilon t} and assume the solution y(t) is a perturbation-series solution dependent both on t and t1, treated as: y ( t ) = Y 0 ( t , t 1 ) + ε Y 1 ( t , t 1 ) + ⋯ . {\displaystyle y(t)=Y_{0}(t,t_{1})+\varepsilon Y_{1}(t,t_{1})+\cdots .} So: d y d t = ( ∂ Y 0 ∂ t + d t 1 d t ∂ Y 0 ∂ t 1 ) + ε ( ∂ Y 1 ∂ t + d t 1 d t ∂ Y 1 ∂ t 1 ) + ⋯ = ∂ Y 0 ∂ t + ε ( ∂ Y 0 ∂ t 1 + ∂ Y 1 ∂ t ) + O ( ε 2 ) , {\displaystyle {\begin{aligned}{\frac {dy}{dt}}&=\left({\frac {\partial Y_{0}}{\partial t}}+{\frac {dt_{1}}{dt}}{\frac {\partial Y_{0}}{\partial t_{1}}}\right)+\varepsilon \left({\frac {\partial Y_{1}}{\partial t}}+{\frac {dt_{1}}{dt}}{\frac {\partial Y_{1}}{\partial t_{1}}}\right)+\cdots \\&={\frac {\partial Y_{0}}{\partial t}}+\varepsilon \left({\frac {\partial Y_{0}}{\partial t_{1}}}+{\frac {\partial Y_{1}}{\partial t}}\right)+{\mathcal {O}}(\varepsilon ^{2}),\end{aligned}}} using dt1/dt = ε. Similarly: d 2 y d t 2 = ∂ 2 Y 0 ∂ t 2 + ε ( 2 ∂ 2 Y 0 ∂ t ∂ t 1 + ∂ 2 Y 1 ∂ t 2 ) + O ( ε 2 ) . {\displaystyle {\frac {d^{2}y}{dt^{2}}}={\frac {\partial ^{2}Y_{0}}{\partial t^{2}}}+\varepsilon \left(2{\frac {\partial ^{2}Y_{0}}{\partial t\,\partial t_{1}}}+{\frac {\partial ^{2}Y_{1}}{\partial t^{2}}}\right)+{\mathcal {O}}(\varepsilon ^{2}).} Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become: ∂ 2 Y 0 ∂ t 2 + Y 0 = 0 , ∂ 2 Y 1 ∂ t 2 + Y 1 = − Y 0 3 − 2 ∂ 2 Y 0 ∂ t ∂ t 1 . {\displaystyle {\begin{aligned}{\frac {\partial ^{2}Y_{0}}{\partial t^{2}}}+Y_{0}&=0,\\{\frac {\partial ^{2}Y_{1}}{\partial t^{2}}}+Y_{1}&=-Y_{0}^{3}-2\,{\frac {\partial ^{2}Y_{0}}{\partial t\,\partial t_{1}}}.\end{aligned}}} === Solution === The zeroth-order problem has the general solution: Y 0 ( t , t 1 ) = A ( t 1 ) e + i t + A ∗ ( t 1 ) e − i t , {\displaystyle Y_{0}(t,t_{1})=A(t_{1})\,e^{+it}+A^{\ast }(t_{1})\,e^{-it},} with A(t1) a complex-valued amplitude to the zeroth-order solution Y0(t, t1) and i2 = −1. Now, in the first-order problem the forcing in the right hand side of the differential equation is [ − 3 A 2 A ∗ − 2 i d A d t 1 ] e + i t − A 3 e + 3 i t + c . c . {\displaystyle \left[-3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1}}}\right]\,e^{+it}-A^{3}\,e^{+3it}+c.c.} where c.c. denotes the complex conjugate of the preceding terms. The occurrence of secular terms can be prevented by imposing on the – yet unknown – amplitude A(t1) the solvability condition − 3 A 2 A ∗ − 2 i d A d t 1 = 0. {\displaystyle -3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1}}}=0.} The solution to the solvability condition, also satisfying the initial conditions y(0) = 1 and dy/dt(0) = 0, is: A = 1 2 exp ⁡ ( 3 8 i t 1 ) . {\displaystyle A={\tfrac {1}{2}}\,\exp \left({\tfrac {3}{8}}\,i\,t_{1}\right).} As a result, the approximate solution by the multiple-scales analysis is y ( t ) = cos ⁡ [ ( 1 + 3 8 ε ) t ] + O ( ε ) , {\displaystyle y(t)=\cos \left[\left(1+{\tfrac {3}{8}}\,\varepsilon \right)t\right]+{\mathcal {O}}(\varepsilon ),} using t1 = εt and valid for εt = O(1). This agrees with the nonlinear frequency changes found by employing the Lindstedt–Poincaré method. This new solution is valid until t = O ( ϵ − 2 ) {\displaystyle t=O(\epsilon ^{-2})} . Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, i.e., t2 = ε2 t, t3 = ε3 t, etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see Kevorkian & Cole 1996; Bender & Orszag 1999). === Coordinate transform to amplitude/phase variables === Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms, as described next. A solution y ≈ r cos ⁡ θ {\displaystyle y\approx r\cos \theta } is sought in new coordinates ( r , θ ) {\displaystyle (r,\theta )} where the amplitude r ( t ) {\displaystyle r(t)} varies slowly and the phase θ ( t ) {\displaystyle \theta (t)} varies at an almost constant rate, namely d θ / d t ≈ 1. {\displaystyle d\theta /dt\approx 1.} Straightforward algebra finds the coordinate transform y = r cos ⁡ θ + 1 32 ε r 3 cos ⁡ 3 θ + 1 1024 ε 2 r 5 ( − 21 cos ⁡ 3 θ + cos ⁡ 5 θ ) + O ( ε 3 ) {\displaystyle y=r\cos \theta +{\frac {1}{32}}\varepsilon r^{3}\cos 3\theta +{\frac {1}{1024}}\varepsilon ^{2}r^{5}(-21\cos 3\theta +\cos 5\theta )+{\mathcal {O}}(\varepsilon ^{3})} transforms Duffing's equation into the pair that the radius is constant d r / d t = 0 {\displaystyle dr/dt=0} and the phase evolves according to d θ d t = 1 + 3 8 ε r 2 − 15 256 ε 2 r 4 + O ( ε 3 ) . {\displaystyle {\frac {d\theta }{dt}}=1+{\frac {3}{8}}\varepsilon r^{2}-{\frac {15}{256}}\varepsilon ^{2}r^{4}+{\mathcal {O}}(\varepsilon ^{3}).} That is, Duffing's oscillations are of constant amplitude r {\displaystyle r} but have different frequencies d θ / d t {\displaystyle d\theta /dt} depending upon the amplitude. More difficult examples are better treated using a time-dependent coordinate transform involving complex exponentials (as also invoked in the previous multiple time-scale approach). A web service will perform the analysis for a wide range of examples. == See also == Method of matched asymptotic expansions WKB approximation Method of averaging Krylov–Bogoliubov averaging method == Notes == == References == == External links == Carson C. Chow (ed.). "Multiple scale analysis". Scholarpedia.
Wikipedia:Multiplicative cascade#0	In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process. == Definition == The plots above are examples of multiplicative cascade multifractals. To create these distributions there are a few steps to take. Firstly, we must create a lattice of cells which will be our underlying probability density field. Secondly, an iterative process is followed to create multiple levels of the lattice: at each iteration the cells are split into four equal parts (cells). Each new cell is then assigned a probability randomly from the set { p 1 , p 2 , p 3 , p 4 } {\displaystyle \lbrace p_{1},p_{2},p_{3},p_{4}\rbrace } without replacement, where p i ∈ [ 0 , 1 ] {\displaystyle p_{i}\in [0,1]} . This process is continued to the Nth level. For example, in constructing such a model down to level 8 we produce a 48 array of cells. Thirdly, the cells are filled as follows: We take the probability of a cell being occupied as the product of the cell's own pi and those of all its parents (up to level 1). A Monte Carlo rejection scheme is used repeatedly until the desired cell population is obtained, as follows: x and y cell coordinates are chosen randomly, and a random number between 0 and 1 is assigned; the (x, y) cell is then populated depending on whether the assigned number is lesser than (outcome: not populated) or greater or equal to (outcome: populated) the cell's occupation probability. == Examples == To produce the plots above we filled the probability density field with 5,000 points in a space of 256 × 256. An example of the probability density field: The fractals are generally not scale-invariant and therefore cannot be considered standard fractals. They can however be considered multifractals. The Rényi (generalized) dimensions can be theoretically predicted. It can be shown that as N → ∞ {\displaystyle N\rightarrow \infty } , D q = log 2 ⁡ ( f 1 q + f 2 q + f 3 q + f 4 q ) 1 − q , {\displaystyle D_{q}={\frac {\log _{2}\left(f_{1}^{q}+f_{2}^{q}+f_{3}^{q}+f_{4}^{q}\right)}{1-q}},} where N is the level of the grid refinement and, f i = p i ∑ i p i . {\displaystyle f_{i}={\frac {p_{i}}{\sum _{i}p_{i}}}.} == See also == Fractal dimension Hausdorff dimension Scale invariance == References ==
Wikipedia:Multiplicative digital root#0	In number theory, the multiplicative digital root of a natural number n {\displaystyle n} in a given number base b {\displaystyle b} is found by multiplying the digits of n {\displaystyle n} together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of n {\displaystyle n} . The multiplicative digital root for the first few positive integers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0. (sequence A031347 in the OEIS) Multiplicative digital roots are the multiplicative equivalent of digital roots. == Definition == Let n {\displaystyle n} be a natural number. We define the digit product for base b > 1 {\displaystyle b>1} F b : N → N {\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} } to be the following: F b ( n ) = ∏ i = 0 k − 1 d i {\displaystyle F_{b}(n)=\prod _{i=0}^{k-1}d_{i}} where k = ⌊ log b ⁡ n ⌋ + 1 {\displaystyle k=\lfloor \log _{b}{n}\rfloor +1} is the number of digits in the number in base b {\displaystyle b} , and d i = n mod b i + 1 − n mod b i b i {\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}} is the value of each digit of the number. A natural number n {\displaystyle n} is a multiplicative digital root if it is a fixed point for F b {\displaystyle F_{b}} , which occurs if F b ( n ) = n {\displaystyle F_{b}(n)=n} . For example, in base b = 10 {\displaystyle b=10} , 0 is the multiplicative digital root of 9876, as F 10 ( 9876 ) = ( 9 ) ( 8 ) ( 7 ) ( 6 ) = 3024 {\displaystyle F_{10}(9876)=(9)(8)(7)(6)=3024} F 10 ( 3024 ) = ( 3 ) ( 0 ) ( 2 ) ( 4 ) = 0 {\displaystyle F_{10}(3024)=(3)(0)(2)(4)=0} F 10 ( 0 ) = 0 {\displaystyle F_{10}(0)=0} All natural numbers n {\displaystyle n} are preperiodic points for F b {\displaystyle F_{b}} , regardless of the base. This is because if n ≥ b {\displaystyle n\geq b} , then n = ∑ i = 0 k − 1 d i b i {\displaystyle n=\sum _{i=0}^{k-1}d_{i}b^{i}} and therefore F b ( n ) = ∏ i = 0 k − 1 d i = d k − 1 ∏ i = 0 k − 2 d i < d k − 1 b k − 1 < ∑ i = 0 k − 1 d i b i = n {\displaystyle F_{b}(n)=\prod _{i=0}^{k-1}d_{i}=d_{k-1}\prod _{i=0}^{k-2}d_{i}<d_{k-1}b^{k-1}<\sum _{i=0}^{k-1}d_{i}b^{i}=n} If n < b {\displaystyle n<b} , then trivially F b ( n ) = n {\displaystyle F_{b}(n)=n} Therefore, the only possible multiplicative digital roots are the natural numbers 0 ≤ n < b {\displaystyle 0\leq n<b} , and there are no cycles other than the fixed points of 0 ≤ n < b {\displaystyle 0\leq n<b} . == Multiplicative persistence == The number of iterations i {\displaystyle i} needed for F b i ( n ) {\displaystyle F_{b}^{i}(n)} to reach a fixed point is the multiplicative persistence of n {\displaystyle n} . The multiplicative persistence is undefined if it never reaches a fixed point. In base 10, it is conjectured that there is no number with a multiplicative persistence i > 11 {\displaystyle i>11} : this is known to be true for numbers n ≤ 10 20585 {\displaystyle n\leq 10^{20585}} . The smallest numbers with persistence 0, 1, ... are: 0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899. (sequence A003001 in the OEIS) The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be sorted, and, except for the first two digits, all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits. Based on these restrictions, the number of candidates for k {\displaystyle k} -digit numbers with record-breaking persistence is only proportional to the square of k {\displaystyle k} , a tiny fraction of all possible k {\displaystyle k} -digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist. == Extension to negative integers == The multiplicative digital root can be extended to the negative integers by use of a signed-digit representation to represent each integer. == Programming example == The example below implements the digit product described in the definition above to search for multiplicative digital roots and multiplicative persistences in Python. == See also == Arithmetic dynamics Digit sum Digital root Sum-product number == References == == Literature == Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 398–399. ISBN 978-0-387-20860-2. Zbl 1058.11001. == External links == What's special about 277777788888899? - Numberphile on YouTube (Mar 21, 2019)
Wikipedia:Multiplicative inverse#0	In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution). Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1). The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation of Euclid's Elements. In the phrase multiplicative inverse, the qualifier multiplicative is often omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba; then "inverse" typically implies that an element is both a left and right inverse. The notation f −1 is sometimes also used for the inverse function of the function f, which is for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin x) = (sin x)−1 is the cosecant of x, and not the inverse sine of x denoted by sin−1 x or arcsin x. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called the bijection réciproque). == Examples and counterexamples == In the real numbers, zero does not have a reciprocal (division by zero is undefined) because no real number multiplied by 0 produces 1 (the product of any number with zero is zero). With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, and reciprocals of every complex number are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no integer other than 1 and −1 has an integer reciprocal, and so the integers are not a field. In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 (mod n). This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 ⋅ 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it. The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements x, y such that xy = 0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring. The linear map that has the matrix A−1 with respect to some base is then the inverse function of the map having A as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, but they still do not coincide, since the multiplicative inverse of Ax would be (Ax)−1, not A−1x. These two notions of an inverse function do sometimes coincide, for example for the function f ( x ) = x i = e i ln ⁡ ( x ) {\displaystyle f(x)=x^{i}=e^{i\ln(x)}} where ln {\displaystyle \ln } is the principal branch of the complex logarithm and e − π < \| x \| < e π {\displaystyle e^{-\pi }<\|x\|<e^{\pi }} : ( ( 1 / f ) ∘ f ) ( x ) = ( 1 / f ) ( f ( x ) ) = 1 / ( f ( f ( x ) ) ) = 1 / e i ln ⁡ ( e i ln ⁡ ( x ) ) = 1 / e i i ln ⁡ ( x ) = 1 / e − ln ⁡ ( x ) = x {\displaystyle ((1/f)\circ f)(x)=(1/f)(f(x))=1/(f(f(x)))=1/e^{i\ln(e^{i\ln(x)})}=1/e^{ii\ln(x)}=1/e^{-\ln(x)}=x} . The trigonometric functions are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; the secant is the reciprocal of the cosine; the cosecant is the reciprocal of the sine. A ring in which every nonzero element has a multiplicative inverse is a division ring; likewise an algebra in which this holds is a division algebra. == Complex numbers == As mentioned above, the reciprocal of every nonzero complex number z = a + b i {\displaystyle z=a+bi} is complex. It can be found by multiplying both top and bottom of 1/z by its complex conjugate z ¯ = a − b i {\displaystyle {\bar {z}}=a-bi} and using the property that z z ¯ = ‖ z ‖ 2 {\displaystyle z{\bar {z}}=\\|z\\|^{2}} , the absolute value of z squared, which is the real number a2 + b2: 1 z = z ¯ z z ¯ = z ¯ ‖ z ‖ 2 = a − b i a 2 + b 2 = a a 2 + b 2 − b a 2 + b 2 i . {\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{\\|z\\|^{2}}}={\frac {a-bi}{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i.} The intuition is that z ¯ ‖ z ‖ {\displaystyle {\frac {\bar {z}}{\\|z\\|}}} gives us the complex conjugate with a magnitude reduced to a value of 1 {\displaystyle 1} , so dividing again by ‖ z ‖ {\displaystyle \\|z\\|} ensures that the magnitude is now equal to the reciprocal of the original magnitude as well, hence: 1 z = z ¯ ‖ z ‖ 2 {\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{\\|z\\|^{2}}}} In particular, if \|\|z\|\|=1 (z has unit magnitude), then 1 / z = z ¯ {\displaystyle 1/z={\bar {z}}} . Consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property. For example, additive and multiplicative inverses of i are −(i) = −i and 1/i = −i, respectively. For a complex number in polar form z = r(cos φ + i sin φ), the reciprocal simply takes the reciprocal of the magnitude and the negative of the angle: 1 z = 1 r ( cos ⁡ ( − φ ) + i sin ⁡ ( − φ ) ) . {\displaystyle {\frac {1}{z}}={\frac {1}{r}}\left(\cos(-\varphi )+i\sin(-\varphi )\right).} == Calculus == In real calculus, the derivative of 1/x = x−1 is given by the power rule with the power −1: d d x x − 1 = ( − 1 ) x ( − 1 ) − 1 = − x − 2 = − 1 x 2 . {\displaystyle {\frac {d}{dx}}x^{-1}=(-1)x^{(-1)-1}=-x^{-2}=-{\frac {1}{x^{2}}}.} The power rule for integrals (Cavalieri's quadrature formula) cannot be used to compute the integral of 1/x, because doing so would result in division by 0: ∫ d x x = x 0 0 + C {\displaystyle \int {\frac {dx}{x}}={\frac {x^{0}}{0}}+C} Instead the integral is given by: ∫ 1 a d x x = ln ⁡ a , {\displaystyle \int _{1}^{a}{\frac {dx}{x}}=\ln a,} ∫ d x x = ln ⁡ x + C . {\displaystyle \int {\frac {dx}{x}}=\ln x+C.} where ln is the natural logarithm. To show this, note that d d y e y = e y {\textstyle {\frac {d}{dy}}e^{y}=e^{y}} , so if x = e y {\displaystyle x=e^{y}} and y = ln ⁡ x {\displaystyle y=\ln x} , we have: d x d y = x ⇒ d x x = d y ⇒ ∫ d x x = ∫ d y = y + C = ln ⁡ x + C . {\displaystyle {\begin{aligned}&{\frac {dx}{dy}}=x\quad \Rightarrow \quad {\frac {dx}{x}}=dy\\[10mu]&\quad \Rightarrow \quad \int {\frac {dx}{x}}=\int dy=y+C=\ln x+C.\end{aligned}}} == Algorithms == The reciprocal may be computed by hand with the use of long division. Computing the reciprocal is important in many division algorithms, since the quotient a/b can be computed by first computing 1/b and then multiplying it by a. Noting that f ( x ) = 1 / x − b {\displaystyle f(x)=1/x-b} has a zero at x = 1/b, Newton's method can find that zero, starting with a guess x 0 {\displaystyle x_{0}} and iterating using the rule: x n + 1 = x n − f ( x n ) f ′ ( x n ) = x n − 1 / x n − b − 1 / x n 2 = 2 x n − b x n 2 = x n ( 2 − b x n ) . {\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}=x_{n}-{\frac {1/x_{n}-b}{-1/x_{n}^{2}}}=2x_{n}-bx_{n}^{2}=x_{n}(2-bx_{n}).} This continues until the desired precision is reached. For example, suppose we wish to compute 1/17 ≈ 0.0588 with 3 digits of precision. Taking x0 = 0.1, the following sequence is produced: x1 = 0.1(2 − 17 × 0.1) = 0.03 x2 = 0.03(2 − 17 × 0.03) = 0.0447 x3 = 0.0447(2 − 17 × 0.0447) ≈ 0.0554 x4 = 0.0554(2 − 17 × 0.0554) ≈ 0.0586 x5 = 0.0586(2 − 17 × 0.0586) ≈ 0.0588 A typical initial guess can be found by rounding b to a nearby power of 2, then using bit shifts to compute its reciprocal. In constructive mathematics, for a real number x to have a reciprocal, it is not sufficient that x ≠ 0. There must instead be given a rational number r such that 0 < r < \|x\|. In terms of the approximation algorithm described above, this is needed to prove that the change in y will eventually become arbitrarily small. This iteration can also be generalized to a wider sort of inverses; for example, matrix inverses. == Reciprocals of irrational numbers == Every real or complex number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have important special properties. Examples include the reciprocal of e (≈ 0.367879) and the golden ratio's reciprocal (≈ 0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; f ( 1 / e ) {\displaystyle f(1/e)} is the global minimum of f ( x ) = x x {\displaystyle f(x)=x^{x}} . The second number is the only positive number that is equal to its reciprocal plus one: φ = 1 / φ + 1 {\displaystyle \varphi =1/\varphi +1} . Its additive inverse is the only negative number that is equal to its reciprocal minus one: − φ = − 1 / φ − 1 {\displaystyle -\varphi =-1/\varphi -1} . The function f ( n ) = n + n 2 + 1 , n ∈ N , n > 0 {\textstyle f(n)=n+{\sqrt {n^{2}+1}},n\in \mathbb {N} ,n>0} gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, f ( 2 ) {\displaystyle f(2)} is the irrational 2 + 5 {\displaystyle 2+{\sqrt {5}}} . Its reciprocal 1 / ( 2 + 5 ) {\displaystyle 1/(2+{\sqrt {5}})} is − 2 + 5 {\displaystyle -2+{\sqrt {5}}} , exactly 4 {\displaystyle 4} less. Such irrational numbers share an evident property: they have the same fractional part as their reciprocal, since these numbers differ by an integer. The reciprocal function plays an important role in simple continued fractions, which have a number of remarkable properties relating to the representation of (both rational and) irrational numbers. == Further remarks == If the multiplication is associative, an element x with a multiplicative inverse cannot be a zero divisor (x is a zero divisor if some nonzero y, xy = 0). To see this, it is sufficient to multiply the equation xy = 0 by the inverse of x (on the left), and then simplify using associativity. In the absence of associativity, the sedenions provide a counterexample. The converse does not hold: an element which is not a zero divisor is not guaranteed to have a multiplicative inverse. Within Z, all integers except −1, 0, 1 provide examples; they are not zero divisors nor do they have inverses in Z. If the ring or algebra is finite, however, then all elements a which are not zero divisors do have a (left and right) inverse. For, first observe that the map f(x) = ax must be injective: f(x) = f(y) implies x = y: a x = a y ⇒ a x − a y = 0 ⇒ a ( x − y ) = 0 ⇒ x − y = 0 ⇒ x = y . {\displaystyle {\begin{aligned}ax&=ay&\quad \Rightarrow &\quad ax-ay=0\\&&\quad \Rightarrow &\quad a(x-y)=0\\&&\quad \Rightarrow &\quad x-y=0\\&&\quad \Rightarrow &\quad x=y.\end{aligned}}} Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the map is necessarily surjective. Specifically, ƒ (namely multiplication by a) must map some element x to 1, ax = 1, so that x is an inverse for a. == Applications == The expansion of the reciprocal 1/q in any base can also act as a source of pseudo-random numbers, if q is a "suitable" safe prime, a prime of the form 2p + 1 where p is also a prime. A sequence of pseudo-random numbers of length q − 1 will be produced by the expansion. == See also == Division (mathematics) Exponential decay Fraction Group (mathematics) Hyperbola Inverse distribution List of sums of reciprocals Repeating decimal 6-sphere coordinates Unit fractions – reciprocals of integers Zeros and poles == Notes == == References == Maximally Periodic Reciprocals, Matthews R.A.J. Bulletin of the Institute of Mathematics and its Applications vol 28 pp 147–148 1992
Wikipedia:Multiplicity (mathematics)#0	In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct". == Multiplicity of a prime factor == In prime factorization, the multiplicity of a prime factor is its p {\displaystyle p} -adic valuation. For example, the prime factorization of the integer 60 is 60 = 2 × 2 × 3 × 5, the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has four prime factors allowing for multiplicities, but only three distinct prime factors. == Multiplicity of a root of a polynomial == Let F {\displaystyle F} be a field and p ( x ) {\displaystyle p(x)} be a polynomial in one variable with coefficients in F {\displaystyle F} . An element a ∈ F {\displaystyle a\in F} is a root of multiplicity k {\displaystyle k} of p ( x ) {\displaystyle p(x)} if there is a polynomial s ( x ) {\displaystyle s(x)} such that s ( a ) ≠ 0 {\displaystyle s(a)\neq 0} and p ( x ) = ( x − a ) k s ( x ) {\displaystyle p(x)=(x-a)^{k}s(x)} . If k = 1 {\displaystyle k=1} , then a is called a simple root. If k ≥ 2 {\displaystyle k\geq 2} , then a {\displaystyle a} is called a multiple root. For instance, the polynomial p ( x ) = x 3 + 2 x 2 − 7 x + 4 {\displaystyle p(x)=x^{3}+2x^{2}-7x+4} has 1 and −4 as roots, and can be written as p ( x ) = ( x + 4 ) ( x − 1 ) 2 {\displaystyle p(x)=(x+4)(x-1)^{2}} . This means that 1 is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of the fundamental theorem of algebra. If a {\displaystyle a} is a root of multiplicity k {\displaystyle k} of a polynomial, then it is a root of multiplicity k − 1 {\displaystyle k-1} of the derivative of that polynomial, unless the characteristic of the underlying field is a divisor of k, in which case a {\displaystyle a} is a root of multiplicity at least k {\displaystyle k} of the derivative. The discriminant of a polynomial is zero if and only if the polynomial has a multiple root. === Behavior of a polynomial function near a multiple root === The graph of a polynomial function f touches the x-axis at the real roots of the polynomial. The graph is tangent to it at the multiple roots of f and not tangent at the simple roots. The graph crosses the x-axis at roots of odd multiplicity and does not cross it at roots of even multiplicity. A non-zero polynomial function is everywhere non-negative if and only if all its roots have even multiplicity and there exists an x 0 {\displaystyle x_{0}} such that f ( x 0 ) > 0 {\displaystyle f(x_{0})>0} . == Multiplicity of a solution of a nonlinear system of equations == For an equation f ( x ) = 0 {\displaystyle f(x)=0} with a single variable solution x ∗ {\displaystyle x_{}} , the multiplicity is k {\displaystyle k} if f ( x ∗ ) = f ′ ( x ∗ ) = ⋯ = f ( k − 1 ) ( x ∗ ) = 0 {\displaystyle f(x_{})=f'(x_{})=\cdots =f^{(k-1)}(x_{})=0} and f ( k ) ( x ∗ ) ≠ 0. {\displaystyle f^{(k)}(x_{})\neq 0.} In other words, the differential functional ∂ j {\displaystyle \partial _{j}} , defined as the derivative 1 j ! d j d x j {\displaystyle {\frac {1}{j!}}{\frac {d^{j}}{dx^{j}}}} of a function at x ∗ {\displaystyle x_{}} , vanishes at f {\displaystyle f} for j {\displaystyle j} up to k − 1 {\displaystyle k-1} . Those differential functionals ∂ 0 , ∂ 1 , ⋯ , ∂ k − 1 {\displaystyle \partial _{0},\partial _{1},\cdots ,\partial _{k-1}} span a vector space, called the Macaulay dual space at x ∗ {\displaystyle x_{}} , and its dimension is the multiplicity of x ∗ {\displaystyle x_{}} as a zero of f {\displaystyle f} . Let f ( x ) = 0 {\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} } be a system of m {\displaystyle m} equations of n {\displaystyle n} variables with a solution x ∗ {\displaystyle \mathbf {x} _{}} where f {\displaystyle \mathbf {f} } is a mapping from R n {\displaystyle R^{n}} to R m {\displaystyle R^{m}} or from C n {\displaystyle C^{n}} to C m {\displaystyle C^{m}} . There is also a Macaulay dual space of differential functionals at x ∗ {\displaystyle \mathbf {x} _{}} in which every functional vanishes at f {\displaystyle \mathbf {f} } . The dimension of this Macaulay dual space is the multiplicity of the solution x ∗ {\displaystyle \mathbf {x} _{}} to the equation f ( x ) = 0 {\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} } . The Macaulay dual space forms the multiplicity structure of the system at the solution. For example, the solution x ∗ = ( 0 , 0 ) {\displaystyle \mathbf {x} _{}=(0,0)} of the system of equations in the form of f ( x ) = 0 {\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} } with f ( x ) = [ sin ⁡ ( x 1 ) − x 2 + x 1 2 x 1 − sin ⁡ ( x 2 ) + x 2 2 ] {\displaystyle \mathbf {f} (\mathbf {x} )=\left[{\begin{array}{c}\sin(x_{1})-x_{2}+x_{1}^{2}\\x_{1}-\sin(x_{2})+x_{2}^{2}\end{array}}\right]} is of multiplicity 3 because the Macaulay dual space span ⁡ { ∂ 00 , ∂ 10 + ∂ 01 , − ∂ 10 + ∂ 20 + ∂ 11 + ∂ 02 } {\displaystyle \operatorname {span} \{\partial _{00},\partial _{10}+\partial _{01},-\partial _{10}+\partial _{20}+\partial _{11}+\partial _{02}\}} is of dimension 3, where ∂ i j {\displaystyle \partial _{ij}} denotes the differential functional 1 i ! j ! ∂ i + j ∂ x 1 i ∂ x 2 j {\displaystyle {\frac {1}{i!j!}}{\frac {\partial ^{i+j}}{\partial x_{1}^{i}\,\partial x_{2}^{j}}}} applied on a function at the point x ∗ = ( 0 , 0 ) {\displaystyle \mathbf {x} _{*}=(0,0)} . The multiplicity is always finite if the solution is isolated, is perturbation invariant in the sense that a k {\displaystyle k} -fold solution becomes a cluster of solutions with a combined multiplicity k {\displaystyle k} under perturbation in complex spaces, and is identical to the intersection multiplicity on polynomial systems. == Intersection multiplicity == In algebraic geometry, the intersection of two sub-varieties of an algebraic variety is a finite union of irreducible varieties. To each component of such an intersection is attached an intersection multiplicity. This notion is local in the sense that it may be defined by looking at what occurs in a neighborhood of any generic point of this component. It follows that without loss of generality, we may consider, in order to define the intersection multiplicity, the intersection of two affines varieties (sub-varieties of an affine space). Thus, given two affine varieties V1 and V2, consider an irreducible component W of the intersection of V1 and V2. Let d be the dimension of W, and P be any generic point of W. The intersection of W with d hyperplanes in general position passing through P has an irreducible component that is reduced to the single point P. Therefore, the local ring at this component of the coordinate ring of the intersection has only one prime ideal, and is therefore an Artinian ring. This ring is thus a finite dimensional vector space over the ground field. Its dimension is the intersection multiplicity of V1 and V2 at W. This definition allows us to state Bézout's theorem and its generalizations precisely. This definition generalizes the multiplicity of a root of a polynomial in the following way. The roots of a polynomial f are points on the affine line, which are the components of the algebraic set defined by the polynomial. The coordinate ring of this affine set is R = K [ X ] / ⟨ f ⟩ , {\displaystyle R=K[X]/\langle f\rangle ,} where K is an algebraically closed field containing the coefficients of f. If f ( X ) = ∏ i = 1 k ( X − α i ) m i {\displaystyle f(X)=\prod _{i=1}^{k}(X-\alpha _{i})^{m_{i}}} is the factorization of f, then the local ring of R at the prime ideal ⟨ X − α i ⟩ {\displaystyle \langle X-\alpha _{i}\rangle } is K [ X ] / ⟨ ( X − α ) m i ⟩ . {\displaystyle K[X]/\langle (X-\alpha )^{m_{i}}\rangle .} This is a vector space over K, which has the multiplicity m i {\displaystyle m_{i}} of the root as a dimension. This definition of intersection multiplicity, which is essentially due to Jean-Pierre Serre in his book Local Algebra, works only for the set theoretic components (also called isolated components) of the intersection, not for the embedded components. Theories have been developed for handling the embedded case (see Intersection theory for details). == In complex analysis == Let z0 be a root of a holomorphic function f, and let n be the least positive integer such that the nth derivative of f evaluated at z0 differs from zero. Then the power series of f about z0 begins with the nth term, and f is said to have a root of multiplicity (or “order”) n. If n = 1, the root is called a simple root. We can also define the multiplicity of the zeroes and poles of a meromorphic function. If we have a meromorphic function f = g h , {\textstyle f={\frac {g}{h}},} take the Taylor expansions of g and h about a point z0, and find the first non-zero term in each (denote the order of the terms m and n respectively) then if m = n, then the point has non-zero value. If m > n , {\displaystyle m>n,} then the point is a zero of multiplicity m − n . {\displaystyle m-n.} If m < n {\displaystyle m<n} , then the point has a pole of multiplicity n − m . {\displaystyle n-m.} == References == Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999. ISBN 0-8176-4011-8.
Wikipedia:Multivalued function#0	In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions, but English Wikipedia currently does, having a separate article for each. A multivalued function of sets f : X → Y is a subset Γ f ⊆ X × Y . {\displaystyle \Gamma _{f}\ \subseteq \ X\times Y.} Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γf. If f is an ordinary function, it is a multivalued function by taking its graph Γ f = { ( x , f ( x ) ) : x ∈ X } . {\displaystyle \Gamma _{f}\ =\ \{(x,f(x))\ :\ x\in X\}.} They are called single-valued functions to distinguish them. == 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 ) {\displaystyle f(z)} in some neighbourhood of a point z = a {\displaystyle z=a} . This is the case for functions defined by the implicit function theorem or by a Taylor series around z = a {\displaystyle z=a} . In such a situation, one may extend the domain of the single-valued function f ( z ) {\displaystyle f(z)} along curves in the complex plane starting at a {\displaystyle a} . In doing so, one finds that the value of the extended function at a point z = b {\displaystyle z=b} depends on the chosen curve from a {\displaystyle a} to b {\displaystyle 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 ) = z {\displaystyle 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 {\displaystyle z=1} in the complex plane, and then further along curves starting at z = 1 {\displaystyle z=1} , so that the values along a given curve vary continuously from 1 = 1 {\displaystyle {\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 ) {\displaystyle 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 ) {\displaystyle f(z)} . == Inverses of functions == If f : X → Y is an ordinary function, then its inverse is the multivalued function Γ f − 1 ⊆ Y × X {\displaystyle \Gamma _{f^{-1}}\ \subseteq \ Y\times X} defined as Γf, viewed as a subset of X × Y. When f is a differentiable function between manifolds, the inverse function theorem gives conditions for this to be single-valued locally in X. For example, the complex logarithm log(z) is the multivalued inverse of the exponential function ez : C → C×, with graph Γ log ⁡ ( z ) = { ( z , w ) : w = log ⁡ ( z ) } ⊆ C × C × . {\displaystyle \Gamma _{\log(z)}\ =\ \{(z,w)\ :\ w=\log(z)\}\ \subseteq \ \mathbf {C} \times \mathbf {C} ^{\times }.} It is not single valued, given a single w with w = log(z), we have log ⁡ ( z ) = w + 2 π i Z . {\displaystyle \log(z)\ =\ w\ +\ 2\pi i\mathbf {Z} .} Given any holomorphic function on an open subset of the complex plane C, its analytic continuation is always a multivalued function. == Concrete 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 4 = ± 2 = { 2 , − 2 } {\displaystyle {\sqrt {4}}=\pm 2=\{2,-2\}} ; although zero has only one square root, 0 = { 0 } {\displaystyle {\sqrt {0}}=\{0\}} . Note that x {\displaystyle {\sqrt {x}}} usually denotes only the principal square root of x {\displaystyle x} . 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 + b i ) {\displaystyle \log(a+bi)} for real numbers a {\displaystyle a} and b {\displaystyle b} are log ⁡ a 2 + b 2 + i arg ⁡ ( a + b i ) + 2 π n i {\displaystyle \log {\sqrt {a^{2}+b^{2}}}+i\arg(a+bi)+2\pi ni} for all integers n {\displaystyle n} . Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have tan ⁡ ( π 4 ) = tan ⁡ ( 5 π 4 ) = tan ⁡ ( − 3 π 4 ) = tan ⁡ ( ( 2 n + 1 ) π 4 ) = ⋯ = 1. {\displaystyle \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 == 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. == See also == Relation (mathematics) Function (mathematics) Binary relation Set-valued function == 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) == References ==
Wikipedia:Muneer Ahmad Rashid#0	Muneer Ahmad Rashid, FPAS (born 1934), also spelled as Munir Ahmad Rashid, is a Pakistani mathematical physicist and emeritus professor of applied and mathematical physics at the Centre for Advanced Mathematics and Physics of the National University of Sciences and Technology. A physicist turned mathematician, Rashid has made numerous contributions in Special unitary group, applied mathematics, theoretical and nuclear physics, SO(2), and dark energy. He was a student of physicist and scientist Abdus Salam. == Education == Rashid was born in Lahore, British India where he had completed his high-school from there in 1950. Rashid attended the Punjab University in 1950, and had received his double BSc (Hons) in physics and mathematics in 1955. In 1957, he did his M.A. in mathematics and taught as a lecturer in mathematics at the Government College University where he stayed there until 1960. Rashid then travelled to United Kingdom in 1961 to attend Imperial College London. During the 1960s, an physicist Salam was also teaching at Imperial College London where he was supervising the doctorate studies of my many Pakistani students. Rashid joined the Salam's group and began working with Salam's group at the Imperial College. In 1964, Munir Ahmad did his PhD in Mathematical physics under the supervision of Salam, where his doctoral thesis were entitled "Generalization of Mass Formula in Unitary Symmetries". Munir Rashid also did his D.Sc. in mathematical physics under the supervision of Salam at the University of London in 1980. == Academic career == Rashid, following his doctorate degree, came back to Pakistan where he joined Quaid-i-Azam University as an associate professor in 1968. In September 1968, he travelled to United States where he joined University of Rochester as a Visiting Senior Research Associate. Rashid stayed in United States until 1970 and, during this time, he had carried out the research in mathematical physics. In 1970, Rashid came back to Pakistan and re-joined Quaid-i-Azam University as a professor. During the 1960s, Rashid had closely worked with Salam's students to the field of SU(3) or Special unitary group. Rashid had closely worked with an Israeli theoretical physicist Harry J. Lipkin, who is also a student of Salam. During the 1960s, Lipkin was working on SU(3) field and had brought his work to Salam to re-check the work, which according to Lipkin, his predictions did not meet the results as new experimental results from were available to them from CERN. Salam travelled back to Pakistan next day and on Salam's recommendation, Lipkin met with Rashid in London. Rashid gained famed when he had independently discovered the error in Lipkin and Salam's work. Imperial College's Physics Department head, Gerry Brown, who was an editor-in-chief, of Physics Letter, accepted the suggestion from Imperial College that Salam's and Rashid's names to be list of authors. The next day, the paper appeared as Unitary Symmetry: A collaboration of three Israelis and two Pakistanis M. A. Rashid returned to Pakistan in 1970, and joined Quaid-i-Azam University as a professor. Rashid, in midst of 1971 Winter war, was serving as a research associate in Quaid-i-Azam University when Salam had asked him to join Pakistan Atomic Energy Commission (PAEC). Rashid, along with Fayyazuddin, attended the famous Multan Meeting in January 1972. It was here Salam had delegated numerous mathematicians at the Quaid-i-Azam University to participate in the nuclear research program. Rashid began to work with Asghar Qadir's group in PAEC and formed mathematical physics research group. He then was transferred to Riazuddin's Theoretical Physics Group (TPG) where he had closely worked with Riazuddin in the development of nuclear weapons. Rashid continued his close association with Salam in PAEC where he had worked on the mathematics problems related to the science of the designing of nuclear weapons. He played a major role in the development of the designing of the atomic bomb, through his calculations on critical mass theory. Rashid had contributed in scattering theory where he had solved the mathematics problems in scattering theory, mainly predicting the scattering of optical waves and the behaviour of the elementary particles in the general process of testing of the nuclear device. Rashid also had applied the Hamiltonian harmonic oscillator theory to approximate the optical wavelengths and the transition amplitudes of the Quantum particles in the tested nuclear device. To approximate the data and the position of the nuclear particles and their effect in an affected nuclear test sites, Rashid used complex mathematical series, Integrals and mathematical permutation where he published his work under the supervision of Salam at the PAEC. Rashid continued his research at the PAEC, and left Pakistan in 1978, to join Salam in London, Great Britain. He joined the London University where, under Salam's supervision, Rashid received his D.Sc. in 1980. After his D.Sc. from London University, Rashid travelled to Nigeria, where he joined the Ahmadu Bello University, Zaria, Kaduna State, as a foreign professor, teaching there until 2005. In 2005, after being requested by the President of Pakistan, Rashid returned to Pakistan, and joined the Centre for Advanced Mathematics and Physics (CAMP) at National University of Science and Technology (NUST), where he teaches applied mathematics. == Research work == Specialized in mathematical physics under Salam, Rashid had developed an early interest in scattering theory where he had published numerous papers. His contribution to scattering theory at Pakistan Atomic Energy Commission had made breaking discoveries to the field of nuclear physics. On 29 March 2009, at the Mathematical conference, Rashid, using the resources and work of Richard Taylor and Andrew Wiles, proved the Fermat's Last Theorem, and the papers containing the published proof of Fermat's Last Theorem were presented at the conference. On 1 April 2006, at the 12th Regional Conference on Mathematical Physics, held by the National Center for Physics, Rashid proof and presented his papers on "Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian" where he had proposed and solved mathematical problems on Hamiltonian matrix Spherical and Cylindrical harmonics by applying the Hamiltonian mechanics. He also made numerous contribution on pure mathematics, Statistical mechanics and physics. == Publications == === Selected research papers === "Wormholes supported by phantom-like modified Chaplygin gas" M. Jamil, M. U. Farooq(2009) Eur. Phys. J Vol:59 pp:907–912 (Journal) "Constraints on coupling constant between dark energy and dark matter" (2009) Eur.Phys.J.C Vol:60 pp:141–147 (Journal) "Generalized Holographic Dark Energy Model" (2009) Eur.Phys.J.C Vol:61 pp:471–476 (Journal) HEC Recognized:Yes "Linear invariants of a cartesian tensor" (2009) Quarterly Journal of Mechanics and Applied Mathematics Vol:62 pp:31–38 (Journal) HEC Recognized:Yes "Constraints on coupling constant between dark energy and dark matter" (2009) Eur. Phys. J Vol:60 pp:141–147 (Journal) HEC Recognized:Yes "Interacting dark energy with inhomogeneous equation of state" (2008) The European Physical Journal C Vol: pp:- (Journal) HEC Recognized:Yes "Interacting modified variable chaplygin gas in a non-flat universe" (2008) The European Physical Journal C Vol: pp:- (Journal) HEC Recognized:Yes "Charged black holes in phantom cosmology" (2008) The European Physical Journal C Vol AO Ajibade "A Strange property of the determinant of the Minors " (2007) Int. J. Math. Educ. Sci. Technol "Factorality in Riecz groups " (2007) accessed for publication in the Journal of Group Theory Vol: == References == == External links == Conference Papers Research Papers of M. A. Rashid CAMP CCIT Harry J. Lipkin tribute to M. A. Rashid Biography
Wikipedia:Murad Taqqu#0	Murad Salman Taqqu (Arabic: مراد طقو) is an Iraqi probabilist and statistician specializing in time series and stochastic processes. His research areas have included long-range dependence, self-similar processes, and heavy tails. Taqqu is a professor emeritus at Boston University Department of Mathematics and Statistics, and a fellow of the American Mathematical Society. == Education and career == Taqqu was born in Baghdad but grew up in Switzerland. As an undergraduate, he studied physics and mathematics at the École Polytechnique Fédérale de Lausanne. He obtained his Ph.D. at Columbia University in 1972, with his dissertation Limit Theorems for Sums of Strongly Dependent Random Variables supervised by Benoit Mandelbrot. Between 1972 and 1974 Taqqu lectured at the Hebrew University of Jerusalem and worked as a post-doctoral fellow at the Weizmann Institute in Rehovot. From 1974 to 1985, he was a faculty member at the School of Operations Research and Industrial Engineering at Cornell University. Since 1985, Taqqu has served as a professor in the Department of Mathematics and Statistics at Boston University, where he is currently professor emeritus. He has published over 250 papers, and co-authored or co-edited 11 books. == Honors and awards == Dr. Taqqu was named a Guggenheim Fellow in 1987. Dr. Taqqu's 1995 paper "On the Self-Similar Nature of Ethernet Traffic," co-authored with Will Leland, Walter Willinger, and Dan Wilson, won the William J. Bennett Award from the IEEE Communications Society. The same group was awarded the IEEE W.R.G. Baker Prize Paper Award in 1996. The same four co-authors were recognized again, 11 years after the original publication, with the ACM/SIGCOMM Test of Time Award. Dr. Taqqu's 2000 paper "Meaningful MRA initialization for discrete time series," co-authored with D. Veitch and P Abry, was named the best paper in signal processing by EURASIP. He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to self-similar random processes and their applications to real world phenomena such as diverse internet traffic and hydrology". == Selected books == Author Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Gennady Samorodnitsky and Murad S. Taqqu, ISBN 0-412-05171-0, Chapman and Hall, New York (1994). Wiener Chaos: Moments, Cumulants and Diagrams: A survey with Computer Implementation. Giovanni Peccati and Murad S. Taqqu. ISBN 88-470-1678-9. Springer, (2011). Editor New Directions in Time Series Analysis, Part I. D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt and M.S. Taqqu, editors. "The IMA Volumes in Mathematics and its Applications". Series. Vol. 45. ISBN 0-387-97896-8. Springer Verlag, New York, (1992). New Directions in Time Series Analysis, Part II. D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt and M.S. Taqqu, editors. "The IMA Volumes in Mathematics and its Applications". Series. Vol. 46. ISBN 0-387-97914-X. Springer Verlag, New York, (1992). Stochastic Processes and Related Topics: In memory of Stamatis Cambanis 1943–1995. Ioannis Karatzas, Balram S. Rajput, Murad S. Taqqu, editors. Trends in Mathematics Series. ISBN 3-7643-3998-5. Birkhäuser, Boston (1998). A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Robert J. Adler, Raisa E. Feldman and Murad S. Taqqu, editors. ISBN 0-8176-3951-9. Birkhäuser, Boston (1998). Theory and Applications of Long-Range Dependence. Paul Doukhan, Georges Oppenheim and Murad S.Taqqu, editors. ISBN 0-8176-4168-8. Birkhäuser, Boston (2003). == References == == External links == Home page
Wikipedia:Murat Tuncali#0	Murat Tuncali (born 1959) is a Mathematics Professor at Nipissing University in North Bay, Ontario. He is also the chair of the Department of Computer Science and Mathematics. He graduated with his Bachelor of Science at Boğaziçi University, in Turkey. He then graduated from University of Saskatchewan with an MSc, and a PhD. He has won many awards over the years, including the Chancellor’s Award for Excellence in Research 1999–2000, Research Achievement Award 2001–2002. He took a break from his position between the months of January and June 2006. == References == == External links == Murat Tuncali at the Mathematics Genealogy Project "Extensional dimension and completion of maps by H. Murat Tuncali, E. D. Tymchatyn and Vesko Valov". At.yorku.ca. Retrieved 2 April 2019. "Atlas: On Generalizations of the Hahn-Mazurkiewicz Theorem by Murat Tuncali". 20 November 2008. Archived from the original on 20 November 2008. Retrieved 2 April 2019. Tuncali, Murat (2 April 2019). "Lifting Paths on Quotient Spaces". At.yorku.ca. Retrieved 2 April 2019.
Wikipedia:Murderous Maths#0	Murderous Maths is a series of British educational books by author Kjartan Poskitt. Most of the books in the series are illustrated by illustrator Philip Reeve, with the exception of "The Secret Life of Codes", which is illustrated by Ian Baker, "Awesome Arithmetricks" illustrated by Daniel Postgate and Rob Davis, and "The Murderous Maths of Everything", also illustrated by Rob Davis. The Murderous Maths books have been published in over 25 countries. The books, which are aimed at children aged 8 and above, teach maths, spanning from basic arithmetic to relatively complex concepts such as the quadratic formula and trigonometry. The books are written in an informal similar style to the Horrible Histories, Horrible Science and Horrible Geography series, involving evil geniuses, gangsters, and a generally comedic tone. == Development == The first two books of the series were originally part of "The Knowledge" (now "Totally") series, itself a spin-off of Horrible Histories. However, these books were eventually redesigned and they, as well as the rest of the titles in the series, now use the Murderous Maths banner. According to Poskitt, "these books have even found their way into schools and proved to be a boost to GCSE studies". The books are also available in foreign editions, including: German, Spanish, Polish, Czech, Greek, Dutch, Norwegian, Turkish, Croatian, Italian, Lithuanian, Korean, Danish, Hungarian, Finnish, Thai and Portuguese (Latin America). In 2009, the books were redesigned again, changing the cover art style and the titles of most of the books in the series. Poskitt's goal, according to the Murderous Maths website, is to write books that are "something funny to read", have "good amusing illustrations", include "tricks", and "explaining the maths involved as clearly as possible". He adds that although he doesn't "work to any government imposed curriculum or any stage achievement levels", he has "been delighted to receive many messages of support and thanks from parents and teachers in the UK, the United States and elsewhere". == Titles == The following are the thirteen books that are available in the series. Guaranteed to Bend Your Brain (previously Murderous Maths) (1997), ISBN 0-439-01156-6 - (addition, subtraction, multiplication, division, percentages, powers, tessellation, Roman numerals, the development of the "10" and the place system, shortcomings of calculators, prime numbers, time - how the year and day got divided, digital/analogue clocks, angles, introduction to real Mathematicians, magic squares, mental arithmetic, card trick with algebra explanation, rounding and symmetry.) Guaranteed to Mash your Mind (previously More Murderous Maths) (1998), ISBN 0-439-01153-1 (the monomino, domino, tromino, tetromino, pentomino, hexomino and heptomino, length area and volume, dimensions, measuring areas and volumes, basic rectangle and triangle formulas, speed, conversion of units, Möbius strip, Pythagoras, right-angled triangles, irrational numbers, pi, area and perimeter, bisecting angles, triangular numbers, topology networks, magic squares.) Awesome Arithmetricks (previously The Essential Arithmetricks: How to + - × ÷) (1998), ISBN 0-439-01157-4 - (counting, odd even and negative numbers, signs of maths, place value and rounding off, manipulating equations, + - x ÷ %, long division, times tables, estimation, decimal signs, QED.) The Mean & Vulgar Bits (previously The Mean & Vulgar Bits: Fractions and Averages) (2000), ISBN 0-439-01270-8 (fractions, converting improper and mixed fractions, adding subtracting multiplying and dividing fractions, primes and prime factors, reducing fractions, highest common factor and lowest common denominators, Egyptian fractions, comparing fractions, cancelling out fractions, converting fractions to decimals, decimal place system, percentages: increase and decrease, averages: mean mode and median.) Desperate Measures (previously Desperate Measures: Length, Area and Volume) (2000), ISBN 0-439-01370-4 (measuring lines: units and accuracy, old measuring systems, the development of metric, the SI system and powers of ten, shapes, measuring areas and area formulas, weight, angles, measuring volume, Archimedes Principle, density, time and how the modern calendar developed.) Do You Feel Lucky? (previously Do You Feel Lucky: The Secrets of Probability) (2001), ISBN 0-439-99607-4 (chance, tree diagrams, mutually exclusive and independent chances, Pascal's Triangle, permutations and combinations, sampling.) Savage Shapes (previously Vicious Circles and Other Savage Shapes) (2002), ISBN 0-439-99747-X (signs in geometric diagrams, Loci, constructions: perpendicular bisectors; dropping perpendiculars; bisecting angles, triangles: similar; congruent; equal areas, polygons: regular; irregular; angle sizes and construction, tessellations and Penrose Tiles, origami, circles: chord; tangent; angle theorems, regular solids, Euler's formula, ellipses, Geometric proof of Pythagoras' Theorem.) The Key To The Universe (previously Numbers: The Key To The Universe) (2002), ISBN 0-439-98116-6 (phi, Fibonacci Series, Golden Ratio, properties of Square, Triangle, Cube, Centred Hexagon and Tetrahedral numbers, "difference of two squares", number superstitions, prime numbers, Mersenne primes, tests to see if a number will divide by anything from 2-13 and 19, finger multiplication, binary, octal, and hexadecimal, perfect numbers, tricks of the nine times table, irrational transcendental and imaginary numbers, infinity.) The Phantom X (previously The Phantom X: Algebra) (2003), ISBN 0-439-97729-0 (variables, elementary algebra, brackets, factorising, expanding, and simplifying expressions, solving quadratics and the quadratic formula, "Think of a number" tricks, difference of two squares, coefficients of (a-b)n, linear graphs: co-ordinates; gradients; y intercept, non-linear function graphs including parabolas, simultaneous equations: substitution and elimination, dividing by zero!.) The Fiendish Angletron (previously The Fiendish Angletron: Trigonometry) (2004), ISBN 0-439-96859-3 (scales and ratios in maps and diagrams, protractor and compass, SIN, COS and TAN ratios in right angled triangles, trig on a calculator; normal and inverse, sine and cosine formulas for non-right-angled triangles, triangulation, parallax angles and parsecs, sin/cos/tan relationships, sin wave, bearings.) The Perfect Sausages (previously The Perfect Sausage and other Fundamental Formulas) (2005), ISBN 0-439-95901-2 (areas and volumes, ellipsoids and toruses, number formulas (e.g. triangle, hexagonal), speed, acceleration, stopping time, distance, force, gravity, projectiles, Money: percentages; simple and compound interest, permutations and combinations.) The 5ecret L1fe of Code5 (previously Codes: How to Make Them and Break Them) (2007), ISBN 978-1-4071-0715-8 (patterns, logic and deduction, prime numbers, high powers, modular arithmetic.) Easy Questions, Evil Answers (2010), ISBN 1-407-11451-4 (formulas, working out square roots by hand, π, Pythagoras, paradoxes, problem solving, metric prefixes, large numbers, vectors.) Related puzzle books have been published also: Professor Fiendish's Book of Diabolical Brain-benders (2002), ISBN 0-439-98226-X (mazes, logic, coin problems, number crosswords, shape cutting/rearranging, number squares.) Professor Fiendish's Book of Brain-benders (a smaller version of the above) (same as above) Sudoku: 100 Fun Number Puzzles (2005), ISBN 0-439-84570-X Kakuro and Other Fiendish Number Puzzles (2006), ISBN 0-439-95164-X One title that covers many different areas of mathematics has also been released: The Most Epic Book of Maths Ever (previously The Murderous Maths of Everything) (2010), ISBN 1-407-10367-9 (prime numbers, Sieve of Eratosthenes, Pythagoras' Theorem, triangular numbers, square numbers, the International Date Line, geometry, geometric constructions, topology, Möbius strips, curves (conic sections and cycloids Golomb Rulers, four-dimensional "Tic Tac Toe", The Golden Ratio, Fibonacci sequence, Logarithmic spirals, musical ratios, Theorems (including Ham sandwich theorem and Fixed point theorem), probability (cards, dice, cluedo etc.), Pascal's Triangle, Sierpinski Triangle, chess board, light years, size and distance of moon and planets, orbit, size of stars, shape of galaxy.) Kjartan has also written a book entitled Everyday Maths for Grown-Ups (2011). == Reviews == A recommendation of the series by Scientific American includes a quote from a Stanford engineer named Stacy F. Bennet, who described the series as "very humorous and engaging introductions to such topics as algebra, geometry and probability". On 22 November 1997, that same publication said of the series, "Have a look at Murderous Maths by Kjartan Poskitt. It is a truly addictive reading book, and was leapt on and devoured by my children. The book is full of awful jokes, fascinating facts, real murders and yes, the maths is good too. This is a brilliant book." The Primary Times released a review of Professor Fiendish's Book of Diabolical Brain-benders on November 25, 2002, describing the title as "intriguing, fun to do, and not at all dry", and adding "I warn you, once you start, you'll be 'hooked'!". The Times Educational Supplement also published a review on the book on December 6, 2002, describing the title as being "action-packed" and reasoning that "behind the non-stop fun, serious mathematical principles are being investigated". Kjartan did a presentation for 350 kids and 10 teachers at Wolfreton School, Hull in June 2004. Reporter Linda Blackbourne described it as a "stand-up maths routine [that] has children - and teachers - in fits of laughter". Carousel issue 16 (the guide to children's books) commented on the event: "...he possesses a prodigious gift of the (Yorkshire) gab, appears to be incapable of not enjoying himself, and plays his audience with the finesse of a maestro. Maths will never seem the same again". The Times Educational Supplement described Murderous Maths as "A stand-up maths routine has children and teachers in fits of laughter... maths has never been so much fun". The Western Gazette said: "It is not often that you see a grown maths teacher cry with laughter...", while The Worthing Gazette said: "The kids went wild, they absolutely loved it...". The Stockton Evening Gazette said: "Headteacher Barry Winter said it was a stroke of genius inviting the quick-witted author to open the resource centre". The GCSE book in the Guardian said: "Those who have experienced Poskitt "live" will recognise his commitment to getting readers involved with the learning process" (Nov 6th 2001), and The Press (York) described it as "...charismatic..." A review by science writer Brian Clegg described his views on Murderous Maths: Desperate Measures: It's the usual clever mix of light historical context − mostly ancient from Israelites and Archimedes to the Romans − and real insights into fascinating aspects of something that sits nicely between maths and practical science. There's plenty to keep the reader and interested, and even adults perusing it will have one or two surprises along the way. Because it is very much applied maths, there is also a lot more opportunity to have fun with practical things to try out than has been the case with some of the Murderous Maths series. All in all this is a great addition to the fold. == Spin-offs == Killer Puzzles (Written by Kjartan Poskitt) The Urgum The Axeman series (by Kjartan Poskitt and illustrated by Philip Reeve) == See also == Horrible Histories Horrible Science == References == == External links == The official Murderous Maths website Horrible Books and Magazines United States archived Daily Telegraph article
Wikipedia:Muriel Kennett Wales#0	Muriel Kennett Wales (9 Jun 1913 – 8 August 2009) was an Irish-Canadian mathematician, and is believed to have been the first Irish-born woman to earn a PhD in pure mathematics. == Life == She was born Muriel Kennett on 9 June 1913 in Belfast. In 1914, her mother moved to Vancouver, British Columbia, and soon remarried; henceforth Muriel was known by her mother's new last name, Wales. She was first educated at the University of British Columbia (BA 1934, MA 1937 with the thesis Determination of Bases for Certain Quartic Number Fields). In 1941 she was awarded the PhD from the University of Toronto for the dissertation Theory Of Algebraic Functions Based On The Use Of Cycles under Samuel Beatty (himself the first person to receive a PhD in mathematics in Canada, in 1915). She spent most of the 1940s working in atomic energy, in Toronto and Montreal, but by 1949 had retired back to Vancouver where she worked in her step-father's shipping company. == References == == External links == Muriel Kennett Wales at the Mathematics Genealogy Project
Wikipedia:Murray S. Klamkin#0	Murray Seymour Klamkin (March 5, 1921 – August 6, 2004) was an American mathematician, known as prolific proposer and editor of professionally-challenging mathematical problems. == Life == Klamkin was born on March 5, 1921, in Brooklyn, New York. He received a bachelor's degree from the Cooper Union in 1942 and, after four years of service in the United States Army, earned a master's degree from the Polytechnic Institute of Brooklyn in 1947, where he taught from 1948 until 1957. After this, Klamkin worked at AVCO, taught at SUNY Buffalo (1962–1964), and served as the principal research scientist at Ford Motor Company (1965–1976). During this period, he was also a visiting professor at the University of Minnesota. After leaving Ford, he became a professor at the University of Waterloo. From 1976 to 1981 Klamkin was the chairman of the Department of Mathematics at the University of Alberta. After 1981 he became an emeritus professor at Alberta. Klamkin died August 6, 2004. == Mathematical problems == Klamkin was known worldwide as a prolific proposer and editor of professionally challenging mathematical problems. He served as problem editor for SIAM Review, the American Mathematical Monthly, Math Horizons, and other journals. He was also known for his work in high-level mathematics competitions, such as the USA Mathematical Olympiad, the International Mathematical Olympiad, and the Putnam Competition. In 1988 the Mathematical Association of America gave him its Award for Distinguished Service to Mathematics, its highest service award. In 1992, the World Federation of National Mathematics Competitions awarded Klamkin the David Hilbert Award for his contributions to mathematics competitions. == Selected bibliography == "The asymmetric propeller", Leon Bankoff, Paul Erdös, and Murray S. Klamkin, Mathematics Magazine 46, #5 (November 1973), pp. 270–272. International Mathematical Olympiads, 1978–1985 and Forty Supplementary Problems, ed. Murray S. Klamkin, pub. Washington, DC: Mathematical Association of America, 1986. ISBN 0-88385-631-X. Mathematical Modelling: Classroom Notes in Applied Mathematics, ed. Murray S. Klamkin, pub. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1987. ISBN 0-89871-204-1. U.S.A. Mathematical Olympiads, 1972–1986, ed. Murray S. Klamkin, pub. Washington, DC: Mathematical Association of America, 1988. ISBN 0-88385-634-4. Problems in Applied Mathematics: Selections from SIAM Review, ed. Murray S. Klamkin, pub. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1990. ISBN 0-89871-259-9. Five Hundred Mathematical Challenges, Edward J. Barbeau, Murray S. Klamkin, and William O. J. Moser, pub. Washington, DC: Mathematical Association of America, 1995. ISBN 0-88385-519-4. Liu, Andy; Shawyer, Bruce, eds. (2008). Problems from Murray Klamkin. Washington: Mathematical Association of America. ISBN 978-0-88385-828-8. A collection of problems posed by Klamkin in Crux Mathematicorum with Mathematical Mayhem. On cooking a roast, Murray S. Klamkin. SIAM Review 3.2 (1961): 167–169. == See also == List of University of Waterloo people == References ==
Wikipedia:MyMaths#0	MyMaths is a subscription-based mathematics website which can be used on interactive whiteboards or by students and teachers at home. It is owned and operated by Oxford University Press, who acquired the site in 2011. As of February 2021, MyMaths has over 4 million student users in over 70 countries worldwide. == Usage and Cost == MyMaths operates a subscription model, where schools must pay to access the service. There is a cost of £392 for primary schools or £695 for secondary schools, per annum and not including VAT. Limited resources are available as a free trial. Schools receive an institution username and password, allowing students to access content on the site, and can set up profiles for individual students, enabling teachers to track the progress and grades achieved on homework. == Content == MyMaths has a wide range of curriculum materials and resources aimed at students in primary and secondary schools, covering content from KS1 foundations to A-Level Further Mathematics. However, it does not cover all topics. Each topic consists of a 'Lesson' which teaches the methods and provides interactive examples, as well as an "Online homework" task which provides and automatically marks practice questions. "Booster packs" for revision purposes and simplistic games are also available. Whilst MyMaths only provides content for UK examinations, the sister site MyiMaths provides content for international qualifications. == Flash == MyMaths was originally constructed in Flash, but most content has now been translated into HTML or other standards as of January 2021. 28 games and some other content was replaced or deleted after widespread support for Flash was ended at the end of 2020. == 'Hacks' == Several websites claim to offer 'hacks' to easily access answers on the site, although these merely involve opening several tabs or checking client side source code. A popular excuse amongst students completing homework was that they had forgotten to click the 'Checkout' tab, which resulted in the results of supposedly completed work not being saved. == Impact == An impact study undertaken by Oxford University Press — the owners of the website — conducted 22 interviews with teachers, finding that using MyMaths saved teachers between 15 minutes and 5 hours a week, with an average of around 2 hours. == References == == External links == Official website
Wikipedia:Mykola Chaikovsky#0	Mykola Chaikovsky or Chaykovskyi (Ukrainian: Чайковський Микола Андрійович; 2 January 1887 – 7 October 1970) was a Ukrainian teacher, mathematician and writer. In 1918 he wrote one of the first works of Ukrainian science fiction (За силу сонця, Za syly sontsia, By the Power of the Sun). == Biography == He was born in Berezhany, Galicia on 2 January 1887. He was the first son of lawyer and writer, Andrii Chaikovsky. He studied in the local gymnasium (the Berezhany Gymnasium) from 1897 to 1905. Next, he entered the in University of Prague, where he studied mechanics and philosophy. After about two years, he moved to the University of Vienna, studying philosophy, although he was increasingly interested in mathematics. He graduated from Vienna university. receiving a PhD degree in 1911. In 1913 he became a full member of the Shevchenko Scientific Society. From 1910 he briefly worked a Ukrainian school in Ternopil. In 1912 he got married. He also briefly worked at the University of Berlin and at a school in Austria. During World War I he was a translator and educator at a camp for Russian prisoners of war in Vienna, and also worked as a teacher at a real school in Rava-Ruska. From 1914 to 1918 he was involved with the Union for the Liberation of Ukraine, and spend some time in Freistadt, Germany. After World War I he became a teacher in Galicia (a Privatdozent at Kamianets-Podilskyi Ivan Ohiienko National University). He also held a number of other positions: a teacher at a girls' gymnasium, and a teacher at a private university in Lviv. From 1924 he was a director of a private gymnasium in Yavoriv, and in 1927 - director of another gymnasium in Rohatyn. In 1923 or 1929 (sources vary) he became a professor at the a high school in Odessa (Institute of People's Education in Odesa - Одеський інститут народної освіти, ОІНО). In the 1930s he was persecuted by the Soviet authorities. In March 1933, he was arrested as a "Polish spy" with alleged ties to the Ukrainian Military Organization and sentenced to 10 years in a labor camp. He spend the next 10 years as a prison laborer in Karelia and Arkhangelsk, surviving the gruesome working conditions during the construction of the White Sea–Baltic Canal. After his release, he worked first Tomsk, then in Semipalatinsk (now Semey, Kazakhstan), where from 1944 to 1947 he was a teacher (or university professor, sources vary) at the pedagogical institute, and then at the Ural State Pedagogical University. In May 1954 he returned to Ukraine. In 1956 he was rehabilitated and became a professor at Lviv Pedagogical Institute, and from 1961 at Lviv University. He died in Lviv on 7 October 1970. He was buried at the Lychakiv Cemetery. == Works == He published his first article (an academic article on mathematics) in 1908. He wrote other academic articles, as well as textbooks on mathematics, and compiled a dictionary of mathematical terminology in Ukrainian (1924), In 1931 he published Ukrainian Scientific Mathematical Bibliography (1931). In 1915 he published a book on solar and lunar eclipses. He wrote 51 articles for the Ukrainian Soviet Encyclopedia. His bibliography includes 193 scientific works in mathematics. As a scientist, his research concerned the topic of geometrization of school algebra. In 1918 he wrote one of the first works of Ukrainian science fiction, За силу сонця (Za syly sontsia, By the Power of the Sun). It was published by the Ukrainian Pedagogical Society seven years later, in Lviv (then Poland). In it, Chaikovskyi imagined Ukraine as a world pioneer in solar power, with an intrigue about spies from other countries trying to steal Ukrainian technology. It has been described as the first Ukrainian-language work of science fiction. The work is also notable as one of the first works that described the concept of a radiotelephone. The novel did not became well known in Ukraine, however; as it was published in Poland, therefore it was neither republished nor mentioned in literary criticism in the Soviet Union, and was effectively sidelined during the formative stage of Ukrainian science fiction in the 1920s and 1930s. In 1926, he published an future studies essay entitled “Technology of Tomorrow”, also in Lviv. == Remembrance == Parliament of Ukraine (Verkhovna Rada) declared January 2, 2017, to be celebrated as the 130th anniversary of his birth. == References == == External links == Biography on fantlab.ru (in Russian)
Wikipedia:Mykola Polyakov#0	Mykola Polyakov (ukr. Микола Вікторович Поляков; 1 May 1946 – 21 September 2020) was a Ukrainian scientist and rector of Dnipropetrovsk National University. == Biography == Mykola Polyakov was born on 1 May 1946 in Dnipropetrovsk, USSR. In 1971 Polyakov graduated from Dnipropetrovsk National University specializing in mathematics. In 1974 he obtained his Candidate of Sciences (PhD) degree, in 1983 – Doctor of Sciences (Habilitation) degree. During his student years, Polyakov was a Komsomol activist and a member of the Communist Party. Between 1989 and 1996 he was the dean of the mechanics-mathematics faculty. Between 1996 and 1998 Polyakov was a vice-rector of the DNU. In 1998 he was promoted to rector. Polyakov was the author of more than 300 scientific articles, books and inventions. Mykola Polyakov was the chief-editor of the scientific magazine "The Bulletin of the Dnipropetrovsk National University". In 2010 Mykola Polyakov joined the Party of Regions. On October 31, 2010 he won the local election. He was a member of Dnipropetrovsk City Council. == External links == http://www.dnu.dp.ua/view/history http://dniprograd.org/ua/articles/sosiety/856 http://www.nbuv.gov.ua/portal/natural/tmekh/polyakov.html
Wikipedia:Mythily Ramaswamy#0	Mythily Ramaswamy (born 6 June 1954) is an Indian mathematician and professor in the Department of Mathematics at the TIFR Centre for Applicable Mathematics of the Tata Institute of Fundamental Research in Bangalore. Her research involves functional analysis and controllability of partial differential equations. == Education == Ramaswamy was born near Mumbai, to a banking family, but moved often to other parts of India as a child. She obtained her doctorate in 1990 from Pierre and Marie Curie University in Paris. Her dissertation, Sur des questions de symetrie dans des problemes elliptiques [On questions of symmetry in elliptic problems] was supervised by Henri Berestycki. == Recognition == Ramaswamy was the 2004 winner of the Kalpana Chawla Award of the Karnataka State Council for Science and Technology, "given to a young woman scientist for achievements in the field of science and technology". She was elected to the Indian Academy of Sciences in 2007. She became a Fulbright Scholar for 2016–2017, funding her to visit Michael Renardy at Virginia Tech. == References ==
Wikipedia:Márta Svéd#0	Márta Svéd (1910 – 30 September 2005) was a Hungarian mathematician who was a teacher of mathematics at the University of Adelaide after moving to Australia in the 1930s. She was 75 years old when she completed her PhD in 1985. She wrote the textbook Journey into Geometries (1991), and won the BH Neumann Award in 1994 for her contributions to mathematics learning in Australia. == Early life == Márta Svéd was in the same high school class in Budapest as Esther Klein. She became interested in mathematics through Középiskolai Matematikai Lapok (KöMaL), a Hungarian magazine for high school mathematicians, and through its problem-solving column, where Paul Erdős was also a regular solver. She took third place in her year's offering of the Hungarian national high school mathematics competition, ahead of Pál Turán but behind her future husband, civil engineer George Svéd. Due to the restrictions placed on Jews in Hungary in the late 1920s, only two students from their class could study science or mathematics at the university in Budapest; Márta took the mathematics position, and Klein studied physics instead. == Later life == Svéd and her husband moved to Australia in 1939 and had one son and one daughter. She became the head of the mathematics department at Wilderness School, a private Adelaide high school for girls, and in the same year helped found Australia's first high school mathematics magazine. Her old friend Klein, meanwhile, had married mathematician George Szekeres and escaped Europe for Shanghai; after World War II, the Szekeres and Svéd families shared a small apartment in Adelaide. Svéd died on 30 September 2005, not long after the death of her friends, George and Esther Szekeres, who died within an hour of each other. == Contributions == In 1985, Svéd completed a PhD at the University of Adelaide, at age 75. Her dissertation, On finite linear and Baer structures, concerned finite geometry, and was supervised by Rey Casse. Her 1991 book, Journey into Geometries (MAA Spectrum), has been described by reviewer David A. Thomas as an "Alice-in-Wonderland-type journey into non-Euclidean geometry", written in a conversational style. Svéd's posthumously-published book Two Lives and a Bonus (Peacock Publications, 2006) documents her early life in Budapest. == Awards and honours == Svéd won the BH Neumann Award of the Australian Mathematics Trust in 1994. The award citation credited her in particular for the flourishing of mathematics competitions in Australia and the success of Australia in international mathematics competitions. The University of Adelaide offers a scholarship for women mathematicians named in memory of Svéd. == References ==
Wikipedia:Márton Balázs#0	Márton Balázs (July 17, 1929 – April 13, 2016) was a Romanian mathematician of Hungarian descent. He was born in Lueta, Odorhei County (now Harghita County), Romania. After graduating from high school in Odorheiu Secuiesc, he got his undergraduate degree a in mathematics and physics from Bolyai University of Cluj-Napoca. He began his professional career there at the Department of General Physics, then continued at the Department of Geometry. He continued his university activities at the merged university, named Babeș-Bolyai University and he received his doctorate degree in 1968. From 1972, he was a lecturer (associate professor) at the Department of Function Theory and Analysis, and from 1990, a university professor. From 1990 to 1992, he was deputy dean. He retired in 1994. His scientific work falls within the field of analysis and numerical analysis (solving equations in abstract spaces). He has published numerous papers in Romanian, English, and French in Romanian and foreign scientific journals. He is the author of several university notes. He was the editor-in-chief of Matematikai Lapok (Mathematical Gazette for high school students) for several years, and for a decade and a half he was the head of the Romanian-language scientific seminar on Mathematics Didactics. == Notes == == References == Erdélyi magyar ki kicsoda (Hungarian Who's Who from Transylvania). Nagyvárad (Oradea, Romania), Ed. RMDSZ–BMC. 2010. p. 61. ISBN 978-973-00725-6-3 ZbMATH profile
Wikipedia:Möbius transformation#0	In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f ( z ) = a z + b c z + d {\displaystyle f(z)={\frac {az+b}{cz+d}}} of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0. Geometrically, a Möbius transformation can be obtained by first applying the inverse stereographic projection from the plane to the unit sphere, moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2, C). Together with its subgroups, it has numerous applications in mathematics and physics. Möbius geometries and their transformations generalize this case to any number of dimensions over other fields. Möbius transformations are named in honor of August Ferdinand Möbius; they are an example of homographies, linear fractional transformations, bilinear transformations, and spin transformations (in relativity theory). == Overview == Möbius transformations are defined on the extended complex plane C ^ = C ∪ { ∞ } {\displaystyle {\widehat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}} (i.e., the complex plane augmented by the point at infinity). Stereographic projection identifies C ^ {\displaystyle {\widehat {\mathbb {C} }}} with a sphere, which is then called the Riemann sphere; alternatively, C ^ {\displaystyle {\widehat {\mathbb {C} }}} can be thought of as the complex projective line C P 1 {\displaystyle \mathbb {C} \mathbb {P} ^{1}} . The Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they are the automorphisms of C P 1 {\displaystyle \mathbb {C} \mathbb {P} ^{1}} as an algebraic variety. Therefore, the set of all Möbius transformations forms a group under composition. This group is called the Möbius group, and is sometimes denoted Aut ⁡ ( C ^ ) {\displaystyle \operatorname {Aut} ({\widehat {\mathbb {C} }})} . The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds. In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory. Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). A particularly important discrete subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations. Möbius transformations can be more generally defined in spaces of dimension n > 2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to Liouville's theorem a Möbius transformation can be expressed as a composition of translations, similarities, orthogonal transformations and inversions. == Definition == The general form of a Möbius transformation is given by f ( z ) = a z + b c z + d , {\displaystyle f(z)={\frac {az+b}{cz+d}},} where a, b, c, d are any complex numbers that satisfy ad − bc ≠ 0. In case c ≠ 0, this definition is extended to the whole Riemann sphere by defining f ( − d c ) = ∞ , f ( ∞ ) = a c . {\displaystyle {\begin{aligned}f\left({\frac {-d}{c}}\right)&=\infty ,\\f(\infty )&={\frac {a}{c}}.\end{aligned}}} If c = 0, we define f ( ∞ ) = ∞ . {\displaystyle f(\infty )=\infty .} Thus a Möbius transformation is always a bijective holomorphic function from the Riemann sphere to the Riemann sphere. The set of all Möbius transformations forms a group under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The Möbius group is then a complex Lie group. The Möbius group is usually denoted Aut ⁡ ( C ^ ) {\displaystyle \operatorname {Aut} ({\widehat {\mathbb {C} }})} as it is the automorphism group of the Riemann sphere. If ad = bc, the rational function defined above is a constant (unless c = d = 0, when it is undefined): a z + b c z + d = a c = b d , {\displaystyle {\frac {az+b}{cz+d}}={\frac {a}{c}}={\frac {b}{d}},} where a fraction with a zero denominator is ignored. A constant function is not bijective and is thus not considered a Möbius transformation. An alternative definition is given as the kernel of the Schwarzian derivative. == Fixed points == Every non-identity Möbius transformation has two fixed points γ 1 , γ 2 {\displaystyle \gamma _{1},\gamma _{2}} on the Riemann sphere. The fixed points are counted here with multiplicity; the parabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity. === Determining the fixed points === The fixed points of the transformation f ( z ) = a z + b c z + d {\displaystyle f(z)={\frac {az+b}{cz+d}}} are obtained by solving the fixed point equation f(γ) = γ. For c ≠ 0, this has two roots obtained by expanding this equation to c γ 2 − ( a − d ) γ − b = 0 , {\displaystyle c\gamma ^{2}-(a-d)\gamma -b=0\ ,} and applying the quadratic formula. The roots are γ 1 , 2 = ( a − d ) ± ( a − d ) 2 + 4 b c 2 c = ( a − d ) ± Δ 2 c {\displaystyle \gamma _{1,2}={\frac {(a-d)\pm {\sqrt {(a-d)^{2}+4bc}}}{2c}}={\frac {(a-d)\pm {\sqrt {\Delta }}}{2c}}} with discriminant Δ = ( tr ⁡ H ) 2 − 4 det H = ( a + d ) 2 − 4 ( a d − b c ) , {\displaystyle \Delta =(\operatorname {tr} {\mathfrak {H}})^{2}-4\det {\mathfrak {H}}=(a+d)^{2}-4(ad-bc),} where the matrix H = ( a b c d ) {\displaystyle {\mathfrak {H}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}} represents the transformation. Parabolic transforms have coincidental fixed points due to zero discriminant. For c nonzero and nonzero discriminant the transform is elliptic or hyperbolic. When c = 0, the quadratic equation degenerates into a linear equation and the transform is linear. This corresponds to the situation that one of the fixed points is the point at infinity. When a ≠ d the second fixed point is finite and is given by γ = − b a − d . {\displaystyle \gamma =-{\frac {b}{a-d}}.} In this case the transformation will be a simple transformation composed of translations, rotations, and dilations: z ↦ α z + β . {\displaystyle z\mapsto \alpha z+\beta .} If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation: z ↦ z + β . {\displaystyle z\mapsto z+\beta .} === Topological proof === Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2: χ ( C ^ ) = 2. {\displaystyle \chi ({\hat {\mathbb {C} }})=2.} Firstly, the projective linear group PGL(2, K) is sharply 3-transitive – for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially dimension counting, as the group is 3-dimensional). Thus any map that fixes at least 3 points is the identity. Next, one can see by identifying the Möbius group with P G L ( 2 , C ) {\displaystyle \mathrm {PGL} (2,\mathbb {C} )} that any Möbius function is homotopic to the identity. Indeed, any member of the general linear group can be reduced to the identity map by Gauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the Lefschetz number of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic. By contrast, the projective linear group of the real projective line, PGL(2, R) need not fix any points – for example ( 1 + x ) / ( 1 − x ) {\displaystyle (1+x)/(1-x)} has no (real) fixed points: as a complex transformation it fixes ±i – while the map 2x fixes the two points of 0 and ∞. This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more. === Normal form === Möbius transformations are also sometimes written in terms of their fixed points in so-called normal form. We first treat the non-parabolic case, for which there are two distinct fixed points. Non-parabolic case: Every non-parabolic transformation is conjugate to a dilation/rotation, i.e., a transformation of the form z ↦ k z {\displaystyle z\mapsto kz} (k ∈ C) with fixed points at 0 and ∞. To see this define a map g ( z ) = z − γ 1 z − γ 2 {\displaystyle g(z)={\frac {z-\gamma _{1}}{z-\gamma _{2}}}} which sends the points (γ1, γ2) to (0, ∞). Here we assume that γ1 and γ2 are distinct and finite. If one of them is already at infinity then g can be modified so as to fix infinity and send the other point to 0. If f has distinct fixed points (γ1, γ2) then the transformation g f g − 1 {\displaystyle gfg^{-1}} has fixed points at 0 and ∞ and is therefore a dilation: g f g − 1 ( z ) = k z {\displaystyle gfg^{-1}(z)=kz} . The fixed point equation for the transformation f can then be written f ( z ) − γ 1 f ( z ) − γ 2 = k z − γ 1 z − γ 2 . {\displaystyle {\frac {f(z)-\gamma _{1}}{f(z)-\gamma _{2}}}=k{\frac {z-\gamma _{1}}{z-\gamma _{2}}}.} Solving for f gives (in matrix form): H ( k ; γ 1 , γ 2 ) = ( γ 1 − k γ 2 ( k − 1 ) γ 1 γ 2 1 − k k γ 1 − γ 2 ) {\displaystyle {\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\begin{pmatrix}\gamma _{1}-k\gamma _{2}&(k-1)\gamma _{1}\gamma _{2}\\1-k&k\gamma _{1}-\gamma _{2}\end{pmatrix}}} or, if one of the fixed points is at infinity: H ( k ; γ , ∞ ) = ( k ( 1 − k ) γ 0 1 ) . {\displaystyle {\mathfrak {H}}(k;\gamma ,\infty )={\begin{pmatrix}k&(1-k)\gamma \\0&1\end{pmatrix}}.} From the above expressions one can calculate the derivatives of f at the fixed points: f ′ ( γ 1 ) = k {\displaystyle f'(\gamma _{1})=k} and f ′ ( γ 2 ) = 1 / k . {\displaystyle f'(\gamma _{2})=1/k.} Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (k) of f as the characteristic constant of f. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant: H ( k ; γ 1 , γ 2 ) = H ( 1 / k ; γ 2 , γ 1 ) . {\displaystyle {\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\mathfrak {H}}(1/k;\gamma _{2},\gamma _{1}).} For loxodromic transformations, whenever \|k\| > 1, one says that γ1 is the repulsive fixed point, and γ2 is the attractive fixed point. For \|k\| < 1, the roles are reversed. Parabolic case: In the parabolic case there is only one fixed point γ. The transformation sending that point to ∞ is g ( z ) = 1 z − γ {\displaystyle g(z)={\frac {1}{z-\gamma }}} or the identity if γ is already at infinity. The transformation g f g − 1 {\displaystyle gfg^{-1}} fixes infinity and is therefore a translation: g f g − 1 ( z ) = z + β . {\displaystyle gfg^{-1}(z)=z+\beta \,.} Here, β is called the translation length. The fixed point formula for a parabolic transformation is then 1 f ( z ) − γ = 1 z − γ + β . {\displaystyle {\frac {1}{f(z)-\gamma }}={\frac {1}{z-\gamma }}+\beta .} Solving for f (in matrix form) gives H ( β ; γ ) = ( 1 + γ β − β γ 2 β 1 − γ β ) {\displaystyle {\mathfrak {H}}(\beta ;\gamma )={\begin{pmatrix}1+\gamma \beta &-\beta \gamma ^{2}\\\beta &1-\gamma \beta \end{pmatrix}}} Note that det H ( β ; γ ) = \| H ( β ; γ ) \| = det ( 1 + γ β − β γ 2 β 1 − γ β ) = 1 − γ 2 β 2 + γ 2 β 2 = 1 {\displaystyle \det {\mathfrak {H}}(\beta ;\gamma )=\|{\mathfrak {H}}(\beta ;\gamma )\|=\det {\begin{pmatrix}1+\gamma \beta &-\beta \gamma ^{2}\\\beta &1-\gamma \beta \end{pmatrix}}=1-\gamma ^{2}\beta ^{2}+\gamma ^{2}\beta ^{2}=1} If γ = ∞: H ( β ; ∞ ) = ( 1 β 0 1 ) {\displaystyle {\mathfrak {H}}(\beta ;\infty )={\begin{pmatrix}1&\beta \\0&1\end{pmatrix}}} Note that β is not the characteristic constant of f, which is always 1 for a parabolic transformation. From the above expressions one can calculate: f ′ ( γ ) = 1. {\displaystyle f'(\gamma )=1.} == Poles of the transformation == The point z ∞ = − d c {\textstyle z_{\infty }=-{\frac {d}{c}}} is called the pole of H {\displaystyle {\mathfrak {H}}} ; it is that point which is transformed to the point at infinity under ⁠ H {\displaystyle {\mathfrak {H}}} ⁠. The inverse pole Z ∞ = a c {\textstyle Z_{\infty }={\frac {a}{c}}} is that point to which the point at infinity is transformed. The point midway between the two poles is always the same as the point midway between the two fixed points: γ 1 + γ 2 = z ∞ + Z ∞ . {\displaystyle \gamma _{1}+\gamma _{2}=z_{\infty }+Z_{\infty }.} These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation. A transform H {\displaystyle {\mathfrak {H}}} can be specified with two fixed points γ1, γ2 and the pole z ∞ {\displaystyle z_{\infty }} . H = ( Z ∞ − γ 1 γ 2 1 − z ∞ ) , Z ∞ = γ 1 + γ 2 − z ∞ . {\displaystyle {\mathfrak {H}}={\begin{pmatrix}Z_{\infty }&-\gamma _{1}\gamma _{2}\\1&-z_{\infty }\end{pmatrix}},\;\;Z_{\infty }=\gamma _{1}+\gamma _{2}-z_{\infty }.} This allows us to derive a formula for conversion between k and z ∞ {\displaystyle z_{\infty }} given γ 1 , γ 2 {\displaystyle \gamma _{1},\gamma _{2}} : z ∞ = k γ 1 − γ 2 1 − k {\displaystyle z_{\infty }={\frac {k\gamma _{1}-\gamma _{2}}{1-k}}} k = γ 2 − z ∞ γ 1 − z ∞ = Z ∞ − γ 1 Z ∞ − γ 2 = a − c γ 1 a − c γ 2 , {\displaystyle k={\frac {\gamma _{2}-z_{\infty }}{\gamma _{1}-z_{\infty }}}={\frac {Z_{\infty }-\gamma _{1}}{Z_{\infty }-\gamma _{2}}}={\frac {a-c\gamma _{1}}{a-c\gamma _{2}}},} which reduces down to k = ( a + d ) + ( a − d ) 2 + 4 b c ( a + d ) − ( a − d ) 2 + 4 b c . {\displaystyle k={\frac {(a+d)+{\sqrt {(a-d)^{2}+4bc}}}{(a+d)-{\sqrt {(a-d)^{2}+4bc}}}}.} The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios λ 1 λ 2 {\textstyle {\frac {\lambda _{1}}{\lambda _{2}}}} of H {\displaystyle {\mathfrak {H}}} (compare the discussion in the preceding section about the characteristic constant of a transformation). Its characteristic polynomial is equal to det ( λ I 2 − H ) = λ 2 − tr ⁡ H λ + det H = λ 2 − ( a + d ) λ + ( a d − b c ) {\displaystyle \det(\lambda I_{2}-{\mathfrak {H}})=\lambda ^{2}-\operatorname {tr} {\mathfrak {H}}\,\lambda +\det {\mathfrak {H}}=\lambda ^{2}-(a+d)\lambda +(ad-bc)} which has roots λ i = ( a + d ) ± ( a − d ) 2 + 4 b c 2 = ( a + d ) ± ( a + d ) 2 − 4 ( a d − b c ) 2 = c γ i + d . {\displaystyle \lambda _{i}={\frac {(a+d)\pm {\sqrt {(a-d)^{2}+4bc}}}{2}}={\frac {(a+d)\pm {\sqrt {(a+d)^{2}-4(ad-bc)}}}{2}}=c\gamma _{i}+d\,.} == Simple Möbius transformations and composition == A Möbius transformation can be composed as a sequence of simple transformations. The following simple transformations are also Möbius transformations: f ( z ) = z + b ( a = 1 , c = 0 , d = 1 ) {\displaystyle f(z)=z+b\quad (a=1,c=0,d=1)} is a translation. f ( z ) = a z ( b = 0 , c = 0 , d = 1 ) {\displaystyle f(z)=az\quad (b=0,c=0,d=1)} is a combination of a homothety (uniform scaling) and a rotation. If \| a \| = 1 {\displaystyle \|a\|=1} then it is a rotation, if a ∈ R {\displaystyle a\in \mathbb {R} } then it is a homothety. f ( z ) = 1 / z ( a = 0 , b = 1 , c = 1 , d = 0 ) {\displaystyle f(z)=1/z\quad (a=0,b=1,c=1,d=0)} (inversion and reflection with respect to the real axis) === Composition of simple transformations === If c ≠ 0 {\displaystyle c\neq 0} , let: f 1 ( z ) = z + d / c {\displaystyle f_{1}(z)=z+d/c\quad } (translation by d/c) f 2 ( z ) = 1 / z {\displaystyle f_{2}(z)=1/z\quad } (inversion and reflection with respect to the real axis) f 3 ( z ) = b c − a d c 2 z {\displaystyle f_{3}(z)={\frac {bc-ad}{c^{2}}}z\quad } (homothety and rotation) f 4 ( z ) = z + a / c {\displaystyle f_{4}(z)=z+a/c\quad } (translation by a/c) Then these functions can be composed, showing that, if f ( z ) = a z + b c z + d , {\displaystyle f(z)={\frac {az+b}{cz+d}},} one has f = f 4 ∘ f 3 ∘ f 2 ∘ f 1 . {\displaystyle f=f_{4}\circ f_{3}\circ f_{2}\circ f_{1}.} In other terms, one has a z + b c z + d = a c + e z + d c , {\displaystyle {\frac {az+b}{cz+d}}={\frac {a}{c}}+{\frac {e}{z+{\frac {d}{c}}}},} with e = b c − a d c 2 . {\displaystyle e={\frac {bc-ad}{c^{2}}}.} This decomposition makes many properties of the Möbius transformation obvious. == Elementary properties == A Möbius transformation is equivalent to a sequence of simpler transformations. The composition makes many properties of the Möbius transformation obvious. === Formula for the inverse transformation === The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions g1, g2, g3, g4 such that each gi is the inverse of fi. Then the composition g 1 ∘ g 2 ∘ g 3 ∘ g 4 ( z ) = f − 1 ( z ) = d z − b − c z + a {\displaystyle g_{1}\circ g_{2}\circ g_{3}\circ g_{4}(z)=f^{-1}(z)={\frac {dz-b}{-cz+a}}} gives a formula for the inverse. === Preservation of angles and generalized circles === From this decomposition, we see that Möbius transformations carry over all non-trivial properties of circle inversion. For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilations and isometries (translation, reflection, rotation), which trivially preserve angles. Furthermore, Möbius transformations map generalized circles to generalized circles since circle inversion has this property. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity. Note that a Möbius transformation does not necessarily map circles to circles and lines to lines: it can mix the two. Even if it maps a circle to another circle, it does not necessarily map the first circle's center to the second circle's center. === Cross-ratio preservation === Cross-ratios are invariant under Möbius transformations. That is, if a Möbius transformation maps four distinct points z 1 , z 2 , z 3 , z 4 {\displaystyle z_{1},z_{2},z_{3},z_{4}} to four distinct points w 1 , w 2 , w 3 , w 4 {\displaystyle w_{1},w_{2},w_{3},w_{4}} respectively, then ( z 1 − z 3 ) ( z 2 − z 4 ) ( z 2 − z 3 ) ( z 1 − z 4 ) = ( w 1 − w 3 ) ( w 2 − w 4 ) ( w 2 − w 3 ) ( w 1 − w 4 ) . {\displaystyle {\frac {(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{2}-z_{3})(z_{1}-z_{4})}}={\frac {(w_{1}-w_{3})(w_{2}-w_{4})}{(w_{2}-w_{3})(w_{1}-w_{4})}}.} If one of the points z 1 , z 2 , z 3 , z 4 {\displaystyle z_{1},z_{2},z_{3},z_{4}} is the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of z 1 , z 2 , z 3 , ∞ {\displaystyle z_{1},z_{2},z_{3},\infty } is ( z 1 − z 3 ) ( z 2 − z 3 ) . {\displaystyle {\frac {(z_{1}-z_{3})}{(z_{2}-z_{3})}}.} The cross ratio of four different points is real if and only if there is a line or a circle passing through them. This is another way to show that Möbius transformations preserve generalized circles. === Conjugation === Two points z1 and z2 are conjugate with respect to a generalized circle C, if, given a generalized circle D passing through z1 and z2 and cutting C in two points a and b, (z1, z2; a, b) are in harmonic cross-ratio (i.e. their cross ratio is −1). This property does not depend on the choice of the circle D. This property is also sometimes referred to as being symmetric with respect to a line or circle. Two points z, z∗ are conjugate with respect to a line, if they are symmetric with respect to the line. Two points are conjugate with respect to a circle if they are exchanged by the inversion with respect to this circle. The point z∗ is conjugate to z when L is the line determined by the vector based upon eiθ, at the point z0. This can be explicitly given as z ∗ = e 2 i θ z − z 0 ¯ + z 0 . {\displaystyle z^{}=e^{2i\theta }\,{\overline {z-z_{0}}}+z_{0}.} The point z∗ is conjugate to z when C is the circle of a radius r, centered about z0. This can be explicitly given as z ∗ = r 2 z − z 0 ¯ + z 0 . {\displaystyle z^{}={\frac {r^{2}}{\overline {z-z_{0}}}}+z_{0}.} Since Möbius transformations preserve generalized circles and cross-ratios, they also preserve the conjugation. == Projective matrix representations == === Isomorphism between the Möbius group and PGL(2, C) === The natural action of PGL(2, C) on the complex projective line CP1 is exactly the natural action of the Möbius group on the Riemann sphere ==== Correspondance between the complex projective line and the Riemann sphere ==== Here, the projective line CP1 and the Riemann sphere are identified as follows: [ z 1 : z 2 ] ∼ z 1 z 2 . {\displaystyle [z_{1}:z_{2}]\ \thicksim {\frac {z_{1}}{z_{2}}}.} Here [z1:z2] are homogeneous coordinates on CP1; the point [1:0] corresponds to the point ∞ of the Riemann sphere. By using homogeneous coordinates, many calculations involving Möbius transformations can be simplified, since no case distinctions dealing with ∞ are required. ==== Action of PGL(2, C) on the complex projective line ==== Every invertible complex 2×2 matrix H = ( a b c d ) {\displaystyle {\mathfrak {H}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}} acts on the projective line as z = [ z 1 : z 2 ] ↦ w = [ w 1 : w 2 ] , {\displaystyle z=[z_{1}:z_{2}]\mapsto w=[w_{1}:w_{2}],} where ( w 1 w 2 ) = ( a b c d ) ( z 1 z 2 ) = ( a z 1 + b z 2 c z 1 + d z 2 ) . {\displaystyle {\begin{pmatrix}w_{1}\\w_{2}\end{pmatrix}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}z_{1}\\z_{2}\end{pmatrix}}={\begin{pmatrix}az_{1}+bz_{2}\\cz_{1}+dz_{2}\end{pmatrix}}.} The result is therefore w = [ w 1 : w 2 ] = [ a z 1 + b z 2 : c z 1 + d z 2 ] {\displaystyle w=[w_{1}:w_{2}]=[az_{1}+bz_{2}:cz_{1}+dz_{2}]} Which, using the above identification, corresponds to the following point on the Riemann sphere : w = [ a z 1 + b z 2 : c z 1 + d z 2 ] ∼ a z 1 + b z 2 c z 1 + d z 2 = a z 1 z 2 + b c z 1 z 2 + d . {\displaystyle w=[az_{1}+bz_{2}:cz_{1}+dz_{2}]\thicksim {\frac {az_{1}+bz_{2}}{cz_{1}+dz_{2}}}={\frac {a{\frac {z_{1}}{z_{2}}}+b}{c{\frac {z_{1}}{z_{2}}}+d}}.} ==== Equivalence with a Möbius transformation on the Riemann sphere ==== Since the above matrix is invertible if and only if its determinant ad − bc is not zero, this induces an identification of the action of the group of Möbius transformations with the action of PGL(2, C) on the complex projective line. In this identification, the above matrix H {\displaystyle {\mathfrak {H}}} corresponds to the Möbius transformation z ↦ a z + b c z + d . {\displaystyle z\mapsto {\frac {az+b}{cz+d}}.} This identification is a group isomorphism, since the multiplication of H {\displaystyle {\mathfrak {H}}} by a non zero scalar λ {\displaystyle \lambda } does not change the element of PGL(2, C), and, as this multiplication consists of multiplying all matrix entries by λ , {\displaystyle \lambda ,} this does not change the corresponding Möbius transformation. === Other groups === For any field K, one can similarly identify the group PGL(2, K) of the projective linear automorphisms with the group of fractional linear transformations. This is widely used; for example in the study of homographies of the real line and its applications in optics. If one divides H {\displaystyle {\mathfrak {H}}} by a square root of its determinant, one gets a matrix of determinant one. This induces a surjective group homomorphism from the special linear group SL(2, C) to PGL(2, C), with ± I {\displaystyle \pm I} as its kernel. This allows showing that the Möbius group is a 3-dimensional complex Lie group (or a 6-dimensional real Lie group), which is a semisimple and non-compact, and that SL(2,C) is a double cover of PSL(2, C). Since SL(2, C) is simply-connected, it is the universal cover of the Möbius group, and the fundamental group of the Möbius group is Z2. === Specifying a transformation by three points === Given a set of three distinct points z 1 , z 2 , z 3 {\displaystyle z_{1},z_{2},z_{3}} on the Riemann sphere and a second set of distinct points ⁠ w 1 , w 2 , w 3 {\displaystyle w_{1},w_{2},w_{3}} ⁠, there exists precisely one Möbius transformation f ( z ) {\displaystyle f(z)} with f ( z j ) = w j {\displaystyle f(z_{j})=w_{j}} for ⁠ j = 1 , 2 , 3 {\displaystyle j=1,2,3} ⁠. (In other words: the action of the Möbius group on the Riemann sphere is sharply 3-transitive.) There are several ways to determine f ( z ) {\displaystyle f(z)} from the given sets of points. ==== Mapping first to 0, 1, ∞ ==== It is easy to check that the Möbius transformation f 1 ( z ) = ( z − z 1 ) ( z 2 − z 3 ) ( z − z 3 ) ( z 2 − z 1 ) {\displaystyle f_{1}(z)={\frac {(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}}} with matrix H 1 = ( z 2 − z 3 − z 1 ( z 2 − z 3 ) z 2 − z 1 − z 3 ( z 2 − z 1 ) ) {\displaystyle {\mathfrak {H}}_{1}={\begin{pmatrix}z_{2}-z_{3}&-z_{1}(z_{2}-z_{3})\\z_{2}-z_{1}&-z_{3}(z_{2}-z_{1})\end{pmatrix}}} maps z 1 , z 2 and z 3 {\displaystyle z_{1},z_{2}{\text{ and }}z_{3}} to ⁠ 0 , 1 , and ∞ {\displaystyle 0,1,\ {\text{and}}\ \infty } ⁠, respectively. If one of the z j {\displaystyle z_{j}} is ∞ {\displaystyle \infty } , then the proper formula for H 1 {\displaystyle {\mathfrak {H}}_{1}} is obtained from the above one by first dividing all entries by z j {\displaystyle z_{j}} and then taking the limit ⁠ z j → ∞ {\displaystyle z_{j}\to \infty } ⁠. If H 2 {\displaystyle {\mathfrak {H}}_{2}} is similarly defined to map w 1 , w 2 , w 3 {\displaystyle w_{1},w_{2},w_{3}} to 0 , 1 , and ∞ , {\displaystyle 0,1,\ {\text{and}}\ \infty ,} then the matrix H {\displaystyle {\mathfrak {H}}} which maps z 1 , 2 , 3 {\displaystyle z_{1,2,3}} to w 1 , 2 , 3 {\displaystyle w_{1,2,3}} becomes H = H 2 − 1 H 1 . {\displaystyle {\mathfrak {H}}={\mathfrak {H}}_{2}^{-1}{\mathfrak {H}}_{1}.} The stabilizer of { 0 , 1 , ∞ } {\displaystyle \{0,1,\infty \}} (as an unordered set) is a subgroup known as the anharmonic group. ==== Explicit determinant formula ==== The equation w = a z + b c z + d {\displaystyle w={\frac {az+b}{cz+d}}} is equivalent to the equation of a standard hyperbola c w z − a z + d w − b = 0 {\displaystyle cwz-az+dw-b=0} in the ( z , w ) {\displaystyle (z,w)} -plane. The problem of constructing a Möbius transformation H ( z ) {\displaystyle {\mathfrak {H}}(z)} mapping a triple ( z 1 , z 2 , z 3 ) {\displaystyle (z_{1},z_{2},z_{3})} to another triple ( w 1 , w 2 , w 3 ) {\displaystyle (w_{1},w_{2},w_{3})} is thus equivalent to finding the coefficients a , b , c , d {\displaystyle a,b,c,d} of the hyperbola passing through the points ⁠ ( z i , w i ) {\displaystyle (z_{i},w_{i})} ⁠. An explicit equation can be found by evaluating the determinant \| z w z w 1 z 1 w 1 z 1 w 1 1 z 2 w 2 z 2 w 2 1 z 3 w 3 z 3 w 3 1 \| {\displaystyle {\begin{vmatrix}zw&z&w&1\\z_{1}w_{1}&z_{1}&w_{1}&1\\z_{2}w_{2}&z_{2}&w_{2}&1\\z_{3}w_{3}&z_{3}&w_{3}&1\end{vmatrix}}\,} by means of a Laplace expansion along the first row, resulting in explicit formulae, a = z 1 w 1 ( w 2 − w 3 ) + z 2 w 2 ( w 3 − w 1 ) + z 3 w 3 ( w 1 − w 2 ) , b = z 1 w 1 ( z 2 w 3 − z 3 w 2 ) + z 2 w 2 ( z 3 w 1 − z 1 w 3 ) + z 3 w 3 ( z 1 w 2 − z 2 w 1 ) , c = w 1 ( z 3 − z 2 ) + w 2 ( z 1 − z 3 ) + w 3 ( z 2 − z 1 ) , d = z 1 w 1 ( z 2 − z 3 ) + z 2 w 2 ( z 3 − z 1 ) + z 3 w 3 ( z 1 − z 2 ) {\displaystyle {\begin{aligned}a&=z_{1}w_{1}(w_{2}-w_{3})+z_{2}w_{2}(w_{3}-w_{1})+z_{3}w_{3}(w_{1}-w_{2}),\\[5mu]b&=z_{1}w_{1}(z_{2}w_{3}-z_{3}w_{2})+z_{2}w_{2}(z_{3}w_{1}-z_{1}w_{3})+z_{3}w_{3}(z_{1}w_{2}-z_{2}w_{1}),\\[5mu]c&=w_{1}(z_{3}-z_{2})+w_{2}(z_{1}-z_{3})+w_{3}(z_{2}-z_{1}),\\[5mu]d&=z_{1}w_{1}(z_{2}-z_{3})+z_{2}w_{2}(z_{3}-z_{1})+z_{3}w_{3}(z_{1}-z_{2})\end{aligned}}} for the coefficients a , b , c , d {\displaystyle a,b,c,d} of the representing matrix ⁠ H {\displaystyle {\mathfrak {H}}} ⁠. The constructed matrix H {\displaystyle {\mathfrak {H}}} has determinant equal to ⁠ ( z 1 − z 2 ) ( z 1 − z 3 ) ( z 2 − z 3 ) ( w 1 − w 2 ) ( w 1 − w 3 ) ( w 2 − w 3 ) {\displaystyle (z_{1}-z_{2})(z_{1}-z_{3})(z_{2}-z_{3})(w_{1}-w_{2})(w_{1}-w_{3})(w_{2}-w_{3})} ⁠, which does not vanish if the z j {\displaystyle z_{j}} resp. w j {\displaystyle w_{j}} are pairwise different thus the Möbius transformation is well-defined. If one of the points z j {\displaystyle z_{j}} or w j {\displaystyle w_{j}} is ⁠ ∞ {\displaystyle \infty } ⁠, then we first divide all four determinants by this variable and then take the limit as the variable approaches ⁠ ∞ {\displaystyle \infty } ⁠. == Subgroups of the Möbius group == If we require the coefficients a , b , c , d {\displaystyle a,b,c,d} of a Möbius transformation to be real numbers with ⁠ a d − b c = 1 {\displaystyle ad-bc=1} ⁠, we obtain a subgroup of the Möbius group denoted as PSL(2, R). This is the group of those Möbius transformations that map the upper half-plane H = {x + iy : y > 0} to itself, and is equal to the group of all biholomorphic (or equivalently: bijective, conformal and orientation-preserving) maps H → H. If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane H2, the Poincaré half-plane model, and PSL(2, R) is the group of all orientation-preserving isometries of H2 in this model. The subgroup of all Möbius transformations that map the open disk D = {z : \|z\| < 1} to itself consists of all transformations of the form f ( z ) = e i ϕ z + b b ¯ z + 1 {\displaystyle f(z)=e^{i\phi }{\frac {z+b}{{\bar {b}}z+1}}} with ϕ {\displaystyle \phi } ∈ R, b ∈ C and \|b\| < 1. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps D → D. By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the Poincaré disk model, and this group is the group of all orientation-preserving isometries of H2 in this model. Since both of the above subgroups serve as isometry groups of H2, they are isomorphic. A concrete isomorphism is given by conjugation with the transformation f ( z ) = z + i i z + 1 {\displaystyle f(z)={\frac {z+i}{iz+1}}} which bijectively maps the open unit disk to the upper half plane. Alternatively, consider an open disk with radius r, centered at r i. The Poincaré disk model in this disk becomes identical to the upper-half-plane model as r approaches ∞. A maximal compact subgroup of the Möbius group M {\displaystyle {\mathcal {M}}} is given by (Tóth 2002) M 0 := { z ↦ u z − v ¯ v z + u ¯ : \| u \| 2 + \| v \| 2 = 1 } , {\displaystyle {\mathcal {M}}_{0}:=\left\{z\mapsto {\frac {uz-{\bar {v}}}{vz+{\bar {u}}}}:\|u\|^{2}+\|v\|^{2}=1\right\},} and corresponds under the isomorphism M ≅ PSL ⁡ ( 2 , C ) {\displaystyle {\mathcal {M}}\cong \operatorname {PSL} (2,\mathbb {C} )} to the projective special unitary group PSU(2, C) which is isomorphic to the special orthogonal group SO(3) of rotations in three dimensions, and can be interpreted as rotations of the Riemann sphere. Every finite subgroup is conjugate into this maximal compact group, and thus these correspond exactly to the polyhedral groups, the point groups in three dimensions. Icosahedral groups of Möbius transformations were used by Felix Klein to give an analytic solution to the quintic equation in (Klein 1913); a modern exposition is given in (Tóth 2002). If we require the coefficients a, b, c, d of a Möbius transformation to be integers with ad − bc = 1, we obtain the modular group PSL(2, Z), a discrete subgroup of PSL(2, R) important in the study of lattices in the complex plane, elliptic functions and elliptic curves. The discrete subgroups of PSL(2, R) are known as Fuchsian groups; they are important in the study of Riemann surfaces. == Classification == In the following discussion we will always assume that the representing matrix H {\displaystyle {\mathfrak {H}}} is normalized such that ⁠ det H = a d − b c = 1 {\displaystyle \det {\mathfrak {H}}=ad-bc=1} ⁠. Non-identity Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic, with the hyperbolic ones being a subclass of the loxodromic ones. The classification has both algebraic and geometric significance. Geometrically, the different types result in different transformations of the complex plane, as the figures below illustrate. The four types can be distinguished by looking at the trace tr ⁡ H = a + d {\displaystyle \operatorname {tr} {\mathfrak {H}}=a+d} . The trace is invariant under conjugation, that is, tr G H G − 1 = tr H , {\displaystyle \operatorname {tr} \,{\mathfrak {GHG}}^{-1}=\operatorname {tr} \,{\mathfrak {H}},} and so every member of a conjugacy class will have the same trace. Every Möbius transformation can be written such that its representing matrix H {\displaystyle {\mathfrak {H}}} has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations H , H ′ {\displaystyle {\mathfrak {H}},{\mathfrak {H}}'} (both not equal to the identity transform) with det H = det H ′ = 1 {\displaystyle \det {\mathfrak {H}}=\det {\mathfrak {H}}'=1} are conjugate if and only if tr 2 ⁡ H = tr 2 ⁡ H ′ . {\displaystyle \operatorname {tr} ^{2}{\mathfrak {H}}=\operatorname {tr} ^{2}{\mathfrak {H}}'.} === Parabolic transforms === A non-identity Möbius transformation defined by a matrix H {\displaystyle {\mathfrak {H}}} of determinant one is said to be parabolic if tr 2 ⁡ H = ( a + d ) 2 = 4 {\displaystyle \operatorname {tr} ^{2}{\mathfrak {H}}=(a+d)^{2}=4} (so the trace is plus or minus 2; either can occur for a given transformation since H {\displaystyle {\mathfrak {H}}} is determined only up to sign). In fact one of the choices for H {\displaystyle {\mathfrak {H}}} has the same characteristic polynomial X2 − 2X + 1 as the identity matrix, and is therefore unipotent. A Möbius transform is parabolic if and only if it has exactly one fixed point in the extended complex plane C ^ = C ∪ { ∞ } {\displaystyle {\widehat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}} , which happens if and only if it can be defined by a matrix conjugate to ( 1 1 0 1 ) {\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}} which describes a translation in the complex plane. The set of all parabolic Möbius transformations with a given fixed point in C ^ {\displaystyle {\widehat {\mathbb {C} }}} , together with the identity, forms a subgroup isomorphic to the group of matrices { ( 1 b 0 1 ) ∣ b ∈ C } ; {\displaystyle \left\{{\begin{pmatrix}1&b\\0&1\end{pmatrix}}\mid b\in \mathbb {C} \right\};} this is an example of the unipotent radical of a Borel subgroup (of the Möbius group, or of SL(2, C) for the matrix group; the notion is defined for any reductive Lie group). === Characteristic constant === All non-parabolic transformations have two fixed points and are defined by a matrix conjugate to ( λ 0 0 λ − 1 ) {\displaystyle {\begin{pmatrix}\lambda &0\\0&\lambda ^{-1}\end{pmatrix}}} with the complex number λ not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex number k = λ2, called the characteristic constant or multiplier of the transformation. === Elliptic transforms === The transformation is said to be elliptic if it can be represented by a matrix H {\displaystyle {\mathfrak {H}}} of determinant 1 such that 0 ≤ tr 2 ⁡ H < 4. {\displaystyle 0\leq \operatorname {tr} ^{2}{\mathfrak {H}}<4.} A transform is elliptic if and only if \|λ\| = 1 and λ ≠ ±1. Writing λ = e i α {\displaystyle \lambda =e^{i\alpha }} , an elliptic transform is conjugate to ( cos ⁡ α − sin ⁡ α sin ⁡ α cos ⁡ α ) {\displaystyle {\begin{pmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{pmatrix}}} with α real. For any H {\displaystyle {\mathfrak {H}}} with characteristic constant k, the characteristic constant of H n {\displaystyle {\mathfrak {H}}^{n}} is kn. Thus, all Möbius transformations of finite order are elliptic transformations, namely exactly those where λ is a root of unity, or, equivalently, where α is a rational multiple of π. The simplest possibility of a fractional multiple means α = π/2, which is also the unique case of tr ⁡ H = 0 {\displaystyle \operatorname {tr} {\mathfrak {H}}=0} , is also denoted as a circular transform; this corresponds geometrically to rotation by 180° about two fixed points. This class is represented in matrix form as: ( 0 − 1 1 0 ) . {\displaystyle {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.} There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: 1 / z , {\displaystyle 1/z,} which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), 1 − z {\displaystyle 1-z} , which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and z / ( z − 1 ) {\displaystyle z/(z-1)} which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2). === Hyperbolic transforms === The transform is said to be hyperbolic if it can be represented by a matrix H {\displaystyle {\mathfrak {H}}} whose trace is real with tr 2 ⁡ H > 4. {\displaystyle \operatorname {tr} ^{2}{\mathfrak {H}}>4.} A transform is hyperbolic if and only if λ is real and λ ≠ ±1. === Loxodromic transforms === The transform is said to be loxodromic if tr 2 ⁡ H {\displaystyle \operatorname {tr} ^{2}{\mathfrak {H}}} is not in [0, 4]. A transformation is loxodromic if and only if \| λ \| ≠ 1 {\displaystyle \|\lambda \|\neq 1} . Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. See the geometric figures below. === General classification === === The real case and a note on terminology === Over the real numbers (if the coefficients must be real), there are no non-hyperbolic loxodromic transformations, and the classification is into elliptic, parabolic, and hyperbolic, as for real conics. The terminology is due to considering half the absolute value of the trace, \|tr\|/2, as the eccentricity of the transformation – division by 2 corrects for the dimension, so the identity has eccentricity 1 (tr/n is sometimes used as an alternative for the trace for this reason), and absolute value corrects for the trace only being defined up to a factor of ±1 due to working in PSL. Alternatively one may use half the trace squared as a proxy for the eccentricity squared, as was done above; these classifications (but not the exact eccentricity values, since squaring and absolute values are different) agree for real traces but not complex traces. The same terminology is used for the classification of elements of SL(2, R) (the 2-fold cover), and analogous classifications are used elsewhere. Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities. == Geometric interpretation of the characteristic constant == The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transformation in the non-parabolic case: The characteristic constant can be expressed in terms of its logarithm: e ρ + α i = k . {\displaystyle e^{\rho +\alpha i}=k.} When expressed in this way, the real number ρ becomes an expansion factor. It indicates how repulsive the fixed point γ1 is, and how attractive γ2 is. The real number α is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about γ1 and clockwise about γ2. === Elliptic transformations === If ρ = 0, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be elliptic. These transformations tend to move all points in circles around the two fixed points. If one of the fixed points is at infinity, this is equivalent to doing an affine rotation around a point. If we take the one-parameter subgroup generated by any elliptic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the same two points. All other points flow along a family of circles which is nested between the two fixed points on the Riemann sphere. In general, the two fixed points can be any two distinct points. This has an important physical interpretation. Imagine that some observer rotates with constant angular velocity about some axis. Then we can take the two fixed points to be the North and South poles of the celestial sphere. The appearance of the night sky is now transformed continuously in exactly the manner described by the one-parameter subgroup of elliptic transformations sharing the fixed points 0, ∞, and with the number α corresponding to the constant angular velocity of our observer. Here are some figures illustrating the effect of an elliptic Möbius transformation on the Riemann sphere (after stereographic projection to the plane): These pictures illustrate the effect of a single Möbius transformation. The one-parameter subgroup which it generates continuously moves points along the family of circular arcs suggested by the pictures. === Hyperbolic transformations === If α is zero (or a multiple of 2π), then the transformation is said to be hyperbolic. These transformations tend to move points along circular paths from one fixed point toward the other. If we take the one-parameter subgroup generated by any hyperbolic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the same two points. All other points flow along a certain family of circular arcs away from the first fixed point and toward the second fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere. This too has an important physical interpretation. Imagine that an observer accelerates (with constant magnitude of acceleration) in the direction of the North pole on his celestial sphere. Then the appearance of the night sky is transformed in exactly the manner described by the one-parameter subgroup of hyperbolic transformations sharing the fixed points 0, ∞, with the real number ρ corresponding to the magnitude of his acceleration vector. The stars seem to move along longitudes, away from the South pole toward the North pole. (The longitudes appear as circular arcs under stereographic projection from the sphere to the plane.) Here are some figures illustrating the effect of a hyperbolic Möbius transformation on the Riemann sphere (after stereographic projection to the plane): These pictures resemble the field lines of a positive and a negative electrical charge located at the fixed points, because the circular flow lines subtend a constant angle between the two fixed points. === Loxodromic transformations === If both ρ and α are nonzero, then the transformation is said to be loxodromic. These transformations tend to move all points in S-shaped paths from one fixed point to the other. The word "loxodrome" is from the Greek: "λοξος (loxos), slanting + δρόμος (dromos), course". When sailing on a constant bearing – if you maintain a heading of (say) north-east, you will eventually wind up sailing around the north pole in a logarithmic spiral. On the mercator projection such a course is a straight line, as the north and south poles project to infinity. The angle that the loxodrome subtends relative to the lines of longitude (i.e. its slope, the "tightness" of the spiral) is the argument of k. Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles. But any loxodromic transformation will be conjugate to a transform that moves all points along such loxodromes. If we take the one-parameter subgroup generated by any loxodromic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the same two points. All other points flow along a certain family of curves, away from the first fixed point and toward the second fixed point. Unlike the hyperbolic case, these curves are not circular arcs, but certain curves which under stereographic projection from the sphere to the plane appear as spiral curves which twist counterclockwise infinitely often around one fixed point and twist clockwise infinitely often around the other fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere. You can probably guess the physical interpretation in the case when the two fixed points are 0, ∞: an observer who is both rotating (with constant angular velocity) about some axis and moving along the same axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points 0, ∞, and with ρ, α determined respectively by the magnitude of the actual linear and angular velocities. === Stereographic projection === These images show Möbius transformations stereographically projected onto the Riemann sphere. Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different from having the fixed points in an arbitrary location. == Iterating a transformation == If a transformation H {\displaystyle {\mathfrak {H}}} has fixed points γ1, γ2, and characteristic constant k, then H ′ = H n {\displaystyle {\mathfrak {H}}'={\mathfrak {H}}^{n}} will have γ 1 ′ = γ 1 , γ 2 ′ = γ 2 , k ′ = k n {\displaystyle \gamma _{1}'=\gamma _{1},\gamma _{2}'=\gamma _{2},k'=k^{n}} . This can be used to iterate a transformation, or to animate one by breaking it up into steps. These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants. And these images demonstrate what happens when you transform a circle under Hyperbolic, Elliptical, and Loxodromic transforms. In the elliptical and loxodromic images, the value of α is 1/10. == Higher dimensions == In higher dimensions, a Möbius transformation is a homeomorphism of ⁠ R n ¯ {\displaystyle {\overline {\mathbb {R} ^{n}}}} ⁠, the one-point compactification of ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠, which is a finite composition of inversions in spheres and reflections in hyperplanes. Liouville's theorem in conformal geometry states that in dimension at least three, all conformal transformations are Möbius transformations. Every Möbius transformation can be put in the form f ( x ) = b + α A ( x − a ) \| x − a \| ε , {\displaystyle f(x)=b+{\frac {\alpha A(x-a)}{\|x-a\|^{\varepsilon }}},} where a , b ∈ R n {\displaystyle a,b\in \mathbb {R} ^{n}} , α ∈ R {\displaystyle \alpha \in \mathbb {R} } , A {\displaystyle A} is an orthogonal matrix, and ε {\displaystyle \varepsilon } is 0 or 2. The group of Möbius transformations is also called the Möbius group. The orientation-preserving Möbius transformations form the connected component of the identity in the Möbius group. In dimension n = 2, the orientation-preserving Möbius transformations are exactly the maps of the Riemann sphere covered here. The orientation-reversing ones are obtained from these by complex conjugation. The domain of Möbius transformations, i.e. ⁠ R n ¯ {\displaystyle {\overline {\mathbb {R} ^{n}}}} ⁠, is homeomorphic to the n-dimensional sphere S n {\displaystyle S^{n}} . The canonical isomorphism between these two spaces is the Cayley transform, which is itself a Möbius transformation of ⁠ R n + 1 ¯ {\displaystyle {\overline {\mathbb {R} ^{n+1}}}} ⁠. This identification means that Möbius transformations can also be thought of as conformal isomorphisms of S n {\displaystyle S^{n}} . The n-sphere, together with action of the Möbius group, is a geometric structure (in the sense of Klein's Erlangen program) called Möbius geometry. == Applications == === Lorentz transformation === An isomorphism of the Möbius group with the Lorentz group was noted by several authors: Based on previous work of Felix Klein (1893, 1897) on automorphic functions related to hyperbolic geometry and Möbius geometry, Gustav Herglotz (1909) showed that hyperbolic motions (i.e. isometric automorphisms of a hyperbolic space) transforming the unit sphere into itself correspond to Lorentz transformations, by which Herglotz was able to classify the one-parameter Lorentz transformations into loxodromic, elliptic, hyperbolic, and parabolic groups. Other authors include Emil Artin (1957), H. S. M. Coxeter (1965), and Roger Penrose, Wolfgang Rindler (1984), Tristan Needham (1997) and W. M. Olivia (2002). Minkowski space consists of the four-dimensional real coordinate space R4 consisting of the space of ordered quadruples (x0, x1, x2, x3) of real numbers, together with a quadratic form Q ( x 0 , x 1 , x 2 , x 3 ) = x 0 2 − x 1 2 − x 2 2 − x 3 2 . {\displaystyle Q(x_{0},x_{1},x_{2},x_{3})=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.} Borrowing terminology from special relativity, points with Q > 0 are considered timelike; in addition, if x0 > 0, then the point is called future-pointing. Points with Q < 0 are called spacelike. The null cone S consists of those points where Q = 0; the future null cone N+ are those points on the null cone with x0 > 0. The celestial sphere is then identified with the collection of rays in N+ whose initial point is the origin of R4. The collection of linear transformations on R4 with positive determinant preserving the quadratic form Q and preserving the time direction form the restricted Lorentz group SO+(1, 3). In connection with the geometry of the celestial sphere, the group of transformations SO+(1, 3) is identified with the group PSL(2, C) of Möbius transformations of the sphere. To each (x0, x1, x2, x3) ∈ R4, associate the hermitian matrix X = [ x 0 + x 1 x 2 + i x 3 x 2 − i x 3 x 0 − x 1 ] . {\displaystyle X={\begin{bmatrix}x_{0}+x_{1}&x_{2}+ix_{3}\\x_{2}-ix_{3}&x_{0}-x_{1}\end{bmatrix}}.} The determinant of the matrix X is equal to Q(x0, x1, x2, x3). The special linear group acts on the space of such matrices via for each A ∈ SL(2, C), and this action of SL(2, C) preserves the determinant of X because det A = 1. Since the determinant of X is identified with the quadratic form Q, SL(2, C) acts by Lorentz transformations. On dimensional grounds, SL(2, C) covers a neighborhood of the identity of SO(1, 3). Since SL(2, C) is connected, it covers the entire restricted Lorentz group SO+(1, 3). Furthermore, since the kernel of the action (1) is the subgroup {±I}, then passing to the quotient group gives the group isomorphism Focusing now attention on the case when (x0, x1, x2, x3) is null, the matrix X has zero determinant, and therefore splits as the outer product of a complex two-vector ξ with its complex conjugate: The two-component vector ξ is acted upon by SL(2, C) in a manner compatible with (1). It is now clear that the kernel of the representation of SL(2, C) on hermitian matrices is {±I}. The action of PSL(2, C) on the celestial sphere may also be described geometrically using stereographic projection. Consider first the hyperplane in R4 given by x0 = 1. The celestial sphere may be identified with the sphere S+ of intersection of the hyperplane with the future null cone N+. The stereographic projection from the north pole (1, 0, 0, 1) of this sphere onto the plane x3 = 0 takes a point with coordinates (1, x1, x2, x3) with x 1 2 + x 2 2 + x 3 2 = 1 {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1} to the point ( 1 , x 1 1 − x 3 , x 2 1 − x 3 , 0 ) . {\displaystyle \left(1,{\frac {x_{1}}{1-x_{3}}},{\frac {x_{2}}{1-x_{3}}},0\right).} Introducing the complex coordinate ζ = x 1 + i x 2 1 − x 3 , {\displaystyle \zeta ={\frac {x_{1}+ix_{2}}{1-x_{3}}},} the inverse stereographic projection gives the following formula for a point (x1, x2, x3) on S+: The action of SO+(1, 3) on the points of N+ does not preserve the hyperplane S+, but acting on points in S+ and then rescaling so that the result is again in S+ gives an action of SO+(1, 3) on the sphere which goes over to an action on the complex variable ζ. In fact, this action is by fractional linear transformations, although this is not easily seen from this representation of the celestial sphere. Conversely, for any fractional linear transformation of ζ variable goes over to a unique Lorentz transformation on N+, possibly after a suitable (uniquely determined) rescaling. A more invariant description of the stereographic projection which allows the action to be more clearly seen is to consider the variable ζ = z:w as a ratio of a pair of homogeneous coordinates for the complex projective line CP1. The stereographic projection goes over to a transformation from C2 − {0} to N+ which is homogeneous of degree two with respect to real scalings which agrees with (4) upon restriction to scales in which z z ¯ + w w ¯ = 1. {\displaystyle z{\bar {z}}+w{\bar {w}}=1.} The components of (5) are precisely those obtained from the outer product [ x 0 + x 1 x 2 + i x 3 x 2 − i x 3 x 0 − x 1 ] = 2 [ z w ] [ z ¯ w ¯ ] . {\displaystyle {\begin{bmatrix}x_{0}+x_{1}&x_{2}+ix_{3}\\x_{2}-ix_{3}&x_{0}-x_{1}\end{bmatrix}}=2{\begin{bmatrix}z\\w\end{bmatrix}}{\begin{bmatrix}{\bar {z}}&{\bar {w}}\end{bmatrix}}.} In summary, the action of the restricted Lorentz group SO+(1,3) agrees with that of the Möbius group PSL(2, C). This motivates the following definition. In dimension n ≥ 2, the Möbius group Möb(n) is the group of all orientation-preserving conformal isometries of the round sphere Sn to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space R1,n+1, there is an isomorphism of Möb(n) with the restricted Lorentz group SO+(1,n+1) of Lorentz transformations with positive determinant, preserving the direction of time. Coxeter began instead with the equivalent quadratic form ⁠ Q ( x 1 , x 2 , x 3 x 4 ) = x 1 2 + x 2 2 + x 3 2 − x 4 2 {\displaystyle Q(x_{1},\ x_{2},\ x_{3}\ x_{4})=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}} ⁠. He identified the Lorentz group with transformations for which {x \| Q(x) = −1} is stable. Then he interpreted the x's as homogeneous coordinates and {x \| Q(x) = 0}, the null cone, as the Cayley absolute for a hyperbolic space of points {x \| Q(x) < 0}. Next, Coxeter introduced the variables ξ = x 1 x 4 , η = x 2 x 4 , ζ = x 3 x 4 {\displaystyle \xi ={\frac {x_{1}}{x_{4}}},\ \eta ={\frac {x_{2}}{x_{4}}},\ \zeta ={\frac {x_{3}}{x_{4}}}} so that the Lorentz-invariant quadric corresponds to the sphere ⁠ ξ 2 + η 2 + ζ 2 = 1 {\displaystyle \xi ^{2}+\eta ^{2}+\zeta ^{2}=1} ⁠. Coxeter notes that Felix Klein also wrote of this correspondence, applying stereographic projection from (0, 0, 1) to the complex plane z = ξ + i η 1 − ζ . {\textstyle z={\frac {\xi +i\eta }{1-\zeta }}.} Coxeter used the fact that circles of the inversive plane represent planes of hyperbolic space, and the general homography is the product of inversions in two or four circles, corresponding to the general hyperbolic displacement which is the product of inversions in two or four planes. === Hyperbolic space === As seen above, the Möbius group PSL(2, C) acts on Minkowski space as the group of those isometries that preserve the origin, the orientation of space and the direction of time. Restricting to the points where Q = 1 in the positive light cone, which form a model of hyperbolic 3-space H3, we see that the Möbius group acts on H3 as a group of orientation-preserving isometries. In fact, the Möbius group is equal to the group of orientation-preserving isometries of hyperbolic 3-space. If we use the Poincaré ball model, identifying the unit ball in R3 with H3, then we can think of the Riemann sphere as the "conformal boundary" of H3. Every orientation-preserving isometry of H3 gives rise to a Möbius transformation on the Riemann sphere and vice versa. == See also == == Notes == == References == Specific General == Further reading == Lawson, M. V. (1998). "The Möbius Inverse Monoid". Journal of Algebra. 200 (2): 428. doi:10.1006/jabr.1997.7242. == External links == "Quasi-conformal mapping", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Conformal maps gallery Weisstein, Eric W. "Linear Fractional Transformation". MathWorld.
Wikipedia:N-ary associativity#0	In algebra, n-ary associativity is a generalization of the associative law to n-ary operations. A ternary operation is ternary associative if one has always ( a b c ) d e = a ( b c d ) e = a b ( c d e ) ; {\displaystyle (abc)de=a(bcd)e=ab(cde);} that is, the operation gives the same result when any three adjacent elements are bracketed inside a sequence of five operands. Similarly, an n-ary operation is n-ary associative if bracketing any n adjacent elements in a sequence of n + (n − 1) operands do not change the result. == References ==
Wikipedia:N-flake#0	An n-flake, polyflake, or Sierpinski n-gon,: 1 is a fractal constructed starting from an n-gon. This n-gon is replaced by a flake of smaller n-gons, such that the scaled polygons are placed at the vertices, and sometimes in the center. This process is repeated recursively to result in the fractal. Typically, there is also the restriction that the n-gons must touch yet not overlap. == In two dimensions == The most common variety of n-flake is two-dimensional (in terms of its topological dimension) and is formed of polygons. The four most common special cases are formed with triangles, squares, pentagons, and hexagons, but it can be extended to any polygon.: 2 Its boundary is the von Koch curve of varying types – depending on the n-gon – and infinitely many Koch curves are contained within. The fractals occupy zero area yet have an infinite perimeter. The formula of the scale factor r for any n-flake is: r = 1 2 ( 1 + ∑ k = 1 ⌊ n / 4 ⌋ cos ⁡ 2 π k n ) {\displaystyle r={\frac {1}{2\left(1+\displaystyle \sum _{k=1}^{\lfloor n/4\rfloor }{\cos {\frac {2\pi k}{n}}}\right)}}} where cosine is evaluated in radians and n is the number of sides of the n-gon. The Hausdorff dimension of a n-flake is log ⁡ m log ⁡ r {\displaystyle \textstyle {\frac {\log m}{\log r}}} , where m is the number of polygons in each individual flake and r is the scale factor. === Sierpinski triangle === The Sierpinski triangle is an n-flake formed by successive flakes of three triangles. Each flake is formed by placing triangles scaled by 1/2 in each corner of the triangle they replace. Its Hausdorff dimension is equal to log ⁡ ( 3 ) log ⁡ ( 2 ) {\displaystyle \textstyle {\frac {\log(3)}{\log(2)}}} ≈ 1.585. The log ⁡ ( 3 ) log ⁡ ( 2 ) {\displaystyle \textstyle {\frac {\log(3)}{\log(2)}}} is obtained because each iteration has 3 triangles that are scaled by 1/2. === Vicsek fractal === If a sierpinski 4-gon were constructed from the given definition, the scale factor would be 1/2 and the fractal would simply be a square. A more interesting alternative, the Vicsek fractal, rarely called a quadraflake, is formed by successive flakes of five squares scaled by 1/3. Each flake is formed either by placing a scaled square in each corner and one in the center or one on each side of the square and one in the center. Its Hausdorff dimension is equal to log ⁡ ( 5 ) log ⁡ ( 3 ) {\displaystyle \textstyle {\frac {\log(5)}{\log(3)}}} ≈ 1.4650. The log ⁡ ( 5 ) log ⁡ ( 3 ) {\displaystyle \textstyle {\frac {\log(5)}{\log(3)}}} is obtained because each iteration has 5 squares that are scaled by 1/3. The boundary of the Vicsek Fractal is a Type 1 quadratic Koch curve. === Pentaflake === A pentaflake, or sierpinski pentagon, is formed by successive flakes of six regular pentagons. Each flake is formed by placing a pentagon in each corner and one in the center. Its Hausdorff dimension is equal to log ⁡ ( 6 ) log ⁡ ( 1 + φ ) {\displaystyle \textstyle {\frac {\log(6)}{\log(1+\varphi )}}} ≈ 1.8617, where φ = 1 + 5 2 {\displaystyle \textstyle {\varphi ={\frac {1+{\sqrt {5}}}{2}}}} (golden ratio). The log ⁡ ( 6 ) log ⁡ ( 1 + φ ) {\displaystyle \textstyle {\frac {\log(6)}{\log(1+\varphi )}}} is obtained because each iteration has 6 pentagons that are scaled by 1 1 + φ {\displaystyle \textstyle {\frac {1}{1+\varphi }}} . The boundary of a pentaflake is the Koch curve of 72 degrees. There is also a variation of the pentaflake that has no central pentagon. Its Hausdorff dimension equals log ⁡ ( 5 ) log ⁡ ( 1 + φ ) {\displaystyle \textstyle {\frac {\log(5)}{\log(1+\varphi )}}} ≈ 1.6723. This variation still contains infinitely many Koch curves, but they are somewhat more visible. Concentric patterns of pentaflake boundary shaped tiles can cover the plane, with the central point being covered by a third shape formed of segments of 72-degree Koch curve, also with 5-fold rotational and reflective symmetry. === Hexaflake === A hexaflake, is formed by successive flakes of seven regular hexagons. Each flake is formed by placing a scaled hexagon in each corner and one in the center. Each iteration has 7 hexagons that are scaled by 1/3. Therefore the hexaflake has 7n−1 hexagons in its nth iteration, and its Hausdorff dimension is equal to log ⁡ ( 7 ) log ⁡ ( 3 ) {\displaystyle \textstyle {\frac {\log(7)}{\log(3)}}} ≈ 1.7712. The boundary of a hexaflake is the standard Koch curve of 60 degrees and infinitely many Koch snowflakes are contained within. Also, the projection of the cantor cube onto the plane orthogonal to its main diagonal is a hexaflake. The hexaflake has been applied in the design of antennas and optical fibers. Like the pentaflake, there is also a variation of the hexaflake, called the Sierpinski hexagon, that has no central hexagon. Its Hausdorff dimension equals log ⁡ ( 6 ) log ⁡ ( 3 ) {\displaystyle \textstyle {\frac {\log(6)}{\log(3)}}} ≈ 1.6309. This variation still contains infinitely many Koch curves of 60 degrees. === Polyflake === n-flakes of higher polygons also exist, though they are less common and usually do not have a central polygon. [If a central polygon is generated, the scale factor differs for odd and even n {\displaystyle n} : R = 1 − 2 r {\displaystyle R=1-2r} for even n {\displaystyle n} and R = 1 − r c o s ( π / n ) − r {\displaystyle \textstyle {R={\frac {1-r}{cos(\pi /n)}}-r}} for odd n {\displaystyle n} .] Some examples are shown below; the 7-flake through 12-flake. While it may not be obvious, these higher polyflakes still contain infinitely many Koch curves, but the angle of the Koch curves decreases as n increases. Their Hausdorff dimensions are slightly more difficult to calculate than lower n-flakes because their scale factor is less obvious. However, the Hausdorff dimension is always less than two but no less than one. An interesting n-flake is the ∞-flake, because as the value of n increases, an n-flake's Hausdorff dimension approaches 1,: 7 == In three dimensions == n-flakes can generalized to higher dimensions, in particular to a topological dimension of three. Instead of polygons, regular polyhedra are iteratively replaced. However, while there are an infinite number of regular polygons, there are only five regular, convex polyhedra. Because of this, three-dimensional n-flakes are also called platonic solid fractals. In three dimensions, the fractals' volume is zero. === Sierpinski tetrahedron === A Sierpinski tetrahedron is formed by successive flakes of four regular tetrahedrons. Each flake is formed by placing a tetrahedron scaled by 1/2 in each corner. Its Hausdorff dimension is equal to log ⁡ ( 4 ) log ⁡ ( 2 ) {\displaystyle \textstyle {\frac {\log(4)}{\log(2)}}} , which is exactly equal to 2. On every face there is a Sierpinski triangle and infinitely many are contained within. === Hexahedron flake === A hexahedron, or cube, flake defined in the same way as the Sierpinski tetrahedron is simply a cube and is not interesting as a fractal. However, there are two pleasing alternatives. One is the Menger Sponge, where every cube is replaced by a three dimensional ring of cubes. Its Hausdorff dimension is log ⁡ ( 20 ) log ⁡ ( 3 ) {\displaystyle \textstyle {\frac {\log(20)}{\log(3)}}} ≈ 2.7268. Another hexahedron flake can be produced in a manner similar to the Vicsek fractal extended to three dimensions. Every cube is divided into 27 smaller cubes and the center cross is retained, which is the opposite of the Menger sponge where the cross is removed. However, it is not the Menger Sponge complement. Its Hausdorff dimension is log ⁡ ( 7 ) log ⁡ ( 3 ) {\displaystyle \textstyle {\frac {\log(7)}{\log(3)}}} ≈ 1.7712, because a cross of 7 cubes, each scaled by 1/3, replaces each cube. === Octahedron flake === An octahedron flake, or sierpinski octahedron, is formed by successive flakes of six regular octahedra. Each flake is formed by placing an octahedron scaled by 1/2 in each corner. Its Hausdorff dimension is equal to log ⁡ ( 6 ) log ⁡ ( 2 ) {\displaystyle \textstyle {\frac {\log(6)}{\log(2)}}} ≈ 2.5849. On every face there is a Sierpinski triangle and infinitely many are contained within. === Dodecahedron flake === A dodecahedron flake, or sierpinski dodecahedron, is formed by successive flakes of twenty regular dodecahedra. Each flake is formed by placing a dodecahedron scaled by 1 2 + φ {\displaystyle \textstyle {\frac {1}{2+\varphi }}} in each corner. Its Hausdorff dimension is equal to log ⁡ ( 20 ) log ⁡ ( 2 + φ ) {\displaystyle \textstyle {\frac {\log(20)}{\log(2+\varphi )}}} ≈ 2.3296. === Icosahedron flake === An icosahedron flake, or sierpinski icosahedron, is formed by successive flakes of twelve regular icosahedra. Each flake is formed by placing an icosahedron scaled by 1 1 + φ {\displaystyle \textstyle {\frac {1}{1+\varphi }}} in each corner. Its Hausdorff dimension is equal to log ⁡ ( 12 ) log ⁡ ( 1 + φ ) {\displaystyle \textstyle {\frac {\log(12)}{\log(1+\varphi )}}} ≈ 2.5819. == See also == List of fractals by Hausdorff dimension == References == == External links == Quadraflakes, Pentaflakes, Hexaflakes and more – includes Mathematica code to generate these fractals Javascript for covering the plane with 5-fold symmetric Pentaflake tiles.
Wikipedia:N. M. H. Lightfoot#0	Nicholas Morpeth Hutchinson Lightfoot FRSE (1902–1962) was a British mathematician and academic administrator. He was an expert on heat conduction. == Life == He was born in Jarrow in north-east England on 14 October 1902, the son of Thomas Lightfoot. He was educated locally, but excelled, winning a place at Cambridge University where he graduated BA in 1923 and continued as a postgraduate, gaining a further MA. In 1929 he began lecturing in mathematics at Heriot-Watt College in Edinburgh. In 1931 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Sir Edmund Taylor Whittaker, James Cameron Smail, Sir Charles Galton Darwin, and Edward Thomas Copson. In 1943 he moved to the south of England to take on the role of Principal of South East Essex Technical College. In 1950 he moved to head Chelsea Polytechnic. He served as President of the Association of Technical Institutions 1955–6. He died in London on 14 November 1962. == Family == In 1929 he married Janet Moulton (d.1961). == Publications == The Solidification of Molten Steel (1929) == References ==
Wikipedia:Nabla symbol#0	The nabla is a triangular symbol resembling an inverted Greek delta: ∇ {\displaystyle \nabla } or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word νάβλα for a Phoenician harp, and was suggested by the encyclopedist William Robertson Smith in an 1870 letter to Peter Guthrie Tait. The nabla symbol is available in standard HTML as ∇ and in LaTeX as \nabla. In Unicode, it is the character at code point U+2207, or 8711 in decimal notation, in the Mathematical Operators block. As an operator, it is often called del. == History == The differential operator given in Cartesian coordinates { x , y , z } {\displaystyle \{x,y,z\}} on three-dimensional Euclidean space by was introduced in 1831 by the Irish mathematician and physicist William Rowan Hamilton, who called it ◁. (The unit vectors { i , j , k } {\displaystyle \{\mathbf {i} ,\mathbf {j} ,\mathbf {k} \}} were originally right versors in Hamilton's quaternions.) The mathematics of ∇ received its full exposition at the hands of P. G. Tait. After receiving Smith's suggestion, Tait and James Clerk Maxwell referred to the operator as nabla in their extensive private correspondence; most of these references are of a humorous character. C. G. Knott's Life and Scientific Work of Peter Guthrie Tait (p. 145): It was probably this reluctance on the part of Maxwell to use the term Nabla in serious writings which prevented Tait from introducing the word earlier than he did. The one published use of the word by Maxwell is in the title to his humorous Tyndallic Ode, which is dedicated to the "Chief Musician upon Nabla", that is, Tait. William Thomson (Lord Kelvin) introduced the term to an American audience in an 1884 lecture; the notes were published in Britain and the U.S. in 1904. The name is acknowledged, and criticized, by Oliver Heaviside in 1891: The fictitious vector ∇ given by is very important. Physical mathematics is very largely the mathematics of ∇. The name Nabla seems, therefore, ludicrously inefficient. Heaviside and Josiah Willard Gibbs (independently) are credited with the development of the version of vector calculus most popular today. The influential 1901 text Vector Analysis, written by Edwin Bidwell Wilson and based on the lectures of Gibbs, advocates the name "del": This symbolic operator ∇ was introduced by Sir W. R. Hamilton and is now in universal employment. There seems, however, to be no universally recognized name for it, although owing to the frequent occurrence of the symbol some name is a practical necessity. It has been found by experience that the monosyllable del is so short and easy to pronounce that even in complicated formulae in which ∇ occurs a number of times, no inconvenience to the speaker or listener arises from the repetition. ∇V is read simply as "del V". This book is responsible for the form in which the mathematics of the operator in question is now usually expressed—most notably in undergraduate physics, and especially electrodynamics, textbooks. == Modern uses == The nabla is used in vector calculus as part of three distinct differential operators: the gradient (∇), the divergence (∇⋅), and the curl (∇×). The last of these uses the cross product and thus makes sense only in three dimensions; the first two are fully general. They were all originally studied in the context of the classical theory of electromagnetism, and contemporary university physics curricula typically treat the material using approximately the concepts and notation found in Gibbs and Wilson's Vector Analysis. The symbol is also used in differential geometry to denote a connection. A symbol of the same form, though presumably not genealogically related, appears in other areas, e.g.: As the all relation, particularly in lattice theory. As the backward difference operator, in the calculus of finite differences. As the widening operator, an operator that permits static analysis of programs to terminate in finite time, in the computer science field of abstract interpretation. As function definition marker and self-reference (recursion) in the APL programming language As an indicator of indeterminacy in philosophical logic. In naval architecture (ship design), to designate the volume displacement of a ship or any other waterborne vessel; the graphically similar delta is used to designate weight displacement (the total weight of water displaced by the ship), thus ∇ = Δ / ρ {\displaystyle \nabla =\Delta /\rho } where ρ {\displaystyle \rho } is the density of seawater. == See also == Del, treating the mathematics of the vector differential operator Del in cylindrical and spherical coordinates grad, div, and curl, differential operators defined using nabla History of quaternions Notation for differentiation Covariant derivative, also known as connection Nevel == Footnotes == == External links == Arnold Neumaier (2004). "History of Nabla". Arnold Neumaier (January 26, 1998). Cleve Moler (ed.). "History of Nabla". NA Digest, Volume 98, Issue 03. netlib.org. Miller, Jeff. "Earliest Uses of Symbols of Calculus". Tai, Chen. A survey of the improper use of ∇ in vector analysis (1994).
Wikipedia:Nachman Aronszajn#0	In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ in which all levels have size less than κ and all branches have height less than κ (so Aronszajn trees are the same as ℵ 1 {\displaystyle \aleph _{1}} -Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by Kurepa (1935). A cardinal κ for which no κ-Aronszajn trees exist is said to have the tree property (sometimes the condition that κ is regular and uncountable is included). == Existence of κ-Aronszajn trees == Kőnig's lemma states that ℵ 0 {\displaystyle \aleph _{0}} -Aronszajn trees do not exist. The existence of Aronszajn trees ( = ℵ 1 {\displaystyle =\aleph _{1}} -Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of Kőnig's lemma does not hold for uncountable trees. The existence of ℵ 2 {\displaystyle \aleph _{2}} -Aronszajn trees is undecidable in ZFC: more precisely, the continuum hypothesis implies the existence of an ℵ 2 {\displaystyle \aleph _{2}} -Aronszajn tree, and Mitchell and Silver showed that it is consistent (relative to the existence of a weakly compact cardinal) that no ℵ 2 {\displaystyle \aleph _{2}} -Aronszajn trees exist. Jensen proved that V = L implies that there is a κ-Aronszajn tree (in fact a κ-Suslin tree) for every infinite successor cardinal κ. Cummings & Foreman (1998) showed (using a large cardinal axiom) that it is consistent that no ℵ n {\displaystyle \aleph _{n}} -Aronszajn trees exist for any finite n other than 1. If κ is weakly compact then no κ-Aronszajn trees exist. Conversely, if κ is inaccessible and no κ-Aronszajn trees exist, then κ is weakly compact. == Special Aronszajn trees == An Aronszajn tree is called special if there is a function f from the tree to the rationals so that f(x) < f(y) whenever x < y. Martin's axiom MA( ℵ 1 {\displaystyle \aleph _{1}} ) implies that all Aronszajn trees are special, a proposition sometimes abbreviated by EATS. The stronger proper forcing axiom implies the stronger statement that for any two Aronszajn trees there is a club set of levels such that the restrictions of the trees to this set of levels are isomorphic, which says that in some sense any two Aronszajn trees are essentially isomorphic (Abraham & Shelah 1985). On the other hand, it is consistent that non-special Aronszajn trees exist, and this is also consistent with the generalized continuum hypothesis plus Suslin's hypothesis (Schlindwein 1994). == Construction of a special Aronszajn tree == A special Aronszajn tree can be constructed as follows. The elements of the tree are certain well-ordered sets of rational numbers with supremum that is rational or −∞. If x and y are two of these sets then we define x ≤ y (in the tree order) to mean that x is an initial segment of the ordered set y. For each countable ordinal α we write Uα for the elements of the tree of level α, so that the elements of Uα are certain sets of rationals with order type α. The special Aronszajn tree T is the union of the sets Uα for all countable α. We construct the countable levels Uα by transfinite induction on α as follows starting with the empty set as U0: If α + 1 is a successor then Uα+1 consists of all extensions of a sequence x in Uα by a rational greater than sup x. Uα + 1 is countable as it consists of countably many extensions of each of the countably many elements in Uα. If α is a limit then let Tα be the tree of all points of level less than α. For each x in Tα and for each rational number q greater than sup x, choose a level α branch of Tα containing x with supremum q. Then Uα consists of these branches. Uα is countable as it consists of countably many branches for each of the countably many elements in Tα. The function f(x) = sup x is rational or −∞, and has the property that if x < y then f(x) < f(y). Any branch in T is countable as f maps branches injectively to −∞ and the rationals. T is uncountable as it has a non-empty level Uα for each countable ordinal α which make up the first uncountable ordinal. This proves that T is a special Aronszajn tree. This construction can be used to construct κ-Aronszajn trees whenever κ is a successor of a regular cardinal and the generalized continuum hypothesis holds, by replacing the rational numbers by a more general η set. == See also == Kurepa tree Aronszajn line == References == Abraham, Uri; Shelah, Saharon (1985), "Isomorphism types of Aronszajn trees", Israel Journal of Mathematics, 50: 75–113, doi:10.1007/BF02761119 Cummings, James; Foreman, Matthew (1998), "The tree property", Advances in Mathematics, 133 (1): 1–32, doi:10.1006/aima.1997.1680, MR 1492784 Kunen, Kenneth (2011), Set theory, Studies in Logic, vol. 34, London: College Publications, ISBN 978-1-84890-050-9, Zbl 1262.03001 Kurepa, G. (1935), "Ensembles ordonnés et ramifiés", Publ. math. Univ. Belgrade, 4: 1–138, JFM 61.0980.01, Zbl 0014.39401 Schlindwein, Chaz (1994), "Consistency of Suslin's Hypothesis, A Nonspecial Aronszajn Tree, and GCH", Journal of Symbolic Logic, 59 (1), The Journal of Symbolic Logic, Vol. 59, No. 1: 1–29, doi:10.2307/2275246, JSTOR 2275246 Schlindwein, Ch. (2001) [1994], "Aronszajn tree", Encyclopedia of Mathematics, EMS Press Schlindwein, Chaz (1989), "Special non-special ℵ 1 {\displaystyle \aleph _{1}} -trees", Set Theory and its Applications, 1401: 160–166, doi:10.1007/BFb0097338 Todorčević, S. (1984), "Trees and linearly ordered sets", Handbook of set-theoretic topology, Amsterdam: North-Holland, pp. 235–293, MR 0776625 == External links == PlanetMath
Wikipedia:Nadia Ghazzali#0	Nadia Ghazzali (born April 3, 1961) is a Canadian statistician, the former president of the Université du Québec à Trois-Rivières, where she continues to work as a professor in the department of mathematics and computer science. As a statistician, she is known for her work on NbClust, a package in the R statistical software system for determining the number of clusters in a data set. == Education and career == Ghazzali was born on April 3, 1961, in Casablanca. After studying at the University of Rennes 1 in France, she came to Canada as a postdoctoral researcher at McGill University, and joined the faculty at the Université Laval in 1993. She was president of the Université du Québec à Trois-Rivières from 2012 until 2015, when she resigned after facing criticism from the Auditor General of Québec over management practices in university construction. Ghazzali is current (2021-2023) Deputy President of INWES, the International Network of Women Engineers and Scientists. == Recognition == At Laval, Ghazzali was given the NSERC Chair for Women in Science and Engineering (CWSE) in 2006. In the same year, she was named a corresponding member of the Hassan II Academy of Sciences and Technologies. == References ==
Wikipedia:Nae Ionescu#0	Nae Ionescu (Romanian: [ˈna.e joˈnesku], born Nicolae C. Ionescu; 16 June [O.S. 4 June] 1890 – 15 March 1940) was a Romanian philosopher, logician, mathematician, professor, and journalist. == Life == Born in Brăila, Ionescu studied Letters at the University of Bucharest until 1912. Upon graduation, he was appointed teacher at the Matei Basarab High School in Bucharest. When World War I began, he traveled to Germany for additional studies at the University of Göttingen. Romania's entry into the war on the Entente side prevented him from returning, but he was awarded a doctorate in philosophy in 1919 from the University of Munich. His thesis was entitled Die Logistik als Versuch einer neuen Begründung der Mathematik ("Formal logic as an attempt at a new foundation of mathematics"). Back in Romania, after another brief stint teaching, Ionescu was appointed assistant to Constantin Rădulescu-Motru at the University of Bucharest's department of Logic and Theory of Knowledge. His life's work had a profound effect on a generation of Romanian thinkers, first for his studies on comparative religion, philosophy, and mysticism, but later for his nationalist and far right sentiment. Some of the figures he influenced include Constantin Noica, Mircea Eliade, Emil Cioran, Haig Acterian, Jeni Acterian, Mihail Sebastian, Mircea Vulcănescu, and Petre Țuțea. The existentialist and partly mystical school of thought Ionescu introduced bore the name Trăirism. Trăirism intersected at several points with the ideology of the Iron Guard; the connection became even more direct when many of its adherents also publicly associated with the latter. Ionescu himself was more reserved in his dealings with the Guard. He was the editor of the highly influential newspaper Cuvântul, which had long backed King Carol II – the major rival of the Guard. However, Ionescu moved away from the monarchy due to Carol's inner circle. Ionescu's antisemitism was a decisive factor in his switching of allegiances: Jewish writer Mihail Sebastian's Journal depicts the interval during which Ionescu's virulence grew, as well as the reasons that were animating his large following. === Mihail Sebastian incident === During the period when Sebastian and Ionescu were still on speaking terms, the latter had agreed to write the preface of Sebastian's book De două mii de ani... ("For two thousand years..."). Ionescu's introduction shocked Sebastian, who "loved and admired Ionescu", as it included several overtly antisemitic statements. Mircea Eliade recalls the incident in his autobiography: "Judah suffers because it must suffer," Nae had written. And he explained why: the Jews had refused to acknowledge Jesus Christ as their Messiah. This suffering in history reflected, in a certain sense, the destiny of the Hebrew people who, precisely because they had rejected Christianity, could not be saved. Extra Ecclesiam nulla salus. Eliade notes that this incident marked a profound departure for Ionescu, who in the late 1920s had suggested to Eliade, who was then his student, that he had been tempted "to give up both journalism and politics and devote myself entirely to Hebraic studies". Sebastian, though dejected by the incident, opted to keep Ionescu's introduction in the book. === Later life === After Carol's crackdown on the Iron Guard, Nae Ionescu and his disciples were rounded up and imprisoned at a makeshift camp in Miercurea-Ciuc. The experience took a toll on his fragile health, and he died soon thereafter, at age 49. Some close sources indicated that he was assassinated by poisoning due to his involvement with the Iron Guard. He was buried at Bellu Cemetery in Bucharest; his coffin was carried by some of his disciples, including Mircea Eliade. === Nae Ionescu's villa in Băneasa === Nae Ionescu was the owner of a luxury villa in Băneasa village (now Băneasa neighborhood, in the northern part of Bucharest), situated at 4 Ion Ionescu de la Brad Avenue (position 44.498604°N 26.075206°E / 44.498604; 26.075206). It was considered one of the finest Bucharest residences at the time. The house remained in collective memory mainly because it is described in his students' writings (Mircea Vulcănescu and others). The house was built by architect George Matei Cantacuzino, under its owner's direct supervision. In his book "Nae Ionescu așa cum l-am cunoscut" ("Nae Ionescu As I Knew Him"), Vulcănescu dedicates many pages in describing this house, in detail. He evokes the fact that a painting of El Greco was displayed there (a Descent from the Cross), on which Ionescu took great pride. Nae Ionescu died in this very house, because of a heart attack, on 15 March 1940, while his girlfriend Cella Delavrancea was present. After his death, the house served as an official residence to Marshal Ion Antonescu (sometime after his rise in power on 6 September 1940). After the Communist regime was installed, the villa was given to the University of Agronomic Sciences (since the land of "Ferma Regală Băneasa" (Băneasa Royal Farm) lies around the building). Today, it is the headquarters of "Stațiunea de Cercetare-Dezvoltare pentru Pomicultură Băneasa" (statiuneabaneasa.ro), a research facility for pomology. == Notes == == External links == Works by or about Nae Ionescu at the Internet Archive Full text of Ionescu's lectures on the Philosophy of Religion (in Romanian) Andrei Oișteanu, "Acuzația de omor ritual (O sută de ani de la pogromul de la Chișinău)(3)", page 4, in Contrafort (in Romanian) The beguiling Nae Ionescu (in Romanian) Nae Ionescu's "Villa on The Road" on Wikimapia Archived 2012-01-11 at the Wayback Machine
Wikipedia:Nagambal Shah#0	Nagambal D. "Swarna" Shah is an American mathematician and statistician known for her mentorship of students at Spelman College. She is the founder of the annual StatFest of the American Statistical Association, a leader of the association's Diversity Mentoring Program, and the former chair of the association's Committee on Minorities in Statistics. == Education and career == Shah is originally from India, where she did her undergraduate studies in mathematics and earned a master's degree in statistics. She completed a Ph.D. in statistics in 1970 at the University of Windsor in Canada. She joined the Spelman College mathematics department in 1972, and retired to become a professor emerita in 2014. == Recognition == In 2001, the Rollins School of Public Health of Emory University gave Shah their Martin Luther King Jr. Community Service Award. Shah was named a Fellow of the American Statistical Association in 2010. The National Association of Mathematicians gave her their Lifetime Achievement Award in 2017.. The American Association for the Advancement of Science gave her their Lifetime Mentor Award in 2023.. == References ==
Wikipedia:Nagata's conjecture#0	In algebra, Nagata's conjecture states that Nagata's automorphism of the polynomial ring k[x,y,z] is wild. The conjecture was proposed by Nagata (1972) and proved by Ualbai U. Umirbaev and Ivan P. Shestakov (2004). Nagata's automorphism is given by ϕ ( x , y , z ) = ( x − 2 Δ y − Δ 2 z , y + Δ z , z ) , {\displaystyle \phi (x,y,z)=(x-2\Delta y-\Delta ^{2}z,y+\Delta z,z),} where Δ = x z + y 2 {\displaystyle \Delta =xz+y^{2}} . For the inverse, let ( a , b , c ) = ϕ ( x , y , z ) {\displaystyle (a,b,c)=\phi (x,y,z)} Then z = c {\displaystyle z=c} and Δ = b 2 + a c {\displaystyle \Delta =b^{2}+ac} . With this y = b − Δ c {\displaystyle y=b-\Delta c} and x = a + 2 Δ y + Δ 2 z {\displaystyle x=a+2\Delta y+\Delta ^{2}z} . == References == Nagata, Masayoshi (1972), On automorphism group of k[x,y], Tokyo: Kinokuniya Book-Store Co. Ltd., MR 0337962 Umirbaev, Ualbai U.; Shestakov, Ivan P. (2004), "The tame and the wild automorphisms of polynomial rings in three variables", Journal of the American Mathematical Society, 17 (1): 197–227, doi:10.1090/S0894-0347-03-00440-5, ISSN 0894-0347, MR 2015334
Wikipedia:Najiba Sbihi#0	Najiba Sbihi (born 1953) is a Moroccan mathematician and operations researcher, known for her contributions to graph theory and graph algorithms. == Education and career == Sbihi earned a degree from the Faculty of Sciences of Mohammed V University in Rabat, Morocco in 1973. She continued her studies in France at Joseph Fourier University in Grenoble, first in computer science in which she earned a bachelor's degree in 1975. Continuing in operations research, she earned a diplôme d'études approfondies in 1976, a doctorat de troisième cycle in 1978 under the supervision of Michel Sakarovitch, and a doctorat d'état in 1987, supervised by Jean Fonlupt. Her doctoral study also included research in Canada with Jack Edmonds at the University of Waterloo and with Václav Chvátal at McGill University. She worked with the Moroccan National Center for Scientific and Technical Research until, in 1992, becoming a professor of industrial engineering in the Mohammadia School of Engineering in Rabat. She headed the Department of Industrial Engineering from 1995 to 1997. == Contributions == Sbihi's contributions to graph theory and graph algorithms include the discovery that the maximum independent set problem can be solved in polynomial time on claw-free graphs.[A] With Chvátal, she proved a special case of the strong perfect graph theorem, for the graphs that have no bull graph as an induced subgraph.[B] Their work in this area introduced a type of graph decomposition that was central to the eventual proof of the full strong perfect graph theorem. She and Chvátal also devised efficient algorithms for recognizing the claw-free perfect graphs,[C] and later she and Bruce Reed showed how to recognize the Bull-free perfect graphs.[D] == Selected publications == == References ==
Wikipedia:Nalin de Silva#0	Thakurartha Devadithya Guardiyawasam Lindamulage Nalin Kumar de Silva (Sinhala: නලින් ද සිල්වා; 20 October 1944 – 1 May 2024) was a Sri Lankan philosopher, polymath and a political analyst. He was the former Sri Lankan ambassador in Myanmar. He was a professor in the department of mathematics, a member of University Grant Commission and the dean of the faculty of science at the University of Kelaniya, Sri Lanka. De Silva openly stated that the so-called 'objective scientific method' was a lie. In June 2011 he stated that information regarding the presence of arsenic in water claiming that the cause of Rajarata chronic kidney disease had been given to him by the god Natha. The Sri Lanka Association for the Advancement of Science, an organisation in which De Silva claimed lifelong membership, has said that they consider it "extremely unfortunate that the Dean of a Science Faculty should make it his publicly stated aim to run down science and bring it into disrepute." However the presence of arsenic was verified in water and vegetation, and the source was found to be agrochemical fertilizers. == Early life and education == Nalin de Silva was born on 20 October 1944 in Kovilagodella, Panadura, Sri Lanka. His father was Daniel De Silva, a principal, and his mother was Jayline Perera, a school teacher. De Silva was the eldest of eight siblings. He had primary education at Bauddhaloka Maha Vidyalaya, Thurstan College Colombo and secondary education at Royal College Colombo. He captained the Sinhala Debating team at Royal College and won the Weerasooriya medal for oratory. He entered the University of Ceylon in 1963 and graduated in mathematics in 1967. De Silva entered University of Sussex in January 1969, where he obtained his doctorate in theoretical physics –cosmology – in 1970. He was married and had two sons and one daughter. == Academic work == === Constructive relativism === Nalin de Silva was formally a member of the Marxist Trotskyist Lanka Sama Samaja Party and the Nava Sama Samaja Party, and as a Marxist studying both disciplines he intensively began to question the foundations of both Marxism and science. As a result, in 1986, he wrote Mage Lokaya (My World), criticising the basis of the established western system of knowledge, and its propagation, which he refers as "domination throughout the world". He explained in this book that mind independent reality is impossible and knowledge is not found but constructed. This has further evolved into a study of epistemology and ontology, and in the process he has introduced and developed the concept of "constructive relativism" as the basis on which knowledge is constructed relative to the sense organs, culture and the mind completely based on Avidya. Though he was identified as a representative of "Jathika Chinthanaya" an idea proposed by Gunadasa Amarasekara based on national thought concept of Erich Fromm, Nalin de Silva's concept Chinthanaya is a different epistemological concept. === Vidyalankara interpretation === Vidyalankara interpretation was presented based on the theoretical and experimental work conducted by Dr. Nalin de Silva and his student Suraj Chandana at University of Kelaniya (Vidyalankara) in which Dr. de Silva rejects the wave nature of particles. According to him what gives rise to a bright and dark fringe pattern in the Young's double slit experiment is not the so called wave nature but the particle nature itself though not necessarily those of ordinary classical particles. Dr. de Silva theorises that each particle that hit the screen on a double slit experiment appears at both the slits simultaneously prior to reaching the screen. According to Janaka Wansapura, Ph.D. this is a paradigm shift in the knowledge of quantum nature. === Accelerating and decelerating Universe === Dr. Nalin de Silva has given a model for the universe with both acceleration and deceleration. === Arsenic controversy === The Rajarata chronic kidney disease (RCKD) has been an ailment prevalent in the North Central Province of Sri Lanka for several decades. In 2010 a team of scientists headed by Priyani Paranagama, head of the chemistry department at Kelaniya University, conducted a study to find the cause of the disease. The team was advised by Nalin de Silva, dean of the faculty of science, Kelaniya University. The team presented their findings in June 2011. They stated that they had found high concentrations of arsenic in drinking water, vegetation, rice and other samples in the region. The team, however, refused to release their data to the public, raising questions about their methodology. Their finding were followed by allegations that the arsenic was being introduced to the system through foreign pesticides and fertilizers. As this statement caused a huge uproar in the agricultural sector in Sri Lanka, the Industrial Technology Institute stated that they had used Hydride Generation Atomic Absorption Spectrometry on rice samples which showed no high levels of arsenic. The ITI study also found that in samples of 28 different pesticides only 3 showed any signs of arsenic, and even then not in the concentrations that De Silva's team alleged. The Kelaniya University team stated that they had used new methods to locate the Arsenic in the samples they collected but did not feel a need to submit their findings for peer-review. De Silva stated that they did not publish their results in a "so-called peer reviewed journal due to our concern regarding the authorship of the paper as the original idea was given by 'samyak drshtika devivaru' also known as Devas. When the authenticity of the findings made by the University of Kelaniya team were questioned by other scientists, De Silva stated that the god "Natha" had first told them about the arsenic through a mystic. The merit (pina) in this regard should go to the "samyak drshtika devivaru" who first told that Arsenic is present in Rajarata drinking water and later revealed to us that Arsenic which is not found naturally in Sri Lanka has found its way to the wells and the wewas in Rajarata through Agrochemicals. It's simply like this. We can see professors. Professors cannot see Arsenic. We cannot see gods. Gods can see Arsenic. I like this because there is certain symmetry in it... I know very well that the so-called objective scientific method is a lie. Paranagama too has stood by De Silva and defended their methods. She said that they had first been told of the arsenic by the wife of "a university lecturer who had developed her mental faculties to communicate with a higher being", but then used scientific methods to deduce it. The Sri Lanka Association for the Advancement of Science released a statement in July 2011 questioning the validity of the findings and the process and individuals involved in the study. Their statement laid the blame for the scandal firmly on Prof Nalin De Silva, and they called for other scientists and researchers in the group to distance themselves from De Silva to have their work taken seriously . There is another serious issue which casts grave doubts on the credibility of the claims made by the Kelaniya group. The press has publicly identified as the leader of this group an individual, who despite holding a responsible position, professes a disdain for 'Western' science. He has publicly claimed supernatural revelations ('samyak drushtika devivaru') as the source of his group's information and even methods. While recognising that many scientists are deeply religious, the SLAAS wishes to state categorically that superstition and the supernatural have no place in science, and that scientific results inspired by such sources are highly suspect because of a probable bias on the part of the investigator. We also note that other researchers in the group have yet to distance themselves from the eccentric statements of their leader, and they need to do this if they wish to be taken seriously as scientists. === Western cultural dominance === De Silva was also a vocal opponent of what he calls western cultural dominance which he claimed is in its final phase. The western Christian modernity (WCM) is in the last phase of its hegemony and is acting like an insane person. It has no respect for the sovereignty of the other countries as has been exemplified from invasions, killings and supporting so called rebel groups in the countries in Asia and Africa. The USA that became the leader of WCM after the so called second world war is trying desperately to hold to the hegemony of WCM. After more than five hundred years of world domination WCM will have to leave the planet without making the people suffer more and more. De Silva also referred to linguist and philosopher Noam Chomsky as a puppet of the Western Christian Modernity. The west also maintains people such as Chomsky who is not known to the average American in order to show to the world that there is freedom for the intellectuals to champion their views heard more by the rest of the world than by the Americans themselves. === Philosophy of science === De Silva, who was influenced by Paul Feyerabend, rejected the existence of scientific method and proposed proliferation of sciences on various chinthanayas. De Silva proposed theories are stories constructed by abductive reasoning to explain phenomena. De Silva refused ideas that science is truth or science is advancing towards the truth. === Sacking from University of Colombo === De Silva was sacked from the University of Colombo for engaging in unlawful political activity within the campus. In subsequent letters he has described the event. When I was in the University of Colombo, not the best place to fight the cultural pentagon, I was asked by the then Vice Chancellor G. L. Peiris why I was teaching Jathika Chinthanaya to a captive audience. The audience was not that captive and had the backing of the Vice Chancellor and some members of the senior staff, and of course the cultural pentagon. Nothing happened to the captive audience but soon I found that I was out of the University system. === Veemansaka Parshadaya === In 2008 the Veemansaka Parshadaya, a student organisation where De Silva is the treasurer and mentor started a campaign to ban western clothing in Kelaniya University. == Associations == Nalin de Silva was a member of the International Astronomical Union. == Death == De Silva died on 1 May 2024 in Fremont, California, at the age of 79. == Bibliography == Mage Lokaya (My World) Ape Pravada (Our Theories) Parisaraya Ha Batahira Akalpa (Environment and Western Attitudes) Ape Pravada – 2 (Our Theories – 2) Jathiya Sanskruthiya Saha Chinthanaya (Nation, Culture and Chinthanaya), Maharagama, 1991 OCLC 27387252 Bududahama Pavadeema (Buddhism Betrayed), Kelaniya, 1993 OCLC 32274198 Apohakaye Roopikaya (Formalism of Dialectics) Marxvadaye Daridrathawaya (Poverty of Marxism) Prabhakaran Ohuge Seeyala Bappala (Prabhakaran and his relatives), Maharagama, 1995 OCLC 35151717 An Introduction to Tamil Racism in Sri Lanka, Maharagama, 1997 OCLC 40331529 (English translation of the previous book) Nidahase Pahantemba (Beacon of Freedom) Nidahasa Dinaganeema (Winning Freedom Thyroideedom) Jathiye Iranama (Fate of the Nation) Sinhala Avurudda (Sinhala New Year) Vidya Kathandara (Stories of Science) Methivaranaya Ha Hamuduruwo (Elections and Bhikkus) == See also == Sri Lankan Non Career Diplomats == References ==
Wikipedia:Namig Nasrullayev#0	Namiq Nasrulla oğlu Nasrullayev (Azerbaijani: Namiq Nəsrulla oğlu Nəsrullayev; 2 February 1945 – 17 January 2023) was an Azerbaijani mathematician and politician. He served as Minister of Economy from 1996 to 2001 and chairman of the Chamber of Accounts from 2001 to 2007. Nasrullayev died in Baku on 17 January 2023, at the age of 77. == References ==
Wikipedia:Namioka's theorem#0	In functional analysis, Namioka's theorem is a result concerning the relationship between separate continuity and joint continuity of functions defined on product spaces. Named after mathematician Isaac Namioka, who proved it in his 1974 paper Separate Continuity and Joint Continuity published in the Pacific Journal of Mathematics, the theorem establishes conditions under which a separately continuous function must be jointly continuous on a topologically large subset of its domain. == Statement == Namioka's theorem. Let X {\displaystyle X} be a Čech-complete topological space (such as a complete metric space), Y {\displaystyle Y} be a compact Hausdorff space, and Z {\displaystyle Z} be a metric space. If f : X × Y → Z {\displaystyle f:X\times Y\rightarrow Z} is separately continuous, meaning that for each fixed x 0 ∈ X {\displaystyle x_{0}\in X} , the function y ↦ f ( x 0 , y ) {\displaystyle y\mapsto f(x_{0},y)} is continuous on Y {\displaystyle Y} , and for each fixed y 0 ∈ Y {\displaystyle y_{0}\in Y} , the function x ↦ f ( x , y 0 ) {\displaystyle x\mapsto f(x,y_{0})} is continuous on X {\displaystyle X} , then there exists a dense G δ {\displaystyle G_{\delta }} -subset G {\displaystyle G} of X {\displaystyle X} such that f {\displaystyle f} is jointly continuous at each point of G × Y {\displaystyle G\times Y} . Namioka's theorem can be equivalently stated in terms of the set C ( f ) {\displaystyle C(f)} of points where f {\displaystyle f} is continuous, stating that the projection of C ( f ) {\displaystyle C(f)} onto X {\displaystyle X} contains a dense G δ {\displaystyle G_{\delta }} subset of X {\displaystyle X} . == History == French mathematician René Baire was among the first to systematically study the relationship between separate and joint continuity in 1899, for real-valued functions of real variables. Austrian mathematician Hans Hahn later extended these investigations in 1932, proving similar results for functions defined on complete metric spaces. Namioka generalized these results to non-metrizable spaces, particularly to Čech-complete spaces, which include all complete metric spaces as a special case. There exists a proof using tools from general topology such as the Arkhangel'skii–Frolík covering theorem and the Kuratowski and Ryll-Nardzewski measurable selection theorem. == See also == Baire space Stone–Čech compactification == References ==
Wikipedia:Nandigrama#0	Nandigrama is the name of a location, place or region somewhere in Western India where a school of astronomers and mathematicians flourished during the thirteenth-eighteenth centuries CE. David Pingree, one of America's leading historians of the exact sciences (primarily mathematics) in antiquity, identified Nandigrama with Nandod in Gujarat. However, modern scholarship has identified Nandigrama as the Nandgaon village in the Raigad district in Maharashtra State. It lies about 64 km south of Mumbai on the Konkan coast. == Astronomers and mathematicians == Ganesa Daivajna was an astronomer born in 1507 in Nandigrama. His father Kesava Daivajna and paternal grandfather Kamalakara (not to be confused with Kamalakara of Golagrama) were also eminent astronomers. Kesava has been considered as one of the best observational astronomers of ancient India. Ganesa authored several important treatises and manuals on astronomy and astrology. Some of them are "Grahalaghava," "Laghu- and Brht-Tithi Chintamani,” a commentary on Bhaskara II’s "Siddharta Siromani," “Buddhi Vilasini," a commentary on Bhaskara II’s "Lilavati," "Sraddha nirnaya," “Patasarani,” and “Parva nimaya.” The “Graha Laghava” is extensively used by the Panchanga makers in Maharashtra, Gujarat, Andhra Pradesh, and Karnataka. In the “Graha Laghava,” planetary positions have been given at the instant of sunrise of the new moon day of Palguna of Saka 1441, which corresponds to March 19, AD 1520. == See also == Jambusagaranagara Dadhigrama Golagrama == References ==
Wikipedia:Nanny Cedercreutz#0	Ebba Louise Nanny Cedercreutz (née Lagerborg; 19 March 1866 in Cannes – 8 December 1950 in Helsinki) was a Finnish author and physicist. == Biography == Cedercreutz's father was an engineer, statesman Alexander Wilhelm Lagerborg, who had travelled with his wife Anna Maria Christina (Nanny) Franzén to southern France to recover from tuberculosis. Nanny was born 19 March 1866 in Cannes, France, but the birth caused her mother's death. Her father married Amalia Tigerhjelm, Nanny's tutor five years later. Strong-willed and independent, Nanny enjoyed mountain climbing and hiking in Norway and Switzerland in her youth. In 1885, she arrived in Geneva, Switzerland at the age of 19 to study French, but soon began to listen to mathematics and physics lectures held at Geneva University. She continued her studies in mathematics and physics at the Stockholm University from 1886 to 1889, and attended in particular the mathematics lessons of Gösta Mittag-Leffler and those of physics of Knut Ångström. She authored her doctoral dissertation on rock salt at the Stockholm University in 1888. After this, she studied mathematics at Sorbonne University in Paris. In 1890, Nanny Cedercreutz presented a paper to the Mathematical Society of France about the movement of a solid around a fixed point, and it was published in the Bulletin of the Society. She was invited as the second woman in the world to become a member of La Société mathématique de France (the first woman was Sofya Kovalevskaya, who was elected a member of the society in 1882). In the same year, Nanny also completed a licencie ès sciences mathématiques degree in mathematics at the Sorbonne. In 1892, Nanny married the doctor Emil Waldemar Cedercreutz (1852–1924), with whom she had five children between 1893 and 1906: Carl Wilhelm Cedercreutz, Kjerstin Cedercreutz (who died a month after his birth), Per Skragge Skragge Cedercreutz, Lars Valdemar Cedercreutz and Eva Margareta Bang. Her husband's nephew of the same name was a well-known visual artist. From 1912, she wrote, sometimes under a pseudonym (Ala or Bengt Ivarson), novels for young people, collections of poetry, a rhyming chronicle and travel and natural history stories. She also wrote an autobiography of her youth, illustrated with drawings and photographs. Nanny Cedercreutz died at 85 on 8 December 1950 in Helsinki, Finland. == Works == Between 1914 and 1948, she wrote thirteen books in Swedish; nine under her own name, one under the pseudonym Ala and three under the male name Bengt Ivarson. Under her own name: Studies on the variation of refractive indices and the density of rock salt under the influence of temperature; fysiikan väitöskirja. Stockholm, 1888 From the Alps and the sea. Söderström, Helsinki, 1914 Fröken Milla Lund och Fiken, Hennes hund: en rimkrönika från nådens år 1916 ; silhouette by Emil Cedercreutz. Schildt, Helsingfors 1919 (Finnish translation: Miss Milla Lund and Ahnas, his dog: a rhyming short story of grace from 1916, Palladium books, Tampere 1996) At home and outside. The author, Helsinki 1934 Wanderlust. Söderström, Helsinki 1935 Travel memories. The author, Helsinki 1940 My joy. The author, Helsinki 1943 Uneven. The author, Helsinki 1945 Alternation. The author, Helsinki 1946 From different times. The author, Helsinki 1948 Under the pseudonym Ala: Arabella's Travels Told for Youth. Mercator, Helsinki 1912 Under the pseudonym Bengt Ivarson: Gifts and old ladies. Schildt, Helsinki 1917 Kind and naughty. Schildt, Helsinki 1925 Distressed and happy. Söderström, Helsinki 1932 == References ==
Wikipedia:Naoki Saito (mathematician)#0	Naoki Saito is an applied mathematician specializing in applied and computational harmonic analysis, and interested in feature extraction, pattern recognition, graph signal processing, statistical signal processing, Laplacian eigenfunctions, and human and machine perception. == Education == Saito studied at the University of Tokyo, receiving his BEng in 1982 and his MEng in 1984. He joined Nippon Schlumberger K.K. in 1984, and in 1986 moved to Schlumberger-Doll Research (SDR), Ridgefield, Connecticut, where he was a research scientist. He continued his studies, receiving his PhD in applied mathematics from Yale University in 1994. == Career == Saito began teaching at the Department of Mathematics at the University of California, Davis in 1997, where he is currently a professor and a director of the UC Davis TETRAPODS Institute of Data Science, one of National Science Foundation's Transdisciplinary Research In Principles Of Data Science (TRIPODS) Institutes, which brings together the theoretical computer science, electrical engineering, mathematics, and statistics communities to develop the theoretical foundations of data science. He was also Chair of the Graduate Group in Applied Mathematics at UC Davis from 2007 to 2012. He is a senior member of IEEE as well as a member of IMS, SIAM, and JSIAM. He served as Chair of the SIAM Activity Group on Imaging Science from 2013 to 2015, and is a member of the editorial board of Applied and Computational Harmonic Analysis, Inverse Problems and Imaging, and Journal of Mathematical Imaging and Vision. Saito received the Best Paper Awards from the SPIE (1994), and JSIAM (2016) as well as the Henri Doll Award from Schlumberger (1997), Young Investigator Award from the ONR (2000), and the Presidential Early Career Award for Scientists and Engineers (PECASE) (2000). Saito has contributed to a number of notable interdisciplinary projects including the development of a noninvasive fetal oxygen monitor as well as the Lake Tahoe Sonification Project in 2014. == References == == External links == Naoki Saito, at University of California, Davis
Wikipedia:Napkin ring problem#0	In geometry, the napkin-ring problem involves finding the volume of a "band" of specified height around a sphere, i.e. the part that remains after a hole in the shape of a circular cylinder is drilled through the center of the sphere. It is a counterintuitive fact that this volume does not depend on the original sphere's radius but only on the resulting band's height. The problem is so called because after removing a cylinder from the sphere, the remaining band resembles the shape of a napkin ring. == Statement == Suppose that the axis of a right circular cylinder passes through the center of a sphere of radius R {\displaystyle R} and that h {\displaystyle h} represents the height (defined as the distance in a direction parallel to the axis) of the part of the cylinder that is inside the sphere. The "band" is the part of the sphere that is outside the cylinder. The volume of the band depends on h {\displaystyle h} but not on R {\displaystyle R} : V = π h 3 6 . {\displaystyle V={\frac {\pi h^{3}}{6}}.} As the radius R {\displaystyle R} of the sphere shrinks, the diameter of the cylinder must also shrink in order that h {\displaystyle h} can remain the same. The band gets thicker, and this would increase its volume. But it also gets shorter in circumference, and this would decrease its volume. The two effects exactly cancel each other out. In the extreme case of the smallest possible sphere, the cylinder vanishes (its radius becomes zero) and the height h {\displaystyle h} equals the diameter of the sphere. In this case the volume of the band is the volume of the whole sphere, which matches the formula given above. An early study of this problem was written by 17th-century Japanese mathematician Seki Kōwa. According to Smith & Mikami (1914), Seki called this solid an arc-ring, or in Japanese kokan or kokwan. == Proof == Suppose the radius of the sphere is R {\displaystyle R} and the length of the cylinder (or the tunnel) is h {\displaystyle h} . By the Pythagorean theorem, the radius of the cylinder is R 2 − ( h 2 ) 2 , ( 1 ) {\displaystyle {\sqrt {R^{2}-\left({\frac {h}{2}}\right)^{2}}},\qquad \qquad (1)} and the radius of the horizontal cross-section of the sphere at height y {\displaystyle y} above the "equator" is R 2 − y 2 . ( 2 ) {\displaystyle {\sqrt {R^{2}-y^{2}}}.\qquad \qquad (2)} The cross-section of the band with the plane at height y {\displaystyle y} is the region inside the larger circle of radius given by (2) and outside the smaller circle of radius given by (1). The cross-section's area is therefore the area of the larger circle minus the area of the smaller circle: π ( larger radius ) 2 − π ( smaller radius ) 2 = π ( R 2 − y 2 ) 2 − π ( R 2 − ( h 2 ) 2 ) 2 = π ( ( h 2 ) 2 − y 2 ) . {\displaystyle {\begin{aligned}&{}\quad \pi ({\text{larger radius}})^{2}-\pi ({\text{smaller radius}})^{2}\\&=\pi \left({\sqrt {R^{2}-y^{2}}}\right)^{2}-\pi \left({\sqrt {R^{2}-\left({\frac {h}{2}}\right)^{2}\,{}}}\,\right)^{2}=\pi \left(\left({\frac {h}{2}}\right)^{2}-y^{2}\right).\end{aligned}}} The radius R does not appear in the last quantity. Therefore, the area of the horizontal cross-section at height y {\displaystyle y} does not depend on R {\displaystyle R} , as long as y ≤ h 2 ≤ R {\displaystyle y\leq {\tfrac {h}{2}}\leq R} . The volume of the band is ∫ − h / 2 h / 2 ( area of cross-section at height y ) d y , {\displaystyle \int _{-h/2}^{h/2}({\text{area of cross-section at height }}y)\,dy,} and that does not depend on R {\displaystyle R} . This is an application of Cavalieri's principle: volumes with equal-sized corresponding cross-sections are equal. Indeed, the area of the cross-section is the same as that of the corresponding cross-section of a sphere of radius h / 2 {\displaystyle h/2} , which has volume 4 3 π ( h 2 ) 3 = π h 3 6 . {\displaystyle {\frac {4}{3}}\pi \left({\frac {h}{2}}\right)^{3}={\frac {\pi h^{3}}{6}}.} == Another Derivation == We can also find the napkin ring's volume V n {\displaystyle V_{n}} using previous results. Specifically, volume V n {\displaystyle V_{n}} must equal the original sphere's volume 4 π R 3 / 3 {\displaystyle 4\pi R^{3}/3} minus the cylinder's volume minus the volume of two spherical caps V n ( R , h , r c , h s ) = { 4 3 π R 3 } − { π ( r c ) 2 h } − 2 { π 3 ( h s ) 2 ( 3 R − h s ) } {\displaystyle V_{n}(R,h,r_{c},h_{s})=\left\{{\frac {4}{3}}\pi R^{3}\right\}-\left\{\pi (r_{c})^{2}\,h\right\}-2\left\{{\frac {\pi }{3}}(h_{s})^{2}(3R-h_{s})\right\}} In the above, the cylinder volume uses its radius r c {\displaystyle r_{c}} which can be written in terms of R {\displaystyle R} and h {\displaystyle h} as shown in (1) r c = R 2 − ( h 2 ) 2 {\displaystyle r_{c}={\sqrt {R^{2}-\left({\frac {h}{2}}\right)^{2}}}} The spherical cap volume used in V n {\displaystyle V_{n}} uses the cap's height h s {\displaystyle h_{s}} . This is found by knowing the height of the sphere 2 R {\displaystyle 2R} also equals the cylinder's height h {\displaystyle h} plus two spherical cap heights 2 R = h + 2 h s ⟹ h s = R − h 2 {\displaystyle 2R=h+2h_{s}\qquad \implies \qquad h_{s}=R-{\frac {h}{2}}} Substituting r c {\displaystyle r_{c}} and h s {\displaystyle h_{s}} into the expression above for V n {\displaystyle V_{n}} one finds all terms containing R {\displaystyle R} cancel and one gets V n = π 6 h 3 {\displaystyle V_{n}={\frac {\pi }{6}}h^{3}} == See also == Visual calculus, an intuitive way to solve this type of problem, originally applied to finding the area of an annulus, given only its chord length String girdling Earth, another problem where the radius of a sphere or circle is counter-intuitively irrelevant == References == == Further reading == Devlin, Keith (2008), The Napkin Ring Problem, Mathematical Association of America, archived from the original on 30 April 2008, retrieved 25 February 2009 Devlin, Keith (2008), Lockhart's Lament, Mathematical Association of America, archived from the original on 10 May 2008, retrieved 25 February 2009 Gardner, Martin (1994), "Hole in the Sphere", My best mathematical and logic puzzles, Dover Publications, p. 8 Jones, Samuel I. (1912), Mathematical Wrinkles for Teachers and Private Learners, Norwood, MA: J. B. Cushing Co. Problem 132 asks for the volume of a sphere with a cylindrical hole drilled through it, but does not note the invariance of the problem under changes of radius. Levi, Mark (2009), "6.3 How Much Gold Is in a Wedding Ring?", The Mathematical Mechanic: Using Physical Reasoning to Solve Problems, Princeton University Press, pp. 102–104, ISBN 978-0-691-14020-9. Levi argues that the volume depends only on the height of the hole based on the fact that the ring can be swept out by a half-disk with the height as its diameter. Lines, L. (1965), Solid geometry: With Chapters on Space-lattices, Sphere-packs and Crystals, Dover. Reprint of 1935 edition. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a "napkin ring" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole. Pólya, George (1990), Mathematics and Plausible Reasoning, Vol. I: Induction and Analogy in Mathematics, Princeton University Press, pp. 191–192. Reprint of 1954 edition. == External links == Weisstein, Eric W., "Spherical Ring", MathWorld
Wikipedia:Narasimhan–Seshadri theorem#0	In mathematics, the Narasimhan–Seshadri theorem, proved by Narasimhan and Seshadri (1965), says that a holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible projective unitary representation of the fundamental group. The main case to understand is that of topologically trivial bundles, i.e. those of degree zero (and the other cases are a minor technical extension of this case). This case of the Narasimhan–Seshadri theorem says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible unitary representation of the fundamental group of the Riemann surface. Donaldson (1983) gave another proof using differential geometry, and showed that the stable vector bundles have an essentially unique unitary connection of constant (scalar) curvature. In the degree zero case, Donaldson's version of the theorem says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it admits a flat unitary connection compatible with its holomorphic structure. Then the fundamental group representation appearing in the original statement is just the monodromy representation of this flat unitary connection. == See also == Nonabelian Hodge correspondence Kobayashi–Hitchin correspondence Stable vector bundle == References == Donaldson, S. K. (1983), "A new proof of a theorem of Narasimhan and Seshadri", Journal of Differential Geometry, 18 (2): 269–277, ISSN 0022-040X, MR 0710055 Narasimhan, M. S.; Seshadri, C. S. (1965), "Stable and unitary vector bundles on a compact Riemann surface", Annals of Mathematics, Second Series, 82: 540–567, doi:10.2307/1970710, ISSN 0003-486X, MR 0184252
Wikipedia:Nash equilibrium#0	In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed). The idea of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly. If each player has chosen a strategy – an action plan based on what has happened so far in the game – and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice choosing A. In a game in which Carol and Dan are also players, (A, B, C, D) is a Nash equilibrium if A is Alice's best response to (B, C, D), B is Bob's best response to (A, C, D), and so forth. John Nash showed that there is a Nash equilibrium, possibly in mixed strategies, for every finite game. == Applications == Game theorists use Nash equilibrium to analyze the outcome of the strategic interaction of several decision makers. In a strategic interaction, the outcome for each decision-maker depends on the decisions of the others as well as their own. The simple insight underlying Nash's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do taking into account what the player expects the others to do. Nash equilibrium requires that one's choices be consistent: no players wish to undo their decision given what the others are deciding. The concept has been used to analyze hostile situations such as wars and arms races (see prisoner's dilemma), and also how conflict may be mitigated by repeated interaction (see tit-for-tat). It has also been used to study to what extent people with different preferences can cooperate (see battle of the sexes), and whether they will take risks to achieve a cooperative outcome (see stag hunt). It has been used to study the adoption of technical standards, and also the occurrence of bank runs and currency crises (see coordination game). Other applications include traffic flow (see Wardrop's principle), how to organize auctions (see auction theory), the outcome of efforts exerted by multiple parties in the education process, regulatory legislation such as environmental regulations (see tragedy of the commons), natural resource management, analysing strategies in marketing, penalty kicks in football (I.e. soccer; see matching pennies), robot navigation in crowds, energy systems, transportation systems, evacuation problems and wireless communications. == History == Nash equilibrium is named after American mathematician John Forbes Nash Jr. The same idea was used in a particular application in 1838 by Antoine Augustin Cournot in his theory of oligopoly. In Cournot's theory, each of several firms choose how much output to produce to maximize its profit. The best output for one firm depends on the outputs of the others. A Cournot equilibrium occurs when each firm's output maximizes its profits given the output of the other firms, which is a pure-strategy Nash equilibrium. Cournot also introduced the concept of best response dynamics in his analysis of the stability of equilibrium. Cournot did not use the idea in any other applications, however, or define it generally. The modern concept of Nash equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible pure strategies (which might put 100% of the probability on one pure strategy; such pure strategies are a subset of mixed strategies). The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior, but their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions. The contribution of Nash in his 1951 article "Non-Cooperative Games" was to define a mixed-strategy Nash equilibrium for any game with a finite set of actions and prove that at least one (mixed-strategy) Nash equilibrium must exist in such a game. The key to Nash's ability to prove existence far more generally than von Neumann lay in his definition of equilibrium. According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes [their] payoff if the strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others." Putting the problem in this framework allowed Nash to employ the Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same purpose. Game theorists have discovered that in some circumstances Nash equilibrium makes invalid predictions or fails to make a unique prediction. They have proposed many solution concepts ('refinements' of Nash equilibria) designed to rule out implausible Nash equilibria. One particularly important issue is that some Nash equilibria may be based on threats that are not 'credible'. In 1965 Reinhard Selten proposed subgame perfect equilibrium as a refinement that eliminates equilibria which depend on non-credible threats. Other extensions of the Nash equilibrium concept have addressed what happens if a game is repeated, or what happens if a game is played in the absence of complete information. However, subsequent refinements and extensions of Nash equilibrium share the main insight on which Nash's concept rests: the equilibrium is a set of strategies such that each player's strategy is optimal given the choices of the others. == Definitions == === Nash equilibrium === A strategy profile is a set of strategies, one for each player. Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing their strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?" For instance if a player prefers "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to the other players' strategies in that equilibrium. Formally, let S i {\displaystyle S_{i}} be the set of all possible strategies for player i {\displaystyle i} , where i = 1 , … , N {\displaystyle i=1,\ldots ,N} . Let s ∗ = ( s i ∗ , s − i ∗ ) {\displaystyle s^{}=(s_{i}^{},s_{-i}^{})} be a strategy profile, a set consisting of one strategy for each player, where s − i ∗ {\displaystyle s_{-i}^{}} denotes the N − 1 {\displaystyle N-1} strategies of all the players except i {\displaystyle i} . Let u i ( s i , s − i ∗ ) {\displaystyle u_{i}(s_{i},s_{-i}^{})} be player i's payoff as a function of the strategies. The strategy profile s ∗ {\displaystyle s^{}} is a Nash equilibrium if u i ( s i ∗ , s − i ∗ ) ≥ u i ( s i , s − i ∗ ) for all s i ∈ S i . {\displaystyle u_{i}(s_{i}^{},s_{-i}^{})\geq u_{i}(s_{i},s_{-i}^{})\ {\text{for all}}\ s_{i}\in S_{i}.} A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be weak: a player might be indifferent among several strategies given the other players' choices. It is unique and called a strict Nash equilibrium if the inequality is strict so one strategy is the unique best response: u i ( s i ∗ , s − i ∗ ) > u i ( s i , s − i ∗ ) for all s i ∈ S i , s i ≠ s i ∗ . {\displaystyle u_{i}(s_{i}^{},s_{-i}^{})>u_{i}(s_{i},s_{-i}^{})\ {\text{for all}}\ s_{i}\in S_{i},s_{i}\neq s_{i}^{}.} The strategy set S i {\displaystyle S_{i}} can be different for different players, and its elements can be a variety of mathematical objects. Most simply, a player might choose between two strategies, e.g. S i = { Yes , No } . {\displaystyle S_{i}=\{{\text{Yes}},{\text{No}}\}.} Or the strategy set might be a finite set of conditional strategies responding to other players, e.g. S i = { Yes ∣ p = Low , No ∣ p = High } . {\displaystyle S_{i}=\{{\text{Yes}}\mid p={\text{Low}},{\text{No}}\mid p={\text{High}}\}.} Or it might be an infinite set, a continuum or unbounded, e.g. S i = { Price } {\displaystyle S_{i}=\{{\text{Price}}\}} such that Price {\displaystyle {\text{Price}}} is a non-negative real number. Nash's existing proofs assume a finite strategy set, but the concept of Nash equilibrium does not require it. == Variants == === Pure/mixed equilibrium === A game can have a pure-strategy or a mixed-strategy Nash equilibrium. In the latter, not every player always plays the same strategy. Instead, there is a probability distribution over different strategies. === Strict/non-strict equilibrium === Suppose that in the Nash equilibrium, each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?" If every player's answer is "Yes", then the equilibrium is classified as a strict Nash equilibrium. If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that gives exactly the same payout (i.e. the player is indifferent between switching and not), then the equilibrium is classified as a weak or non-strict Nash equilibrium. === Equilibria for coalitions === The Nash equilibrium defines stability only in terms of individual player deviations. In cooperative games such a concept is not convincing enough. Strong Nash equilibrium allows for deviations by every conceivable coalition. Formally, a strong Nash equilibrium is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. However, the strong Nash concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be weakly Pareto efficient. As a result of these requirements, strong Nash is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium. A refined Nash equilibrium known as coalition-proof Nash equilibrium (CPNE) occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate. Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k. CPNE is related to the theory of the core. == Existence == === Nash's existence theorem === Nash proved that if mixed strategies (where a player chooses probabilities of using various pure strategies) are allowed, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium, which might be a pure strategy for each player or might be a probability distribution over strategies for each player. Nash equilibria need not exist if the set of choices is infinite and non-compact. For example: A game where two players simultaneously name a number and the player naming the larger number wins does not have a NE, as the set of choices is not compact because it is unbounded. Each of two players chooses a real number strictly less than 5 and the winner is whoever has the biggest number; no biggest number strictly less than 5 exists (if the number could equal 5, the Nash equilibrium would have both players choosing 5 and tying the game). Here, the set of choices is not compact because it is not closed. However, a Nash equilibrium exists if the set of choices is compact with each player's payoff continuous in the strategies of all the players. === Rosen's existence theorem === Rosen extended Nash's existence theorem in several ways. He considers an n-player game, in which the strategy of each player i is a vector si in the Euclidean space Rmi. Denote m:=m1+...+mn; so a strategy-tuple is a vector in Rm. Part of the definition of a game is a subset S of Rm such that the strategy-tuple must be in S. This means that the actions of players may potentially be constrained based on actions of other players. A common special case of the model is when S is a Cartesian product of convex sets S1,...,Sn, such that the strategy of player i must be in Si. This represents the case that the actions of each player i are constrained independently of other players' actions. If the following conditions hold: T is convex, closed and bounded; Each payoff function ui is continuous in the strategies of all players, and concave in si for every fixed value of s−i. Then a Nash equilibrium exists. The proof uses the Kakutani fixed-point theorem. Rosen also proves that, under certain technical conditions which include strict concavity, the equilibrium is unique. Nash's result refers to the special case in which each Si is a simplex (representing all possible mixtures of pure strategies), and the payoff functions of all players are bilinear functions of the strategies. == Rationality == The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily Pareto optimal. Nash equilibrium may also have non-rational consequences in sequential games because players may "threaten" each other with threats they would not actually carry out. For such games the subgame perfect Nash equilibrium may be more meaningful as a tool of analysis. == Examples == === Coordination game === The coordination game is a classic two-player, two-strategy game, as shown in the example payoff matrix to the right. There are two pure-strategy equilibria, (A,A) with payoff 4 for each player and (B,B) with payoff 2 for each. The combination (B,B) is a Nash equilibrium because if either player unilaterally changes their strategy from B to A, their payoff will fall from 2 to 1. A famous example of a coordination game is the stag hunt. Two players may choose to hunt a stag or a rabbit, the stag providing more meat (4 utility units, 2 for each player) than the rabbit (1 utility unit). The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, the stag hunter will totally fail, for a payoff of 0, whereas the rabbit hunter will succeed, for a payoff of 1. The game has two equilibria, (stag, stag) and (rabbit, rabbit), because a player's optimal strategy depends on their expectation on what the other player will do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they think the other will hunt the rabbit, they too will hunt the rabbit. This game is used as an analogy for social cooperation, since much of the benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation. Driving on a road against an oncoming car, and having to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix: In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%, 100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player are (50%, 50%). === Network traffic === An application of Nash equilibria is in determining the expected flow of traffic in a network. Consider the graph on the right. If we assume that there are x {\displaystyle x} "cars" traveling from A to D, what is the expected distribution of traffic in the network? This situation can be modeled as a "game", where every traveler has a choice of 3 strategies and where each strategy is a route from A to D (one of ABD, ABCD, or ACD). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via ABD experiences travel time of 1 + x 100 + 2 {\displaystyle 1+{\frac {x}{100}}+2} , where x {\displaystyle x} is the number of cars traveling on edge AB. Thus, payoffs for any given strategy depend on the choices of the other players, as is usual. However, the goal, in this case, is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same. When that happens, no single driver has any incentive to switch routes, since it can only add to their travel time. For the graph on the right, if, for example, 100 cars are travelling from A to D, then equilibrium will occur when 25 drivers travel via ABD, 50 via ABCD, and 25 via ACD. Every driver now has a total travel time of 3.75 (to see this, a total of 75 cars take the AB edge, and likewise, 75 cars take the CD edge). Notice that this distribution is not, actually, socially optimal. If the 100 cars agreed that 50 travel via ABD and the other 50 through ACD, then travel time for any single car would actually be 3.5, which is less than 3.75. This is also the Nash equilibrium if the path between B and C is removed, which means that adding another possible route can decrease the efficiency of the system, a phenomenon known as Braess's paradox. === Competition game === This can be illustrated by a two-player game in which both players simultaneously choose an integer from 0 to 3 and they both win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then they have to give up two points to the other. This game has a unique pure-strategy Nash equilibrium: both players choosing 0 (highlighted in light red). Any other strategy can be improved by a player switching their number to one less than that of the other player. In the adjacent table, if the game begins at the green square, it is in player 1's interest to move to the purple square and it is in player 2's interest to move to the blue square. Although it would not fit the definition of a competition game, if the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria: (0,0), (1,1), (2,2), and (3,3). === Nash equilibria in a payoff matrix === There is an easy numerical way to identify Nash equilibria on a payoff matrix. It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the payoff pair of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell – then the cell represents a Nash equilibrium. We can apply this rule to a 3×3 matrix: Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash equilibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A), 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B), 25 is the maximum of the second column and 40 is the maximum of the first row; the same applies for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns. This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the pair is the maximum of the row. If these conditions are met, the cell represents a Nash equilibrium. Check all columns this way to find all NE cells. An N×N matrix may have between 0 and N×N pure-strategy Nash equilibria. == Stability == The concept of stability, useful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria. A Nash equilibrium for a mixed-strategy game is stable if a small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold: the player who did not change has no better strategy in the new circumstance the player who did change is now playing with a strictly worse strategy. If these cases are both met, then a player with the small change in their mixed strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed. In the "driving game" example above there are both stable and unstable equilibria. The equilibria involving mixed strategies with 100% probabilities are stable. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn. The (50%,50%) equilibrium is unstable. If either player changes their probabilities (which would neither benefit or damage the expectation of the player who did the change, if the other player's mixed strategy is still (50%,50%)), then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%). Stability is crucial in practical applications of Nash equilibria, since the mixed strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium. Finally in the eighties, building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept. Mertens stable equilibria satisfy both forward induction and backward induction. In a game theory context stable equilibria now usually refer to Mertens stable equilibria. == Occurrence == If a game has a unique Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are: The players all will do their utmost to maximize their expected payoff as described by the game. The players are flawless in execution. The players have sufficient intelligence to deduce the solution. The players know the planned equilibrium strategy of all of the other players. The players believe that a deviation in their own strategy will not cause deviations by any other players. There is common knowledge that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on. === Where the conditions are not met === Examples of game theory problems in which these conditions are not met: The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy. For instance, the prisoner's dilemma is not a dilemma if either player is happy to be jailed indefinitely. Intentional or accidental imperfection in execution. For example, a computer capable of flawless logical play facing a second flawless computer will result in equilibrium. Introduction of imperfection will lead to its disruption either through loss to the player who makes the mistake, or through negation of the common knowledge criterion leading to possible victory for the player. (An example would be a player suddenly putting the car into reverse in the game of chicken, ensuring a no-loss no-win scenario). In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess. Or, if known, it may not be known to all players, as when playing tic-tac-toe with a small child who desperately wants to win (meeting the other criteria). The criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria. Players wrongly distrusting each other's rationality may adopt counter-strategies to expected irrational play on their opponents’ behalf. This is a major consideration in "chicken" or an arms race, for example. === Where the conditions are met === In his Ph.D. dissertation, John Nash proposed two interpretations of his equilibrium concept, with the objective of showing how equilibrium points can be connected with observable phenomenon. (...) One interpretation is rationalistic: if we assume that players are rational, know the full structure of the game, the game is played just once, and there is just one Nash equilibrium, then players will play according to that equilibrium. This idea was formalized by R. Aumann and A. Brandenburger, 1995, Epistemic Conditions for Nash Equilibrium, Econometrica, 63, 1161-1180 who interpreted each player's mixed strategy as a conjecture about the behaviour of other players and have shown that if the game and the rationality of players is mutually known and these conjectures are commonly known, then the conjectures must be a Nash equilibrium (a common prior assumption is needed for this result in general, but not in the case of two players. In this case, the conjectures need only be mutually known). A second interpretation, that Nash referred to by the mass action interpretation, is less demanding on players: [i]t is unnecessary to assume that the participants have full knowledge of the total structure of the game, or the ability and inclination to go through any complex reasoning processes. What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations. If there is a stable average frequency with which each pure strategy is employed by the average member of the appropriate population, then this stable average frequency constitutes a mixed strategy Nash equilibrium. For a formal result along these lines, see Kuhn, H. and et al., 1996, "The Work of John Nash in Game Theory", Journal of Economic Theory, 69, 153–185. Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in economics and evolutionary biology, the NE has explanatory power. The payoff in economics is utility (or sometimes money), and in evolutionary biology is gene transmission; both are the fundamental bottom line of survival. Researchers who apply games theory in these fields claim that strategies failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the "stability" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research. == NE and non-credible threats == The Nash equilibrium is a superset of the subgame perfect Nash equilibrium. The subgame perfect equilibrium in addition to the Nash equilibrium requires that the strategy also is a Nash equilibrium in every subgame of that game. This eliminates all non-credible threats, that is, strategies that contain non-rational moves in order to make the counter-player change their strategy. The image to the right shows a simple sequential game that illustrates the issue with subgame imperfect Nash equilibria. In this game player one chooses left(L) or right(R), which is followed by player two being called upon to be kind (K) or unkind (U) to player one, However, player two only stands to gain from being unkind if player one goes left. If player one goes right the rational player two would de facto be kind to her/him in that subgame. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L, (U,U)) Nash equilibrium. Therefore, if rational behavior can be expected by both parties the subgame perfect Nash equilibrium may be a more meaningful solution concept when such dynamic inconsistencies arise. == Proof of existence == === Proof using the Kakutani fixed-point theorem === Nash's original proof (in his thesis) used Brouwer's fixed-point theorem (e.g., see below for a variant). This section presents a simpler proof via the Kakutani fixed-point theorem, following Nash's 1950 paper (he credits David Gale with the observation that such a simplification is possible). To prove the existence of a Nash equilibrium, let r i ( σ − i ) {\displaystyle r_{i}(\sigma _{-i})} be the best response of player i to the strategies of all other players. r i ( σ − i ) = a r g m a x σ i ⁡ u i ( σ i , σ − i ) {\displaystyle r_{i}(\sigma _{-i})=\mathop {\underset {\sigma _{i}}{\operatorname {arg\,max} }} u_{i}(\sigma _{i},\sigma _{-i})} Here, σ ∈ Σ {\displaystyle \sigma \in \Sigma } , where Σ = Σ i × Σ − i {\displaystyle \Sigma =\Sigma _{i}\times \Sigma _{-i}} , is a mixed-strategy profile in the set of all mixed strategies and u i {\displaystyle u_{i}} is the payoff function for player i. Define a set-valued function r : Σ → 2 Σ {\displaystyle r\colon \Sigma \rightarrow 2^{\Sigma }} such that r = r i ( σ − i ) × r − i ( σ i ) {\displaystyle r=r_{i}(\sigma _{-i})\times r_{-i}(\sigma _{i})} . The existence of a Nash equilibrium is equivalent to r {\displaystyle r} having a fixed point. Kakutani's fixed point theorem guarantees the existence of a fixed point if the following four conditions are satisfied. Σ {\displaystyle \Sigma } is compact, convex, and nonempty. r ( σ ) {\displaystyle r(\sigma )} is nonempty. r ( σ ) {\displaystyle r(\sigma )} is upper hemicontinuous r ( σ ) {\displaystyle r(\sigma )} is convex. Condition 1. is satisfied from the fact that Σ {\displaystyle \Sigma } is a simplex and thus compact. Convexity follows from players' ability to mix strategies. Σ {\displaystyle \Sigma } is nonempty as long as players have strategies. Condition 2. and 3. are satisfied by way of Berge's maximum theorem. Because u i {\displaystyle u_{i}} is continuous and compact, r ( σ i ) {\displaystyle r(\sigma _{i})} is non-empty and upper hemicontinuous. Condition 4. is satisfied as a result of mixed strategies. Suppose σ i , σ i ′ ∈ r ( σ − i ) {\displaystyle \sigma _{i},\sigma '_{i}\in r(\sigma _{-i})} , then λ σ i + ( 1 − λ ) σ i ′ ∈ r ( σ − i ) {\displaystyle \lambda \sigma _{i}+(1-\lambda )\sigma '_{i}\in r(\sigma _{-i})} . i.e. if two strategies maximize payoffs, then a mix between the two strategies will yield the same payoff. Therefore, there exists a fixed point in r {\displaystyle r} and a Nash equilibrium. When Nash made this point to John von Neumann in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a fixed-point theorem." (See Nasar, 1998, p. 94.) === Alternate proof using the Brouwer fixed-point theorem === We have a game G = ( N , A , u ) {\displaystyle G=(N,A,u)} where N {\displaystyle N} is the number of players and A = A 1 × ⋯ × A N {\displaystyle A=A_{1}\times \cdots \times A_{N}} is the action set for the players. All of the action sets A i {\displaystyle A_{i}} are finite. Let Δ = Δ 1 × ⋯ × Δ N {\displaystyle \Delta =\Delta _{1}\times \cdots \times \Delta _{N}} denote the set of mixed strategies for the players. The finiteness of the A i {\displaystyle A_{i}} s ensures the compactness of Δ {\displaystyle \Delta } . We can now define the gain functions. For a mixed strategy σ ∈ Δ {\displaystyle \sigma \in \Delta } , we let the gain for player i {\displaystyle i} on action a ∈ A i {\displaystyle a\in A_{i}} be Gain i ( σ , a ) = max { 0 , u i ( a , σ − i ) − u i ( σ i , σ − i ) } . {\displaystyle {\text{Gain}}_{i}(\sigma ,a)=\max\{0,u_{i}(a,\sigma _{-i})-u_{i}(\sigma _{i},\sigma _{-i})\}.} The gain function represents the benefit a player gets by unilaterally changing their strategy. We now define g = ( g 1 , … , g N ) {\displaystyle g=(g_{1},\dotsc ,g_{N})} where g i ( σ ) ( a ) = σ i ( a ) + Gain i ( σ , a ) {\displaystyle g_{i}(\sigma )(a)=\sigma _{i}(a)+{\text{Gain}}_{i}(\sigma ,a)} for σ ∈ Δ , a ∈ A i {\displaystyle \sigma \in \Delta ,a\in A_{i}} . We see that ∑ a ∈ A i g i ( σ ) ( a ) = ∑ a ∈ A i σ i ( a ) + Gain i ( σ , a ) = 1 + ∑ a ∈ A i Gain i ( σ , a ) > 0. {\displaystyle \sum _{a\in A_{i}}g_{i}(\sigma )(a)=\sum _{a\in A_{i}}\sigma _{i}(a)+{\text{Gain}}_{i}(\sigma ,a)=1+\sum _{a\in A_{i}}{\text{Gain}}_{i}(\sigma ,a)>0.} Next we define: { f = ( f 1 , ⋯ , f N ) : Δ → Δ f i ( σ ) ( a ) = g i ( σ ) ( a ) ∑ b ∈ A i g i ( σ ) ( b ) a ∈ A i {\displaystyle {\begin{cases}f=(f_{1},\cdots ,f_{N}):\Delta \to \Delta \\f_{i}(\sigma )(a)={\frac {g_{i}(\sigma )(a)}{\sum _{b\in A_{i}}g_{i}(\sigma )(b)}}&a\in A_{i}\end{cases}}} It is easy to see that each f i {\displaystyle f_{i}} is a valid mixed strategy in Δ i {\displaystyle \Delta _{i}} . It is also easy to check that each f i {\displaystyle f_{i}} is a continuous function of σ {\displaystyle \sigma } , and hence f {\displaystyle f} is a continuous function. As the cross product of a finite number of compact convex sets, Δ {\displaystyle \Delta } is also compact and convex. Applying the Brouwer fixed point theorem to f {\displaystyle f} and Δ {\displaystyle \Delta } we conclude that f {\displaystyle f} has a fixed point in Δ {\displaystyle \Delta } , call it σ ∗ {\displaystyle \sigma ^{}} . We claim that σ ∗ {\displaystyle \sigma ^{}} is a Nash equilibrium in G {\displaystyle G} . For this purpose, it suffices to show that ∀ i ∈ { 1 , ⋯ , N } , ∀ a ∈ A i : Gain i ( σ ∗ , a ) = 0. {\displaystyle \forall i\in \{1,\cdots ,N\},\forall a\in A_{i}:\quad {\text{Gain}}_{i}(\sigma ^{},a)=0.} This simply states that each player gains no benefit by unilaterally changing their strategy, which is exactly the necessary condition for a Nash equilibrium. Now assume that the gains are not all zero. Therefore, ∃ i ∈ { 1 , ⋯ , N } , {\displaystyle \exists i\in \{1,\cdots ,N\},} and a ∈ A i {\displaystyle a\in A_{i}} such that Gain i ( σ ∗ , a ) > 0 {\displaystyle {\text{Gain}}_{i}(\sigma ^{},a)>0} . Then ∑ a ∈ A i g i ( σ ∗ , a ) = 1 + ∑ a ∈ A i Gain i ( σ ∗ , a ) > 1. {\displaystyle \sum _{a\in A_{i}}g_{i}(\sigma ^{},a)=1+\sum _{a\in A_{i}}{\text{Gain}}_{i}(\sigma ^{},a)>1.} So let C = ∑ a ∈ A i g i ( σ ∗ , a ) . {\displaystyle C=\sum _{a\in A_{i}}g_{i}(\sigma ^{},a).} Also we shall denote Gain ( i , ⋅ ) {\displaystyle {\text{Gain}}(i,\cdot )} as the gain vector indexed by actions in A i {\displaystyle A_{i}} . Since σ ∗ {\displaystyle \sigma ^{}} is the fixed point we have: σ ∗ = f ( σ ∗ ) ⇒ σ i ∗ = f i ( σ ∗ ) ⇒ σ i ∗ = g i ( σ ∗ ) ∑ a ∈ A i g i ( σ ∗ ) ( a ) ⇒ σ i ∗ = 1 C ( σ i ∗ + Gain i ( σ ∗ , ⋅ ) ) ⇒ C σ i ∗ = σ i ∗ + Gain i ( σ ∗ , ⋅ ) ⇒ ( C − 1 ) σ i ∗ = Gain i ( σ ∗ , ⋅ ) ⇒ σ i ∗ = ( 1 C − 1 ) Gain i ( σ ∗ , ⋅ ) . {\displaystyle {\begin{aligned}\sigma ^{}=f(\sigma ^{})&\Rightarrow \sigma _{i}^{}=f_{i}(\sigma ^{})\\&\Rightarrow \sigma _{i}^{}={\frac {g_{i}(\sigma ^{})}{\sum _{a\in A_{i}}g_{i}(\sigma ^{})(a)}}\\[6pt]&\Rightarrow \sigma _{i}^{}={\frac {1}{C}}\left(\sigma _{i}^{}+{\text{Gain}}_{i}(\sigma ^{},\cdot )\right)\\[6pt]&\Rightarrow C\sigma _{i}^{}=\sigma _{i}^{}+{\text{Gain}}_{i}(\sigma ^{},\cdot )\\&\Rightarrow \left(C-1\right)\sigma _{i}^{}={\text{Gain}}_{i}(\sigma ^{},\cdot )\\&\Rightarrow \sigma _{i}^{}=\left({\frac {1}{C-1}}\right){\text{Gain}}_{i}(\sigma ^{},\cdot ).\end{aligned}}} Since C > 1 {\displaystyle C>1} we have that σ i ∗ {\displaystyle \sigma _{i}^{}} is some positive scaling of the vector Gain i ( σ ∗ , ⋅ ) {\displaystyle {\text{Gain}}_{i}(\sigma ^{},\cdot )} . Now we claim that ∀ a ∈ A i : σ i ∗ ( a ) ( u i ( a i , σ − i ∗ ) − u i ( σ i ∗ , σ − i ∗ ) ) = σ i ∗ ( a ) Gain i ( σ ∗ , a ) {\displaystyle \forall a\in A_{i}:\quad \sigma _{i}^{}(a)(u_{i}(a_{i},\sigma _{-i}^{})-u_{i}(\sigma _{i}^{},\sigma _{-i}^{}))=\sigma _{i}^{}(a){\text{Gain}}_{i}(\sigma ^{},a)} To see this, first if Gain i ( σ ∗ , a ) > 0 {\displaystyle {\text{Gain}}_{i}(\sigma ^{},a)>0} then this is true by definition of the gain function. Now assume that Gain i ( σ ∗ , a ) = 0 {\displaystyle {\text{Gain}}_{i}(\sigma ^{},a)=0} . By our previous statements we have that σ i ∗ ( a ) = ( 1 C − 1 ) Gain i ( σ ∗ , a ) = 0 {\displaystyle \sigma _{i}^{}(a)=\left({\frac {1}{C-1}}\right){\text{Gain}}_{i}(\sigma ^{},a)=0} and so the left term is zero, giving us that the entire expression is 0 {\displaystyle 0} as needed. So we finally have that 0 = u i ( σ i ∗ , σ − i ∗ ) − u i ( σ i ∗ , σ − i ∗ ) = ( ∑ a ∈ A i σ i ∗ ( a ) u i ( a i , σ − i ∗ ) ) − u i ( σ i ∗ , σ − i ∗ ) = ∑ a ∈ A i σ i ∗ ( a ) ( u i ( a i , σ − i ∗ ) − u i ( σ i ∗ , σ − i ∗ ) ) = ∑ a ∈ A i σ i ∗ ( a ) Gain i ( σ ∗ , a ) by the previous statements = ∑ a ∈ A i ( C − 1 ) σ i ∗ ( a ) 2 > 0 {\displaystyle {\begin{aligned}0&=u_{i}(\sigma _{i}^{},\sigma _{-i}^{})-u_{i}(\sigma _{i}^{},\sigma _{-i}^{})\\&=\left(\sum _{a\in A_{i}}\sigma _{i}^{}(a)u_{i}(a_{i},\sigma _{-i}^{})\right)-u_{i}(\sigma _{i}^{},\sigma _{-i}^{})\\&=\sum _{a\in A_{i}}\sigma _{i}^{}(a)(u_{i}(a_{i},\sigma _{-i}^{})-u_{i}(\sigma _{i}^{},\sigma _{-i}^{}))\\&=\sum _{a\in A_{i}}\sigma _{i}^{}(a){\text{Gain}}_{i}(\sigma ^{},a)&&{\text{ by the previous statements }}\\&=\sum _{a\in A_{i}}\left(C-1\right)\sigma _{i}^{}(a)^{2}>0\end{aligned}}} where the last inequality follows since σ i ∗ {\displaystyle \sigma _{i}^{}} is a non-zero vector. But this is a clear contradiction, so all the gains must indeed be zero. Therefore, σ ∗ {\displaystyle \sigma ^{*}} is a Nash equilibrium for G {\displaystyle G} as needed. == Computing Nash equilibria == If a player A has a dominant strategy s A {\displaystyle s_{A}} then there exists a Nash equilibrium in which A plays s A {\displaystyle s_{A}} . In the case of two players A and B, there exists a Nash equilibrium in which A plays s A {\displaystyle s_{A}} and B plays a best response to s A {\displaystyle s_{A}} . If s A {\displaystyle s_{A}} is a strictly dominant strategy, A plays s A {\displaystyle s_{A}} in all Nash equilibria. If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays their strictly dominant strategy. In games with mixed-strategy Nash equilibria, the probability of a player choosing any particular (so pure) strategy can be computed by assigning a variable to each strategy that represents a fixed probability for choosing that strategy. In order for a player to be willing to randomize, their expected payoff for each (pure) strategy should be the same. In addition, the sum of the probabilities for each strategy of a particular player should be 1. This creates a system of equations from which the probabilities of choosing each strategy can be derived. === Examples === In the matching pennies game, player A loses a point to B if A and B play the same strategy and wins a point from B if they play different strategies. To compute the mixed-strategy Nash equilibrium, assign A the probability p {\displaystyle p} of playing H and ( 1 − p ) {\displaystyle (1-p)} of playing T, and assign B the probability q {\displaystyle q} of playing H and ( 1 − q ) {\displaystyle (1-q)} of playing T. E [ payoff for A playing H ] = ( − 1 ) q + ( + 1 ) ( 1 − q ) = 1 − 2 q , E [ payoff for A playing T ] = ( + 1 ) q + ( − 1 ) ( 1 − q ) = 2 q − 1 , E [ payoff for A playing H ] = E [ payoff for A playing T ] ⟹ 1 − 2 q = 2 q − 1 ⟹ q = 1 2 . E [ payoff for B playing H ] = ( + 1 ) p + ( − 1 ) ( 1 − p ) = 2 p − 1 , E [ payoff for B playing T ] = ( − 1 ) p + ( + 1 ) ( 1 − p ) = 1 − 2 p , E [ payoff for B playing H ] = E [ payoff for B playing T ] ⟹ 2 p − 1 = 1 − 2 p ⟹ p = 1 2 . {\displaystyle {\begin{aligned}&\mathbb {E} [{\text{payoff for A playing H}}]=(-1)q+(+1)(1-q)=1-2q,\\&\mathbb {E} [{\text{payoff for A playing T}}]=(+1)q+(-1)(1-q)=2q-1,\\&\mathbb {E} [{\text{payoff for A playing H}}]=\mathbb {E} [{\text{payoff for A playing T}}]\implies 1-2q=2q-1\implies q={\frac {1}{2}}.\\&\mathbb {E} [{\text{payoff for B playing H}}]=(+1)p+(-1)(1-p)=2p-1,\\&\mathbb {E} [{\text{payoff for B playing T}}]=(-1)p+(+1)(1-p)=1-2p,\\&\mathbb {E} [{\text{payoff for B playing H}}]=\mathbb {E} [{\text{payoff for B playing T}}]\implies 2p-1=1-2p\implies p={\frac {1}{2}}.\end{aligned}}} Thus, a mixed-strategy Nash equilibrium in this game is for each player to randomly choose H or T with p = 1 2 {\displaystyle p={\frac {1}{2}}} and q = 1 2 {\displaystyle q={\frac {1}{2}}} . == Oddness of equilibrium points == In 1971, Robert Wilson came up with the "oddness theorem", which says that "almost all" finite games have a finite and odd number of Nash equilibria. In 1993, Harsanyi published an alternative proof of the result. "Almost all" here means that any game with an infinite or even number of equilibria is very special in the sense that if its payoffs were even slightly randomly perturbed, with probability one it would have an odd number of equilibria instead. The prisoner's dilemma, for example, has one equilibrium, while the battle of the sexes has three—two pure and one mixed, and this remains true even if the payoffs change slightly. The free money game is an example of a "special" game with an even number of equilibria. In it, two players have to both vote "yes" rather than "no" to get a reward and the votes are simultaneous. There are two pure-strategy Nash equilibria, (yes, yes) and (no, no), and no mixed strategy equilibria, because the strategy "yes" weakly dominates "no". "Yes" is as good as "no" regardless of the other player's action, but if there is any chance the other player chooses "yes" then "yes" is the best reply. Under a small random perturbation of the payoffs, however, the probability that any two payoffs would remain tied, whether at 0 or some other number, is vanishingly small, and the game would have either one or three equilibria instead. == See also == == Notes == == References == == Bibliography == === Game theory textbooks === === Original Nash papers === === Other references === == External links == "Nash theorem (in game theory)", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Complete Proof of Existence of Nash Equilibria Simplified Form and Related Results Archived 2021-07-31 at the Wayback Machine
Wikipedia:Nassif Ghoussoub#0	Nassif A. Ghoussoub is a Canadian mathematician working in the fields of non-linear analysis and partial differential equations. He is a Professor of Mathematics and a Distinguished University Scholar at the University of British Columbia. == Early life and education == Ghoussoub was born to Lebanese parents in Western Africa (now Mali). He completed his doctorat 3ème cycle (PhD) in 1975, and a Doctorat d'Etat in 1979 at the Pierre and Marie Curie University, where his advisors were Gustave Choquet and Antoine Brunel. == Career == Ghoussoub completed his post-doctoral fellowship at the Ohio State University during 1976–77. He then joined the University of British Columbia, where he currently holds a position of Professor of Mathematics and a Distinguished University Scholar. Ghoussoub is known for his work in functional analysis, non-linear analysis, and partial differential equations. He was vice-president of the Canadian Mathematical Society from 1994 to 1996, the founding director of the Pacific Institute for the Mathematical Sciences (PIMS) for the period 1996–2003, the co-editor-in-chief of the Canadian Journal of Mathematics during 1993–2002, a co-founder of the MITACS Network of Centres of Excellence, and is the founder and scientific director (2001 - 2020) of the Banff International Research Station (BIRS). In 1994, Ghoussoub became a fellow of the Royal Society of Canada, and in 2012, a fellow of the American Mathematical Society. Ghoussoub has been awarded multiple awards and distinctions, including the Coxeter-James prize in 1990, and the Jeffrey-Williams prize in 2007. He holds honorary doctorates from the Université Paris-Dauphine (France), and the University of Victoria (Canada). He was awarded the Queen Elizabeth II Diamond Jubilee Medal in 2012, and appointed to the Order of Canada in 2015, with the grade of officer for contributions to mathematics, research, and education. In 2018, Ghoussoub was elected a faculty representative on the University of British Columbia's Board of Governors. He will serve until February 29, 2020. Ghoussoub has previously served two consecutive terms in this role from 2008 to 2014. Ghoussoub's scholarly work has been cited over 5,900 times and has an h-index of 40. == Awards == Coxeter-James Prize, Canadian Mathematical Society (1990) Killam Senior Research Fellowship, UBC (1992) Fellow of the Royal Society of Canada (1994) Distinguished University Scholar, UBC (2003) Doctorat Honoris Causa, Paris Dauphine University Jeffery–Williams Prize, Canadian mathematical Society (2007) Faculty of Science Achievement Award for outstanding service and leadership, UBC (2007) David Borwein Distinguished Career Award, Canadian Mathematical Society (2010) Fellow of the American Mathematical Society (2012) Queen Elizabeth II Diamond Jubilee Medal (2012) Honorary Doctor of Science-University of Victoria (June 2015) Officer of the Order of Canada (December 2015) Inaugural fellow of the Canadian Mathematical Society, 2018 == Bibliography == === Selected Academic Publications === Ghoussoub, N.; Preiss, D. (1989). "A general mountain pass principle for locating and classifying critical points". Annales de l'Institut Henri Poincaré C. 6 (5): 321–330. Bibcode:1989AIHPC...6..321G. doi:10.1016/s0294-1449(16)30313-4. ISSN 0294-1449. S2CID 53449063. Ghoussoub, N.; Gui, C. (1 July 1998). "On a conjecture of De Giorgi and some related problems". Mathematische Annalen. 311 (3): 481–491. doi:10.1007/s002080050196. ISSN 0025-5831. S2CID 120215307. Ghoussoub, N.; Yuan, C. (6 July 2000). "Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents". Transactions of the American Mathematical Society. 352 (12): 5703–5743. doi:10.1090/s0002-9947-00-02560-5. ISSN 0002-9947. Ekeland, Ivar; Ghoussoub, Nassif (4 January 2002). "Selected new aspects of the calculus of variations in the large". Bulletin of the American Mathematical Society. 39 (2): 207–265. doi:10.1090/s0273-0979-02-00929-1. ISSN 0273-0979. Ghoussoub, Nassif; Guo, Yujin (2007). "On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case". SIAM Journal on Mathematical Analysis. 38 (5): 1423–1449. arXiv:math/0509534. doi:10.1137/050647803. ISSN 0036-1410. S2CID 10221543. === Books === Ghoussoub, N.; Moradifam, Amir (2013). Functional inequalities : new perspectives and new applications. Providence, Rhode Island. ISBN 978-0-8218-9152-0. OCLC 823209440.{{cite book}}: CS1 maint: location missing publisher (link) Ghoussoub, Nassif; Guo, Yujin (2010). Mathematical analysis of partial differential equations modeling electrostatic MEMS. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-4957-6. OCLC 466771715. Ghoussoub, N. (2008). Self-dual partial differential systems and their variational principles. New York: Springer. ISBN 978-0-387-84896-9. OCLC 258078803. Ghoussoub, N.; Cambridge University Press (1993). Duality and Perturbation Methods in Critical Point Theory. Cambridge: Cambridge University Press. ISBN 978-0-511-55170-3. OCLC 668203489. == See also == Banff International Research Station == References == == External links == Nassif Ghoussoub's homepage Piece of Mind, Nassif's personal blog A biography
Wikipedia:Natalia Komarova#0	Natalia L. Komarova (born 1971) is a Russian-American applied mathematician whose research concerns the mathematical modeling of cancer, the evolution of language, gun control, pop music, and other complex systems. She is a Professor of Mathematics and Dean's Scholar at the University of California, San Diego. == Education and career == Komarova studied physics at Moscow State University, earning a master's degree there in 1993. She completed her Ph.D. in 1998 at the University of Arizona. Her dissertation, Essays on Nonlinear Waves: Patterns under Water; Pulse Propagation through Random Media, was supervised by Alan C. Newell. After postdoctoral research at the University of Warwick, the Institute for Advanced Study, and the University of Chicago, Komarova became a lecturer at the University of Leeds in 2000. She moved to Rutgers University in 2003 and to the University of California, Irvine in 2004. At UC Irvine, she was named a Chancellor's Professor in 2017. In 2024 she moved to University of California, San Diego. == Recognition == Komarova won a Sloan Research Fellowship in 2005. In 2023, Komarova was elected a Fellow of the American Association for the Advancement of Science. == Books == Komarova is married to UC Irvine evolutionary biologist Dominik Wodarz. She has written three books with Wodarz: Computational Biology of Cancer: Lecture Notes and Mathematical Modeling (World Scientific, 2005) Dynamics Of Cancer: Mathematical Foundations Of Oncology (World Scientific, 2014) Targeted Cancer Treatment in Silico: Small Molecule Inhibitors and Oncolytic Viruses (Birkhäuser, 2014) == References == == External links == Home page Natalia Komarova publications indexed by Google Scholar
Wikipedia:Nataliya Kalashnykova#0	Nataliya Ivanovna Kalashnykova is a Soviet and Mexican mathematician specializing in mathematical optimization, and especially bilevel optimization, with applications in modeling human migration and in the pricing of natural gas and toll roads. She is a professor at the Autonomous University of Nuevo León, in the Facultad de Ciencias Físico Matemáticas. == Education and career == Kalashnykova earned a master's degree in mathematical sciences from Novosibirsk State University in 1978. She completed a doctorate there in 1989, through the Siberian Division of the Academy of Sciences of the USSR. Her dissertation, Control of Accuracy in Bi-Level Iteration Processes, was supervised by Vladimir Aleksandrovich Bulavsky. She also earned a second master's degree in economics from Sumy State University in Ukraine in 1999. She became a faculty member at the Altai State Technical University, at the Siberian State University of Telecommunications and Informatics in Novosibirsk, and at Sumy State University, and a postdoctoral researcher at the Central Economic Mathematical Institute. She moved to her present position in Mexico at the Autonomous University of Nuevo León in 2001. == Recognition == Kalashnykova is a member of the Mexican Academy of Sciences. == Personal life == Kalashnykova is married to Vyacheslav Kalashnikov Polishchuk, another former Soviet mathematician in Mexico. == Books == Kalashnykova is a coauthor of books including: Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks (with Stephan Dempe, Vyacheslav Kalashnikov, and Gerardo A. Pérez-Valdés, Springer, 2015) Public Interest and Private Enterprize [sic]: New Developments: Theoretical Results and Numerical Algorithms (with José Guadalupe Flores Muñiz, Viacheslav V. Kalashnikov, and Vladik Kreinovich, Lecture Notes in Networks and Systems 138, Springer, 2021) == References == == External links == Nataliya Kalashnykova publications indexed by Google Scholar
Wikipedia:Nataša Jonoska#0	Nataša Jonoska (Macedonian: Наташа Јоноска, pronounced [na'taʃa jɔ'noska]; born 1961, also spelled Natasha Jonoska) is a Macedonian mathematician and professor at the University of South Florida known for her work in DNA computing. Her research is about how biology performs computation, "in particular using formal models such as cellular or other finite types of automata, formal language theory symbolic dynamics, and topological graph theory to describe molecular computation". She received her bachelor's degree in mathematics and computer science from Ss. Cyril and Methodius University of Skopje in Yugoslavia (now North Macedonia) in 1984. She earned her PhD in mathematics from the State University of New York at Binghamton in 1993 with the dissertation "Synchronizing Representations of Sofic Systems". Her dissertation advisor was Tom Head. In 2007, she won the Rosenberg Tulip Award in DNA Computing for her work in applications of Automata theory and graph theory to DNA nanotechnology. She was elected a AAAS Fellow in 2014 for advancements in understanding information processing in molecular self-assembly. She is a board member for many journals including Theoretical Computer Science, the International Journal of Foundations of Computer Science, Computability, and Natural Computing. In 2022 she was awarded a Simons Fellowship. == Select publications == J. Chen, N. Jonoska, G. Rozenberg, (eds). Nanotechnology: Science and Computing, Springer- Verlag 2006. N. Jonoska, Gh. Paun, G. Rozenberg, (eds.). Aspects of Molecular Computing LNCS 2950, Springer-Verlag 2004. N. Jonoska, N.C. Seeman, (eds.). DNA Computing, Revised papers from the 7th International Meeting on DNA-Based Computers, LNCS 2340, Springer-Verlag 2002. == References == == External links == dblp computer science bibliography
Wikipedia:Nataša Pavlović#0	Nataša Pavlović is a Serbian mathematician who works as a professor of mathematics at the University of Texas at Austin. Her research concerns fluid dynamics and nonlinear dispersive partial differential equations. She is known for her work with Nets Katz pioneering an approach to constructing singularities in equations resembling the Navier–Stokes equations, by transferring a finite amount of energy through an infinitely decreasing sequence of time and length scales. Pavlović earned a bachelor's degree in mathematics from the University of Belgrade in 1996, and completed her doctorate from the University of Illinois at Chicago in 2002 under the joint supervision of Susan Friedlander and Nets Katz. After temporary positions at the Clay Mathematics Institute, Princeton University, Institute for Advanced Study, and Mathematical Sciences Research Institute, she joined the Princeton faculty in 2005, and moved to the University of Texas in 2007. She was a Sloan Research Fellow from 2008 to 2012. In 2015 she was elected as a fellow of the American Mathematical Society. From 2013-2015, Pavlović served as a Council Member at Large for the American Mathematical Society. == References == == External links == Home page
Wikipedia:Nathalie Eisenbaum#0	Nathalie Eisenbaum is a French mathematician, statistician, and probability theorist. She works as a director of research with the Centre national de la recherche scientifique, associated with the laboratory for applied mathematics at Paris Descartes University and was previously a researcher in the Laboratoire de Probabilités, Statistique et Modélisation (laboratory for probability, statistics, and modeling) at Pierre and Marie Curie University. Eisenbaum completed her doctorate at Pierre and Marie Curie University in 1989. Her dissertation, Temps locaux, excursions et lieu le plus visité par un mouvement brownien linéaire, was supervised by Marc Yor. She is a Fellow of the Institute of Mathematical Statistics. In 2011, Eisenbaum and Haya Kaspi shared the Itô Prize of the Bernoulli Society for Mathematical Statistics and Probability for their joint work on permanental point processes (processes whose joint intensity can be represented as a permanent). == References ==
Wikipedia:Nathalie Wahl#0	Nathalie Wahl (born 1976) is a Belgian mathematician specializing in topology, including algebraic topology, homotopy theory, and geometric topology. She is a professor of mathematics at the University of Copenhagen, where she directs the Copenhagen Center for Geometry and Topology. == Education and career == Wahl was born in Brussels, and earned a license in mathematics in 1998 at the Université libre de Bruxelles, advised by Jean-Paul Doignon. Her undergraduate thesis concerned infinite antimatroids, and she published the same material in 2001 as her first journal paper. She completed a Ph.D. at the University of Oxford in 2001, with a dissertation Ribbon Graphs and Related Operads in algebraic topology supervised by Ulrike Tillmann. After short-term positions at Northwestern University, Aarhus University, and the University of Chicago, she joined the Department of Mathematical Sciences at the University of Copenhagen in 2006, and was promoted to full professor there in 2010. In 2020 she became Center Leader of the Copenhagen Center for Geometry and Topology. == Recognition == In 2008, Wahl won the Young Elite Researcher Award (Ung Eliteforskerprisen) of the Independent Research Fund Denmark (Danmarks Frie Forskningsfond). In 2016, she was elected to the Danish Academy of Natural Sciences. == References == == External links == Home page Nathalie Wahl publications indexed by Google Scholar
Wikipedia:Nathan Divinsky#0	Nathan Joseph Harry Divinsky (October 29, 1925 – June 17, 2012) was a Canadian mathematician, university professor, chess master, writer, and politician. Divinsky was also known for being the former husband of the 19th prime minister of Canada, Kim Campbell. Divinsky and Campbell were married from 1972 to 1983. == Early life, education, and academic career == He was born in Winnipeg, Manitoba, in 1925, and was a contemporary and friend of Canadian Grandmaster and lawyer Daniel Yanofsky. Divinsky received a Bachelor of Science from the University of Manitoba in 1946. He received a Master of Science in 1947, and a PhD in Mathematics under A. A. Albert in 1950 from the University of Chicago after which he returned to Winnipeg and was on the staff of the Mathematics Department of the University of Manitoba for most of the '50s. Divinsky then moved to Vancouver where he served as a mathematics professor, and also as an assistant dean of science, at the University of British Columbia in Vancouver, where he spent the remainder of his professional career. He was featured in many segments relating to mathematics and chess on the Discovery Channel Canada program @discovery.ca, later called Daily Planet. During the first two seasons of the show, he presented a weekly contest segment emphasizing math puzzles. == Political career == Divinsky was a Vancouver School Board trustee from 1974 to 1980, and was the Chair from 1978 to 1980. He served as an alderman on Vancouver's city council from 1981 to 1982. == Chess life == Divinsky learned his early chess as a teenager at the Winnipeg Jewish Chess Club, along with Yanofsky. He tied for 3rd–4th places in the Closed Canadian Chess Championship, held at Saskatoon 1945, with 9.5/12, along with John Belson; the joint winners were Yanofsky and Frank Yerhoff at 10.5/12. In the 1951 Closed Canadian Chess Championship, held at Vancouver, Divinsky scored 6/12 to tie for 5th–7th places. He won the Manitoba Championship in both 1946 and 1952, and finished runner-up in 1945. He tied for first place in the 1959 Manitoba Open. Divinsky scored 7.5/11 at Bognor Regis 1966, finishing in a tie for 7–13th places. He represented Canada twice at the Chess Olympiads, in 1954 at Amsterdam (second reserve board, 0.5/1), and in 1966 at Havana (second reserve board, 4.5/8). Divinsky served as playing captain for both teams, and was the non-playing captain for the 1988 Canadian Olympiad team. Divinsky attained the playing level of National Master in Canada, and received through the Commonwealth Chess Association (founded by English Grandmaster Raymond Keene) the honorary title of International Master (although he did not receive this title officially from FIDE, the World Chess Federation). Divinsky was also a Life Master at Bridge from 1972. Divinsky served for 15 years, from 1959 to 1974, as editor of the magazine Canadian Chess Chat, and contributed occasionally to other Canadian chess magazines. He played an important role in chess organization in Canada from the 1950s. He first served as Canada's representative to FIDE (the World Chess Federation), from 1987 to 1994, and served again in this post in 2007. During both terms, he served as a member of the FIDE General Assembly, since Canada is a zone of FIDE. He is a member of the Canadian Chess Hall of Fame, served as President of the Chess Federation of Canada in 1954, and was a Life Governor of the CFC. He wrote several books on chess (see bibliography below). Chess historian Edward Winter in a 1992 review was very critical of Divinsky's The Batsford Chess Encyclopedia, calling it "A Catastrophic Encyclopedia". Winter in 2008 selected it as one of the five worst chess books in English from the past two decades. Winter's 1989 review of Divinsky and Raymond Keene's book Warriors of the Mind was also negative. In this book, the authors compared great chess champions throughout history using an advanced mathematical treatment; while necessarily imperfect due to generational evolution in chess, it was in fact the pioneering work in this field. == Family, and marriage to Kim Campbell == Divinsky was married three times. He had three daughters from his first marriage: Judy, Pamela, and Mimi. Divinsky met Kim Campbell, 22 years younger, while she was an undergraduate student at the University of British Columbia in the late 1960s. Their relationship continued while Campbell did graduate work at the London School of Economics, and the two were married in 1972. It was his second marriage and her first. Divinsky was a strong influence in interesting Campbell in political activity. The two divorced in 1983, but they remained on good terms. Their marriage produced no children. He died, aged 86, in Vancouver, survived by his third wife Marilyn Goldstone. == Selected bibliography == Rings and Radicals, University of Toronto Press, 1965. Linear Algebra, 1975. Around the Chess World in 80 Years. The Batsford Encyclopedia of Chess, 1990. ISBN 0-7134-6214-0 Life Maps of the Great Chess Masters, 1994, Seattle, International Chess Enterprises. Warriors of the Mind: A Quest for the Supreme Genius of the Chess Board (with Raymond Keene), 1989, 2002. ISBN 0-9513757-2-5 == Notes == == References == "Canadian Who's Who 1997 entry". Retrieved February 19, 2009. == External links == Nathan Divinsky player profile and games at Chessgames.com Canadian Chess Hall of Fame Inductee 2001 Archived July 17, 2018, at the Wayback Machine Nathan Divinsky at the Mathematics Genealogy Project
Wikipedia:Nathan Mendelsohn#0	Nathan Saul Mendelsohn, (April 14, 1917 – July 4, 2006) was an American-born mathematician who lived and worked in Canada. Mendelsohn was a researcher in several areas of discrete mathematics, including group theory and combinatorics. == Early life and education == Mendelsohn was born in 1917 in Brooklyn, New York City, the eldest of four children of Samuel Mendelsohn (1880–1959) and Sylvia, née Kirschenbaum (1895–1984). His paternal grandparents, Hyman Mendelsohn (1846–1928) and Hinda, née Silverstone (1859–1942) had originally immigrated to Montreal from Romania in 1898. His uncle was noted open water swimmer, Shier Mendelson. In 1918, he and his family moved to Toronto, Ontario, Canada, after a fire destroyed the tenement they were living in. Mendelsohn and his family lived in a house at 13 Euclid Avenue. Mendelsohn completed all his education at the University of Toronto. He would have been unable to attend university had he not won a four years' tuition and books scholarship. In 1938, he was on the University of Toronto team for the first Putnam Competition, along with Irving Kaplansky and John Coleman. The team placed first and each of the three team members won fifty dollars. Mendelsohn was a junior, the other two were seniors. The subsequent year Mendelsohn was barred from competition as at that time the winning university set the examination for the next year and its students were barred from competition. Mendelsohn completed his Ph.D. dissertation in 1941. It was titled "A Group-Theoretic Characterization of the General Projective Collineation Group", and summarized in the Proceedings of the National Academy of Sciences in 1944. His supervisor was Gilbert de Beauregard Robinson. Mendelsohn also began practising magic tricks in high school as a means of steadying a tremor in his hands. He placed second in the 1953 International Brotherhood of Magicians contest, behind Johnny Carson. He could memorize a shuffled deck of cards seeing each card only once briefly, or a list of fifty objects called out in any order. He could identify the position of each card or name the card in any position. == Career == During the Second World War, Mendelsohn worked on simulations of artillery and code breaking. As with much of the mathematical work for military purposes during the time, it was classified. Although others related after fifty years what their exact role was, Nathan Mendelsohn strictly followed the Official Secrets Act and never revealed exact details of what he had done. We now know that when Norway fell to the Nazis, he worked on a team recomputing ballistics tables for Canadian wood as TNT is made from wood. Then he went on to break code at Camp X, which was Canada’s equivalent of Bletchley Park. From 1945 to 1947, Mendelsohn was a professor at Queen's University in Kingston, Ontario, Canada. Mendelsohn's son later remarked that Mendelsohn "understood that, as a Jew, he would never get a permanent position" at Queen's, as the university "already had a Jewish professor in the department." In 1947, Mendelsohn moved to the University of Manitoba in Winnipeg, Manitoba. Mendelsohn stayed at the University of Manitoba until his retirement in 2005. During early summers at the University of Manitoba, Mendelsohn would travel to Quebec City to teach to supplement his $3,000 annual salary at the University of Manitoba. In 1958, Mendelsohn and Dulmage published the paper "Coverings of biparte graphs", in which the Dulmage–Mendelsohn decomposition is described. Mendelsohn is also remembered for Mendelsohn triple systems. Mendelsohn was head of the department of mathematics at the University of Manitoba for almost a quarter of a century. In the early 1960s, Mendelsohn returned to classified mathematics, this time at the RAND Corporation. From 1969 to 1971, Mendelsohn was the president of the Canadian Mathematical Society. In 1985, Mendelsohn was the subject of a short film form the National Film Board of Canada, titled "An Aesthetic Indulgence". == Retirement == Mendelsohn retired from the University of Manitoba in 2005. He died on July 4, 2006, from hepatitis C obtained through tainted blood. == Awards == In 1957, Mendelsohn was made a fellow of the Royal Society of Canada. He won the Henry Marshall Tory Medal in 1979. On April 15, 1999, Mendelsohn was made a member of the Order of Canada. His citation reads, in part, that Mendelsohn is "known throughout the world as an authority in combinatorics, classical geometry and finite groups". == Nathan Mendelsohn Prize == In 2008 the Nathan Mendelsohn Prize was established by his family at the University of Manitoba for the highest ranking student at a Canadian University in Putnam Competition. == References == == External links == An Aesthetic Indulgence. A short film about Mendelsohn. An Aesthetic Indulgence. National Film Board of Canada webpage. O'Connor, John J.; Robertson, Edmund F., "Nathan Mendelsohn", MacTutor History of Mathematics Archive, University of St Andrews Nathan Mendelsohn at the Mathematics Genealogy Project
Wikipedia:Nati Linial#0	Nathan (Nati) Linial (Hebrew: נתן (נתי) ליניאל; born 1953 in Haifa, Israel) is an Israeli mathematician and computer scientist, a professor in the Rachel and Selim Benin School of Computer Science and Engineering at the Hebrew University of Jerusalem, and an ISI highly cited researcher. Linial did his undergraduate studies at the Technion, and received his PhD in 1978 from the Hebrew University under the supervision of Micha Perles. He was a postgraduate researcher at the University of California, Los Angeles before returning to the Hebrew University as a faculty member. In 2012 he became a fellow of the American Mathematical Society. In 2019 he won the FOCS Test of Time Award for the paper "Constant Depth Circuits, Fourier Transform, and Learnability", co-authored with Yishay Mansour and Noam Nisan. == Selected publications == Linial, Nati (1992), "Locality in Distributed Graph Algorithms", SIAM J. Comput., 21 (1): 193–201, CiteSeerX 10.1.1.471.6378, doi:10.1137/0221015. The paper won the 2013 Dijkstra Prize. In the words of the prize committee: "This paper has had a major impact on distributed message-passing algorithms. It focused a spotlight on the notion of locality in distributed computation and raised interesting questions concerning the locality level of various distributed problems, in terms of their time complexity on different classes of networks. Towards that goal, in this paper, Linial developed a model particularly suitable for studying locality, which ignores message sizes, asynchrony and failures. This clean model allowed researchers to isolate the effects of locality and study the roles of distances and neighborhoods, as graph theoretic notions, and their interrelations with algorithmic and complexity-theoretic problems in distributed computing." Borodin, Allan; Linial, Nathan; Saks, Michael E. (1992), "An optimal on-line algorithm for metrical task system", J. ACM, 39 (4): 745–763, doi:10.1145/146585.146588, S2CID 18783826. This paper on competitive analysis of online algorithms studies metrical task systems, a very general model of tasks where decisions on how to service a sequence of requests must be made without knowledge of future requests. It introduces the metrical task system model, describes how to use it to model various scheduling problems, and develops an algorithm that in many situations can be shown to perform optimally. Linial, Nathan; Mansour, Yishay; Nisan, Noam (1993), "Constant depth circuits, Fourier transform, and learnability", J. ACM, 40 (3): 607–620, doi:10.1145/174130.174138, S2CID 16978276. By performing harmonic analysis on functions in the complexity class AC0 (a class representing highly parallelizable computational problems), Linial and his co-authors show that these functions behave poorly as pseudorandom number generators, can be approximated well by polynomials, and can be learned efficiently by machine learning systems. Linial, Nathan; London, Eran; Rabinovich, Yuri (1995), "The geometry of graphs and some of its algorithmic applications", Combinatorica, 15 (2): 215–245, doi:10.1007/BF01200757, S2CID 5071936. Linial's most-cited paper according to Google scholar, this paper explores connections between graph-theoretic problems such as the multi-commodity flow problem and low-distortion embeddings of metric spaces into low-dimensional spaces such as those given by the Johnson–Lindenstrauss lemma. Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications", Bulletin of the American Mathematical Society, 43 (4): 439–561, doi:10.1090/S0273-0979-06-01126-8, MR 2247919. In 2008 Linial and his co-authors won the Levi L. Conant Prize of the American Mathematical Society for best mathematical exposition for this article, a survey on expander graphs. == References ==
Wikipedia:National Centre for Excellence in the Teaching of Mathematics#0	The National Centre for Excellence in the Teaching of Mathematics (NCETM) is an institution set up in the wake of the Smith Report to improve mathematics teaching in England. It provides strategic leadership for mathematics-specific CPD and aims to raise the professional status of all those engaged in the teaching of mathematics so that the mathematical potential of learners will be fully realised. == Structure == Its Director until March 2013 was Dame Celia Hoyles, Professor of Mathematics Education at the Institute of Education, University of London and former chief adviser on mathematics education for the government. She was succeeded by the current Director, Charlie Stripp. An innovative NCETM development is the MatheMaPedia project, masterminded by John Mason, which is a "maths teaching wiki". Initially headquartered in London, it is headquartered in the south of Sheffield city centre; it is the headquarters of Tribal Education. It is run by Mathematics in Education and Industry (MEI) and Tribal Education. === Online discussions === Special online events have included the world’s first online discussion of proof. == See also == Centre for Industry Education Collaboration and National Centre for Computing Education, also at York Count On - maths education initiative Mathematics education in the United Kingdom International Congress on Mathematical Education == References ==
Wikipedia:National Cipher Challenge#0	The National Cipher Challenge is an annual cryptographic competition organised by the University of Southampton School of Mathematics. Competitors attempt to break cryptograms published on the competition website. In the 2017, more than 7,500 students took part in the competition. Participants must be in full-time school level education in order to qualify for prizes. == Format == The competition is organised into eight to ten challenges, which are further subdivided into parts A and B. The part A challenge consists of a comparatively simpler cryptogram, and usually provides some useful information to assist in the solving of part B. Part B is usually more complex. In later challenges the cryptograms become harder to break. In the past, part A cryptograms have been encrypted with the Caesar cipher, the Affine cipher, the Keyword cipher, the Transposition cipher, the Vigenère cipher and the 2x2 Hill cipher. The part B challenges are intended to be harder. These begin with relatively simple substitution ciphers, including the Bacon cipher and Polybius square, before moving on to transposition ciphers, Playfair ciphers and polyalphabetic ciphers such as the Vigenère cipher, the Autokey cipher and the Alberti cipher. In the later stages of the competition, the ADFGVX cipher, the Solitaire cipher, the Double Playfair cipher, the Hill cipher, the Book cipher and versions of the Enigma and Fialka cipher machines have all been used. The 2009 challenge ended with a Jefferson Disk cipher, the 2012 challenge ended with the ADFGVX Cipher, the 2014 with the Playfair Cipher, and the most recent challenge ended with a sectioned Cadenus transposition. == Prizes == £25 cash prizes are awarded to eight random entrants who submit a correct solution for each part A of the challenge. Leaderboards for the part B challenges are also compiled, based on how accurate solutions are and how quickly the entrant broke the cipher. Prizes are awarded to the top three entrants at the end of the challenge. In the 2009/10 challenge, the sponsors provided several prizes: IBM provided iPod Touches to each member of the team winning the Team Prize, Trinity College provided a cash prize of £700, and GCHQ provided a cash prize of £1000. In previous years prizes such as an IBM Thinkpad laptop have been awarded. After the challenge the winners of the top prizes and other randomly selected entrants are invited to a day held at Bletchley Park consisting of lectures (with subjects such as the Semantic Web, World War II cryptography and computer programming) and the prize-giving ceremony. Current sponsors of the competition include GCHQ, IBM, British Computer Society, Trinity College, Cambridge, Cambridge University Press, Winton Capital Management and EPSRC. == References == == External links == The official challenge website The website of the 2016/17 challenge (slightly broken) The website of the 2015/16 challenge The websites for the challenges earlier than this are no longer available.
Wikipedia:National Mathematics Day (India)#0	The 2012 Indian stamp featured Srinivasa Ramanujan. The Indian government declared 22 December to be celebrated as National Mathematics Day every year to mark the birth anniversary of the Indian mathematician Srinivasa Ramanujan. It was introduced by Prime Minister Manmohan Singh on 26 December 2011 at Madras University, to mark the 125th birth anniversary of the Indian mathematician Srinivasa Ramanujan. On this occasion Prime minister Manmohan Singh also announced that 2012 would be celebrated as the National Mathematics Year. Since then, India's National Mathematics Day is celebrated on 22 December every year with numerous educational events held at schools and universities throughout the country. In 2017, the day's significance was enhanced by the opening of the Ramanujan Math Park in Kuppam, in Chittoor, Andhra Pradesh. National Mathematics Day is celebrated in all schools and universities throughout the country. == See also == International Day of Mathematics (Pi Day) == References ==
Wikipedia:National Mathematics Talent Contest#0	The National Mathematics Talent Contest or NMTC is a national-level mathematics contest conducted by the Association of Mathematics Teachers of India (AMTI). It is strongest in Tamil Nadu, which is the operating base of the AMTI. The AMTI is a pioneer organisation in promoting, and conducting, Maths Talent Tests in India. In the National level tests 66,066 students, from 332 institutions spread all over India, participated at the screening level. Of these, 10% were selected for the final test. For the benefit of final level contestants, and the chosen few for INMO, special orientation camps were conducted. Merit certificates and prizes were awarded to the deserving students. Thirty-five among them from Tamil Nadu and Puduchery at the Junior and Inter Levels have been sponsored to write the Indian National Mathematics Olympiad (INMO 2013). From among them 2 have been selected at the national level. == Levels == Primary level: Standard 5 and 6, is called the Gauss Contest Sub-junior level: Standards 7 and 8, is called the Kaprekar Contest Junior level: Standards 9 and 10, is called the Bhaskara Contest Inter level: Standards 11 and 12, is called the Ramanujan Contest Senior level: B.Sc. students, is called the Aryabhata Contest == Stages == For all levels except the Senior level, there is a preliminary examination comprising multiple choice questions. The preliminary examination is held in the end of August. Students qualifying in the preliminary examination are eligible to sit for the main examination, which is held around the last week of October. A week before the main examination, students are invited for a two-day orientation camp. == Fee == The fee for the preliminary examination is Rs. 50 in India. No further fee is required for the main examination. Rs. 75/- per candidate (out of which Rs.15/- will be retained by the institution only for all expenses and Rs.60/- to be sent to AMTI). == Syllabus == No special knowledge of curriculum material is required. Good knowledge of curriculum at the next lower level would be helpful. The syllabus for Mathematics Olympiad (Regional, National and International) is pre-degree college mathematics. The areas covered are, mainly – a)Algebra, b) Geometry, c) Number theory and d) Graph theory & combinatorics. Algebra: Polynomials, Solving equations, inequalities, and complex numbers. Geometry: Geometry of triangles and circles. (Trigonometric methods, vector methods, complex number methods, transformation geometry methods can also be used to solve problems) Number Theory: Divisibility, Diophantine equations, congruence relations, prime numbers and elementary results on prime numbers. Combinatorics & Graph Theory: Counting techniques, pigeon hole principle, the principle of inclusion and exclusion, basic graph theory. == External links == MATH -n-CODING TECH COMPETITION (MCTC) - 2022 AMTI page on the NMTC
Wikipedia:National Mathematics Year#0	In India and in Nigeria the year 2012 CE was celebrated as National Mathematics Year. In India, the National Mathematics Year was a tribute to the mathematical genius Srinivasa Ramanujan who was born on 22 December 1887 and whose 125th birthday falls on 22 December 2012. In Nigeria, the year 2012 was observed as National Mathematics Year as part of the federal government's effort to promote and popularize the study of mathematics. == National Mathematics Year in India == The decision to designate the year 2012 CE as National Mathematics Year was announced by Dr Manmohan Singh, Prime Minister of India , during the inaugural ceremony of the celebrations to mark the 125th birth anniversary of Srinivasa Ramanujan held at the Madras University Centenary Auditorium on 26 February 2012. The Prime Minister also announced that December 22 would be celebrated as National Mathematics Day from 2012 onwards. An Organising Committee with Professor M.S. Raghunathan, President of the Ramanujan Mathematical Society as chair, and Professor Dinesh Singh, Secretary of the Ramanujan Mathematical Society as secretary, has been formed to formulate and implement programmes and projects as part of the observance of the National Mathematics Year. A National Committee with Minister for Kapil Sibal as the chair supervises the activities of the Organising Committee. == National Mathematics Year in Nigeria == In Nigeria, the various activities planned as part of the celebration of National Mathematics Year would be centred on the theme Mathematics: The Key to Transformation. The events were inaugurated on 1 March 2012 at a function in Musa Yar’adua Dome, Abuja. Thirteen projects of national importance are planned as part of the celebrations. == References ==
Wikipedia:National Numeracy#0	Numeracy is the ability to understand, reason with, and apply simple numerical concepts; it is the numerical counterpart of literacy. The charity National Numeracy states: "Numeracy means understanding how mathematics is used in the real world and being able to apply it to make the best possible decisions...It's as much about thinking and reasoning as about 'doing sums'". Basic numeracy skills consist of comprehending fundamental arithmetical operations like addition, subtraction, multiplication, and division. For example, if one can understand simple mathematical equations such as 2 + 2 = 4, then one would be considered to possess at least basic numeric knowledge. Substantial aspects of numeracy also include number sense, operation sense, computation, measurement, geometry, probability and statistics. A numerically literate person can manage and respond to the mathematical demands of life. By contrast, innumeracy (the lack of numeracy) can have a negative impact. Numeracy has an influence on healthy behaviors, financial literacy, and career decisions. Therefore, innumeracy may negatively affect economic choices, financial outcomes, health outcomes, and life satisfaction. It also may distort risk perception in health decisions. Greater numeracy has been associated with reduced susceptibility to framing effects, less influence of nonnumerical information such as mood states, and greater sensitivity to different levels of numerical risk. Ellen Peters and her colleagues argue that achieving the benefits of numeric literacy, however, may depend on one's numeric self-efficacy or confidence in one's skills. == Representation of numbers == Humans have evolved to mentally represent numbers in two major ways from observation (not formal math). These representations are often thought to be innate (see Numerical cognition), to be shared across human cultures, to be common to multiple species, and not to be the result of individual learning or cultural transmission. They are: Approximate representation of numerical magnitude, and Precise representation of the quantity of individual items. Approximate representations of numerical magnitude imply that one can relatively estimate and comprehend an amount if the number is large (see Approximate number system). For example, one experiment showed children and adults arrays of many dots. After briefly observing them, both groups could accurately estimate the approximate number of dots. However, distinguishing differences between large numbers of dots proved to be more challenging. Precise representations of distinct items demonstrate that people are more accurate in estimating amounts and distinguishing differences when the numbers are relatively small (see Subitizing). For example, in one experiment, an experimenter presented an infant with two piles of crackers, one with two crackers the other with three. The experimenter then covered each pile with a cup. When allowed to choose a cup, the infant always chose the cup with more crackers because the infant could distinguish the difference. Both systems—approximate representation of magnitude and precise representation quantity of individual items—have limited power. For example, neither allows representations of fractions or negative numbers. More complex representations require education. However, achievement in school mathematics correlates with an individual's unlearned approximate number sense. == Definitions and assessment == Fundamental (or rudimentary) numeracy skills include understanding of the real number line, time, measurement, and estimation. Fundamental skills include basic skills (the ability to identify and understand numbers) and computational skills (the ability to perform simple arithmetical operations and compare numerical magnitudes). More sophisticated numeracy skills include understanding of ratio concepts (notably fractions, proportions, percentages, and probabilities), and knowing when and how to perform multistep operations. Two categories of skills are included at the higher levels: the analytical skills (the ability to understand numerical information, such as required to interpret graphs and charts) and the statistical skills (the ability to apply higher probabilistic and statistical computation, such as conditional probabilities). A variety of tests have been developed for assessing numeracy and health numeracy. Different tests have been developed to evaluate health numeracy. Two of these tests that have been found to be "reliable and valid" are the GHNT-21 and GHNT-6. == Childhood influences == The first couple of years of childhood are considered to be a vital part of life for the development of numeracy and literacy. There are many components that play key roles in the development of numeracy at a young age, such as Socioeconomic Status (SES), parenting, Home Learning Environment (HLE), and age. == Socioeconomic status == Children who are brought up in families with high SES tend to be more engaged in developmentally enhancing activities. These children are more likely to develop the necessary abilities to learn and to become more motivated to learn. More specifically, a mother's education level is considered to have an effect on the child's ability to achieve in numeracy. That is, mothers with a high level of education will tend to have children who succeed more in numeracy. A number of studies have, moreover, proved that the education level of the mother is strongly correlated with the average age of getting married. More precisely, females who entered the marriage later, tend to have greater autonomy, chances for skills premium and level of education (i.e. numeracy). Hence, they were more likely to share this experience with children. == Parenting == Parents are advised to collaborate with their child in simple learning exercises, such as reading a book, painting, drawing, and playing with numbers. On a more expressive note, the act of using complex language, being more responsive towards the child, and establishing warm interactions are recommended to parents with the confirmation of positive numeracy outcomes. When discussing beneficial parenting behaviors, a feedback loop is formed because pleased parents are more willing to interact with their child, which in essence promotes better development in the child. == Home-learning environment == Along with parenting and SES, a strong home-learning environment increases the likelihood of the child being prepared for comprehending complex mathematical schooling. For example, if a child is influenced by many learning activities in the household, such as puzzles, coloring books, mazes, or books with picture riddles, then they will be more prepared to face school activities. == Age == Age is accounted for when discussing the development of numeracy in children. Children under the age of 5 have the best opportunity to absorb basic numeracy skills. After the age of seven, achievement of basic numeracy skills become less influential. For example, a study was conducted to compare the reading and mathematical abilities between children of ages five and seven, each in three different mental capacity groups (underachieving, average, and overachieving). The differences in the amount of knowledge retained were greater between the three different groups aged five than between the groups aged seven. This reveals that those of younger ages have an opportunity to retain more information, like numeracy. According to Gelman and Gallistel in The Child's Understanding of Number, 'children as young as 2 years can accurately judge numerosity provided that the numerosity is not larger than two or three'. Children as young as three have been found to understand elementary mathematical concepts. Kilpatrick and his colleagues state 'most preschoolers show that they can understand and perform simple addition and subtraction by at least 3 years of age'. Lastly, it has been observed that pre-school children benefit from their basic understanding of 'counting, reading and writing of numbers, understanding of simple addition and subtraction, numerical reasoning, classifying of objects and shapes, estimating, measuring, [and the] reproduction of number patterns'. == Literacy == There seems to be a relationship between literacy and numeracy, which can be seen in young children. Depending on the level of literacy or numeracy at a young age, one can predict the growth of literacy and/ or numeracy skills in future development. There is some evidence that humans may have an inborn sense of number. In one study for example, five-month-old infants were shown two dolls, which were then hidden with a screen. The babies saw the experimenter pull one doll from behind the screen. Without the child's knowledge, a second experimenter could remove, or add dolls, unseen behind the screen. When the screen was removed, the infants showed more surprise at an unexpected number (for example, if there were still two dolls). Some researchers have concluded that the babies were able to count, although others doubt this and claim the infants noticed surface area rather than number. == Employment == Numeracy has a huge impact on employment. In a work environment, numeracy can be a controlling factor affecting career achievements and failures. Many professions require individuals to have well-developed numerical skills: for example, mathematician, physicist, accountant, actuary, financial analyst, engineer, and architect. This is why a major target of the UN's Sustainable Development Goal 4 is to substantially increase the number of youths who have relevant skills for decent work and employment because, even outside these specialized areas, the lack of numeracy skills can reduce employment opportunities and promotions, resulting in unskilled manual careers, low-paying jobs, and even unemployment. For example, carpenters and interior designers need to be able to measure, use fractions, and handle budgets. Another example of numeracy influencing employment was demonstrated at the Poynter Institute. The Poynter Institute has recently included numeracy as one of the skills required by competent journalists. Max Frankel, former executive editor of The New York Times, argues that "deploying numbers skillfully is as important to communication as deploying verbs". Unfortunately, it is evident that journalists often show poor numeracy skills. In a study by the Society of Professional Journalists, 58% of job applicants interviewed by broadcast news directors lacked an adequate understanding of statistical materials. To assess job applicants, psychometric numerical reasoning tests have been created by occupational psychologists, who are involved in the study of numeracy. These tests are used to assess ability to comprehend and apply numbers. They are sometimes administered with a time limit, so that the test-taker must think quickly and concisely. Research has shown that these tests are very useful in evaluating potential applicants because they do not allow the applicants to prepare for the test, unlike interview questions. This suggests that an applicant's results are reliable and accurate These tests first became prevalent during the 1980s, following the pioneering work of psychologists, such as P. Kline, who published a book in 1986 entitled A handbook of test construction: Introduction to psychometric design, which explained that psychometric testing could provide reliable and objective results, which could be used to assess a candidate's numerical abilities. == Innumeracy and dyscalculia == The term innumeracy is a neologism, coined by analogy with illiteracy. Innumeracy refers to a lack of ability to reason with numbers. The term was coined by cognitive scientist Douglas Hofstadter; however, it was popularized in 1989 by mathematician John Allen Paulos in his book Innumeracy: Mathematical Illiteracy and its Consequences. Developmental dyscalculia refers to a persistent and specific impairment of basic numerical-arithmetical skills learning in the context of normal intelligence. == Patterns and differences == The root causes of innumeracy vary. Innumeracy has been seen in those suffering from poor education and childhood deprivation of numeracy. Innumeracy is apparent in children during the transition between numerical skills obtained before schooling and the new skills taught in the education departments because of their memory capacity to comprehend the material. Patterns of innumeracy have also been observed depending on age, gender, and race. Older adults have been associated with lower numeracy skills than younger adults. Men have been identified to have higher numeracy skills than women. Some studies seem to indicate young people of African heritage tend to have lower numeracy skills. The Trends in International Mathematics and Science Study (TIMSS) in which children at fourth-grade (average 10 to 11 years) and eighth-grade (average 14 to 15 years) from 49 countries were tested on mathematical comprehension. The assessment included tests for number, algebra (also called patterns and relationships at fourth grade), measurement, geometry, and data. The latest study, in 2003, found that children from Singapore at both grade levels had the highest performance. Countries like Hong Kong SAR, Japan, and Taiwan also shared high levels of numeracy. The lowest scores were found in countries like South Africa, Ghana, and Saudi Arabia. Another finding showed a noticeable difference between boys and girls, with some exceptions. For example, girls performed significantly better in Singapore, and boys performed significantly better in the United States. == Theory == There is a theory that innumeracy is more common than illiteracy when dividing cognitive abilities into two separate categories. David C. Geary, a notable cognitive developmental and evolutionary psychologist from the University of Missouri, created the terms "biological primary abilities" and "biological secondary abilities". Biological primary abilities evolve over time and are necessary for survival. Such abilities include speaking a common language or knowledge of simple mathematics. Biological secondary abilities are attained through personal experiences and cultural customs, such as reading or high level mathematics learned through schooling. Literacy and numeracy are similar in the sense that they are both important skills used in life. However, they differ in the sorts of mental demands each makes. Literacy consists of acquiring vocabulary and grammatical sophistication, which seem to be more closely related to memorization, whereas numeracy involves manipulating concepts, such as in calculus or geometry, and builds from basic numeracy skills. This could be a potential explanation of the challenge of being numerate. == Innumeracy and risk perception in health decision-making == Health numeracy has been defined as "the degree to which individuals have the capacity to access, process, interpret, communicate, and act on numerical, quantitative, graphical, biostatistical, and probabilistic health information needed to make effective health decisions". The concept of health numeracy is a component of the concept of health literacy. Health numeracy and health literacy can be thought of as the combination of skills needed for understanding risk and making good choices in health-related behavior. Health numeracy requires basic numeracy but also more advanced analytical and statistical skills. For instance, health numeracy also requires the ability to understand probabilities or relative frequencies in various numerical and graphical formats, and to engage in Bayesian inference, while avoiding errors sometimes associated with Bayesian reasoning (see Base rate fallacy, Conservatism (Bayesian)). Health numeracy also requires understanding terms with definitions that are specific to the medical context. For instance, although 'survival' and 'mortality' are complementary in common usage, these terms are not complementary in medicine (see five-year survival rate). Innumeracy is also a very common problem when dealing with risk perception in health-related behavior; it is associated with patients, physicians, journalists and policymakers. Those who lack or have limited health numeracy skills run the risk of making poor health-related decisions because of an inaccurate perception of information. For example, if a patient has been diagnosed with breast cancer, being innumerate may hinder her ability to comprehend her physician's recommendations, or even the severity of the health concern or even the likelihood of treatment benefits. One study found that people tended to overestimate their chances of survival or even to choose lower-quality hospitals. Innumeracy also makes it difficult or impossible for some patients to read medical graphs correctly. Some authors have distinguished graph literacy from numeracy. Indeed, many doctors exhibit innumeracy when attempting to explain a graph or statistics to a patient. A misunderstanding between a doctor and patient, due to either the doctor, patient, or both being unable to comprehend numbers effectively, could result in serious harm to health. Different presentation formats of numerical information, for instance natural frequency icon arrays, have been evaluated to assist both low-numeracy and high-numeracy individuals. Other data formats provide more assistance to low-numeracy people. == Evolution of numeracy == In the field of economic history, numeracy is often used to assess human capital at times when there was no data on schooling or other educational measures. Using a method called age-heaping, researchers like Professor Jörg Baten study the development and inequalities of numeracy over time and throughout regions. For example, Baten and Hippe find a numeracy gap between regions in western and central Europe and the rest of Europe for the period 1790–1880. At the same time, their data analysis reveals that these differences as well as within country inequality decreased over time. Taking a similar approach, Baten and Fourie find overall high levels of numeracy for people in the Cape Colony (late 17th to early 19th century). In contrast to these studies comparing numeracy over countries or regions, it is also possible to analyze numeracy within countries. For example, Baten, Crayen and Voth look at the effects of war on numeracy in England, and Baten and Priwitzer find a "military bias" in what is today western Hungary: people opting for a military career had - on average - better numeracy indicators (1 BCE to 3CE). == See also == == Notes == == External links == The Berlin Numeracy Test CDC Health Literacy Resources Agency for Healthcare Research and Quality Health Literacy Measurement tools Australian blog post reviewing the increasing importance of teaching numeracy skills
Wikipedia:National Numeracy Strategy#0	The National Numeracy Strategy was designed to facilitate a sound grounding in maths for all primary school pupils. It arose out of the National Numeracy Project in 1996, led by a Numeracy Task Force in England, and was launched in 1998 and implemented in schools in 1999. The strategy included an outline of expected teaching in mathematics for all pupils from Reception to Year 6. In 2003, the strategy, including the framework for teaching, was absorbed into the broader Primary National Strategy. The framework for teaching was then updated in 2006, but ceased to operate in 2011. == See also == National Curriculum (England, Wales and Northern Ireland) Key Stage Chunking (division) Grid method multiplication Number bond == References == == Further reading == Department for Education and Employment (1998), The implementation of the National Numeracy Strategy: The final report of the Numeracy Task Force, London: DfEE Department for Education and Employment (1999), The National Numeracy Strategy: framework for teaching mathematics from reception to Year 6, London: DfEE. ISBN 0-85522-922-5 QCA (1999), Standards in mathematics: exemplification of key learning objectives from reception to year 6 Rob Eastaway, Why parents can't do maths today, BBC News, 10 September 2010 Ian Thompson (2000), Is the National Numeracy Strategy evidence based?, Mathematics Teaching, 171, 23–27 Dylan V. Jones (2002), National numeracy initiatives in England and Wales: a comparative study of policy, The Curriculum Journal, 13 (1), 5–23. Chris Kyriacou and Maria Goulding (2004), A systematic review of the impact of the Daily Mathematics Lesson in enhancing pupil confidence and competence in early mathematics, Evidence for Policy and Practice Information and Co-ordinating Centre (EPPI), Institute of Education, London. == External links == Government Primary Frameworks website (Archived, via the National Archives)
Wikipedia:Naum Akhiezer#0	Naum Ilyich Akhiezer (Ukrainian: Нау́м Іллі́ч Ахіє́зер; Russian: Нау́м Ильи́ч Ахие́зер; 6 March 1901 – 3 June 1980) was a Soviet and Ukrainian mathematician of Jewish origin, known for his works in approximation theory and the theory of differential and integral operators. He is also known as the author of classical books on various subjects in analysis, and for his work on the history of mathematics. He is the brother of the theoretical physicist Aleksandr Akhiezer. == Biography == Naum Akhiezer was born in Cherykaw (now in Belarus). His father was the medical doctor for the district in and around Cherykaw. Naum Akhiezer studied in the Kyiv Institute of Public Education (now Taras Shevchenko National University of Kyiv). In 1928, he defended his PhD thesis "Aerodynamical Investigations" under the supervision of Dmitry Grave. From 1928 to 1933, he worked at the Kyiv University and at the Kyiv Aviation Institute. In 1933, Naum Akhiezer moved to Kharkiv. From 1933 to his death, except for the years of war and evacuation, he was a professor at Kharkov University and at other institutes in Kharkiv. From 1935 to 1940 and from 1947 to 1950 he was director of the Kharkov Institute of Mathematics and Mechanics. For many years he headed the Kharkov Mathematical Society. == Work == Akhiezer obtained important results in approximation theory (in particular, on extremal problems, constructive function theory, and the problem of moments), where he masterly applied the methods of the geometric theory of functions of a complex variable (especially, conformal mappings and the theory of Riemann surfaces) and of functional analysis. He found the fundamental connection between the inverse problem for important classes of differential and finite difference operators of the second order with a finite number of gaps in the spectrum, and the Jacobi inversion problem for Abelian integrals. This connection led to explicit solutions of the inverse problem for the so-called finite-gap operators. == Some publications == === Books in analysis === Ахиезер, Н.И. (2001). Избранные труды по теории функций и математической физике (vol. 1–2) [Selected works in function theory and mathematical physics]. Kharkiv: Acta. Ахиезер, Н.И. (1984). Лекции об интегральных преобразованиях. Kharkov: Vishcha Shkola. English translation: Akhiezer, N. I. (1988). Lectures on Integral Transforms. Providence, RI: American Mathematical Society. MR 0971981. Ахиезер, Н.И. (1981). Вариационное исчисление. Kharkov: Vishcha Shkola.. English translation: Akhiezer, N. I. (1988). The calculus of variations. Chur: Harwood Academic Publishers. ISBN 3-7186-4805-9. MR 0949441. Ахиезер, Н.И. (1977–1978). Теория линейных операторов в Гильбертовом пространстве (vol. 1–2) (3rd ed.). Kharkov: Vishcha Shkola.. English translation: Akhiezer, N.I.; Glazman, I.M. (1981). Theory of Linear Operators in Hilbert Space (vol. 1 – 2) (3rd ed.). Boston, Mass. – London: Pitman (Advanced Publishing Program). MR 0615736. Ахиезер, Н.И. (1970). Элементы теории эллиптических функций. Moscow: Nauka. English translation: Akhiezer, N. I. (1990). Elements of the theory of elliptic functions. Providence, RI: American Mathematical Society. ISBN 0-8218-4532-2. MR 1054205. Ахиезер, Н.И. (1961). Классическая проблема моментов и некоторые вопросы анализа, связанные с нею. Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit. English translation: Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd. Ахиезер, Н.И. (1965). Лекции по теории аппроксимации (2nd ed.). Moscow: Nauka. English translation (of the 1st edition): Achiezer, N. I. (1956). Theory of approximation. Translated by Charles J. Hyman. New York: Frederick Ungar Publishing. Ахиезер, Н.И.; Крейн, М.Г. (1938). О некоторых вопросах теории моментов. Kharkov: GONTI. English translation: Akhiezer, N.I.; Krein, M.G. (1962). Some questions in the theory of moments. Providence, R.I.: American Mathematical Society. MR 0167806. === History of mathematics === Ахиезер, Н.И. (1955). Академик С. Н. Бернштейн и его работы по конструктивноĭ теории функций [Academician S. N. Bernstein and his work on the constructive theory of functions]. Kharkov: Izdat. Harʹkov. Gosudarstv. Univ. German translation: Akhiezer, N.I. (2000). "Das Akademiemitglied S.N.Bernstein und seine Arbeiten zur konstruktiven Funktionentheorie". Mitt. Math. Sem. Giessen. 240. MR 1755757. Ahiezer, N.I.; Petrovskiĭ, I.G. (1961). "The contribution of S. N. Bernšteĭn to the theory of partial differential equations". Uspekhi Mat. Nauk (in Russian). 16 (2 (98)): 5–20. MR 0130816. Ahiezer, N.I. (1978). "On the spectral theory of Lamé's equation". Istor.-Mat. Issled. (in Russian). 23: 77–86. MR 0517631. Akhiezer, N.I. (1998). "Function theory according to Chebyshev". Mathematics of the 19th century. Basel: Birkhäuser. pp. 1–81. MR 1634233. == References == == External links == Naum Akhiezer at the Mathematics Genealogy Project History of Approximation Theory (HAT) page Author profile in the database zbMATH A. N. Kolmogorow, M. G. Krein, B. Ja. Lewin et al.: . In: . Band 36, Nr. 4, 1981, S. 183–184
Wikipedia:Naum Krasner#0	Naum Krasner (21 February 1924 – 5 March 1999) was a Russian mathematician and economist. == Biography == Born on 21 February 1924 in Vinnitsa, Ukrainian SSR. In 1941 he graduated from high school. In autumn 1941 he evacuated to Kuibyshev region, worked as a teacher in a rural school. A former colonel in the Soviet Army, he joined Voronezh State University as a student in 1957 and, on graduating in 1961, joined the faculty there. In 1969 he became Methods of Operational Research Chair and, later, vice-dean of the Faculty of Applied Mathematics and Mechanics. Died on 5 March 1999 of lung cancer. Buried at the Kominternovsky cemetery in Voronezh. == References == == External links == In memory of Naum Krasner
Wikipedia:Naum Meiman#0	Naum Natanovich (Nokhim Sanalevich) Meiman (Russian: Нау́м Ната́нович (Но́хим Са́нелевич) Ме́йман, 12 May 1912, Bazar, Ukraine – 31 March 2001, Tel Aviv) was a Soviet mathematician, and dissident. He is known for his work in complex analysis, partial differential equations, and mathematical physics, as well as for his dissident activity, in particular, for being a member of the Moscow Helsinki Group. == Life == He was born in Bazar, Ukraine on 12 May 1912. In 1932 he graduated from Kazan State University as an extern. In 1937 he submitted his Ph.D. under the supervision of Nikolai Chebotaryov and was awarded the degree Doktor nauk. In 1939 he became a full professor at Kazan State University. He worked for two years in the Mathematics Institute at the University of Kharkiv, where he became friends with Lev Landau with whom he collaborated for many years. After the Second World War, he went to Moscow and worked at the Institute for Physical Problems, where he was a head of the mathematics lab. Then he worked in the Institute for Theoretical and Experimental Physics. In 1953, he was awarded a Stalin prize for his work in theoretical physics. He made important contributions in the development of nuclear weapons in the USSR. Starting in 1968, Meiman became active in politics and signed several letters of protest against political trials in the USSR. In 1971, he retired and applied for permission to emigrate to Israel. Denied on grounds of knowing state secrets, he soon became a refusenik. Gradually he became more active in politics, and was a member of the Moscow Helsinki Group beginning in 1977. Later he became deputy chairman and the last active free member, writing hundreds of the group's documents. He also participated in a Refusenik scientific seminar. He was permanently under surveillance by the KGB, who also bugged his telephone and searched his home. In 1982, Naum Meiman and Andrei Dmitrievich Sakharov published a letter in defence of Yuri Fyodorovich Orlov. Meiman also struggled for the right of his wife Inna Meiman-Kitrossky to go to the USA for medical treatment since she had been diagnosed with cancer. After several years of struggle, she was allowed to go to the US and she died in February 1987 in Georgetown (Washington, D.C.). Meiman was not allowed to attend her funeral in Washington D.C. In 1988 Meiman was finally allowed to emigrate to Israel, where he became a professor emeritus in Tel Aviv University. In 1992, in Tel-Aviv, there was a conference in his honor dedicated to his 80th birthday. Meiman died there in 2001. == References == == Further reading == Mydans, Seth (29 July 1985). "Soviet human rights battle: only isolated voices remain". The New York Times. Sakharov, Andrei (February 1978). "Letter from Sakharov and Meiman". Nature. 271 (5645): 499. Bibcode:1978Natur.271..499S. doi:10.1038/271499c0. Saxon, Wolfgang (10 February 1987). "Inna Meiman, emigre, dies at 53; left Soviet for cancer treatment". The New York Times. Taubman, Philip (26 February 1987). "Old and alone, Soviet dissident looks to exit". The New York Times. == Online journals == Московская Хельсинкская группа (Moscow Helsinki group). Мейман Наум Натанович (Naum Meiman) (in Russian) Naum Meiman. List of works
Wikipedia:Naum Z. Shor#0	Naum Zuselevich Shor (Russian: Наум Зуселевич Шор) (1 January 1937 – 26 February 2006) was a Soviet and Ukrainian mathematician specializing in optimization. He made significant contributions to nonlinear and stochastic programming, numerical techniques for non-smooth optimization, discrete optimization problems, matrix optimization, dual quadratic bounds in multi-extremal programming problems. Shor became a full member of the National Academy of Science of Ukraine in 1998. == Subgradient methods == N. Z. Shor is well known for his method of generalized gradient descent with space dilation in the direction of the difference of two successive subgradients (the so-called r-algorithm), that was created in collaboration with Nikolay G. Zhurbenko. The ellipsoid method was re-invigorated by A.S. Nemirovsky and D.B. Yudin, who developed a careful complexity analysis of its approximation properties for problems of convex minimization with real data. However, it was Leonid Khachiyan who provided the rational-arithmetic complexity analysis, using an ellipsoid algorithm, that established that linear programming problems can be solved in polynomial time. It has long been known that the ellipsoidal methods are special cases of these subgradient-type methods. == R-algorithm == Shor's r-algorithm is for unconstrained minimization of (possibly) non-smooth functions, which has been somewhat popular despite an unknown convergence rate. It can be viewed as a quasi-Newton method, although it does not satisfy the secant equation. Although the method involves subgradients, it is distinct from his so-called subgradient method described above. == References == === Notes === === Bibliography === "Congratulations to Naum Shor on his 65th birthday", Journal of Global Optimization, 24 (2): 111–114, 2002, doi:10.1023/A:1020215832722, S2CID 195226482. == External links == ORB Newsletter Issue 5 contains an article with a short biography Video on YouTube
Wikipedia:Nazım Terzioğlu#0	Nazım Terzioğlu (1912 – September 20, 1976) was a Turkish mathematician. He was one of the first mathematicians in Turkish academia. His son, Tosun Terzioğlu, was also a mathematician. == Early life == Nazım Terzioğlu completed his primary education in his place of birth, Kayseri. He started his secondary education in Istanbul and then continued in İzmir until his graduation from İzmir High School in 1930. At that time, some of Turkey's most qualified mathematics teachers worked at İzmir High School. Alumni of that school included mathematicians such as Cahit Arf (1910–1997) and Tevfik Oktay Kabakcıoğlu (1910–1971). In those years, the successful young people were sent abroad by the government to be trained as qualified workforce in various fields needed for the country. Terzioğlu passed the relevant exam and left for Germany to study mathematics on behalf of the Ministry of Education of Turkey. He pursued his higher education in the University of Göttingen and Munich University. He completed his Ph.D. under the supervision of the famous mathematician of that period, Constantin Carathéodory (1873–1950), who was a member of a Greek family in Fener, Istanbul. == Career == Upon completion of his education in Germany, Terzioğlu began to work as an assistant of Mathematical Mechanics and Advanced Geometry in the Institute of Mathematics of the Faculty of Science of Istanbul University in 1937. He became associate professor in 1942 and the following year, he was appointed to professorship in the newly established Institute of Mathematics of the Faculty of Science of Ankara University (1943). After spending two years in this faculty, he returned to Istanbul University as a professor (1944). At Istanbul University, he worked as the Dean of the Faculty of Science in 1950–1952. During the same period, Terzioğlu established some of the scientific institutions for which Turkey had felt the major need until those years. These are the Institute for Geophysics of Istanbul University, the Institute for Hydrobiology in Istanbul Baltalimanı and the Cosmic Ray Institute which Terzioğlu founded at Uludağ, Bursa in cooperation with Adnan Sokullu and Sait Akpınar. After his deanship in the Faculty of Science, he became the Chairman of the Analysis Division of the Institute of Mathematics in the same faculty (1953). In 1965–1967, Terzioğlu, in addition to his responsibilities in Istanbul University, worked first by proxy then acting as the principal founder-rector of Karadeniz Technical University (KTU). It is his honour to establish the first Faculty of Fundamental Sciences of Turkey in KTU. In 1967, Terzioğlu returned to his mission in the Faculty of Science of Istanbul University. In 1969 and 1971, he was elected as the rector of Istanbul University. He had maintained this position for two periods (28 October 1969 – 28 October 1971 and 28 October 1971 – 31 May 1974). In his first years as a rector, he restored the building of a historical soup kitchen which was assigned by Wakfs to the university as a part of the Sehzade Mosque. On 6 August 1971, by setting up a new printing system in it, he put the same building into service with the name of Research Institute for Mathematics of Faculty of Science. Terzioğlu also established a mathematics library within this institute with a capacity of 2000 books which he provided through donations and purchase from foreign countries. After his death, the institute was named on the proposal of the Faculty of Science as Nazim Terzioğlu Mathematics Research Institute. As an outcome of the negotiations with Silivri Municipality, Terzioğlu provided Istanbul University with 35 acres of land to be donated in Silivri. In a part of this land, 18 study rooms, 3 large conference halls, a library and a guest house to accommodate scientists coming from abroad were constructed in accord with his order. Terzioğlu considered graduate education very seriously. He believed that talented young people ought to be trained in a particular way. To provide such an environment, he invited foreign scientists and organized congresses, seminars, colloquia, summer and progress courses in Silivri facilities which was opened into use on September 3, 1973. Thanks to these activities, he made significant contributions to the education of young generations. The scientific meetings organized by Terzioğlu in Silivri facilities are: February 10–14, 1973: First National Meeting of Mathematicians; July 9–14, 1973: the preparatory course related to the Summer Seminar on International Display Theory of Finite Groups; July 15–28, 1973: Summer Seminar on International Display Theory of Finite Groups; August 20 – September 9, 1973: International Symposium on Functional Analysis; September 8–21, 1975: the preparatory course related to the International Symposium on Algebraic Number Theory; September 22–27, 1975: International Symposium on Algebraic Number Theory; April 23–26, 1976: Second National Meeting of Mathematicians; August 1976: Ultrasound Congress (joint with physicists); September 5–11, 1976: International Congress of Functional Analysis; September 20–25, 1976: Rolf Nevanlinna International Symposium. == Death == Terzioğlu died as a result of a heart attack in the morning of the opening day of the International Symposium organized to tribute Rolf Nevanlinna, who had been a teacher of Terzioğlu. Albeit his unexpected loss, the symposium was completed after some rearrangements were made in the program. The guest mathematicians also attended the funeral ceremony on September 22 and the symposium began on September 23. Terzioğlu was elected as the honorary guest of this symposium and the title doctoris honoris causa was awarded to Rolf Nevanlinna by Istanbul University. == Legacy == One of the contributions of Terzioğlu as the director of the Mathematics Research Institute to Turkey's mathematical culture and the history of science was the systematic scan of the Islamic literature relevant to mathematics and the presentation of the information related to conic sections in ancient mathematics to the scientific community. As a result of these efforts, the facsimile of two ancient texts of mathematics originally written in Arabic were realized. The first one is the preface of Mecmuatu'r-risail, the Arabic translation by Beni Musa b. Sakir (died in 873) of Conica, which is the work of Apollonius of Perga (BC 262–190) on the conic sections. This preface, published with the title Das Vorwort des Astronomen Bani Musa b. Sakir, describes how the Apollonius' Conica was acquired by the Islamic world. After that, Terzioğlu published the facsimile of the copy of the lost 8th book of Apollonius' Conica which was rewritten by Ibnu'l-Heysem (965–1039) with the help from other sources. In the introduction part of this book with the title Das Achte Buch zu den Conica des Apollonios von Perge, the following information is provided in summary: In ancient mathematics, the interest for conics starts with Menaechmus (BC IV. Century) and reaches the summit with Apollonius of Perga. Apollonius wrote his famous work Conica by processing previous information and adding up his own inventions. The first 7 volumes of this work consisting of 8 volumes in total are known whereas the 8th volume is missing. The Islamic and Western mathematicians working in this field took place in the reconstruction of the 8th volume. The most successful one of these works is that of Edmund Halley's (1656–1742) Apollonii Per-gaei conicorum (Oxoniae, 1710). The 8th book of Conica reconstructed by Ibn el-Heysem is the 4th manuscript with the name Makalatu'l-Hasan b.el-Hasan b.el Heysem fi el-kitabu'l-mahrutat in the Mecmu'atu'r-risail, which is recorded under no. 1796 in Manisa Library. The fact that Ibn el-Heysem completed this work nearly 700 years before Halley is interesting. Within the framework of this program, Terzioğlu was preparing for publication the first 7 books of Conica, which were translated into Arabic in 415/1024 AD by Ibnu'l-Heysem who had also examined the previous translations of his time. Terzioğlu's death coincides with the time when the facsimile of the manuscript located at No. 2762 of Suleymaniye Library, Ayasofya had been completed. As the part of the book he wanted to include related to the history of conics remained incomplete, it was removed from press and was published later with the title Kitab al-Mahrutat Das Buch der Kegelschnitte des Apollonios von Perge by the Research Institute for Mathematics. It includes a part in which the description of the manuscript and the direct translation of its preface are given in Turkish and German. One of the most important services of Terzioğlu to the Turkish history of science is to make translate into Turkish in Latin letters the published first two volumes and the third volume as a manuscript (see Istanbul University Library TY. 903, 904, 905 for copies of manuscripts) of Asar-i Bakiye} (Vol. I–II, Istanbul, 1329/1913) by Salih Zeki Bey (1863–1921) during his presidency of the Turkish Mathematics Association. His aim was to offer such an old source to the benefit of young generations. === Positions and awards === Terzioğlu, who had an important role in the revival of the Union of Balkan Mathematicians (French: Union Balkanique des Mathematiciens) which was founded before World War II, had been the president of that organization for two periods (1966–1971). He was also selected as the chairman of the IV. Congress of Balkan Mathematicians organized in Istanbul on August 29, 1972. Among his other international activities, the role he played in providing Turkey with the membership of the International Mathematical Union is an unforgettable service. In 1973, Terzioğlu was selected as a member of Hahnemann Medical Society of America. In 1974, he has been awarded the Medal of Merit of Federal Republic of Germany by the German President on his endeavor for the development of Turkish–German relations. He also has two medals given by the Charles University in Prague and the Finland University of Jyväskylä. Nazim Terzioğlu has been awarded on December 2, 1982 the TÜBİTAK Service Award thanks to his contributions to the development of mathematics in our country. His family established a Mathematics Research Award on behalf of Terzioğlu who gave efforts during his life for the development of mathematics and the creation of a research potential. For the first time, this award had been given to three young mathematicians in a ceremony at the Faculty of Science of Istanbul University on September 20, 1981, which is the fifth year of his death. The second award in 1982 was given to a young mathematician in the opening ceremony of the International Symposium on Mathematics that was held on 14–24 September 1982 at the Karadeniz Technical University where Terzioğlu served as the founder-rector. His son, Tosun Terzioğlu (1942–2016) was a Turkish mathematician and academic administrator. == Books == The books written by Terzioğlu, who has many published articles in his own field, are: Über Finslersche Raume (Doktorarbeit), München, 1936 (On Finsler Spaces (Ph.D. Thesis), Munich, 1936.) Fonksiyonlar Teorisine Baslangic. Fonksiyonlar Teorisi. 2 Cilt. (Konrad Knopp'dan ceviri), Istanbul, 1938–1939. (Introduction to the Theory of Functions. Theory of Functions by Konrad Knopp, 2 Volumes (translated), Istanbul, 1938–1939.) Finsler Uzay\i nda Gauss–Bonnet Teoremi, Istanbul 1948. (Gauss–Bonnet theorem in Finsler Spaces, Istanbul 1948.) Lise Fen Kolu Icin Modern Geometri: Konikler, (Ahmet Nazmi Ilker ile), Istanbul, 1960. (Modern Geometry for the Science Sections of High Schools: Conics, (with Ahmet Nazmi Ilker), Istanbul, 1960.) Liseler Icin Cebir Temrinleri (P. Aubert ve G. Papelier'den ceviri), Istanbul, 1960. (Exercises in Algebra for High Schools by P. Aubert and G. Papelier (translated), Istanbul, 1960.) Diferansiyel ve Integral Hesap, (Edmund Landau'dan ceviri), Istanbul, 1961. (Differential and Integral Calculus by Edmund Landau (translated), Istanbul, 1961.) Lise Fen Kolu Icin Modern Geometri. Fasikül I-Kesenler; Fasikül II-Harmonik Bolme, Harmonik Demet, Daireye Göre Kuvvet; Fasikül III-Daireye Göre Kutup ve Kutup Dogrusu (G. Papelier'den ceviri), Istanbul, 1968. (Modern Geometry for the Science Sections of High Schools. Fascicle I: Secants; Fascicle II: Harmonic Division, Harmonic Pencil, Power with respect to the sphere; Fascicle III: Pole and polar line with respect to the sphere by G. Papelier (translated), Istanbul, 1968.) Analiz Problemleri, Istanbul, 1973. (Problems in Analysis, Istanbul, 1973.) Das Vorwort des Astronomen Bani Musa b. Sakir zu den Conica des Apollonios von Perge, Istanbul, 1974. (The foreword of the Astronomer Bani Musa b. Sakir to the Conics of Apollonius of Perga, Istanbul, 1974.) Das achte Buch zu den Conica des Apollonios von Perge re-konstruiert von Ibn al-Haysam, Istanbul, 1974. (The Eighth Book to the Conics of Apollonius of Perga Reconstructed by Ibn-Haysam, Istanbul, 1974.) Kitab al-Mahrutat. Das Buch der Kegelschnitte des Apollonios von Perge, Istanbul, 1981. (Kitab al-Mahrutat. The Book of Conic Sections of Apollonius of Perga, Istanbul, 1981.) == References == == Further reading == Nazim Terzioğlu, History of Science (Monthly Journal), February 1993, Number 16, 11–19. International Symposium on Analysis and Theory of Functions, ATF2009 (Dedicated to Nazim Terzioğlu) Abstract Book.
Wikipedia:Near sets#0	In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. Descriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjoint sets. Spatially near sets are also descriptively near sets. The underlying assumption with descriptively close sets is that such sets contain elements that have location and measurable features such as colour and frequency of occurrence. The description of the element of a set is defined by a feature vector. Comparison of feature vectors provides a basis for measuring the closeness of descriptively near sets. Near set theory provides a formal basis for the observation, comparison, and classification of elements in sets based on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as engineering and science problems. Near sets have a variety of applications in areas such as topology, pattern detection and classification, abstract algebra, mathematics in computer science, and solving a variety of problems based on human perception that arise in areas such as image analysis, image processing, face recognition, ethology, as well as engineering and science problems. From the beginning, descriptively near sets have proved to be useful in applications of topology, and visual pattern recognition , spanning a broad spectrum of applications that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analysis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, and topological psychology. As an illustration of the degree of descriptive nearness between two sets, consider an example of the Henry colour model for varying degrees of nearness between sets of picture elements in pictures (see, e.g., §4.3). The two pairs of ovals in Fig. 1 and Fig. 2 contain coloured segments. Each segment in the figures corresponds to an equivalence class where all pixels in the class have similar descriptions, i.e., picture elements with similar colours. The ovals in Fig.1 are closer to each other descriptively than the ovals in Fig. 2. == History == It has been observed that the simple concept of nearness unifies various concepts of topological structures inasmuch as the category Near of all nearness spaces and nearness preserving maps contains categories sTop (symmetric topological spaces and continuous maps), Prox (proximity spaces and δ {\displaystyle \delta } -maps), Unif (uniform spaces and uniformly continuous maps) and Cont (contiguity spaces and contiguity maps) as embedded full subcategories. The categories ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} and ε A M e r {\displaystyle {\boldsymbol {\varepsilon {AMer}}}} are shown to be full supercategories of various well-known categories, including the category s T o p {\displaystyle {\boldsymbol {sTop}}} of symmetric topological spaces and continuous maps, and the category M e t ∞ {\displaystyle {\boldsymbol {Met^{\infty }}}} of extended metric spaces and nonexpansive maps. The notation A ↪ B {\displaystyle {\boldsymbol {A}}\hookrightarrow {\boldsymbol {B}}} reads category A {\displaystyle {\boldsymbol {A}}} is embedded in category B {\displaystyle {\boldsymbol {B}}} . The categories ε A M e r {\displaystyle {\boldsymbol {\varepsilon AMer}}} and ε A N e a r {\displaystyle {\boldsymbol {\varepsilon ANear}}} are supercategories for a variety of familiar categories shown in Fig. 3. Let ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} denote the category of all ε {\displaystyle \varepsilon } -approach nearness spaces and contractions, and let ε A M e r {\displaystyle {\boldsymbol {\varepsilon AMer}}} denote the category of all ε {\displaystyle \varepsilon } -approach merotopic spaces and contractions. Among these familiar categories is s T o p {\displaystyle {\boldsymbol {sTop}}} , the symmetric form of T o p {\displaystyle {\boldsymbol {Top}}} (see category of topological spaces), the category with objects that are topological spaces and morphisms that are continuous maps between them. M e t ∞ {\displaystyle {\boldsymbol {Met^{\infty }}}} with objects that are extended metric spaces is a subcategory of ε A P {\displaystyle {\boldsymbol {\varepsilon AP}}} (having objects ε {\displaystyle \varepsilon } -approach spaces and contractions) (see also). Let ρ X , ρ Y {\displaystyle \rho _{X},\rho _{Y}} be extended pseudometrics on nonempty sets X , Y {\displaystyle X,Y} , respectively. The map f : ( X , ρ X ) ⟶ ( Y , ρ Y ) {\displaystyle f:(X,\rho _{X})\longrightarrow (Y,\rho _{Y})} is a contraction if and only if f : ( X , ν D ρ X ) ⟶ ( Y , ν D ρ Y ) {\displaystyle f:(X,\nu _{D_{\rho _{X}}})\longrightarrow (Y,\nu _{D_{\rho _{Y}}})} is a contraction. For nonempty subsets A , B ∈ 2 X {\displaystyle A,B\in 2^{X}} , the distance function D ρ : 2 X × 2 X ⟶ [ 0 , ∞ ] {\displaystyle D_{\rho }:2^{X}\times 2^{X}\longrightarrow [0,\infty ]} is defined by D ρ ( A , B ) = { inf { ρ ( a , b ) : a ∈ A , b ∈ B } , if A and B are not empty , ∞ , if A or B is empty . {\displaystyle D_{\rho }(A,B)={\begin{cases}\inf {\{\rho (a,b):a\in A,b\in B\}},&{\text{if }}A{\text{ and }}B{\text{ are not empty}},\\\infty ,&{\text{if }}A{\text{ or }}B{\text{ is empty}}.\end{cases}}} Thus ε {\displaystyle {\boldsymbol {\varepsilon }}} AP is embedded as a full subcategory in ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} by the functor F : ε A P ⟶ ε A N e a r {\displaystyle F:{\boldsymbol {\varepsilon {AP}}}\longrightarrow {\boldsymbol {\varepsilon {ANear}}}} defined by F ( ( X , ρ ) ) = ( X , ν D ρ ) {\displaystyle F((X,\rho ))=(X,\nu _{D_{\rho }})} and F ( f ) = f {\displaystyle F(f)=f} . Then f : ( X , ρ X ) ⟶ ( Y , ρ Y ) {\displaystyle f:(X,\rho _{X})\longrightarrow (Y,\rho _{Y})} is a contraction if and only if f : ( X , ν D ρ X ) ⟶ ( Y , ν D ρ Y ) {\displaystyle f:(X,\nu _{D_{\rho _{X}}})\longrightarrow (Y,\nu _{D_{\rho _{Y}}})} is a contraction. Thus ε A P {\displaystyle {\boldsymbol {\varepsilon {AP}}}} is embedded as a full subcategory in ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} by the functor F : ε A P ⟶ ε A N e a r {\displaystyle F:{\boldsymbol {\varepsilon {AP}}}\longrightarrow {\boldsymbol {\varepsilon {ANear}}}} defined by F ( ( X , ρ ) ) = ( X , ν D ρ ) {\displaystyle F((X,\rho ))=(X,\nu _{D_{\rho }})} and F ( f ) = f . {\displaystyle F(f)=f.} Since the category M e t ∞ {\displaystyle {\boldsymbol {Met^{\infty }}}} of extended metric spaces and nonexpansive maps is a full subcategory of ε A P {\displaystyle {\boldsymbol {\varepsilon {AP}}}} , therefore, ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} is also a full supercategory of M e t ∞ {\displaystyle {\boldsymbol {Met^{\infty }}}} . The category ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} is a topological construct. The notions of near and far in mathematics can be traced back to works by Johann Benedict Listing and Felix Hausdorff. The related notions of resemblance and similarity can be traced back to J.H. Poincaré, who introduced sets of similar sensations (nascent tolerance classes) to represent the results of G.T. Fechner's sensation sensitivity experiments and a framework for the study of resemblance in representative spaces as models of what he termed physical continua. The elements of a physical continuum (pc) are sets of sensations. The notion of a pc and various representative spaces (tactile, visual, motor spaces) were introduced by Poincaré in an 1894 article on the mathematical continuum, an 1895 article on space and geometry and a compendious 1902 book on science and hypothesis followed by a number of elaborations, e.g.,. The 1893 and 1895 articles on continua (Pt. 1, ch. II) as well as representative spaces and geometry (Pt. 2, ch IV) are included as chapters in. Later, F. Riesz introduced the concept of proximity or nearness of pairs of sets at the International Congress of Mathematicians (ICM) in 1908. During the 1960s, E.C. Zeeman introduced tolerance spaces in modelling visual perception. A.B. Sossinsky observed in 1986 that the main idea underlying tolerance space theory comes from Poincaré, especially. In 2002, Z. Pawlak and J. Peters considered an informal approach to the perception of the nearness of physical objects such as snowflakes that was not limited to spatial nearness. In 2006, a formal approach to the descriptive nearness of objects was considered by J. Peters, A. Skowron and J. Stepaniuk in the context of proximity spaces. In 2007, descriptively near sets were introduced by J. Peters followed by the introduction of tolerance near sets. Recently, the study of descriptively near sets has led to algebraic, topological and proximity space foundations of such sets. == Nearness of sets == The adjective near in the context of near sets is used to denote the fact that observed feature value differences of distinct objects are small enough to be considered indistinguishable, i.e., within some tolerance. The exact idea of closeness or 'resemblance' or of 'being within tolerance' is universal enough to appear, quite naturally, in almost any mathematical setting (see, e.g.,). It is especially natural in mathematical applications: practical problems, more often than not, deal with approximate input data and only require viable results with a tolerable level of error. The words near and far are used in daily life and it was an incisive suggestion of F. Riesz that these intuitive concepts be made rigorous. He introduced the concept of nearness of pairs of sets at the ICM in Rome in 1908. This concept is useful in simplifying teaching calculus and advanced calculus. For example, the passage from an intuitive definition of continuity of a function at a point to its rigorous epsilon-delta definition is sometime difficult for teachers to explain and for students to understand. Intuitively, continuity can be explained using nearness language, i.e., a function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } is continuous at a point c {\displaystyle c} , provided points { x } {\displaystyle \{x\}} near c {\displaystyle c} go into points { f ( x ) } {\displaystyle \{f(x)\}} near f ( c ) {\displaystyle f(c)} . Using Riesz's idea, this definition can be made more precise and its contrapositive is the familiar definition. == Generalization of set intersection == From a spatial point of view, nearness (a.k.a. proximity) is considered a generalization of set intersection. For disjoint sets, a form of nearness set intersection is defined in terms of a set of objects (extracted from disjoint sets) that have similar features within some tolerance (see, e.g., §3 in). For example, the ovals in Fig. 1 are considered near each other, since these ovals contain pairs of classes that display similar (visually indistinguishable) colours. == Efremovič proximity space == Let X {\displaystyle X} denote a metric topological space that is endowed with one or more proximity relations and let 2 X {\displaystyle 2^{X}} denote the collection of all subsets of X {\displaystyle X} . The collection 2 X {\displaystyle 2^{X}} is called the power set of X {\displaystyle X} . There are many ways to define Efremovič proximities on topological spaces (discrete proximity, standard proximity, metric proximity, Čech proximity, Alexandroff proximity, and Freudenthal proximity), For details, see § 2, pp. 93–94 in. The focus here is on standard proximity on a topological space. For A , B ⊂ X {\displaystyle A,B\subset X} , A {\displaystyle A} is near B {\displaystyle B} (denoted by A δ B {\displaystyle A\ \delta \ B} ), provided their closures share a common point. The closure of a subset A ∈ 2 X {\displaystyle A\in 2^{X}} (denoted by cl ( A ) {\displaystyle {\mbox{cl}}(A)} ) is the usual Kuratowski closure of a set, introduced in § 4, p. 20, is defined by cl ( A ) = { x ∈ X : D ( x , A ) = 0 } , where D ( x , A ) = inf { d ( x , a ) : a ∈ A } . {\displaystyle {\begin{aligned}{\mbox{cl}}(A)&=\left\{x\in X:D(x,A)=0\right\},\ {\mbox{where}}\\D(x,A)&=\inf \left\{d(x,a):a\in A\right\}.\end{aligned}}} I.e., cl ( A ) {\displaystyle {\mbox{cl}}(A)} is the set of all points x {\displaystyle x} in X {\displaystyle X} that are close to A {\displaystyle A} ( D ( x , A ) {\displaystyle D(x,A)} is the Hausdorff distance (see § 22, p. 128, in) between x {\displaystyle x} and the set A {\displaystyle A} and d ( x , a ) = \| x − a \| {\displaystyle d(x,a)=\left\|x-a\right\|} (standard distance)). A standard proximity relation is defined by δ = { ( A , B ) ∈ 2 X × 2 X : cl ( A ) ∩ cl ( B ) ≠ ∅ } . {\displaystyle \delta =\left\{(A,B)\in 2^{X}\times 2^{X}:{\mbox{cl}}(A)\ \cap \ {\mbox{cl}}(B)\neq \emptyset \right\}.} Whenever sets A {\displaystyle A} and B {\displaystyle B} have no points in common, the sets are farfrom each other (denoted A δ _ B {\displaystyle A\ {\underline {\delta }}\ B} ). The following EF-proximity space axioms are given by Jurij Michailov Smirnov based on what Vadim Arsenyevič Efremovič introduced during the first half of the 1930s. Let A , B , E ∈ 2 X {\displaystyle A,B,E\in 2^{X}} . EF.1 If the set A {\displaystyle A} is close to B {\displaystyle B} , then B {\displaystyle B} is close to A {\displaystyle A} . EF.2 A ∪ B {\displaystyle A\cup B} is close to E {\displaystyle E} , if and only if, at least one of the sets A {\displaystyle A} or B {\displaystyle B} is close to E {\displaystyle E} . EF.3 Two points are close, if and only if, they are the same point. EF.4 All sets are far from the empty set ∅ {\displaystyle \emptyset } . EF.5 For any two sets A {\displaystyle A} and B {\displaystyle B} which are far from each other, there exists C , D ∈ 2 X {\displaystyle C,D\in 2^{X}} , C ∪ D = X {\displaystyle C\cup D=X} , such that A {\displaystyle A} is far from C {\displaystyle C} and B {\displaystyle B} is far from D {\displaystyle D} (Efremovič-axiom). The pair ( X , δ ) {\displaystyle (X,\delta )} is called an EF-proximity space. In this context, a space is a set with some added structure. With a proximity space X {\displaystyle X} , the structure of X {\displaystyle X} is induced by the EF-proximity relation δ {\displaystyle \delta } . In a proximity space X {\displaystyle X} , the closure of A {\displaystyle A} in X {\displaystyle X} coincides with the intersection of all closed sets that contain A {\displaystyle A} . Theorem 1 The closure of any set A {\displaystyle A} in the proximity space X {\displaystyle X} is the set of points x ∈ X {\displaystyle x\in X} that are close to A {\displaystyle A} . == Visualization of EF-axiom == Let the set X {\displaystyle X} be represented by the points inside the rectangular region in Fig. 5. Also, let A , B {\displaystyle A,B} be any two non-intersection subsets (i.e. subsets spatially far from each other) in X {\displaystyle X} , as shown in Fig. 5. Let C c = X ∖ C {\displaystyle C^{c}=X\backslash C} (complement of the set C {\displaystyle C} ). Then from the EF-axiom, observe the following: A δ _ B , B ⊂ C , D = C c , X = D ∪ C , A ⊂ D , hence, we can write A δ _ B ⇒ A δ _ C and B δ _ D , for some C , D in X so that C ∪ D = X . ◼ {\displaystyle {\begin{aligned}A&{}\mathrel {\underline {\delta }} B,\\B&\subset C,\\D&=C^{c},\\X&=D\cup C,\\A&\subset D,\ {\mbox{hence, we can write}}\\A\ {\underline {\delta }}\ B\ &\Rightarrow \ A\ {\underline {\delta }}\ C\ {\mbox{and}}\ B\ {\underline {\delta }}\ D,\ {\mbox{for some}}\ C,D\ {\mbox{in}}\ X{\mbox{ so that }}C\cup D=X.\qquad \blacksquare \end{aligned}}} == Descriptive proximity space == Descriptively near sets were introduced as a means of solving classification and pattern recognition problems arising from disjoint sets that resemble each other. Recently, the connections between near sets in EF-spaces and near sets in descriptive EF-proximity spaces have been explored in. Again, let X {\displaystyle X} be a metric topological space and let Φ = { ϕ 1 , … , ϕ n } {\displaystyle \Phi =\left\{\phi _{1},\dots ,\phi _{n}\right\}} a set of probe functions that represent features of each x ∈ X {\displaystyle x\in X} . The assumption made here is X {\displaystyle X} contains non-abstract points that have measurable features such as gradient orientation. A non-abstract point has a location and features that can be measured (see § 3 in ). A probe function ϕ : X → R {\displaystyle \phi :X\rightarrow \mathbb {R} } represents a feature of a sample point in X {\displaystyle X} . The mapping Φ : X ⟶ R n {\displaystyle \Phi :X\longrightarrow \mathbb {R} ^{n}} is defined by Φ ( x ) = ( ϕ 1 ( x ) , … , ϕ n ( x ) ) {\displaystyle \Phi (x)=(\phi _{1}(x),\dots ,\phi _{n}(x))} , where R n {\displaystyle \mathbb {R} ^{n}} is an n-dimensional real Euclidean vector space. Φ ( x ) {\displaystyle \Phi (x)} is a feature vector for x {\displaystyle x} , which provides a description of x ∈ X {\displaystyle x\in X} . For example, this leads to a proximal view of sets of picture points in digital images. To obtain a descriptive proximity relation (denoted by δ Φ {\displaystyle \delta _{\Phi }} ), one first chooses a set of probe functions. Let Q : 2 X ⟶ 2 R n {\displaystyle {\mathcal {Q}}:2^{X}\longrightarrow 2^{R^{n}}} be a mapping on a subset of 2 X {\displaystyle 2^{X}} into a subset of 2 R n {\displaystyle 2^{R^{n}}} . For example, let A , B ∈ 2 X {\displaystyle A,B\in 2^{X}} and Q ( A ) , Q ( B ) {\displaystyle {\mathcal {Q}}(A),{\mathcal {Q}}(B)} denote sets of descriptions of points in A , B {\displaystyle A,B} , respectively. That is, Q ( A ) = { Φ ( a ) : a ∈ A } , Q ( B ) = { Φ ( b ) : b ∈ B } . {\displaystyle {\begin{aligned}{\mathcal {Q}}(A)&=\left\{\Phi (a):a\in A\right\},\\{\mathcal {Q}}(B)&=\left\{\Phi (b):b\in B\right\}.\end{aligned}}} The expression A δ Φ B {\displaystyle A\mathrel {\delta _{\Phi }} B} reads A {\displaystyle A} is descriptively near B {\displaystyle B} . Similarly, A δ _ Φ B {\displaystyle A\mathrel {{\underline {\delta }}_{\Phi }} B} reads A {\displaystyle A} is descriptively far from B {\displaystyle B} . The descriptive proximity of A {\displaystyle A} and B {\displaystyle B} is defined by A δ Φ B ⇔ Q ( cl ( A ) ) δ Q ( cl ( B ) ) ≠ ∅ . {\displaystyle A\mathrel {\delta _{\Phi }} B\Leftrightarrow {\mathcal {Q}}({\mbox{cl}}(A))\mathrel {\delta } {\mathcal {Q}}({\mbox{cl}}(B))\neq \emptyset .} The descriptive intersection ∩ Φ {\displaystyle \mathop {\cap } _{\Phi }} of A {\displaystyle A} and B {\displaystyle B} is defined by A ∩ Φ B = { x ∈ A ∪ B : Q ( A ) δ Q ( B ) } . {\displaystyle A\mathbin {\mathop {\cap } _{\Phi }} B=\left\{x\in A\cup B:{\mathcal {Q}}(A)\mathrel {\delta } {\mathcal {Q}}(B)\right\}.} That is, x ∈ A ∪ B {\displaystyle x\in A\cup B} is in A ∩ Φ B {\displaystyle A\mathbin {\mathop {\cap } _{\Phi }} B} , provided Φ ( x ) = Φ ( a ) = Φ ( b ) {\displaystyle \Phi (x)=\Phi (a)=\Phi (b)} for some a ∈ A , b ∈ B {\displaystyle a\in A,b\in B} . Observe that A {\displaystyle A} and B {\displaystyle B} can be disjoint and yet A ∩ Φ B {\displaystyle A\mathbin {\mathop {\cap } _{\Phi }} B} can be nonempty. The descriptive proximity relation δ Φ {\displaystyle \delta _{\Phi }} is defined by δ Φ = { ( A , B ) ∈ 2 X × 2 X : cl ( A ) ∩ Φ cl ( B ) ≠ ∅ } . {\displaystyle \delta _{\Phi }=\left\{(A,B)\in 2^{X}\times 2^{X}:{\mbox{cl}}(A)\mathbin {\mathop {\cap } _{\Phi }} {\mbox{cl}}(B)\neq \emptyset \right\}.} Whenever sets A {\displaystyle A} and B {\displaystyle B} have no points with matching descriptions, the sets are descriptively far from each other (denoted by A δ _ Φ B {\displaystyle A\ {\underline {\delta }}_{\Phi }\ B} ). The binary relation δ Φ {\displaystyle \delta _{\Phi }} is a descriptive EF-proximity, provided the following axioms are satisfied for A , B , E ⊂ X {\displaystyle A,B,E\subset X} . dEF.1 If the set A {\displaystyle A} is descriptively close to B {\displaystyle B} , then B {\displaystyle B} is descriptively close to A {\displaystyle A} . dEF.2 A ∪ B {\displaystyle A\cup B} is descriptively close to E {\displaystyle E} , if and only if, at least one of the sets A {\displaystyle A} or B {\displaystyle B} is descriptively close to E {\displaystyle E} . dEF.3 Two points x , y ∈ X {\displaystyle x,y\in X} are descriptively close, if and only if, the description of x {\displaystyle x} matches the description of y {\displaystyle y} . dEF.4 All nonempty sets are descriptively far from the empty set ∅ {\displaystyle \emptyset } . dEF.5 For any two sets A {\displaystyle A} and B {\displaystyle B} which are descriptively far from each other, there exists C , D ∈ 2 X {\displaystyle C,D\in 2^{X}} , C ∪ D = X {\displaystyle C\cup D=X} , such that A {\displaystyle A} is descriptively far from C {\displaystyle C} and B {\displaystyle B} is descriptively far from D {\displaystyle D} (Descriptive Efremovič axiom). The pair ( X , δ Φ ) {\displaystyle (X,\delta _{\Phi })} is called a descriptive proximity space. == Proximal relator spaces == A relator is a nonvoid family of relations R {\displaystyle {\mathcal {R}}} on a nonempty set X {\displaystyle X} . The pair ( X , R ) {\displaystyle (X,{\mathcal {R}})} (also denoted X ( R ) {\displaystyle X({\mathcal {R}})} ) is called a relator space. Relator spaces are natural generalizations of ordered sets and uniform spaces. With the introduction of a family of proximity relations R δ {\displaystyle {\mathcal {R}}_{\delta }} on X {\displaystyle X} , we obtain a proximal relator space ( X , R δ ) {\displaystyle (X,{\mathcal {R}}_{\delta })} . For simplicity, we consider only two proximity relations, namely, the Efremovič proximity δ {\displaystyle \delta } and the descriptive proximity δ Φ {\displaystyle \delta _{\Phi }} in defining the descriptive relator R δ Φ {\displaystyle {\mathcal {R}}_{\delta _{\Phi }}} . The pair ( X , R δ Φ ) {\displaystyle (X,{\mathcal {R}}_{\delta _{\Phi }})} is called a proximal relator space. In this work, X {\displaystyle X} denotes a metric topological space that is endowed with the relations in a proximal relator. With the introduction of ( X , R δ Φ ) {\displaystyle (X,{\mathcal {R}}_{\delta _{\Phi }})} , the traditional closure of a subset (e.g., ) can be compared with the more recent descriptive closure of a subset. In a proximal relator space X {\displaystyle X} , the descriptive closure of a set A {\displaystyle A} (denoted by cl Φ ( A ) {\displaystyle {\mbox{cl}}_{\Phi }(A)} ) is defined by cl Φ ( A ) = { x ∈ X : Φ ( x ) δ Q ( cl ( A ) ) } . {\displaystyle {\mbox{cl}}_{\Phi }(A)=\left\{x\in X:{\Phi (x)}\mathrel {\delta } {\mathcal {Q}}({\mbox{cl}}(A))\right\}.} That is, x ∈ X {\displaystyle x\in X} is in the descriptive closure of A {\displaystyle A} , provided the closure of Φ ( x ) {\displaystyle \Phi (x)} and the closure of Q ( cl ( A ) ) {\displaystyle {\mathcal {Q}}({\mbox{cl}}(A))} have at least one element in common. Theorem 2 The descriptive closure of any set A {\displaystyle A} in the descriptive EF-proximity space ( X , R δ Φ ) {\displaystyle (X,{\mathcal {R}}_{\delta _{\Phi }})} is the set of points x ∈ X {\displaystyle x\in X} that are descriptively close to A {\displaystyle A} . Theorem 3 Kuratowski closure of a set A {\displaystyle A} is a subset of the descriptive closure of A {\displaystyle A} in a descriptive EF-proximity space. Theorem 4 Let ( X , R δ Φ ) {\displaystyle (X,{\mathcal {R}}_{\delta _{\Phi }})} be a proximal relator space, A ⊂ X {\displaystyle A\subset X} . Then cl ( A ) ⊆ cl Φ ( A ) {\displaystyle {\mbox{cl}}(A)\subseteq {\mbox{cl}}_{\Phi }(A)} . Proof Let Φ ( x ) ∈ Q ( X ∖ cl ( A ) ) {\displaystyle \Phi (x)\in {\mathcal {Q}}(X\setminus {\mbox{cl}}(A))} such that Φ ( x ) = Φ ( a ) {\displaystyle \Phi (x)=\Phi (a)} for some a ∈ cl A {\displaystyle a\in {\mbox{cl}}A} . Consequently, Φ ( x ) ∈ Q ( cl Φ ( A ) ) {\displaystyle \Phi (x)\in {\mathcal {Q}}({\mbox{cl}}_{\Phi }(A))} . Hence, cl ( A ) ⊆ cl Φ ( A ) {\displaystyle {\mbox{cl}}(A)\subseteq {\mbox{cl}}_{\Phi }(A)} In a proximal relator space, EF-proximity δ {\displaystyle \delta } leads to the following results for descriptive proximity δ Φ {\displaystyle \delta _{\Phi }} . Theorem 5 Let ( X , R δ Φ ) {\displaystyle (X,{\mathcal {R}}_{\delta _{\Phi }})} be a proximal relator space, A , B , C ⊂ X {\displaystyle A,B,C\subset X} . Then 1° A δ B implies A δ Φ B {\displaystyle A\ \delta \ B\ {\mbox{implies}}\ A\ \delta _{\Phi }\ B} . 2° ( A ∪ B ) δ C implies ( A ∪ B ) δ Φ C {\displaystyle (A\cup B)\ \delta \ C\ {\mbox{implies}}\ (A\cup B)\ \delta _{\Phi }\ C} . 3° cl A δ cl B implies cl A δ Φ cl B {\displaystyle {\mbox{cl}}A\ \delta \ {\mbox{cl}}B\ {\mbox{implies}}\ {\mbox{cl}}A\ \delta _{\Phi }\ {\mbox{cl}}B} . Proof 1° A δ B ⇔ A ∩ B ≠ ∅ {\displaystyle A\ \delta \ B\Leftrightarrow A\cap B\neq \emptyset } . For x ∈ A ∩ B , Φ ( x ) ∈ Q ( A ) {\displaystyle x\in A\cap B,\Phi (x)\in {\mathcal {Q}}(A)} and Φ ( x ) ∈ Q ( B ) {\displaystyle \Phi (x)\in {\mathcal {Q}}(B)} . Consequently, A δ Φ B {\displaystyle A\ \delta _{\Phi }\ B} . 1° ⇒ 2° 3° cl A δ cl B {\displaystyle {\mbox{cl}}A\ \delta \ {\mbox{cl}}B} implies that cl A {\displaystyle {\mbox{cl}}A} and cl A {\displaystyle {\mbox{cl}}A} have at least one point in common. Hence, 1° ⇒ 3°. ◼ {\displaystyle \qquad \blacksquare } == Descriptive 𝛿-neighbourhoods == In a pseudometric proximal relator space X {\displaystyle X} , the neighbourhood of a point x ∈ X {\displaystyle x\in X} (denoted by N x , ε {\displaystyle N_{x,\varepsilon }} ), for ε > 0 {\displaystyle \varepsilon >0} , is defined by N x , ε = { y ∈ X : d ( x , y ) < ε } . {\displaystyle N_{x,\varepsilon }=\left\{y\in X:d(x,y)<\varepsilon \right\}.} The interior of a set A {\displaystyle A} (denoted by int ( A ) {\displaystyle {\mbox{int}}(A)} ) and boundary of A {\displaystyle A} (denoted by bdy ( A ) {\displaystyle {\mbox{bdy}}(A)} ) in a proximal relator space X {\displaystyle X} are defined by int ( A ) = { x ∈ X : N x , ε ⊆ A } . bdy ( A ) = cl ( A ) ∖ int ( A ) . {\displaystyle {\begin{aligned}{\mbox{int}}(A)&=\left\{x\in X:N_{x,\varepsilon }\subseteq A\right\}.\\{\mbox{bdy}}(A)&={\mbox{cl}}(A)\setminus {\mbox{int}}(A).\end{aligned}}} A set A {\displaystyle A} has a natural strong inclusion in a set B {\displaystyle B} associated with δ {\displaystyle \delta } } (denoted by A ≪ δ B {\displaystyle A\ll _{\delta }B} ), provided A ⊂ int ( B ) {\displaystyle A\subset {\mbox{int}}(B)} ; i.e., A δ _ X ∖ int ( B ) {\displaystyle A\mathrel {\underline {\delta }} X\setminus {\mbox{int}}(B)} ( A {\displaystyle A} is far from the complement of int ( B ) {\displaystyle {\mbox{int}}(B)} ). Correspondingly, a set A {\displaystyle A} has a descriptive strong inclusion in a set B {\displaystyle B} associated with δ Φ {\displaystyle \delta _{\Phi }} (denoted by A ≪ Φ B {\displaystyle A\mathrel {\mathop {\ll } _{\Phi }} B} ), provided Q ( A ) ⊂ Q ( int ( B ) ) {\displaystyle {\mathcal {Q}}(A)\subset \ {\mathcal {Q}}({\mbox{int}}(B))} ; i.e., A δ _ Φ X ∖ int ( B ) {\displaystyle A\ {\underline {\delta }}_{\Phi }\ X\setminus {\mbox{int}}(B)} ( Q ( A ) {\displaystyle {\mathcal {Q}}(A)} is far from the complement of int B {\displaystyle {\mbox{int}}B} ). Let ≪ Φ {\displaystyle \mathop {\ll } _{\Phi }} be a descriptive δ {\displaystyle \delta } -neighbourhood relation defined by ≪ Φ = { ( A , B ) ∈ 2 X × 2 X : Q ( A ) ⊂ Q ( int ( B ) ) } . {\displaystyle \mathop {\ll } _{\Phi }=\left\{(A,B)\in 2^{X}\times 2^{X}:{\mathcal {Q}}(A)\subset {\mathcal {Q}}({\mbox{int}}(B))\right\}.} That is, A ≪ Φ B {\displaystyle A\mathrel {\mathop {\ll } _{\Phi }} B} , provided the description of each a ∈ A {\displaystyle a\in A} is contained in the set of descriptions of the points b ∈ int ( B ) {\displaystyle b\in {\mbox{int}}(B)} . Now observe that any A , B {\displaystyle A,B} in the proximal relator space X {\displaystyle X} such that A δ _ Φ B {\displaystyle A\mathrel {{\underline {\delta }}_{\Phi }} B} have disjoint δ Φ {\displaystyle \delta _{\Phi }} -neighbourhoods; i.e., A δ _ Φ B ⇔ A ≪ Φ E 1 , B ≪ Φ E 2 , for some E 1 , E 2 ⊂ X (See Fig. 6). {\displaystyle A\mathrel {{\underline {\delta }}_{\Phi }} B\Leftrightarrow A\mathrel {\mathop {\ll } _{\Phi }} E1,B\mathrel {\mathop {\ll } _{\Phi }} E2,\ {\mbox{for some}}\ E1,E2\subset X\ {\mbox{(See Fig. 6).}}} Theorem 6 Any two sets descriptively far from each other belong to disjoint descriptive δ Φ {\displaystyle \delta _{\Phi }} -neighbourhoods in a descriptive proximity space X {\displaystyle X} . A consideration of strong containment of a nonempty set in another set leads to the study of hit-and-miss topologies and the Wijsman topology. == Tolerance near sets == Let ε {\displaystyle \varepsilon } be a real number greater than zero. In the study of sets that are proximally near within some tolerance, the set of proximity relations R δ Φ {\displaystyle {\mathcal {R}}_{\delta _{\Phi }}} is augmented with a pseudometric tolerance proximity relation (denoted by δ Φ , ε {\displaystyle \delta _{\Phi ,\varepsilon }} ) defined by D Φ ( A , B ) = inf { d ( Φ ( a ) , Φ ( a ) ) : Φ ( a ) ∈ Q ( A ) , Φ ( a ) ∈ Q ( B ) } , d ( Φ ( a ) , Φ ( a ) ) = ∑ i = 1 n ⁡ \| ϕ i ( a ) − ϕ i ( b ) \| , δ Φ , ε = { ( A , B ) ∈ 2 X × 2 X : \| D ( cl ( A ) , cl ( B ) ) \| < ε } . {\displaystyle {\begin{aligned}D_{\Phi }(A,B)&=\inf \left\{d(\Phi (a),\Phi (a)):\Phi (a)\in {\mathcal {Q}}(A),\Phi (a)\in {\mathcal {Q}}(B)\right\},\\d(\Phi (a),\Phi (a))&=\mathop {\sum } _{i=1}^{n}\|\phi _{i}(a)-\phi _{i}(b)\|,\\\delta _{\Phi ,\varepsilon }&=\left\{(A,B)\in 2^{X}\times 2^{X}:\|D({\mbox{cl}}(A),{\mbox{cl}}(B))\|<\varepsilon \right\}.\end{aligned}}} Let R δ Φ , ε = R δ Φ ∪ { δ Φ , ε } {\displaystyle {\mathcal {R}}_{\delta _{\Phi ,\varepsilon }}={\mathcal {R}}_{\delta _{\Phi }}\cup \left\{\delta _{\Phi ,\varepsilon }\right\}} . In other words, a nonempty set equipped with the proximal relator R δ Φ , ε {\displaystyle {\mathcal {R}}_{\delta _{\Phi ,\varepsilon }}} has underlying structure provided by the proximal relator R δ Φ {\displaystyle {\mathcal {R}}_{\delta _{\Phi }}} and provides a basis for the study of tolerance near sets in X {\displaystyle X} that are near within some tolerance. Sets A , B {\displaystyle A,B} in a descriptive pseudometric proximal relator space ( X , R δ Φ , ε ) {\displaystyle (X,{\mathcal {R}}_{\delta _{\Phi ,\varepsilon }})} are tolerance near sets (i.e., A δ Φ , ε B {\displaystyle A\ \delta _{\Phi ,\varepsilon }\ B} ), provided D Φ ( A , B ) < ε . {\displaystyle D_{\Phi }(A,B)<\varepsilon .} == Tolerance classes and preclasses == Relations with the same formal properties as similarity relations of sensations considered by Poincaré are nowadays, after Zeeman, called tolerance relations. A tolerance τ {\displaystyle \tau } on a set O {\displaystyle O} is a relation τ ⊆ O × O {\displaystyle \tau \subseteq O\times O} that is reflexive and symmetric. In algebra, the term tolerance relation is also used in a narrow sense to denote reflexive and symmetric relations defined on universes of algebras that are also compatible with operations of a given algebra, i.e., they are generalizations of congruence relations (see e.g.,). In referring to such relations, the term algebraic tolerance or the term algebraic tolerance relation is used. Transitive tolerance relations are equivalence relations. A set O {\displaystyle O} together with a tolerance τ {\displaystyle \tau } is called a tolerance space (denoted ( O , τ ) {\displaystyle (O,\tau )} ). A set A ⊆ O {\displaystyle A\subseteq O} is a τ {\displaystyle \tau } -preclass (or briefly preclass when τ {\displaystyle \tau } is understood) if and only if for any x , y ∈ A {\displaystyle x,y\in A} , ( x , y ) ∈ τ {\displaystyle (x,y)\in \tau } . The family of all preclasses of a tolerance space is naturally ordered by set inclusion and preclasses that are maximal with respect to set inclusion are called τ {\displaystyle \tau } -classes or just classes, when τ {\displaystyle \tau } is understood. The family of all classes of the space ( O , τ ) {\displaystyle (O,\tau )} is particularly interesting and is denoted by H τ ( O ) {\displaystyle H_{\tau }(O)} . The family H τ ( O ) {\displaystyle H_{\tau }(O)} is a covering of O {\displaystyle O} . The work on similarity by Poincaré and Zeeman presage the introduction of near sets and research on similarity relations, e.g.,. In science and engineering, tolerance near sets are a practical application of the study of sets that are near within some tolerance. A tolerance ε ∈ ( 0 , ∞ ] {\displaystyle \varepsilon \in (0,\infty ]} is directly related to the idea of closeness or resemblance (i.e., being within some tolerance) in comparing objects. By way of application of Poincaré's approach in defining visual spaces and Zeeman's approach to tolerance relations, the basic idea is to compare objects such as image patches in the interior of digital images. === Examples === Simple example The following simple example demonstrates the construction of tolerance classes from real data. Consider the 20 objects in the table below with \| Φ \| = 1 {\displaystyle \|\Phi \|=1} . Let a tolerance relation be defined as ≅ ε ⇔ { ( x , y ) ∈ O × O : ∥ Φ ( x ) − Φ ( y ) ∥ 2 ≤ ε } {\displaystyle \cong _{\varepsilon }{}\Leftrightarrow \{(x,y)\in O\times O:\;\parallel \Phi (x)-\Phi (y)\parallel _{_{2}}\leq \varepsilon \}} Then, setting ε = 0.1 {\displaystyle \varepsilon =0.1} gives the following tolerance classes: H ≅ ε ( O ) = { { x 1 , x 8 , x 10 , x 11 } , { x 1 , x 9 , x 10 , x 11 , x 14 } , { x 2 , x 7 , x 18 , x 19 } , { x 3 , x 12 , x 17 } , { x 4 , x 13 , x 20 } , { x 4 , x 18 } , { x 5 , x 6 , x 15 , x 16 } , { x 5 , x 6 , x 15 , x 20 } , { x 6 , x 13 , x 20 } } . {\displaystyle {\begin{aligned}H_{\cong _{\varepsilon }}(O)={}&\{\{x_{1},x_{8},x_{10},x_{11}\},\{x_{1},x_{9},x_{10},x_{11},x_{14}\},\\&\{x_{2},x_{7},x_{18},x_{19}\},\\&\{x_{3},x_{12},x_{17}\},\\&\{x_{4},x_{13},x_{20}\},\{x_{4},x_{18}\},\\&\{x_{5},x_{6},x_{15},x_{16}\},\{x_{5},x_{6},x_{15},x_{20}\},\\&\{x_{6},x_{13},x_{20}\}\}.\end{aligned}}} Observe that each object in a tolerance class satisfies the condition ∥ Φ ( x ) − Φ ( y ) ∥ 2 ≤ ε {\displaystyle \parallel \Phi (x)-\Phi (y)\parallel _{2}\leq \varepsilon } , and that almost all of the objects appear in more than one class. Moreover, there would be twenty classes if the indiscernibility relation was used since there are no two objects with matching descriptions. Image processing example The following example provides an example based on digital images. Let a subimage be defined as a small subset of pixels belonging to a digital image such that the pixels contained in the subimage form a square. Then, let the sets X {\displaystyle X} and Y {\displaystyle Y} respectively represent the subimages obtained from two different images, and let O = { X ∪ Y } {\displaystyle O=\{X\cup Y\}} . Finally, let the description of an object be given by the Green component in the RGB color model. The next step is to find all the tolerance classes using the tolerance relation defined in the previous example. Using this information, tolerance classes can be formed containing objects that have similar (within some small ε {\displaystyle \varepsilon } ) values for the Green component in the RGB colour model. Furthermore, images that are near (similar) to each other should have tolerance classes divided among both images (instead of a tolerance classes contained solely in one of the images). For example, the figure accompanying this example shows a subset of the tolerance classes obtained from two leaf images. In this figure, each tolerance class is assigned a separate colour. As can be seen, the two leaves share similar tolerance classes. This example highlights a need to measure the degree of nearness of two sets. == Nearness measure == Let ( U , R δ Φ , ε ) {\displaystyle (U,{\mathcal {R}}_{\delta _{\Phi ,\varepsilon }})} denote a particular descriptive pseudometric EF-proximal relator space equipped with the proximity relation δ Φ , ε {\displaystyle \delta _{\Phi ,\varepsilon }} and with nonempty subsets X , Y ∈ 2 U {\displaystyle X,Y\in 2^{U}} and with the tolerance relation ≅ Φ , ε {\displaystyle \cong _{\Phi ,\varepsilon }} defined in terms of a set of probes Φ {\displaystyle \Phi } and with ε ∈ ( 0 , ∞ ] {\displaystyle \varepsilon \in (0,\infty ]} , where ≃ Φ , ε = { ( x , y ) ∈ U × U ∣ \| Φ ( x ) − Φ ( y ) \| ≤ ε } . {\displaystyle \simeq _{\Phi ,\varepsilon }=\{(x,y)\in U\times U\mid \ \|\Phi (x)-\Phi (y)\|\leq \varepsilon \}.} Further, assume Z = X ∪ Y {\displaystyle Z=X\cup Y} and let H τ Φ , ε ( Z ) {\displaystyle H_{\tau _{\Phi ,\varepsilon }}(Z)} denote the family of all classes in the space ( Z , ≃ Φ , ε ) {\displaystyle (Z,\simeq _{\Phi ,\varepsilon })} . Let A ⊆ X , B ⊆ Y {\displaystyle A\subseteq X,B\subseteq Y} . The distance D t N M : 2 U × 2 U :⟶ [ 0 , ∞ ] {\displaystyle D_{_{tNM}}:2^{U}\times 2^{U}:\longrightarrow [0,\infty ]} is defined by D t N M ( X , Y ) = { 1 − t N M ( A , B ) , if X and Y are not empty , ∞ , if X or Y is empty , {\displaystyle D_{_{tNM}}(X,Y)={\begin{cases}1-tNM(A,B),&{\mbox{if }}X{\mbox{ and }}Y{\mbox{ are not empty}},\\\infty ,&{\mbox{if }}X{\mbox{ or }}Y{\mbox{ is empty}},\end{cases}}} where t N M ( A , B ) = ( ∑ C ∈ H τ Φ , ε ( Z ) \| C \| ) − 1 ⋅ ∑ C ∈ H τ Φ , ε ( Z ) \| C \| min ( \| C ∩ A \| , \| [ C ∩ B \| ) max ( \| C ∩ A \| , \| C ∩ B \| ) . {\displaystyle tNM(A,B)={\Biggl (}\sum _{C\in H_{\tau _{\Phi ,\varepsilon }}(Z)}\|C\|{\Biggr )}^{-1}\cdot \sum _{C\in H_{\tau _{\Phi ,\varepsilon }}(Z)}\|C\|{\frac {\min(\|C\cap A\|,\|[C\cap B\|)}{\max(\|C\cap A\|,\|C\cap B\|)}}.} The details concerning t N M {\displaystyle tNM} are given in. The idea behind t N M {\displaystyle tNM} is that sets that are similar should have a similar number of objects in each tolerance class. Thus, for each tolerance class obtained from the covering of Z = X ∪ Y {\displaystyle Z=X\cup Y} , t N M {\displaystyle tNM} counts the number of objects that belong to X {\displaystyle X} and Y {\displaystyle Y} and takes the ratio (as a proper fraction) of their cardinalities. Furthermore, each ratio is weighted by the total size of the tolerance class (thus giving importance to the larger classes) and the final result is normalized by dividing by the sum of all the cardinalities. The range of t N M {\displaystyle tNM} is in the interval [0,1], where a value of 1 is obtained if the sets are equivalent (based on object descriptions) and a value of 0 is obtained if they have no descriptions in common. As an example of the degree of nearness between two sets, consider figure below in which each image consists of two sets of objects, X {\displaystyle X} and Y {\displaystyle Y} . Each colour in the figures corresponds to a set where all the objects in the class share the same description. The idea behind t N M {\displaystyle tNM} is that the nearness of sets in a perceptual system is based on the cardinality of tolerance classes that they share. Thus, the sets in left side of the figure are closer (more near) to each other in terms of their descriptions than the sets in right side of the figure. == Near set evaluation and recognition (NEAR) system == The Near set Evaluation and Recognition (NEAR) system, is a system developed to demonstrate practical applications of near set theory to the problems of image segmentation evaluation and image correspondence. It was motivated by a need for a freely available software tool that can provide results for research and to generate interest in near set theory. The system implements a Multiple Document Interface (MDI) where each separate processing task is performed in its own child frame. The objects (in the near set sense) in this system are subimages of the images being processed and the probe functions (features) are image processing functions defined on the subimages. The system was written in C++ and was designed to facilitate the addition of new processing tasks and probe functions. Currently, the system performs six major tasks, namely, displaying equivalence and tolerance classes for an image, performing segmentation evaluation, measuring the nearness of two images, performing Content Based Image Retrieval (CBIR), and displaying the output of processing an image using a specific probe function. == Proximity System == The Proximity System is an application developed to demonstrate descriptive-based topological approaches to nearness and proximity within the context of digital image analysis. The Proximity System grew out of the work of S. Naimpally and J. Peters on Topological Spaces. The Proximity System was written in Java and is intended to run in two different operating environments, namely on Android smartphones and tablets, as well as desktop platforms running the Java Virtual Machine. With respect to the desktop environment, the Proximity System is a cross-platform Java application for Windows, OSX, and Linux systems, which has been tested on Windows 7 and Debian Linux using the Sun Java 6 Runtime. In terms of the implementation of the theoretical approaches, both the Android and the desktop based applications use the same back-end libraries to perform the description-based calculations, where the only differences are the user interface and the Android version has less available features due to restrictions on system resources. == See also == == Notes == == References == == Further reading ==

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