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In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics. Originally known as the upper bound conjecture, this statement was formulated by Theodore Motzkin, proved in 1970 by Peter McMullen, and strengthened from polytopes to subdivisions of a sphere in 1975 by Richard P. Stanley.
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https://en.wikipedia.org/wiki/Upper_bound_theorem
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In mathematics, the upper half-plane, H , {\displaystyle \,{\mathcal {H}}\,,} is the set of points ( x , y ) {\displaystyle (x,y)} in the Cartesian plane with y > 0 {\displaystyle y>0} . The lower half-plane is defined similarly, by requiring that y {\displaystyle y} be negative instead. Each is an example of two-dimensional half-space.
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https://en.wikipedia.org/wiki/Upper_half_plane
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In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton { a } {\displaystyle \{a\}} is the order section a ] = { x ≤ a } {\displaystyle a]=\{x\leq a\}} for each a ∈ X . {\displaystyle a\in X.} If ≤ {\displaystyle \leq } is a partial order, the upper topology is the least order consistent topology in which all open sets are up-sets.
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https://en.wikipedia.org/wiki/Lower_topology
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However, not all up-sets must necessarily be open sets. The lower topology induced by the preorder is defined similarly in terms of the down-sets. The preorder inducing the upper topology is its specialization preorder, but the specialization preorder of the lower topology is opposite to the inducing preorder.
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https://en.wikipedia.org/wiki/Lower_topology
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The real upper topology is most naturally defined on the upper-extended real line ( − ∞ , + ∞ ] = R ∪ { + ∞ } {\displaystyle (-\infty ,+\infty ]=\mathbb {R} \cup \{+\infty \}} by the system { ( a , + ∞ ]: a ∈ R ∪ { ± ∞ } } {\displaystyle \{(a,+\infty ]:a\in \mathbb {R} \cup \{\pm \infty \}\}} of open sets. Similarly, the real lower topology { . {\displaystyle {(-\infty ,+\infty ]}.}
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https://en.wikipedia.org/wiki/Lower_topology
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In mathematics, the usual convention for any Riemannian manifold is to use a positive-definite metric tensor (meaning that after diagonalization, elements on the diagonal are all positive). In theoretical physics, spacetime is modeled by a pseudo-Riemannian manifold. The signature counts how many time-like or space-like characters are in the spacetime, in the sense defined by special relativity: as used in particle physics, the metric has an eigenvalue on the time-like subspace, and its mirroring eigenvalue on the space-like subspace. In the specific case of the Minkowski metric, d s 2 = c 2 d t 2 − d x 2 − d y 2 − d z 2 {\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}} ,the metric signature is ( 1 , 3 , 0 ) + {\displaystyle (1,3,0)^{+}} or (+, −, −, −) if its eigenvalue is defined in the time direction, or ( 1 , 3 , 0 ) − {\displaystyle (1,3,0)^{-}} or (−, +, +, +) if the eigenvalue is defined in the three spatial directions x, y and z. (Sometimes the opposite sign convention is used, but with the one given here s directly measures proper time.)
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https://en.wikipedia.org/wiki/Lorentz_signature
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In mathematics, the value distribution theory of holomorphic functions is a division of mathematical analysis. It tries to get quantitative measures of the number of times a function f(z) assumes a value a, as z grows in size, refining the Picard theorem on behaviour close to an essential singularity. The theory exists for analytic functions (and meromorphic functions) of one complex variable z, or of several complex variables.
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https://en.wikipedia.org/wiki/Value_distribution_theory
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In the case of one variable the term Nevanlinna theory, after Rolf Nevanlinna, is also common. The now-classical theory received renewed interest, when Paul Vojta suggested some analogies with the problem of integral solutions to Diophantine equations. These turned out to involve some close parallels, and to lead to fresh points of view on the Mordell conjecture and related questions.
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https://en.wikipedia.org/wiki/Value_distribution_theory
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In mathematics, the values of the trigonometric functions can be expressed approximately, as in cos ( π / 4 ) ≈ 0.707 {\displaystyle \cos(\pi /4)\approx 0.707} , or exactly, as in cos ( π / 4 ) = 2 / 2 {\displaystyle \cos(\pi /4)={\sqrt {2}}/2} . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots.
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https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals
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In mathematics, the van der Corput inequality is a corollary of the Cauchy–Schwarz inequality that is useful in the study of correlations among vectors, and hence random variables. It is also useful in the study of equidistributed sequences, for example in the Weyl equidistribution estimate. Loosely stated, the van der Corput inequality asserts that if a unit vector v {\displaystyle v} in an inner product space V {\displaystyle V} is strongly correlated with many unit vectors u 1 , … , u n ∈ V {\displaystyle u_{1},\dots ,u_{n}\in V} , then many of the pairs u i , u j {\displaystyle u_{i},u_{j}} must be strongly correlated with each other. Here, the notion of correlation is made precise by the inner product of the space V {\displaystyle V}: when the absolute value of ⟨ u , v ⟩ {\displaystyle \langle u,v\rangle } is close to 1 {\displaystyle 1} , then u {\displaystyle u} and v {\displaystyle v} are considered to be strongly correlated. (More generally, if the vectors involved are not unit vectors, then strong correlation means that | ⟨ u , v ⟩ | ≈ ‖ u ‖ ‖ v ‖ {\displaystyle |\langle u,v\rangle |\approx \|u\|\|v\|} .)
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https://en.wikipedia.org/wiki/Van_der_Corput_inequality
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In mathematics, the variation diminishing property of certain mathematical objects involves diminishing the number of changes in sign (positive to negative or vice versa).
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https://en.wikipedia.org/wiki/Variation_diminishing_property_of_Bézier_curves
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In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory. These related concepts are explored in a spectrum of articles: exponential map (Riemannian geometry) matrix exponential exponential function infinitesimal generator (→ Lie group) integral curve (→ vector field) one-parameter subgroup flow (geometry) geodesic flow Hamiltonian flow Ricci flow Anosov flow injectivity radius (→ glossary)
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https://en.wikipedia.org/wiki/Vector_flow
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In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope specified by a set of linear inequalities: A x ≤ b {\displaystyle Ax\leq b} where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. The inverse (dual) problem of finding the bounding inequalities given the vertices is called facet enumeration (see convex hull algorithms).
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https://en.wikipedia.org/wiki/Vertex_enumeration_problem
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In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π: E → B {\displaystyle \pi \colon E\to B} , the vertical bundle V E {\displaystyle VE} and horizontal bundle H E {\displaystyle HE} are subbundles of the tangent bundle T E {\displaystyle TE} of E {\displaystyle E} whose Whitney sum satisfies V E ⊕ H E ≅ T E {\displaystyle VE\oplus HE\cong TE} . This means that, over each point e ∈ E {\displaystyle e\in E} , the fibers V e E {\displaystyle V_{e}E} and H e E {\displaystyle H_{e}E} form complementary subspaces of the tangent space T e E {\displaystyle T_{e}E} . The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle requires some choice of complementary subbundle.
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https://en.wikipedia.org/wiki/Vertical_and_horizontal_bundles
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To make this precise, define the vertical space V e E {\displaystyle V_{e}E} at e ∈ E {\displaystyle e\in E} to be ker ( d π e ) {\displaystyle \ker(d\pi _{e})} . That is, the differential d π e: T e E → T b B {\displaystyle d\pi _{e}\colon T_{e}E\to T_{b}B} (where b = π ( e ) {\displaystyle b=\pi (e)} ) is a linear surjection whose kernel has the same dimension as the fibers of π {\displaystyle \pi } . If we write F = π − 1 ( b ) {\displaystyle F=\pi ^{-1}(b)} , then V e E {\displaystyle V_{e}E} consists of exactly the vectors in T e E {\displaystyle T_{e}E} which are also tangent to F {\displaystyle F} .
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https://en.wikipedia.org/wiki/Vertical_and_horizontal_bundles
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The name is motivated by low-dimensional examples like the trivial line bundle over a circle, which is sometimes depicted as a vertical cylinder projecting to a horizontal circle. A subspace H e E {\displaystyle H_{e}E} of T e E {\displaystyle T_{e}E} is called a horizontal space if T e E {\displaystyle T_{e}E} is the direct sum of V e E {\displaystyle V_{e}E} and H e E {\displaystyle H_{e}E} . The disjoint union of the vertical spaces VeE for each e in E is the subbundle VE of TE; this is the vertical bundle of E. Likewise, provided the horizontal spaces H e E {\displaystyle H_{e}E} vary smoothly with e, their disjoint union is a horizontal bundle.
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https://en.wikipedia.org/wiki/Vertical_and_horizontal_bundles
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The use of the words "the" and "a" here is intentional: each vertical subspace is unique, defined explicitly by ker ( d π e ) {\displaystyle \ker(d\pi _{e})} . Excluding trivial cases, there are an infinite number of horizontal subspaces at each point.
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https://en.wikipedia.org/wiki/Vertical_and_horizontal_bundles
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Also note that arbitrary choices of horizontal space at each point will not, in general, form a smooth vector bundle; they must also vary in an appropriately smooth way. The horizontal bundle is one way to formulate the notion of an Ehresmann connection on a fiber bundle. Thus, for example, if E is a principal G-bundle, then the horizontal bundle is usually required to be G-invariant: such a choice is equivalent to a connection on the principal bundle. This notably occurs when E is the frame bundle associated to some vector bundle, which is a principal GL n {\displaystyle \operatorname {GL} _{n}} bundle.
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https://en.wikipedia.org/wiki/Vertical_and_horizontal_bundles
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In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x. If a vertical line intersects a curve on an xy-plane more than once then for one value of x the curve has more than one value of y, and so, the curve does not represent a function. If all vertical lines intersect a curve at most once then the curve represents a function.
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https://en.wikipedia.org/wiki/Vertical_line_test
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In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games. The classical concept was that a PDE F ( x , u , D u , D 2 u ) = 0 {\displaystyle F(x,u,Du,D^{2}u)=0} over a domain x ∈ Ω {\displaystyle x\in \Omega } has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that x {\displaystyle x} , u {\displaystyle u} , D u {\displaystyle Du} , D 2 u {\displaystyle D^{2}u} satisfy the above equation at every point. If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution.
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https://en.wikipedia.org/wiki/Viscosity_solution
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Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either D u {\displaystyle Du} or D 2 u {\displaystyle D^{2}u} does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.
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https://en.wikipedia.org/wiki/Viscosity_solution
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In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980. In 1929, during his work on the Banach–Tarski paradox, John von Neumann defined the concept of amenable groups and showed that no amenable group contains a free subgroup of rank 2. The suggestion that the converse might hold, that is, that every non-amenable group contains a free subgroup on two generators, was made by a number of different authors in the 1950s and 1960s.
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https://en.wikipedia.org/wiki/Von_Neumann_conjecture
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Although von Neumann's name is popularly attached to the conjecture, its first written appearance seems to be due to Mahlon Marsh Day in 1957. The Tits alternative is a fundamental theorem which, in particular, establishes the conjecture within the class of linear groups. The historically first potential counterexample is Thompson group F. While its amenability is a wide open problem, the general conjecture was shown to be false in 1980 by Alexander Ol'shanskii; he demonstrated that Tarski monster groups, constructed by him, which are easily seen not to have free subgroups of rank 2, are not amenable.
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https://en.wikipedia.org/wiki/Von_Neumann_conjecture
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Two years later, Sergei Adian showed that certain Burnside groups are also counterexamples. None of these counterexamples are finitely presented, and for some years it was considered possible that the conjecture held for finitely presented groups.
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https://en.wikipedia.org/wiki/Von_Neumann_conjecture
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However, in 2003, Alexander Ol'shanskii and Mark Sapir exhibited a collection of finitely-presented groups which do not satisfy the conjecture. In 2013, Nicolas Monod found an easy counterexample to the conjecture. Given by piecewise projective homeomorphisms of the line, the group is remarkably simple to understand.
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https://en.wikipedia.org/wiki/Von_Neumann_conjecture
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Even though it is not amenable, it shares many known properties of amenable groups in a straightforward way. In 2013, Yash Lodha and Justin Tatch Moore isolated a finitely presented non amenable subgroup of Monod's group. This provides the first torsion-free finitely presented counterexample, and admits a presentation with 3 generators and 9 relations. Lodha later showed that this group satisfies the property F ∞ {\displaystyle F_{\infty }} , which is a stronger finiteness property.
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https://en.wikipedia.org/wiki/Von_Neumann_conjecture
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In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, assuming the axiom of choice. It is based on the earlier Banach–Tarski paradox, which is in turn based on the Hausdorff paradox.
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https://en.wikipedia.org/wiki/Von_Neumann_paradox
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Banach and Tarski had proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations (whether the special linear group or the special affine group) contains such subgroups, and this opens the possibility of performing paradoxical decompositions using them.
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https://en.wikipedia.org/wiki/Von_Neumann_paradox
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In mathematics, the walk-on-spheres method (WoS) is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value problem for partial differential equations (PDEs). The WoS method was first introduced by Mervin E. Muller in 1956 to solve Laplace's equation, and was since then generalized to other problems. It relies on probabilistic interpretations of PDEs, and simulates paths of Brownian motion (or for some more general variants, diffusion processes), by sampling only the exit-points out of successive spheres, rather than simulating in detail the path of the process. This often makes it less costly than "grid-based" algorithms, and it is today one of the most widely used "grid-free" algorithms for generating Brownian paths.
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https://en.wikipedia.org/wiki/Walk-on-spheres_method
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In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices.
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https://en.wikipedia.org/wiki/Weakly_chained_diagonally_dominant
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In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which x {\displaystyle x} precedes y {\displaystyle y} if and only if y {\displaystyle y} is either x {\displaystyle x} or the sum of x {\displaystyle x} and some positive integer (other orderings include the ordering 2 , 4 , 6 , . . .
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https://en.wikipedia.org/wiki/Well-ordering_principle
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{\displaystyle 2,4,6,...} ; and 1 , 3 , 5 , . . .
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https://en.wikipedia.org/wiki/Well-ordering_principle
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{\displaystyle 1,3,5,...} ). The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem". On other occasions it is understood to be the proposition that the set of integers { … , − 2 , − 1 , 0 , 1 , 2 , 3 , … } {\displaystyle \{\ldots ,-2,-1,0,1,2,3,\ldots \}} contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element.
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https://en.wikipedia.org/wiki/Well-ordering_principle
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In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).
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https://en.wikipedia.org/wiki/Zermelo's_well-ordering_theorem
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Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. One famous consequence of the theorem is the Banach–Tarski paradox.
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https://en.wikipedia.org/wiki/Zermelo's_well-ordering_theorem
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In mathematics, the wholeness axiom is a strong axiom of set theory introduced by Paul Corazza in 2000.
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https://en.wikipedia.org/wiki/Wholeness_axiom
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In mathematics, the winding number of a curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The notion of winding number is important in the mathematical description of T-duality where it is used to measure the winding of strings around compact extra dimensions. For example, the image below shows several examples of curves in the plane, illustrated in red. Each curve is assumed to be closed, meaning it has no endpoints, and is allowed to intersect itself.
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https://en.wikipedia.org/wiki/T-duality
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Each curve has an orientation given by the arrows in the picture. In each situation, there is a distinguished point in the plane, illustrated in black. The winding number of the curve around this distinguished point is equal to the total number of counterclockwise turns that the curve makes around this point.
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https://en.wikipedia.org/wiki/T-duality
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When counting the total number of turns, counterclockwise turns count as positive, while clockwise turns counts as negative. For example, if the curve first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three.
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https://en.wikipedia.org/wiki/T-duality
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According to this scheme, a curve that does not travel around the distinguished point at all has winding number zero, while a curve that travels clockwise around the point has negative winding number. Therefore, the winding number of a curve may be any integer. The pictures above show curves with winding numbers between −2 and 3:
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https://en.wikipedia.org/wiki/T-duality
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In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be non-integer. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise. Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory).
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https://en.wikipedia.org/wiki/Winding_number
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In mathematics, the witch of Agnesi (Italian pronunciation: ) is a cubic plane curve defined from two diametrically opposite points of a circle. It gets its name from Italian mathematician Maria Gaetana Agnesi, and from a mistranslation of an Italian word for a sailing sheet. Before Agnesi, the same curve was studied by Fermat, Grandi, and Newton.
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https://en.wikipedia.org/wiki/Witch_of_Agnesi
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The graph of the derivative of the arctangent function forms an example of the witch of Agnesi. As the probability density function of the Cauchy distribution, the witch of Agnesi has applications in probability theory. It also gives rise to Runge's phenomenon in the approximation of functions by polynomials, has been used to approximate the energy distribution of spectral lines, and models the shape of hills.
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https://en.wikipedia.org/wiki/Witch_of_Agnesi
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The witch is tangent to its defining circle at one of the two defining points, and asymptotic to the tangent line to the circle at the other point. It has a unique vertex (a point of extreme curvature) at the point of tangency with its defining circle, which is also its osculating circle at that point. It also has two finite inflection points and one infinite inflection point. The area between the witch and its asymptotic line is four times the area of the defining circle, and the volume of revolution of the curve around its defining line is twice the volume of the torus of revolution of its defining circle.
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https://en.wikipedia.org/wiki/Witch_of_Agnesi
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In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings: A fixed and well-defined number or other non-changing mathematical object. The terms mathematical constant or physical constant are sometimes used to distinguish this meaning. A function whose value remains unchanged (i.e., a constant function).
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https://en.wikipedia.org/wiki/Constant_(mathematics)
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Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question.For example, a general quadratic function is commonly written as: a x 2 + b x + c , {\displaystyle ax^{2}+bx+c\,,} where a, b and c are constants (coefficients or parameters), and x a variable—a placeholder for the argument of the function being studied. A more explicit way to denote this function is x ↦ a x 2 + b x + c , {\displaystyle x\mapsto ax^{2}+bx+c\,,} which makes the function-argument status of x (and by extension the constancy of a, b and c) clear. In this example a, b and c are coefficients of the polynomial.
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https://en.wikipedia.org/wiki/Constant_(mathematics)
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Since c occurs in a term that does not involve x, it is called the constant term of the polynomial and can be thought of as the coefficient of x0. More generally, any polynomial term or expression of degree zero (no variable) is a constant. : 18
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https://en.wikipedia.org/wiki/Constant_(mathematics)
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In mathematics, the word null (from German: null meaning "zero", which is from Latin: nullus meaning "none") is often associated with the concept of zero or the concept of nothing. It is used in varying context from "having zero members in a set" (e.g., null set) to "having a value of zero" (e.g., null vector).In a vector space, the null vector is the neutral element of vector addition; depending on the context, a null vector may also be a vector mapped to some null by a function under consideration (such as a quadratic form coming with the vector space, see null vector, a linear mapping given as matrix product or dot product, a seminorm in a Minkowski space, etc.). In set theory, the empty set, that is, the set with zero elements, denoted "{}" or "∅", may also be called null set. In measure theory, a null set is a (possibly nonempty) set with zero measure.
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https://en.wikipedia.org/wiki/Null_(mathematics)
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A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element). For example, in linear algebra, the null space of a linear mapping, also known as kernel, is the set of vectors which map to the null vector under that mapping. In statistics, a null hypothesis is a proposition that no effect or relationship exists between populations and phenomena. It is the hypothesis which is presumed true—unless statistical evidence indicates otherwise.
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https://en.wikipedia.org/wiki/Null_(mathematics)
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In mathematics, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism in non-orientable surfaces. It can be constructed by sliding a Möbius band included on the surface around an essential 1-sided closed curve until the original position; thus it is necessary that the surfaces have genus greater than one. The projective plane R P 2 {\displaystyle {\mathbb {R} P}^{2}} has no y-homeomorphism.
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https://en.wikipedia.org/wiki/Y-homeomorphism
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In mathematics, the zero ideal in a ring R {\displaystyle R} is the ideal { 0 } {\displaystyle \{0\}} consisting of only the additive identity (or zero element). The fact that this is an ideal follows directly from the definition.
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https://en.wikipedia.org/wiki/List_of_zero_terms
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In mathematics, the zero module is the module consisting of only the additive identity for the module's addition function. In the integers, this identity is zero, which gives the name zero module. That the zero module is in fact a module is simple to show; it is closed under addition and multiplication trivially.
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https://en.wikipedia.org/wiki/List_of_zero_terms
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In mathematics, the zero tensor is a tensor, of any order, all of whose components are zero. The zero tensor of order 1 is sometimes known as the zero vector. Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation.
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https://en.wikipedia.org/wiki/Zero_vector
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In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.
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https://en.wikipedia.org/wiki/ℓ-adic_cohomology
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In mathematics, the étale topos of a scheme X is the category of all étale sheaves on X. An étale sheaf is a sheaf on the étale site of X.
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https://en.wikipedia.org/wiki/Étale_topos
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In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century.It states that if a n {\displaystyle a_{n}} , b n {\displaystyle b_{n}} , for n = 1 , 2 , 3 , … {\displaystyle n=1,2,3,\ldots } are real numbers and | b n | ≥ | a n | + 1 {\displaystyle |b_{n}|\geq |a_{n}|+1} for all n {\displaystyle n} , then a 1 b 1 + a 2 b 2 + a 3 b 3 + ⋱ {\displaystyle {\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\ddots }}}}}}} converges absolutely to a number f {\displaystyle f} satisfying 0 < | f | < 1 {\displaystyle 0<|f|<1} , meaning that the series f = ∑ n { A n B n − A n − 1 B n − 1 } , {\displaystyle f=\sum _{n}\left\{{\frac {A_{n}}{B_{n}}}-{\frac {A_{n-1}}{B_{n-1}}}\right\},} where A n / B n {\displaystyle A_{n}/B_{n}} are the convergents of the continued fraction, converges absolutely.
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https://en.wikipedia.org/wiki/Śleszyński–Pringsheim_theorem
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In mathematics, the Ξ function (named for the Greek letter Ξ or Xi) may refer to: Riemann Xi function, a variant of the Riemann zeta function with a simpler functional equation Harish-Chandra's Ξ function, a special spherical function on a semisimple Lie group
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https://en.wikipedia.org/wiki/Ξ_function
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In mathematics, there are a few topological spaces named after M. K. Fort, Jr.
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https://en.wikipedia.org/wiki/Fort_space
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In mathematics, there are different results that share the common name of the Ky Fan inequality. The Ky Fan inequality presented here is used in game theory to investigate the existence of an equilibrium. Another Ky Fan inequality is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval.
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https://en.wikipedia.org/wiki/Ky_Fan_inequality_(game_theory)
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In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.
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https://en.wikipedia.org/wiki/Trace_inequalities
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In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, see Modes of convergence (annotated index) Note that each of the following objects is a special case of the types preceding it: sets, topological spaces, uniform spaces, TAGs (topological abelian groups), normed spaces, Euclidean spaces, and the real/complex numbers. Also, note that any metric space is a uniform space.
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https://en.wikipedia.org/wiki/Modes_of_convergence
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In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms. Another branch of mathematics known as universal algebra studies algebraic structures in general.
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https://en.wikipedia.org/wiki/Outline_of_algebraic_structures
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From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition. Concrete examples of each structure will be found in the articles listed.
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https://en.wikipedia.org/wiki/Outline_of_algebraic_structures
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Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors.
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https://en.wikipedia.org/wiki/Outline_of_algebraic_structures
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Most structures appearing on this page will be common ones which most authors agree on. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.
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https://en.wikipedia.org/wiki/Outline_of_algebraic_structures
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In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition. The article presents several such constructions.
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https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
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They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.
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https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
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In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer.
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https://en.wikipedia.org/wiki/Kummer's_function
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Kummer's function is defined by Λ n ( z ) = ∫ 0 z log n − 1 | t | 1 + t d t . {\displaystyle \Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.} The duplication formula is Λ n ( z ) + Λ n ( − z ) = 2 1 − n Λ n ( − z 2 ) {\displaystyle \Lambda _{n}(z)+\Lambda _{n}(-z)=2^{1-n}\Lambda _{n}(-z^{2})} .Compare this to the duplication formula for the polylogarithm: Li n ( z ) + Li n ( − z ) = 2 1 − n Li n ( z 2 ) .
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https://en.wikipedia.org/wiki/Kummer's_function
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{\displaystyle \operatorname {Li} _{n}(z)+\operatorname {Li} _{n}(-z)=2^{1-n}\operatorname {Li} _{n}(z^{2}).} An explicit link to the polylogarithm is given by Li n ( z ) = Li n ( 1 ) + ∑ k = 1 n − 1 ( − ) k − 1 log k | z | k ! Li n − k ( z ) + ( − ) n − 1 ( n − 1 ) !
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https://en.wikipedia.org/wiki/Kummer's_function
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. {\displaystyle \operatorname {Li} _{n}(z)=\operatorname {Li} _{n}(1)\;\;+\;\;\sum _{k=1}^{n-1}(-)^{k-1}\;{\frac {\log ^{k}|z|}{k! }}\;\operatorname {Li} _{n-k}(z)\;\;+\;\;{\frac {(-)^{n-1}}{(n-1)!}}\;\left.}
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https://en.wikipedia.org/wiki/Kummer's_function
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In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line: This integral is not absolutely convergent, meaning | sin x x | {\displaystyle \left|{\frac {\sin x}{x}}\right|} is not Lebesgue-integrable, because the Dirichlet integral is infinite in the sense of Lebesgue integration. It is, however, finite in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral. This can be seen by using Dirichlet's test for improper integrals.
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https://en.wikipedia.org/wiki/Dirichlet_integral
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It is a good illustration of special techniques for evaluating definite integrals. The sine integral, an antiderivative of the sinc function, is not an elementary function. However the improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign, contour integration, and the Dirichlet kernel.
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https://en.wikipedia.org/wiki/Dirichlet_integral
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In mathematics, there are several theorems basic to algebraic K-theory. Throughout, for simplicity, we assume when an exact category is a subcategory of another exact category, we mean it is strictly full subcategory (i.e., isomorphism-closed.)
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https://en.wikipedia.org/wiki/Basic_theorems_in_algebraic_K-theory
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In mathematics, there are two competing definitions for a chiral polytope. One is that it is a polytope that is chiral (or "enantiomorphic"), meaning that it does not have mirror symmetry. By this definition, a polytope that lacks any symmetry at all would be an example of a chiral polytope. The other, competing definition of a chiral polytope is that it is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms of the action of the symmetry group of the polytope on its flags. By this definition, even highly-symmetric and enantiomorphic polytopes such as the snub cube are not chiral. Much of the study of symmetric but chiral polytopes has been carried out in the framework of abstract polytopes, because of the paucity of geometric examples.
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https://en.wikipedia.org/wiki/Chiral_polytope
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In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of sets R {\displaystyle {\mathcal {R}}} is called a ring (of sets) if it is closed under union and intersection. That is, the following two statements are true for all sets A {\displaystyle A} and B {\displaystyle B} , A , B ∈ R {\displaystyle A,B\in {\mathcal {R}}} implies A ∪ B ∈ R {\displaystyle A\cup B\in {\mathcal {R}}} and A , B ∈ R {\displaystyle A,B\in {\mathcal {R}}} implies A ∩ B ∈ R . {\displaystyle A\cap B\in {\mathcal {R}}.}
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https://en.wikipedia.org/wiki/Ring_of_sets
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In measure theory, a nonempty family of sets R {\displaystyle {\mathcal {R}}} is called a ring (of sets) if it is closed under union and relative complement (set-theoretic difference). That is, the following two statements are true for all sets A {\displaystyle A} and B {\displaystyle B} , A , B ∈ R {\displaystyle A,B\in {\mathcal {R}}} implies A ∪ B ∈ R {\displaystyle A\cup B\in {\mathcal {R}}} and A , B ∈ R {\displaystyle A,B\in {\mathcal {R}}} implies A ∖ B ∈ R . {\displaystyle A\setminus B\in {\mathcal {R}}.} This implies that a ring in the measure-theoretic sense always contains the empty set. Furthermore, for all sets A and B, A ∩ B = A ∖ ( A ∖ B ) , {\displaystyle A\cap B=A\setminus (A\setminus B),} which shows that a family of sets closed under relative complement is also closed under intersection, so that a ring in the measure-theoretic sense is also a ring in the order-theoretic sense.
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https://en.wikipedia.org/wiki/Ring_of_sets
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In mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal with the second, called a L-semi-inner product or semi-inner product in the sense of Lumer, which is an inner product not required to be conjugate symmetric. It was formulated by Günter Lumer, for the purpose of extending Hilbert space type arguments to Banach spaces in functional analysis. Fundamental properties were later explored by Giles.
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https://en.wikipedia.org/wiki/L-semi-inner_product
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In mathematics, there are two different results that share the common name of the Ky Fan inequality. One is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval. The result was published on page 5 of the book Inequalities by Edwin F. Beckenbach and Richard E. Bellman (1961), who refer to an unpublished result of Ky Fan. They mention the result in connection with the inequality of arithmetic and geometric means and Augustin Louis Cauchy's proof of this inequality by forward-backward-induction; a method which can also be used to prove the Ky Fan inequality. This Ky Fan inequality is a special case of Levinson's inequality and also the starting point for several generalizations and refinements; some of them are given in the references below. The second Ky Fan inequality is used in game theory to investigate the existence of an equilibrium.
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https://en.wikipedia.org/wiki/Ky_Fan_inequality
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In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all k-dimensional affine subspaces of Rn (described on this page), while in the other the affine Grassmannian is a quotient of a group-ring based on formal Laurent series.
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https://en.wikipedia.org/wiki/Affine_Grassmannian_(manifold)
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In mathematics, there are two natural interpretations of the place-permutation action of symmetric groups, in which the group elements act on positions or places. Each may be regarded as either a left or a right action, depending on the order in which one chooses to compose permutations. There are just two interpretations of the meaning of "acting by a permutation σ {\displaystyle \sigma } " but these lead to four variations, depending whether maps are written on the left or right of their arguments. The presence of so many variations often leads to confusion. When regarding the group algebra of a symmetric group as a diagram algebra it is natural to write maps on the right so as to compute compositions of diagrams from left to right.
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https://en.wikipedia.org/wiki/Place-permutation_action
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In mathematics, there are two types of Euler integral: The Euler integral of the first kind is the beta function The Euler integral of the second kind is the gamma function For positive integers m and n, the two integrals can be expressed in terms of factorials and binomial coefficients:
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https://en.wikipedia.org/wiki/Euler_integral
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In mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1 factor. There are an uncountable number of other factors of type II1. Connes proved that the infinite one is also unique.
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https://en.wikipedia.org/wiki/Hyperfinite_type_II-1_factor
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In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces), but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.
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https://en.wikipedia.org/wiki/Tensor_norm
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In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras. A magma which is both commutative and associative is a commutative semigroup.
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https://en.wikipedia.org/wiki/Commutative_magma
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In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up.
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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Examples of the most intriguing of these formulae include infinite series for π, one of which is given below: 1 π = 2 2 9801 ∑ k = 0 ∞ ( 4 k ) ! ( 1103 + 26390 k ) ( k ! )
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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4 396 4 k . {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.} This result is based on the negative fundamental discriminant d = −4 × 58 = −232 with class number h(d) = 2.
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962, which is related to the fact that e π 58 = 396 4 − 104.000000177 … . {\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .} This might be compared to Heegner numbers, which have class number 1 and yield similar formulae.
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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Ramanujan's series for π converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 9801√2/4412 for π, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places. See also the more general Ramanujan–Sato series. One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed a problem: Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right.
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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If n is between 50 and 500, what are n and x?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction.
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple.
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied."
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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His intuition also led him to derive some previously unknown identities, such as ( 1 + 2 ∑ n = 1 ∞ cos ( n θ ) cosh ( n π ) ) − 2 + ( 1 + 2 ∑ n = 1 ∞ cosh ( n θ ) cosh ( n π ) ) − 2 = 2 Γ 4 ( 3 4 ) π = 8 π 3 Γ 4 ( 1 4 ) {\displaystyle {\begin{aligned}&\left(1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right)^{-2}+\left(1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right)^{-2}\\={}&{\frac {2\Gamma ^{4}{\bigl (}{\frac {3}{4}}{\bigr )}}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}{\bigl (}{\frac {1}{4}}{\bigr )}}}\end{aligned}}} for all θ such that | ℜ ( θ ) | < π {\displaystyle |\Re (\theta )|<\pi } and | ℑ ( θ ) | < π {\displaystyle |\Im (\theta )|<\pi } , where Γ(z) is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of θ0, θ4, and θ8 gives some deep identities for the hyperbolic secant. In 1918, Hardy and Ramanujan studied the partition function P(n) extensively.
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method.In the last year of his life, Ramanujan discovered mock theta functions. For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.
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https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
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In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics. This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail.
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https://en.wikipedia.org/wiki/Stone's_duality
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In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.
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https://en.wikipedia.org/wiki/Sobolev_inequality
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In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".Throughout this article, ( e π i τ ) α {\displaystyle (e^{\pi i\tau })^{\alpha }} should be interpreted as e α π i τ {\displaystyle e^{\alpha \pi i\tau }} (in order to resolve issues of choice of branch).
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https://en.wikipedia.org/wiki/Theta_series
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In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator.
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https://en.wikipedia.org/wiki/Time-scale_calculus
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In mathematics, tiny and miny are operators that yield infinitesimal values when applied to numbers in combinatorial game theory. Given a positive number G, tiny G (denoted by ⧾G in many texts) is equal to {0|{0|-G}} for any game G, whereas miny G (analogously denoted ⧿G) is tiny G's negative, or {{G|0}|0}. Tiny and miny aren't just abstract mathematical operators on combinatorial games: tiny and miny games do occur "naturally" in such games as toppling dominoes. Specifically, tiny n, where n is a natural number, can be generated by placing two black dominoes outside n + 2 white dominoes.
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https://en.wikipedia.org/wiki/Tiny_and_miny
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