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In mathematics, a Swiss cheese is a compact subset of the complex plane obtained by removing from a closed disc some countable union of open discs, usually with some restriction on the centres and radii of the removed discs. Traditionally the deleted discs should have pairwise disjoint closures which are subsets of the interior of the starting disc, the sum of the radii of the deleted discs should be finite, and the Swiss cheese should have empty interior. This is the type of Swiss cheese originally introduced by the Swiss mathematician Alice Roth. More generally, a Swiss cheese may be all or part of Euclidean space Rn – or of an even more complicated manifold – with "holes" in it.
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https://en.wikipedia.org/wiki/Swiss_cheese_(mathematics)
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In mathematics, a Sylvester domain, named after James Joseph Sylvester by Dicks & Sontag (1978), is a ring in which Sylvester's law of nullity holds. This means that if A is an m by n matrix, and B is an n by s matrix over R, then ρ(AB) ≥ ρ(A) + ρ(B) – nwhere ρ is the inner rank of a matrix. The inner rank of an m by n matrix is the smallest integer r such that the matrix is a product of an m by r matrix and an r by n matrix. Sylvester (1884) showed that fields satisfy Sylvester's law of nullity and are, therefore, Sylvester domains.
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https://en.wikipedia.org/wiki/Sylvester_domain
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In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their resultant, which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain). Sylvester matrices are named after James Joseph Sylvester.
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https://en.wikipedia.org/wiki/Sylvester_matrix
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In mathematics, a Szegő polynomial is one of a family of orthogonal polynomials for the Hermitian inner product ⟨ f | g ⟩ = ∫ − π π f ( e i θ ) g ( e i θ ) ¯ d μ {\displaystyle \langle f|g\rangle =\int _{-\pi }^{\pi }f(e^{i\theta }){\overline {g(e^{i\theta })}}\,d\mu } where dμ is a given positive measure on . Writing ϕ n ( z ) {\displaystyle \phi _{n}(z)} for the polynomials, they obey a recurrence relation ϕ n + 1 ( z ) = z ϕ ( z ) + ρ n + 1 ϕ ∗ ( z ) {\displaystyle \phi _{n+1}(z)=z\phi (z)+\rho _{n+1}\phi ^{*}(z)} where ρ n + 1 {\displaystyle \rho _{n+1}} is a parameter, called the reflection coefficient or the Szegő parameter.
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https://en.wikipedia.org/wiki/Szegő_polynomial
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In mathematics, a Takiff algebra is a Lie algebra over a truncated polynomial ring. More precisely, a Takiff algebra of a Lie algebra g over a field k is a Lie algebra of the form g/(xn+1) = g⊗kk/(xn+1) for some positive integer n. Sometimes these are called generalized Takiff algebras, and the name Takiff algebra is used for the case when n = 1. These algebras (for n = 1) were studied by Takiff (1971), who in some cases described the ring of polynomials on these algebras invariant under the action of the adjoint group.
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https://en.wikipedia.org/wiki/Takiff_algebra
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In mathematics, a Tamari lattice, introduced by Dov Tamari (1962), is a partially ordered set in which the elements consist of different ways of grouping a sequence of objects into pairs using parentheses; for instance, for a sequence of four objects abcd, the five possible groupings are ((ab)c)d, (ab)(cd), (a(bc))d, a((bc)d), and a(b(cd)). Each grouping describes a different order in which the objects may be combined by a binary operation; in the Tamari lattice, one grouping is ordered before another if the second grouping may be obtained from the first by only rightward applications of the associative law (xy)z = x(yz). For instance, applying this law with x = a, y = bc, and z = d gives the expansion (a(bc))d = a((bc)d), so in the ordering of the Tamari lattice (a(bc))d ≤ a((bc)d).
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https://en.wikipedia.org/wiki/Tamari_lattice
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In this partial order, any two groupings g1 and g2 have a greatest common predecessor, the meet g1 ∧ g2, and a least common successor, the join g1 ∨ g2. Thus, the Tamari lattice has the structure of a lattice. The Hasse diagram of this lattice is isomorphic to the graph of vertices and edges of an associahedron.
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https://en.wikipedia.org/wiki/Tamari_lattice
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The number of elements in a Tamari lattice for a sequence of n + 1 objects is the nth Catalan number Cn. The Tamari lattice can also be described in several other equivalent ways: It is the poset of sequences of n integers a1, ..., an, ordered coordinatewise, such that i ≤ ai ≤ n and if i ≤ j ≤ ai then aj ≤ ai (Huang & Tamari 1972). It is the poset of binary trees with n leaves, ordered by tree rotation operations. It is the poset of ordered forests, in which one forest is earlier than another in the partial order if, for every j, the jth node in a preorder traversal of the first forest has at least as many descendants as the jth node in a preorder traversal of the second forest (Knuth 2005). It is the poset of triangulations of a convex n-gon, ordered by flip operations that substitute one diagonal of the polygon for another.
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https://en.wikipedia.org/wiki/Tamari_lattice
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In mathematics, a Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory. The name is taken from Tadao Tannaka and Tannaka–Krein duality, a theory about compact groups G and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some simplifications made.
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https://en.wikipedia.org/wiki/Tannakian_duality
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The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups G which are profinite groups. The gist of the theory is that the fiber functor Φ of the Galois theory is replaced by a tensor functor T from C to K-Vect. The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group (a priori only a monoid) of natural transformations of T into itself, that respect the tensor structure. This is by nature not an algebraic group, but an inverse limit of algebraic groups (pro-algebraic group).
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https://en.wikipedia.org/wiki/Tannakian_duality
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In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G(Ks) (the Ks-valued points of G). In this case, the Tate module of A is equipped with an action of the absolute Galois group of K, and it is referred to as the Tate module of G.
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https://en.wikipedia.org/wiki/Tate_module_of_a_number_field
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In mathematics, a Tate vector space is a vector space obtained from finite-dimensional vector spaces in a way that makes it possible to extend concepts such as dimension and determinant to an infinite-dimensional situation. Tate spaces were introduced by Alexander Beilinson, Boris Feigin, and Barry Mazur (1991), who named them after John Tate.
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https://en.wikipedia.org/wiki/Tate_vector_space
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In mathematics, a Teichmüller modular form is an analogue of a Siegel modular form on Teichmüller space.
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https://en.wikipedia.org/wiki/Teichmüller_modular_form
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In mathematics, a Thue equation is a Diophantine equation of the form ƒ(x,y) = r,where ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number. It is named after Axel Thue, who in 1909 proved that a Thue equation can have only finitely many solutions in integers x and y, a result known as Thue's theorem, The Thue equation is solvable effectively: there is an explicit bound on the solutions x, y of the form ( C 1 r ) C 2 {\displaystyle (C_{1}r)^{C_{2}}} where constants C1 and C2 depend only on the form ƒ. A stronger result holds: if K is the field generated by the roots of ƒ, then the equation has only finitely many solutions with x and y integers of K, and again these may be effectively determined.
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https://en.wikipedia.org/wiki/Thue_equation
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In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683.Simply, it is a method for transforming a polynomial equation of degree n ≥ 2 {\displaystyle n\geq 2} with some nonzero intermediate coefficients, a 1 , . . .
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https://en.wikipedia.org/wiki/Tschirnhaus_transformation
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, a n − 1 {\displaystyle a_{1},...,a_{n-1}} , such that some or all of the transformed intermediate coefficients, a 1 ′ , . . .
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https://en.wikipedia.org/wiki/Tschirnhaus_transformation
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, a n − 1 ′ {\displaystyle a'_{1},...,a'_{n-1}} , are exactly zero. For example, finding a substitutionfor a cubic equation of degree n = 3 {\displaystyle n=3} ,such that substituting x = x ( y ) {\displaystyle x=x(y)} yields a new equationsuch that a 1 ′ = 0 {\displaystyle a'_{1}=0} , a 2 ′ = 0 {\displaystyle a'_{2}=0} , or both. More generally, it may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root.
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https://en.wikipedia.org/wiki/Tschirnhaus_transformation
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In mathematics, a Tutte–Grothendieck (TG) invariant is a type of graph invariant that satisfies a generalized deletion–contraction formula. Any evaluation of the Tutte polynomial would be an example of a TG invariant.
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https://en.wikipedia.org/wiki/Tutte–Grothendieck_invariant
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In mathematics, a V-ring is a ring R such that every simple R-module is injective. The following three conditions are equivalent: Every simple left (resp. right) R-module is injective The radical of every left (resp. right) R-module is zero Every left (resp.
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https://en.wikipedia.org/wiki/V-ring_(ring_theory)
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right) ideal of R is an intersection of maximal left (resp. right) ideals of RA commutative ring is a V-ring if and only if it is Von Neumann regular. == References ==
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https://en.wikipedia.org/wiki/V-ring_(ring_theory)
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In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988), with a basis of elements φλ corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants Nνλμ describe fusion of primary fields.
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https://en.wikipedia.org/wiki/Verlinde_formula
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In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. In 1970, Robert Solovay constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable, assuming the existence of an inaccessible cardinal (see Solovay model).
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https://en.wikipedia.org/wiki/Vitali_set
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In mathematics, a Vogan diagram, named after David Vogan, is a variation of the Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Satake diagrams they are a different way of classifying simple Lie algebras.
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https://en.wikipedia.org/wiki/Vogan_diagram
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In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation.
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https://en.wikipedia.org/wiki/Voronoi_decomposition
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The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art.
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https://en.wikipedia.org/wiki/Voronoi_decomposition
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In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups. The Voronoi (summation) formula for GL(2) has long been a standard tool for studying analytic properties of automorphic forms and their L-functions. There have been numerous results coming out the Voronoi formula on GL(2). The concept is named after Georgy Voronoy.
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https://en.wikipedia.org/wiki/Voronoi_formula
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In mathematics, a Waldhausen category is a category C equipped with some additional data, which makes it possible to construct the K-theory spectrum of C using a so-called S-construction. It's named after Friedhelm Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to categories not necessarily of algebraic origin, for example the category of topological spaces.
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https://en.wikipedia.org/wiki/Waldhausen_category
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In mathematics, a Wall polynomial is a polynomial studied by Wall (1963) in his work on conjugacy classes in classical groups, and named by Andrews (1984).
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https://en.wikipedia.org/wiki/Wall_polynomial
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In mathematics, a Walsh matrix is a specific square matrix of dimensions 2n, where n is some particular natural number. The entries of the matrix are either +1 or −1 and its rows as well as columns are orthogonal, i.e. dot product is zero. The Walsh matrix was proposed by Joseph L. Walsh in 1923. Each row of a Walsh matrix corresponds to a Walsh function.
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https://en.wikipedia.org/wiki/Walsh_matrix
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The Walsh matrices are a special case of Hadamard matrices. The naturally ordered Hadamard matrix is defined by the recursive formula below, and the sequency-ordered Hadamard matrix is formed by rearranging the rows so that the number of sign changes in a row is in increasing order. Confusingly, different sources refer to either matrix as the Walsh matrix. The Walsh matrix (and Walsh functions) are used in computing the Walsh transform and have applications in the efficient implementation of certain signal processing operations.
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https://en.wikipedia.org/wiki/Walsh_matrix
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In mathematics, a Weierstrass point P {\displaystyle P} on a nonsingular algebraic curve C {\displaystyle C} defined over the complex numbers is a point such that there are more functions on C {\displaystyle C} , with their poles restricted to P {\displaystyle P} only, than would be predicted by the Riemann–Roch theorem. The concept is named after Karl Weierstrass. Consider the vector spaces L ( 0 ) , L ( P ) , L ( 2 P ) , L ( 3 P ) , … {\displaystyle L(0),L(P),L(2P),L(3P),\dots } where L ( k P ) {\displaystyle L(kP)} is the space of meromorphic functions on C {\displaystyle C} whose order at P {\displaystyle P} is at least − k {\displaystyle -k} and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on C {\displaystyle C} ; it is non-decreasing; and from the Riemann–Roch theorem the dimension eventually increments by exactly 1 as we move to the right.
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https://en.wikipedia.org/wiki/Weierstrass_point
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In fact if g {\displaystyle g} is the genus of C {\displaystyle C} , the dimension from the k {\displaystyle k} -th term is known to be l ( k P ) = k − g + 1 , {\displaystyle l(kP)=k-g+1,} for k ≥ 2 g − 1. {\displaystyle k\geq 2g-1.} Our knowledge of the sequence is therefore 1 , ?
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https://en.wikipedia.org/wiki/Weierstrass_point
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, ? , … , ?
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https://en.wikipedia.org/wiki/Weierstrass_point
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, g , g + 1 , g + 2 , … . {\displaystyle 1,?,?,\dots ,?,g,g+1,g+2,\dots .} What we know about the ?
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https://en.wikipedia.org/wiki/Weierstrass_point
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entries is that they can increment by at most 1 each time (this is a simple argument: L ( n P ) / L ( ( n − 1 ) P ) {\displaystyle L(nP)/L((n-1)P)} has dimension as most 1 because if f {\displaystyle f} and g {\displaystyle g} have the same order of pole at P {\displaystyle P} , then f + c g {\displaystyle f+cg} will have a pole of lower order if the constant c {\displaystyle c} is chosen to cancel the leading term). There are 2 g − 2 {\displaystyle 2g-2} question marks here, so the cases g = 0 {\displaystyle g=0} or 1 {\displaystyle 1} need no further discussion and do not give rise to Weierstrass points. Assume therefore g ≥ 2 {\displaystyle g\geq 2} .
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https://en.wikipedia.org/wiki/Weierstrass_point
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There will be g − 1 {\displaystyle g-1} steps up, and g − 1 {\displaystyle g-1} steps where there is no increment. A non-Weierstrass point of C {\displaystyle C} occurs whenever the increments are all as far to the right as possible: i.e. the sequence looks like 1 , 1 , … , 1 , 2 , 3 , 4 , … , g − 1 , g , g + 1 , … . {\displaystyle 1,1,\dots ,1,2,3,4,\dots ,g-1,g,g+1,\dots .}
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https://en.wikipedia.org/wiki/Weierstrass_point
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Any other case is a Weierstrass point. A Weierstrass gap for P {\displaystyle P} is a value of k {\displaystyle k} such that no function on C {\displaystyle C} has exactly a k {\displaystyle k} -fold pole at P {\displaystyle P} only. The gap sequence is 1 , 2 , … , g {\displaystyle 1,2,\dots ,g} for a non-Weierstrass point.
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https://en.wikipedia.org/wiki/Weierstrass_point
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For a Weierstrass point it contains at least one higher number. (The Weierstrass gap theorem or Lückensatz is the statement that there must be g {\displaystyle g} gaps.) For hyperelliptic curves, for example, we may have a function F {\displaystyle F} with a double pole at P {\displaystyle P} only.
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https://en.wikipedia.org/wiki/Weierstrass_point
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Its powers have poles of order 4 , 6 {\displaystyle 4,6} and so on. Therefore, such a P {\displaystyle P} has the gap sequence 1 , 3 , 5 , … , 2 g − 1. {\displaystyle 1,3,5,\dots ,2g-1.}
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https://en.wikipedia.org/wiki/Weierstrass_point
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In general if the gap sequence is a , b , c , … {\displaystyle a,b,c,\dots } the weight of the Weierstrass point is ( a − 1 ) + ( b − 2 ) + ( c − 3 ) + … . {\displaystyle (a-1)+(b-2)+(c-3)+\dots .} This is introduced because of a counting theorem: on a Riemann surface the sum of the weights of the Weierstrass points is g ( g 2 − 1 ) .
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https://en.wikipedia.org/wiki/Weierstrass_point
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{\displaystyle g(g^{2}-1).} For example, a hyperelliptic Weierstrass point, as above, has weight g ( g − 1 ) / 2. {\displaystyle g(g-1)/2.}
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https://en.wikipedia.org/wiki/Weierstrass_point
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Therefore, there are (at most) 2 ( g + 1 ) {\displaystyle 2(g+1)} of them. The 2 g + 2 {\displaystyle 2g+2} ramification points of the ramified covering of degree two from a hyperelliptic curve to the projective line are all hyperelliptic Weierstrass points and these exhausts all the Weierstrass points on a hyperelliptic curve of genus g {\displaystyle g} . Further information on the gaps comes from applying Clifford's theorem.
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https://en.wikipedia.org/wiki/Weierstrass_point
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Multiplication of functions gives the non-gaps a numerical semigroup structure, and an old question of Adolf Hurwitz asked for a characterization of the semigroups occurring. A new necessary condition was found by R.-O. Buchweitz in 1980 and he gave an example of a subsemigroup of the nonnegative integers with 16 gaps that does not occur as the semigroup of non-gaps at a point on a curve of genus 16 (see ). A definition of Weierstrass point for a nonsingular curve over a field of positive characteristic was given by F. K. Schmidt in 1939.
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https://en.wikipedia.org/wiki/Weierstrass_point
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In mathematics, a Weierstrass ring, named by Nagata after Karl Weierstrass, is a commutative local ring that is Henselian, pseudo-geometric, and such that any quotient ring by a prime ideal is a finite extension of a regular local ring.
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https://en.wikipedia.org/wiki/Weierstrass_ring
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In mathematics, a Weil group, introduced by Weil (1951), is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is generally denoted WF. There also exists "finite level" modifications of the Galois groups: if E/F is a finite extension, then the relative Weil group of E/F is WE/F = WF/W cE (where the superscript c denotes the commutator subgroup). For more details about Weil groups see (Artin & Tate 2009) or (Tate 1979) or (Weil 1951).
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https://en.wikipedia.org/wiki/Weil_group_of_a_class_formation
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In mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl:The sequence of all multiples of an irrational α, 0, α, 2α, 3α, 4α, ... is equidistributed modulo 1.In other words, the sequence of the fractional parts of each term will be uniformly distributed in the interval [0, 1).
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https://en.wikipedia.org/wiki/Weyl_sequence
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In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R). Whittaker's equation is d 2 w d z 2 + ( − 1 4 + κ z + 1 / 4 − μ 2 z 2 ) w = 0.
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https://en.wikipedia.org/wiki/Whittaker_function
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{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.} It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by M κ , μ ( z ) = exp ( − z / 2 ) z μ + 1 2 M ( μ − κ + 1 2 , 1 + 2 μ , z ) {\displaystyle M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right)} W κ , μ ( z ) = exp ( − z / 2 ) z μ + 1 2 U ( μ − κ + 1 2 , 1 + 2 μ , z ) .
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https://en.wikipedia.org/wiki/Whittaker_function
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{\displaystyle W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right).} The Whittaker function W κ , μ ( z ) {\displaystyle W_{\kappa ,\mu }(z)} is the same as those with opposite values of μ, in other words considered as a function of μ at fixed κ and z it is even functions. When κ and z are real, the functions give real values for real and imaginary values of μ. These functions of μ play a role in so-called Kummer spaces.Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker models.
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https://en.wikipedia.org/wiki/Whittaker_function
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In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2)Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (formerly known as Catalan's conjecture).
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https://en.wikipedia.org/wiki/Wieferich_pair
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In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
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https://en.wikipedia.org/wiki/Witt–Grothendieck_ring
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In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space V ⊗ n {\displaystyle V^{\otimes n}} obtained from the action of S n {\displaystyle S_{n}} on V ⊗ n {\displaystyle V^{\otimes n}} by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young.
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https://en.wikipedia.org/wiki/Young_symmetrizer
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In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley.
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https://en.wikipedia.org/wiki/Young_tableau
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In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular.
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https://en.wikipedia.org/wiki/Zariski_geometry
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In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.
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https://en.wikipedia.org/wiki/Zimmert_set
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In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity: ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) + a ∘ ( c ∘ b ) . {\displaystyle (a\circ b)\circ c=a\circ (b\circ c)+a\circ (c\circ b).} Zinbiel algebras were introduced by Jean-Louis Loday (1995).
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https://en.wikipedia.org/wiki/Zinbiel_algebra
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The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.In any Zinbiel algebra, the symmetrised product a ⋆ b = a ∘ b + b ∘ a {\displaystyle a\star b=a\circ b+b\circ a} is associative. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over V is the tensor algebra with product ( x 0 ⊗ ⋯ ⊗ x p ) ∘ ( x p + 1 ⊗ ⋯ ⊗ x p + q ) = x 0 ∑ ( p , q ) ( x 1 , … , x p + q ) , {\displaystyle (x_{0}\otimes \cdots \otimes x_{p})\circ (x_{p+1}\otimes \cdots \otimes x_{p+q})=x_{0}\sum _{(p,q)}(x_{1},\ldots ,x_{p+q}),} where the sum is over all ( p , q ) {\displaystyle (p,q)} shuffles.
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https://en.wikipedia.org/wiki/Zinbiel_algebra
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In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent x there exists an element y such that xy is a non-zero idempotent (Kaplansky 1968, pages 19, 25). Kaplansky (1951) named them after Max August Zorn, who studied a similar condition in (Zorn 1941). For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J(R) is a nil ideal and every right ideal of R which is not contained in J(R) contains a nonzero idempotent. Replacing "right ideal" with "left ideal" yields an equivalent definition. Left or right Artinian rings, left or right perfect rings, semiprimary rings and von Neumann regular rings are all examples of associative Zorn rings.
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https://en.wikipedia.org/wiki/Zorn_ring
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In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernstein functor is closely related.
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https://en.wikipedia.org/wiki/Zuckerman_functor
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In mathematics, a balanced matrix is a 0-1 matrix (a matrix where every entry is either zero or one) that does not contain any square submatrix of odd order having all row sums and all column sums equal to 2. Balanced matrices are studied in linear programming. The importance of balanced matrices comes from the fact that the solution to a linear programming problem is integral if its matrix of coefficients is balanced and its right hand side or its objective vector is an all-one vector. In particular, if one searches for an integral solution to a linear program of this kind, it is not necessary to explicitly solve an integer linear program, but it suffices to find an optimal vertex solution of the linear program itself. As an example, the following matrix is a balanced matrix: {\displaystyle {\begin{bmatrix}1&1&1&1\\1&1&0&0\\1&0&1&0\\1&0&0&1\\\end{bmatrix}}}
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https://en.wikipedia.org/wiki/Balanced_matrix
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In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere.
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https://en.wikipedia.org/wiki/4-ball_(mathematics)
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Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere.
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https://en.wikipedia.org/wiki/4-ball_(mathematics)
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In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed n {\displaystyle n} -dimensional ball is often denoted as B n {\displaystyle B^{n}} or D n {\displaystyle D^{n}} while the open n {\displaystyle n} -dimensional ball is Int B n {\displaystyle \operatorname {Int} B^{n}} or Int D n {\displaystyle \operatorname {Int} D^{n}} .
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https://en.wikipedia.org/wiki/4-ball_(mathematics)
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In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by A. H. Clifford (1954). The lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and Gerhard. Semilattices, left-zero bands, right-zero bands, rectangular bands, normal bands, left-regular bands, right-regular bands and regular bands are specific subclasses of bands that lie near the bottom of this lattice and which are of particular interest; they are briefly described below.
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https://en.wikipedia.org/wiki/Band_(algebra)
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In mathematics, a base (or basis; PL: bases) for the topology τ of a topological space (X, τ) is a family B {\displaystyle {\mathcal {B}}} of open subsets of X such that every open set of the topology is equal to the union of some sub-family of B {\displaystyle {\mathcal {B}}} . For example, the set of all open intervals in the real number line R {\displaystyle \mathbb {R} } is a basis for the Euclidean topology on R {\displaystyle \mathbb {R} } because every open interval is an open set, and also every open subset of R {\displaystyle \mathbb {R} } can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called basic open sets, are often easier to describe and use than arbitrary open sets.
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https://en.wikipedia.org/wiki/Countable_base
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Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X {\displaystyle X} form a base for a topology on X {\displaystyle X} .
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https://en.wikipedia.org/wiki/Countable_base
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Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on X {\displaystyle X} , obtained by taking all possible unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Bases for topologies are also closely related to neighborhood bases.
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https://en.wikipedia.org/wiki/Countable_base
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In mathematics, a base-orderable matroid is a matroid that has the following additional property, related to the bases of the matroid. For any two bases A {\displaystyle A} and B {\displaystyle B} there exists a feasible exchange bijection, defined as a bijection f {\displaystyle f} from A {\displaystyle A} to B {\displaystyle B} , such that for every a ∈ A ∖ B {\displaystyle a\in A\setminus B} , both ( A ∖ { a } ) ∪ { f ( a ) } {\displaystyle (A\setminus \{a\})\cup \{f(a)\}} and ( B ∖ { f ( a ) } ) ∪ { a } {\displaystyle (B\setminus \{f(a)\})\cup \{a\}} are bases.The property was introduced by Brualdi and Scrimger. A strongly-base-orderable matroid has the following stronger property:For any two bases A {\displaystyle A} and B {\displaystyle B} , there is a strong feasible exchange bijection, defined as a bijection f {\displaystyle f} from A {\displaystyle A} to B {\displaystyle B} , such that for every X ⊆ A {\displaystyle X\subseteq A} , both ( A ∖ X ) ∪ f ( X ) {\displaystyle (A\setminus X)\cup f(X)} and ( B ∖ f ( X ) ) ∪ X {\displaystyle (B\setminus f(X))\cup X} are bases.
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https://en.wikipedia.org/wiki/Base-orderable_matroid
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In mathematics, a basic algebraic operation is any one of the common operations of arithmetic, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields.
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https://en.wikipedia.org/wiki/Algebraic_operation
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An algebraic operation may also be defined simply as a function from a Cartesian power of a set to the same set.The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation that is not algebraic.
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https://en.wikipedia.org/wiki/Algebraic_operation
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In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers.
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https://en.wikipedia.org/wiki/Semi-algebraic_sets
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In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).
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https://en.wikipedia.org/wiki/Blending_function
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In mathematics, a basis of a matroid is a maximal independent set of the matroid—that is, an independent set that is not contained in any other independent set.
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https://en.wikipedia.org/wiki/Basis_of_a_matroid
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In mathematics, a bi-directional delay line is a numerical analysis technique used in computer simulation for solving ordinary differential equations by converting them to hyperbolic equations. In this way an explicit solution scheme is obtained with highly robust numerical properties. It was introduced by Auslander in 1968. It originates from simulation of hydraulic pipelines where wave propagation was studied.
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https://en.wikipedia.org/wiki/Bi-directional_delay_line
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It was then found that it could be used as an efficient numerical technique for numerically insulating different parts of a simulation model in each times step. It is used in the HOPSAN simulation package (Krus et al. 1990). It is also known as the Transmission Line Modelling (TLM) from an independent development by Johns and O'Brian 1980. This is also extended to partial differential equations.
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https://en.wikipedia.org/wiki/Bi-directional_delay_line
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In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.)
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https://en.wikipedia.org/wiki/Bialgebra
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Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism. As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is self-dual, so if one can define a dual of B (which is always possible if B is finite-dimensional), then it is automatically a bialgebra.
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https://en.wikipedia.org/wiki/Bialgebra
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In mathematics, a biased graph is a graph with a list of distinguished circles (edge sets of simple cycles), such that if two circles in the list are contained in a theta graph, then the third circle of the theta graph is also in the list. A biased graph is a generalization of the combinatorial essentials of a gain graph and in particular of a signed graph. Formally, a biased graph Ω is a pair (G, B) where B is a linear class of circles; this by definition is a class of circles that satisfies the theta-graph property mentioned above. A subgraph or edge set whose circles are all in B (and which contains no half-edges) is called balanced.
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https://en.wikipedia.org/wiki/Biased_graph
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For instance, a circle belonging to B is balanced and one that does not belong to B is unbalanced. Biased graphs are interesting mostly because of their matroids, but also because of their connection with multiary quasigroups. See below.
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https://en.wikipedia.org/wiki/Biased_graph
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In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou. Bicategories may be considered as a weakening of the definition of 2-categories. A similar process for 3-categories leads to tricategories, and more generally to weak n-categories for n-categories.
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https://en.wikipedia.org/wiki/Bicategory
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In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix. When the diagonal above the main diagonal has the non-zero entries the matrix is upper bidiagonal.
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https://en.wikipedia.org/wiki/Bidiagonal_matrix
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When the diagonal below the main diagonal has the non-zero entries the matrix is lower bidiagonal. For example, the following matrix is upper bidiagonal: ( 1 4 0 0 0 4 1 0 0 0 3 4 0 0 0 3 ) {\displaystyle {\begin{pmatrix}1&4&0&0\\0&4&1&0\\0&0&3&4\\0&0&0&3\\\end{pmatrix}}} and the following matrix is lower bidiagonal: ( 1 0 0 0 2 4 0 0 0 3 3 0 0 0 4 3 ) . {\displaystyle {\begin{pmatrix}1&0&0&0\\2&4&0&0\\0&3&3&0\\0&0&4&3\\\end{pmatrix}}.}
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https://en.wikipedia.org/wiki/Bidiagonal_matrix
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In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set; there are no unpaired elements between the two sets. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. The term one-to-one correspondence must not be confused with one-to-one function (an injective function; see figures). A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements.
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https://en.wikipedia.org/wiki/Bijective_relation
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For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms the symmetric group. Bijective functions are essential to many areas of mathematics including the definitions of isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and projective maps.
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https://en.wikipedia.org/wiki/Bijective_relation
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In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B: V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v)The dot product on R n {\displaystyle \mathbb {R} ^{n}} is an example of a bilinear form.The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.
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https://en.wikipedia.org/wiki/Radical_of_a_quadratic_space
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In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
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https://en.wikipedia.org/wiki/Separate_continuity
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In mathematics, a bilinear program is a nonlinear optimization problem whose objective or constraint functions are bilinear. An example is the pooling problem.
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https://en.wikipedia.org/wiki/Bilinear_program
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In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs. Precisely stated, a function f {\displaystyle f} is binary if there exists sets X , Y , Z {\displaystyle X,Y,Z} such that f: X × Y → Z {\displaystyle \,f\colon X\times Y\rightarrow Z} where X × Y {\displaystyle X\times Y} is the Cartesian product of X {\displaystyle X} and Y . {\displaystyle Y.}
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https://en.wikipedia.org/wiki/Binary_functions
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In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations.
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https://en.wikipedia.org/wiki/Commutative_operation
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The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.
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https://en.wikipedia.org/wiki/Commutative_operation
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In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation on a set is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication.
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https://en.wikipedia.org/wiki/Binary_operation
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Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a binary operation. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.
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https://en.wikipedia.org/wiki/Binary_operation
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In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables q ( x , y ) = a x 2 + b x y + c y 2 , {\displaystyle q(x,y)=ax^{2}+bxy+cy^{2},\,} where a, b, c are the coefficients. When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form. This article is entirely devoted to integral binary quadratic forms.
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https://en.wikipedia.org/wiki/Binary_quadratic_form
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This choice is motivated by their status as the driving force behind the development of algebraic number theory. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to quadratic and more general number fields, but advances specific to binary quadratic forms still occur on occasion. Pierre Fermat stated that if p is an odd prime then the equation p = x 2 + y 2 {\displaystyle p=x^{2}+y^{2}} has a solution iff p ≡ 1 ( mod 4 ) {\displaystyle p\equiv 1{\pmod {4}}} , and he made similar statement about the equations p = x 2 + 2 y 2 {\displaystyle p=x^{2}+2y^{2}} , p = x 2 + 3 y 2 {\displaystyle p=x^{2}+3y^{2}} , p = x 2 − 2 y 2 {\displaystyle p=x^{2}-2y^{2}} and p = x 2 − 3 y 2 {\displaystyle p=x^{2}-3y^{2}} .
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https://en.wikipedia.org/wiki/Binary_quadratic_form
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x 2 + y 2 , x 2 + 2 y 2 , x 2 − 3 y 2 {\displaystyle x^{2}+y^{2},x^{2}+2y^{2},x^{2}-3y^{2}} and so on are quadratic forms, and the theory of quadratic forms gives a unified way of looking at and proving these theorems. Another instance of quadratic forms is Pell's equation x 2 − n y 2 = 1 {\displaystyle x^{2}-ny^{2}=1} . Binary quadratic forms are closely related to ideals in quadratic fields, this allows the class number of a quadratic field to be calculated by counting the number of reduced binary quadratic forms of a given discriminant. The classical theta function of 2 variables is ∑ ( m , n ) ∈ Z 2 q m 2 + n 2 {\displaystyle \sum _{(m,n)\in \mathbb {Z} ^{2}}q^{m^{2}+n^{2}}} , if f ( x , y ) {\displaystyle f(x,y)} is a positive definite quadratic form then ∑ ( m , n ) ∈ Z 2 q f ( m , n ) {\displaystyle \sum _{(m,n)\in \mathbb {Z} ^{2}}q^{f(m,n)}} is a theta function.
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https://en.wikipedia.org/wiki/Binary_quadratic_form
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In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is, an element m ∈ S not related by s R m (for instance, "s is not smaller than m") for any s ∈ S. In other words, a relation is well founded if Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set. Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, which can be proved when there is no infinite sequence x0, x1, x2, ... of elements of X such that xn+1 R xn for every natural number n.In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order.
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https://en.wikipedia.org/wiki/Well-founded_order
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In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded. A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R−1 is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewriting systems, a Noetherian relation is also called terminating.
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https://en.wikipedia.org/wiki/Well-founded_order
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In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself.An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
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https://en.wikipedia.org/wiki/Irreflexive_kernel
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In mathematics, a binary relation R {\displaystyle R} on a set X {\displaystyle X} is antisymmetric if there is no pair of distinct elements of X {\displaystyle X} each of which is related by R {\displaystyle R} to the other. More formally, R {\displaystyle R} is antisymmetric precisely if for all a , b ∈ X , {\displaystyle a,b\in X,} or equivalently, The definition of antisymmetry says nothing about whether a R a {\displaystyle aRa} actually holds or not for any a {\displaystyle a} . An antisymmetric relation R {\displaystyle R} on a set X {\displaystyle X} may be reflexive (that is, a R a {\displaystyle aRa} for all a ∈ X {\displaystyle a\in X} ), irreflexive (that is, a R a {\displaystyle aRa} for no a ∈ X {\displaystyle a\in X} ), or neither reflexive nor irreflexive. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
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https://en.wikipedia.org/wiki/Anti-symmetric_relation
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In mathematics, a binary relation R ⊆ X×Y between two sets X and Y is total (or left total) if the source set X equals the domain {x: there is a y with xRy }. Conversely, R is called right total if Y equals the range {y: there is an x with xRy }. When f: X → Y is a function, the domain of f is all of X, hence f is a total relation. On the other hand, if f is a partial function, then the domain may be a proper subset of X, in which case f is not a total relation. "A binary relation is said to be total with respect to a universe of discourse just in case everything in that universe of discourse stands in that relation to something else."
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https://en.wikipedia.org/wiki/Left-total_relation
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