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{\displaystyle \mathbb {R} ^{+}.} There is, however, exactly one infimum of the positive real numbers relative to the real numbers: 0 , {\displaystyle 0,} which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.
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https://en.wikipedia.org/wiki/Infimum_and_supremum
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In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, p1m1, seen as an infinite set of parallel reflections along an axis.
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https://en.wikipedia.org/wiki/Infinite_dihedral_group
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In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, also written ∑ n = 0 ∞ ( − 1 ) n {\displaystyle \sum _{n=0}^{\infty }(-1)^{n}} is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it does not have a sum. However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. For example, the Cesàro summation and the Ramanujan summation of this series is 1/2.
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https://en.wikipedia.org/wiki/Grandi's_series
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In mathematics, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as 1 2 + 1 4 + 1 8 + 1 16 + ⋯ = ∑ n = 1 ∞ ( 1 2 ) n = 1. {\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1.} The series is related to philosophical questions considered in antiquity, particularly to Zeno's paradoxes.
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https://en.wikipedia.org/wiki/1/2_+_1/4_+_1/8_+_1/16_+_⋯
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In mathematics, the infinite series 1/4 + 1/16 + 1/64 + 1/256 + ⋯ is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC. As it is a geometric series with first term 1/4 and common ratio 1/4, its sum is ∑ n = 1 ∞ 1 4 n = 1 4 1 − 1 4 = 1 3 . {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{4^{n}}}={\frac {\frac {1}{4}}{1-{\frac {1}{4}}}}={\frac {1}{3}}.}
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https://en.wikipedia.org/wiki/1/4_+_1/16_+_1/64_+_1/256_+_⋯
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In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group G on a vector space V is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonalizing the representation. It therefore is a way of extracting something essential from the representation ρ by two successive linearizations.
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https://en.wikipedia.org/wiki/Infinitesimal_character
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In mathematics, the infinity Laplace (or L ∞ {\displaystyle L^{\infty }} -Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated Δ ∞ {\displaystyle \Delta _{\infty }} . It is alternately defined by Δ ∞ u ( x ) = ⟨ D u , D 2 u D u ⟩ = ∑ i , j ∂ 2 u ∂ x i ∂ x j ∂ u ∂ x i ∂ u ∂ x j {\displaystyle \Delta _{\infty }u(x)=\langle Du,D^{2}u\,Du\rangle =\sum _{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\,\partial x_{j}}}{\frac {\partial u}{\partial x_{i}}}{\frac {\partial u}{\partial x_{j}}}} or Δ ∞ u ( x ) = ⟨ D u , D 2 u D u ⟩ | D u | 2 = 1 | D u | 2 ∑ i , j ∂ 2 u ∂ x i ∂ x j ∂ u ∂ x i ∂ u ∂ x j . {\displaystyle \Delta _{\infty }u(x)={\frac {\langle Du,D^{2}u\,Du\rangle }{|Du|^{2}}}={\frac {1}{|Du|^{2}}}\sum _{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\,\partial x_{j}}}{\frac {\partial u}{\partial x_{i}}}{\frac {\partial u}{\partial x_{j}}}.} The first version avoids the singularity which occurs when the gradient vanishes, while the second version is homogeneous of order zero in the gradient.
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https://en.wikipedia.org/wiki/Infinity_Laplacian
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Verbally, the second version is the second derivative in the direction of the gradient. In the case of the infinity Laplace equation Δ ∞ u = 0 {\displaystyle \Delta _{\infty }u=0} , the two definitions are equivalent. While the equation involves second derivatives, usually (generalized) solutions are not twice differentiable, as evidenced by the well-known Aronsson solution u ( x , y ) = | x | 4 / 3 − | y | 4 / 3 {\displaystyle u(x,y)=|x|^{4/3}-|y|^{4/3}} .
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https://en.wikipedia.org/wiki/Infinity_Laplacian
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For this reason the correct notion of solutions is that given by the viscosity solutions. Viscosity solutions to the equation Δ ∞ u = 0 {\displaystyle \Delta _{\infty }u=0} are also known as infinity harmonic functions. This terminology arises from the fact that the infinity Laplace operator first arose in the study of absolute minimizers for ‖ D u ‖ L ∞ {\displaystyle \|Du\|_{L^{\infty }}} , and it can be viewed in a certain sense as the limit of the p-Laplacian as p → ∞ {\displaystyle p\rightarrow \infty } . More recently, viscosity solutions to the infinity Laplace equation have been identified with the payoff functions from randomized tug-of-war games. The game theory point of view has significantly improved the understanding of the partial differential equation itself.
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https://en.wikipedia.org/wiki/Infinity_Laplacian
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In mathematics, the infinity symbol is used more often to represent a potential infinity, rather than an actually infinite quantity as included in the cardinal numbers and the ordinal numbers (which use other notations, such as ℵ 0 {\displaystyle \,\aleph _{0}\,} and ω, for infinite values). For instance, in mathematical expressions with summations and limits such as the infinity sign is conventionally interpreted as meaning that the variable grows arbitrarily large towards infinity, rather than actually taking an infinite value, although other interpretations are possible.The infinity symbol may also be used to represent a point at infinity, especially when there is only one such point under consideration. This usage includes, in particular, the infinite point of a projective line, and the point added to a topological space to form its one-point compactification.
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https://en.wikipedia.org/wiki/Symbol_of_infinity
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In mathematics, the inflation-restriction exact sequence is an exact sequence occurring in group cohomology and is a special case of the five-term exact sequence arising from the study of spectral sequences. Specifically, let G be a group, N a normal subgroup, and A an abelian group which is equipped with an action of G, i.e., a homomorphism from G to the automorphism group of A. The quotient group G/N acts on AN = { a ∈ A: na = a for all n ∈ N}. Then the inflation-restriction exact sequence is: 0 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A) In this sequence, there are maps inflation H 1(G/N, AN) → H 1(G, A) restriction H 1(G, A) → H 1(N, A)G/N transgression H 1(N, A)G/N → H 2(G/N, AN) inflation H 2(G/N, AN) →H 2(G, A)The inflation and restriction are defined for general n: inflation Hn(G/N, AN) → Hn(G, A) restriction Hn(G, A) → Hn(N, A)G/NThe transgression is defined for general n transgression Hn(N, A)G/N → Hn+1(G/N, AN)only if Hi(N, A)G/N = 0 for i ≤ n − 1.The sequence for general n may be deduced from the case n = 1 by dimension-shifting or from the Lyndon–Hochschild–Serre spectral sequence.
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https://en.wikipedia.org/wiki/Inflation-restriction_exact_sequence
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In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y {\displaystyle Y} without any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to Y {\displaystyle Y} -valued functions.
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https://en.wikipedia.org/wiki/Injective_tensor_product
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In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the X-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons.
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https://en.wikipedia.org/wiki/Lebesgue-integrable_function
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However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the necessary abstractions for this.
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https://en.wikipedia.org/wiki/Lebesgue-integrable_function
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The Lebesgue integral plays an important role in probability theory, real analysis, and many other fields in mathematics. It is named after Henri Lebesgue (1875–1941), who introduced the integral (Lebesgue 1904). It is also a pivotal part of the axiomatic theory of probability. The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the Lebesgue measure.
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https://en.wikipedia.org/wiki/Lebesgue-integrable_function
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In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.
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https://en.wikipedia.org/wiki/Maclaurin–Cauchy_test
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In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product ι X ω {\displaystyle \iota _{X}\omega } is sometimes written as X ⌟ ω . {\displaystyle X\mathbin {\lrcorner } \omega .}
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https://en.wikipedia.org/wiki/Interior_multiplication
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In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by putting a complex structure on the torus H n ( M , R ) / H n ( M , Z ) {\displaystyle H^{n}(M,\mathbb {R} )/H^{n}(M,\mathbb {Z} )} for n odd. There are several different natural ways to put a complex structure on this torus, giving several different sorts of intermediate Jacobians, including one due to André Weil (1952) and one due to Phillip Griffiths (1968, 1968b). The ones constructed by Weil have natural polarizations if M is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations.
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https://en.wikipedia.org/wiki/Intermediate_Jacobian
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A complex structure on a real vector space is given by an automorphism I with square − 1 {\displaystyle -1} . The complex structures on H n ( M , R ) {\displaystyle H^{n}(M,\mathbb {R} )} are defined using the Hodge decomposition H n ( M , R ) ⊗ C = H n , 0 ( M ) ⊕ ⋯ ⊕ H 0 , n ( M ) . {\displaystyle H^{n}(M,{\mathbb {R} })\otimes {\mathbb {C} }=H^{n,0}(M)\oplus \cdots \oplus H^{0,n}(M).} On H p , q {\displaystyle H^{p,q}} the Weil complex structure I W {\displaystyle I_{W}} is multiplication by i p − q {\displaystyle i^{p-q}} , while the Griffiths complex structure I G {\displaystyle I_{G}} is multiplication by i {\displaystyle i} if p > q {\displaystyle p>q} and − i {\displaystyle -i} if p < q {\displaystyle p
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https://en.wikipedia.org/wiki/Intermediate_Jacobian
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In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of G need be considered, rather than two. This was certainly familiar before Hilbert.
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https://en.wikipedia.org/wiki/Decomposition_group
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In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd (co)homology group of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a smooth structure.
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https://en.wikipedia.org/wiki/Intersection_form_(4-manifold)
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In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space.
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https://en.wikipedia.org/wiki/Intersection
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Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident to each of original objects.
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https://en.wikipedia.org/wiki/Intersection
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In this approach an intersection can be sometimes undefined, such as for parallel lines. In both cases the concept of intersection relies on logical conjunction. Algebraic geometry defines intersections in its own way with intersection theory.
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https://en.wikipedia.org/wiki/Intersection
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In mathematics, the interval chromatic number X<(H) of an ordered graph H is the minimum number of intervals the (linearly ordered) vertex set of H can be partitioned into so that no two vertices belonging to the same interval are adjacent in H.
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https://en.wikipedia.org/wiki/Interval_chromatic_number_of_an_ordered_graph
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In mathematics, the intrinsic flat distance is a notion for distance between two Riemannian manifolds which is a generalization of Federer and Fleming's flat distance between submanifolds and integral currents lying in Euclidean space.
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https://en.wikipedia.org/wiki/Intrinsic_flat_distance
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In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property: L { f } ( s ) = L { f ( t ) } ( s ) = F ( s ) , {\displaystyle {\mathcal {L}}\{f\}(s)={\mathcal {L}}\{f(t)\}(s)=F(s),} where L {\displaystyle {\mathcal {L}}} denotes the Laplace transform. It can be proven that, if a function F(s) has the inverse Laplace transform f(t), then f(t) is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem.The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamical systems.
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https://en.wikipedia.org/wiki/Post's_inversion_formula
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In mathematics, the inverse bundle of a fibre bundle is its inverse with respect to the Whitney sum operation. Let E → M {\displaystyle E\rightarrow M} be a fibre bundle. A bundle E ′ → M {\displaystyle E'\rightarrow M} is called the inverse bundle of E {\displaystyle E} if their Whitney sum is a trivial bundle, namely if E ⊕ E ′ ≅ M × R n . {\displaystyle E\oplus E'\cong M\times \mathbb {R} ^{n}.\,} Any vector bundle over a compact Hausdorff base has an inverse bundle.
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https://en.wikipedia.org/wiki/Inverse_bundle
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In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by f − 1 . {\displaystyle f^{-1}.} For a function f: X → Y {\displaystyle f\colon X\to Y} , its inverse f − 1: Y → X {\displaystyle f^{-1}\colon Y\to X} admits an explicit description: it sends each element y ∈ Y {\displaystyle y\in Y} to the unique element x ∈ X {\displaystyle x\in X} such that f(x) = y. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result.
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https://en.wikipedia.org/wiki/Partial_inverse
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To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of f is the function f − 1: R → R {\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} } defined by f − 1 ( y ) = y + 7 5 . {\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}
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https://en.wikipedia.org/wiki/Partial_inverse
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In mathematics, the inverse gamma function Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inverse function of the gamma function. In other words, y = Γ − 1 ( x ) {\displaystyle y=\Gamma ^{-1}(x)} whenever Γ ( y ) = x {\textstyle \Gamma (y)=x} . For example, Γ − 1 ( 24 ) = 5 {\displaystyle \Gamma ^{-1}(24)=5} . Usually, the inverse gamma function refers to the principal branch with domain on the real interval [ β , + ∞ ) {\displaystyle \left[\beta ,+\infty \right)} and image on the real interval [ α , + ∞ ) {\displaystyle \left[\alpha ,+\infty \right)} , where β = 0.8856031 … {\displaystyle \beta =0.8856031\ldots } is the minimum value of the gamma function on the positive real axis and α = Γ − 1 ( β ) = 1.4616321 … {\displaystyle \alpha =\Gamma ^{-1}(\beta )=1.4616321\ldots } is the location of that minimum.
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https://en.wikipedia.org/wiki/Inverse_gamma_function
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In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-. For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding hyperbolic angle measure, for example arsinh ( sinh a ) = a {\displaystyle \operatorname {arsinh} (\sinh a)=a} and sinh ( arsinh x ) = x .
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https://en.wikipedia.org/wiki/Area_hyperbolic_tangent
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{\displaystyle \sinh(\operatorname {arsinh} x)=x.} Hyperbolic angle measure is the length of an arc of a unit hyperbola x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1} as measured in the Lorentzian plane (not the length of a hyperbolic arc in the Euclidean plane), and twice the area of the corresponding hyperbolic sector. This is analogous to the way circular angle measure is the arc length of an arc of the unit circle in the Euclidean plane or twice the area of the corresponding circular sector.
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https://en.wikipedia.org/wiki/Area_hyperbolic_tangent
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Alternately hyperbolic angle is the area of a sector of the hyperbola x y = 1. {\displaystyle xy=1.}
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https://en.wikipedia.org/wiki/Area_hyperbolic_tangent
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Some authors call the inverse hyperbolic functions hyperbolic area functions.Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. It also occurs in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
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https://en.wikipedia.org/wiki/Area_hyperbolic_tangent
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In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory. By working in the dual category, that is by reverting the arrows, an inverse limit becomes a direct limit or inductive limit, and a limit becomes a colimit.
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https://en.wikipedia.org/wiki/Inverse_limit
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In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function. There has been a great deal of activity in the study of this problem since the early 20th century. A notable advance in this field was a 1941 paper by the American mathematician Jesse Douglas, in which he provided necessary and sufficient conditions for the problem to have a solution; these conditions are now known as the Helmholtz conditions, after the German physicist Hermann von Helmholtz.
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https://en.wikipedia.org/wiki/Inverse_problem_for_Lagrangian_mechanics
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In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential. The inverse scattering transform may be applied to many of the so-called exactly solvable models, that is to say completely integrable infinite dimensional systems.
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https://en.wikipedia.org/wiki/Inverse_scattering_theory
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In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
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https://en.wikipedia.org/wiki/Arc_(function_prefix)
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In mathematics, the irrational base discrete weighted transform (IBDWT) is a variant of the fast Fourier transform using an irrational base; it was developed by Richard Crandall (Reed College), Barry Fagin (Dartmouth College) and Joshua Doenias (NeXT Software) in the early 1990s using Mathematica.The IBDWT is used in the Great Internet Mersenne Prime Search's client Prime95 to perform FFT multiplication, as well as in other programs implementing Lucas–Lehmer test, such as CUDALucas and Glucas.
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https://en.wikipedia.org/wiki/Irrational_base_discrete_weighted_transform
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In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two.
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https://en.wikipedia.org/wiki/Incommensurable_magnitudes
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In fact, all square roots of natural numbers, other than of perfect squares, are irrational.Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat.
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https://en.wikipedia.org/wiki/Incommensurable_magnitudes
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Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. Irrational numbers can also be expressed as non-terminating continued fractions and many other ways. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.
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https://en.wikipedia.org/wiki/Incommensurable_magnitudes
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In mathematics, the irregularity of a complex surface X is the Hodge number h 0 , 1 = dim H 1 ( O X ) {\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})} , usually denoted by q. The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in characteristic 0 but can be smaller in positive characteristic.The name "irregularity" comes from the fact that for the first surfaces investigated in detail, the smooth complex surfaces in P3, the irregularity happens to vanish. The irregularity then appeared as a new "correction" term measuring the difference p g − p a {\displaystyle p_{g}-p_{a}} of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes. For a complex analytic manifold X of general dimension, the Hodge number h 0 , 1 = dim H 1 ( O X ) {\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})} is called the irregularity of X {\displaystyle X} , and is denoted by q.
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https://en.wikipedia.org/wiki/Irregularity_of_a_surface
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In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal.The terminology arises from the connection with algebraic geometry. If R = k (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) graded with respect to degree, there is a bijective correspondence between projective algebraic sets in projective n-space over k and homogeneous, radical ideals of R not equal to the irrelevant ideal. More generally, for an arbitrary graded ring R, the Proj construction disregards all irrelevant ideals of R.
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https://en.wikipedia.org/wiki/Irrelevant_ideal
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In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function. The elements of the isometry group are sometimes called motions of the space.
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https://en.wikipedia.org/wiki/Isometry_group
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Every isometry group of a metric space is a subgroup of isometries. It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space. See symmetry group. A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form; transformations preserving this form are sometimes called "isometries", and the collection of them is then said to form an isometry group of the pseudo-Euclidean space.
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https://en.wikipedia.org/wiki/Isometry_group
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In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large-scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space). In the Euclidean space, the isoperimetric inequality says that of all bodies with the same volume, the ball has the smallest surface area. In other manifolds it is usually very difficult to find the precise body minimizing the surface area, and this is not what the isoperimetric dimension is about. The question we will ask is, what is approximately the minimal surface area, whatever the body realizing it might be.
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https://en.wikipedia.org/wiki/Isoperimetric_dimension
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In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} the inequality lower bounds the surface area or perimeter per ( S ) {\displaystyle \operatorname {per} (S)} of a set S ⊂ R n {\displaystyle S\subset \mathbb {R} ^{n}} by its volume vol ( S ) {\displaystyle \operatorname {vol} (S)} , per ( S ) ≥ n vol ( S ) ( n − 1 ) / n vol ( B 1 ) 1 / n {\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{(n-1)/n}\,\operatorname {vol} (B_{1})^{1/n}} ,where B 1 ⊂ R n {\displaystyle B_{1}\subset \mathbb {R} ^{n}} is a unit sphere. The equality holds only when S {\displaystyle S} is a sphere in R n {\displaystyle \mathbb {R} ^{n}} . On a plane, i.e. when n = 2 {\displaystyle n=2} , the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses.
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https://en.wikipedia.org/wiki/Spherical_isoperimetric_inequality
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Isoperimetric literally means "having the same perimeter". Specifically in R 2 {\displaystyle \mathbb {R} ^{2}} , the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that L 2 ≥ 4 π A , {\displaystyle L^{2}\geq 4\pi A,} and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length.
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https://en.wikipedia.org/wiki/Spherical_isoperimetric_inequality
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The closely related Dido's problem asks for a region of the maximal area bounded by a straight line and a curvilinear arc whose endpoints belong to that line. It is named after Dido, the legendary founder and first queen of Carthage.
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https://en.wikipedia.org/wiki/Spherical_isoperimetric_inequality
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The solution to the isoperimetric problem is given by a circle and was known already in Ancient Greece. However, the first mathematically rigorous proof of this fact was obtained only in the 19th century. Since then, many other proofs have been found.
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https://en.wikipedia.org/wiki/Spherical_isoperimetric_inequality
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The isoperimetric problem has been extended in multiple ways, for example, to curves on surfaces and to regions in higher-dimensional spaces. Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere.
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https://en.wikipedia.org/wiki/Spherical_isoperimetric_inequality
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In mathematics, the ith Bass number of a module M over a local ring R with residue field k is the k-dimension of Ext R i ( k , M ) {\displaystyle \operatorname {Ext} _{R}^{i}(k,M)} . More generally the Bass number μ i ( p , M ) {\displaystyle \mu _{i}(p,M)} of a module M over a ring R at a prime ideal p is the Bass number of the localization of M for the localization of R (with respect to the prime p). Bass numbers were introduced by Hyman Bass (1963, p.11). The Bass numbers describe the minimal injective resolution of a finitely-generated module M over a Noetherian ring: for each prime ideal p there is a corresponding indecomposable injective module, and the number of times this occurs in the ith term of a minimal resolution of M is the Bass number μ i ( p , M ) {\displaystyle \mu _{i}(p,M)}
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https://en.wikipedia.org/wiki/Bass_number
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In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years this notion has found applications in a large number of engineering fields and is still a topic of active research.
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https://en.wikipedia.org/wiki/Joint_spectral_radius
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In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L: V → W between two vector spaces V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W, or more symbolically: ker ( L ) = { v ∈ V ∣ L ( v ) = 0 } = L − 1 ( 0 ) . {\displaystyle \ker(L)=\left\{\mathbf {v} \in V\mid L(\mathbf {v} )=\mathbf {0} \right\}=L^{-1}(0).}
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https://en.wikipedia.org/wiki/Null_Space
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In mathematics, the knot complement of a tame knot K is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that K is a knot in a three-manifold M (most often, M is the 3-sphere). Let N be a tubular neighborhood of K; so N is a solid torus.
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https://en.wikipedia.org/wiki/Link_complement
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The knot complement is then the complement of N, X K = M − interior ( N ) . {\displaystyle X_{K}=M-{\mbox{interior}}(N).} The knot complement XK is a compact 3-manifold; the boundary of XK and the boundary of the neighborhood N are homeomorphic to a two-torus.
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https://en.wikipedia.org/wiki/Link_complement
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Sometimes the ambient manifold M is understood to be the 3-sphere. Context is needed to determine the usage. There are analogous definitions for the link complement.
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https://en.wikipedia.org/wiki/Link_complement
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Many knot invariants, such as the knot group, are really invariants of the complement of the knot. When the ambient space is the three-sphere no information is lost: the Gordon–Luecke theorem states that a knot is determined by its complement. That is, if K and K′ are two knots with homeomorphic complements then there is a homeomorphism of the three-sphere taking one knot to the other.
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https://en.wikipedia.org/wiki/Link_complement
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In mathematics, the lakes of Wada (和田の湖, Wada no mizuumi) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of one of the lakes, the other two lakes' boundaries also contain that point. More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems.
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https://en.wikipedia.org/wiki/Lakes_of_Wada
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This property is rare in real-world systems. The lakes of Wada were introduced by Kunizō Yoneyama (1917, page 60), who credited the discovery to Takeo Wada. His construction is similar to the construction by Brouwer (1910) of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum.
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https://en.wikipedia.org/wiki/Lakes_of_Wada
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In mathematics, the language of ordered abelian groups has one constant symbol 0, one unary function symbol −, one binary function symbol +, and one binary relation symbol ≤. Then: The expressions +(x, y) and +(x, +(y, −(z))) are terms. These are usually written as x + y and x + y − z. The expressions +(x, y) = 0 and ≤(+(x, +(y, −(z))), +(x, y)) are atomic formulas. These are usually written as x + y = 0 and x + y − z ≤ x + y. The expression ( ∀ x ∀ y {\displaystyle (\forall x\forall y\,} is a formula, which is usually written as ∀ x ∀ y ( x + y ≤ z ) → ∀ x ∀ y ( x + y = 0 ) .
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https://en.wikipedia.org/wiki/Predicate_calculus
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{\displaystyle \forall x\forall y(x+y\leq z)\to \forall x\forall y(x+y=0).} This formula has one free variable, z.The axioms for ordered abelian groups can be expressed as a set of sentences in the language. For example, the axiom stating that the group is commutative is usually written ( ∀ x ) ( ∀ y ) . {\displaystyle (\forall x)(\forall y).}
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https://en.wikipedia.org/wiki/Predicate_calculus
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In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are ordinal collapsing functions. The large Veblen ordinal is sometimes denoted by ϕ Ω Ω ( 0 ) {\displaystyle \phi _{\Omega ^{\Omega }}(0)} or θ ( Ω Ω ) {\displaystyle \theta (\Omega ^{\Omega })} or ψ ( Ω Ω Ω ) {\displaystyle \psi (\Omega ^{\Omega ^{\Omega }})} . It was constructed by Veblen using an extension of Veblen functions allowing infinitely many arguments.
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https://en.wikipedia.org/wiki/Large_Veblen_ordinal
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In mathematics, the lattice of subgroups of a group G {\displaystyle G} is the lattice whose elements are the subgroups of G {\displaystyle G} , with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.
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https://en.wikipedia.org/wiki/Lattice_of_subgroups
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In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space. The law encodes a lot of information about the process; in the case of a random walk, for example, the law is the probability distribution of the possible trajectories of the walk.
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https://en.wikipedia.org/wiki/Law_(stochastic_processes)
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In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exactly one of xRy, yRx and x = y holds. Writing R as <, this is stated in formal logic as: ∀ x ∈ X ∀ y ∈ X ( ∨ ∨ ) . {\displaystyle \forall x\in X\,\forall y\in X\,([x
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https://en.wikipedia.org/wiki/Law_of_trichotomy
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In mathematics, the layer cake representation of a non-negative, real-valued measurable function f {\displaystyle f} defined on a measure space ( Ω , A , μ ) {\displaystyle (\Omega ,{\mathcal {A}},\mu )} is the formula f ( x ) = ∫ 0 ∞ 1 L ( f , t ) ( x ) d t , {\displaystyle f(x)=\int _{0}^{\infty }1_{L(f,t)}(x)\,\mathrm {d} t,} for all x ∈ Ω {\displaystyle x\in \Omega } , where 1 E {\displaystyle 1_{E}} denotes the indicator function of a subset E ⊆ Ω {\displaystyle E\subseteq \Omega } and L ( f , t ) {\displaystyle L(f,t)} denotes the super-level set L ( f , t ) = { y ∈ Ω ∣ f ( y ) ≥ t } . {\displaystyle L(f,t)=\{y\in \Omega \mid f(y)\geq t\}.} The layer cake representation follows easily from observing that 1 L ( f , t ) ( x ) = 1 ( t ) {\displaystyle 1_{L(f,t)}(x)=1_{}(t)} and then using the formula f ( x ) = ∫ 0 f ( x ) d t . {\displaystyle f(x)=\int _{0}^{f(x)}\,\mathrm {d} t.}
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https://en.wikipedia.org/wiki/Layer_cake_representation
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The layer cake representation takes its name from the representation of the value f ( x ) {\displaystyle f(x)} as the sum of contributions from the "layers" L ( f , t ) {\displaystyle L(f,t)}: "layers"/values t {\displaystyle t} below f ( x ) {\displaystyle f(x)} contribute to the integral, while values t {\displaystyle t} above f ( x ) {\displaystyle f(x)} do not. It is a generalization of Cavalieri's principle and is also known under this name. : cor. 2.2.34 An important consequence of the layer cake representation is the identity ∫ Ω f ( x ) d μ ( x ) = ∫ 0 ∞ μ ( { x ∈ Ω ∣ f ( x ) > t } ) d t , {\displaystyle \int _{\Omega }f(x)\,\mathrm {d} \mu (x)=\int _{0}^{\infty }\mu (\{x\in \Omega \mid f(x)>t\})\,\mathrm {d} t,} which follows from it by applying the Fubini-Tonelli theorem. An important application is that L p {\displaystyle L^{p}} for 1 ≤ p < + ∞ {\displaystyle 1\leq p<+\infty } can be written as follows ∫ Ω | f ( x ) | p d μ ( x ) = p ∫ 0 ∞ s p − 1 μ ( { x ∈ Ω ∣ | f ( x ) | > s } ) d s , {\displaystyle \int _{\Omega }|f(x)|^{p}\,\mathrm {d} \mu (x)=p\int _{0}^{\infty }s^{p-1}\mu (\{x\in \Omega \mid \,|f(x)|>s\})\mathrm {d} s,} which follows immediately from the change of variables t = s p {\displaystyle t=s^{p}} in the layer cake representation of | f ( x ) | p {\displaystyle |f(x)|^{p}} .
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https://en.wikipedia.org/wiki/Layer_cake_representation
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In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X. Not every (partially) ordered set has the least upper bound property. For example, the set Q {\displaystyle \mathbb {Q} } of all rational numbers with its natural order does not have the least upper bound property.
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https://en.wikipedia.org/wiki/Least-upper-bound_property
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The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. It can be used to prove many of the fundamental results of real analysis, such as the intermediate value theorem, the Bolzano–Weierstrass theorem, the extreme value theorem, and the Heine–Borel theorem. It is usually taken as an axiom in synthetic constructions of the real numbers, and it is also intimately related to the construction of the real numbers using Dedekind cuts. In order theory, this property can be generalized to a notion of completeness for any partially ordered set. A linearly ordered set that is dense and has the least upper bound property is called a linear continuum.
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https://en.wikipedia.org/wiki/Least-upper-bound_property
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In mathematics, the lemniscate constant ϖ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of π for the circle. Equivalently, the perimeter of the lemniscate ( x 2 + y 2 ) 2 = x 2 − y 2 {\displaystyle (x^{2}+y^{2})^{2}=x^{2}-y^{2}} is 2ϖ. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. The symbol ϖ is a cursive variant of π; see Pi § Variant pi. Gauss's constant, denoted by G, is equal to ϖ /π ≈ 0.8346268.John Todd named two more lemniscate constants, the first lemniscate constant A = ϖ/2 ≈ 1.3110287771 and the second lemniscate constant B = π/(2ϖ) ≈ 0.5990701173.Sometimes the quantities 2ϖ or A are referred to as the lemniscate constant.
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https://en.wikipedia.org/wiki/Lemniscate_constant
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In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead), are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle x 2 + y 2 = x , {\displaystyle x^{2}+y^{2}=x,} the lemniscate sine relates the arc length to the chord length of a lemniscate ( x 2 + y 2 ) 2 = x 2 − y 2 . {\displaystyle {\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}.}
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https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions
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The lemniscate functions have periods related to a number ϖ = {\displaystyle \varpi =} 2.622057... called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the (quadratic) π = {\displaystyle \pi =} 3.141592..., ratio of perimeter to diameter of a circle. As complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods { ( 1 + i ) ϖ , ( 1 − i ) ϖ } , {\displaystyle \{(1+i)\varpi ,(1-i)\varpi \},} and are a special case of two Jacobi elliptic functions on that lattice, sl z = sn ( z ; i ) , {\displaystyle \operatorname {sl} z=\operatorname {sn} (z;i),} cl z = cd ( z ; i ) {\displaystyle \operatorname {cl} z=\operatorname {cd} (z;i)} .
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https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions
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Similarly, the hyperbolic lemniscate sine slh and hyperbolic lemniscate cosine clh have a square period lattice with fundamental periods { 2 ϖ , 2 ϖ i } . {\displaystyle {\bigl \{}{\sqrt {2}}\varpi ,{\sqrt {2}}\varpi i{\bigr \}}.} The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function ℘ ( z ; a , 0 ) {\displaystyle \wp (z;a,0)} .
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https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions
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In mathematics, the length of an element w in a Weyl group W, denoted by l(w), is the smallest number k so that w is a product of k reflections by simple roots. (So, the notion depends on the choice of a positive Weyl chamber.) In particular, a simple reflection has length one. The function l is then an integer-valued function of W; it is a length function of W. It follows immediately from the definition that l(w−1) = l(w) and that l(ww'−1) ≤ l(w) + l(w' ).
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https://en.wikipedia.org/wiki/Length_of_a_Weyl_group_element
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In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements. Another variant, widely used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences, to which the lexicographical order is applied. A generalization defines an order on a Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered.
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https://en.wikipedia.org/wiki/Lexicographical_ordering
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In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.
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https://en.wikipedia.org/wiki/Limit_comparison_test
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In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively.
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https://en.wikipedia.org/wiki/Limit_inferior_(topological_space)
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In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. The limit inferior of a sequence ( x n ) {\displaystyle (x_{n})} is denoted by and the limit superior of a sequence ( x n ) {\displaystyle (x_{n})} is denoted by
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https://en.wikipedia.org/wiki/Limit_inferior_(topological_space)
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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist. The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
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https://en.wikipedia.org/wiki/Algebraic_limit_theorem
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In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the lim {\displaystyle \lim } symbol (e.g., lim n → ∞ a n {\displaystyle \lim _{n\to \infty }a_{n}} ). If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.
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https://en.wikipedia.org/wiki/Convergent_sequence
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In mathematics, the limit of a sequence of sets A 1 , A 2 , … {\displaystyle A_{1},A_{2},\ldots } (subsets of a common set X {\displaystyle X} ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves real-valued. As is the case with sequences of other objects, convergence is not necessary or even usual. More generally, again analogous to real-valued sequences, the less restrictive limit infimum and limit supremum of a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit supremum are identical. (See below).
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https://en.wikipedia.org/wiki/Set-theoretic_limit
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Such set limits are essential in measure theory and probability. It is a common misconception that the limits infimum and supremum described here involve sets of accumulation points, that is, sets of x = lim k → ∞ x k , {\displaystyle x=\lim _{k\to \infty }x_{k},} where each x k {\displaystyle x_{k}} is in some A n k . {\displaystyle A_{n_{k}}.}
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https://en.wikipedia.org/wiki/Set-theoretic_limit
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This is only true if convergence is determined by the discrete metric (that is, x n → x {\displaystyle x_{n}\to x} if there is N {\displaystyle N} such that x n = x {\displaystyle x_{n}=x} for all n ≥ N {\displaystyle n\geq N} ). This article is restricted to that situation as it is the only one relevant for measure theory and probability. See the examples below. (On the other hand, there are more general topological notions of set convergence that do involve accumulation points under different metrics or topologies.)
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https://en.wikipedia.org/wiki/Set-theoretic_limit
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In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolvent near the essential spectrum. The term is often used to indicate that the resolvent, when considered not in the original space (which is usually the L 2 {\displaystyle L^{2}} space), but in certain weighted spaces (usually L s 2 {\displaystyle L_{s}^{2}} , see below), has a limit as the spectral parameter approaches the essential spectrum. This concept developed from the idea of introducing complex parameter into the Helmholtz equation ( Δ + k 2 ) u ( x ) = − F ( x ) {\displaystyle (\Delta +k^{2})u(x)=-F(x)} for selecting a particular solution.
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https://en.wikipedia.org/wiki/Limiting_absorption_principle
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This idea is credited to Vladimir Ignatowski, who was considering the propagation and absorption of the electromagnetic waves in a wire. It is closely related to the Sommerfeld radiation condition and the limiting amplitude principle (1948). The terminology – both the limiting absorption principle and the limiting amplitude principle – was introduced by Aleksei Sveshnikov.
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https://en.wikipedia.org/wiki/Limiting_absorption_principle
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In mathematics, the limiting amplitude principle is a concept from operator theory and scattering theory used for choosing a particular solution to the Helmholtz equation. The choice is made by considering a particular time-dependent problem of the forced oscillations due to the action of a periodic force. The principle was introduced by Andrey Nikolayevich Tikhonov and Alexander Andreevich Samarskii. It is closely related to the limiting absorption principle (1905) and the Sommerfeld radiation condition (1912). The terminology -- both the limiting absorption principle and the limiting amplitude principle -- was introduced by Aleksei Sveshnikov.
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https://en.wikipedia.org/wiki/Limiting_amplitude_principle
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In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. This requires the property that the field trace TrL/K provides a non-degenerate quadratic form over K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect field, and hence in the cases where K is finite, or of characteristic zero. A dual basis () is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using a second basis for computations.
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https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension
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Consider two bases for elements in a finite field, GF(pm): B 1 = α 0 , α 1 , … , α m − 1 {\displaystyle B_{1}={\alpha _{0},\alpha _{1},\ldots ,\alpha _{m-1}}} and B 2 = γ 0 , γ 1 , … , γ m − 1 {\displaystyle B_{2}={\gamma _{0},\gamma _{1},\ldots ,\gamma _{m-1}}} then B2 can be considered a dual basis of B1 provided Tr ( α i ⋅ γ j ) = { 0 , if i ≠ j 1 , otherwise {\displaystyle \operatorname {Tr} (\alpha _{i}\cdot \gamma _{j})=\left\{{\begin{matrix}0,&\operatorname {if} \ i\neq j\\1,&\operatorname {otherwise} \end{matrix}}\right.} Here the trace of a value in GF(pm) can be calculated as follows: Tr ( β ) = ∑ i = 0 m − 1 β p i {\displaystyle \operatorname {Tr} (\beta )=\sum _{i=0}^{m-1}\beta ^{p^{i}}} Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with multiplication by the multiplicative identity (usually 1).
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https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension
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In mathematics, the linear span (also called the linear hull or just span) of a set S of vectors (from a vector space), denoted span(S), is defined as the set of all linear combinations of the vectors in S. For example, two linearly independent vectors span a plane. The linear span can be characterized either as the intersection of all linear subspaces that contain S, or as the smallest subspace containing S. The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules. To express that a vector space V is a linear span of a subset S, one commonly uses the following phrases—either: S spans V, S is a spanning set of V, V is spanned/generated by S, or S is a generator or generator set of V.
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https://en.wikipedia.org/wiki/Linear_span
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In mathematics, the lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent each line in S lie on a quadric, Q known as the Klein quadric. If the underlying vector space of S is the 4-dimensional vector space V, then T has as the underlying vector space the 6-dimensional exterior square Λ2V of V. The line coordinates obtained this way are known as Plücker coordinates. These Plücker coordinates satisfy the quadratic relation p 12 p 34 + p 13 p 42 + p 14 p 23 = 0 {\displaystyle p_{12}p_{34}+p_{13}p_{42}+p_{14}p_{23}=0} defining Q, where p i j = u i v j − u j v i {\displaystyle p_{ij}=u_{i}v_{j}-u_{j}v_{i}} are the coordinates of the line spanned by the two vectors u and v. The 3-space, S, can be reconstructed again from the quadric, Q: the planes contained in Q fall into two equivalence classes, where planes in the same class meet in a point, and planes in different classes meet in a line or in the empty set. Let these classes be C {\displaystyle C} and C ′ {\displaystyle C'} . The geometry of S is retrieved as follows: The points of S are the planes in C. The lines of S are the points of Q. The planes of S are the planes in C’.The fact that the geometries of S and Q are isomorphic can be explained by the isomorphism of the Dynkin diagrams A3 and D3.
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https://en.wikipedia.org/wiki/Klein_quadric
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In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves (this is not true for curves in most 3-manifolds, where linking numbers can also be fractions or just not exist at all). The linking number was introduced by Gauss in the form of the linking integral. It is an important object of study in knot theory, algebraic topology, and differential geometry, and has numerous applications in mathematics and science, including quantum mechanics, electromagnetism, and the study of DNA supercoiling.
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https://en.wikipedia.org/wiki/Linking_coefficient
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In mathematics, the little q-Jacobi polynomials pn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Hahn (1949). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
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https://en.wikipedia.org/wiki/Little_q-Jacobi_polynomials
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In mathematics, the little q-Laguerre polynomials pn(x;a|q) or Wall polynomials Wn(x; b,q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme closely related to a continued fraction studied by Wall (1941). (The term "Wall polynomial" is also used for an unrelated Wall polynomial in the theory of classical groups.) Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
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https://en.wikipedia.org/wiki/Little_q-Laguerre_polynomials
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In mathematics, the local Heun function H ℓ ( a , q ; α , β , γ , δ ; z ) {\displaystyle H\ell (a,q;\alpha ,\beta ,\gamma ,\delta ;z)} (Karl L. W. Heun 1889) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0. The local Heun function is called a Heun function, denoted Hf, if it is also regular at z = 1, and is called a Heun polynomial, denoted Hp, if it is regular at all three finite singular points z = 0, 1, a.
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https://en.wikipedia.org/wiki/Heun_function
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In mathematics, the local Langlands conjectures, introduced by Robert Langlands (1967, 1970), are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.
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https://en.wikipedia.org/wiki/Local_Langlands_conjectures
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In mathematics, the local trace formula (Arthur 1991) is a local analogue of the Arthur–Selberg trace formula that describes the character of the representation of G(F) on the discrete part of L2(G(F)), for G a reductive algebraic group over a local field F.
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https://en.wikipedia.org/wiki/Local_trace_formula
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In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering.
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https://en.wikipedia.org/wiki/Logarithm_function
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