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c_xbdayvwwijp8 | {\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 on... | Infimum and supremum |
c_wr1wozpwaj86 | 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. | Infinite dihedral group |
c_10c99z0c25c6 | 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, me... | Grandi's series |
c_e73p2bnlmo99 | 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... | 1/2 + 1/4 + 1/8 + 1/16 + ⋯ |
c_riecrwx5cjc4 | 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... | 1/4 + 1/16 + 1/64 + 1/256 + ⋯ |
c_6t4cr11kypmt | 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 f... | Infinitesimal character |
c_yj1yyn6vzue8 | 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 {\displays... | Infinity Laplacian |
c_fm4eco25010a | 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 differen... | Infinity Laplacian |
c_x3fqo2eheixu | 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... | Infinity Laplacian |
c_h9pu629kqh8j | 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 mathematic... | Symbol of infinity |
c_z6jq3pg6cph3 | 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 ... | Inflation-restriction exact sequence |
c_87synyf8fded | 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 t... | Injective tensor product |
c_hodnm2x7x1pb | 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 ... | Lebesgue-integrable function |
c_el406tqvhv36 | 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 space... | Lebesgue-integrable function |
c_md25scj8522q | 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 ei... | Lebesgue-integrable function |
c_9v6nbbswkvwx | 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. | Maclaurin–Cauchy test |
c_pe8tlf1q3iy0 | 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... | Interior multiplication |
c_so4ke3lg54ma | 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 ) {\disp... | Intermediate Jacobian |
c_zfp0y7sej1jy | 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... | Intermediate Jacobian |
c_b4qsjncmpie8 | 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... | Decomposition group |
c_b35ur7cw7fg7 | 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. | Intersection form (4-manifold) |
c_psr4ah2z5x5z | 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, th... | Intersection |
c_vkdf31up0znl | 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. | Intersection |
c_mo2e7nbqbjjc | 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. | Intersection |
c_folc6rxkz3i2 | 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. | Interval chromatic number of an ordered graph |
c_0hnub4n98qne | 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. | Intrinsic flat distance |
c_ewi68bxmen2e | 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}... | Post's inversion formula |
c_1qrrrfe8f0kw | 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... | Inverse bundle |
c_pcbwq3pdny1g | 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 ... | Partial inverse |
c_82uikma0u7mw | 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}}.} | Partial inverse |
c_9fxmdzgaj6vy | 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} . Usua... | Inverse gamma function |
c_4s2fgqmasr5t | 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 hyperbo... | Area hyperbolic tangent |
c_5xncg9vbh10u | {\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. ... | Area hyperbolic tangent |
c_er4rwv0wd05s | Alternately hyperbolic angle is the area of a sector of the hyperbola x y = 1. {\displaystyle xy=1.} | Area hyperbolic tangent |
c_bnyr9db4e0dp | 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 equ... | Area hyperbolic tangent |
c_gbqbfbknhoys | 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 cate... | Inverse limit |
c_szwmco72r0ql | 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. ... | Inverse problem for Lagrangian mechanics |
c_at8z3uonxou0 | 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 scatter... | Inverse scattering theory |
c_l2zh4oc1dqk0 | 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, s... | Arc (function prefix) |
c_qeiyfadci51t | 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 t... | Irrational base discrete weighted transform |
c_7xlapnrwuzyp | 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... | Incommensurable magnitudes |
c_rdq7ilalhpl9 | 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 ... | Incommensurable magnitudes |
c_mm4k1qjknbbd | 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 ... | Incommensurable magnitudes |
c_fsc9m5lk07wy | 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 P... | Irregularity of a surface |
c_0oek82inpvro | 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 ... | Irrelevant ideal |
c_34vzq6zzoax9 | 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 ... | Isometry group |
c_5mqcyp8xwphn | 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 ... | Isometry group |
c_h2w4vhircm3p | 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 sp... | Isoperimetric dimension |
c_pwoyvryr8m09 | 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 ... | Spherical isoperimetric inequality |
c_sgzvarfze1m9 | 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 ... | Spherical isoperimetric inequality |
c_f7fd6gy6ulka | 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. | Spherical isoperimetric inequality |
c_vemxh1wnykd6 | 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. | Spherical isoperimetric inequality |
c_2sr4ap7xn6ca | 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 r... | Spherical isoperimetric inequality |
c_h03adbplveg0 | 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 n... | Bass number |
c_767oy6hk14bw | 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. | Joint spectral radius |
c_h8lzhsctxwez | 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... | Null Space |
c_juk8vfpn32iw | 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 neig... | Link complement |
c_yhebvs8z2kgb | 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. | Link complement |
c_ls86a3ca1g1j | 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. | Link complement |
c_r1on5w2azgof | 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 ... | Link complement |
c_xe0sj0q5590i | 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 cont... | Lakes of Wada |
c_18yu4542kxk0 | 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 s... | Lakes of Wada |
c_u4r97gojixv1 | 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 ≤... | Predicate calculus |
c_r1tjza156knz | {\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 ) . {\displaystyl... | Predicate calculus |
c_olwzvqyl3hia | 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 ... | Large Veblen ordinal |
c_znivm6dskmqb | 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 thei... | Lattice of subgroups |
c_b02qq7fi53gd | 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 traje... | Law (stochastic processes) |
c_n5xuxg9bkg5w | 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 ( ∨ ∨ ) . {\dis... | Law of trichotomy |
c_mc7l0cmj6iet | 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)\,\math... | Layer cake representation |
c_l7vj1bio2cs8 | 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 {\displaystyl... | Layer cake representation |
c_onsgj1zcbsm4 | 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 (s... | Least-upper-bound property |
c_jciudhry9ayj | 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, ... | Least-upper-bound property |
c_e1wd99vwsbm3 | 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^... | Lemniscate constant |
c_a96tjj0y8yrm | 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 wit... | Lemniscate elliptic functions |
c_hydesv4f2qud | 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. ... | Lemniscate elliptic functions |
c_8p77nnl8hudj | 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 Weier... | Lemniscate elliptic functions |
c_8qu3ssxitjza | 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 ... | Length of a Weyl group element |
c_sb1h28gbm38j | 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 |
c_8fgdcdkbpkj1 | 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. | Limit comparison test |
c_zg7ds5oantia | 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, respecti... | Limit inferior (topological space) |
c_z35uphsmipkg | 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 infi... | Limit inferior (topological space) |
c_yo5orrrsqutx | 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 func... | Algebraic limit theorem |
c_akb7feasx8pe | 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 s... | Convergent sequence |
c_xtgz88y4xxmy | 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-theoretic limit |
c_r8pjtmc1f07f | 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 .... | Set-theoretic limit |
c_lwc0ytts0sd6 | 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 th... | Set-theoretic limit |
c_d93jgztper8e | 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... | Limiting absorption principle |
c_ru7z9cw8g8ad | 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 ampl... | Limiting absorption principle |
c_xa5nvxv4131w | 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 principl... | Limiting amplitude principle |
c_5a1gmbh5i2xf | 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 tru... | Dual basis in a field extension |
c_ilk7od8dfvs2 | 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 ... | Dual basis in a field extension |
c_p124fsi3s3le | 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 in... | Linear span |
c_ifa24jk4wnzs | 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 ... | Klein quadric |
c_lnkanm7kox0t | 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 ... | Linking coefficient |
c_b2oy97hpsmlc | 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. | Little q-Jacobi polynomials |
c_7zxduzu8culn | 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 th... | Little q-Laguerre polynomials |
c_rnz1ddio0oad | 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 ... | Heun function |
c_8qy23iudtj4x | 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... | Local Langlands conjectures |
c_bmqkerexkj9x | 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. | Local trace formula |
c_h4c941uutbhb | 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 lo... | Logarithm function |
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