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wiki_2200_chunk_0 | Miracle flatness | In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University (Nagata 1962, p.217... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2201_chunk_0 | Hodge algebra | In mathematics, a Hodge algebra or algebra with straightening law is a commutative algebra that is a free module over some ring R, together with a given basis similar to the basis of standard monomials of the coordinate ring of a Grassmannian. Hodge algebras were introduced by Corrado De Concini, David Eisenbud, and Cl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2202_chunk_0 | Hodge filtration | In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties (even if they are singula... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2203_chunk_0 | Quasitriangular Hopf algebra | Also as a consequence of the properties of quasitriangularity, ( ϵ ⊗ 1 ) R = ( 1 ⊗ ϵ ) R = 1 ∈ H {\displaystyle (\epsilon \otimes 1)R=(1\otimes \epsilon )R=1\in H} ; moreover R − 1 = ( S ⊗ 1 ) ( R ) {\displaystyle R^{-1}=(S\otimes 1)(R)} , R = ( 1 ⊗ S ) ( R − 1 ) {\displaystyle R=(1\otimes S)(R^{-1})} , and ( S ⊗ S ) (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2204_chunk_0 | Hopf algebra | In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2205_chunk_0 | Hurwitz matrix | In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2206_chunk_0 | Hurwitz polynomial | In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. Such a polynomial must have coefficients that are positive real numbers. The te... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2207_chunk_0 | Hurwitz polynomial | P(s) is real when s is real.2. The roots of P(s) have real parts which are zero or negative.Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2208_chunk_0 | Integral quaternion | In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is H = { a + b i + c j + d k ∈ H ∣ a , b , c , d ∈ Z or a , b , c , d ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2209_chunk_0 | Integral quaternion | That is, either a, b, c, d are all integers, or they are all half-integers. H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by Adolf Hurwitz (1919). A Lipschitz quaternion (or Lipschitz integer) is a quaternion whos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2210_chunk_0 | J-structure | In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by Springer (1973) to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2211_chunk_0 | Jackson q-Bessel function | In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1906a, 1906b, 1905a, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2212_chunk_0 | Jacobi sum | In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime number p, defined by J ( χ , ψ ) = ∑ χ ( a ) ψ ( 1 − a ) , {\displaystyle J(\chi ,\psi )=\sum \chi (a)\psi (1-a)\,,} where the summation run... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2213_chunk_0 | Jacobi sum | Jacobi sums J can be factored generically into products of powers of Gauss sums g. For example, when the character χψ is nontrivial, J ( χ , ψ ) = g ( χ ) g ( ψ ) g ( χ ψ ) , {\displaystyle J(\chi ,\psi )={\frac {g(\chi )g(\psi )}{g(\chi \psi )}}\,,} analogous to the formula for the beta function in terms of gamma func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2214_chunk_0 | Jacobi sum | Like Gauss sums, Jacobi sums have known prime ideal factorisations in their cyclotomic fields; see Stickelberger's theorem. When χ is the Legendre symbol, J ( χ , χ ) = − χ ( − 1 ) = ( − 1 ) p + 1 2 . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2215_chunk_0 | Jacobi sum | {\displaystyle J(\chi ,\chi )=-\chi (-1)=(-1)^{\frac {p+1}{2}}\,.} In general the values of Jacobi sums occur in relation with the local zeta-functions of diagonal forms. The result on the Legendre symbol amounts to the formula p + 1 for the number of points on a conic section that is a projective line over the field o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2216_chunk_0 | Jacobi sum | A paper of André Weil from 1949 very much revived the subject. Indeed, through the Hasse–Davenport relation of the late 20th century, the formal properties of powers of Gauss sums had become current once more. As well as pointing out the possibility of writing down local zeta-functions for diagonal hypersurfaces by mea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2217_chunk_0 | Jacobian | In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: Jacobian matrix and determinant Jacobian elliptic functions Jacobian variety Intermediate Jacobian | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2218_chunk_0 | Jaffard ring | A Jaffard ring that is also an integral domain is called a Jaffard domain. The Jaffard property is satisfied by any Noetherian ring R, and examples of non-Noetherian rings might appear to be quite difficult to find, however they do arise naturally. For example, the ring of (all) algebraic integers, or more generally, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2219_chunk_0 | Jónsson–Tarski algebra | In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X×X. They were introduced by Bjarni Jónsson and Alfred Tarski (1961, Theorem 5). Smirnov (1971), named them after Georg Cantor because of Cantor's pairing function and Cantor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2220_chunk_0 | Kac algebra | In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2221_chunk_0 | Kempe chain | In mathematics, a Kempe chain is a device used mainly in the study of the four colour theorem. Intuitively, it is a connected chain of points on a graph with alternating colors. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2222_chunk_0 | Killing tensor | In mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields. It is a concept in pseudo-Riemannian geometry, and is mainly used in the theory of general relativity. Killing tensors satisfy an equation similar to Killing's equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2223_chunk_0 | Killing tensor | Like Killing vectors, every Killing tensor corresponds to a quantity which is conserved along geodesics. However, unlike Killing vectors, which are associated with symmetries (isometries) of a manifold, Killing tensors generally lack such a direct geometric interpretation. Killing tensors are named after Wilhelm Killin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2224_chunk_0 | Killing vectors | In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2225_chunk_0 | Regular algebra | In mathematics, a Kleene algebra ( KLAY-nee; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2226_chunk_0 | Klein geometry | In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry. For background and motivation see the article on ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2227_chunk_0 | Kleinian group | In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2228_chunk_0 | Kline sphere characterization | In mathematics, a Kline sphere characterization, named after John Robert Kline, is a topological characterization of a two-dimensional sphere in terms of what sort of subset separates it. Its proof was one of the first notable accomplishments of R. H. Bing; Bing gave an alternate proof using brick partitioning in his p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2229_chunk_0 | Kochanek–Bartels spline | In mathematics, a Kochanek–Bartels spline or Kochanek–Bartels curve is a cubic Hermite spline with tension, bias, and continuity parameters defined to change the behavior of the tangents. Given n + 1 knots, p0, ..., pn,to be interpolated with n cubic Hermite curve segments, for each curve we have a starting point pi an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2230_chunk_0 | Koecher–Maass series | In mathematics, a Koecher–Maass series is a type of Dirichlet series that can be expressed as a Mellin transform of a Siegel modular form, generalizing Hecke's method of associating a Dirichlet series to a modular form using Mellin transforms. They were introduced by Koecher (1953) and Maass (1950). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2231_chunk_0 | Kolmogorov system | In mathematics, a Kolmogorov automorphism, K-automorphism, K-shift or K-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law. All Bernoulli automorphisms are K-automorphisms (one says they have the K-property), but not vice versa. Many erg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2232_chunk_0 | Koszul–Tate resolution | In mathematics, a Koszul–Tate resolution or Koszul–Tate complex of the quotient ring R/M is a projective resolution of it as an R-module which also has a structure of a dg-algebra over R, where R is a commutative ring and M ⊂ R is an ideal. They were introduced by Tate (1957) as a generalization of the Koszul resolutio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2233_chunk_0 | Lagrangian system | In mathematics, a Lagrangian system is a pair (Y, L), consisting of a smooth fiber bundle Y → X and a Lagrangian density L, which yields the Euler–Lagrange differential operator acting on sections of Y → X. In classical mechanics, many dynamical systems are Lagrangian systems. The configuration space of such a Lagrangi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2234_chunk_0 | Laguerre plane | In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre. The classical Laguerre plane is an incidence structure that describes the incidence behaviour of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2235_chunk_0 | Lambert series | In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form S ( q ) = ∑ n = 1 ∞ a n q n 1 − q n . {\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}.} It can be resumed formally by expanding the denominator: S ( q ) = ∑ n = 1 ∞ a n ∑ k = 1 ∞ q n k = ∑ m = 1 ∞ b m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2236_chunk_0 | Lamé function | In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper (Gabriel Lamé 1837). Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2237_chunk_0 | Laurent polynomials | In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field F {\displaystyle \mathbb {F} } is a linear combination of positive and negative powers of the variable with coefficients in F {\displaystyle \mathbb {F} } . Laurent polynomials in X form a ring denoted F {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2238_chunk_0 | Lebrun manifold | In mathematics, a LeBrun manifold is a connected sum of copies of the complex projective plane, equipped with an explicit self-dual metric. Here, self-dual means that the Weyl tensor is its own Hodge star. The metric is determined by the choice of a finite collection of points in hyperbolic 3-space. These metrics were ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2239_chunk_0 | Leavitt path algebra | In mathematics, a Leavitt path algebra is a universal algebra constructed from a directed graph. Leavitt path algebras generalize Leavitt algebras and may be considered as algebraic analogues of graph C*-algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2240_chunk_0 | Lefschetz manifold | In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold ( M 2 n , ω ) {\displaystyle (M^{2n},\omega )} , sharing a certain cohomological property with Kähler manifolds, that of satisfying the conclusion of the Hard Lefschetz theorem. More precisely, the strong Lefschetz property asks that for k... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2241_chunk_0 | Lefschetz pencil | In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2242_chunk_0 | Leray cover | On the other hand, Čech cohomology with respect to an open cover is well-suited to calculation, but of limited usefulness because it depends on the open cover chosen, not only on the sheaves and the space. By taking a direct limit of Čech cohomology over arbitrarily fine covers, we obtain a Čech cohomology theory that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2243_chunk_0 | Lidstone series | In mathematics, a Lidstone series, named after George James Lidstone, is a kind of polynomial expansion that can express certain types of entire functions. Let ƒ(z) be an entire function of exponential type less than (N + 1)π, as defined below. Then ƒ(z) can be expanded in terms of polynomials An as follows: f ( z ) = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2244_chunk_0 | Lidstone series | {\displaystyle f(z)=\sum _{n=0}^{\infty }\left+\sum _{k=1}^{N}C_{k}\sin(k\pi z).} Here An(z) is a polynomial in z of degree n, Ck a constant, and ƒ(n)(a) the nth derivative of ƒ at a. A function is said to be of exponential type of less than t if the function h ( θ ; f ) = lim sup r → ∞ 1 r log | f ( r e i θ ) | {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2245_chunk_0 | Center of a Lie algebra | In mathematics, a Lie algebra (pronounced LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an alternating bilinear map g × g → g {\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}} , that satisfies the Jacobi identity. Otherwise sai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2246_chunk_0 | Center of a Lie algebra | Given an associative algebra (like for example the space of square matrices), a Lie bracket can be and is often defined through the commutator, namely defining = x y − y x {\displaystyle =xy-yx} correctly defines a Lie bracket in addition to the already existing multiplication operation. Lie algebras are closely relat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2247_chunk_0 | Center of a Lie algebra | This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2248_chunk_0 | Center of a Lie algebra | Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example (that is not derived from an associative algebra) is the space of three dimensional vectors g = R 3 {\displaystyle {\mathfrak {g}}=\mathbb {R} ^{3}} with the Lie bracket ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2249_chunk_0 | Center of a Lie algebra | This is skew-symmetric since x × y = − y × x {\displaystyle x\times y=-y\times x} , and instead of associativity it satisfies the Jacobi identity: x × ( y × z ) = ( x × y ) × z + y × ( x × z ) . {\displaystyle x\times (y\times z)\ =\ (x\times y)\times z\ +\ y\times (x\times z).} This is the Lie algebra of the Lie group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2250_chunk_0 | Nilpotent Lie algebra | In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras g ≥ ≥ ] ≥ ] ] ≥ . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2251_chunk_0 | Nilpotent Lie algebra | . {\displaystyle {\mathfrak {g}}\geq \geq ]\geq ]]\geq ...} We write g 0 = g {\displaystyle {\mathfrak {g}}_{0}={\mathfrak {g}}} , and g n = {\displaystyle {\mathfrak {g}}_{n}=} for all n > 0 {\displaystyle n>0} . If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nil... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2252_chunk_0 | Nilpotent Lie algebra | The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups. The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions. Note that the definition means ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2253_chunk_0 | Derived Lie algebra | In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra g {\displaystyle {\mathfrak {g}}} is the subalgebra of g {\displaystyle {\mathfrak {g}}} , denoted {\displaystyle } that consists of all linear... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2254_chunk_0 | Derived Lie algebra | . {\displaystyle {\mathfrak {g}}\geq \geq ,]\geq ,],,]]\geq ...} If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analog... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2255_chunk_0 | Derived Lie algebra | Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. The solvable Lie algebras are precisely those that can be obtained from semidirect products,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2256_chunk_0 | Reductive Lie algebra | In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: g = s ⊕ a ; {\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};} t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2257_chunk_0 | Semi-simple Lie group | In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2258_chunk_0 | Lie algebroid | In mathematics, a Lie algebroid is a vector bundle A → M {\displaystyle A\rightarrow M} together with a Lie bracket on its space of sections Γ ( A ) {\displaystyle \Gamma (A)} and a vector bundle morphism ρ: A → T M {\displaystyle \rho :A\rightarrow TM} , satisfying a Leibniz rule. A Lie algebroid can thus be thought o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2259_chunk_0 | Lie bialgebra | In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2260_chunk_0 | Infinite dimensional Lie group | Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group SO ( 3 ) {\displaystyle {\text{SO}}(3)} ). Lie groups are widely used in many parts of modern mathematics and physics. Lie groups... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2261_chunk_0 | Lie superalgebra | In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2‑grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2262_chunk_0 | Lie-* algebra | In mathematics, a Lie-* algebra is a D-module with a Lie* bracket. They were introduced by Alexander Beilinson and Vladimir Drinfeld (Beilinson & Drinfeld (2004, section 2.5.3)), and are similar to the conformal algebras discussed by Kac (1998) and to vertex Lie algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2263_chunk_0 | Lipschitz domain | In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The term is named after the German mathematician Rudolf Lipschitz. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2264_chunk_0 | Lucas chain | In mathematics, a Lucas chain is a restricted type of addition chain, named for the French mathematician Édouard Lucas. It is a sequence a0, a1, a2, a3, ...that satisfies a0=1,and for each k > 0: ak = ai + aj, and either ai = aj or |ai − aj| = am, for some i, j, m < k.The sequence of powers of 2 (1, 2, 4, 8, 16, ...) a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2265_chunk_0 | Luzin set | In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense subset is countable. There are many minor variations of this definition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a counta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2266_chunk_0 | Macbeath region | In mathematics, a Macbeath region is an explicitly defined region in convex analysis on a bounded convex subset of d-dimensional Euclidean space R d {\displaystyle \mathbb {R} ^{d}} . The idea was introduced by Alexander Macbeath (1952) and dubbed by G. Ewald, D. G. Larman and C. A. Rogers in 1970. Macbeath regions hav... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2267_chunk_0 | Madhava series | In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century Kerala by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2268_chunk_0 | Madhava series | + θ 5 5 ! − θ 7 7 ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2269_chunk_0 | Madhava series | + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! θ 2 k + 1 , cos θ = 1 − θ 2 2 ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2270_chunk_0 | Madhava series | + θ 4 4 ! − θ 6 6 ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2271_chunk_0 | Madhava series | θ 2 k , arctan x = x − x 3 3 + x 5 5 − x 7 7 + ⋯ = ∑ k = 0 ∞ ( − 1 ) k 2 k + 1 x 2 k + 1 where | x | ≤ 1. {\displaystyle {\begin{alignedat}{3}\sin \theta &=\theta -{\frac {\theta ^{3}}{3! }}+{\frac {\theta ^{5}}{5! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2272_chunk_0 | Madhava series | }}-{\frac {\theta ^{7}}{7! }}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)! }}\theta ^{2k+1},\\\cos \theta &=1-{\frac {\theta ^{2}}{2! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2273_chunk_0 | Madhava series | }}+{\frac {\theta ^{4}}{4! }}-{\frac {\theta ^{6}}{6! }}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2274_chunk_0 | Madhava series | }}\theta ^{2k},\\\arctan x&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}x^{2k+1}\quad {\text{where }}|x|\leq 1.\end{alignedat}}} All three series were later independently discovered in 17th century Europe. The series for sine and cosine were rediscov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2275_chunk_0 | Madhava series | In recognition of Madhava's priority, in recent literature these series are sometimes called the Madhava–Newton series, Madhava–Gregory series, or Madhava–Leibniz series (among other combinations).No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2276_chunk_0 | Maharam algebra | In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam (1947). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2277_chunk_0 | Malcev Lie algebra | In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar. Both were introduced by Quillen (1969, Appendix A3), based on the work of (Mal'cev 1949). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2278_chunk_0 | Malcev algebra | In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that x y = − y x {\displaystyle xy=-yx} and satisfies the Malcev identity ( x y ) ( x z ) = ( ( x y ) z ) x + ( ( y z ) x ) x + ( ( z x ) x ) y . {\displaystyle (xy)(xz)=((xy)z... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2279_chunk_0 | Malcev algebra | Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2280_chunk_0 | Markov Decision Process | In mathematics, a Markov decision process (MDP) is a discrete-time stochastic control process. It provides a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control of a decision maker. MDPs are useful for studying optimization problems solved via ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2281_chunk_0 | Markov Decision Process | The process responds at the next time step by randomly moving into a new state s ′ {\displaystyle s'} , and giving the decision maker a corresponding reward R a ( s , s ′ ) {\displaystyle R_{a}(s,s')} . The probability that the process moves into its new state s ′ {\displaystyle s'} is influenced by the chosen action. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2282_chunk_0 | Markov Decision Process | Thus, the next state s ′ {\displaystyle s'} depends on the current state s {\displaystyle s} and the decision maker's action a {\displaystyle a} . But given s {\displaystyle s} and a {\displaystyle a} , it is conditionally independent of all previous states and actions; in other words, the state transitions of an MDP s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2283_chunk_0 | Markov odometer | In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation is orbit-equivalent to a Markov odometer.The basic examp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2284_chunk_0 | Mersenne numbers | Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2285_chunk_0 | Metzler matrix | In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): ∀ i ≠ j x i j ≥ 0. {\displaystyle \forall _{i\neq j}\,x_{ij}\geq 0.} It is named after the American economist Lloyd Metzler. Metzler matrices appear in stability analysis of time delaye... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2286_chunk_0 | Moishezon manifold | {\displaystyle \dim _{\mathbf {C} }M=a(M)=\operatorname {tr.deg.} _{\mathbf {C} }\mathbf {C} (M).} Complex algebraic varieties have this property, but the converse is not true: Hironaka's example gives a smooth 3-dimensional Moishezon manifold that is not an algebraic variety or scheme. Moishezon (1967, Chapter I, Theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2287_chunk_0 | Algebraic hyperbolicity | In mathematics, a Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge Lang to enunciate a range of conjectures linking the geometry of varieties to their Diophantine properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2288_chunk_0 | Moufang loop | In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang (1935). Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2289_chunk_0 | Nekrasov matrix | In mathematics, a Nekrasov matrix or generalised Nekrasov matrix is a type of diagonally dominant matrix (i.e. one in which the diagonal elements are in some way greater than some function of the non-diagonal elements). Specifically if A is a generalised Nekrasov matrix, its diagonal elements are non-zero and the diago... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2290_chunk_0 | Table of Newtonian series | In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a n {\displaystyle a_{n}} written in the form f ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( s n ) a n = ∑ n = 0 ∞ ( − s ) n n ! a n {\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n}}{n! }}a_{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2291_chunk_0 | Niemeier lattice | In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Hans-Volker Niemeier (1973). Venkov (1978) gave a simplified proof of the classification. In the 1970s, Witt (1941) has a sentence mentioning that he found more than 10 such lattices in th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2292_chunk_0 | Noetherian rings | A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields),... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2293_chunk_0 | Nori-semistable vector bundle | In mathematics, a Nori semistable vector bundle is a particular type of vector bundle whose first definition has been first implicitly suggested by Madhav V. Nori, as one of the main ingredients for the construction of the fundamental group scheme. The original definition given by Nori was obviously not called Nori sem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2294_chunk_0 | P-matrix | In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P 0 {\displaystyle P_{0}} -matrices, which are the closure of the class of P-matrices, with every principal minor ≥ {\displaystyle \geq } 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2295_chunk_0 | Padé approximant | In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2296_chunk_0 | Padé approximant | For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since Padé approxima... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2297_chunk_0 | Paley–Wiener theorem | In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2298_chunk_0 | Perron number | In mathematics, a Perron number is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than α in absolute value. For example, the larger of the two roots of the irreducible polynomial x 2 − 3 x + 1 {\displaystyle x^{2}-3x+1} is a Perron number. Perron numbers are named ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2299_chunk_0 | Petersson algebra | In mathematics, a Petersson algebra is a composition algebra over a field constructed from an order-3 automorphism of a Hurwitz algebra. They were first constructed by Petersson (1969). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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