id stringlengths 14 19 | title stringlengths 1 124 | text stringlengths 12 2.83k | source stringclasses 1
value |
|---|---|---|---|
wiki_7000_chunk_0 | Maximum modulus principle | In mathematics, the maximum modulus principle in complex analysis states that if f {\displaystyle f} is a holomorphic function, then the modulus | f | {\displaystyle |f|} cannot exhibit a strict local maximum that is properly within the domain of f {\displaystyle f} . In other words, either f {\displaystyle f} is local... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7001_chunk_0 | Max–min inequality | In mathematics, the max–min inequality is as follows: For any function f: Z × W → R , {\displaystyle \ f:Z\times W\to \mathbb {R} \ ,} sup z ∈ Z inf w ∈ W f ( z , w ) ≤ inf w ∈ W sup z ∈ Z f ( z , w ) . {\displaystyle \sup _{z\in Z}\inf _{w\in W}f(z,w)\leq \inf _{w\in W}\sup _{z\in Z}f(z,w)\ .} When equality holds one ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7002_chunk_0 | Mean dimension | In mathematics, the mean (topological) dimension of a topological dynamical system is a non-negative extended real number that is a measure of the complexity of the system. Mean dimension was first introduced in 1999 by Gromov. Shortly after it was developed and studied systematically by Lindenstrauss and Weiss. In par... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7003_chunk_0 | Mean dimension | For various topological dynamical systems with infinite topological entropy, the mean dimension can be calculated or at least bounded from below and above. This allows mean dimension to be used to distinguish between systems with infinite topological entropy. Mean dimension is also related to the problem of embedding t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7004_chunk_0 | Mean curvature | In mathematics, the mean curvature H {\displaystyle H} of a surface S {\displaystyle S} is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7005_chunk_0 | Mean value problem | In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks: For a given complex polynomial f {\displaystyle f} of degree d ≥ 2 {\displaystyle d\geq 2} A and a complex number z {\displaystyle z} , is there a critical point c {\displaystyle c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7006_chunk_0 | Mean-value theorem | In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7007_chunk_0 | Measurable Riemann mapping theorem | In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct generalization of the Riemann mapping theorem, but instead a result concerning quasiconformal mappings and soluti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7008_chunk_0 | Mediant (mathematics) | However, if the fraction 1/1 is replaced by the fraction 2/2, which is an equivalent fraction denoting the same rational number 1, the mediant of the fractions 2/2 and 1/2 is 3/4. For a stronger connection to rational numbers the fractions may be required to be reduced to lowest terms, thereby selecting unique represen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7009_chunk_0 | Fuzzy membership | In mathematics, the membership function of a fuzzy set is a generalization of the indicator function for classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7010_chunk_0 | Metaplectic group | In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infini... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7011_chunk_0 | Indicial equation | In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form with u ′ ≡ d u d z {\textstyle u'\equiv {\frac {du}{dz}}} and u ″ ≡ d 2 u d z 2 {\textstyle u''\equiv {\frac {d^{2}u}{dz^{2}}}} . in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7012_chunk_0 | Charpit method | In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7013_chunk_0 | Clearing denominators | In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7014_chunk_0 | Hadamard's method of descent | In mathematics, the method of descent is the term coined by the French mathematician Jacques Hadamard as a method for solving a partial differential equation in several real or complex variables, by regarding it as the specialisation of an equation in more variables, constant in the extra parameters. This method has be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7015_chunk_0 | Method of dominant balance | In mathematics, the method of dominant balance is used to determine the asymptotic behavior of solutions to an ordinary differential equation without fully solving the equation. The process is iterative, in that the result obtained by performing the method once can be used as input when the method is repeated, to obtai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7016_chunk_0 | Equating coefficients | In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters. It relies on the fact that two expressions are identical precisely when corresponding coefficients are equal for each different type of term. The m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7017_chunk_0 | Method of matched asymptotic expansions | In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used when solving singularly perturbed differential equations. It involves finding several different approximate solutions, eac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7018_chunk_0 | Saddle-point method | In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7019_chunk_0 | Saddle-point method | One version of the method of steepest descent deforms the contour of integration C into a new path integration C′ so that the following conditions hold: C′ passes through one or more zeros of the derivative g′(z), the imaginary part of g(z) is constant on C′.The method of steepest descent was first published by Debye (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7020_chunk_0 | Method of undetermined coefficients | In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7021_chunk_0 | Metric derivative | In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a notion of distance (i.e. metric spaces) but not direction (such as vector spaces). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7022_chunk_0 | Mex (mathematics) | Minimum excluded values of subclasses of the ordinal numbers are used in combinatorial game theory to assign nim-values to impartial games. According to the Sprague–Grundy theorem, the nim-value of a game position is the minimum excluded value of the class of values of the positions that can be reached in a single move... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7023_chunk_0 | Tensor (intrinsic definition) | In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7024_chunk_0 | Tensor (intrinsic definition) | The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally. Note: This article assumes an understanding of the tensor product of vector spaces without chosen base... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7025_chunk_0 | Elliptic modulus | In mathematics, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7026_chunk_0 | Modulus and characteristic of convexity | In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7027_chunk_0 | Moment (statistics) | In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7028_chunk_0 | Moment (statistics) | For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from 0 to ∞) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7029_chunk_0 | Monkey saddle | In mathematics, the monkey saddle is the surface defined by the equation z = x 3 − 3 x y 2 , {\displaystyle z=x^{3}-3xy^{2},\,} or in cylindrical coordinates z = ρ 3 cos ( 3 φ ) . {\displaystyle z=\rho ^{3}\cos(3\varphi ).} It belongs to the class of saddle surfaces, and its name derives from the observation that a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7030_chunk_0 | Monkey saddle | The point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} on the monkey saddle corresponds to a degenerate critical point of the function z ( x , y ) {\displaystyle z(x,y)} at ( 0 , 0 ) {\displaystyle (0,0)} . The monkey saddle has an isolated umbilical point with zero Gaussian curvature at the origin, while the curvature is str... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7031_chunk_0 | Monkey saddle | {\displaystyle z=x^{3}-3xy^{2}=\operatorname {Re} =\operatorname {Re} =r^{3}\cos(3\varphi ).} By replacing 3 in the cylindrical equation with any integer k ≥ 1 , {\displaystyle k\geq 1,} one can create a saddle with k {\displaystyle k} depressions. Another orientation of the monkey saddle is the Smelt petal defined by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7032_chunk_0 | Monster Lie algebra | In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7033_chunk_0 | Mountain climbing problem | In mathematics, the mountain climbing problem is a mathematical problem that considers a two-dimensional mountain range (represented as a continuous function), and asks whether it is possible for two mountain climbers starting at sea level on the left and right sides of the mountain to meet at the summit, while maintai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7034_chunk_0 | Multicomplex number | In the multicomplex number systems one also requires that i n i m = i m i n {\displaystyle i_{n}i_{m}=i_{m}i_{n}} (commutativity). Then C 1 {\displaystyle \mathbb {C} _{1}} is the complex number system, C 2 {\displaystyle \mathbb {C} _{2}} is the bicomplex number system, C 3 {\displaystyle \mathbb {C} _{3}} is the tric... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7035_chunk_0 | Multicomplex number | {\displaystyle \mathbb {C} _{n}.} The multicomplex number systems are not to be confused with Clifford numbers (elements of a Clifford algebra), since Clifford's square roots of −1 anti-commute ( i n i m + i m i n = 0 {\displaystyle i_{n}i_{m}+i_{m}i_{n}=0} when m ≠ n for Clifford). Because the multicomplex numbers hav... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7036_chunk_0 | Multinomial formula | In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7037_chunk_0 | Multiple gamma function | In mathematics, the multiple gamma function Γ N {\displaystyle \Gamma _{N}} is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by Barnes (1901). At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied thes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7038_chunk_0 | Multiple orthogonal polynomials | In mathematics, the multiple orthogonal polynomials (MOPs) are orthogonal polynomials in one variable that are orthogonal with respect to a finite family of measures. The polynomials are divided into two classes named type 1 and type 2.In the literature, MOPs are also called d {\displaystyle d} -orthogonal polynomials,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7039_chunk_0 | Multiple zeta values | In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by ζ ( s 1 , … , s k ) = ∑ n 1 > n 2 > ⋯ > n k > 0 1 n 1 s 1 ⋯ n k s k = ∑ n 1 > n 2 > ⋯ > n k > 0 ∏ i = 1 k 1 n i s i , {\displaystyle \zeta (s_{1},\ldots ,s_{k})=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ {\frac {1}{n_{1}^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7040_chunk_0 | Multiplication theorem | In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7041_chunk_0 | Oseledets theorem | In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets (also spelled "Oseledec") in 1965 and reported at the International Mathematical Congress in Moscow in 196... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7042_chunk_0 | Multiplier algebra | In mathematics, the multiplier algebra, denoted by M(A), of a C*-algebra A is a unital C*-algebra that is the largest unital C*-algebra that contains A as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification. Multiplier algebras were introduced by Busby (1968). For ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7043_chunk_0 | Multivariate gamma function | In mathematics, the multivariate gamma function Γp is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate beta distribution.It has two equivalent definitions. One is given ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7044_chunk_0 | Multivariate gamma function | The other one, more useful to obtain a numerical result is: Γ p ( a ) = π p ( p − 1 ) / 4 ∏ j = 1 p Γ ( a + ( 1 − j ) / 2 ) . {\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+(1-j)/2).} In both definitions, a {\displaystyle a} is a complex number whose real part satisfies ℜ ( a ) > ( p − 1 ) / 2 {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7045_chunk_0 | Multivariate gamma function | Note that Γ 1 ( a ) {\displaystyle \Gamma _{1}(a)} reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for p ≥ 2 {\displaystyle p\geq 2}: Γ p ( a ) = π ( p − 1 ) / 2 Γ ( a ) Γ p − 1 ( a − 1 2 ) = π ( p − 1 ) / 2 Γ p − 1 ( a ) Γ ( a + ( 1 − p ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7046_chunk_0 | Multivariate gamma function | Thus Γ 2 ( a ) = π 1 / 2 Γ ( a ) Γ ( a − 1 / 2 ) {\displaystyle \Gamma _{2}(a)=\pi ^{1/2}\Gamma (a)\Gamma (a-1/2)} Γ 3 ( a ) = π 3 / 2 Γ ( a ) Γ ( a − 1 / 2 ) Γ ( a − 1 ) {\displaystyle \Gamma _{3}(a)=\pi ^{3/2}\Gamma (a)\Gamma (a-1/2)\Gamma (a-1)} and so on. This can also be extended to non-integer values of p {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7047_chunk_0 | Complex coordinate space | In mathematics, the n-dimensional complex coordinate space (or complex n-space) is the set of all ordered n-tuples of complex numbers. It is denoted C n {\displaystyle \mathbb {C} ^{n}} , and is the n-fold Cartesian product of the complex plane C {\displaystyle \mathbb {C} } with itself. Symbolically, or The variables ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7048_chunk_0 | Complex coordinate space | The real and imaginary parts of the coordinates set up a bijection of C n {\displaystyle \mathbb {C} ^{n}} with the 2n-dimensional real coordinate space, R 2 n {\displaystyle \mathbb {R} ^{2n}} . With the standard Euclidean topology, C n {\displaystyle \mathbb {C} ^{n}} is a topological vector space over the complex nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7049_chunk_0 | Complex coordinate space | A function on an open subset of complex n-space is holomorphic if it is holomorphic in each complex coordinate separately. Several complex variables is the study of such holomorphic functions in n variables. More generally, the complex n-space is the target space for holomorphic coordinate systems on complex manifolds. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7050_chunk_0 | Symmetric product of an algebraic curve | Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients. For C the projective line (say the Riemann sphere C {\displaystyle \mathbb {C} } ∪ {∞} ≈ S2), its nth symmetric produc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7051_chunk_0 | Symmetric product of an algebraic curve | That means that at the level of function fields it is possible to construct J by taking linearly disjoint copies of the function field of C, and within their compositum taking the fixed subfield of the symmetric group. This is the source of André Weil's technique of constructing J as an abstract variety from 'birationa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7052_chunk_0 | Harmonic number | In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: Starting from n = 1, the sequence of harmonic numbers begins: Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n posit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7053_chunk_0 | Harmonic number | The harmonic numbers roughly approximate the natural logarithm function: 143 and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7054_chunk_0 | Harmonic number | When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value. The Bertrand-Chebyshev theorem implies that, exc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7055_chunk_0 | Symmetric power | In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product X n := X × ⋯ × X {\displaystyle X^{n}:=X\times \cdots \times X} by the permutation action of the symmetric group S n {\displaystyle {\mathfrak {S}}_{n}} . More precisely, the notion exists at least in the following three areas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7056_chunk_0 | Symplectic group | In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7057_chunk_0 | Counting numbers | This chain of extensions canonically embeds the natural numbers in the other number systems. Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7058_chunk_0 | Nil-Coxeter algebra | In mathematics, the nil-Coxeter algebra, introduced by Fomin & Stanley (1994), is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7059_chunk_0 | Nilpotent cone | In mathematics, the nilpotent cone N {\displaystyle {\mathcal {N}}} of a finite-dimensional semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} is the set of elements that act nilpotently in all representations of g . {\displaystyle {\mathfrak {g}}.} In other words, N = { a ∈ g: ρ ( a ) is nilpotent for all repres... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7060_chunk_0 | Nimber | In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7061_chunk_0 | Nine lemma | 201) offers a satirical view of the nine lemma: "Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but not in the squares; put them at both ends of the horizontal and vertical ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7062_chunk_0 | No-wandering-domain theorem | In mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985. The theorem states that a rational map f: Ĉ → Ĉ with deg(f) ≥ 2 does not have a wandering domain, where Ĉ denotes the Riemann sphere. More precisely, for every component U in the Fatou set of f, the sequ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7063_chunk_0 | No-wandering-domain theorem | {\displaystyle f^{n}=\underbrace {f\circ f\circ \cdots \circ f} _{n}.} The theorem does not hold for arbitrary maps; for example, the transcendental map f ( z ) = z + 2 π sin ( z ) {\displaystyle f(z)=z+2\pi \sin(z)} has wandering domains. However, the result can be generalized to many situations where the functions ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7064_chunk_0 | Noncommutative symmetric function | In mathematics, the noncommutative symmetric functions form a Hopf algebra NSymm analogous to the Hopf algebra of symmetric functions. The Hopf algebra NSymm was introduced by Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir Retakh, and Jean-Yves Thibon. It is noncommutative but cocommutative gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7065_chunk_0 | Nonmetricity tensor | In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a tensor field of order three. It vanishes for the case of Riemannian geometry and can be used to study non-Riemannian spacetimes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7066_chunk_0 | Merkurjev–Suslin theorem | In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7067_chunk_0 | Merkurjev–Suslin theorem | John Milnor speculated that this theorem might be true for ℓ = 2 {\displaystyle \ell =2} and all n {\displaystyle n} , and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjectu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7068_chunk_0 | Cyclic and separating vector | In mathematics, the notion of a cyclic and separating vector is important in the theory of von Neumann algebras, and in particular in Tomita–Takesaki theory. A related notion is that of a vector which is cyclic for a given operator. The existence of cyclic vectors is guaranteed by the Gelfand–Naimark–Segal (GNS) constr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7069_chunk_0 | Quasi-continuous function | In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the converse is not true in general. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7070_chunk_0 | Split form | In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0: g ≃ g 0 ⊗ R C . {\displaystyle {\mathfrak {g}}\simeq {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7071_chunk_0 | Dimension function | In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple "diameter to the dimension" power law used in the construction of s-dimensional Hausdorff measure. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7072_chunk_0 | Cylindric algebra | In mathematics, the notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equipped with additional cylindrification operation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7073_chunk_0 | Expansivity constant | In mathematics, the notion of expansivity formalizes the notion of points moving away from one another under the action of an iterated function. The idea of expansivity is fairly rigid, as the definition of positive expansivity, below, as well as the Schwarz–Ahlfors–Pick theorem demonstrate. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7074_chunk_0 | Exterior space | In mathematics, the notion of externology in a topological space X generalizes the basic properties of the family εXcc = {E ⊆ X: X\E is a closed compact subset of X}of complements of the closed compact subspaces of X, which are used to construct its Alexandroff compactification. An externology permits to introduce a no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7075_chunk_0 | Automorphic function | In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group G {\displaystyle G} acts on a complex-analytic manifold X {\displaystyle X} . Then, G {\displaystyle G} also acts on the space of holomorphic functions from X {\displaystyle X} to the complex num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7076_chunk_0 | Automorphic function | A function f {\displaystyle f} is termed an automorphic form if the following holds: f ( g . x ) = j g ( x ) f ( x ) {\displaystyle f(g.x)=j_{g}(x)f(x)} where j g ( x ) {\displaystyle j_{g}(x)} is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant unde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7077_chunk_0 | Automorphic function | The factor of automorphy for the automorphic form f {\displaystyle f} is the function j {\displaystyle j} . An automorphic function is an automorphic form for which j {\displaystyle j} is the identity. Some facts about factors of automorphy: Every factor of automorphy is a cocycle for the action of G {\displaystyle G} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7078_chunk_0 | Automorphic function | The factor of automorphy is a coboundary if and only if it arises from an everywhere nonzero automorphic form. For a given factor of automorphy, the space of automorphic forms is a vector space. The pointwise product of two automorphic forms is an automorphic form corresponding to the product of the corresponding facto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7079_chunk_0 | Automorphic cuspidal representation | In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group G {\displaystyle G} acts on a complex-analytic manifold X {\displaystyle X} . Then, G {\displaystyle G} also acts on the space of holomorphic functions from X {\displaystyle X} to the complex num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7080_chunk_0 | Automorphic cuspidal representation | Equivalently, an automorphic form is a function whose divisor is invariant under the action of G {\displaystyle G} . The factor of automorphy for the automorphic form f {\displaystyle f} is the function j {\displaystyle j} . An automorphic function is an automorphic form for which j {\displaystyle j} is the identity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7081_chunk_0 | Automorphic cuspidal representation | An automorphic form is a function F on G (with values in some fixed finite-dimensional vector space V, in the vector-valued case), subject to three kinds of conditions: to transform under translation by elements γ ∈ Γ {\displaystyle \gamma \in \Gamma } according to the given factor of automorphy j; to be an eigenfuncti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7082_chunk_0 | Automorphic cuspidal representation | The third condition is to handle the case where G/Γ is not compact but has cusps. The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1-cocycle in the language of group cohomology. The values of j may be complex numbers, or in fact complex square matrices, corresponding to th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7083_chunk_0 | Automorphic cuspidal representation | The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when j is derived from a Jacobian matrix, by means of the chain rule. A more straightforward but technically advanced definition using class field theory, constructs automorphic forms and their correspondent functions ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7084_chunk_0 | Automorphic cuspidal representation | Herein, the analytical structure of its L-function allows for generalizations with various algebro-geometric properties; and the resultant Langlands program. To oversimplify, automorphic forms in this general perspective, are analytic functionals quantifying the invariance of number fields in a most abstract sense, the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7085_chunk_0 | Automorphic cuspidal representation | Allowing a powerful mathematical tool for analyzing the invariant constructs of virtually any numerical structure. Examples of automorphic forms in an explicit unabstracted state are difficult to obtain, though some have directly analytical properties: - The Eisenstein series (which is a prototypical modular form) over... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7086_chunk_0 | Automorphic cuspidal representation | - Generally any harmonic analytic object as a functor over Galois groups which is invariant on its ideal class group (or idele). As a general principle, automorphic forms can be thought of as analytic functions on abstract structures, which are invariant with respect to a generalized analogue of their prime ideal (or a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7087_chunk_0 | Polyconvex function | In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. Let Mm×n(K) denote the space of all m × n matrices over the field K, which may be either the real numbers R, or the complex numbers C. A function f: Mm×n(K) → R ∪ {±∞} is said to be po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7088_chunk_0 | Hemicontinuity | In mathematics, the notion of the continuity of functions is not immediately extensible to set-valued functions between two sets A and B. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A set-valued function that has both properties is said to be continuous in an analogy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7089_chunk_0 | Motzkin number | In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory. The M... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7090_chunk_0 | Nth-term test | In mathematics, the nth-term test for divergence is a simple test for the divergence of an infinite series:If lim n → ∞ a n ≠ 0 {\displaystyle \lim _{n\to \infty }a_{n}\neq 0} or if the limit does not exist, then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} diverges.Many authors do not name this test or giv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7091_chunk_0 | Cayley numbers | In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold O {\displaystyle \mathbb {O} } . Octonions have eight dimensions; twice the number of dimensions of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7092_chunk_0 | Cayley numbers | Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson construction to the octonions produces the sedenions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7093_chunk_0 | Norm topology | In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ‖ T ‖ {\displaystyle \|T\|} of a linea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7094_chunk_0 | Kirillov orbit theory | In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of the group on the dual space of its Lie algebra. The th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7095_chunk_0 | Finite order | The order of a group G is denoted by ord(G) or |G|, and the order of an element a is denoted by ord(a) or |a|, instead of ord ( ⟨ a ⟩ ) , {\displaystyle \operatorname {ord} (\langle a\rangle ),} where the brackets denote the generated group. Lagrange's theorem states that for any subgroup H of a finite group G, the o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7096_chunk_0 | Cobordism ring | In mathematics, the oriented cobordism ring is a ring where elements are oriented cobordism classes of manifolds, the multiplication is given by the Cartesian product of manifolds and the addition is given as the disjoint union of manifolds. The ring is graded by dimensions of manifolds and is denoted by Ω ∗ S O = ⊕ 0 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7097_chunk_0 | Origin (mathematics) | In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7098_chunk_0 | Complex orthogonal group | In mathematics, the orthogonal group in dimension n {\displaystyle n} , denoted O ( n ) {\displaystyle \operatorname {O} (n)} , is the group of distance-preserving transformations of a Euclidean space of dimension n {\displaystyle n} that preserve a fixed point, where the group operation is given by composing transfo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7099_chunk_0 | Complex orthogonal group | By extension, for any field F {\displaystyle F} , an n × n {\displaystyle n\times n} matrix with entries in F {\displaystyle F} such that its inverse equals its transpose is called an orthogonal matrix over F {\displaystyle F} . The n × n {\displaystyle n\times n} orthogonal matrices form a subgroup, denoted O ( n , ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.