id stringlengths 14 19 | title stringlengths 1 124 | text stringlengths 12 2.83k | source stringclasses 1
value |
|---|---|---|---|
wiki_5400_chunk_0 | Superfunction | In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the superfunction could be defined by the expression S ( z ; x ) = f ( f ( … f ( x ) … ) ) ⏟ z evaluations of the function f . {\displaystyle S(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5401_chunk_0 | Superfunction | Such a definition is valid only for a positive integer index z. The variable x is often omitted. Much study and many applications of superfunctions employ various extensions of these superfunctions to complex and continuous indices; and the analysis of existence, uniqueness and their evaluation. The Ackermann functions... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5402_chunk_0 | Global spherical shell | The name "class VII" comes from (Kodaira 1964, theorem 21), which divided minimal surfaces into 7 classes numbered I0 to VII0. However Kodaira's class VII0 did not have the condition that the Kodaira dimension is −∞, but instead had the condition that the geometric genus is 0. As a result, his class VII0 also included ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5403_chunk_0 | Jordan frame (Jordan algebra) | In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5404_chunk_0 | Symmetric convolution | In mathematics, symmetric convolution is a special subset of convolution operations in which the convolution kernel is symmetric across its zero point. Many common convolution-based processes such as Gaussian blur and taking the derivative of a signal in frequency-space are symmetric and this property can be exploited ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5405_chunk_0 | Symmetric map | In mathematics, symmetrization is a process that converts any function in n {\displaystyle n} variables to a symmetric function in n {\displaystyle n} variables. Similarly, antisymmetrization converts any function in n {\displaystyle n} variables into an antisymmetric function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5406_chunk_0 | Synthetic differential geometry | In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5407_chunk_0 | Synthetic differential geometry | The third insight is that over a certain category, these are representable functors. Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used. Synthetic differential geometry can serve as a platform for formulating certain otherwise obscure or con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5408_chunk_0 | Systolic geometry | In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5409_chunk_0 | Systoles of surfaces | In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5410_chunk_0 | Trigonometric tables | In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of study, which led to the development of the first mec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5411_chunk_0 | Trigonometric tables | Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of trigonometric functions is still used in computer graphics, where only modest accuracy may be required and speed is often paramount. Another... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5412_chunk_0 | Trigonometric tables | In this case, calling generic library routines every time is unacceptably slow. One option is to call the library routines once, to build up a table of those trigonometric values that will be needed, but this requires significant memory to store the table. The other possibility, since a regular sequence of values is re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5413_chunk_0 | Seventh root | When complex nth roots are considered, it is often useful to choose one of the roots, called principal root, as a principal value. The common choice is to choose the principal nth root of x as the nth root with the greatest real part, and when there are two (for x real and negative), the one with a positive imaginary p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5414_chunk_0 | Seventh root | Roots can also be defined as special cases of exponentiation, where the exponent is a fraction: x n = x 1 / n . {\displaystyle {\sqrt{x}}=x^{1/n}.} Roots are used for determining the radius of convergence of a power series with the root test. The nth roots of 1 are called roots of unity and play a fundamental role in v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5415_chunk_0 | Rummler–Sullivan theorem | In mathematics, tautness is a rigidity property of foliations. A taut foliation is a codimension 1 foliation of a closed manifold with the property that every leaf meets a transverse circle. : 155 By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. If the foliat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5416_chunk_0 | Rummler–Sullivan theorem | Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Furthermore, for compact manifolds the existence, for every leaf L {\displaystyle L} , of a transverse circle meeting L {\displaystyle L} , implies the existence of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5417_chunk_0 | Tensor calculus | In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5418_chunk_0 | Tensor calculus | Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5419_chunk_0 | Tensor calculus | In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5420_chunk_0 | Iterated exponentiation | In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though ↑↑ {\displaystyle \uparrow \uparrow } and the left-exponent xb are common. Under the definition as repeated exponentiation, n a {\displaystyle {^{n}a}} means a a ⋅ ⋅... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5421_chunk_0 | Iterated exponentiation | Tetration is also defined recursively as a ↑↑ n := { 1 if n = 0 , a a ↑↑ ( n − 1 ) if n > 0 , {\displaystyle {a\uparrow \uparrow n}:={\begin{cases}1&{\text{if }}n=0,\\a^{a\uparrow \uparrow (n-1)}&{\text{if }}n>0,\end{cases}}} allowing for attempts to extend tetration to non-natural numbers such as real and complex numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5422_chunk_0 | Erdős–Szekeres conjecture | In mathematics, the "happy ending problem" (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther Klein) is the following statement: This was one of the original results that led to the development of Ramsey theory. The happy ending theorem can be proven by a simple case analysis: if four ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5423_chunk_0 | Erdős–Szekeres conjecture | See Peterson (2000) for an illustrated explanation of this proof, and Morris & Soltan (2000) for a more detailed survey of the problem. The Erdős–Szekeres conjecture states precisely a more general relationship between the number of points in a general-position point set and its largest subset forming a convex polygon,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5424_chunk_0 | Extension (semantics) | In mathematics, the 'extension' of a mathematical concept C {\displaystyle C} is the set that is specified by C {\displaystyle C} . (That set might be empty, currently.) For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5425_chunk_0 | Extension (semantics) | The extension of a set is the set itself. That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory. This kind of extension is used so constantly in contemporary mathematics based on set theory that it can be called an implicit assumption. A ty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5426_chunk_0 | Shift theorem | In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (D-operators) and exponential functions. It permits one to eliminate, in certain cases, the exponential from under the D-operators. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5427_chunk_0 | Peetre theorem | In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finite... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5428_chunk_0 | Lah number | In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were discovered by Ivo Lah in 1954. Explicitly, the unsigned Lah numbers L ( n , k ) {\displaystyle L(n,k)} are given by the formula involving the binomial coefficient ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5429_chunk_0 | 15 and 290 theorems | In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated, and was ne... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5430_chunk_0 | 2π theorem | In mathematics, the 2π theorem of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let M be a cusped hyperbolic 3-manifold. Disjoint horoball neighborhoods of each cusp can be selected. The boundaries of these neighborhoods... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5431_chunk_0 | 2π theorem | A slope, i.e. unoriented isotopy class of simple closed curves on these boundaries, thus has a well-defined length by taking the minimal Euclidean length over all curves in the isotopy class. The 2π theorem states: a Dehn filling of M with each filling slope greater than 2π results in a 3-manifold with a complete metri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5432_chunk_0 | 2π theorem | The basic idea of the proof is to explicitly construct a negatively curved metric inside each horoball neighborhood that matches the metric near the horospherical boundary. This construction, using cylindrical coordinates, works when the filling slope is greater than 2π. See Bleiler & Hodgson (1996) for complete detail... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5433_chunk_0 | 2π theorem | According to the geometrization conjecture, these negatively curved 3-manifolds must actually admit a complete hyperbolic metric. A horoball packing argument due to Thurston shows that there are at most 48 slopes to avoid on each cusp to get a hyperbolic 3-manifold. For one-cusped hyperbolic 3-manifolds, an improvement... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5434_chunk_0 | 2π theorem | This result was later improved independently by Ian Agol (2000) and Marc Lackenby (2000) with the 6 theorem. The "6 theorem" states that Dehn filling along slopes of length greater than 6 results in a hyperbolike 3-manifold, i.e. an irreducible, atoroidal, non-Seifert-fibered 3-manifold with infinite word hyperbolic fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5435_chunk_0 | 57-cell | In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. The symmetry order is 3420, from the product of the number of cells (57) and the symmetry of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5436_chunk_0 | 57-cell | The symmetry abstract structure is the projective special linear group, L2(19). It has Schläfli type {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by H. S. M. Coxeter (1982). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5437_chunk_0 | ATS theorem | In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5438_chunk_0 | Abel transform | In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by F ( y ) = 2 ∫ y ∞ f ( r ) r r 2 − y 2 d r . {\displaystyle F(y)=2\int _{y}^{\infty }{\frac {f(r)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5439_chunk_0 | Abel transform | {\displaystyle f(r)=-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}\,{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.} In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5440_chunk_0 | Abel–Jacobi theorem | In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are lin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5441_chunk_0 | Abel–Plana formula | In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It states that ∑ n = 0 ∞ f ( a + n ) = ∫ a ∞ f ( x ) d x + f ( a ) 2 + ∫ 0 ∞ f ( a − i x ) − f ( a + i x ) i ( e 2 π x − 1 ) d x {\displaystyle \sum _{n=0}^{\infty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5442_chunk_0 | Abel–Ruffini theorem | In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5443_chunk_0 | Abel–Ruffini theorem | This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that x 5 − x − 1 = 0 {\displaystyle x^{5}-x-1=0} is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degree five or higher cannot be solved in radicals. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5444_chunk_0 | Abhyankar–Moh theorem | In mathematics, the Abhyankar–Moh theorem states that if L {\displaystyle L} is a complex line in the complex affine plane C 2 {\displaystyle \mathbb {C} ^{2}} , then every embedding of L {\displaystyle L} into C 2 {\displaystyle \mathbb {C} ^{2}} extends to an automorphism of the plane. It is named after Shreeram Shan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5445_chunk_0 | Ackermann ordinal | Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The last one is an extension of the Veblen functions for more than 2 arguments. The smaller ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5446_chunk_0 | Adams–Novikov spectral sequence | In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5447_chunk_0 | Ahlfors measure conjecture | In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0. The conjecture was introduced by Ahlfors (1966), who proved it in the case that the Kleinian group has a fundamental domain with a finite number o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5448_chunk_0 | Almgren–Pitts min-max theory | In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the efforts for generalizing George David Birkhoff's method for the construction of simple closed geodesics on the sphere, to al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5449_chunk_0 | Alperin–Brauer–Gorenstein theorem | In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective special unitary groups over a finite field of odd order, depending on a certain c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5450_chunk_0 | Andreotti–Frankel theorem | In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if V {\displaystyle V} is a smooth, complex affine variety of complex dimension n {\displaystyle n} or, more generally, if V {\displaystyle V} is any Stein manifold of dimension n {\displaystyle n} , the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5451_chunk_0 | Andreotti–Grauert theorem | In mathematics, the Andreotti–Grauert theorem, introduced by Andreotti and Grauert (1962), gives conditions for cohomology groups of coherent sheaves over complex manifolds to vanish or to be finite-dimensional. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5452_chunk_0 | Andreotti–Vesentini theorem | In mathematics, the Andreotti–Vesentini separation theorem, introduced by Aldo Andreotti and Edoardo Vesentini (1965, 1965b) states that certain cohomology groups of coherent sheaves are separated. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5453_chunk_0 | André–Oort conjecture | In mathematics, the André–Oort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the Manin–Mumford conjecture, which is now a theorem (proven in several different ways). The conjecture concerns itself with a characterization of the Zariski closure ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5454_chunk_0 | Angelescu polynomials | In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Angelescu (1938). The polynomials can be given by the generating functionBoas & Buck (1958, p.41) They can also be defined by the equation where A n ( x ) n ! {\displaystyle {\frac {A_{n}(x)}{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5455_chunk_0 | Anger function | In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as J ν ( z ) = 1 π ∫ 0 π cos ( ν θ − z sin θ ) d θ {\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta } with complex parameter v and complex variable x. It is clos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5456_chunk_0 | Appell–Humbert theorem | In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety. It was proved for 2-dimensional tori by Appell (1891) and Humbert (1893), and in general by Lefschetz (1921) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5457_chunk_0 | Arason invariant | In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by (Arason 1975, Theorem 5.7). The Rost invariant is a generalization of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5458_chunk_0 | Arthur's conjectures | In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local fields made by James Arthur (1989), motivated by the Arthur–Selberg trace formula. Arthur's conjectures imply the generalized Raman... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5459_chunk_0 | Relative trace formula | In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of G(A) on the discrete part L20... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5460_chunk_0 | Relative trace formula | The simple trace formula (Flicker & Kazhdan 1988) is less general but easier to prove. The local trace formula is an analogue over local fields. Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5461_chunk_0 | Artin approximation theorem | In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin (1969) in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k. More precisely, Artin proved two such theorems: one, in 1968, on approximatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5462_chunk_0 | Artin conductor | In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by Emil Artin (1930, 1931) as an expression appearing in the functional equation of an Artin L-function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5463_chunk_0 | Dwork's lemma | In mathematics, the Artin–Hasse exponential, introduced by Artin and Hasse (1928), is the power series given by E p ( x ) = exp ( x + x p p + x p 2 p 2 + x p 3 p 3 + ⋯ ) . {\displaystyle E_{p}(x)=\exp \left(x+{\frac {x^{p}}{p}}+{\frac {x^{p^{2}}}{p^{2}}}+{\frac {x^{p^{3}}}{p^{3}}}+\cdots \right).} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5464_chunk_0 | Artin–Mazur zeta function | In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals. It is defined from a given function f {\displaystyle f} as the formal power series ζ f ( z ) = exp ( ∑ n = 1 ∞ | Fix ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5465_chunk_0 | Krull's intersection theorem | In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work. An intuitive ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5466_chunk_0 | Krull's intersection theorem | One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion. The lemma also plays a key role in the study of ℓ-adic sheaves. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5467_chunk_0 | Artin–Zorn theorem | In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin.The Artin–Zorn theorem is a generalization of the Wedderburn theorem, whic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5468_chunk_0 | Askey–Gasper inequality | In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976) and used in the proof of the Bieberbach conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5469_chunk_0 | Askey–Wilson polynomial | In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Askey and Wilson (1985) as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme. Ask... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5470_chunk_0 | Atiyah conjecture on configurations | In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by Atiyah (2000, 2001) stating that a certain n by n matrix depending on n points in R3 is always non-singular. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5471_chunk_0 | Atiyah-Bott fixed point theorem | In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M, which uses an elliptic complex on M. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5472_chunk_0 | Atiyah-Hirzebruch spectral sequence | It can be derived from an exact couple that gives the E 1 {\displaystyle E_{1}} page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E {\displaystyle E} . In detail, assume X {\displaystyle X} to be the total space of a Serre fibration with fibre F {\displaystyle F} and base spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5473_chunk_0 | Aubin–Lions lemma | In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of nonlinear evolutionary partial differential equations. Typically, to prove the existence of solutions one first constru... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5474_chunk_0 | Auslander algebra | In mathematics, the Auslander algebra of an algebra A is the endomorphism ring of the sum of the indecomposable modules of A. It was introduced by Auslander (1974). An Artin algebra Γ is called an Auslander algebra if gl dim Γ ≤ 2 and if 0→Γ→I→J→K→0 is a minimal injective resolution of Γ then I and J are projective Γ-m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5475_chunk_0 | Ax-Grothendieck theorem | In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck.The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5476_chunk_0 | Babenko–Beckner inequality | In mathematics, the Babenko–Beckner inequality (after K. Ivan Babenko and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of Lp spaces. The (q, p)-norm of the n-dimensional Fourier transform is defined to be ‖ F ‖ q , p = su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5477_chunk_0 | Babuška–Lax–Milgram theorem | In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Ivo Babuška,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5478_chunk_0 | Bachmann–Howard ordinal | In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5479_chunk_0 | Backus–Gilbert method | In mathematics, the Backus–Gilbert method, also known as the optimally localized average (OLA) method is named for its discoverers, geophysicists George E. Backus and James Freeman Gilbert. It is a regularization method for obtaining meaningful solutions to ill-posed inverse problems. Where other regularization methods... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5480_chunk_0 | Backus–Gilbert method | Given a data array X, the basic Backus-Gilbert inverse is: H θ = C − 1 G θ G θ T C − 1 G θ {\displaystyle \mathbf {H} _{\theta }={\frac {\mathbf {C} ^{-1}\mathbf {G} _{\theta }}{\mathbf {G} _{\theta }^{T}\mathbf {C} ^{-1}\mathbf {G} _{\theta }}}} where C is the covariance matrix of the data, and Gθ is an a priori const... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5481_chunk_0 | Baker-Campbell-Hausdorff formula | In mathematics, the Baker–Campbell–Hausdorff formula is the solution for Z {\displaystyle Z} to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z {\displaystyle Z} in Lie algebraic terms, that i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5482_chunk_0 | Baker-Campbell-Hausdorff formula | Meanwhile, every element g {\displaystyle g} sufficiently close to the identity in G {\displaystyle G} can be expressed as g = e X {\displaystyle g=e^{X}} for a small X {\displaystyle X} in g {\displaystyle {\mathfrak {g}}} . Thus, we can say that near the identity the group multiplication in G {\displaystyle G} —writt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5483_chunk_0 | Baker-Campbell-Hausdorff formula | If X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small n × n {\displaystyle n\times n} matrices, then Z {\displaystyle Z} can be computed as the logarithm of e X e Y {\displaystyle e^{X}e^{Y}} , where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Haus... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5484_chunk_0 | Balian–Low theorem | In mathematics, the Balian–Low theorem in Fourier analysis is named for Roger Balian and Francis E. Low. The theorem states that there is no well-localized window function (or Gabor atom) g either in time or frequency for an exact Gabor frame (Riesz Basis). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5485_chunk_0 | Banach fixed point theorem | In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach-Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5486_chunk_0 | Banach–Stone theorem | In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff space X from the Banach space structure of the space C... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5487_chunk_0 | Bareiss Algorithm | In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder). The method can also be used to compute t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5488_chunk_0 | Barnes G-function | In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5489_chunk_0 | Barnes G-function | Formally, the Barnes G-function is defined in the following Weierstrass product form: G ( 1 + z ) = ( 2 π ) z / 2 exp ( − z + z 2 ( 1 + γ ) 2 ) ∏ k = 1 ∞ { ( 1 + z k ) k exp ( z 2 2 k − z ) } {\displaystyle G(1+z)=(2\pi )^{z/2}\exp \left(-{\frac {z+z^{2}(1+\gamma )}{2}}\right)\,\prod _{k=1}^{\infty }\left\{\left(1+... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5490_chunk_0 | Barnes–Wall lattice | In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Barnes & Wall (1959)), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 2, and i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5491_chunk_0 | Bass–Quillen conjecture | In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring A {\displaystyle A} . The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5492_chunk_0 | Bateman function | In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by k n ( x ) = 2 π ∫ 0 π / 2 cos ( x tan θ − n θ ) d θ {\displaystyle \displaystyle k_{n}(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan \theta -n\thet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5493_chunk_0 | Bateman polynomials | Pasternack (1939) generalized the Bateman polynomials to polynomials Fmn with F n m ( d d x ) sech m + 1 ( x ) = sech m + 1 ( x ) P n ( tanh ( x ) ) {\displaystyle F_{n}^{m}\left({\frac {d}{dx}}\right)\operatorname {sech} ^{m+1}(x)=\operatorname {sech} ^{m+1}(x)P_{n}(\tanh(x))} These generalized polynomials also ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5494_chunk_0 | Bauer–Fike theorem | In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix. Informally s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5495_chunk_0 | Beauville–Laszlo theorem | In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by Arnaud Beauville and Yves Laszlo (1995). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5496_chunk_0 | Beck–Fiala theorem | In mathematics, the Beck–Fiala theorem is a major theorem in discrepancy theory due to József Beck and Tibor Fiala. Discrepancy is concerned with coloring elements of a ground set such that each set in a certain set system is as balanced as possible, i.e., has approximately the same number of elements of each color. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5497_chunk_0 | Bell series | In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function f {\displaystyle f} and a prime p {\displaystyle p} , define the formal power series f p ( x ) {\displaystyle f_{p}(x)}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5498_chunk_0 | Bell series | Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem: given multiplicative functions f {\displaystyle f} and g {\displaystyle g} , one has f = g {\displaystyle f=g} if and only if: f p ( x ) = g p ( x ) {\displaystyle f_{p}(x)=g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5499_chunk_0 | Beltrami differential equation | In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ∂ w ∂ z ¯ = μ ∂ w ∂ z . {\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}.} for w a complex distribution of the complex variable z in some open set U, with derivatives tha... | 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.