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wiki_6300_chunk_0 | Russo–Dye theorem | In mathematics, the Russo–Dye theorem is a result in the field of functional analysis. It states that in a unital C*-algebra, the closure of the convex hull of the unitary elements is the closed unit ball. : 44 The theorem was published by B. Russo and H. A. Dye in 1966. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6301_chunk_0 | Samuelson–Berkowitz algorithm | In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an n × n {\displaystyle n\times n} matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVerrier algorithm, it performs no divisions, so may be applied to a wider range of algebrai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6302_chunk_0 | Geometric Satake correspondence | In mathematics, the Satake isomorphism, introduced by Ichirō Satake (1963), identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by Ivan Mirković and Kari Vilonen (2007)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6303_chunk_0 | Lang–Trotter conjecture | In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all prime numbers p. Mikio Sato and John Tate independently posed the conjecture around 1960. If Np denotes the number of poin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6304_chunk_0 | Schneider–Lang theorem | In mathematics, the Schneider–Lang theorem is a refinement by Lang (1966) of a theorem of Schneider (1949) about the transcendence of values of meromorphic functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems, and implies the transcendence of some values of elliptic functions and ell... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6305_chunk_0 | Schönflies problem | In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6306_chunk_0 | Schoen–Yau conjecture | In mathematics, the Schoen–Yau conjecture is a disproved conjecture in hyperbolic geometry, named after the mathematicians Richard Schoen and Shing-Tung Yau. It was inspired by a theorem of Erhard Heinz (1952). One method of disproof is the use of Scherk surfaces, as used by Harold Rosenberg and Pascal Collin (2006). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6307_chunk_0 | Schottky form | In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form J of degree 4 and weight 8, introduced by Friedrich Schottky (1888, 1903) as a degree 16 polynomial in the Thetanullwerte of genus 4. He showed that it vanished at all Jacobian points (the points of the degree 4 Siegel upper half-space corr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6308_chunk_0 | Schottky problem | In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6309_chunk_0 | Schreier refinement theorem | In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups that sends each factor group to an isomorphic one. The theorem is named afte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6310_chunk_0 | Schuette–Nesbitt formula | In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after Donald R. Schuette and Cecil J. Nesbitt. The probabilistic version of the Schuette–Nesbitt formula has practical applications in actuarial science, where it is used to calculate the net single premiu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6311_chunk_0 | Schwartz kernel theorem | In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6312_chunk_0 | Schwarz lantern | In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. It is formed by stacked rings of isosceles triangles, arranged within each ring in the same pattern ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6313_chunk_0 | Schwarz lantern | It is also known as Schwarz's boot, Schwarz's polyhedron, or the Chinese lantern.As Schwarz showed, for the surface area of a polyhedron to converge to the surface area of a curved surface, it is not sufficient to simply increase the number of rings and the number of isosceles triangles per ring. Depending on the relat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6314_chunk_0 | Schwarz's lemma | In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest resu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6315_chunk_0 | Schwarz reflection principle | In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6316_chunk_0 | Schwarz reflection principle | Suppose that F is a continuous function on the closed upper half plane { z ∈ C ∣ Im ( z ) ≥ 0 } {\displaystyle \left\{z\in \mathbb {C} \mid \operatorname {Im} (z)\geq 0\right\}} , holomorphic on the upper half plane { z ∈ C ∣ Im ( z ) > 0 } {\displaystyle \left\{z\in \mathbb {C} \mid \operatorname {Im} (z)>0\right\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6317_chunk_0 | Schwarz reflection principle | In fact Morera's theorem is well adapted to proving such statements. Contour integrals involving the extension of F clearly split into two, using part of the real axis. So, given that the principle is rather easy to prove in the special case from Morera's theorem, understanding the proof is enough to generate other res... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6318_chunk_0 | Schwarzian derivative | In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6319_chunk_0 | Schwarz–Ahlfors–Pick theorem | In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane H to itself, will not increase the Poincaré dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6320_chunk_0 | Schwarz–Ahlfors–Pick theorem | Let U be the unit disk with Poincaré metric ρ {\displaystyle \rho } ; let S be a Riemann surface endowed with a Hermitian metric σ {\displaystyle \sigma } whose Gaussian curvature is ≤ −1; let f: U → S {\displaystyle f:U\rightarrow S} be a holomorphic function. Then σ ( f ( z 1 ) , f ( z 2 ) ) ≤ ρ ( z 1 , z 2 ) {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6321_chunk_0 | Schwarz–Ahlfors–Pick theorem | {\displaystyle z_{1},z_{2}\in U.} A generalization of this theorem was proved by Shing-Tung Yau in 1973. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6322_chunk_0 | Scott core theorem | In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott, (Scott 1973). The precise statement is as follows: Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6323_chunk_0 | Segre class | In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6324_chunk_0 | Segre mapping | In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6325_chunk_0 | Seifert conjecture | In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase non-existence as a conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6326_chunk_0 | Seifert–Van Kampen theorem | In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X {\displaystyle X} in terms of the fundamental groups of two open, path-connected... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6327_chunk_0 | Selberg's zeta function conjecture | In mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ(1/2 + it). It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. Results on this can be for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6328_chunk_0 | Selberg integral | In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6329_chunk_0 | Selberg trace formula | In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group. The character is given by the trace of certain functions on G. The simples... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6330_chunk_0 | Selberg trace formula | The case when Γ\G is not compact is harder, because there is a continuous spectrum, described using Eisenstein series. Selberg worked out the non-compact case when G is the group SL(2, R); the extension to higher rank groups is the Arthur–Selberg trace formula. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6331_chunk_0 | Selberg trace formula | When Γ is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in terms of geometric data involving the lengths of geodesics on the Riemann surface. In this case the Selberg trace formula is formally similar to the explicit formulas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6332_chunk_0 | Serre spectral sequence | In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space X of a (Serre) fib... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6333_chunk_0 | Shapiro inequality | In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6334_chunk_0 | Shapiro polynomials | In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums. In signal processing, the Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6335_chunk_0 | Sherman–Takeda theorem | In mathematics, the Sherman–Takeda theorem states that if A is a C*-algebra then its double dual is a W*-algebra, and is isomorphic to the weak closure of A in the universal representation of A. The theorem was announced by Sherman (1950) and proved by Takeda (1954). The double dual of A is called the universal envelop... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6336_chunk_0 | Shimizu L-function | In mathematics, the Shimizu L-function, introduced by Hideo Shimizu (1963), is a Dirichlet series associated to a totally real algebraic number field. Michael Francis Atiyah, H. Donnelly, and I. M. Singer (1983) defined the signature defect of the boundary of a manifold as the eta invariant, the value as s=0 of their e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6337_chunk_0 | Siegel G-function | In mathematics, the Siegel G-functions are a class of functions in transcendental number theory introduced by C. L. Siegel. They satisfy a linear differential equation with polynomial coefficients, and the coefficients of their power series expansion lie in a fixed algebraic number field and have heights of at most exp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6338_chunk_0 | Siegel upper half-space | Writing a generic matrix Z in the Siegel upper half-space in terms of its real and imaginary parts as Z = X + iY, all metrics with isometry group Sp(2g, R) are proportional to d s 2 = tr ( Y − 1 d Z Y − 1 d Z ¯ ) . {\displaystyle ds^{2}={\text{tr}}(Y^{-1}dZY^{-1}d{\bar {Z}}).} The Siegel upper half-plane can be identif... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6339_chunk_0 | Siegel–Weil formula | In mathematics, the Siegel–Weil formula, introduced by Weil (1964, 1965) as an extension of the results of Siegel (1951, 1952), expresses an Eisenstein series as a weighted average of theta series of lattices in a genus, where the weights are proportional to the inverse of the order of the automorphism group of the lat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6340_chunk_0 | Silverman–Toeplitz theorem | In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit.An infinite matrix ( a i , j ) i ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6341_chunk_0 | Simon problems | In mathematics, the Simon problems (or Simon's problems) are a series of fifteen questions posed in the year 2000 by Barry Simon, an American mathematical physicist. Inspired by other collections of mathematical problems and open conjectures, such as the famous list by David Hilbert, the Simon problems concern quantum ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6342_chunk_0 | Sims conjecture | In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims. He conjectured that if G {\displaystyle G} is a primitive permutation group on a finite set S {\displaystyle S} and G α {\displaystyle G_{\alpha }} denotes the stabilizer of the point α {\displaystyle \alpha } in S {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6343_chunk_0 | Sims conjecture | Thus, in a primitive permutation group with "large" stabilizers, these stabilizers cannot have any small orbit. A consequence of their proof is that there exist only finitely many connected distance-transitive graphs having degree greater than 2. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6344_chunk_0 | Skolem problem | In mathematics, the Skolem problem is the problem of determining whether the values of a constant-recursive sequence include the number zero. The problem can be formulated for recurrences over different types of numbers, including integers, rational numbers, and algebraic numbers. It is not known whether there exists a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6345_chunk_0 | Skolem problem | This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly. Skolem proved this for recurrences over the rational numbers, and Mahler and Lech extended it to other systems of numbers. However, the proofs of the theorem do not show how to test wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6346_chunk_0 | Skolem problem | There does exist an algorithm to test whether a constant-recursive sequence has infinitely many zeros, and if so to construct a decomposition of the positions of those zeros into periodic subsequences, based on the algebraic properties of the roots of the characteristic polynomial of the given recurrence. The remaining... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6347_chunk_0 | Smith conjecture | In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot. Paul A. Smith (1939, remark after theorem 4) showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6348_chunk_0 | Smith conjecture | The proof of the general case was described by John Morgan and Hyman Bass (1984) and depended on several major advances in 3-manifold theory, In particular the work of William Thurston on hyperbolic structures on 3-manifolds, and results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, with some ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6349_chunk_0 | Smith normal form | In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6350_chunk_0 | Siegel mass formula | In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6351_chunk_0 | Siegel mass formula | It was rediscovered by H. Minkowski (1885), and an error in Minkowski's paper was found and corrected by C. L. Siegel (1935). Many published versions of the mass formula have errors; in particular the 2-adic densities are difficult to get right, and it is sometimes forgotten that the trivial cases of dimensions 0 and 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6352_chunk_0 | Grothendieck–Springer simultaneous resolution | In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra, or the unipotent elements of a reductive algebraic group, introduced by Tonny Albert Springer in 1969. The fibers of this resolution are called Springer fibers.If U is the variety of unipotent eleme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6353_chunk_0 | Stallings–Zeeman theorem | In mathematics, the Stallings–Zeeman theorem is a result in algebraic topology, used in the proof of the Poincaré conjecture for dimension greater than or equal to five. It is named after the mathematicians John R. Stallings and Christopher Zeeman. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6354_chunk_0 | Stein–Strömberg theorem | In mathematics, the Stein–Strömberg theorem or Stein–Strömberg inequality is a result in measure theory concerning the Hardy–Littlewood maximal operator. The result is foundational in the study of the problem of differentiation of integrals. The result is named after the mathematicians Elias M. Stein and Jan-Olov Ström... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6355_chunk_0 | Stieltjes constants | In mathematics, the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function: ζ ( 1 + s ) = 1 s + ∑ n = 0 ∞ ( − 1 ) n n ! γ n s n . {\displaystyle \zeta (1+s)={\frac {1}{s}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n! }}\gamma _{n}s^{n}.} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6356_chunk_0 | Stieltjes moment problem | In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (m0, m1, m2, ...) to be of the form m n = ∫ 0 ∞ x n d μ ( x ) {\displaystyle m_{n}=\int _{0}^{\infty }x^{n}\,d\mu (x)} for some measure μ. If such a function μ exists, one asks wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6357_chunk_0 | Stieltjes polynomials | In mathematics, the Stieltjes polynomials En are polynomials associated to a family of orthogonal polynomials Pn. They are unrelated to the Stieltjes polynomial solutions of differential equations. Stieltjes originally considered the case where the orthogonal polynomials Pn are the Legendre polynomials. The Gauss–Kronr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6358_chunk_0 | Stirling polynomial | In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6359_chunk_0 | Stolarsky mean | In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6360_chunk_0 | Stolz–Cesàro theorem | In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital'... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6361_chunk_0 | Stone functor | In mathematics, the Stone functor is a functor S: Topop → Bool, where Top is the category of topological spaces and Bool is the category of Boolean algebras and Boolean homomorphisms. It assigns to each topological space X the Boolean algebra S(X) of its clopen subsets, and to each morphism fop: X → Y in Topop (i.e., a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6362_chunk_0 | Strahler stream order | In mathematics, the Strahler number or Horton–Strahler number of a mathematical tree is a numerical measure of its branching complexity. These numbers were first developed in hydrology, as a way of measuring the complexity of rivers and streams, by Robert E. Horton (1945) and Arthur Newell Strahler (1952, 1957). In thi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6363_chunk_0 | Sturm theorem | In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with p and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6364_chunk_0 | Sturm theorem | By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univariate polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6365_chunk_0 | Sturm theorem | For computing over the reals, Sturm's theorem is less efficient than other methods based on Descartes' rule of signs. However, it works on every real closed field, and, therefore, remains fundamental for the theoretical study of the computational complexity of decidability and quantifier elimination in the first order ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6366_chunk_0 | Sturm series | In mathematics, the Sturm series associated with a pair of polynomials is named after Jacques Charles François Sturm. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6367_chunk_0 | Sugeno integral | In mathematics, the Sugeno integral, named after M. Sugeno, is a type of integral with respect to a fuzzy measure. Let ( X , Ω ) {\displaystyle (X,\Omega )} be a measurable space and let h: X → {\displaystyle h:X\to } be an Ω {\displaystyle \Omega } -measurable function. The Sugeno integral over the crisp set A ⊆ X {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6368_chunk_0 | Suita conjecture | In mathematics, the Suita conjecture is a conjecture related to the theory of the Riemann surface, the boundary behavior of conformal maps, the theory of Bergman kernel, and the theory of the L2 extension. The conjecture states the following: Suita (1972): Let R be an Riemann surface, which admits a nontrivial Green fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6369_chunk_0 | Suita conjecture | Let c β ( z ) {\displaystyle c_{\beta }(z)} be the logarithmic capacity which is locally defined by c β ( z 0 ) := exp lim ξ → z ( G R ( z , z 0 ) − log | ω ( z ) | ) {\displaystyle c_{\beta }(z_{0}):=\exp \lim _{\xi \to z}(G_{R}(z,z_{0})-\log |\omega (z)|)} on R. Then, the inequality ( c β ( z 0 ) ) 2 ≤ π B R ( z ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6370_chunk_0 | Singular homology of abstract algebraic varieties | In mathematics, the Suslin homology is a homology theory attached to algebraic varieties. It was proposed by Suslin in 1987, and developed by Suslin and Voevodsky (1996). It is sometimes called singular homology as it is analogous to the singular homology of topological spaces. By definition, given an abelian group A a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6371_chunk_0 | Séminaire de Géométrie Algébrique du Bois Marie | In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie (SGA) was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris. (The name came from the small wood on th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6372_chunk_0 | T(1) theorem | In mathematics, the T(1) theorem, first proved by David & Journé (1984), describes when an operator T given by a kernel can be extended to a bounded linear operator on the Hilbert space L2(Rn). The name T(1) theorem refers to a condition on the distribution T(1), given by the operator T applied to the function 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6373_chunk_0 | Tamagawa number | In mathematics, the Tamagawa number τ ( G ) {\displaystyle \tau (G)} of a semisimple algebraic group defined over a global field k is the measure of G ( A ) / G ( k ) {\displaystyle G(\mathbb {A} )/G(k)} , where A {\displaystyle \mathbb {A} } is the adele ring of k. Tamagawa numbers were introduced by Tamagawa (1966), ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6374_chunk_0 | Tarski–Seidenberg theorem | In mathematics, the Tarski–Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities. The theorem—also known as the Tarski–... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6375_chunk_0 | Tarski–Seidenberg theorem | An important consequence is the decidability of the theory of real-closed fields. Although the original proof of the theorem was constructive, the resulting algorithm has a computational complexity that is too high for using the method on a computer. George E. Collins introduced the algorithm of cylindrical algebraic d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6376_chunk_0 | Tate curve | In mathematics, the Tate curve is a curve defined over the ring of formal power series Z ] {\displaystyle \mathbb {Z} ]} with integer coefficients. Over the open subscheme where q is invertible, the Tate curve is an elliptic curve. The Tate curve can also be defined for q as an element of a complete field of norm less... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6377_chunk_0 | Tate topology | In mathematics, the Tate topology is a Grothendieck topology of the space of maximal ideals of a k-affinoid algebra, whose open sets are the admissible open subsets and whose coverings are the admissible open coverings. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6378_chunk_0 | Taylor polynomial | In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6379_chunk_0 | Taylor polynomial | The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6380_chunk_0 | Taylor polynomial | If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6381_chunk_0 | Teichmüller cocycle | In mathematics, the Teichmüller cocycle is a certain 3-cocycle associated to a simple algebra A over a field L which is a finite Galois extension of a field K and which has the property that any automorphism of L over K extends to an automorphism of A. The Teichmüller cocycle, or rather its cohomology class, is the obs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6382_chunk_0 | Tukey's lemma | In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma), named after John Tukey and Oswald Teichmüller, is a lemma that states that every nonempty collection of finite character has a maximal element with respect to inclusion. Over Zermelo–Fraenkel set theory, the Teichmüller–Tukey lemma is equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6383_chunk_0 | Thom spectrum | In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6384_chunk_0 | Thomas–Fermi equation | In mathematics, the Thomas–Fermi equation for the neutral atom is a second order non-linear ordinary differential equation, named after Llewellyn Thomas and Enrico Fermi, which can be derived by applying the Thomas–Fermi model to atoms. The equation reads d 2 y d x 2 = 1 x y 3 / 2 {\displaystyle {\frac {d^{2}y}{dx^{2}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6385_chunk_0 | Thurston norm | In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of homology classes represented by surfaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6386_chunk_0 | Tits alternative | In mathematics, the Tits alternative, named after Jacques Tits, is an important theorem about the structure of finitely generated linear groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6387_chunk_0 | Todd class | In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6388_chunk_0 | Tonelli–Hobson test | In mathematics, the Tonelli–Hobson test gives sufficient criteria for a function ƒ on R2 to be an integrable function. It is often used to establish that Fubini's theorem may be applied to ƒ. It is named for Leonida Tonelli and E. W. Hobson. More precisely, the Tonelli–Hobson test states that if ƒ is a real-valued meas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6389_chunk_0 | Tor functor | In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie alge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6390_chunk_0 | Tor functor | In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book Homological Al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6391_chunk_0 | Torelli's theorem | In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6392_chunk_0 | Torelli's theorem | Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the local Torelli theorem. Secondly, to other period mappings. A case that has been investigated deeply is for K3 surfaces (by Viktor S. Kulikov, Ilya Pyatetskii-Shapiro, Igor Shafarevich and Fedor Bogomolov) and hyp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6393_chunk_0 | Trefftz method | In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician Erich Trefftz(de) (1888–1937). It falls within the class of finite element methods. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6394_chunk_0 | Trombi–Varadarajan theorem | In mathematics, the Trombi–Varadarajan theorem, introduced by Trombi and Varadarjan (1971), gives an isomorphism between a certain space of spherical functions on a semisimple Lie group, and a certain space of holomorphic functions defined on a tubular neighborhood of the dual of a Cartan subalgebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6395_chunk_0 | Tutte homotopy theorem | In mathematics, the Tutte homotopy theorem, introduced by Tutte (1958), generalises the concept of "path" from graphs to matroids, and states roughly that closed paths can be written as compositions of elementary closed paths, so that in some sense they are homotopic to the trivial closed path. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6396_chunk_0 | Veblen functions | In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in Veblen (1908). If φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common fixed points of φβ for β<α... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6397_chunk_0 | Veblen–Young theorem | In mathematics, the Veblen–Young theorem, proved by Oswald Veblen and John Wesley Young (1908, 1910, 1917), states that a projective space of dimension at least 3 can be constructed as the projective space associated to a vector space over a division ring. Non-Desarguesian planes give examples of 2-dimensional projecti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6398_chunk_0 | Virasoro algebra | In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6399_chunk_0 | Vitali covering lemma | In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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