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3-manifold Seifert–Weber space 3-manifold > Important examples of 3-manifolds > Seifert–Weber space It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
3-manifold Seifert–Weber space 3-manifold > Important examples of 3-manifolds > Seifert–Weber space Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space. With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Selberg conjecture Summary Selberg_conjecture In mathematics, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by Selberg (1965, p. 13), states that the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1/4. Selberg showed that the eigenvalues a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Selberg conjecture Summary Selberg_conjecture The generalized Ramanujan conjecture for the general linear group implies Selberg's conjecture. More precisely, Selberg's conjecture is essentially the generalized Ramanujan conjecture for the group GL2 over the rationals at the infinite place, and says that the component a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sendov's conjecture Summary Sendov's_conjecture In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov. The conjecture states that for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sendov's conjecture Summary Sendov's_conjecture The Gauss–Lucas theorem says that all of the critical points lie within the convex hull of the roots. It follows that the critical points must be within the unit disk, since the roots are. The conjecture has been proven for n < 9 by Brown-Xiang and for n sufficiently larg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Localizing subcategory Summary Localizing_subcategory In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Serre modularity conjecture Summary Serre's_modularity_conjecture In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre (1975, 1987), states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture spec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Serre's intersection formula Summary Serre's_multiplicity_conjectures In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection number... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Serre's intersection formula Summary Serre's_multiplicity_conjectures {\displaystyle \chi (R/P,R/Q):=\sum _{i=0}^{\infty }(-1)^{i}\ell _{R}(\operatorname {Tor} _{i}^{R}(R/P,R/Q)).} This requires the concept of the length of a module, denoted here by ℓ R {\displaystyle \ell _{R}} , and the assumption that ℓ R ( ( R / P ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Serre's intersection formula Summary Serre's_multiplicity_conjectures If this idea were to work, however, certain classical relationships would presumably have to continue to hold. Serre singled out four important properties. These then became conjectures, challenging in the general case. (There are more general statem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Shafarevich's theorem on solvable Galois groups Summary Shafarevich's_theorem_on_solvable_Galois_groups In mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational numbers. It was first proved by Igor Shafarevich (1954), though Alexander Schmi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sharafutdinov's retraction Summary Sharafutdinov's_retraction In mathematics, Sharafutdinov's retraction is a construction that gives a retraction of an open non-negatively curved Riemannian manifold onto its soul. It was first used by Sharafutdinov to show that any two souls of a complete Riemannian manifold with non-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sharkovsky's theorem Summary Sharkovskii's_theorem In mathematics, Sharkovskii's theorem (also spelled Sharkovsky's theorem, Sharkovskiy's theorem, Šarkovskii's theorem or Sarkovskii's theorem), named after Oleksandr Mykolayovych Sharkovsky, who published it in 1964, is a result about discrete dynamical systems. One of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Shimura's reciprocity law Summary Shimura's_reciprocity_law In mathematics, Shimura's reciprocity law, introduced by Shimura (1971), describes the action of ideles of imaginary quadratic fields on the values of modular functions at singular moduli. It forms a part of the Kronecker Jugendtraum, explicit class field theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Shintani's unit theorem Summary Shintani's_unit_theorem In mathematics, Shintani's unit theorem introduced by Shintani (1976, proposition 4) is a refinement of Dirichlet's unit theorem and states that a subgroup of finite index of the totally positive units of a number field has a fundamental domain given by a rational... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Siegel modular form Summary Siegel_modular_form In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Siegel modular form Summary Siegel_modular_form Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables. Siegel modular forms were first investigated by Carl Ludwig Siegel (1939) for the purpose of studying quadratic forms analytically. These primari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Siegel identity Summary Siegel_identity In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Siegel's theorem on integral points Summary Siegel's_theorem_on_integral_points In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sister Celine's polynomials Summary Sister_Celine's_polynomials In mathematics, Sister Celine's polynomials are a family of hypergeometric polynomials introduced by Mary Celine Fasenmyer (1947). They include Legendre polynomials, Jacobi polynomials, and Bateman polynomials as special cases. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axiom A Summary Axiom_A In mathematics, Smale's axiom A defines a class of dynamical systems which have been extensively studied and whose dynamics is relatively well understood. A prominent example is the Smale horseshoe map. The term "axiom A" originates with Stephen Smale. The importance of such systems is demonstra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sobolev orthogonal polynomials Summary Sobolev_orthogonal_polynomials In mathematics, Sobolev orthogonal polynomials are orthogonal polynomials with respect to a Sobolev inner product, i.e. an inner product with derivatives. By having conditions on the derivatives, the Sobolev orthogonal polynomials in general no longe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sobolev spaces for planar domains Summary Sobolev_spaces_for_planar_domains In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Solèr theorem Summary Solèr_theorem In mathematics, Solèr's theorem is a result concerning certain infinite-dimensional vector spaces. It states that any orthomodular form that has an infinite orthonormal sequence is a Hilbert space over the real numbers, complex numbers or quaternions. Originally proved by Maria Pia S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Solèr theorem Summary Solèr_theorem In particular, Solèr's theorem helps to fill a gap in the effort to use Gleason's theorem to rederive quantum mechanics from information-theoretic postulates. It is also an important step in the Heunen-Kornell axiomatisation of the category of Hilbert spaces. Physicist John C. Baez n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Somos' quadratic recurrence constant Summary Somos'_quadratic_recurrence_constant In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number σ = 1 2 3 ⋯ = 1 1 / 2 2 1 / 4 3 1 / 8 ⋯ . {\displaystyle \sigma ={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots .\,} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Somos' quadratic recurrence constant Summary Somos'_quadratic_recurrence_constant This sequence can be shown to have asymptotic behaviour as follows: g n ∼ σ 2 n n + 2 + O ( 1 n ) . {\displaystyle g_{n}\sim {\frac {\sigma ^{2^{n}}}{n+2+O({\frac {1}{n}})}}.} Guillera and Sondow give a representation in terms of the deri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sonine formula Summary Sonine_formula In mathematics, Sonine's formula is any of several formulas involving Bessel functions found by Nikolay Yakovlevich Sonin. One such formula is the following integral formula involving a product of three Bessel functions: ∫ 0 ∞ J z ( a t ) J z ( b t ) J z ( c t ) t 1 − z d t = 2 z −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sophie Germain's identity Summary Sophie_Germain's_identity In mathematics, Sophie Germain's identity is a polynomial factorization named after Sophie Germain stating that Beyond its use in elementary algebra, it can also be used in number theory to factorize integers of the special form x 4 + 4 y 4 {\displaystyle x^{4... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
S-duality (homotopy theory) Summary S-duality_(homotopy_theory) In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space X may be considered as dual to its complement in the n-sphere, where n is large enough. Its origins lie in Alexander dual... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
S-duality (homotopy theory) Summary S-duality_(homotopy_theory) It is named for Edwin Spanier and J. H. C. Whitehead, who developed it in papers from 1955. The basic point is that sphere complements determine the homology, but not the homotopy type, in general. What is determined, however, is the stable homotopy type, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Specht's theorem Summary Specht's_theorem In mathematics, Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940.Two matrices A and B with complex number entries are said to be unitarily equivalent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Specht's theorem Summary Specht's_theorem If A and B are unitarily equivalent, then tr AA* = tr BB*, where tr denotes the trace (in other words, the Frobenius norm is a unitary invariant). This follows from the cyclic invariance of the trace: if B = U *AU, then tr BB* = tr U *AUU *A*U = tr AUU *A*UU * = tr AA*, where t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Specht's theorem Summary Specht's_theorem The formulation of the theorem uses the following definition. A word in two variables, say x and y, is an expression of the form W ( x , y ) = x m 1 y n 1 x m 2 y n 2 ⋯ x m p , {\displaystyle W(x,y)=x^{m_{1}}y^{n_{1}}x^{m_{2}}y^{n_{2}}\cdots x^{m_{p}},} where m1, n1, m2, n2, …,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Specht's theorem Summary Specht's_theorem {\displaystyle m_{1}+n_{1}+m_{2}+n_{2}+\cdots +m_{p}.} Specht's theorem: Two matrices A and B are unitarily equivalent if and only if tr W(A, A*) = tr W(B, B*) for all words W.The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Le... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spence's function Summary Spence's_function In mathematics, Spence's function, or dilogarithm, denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: Li 2 ( z ) = − ∫ 0 z ln ( 1 − u ) u d u , z ∈ C {\displaystyle \ope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spence's function Summary Spence's_function Alternatively, the dilogarithm function is sometimes defined as ∫ 1 v ln t 1 − t d t = Li 2 ( 1 − v ) . {\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v).} In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spence's function Summary Spence's_function Specifically, a simplex whose vertices have cross ratio z has hyperbolic volume D ( z ) = Im Li 2 ( z ) + arg ( 1 − z ) log | z | . {\displaystyle D(z)=\operatorname {Im} \operatorname {Li} _{2}(z)+\arg(1-z)\log |z|.} The function D(z) is sometimes called the Bloch-Wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spence's function Summary Spence's_function Lobachevsky's function and Clausen's function are closely related functions. William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century. He was at school with John Galt, who later wrote... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spencer cohomology Summary Spencer_cohomology In mathematics, Spencer cohomology is cohomology of a manifold with coefficients in the sheaf of solutions of a linear partial differential operator. It was introduced by Donald C. Spencer in 1969. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sperner coloring Summary Sperner_coloring In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an n {\displaystyle n} -dimensional s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sperner coloring Summary Sperner_coloring According to the Soviet Mathematical Encyclopaedia (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had also become known as the Sperner lemma – this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spijker's lemma Summary Spijker's_lemma In mathematics, Spijker's lemma is a result in the theory of rational mappings of the Riemann sphere. It states that the image of a circle under a complex rational map with numerator and denominator having degree at most n has length at most 2nπ. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spouge's approximation Summary Spouge's_approximation In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper. The formula is a modification of Stirling's approximation, and has the form Γ ( z + ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spouge's approximation Summary Spouge's_approximation {\displaystyle {\begin{aligned}c_{0}&={\sqrt {2\pi }}\\c_{k}&={\frac {(-1)^{k-1}}{(k-1)! }}(-k+a)^{k-{\frac {1}{2}}}e^{-k+a}\qquad k\in \{1,2,\dots ,a-1\}.\end{aligned}}} Spouge has proved that, if Re(z) > 0 and a > 2, the relative error in discarding εa(z) is bound... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Spouge's approximation Summary Spouge's_approximation The formula is similar to the Lanczos approximation, but has some distinct features. Whereas the Lanczos formula exhibits faster convergence, Spouge's coefficients are much easier to calculate and the error can be set arbitrarily low. The formula is therefore feasib... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Steinhaus–Moser notation Summary Moser_polygon_notation In mathematics, Steinhaus–Moser notation is a notation for expressing certain large numbers. It is an extension (devised by Leo Moser) of Hugo Steinhaus's polygon notation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stickelberger's theorem Summary Stickelberger's_theorem In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stinespring factorization theorem Summary Stinespring_factorization_theorem In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra A as a composition... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Karamata notation Summary Stirling_number In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus differentialis (1730). They were rediscovered and given a combinatorial meanin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Karamata notation Summary Stirling_number Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind. Each kind is detailed in its respective article, this one serving as a description of relations between them. A common property of all three kinds is that they describe coefficients relat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stirling's approximation Summary Stirling_approximation In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n {\displaystyle n} . It is named after James Stirling, though a related but less ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stirling's approximation Summary Stirling_approximation and n ln n − n {\displaystyle n\ln n-n} will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to use instead the binary logarithm, giving the equivalent form Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Representation theorem for Boolean algebras Summary Stone's_representation_theorem In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stone's theorem on one-parameter unitary groups Summary Stone's_theorem_on_one-parameter_unitary_groups In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stone's theorem on one-parameter unitary groups Summary Stone's_theorem_on_one-parameter_unitary_groups The theorem was proved by Marshall Stone (1930, 1932), and John von Neumann (1932) showed that the requirement that ( U t ) t ∈ R {\displaystyle (U_{t})_{t\in \mathbb {R} }} be strongly continuous can be relaxed to s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strassmann's theorem Summary Strassmann's_theorem In mathematics, Strassmann's theorem is a result in field theory. It states that, for suitable fields, suitable formal power series with coefficients in the valuation ring of the field have only finitely many zeroes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sullivan conjecture Summary Sullivan_conjecture In mathematics, Sullivan conjecture or Sullivan's conjecture on maps from classifying spaces can refer to any of several results and conjectures prompted by homotopy theory work of Dennis Sullivan. A basic theme and motivation concerns the fixed point set in group actions... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sullivan conjecture Summary Sullivan_conjecture Roughly speaking, it is difficult to map such a space B G {\displaystyle BG} continuously into a finite CW complex X {\displaystyle X} in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by Haynes Miller. Specifically, in 1984, Miller prove... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sullivan conjecture Summary Sullivan_conjecture This is equivalent to the statement that the map X {\displaystyle X} → F ( B G , X ) {\displaystyle F(BG,X)} from X to the function space of maps B G {\displaystyle BG} → X {\displaystyle X} , not necessarily preserving the base point, given by sending a point x {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sullivan conjecture Summary Sullivan_conjecture In general, for a group G {\displaystyle G} acting on a space X {\displaystyle X} , the homotopy fixed points are the fixed points F ( E G , X ) G {\displaystyle F(EG,X)^{G}} of the mapping space F ( E G , X ) {\displaystyle F(EG,X)} of maps from the universal cover E G {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sullivan conjecture Summary Sullivan_conjecture An important ingredient and motivation for his proof is a result of Gunnar Carlsson on the homology of B Z / 2 {\displaystyle BZ/2} as an unstable module over the Steenrod algebra.Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on X ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface fairing Summary Surface_fairing In mathematics, Surface fairing is an aspect of mesh smoothing. The goal of surface fairing is to compute shapes that are as smooth as possible. On an abstract level, mesh smoothing is concerned with the design and computation of smooth functions f: S → R d {\displaystyle f:S\rig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface fairing Summary Surface_fairing It also smooths the function as much as possible in order to obtain, e.g., an as-smooth-as-possible surface patch or an as-smooth-as-possible shape deformation.How to actually measure smoothness or fairness obviously depends on the application, but in general fair surfaces should... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface fairing Summary Surface_fairing A minimization of this fairness energy—subject to user-defined constraints—eventually yields the desired shape. Example applications include the construction of smooth blend surfaces and hole filling by smooth patches. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Suslin's hypothesis Summary Suslin_hypothesis In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin (1920) and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; Solovay & Tennenbaum (1971) show... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sylvester's criterion Summary Sylvester's_criterion In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetry-preserving filter Summary Symmetry-preserving_filter In mathematics, Symmetry-preserving observers, also known as invariant filters, are estimation techniques whose structure and design take advantage of the natural symmetries (or invariances) of the considered nonlinear model. As such, the main benefit is an ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Szymanski's conjecture Summary Szymanski's_conjecture In mathematics, Szymanski's conjecture, named after Ted H. Szymanski (1989), states that every permutation on the n-dimensional doubly directed hypercube graph can be routed with edge-disjoint paths. That is, if the permutation σ matches each vertex v to another ver... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tait's conjecture Summary Tait's_conjecture In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by P. G. Tait (1884) and disproved by W. T. Tutte (1946), who constructed a counterexample with 25 faces, 6... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tait's conjecture Summary Tait's_conjecture The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an ed... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Takeuti conjecture Summary Takeuti_conjecture In mathematics, Takeuti's conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-order logic has cut-elimination (Takeuti 1953). It was settled positively: By Tait, using a semantic technique for proving cut-elimination, based on work by Schütt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tanaka equation Summary Tanaka_equation In mathematics, Tanaka's equation is an example of a stochastic differential equation which admits a weak solution but has no strong solution. It is named after the Japanese mathematician Hiroshi Tanaka (Tanaka Hiroshi). Tanaka's equation is the one-dimensional stochastic differe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tanaka equation Summary Tanaka_equation The signum function does not satisfy the Lipschitz continuity condition required for the usual theorems guaranteeing existence and uniqueness of strong solutions. The Tanaka equation has no strong solution, i.e. one for which the version B of Brownian motion is given in advance a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tanaka equation Summary Tanaka_equation In this case, simply choose X to be any Brownian motion B ^ {\displaystyle {\hat {B}}} and define B ~ {\displaystyle {\tilde {B}}} by B ~ t = ∫ 0 t sgn ( B ^ s ) d B ^ s = ∫ 0 t sgn ( X s ) d X s , {\displaystyle {\tilde {B}}_{t}=\int _{0}^{t}\operatorname {sgn} {\big (}{\hat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tannaka–Krein duality Summary Tannaka–Krein_duality In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tannaka–Krein duality Summary Tannaka–Krein_duality In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(G) with some additional structure, formed by the finite-dimensional representations of G. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tarski's plank problem Summary Tarski's_plank_problem In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tarski's theorem about choice Summary Tarski's_theorem_about_choice In mathematics, Tarski's theorem, proved by Alfred Tarski (1924), states that in ZF the theorem "For every infinite set A {\displaystyle A} , there is a bijective map between the sets A {\displaystyle A} and A × A {\displaystyle A\times A} " implies th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tate cohomology group Summary Tate_cohomology_group In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by John Tate (1952, p. 297), and are used in class field theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Poitou-Tate duality Summary Poitou-Tate_duality In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by John Tate (1962) and Georges Poitou (1967). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tate pairing Summary Tate_pairing In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by Tate (1958, 1963) and extended by Lichtenbaum (1969). Rück & Frey (1994... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tate's isogeny theorem Summary Tate's_isogeny_theorem In mathematics, Tate's isogeny theorem, proved by Tate (1966), states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thaine's theorem Summary Thaine's_theorem In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by Thaine (1988). Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem (Washington 1997), to prove that some Tate–Shafarevich groups are finit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fundamental theorem of topos theory Summary Fundamental_theorem_of_topos_theory In mathematics, The fundamental theorem of topos theory states that the slice E / X {\displaystyle \mathbf {E} /X} of a topos E {\displaystyle \mathbf {E} } over any one of its objects X {\displaystyle X} is itself a topos. Moreover, if the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Theory of Lie groups Summary Theory_of_Lie_groups In mathematics, Theory of Lie groups is a series of books on Lie groups by Claude Chevalley (1946, 1951, 1955). The first in the series was one of the earliest books on Lie groups to treat them from the global point of view, and for many years was the standard text on L... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thomae's formula Summary Thomae's_formula In mathematics, Thomae's formula is a formula introduced by Carl Johannes Thomae (1870) relating theta constants to the branch points of a hyperelliptic curve (Mumford 1984, section 8). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nielsen–Thurston classification Summary Nielsen-Thurston_classification In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by Jakob Nielsen (1944). Given a homeomorphism f: S → S, there is a map g isotop... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nielsen–Thurston classification Summary Nielsen-Thurston_classification In what follows, we assume S has genus at least two, as this is the case Thurston considered. (Note, however, that the cases where S has boundary or is not orientable are definitely still of interest.) The three types in this classification are not... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometrisation conjecture Summary Geometrization_conjecture In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometrisation conjecture Summary Geometrization_conjecture The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometrisation conjecture Summary Geometrization_conjecture Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server. Perelman's papers were studied by several independent groups that produced books and online ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thābit ibn Qurra Mathematics Thābit_ibn_Qurra > Mathematics In mathematics, Thābit derived an equation for determining amicable numbers. His proof of this rule is presented in the Treatise on the Derivation of the Amicable Numbers in an Easy Way. This was done while writing on the theory of numbers, extending their use... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thābit ibn Qurra Mathematics Thābit_ibn_Qurra > Mathematics He provided a strengthened extension of Pythagoras' proof which included the knowledge of Euclid's fifth postulate. This postulate states that the intersection between two straight line segments combine to create two interior angles which are less than 180 deg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thābit ibn Qurra Mathematics Thābit_ibn_Qurra > Mathematics Thābit believed that geometry was tied with the equality and differences of magnitudes of lines and angles, as well as that ideas of motion (and ideas taken from physics more widely) should be integrated in geometry.The continued work done on geometric relatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thābit ibn Qurra Mathematics Thābit_ibn_Qurra > Mathematics His work with conic sections and the calculation of a paraboloid shape (cupola) show his proficiency as an Archimedean geometer. This is further embossed by Thābit's use of the Archimedean property in order to produce a rudimentary approximation of the volume ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Toda–Smith complex Summary Toda–Smith_complex In mathematics, Toda–Smith complexes are spectra characterized by having a particularly simple BP-homology, and are useful objects in stable homotopy theory. Toda–Smith complexes provide examples of periodic self maps. These self maps were originally exploited in order to c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tonelli's theorem (functional analysis) Summary Tonelli's_theorem_(functional_analysis) In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weak lower semicontinuity of nonlinear functionals on Lp spaces. As such, it has major implications for functional analysis and the calculus of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tsen's theorem Summary Tsen's_theorem In mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes, and more generally that all the Galois cohomology groups H... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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