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wiki_1900_chunk_0 | Scheinerman's conjecture | Scheinerman also conjectured that segments with only three directions would be sufficient to represent 3-colorable graphs, and West (1991) conjectured that analogously every planar graph could be represented using four directions. If a graph is represented with segments having only k directions and no two segments belo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1901_chunk_0 | Scheinerman's conjecture | Hartman, Newman & Ziv (1991) and de Fraysseix, Ossona de Mendez & Pach (1991) proved that every bipartite planar graph can be represented as an intersection graph of horizontal and vertical line segments; for this result see also Czyzowicz, Kranakis & Urrutia (1998). De Castro et al. (2002) proved that every triangle-f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1902_chunk_0 | Scheinerman's conjecture | Chalopin, Gonçalves & Ochem (2007) proved that planar graphs are in 1-STRING, the class of intersection graphs of simple curves in the plane that intersect each other in at most one crossing point per pair. This class is intermediate between the intersection graphs of segments appearing in Scheinerman's conjecture and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1903_chunk_0 | Schilder's theorem | In mathematics, Schilder's theorem is a generalization of the Laplace method from integrals on R n {\displaystyle \mathbb {R} ^{n}} to functional Wiener integration. The theorem is used in the large deviations theory of stochastic processes. Roughly speaking, out of Schilder's theorem one gets an estimate for the proba... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1904_chunk_0 | Schinzel's hypothesis H | In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalization of widely open conjectures such as the twin prime conjecture. The hypothesis is named after Andrzej Schinzel. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1905_chunk_0 | Schreier's lemma | In mathematics, Schreier's lemma is a theorem in group theory used in the Schreier–Sims algorithm and also for finding a presentation of a subgroup. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1906_chunk_0 | Schubert's enumerative calculus | In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1907_chunk_0 | Schubert's enumerative calculus | The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1908_chunk_0 | Schur algebra | In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of those two groups. Their use was promoted by the influential monograph of J. A. G... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1909_chunk_0 | Skew Schur function | In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible represe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1910_chunk_0 | Schur's Inequality | In mathematics, Schur's inequality, named after Issai Schur, establishes that for all non-negative real numbers x, y, z, and t>0, ∑ c y c x t ( x − y ) ( x − z ) = x t ( x − y ) ( x − z ) + y t ( y − z ) ( y − x ) + z t ( z − x ) ( z − y ) ≥ 0 {\displaystyle \sum _{cyc}x^{t}(x-y)(x-z)=x^{t}(x-y)(x-z)+y^{t}(y-z)(y-x)+z^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1911_chunk_0 | Schwartz function | In mathematics, Schwartz space S {\displaystyle {\mathcal {S}}} is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1912_chunk_0 | 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 are at least 3/16. Subsequent works improved th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1913_chunk_0 | 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 at infinity of the corresponding representation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1914_chunk_0 | 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 a polynomial f ( z ) = ( z − r 1 ) ⋯ ( z − r n )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1915_chunk_0 | 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 large by Tao. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1916_chunk_0 | Serre 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 specifies the weight and level of the modular form. The conjecture in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1917_chunk_0 | Serre's intersection formula | 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 numbers, around 1949, there had been a question of how to provide a more fle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1918_chunk_0 | Serre's intersection formula | {\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 ) ⊗ ( R / Q ) ) < ∞ . {\displaystyle \ell _{R}((R/P)\otimes (R/Q))<\in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1919_chunk_0 | Serre's intersection formula | 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 statements of these conjectures where R/P and R/Q are replaced by finitely g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1920_chunk_0 | 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 Schmidt later pointed out a gap in the proof, which was fixed by Shafarevich (1989). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1921_chunk_0 | Sharkovsky'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 the implications of the theorem is that if a discr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1922_chunk_0 | 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 polyhedric cone in the Minkowski space of the field (Ne... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1923_chunk_0 | 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 primarily arise in various branches of number theory, s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1924_chunk_0 | 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 coordinates in the ring of integers O of K, provided g > 0. The theorem was first... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1925_chunk_0 | 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 demonstrated by the chaotic hypot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1926_chunk_0 | 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 Solèr, the result is significant for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1927_chunk_0 | 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 notes,Nothing in the assumptions ment... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1928_chunk_0 | 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 derivative of the Lerch transcendent: ln σ = − 1 2 ∂ Φ ∂ s ( 1 2 , 0 , 1 ) {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1929_chunk_0 | 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 − 1 Δ ( a , b , c ) 2 z − 1 π 1 / 2 Γ (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1930_chunk_0 | 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}+4y^{4}} , and it frequently forms the basis of problems in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1931_chunk_0 | 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 if there exists a unitary matrix U such t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1932_chunk_0 | 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 the second equality is cyclic invariance.Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1933_chunk_0 | 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, …, mp are non-negative integers. The degree ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1934_chunk_0 | 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. Let n denote the size of the matrices A and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1935_chunk_0 | 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 \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1936_chunk_0 | 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 simplex. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1937_chunk_0 | 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-Wigner function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1938_chunk_0 | 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 a biographical essay on Spence. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1939_chunk_0 | 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 simplex contains a cell whose vertices all ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1940_chunk_0 | 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 as the Knaster–Kuratowski–Mazurkiewicz lemm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1941_chunk_0 | 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 + 1 ) = ( z + a ) z + 1 2 e − z − a ( c 0 + ∑ k = 1 a − ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1942_chunk_0 | 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 feasible for arbitrary-precision evaluation of the gamma fun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1943_chunk_0 | 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 due to Ludwig Stickelberger (1890). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1944_chunk_0 | 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 of two completely positive maps each of which has a special form: A *-repre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1945_chunk_0 | Karamata notation | 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 relating three different sequences of polynomia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1946_chunk_0 | Stirling's 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 precise result was first stated by Abraham de Moivre.One... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1947_chunk_0 | Stirling's 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 The error term in either base can be expressed more precis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1948_chunk_0 | Representation theorem for Boolean algebras | 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 first half of the 20th century. The theorem was first proved by Marshall H. Stone.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1949_chunk_0 | 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 {\displaystyle {\mathcal {H}}} and one-parameter families ( U t ) t ∈ R {\displaystyle (U_{t})_{t\in \mathbb {R} }} o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1950_chunk_0 | 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 say that it is merely weakly measurable, at least when the Hilbert space is separable. This is an impress... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1951_chunk_0 | 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 |
wiki_1952_chunk_0 | 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 proved that the function space, carrying the compact-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1953_chunk_0 | 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 {\displaystyle x} of X {\displaystyle X} to the constant ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1954_chunk_0 | 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 {\displaystyle EG} of B G {\displaystyle BG} to X... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1955_chunk_0 | 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 {\displaystyle X} is allowed to be non-trivial. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1956_chunk_0 | 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\rightarrow \mathbb {R} ^{d}} on a triangle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1957_chunk_0 | 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 follow the principle of simplest shape:... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1958_chunk_0 | 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 |
wiki_1959_chunk_0 | Suslin's 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) showed that the statement can neither be proven no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1960_chunk_0 | 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 and only if all the following matrices have a positiv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1961_chunk_0 | 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 edge coloring is easy to find: use two colors ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1962_chunk_0 | 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 Schütte (Tait 1966); Independently by Prawitz (Prawi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1963_chunk_0 | 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 differential equation d X t = sgn ( X t ) d B... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1964_chunk_0 | 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 and the solution X is adapted to the filt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1965_chunk_0 | 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 {B}}_{s}{\big )}\,\mathrm {d} {\hat {B}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1966_chunk_0 | 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 the axiom of choice. The opposite direction was already known, thus th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1967_chunk_0 | 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 |
wiki_1968_chunk_0 | 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 |
wiki_1969_chunk_0 | 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 finite, and in the proof of Mihăilescu's theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1970_chunk_0 | 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 there is a morphism f: A → B {\displaystyle f:A\rightarrow B} in E {\displaystyle \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1971_chunk_0 | 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 Lie groups. The second and third volumes, on algebr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1972_chunk_0 | 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 |
wiki_1973_chunk_0 | 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 isotopic to f such that at least one of the following holds: g is periodic, i.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1974_chunk_0 | Geometrisation 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, which states that every simply connected Riemann surface ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1975_chunk_0 | Geometrisation 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 conjecture. Thurston announced a proof in the 1980s and since... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1976_chunk_0 | Geometrisation 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 manuscripts filling in the complete details of his arguments. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1977_chunk_0 | Thābit ibn Qurra | 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 to describe the ratios between geometrical quantities, a st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1978_chunk_0 | Thābit ibn Qurra | 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 degrees. The method of reduction and composition used by Thābit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1979_chunk_0 | Thābit ibn Qurra | 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 relations and the resulting exponential series allowed Thābit to ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1980_chunk_0 | Thābit ibn Qurra | 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 of a paraboloid. The use of uneven sections, while relativel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1981_chunk_0 | 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 variations. Roughly, it shows that weak lower semicontinuity for integral functionals is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1982_chunk_0 | 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 i(K, K*) vanish for i ≥ 1. This resul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1983_chunk_0 | Tucker decomposition | In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1984_chunk_0 | Tucker decomposition | In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1985_chunk_0 | Tucker's lemma | In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker. Let T be a triangulation of the closed n-dimensional ball B n {\displaystyle B_{n}} . Assume T is antipodally symmetric on the boundary sphere S n − 1 {\displaystyle S_{n-1}} . That means that the subset o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1986_chunk_0 | Tucker's lemma | . , + n , − n } {\displaystyle L:V(T)\to \{+1,-1,+2,-2,...,+n,-n\}} be a labeling of the vertices of T which is an odd function on S n − 1 {\displaystyle S_{n-1}} , i.e, L ( − v ) = − L ( v ) {\displaystyle L(-v)=-L(v)} for every vertex v ∈ S n − 1 {\displaystyle v\in S_{n-1}} . Then Tucker's lemma states that T contai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1987_chunk_0 | Tychonoff theorem | In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1988_chunk_0 | Tychonoff theorem | (This reference is mentioned in "Topology" by Hocking and Young, Dover Publications, Ind.) Tychonoff's theorem is often considered as perhaps the single most important result in general topology (along with Urysohn's lemma). The theorem is also valid for topological spaces based on fuzzy sets. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1989_chunk_0 | Varadhan's lemma | In mathematics, Varadhan's lemma is a result from the large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic φ(Zε) of a family of random variables Zε as ε becomes small in terms of a rate function for the variables. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1990_chunk_0 | Veblen's theorem | In mathematics, Veblen's theorem, introduced by Oswald Veblen (1912), states that the set of edges of a finite graph can be written as a union of disjoint simple cycles if and only if every vertex has even degree. Thus, it is closely related to the theorem of Euler (1736) that a finite graph has an Euler tour (a single... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1991_chunk_0 | Veblen's theorem | However, Veblen's theorem applies also to disconnected graphs, and can be generalized to infinite graphs in which every vertex has finite degree (Sabidussi 1964). If a countably infinite graph G has no odd-degree vertices, then it may be written as a union of disjoint (finite) simple cycles if and only if every finite ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1992_chunk_0 | Verdier duality | In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by Jean-Louis Verdier (1995) as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1993_chunk_0 | Viennot's geometric construction | In mathematics, Viennot's geometric construction (named after Xavier Gérard Viennot) gives a diagrammatic interpretation of the Robinson–Schensted correspondence in terms of shadow lines. It has a generalization to the Robinson–Schensted–Knuth correspondence, which is known as the matrix-ball construction. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1994_chunk_0 | Vieta's theorem | In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta"). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1995_chunk_0 | Vinberg's algorithm | In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. Conway (1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1996_chunk_0 | Vincent's theorem | In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent—is a theorem that isolates the real roots of polynomials with rational coefficients. Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, havin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1997_chunk_0 | Vinogradov's mean-value theorem | In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers. It is an important inequality in analytic number theory, named for I. M. Vinogradov. More specifically, let J s , k ( X ) {\displaystyle J_{s,k}(X)} count the number of solutions to the system of k {\displaystyle k} s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1998_chunk_0 | Vinogradov's mean-value theorem | {\displaystyle f_{k}(\mathbf {\alpha } ;X)=\sum _{1\leq x\leq X}\exp(2\pi i(\alpha _{1}x+\cdots +\alpha _{k}x^{k})).} Vinogradov's mean-value theorem gives an upper bound on the value of J s , k ( X ) {\displaystyle J_{s,k}(X)} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1999_chunk_0 | Vinogradov's mean-value theorem | A strong estimate for J s , k ( X ) {\displaystyle J_{s,k}(X)} is an important part of the Hardy-Littlewood method for attacking Waring's problem and also for demonstrating a zero free region for the Riemann zeta-function in the critical strip. Various bounds have been produced for J s , k ( X ) {\displaystyle J_{s,k}(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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