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57-cell Summary 57-cell In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. The symmetry order is 3420, from the product of the number of cells ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
57-cell Summary 57-cell The symmetry abstract structure is the projective special linear group, L2(19). It has Schläfli type {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by H. S. M. Coxeter (1982). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Buckyball surface Summary ADE_classification In mathematics, the ADE classification (originally A-D-E classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, rather than a posteriori verifica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Buckyball surface Summary ADE_classification Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of π / 2 = 90 ∘ {\displaystyle \pi /2=90^{\circ }} (no edge between the vertices) or 2 π / 3 = 120 ∘ {\displaystyle 2\pi /3=120^{\circ }} (sing... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Buckyball surface Summary ADE_classification {\displaystyle D_{n}.} If one extends the families to include redundant terms, one obtains the exceptional isomorphisms D 3 ≅ A 3 , E 4 ≅ A 4 , E 5 ≅ D 5 , {\displaystyle D_{3}\cong A_{3},E_{4}\cong A_{4},E_{5}\cong D_{5},} and corresponding isomorphisms of classified object... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
AKNS system Summary AKNS_system In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur from their publication in Studies in Applied Mathematics: Ablowitz, Kaup, and Newell et al. (1974). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
ATS theorem Summary ATS_theorem In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abel transform Summary Abel_transform In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by F ( y ) = 2 ∫ y ∞ f ( r ) r r 2 − y 2 d r . {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abel transform Summary Abel_transform {\displaystyle f(r)=-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}\,{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.} In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is use... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abel transform Summary Abel_transform In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the cen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abel–Jacobi theorem Summary Abel–Jacobi_theorem In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abel–Plana formula Summary Abel–Plana_formula In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It states that ∑ n = 0 ∞ f ( a + n ) = ∫ a ∞ f ( x ) d x + f ( a ) 2 + ∫ 0 ∞ f ( a − i x ) − f ( a + i x ) i ( e 2 π ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abel–Ruffini theorem Summary Abel–Ruffini_theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abel–Ruffini theorem Summary Abel–Ruffini_theorem This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that x 5 − x − 1 = 0 {\displaystyle x^{5}-x-1=0} is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abhyankar–Moh theorem Summary Abhyankar–Moh_theorem In mathematics, the Abhyankar–Moh theorem states that if L {\displaystyle L} is a complex line in the complex affine plane C 2 {\displaystyle \mathbb {C} ^{2}} , then every embedding of L {\displaystyle L} into C 2 {\displaystyle \mathbb {C} ^{2}} extends to an automo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ackermann ordinal Summary Ackermann_ordinal In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal. Unfortunately there is no standard notation for ordinals ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ackermann ordinal Summary Ackermann_ordinal Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The last one is an extension of the Veblen funct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Adams–Novikov spectral sequence Summary Adams–Novikov_spectral_sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ahlfors measure conjecture Summary Ahlfors_measure_conjecture In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0. The conjecture was introduced by Ahlfors (1966), who proved it in the case that the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Al-Salam–Ismail polynomials Summary Al-Salam–Ismail_polynomials In mathematics, the Al-Salam–Ismail polynomials are a family of orthogonal polynomials introduced by Al-Salam and Ismail (1983). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Albanese variety Summary Albanese_variety In mathematics, the Albanese variety A ( V ) {\displaystyle A(V)} , named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Skein module Summary Alexander_polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now cal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Almgren–Pitts min-max theory Summary Almgren–Pitts_min-max_theory In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the efforts for generalizing George David Birkhoff's method fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alperin–Brauer–Gorenstein theorem Summary Alperin–Brauer–Gorenstein_theorem In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective speci... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alvis–Curtis dual Summary Alvis–Curtis_dual In mathematics, the Alvis–Curtis duality is a duality operation on the characters of a reductive group over a finite field, introduced by Charles W. Curtis (1980) and studied by his student Dean Alvis (1979). Kawanaka (1981, 1982) introduced a similar duality operation for Li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Andreotti–Frankel theorem Summary Andreotti–Frankel_theorem In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if V {\displaystyle V} is a smooth, complex affine variety of complex dimension n {\displaystyle n} or, more generally, if V {\displaystyle V} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Andreotti–Grauert theorem Summary Andreotti–Grauert_theorem In mathematics, the Andreotti–Grauert theorem, introduced by Andreotti and Grauert (1962), gives conditions for cohomology groups of coherent sheaves over complex manifolds to vanish or to be finite-dimensional. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Andreotti–Vesentini theorem Summary Andreotti–Vesentini_theorem In mathematics, the Andreotti–Vesentini separation theorem, introduced by Aldo Andreotti and Edoardo Vesentini (1965, 1965b) states that certain cohomology groups of coherent sheaves are separated. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Andrews–Curtis conjecture Summary Andrews–Curtis_conjecture In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named aft... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
André–Oort conjecture Summary André–Oort_conjecture In mathematics, the André–Oort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the Manin–Mumford conjecture, which is now a theorem (proven in several different ways). The conjecture concerns it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Angelescu polynomials Summary Angelescu_polynomials In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Angelescu (1938). The polynomials can be given by the generating functionBoas & Buck (1958, p.41) They can also be defined by the equation w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Anger function Summary Anger_function In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as J ν ( z ) = 1 π ∫ 0 π cos ( ν θ − z sin θ ) d θ {\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta } with complex paramete... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Appell–Humbert theorem Summary Appell–Humbert_theorem In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety. It was proved for 2-dimensional tori by Appell (1891) and Humbert (1893), and in general by Lefschetz (1921) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arason invariant Summary Arason_invariant In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by (Arason 1975, Theorem 5.7). Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arens square Summary Arens_square In mathematics, the Arens square is a topological space, named for Richard Friederich Arens. Its role is mainly to serve as a counterexample. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arens–Fort space Summary Arens–Fort_space In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arf invariant Summary Arf_invariant In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf (1941) when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arf invariant Summary Arf_invariant Two nonsingular quadratic forms over F2 are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to Leonard Dickson (1901), even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arthur's conjectures Summary Arthur's_conjectures In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local fields made by James Arthur (1989), motivated by the Arthur–Selberg trace formula... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relative trace formula Summary Relative_trace_formula In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relative trace formula Summary Relative_trace_formula The simple trace formula (Flicker & Kazhdan 1988) is less general but easier to prove. The local trace formula is an analogue over local fields. Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artin approximation theorem Summary Artin_approximation_theorem In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin (1969) in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k. More precisel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artin conductor Summary Swan_representation In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by Emil Artin (1930, 1931) as an expression appearing in the functional equation of an Artin L-function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dwork's lemma Summary Artin–Hasse_exponential In mathematics, the Artin–Hasse exponential, introduced by Artin and Hasse (1928), is the power series given by E p ( x ) = exp ( x + x p p + x p 2 p 2 + x p 3 p 3 + ⋯ ) . {\displaystyle E_{p}(x)=\exp \left(x+{\frac {x^{p}}{p}}+{\frac {x^{p^{2}}}{p^{2}}}+{\frac {x^{p^{3}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artin–Mazur zeta function Summary Artin–Mazur_zeta_function In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals. It is defined from a given function f {\displaystyle f} as th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Krull's intersection theorem Summary Artin-Rees_lemma In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was kno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Krull's intersection theorem Summary Artin-Rees_lemma One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion. The lemma also plays a key role in the study of ℓ-adic sheaves. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artin–Zorn theorem Summary Artin–Zorn_theorem In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin.The Artin–Zorn theorem is a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Askey scheme Summary Askey_scheme In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in Andrews & Askey (1985), the Askey scheme was first drawn by Labelle (1985) and by Askey ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Askey–Gasper inequality Summary Askey–Gasper_inequality In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976) and used in the proof of the Bieberbach conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Askey–Wilson polynomial Summary Askey–Wilson_polynomials In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Askey and Wilson (1985) as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as spec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Assouad–Nagata dimension Summary Assouad–Nagata_dimension In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces, introduced by Jun-iti Nagata in 1958 and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Atiyah conjecture Summary Atiyah_conjecture In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of l 2 {\displaystyle l^{2}} -Betti numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Atiyah conjecture on configurations Summary Atiyah_conjecture_on_configurations In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by Atiyah (2000, 2001) stating that a certain n by n matrix depending on n points in R3 is always non-singular. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Atiyah-Bott fixed point theorem Summary Atiyah-Bott_fixed_point_theorem In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M, which uses an elliptic complex on M. This is a system of ellip... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Atiyah-Hirzebruch spectral sequence Summary Atiyah–Hirzebruch_spectral_sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by Michael Atiyah and Friedrich Hirzebruch (1961) in the special case of topological K-theory. For a CW comple... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Atiyah-Hirzebruch spectral sequence Summary Atiyah–Hirzebruch_spectral_sequence It can be derived from an exact couple that gives the E 1 {\displaystyle E_{1}} page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E {\displaystyle E} . In detail, assume X {\displaystyle X} to be ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Atiyah–Jones conjecture Summary Atiyah–Jones_conjecture In mathematics, the Atiyah–Jones conjecture is a conjecture about the homology of the moduli spaces of instantons. The original form of the conjecture considered instantons over a 4-dimensional sphere. It was introduced by Michael Francis Atiyah and John D. S. Jon... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Atiyah–Jones conjecture Summary Atiyah–Jones_conjecture The more general version of the Atiyah–Jones conjecture is a question about the homology of the moduli spaces of instantons on any 4-dimensional real manifold, or on a complex surface. The Atiyah–Jones conjecture has been proved for ruled surfaces by R. J. Milgram... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Aubin–Lions lemma Summary Aubin–Lions_lemma In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of nonlinear evolutionary partial differential equations. Typically, to prove ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Auslander algebra Summary Auslander_algebra In mathematics, the Auslander algebra of an algebra A is the endomorphism ring of the sum of the indecomposable modules of A. It was introduced by Auslander (1974). An Artin algebra Γ is called an Auslander algebra if gl dim Γ ≤ 2 and if 0→Γ→I→J→K→0 is a minimal injective res... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ax-Grothendieck theorem Summary Ax–Grothendieck_theorem In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck.The theorem is often given as this special case: If P is an injective polynomial function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Babenko–Beckner inequality Summary Babenko–Beckner_inequality In mathematics, the Babenko–Beckner inequality (after K. Ivan Babenko and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of Lp spaces. The (q, p)-norm of the n-d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Babuška–Lax–Milgram theorem Summary Babuška–Lax–Milgram_theorem In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bachmann–Howard ordinal Summary Bachmann–Howard_ordinal In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Backus–Gilbert method Summary Backus–Gilbert_method In mathematics, the Backus–Gilbert method, also known as the optimally localized average (OLA) method is named for its discoverers, geophysicists George E. Backus and James Freeman Gilbert. It is a regularization method for obtaining meaningful solutions to ill-posed ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Backus–Gilbert method Summary Backus–Gilbert_method Given a data array X, the basic Backus-Gilbert inverse is: H θ = C − 1 G θ G θ T C − 1 G θ {\displaystyle \mathbf {H} _{\theta }={\frac {\mathbf {C} ^{-1}\mathbf {G} _{\theta }}{\mathbf {G} _{\theta }^{T}\mathbf {C} ^{-1}\mathbf {G} _{\theta }}}} where C is the covari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Baily–Borel compactification Summary Minimal_compactification In mathematics, the Baily–Borel compactification is a compactification of a quotient of a Hermitian symmetric space by an arithmetic group, introduced by Walter L. Baily and Armand Borel (1964, 1966). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Baker-Campbell-Hausdorff formula Summary Baker–Campbell–Hausdorff_formula In mathematics, the Baker–Campbell–Hausdorff formula is the solution for Z {\displaystyle Z} to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Baker-Campbell-Hausdorff formula Summary Baker–Campbell–Hausdorff_formula Meanwhile, every element g {\displaystyle g} sufficiently close to the identity in G {\displaystyle G} can be expressed as g = e X {\displaystyle g=e^{X}} for a small X {\displaystyle X} in g {\displaystyle {\mathfrak {g}}} . Thus, we can say tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Baker-Campbell-Hausdorff formula Summary Baker–Campbell–Hausdorff_formula If X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small n × n {\displaystyle n\times n} matrices, then Z {\displaystyle Z} can be computed as the logarithm of e X e Y {\displaystyle e^{X}e^{Y}} , where the exponentials and the logar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Balian–Low theorem Summary Balian–Low_theorem In mathematics, the Balian–Low theorem in Fourier analysis is named for Roger Balian and Francis E. Low. The theorem states that there is no well-localized window function (or Gabor atom) g either in time or frequency for an exact Gabor frame (Riesz Basis). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Banach fixed point theorem Summary Contractive_mapping_theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach-Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Banach game Summary Banach_game In mathematics, the Banach game is a topological game introduced by Stefan Banach in 1935 in the second addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game.Given a subset X {\displaystyle X} of real numbers, two players alternatively write down arbitrary (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Banach–Stone theorem Summary Banach–Stone_theorem In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bareiss Algorithm Summary Bareiss_Algorithm In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Barnes G-function Summary Barnes_G-function In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Barnes G-function Summary Barnes_G-function Formally, the Barnes G-function is defined in the following Weierstrass product form: G ( 1 + z ) = ( 2 π ) z / 2 exp ( − z + z 2 ( 1 + γ ) 2 ) ∏ k = 1 ∞ { ( 1 + z k ) k exp ( z 2 2 k − z ) } {\displaystyle G(1+z)=(2\pi )^{z/2}\exp \left(-{\frac {z+z^{2}(1+\gamma )}{2}}\r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Barnes–Wall lattice Summary Barnes–Wall_lattice In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Barnes & Wall (1959)), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Barnes–Wall lattice Summary Barnes–Wall_lattice There are 4320 vectors of norm 4 in the Barnes–Wall lattice (the shortest nonzero vectors in this lattice). The genus of the Barnes–Wall lattice was described by Scharlau & Venkov (1994) and contains 24 lattices; all the elements other than the Barnes–Wall lattice have ro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bass–Quillen conjecture Summary Bass–Quillen_conjecture In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring A {\displaystyle A} . The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bateman function Summary Bateman_function In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by k n ( x ) = 2 π ∫ 0 π / 2 cos ( x tan θ − n θ ) d θ {\displaystyle \displaystyle k_{n}(x)={\frac {2}{\pi }}\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bateman polynomials Summary Bateman_polynomials In mathematics, the Bateman polynomials are a family Fn of orthogonal polynomials introduced by Bateman (1933). The Bateman–Pasternack polynomials are a generalization introduced by Pasternack (1939). Bateman polynomials can be defined by the relation F n ( d d x ) sech ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bateman polynomials Summary Bateman_polynomials where Pn is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by F n ( x ) = 3 F 2 ( − n , n + 1 , 1 2 ( x + 1 ) 1 , 1 ; 1 ) . {\displaystyle F_{n}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+1)\\1,~1\end{array}};1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bateman polynomials Summary Bateman_polynomials Pasternack (1939) generalized the Bateman polynomials to polynomials Fmn with F n m ( d d x ) sech m + 1 ( x ) = sech m + 1 ( x ) P n ( tanh ( x ) ) {\displaystyle F_{n}^{m}\left({\frac {d}{dx}}\right)\operatorname {sech} ^{m+1}(x)=\operatorname {sech} ^{m+1}(x)P_{n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bauer–Fike theorem Summary Bauer–Fike_theorem In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chose... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Beauville–Laszlo theorem Summary Beauville–Laszlo_theorem In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by Arnaud Beauville and Yves Laszlo (1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Beck–Fiala theorem Summary Beck–Fiala_theorem In mathematics, the Beck–Fiala theorem is a major theorem in discrepancy theory due to József Beck and Tibor Fiala. Discrepancy is concerned with coloring elements of a ground set such that each set in a certain set system is as balanced as possible, i.e., has approximately... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bell series Summary Bell_series In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function f {\displaystyle f} and a prime p {\displaystyle p} , define the formal power series f ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bell series Summary Bell_series Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem: given multiplicative functions f {\displaystyle f} and g {\displaystyle g} , one has f = g {\displaystyle f=g} if and only if: f p ( x ) = g p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Aitken's array Summary Aitken's_array In mathematics, the Bell triangle is a triangle of numbers analogous to Pascal's triangle, whose values count partitions of a set in which a given element is the largest singleton. It is named for its close connection to the Bell numbers, which may be found on both sides of the tri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Beltrami differential equation Summary Beltrami_equation In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ∂ w ∂ z ¯ = μ ∂ w ∂ z . {\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}.} for w a complex distribution of the co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Beltrami differential equation Summary Beltrami_equation Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 19... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Beltrami differential equation Summary Beltrami_equation The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Rie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bendixson–Dulac theorem Summary Bendixson–Dulac_theorem In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a C 1 {\displaystyle C^{1}} function φ ( x , y ) {\displaystyle \varphi (x,y)} (called the Dulac function) such that the expression ∂ ( φ f ) ∂ x + ∂ ( φ g ) ∂ y {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Benjamin–Ono equation Summary Benjamin–Ono_equation In mathematics, the Benjamin–Ono equation is a nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water. It was introduced by Benjamin (1967) and Ono (1975). The Benjamin–Ono equation is u t + u u x + H u x x = 0 {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bergman–Weil formula Summary Weil_domain In mathematics, the Bergman–Weil formula is an integral representation for holomorphic functions of several variables generalizing the Cauchy integral formula. It was introduced by Bergmann (1936) and Weil (1935). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Seidel triangle Summary Bernoulli_Numbers In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Seidel triangle Summary Bernoulli_Numbers For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Seidel triangle Summary Bernoulli_Numbers The Bernoulli numbers are special values of the Bernoulli polynomials B n ( x ) {\displaystyle B_{n}(x)} , with B n − = B n ( 0 ) {\displaystyle B_{n}^{-{}}=B_{n}(0)} and B n + = B n ( 1 ) {\displaystyle B_{n}^{+}=B_{n}(1)} .The Bernoulli numbers were discovered around the same... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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