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c_tzlx1fwmps0t | In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. The symmetry order is 3420, from the product of the number of cells (57) and the symmetry of... | 57-cell |
c_mw7da45705vq | 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). | 57-cell |
c_rr6gjxewp9u3 | 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 verification of a parallelism, was posed in (Arnold 1... | Buckyball surface |
c_xlkapcd9orhw | 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 }} (single edge between the vertices). These are two ... | Buckyball surface |
c_yc5h1d1dydpe | {\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 objects. The A, D, E nomenclature also yields the s... | Buckyball surface |
c_0cklfwh2cinn | 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). | AKNS system |
c_yrdb9bawxsvl | 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. | ATS theorem |
c_00k3xqev9nuo | In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by F ( y ) = 2 ∫ y ∞ f ( r ) r r 2 − y 2 d r . {\displaystyle F(y)=2\int _{y}^{\infty }{\frac {f(r)... | Abel transform |
c_da9hea28lfxe | {\displaystyle f(r)=-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}\,{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.} In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function g... | Abel transform |
c_onnu5kof633p | 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 center. Abel transform is limited to appl... | Abel transform |
c_jjyguxeiy1hm | In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are lin... | Abel–Jacobi theorem |
c_azb6iag7msrz | In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It states that ∑ n = 0 ∞ f ( a + n ) = ∫ a ∞ f ( x ) d x + f ( a ) 2 + ∫ 0 ∞ f ( a − i x ) − f ( a + i x ) i ( e 2 π x − 1 ) d x {\displaystyle \sum _{n=0}^{\infty... | Abel–Plana formula |
c_wiyy8drlk858 | In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.... | Abel–Ruffini theorem |
c_z19kw682d8yo | This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that x 5 − x − 1 = 0 {\displaystyle x^{5}-x-1=0} is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degree five or higher cannot be solved in radicals. The... | Abel–Ruffini theorem |
c_1uazh0jjjqt4 | In mathematics, the Abhyankar–Moh theorem states that if L {\displaystyle L} is a complex line in the complex affine plane C 2 {\displaystyle \mathbb {C} ^{2}} , then every embedding of L {\displaystyle L} into C 2 {\displaystyle \mathbb {C} ^{2}} extends to an automorphism of the plane. It is named after Shreeram Shan... | Abhyankar–Moh theorem |
c_dcxaijvx72pu | 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 beyond the Feferman–Schütte ordinal Γ0. | Ackermann ordinal |
c_uvm5b8pzjba9 | Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The last one is an extension of the Veblen functions for more than 2 arguments. The smaller ... | Ackermann ordinal |
c_ojhmqc0edszv | In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation ... | Adams–Novikov spectral sequence |
c_grukg5kc0zn7 | In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0. The conjecture was introduced by Ahlfors (1966), who proved it in the case that the Kleinian group has a fundamental domain with a finite number o... | Ahlfors measure conjecture |
c_feiw30u5l0ku | In mathematics, the Al-Salam–Ismail polynomials are a family of orthogonal polynomials introduced by Al-Salam and Ismail (1983). | Al-Salam–Ismail polynomials |
c_tgo7k87urae4 | In mathematics, the Albanese variety A ( V ) {\displaystyle A(V)} , named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. | Albanese variety |
c_9qsptilvfwcm | 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 called the Alexander–Conway polynomial, could... | Skein module |
c_tpce8jeqr00u | In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the efforts for generalizing George David Birkhoff's method for the construction of simple closed geodesics on the sphere, to al... | Almgren–Pitts min-max theory |
c_tkux3jb9codx | In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective special unitary groups over a finite field of odd order, depending on a certain c... | Alperin–Brauer–Gorenstein theorem |
c_zln36yr8p6y8 | 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 Lie algebras. Alvis–Curtis duality has order 2... | Alvis–Curtis dual |
c_y8wfm1aytot2 | In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if V {\displaystyle V} is a smooth, complex affine variety of complex dimension n {\displaystyle n} or, more generally, if V {\displaystyle V} is any Stein manifold of dimension n {\displaystyle n} , the... | Andreotti–Frankel theorem |
c_pti6jkuba8cv | 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. | Andreotti–Grauert theorem |
c_qng0h4keg6sr | 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. | Andreotti–Vesentini theorem |
c_r7e9h750le97 | 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 after James J. Andrews and Morton L. Curtis who proposed it in ... | Andrews–Curtis conjecture |
c_v6hwj22178f3 | In mathematics, the André–Oort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the Manin–Mumford conjecture, which is now a theorem (proven in several different ways). The conjecture concerns itself with a characterization of the Zariski closure ... | André–Oort conjecture |
c_k86gu6f1rixs | In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Angelescu (1938). The polynomials can be given by the generating functionBoas & Buck (1958, p.41) They can also be defined by the equation where A n ( x ) n ! {\displaystyle {\frac {A_{n}(x)}{... | Angelescu polynomials |
c_jdx6qum0pr34 | In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as J ν ( z ) = 1 π ∫ 0 π cos ( ν θ − z sin θ ) d θ {\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta } with complex parameter v and complex variable x. It is clos... | Anger function |
c_3yqyqwasgusv | 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) | Appell–Humbert theorem |
c_omwdeauyd6jt | In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by (Arason 1975, Theorem 5.7). The Rost invariant is a generalization of th... | Arason invariant |
c_odmz6gqdc457 | In mathematics, the Arens square is a topological space, named for Richard Friederich Arens. Its role is mainly to serve as a counterexample. | Arens square |
c_a5hh5laklraf | 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. | Arens–Fort space |
c_vwchyxpdi2gv | 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 substitute, in characteristic 2, for the di... | Arf invariant |
c_ykbo00am4w79 | 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. The Arf invariant is particularly app... | Arf invariant |
c_ckxa3wtuj41u | In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local fields made by James Arthur (1989), motivated by the Arthur–Selberg trace formula. Arthur's conjectures imply the generalized Raman... | Arthur's conjectures |
c_d9ve481093ne | In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of G(A) on the discrete part L20... | Relative trace formula |
c_5qxo8m3ryyi2 | 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. | Relative trace formula |
c_pgcejvu4gxe4 | In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin (1969) in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k. More precisely, Artin proved two such theorems: one, in 1968, on approximatio... | Artin approximation theorem |
c_x2lzcpiqsaqn | 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. | Artin conductor |
c_icji78gel4cj | In mathematics, the Artin–Hasse exponential, introduced by Artin and Hasse (1928), is the power series given by E p ( x ) = exp ( x + x p p + x p 2 p 2 + x p 3 p 3 + ⋯ ) . {\displaystyle E_{p}(x)=\exp \left(x+{\frac {x^{p}}{p}}+{\frac {x^{p^{2}}}{p^{2}}}+{\frac {x^{p^{3}}}{p^{3}}}+\cdots \right).} | Dwork's lemma |
c_5hymfai65ia8 | In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals. It is defined from a given function f {\displaystyle f} as the formal power series ζ f ( z ) = exp ( ∑ n = 1 ∞ | Fix ... | Artin–Mazur zeta function |
c_u5bjsocvrndc | In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work. An intuitive ... | Krull's intersection theorem |
c_vaaldnq4ji1q | 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. | Krull's intersection theorem |
c_k244vi1qobjk | In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin.The Artin–Zorn theorem is a generalization of the Wedderburn theorem, whic... | Artin–Zorn theorem |
c_jw0b0o898qkl | 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 and Wilson (1985), and has since b... | Askey scheme |
c_7a7p6ra8wy1z | 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. | Askey–Gasper inequality |
c_rh52ohklnge6 | In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Askey and Wilson (1985) as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme. Ask... | Askey–Wilson polynomial |
c_doomvpw58gbe | 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. | Assouad–Nagata dimension |
c_87eyaethkx5l | 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. | Atiyah conjecture |
c_dos14tzsjt2f | 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. | Atiyah conjecture on configurations |
c_oyru6emfr09w | In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M, which uses an elliptic complex on M. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham c... | Atiyah-Bott fixed point theorem |
c_8w46tb63g94i | 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 complex X {\displaystyle X} and a generalized cohomology theory E ∙ {\displaystyle E^{... | Atiyah-Hirzebruch spectral sequence |
c_z2ryupzfcasa | It can be derived from an exact couple that gives the E 1 {\displaystyle E_{1}} page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E {\displaystyle E} . In detail, assume X {\displaystyle X} to be the total space of a Serre fibration with fibre F {\displaystyle F} and base spa... | Atiyah-Hirzebruch spectral sequence |
c_o5pkrmf25gy4 | 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. Jones (1978) and proved by Charles P. Boyer, Jacques C. Hur... | Atiyah–Jones conjecture |
c_250f3fptgm6j | 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 and J. Hurtubise, and for rational surfaces by Elizabet... | Atiyah–Jones conjecture |
c_asssauy6uebw | In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of nonlinear evolutionary partial differential equations. Typically, to prove the existence of solutions one first constru... | Aubin–Lions lemma |
c_ijb2j9mfaed3 | In mathematics, the Auslander algebra of an algebra A is the endomorphism ring of the sum of the indecomposable modules of A. It was introduced by Auslander (1974). An Artin algebra Γ is called an Auslander algebra if gl dim Γ ≤ 2 and if 0→Γ→I→J→K→0 is a minimal injective resolution of Γ then I and J are projective Γ-m... | Auslander algebra |
c_oebkcyu8logq | In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck.The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself th... | Ax-Grothendieck theorem |
c_yncy70iwerpy | In mathematics, the Babenko–Beckner inequality (after K. Ivan Babenko and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of Lp spaces. The (q, p)-norm of the n-dimensional Fourier transform is defined to be ‖ F ‖ q , p = su... | Babenko–Beckner inequality |
c_qcttvo0o2gnp | In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Ivo Babuška,... | Babuška–Lax–Milgram theorem |
c_384mk2vcacuz | In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was... | Bachmann–Howard ordinal |
c_071yduxcv264 | In mathematics, the Backus–Gilbert method, also known as the optimally localized average (OLA) method is named for its discoverers, geophysicists George E. Backus and James Freeman Gilbert. It is a regularization method for obtaining meaningful solutions to ill-posed inverse problems. Where other regularization methods... | Backus–Gilbert method |
c_kjj8emve3un9 | Given a data array X, the basic Backus-Gilbert inverse is: H θ = C − 1 G θ G θ T C − 1 G θ {\displaystyle \mathbf {H} _{\theta }={\frac {\mathbf {C} ^{-1}\mathbf {G} _{\theta }}{\mathbf {G} _{\theta }^{T}\mathbf {C} ^{-1}\mathbf {G} _{\theta }}}} where C is the covariance matrix of the data, and Gθ is an a priori const... | Backus–Gilbert method |
c_8rhmxsnwzw0k | 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). | Baily–Borel compactification |
c_mejrimhlvmbf | In mathematics, the Baker–Campbell–Hausdorff formula is the solution for Z {\displaystyle Z} to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z {\displaystyle Z} in Lie algebraic terms, that i... | Baker-Campbell-Hausdorff formula |
c_u6pzk01lxjwc | Meanwhile, every element g {\displaystyle g} sufficiently close to the identity in G {\displaystyle G} can be expressed as g = e X {\displaystyle g=e^{X}} for a small X {\displaystyle X} in g {\displaystyle {\mathfrak {g}}} . Thus, we can say that near the identity the group multiplication in G {\displaystyle G} —writt... | Baker-Campbell-Hausdorff formula |
c_az9yc0ux70vt | If X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small n × n {\displaystyle n\times n} matrices, then Z {\displaystyle Z} can be computed as the logarithm of e X e Y {\displaystyle e^{X}e^{Y}} , where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Haus... | Baker-Campbell-Hausdorff formula |
c_vy5p7kay5ox4 | 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). | Balian–Low theorem |
c_9yir56p0sf26 | In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach-Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides ... | Banach fixed point theorem |
c_gs7j14ag5lv2 | 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 (not necessarily in X {\displayst... | Banach game |
c_7z00o534ads3 | In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff space X from the Banach space structure of the space C... | Banach–Stone theorem |
c_tlnu2x4wp9do | In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder). The method can also be used to compute t... | Bareiss Algorithm |
c_ls1t04ei8de3 | In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma functio... | Barnes G-function |
c_5u6nzdwaexy9 | Formally, the Barnes G-function is defined in the following Weierstrass product form: G ( 1 + z ) = ( 2 π ) z / 2 exp ( − z + z 2 ( 1 + γ ) 2 ) ∏ k = 1 ∞ { ( 1 + z k ) k exp ( z 2 2 k − z ) } {\displaystyle G(1+z)=(2\pi )^{z/2}\exp \left(-{\frac {z+z^{2}(1+\gamma )}{2}}\right)\,\prod _{k=1}^{\infty }\left\{\left(1+... | Barnes G-function |
c_jks17odznklv | In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Barnes & Wall (1959)), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 2, and i... | Barnes–Wall lattice |
c_plygrh7wymnt | 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 root system of maximal rank 16. The Barnes–Wall la... | Barnes–Wall lattice |
c_adt8nmygmdsb | 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. | Bass–Quillen conjecture |
c_e5hyzj47o8io | In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by k n ( x ) = 2 π ∫ 0 π / 2 cos ( x tan θ − n θ ) d θ {\displaystyle \displaystyle k_{n}(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan \theta -n\thet... | Bateman function |
c_x1tq59vbyapq | 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 ( x ) = sech ( x ) P n ( tanh ( x ) ) . {\d... | Bateman polynomials |
c_59szn6lwo4rp | 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\right).} | Bateman polynomials |
c_tz8mnx84j3bp | Pasternack (1939) generalized the Bateman polynomials to polynomials Fmn with F n m ( d d x ) sech m + 1 ( x ) = sech m + 1 ( x ) P n ( tanh ( x ) ) {\displaystyle F_{n}^{m}\left({\frac {d}{dx}}\right)\operatorname {sech} ^{m+1}(x)=\operatorname {sech} ^{m+1}(x)P_{n}(\tanh(x))} These generalized polynomials also ... | Bateman polynomials |
c_zc0ppnedqiw5 | In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix. Informally s... | Bauer–Fike theorem |
c_3x40hj2qsgy8 | In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by Arnaud Beauville and Yves Laszlo (1995). | Beauville–Laszlo theorem |
c_7vwh1h0c9hvw | In mathematics, the Beck–Fiala theorem is a major theorem in discrepancy theory due to József Beck and Tibor Fiala. Discrepancy is concerned with coloring elements of a ground set such that each set in a certain set system is as balanced as possible, i.e., has approximately the same number of elements of each color. Th... | Beck–Fiala theorem |
c_2nijkz4ar35v | In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function f {\displaystyle f} and a prime p {\displaystyle p} , define the formal power series f p ( x ) {\displaystyle f_{p}(x)}... | Bell series |
c_8lih04g3amqk | Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem: given multiplicative functions f {\displaystyle f} and g {\displaystyle g} , one has f = g {\displaystyle f=g} if and only if: f p ( x ) = g p ( x ) {\displaystyle f_{p}(x)=g... | Bell series |
c_wfzz4d4r89ug | 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 triangle, and which are in turn named aft... | Aitken's array |
c_gc7639x6qibw | In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ∂ w ∂ z ¯ = μ ∂ w ∂ z . {\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}.} for w a complex distribution of the complex variable z in some open set U, with derivatives tha... | Beltrami differential equation |
c_c161hvgei9r4 | 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 1950s, provides global solutions of the equation on C and r... | Beltrami differential equation |
c_wuczjzzm6r9z | 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 Riemann mapping theorem and the simultaneous uniformization ... | Beltrami differential equation |
c_7i2sw9s4vsch | 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 {\displaystyle {\frac {\partial (\varphi f)}{\partial x}}+{\frac ... | Bendixson–Dulac theorem |
c_06jesdbp88n9 | 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 {\displaystyle u_{t}+uu_{x}+Hu_{xx}=0} where H is the Hi... | Benjamin–Ono equation |
c_r5bsnlua98rz | 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). | Bergman–Weil formula |
c_ex9ptij8bftg | 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 sum of m-th powers of the first n positive ... | Seidel triangle |
c_8m54qrreat5s | For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise. | Seidel triangle |
c_i7vog1x3b2hu | 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 time by the Swiss mathematician Jacob Ber... | Seidel triangle |
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