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c_va6oirrxpzz0 | In mathematics, the Dirichlet space on the domain Ω ⊆ C , D ( Ω ) {\displaystyle \Omega \subseteq \mathbb {C} ,\,{\mathcal {D}}(\Omega )} (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space H 2 ( Ω ) {\displaystyle H^{2}(\Omega... | Dirichlet space |
c_fvfo8dxa8lwq | It is not a norm in general, since D ( f ) = 0 {\displaystyle {\mathcal {D}}(f)=0} whenever f is a constant function. For f , g ∈ D ( Ω ) {\displaystyle f,\,g\in {\mathcal {D}}(\Omega )} , we define D ( f , g ) := 1 π ∬ Ω f ′ ( z ) g ′ ( z ) ¯ d A ( z ) . {\displaystyle {\mathcal {D}}(f,\,g):={1 \over \pi }\iint _{\Ome... | Dirichlet space |
c_9bmgzwvabuj6 | This is a semi-inner product, and clearly D ( f , f ) = D ( f ) {\displaystyle {\mathcal {D}}(f,\,f)={\mathcal {D}}(f)} . We may equip D ( Ω ) {\displaystyle {\mathcal {D}}(\Omega )} with an inner product given by ⟨ f , g ⟩ D ( Ω ) := ⟨ f , g ⟩ H 2 ( Ω ) + D ( f , g ) ( f , g ∈ D ( Ω ) ) , {\displaystyle \langle f,g\ra... | Dirichlet space |
c_2i8azjtk3c0t | The corresponding norm ‖ ⋅ ‖ D ( Ω ) {\displaystyle \|\cdot \|_{{\mathcal {D}}(\Omega )}} is given by ‖ f ‖ D ( Ω ) 2 := ‖ f ‖ H 2 ( Ω ) 2 + D ( f ) ( f ∈ D ( Ω ) ) . {\displaystyle \|f\|_{{\mathcal {D}}(\Omega )}^{2}:=\|f\|_{H^{2}(\Omega )}^{2}+{\mathcal {D}}(f)\;\;\;\;\;(f\in {\mathcal {D}}(\Omega )).} Note that this... | Dirichlet space |
c_684ri4fzshl2 | The Dirichlet space is not an algebra, but the space D ( Ω ) ∩ H ∞ ( Ω ) {\displaystyle {\mathcal {D}}(\Omega )\cap H^{\infty }(\Omega )} is a Banach algebra, with respect to the norm ‖ f ‖ D ( Ω ) ∩ H ∞ ( Ω ) := ‖ f ‖ H ∞ ( Ω ) + D ( f ) 1 / 2 ( f ∈ D ( Ω ) ∩ H ∞ ( Ω ) ) . {\displaystyle \|f\|_{{\mathcal {D}}(\Omega )... | Dirichlet space |
c_yociy8bpzubm | {\displaystyle \|f\|_{\mathcal {D}}^{2}=\sum _{n\geq 0}(n+1)|a_{n}|^{2}.} Clearly, D {\displaystyle {\mathcal {D}}} contains all the polynomials and, more generally, all functions f {\displaystyle f} , holomorphic on D {\displaystyle \mathbb {D} } such that f ′ {\displaystyle f'} is bounded on D {\displaystyle \mathbb ... | Dirichlet space |
c_c8yp2dp7o81s | In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the disco... | Dirichlet conditions |
c_ws2jwei5ul1y | In mathematics, the Dixmier mapping describes the space Prim(U(g)) of primitive ideals of the universal enveloping algebra U(g) of a finite-dimensional solvable Lie algebra g over an algebraically closed field of characteristic 0 in terms of coadjoint orbits. More precisely, it is a homeomorphism from the space of orbi... | Dixmier mapping |
c_zc2lwwcqr9cf | In mathematics, the Dixmier trace, introduced by Jacques Dixmier (1966), is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces. Some applications of Dixmier traces to noncommutative geometry are described in... | Dixmier trace |
c_y9icuo0sszee | In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these functions satisfy the identity cm 3 z + sm 3 z = 1 {\displaystyle \operato... | Dixon elliptic functions |
c_dr87fxdfna2y | In mathematics, the Dottie number is a constant that is the unique real root of the equation cos x = x {\displaystyle \cos x=x} ,where the argument of cos {\displaystyle \cos } is in radians. The decimal expansion of the Dottie number is 0.739085... {\displaystyle 0.739085...} .Since cos ( x ) − x {\displaystyle \c... | Dottie number |
c_0pd45cvv6u5v | In mathematics, the Double extension set theory (DEST) is an axiomatic set theory proposed by Andrzej Kisielewicz consisting of two separate membership relations on the universe of sets, denoted here by ∈ {\displaystyle \in } and ε {\displaystyle \varepsilon } , and a set of axioms relating the two. The intention behin... | Double extension set theory |
c_mq0447iwj0ca | Then, the axioms of DEST posit a set A = { x | ϕ ( x ) } {\displaystyle A=\{x|\phi (x)\}} such that x ε A ⟺ ϕ ( x ) {\displaystyle x\varepsilon A\iff \phi (x)} . For instance, x ∉ x {\displaystyle x\notin x} is a formula involving only ∈ {\displaystyle \in } , and thus DEST posits the Russell set R = { x | x ∉ x } {\di... | Double extension set theory |
c_70yr4213eqlu | Since the membership relations are different, we thus avoid the Russell's paradox. The focus in DEST is on regular sets, which are sets whose extensions under the two membership relations coincide, i.e., sets A {\displaystyle A} for which it holds that ∀ x . | Double extension set theory |
c_xqvz0wf1nx47 | x ∈ A ⟺ x ε A {\displaystyle \forall x.x\in A\iff x\varepsilon A} . The preceding discussion suggests that the Russell set R = { x | x ∉ x } {\displaystyle R=\{x|x\notin x\}} cannot be regular, as otherwise it leads to the Russell's paradox. == References == | Double extension set theory |
c_j712voya8mwe | In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix. Let A be a square matrix. The index of A is the least nonnegative integer k such that rank(Ak+1) = rank(Ak). | Drazin inverse |
c_ucoqmkuolbtp | The Drazin inverse of A is the unique matrix AD that satisfies A k + 1 A D = A k , A D A A D = A D , A A D = A D A . {\displaystyle A^{k+1}A^{\text{D}}=A^{k},\quad A^{\text{D}}AA^{\text{D}}=A^{\text{D}},\quad AA^{\text{D}}=A^{\text{D}}A.} It's not a generalized inverse in the classical sense, since A A D A ≠ A {\displa... | Drazin inverse |
c_xvbo40ywtlir | If A is invertible with inverse A − 1 {\displaystyle A^{-1}} , then A D = A − 1 {\displaystyle A^{\text{D}}=A^{-1}} . If A is a block diagonal matrix A = {\displaystyle A={\begin{bmatrix}B&0\\0&N\end{bmatrix}}} where B {\displaystyle B} is invertible with inverse B − 1 {\displaystyle B^{-1}} and N {\displaystyle N} is... | Drazin inverse |
c_3z089whvf5kg | The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or {1,2,5}-inverse and denoted A#. The group inverse can be defined, equivalently, by the properties AA#A = A, A#AA# = A#, and AA# = A#A. A projection matrix P, defined as a matrix such that P2 = P, has index 1 (or 0) and has Drazin inverse PD =... | Drazin inverse |
c_q3ayjd594ihp | {\displaystyle A^{\text{D}}=0.} The hyper-power sequence is A i + 1 := A i + A i ( I − A A i ) ; {\displaystyle A_{i+1}:=A_{i}+A_{i}\left(I-AA_{i}\right);} for convergence notice that A i + j = A i ∑ k = 0 2 j − 1 ( I − A A i ) k . {\displaystyle A_{i+j}=A_{i}\sum _{k=0}^{2^{j}-1}\left(I-AA_{i}\right)^{k}.} For A 0 := ... | Drazin inverse |
c_ocqi3yhhyubh | In mathematics, the Drinfeld upper half plane is a rigid analytic space analogous to the usual upper half plane for function fields, introduced by Drinfeld (1976). It is defined to be P1(C)\P1(F∞), where F is a function field of a curve over a finite field, F∞ its completion at ∞, and C the completion of the algebraic ... | Drinfeld upper half plane |
c_2mnow3yo7fss | In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra and the invariants of its symmetric algebra. It was introduced by Michel Duflo (1977) and later generalized to arbitrary finite-dimensional Lie algebras by Kontsevich. The P... | Duflo isomorphism |
c_98h1tgcxs3sx | It is equivariant with respect to the natural representation of g {\displaystyle {\mathfrak {g}}} on these spaces, so it restricts to a vector space isomorphism F: S ( g ) g → U ( g ) g {\displaystyle F\colon S({\mathfrak {g}})^{\mathfrak {g}}\to U({\mathfrak {g}})^{\mathfrak {g}}} where the superscript indicates the s... | Duflo isomorphism |
c_actbtpy9ywdx | {\displaystyle F\circ G\colon S({\mathfrak {g}})^{\mathfrak {g}}\to U({\mathfrak {g}})^{\mathfrak {g}}.} Later, using the Kontsevich formality theorem, Kontsevich showed that this works for all finite-dimensional Lie algebras. | Duflo isomorphism |
c_8dkmoju2lnry | Following Calaque and Rossi, the map G {\displaystyle G} can be defined as follows. The adjoint action of g {\displaystyle {\mathfrak {g}}} is the map g → E n d ( g ) {\displaystyle {\mathfrak {g}}\to \mathrm {End} ({\mathfrak {g}})} sending x ∈ g {\displaystyle x\in {\mathfrak {g}}} to the operation {\displaystyle } ... | Duflo isomorphism |
c_1cgr0n0toxvw | Call this element a d ∈ S ( g ∗ ) ⊗ E n d ( g ) {\displaystyle \mathrm {ad} \in S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})} Both S ( g ∗ ) {\displaystyle S({\mathfrak {g}}^{\ast })} and E n d ( g ) {\displaystyle \mathrm {End} ({\mathfrak {g}})} are algebras so their tensor product is as well. Th... | Duflo isomorphism |
c_r9ynt844l347 | As a result, the algebra S ( g ∗ ) {\displaystyle S({\mathfrak {g}}^{\ast })} acts on as differential operators on S ( g ) {\displaystyle S({\mathfrak {g}})} , and this extends to an action of S ( g ) {\displaystyle S({\mathfrak {g}})} on S ( g ) {\displaystyle S({\mathfrak {g}})} . We can thus define a linear map G: S... | Duflo isomorphism |
c_lrtcfmn2icvk | In mathematics, the Dushnik–Miller theorem is a result in order theory stating that every infinite linear order has a non-identity order embedding into itself. It is named for Ben Dushnik and E. W. Miller, who published this theorem for countable linear orders in 1940. More strongly, they showed that in the countable c... | Dushnik–Miller theorem |
c_jfec4s2qgzu7 | In mathematics, the Dwork unit root zeta function, named after Bernard Dwork, is the L-function attached to the p-adic Galois representation arising from the p-adic etale cohomology of an algebraic variety defined over a global function field of characteristic p. The Dwork conjecture (1973) states that his unit root ze... | Dwork conjecture on unit root zeta functions |
c_reji7xwjrkol | In mathematics, the Dynkin index I ( λ ) {\displaystyle I({\lambda })} of a finite-dimensional highest-weight representation of a compact simple Lie algebra g {\displaystyle {\mathfrak {g}}} with highest weight λ {\displaystyle \lambda } is defined by where V 0 {\displaystyle V_{0}} is the 'defining representation', wh... | Dynkin index |
c_donwzbbhwxpu | Since the trace forms are bilinear forms, we can take traces to obtain I ( λ ) = dim V λ 2 dim g ( λ , λ + 2 ρ ) {\displaystyle I(\lambda )={\frac {\dim V_{\lambda }}{2\dim {\mathfrak {g}}}}(\lambda ,\lambda +2\rho )} where the Weyl vector ρ = 1 2 ∑ α ∈ Δ + α {\displaystyle \rho ={\frac {1}{2}}\sum _{\alpha \in \De... | Dynkin index |
c_3hu5h15oyi9n | In mathematics, the Dyson conjecture (Freeman Dyson 1962) is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Wilson and Gunson. Andrews generalized it to the q-Dyson conjecture, proved by Zeilberger and Bressoud and sometimes called the Zeilberger–Bressoud theorem. M... | Dyson conjecture |
c_yqrwwyy7t6vm | In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E8 root system. The norm of the E8 lattice (divided by 2) is a positive definite even unimodular quadr... | E8 lattice |
c_qq0bxf9xqgzq | In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice. | E8 manifold |
c_13dz8hq7jj1x | In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime p. It is described in more detail in Ravenel (2003, chapter 1.5) and Mahowald (2001). It is related to the EHP long exact sequence of Whitehead (1953); the name "EHP" ... | EHP spectral sequence |
c_dp369vvpeyto | The spectral sequence has the advantage that the input is previously calculated homotopy groups. It was used by Oda (1977) to calculate the first 31 stable homotopy groups of spheres. For arbitrary primes one uses some fibrations found by Toda (1962): S ^ 2 n ( p ) → Ω S 2 n + 1 ( p ) → Ω S 2 p n + 1 ( p ) {\displaysty... | EHP spectral sequence |
c_3tkid5va06ni | In mathematics, the ELSV formula, named after its four authors Torsten Ekedahl, Sergei Lando, Michael Shapiro, Alek Vainshtein, is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves. Several fundamental results in the intersection theo... | ELSV formula |
c_6rd22t9bhja5 | In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have a fixed point. The result was proved in 1968 by Clifford Earle and Richard S. Hamilton by showing that, w... | Earle–Hamilton fixed-point theorem |
c_oh8szd12nxzz | In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two unital magma structures on a set where one is a homomorphism for the other. Given this, the structures are the same, and the resulting magma is a commutative monoid. This can then be used to prov... | Eckmann–Hilton argument |
c_fnsxlx3rkflh | In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. Applications of Edwards curves to cryptography were developed by Daniel J. Bernstein and Tanja Lange: they pointed out se... | Edwards curves |
c_q54u5qog2isk | In mathematics, the Ehrenpreis conjecture of Leon Ehrenpreis states that for any K greater than 1, any two closed Riemann surfaces of genus at least 2 have finite-degree covers which are K-quasiconformal: that is, the covers are arbitrarily close in the Teichmüller metric. A proof was announced by Jeremy Kahn and Vladi... | Ehrenpreis conjecture |
c_7bknxcvtlsk0 | In mathematics, the Eisenstein ideal is an ideal in the endomorphism ring of the Jacobian variety of a modular curve, consisting roughly of elements of the Hecke algebra of Hecke operators that annihilate the Eisenstein series. It was introduced by Barry Mazur (1977), in studying the rational points of modular curves. ... | Eisenstein ideal |
c_tu4c4nh6zmwd | In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form z = a + b ω , {\displaystyle z=a+b\omega ,} where a and b are integers and ω = − 1 + i 3 2 = e i 2 π / 3 {\displaystyle \omega ={\frac {-1+i... | Eisenstein integer |
c_pkzxgec1b1dj | In mathematics, the Ellis–Numakura lemma states that if S is a non-empty semigroup with a topology such that S is compact and the product is semi-continuous, then S has an idempotent element p, (that is, with pp = p). The lemma is named after Robert Ellis and Katsui Numakura. | Ellis–Numakura lemma |
c_u3o1wrz4wpsd | In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type t... | Enriques–Kodaira classification |
c_mafiili32uzz | Federigo Enriques (1914, 1949) described the classification of complex projective surfaces. Kunihiko Kodaira (1964, 1966, 1968a, 1968b) later extended the classification to include non-algebraic compact surfaces. The analogous classification of surfaces in positive characteristic was begun by David Mumford (1969) and c... | Enriques–Kodaira classification |
c_chrfpoezdsgv | In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the theorem in 1938, but did not publish it until 1961. It is part of the field of combinatorics, and one of the central resu... | Erdős–Ko–Rado theorem |
c_qcwg7jo3l72r | One way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. The Erdős–Ko–Rado theorem states that when n {\displaystyle n} is large enough for the problem to be non... | Erdős–Ko–Rado theorem |
c_szyopbtk9df4 | When n = 2 r {\displaystyle n=2r} there are other equally-large families, but for larger values of n {\displaystyle n} only the families constructed in this way can be largest. The Erdős–Ko–Rado theorem can also be described in terms of hypergraphs or independent sets in Kneser graphs. Several analogous theorems apply ... | Erdős–Ko–Rado theorem |
c_2l1dsuprmnyw | In mathematics, the Erdős–Szekeres theorem asserts that, given r, s, any sequence of distinct real numbers with length at least (r − 1)(s − 1) + 1 contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s. The proof appeared in the same 1935 paper that mentions th... | Erdős–Szekeres theorem |
c_iz5gg23deglh | In mathematics, the Erdős–Turán inequality bounds the distance between a probability measure on the circle and the Lebesgue measure, in terms of Fourier coefficients. It was proved by Paul Erdős and Pál Turán in 1948.Let μ be a probability measure on the unit circle R/Z. The Erdős–Turán inequality states that, for any ... | Erdős–Turán inequality |
c_nmmxetoeertq | In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam. | Erdős–Ulam problem |
c_4ohl4m31e96v | In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über neuere geometrische Forschungen. It is named after the University Erlangen-Nürnberg, where Klein worked. | Erlangen program |
c_iupktf64mbys | By 1872, non-Euclidean geometries had emerged, but without a way to determine their hierarchy and relationships. Klein's method was fundamentally innovative in three ways: Projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was more rest... | Erlangen program |
c_jm9qu64fn70c | In mathematics, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist Frederick J. Ernst. | Ernst equation |
c_0zdew2tzkxtg | In mathematics, the Euclidean algorithm or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements... | Algorithm |
c_s4fx4kz74dlg | It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number a multitude composed of units": a counting num... | Algorithm |
c_hhqlhfgpnaxr | To "measure" is to place a shorter measuring length s successively (q times) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q×s, q being the quotient, or remainder r is the "modulus", the integer-fractional part left over after the division.For ... | Algorithm |
c_l49bknjvkuow | In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Element... | Euclid's algorithm |
c_hd3qr4rb74w1 | It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidea... | Euclid's algorithm |
c_txwa1ectc2sx | For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that... | Euclid's algorithm |
c_gsv4ht1wg2jn | By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known a... | Euclid's algorithm |
c_4wmklr3welvv | A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two (with this version, the algorithm stops when reaching a zero remainder). With this improvement, the algorithm never requires more steps than five times... | Euclid's algorithm |
c_gcn5o223uvlc | Additional methods for improving the algorithm's efficiency were developed in the 20th century. The Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic. | Euclid's algorithm |
c_exj269i8pgag | Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multi... | Euclid's algorithm |
c_8nlpgei1aqht | In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the an... | Distance formula |
c_9s5pdebbks2c | The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has... | Distance formula |
c_qxok80exqs57 | In mathematics, the Euler function is given by ϕ ( q ) = ∏ k = 1 ∞ ( 1 − q k ) , | q | < 1. {\displaystyle \phi (q)=\prod _{k=1}^{\infty }(1-q^{k}),\quad |q|<1.} Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics and complex analysis... | Euler function |
c_sia8bv7gafdb | In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion 1 cosh t = 2 e t + e − t = ∑ n = 0 ∞ E n n ! ⋅ t n {\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n! }}\cdot t^{n}} ,where cosh ( t... | Euler numbers |
c_rvlt8hdgsib8 | The Euler numbers are related to a special value of the Euler polynomials, namely: E n = 2 n E n ( 1 2 ) . {\displaystyle E_{n}=2^{n}E_{n}({\tfrac {1}{2}}).} The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also oc... | Euler numbers |
c_rfvntst01606 | In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an ( n + 1 ) {\displaystyle (n+1)} -fold sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to th... | Euler sequence |
c_orlayzksjq0r | In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansi... | Euler-Maclaurin formula |
c_y9kdb1gnthhf | The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. It was later generalized to Darboux's formula. | Euler-Maclaurin formula |
c_s7f5nafo4xw0 | In mathematics, the Euler–Poisson–Darboux equation is the partial differential equation u x , y + N ( u x + u y ) x + y = 0. {\displaystyle u_{x,y}+{\frac {N(u_{x}+u_{y})}{x+y}}=0.} This equation is named for Siméon Poisson, Leonhard Euler, and Gaston Darboux. It plays an important role in solving the classical wave eq... | Euler–Poisson–Darboux equation |
c_7aluz23u7bok | In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi. u x x + x u y y = 0. | Tricomi equation |
c_m4bkt24746lc | {\displaystyle u_{xx}+xu_{yy}=0.\,} It is elliptic in the half plane x > 0, parabolic at x = 0 and hyperbolic in the half plane x < 0. Its characteristics are x d x 2 + d y 2 = 0 , {\displaystyle x\,dx^{2}+dy^{2}=0,\,} which have the integral y ± 2 3 x 3 / 2 = C , {\displaystyle y\pm {\frac {2}{3}}x^{3/2}=C,} where C i... | Tricomi equation |
c_7ludgfu9vbcg | In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras... | Ext functor |
c_94l4lyxscddy | In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book Homolog... | Ext functor |
c_xi0grz0e51bh | In mathematics, the F. and M. Riesz theorem is a result of the brothers Frigyes Riesz and Marcel Riesz, on analytic measures. It states that for a measure μ on the circle, any part of μ that is not absolutely continuous with respect to the Lebesgue measure dθ can be detected by means of Fourier coefficients. More preci... | F. and M. Riesz theorem |
c_0pmyqi9ix682 | The formulation here is as in Walter Rudin, Real and Complex Analysis, p. 335. The proof given uses the Poisson kernel and the existence of boundary values for the Hardy space H1. Expansions to this theorem were made by James E. Weatherbee in his 1968 dissertation: Some Extensions Of The F. And M. Riesz Theorem On Abso... | F. and M. Riesz theorem |
c_btd4a9ztnpn6 | In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicists Jacques Bros and Daniel Iagolnitzer in order to characterise the local analyticity of functions (or distributions) on Rn. The transform provides an alterna... | Fourier–Bros–Iagolnitzer transform |
c_qzqslyb2zhmh | In mathematics, the FEE method, or fast E-function evaluation method, is the method of fast summation of series of a special form. It was constructed in 1990 by Ekaterina Karatsuba and is so-named because it makes fast computations of the Siegel E-functions possible, in particular of e x {\displaystyle e^{x}} . A class... | FEE method |
c_5f85y2e9o72f | Using the FEE, it is possible to prove the following theorem: Theorem: Let y = f ( x ) {\displaystyle y=f(x)} be an elementary transcendental function, that is the exponential function, or a trigonometric function, or an elementary algebraic function, or their superposition, or their inverse, or a superposition of the ... | FEE method |
c_3mepxv2je1hp | The algorithms based on the method FEE include the algorithms for fast calculation of any elementary transcendental function for any value of the argument, the classical constants e, π , {\displaystyle \pi ,} the Euler constant γ , {\displaystyle \gamma ,} the Catalan and the Apéry constants, such higher transcendental... | FEE method |
c_idfvckbky6dg | In mathematics, the Faber polynomials Pm of a Laurent series f ( z ) = z − 1 + a 0 + a 1 z + ⋯ {\displaystyle \displaystyle f(z)=z^{-1}+a_{0}+a_{1}z+\cdots } are the polynomials such that P m ( f ) − z − m {\displaystyle \displaystyle P_{m}(f)-z^{-m}} vanishes at z=0. They were introduced by Faber (1903, 1919) and stud... | Faber polynomials |
c_nflqnr4zwjgb | In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). It was also written down as the Fourier transform of f ^ ( z ) = ∏ m = 1 ∞ ( cos π z 2 m ) m {\displaystyle {\hat {f}}(z)=\prod _{m=1}^{\infty }\left(\cos {\frac {\pi z}{2^... | Fabius function |
c_z7dd8th90xg8 | {\displaystyle 0\leq x\leq 1/2.} It follows that f ( x ) {\displaystyle f(x)} is monotone increasing for 0 ≤ x ≤ 1 , {\displaystyle 0\leq x\leq 1,} with f ( 1 / 2 ) = 1 / 2 {\displaystyle f(1/2)=1/2} and f ( 1 ) = 1. {\displaystyle f(1)=1.} There is a unique extension of f to the real numbers that satisfies the same di... | Fabius function |
c_jqkmjnb52z1k | In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its d... | Fabry gap theorem |
c_lwoiwavvkc5e | In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. With the restricted definition, each Farey sequence starts with ... | Farey graph |
c_0bs8dwzvdwc5 | In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms. The motivation is the interest in the target of the assembly maps; this may be, for instance, the algebraic K-theory of a gr... | Farrell-Jones conjecture |
c_6yhc67ncnph8 | In mathematics, the Farrell–Markushevich theorem, proved independently by O. J. Farrell (1899–1981) and A. I. Markushevich (1908–1979) in 1934, is a result concerning the approximation in mean square of holomorphic functions on a bounded open set in the complex plane by complex polynomials. It states that complex polyn... | Farrell–Markushevich theorem |
c_kgnav87msnpu | In mathematics, the Fatou conjecture, named after Pierre Fatou, states that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters. | Fatou conjecture |
c_zi2ejc25hfhh | In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these functions. The theorem is named after Pierre Fatou a... | Fatou–Lebesgue theorem |
c_h72p7ls561ov | In mathematics, the Favard constant, also called the Akhiezer–Krein–Favard constant, of order r is defined as K r = 4 π ∑ k = 0 ∞ r + 1 . {\displaystyle K_{r}={\frac {4}{\pi }}\sum \limits _{k=0}^{\infty }\left^{r+1}.} This constant is named after the French mathematician Jean Favard, and after the Soviet mathematicia... | Favard constant |
c_gkfli1c02d5s | In mathematics, the Faxén integral (also named Faxén function) is the following integral Fi ( α , β ; x ) = ∫ 0 ∞ exp ( − t + x t α ) t β − 1 d t , ( 0 ≤ Re ( α ) < 1 , Re ( β ) > 0 ) . {\displaystyle \operatorname {Fi} (\alpha ,\beta ;x)=\int _{0}^{\infty }\exp(-t+xt^{\alpha })t^{\beta -1}\mathrm {d} t,\qquad ... | Faxén integral |
c_ehnj45dcglpd | In mathematics, the Federer–Morse theorem, introduced by Federer and Morse (1943), states that if f is a surjective continuous map from a compact metric space X to a compact metric space Y, then there is a Borel subset Z of X such that f restricted to Z is a bijection from Z to Y. Moreover, the inverse of that restrict... | Federer–Morse theorem |
c_uq4qjtqzarnd | In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte, the former of whom suggested the name Γ0.There is no standard notation for or... | Feferman–Schütte ordinal |
c_mdcli12wmnsf | In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by Walter Feit and John G. Thompson (1962). The conjecture states that there are no distinct prime numbers p and q such that p q − 1 p − 1 {\displaystyle {\frac {p^{q}-1}{p-1}}} divides q p − 1 q − 1 {\displaystyle {\frac {q^{p}-1}... | Feit–Thompson conjecture |
c_z8da00qwqhfu | In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963). | Odd order theorem |
c_qlnlzs0fgor6 | In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959). | Fejér kernel |
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