• In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative f ′ ( a ) {\textstyle...
    3 KB (583 words) - 15:12, 27 December 2024
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    formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. If all the partial derivatives...
    13 KB (1,884 words) - 01:18, 9 June 2025
  • containing 1". Hensel's lemma is fundamental in p-adic analysis, a branch of analytic number theory. The proof of Hensel's lemma is constructive, and leads...
    42 KB (9,044 words) - 20:58, 24 May 2025
  • on a parameter, the increment Δ x {\displaystyle \Delta x} of the independent variable. If we so choose, we can make the increment smaller and smaller...
    39 KB (6,577 words) - 14:32, 2 June 2025
  • of equations List of fundamental theorems List of hypotheses List of inequalities Lists of integrals List of laws List of lemmas List of limits List of...
    78 KB (6,296 words) - 20:31, 6 July 2025
  • may be Lebesgue integrable, but the integral of f ′ differs from the increment of f (how much f changes over an interval). This happens for example with...
    19 KB (2,685 words) - 08:58, 28 May 2025
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    1007/978-0-8176-4789-6_14, ISBN 978-0-8176-4788-9, MR 2777924; see proof of Lemma 5, p. 361 Archived 2022-01-08 at the Wayback Machine Read, Ronald C. (1968)...
    30 KB (3,443 words) - 23:25, 29 June 2025
  • Thumbnail for Laplace's equation
    _{y}=\varphi _{x}.} This relation does not determine ψ, but only its increments: d ψ = − φ y d x + φ x d y . {\displaystyle d\psi =-\varphi _{y}\,dx+\varphi...
    33 KB (5,075 words) - 15:19, 13 April 2025
  • Thumbnail for Fermat's Last Theorem
    descent contained a major gap. However, since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited...
    103 KB (11,700 words) - 20:03, 9 July 2025
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    known as prime factorization. A fundamental property of primes is shown in Euclid's lemma. It is a consequence of the lemma that if a prime divides a product...
    81 KB (9,977 words) - 15:36, 28 June 2025
  • when the limits of integration are in the opposite order (b < a), the increment dx is negative in the direction of integration. More generally, an m-form...
    67 KB (10,058 words) - 14:15, 26 June 2025
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    differentiable function defined on the real line, a consequence of Borel's lemma. As a result, the radius of convergence of a Taylor series can be zero....
    48 KB (8,229 words) - 17:42, 2 July 2025
  • logical systems that deviate from classical logic. diagonalization lemma A lemma used in the proof of Gödel's incompleteness theorems, stating that for...
    271 KB (30,237 words) - 15:11, 3 July 2025
  • reductions produce zero. The algorithm terminates always because of Dickson's lemma or because polynomial rings are Noetherian (Hilbert's basis theorem). Condition...
    63 KB (10,037 words) - 22:27, 19 June 2025
  • Thumbnail for History of mathematics
    program has progressed significantly, and there have been proofs of the fundamental lemma (2008), as well as a proposed proof of the geometric Langlands correspondence...
    148 KB (17,050 words) - 18:16, 8 July 2025
  • has been studied, including Lévy processes, stochastic networks (Kelly's lemma), birth and death processes, Markov chains, and piecewise deterministic...
    8 KB (1,062 words) - 04:49, 22 June 2025
  • algebraic systems. The first theorem given in the text, Euclid's lemma, captures a fundamental property of prime numbers. The text was ubiquitous in the quadrivium...
    98 KB (11,251 words) - 14:26, 22 June 2025
  • does not consider the possibility of the surface deforming as it moves. Lemma. One has: ∂ ∂ b ( ∫ a b f ( x ) d x ) = f ( b ) , ∂ ∂ a ( ∫ a b f ( x )...
    53 KB (11,253 words) - 03:22, 22 June 2025
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    ultrafilters that do not contain any finite sets. (The good news is that Zorn's lemma guarantees the existence of many such U; the bad news is that they cannot...
    33 KB (4,924 words) - 08:45, 23 June 2025
  • of y = f ( x ) {\displaystyle y=f(x)} as follows: an infinitely small increment α {\displaystyle \alpha } of the independent variable x always produces...
    63 KB (9,324 words) - 15:49, 8 July 2025
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    honours and reputations. Some are Kitaw Ejigu, Mulugeta Bekele, Aklilu Lemma, Gebisa Ejeta and Melaku Worede. Computer scientist Timnit Gebru, named...
    229 KB (22,340 words) - 22:19, 9 July 2025
  • explicit NULL test). As an example of the use of pairs, the shift-and-increment function that maps (m, n) to (n, n + 1) can be defined as Φ := λx.PAIR...
    90 KB (12,115 words) - 02:49, 7 July 2025
  • K_{t+1}} , is K t + I t − D t {\displaystyle K_{t}+I_{t}-D_{t}} . The net increment to the capital stock is the difference between gross investment and depreciation...
    229 KB (25,481 words) - 21:54, 19 June 2025
  • Thumbnail for Path integral formulation
    defined, the easiest way is to add a small imaginary part to the time increment ε. This is closely related to Wick rotation. Then the same convolution...
    86 KB (14,277 words) - 08:57, 19 May 2025
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    relativity or general relativity. Astronomy portal Canonical coordinates Fundamental lemma of the calculus of variations Functional derivative Generalized coordinates...
    96 KB (15,276 words) - 06:38, 28 June 2025
  • Thumbnail for Quran
    God to the final Islamic prophet Muhammad through the angel Gabriel incrementally over a period of some 23 years, beginning on the Laylat al-Qadr, when...
    198 KB (22,507 words) - 20:43, 25 June 2025
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    Barn Cases Ichikawa & Steup 2018, § 3. The Gettier Problem, § 4. No False Lemmas, § 5. Modal Conditions, § 6. Doing Without Justification? Steup & Neta 2020...
    187 KB (19,036 words) - 06:23, 7 July 2025
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    Charles University. p. 1. Archived from the original (PDF) on 2013-11-02. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric"...
    62 KB (8,757 words) - 16:57, 9 July 2025
  • Thumbnail for Derivations of the Lorentz transformations
    ϕ is small, so dividing by a positive integer N gives an even smaller increment of rapidity ϕ/N, and N of these infinitesimal boosts will give the original...
    71 KB (13,127 words) - 04:34, 25 June 2025
  • Thumbnail for Taylor's theorem
    It is the starting point of the study of analytic functions, and is fundamental in various areas of mathematics, as well as in numerical analysis and...
    54 KB (9,632 words) - 05:41, 2 June 2025