text stringlengths 42 3.65k | source stringclasses 1
value |
|---|---|
Carathéodory function Summary Carathéodory_function In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nev... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermitian function Summary Hermitian_symmetry In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: f ∗ ( x ) = f ( − x ) {\displaystyle f^{*}(x)=f(-x)} (where the ∗ {\displaystyle ^{*}} indi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Young measure Summary Young_measure In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limit. Young measures have applications in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bounded domain Summary Closed_region In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rn or the complex coordinate space Cn. A connected open subset of coordinate space is frequently used... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equicontinuous linear maps Summary Equicontinuity In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equicontinuous linear maps Summary Equicontinuity As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Function of bounded variation Summary Bv_space In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Function of bounded variation Summary Bv_space Another characterization states that the functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone. In particular, a BV function may have discontinuities, but at most countab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Function of bounded variation Summary Bv_space In the case of several variables, a function f defined on an open subset Ω of R n {\displaystyle \mathbb {R} ^{n}} is said to have bounded variation if its distributional derivative is a vector-valued finite Radon measure. One of the most important aspects of functions of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Metric differential Summary Metric_differential In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete metric space Summary Completion_(topology) In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Modulus of continuity Summary Modulus_of_continuity In mathematical analysis, a modulus of continuity is a function ω: → used to measure quantitatively the uniform continuity of functions. So, a function f: I → R admits ω as a modulus of continuity if and only if | f ( x ) − f ( y ) | ≤ ω ( | x − y | ) , {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Modulus of continuity Summary Modulus_of_continuity In general, the role of ω is to fix some explicit functional dependence of ε on δ in the (ε, δ) definition of uniform continuity. The same notions generalize naturally to functions between metric spaces. Moreover, a suitable local version of these notions allows to de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Modulus of continuity Summary Modulus_of_continuity A special role is played by concave moduli of continuity, especially in connection with extension properties, and with approximation of uniformly continuous functions. For a function between metric spaces, it is equivalent to admit a modulus of continuity that is eith... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Measure zero Summary Null_set In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Measure zero Summary Null_set Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = ( X , Σ , μ ) {\displaystyle M=(X,\Sigma ,\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive invariant set Summary Positive_invariant_set In mathematical analysis, a positively (or positive) invariant set is a set with the following properties: Suppose x ˙ = f ( x ) {\displaystyle {\dot {x}}=f(x)} is a dynamical system, x ( t , x 0 ) {\displaystyle x(t,x_{0})} is a trajectory, and x 0 {\displaystyle x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Space-filling curves Summary Space-filling_curve In mathematical analysis, a space-filling curve is a curve whose range reaches every point in a higher dimensional region, typically the unit square (or more generally an n-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strong measure zero set Summary Strong_measure_zero_set In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that |In| < εn for all n and A is contained in the union of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strong measure zero set Summary Strong_measure_zero_set The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero.Borel's conjecture states that every strong measure zero set is countable. It is now known that this statement is independent of ZFC (the Zermelo–Fraenkel ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strong measure zero set Summary Strong_measure_zero_set Sierpiński proved in 1928 that the continuum hypothesis (which is now also known to be independent of ZFC) implies the existence of uncountable strong measure zero sets. In 1976 Laver used a method of forcing to construct a model of ZFC in which Borel's conjecture... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strong measure zero set Summary Strong_measure_zero_set The following characterization of strong measure zero sets was proved in 1973: A set A ⊆ R has strong measure zero if and only if A + M ≠ R for every meagre set M ⊆ R.This result establishes a connection to the notion of strongly meagre set, defined as follows: A ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thin set (analysis) Summary Thin_set_(analysis) In mathematical analysis, a thin set is a subset of n-dimensional complex space Cn with the property that each point has a neighbourhood on which some non-zero holomorphic function vanishes. Since the set on which a holomorphic function vanishes is closed and has empty in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Osgood curve Summary Osgood_curve In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area. Despite its area, it is not possible for such a curve to cover any two-dimensional region, distinguishing them from space-filling curves. Osgood curves are named after William Fogg Osgood... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Improper integrals Summary Improper_integrals In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unbo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Improper integrals Summary Improper_integrals If a regular definite integral (which may retronymically be called a proper integral) is worked out as if it is improper, the same answer will result. In the simplest case of a real-valued function of a single variable integrated in the sense of Riemann (or Darboux) over a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Improper integrals Summary Improper_integrals Such an integral is sometimes described as being of the "first" type or kind if the integrand otherwise satisfies the assumptions of integration. Integrals in the fourth form that are improper because f ( x ) {\displaystyle f(x)} has a vertical asymptote somewhere on the in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Improper integrals Summary Improper_integrals For example, in case 1, if f ( x ) {\displaystyle f(x)} is continuous on the entire interval [ a , ∞ ) {\displaystyle [a,\infty )} , then ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Improper integrals Summary Improper_integrals If f ( x ) {\displaystyle f(x)} is only continuous on ( a , ∞ ) {\displaystyle (a,\infty )} and not at a {\displaystyle a} itself, then typically this is rewritten as ∫ a ∞ f ( x ) d x = lim t → a + ∫ t c f ( x ) d x + lim b → ∞ ∫ c b f ( x ) d x , {\displaystyle \int _{a}^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Improper integrals Summary Improper_integrals Alternatively, an iterated limit could be used or a single limit based on the Cauchy principal value. If f ( x ) {\displaystyle f(x)} is continuous on [ a , d ) {\displaystyle [a,d)} and ( d , ∞ ) {\displaystyle (d,\infty )} , with a discontinuity of any kind at d {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Improper integrals Summary Improper_integrals The function f ( x ) {\displaystyle f(x)} can have more discontinuities, in which case even more limits would be required (or a more complicated principal value expression). Cases 2–4 are handled similarly. See the examples below. Improper integrals can also be evaluated in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Function of a real variable Summary Function_of_a_real_variable In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers R {\displaystyle \mathbb {R} } , or a subset of R {\displaystyle \m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Function of a real variable Summary Function_of_a_real_variable However, it is often assumed to have a structure of R {\displaystyle \mathbb {R} } -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an R {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Function of a real variable Summary Function_of_a_real_variable If the codomain has a structure of R {\displaystyle \mathbb {R} } -algebra, the same is true for the functions. The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous functions on a compact Hausdorff space Summary Continuous_functions_on_a_compact_Hausdorff_space In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous functions on a compact Hausdorff space Summary Continuous_functions_on_a_compact_Hausdorff_space {\displaystyle X.} The space C ( X ) {\displaystyle {\mathcal {C}}(X)} is a Banach algebra with respect to this norm. (Rudin 1973, §11.3) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Function (mathematics) Function space Functional_notation > Function space In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Asymptotic theory Summary Asymptotically_equal In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f (n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes ver... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Asymptotic theory Summary Asymptotically_equal This is often written symbolically as f (n) ~ n2, which is read as "f(n) is asymptotic to n2". An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Constructive function theory Summary Constructive_function_theory In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernstein. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Epi-convergence Summary Epi-convergence In mathematical analysis, epi-convergence is a type of convergence for real-valued and extended real-valued functions. Epi-convergence is important because it is the appropriate notion of convergence with which to approximate minimization problems in the field of mathematical opt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power series ring Interpreting formal power series as functions Non-commuting_formal_power_series > Interpreting formal power series as functions In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers. Formal power series over certain special rings can also... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power series ring Interpreting formal power series as functions Non-commuting_formal_power_series > Interpreting formal power series as functions {\displaystyle f(x)=\sum _{n\geq 0}a_{n}x^{n}.} This series is guaranteed to converge in S {\displaystyle S} given the above assumptions on x {\displaystyle x} . Furthermore,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power series ring Interpreting formal power series as functions Non-commuting_formal_power_series > Interpreting formal power series as functions {\displaystyle (fg)(x)=f(x)g(x).} Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved. Since the topology on R ] {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power series ring Interpreting formal power series as functions Non-commuting_formal_power_series > Interpreting formal power series as functions {\displaystyle f\in R].} With this formalism, we can give an explicit formula for the multiplicative inverse of a power series f {\displaystyle f} whose constant coefficient ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Idempotent analysis Summary Idempotent_analysis In mathematical analysis, idempotent analysis is the study of idempotent semirings, such as the tropical semiring. The lack of an additive inverse in the semiring is compensated somewhat by the idempotent rule A ⊕ A = A {\displaystyle A\oplus A=A} . == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proper convex function Summary Proper_convex_function In mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain, that never takes on the value − ∞ {\displaystyle -\infty } and also is not identic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proper convex function Summary Proper_convex_function Such a point, if it exists, is called a global minimum point of the function and its value at this point is called the global minimum (value) of the function. If the function takes − ∞ {\displaystyle -\infty } as a value then − ∞ {\displaystyle -\infty } is necessar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Proper convex function Summary Proper_convex_function Extended real-valued function for which the minimization problem is not solved by any one of these three trivial cases are exactly those that are called proper. Many (although not all) results whose hypotheses require that the function be proper add this requirement... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parameter Mathematical analysis Parameter > Mathematical functions > Mathematical analysis In mathematical analysis, integrals dependent on a parameter are often considered. These are of the form F ( t ) = ∫ x 0 ( t ) x 1 ( t ) f ( x ; t ) d x . {\displaystyle F(t)=\int _{x_{0}(t)}^{x_{1}(t)}f(x;t)\,dx.} In this formul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parameter Mathematical analysis Parameter > Mathematical functions > Mathematical analysis When evaluating the integral, t is held constant, and so it is considered to be a parameter. If we are interested in the value of F for different values of t, we then consider t to be a variable. The quantity x is a dummy variabl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Limit inferior The case of sequences of real numbers Limit_supremum > The case of sequences of real numbers In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized Fourier series Summary Generalized_Fourier_series In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Microlocalization functor Summary Microlocal_analysis In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, ps... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Wavefront set Summary Wave_front_set In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmande... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nested sequence of closed intervals The construction of the real numbers Nested_sequences_of_intervals > The construction of the real numbers In mathematical analysis, nested intervals provide one method of axiomatically introducing the real numbers as the completion of the rational numbers, being a necessity for discu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nullcline Summary Nullcline In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations x 1 ′ = f 1 ( x 1 , … , x n ) {\displaystyle x_{1}'=f_{1}(x_{1},\ldots ,x_{n})} x 2 ′ = f 2 ( x 1 , … , x n ) {\displaystyle x_{2}'=f_{2}(x_{1},\ldots ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-variation Summary P-variation In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1 {\displaystyle p\geq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f: I → ( M , d ) {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Upper semi-continuous Summary Upper_semi-continuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively, lower) semicontinuous at a point x 0 {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Agranovich–Dynin formula Summary Agranovich–Dynin_formula In mathematical analysis, the Agranovich–Dynin formula is a formula for the index of an elliptic system of differential operators, introduced by Agranovich and Dynin (1962). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alexandrov theorem Summary Alexandrov_theorem In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if U is an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f: U → R m {\displaystyle f\colon U\to \mathbb {R} ^{m}} is a convex function, then f {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bessel–Clifford function Summary Bessel–Clifford_function In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions. If π ( x )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bohr–Mollerup theorem Summary Bohr–Mollerup_theorem In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for x > 0 by Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t {\displaystyle \Gamma (x)=\int ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Brezis–Gallouët inequality Summary Brezis–Gallouët_inequality In mathematical analysis, the Brezis–Gallouët inequality, named after Haïm Brezis and Thierry Gallouët, is an inequality valid in 2 spatial dimensions. It shows that a function of two variables which is sufficiently smooth is (essentially) bounded, and provi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Brezis–Gallouët inequality Summary Brezis–Gallouët_inequality Let Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} be the exterior or the interior of a bounded domain with regular boundary, or R 2 {\displaystyle \mathbb {R} ^{2}} itself. Then the Brezis–Gallouët inequality states that there exists a real C {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cauchy index Summary Cauchy_index In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of r(x) = p(x)/q(x)over the real line is the difference between the number of roots of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Chebyshev–Markov–Stieltjes inequalities Summary Chebyshev–Markov–Stieltjes_inequalities In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dirichlet kernel Summary Dirichlet_kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as where n is any nonnegative integer. The kernel functions are periodic with period 2 π {\displaystyle 2\pi } . ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Foias constant Summary Foias_constant In mathematical analysis, the Foias constant is a real number named after Ciprian Foias. It is defined in the following way: for every real number x1 > 0, there is a sequence defined by the recurrence relation x n + 1 = ( 1 + 1 x n ) n {\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Haar integral Summary Haar's_theorem In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Karamata's tauberian theorem Summary Hardy–Littlewood_Tauberian_theorem In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence a n ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Karamata's tauberian theorem Summary Hardy–Littlewood_Tauberian_theorem The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood. : 226 In 1930, Jovan Karamata gave a new and much simpler proof. : 226 | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hardy–Littlewood inequality Summary Hardy–Littlewood_inequality In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f {\displaystyle f} and g {\displaystyle g} are nonnegative measurable real functions vanishing at infinity that are defined on n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert–Schmidt theorem Summary Hilbert–Schmidt_theorem In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in so... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kakutani fixed-point theorem Summary Kakutani_fixed-point_theorem In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kakutani fixed-point theorem Summary Kakutani_fixed-point_theorem Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lagrange–Bürmann formula Summary Lagrange–Bürmann_formula In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Minkowski inequality Summary Minkowski_inequality In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S {\displaystyle S} be a measure space, let 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } and let f {\displaystyle f} and g {\displaystyle g} be elements of L p (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pólya–Szegő inequality Summary Pólya–Szegő_inequality In mathematical analysis, the Pólya–Szegő inequality (or Szegő inequality) states that the Sobolev energy of a function in a Sobolev space does not increase under symmetric decreasing rearrangement. The inequality is named after the mathematicians George Pólya and G... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rademacher–Menchov theorem Summary Rademacher–Menchov_theorem In mathematical analysis, the Rademacher–Menchov theorem, introduced by Rademacher (1922) and Menchoff (1923), gives a sufficient condition for a series of orthogonal functions on an interval to converge almost everywhere. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Schur test Summary Schur_test In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L 2 → L 2 {\displaystyle L^{2}\to L^{2}} operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version. Let X , Y {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Schur test Summary Schur_test {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.} If there exist real functions p ( x ) > 0 {\displaystyle \,p(x)>0} and q ( y ) > 0 {\displaystyle \,q(y)>0} and numbers α , β > 0 {\displaystyle \,\alpha ,\beta >0} such that ( 1 ) ∫ Y K ( x , y ) q ( y ) d y ≤ α p ( x ) {\displaystyle (1)\qqu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Szegő limit theorems Summary Szegő_limit_theorems In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weierstrass approximation theorem Summary Stone-Weierstrass_theorem In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weierstrass approximation theorem Summary Stone-Weierstrass_theorem His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact Hausdorff space X is considered, and instead of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Whitney covering lemma Summary Whitney_covering_lemma In mathematical analysis, the Whitney covering lemma, or Whitney decomposition, asserts the existence of a certain type of partition of an open set in a Euclidean space. Originally it was employed in the proof of Hassler Whitney's extension theorem. The lemma was su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Whitney covering lemma Summary Whitney_covering_lemma More precisely: Whitney Covering Lemma (Grafakos 2008, Appendix J) Let Ω {\displaystyle \Omega } be an open non-empty proper subset of R n {\displaystyle \mathbb {R} ^{n}} . Then there exists a family of closed cubes { Q j } j {\displaystyle \{Q_{j}\}_{j}} such that... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Whitney covering lemma Summary Whitney_covering_lemma {\displaystyle {\sqrt {n}}\ell (Q_{j})\leq \mathrm {dist} (Q_{j},\Omega ^{c})\leq 4{\sqrt {n}}\ell (Q_{j}).} If the boundaries of two cubes Q j {\displaystyle Q_{j}} and Q k {\displaystyle Q_{k}} touch then 1 4 ≤ ℓ ( Q j ) ℓ ( Q k ) ≤ 4. {\displaystyle {\frac {1}{4}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Young's inequality for integral operators Summary Young's_inequality_for_integral_operators In mathematical analysis, the Young's inequality for integral operators, is a bound on the L p → L q {\displaystyle L^{p}\to L^{q}} operator norm of an integral operator in terms of L r {\displaystyle L^{r}} norms of the kernel ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternating series test Summary Alternating_series_test In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes kno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characteristic variety Summary Characteristic_variety In mathematical analysis, the characteristic variety of a microdifferential operator P is an algebraic variety that is the zero set of the principal symbol of P in the cotangent bundle. It is invariant under a quantized contact transformation. The notion is also def... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mean-periodic function Summary Mean-periodic_function In mathematical analysis, the concept of a mean-periodic function is a generalization of the concept of a periodic function introduced in 1935 by Jean Delsarte. Further results were made by Laurent Schwartz. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Final value theorem Summary Final_value_theorem In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if f ( t ) {\displaystyle f(t)} in continuous time has (unilater... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Initial value theorem Summary Initial_value_theorem In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.Let F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt} be the (one... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Intermediate Value Theorem Summary Bolzano's_theorem In mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval , then it takes on any given value between f ( a ) {\displaystyle f(a)} and f ( b ) {\displaystyle f(b)} at some po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Local maxima Summary Maximum_and_minimum In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. Known generically as extremum, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the gl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Local maxima Summary Maximum_and_minimum As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mean value theorem for divided differences Summary Mean_value_theorem_(divided_differences) In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rising sun lemma Summary Rising_sun_lemma In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma.The lemma is stated as follows: Suppose g is a real-valued continu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differentiability class Summary Parametric_continuity In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.