Mathematical analysis: vocabulary and definitions (129 words)

This page gathers the 129 dictionary entries belonging to the domain “Mathematical analysis”, as labelled by Wiktionary. Each word leads to its full entry: definitions, etymology, pronunciation, examples.

Abel sum noun Given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is…
absolute differential calculus noun Ricci calculus; the rules of index notation and manipulation for tensors and tensor fields, as developed by Gregorio Ricci-Curbastro.
adjoint noun Hermitian conjugate.
affine arithmetic noun A model for self-validated numerical analysis in which quantities are represented as affine combinations (called affine forms) of certain…
algebra noun A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and…
analytic adj Being defined in terms of objects of differential calculus such as derivatives.
analytic continuation noun The practice of extending analytic functions.
analytic function noun Any smooth (infinitely differentiable) function f, defined on an open set D⊆ℂ( textit or⊆ℝ), whose value in some neighbourhood of any given…
annihilator method noun A procedure used to find a particular solution to certain types of inhomogeneous ordinary differential equations.
approach verb Used when defining limits, preceded by as: To become arbitrarily close to some value, be it a number, vector or infinity and have an effect…
Archimedean property noun A property of the set of real numbers, that for any real number there is always a natural number greater than that real number.
arithmetic function noun Any function that is defined for all positive integers, and has values that are either real or complex.
arithmetic progression noun A sequence in which each term except the first is obtained from the previous by adding a constant value, known as the common difference of…
asymptote noun A straight line which a curve approaches arbitrarily closely as it goes to infinity. The limit of the curve; its tangent "at infinity".
asymptotic adj Coming into consideration as a variable tends to a limit, usually infinity.
Bernoulli number noun Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and…
Bernstein basis polynomial noun Any polynomial of the form b_i,n(t)=n choose itⁱ(1-t)ⁿ⁻ⁱ!!, being the iᵗʰ term of the binomial expansion of (t+(1-t))ⁿ!.
beta function noun A mathematical function that is also called the Euler integral of the first kind; it is a symmetric special function.
big O noun Upper bound function; see big O notation.
Borel adj being a member of a Borel σ-algebra; being a Borel set
bounded adj That can be enclosed within a ball of finite radius.
Cantor set noun A subset of an interval formed by recursively removing an interval in the middle of every connected component of the set.
Cauchy problem noun For a given m-order partial differential equation, the problem of finding a solution function u on ℝⁿ that satisfies the boundary…
Cauchy sequence noun Any sequence x_n in a metric space with metric d such that for every ϵ>0 there exists a natural number N such that for all k,m>N…
centroid noun An analogue of the centre of gravity of a nonuniform body in which local density is replaced by a specified function (which can take…
characteristic function noun A function which is equal to 1 for all points in its domain which belong to a given set, and is equal to 0 for all points in the domain…
codomain noun The target set into which a function is formally defined to map elements of its domain; the set denoted Y in the notation f : X → Y.
coincidence noun A coincidence point.
collection noun A set of sets; used because such a thing is in general too large to comply with the formal definition of a set.
complete adj In which every Cauchy sequence converges to a point within the space.
complete measure noun A measure such that, for every set of measure zero belonging to its domain, all subsets of that set are also assigned measure zero by the…
complex adj Whose range is a subset of the complex numbers.
complex measure noun A function whose domain is a σ-algebra, whose codomain is the set of complex numbers, and which is countably additive.
continuous adj Such that, for every x in the domain, for each small open interval D about f(x), there's an interval containing x whose image is in D.
continuous function noun a function whose value at any point in its domain is equal its limit at the same point
convergent adj Of a (typically infinite) sequence or series in a topological space, converging to a (finite, proper) limit.
convex envelope noun Convex hull.
coordinate system noun A method of representing the position of any individual point in a space of given dimensionality as a tuple of numerical coordinates.
countable additivity noun The following property of a measure: that the measure of a countable union of disjoint sets is equal to the sum of their measures.
counting measure noun A positive measure which, for each subset of a given measure space, assigns a value equal to the number of points in that subset.
del noun The symbol ∇ used to denote the gradient operator.
difference quotient noun The average rate of change of the function f(x) over the interval [a,,a+h]. As h approaches zero, the value of (f(a+h)-f(a))/h approaches…
differential geometry noun The study of geometry, especially geometric structures on differentiable manifolds, using techniques from calculus, linear algebra and…
differential operator noun An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives).
Dirichlet energy noun A quadratic functional which, given a real function defined on an open subset of ℝⁿ, yields a real number that is a measure of how variable…
domain noun An open and connected set in some topology. For example, the interval (0,1) as a subset of the real numbers.
embedding noun A map between metric spaces which preserves distances up to some scaling factor (called the distortion).
equiregular adj Containing elements that are both outer regular and inner regular and which have inverses that are also both outer regular and inner…
essential infimum noun The infimum (greatest lower bound) of a function which holds almost everywhere. In symbols, ess inf f= sup m:μ(x:f(x)<m)=0.
essential supremum noun The supremum (least upper bound) of a function which holds almost everywhere. In symbols, ess sup f= inf M:μ(x:f(x)>M)=0
Fourier series noun Any series resulting from the decomposition of a periodic function into terms involving cosines and sines (or, equivalently, complex…
Fourier transform noun A particular integral transform that when applied to a function of time (such as a signal), converts the function to one that plots the…
fractional calculus noun The branch of mathematics that studies generalisations of calculus to allow noninteger (i.e., real or complex) powers of the…
gamma function noun A meromorphic function which generalizes the notion of factorial to complex numbers and has singularities at the nonpositive integers; any…
gauge noun A semi-norm; a function that assigns a non-negative size to all vectors in a vector space.
geometric analysis noun A branch of mathematics in which techniques from differential geometry are used in the analysis and solution of partial differential…
geometric progression noun A sequence in which each term except the first is obtained from the previous by multiplying it by a constant value, known as the common…
geometric series noun Any infinite series whose terms are in geometric progression; any series of the form a+ar+ar²+ar³⋯=a∑ᵢ₌₀ ᪲rⁱ, where a≠0, r≠0, and r≠1.
harmonic analysis noun A study of the representation of functions or signals as the superposition of basic waves, involving the notions of harmonic functions…
harmonic series noun The divergent series whose terms are the reciprocals of the positive integers; the series ∑ₙ₌₁ ᪲1/n.
Hausdorff dimension noun A type of fractal dimension, a real-valued measure of a geometric object that assigns 1 to a line segment, 2 to a square and 3 to a cube.…
Hausdorff metric noun In the abstract metric space of all compact subsets of ℝⁿ, given a pair of compact sets A and B, the Hausdorff metric is h(A,B)= mbox…
homogeneous adj Such that if each of f 's inputs are multiplied by the same scalar, f 's output is multiplied by the same scalar to some fixed power…
implicit function noun A function defined by a (multivariable) implicit equation when one of the variables is regarded as the value of the function, especially…
indeterminate adj Not definitively or precisely determined, because of the presence of infinity or zero symbols used in any of several improper combinations.
infinite series noun Any expression that represents the addition a countably infinite number of ordered summands, often explicitly indexed by the positive or…
infinitesimal analysis noun a systematic employment of infinitesimals that reduces calculus to algebra; nonstandard analysis.
infinity noun An idealised point which is said to be approached by sequences of values whose magnitudes increase without bound.
inner regular adj Whose measure is equal to the supremum of the measures of all compact sets which are contained by it.
integrability noun The quality of being integrable (having an antidifference or antiderivative).
integration by parts noun A method of integration directly related to the rule for differentiation of products; can be written as ∫udv=uv-∫vdu.
k-cell noun A subset of k-dimensional Euclidean space which is obtained by taking the Cartesian product of k intervals (closed, open, or half-closed)…
Lebesgue integral noun An integral which has more general application than that of the Riemann integral, because it allows the region of integration to be…
Lebesgue measurable adj Of a set in some k-dimensional Euclidean space, That it is an element of the domain of the Lebesgue measure of the ambient Euclidean space.
Lebesgue measure noun A unique complete translation-invariant measure for the σ-algebra which contains all k-cells in a given Euclidean space, and which assigns…
level set noun The set of values x for which a real-valued function f(x) is equal to a given constant.
Lipschitz condition noun A property which can be said to be held by some point in the domain of a real-valued function if there exists a neighborhood of that point…
locally adv Within a sufficiently small sphere (or circle or interval) around a given point (sometimes, around any point).
logarithmic derivative noun Given a real or complex function f, the ratio of the value of the derivative to the value of the function, (f')/f, regarded as a function.
maximum noun An upper bound of a set which is also an element of that set.
measurable function noun A function whose codomain is a topological space and domain is a measurable space, with the property that the inverse image of every open…
Mellin transform noun An integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.
metric noun A function which satisfies a particular set of formal conditions, created to generalize the notion of the distance between two points.…
minimum noun A lower bound of a set which is also an element of that set.
monotone function noun A function f : X→Y (where X and Y are posets with partial order "≤") with either: (1) the property that x ≤ y implies f(x) ≤ f(y), or (2)…
monotonicity noun Said of a positive measure: the property of a positive measure of a measure space, that given two measurable sets where the first set is…
nabla noun The symbol ∇, used to denote the gradient operator.
nonstandard analysis noun Analysis founded upon the hyperreal number system, i.e., which formally defines and uses infinitesimals, especially differentials.
norm verb To endow (a vector space, etc.) with a norm.
normed adj Of a mathematical structure, endowed with a norm.
open ball noun The set of all points in a metric space whose distance from a given point ("the open ball's center") is less than a given length ("the open…
open set noun A set which can be described as an (arbitrary) union of open balls. Equivalently, a set such that for every point in it, there is an open…
orientation noun The choice of which ordered bases are "positively" oriented and which are "negatively" oriented on a real vector space.
outer measure noun A measure-like function derived from some premeasure as follows: when it is applied to a set E, it yields the infimum of the premeasures of…
outer regular adj (of a Borel set) That its measure is equal to the infimum of the measures of all open sets which contain it.
perfect adj Equal to its set of limit points, i.e. set A is perfect if A=A'.
positive linear functional noun A kind of linear functional which yields a non-negative scalar when given a non-negative function as parameter.
principal part noun A polynomial approximation of a power series, made up of monomials whose indices lie in the Newton diagram of the power series and which…
proper adj Such that every closed ball is compact.
quadrature noun The act or process of solving an indefinite integral by symbolic means.
quasiregular adj Having certain properties in common with holomorphic functions of a single complex variable.
real valued adj Having only real values: having as its codomain the set of real numbers or a subset thereof.
regular adj Such that every set in its domain is both outer regular and inner regular.
Riemann integral noun A type of integral whose computation involves dividing the interval of integration into smaller sub-intervals, summing sample values of the…
Riemann-Lebesgue lemma noun A lemma, of importance in harmonic analysis and asymptotic analysis, stating that the Fourier transform or Laplace transform of an L1…
ring noun A family of sets that is closed under finite unions and set-theoretic differences.
root noun A zero (of an equation).
semi-norm noun A function denoted ∥v∥ that maps a vector v to a non-negative value such that ∥cv∥ = |c|.∥v∥, where c is a scalar, and ∥v + w∥ ≤ ∥v∥ + ∥w∥…
separable adj Having a countable dense subset.
sequential continuity noun The property of a function f between metric spaces, that given a convergent sequence x_n→x^*, then f(x_n)→f(x^*), i.e. the property of a…
sigmoid function noun Any of various real functions whose graph resembles an elongated letter "S"; specifically, the logistic function y=(eˣ)/(eˣ+1)=1/(1+e⁻ˣ).
smoothness noun The highest order of derivative (the differentiability class) over a given domain.
subduction noun A surjection between diffeological spaces such that the target is identified as the pushforward of the source.
subscalar adj restricted to an invariant subspace (of a scalar operator)
Tauberian adj Being or relating to Tauberian theorems, a class of theorems that are partial converses to Abelian theorems.
Tauberian theorem noun Any of a class of theorems which, for a given Abelian theorem, specifies conditions such that any series whose Abel sums converge (as…
transcendental function noun Any function that is algebraically independent of its variable(s); a function which does not satisfy a polynomial equation whose…
transform noun An operation (often an integration) that converts one function into another.
translate noun In Euclidean spaces: a set of points obtained by adding a given fixed vector to each point of a given set.
transmission function noun The mathematical function of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in…
triangle inequality noun The inequality that states that the magnitude of the sum of two vectors is less than or equal to the sum of the magnitudes of the vectors.
trigonometry noun The branch of mathematics that deals with the relationships between the sides and angles of (in particular) right-angled triangles, as…
turning number noun A version of winding number in which the number of rotations is counted with respect to the tangent of the curve rather than a fixed point.
uniformly continuous adj That for every real ε > 0 there exists a real δ > 0 such that for all pairs of points x and y in X for which D_X(x,y)<δ, it must be the…
unitary adj Whose inverse is equal to its adjoint.
weight noun A variable which multiplies a value for ease of statistical manipulation.
δ-box noun A k-cell which is a Cartesian product of half-closed intervals which are closed at their infima and open at their suprema, and such that…
μ-completion noun A σ-algebra which is obtained as a "completion" of a given σ-algebra, which includes all subsets of the given measure space which…
σ-algebra noun A collection of subsets of a given set, such that the empty set is part of this collection, the collection is closed under complements…

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