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and proved this by proving bounds on a circular maximal function analogous to the Kakeya maximal function. It was conjectured that there existed sets of
Kakeya_set
Maximal functions appear in many forms in harmonic analysis (an area of mathematics). One of the most important of these is the Hardy–Littlewood maximal
Maximal_function
Mathematical operator in real and harmonic analysis
maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis. The operator takes a locally integrable function
Hardy–Littlewood maximal function
Hardy–Littlewood_maximal_function
Open cover in mathematical analysis
r)}|f(y)|\,d\mu (y){\Bigr )}.} This maximal function is lower semicontinuous, hence measurable. The following maximal inequality is satisfied for every
Besicovitch_covering_theorem
Concept within complex analysis
such that for some Schwartz function Φ {\displaystyle \Phi } with ∫ Φ = 1 {\displaystyle \int \Phi =1} , the maximal function ( M Φ f ) ( x ) = sup t >
Hardy_space
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
Area of mathematical analysis
the Fourier transform, while modern harmonic analysis also studies maximal functions, singular integrals, oscillatory integrals, Fourier multipliers, Littlewood–Paley
Harmonic_analysis
Test to evaluate respiratory system
Maximal inspiratory pressure (MIP) is the maximal pressure that can be produced by the patient trying to inhale through a blocked mouthpiece. Maximal
Pulmonary_function_testing
Class of mathematical functions
plane containing the closed unit disc D(0, 1). The radial maximal function for the function φ (restricted to the unit disc) is defined on the unit circle
Subharmonic_function
is prime. A maximal order for Ω(n) is ln n / ln 2 It is conjectured that the Mertens function, or summatory function of the Möbius function, satisfies
Extremal orders of an arithmetic function
Extremal_orders_of_an_arithmetic_function
Mathematical theorem in real analysis
a locally integrable function f—can be proved as a consequence of the weak–L1 estimates for the Hardy–Littlewood maximal function. The proof below follows
Lebesgue differentiation theorem
Lebesgue_differentiation_theorem
Maximum rate of oxygen consumption as measured during incremental exercise
V̇O2 max (also maximal oxygen consumption, maximal oxygen uptake or maximal aerobic capacity) is the maximum rate of oxygen consumption attainable during
VO2_max
Mathematical functions
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied
Lemniscate_elliptic_functions
Art movement
In the arts, maximalism is an aesthetic characterized by excess and abundance, serving as a reaction against minimalism. The philosophy can be summarized
Maximalism
Set of functions between two fixed sets
interpolation inequality, the Rellich–Kondrachov theorem, the Hardy–Littlewood maximal function, etc. Let Ω ⊆ R n {\displaystyle \Omega \subseteq \mathbb {R} ^{n}}
Function_space
Mathematical proposition equivalent to the axiom of choice
(that is, every totally ordered subset) necessarily contains at least one maximal element. The lemma was proven (assuming the axiom of choice) by Kazimierz
Zorn's_lemma
Hardy–Littlewood maximal operator is bounded on Lp(dω). Specifically, we consider functions f on Rn and their associated maximal functions M( f ) defined
Muckenhoupt_weights
Mathematical result or axiom on order relations
In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914. It states that
Hausdorff_maximal_principle
British mathematician (1877–1947)
Hardy–Littlewood inequality Hardy–Littlewood maximal function Hardy–Littlewood tauberian theorem Hardy–Littlewood zeta function conjectures Hardy–Ramanujan Journal
G._H._Hardy
British mathematician (1885–1977)
definition Hardy–Littlewood inequality Hardy–Littlewood maximal function Hardy–Littlewood zeta function conjectures Hardy–Littlewood tauberian theorem First
John_Edensor_Littlewood
Association of one output to each input
}}-{\sqrt {3}}} for x. By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval [−2, 2] and
Function_(mathematics)
Mathematical concept
of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
Mathematical description of quantum state
choice of maximal commuting sets of observables for the abstract state space, there is a corresponding representation that is associated to a function space
Wave_function
Maximal compact connected Abelian Lie subgroup
isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T′ containing
Maximal_torus
Graph with at most one cycle per component
direction, any maximal directed pseudoforest determines a function ƒ such that ƒ(x) is the target of the edge that goes out from x, and any non-maximal directed
Pseudoforest
Mathematical theory by discovered by Józef Marcinkiewicz
p equal to 1 or ∞. Another famous example is the Hardy–Littlewood maximal function, which is only sublinear operator rather than linear. While L p {\displaystyle
Marcinkiewicz interpolation theorem
Marcinkiewicz_interpolation_theorem
Probabilistic Condorcet method
all maximal lotteries is the only rule satisfying reinforcement, Condorcet-consistency, and independence of clones. The social welfare function that
Maximal_lotteries
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
American mathematician (1931–2018)
Stein maximal principle (showing that under many circumstances, almost everywhere convergence is equivalent to the boundedness of a maximal function), Stein
Elias_M._Stein
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Pulmonary function test
measuring of breath) is the most common of the pulmonary function tests (PFTs). It measures lung function, specifically the amount (volume) and/or speed (flow)
Spirometry
Largest and smallest value taken by a function at a given point
analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extrema, they
Maximum_and_minimum
Fast-growing function
sequence with maximal length. The function SSCG ( k ) {\displaystyle {\text{SSCG}}(k)} denotes that length for simple subcubic graphs. The function SCG ( k
Friedman's_SSCG_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
fact examples had been found earlier of functions that were nowhere differentiable (see Weierstrass function). According to Weierstrass in his paper,
List_of_conjectures
Function with a multiplicative scaling behaviour
Conversely, every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree k, defined
Homogeneous_function
Hypercube partition of Euclidean space
integrable function and |B(x, r)| denotes the measure of the ball B(x, r). The Hardy–Littlewood maximal inequality states that for an integrable function f, |
Dyadic_cubes
Integral transform
0^{+}}\|I^{\alpha }f-f\|_{p}=0} for all p ≥ 1. Moreover, by estimating the maximal function of I, one can show that the limit Iα f → f holds pointwise almost everywhere
Riemann–Liouville_integral
Device used to measure respiration rate
impact of chronic diseases or their treatment on the respiratory muscles. Maximal inspiratory pressure (MIP), also known as negative inspiratory force (NIF)
Respiratory_pressure_meter
Function specifying the behavior of a component in an electronic or control system
described by some transfer function, "families" of special transfer functions are commonly used: Butterworth filter – maximally flat in passband and stopband
Transfer_function
Concentration of a compound where 50% of its maximal effect is observed
Half maximal effective concentration (EC50) is a measure of the concentration of a drug, antibody or toxicant which induces a biological response halfway
EC50
American mathematician
from Rutgers University, with thesis Inequalities for the Ergodic Maximal Function written under the direction of Richard Floyd Gundy. He has recently
Roger_Jones_(mathematician)
S-shaped curve
noted above) and then reaching a maximal limit. A logistic function, or related functions (e.g. the Gompertz function) are usually used in a descriptive
Logistic_function
Topology on prime ideals and algebraic varieties
field and the maximal ideals of the ring of its regular functions. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative
Zariski_topology
Negative of a convex function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to
Concave_function
Medical condition
difference between typical functioning, that is – the normal level of functioning for daily life, and maximal functioning, that is – what cognitive tests
Cognitive_impairment
In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev
Christ–Kiselev maximal inequality
Christ–Kiselev_maximal_inequality
Type of signal processing filter
that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer
Butterworth_filter
Type of function space
which arises in the study of Hardy–Littlewood maximal functions, consisting of measurable functions f {\displaystyle f} such that ∫ R n | f ( x ) |
Orlicz_space
Half maximal inhibitory concentration
Half maximal inhibitory concentration (IC50) is a measure of the potency of a substance in inhibiting a specific biological or biochemical function. IC50
IC50
Manifold upon which it is possible to perform calculus
point. A differentiable function "usually" has maximal rank, in a precise sense given by Sard's theorem. Functions of maximal rank at a point are called
Differentiable_manifold
American mathematician (b. 1949)
Coifman, R.; Fefferman, C. (1974), "Weighted norm inequalities for maximal functions and singular integrals", Studia Mathematica, 51 (3): 241–250, doi:10
Charles_Fefferman
Topological space that locally resembles Euclidean space
atlas is called the maximal atlas (i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold
Manifold
Set disjoint from its sumset with itself
sum-free set that an abelian group G contains? A sum-free set is said to be maximal if it is not a proper subset of another sum-free set. Let f : [ 1 , ∞ )
Sum-free_set
Concept in music theory
is fixed and the bracket pair is the floor function. Jack Douthett and Richard Krantz introduced maximally even sets to the mathematics literature. A
Maximal_evenness
Family of functions
→ ∞ {\displaystyle n\to \infty } . This uses the Hardy–Littlewood maximal function. If ( k n ) {\displaystyle (k_{n})} is not radially decreasing symmetric
Summability_kernel
Difference between two successive prime numbers
is a maximal gap, if g m < g n {\displaystyle g_{m}<g_{n}} for all m < n {\displaystyle m<n} . As of May 2026[update], the largest known maximal prime
Prime_gap
Study of computable functions and Turing degrees
then every maximal set is mapped to another maximal set. In 1974, Soare showed that also the converse holds, that is, every two maximal sets are automorphic
Computability_theory
Maximal smooth atlas for a topological manifold
and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa. In general, computations with the maximal atlas
Smooth_structure
American physical therapist and dance therapist (1900–1981)
mechanical anatomical activity of physical therapy, in order to enhance maximal functioning. In physical therapy, that meant thinking in terms of movement in
Irmgard_Bartenieff
(Mathematical) ring with a unique maximal ideal
Both (p) and (q) are maximal ideals here. To motivate the name "local" for these rings, we consider real-valued continuous functions defined on some open
Local_ring
Concept in theoretical computer science
performance of Turing machines in other ways than time or maximal number of ones. For example: The function num ( n ) {\displaystyle {\text{num}}(n)} is defined
Busy_beaver
n-dimensional Euclidean space Rn and let M denote the Hardy–Littlewood maximal operator: for a function f : Rn → R, Mf : Rn → R is defined by M f ( x ) = sup r > 0
Stein–Strömberg_theorem
Set of edges without common vertices
unsaturated). A maximal matching is a matching M of a graph G that is not a subset of any other matching. A matching M of a graph G is maximal if every edge
Matching_(graph_theory)
zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup)
Zonal_spherical_function
Grauert's approximation theorem. Hardy-Littlewood maximal inequality The Hardy-Littlewood maximal function of f ∈ L 1 ( R n ) {\displaystyle f\in L^{1}(\mathbb
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
transforms and maximal functions" (PDF). Studia Math. 72 (1): 9–26. doi:10.4064/sm-72-1-9-26. Ariño, Miguel A.; —— (1990). "Maximal functions on classical
Benjamin_Muckenhoupt
Function in Boolean algebra
Boolean function. The n-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number
Parity_function
Branch of mathematics
cancerous tumor grows. In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal
Calculus
Type of shift register in computing
shift register whose input bit is a linear function of its previous state. The most commonly used linear function of single bits is exclusive-or (XOR). Thus
Linear-feedback shift register
Linear-feedback_shift_register
3-smooth number (29×3), number of threshold functions of exactly 4 variables 1537 = Keith number, 1539 = maximal number of pieces that can be obtained by
1000_(number)
Partially ordered set equipped with a rank function
and only if all maximal chains in P have the same length: setting the rank of the least element to 0 then determines the rank function completely. This
Graded_poset
Optimizing objective functions that have constrained variables
objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy
Constrained_optimization
Specific values of a multivalued function
refers value specifically to such a maximal branch. The principal branch of a multivariate function is one of these maximal branches that is selected once
Principal_value
Axiom of set theory
subcollection is precisely one that is maximal with respect to set inclusion.) For any set X {\displaystyle X} , there exists a maximal (under set inclusion) collection
Axiom_of_choice
Mathematical function
mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane
Dedekind_eta_function
In computability theory, a maximal set is a coinfinite computably enumerable subset A of the natural numbers such that for every further computably enumerable
Maximal set (computability theory)
Maximal_set_(computability_theory)
terms, a complete global analytic function is a path-connected sheaf of germs of analytic functions which is maximal in the sense that it is not contained
Global_analytic_function
Function returning one of only two values
switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the
Boolean_function
Mathematical logic concept
complement of the function which enumerates any maximal recursively enumerable set dominates every general recursive function. There exists maximal recursively
Computably_enumerable_set
Argument for the existence of God
require lower-order designs of individual organisms to fall short of maximal function. — William A. Dembski, The Design Revolution: Answering the Toughest
Teleological_argument
Azerbaijani mathematician
partial differential equations on Lie groups Singular integrals, maximal functions, and other integral operators, generated by Bessel differential operators
Vagif_Guliyev
Probability distribution
real-valued random variable. The general form of its probability density function is f ( x ) = 1 2 π σ 2 exp ( − ( x − μ ) 2 2 σ 2 ) . {\displaystyle f(x)={\frac
Normal_distribution
Economics theory
aversion expressed by those given utility function. Several functional forms often used for utility functions are represented by these measures. The higher
Risk_aversion
Exponential function of an exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f ( x ) = a b x = a ( b x ) {\displaystyle
Double_exponential_function
Measure of relative response to a drug
Intrinsic activity (IA) and maximal efficacy (Emax) refer to the relative ability of a drug-receptor complex to produce a maximum functional response
Intrinsic_activity
Conjecture on zeros of the zeta function
order n. For an example from group theory, if g(n) is Landau's function given by the maximal order of elements of the symmetric group Sn of degree n, then
Riemann_hypothesis
Matrix of partial derivatives of a vector-valued function
Rn → Rm is a differentiable function, a critical point of f is a point where the rank of the Jacobian matrix is not maximal. This means that the rank at
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Surface with vanishing affine mean curvature
In affine differential geometry, an affine maximal surface is a locally strongly convex hypersurface in an equiaffine manifold whose affine mean curvature
Affine_maximal_surface
American award for mathematical analysis
Univ. Press, Princeton, NJ, 1995 An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–674. A Kakeya-type
Bôcher_Memorial_Prize
Theorem in complex analysis
utilizes the symmetry of the Poisson kernel using the Hardy–Littlewood maximal function for the circle. The analogous theorem is frequently defined for the
Fatou's_theorem
Type of protein
D-stem of the Selenocysteine tRNA Provides Resilience at the Expense of Maximal Function". Journal of Biological Chemistry. 288 (19): 13337–13344. doi:10.1074/jbc
Selenoprotein
Mathematical function, used to describe magnetization
Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics. These functions are
Brillouin and Langevin functions
Brillouin_and_Langevin_functions
Extension of the domain of an analytic function (mathematics)
analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the
Analytic_continuation
Type of pseudorandom binary sequence
of pseudorandom binary sequence. They are bit sequences generated using maximal linear-feedback shift registers and are so called because they are periodic
Maximum-length_sequence
Task of computing complete subgraphs
vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem
Clique_problem
Point where the derivative of a function is zero or undefined (in certain cases)
being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds
Critical_point_(mathematics)
ρ d k , {\displaystyle \Xi (g)=\int _{K}a(kg)^{\rho }dk,} where K is a maximal compact subgroup of a semisimple Lie group with Iwasawa decomposition G=NAK
Harish-Chandra's_Ξ_function
Physical function
convenient set of Wannier functions. In practice, this is usually the maximally-localized set, in which the Wannier function ϕR is localized around the
Wannier_function
Mathematical function
In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric
Landau's_function
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MAXIMAL FUNCTION