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Type of function
In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties: f is continuous on [a, b]. (**)
Singular_function
Class of discontinuous functions
Singularity functions are a class of discontinuous functions that contain singularities, i.e., they are discontinuous at their singular points. Singularity
Singularity_function
Continuous function that is not absolutely continuous
the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue
Cantor_function
Point where a mathematical object behaves irregularly
reciprocal function f ( x ) = 1 / x {\displaystyle f(x)=1/x} has a singularity at x = 0 {\displaystyle x=0} , where the value of the function is not defined
Singularity_(mathematics)
Concept in differential equation mathematics
coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction
Regular_singular_point
Inputs for which a function's value is non-zero
distribution has singular support { 0 } {\displaystyle \{0\}} : it cannot accurately be expressed as a function in relation to test functions with support
Support_(mathematics)
Advertising executive, published author and speaker
hotel that decides to replace its doorman. One may consider their singular function to be opening and closing the door which is easily replaceable by
Rory Sutherland (advertising executive)
Rory_Sutherland_(advertising_executive)
Hypothetical event
The technological singularity, often simply called the singularity, is a hypothetical event in which technological growth accelerates beyond human control
Technological_singularity
Smooth approximation of one-hot arg max
max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The reason it
Softmax_function
problem fails to have a unique solution need not be singular functions. In some cases, the term singular solution is used to mean a solution at which there
Singular_solution
Function with unusual fractal properties
the Minkowski question mark function to ?:[0,1] → [0,1], it can be used as the cumulative distribution function of a singular distribution on the unit interval
Minkowski's question-mark function
Minkowski's_question-mark_function
Solution of a confluent hypergeometric equation
differential equation as the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit of
Confluent hypergeometric function
Confluent_hypergeometric_function
Undefined point on a holomorphic function which can be made regular
removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point
Removable_singularity
Invariant that plays a role in algebraic geometry and singularity theory
particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ
Milnor_number
Has no other singularities close to it
a complex number z 0 {\displaystyle z_{0}} is an isolated singularity of a function f {\displaystyle f} if there exists an open disk D {\displaystyle
Isolated_singularity
Mathematical function having a characteristic S-shaped curve or sigmoid curve
for sigmoid functions not evident or intuitive M1: Inverse of singularity functions M2: Sigmoid functions of embedded positive functions M3: Rising a
Sigmoid_function
Functions in harmonic analysis mathematics
dy,} whose kernel function K : R n × R n → R {\displaystyle K:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} } is singular along the diagonal
Singular_integral
Probability distribution in measure theory
is called singular, if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure.
Singular_measure
Function in quantum field theory showing probability amplitudes of moving particles
Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. These functions are most
Propagator
Matrix decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a scaling, followed
Singular_value_decomposition
Function defined by a hypergeometric series
regular singularities. The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list). The hypergeometric function is
Hypergeometric_function
Branch of mathematics studying functions of a complex variable
on the details of the contour, only how it winds around the singularities of the function. Being able to move a given contour to a more suitable one often
Complex_analysis
sequences of functions. Singular function: continuous, with zero derivative almost everywhere, but non-constant. Integrable function: has an integral (finite)
List_of_types_of_functions
Mathematical theory
branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where
Singularity_theory
Function for Heun's differential equation
equation that is holomorphic and 1 at the singular point z = 0. The local Heun function is called a Heun function, denoted Hf, if it is also regular at z = 1
Heun_function
Distribution concentrated on a set of measure zero
distribution (with a probability mass function), an absolutely continuous distribution (with a probability density), a singular distribution (with neither), or
Singular_distribution
Functions in mathematics
entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however
Harmonic_function
Objects extending the notion of functions
F_{\rm {smooth}}} and its singular F s i n g u l a r {\displaystyle F_{\rm {singular}}} parts. The product of generalized functions F {\displaystyle F} and
Generalized_function
Topics referred to by the same term
production by Santa Clara Vanguard Drum and Bugle Corps a singular function in mathematics Cantor function Baguenaudier, a disentanglement puzzle This disambiguation
Devil's_staircase
Concept in algebraic topology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X {\displaystyle X} , the
Singular_homology
Point on a curve not given by a smooth embedding of a parameter
geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends
Singular_point_of_a_curve
Point where the derivative of a function is zero or undefined (in certain cases)
connected components by a function of the degrees of the polynomials that define the variety. Singular point of a curve Singularity theory Nullcline Milnor
Critical_point_(mathematics)
Point without a tangent space
, y ) = 0 , {\displaystyle F(x,y)=0,} where F is a smooth function is said to be singular at a point if the Taylor series of F has order at least 2 at
Singular point of an algebraic variety
Singular_point_of_an_algebraic_variety
Technique in analytic number theory
coefficients). Technically, the generating function is scaled to have radius of convergence 1, so it has singularities on the unit circle – thus one cannot
Hardy–Ramanujan–Littlewood circle method
Hardy–Ramanujan–Littlewood_circle_method
Theorem in axiomatic set theory
theorem says that very little about this function can be determined in ZFC without additional axioms. For singular κ {\displaystyle \kappa } , upper bounds
Gimel_function
Integral transform and linear operator
Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t).
Hilbert_transform
Concept in algebraic geometry
resolution of singularities was to find a nonsingular model for the function field of a variety X, in other words a complete non-singular variety X′ with
Resolution_of_singularities
Type of function in mathematics
unit circle. Thus, even for real-valued functions, the role of complex singularities is important: a function can be infinitely differentiable on the
Analytic_function
Association of one output to each input
to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows
Function_(mathematics)
Area of mathematical analysis
Fourier transform, while modern harmonic analysis also studies maximal functions, singular integrals, oscillatory integrals, Fourier multipliers, Littlewood–Paley
Harmonic_analysis
Concept in complex analysis
type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential
Zeros_and_poles
Special mathematical function defined as sin(x)/x
cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic everywhere
Sinc_function
Set theory concept
implies κcf(κ) = κ+, where cf denotes the cofinality function. Note that κcf(κ)= 2κ for all singular strong limit cardinals κ. The second formulation of
Singular_cardinals_hypothesis
Solutions of Legendre's differential equation
separately as Legendre's function of the second kind, and denoted Qn. This is a second order linear equation with three regular singular points (at 1, −1, and
Legendre_function
Location around which a function displays irregular behavior
essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category essential singularity is a "left-over"
Essential_singularity
Conformal mappings in complex analysis
where Γ ( x ) {\textstyle \Gamma (x)} is the gamma function. Near each singular point, the function may be approximated as s 0 ( z ) = z α ( 1 + O ( z
Schwarz_triangle_function
General relativity model near spacetime singularities
relativity has a page on the topic of: BKL singularity A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe
BKL_singularity
Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals and
Maximal_function
2005 non-fiction book by Ray Kurzweil
The Singularity Is Near: When Humans Transcend Biology is a 2005 non-fiction book about artificial intelligence and the future of humanity by inventor
The_Singularity_Is_Near
Extension of the domain of an analytic function (mathematics)
to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points
Analytic_continuation
Noncommutative geometric structure
of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators
Singular_trace
Curve defined as zeros of polynomials
those that lack any singularities. Two nonsingular projective curves over a field are isomorphic if and only if their function fields are isomorphic
Algebraic_curve
Family of solutions to related differential equations
The Bessel function of the first kind is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero. The
Bessel_function
Phenomenon within general relativity
curvature singularity at the Cauchy horizon known as the mass-inflation singularity, the Cauchy horizon singularity, the infalling singularity, or the "fat
Mass_inflation
Attribute of a mathematical function
function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f : C ∖ { a k } k → C {\displaystyle
Residue_(complex_analysis)
Point on a curve where motion must move backwards
type A2-singularity. Let f (x, y) be a smooth function of x and y and assume, for simplicity, that f (0, 0) = 0. Then a type A2-singularity of f at (0
Cusp_(singularity)
Words in English that substitute for a noun or noun phrase
relatively small category of words in Modern English whose primary semantic function is that of a pro-form for a noun phrase. Traditional grammars consider
Pronouns_in_English
Group of West Germanic languages
standard Dutch), the historical second person plural form has acquired a singular function (e.g. standard Dutch jij maakt 'you (sg.) make'), and a new plural
North_Sea_Germanic
in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually
Splitting_lemma_(functions)
Stochastic process generalizing Brownian motion
a function of two variables x and t, the local time is still continuous. Treated as a function of t (while x is fixed), the local time is a singular function
Wiener_process
Type of singularity analysis
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
Wave_front_set
Mathematical theorem in set theory
2^{\lambda }} for singular cardinals λ {\displaystyle \lambda } . PCF theory shows that the values of the continuum function on singular cardinals are strongly
Easton's_theorem
Domain of convergence of power series
Taylor series of the analytic function to which it converges. In case of multiple singularities of a function (singularities are those values of the argument
Radius_of_convergence
`smooth' part of the solution after adding global singular functions to take care of corner singularities. The method can be extended to variable coefficient
Fokas_method
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
Concept in mathematics
In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to
Singular_perturbation
Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood
Holomorphic_function
In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. Hirzebruch (1973)
Signature_defect
{\mathcal {A}}} contains the higher-order singular values. HOSVD (M-moe SVD) canonical form of TP function f ( x ) = A ⊠ n = 1 N w n ( x n ) , {\displaystyle
HOSVD-based canonical form of TP functions and qLPV models
HOSVD-based_canonical_form_of_TP_functions_and_qLPV_models
Class of mathematical function
singularity. The function f ( z ) = sin 1 z {\displaystyle f(z)=\sin {\frac {1}{z}}} is not meromorphic either, as it has an essential singularity at
Meromorphic_function
Growth function exhibiting a singularity at a finite time
singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function 1
Hyperbolic_growth
Analysis theorem
analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function f : Rd → C, where
Calderón–Zygmund_lemma
Theorem
to the nearest non-removable singularity; if there are no singularities (i.e., if f {\displaystyle f} is an entire function), then the radius of convergence
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Generalization of the hypergeometric differential equation
generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0
Riemann's differential equation
Riemann's_differential_equation
Family of mathematical functions
and gives all parameter values for which the resulting function has degenerate singularities. Sometimes unfoldings are called deformations, versal unfoldings
Unfolding_(functions)
Mathematical function
The singularity spectrum is a function used in multifractal analysis to describe the fractal dimension of a subset of points of a function belonging to
Singularity_spectrum
Operation in differential geometry
fact a fibre bundle, it suffices to establish that it has non-singular transition functions under a change of coordinates. Let ( y i ) : M → R n {\displaystyle
Jet_(mathematics)
summation) and treats analytic functions with isolated singularities. He introduced the term in the late 1970s. Resurgent functions have applications in asymptotic
Resurgent_function
Generalized function whose value is zero everywhere except at zero
it is a singular measure. Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which
Dirac_delta_function
Analytic function in mathematics
infinity on the Riemann sphere the zeta function has an essential singularity. For sums involving the zeta function at integer and half-integer values, see
Riemann_zeta_function
Sigmoid shape special function
± i ∞ {\displaystyle \pm i\infty } . The error function is an entire function; it has no singularities (except at infinity) and its Taylor expansion always
Error_function
Singularities of holomorphic functions extend infinitely outward
states that the support of the singularities of such functions cannot be compact, therefore the singular set of a function of several complex variables
Hartogs's_extension_theorem
interpolation function develops a sharp peak as the field point approaches the boundary. Consequently, the kernels become “nearly singular” and can not
Singular_boundary_method
Analyzes the topology of a manifold by studying differentiable functions on that manifold
independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular Betti numbers agree
Morse_theory
Family of transcription factors
functional assays. The developmentally important Sox family has no singular function, and many members possess the ability to regulate several different
SOX_gene_family
Pronoun that is associated with a particular grammatical person
Personal pronouns may also take different forms depending on number (usually singular or plural), grammatical or natural gender, case, and formality. The term
Personal_pronoun
Subdivision of the United Arab Emirates
United Arab Emirates consists of seven emirates (Arabic: إمارات ʾimārāt; singular: إمارة ʾimārah), which were historically known as the Trucial States. Each
Emirates of the United Arab Emirates
Emirates_of_the_United_Arab_Emirates
Hypothetical phenomenon
In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon. When there exists at least one causal
Naked_singularity
On when a manifold that admits a singular foliation is homeomorphic to the sphere
Morse function, being the singularity a critical point of the function. The singularity is a center if it is a local extremum of the function; otherwise
Reeb_sphere_theorem
Language where one kind of inflection indicates multiple changes of aspect
ending -um denotes masculine accusative singular, neuter accusative singular, or neuter nominative singular. Many Indo-European languages feature fusional
Fusional_language
Key concept in higher-order singular value decomposition of functions
and Yam as key concept for higher-order singular value decomposition of functions. It transforms a function (which can be given via closed formulas or
Tensor product model transformation
Tensor_product_model_transformation
Analytic function in mathematics
In analysis, a lacunary function or series is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within
Lacunary_function
Algebraic structure used in topology
-cochain on X {\displaystyle X} can be identified with a function from the set of singular i {\displaystyle i} -simplices in X {\displaystyle X} to A
Cohomology
Study of smooth real-valued functions on manifold and their singularities
at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions f : M → R {\displaystyle
Cerf_theory
Method for load calculation in construction
A well organized family of functions called singularity functions are often used as a shorthand for the Dirac function, its derivative, and its antiderivatives
Euler–Bernoulli_beam_theory
Inverse functions of sin, cos, tan, etc.
trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under
Inverse trigonometric functions
Inverse_trigonometric_functions
Mathematical theorem
Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and
Casorati–Weierstrass_theorem
Description of the degeneracy of a function
and in particular singularity theory, an Ak singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced
Ak_singularity
Theorem about the range of an analytic function
lacunary value of the function. Great Picard's Theorem: If an analytic function f {\textstyle f} has an essential singularity at a point w {\textstyle
Picard_theorem
Signal processing technique
Generalized pencil-of-function method (GPOF), also known as matrix pencil method, is a signal processing technique for estimating a signal or extracting
Generalized pencil-of-function method
Generalized_pencil-of-function_method
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