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Objects extending the notion of functions
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory
Generalized_function
Family of power series in mathematics
mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The
Generalized hypergeometric function
Generalized_hypergeometric_function
Class of statistical models
In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing
Generalized_linear_model
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
{\displaystyle 1/p!} in § Properties of the generalized Kronecker delta below disappearing. In terms of the indices, the generalized Kronecker delta is defined as:
Kronecker_delta
Function returning minus 1, zero or plus 1
(\operatorname {sgn} 0)^{2}=0} . This generalized signum allows construction of the algebra of generalized functions, but the price of such generalization
Sign_function
Operation on mathematical functions
vector/tuple-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition. A set of finitary
Function_composition
Method of solution to differential equations
into account the modern language of the theory of distributions or generalized functions. Building off of the superposition principle in many-body theory
Green's_function
Integration kernels for smoothing out sharp features
smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution
Mollifier
Order-preserving mathematical function
arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus, a function f {\displaystyle f} defined on a subset
Monotonic_function
Indicator function of positive numbers
The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function named after Oliver Heaviside
Heaviside_step_function
Generalized version of classical Green's function
Multiscale Green's function (MSGF) is a generalized and extended version of the classical Green's function (GF) technique for solving mathematical equations
Multiscale_Green's_function
On converting relations to functions of several real variables
Ulisse Dini (1845–1918) generalized the real-variable version of the implicit function theorem to the context of functions of any number of real variables
Implicit_function_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
Theorem
the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz
Schwartz_kernel_theorem
Sequence of differential equation solutions
+1-x\right)y'+n\,y=0} are called generalized Laguerre polynomials, or associated Laguerre polynomials. One can also define the generalized Laguerre polynomials recursively
Laguerre_polynomials
'algebra of continuous (set-valued) functions' on a generalized space, not the generalized space itself." A generalized space should not be confused with
Generalized_space
Matrix of partial derivatives of a vector-valued function
vector-valued function in several variables generalizes the gradient of a scalar-valued function in several variables, which in turn generalizes the derivative
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
to spaces of white noise test functions φ {\displaystyle \varphi } , and, by duality, to spaces of generalized functions Ψ {\displaystyle \Psi } of white
White_noise_analysis
Periodic distribution ("function") of "point-mass" Dirac delta sampling
mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic generalized function with the formula Ш T ( t ) := ∑ k
Dirac_comb
Formulation of classical mechanics
}}=0} so the new generalized coordinates and momenta are constants of motion. As they are constants, in this context the new generalized momenta P {\displaystyle
Hamilton–Jacobi_equation
Objects that generalize functions
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Family of probability distributions often used to model tails or extreme values
In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another
Generalized Pareto distribution
Generalized_Pareto_distribution
Analytic function that does not satisfy a polynomial equation
Complex function Function (mathematics) Generalized function List of special functions and eponyms List of types of functions Rational function Special
Transcendental_function
Study of rates of change
(after Laurent Schwartz) extended derivation to generalized functions (e.g., the Dirac delta function previously introduced in Quantum Mechanics) and
Differential_calculus
Class of discontinuous functions
points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution
Singularity_function
the generalized-Ozaki (GO) cost function is a general description of the cost of production proposed by Shinichiro Nakamura. The GO cost function is notable
Generalized Ozaki cost function
Generalized_Ozaki_cost_function
In mathematics, the crenel function is a periodic discontinuous function P(x) defined as 1 for x belonging to a given interval and 0 outside of it. It
Crenel_function
Mathematical theorem
of distributions (generalized functions) eliminates analytic problems with the symmetry. The derivative of an integrable function can always be defined
Symmetry of second derivatives
Symmetry_of_second_derivatives
for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive
Weil's_criterion
Multivariate derivative (mathematics)
scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle \nabla
Gradient
Construction for adding objects to a Hilbert space
paragraphs 23.8 and 23.32) Gel'fand, I. M.; Vilenkin, N. Ya (1964). Generalized Functions: Applications of Harmonic Analysis. Burlington: Elsevier Science
Rigged_Hilbert_space
Mathematical function with no sudden changes
where arguments and values of functions are real numbers and complex numbers. The concept has been generalized to functions between metric spaces and between
Continuous_function
Mathematical conjecture about zeros of L-functions
special case of Dirichlet L-functions.) The Generalized Riemann hypothesis asserts that all nontrivial zeros of Dirichlet L-function L ( χ , s ) {\textstyle
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Theorem in mathematics
complex-valued functions of a complex variable. It generalizes to functions from n-tuples (of real or complex numbers) to n-tuples, and to functions between
Inverse_function_theorem
Associative algebra used in combinatorics
examples can be unified and generalized by considering a multiset E, and finite sub-multisets S and T of E. The Möbius function is μ ( S , T ) = { 0 if
Incidence_algebra
Generalization of a mathematical distribution
an ultra-distribution) is a generalized function that extends the concept of a distributions by allowing test functions whose Fourier transforms have
Ultradistribution
N-th root of the arithmetic mean of the given numbers raised to the power n
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include
Generalized_mean
Mathematics of real numbers and real functions
Distributions (or generalized functions) are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives
Real_analysis
In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches
Limit_of_distributions
Integral of sin(x)/x from 0 to infinity
however, integrable in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral. This can be seen by using Dirichlet's
Dirichlet_integral
Association of one output to each input
logicians, give precise definitions for these weakly specified functions. These generalized functions may be critical in the development of a formalization of
Function_(mathematics)
In mathematical heat conduction, the Green's function number is used to uniquely categorize certain fundamental solutions of the heat equation to make
Green's_function_number
Moll. It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders. The generalized polygamma
Balanced_polygamma_function
S-shaped curve
logistic function and generalizations. In growth modeling, numerous generalizations exist, including the generalized logistic curve, the Gompertz function, the
Logistic_function
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Mathematical approximation of a function
z-a} is known as a Puiseux series. The Taylor series may also be generalized to functions of more than one variable with T ( x 1 , … , x d ) = ∑ n 1 = 0
Taylor_series
Formula for the derivative of a product
v+u\cdot {\frac {dv}{dx}}.} The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product
Product_rule
Statement about integration on manifolds
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called
Generalized_Stokes_theorem
Branch of mathematics
called the derivative function or just the derivative of the original function. Geometrically speaking, the derivative generalizes the idea of the slope
Calculus
Method for assigning values to integrals
a<b<c} and where b is the difficult point, at which the behavior of the function f is such that ∫ a b f ( x ) d x = ± ∞ {\displaystyle \int _{a}^{b}f(x)\
Cauchy_principal_value
Mathematical relation consisting of a multi-variable function equal to zero
multivariable functions that are continuously differentiable. A common type of implicit function is an inverse function. Not all functions have a unique
Implicit_function
Type of singularity 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
Wave_front_set
Statistics models class
a generalized additive model (GAM) is a generalized linear model in which the linear response variable depends linearly on unknown smooth functions of
Generalized_additive_model
Instantaneous rate of change (mathematics)
the second derivative is its acceleration. Derivatives can be generalized to functions of several real variables. In this case, the derivative is reinterpreted
Derivative
Operation in calculus
sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on the type of the function as well as the domain over
Integral
Topological vector spaces
(2001) [1994], "Generalized function", Encyclopedia of Mathematics, EMS Press. Vladimirov, V.S. (2001) [1994], "Generalized functions, space of", Encyclopedia
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Topics referred to by the same term
Wiktionary, the free dictionary. Distribution (mathematical analysis), generalized function used to formulate solutions of partial differential equations Distribution
Distribution
Matrix of second derivatives
partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix
Hessian_matrix
Mapping involving integration between function spaces
maps a function from its original function space into another function space via integration, where some of the properties of the original function might
Integral_transform
Equation in Fourier analysis
summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined
Poisson_summation_formula
Formula for the derivative of an inverse function
calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the
Inverse_function_rule
Probability distribution
{\displaystyle a>0} . For non-negative x from a generalized gamma distribution, the probability density function is f ( x ; a , d , p ) = ( p / a d ) x d −
Generalized gamma distribution
Generalized_gamma_distribution
Technique in integral evaluation
in 1769. Although generalized to triple integrals by Lagrange in 1773, and used by Legendre, Laplace, and Gauss, and first generalized to n variables by
Integration_by_substitution
Function with a multiplicative scaling behaviour
∈ V . {\displaystyle v\in V.} This definition is often further generalized to functions whose domain is not V, but a cone in V, that is, a subset C of
Homogeneous_function
Probability distribution
The exponential generalized beta (EGB) distribution follows directly from the GB and generalizes other common distributions. A generalized beta random variable
Generalized_beta_distribution
"Smoothing" integral transform
be given by the function F {\displaystyle F} . By using values of t {\displaystyle t} different from 1, we can define the generalized Weierstrass transform
Weierstrass_transform
mathematics, the Bessel–Maitland function, or Wright generalized Bessel function, is a generalization of the Bessel function, introduced by Edward Maitland
Bessel–Maitland_function
Fundamental object of geometry
points with non-zero charge). The Dirac delta function, or δ function, is (informally) a generalized function on the real number line that is zero everywhere
Point_(geometry)
Function used as a performance test problem for optimization algorithms
Theory and Applications. 80: 175–179. doi:10.1007/BF02196600. "Generalized Rosenbrock's function". Retrieved 2008-09-16. Kok, Schalk; Sandrock, Carl (2009)
Rosenbrock_function
Concept in statistical mechanics
motion to d time (but still one space) dimensions: it is a random (generalized) function from Rd to R. In particular, the one-dimensional continuum GFF is
Gaussian_free_field
degenerate distribution — it is a Distribution (mathematics) in the generalized function sense; but the notation treats it as if it were a continuous distribution
List of probability distributions
List_of_probability_distributions
Approximation of a function by a polynomial
theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle k} , called
Taylor's_theorem
Relationship between derivatives and integrals
differentiating a function (calculating its slopes, or rate of change at every point on its domain) with the concept of integrating a function (calculating
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Mathematical solution
a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not
Weak_solution
Mathematical theorem about functions
Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed
Fourier_inversion_theorem
Mathematical function
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows
Generalised_logistic_function
Model of an energy potential in quantum mechanics
potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero
Delta_potential
Conditions for switching order of integration in calculus
distributions - version of Fubini's theorem for distributions, that is, generalized functions Kuratowski–Ulam theorem – analog of Fubini's theorem for arbitrary
Fubini's_theorem
Type of function
singularity Generalized function Distribution Minkowski's question-mark function (**) This condition depends on the references "Singular function", Encyclopedia
Singular_function
Techniques in mathematical analysis
analysis is a branch of mathematical analysis that studies functions, generalized functions and partial differential equations by localizing them both
Microlocal_analysis
{\displaystyle S_{\alpha }^{\beta }} is a space of test functions for the theory of generalized functions, introduced by Gelfand and Shilov (1968, Chapter IV)
Gelfand–Shilov_space
Nonspecific long-lasting anxiety
Generalized anxiety disorder (GAD) is an anxiety disorder characterized by excessive, uncontrollable, and often irrational worry about events or activities
Generalized_anxiety_disorder
Notion in calculus
calculus, the differential represents the principal part of the change in a function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the independent
Differential_of_a_function
Method of mathematical integration
introduce the Lebesgue integral is to use so-called simple functions, which generalize the step functions of Riemann integration. Consider, for example, determining
Lebesgue_integral
Method of differentiating single-term polynomials
In calculus, the power rule is used to differentiate functions of the form f ( x ) = x r {\displaystyle f(x)=x^{r}} , whenever r {\displaystyle r} is
Power_rule
Mathematical concept
numbers. Non-function elements of our new space can be thought of as "operators", or generalized functions, whose algebraic action on functions is always
Convolution_quotient
Type of computer arithmetic
{\displaystyle x=\ell +f=3+0.9711308=3.9711308.} The mapping function is called the generalized logarithm function. It is defined as ψ ( X ) = { X if 0 ≤ X < 1 ,
Symmetric level-index arithmetic
Symmetric_level-index_arithmetic
Mathematical function, denoted exp(x) or e^x
current value of f ( x ) {\displaystyle f(x)} . The exponential function can be generalized to accept complex numbers as arguments. This reveals relations
Exponential_function
Type of derivative in mathematics
multivariable calculus, the same property is generalized to define the derivative of a vector-valued function or function of a vector argument. Sometimes called
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Indefinite integral
function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function
Antiderivative
Distributions on spaces of differential forms
integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out
Current_(mathematics)
Method for solving continuous operator problems (such as differential equations)
structures. They showed that the generalized function, namely unit-step function, Dirac’s delta function, and the doublet function are needed for obtaining accurate
Galerkin_method
Boundary condition for generalized functions
extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important
Trace_operator
Probability distribution
The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two parametric families of continuous probability distributions
Generalized normal distribution
Generalized_normal_distribution
Technique of studying linear partial differential equations
real-analytic functions on M is the restriction of the sheaf of holomorphic functions on X to M. Hyperfunction D-module Microlocal analysis Generalized function Edge-of-the-wedge
Algebraic_analysis
Mathematical theorem
a Paley–Wiener theorem is a theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform.
Paley–Wiener_theorem
Formula in calculus
theorem), generalized to an appropriate class of functions.[citation needed] The full generalization of the chain rule to multi-variable functions (such as
Chain_rule
Calculus of vector-valued functions
eigenvalues of the Hessian matrix at these zeros. Vector calculus can also be generalized to other 3-manifolds and higher-dimensional spaces. Vector calculus is
Vector_calculus
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized
Rectangular_function
GENERALIZED FUNCTION
GENERALIZED FUNCTION
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, a great functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Celtic
, great justiciary, or functionary.
Male
Egyptian
, the son of the functionary Heknofre.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Male
Egyptian
, an Egyptian functionary.
Biblical
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Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, an Egyptian functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
Male
Egyptian
, a high Egyptian functionary.
GENERALIZED FUNCTION
GENERALIZED FUNCTION
Girl/Female
Hindu
Boy/Male
Indian, Sanskrit
Growing; Thriving
Boy/Male
Scottish American Teutonic
From the island of the lime tree. Although in the past, Lindsay was a common boys' name, today...
Girl/Female
Tamil
Ashrika | à®…à®·à¯à®°à¯€à®•à®¾
Someone gives shelter
Boy/Male
Biblical
Sitting, or captivity, of the father'.
Girl/Female
Indian, Telugu
Goddess Sita
Girl/Female
Indian
Happy, Precious, Generous
Girl/Female
Hindu
Melodious
Surname or Lastname
English and Scottish
English and Scottish : occupational name for an archer, Middle English bow(e)man, bouman (from Old English boga ‘bow’ + mann ‘man’). This word was distinguished from Bowyer, which denoted a maker or seller of the articles. It is possible that in some cases the surname referred originally to someone who untangled wool with a bow. This process, which originated in Italy, became quite common in England in the 13th century. The vibrating string of a bow was worked into a pile of tangled wool, where its rapid vibrations separated the fibers, while still leaving them sufficiently entwined to produce a fine, soft yarn when spun.Americanized form of German Baumann (see Bauer) or the Dutch cognate Bouman.
Boy/Male
British, English
From the Field Estate
GENERALIZED FUNCTION
GENERALIZED FUNCTION
GENERALIZED FUNCTION
GENERALIZED FUNCTION
GENERALIZED FUNCTION
v. i.
To form into a genus; to view objects in their relations to a genus or class; to take general or comprehensive views.
imp. & p. p.
of Federalize
v. t.
To bring under a genus or under genera; to view in relation to a genus or to genera.
a.
Capable of being generalized, or reduced to a general form of statement, or brought under a general rule.
p. pr. & vb. n.
of Generalize
a.
Comprising structural characters which are separated in more specialized forms; synthetic; as, a generalized type.
v. t.
To derive or deduce (a general conception, or a general principle) from particulars.
n.
A generalized concept of magnitude.
n.
The system by which power is centralized, as in a government.
v. t.
To impregnate with a mineral; as, mineralized water.
imp. & p. p.
of Centralize
n.
One who takes general or comprehensive views.
imp. & p. p.
of Generalize
v. t.
To apply to other genera or classes; to use with a more extensive application; to extend so as to include all special cases; to make universal in application, as a formula or rule.
v. t.
To generalize or conclude as an inference from all the particulars; -- the opposite of deduce.
a.
Destitute of function, or of an appropriate organ. Darwin.
n.
A fishlike creature (Amphioxus lanceolatus), two or three inches long, found in temperature seas; -- also called the lancelet. Its body is pointed at both ends. It is the lowest and most generalized of the vertebrates, having neither brain, skull, vertebrae, nor red blood. It forms the type of the group Acrania, Leptocardia, etc.
n.
The act or process of centralizing, or the state of being centralized; the act or process of combining or reducing several parts into a whole; as, the centralization of power in the general government; the centralization of commerce in a city.
v. t.
To make universal; to generalize.
imp. & p. p.
of Mineralize