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representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes of G. In other words, it is invariant
Class_function
Programming language feature
first-class functions if it treats functions as first-class citizens. This means the language supports passing functions as arguments to other functions, returning
First-class_function
Degree of differentiability of a function or map
of a function describes the number of times it can be differentiated without producing discontinuities. The smoothness, or differentiability class, is
Smoothness
Collection of sets in mathematics that can be defined based on a property of its members
be thought of as "classes". In ZF, the concept of a function can also be generalised to classes. A class function is not a function in the usual sense
Class_(set_theory)
Function definition that is not bound to an identifier
with first-class functions, where they fulfil the same role for the function type as literals do for other data types. Anonymous functions originate in
Anonymous_function
Technique for creating lexically scoped first class functions
lexical closure or function closure, is a technique for implementing lexically scoped name binding in a language with first-class functions. Operationally
Closure (computer programming)
Closure_(computer_programming)
special comparison functions. Class K {\displaystyle {\mathcal {K}}} functions belong to this family: Definition: a continuous function α : [ 0 , a ) → [
Class_kappa_function
In statistical learning theory, a learnable function class is a set of functions for which an algorithm can be devised to asymptotically minimize the expected
Learnable_function_class
Concept in programming language design
passed as an argument, returned from a function, and assigned to a variable. The concept of first- and second-class objects was introduced by Christopher
First-class_citizen
Programming construct
first-class function objects and may even make significant use of them. Functional programming languages additionally support closures, i.e. first-class functions
Function_object
Software design pattern
be achieved by mechanisms such as the native function pointer, the first-class function, classes or class instances in object-oriented programming languages
Strategy_pattern
(exponentially many) set-theoretic functions mappings A to B in the category of sets. The class of such maps or functions is called the exponential object
Function_type
Duality between the process of restricting and inducting in representation theory
class function of G to H, and let Ind H G {\displaystyle \operatorname {Ind} _{H}^{G}} denote the induced class function of a given class function on
Frobenius_reciprocity
Type of data structure
collectively referred to as non-union classes) that has data and functions (also called member variables and member functions) as its members whose access is
C++_classes
Function that takes one or more functions as an input or that outputs a function
computer science, a higher-order function (HOF) is a function that does at least one of the following: takes one or more functions as arguments (i.e. a procedural
Higher-order_function
Concept within complex analysis
complex analysis, the Hardy spaces (or Hardy classes) H p {\displaystyle H^{p}} are spaces of holomorphic functions on the unit disk or upper half plane. They
Hardy_space
Composite data type
such as System F-sub. Since type-theoretical records may contain first-class function-typed fields in addition to data, they can express many features of
Record_(computer_science)
Negative of a convex function
a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex
Concave_function
Book by Thorstein Veblen
and sociology, and a critique of conspicuous consumption as a function of social class and of consumerism, which are social activities derived from the
The Theory of the Leisure Class
The_Theory_of_the_Leisure_Class
Concept in mathematical group theory
homomorphism in general. Characters are class functions, that is, they each take a constant value on a given conjugacy class. More precisely, the set of irreducible
Character_theory
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Mathematical function whose derivative exists
differentiable functions are sometimes said to be of class C1. A function is of class C2 if the first and second derivative of the function both exist and
Differentiable_function
Basic result in harmonic analysis on compact topological groups
basis of L 2 ( G ) . {\displaystyle L^{2}(G).} A function f {\displaystyle f} on G is called a class function if f ( h g h − 1 ) = f ( g ) {\displaystyle f(hgh^{-1})=f(g)}
Peter–Weyl_theorem
Representations of finite groups, particularly on vector spaces
of functions on groups: Definition (Class functions). A function φ : G → C {\displaystyle \varphi :G\to \mathbb {C} } is called a class function if it
Representation theory of finite groups
Representation_theory_of_finite_groups
Set of rules defining correctly structured programs
built-in data types and structures, control flow mechanisms, first-class functions, and modules for better code reusability and organization. Python also
Python_syntax_and_semantics
System with weakly-associated components
object). See First-class function for further insight into functions as objects, which qualifies as one form of first-class function. For example, in an
Loose_coupling
Smooth approximation of one-hot arg max
network to a probability distribution over predicted output classes. The softmax function takes as input a tuple z of K real numbers and normalizes it
Softmax_function
Two-dimensional group theory table
conjugacy class of G {\displaystyle G} (because characters are class functions). The columns are labelled by (representatives of) the conjugacy classes of G
Character_table
2011 edition of the C++ programming language standard
by std::unique_ptr. Function object base classes (std::unary_function, std::binary_function), adapters to pointers to functions and adapters to pointers
C++11
Programming mechanism
later. Such a function can always be defined but languages with first-class functions make it easier. The ability to easily compose functions encourages
Function composition (computer science)
Function_composition_(computer_science)
the classes slightly differently, by removing all functions of class less than α from the functions of class α. This means that each Baire function has
Baire_function
Variable defined in a class whose objects all possess the same copy
and class members applies to methods ("member functions") as well; a class may have both instance methods and class methods. In some languages, class variables
Class_variable
Type of function in complex analysis
mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis
Plurisubharmonic_function
Type of continuity of a complex-valued function
In mathematics, we say that a function satisfies a Hölder condition, or is α {\displaystyle \alpha } -Hölder continuous or simply Hölder continuous, if
Hölder_condition
Function that is continuous everywhere but differentiable nowhere
smoothness. These types of functions were disliked by contemporaries. For instance, Charles Hermite, on finding that one class of function he was working on had
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Mathematical concept
Given a class function G: V → V (where V is the class of all sets), there exists a unique transfinite sequence F: Ord → V (where Ord is the class of all
Transfinite_induction
Topological group with compact topology
(x)=\operatorname {trace} (\Pi (x)),\quad x\in K} . This function is easily seen to be a class function, i.e., X ( x y x − 1 ) = X ( y ) {\displaystyle \mathrm
Compact_group
Lightweight programming language
programmers to implement namespaces, classes and other related features using its single table implementation; first-class functions allow the employment of many
Lua
Type of Dirichlet series associated to number field extensions
absolute norms of ideals. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. They play an important
Artin_L-function
special comparison functions. Class K L {\displaystyle {\mathcal {KL}}} functions belong to this family: Definition: A continuous function β : [ 0 , a ) ×
Class_kappa-ell_function
High-level programming language
has dynamic typing, prototype-based object-orientation, and first-class functions. It is multi-paradigm, supporting event-driven, functional, and imperative
JavaScript
Indicator function of positive numbers
of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally
Heaviside_step_function
Concept in set theory
schema of replacement states that if F {\displaystyle F} is a definable class function, as above, and A {\displaystyle A} is any set, then the image F [ A
Axiom_schema_of_replacement
quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on an
Quasi-analytic_function
Programming paradigm based on applying and composing functions
running state of the program. In functional programming, functions are treated as first-class entities, meaning that they can be bound to names (including
Functional_programming
Theorem in harmonic analysis
characters. From basic character theory, if f {\displaystyle f} is a class function, we have the Parseval formula ‖ f ‖ G 2 = ‖ f ^ ‖ G ^ 2 {\displaystyle
Plancherel_theorem
Class of mathematical functions
elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred
Weierstrass_elliptic_function
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
with first-class functions, where they fulfil the same role for the function type as literals do for other data types. Anonymous functions originate in
Examples of anonymous functions
Examples_of_anonymous_functions
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
Function whose domain is the positive integers
prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value
Arithmetic_function
Finalizer First-class function Fragile base class Function composition Generic programming God object Heap-based memory allocation Helper class Hybrid language
Index of object-oriented programming articles
Index_of_object-oriented_programming_articles
Collection of objects and morphisms
{mor} ({\mathcal {C}})} of morphisms or arrows, a domain or source class function dom : mor ( C ) → ob ( C ) {\displaystyle \operatorname {dom} :\operatorname
Category_(mathematics)
Inheritable and overridable function or method for which dynamic dispatch is facilitated
being overridden by derived classes (such as the final and private keywords in Java and PHP). The concept of the virtual function solves the following problem:
Virtual_function
Topics referred to by the same term
programming First class (computing), a concept in data modeling First-class function First-class object, a concept in programming FirstClass, e-mail and online
First_class
l-values, as appropriate. R, S and T stand for a data type, and K for a class or enumeration type. Some operators have alternative spellings using digraphs
Operators_in_C_and_C++
Mathematical formula
says there exists a real-valued continuous function u on T such that for every class function f on G (function invariant under conjugation by G): ∫Gf(g)dg=∫Tf(t)u(t)dt
Weyl_integration_formula
Mapping which preserves all topological properties of a given space
equivalence classes are called homeomorphism classes. The third requirement, that f−1 be continuous, is essential. Consider for instance the function f:[0,2π)→S1
Homeomorphism
Mathematical result on ordinals
proved by Oswald Veblen in 1908. A normal function is a class function f {\displaystyle f} from the class Ord of ordinal numbers to itself such that:
Fixed-point lemma for normal functions
Fixed-point_lemma_for_normal_functions
Programming language
facilities. Iterators are supported and can be used as first class entities, as can functions, allowing for the use of functional programming methods. Object-oriented
Nim_(programming_language)
Software design pattern
member functions in the derived class. However, if base class member functions use CRTP for all member function calls, the overridden functions in the
Curiously recurring template pattern
Curiously_recurring_template_pattern
Economic and social class in the United States
OCLC 886497772. Retrieved August 21, 2018→ Values of the Working Class: function of their resources, position in occupational hierarchy, & their historical
Working class in the United States
Working_class_in_the_United_States
Function space of all functions whose derivatives are rapidly decreasing
mathematics, Schwartz space S {\displaystyle {\mathcal {S}}} is the function space of all functions whose derivatives of all orders are rapidly decreasing. This
Schwartz_space
H ≤ G. More generally, there is also a notion of the induction of a class function f on H, Ind ( f ) {\displaystyle \operatorname {Ind} (f)} , given
Induced_character
Axiomatic definition of a class of L-functions
In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms
Selberg_class
Fundamental result in the branch of mathematics known as character theory
here is that a Z [ ω ] {\displaystyle \mathbb {Z} [\omega ]} -valued class function lies in the ideal I*(G) if its values are all divisible (in Z [ ω ]
Brauer's theorem on induced characters
Brauer's_theorem_on_induced_characters
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
About mathematical functions
The mathematical concept of a function dates from the 17th century in connection with the development of calculus; for example, the slope d y / d x {\displaystyle
History of the function concept
History_of_the_function_concept
Extension of the factorial function
Funktionen, die der Gammafunktion analog sind" [On a class of functions analogous to the gamma function]. Leipzig Weidmannsche Buchhandlung. 46: 268–275.
Gamma_function
Generic type features in C++
programming language that allow functions and classes to operate with generic types. This allows a function or class declaration to reference via a generic
Template_(C++)
In functional programming
application (or partial function application) refers to the process of fixing a number of arguments of a function, producing another function of smaller arity
Partial_application
Theorem in mathematics
determinant". If the function of the theorem belongs to a higher differentiability class, the same is true for the inverse function. There are also versions
Inverse_function_theorem
C/C++ library offered in pure header files
library is called header-only if the full definitions of all macros, functions and classes comprising the library are visible to the compiler in a header file
Header-only
Set of rules defining correctly structured C++ program
a function in a derived class overrides a function in a base class, the function to call is determined by the type of the object. A given function is
C++_syntax
Function defined by multiple sub-functions
mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned
Piecewise_function
Style of computer programming
{$endif} interface type TGenericClass<T> = class function Foo(const AValue: T): T; end; implementation function TGenericClass<T>.Foo(const AValue: T): T;
Generic_programming
In object-oriented programming, a friend function, that is a "friend" of a given class, is a function that is given the same access as methods to private
Friend_function
Meromorphic function on the complex plane
An L-function is a meromorphic function on the complex plane, and one out of several categories of mathematical objects studied in analytic number theory
L-function
Generalization of "n-th" to infinite cases
cardinals, the class of limit cardinals, and the class of fixed points of the ℵ {\displaystyle \aleph } -function. In contrast, the class of regular cardinals
Ordinal_number
Mechanism for supporting dynamic dispatch
method binding). Whenever a class defines a virtual function (or method), most compilers add a hidden member variable to the class that points to an array
Virtual_method_table
Pointer that points to a function
} Functors, or function objects, are similar to function pointers, and can be used in similar ways. A functor is an object of a class type that implements
Function_pointer
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Complexity class
complexity class FP is the set of function problems that can be solved by a deterministic Turing machine in polynomial time (and for which the function problem
FP_(complexity)
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
Indicator function of rational numbers
shows that the Dirichlet function is a Baire class 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous
Dirichlet_function
Programming language implementation problem
funarg problem (function argument problem) refers to the difficulty in implementing first-class functions (functions as first-class objects) in programming
Funarg_problem
Behavioral design pattern
first-class functions and higher-order functions in functional programming languages. Specifically, the invoker object is a higher-order function of which
Command_pattern
Functions in mathematics
commute on this class of functions. In several ways, the harmonic functions are real analogues to holomorphic functions. All harmonic functions are analytic
Harmonic_function
List data structure
proportional to its length. Difference lists can be implemented using first-class functions or using unification. Whether a difference list is more efficient than
Difference_list
One of several equivalent definitions of a computable function
Ackermann function. Other equivalent classes of functions are the functions of lambda calculus and the functions that can be computed by Markov algorithms
General_recursive_function
Programming language concept
For example, an Action<T> represents a first-class function expecting an argument of type T, and a function that can handle any type of animal can always
Type_variance
Term for an idiom in the C++ language
expansion. The idiom is characterized by an in-class friend function definition appearing in the base class template component of the curiously recurring
Barton–Nackman_trick
Maximal compact connected Abelian Lie subgroup
important special case of this result occurs when f is a class function, that is, a function invariant under conjugation. In that case, we have ∫ G f
Maximal_torus
Set of all things that may be the input of a mathematical function
the unknown function(s) sought. For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which
Domain_of_a_function
Subtype of white blood cell
primarily responsible for humoral immunity (relating to antibodies). The function of T cells and B cells is to recognize specific "non-self" antigens, during
Lymphocyte
Formula in number theory
theory, the class number formula relates many important invariants of an algebraic number field to a special value of its Dedekind zeta function. We start
Class_number_formula
Set of problems in computational complexity theory
Other types of problems that certain complexity classes are defined in terms of include: Function problems (e.g. FP) Counting problems (e.g. #P) Optimization
Complexity_class
Type of character in number theory
Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which
Hecke_character
Concept in computability theory
He defined the class of elementary recursive functions ("Kalmár elementary functions") as a subset of the primitive recursive functions — specifically
Elementary_recursive_function
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