Operator

This article is about operators in mathematics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For other uses, see operator (disambiguation).

In mathematics, an operator is a function which operates on (or modifies) another function. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Often, an "operator" is a function which acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which are solutions of differential equations. Linear algebra is the branch of Mathematics concerned with A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the An operator can perform a function on any number of operands (inputs) though most often there is only one operand.

An operator might also be called an operation, but the point of view is different. In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values For instance, one can say "the operation of addition" (but not the "operator of addition") when focusing on the operands and result. One says "addition operator" when focusing on the process of addition, or from the more abstract viewpoint, the function +: S×S → S.

Notation

An operator name or operator symbol is a notation which denotes a particular operator. When there is no danger of confusion, an operator name or operator symbol may be referred to more briefly as an "operator". Strictly speaking, however, the operator is a mathematical object and not the syntactic entity which denotes it. The reason for identifying it with its notation is that there are some operators which have come to have standard notations.

Simple examples of operators

In linear algebra an "operator" is a linear operator. Linear algebra is the branch of Mathematics concerned with In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In analysis an "operator" may be a differential operator, to perform ordinary differentiation, or an integral operator, to perform ordinary integration. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator In Mathematics, an integral transform is any transform T of the following form (Tf(u = \int_{t_1}^{t_2} K(t u\ f(t\ dt

One example of a differential operator is the derivative itself. In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator The corresponding operator name D, when placed before a differentiable function f, indicates that the function is to be differentiated with respect to the variable.

Operators versus functions

The word operator can in principle be applied to any function. However, in practice it is most often applied to functions which operate on mathematical entities of higher complexity than real numbers, such as vectors, random variables, or mathematical expressions. An entity is something that has a distinct separate Existence, though it need not be a material existence In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In mathematics the word expression is a term for any well-formed combination of mathematical symbols The differential and integral operators, for example, have domains and codomains whose elements are mathematical expressions of indefinite complexity. In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator In Mathematics, an integral transform is any transform T of the following form (Tf(u = \int_{t_1}^{t_2} K(t u\ f(t\ dt In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the codomain, or target, of a function f: X → Y is the set In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the In contrast, functions with vector-valued domains but scalar ranges are called functionals and forms. In Mathematics, a functional is traditionally a map from a Vector space to the field underlying the vector space which is usually the Real In Multilinear algebra, a multilinear form is a map of the type f V^N \to K where V is a Vector space

In general, if either the domain or codomain (or both) of a function contains elements significantly more complex than real numbers, that function is referred to as an operator. Conversely, if neither the domain nor the codomain of a function contain elements more complicated than real numbers, that function is likely to be referred to simply as a function. Trigonometric functions such as cosine are examples of the latter case.

Additionally, when functions are used so often that they have evolved faster or easier notations than the generic F(x,y,z,. . . ) form, the resulting special forms are also called operators. Examples include infix operators such as addition "+" and division "/", and postfix operators such as factorial "!". Infix notation is the common arithmetic and logical formula notation in which Operators are written Infix -style between the Operands they act on (e Reverse Polish notation (or just RPN) by analogy with the related Polish notation, a prefix notation introduced in 1920 by the Polish mathematician This usage is unrelated to the complexity of the entities involved.

Influences from other disciplines

Concepts from other disciplines, including in physics and to a lesser degree computer science, have influenced the ways in which operators are perceived and used. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

Physics

The mutual influence between physics and mathematics regarding the concept of operators has been long-term, beginning in the early 1900s, and profound in both directions. Quantum mechanics in particular was forced to move from classical measurement strategies involving only simple numeric values to the use of operators which transformed and manipulated far less intuitive entities. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons These included vectors in both real space and in generalizations of real space called Hilbert spaces, spinors, and various forms of matrices. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally The great physicist P.A.M. Dirac captured the importance of the relationship between quantum physics and mathematics by saying "Physical laws should have mathematical beauty and simplicity. "

Examples of mathematical operators

This section concentrates on illustrating the expressive power of the operator concept in mathematics. Please refer to individual topics pages for further details.

Linear operators

Main article: Linear transformation

The most common kind of operator encountered are linear operators. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In talking about linear operators, the operator is signified generally by the letters T or L. Linear operators are those which satisfy the following conditions; take the general operator T, the function acted on under the operator T, written as f(x), and the constant a:

T (f (x) + g (x)) = T (f (x)) + T (g (x))

T (a f (x)) = a T (f (x))

Many operators are linear. For example, the differential operator and Laplacian operator, which we will see later.

Linear operators are also known as linear transformations or linear mappings. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Many other operators one encounters in mathematics are linear, and linear operators are the most easily studied (Compare with nonlinearity). This article describes the use of the term nonlinearity in mathematics

Such an example of a linear transformation between vectors in R² is reflection: given a vector x = (x₁, x₂)

Q(x₁, x₂) = (−x₁, x₂)

We can also make sense of linear operators between generalisations of finite-dimensional vector spaces. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it For example, there is a large body of work dealing with linear operators on Hilbert spaces and on Banach spaces. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis See also operator algebra. In Functional analysis, an operator algebra is an algebra of continuous Linear operators on a Topological vector space with the multiplication

Operators in probability theory

Main article: Probability theory

Operators are also involved in probability theory, such as expectation, variance, covariance, factorials, etc. Probability theory is the branch of Mathematics concerned with analysis of random phenomena In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance Definition The factorial function is formally defined by n!=\prod_{k=1}^n k

Operators in calculus

Calculus is, essentially, the study of two particular operators: the differential operator D = d/dt, and the indefinite integral operator $\int_0^t$ . Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator In Mathematics, in the area of Functional analysis and Operator theory, the Volterra operator represents the operation of Indefinite integration These operators are linear, as are many of the operators constructed from them. In more advanced parts of mathematics, these operators are studied as a part of functional analysis. For functional analysis as used in psychology see the Functional analysis (psychology article

The differential operator

Main article: Differential operator

The differential operator is an operator which is fundamentally used in calculus to denote the action of taking a derivative. In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator Common notations are dy/dx, and y'(x) to denote the derivative of y(x). Here, however, we will use the notation which is closest to the operator notation we have been using; that is, using Df to represent the action of taking the derivative of f.

Integral operators

Given that integration is an operator as well (inverse of differentiation), we have some important operators we can write in terms of integration.

Convolution

Main article: Convolution

The convolution $*\,$ is a mapping from two functions f(t) and g(t) to another function, defined by an integral as follows:

$(f * g)(t) = \int_0^t f(\tau) g(t - \tau) \,d\tau.$

Fourier transform

Main article: Fourier transform

The Fourier transform is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and It is another integral operator; it is useful mainly because it converts a function on one (spatial) domain to a function on another (frequency) domain, in a way which is effectively invertible. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to Nothing significant is lost, because there is an inverse transform operator. In the simple case of periodic functions, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves and cosine waves:

$f(t) = {a_0 \over 2} + \sum_{n=1}^{\infty}{ a_n \cos ( \omega n t ) + b_n \sin ( \omega n t ) }$

When dealing with general function R → C, the transform takes on an integral form:

$f(t) = {1 \over \sqrt{2 \pi}} \int_{- \infty}^{+ \infty}{g( \omega )e^{ i \omega t } \,d\omega }.$

Laplacian transform

Main article: Laplace transform

The Laplace transform is another integral operator and is involved in simplifying the process of solving differential equations. In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic

Given f = f(s), it is defined by:

$F(s) = (\mathcal{L}f)(s) =\int_0^\infty e^{-st} f(t)\,dt.$

Fundamental operators on scalar and vector fields

Main articles: vector calculus, vector field, scalar field, gradient, divergence, and curl

Three operators are key to vector calculus:

∇, known as gradient, assigns a vector at every point in a scalar field which points in the direction of greatest change of that field. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the cURL is a Command line tool for transferring files with URL syntax. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar
Divergence is an operator which measures a vector field's tendency to originate from or converge upon a given point. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the
Curl is a vector operator which shows a vector field's tendency to rotate about a point. cURL is a Command line tool for transferring files with URL syntax.

Relation to type theory

Main article: Type theory

In type theory, an operator itself is a function, but has an attached type indicating the correct operand, and the kind of function returned. In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory Functions can therefore conversely be considered operators, for which we forget some of the type baggage, leaving just labels for the domain and codomain.

Operators in physics

Main article: Operator (physics)

In physics, an operator often takes on a more specialized meaning than in mathematics. In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Operators as observables are a key part of the theory of quantum mechanics. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In that context operator often means a linear transformation from a Hilbert space to another, or (more abstractly) an element of a C*-algebra. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This article assumes some familiarity with Analytic geometry and the concept of a limit. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics.

Operators in computer programming languages

Main article: Operator (programming)

In general, the term 'operator' in computer programming languages has the same meaning as in mathematics. Programming languages generally support a set of operators that are similar to operators in mathematics. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. This is particularly true in functional programming languages, where an operator is also a function. In Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and

Operators as primitives

However, most programming languages distinguish between operators and functions in that operators are a special primitive part of the language, both syntactically and in terms of functionality. For example, most languages provide a '+' (addition) operator, which adds two numbers without making a function call. The plus and minus signs ( + and &minus) are Mathematical symbols used to represent the notions of positive and negative as well as the operations

In many languages, this behaviour is totally different from that of a function call. For example, in C (and many derivatives such as Java), the arithmetic operators can act on any numeric data type, while functions are only allowed to act on a single explicit type. tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured A data type in Programming languages is an attribute of a datum which tells the computer (and the programmer something about the kind of datum it is However in C++ the distinction is blurred, since Operator overloading allows operators to be defined as functions, albeit only for data types that are not built-in. C++ (" C Plus Plus " ˌsiːˌplʌsˈplʌs is a general-purpose Programming language. In Computer programming, operator overloading (less commonly known as operator Ad-hoc polymorphism) is a specific case of polymorphism in

Other languages (primarily older ones) do not have functions which return values at all. However, they often still have operators which do return values, widening the distinction between operators and functions.

Non-mathematical operators

Programming languages often feature non-mathematical operators. These may include operators which reference or dereference pointers, which access array elements, or get the size of a data type. In Computer science an array is a Data structure consisting of a group of elements that are accessed by indexing. In the programming languages C and C++, the Unary operator ' sizeof' is used to calculate the sizes of datatypes. They may also include compound operators such as "+=", which increments a variable by a given value.

Operators in assembly language

In assembly language programming, the term "operator" may refer to the opcode of a given instruction. See the terminology section below for information regarding inconsistent use of the terms assembly and assembler In computer technology an opcode ( op eration code) is the portion of a Machine language instruction that specifies the operation to be performed This is very similar to the primitive concept of an operator in a higher-level language.

Dictionary

-noun

One who operates.
A telecommunications facilitator whose job is to establish temporary network connections.
(mathematics) A function or other mapping that carries variables defined on a domain into another variable or set of variables in a defined range.
Another name for Chinese whispers.
(informal) A person who is adept at making deals or getting results, especially one who uses questionable methods.
A member of a military Special Operations unit.
(computing) The administrator of a channel or network on IRC.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents