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Encyclopedia > Operator

In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. Addition and multiplication are simple examples of operators. Another example is the D operator, when placed before a differentiable function f(t), indicates that the function is to be differentiated with respect to the variable t. Euclid, detail from The School of Athens by Raphael. ... Partial plot of a function f. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ...


Most operators perform a function on two inputs called operands. Although this is the case, an operator can perform a function on any number of inputs - though more than two inputs can make for a confusing operator. In mathematics, an operand is one of the inputs of an operator. ...

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Operators and levels of abstraction

To begin with, the usage of operator in mathematics is subsumed in the usage of function: an operator can be taken to be some special kind of function. The word is generally used to call attention to some aspect of its nature as a function. Since there are several such aspects that are of interest, there is no completely consistent terminology. Common are these: Partial plot of a function f. ...

  • To draw attention to the function domain, which may itself consist of vectors or functions, rather than just numbers. The expectation operator in probability theory, for example, has random variables as domain (and is also a functional).
  • To draw attention to the fact that the domain consists of pairs or tuples of some sort, in which case operator is synonymous with the usual mathematical sense of operation.
  • To draw attention to the function codomain; for example a vector-valued function might be called an operator.

A single operator might conceivably qualify under all three of these. Other important ideas are: In mathematics, the domain of a function is the set of all input values to the function. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Partial plot of a function f. ... expectation in the context of probability theory and statistics, see expected value. ... Probability theory is the mathematical study of probability. ... A random variable is a term used in mathematics and statistics. ... In mathematics, the term functional is applied to certain functions. ... In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects (an infinite sequence is a family). ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ...

  • Overloading, in which for example addition, say, is thought of as a single operator able to act on numbers, vectors, matrices…
  • Operators are often in practice just partial functions, a common phenomenon in the theory of differential equations since there is no guarantee that the derivative of a function exists.
  • Use of higher operations on operators, meaning that operators are themselves combined.

These are abstract ideas from mathematics, and computer science. They may however also be encountered in quantum mechanics. There Dirac drew a clear distinction between q-number or operator quantities, and c-numbers which are conventional complex numbers. The manipulation of q-numbers from that point on became basic to theoretical physics. In computer science, overloading is a type of polymorphism where different functions with the same name are invoked based on the data types of the parameters passed. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ... Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Dirac is a prototype algorithm for the encoding and decoding (see codec) of raw video and sound. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = −1. ...


Describing operators

Operators are described usually by the number of operands:

The number of operands is also called the arity of the operator. If an operator has an arity given as n-ary (or n-adic), then it takes n arguments. In programming, other than functional programming, the -ary terms are more often used than the other variants. See arity for an extensive list of the -ary endings. The field of universal algebra also includes the study of operators and their arities. In mathematics, a unary operation is an operation with only one operand. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a ternary operation is any operation of arity three, that is, that takes three arguments. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions. ... In mathematics and computer programming the arity of a function or an operator is the number of arguments or operands it takes (arity is sometimes referred to as valency, although that actually refers to another meaning of valency in mathematics). ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ...


Notations

There are five major systematic ways of writing operators and their arguments. These are

  • prefix: where the operator name comes first and the arguments follow, for example:
Q(x1, x2,...,xn).
In prefix notation, the brackets are sometimes omitted if it is known that Q is an n-ary operator.
  • postfix: where the operator name comes last and the arguments precede, for example:
(x1, x2,...,xn) Q
In postfix notation, the brackets are sometimes omitted if it is known that Q is an n-ary operator.
  • infix: where the operator name comes between the arguments. This is awkward and not commonly used for operators other than binary operators. Infix style is written, for example, as
x1 Q x2.
  • a superscript or subscript on the right or on the left; the main uses are selection (an index), such as a coordinate of a vector, and, in the case of a superscript on the right, for exponentiation of numbers and square matrices, and multiple function composition.

For operators on a single argument, prefix notation such as −7 is most common, but postfix such as 5! (factorial) or x* is also usual. Polish notation, also known as prefix notation was created by Jan Łukasiewicz. ... Reverse Polish notation (RPN) , also known as postfix notation, is an arithmetic formula notation, derived from the Polish notation introduced in 1920 by the Polish mathematician Jan Łukasiewicz. ... Infix has meanings in linguistics, mathematics and computer science, and chemistry. ... Juxtaposition (noun) is an act or instance of placing two things close together or side by side. ... In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ... This article gives an overview of the various ways to multiply matrices. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. ... A superscript is a number, figure, or symbol that appears above the normal line of type, at the right or left of another symbol or text. ... A subscript is a number, figure or indicator, that appears below the normal line of type, when used in a formula, mathematical expression or description of a chemical compound. ... In mathematics, an index is a superscript or subscript to a symbol. ... In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ...


There are other notations commonly met. Writing exponents such as 28 is really a law unto itself, since it is postfix only as a unary operator applied to 2, but on a slant as binary operator. In some literature, a circumflex is written over the operator name. In certain circumstances, they are written unlike functions, when an operator has a single argument or operand. For example, if the operator name is Q and the operand a function f, we write Qf and not usually Q(f); this latter notation may however be used for clarity if there is a product — for instance, Q(fg). Later on we will use Q to denote a general operator, and xi to denote the i-th argument. In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... The circumflex ( ˆ ) (more commonly known as an uppen) is a diacritic mark used in written Greek, French, Esperanto, Norwegian, Romanian, Slovak, Vietnamese, Japanese romaji, Welsh, Portuguese, Italian, Afrikaans, and other languages. ... A parameter is a measurement or value on which something else depends. ...


Notations for operators include the following. If f(x) is a function of x and Q is the general operator we can write Q acting on f as (Qf)(x) also.


Operators are often written in calligraphy to differentiate them from standard functions. For instance, the Fourier transform (an operator on functions) of f(t) (a function of t), which produces another function F(ω) (a function of ω), would be represented as mathcal{F} { f(t) } = F(omega). Calligraphy in a Latin Bible of AD 1407 on display in Malmesbury Abbey, Wiltshire, England. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...


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 In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...


The most common kind of operator encountered are linear operators. 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(af(x)) = aT(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. Many other operators one encounters in mathematics are linear, and linear operators are the most easily studied (Compare with nonlinearity). In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. ...


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. For example, there is a large body of work dealing with linear operators on Hilbert spaces and on Banach spaces. See also operator algebra. 2-dimensional renderings (ie. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. ...


Operators in probability theory

Main article: Probability theory Probability theory is the mathematical study of probability. ...


Operators are also involved in probability theory. Such operators as expectation, variance, covariance, factorials, etc. expectation in the context of probability theory and statistics, see expected value. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ...


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. 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. Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... In mathematics, a differential operator is a linear 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, viewed as a bounded linear operator on the space L2(0,1) of complex-valued square integrable functions on the interval (0,1). ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...


The differential operator

Main article: Differential operator In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...


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


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 For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...


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) dtau.

Fourier transform

Main article: Fourier transform The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...


The Fourier transform is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few. 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 that is effectively invertible. 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: In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C is the...

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 RC, the transform takes on an integral form: In calculus, the integral of a function is a generalization of area, mass, volume and total. ...

f(t) = {1 over sqrt{2 pi}} int_{- infty}^{+ infty}{g( omega ) cdot exp {( i omega t )} cdot 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, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...


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, scalar field, gradient, divergence, and curl Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... This article is about the cURL command line tool. ...


Three main operators are key to vector calculus, the operator ∇, known as gradient, where at a certain point in a scalar field forms a vector which points in the direction of greatest change of that scalar field. In a vector field, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point. Curl, in a vector field, is a vector operator that shows a vector field's tendency to rotate about a point. Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... This article is about the cURL command line tool. ...


Relation to type theory

Main article: Type theory At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ...


In type theory, an operator itself is a function, but has an attached type indicating the correct operand, and the kind of function returned. Functions can therefore conversely be considered operators, for which we forget some of the type baggage, leaving just labels for the domain and codomain. At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ...


Operators in physics

Main article: Operator (physics) In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ...


In physics, an operator often takes on a more specialized meaning than in mathematics. Operators as observables are a key part of the theory of quantum mechanics. In that context operator often means a linear transformation from a Hilbert space to another, or (more abstractly) an element of a C*-algebra. A Superconductor demonstrating the Meissner Effect. ... 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 operations. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... C*-algebras are an important area of research in functional analysis. ...


See also


  Results from FactBites:
 
Operator - Wikipedia, the free encyclopedia (1577 words)
In mathematics, an operator name or an operator symbol is a notation that denotes the type of mathematical object that is known as an operation.
Operators are often in practice just partial functions, a common phenomenon in the theory of differential equations since there is no guarantee that the derivative of a function exists.
Three main operators are key to vector calculus, the operator ∇, known as gradient, where at a certain point in a scalar field forms a vector which points in the direction of greatest change of that scalar field.
Operator algebra page of N. C. Phillips (1256 words)
Noncommutative geometry and operator algebras at Cardiff University.
Operator algebras at the University of Southern Denmark (Odense).
Noncommutative geometry and operator algebras at Vanderbilt University.
  More results at FactBites »

 
 

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