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Operator

In mathematics, an operator generally is a symbolism to show a certain mapping, usually from one or more given functions to another (between function spaces), however, operators can refer to mappings between vector spaces in general as well.

Operators in mathematics

Operators generally transform functions into other functions. We also say an operator maps a function to another. In some literature, they are designated by showing a small uphat 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 called Q and operates on a function f, we write Qf and not usually Q(f), however this latter notation may be used for clarity if there is a product for instance, eg. Q(fg). Throughout this article we will use Q to denote a general operator, and xi to denote the i-th argument.

Notations for operations on functions may also be notated as 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.

Functions can be considered operators, but are generally thought of differently conceptually. "Numbers" can be considered functions too, if f(x)=x0, this represents the number 1. Similarly after multiplication by a constant, any number can be defined. When an operator takes some numbers as arguments, we can consistenly regard the operator as still transforming functions, since we have seen that numbers can be considered as functions.

Notations and ideas

Describing operators

Operators are described usually by the number of operands:
• monodic, or unary: one argument
• dyadic, or binary: two arguments
• triadic, or ternary: three arguments
and so on.

Notating operators

There are three major 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 a 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 a n-ary operator.
• infix: where the operator name comes between the arguments. This is not commonly used for operators taking greater than 2 arguments, ie binary operators. Trivially for an operator taking 1 argument, writing infix is equivalent to writing prefix. Infix style is written, for example:
x1 Q x2

Linear operators

A key concept is the concept of the linear operator. Linear operators are those which satisfy the following conditions; take the general operator Q, the function acted on under the operator Q, written as f(x), and the constant a:
Such examples of linear operators are the differential operator and Laplacian operator, which we will see later.

Linear operators with respect to mappings between vector spaces are known more commonly as linear transformations or linear mappings.

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

Q(x1, x2)=(-x1, x2)

abstract algebra, may satisfy the commutative and associative laws. If there is also a predefined multiplicative operator then the operator must satisfy the distributive law.

Multiplicative operators

A multiplcative operator, in abstract algebra, may satisfy the associative law. If there is also a predefined multiplicative operator the operator must satisfy the distributive law.

Standard operators

Arithmetic operators are binary operators that perform simple transformations that many would find familiar. It is not obvious, but addition, subtraction, etc. are in fact operators. Many of these standard arithmetic operators use symbols to denote what operations are being performed.

Addition is written using the symbol +. It transforms two numbers x1 and x2 into their sum. For example:
3 + 5 = 8

It is written most commonly as x1+x2. In prefix notation, it may be written as + x1 x2, or +(x1,x1), or even with + changed to a word, such as
plus x1 x1.

Subtraction

Subtraction is written using the symbol -. It transforms two numbers x1 and x2 into their difference. For example:
11 - 4 = 7

It is written most commonly as x1-x2. In prefix notation, it may be written as - x1 x2, or -(x1,x1), or even with - changed to a word, such as
sub x1 x1.

Subtraction is equivalent to addition. The identity is that:
x1 - x2x1 + (-x2)
where - as a unary operator represents negation (see next section)

Negation

Negation is written also using the symbol -, however, it is only a unary operator. Given a number α, we denote the transformation of α to its additive inverse by -α. The additive inverse of a number k is an element k', such that k+k'=0.

Multiplication

Multiplication is written using the symbol ×. In certain circumstances, the operator symbol is omitted usually when the arguments to × are variable quantities, eg xy. Less commonly when representing the product of two numbers, they are placed in brackets and placed adjacently, eg. (2)(3)=6. Less commoner still, a small dot is used infix instead of ×, eg 2·3=6

Multiplication transforms two numbers x1 and x2 into their product. For example:

6 × 2 = 12

It is written most commonly as x1x2. In prefix notation (using ×) it may be written as × x1 x2, or ×(x1,x1), or even with × changed to a word, such as
mul x1 x1.

Multiplication is equivalent to repeated addition. The identity is that:
x1x2x1 + x1 + ...( x2 times)...+x1

Division

Division is written using the symbol /. Like multiplication, there are several ways to denote this, other than using /. If there is not much room on a page, or when typeset on a single line, the two arguments are written infix, eg 3 / 4, or x1/x2. If there is room on a page, the two arguments are usually written atop each other and a line seperating them, eg:

Division transforms two numbers x1 and x2 into their quotient. For example:
8 / 2 = 4

It is written most commonly as x1/x2. In prefix notation (using /) it may be written as / x1 x2, or /(x1,x1), or even with / changed to a word, such as
div x1 x1.

Division is equivalent to repeated subtraction. x2 is subtracted from x1 until there only is a positive remainder left. When, after application of this algorithm, there is zero remainder, we call the amount of subtractions we have performed the quotient. If not, we can write the result of this operation as either a fraction or as a decimal number (See those articles for further information).

Exponentiation

Exponentiation is most generally not written using a symbol, but with the second argument written as a superscript, for example . In certain circumstances, as in representing this operation in programming, the symbol ^ is used.

Exponentiation transforms two numbers x1 and x2 into their repeated product. For example:

62 = 6 × 6 = 36

In prefix notation (using ^) it may be written as ^ x1 x2, or ^(x1,x1), or even with ^ changed to a word, such as
pow x1 x1.

Exponentiation is equivalent to repeated multiplication. The identity is that:
x1x2x1x1 ...( x2 times)...x1.

Generalizations

With addition as a basic operator, we can see that the extension of multiplication is an iterated addition. Exponentiation is an iterated multiplication.

We have a notation we can use to show an extension of this generality.

hyper4 is the operator that is defined as repeated exponentiation. If we define Q to be a binary operator, Q x1 x2 =

where x1 is exponentiated x2 times.

This operation has several names, viz., tetration, superpower, superdegree, or powerlog. The two most common notations for this is Knuth's up-arrow notation as x1 ↑↑ x2, and hyper4. Less commonly seen, though somewhat more convenient notations are x1(4)x2.

Only the hyper4, definition is technically a different operator, since this operation can be reduced to exponentiated exponentiation (iterated exponentiation). If we again define Q x1 x2 = : as before, then we define x1 ↑↑↑ x2 or hyper5(x1,x2) as being:

nesting x2 times.

Further generalizations can be taken similarly ad infinitum.

We can generalize back addition, multiplication, and exponentiation in terms using the notations we have just described, ie.,

• hyper1 x1 x2 = x1 + x2
• hyper2 x1 x2 = x1x2
• hyper3 x1 x2 =

Similar behaviors

Some operators aforementioned can also have other behaviours than what was previously described. In programming terms, this is known as overloading, however in mathematics the meaning of an operation is understood from the context by generally the subject matter or what the arguments are. Some examples follow.

The concept of the addition operator + has been extended to cover addition of sets, vectorss and matrices.

Matrix multiplication
Multiplication of a vector by a particular matrix is a unary operator or transformation. We can regard the multiplication of the matrix to be an operator (see below).

Elementary function operators
We have seen that an operator transforms one function to another. So, we can define + to be the sum of the two functions, x1 and x2. resulting in another function. For example, if we define Q this way;
Q (x2+3x) (5x2+9) = 6x2+3x+9
We can define multiplication, division, etc. in the same way.

Function composition
Additionally, we have some other operators which we can define on functions. One such fundamental operator is that of function composition. Given two functions x1=f(t) and x2=g(t), define the operator Q:
Q x1 x2 = f(g(t))
We write this operator infix using a small circle. So, with the same definitions as before,
f(g(t))=(fog)(x)=x1 o x2

Probability theory

Operators are also involved in probability theory. Such operators as expectation, variance, covariance, factorials, et al.

Factorials are essential to the combination and permutation functions of probability and combinatorics, and are also the most commonly known postfix operator, being denoted by a ! placed after the number it expands. Its expansion follows the pattern,

x! = 1 * 2 * ... * (x-1) * x

Calculus and operators

Calculus is, essentially, the study of one particular operator, and its behavior embodies and exemplifies the idea of the operator in great clarity. This key operator we study in Calculus is the differential operator.

The differential operator

The differential operator is the symbolism used in Calculus to denote the action of taking a derivative. Common notations are such d/dx, y'(x) to denote the derivative of y(x). However here 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.

Notations
If f is a function of n variables t1,t1,...,tn, we write
to represent the action of differentiating f with respect to ti. If we differentiate f, k times, we write

How does the differential operator exemplify the idea of the operator? Consider the function f=x2. Elementary calculus tells us that D f = 2x, futhermore if f=xα, D f = αxα-1. So we see clearly that the differential operator maps, or transforms, functions of the form xα to functions αxα-1.

The act of integration is also equivalent somewhat to taking the derivative backwards. So, in a sense it is differentiating -1 times, so we have integration in terms of the differential operator:

It is clear that integration thus is equivalent to differentiation, so integration acts just like an operator as well -- mapping functions to functions.

Integral operators

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

Convolution
The convolution of two functions is a mapping from two functions to one other, defined by an integral as follows:

If x1=f(t) and x2=g(t), define the operator Q such that;

which we write as .

Fourier transform
The Fourier transform is another integral operator, and is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few.

It is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves:

f(x) = ∑ A1 sin ω + A2 sin ω/2 + A3 sin ω/3 + ...

Laplacian transform
The Laplacian transform is another integral operator and is involved in simplifying the process of solving differential equations.

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

See

Operators in physics

In physics, an operator often takes on a more specialized meaning that in mathematics. It often means a linear transformation from a Hilbert space to another or an element of a C* algebra. See Operator (physics).

Operators are also a key part of the theory of quantum mechanics.

Operators in programming

Programming languages, being that computers are mathematical devices, have a set of operators that perform various functions.

The arithmetic operators are the same as the mathematical ones while the bit (binary digit) operations deal with the binary number system. The logical operators determine boolean values. The string operators manipulate strings of text and there are operators which allocate segments of memory for use.

Operators are also terms for some functionality in programming languages. Consider the C programming language syntax for pointers, using the operators * and &. sizeof is sometimes considered an operator, and in C++, new and delete are also operators.

In object oriented languages, like C++, you can define your own uses for operators.

Operators in telecommunications

Operators in telecommunications, who are usually women, aid telephone users in various ways including long distance calling, directory assistance and telephone repair. As technology advances, human operators are becoming more often replaced by a computerized system, and the idiom is turning over to mean a secret agent.