owner
operator
Owner
n. 1. One who owns; a rightful proprietor; one who has the legal
or rightful title, whether he is the possessor or not.
Noun 1. owner  (law) someone who owns (is legal possessor of) a
business; "he is the owner of a chain of restaurants"
Synonyms: proprietor Op´er`a`tor
n. 1. One who, or that which, operates or produces an effect.
2. (Surg.) One who performs some act upon the human body by means
of the hand, or with instruments.
3. A dealer in stocks or any commodity for speculative purposes;
a speculator.
4. (Math.) The symbol that expresses the operation to be performed;
 called also facient.
5. A person who operates a telephone switchboard.
6. A person who schemes and maneuvers adroitly or deviously to achieve
his/her purposes.
Noun 1. operator  (mathematics) a symbol that represents a function
from functions to functions; "the integral operator"
2. operator  an agent that operates some apparatus or machine;
"the operator of the switchboard"
Synonyms: manipulator
3. operator  someone who owns or operates a business; "who
is the operator of this franchise?"
4. operator  a shrewd or unscrupulous person who knows how to circumvent
difficulties
Synonyms: wheeler dealer, hustler
5. operator  a speculator who trades aggressively on stock or commodity
markets
Dictionary of
Computing
(programming) operator  A symbol used as a function, with infix
syntax if it has two arguments (e.g. "+") or prefix syntax
if it has only one (e.g. Boolean NOT). Many languages use operators
for builtin functions such as arithmetic and logic.
This article is about operators in mathematics, for other kinds
of operators see operator (disambiguation).
In mathematics, an operator is some kind of function; if it comes
with a specified type of operand as function domain, it is no more
than another way of talking of functions of a given type. The most
frequently met usage is a mapping between vector spaces; this kind
of operator is distinguished by taking one vector and returning
another. For example, consider an enlargement, say by a factor of
v2; such as is required to take one size of paper to another. It
can also be applied geometrically to vectors as operands.
In many important
cases, operators transform functions into other functions. We also
say an operator maps a function to another. The operator itself
is a function, but has an attached type indicating the correct operand,
and the kind of function returned. This extra data can be defined
formally, using type theory; but in everyday usage saying operator
flags its significance. 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 and levels of abstractionTo 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 probably
used to call attention to some aspect of its nature as function.
Since there are several such aspects that are of interest, there
is no completely consistent terminology. Common are these:
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 vectorvalued
function might be called an operator.
A single operator might conceivably qualify under all three of these.
Other important ideas are:
Overloading,
in which 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 qnumber or operator quantities,
and cnumbers which are conventional complex numbers. The manipulation
of qnumbers from that point on became basic to theoretical physics.
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.
Notations
There are three 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 a nary 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 nary 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
There are other notations commonly met. In some literature, a small
uphat 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 ith argument.
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.
Examples of mathematical operatorsThis 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 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:
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).
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 finitedimensional
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.
Operators in probability theory
Main article: Probability theory
Operators are also involved in probability theory. Such operators
as expectation, variance, covariance, factorials, et al.
Operators in calculus
Calculus is, essentially, the study of one particular operator,
and its behavior embodies and exemplifies the idea of the operator
very clearly. The key operator studied is the differential operator.
It is linear, as are many of the operators constructed from it.
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. 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.
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
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
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. 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:
When dealing
with general function R>C, the transform takes up an integral
form:
Laplacian transform
Main article: Laplace transform The Laplace transform is another
integral operator and is involved in simplifying the process of
solving differential equations.
Given f=f(s),
it is defined by:
Fundamental
operators on scalar and vector fields
Main articles: vector calculus, scalar field, gradient, divergence,
and curl
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.
Operators in physicsMain article: Operator (physics)
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.
See alsofunction (mathematics),
unary operation
binary operation
ternary operation
common operator notation.
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