owner
operator truck driver
Owner
n. 1. One who owns; a rightful proprietor; one who has the legal
or rightful title, whether he is the possessor or not.
owner
operator truck driver
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.
owner operator truck
driver 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 built-in 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 vector-valued
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 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.
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 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
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 i-th 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 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.
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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