Order
The order of an algebraic curve , where
is a polynomial in
and
, is the highest degree of the terms of this polynomial. For instance, the ellipse
is a curve of order two, and the lemniscate
is a curve of order four (cf. Algebraic curve).
The order of an infinitesimal quantity with respect to an infinitesimal quantity
is (if it exists) the number
such that the limit
exists and is not infinite or equal to zero. For instance,
as
is an infinitesimal of order two with respect to
since
. One says that
is an infinitesimal of higher order than
if
, and of lower order than
if
. Analogously one can define the orders of infinitely large quantities (cf. Infinitesimal calculus).
The order of a zero (respectively, a pole) of a function
is the number
such that the limit
(respectively,
) exists and is not infinite or equal to zero (cf. e.g. Analytic function; Meromorphic function; Pole (of a function); Rational function).
The order of a derivative is the number of times one has to differentiate a function to obtain this derivative. For instance, is a derivative of order two,
is a derivative of order four. Similarly the order of a differential is defined (cf. Differential calculus).
The order of a differential equation is the highest order of the derivatives in it. For example, is an equation of order three,
is an equation of order two (cf. Differential equation, ordinary).
The order of a square matrix is the number of its rows or columns (cf. Matrix).
The order of a finite group is the number of elements in the group (cf. Finite group). If the group is infinite, one says that it is a group of infinite order. One should not confuse the order of a group with an order on a group (see Ordered group; Partially ordered group).
The order of an element of a group is the positive integer equal to the number of elements of the cyclic subgroup generated by this element, or to if this subgroup is infinite (cf. also Cyclic group). In the last case the element is of infinite order. If the order of an element
is finite and equal to
, then
is the least among the numbers for which
.
A right order in a ring is a subring
of
such that for any
there are
such that
is invertible in
and
. In other words,
is a subring of
such that
is a classical right ring of fractions of
(see Fractions, ring of).
If in some studies or calculations all powers starting with the -st of some small quantity are neglected, one says that this study or calculation is carried out up to quantities of order
. For example, in studies of small oscillations of a string the terms with second and higher degrees of deflection and its derivatives are neglected, as a result one obtains a linear equation (linearization of the problem).
The word "order" is also used in the calculus of differences (differences of different order, cf. Finite-difference calculus), in the theory of many special functions (e.g. cylinder functions of order ), etc.
In measurements one speaks about a quantity of order , which means that it is included between
and
.
Comments
The above does not exhaust the many meanings in which the word "order" is used in mathematics.
If is a balanced incomplete block design, or design with parameters
,
,
,
,
(see Block design), then
is called the order of the design.
A finite projective plane is of order if each line has precisely
points (and there are (hence) precisely
points and
lines).
Let ,
, be a covering of a subset
, i.e.
. The covering is said to be of order
if
is the least integer such that any subfamily of
consisting of
elements has empty intersection.
Let be a transcendental entire function (cf. Entire function). For each real number
, let
. Then the order of the transcendental entire function
is defined as
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The function is called of finite order if is finite and of infinite order otherwise.
The order of an elliptic function is the number of times it takes each value in its period parallelogram, cf. Elliptic function.
Let be a meromorphic function in
. For each possible value
, including
, let
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where is the number of
-points of
in
, i.e. the points with
, counted with multiplicity. The functions
and
are called the counting function and proximity function, respectively. The function
is called the order function or characteristic function of
. One has
(Nevanlinna's first theorem), as
, for all
. One has also
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where, as in 16) above, . The order of the meromorphic function
is defined as
.
The -th order modulus of continuity of a continuous function
on
is defined by
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See also Continuity, modulus of; Smoothness, modulus of.
Consider a system of ordinary differential equations on an interval
and a numerical solution method which calculates the
at mesh points
, so that
is the stepsize. Let
be the calculated value at
of
,
the "true value" ,
. If
as
, then the solution process is of order
.
Consider an ordinary curve in
, i.e.
is the union of a finite number of simple arcs meeting at a finite number of points. For a point
the boundary of a sufficiently small neighbourhood of
meets
at a finite number of points, which is independent of the neighbourhood. This number is called the order of
on
. A point of order 1 is an end point, one of order 2 an ordinary point, and one of order
a branch point.
Let be an
-dimensional manifold and
an
-dimensional cycle in
which is a boundary. The linking coefficient
of a point
not in
, the underlying space of
, with
is called the order of the point
with respect to
. In the case
, and
a closed curve
,
, this is the rotation number around
of
.
The word "order" also occurs as a synonym for an order relation on a set, or an ordering (cf. also Order (on a set)).
For the concept of order of magnitude of a function at a point (including ) and related concepts cf. Order relation.
Consider a Dirichlet series , and let
be the abscissa of convergence of
. I.e. the series converges for
and diverges for
. If
, then
as
. In his thesis, H. Bohr introduced
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and called it the order of over the line
. The function
is non-negative, convex, continuous, and monotone decreasing. Bohr found that there is a kind of periodicity for the values of
over this line; this started the theory of almost-periodic functions (cf. Almost-periodic function).
Let be a Dedekind domain, i.e. a (not necessarily commutative) integral domain in which every ideal is uniquely decomposed into prime ideals (cf. also Dedekind ring). Let
be a separable algebra of finite degree over
, the quotient field of
. An
-lattice
in
is a finitely-generated submodule (over
) of
such that
. An
-lattice that is a subring of
and which contains
is called an
-order. A maximal order is one that is not contained in any order. Such a maximal order always exists. If
is commutative it is unique.
In the case is a global or local field,
its ring of integers,
a finite field extension of
, the maximal order is the ring of integers of
, which is the integral closure of
in
(cf. Integral extension of a ring). It is also called the principal order.
In some, mainly physics literature, one speaks of the order of a Lie group as the number of parameters needed to parametrize it, i.e. the order of the Lie group in this sense is the dimension of
(cf. also Lie group).
For references see the various articles directly or indirectly referred to.
Order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Order&oldid=16876