# Order

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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 .

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

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

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

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

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

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.

How to Cite This Entry:
Order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Order&oldid=16876
This article was adapted from an original article by Material from the article "Order" in BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article