# Order

The order of an algebraic curve $F ( x , y ) = 0$, where $F ( x , y )$ is a polynomial in $x$ and $y$, is the highest degree of the terms of this polynomial. For instance, the ellipse $x ^ {2} / a ^ {2} + y ^ {2} / b ^ {2} = 1$ is a curve of order two, and the lemniscate $( x ^ {2} + y ^ {2} ) ^ {2} = a ^ {2} ( x ^ {2} - y ^ {2} )$ is a curve of order four (cf. Algebraic curve).

The order of an infinitesimal quantity $\alpha$ with respect to an infinitesimal quantity $\beta$ is (if it exists) the number $n$ such that the limit $\lim\limits \alpha / \beta ^ {n}$ exists and is not infinite or equal to zero. For instance, $\sin ^ {2} 3 x$ as $x \rightarrow 0$ is an infinitesimal of order two with respect to $x$ since $\lim\limits _ {x \rightarrow 0 } ( \sin ^ {2} ( 3 x ) / x ^ {2} ) = 9$. One says that $\alpha$ is an infinitesimal of higher order than $\beta$ if $\lim\limits \alpha / \beta = 0$, and of lower order than $\beta$ if $\lim\limits \alpha / \beta = \infty$. Analogously one can define the orders of infinitely large quantities (cf. Infinitesimal calculus).

The order of a zero (respectively, a pole) $a$ of a function $f$ is the number $n$ such that the limit $\lim\limits _ {x \rightarrow a } f ( x) / ( x - a ) ^ {n}$( respectively, $\lim\limits ( x - a ) ^ {n} f ( x)$) 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, $y ^ {\prime\prime}$ is a derivative of order two, $\partial ^ {4} z / \partial ^ {2} x \partial ^ {2} y$ 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, $y ^ {\prime\prime\prime} y ^ \prime - ( y ^ {\prime\prime} ) ^ {2} = 1$ is an equation of order three, $y ^ {\prime\prime} - 3 y ^ \prime + y = 0$ 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 $G$ 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 $\infty$ 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 $a$ is finite and equal to $n$, then $n$ is the least among the numbers for which $a ^ {n} = 1$.

A right order in a ring $Q$ is a subring $R$ of $Q$ such that for any $x \in Q$ there are $a , b \in R$ such that $b$ is invertible in $Q$ and $x = ab ^ {-} 1$. In other words, $R$ is a subring of $Q$ such that $Q$ is a classical right ring of fractions of $R$( see Fractions, ring of).

If in some studies or calculations all powers starting with the $( n + 1 )$- st of some small quantity are neglected, one says that this study or calculation is carried out up to quantities of order $n$. 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 $n$), etc.

In measurements one speaks about a quantity of order $10 ^ {n}$, which means that it is included between $0. 5 \cdot 10 ^ {n}$ and $5 \cdot 10 ^ {n}$.

The above does not exhaust the many meanings in which the word "order" is used in mathematics.

If $( V, B)$ is a balanced incomplete block design, or design with parameters $v$, $b$, $r$, $k$, $\lambda$( see Block design), then $n = r - \lambda$ is called the order of the design.

A finite projective plane is of order $k$ if each line has precisely $k+ 1$ points (and there are (hence) precisely $k ^ {2} + k + 1$ points and $k ^ {2} + k+ 1$ lines).

Let $\mathfrak M = \{ M _ \lambda \} _ {\lambda \in \Lambda }$, $M _ \lambda \subset S$, be a covering of a subset $A \subset S$, i.e. $A \subset \cup _ \lambda M _ \lambda$. The covering is said to be of order $k$ if $k$ is the least integer such that any subfamily of $\mathfrak M$ consisting of $k+ 1$ elements has empty intersection.

Let $f( z)$ be a transcendental entire function (cf. Entire function). For each real number $r > 0$, let $M( r) = \max _ {| z| = r } | f( z) |$. Then the order of the transcendental entire function $f( z)$ is defined as

$$\rho = {\lim\limits \sup } _ {r \rightarrow \infty } \ \frac{ \mathop{\rm log} \mathop{\rm log} M( r) }{ \mathop{\rm log} r } .$$

The function is called of finite order if $\rho$ 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 $f( z)$ be a meromorphic function in $| z | < R \leq \infty$. For each possible value $\alpha$, including $\infty$, let

$$N( r , \alpha ) = \int\limits _ { 0 } ^ { r } \frac{n( t, \alpha ) - n( 0 , \alpha ) }{t} \ dt + n( 0 , \alpha ) \mathop{\rm log} r ,$$

$$m( r, \alpha ) = \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm log} ^ {+} \left | \frac{1}{f( re ^ {i \theta } ) - \alpha } \ \right | d \theta \ \textrm{ if } \alpha \neq \infty ,$$

$$m( r, \infty ) = \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm log} ^ {+} | f( re ^ {i \theta } ) | d \theta ,$$

where $n( r, \alpha )$ is the number of $\alpha$- points of $f( z)$ in $| z | \leq r$, i.e. the points with $f( z) = \alpha$, counted with multiplicity. The functions $N$ and $m$ are called the counting function and proximity function, respectively. The function $T( r) = m( r, \infty ) + N( r, \infty )$ is called the order function or characteristic function of $f( z)$. One has $T( r) = m( r, \alpha ) + N( r, \alpha ) + O( 1)$( Nevanlinna's first theorem), as $r \rightarrow \infty$, for all $\alpha$. One has also

$${\lim\limits \sup } _ {r \rightarrow \infty } \ \frac{ \mathop{\rm log} T( r) }{ \mathop{\rm log} r } = \ {\lim\limits \sup } _ {r \rightarrow \infty } \ \frac{ \mathop{\rm log} \mathop{\rm log} M( r) }{ \mathop{\rm log} r } ,$$

where, as in 16) above, $M( r) = \max _ {| z| = r } | f( z) |$. The order of the meromorphic function $f( z)$ is defined as ${\lim\limits \sup } _ {r \rightarrow \infty } ( \mathop{\rm log} r ) ^ {-} 1 \mathop{\rm log} T( r)$.

The $k$- th order modulus of continuity of a continuous function $f$ on $[ a, b]$ is defined by

$$\omega _ {k} ( f; t) = \sup _ { {\begin{array}{c} {| h| \leq t } \\ {a \leq x \leq b } \\ {a \leq x+ kh \leq b } \end{array} } } \ \left | \sum _ { i= } 0 ^ { k } (- 1) ^ {k-} i \left ( \begin{array}{c} k \\ i \end{array} \right ) f( x+ ih) \right | .$$

Consider a system of ordinary differential equations ${dy ^ {i} } / dx = f ^ { i } ( x, y ^ {1} ( x) \dots y ^ {n} ( x))$ on an interval $[ a, b]$ and a numerical solution method which calculates the $y ^ {i}$ at mesh points $x _ {k} = a+ kh$, so that $h$ is the stepsize. Let $y _ {k} ^ {i}$ be the calculated value at $x _ {k}$ of $y ^ {i}$, $y ^ {i} ( x _ {k} )$ the "true value" , $e _ {k} ^ {i} = y _ {k} ^ {i} - y ^ {i} ( x _ {k} )$. If $e _ {k} ^ {i} = O( h ^ {r} )$ as $h \rightarrow 0$, then the solution process is of order $r$.

Consider an ordinary curve $C$ in $E ^ {2}$, i.e. $C$ is the union of a finite number of simple arcs meeting at a finite number of points. For a point $p \in C$ the boundary of a sufficiently small neighbourhood of $p$ meets $C$ at a finite number of points, which is independent of the neighbourhood. This number is called the order of $p$ on $C$. A point of order 1 is an end point, one of order 2 an ordinary point, and one of order $\geq 3$ a branch point.

Let $M ^ {n}$ be an $n$- dimensional manifold and $Z ^ {n-} 1$ an $( n - 1)$- dimensional cycle in $M ^ {n}$ which is a boundary. The linking coefficient $\mathop{\rm Lk} ( P, Z ^ {n-} 1 )$ of a point $P$ not in $| Z ^ {n-} 1 |$, the underlying space of $Z ^ {n-} 1$, with $Z ^ {n-} 1$ is called the order of the point $P$ with respect to $Z ^ {n-} 1$. In the case $M ^ {n} = \mathbf R ^ {2}$, and $Z ^ {n-} 1$ a closed curve $\{ {f( t) } : {0 \leq t \leq t } \}$, $f( 0) = f( 1)$, this is the rotation number around $P$ of $f$.

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 $\infty$) and related concepts cf. Order relation.

Consider a Dirichlet series $f( z) = \sum _ {n=} 1 ^ \infty a _ {n} \mathop{\rm exp} (- \lambda _ {n} z)$, and let $S$ be the abscissa of convergence of $f$. I.e. the series converges for $\mathop{\rm Re} ( z) > S$ and diverges for $\mathop{\rm Re} ( z) < S$. If $x = \mathop{\rm Re} ( z) > S$, then $f( z) = o(| y |)$ as $| y | \rightarrow \infty$. In his thesis, H. Bohr introduced

$$\mu ( x) = {\lim\limits \sup } _ {| y| \rightarrow \infty } \ \frac{ \mathop{\rm log} | f( x+ iy) | }{ \mathop{\rm log} | y | } ,$$

and called it the order of $f$ over the line $\mathop{\rm Re} ( z) = x$. The function $\mu ( x)$ is non-negative, convex, continuous, and monotone decreasing. Bohr found that there is a kind of periodicity for the values of $f$ over this line; this started the theory of almost-periodic functions (cf. Almost-periodic function).

Let $A$ 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 $B$ be a separable algebra of finite degree over $F$, the quotient field of $A$. An $A$- lattice $L$ in $B$ is a finitely-generated submodule (over $A$) of $B$ such that $FL = B$. An $A$- lattice that is a subring of $B$ and which contains $A$ is called an $A$- order. A maximal order is one that is not contained in any order. Such a maximal order always exists. If $B$ is commutative it is unique.

In the case $F$ is a global or local field, $A$ its ring of integers, $B$ a finite field extension of $F$, the maximal order is the ring of integers of $B$, which is the integral closure of $A$ in $B$( 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 $G$ in this sense is the dimension of $G$( 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=48065
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