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} $.

Comments

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

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

See also Continuity, modulus of; Smoothness, modulus of.

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.