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The order of an algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700401.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700402.png" /> is a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700404.png" />, is the highest degree of the terms of this polynomial. For instance, the ellipse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700405.png" /> is a curve of order two, and the lemniscate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700406.png" /> is a curve of order four (cf. [[Algebraic curve|Algebraic curve]]).
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The order of an infinitesimal quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700407.png" /> with respect to an infinitesimal quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700408.png" /> is (if it exists) the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o0700409.png" /> such that the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004010.png" /> exists and is not infinite or equal to zero. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004011.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004012.png" /> is an infinitesimal of order two with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004013.png" /> since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004014.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004015.png" /> is an infinitesimal of higher order than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004016.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004017.png" />, and of lower order than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004018.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004019.png" />. Analogously one can define the orders of infinitely large quantities (cf. [[Infinitesimal calculus|Infinitesimal calculus]]).
+
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{{TEX|done}}
  
The order of a zero (respectively, a pole) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004020.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004021.png" /> is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004022.png" /> such that the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004023.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004024.png" />) exists and is not infinite or equal to zero (cf. e.g. [[Analytic function|Analytic function]]; [[Meromorphic function|Meromorphic function]]; [[Pole (of a function)|Pole (of a function)]]; [[Rational function|Rational function]]).
+
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|Algebraic curve]]).
  
The order of a derivative is the number of times one has to differentiate a function to obtain this derivative. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004025.png" /> is a derivative of order two, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004026.png" /> is a derivative of order four. Similarly the order of a differential is defined (cf. [[Differential calculus|Differential calculus]]).
+
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|Infinitesimal calculus]]).
  
The order of a differential equation is the highest order of the derivatives in it. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004027.png" /> is an equation of order three, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004028.png" /> is an equation of order two (cf. [[Differential equation, ordinary|Differential equation, ordinary]]).
+
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|Analytic function]]; [[Meromorphic function|Meromorphic function]]; [[Pole (of a function)|Pole (of a function)]]; [[Rational function|Rational function]]).
  
The order of a square matrix is the number of its rows or columns (cf. [[Matrix|Matrix]]).
+
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|Differential calculus]]).
  
The order of a finite group is the number of elements in the group (cf. [[Finite group|Finite group]]). If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004029.png" /> 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|Ordered group]]; [[Partially ordered group|Partially ordered group]]).
+
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|Differential equation, ordinary]]).
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004030.png" /> if this subgroup is infinite (cf. also [[Cyclic group|Cyclic group]]). In the last case the element is of infinite order. If the order of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004031.png" /> is finite and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004033.png" /> is the least among the numbers for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004034.png" />.
+
The order of a square matrix is the number of its rows or columns (cf. [[Matrix|Matrix]]).
  
A right order in a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004035.png" /> is a subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004037.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004038.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004040.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004042.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004043.png" /> is a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004045.png" /> is a classical right ring of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004046.png" /> (see [[Fractions, ring of|Fractions, ring of]]).
+
The order of a finite group is the number of elements in the group (cf. [[Finite group|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|Ordered group]]; [[Partially ordered group|Partially ordered group]]).
  
If in some studies or calculations all powers starting with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004047.png" />-st of some small quantity are neglected, one says that this study or calculation is carried out up to quantities of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004048.png" />. 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 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|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 $.
  
The word "order" is also used in the calculus of differences (differences of different order, cf. [[Finite-difference calculus|Finite-difference calculus]]), in the theory of many special functions (e.g. [[Cylinder functions|cylinder functions]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004049.png" />), etc.
+
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|Fractions, ring of]]).
  
In measurements one speaks about a [[Quantity|quantity]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004050.png" />, which means that it is included between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004052.png" />.
+
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|Finite-difference calculus]]), in the theory of many special functions (e.g. [[Cylinder functions|cylinder functions]] of order  $  n $),
 +
etc.
  
 +
In measurements one speaks about a [[Quantity|quantity]] of order  $  10  ^ {n} $,
 +
which means that it is included between  $  0. 5 \cdot 10  ^ {n} $
 +
and  $  5 \cdot 10  ^ {n} $.
  
 
====Comments====
 
====Comments====
 
The above does not exhaust the many meanings in which the word  "order"  is used in mathematics.
 
The above does not exhaust the many meanings in which the word  "order"  is used in mathematics.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004053.png" /> is a balanced incomplete block design, or design with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004058.png" /> (see [[Block design|Block design]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004059.png" /> is called the order of the design.
+
If $  ( V, B) $
 +
is a balanced incomplete block design, or design with parameters $  v $,  
 +
$  b $,  
 +
$  r $,  
 +
$  k $,  
 +
$  \lambda $(
 +
see [[Block design|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).
  
A finite projective plane is of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004060.png" /> if each line has precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004061.png" /> points (and there are (hence) precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004062.png" /> points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004063.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004065.png" />, be a covering of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004066.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004067.png" />. The covering is said to be of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004068.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004069.png" /> is the least integer such that any subfamily of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004070.png" /> consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004071.png" /> elements has empty intersection.
+
Let $  f( z) $
 +
be a transcendental entire function (cf. [[Entire function|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
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004072.png" /> be a transcendental entire function (cf. [[Entire function|Entire function]]). For each real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004073.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004074.png" />. Then the order of the transcendental entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004075.png" /> is defined as
+
$$
 +
\rho  = {\lim\limits  \sup } _ {r \rightarrow \infty } \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004076.png" /></td> </tr></table>
+
\frac{ \mathop{\rm log}  \mathop{\rm log}  M( r) }{ \mathop{\rm log}  r }
 +
.
 +
$$
  
The function is called of finite order if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004077.png" /> is finite and of infinite order otherwise.
+
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|Elliptic function]].
 
The order of an elliptic function is the number of times it takes each value in its period parallelogram, cf. [[Elliptic function|Elliptic function]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004078.png" /> be a [[Meromorphic function|meromorphic function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004079.png" />. For each possible value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004080.png" />, including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004081.png" />, let
+
Let $  f( z) $
 +
be a [[Meromorphic function|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 ,
 +
$$
 +
 
 +
$$
 +
m( r, \alpha )  =
 +
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi }  \mathop{\rm log}  ^ {+}
 +
\left |
 +
\frac{1}{f( re ^ {i \theta } ) - \alpha }
 +
\
 +
\right |  d \theta \  \textrm{ if }  \alpha \neq \infty ,
 +
$$
 +
 
 +
$$
 +
m( r, \infty )  =
 +
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
\mathop{\rm log}  ^ {+}  | f( re ^ {i \theta } ) |  d \theta ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004082.png" /></td> </tr></table>
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004083.png" /></td> </tr></table>
+
$$
 +
{\lim\limits  \sup } _ {r \rightarrow \infty } \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004084.png" /></td> </tr></table>
+
\frac{ \mathop{\rm log}  T( r) }{ \mathop{\rm log}  r }
 +
  = \
 +
{\lim\limits  \sup } _ {r \rightarrow \infty } \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004085.png" /> is the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004087.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004088.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004089.png" />, i.e. the points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004090.png" />, counted with multiplicity. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004092.png" /> are called the counting function and proximity function, respectively. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004093.png" /> is called the order function or characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004094.png" />. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004095.png" /> (Nevanlinna's first theorem), as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004096.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004097.png" />. One has also
+
\frac{ \mathop{\rm log}  \mathop{\rm log}  M( r) }{ \mathop{\rm log}  r }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004098.png" /></td> </tr></table>
+
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) $.
  
where, as in 16) above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o07004099.png" />. The order of the meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040100.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040101.png" />.
+
The $  k $-
 +
th order modulus of continuity of a continuous function $  f $
 +
on  $  [ a, b] $
 +
is defined by
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040103.png" />-th order modulus of continuity of a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040104.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040105.png" /> 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}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040106.png" /></td> </tr></table>
+
} } \
 +
\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|Continuity, modulus of]]; [[Smoothness, modulus of|Smoothness, modulus of]].
 
See also [[Continuity, modulus of|Continuity, modulus of]]; [[Smoothness, modulus of|Smoothness, modulus of]].
  
Consider a system of ordinary differential equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040107.png" /> on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040108.png" /> and a numerical solution method which calculates the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040109.png" /> at mesh points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040110.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040111.png" /> is the stepsize. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040112.png" /> be the calculated value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040113.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040115.png" /> the  "true value" , <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040116.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040117.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040118.png" />, then the solution process is of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040119.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040120.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040121.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040122.png" /> is the union of a finite number of simple arcs meeting at a finite number of points. For a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040123.png" /> the boundary of a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040124.png" /> meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040125.png" /> at a finite number of points, which is independent of the neighbourhood. This number is called the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040126.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040127.png" />. A point of order 1 is an end point, one of order 2 an ordinary point, and one of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040128.png" /> a branch point.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040129.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040130.png" />-dimensional manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040131.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040132.png" />-dimensional cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040133.png" /> which is a boundary. The [[Linking coefficient|linking coefficient]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040134.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040135.png" /> not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040136.png" />, the underlying space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040137.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040138.png" /> is called the order of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040139.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040140.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040141.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040142.png" /> a closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040144.png" />, this is the [[Rotation number|rotation number]] around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040145.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040146.png" />.
+
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|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|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)|Order (on a set)]]).
 
The word  "order"  also occurs as a synonym for an order relation on a set, or an ordering (cf. also [[Order (on a set)|Order (on a set)]]).
  
For the concept of order of magnitude of a function at a point (including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040147.png" />) and related concepts cf. [[Order relation|Order relation]].
+
For the concept of order of magnitude of a function at a point (including $  \infty $)  
 +
and related concepts cf. [[Order relation|Order relation]].
 +
 
 +
Consider a [[Dirichlet series|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
  
Consider a [[Dirichlet series|Dirichlet series]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040148.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040149.png" /> be the abscissa of convergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040150.png" />. I.e. the series converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040151.png" /> and diverges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040152.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040153.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040154.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040155.png" />. In his thesis, H. Bohr introduced
+
$$
 +
\mu ( x)  =  {\lim\limits  \sup } _ {| y| \rightarrow \infty } \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040156.png" /></td> </tr></table>
+
\frac{ \mathop{\rm log}  | f( x+ iy) | }{ \mathop{\rm log}  | y | }
 +
,
 +
$$
  
and called it the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040157.png" /> over the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040158.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040159.png" /> is non-negative, convex, continuous, and monotone decreasing. Bohr found that there is a kind of periodicity for the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040160.png" /> over this line; this started the theory of almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]).
+
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|Almost-periodic function]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040161.png" /> 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|Dedekind ring]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040162.png" /> be a separable algebra of finite degree over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040163.png" />, the quotient field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040164.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040165.png" />-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040166.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040167.png" /> is a finitely-generated submodule (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040168.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040169.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040170.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040171.png" />-lattice that is a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040172.png" /> and which contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040173.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040175.png" />-order. A maximal order is one that is not contained in any order. Such a maximal order always exists. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040176.png" /> is commutative it is unique.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040177.png" /> is a global or local field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040178.png" /> its ring of integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040179.png" /> a finite field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040180.png" />, the maximal order is the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040181.png" />, which is the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040182.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040183.png" /> (cf. [[Integral extension of a ring|Integral extension of a ring]]). It is also called the principal order.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040184.png" /> in this sense is the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070040/o070040185.png" /> (cf. also [[Lie group|Lie group]]).
+
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|Lie group]]).
  
 
For references see the various articles directly or indirectly referred to.
 
For references see the various articles directly or indirectly referred to.

Latest revision as of 08:04, 6 June 2020


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.

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