Search results
Results From The WOW.Com Content Network
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A series with the additional property that A i ≠ A i +1 for all i is called a series without repetition; equivalently, each A i is a proper subgroup of A i +1. The length of a series is the number of strict inclusions A i < A i +1. If the series has no repetition then the length is n.
For example, the subgroup Z 7 of the non-abelian group of order 21 is normal (see List of small non-abelian groups and Frobenius group#Examples). An alternative proof of the result that a subgroup of index lowest prime p is normal, and other properties of subgroups of prime index are given in ( Lam 2004 ).
This is the group obtained from the orthogonal group in dimension 2n + 1 by taking the kernel of the determinant and spinor norm maps. B 1 (q) also exists, but is the same as A 1 (q). B 2 (q) has a non-trivial graph automorphism when q is a power of 2. This group is obtained from the symplectic group in 2n dimensions by quotienting out the center.
In fact, every real number can be written as the limit of a sequence of rational numbers (e.g. via its decimal expansion, also see completeness of the real numbers). As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing.
Equivalently, a composition series is a subnormal series such that each factor group H i+1 / H i is simple. The factor groups are called composition factors. A subnormal series is a composition series if and only if it is of maximal length. That is, there are no additional subgroups which can be "inserted" into a composition series.
The manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that associates every pair of elements of the set to an element of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f ( v ) to each vertex v , a graph G is f -choosable (or f -list-colorable ) if it has a list coloring no matter how one assigns a list of f ( v ...