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Appendix A: Common Mathematical Terms
Basic mathematical terms are reviewed. The readers' familiarity with
sets and subsets [2] is required in the
following. All sets which contain only one element (singleton) can be
used to describe a vertex.
Figure:
A relation
of a set
and a set
.

A permutation of a finite set is usually given by its action on the
elements of the set. An example is given by the basic set
with a transformation into
where the notation states that the permutation
maps
,
and so on. Thereby all
permutations of a finite set can be placed into two sets.
Definition 46 (Parity)
A permutation of a finite set can be expressed as either an even or an
odd number of transpositions, but not both. In the former case, the
permutation is called even, in the latter it is odd.
Figure:
The image of
under
.

Definition 49 (Preimage)
The preimage (fiber, inverse image) of a set
under
is the subset of
defined by
The preimage of a singleton,
, is a fiber or a level set.
A fiber can also be the empty set
. The union set of all
fibers of a function is called the total space, E.
Definition 51 (Abelian)
A group
is Abelian, if its binary operation
is commutative.
Definition 56 (Vector Space)
A vector space
over a field
is a set associated
with two binary operations. The elements of
are called
vectors, while the elements of
are called scalars. The
binary operations are addition

(11.3) 
which is commutative as well as associative and which possesses a
neutral as well as an inverse element, and scalar multiplication

(11.4) 
which is distributive with addition.
is closed under
its two binary operations.
Definition 57 (Linear Mapping)
A mapping
is called linear,
if the following relation holds


(11.5) 
A mapping

(11.6) 
that is linear in both of its arguments, as in the following relations
is called bilinear.
The extension of this notion to more than two arguments is called a multilinear mapping.
Figure:
The kernel of
.

Next: 12. STL Iterator Analysis
Up: 4 Applied Concepts
Previous: 10. Summary and Outlook
R. Heinzl: Concepts for Scientific Computing