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Commit 3c208e74 by Michael Kohlhase

### bla

parents 0301f4a2 c989e73f


[creators=miko]dgraphde [id=dgraph.de.def] Ein [dgraph]gerichteterGraph (auch [dgraph]Digraph) ist ein [structure]Paarstructure \structure aus einer Menge und einer Menge \sseteq\pair geordneter Paare ”uber . Wir nennen die [vertex]Knoten und die [edge]KantenKante (auch [edge]B”ogenBogen) von .



[creators=miko]dgraphen [id=dgraph.def] A directedgraph (also called [directed-graph]digraph or [directed-graph]orientedgraph) is a [structure]pairstructure \structure such that is a set and \sseteq\pairs. We call the verticesvertex (or [vertex]nodes) and the edges of .



[creators=miko]dgraph [smglom/mv]structure [smglom/sets]pair subsupset *dgraph vertex edge directedgraph node


branch

1. (a)

Module graph
syno- hyper- hypo- mero-nyms de ro zh
A graph is a structure such that is a set and is a subset of the set of pairs from . We call the vertices (or nodes, points, junctions) and the edges (or lines, branches, arcs) of .

2. (b)

Module inverse function
syno-
hyper- hypo- mero-nyms de ro zh
A branch of a multivalued function is a univalent sub-relation .

3. (c)

branch curve


[creators=miko]norm-metricennormmetric-space

[type=obligation,id=obl.norm-metric.en] \defeq\norm is a [metric-space]distancefunction [for=obl.norm-metric.en]we prove the three conditions for a distance function: 0, iff \norm0, iff 0 (as \normOp separates points), iff . \norm\norm by absolute homogeneity of \normOp. the triangle inequality for follows from that for \normOp.



[creators=miko]norm-metricnormmetric-space base-set \funcdotText node: x,y\normx-y [by=norm-metric]metric-spacenormed-vector-space [by=norm-metric]distance-functionnorm



[creators=miko]metric-space [smglom/numberfields]realnumbers [smglom/numberfields]numbers-orders [smglom/sets]functions

[creators=miko]metric-spaceen Let be a set, then we call a function \fun\pairs\RealNumbers a distancefunction (or [distance-function]metric) on , iff for all \minset the following three identities hold:

1. 1.

0 iff (identityofindiscernibles),

2. 2.

(symmetry), and

3. 3.

\lethan (triangleinequality).

We call \structure a metricspace with baseset and metric .



[creators=miko]norm [smglom/numberfields]complexnumbers [smglom/linear-algebra]vector-space [smglom/numberfields]absolutevalue [smglom/numberfields]numbers-orders [smglom/sets]functions

||||||||

[creators=miko]normen Given a [vector-space]vectorspace over a [subfield]subfield of the [complexnumbers]complexnumbers, a norm on is a function \fun\normOp\RealNumbers such that for all \inset and \minset

1. 1.

\norm\atimes\absolutevalue\norm (absolutehomogeneity or [absolute-homogeneity]absolutescalability).

2. 2.

3. 3.

If \norm0, then is the zero vector (separatespoints).

We call the pair \structure\normOp a normedvectorspace with norm \normOp

 SL ( , )
 special linear group of order over


[creators=miko]special-linear-groupen

special linear group of order  over

The speciallineargroup \SLgroup of degree over a [field]field is the set of matrices with [determinant]determinant 1, with the [group]groupoperations of ordinary [matrix]matrixmultiplication and [matrix]matrixinversion.



[creators=miko]special-linear-group [smglom/linear-algebra]determinant [smglom/algebra]field 18F25,20Gxx speciallineargroup

SL\SLgroupOp(,)
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 ... ... @@ -78,7 +78,7 @@ \input{intro} \input{preliminaries} \input{datamodel} %\input{implementation} missing files \input{implementation}% missing files \ednote{\texttt{implementation.tex} has missing files, commit them from the old laptop} \input{encoding} \input{concl} ... ...
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