Difference between revisions of "Incidence coefficient"
m (link) |
m (fix tex) |
||
(One intermediate revision by one other user not shown) | |||
Line 1: | Line 1: | ||
+ | <!-- | ||
+ | i0504001.png | ||
+ | $#A+1 = 44 n = 0 | ||
+ | $#C+1 = 44 : ~/encyclopedia/old_files/data/I050/I.0500400 Incidence coefficient | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
+ | |||
+ | {{TEX|auto}} | ||
+ | {{TEX|done}} | ||
+ | |||
A number characterizing the coherence of the orientations of incident elements of simplicial, polyhedral (CW-) and other complexes. The concept of the incidence coefficient and its properties necessarily enter into the definition of an arbitrary abstract complex (cf. [[Complex (in homological algebra)|Complex (in homological algebra)]]). | A number characterizing the coherence of the orientations of incident elements of simplicial, polyhedral (CW-) and other complexes. The concept of the incidence coefficient and its properties necessarily enter into the definition of an arbitrary abstract complex (cf. [[Complex (in homological algebra)|Complex (in homological algebra)]]). | ||
− | Let | + | Let $ t ^ {n} = ( a _ {0} \dots a _ {n} ) $ |
+ | be an oriented simplex in $ \mathbf R ^ {N} $, | ||
+ | i.e. a simplex in which a definite order of its vertices $ a _ {i} $ | ||
+ | has been chosen, and let $ t _ {i} ^ {n-1} = ( a _ {0} \dots a _ {i-1} , a _ {i+1} \dots a _ {n} ) $ | ||
+ | be its oriented face opposite to $ a _ {i} $. | ||
+ | If $ i $ | ||
+ | is even, then $ t ^ {n} $ | ||
+ | and $ t _ {i} ^ {n-1} $ | ||
+ | are coherently oriented, and the orientation of $ t _ {i} ^ {n-1} $ | ||
+ | is induced by the orientation of $ t ^ {n} $; | ||
+ | in this case they are assigned the incidence coefficient $ [ t ^ {n} : t _ {i} ^ {n-1} ] = + 1 $. | ||
+ | If $ i $ | ||
+ | is odd, then $ t ^ {n} $ | ||
+ | and $ t _ {i} ^ {n-1} $ | ||
+ | are non-coherently oriented, and they are assigned the incidence coefficient $ [ t ^ {n} : t _ {i} ^ {n-1} ] = - 1 $. | ||
− | Suppose now that | + | Suppose now that $ t ^ {n} $ |
+ | and $ t ^ {n-1} $ | ||
+ | are elements (simplices) of a [[Simplicial complex|simplicial complex]] in $ \mathbf R ^ {N} $. | ||
+ | Then their incidence coefficient is defined as follows. If $ t ^ {n} $ | ||
+ | and $ t ^ {n-1} $ | ||
+ | are not incident, then $ [ t ^ {n} : t ^ {n-1} ] = 0 $; | ||
+ | if $ t ^ {n} $ | ||
+ | and $ t ^ {n-1} $ | ||
+ | are incident, then $ [ t ^ {n} : t ^ {n-1} ] = 1 $ | ||
+ | or $ - 1 $, | ||
+ | depending on whether they are coherently oriented or not. | ||
Properties of incidence coefficients. | Properties of incidence coefficients. | ||
− | + | $$ \tag{1 } | |
+ | [ - t ^ {n} : t ^ {n-1} ] = \ | ||
+ | [ t ^ {n} : - t ^ {n-1} ] = - | ||
+ | [ t ^ {n} : t ^ {n-1} ] , | ||
+ | $$ | ||
− | where | + | where $ - t ^ {n} $ |
+ | is the oppositely-oriented simplex, i.e. the simplex oriented by an [[odd permutation]] of the vertices of $ t ^ {n} $; | ||
− | + | $$ \tag{2 } | |
+ | \sum _ { k } | ||
+ | [ t ^ {n} : t _ {k} ^ {n-1} ] | ||
+ | [ t _ {k} ^ {n-1} : t ^ {n-2} ] | ||
+ | = 0 , | ||
+ | $$ | ||
− | where the summation extends over all oriented simplices | + | where the summation extends over all oriented simplices $ t _ {k} ^ {n-1} $ |
+ | (for some definitions of a simplicial complex (2) holds only if completeness is required). | ||
− | Analogously, for a suitable definition of coherence of orientation the incidence coefficient of two elements of a [[Polyhedral complex|polyhedral complex]] can be defined. Let | + | Analogously, for a suitable definition of coherence of orientation the incidence coefficient of two elements of a [[Polyhedral complex|polyhedral complex]] can be defined. Let $ \mathbf R ^ {n-1} $ |
+ | be a subspace in $ \mathbf R ^ {n} $, | ||
+ | let $ \mathbf R _ {1} ^ {n} $ | ||
+ | be one of the half-spaces bounded by $ \mathbf R ^ {n-1} $, | ||
+ | and let in $ \mathbf R ^ {n} $ | ||
+ | be chosen an oriented vector basis $ ( e _ {1} \dots e _ {n} ) $. | ||
+ | Then $ \mathbf R _ {1} ^ {n} $ | ||
+ | and $ \mathbf R ^ {n-1} $ | ||
+ | are called coherently oriented if $ ( e _ {2} \dots e _ {n} ) $ | ||
+ | is a basis in $ \mathbf R ^ {n-1} $ | ||
+ | and $ e _ {1} $ | ||
+ | is directed into $ \mathbf R _ {1} ^ {n} $. | ||
+ | Two cells $ \sigma ^ {r} $ | ||
+ | and $ \sigma ^ {r-1} $ | ||
+ | are coherently oriented if they are contained in a certain coherently-oriented half-space and subspace, respectively. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980)</TD></TR></table> |
Latest revision as of 20:44, 22 December 2020
A number characterizing the coherence of the orientations of incident elements of simplicial, polyhedral (CW-) and other complexes. The concept of the incidence coefficient and its properties necessarily enter into the definition of an arbitrary abstract complex (cf. Complex (in homological algebra)).
Let $ t ^ {n} = ( a _ {0} \dots a _ {n} ) $ be an oriented simplex in $ \mathbf R ^ {N} $, i.e. a simplex in which a definite order of its vertices $ a _ {i} $ has been chosen, and let $ t _ {i} ^ {n-1} = ( a _ {0} \dots a _ {i-1} , a _ {i+1} \dots a _ {n} ) $ be its oriented face opposite to $ a _ {i} $. If $ i $ is even, then $ t ^ {n} $ and $ t _ {i} ^ {n-1} $ are coherently oriented, and the orientation of $ t _ {i} ^ {n-1} $ is induced by the orientation of $ t ^ {n} $; in this case they are assigned the incidence coefficient $ [ t ^ {n} : t _ {i} ^ {n-1} ] = + 1 $. If $ i $ is odd, then $ t ^ {n} $ and $ t _ {i} ^ {n-1} $ are non-coherently oriented, and they are assigned the incidence coefficient $ [ t ^ {n} : t _ {i} ^ {n-1} ] = - 1 $.
Suppose now that $ t ^ {n} $ and $ t ^ {n-1} $ are elements (simplices) of a simplicial complex in $ \mathbf R ^ {N} $. Then their incidence coefficient is defined as follows. If $ t ^ {n} $ and $ t ^ {n-1} $ are not incident, then $ [ t ^ {n} : t ^ {n-1} ] = 0 $; if $ t ^ {n} $ and $ t ^ {n-1} $ are incident, then $ [ t ^ {n} : t ^ {n-1} ] = 1 $ or $ - 1 $, depending on whether they are coherently oriented or not.
Properties of incidence coefficients.
$$ \tag{1 } [ - t ^ {n} : t ^ {n-1} ] = \ [ t ^ {n} : - t ^ {n-1} ] = - [ t ^ {n} : t ^ {n-1} ] , $$
where $ - t ^ {n} $ is the oppositely-oriented simplex, i.e. the simplex oriented by an odd permutation of the vertices of $ t ^ {n} $;
$$ \tag{2 } \sum _ { k } [ t ^ {n} : t _ {k} ^ {n-1} ] [ t _ {k} ^ {n-1} : t ^ {n-2} ] = 0 , $$
where the summation extends over all oriented simplices $ t _ {k} ^ {n-1} $ (for some definitions of a simplicial complex (2) holds only if completeness is required).
Analogously, for a suitable definition of coherence of orientation the incidence coefficient of two elements of a polyhedral complex can be defined. Let $ \mathbf R ^ {n-1} $ be a subspace in $ \mathbf R ^ {n} $, let $ \mathbf R _ {1} ^ {n} $ be one of the half-spaces bounded by $ \mathbf R ^ {n-1} $, and let in $ \mathbf R ^ {n} $ be chosen an oriented vector basis $ ( e _ {1} \dots e _ {n} ) $. Then $ \mathbf R _ {1} ^ {n} $ and $ \mathbf R ^ {n-1} $ are called coherently oriented if $ ( e _ {2} \dots e _ {n} ) $ is a basis in $ \mathbf R ^ {n-1} $ and $ e _ {1} $ is directed into $ \mathbf R _ {1} ^ {n} $. Two cells $ \sigma ^ {r} $ and $ \sigma ^ {r-1} $ are coherently oriented if they are contained in a certain coherently-oriented half-space and subspace, respectively.
References
[1] | P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian) |
[2] | P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) |
[3] | A. Dold, "Lectures on algebraic topology" , Springer (1980) |
Incidence coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Incidence_coefficient&oldid=39856