SUO: Differential And Riemannian Manifolds
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DARM. Commentary Note 1
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I will now try to say, in a very tentative way, what I think that
the themes and variations of manifold theory, if suitably adapted,
might have to do with the business of inquiry, modeling, semantics,
semiotics, and sign relations in general, especially in the light of
many compelling questions about change and diversity in our conceptual
and symbolic systems, including the problems of designing interoperable
perspectives and mutually intelligible codes for the worlds we construe
to exist and the worlds we have come to inhabit.
Let's view our archetype of a manifold, the Figure of a space X
and a couple of charts (U_i, q_i) and (U_j, q_j) from its atlas:
o-----------------------------------------------------------o
| X |
| |
| o-------------o o-------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | U_i | U_ij | U_j | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ | / \ | / |
| o------|------o o------|------o |
| | | |
| | | |
o--------------------|-----------------|--------------------o
| |
q_i | | q_j
| |
o--------------------|-----o o-----|--------------------o
| E_i v | | v E_j |
| | | |
| o----------o | | o----------o |
| / \ | | / \ |
| / o | | o \ |
| / / \ | | / \ \ |
| / / \ | | / \ \ |
| o o o | | o o o |
| | | | | q_ij| | | | |
| | | ------------------> | | |
| | | | | | | | | |
| | q_i U_ij ----- | | | | ----- q_j U_ij | |
| | | | | | | | | |
| o o o | | o o o |
| \ \ / | | \ / / |
| \ \ / | | \ / / |
| \ o | | o / |
| \ / | | \ / |
| o----------o | | o----------o |
| | | |
| | | |
o--------------------------o o--------------------------o
Figure 1. Manifold X with Charts (U_i, q_i) and (U_j, q_j)
| Let X be a set. An "atlas" of class C^p (p >= 0) on X
| is a collection of pairs (U_i, q_i) (i ranging in some
| indexing set), satisfying the following conditions:
|
| AT 1. Each U_i is a subset of X, and the U_i cover X.
|
| AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i
| of some Banach space E_i, and for each i, j it is the case
| that q_i (U_i |^| U_j) is open in E_i.
|
| AT 3. The map:
|
| (q_j) o (q_i)^(-1) : q_i (U_i |^| U_j) -> q_j (U_i |^| U_j)
|
| is a C^p-isomorphism for each pair of indices i, j.
|
| Each pair (U_i, q_i) will be called a "chart" of the atlas.
| If a point x of X lies in U_i, then we say that (U_i, q_i)
| is a "chart at x".
|
| Lang, DARM, pp. 20-21.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
Let us now back away from the picture and view it more impressionistically.
We may view X as being the "object space" or the "real" space in which all
of us are really the most interested, at least, if we know what's good for
us, and regard E_i and E_j to be the spaces of, let us say, my impressions,
lexicon, measurements, nomenclature, senses, signs, symbology, terminology,
utterances, vocabulary, whatever it happens to be, and yours, respectively.
Focus on the subsets of X, E_i, E_j that are defined and marked as follows:
U_ij = U_i |^| U_j c X
E_ij = q_i U_ij c E_i
E_ji = q_j U_ij c E_j
The mapping of the form (q_j) o (q_i)^(-1) is what does the work
of partially translating my code into yours, to the extent that
it is possible to do so by flipping charts. This is easier to
see if one lays out the maps in a straight-line presentation:
(q_i)^(-1) q_j
E_ij ------------> U_ij ------------> E_ji
Naturally enough, maps of the form (q_j) o (q_i)^(-1),
that change coordinates from chart to chart within the
same atlas, are known as "transition" or "translation"
maps. As a short form, let q_ij = (q_j) o (q_i)^(-1).
Here are a couple of helpful hints about
reading these brands of translation maps:
Reading 1.
If E_i is my code space and E_j is your code space,
then I may read the application of the translation
q_ji (w) = ((q_i) o (q_j)^(-1))(w) in this fashion:
| q_ji (w) = ((q_i) o (q_j)^(-1))(w)
|
| = my name for what you call w.
Reading 2.
If E_i is a new code space and E_j is an old code space,
then we may interpret the application of the translation
q_ji (w) = ((q_i) o (q_j)^(-1))(w) in the following way:
| q_ji (w) = ((q_i) o (q_j)^(-1))(w)
|
| = the new name for what we used to call w.
In other words, as one says, we are talking about an
objective interpretive situation, where the sign w and
the interpretant sign w' = ((q_i) o (q_j)^(-1))(w) both
denote the shared object x.
Next question: Does this manifold picture capture the
most generic brand of objective interpretive situation?
Jon Awbrey
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DARM. Differential And Riemannian Manifolds
01. http://suo.ieee.org/ontology/msg04770.html
02. http://suo.ieee.org/ontology/msg04771.html
03. http://suo.ieee.org/ontology/msg04772.html
04. http://suo.ieee.org/ontology/msg04773.html
05. http://suo.ieee.org/ontology/msg04774.html
06. http://suo.ieee.org/ontology/msg04775.html
07. http://suo.ieee.org/ontology/msg04776.html
08. http://suo.ieee.org/ontology/msg04777.html
09. http://suo.ieee.org/ontology/msg04778.html
10. http://suo.ieee.org/ontology/msg04779.html
11. http://suo.ieee.org/ontology/msg04780.html
12. http://suo.ieee.org/ontology/msg04781.html
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