SUO: RE: FW: ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
Jim,
. Thanks for this.
. What are the specifications for the Ontology web address /
site:
* scope,
* purpose,
* content,
* etc.?
. Who manages / administers it, and ensures compliance, etc.?
. How does one unsubscribe, should I decide that is the way to
go?
Cheers Graham Horn
National Data Standards Unit
Australian Institute of Health and Welfare
================================================
Phone: 02.6244.1094
Fax: 02.6244.1199
Email: Graham.Horn@aihw.gov.au <mailto:graham.horn@aihw.gov.au>
-----Original Message-----
From: jim.s3@juno.com [mailto:jim.s3@juno.com]
Sent: Tuesday, 7 August 2001 10:12
To: graham.horn@aihw.gov.au
Subject: Re: FW: ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
Graham,
I don't subscribe to the Ontology list.
Also, I did not get a copy of this message at my office address,
so best to send to this address.
Jim
On Mon, 6 Aug 2001 17:26:15 +1000 "Horn, Graham"
<graham.horn@aihw.gov.au> writes:
> Jim,
> . What is the traffic on the Ont address apart from
> Jon
> Awbrey?
>
>
>
> Cheers Graham Horn
> National Data Standards Unit
> Australian Institute of Health and Welfare
> ================================================
> Phone: 02.6244.1094
> Fax: 02.6244.1199
> Email: Graham.Horn@aihw.gov.au
> <mailto:graham.horn@aihw.gov.au>
>
>
> -----Original Message-----
> From: Jon Awbrey [SMTP:jawbrey@oakland.edu]
> <mailto:[SMTP:jawbrey@oakland.edu]>
> Sent: Saturday, 4 August 2001 1:44
> To: Arisbe; Generic Ontology Group; Organization Complexity
> Autonomy
> Subject: ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
>
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> Manifold SIG,
>
> It will be useful to keep at the ready a compendium
> of the most essential elements of our subject, and
> so I will maintain what is needed in a cumulative
> appendix to these notes.
>
> I continue the story with further readings from Lang's DARM,
> knitting up a
> few more strands of terminology into our yarn.
> Recall the definition of an atlas:
> An "atlas of class C^p (p >= 0)" on a set X is a collection of pairs
> (U_i,
> q_i), satisfying the conditions AT 1, AT 2, AT 3, (vide syllabus at
> end of
> this note).
> Naturally enough, however much artifice may have gone into its
> natural
> naming, an atlas is conceived and executed all in order to collect a
> number
> of charts:
> | 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.
>
> We find next the need for a notion of "compatibility" among and
> between
> different atlases and their charts:
> | Suppose that we are given an open subset U of X and a topological
> isomorphism
> | q : U -> U' onto an open subset of some Banach space E. We shall
> say that
> | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i
> q^-1
> | (defined on a suitable intersection as in AT 3) is a
> C^p-isomorphism.
> |
> | Two atlases are said to be "compatible" if each chart of one is
> compatible
> with
> | the other atlas. One verifies immediately that the relation of
> compatibility
> | between atlases is an equivalence relation. An equivalence class
> of
> atlases
> | of class C^p on X is said to define a structure of "C^p-manifold"
> on X.
> |
> | If all the vector spaces E_i in some atlas are toplinearly
> isomorphic,
> then we can always
> | find an equivalent atlas for which they are all equal, say to the
> vector
> space E. We then
> | say that X is an "E-manifold" or that X is "modeled" on E.
> |
> | Lang, DARM, page 21
>
> E-nough For E-nonce,
> Jon Awbrey
> ¤~~~~~~~~~¤~~~~~~~~~¤~SYLLABUS~¤~~~~~~~~~¤~~~~~~~~~¤
> | 2. Manifolds
> |
> | 2.1. Atlases, Charts, Morphisms
> |
> | 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 any i, j, [it holds 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.
> |
> | Lang, DARM, page 20
>
> | o---------------------------------------o
> o-------------------o
> | | X | | E_i
> |
> | |
> | |
> |
> | | | |
> o
> |
> | | | |
> / \
> |
> | | o | | /
> \
> |
> | | / \ | | /
> \
> |
> | | / \ | | /
> \
> |
> | | / \ q_i | | / q_i
> U_i \
> |
> | | / o---------------------->| o o
> o
> |
> | | / \ | | \ / \
> /
> |
> | | / \ | | \ /
> \ /
> |
> | | / U_i \ | | o
> o
> |
> | | / \ | | \
> /
> |
> | | / \ | | \
> /
> |
> | | o o o | | o
> |
> | | \ / \ / | |
> |
> | | \ / \ / | |
> |
> | | \ / U_i \ / |
> o---------|---------o
> | | \ / \ / | |
> | | o |^| o | q_j o
> q_i^-1
> | | / \ / \ | |
> | | / \ U_j / \ |
> o---------v---------o
> | | / \ / \ | | E_j
> |
> | | / \ / \ | |
> |
> | | o o o | | o
> |
> | | \ / | | /
> \
> |
> | | \ / | | /
> \
> |
> | | \ U_j / | | o
> o
> |
> | | \ / | | / \
> / \
> |
> | | \ / | | / \ /
> \
> |
> | | \ o---------------------->| o o
> o
> |
> | | \ / q_j | | \ q_j
> U_j /
> |
> | | \ / | | \
> /
> |
> | | \ / | | \
> /
> |
> | | o | | \
> /
> |
> | | | |
> \ /
> |
> | | | |
> o
> |
> | |
> | |
> |
> | |
> | |
> |
> | o---------------------------------------o
> o-------------------o
> |
> | Figure 1. Manifold Of Coordinated Impressions
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~SUBALLYS~¤~~~~~~~~~¤~~~~~~~~~¤
>