[Arisbe] Other sides of the conversation?

[elijah wright]

i dropped this message from the list, as it came from a non-subscribed
address (hi wendy, nice to meet you - feel free to subscribe to the arisbe
list, if you're not already here!) but i think that it bears some thought
and response.

jon, if it would help you, i would be DELIGHTED to help you with setting
up a forum in which to post your musings.  they're typically fringe for
both the arisbe list and a number of the other lists that you crosspost
them to, and it is really hard to follow just exactly what lists are
providing the content that you're running through so much of.  i can't
track the threads, like wendy, because i'm not on the same lists that you
are.

anyhow - will you let me know if there's something i can do to make your
life easier?  i know you're a smart cookie, jon, and it would be great if
more of us could manage to follow along and get a grip on what you're
talking about.

thanks, 

elijah  =)



On Fri, 24 Aug 2001, Wendy Petzall wrote:

> Date: Fri, 24 Aug 2001 08:53:41 -0400
> From: Wendy Petzall <wmpetzall@hotmail.com>
> To: arisbe@eckhart.stderr.org
> Subject: Re: [Arisbe] Other sides of the conversation?
> Resent-Date: Sat, 25 Aug 2001 20:36:02 -0400
> Resent-From: arisbe-admin@stderr.org
> Resent-To: arisbe-owner@eckhart.stderr.org
> 
> Dear All,
> 
> I have just received SIXTY (60!) messages from the list, all at the same
> time. The first one was from Elijah, explaining some problems with the
> server, which have now been corrected.
> 
> All the other messages came from Jon Awbrey. Some of them were just
> comments on various aspects of the points under discussion, but several
> were clearly answers to other messages...
> 
> I would like to re-pose the question I asked some time ago:  Is there any
> way of actually receiving those messages?    'Cause as the situation now
> stands, it's like listening to one side of a conversation, and having to
> guess at whatever is being said by the other party (or parties, in this
> case). And the exercise loses its attraction after a time.
> 
> Thanks!
> 
> Wendy
> 
> >From: Jon Awbrey
> >Reply-To: arisbe@stderr.org
> >To: Arisbe
> >Subject: [Arisbe] Re: Manifolds Of Sensuous Impressions (MOSI's)
> >Date: Fri, 24 Aug 2001 00:31:01 -0400
> >
> >¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
> >
> >| For the feeling to be cognitive in the specific sense, then,
> >| it must be self-transcendent; and we must prevail upon the
> >| god to 'create a reality outside of it' to correspond to its
> >| intrinsic quality 'q'. Thus only can it be redeemed from the
> >| condition of being a solipsism. If now the new-created reality
> >| 'resemble' the feeling's quality 'q', I say that the feeling may
> >| be held by us 'to be cognizant of that reality'.
> >|
> >| William James, 'The Meaning of Truth',
> >| Longmans, Green, & Co., London, 1909,
> >| page 6.
> >
> >¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
> >
> >| A morphism f : X -> Y will be called a "submersion" at a point x in X
> >| if there exists a chart (U, q) at x and a chart (V, r) at f(x) such
> that
> >| q gives an isomorphism of U on a product U_1 x U_2 (U_1 and U_2 open
> in
> >| some Banach spaces), and such that the map
> >|
> >| rfq^-1 = f_V,U : U_1 x U_2 -> V
> >|
> >| is a projection. One sees then that the image of a submersion is
> >| an open subset (a submersion is in fact an open mapping). We say
> >| that f is a "submersion" if it is a submersion at every point.
> >|
> >| For manifolds modeled on Banach spaces, we have the usual criterion
> >| for immersions and submersions in terms of the derivative.
> >|
> >| Proposition 2.2. Let X, Y be manifolds of class C^p (p >= 1) modeled
> on Banach spaces.
> >|
> >| Let f : X -> Y be a C^p-morphism. Let x be in X. Then:
> >|
> >| 1. f is an immersion at x if and only if
> >|
> >| there exists a chart (U, q) at x and (V, r) at f(x)
> >|
> >| such that f'_V,U(qx) is injective and splits.
> >|
> >| 2. f is a submersion at x if and only if
> >|
> >| there exists a chart (U, q) at x and (V, r) at f(x)
> >|
> >| such that f'_V,U(qx) is surjective and its kernel splits.
> >|
> >| Proof. This is an immediate consequence
> >| of Corollaries 5.4 and 5.6 of
> >| the inverse mapping theorem.
> >|
> >| The conditions expressed in [Propositions 2.2.1 and 2.2.2] depend only
> on the
> >| derivative [f'], and if they hold for one choice of charts (U, q) and
> (V, r),
> >| respectively, then they hold for every choice of such charts. It is
> therefore
> >| convenient to introduce a terminology in order to deal with such
> properties.
> >|
> >| Lang, DARM, page 25.
> >
> >¤~~~~~~~~~¤~~~~~~~~~¤~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.
> >|
> >| 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.
> >|
> >| 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.
> >|
> >| If E = R^n for some fixed n, then we say that the manifold is
> "n-dimensional".
> >| In this case, a chart
> >|
> >| q : U -> R^n
> >|
> >| is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a
> point of U,
> >| these functions are often written
> >|
> >| x_1(P), ..., x_n(P),
> >|
> >| or simply x_1, ..., x_n. They are called "local coordinates" on the
> manifold.
> >|
> >| If the integer p (which may also be infinity) is fixed throughout a
> discussion,
> >| we also say that X is a manifold.
> >|
> >| Lang, DARM, page 21.
> >|
> >| Let X be manifold, and U an open subset of X. Then it is possible,
> >| in the obvious way, to induce a manifold structure on U, by taking
> >| as charts the intersections
> >|
> >| (U_i |^| U, q_i | (U_i |^| U)).
> >|
> >| [Notation. "f | S" indicates the function f as restricted to the set
> S.]
> >|
> >| If X is a topological space, covered by open subsets V_j, and if we
> are
> >| given on each V_j a manifold structure such that for each pair j, j'
> the
> >| induced structure on V_j |^| V_j' coincides, then it is clear that we
> can
> >| give to X a unique manifold structure inducing the given ones on each
> V_j.
> >|
> >| If X, Y are two manifolds, then one can give the
> >| product X x Y a manifold structure in the obvious way.
> >| If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y
> >| respectively, then
> >|
> >| {(U_i x V_j, q_i x r_j)}
> >|
> >| is an atlas for the product, and the product of compatible
> >| atlases gives rise to compatible atlases, so that we do get
> >| a well-defined product structure.
> >|
> >| Let X, Y be two manifolds. Let f : X -> Y be a map.
> >| We shall say that f is a "C^p-morphism" if, given x in X,
> >| there exists a chart (U, q) at x and a chart (V, r) at f(x)
> >| such that f(U) c V, and the map
> >|
> >| r o f o q^-1 : qU -> rV
> >|
> >| is a C^p-morphism in the sense of Chapter 1, Section 3.
> >| One sees then immediately that this same condition holds
> >| for any choice of charts (U, q) at x and (V, r) at f(x)
> >| such that F(U) c V.
> >|
> >| It is clear that the composite of two C^p-morphisms is itself
> >| a C^p-morphism (because it is true for open subsets of vector
> >| spaces). The C^p-manifolds and C^p-morphisms form a category.
> >| The notion of isomorphism is therefore defined ...
> >|
> >| If f : X -> Y is a morphism, and (U, q) is a chart
> >| at a point x in X, while (V, r) is a chart at f(x),
> >| then we shall also denote by
> >|
> >| f_V,U : qU -> rV
> >|
> >| the map rfq^-1 [that is, r o f o q^-1].
> >|
> >| Lang, DARM, page 22.
> >|
> >| It is also convenient to have a local terminology.
> >| Let U be an open set (of a manifold or a Banach space)
> >| containing a point x_0. By a "local isomorphism" at x_0
> >| we mean an isomorphism
> >|
> >| f : U_1 -> V
> >|
> >| from some open set U_1 containing x_0 (and contained in U)
> >| to an open set V (in some manifold or some Banach space).
> >| Thus a local isomorphism is essentially a change of chart,
> >| locally near a given point.
> >|
> >| 2.2. Submanifolds, Immersions, Submersions
> >|
> >| Let X be a topological space, and Y a subset of X.
> >| We say that Y is "locally closed" in X if every point
> >| y in Y has an open neighborhood U in X such that Y |^| U
> >| is closed in U. One verifies easily that a locally closed
> >| subset is the intersection of an open set and a closed set.
> >| For instance, any open subset of X is locally closed, and
> >| any open interval is locally closed in the plane.
> >|
> >| Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X
> >| and assume that for each point y in Y there exists a chart (V, r) at y
> >| such that r gives an isomorphism of V with a product V_1 x V_2 where
> >| V_1 is open in some space E_1 and V_2 is open in some space E_2,
> >| and such that
> >|
> >| r(Y |^| V) = V_1 x a_2
> >|
> >| for some point a_2 in V_2 (which we could take to be 0). Then it is
> clear
> >| that Y is locally closed in X. Furthermore, the map r induces a
> bijection
> >|
> >| r_1 : Y |^| V -> V_1.
> >|
> >| The collection of pairs (Y |^| V, r_1) obtained in the above manner
> constitues
> >| an atlas for Y, of class C^p. The verification of this assertion,
> whose formal
> >| details we leave to the reader, depends on the following obvious fact.
> >|
> >| Lang, DARM, page 23.
> >|
> >| Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces,
> >|
> >| and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism.
> >|
> >| Let a_2 be in U_2 and b_2 be in V_2
> >|
> >| and assume that g maps U_1 x a_2 into V_1 x b_2.
> >|
> >| Then the induced map
> >|
> >| g_1 : U_1 -> V_1
> >|
> >| is also a morphism.
> >|
> >| Indeed, it is obtained as a composite map
> >|
> >| U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1,
> >|
> >| the first map being an inclusion and the third a projection.
> >|
> >| We have therefore defined a C^p-structure on Y which will be called
> >| a "submanifold" of X. This structure satisfies a universal mapping
> >| property, which characterizes it, namely:
> >|
> >| | Given any map f : Z -> X from a manifold Z into X such that
> >| | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map.
> >| | Then f is a morphism if and only if f_Y is a morphism.
> >|
> >| The proof of this assertion depends on Lemma 2.1, and is trivial.
> >|
> >| Finally, we note that the inclusion of Y into X is a morphism.
> >|
> >| If Y is also a closed subspace of X, then
> >| we say that it is a "closed submanifold".
> >|
> >| Suppose that X is finite dimensional of dimension n, and that Y is a
> submanifold of dimension m.
> >| Then from the definition we see that the local product structure in
> the neighborhood of a point
> >| of Y can be expressed in terms of local coordinates as follows. Each
> point P of Y has an open
> >| neighborhood U in X with local coordinates (x_1, ..., x_n) such that
> the points of Y in U are
> >| precisely those whose last n - m coordinates are 0, that is, those
> points having coordinates
> >| of type
> >|
> >| (x_1, ..., x_m, 0, ..., 0).
> >|
> >| Let f : Z -> X be a morphism, and let z be in Z. We shall say that f
> is
> >| an "immersion" at z if there exists an open neighborhood Z_1 of z in Z
> >| such that the restriction of f to Z_1 induces an isomorphism of Z_1
> >| onto a submanifold of X. We say that f is an "immersion" if it is
> >| an immersion at every point.
> >|
> >| Note that there exist injective immersions
> >| which are not isomorphisms onto submanifolds,
> >| as given by the following example:
> >| ________
> >| / \
> >| / \
> >| | |
> >| | |
> >| \ V
> >| \___________________________________________
> >|
> >| (The arrow means that the line approaches itself without touching.)
> >| An immersion which does give an isomorphism onto a submanifold is
> >| called an "embedding", and it is called a "closed embedding" if
> >| this submanifold is closed.
> >|
> >| Lang, DARM, pages 24-25.
> >|
> >| Serge Lang,
> >|'Differential & Riemannian Manifolds',
> >| Springer-Verlag, New York, NY, 1995.
> >
> >¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
> >
> >| 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 Cognitive Impressions
> >
> >¤~~~~~~~~~¤~~~~~~~~~¤~SUBALLYS~¤~~~~~~~~~¤~~~~~~~~~¤
> >
> >References And Incidental Nuances (RAIN)
> >
> >http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html
> >http://www.philosophy.ru/library/kant/01/cr_pure_reason.html
> >http://ez2www.com/go.php3?site=book&go=0387943382
> >http://hallmathematics.com/mathematics/1433.shtml
> >http://hallmathematics.com/mathematics/630.shtml
> >
> >¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
> >_______________________________________________
> >Arisbe mailing list
> >Arisbe@stderr.org
> >http://stderr.org/cgi-bin/mailman/listinfo/arisbe
> 
> ________________________________________________________________________________
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> 
> 

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