ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
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| But every name, as students of logic know, has its "denotation"; and the
| denotation always means some reality or content, relationless 'ab extra'
| or with its internal relations unanalyzed, like the 'q' which our
| primitive sensation is supposed to know. No relation-expressing
| proposition is possible except on the basis of a preliminary
| acquaintance with such "facts", with such contents, as this.
| Let the 'q' be fragrance, let it be toothache, or let it be
| a more complex kind of feeling, like that of the full-moon
| swimming in her blue abyss, it must first come in that
| simple shape, and be held fast in that first intention,
| before any knowledge 'about' it can be attained.
| The knowledge 'about' it is 'it' with a context
| added. Undo 'it', and what is added cannot
| be 'con'-text.
|
| James, "Func of Cog", pages 14-15.
|
| William James, "The Function Of Cognition",
| Read before the Aristotelian Society, 1 Dec 1884.
| First published in 'Mind', 10 (1885). Reprinted in
|'The Meaning Of Truth, A Sequel To "Pragmatism"',
| Longmans, Green, & Company, London, UK, 1909.
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| If E is a Banach space, then the diagonal @D@ in ExE is
| a closed subspace and splits: Either factor Ex0 or 0xE
| is a closed complement. Consequently, the diagonal is
| a closed submanifold of ExE. If X is any manifold of
| class C^p, p >= 1, then the diagonal is therefore also
| a submanifold. (It is closed of course if and only if
| X is Hausdorff.)
|
| Let f : X -> Z and g : Y -> Z be two C^p-morphisms, p >= 1.
| We say that they are 'transversal' if the morphism
|
| f x g : X x Y -> Z x Z
|
| is transversal over the diagonal. We remark right away
| that the surjectivity of the map in Proposition 2.4 can be
| expressed in two ways. Given two points x in X and y in Y
| such that f(x) = g(y) = z, the condition
|
| Im(T_x f) + Im(T_y g) = T_z(Z)
|
| is equivalent to the condition
|
| Im(T_(x,y)(f x g)) + T_(z,z)(@D@) = T_(z,z)(Z x Z).
|
| Thus in the finite dimensional case, we could
| take it as the definition of transversality.
|
| Lang, DARM, pages 28-29.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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| o---------------------------------------o o-------------------o
| | X | | E_i |
| | | | |
| | | | o |
| | | | / \ |
| | o | | / \ |
| | / \ | | / \ |
| | / \ | | / \ |
| | / \ q_i | | / q_i U_i \ |
| | / o---------------------->| o o o |
| | / \ | | \ / \ / |
| | / \ | | \ / \ / |
| | / U_i \ | | o Eij o |
| | / \ | | \ / |
| | / \ | | \ / |
| | o o o | | o |
| | \ / \ / | | |
| | \ / \ / | | |
| | \ / \ / | o---------|---------o
| | \ / \ / | |
| | o Uij o | q_j o q_i^-1
| | / \ / \ | |
| | / \ / \ | o---------v---------o
| | / \ / \ | | E_j |
| | / \ / \ | | |
| | o o o | | o |
| | \ / | | / \ |
| | \ / | | / \ |
| | \ U_j / | | o Eji o |
| | \ / | | / \ / \ |
| | \ / | | / \ / \ |
| | \ o---------------------->| o o o |
| | \ / q_j | | \ q_j U_j / |
| | \ / | | \ / |
| | \ / | | \ / |
| | o | | \ / |
| | | | \ / |
| | | | o |
| | | | |
| | | | |
| o---------------------------------------o o-------------------o
|
| Figure 1. Manifold Of Contextural Impressions
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References And Incidental Nuances (RAIN)
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html
http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm
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
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