By Pollack J.

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**Extra resources for Connectionism - Past, Present, and Future [jnl article]**

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II. Variation with ﬁxed point x ∈ X. Let T be the base of a family of bundles on X, that is, a bundle X ×T is given on W . We can then construct the new family {elmkx (g)(Wt )} of all elementary transformations with ﬁxed point x ∈ X. The base of the new family is Gk (Wx ) and the bundle on X × Gk (Wx ) deﬁning the family is denoted by ELMkx (W ). When k = n − 1, the operation is simply denoted by ELMx (W ). We now take the most non-trivial family consisting of one trivial n-dimensional bundle In on X, and apply the operation ELMx to it (n + 1)(g − 1) times: ELMx ( · · · (ELMx (In )) · · · ) (n + 1)(g − 1) times We obtain a family of bundles on X with determinant (n + 1) × (g − 1)x.

The operation of multiplication by 1 0 is deﬁned on assignments, but not on classes of assignments of matrix 0 τx0 divisors; it does not commute with multiplication on the left by Ax , a regular 1 0 and regularly invertible matrix. But on the set of left cosets Ax0 the 0 τx0 operation is well-deﬁned. 1 α A choice of coset , α ∈ C, is a choice of g ∈ Vx , and determines 0 τ elmx0 (g) (see [16]). § 2 Variations of elementary operations. The symbol for an elementary operation elmkx0 (g)V is adorned with three suﬃxes, three continuous parameters: the point x ∈ X, the subspace g ∈ Gk (Vx ) and the bundle V ∈ S.

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