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Extra info for 1-Convex Manifolds are p-Kahler

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Generally, for a given application, the architecture of an RNN is either chosen randomly or based on user’s experience. The user then tries a large number of structures for the selected architecture and the parameters for each of these structures. The difficulty in making optimal choices for these properties has limited the application of RNNs in spite of its vast potential. Some details about the determination of these properties of an RNN will be provided in the coming sections along with possible issues for future work.

Xn Þ=ðx1 , x2 , . . , xn Þ (2:28) X1 ÂX2 ÂÁÁÁÂXn where mR (x1, x2, . . , xn) is a MF of the relation R, which represents the degree of association (correlation) among the elements of the different domain Xi. It is a mapping from the Cartesian space X1  X2  Á Á Á  Xn onto a continuous unit interval [0, 1] as R: X1  X2  Á Á Á  Xn ! ½0, 1Š (2:29) Note that a fuzzy relation is in fact a fuzzy set. In general, the MF mR (x1, x2, . . , xn) is a hyperplane in the (n þ 1)-dimensional space.

Snj 2 Rn , and the center vector, m j ¼ m1j , m2j , . . , mnj 2 Rn . For N given input–output training pairs, (x(t), d(t)), t ¼ 1, 2, . . , N, the task of training a RBFN is to design an RBFN f(x) such that d(k) ¼ f ½x(k)Š þ e(k) ¼ w0 þ M X fj ½x(k)Šwj þ e(k) (2:93) j¼1 where fj(Á) ¼ f(Ájlj). , 1991): d ¼ Fw þ e (2:94) where d ¼ [d(1), . . , d(N)]T 2 RN F ¼ [f0, f1, . . , fM] 2 RN Â(Mþ1) with f0 ¼ [1, . . , 1]T, fj ¼ {f[x(1)jlj], . . , f [x(N)jlj]}T w ¼ (w0, w1, . . , wM)T 2 RMþ1 e ¼ [e(1), .

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1-Convex Manifolds are p-Kahler by Alessandrini L.

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