We show that a real binary form f of degree n has n distinct real roots if and only if for any (alpha, beta) is an element of R(2)\{0} all the forms alpha f(x) + beta f(y) have n - 1 distinct real roots. This answers to a question of Comon and Ottaviani (On the typical rank of real binary forms, available at arXiv:math/0909.4865, 2009), and allows to complete their argument to show that f has symmetric rank n if and only if it has a distinct real roots
We consider graded artinian complete intersection algebras A = C[x 0 , . . . , xm]/I with I generated by homogeneous forms of degree d ≥ 2. We show that the general multiplication by a linear form µ L : A d−1 → A d is injective. We prove that the Weak Lefschetz Property for holds for any c.i. algebra A as above with d = 2 and m ≤ 4.
In the paper 95 339535, Fax 39 95 338887 the capability of neural networks with complex neurons to approximate complex valued functions, is investigated. A density theorem for complex MLPs with non-analytic activation function and one hidden layer is proved. The Back Propagation algorithms for the MLPs with real, complex analytic and complex non-analytic activation function are compared with a numerical example.
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