By Toshinori Deguchi, Junya Fukuta, Naohiro Ishii (auth.), Marco Tomassini, Alberto Antonioni, Fabio Daolio, Pierre Buesser (eds.)

The ebook constitutes the refereed court cases of the eleventh foreign convention on Adaptive and common Computing Algorithms, ICANNGA 2013, held in Lausanne, Switzerland, in April 2013.

The fifty one revised complete papers provided have been rigorously reviewed and chosen from a complete of ninety one submissions. The papers are equipped in topical sections on neural networks, evolutionary computation, gentle computing, bioinformatics and computational biology, complicated computing, and applications.

**Read Online or Download Adaptive and Natural Computing Algorithms: 11th International Conference, ICANNGA 2013, Lausanne, Switzerland, April 4-6, 2013. Proceedings PDF**

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**Additional info for Adaptive and Natural Computing Algorithms: 11th International Conference, ICANNGA 2013, Lausanne, Switzerland, April 4-6, 2013. Proceedings**

**Example text**

It is easy to show that the communication matrix M (β¯d ) is a Hadamard matrix. Indeed, its rows multiplied by 2−d/2 form the Fourier basis Fd/2 of B({0, 1}d/2). Let α := card(β¯d−1 ({1}) and β := card(β¯d−1 ({−1}). So by the deﬁnition, α+ β = 2d . As the 2d entries of 2d/2 × 2d/2 matrix M (β¯d ) represent values of β¯d (u) for all u ∈ {0, 1}d, by Lemma 1 we have |α − β| ≤ 23d/4 . On the other hand, α+β = 2 min(α, β)+|α−β| = 2d . Hence, |α−β| = 2d −2 min(α, β) ≤ 23d/4 . Thus 2d − 23d/4 ≤ 2 min(α, β), hence α and β must be exponential in d.

5 Conclusion The neural networks are analyzed to make clear functions of the layered network of V1 followed by MT in the brain cortical area. In this paper, the structure and function of the nonlinear biological asymmetrical network, are analyzed first to detect Vector Generation and Operations in Neural Networks Computations 19 the movement direction and its speed of the stimulus from the point of the neural computation. The conditions of the movement and its direction of the stimulus, are derived from these analyses.

In the proof of Proposition 2, it was veriﬁed that β¯d has a Hadamard communication matrix. So we get the next Corollary. m Corollary 3. Let d be an even integer and β¯d (x) = i=1 wi ϑ(ei · x + bi ) a m 2d/3 ¯ representation of the function βd . Then i=1 |wi | = Ω(2 ). So in a representation of the d-dimensional function inner product mod 2 by a one-hidden-layer Heaviside perceptron network, the number of units must be exponential in d or some output weights must have absolute values of sizes exponential in d.