Download PDF by A. I. Markushevich: Curvas Maravillosas, Numeros Complejos y Representaciones

By A. I. Markushevich

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Extra resources for Curvas Maravillosas, Numeros Complejos y Representaciones Conformes, Funciones Maravillosas

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Let K, = (Kc — K)/Kc and take \ a s the 'correlation length' defining the average domain in r-space for which the information source is primarily dominated by 'strong' ties. The first step is to average across r-space in terms of 'clumps' of length R=. Then H[J, K, X] -> H[JR, KR, X]. 1) with / ( I ) = 1 and J\ = J,K\ — K. The first equation significantly extends Wilson's treatment. It states that 'processing capacity,' as indexed by the source uncertainty of the system, representing the 'richness' of the generalized language, grows monotonically as f(R)9 which must itself be a dimensionless function in R, since both H[KR, JR] and H[K, J] are themselves dimensionless.

We will return to this point repeatedly. 2. 'Biological' phase transitions Now the mathematical detail concealed by the invocation of the asymptotic limit theorems emerges with a vengeance. 1) states that the information source and the correlation length, the degree of coherence on the underlying network, scale under renormalization clustering in chunks of size R as H[KR,JR]/f(R) = with / ( I ) = 1, K\ = K^J\ — J, where we have slightly rearranged terms. 4) The Ui, Vi, i — 1, 2 are functions of KR, JR, but not explicitly of R itself.

They can be tuned continuously... Suppose that a structured external environment, itself an appropriately regular information source Y, 'engages' a modifiable cognitive system. The environment begins to write an image of itself on the cognitive system in a distorted manner permitting definition of a mutual information I[K] splitting criterion according to the Rate Distortion or Joint Asymptotic Equipartition Theorems. K is an inverse coupling parameter between system and environment (Wallace, 2002a, b).

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Curvas Maravillosas, Numeros Complejos y Representaciones Conformes, Funciones Maravillosas by A. I. Markushevich


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