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ЖУРНАЛЫ // Алгебра и анализ // Архив

Алгебра и анализ, 1994, том 6, выпуск 1, страницы 127–131 (Mi aa427)

Эта публикация цитируется в 5 статьях

Статьи

Deviation theorems for pfaffian sigmoids

D. Yu. Grigorievab

a On leave from Mathematical Institute, St. Petersburg, RUSSIA
b Departments Computer Science and Mathematics, Penn State University, State College, PA, USA

Аннотация: By a Pfaffian sigmoid of depth $d$ we mean a circuit with $d$ layers in which rational operations are admitted at each layer, and to jump to the next layer one solves an ordinary differential equation of the type $v'=p(v)$ where $p$ is a polynomial whose coefficients are functions computed at the previous layers of the sigmoid. Thus, a Pfaffian sigmoid computes Pfaffian functions (in the sense of A. Khovanskii). A deviation theorem is proved which states that for a real function $f$, $f\not\equiv 0$, computed by a Pfaffian sigmoid of depth (or parallel complexity) $d$ there exists an integer $n$ such that for a certain $x_0$ the inequalities $(\exp(\dots(\exp(|x|^n))\dots))^{-1}\leq|f(x)|\leq\exp(\dots(\exp(|x|^n))\dots)$ hold for all $|x|\geq x_0$, where the iteration of the exponential function is taken $d$ times. One can treat the deviation theorem as an analogue of the Liouville theorem (on algebraic numbers) for Pfaffian functions.

Ключевые слова: Pfaffian sigmoid, deviation theorems, parallel complexity.

Поступила в редакцию: 13.04.1993

Язык публикации: английский


 Англоязычная версия: St. Petersburg Mathematical Journal, 1995, 6:1, 107–111

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