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Algebra i Analiz, 1994, Volume 6, Issue 1, Pages 127–131 (Mi aa427)  

This article is cited in 5 scientific papers (total in 5 papers)

Research Papers

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
Full-text PDF (501 kB) Citations (5)
Abstract: 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.
Keywords: Pfaffian sigmoid, deviation theorems, parallel complexity.
Received: 13.04.1993
Bibliographic databases:
Document Type: Article
Language: English
Citation: D. Yu. Grigoriev, “Deviation theorems for pfaffian sigmoids”, Algebra i Analiz, 6:1 (1994), 127–131; St. Petersburg Math. J., 6:1 (1995), 107–111
Citation in format AMSBIB
\Bibitem{Gri94}
\by D.~Yu.~Grigoriev
\paper Deviation theorems for pfaffian sigmoids
\jour Algebra i Analiz
\yr 1994
\vol 6
\issue 1
\pages 127--131
\mathnet{http://mi.mathnet.ru/aa427}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1274967}
\zmath{https://zbmath.org/?q=an:0844.05090}
\transl
\jour St. Petersburg Math. J.
\yr 1995
\vol 6
\issue 1
\pages 107--111
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Алгебра и анализ St. Petersburg Mathematical Journal
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