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Hiblert's 13th Problem

Feng, Ziqin (2010) Hiblert's 13th Problem. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

The 13th Problem from Hilbert's famous list [16] asks whether every continuous function of three variables can be written as a superposition (in other words, composition) of continuous functions of two variables. Let Χ be a space. A family Φ ⊆ C(Χ) is said to be basic for Χ if each f in C(Χ) can be written as linear superposition for some functions from in Φ and some one-variable real functions. A family Ψ is elementary in dimension m if the family of maps generated by Ψ by addition is basic for Χ*…*Χ . Kolmogorov and Arnold [18, 4] showed that the closed unit interval has a finite elementary family in every dimension, thereby solving Hilbert's 13th Problem.Define a new cardinal invariant basic(Χ ) = min {|Φ|: Φ is a basic family for Χ}. It is established that a space has a finite basic family if and only if it is finite dimensional, locally compact and separable metrizable (or equivalently, homeomorphic to a closed subspace of Euclidean space).Such a space has dim(Χ) ≤ n if and only if basic(Χ) ≤ 2n+1. Separable metrizable spaces either have finite basic(Χ) or basic(Χ) equal to the continuum. The value of basic(K) for a compact space K is closely connected with the cofinality of the countable subsets of a basis B for K of minimal size ordered by set inclusion.It is proved that a space has a finite elementary family in every dimension m if and only if it is homeomorphic to a closed subspace of Euclidean space. It is further shown that there is a finite elementary family for the reals in each dimension m consisting of effectively computable functions, and effective procedures for representing any continuous function of m real variables as a superposition of these elementary functions and other univariate maps.


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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
CreatorsEmailPitt UsernameORCID
Feng, Ziqinzif1@pitt.eduZIF1
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairGartside, Paulgartside@math.pitt.eduPMG20
Committee MemberEspinoza, Benjaminbee1@pitt.eduBEE1
Committee MemberLennard, Christopher Jlennard@pitt.eduLENNARD
Committee MemberHeath, Robertrwheath@pitt.eduRWHEATH
Committee MemberUspenskiy, Vladimiruspensk@math.ohiou.edu
Date: 30 September 2010
Date Type: Completion
Defense Date: 5 February 2010
Approval Date: 30 September 2010
Submission Date: 7 July 2010
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Elementary Family; Neural Networks; Basic Family; Komogorov's Superposition Theorem
Other ID: http://etd.library.pitt.edu/ETD/available/etd-07072010-152705/, etd-07072010-152705
Date Deposited: 10 Nov 2011 19:50
Last Modified: 15 Nov 2016 13:45
URI: http://d-scholarship.pitt.edu/id/eprint/8300

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