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(Unpublished)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Unpublished 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

ETD Committee: 

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/etd07072010152705/, etd07072010152705 
Date Deposited: 
10 Nov 2011 19:50 
Last Modified: 
15 Nov 2016 13:45 
URI: 
http://dscholarship.pitt.edu/id/eprint/8300 
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