Link to the University of Pittsburgh Homepage
Link to the University Library System Homepage Link to the Contact Us Form

Hiblert's 13th Problem

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 (Unpublished)

Primary Text

Download (1MB) | Preview


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.


Social Networking:
Share |


Item Type: University of Pittsburgh ETD
Status: Unpublished
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,
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:, etd-07072010-152705
Date Deposited: 10 Nov 2011 19:50
Last Modified: 15 Nov 2016 13:45


Monthly Views for the past 3 years

Plum Analytics

Actions (login required)

View Item View Item