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AuthorRepovš, Dušan, 1954-
Željko, Matjaž
Title On basic embeddings into the plane / Dušan Repovš, Matjaž Željko
Other titlesO bazičnih vložitvah v ravnino
Type/contenttype of material article - component part
LanguageEnglish
Publication date2006
Physical descriptionstr. 1665-1677
NotesBibliografija: str. 1676-1677
Uncontrolled subject headingsmatematika / topologija / bazična vložitev / linearna relacija / zvezna funkcija / mathematics / topology / basic embedding / linear relation / continuous function / array
UDC515.125.5
Other class numbers
54F50
54C25
46J10
54C30 (MSC 2000)
URLhttp://rmmc.eas.asu.edu/rmj/rmj.html
SummaryPodmnožica $K \subset \RR^2$ je bazična, če za vsako funkcijo $f \colon K \to \RR$ obstajata taki funkciji $g,h \colon \RR \to \RR$, da je $f(x,y) = g(x)+h(y)$ za vsako točko $(x,y) \in K$. Če so vse tri funkcije v tej definiciji zvezne} ({\it differenciabilne}), je vložitev $C^0$-bazična ($C^1$-bazična). Pojem bazične vložitve se pojavi pri preučevanju Hilbertovega 13. problema. V članku dokažemo, da je vsak končen graf, ki je $C^0$-bazično vložljiv v ravnino, tudi $C^1$-bazično vložljiv v ravnino. V dokazu konstruiramo eksplicitno $C^1$-bazično vložitev in uporabimo Skopenkovo karakterizacijo $C^0$-bazično vložljivih grafov v ravnino. Dobljeni rezultat je netrivialen, saj ravnina vsebuje grafe, ki so $C^0$-bazični vendar ne $C^1$-bazični in tudi grafe, ki so $C^1$-bazični, a niso $C^0$-bazični (Baran-Skopenkov). Dokažemo še, da za vsako celo število $k \ge 0$ obstaja podmnožica ravnine, ki je $C^r$-bazična za vsak $0 \le r \le k$, a ni $C^r$-bazična za noben $k<r \le \omega$.
A subset $K \subset \RR^2$ is said to be basic if for each function $f:K \to \RR$ there exist functions $g,h: \RR \to \RR$ such that $f(x, y) = g(x) + h(y)$ for each point $(x,y) \in K$. If all the three functions in this definition are assumed to be continuous (differentiable), then the embedding is $C^0$-basic ($C^1$-basic). This notion appeared in studies of Hilbert¡ s 13th problem on superpositions. We prove that if a finite graph is $C^0$-basically embeddable in the plane, then it is $C^1$-basically embeddable in the plane. In our proof we construct an explicit $C^1$-basic embedding and use the Skopenkov characterization of graphs $C^0$-basically embeddable in the plane. Our result is nontrivial because the plane contains graphs which are $C^0$-basic but not $C^1$-basic and graphs which are $C^1$-basic but not $C^0$-basic (Baran-Skopenkov). We also prove that given any integer $k \ge 0$, there is a subset of the plane which is $C^r$-basic for each $0 \le r \le k$ but not $C^r$-basic for each $k < r \le \omega$.
COBISS.SI-ID14140505
See publication: TI=Rocky Mountain journal of mathematics ISSN: 0035-7596.- Vol. 36, no. 5 (2006), str. 1665-1677

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