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A remark on Vapnik-Chervonienkis classes

100%

Smoktunowicz A.

Colloquium Mathematicum

1997

tom 74

nr 1

93-98

We show that the family of all lines in the plane which is a VC class of index 2 cannot be obtained in a finite number of steps starting with VC classes of index 1 and applying the operations of intersection and union. This confirms a common belief among specialists and solves a question asked by several authors.

Centers in domains with quadratic growth

100%

Smoktunowicz A.

Open Mathematics

2005

tom 3

nr 4

644-653

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Open Mathematics

2 Smoktunowicz A.

1 2005

1 1997

Wyniki wyszukiwania

A remark on Vapnik-Chervonienkis classes

Centers in domains with quadratic growth