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1
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The constants of the Volterra derivation

100%
Hegedűs P.
Open Mathematics
|
2012
|
tom 10
|
nr 3
969-973
EN
The ring of constants of the Volterra derivation is found. Confirming a conjecture of Zielinski, it is always a polynomial ring.
2
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Border bases and kernels of homomorphisms and of derivations

84%
Zieliński J.
Open Mathematics
|
2010
|
tom 8
|
nr 4
780-785
EN
Border bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.
3
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The five-variable Volterra system

84%
Zieliński J.
Open Mathematics
|
2011
|
tom 9
|
nr 4
888-896
EN
We give a description of all polynomial constants of the five-variable Volterra derivation, hence of all polynomial first integrals of its corresponding Volterra system of differential equations. The Volterra system plays a significant role in plasma physics and population biology.
4
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Rings of constants of four-variable Lotka-Volterra systems

84%
Zieliński J.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1923-1931
EN
Lotka-Volterra systems appear in population biology, plasma physics, laser physics and derivation theory, among many others. We determine the rings of constants of four-variable Lotka-Volterra derivations with four parameters C 1, C 2, C 3, C 4 ∈ k, where k is a field of characteristic zero. Thus, we give a full description of polynomial first integrals of the respective systems of differential equations.
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