Displaying similar documents to “Commutants and derivation ranges”

Partial differential equations in Banach spaces involving nilpotent linear operators

Antonia Chinnì, Paolo Cubiotti (1996)

Annales Polonici Mathematici

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Let E be a Banach space. We consider a Cauchy problem of the type ⎧ D t k u + ∑ j = 0 k - 1 ∑ | α | ≤ m A j , α ( D t j D x α u ) = f in ℝ n + 1 , ⎨ ⎩ D t j u ( 0 , x ) = φ j ( x ) in ℝ n , j=0,...,k-1, where each A j , α is a given continuous linear operator from E into itself. We prove that if the operators A j , α are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions u ∈ C ∞ ( ℝ n + 1 , E ) whose derivatives are equi-bounded on each bounded subset of ℝ n + 1 .

Images of locally nilpotent derivations of bivariate polynomial algebras over a domain

Xiaosong Sun, Beini Wang (2024)

Czechoslovak Mathematical Journal

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We study the LND conjecture concerning the images of locally nilpotent derivations, which arose from the Jacobian conjecture. Let R be a domain containing a field of characteristic zero. We prove that, when R is a one-dimensional unique factorization domain, the image of any locally nilpotent R -derivation of the bivariate polynomial algebra R [ x , y ] is a Mathieu-Zhao subspace. Moreover, we prove that, when R is a Dedekind domain, the image of a locally nilpotent R -derivation of R [ x , y ] with some...

Distributivity of strong implications over conjunctive and disjunctive uninorms

Daniel Ruiz-Aguilera, Joan Torrens (2006)

Kybernetika

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This paper deals with implications defined from disjunctive uninorms U by the expression I ( x , y ) = U ( N ( x ) , y ) where N is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a t -norm or a t -conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works...

Retracts that are kernels of locally nilpotent derivations

Dayan Liu, Xiaosong Sun (2022)

Czechoslovak Mathematical Journal

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Let k be a field of characteristic zero and B a k -domain. Let R be a retract of B being the kernel of a locally nilpotent derivation of B . We show that if B = R ⊕ I for some principal ideal I (in particular, if B is a UFD), then B = R [ 1 ] , i.e., B is a polynomial algebra over R in one variable. It is natural to ask that, if a retract R of a k -UFD B is the kernel of two commuting locally nilpotent derivations of B , then does it follow that B ≅ R [ 2 ] ? We give a negative answer to this question. The interest in...

𝒟 n , r is not potentially nilpotent for n ≥ 4 r - 2

Yan Ling Shao, Yubin Gao, Wei Gao (2016)

Czechoslovak Mathematical Journal

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An n × n sign pattern 𝒜 is said to be potentially nilpotent if there exists a nilpotent real matrix B with the same sign pattern as 𝒜 . Let 𝒟 n , r be an n × n sign pattern with 2 ≤ r ≤ n such that the superdiagonal and the ( n , n ) entries are positive, the ( i , 1 ) ( i = 1 , ⋯ , r ) and ( i , i - r + 1 ) ( i = r + 1 , ⋯ , n ) entries are negative, and zeros elsewhere. We prove that for r ≥ 3 and n ≥ 4 r - 2 , the sign pattern 𝒟 n , r is not potentially nilpotent, and so not spectrally arbitrary.