Displaying similar documents to “Parallelepipeds, nilpotent groups and Gowers norms”

Hurewicz-Serre theorem in extension theory

M. Cencelj, J. Dydak, A. Mitra, A. Vavpetič (2008)

Fundamenta Mathematicae

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The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that X τ K ( H k ( L ) , k ) for all k ≥ 1, then X τ K ( π k ( F ) , k ) and X τ K ( π k ( L ) , k ) for all k ≥ . Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈...

Convergence of formal solutions of first order singular partial differential equations of nilpotent type

Masatake Miyake, Akira Shirai (2012)

Banach Center Publications

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Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: P u ( x , y , z ) : = ( y x - z y ) u ( x , y , z ) = f ( x , y , z ) x , y , z , where P = y x - z y : x , y , z x , y , z . For this equation, our aim is to characterize the solvability on x , y , z by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.

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...

Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center

Bin Ren, Lin Sheng Zhu (2017)

Czechoslovak Mathematical Journal

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A Lie algebra L is called 2-step nilpotent if L is not abelian and [ L , L ] lies in the center of L . 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.

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 n + 1 , ⎨ ⎩ D t j u ( 0 , x ) = φ j ( x ) in 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 ( n + 1 , E ) whose derivatives are equi-bounded on each bounded subset of n + 1 .

Leibniz's rule on two-step nilpotent Lie groups

Krystian Bekała (2016)

Colloquium Mathematicae

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Let be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows us to define a generalized multiplication f g = ( f g ) of two functions in the Schwartz class (*), where and are the Abelian Fourier transforms on the Lie algebra and on the dual * and ∗ is the convolution on the group . In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order...

On nilpotent operators

Laura Burlando (2005)

Studia Mathematica

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We give several necessary and sufficient conditions in order that a bounded linear operator on a Banach space be nilpotent. We also discuss three necessary conditions for nilpotency. Furthermore, we construct an infinite family (in one-to-one correspondence with the square-summable sequences ( ε ) n of strictly positive real numbers) of nonnilpotent quasinilpotent operators on an infinite-dimensional Hilbert space, all the iterates of each of which have closed range. Each of these operators...

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.

Sub-Laplacian with drift in nilpotent Lie groups

Camillo Melzi (2003)

Colloquium Mathematicae

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We consider the heat kernel ϕ t corresponding to the left invariant sub-Laplacian with drift term in the first commutator of the Lie algebra, on a nilpotent Lie group. We improve the results obtained by G. Alexopoulos in [1], [2] proving the “exact Gaussian factor” exp(-|g|²/4(1+ε)t) in the large time upper Gaussian estimate for ϕ t . We also obtain a large time lower Gaussian estimate for ϕ t .

On finite commutative loops which are centrally nilpotent

Emma Leppälä, Markku Niemenmaa (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let Q be a finite commutative loop and let the inner mapping group I ( Q ) C p n × C p n , where p is an odd prime number and n 1 . We show that Q is centrally nilpotent of class two.

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

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Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to k , k>1. We consider a class of second order left-invariant differential operators on S of the form α = L a + Δ α , where α k , and for each a k , L a is left-invariant second order differential operator on N and Δ α = Δ - α , , where Δ is the usual Laplacian on k . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively)...

On the nilpotent residuals of all subalgebras of Lie algebras

Wei Meng, Hailou Yao (2018)

Czechoslovak Mathematical Journal

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Let 𝒩 denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field 𝔽 , there exists a smallest ideal I of L such that L / I 𝒩 . This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by L 𝒩 . In this paper, we define the subalgebra S ( L ) = H L I L ( H 𝒩 ) . Set S 0 ( L ) = 0 . Define S i + 1 ( L ) / S i ( L ) = S ( L / S i ( L ) ) for i 1 . By S ( L ) denote the terminal term of the ascending series. It is proved that L = S ( L ) if and only if L 𝒩 is nilpotent. In addition, we investigate the basic properties of a...

A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent

Fares Gherbi, Nadir Trabelsi (2024)

Czechoslovak Mathematical Journal

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Let 𝔐 be the class of groups satisfying the minimal condition on normal subgroups and let Ω be the class of groups of finite lower central depth, that is groups G such that γ i ( G ) = γ i + 1 ( G ) for some positive integer i . The main result states that if G is a finitely generated hyper-(Abelian-by-finite) group such that for every x G , there exists a normal subgroup H x of finite index in G satisfying x , x h 𝔐 Ω for every h H x , then G is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated...

A new family of spectrally arbitrary ray patterns

Yinzhen Mei, Yubin Gao, Yan Ling Shao, Peng Wang (2016)

Czechoslovak Mathematical Journal

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An n × n ray pattern 𝒜 is called a spectrally arbitrary ray pattern if the complex matrices in Q ( 𝒜 ) give rise to all possible complex polynomials of degree n . In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an n × n irreducible spectrally arbitrary ray pattern is 3 n - 1 . In this paper, we introduce a new family of spectrally arbitrary ray patterns of order n with exactly 3 n - 1 nonzeros.

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...

Naive boundary strata and nilpotent orbits

Matt Kerr, Gregory Pearlstein (2014)

Annales de l’institut Fourier

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We give a Hodge-theoretic parametrization of certain real Lie group orbits in the compact dual of a Mumford-Tate domain, and characterize the orbits which contain a naive limit Hodge filtration. A series of examples are worked out for the groups S U ( 2 , 1 ) , S p 4 , and G 2 .