Complex structures in algebra, topology and differential equations

Balanov, Zalman; Krasnov, Yakov

doi:10.1515/gmj-2014-0037

Cited by 3 publications

(5 citation statements)

References 14 publications

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“…On the other hand, the condition n 2 ∈ A λ2 (u) is equivalent to aA 1 12 + cβ 1 = 0, aA 3 12 + cβ 3 = 0, and we obtain nontrivial solutions of and only if Then we compute u 2 and the third component should be zero. A direct computation gives the necessary and sufficient condition…”

Section: The Casementioning

confidence: 86%

“…with no condition on scalars, except A 1 12 , A 2 12 = (0, 0). We notice that the condition A 1 12 , A 2 12 = (0, 0) corresponds to the case studied in [28] for which instability of (0, 0, 0) and any nilpotent in N 2 holds.…”

Section: Applications In Dimensionmentioning

confidence: 99%

“…Applications in dimension 3. We consider the general family of algebras A ′ defined by (2.1) with, this time, n 1 n 2 = A 1 12 n 1 + A 2 12 n 2 + A 3 12 e 3 , where A 3 12 = 0 since N 2 N 2 ⊂ N 2 . Let u = an 1 + bn 2 + ce 3 be a vector not in N 2 (i.e.…”

Section: The Casementioning

confidence: 99%

“…There are also some papers about (algebraic) structure(s) and dynamics in systems (1.1), e.g. [1,4,5,8] and applications beyond ODEs (see e.g. [12,13,14,15,26,16,21]).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On a generalization of some instability results for Riccati equations via nonassociative algebras

Boujemaa¹,

Ferčec²

2022

Glas. Mat. Ser. III

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In [28], for any real non associative algebra of dimension \(m\geq2\), having \(k\) linearly independent nilpotent elements \(n_{1}\), \(n_{2}\), …, \(n_{k},\) \(1\leq k\leq m-1\), Mencinger and Zalar defined near idempotents and near nilpotents associated to \(n_{1}\), \(n_{2}\), …, \(n_{k}\). Assuming \(\mathcal{N}_{k}\mathcal{N}_{k}=\left\{ 0\right\}\), where \(\mathcal{N} _{k}=\operatorname*{span}\left\{ n_{1},n_{2},\ldots,n_{k}\right\} \), they showed that if there exists a near idempotent or a near nilpotent, called \(u\), associated to \(n_{1},n_{2},\ldots,n_{k}\) verifying \(n_{i}u\in\mathbb{R}n_{i},\) for \(1\leq i\leq k\), then any nilpotent element in \(\mathcal{N}_{k}\) is unstable. They also raised the question of extending their results to cases where \(\mathcal{N}_{k}\mathcal{N}_{k}\not =\left\{ 0\right\} \) with \(\mathcal{N}_{k}\mathcal{N}_{k}\subset\mathcal{N}_{k}\mathcal{\ }\)and to cases where \(\mathcal{N}_{k}\mathcal{N}_{k} \not\subset \mathcal{N}_{k}.\) In this paper, positive answers are emphasized and in some cases under the weaker conditions \(n_{i}u\in\mathcal{N}_{k}\). In addition, we characterize all such algebras in dimension 3.

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Section: The Casementioning

confidence: 86%

Section: Applications In Dimensionmentioning

confidence: 99%

Section: The Casementioning

confidence: 99%

“…There are also some papers about (algebraic) structure(s) and dynamics in systems (1.1), e.g. [1,4,5,8] and applications beyond ODEs (see e.g. [12,13,14,15,26,16,21]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a generalization of some instability results for Riccati equations via nonassociative algebras

Boujemaa¹,

Ferčec²

2022

Glas. Mat. Ser. III

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“…This idea was considered by many authors (e.g. [5,6,3,[10][11][12]14,2,[19][20][21][23][24][25] to mention just few) and is applicable also for partial differential equations ( [13]) and even for difference systems ( [22,15]). Note that the process called homogenization ( [26, p. 22]) transforms any Riccati equation with constant coefficients into the form (1.3).…”

Section: Introductionmentioning

confidence: 99%

Near-idempotents, near-nilpotents and stability of critical points for Riccati equations

Zalar¹,

Mencinger²

2018

Glas. Mat. Ser. III

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The paper introduces two algebraic concepts, near-idempotents and near-nilpotents associated to subspaces N of critical points, which can be used to re-phrase a theorem due to Boujemaa, El Qotbi and Rouiouih on stability for the Ricatti equation,ẋ = x(t) 2 , associated to algebra A ≈ R d. Using this concepts their result corresponds to the case dim N = 1. Our main results are a generalization of the above mentioned theorem to N of arbitrary dimension and a counterexample which shows, even in the general setting, that the essential condition that critical points must be eigenvectors of a suitable multiplication operator cannot be omitted from the formulation due to Boujemaa et al.

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On Topological Aspects of Numerical Range

Giorgadze,

Khimshiashvili

2024

Analysis Without Borders

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Complex structures in algebra, topology and differential equations

Cited by 3 publications

References 14 publications

On a generalization of some instability results for Riccati equations via nonassociative algebras

On a generalization of some instability results for Riccati equations via nonassociative algebras

Near-idempotents, near-nilpotents and stability of critical points for Riccati equations

On Topological Aspects of Numerical Range

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