On relation between the ideal core and ideal cluster points

Kadets, Vladimir; Seliutin, Dmytro

doi:10.1016/j.jmaa.2020.124430

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…In other words, K-core(x) = co L x , where co S is the convex hull of S ⊆ R k and L x is the set of accumulation points of x. The ideal version of the Knopp core has been studied in [16,18]. The classical Knopp theorem states that, if k = 2 and A is a nonnegative regular matrix, then K-core(Ax) ⊆ K-core(x) (5) for all bounded sequences x, where Ax = (A n x : n ≥ 1), see [17, p. 115]; cf.…”

Section: •2 Regular Matricesmentioning

confidence: 99%

Most numbers are not normal

Aveni¹,

Leonetti²

2022

Math. Proc. Camb. Phil. Soc.

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We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$ , the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.

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Section: •2 Regular Matricesmentioning

confidence: 99%

Most numbers are not normal

Aveni¹,

Leonetti²

2022

Math. Proc. Camb. Phil. Soc.

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“…It is worth noting that -cluster points have been studied much before under a different name. Indeed, as it follows by [16, Theorem 4.2] and [14, Lemma 2.2], they correspond to classical “cluster points” of a filter (depending on ) on the underlying space (cf. [2, Definition 2, p. 69]).…”

Section: Introductionmentioning

confidence: 99%

Turnpike in infinite dimension

Leonetti

Caprio²

2021

Can. Math. Bull.

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Let $\Phi $ be a correspondence from a normed vector space X into itself, let $u: X\to \mathbf {R}$ be a function, and let $\mathcal {I}$ be an ideal on $\mathbf {N}$ . In addition, assume that the restriction of u on the fixed points of $\Phi $ has a unique maximizer $\eta ^\star $ . Then, we consider feasible paths $(x_0,x_1,\ldots )$ with values in X such that $x_{n+1} \in \Phi (x_n)$ , for all $n\ge 0$ . Under certain additional conditions, we prove the following turnpike result: every feasible path $(x_0,x_1,\ldots )$ which maximizes the smallest $\mathcal {I}$ -cluster point of the sequence $(u(x_0),u(x_1),\ldots )$ is necessarily $\mathcal {I}$ -convergent to $\eta ^\star $ . We provide examples that, on the one hand, justify the hypotheses of our result and, on the other hand, prove that we are including new cases which were previously not considered in the related literature.

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“…It is worth noting that I-cluster points have been studied much before under a different name. Indeed, as it follows by [14,Theorem 4.2] and [12,Lemma 2.2], they correspond to classical "cluster points" of a filter (depending on x) on the underlying space, cf. [2, Definition 2, p.69].…”

Section: Introductionmentioning

confidence: 99%

Turnpike in infinite dimension

Leonetti,

Caprio

2020

Preprint

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Let Φ be a correspondence from a normed vector space X into itself, let u : X → R be a function, and I be an ideal on N. Also, assume that the restriction of u on the fixed points of Φ has a unique maximizer η ⋆ . Then, we consider feasible paths (x 0 , x 1 , . . .) with values in X such that x n+1 ∈ Φ(x n ) for all n ≥ 0. Under certain additional conditions, we prove the following turnpike result: every feasible path (x 0 , x 1 , . . .) which maximizes the smallest I-cluster point of the sequence (u(x 0 ), u(x 1 ), . . .) is necessarily I-convergent to η ⋆ .We provide examples that, on the one hand, justify the hypotheses of our result and, on the other hand, prove that we are including new cases which were previously not considered in the related literature.

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On relation between the ideal core and ideal cluster points

Cited by 4 publications

References 8 publications

Most numbers are not normal

Most numbers are not normal

Turnpike in infinite dimension

Turnpike in infinite dimension

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