Grothendieck spaces: the landscape and perspectives

Kania, Tomasz

doi:10.1007/s11537-021-2116-3

Cited by 12 publications

(3 citation statements)

References 126 publications

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“…Then, with the help of the advantage of genetic algorithm in searching the optimal solution, the final scheme of landscape architecture design can be matched with the personal concept of landscape architects. Affected by the number of population, in order to search the optimal solution comprehensively, quickly, and accurately, genetic algorithm usually requires individuals to provide easily perceived parameters to designers [23]. Therefore, layer-by-layer solution method can be adopted.…”

Section: Landscape Environment Optimization Based On Interactive Gene...mentioning

confidence: 99%

The Application of Genetic Algorithm in the Optimal Design of Landscape Space Environment

Liu

2022

Mathematical Problems in Engineering

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Garden landscape not only provides people with places of rest and entertainment, but also protects the natural environment and maintained ecological balance. Although the traditional garden architectural style could retain the classical landscape style, the modern garden facilities and conditions had been greatly improved, and people’s expectations for the construction level of garden landscape continued to improve. Therefore, the effect of traditional landscape design could no longer meet the requirements of social development. This article proposed an interactive genetic algorithm-based landscape space environment optimization design method, in order to provide a certain theoretical reference for landscape design. Firstly, by analyzing the relationship between landscape and buildings, the change in people’s demand for landscape space environment and the relevant characteristics of landscape, this article expounded on the basic principles and methods that landscape design should follow and gave the problems existing in landscape design. Secondly, the interactive genetic algorithm and its innovative design theory were summarized, and the optimization design method of garden landscape space environment based on interactive genetic algorithm was proposed. Finally, the evaluation index system of landscape spatial environment was constructed, and experimental analysis was carried out with a landscape design as a case. The results showed that compared with the traditional landscape design methods, the design scheme proposed in this article could achieve better evaluation results. The optimization design method of landscape space environment proposed in this article could provide some technical support and theoretical reference for landscape architecture and design.

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Section: Landscape Environment Optimization Based On Interactive Gene...mentioning

confidence: 99%

The Application of Genetic Algorithm in the Optimal Design of Landscape Space Environment

Liu

2022

Mathematical Problems in Engineering

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“…It follows from Mazur's lemma that Grothendieck spaces, in particular, reflexive spaces, have property (𝐾). (A Banach space 𝑋 is a Grothendieck space, whenever every weak* convergent sequence in 𝑋 * converges weakly, see [19]. ) Figiel, Johnson, and Pełczyński [17] refined property (𝐾) by introducing a weaker property that they call property (𝑘); this property appeared implicitly also in [22].…”

Section: Introductionmentioning

confidence: 99%

“…It follows from Mazur's lemma that Grothendieck spaces, in particular, reflexive spaces, have property ( K ). (A Banach space X is a Grothendieck space , whenever every weak* convergent sequence in

X^{*}

converges weakly, see [19]. )…”

Section: Introductionmentioning

confidence: 99%

Quantifying properties (K) and (μs$\mu ^{s}$)

Chen

Kania

Ruan

2022

Mathematische Nachrichten

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A Banach space 𝑋 has property (𝐾), whenever every weak* null sequence in the dual space admits a convex block subsequence (𝑓 𝑛 ) ∞ 𝑛=1 so that ⟨𝑓 𝑛 , 𝑥 𝑛 ⟩ → 0 as 𝑛 → ∞ for every weakly null sequence (𝑥 𝑛 ) ∞ 𝑛=1 in 𝑋; 𝑋 has property (𝜇 𝑠 ) if every weak * null sequence in 𝑋 * admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property (𝜇 𝑠 ) and reflexivity (or even the Grothendieck property) imply property (𝐾). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.

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On Weak$$^*$$-Extensible Subspaces of Banach Spaces

Martínez-Cervantes

Rodríguez

2022

Mediterr. J. Math.

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Let X be a Banach space and $$Y \subseteq X$$ Y ⊆ X be a closed subspace. We prove that if the quotient X/Y is weakly Lindelöf determined or weak Asplund, then for every $$w^*$$ w ∗ -convergent sequence $$(y_n^*)_{n\in \mathbb N}$$ ( y n ∗ ) n ∈ N in $$Y^*$$ Y ∗ there exist a subsequence $$(y_{n_k}^*)_{k\in \mathbb N}$$ ( y n k ∗ ) k ∈ N and a $$w^*$$ w ∗ -convergent sequence $$(x_k^*)_{k\in \mathbb N}$$ ( x k ∗ ) k ∈ N in $$X^*$$ X ∗ such that $$x_k^*|_Y=y_{n_k}^*$$ x k ∗ | Y = y n k ∗ for all $$k\in \mathbb N$$ k ∈ N . As an application, we obtain that Y is Grothendieck whenever X is Grothendieck and X/Y is reflexive, which answers a question raised by González and Kania.

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Grothendieck spaces: the landscape and perspectives

Cited by 12 publications

References 126 publications

The Application of Genetic Algorithm in the Optimal Design of Landscape Space Environment

The Application of Genetic Algorithm in the Optimal Design of Landscape Space Environment

Quantifying properties (K) and (μs$\mu ^{s}$)

On Weak$$^*$$-Extensible Subspaces of Banach Spaces

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