Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian

Abugirda, Hussien; Ayanbayev, Birzhan; Katzourakis, Nikos

doi:10.1216/rmj.2020.50.383

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…The notion of generalised solution of Definitions 2 & 3 will be the central notion of solution for our fully nonlinear PDE (1.2). For more on the theory of D-solutions for general systems, analytic properties, existence/uniqueness/partial regularity results see [K8]- [K11] and [2,1,AK,CKP].…”

Section: Introductionmentioning

confidence: 99%

Second-order L ^∞ variational problems and the ∞-polylaplacian

Katzourakis

Pryer

2018

Advances in Calculus of Variations

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In this paper we initiate the study of 2nd order variational problems in L ∞ , seeking to minimise the L ∞ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. Given H ∈ C 1 (R n×n s ), for the functionalthe associated equation is the fully nonlinear 3rd order PDESpecial cases arise when H is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞-Polylaplacian and the ∞-Bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Second-order L ^∞ variational problems and the ∞-polylaplacian

Katzourakis

Pryer

2018

Advances in Calculus of Variations

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“…In future studies, one may try to consider these new definitions instead of the previous studies that include the 𝐿 p -spaces to see and get more important results. For example, it may be used to extend the results of [15], and [16].…”

Section: Discussionmentioning

confidence: 99%

The L^P-spaces of functions from the n-dimensional real space to the N-dimensional quaternionic space

Abugirda,

Abdullah,

Al-Yasiri

2024

Iraqi Journal of Science

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In this paper, we introduce new definitions of the - spaces namely the - spaces Here, and are natural numbers that are not necessarily equal, such that . The space refers to the n-dimensional Euclidean space, refers to the quaternions set and refers to the N-dimensional quaternionic space. Furthermore, we establish and prove some properties of their elements. These elements are quaternion-valued N-vector functions defined on , and the spaces have never been introduced in this way before.

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“…In this way, the problem is simply transformed into the solution of the characteristic function of the Laplace operator. LE algorithm is not sensitive to noise, as it maintains these local characteristics (Abugirda et al, 2017). The objective function of the Laplace feature-map optimization is as follows…”

Section: Lle and Lementioning

confidence: 99%

A nonlinear method for monitoring industrial process

Feng

2020

Transactions of the Institute of Measurement and Control

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Aiming at fault detection in industrial processes with nonlinear or high dimensions, a novel method based on locally linear embedding preserve neighborhood for fault detection is proposed in this paper. Locally linear embedding preserve neighborhood is a feature-mapping method that combines Locally linear embedding and Laplacian eigenmaps algorithms. First, two weight matrices are obtained by the Locally linear embedding and Laplacian eigenmaps, respectively. Subsequently, the two weight matrices are combined by a balance factor to obtain the objective function. Locally linear embedding preserve neighborhood method can effectively maintain the characteristics of data in high-dimensional space. The purpose of dimension reduction is to map the high-dimensional data to low-dimensional space by optimizing the objective function. Process monitoring is performed by constructing T2 and Q statistics. To demonstrate its effectiveness and superiority, the proposed locally linear embedding preserve neighborhood for fault detection method is tested under the Swiss Roll dataset and an industrial case study. Compared with traditional fault detection methods, the proposed method in this paper effectively improves the detection rate and reduces the false alarm rate.

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Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian

Cited by 4 publications

References 24 publications

Second-order L ^∞ variational problems and the ∞-polylaplacian

Second-order L ^∞ variational problems and the ∞-polylaplacian

The L^P-spaces of functions from the n-dimensional real space to the N-dimensional quaternionic space

A nonlinear method for monitoring industrial process

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Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian

Cited by 4 publications

References 24 publications

Second-order L ∞ variational problems and the ∞-polylaplacian

Second-order L ∞ variational problems and the ∞-polylaplacian

The L^P-spaces of functions from the n-dimensional real space to the N-dimensional quaternionic space

A nonlinear method for monitoring industrial process

Contact Info

Product

Resources

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Second-order L ^∞ variational problems and the ∞-polylaplacian

Second-order L ^∞ variational problems and the ∞-polylaplacian