scholarly journals Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics

2021 ◽  
Vol 41 (8) ◽  
pp. 3985
Author(s):  
José A. Carrillo ◽  
Bertram Düring ◽  
Lisa Maria Kreusser ◽  
Carola-Bibiane Schönlieb
Keyword(s):  
Nonlocal Interaction ◽  
Equation Analysis ◽  
Interaction Equation

scholarly journals Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

2021 ◽  
Author(s):  
Antonio Esposito ◽  
Francesco S. Patacchini ◽  
André Schlichting ◽  
Dejan Slepčev
Keyword(s):  
Gradient Flow ◽  
Wasserstein Distance ◽  
Convergence Result ◽  
Existence Theory ◽  
Nonlocal Interaction ◽  
Interaction Energies ◽  
Gradient Flows ◽  
Empirical Measures ◽  
Interaction Equation ◽  
The Continuum

AbstractWe consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

scholarly journals The nonlocal-interaction equation near attracting manifolds

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Francesco S. Patacchini ◽  
Dejan Slepčev
Keyword(s):  
Numerical Experiments ◽  
Compact Manifold ◽  
Attractive Potential ◽  
Nonlocal Interaction ◽  
Well Posedness ◽  
Gradient Flows ◽  
Compact Sets ◽  
Interaction Equation ◽  
Sets With Positive Reach ◽  
Regular Sets

<p style='text-indent:20px;'>We study the approximation of the nonlocal-interaction equation restricted to a compact manifold <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> embedded in <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula>, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> can be approximated by the classical nonlocal-interaction equation on <inline-formula><tex-math id="M4">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> by adding an external potential which strongly attracts to <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. The proof relies on the Sandier–Serfaty approach [<xref ref-type="bibr" rid="b23">23</xref>,<xref ref-type="bibr" rid="b24">24</xref>] to the <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on <inline-formula><tex-math id="M7">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, which was shown [<xref ref-type="bibr" rid="b10">10</xref>]. We also provide an another approximation to the interaction equation on <inline-formula><tex-math id="M8">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, based on iterating approximately solving an interaction equation on <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> and projecting to <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.</p>

scholarly journals Does Emotional Intelligence Influence Academic Performance? The Role of Compassion and Engagement in Education for Sustainable Development

2021 ◽  
Vol 13 (4) ◽  
pp. 1721
Author(s):  
Marta Estrada ◽  
Diego Monferrer ◽  
Alma Rodríguez ◽  
Miguel Ángel Moliner
Keyword(s):  
Sustainable Development ◽  
Emotional Intelligence ◽  
Academic Performance ◽  
Structural Equation ◽  
The Sustainable Development ◽  
Equation Analysis ◽  
Emotional Aspects ◽  
Development Goals ◽  
Sustainable Societies ◽  
Social Environmental

Education must guide students’ emotional development, not only to improve their skills and help them achieve their maximum performance, but to establish the foundations of a more cooperative and compassionate society. Achieving the Sustainable Development Goals, therefore, implies focusing on emotional aspects as well as financial, social, environmental, and scientific objectives. In this line, the goal of this study is to show how emotional intelligence, which is an essential dimension in the development and management of emotional competences required to build sustainable societies, plays a key role in optimising student’s academic performance in the classroom through compassion and academic commitment. The research model was tested with a questionnaire addressed to 550 students from four higher education institutions and one secondary school. The results of a structural equation analysis confirmed the study hypotheses. Emotional intelligence was shown to be positively related to compassion and higher levels of commitment, which, consequently, led to better academic performance. This finding will encourage interest in developing emotional intelligence, not only for its long-term value in training healthy citizens, but also for its short-term results in the classroom.

scholarly journals Integral Equation Analysis of Plane Wave Scattering by Coplanar Graphene-Strip Gratings in the THz Range

2013 ◽  
Vol 3 (5) ◽  
pp. 666-674 ◽  
Author(s):  
Olga V. Shapoval ◽  
Juan Sebastian Gomez-Diaz ◽  
Julien Perruisseau-Carrier ◽  
Juan R. Mosig ◽  
Alexander I. Nosich
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Wave Scattering ◽  
Equation Analysis ◽  
Thz Range ◽  
Plane Wave Scattering ◽  
Graphene Strip
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