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A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce measure-theoretic Reeb graphs and Reeb spaces when the domain or the range is modeled as a metric measure space (i.e., a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph and Reeb space in representing the topological features of a scalar field while accounting for the distribution of the measure. We first introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively. 

Implicit graph neural networks (IGNNs) – that solve a fixed-point equilibrium equation using Picard iteration for representation learning – have shown remarkable performance in learning longrange dependencies (LRD) in the underlying graphs. However, IGNNs suffer from several issues, including 1) their expressivity is limited by their parameterizations for the well-posedness guarantee, 2) IGNNs are unstable in learning LRD, and 3) IGNNs become computationally inefficient when learning LRD. In this paper, we provide a new well-posedness characterization for IGNNs leveraging monotone operator theory, resulting in a much more expressive parameterization than the existing one. We also propose an orthogonal parameterization for IGNN based on Cayley transform to stabilize learning LRD. Furthermore, we leverage Andersonaccelerated operator splitting schemes to efficiently solve for the fixed point of the equilibrium equation of IGNN with monotone or orthogonal parameterization. We verify the computational efficiency and accuracy of the new models over existing IGNNs on various graph learning tasks at both graph and node levels. Code is available at https://github.com/ Utah-Math-Data-Science/MIGNN 

https://arxiv.org/abs/2301.00246 

Non-stoichiometric perovskite oxides have been studied as a new family of redox oxides for solar thermochemical hydrogen (STCH) production owing to their favourable thermodynamic properties. However, conventional perovskite oxides suffer from limited phase stability and kinetic properties, and poor cyclability. Here, we report a strategy of introducing A-site multi-principal-component mixing to develop a high-entropy perovskite oxide, (La1/6Pr1/6Nd1/6Gd1/6Sr1/6Ba1/6)MnO3 (LPNGSB_Mn), which shows desirable thermodynamic and kinetics properties as well as excellent phase stability and cycling durability. LPNGSB_Mn exhibits enhanced hydrogen production (∼77.5 mmol/mol-oxide) compared to (La2/3Sr1/3)MnO3 (∼53.5 mmol / mol-oxide) in a short 1 hour redox duration and high STCH and phase stability for 50 cycles. LPNGSB_Mn possesses a moderate enthalpy of reduction (252.51–296.32 kJ / mol-oxide), a high entropy of reduction (126.95–168.85 J / mol-oxide), and fast surface oxygen exchange kinetics. All A-site cations do not show observable valence changes during the reduction and oxidation processes. This research preliminarily explores the use of one A-site high-entropy perovskite oxide for STCH.