A Double Saddle-Node Bifurcation Theorem

In this paper, we consider an abstract equation F(lambda, u) = 0 with one parameter lambda, where F epsilon C-P(R x X, Y), p \u3e= 2, is a nonlinear differentiable mapping, and X, Y are Banach spaces. We apply Lyapunov-Schmidt procedure and Morse Lemma to obtain a double saddle-node bifurcation theorem with a two-dimensional kernel. Applications include a perturbed problem and a semilinear elliptic equation

Bifurcation from a degenerate simple eigenvalue

It is proved that a symmetry-breaking bifurcation occurs at a simple eigenvalue despite the usual transversality condition fails, and this bifurcation from a degenerate simple eigenvalue result complements the classical one with the transversality condition. The new result is applied to an imperfect pitchfork bifurcation, in which a forward transcritical bifurcation changes to a backward one when the perturbation parameter changes. Several applications in ecological and genetics models are shown. (C) 2013 Elsevier Inc. All rights reserved

EDEN: A Plug-in Equivariant Distance Encoding to Beyond the 1-WL Test

The message-passing scheme is the core of graph representation learning. While most existing message-passing graph neural networks (MPNNs) are permutation-invariant in graph-level representation learning and permutation-equivariant in node- and edge-level representation learning, their expressive power is commonly limited by the 1-Weisfeiler-Lehman (1-WL) graph isomorphism test. Recently proposed expressive graph neural networks (GNNs) with specially designed complex message-passing mechanisms are not practical. To bridge the gap, we propose a plug-in Equivariant Distance ENcoding (EDEN) for MPNNs. EDEN is derived from a series of interpretable transformations on the graph's distance matrix. We theoretically prove that EDEN is permutation-equivariant for all level graph representation learning, and we empirically illustrate that EDEN's expressive power can reach up to the 3-WL test. Extensive experiments on real-world datasets show that combining EDEN with conventional GNNs surpasses recent advanced GNNs

Recent Advances in Heterogeneous Catalytic Hydrogenation of CO2 to Methane

With the accelerating industrialization, urbanization process, and continuously upgrading of consumption structures, the CO2 from combustion of coal, oil, natural gas, and other hydrocarbon fuels is unbelievably increased over the past decade. As an important carbon resource, CO2 gained more and more attention because of its converting properties to lower hydrocarbon, such as methane, methanol, and formic acid. Among them, CO2 methanation is considered to be an extremely efficient method due to its high CO2 conversion and CH4 selectivity. However, the CO2 methanation process requires high reaction temperatures (300–400°C), which limits the theoretical yield of methane. Thus, it is desirable to find a new strategy for the efficient conversion of CO2 to methane at relatively low reaction temperature, and the key issue is using the catalysts in the process. The advances in the noble metal catalysts, Ni-based catalysts, and Co-based catalysts, for catalytic hydrogenation CO2 to methane are reviewed in this paper, and the effects of the supports and the addition of second metal on CO2 methanation as well as the reaction mechanisms are focused

Position-Aware Subgraph Neural Networks with Data-Efficient Learning

Data-efficient learning on graphs (GEL) is essential in real-world applications. Existing GEL methods focus on learning useful representations for nodes, edges, or entire graphs with ``small'' labeled data. But the problem of data-efficient learning for subgraph prediction has not been explored. The challenges of this problem lie in the following aspects: 1) It is crucial for subgraphs to learn positional features to acquire structural information in the base graph in which they exist. Although the existing subgraph neural network method is capable of learning disentangled position encodings, the overall computational complexity is very high. 2) Prevailing graph augmentation methods for GEL, including rule-based, sample-based, adaptive, and automated methods, are not suitable for augmenting subgraphs because a subgraph contains fewer nodes but richer information such as position, neighbor, and structure. Subgraph augmentation is more susceptible to undesirable perturbations. 3) Only a small number of nodes in the base graph are contained in subgraphs, which leads to a potential ``bias'' problem that the subgraph representation learning is dominated by these ``hot'' nodes. By contrast, the remaining nodes fail to be fully learned, which reduces the generalization ability of subgraph representation learning. In this paper, we aim to address the challenges above and propose a Position-Aware Data-Efficient Learning framework for subgraph neural networks called PADEL. Specifically, we propose a novel node position encoding method that is anchor-free, and design a new generative subgraph augmentation method based on a diffused variational subgraph autoencoder, and we propose exploratory and exploitable views for subgraph contrastive learning. Extensive experiment results on three real-world datasets show the superiority of our proposed method over state-of-the-art baselines.Comment: 9 pages, 7 figures, accepted by WSDM 2

Sparse Group Variable Selection for Gene-Environment Interactions in the Longitudinal Stud

Recently, regularized variable selection has emerged as a powerful tool to iden- tify and dissect gene-environment interactions. Nevertheless, in longitudinal studies with high di- mensional genetic factors, regularization methods for G×E interactions have not been systemati- cally developed. In this package, we provide the implementation of sparse group variable selec- tion, based on both the quadratic inference function (QIF) and generalized estimating equa- tion (GEE), to accommodate the bi-level selection for longitudinal G×E studies with high dimen- sional genomic features. Alternative methods conducting only the group or individual level se- lection have also been included. The core modules of the package have been developed in C++