Blow-Up Phenomena for Porous Medium Equation with Nonlinear Flux on the Boundary

We investigate the blow-up phenomena for nonnegative solutions of porous medium equation with Neumann boundary conditions. We find that the absorption and the nonlinear flux on the boundary have some competitions in the blow-up phenomena

Global exponential convergence of delayed inertial Cohen–Grossberg neural networks

In this paper, the exponential convergence of delayed inertial Cohen–Grossberg neural networks (CGNNs) is studied. Two methods are adopted to discuss the inertial CGNNs, one is expressed as two first-order differential equations by selecting a variable substitution, and the other does not change the order of the system based on the nonreduced-order method. By establishing appropriate Lyapunov function and using inequality techniques, sufficient conditions are obtained to ensure that the discussed model converges exponentially to a ball with the prespecified convergence rate. Finally, two simulation examples are proposed to illustrate the validity of the theorem results

Atomic Bose-Einstein condensate in a twisted-bilayer optical lattice

Observation of strong correlations and superconductivity in twisted-bilayer-graphene have stimulated tremendous interest in fundamental and applied physics. In this system, the superposition of two twisted honeycomb lattices, generating a Moir$\acute{\mathrm{e}}$ pattern, is the key to the observed flat electronic bands, slow electron velocity and large density of states. Despite these observations, a full understanding of the emerging superconductivity from the coupled insulating layers and the appearance of a small magic angle remain a hot topic of research. Here, we demonstrate a quantum simulation platform to study superfluids in twisted bilayer lattices based on Bose-Einstein condensates loaded into spin-dependent optical lattices. The lattices are made of two sets of laser beams that independently address atoms in different spin states, which form the synthetic dimension of the two layers. The twisted angle of the two lattices is controlled by the relative angle of the laser beams. We show that atoms in each spin state only feel one set of the lattice and the interlayer coupling can be controlled by microwave coupling between the spin states. Our system allows for flexible control of both the inter- and intralayer couplings. Furthermore we directly observe the spatial Moir$\acute{\mathrm{e}}$ pattern and the momentum diffraction, which confirm the presence of atomic superfluid in the bilayer lattices. Our system constitutes a powerful platform to investigate the physics underlying the superconductivity in twisted-bilayer-graphene and to explore other novel quantum phenomena difficult to realize in materials.Comment: 6 pages, 5 figure

Graph-based Alignment and Uniformity for Recommendation

Collaborative filtering-based recommender systems (RecSys) rely on learning representations for users and items to predict preferences accurately. Representation learning on the hypersphere is a promising approach due to its desirable properties, such as alignment and uniformity. However, the sparsity issue arises when it encounters RecSys. To address this issue, we propose a novel approach, graph-based alignment and uniformity (GraphAU), that explicitly considers high-order connectivities in the user-item bipartite graph. GraphAU aligns the user/item embedding to the dense vector representations of high-order neighbors using a neighborhood aggregator, eliminating the need to compute the burdensome alignment to high-order neighborhoods individually. To address the discrepancy in alignment losses, GraphAU includes a layer-wise alignment pooling module to integrate alignment losses layer-wise. Experiments on four datasets show that GraphAU significantly alleviates the sparsity issue and achieves state-of-the-art performance. We open-source GraphAU at https://github.com/YangLiangwei/GraphAU.Comment: 4 page