Homoclinic orbits for second order self-adjoint difference equations

AbstractIn this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equationsΔ[p(t)Δu(t−1)]+q(t)u(t)=f(t,u(t)),t∈Z, without any periodicity assumptions on p(t), q(t) and f, providing that f(t,x) grows superlinearly both at origin and at infinity or is an odd function with respect to x∈R, and satisfies some additional assumptions

Contagion processes on the static and activity driven coupling networks

The evolution of network structure and the spreading of epidemic are common coexistent dynamical processes. In most cases, network structure is treated either static or time-varying, supposing the whole network is observed in a same time window. In this paper, we consider the epidemic spreading on a network consisting of both static and time-varying structures. At meanwhile, the time-varying part and the epidemic spreading are supposed to be of the same time scale. We introduce a static and activity driven coupling (SADC) network model to characterize the coupling between static (strong) structure and dynamic (weak) structure. Epidemic thresholds of SIS and SIR model are studied on SADC both analytically and numerically with various coupling strategies, where the strong structure is of homogeneous or heterogeneous degree distribution. Theoretical thresholds obtained from SADC model can both recover and generalize the classical results in static and time-varying networks. It is demonstrated that weak structures can make the epidemics break out much more easily in homogeneous coupling but harder in heterogeneous coupling when keeping same average degree in SADC networks. Furthermore, we show there exists a threshold ratio of the weak structure to have substantive effects on the breakout of the epidemics. This promotes our understanding of why epidemics can still break out in some social networks even we restrict the flow of the population

Shape Hessian for generalized Oseen flow by differentiability of a minimax: A Lagrangian approach

summary:The goal of this paper is to compute the shape Hessian for a generalized Oseen problem with nonhomogeneous Dirichlet boundary condition by the velocity method. The incompressibility will be treated by penalty approach. The structure of the shape gradient and shape Hessian with respect to the shape of the variable domain for a given cost functional are established by an application of the Lagrangian method with function space embedding technique

A Group Symmetric Stochastic Differential Equation Model for Molecule Multi-modal Pretraining

Molecule pretraining has quickly become the go-to schema to boost the performance of AI-based drug discovery. Naturally, molecules can be represented as 2D topological graphs or 3D geometric point clouds. Although most existing pertaining methods focus on merely the single modality, recent research has shown that maximizing the mutual information (MI) between such two modalities enhances the molecule representation ability. Meanwhile, existing molecule multi-modal pretraining approaches approximate MI based on the representation space encoded from the topology and geometry, thus resulting in the loss of critical structural information of molecules. To address this issue, we propose MoleculeSDE. MoleculeSDE leverages group symmetric (e.g., SE(3)-equivariant and reflection-antisymmetric) stochastic differential equation models to generate the 3D geometries from 2D topologies, and vice versa, directly in the input space. It not only obtains tighter MI bound but also enables prosperous downstream tasks than the previous work. By comparing with 17 pretraining baselines, we empirically verify that MoleculeSDE can learn an expressive representation with state-of-the-art performance on 26 out of 32 downstream tasks

Exploring crowd persistent dynamism from pedestrian crossing perspective: An empirical study

Crowd studies have gained increasing relevance due to the recurring incidents of crowd crush accidents. In addressing the issue of the crowd's persistent dynamism, this paper explored the macroscopic and microscopic features of pedestrians crossing in static and dynamic contexts, employing a series of systematic experiments. Firstly, empirical evidence has confirmed the existence of crowd's persistent dynamism. Subsequently, the research delves into two aspects, qualitative and quantitative, to address the following questions:(1) Cross pedestrians tend to avoid high-density areas when crossing static crowds and particularly evade pedestrians in front to avoid deceleration, thus inducing the formation of cross-channels, a self-organization phenomenon.(2) In dynamic crowds, when pedestrian suffers spatial constrained, two patterns emerge: decelerate or detour. Research results indicate the differences in pedestrian crossing behaviors between static and dynamic crowds, such as the formation of crossing channels, backward detours, and spiral turning. However, the strategy of pedestrian crossing remains consistent: utilizing detours to overcome spatial constraints. Finally, the empirical results of this study address the final question: pedestrians detouring causes crowds' persistent collective dynamism. These findings contribute to an enhanced understanding of pedestrian dynamics in extreme conditions and provide empirical support for research on individual movement patterns and crowd behavior prediction.Comment: 31pages, 17figure

Predictive algorithm for run-in value of warp knitting based on weave matrix

To predict the run-in values of single-needle-bar warp-knitted fabrics, three-dimensional weave matrixes have been established by considering main parameters of shogging movement, take-up density and total bar number. Length of a stitch has been deduced from the parameters in weave matrixes, and a new predictive algorithm model is developed. Moreover, to validate the accuracy of the proposed predictive algorithm, 30 samples with different parameters are knitted on HKS4-EL warp-knitting machine, and the predicted run-in values and measured run-in values of the samples are compared. It can be deduced from the comparison that the predictive algorithm model can provide high prediction accuracy with a relative error of < 4.26%