Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs

Let (𝒳,𝒴) be a complete and hereditary cotorsion pair in a bicomplete abelian category 𝒜. We introduce a Gorenstein category 𝒢(𝒳) and 𝒢(𝒳)-resolution dimension of complexes with respect to (𝒳,𝒴). For complexes with finite 𝒢(𝒳)-resolution dimension, Tate 𝒳-resolutions are constructed. Furthermore, we study relative Tate cohomology of complexes, which is useful for detecting the finiteness of the relative homological dimensions of complexes.     

Heterogeneous virus propagation in networks: a theoretical study

The node‐based epidemic modeling is an effective approach to the understanding of the impact of the structure of the propagation network on the epidemics of electronic virus. In view of the heterogeneity of the propagation network, a heterogeneous node‐based SIRS model is proposed. Theoretical analysis shows that the maximum eigenvalue of a matrix related to the model determines whether viruses tend to extinction or persist. When viruses persist, the connectedness of the propagation network implies the existence and uniqueness of a viral equilibrium, and a set of sufficient conditions for the global stability of the viral equilibrium are given. Numerical examples verify the correctness of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

Synchronization of complex networks with time‐varying inner coupling and outer coupling matrices

Synchronization of complex networks with time‐varying coupling matrices is studied in this paper. Two kinds of time‐varying coupling are taken into account. One is the time‐varying inner coupling in the node state space and the other is the time‐varying outer coupling in the network topology space. By respectively setting linear controllers and adaptive controllers, time‐varying complex networks can be synchronized to a desired state. Meanwhile, different influences of the control parameters of linear controllers and adaptive controllers on the synchronization have also been investigated. Based on the Lyapunov function theory, we construct appropriate positive‐definite functions, and several sufficient synchronization criteria are obtained. Numerical simulations further illustrate the effectiveness of conclusions. Copyright © 2017 John Wiley & Sons, Ltd.     

Probing the Charge Transfer in a Frustrated Lewis Pair by Resonance Raman Spectroscopy and DFT Calculations

A classical Lewis adduct derives from a covalent bond between a Lewis acid and a base. When the adduct formation is precluded by means of steric hindrance the association of the respective acid-base molecular system is defined as a frustrated Lewis pair (FLP). In this work, the archetypal FLP Mes₃P/B(C₆F₅)₃ was characterized for the first time by resonance Raman spectroscopy, and the results were supported by density functional theory (DFT) calculations. The charge transfer nature of the lowest energy electronic transition, from phosphine to borane, was confirmed by the selective enhancement of the Raman bands associated to the FLP chromophore at resonance condition. Herein, we demonstrate the use of resonance Raman spectroscopy as a distinguished technique to probe the weak interaction involved in FLP chemistry.     

Three-Dimensional Diabatic Potential Energy Surfaces of Thiophenol with Neural Networks

Three-dimensional (3D) diabatic potential energy surfaces (PESs) of thiophenol involving the S\begin{document}$_0$\end{document}, and coupled \begin{document}$^1$\end{document}\begin{document}$\pi\pi^*$\end{document} and \begin{document}$^1$\end{document}\begin{document}$\pi\sigma^*$\end{document} states were constructed by a neural network approach. Specifically, the diabatization of the PESs for the \begin{document}$^1$\end{document}\begin{document}$\pi\pi^*$\end{document} and \begin{document}$^1\pi\sigma^*$\end{document} states was achieved by the fitting approach with neural networks, which was merely based on adiabatic energies but with the correct symmetry constraint on the off-diagonal term in the diabatic potential energy matrix. The root mean square errors (RMSEs) of the neural network fitting for all three states were found to be quite small (\begin{document}$<$\end{document}4 meV), which suggests the high accuracy of the neural network method. The computed low-lying energy levels of the S\begin{document}$_0$\end{document} state and lifetime of the 0\begin{document}$^0$\end{document} state of S\begin{document}$_1$\end{document} on the neural network PESs are found to be in good agreement with those from the earlier diabatic PESs, which validates the accuracy and reliability of the PESs fitted by the neural network approach.     

Electron and positron pair emission by low energy positron impact on surfaces

The emission of electron pairs from surfaces has the power to reveal details about the electron–electron interaction in condensed matter. This process, stimulated by a primary electron or photon beam, has been studied both in experiment and theory over the last two decades. An additional pathway, namely positron–electron pair emission, holds the promise to provide additional information. It is based on the notion that the Pauli exclusion principle does not need to be considered for this process.We have commissioned a laboratory based positron source and performed a systematic study on a variety of solid surfaces. In a symmetric emission geometry we can explore the fact that positron and electron are distinguishable particles. Following fundamental symmetry arguments we have to expect that the available energy is shared unequally among positron and electron. Experimentally we observe such a behavior for all materials studied. We find an universal feature for all materials in the sense that on average the positron carries a larger fraction of the available energy. This is qualitatively accounted for by a simplified scattering model. Numerical results, which we obtained by a microscopic theory of positron–electron emission from surfaces, reveal however that there are also cases in which the electron carries more energy. Whether the positron or the electron is more energetic depends on details of the bound electron state and of the emission geometry. The coincidence intensity is strongly material dependent and there exists an almost monotonic relation between the singles and coincidence intensity. These results resemble the findings obtained in electron and photon stimulated electron pair emission. An additional reaction channel is the emission of an electron pair upon positron impact. We will discuss the energy distributions and the material dependence of the coincidence signal which shows similar features as those for positron–electron pairs.     

A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters

We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points lines, where three interfaces meet, and at the boundary points lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.     

A deep neural network-based method for solving obstacle problems

In this paper, we propose a method based on deep neural networks to solve obstacle problems. By introducing penalty terms, we reformulate the obstacle problem as a minimization optimization problem and utilize a deep neural network to approximate its solution. The convergence analysis is established by decomposing the error into three parts: approximation error, statistical error and optimization error. The approximate error is bounded by the depth and width of the network, the statistical error is estimated by the number of samples, and the optimization error is reflected in the empirical loss term. Due to its unsupervised and meshless advantages, the proposed method has wide applicability. Numerical experiments illustrate the effectiveness and robustness of the proposed method and verify the theoretical proof.