Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation

The integrable Novikov equation can be regarded as one of the Camassa-Holm-type equations with cubic nonlinearity. In this paper, we prove the global existence and uniqueness of the H\"older continuous energy conservative solutions for the Cauchy problem of the Novikov equation

Stability of solitary waves of a generalized two-component Camassa-Holm system

We study here the existence of solitary wave solutions of a generalized two-component Camassa-Holm system. In addition to those smooth solitary-wave solutions, we show that there are solitary waves with singularities: peaked and cusped solitary waves. We also demonstrate that all smooth solitary waves are orbitally stable in the energy space. We finally give a sufficient condition for global strong solutions to the equation without certain parameters.Comment: 23 pages, 1 figur

Searching for axionlike particle at future epep colliders

We explore the possibility of searching for axionlike particle (ALP) by $ep$ collisions via the subprocess $e^{-}\gamma \rightarrow e^{-}a \rightarrow e^{-}\gamma\gamma$. Sensitivities to the effective ALP-photon coupling $g_{a\gamma\gamma}$ for its mass in the range of 10 GeV $< M_{a}< 3$ TeV are obtained for the LHeC and its high-energy upgrade, FCC-eh. Comparing to existing bounds on the ALP free parameters, we find that the bounds given by $ep$ colliders are competitive and complementary to other colliders.Comment: 11 pages, 3 figure

On the well-posedness of a weakly dispersive Boussinesq system

We study the Cauchy problem for one-dimensional dispersive system of Boussinesq type which models weakly nonlinear long wave surface waves. We establish the local well-posedness and ill-posedness of solutions to the system. We also provide criteria for the formation of singularities.Comment: 20 page

Finite Sample Analysis of the GTD Policy Evaluation Algorithms in Markov Setting

In reinforcement learning (RL) , one of the key components is policy evaluation, which aims to estimate the value function (i.e., expected long-term accumulated reward) of a policy. With a good policy evaluation method, the RL algorithms will estimate the value function more accurately and find a better policy. When the state space is large or continuous \emph{Gradient-based Temporal Difference(GTD)} policy evaluation algorithms with linear function approximation are widely used. Considering that the collection of the evaluation data is both time and reward consuming, a clear understanding of the finite sample performance of the policy evaluation algorithms is very important to reinforcement learning. Under the assumption that data are i.i.d. generated, previous work provided the finite sample analysis of the GTD algorithms with constant step size by converting them into convex-concave saddle point problems. However, it is well-known that, the data are generated from Markov processes rather than i.i.d. in RL problems.. In this paper, in the realistic Markov setting, we derive the finite sample bounds for the general convex-concave saddle point problems, and hence for the GTD algorithms. We have the following discussions based on our bounds. (1) With variants of step size, GTD algorithms converge. (2) The convergence rate is determined by the step size, with the mixing time of the Markov process as the coefficient. The faster the Markov processes mix, the faster the convergence. (3) We explain that the experience replay trick is effective by improving the mixing property of the Markov process. To the best of our knowledge, our analysis is the first to provide finite sample bounds for the GTD algorithms in Markov setting

The overshoot and phenotypic equilibrium in characterizing cancer dynamics of reversible phenotypic plasticity

The paradigm of phenotypic plasticity indicates reversible relations of different cancer cell phenotypes, which extends the cellular hierarchy proposed by the classical cancer stem cell (CSC) theory. Since it is still question able if the phenotypic plasticity is a crucial improvement to the hierarchical model or just a minor extension to it, it is worthwhile to explore the dynamic behavior characterizing the reversible phenotypic plasticity. In this study we compare the hierarchical model and the reversible model in predicting the cell-state dynamics observed in biological experiments. Our results show that the hierarchical model shows significant disadvantages over the reversible model in describing both long-term stability (phenotypic equilibrium) and short-term transient dynamics (overshoot) of cancer cells. In a very specific case in which the total growth of population due to each cell type is identical, the hierarchical model predicts neither phenotypic equilibrium nor overshoot, whereas thereversible model succeeds in predicting both of them. Even though the performance of the hierarchical model can be improved by relaxing the specific assumption, its prediction to the phenotypic equilibrium strongly depends on a precondition that may be unrealistic in biological experiments, and it also fails to capture the overshoot of CSCs. By comparison, it is more likely for the reversible model to correctly describe the stability of the phenotypic mixture and various types of overshoot behavior.Comment: 24 pages, 6 figure

Convergence Analysis of Distributed Stochastic Gradient Descent with Shuffling

When using stochastic gradient descent to solve large-scale machine learning problems, a common practice of data processing is to shuffle the training data, partition the data across multiple machines if needed, and then perform several epochs of training on the re-shuffled (either locally or globally) data. The above procedure makes the instances used to compute the gradients no longer independently sampled from the training data set. Then does the distributed SGD method have desirable convergence properties in this practical situation? In this paper, we give answers to this question. First, we give a mathematical formulation for the practical data processing procedure in distributed machine learning, which we call data partition with global/local shuffling. We observe that global shuffling is equivalent to without-replacement sampling if the shuffling operations are independent. We prove that SGD with global shuffling has convergence guarantee in both convex and non-convex cases. An interesting finding is that, the non-convex tasks like deep learning are more suitable to apply shuffling comparing to the convex tasks. Second, we conduct the convergence analysis for SGD with local shuffling. The convergence rate for local shuffling is slower than that for global shuffling, since it will lose some information if there's no communication between partitioned data. Finally, we consider the situation when the permutation after shuffling is not uniformly distributed (insufficient shuffling), and discuss the condition under which this insufficiency will not influence the convergence rate. Our theoretical results provide important insights to large-scale machine learning, especially in the selection of data processing methods in order to achieve faster convergence and good speedup. Our theoretical findings are verified by extensive experiments on logistic regression and deep neural networks

On a shallow-water approximation to the Green-Naghdi equations with the Coriolis effect

We consider an asymptotic 1D (in space) rotation-Camassa-Holm (R-CH) model, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with allowance for the weak Coriolis effect due to the Earth's rotation. This model equation has similar wave-breaking phenomena as the Camassa-Holm equation. It is analogous to the rotation-Green-Naghdi (R-GN) equations with the weak Earth's rotation effect, modeling the propagation of wave allowing large amplitude in shallow water. We provide here a rigorous justification showing that solutions of the R-GN equations tend to associated solution of the R-CH model equation in the Camassa-Holm regime with the small amplitude and the larger wavelength. Furthermore, we demonstrate that the R-GN model equations are locally well-posed in a Sobolev space by the refined energy estimates

Symmetry Enforced Chiral Hinge States and Surface Quantum Anomalous Hall Effect in Magnetic Axion Insulator Bi2−xSmxSe3\text{Bi}_{2-x}\text{Sm}_x\text{Se}_3

A universal mechanism to generate chiral hinge states in the ferromagnetic axion insulator phase is proposed, which leads to an exotic transport phenomena, the quantum anomalous Hall effect (QAHE) on some particular surfaces determined by both the crystalline symmetry and the magnetization direction. A realistic material system Sm doped $\text{Bi}_2\text{Se}_3$ is then proposed to realize such exotic hinge states by combing the first principle calculations and the Green's function techniques. A physically accessible way to manipulate the surface QAHE is also proposed, which makes it very different from the QAHE in ordinary 2D systems.Comment: 8 pages, 5 figure

Asymmetrically interacting dynamics with mutual confirmation from multi-source on multiplex networks

In the early stage of epidemics, individuals' determination on adopting protective measures, which can reduce their risk of infection and suppress disease spreading, is likely to depend on multiple information sources and their mutual confirmation due to inadequate exact information. Here we introduce the inter-layer mutual confirmation mechanism into the information-disease interacting dynamics on multiplex networks. In our model, an individual increases the information transmission rate and willingness to adopt protective measures once he confirms the authenticity of news and severity of disease from neighbors status in multiple layers. By using the microscopic Markov chain approach, we analytically calculate the epidemic threshold and the awareness and infected density in the stationary state, which agree well with simulation results. We find that the increment of epidemic threshold when confirming the aware neighbors on communication layer is larger than that of the contact layer. On the contrary, the confirmation of neighbors' awareness and infection from the contact layer leads to a lower final infection density and a higher awareness density than that of the communication layer. The results imply that individuals' explicit exposure of their infection and awareness status to neighbors, especially those with real contacts, is helpful in suppressing epidemic spreading.Comment: 13 pages,8 figure