Symmetric integrators with improved uniform error bounds and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

In this paper, we are concerned with symmetric integrators for the nonlinear relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation has strong nonlinearity when \eps is small. There two aspects bring significantly numerical burdens in designing numerical methods. We propose and analyze a novel class of symmetric integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the proposed symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy in time is improved to be \mathcal{O}(\varepsilon^{3} \hh^2) and \mathcal{O}(\varepsilon^{4} \hh^4) for the time stepsize \hh. The near energy conservation over long times is established for the multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on a NRKG equation show that the proposed integrators have improved uniform error bounds, excellent long time energy conservation and competitive efficiency

Dynamic Behaviors of Holling Type II Predator-Prey System with Mutual Interference and Impulses

A class of Holling type II predator-prey systems with mutual interference and impulses is presented. Sufficient conditions for the permanence, extinction, and global attractivity of system are obtained. The existence and uniqueness of positive periodic solution are also established. Numerical simulations are carried out to illustrate the theoretical results. Meanwhile, they indicate that dynamics of species are very sensitive with the period matching between species’ intrinsic disciplinarians and the perturbations from the variable environment. If the periods between individual growth and impulse perturbations match well, then the dynamics of species periodically change. If they mismatch each other, the dynamics differ from period to period until there is chaos

Improved Bound on Vertex Degree Version of Erd\H{o}s Matching Conjecture

For a $k$-uniform hypergraph $H$, let $\delta_1(H)$ denote the minimum vertex degree of $H$, and $\nu(H)$ denote the size of a maximum matching in $H$. In this paper, we show that for sufficiently large integer $n$ and integers $k\geq 3$ and $m\ge 1$, there is a positive $\beta=\beta(k)<1/(3^{k}2k^{5}(k!))^4$ such that if $H$ is a $n$-vertex $k$-graph with $1\leq m\leq (\frac{k}{2(k-1)}-\beta)\frac{n}{k}$ and $\delta_1(H)>{{n-1}\choose {k-1}}-{{n-m}\choose {k-1}},$ then $\nu(H)\geq m$. This improves upon earlier results of Bollob\'{a}s, Daykin and Erd\H{o}s (1976) for the range $n> 2k^3(m+1)$ and Huang and Zhao (2017) for the range $n\geq 3k^2 m$

Semi-discretization and full-discretization with improved accuracy for charged-particle dynamics in a strong nonuniform magnetic field

The aim of this paper is to formulate and analyze numerical discretizations of charged-particle dynamics (CPD) in a strong nonuniform magnetic field. A strategy is firstly performed for the two dimensional CPD to construct the semi-discretization and full-discretization which have improved accuracy. This accuracy is improved in the position and in the velocity when the strength of the magnetic field becomes stronger. This is a better feature than the usual so called ``uniformly accurate methods”. To obtain this refined accuracy, some reformulations of the problem and two-scale exponential integrators are incorporated, and the improved accuracy is derived from this new procedure. Then based on the strategy given for the two dimensional case, a new class of uniformly accurate methods with simple scheme is formulated for the three dimensional CPD in maximal ordering case. All the theoretical results of the accuracy are numerically illustrated by some numerical tests

Least squares support vector machine with parametric margin for binary classification

In this paper, we propose a least squares support vector machine with parametric margin (Par-LSSVM) for binary classification, which only needs to solve a system of linear equation. Par-LSSVM is able to handle the datasets with heteroscedastic noise. And the closer hyperplane to the test data point gives the class label, and this makes Par-LSSVM capable of dealing with &quot;Cross Planes&quot; datasets. The experimental results on several artificial, benchmark and USPS datasets indicate that our proposed algorithm outperforms Par-nu-SVM for binary classification problem