We consider the system $\dot {x}=(A+ P(t,\epsilon )) x, x\in \mathbb {R}^{3}, $ where $A, P$ are both three-dimensional skew symmetric matrices, A is a constant matrix with eigenvalues $\pm i\unicode{x3bb} $ and 0, $P(t,\epsilon )$ is $C^{m}(m=0,1)$-smooth in $\epsilon $ and analytic quasi-periodic with respect to t with basic frequencies $\omega =(1,\alpha )$, with $\alpha $ being irrational, and $\epsilon $ is a small parameter. Under some non-resonant conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy condition, it is proved that for many sufficiently small parameters, this system can be reduced to a rotation system. Furthermore, if the basic frequencies satisfy that $ 0\leq \beta (\alpha ) < r,$ where $\beta (\alpha )=\limsup \nolimits _{n\rightarrow \infty } {\ln q_{n+1}}/{q_{n}},$ $q_{n}$ is the sequence of denominations of the best rational approximations for $\alpha \in \mathbb {R} \setminus \mathbb {Q},$ r is the initial radius of analytic domain, it is proved that for many sufficiently small parameters, this system can be reduced to a constant system.

In order to improve the speed of motion planning, this paper proposes an improved RRTConnect algorithm (SDPS-RRTConnect) based on sparse dead point saved strategy. Combining sparse expansion strategy and dead point saved strategy, the algorithm can reduce the number of collision detection, fast convergence, avoid falling into local minimum, and ensure the completeness of search space. The algorithm is simulated in different environments. The results show that in complex environments, the sparse dead point saved strategy plays a good effect. In simple environments, the greedy connection strategy works well. Compared with the standard RRT algorithm, the standard RRTConnect algorithm, and the SDPS-RRT algorithm, the SDPS-RRTConnect algorithm has the shortest planning time, and it is suitable for both simple and complex environments. The 500 experiments show that the algorithm has strong robustness. The actual robot experiments show that the path planned by SDPS-RRTConnect algorithm is effective.

In this paper we consider the following nonlinear quasi-periodic system:

$$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$

$A$

$d\times d$

$\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$

$\unicode[STIX]{x1D716}$

$h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$

$x\rightarrow 0$

$P,g$

$h$

$t$

$\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$

$\unicode[STIX]{x1D6FC}$

$\unicode[STIX]{x1D716}$

$$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$

$h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$

$\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$

$\unicode[STIX]{x1D716}$

China’s current basic health insurance reform aims at promoting equity in the economic accessibility of health services for all citizens, to better ensure healthcare justice. Therefore, it is important to assess equity not only from a socioeconomic perspective but also from an ethical angle. This article investigates the basic health insurance system of Hunan Province in China by focusing on insurance types as well as their classification standards, mechanisms, and utilization according to local policy documents and data. This study demonstrates the reforming achievements and the inequity of institutional design according to two interrelated dimensions: equal opportunity of access to healthcare insurance and reducing inequality in insurance benefits. The article concludes that to achieve opportunity equity and outcome fairness, the reform should focus on designing the system to promote equity with respect to procedures and rules and to be more attentive to the interests of vulnerable groups and especially to rural residents.

This work focuses on the persistence of lower-dimensional tori with prescribed frequencies and singular normal matrices in reversible systems. By the Kolmogorov–Arnold–Moser theory and the special structure of unperturbed nonlinear terms in the differential equation, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of X. Wang et al [Degenerate lower dimensional tori in reversible systems. J. Math. Anal. Appl.387 (2012), 776–790].

In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the small parameter, we prove that the system is reducible for most of the sufficiently small parameters in the sense of the Lebesgue measure.

This paper is concerned with the following semilinear elliptic equations of the form

where ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε > 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

In this paper, using the Kolmogorov–Arnold–Moser method we prove reducibility of a class of nonlinear quasi-periodic differential equation with degenerate equilibrium point under small perturbation and obtain a quasi-periodic solution near the equilibrium point.

The system $-d^2y/dt^2+u(t)y=Ey\ (y\in R^n)$ is considered, where $E=\text{diag} (\lambda_1^2,\, \lambda_2^2,\dots,\lambda_n^2)$ is a diagonal matrix with $\lambda_j>0\ (j=1,2,\dots, n)$ being regarded as parameters, and u a real analytic quasi-periodic symmetric matrix. Suppose that the basic frequencies of u satisfy a Diophantine condition. Then, for most of sufficiently large $\lambda_j\ (j=1,2,\dots,n)$, the system has n pairs of conjugate quasi-periodic solutions.

This paper treats the reducibility of the quasiperiodic linear differential equations

where A is a constant matrix with multiple eigenvalues, Q(t) is a quasiperiodic matrix with respect to time t, and ε is a small perturbation parameter. Under some non-resonant conditions, rapidly convergent methods prove that, for most sufficiently small ε, the differential equations are reducible to a constant coefficient differential equation by means of a quasiperiodic change of variables with the same frequencies as Q(t).