This paper proposes an iterative algorithm to solve the inverse displacement for a hyper-redundant elephant’s trunk robot (HRETR). In this algorithm, each parallel module is regarded as a geometric line segment and point model. According to the forward approximation and inverse pose adjustment principles, the iteration process can be divided into forward and backward iteration. This iterative algorithm transforms the inverse displacement problem of the HRETR into the parallel module’s inverse displacement problem. Considering the mechanical joint constraints, multiple iterations are carried out to ensure that the robot satisfies the required position error. Simulation results show that the algorithm is effective in solving the inverse displacement problem of HRETR.

For $p\geq 2$, let $E$ be a 2-uniformly smooth and $p$-uniformly convex real Banach space and let $A:E\rightarrow E^{\ast }$ be a Lipschitz and strongly monotone mapping such that $A^{-1}(0)\neq \emptyset$. For given $x_{1}\in E$, let $\{x_{n}\}$ be generated by the algorithm $x_{n+1}=J^{-1}(Jx_{n}-\unicode[STIX]{x1D706}Ax_{n})$, $n\geq 1$, where $J$ is the normalized duality mapping from $E$ into $E^{\ast }$ and $\unicode[STIX]{x1D706}$ is a positive real number in $(0,1)$ satisfying suitable conditions. Then it is proved that $\{x_{n}\}$ converges strongly to the unique point $x^{\ast }\in A^{-1}(0)$. Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.

We present the first exact simulation method for multidimensional reflected Brownian motion (RBM). Exact simulation in this setting is challenging because of the presence of correlated local-time-like terms in the definition of RBM. We apply recently developed so-called ε-strong simulation techniques (also known as tolerance-enforced simulation) which allow us to provide a piecewise linear approximation to RBM with ε (deterministic) error in uniform norm. A novel conditional acceptance–rejection step is then used to eliminate the error. In particular, we condition on a suitably designed information structure so that a feasible proposal distribution can be applied.

The numerical stability of the explicit precise algorithm, which was developed for the viscoplastic materials, was analyzed. It was found that this algorithm is not absolutely stable. A necessary but not sufficient condition for the numerical stability was deduced. It showed that the time step in numerical calculation should be less than a certain value to guarantee the stability of explicit precise algorithm. Through a series of numerical examples, the stability analysis on the explicit precise algorithm was proved to be reliable. At last, an iterative algorithm was presented for viscoplastic materials. Both of the theoretical and numerical results showed that the iterative algorithm is unconditionally stable and its convergence rate is rapid. In practice, the explicit precise algorithm and iterative algorithm can be combined to obtain reliable results with the minimum computing costs.

In this paper we will show how to set up a practical bonus-malus system with a finite number of classes. We will use the actual claim amount and claims frequency distributions in order to predict the future observed claims frequency when the new bonus-malus system will be in use. The future observed claims frequency is used to set up an optimal bonus-malus system as well as the transient and stationary distributions of the drivers in the new bonus-malus system. When the number of classes as well as the transition rules of the new bonus-malus system have been adopted, the premium levels are obtained by minimizing a certain distance between the levels of the practical bonus-malus system and the corresponding optimal bonus-malus system. Some iterations are necessary in order to reach stabilization of the future observed claims frequency and the levels of the practical bonus-malus system.