This paper, an addendum to “Dialectical Thermodynamics’ solution to the conceptual imbroglio that is the reversible path”, this journal, 10, 775-799, was written in response to the requests of several readers to provide further evidence of the said “imbroglio”. The evidence here presented relates to the incompatibility existing between the total-entropy and the Gibbs energy prescriptions for the reversible path. The previously published proof of the negentropic nature of the transformation of heat into work is here included to validate out conclusions about the Gibbs energy perspective.
In 2012, Ponraj et al. defined a concept of k-product cordial labeling as follows: Let f be a map from V(G)to { 0,1,⋯,k−1 }where k is an integer, 1≤k≤| V(G) |. For each edge uvassign the label f(u)f(v)(modk). f is called a k-product cordial labeling if | vf(i)−vf(j) |≤1, and | ef(i)−ef(j) |≤1, i,j∈{ 0,1,⋯,k−1 }, where vf(x)and ef(x)denote the number of vertices and edges respectively labeled with x (x=0,1,⋯,k−1). Motivated by this concept, we further studied and established that several families of graphs admit k-product cordial labeling. In this paper, we show that the path graphs Pnadmit k-product cordial labeling.
针对RRT(rapidly exploring random tree)路径规划算法搜索范围大、目标导向差、容易陷入局部最小值以及路径曲折等问题,提出了一种限制自适应采样区域的改进RRT路径规划算法。将整个搜索空间划分成均匀的等级,根据新节点所在等级和该等级内采样点数量动态调整采样区域,减小搜索范围;利用新节点改进策略使随机树根据环境信息自适应地向目标点调整,并改变扩展步长生成新节点;利用障碍物躲避策略提高算法的目标导向性和躲避障碍物的性能;利用改进的逆向寻优和插入节点并减小转向角的三次B样条曲线对路径进行优化处理。该算法在不同的路径环境中相较于RRT算法的搜索时间和迭代次数均减少了70%以上,且经过优化的路径更短、更平滑。
We examined the fractional second-order singular Lagrangian systems. We wrote the action principal function and equations of motion as fractional total differential equations. Also, we constructed the set of Hamilton-Jacobi partial differential equations (HJPDEs) within fractional calculus. We formulated the fractional path integral quantization for these systems. A mathematical example is examined with first- and second-class constraints.
An improved version of the sparse A^(*)algorithm is proposed to address the common issue of excessive expansion of nodes and failure to consider current ship status and parameters in traditional path planning algorithms.This algorithm considers factors such as initial position and orientation of the ship,safety range,and ship draft to determine the optimal obstacle-avoiding route from the current to the destination point for ship planning.A coordinate transformation algorithm is also applied to convert commonly used latitude and longitude coordinates of ship travel paths to easily utilized and analyzed Cartesian coordinates.The algorithm incorporates a hierarchical chart processing algorithm to handle multilayered chart data.Furthermore,the algorithm considers the impact of ship length on grid size and density when implementing chart gridification,adjusting the grid size and density accordingly based on ship length.Simulation results show that compared to traditional path planning algorithms,the sparse A^(*)algorithm reduces the average number of path points by 25%,decreases the average maximum storage node number by 17%,and raises the average path turning angle by approximately 10°,effectively improving the safety of ship planning paths.
Yongjian ZhaiJianhui CuiFanbin MengHuawei XieChunyan HouBin Li
