A nonlocal curve flow in centro-affine geometry

In this paper, the isoperimetric inequality in centro-affine plane geometry is obtained. We also investigate the long-term behavior of an invariant plane curve flow, whose evolution process can be expressed as a second-order nonlinear parabolic equation with respect to centro-affine curvature. The forward and backward limits in time are discussed, which shows that a closed convex embedded curve may converge to an ellipse when evolving according to this flow

A Study on the Vehicle Routing Problem Considering Infeasible Routing Based on the Improved Genetic Algorithm

The study aims to optimize the vehicle routing problem, considering infeasible routing, to minimize losses for the company. Firstly, a vehicle routing model with hard time windows and infeasible route constraints is established, considering both the minimization of total vehicle travel distance and the maximization of customer satisfaction. Subsequently, a Floyd-based improved genetic algorithm that incorporates local search is designed. Finally, the computational experiment demonstrates that compared with the classic genetic algorithm, the improved genetic algorithm reduced the average travel distance by 20.6% when focusing on travel distance and 18.4% when prioritizing customer satisfaction. In both scenarios, there was also a reduction of one in the average number of vehicles used. The proposed method effectively addresses the model introduced in this study, resulting in a reduction in total distance and an enhancement of customer satisfaction

Poly[[(μ3-5,6-dicarboxy­bicyclo­[2.2.2]oct-7-ene-2,3-dicarboxyl­ato)(1,10-phenanthroline)copper(II)] monohydrate]

In the title compound, {[Cu(C12H10O8)(C12H8N2)]·H2O}n, the CuII ion is five-coordinated by two N atoms from one phenanthroline ligand and three O atoms from three different H2 L 2− anions (H4 L is bicyclo­[2.2.2]oct-7-ene-2,3,5,6-tetra­carboxylic acid) in a distorted square-pyramidal geometry. Each H2 L 2− ion bridges three CuII atoms to form a zigzag sheet parallel to the ab plane. The crystal structure is consolidated by O—H⋯O hydrogen bonds

CDIO-CT collaborative strategy for solving complex STEM problems in system modeling and simulation: an illustration of solving the period of mathematical pendulum

The problem-project-oriented STEM education plays a significant role in training students' ability of innovation. Although the conceive-design-implement-operate (CDIO) approach and the computational thinking (CT) are hot topics in recent decade, there are still two deficiencies: the CDIO approach and CT are discussed separately and a general framework of coping with complex STEM problems in system modeling and simulation is missing. In this paper, a collaborative strategy based on the CDIO and CT is proposed for solving complex STEM problems in system modeling and simulation with a general framework, in which the CDIO is about ``how to do", CT is about ``how to think", and the project means ``what to do". As an illustration, the problem of solving the period of mathematical pendulum (MP) is discussed in detail. The most challenging task involved in the problem is to compute the complete elliptic integral of the first kind (CEI-1). In the philosophy of STEM education, all problems have more than one solutions. For computing the CEI-1, four methods are discussed with a top-down strategy, which includes the infinite series method, arithmetic-geometric mean (AGM) method, Gauss-Chebyshev method and Gauss-Legendre method. The algorithms involved can be utilized for R & D projects of interest and be reused according to the requirements encountered. The general framework for solving complex STEM problem in system modeling and simulation is worth recommending to the college students and instructors.Comment: 27 pages, 12 figures, 11 table