通过对高考数学试卷中涉及平面向量数量积问题的解题分析,本文总结了几个运用三角函数解决此类问题的常见方法,同时,深入探讨了高考中平面向量数量积问题的命题意图,并据此提出了相应的教学建议。Through the analysis of solving the p...
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通过对高考数学试卷中涉及平面向量数量积问题的解题分析,本文总结了几个运用三角函数解决此类问题的常见方法,同时,深入探讨了高考中平面向量数量积问题的命题意图,并据此提出了相应的教学建议。Through the analysis of solving the problem of the product of plane vectors in the mathematics test paper of the college entrance examination, this article summarizes several common methods of using trigonometric functions to solve such problems. At the same time, it deeply explores the proposition intention of the product of plane vectors in the college entrance examination and proposes corresponding teaching suggestions based on this.
本文以2013~2022年我国30个省份为研究对象,构建了中国省域新质生产力水平指标体系,并基于熵值法进行了测算,采用东中西三大区域划分视角,利用了核密度估计方法和Dagum基尼系数分解法分析了中国省域新质生产力水平的区域差异及时空演变。结果表明,东部地区新质生产力水平远高于其他地区的新质生产力水平;全国的新质生产力水平差异整体呈扩大趋势,中国新质生产力区域差异主要是地区间差异造成的;东北地区新质生产力水平显著存在两极分化特征,存在明显的空间极化现象。本文旨在为提高新质生产力的相关政策制定提供了科学依据。This study focuses on the 30 provinces in China from 2013 to 2022, constructing a provincial-level index system of new quality productivity and calculating it based on the entropy method. Adopting the perspective of dividing the country into eastern, central, and western regions, this study utilizes kernel density estimation method and Dagum Gini coefficient decomposition method to analyze the regional differences and spatial-temporal evolution of new quality productivity in Chinese provincial areas. The results indicate that the new quality productivity level in the eastern region is significantly higher than that in other regions;the overall difference in new quality productivity level in the country is expanding, mainly caused by inter-regional differences. The northeastern region shows significant polarization in new quality productivity level, highlighting spatial polarization phenomena. These findings provide a scientific basis for policy-making.
混合整数最优控制问题(Mixed-Integer Optimal Control Problem, MIOCP)因其包含整数变量而更加复杂,但更能贴近实际应用需求。Sager等人(Math. Program. Vol. 133, 2012)提出了一种松弛–取整策略,将MICOP问题凸松弛为经典最优控制,再...
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混合整数最优控制问题(Mixed-Integer Optimal Control Problem, MIOCP)因其包含整数变量而更加复杂,但更能贴近实际应用需求。Sager等人(Math. Program. Vol. 133, 2012)提出了一种松弛–取整策略,将MICOP问题凸松弛为经典最优控制,再对凸松弛的最优解进行取整近似(Sum Up Rounding, SUR),得到原问题的近似解,并证明了近似误差为时间步长的同阶无穷小。然而,该近似误差估计的同阶无穷小的系数项是时间区间总长度的指数函数,当控制问题的时间区间较大时,这个误差可能会非常大。针对这一问题,本文对SUR策略进行改进,提出一个新的控制取整策略,证明了新控制策略的收敛性,并通过数值例子验证了本文的策略显著提高了MIOCP的求解精度。Mixed-Integer Optimal Control Problems (MIOCP) are more complex due to the inclusion of integer variables but are more aligned with practical application needs. Sager et al. (Math. Program. Vol. 133, 2012) proposed a relaxation-rounding strategy that convexly relaxes the MIOCP to a classical optimal control problem and then approximates the optimal solution of the convex relaxation (Sum Up Rounding, SUR) to obtain an approximate solution to the original problem, proving that the approximation error is of the same order as the infinitesimal of the time step. However, the coefficient of this infinitesimal error estimate is an exponential function of the total length of the time interval, and when the time interval of the control problem is large, this error can be very significant. To address this issue, this paper improves the SUR strategy and proposes a new control rounding strategy, proving the convergence of the new control strategy and verifying through numerical examples that it significantly improves the solution accuracy of MIOCP.
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