作为一种传统民俗活动，“板凳龙”对我国具有重要的文化意义。本文旨在研究“板凳龙”舞龙队在不同含螺线 轨迹下的运动状态，并对轨迹和速度进行优化。运动状态主要包括位置和速度。

针对问题一：本文首先研究舞龙队在等距螺旋线上的运动状态。本文将对整个舞龙队的求解分为时间和空间两个 维度，在时间维度上，基于等距螺旋线轨迹建立运动学微分方程，通过基于牛顿迭代法的fsolve方法求解龙头在不同时 刻的位置；在空间维度上，基于螺线轨迹和板凳几何长度建立位置方程组，从龙头逐步迭代计算龙身龙尾把手位置，并 根据角速度约束得到速度。本文通过Python实现上述算法，得到0至300s舞龙队各时刻把手的运动状态。

针对问题二：考虑碰撞问题，本文研究舞龙队盘入终止时刻及其运动状态。基于问题一模型，本文对板凳之间的 几何位置关系建立板凳碰撞模型，通过对板凳矩形空间进行检验，判断板凳是否碰撞。本文通过Matlab对舞龙队盘入 过程进行模拟计算，得出盘入终止时刻为432.877981s，并得到此时刻下舞龙队各把手的运动状态。

针对问题三：引入掉头空间，本文基于前两个模型对螺距进行优化。根据是否讨论空间高度的两种情况，本文得出 不同结果。根据题面图示的各节板凳高度均不同的理想状态下，通过板凳与人碰撞时的几何关系，求解出最小螺距为0.280618m； 若忽略高度，板凳在平面投影重叠即视为碰撞，本文通过二分寻优法对螺距搜索，基于GJK算法，通过计算原点与闵可夫 斯基差集的的关系对板凳碰撞进行检测，模拟运动过程得到最小螺距为0.469873m。

针对问题四：考虑更复杂的可掉头螺线-圆弧轨迹，本文研究舞龙队在一段时间上的运动状态。对于引入的连接处 相切的S型曲线，舞龙队进入掉头空间的速度方向已经固定，无论两圆弧半径如何调整，根据几何关系S型曲线的弧长都 恒定不变，因此掉头曲线长度无法变化。在两圆弧半径关系约束下，可求出两圆弧的轨迹方程，与两螺线轨迹结合带入 问题一模型中进行分段求解，并根据运动学规律，得到-100s至100s舞龙队各时刻把手的运动状态。

针对问题五：本文在问题四基础上对龙头速度进行优化，以满足对各把手速度不超过2m/s的约束条件。基于遗传 算法，本文逐步迭代优化不同龙头速度，对整个舞龙队在可调头螺线-圆弧轨迹运动过程进行模拟，求解得到龙头的最大 行进速度0.544437m/s。

最后，经检验，模型准确性高，并评价模型的优缺点和推广改进。

Personal evaluation: The mathematical model established in this contest is quite complex, resulting in high programming difficulty. Our assumptions and modeling of speed in the contest were significantly different from the standards and reality, resulting in significant errors in the calculation of key variables and thus failing to win the award.

个人评价：本次比赛所建立的数学模型较为复杂，因此导致编程难度很高。在赛程中我们对速度的假设和建模与 标准和实际相差较大，使得对关键变量的计算出现较大错误，因而未能获奖。

As a species that can change sex ratios based on RA(Resource Availability), lampreys play an important role in ecosystems. To take a research of the effects of VSR(Variable Sex Ratios) of lampreys on the ecosystem and itself, this paper develops a mathematical model, providing insights into the resulting interactions in an ecosystem.

For tasks 1, we establish a model of the interaction between lampreys and larger ecosystems. Firstly, according to the actual growth mechanism of the lamprey, we divide the lamprey population into three parts: larvae, females, and males. Based on Lotka-Volterra equations, we establish a model about the population of three parts, where larvae grow by consuming humus and adults rely on parasitism to the lower trophic level. We measure the probability of larvae growing into females and males by introducing the Sigmoid function. To consider a larger ecosystem, we add the upper and lower trophic levels of lampreys and define RA based on the population’s own capabilities. We implemented the model in Python and simulated the results of various resource situations including the absence of natural enemies by Monte Carlo method. Comparing VSR and fixed probability, we find that the impact of VSR varies under different resource conditions.

For task 2, we proposed the advantages and disadvantages of the lamprey population based on the experimental results of the model. The advantages lie in its ability to enhance adaptability to environmental changes, optimize resource utilization, and balance dynamic ecosystems; the disadvantages are that it may lead to population flooding, intensified resource competition, and risk of ecosystem instability.

For task 3, we analyze the impact of VSR on ecosystem stability. We analyze the impact of VSR on structural stability through biodiversity and total biomass; Based on the sensitivity analysis method, we quantify the stability of resistance and resilience by the changing amplitude of population, and then analyze the impact of VSR on functional stability. Experimental results have proven that VSR can enhance the stability of ecosystems in most cases.

For task 4, we study the impact of VSR on others in the ecosystem. For parasites of the same trophic level, VSR will increase their disadvantages in competition with lampreys; For further trophic levels, we analyzed the impact of VSR based on ecosystem stability; For humans, we quantified the total economic value brought by fishing and found through the comparison of calculation results that VSR always lead to greater benefits for humans.

Furthermore, we analyze the advantages and disadvantages of the model and discuss its future development. Finally, we summarize the conclusions of this paper and complete all tasks.

Personal evaluation: Compared to previous contests, this contest is quite relaxed, and the innovative points proposed in this work are rich and novel, with rich and comprehensive content, thanks to the engineering problem-solving methods and processes. However, this work did not receive the expected award, which may be due to the subjective evaluation of method innovation and performance forms.

个人评价：相较于以往的比赛而言，这场比赛是相当轻松的，而且本篇工作所提出的创新点丰富且新颖， 内容充实而全面，这得益于工程化解决问题的方法和流程。但是，这篇工作并未取得预期的奖项， 这可能是因为对方法创新和表现形式的评价具有较大的主观性。

本文研究了定日镜场的输出热功率和优化布局。通过对逐个定日镜的余弦效率、集 热器截断效率和阴影遮挡效率进行计算，得到每个定日镜的光学效率，进而总体计算出 整个定日镜场的输出热功率，进而在此基础上，我们运用模拟退火算法和GWO 算法对 单位镜面面积年平均输出热功率进行最大化，研究了在各个定日镜尺寸、高度相同和不 同两种情况下定日镜的尺寸和高度、定日镜的数目和位置、吸收塔的位置等决策变量的 影响，得到了定日镜场的最优化布局。

针对问题1：首先，我们根据给定定日镜场的各项参数，逐步解析影响定日镜场光 学效率的因素并依次建立数学模型，通过坐标系的转换，基于光线反射定律用几何矢量 法计算余弦效率的公式；对于集热器截断效率，我们参考HFCAL 模型，利用圆高斯分 布来表示吸热器表面的热流密度函数，并进行二重积分来进行计算；我们选用更简便快 捷的几何投影法计算阴影遮挡效率，通过Arsl 等的一种基于Sassi 中心点投影法的方法 来逐级计算阴影因素和遮挡因素，并以面积比的形式来表示。我们应用Python 实现并 整合了上述过程，对整个定日镜场的年平均光学效率等参数进行总体计算，得到年平均 输出热功率为41414.6512MW，单位面积镜面年平均输出热功率为0.659259kW/m2。

针对问题2：在每个定日镜尺寸、高度相同的情况下，限定额定年平均输出热功率 为60MW，对于单位镜面面积年平均输出热功率这一优化目标，我们将所有决策变量 分为宏观和微观两类。我们先考虑微观层面的决策变量，研究每个定日镜的尺寸和高度 对目标的影响，采用模拟退火算法，初始化原始解并逐步迭代计算新解的目标函数值 并以一定的保留概率对解进行不断更新，得到全局最优且精度较高的最佳定日镜长度 Lx = 2.0001m，宽度Ly = 2.5423m，高度z = 2.1273m。再考虑宏观层面的决策变量， 定日镜的数目和位置、吸收塔的位置通过影响整个定日镜场的布局规则来影响目标函 数，我们基于改进的GWO 算法，通过集中效率这一函数逐步迭代定日镜的位置，进而 得到整个定日镜场的整体最优布局，单位面积镜面年平均输出热功率为0.6352kW/m2。

针对问题3：在问题2 的基础上取消每个定日镜尺寸、高度相同的限制，考虑微观 效率和宏观布局之间影响较小，我们在问题2 已经得到的高效布局的基础上，对整个定 日镜场的每个定日镜考虑最优尺寸和高度。经过对定日镜尺寸和高度的灵敏度分析，我 们发现尺寸主要通过集热器截断效率来影响目标，高度主要通过余弦效率来影响目标。 进而，我们分别单独研究两种决策变量与相应效率的函数关系，通过改进的L-BFGS 算 法计算出每个定日镜的最优尺寸和高度，得到新的定日镜场的最优布局，单位面积镜面 年平均输出热功率为0.6549kW/m2。

Personal evaluation: The workload of this game is quite substantial. Most problems cannot be directly solved through the basic methods of mathematical modeling. It is necessary to extensively consult relevant literature in the field, seek the most suitable solution, and ensure a balance between time and accuracy. Thanks to the engineering solution approach, our team did not waste any time in the past 3 days and successfully completed all tasks. However, due to the inaccurate results, we unfortunately did not win the national award.

个人评价：这场比赛的工作量相当庞大。大部分问题无法通过数学建模的基本方法直接解决， 必须广泛查阅相关领域的文献，寻求最合适的解决方案，并要保证时间和精度的平衡。得益于工程化的解决思路， 我们团队没有浪费3天中的任何时间，成功完成了全部任务，但由于结果不够准确，遗憾未能获得国奖。

本文以有机污染物多氯联苯（PCBs）为例，基于Lotka-Volterra 种间捕食模型和脂 水系统中有机化合物的分配系数及其与鱼类生物富集因子的关系，研究了污染环境中的 水生态系统在不同污染物浓度、不同食物链（网）条件下，不同物种的种群数量变化趋 势，并得到了生态系统崩溃的PCBs 临界初始浓度，最后基于此提出一些有效的污染治 理的建议。

针对问题一，考虑在不同污染物浓度下的三个物种捕食食物链，我们基于Lotka- Volterra 种间捕食模型建立物种数量变化模型。为了分析污染物浓度对食物链的影响， 我们基于拉乌尔定律和弗洛里－哈金斯溶液理论得到的脂水系统中有机化合物的分配 系数及其与鱼类生物富集因子的关系，对生物对污染物的富集过程建立动力学方程，并 通过四五阶龙格-库塔方法，基于文献中的初始条件数据，对基于微分方程的模型求解。 进而，我们假定污染物通过影响物种的自然增长率来影响种群变化过程，建立污染物与 种群数量变化的关系。为了对模型结果进行可视化，我们采用元胞自动机模型对系统的 变化过程进行模拟，直观展示了系统在真实情况下各个物种数量的随机变化结果，与理 论模型结果相吻合。

针对问题二，考虑引入入侵物种，我们在问题一模型的基础上，增加了新物种并引 入了竞争关系，将食物链扩充为食物网，修改了相关变量和参数，采用同样的方法和元 胞自动机模型计算种群数量变化并对进行可视化。

针对问题三，考虑第三营养级瞬间灭绝，我们在问题一模型的基础上，去除了第三 营养级物种，分析在给定污染条件下只有两个物种的捕食食物链的各个种群数量变化 情况，继续采用同样的方法和元胞自动机模型求解，并通过不断调整PCBs 初始浓度参 数，得到了生态系统崩溃的污染物浓度阈值为35mmol/L。

针对问题四，我们提出了一些有效的污染治理的建议。根据得出的研究结果和查阅 文献得到的有关知识，我们从减少排放、物种保护、化学降解、生物降解和集中处理五 个方面，提出了五条生态治理的建议。

最后，我们对本模型进行了分析与检验。我们对不同种群的自然增长率和捕食转化 率进行了模型灵敏度分析，并对微分方程模型和元胞自动机模型的结果进行了误差检 验，得到的误差很小，证明模型的可靠性强。总而言之，该模型适用性强，结果合理有 效，具有很大的社会价值和生态价值。

Personal evaluation: This contest is the starting point of Engineering of Mathematical Contest in Modeling(EMCM). Our team has gradually solved mathematical modeling problems that tend to be homogenized with a clear and explicit approach, efficiently completing the entire process of solving mathematical modeling problems, and forming a standardized set of experience in completing mathematical modeling contests. Therefore, even in the case of the spread of influenza, our team still successfully completed the task and won the prize.

个人评价：这次比赛是数学建模竞赛工程(EMCM)的起点。我们团队以清晰明确的思路逐步解决趋于同质化的数学建模问题， 高效地完成了解决数学建模问题的全流程，形成了一套标准化的完成数学建模竞赛的经验。因此，即使是在流行性感冒蔓延的情况下， 我们团队依然顺利完成任务并获奖。

Nowadays, hosting the Olympics has become a thorny issue. Some countries which have hosted the Olympic Games before are facing a variety of short- and long-term negative impacts. However, there are still many countries obtained good returns after hosting the Olympics. What we need to do is measure the impact of hosting the Olympics from different perspectives and propose potential strategies for improvement. Based on this, we have established two models to achieve the above goals. By utilizing this model, we can provide recommendations and solutions for future Olympic Games.

In Model 1, We are expected to analyze the impact of hosting the Olympics from different perspectives. To achieve it, we collected data on various aspects of the 11 Olympic Games since the 21st century which include economic, land use, human satisfaction, travel, opportunity for future improvements, host city/nation prestige and ecological environment. Then we conducted a validation of the data and got the KMO value and p-value of the Bartlett’s test, which proved the suitability and independence of the data. After that we used factor analysis method to synthesize 15 indicators and obtained a weight matrix for six factors. Based on this, we calculated the final score for each Olympic Games and defined it as degree of suitability for hosting the Olympic. Finally, to specifically describe the effectiveness of hosting the Olympics, we classified the effects of the Olympics into five categories by K-means cluster, which are named excellent, good, average, fair and poor.

In Model 2, We want to analyze the factors that affect the effectiveness of the Olympic Games and calculate their importance, and then use this model to evaluate the expected Olympic performance of other countries around the world. Therefore, we selected 14 indicators from 4 aspects and distinguished between winter and summer. They are climate, economic, traffic and human population. For intermediate indicators such as temperature and population growth rate, we use the fuzzy comprehensive evaluation method to process these data uniformly. After further data processing, we determined the weight of each indicator through the entropy weight method, to analyze the degree of influence of different factors on the effectiveness of hosting the Olympics. Then we expand our perspective to the world, using the same data processing methods to obtain the same indicators from more than 200 countries around the world. Based on the existing data from 11 countries, we will perform Gaussian naive Bayesian classification on them to analyse the effect of hosting the Olympic Games for each country on Earth.

Based on these two models, we proposed potential strategies and policies for improving the future of the Olympics, and wrote a memorandum which summary our key points and timeline to the IOC. Hopefully the Olympics can be better and better hosted in the future under our recommendations.

Personal evaluation: This contest is my first attempt to solve problems related to big data analysis and comprehensive evaluation. The workload of this game is quite large, coupled with some scheduling issues, resulting in incomplete description of the entire solution, unclear presentation of results, and relatively limited textual content. This may be a significant reason for not winning the award. However, this work still has sufficient technical skills and workload, which has allowed me to accumulate a lot of experience.

个人评价：这场比赛是我首次尝试解决大数据分析和综合评价类问题。这场比赛的工作量相当庞大，再加上时间安排出现了一些问题， 导致对于整个解决方案的描述不够完整，结果呈现不够清晰，文字内容相对较少。这可能是导致未能获奖的重要原因。 但是这项工作仍是有足够技术水平和工作量的，这也让我积累了很多经验。

Nowadays, environmental protection has become a hot topic in the world. It is of great significance for environmental protection to explore the impact of climate and drought on different plant communities. In this paper, we propose a grassland mechanism model based on ordinary differential equations to analyze the impact of different factors. Our model includes three modules and an assessment method for the environment.

To begin with, we divided plant communities into three types which are tree, bush and herb. In order to research the interaction between plant populations, we developed the Module I based on logistic model named Three Types of Plant Community Interaction Model. Due to the harsh living environment leading to intense competition for survival of plants, we mainly took into account their interspecific competition and drought. Based on it, we obtained the quantitative change curves of three types of plant populations under different conditions.

Next, we’d like to explore the relationship between biomass and population change within three types of plants. Kinds of factors directly affecst the biomass of plant communities, but there is no direct relationship between them and the population. In order to express the relationship between population and biomass, we developed the Module II of our model named Community Plant Species Number Change Model based on probability and logit transformation. In this module, we focus on the influence of environmental capacity and suitable species number.

Then, in consideration of various irregular weather cycles, we defined Degree of Drought to describe precipitation and drought area in Module III of our model named Drought Degree and Precipitation Model. In the definition, we also consider the water demand of plants. Weather conditions affect plant communities through drought degree.

Later, considering the impact of other factors such as pollution and habitat reduction, we revised the Module I of our model and added the impact of pollution index. The change of habitat size can directly affect the plant communities through the area. It is a part of environmental capacity.

Furthermore, in order to evaluate the environment well, we defined Environmental Assessment Indicators based on biodiversity and drought degree in our environmental assessment model. In this definition, we fully considered the problems of species scarcity and species flooding.

Finally we implemented our model with MATLAB and made qualitative and quantitative analysis of the whole problem. We obtained the results and the curve of change with time under the influence of different factors. As a conclusion, we propose five measures to give our contribution to long-term viability of a plant community and the larger environment. Hopefully our model can be useful for environmental protection.

Personal evaluation: This contest is of great significance to us, as it is the first time our team has won an award in a mathematical modeling contest. The work of this contest is also the basis for our subsequent papers published in ordinary journals. This proves that our mathematical modeling knowledge and ability have been recognized. The workload this time is not large, but it also includes some small innovative points, which have a certain practical significance.

个人评价：这场比赛对于我们有着重要的意义，因为这是我们团队首次取得数学建模竞赛的奖项。这场比赛的工作也是 我们后续在普刊上发表的论文的基础。这证明了我们的数学建模知识和能力得到了认可。这次的工作量并不大，但是也包含了 一些小小的创新点，具有一定的现实作用。

本文针对包含PTO 系统的波浪能装置的研究，对波浪能在垂荡和纵摇情况下PTO 系 统的输出功率建立微分方程模型，分析在直线阻尼器和旋转阻尼器的阻尼系数影响下输 出功率的变化，并计算出最大输出功率。

针对问题一，在仅做垂荡运动的情况下，首先对浮子和振子进行受力分析，根据牛 顿第二定律对浮子和振子建立微分方程模型，在直线阻尼器的阻尼系数固定和与浮子和 振子的相对速度的幂成正比两种条件下，分别对浮子和振子的速度和位移建立模型，并 通过计算机程序对浮子和振子在时间轴上运动状态的变化进行模拟，得到不同时间、不 同条件下浮子和振子的速度和位移。

针对问题二，在问题一的基础上，根据功率公式对两种条件下的输出功率建立模型， 以阻尼系数和相对速度的幂指数为参数，探究两参数在输出功率模型中的相关关系，然 后分别分析两参数对输出功率的影响，进而进行汇总，计算出最大输出功率要求下对应 的阻尼系数和幂指数，得到最大输出功率。

针对问题三，在垂荡运动和纵摇运动共同作用的影响下，首先对浮子和振子进行受 力分析，根据牛顿第二定律和刚体的定轴转动定律对浮子和振子建立微分方程模型，研 究浮子和振子的运动状态参数关于时间的变化情况，对位移、速度、角位移和角速度建 立模型，然后通过计算机程序对浮子和振子的运动状态进行模拟，得到不同时间下浮子 和振子的运动状态参数。

针对问题四，在问题三的基础上，根据功率公式分别对直线阻尼器功率和旋转阻尼 器功率建立模型，分析两种阻尼器的阻尼系数对总输出功率的影响方式，并计算在最大 总输出功率要求下两种阻尼器的阻尼系数，进而得到PTO 系统总最大输出功率。

Personal evaluation: This contest was completed by our team during the preparation and participation period for the delayed final exam due to the epidemic. It was also the first time our team has used advanced mathematical knowledge and modeling methods to solve mechanism modeling problems. It is a comprehensive reflection of our team's learning and application ability, time management ability, and stress resistance ability. This also laid the foundation for my strongest area of expertise in mathematical modeling: mechanism modeling.

个人评价：这场比赛是我们团队在准备和参加因为疫情延期的期末考试的时间段里进行的， 也是我们团队首次利用高等数学知识和建模方法解决机理模型问题，是我们团队学习和应用能力、 时间管理能力、抗压能力等的综合体现。这也为我在数学建模领域最擅长的方向——机理模型方向奠定了基础。

As the earth's resources are increasingly exhausted, people have to explore outer space for more resources. Asteroid mining has become an important means for people to obtain resources. In the future, asteroid mining will have a great impact on human society, especially on global equity .In this paper, we address the following questions: how to assess global equity, what is the future vision of asteroid mining, how will asteroid mining affect global equity, and what policies should be adopted to make asteroid mining benefit all of humanity.

For task 1, we defined global equity and built a model to assess global equity based on the definition. We first identified a system of indicators with GDP and National Financial Investment in Scientific Research (NFSR) as indicators, and then used the entropy weighting method (EMW) to calculate the weights of each indicator to determine the National Development Level (NDL), and the standard deviation of NDL is used to assess the global equity level.

For task 2, we conceptualized the future picture of asteroid mining from the perspective of countries and companies respectively, including which resources to be extracted, by what specific means, and who will provide the funding and so on. Then we analyse the influence of asteroid mining on global equity according to previous model. In a rigorous proof we obviously find that NDL has a positive correlation with GDP and NFSR. So we model to show the influence of asteroid mining on global equity.

For task 3, in order to analyse the impact of condition changes on global equity, we develop Multivariate Analysis that Combines Multi Objects with Environment to conduct a comprehensive analysis from different angles. We focus on there main objects which are miner, market and tax and get a specific impact on global equity combined with the environment. Then we extend our thinking to see the asteroid mining from the universal angle of view and understand the meaning of it.

For task 4, we propose policy recommendations for such impacts to make asteroid mining truly beneficial to all of humanity, and analyze the strengths and weaknesses of our modeling process.

Personal evaluation: For my team, who have little modeling knowledge, most of the work in this contest is based on elementary mathematical knowledge and logical reasoning statements. Thanks to the openness of the ICM, we successfully completed the contest without the need for extensive and advanced workload. This is an experiential exploration and a small attempt.

个人评价：对于基本没有建模知识的我们团队而言，本次比赛的大部分工作都是 基于初等数学知识和逻辑推理陈述的。得益于美赛的开放性，我们不需要很丰富和 高级的工作量也成功完成了比赛。这是一次经验探索和小小的尝试。

1. Color 颜色 : Gold - Advanced (National) 金色-高级(国家级) Blue - Intermediate (Provincial) 蓝色-中级(省级) Green - Primary 绿色-初级

2. Pre-term Symbols 题前符号 : The first symbol represents the actual competition situation, and the second symbol represents the competition requirements 第一个符号代表实际参赛情况，第二个符号代表比赛要求

3. Symbol Meaning 符号含义 : ~ solo 单人参加 # team with three 三人组队 $ team with multi 多人团队