美赛优秀论文

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美赛论文

注：LEO 低地球轨道MEO中地球轨道GeO 同步卫星轨道risk-profit 风险利润率fixed-profit rate 固定利润率提出一个合理的商业计划，可以使我们抓住商业机会，我们建立四个模型来分析三个替代方案（水射流，激光，卫星）和组合，然后确定是否存在一个经济上有吸引力的机会，从而设计了四种模型分析空间碎片的风险、成本、利润和预测。

首先，我们建立了利润模型基于净现值（NPV）模型，并确定三个最佳组合的替代品与定性分析：1）考虑了三个备选方案的组合时，碎片的量是巨大的；2）考虑了水射流和激光的结合，认为碎片的大小不太大；3）把卫星和激光的结合当尺寸的这些碎片足够大。

其次，建立风险定性分析模型，对影响因素进行分析在每一种替代的风险，并得出一个结论，风险将逐渐下降直到达到一个稳定的数字。

在定量分析技术投入和对设备的影响投资中，我们建立了双重技术的学习曲线模型,找到成本的变化规律与时间的变化。

然后，我们开发的差分方程预测模型预测的量在未来的四年内每年发射的飞机。

结合结果我们从预测中，我们可以确定最佳的去除选择。

最后，分析了模型的灵敏度，讨论了模型的优势和我们的模型的弱点，目前的非技术性的信，指出了未来工作。

目录1，简介1.1问题的背景1.2可行方案1.3一般的假设1.4我们的思想的轮廓2，我们的模型2.1 时间---利润模型2.1.1 模型的符号2.1.2 模型建立2.1.3 结果与分析2.2 . 差分方程的预测模型2.2.1 模型建立2.2.2 结果分析2.3 双因子技术-学习曲线模型2.3.1 模型背景知识2.3.2 模型的符号2.3.3 模型建立2.3.4 结果分析2.4风险定性分析模型2.4.1 模型背景2.4.2 模型建立2.4.3 结果与分析3．在我们模型的灵敏度分析3.1 差分方程的预测模型。

3.1.1 稳定性分析3.1.2 敏感性分析3.2 双因子技术学习曲线模型3.2.1 稳定性分析3.2.2 敏感性分析4 优点和缺点查分方程预测模型优点缺点双因子技术学习曲线模型优点缺点时间---利润模型优点缺点5..结论6..未来的工作7．参考双赢模式：拯救地球，抓住机遇1..简介问题的背景空间曾经很干净整洁。

美国大学生数学建模竞赛优秀论文

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number7018Problem ChosencFor office use onlyF1________________F2________________F3________________F4________________ SummaryThe article is aimed to research the potential impact of the marine garbage debris on marine ecosystem and human beings，and how we can deal with the substantial problems caused by the aggregation of marine wastes.In task one，we give a definition of the potential long-term and short-term impact of marine plastic garbage. Regard the toxin concentration effect caused by marine garbage as long-term impact and to track and monitor it. We etablish the composite indicator model on density of plastic toxin，and the content of toxin absorbed by plastic fragment in the ocean to express the impact of marine garbage on ecosystem. Take Japan sea as example to examine our model.In ask two, we designe an algorithm, using the density value of marine plastic of each year in discrete measure point given by reference，and we plot plastic density of the whole area in varies locations. Based on the changes in marine plastic density in different years， we determine generally that the center of the plastic vortex is East—West140°W—150°W, South—North30°N—40°N. According to our algorithm, we can monitor a sea area reasonably only by regular observation of part of the specified measuring pointIn task three，we classify the plastic into three types，which is surface layer plastic，deep layer plastic and interlayer between the two. Then we analysis the the degradation mechanism of plastic in each layer. Finally，we get the reason why those plastic fragments come to a similar size.In task four， we classify the source of the marine plastic into three types，the land accounting for 80%，fishing gears accounting for 10%，boating accounting for 10%，and estimate the optimization model according to the duel-target principle of emissions reduction and management. Finally, we arrive at a more reasonable optimization strategy.In task five，we first analyze the mechanism of the formation of the Pacific ocean trash vortex， and thus conclude that the marine garbage swirl will also emerge in south Pacific，south Atlantic and the India ocean. According to the Concentration of diffusion theory, we establish the differential prediction model of the future marine garbage density，and predict the density of the garbage in south Atlantic ocean. Then we get the stable density in eight measuring point .In task six, we get the results by the data of the annual national consumption ofpolypropylene plastic packaging and the data fitting method, and predict the environmental benefit generated by the prohibition of polypropylene take-away food packaging in the next decade. By means of this model and our prediction，each nation will reduce releasing 1.31 million tons of plastic garbage in next decade.Finally, we submit a report to expediction leader，summarize our work and make some feasible suggestions to the policy- makers.Task 1:Definition:●Potential short-term effects of the plastic: the hazardeffects will be shown in the short term.●Potential long-term effects of the plastic: thepotential effects, of which hazards are great, willappear after a long time.The short- and long-term effects of the plastic on the ocean environment:In our definition, the short-term and long-term effects of the plastic on the ocean environment are as follows.Short-term effects:1）The plastic is eaten by marine animals or birds.2） Animals are wrapped by plastics, such as fishing nets, which hurt or even kill them.3）Deaden the way of the passing vessels.Long-term effects:1）Enrichment of toxins through the food chain: the waste plastic in the ocean has no natural degradation in theshort-term, which will first be broken down into tinyfragments through the role of light, waves,micro-organisms, while the molecular structure has notchanged. These "plastic sands", easy to be eaten byplankton, fish and other, are Seemingly very similar tomarine life’s food,causing the enrichment and delivery of toxins.2）Accelerate the greenhouse effect: after a long-term accumulation and pollution of plastics, the waterbecame turbid, which will seriously affect the marineplants (such as phytoplankton and algae) inphotosynthesis. A large number of plankton’s deathswould also lower the ability of the ocean to absorbcarbon dioxide, intensifying the greenhouse effect tosome extent.To monitor the impact of plastic rubbish on the marine ecosystem:According to the relevant literature, we know that plastic resin pellets accumulate toxic chemicals , such as PCBs、DDE , and nonylphenols , and may serve as a transport medium and soure of toxins to marine organisms that ingest them[]2. As it is difficult for the plastic garbage in the ocean to complete degradation in the short term, the plastic resin pellets in the water will increase over time and thus absorb more toxins, resulting in the enrichment of toxins and causing serious impact on the marine ecosystem.Therefore, we track the monitoring of the concentration of PCBs, DDE, and nonylphenols containing in the plastic resin pellets in the sea water, as an indicator to compare the extent of pollution in different regions of the sea, thus reflecting the impact of plastic rubbish on ecosystem.To establish pollution index evaluation model: For purposes of comparison, we unify the concentration indexes of PCBs, DDE, and nonylphenols in a comprehensive index.Preparations:1）Data Standardization2）Determination of the index weightBecause Japan has done researches on the contents of PCBs,DDE, and nonylphenols in the plastic resin pellets, we illustrate the survey conducted in Japanese waters by the University of Tokyo between 1997 and 1998.To standardize the concentration indexes of PCBs, DDE,and nonylphenols. We assume Kasai Sesside Park, KeihinCanal, Kugenuma Beach, Shioda Beach in the survey arethe first, second, third, fourth region; PCBs, DDE, andnonylphenols are the first, second, third indicators.Then to establish the standardized model:j j jij ij V V V V V min max min --= （1,2,3,4;1,2,3i j ==）wherej V max is the maximum of the measurement of j indicator in the four regions.j V min is the minimum of the measurement of j indicatorstandardized value of j indicator in i region.According to the literature [2], Japanese observationaldata is shown in Table 1.Table 1. PCBs, DDE, and, nonylphenols Contents in Marine PolypropyleneTable 1 Using the established standardized model to standardize, we have Table 2.In Table 2，the three indicators of Shioda Beach area are all 0, because the contents of PCBs, DDE, and nonylphenols in Polypropylene Plastic Resin Pellets in this area are the least, while 0 only relatively represents the smallest. Similarly, 1 indicates that in some area the value of a indicator is the largest.To determine the index weight of PCBs, DDE, and nonylphenolsWe use Analytic Hierarchy Process (AHP) to determine the weight of the three indicators in the general pollution indicator. AHP is an effective method which transforms semi-qualitative and semi-quantitative problems into quantitative calculation. It uses ideas of analysis and synthesis in decision-making, ideally suited for multi-index comprehensive evaluation.Hierarchy are shown in figure 1.Fig.1 Hierarchy of index factorsThen we determine the weight of each concentrationindicator in the generall pollution indicator, and the process are described as follows:To analyze the role of each concentration indicator, we haveestablished a matrix P to study the relative proportion.⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=111323123211312P P P P P P P Where mn P represents the relative importance of theconcentration indicators m B and n B . Usually we use 1，2，…，9 and their reciprocals to represent different importance. The greater the number is, the more important it is. Similarly, the relative importance of m B and n B is mn P /1（3,2,1,=n m ）.Suppose the maximum eigenvalue of P is m ax λ, then theconsistency index is1max --=n nCI λThe average consistency index is RI , then the consistencyratio isRICI CR = For the matrix P of 3≥n , if 1.0<CR the consistency isthougt to be better, of which eigenvector can be used as the weight vector.We get the comparison matrix accoding to the harmful levelsof PCBs, DDE, and nonylphenols and the requirments ofEPA on the maximum concentration of the three toxins inseawater as follows:⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=165416131431P We get the maximum eigenvalue of P by MATLAB calculation0012.3max =λand the corresponding eigenvector of it is()2393.02975.09243.0，，=W1.0042.012.1047.0<===RI CI CR Therefore,we determine the degree of inconsistency formatrix P within the permissible range. With the eigenvectors of p as weights vector, we get thefinal weight vector by normalization ()1638.02036.06326.0'，，=W . Defining the overall target of pollution for the No i oceanis i Q , among other things the standardized value of threeindicators for the No i ocean is ()321,,i i i i V V V V = and the weightvector is 'W ,Then we form the model for the overall target of marine pollution assessment, （3,2,1=i ）By the model above, we obtained the Value of the totalpollution index for four regions in Japanese ocean in Table 3T B W Q '=In Table3, the value of the total pollution index is the hightest that means the concentration of toxins in Polypropylene Plastic Resin Pellets is the hightest, whereas the value of the total pollution index in Shioda Beach is the lowest(we point up 0 is only a relative value that’s not in the name of free of plastics pollution)Getting through the assessment method above, we can monitor the concentration of PCBs, DDE and nonylphenols in the plastic debris for the sake of reflecting the influence to ocean ecosystem.The highter the the concentration of toxins,the bigger influence of the marine organism which lead to the inrichment of food chain is more and more dramatic.Above all, the variation of toxins’ concentration simultaneously reflects the distribution and time-varying of marine litter. We can predict the future development of marine litter by regularly monitoring the content of these substances, to provide data for the sea expedition of the detection of marine litter and reference for government departments to make the policies for ocean governance.Task 2:In the North Pacific, the clockwise flow formed a never-ending maelstrom which rotates the plastic garbage. Over the years, the subtropical eddy current in North Pacific gathered together the garbage from the coast or the fleet, entrapped them in the whirlpool, and brought them to the center under the action of the centripetal force, forming an area of 3.43 million square kilometers (more than one-third of Europe) .As time goes by, the garbage in the whirlpool has the trend of increasing year by year in terms of breadth, density, and distribution. In order to clearly describe the variability of the increases over time and space, according to “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999—2008”, we analyze the data, exclude them with a great dispersion, and retain them with concentrated distribution, while the longitude values of the garbage locations in sampled regions of years serve as the x-coordinate value of a three-dimensional coordinates, latitude values as the y-coordinate value, the Plastic Count per cubic Meter of water of the position as the z-coordinate value. Further, we establish an irregular grid in the yx plane according to obtained data, and draw a grid line through all the data points. Using the inverse distance squared method with a factor, which can not only estimate the Plastic Count per cubic Meter of water of any position, but also calculate the trends of the Plastic Counts per cubic Meter of water between two original data points, we can obtain the unknown grid points approximately. When the data of all the irregular grid points are known (or approximately known, or obtained from the original data), we can draw the three-dimensional image with the Matlab software, which can fully reflect the variability of the increases in the garbage density over time and space.Preparations:First, to determine the coordinates of each year’s sampled garbage.The distribution range of garbage is about the East - West 120W-170W, South - North 18N-41N shown in the “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”, we divide a square in the picture into 100 grids in Figure (1) as follows:According to the position of the grid where the measuring point’s center is, we can identify the latitude and longitude for each point, which respectively serve as the x- and y- coordinate value of the three-dimensional coordinates.To determine the Plastic Count per cubic Meter of water. As the “Plastic Count per cubic Meter of water” provided by “Count Densities of P lastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”are 5 density interval, to identify the exact values of the garbage density of one year’s different measuring points, we assume that the density is a random variable which obeys uniform distribution in each interval.Uniform distribution can be described as below:()⎪⎩⎪⎨⎧-=01a b x f ()others b a x ,∈We use the uniform function in Matlab to generatecontinuous uniformly distributed random numbers in each interval, which approximately serve as the exact values of the garbage density andz-coordinate values of the three-dimensional coordinates of the year’s measuring points.Assumptions(1)The data we get is accurate and reasonable.(2)Plastic Count per cubic Meter of waterIn the oceanarea isa continuous change.(3)Density of the plastic in the gyre is a variable by region.Density of the plastic in the gyre and its surrounding area is interdependent , However, this dependence decreases with increasing distance . For our discussion issue, Each data point influences the point of each unknown around and the point of each unknown around is influenced by a given data point. The nearer a given data point from the unknown point, the larger the role.Establishing the modelFor the method described by the previous,we serve the distributions of garbage density in the “Count Pensities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”as coordinates ()z y,, As Table 1:x,Through analysis and comparison, We excluded a number of data which has very large dispersion and retained the data that is under the more concentrated the distribution which, can be seen on Table 2.In this way, this is conducive for us to get more accurate density distribution map.Then we have a segmentation that is according to the arrangement of the composition of X direction and Y direction from small to large by using x co-ordinate value and y co-ordinate value of known data points n, in order to form a non-equidistant Segmentation which has n nodes. For the Segmentation we get above,we only know the density of the plastic known n nodes, therefore, we must find other density of the plastic garbage of n nodes.We only do the sampling survey of garbage density of the north pacificvortex,so only understand logically each known data point has a certain extent effect on the unknown node and the close-known points of density of the plastic garbage has high-impact than distant known point.In this respect,we use the weighted average format, that means using the adverse which with distance squared to express more important effects in close known points. There're two known points Q1 and Q2 in a line ,that is to say we have already known the plastic litter density in Q1 and Q2, then speculate the plastic litter density's affects between Q1、Q2 and the point G which in the connection of Q1 and Q2. It can be shown by a weighted average algorithm22212221111121GQ GQ GQ Z GQ Z Z Q Q G +*+*=in this formula GQ expresses the distance between the pointG and Q.We know that only use a weighted average close to the unknown point can not reflect the trend of the known points, we assume that any two given point of plastic garbage between the changes in the density of plastic impact the plastic garbage density of the unknown point and reflecting the density of plastic garbage changes in linear trend. So in the weighted average formula what is in order to presume an unknown point of plastic garbage density, we introduce the trend items. And because the greater impact at close range point, and thus the density of plastic wastes trends close points stronger. For the one-dimensional case, the calculation formula G Z in the previous example modify in the following format:2212122212212122211111112121Q Q GQ GQ GQ Q Q GQ Z GQ Z GQ Z Z Q Q Q Q G ++++*+*+*=Among them, 21Q Q known as the separation distance of the known point, 21Q Q Z is the density of plastic garbage which is the plastic waste density of 1Q and 2Q for the linear trend of point G . For the two-dimensional area, point G is not on the line 21Q Q , so we make a vertical from the point G and cross the line connect the point 1Q and 2Q , and get point P , the impact of point P to 1Q and 2Q just like one-dimensional, and the one-dimensional closer of G to P , the distant of G to P become farther, the smaller of the impact, so the weighting factor should also reflect the GP in inversely proportional to a certain way, then we adopt following format:221212222122121222211111112121Q Q GQ GP GQ GQ Q Q GQ GP Z GQ Z GQ Z Z P Q Q Q Q G ++++++*+*+*=Taken together, we speculated following roles:(1) Each known point data are influence the density of plastic garbage of each unknown point in the inversely proportional to the square of the distance;(2) the change of density of plastic garbage between any two known points data, for each unknown point are affected, and the influence to each particular point of their plastic garbage diffuse the straight line along the two known particular point; (3) the change of the density of plastic garbage between any two known data points impact a specific unknown points of the density of plastic litter depends on the three distances: a. the vertical distance to a straight line which is a specific point link to a known point;b. the distance between the latest known point to a specific unknown point;c. the separation distance between two known data points.If we mark 1Q ，2Q ，…，N Q as the location of known data points,G as an unknown node, ijG P is the intersection of the connection of i Q ，j Q and the vertical line from G to i Q ，j Q()G Q Q Z j i ,,is the density trend of i Q ，j Q in the of plasticgarbage points and prescribe ()G Q Q Z j i ,,is the testing point i Q ’ s density of plastic garbage ,so there are calculation formula:()()∑∑∑∑==-==++++*=Ni N ij ji i ijGji i ijG N i Nj j i G Q Q GQ GPQ Q GQ GP G Q Q Z Z 11222222111,,Here we plug each year’s observational data in schedule 1 into our model, and draw the three-dimensional images of the spatial distribution of the marine garbage ’s density with Matlab in Figure (2) as follows:199920002002200520062007－2008(1)It’s observed and analyzed that, from 1999 to 2008, the density of plastic garbage is increasing year by year and significantly in the region of East – West 140W-150W, south - north 30N-40N. Therefore, we can make sure that this region is probably the center of the marine litter whirlpool. Gathering process should be such that the dispersed garbage floating in the ocean move with the ocean currents and gradually close to the whirlpool region. At the beginning, the area close to the vortex will have obviously increasable about plastic litter density, because of this centripetal they keeping move to the center of the vortex ,then with the time accumulates ,the garbage density in the center of the vortex become much bigger and bigger , at last it becomes the Pacific rubbish island we have seen today.It can be seen that through our algorithm, as long as the reference to be able to detect the density in an area which has a number of discrete measuring points,Through tracking these density changes ,we Will be able to value out all the waters of the density measurement through our models to determine,This will reduce the workload of the marine expedition team monitoring marine pollution significantly, and also saving costs .Task 3:The degradation mechanism of marine plasticsWe know that light, mechanical force, heat, oxygen, water, microbes, chemicals, etc. can result in the degradation of plastics . In mechanism ,Factors result in the degradation can be summarized as optical ，biological，and chemical。

数学建模美赛获奖论文

Some players believe that “corking” a bat enhances the “sweet spot” effect. There are some arguments about that .Such asa corked bat has (slightly) less mass.,less mass (lower inertia) means faster swing speed and less mass means a less effective collision. These are just some people’s views, other people may have different opinions. Whethercorking is helpful in the baseball game has not been strongly confirmed yet. Experiments seem to have inconsistent results.
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2010 Mathematical Contest in Modeling (MCM) Summary Sheet
(Attach a copy of this page to each copy of your solution paper.)
Keywords:simple harmonic motion system , differential equations model , collision system

2022年美赛C题论文

2022年美赛C题论文基于相关性分析及线性规划下的交易策略的最优问题本文针对每日价格流来确定交易员是否应该购买、持有或出售其投资组合中的资产问题，主要通过建立具有不同投资项目的未来产值预测模型及其损失定量分析和敏感性评价模型，以此来对美国的黄金、现金、比特币三种投资进行组合并评价其未来的价值。

对于问题 1 中对美国的黄金和比特币的价值年变化特征进行相关分析，筛选出价值变化较高的年份以及对美国 2021 年的 10月 9 日进行具体的定量分析，这里选取美国 2016 年不同短历时黄金和比特币资料,采用频率分析、小波分析和极值分析等方法, 揭示了美国市场投资的演变规律，之后我们对于 2021 年的具体数据通过上述方法进行了具体分析。

选取价值变化趋势、中位数、成本、回报率等作为分析因素，确定各因素影响权重，从而确定现金、比特币和黄金的持有率，以达到最优的投资结果。

对于问题 2 评价模型的敏感性，根据五项指标数据，同时参照第一问求解得到的各指标对排名的的贡献度，我们综合考虑各种因素，对投资回报的指标进行赋值。

结合中国的各项数据，我们将估值水平，也就是总价值从 0 至 10 划分为十个不同的层次阶段，0-1 阶段为一等高价值，其余水平依次类推。

采用二次多项式拟合提取黄金、比特币的趋势分量，采用谐波分析法提取的周期成分，利用线性回归模型求解随机成分，最后将三者叠加，构建了各投资项目的预报模型。

模型计算结果与实测数据对比可知，应用预报模型对投资回报进行预报精度较高。

一、问题重述 1.1 背景资料市场交易员经常买卖波动性资产，其目标是使其总回报最大化。

每一次买卖通常都有一笔佣金。

其中两种资产是黄金和比特币。

1.2 需要解决的问题交易员要求开发一个模型，该模型仅使用迄今为止过去的每日价格流来确定交易员每天是否应该购买、持有或出售其投资组合中的资产。

二、问题分析2.1 问题 1 的分析通过对比美国的黄金和比特币的价值年变化特征进行分析，发现无论是黄金还是比特币，每日的资金流都处于波动之中。

2016 美国大学生数学竞赛优秀论文AB

2016年美赛A题热水澡一个人用热水通过一个水龙头来注满一个浴缸，然后坐在在浴缸中，清洗和放松。

不幸的是，浴缸不是一个带有二次加热系统和循环喷流的温泉式浴缸，而是一个简单的水容器。

过一会儿，洗澡水就会明显地变凉，所以洗澡的人需要不停地将热水从水龙头注入，以加热洗浴水。

该浴缸的设计是以这样一种方式，当浴缸里的水达到容量极限，多余的水通过溢流口泄流。

考虑空间和时间等因素，建立一个浴缸的水温模型，以确定最佳的策略，使浴缸里的人可以用这个模型来让整个浴缸保持或尽可能接近初始的温度，而不浪费太多的水。

使用你的模型来确定你的策略对浴缸的形状和体积，浴缸里的人的形状、体积、温度，以及浴缸中的人的运动等因素的依赖程度。

如果这个人一开始用了一种泡泡浴剂加入浴缸，以协助清洗，这会怎样影响你的模型的结果？除了要求的一页MCM摘要提交之外，你的报告必须包括一页的为浴缸用户准备的非技术性的说明书来阐释你的策略，同时解释为什么洗澡水的温度得到均衡地保持是如此之难。

2016年美赛B题太空垃圾在地球轨道上的小碎片的数量已引起越来越多的关注。

据估计，目前有超过500，000块的空间碎片，也被称为轨道碎片，由于被认为对空间飞行器是潜在的威胁而正在被跟踪。

2009年2月10日，俄罗斯卫星kosmos-2251和美国卫星iridium-33相撞之后，该问题受到了新闻媒体更广泛的讨论。

一些消除碎片方法已经被提出。

这些方法包括使用微型的基于太空的喷水飞机和高能量的激光来针对一些特定的碎片和设计大型卫星来清扫碎片。

碎片按照大小和质量分步，从刷了油漆的薄片到废弃的卫星都有。

碎片在轨道上的高速度飞行使得捕捉十分困难。

建立一个以时间为考量的模型，以确定最佳的方法或系列方法，为一个私营企业提供商机，以解决空间碎片问题。

你的模型应该包括定量和定性的对成本，风险，收益的估计，并考虑其他的一些重要因素。

你的模型应该能够评估某种方法，以及组合的系列方法，并能够研究各种重要的假设情况。

美赛金奖论文

1
Team # 14604
Catalogue
Abstracts ........................................................................................................................................... 1 Contents ............................................................................................................................................ 3 1. Introduction ................................................................................................................................... 3 1.1 Restatement of the Problem ................................................................................................ 3 1.2 Survey of the Previous Research......................................................................................... 3 2. Assumptions .................................................................................................................................. 4 3. Parameters ..................................................................................................................................... 4 4. Model A ----------Package model .................................................................................................. 6 4.1 Motivation ........................................................................................................................... 6 4.2 Development ....................................................................................................................... 6 4.2.1 Module 1: Introduce of model A .............................................................................. 6 4.2.2 Module 2: Solution of model A .............................................................................. 10 4.3 Conclusion ........................................................................................................................ 11 5. Model B----------Optional model ................................................................................................ 12 5.1 Motivation ......................................................................................................................... 12 5.2 Development ..................................................................................................................... 12 5.2.1 Module B: Choose oar- powered rubber rafts or motorized boats either ............... 12 5.2.2 Module 2: Choose mix of oar- powered rubber rafts and motorized boats ............ 14 5.3 Initial arrangement ............................................................................................................ 17 5.4. Deepened model B ........................................................................................................... 18 5.4.1 Choose the campsites allodium .............................................................................. 18 5.4.2 Choose the oar- powered rubber rafts or motorized boats allodium ...................... 19 5.5 An example of reasonable arrangement ............................................................................ 19 5.6 The strengths and weakness .............................................................................................. 20 6. Extensions ................................................................................................................................... 21 7. Memo .......................................................................................................................................... 25 8. References ................................................................................................................................... 26 9. Appendices .................................................................................................................................. 27 9.1 Appendix I .................................................................................................. 27 9.2 Appendix II ....................................................................................................................... 29

美赛一等奖论文-中文翻译版

目录问题回顾 (3)问题分析： (4)模型假设： (6)符号定义 (7)4.1---------- (8)4.2 有热水输入的温度变化模型 (17)4.2.1模型假设与定义 (17)4.2.2 模型的建立The establishment of the model (18)4.2.3 模型求解 (19)4.3 有人存在的温度变化模型Temperature model of human presence (21)4.3.1 模型影响因素的讨论Discussion influencing factors of the model (21)4.3.2模型的建立 (25)4.3.3 Solving model (29)5.1 优化目标的确定 (29)5.2 约束条件的确定 (31)5.3模型的求解 (32)5.4 泡泡剂的影响 (35)5.5 灵敏度的分析 (35)8 non-technical explanation of the bathtub (37)Sｕmｍary人们经常在充满热水的浴缸里得到清洁和放松。

本文针对只有一个简单的热水龙头的浴缸，建立一个多目标优化模型，通过调整水龙头流量大小和流入水的温度来使整个泡澡过程浴缸内水温维持基本恒定且不会浪费太多水。

首先分析浴缸中水温度变化的具体情况。

根据能量转移的特点将浴缸中的热量损失分为两类情况:沿浴缸四壁和底面向空气中丧失的热量根据傅里叶导热定律求出；沿水面丧失的热量根据水由液态变为气态的焓变求出。

因涉及的参数过多,将系数进行回归分析的得到一个一元二次函数。

结合两类热量建立了温度关于时间的微分方程。

加入阻滞因子考虑环境温湿度升高对水温的影响,最后得到水温度随时间的变化规律(见图＊*)。

优化模型考虑保持水龙头匀速流入热水的情况。

将过程分为浴缸未加满和浴缸加满而水从排水口溢出的两种情况，根据能量守恒定律优化上述微分方程,建立一个有热源的情况下水的温度随时间变化的分段模型，（见图*＊)接下来考虑人在浴缸中对水温的影响。

数学建模美赛一等奖优秀专业论文

For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number52888Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________Mathematical Contest in Modeling (MCM/ICM) Summary SheetSummaryIt’s pleasant t o go home to take a bath with the evenly maintained temperature of hot water throughout the bathtub. This beautiful idea, however, can not be always realized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperature even and as close as possible to the initial temperature without wasting too much water. This paper proposes a partial differential equation of the heat conduction of the bath water temperature, and an object programming model. Based on the Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), this paper illustrates the best strategy the person in the bathtub can adopt to satisfy his desires. First, a spatiotemporal partial differential equation model of the heat conduction of the temperature of the bath water is built. According to the priority, an object programming model is established, which takes the deviation of temperature throughout the bathtub, the deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objective—homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results all imply that the top priority strategy is: The proper motions of the person making the temperature well-distributed throughout the bathtub. Therefore, the Partial Differential Equation model (PDE) can be simplified to the ordinary differential equation model.Second, the weights for the remaining three objectives are determined based on the tolerance of temperature and the hobby of the person by applying Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Therefore, the evaluation model of the synthesis score of the strategy is proposed to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as close as the initial condition results in the fewer number of switching faucet while attention to water consumption gives rise to the more number. Third, the paper conducts the analysis of the diverse parameters in the model to determine the best strategy, respectively, by controlling the other parameters constantly, and adjusting the parameters of the volume, shape of the bathtub and the shape, volume, temperature and the motions and other parameters of the person in turns. All results indicate that the differential model and the evaluation model developed in this paper depends upon the parameters therein. When considering the usage of a bubble bath additive, it is equal to be the obstruction between water and air. Our results show that this strategy can reduce the dropping rate of the temperatureeffectively, and require fewer number of switching.The surface area and heat transfer coefficient can be increased because of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stochastic one. With the above evaluation model, this paper present the stochastic optimization model to determine the best strategy. Taking the disparity from the initial temperature as the suboptimum objectives, the result of the model reveals that it is very difficult to keep the temperature constant even wasting plentiful hot water in reality.Finally, the paper performs sensitivity analysis of parameters. The result shows that the shape and the volume of the tub, different hobbies of people will influence the strategies significantly. Meanwhile, combine with the conclusion of the paper, we provide a one-page non-technical explanation for users of the bathtub.Fall in love with your bathtubAbstractIt’s pleasant t o go home to take a bath with the evenly maintained temperature of hot water throughout the bathtub. This beautiful idea, however, can not be always realized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperature even and as close as possible to the initial temperature without wasting too much water. This paper proposes a partial differential equation of the heat conduction of the bath water temperature, and an object programming model. Based on the Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), this paper illustrates the best strategy the person in the bathtub can adopt to satisfy his desires. First, a spatiotemporal partial differential equation model of the heat conduction of the temperature of the bath water is built. According to the priority, an object programming model is established, which takes the deviation of temperature throughout the bathtub, the deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objective—homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results all imply that the top priority strategy is: The proper motions of the person making the temperature well-distributed throughout the bathtub. Therefore, the Partial Differential Equation model (PDE) can be simplified to the ordinary differential equation model.Second, the weights for the remaining three objectives are determined based on the tolerance of temperature and the hobby of the person by applying Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Therefore, the evaluation model of the synthesis score of the strategy is proposed to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as close as the initial condition results in the fewer number of switching faucet while attention to water consumption gives rise to the more number. Third, the paper conducts the analysis of the diverse parameters in the model to determine the best strategy, respectively, by controlling the other parameters constantly, and adjusting the parameters of the volume, shape of the bathtub and the shape, volume, temperature and the motions and other parameters of the person in turns. All results indicate that the differential model and the evaluation model developed in this paper depends upon the parameters therein. When considering the usage of a bubble bath additive, it is equal to be the obstruction between water and air. Our results show that this strategy can reduce the dropping rate of the temperature effectively, and require fewer number of switching.The surface area and heat transfer coefficient can be increased because of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stochastic one. With the above evaluation model, this paper present the stochastic optimization model to determine the best strategy. Taking the disparity from the initial temperature as the suboptimum objectives, the result of the model reveals that it is very difficult to keep the temperature constant even wasting plentiful hotwater in reality.Finally, the paper performs sensitivity analysis of parameters. The result shows that the shape and the volume of the tub, different hobbies of people will influence the strategies significantly. Meanwhile, combine with the conclusion of the paper, we provide a one-page non-technical explanation for users of the bathtub.Keywords:Heat conduction equation; Partial Differential Equation model (PDE Model); Objective programming; Strategy; Analytical Hierarchy Process (AHP) Problem StatementA person fills a bathtub with hot water and settles into the bathtub to clean and relax. However, the bathtub is not a spa-style tub with a secondary hearing system, as time goes by, the temperature of water will drop. In that conditions,we need to solve several problems:(1) Develop a spatiotemporal model of the temperature of the bathtub water to determine the best strategy to keep the temperature even throughout the bathtub and as close as possible to the initial temperature without wasting too much water;(2) Determine the extent to which your strategy depends on the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, and the motions made by the person in the bathtub.(3)The influence of using b ubble to model’s results.(4)Give a one-page non-technical explanation for users that describes your strategyGeneral Assumptions1.Considering the safety factors as far as possible to save water, the upper temperature limit is set to 45 ℃;2.Considering the pleasant of taking a bath, the lower temperature limit is set to 33℃;3.The initial temperature of the bathtub is 40℃.Table 1Model Inputs and SymbolsSymbols Definition UnitT Initial temperature of the Bath water ℃℃T∞Outer circumstance temperatureT Water temperature of the bathtub at the every moment ℃t Time hx X coordinates of an arbitrary point my Y coordinates of an arbitrary point mz Z coordinates of an arbitrary point mαTotal heat transfer coefficient of the system 2（）⋅/W m K1SThe surrounding-surface area of the bathtub 2m 2S The above-surface area of water2m 1H Bathtub’s thermal conductivity/W m K ⋅（） D The thickness of the bathtub wallm 2H Convection coefficient of water2/W m K ⋅（） a Length of the bathtubm b Width of the bathtubm h Height of the bathtubm V The volume of the bathtub water3m c Specific heat capacity of water/()J kg ⋅℃ ρ Density of water3/kg m ()v t Flooding rate of hot water3/m s r TThe temperature of hot water ℃Temperature ModelBasic ModelA spatio-temporal temperature model of the bathtub water is proposed in this paper. It is a four dimensional partial differential equation with the generation and loss of heat. Therefore the model can be described as the Thermal Equation.The three-dimension coordinate system is established on a corner of the bottom of the bathtub as the original point. The length of the tub is set as the positive direction along the x axis, the width is set as the positive direction along the y axis, while the height is set as the positive direction along the z axis, as shown in figure 1.Figure 1. The three-dimension coordinate systemTemperature variation of each point in space includes three aspects: one is the natural heat dissipation of each point in space; the second is the addition of exogenous thermal energy; and the third is the loss of thermal energy . In this way , we build the Partial Differential Equation model as follows:22212222(,,,)(,,,)()f x y z t f x y z t T T T T t x y z c Vαρ-∂∂∂∂=+++∂∂∂∂ (1) Where● t refers to time;● T is the temperature of any point in the space;● 1f is the addition of exogenous thermal energy;● 2f is the loss of thermal energy.According to the requirements of the subject, as well as the preferences of people, the article proposes these following optimization objective functions. A precedence level exists among these objectives, while keeping the temperature even throughout the bathtub must be ensured.Objective 1(.1O ): keep the temperature even throughout the bathtub;22100min (,,,)(,,,)t t V V F t T x y z t dxdydz dt t T x y z t dxdydz dt ⎡⎤⎡⎤⎛⎫=-⎢⎥ ⎪⎢⎥⎢⎥⎣⎦⎝⎭⎣⎦⎰⎰⎰⎰⎰⎰⎰⎰ (2) Objective 2(.2O ): keep the temperature as close as possible to the initial temperature;[]2200min (,,,)tV F T x y z t T dxdydz dt ⎛⎫=- ⎪⎝⎭⎰⎰⎰⎰ (3) Objective 3(.3O ): do not waste too much water;()30min tF v t dt =⋅⎰ (4) Objective 4(.4O ): fewer times of switching.4min F n = (5)Since the .1O is the most crucial, we should give priority to this objective. Therefore, the highest priority strategy is given here, which is homogenization of temperature.Strategy 0 – Homogenization of T emperatureThe following three reasons are provided to prove the importance of this strategy. Reason 1-SimulationIn this case, we use grid algorithm to make discretization of the formula (1), and simulate the distribution of water temperature.(1) Without manual intervention, the distribution of water temperature as shown infigure 2. And the variance of the temperature is 0.4962. 00.20.40.60.8100.51 1.5200.5Length WidthH e i g h t 4242.54343.54444.54545.5Distribution of temperature at the length=1Distribution of temperatureat the width=1Hot water Cool waterFigure 2. Temperature profiles in three-dimension space without manual intervention(2) Adding manual intervention, the distribution of water temperature as shown infigure 3. And the variance of the temperature is 0.005. 00.5100.51 1.5200.5 Length WidthH e i g h t 44.744.7544.844.8544.944.9545Distribution of temperatureat the length=1Distribution of temperature at the width=1Hot water Cool waterFigure 3. Temperature profiles in three-dimension space with manual interventionComparing figure 2 with figure 3, it is significant that the temperature of water will be homogeneous if we add some manual intervention. Therefore, we can assumed that222222()0T T T x y zα∂∂∂++≠∂∂∂ in formula (1). Reason 2-EstimationIf the temperature of any point in the space is different, then222222()0T T T x y zα∂∂∂++≠∂∂∂ Thus, we find two points 1111(,,,)x y z t and 2222(,,,)x y z t with:11112222(,,,)(,,,)T x y z t T x y z t ≠Therefore, the objective function 1F could be estimated as follows:[]2200200001111(,,,)(,,,)(,,,)(,,,)0t t V V t T x y z t dxdydz dt t T x y z t dxdydz dt T x y z t T x y z t ⎡⎤⎡⎤⎛⎫-⎢⎥ ⎪⎢⎥⎢⎥⎣⎦⎝⎭⎣⎦≥->⎰⎰⎰⎰⎰⎰⎰⎰ (6) The formula (6) implies that some motion should be taken to make sure that the temperature can be homogeneous quickly in general and 10F =. So we can assumed that: 222222()0T T T x y zα∂∂∂++≠∂∂∂. Reason 3-Analytical analysisIt is supposed that the temperature varies only on x axis but not on the y-z plane. Then a simplified model is proposed as follows:()()()()()()()2sin 000,0,,00,000t xx x T a T A x l t l T t T l t t T x x l π⎧=+≤≤≤⎪⎪⎪==≤⎨⎪⎪=≤≤⎪⎩ (7)Then we use two ways, Fourier transformation and Laplace transformation, in solving one-dimensional heat equation [Qiming Jin 2012]. Accordingly, we get the solution:()()2222/22,1sin a t l Al x T x t e a l πππ-=- (8) Where ()0,2x ∈, 0t >, ()01|x T f t ==(assumed as a constant), 00|t T T ==.Without general assumptions, we choose three specific value of t , and gain a picture containing distribution change of temperature in one-dimension space at different time.00.20.40.60.811.2 1.4 1.6 1.8200.511.522.533.54Length T e m p e r a t u r e time=3time=5time=8Figure 4. Distribution change of temperature in one-dimension space at different timeT able 2.V ariance of temperature at different timet3 5 8 variance0.4640 0.8821 1.3541It is noticeable in Figure 4 that temperature varies sharply in one-dimensional space. Furthermore, it seems that temperature will vary more sharply in three-dimension space. Thus it is so difficult to keep temperature throughout the bathtub that we have to take some strategies.Based on the above discussion, we simplify the four dimensional partial differential equation to an ordinary differential equation. Thus, we take the first strategy that make some motion to meet the requirement of homogenization of temperature, that is 10F =.ResultsTherefore, in order to meet the objective function, water temperature at any point in the bathtub needs to be same as far as possible. We can resort to some strategies to make the temperature of bathtub water homogenized, which is (,,)x y z ∀∈∀. That is,()(),,,T x y z t T t =Given these conditions, we improve the basic model as temperature does not change with space.112213312()()()()/()p r H S dT H S T T H S T T c v T T c V V dt D μρρ∞⎡⎤=++-+-+--⎢⎥⎣⎦(9) Where● 1μis the intensity of people’s movement ;● 3H is convection between water and people;● 3S is contact area between water and people;● p T is body surface temperature;● 1V is the volume of the bathtub;● 2V is the volume of people.Where the μ refers to the intensity of people ’s movement. It is a constant. However , it is a random variable in reality, which will be taken into consideration in the following.Model T estingWe use the oval-shaped bathtub to test our model. According to the actual situation, we give initial values as follows:0.19λ=,0.03D =,20.54H =,25T ∞=,040T =00.20.40.60.8125303540Time T e m p e r a t u r eFigure 5. Basic modelThe Figure 5 shows that the temperature decreases monotonously with time. And some signs of a slowing down in the rate of decrease are evident in the picture. Reaching about two hours, the water temperature does not change basically and be closely to the room temperature. Obviously , it is in line with the actual situation, indicating the rationality of this model.ConclusionOur model is robust under reasonable conditions, as can be seen from the testing above. In order to keep the temperature even throughout the bathtub, we should take some strategies like stirring constantly while adding hot water to the tub. Most important of all, this is the necessary premise of the following question.Strategy 1 – Fully adapted to the hot water in the tubInfluence of body surface temperatureWe select a set of parameters to simulate two kinds of situation separately.The first situation is that do not involve the factor of human1122()()/H S dT H S T T cV dt D ρ∞⎡⎤=+-⎢⎥⎣⎦(10) The second situation is that involves the factor of human112213312()()()/()p H S dT H S T T H S T T c V V dt D μρ∞⎡⎤=++-+--⎢⎥⎣⎦(11) According to the actual situation, we give specific values as follows, and draw agraph of temperature of two functions.33p T =,040T =204060801001201401601803838.53939.540TimeT e m p e r a t u r eWith body Without bodyFigure 6a. Influence of body surface temperature50010001500200025003000350025303540TimeT e m p e r a t u r eWith body Without bodyCoincident pointFigure 6b. Influence of body surface temperatureThe figure 6 shows the difference between two kinds of situation in the early time (before the coincident point ), while the figure 7 implies that the influence of body surface temperature reduces as time goes by . Combing with the degree of comfort ofbath and the factor of health, we propose the second optimization strategy: Fully adapted to the hot water after getting into the bathtub.Strategy 2 –Adding water intermittentlyInfluence of adding methods of waterThere are two kinds of adding methods of water. One is the continuous; the other is the intermittent. We can use both different methods to add hot water.1122112()()()/()r H S dT H S T T c v T T c V V dt D μρρ∞⎡⎤=++-+--⎢⎥⎣⎦(12) Where r T is the temperature of the hot water.To meet .3O , we calculated the minimum water consumption by changing the flow rate of hot water. And we compared the minimum water consumptions of the continuous with the intermittent to determine which method is better.A . Adding water continuouslyAccording to the actual situation, we give specific values as follows and draw a picture of the change of temperature.040T =, 37d T =, 45r T =5001000150020002500300035003737.53838.53939.54040.5TimeT e m p e r a t u r eadd hot waterFigure 7. Adding water continuouslyIn most cases, people are used to have a bath in an hour. Thus we consumed that deadline of the bath: 3600final t =. Then we can find the best strategy in Figure 5 which is listed in Table 2.T able 3Strategy of adding water continuouslystart t final tt ∆ vr T varianceWater flow 4 min 1 hour56 min537.410m s -⨯45℃31.8410⨯0.2455 3mB . Adding water intermittentlyMaintain the values of 0T ,d T ,r T ,v , we change the form of adding water, and get another graph.5001000150020002500300035003737.53838.53939.540TimeT e m p e r a t u r et1=283(turn on)t3=2107(turn on)t2=1828(turn off)Figure 8. Adding water intermittentlyT able 4.Strategy of adding water intermittently()1t on ()2t off 3()t on vr T varianceWater flow 5 min 30 min35min537.410m s -⨯45℃33.610⨯0.2248 3mConclusionDifferent methods of adding water can influence the variance, water flow and the times of switching. Therefore, we give heights to evaluate comprehensively the methods of adding hot water on the basis of different hobbies of people. Then we build the following model:()()()2213600210213i i n t t i F T t T dtF v t dtF n -=⎧=-⎪⎪⎪=⎨⎪⎪=⎪⎩⎰∑⎰ (13) ()112233min F w F w F w F =++ (14)12123min ..510mini i t s t t t +>⎧⎨≤-≤⎩Evaluation on StrategiesFor example: Given a set of parameters, we choose different values of v and d T , and gain the results as follows.Method 1- AHPStep 1:Establish hierarchy modelFigure 9. Establish hierarchy modelStep 2: Structure judgment matrix153113511133A ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦Step 3: Assign weight1w 2w3w 0.650.220.13Method 2-TopsisStep1 ：Create an evaluation matrix consisting of m alternatives and n criteria, with the intersection of each alternative and criteria given as ij x we therefore have a matrixStep2：The matrix ij m n x ⨯（）is then normalised to form the matrix ij m n R r ⨯=（）, using thenormalisation method21r ,1,2,,;1,2,ijij mij i x i n j m x====∑…………，Step3：Calculate the weighted normalised decision matrix()(),1,2,,ij j ij m n m nT t w r i m ⨯⨯===⋅⋅⋅where 1,1,2,,nj j jj w W Wj n ===⋅⋅⋅∑so that11njj w==∑, and j w is the original weight given to the indicator,1,2,,j v j n =⋅⋅⋅.Step 4： Determine the worst alternative ()w A and the best alternative ()b A()(){}{}()(){}{}max 1,2,,,min 1,2,,1,2,,n ,min 1,2,,,max 1,2,,1,2,,n ,w ij ij wjbijij bjA t i m j J t i m j J t j A t i m j J t i m j J tj -+-+==∈=∈====∈=∈==where, {}1,2,,J j n j +==⋅⋅⋅ associated with the criteria having a positive impact, and {}1,2,,J j n j -==⋅⋅⋅associated with the criteria having a negative impact. Step 5： Calculate the L2-distance between the target alternative i and the worst condition w A()21,1,2,,m niw ij wj j d tt i ==-=⋅⋅⋅∑and the distance between the alternative i and the best condition b A()21,1,2,,m nib ij bj j d t t i ==-=⋅⋅⋅∑where iw d and ib d are L2-norm distances from the target alternative i to the worst and best conditions, respectively .Step 6 ：Calculate the similarity to the worst condition Step 7 ： Rank the alternatives according to ()1,2,,iw s i m =⋅⋅⋅ Step 8 : Assign weight1w2w 3w 0.55 0.170.23ConclusionAHP gives height subjectively while TOPSIS gives height objectively. And the heights are decided by the hobbies of people. However, different people has different hobbies, we choose AHP to solve the following situations.Impact of parametersDifferent customers have their own hobbies. Some customers prefer enjoying in the bath, so the .2O is more important . While other customers prefer saving water, the .3O is more important. Therefore, we can solve the problem on basis of APH . 1. Customers who prefer enjoying: 20.83w =,30.17w =According to the actual situation, we give initial values as follows:13S =,11V =,2 1.4631S =,20.05V =,33p T =,110μ=Ensure other parameters unchanged, then change the values of these parameters including 1S ,1V ,2S ,2V ,d T ,1μ. So we can obtain the optimal strategies under different conditions in Table 4.T able 5.Optimal strategies under different conditions2.Customers who prefer saving: 20.17w =,30.83w =Just as the former, we give the initial values of these parameters including1S ,1V ,2S ,2V ,d T ,1μ, then change these values in turn with other parameters unchanged. So we can obtain the optimal strategies as well in these conditions.T able 6.Optimal strategies under different conditionsInfluence of bubbleUsing the bubble bath additives is equivalent to forming a barrier between the bath water and air, thereby slowing the falling velocity of water temperature. According to the reality, we give the values of some parameters and gain the results as follows:5001000150020002500300035003334353637383940TimeT e m p e r a t u r eWithour bubbleWith bubbleFigure 10. Influence of bubbleT able 7.Strategies (influence of bubble)Situation Dropping rate of temperature (the larger the number, the slower)Disparity to theinitial temperatureWater flow Times of switchingWithout bubble 802 1.4419 0.1477 4 With bubble 34499.85530.01122The Figure 10 and the Table 7 indicates that adding bubble can slow down the dropping rate of temperature effectively . It can decrease the disparity to the initial temperature and times of switching, as well as the water flow.Improved ModelIn reality , human ’s motivation in the bathtub is flexible, which means that the parameter 1μis a changeable measure. Therefore, the parameter can be regarded as a random variable, written as ()[]110,50t random μ=. Meanwhile, the surface of water will come into being ripples when people moves in the tub, which will influence the parameters like 1S and 2S . So, combining with reality , we give the range of values as follows:()[]()[]111222,1.1,1.1S t random S S S t random S S ⎧=⎪⎨=⎪⎩Combined with the above model, the improved model is given here:()[]()[]()[]11221121111222()()()/()10,50,1.1,1.1a H S dT H S T T c v T T c V V dt D t random S t random S S S t random S S μρρμ∞⎧⎡⎤=++-+--⎪⎢⎥⎣⎦⎨⎪===⎩(15)Given the values, we can get simulation diagram:050010001500200025003000350039.954040.0540.140.15TimeT e m p e r a t u r eFigure 11. Improved modelThe figure shows that the variance is small while the water flow is large, especially the variance do not equals to zero. This indicates that keeping the temperature of water is difficult though we regard .2O as the secondary objective.Sensitivity AnalysisSome parameters have a fixed value throughout our work. By varying their values, we can see their impacts.Impact of the shape of the tub0.70.80.91 1.1 1.2 1.3 1.433.23.43.63.84Superficial areaT h e t i m e sFigure 12a. Times of switching0.70.80.91 1.11.21.31.43890390039103920393039403950Superficial areaV a r i a n c eFigure 12b. V ariance of temperature0.70.80.91 1.1 1.2 1.3 1.40.190.1950.20.2050.21Superficial areaW a t e r f l o wFigure 12c. Water flowBy varying the value of some parameters, we can get the relationships between the shape of tub and the times of switching, variance of temperature, and water flow et. It is significant that the three indexes will change as the shape of the tub changes. Therefore the shape of the tub makes an obvious effect on the strategies. It is a sensitive parameter.Impact of the volume of the tub0.70.80.91 1.1 1.2 1.3 1.4 1.533.544.55VolumeT h e t i m e sFigure 13a. Times of switching。

数学建模美赛优秀论文

A Summary
Our solution consists of three mathematical models, offering a thorough perspective of the leaf. In the weight evaluation model, we consider the tree crown to be spherical, and leaves reaching photosynthesis saturation will let sunlight pass through. The Fibonacci number is helping leaves to minimize overlapping each other. Thus, we obtain the total leaf area and by multiplying it to the leaf area ratio we will get the leaf weight. Furthermore, a Logistic model is applied to depict the relationship between the leaf weight and the physical characteristic of a tree, making it easy to estimate the leaf weight by simply measure the circumstance of the trunk. In the shape correlation model, the shape of a leaf is represented by its surface area. Trees living in different habitats have different sizes of leaves. Mean annual temperature(T) and mean annual precipitation(P) are supposed to be significant in determining the leaf area. We have also noticed that the density of leaves and the density of branches greatly affect the size of leaf. To measure the density, we adopt the number of leaves per unit-length branch(N) and the length of intervals between two leaf branches(L) in the model. By applying multiple linear regression to data of six tree species in different habitats, we lately discovered that leaf area is positively correlated with T, P and L. In the leaf classification model, a matter-element model is applied to evaluate the leaf, offering a way of classifying leaf according to preset criteria. In this model, the parameters in the previous model are applied to classify the leaf into three categories: Large, Medium, and Small. Data of a tree species is tested for its credit, proving the model to be an effective model of classification especially suitable for computer standardized evaluation. In sum, our models unveil the facts concerning how leaves increase as the tree grows, why different kinds of trees have different shapes of leaves, and how to classify leaves. The imprecision of measurement and the limitedness of data are the main impediment of our modeling, and some correlation might be more complicated than our hypotheses.

美赛数学建模比赛论文实用模板

The Keep-Right-Except-To-Pass RuleSummaryAs for the first question, it provides a traffic rule of keep right except to pass, requiring us to verify its effectiveness. Firstly, we define one kind of traffic rule different from the rule of the keep right in order to solve the problem clearly; then, we build a Cellular automaton model and a Nasch model by collecting massive data; next, we make full use of the numerical simulation according to several influence factors of traffic flow; At last, by lots of analysis of graph we obtain, we indicate a conclusion as follow: when vehicle density is lower than 0.15, the rule of lane speed control is more effective in terms of the factor of safe in the light traffic; when vehicle density is greater than 0.15, so the rule of keep right except passing is more effective In the heavy traffic.As for the second question, it requires us to testify that whether the conclusion we obtain in the first question is the same apply to the keep left rule. First of all, we build a stochastic multi-lane traffic model; from the view of the vehicle flow stress, we propose that the probability of moving to the right is 0.7and to the left otherwise by making full use of the Bernoulli process from the view of the ping-pong effect, the conclusion is that the choice of the changing lane is random. On the whole, the fundamental reason is the formation of the driving habit, so the conclusion is effective under the rule of keep left.As for the third question, it requires us to demonstrate the effectiveness of the result advised in the first question under the intelligent vehicle control system. Firstly, taking the speed limits into consideration, we build a microscopic traffic simulator model for traffic simulation purposes. Then, we implement a METANET model for prediction state with the use of the MPC traffic controller. Afterwards, we certify that the dynamic speed control measure can improve the traffic flow .Lastly neglecting the safe factor, combining the rule of keep right with the rule of dynamical speed control is the best solution to accelerate the traffic flow overall.Key words：Cellular automaton model Bernoulli process Microscopic traffic simulator model The MPC traffic controlContentContent (2)1. Introduction (3)2. Analysis of the problem (3)3. Assumption (3)4. Symbol Definition (3)5. Models (4)5.1 Building of the Cellular automaton model (4)5.1.1 Verify the effectiveness of the keep right except to pass rule (4)5.1.2 Numerical simulation results and discussion (5)5.1.3 Conclusion (8)5.2 The solving of second question (8)5.2.1 The building of the stochastic multi-lane traffic model (9)5.2.2 Conclusion (9)5.3 Taking the an intelligent vehicle system into a account (9)5.3.1 Introduction of the Intelligent Vehicle Highway Systems (9)5.3.2 Control problem (9)5.3.3 Results and analysis (9)5.3.4 The comprehensive analysis of the result (10)6. Improvement of the model (11)6.1 strength and weakness (11)6.1.1 Strength (11)6.1.2 Weakness (11)6.2 Improvement of the model (11)7. Reference (13)1. IntroductionAs is known to all, it’s essential for us to drive automobiles, thus the driving rules is crucial important. In many countries like USA, China, drivers obey the rules which called “The Keep-Right-Except-To-Pass (that is, when driving automobiles, the rule requires drivers to drive in the right-most unless theyare passing another vehicle)”.2. Analysis of the problemFor the first question, we decide to use the Cellular automaton to build models,then analyze the performance of this rule in light and heavy traffic. Firstly,we mainly use the vehicle density to distinguish the light and heavy traffic; secondly, we consider the traffic flow and safe as the represent variable which denotes the light or heavy traffic; thirdly, we build and analyze a Cellular automaton model; finally, we judge the rule through two different driving rules,and then draw conclusions.3. AssumptionIn order to streamline our model we have made several key assumptions●The highway of double row three lanes that we study can representmulti-lane freeways.●The data that we refer to has certain representativeness and descriptive●Operation condition of the highway not be influenced by blizzard oraccidental factors●Ignore the driver's own abnormal factors, such as drunk driving andfatigue driving●The operation form of highway intelligent system that our analysis canreflect intelligent system●In the intelligent vehicle system, the result of the sampling data hashigh accuracy.4. Symbol Definitioni The number of vehiclest The time5. ModelsBy analyzing the problem, we decided to propose a solution with building a cellular automaton model.5.1 Building of the Cellular automaton modelThanks to its simple rules and convenience for computer simulation, cellular automaton model has been widely used in the study of traffic flow in recent years. Let )(t x i be the position of vehicle i at time t , )(t v i be the speed of vehicle i at time t , p be the random slowing down probability, and R be the proportion of trucks and buses, the distance between vehicle i and the front vehicle at time t is:1)()(1--=-t x t x gap i i i , if the front vehicle is a small vehicle.3)()(1--=-t x t x gap i i i , if the front vehicle is a truck or bus.5.1.1 Verify the effectiveness of the keep right except to pass ruleIn addition, according to the keep right except to pass rule, we define a new rule called: Control rules based on lane speed. The concrete explanation of the new rule as follow:There is no special passing lane under this rule. The speed of the first lane (the far left lane) is 120–100km/h (including 100 km/h)；the speed of the second lane (the middle lane) is 100–80km8/h (including80km/h)；the speed of the third lane (the far right lane) is below 80km/ h. The speeds of lanes decrease from left to right.● Lane changing rules based lane speed controlIf vehicle on the high-speed lane meets control v v <, ),1)(min()(max v t v t gap i f i +≥, safe b i gap t gap ≥)(, the vehicle will turn into the adjacent right lane, and the speed of the vehicle after lane changing remains unchanged, where control v is the minimum speed of the corresponding lane.● The application of the Nasch model evolutionLet d P be the lane changing probability (taking into account the actual situation that some drivers like driving in a certain lane, and will not takethe initiative to change lanes), )(t gap f i indicates the distance between the vehicle and the nearest front vehicle, )(t gap b i indicates the distance between the vehicle and the nearest following vehicle. In this article, we assume that the minimum safe distance gap safe of lane changing equals to the maximum speed of the following vehicle in the adjacent lanes.Lane changing rules based on keeping right except to passIn general, traffic flow going through a passing zone (Fig. 5.1.1) involves three processes: the diverging process (one traffic flow diverging into two flows), interacting process (interacting between the two flows), and merging process (the two flows merging into one) [4].Fig.5.1.1 Control plan of overtaking process(1) If vehicle on the first lane (passing lane) meets ),1)(min()(max v t v t gap i f i +≥ and safe b i gap t gap ≥)(, the vehicle will turn into the second lane, the speed of the vehicle after lane changing remains unchanged.5.1.2 Numerical simulation results and discussionIn order to facilitate the subsequent discussions, we define the space occupation rate as L N N p truck CAR ⨯⨯+=3/)3(, where CAR N indicates the number ofsmall vehicles on the driveway,truck N indicates the number of trucks and buses on the driveway, and L indicates the total length of the road. The vehicle flow volume Q is the number of vehicles passing a fixed point per unit time,T N Q T /=, where T N is the number of vehicles observed in time duration T .The average speed ∑∑⨯=T it i a v T N V 11)/1(, t i v is the speed of vehicle i at time t . Take overtaking ratio f p as the evaluation indicator of the safety of traffic flow, which is the ratio of the total number of overtaking and the number of vehicles observed. After 20,000 evolution steps, and averaging the last 2000 steps based on time, we have obtained the following experimental results. In order to eliminate the effect of randomicity, we take the systemic average of 20 samples [5].Overtaking ratio of different control rule conditionsBecause different control conditions of road will produce different overtaking ratio, so we first observe relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.(a) Based on passing lane control (b) Based on speed control Fig.5.1.3Fig.5.1.3 Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.It can be seen from Fig. 5.1.3:(1) when the vehicle density is less than 0.05, the overtaking ratio will continue to rise with the increase of vehicle density; when the vehicle density is larger than 0.05, the overtaking ratio will decrease with the increase of vehicle density; when density is greater than 0.12, due to the crowding, it willbecome difficult to overtake, so the overtaking ratio is almost 0.(2) when the proportion of large vehicles is less than 0.5, the overtaking ratio will rise with the increase of large vehicles; when the proportion of large vehicles is about 0.5, the overtaking ratio will reach its peak value; when the proportion of large vehicles is larger than 0.5, the overtaking ratio will decrease with the increase of large vehicles, especially under lane-based control condition s the decline is very clear.● Concrete impact of under different control rules on overtaking ratioFig.5.1.4Fig.5.1.4 Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions. (Figures in left-hand indicate the passing lane control, figures in right-hand indicate the speed control. 1f P is the overtaking ratio of small vehicles over large vehicles, 2f P is the overtaking ratio of small vehicles over small vehicles, 3f P is the overtaking ratio of large vehicles over small vehicles, 4f P is the overtaking ratio of large vehicles over large vehicles.). It can be seen from Fig. 5.1.4:(1) The overtaking ratio of small vehicles over large vehicles under passing lane control is much higher than that under speed control condition, which is because, under passing lane control condition, high-speed small vehicles have to surpass low-speed large vehicles by the passing lane, while under speed control condition, small vehicles are designed to travel on the high-speed lane, there is no low- speed vehicle in front, thus there is no need to overtake.● Impact of different control rules on vehicle speedFig. 5.1.5 Relationships among vehicle density, proportion of large vehicles and average speed under different control conditions. (Figures in left-hand indicates passing lane control, figures in right-hand indicates speed control.a X is the average speed of all the vehicles, 1a X is the average speed of all the small vehicles, 2a X is the average speed of all the buses and trucks.).It can be seen from Fig. 5.1.5:(1) The average speed will reduce with the increase of vehicle density and proportion of large vehicles.(2) When vehicle density is less than 0.15,a X ,1a X and 2a X are almost the same under both control conditions.Effect of different control conditions on traffic flowFig.5.1.6Fig. 5.1.6 Relationships among vehicle density, proportion of large vehicles and traffic flow under different control conditions. (Figure a1 indicates passing lane control, figure a2 indicates speed control, and figure b indicates the traffic flow difference between the two conditions.It can be seen from Fig. 5.1.6：(1) When vehicle density is lower than 0.15 and the proportion of large vehicles is from 0.4 to 1, the traffic flow of the two control conditions are basically the same.(2) Except that, the traffic flow under passing lane control condition is slightly larger than that of speed control condition.5.1.3 ConclusionIn this paper, we have established three-lane model of different control conditions, studied the overtaking ratio, speed and traffic flow under different control conditions, vehicle density and proportion of large vehicles.5.2 The solving of second question5.2.1 The building of the stochastic multi-lane traffic model5.2.2 ConclusionOn one hand, from the analysis of the model, in the case the stress is positive, we also consider the jam situation while making the decision. More specifically, if a driver is in a jam situation, applying ))(,2(x P B R results with a tendency of moving to the right lane for this driver. However in reality, drivers tend to find an emptier lane in a jam situation. For this reason, we apply a Bernoulli process )7.0,2(B where the probability of moving to the right is 0.7and to the left otherwise, and the conclusion is under the rule of keep left except to pass, So, the fundamental reason is the formation of the driving habit.5.3 Taking the an intelligent vehicle system into a accountFor the third question, if vehicle transportation on the same roadway was fully under the control of an intelligent system, we make some improvements for the solution proposed by us to perfect the performance of the freeway by lots of analysis.5.3.1 Introduction of the Intelligent Vehicle Highway SystemsWe will use the microscopic traffic simulator model for traffic simulation purposes. The MPC traffic controller that is implemented in the Matlab needs a traffic model to predict the states when the speed limits are applied in Fig.5.3.1. We implement a METANET model for prediction purpose[14].5.3.2 Control problemAs a constraint, the dynamic speed limits are given a maximum and minimum allowed value. The upper bound for the speed limits is 120 km/h, and the lower bound value is 40 km/h. For the calculation of the optimal control values, all speed limits are constrained to this range. When the optimal values are found, they are rounded to a multiplicity of 10 km/h, since this is more clear for human drivers, and also technically feasible without large investments.5.3.3 Results and analysisWhen the density is high, it is more difficult to control the traffic, since the mean speed might already be below the control speed. Therefore, simulations are done using densities at which the shock wave can dissolve without using control, and at densities where the shock wave remains. For each scenario, five simulations for three different cases are done, each with a duration of one hour. The results of the simulations are reported in Table 5.1, 5.2, 5.3. Table.5.1 measured results for the unenforced speed limit scenariodem q case#1 #2 #3 #4 #5 TTS:mean(std ) TPN 4700no shock 494.7452.1435.9414.8428.3445.21(6.9%) 5:4wave 3 5 8 8 0 14700nocontrolled520.42517.48536.13475.98539.58517.92(4.9%)6:364700 controlled 513.45488.43521.35479.75-486.5500.75(4.0%)6:244700 no shockwave493.9472.6492.78521.1489.43493.96(3.5%)6:034700 uncontrolled635.1584.92643.72571.85588.63604.84(5.3%)7:244700 controlled 575.3654.12589.77572.15586.46597.84(6.4%)7:19●Enforced speed limits●Intelligent speed adaptationFor the ISA scenario, the desired free-flow speed is about 100% of the speed limit. The desired free-flow speed is modeled as a Gaussian distribution, with a mean value of 100% of the speed limit, and a standard deviation of 5% of the speed limit. Based on this percentage, the influence of the dynamic speed limits is expected to be good[19].5.3.4 The comprehensive analysis of the resultFrom the analysis above, we indicate that adopting the intelligent speed control system can effectively decrease the travel times under the control of an intelligent system, in other words, the measures of dynamic speed control can improve the traffic flow.Evidently, under the intelligent speed control system, the effect of the dynamic speed control measure is better than that under the lane speed control mentioned in the first problem. Because of the application of the intelligent speed control system, it can provide the optimal speed limit in time. In addition, it can guarantee the safe condition with all kinds of detection device and the sensor under the intelligent speed system.On the whole, taking all the analysis from the first problem to the end into a account, when it is in light traffic, we can neglect the factor of safe with the help of the intelligent speed control system.Thus, under the state of the light traffic, we propose a new conclusion different from that in the first problem: the rule of keep right except to pass is more effective than that of lane speed control.And when it is in the heavy traffic, for sparing no effort to improve the operation efficiency of the freeway, we combine the dynamical speed control measure with the rule of keep right except to pass, drawing a conclusion that the application of the dynamical speed control can improve the performance ofthe freeway.What we should highlight is that we can make some different speed limit as for different section of road or different size of vehicle with the application of the Intelligent Vehicle Highway Systems.In fact, that how the freeway traffic operate is extremely complex, thereby, with the application of the Intelligent Vehicle Highway Systems, by adjusting our solution originally, we make it still effective to freeway traffic.6. Improvement of the model6.1 strength and weakness6.1.1 Strength●it is easy for computer simulating and can be modified flexibly to consideractual traffic conditions ,moreover a large number of images make the model more visual.●The result is effectively achieved all of the goals we set initially, meantimethe conclusion is more persuasive because of we used the Bernoulli equation.●We can get more accurate result as we apply Matlab.6.1.2 Weakness●The relationship between traffic flow and safety is not comprehensivelyanalysis.●Due to there are many traffic factors, we are only studied some of the factors,thus our model need further improved.6.2 Improvement of the modelWhile we compare models under two kinds of traffic rules, thereby we come to the efficiency of driving on the right to improve traffic flow in some circumstance. Due to the rules of comparing is too less, the conclusion is inadequate. In order to improve the accuracy, We further put forward a kinds of traffic rules: speed limit on different type of cars.The possibility of happening traffic accident for some vehicles is larger, and it also brings hidden safe troubles. So we need to consider separately about different or specific vehicle types from the angle of the speed limiting in order to reduce the occurrence of traffic accidents, the highway speed limit signs is in Fig.6.1.Fig .6.1Advantages of the improving model are that it is useful to improve the running condition safety of specific type of vehicle while considering the difference of different types of vehicles. However, we found that the rules may be reduce the road traffic flow through the analysis. In the implementation it should be at the 85V speed of each model as the main reference basis. In recent years, the 85V of some researchers for the typical countries from Table 6.1[ 21]: Table 6.1 Operating speed prediction modeAuthorCountry Model Ottesen andKrammes2000America LC DC L DC V C ⨯---=01.0012.057.144.10285Andueza2000Venezuel a ].[308.9486.7)/894()/2795(25.9885curve horizontal L DC Ra R V T ++--= ].[tan 819.27)/3032(69.10085gent L R V T +-= Jessen2001 America ][00239.0614.0279.080.86185LSD ADT G V V P --+=][00212.0432.010.7285NLSD ADT V V P -+=Donnell2001 America 22)2(8500724.040.10140.04.78T L G R V --+=22)3(85008369.048.10176.01.75T L G R V --+= 22)4(8500810.069.10176.05.74T L G R V --+=22)5(8500934.008.21.83T L G V --=BucchiA.BiasuzziK.And SimoneA.2005Italy DC V 124.0164.6685-= DC E V 4.046.3366.5585--= 2855.035.1119.0745.65DC E DC V ---= Fitzpatrick America KV 98.17507.11185-= Meanwhile, there are other vehicles driving rules such as speed limit in adverseweather conditions. This rule can improve the safety factor of the vehicle to some extent. At the same time, it limits the speed at the different levels.7. Reference[1] M. Rickert, K. Nagel, M. Schreckenberg, A. Latour, Two lane trafﬁ csimulations using cellular automata, Physica A 231 (1996) 534–550.[20] J.T. Fokkema, Lakshmi Dhevi, Tamil Nadu Trafﬁ c Management and Control inIntelligent Vehicle Highway Systems,18(2009).[21] Yang Li, New Variable Speed Control Approach for Freeway. (2011) 1-66。

美赛论文模板(超实用)

TitileSummaryDuring cell division, mitotic spindles are assembled by microtubule-based motor proteins1, 2. The bipolar organization of spindles is essential for proper segregation of chromosomes, and requires plus-end-directed homotetrameric motor proteins of the widely conserved kinesin-5 (BimC) family3. Hypotheses for bipolar spindle formation include the 'push−pull mitotic muscle' model, in which kinesin-5 and opposing motor proteins act between overlapping microtubules2, 4, 5. However, the precise roles of kinesin-5 during this process are unknown. Here we show that the vertebrate kinesin-5 Eg5 drives the sliding of microtubules depending on their relative orientation. We found in controlled in vitro assays that Eg5 has the remarkable capability of simultaneously moving at 20 nm s-1 towards the plus-ends of each of the two microtubules it crosslinks. For anti-parallel microtubules, this results in relative sliding at 40 nm s-1, comparable to spindle pole separation rates in vivo6. Furthermore, we found that Eg5 can tether microtubule plus-ends, suggesting an additional microtubule-binding mode for Eg5. Our results demonstrate how members of the kinesin-5 family are likely to function in mitosis, pushing apart interpolar microtubules as well as recruiting microtubules into bundles that are subsequently polarized by relative sliding. We anticipate our assay to be a starting point for more sophisticated in vitro models of mitotic spindles. For example, the individual and combined action of multiple mitotic motors could be tested, including minus-end-directed motors opposing Eg5 motility. Furthermore, Eg5 inhibition is a major target of anti-cancer drug development, and a well-defined and quantitative assay for motor function will be relevant for such developmentsContentTitile (1)Summary (1)1Introduction (1)1.1Restatement of the Problem (1)1.2Background (1)1.1.1Common Solving Technique (1)1.1.2Previous Works (1)1.3Example (1)2Analysis of the Problem (1)2.1Outline of the Approach (1)2.2Basic Assumptions (2)2.3Definitions and Key Terms (2)3Calculating and Simplifying the Model (2)4The Model Results (3)5Validating the Model (3)6Strengths and Weaknesses (3)6.1Strengths (3)6.2Weaknesses (3)7Food for Thought (3)8Conclusion (3)References (4)Appendices (4)Appendix A Source Code (4)Appendix B (4)1Introduction1.1Restatement of the Problem …1.2Background…1.1.1Common Solving Technique…1.1.2Previous Works…1.3Example…2Analysis of the Problem …2.1Outline of the Approach…2.2Basic Assumptions●●●●●2.3Definitions and Key Terms●●●●Table 1.…Symbol Meaning Unit3Calculating and Simplifying the Model …4The Model Results……5Validating the Model…6Strengths and Weaknesses6.1S trengths●●●●6.2W eaknesses●●●●7Food for Thought…8Conclusion….References…AppendicesAppendix A Source CodeHere are the simulation programmes we used in our model as follow. Input matlab source:……….Appendix B…….Input C++ source：…………..…………..。

数学建模美赛特等奖论文(中文版)分析溃坝：针对南卡罗来纳州大坝坍塌建立模型

分析溃坝：针对南卡罗来纳州大坝坍塌建立模型摘要萨鲁达大坝建立在卡罗莱纳州的墨累湖与萨鲁达河之间，如果发生地震大坝就会坍塌。

本文通过建立模型来分析以下四种大坝决口时水的流量以及洪水泛滥时水的流量：● 大坝的绝大部分被瞬间侵蚀看成是大坝瞬间彻底坍塌；● 大坝的绝大部分被缓慢侵蚀看成是大坝延期彻底坍塌；● 管涌就是先形成一个小孔，最终形成一个裂口；● 溢出就是大坝被侵蚀后，形成一个梯形的裂口。

本文建立了两个模型来描述下游洪水的泛滥情况。

两个模型都采用离散网格的方法，将一个地区看成是一个网格，每个网格都包含洪水的深度和体积。

复力模型运用了网格的速度、重力以及邻近网格的压力来模拟水流。

下坡模型假定水流速度与邻近网格间水位高度的成正比例。

下坡模型是高效率的、直观的、灵活的，可以适用于已知海拔的任何地区。

它的两个参数稳定并限制了水流，但该模型的预测很少依赖于它们的静态值。

对于萨鲁达溃坝，洪水总面积为25.106km ；它还没有到达国会大厦。

罗威克里克的洪水向上游延伸了km 4.4，覆盖面积达24.26.1km -变量及假设表1说明了用来描述和模拟模型的变量，表2列出了模拟程序中的参数。

表 1模型中的变量.变量定义溃坝时的水流量速率1TF Q 瞬间彻底坍塌2TF Q 延期彻底坍塌PIPE Q 管涌OT Q 溢出peak Q 最大流速溃坝时水流出到停止所用时间1TF t 瞬间彻底坍塌2TF t 延期彻底坍塌PIPE t 管涌OT t 溢出V ∆ 溃坝后从墨累湖里流出的水的总体积Lm Vol 墨累湖的原来体积LM Area 墨累湖的原来面积breach d 从裂口到坝顶距离breach t 从裂口开始到溃坝形成的时间近似圆锥的墨累湖的侧面一般假设● 正常水位是在溃坝前的湖水位置。

● 河道中的水流不随季节变化而变动。

● 墨累湖里的水的容积可以看作为一个正圆锥（图1 ）。

表2 模拟程序中的参数参数所取值意义BREACH_TYPE 变量瞬间彻底坍塌，延期彻底坍，管涌，溢出模型中的一种 T ∆ 0.10 时间不长的长度(s)MIN_DEPTH 0001.0 网格空时的水的深度(m) FINAT T 100000 大坝彻底决口所用时间 b T 3600 溃坝达最大值的时间(s) peak Q 25000 溃坝的最大流速(m 3/s) breach d 30 蓄水池的最初深度(m) LM Volume 910714.2⨯ 墨累湖的总体积(m 3) LM Area 610202⨯ 墨累湖的总面积(m 2)k 504.0 扩散因素 (控制两网格间交换的水的数量) MAX_LOSS_FRAC 25.0 单位网格中水的最大流失量图 1. 水库近似一个正圆锥.大坝假设● 萨鲁达大坝在以下四种方式之一坍塌：-瞬间彻底坍塌，-延期彻底坍塌，-管涌，-溢出。

美国大学生数学建模大赛优秀论文一等奖摘要

SummaryChina is the biggest developing country. Whether water is sufficient or not will have a direct impact on the economic development of our country. China's water resources are unevenly distributed. Water resource will critically restrict the sustainable development of China if it can not be properly solved.First, we consider a greater number of Chinese cities so that China is divided into 6 areas. The first model is to predict through division and classification. We predict the total amount of available water resources and actual water usage for each area. And we conclude that risk of water shortage will exist in North China, Northwest China, East China, Northeast China, whereas Southwest China, South China region will be abundant in water resources in 2025.Secondly, we take four measures to solve water scarcity: cross-regional water transfer, desalination, storage, and recycling. The second model mainly uses the multi-objective planning strategy. For inter-regional water strategy, we have made reference to the the strategy of South-to-North Water Transfer[5]and other related strategies, and estimate that the lowest cost of laying the pipeline is about 33.14 billion yuan. The program can transport about 69.723 billion cubic meters water to the North China from the Southwest China region per year. South China to East China water transfer is about 31 billion cubic meters. In addition, we can also build desalination mechanism program in East China and Northeast China, and the program cost about 700 million and can provide 10 billion cubic meters a year.Finally, we enumerate the east China as an example to show model to improve. Other area also can use the same method for water resources management, and deployment. So all regions in the whole China can realize the water resources allocation.In a word, the strong theoretical basis and suitable assumption make our model estimable for further study of China's water resources. Combining this model with more information from the China Statistical Yearbook will maximize the accuracy of our model.。

美赛论文优秀模版可编辑

2015Mathematical Contest in Modeling (MCM/ICM) Summary SheetIn order to evaluate the performanee of a coach, we describe metrics in five aspects: historicalrecord, game gold content, playoff performanee, honors and contribution to the sports. Moreover, each aspect is subdivided into several sec on dary metrics. Take playoff performa nce as example, we collect postseas on result (Sweet Sixtee n, Final Four, etc.) per year from NCAA official website, Wikimedia and so on.First, ****grade.To eval*** , in turn, are John Wooden, Mike Krzyzewski, Adolph Rupp, Dean Smith and Bob Kni ght.Time line horizon does make a difference. According to turning points in NCAA history, we divide theprevious century into six periods with different time weights which lead to the cha nge of ranking.We con duct sen sitivity an alysis on FSE to find best membership fun cti on and calculati on rule. Sen sitivity an alysis on aggregation weight is also performed. It proves AM performs better than single model. As a creative use,top 3 presidents (U.S.) are picked out: Abraham Lincoln, George Washi ngton, Fran kli n D. RooseveltAt last, the stre ngth and weak ness of our mode are discussed, non-tech ni cal expla nati on is prese nted and the future work is poin ted as well.Key words: Ebola virus diseaseEpidemiology West Africa; ******ContentsI. I ntroduct ion (2)1.1 ............................................................................................................................................................................... 2 For office use only Team Con trol Number For office use only T1 _____________________ 11111 T2 ______________________ F1 _____________________ F2 _____________________ T3 ____________________ T4 ____________________Problem Chose nABCD F3 _____________________ F4 _____________________1.2 (2)1.3 (2)1.4 (2)1.5 (2)1.6 (2)II.The Descripti on of the Problem (2)2.1How do we approximate the whole course of pay ing toll? (2)2.2How do we defi ne the optimal configuration? (3)2.3The local optimizati on and the overall optimization (3)2.4The differe nces in weights and sizes of vehicles (3)2.5What if there is no data available? (3)III....................................................................................................................................................................................... M odels (3)3.1Basic Model (3)3.1.1Terms, Definitions and Symbols (3)3.1.2Assumpti ons (3)3.1.3The Foun datio n of Model (4)3.1.4Soluti on and Result (4)3.1.5An alysis of the Result (4)3.1.6Stren gth and Weakness (4)3.2Improved Model (4)3.2.1Extra Symbols (5)3.2.2Additi onal Assumpti ons (5)3.2.3The Foun datio n of Model (5)3.2.4Soluti on and Result (5)3.2.5An alysis of the Result (6)3.2.6Stren gth and Weakness (6)IV.Con clusi ons (6)4.1Con clusi ons of the problem (6)4.2Methods used in our models (6)4.3Applicati ons of our models (6)V.Future Work (6)5.1Ano ther model (6)5.1.1The limitations of queuing theory (6)5.1.2 (6)5.1.3 (7)5.1.4 (7)5.2Ano ther layout of toll plaza (7)5.3The n ewly- adopted chargi ng methods (7)VI.References (8)VII.Appe ndix (8)I.I ntroductionIn order to indicate the origin of the toll way problems, the following background is worth men ti oning.1.11.21.31.41.51.6II.The Description of the Problem2.1How d o we approximate the whole course of pay ing toll?2.2How d o we defi ne the optimal con figurati on?1)From the perspective of motorist:2)From the perspective of the toll plaza:3)Compromise:2.3The l ocal optimization and the overall optimizationVirtually:2.4The differe nces in weights and sizes of vehicl es2.5What if there is no data availabl e?III.Models3.1Basic Model3.1.1Terms, Definitions and SymbolsThe sig ns and defi niti ons are mostly gen erated fronqueu ing theory.3.1.2Assumpti ons3.1.3The Foun dation of Model1)The utility functionThe cost of toll plaza:The loss of motorist:The weight of each aspect:Compromise:2)The integer programmingAccord ing to queu ing theory, we can calculate the statistical properties as follows.3)The overall optimization and the local optimizationThe overall optimization:The local optimization:The optimal number of tollbooths:3.1.4Solution and Result1)The solution of the integer programming:2)Results:3.1.5An alysis of the ResultLocal optimization and overalloptimization:Sensitivity: The result is quite sen sitive to the cha nge of thehree parametersTrend:Comparison:3.1.6Stre ngth and Weak nessStrength: In despite of this, the model has provedhat . Moreover, we have draw n some usefulcon clusi on sabout . The model is fit for, such asWeakness:This model just applies to . As we havestated, . That' s just whatwe should do in the improved model.3.2 Improved Model3.2.1Extra SymbolsSigns and defi niti ons in dicated above are still valid. Here are some extra sig ns and defi niti ons.3.2.2Additi onal Assumpti onsAssumpti ons concerning the an terior process are the same as the Basic Model.3.2.3The Foun dation of Model1) How do we determine the optimal number?As we have con cluded from the Basic Model,3.2.4Solution and Result1)Simulation algorithmBased on the analysis above, we design our simulation arithmetic as follows.Step1:Step2:Step3:Step4:Step5:Step6:Step7:Step8:Step9:2)Flow chartThe figure below is the flow chart of the simulatio n.3)Solution3.2.5 Analysis of the Result3.2.6 Stre ngth and Weak nessStrength: The Improved Model aims to make up forthe n eglect of . The result seems to declare that this model is more reas on able tha n the Basic Model and much more effective tha n the existi ng desig n.Weakness: . Thus the model is still an approximate on a large scale. This hasdoomed to limit the applications of it.IV. Conclusions4.1 Con clusi ons of the probl em4.2 Methods used in our mod els4.3 Applicati ons of our mod elsV. Future Work5.1 Ano ther model5.1.1 The limitations of queuing theory5.1.25.1.35.1.41)2)3)4)5.2 Ano ther layout of toll plaza5.3 The newly- ad opted charging methodsVI. References[1][2][3][4]VII. Appe ndix。

2020年美赛C题论文

2020年美赛C题论文引言在2020年的美赛C题中，我们将研究某城市的停车问题。

停车问题在现代城市中非常普遍，而且经常引起交通拥堵和资源浪费。

因此，寻找一种合理的停车方案对于城市的可持续发展至关重要。

本文将介绍我们对该停车问题的建模过程、假设和模型结果。

问题描述该城市位于一个山区，拥有许多旅游景点，吸引了大量游客。

然而，停车场的数量有限，传统的交通管理方式导致了拥堵和停车困难。

因此，我们需要提出一种新的停车方案，以改善交通状况和资源利用。

我们需要回答以下问题：1.如何确定合理的停车位价格以确保公平性和减少拥堵？2.如何确定合理的停车位数量以满足游客的需求？3.如何指导游客选择合适的停车场？数据处理和建模为了解决上述问题，我们从该城市收集了大量的交通数据和停车场信息。

首先，我们对数据进行处理，包括数据清洗、整理和校验。

然后，我们使用Python编程语言对数据进行分析和建模。

下面是我们的建模过程：1.确定停车需求模型：我们将游客的停车需求建模为一个随机变量，可以以概率密度函数的形式表示。

为了准确地估计需求模型，我们使用了大量的历史停车数据和游客统计数据。

2.确定停车位定价模型：我们考虑了停车位价格对停车需求的影响，并建立了一个停车位定价模型。

该模型将考虑停车位的成本、游客的支付意愿和其他相关因素。

3.确定停车场选择模型：我们使用了多属性决策分析方法来确定游客选择停车场的因素和权重。

通过评估每个停车场的特点和游客的偏好，我们可以为游客提供选择停车场的指导。

假设为了简化问题和建立数学模型，我们做出了以下假设：1.停车需求是服从某种概率分布的随机变量。

2.停车位定价的主要目标是确保公平性和减少拥堵。

3.游客的停车选择主要受停车位价格和距离的影响。

4.停车场之间没有容量限制。

这些假设可以帮助我们建立合理的模型和解决方案，但也需要在实际应用中考虑其他可能的因素。

模型结果基于我们的建模过程和假设，我们得到了以下模型结果：1.停车需求模型：通过对历史停车数据和游客统计数据的分析，我们得到了停车需求的概率密度函数模型。

历年美赛数学建模优秀论文大全

2008国际大学生数学建模比赛参赛作品---------WHO所属成员国卫生系统绩效评估作品名称：Less Resources, more outcomes参赛单位：重庆大学参赛时间：2008年2月15日至19日指导老师：何仁斌参赛队员：舒强机械工程学院05级罗双才自动化学院05级黎璨计算机学院05级ContentLess Resources, More Outcomes (4)1. Summary (4)2. Introduction (5)3. Key Terminology (5)4. Choosing output metrics for measuring health care system (5)4.1 Goals of Health Care System (6)4.2 Characteristics of a good health care system (6)4.3 Output metrics for measuring health care system (6)5. Determining the weight of the metrics and data processing (8)5.1 Weights from statistical data (8)5.2 Data processing (9)6. Input and Output of Health Care System (9)6.1 Aspects of Input (10)6.2 Aspects of Output (11)7. Evaluation System I : Absolute Effectiveness of HCS (11)7.1Background (11)7.2Assumptions (11)7.3Two approaches for evaluation (11)1. Approach A : Weighted Average Evaluation Based Model (11)2. Approach B: Fuzzy Comprehensive Evaluation Based Model [19][20] (12)7.4 Applying the Evaluation of Absolute Effectiveness Method (14)8. Evaluation system II: Relative Effectiveness of HCS (16)8.1 Only output doesn’t work (16)8.2 Assumptions (16)8.3 Constructing the Model (16)8.4 Applying the Evaluation of Relative Effectiveness Method (17)9. EAE VS ERE: which is better? (17)9.1 USA VS Norway (18)9.2 USA VS Pakistan (18)10. Less Resources, more outcomes (19)10.1Multiple Logistic Regression Model (19)10.1.1 Output as function of Input (19)10.1.2Assumptions (19)10.1.3Constructing the model (19)10.1.4. Estimation of parameters (20)10.1.5How the six metrics influence the outcomes? (20)10.2 Taking USA into consideration (22)10.2.1Assumptions (22)10.2.2 Allocation Coefficient (22)10.3 Scenario 1: Less expenditure to achieve the same goal (24)10.3.1 Objective function： (24)10.3.2 Constraints (25)10.3.3 Optimization model 1 (25)10.3.4 Solutions of the model (25)10.4. Scenario2: More outcomes with the same expenditure (26)10.4.1Objective function (26)10.4.2Constraints (26)10.4.3 Optimization model 2 (26)10.4.4Solutions to the model (27)15. Strengths and Weaknesses (27)Strengths (27)Weaknesses (27)16. References (28)Less Resources, More Outcomes1. SummaryIn this paper, we regard the health care system (HCS) as a system with input and output, representing total expenditure on health and its goal attainment respectively. Our goal is to minimize the total expenditure on health to archive the same or maximize the attainment under given expenditure.First, five output metrics and six input metrics are specified. Output metrics are overall level of health, distribution of health in the population,etc. Input metrics are physician density per 1000 population, private prepaid plans as % private expenditure on health, etc.Second, to evaluate the effectiveness of HCS, two evaluation systems are employed in this paper:●Evaluation of Absolute Effectiveness(EAE)This evaluation system only deals with the output of HCS，and we define Absolute Total Score (ATS) to quantify the effectiveness. During the evaluation process, weighted average sum of the five output metrics is defined as ATS, and the fuzzy theory is also employed to help assess HCS.●Evaluation of Relative Effectiveness(ERE)This evaluation system deals with the output as well as its input, and also we define Relative Total Score (RTS) to quantify the effectiveness. The measurement to ATS is units of output produced by unit of input.Applying the two kinds of evaluation system to evaluate HCS of 34 countries (USA included), we can find some countries which rank in a higher position in EAE get a relatively lower rank in ERE, such as Norway and USA, indicating that their HCS should have been able to archive more under their current resources .Therefore, taking USA into consideration, we try to explore how the input influences the output and archive the goal: less input, more output. Then three models are constructed to our goal:●Multiple Logistic RegressionWe model the output as function of input by the logistic equation. In more detains, we model ATS (output) as the function of total expenditure on health system. By curve fitting, we estimate the parameters in logistic equation, and statistical test presents us a satisfactory result.●Linear Optimization Model on minimizing the total expenditure on healthWe try to minimize the total expenditure and at the same time archive the same, that is to get a ATS of 0.8116. We employ software to solve the model, and by the analysis of the results. We cut it to 2023.2 billion dollars, compared to the original data 2109.8 billion dollars.●Linear Optimization Model on maximizing the attainment. We try to maximize the attainment (absolute total score) under the same total expenditure in2007.And we optimize the ATS to 0.8823, compared to the original data 0.8116.Finally, we discuss strengths and weaknesses of our models and make necessary recommendations to the policy-makers。

美赛数学建模优秀论文

Why Crime Doesn’t Pay:Locating Criminals Through Geographic ProﬁlingControl Number:#7272February22,2010AbstractGeographic proﬁling,the application of mathematics to criminology, has greatly improved police eﬀorts to catch serial criminals byﬁnding their residence.However,many geographic proﬁles either generate an extremely large area for police to cover or generates regions that are unstable with respect to internal parameters of the model.We propose,formulate,and test the Gaussian Rossmooth(GRS)Method,which takes the strongest elements from multiple existing methods and combines them into a more stable and robust model.We also propose and test a model to predict the location of the next crime.We tested our models on the Yorkshire Ripper case.Our results show that the GRS Method accurately predicts the location of the killer’s residence.Additionally,the GRS Method is more stable with respect to internal parameters and more robust with respect to outliers than the existing methods.The model for predicting the location of the next crime generates a logical and reasonable region where the next crime may occur.We conclude that the GRS Method is a robust and stable model for creating a strong and eﬀective model.1Control number:#72722Contents1Introduction4 2Plan of Attack4 3Deﬁnitions4 4Existing Methods54.1Great Circle Method (5)4.2Centrography (6)4.3Rossmo’s Formula (8)5Assumptions8 6Gaussian Rossmooth106.1Properties of a Good Model (10)6.2Outline of Our Model (11)6.3Our Method (11)6.3.1Rossmooth Method (11)6.3.2Gaussian Rossmooth Method (14)7Gaussian Rossmooth in Action157.1Four Corners:A Simple Test Case (15)7.2Yorkshire Ripper:A Real-World Application of the GRS Method167.3Sensitivity Analysis of Gaussian Rossmooth (17)7.4Self-Consistency of Gaussian Rossmooth (19)8Predicting the Next Crime208.1Matrix Method (20)8.2Boundary Method (21)9Boundary Method in Action21 10Limitations22 11Executive Summary2311.1Outline of Our Model (23)11.2Running the Model (23)11.3Interpreting the Results (24)11.4Limitations (24)12Conclusions25 Appendices25 A Stability Analysis Images252Control number:#72723List of Figures1The eﬀect of outliers upon centrography.The current spatial mean is at the red diamond.If the two outliers in the lower leftcorner were removed,then the center of mass would be locatedat the yellow triangle (6)2Crimes scenes that are located very close together can yield illog-ical results for the spatial mean.In this image,the spatial meanis located at the same point as one of the crime scenes at(1,1)..7 3The summand in Rossmo’s formula(2B=6).Note that the function is essentially0at all points except for the scene of thecrime and at the buﬀer zone and is undeﬁned at those points..9 4The summand in smoothed Rossmo’s formula(2B=6,φ=0.5, and EPSILON=0.5).Note that there is now a region aroundthe buﬀer zone where the value of the function no longer changesvery rapidly (13)5The Four Corners Test Case.Note that the highest hot spot is located at the center of the grid,just as the mathematics indicates.15 6Crimes and residences of the Yorkshire Ripper.There are two residences as the Ripper moved in the middle of the case.Someof the crime locations are assaults and others are murders (16)7GRS output for the Yorkshire Ripper case(B=2.846).Black dots indicate the two residences of the killer (17)8GRS method run on Yorkshire Ripper data(B=2).Note that the major diﬀerence between this model and Figure7is that thehot zones in thisﬁgure are smaller than in the original run (18)9GRS method run on Yorkshire Ripper data(B=4).Note that the major diﬀerence between this model and Figure7is that thehot zones in thisﬁgure are larger than in the original run (19)10The boundary region generated by our Boundary Method.Note that boundary region covers many of the crimes committed bythe Sutcliﬀe (22)11GRS Method onﬁrst eleven murders in the Yorkshire Ripper Case25 12GRS Method onﬁrst twelve murders in the Yorkshire Ripper Case263Control number:#727241IntroductionCatching serial criminals is a daunting problem for law enforcement oﬃcers around the world.On the one hand,a limited amount of data is available to the police in terms of crimes scenes and witnesses.However,acquiring more data equates to waiting for another crime to be committed,which is an unacceptable trade-oﬀ.In this paper,we present a robust and stable geographic proﬁle to predict the residence of the criminal and the possible locations of the next crime.Our model draws elements from multiple existing models and synthesizes them into a uniﬁed model that makes better use of certain empirical facts of criminology.2Plan of AttackOur objective is to create a geographic proﬁling model that accurately describes the residence of the criminal and predicts possible locations for the next attack. In order to generate useful results,our model must incorporate two diﬀerent schemes and must also describe possible locations of the next crime.Addi-tionally,we must include assumptions and limitations of the model in order to ensure that it is used for maximum eﬀectiveness.To achieve this objective,we will proceed as follows:1.Deﬁne Terms-This ensures that the reader understands what we aretalking about and helps explain some of the assumptions and limitations of the model.2.Explain Existing Models-This allows us to see how others have at-tacked the problem.Additionally,it provides a logical starting point for our model.3.Describe Properties of a Good Model-This clariﬁes our objectiveand will generate a sketelon for our model.With this underlying framework,we will present our model,test it with existing data,and compare it against other models.3DeﬁnitionsThe following terms will be used throughout the paper:1.Spatial Mean-Given a set of points,S,the spatial mean is the pointthat represents the middle of the data set.2.Standard Distance-The standard distance is the analog of standarddeviation for the spatial mean.4Control number:#727253.Marauder-A serial criminal whose crimes are situated around his or herplace of residence.4.Distance Decay-An empirical phenomenon where criminal don’t traveltoo far to commit their crimes.5.Buﬀer Area-A region around the criminal’s residence or workplacewhere he or she does not commit crimes.[1]There is some dispute as to whether this region exists.[2]In our model,we assume that the buﬀer area exists and we measure it in the same spatial unit used to describe the relative locations of other crime scenes.6.Manhattan Distance-Given points a=(x1,y1)and b=(x2,y2),theManhattan distance from a to b is|x1−x2|+|y1−y2|.This is also known as the1−norm.7.Nearest Neighbor Distance-Given a set of points S,the nearestneighbor distance for a point x∈S ismin|x−s|s∈S−{x}Any norm can be chosen.8.Hot Zone-A region where a predictive model states that a criminal mightbe.Hot zones have much higher predictive scores than other regions of the map.9.Cold Zone-A region where a predictive model scores exceptionally low. 4Existing MethodsCurrently there are several existing methods for interpolating the position of a criminal given the location of the crimes.4.1Great Circle MethodIn the great circle method,the distances between crimes are computed and the two most distant crimes are chosen.Then,a great circle is drawn so that both of the points are on the great circle.The midpoint of this great circle is then the assumed location of the criminal’s residence and the area bounded by the great circle is where the criminal operates.This model is computationally inexpensive and easy to understand.[3]Moreover,it is easy to use and requires very little training in order to master the technique.[2]However,it has certain drawbacks.For example,the area given by this method is often very large and other studies have shown that a smaller area suﬃces.[4]Additionally,a few outliers can generate an even larger search area,thereby further slowing the police eﬀort.5Control number:#727264.2CentrographyIn centrography ,crimes are assigned x and y coordinates and the “center of mass”is computed as follows:x center =n i =1x i ny center =n i =1y i nIntuitively,centrography ﬁnds the mean x −coordinate and the mean y -coordinate and associates this pair with the criminal’s residence (this is calledthe spatial mean ).However,this method has several ﬂaws.First,it can be unstablewith respect to outliers.Consider the following set of points (shown in Figure 1:Figure 1:The eﬀect of outliers upon centrography.The current spatial mean is at the red diamond.If the two outliers in the lower left corner were removed,then the center of mass would be located at the yellow triangle.Though several of the crime scenes (blue points)in this example are located in a pair of upper clusters,the spatial mean (red point)is reasonably far away from the clusters.If the two outliers are removed,then the spatial mean (yellow point)is located closer to the two clusters.A similar method uses the median of the points.The median is not so strongly aﬀected by outliers and hence is a more stable measure of the middle.[3]6Control number:#72727 Alternatively,we can circumvent the stability problem by incorporating the 2-D analog of standard deviation called the standard distance:σSD=d center,iNwhere N is the number of crimes committed and d center,i is the distance from the spatial center to the i th crime.By incorporating the standard distance,we get an idea of how“close together”the data is.If the standard distance is small,then the kills are close together. However,if the standard distance is large,then the kills are far apart. Unfortunately,this leads to another problem.Consider the following data set (shown in Figure2):Figure2:Crimes scenes that are located very close together can yield illogical results for the spatial mean.In this image,the spatial mean is located at the same point as one of the crime scenes at(1,1).In this example,the kills(blue)are closely clustered together,which means that the centrography model will yield a center of mass that is in the middle of these crimes(in this case,the spatial mean is located at the same point as one of the crimes).This is a somewhat paradoxical result as research in criminology suggests that there is a buﬀer area around a serial criminal’s place of residence where he or she avoids the commission of crimes.[3,1]That is,the potential kill area is an annulus.This leads to Rossmo’s formula[1],another mathematical model that predicts the location of a criminal.7Control number:#727284.3Rossmo’s FormulaRossmo’s formula divides the map of a crime scene into grid with i rows and j columns.Then,the probability that the criminal is located in the box at row i and column j isP i,j=kTc=1φ(|x i−x c|+|y j−y c|)f+(1−φ)(B g−f)(2B−|x i−x c|−|y j−y c|)gwhere f=g=1.2,k is a scaling constant(so that P is a probability function), T is the total number of crimes,φputs more weight on one metric than the other,and B is the radius of the buﬀer zone(and is suggested to be one-half the mean of the nearest neighbor distance between crimes).[1]Rossmo’s formula incorporates two important ideas:1.Criminals won’t travel too far to commit their crimes.This is known asdistance decay.2.There is a buﬀer area around the criminal’s residence where the crimesare less likely to be committed.However,Rossmo’s formula has two drawbacks.If for any crime scene x c,y c,the equality2B=|x i−x c|+|y j−y c|,is satisﬁed,then the term(1−φ)(B g−f)(2B−|x i−x c|−|y j−y c|)gis undeﬁned,as the denominator is0.Additionally,if the region associated withij is the same region as the crime scene,thenφi c j c is unde-ﬁned by the same reasoning.Figure3illustrates this:This“delta function-like”behavior is disconcerting as it essentially states that the criminal either lives right next to the crime scene or on the boundary deﬁned by Rossmo.Hence,the B-value becomes exceptionally important and needs its own heuristic to ensure its accuracy.A non-optimal choice of B can result in highly unstable search zones that vary when B is altered slightly.5AssumptionsOur model is an expansion and adjustment of two existing models,centrography and Rossmo’s formula,which have their own underlying assumptions.In order to create an eﬀective model,we will make the following assumptions:1.The buﬀer area exists-This is a necessary assumption and is the basisfor one of the mathematical components of our model.2.More than5crimes have occurred-This assumption is importantas it ensures that we have enough data to make an accurate model.Ad-ditionally,Rossmo’s model stipulates that5crimes have occurred[1].8Control number:#72729Figure3:The summand in Rossmo’s formula(2B=6).Note that the function is essentially0at all points except for the scene of the crime and at the buﬀer zone and is undeﬁned at those points3.The criminal only resides in one location-By this,we mean thatthough the criminal may change residence,he or she will not move toa completely diﬀerent area and commit crimes there.Empirically,thisassumption holds,with a few exceptions such as David Berkowitz[1].The importance of this assumption is it allows us to adapt Rossmo’s formula and the centrography model.Both of these models implicitly assume that the criminal resides in only one general location and is not nomadic.4.The criminal is a marauder-This assumption is implicitly made byRossmo’s model as his spatial partition method only considers a small rectangular region that contains all of the crimes.With these assumptions,we present our model,the Gaussian Rossmooth method.9Control number:#7272106Gaussian Rossmooth6.1Properties of a Good ModelMuch of the literature regarding criminology and geographic proﬁling contains criticism of existing models for catching criminals.[1,2]From these criticisms, we develop the following criteria for creating a good model:1.Gives an accurate prediction for the location of the criminal-This is vital as the objective of this model is to locate the serial criminal.Obviously,the model cannot give a deﬁnite location of the criminal,but it should at least give law enforcement oﬃcials a good idea where to look.2.Provides a good estimate of the location of the next crime-Thisobjective is slightly harder than theﬁrst one,as the criminal can choose the location of the next crime.Nonetheless,our model should generate a region where law enforcement can work to prevent the next crime.3.Robust with respect to outliers-Outliers can severely skew predic-tions such as the one from the centrography model.A good model will be able to identify outliers and prevent them from adversely aﬀecting the computation.4.Consitent within a given data set-That is,if we eliminate data pointsfrom the set,they do not cause the estimation of the criminal’s location to change excessively.Additionally,we note that if there are,for example, eight murders by one serial killer,then our model should give a similar prediction of the killer’s residence when it considers theﬁrstﬁve,ﬁrst six,ﬁrst seven,and all eight murders.5.Easy to compute-We want a model that does not entail excessivecomputation time.Hence,law enforcement will be able to get their infor-mation more quickly and proceed with the case.6.Takes into account empirical trends-There is a vast amount ofempirical data regarding serial criminals and how they operate.A good model will incorporate this data in order to minimize the necessary search area.7.Tolerates changes in internal parameters-When we tested Rossmo’sformula,we found that it was not very tolerant to changes of the internal parameters.For example,varying B resulted in substantial changes in the search area.Our model should be stable with respect to its parameters, meaning that a small change in any parameter should result in a small change in the search area.10Control number:#7272116.2Outline of Our ModelWe know that centrography and Rossmo’s method can both yield valuable re-sults.When we used the mean and the median to calculate the centroid of a string of murders in Yorkshire,England,we found that both the median-based and mean-based centroid were located very close to the home of the criminal. Additionally,Rossmo’s method is famous for having predicted the home of a criminal in Louisiana.In our approach to this problem,we adapt these methods to preserve their strengths while mitigating their weaknesses.1.Smoothen Rossmo’s formula-While the theory behind Rossmo’s for-mula is well documented,its implementation isﬂawed in that his formula reaches asymptotes when the distance away from a crime scene is0(i.e.point(x i,y j)is a crime scene),or when a point is exactly2B away froma crime scene.We must smoothen Rossmo’s formula so that idea of abuﬀer area is mantained,but the asymptotic behavior is removed and the tolerance for error is increased.2.Incorporate the spatial mean-Using the existing crime scenes,we willcompute the spatial mean.Then,we will insert a Gaussian distribution centered at that point on the map.Hence,areas near the spatial mean are more likely to come up as hot zones while areas further away from the spatial mean are less likely to be viewed as hot zones.This ensures that the intuitive idea of centrography is incorporated in the model and also provides a general area to search.Moreover,it mitigates the eﬀect of outliers by giving a probability boost to regions close to the center of mass,meaning that outliers are unlikely to show up as hot zones.3.Place more weight on theﬁrst crime-Research indicates that crimi-nals tend to commit theirﬁrst crime closer to their home than their latter ones.[5]By placing more weight on theﬁrst crime,we can create a model that more eﬀectively utilizes criminal psychology and statistics.6.3Our Method6.3.1Rossmooth MethodFirst,we eliminated the scaling constant k in Rossmo’s equation.As such,the function is no longer a probability function but shows the relative likelihood of the criminal living in a certain sector.In order to eliminate the various spikes in Rossmo’s method,we altered the distance decay function.11Control number:#727212We wanted a distance decay function that:1.Preserved the distance decay eﬀect.Mathematically,this meant that thefunction decreased to0as the distance tended to inﬁnity.2.Had an interval around the buﬀer area where the function values wereclose to each other.Therefore,the criminal could ostensibly live in a small region around the buﬀer zone,which would increase the tolerance of the B-value.We examined various distance decay functions[1,3]and found that the func-tions resembled f(x)=Ce−m(x−x0)2.Hence,we replaced the second term in Rossmo’s function with term of the form(1−φ)×Ce−k(x−x0)2.Our modiﬁed equation was:E i,j=Tc=1φ(|x i−x c|+|y j−y c|)f+(1−φ)×Ce−(2B−(|x i−x c|+|y j−y c|))2However,this maintained the problematic region around any crime scene.In order to eliminate this problem,we set an EPSILON so that any point within EPSILON(deﬁned to be0.5spatial units)of a crime scene would have a weighting of a constant cap.This prevented the function from reaching an asymptote as it did in Rossmo’s model.The cap was deﬁned asCAP=φEPSILON fThe C in our modiﬁed Rossmo’s function was also set to this cap.This way,the two maximums of our modiﬁed Rossmo’s function would be equal and would be located at the crime scene and the buﬀer zone.12Control number:#727213This function yielded the following curve (shown in in Figure4),which ﬁt both of our criteria:Figure 4:The summand in smoothed Rossmo’s formula (2B =6,φ=0.5,and EPSILON =0.5).Note that there is now a region around the buﬀer zone where the value of the function no longer changes very rapidly.At this point,we noted that E ij had served its purpose and could be replaced in order to create a more intuitive idea of how the function works.Hence,we replaced E i,j with the following sum:Tc =1[D 1(c )+D 2(c )]where:D 1(c )=min φ(|x i −x c |+|y j −y c |),φEPSILON D 2(c )=(1−φ)×Ce −(2B −(|x i −x c |+|y j −y c |))2For equal weighting on both D 1(c )and D 2(c ),we set φto 0.5.13Control number:#7272146.3.2Gaussian Rossmooth MethodNow,in order to incorporate the inuitive method,we used centrography to locate the center of mass.Then,we generated a Gaussian function centered at this point.The Gaussian was given by:G=Ae −@(x−x center)22σ2x+(y−y center)22σ2y1Awhere A is the amplitude of the peak of the Gaussian.We determined that the optimal A was equal to2times the cap deﬁned in our modiﬁed Rossmo’s equation.(A=2φEPSILON f)To deal with empirical evidence that theﬁrst crime was usually the closest to the criminal’s residence,we doubled the weighting on theﬁrst crime.However, the weighting can be represented by a constant,W.Hence,ourﬁnal Gaussian Rosmooth function was:GRS(x i,y j)=G+W(D1(1)+D2(1))+Tc=2[D1(c)+D2(c)]14Control number:#7272157Gaussian Rossmooth in Action7.1Four Corners:A Simple Test CaseIn order to test our Gaussain Rossmooth(GRS)method,we tried it against a very simple test case.We placed crimes on the four corners of a square.Then, we hypothesized that the model would predict the criminal to live in the center of the grid,with a slightly higher hot zone targeted toward the location of the ﬁrst crime.Figure5shows our results,whichﬁts our hypothesis.Figure5:The Four Corners Test Case.Note that the highest hot spot is located at the center of the grid,just as the mathematics indicates.15Control number:#727216 7.2Yorkshire Ripper:A Real-World Application of theGRS MethodAfter the model passed a simple test case,we entered the data from the Yorkshire Ripper case.The Yorkshire Ripper(a.k.a.Peter Sutcliﬀe)committed a string of13murders and several assaults around Northern England.Figure6shows the crimes of the Yorkshire Ripper and the locations of his residence[1]:Figure6:Crimes and residences of the Yorkshire Ripper.There are two res-idences as the Ripper moved in the middle of the case.Some of the crime locations are assaults and others are murders.16Control number:#727217 When our full model ran on the murder locations,our data yielded the image show in Figure7:Figure7:GRS output for the Yorkshire Ripper case(B=2.846).Black dots indicate the two residences of the killer.In this image,hot zones are in red,orange,or yellow while cold zones are in black and blue.Note that the Ripper’s two residences are located in the vicinity of our hot zones,which shows that our model is at least somewhat accurate. Additionally,regions far away from the center of mass are also blue and black, regardless of whether a kill happened there or not.7.3Sensitivity Analysis of Gaussian RossmoothThe GRS method was exceptionally stable with respect to the parameter B. When we ran Rossmo’s model,we found that slight variations in B could create drastic variations in the given distribution.On many occassions,a change of 1spatial unit in B caused Rossmo’s method to destroy high value regions and replace them with mid-level value or low value regions(i.e.,the region would completely dissapper).By contrast,our GRS method scaled the hot zones.17Control number:#727218 Figures8and9show runs of the Yorkshire Ripper case with B-values of2and 4respectively.The black dots again correspond to the residence of the criminal. The original run(Figure7)had a B-value of2.846.The original B-value was obtained by using Rossmo’s nearest neighbor distance metric.Note that when B is varied,the size of the hot zone varies,but the shape of the hot zone does not.Additionally,note that when a B-value gets further away from the value obtained by the nearest neighbor distance metric,the accuracy of the model decreases slightly,but the overall search areas are still quite accurate.Figure8:GRS method run on Yorkshire Ripper data(B=2).Note that the major diﬀerence between this model and Figure7is that the hot zones in this ﬁgure are smaller than in the original run.18Control number:#727219Figure9:GRS method run on Yorkshire Ripper data(B=4).Note that the major diﬀerence between this model and Figure7is that the hot zones in this ﬁgure are larger than in the original run.7.4Self-Consistency of Gaussian RossmoothIn order to test the self-consistency of the GRS method,we ran the model on theﬁrst N kills from the Yorkshire Ripper data,where N ranged from6to 13,inclusive.The self-consistency of the GRS method was adversely aﬀected by the center of mass correction,but as the case number approached11,the model stabilized.This phenomenon can also be attributed to the fact that the Yorkshire Ripper’s crimes were more separated than those of most marauders.A selection of these images can be viewed in the appendix.19Control number:#7272208Predicting the Next CrimeThe GRS method generates a set of possible locations for the criminal’s resi-dence.We will now present two possible methods for predicting the location of the criminal’s next attack.One method is computationally expensive,but more rigorous while the other method is computationally inexpensive,but more intuitive.8.1Matrix MethodGiven the parameters of the GRS method,the region analyzed will be a square with side length n spatial units.Then,the output from the GRS method can be interpreted as an n×n matrix.Hence,for any two runs,we can take the norm of their matrix diﬀerence and compare how similar the runs were.With this in mind,we generate the following method.For every point on the grid:1.Add crime to this point on the grid.2.Run the GRS method with the new set of crime points.pare the matrix generated with these points to the original matrix bysubtracting the components of the original matrix from the components of the new matrix.4.Take a matrix norm of this diﬀerence matrix.5.Remove the crime from this point on the grid.As a lower matrix norm indicates a matrix similar to our original run,we seek the points so that the matrix norm is minimized.There are several matrix norms to choose from.We chose the Frobenius norm because it takes into account all points on the diﬀerence matrix.[6]TheFrobenius norm is:||A||F=mi=1nj=1|a ij|2However,the Matrix Method has one serious drawback:it is exceptionally expensive to compute.Given an n×n matrix of points and c crimes,the GRS method runs in O(cn2).As the Matrix method runs the GRS method at each of n2points,we see that the Matrix Method runs in O(cn4).With the Yorkshire Ripper case,c=13and n=151.Accordingly,it requires a fairly long time to predict the location of the next crime.Hence,we present an alternative solution that is more intuitive and eﬃcient.20Control number:#7272218.2Boundary MethodThe Boundary Method searches the GRS output for the highest point.Then,it computes the average distance,r,from this point to the crime scenes.In order to generate a resonable search area,it discards all outliers(i.e.,points that were several times further away from the high point than the rest of the crimes scenes.)Then,it draws annuli of outer radius r(in the1-norm sense)around all points above a certain cutoﬀvalue,deﬁned to be60%of the maximum value. This value was chosen as it was a high enough percentage value to contain all of the hot zones.The beauty of this method is that essentially it uses the same algorithm as the GRS.We take all points on the hot zone and set them to“crime scenes.”Recall that our GRS formula was:GRS(x i,y j)=G+W(D1(1)+D2(1))+Tc=2[(D1(c)+D2(c))]In our boundary model,we only take the terms that involve D2(c).However, let D 2(c)be a modiﬁed D2(c)deﬁned as follows:D 2(c)=(1−φ)×Ce−(r−(|x i−x c|+|y j−y c|))2Then,the boundary model is:BS(x i,y j)=Tc=1D 2(c)9Boundary Method in ActionThis model generates an outer boundary for the criminal’s next crime.However, our model does notﬁll in the region within the inner boundary of the annulus. This region should still be searched as the criminal may commit crimes here. Figure10shows the boundary generated by analyzing the Yorkshire Ripper case.21。

大学生美赛建模论文

For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number27328Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________ 2014Mathematical Contest in Modeling (MCM/ICM) Summary Sheet(Attach a copy of this page to your solution paper.)Type a summary of your results on this page. Do not includethe name of your school, advisor, or team members on this page.2014 A: The Keep-Right-Except-To-Pass RuleIt is very important to obey the traffic rules when you drive, especially in which direction do you drive. The rules can avoid the chaos and traffic accidents.To judge whether the rule is good, we can consider traffic flow and safety. So, how to assess the performance of the role? How to improve and perfect it?We build this model mainly from four aspects.For the first question we mainly take relevant factors into consideration. At the beginning, we divide the traffic condition into four categories. That is: Whether obeying the rule in heavy or light traffic. Then for the four categories, we analyze the problem in six aspects. At last,we get some traffic performance evaluation scores. Therefore we can evaluate the rule in a mathematical point of view. For the second question we think about the relationship between whether obeying the rule, the distance between two vehicles and the amount of vehicles passed by in an hour. We calculate the traffic fl ow based on the previous research’s formula. With the help of the statistics we measured before, we can calculate the score under the four circumstances. For the third question, we think about the impact of driving direction on vehicle’s speed. If the spee d changes, then the score will change, too. Our model also need to modify correspondingly. For the fourth question, We analyze the difference between artificial system and intelligent system. We adopt the intelligent system to modify our model because the traffic flow and safety index change. To sum up, when the traffic is heavy the rule performs well and is effective to promote traffic flow. However, when the traffic is light the rule performs weaker and has no significant effect to promote traffic flow. But the conclusion is exactly opposite in the left-driving countries. Intelligent system doesn't have a big impact on our conclusion. Our advise is when the traffic is light, we can leave one lane just for passing vehicles.COVERCOVER 0Summary (2)Key word (3)Problem background (3)Problem analysis (4)Assumptions (5)Symbols description (5)Model design and solving (6)Model 1: Introduction of vehicle length, weather, frictions and actual speed (6)Model 2: Calculating the distance between two vehicles (8)Model 3: Scores considering S(the traffic flow within an hour) and I(the number of traffic accident every 100000 kilometers) .. 9 Model 4: Improving traffic flow (13)Model 5: Can our new model be applied to driving-in-the-left countries ? (14)Model 6: Intelligent system model (14)Weaknesses and strengths of the model (15)Reference (15)Appendix (16)2014 A: The Keep-Right-Except-To-Pass RuleSummaryIt is very important to obey the traffic rules when you drive, especially in which direction do you drive. The rules can avoid the chaos and traffic accidents.To judge whether the rule is good, we can consider traffic flow and safety. So, how to assess the performance of the role? How to improve and perfect it?We build this model mainly from four aspects.For the first question we mainly take relevant factors into consideration. At the beginning, we divide the traffic condition into four categories. That is: Whether obeying the rule in heavy or light traffic. Then for the four categories, we analyze the problem in six aspects. At last,we get some traffic performance evaluation scores. Therefore we can evaluate the rule in a mathematical point of view. For the second question we think about the relationship between whether obeying the rule, the distance between two vehicles and the amount of vehicles passed by in an hour. We calculate the traffic flow based on the previous research’s formula. With the help of the statistics we measured before, we can calculate the score under the four circumstances. For the third question, we think about the impact of driving direction on vehicle’s speed. If the speed changes, then the score will change, too. Our model also need to modify correspondingly. For the fourth question, We analyze the difference between artificial system and intelligent system. We adopt the intelligent system to modify our model because the traffic flow and safety index change. To sum up, when the traffic is heavy the rule performs well and is effective to promote traffic flow. However, when the traffic is light the rule performs weaker and has no significant effect to promote traffic flow. But the conclusion is exactly opposite in the left-driving countries. Intelligent system doesn't have a big impact on our conclusion. Our advise is when the traffic is light, we can leave one lane just for passing vehicles.Key wordfactor analysis, optimization methods, traffic flow, the number of traffic accidents, scoreProblem backgroundIt is very important to obey the traffic rules when you drive, especially in which direction do you drive. The rules can avoid the chaos and traffic accidents. We can drive on the left or on the right.34% countries drive on the left and 66% countries drive on the right. The biggest advantage of driving on the left is human’s instinct to avoid its evils. In the case of fast movement, when you find it’s dangerous in the front, you will instinctively tilt to the left to protect your heart. The advantage of driving on the right is that drivers can take charge of the steering wheel by the left hand and can change the shift flexibly. Most of the countries adopt the rule that driving on the right. Drivers accustomed to the right don’t need to spare time on learning the left rules. At the same time, the seats drivers sit are usually on the left all over the world. If the car is the same, it is cheaper to buy the car that the seat is on the left than on the right. In our country, as the result of politics, economy and culture, we obey to the rule that drive on the right unless passing anther car.To judge whether the rule is good, we can consider traffic flow and safety. Generally speaking, the traffic rule will have good to promoting traffic flow, but when the vehicles are keeping increasing, the traffic must be heavy ,so the number of traffic accidents must increase, too. So, if we want to find a better way ,we must make the rules considering traffic flow and safety.So, regardless of the different driving directions, when we make the traffic rules, we must consider these factors. To ensure safety, some roads are under the control of intelligent systems. This can avoid driving after drinking, driving with tiredness. Theintelligent system can avoid some human judgment and forecast the traffic condition. The road will not be too crowded under the control of it. So this system can be good to the traffic.Based on this background, we build a mathematical model to check if the rule is rather good. At the same time ,we will answer the questions below:When we take safety, weather, road condition, traffic flow into consideration, how is the performance of this rule in light and heavy traffic?Is this rule effective in promoting better traffic flow? If not, how to check our model? Is our model applying to the driving on the left countries?If vehicle transportation on the same roadway was fully under the control of an intelligent system –either part of the road network or imbedded in the design of all vehicles using the roadway–to what extent would this change the results of your earlier analysis?Problem analysisAfter reading the question, we will split the problem into some small problems to solve.●Consider the road friction, weather, traffic condition, weather obey the rule’simpact on the vehicle’s speed.●Consider the relationship between the distance of two vehicles and speed.●Consider the relationship between traffic flow and the number of traffic accidents.We have two conditions: heavy traffic and light traffic. We calculate the score in different conditions when obeying the rule or not obeying the rule.●If our model can be applied to other countries?●Consider the intelligent system’s effect. Under the intelligent system, do we needto check our model?In question 1,2,3,4,we build a score to discuss the rule’s impact on traffic condition in heavy or light traffic. In the score, the traffic flow index takes up 50%,the safety index takes up 50%.In the same way, we can calculate the score when not obeying the rule.After preliminary analysis, when the traffic is light, if we don’t obey the rule, the score is higher. So, we take three road lanes for instance, when the traffic is heavy, we obey the rule. when the traffic is light, we don’t obey the rule. In question 5,We take human body’s inflexibility into consideration because it will affect the speed. In question 6,with the help of intelligent system, we can avoid heavy traffic and traffic accidents, we will have a new score and then check the rules.AssumptionsIn the optimization methods, we just think the road is straight to build a model.We set medium size vehicle as a standard. The stable speed of the vehicle in the free state(medium-car standard model)is X.Speed conversion coefficient under the standard of medium size car’s speed is A.We consider the vehicles all drive on the right, then we have four conditions, obey the rules or not, heavy traffic or light.The traffic flow is within an our. We take one normal day for instance. From 7a.m. to 9a.m and4:30p.m to 6p.m it is heavy traffic. From 23p.m to 4a.m it is light traffic. On this basis, we can build our model.Symbols descriptionModel design and solvingModel 1: Introduction of vehicle length, weather, frictions and actual speedIt covers 6 variable: vehicle length(c),the weather, road conditions and friction impact on the speed(K),the traffic impact on the speed(α),whether obeying rule impact on the speed(β),The stable speed of the vehicle in the free state(medium-car standard model)(X) and Speed conversion coefficient under the standard of medium size car’s speed(A).According to the research, we can know K and A.To K, we have five conditions:Level one: The road condition is good and vehicles can pass away normally.Level two: The road condition is slightly bad and vehicles can pass slowly.Level three: The road condition is bad and cars can pass slowly with a reasonable distance.Level four: The road condition is worse and vehicles can’t pass safely.Level five: The road condition is worst and no vehicles can pass.（Sheet 1）To A, we divide vehicles into 8 kinds. They are cars, van, motor coach, small trucks, freight train, large trucks, tractor and trailer. The stable speed of the vehicle in the free state(medium-car standard model) is X and Speed conversion coefficient under the standard of medium size car’s speed is A.Sheet 2 shows the vehicle speed Conversion Factor A：（Sheet 2）Considering these factors, the speed X is affected by α，β，A and K. According to the optimization methods, we just think these factors are proportional to X. Above all, we can define the actual speed of the vehicle.Model 2: Calculating the distance between two vehiclesWe think the least distance between two vehicles can ensure safety when braking.The vehicle brake deceleration process can be seen as linear motion.The acceleration is K ·g(g is acceleration due to gravity ). Then:22220122V kgt Vt kgt L V KXKX L gβαβα-=⎧⎪⎨-=⎪⎩=∴=Model 3: Scores considering S(the traffic fl ow within an hour) and I(thenumber of traffic accident every 100000 kilometers)Our model will give us a score considering S(the traffic flow within an hour) and I(thenumber of traffic accident every 100000 kilometers).If the score is big enough, thenthe rule is effective in promoting traffic condition.Firstly, we deduce the formula of the traffic flow. The most widely used formula in ourcountry is: time*V/(C+L).Considering heavy and light traffic, we just need to chooseone our in heavy time and light time.Secondly, we deduce the formula of the number of traffic accident every 100000kilometers(I).Obviously, the actual speed affects I. Here are some reasons below:(1)affecting eyesight. When speed is increasing, the drivers eyesight will be poorerthan the still eyesight.(2)affecting visual range. When speed is increasing, the visual range will be small andnarrow.(3)affecting recognizable sight. When driving, the driver needs to identify varioustraffic signs or traffic environment. The identification distance will be different under the different speed. When the speed is too high, the identification distance will be small. It will be hard to be aware of the road condition.(4)affecting judgment. It is according to the position change that drivers recognizethe object.When the vehicle is in a traveling state , the objects outside the vehicle change the position slowly and small.If the speed is high, then the change is slow and small. It is really hard to identify. Therefore, when the driver is driving, his ability to distinguish an object will fall.(5)impact on the braking distance and security region. As picture 3 shows,a vehiclewith the speed of 30km / h can stop his car in 13 meters. The passers-by will be safe outside the 13 meters. The damage is zero and this area is safe to passers-by.However, a vehicle with the speed of 50km / h can stop his car in 26 meters. If the passers-by still walk in 13metres,it will be dangerous. The safety index is zero.(picture 3)Sheet 4 is an ordinary road braking distances on different speed：Sheet 5 :stopping sight distance in a highway：In 2004, after analyzing the traffic accidents, Chinese scholars think accident rate increases when the standard deviation of speed increases. The bigger the standard deviation is, the higher the accident rate is. At the same time, research indicates that speed standard deviation and the accident rate are exponential relationship.With the improvement in the level of speed’s discrete, accident rate will be growing exponentially .Below are the picture 6:(picture 6)In 1993,Monash University Accident Research Centre summed a function model to tell the relationship between the speed level and traffic accidents rate.The function is :23=+∆+∆5000.80.014I V VIn the formula,I means the number of traffic accident every 100000 kilometers.-,Our model is to use this formula to calculate the number of V∆means V Xaccidents .At last, how do we calculate the score P ? That is the traffic flow index takes up 50% and the safety index takes up 50%.111V L + 1S )221V L +2S )331V L +S )10.8(V )X -(1I ) By linear regression, we can calculate αand β.Under the circumstance of heavy traffic, we take the scores when obeying the rule or not obeying the rule for comparison. We can have a conclusion that in this case the score is higher when obeying the rule. Similarly, under the circumstance of light traffic, we take the scoreswhen obeying the rule or not obeying the rule for comparison. We can have a conclusion that in this case the score is higher when not obeying the rule. Therefore, we can know the rule performs well in heavy traffic but performs bad in light traffic.Model 4: Improving traffic flowThe model here is similar the model above when calculating traffic flow./(C )S t V L =+.Similarly, we have to conditions. That is heavy traffic and light traffic, then we calculate the traffic flow when obeying the rule or not obeying rule to see if the rule is effective to promote the traffic flow.111V L +1S ) 221V L +2S ) 331V L +3S )According to the α、β ,we can calculate the traffic flow S easily.We find it obviously when the traffic is heavy, we will have a bigger traffic flow when obeying the rule. When the traffic is light, the traffic flow is bigger if not obeying the rule.Therefore, We promote When the traffic is heavy, we need to obey the previous rule--- drivers need to drive in the right-most lane unless they are passing another vehicle. When the traffic is light, we take two lanes in the right side for driving and the left one side for passing vehicles.(3 lanes for instance)At this time we control the vehicle’s speed in a reasonable range and then insureβ1βin a flexible range. Then the traffic flow is bigger when not obeying the and2previous rule.Model 5: Can our new model be applied to driving-in-the-left countries ? Whether our new model can be applied to driving-in-the-left country depends on the traffic flow and safety index. At this moment ,our model becomes that keep left when driving unless passing vehicles in heavy traffic, while in light traffic, we retained two lanes on the left as travel lanes and one left lane as a passing lane. Similarly, taking score into consideration, which is composed of traffic and security Index. In fact, it is easy to find that β1 and β2 exchanged themselves, so w e have enacted new rules about reciprocity. It is that we retained two lanes on the left as travel lanes and one remaining lane as a passing lane when the traffic is heavy. In addition, maintaining the previous rule that keep left except passing in light traffic.Model 6: Intelligent system modelIf the intelligent system replace the artificial system, the safety index will increase in a big extent. At the same time, the traffic flow will increase, too. However,βwill not change because this factor depends on human’s performance. So we need to modify our model partly. When the traffic is heavy, we keep the previous rule. When the traffic is light, we make one lane for passing vehicles. With the help of intelligent system, we can make two lanes in the most side for passing and the lanes left for driving.Weaknesses and strengths of the modelAt the very beginning, we analyze some factors that will affect the result. We just think the problem is not very complex, so we simplify our work. We give all the index a reasonable flexible range. We also consult some authority research.However, we also have some disadvantages. We don’t take the traffic lights and some more complex road condition into consideration.Time is urgent, we don’t find more statistics to test our model accurately. Reference[1] [J] The model of the relationship between vehicle speed and traffic flow, Wei Wang, Nan Jing ,Jiangsu province[2] [D] The research of intelligent system and traffic flow, Jianhui Dai, Tian Jin[3] Addison, Low Paul S, David J. Order and chaos in the dynamics of vehicle platoons[ J] . Traffic Engineering & Control,1996, 37( 7 8) : 456 -459.[4] Daganzo C F, Cassidy M J, Bertini R L. Possible explanations of phase transitions in highway traffic[ J] . Transportation Research , Part A, 1999, 33( 5) : 365- 379.[ 5] Jiang Y. Traffic capacity speed and queue- discharge rate of Indiana. s four- lane freeway work zones[ A] . In: Transportation Research Record 1657, TRB , National Research Council [ C] . Washington D C, 1999. 39 -44.[6] Schonfeld P, Chien S. Optimal work zone lengths for two lane highways [ J] . Journal of Transportation Engineering , Urban Transportation Division, ASCE , 1999, 125( 1) : 21-29.[ 7] Nam D D, Drew D R. Analyzing freeway traffic under congestion: traffic dynamics approach [ J] . Journal of Transportation Engineering, Urban Transportation Division, ASCE,1998, 124( 3) : 208 -212.Appendix[1]alpha1=0.56;alpha2=1.76;beta1=1.16;beta2=0.98;X=20;K=0.65;g=10;C=5;V1=beta1*alpha1*K*X;V2=beta1*alpha2*K*X;V3=beta2*alpha1*K*X;V4=beta2*alpha2*K*X;L1=((beta1)^2*(alpha1)^2*K*X^2)/(2*g);L2=((beta1)^2*(alpha2)^2*K*X^2)/(2*g);L3=((beta2)^2*(alpha1)^2*K*X^2)/(2*g);L4=((beta2)^2*(alpha2)^2*K*X^2)/(2*g);S1=((1*V1)/(C+L1))*3600S2=((1*V2)/(C+L2))*3600S3=((1*V3)/(C+L3))*3600S4=((1*V4)/(C+L4))*3600W1=S1/(S1+S3);W2=S2/(S2+S4);W3=S3/(S1+S3);W4=S4/(S2+S4);I1=500+0.8*(abs(V1-X))^2*3.6^2+0.014*(abs(V1-X))^3*3.6^3;I2=500+0.8*(abs(V2-X))^2*3.6^2+0.014*(abs(V2-X))^3*3.6^3;I3=500+0.8*(abs(V3-X))^2*3.6^2+0.014*(abs(V3-X))^3*3.6^3;I4=500+0.8*(abs(V4-X))^2*3.6^2+0.014*(abs(V4-X))^3*3.6^3;Q1=1-(I1/(I1+I3));Q2=1-(I2/(I2+I4));Q3=1-(I3/(I1+I3));Q4=1-(I4/(I2+I4));P1=(W1+Q1)/2P2=(W2+Q2)/2P3=(W3+Q3)/2P4=(W4+Q4)/2S1 =2.8993e+003S2 =1.6144e+003S3 =2.8809e+003S4 =1.8482e+003P1 =0.5283P2 =0.4011P3 =0.4717P4 = 0.5989[2]alpha1=0.56;alpha2=1.76;beta1=1.16;beta2=0.98;X=8;K=0.65;g=10;C=5; V1=beta1*alpha1*K*X;V2=beta1*alpha2*K*X;V3=beta2*alpha1*K*X;V4=beta2*alpha2*K*X;L1=((beta1)^2*(alpha1)^2*K*X^2)/(2*g);L2=((beta1)^2*(alpha2)^2*K*X^2)/(2*g);L3=((beta2)^2*(alpha1)^2*K*X^2)/(2*g);L4=((beta2)^2*(alpha2)^2*K*X^2)/(2*g);S1=((1*V1)/(C+L1))*3600S2=((1*V2)/(C+L2))*3600S3=((1*V3)/(C+L3))*3600S4=((1*V4)/(C+L4))*3600W1=S1/(S1+S3);W2=S2/(S2+S4);W3=S3/(S1+S3);W4=S4/(S2+S4);I1=500+0.8*(abs(V1-X))^2*3.6^2+0.014*(abs(V1-X))^3*3.6^3;I2=500+0.8*(abs(V2-X))^2*3.6^2+0.014*(abs(V2-X))^3*3.6^3;I3=500+0.8*(abs(V3-X))^2*3.6^2+0.014*(abs(V3-X))^3*3.6^3;I4=500+0.8*(abs(V4-X))^2*3.6^2+0.014*(abs(V4-X))^3*3.6^3;Q1=1-(I1/(I1+I3));Q2=1-(I2/(I2+I4));Q3=1-(I3/(I1+I3));Q4=1-(I4/(I2+I4));P1=(W1+Q1)/2P2=(W2+Q2)/2P3=(W3+Q3)/2P4=(W4+Q4)/2S1 =2.0689e+003S2 = 2.7959e+003S3 =1.8259e+003S4 =2.8860e+003P1 = 0.5274P2 = 0.4795P3 = 0.4726P4 = 0.5205[3]alpha1=0.56;alpha2=1.76;beta1=1.16;beta2=0.98;X=30;K=0.65;g=10;C=5;V1=beta1*alpha1*K*XV2=beta1*alpha2*K*XV3=beta2*alpha1*K*XV4=beta2*alpha2*K*XL1=((beta1)^2*(alpha1)^2*K*X^2)/(2*g)L2=((beta1)^2*(alpha2)^2*K*X^2)/(2*g)L3=((beta2)^2*(alpha1)^2*K*X^2)/(2*g)L4=((beta2)^2*(alpha2)^2*K*X^2)/(2*g)S1=((1*V1)/(C+L1))*3600S2=((1*V2)/(C+L2))*3600S3=((1*V3)/(C+L3))*3600S4=((1*V4)/(C+L4))*3600W1=S1/(S1+S3)W2=S2/(S2+S4)W3=S3/(S1+S3)W4=S4/(S2+S4)I1=500+0.8*(abs(V1-X))^2*3.6^2+0.014*(abs(V1-X))^3*3.6^3 I2=500+0.8*(abs(V2-X))^2*3.6^2+0.014*(abs(V2-X))^3*3.6^3 I3=500+0.8*(abs(V3-X))^2*3.6^2+0.014*(abs(V3-X))^3*3.6^3 I4=500+0.8*(abs(V4-X))^2*3.6^2+0.014*(abs(V4-X))^3*3.6^3 Q1=1-(I1/(I1+I3))Q2=1-(I2/(I2+I4))Q3=1-(I3/(I1+I3))Q4=1-(I4/(I2+I4))P1=(W1+Q1)/2P2=(W2+Q2)/2P3=(W3+Q3)/2P4=(W4+Q4)/2S1 =2.6294e+003S2 =1.1292e+003S3 =2.7898e+003S4 =1.3159e+003P1 =0.5243P2 =0.3510P3 =0.4757P4 =0.6490。

数学建模美赛三等奖论文

Water, Water, EverywhereSummaryDue to population growth, economic development, rapid urbanization, large-scale industrialization and environmental concerns water stress has emerged as a real threat. [1]This paper was motivated by the increasing awareness of the need for fresh water since fresh water crisis is already evident in many areas on the world, varying in scale and intensity.Firstly, we testify water demand and supply sequence are stable by means of unit root test, then predict the freshwater demand and supply in 2025 by using ARMA Model and Malthus Population Model .Secondly, we give more concern on four aspects: Diversion Project, Desalinization, Sewage treatment and Conservation of water resources, building some models such as Cost-benefits analysis and Tiered water pricing model. Comparing the cost-benefit ratio, the sewage treatment cost-benefitratio is the smallest--0.142, that is to say it is more cost-efficient.Finally, we use our models to analyze the impacts of these strategies, we can conclude that conservation of water resources is the most feasible.Keywords:Cost-benefit analysis ARMA ModelTiered water pricing modelA Letter to a governmental leadershipFebruary 4, 2013Dear Sir,During the four days working, our team spares no effort using cost and benefits analysis determine water strategy for 2013 about how to use water efficiently to meet the need in 2025. Now, we outline our conclusion to you.z Diversion ProjectThe South-North Water Transfer Project is a multi-decade infrastructure project solved the unbalance of water resource. The cost is 6.2yuan/3m, and it will much higher while the distance is more than 40 kilometers.z DesalinizationDesalinization utilizes the enormous seawater and provides freshwater in a cheaper price. However, interior regions with water scarcity can hardly benefit from it as most desalinization manufacturers located on eastern coastal areas. The cost of production is 5.446 yuan/t, but the transport costs less the cost-efficient competitiveness. The cost can be decreased by using more advanced technology.z Sewage treatmentSewage treatment can relief the environmental impact of water pollution by removing contaminants from water, the cost of Sewage treatment is 0.5yuan/t. z Conservation of water resourcesConservation makes sure of the source of rational use of water. There are several approaches on water resources conservation, the main problem is the lack of supervision. The benefit-cost ratio is between 0.95 and 3.23, and it has a high return-investment ratio.z Each of the above water strategy has its own advantages and disadvantages, we should consider the aspects of economic, physical, environmental, geographical, and technique factors overall, then choose the optimal strategy for different area.Yours sincerely,COMAP #23052ContentI Introduction (2)II Assumptions (3)III Models (3)3.1 The prediction of freshwater shortage in 2025 (3)3.1.1 The prediction of freshwater demand (3)3.1.1.1 The description of basic model (3)3.1.1.2 Model building (4)3.1.1.3 Model prediction (5)3.1.2 The prediction of freshwater supply (7)3.1.2.1 Model building (7)3.1.2.2 Model prediction (8)3.1.3. Conclusion (9)3.2Water strategy (9)3.2.1 Diversion Project (9)3.2.2 Desalinization (14)3.2.3 Sewage Treatment (16)3.2.4 Conservation of water resources (19)3.2.4.1 Agricultural water saving (20)3.2.4.2 Life water saving (21)IV The influence of our strategy (25)4.1 The influence of Water Diversion Project (25)4.2 The influence of desalination (25)4.3 The influence of sewage treatment (26)4.4 Water-saving society construction (26)V References (27)VI Appendix (28)I IntroductionAccording to relevant data shows that 99 percent of all water on earth is unusable, which is located in oceans, glaciers, atmospheric water and other saline water. And even of the remaining fraction of 1 percent, much of that is not available for our uses. For a detailed explanation, the following bar charts show the distribution of Earth's water: The left-side bar shows where the water on Earth exists; about 97 percent of all water is in the oceans. The middle bar shows the distribution of that 3 percent of all Earth's water that is fresh water. The majority, about 69 percent, is locked up in glaciers and icecaps, mainly in Greenland and Antarctica.[2] Except for the deep groundwater which is difficult to extract, what can be really used in our daily life is just 0.26 percent of all water on earth.Figure 1 The distribution of Earth's waterFreshwater is an important natural resource necessary for the survival of all ecosystems. There is a variety of unexpected consequence due to the lack of freshwater: 6,000 children die every day from diseases associated with unsafe water and poor sanitation and hygiene; Unsafe water and sanitation leads to 80% of all the diseases in the developing world;[3]Species which live in freshwater may be extinct, thus, breaking the food chain balance severely; The development of economic slow down in no small measure.It is with these thoughts in mind, many people think freshwater is very important than ever before.So, how to use freshwater efficiently? What is the best water strategy? Readmore and you will find more.II AssumptionsIn order to streamline our model we have made several key assumptions ：1. We chose China as the object study.2. The water consumption of the whole nation could be approximate regardedas the demand of water .3. The Precipitation is in accordance with the supply of water .4. No considering about sea level rising because of global warmingIII Models3.1 The prediction of freshwater shortage in 2025How much freshwater should our strategy supply? Firstly, our work is to predict the gap between freshwater demand and supply in 2025. We obtain thefreshwater consumption data from China Statistical Yearbook. 3.1.1 The prediction of freshwater demandWe forecast the per capita demand for freshwater by building the ARMA Model .3.1.1.1 The description of basic modelThe notation ARMA(p, q) refers to the model with p autoregressive termsand q moving-average terms. This model contains the AR(p) and MA(q) models,mathematical formula is:qt q t t t p t p t t t y y y y −−−−−−−−−−+++=εθεθεθεφφφ......22112211 (1) AR(p) modelt p t p t t t y y y y εφφφ+++=−−−...2211 (2) MA(q) model q t q t t t t y −−−−−−−=εθεθεθε....2211 (3)),.....,2,1(p i i =φ ,),.....,2,1(q j j =θare undetermined coefficients of themodel, t ε is error term, t y is a stationary time series.3.1.1.2 Model buildingAll steps achieved by using EviewsStep1: ADF test stability of sequenceNull hypothesis:1:0=ρH , 1:1≠ρH , ρis unit root.Table 1 Null Hypothesis: Y has a unit root Exogenous: Constant Lag Length: 3 (Automatic based on SIC, MAXLAG=3) t-Statistic Prob. Augmented Dickey-Fuller test statistic -5.3783580.0040 Test critical values: 1% level-4.582648 5% level -3.32096910% level -2.801384We know Prob=0.0040 that we can reject the null hypothesis, and thenydoesn’t have a unit root, in other words, is stationary series. Step 2: Building the ARMA ModelThen we try to make sure of p and q by using the stationary series y .Table 2Date: 02/02/13 Time: 11:08Sample(adjusted): 2001 2011Included observations: 11 after adjusting endpointsConvergence achieved after 12 iterationsBackcast: 1998 2000Variable Coefficie nt Std. Error t-StatisticProb.AR(1) 1.0105040.005813173.8325 0.0000MA(3) 0.9454040.03650725.89639 0.0000R-squared 0.831422 Mean dependent varAdjustedR-squared 0.812692 S.D. dependent varS.E. of regression 5.085256 Akaike info criterionSo, we can get the final model, is:310.9454041.010504−−+=t t t d y y ε (4)3.1.1.3 Model predictionStep 1: The prediction of per capita freshwater demandWe use model (4) to predict the per capita demand of freshwater in the year2025, the result as Figure3.Figure 2 sequence diagram of dynamic predictionFrom the diagram, we can see the per capita freshwater demand is raising.The detailed data as Table3: Table 3 2010 2011 2012 2013 2014 2015 2016 2017 483.3584 488.4357 493.5662 498.7507503.9896509.2836514.6332 520.03892018 2019 2020 2021 2022 2023 2024 2025 525.5015 531.0214 536.5993 542.2358547.9315553.6871559.503 565.3801(cu.m/person)Through the above efforts, we get the 2025 per capita freshwater demand is565.3801 cu.mStep 2: The prediction of the whole freshwater demandThe relationship among d Q ,t N ,daverage Q is: daverage t d Q N Q ×= (5)d Q is the whole demand of freshwater, t N is the total population ,daverage Q is per capita of freshwater demand.Then we etimate the total population by the Malthus Population Model . rt e N t N 0)(=[4] (6))(t N is the population at time t,0N is the population at time 0,r is net relative growth rate of the populationrt e N N 2011)2025(= (7)By calculating, we get:(billion)42.11.347)2025(1500479.0≈=×e N (8)At last,we could get the whole demand of freshwater while the time is 2025.38.5652.14)2025(×=×=daverage d Q N Q ()cu.m million 100 8028.396= (9)3.1.2 The prediction of freshwater supplySimilarily,we predict freshwater supply using the ARMA Model. 3.1.2.1 Model buildingStep1: ADF test stability of sequenceNull hypothesis:1:0=ρH , 1:1≠ρH , ρis unit root. Table 4 Null Hypothesis: D(Y) has a unit root Exogenous: Constant Lag Length: 2 (Automatic based on SIC, MAXLAG=3)t-Statistic Prob. Augmented Dickey-Fuller test statistic-9.433708 0.0002 Test critical values: 1% level -4.803492 5% level -3.40331310% level -2.841819From the table, we find that first difference of supply data is smooth, we canreject the null hypothesis, that is ()y D is a smooth series.Step 2: Building the ARMA ModelWe use the smooth series ()y D to make sure the number of order.Table 5Date: 02/02/13 Time: 14:16Sample(adjusted): 2002 2010 Backcast: 1999 2001Variable CoefficientStd. Error t-Statistic Prob. AR(1) 0.6351030.158269 4.012804 0.0051 MA(3) -0.9923370.069186-14.34306 0.0000 R-squared 0.812690 Mean dependent var 50.51111Adjusted R-squared 0.785931 S.D. dependent var 119.1793S.E. of regression 55.14139 Akaike info criterion 11.05081Sum squared resid 21284.01 Schwarz criterion11.09464 Log likelihood -47.72864 Durbin-Watson stat 2.895553Then ,we get the final model is:)0.992337D(-)0.635103D()(31−−=t t t s y y D ε (10) 3.1.2.2 Model predictionWe use the effective model to predict freshwater supply in short-term until theyear 2025.Figure 3 sequence diagram of dynamic predictionFrom the diagram, we can see the supply remains unchanged basically .T The detailed data as Table6: Table 6 2010 2011 2012 2013 2014 2015 2016 2017 5630.203 5630.594 5630.843 5631.0015631.1025631.1655631.206 5631.2322018 2019 2020 2021 2022 2023 2024 2025 5631.248 5631.258 5631.265 5631.2695631.2725631.2735631.275 5631.275(100 million cu.m)According to the above data,we gain the supply of freshwater 2025, is5631.275(100 million cu.m)3.1.3. ConclusionFrom the above result,we find a serious issue:Table 7Year Demand offreshwater Supply of freshwater Net demand Unit2025 8028.396 5631.275 2397.121(100 million cu.m)In the year 2025, China will face the serious situation of freshwater shortage, the gap will reach 2397.121(100 million cu.m), therefore, in order to avoid this, we need to determine a series strategy to utilize freshwater efficiently.3.2Water strategy3.2.1 Diversion ProjectOn one hand, in view of Figure4, we can get information: Southeast coast is of the maximum precipitation, followed by the northern region, the western least.Figure 4 Precipitation Allocation Map of Major CitiesOn the other hand, in view of Figure 5, we can get information: The northern region and the southern coastal areas have the most water consumption, the western use less.Figure 5 Water Use MapDetailed data see to attached Table8 and Table9.South-to-North Water Diversion ProjectThe South–North Water Transfer Project is a multi-decade infrastructure project of China to better utilize water resources. This is because heavily industrialized Northern China has a much lower rainfall and its rivers are running dry. The project includes a Eastern, a Central and a Western route.Figure 6 The route of South-to-North Water Diversion ProjectHere, we take Western Route Project (WRP) as a representative, analysis the cost and benefits. As the strategic project to solve the problem of poorer water Northwest and North China, WRP will divert water from the upper reach of Yangtze River into Yellow Rive.Cost and benefits analysisThe direct quantitative economic benefits include urban water supply economic benefits, ecological environment water supply economic benefits, and the Yellow River mainstream hydroelectric economic benefits.[5]Urban water supply economic benefits:(1) Calculation MethodIn view of the water shadow price is difficult to determine, the equivalent engineering is not easy to choose, and the lack of water loss index is unpredictable, combined with the stage job characteristics, we select the method of sharing coefficient to calculate the urban water supply economic benefits.(2) Calculation ParametersThe Water consumption quota of per ten thousand yuan industrial output value is based on status quota, the predicted water consumption quota of per ten thousand yuan output value according to reach in 2 0 2 0 is :Lanzhou tom/ ten thousand yuan, gantry to Sanmenxia HeKouZhen river section for 26 3m/ ten thousand yuan. After a comprehensive analysis, set the reach for 20 3industrial water supply benefit allocation coefficient values 2.0 %.(3) Calculation ResultsAccording to (1) and (2), get table 10:Table 10water supply 3.2 billion 3.mproject benefits 20 billion yuan.8yuan /3maverage economic benefit 70z Ecological environment water supply economic benefits:(1) Calculation methodTake Forestry and animal husbandry as the representative, calculate whoseirrigation Economic benefits, and consider the allocation function of water supply. Analyse forestry benefits in reference with the increased wood savings, Animal husbandry in reference with the increased output of animals which were feeded by the incresed irrigation pasture (represented by sheep), both Forestry and animal husbandry account for half of the Ecological environment water supply.(2) Calculation parameters Set the water consumption quotas of Forestry irrigation unified as 233750hm m , the water supply sharing coefficient of Xiang irrigation as 0.60. In the calculation of forestry benefit, the increase of accumulated timber amount is ()a hm m ⋅235.22, timber price is 3300m yuan ; in the calculation of animal husbandry benefit , the increased stocking rates of unit pasture area is 25.22hm , taken a standard sheep price as yuan 260.(3) Calculation ResultsAccording to (1) and (2), the ecological environment water supply economic benefits is 714.1 billion, in which, The Yellow River replenishment economic benefits is 008.1billion yuan.z Hydroelectric economic benefits.(1) Diversion increased energy indicators:The increased electricity indicators is 306.9billion h kw ⋅, capacity enlargement the scale of 241 ten thousand kw .(2) Calculation methodTake the Optimal equivalent alternative engineering cost method, chosen fire electricity as an alternative project which can meet the power requirements of grid electricity equally. The sum of alternative engineering required annualinvestment translation and the annual running costs is increased annual power generating efficiency of the Yellow River cascade hydropower stations. (3) Calculation parametersThe power plant construction investment of kw $450, duration of five years, the investment proportion were 10%, 25%, 35%, 25%, 5%. Both the economic life of mechanical and electrical equipment and the metal structures equipment are taken as 20 years, considering the update ratio as 80% of the original investment. Standard coal price is taken as 160 dollars, standard coal consumption is taken as ()h kw g ⋅350. The fixed run rates take 4.5%, thesocial discount rate is 12%, the hydropower economic useful life of 5 years.(4) Calculation ResultsBy analysis and calculation, the first phase of water regulation produce the hydropower economic benefit is 3.087 billion.z Total economic benefits:Preliminary cost estimates of the project diversionOn the basis of economic nature classification, the total cost includes themachinery depreciation charges, wages and welfare costs, repair costs, thecost of materials,water district maintenance fees, management fees, water fees, interest expense and other . Analysis in the light of various estimates condition, the cost of water diverted into the Yellow River is 31~7.0m yuan c =The cost-benefit rate ()85.8~2.61∈=rc ω (11) 3.2.2 DesalinizationThough diversion project can balance water supply between places one has enough water and the other has water shortage, the costs will higher than desalinization when the distance more than 40 kilometers.Desalinization and comprehensive utilization of the work are increasingly taking centre stage on the problem of solving freshwater scarcity. Many countries and areas devote to optimize an effective way by enhancing the development of science and technology.According to the International Desalination Association, in 2009,14,451 desalination plants operated worldwide, producing 59.9 million cubic meters per day, a year-on-year increase of 12.3%.[6] The production was 68 million 3m in 2010, and expected to reach 120million 3m by 2020; some 40 million 3m is planned for the Middle East.[7]China has built more than 70 sets of sea water desalinization device with the design capacity of 600,000m3 and an average annual growth rate of more than 60%; technology with independent intellectual property rights of a breakthrough in the reverse osmosis seawater membranes, high pressure pumps, devices for energy recovery achieved significant progress, the desalinization rate raises from 99.2% to 99.7%; conditions of industrial development and the desalination market has been basically formed.MethodsDe-salinization refers to any of several processes that remove some amount of salt and other minerals from saline water. More generally, desalination may also refer to the removal of salts and minerals.[8] Most of the modern interest in desalination is focused on developing cost-effective ways of providing fresh water for human use.There are two main methods of desalinization:1. Extract freshwater from saline water: Distillation (Multi-stage flash distillation, Vapor compression distillation, Low temperature multi-effect distillation), Reverse osmosis, Hydrate formation process, Solvent extraction, Freezing.2. Remove salt from saline water: Ion exchange process, Pressure infiltration method, Electroosmosis demolition method.For desalination, energy consumption directly determines the level of the cost of the key. Among the above methods, reverse osmosis is more cost-effective than the other ways of providing fresh water for human use. So, reverse osmosis technology has become the dominant technology in international desalinization of seawater.The following two figures show the working principle diagram of a reverse osmosis system.Figure7 working principle diagram of a reverse osmosis systemCost and benefits analysisTable 12 general costs for a reverse osmosis systemItem Unitprice(yuan/t)Chemicals cost 0.391electric charge 2.85Wages 0.034 Labor costWelfare 0.04 Administrative expenses 0.0008maintenance costs 0.23Membrane replacement cost 0.923Depreciationexpense Fixed assets depreciation0.97expenseTotal costs 5.446Table 13 general benefit for a reverse osmosis systemItem ValueHourly output(t) 10Working hours/day24 Daily output(t)240 Working days/year 365 Yearly output(t)87600 Yearly other benefits(yuan)310980 Unit water other benefits3.55 Water Price(yuan/t)8 Unit water total benefits11.55 Unit water total benefit 55.11=rWater cost-benefit ratio 4715.055.11446.52===r c w (12) 3.2.3 Sewage TreatmentSewage treatment is an important process of water pollution treatment. It uses physical, chemical, and biological ways removing contaminants from water . Its objective is to relief the environment impact of water pollution.This diagram shows a typical sewage treatment process.Figure 8 Sewage treatment flow mapTake Sewage Treatment Plant in east china as an example to analysis the cost and benefit of sewage treatment.Suppose:Sewage treatment scale d t x 100001=,The Sewage Treatment Plant workdays in a year 300=d ,Concession period is twenty to thirty years, generally 251=t years, Construction period is one to three years, generally 32=t years.Operation period = Concession period - Construction period.Cost estimation Table 14 fixed investment estimate c1(ten thousand Yuan)number project ConstructioninvestmentEquipment investment 1 Preprocessing stage38 27 2Biological treatment section 42 134 3End-product stage 11 44 4 Sludge treatment section 6323 5 accessory equipment 456 Line instrument 687 Construction investment 3008 Unexpected expense 809 Other expense 10010 Total investment975 Table 15 Operating expense estimate c2 (ten thousand Yuan)[9]number project expenses1 maintenance expenses 6.52 wages 103 Power Consumption 404Agent cost 10 5 Small meter operating cost 66.56 Amortization of intangibles 127Amortization of Construction 6.6 8Amortization of Equipment 19.8 9Annual total cost 104.9 10 Tons of water operation cost 0.29Annual total investment 15022213=+÷=c c c ten thousand YuanAnnual amount of sewage treatment t x x 3000000100003003001=×=×= Unit sewage investment t yuan t yuan x c c 5.03000000150000034=÷=÷= Benefit analysisSewage mainly comes from domestic sewage(40%), industrial sewage(30%), and the others(including stormwater , 30%)Sewage treatment price: domestic sewage is about t yuan 8.0, industrialsewage is about t yuan 5.1, and other is about t yuan 5.2.Unit sewage treatment approximate price t yuan t yuan t yuan tyuan p 52.1%302%305.1%408.01=×+×+×=Unit Sewage treatment benefit:t yuan p p r 52.321=+= Cost-benefit ratio 142.052.35.043===r c ω (13) 3.2.4 Conservation of water resourcesTo realize the sustainable development of water resources, one of the important aspects is the conservation of water resources. Saving water is thekey of conservation, so, we the construction of water-saving society is the keyof water resources conservation strategy.To construct the water-saving society, we give more concern about two aspects:agricultural water saving and life water saving. Finally, we analysis the cost andbenefit about water-saving society by building model.3.2.4.1 Agricultural water savingStrategic suggestions of water-saving agriculture1. Strengthen the government policies and public finance support2. Mobilizing all social forces to promote water-saving agriculture development3. Innovating enterprises to improve the science and technology4. Suggesting countries to regard water saving as a basic state policy5. Implement the strategy of science and technology innovationwater saving function product research and development as the key point, the research and development of a batch of suitable for high efficiency and low energy consumption, low investment, multi-function water saving and high efficient agriculture key technology and major equipment. Micro sprinkler irrigation water saving technology and equipment is the typical technology.[10] Typical analysis: drip irrigation technologyIrrigation uniformity DU and field irrigation water utilization αE can be expressed as the technical elements of the function ：[11]),,,,,,(01co c in t F I S n L q f DU α=),,,,,,,(0SMD t F I S n L q f E co c in αα=RD SMD fc )(θθ−=in q is single discharge into earth,L is (channel) long,n is manning coefficient,0S is tiny terrain conditions,c I is soil infiltration parameters,αF is (channel) cross-sectional parameters,co t is irrigation water supply time,SMD is irrigation soil water deficit value,fc θ is the soil field capacity,θis the soil moisture content,RD is the root zone depth.According to the study we found that the use of modern surface irrigation technology such as sprinkler irrigation, micro spray irrigation and pressure irrigation system, can improve the utilization rate of water to 95%, better than common ground water saving irrigation mode, more than 1/2 ~ 2/3 of water-saving irrigation mode, therefore, advanced water saving technology is very important. 3.2.4.2 Life water saving China is a country with a large population and scarce water , so we should use water more reasonably and effectively.Tiered water pricing modelThe model is for all types of users in certain period to regulate basic water consumption, in the basic consumption, we collect fees by the basic price standard, when actual consumption beyond basic consumption, the beyond part will introduce penalty factor: the more water exceed, the higher punishment rate will be. For actual consumption is less than basic consumption, the user can get additional incentives, encouraging people to save water .[12] Three ladder water price modelWe assume that urban resident’s basic water consumption is 1q , the first stage water price is 1P , the second stage water price is 2P , by analogy, q P is used to express the water price in stage q , model formula is()()⎪⎪⎩⎪⎪⎨⎧−++−+−+=−)(11211121111m m q q q p q q p q p q q p q p q p p L L L (14) From the equation (14), that in the tiered water pricing system, as more price levers are divided, it will be more able to reflect the city water supply’s public property and public welfare, be much beneficial to motive users to save water . On the other hand, much more price levers will be bound to increase the transaction cost of both the water supplier and the water user . Seeing from practical application effects of the current step water price model , Three ladder water price model much meets the actual functional requirements of urban water supply system in our country, the specific pricing method see Figure 9.Figure 9 Taking three step level water price model, can to some extent, Contain people waste the limited water resources , promote enterprises into taking all kinds of advanced technologies to improve the Comprehensive utilization of water resources, and realize the goal of urban water conservation and limited water resources Sustainable and high-efficiency using and saving. In conclude, it’s an effective and feasible strategy at present.Cost-Benefit Analysis of water-saving society construction1. Cost-Benefit Analyses ModelThe benefit of the water-saving society construction n s B B B −= (15) :s B water use benefit of the whole society in Water-saving condition。

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The Design of Snowboard HalfpipeAbstract: Based on the snowboard movement theory, the flight height depends on the out- velocity. We take the technical parameters of four sites and five excellent snowboarders for statistical analysis. As results show that the size of halfpipe (length, width and depth, halfpipe slope) influence the in- velocity and out- velocity. Help ramp, the angle between the snowboard’s direction and speed affect velocity ’s loss.For the halfpipe, we established the differential equation model, based on weight, friction, air density, resistance coefficient, the area of resistance, and other factors and the law of energy conservation. the model’s results show that the snowboarders’ energy lose from four aspects(1) the angle between the direction of snowboard and the speed, which formed because of the existing halfpipe(2) The friction between snowboard and the surface(3) the air barrier(4) the collision with the wall for getting vertical speed before sliping out of halfpipe.Therefore, we put forward an improving model called L-halfpipe，so as to eliminate or reduce the angle between the snowboard and the speed .Smaller radius can also reduce the energy absorption by the wall.At last, we put forward some conception to optimize the design of the halfpipe in the perspective of safety and producing torsion.Key words:snowboard; halfpipe; differential equation model;L-halfpipeContents1. Introduction (3)简介1.1the origin of the snowboard course problems (3)滑雪课程的起源问题。

1.2 the background (3)背景2. The Description of Problem (3)问题的描述2.1Practical halfpipe’s requirements (3)实用halfpipe的需求2.1.1 the maximum vertical and the largest body twist (3)最大垂直和最大的身体扭曲2.1.2 Speed analysis (3)速度分析2.2 Halfpipe’s own conditions (4)Halfpipe自身的条件2.2.1 Friction (4)摩擦2.2.2 the size of halfpipe (4)halfpipe的大小3. Model (4)模型3.1 Definitions and Symbols (4)定义和符号3.2 Assumptions (5)假设3.3 the simple analysis of gravity and friction when sliding in the halfpipe (5)简单的分析重力和摩擦力的halfpipe时滑动3.4 in-velocity of factors (6)速度的因素3.4.1 the snowboarder’ angle when in and the speed loss (6)滑雪在角和速度上的损失3.5 out-velocity of factors (8)初速度的因素3.5.1 Help ramp (8)帮助坡道3.5.2 the force point and the plate angle when out (9)力的点和板角3.5.3 the snowboarder’ angle when out and the speed loss (9)滑雪在角和速度上的损失3.5.4 H alfpipe’s Radius (11)Halfpipe的半径3.6 the in-velocity comparison with the out- velocity (14)速度与速率的对比3.7Snowboarder’s position impact on the speed (14)滑雪的位置影响速度3.8 the entire movement of the energychange in the halfpipe (15)在halfpipe中整个运动的能量变化3.9 the balance of speed after considering the air resistance (18)后速度的平衡考虑空气阻力3.10 L-halfpipe (19)左halfpipe3.11 Solution and Result (20)解决方案和结果4. Conclusions (21)总结4.1 Conclusions of the problem (21)结论的问题5. Future Work (21)工作展望5.1 other models (21)其他模型5.1.1 H alfpipe’s location outdoor (22)Halfpipe位置的户外5.1.2 H alfpipe’s material (22)Halfpipe的材料6. References (22)参考文献1. IntroductionIn order to indicate the origin of the snowboard course problems, the following background is worth mentioning.1.1 The origin of the snowboard course problemsIn the past, a significant amount of half pipe anxiety was due to the learning curve of a new sport, and educating resorts and pipe construction person nelson how to prepare the best shapes with basic resort equipment. This mode of operation is changing with the advent of new snowboard specific technology both in machine and hand tools. As technology has made half pipes better, the standards have also been proved. Most half pipe riders have a vision of what an ideal pipe should look like, but shifting that vision into reality seems to be a quantum leap.1.2 The backgroundThe problem lies in the fact that too many people who control the decision making process view of the half pipe as a fixed and static feature, and that once built, a pipe is left to the forces of nature. A severe change of opinions needed, as the half pipe needs to be thought of as an elastic form (almost lifelike) that changes daily and which needs continual maintenance. Another huge factor in developing consistent half pipes is a set of standards. Over the years, the NASBA, OP, USASA, USSA, ISF, and FIS have given differing pipe dimensions to resorts. All this help from various organizations has left pipe building more of an art than a science. Both the ISF and the FIS are now promoting similar versions of half pipe dimensions. So we need to redesign the shape of a snowboard course to maximize the production of vertical air by a skilled snowboarder.2. The Description of the Problem2.1 Practical halfpipe’s requirements2.1.1 the maximum vertical and the largest body twistSnowboarders’ greatest height, the number of rotations (the larges t body twist) and the beautiful action will affect the athlete's score. the longer the spare time left, the more rotations to do for snowboarders. The basic physics principle at work here is the conservation of angular momentum. The angular momentum of the snowboarder is determined at takeoff, and cannot be changed once the snowboarder is airborne. So to make turns in the air the snowboarder must give himself initial rotation upon takeoff. In order to reach the maximum height, the maximum out-velocity would be required.so we analyzed the in- velocity and the out-velocity, and the shape of space (length, width, depth, field gradient) affect the in- velocity and the out-velocity obviously.But the height can not be too high, because too high speed would be a big threat to the safety of snowboarders. Therefore, in order to control the maximum speed, we need to redesign the halfpipe.2.1.2 Speed analysisWhether to reach the maximum vertical height or to produce the largest body twist speed is is a reflection of practical indicators to the halfpipe design.The composition of the factors in the action.Including the fly height, difficulty, diversity, qualitycompletion of the action, Site use and landing conditions and so on because the height have an limit effect on difficulty, diversity, quality of action completement, so the fly height is the core elements of many factors.To conclude,no height,no no flight time and no flight time,no difficult action.As the free fall shows:V=.The height snowboarders can reach have a veryyclose relationship with the speed.2.2 Half pipe’s own conditions2.2.1 FrictionFriction, including friction between the board and the snow as well as air friction.The dynamic friction coefficient between Snow and the board changes from 0.03 to 0.2.Take 0.2 for example, the maximum friction coefficient and the full effect of body weight to calculate the vertical friction0.2Wf=, that the acceleration less due to friction is generated to accelerate the role of body weight 0.2 times, much smaller than resulting in the acceleration of gravity effect. Air friction 2f C Av, in our model, we do not consider the influence of air friction.0.5a a d2.2.2 the size of halfpipeUnder certain circumstances,as the length, depth, tilt angle increases, the speed will be. In view of snowboard safety, speed can not be infinite, which has some of the value of the constraints.3. Models3.1 Definitions and SymbolsFlat：the bottom ground of U grooveTransitions：the transition zone between Horizontal and vertical groove bottom wall Verticals：the vertical parts of the walls between the Lip and the Transitions Platform：the level platform on the snow wall surfaceEntry Ramp：the slippery position of U-shaped slotm：Athlete's qualityg：Gravity accelerationV：Athletes’ speed when first enter u-shaped slot1V：Athletes’ speed when last sliding out u-shaped slottl：under side rectangular width of U-shaped slot1l：the length of U-shaped slot2R：the deep of U-shaped slotn：Athletes emptied timesβ：Angle between Athletes’ speed and slot edge horizontal when first enteru-shaped slotu ： the frictional factor between Skateboarding and snowf A ：how much work friction do when Athletes vertically into a u-shaped slot in arc d C ：Air resistance coefficienta ρ：Air densityν：Athletes’ speed relative air movementA ：Corresponding to the projective area of v3.2 Assumptions1.Assuming frictional factor is a constant when athletes are in taxiing process2.Assuming no melting snow when athletes are in taxiing process)3.Assuming the maximizing friction is gravity, frictional factor as the biggest 0.2, when compared friction work and gravity work4.Assuming the loss of speed is 2 meters per second because of the Angle between the speed and direction of existence with blade when athletes come into (out) the slots every time3.3 the simple analysis of gravity and friction when sliding in the halfpipeIf the athlete slip into the half pipe with a certain speed. Athletes in motion of constantly falling in vertical direction Increasing gravitational potential energy. The process in motion need to overcome the frictional resistance acting between the skate and snow acting must also overcome the air resistance acting. We use all ski areas in China to analyze the data[1] as follows in Table1:Table 2 National snowboard half pipe skiing skill to the situation Championship Series17 ° slope of more than 100 meters along the length of glide in the groove The competition is in the17 ° slope and along the length of more than 100 meters slide in the grooves and do all kinds of flip, twist, grasp the difficulty of board action, the action is completed in a certain vertical height of drop. The standards of international competition venues, can be obtained by calculating the U-groove vertical drop 150*sin17h =.Those athletes complete the maneuver in the vertical direction to produce the height of 40 meters gap. A gap of more than 40 meters in the vertical direction athletes can have a very substantial increase in the rate. A gap of more than 40 meters in the vertical direction athletes can have a very substantial increase in the rate. In terms of free fall calculations 22104020y V hg =≈⨯⨯≈m / s, However the snow and the board’s dynamic friction coefficient between 0.03 to 0.2, the maximum friction coefficient and the full body weight to calculate the friction force acting perpendicular 0.2W of t =.That the speed less is due to the friction resistance, it is weight generated to accelerate the role of body weight 0.2 times, far less than the acceleration of gravity produces results. Therefore, venue’s height of fall is an important way for athletes obtained the vertical velocity. Athletes can complete the vertical velocity and level velocity conversion with a reasonable technology, So that Athletes most likely to get to the maximum vacate height at the last vacate.3.4 in-velocity of factors[1]3.4.1 the snowboarder’ angle when in and the speed lossPlayers control the skis taxiing around the edge of the board into the slot ，both the before and the after of snowboard have the effect of braking, so in order to reduce the loss of speed, so that ，the speed of the body center of gravity in the same direction with the board's longitudinal axis as far as possible ，to reduce the braking effect when the snowboard have instant contact with the snow, and homeopathic slide, taking fulladvantage of wall height difference obtained acceleration. It can be seen the speed of full contact is less than the speed of front panel from Table 3, indicating that the human body has a loss of speed when completely into the slot, Since the existence of wall resistance, the speed loss is normal. However, if the speed of body center of gravity has the same direction with the blade, the speed of the losses will be reduced. As can be seen from Table 3, the athlete’ gravity speed direction has an angle with direction of blade center, the minimum is 1.2, and the maximum is 5.4, the speed ofdirection and the direction with the blade did not reach exactly the same. Decrease the maximum rate reached 27.5%, a minimum rate of 6.8%.Figure 1 the angle between the rate of speed loss and direction with the blade when into the slotIt can be seen that the speed loss rate and direction with the blade angle has not exactly the same trend from Figure 1, there may be several reasons as follows:(1)players is not very skilled when sliding into the slot, the ability of controlling board is not strong(2) It may require different sliding speed for the different air movement in the next time, resulting in players want to control taxi speed on purpose (3)the center of gravity is too forward, the gravity torque is too large, have Side effect, So the technology will have a major impact in speed.3.5 out-velocity of factors [1]3.5.1 help rampAthletes for the first time into the slot before sliding into the slot with help, Athletes should be actively obtained the speed of access to controlled, If the snowboarder into the slot before , after slide a certain distance at the edge of the slot, Obtain a certain speed. and before leaping into the slot and in a certain height 0E , you'll get some initial energy reserves 000E E E 动势(0E Representative athlete ofthe initial energy, 0动E representing athletes initial kinetic energy,0势E .Representing athletes Initial potential) With the completion of the action into the groove, getting smaller and smaller potential energy athletes to complete, in the case of gravity does positive work, the potential energy of the players is correspondingincrease, that the athletes will get the vertical speed by energy transfer. After get some of the vertical velocity into the tank, the athletes have a certain amount of kinetic energy reserves; athletes using the kinetic energy reserves, transformation to the potential when out the half pipe, it can achieve the purpose of improving flight altitude; flight altitude do reserve for potential of the next action into the half pipe for the next action to provide time and space to ensure the successful completion However, athletes in the kinetic and potential energy conversion, to achieve the speed must be controllable. If the speed is not contro llable, it will affect the athlete’s performance, Otherwise it will lead to serious accidents. From Table 4, it can be seen that the athletes Lei Pan rear positive blade rate of 540movement into the tank thelargest; is s.14, the minimum Shi wan Cheng's anti-blade rate of om93720front foot movement, is s11. The actions are successful action, but also a national athlete,.m06so you can give a preliminary conclusion: the speed of athletes in the following speed control 15 meters per second.3.5.2 the force point and the plate angle when outIn the trench wall of the moment, because of losing the support of the front skis, then, the stress point should be to leave the center of board, and gradually transition back to the board, so that the stress point is always forcing plate wall, front foot homeopathic slide, back foot should be gradually forced pedal. When reaction force in sufficient, maintain parabolic path smooth, increasing the speed, and maintain a reasonable angle of the slot. At the same time of achieving the goal of increasing height highly effective, also get into the appropriate slot speed and angle of twist. Reaching movements while floating high, reducing the level of speed and the effect of resistance into the half pipe, reasonably come into the groove; do energy reserves for the next the action.3.5.3 the snowboarder’ angle when out and the speed lossCheng（20）Sun ZhiFeng front o72010.20 8.24 4.0 19.2Huang Shi Ying Anti-fronto72013.73 12.09 3.9 11.9ZenXiao Hua front o72011.65 11．55 0.3 0.9Liu JiaYu behind o54011.209.82 4.1 12.3Pan Lei behind o54012.00 9.11 3.0 24.0 Table 4 is part of the e lite athlete’s slotting board kinematic parameters. By comparing the data in Table 4, we can find what the speed of completely clear out the slot is less than the speed of the front panel instantaneous slip out the slot. It can be seen that five players’ speed and the direction of blade angle have positively correlated with the loss rate in Figure 2, indicating that the greater of angle between speed and direction with the blade, the greater of loss speed, so you need to control the sliding board direction, letting the long axis have the same direction with the speed of human body.Figure 2 the angle between the rate of speed loss and direction with the snowboardwhen out of the halfpipe3.5.4 H alfpipe’s RadiusAppropriate reduced orbit radius can increase the speed when athletes slip out half pipe, and favor the athletes to make various actions in the air. Sides rail identifiable by two 1/4arcs, we can deduces the formulaof tfti o i oi f I dt M w I w ∑⎰+=)(Then taking orbit design into consideration, the optimal speedup method is to reduce the rail depth (by our hypothesis know depth and arc radius is equal), namely decreases of r , and so can reduce of I , effectively increase f w . But, taking the athlete's safety into consideration, the athletes' speed may not excessive, namely orbit radius cannot be too small. General provisions half pipe orbit radius scope for 3-4.5m, guarantee the slot speed are not more than 15, also ensures the athlete's safety.The basic snowboarding physics behind this phenomenon can be understood by applying the principle of angular impulse and momentum.The schematic of the physics of snowboarding in this analysis is given below.Figure 3 the analysis of forceWhere:i w is the initial angular velocity of the body (consisting of snowboarder plusboard), at position (1)f w is the final angular velocity of the body, at position (2), which is the point at which the snowboarder exits the half-pipei V is the initial velocity of the center of mass G of the body, at position (1)f V is the final velocity of the center of mass G of the body, at position (2)i r is the initial distance from the center of rotation o to the body's center of massG, at position (1)f r is the final distance from the center of rotation o to the body's center of mass G , at position (2)g is the acceleration due to gravityN is the normal force acting on the snowboard, as shownF is the friction force acting on the snowboard, as shownIt is assumed that the half-pipe is a perfect circle with center at o. The physics of snowboarding in this analysis can be treated as a two-dimensional problem. Now, apply the equation for angular impulse and momentum to the system (consisting of snowboarder plus board):tfoi i o of fti I w M dt I w +=∑⎰Where:oi I is the initial moment of inertia of the body (consisting of snowboarder plus board) about an axis passing through point o and pointing out of the page, at position(1)of I is the final moment of inertia of the body (consisting of snowboarder plus board) about an axis passing through point o and pointing out of the page, at position(2)o M ∑ is the sum of the moments about point o. These moments are integrated between an initial time i t (at position 1) and a final time f t (at position 2)Here we are assuming that the body can be treated as rigid at positions (1) and (2), even though the snowboarder does in fact change his moment of inertia between thesetwo positions. But as it turns out, when using this equation we only need to know the initial and final values of the moment of inertia of the body.The line of action of the normal force N passes through point o, so it does not exert a moment on the body about point o. The friction force F is small so it can be neglected in terms of its moment contribution. This leaves only the gravitational force which exerts a moment on the body about point o. (Note that the gravitational force acts through the center of mass of the body, consisting of snowboarder plus board). In the above equation isolate f w . Thus,tfoi i o tif of I w M dtw I +=∑⎰Now,22oi Gi i of Gf f I I mr I I mr =+=+Where:Gi I is the initial moment of inertia of the body about an axis passing through point G and pointing out of the page, at position (1)Gf I is the final moment of inertia of the body about an axis passing through point G and pointing out of the page, at position (2)m is the mass of the bodyIn the above equation for f w , if we decrease of I the angular velocity f w will increase beyond the value it would be if we did not decrease of I . In practice this can be accomplished by sufficiently reducing the distance from the center of mass of the body G to the point o. In other words, make f r small enough and f w will increase. Note also that the terms Gf I and o M ∑ may also change somewhat. But the dominant effect will be that of reducing f r .At positions (1) and (2), the velocity of the center of mass G is given byi i if f fV w r V w r ==These two velocities are parallel to the half-pipe since the body is rigid at positions (1) and (2).r small appropriate, Looking at the above equations for velocity, if we makesfw. This in turn will result in his velocity the snowboarder will significantly increasefV) being greater than otherwise.exiting the pipe (f3.6 The in-velocity comparison with the out- velocity [1]It can be seen that the speed of athletes when athletes slip out half pipe is less than the speed of athletes when athletes slip out half pipe from Figure 4. The biggest difference between the two is the Shi wan Cheng, the smallest difference between the two is that Zen Xiao Ye. The average speed is 11.69m s when slip into half pipe, the average down is1.94m s,the speed decline will lead to altitude declining when slip out half pipe, having effect on the speed of slipping into half pipe next time, which restricts movements of athletes and sports techniques to improve the difficulty level of play, but also make the action quality greatly reduced, so the players should pay attention to the completion of a continuous action of the hair lower limb muscle strength.Figure 4 the chart of comparison about speed change when into (out of)half pipe 3.7 Snowb oarder’s position impact on the speedPumping on a half-pipe is used by snowboarders to increase their vertical take-off speed when they exit the pipe. This enables them to reach greater height and performmore aerial tricks, while airborne. The principle is exactly the same as for skateboarders pumping on a half-pipe.The snowboarder is able to increase his speed on the half-pipe with his feet remaining firmly on the board. This begs the question, what is the physics of snowboarding taking place that enables the snowboarder to increase his speed on the half-pipe?To increase his speed, the snowboarder crouches down in the straight part of the half-pipe. Then when he enters the curved portion of the half-pipe he lifts his body and arms up, which results in him exiting the pipe at greater speed than he would otherwise.r small Looking at the above equations for velocity, if the snowboarder makesfw. This in enough (by lifting his body and arms up), he will significantly increasefV being greater than if he did not lift turn will result in his velocity exiting the pipe (fhis body and arms up.By continually pumping his body (by crouching down and lifting his body and arms up in the curved portion of the half-pipe), the snowboarder is able to continually increase his velocity, eventually allowing sufficient height to be reached (upon exiting the half-pipe) to perform a variety of mid-air tricks.A more intuitive (non-mathematical) explanation of the physics of snowboarding taking place here is that pumping adds energy to the system in the same way that a child pumping on a swing adds energy, and results in him swinging higher. Therefore, the physics of snowboarding related to pumping on a half-pipe is similar to pumping on a swing.As a snowboarder lifts his arms and body up he feels resistance due to the force of centripetal acceleration which tends to push his body away from the center of rotation o. This resistance is proof that work is being done, and therefore energy is being added to the system.3.8 the entire movement of the energy change in the halfpipeHow the energy change during the Athletes’ entire movement in the half pipe.Figure 5 3-D half pipeFigure 6 halfpipe’s cr oss-sectionFrom Figure 6, we can know both sides of the curved part is the 1 / 4 cylinder in the side, the middle is rectangle.As shown, we assume that the depth of half pipe is R, the middle part length is 1l , the width of half pipe is 12l R , the half pipe ’s length is 2l , half pipe ’s inclination angle is α.When the athletes straight down into the tank by the vertical speed, we analysis the friction’s work in this process.When the athletes straight down into the tank where has friction, the friction’ work can be applied to functional principle, considering the given state can find out friction ’s work, But this does not consider the specific forms of friction force. By the analysis of analytical solution, we can describe its distribution characteristics.[2] α1l R 2l βAs shown in Fig 7, Objects satisfied Newton equations, the tangent of the form and normal directions form is (considered f uN =),dt dv m uN mg =-θcos ，————————————————（1）R v m mg N 2sin =-θ，————————————————（2）Figure 7 objects in circular orbit forcePray for (2) a derivative timedt dv v R m dt d mg dt dN 2cos =-θθ， dtd R v θ= and (1) David into the type dtd uN mg dt d mg dt dN θθθθ)2cos 2(cos -=-， θθcos 32mg uN d dN =+—————————————————— (3) Solving (3) type is the key to solve the solution of friction()()[]()[]⎰⎰⎰⎰+-=+-=c d u mg u c d ud mg ud N θθθθθθθθ*2exp *cos 3)2exp(*2exp *cos 32exp ——————————（4） Among them;()()()()()θθθθθθθθθθθθθθθd u u u u u u d u u u u d u *2exp *cos 412exp 4sin 2exp 2cos 2exp 2sin )2exp(2cos 2exp *cos 22⎰⎰⎰-+=+=namely ())sin cos 2(14)2exp(*2exp *cos 2θθθθθθ++=⎰u u u d u ， So:)2exp()sin cos 2(4132θθθu C u umg N -+++=， fBecause of: 0,0==N θ，will,24132U mg uC +-=. So:))2exp(2sin cos 2(4132θθθu u u umg N --++=———————————（5） And:))2exp(*2sin cos 2(4132θθθu u u umg uN f --++== ())cos sin 2(4132sin cos 241322202θθθθθθθθθu u f e u u umgR d ue u u umgR uNRd uNds A --+-+-=-++-=-=-=⎰⎰⎰3.9 the balance of speed after considering the air resistance [3]If in the process of straight downhill snow is flat and snowboard does not leave the ground can be approximately described by plane hinged to the relationship between ski and snow we watch skis and skiers as a whole force people ski and snowboard in the force of both concentration a nd reduced to a couple Torques’s’. At this point slide in the snow is equivalent to a single degree of freedom motion system as Figure 8When the system is in static equilibrium with⎩⎨⎧=-=+-0cos 0)(sin ααmg f f f mg sr a Which 20.5a a d f C Av ρ=,r s f uf =.Joint Solution available ：cos r f umg α=Can be seen, friction and gravity components is balance at the balance.Figure8 single degree of freedom motion systemInto the above equation can be obtained:0)cos 5.0(sin 2=+-αραumg Av C mg d aAC u mg v d a ραα)cos (sin 2-= suppose 12sin (1)u u β-=+，。