2013年数学建模美赛B题论文

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2013美国大学生数学建模竞赛论文

summaryOur solution paper mainly deals with the following problems:·How to measure the distribution of heat across the outer edge of pans in differentshapes and maximize even distribution of heat for the pan·How to design the shape of pans in order to make the best of space in an oven·How to optimize a combination of the former two conditions.When building the mathematic models, we make some assumptions to get themto be more reasonable. One of the major assumptions is that heat is evenly distributedwithin the oven. We also introduce some new variables to help describe the problem.To solve all of the problems, we design three models. Based on the equation ofheat conduction, we simulate the distribution of heat across the outer edge with thehelp of some mathematical softwares. In addition, taking the same area of all the pansinto consideration, we analyze the rate of space utilization ratio instead of thinkingabout maximal number of pans contained in the oven. What’s more, we optimize acombination of conditions (1) and (2) to find out the best shape and build a function toshow the relation between the weightiness of both conditions and the width to lengthratio, and to illustrate how the results vary with different values of W/L and p.To test our models, we compare the results obtained by stimulation and our models, tofind that our models fit the truth well. Yet, there are still small errors. For instance, inModel One, the error is within 1.2% .In our models, we introduce the rate of satisfaction to show how even thedistribution of heat across the outer edge of a pan is clearly. And with the help ofmathematical softwares such as Matlab, we add many pictures into our models,making them more intuitively clear. But our models are not perfect and there are someshortcomings such as lacking specific analysis of the distribution of heat across theouter edge of a pan of irregular shapes. In spite of these, our models can mainlypredict the actual conditions, within reasonable range of error.For office use onlyT1 ________________T2 ________________T3 ________________T4 ________________ Team Control Number18674 Problem Chosen AFor office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________2013 Mathematical Contest in Modeling (MCM) 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.The Ultimate Brownie PanAbstractWe introduce three models in the paper in order to find out the best shape for the Brownie Pan, which is beneficial to both heat conduction and space utility.The major assumption is that heat is evenly distributed within the oven. On the basis of this, we introduce three models to solve the problem.The first model deals with heat distribution. After simulative experiments and data processing, we achieve the connection between the outer shape of pans and heat distribution.The second model is mainly on the maximal number of pans contained in an oven. During the course, we use utility rate of space to describe the number. Finally, we find out the functional relation.Having combined both of the conditions, we find an equation relation. Through mathematical operation, we attain the final conclusion.IntroductionHeat usage has always been one of the most challenging issues in modern world. Not only does it has physic significance, but also it can influence each bit of our daily life. Likewise,space utilization, beyond any doubt, also contains its own strategic importance. We build three mathematic models based on underlying theory of thermal conduction and tip thermal effects.The first model describes the process and consequence of heat conduction, thus representing the temperature distribution. Given the condition that regular polygons gets overcooked at the corners, we introduced the concept of tip thermal effects into our prediction scheme. Besides, simulation technique is applied to both models for error correction to predict the final heat distribution.Assumption• Heat is distributed evenly in the oven.Obviously, an oven has its normal operating temperature, which is gradually reached actually. We neglect the distinction of temperature in the oven and the heating process, only to focus on the heat distribution of pans on the basis of their construction.Furthermore, this assumption guarantees the equivalency of the two racks.• Thermal conductivity is temperature-invariant.Thermal conductivity is a physical quantity, symbolizing the capacity of materials. Always, the thermal conductivity of metal material usually varies with different temperatures, in spite of tiny change in value. Simply, we suppose the value to be a constant.• Heat flux of boundaries keeps steady.Heat flux is among the important indexes of heat dispersion. In this transference, we give it a constant value.• Heat conduction dom inates the variation of temperature, while the effects ofheat radiation and heat convection can be neglected.Actually, the course of heat conduction, heat radiation and heat convectiondecide the variation of temperature collectively. Due to the tiny influence of other twofactors, we pay closer attention to heat conduction.• The area of ovens is a constant.I ntroduction of mathematic modelsModel 1: Heat conduction• Introduction of physical quantities:q: heat fluxλ: Thermal conductivityρ: densityc: specific heat capacityt: temperature τ: timeV q : inner heat sourceW q : thermal fluxn: the number of edges of the original polygonsM t : maximum temperaturem t : minimum temperatureΔt: change quantity of temperatureL: side length of regular polygon• Analysis:Firstly, we start with The Fourier Law:2(/)q gradt W m λ=- . (1) According to The Fourier Law, along the direction of heat conduction, positionsof a larger cross-sectional area are lower in temperature. Therefore, corners of panshave higher temperatures.Secondly, let’s analyze the course of heat conduction quantitatively.To achieve this, we need to figure out exact temperatures of each point across theouter edge of a pan and the variation law.Based on the two-dimension differential equation of heat conduction:()()V t t t c q x x y yρλλτ∂∂∂∂∂=++∂∂∂∂∂. (2) Under the assumption that heat distribution is time-independent, we get0t τ∂=∂. (3)And then the heat conduction equation (with no inner heat source)comes to:20t ∇=. (4)under the Neumann boundary condition: |W s q t n λ∂-=∂. (5)Then we get the heat conduction status of regular polygons and circles as follows:Fig 1In consideration of the actual circumstances that temperature is higher at cornersthan on edges, we simulate the temperature distribution in an oven and get resultsabove. Apparently, there is always higher temperature at corners than on edges.Comparatively speaking, temperature is quite more evenly distributed around circles.This can prove the validity of our model rudimentarily.From the figure above, we can get extreme values along edges, which we callM t and m t . Here, we introduce a new physical quantity k , describing the unevennessof heat distribution. For all the figures are the same in area, we suppose the area to be1. Obviously, we have22sin 2sin L n n n ππ= (6) Then we figure out the following results.n t M t m t ∆ L ksquare 4 214.6 203.3 11.3 1.0000 11.30pentagon 5 202.1 195.7 6.4 0.7624 8.395hexagon 6 195.7 191.3 4.4 0.6204 7.092heptagon 7 193.1 190.1 3.0 0.5246 5.719octagon 8 191.1 188.9 2.2 0.4551 4.834nonagon 9 188.9 187.1 1.8 0.4022 4.475decagon 10 189.0 187.4 1.6 0.3605 4.438Table 1It ’s obvious that there is negative correlation between the value of k and thenumber of edges of the original polygons. Therefore, we can use k to describe theunevenness of temperature distribution along the outer edge of a pan. That is to say, thesmaller k is, the more homogeneous the temperature distribution is.• Usability testing:We use regular hendecagon to test the availability of the model.Based on the existing figures, we get a fitting function to analyze the trend of thevalue of k. Again, we introduce a parameter to measure the value of k.Simply, we assume203v k =, (7) so that100v ≤. (8)n k v square 4 11.30 75.33pentagon 5 8.39 55.96hexagon 6 7.09 47.28heptagon 7 5.72 38.12octagon 8 4.83 32.23nonagon9 4.47 29.84 decagon 10 4.44 29.59Table 2Then, we get the functional image with two independent variables v and n.Fig 2According to the functional image above, we get the fitting function0.4631289.024.46n v e -=+.(9) When it comes to hendecagons, n=11. Then, v=26.85.As shown in the figure below, the heat conduction is within our easy access.Fig 3So, we can figure out the following result.vnActually,2026.523tvL∆==.n ∆t L k vhendecagons 11 187.1 185.8 1.3 0.3268 3.978 26.52Table 3Easily , the relative error is 1.24％.So, our model is quite well.• ConclusionHeat distribution varies with the shape of pans. To put it succinctly, heat is more evenly distributed along more edges of a single pan. That is to say, pans with more number of peripheries or more smooth peripheries are beneficial to even distribution of heat. And the difference in temperature contributes to overcooking. Through calculation, the value of k decreases with the increase of edges. With the help of the value of k, we can have a precise prediction of heat contribution.Model 2: The maximum number• Introduction of physical quantities:n: the number of edges of the original polygonsα: utility rate of space• Analysis:Due to the fact that the area of ovens and pans are constant, we can use the area occupied by pans to describe the number of pans. Further, the utility rate of space can be used to describe the number. In the following analysis, we will make use of the utility rate of space to pick out the best shape of pans. We begin with the best permutation devise of regular polygon. Having calculated each utility rate of space, we get the variation tendency.• Model Design:W e begin with the scheme which makes the best of space. Based on this knowledge, we get the following inlay scheme.Fig 4Fig 5According to the schemes, we get each utility rate of space which is showed below.n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 shape square pentagon hexagon heptagon octagon nonagon decagon hendecagon utility rate(％)100.00 85.41 100.00 84.22 82.84 80.11 84.25 86.21Table 4Using the ratio above, we get the variation tendency.Fig 6 nutility rate of space• I nstructions:·The interior angle degrees of triangles, squares, and regular hexagon can be divided by 360, so that they all can completely fill a plane. Here, we exclude them in the graph of function.·When n is no more than 9, there is obvious negative correlation between utility rate of space and the value of n. Otherwise, there is positive correlation.·The extremum value of utility rate of space is 90.69％,which is the value for circles.• Usability testing:We pick regular dodecagon for usability testing. Below is the inlay scheme.Fig 7The space utility for dodecagon is 89.88％, which is around the predicted value. So, we’ve got a rather ideal model.• Conclusion:n≥), the When the number of edges of the original polygons is more than 9(9 space utility is gradually increasing. Circles have the extreme value of the space utility. In other words, circles waste the least area. Besides, the rate of increase is in decrease. The situation of regular polygon with many sides tends to be that of circles. In a word, circles have the highest space utility.Model 3: Rounded rectangle• Introduction of physical quantities:A: the area of the rounded rectanglel: the length of the rounded rectangleα: space utilityβ: the width to length ratio• Analysis:Based on the combination of consideration on the highest space utility of quadrangle and the even heat distribution of circles, we invent a model using rounded rectangle device for pans. It can both optimize the cooking effect and minimize the waste of space.However, rounded rectangles are exactly not the same. Firstly, we give our rounded rectangle the same width to length ratio (W/L) as that of the oven, so that least area will be wasted. Secondly, the corner radius can not be neglected as well. It’ll give the distribution of heat across the outer edge a vital influence. In order to get the best pan in shape, we must balance how much the two of the conditions weigh in the scheme.• Model Design:To begin with, we investigate regular rounded rectangle.The area224r ar a A π++= (10) S imilarly , we suppose the value of A to be 1. Then we have a function between a and r :21(4)2a r r π=+--(11) Then, the space utility is()212a r α=+ (12) And, we obtain()2114rαπ=+- (13)N ext, we investigate the relation between k and r, referring to the method in the first model. Such are the simulative result.Fig 8Specific experimental results arer a ∆t L k 0.05 0.90 209.2 199.9 9.3 0.98 9.49 0.10 0.80 203.8 196.4 7.4 0.96 7.70 0.15 0.71 199.6 193.4 6.2 0.95 6.56 0.20 0.62 195.8 190.5 5.3 0.93 5.69 0.25 0.53 193.2 189.1 4.1 0.92 4.46Table 5According to the table above, we get the relation between k and r.Fig 9So, we get the function relation3.66511.190.1013r k e -=+. (14) After this, we continue with the connection between the width to length ratioW Lβ=and heat distribution. We get the following results.krFig 10From the condition of heat distribution, we get the relation between k and βFig 11And the function relation is4.248 2.463k β=+ (15)Now we have to combine the two patterns together:3.6654.248 2.463(11.190.1013)4.248 2.463r k e β-+=++ (16)Finally, we need to take the weightiness (p) into account,(,,)()(,)(1)f r p r p k r p βαβ=⋅+⋅- (17)To standard the assessment level, we take squares as criterion.()(,)(1)(,,)111.30r p k r p f r p αββ⋅⋅-=+ (18) Then, we get the final function3.6652(,,)(1)(0.37590.2180)(1.6670.0151)1(4)r p f r p p e rββπ-=+-⋅+⋅++- (19) So we get()()3.6652224(p 1)(2.259β 1.310)14r p f e r r ππ--∂=-+-+∂⎡⎤+-⎣⎦ (20) Let 0f r∂=∂,we can get the function (,)r p β. Easily,0r p∂<∂ and 0r β∂>∂ (21) So we can come to the conclusion that the value of r decreases with the increase of p. Similarly, the value of r increases with the increase of β.• Conclusion:Model 3 combines all of our former analysis, and gives the final result. According to the weightiness of either of the two conditions, we can confirm the final best shape for a pan.• References:[1] Xingming Qi. Matlab 7.0. Beijing: Posts & Telecom Press, 2009: 27-32[2] Jiancheng Chen, Xinsheng Pang. Statistical data analysis theory and method. Beijing: China's Forestry Press, 2006: 34-67[3] Zhengshen Fan. Mathematical modeling technology. Beijing: China Water Conservancy Press, 2003: 44-54Own It NowYahoo! Ladies and gentlemen, please just have a look at what a pan we have created-the Ultimate Brownie Pan.Can you imagine that just by means of this small invention, you can get away of annoying overcookedchocolate Brownie Cake? Pardon me, I don’t want to surprise you, but I must tell you , our potential customers, that we’ve made it! Believing that it’s nothing more than a common pan, some people may think that it’s not so difficult to create such a pan. To be honest, it’s not just a simple pan as usual, and it takes a lot of work. Now let me show you how great it is. Here we go!Believing that it’s nothing more than a common pan, some people may think that it’s not so difficult to create such a pan. To be honest, it’s not just a simple pan as usual, and it takes a lot of work. Now let me show you how great it is. Here we go!Maybe nobody will deny this: when baked in arectangular pan, cakes get easily overcooked at thecorners (and to a lesser extent at the edges).But neverwill this happen in a round pan. However, round pansare not the best in respects of saving finite space in anoven. How to solve this problem? This is the key pointthat our work focuses on.Up to now, as you know, there have been two factors determining the quality of apan -- the distribution of heat across the outer edge of and thespace occupied in an oven. Unfortunately, they cannot beachieved at the same time. Time calls for a perfect pan, andthen our Ultimate Brownie Pan comes into existence. TheUltimate Brownie Pan has an outstandingadvantage--optimizing a combination of the two conditions. As you can see, it’s so cute. And when you really begin to use it, you’ll find yourself really enjoy being with it. By using this kind of pan, you can use four pans in the meanwhile. That is to say you can bake more cakes at one time.So you can see that our Ultimate Brownie Pan will certainly be able to solve the two big problems disturbing so many people. And so it will! Feel good? So what are you waiting for? Own it now!。

2013年美国数学建模竞赛B题

水资源计划摘要本文是要设计一个有效的，可行的，低成本的用水计划，来满足某国2025年的用水需求。

我们选择中国为研究对象，根据中国各地区历年的水资源总量并求出其均值，参考各地区历年用水总量来预测2025年的用水总量，将两者相减得出差值，并以此为依据将中国各地区分为缺水地区，不缺水地区，水资源丰富地区三类。

经研究分析有两种可行性高的方案。

第一种，由水资源丰富地区向缺水地区提供水。

第二种，是由沿海缺水城市进行海水淡化并运往其他缺水城市。

我们主要考虑经济因素对两种方案进行分析研究，最终得出结论由水资源丰富地区铺设管道向缺水地区提供水为最优方案。

并以各省的省会作为核心城市，说明全省的需水和调水情况，并以省会城市或直辖市为顶点构成一个赋权图，即把问题转换为求水资源丰富地区到缺水地区的最短路问题，并用图论的知识来解决问题。

在此基础上考虑到此方案会改变就业，生产力，水资源利用等因素，从而对经济，物理，环境产生不同程度的影响，并用层次分析加以研究，最终以报告的方式向政府反映。

关键词：回归分析最小生成树层次分析法一、问题重述淡水是世界大部分地区的发展限制。

试建立一个数学模型，用来确定一个有效的、可行的和低成本的水资源战略，以满足2025年预计的用水需求，特别是，您的数学模型必须解决存储和输送，去盐碱化和环境保护等问题。

如果可能的话，用你的模型探讨此战略在经济，物理和环境等方面的影响。

试提供一个非技术性的文件，向政府相关部门介绍你的方法以及其可行性和成本，并说明为什么它是“最好的水战略”。

二、符号说明ˆy：预测得出的2025年用水量；S：输水的造价；1S：海水淡化的造价；2d1: 输水工程的单位造价；d2：海水淡化的单位造价；2R：拟合度.三、模型假设1.从2013年到2025年各外部因素对水资源总量无影响，例如：雪灾、地震、洪水、战争等对环境的影响；2.各地区海水淡化单位费用相同；3.不同地区淡水转移的单位费用相同；4.人们的消费水平及劳动力费用不会随意外事故发生明显改变。

2013全国数学建模竞赛B题优秀论文.

基于最小二乘法的碎纸片拼接复原数学模型摘要首先对图片进行灰度化处理,然后转化为0-1二值矩阵，利用矩阵行（列）偏差函数，建立了基于最小二乘法的碎纸片拼接数学模型，并利用模型对图片进行拼接复原。

针对问题一，当两个数字矩阵列向量的偏差函数最小时，对应两张图片可以左右拼接。

经计算，得到附件1的拼接结果为：08,14,12,15,03,10,02,16,01,04,05,09,13,18,11,07,17,00,06。

附件2的拼接结果为:03,06,02,07,15,18,11,00,05,01,09,13,10,08,12,14,17,16,04。

针对问题二，首先根据每张纸片内容的不同特性，对图片进行聚类分析，将209张图片分为11类；对于每一类图片，按照问题一的模型与算法，即列偏差函数最小则进行左右拼接,对于没有拼接到组合里的碎纸片进行人工干预，我们得到了11组碎纸片拼接而成的图片；对于拼接好的11张图片，按照问题一的模型与算法，即行偏差函数最小则进行上下拼接,对于没有拼接到组合里的碎纸片进行人工干预。

我们最终经计算，附件3的拼接结果见表9，附件4的拼接结果见表10。

针对问题三，由于图片区分正反两面，在问题二的基础上，增加图片从下到上的裁截距信息，然后进行两次聚类，从而将所有图片进行分类，利用计算机自动拼接与人工干预相结合，对所有图片进行拼接复原。

经计算，附件5的拼接结果见表14和表15该模型的优点是将图片分为具体的几类，大大的减少了工作量，缺点是针对英文文章的误差比较大。

关键字：灰度处理，图像二值化，最小二乘法，聚类分析，碎纸片拼接一、问题重述碎纸片的拼接复原技术在司法鉴定、历史文献修复与研究、军事情报获取以及故障分析等领域都有着广泛的应用。

近年来，随着德国“斯塔西”文件的恢复工程的公布，碎纸文件复原技术的研究引起了人们的广泛关注。

传统上，拼接复原工作需由人工完成，准确率较高，但效率很低。

特别是当碎片数量巨大，人工拼接很难在短时间内完成任务。

(完整word版)2013年数学建模b题

精心整理碎纸片的拼接复原【摘要】：碎纸片拼接技术是数字图像处理领域的一个重要研究方向，把计算机视觉和程序识别应用于碎纸片的复原，在考古、司法、古生物学等方面具有广泛的应用，具有重要的现实意义。

本文主要结合各种实际应用背景，针对碎纸机绞碎的碎纸片，基于计算机辅助对碎纸片进行自动拼接复原研究。

针对问题1，依据图像预处理理论，通过matlab程序处理图像，将图像转化成适合于计算机处理的数字图像，进行灰度分析，提取灰度矩阵。

对于仅纵切的碎纸片，根据矩阵的行提取理论，将。

建中的任一列与矩阵值，序列号。

将程序进行循环操作，得到最终的碎片自动拼接结果。

、；分别作为新生成的矩阵、。

，将矩阵中的任一列分别与矩阵中每一列代入模型，所得p值对应的值即为横排序；将矩阵中的任一行分别于矩阵中的任一行代入模型，所得q值对应的值即为列排序。

循环进行此程序，得计算机的最终运行结果。

所得结果有少许误差，需人工调制，更正排列顺序，得最终拼接结果。

针对问题3，基于碎纸片的文字行列特征，采用遗传算法，将所有的可能性拼接进行比较，进行择优性选择。

反面的排序结果用于对正面排序的检验，发现结果有误差，此时，进行人工干预，调换碎纸片的排序。

【关键词】:灰度矩阵欧式距离图像匹配自动拼接人工干预一、问题重述破碎文件的拼接在司法物证复原、历史文献修复以及军事情报获取等领域都有着重要的应用。

传统上，大量的纸质物证复原工作都是以人工的方式完成的，准确率较高，但效率很低。

特别是当碎片数量巨大，人工拼接不但耗费大量的人力、物力，而且还可能对物证造成一定的损坏。

随着计算机技术的发展，人们试图把计算机视觉和模式识别应用于碎纸片复原，开展对碎纸片自动拼接技术的研究，以提高拼接复原效率。

试讨论一下问题，并根据题目要求建立相应的模型和算法：、附件4(1)(2)(3)(4)纸片的自动拼接。

问题1：根据图像预处理理论，通过程序语言将图像导入matlab程序，对图像进行预处理，将碎纸片转换成适合于计算机处理的数字图像形式，并对数字图像进行灰度分析，提取灰度矩阵。

2013年深圳杯数学建模B题参考论文

2013年深圳杯数学建模B题参考论文2013年山西赛区数学建模联赛承诺书我们仔细阅读了山西赛区数学建模联赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。

我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。

我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。

如有违反竞赛规则的行为，我们将受到严肃处理。

我们参赛选择的题号是（从A/B/C/D中选择一项填写）：B我们的参赛报名号为：参赛队员(打印并签名) ：目录一、摘要 (3)二、关键词 (3)三、问题重述与分析 (3)四、模型假设 (3)五、符号说明 (9)六、模型的建立与求解 (9)七、模型的检验 (10)八、模型的优缺点分析 (10)九、求解 (10)十、合理化建议 (12)参考文献 (13)深圳关内外交通拥堵探究与治理一、摘要交通拥堵是目前中国各大城市面临的共同难题，但拥堵的成因各不相同，因而需要在摸清规律的基础上有针对性地提出解决方案。

就深圳而言，交通堵塞对人们出行造成了较大的影响,造成深圳城市交通拥堵主要原因:1、城市功能区构建。

2、城市公交发展相对迟缓，市民出行系统结构不合理。

随着深圳城市的不断扩大，市民出行距离的加大，公交车辆还远不能满足市民上下班或者出行的需要，城市公交发展却相对迟缓，市民出行系统结构不合理，导致了交通环境的进一步恶化。

3、城市道路交通规划滞后。

例如老城区路幅不宽，支路多、小街小巷多而且与主干道衔接相通，车辆交汇频繁致使交通不畅，成为“瓶颈”路口。

4、城市道路资源时常被侵占。

5、部分路段红绿灯过多。

6、市内停车供求矛盾突出。

7、市民交规意识薄弱。

公共空间中以各种方式进行的空间移动（即交通）的需求，它具有需求时间和空间的不均匀性、需求目的的差异性等特征。

2013年美赛数学建模b题的分析

Байду номын сангаас
美国2025年可提供的实际淡水量怎样预测？
（可以先预测不同地区不同领域供水量，再相加）
如果实际淡水量小于需求量，可通过那些方法提供不足的淡水？（例如：海水淡化、不同地区淡水转移）
这些方法那个更好？（比较的依据是什么？例如费用，环境）需要做出那些假设？
1. 从现在到2025年之前不会有战争，自然灾害等影响因素 2.同一个地区的居民年用水量相同 3.同一个地区的居民量的年增长率为常数 4.不同地区海水淡化的单位费用相同 5.不同地区淡水转移的单位费用相同
注：模型假设一般是在问题分析中根据需要提出的，所以可以先提出基本假设，以后再补充
Vx pre S
1.预测2025年供水量的模型（以一个州的降水量为例）
其中，表示州i 的可用降水量，表示该州的单位面积平均年降水量， Si 表示该州的淡水地表面积
Vx pre S
2.预测2025年需水量的模型（以一个州的居民用水量为例）
若 V大于0，则说明2025年美国的淡水量是足够的，然后再看每个州需要补充的淡水量，即 V V ，
si ri
若每个州需要补充的淡水量均大于0，则不需要采取任何措施，
否则，需要在不通州之间进行淡水调度。
若 V小于0，则说明2025年美国的淡水量是不足的，需要采取海水淡化等措施
注：为使表达更清晰，可以在论文中采用图、表等形式
其中，表示州i 在2025年人口数量，表示该州在2010年人口数量，表示该州的人口平均年增长率则该州在2025年的居民用水量为
i Vri w N 2 0 0 5
其中，w 表示该州居民的平均年用水量
3. 2025年美国需要补充的淡水量

2013年美国数学建模大赛B题

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number2222463463Problem ChosenBFor office use onlyF1________________F2________________F3________________F4________________ 2013Mathematical Contest in Modeling(MCM/ICM)Summary Sheet(Attach a copy of this page to your solution paper.)AbstractRecently,more and more information about freshwater resource comes into people’s view.Actually,the freshwater resource,which is originally small in reserve,becomes more and more precious with the pollution of wastewater discharged by lawbreakers. So it’s not surprising that many areas in China is short of freshwater.This paper aims at formulating a comprehensive planning of freshwater use.Initially,we make a prediction in model1to find which provinces are short of water by applying polynomial fitting toolbox of MATLAB.Eventually,our results indicate that10provinces in China are short of freshwater,which are as follows:Jiangsu, Shandong,Ningxia,Hebei,Henan,Tianjin,Beijing,Inner Mongolia and Anhui.Then,in our model2,employing the Principal Component Analysis method and the statistical software SPSS,we finally conclude that the major factors influence the sustainable use of freshwater are rainfall,polluted water treatment ratio,GDP growth ratio and direct discharge quantity.Finally,according to the results and conclusions draw in the former analysis,we construct an optimization model,in which we provide the detailed water useplannings from2013to2025to the governments of every area.Key words:freshwater resource,polluted water treatment,desalination,sustainable developmentWater,Water everywhereMathematic Models of National Water Strategy in China1.IntroductionIn recent years,with the dramatic growth of the world’s population and the development of industry,more and more challenges emerge in the process of people dealing with the world,of which,water shortage is a serious problem threatens humanity.As we all know,water is indispensable in guaranteeing life and health. However,the freshwater,which can be directly used by human,is extremely limited though the fact that water covers70.8percent of the earth’s surface,in which,97.5 percent is undrinkable salt water while the rest2.5percent is polar ice caps,mountain glaciers and the snow and ice of the permafrost zone that are difficult to exploit.In fact,only0.26percent of the water reserved in the earth can be accessible,which are mainly rivers,lakes and a part of the groundwater(See,fig.1).Given the current situation of freshwater on the earth showed previously,it is not surprising that more than1billion people have no access to safe water to meet their basic need for life[1].Fig.1.Distribution of water resources in worldAs the country with the most population but less abundant water resource,China will be more nervous in dealing with water problems.The total water reserve of China in 20th,October,2009is2800billion cubic meters,of which surface water reserve is 2700billion cubic meters and underground water reserve is830billion cubic meters. Considering the mutual interconversion and supply between surface water and groundwater,we deduct the double-counting amount730billion cubic meters and then the underground water reserve is about100billion cubic meters.According to the international criterion,water reserve per capita(WRPC)is classified into four grades:(1)the WRPC lower than3000cubic meters is defined as mild water shortage;(2)the WRPC lower than2000cubic meters is defined as moderate water shortage;(3)the WRPC lower than1000cubic meters is defined as severe water shortage;(4)the WRPC lower than500cubic meters is defined as extreme water shortage. (1)-(4)will be the criterion to evaluate the grade of water shortage.In this paper,first,we make a prediction in model1to find which provinces are short of water by applying polynomial fitting toolbox of MATLAB.Second,employing the Principal Component Analysis method and the statistical software SPSS,we finally conclude that the major factors influence the sustainable use of freshwater are rainfall, polluted water treatment ratio,GDP growth ratio and direct discharge quantity. Finally,according to the results and conclusions draw in the former analysis,we construct an optimization model,in which we provide the detailed water use plannings from2013to2025to the governments of every area.2.Assumptions(1)The influence of the thaw of glaciers caused by climate change and global warming on freshwater use and storage is not considered.(2)Sweeping reforms will not be implemented in terms of water use,industry, economy and environment-protection.(3)The population change follows the current demographic trends and the water use situation will not change dramatically with mutations in the population.(4)The natural purification of water is neglected in the model.3.MODEL1(1)Prediction of the water shortage in every province of China in2025After experimenting and comparing various predicting methods like Grey Forecasting Model and Gaussian Interpolation Fitting Forecast,we finally choose Linear Fit, which is proved to show the best effect,to complete our prediction model.The predicting result of water storage and water need of every province in China in 2025is showed in the Table1(the calculating results are from MATLAB).Table1.The predicting result of water storage and water need of every province in China in2025[2]In the sheet,Water Shortage=Water Need-Water Storage;Water Utilization=Water Need/Water Storage.From the sheet,we can conclude that Jiangsu,Shandong,Ningxia,Hebei,Henan, Tianjin,Beijing,Inner Mongolia and Anhui will be seriously shortage of water in 2025.Because the water storage can not be totally used every year,shortage of water is routinely defined as the water need is less than75percent of the water storage.Validation of the M odel1(2)Validation(2)Fig.2.The freshwater distribution map of ChinaCompared with the freshwater distribution map of China displayed in the following picture,our conclusion of prediction is matches with the actual situation[3](See,fig.2).MODEL2：Sustainable Water Use Plans For The Provinces Which Are 4.4.MODELShort of Water Based on Principal Component AnalysisBefore making a sustainable and efficient planning for the provinces that are short of water,we must know which kinds of factors influences the water use system most.And then our planning can be mainly concentrated on the key factor,which will greatly simplify the evaluation model.Take into consideration that the evaluation to the utilization of water resource system is a complex process,which involves the mutual coordination relationship among water reserve,water distribution,water use, water environment protection and polluted water treatment,we are supposed to apply a variety of indexes to construct a comprehensive evaluation system to describe the water use situation.Additionally,diffident indexes,being dimensioned in variable criteria,may lead to different evaluation results which may even conflict.Principal component analysis is appropriate when you have obtained measures on a number of observed variables and wish to develop a smaller number of artificial variables(called principal components)that will account for most of the variance in the observed variables.The principal components may then be used as predictor or criterion variables in subsequent analyses.So PCA has great superiority in constructing a comprehensive evaluating system,in which water use evaluation indexes are observed variables while our task is to develop some principal unobserved components and find the relationship between the components and evaluation system.(1)Principal Component AnalysisPrincipal component analysis is a variable reduction procedure.It is useful when you have obtained data on a number of variables(possibly a large number of variables), and believe that there is some redundancy in those variables.In this case,redundancy means that some of the variables are correlated with one another,possibly because they are measuring the same construct.Because of this redundancy,you believe that it should be possible to reduce the observed variables into a smaller number of principal components(artificial variables)that will account for most of the variance in the observed variables.The mathematical technique used in PCA is called eigen analysis:we solve for the eigenvalues and eigenvectors of a square symmetric matrix with sums of squares and cross products.The eigenvector associated with the largest eigenvalue has the same direction as the first principal component.The eigenvector associated with the second largest eigenvalue determines the direction of the second principal component.The sum of the eigenvalues equals the trace of the square matrix and the maximum number of eigenvectors equals the number of rows (or columns)of this matrix.PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by any projection of the data comes to lie on the first coordinate (called the first principal component),the second greatest variance on the second coordinate,and so on.When the analysis is complete,the resulting components will display varying degrees of correlation with the observed variables,but are completely uncorrelated with one another [4].(2)Procedure of PCA StandardizationStandardizing the original data is inevitable because of their different dimension and criteria.Given n samples and p indexes,we can get the data matrix:X =()ij n p x ×.Then standardize the data by:*()/ij ij i i x x x σ=−，whereij X :the value of index j in sample i;ij x :the original data of index j in sample i;i x :the mean of the original data of one index in sample i;i σ:the standard deviation of the original data of one index in sample i.(3)(3)CalculateCalculate the correlation coefficient matrix Calculate the correlation coefficient matrix R=()ij n n r ×according to the standardizeddata matrix *()ij n p x ×.11()()/.nij ki i kj j i j k r X X x x n σσ==−−∑(4)(4)CalculateCalculate the eigenvalues and eigenvectors On the basis of characteristic equation 0I R λ−=,we can get the eigenvalues i λand eigenvectors i e .(1i e =)(5)(5)CalculateCalculate the contribution rate of the principal component The contribution rate of the principal component i z is 1/pi k k λλ=∑,the cumulativecontribution rate is11ikk pkk λλ==∑∑.Generally,we choose the principal components 1,2,.....mcorresponds to the eigenvalues of 12,,...m λλλwhich make up the cumulative contribution rate of 85%(or higher than 85%).(6)Calculating Principal C omponent L oadingsThe principal component loading is calculatedby following formula ：(,)ij i i ija p z x ==（i,j=1,2,3,...p ）With the former analyzing process,we can make a rational simplification to the numerous variables under the principle of guaranteeing the least loss of infor mation.(7)PCA Applied To The Evaluation SystemTo apply the PCA to our water use evaluation model,we divide the final evaluation target into three subsystems:water utilization system,economic system and water environment system.Then we determine a variety of indexes(as follows)to indicate and quantify these three abstract subsystems.Fig.3.Water utilization systemFig.4.Economic systemFig.5.Water environment systemAn comprehensive water use evaluation system is showed in the follow picture:prehensive water use evaluation systemBecause of the heavy calculating tasks caused by numerous indexes and data,we use SPSS,an accurate and popular statistical analysis software,to simplify the complex computing task.Enter the data and we get the result as follow:urate and popular statistical analysis resultAfter several rounds of PCA experiments and then,from the data and sheets showed in the analysis results,we can conclude that the most-influenced factors are as follows:RainfallPolluted Water Treatment RatioGDP Growth RatioDirect Discharge QuantityConsidering that the unstable economic development environment leads to the change of GDP growth ratio undulates dramatically in the recent years,we find it difficult to evaluate the influence of GDP Growth Ratio on the Water Use Evaluation System.So the GDP Growth Ratio factor may need more data,assumptions and professional knowledge to deal with,which may be out of our capacity.We decide to focus more on the other3factors.According to the statistics，the drainage area of Yangtze River and its southernregions possess36.5percent of the land area but81percent of the freshwater while the drainage area of Huaihe River and its northern regions possess63.5percent of the land area but19percent of the freshwater.The Yellow River drainage,the Huaihe River drainage and the Haihe River drainage are most shortage of freshwater,with only7.7percent of freshwater.Currently,16provinces in China are under the line of severe water shortage(1000cubic meter per capita),and6provinces Ningxia,Hebei, Shandong,Henan,Shanxi and Jiangsu are under the line of extreme water shortage (500cubic meter).Though the shortage of water mainly highlights in the northern areas in the term of water reserve per capita,water pollution is a nationwide problem.Moreover,the more water an area possesses and the more intensive the population is,the more serious the pollution is.As a result,the water of the areas which are abundant of water is in poor quality,which is a more serious problem.Based on the survey,about90percent of city freshwater reserve is polluted to some extent,and the water pollution has been spreading from the rivers and tributaries to the main stream,penetrating from surface to underground,stretching from the land to the sea,spreading from urban to rural areas.Recently,the polluted water emissions are increasing by1.8billion tons,and the industrial wastewater and domestic sewage emissions have reached to0.16billion tons,of which80percent is directly eliminated without treatment.The third point is that both low efficient water use and over-exploit exist.Firstly,it is low efficient water use.And the more it lacks water,the lower efficiency it is.For example,reaches of yellow river lack water severely but agricultural irrigation still adopts large area flooding.In Ningxia,Inner Mongolia irrigated areas,each unit of land uses water of over1,000m³on average,several times and even over ten times higher than water-efficient irrigated area;water use ratio of agriculture is lower generally.At present,water use amount for producing one unit of food is2~2.5times as much as that in developed countries.Water use for agriculture is like that,and water use for industry is also like that. Presently water recycling use ratio for industry of china is much lower than75%of that in developed countries.GDP water use amount per unit is as over10times todozens of tiimes as that in developed countries.Water deprivation per unit for some important products is also several times,even dozens of times higher than that in devloped countries.Secondly,it is more severe to exploit water source excessively.take haihe river drainage basin for an example.It is one of the densest populations in china,including most areas in Beijing,Tianjin,Hebei and some parts in Shandong,Shanxi,Neimeng and Henan.Social economics here change a lot in recent pared to1950s, the population increases as much as before;the irrigated areas increase6times as much as before;gdp increases over30times as much as before;and these make total water use amount increase4times as much as before which exceed bearing capacity of water source.In result,surface water and ground water are over exploited for long time.The exploitation ratio reaches98%and it is much higher than40%of warning line.As the data from ministry of water resources shows,the ground water over-exploitation extends from56areas in1980s to present164areas,from87,000 square km to180,000square km and is more than10billion m³each year on average. The ground of over60,000square km descend to some extend.(China Ministry of Water Resources)According to these situations,we plan to take the following measures.(1)No flooding irritation for agriculture irritation;try little sprairritation;it is best to the best practical irritation method like drip irritation etc.(2)Take more invest in swage treatment to get100%of industrial and life swage treatment ratio;no discharging without life swage treatment in order for recycling use.(3)Increase management of contaminated river;reorganize severely contaminated factories and mines;the state offers some finance and subsidy to those who are not able to deal with pollution.(4)Try to use river fresh water;reduce exploiting the ground water.(5)For investment of seawater desalting,build seawater desalting factories;midland builds disposal factories for salt and soda water.In the PCA method,we know what affect more for water resource recycling use arerainfall,swage treatment ratio,swage direct discharging amount and water use amount per unit of farmland.(1)RainfallThe data and conclusions from China Meteorological Research Institute reveals that precipitation in China varies with locations and the precipitation change is cyclical(with a fixed period).In other words,the precipitation trends to be stable.So we believe that in a long period(until2025in this model),the influence of rainfall on the freshwater stabilizes and can be out of our consideration in the sustainable water use planning.(2)(2)TheThe A nalysis On The Conservation Of Farmland IrrigationAgricultural water consumption of the ten provinces which are shortage of water[5]: Table2.Agricultural water consumption of the ten provinces which are shortage of waterRegions 201120102009200820072006200520042003 Agricultural Water Consumption(0.1billion cubic meter)nation37433689.137233663.53599.53664.453580.3585.73432.8Beijing1010.811.411.411.712.0512.713.012.9 Tianjin1211.012.813.013.813.4313.612.011.2 Hebei140143.8143.9143.2151.6152.57150.2147.1149.6 InnerMongolia136134.5138.7134.1141.8142.18143.9149.4146.1Shanghai1616.816.816.716.218.3718.518.816.3Jiangsu308304.2300.1287.3268.5270.69263.8288.5223.1 Anhui168166.7167.2151.9120.6136.44113.6121.793.8 Shandong149154.8156.4157.6159.7169.40156.3154.3157.0Henan125125.6138.1133.5120.1140.15114.5124.5113.3 Ningxi6665.165.368.064.871.7372.368.658.4aThe following is the nationwide agricultural water consumption per mu from2006to 2007,Table3.The nationwide agricultural water consumption per mu from2006to20072006449m32007434m32008435m32009431m32010421m32011415m3mean430.8m3According to the calculation,we can conclude that we will save about40billion cubic meters water every year if the agricultural water consumption make a10%reduction. Agricultural water consumption accounts for about10percent of the total water consumption in the ten provinces.Assume that the national agricultural water consumption reduces by half in2025,then the amount of water saved will be surprising(show in the following sheet):Table4.Regions Mean Water Consumption Saving WaterNation3631.291102.536518Beijing11.77 3.57362117Tianjin12.54 3.807409471Hebei146.8944.59891365 Inner Mongolia140.7442.73164345 Shanghai17.16 5.210139276Jiangsu279.3584.81657382Anhui137.7741.82988858Shandong157.1747.72013928Henan126.0838.2805571Ningxia66.6920.24849582 Analysis and planning of polluted water treatment(3)Analysis(3)Polluted water emissions are very large in China every year,we make a prediction onthe industrial polluted water emissions in2025with the data from2003to2010by linear fitting toolbox in MATLAB.The results are as follows:Table5.YearIndustrial Polluted WaterEmissions(ten thousand cubic meter)Polluted Water TreatmentRatio200322440600.93917200422732100.9397200523023600.94023200623315000.94076200723606500.941289200823898000.941819200924189400.942349201024480900.942879201124772400.943409201225063800.943939201325355300.944468201425646800.944998201525938200.945528201626229700.946058201726521200.946588201826812600.947118201927104100.947648202027395500.948177202127687000.948707202227978500.949237202328269900.949767202428561400.950297202528852900.950827The results of domestic wastewater emissions are as follows:Table6.The results of domestic wastewater emissionsYear domestic wastewater emissions2003242793020042613170200527984202006298366020073168900200833541402009353939020103724630201139098702012409511020134280360201444656002015465084020164836090201750213302018520657020195391810202055770602021576230020225947540202361327802024631803020256503270From the results,it’s not difficult to find that the polluted water emissions are increasing sharply,reaching933888.17million cubic meters.Polluted water recycling can greatly alleviate the circumstance of water shortage,make contribution to environment protection and sustainable water use.The sheet also shows that the polluted water treatment ratio exceeds95%in2025 while the rest5%polluted water is difficult to purify.This rest part seriously harms the environment.So proposals of dealing with refractory polluted water are expected to implement before dry season in2025approaches to guarantee abundant freshwater. Constructing polluted water treatment factories to purify the polluted water eliminated by factories is executable in the process.Domestic wastewater emissions are not strictly limited by legislation,which leads tothe wastewater emissions without treatment.Our preliminary plans to deal with domestic wastewater emissions are constructing small-scale purifying factories which are capable of dealing with the99%domestic wastewater produced by the cities where the factories locate.This planning aims at dealing with the domestic wastewater of which the emissions are not strict and recycling the freshwater.We can know from the previous data that the expense of treating one cubic meter polluted water is0.6yuan.Assume that the polluted water emissions are900cubic meters a year,and we can get the expense a year54billion pared to the GDP(39798.3billion yuan in2010)and revenue(8308billion yuan),the investment on dealing with the polluted water is acceptable.However,the investment on the polluted water treating factories is high,1500yuan a square meter,so investment can be accomplished year by year.The polluted water treatment ratio is above94percent, so the late investment is810billion yuan(dealing with0.54billion cubic meters). Though the investment is high but the benefit is low,it is inevitable for recycling freshwater and protecting environment.In summary,the polluted water treatment quantity will reach90billion cubic meters until2025while the saving quantity of water in agriculture reaching110.25billion cubic meters.5.Analysis of the water diversionIt’s exactly the greatest characteristic of the distribution of the freshwater in China, seriously unbalanced in south and north,that mainly leads to the water shortage in the northern areas.The total freshwater resources is relatively stable in China every year.The south-to-north water diversion project dramatically improves the unbalanced situation.The project eventually will achieve a44.8billions cubic meters water diversion,in which the eastern route contributes an amount of14.8billion cubic meters while the central route13billion and the western route17billion.Now that this project is so important,we will devote to analyzing the project in detail and working out a rational water transfer planning to deal with the current water use situation.(1)Eastern routeThe Eastern route is the most advanced in terms of construction.It consists of an upgrade of the Grand Canal.Water from the Yangtze River will be drawn into the canal in Jiangdu,where a giant400m³/s.(12.6Billion m3/year if operated continuously)pumping station was built already in the1980s,and is then fed uphill by pumping stations along the Grand Canal and through a tunnel under the Yellow River,from where it can flow downhill to reservoirs near Tianjin.Construction on the Eastern Route officially began on December27,2002,and water was supposed to reach Tianjin by2012.However,water pollution has affected the viability of this project.The completed line will be slightly over716miles(1,152km)long,equipped with23pumping stations with a power capacity of454megawatts.It includes two 9.3m diameter horizontal tunnels70m under the riverbed of the Yellow River.(2)Central routeThe central route is from Danjiangkou Reservoir on the Han river,a tributary of the Yangtse River,to Beijing.This route is built on the North China Plainand,once the Yellow River has been crossed,water can flow all the way to Beijing by gravity.The main engineering challenge is to build a tunnel under theYellow River.Construction on the central route began in2004.In2008the307km-long Northern stretch of the central route was completed at a cost of US$2billion.Water in that stretch of the canal does not yet come from the Han River,but from various reservoirs in Hebei Province south of Beijing.Farmers and industries in Hebei had to cut back water consumption to allow for water to be transferred to Beijing.The whole project was expected to be completed around2010.This has recently been set back to2014to allow for more environmental protections to be built.A problem is the influence on the Han River,where~1/3of the water is diverted.One long-term consideration is to build another canal to divert water from the Three Gorges Dam to Danjiangkou Reservoir.Another major difficulty is the resettlement of~330,000 persons around Danjiangkou Reservoir and along the route.On October18,2009, Chinese officials began to relocate residents from the areas of the Hubei and Henan provinces that will be affected by the reservoir.The completed line will be approximately1,264km long,initially providing9.5billion m3of water annually.By 2030,it is expected to increase its water transfer to12to13billion m3annually. Industries are prohibited to locate in the watershed of the reservoir in order to keep its water drinkable.(3)Western routeThe western route,called the Big Western Line,is still at the planning stage.It aims to divert water from the headwaters of the Yangtze River(the Tontian,Yalong and Dadu Rivers)into the headwaters of the Yellow River.In order to move the water through the drainage divide between these rivers,huge dams and long tunnels are needed to be built to cross the Qinghai-Tibetan Plateau and Western Yunnan Plateaus. This route is designed to bring3.8billion m3of water from three tributaries of the Yangtze River about450km across the Bayankala Mountains to northwest China.The Tongtian diversion line would be289km in length,the Yalong131km,and the Dadu 30km.The feasibility of this route is still under study and this project won't start in the near future.Environmentalists have raised concerns about potential flooding that could result.[5]The respective rivers are entirely within China.In addition,there are long-standing plans to divert about200billion cubic metres of water annually from the upstream sections of six rivers in southwesternChina, including the Mekong(Lancang River),the Yarlung Zangbo(called Brahmaputra further downstream)and the Salween(Nu River),to the Yangtze River,the Yellow River and ultimately to the dry areas of northern China through a system of reservoirs, tunnels and natural rivers.The project was considered too immense and costly to be undertaken at the time.The respective rivers are transboundary and a diversion would affect India,Bangladesh,Myanmar,Laos,Thailand,Cambodia and Vietnam.The diversion map is showed in the following picture:Fig.8.The diversion map。

2013年美国数学建模B题

PROBLEM B: Water, Water, EverywhereFresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feasible, and cost-efficient water strategy for 2013 to meet the projected water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical model must address storage and movement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to governmental leadership outlining your approach, its feasibility andcosts, and why it is the “best water strategy choice.”Countries: United States, China, Russia, Egypt, or Saudi Arabia水, 水, 无处不在(美国竞赛2013年B题)淡水资源逐渐成为这个世界大多数国家发展的极限约束。

【7A版】2013全国大学生数学建模比赛B题-答案

承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。

我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。

如有违反竞赛规则的行为，我们将受到严肃处理。

我们参赛选择的题号是（从A/B/C/D中选择一项填写）： B我们的参赛报名号为（如果赛区设置报名号的话）：所属学校（请填写完整的全名）：重庆邮电大学参赛队员(打印并签名) ：1.2.3.指导教师或指导教师组负责人(打印并签名)：日期：20GG 年9 月13 日赛区评阅编号（由赛区组委会评阅前进行编号）：编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：碎纸片的拼接复原摘要本文研究的是碎纸片的拼接复原问题。

由于人工做残片复原虽然准确度高，但有着效率低的缺点，仅由计算机处理复原，会由于各类条件的限制造成误差与错误，所以为了解决题目中给定的碎纸片复原问题，我们采用人机结合的方法建立碎纸片的计算机复原模型解决残片复原问题，并把计算机通过算法复原的结果优劣情况作为评价复原模型好坏的标准，通过人工后期的处理得到最佳结果。

面对题目中给出的BMP格式的黑白文字图片，我们使用matlab软件的图像处理功能把图像转化为矩阵形式，矩阵中的元素表示图中该位置像素的灰度值，再对元素进行二值化处理得到新的矩阵。

题目每一个附件中的碎纸片均为来自同一页的文件，所以不需考虑残片中含有未知纸张的残片以及残片中不会含有公共部分。

鉴于残片形状分为“长条形”与“小长方形”，残片内容分为中文、英文，纸张的打印类型分为“单面型”、“双面型”，所以我们根据残片的类型对矩阵做不同处理。

2013年全国数学建模B题一等奖论文

（由由由由由由）第十届华为杯全国研究生数学建模竞参学校南京师范大学参参队号103190031.佟德宇队员姓名2.顾燕3.贾泽慧(由由由由由由)第十届华为杯全国研究生数学建模竞参题目功率放大器非线性特性及预失真建模摘要针对问题一中求解输入输出信号之间的非线性功放特性函数问题, 采用了不同的多项式函数, 运用最小二乘法或正则化后的最小二乘法进行拟合求解. 并用参数NMSE 来评价所建模型的准确度. 结果发现在逼近函数选为函数基的情况下, 采用正则化后的最小二乘法得出的模型准确度最好, 其对应的参数NMSE=-68.6294.同时考虑计算量和模型准确度, 在由多项式变形函数逼近功放的模型基础上, 来进行预失真模型的建立. 根据题中给出的原则和约束, 可知预失真模型的表达式与功放模型的表达式是类似的, 从而可建立相应的预失真模型.：-11()()()K k k k z t h x t x t ==∑K=4时, 整体模型的放大倍数g=1.8693, 参数NMSE=-32.5819, EVM=2.3491; K=5时, g=1.8473, 参数NMSE=-37.1398, EVM=1.3900; K=7时, g=1.8326, 参数NMSE=-46.0624, EVM=0.4976.针对问题二, 直接将功放的输入输出与题目中所提的“和记忆多项式”模型进行拟合, 运用正则化后的最小二乘法进行求解, 这很好的保证了模型的可解性. 本题只考虑功放模型次数为5的情形. 当记忆深度为7时, 得NMSE=-45.8394; 当记忆深度为3时, 得NMSE=-44.5315. 预失真模型的建立与问题一类似, 文中以框图的方式建立了预失真处理的模型实现示意图, 并对次数为5、记忆深度为3的情形, 求解出整体模型的放大倍数g=9.4908, 参数NMSE=-37.8368, EVM=0.0128.针对问题三, 将所给的离散的、有限的输入输出数据作为随机过程的样本函数,通过其傅立叶变换得到功率谱参度函数. 文中分别给出了输入信号、无预失真补偿的功率放大器输出信号、采用预失真补偿的功率放大器输出信号的功率谱参度图形. 可解出它们的ACPR 分别为-155.6610、-74.3340、-104.4904, 最后对结果进行分析评价, 得出采用预失真补偿的功率放大器的输出信号效果比无预失真补偿的效果好. 关键字：最小二乘法、Tikhonov正则化、Fourier变换一、问题重述信号的功率放大是电子通信系统的关键功能之一, 其实现模块称为功率放大器( PA, Power Amplifier), 简称功放. 功放的输出信号相对于输入信号可能产生非线性变形, 这将带来无益的干扰信号, 影响信信息的正确传递和接收, 此现象称为非线性失真.功放非线性属于有源电子器件的固有特性, 研究其机理并采取措施改善, 具有重要意义. 目前已经提出了各种技术来克服功放的非线性失真, 其中预失真技术是被研究的较多的一项技术, 其最新的研究成果已经被运用于实际的产品中, 但在新算法、实现复杂度、计算速度、效果精度等方面仍有相当的研究价值.预失真的基本原理是：在功放前设置一个预失真处理模块, 这两个模块的合成总效果使整体输入-输出特性线性化, 输出功率得到充分利用.文中给出了NMSE 、EVM 等参数评价所建模型其准确度, 以及ACPR 表示信道的带外失真的参数.根据数据文件中给出的某功放无记忆效应、有记忆效应的复输入输出测试数据：(1)我们建立此功放的非线性数学模型()G ⋅, 并用NMSE 来评价所建模型的准确度.(2)根据线性化原则以及“输出幅度限制”和“功率最大化”约束, 计算线性化后最大可能的幅度放大倍数, 建立预失真模型. 并运用评价指标参数NMSE/EVM 评价预失真补偿的计算结果.(3)应用问题二中所给的数据, 计算功放预失真补偿前后的功率谱参度(输入信号、无预失真补偿的功率放大器输出信号、采用预失真补偿的功率放大器输出信号), 并用图形的方式表示了这三类信号的功率谱参度. 最后用相邻信道功率比ACPR 对结果进行分析.二、模型假设1、假设题中所给的功放输入输出数据采样误差为0.2、假设题中所给的功放输入输出数据具有代表性、一般性.3、假设存在这样的预失真处理器, 能够做到将输入数据变为模型求解所得的预失真处理输出结果.三、基本知识§3.1 最小二乘方法最小二乘方法[][]12产生于数据拟合问题, 它是一种基于观测数据与模型数据之间的差的平方和最小来估计数学模型中参数的方法. 输入数据t 与输出数据y 之间大致服从如下函数关系(,)y x t φ=,式中n x R ∈为待定参数. 为估计参数x 的值, 要先经过多次试验取得观测数据1122(,),(,),,(,)m m t y t y t y , 然后基于模型输出值和实际观测值的误差平方和21((,))m i ii y x t φ=−∑最小来求参数x 的值, 这就是最小二乘问题. 一般地, m n .引入函数()(,), 1,2,,i i i r x y x t i m φ=−= ,并记12()((), (), , ())m r x r x r x r x = ,则最小二乘问题即为n min ()()T x Rr x r x ∈. 如果最小二乘问题中的模型函数估计准确, 那么最小二乘问题的最优值是很靠近零的. 因此()r x 常称作残量函数.对于线性最小二乘问题, 残量函数可以表示为()r x b Ax =−,从而线性最小二乘问题可以表示为2min n x R b Ax ∈−. (3.1.1) 若A 是列满秩的, 且考虑到二次凸函数的稳定点即为最小值点, 可以直接得到x 的求解公式, 即()1T T x A A A b −=. (3.1.2) 而对于复数域上的线性最小二乘问题n 2min x C b Ax ∈−, 也可以直接得到x 的求解公式, 即为()-1T x A A A b =, (3.1.3) 其中, T A 表示A 的共轭转置.§3.2 Tikhonov 正则化在使用最小二乘方法进行参数估计的时候, 由于A 不一定是列满秩的, 故T A A 不一定是可逆的, 此时就不能够用上面所推得的公式进行直接的求解了. 为了克服这个困难,考虑Tikhonov 正则化[]3方法, 即给目标函数加上一个正则项(即一个邻近项)2k k x x λ−.此时, 最小二乘问题转化为n 221min +k k k x C x b Ax x x λ+∈=−−.其中k x 是第k 步迭代得到的解, k λ可以选为一个常数或一个单调下降趋于0的数列. 迭代的终止准则为1k k x x ε+−≤,其中ε是一个给定的误差上界.考虑到二次凸函数的稳定点即为最小值点, 这时问题22min n k k x C b Ax x x λ∈−+− 是可以直接求解的, 给出x 的求解公式为()()1T k k k x A A I A b x λλ−=++.显然, 此时即使A 非列满秩, 问题也是可以求解的.四、问题分析问题一题中已给出了某功放无记忆效应的复输入输出测试数据, 现需要建立此功放的非线性特性数学模型, 拟合出功放的特性函数()G⋅. 根据函数逼近理论, 功放的特性函数可以用多项式来表示, 也可以用空间中的一由正交函数基来表示. 然后采用最小二乘法或正则化后的最小二乘法, 将这些情况都进行求解, 得出功放的特性函数()G⋅. 并在最后用参数NMSE(归一化均方误差)来评价所建模型的准确度.接着, 在前面所建模型的基础上, 选择一个计算量适当, 且准确度较好的()G⋅的一个拟合模型. 然后根据线性化原则以及“输出幅度限制”和“功率最大化”约束, 建立预失真模型, 使得整体模型线性化后放大倍数尽可能的大. 通过对优化模型的分析可知, 对预失真特性函数()F⋅的求解可以转化为对1Gg−⎛⎞⎜⎟⎝⎠的求解, 且预失真模型的表达式与功放模型的表达式是类似的. 在求解1Gg−⎛⎞⎜⎟⎝⎠时, 可以对求解所用模型的次数进行不同的选取,分别得出整体模型的g和NMSE、EVM的值, 用来评价预失真补偿的结果.问题二题中已给出了某功放有记忆效应的复输入输出测试数据, 现需要建立此功放的非线性特性数学模型, 拟合出功放的特性函数()G⋅. 根据函数逼近理论, 本文直接将功放的输入输出与题目中所提的“和记忆多项式”模型来进行拟合, 在使用最小二乘方法求解时, 我们对目标函数加了一个正则项, 以保证求解的可实现性.预失真处理器模型的建立与问题一类似, 且给出了以框图的方式建立的预失真处理的模型实现示意图.问题三问题二中所给的输入输出数据是离散的、有限的, 在这种情况下计算功率谱参度的函数可以用自相关函数法或对随机过程{}()x t的样本函数作傅立叶变换得到, 文中采取第二种方法来求解.五、模型建立与求解§5.1 问题一的模型与求解§5.1.1 无记忆功放的特性函数()G⋅模型建立文章中已给出某功放无记忆效应的复输入输出测试数据, 这些数据是对功放输入)(tx/输出)(t z进行离散采样后得到的, 它们的值为分别为()x n/()z n(采样过程符合Nyquist采样定理要求).对于问题一, 根据文章中所给的某功放无记忆效应的复输入输出测试数据, 首先需要建立此功放的非线性特性数学模型, 拟合出功放的特性函数()G⋅. 根据函数逼近理论,可以采用1、多项式的形式2、多项式的变形的形式3、空间中的一由正交函数基的线性由合来表示4、正则化下, 空间中的一由正交函数基的线性由合来表示下面将这些情况都进行建模, 来拟合功放的特性函数()G ⋅, 并在最后进行比较选择优者.所求得的模型的数值计算结果业界常用NMSE 、EVM 等参数评价其准确度, NMSE 的具体定义如下. 采用归一化均方误差 (Normalized Mean Square Error, NMSE) 来表征计算精度, 其表达式为211021ˆ|()()|NMSE 10log |()|N n N n z n z n z n ==−=∑∑ . (5.1.1) 如果用z 表示实际信号值, ˆz表示通过模型计算的信号值, NMSE 就反映了模型与实际模块的接近程度. 显然NMSE 的值越小, 模型的数值计算结果就越准确.误差矢量幅度 (Error Vector Magnitude, EVM)定义为误差矢量信号平均功率的均方根和参照信号平均功率的均方根的比值, 以百分数形式表示. 如果用X 表示理想的信号输出值, e 表示理想输出与整体模型输出信号的误差, 可用EVM 衡量整体模型对信号的幅度失真程度:EVM 100%= . (5.1.2)模型一多项式的形式首先根据函数逼近的Weierstrass 定理, 对解析函数采用简单的多项式来表示, 可表示为∑==Kk k k t x h t z 1)()(. (5.1.3)因为此时是要将观测数据与形式已经固定的函数(5.1.3)进行拟合, 而目的是求解该函数的各项系数, 所以该问题其实就是最简单的线性最小二乘问题.模型建立()n 211min ()N K k k h C n k z n h x n ∈==−∑∑, (5.1.4) 其中, ()x n 和()z n 为文章中所给的输入和输出测试数据, 这些数据是对功放输入()x t 、输出()z t 进行离散采样后得到的(采样过程符合Nyquist 采样定理要求),N 为功放输入输出数据的总个数.将问题(5.1.4)与( 3.1.1)进行对应, 由( 3.1.3)可以直接得到系数的表达式为()-1T h A A A z = 其中232323 (1) (1) (1) (1) (2) (2) (2) (2) () () () ()K K K x x x x x x x x A x N x N x N x N ⎡⎤…⎢⎥…⎢⎥=⎢⎥⎢⎥⎢…⎥⎣⎦, ()12,,,TK h h h h =…, ()()()()1,2,,Tz z z z N =….结果当3K =时, (见附录2.1.1)该表达式中的系数为123 2.908532278399690.060653883258900.213775998314930.43417026083854 0.198185637666730.27826757408010h ih i h i=−=−=+.根据模型一以及(5.1.1)式, 可以求出NMSE 的值如下：()NMSE 13.4414169873254 3k =−=.当5k =时, (见附录2.1.2 )表达式中的系数为12345 2.908037719327826 - 0.063527494375989i0.343519806629302 - 0.388942747664566i0.541211413428411 - 0.144422960285135i -0.399744749427209 - 0.558463329513045i-0.271952185146638 + 0.1205591h h h h h =====40060622i根据模型一以及(5.1.1)式, 可以求出NMSE 的值如下：()NMSE -21.544782705381238 5k ==.模型二多项式的变形同时我们也考虑了多项式变形[]4的情形来对其进行表示, 其表示式为-11()()()K k k k z t h x t x t ==∑. (5.1.5)因为此时是要将观测数据与形式已经固定的函数(5.1.5)进行拟合, 而目的是求解该函数的各项系数, 所以该问题其实就是最简单的线性最小二乘问题.模型建立()n 2-111min ()()N K k k h C n k z n h x n x n ∈==−∑∑ (5.1.6)其中N 为所给功放输入输出数据的总个数, K 为表达式的次数. 将问题(5.1.6)与(3.1.1)进行对应, 由(3.1.3)可以直接得到系数的表达式为()-1T h A A A z = 其中212121(1) (1)(1) (1)(1) (1)(1)(2) (2)(2) (2)(2) (2)(2) () ()() ()() ()()K K K x x x x x x x x x x x x x x A x N x N x N x N x N x N x N −−−⎡⎤…⎢⎥⎢⎥…=⎢⎥⎢⎥⎢⎥…⎢⎥⎣⎦,()123,,,,TK h h h h h =…, ()()()()()1,2,3,,Tz z z z z N =…. 分别考虑当3k =, 5k =时, 该表达式的具体形式(即确定表达式的系数).结果当3k =时, (见附录2.1.3 )表达式中的系数为123 3.051183005392040.00000000000001 0.006071903393980.00000000000005 1.170159412626470.00000000000004h ih i h i=−=+=−−.根据上面所建立的模型以及(5.1.1)式, 可以求出NMSE 的值如下：()NMSE 29.7446547565428 3k =−=.当5k =时, (见附录2.1.4 )表达式中的系数为12345 2.967983597251020.00000000000080 0.309931644197600.00000000000873 0.153664636905190.00000000002804 3.424500445954250.00000000003458 2.208212395486470.00000000001446h ih ih i h ih i=−=+=−−=−+=−.根据上面所建立的模型以及(5.1.1)式, 可以求出NMSE 的值如下：()NMSE 45.379717608769994 5k =−=模型三空间中的一由正交函数基的线性由合最后根据函数逼近理论, 可采用空间中的一由正交函数基[]4的线性由合来表示该特性函数(参考文献3中的方法), 其表达式为()z t h =Ψ, (5.1.7)其中正交矩阵12[() () ()]k x x x ψψψΨ= ,11()!()(1)(1)!(1)!()!kl l k k l k l x x x l l k l ψ−+=+=−−+−∑. 因为此时是要将观测数据与形式已经固定的函数(5.1.7)进行拟合, 而目的是求解该函数的各项系数, 所以该问题其实就是最简单的线性最小二乘问题.模型建立 n 2min h C z h ∈−Ψ (5.1.8) 其中()123,,,,TK h h h h h =…, ()()()()()1,2,3,,T z z z z z N =…, ()()()12[() ()()]k x n x n x n ψψψΨ= ,()()()11()!()(1)(1)!(1)!()!k l l kk l k l x n x n x n l l k l ψ−+=+=−−+−∑, N 为功放的输入输出数据的总个数. 将问题(5.1.8)与(3.1.1)进行对应, 由(3.1.3)可以直接得到系数的表达式为 ()-1T T h z =ΨΨΨ. 由于计算量较大, 我们选取7=k 来进行拟合, 得出表达式中的系数.结果(见附录2.1.5)当7=k 时, 表达式中的系数为12345 3.287412936081622-7.322701472967097-015-0.091488124421954-2.16460963736731-015-0.066219774105875 5.035305939565804-0160.038056322596937 2.726632938529483-0160.01014165858755-1.2h e ih e ih e ih e i h ===+=+=6758894247527231-016-0.005283612035716-2.653720342429833-016-0.001265433154276-1.923256069376669-016e ih e ih e i==.根据上面所建立的模型以及(5.1.1)式, 可以求出NMSE 的值如下：()NMSE -60.5675309366592 7k ==模型四模型三正则化模型建立对于模型三, 由于所给的数据较多, 很难避免本文3.2节中所提到的T ΨΨ奇异的情况, 故对(5.1.8)再进行一个Tikhonov 正则化. 即对(5.1.8)加一个正则项2k k h h λ−.问题转变为()1221min K M k k k h C h z h h h λ⋅×+∈=−Ψ+−. (5.1.9) 其中k h 是第k 步迭代得到的解(计算机运行求解时是要给其赋一个初始值的), 而k λ可以选为一个常数或一个单调下降趋于0的数列. 而迭代的终止准则为1k k h h ε+−≤,其中ε是一个给定的误差上界.考虑到二次凸函数的稳定点即为最小值点, 问题(5.1.9)是可以直接求解的, 得到h 的求解公式为()()()1T Tk k k h I z n h λλ−=ΨΨ+Ψ+. (5.1.10)此处, 我们仍选取7=k 来进行拟合, 其中一些参数选取为800111, 1, 0.8, 10k k h i λλλε−+=+===.则可得出表达式(5.1.7)中的系数.结果(见附录2.1.6)123456 3.2873994140515280.000008426827987-0.0914922453118830.000002568107767-0.066218825186175-0.000000591359660.038056824724197-0.0000003129219510.010141412616440.000000153287355-0h ih ih ih i h ih =+=+===+=7.0052839775157310.000000227764411-0.0012655686759970.000000084456122ih i+=+根据上面所建立的模型以及(5.1.1)式, 可以求出NMSE 的值如下：()NMSE -68.6293523598994 7k ==模型一~模型四的总评价对四种模型下参数NMSE 的大小进行比较发现, 当选用一由正交函数基, 并运用正则化后的最小二乘方法来对功放特性函数进行拟合时(即模型四), NMSE 的值是最小的. 也就是说2121ˆ|()()||()|Nn Nn z n zn z n ==−∑∑在模型四下是最靠近0的, 故模型四是逼近效果最好的.但模型四的计算复杂度是很大, 由所得的NMSE 参数可发现模型二的计算精度也是不错的, 但其计算的复杂度比模型四要小很多, 故选择模型二来求解功放特性函数. 且在下面的无记忆功放模型的预失真处理建模中, 功放特性函数是由模型二得出的.§5.1.2四种模型的输入输出幅度比较图与评价下面将实际的与拟合的复输入输出幅度值进行作图, 以便更直观的看出模型的逼近效果.图5.1 模型一k=3实际与拟合功放输入/输出幅度散点图图5.1模型一k=5实际与拟合功放输入/输出幅度散点图图5.3模型二k=3实际与拟合功放输入/输出幅度散点图图5.4 模型二k=5实际与拟合功放输入/输出幅度散点图图5.5 模型三实际与拟合的功放输入/输出幅度散点图图5.6模型四实际与拟合的功放输入/输出幅度散点图根据观察比较发现, 当用正交的函数基或对其实行一个正则化(即模型三和模型四), 来对功放特性函数进行拟合的时候, 拟合情形的输入输出幅度散点图与实际的输入输出幅度散点图的逼近效果是最佳的.k=时, 其散点图的逼近效果也是很好的.同时可观察到但模型二中的次数5§5.1.3 预失真处理模型建立选定-11():()()()Kk k k G z n b x n x n =⋅=∑的阶数5K =, 通过上面的算法可以得到当F 取不同阶数的情况下, g, NMSE, EVM 的结果及图像表5.1 F 取不同阶数情况下g, NMSE, EVM 的结果F 的阶数Kg NMSE EVM 4 1.86932497973065-32.5819077399852 2.34911681195961% 5 1.84730161996524-37.1398119663279 1.38998272147897% 7 1.83264461869445-46.06241433950440.497598752653887%由表5.1的结果可以看出当F 的阶数越高时, 得到的g 的值越小(说明线性化后的幅度放大倍数越小), NMSE 、EVM 的值越小(说明模型的计算精度越高, 整体模型对信号的幅度失真程度越小).图5.7理想信号与所建模型得到的输出信号对比(K=4) 图5.8理想信号与所建模型得到的输出信号对比(K=5)图5.9理想信号与所建模型得到的输出信号对比(K=7)根据观察发现, 当K 的取值越大时, 所建模型的输入输出幅度散点图与理想的输入输出幅度散点图的逼近效果越好.§5.2 问题二的模型与求解§5.2.1 有记忆功放的特性函数()G ⋅模型建立对于问题二, 根据文章中所给的某功放有记忆效应的复输入输出测试数据, 首先需要建立此功放的非线性特性数学模型, 拟合出功放的特性函数()G ⋅. 此时功放不仅与此时刻输入有关, 而且与此前某一时间段的输入有关, 其可以由为101111022220212()()()(1)()()(1)()K Mk km M k m M z n h x n m h x n h x n h x n M h x n h x n h x n M ===−=+−++−++−++−+∑∑ 01 ()(1)()K K K K K KM h x n h x n h x n M ++−++− , 0,1,2,,n N = .式中M 表示记忆深度, km h 为系数. 具有记忆效应的功放模型也可以用更一般的V olterra级数[][]56表示, 由于V olterra 级数太复杂, 简化模型有Wiener 、Hammersteint 等[][]47. 由于常用复值输入-输出信号, 上式也可表示为便于计算的“和记忆多项式”模型-110()(-)|(-)|K Mk km k m z n h x n m x n m ===∑∑ 0,1,2,,n N = (5.2.1)模型建立本文采用“和记忆多项式”模型(5.2.1)式来进行拟合. 我们用最小二乘法来求解, 由于本问中所给的输入输出的数据个数非常大, 故现在选取其中的一部分来进行拟合, 求得功放过程的模型. 我们选取输入输出数据的次数n 为1M +的倍数的数据来进行拟合, 最小二乘公式即为()()12-1(1)|10min (-)|(-)|K M K Mk km h CM nk m n Nz n h x n m x n m ××∈+==≤−∑∑∑ (5.2.2) 其中N 是指所有的功放的输入数据总个数, K 表示所选模型的最高次数, M 表示记忆深度(本文在求解模型时是事先给定的), ()x n 是第n 个复输入值, ()z n 是第n 个复输出值, km h 为系数, ()102001222212,,,,,,,, ,,,,TK K M M KM h h h h h h h h h h =…………….由于所给的数据较多, 即便是选取了部分数据进行拟合,但仍很难避免3.2节中所提到的A A 奇异的情况, 故对(5.2.2)再进行一个Tikhonov 正则化. 即对(5.2.2)加一个正则项2k k h h λ−,则问题转变为()()122-11(1)|10min (-)|(-)|K M K Mk k km k k h CM nk m n Nh z n h x n m x n m h h λ××+∈+==≤=−+−∑∑∑ (5.2.3) 其中k h 是第k 步迭代得到的解, 而k λ可以选为一个常数或一个单调下降趋于0的数列. 而迭代的终止准则为1k k h h ε+−≤,其中ε是一个给定的误差上界.当给定一个记忆深度M 后, 我们可以将问题(5.2.3)化成如下形式的问题, 即()22min nk k h Cz n Ah h h λ∈−+− (5.2.4) 其中A 是一个()()()()/11N M K M +×⋅+的复矩阵, 即1111(1) (1)(1) (1)(1) (1) (1)(1) (22) (22)(22) (22)(22) (2) (1)(1) K K K K x M x M x M x M x M x x x x M x M x M x M x M x M x x A −−−−+++++++++++=……………… ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦而()102001121112,,,,,,,, ,,,,TK K M M KM h h h h h h h h h h =…………….考虑到二次凸函数的稳定点即为最小值点, 问题(5.2.4)是可以直接求解的, h 的求解公式为()()()1Tk kk h A A I A z n h λλ−=++. (5.2.5)本题中已给出有记忆功放输入输出数据的总个数为73920N =, 并分别取 87, 5, 10M K ε−===和 83, 5, 10M K ε−===这两种情况. 这样就可以根据(5.2.5)求得h .结果(见附录2.2.1、2.2.2)当7,5M K ==时, 由于系数共有40个, 即h 是一个401×的大向量, 故将该结果放到附录中. 再根据上面所建立的模型及(5.1.1)式, 求出该模型的NMSE 值如下：NMSE -45.839408840847 7,5M K ===.当3,5M K ==时, 由于系数共有20个, 即h 是一个201×的大向量, 故将该结果放到附录中. 再根据上面所建立的模型及(5.1.1)式, 求出该模型的NMSE 值如下：NMSE 44.5315001961471 3,5M K =−==.§5.2.2有记忆功放模型的输入输出幅度图下面将实际与拟合的复输入输出幅度进行作图, 以便更直观的看出模型的逼近效果.图5.10 M=7实际与拟合功放输入/输出幅度散点图图5.11 M=3实际与拟合功放输入/输出幅度散点图总评价根据观察比较发现, 尽管在用“和记忆多项式”模型进行拟合时, 我们只选取了一部分输入输出测量数据进行模型的建构. 但通过对上面两图的观察, 当对所有的输入测量数据进行作图时, 可发现拟合得到的输入输出幅度散点图与实际的输入输出幅度散点图的逼近效果还是很好的.§5.2.3 预失真处理模型建立上面已求得功放特性函数()G ⋅的模型, 采用“和记忆多项式”模型-110()(-)|(-)|K Mk kmk m z n hx n m x n m ===∑∑建立的功放模型. 下面建模的总体原则是使预失真和功放的联合模型呈线性后误差最小. 在此模型中, 有两个约束需要考虑：(1)输出幅度限制：即模型中的预失真处理的输出幅度不大于给出的功放输入幅度最大值.(2)功率最大化：即模型的建立必需考虑尽可能使功放的信号平均输出功率最大, 因此预失真处理后的输出幅度需尽可能提高.0≤下面我们将给出解决该优化问题的算法：给定判断容限step1选定-110(): ()(-)|(-)|KMk km k m G z n h x n m x n m ==⋅=∑∑的阶数为5K =. 因数据量很大且算法较复杂, 本文对F 进行多次计算, 发现当阶数为5K =的时候与更高阶相比, 效果就已经很好了, 故下面只给出阶数为5K =时g, NMSE, EVM 的结果.本文取定记忆深度为 3M =, 现根据算法5.2可求得9.490829228013789g =,由于系数一共有20个, 即h 是一个201×的向量, 故将此结果放到附录中.根据上面所建模型以及(5.1.1)、(5.1.2)式, 可求出该模型的NMSE 、EVM 值如下：.NMSE -37.836849855461956EVM 0.012827957346961== 3,5M K ==由所得数据, 可以发现在该算法下, 得到的g 的值比较大(说明线性化后的幅度放大倍数大), NMSE 、EVM 的值较小(说明模型的计算精度越高, 整体模型对信号的幅度失真程度越小).图5.13 M=3, K=5实际与拟合功放输入/输出幅度散点图观察图5.13发现, 该情况下所建模型的输入输出幅度散点图与理想的输入输出幅度散点图逼近效果还是较好的. 故该模型是可行的.§5.3 问题三的模型与求解 §5.3.1背景知识功率谱的概念是针对功率有限信号的, 所表现的是单位频带内信号功率随频率的变化情况. 保留了频谱的幅度信息, 但是丢掉了相位信息, 所以频谱不同的信号其功率谱是可能相同的. 功率谱是随机过程的统计平均概念, 平稳随机过程的功率谱是一个确定函数；而频谱是随机过程样本的Fourier 变换, 对于一个随机过程而言, 频谱也是一个“随机过程”(随机的频域序列).功率谱参度(PSD), 它定义了信号或者时间序列的功率如何随频率分布. 这里功率可能是实际物理上的功率, 或者更经常便于表示抽象的信号, 被定义为信号数值的平方, 也就是当信号的负载为1欧姆(ohm)时的实际功率.由于平均值不为零的信号不是平方可积的, 所以在这种情况下就没有傅立叶变换. 维纳-辛钦定理(Wiener-Khinchin theorem)提供了一个简单的替换方法. 如果信号可以看作是平稳随机过程, 那么功率谱参度就是信号自相关函数的傅立叶变换. 信号的功率谱参度当且仅当信号是广义的平稳过程的时候才存在; 如果信号不是平稳过程, 那么自相关函数一定是两个变量的函数, 这样就不存在功率谱参度, 但是可以使用类似的技术估计时变谱参度. 随机信号是时域无限信号, 不具备可积分条件, 因此不能直接进行傅氏变换. 一般用具有统计特性的功率谱来作为谱分析的依据. 功率谱与自相关函数是一个傅氏变换对.一般的功率谱参度都是针对平稳随机过程的, 由于平稳随机过程的样本函数一般不是绝对可积的, 因此不能直接对它进行傅立叶分析. 可以有三种办法来重新定义谱参度,来克服上述困难.1. 用相关函数的傅立叶变换来定义谱参度；2. 用随机过程的有限时间傅立叶变换来定义谱参度；3. 用平稳随机过程的谱分解来定义谱参度.§5.3.2 模型建立计算功率谱参度函数通常有两种方法[]8. 一种叫做标准的自相关函数法, 其表达式为：(1)0()4()cos 2d x x G f R f τπττ∞=∫ (5.3.1)其中()x R τ表示某个各态历经的随机过程{}()x t 的自相关函数；另一种叫做直接法, 即是直接对随机过程{}()x t 的样本函数作傅立叶变换得到功率谱参度函数, 其表达式为：2(2)202()lim ()d T j ftx T G f x t e t Tπ−→∞=∫ (5.3.2)在计算机上计算功率谱参度函数时, 要求输入的数据必须是离散数值, 所以要对连续观测的数据记录必须做离散化处理. 这叫做数据采样. 离散化的数据值叫做采样数据. 实际计算时, 要求参加运算的采样数据的个数是有限的(即是说, 在有限的时间区段0-T 上进行计算). 在记录是离散的、有限的情况下, 计算功率谱参度函数的公式可以分别近似地表示为：1(1)01()22cos 2cos 2M x r M r G f t R R fr t R fM t ππ−=⎡⎤=Δ+Δ+Δ⎢⎥⎣⎦∑ (5.3.3)和21(2)202()N j fi t x i i G f t x e N t π−−Δ==ΔΔ∑ (5.3.4)这里, 将(5.3.4)式整理为()()21P f X f N=(5.3.5) 其中()X f 是()x n 的傅里叶变换, 在计算过程中可以直接调用FFT 函数.另外由题意可设出, per F 表示每个点上的频率, 其表达式为sper F F N=. M 表示每个信道所含的点的个数, 其表达式为0perF M F =.其中0F 表示每个传输信道上的频率. 故传输信道就只包含M 个点, 相邻信道也只包含M 个点.由于非线性效应产生的新频率分量由对邻道信号有一定的影响, 现用相邻信道功率比(Adjacent Channel Power Ratio, ACPR)表示信道的带外失真的参数, 衡量由于非线性效应所产生的新频率分量对邻道信号的影响程度. 其定义为。

美国大学生数学建模竞赛2013 获奖论文

Team #111111
Page 3 of 22
General Assumptions
The heat can only transfer to the pan from its outer edge through the air. Since the food placed on it prevents the heat from conducting to it, this is a reasonable. The temperature in oven is even since the air is flowing. That there is only one kind of pans in the ovens. Initially there are two racks in the oven, evenly spaced. We suppose that the temperature and heat are equivalent and constant, so we just considerate one rack and the other one is the same with it The ratio of the oven plane’s width and length is W/L. Every pan shares the same area of A. The data we cited in the models are true. The area of the oven is S 750cm 2 , and the ratio between width and length is W / L 22 : 34 . Moreover, the area of pan is A 100cm2 .[2]

2013年全国大学生电工杯数学建模竞赛一等奖论文(B题)

%
（1-2b）
化学不完全燃烧热损失是由于烟气中残留有诸如 CO ，H 2 ，CH 4 等可燃气体成分而未释放出燃烧热就随烟气排出所造成的热损失。气体不完全燃烧产物为 CO ， H 2 ， CH 4 等可燃气体，则其热损失应为烟气中各可燃气体体积与它们的体积发热量乘积的总和。题中说明过量空气系数对化学不完全燃烧热损失影响较小，故可视为常数处理。所以，化学不完全燃烧热损失与过量空气系数没有直接关系，故可以假设化学不完全燃烧热损失 q3 为一常数，即： q3 K （1-3） 5.1.4 机械不完全燃烧热损失 q4 的计算机械不完全燃烧热损失是由于进入炉膛的燃料中，有一部分没有参与燃烧或未燃尽而被排出炉外引起的热损失。论其实质，是包含在灰渣（包括灰渣、漏煤、烟道灰、飞灰以及溢流灰、冷灰渣等）中的未燃尽的碳造成的热量的损失。对层燃炉而言，主要由灰渣、漏煤、和飞灰三项组成。在实际中因为漏煤的含量相对较少所以本文不考虑漏煤的量，对于运行中的锅炉，分别收集它的每小时的灰渣和飞灰的质量 Ghz 和 G fh （kg/h），同时分析出它们所含可燃物质的质量百分数 Chz 和 C fh （%）和可燃烧的发热量 Qhz 和 Q fh （kJ/kg）则灰渣和飞灰损
q2 q3 q4 q5 q6 I py
Qgy Qr H Wy Ghz G fh ahz a fh ahz
y
py hz
Ay (c ) hz
hz gl
5.模型的建立和求解
5.1 问题一：确定锅炉运行的最佳过量空气系数 5.1.1 问题的分析因为 q 2 q3 q 4 先减少后增加，有一个最小值，与此最小值对应的空气系数称为最佳过量空气系数。所以首先要求出 q2 、q3 和 q4 的表达式。然后求得 q 2 q3 q 4 的表达式，在对这个表达式进行求导，让导数等于 0 这就是最佳过量空气系数。 5.1.2 排烟热损失 q2 的计算由于技术经济条件的限制，烟气离开锅炉排入大气时，烟气温度比进入锅炉的空气温度要高得多，排烟所带走的热量损失简称为排烟热损失。排烟热损失可按如下公式计算[3]：（1-1） Q2 I py pyVk0 (ct ) amb kJ / kg

2013年MCM_B题,水资源的调度案例分析

这是一篇剑走偏锋的文章，不是正统，风险性很大，不建议大家模仿他，因为我也把握能获奖。
Байду номын сангаас
当然，如果比赛的时候愿意搏一搏单车变摩托，也是可以的。
本文的创新点在于，用Cobb-Douglas函数模拟淡化海水的数量。前面两篇文章需要的海水数量直接得到为什么需要这样做，因为沙特阿拉伯必须依靠海水淡化模拟淡化海水的数量，实际上就是预测需求。
II-1. Water Supply and Demand
2025年的淡水供应：假设水资源稳定，用过去几年的均值来代替，作为2025年的可用水资源。（可用水资源=查到的水资源*利用率）
2025年的淡水需求：民用水+农业水+工业水（因为增长率不同）。
评分要点：如果使用多种信息预测淡水需求，将给评委留下审核的印象
Model II：An Outstanding Water Strategy
图论模型（优化模型的一个分支）
将30个省份看成30个节点（为什么不是31个），建立包含33个节点的网络图。剩下的3个节点分别表示总体供水、海水淡化水和总体用水需求用回归分析确定各省份2025年水资源的供给与需求用模拟退火算法求解最低成本最大流量问题仅有一个模型：先仅考虑成本最低，再考虑环境对水运输的影响（间接成本），最后考虑了管道的最小结合距离
How to Approach the Problem
——Creating a Model
一个合理、简单的模型（Basic model）应该包含淡水调配过程。
输送费用=管道建设费用+运输费用+污染物费用（可选）一个优化模型
一个更复杂、更合乎实际的模型（Improved model）应该考虑从沿海地区淡化海水以满足淡水供应。此时淡水供应无上限，但需要考虑淡化成本。应该在模型中考虑节水因素：

2013年数学建模b题纸片拼接

2013年数学建模b题纸片拼接1. 引言2013年数学建模比赛中的B题，是一道关于纸片拼接的问题。

纸片拼接这一主题，在数学建模的题目中并不常见，但却涉及了许多有趣的数学和几何问题。

在接下来的文章中，我将从不同的角度和深度来探讨这一主题，希望能够对你的理解和思考有所启发。

2. 纸片拼接的基本概念让我们来了解一下纸片拼接的基本概念。

在这个问题中，我们需要将大量的纸片按照一定的规则进行拼接，以得到一个特定的形状或图案。

这涉及到对纸片的形状、尺寸和拼接方式的研究和分析。

还需要考虑到纸片的变形和叠放等因素，这是一个具有挑战性的问题。

3. 纸片拼接的数学模型在解决纸片拼接的问题时，我们需要建立相应的数学模型来描述和分析。

这包括对纸片的几何形状进行建模，考虑到其尺寸、边界和变形等因素；同时需要建立拼接规则和约束条件，以确保拼接的合理性和有效性。

通过建立数学模型，可以更好地理解纸片拼接问题的本质，并为后续的求解和优化提供基础。

4. 深入探讨纸片拼接的几何特性在纸片拼接的过程中，我们不仅需要考虑到其形状和尺寸，还需要深入研究其几何特性。

这涉及到对纸片的曲率、折叠和叠放等几何特征的分析，以便更好地理解和控制拼接的过程。

还需要考虑到纸片的叠放和叠合时可能出现的奇异现象，这对于拼接的成功至关重要。

5. 数学建模与实际应用让我们来谈谈纸片拼接的数学建模与实际应用。

纸片拼接这一看似抽象的问题，实际上与现实生活中的许多工程和制造过程有着密切的联系。

在纺织、纸品和航空航天等领域，都存在着类似的拼接和叠放问题。

通过对纸片拼接问题的研究和建模，可以为这些实际应用提供理论支持和技术指导。

6. 总结回顾通过以上的探讨，我们可以看到，纸片拼接这一看似简单的问题，实际上涉及到许多有趣的数学和几何问题。

从纸片的基本概念、数学建模到几何特性和实际应用，我们可以更加全面、深刻和灵活地理解这一主题。

我个人认为，纸片拼接问题不仅具有学术研究的价值，还具有实际应用的潜力，希望能够引起更多人的关注和研究。

2013全国大学生数学建模比赛B题-答案

2013高教社杯全国大学生数学建模竞赛承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。

我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。

如有违反竞赛规则的行为，我们将受到严肃处理。

我们参赛选择的题号是（从A/B/C/D中选择一项填写）： B我们的参赛报名号为（如果赛区设置报名号的话）：所属学校（请填写完整的全名）：重庆邮电大学参赛队员(打印并签名) ：1.2.3.指导教师或指导教师组负责人(打印并签名)：日期： 2013 年 9 月 13 日赛区评阅编号（由赛区组委会评阅前进行编号）：2013高教社杯全国大学生数学建模竞赛编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：碎纸片的拼接复原摘要本文研究的是碎纸片的拼接复原问题。

题目每一个附件中的碎纸片均为来自同一页的文件，所以不需考虑残片中含有未知纸张的残片以及残片中不会含有公共部分。

2013年数学建模b题

本文主要结合各种实际应用背景，针对碎纸机绞碎的碎纸片，基于计算机辅助对碎纸片进行自动拼接复原研究。

针对问题1，依据图像预处理理论，通过matlab程序处理图像，将图像转化成适合于计算机处理的数字图像，进行灰度分析，提取灰度矩阵。

对于仅纵切的碎纸片，根据矩阵的行提取理论，将每个灰度矩阵的第一列提取，作为新矩阵，提取每个灰度矩阵的最后一列，生成新矩阵。

建立碎纸片匹配模型:将矩阵中的任一列与矩阵中的每一列带入模型，所得p值对应的值，即为所拼接的碎片序列号。

将程序进行循环操作，得到最终的碎片自动拼接结果。

针对问题2，首先将图像信息进行灰度分析，提取灰度矩阵。

基于既纵切又横切的碎纸片，根据矩阵的行列提取理论，分别提取每个灰度矩阵的第一列和最后一列，分别生成新矩阵、；提取所有灰度矩阵的第一行和最后一行，分别作为新生成的矩阵、。

由于纸质文件边缘空白处的灰度值为常量，通过对灰度矩阵的检验提取，确定最左列的碎纸片排序。

在此基础上，采用从局部到整体，从左到右的方法，建立匹配筛选模型：，将矩阵中的任一列分别与矩阵中每一列代入模型，所得p值对应的值即为横排序；将矩阵中的任一行分别于矩阵中的任一行代入模型，所得q值对应的值即为列排序。

循环进行此程序，得计算机的最终运行结果。

所得结果有少许误差，需人工调制，更正排列顺序，得最终拼接结果。

针对问题3，基于碎纸片的文字行列特征，采用遗传算法，将所有的可能性拼接进行比较，进行择优性选择。

反面的排序结果用于对正面排序的检验，发现结果有误差，此时，进行人工干预，调换碎纸片的排序。

2013美国大学生数学建模论文终稿

本文建立了三个模型(model)，模型一（model 1 ）用于解释不同形状的pan（从矩形到圆形中的任一形状）在其外围边沿的热量分布（原文：the distribution of heat across the outer edge of a pan for pans of different shapes --rectangular to circular and other shapes in between ），模型二用于在一定条件下选取最优形状的pan（the best type of pan (shape)），第三个模型为对问题一、二的优化（optimize（ .））方案。

In this paper, we formulate 3 relevant models. Through model 1, we display the distribution of heat across the outer edge of a pan for pans of different shapes -rectangular to circular and other shapes in between. While in model 2, we can select out the best type of pan in certain condition. Optimize a combination of model 1 and 2, then we get model 3.首先对于模型一，我们将求解pan的外沿热量分布（the distribution of heat across the outer edge of a pan ）转化为求解pan所在平面的温度场（temperature field），根据热力学理论（thermodynamic theory ）写出温度分布方程（Temperature distribution function），同时由假设条件（assumed condition）确定方程的初始条件（initial condition）和边界条件（boundary condition），利用Matlab（软件）求其数值解（numerical solution）。

2013美国数学建B题,模水资源利用论文

我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。

如有违反竞赛规则的行为，我们将受到严肃处理。

我们授权全国大学生数学建模竞赛组委会，可将我们的论文以任何形式进行公开展示（包括进行网上公示，在书籍、期刊和其他媒体进行正式或非正式发表等）。

我们参赛选择的题号是（从A/B/C/D中选择一项填写）： B我们的参赛报名号为（如果赛区设置报名号的话）：所属学校（请填写完整的全名）河南理工大学参赛队员(打印并签名) 1.2.3.指导教师或指导教师组负责人(打印并签名)：日期： 2013 年 07 月 29 日赛区评阅编号（由赛区组委会评阅前进行编号）：编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：赛区评阅记录（可供赛区评阅时使用）：评阅人评分备注全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：新能源ne-37与pw2u 路线铺设摘要本文通过最小生成树方法、最优化算法、网络规划方法等解决了关于新能源ne-37与pw2u 路线铺设问题。

对于问题一，首先将所给的交通图抽象成一张无向图，以各管道在各边铺设的长度为权值赋给各边。

然后我们运用了两种方法最小生成树算法和Dijkstra 算法求出连接各能源消耗单位之间的最短路径，滤去无用数据，找出我们所需要的满足每个单位至少有一点经过，且尽可能短的路径，其结果分别为24.3公里和23.6公里。

比较后发现两种方法结果相近，都能满足所得结果即为问题1所要求的最优公路交通网数学模型。