2010年美国大学生数学建模竞赛B题一等奖

2010年美国数学建模邀请赛试题2010-02-19 09:09PROBLEM A: The Sweet SpotExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe th at “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?PROBLEM B: CriminologyIn 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a “center of mass” of the locations of the attacks. In the end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the “geographical profile” of a suspected serial criminal ba sed on the locations of the crimes.Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for lawenforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings.In addition to the required one-page summary, your report should include an additional two-page executive summary. The executive summary should provide a broad overview of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool. The executive summary will be read by a chief of police and should include technical details appropriate to the intended audience.。

2010年上海世博会影响力的定量评估摘要上海世博会的举办对中国乃至世界的快速发展都产生了深远的影响。

本文着眼于上海世博会促进旅游业这一侧面，提出并解决三个重要的问题。

一：上海世博会对上海入境人数的贡献旅游业发展具有趋势性、周期性、随机性，根据这一规律建立入境人数本底趋势模型，使用内插法处理过的2007年7月至2010年3月数据拟合出本底趋势线，进而求出2010年4、5、6、7月的本底值，与实际人数相比得到世博会对入境人数平均贡献率为22.41%。

二：未来几个月上海入境人数走势利用第一问模型得出08-11月世博会给上海带来544815人的额外入境人数。

三：世博会给上海带来的直接利益通过对已知的世博会入园人数进行分析，建立每天入园人数的时间序列分析模型，由于人员的变动性，故采用时间序列分解法求解。

运用趋势外推法加权拟合出长期趋势直线，综合考虑影响参观人数的随机因素，预测出上海世博会最终入园人数为7010.82万人。

门票总收入达103.88亿元。

关键字：本底趋势线内插法定量评估时间序列分析模型趋势外推法一、问题重述题目背景：2010年上海世博会是首次在中国举办的世界博览会。

从1851年伦敦的“万国工业博览会”开始，世博会正日益成为各国人民交流历史文化、展示科技成果、体现合作精神、展望未来发展等的重要舞台。

请你们选择感兴趣的某个侧面，建立数学模型，利用互联网数据，定量评估2010年上海世博会的影响力。

问题提出：上海世博会促进旅游业这一侧面，提出了三个重要的问题。

问题一：上海世博会期间，上海的入境人数有什么变化，给出相应的数学模型，并计算世博会对入境人数的平均贡献率。

问题二：未来几个月上海入境人数走势。

问题三：从互联网获取每天入园参观人数，建立每天的参观人数的预测模型，并预测最终入园人数并估算世博会的门票总收入。

二、符号约定三、模型假设1、忽略国家政策、军事、节假日等方面对上海入境人数的影响。

2、将世博会期间的天气情况影响限制在一定波动范围内。

Team#8254page1of16 Play your best hit:The mystery in baseball batMCM Contest Question ATeam#8254February22,2010Contents1Introduction (2)1.1Problem Restatement (2)1.2What is the sweet spot on earth? (2)1.3How to cork a bat? (2)1.4The material of baseball bat matters? (2)2Assumption (3)3A Simplified Model (3)3.1Model Description (3)3.2Finding the”sweet spot” (5)4An Augmented Model (6)4.1Model Description (6)4.2Why no corked bats and metal bats? (8)4.2.1Qualitative Analysis (8)4.2.2Quantitative Analysis (9)5Sensitivity (12)6Conclusion (13)6.1Strength (13)6.2Weakness (13)6.3Recommendation (13)7References (13)8Appendix (14)Team#8254page2of16 1Introduction1.1Problem RestatementEvery experienced hitter knows that there is a spot on the baseball bat that when hitting with this spot,the hand of the hitter feels no pain and the ball can be hit farther.This is the”sweet spot”.According to the theory of torque,this spot should be at the end of the bat,but the experience of the hitters proves it wrong.The location of”sweet spot”on a given baseball bat is approximately 6-1/2”from the end of the bat[1].Our purpose is to establish a model to give a scientific explanation.In addition,some players believe that drilling a cylinder in the end of the bat andfilling it with cork or rubber,namely,to cork the bat [2],enhances the”sweet spot”effect.We would like to found our model and use it to interpret it.What’s more,the question that whether the material of bats matters is also a part discussed in this paper.1.2What is the sweet spot on earth?There are many definitions of”sweet spot”[3].Here in this paper,we define it as the location where maximum energy is transferred to the ball.1.3How to cork a bat?”Corking”is to drill a1inch diameter hole6inches longitudinally into the bat’s barrel end.The structure of the real bat and the corked bat can be showed in Fig1.In our augmented model,we will analyze the effect of corking.1.4The material of baseball bat matters?Compared to the wood bats,aluminum bats is not easily be broke,and as the aluminum bats are hollow,the thickness of the shell can be manipulated so that the center of mass may be more closer to the handle,and consequently reducing the perceived weight while swinging.In this way,it increases the mobility of(a)Normal baseball bat(b)Corked baseball batFig1:Normal baseball bat and corked batTeam#8254page3of16 the hitter.What’s more,using aluminum bat can largely improve the energy of the ejecting ball,which may be too hard to catch or even dangerous for players. All those features lead to the prohibition of the using of aluminum bat[4].We will explain it in details later in this paper.Fig2:Cross section of aluminum bat2AssumptionAs the hitting process is too complex,we make the following assumptions in order to simplify the problems and establish our models:1.The better hitting effect means a larger exit speed of the ball and a shorteracceleration time of the bat.2.When the hitter swings,hitter-bat system rotates around a vertical axis that”penetrates”the hitter.3.The collision of the ball and the bat is one-dimensional.4.The hitter always hits the ball at the”sweet spot”.3A Simplified Model3.1Model DescriptionIn our simplified model,we omit the situations that the player might modify the baseball bat and concentrate on explaining why the”sweet spot”is not at the end of the bat but a few distances from the end.When you hit a ball,the bat vibrates in response.These vibrations travel in waves up and down the length of the bat.At one point,called”the node”, the waves always cancel each other out.If you hit the ball on the bat’s node, the vibrations from the impact will cancel out,and you won’t feel any stinging or shaking in your hand.Since little of the bat’s energy is lost to vibrations when this spot is hit,more can go to the ball[5].We will found a model based onTeam#8254page4of16 the characteristics showed as italics.In our model,we ignore the swing of the hitter’s arms when hit.We denote the spot where the hitter holds the bat as pivot, assuming that the pivot isfixed,and the bat is mounted on the pivot so that it can swing around the pivot freely.The parameters we use are given in the tabular Notation,and the force and motion diagram is showed in Fig3.NotationSymbols MeaningF ball the applied force in the collisionF px,F py the vertical and horizontal component forces of the force in the pivotcm the center of mass of the batp the impact pointpivot the spot where the hitter holds the batI pivot the moment of inertia of the bat with respect to pivotL x−y the distances between x and yαthe rotation angular acceleration of the batm bat the mass of the batL sum the whole length of the batFig3:The force and motion diagram of the batNext,we will analyze the model above with the knowledge of kinetics in order tofind the”sweet spot”.Team #8254page 5of 163.2Finding the ”sweet spot”According to the theorem of moment of momentum we have:F ball ·L pivot =I pivot ·α(1)and based on the theorem of motion of center of mass:∑F =ma c we have:F ball +F px =m bat ·L pivot −cm ·α(2)Therefore,with equation (1)and (2)we have:F hand =F ball m bat ·L pivot −cm ·L pivot −p I pivot−1 Thus we know that the horizontal component force is 0whenL pivot −p =I pivotm bat L pivot −cmIn other words,when the distance between the pivot and the p is I pivot m bat ·L pivot −cm ,the p is the sweet spot.In order to find the specific location of the ”sweet spot”on a certain bat,we quote the data from [6]and choose a C243wooden bat,thus we get the parame-ters L pivot −cm =0.42m ,I pivot =0.208kg ·m 2,L knob −pivot =0.15m ,m bat =0.905kg ,L sum =0.86m .Then we calculate L pivot −p =0.55m ,so that L knob −p =0.70m .From the re-sult we can obviously see that the ”sweet spot”is about 0.16m far from the end of the fat part of the bat.The forces on different location of the bat can be showed in Fig4.Fig 4:The rotation system in a hitting processIn this way,we have successfully demonstrated that the experience of the hitters is right.Team#8254page6of16 4An Augmented Model4.1Model DescriptionThe previous model fails to take the situation that some hitter may modify the bat into account.What’s more,in real baseball game it is impossible that the arms of the hitter are stationary when hit.So we augmented ourfirst model. In this augmented model,we assume that the hitter and the bat form a rotation system in which the hitter stays vertical and the bat stays horizontal.The hitting process can thus be modeled as:the system gets a torque T generated by the hitter and rotates around the axis which vertically”penetrates”the hitter and then hits a ball.After hitting process,the ball ejects out with a velocity of v f. The distance between the impact spot and the axis is r.The process is described in Fig5.Fig5:The rotation system in a hitting processWe notice that the ball gets maximum velocity after hitting means the effect of hitting is optimal,so we should focus on the exit speed of the ball.Two processes are displayed as follows:Process1:The hitter swings the bat and accelerates the bat to an angular velocity ofω.Process2:After the imperfect inelastic collision of bat and ball,the ball ejects out with a velocity of v f.In process1,the moment of inertia of bat and the air resistance will hamper the rotation of the system of hitter-bat.It is easy to calculate the liner velocity of any spot on the bat by the kinematics formula v=ωr,and according to the formulas given by Keith Koenig in reference[7],we know that the angular ve-Team#8254page7of16 locity and the linear velocity of the impact spot on the bat before the collision in process1are:ω=TK Dtanhcosh−1expK DI hitter+I batθv bat=rω(3) where•r is the rotation radius,equals the length from the impact spot to the axis.•T is the torque applied to the hitter-bat system which is generated by the hitter.•K D is an aerodynamic parameter,which is given byK D=12ρC D DL44(for more please read reference[7])•tanh is hyperbolic tangent function.•cosh−1is anti-hyperbolic cosine function.•exp is exponential function.•θis the angle that the bat rotates.•I hitter is the hitter’s moment of inertia with respect to the axis.•I bat is the bat’s moment of inertia with respect to the axis.In our case,we treat T,K D,I hitter,θas constant parameters while r and I bat are the only variables.In process2,we assume that the bat and ball have an one-dimensional col-lision and they both have some extent of deformation.The deformation trans-forms a small part of the kinetic energy(about25%,reference[8])into potential energy stored in bat and ball,while most part(about75%)of it dismisses due to the friction and the oscillation of the bat.And referring to reference[7]we havev f=COR−r M1+r Mv ball+COR+11+r Mv bat(4)where•COR is the coefficient of restitution,it will be discussed in the next part.•v ball is the velocity of the ball right before the collision with the bat.Team #8254page 8of 16•v bat is the velocity of the bat right before the collision with the bat.by equation (3)and (4),we getv f =COR −r M 1+r M v ball +COR +11+r M r T K D tanh cosh −1 exp K D I hitter +I batθ (5)The parameters:COR ,I bat ,r M are explained in Appendix .4.2Why no corked bats and metal bats?In order to reveal the influence of each parameter on the velocity of the ball thus to figure out the effect of the corking behavior and predict different be-havior for wood and metal bats,we make some analysis about the application procedure of our augmented model.4.2.1Qualitative AnalysisWe assume that the velocity of the ball is constant before the collision with the bat,and the impact spots are on the same position of the bat,namely,r is a constant.The corking has two aspects of influences:one is the decrease of mass,the other is the deviation of the center of mass.1.Corked bat•The influence on v fDue to the lower density of material corked in the barrel,the whole mass of bat inclines and the center of mass moves to the rotation axis a little.On one hand,it will cause the decrease of I bat as I bat = r 2dm ,and according to the expression of v f :v f =COR −r M 1+r M v ball +COR +11+r M r T K D tanh cosh −1 exp K D I hitter +I batθ we know that the value of r T K D tanh cosh −1 exp K D I hitter +I batθ will increase.On the other hand,as we haver M =m ball (z −z p )2I cm +m bat L 2pivot −cmTeam#8254page9of16 we know that if m bat decreases and the center of mass deviates toknob,I cm and L pivot−cm will both decline,and that will make r M in-crease and lead to the decline of both COR−r M1+r M v ball and COR+11+r M.How-ever,it is impossible for us to ensure whether the value of v f willincrease or decrease without specific data.We will explain it later inthe Quantitative Analysis section.•The influence onflexibilityWe regard the procedure that the hitter-bat system is accelerated fromstationary situation to rotating with an angular velocity ofωas auniformly accelerated motion procedure.By kinematic formulas weknow that the acceleration time is t=2θω,and from the expression ofωmentioned in page6,section4.1,we know that as the I bat decreasesdue to corking,the value ofωwill increase,so that the accelerationtime will be shorten and in this way theflexibility develops.2.Metal bats vs.wood batsUsing our augmented model we are able to analyze the behavior of bats that have different composition material.•The influence on v fMetal bats(usually aluminum bats)have trampoline effect due togood elasticity,so that it has larger COR than wood bats(usually ashbats).In addition,as the aluminum bats have less mass,the influenceof material is just like that in discussing corked bats.It is obvious thatin order tofind which kind of bat’s v f is larger we need the supportof data.•The influence onflexibilityThe aluminum bats are lighter than wood bats so that the influenceonflexibility is the same as corked bats.4.2.2Quantitative Analysis1.Data usingAs the qualitative analysis cannot show the exact differences between wood bats,corked wood bats(corked bats)and aluminum bats,we quote the data from reference[7][12][13][14]to verify our augmented model.Table1:Value of ConstantsTeam#8254page10of16Constants Valuem ball/(g)134.2v ball/(m/s)25(assumption)T316N·mK D/(N·m·s2)0.00544θ 2.36radr/(m)0.762R hitter−pivot/(m)0.305I hitter/(kg·m2)0.444Table2:Value of VariablesValues Wood bat Corked bat Aluminum batCOR0.500.500.58mass/(g)876834827I p/(kg·m2)0.2110.1950.17L cm−pivot/(cm)424039.5I bat/(kg·m2)0.5160.4760.446z−z p/(m)0.470.460.45r M0.14030.14540.1596 Using the data above we can calculate the three relative values of v f that are47.1m/s,47.6m/s and51.3m/s.The I bat is calculated based on parallel axis theorem.For a better compare of the hitting effects of the three bats,we use com-puter to make a hitting experiment simulation about theflying trajectories of the ball after hitting:Assuming that the ball dose a slanting parabolic motion and the launch angle of the ball after hitting with the bat isθ=30◦.Fig6shows the trajec-tories of balls that are hit by the three kinds of bats.2.The influence of three kinds of bats onflexibilityCombine the formula t=2θωwith the expression ofω,we have the expres-sion of the acceleration time t:t=2θTK Dtanhcosh−1expK DI hitter+I batθAfter using the values given in the table1and table2,we get:Kinds of bats t/(s)Wood Bats0.121Corked Bats0.118Aluminum Bats0.116Team#8254page11of16Fig6:The trajectories of the balls hit by the three kinds of bats It is easy to see that the acceleration time of corked bat is0.003s shorter than that of wood bat.During this time,the incoming ball moves0.075m farther,that is to say,theflying distance of the incoming ball from the ser-vice point to the collision point when using a corked bat is0.075m farther than that of wood bat,so that the hitter would feel more easy to deal with the incoming ball as he or she has more time to react and accelerate the bat when using a corked bat.In the same way we know that when usinga aluminum bat rather than a wood bat,the hitter has0.005s longer and0.125m farther to handle the bat.3.Summary of our analysisAt this point,we are able to answer the second and third questions about the corking behavior and the matters of different materials.•Why does Major League Baseball prohibit”corking”?From the analysis above we know that for a baseball bat used in ourmodel,if it is corked,the swinging velocity of bat right before hittingthe ball will increase0.5m/s and theflexibility of swinging can alsobe developed,as the mass of the bat decreases and the center of massof it deviates.•Why does Major League Baseball prohibit metal bats?According to our model,the hitting effect is closely linked with thematerials.We know from our qualitative verification that the hittingvelocity of using a aluminum bat is4.2m/s more than that of woodbat,and the deviation of the center of mass further gains0.005s reac-tion time for the hitter.As the ball hit by aluminum bat has a muchbigger velocity,it makes the catching of the ball difficult and evendangerous.Team#8254page12of16 5SensitivityAfter analyzing the application of our augmented model,wefind it neces-sary to analyze the sensitivity of our model,aiming at implementing it more effectively.We choose to research the exit speed of normal wood bat with differ-ent parameters.1.The influence of massThe change of mass will influence both r M and I bat,but as the value of them are too hard to obtain by direct calculating,we make some simplificationsbelow:r M=m ball (z−z p)2I p=m ball(z−z p)2I cm+m bat L2pivot−cmIn this expression,as I cm is very small(about0.04kg·m2),we treat it as a constant one.It is the same to z−z p(about0.47m)and L pivot−cm(about 0.42m).Hence,the value of r M only correlates to m bat,and we haver M=0.1340.4720.04+m bat×0.422then in I bat=I cm+m bat·L2hitter−cm,we also treat L hitter−cm(about0.725m) as a constant,so we haveI bat=0.04+m bat×0.7252Relating to equation(5),we denote COR=0.5and get:m bat(kg)0.830.850.870.89v f(m/s)46.846.7646.7146.672.The influence of materialThe major influence of different materials is they have different COR s.When the mass is constant,we have I bat=0.516kg·m2,and r M=0.1403, andfinally we get:COR0.40.450.50.550.6v f/(m/s)42.344.747.149.551.9From the results above,we can see that COR seems a more prominent influence.Team#8254page13of16 6Conclusion6.1StrengthFirst of all,this paper solves the problem of”sweet spot”,and we give an easy formula to calculate the position of”sweet spot”.Secondly,we analyze the hitting process and divide it into two stages,and discuss the factors that affect the exit speed in details while giving a formula that can describe this stage.Then we answer the questions through qualitative analysis and quantitative calculation.Finally,we make a analysis about the sensitivity and prove the rationality by comparing the results.6.2WeaknessFirst of all,as the real hitting process is too complex to analyze,we make several simplifications in order to facilitate the founding of model.In model 1,we just regard the bat as a pendulum rod with one endfixed,which is a little different from the real situation.And in model2,we simplify the complex process of hitting into two stages.Secondly,the data wefind are not precise,especially for the value of COR which we regard as constant.Additionally,the calculations in this paper are also simplified,thus the accu-racy of our results declines.6.3RecommendationThe biggest disadvantage of our model is lacking experiments,and if we have time and facilities to do some experiments,the result must be more reliable.For example,the equation(6)in Appendix can be used to measure COR,and in order to measure the value of I pivot,we could refer to[15]and use the method to obtain data.With this data we can verify our model in a better way.7References[1]/wiki/Sweet spot[2]/drussell/bats-new/corkedbat.html[3]/drussell/bats-new/sweetspot.html[4]/wiki/Aluminum Bats vs.Wood Bats[5]/baseball/sweetspot.htmlTeam#8254page14of16[6]/sysengr/slides/baseballBat.ppt[7]Keith Koenig,Nan Davis Mitchell,Thomas E.Hannigan,J.Keith Clutter.The influence of moment of inertia on baseball/softball bat swing speed.SportsEngineedng(2004)7,105-117.[8]Alan M.Nathana.Characterizing the performance of baseball bats.Am.J.Phys.,Vol.71,No.2,February2003134-143[9]/wiki/Coefficient of restitution[10]Lv ZhongjieHuang Fenglei.Coefficient of Restitution of a Circular PlateDuring Inelastic Collision.Transactions of Beijing Institute of Technology.Vol.28No.4.[11]P.J.Drane and J.A.Sherwood.Characterization of the effect of temperatureon baseball COR performance.[12]/sysengr/slides/baseballBat.ppt[13]/docs/621958/How-Does-a-Baseball-Bat-Work[14]/wiki/moment of inertia[15]/drussell/bats-new/bat-moi.html8AppendixParameters ExplanationIn expression(5),page8,section4.1,there are several parameters that will influence thefinal velocity of the ball:1.COR•What is COR?The coefficient of restitution(COR),or bounciness of an object is afractional value representing the ratio of velocities after and before animpact[9].Fig7:The one-dimensional collision processTeam #8254page 15of 16The coefficient of restitution is given byCOR =v 1f −v 2f v 1−v 2(6)where–v 1is the velocity of object 1before the collision.–v 2is the velocity of object 2before the collision.–v 1f is the velocity of object 1after the collision.–v 2f is the velocity of object 2after the collision.All the parameters above are scalars.In the ideal situations,we may have a so-called plastic collision when COR =0,namely the deformation of the material cannot re-store.And when COR =1,called perfectly elastic collision,is a situa-tion that the deformation can restore entirely.In general,the value of COR varies from (0,1).•What factors affect COR ?MaterialCOR represents the deformation recovery ability of the material.Gen-erally speaking,the more elastic the material is,the higher the value of COR will be.Impact velocityCOR decreases when the impact velocity increases.[10]The Temperature and Relative Humility of The EnvironmentCompared to the factors above,another two factors,the tempera-ture and relative humility have a relatively smaller influence.COR decreases when the temperature decreases and it decreases when the relative humility increases.[11]2.I bat•MOI (moment of inertia)[14]Moment of inertia is a measure of an object’s resistance to changes in its rotation rate.It is the rotational analog of mass,the inertia of a rigid rotating body with respect to its rotation.The moment of iner-tia plays much the same role in rotational dynamics as mass does in linear dynamics,determining the relationship between angular mo-mentum and angular velocity,torque and angular acceleration,and several other quantities.It is denoted asI = r 2dmwhere m is mass and r is the perpendicular distance to the axis of rotation.Team#8254page16of16•Parallel Axis TheoremI z=I cm+mL2where I cm is the moment of inertia of the rotor with respect to thecenter of mass,and the L is the distance from the center of mass toaxis z.•The factors affect MOIFigure:It influences the location of the center of mass,thereby affectsthe distance from the center of mass to rotation axis.Mass:Its increase is proportional to the increase of MOI.3.r MIn reference[8],Alan M.Nathan develops a formula relating v f to the initial speed of the ball v ball and the initial speed of the bat at the impact pointv bat as:r M=m ball (z−z p)2I pwhere•m ball and m bat are the ball and the bat’s mass respectively.•Z is the location of the impact point.•Z p is the location of the pivot point.•I p is the moment of inertia of the bat with respect to the pivot point. From the expression above we can see that reducing the mass of the bat m bat while keeping the other parameters constant will lead to a augment of r m.。

上海世博会影响力的定量评估上海世博会影响力的定量评估摘 要2010上海世博会备受世界瞩目，也影响我国各个区域，尤其对上海。

本文首先根据上海市1978年至上海申报世博会成功前期的GDP 数据，运用时间序列预测模型,预测出后期GDP 数据，并与实际GDP 作对比，接着分析与经济增长具有密切关系的九项指标数据，讨论各项经济指标的关联度，最后利用主成分分析法找出这九项指标中的主要影响经济增长的因子，即可反映上海世博对经济增长的影响力。

对于模型一，由于经济数据具有较强的自相关性，先采用ADF 检验时间序列的平衡性，接着通过对时间序列取对数、二次差分进行平衡化，然后运用SPSS 得到了拟合函数2(10.231)(1)ln (10.981)0.02t t B B x B ε−−=++，并通过拟合统计值检验了时间序列模型的可靠性，从而对后期数据作出较准确预测，最后通过比较预测值与实际时，得到了世博对上海的经济起到了明显的作用。

对于模型二，为了分析世博对经济增长产生的影响，是如何通过影响经济增长的各项指标来实现，找出引起经济增长的具有代表性受世博影响的九项指标。

利用SPSS 软件九项指标的数据做回归分析，并结合MATLAB 软件拟合得到单个指标随时间变化的函数。

再综合对九项指标数据进行分析，利用灰色系统理论对这九项指标进行关联度分析，并MATLAB 编程得到九项指标的关联系数如下图：1r2r3r4r5r6r7r8r9r0.8822 0.7667 0.8981 0.7124 0.6897 0.7417 0.6482 0.6663 0.6436 对于模型三，采用主成分分析法，通过SPSS 处理，到得影响经济的三个主成分，其方差贡献率之和达到95.517%，主成分与各因子的关系如下234567891010.9890.9850.9930.9440.9870.9930.1860.5330.647F x x x x x x x x x =+++++−−−234567891020.0120.0180.0120.0610.0510.0140.9160.7420.286F x x x x x x x x x =−−+−+−−−−234567891030.0970.0200.0700.2380.0470.0630.0780.1540.698F x x x x x x x x x =+++++−−− 从其表达式分析可以得到，世博导致了外来总投资，社会固定资产投资总额，消费受额，外贸总额，交通，旅游总收入等有较大幅度的增加。

2010 AMC10美国数学竞赛B卷1. What is 10010031001003（）（）?---(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 02. Makarla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?(A) 15 (B) 20 (C) 25 (D) 30 (E) 353. A drawer contains red, green, blue, and white socks with at least 2 of each color, What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?(A) 3 (B) 4 (C) 5 (D) 8 (E) 94. For a real number x, define ♡(x) to be the average of x and x2. What is♡(1)+ ♡(2)+ ♡(3)?(A) 3 (B) 6 (C) 10 (D) 12 (E) 205. A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?(A) 2 (B) 3 (C) 4 (D) 5 (E) 66. A circle is centered at O, AB is a diameter and C is a point on the circle with∠COB=50°. What is the degree measure of ∠CAB?(A) 20 (B) 25 (C) 45 (D) 50 (E) 657. A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?(A) 16 (B) 24 (C) 28 (D) 32 (E) 368. A ticket to a school play cost x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for x are possible?(A) 1 (B) 2 (C) 3 (D) 4 (E) 59. Lucky Larry’s teacher asked him to substitute numbers for a, b, c, d, and e in the expression ((()))---+and evaluate the result. Larry ignored the parent thesea b c d ebut added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for a, b, c, and d were 1, 2, 3, and 4, respectively. What number did Larry substitute for e?(A) -5 (B) -3 (C) 0 (D) 3 (E) 510. Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rainin the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?(A) 18 (B) 21 (C) 24 (D) 27 (E) 3011. A shopper plans to purchase an item that has a listed price greater than $100 and can use any one of the three coupons. Coupon A gives 15% off the listed price, Coupon B gives $30 off the listed price, and Coupon C gives 25% off the amount by which the listed price exceeds $100.Let x and y be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is y-x?(A) 50 (B) 60 (C) 75 (D) 80 (E) 10012. At the beginning of the school year, 50% of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and 50% answered “No”. At the end of the school year, 70% answered “Yes” and 30% answered “No”. Altogether, x% of the students gave a difference between the maximum and the minimum possible values of x?(A) 0 (B) 20 (C) 40 (D) 60 (E) 8013. What is the sum of all the solutions of 2602=--?x x x(A) 32 (B)60 (C)92 (D) 120 (E) 12414. The average of the numbers 1, 2, 3, …, 98, 99, and x is 100x. What is x? (A)49101 (B) 50101 (C) 12 (D) 51101 (E) 509915. On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse ’s total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?(A) 25(B) 27 (C) 29 (D) 31 (E) 3316. A square of side length 1 and a circle of radiusshare the same center. What is the area inside the circle, but outside the square?(A)13π- (B) 29π- (C) 18π (D) 14(E) 29π17. Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea ’s score was the median among all students, and hers was the highest score on here team. Andrea ’s teammates Beth and Carla placed 37th and 64th , respectively. How many schools are in the city?(A) 22(B) 23 (C) 24 (D) 25 (E) 2618. Positive integers a, b, and c are randomly and independently selected withreplacement from the set {1, 2, 3, …, 2010}. What is the probability that abc ab a++ is divisible by 3?(A) 13(B) 2981(C) 3181(D) 1127(E) 132719. A circle with center O has area 156π. Triangle ABC is equilateral, BC is a chordon the circle, OA=, and point O is outside △ABC. What is the side length of △ABC?(A)(B) 64 (C) (D) 12 (E) 1820. Two circles lie outside regular hexagon ABCDEF. The first is tangent to AB, and the second is tangent to DE, Both are tangent to lines BC and FA. What is the ratio of the area of the second circle to that of the first circle?(A) 18 (B) 27 (C) 36 (D) 81 (E) 10821. A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?(A) 1/10 (B) 1/9 (C) 1/7 (D) 1/6 (E) 1/522. Seven distinct pieces of candy are to be distributed among three bags. The red bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?(A) 1930 (B) 1931 (C) 1932 (D) 1933 (E) 193423. The entries in a 3×3 array include all the digits from 1 through 9, arranged so thatthe entries in every row and column are in increasing order. How many such arrays are there?(A) 18(B) 24 (C) 36 (D) 42 (E) 6024. A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?(A) 30(B) 31 (C) 32 (D) 33 (E) 3425. Let a>0, and let P(x) be a polynomial with integer coefficients such that(1)(3)(5)(7),(2)(4)(6)(8).P P P P a and P P P P a ========- What is the smallest possible value of a?(A) 105(B) 315 (C) 945 (D) 7! (E) 8!。

2010年上海世博会对居民消费结构影响力的定量评估摘要本文从世博会的筹备期间（2003年---2009年）对上海居民消费结构的影响进行定量评估研究。

消费结构是一项反映居民消费水平的重要指标，包含居民的收入水平、消费支出、消费分类三部分[1]。

为了全面反映和研究居民的生活消费状况，我们采取了一系列相互联系的统计指标对上海居民的消费结构进行定量研究。

在对大量的数据分析基础上，研究了上海市居民的收入水平的变化；并且从上海市的几个主要消费群体来分析上海市居民的收入与支出的变化情况；对消费分类的研究，我们选取了食品、衣着、居住、家庭设备用品及服务、交通和通信、文教娱乐用品及服务、医疗保健、商品和服务作为消费分类的八项指标，利用主成分分析的方法对各个主成分进行了详细的定量分析，并运用matlab编程利用曲线拟合的方法做了假设不存在世博会时的预测，再将所搜集到的实际值与预测值作差，我们定义该差值为影响力指数，通过影响力指数的大小来说明上海世博会对上海市居民消费分类的影响，影响力指数越大，说明世博会对上海居民消费结构的影响越深，进而定量评估了上海世博会对上海市居民的消费结构的影响情况。

消费结构的升级产生的经济势力是持久强大的，了解了上海世博会的对上海居民消费结构的影响后，若能顺势调控，则能充分带动经济的发展，为支撑我国国民经济的稳定快速发展提供动力。

关键词：消费结构主成分分析定量评估预测曲线拟合 matlab一 问题的提出2010年上海世博会是首次在中国举办的世界博览会.从1851年伦敦的“万国工业博览会”开始，世博会正日益成为各国人民交流历史文化、展示科技成果、体现合作精神、展望未来发展等的重要舞台.请你们选择感兴趣的某个侧面，建立数学模型，利用互联网数据，定量评估2010年上海世博会的影响力.二 符号说明np x 第n 个样品的第p 个指标 X标准化数据矩阵R 变量的关系矩阵p λ 关系矩阵的特征值p μ p λ所对应的单位特征向量i y 第i 个主成分y 1995年到2002年dy 影响力指数三 模型的假设1、本文所作的影响力评估是针对上海市居民的消费结构.2、本文所作的影响力评估仅限于世博会筹备期间及召开期间的居民消费结构.3、消费结构是一项反映居民消费水平的重要指标，要全面反映和研究居民的生活消费状况，包含居民的收入水平、消费支出、消费分类三部分.4、上海居民消费由食品、衣着、居住、家庭设备用品及服务、交通和通信、文教娱乐用品及服务、医疗保健、商品和服务八部分组成.5、居民的收入是决定居民的消费水平和消费结构的主要因素，收入水平的高低直接决定消费水平的高低.四 模型的建立及求解上海是我国最大的经济中心城市,随着2010年上海世博会的日益临近,将对上海经济发展发挥巨大的作用.投资、消费和出口被称为经济发展的三架马车，2010年的世博会为上海经济发展提供了会前的投资拉动和会后的需求拉动两个方面的刺激，消费是需求的基础，有效地投资必须准确的把握需求的变化.消费是人们为了满足生活需要而消耗产品和服务的行为和过程, 是满足人们生存、发展和生活享受所必需的行为.人们基本的消费状况, 既能反映需求规律, 又成为其他需求的基础,因此, 评估消费状况和需求趋向便成为政府和企业了解市场的起点.据中国社科院的研究, 2001年投资、消费和出口对国内GDP 增长的贡献分别是77%、34%和-11%. 从2002 年上海的统计数据来看, 同样是外需下降、出口下滑, 依靠增幅达31.7%的社会固定资产投资和9.8%的社会消费品零售总额的增长, 才保证了上海经济10.4%的高速增长.由此也可以看到投资和消费是推动上海经济发展的两个最基本因素。

2010高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规范”）A 题 储油罐的变位识别与罐容表标定通常加油站都有若干个储存燃油的地下储油罐，并且一般都有与之配套的“油位计量管理系统”，采用流量计和油位计来测量进/出油量与罐内油位高度等数据，通过预先标定的罐容表（即罐内油位高度与储油量的对应关系）进行实时计算，以得到罐内油位高度和储油量的变化情况。

许多储油罐在使用一段时间后，由于地基变形等原因，使罐体的位置会发生纵向倾斜和横向偏转等变化（以下称为变位），从而导致罐容表发生改变。

按照有关规定，需要定期对罐容表进行重新标定。

图1是一种典型的储油罐尺寸及形状示意图，其主体为圆柱体，两端为球冠体。

图2是其罐体纵向倾斜变位的示意图，图3是罐体横向偏转变位的截面示意图。

请你们用数学建模方法研究解决储油罐的变位识别与罐容表标定的问题。

（1）为了掌握罐体变位后对罐容表的影响，利用如图4的小椭圆型储油罐（两端平头的椭圆柱体），分别对罐体无变位和倾斜角为α=4.10的纵向变位两种情况做了实验，实验数据如附件1所示。

请建立数学模型研究罐体变位后对罐容表的影响，并给出罐体变位后油位高度间隔为1cm 的罐容表标定值。

（2）对于图1所示的实际储油罐，试建立罐体变位后标定罐容表的数学模型，即罐内储油量与油位高度及变位参数（纵向倾斜角度α和横向偏转角度β ）之间的一般关系。

请利用罐体变位后在进/出油过程中的实际检测数据（附件2），根据你们所建立的数学模型确定变位参数，并给出罐体变位后油位高度间隔为10cm 的罐容表标定值。

进一步利用附件2中的实际检测数据来分析检验你们模型的正确性与方法的可靠性。

草根分享论坛 图吧 分享吧 海之南 分享斑竹 share8 好学吧附件1：小椭圆储油罐的实验数据 附件2：实际储油罐的检测数据油油浮子 出油管油位探测装置 注油口 检查口地平线 2m 6m 1m 1m3 m油位高度 图1 储油罐正面示意图 油位探针油位探针α地平线 图2 储油罐纵向倾斜变位后示意图油油浮子出油管油位探测装置注油口 检查口水平线α油油浮子出油管油位探针注油口水平线1.2m1.2m1.78m图3 储油罐截面示意图（b ）横向偏转倾斜后正截面图地平线β地平线垂直线油位探针（a ）无偏转倾斜的正截面图油位探针油位探测装置地平线 油3m油2010高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规范”）B题2010年上海世博会影响力的定量评估2010年上海世博会是首次在中国举办的世界博览会。