2010美赛数学建模
数学建模美赛翻译-关于2010年数学建模美国赛b题的翻译
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美国数学建模B题论文
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.。
2010年北美数学建模竞赛中英文A、B题目
2010年北美数学建模比赛中英文题目(MCM)2010 MCM题目A题：棒球棒上的最佳击球点Explain 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 that “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 prohibit s “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?中文翻译：解释棒球棒上的“最佳击球点”。
2010MCM(数学建模美赛)
2010 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to each copy of 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.SummaryTo reduce the hunting time for investigation and ensure safety of the general public, we optimized the Circle Principle by Golden-section search, so that an accurate geographical profile was able to be drawn in a short time, through error testing by the title given case of Peter Sutcliffe, our models are really useful, time-saving and accurate.Task I: we use the basic Circle Principle confirming a primary geographical profile, combining and the method of probability. Then, a smaller profile appeared, and after computer programming, the result is very reasonable.Task II: in this question, we mainly take Gray Forecast and GM[1,1] predicting the next location. But the highlight is the conjunction with our balance theory, factors influencing the prediction were all set as weight, How to keep balance is the special way to forecast the next spot.Task III: same as Task II, the only different is the number of weight was raised.Task IV: the error is the distance between the realistic location and predicting location, in addition. it’s quite low.Key word: Golden-section Search Gray Forecast Balance TheoryThe Center of Geographical Profile Primary Geographical ProfileCircle PrinciplePROBLEM B: An Easy Profiling ModelContent1. Introduction & Assumption (2)1.1 Introduction (2)1.2 Assumption (3)2. Solution of Task I (3)2.1 Analysis (3)2.2 Profiling model (3)2.2.1 Primary scheme (3)2.2.2 Target scheme (4)2.3 Model Testing (4)3. Solution of Task II (6)3.1 Predicting model (6)4. Solution of Task III (9)4.1 Intensive model (9)5. Solution of Task IV (9)5. Sensitivity Analysis (9)6. Executive summary (13)7. Reference (14)8. Appendix (15)1. Introduction & Assumption1.1 IntroductionCriminology helps the police dealing with cases more effectually, especially for arresting suspect of serial criminal. Reducing hunt time is very imperative for the public safety, consequently a new word was born among a long period of criminal investigation, which is “geographical profile” [1]. And it is an investigative methodology that makes use of the locations of a connected series ofcrimes to determine the most probable area of offender residence. Challenge to predict the residence of a offender accurately and quickly is the most important problemIn our paper, we mainly try to put forward a method to ascertain the geographical profile easily and credibly. In addition, combining the results of profiling model, factors and extra information, we can get a better result by verification.1.2 Assumption·Suppose a continues offend will not change his criminal type .2. Solution of Task I2.1 AnalysisAccording to request of this problem, at least two different schemes were proposed to generate the geographical profile. In order to facilitate Task I, we put forward two schemes, the former one is mainly about to confirm a wide area which has the maximum possibility to find out the suspect, in other words, we could get a primary geographical profile; the latter one could dwindle the range of the profile. Through the schemes above, the final geographical profile has been generated. The schemes above will involve the following a principle of geographical profile.Theoretical Principles of geographical profile1. Circle hypothesis theoryMark all the crime on the map. Suppose that these crimes were committed by a single person who has not yet been found. And to find the distance between two furthest location of crime, Make a circle including all the crime sites, which diameter is the distance between the two locations. Assumption is that the offenders live in the circle. May be somewhere near the center of the circle. People will find that 80% of rapists truly live within the circumference. More than 60% offenders live in this center area of the circle.2.2 Profiling model2.2.1 Primary schemeThis part is primarily about drawing a wide profile of the potential suspect’s residence, and the method that we hold is based on the theory of Circle hypothesis, the process presents as followStep 1. Confirming each past criminal geographical position and figuring all the positions aspoints , then we need to find out the longest distance among each pair of two points in the whole figure so that we could get the criminal center of circle based on circle hypothesis[2].Step 2. Expressing the method in Step 1 into the form of mathematics, we could transform the geographical coordinate into a planar plane. Let x denotes the longitude, y denotes the latitude. To simplify the type of value, if '''o x a b c =,'''o y d e f =,then we convert degree to united form by formula ,603600603600o o b c e f X a Y d =++=++. so that we get all the planar coordinates of criminal locations, due to circle hypothesis, the longest distance of every two points among the figure is able to be found. Getting the coordinates of the two points, we eventually reach the criminal center of circle. For instance, the coordinates of the two points are respectively 11(,)x y and 22(,)x y So the center of circle is1212(,)22x x y y C ++=. Step 3. Drawing a circle of radius with half of the maximum distance above, the primary geographical profile has been made.2.2.2 Target schemeStep 1. As we know that suspect may appear everywhere in this circle, but it ’s boring and time-consuming to check out every place. To find out the residence quickly and accurately, we want to confirm a smaller circle in the primary circle based on the Golden-section search [3]. Then, drawing the smaller circle of 0.618 times the radius, we finally get a smaller geographical profile.Step 2. Determining the final profile, we decide to use a special method that the inspiration is from a TV Series which called Numb3rs. First, supposing (,)E x y is any point on the smaller circle and there are n points on the figure. Second, we require all the distances between the n points and E . At last, through matlab programming, the minimum standard deviation of distances is able to calculate so that we could get the best point.Step 3. Taking the best point as a new center of circle, draw another circle of 0.618 times the radius, In the end, the final circle is our accurate geographical profile.2.3 Model TestingIn order to check out whether the accuracy of our profiling model, we decided to use the case ofPeter Sutcliffe testing our method model. Through the above steps, we have gotten the primary geographical profile which is the big circle in the follow drawing. Then，by means of Target scheme, the circle has been curtailed gradually. And finally, we get the prediction location which is the center of last geographical profile.Flowchart of geographic profile generate algorithm23456789102345678910Geographical ProfilexyFig 1 geographical profile by two schemes3. Solution of Task II3.1 Predicting modelFirst of all, we need to confirm that our purpose is predicting the next distance between criminal location and the center of circle by the method of circle hypothesis and locations of serial past criminal cases. Ronald M. and others have assessed more than 800 murders, additionally, an amusing finding about walking distance of the criminal perpetrators was found. With the increase of the criminal number, his guts and experience have been improved a lot so that his walking distance and the scope of victims are all on the climb. [4]Consequently, as time goes by, the distances between criminal locations and center of circle is an increasing sequence, gray time series forecasting using the system the quantity of the time series forecast. That is the case with the known locations to the center of a circle from the time series to construct the gray prediction model to predict the next location of the crime to the distance from the center of the circle.To predict the next case centered on the distance to pitch again after the next place is to determine the exact location.According to the ecological balance of the biosphere principles of energy and matter in the biosphere constant flow and circulation. Solar energy is the energy source of all life activities, throughout the biosphere at the center.For the next location prediction of crime in order to ensure that the entire system is in equilibrium principle, be determined through experiments. The initial outline of the offender can be seen as activities of the criminals circle, if the offender to a continuous crime, and criminal activities on the need to ensure that insiders are in a state of equilibrium flows. The center circle of the sun is similar to the biosphere.To center of a circle as a fulcrum, with locations in every known case under the hanging weight balance of the whole circle. Weights left on behalf of a criminal information, the more information weights have increased, suggesting that this crime for the entire system, the greater the proportion and significance.Known place of the incident location and the weight each time the number of cases, but also determined the next case of the distance from the center circle, the very next case to determine location. The algorithm in detail is as follow(0)(0)(0)(0),...,(1),(2)(1).d d d d n +(n)，computingInitial Sequence (0)(0)(0)(0){(1),(2),...,()}D d d d n =Normalized (0)(0)(0)(0)(0)(0)(0)(2)(3)(){1,,,...,}(1)(1)(1)d d d n UDd dd = Cumulative (1)(0)(1)(0)(0)(0)(1)(0)(0)(0)(0)(0)(1)(0)(0)(0)(1)1,(2)(2)1,(1)(2)(3)(3)1,(1)(1)......(2)(3)()()1....(1)(1)(1)d d d d d d d d d d d d n d n d d d ==+=++=++++ )1(z means value for the (1)d series,(1)(1)(1)1()()(1),2,3,4,...,2z k d k d k k n ⎡⎤=+-=⎣⎦ then(1)(0)(1)(1)(1)((2),(3),(4),...,()).z z z z z n =Define Gray differential equation model GM(1,1) as(0)(1)()()d k az k b +=let 2,3,4,...,k n = substitute into GM(1,1) model(0)(1)(0)(1)(0)(1)(0)(1)(2)(2),(3)(3),(4)(4),......()(),d az b d az b d az b d n az n b +=+=+=+=let (0)(0)(0)(0)((2),(3),(4),...,())T N Y d d d d n =，T b a u ),(=，(1)(1)(1)(2)1(3)1......()1z z B z n ⎡⎤-⎢⎥-⎢⎥=⎢⎥⎢⎥-⎣⎦then express GM(1,1) model as a matrix equation N Y B u =⋅the Least-squares method is used to conform the value of u .^1^ˆ()T T N u B B B Y a b -⎡⎤⎢⎥==⎢⎥⎢⎥⎣⎦，^^,.a b a b == Discrete solution of differential equations gray concrete expression is(1)(0)(1)((1))ak b bd k de a a -+=-⋅+k n =，st. (1)(1)d n +，calculate (0)(1)(1)(1)(1)().d n d n d n +=+- working out (0)(1)d n +。
2010 美赛 MCM 优秀论文
3 Center of Minimum Distance Model.................................. 5
数学建模竟赛的情况简介
第1教案数学建模及竞赛知识介绍目的要求:1. 了解数学建模的基础知识、相关的基本概念；2. 了解数学模型的特点和学习方法；3. 掌握数学建模的具体过程和步骤，教学重点及难点:重点：了解数学建模的一般步骤和方法，体会如何用数学的语言和方法表述和解决实际问题。
难点：体会如何用数学的语言和方法表述和解决实际问题。
教学方法手段:讲授法，案例教学法，多媒体创新点:应用和创新是数学建模的特点，也是素质教育的灵魂；不论用数学方法解决哪类实际问题，还是与其他学科想结合形成交叉学科，首先的和关键的一步是用数学的语言表述所研究的对象，即建立数学模型。
在高科技，特别是计算机技术迅速发展的今天，计算和建模正成为数学科学技术转化的主要途径。
教学过程:1．1 从现实对象到数学模型本节先讨论原型和模型，特别是数学模型的关系，再介绍数学模型的意义。
原型和模型原型（prototype）和模型（model）是一对对偶体。
原型指人们在现实世界里关心、研究或者从事生产、管理的实际对象。
在科技领域通常使用系统（system）、过程（process）等词汇，如机械系统、电力系统、生态系统、生命系统、社会经济系统，又如钢铁冶炼过程、导弹飞行过程、化学反应过程、污染扩散过程、生产销售过程、计划决策过程等。
本书所述的现实对象、研究对象、实际问题等均指原型。
模型则是指为某个特定目的将原型的某一部分信息减缩、提炼而构成的原型替代物。
特别强调构造模型的目的性。
模型不是原形原封不动的复制品，原型有各个方面和各种层次的特征，而模型只要求反映与某种目的有关的那些方面和层次。
一个原型，为了不同的目的可以有很多不同的模型，模型的基本特征是由构造模型的目的决定的。
例如：展厅里的飞机模型：外形上逼真，但是不一定会飞；航模竞赛的模型飞机：具有良好的飞行性能，在外观上不必苛求；飞机设计、试制过程中用大的数学模型和计算机模拟：要求在数量规律上真实反映飞机的飞行动态特征，毫不涉及飞机的实体。
2010数学建模竞赛B题分析
选取近3次综合世博会作为上海世博会纵向影响力的参考，它们分 别是1992年西班牙塞维利亚世博会，2000德国汉诺威世博会以及2005 年日本爱知世博会，将从上述城市竞争力的各个方面进行讨论，表现上 海世博会对城市竞争力影响的大小。其数据如下表：
年份
主办方
世博投资 （亿元）
世博收益 （亿元）
举办市第三产 业产值增长率
C
I f In If
100%
If
In
正常举行世博会上海净投资额 假设不举办世博会上海净投资额
通过查找相关资料我们得出本次上海市由亍世博会而进行的基础配 套设施投资（丌包括园区投资）总额预测值为3200亿元左右[1]。而丌丼 办世博会的投资额预测值为2346.8亿元（见表2）。利用上述公式计算得 到世博会的相关净投资率约为0.363。 带动投资在上海丌丼办世博会情况下和丼办世博会情况下的比较值 为 0.637:1。
灰色系统(Grey System)理论是我国著名学者邓聚龙教授2O世纪8O年代初 创立的一种兼备软硬科学特性的新理论．该理论将信息完全明确的系统定义为 白色系统，将信息完全丌明确的系统定义为黑色系统，将信息部分明确、部分 丌明确的系统定义为灰色系统．由亍客观世界中，诸如工程技术、社会、经济 、农业、环境、军事等许多领域，大量存在着信息丌完全的情况．要么系统因 素戒参数丌完全明确，因素关系丌完全清楚；要么系统结构丌完全知道，系统 的作用原理丌完全明了等，从而使得客观实际问题需要用灰色系统理论来解决 。
现在来比较世博会的海外影响力，首先注意到在世博会开幕乊前， 关键字Shanghai expo的点击指数较为稳定，保持在7左右。为了比 较世博会前后点击率的差别，我们对世博会期间点击率的积分结果取 平均数，得到世博会当年，关键字Shanghai expo的平均点击指数 24.5657。 上述结果表明：上海世博对上海的知名度影响力贡献比达 1:3=0.3333:1。
2010A美赛特等奖
Every hitter will have such feeling of vibration in hand at the bat gripping part, when the hitting spot is Spot A at the end of bat or at an upper Spot C. And sometimes it is a strong vibration but the ball can not go very far. And when the ball and the bat hit directly at the Spot B at the proper position of the bat (as shown in the following Fig. 1), the ball can go faster and farther without giving the hitter the feeling of strong vibration.
To simplify the problem, the ball is still before hitting, and then:
v=
(1 + e)ω 0 x p 1+ mx 2 p Jz
The momentum
P of the ball after being bit is: (1 + e ) m ω 0 x p P = mv = mx 2 p 1+ Jz
ω0
are the revolving angular velocity of
the bat on Point O before and after hitting respectively, v0 and v are the tangential velocity of the ball along the bat before and after hitting respectively, u0 and u are the linear velocity of the ball before and after hitting respectively. M and m are the mass of the bat and the ball respectively. Taking axle Z as the revolving axle, x p = H as the distance from the center point O to the contact point of the bat and the ball, xc = R as the distance from the center point O to the mass center of the bat, the following equation can be obtained according to the theorem of moment of momentum of collision. That is Use
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连环作案嫌疑人“地理轮廓”估计的统计模型摘要：在不同地点发生的系列谋杀案对社会的危害极大，如何确定其犯罪嫌疑人的住所是破案的关键。
对犯罪嫌疑人住所的估计，给出了3种方法，即圆周假设、质心法和最小距离法，通过对比验证表明，最小距离法较优；一般犯罪分子都会选择”不近不远”的作案地点，基本上服从二维正态分布。
同时考虑犯罪分子的心理特征，作案方式等进一步对模型进行优化，得出更合理的模型，提高对连环作案的案件的效率。
关键字：连环作案、圆周假设、质心法、最小距离、概率分布一．问题叙述：1.1问题重述1981年Peter Sutcliffe(萨克利夫)被判刑因为他参与了十三起谋杀和对其他人的恶毒攻击。
缩小搜索Sutcliffe的方法之一是发现一个攻击位置的“质心”.最终犯罪嫌疑人恰好生活在该方法预测的同一个小镇。
从那时起，已经发展出一系列更加复杂的技术用来预测基于犯罪地点的具有地理效应（地理轮廓）的系列犯罪行为。
你的团队被一个当地警察局要求发展出一种方法用来帮助他们的系列犯罪调查。
（1）你们的方法应该至少需要利用两种不同的情景以生成地理效应（地理轮廓），进而根据不同情况下的分析结果对执法人员提供有效的预测。
（2）基于以往犯罪的时间和位置，预测信息应该提供一些估计或指导下次可能的犯罪地点。
如果在预测中用到了其它的信息，必须提供特别的细节说明告诉我们这些信息是如何被整合的。
（3）你们的方法中也应该包括在给定条件下（包括适当警告信息）下预测的可靠性估计。
1.2“地理画像”地理轮廓是是一项刑事调查方法,分析确定最有可能的罪犯居住面积确定的罪行的连接一连串的位置。
通过采用定性和定量的方法，它有助于理解空间的违法行为和针对较小的区域的社会调查。
通常用于在连环谋杀或强奸（但还纵火，轰炸，抢劫及其他犯罪）的情况下，技术可帮助警方侦探优先考虑大规模的重大犯罪调查，往往需要成百上千个嫌疑人和提示中的信息。
基本原则是犯罪相关的地点提供关于受害人的资料和罪犯的与地理环境的相互作用。
它甚至可以显示罪犯对周边地理情况的熟悉程度和罪犯对安全距离的界定，可以反映了他的非刑事空间生活方式方面即其居住地的规划。
二．模型假设：由于犯罪活动的巨大变动以及几乎所有连环杀人案的凶手通常患有心理疾病，所以采用相对简单的计算机模型去预测连环杀人案一般都面临几个障碍。
以下是应用我们模型的对犯罪行为所采取的假设：•犯罪是单独作案的。
我们假设模型中的案件都是由单独个体作案的，我们的模型不对有组织的犯罪，团伙犯罪和暴动进行分析。
•案件的性质为连环杀人案件;•作案地点在被罪犯被发现地点的附近。
在凶杀案中这意味着尸体的发现地点就是作案地点。
这不是不合理的假设，因为多数连环案件的凶手在作案后会把受害人尸体丢到作案地点附近。
在连环案件中，比如连环强奸案，爆窃或纵火案，作案地点与警方发现地点是无区别的。
•假设罪犯作案动机相对稳定。
•犯罪发生在一个狭小的区域，比如一个市或县。
这里忽略案件发生在州或国家之间的情况。
在那些案件中，我们的模型必须被应用于每一个犯罪群体。
•假设案件的作案地点服从正态分布。
四.犯罪分子的居住地范围的预测4.1圆周假设“犯罪圆周假设”基本原理：假设一个犯罪嫌疑人连续作案，其作案地点分布广泛，若找出两个最远的犯罪位置，将二者连接起来，并以此连线为直径，画出一个包括所有犯罪地点的圆周，多数情况下，犯罪嫌疑人就住在圆周里，而且可能就住在靠近圆周中心的地区，这就是西方犯罪情报分析理论著名的“圆周假设”。
将犯罪嫌疑人两次最远处作案地点之间连线，并以连线为直径,画一个圆周，就可以把所选的作案地点包括在内。
犯罪嫌疑人住所大致处于圆周的中的地带，这与“圆周假设”理论基本吻合。
若以犯罪分子的居住地为起点，根据大致的方位，将发案地点连接起来，就会发现，在早期，犯罪分子主要围绕自己的居住地作案。
到了后来，有了较大的改变，基本上是从居住地出发，放射状向四周散开，由此可以推测出犯罪分子是沿着公路或铁路远行，选择到离家较远的地方作案，然后再按原路返回居住地。
首先，第一次作案在Leeds（利兹），这是值得关注的重点。
其次，以第一次发案点为圆点，向其他发案地区连线就会发现，案件大多发生在两条红线构成的扇区内，其中多数案件都沿公路展开。
再次，曼彻斯特最远处作案点。
最后，将两个最远的作案地点连线，并以最长线的中点为圆心画圆，则可将所有的作案地点全部包括在内，符合犯罪的“圆周假说”理论，犯罪分子有可能居住在圆周内。
结论：两条红线的交叉点，又是第一次发案地点，据此可以预测，以Bradford(布拉德福德)、Leeds （利兹）为中心的地区，应是犯罪嫌疑人的原始居住地，这符合犯罪人员首次作案的心理，即犯罪人员早期作案大多选择其居住地附近。
这符合事实，查资料知道彼得萨克利夫（Peter Sutcliffe ）是居住在Bradford(布拉德福德)的。
且圆心的纬度和经度为（53.77255，1.6869）与实际居住地距离相差6724.8米其中计算不同经纬度之间的公式对上面的公式解释如下：t1 Lung1 表示A 点经纬度，Lat2 Lung2 表示B 点经纬度；2.a=Lat1 – Lat2 为两点纬度之差 b=Lung1 -Lung2 为两点经度之差；3.6378.137为地球半径，单位为千米； 计算出来的结果单位为千米。
4.2.质心求法在一个陌生的地方作案比在一个自己熟悉的地方作案危险; 在自己藏匿的地方周围作案比在相对较远的地方作案更容易暴露自己的行踪以及在在一个地方连续作案比较危险等因素，可以初步利用质心法求得罪犯居住地大致的犯罪在哪个区域内。
(W N ,)=()∑=ni i i W N n 1,1纬度N 经度W 犯罪顺序53.8178 1.5428 1 53.8083 1.5311 2 53.8334 1.5002 3 53.8109 1.7633 4 53.8179 1.5325 5 53.4324 2.2506 6 53.8001 1.7721 753.6544 1.78 8 53.4599 2.2225 9 53.7117 1.8736 10 53.79 1.764 11 53.8085 1.6715 12 53.82281.5781 13纬度的均值，经度的均值,知道PeterSutcliffe 的居住地在布拉德福德伊顿区 6 花园巷且查资料得出其纬经度为（53.81307，1.7893）计算出质点与其居住地的距离相差4902米。
4.3.最近距离法由犯罪心理学，可以对犯罪分子在选择连环作案地点时的心理特征进行分析，他们通常考虑考虑三点：一是对作案地点要熟悉；二是要避开熟人；三是不能离居住地太远。
所以犯罪分子选择的作案地点一般会是离居住地不近不远。
因此可以利用最小距离法()()∑==ni i z d z d 1。
又因犯罪分子以往作案大致在leeds ，manchester ，halifax ， bradford ， huddersfield 这五个市区内，可分别计算这五个市区到13个犯罪地点的最小距离，从而确定犯罪分子的最小距离，即可初步确定犯罪分子在哪个市区。
地区 leeds manchester halifax bradford huddersfield纬度 经度53.80128 1.548567 53.47925 2.247926 53.72702 1.85754 53.79598 1.759398 53.64579 1.785635与第1，2，3，4，5，6，7，8，9，10，11，12，13犯罪地点的距离1874.441 1386.546 4775.786 14131.25 2126.655 61818.64 14670.67 22313.34 58404.56 23559.96 14195.64 8106.886 3077.934 59768.5 59721.34 63034.16 48757.02 60304.06 5209.192 47474.31 36501.85 2729.908 35729.76 47000.25 52729.28 58340.35 23000 23265.3 26272.69 11188.81 23614.48 41764.43 9871.416 9547.353 38215.59 2003.334 9313.444 15208.05 21213.62 14418.12 15044.67 17505.95 1677.162 15086.57 51776.22 950.8893 15791.32 48229.25 12000.73 730.3533 5933.683 12263.35 24902.97 24619.63 28045.43 18405.8325350.6838795.2117170.121026.01935474.49335.956 16087.74 19575.69 23937.12距离总和 230442.3 577300 254478.5 211408.3 282726.8 由以上数据可以看出，犯罪距离之和最小值是211408.3米，初步认定在bradford市区，知道Peter Sutcliffe的居住地在布拉德福德伊顿区 6 花园巷且查资料得出其纬经度为（53.81307，1.7893）计算出质点与其居住地的距离相差2730.7米。
4.4模型的改进由于以上三种方法都有各自的优缺点，且忽略了一些因素如时间间隔因素，距离波动因素等，所以我们可以对其模型进行优化。
4.4.1时间间隔因素得到z2•基于理论：两次作案时间间隔越小，两次作案点越有价值。
•所谓时间间隔：每个作案点时间间隔为它与前一次的间隔•处理权重时间时间间隔30 October 1975020 January 1976825 February 197738123 April 19777826 June 1977641 October 19779221 January 197811231 January 19781016 May 19781044 April 19793222 September 197915020 August 198035217 November 198088由时间间隔可知案件7，8时间间隔最小，两次作案点最有价值。
取两者的中心为z2。
4.4.2距离波动因素得到z3计算初始z 初始a•每个点的距离波动定义为|d-a|•权重处理办法与时间的相同•加权平均得到新的z根据最小距离法得到z3的纬度和经度分别是53.79598和1.759398 4.4.3最大概率圆的算法•认为初始点包含某些重要信息，并且可能在加权之后有信息丢失所以不舍弃z1•用z1，z2，z3确定最小覆盖圆•缺点：•改进：对犯罪地点的纬度经度进行分析，找出质心点，时间间隔因素，距离波动因素的点再次寻找最小覆盖园，使得预测犯罪分子居住地的范围更加精确。