2011美赛题目

AbstractThis paper presents one case study to illustrate how probability distribution and genetic algorithm and geographical analysis of serial crime conducted within a geographic information system can assist crime investigation.Techniques are illustrated for predicting the location of future crimes and for determining the possible residence of offenders based on the geographical pattern of the existing crimes and quantitative method,which is PSO.It is found that such methods are relatively easy to implement within GIS given appropriate data but rely on many assumptions regarding offenders’behaviour.While some success has been achieved in applying the techniques it is concluded that the methods are essentially theory-less and lack evaluation.Future research into the evaluation of such methods and in the geographic behaviour of serial offenders is required in order to apply such methods to investigations with confidence in their reliability.1.IntroductionThis series of armed robberies occurred in Phoenix,Arizona between13September and5December1999and included35robberies of fast food restaurants,hotels and retail businesses.The offenders were named the“Supersonics”by the Phoenix Police Department Robbery Detail as the first two robberies were of Sonic Drive-In restaurants.After the35th robbery,the offenders appear to have desisted from their activity and at present the case remains unsolved.The MO was for the offenders to target businesses where they could easily gain entry,pull on a ski mask or bandanna, confront employees with a weapon,order them to the ground,empty the cash from a safe or cash register into a bag and flee on foot most likely to a vehicle waiting nearby. While it appears that the offenders occasionally worked alone or in pairs,the MO, weapons and witness descriptions tend to suggest a group of at least three offenders. The objective of the analysis was to use the geographic distribution of the crimes to predict the location of the next crime in an area that was small enough to be suitable for the Robbery Detail to conduct stakeouts and surveillance.After working with a popular crime analysis manual(Gottleib,Arenberg and Singh,1994)it was found that the prescribed method produced target areas so large that they were not operationally useful.However,the approach was attractive as it required only basic information and relied on simple statistical analysis.To identify areas that were more useful for the Robbery Detail,it was decided to use a similar approach combined with other measurable aspects of the spatial distribution of the crimes.As this was a“live”case, new crimes and information were integrated into the analysis as it came to hand.2.AssumptionIn order to modify the model existed,we apply serial new assumptions to the principle so that our rectified model can be much more practical.Below are the assumptions:1.C riminals prefer something about the locations where previous crimes werecommitted committed..We supposed the criminals have a greater opportunity to ran away if they choose to crime in the site they are familiar with.In addition,the criminals probably choose previous kill sites where their target potential victims live and work.2.Offenders regard it safer to crime in their previous kill site as time went by.This is true that the site would be severely monitored by police when a short term crime happened and consequently the criminal would suffer a risk of being arrested in that site.And as mentioned above ,the police would reduce the frequency of examining the previous kill sites as time went by.3.Criminals are likely to choose the site that have optimal distance .This is a reasonable assumption since it is probably insecure to crime in the site that stays far away and that costs an amount of energy to escape and adds the opportunity to be arrested in such an unfamiliar terrain.And it is also impossible to crime in the site nearby since it increases the probability of being recognized or being trapped.As a result,we can measure a optimal distance in series perpetrations.4.Crimes are committed by individual.We assume that all the case in the model are committed by individuals instead of by organized members.In this way the criminal is subject to the assumptions mentioned above due to his insufficient preparation.5.Criminals Criminals''movements unconstrained.Because of the difficulty of finding real-world distance data,we invoke the “Manhattan assumption”:There are enough streets and sidewalks in a sufficiently grid-like pattern that movements along real-world movement routes is the same as “straight-line”movement in a space be discrete into city blocks.It is demonstrated that across several types of serial crime,the Euclidean and Manhattan distances are essentially interchangeable in predicting anchor points.3.The prediction of the next crime site3.1The measure of the optimal distanceDue to the fact that the mental optimal distance of the criminal is related to whether he is a careful person or not,it is impossible for him to make a fixed constant.Besides,the optimal distance will change in different moment.However,such distance should be reflected on the distances of the former crime sites.Presume that the coordinates of the n crime sites is respectively ),(11y x 、),(22y x 、……、),(n n y x ,and define the distance between the th i crime site and the th j one as j D ,i .The distance above we first consider it as Euclid distance,which is:22,)()(j i j i j i y y x x D −+−=With that,we are able to measure the distance between the th n crime site and the th 1-n one respectively.According to the assumption 2,the criminal believes that the earlier crime sites have became saferfor him to commit a crime again,so we can define his mental optimal distance,giving the sites the weights from little to much according to when the offenses happened in time sequence,as:∑−==11,n i ni i D w SD Satisfying 121......−<<<n w w w ，111=∑−=n i i w .Presuming the th i crime happens in i t ,whichis measured by week,we can have ∑−==11n i i kk t t w .SD can reflect the criminal's mental condition to some extent,so we can use it to predict the mental optimal distance of the criminal in the th n 1+case.While referring to the th n crime site,the criminal is able to use SD to estimate the optimal distance in the next time,and while referring to the rest crime sites,the optimal distances reduce as time goes back.Thus,the optimal security of the th i crime site can be measured as the following:n ni i SD t t SD *=3.2The measure of the probability distributionGiven the crime sites and location,we can estimate tentatively the probability density distribution of the future crimes,which equals to that we add some small normal distribution to every scene of crime to produce a probability distribution estimate function.The small normal distribution uses the SD mentioned above as the mean,which is:∑=−−=n i i i SD r n y x f 122)2)(exp(211),(σσπi r is defined as the Euclid distance between the site to the th i crime site,and the standard difference of the deviation of the criminal's mental optimal distance is defined as σ,which also reflects the uncertainty of the deviation of the criminal's mental optimal distance,involves the impacts of many factors and can not be measured quantitatively.The discussion of the standard difference is as following:3.3The quantization of the standard differenceThe standard difference is identified according to the following goal,which is,every prediction of the next crime site according to the crime sites where the crimes were committed before should have the highest rate of success.When having to satisfying such optimization objective,it isimpossible to make the direct analysis and exhaustivity.Instead,we have to use the optimized solutions searching algorithm,which is genetic algorithm.\Figure1:The Distribution of the Population of the Last GenerationAccording to the figure,the population of the last generation is mostly concentrated near80, which is used as the standard distance and substituted to the*formula.With the*formula,we are able to predict the probability density of Whether the zones will be the next crime site.Case analysis:5crime site according to the4ones happened before Figure2:The prediction of theth6crime site according to the5ones happened before Figure3:The prediction of theth6crime site according to the5ones happened before Figure4:The prediction of thethAccording to the predictions happened before,the predictions of the outputs based on the models are accurate relatively,and they are able to be the references of the criminal investigations to some extent.However,when is frequency of such crime increases,the predictions of the outputs23crime site according deviated the actual sites more and more,such as the prediction of thethto the22ones happened before,which is:23crime site according to the22ones happened before Figure5:the prediction of thethConclusion according to analysis:It may not be able to predict the next crime site accurately if we use Euclid distance to measure the probability directly.So,we should analyze according to the actual related conditions.For example,we can consider the traffic commutes comprehensively based on the conveniences of the escapes,such as the facilities of the express ways network and the tunnels.According to the hidden security of the commitments,we should consider the population of the area and the distance from the police department.Thus,we should give more weights to the commute convenience,hidden security and less population.In addition,when the commitments increases,the accuracy of the model may decrease,resulted from the fact that when the criminal has more experience,he will choose the next crime sites more randomly.4.Problems and further improvementsWith23crimes in the series the predictions tended to provide large areas that included the target crime but were too large to be useful given the limited resources the police had at their disposal.At this stage,a more detailed look was taken at the directionality and distances between crimes.No significant trends could be found in the sequential distance between crimes so an attempt was made to better quantify the relationship between crimes in terms of directionality.The methodology began by calculating the geographic center of the existing crimes. The geographic center is a derived point that identifies the position at which the distance to each crime is minimized.For applications of the geographic center to crime analysis.Once constructed,the angle of each crime from the north point of the geographic center was calculated.From this it was possible to calculate the change indirection for the sequential crimes.It was found that the offenders were tending to pattern their crimes by switching direction away from the last crime.It appears that the offenders were trying to create a random pattern to avoid detection but unwittingly created a uniform pattern based upon their choice of locations.This relationship was quantified and a simple linear regression used to predict what the next direction would be.The analysis was once again applied to the data.While the area identified was reduced from previous versions and prioritized into sub-segments,the problem remained that the areas predicted were still too large to be used as more than a general guide to resource deployment.A major improvement to the methodology was to include individual targets.By this stage of the series,hotels and auto parts retailers had become the targets of choice.A geo-coded data set became available that allowed hotels and retail outlets to be plotted and compared to the predicted target areas.Ideally those businesses falling within the target areas could be prioritized as more likely targets.However,in some cases the distribution of the likely businesses appeared to contradict the area predicted.For example,few target hotels appeared in the target zone identified by the geographic analysis.In this case,more reliance was placed upon the location of individual targets. From this analysis it was possible to identify a prioritized list of individual commercial targets,which was of more use operationally.Maps were also provided to give an indication of target areas.Figure6demonstrates a map created using this methodology.It is apparent from the above discussion that the target areas identified were often too large to be used as more than a general guide by the Robbery Detail.However,by including the individual targets,it was possible to restrict the possible target areas to smaller,more useful areas,and a few prioritized targets.However,such an approach has the danger of being overly restrictive and it is not the purpose of the analysis to restrict police operations but to suggest priorities.This problem was somewhat dealt with by involving investigators in the analysis and presenting the results in an objective manner,such that investigators could make their own judgments about the results.To be more confident in using this kind of analysis a stronger theoretical background to the methods is required.What has been applied here is to simply exploit the spatial relationships in the information available without considering what the connection is to the actual behaviour of the offenders.For example,what is the reason behind a particular trend observed in the distance between crimes?Why would such a trend be expected between crimes that occur on different days and possibly involve different individuals?While some consideration was given to identifying the reason behind the pattern of directionality and while it seems reasonable to expect offender’s to look for freeway access,such reasoning has tended to follow the analysis rather than substantiate it.Without a theoretical background the analysis rests only on untested statistical relationships that do not provide an answer to the basic question:why this pattern?So next we will apply a quantitative method,which is PSO,based on a theoretical background,to locate the residence of the criminal's residence.5.The prediction of the residenceParticle Swarm Optimization is a evolutionary computation,invented by Dr.Eberhart and Dr.Kennedy.It is a tool of optimization based on iteration,resulted from the research on the behaviors of the bird predation.Initiating a series of random number,the PSO is able to catch the optimization with iteration.Like PSO,the resolution of our residence search problem is the criminal,whose serial crime sites have been abstracted into 23particles without volume and weight and extended to the 2-D space.Like bird,the criminal is presumed to go directly home when he committed a crime.So,there are 23criminals who commit the crimes in the 23sites mention before and then they will go home directly.The criminals are defined as a vector,so are their speed.All criminals have a fittness decided by the optimized functions,and every of them has a according speed which can decide their direction and distance.All the criminals know the best position (pbest,defined as the residence known by the individual),which has been discovered so far,and where they are now.Besides,every criminals also know the best position which has been found by the group (gbest,defined as the residence known by the group).Such search can be regarded as the experience of other criminals.The criminals are able to locate the residence by the experience of itself and the whole criminals.PSO computation initiates the 23criminals and then the offenders will pursue the optimized one to search in the space.In other words,they find the optimized solutions by iteration.Presume that in the 2-D space the location and speed of the ith crime site is relatively ),(2,1,i i i x x X =and ),(2,1,i i i v v V =.In every iteration,the criminals will pursue the two best positions to update themselves.The two best positions are relatively the individual peak (pbest),),(2,1,i i i p p P =,which is found by the criminal himself,and the group optimized solution (gbest),g P ,which has been found to be the optimized solution by the whole group so far.When the criminals found the two optimized solutions,they will update their speed and new position based on the following formulas.2,1),1()()1()]([)]([)()1(,,,,,22,,11,,=++=+−+−+=+j t v t x t x t x p r c t x p r c t wv t V j i j i j i j i j g j i j i j i j i In the above,the w is inertial weighted factor,21c andc are positive learning factors,21r andr are random number which are distributed uniformly between 0and 1.The learning factor can make the criminals have self-conclude ability and ability of learning from others.Here we make both of them be 2,as what they always are in PSO.The inertial weighted factor w decides the extent of the inheritance of the current speed of the crime sites.The appropriate choice can make them have balanced searching and exploring ability.For balancing the global searching ability and the local improving ability of the criminal in the PSO algorithm,here we adopt one of the self-adapted methods,which is Non-linear Dynamic Inertial Weight Coefficient to choose the inertial weight.The expression is as following:⎪⎩⎪⎨⎧=≤−−−−>avg avg avg f f f f f f w w w f f w w ,))*((,minmin min max min max In the above,the max w and min w are defined respectively as the maximum and minimum of w,f means the current functional value of the criminal,and the avg f and min f respectively means the average value and minimum value of all the current criminals.In addition,the inertial weight will change automatically according to the objective value,which gives the name self-adapted method.When the final values,which are estimations of the criminal's residence,become consistent,it will make the inertial weight increase.When they become sparser,it will make the inertial weight decrease.In the meantime,referring to the criminals whose final values are worse than the average value,its according inertial weighted factor will become smaller,which protect the crime site.Oppositely,when referring to the criminals whose final values are better than the average value,its according inertial weighted factor will become bigger,which makes the criminal nearer to the searching zone.So now,with the PSO of Non-linear Dynamic Inertial Weight Coefficient,we can calculate the minimum value of22,)()(j j j i y y x x R −+−=,j=1,2,3 (23)In the above,j ,i R is the residence of the criminal.Thus,we have the output (x,y)as(2.368260870656715,3.031739124610613).We can see the residence in the figure 7.Figure7:The residence in the map6.ConclusionThis paper has presented one case study to illustrate how probability distribution and geographical analysis of serial crime conducted can assist crime investigation. Unfortunately,in the Supersonic armed robbery investigation the areas identified were too large to have been of much use to investigators.Further,because of the number of assumptions applied the method does not inspire enough confidence to dedicate resources to comparing its results to the enormous amount of suspect data collected on the case.While the target areas predicted tended to be large,the mapping of individual commercial targets appears to offer a significant improvement to the method.However,as they stand,these methods lack a theoretical basis that would allow the results to be judged and applied in investigations.Limitations such as these can be offset to some degree by the involvement of investigators in the analysis.In the end,we used a quantitative method to locate the residence of the criminal to make the identified areas smaller.So,due to the advantages and drawbacks of the above methods,we suggest that we should use different methods to help us fight again the crimes comprehensively.。

Putting the Spark Back in theElectric CarTeam#11422February14,20111Team#11422Page2of20Contents1Clarification of Problem3 2Plan of Attack3 3Assumptions3 4On Types of Cars4 5Model for Number of Cars4 6Microeconomic Model56.1Global Influence Model (5)6.1.1Strengths&Weaknesses (7)6.2Localized Behavior (8)6.2.1Cellular automata (8)6.3Strengths and Weaknesses (9)7Macroeconomic Model:Meeting the Energy Demand107.1Current Energy Source and Demand (10)7.2Current Pollution rates (11)7.3Quantizing Pollution (11)7.3.1Health (12)7.4Quantizing Cost (12)7.5αParameter (13)7.6Minimizing X:Genetic and Nelder-Mead Methods (14)7.7Using Alpha to Determine Cost,and Vice Versa (14)7.8Example Calculation (16)7.9Fossil Fuels Saved (17)7.10Strengths and Weaknesses (17)8Meshing the Micro and Macro Models18 9Conclusion19Team#11422Page3of201Clarification of ProblemWith the recent introduction of the Nissan Leaf and Chevy Volt to the world carfleet and the fading supply of petroleum,the possibility of electric vehicles replacing standard petroleum cars is increasing.Questions arise concerning the feasibility of such vehicles,specifically regarding the amount of fossil fuels saved through widespread use of electric cars,and the economic feasibility.It is of great concern to auto-manufacturers and environmentalists alike to determine how to cause electric cars to’catch on,’and of equally great concern to govern-ments to determine how to augment the power grid to meet the demand of the electric carfleet.The models proposed within this paper will offer an insight to these problems.2Plan of AttackOur objective is to model the effects of electric vehicles on the environment, public health,and economy.We need to determine which methods would be most effective in causing widespread use of electric vehicles,within a reasonable time rge scale use of electric vehicles would also put an increased strain on the power grid,which would have to be corrected.To determine the most efficient way to do this,we will proceed as follows:1.Create a model for the amount of electric cars at any given point in time.(Micro)2.Create a model that gives a single value to the effect electric cars have onthe environment,health,and economy.(Macro)3.Connect these models so that,by giving setup conditions,we can deter-mine the cost to minimize the pollution values.3AssumptionsDue to the extrapolative nature of our model,and the difficulty in obtaining reliable global information,several assumptions were made in order to complete our model.These simplifying assumptions will be used throughout the paper and could feasibly be replaced with reliable data when it becomes available.•The cost of building more coal,oil,and natural gas plants is negligible to the cost of yearly fossil fuel production.That is to say,energy costs simply rely on production prices from the plants themselves,and not creation of the plants.•We assume100%efficiency of converting fossil fuels to energy for electric-ity.This makes the calculations for energy easier,removing the need to know the electric energy conversion rate for electric generators.•World Governments can control addition of power plants to determine the proportions of energy from each source.This is essential in changing theTeam#11422Page4of20 makeup of the power grid.By being able to change the ratio of the energy sources of our electricity production can we can change the ratio of the pollutants produced for each unit of electrical energy.•Ratios of energy sources into demand sectors for the US in2009is the same as the ratios across the world.This allowed us to generalize the information that we had to the world-wide energy system.•Price increases quadratically as demand increases for fossil fuels.This allows us to extrapolate the past data,allowing us to produce a prediction of the cost of fossil fuels in the future.•Population within the next50years can be modeled with a cubicfit.This allows us to extrapolate the past data as well,ensuring that we know the amount of cars in a given year.•A major factor in choosing which car to buy is what the people around you own.The movie”Who Killed the Electric Car?”suggested that the main reason that electric cars did not become popular was because many people did not know about them or their properties.This assumption is the basis in the models for the spread of electric cars throughout a population.[9]•It is economically and environmentally infeasible to increase current en-ergy contribution to the electric power grid for each power source by more than%25percent.This is to establish upper bounds for the Nelder-Mead methods,and can be replaced with projected maximum contribution for 2060if/when these values become available.4On Types of CarsWe have decided to base our model solely on electric vehicles versus gasoline vehicles,instead of including hybrid vehicles.We have chosen to do this because we are concerned with the widespread usage of electric vehicles.If electric vehicle usage is widespread,then the idea of a hybrid car is useless,since electric cars can be used for most transportation usages and gas cars can be used for any transportation that electric cars cannot do.Hybrid models were created to transition from gas cars to electric cars.However,if we are to consider widespread usage of electric vehicles,hybrids won’t be necessary.It is worth noting that electric vehicles do have limited range,causing some range anxiety. Modern estimates suggest that90%of automobile users do not have needs that exceed the limitations of electric cars,however,so the range anxiety will only affect10%of the population[9].5Model for Number of CarsIn order to model the change towards electric cars and its impact on the environment,we need a model for the number of cars in the future.We found an estimated134motor vehicles per1000people in the top130developed countries. From this,an estimate of120vehicles per1000people in the world can be made.Team#11422Page5of20 Using this and population data,we can expect to have C cars in t years after 1950based on the following equation[10]:C(t)=.002t(2.55·109+3.91·107t+1.1·106t2−9.38·103t3)We decided upon a cubicfit to model the population because itfitted popu-lation data very well.However,thisfit will only accurately model population data until2060,due to the cubic nature of the function.We have a multiplier of .002t because the number of cars per capita will increase over time,as data has shown[1].We are going to let E equal the proportion of cars which are electric. The following2equations give the number of electric and gasoline vehicles over time in terms of E.E(t)=E·C(t)G(t)=(1−E)·C(t)In our microeconomic model,we examine how E will change50years in the future based on an initial proportion of electric cars.This will allow us to see what must be done to make electric vehicle usage widespread.For our macroeconomic model,we let E=.9since we wish to examine the effects of widespread electric vehicle usage.6Microeconomic ModelThrough examination of how individuals react to electric car usage we can model the change from petroleum vehicles to electric vehicles.Our small-scale model needs to be based on the likelihood that individuals will switch to electric vehicles.We propose two models.Both of these models require a government subsidy for electric cars in order to”jumpstart”their production,and explosion in popularity.Thefirst model is based on coupled differential equations for how one might expect the number of electric vehicles and the number of gasoline vehicles will change over a continuous time interval.The second model is a 2D cellular automata simulation to model the local influence as well as global influence of the number of electric vehicles,over a discrete time interval.6.1Global Influence ModelThis model assumes that individuals are influenced by the global proportion of people who have electric cars.An individual who is going to buy a new car or replace a broken down gas car will buy an electric car with a probability equal to the proportion of people who have electric cars.This is because the more people who have electric cars,the more likely an individual is to hear about electric cars and be persuaded to switch to an electric car.This allows us to define the following coupled differential equations where BDE and BDG is the probability that an electric and gas car will break down during one year.Since gas cars last for about8years and electric cars last for about20years,we letBDE=120and BDG=18[4].E (t)=E(t)E(t)+G(t)·(C (t)+BDG·G(t))−BDE·E(t)·1−E(t)E(t)+G(t)Team#11422Page6of20G (t)=1−E(t)E(t)+G(t)·(C (t)+BDE·E(t))−BDG·G(t)·E(t)E(t)+G(t)Since our C(t)and is only valid forfifty years in the future,solving these equations outright is unnecessary.We use Euler’s Method to approximate E(t) and G(t).To do this,we need two points,one for E(t)and one for G(t).Since t is measured in years after1950,we let G(60)=C(60).We cannot let E(60)=0 since the only way for the number of gas cars to grow is from the probabilityE(t) C(t).This model requires that a certain number of electric cars be seeded intothe population to jump start the growth of electric cars.In order to seed these cars,the government could pay the difference in cost between an electric vehicle and an average gas car to give people an incentive to buy an electric car.By spending this money to encourage people to use electric cars,the government would save money later by spending less money for fossil fuels,such as oil.We will examine how this works after we have built our macroeconomic model.To determine the seeding cost,we assume that the government will pay the differ-ence between the cost of an electric vehicle and a gasoline vehicle.We decided this cost per car would be$41,000−$28,400=$12,600.The following graphs demonstrate the rate at which the proportion of electric vehicles grows(with seeding values of.05and.3)and the following table summarizes this data with varying proportions of seeding in2011.Team#11422Page7of20Seed Proportion Seeding Cost E in20600.011.0327·10110.2670.055.16348·10110.6670.11.0327·10120.8150.151.54854·10120.8770.22.06514·10120.9120.252.58174·10120.9330.33.09834·10120.948The more money spent on jumpstarting electric vehicles,the larger E will be50years in the future.In order to determine which proportion of seeded cars would be most profitable in the future,we would need to know the make up of the power grid,which we determine in our macroeconomic model.We will connect this model to the macroeconomic model later.6.1.1Strengths&WeaknessesA strength of this model is that it allows the government to see what would need to be done in order for people to want to buy electric cars.By basing this model on the proportion of people who have electric cars,this model can realistically model an individual’s likelihood of switching to an electric car.A weakness in this model is that seeding only occurs in one year,instead of a range of years.Another weakness of this model is that it does not include locality,which misses out on what seems to be a crucial point in the rise of electric vehicles.Another weakness of this model is it does not consider current sources of energy.Currently,electric cars are not better for the environment because the largest source of electrical energy is coal;this will be considered and changed in the macroeconomic model.Team#11422Page8of206.2Localized BehaviorThe previous model assumes that the total percentage of electric cars influences the chance of a single person purchasing one.However,a person is affected by the people closest to them in addition to the global behavior.This is why we decided to model the spread of electric cars using2-dimensional cellular automata.First,we decided that a cell’s percentage to pick either electric or gas is based on the8cells that are adjacent,known as the Moore Neighborhood. The influence from locality is converted into a chance of buying an electric car based on the number of your neighbors who are electric cars(N)according to the following equation:P(if electric stay electric)=18·(.1N+.1)P(if gas become electric)=120·(.1N+.1)Global influence is also considered with this model,and is incorporated with what we call the”snowball constant.”The differential equations given above can be applied to cellular automata rules by setting C (t)=0,since the number ofcells is constant,which allows us to substitute E(t)E(t)+G(t)with E p t,the proportionof electric vehicles.Ultimately we can rewrite our differential equations as:E pn+1(t)−E pn(t)=3E p(t)40·(1−E p(t))G pn+1(t)−G pn(t)=3E p(t)40·(E p(t)−1)To consider both the local and globalized behavior(L and W),we can simply weight these with the relative importance of localized behavior(due to the snowball effect)with that of global behavior.Exact values of snowball constants will have to be determined through real world observations,and will likely vary throughout the population.For our data,we used a snowball constant k=4, assuming that localized behavior is responsible for80%of buying patterns.Our final proportion looks like this:P=k·L+W 1+kThe amount of electric cars that are placed initially is changed in order to model the seeding program that the government has put in place.The output of the model gives the percentage of electric cars out of the total population of cars.This percentage can be traced from year to year to give you the effect of governmental electric car seeding,both in thefinal percent as well as the year when government seeding no longer plays a role in the percentage of cars.6.2.1Cellular automataInitial seeding is important as there is no point to seeding more cars if fewer cars will get you to your goal percentage of electric cars on the road by a certain year.By using both the global and the local model,we determined that thefinal percentage of cars that are electric,for a given number ofTeam#11422Page9of20 seeding,is less in the local model than in the global model,meaning that these local interactions seem to slow down the distribution of cars.Thefinal state of one simulation and a chart of the proportion of electric vehicles versus timeare shown below:6.3Strengths and WeaknessesThe benefits of a cellular automata model are many:this model differs from all others in this report int that there is no population increase,which means that this model is independent of theflawed population model,and is free from allflaws that come with that.This means that this model can more accurately model years after1960.Modifications can be made to increase the chances of buying an electric car as time goes on,due to improvements in technology and the decrease in electric car cost.Furthermore,the effects of localized behav-ior are well documented,and completely overlooked with differential equation models.The effects of these localized behavior can be combined with the differ-ential equation model with the snowball constant–an option unavailable to theTeam#11422Page10of20 differential equation model.This localized model is not without it’s weakness. Because of thefinite number of cells,it is difficult to incorporate growth of the population(of total cars)into the CA model.Much of our values for proba-bilities rely on rough probabilities and assumptions of the snowball constant. These can be adjusted on a product-by-product,or even a cell-by-cell basis,but it complicates the model greatly.We also could have overlooked crucial values in our probability models percentages needed to factor into the lifetime cost of a car,relative usability values like the average range an electric car can go without recharging,and more qualitative values like sticker shock.7Macroeconomic Model:Meeting the Energy DemandWhereas our microeconomic model focused on the necessary parameters to en-sure a large number of electric cars in the future,our macroeconomic model focuses on the changes that need to be made to accommodate the increased demand of electricity.It is important to consider both the costs required to produce these new amounts of electric energy,and the”hidden”costs of pollu-tion.Without considering these”hidden”costs,our model would simply gener-ate the cheapest solution to the increased power demand,which could possibly just trade one fossil fuel(gasoline)for another(coal,etc.).Thus,we’ve gener-ated an equation to determine the cost associated with the increased electricity demand,depending upon how much of each energy source we utilize,and a variable parameter equating pollution to cost.7.1Current Energy Source and DemandData from the EIA has shown that38.075quadrillion BTU was used for elec-tricity in2009,producing11.159trillion kWh[7].The breakdown of energy sources contributing to this statistic is summarized in the following chart.Since it is assumed that this breakdown is roughly equal for all highly-developed coun-tries,the countries who will have the largest number of electric cars,and thus increased demand for electricity.[7]Team#11422Page11of20 The most recent electric cars from Li-ion Motors Corp have a range of120 miles and require8hours of charging from a110V source[4].This means that 52.8kWh are needed for a full charge,or.44kWh per mile of travel.This translates to1501.3BTU needed a mile in an electric vehicle.The average pas-senger car can travel31miles on a gallon of gasoline.Since a gallon of gasoline contains roughly116,090BTU,the average gas car runs on3744.84BTU per mile[1].This is more than twice as much energy required by electric cars,so switching to electric cars decreases the amount of energy required worldwide for transportation purposes,but also requires a switch from the100%gasoline power source for gas cars,to the medley of power sources used for electricity.7.2Current Pollution ratesBurning fossil fuels creates pollutants that damage the environment,increasing acid rain,respiratory illness,and photochemical ing current energy source quantities,the pounds per mile of use of electric and gas cars is sum-marized in the following table.Though nuclear power sources have not gained widespread popularity due to social fears of nuclear accidents and the relative cost of creating a network of reactors,their carbon footprints are negligible.It also goes without saying that the footprint of renewable energy sources are also negligible.Pollutant Electric Car Gasoline Car Difference Carbon Dioxide0.1839693020.614153760.430184458Carbon Monoxide0.0001611950.00012358−3.76149·10−5Nitrogen Oxides0.0003609130.0016776880.001316776Sulfur Dioxide0.0018842520.004201710.002317459Particulates0.0019805450.000314567-0.001665978Mercury1.16351·10−82.62139·10−81.45788·10−8 From our equation of the number of cars in terms of years since1950,C(t),and the proportion of those which are electric E,we create the following equations for the pounds of pollutant reduced each year.Pollutant Pounds SavedCarbon Dioxide0.430·12,200·E·C(t)Carbon Monoxide−3.76·10−5·12,200·E·C(t)Nitrogen Oxides0.00132·12,200·E·C(t)Sulfur Dioxide0.00232·12,200·E·C(t)Particulates-0.00167·12,200·E·C(t)Mercury1.46·10−8·12,200·E·C(t)7.3Quantizing PollutionThe pollution value is a metric that determines how good for the environment having a certain percentage of electric cars are.Thefirst value that determines this metric is the amount of pollution that is being saved in pounds per year. The second value is the percent pollutant decrease,which describes how muchTeam#11422Page12of20 control of that pollutant is had.For example,if you could either cut50%of the total carbon monoxide emissions or25%of the total carbon dioxide emis-sions,that50%decrease is weighted heavier,regardless of the actual pounds of emissions you are eliminating.A third,hypothetical,value would be the bad-ness of each pollutant.Since not all pollutants damage the environment and peoples health as much as others,this badness relates to the degree to which the current yearly amounts of pollutants are damaging the environment.In the trials we ran,we assumed that the total yearly emissions of every pollutant were equally bad,so each badness value was set at1.Given the set of pollutants, {CO2,CO,NO x,SO2,P articulate},where the subscripts,G,E,and T corre-spond to the amount emitted from gas cars,the amount saved by electric cars, and total emissions,respectively,our pollution can be defined as follows:P ollution=pollutantsjj G+E·j Ej T7.3.1HealthIn examining the effects of pollution,we should also consider the effects on health.This is incorporated in our Pollution function because if we can only change a small percent of the quantity of a pollutant,it will have a smaller effect on health,whereas if we can change a larger percent of the quantity of a pollutant,it will have a larger effect on health.However,some pollutants may be more damaging to the environment than others,meaning that eliminating 50%of one pollutant would not be equivalent to eliminating50%of another. By analyzing data concerning the effects of the pollutants on health and the environment,a badness factor could be determined by which each pollution percentage change could be multiplied with.By minimizing X,which is a func-tion for cost and pollution,we will also be minimizing the effects.However in this model,we assumed that the badness factor,or the relative damage each pollutant causes to the environment and health,of each pollutant is the same.7.4Quantizing CostOnline sources can be used to estimate the small-scale cost of each BTU of each power source,in addition to the current production in the US[3].Bereft of data of the maximum production limits of each power source,it can be assumed that it would be economically infeasible to increase the current production limits for each power source to electricity by a factor greater than25%.This can be modified if more accurate statistics were obtained.Since widespread use of electric cars will require a major revamping of the power grid,demand will rise dramatically,potentially with no increase in supply.The prices of commodities increase with their scarcity,as seen by supply and demand curves.Again lacking data of supply and demand curves for power sources,we’ll be forced to make several assumptions.Given our maximum production limits,m,and our current production limits(defined to be0here)and prices,i,we can define the price of a commoditiy to be ten times it’s current cost when we reach maximum production.We’ll also define the price to be2.5times current cost when weTeam#11422Page13of20 are halfway between current and maximum production ing these data points,we can set up a quadraticfit to model the price p(L)of one quadrillion BTU’s of a particular energy source per quadrillion BTU(L)more than current production as:p(L)=i−3i·Lm+12i·L2mto determine the total cost to increase production from current values to pro-duction l,we can simply integrate from0to l:P tot(l)=l0P(l)dl=l−3i·l22m+4i·l3m2So total cost for all power sources is equivalent to:Cost=P ower Sourcesjl j−3i j·l2j2m j+4i j·l3jm2jThe current production,cost,and maximum values are shown in the following table,where all productions are in QBTU,and cost in dollars per QBTU[3].Power Source Current Production Max Production Current Cost Petroleum0.383.4793.63∗1010 Natural Gas 6.8948.61751.47∗1010 Coal18.38422.988.7∗109 Renewable Energy 4.213 5.2662.2∗1010 Nuclear Energy8.42610.53255.9∗109 7.5αParameterWith cost and pollution both quantized,we can define an objective function asX=Cost+α·P ollutionWhere the number of electric cars,and hence their energy demand,is held constant.X is dependent on the number of quadrillion BTU’s we add to each power source and the alpha value,because cost is dependent on the power sources,and pollution is dependent on both power sources andα.Since wewant to minimize both cost and pollution,our goal is to minimize X.Theαvalue simply serves as a constant defining how much the government values costto environment.For example,ifα=0,damage to the environment is not takeninto effect and minimizing X is simply minimizing Cost.In and of itself,αis a relative value,as the relationship between it and pollution is very messy (again,dependent on all power sources).However,given a maximum amountof allowable pollutants,anαcan be determined.Possible values forαand their meaning will be discussed further in this report.Team#11422Page14of207.6Minimizing X:Genetic and Nelder-Mead Methods With X being a function of six variables(five power sources,and alpha),there are several methods that we can use search for global minimum,subject to the constraints that each power source is never to decrease from current pro-duction standards(under the assumption that removal of production facilities is both costly and creates largescale unemployment),and is never to exceed previously defined maximum production standards.However,the nonlinear of nature eliminates the possibility of linear algebra techniques.Instead,we’ll rely heavily upon a Nelder-Mead iterative search technique and a genetic algorithm to define global minima.Though both of our techniques warrant equivalent so-lutions,we found that the Nelder-Mead search was much more computationally efficient,so the genetic methods were ruled out.Thus,we run a minimiza-tion of X=Cost+α·P ollution subject to the following constraints,with variables{P etr,Nat,Coal,Ren,Nuc}defining the amount of qBTU’s added to the power grid for petroleum,natural gas,coal,renewable energy and nuclear power,respectively:Minimize X=Cost+α·P ollution Subject toP etr+Nat+Coal+Ren+Nuc≤T otal Energy DemandCurrent P roduction≤P etr≤Current P roduction·1.25With thefinal constraint repeated for all power sources.7.7Using Alpha to Determine Cost,and Vice Versa Sinceαsimply refers to the amount to which we care about the environment, something that is difficult to assign a concrete value to,we’ve allowed forαto vary.By iterating the Nelder-Mead optimization for a range ofαvalues,we can generate plots of each of the Power sources versus alpha.In plainspeak,that is to say that by choosing someαvalue(e.g.,we care x much about damage to the environment),we can locate the values of each power source qBTU by simply reading the graph.Since Cost is simply a function of the power sources and is monotonically increasing withα,we can generate a graph showing cost versus alpha,by simply repeating the above procedure,and then calculating the cost from the values of each power source,plotting this to a particular alpha value. Shown below are graphs of’cost Vs.α’and’Power Sources Vs.α’with90%of the motorfleet being electric cars,50years from today:Team#11422Page15of20These graphs allow us to offer some insight into the behavior of the relation-ship betweenα(how much we care about the environment)and how we should augment the power grid.As is intuitive,a high reliance on coal and natural gas are necessary withα=0.Nuclear power seems constantly limited by our maxi-mum production value,suggesting that nuclear power,if production levels could be raised high enough,could be utilized in generating a low-cost,low-footprint power grid.Also intuitive is the monotonic behavior of the cost vsαgraph. The piecewise behavior is likely a result of certain’feasible pockets’within the polytope scanned with the Nelder-Mead method.。

Minimizing the Number of repeatersIntroductionVery high frequency (VHF) is the radio spectrum，whose frequency band ranges from 30MHz to 300MHz. VHF is always used for radio stations and television broadcasts. In addition, it is also used by signal transmission of sea navigation and aviation. Because the radio spectrum of VHF is transmitted through straight lines, a signal is easily influenced by geographical factors easily. Thus, signals become weak when it is transmitted and some low-power users need repeaters to amplify them and increase the transmission distance. We consider the situation in which every two repeaters are too close or the separate frequency is not far enough apart which can interference with each other. In order to mitigate the interference caused by the nearby repeaters, this paper employs a continuous tone-coded squelch system (CTCSS). We associate to each repeater a separate subaudible tone，that is, the subaudible tone (67Hz-250.3Hz) is added to VHF. In this way, repeaters recognize signals attached to the same subaudible tones just like secret keys. In this system, the nearby repeaters can share the same frequency pair. When users send the signals at one frequency, different repeaters with subaudible tones can recognize signals from the users the same subaudible tone. If the users in a certain area contact with each other, we should consider the signal’ s coverage area of the users and the repeaters. As long as the users’ signals are accepted by repeaters, the signals could be amplified to transmit farther. At the same time, the repeaters attached with the subaudible tones could only recognize the users with the same subaudible tones. Hence, we can consider repeaters corresponding to the number of the users, which leads to the problem of frequency channel. When the number of users in this area increases, we can add repeaters. If two repeaters have different subaudible tones, they would not communicate with each other. Thus, we should consider the problem of how the repeaters communicate with each other when they have different subaudible tones. In the mobile communication system，the spectrum is influenced by many factors such as reflex，diffraction and dispersion. Therefore, when the radio spectrum transmits in the mountainous area，we should still consider the factors above.Repeaters[4]Repeaters are a type of equipment which can amplify signals，make up the deamplification signals and support far distance communication.CTCSS[5]CTCSS(Continuous Tone Controlled Squelch System ) is short for subaudible tones, whose frequency ranges from 67Hz to 250.3Hz. It is added to the radio spectrum to make the signal carry with a unique secret key.AssumptionThe users in the area is uniform distributedThe signal of the radio spectrum in the area can’t be effected by environmentIn a certain period of time there are a small number of users removingAll repeaters have the same standardAnalysis and solution of the model to the first problemThe problem is to find a least number of repeaters in an area of radius 40 miles so that the users in this area can communicate with each other. Considering that the given area is flat, we assume that the signal ofeach repeater covers a circular area and the repeater lies in the center of the circle. The following Figure 1 shows the relationship of three adjacent repeaters.CFor case B of Figure 1, if three circles are tangent to each other, then we find that the center area cannot be covered by the singles. In order to make the signal cover the triangle area, we have to consider adding a For case C, if the intersection of three circles is not null, similar to case B, we also have to add another repeater. Thus, it is easy to find that case A, comparing with cases B and C, is optimal. Thus, we obtain the largest covering area When linked hexagons, as shown in Figure 2. Obviously, it looks like a honeycomb structure. In fact, the honeycomb pattern is one of the most efficient arrangement for radio spectrum. It transmits by the wireless medium of microwave, satellites and radiation. The structure has a feature of point-to-point transmission or multicast. It is widely used in UN Urban Network, Campus Network and Enterprise Network.Figure 2. some circles intersecting together form the closely linked hexagons Now we have a circle with radius of 40 miles. Then we analyze the distances of signals from users and repeaters covering in the circle. Because the differences for the users and repeaters in energy and height, they have different covering distances. We calculate the distances with the theory of space loss. The formula[6]is1288.120lg 20lg 40lg LM F h h d =+-+,LM the wireless lossF the communication working frequency(MHz)1h the height of the repeater (m)2h the height of the user(m)d the distance between the user and repeater(km) We assume that 150F MHZ =,1 1.5h m = and 230h m =, under the condition of the cable loss and antenna gain, we obtain the system gain()(1,21,2)i j SG Pt PA RA CL RR i j =+-++==.The system gain is the allowed decay maximum of the signal from the users to repeaters. If the system gain value is higher than the wireless loss, the users could communicate with each other. Reversely, the users could not communicate. We make the system gain value equals to the wireless loss, thus, we get the extremity distance between the user and repeater. Then we haveSG LM =We choose a typical repeater and the user facility. Thus, the parameters [6] and data of the repeaters are as followsThe transmitting power 120(43)Pt W dBm =The receiving sensitivity 1116RR dBm =-The antenna gain of the repeaters 9.8RA dB =The cable loss 2CL dB =The parameters of the interphoneThe transmitting power 24(36)Pt W dBm =The receiving sensitivity 2116RR dBm =-The antenna gain of the interphone 0PA dB =The system gain of the system from users to repeaters 1144.2SG dB =. Thus, we get the sending distance from the users to repeaters 113.8d km =. Prove in the same way, we have the system gain of the system from the repeaters to users 2151.2SG dB =, the sending distance from the repeaters to the users 220.7d km =According to the sending distance 113.8D km = between user and repeater as well as the property of regular hexagon, we calculate the distance between two repeaters. We obtain that 223.09D km =, which is described in Figure 3. Because 2D is shorter than 2'D , users in this area cannot communicate with each other. Thus, we consider the sending distance 2'D between two repeaters firstly. Then we calculate the distance between the user and the repeater again shown in Figure 4. Finally, we get that 1'12.4D km =.Figure 3. the calculation distance according to the sending distance from users to repeaters.Figure 4. the calculation distance according to the sending distance from repeaters to the users.According to the calculated distance 12'12.4'21.45D km D km ==, we know that the given circle has a radius of 40 miles. We firstly consider the signals ’ covered area of the repeaters. Thus, we get the distribution of the repeater stations in this area showed in Figure 5. The number of repeater stations is 37. However, we need to decide the amount of repeaters distributing in one station.channel (the signaling channel between two points to transmit and receive signals) to transmit signals. Hence, we need 27 frequency channels [2] to maintain the normal communication.In order to avoid the interference about the close frequency between two repeaters, we arrange each repeater 10 frequency channels. We have121145.0145.03145.06145.09145.6145.63145.66145.69146.2146.23146.26146.29146.8146.83146.86146.89147.4147.43147.46147.49Mhz Mhz MHz MHz Mhz Mhz MHz MHz pl r Mhz Mhz MHz MHz Mhz Mhz MHz MHz Mhz Mhz MHz MHz r ⎧⎧⎪⎪⎪⎪⎪⎪⎨⎨⎪⎪⎪⎪⎩⎩（）233145.12145.15145.72145.75()()146.32146.35146.92146.95147.52147.55MHz MHzMHz MHz pl r pl MHz MHz MHz MHz MHz MHz ⎧⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ Here, n is the number of repeaters.In this method of distribution ，we ensure that the signal could still be recognized after transmission. We associate to each repeater a subaudible tone and the users need to use the same tone to receive the corresponding signal. We suppose each repeater station have the same repeaters attached with different subaudible tones. In this way, we guarantee the signals transmitting in this zone without interference. Because when one user sends a signal with a specific frequency, the repeater could send the signal after adding or subtracting 600 KHz. However, our frequency channels cover the whole scope of the frequency. Thus, the signal can be transmitted in this zone.Finally, we calculate the number of the repeaters in a repeater station and obtain the number is 3. Thus, the total number of the repeaters is 3*37111=.When the number of users in this zone increases to 10000, we consider the problem as the first model. In this situation, each repeater station should cover 10000/37270.3= users. Hence, we need 270 frequency channels to maintain the normal communication. Since the number of the channels is too large, it is wasteful to use 10 frequency channels for the first problem. Thus, we consider assigning each repeater station 30 channels. Furthermore, we get 9 repeaters. However, for the frequency rand ranging from145MHz to 148MHz, the channel changes to 11.1KHz, which leads to the channels interfering with each other. Hence, we make use of the CTCSS system to distribute the 9 repeaters different PL tones. We can build the repeaters which can transmit the same frequency and have different tones.11145145.03145.06145.09145.012145.015145.6145.63145.66145.69145.72145.75()146.2146.23146.26146.29146.32146.35146.8146.83146.86146.89146.92146.95147.4147.43147.46147.49147.52147.55r mhz pl ⎧⎪⎪⎪⎨⎪⎪⎪⎩1'1'145145.03145.06145.09145.012145.015145.6145.63145.66145.69145.72145.75()146.2146.23146.26146.29146.32146.35146.8146.83146.86146.89146.92146.95147.4147.43147.46147.49147.52147.55r mhz pl ⎧⎪⎪⎪⎨⎪⎪⎪⎩Thus, we calculate the number of the repeaters in a repeater station is 270/309=. Then the total number of the repeaters is 9*37333=.The model of the line-of-sight propagation considering the effect ofthe mountainsWe search some information on how to build the repeaters at the top of the mountains. According to the factors influencing the positions of the repeaters, we establish a model to simulate these impact factors of transmission of VHF radio spectrum.When repeaters are installed at the tops of the mountainous, the positions of the repeaters are related to the height of the antenna, its coverage radius, the repeater power and antenna gain. Thus, it is difficult to build the communication network. In order to build communication network well, we should do lots of experiments to ensure the positions of the repeaters according to actual geomorphic environment.Since mountains have different heights, we mainly consider three cases. Case 1 is that the heights of the mountains are 15m below, case 2 requires that the heights ranges from 15 to 30m and the last one is 30m above.The Egli modelThis model considers the height of the mountains below 15m. We assume that the mountains in this zone have no larger peaks, that is, this zone is a medium rolling terrain.This model is based on the data of the mobile communication, which is established by Federal Communications Commission (FCC). It is an empirical equation which is summarized from the data of the irregular terrain. This model based on the barrier height is applied to the VHF radio spectrum and the irregular terrain. It demands the barrier height above 15m. When the barrier height is under 15m ，we modify the model to verify the modified factor T C . The loss of the spectrum [1] equation is218820lg 40lg 20lg 20lg T LM F d h h C =++---.Here, we assume that d is the distance between the two antennas (m), h ∆is the height of thetopography. If we use b h to denote the practical height of the sending signal antenna, o h to denote the least effective height of the antenna and m h the practical height of the receiving signal antenna, then theeffective height of the sending signal antenna 1h satisfies1()2b o h h h m +=, and the effective height of the receiving signal antenna 2h satisfies2()2m o h h h m +=, 100-10-20-301020305070100200300500t h e m o d i f y i n g f a c t o r s K /d B /h mFigure 6[1]. the range of the modifying factor. We obtain the relationship between the height of the topography and the modifying factor from the empirical data. Furthermore, we get the equation with respect to h ∆and T C .C 1.6670.1094h25150T MHz F MHz =-∆<< C 2.250.1476h150162T MHz F MHz =-∆<< C 3.750.2461h 450470T MHz F MHz =-∆<<This model for irregular area is fit for the frequency ranging from 40 to 450MHz. When the frequency is higher than 25MHz or lower than 400MHz and the distance between two antennas is less than 64km, the error would be very small. Through the model we can evaluate the value of the wireless loss and the number of the repeaters.Figure 7 describes the positions of the mobile station, repeater and the barrier. Next, we introduce the concept of the clearance.Figure 7．The schematic of the clearanceT the position of the mobile stationR the position of the repeater1d the distance between the mobile station and the barrier2d the distance between the repeaters and the barrierAssume that the line HD is perpendicular to line RT, which is called clearance showed in Figure 7. Because the distance between the two antennas is very far, thus, the HD is short. Then we can substitute the hd for HC . If the radius of the first Fresnel region (the region is used to evaluate the transmission energy of the video spectrum.) is 1F , we regard 1/HC F as the relative clearance.The equation [2] of the radius of the first Fresnel region is12112d d F d d λ=+where λ is a parameter.When the radio spectrum transmits ，there are always many barriers such as constructions, trees and peaks blocking the spectrum. If the height of the barrier has not reached the first Fresnel zone ，the barrier would have little influence to the receiving frequency level. However, when it is in the zone, it will cause the added losses (the power losses of the sending power relative to the receiving power) to decrease the receiving electrical level. The diffraction losses /dB T h e d i f f r a c t i o n l o s s e s /d BFigure 7. The relationship between diffraction losses and clearance [1].The relationship between the added losses and the clearance caused by the barriers is showed in Figure 7. When the height of the barrier is under the line RT and the relative clearance is larger than 0.5，the added losses changes around 0db. In this situation，the practical receiving electrical level approaches the value of the space loss. We can get the value of the clearance HC is less than0.557F or a negative value. It may1hinder the transmission of direct wave. Thus, we should make the barriers lie below the line RT. Strengths●In the first model, we distribute each repeater 5 frequency channels, meanwhile the different repeatershave different PL tones. Thus, under the condition of avoiding the interference of repeaters with each other, we control the number of frequency channels least to make the transmission more efficient.●The model is established when the users are uniformly distributed. When the number of users increases,the number of repeaters increases. Thus, this model applies the zone where the users are unevenlydistributed.●The Egli model is a model considering the modifying factors, which make the mountains areas problembe easily understood.Weaknesses●In the signal’s coverage area of the repeaters, we assume that each channel only has one user. However,in the practical situation, there may not be one user. That is to say, we have wasted the channel.●Our model belongs to fixed channels distribution strategies, the larger number of the users, the largernumber of the channels. It leads to channel interference with each other when channel bandwidth is less than 8.3MHz. Thus, our model only suits for less number of users.●Considering the mountains environment is complex, in our model, we only consider one mountaineffecting the transmission of radio spectrum.References[1] Yao Dongping, Huang Qing and Zhao Hongli, Digital Microwave Communication, Beijing: Beijing Jiaotong University Press, 2004.7.[2] Theodore S. Rappaport, Wireless Communications: Principles and Practice, Second Edition, Prentice Hall PTR，2006.7[3] DeWitt H.Scott, Michael Krigline, Successful Writing for the Real World, Foreign Language Teaching and Research Press, 2009.2[4] /wiki/Repeater, 2011.2.12[5] /wiki/CTCSS, 2011.2.12[6] /view/2074265.htm,2012.2.14。

近几年美国大学生数学建模竞赛（USMCM）的题目包括：
2019年：建立一个模型来模拟东海和黄海的湍流。

2018年：预测联合国安理会和联合国大会决策结果及党派之间的关系。

2017年：建立一个模型来识别投资者风险偏好并帮助他们优化投资组合。

2016年：建立一个模型来识别用户a浏览网页时的行为特征，以便更好地理解和预测用户的行为。

2015年：建立一个模型，根据通信终端的传输速率，识别用户的实时视听传输需求。

2014年：建立一个模型来模拟社会文化传播的影响。

2013年：建立一个模型，根据用户的行为来预测新闻传播的趋势，并建议相关策略。

2012年：建立一个模型来优化公共汽车系统，以满足不同地区乘客的旅行需求。

2011年：建立一个模型，根据居民就医环境的不同，构建卫生保健系统的合理结构。

2010年：建立一个模型，预测印度洋及其邻近海域的风暴强度，以及其对当地的影响。

Ground PollutionBackgroundSeveral practically important but theoretically difficult mathematical problems pertain to the assessment of pollution. One such problem consists in deriving accurate estimates of the location and amount of pollutants seeping inaccessibly underground, and the location of their source, on the basis of very few measurements taken only around, but not necessarily directly in, the suspected polluted region.ExampleThe data set (an Excel file named procdata.xls) shows measurements of pollutants in underground water from 10 monitoring wells (MW) from 1990 to 1997. The units are micrograms per liter (μg/1). The location and elevation for eight wells are known and given in Table 1. The first two numbers are the coordinates of the location of the well on a Cartesian grid on a map. The third number is the altitude in feet above Mean Sea Level of the water level in the well.Table 1.Well Number x-Coordinate(ft) y-Coordinate(ft) Elevation(ft)MW-1 4187.5 6375.0 1482.23MW-3 9062.5 4375.0 1387.92MW-7 7625.0 5812.5 1400.19MW-9 9125.0 4000.0 1384.53MW-11 9062.5 5187.5 1394.26MW-12 9062.5 4562.5 1388.94MW-13 9062.5 5000.0 1394.25MW-14 4750.0 2562.5 1412.00The locations and elevations of the other two wells in the data set (MW-27 and MW-33) are not known. In the data set, you will also see the letter T, M, or B after the well number, indicating that the measurements were taken at the Top,. Middle, or Bottom of the aquifer in the well. Thus, MW-7B and MW-7M are from the same well, but from the bottom and from the middle.Also, other measurements indicate that water tends to flow toward well MW-9in this area.Problem OneBuild a mathematical model to determine whether any new pollution has begun during this time period in the area represented by the data set.If so, identify the new pollutants and estimate the location and time of their source.根据所给数据，建立一个数学模型，来判断在这段时间内这里是否有任何新的污染产生。

PROBLEM A: Snowboard Course.问题A：单板滑雪课程。

[修改]Determine the shape of a snowboard course (currently known as a “halfpipe”) to maximize the production of “vertical air”by a skilled snowboarder.确定一个滑雪课程形状（现为“半管”之称），以最大限度地利用熟练的滑雪板“垂直空气”的生产。

[修改]"Vertical air" is the maximum vertical distance above the edge of the halfpipe.“垂直的空气”是最大的半管以上的边缘的垂直距离。

[修改]Tailor the shape to optimize other possible requirements, such as maximum twist in the air.定制优化其他可能的要求，如空气中的最大扭曲，形状。

[修改]What tradeoffs may be required to develop a “practical” course?什么权衡可能需要制定一个“实用”的课程？[修改]PROBLEM B: Repeater Coordination.B题：直放站协调。

[修改]The VHF radio spectrum involves line-of-sight transmission and reception.甚高频无线电频谱涉及线路的视线传输和接收。

[修改]This limitation can be overcome by “repeaters,” which pick up weak signals, amplify them, and retransmit them on a different frequency.这种限制是可以克服“中继器”接收微弱信号，将其放大后频繁发送。

2011F=ma Contest25QUESTIONS-75MINUTESINSTRUCTIONSDO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN•Use g=10N/kg throughout this contest.•You may write in this booklet of questions.However,you will not receive any credit for anything written in this booklet.•Your answer to each question must be marked on the optical mark answer sheet.•Select the single answer that provides the best response to each question.Please be sure to use a No.2pencil and completelyfill the box corresponding to your choice.If you change an answer,the previous mark must be completely erased.•Correct answers will be awarded one point;incorrect answers will result in a deduction of14point.There isno penalty for leaving an answer blank.•A hand-held calculator may be used.Its memory must be cleared of data and programs.You may use only the basic functions found on a simple scientific calculator.Calculators may not be shared.Cell phones may not be used during the exam or while the exam papers are present.You may not use any tables,books,or collections of formulas.•This test contains25multiple choice questions.Your answer to each question must be marked on the optical mark answer sheet that accompanies the test.Only the boxes preceded by numbers1through25are to be used on the answer sheet.•All questions are equally weighted,but are not necessarily the same level of difficulty.•In order to maintain exam security,do not communicate any information about the questions (or their answers or solutions)on this contest until after February20,2011.•The question booklet and answer sheet will be collected at the end of this exam.You may not use scratch paper.DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN1.A cyclist travels at a constant speed of 22.0km/hr except for a 20minute stop.The cyclist’s average speed was 17.5km/hr.How far did the cyclist travel?(A)28.5km (B)30.3km (C)31.2km (D)36.5km(E)38.9kmQuestions 2to 4refer to the three graphs below which show velocity of three objects as a function of time.Each object is moving only in one dimension.246810−20+2+4v e l o c i t y (m /s )time (s)246810−20+2+4v e l o c i t y (m /s )time (s)246810−20+2+4v e l o c i t y (m /s )time (s)Object I Object II Object III2.Rank the magnitudes of the average acceleration during the ten second interval.(A)I >II >III (B)II >I >III (C)III >II >I (D)I >II =III (E)I =II =III3.Rank the magnitudes of the maximum velocity achieved during the ten second interval.(A)I >II >III (B)II >I >III (C)III >II >I (D)I >II =III (E)I =II =III4.Rank the magnitudes of the distance traveled during the ten second interval.(A)I >II >III (B)II >I >III (C)III >II >I (D)I =II >III (E)I =II =III5.A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed,at a distanceof150,000,000km from the Sun.Which of the following is the closest for the acceleration of the Earth in this orbit?(A)exactly0m/s2(B)0.006m/s2(C)0.6m/s2(D)6m/s2(E)10m/s26.A child is sliding out of control with velocity v c across a frozen lake.He runs head-on into another child,initiallyat rest,with3times the mass of thefirst child,who holds on so that the two now slide together.What is the velocity of the couple after the collision?(A)2v c(B)v c(C)v c/2(D)v c/3(E)v c/47.An ice skater can rotate about a vertical axis with an angular velocityω0by holding her arms straight out.Shecan then pull in her arms close to her body so that her angular velocity changes to2ω0,without the application of any external torque.What is the ratio of herfinal rotational kinetic energy to her initial rotational kinetic energy?(A)√2(B)2(C)2√2(D)4(E)88.When a block of wood with a weight of30N is completely submerged under water the buoyant force on the blockof wood from the water is50N.When the block is released itfloats at the surface.What fraction of the block will then be visible above the surface of the water when the block isfloating?(A)1/15(B)1/5(C)1/3(D)2/5(E)3/59.A spring has an equilibrium length of2.0meters and a spring constant of10newtons/meter.Alice is pulling onone end of the spring with a force of3.0newtons.Bob is pulling on the opposite end of the spring with a force of3.0newtons,in the opposite direction.What is the resulting length of the spring?(A)1.7m(B)2.0m(C)2.3m(D)2.6m(E)8.0m10.Which of the following changes will result in an increase in the period of a simple pendulum?(A)Decrease the length of the pendulum (B)Increase the mass of the pendulum(C)Increase the amplitude of the pendulum swing(D)Operate the pendulum in an elevator that is accelerating upward(E)Operate the pendulum in an elevator that is moving downward at constant speed.11.A large metal cylindrical cup floats in a rectangular tub half-filled with water.The tap is placed over the cup andturned on,releasing water at a constant rate.Eventually the cup sinks to the bottom and is completely submerged.Which of the following five graphs could represent the water level in the sink as a function of time?w a t e r l e v e ltime w a t e r l e v e ltime w a t e r l e v e ltime (A)(B)(C)w a t e r l e v e ltime w a t e r l e v e ltime (D)(E)12.You are given a large collection of identical heavy balls and lightweight rods.When two balls are placed at the endsof one rod and interact through their mutual gravitational attraction (as is shown on the left),the compressive force in the rod is F .Next,three balls and three rods are placed at the vertexes and edges of an equilateral triangle (as is shown on the right).What is the compressive force in each rod in the latter case?(A)1√3F (B)√32F(C)F(D)√3F (E)2F13.The apparatus in the diagram consists of a solid cylinder of radius 1cm attached at the center to two disks ofradius 2cm.It is placed on a surface where it can roll,but will not slip.A thread is wound around the central cylinder.When the thread is pulled at the angle θ=90◦to the horizontal (directly up),the apparatus rolls to the right.Which below is the largest value of θfor which it will not roll to the right when pulling on the thread?(A)θ=15◦(B)θ=30◦(C)θ=45◦(D)θ=60◦(E)None,the apparatus will always roll to the right14.You have5different strings with weights tied at various point,all hanging from the ceiling,and reaching down tothefloor.The string is released at the top,allowing the weights to fall.Which one will create a regular,uniform beating sound as the weights hit thefloor?(A)(B)(C)(D)(E)15.A vertical mass-spring oscillator is displaced2.0cm from equilibrium.The100g mass passes through the equilib-rium point with a speed of0.75m/s.What is the spring constant of the spring?(A)90N/m(B)100N/m(C)110N/m(D)140N/m(E)160N/mQuestions16and17refer to the information and diagram below. Jonathan is using a rope to lift a box with Beckyin it;the box is hanging offthe side of a bridge,Jonathan is on top.A rope is hooked up fromthe box and passes afixed railing;Jonathan holdsthe box up by pressing the rope against the rail-ing with a massless,frictionless physics textbook.The static friction coefficient between the rope andrailing isµs;the kinetic friction coefficient betweenthe rope and railing isµk<µs;the mass of the box and Becky combined is M;and the initial height of the bottom of the box above the ground is h. Assume a massless rope.BeckyLoose ropeFloorJonathan, pushesropefixed hardrailingon book against16.What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping?(A)Mg(B)µk Mg(C)µk Mg/µs(D)(µs+µk)Mg(E)Mg/µs17.Jonathan applies a normal force that is just enough to keep the rope from slipping.Becky makes a small jump,barely leaving contact with thefloor of the box.Upon landing on the box,the force of the impact causes the rope to start slipping from Jonathan’s hand.At what speed does the box smash into the ground?Assume Jonathan’s normal force does not change.(A)√2gH(µk/µs)(B)√2gH(1−µk/µs)(C)√2gHk s(D)√2gHk s(E)√2gH(µs−µk)18.A block of mass m=3.0kg slides down one ramp,and then up a second ramp.The coefficient of kinetic frictionbetween the block and each ramp isµk=0.40.The block begins at a height h1=1.0m above the horizontal.Both ramps are at a30◦incline above the horizontal.To what height above the horizontal does the block rise on the second ramp?(A)0.18m(B)0.52m(C)0.59m(D)0.69m(E)0.71mQuestions19and20refer to the following informationA particle of mass2.00kg moves under a force given byF=−(8.00N/m)(xˆi+yˆj)whereˆi andˆj are unit vectors in the x and y directions.The particle is placed at the origin with an initial velocity v=(3.00m/s)ˆi+(4.00m/s)ˆj.19.After how much time will the particlefirst return to the origin?(A)0.785s(B)1.26s(C)1.57s(D)2.00s(E)3.14s20.What is the maximum distance between the particle and the origin?(A)2.00m(B)2.50m(C)3.50m(D)5.00m(E)7.00m21.An engineer is given afixed volume V m of metal with which to construct a spherical pressure vessel.Interestingly,assuming the vessel has thin walls and is always pressurized to near its bursting point,the amount of gas the vessel can contain,n(measured in moles),does not depend on the radius r of the vessel;instead it depends only on V m (measured in m3),the temperature T(measured in K),the ideal gas constant R(measured in J/(K·mol)),and the tensile strength of the metalσ(measured in N/m2).Which of the following gives n in terms of these parameters?(A)n=23V mσRT(B)n=233√V mσRT(C)n=233√V mσ2 RT(D)n=233√V m2σRT(E)n=233V mσ2RT22.This graph depicts the torque output of a hypothetical gasoline engine as a function of rotation frequency.Theengine is incapable of running outside of the graphed range.IIIIIIEngine Revolutions per Minute0102030O u t p u t T o r q u e (N m )1,0002,000At what engine RPM (revolutions per minute)does the engine produce maximum power?(A)I(B)At some point between I and II (C)II(D)At some point between II and III (E)III23.A particle is launched from the surface of a uniform,stationary spherical planet at an angle to the vertical.Theparticle travels in the absence of air resistance and eventually falls back onto the planet.Spaceman Fred describes the path of the particle as a parabola using the laws of projectile motion.Spacewoman Kate recalls from Kepler’s laws that every bound orbit around a point mass is an ellipse (or circle),and that the gravitation due to a uniform sphere is identical to that of a point mass.Which of the following best explains the discrepancy?(A)Because the experiment takes place very close to the surface of the sphere,it is no longer valid to replacethe sphere with a point mass.(B)Because the particle strikes the ground,it is not in orbit of the planet and therefore can follow a non-elliptical path.(C)Kate disregarded the fact that motions around a point mass may also be parabolas or hyperbolas.(D)Kepler’s laws only hold in the limit of large orbits.(E)The path is an ellipse,but is very close to a parabola due to the short length of the flight relative to thedistance from the center of the planet.24.A turntable is supported on a Teflon ring of inner radius R and outer radius R+δ(δ R),as shown in the diagram.To rotate the turntable at a constant rate,power must be supplied to overcome friction.The manufacturer of the turntable wishes to reduce the power required without changing the rotation rate,the weight of the turntable,or the coefficient of friction of the Teflon surface.Engineers propose two solutions:increasing the width of the bearing (increasingδ),or increasing the radius(increasing R).What are the effects of these proposed changes?(A)Increasingδhas no significant effect on the required power;increasing R increases the required power.(B)Increasingδhas no significant effect on the required power;increasing R decreases the required power.(C)Increasingδincreases the required power;increasing R has no significant effect on the required power.(D)Increasingδdecreases the required power;increasing R has no significant effect on the required power.(E)Neither change has a significant effect on the required power.25.A hollow cylinder with a very thin wall(like a toilet paper tube)and a block are placed at rest at the top of aplane with inclinationθabove the horizontal.The cylinder rolls down the plane without slipping and the block slides down the plane;it is found that both objects reach the bottom of the plane simultaneously.What is the coefficient of kinetic friction between the block and the plane?(A)0tanθ(B)13tanθ(C)12(D)2tanθ3(E)tanθ。

2002年美国大学生数学建模竞赛题目2002 Mathematical Contest in Modeling （MCM）Problems问题A作者：Tjalling Ypma标题：风和喷水池在一个楼群环绕的宽阔的露天广场上，装饰喷泉把水喷向高空。

刮风的日子，风把水花从喷泉吹向过路行人。

喷泉射出的水流受到一个与风速计（用于测量风的速度和方向）相连的机械装置控制，前者安装在一幢邻近楼房的顶上。

这个控制的实际目标，是要为行人在赏心悦目的景象和淋水浸湿之间提供可以接受的平衡：风刮得越猛，水量和喷射高度就越低，从而较少的水花落在水池范围以外。

你的任务是设计一个算法，随着风力条件的变化，运用风速计给出的数据来调整由喷泉射出的水流。

Problem AAuthors: Tjalling YpmaTitle: Wind and WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area.Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.问题B作者：Bill Fox 和 Rich West标题：航空公司超员订票你备好行装准备去旅行，访问New York城的一位挚友。

2011 National English Contest for College Student( Level C —Preliminary)Suggested AnswersPart I Listening Comprehension (30 marks)Section A1. C2. A3. C4. B5. ASection B (5 marks)6. A7. B8. C9. B 10. A 11. C 12. C 13. A 14. C 15. BSection C (5 marks)16. A 17. B 18. C 19. A 20. CSection D (10 marks)21. dilemma 22. candidates 23. institution of first choice 24. weak economy 25. undergraduate 26. degree course 27. studying abroad 28. apprenticeships29. practical work experience 30. job prospects II VocabularyPartⅡ Vocabulary and Structure (15 marks)31. A 32. B 33. C 34. B 35. C 36. D 37. C 38. B 39. A 40. B 41. C 42. A 43. C 44. D 45. BPart III Cloze (15 marks)46. complaints 47. remain 48. where 49. resemble 50. wretched 51. difference52. assumption 53. created 54. out 55. finally 56. lending 57. borrowing58. collaborative59. shared 60. publishersPart IV Reading Comprehension (40 marks)Section A (10 marks)61. False 62. False 63. True 64. False 65. FalseSection B (10 marks)66. Standing up on a surfboard requires good balance.67. In the Pacific Islands.68. In the 1950s and 60s.69. To help their feet stick to the board.70. In the southwest of England.Section C (10 marks)71. average income; life expectancy; level of education72. better health and education systems than others73. The financial crisis74. climate change75. economic growth; greenhouse gas emissionsSection D (10 marks)76. A77. B78. D79. 我认为，父亲常常低估了他们十几岁的孩子，而且还忘记了他们自己小时候博得感受。

2011 ICM C题翻译从环保和经济上说，电动汽车究竟有多少优势？它们的普及是可行和实用的吗？这里有一些问题需要考虑，但是，当然还有更多问题，你将无法在模型中考虑所有的问题：•电动汽车的广泛使用会节约化石燃料吗？或者实际上仅仅是给了化石燃料另一个交易使用的方式，电力目前主要是燃烧化石燃料产生的。

要想通过使用电动汽车来最多地节约能源，需要什么条件到位才可以？•在二十一世纪要想广泛使用可行和有利于环境的电动汽车，需要多少的电力的替代品，如风力和太阳能。

评估是否这些替代电力来源的增长是可能和可行的。

在非高峰时间给电池充电是否是有益的？能增加广泛使用电动汽车的可行性吗？电池需要以多快的速度充电才能最大限度地提高电动车的效率和的实用性？在环境节约和广泛使用电动车辆的可行性方面，这些地区还需要做多大程度的改进？•什么样的基础交通工具的方法是最有效的？不同方法的有效性是基于使用它的国家或地区的情况吗？•直接由电动汽车造成的污染是很低的，但是与电动汽车相联系的有隐性污染源吗？汽油和柴油在引擎内部燃烧产生含亚硝酸盐的氧气，车辆产生一氧化碳和二氧化碳污染，但这些双向产品是我们真正应该担心的吗？在气候和我们的健康方面，什么是这些物质的短期和长期的影响？•日益增加的需要处理的大容量电池所造成的污染究竟有多大？比较一下电动汽车对于环境的影响和化石燃料燃烧的车辆的影响。

•你还应该考虑诸如经济和人口问题，诸如电动车辆的便利性。

电池可以足够快速地充电或更换吗？以便能满足大多数运输的需要或者它们的使用范围受到限制吗？在交通运输中，电动汽车只有有限的作用吗？只对于短途运输（短途客运或轻型车辆）有作用吗？或他们实际上可以用于高负荷、长途的交通运输？政府应当给予补贴来发展电动车技术吗？如果需要，为什么？需要补贴多少，以何种形式？要求•建立电动汽车广泛使用对于环境，社会，经济和健康影响的模型，并且详细阐述政府和电动汽车生产商是否应该支持电动汽车的广泛使用，如果支持的话，应该考虑的关键点。

中继站的协调方案摘要（Abstract ）中继站是将信号进行再生、放大处理后，再转发给下一个中继站，以确保传输信号的质量。

低功耗的用户，例如移动电话用户，在不能直接与其他用户联系的地方可以通过中继站来保持联系。

然而,中继站之间会互相影响,除非彼此之间有足够远的距离或通过充分分离的频率来传送。

为了排除信号间的干扰，实现某一区域内（题中以40英里为半径的圆形区域）通信设备正常的发射和接收信号，需要利用PL 技术对中继站作合理的协调和分配。

首先本文结合香农理论的相关算法，考虑了信号供给系统的损耗、天线增益、信号的传播损耗、辐射效率因素的影响，得到中继站的辐射范围半径公式为：,10,10log ()37.23282010r outr inp P d -=在供给对象为低功率消耗设备，查资料一般发射功率为3.2W ，中继站能接收到的最弱的信号1W μ,代入数据得到每个中继站的辐射半径为15.28m iles 。

同时本文在不考虑其他因素（包括：地形、大雾、山川、建筑物等）对辐射范围和辐射强度的影响下，结合相关知识和题目中给出的条件，在不引入PL 技术时得出每个中继站所服务的用户数量为39个。

对于问题一， 我们首先定义了均衡覆盖、覆盖效率，在均衡覆盖中即用圆覆盖圆形区域，我们根据式子2(2)n k n ππ-=，得出(,)k n 的可能值有(3,6),(4,4),三种，即等效三角形、正方形、正六边形覆盖，并通过覆盖效率的比较，最终得出正六边形覆盖是最好的覆盖方法，即蜂窝拓扑网络。

在这种覆盖情况下我们，我结合中继站覆盖半径15.28m iles ，根据式子m i n 3(1)1,0,1,2,3,N K K K =++=……，求出最少需要19个中继站，并在满足单位面积覆盖同时在线人数的情况下引入PL 技术，得出此时中继站在该区域可同时服务在限人数为1292人。

对于问题二，我们在问题一模型基础上从提高中继站服务人数和减少中继站半径两方面考虑，得出在将PL 分为18层，即中继站同时在线服务人数为702的情况下，结合单位面积同时在线服务人数，得出在中继站最少的情况下，中继站半径在[]11.094，,11.68范围内都可，我们为了让同时在线服务人数最大，取11.094英里，得出服务人数为11305。

11A 设计单板滑雪场摘要本文研究的是单板滑雪轨道的设计问题。

首先，我们考虑使运动员的腾空 高度尽量达到最大，建立以腾空高度为目标函数的模型， 针对问题一， 要使滑雪者的垂直距离最长，通过对滑雪者运动过程的分析， 我们选取滑雪者运动的一个周期进行研究， P 平台通过 F 平台再到到 P 平台的 从 另一端，即从 A—B—C—D(如图 2)。

首先， 结合给定的滑雪路线 A—B—C—D 运用物理学知识对滑雪过程进行分 析，建立以最大腾空高度为目标函数，滑雪道曲率半径 r 、倾斜角  、底部平台 宽度 d 以及运动员出槽时与槽边缘的夹角  为自变量的微分方程， 根据搜索相关 资料得知，滑雪道长度、曲率半径、倾斜角及底部平台宽度的设计都有一定的参 考范围限制， 我们结合 U 型单板滑雪道设计数据。

其次， 对于滑雪道长度的求解， 我们是根据运动员的平均腾空次数及每次腾空时所行距离 l 求解得出，而腾空 次数的确定是取 5 次腾空为标准。

关键词： 受力分析 能量守恒 微分方程一、问题重述单板滑雪场地主要由 Flat 平台、过渡区、垂直区、Platform 平台、入口坡组 成。

运动员的滑雪技巧、身体素质，滑雪场地的坡度、深度、宽度、长度等成为 影响运动员成绩的诸多因素。

在单板滑雪比赛中，当滑雪运动员最大限度地产生 垂直腾空后就能够做出各种动作，那么应该如何考虑哪些权衡因素，从而设计出 比较优化的单板滑雪场地。

确定一个滑雪场的形状，使得滑雪选手垂直腾空高度最大化。

“垂直腾空” 最大化就是指最大的垂直腾空距离在半管边缘以上的距离。

定制形状时要优化其他可能的要求， 如空气中的最大扭曲等。

综合各种条件， 选择最优的滑雪场模型。

初步观察得到U型管道的合理设计有利于运动员水平的发挥，因此我们从设 计U型槽的坡度，弧长，场地长度等角度入手。

需要建立数学模型解决的问题： （1）设计出滑雪道的形状（现在一般为半圆柱管内侧面） ，以使得滑雪者的垂 直距离（滞空时间）最长。

A B2018多跳HF无线电传播语言传播趋势2017管理赞比西河高速路收费合并2016浴缸的水温模型解决空间碎片问题2015根除病毒寻找失踪的飞机2014（交通流、路况）优化（体育教练）综合评价2013平底锅受热，热力学、几何（大模型解答所有题目）可利用淡水资源的匮乏，（水资源）预测、最优化2012一棵树的叶子沿着BigLongRiver野营，（流程）优化2011单板滑雪场地，（物理模型）多目标规划中继站的协调，（信号传输中继站）目标位置确定2010力学（棒球棒最佳击球点）（系列犯罪地理效应）预测模型2009（环岛交通流）优化（能源和手机）多目标规划、预测2008（北极冰融化影响）预测算法设计2007（选取划分）多目标规划（登机时间）优化、多目标规划2006（灌溉喷头位置确定）优化，多目标规划（机场轮椅调度）优化、（预算）预测2005（洪流量）预测（收费亭数量和位置确定）目标规划2004（指纹识别）模型、模型比较（时间调度）优化2003（跳伞）目标规划多目标规划2002(算法设计)预测模型（飞机Overbooking）优化2001（自行车轮胎）优化，多目标规划对模型进行分析和优化2000空间几何，模型分析图论1999（行星碰撞南极后果）预测（房屋最大人数）多目标规划1998核磁共振算法设计成绩排名算法设计C D E F能源配置与预测从汽油驾驶到E（电）驾驶气候变化如何影响区域不稳定？隐私成本问题车辆合作与导航在机场安检站优化乘客吞吐量规划可持续城市迁移到火星：2100城市社会的乌托邦劳动力优质基金挑战。

Team #9262Perfect Half-pipe: The Think ofSnowboard CourseAbstractWith the continuous progress and development, People are actively involved in sports and exploring in it continually. Skiing is popular with the majority of sports fans gradually under this condition. Especially，Snowboarding with good view, challenge and the basis of the masses develops rapidly and has become a major Olympic projects. In this paper, how to design and optimize the snowboard course of half pipe is discussed in detail. We strive to get the perfect course so that snowboarders can achieve the best motion state in the established physical conditions. What’s more, it may promote the development of the sport.This problem can be divided into three modules to discuss and solve. For the first problem of the design of half pipe, it can be based on the point of the energy conservation law. The method of functional analysis (Variation principle and Euler differential equation) is used to set up equations, when the secondary cause is ignored and the boundary conditions are taken into consideration. The curve equation is obtained by the above equation, that is, a skilled snowboarder can make the maximum production of “vertical air”.For the second question, athletes’ maximum twist in the air and some other factors need to be taken into account when to optimize the previous model, so that curve can meet the actual game conditions and appraisal requirements as much as possible. Ultimately, a satisfying curve will be got. The third problem is a problem relatively close contact with the actual, which is to setting down a series of tradeoffs that may be required to develop a “practical” course. In this paper, for the formulation of these factors, the main discussions are the thickness of snow on half pipe and the aspect of economy for the construction.After discussing these three aspects, the paper finally summarizes a construction program and evaluation criteria of the course in current conditions. Finally, by evaluating the advantages and disadvantages of the whole model，we put forward the advanced nature of the model, but also point out some limitations of the model.Key words: Snowboard course, Half-pipe,Functional, Euler equation, Fitted curve,Numerical differentiationTable of contentsTable of contents (1)I. Introduction (2)1.1 Half pipe structure (2)1.2 Background problem (3)1.3 Athletes aerials (3)1.4 A ssume (3)II. Models (5)2.1 problem one (5)2.2 P roblem two (11)2.3 Problem three (13)III. Conclusions (14)I V. Future Work (15)V. Model evaluation (15)5.1 Model Advantages (15)5.2 Model disadvantage s (16)VI. References (17)I. IntroductionSnowboarding is a popular pool game with the world of sports. The U-Snowboard’ length is generally 100-140m , U-type with a width of 14-18m,U-type Depth of 3-4.5m.the slope is 14°-18 °. In competition U-athletes Skate within the taxi ramp edge making the use of slide to do all sorts of spins and jumps action. The referees score according to the athletes’ performance as the Vertical air and the difficulty and effectiveness of action. The actions Consist mainly of the leaping grab the board, leaping catch of non-board , rotating leaping upside down and so on.1.1 Half pipe structureHalf pipe structure contains: steel body frame, slide board, steps to help slide and rails.1.2 Background problemIn order to improve the movement of the watch, it can be improved from two aspects: orbit and the athletes themselves. Now according to the problem the orbit can be designed as a curve. on the curve the athletes can get a maximum speed. The design of orbit includes a wide range of content, such as the shape of U-groove design, track gradient, width and length designed to help the design of sliding section, and so on. The rational design of half pipe can be achieved to transform the energy to efficiency power, make the athletes achieve the best performance in the initial state of the air. This paper discusses the rational design of half pipe to these issues.1.3 Athletes aerialsAthletes on the hillside covering with thick snow skill down with the inertia of the platform, jump into the air, and complete a variety of twists or somersault. Rating criteria: vacated, takeoff, height and distance accounted for 20%; body posture and the level of skill accounted for50%; landing 30%. According to the provisions the difficulty of movements are ranged into small, medium and large. The athletes option the actions. However, the ground must have a slope of about 37 ° and 60 cm above the soft snow layer.1.4 AssumeIn order to simplify the model and can come to a feasible solution,making the following assumptions:1• the shape of a snowboard course has a lowest point, the wide and the length of the snowboard course.2• air resistance can be negligible.3• it is assumed that the athletes themselve s have no influence.II. The Description of the ProblemThis problem is a typical engineering design, involving a lot of disciplines, such as advanced mathematics, engineering mathematics, mechanical dynamics and biological dynamics, as well as the relevant provisions of sports competition and judging standards, and so on. According to the requirement of the problem, determine the shape of a snowboard course to maximize the production of “vertical air” by a skilled snowboarder. For this problem, we can use the boundary conditions and site properties (e.g. symmetry) and other requirements to establish functional combining with the variation principle Euler equations. The original equation can be changed into a functional extremum problem.Secondly, we optimize the model boundary and determine the appropriate snowboard course’s slope toe to make the athletes perform maximum twist or do more difficult action.Finally a practical model should meet the requirements of safety, sustainability and economic. According to the high degree of humansecurity, the source of the snow and the topography the model will be optimized more reasonable.II. Models2.1 problem oneAs shown （3.1）A is the lowest point of the snowboard course.From A to B we want to find a curve to make the athletes get the maximum vertical distance above the edge of the snowboard course.Figure 1 Half-pipeSet A as origin of coordinate.Awing of conservation energy and neglecting air resistance, the mathematical function is,f A mgh mv W mv ++=+2202121 (1)Wherev 0 is the initial velocity （m/h ）,v is the velocity towards destination （m/h ）,m is the mass of an athlete (kg),w is the energy which is made by the athlete (J),h is the Vertical height (m),Af is the friction work （J ），According to mechanical analysis ：rv m mg N 2cos =-θ （2）Where θ is the angle the angle between the tangent and the horizontal line, N is the pressure on the object,r is the radius of curvature,According to friction formula ：⎪⎪⎭⎫ ⎝⎛+==r v m mg N f 2cos θμμ （3） To (3) into equation (1), combined with calculus ：⎰⎪⎪⎭⎫ ⎝⎛+++=+s t dl r v m mg mgh mv W mv 02220cos 2121θμ （4） Friction acting A f ：⎰⎰+=⎪⎪⎭⎫ ⎝⎛+=st st f dl r v m mgB dl r v m mg A 02022cos μμθμ （5）Use higher mathematics ：()'''1232y y r += （6）dx y dl 2'1+= （7）According to the nature of the curve, the speed can be assumed to satisfy this expression ：kx e v v 10= （8）Put all these formulas in order and suppose the expression for the functional ：dx y e y mv dl r v m B kx s t ⎰⎰+==∏20222002'1''μμ （9） Set 2220'1''y e y mv F kx +=μ （10）Reference Euler equation ：0'''22=⎪⎪⎭⎫ ⎝⎛∂∂+⎪⎪⎭⎫ ⎝⎛∂∂-∂∂y F dx d y F dx d y F （11） Obtained ：0=∂∂yF （12）()232220'1''''y e y y mv y F kx +-=∂∂μ （13）()212220'1''y e mv y F kx +=∂∂μ （14） To （12）-（14）into equation （11），Obtained ：()()⎪⎪⎪⎭⎫ ⎝⎛+=⎪⎪⎪⎭⎫⎝⎛+-21222022232220'1'1'''y e mv dx d y e y y mv dx d kx kx μμ（15） Integrate it ：()()C y e mv dx d y e y y mv kx kx +⎪⎪⎪⎭⎫⎝⎛+=+-212220232220'1'1'''μμ（16） Simplified ：()()()C y e y ymv y e mv k y e y y mv kx kx kx ++++=+252220232220232220'1'''3'12'1'''μμμ（17） ()'2''12''32y y y k y -+=（18） Suppose ： ()y p dx dy=So ： dy dpp dx dy dy dp dx y d =⋅=22Substituted into the above equation ：()p p p k dy dp p 21232-+= （19）()kdy p dp p p212224=+- （20） Integrate it ： ()⎰⎰=+-kdy p dp p p 212224 （21） Obtained the final results ：02arctan 33313=+-+-c ky p p p （22） For the difficult equation, we obtain numerical solutions by numerical differentiation, and then obtained function equation by numerical fitting method ：Discrete interval [0,8],Wheretake steps ：h = 1.Each point xi, i = 0,1, …… 8.Every interval [x i , x i +1],the boundary conditions ： y (0) = 0, y '(0) = 0.Into the formula （22） for the boundary conditions ：C = 0Put h y y y i i -=+1' into formula （22）：02arctan 33311131=-⎪⎭⎫ ⎝⎛-+⎪⎭⎫ ⎝⎛--⎪⎭⎫ ⎝⎛-+++i i i i i i i ky h y y h y y h y yNumerical Solution of each point is obtained in turn：x 0 1 2 3 4 5 6 7 8 y 0 0.0069 0.0094 0.0336 0.1329 0.3669 0.7872 1.4127 4.0382 Functions images and function equation are obtained by numericalfitting on Excel：Figure 2 The results of the numerical solution of the fitting imageAfter fitting the equation：y = 0.0003x6 - 0.0063x5 + 0.0435x4 - 0.1289x3 + 0.1594x2 - 0.0571x -0.0008 （23）Then the entire image can be got by symmetry along the y-axis. This models of problem one can be solved.2.2 Problem twoOn question 2, its main purpose is to improve the model in problemone under the condition of meeting the requirements of other possiblecases. Analyze other possible requirements which include a number ofaspects, such as the maximum twist in the air, players’ safety when they leave the ground and the stability of athletes when they land. Among them, we mainly consider the maximum twist of snowboarder in the air.When players leave the ground, they are only affected by gravity and air resistance. We ignore the players’ adjust ment in the air. After the project flying out of the ground, in order to analyzing simply and thinking clearly, the velocity of the object is divided into lines velocity and angular velocity. Velocity contains components of three different directions: horizontal, vertical and longitudinal. Angular velocity consists of somersault angular velocity and twist angular velocity.Figure 3 Flip velocity analysisAfter athletes flying out of the course, the velocity of longitudinal depends on a rational allocation of their own energy when they ski, so the design of course can not be considered. Vertical speed determines the maximum height with which athletes fly out of the course. So it is therequirement for design of the maximum “vertical air ”. For the horizontal velocity, players’ reaction force when they leave the course should be taken into account. And the horizontal velocity generated by reaction force must satisfy the equation below.x V V ≥' （24）Thus it can ensure that athletes fall back to ground safely after flying out, as the same time, it also meet appreciation, technical and safety requirements. On the problem that athletes reverse in the air, Conservation of energy can be used in the cross section.22222222112121212121212121y x af p y x mv mv w J w J A W m mv ++++=++（25）WhereV is the velocity of each state,Wf is the effective bio-energy an athlete release,J is the moment of inertia under different rotations,Aaf is the energy dissipated by air resistance.By checking the literature, moment of inertia J1 is 1.1(2m kg ⋅) and J2 is84.3(2m kg ⋅ ). Combining with the known data, we get the relationshipof w1, w2,y V . According to the value of V , the relationship of x V and y V will be got. The boundary angle is α.Fromtan∠α= y V/x V∠α=83.30So the boundary angle is 83.302.3 Problem threeIn practice, there are many factors to consider, for example, the thickness of snow covering and the construction of the economy, in addition to shape. The topography should be made the best use of to save project cost. Climate also is a constraint. Snow can be smoother and be used longer when the weather is cold.III. ConclusionsThe basis of this model is snowboarding skilled players can generate the maximum vertical air. Awing of numsolve and fitted the mathematical function is,y = 0.0003x6- 0.0063x5+ 0.0435x4- 0.1289x3+ 0.1594x2- 0.0571x -0.0008h=4(m) x0=8(m)∠α=83.30slope angle ∠θ= 180 （International recommended values）Figure 4 Half-pipeThe ultimate resolution of model takes various factors into account. The model can be applied to other similar improvements similar problems, such as the design of emergency chute.IV. Future WorkAlthough this paper considered a wide variety, but only one purpose getting the best track shape. However, in the actual construction process the aims to be achieved are complex and the design aspects are various. If you want to continue the track design, the following areas to be discussed，1. The run-up route’s height and inclination.2. Design of the best athletes’ running track. In the process you need to consider artistic, challenging and security.3. Design the length of the orbit to make athletes can efficiently complete the 5-8 vacated performances.V. Model evaluation5.1 Model Advantages(1). this model is infusion and the result is intuitive.(2). this paper has Strong theory with calculating the best shape theory.(3). This Problem is close to the real life situation, because of considered comprehensive.5.2 Model DisadvantagesSolving the model is complicated and some factors only have the qualitative analysis and not quantitative discussion.VI. References[1] Jason W. Harding , Kristine Toohey, David T. Martin1, Allan G. Hahn, Daniel A. James . 6/2008. TECHNOLOGY AND HALF-PIPE SNOWBOARD COMPETITION –INSIGHT FROM ELITE-LEVEL JUDGES. ISEA.[2] Wu Wei,Xia Xiujun. 2006. Half-pipe snow-board skiing skill training field in summer Explore and Design. China.[3]Xiao Ningning,Gao Jun.2009. Research of the Technical Characteristics of Half-pipe Snowboarding.China.[4] Building A Zaugg Half-Pipe.America. /resort/pipegroomers/pipe.shtml[5] Olympic Half Pipe Snowboarding ./way_5150384_olympic-half-pipe-snowboardi ng-rules.html[6]The Physics Of Snowboarding./physics-of-snowboard ing.html。

MCM 2013 A题：最佳巧克力蛋糕烤盘当你使用一个矩形的烤盘烘烤食物时，热量会集中在烤盘的四个角落，于是角落处的食物就会被烤糊（烤盘边缘处也有类似情形，但程度轻一些）。

当使用一个圆形烤盘时，热量会均匀地分布在整个边缘上，就不会再有边缘上烤糊的现象发生。

然而，由于大多数烤箱内部是矩形的，如果使用圆形烤盘，就不能充分利用烤箱的内部空间了。

建立一个模型，来描述热量在不同形状的烤盘表面的分布。

这些形状包括矩形、圆形以及两者之间的过渡形状。

假设，1、矩形烤箱的宽长比为 W/L。

2、每个烤盘的面积为A。

3、先考虑烤箱内有两个搁架且间隔均匀的情形。

建立一个模型用以选择满足下列条件的最佳烤盘的形状：(1)、使得烤箱中可以容纳的烤盘数量(N)最大。

(2)、使得烤盘上的热量分布(H)最均匀。

3、综合(1)、(2)两个条件，并且为(1)、(2)分别设置权值p和(1-p)，寻求最优。

然后描述结果随着 W/L 和 p 的值的变化是如何变化的。

除了撰写 MCM 论文之外，你还要为新的一期巧克力蛋糕美食杂志准备一个一至两页的广告，阐述你的设计和结果的亮点所在。

MCM 2013 B题：水，水，无处不在淡水资源是世界上许多地方持续发展的限制因素。

建立数学模型来确定一个有效的，可行的，低成本的2013年用水计划，来满足某国(从下方的列表中选择一个国家)未来(2025年)的用水需求，并确定最优的淡水分配计划。

特别的，你的数学模型必须包括储存、运输、淡化和节水等环节。

如果可能的话，用你的模型来讨论你的计划对经济，自然和环境的影响。

提供一个非技术性的意见书给政府领导概述你的方法，以及方法的可行性和成本，以及它为什么是“最好的用水计划的选择”。

国家：美国、中国、俄罗斯、埃及或者沙特阿拉伯。

ICM 2013 C题：地球健康的网络建模背景：全社会都在关注如何研究与应用模型来预测我们地球的生物和环境的健康状况。

许多科学研究表明地球的环境和生物系统所面对的压力正在增加,但是能够验证这一观点的全局性模型却很少。

2011年美国竞赛第2次模拟题Problem A: Big Brother Is Watching YouThe year is 2084 and the security situation is as grim as ever:An overabundance of “jaywalkers” endangers the safety of law-abiding drivers throughout Gotham City. The municipal government is finally ready to deal with this problem decisively and hires you to design a surveillance plan for the entire borough of Manhattan.The Mayor is adamant that all surveillance should be conducted by Micro Unmanned Aerial Vehicles (MAVs) alone. City Hall has a bin ding contract with “Batman & Robin Unlimited”, a conglomerate manufacturing overpriced and antiquated quadrotor mini-helicopters, similar to those that emerged 73 years earlier. But unlike the mini-helicopter drones of 2011, the current MAVs are relatively robust both indoors & outdoors, can fly up to 5 hours without need to recharge or refuel, and require no human being to monitor each of them – instead, a sophisticated computerized controller can be programmed to follow any patrol strategy of your choice.The Mayor actually needs four (4) different plans for deploying these MAVs, but he’ll be grateful for anything you can prepare on such a short notice.Gotta watch them: no geographic point in the city should remain unobserved from the air for more than 15 minutes in a row. How many MAVs will you need to guarantee this?Plan for contingencies: Note that any flight plan involving frequent sharp turns will require more frequent recharging/refueling stops. Moreover, these outdated MAVs are not very reliable and a significant proportion of them might be briefly grounded for repairs, interrupting their patrol activities throughout the day. Despite this, all areas of the city should remain regularly observed (even if somewhat less often). An ideal plan will accomplish this even without reprogramming the remaining drones. What kind of surveillance coverage will your plan provide if 30% of yourdrones become unusable?All areas are equal, but some are more equal than the others: Some parts of the city have a higher density of jaywalkers; e.g., the neighborhood of Gotham University and the financial district are particularly dangerous for drivers. Such areas should be observed at least once in each 5 minute interval. On the other hand, Gotham Central Park has only a few roads passing through it, and there is no need to observe it more than once in 20 minutes. How many MAVs will you need to provide the requested variable level of coverage?Everyone is equal in the eyes of the drones: The troublemakers from Gotham Jaywalking Liberties Union complain that you & all other municipal employees involved in programming the drones have an unfair advantage: even if you don’t know the current position of all MAVs, your insider knowledge (of the surveillance strategy/schedule & the current number of operational drones) may be used to craft a significantly less drone-observable jaywalking path through the city. If necessary, modify your plan to assuage these fears. How many drones will you need to provide a comparable surveillance coverage under the new plan?Problem B: Who Moved the Presidential Candidate's Cheese?New media is rapidly becoming an important part of a presidential candidate's media strategy. The use of the new media by most of the candidates had been unprecedentedly frequent and overwhelming in 2008 American president election cycle. It is most believed that U.S President Barack Obama, who had massively used new media as one of his campaign strategies, not only won the election, but also drawn the world's attention and gained himself the reputation as a "Internet President".Requirement 1: Model the effectiveness of the New media for president election. Requirement 2:Verify your model.Requirement 3: Suggestion for the presidential candidate.2010 Contest MCM PROBLEMSPROBLEM 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 mightseem 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 (hol lowing 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 based 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 law enforcement 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 anappropriate 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.。

2010Semifinal Exam1Semifinal ExamDO NOT DISTRIBUTE THIS PAGEImportant Instructions for the Exam Supervisor •This examination consists of two parts.•Part A has four questions and is allowed90minutes.•Part B has two questions and is allowed90minutes.•Thefirst page that follows is a cover sheet.Examinees may keep the cover sheet for both parts of the exam.•The parts are then identified by the center header on each page.Examinees are only allowed to do one part at a time,and may not work on other parts,even if they have time remaining.•Allow90minutes to complete Part A.Do not let students look at Part B.Collect the answers to Part A before allowing the examinee to begin Part B.Examinees are allowed a10to15 minutes break between parts A and B.•Allow90minutes to complete Part B.Do not let students go back to Part A.•Ideally the test supervisor will divide the question paper into3parts:the cover sheet(page2),Part A(pages3-4),and Part B(pages6-7).Examinees should be provided parts A andB individually,although they may keep the cover sheet.•The supervisor must collect all examination questions,including the cover sheet,at the end of the exam,as well as any scratch paper used by the examinees.Examinees may not take the exam questions.The examination questions may be returned to the students after March 31,2010.•Examinees are allowed calculators,but they may not use symbolic math,programming,or graphic features of these calculators.Calculators may not be shared and their memory must be cleared of data and programs.Cell phones,PDA’s or cameras may not be used during the exam or while the exam papers are present.Examinees may not use any tables,books, or collections of formulas.2010Semifinal Exam Cover Sheet2Semifinal ExamINSTRUCTIONSDO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN •Work Part Afirst.You have90minutes to complete all four problems.Each question is worth25points.Do not look at Part B during this time.•After you have completed Part A you may take a break.•Then work Part B.You have90minutes to complete both problems.Each question is worth 50points.Do not look at Part A during this time.•Show all your work.Partial credit will be given.Do not write on the back of any page.Do not write anything that you wish graded on the question sheets.•Start each question on a new sheet of paper.Put your school ID number,your name,the question number and the page number/total pages for this problem,in the upper right hand corner of each page.For example,School ID#Doe,JamieA1-1/3•A hand-held calculator may be used.Its memory must be cleared of data and programs.You may use only the basic functions found on a simple scientific calculator.Calculators may not be shared.Cell phones,PDA’s or cameras may not be used during the exam or while the exam papers are present.You may not use any tables,books,or collections of formulas.•Questions with the same point value are not necessarily of the same difficulty.•In order to maintain exam security,do not communicate any information about the questions(or their answers/solutions)on this contest until after March31, 2010.Possibly Useful Information.You may use this sheet for both parts of the exam.g=9.8N/kg G=6.67×10−11N·m2/kg2k=1/4π 0=8.99×109N·m2/C2k m=µ0/4π=10−7T·m/Ac=3.00×108m/s k B=1.38×10−23J/KN A=6.02×1023(mol)−1R=N A k B=8.31J/(mol·K)σ=5.67×10−8J/(s·m2·K4)e=1.602×10−19C1eV=1.602×10−19J h=6.63×10−34J·s=4.14×10−15eV·sm e=9.109×10−31kg=0.511MeV/c2(1+x)n≈1+nx for|x| 1sinθ≈θ−16θ3for|θ| 1cosθ≈1−12θ2for|θ| 1Part AQuestion A1An object of mass m is sitting at the northernmost edge of a stationary merry-go-round of radius R.The merry-go-round begins rotating clockwise(as seen from above)with constant angular acceleration ofα.The coefficient of static friction between the object and the merry-go-round is µs.a.Derive an expression for the magnitude of the object’s velocity at the instant when it slidesoffthe merry-go-round in terms ofµs,R,α,and any necessary fundamental constants.b.For this problem assume thatµs=0.5,α=0.2rad/s2,and R=4m.At what angle,asmeasured clockwise from north,is the direction of the object’s velocity at the instant when it slides offthe merry-go-round?Report your answer to the nearest degree in the range0to 360◦.Question A2A spherical shell of inner radius a and outer radius b is made of a material of resistivityρand negligible dielectric activity.A single point charge q0is located at the center of the shell.At time t=0all of the material of the shell is electrically neutral,including both the inner and outer surfaces.What is the total charge on the outer surface of the shell as a function of time for t>0? Ignore any effects due to magnetism or radiation;do not assume that b−a is small.Question A3A cylindrical pipe contains a movable piston that traps2.00mols of air.Originally,the air is at one atmosphere of pressure,a volume V0,and at a temperature of T0=298K.First(process A)the air in the cylinder is compressed at constant temperature to a volume of14V0.Then(processB)the air is allowed to expand adiabatically to a volume of V=15.0L.After this(process C) this piston is withdrawn allowing the gas to expand to the original volume V0while maintaining a constant temperature.Finally(process D)while maintaining afixed volume,the gas is allowed to return to the original temperature T0.Assume air is a diatomic ideal gas,no airflows into,or out of,the pipe at any time,and that the temperature outside the remains constant always.Possiblyuseful information:C p=72R,C v=52R,1atm=1.01×105Pa.a.Draw a P-V diagram of the whole process.b.How much work is done on the trapped air during process A?c.What is the temperature of the air at the end of process B?Question A4The energy radiated by the Sun is generated primarily by the fusion of hydrogen into helium-4.In stars the size of the Sun,the primary mechanism by which fusion takes place is the proton-proton chain.The chain begins with the following reactions:2p→X1+e++X2(0.42MeV)(A4-1)p+X1→X3+γ(5.49MeV)(A4-2) The amounts listed in parentheses are the total kinetic energy carried by the products,including gamma rays.p is a proton,e+is a positron,γis a gamma ray,and X1,X2,and X3are particles for you to identify.The density of electrons in the Sun’s core is sufficient that the positron is annihilated almost immediately,releasing an energy x:e++e−→2γ(x)(A4-3) Subsequently,two major processes occur simultaneously.The“pp I branch”is the single reaction2X3→4He+2X4(y),(A4-4) which releases an energy y.The“pp II branch”consists of three reactions:X3+4He→X5+γ(A4-5)X5+e−→X6+X7(z)(A4-6)X6+X4→24He(A4-7) where z is the energy released in step A4-6.a.Identify X1through X7.X2and X7are neutral particles of negligible mass.It is useful toknow that thefirst few elements,in order of atomic number,are H,He,Li,Be,B,C,N,O.b.The mass of the electron is0.51MeV/c2,the mass of the proton is938.27MeV/c2,andthe mass of the helium-4nucleus is3727.38MeV/c2.Find the energy released during the production of one helium-4nucleus,including the kinetic energy of all products and all energy carried by gamma rays.c.Find the unknown energies x and y above.d.Step(A4-6)does not proceed as follows because there is insufficient energy.X5→X6+e++X7What constraint does this fact place on z?e.In which of the reaction steps is the energy carried by any given product the same every timethe step occurs?Assume that the kinetic energy carried in by the reactants in each step is negligible,and that the products are in the ground state.STOP:Do Not Continue to Part B If there is still time remaining for Part A,you should review your work for Part A,but do not continue to Part B until instructed by your examsupervisor.Part BQuestion B1A thin plank of mass M and length L rotates about a pivot at its center.A block of mass m M slides on the top of the plank.The system moves without friction.Initially,the plank makes an angleθ0with the horizontal,the block is at the upper end of the plank,and the system is at rest. Throughout the problem you may assume thatθ 1,and that the physical dimensions of theplank.block are much,much smaller than the length of thea.For a certain value ofθ0,x=kθthroughout the motion,where k is a constant.What is thisvalue ofθ0?Express your answer in terms of M,m,and any fundamental constants that you require.b.Given thatθ0takes this special value,what is the period of oscillation of the system?Expressyour answer in terms of M,m,and any fundamental constants that you require.c.Determine the maximum value of the ratio between the centripetal acceleration of the blockand the linear acceleration of the block along the plank,writing your answer in terms of m and M,therefore justifying our approximation.Question B2These three parts can be answered independently.a.One pair of ends of two long,parallel wires are connected by a resistor,R=0.25Ω,and afuse that will break instantaneously if5amperes of current pass through it.The other pair of ends are unconnected.A conducting rod of mass m is free to slide along the wires under the influence of gravity.The wires are separated by30cm,and the rod starts out10cm from the resistor and fuse.The whole system is placed in a uniform,constant magneticfield of B=1.2T as shown in thefigure.The resistance of the rod and the wires is negligible.When the rod is released is falls under the influence of gravity,but never loses contact with the long parallel wires.i.What is the smallest mass needed to break the fuse?ii.How fast is the mass moving when the fuse breaks?b.A fuse is composed of a cylindrical wire with length L and radius r L.The resistivity(not resistance!)of the fuse is small,and given byρf.Assume that a uniform current Iflows through the fuse.Write your answers below in terms of L,r,ρf,I,and any fundamental constants.i.What is the magnitude and direction of the electricfield on the surface of the fuse wire?ii.What is the magnitude and direction of the magneticfield on the surface of the fuse wire?iii.The Poynting vector, S is a measure of the rate of electromagnetic energyflow throughE× B,a unit surface area;the vector gives the direction of the energyflow.Since S=1µ0 whereµ0is the permeability of free space and and E and B are the electric and magneticfield vectors,find the magnitude and direction of the Poynting vector associated withthe current in the fuse wire.c.A fuse will break when it reaches its melting point.We know from modern physics that ahot object will radiate energy(approximately)according to the black body law P=σAT4, where T is the temperature in Kelvin,A the surface area,andσis the Stefan-Boltzmann constant.If T f=500K is the melting point of the metal for the fuse wire,with resistivity ρf=120nΩ·m,and I f=5A is the desired breaking current,what should be the radius of the wire r?。