美国大学生数学建模论文-Rolling in the Deep
美国大学生数学建模竞赛优秀论文
For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number7018Problem ChosencFor office use onlyF1________________F2________________F3________________F4________________ SummaryThe article is aimed to research the potential impact of the marine garbage debris on marine ecosystem and human beings，and how we can deal with the substantial problems caused by the aggregation of marine wastes.In task one，we give a definition of the potential long-term and short-term impact of marine plastic garbage. Regard the toxin concentration effect caused by marine garbage as long-term impact and to track and monitor it. We etablish the composite indicator model on density of plastic toxin，and the content of toxin absorbed by plastic fragment in the ocean to express the impact of marine garbage on ecosystem. Take Japan sea as example to examine our model.In ask two, we designe an algorithm, using the density value of marine plastic of each year in discrete measure point given by reference，and we plot plastic density of the whole area in varies locations. Based on the changes in marine plastic density in different years， we determine generally that the center of the plastic vortex is East—West140°W—150°W, South—North30°N—40°N. According to our algorithm, we can monitor a sea area reasonably only by regular observation of part of the specified measuring pointIn task three，we classify the plastic into three types，which is surface layer plastic，deep layer plastic and interlayer between the two. Then we analysis the the degradation mechanism of plastic in each layer. Finally，we get the reason why those plastic fragments come to a similar size.In task four， we classify the source of the marine plastic into three types，the land accounting for 80%，fishing gears accounting for 10%，boating accounting for 10%，and estimate the optimization model according to the duel-target principle of emissions reduction and management. Finally, we arrive at a more reasonable optimization strategy.In task five，we first analyze the mechanism of the formation of the Pacific ocean trash vortex， and thus conclude that the marine garbage swirl will also emerge in south Pacific，south Atlantic and the India ocean. According to the Concentration of diffusion theory, we establish the differential prediction model of the future marine garbage density，and predict the density of the garbage in south Atlantic ocean. Then we get the stable density in eight measuring point .In task six, we get the results by the data of the annual national consumption ofpolypropylene plastic packaging and the data fitting method, and predict the environmental benefit generated by the prohibition of polypropylene take-away food packaging in the next decade. By means of this model and our prediction，each nation will reduce releasing 1.31 million tons of plastic garbage in next decade.Finally, we submit a report to expediction leader，summarize our work and make some feasible suggestions to the policy- makers.Task 1:Definition:●Potential short-term effects of the plastic: the hazardeffects will be shown in the short term.●Potential long-term effects of the plastic: thepotential effects, of which hazards are great, willappear after a long time.The short- and long-term effects of the plastic on the ocean environment:In our definition, the short-term and long-term effects of the plastic on the ocean environment are as follows.Short-term effects:1）The plastic is eaten by marine animals or birds.2） Animals are wrapped by plastics, such as fishing nets, which hurt or even kill them.3）Deaden the way of the passing vessels.Long-term effects:1）Enrichment of toxins through the food chain: the waste plastic in the ocean has no natural degradation in theshort-term, which will first be broken down into tinyfragments through the role of light, waves,micro-organisms, while the molecular structure has notchanged. These "plastic sands", easy to be eaten byplankton, fish and other, are Seemingly very similar tomarine life’s food,causing the enrichment and delivery of toxins.2）Accelerate the greenhouse effect: after a long-term accumulation and pollution of plastics, the waterbecame turbid, which will seriously affect the marineplants (such as phytoplankton and algae) inphotosynthesis. A large number of plankton’s deathswould also lower the ability of the ocean to absorbcarbon dioxide, intensifying the greenhouse effect tosome extent.To monitor the impact of plastic rubbish on the marine ecosystem:According to the relevant literature, we know that plastic resin pellets accumulate toxic chemicals , such as PCBs、DDE , and nonylphenols , and may serve as a transport medium and soure of toxins to marine organisms that ingest them[]2. As it is difficult for the plastic garbage in the ocean to complete degradation in the short term, the plastic resin pellets in the water will increase over time and thus absorb more toxins, resulting in the enrichment of toxins and causing serious impact on the marine ecosystem.Therefore, we track the monitoring of the concentration of PCBs, DDE, and nonylphenols containing in the plastic resin pellets in the sea water, as an indicator to compare the extent of pollution in different regions of the sea, thus reflecting the impact of plastic rubbish on ecosystem.To establish pollution index evaluation model: For purposes of comparison, we unify the concentration indexes of PCBs, DDE, and nonylphenols in a comprehensive index.Preparations:1）Data Standardization2）Determination of the index weightBecause Japan has done researches on the contents of PCBs,DDE, and nonylphenols in the plastic resin pellets, we illustrate the survey conducted in Japanese waters by the University of Tokyo between 1997 and 1998.To standardize the concentration indexes of PCBs, DDE,and nonylphenols. We assume Kasai Sesside Park, KeihinCanal, Kugenuma Beach, Shioda Beach in the survey arethe first, second, third, fourth region; PCBs, DDE, andnonylphenols are the first, second, third indicators.Then to establish the standardized model:j j jij ij V V V V V min max min --= （1,2,3,4;1,2,3i j ==）wherej V max is the maximum of the measurement of j indicator in the four regions.j V min is the minimum of the measurement of j indicatorstandardized value of j indicator in i region.According to the literature [2], Japanese observationaldata is shown in Table 1.Table 1. PCBs, DDE, and, nonylphenols Contents in Marine PolypropyleneTable 1 Using the established standardized model to standardize, we have Table 2.In Table 2，the three indicators of Shioda Beach area are all 0, because the contents of PCBs, DDE, and nonylphenols in Polypropylene Plastic Resin Pellets in this area are the least, while 0 only relatively represents the smallest. Similarly, 1 indicates that in some area the value of a indicator is the largest.To determine the index weight of PCBs, DDE, and nonylphenolsWe use Analytic Hierarchy Process (AHP) to determine the weight of the three indicators in the general pollution indicator. AHP is an effective method which transforms semi-qualitative and semi-quantitative problems into quantitative calculation. It uses ideas of analysis and synthesis in decision-making, ideally suited for multi-index comprehensive evaluation.Hierarchy are shown in figure 1.Fig.1 Hierarchy of index factorsThen we determine the weight of each concentrationindicator in the generall pollution indicator, and the process are described as follows:To analyze the role of each concentration indicator, we haveestablished a matrix P to study the relative proportion.⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=111323123211312P P P P P P P Where mn P represents the relative importance of theconcentration indicators m B and n B . Usually we use 1，2，…，9 and their reciprocals to represent different importance. The greater the number is, the more important it is. Similarly, the relative importance of m B and n B is mn P /1（3,2,1,=n m ）.Suppose the maximum eigenvalue of P is m ax λ, then theconsistency index is1max --=n nCI λThe average consistency index is RI , then the consistencyratio isRICI CR = For the matrix P of 3≥n , if 1.0<CR the consistency isthougt to be better, of which eigenvector can be used as the weight vector.We get the comparison matrix accoding to the harmful levelsof PCBs, DDE, and nonylphenols and the requirments ofEPA on the maximum concentration of the three toxins inseawater as follows:⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=165416131431P We get the maximum eigenvalue of P by MATLAB calculation0012.3max =λand the corresponding eigenvector of it is()2393.02975.09243.0，，=W1.0042.012.1047.0<===RI CI CR Therefore,we determine the degree of inconsistency formatrix P within the permissible range. With the eigenvectors of p as weights vector, we get thefinal weight vector by normalization ()1638.02036.06326.0'，，=W . Defining the overall target of pollution for the No i oceanis i Q , among other things the standardized value of threeindicators for the No i ocean is ()321,,i i i i V V V V = and the weightvector is 'W ,Then we form the model for the overall target of marine pollution assessment, （3,2,1=i ）By the model above, we obtained the Value of the totalpollution index for four regions in Japanese ocean in Table 3T B W Q '=In Table3, the value of the total pollution index is the hightest that means the concentration of toxins in Polypropylene Plastic Resin Pellets is the hightest, whereas the value of the total pollution index in Shioda Beach is the lowest(we point up 0 is only a relative value that’s not in the name of free of plastics pollution)Getting through the assessment method above, we can monitor the concentration of PCBs, DDE and nonylphenols in the plastic debris for the sake of reflecting the influence to ocean ecosystem.The highter the the concentration of toxins,the bigger influence of the marine organism which lead to the inrichment of food chain is more and more dramatic.Above all, the variation of toxins’ concentration simultaneously reflects the distribution and time-varying of marine litter. We can predict the future development of marine litter by regularly monitoring the content of these substances, to provide data for the sea expedition of the detection of marine litter and reference for government departments to make the policies for ocean governance.Task 2:In the North Pacific, the clockwise flow formed a never-ending maelstrom which rotates the plastic garbage. Over the years, the subtropical eddy current in North Pacific gathered together the garbage from the coast or the fleet, entrapped them in the whirlpool, and brought them to the center under the action of the centripetal force, forming an area of 3.43 million square kilometers (more than one-third of Europe) .As time goes by, the garbage in the whirlpool has the trend of increasing year by year in terms of breadth, density, and distribution. In order to clearly describe the variability of the increases over time and space, according to “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999—2008”, we analyze the data, exclude them with a great dispersion, and retain them with concentrated distribution, while the longitude values of the garbage locations in sampled regions of years serve as the x-coordinate value of a three-dimensional coordinates, latitude values as the y-coordinate value, the Plastic Count per cubic Meter of water of the position as the z-coordinate value. Further, we establish an irregular grid in the yx plane according to obtained data, and draw a grid line through all the data points. Using the inverse distance squared method with a factor, which can not only estimate the Plastic Count per cubic Meter of water of any position, but also calculate the trends of the Plastic Counts per cubic Meter of water between two original data points, we can obtain the unknown grid points approximately. When the data of all the irregular grid points are known (or approximately known, or obtained from the original data), we can draw the three-dimensional image with the Matlab software, which can fully reflect the variability of the increases in the garbage density over time and space.Preparations:First, to determine the coordinates of each year’s sampled garbage.The distribution range of garbage is about the East - West 120W-170W, South - North 18N-41N shown in the “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”, we divide a square in the picture into 100 grids in Figure (1) as follows:According to the position of the grid where the measuring point’s center is, we can identify the latitude and longitude for each point, which respectively serve as the x- and y- coordinate value of the three-dimensional coordinates.To determine the Plastic Count per cubic Meter of water. As the “Plastic Count per cubic Meter of water” provided by “Count Densities of P lastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”are 5 density interval, to identify the exact values of the garbage density of one year’s different measuring points, we assume that the density is a random variable which obeys uniform distribution in each interval.Uniform distribution can be described as below:()⎪⎩⎪⎨⎧-=01a b x f ()others b a x ,∈We use the uniform function in Matlab to generatecontinuous uniformly distributed random numbers in each interval, which approximately serve as the exact values of the garbage density andz-coordinate values of the three-dimensional coordinates of the year’s measuring points.Assumptions(1)The data we get is accurate and reasonable.(2)Plastic Count per cubic Meter of waterIn the oceanarea isa continuous change.(3)Density of the plastic in the gyre is a variable by region.Density of the plastic in the gyre and its surrounding area is interdependent , However, this dependence decreases with increasing distance . For our discussion issue, Each data point influences the point of each unknown around and the point of each unknown around is influenced by a given data point. The nearer a given data point from the unknown point, the larger the role.Establishing the modelFor the method described by the previous,we serve the distributions of garbage density in the “Count Pensities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”as coordinates ()z y,, As Table 1:x,Through analysis and comparison, We excluded a number of data which has very large dispersion and retained the data that is under the more concentrated the distribution which, can be seen on Table 2.In this way, this is conducive for us to get more accurate density distribution map.Then we have a segmentation that is according to the arrangement of the composition of X direction and Y direction from small to large by using x co-ordinate value and y co-ordinate value of known data points n, in order to form a non-equidistant Segmentation which has n nodes. For the Segmentation we get above,we only know the density of the plastic known n nodes, therefore, we must find other density of the plastic garbage of n nodes.We only do the sampling survey of garbage density of the north pacificvortex,so only understand logically each known data point has a certain extent effect on the unknown node and the close-known points of density of the plastic garbage has high-impact than distant known point.In this respect,we use the weighted average format, that means using the adverse which with distance squared to express more important effects in close known points. There're two known points Q1 and Q2 in a line ,that is to say we have already known the plastic litter density in Q1 and Q2, then speculate the plastic litter density's affects between Q1、Q2 and the point G which in the connection of Q1 and Q2. It can be shown by a weighted average algorithm22212221111121GQ GQ GQ Z GQ Z Z Q Q G +*+*=in this formula GQ expresses the distance between the pointG and Q.We know that only use a weighted average close to the unknown point can not reflect the trend of the known points, we assume that any two given point of plastic garbage between the changes in the density of plastic impact the plastic garbage density of the unknown point and reflecting the density of plastic garbage changes in linear trend. So in the weighted average formula what is in order to presume an unknown point of plastic garbage density, we introduce the trend items. And because the greater impact at close range point, and thus the density of plastic wastes trends close points stronger. For the one-dimensional case, the calculation formula G Z in the previous example modify in the following format:2212122212212122211111112121Q Q GQ GQ GQ Q Q GQ Z GQ Z GQ Z Z Q Q Q Q G ++++*+*+*=Among them, 21Q Q known as the separation distance of the known point, 21Q Q Z is the density of plastic garbage which is the plastic waste density of 1Q and 2Q for the linear trend of point G . For the two-dimensional area, point G is not on the line 21Q Q , so we make a vertical from the point G and cross the line connect the point 1Q and 2Q , and get point P , the impact of point P to 1Q and 2Q just like one-dimensional, and the one-dimensional closer of G to P , the distant of G to P become farther, the smaller of the impact, so the weighting factor should also reflect the GP in inversely proportional to a certain way, then we adopt following format:221212222122121222211111112121Q Q GQ GP GQ GQ Q Q GQ GP Z GQ Z GQ Z Z P Q Q Q Q G ++++++*+*+*=Taken together, we speculated following roles:(1) Each known point data are influence the density of plastic garbage of each unknown point in the inversely proportional to the square of the distance;(2) the change of density of plastic garbage between any two known points data, for each unknown point are affected, and the influence to each particular point of their plastic garbage diffuse the straight line along the two known particular point; (3) the change of the density of plastic garbage between any two known data points impact a specific unknown points of the density of plastic litter depends on the three distances: a. the vertical distance to a straight line which is a specific point link to a known point;b. the distance between the latest known point to a specific unknown point;c. the separation distance between two known data points.If we mark 1Q ，2Q ，…，N Q as the location of known data points,G as an unknown node, ijG P is the intersection of the connection of i Q ，j Q and the vertical line from G to i Q ，j Q()G Q Q Z j i ,,is the density trend of i Q ，j Q in the of plasticgarbage points and prescribe ()G Q Q Z j i ,,is the testing point i Q ’ s density of plastic garbage ,so there are calculation formula:()()∑∑∑∑==-==++++*=Ni N ij ji i ijGji i ijG N i Nj j i G Q Q GQ GPQ Q GQ GP G Q Q Z Z 11222222111,,Here we plug each year’s observational data in schedule 1 into our model, and draw the three-dimensional images of the spatial distribution of the marine garbage ’s density with Matlab in Figure (2) as follows:199920002002200520062007－2008(1)It’s observed and analyzed that, from 1999 to 2008, the density of plastic garbage is increasing year by year and significantly in the region of East – West 140W-150W, south - north 30N-40N. Therefore, we can make sure that this region is probably the center of the marine litter whirlpool. Gathering process should be such that the dispersed garbage floating in the ocean move with the ocean currents and gradually close to the whirlpool region. At the beginning, the area close to the vortex will have obviously increasable about plastic litter density, because of this centripetal they keeping move to the center of the vortex ,then with the time accumulates ,the garbage density in the center of the vortex become much bigger and bigger , at last it becomes the Pacific rubbish island we have seen today.It can be seen that through our algorithm, as long as the reference to be able to detect the density in an area which has a number of discrete measuring points,Through tracking these density changes ,we Will be able to value out all the waters of the density measurement through our models to determine,This will reduce the workload of the marine expedition team monitoring marine pollution significantly, and also saving costs .Task 3:The degradation mechanism of marine plasticsWe know that light, mechanical force, heat, oxygen, water, microbes, chemicals, etc. can result in the degradation of plastics . In mechanism ,Factors result in the degradation can be summarized as optical ，biological，and chemical。
2013年美国数学建模比赛提交论文-二等奖
Print This Page Close This WindowFor office use only T1________________T2________________T3________________T4________________Team Control Number22940Problem ChosenAFor office use only F1________________F2________________F3________________F4________________2013Mathematical Contest in Modeling (MCM)Summary Sheet(Attach a copy of this page to your solution paper.)Type a summary of your results on this page.Do not include the name of your school,advisor,or team members on this page.SummaryThis paper aims to design the optimal shape of Brownie pan on the given conditions.The influencing factors include the shape of pan,the ratio of width and the length of the oven,and the weight be conferred to the quantity of Brownie pans that can be put in a oven and evenness of heat distribution.We have a scientific study about the three typical shapes,namely rectangular circular and oval,and we get the following conclusion that the time required by circular pans is the shortest,while rectangular pan can maximize the use of space.To solve the problem,we simplify the influencing factors and assume that the area of being effective baking inside the oven can be exactly fulfilled by rectangular pans.So that we can easily solve the condition 1,and be aware of that the rectangular pans is the optimal shape;and the same to the condition 2,we believe that the temperature distribution on the outer edge of circular pans is completely uniform,so the optimal solution is circular.When it comes to the third condition,it’s a typical linear programming problem.We use the idea of normalizing to construct design objectivefunction:(1)s A nZ p p N A=+−.In the formula above,the calculation of area being effective heated adopt the view of discrete,which is very innovative (formula (k)).By using the above objective function,whose results is roughly the same with the results of MATLAB,This can be used to confirm each other.We obtain the desired optimal solution to a maximum of 1.215,which is larger than other shapes’maximum by 21.5%.Its advantages are obvious,so this further illustrates the correctness of the ideal optimal shape.So we can say the pan we design is the Ultimate Brownie Pan!ContentsI.Introduction (4)II.Assumptions (4)III.Analysis&Models (5)IV.Solutions (12)4.1rectangular Brownie Pans (12)4.2circular Brownie Pans (14)4.3oval Brownie Pans (15)V.Optimization of the model (12)5.1rectangular Brownie Pans (16)5.2circular Brownie Pans (16)5.3oval Brownie Pans (17)References (18)Advertising sheet (19)Appendix (20)List of Figures1.Region of discrete (8)2.Volumetric controlled by internal node (8)3.The simulation isotherms of rectangular pan (10)4.The simulation isotherms of circular pan (10)5.The simulation isotherms of oval pan (10)6.Temperature curve of the rectangular pan center (11)7.Temperature curve of the circular pan center (11)8.Temperature curve of the oval pan center (11)9.How the rectangular pans are placed in the oven (14)10.The placement of round Brownie Pan in the oven (14)11.The ideal shape of the Brownie Pan (15)12.The distribution of the pans in the oven (15)13.The placement of rectangular Brownie Pan in the oven (16)14.The zoning figure of the extended the oven (17)15.The arrangement of Brownie Pans in theoretical optimal shape in the oven (18)16.Distribution of temperature of rectangular pan (20)17.Distribution of temperature of circular pan (20)18.Distribution of temperature of oval pan (20)1Introduction:When baking in a rectangular pan heat is concentrated in the4corners and the product gets overcooked at the corners(and to a lesser extent at the edges).In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges.However,since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven.Since both have their own advantages and can complement each other,we consider that a combination of the two will be a better choice.The purpose of this article is to identify the pan in best shape to satisfy various needs.2Assumption:1.The heat inside the oven is Changeless,After a Brownie Pan have been put in the oven,the heat is conducted to the center from the edge.2.The initial temperature of the food is25C o.3.The coefficient of thermal conductivity of the food is0.1.4.To make full use of the area of a rectangular oven,We assume that the area of being effective baking inside the oven can just be fulfilled by rectangular Brownie Pans.Table 1NotationParameter MeaningFThe heat of per unit time through a given area λThermal conductivity TTemperaturetThe temperature field in the Cartesian coordinate system A Cross-sectional areaΔx,ΔyThe amount of change of the x-direction,y-direction τ∆Time variation amount ∆o F The number of grid FourierT ij The temperature of the arbitrary point (i,j)T oThe temperature of the center of the pant j i T ,The temperature of the arbitrary point (i,j)at time t 0div ∇The gradient value at a certain point3Analysis &ModelsWe solved this problem using numerical solution of heat conduction.The numerical solution is based on the basic knowledge of heat transfer,and a method of using partial differential equations solving a heat conduction problem.Heat conduction partial differential equations:⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=∇=∂∂conditions Initial conditions dary B z T y T x T T t T oun 2222222ααThe basic idea:putting the problem of the temperature within the object continuously changing with time and space transformed into the problem of the temperature value in a finite number of discrete points within the area of the field of time and space.Further,using the temperature value of these discrete units to approximate the continuous temperature distribution.Models:1.The establishment of the physical modelAccording to the meaning of the questions,we construct three models,which are rectangular pan,circular pan and oval pan.2.The establishment of the mathematical modelOne of the main purpose of the study heat conduction problem is obtained under certain boundary conditions the state of objects within the temperature distribution with spatial location and time.⑴.Equations of heat conductionThe equations of the temperature distribution in the description of thermal conductivity inside the object is called the partial differential equation of heat conduction,which also known as the heat diffusion w of conservation of energy and the Fourier law are its fundamental basis.Once solving the temperature distribution,the heat conduction rate of inside the object or surface at any point on can be obtained by Fourier law.That is the heat flux density.Unsteady heat conduction problem for the temperature field varies with time,by obtaining the temperature distribution inside the object at different times,can determine the thermal stress and thermal deformation of the various parts.The general form of the equation of heat conduction in the Cartesian coordinate system:v ztz y t y x t x t cφλλλτρ+∂∂∂∂+∂∂∂∂+∂∂∂∂=∂∂)(((................................(a)This formula reflects the relationship between the thermal conductivity of the object within the total energy conservation.In the calculation,the thermal conductivity is seen as a constant,so the above equation reduces to:c zty t x t t v ρφατ+∂∂+∂∂+∂∂=∂∂)(222222...............................................................(b)It is composed by three physical properties of a physical parameter,called thermal diffusivity,can also be referred to the coefficient of thermal conductive,whose unit iss m /2.It indicates the temperature tends to be uniform capacity of the object in the heating or cooling process.This comprehensive parameters in the non-steady-state heat transfer process is a very important parameter.⑵.Single -Valued Property conditionsAll pure heat conduction problem can be described with the equation of heat conduction in the corresponding coordinates,including one-dimensional and multidimensional,steady-state and non-steady-state,constant material properties and Variable physical properties,the internal heat source and no internal heat source heat conduction problems.Differential equations (The mathematical said general solution)must contain the pending integration constant.In addition to the differential equation,for these constants to be determined uniquely determined must be attached to a certain characteristics of solving this particular heat conduction problem and limited of external environment or supplementary explanation.These additional instructions and restriction condition is Single -Valued Property conditions,and mathematics known as boundary conditions.Mathematical model of any one specific full heat conduction problems,in addition to the equation of heat conduction in the appropriately selected coordinates,must be given the corresponding Single -Valued Property conditions.In terms of the general problem of thermal conductivity,the Single -Valued property conditions consist of the following two parts:①.Time conditionsFor unsteady-state heat conduction,the temperature field of the object at the start time must be given,so the time conditions are known as initial conditions.In the three-dimensional Cartesian coordinate system,the initial conditions can generally be expressed as the following form:),,(0z y x f t ==τ....................................................................(c)Under normal circumstances,the object having a completely uniform temperature at the beginning of the process,so we can think that the distribution function (),,(z y x f )is a constant in the above formula.②.Boundary conditionsThe boundary conditions refers to the contact and interaction in the heat exchange of hot objects between the boundary surface and the external environment.For unsteady heat conduction,it is often the external driving force making process take place and develop.Here we consider the first category.Stipulate:The surface temperature is a room temperature.),(y x f t w =,The boundary temperature.Notes:tcons t w tan =3.Discrete the solution domainDiscretization is divided the substantially continuous object into a series of tiny unit,while the center of the unit is called a node.Divided the spatial domain x into n subparagraph,the step is Δx,get 0,l,2,...,i,...,n x node of the n +1;divided the spatial domain y into n subparagraph,while the step is Δy,get 0,l,2,...,i,...,n y node of the n +1.Intersection point (i,j)of the separation line represents the spatial domain position.The size of space and time step depends on the specific circumstances of the problem,and sometimes cannot bearbitrarily selected,needing to consider the node temperature equation solving stability problems which are as shown in the following figure:Figure 1Region of discreteTime can also be divided into many inter-cell.The physical significance of the zone discretization is that we can think a node focus the heat capacity around its tiny area.Thus the node temperature is the average temperature of this tiny area.Thus,the temperature of all the nodes together represents the distribution of the temperature within this continuous region.4.Establish the difference equation of nodes’temperatureUsing difference quotient instead of the derivative,and then on the basis of the relationship of the heat balance we can a establish differential equation.The same as the steady heat conduction,using the method of heat balance,you can establish the temperature difference equations of objects internal nodes and boundary nodes at unsteady heat conduction.As shown in the following figure:Figure 2V olumetric controlled by internal nodeFor the two-dimensional unsteady heat conduction problems of constant material properties and no internal heat source,the internal nodes (i,j)represents the thermal eq uilibrium of the control volume is expressed as:In unit time,the heat flow rate jQ λand iQ λcoming from the adjacent control volume ),1()1j i j i +−，，（and)1,()1,(+−j i j i ，is equal to the control volume thermodynamic energy increase dU .dU Q Q j i =+λλ.Therefore,for each node Differential established as follows:tj,i 2222j,i y T x T t T ⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂α=⎟⎠⎞⎜⎝⎛∂∂................................................(d)for the left side:tT T t T (t j,i t t j ,i j ,i ∆−=∂∂∆+.................................................................(e)for the right side:2,1,,1,222)(x T T T x Tt j i t j i t j i j i ∆+−=∂∂−+........................................................(f)21,,1,,222(y T T T y Tt j i t j i t j i j i ∆+−=∂∂−+...................................................................(g)Making y x ∆=∆，2xF o ∆∆=∆τα，∆o F :called grid Fourier number.(e),(f)and (g)substituted into the formula (d).After finishing,the following equations can be obtained:t j i o t j i t j i t j i t j i o t t j i T F T T T T F T ,1,1,,1,1,)41()(∆+−+−∆∆+−++++=......(h)This equation can draw two conclusions:1).Any node temperature of an internal node at a particular moment can be obtained directly from the temperature of the node and its collar node at the former time.Do not have the simultaneous solution of the equations.This is the advantages of the explicit difference scheme.So we can start from the initial temperature sequentially obtainsτ∆,τ∆2,...,τ∆k each time nodes temperature.2).In front of the t j i T ,coefficient in the formula (h)can not be negative.So041≥−∆o F ,which is equivalent to 41≤∆o F ................................（i ）This is the stability conditions of two-dimensional non-steady-state thermal conductivity inside the object node temperature equation explicit difference scheme.Physical meaning:The temperature of any internal node at time t t ∆+all dependson the temperature of this node and its surrounding nodes at time t .When the temperature of surrounding nodes to are known,the higher the temperature of the node is,the higher the temperature of the node will be at time t t ∆+.Therefore,the coefficient in front of the formula can not be negative,which must meet the formula (i)that represents the stability condition.5.Solve temperature algebra equation and obtain the temperature of all nodes Assuming the area of each Brownie Pan is 2600cm A =and it will be heated for ing Matlab programming solve the problem of rectangular,round and oval pan heat conduct to the center from all sides into the incubator.As shown in the following figures:Figure 6The simulation isothermsFigure 7The simulation isothermsof rectangular panof circular panFigure 8The simulation isothermsof oval panyy102030405060708090100yWe can know from the figures above:whatever the shape of the pan is,the isothermal lines are arcuate curve,and they are nearly circular in its center position.Therefore,in the design of the shape of baking pan,we should try to design it in arc-shape in order to make the baking pan to be evenly heated.Temperature curve of the pan centerFigure 9Temperature curve of Figure 10Temperature curve ofthe rectangular pan centerthe circular pan centerFigure 11Temperature curve of oval pan center6.Conclusion⑴.The circular baking pan temperature rises very quickly.When the food reaches a certain temperature,the time required is the shortest.However,the space utilization of the pan in the rectangular oven is low,not efficient of using energy;⑵.Rectangular baking pan to maximize the use of space,effective use of energy.However,the temperature difference between the center and the periphery will cause the food being heated unevenly;⑶.The oval pan space utilization and degree of heat evenly is between the circular baking pan and rectangular baking pan.T e m p e r a t u r e /T e m p e r a t u r e /T e m p e r a t u r e /4Solutions Solutions::For the condition 1:Based on the assumption 4,it’s obvious that if we use a rectangular Brownie Pan,The Pans can exactly fulfill the oven.That’s to say,the number of pans in rectangular shape used is maximum.For the condition 2:①.How to determine the area where temperature is uniformly distributedWe use temperature variance of all the discrete points as a threshold value to determine whether a point is in the area where temperature is uniformly ly,whether its temperature can satisfy the following inequality:00ij T T div −≤∇..............................(j)If the above equation can be satisfied,we believe that this point is at the place where temperature is evenly distributed.②.The area of the region where temperature is uniformly distributedWhen we calculating the area of the region where temperature is uniformly distributed,We apply the previous concept of discretization.Firstly,we use MATLAB to calculate the number of points(N)in a given range,then,we can know that the area occupied by each discrete point isyx A d d d ×=...........................................(k)so,the total area occupied by all the discrete point isyx a d d N A ××=............................................(l)According to the temperature distribution figure of various Brownie Pan in geometric shapes in the first question,we measure the uniformity of the temperature distribution with the temperature variance22211((,))n navg i j T i j T n σ==−=∑∑..............................(m)and it’s obvious that:2rectangle22circle σσσ>>oval And we can know that only when Brownie Pan is round,can the heat on outer edge ofthe Brownie Pan be completely evenly distributed.For the condition 3:Based on the above two considerations,We think the combination of the above two optimization conditions we can use the following normalized method and get the Objective function ：(1)sA nZ p p N A=+−...................................(n)In addition,there are other restrictions:1.Different kinds of pan share the same area;2.The temperature reached after a specific time(6mins);3.The coefficient of thermal conductivity of the food is similar.(0.1)4.1Firstly,for rectangular Brownie Pans whose heating time is 6mins,we know in the objective function above,n=N,while the area where temperature is uniformly distributed ,A s ,need to be worked out.In this paper we can use the MATLAB program programmed in the Question 1,and we can get the area where temperature is uniformly distributed by directly substituting into the length and width of a rectangle,and the objective function turns into:(1)s AZ p p A=+−........................(o)the equation shows objective function becomes a single-valued function of p,and ithas no relationship with WL,while 'L NL =By using the MATLAB program written in the Analysis &Models,we can get we can obtain the output of 12sets of data(in the table 2),Fitting the relationship between a and b,and we can get the following function:2234.49()469.05599.16s W WA L L=−+Notes:to obtain their solutions,we assume that the area of a rectangle is 600cm 2,And the objective function turns into600/)16.59905.469(49.234)(1(2+−−+=LWL W p p Z With the constrains:..(0,1)[0,1]s t p W L∈∈And the optimal solution is Z=1,if and only if 1&Wp L=it can be satisfied Table 2Scatter values calculated by the MATLABL W 0.800.760.630.600.580.55A s /cm 2374.78378.69397.52402.94406.01412.92LW 0.490.450.420.380.350.32A s /cm 2426.43436.38444.33455.59464.53473.89Figure 12How the rectangular pans are placed in the oven4.2Secondly,for circular Brownie Pan which have been heated in the oven for 6mins,We already know from the question1that heat is evenly distributed on the outer edge of the round pan.Therefore,we can think in round Brownie Pans A A=and the objective function turns into:1nZ p p N=+−........................................(p)Figure 13The placement of round Brownie Pan in the ovenTo simplify the calculation,We make'W L for the fixed value k,namely 'Wk L =,And we know that by calculating the horizontal distance between two round pan is(n −............................(q)And we can obtain constraint condition of circular as following:(1n L −+≤.....................(r)4.3thirdly,according to the above calculation results,we think it will be a good ideato combine the advantages of round Brownie Pan and rectangle Brownie Pan.And we can draw the following diagram of ideal shape of Brownie Pan:We will have a study on the ideal Brownie Pan.In order to ensure the area of the pan unchanged,we can see how Brownie Pan's width changes:224'''2r r L L W rπ−=+−.............(s)To ensure the pan will not exceed the scope of the oven,which is impossible,we can get a restriction on the oven:224''2r r n L NL W r π⎛⎞−+≤⎜⎟−⎝⎠..........(t)Figure 14The ideal shape of the Brownie Pan And the distribution of the pans in the oven as follows.Figure 15The distribution of the pans in ideal shape in ovenAnd the objective function turns into:(1)s A nZ p p N A=+−........................................................(u)By using the MATLAB as well,we know that the optimal solution isWhen p=0.75and 0.8WL=,Z can get the optimal solution,andZ=1.215Taking that the oven has two racks into account ,all of the above data required to be multiplied by a expansion of the coefficient α=2,which has no influence on the relative difference among different shape.5Optimization of the modelIn the optimized design above,we consider that the Brownie Pans can only be placed in a row in the oven.Next,we intend to expand the previously conclusion.And we consider that the Brownie Pans can be placed in N rows in the oven.5.1Firstly,for rectangular Brownie PansFigure 16The placement of rectangular Brownie Pan in the ovenAt this point,we know that:the specifications of Brownie pan is:L W N M×It’s obvious that the Brownie Pan optimization function does not change,which still is:**(1)(1)**ss A A M N Z p p p p M N A A =+−=+− (v)Namely max 1Z =.5.2Secondly,for round Brownie Pans,we divide the extended the oven into threeparts,which is as follows:Figure 17The zoning figure of the extended ovenThe reasons why we adopt the above area dividing method are based on the following two considerations:1.To Simplify the calculation;2.To take advantage of the inequality (r)above.And we can obtain constraint condition of circular are as follows:1.Region 1:the number of the circulars is1([1)([]1)W Ln d d=−×−...........................................(w)Notes:[]tan W Ws ds for the Gauss number ofd d2.Region 2:We assume the number of the circulars is n 2and the constraint condition is(21n L −≤..............................(x)Notes:L L =',d dWW W )1(['−−=3.Region 2:We assume the number of the circulars is n 3and the constraint condition is(31'n L −≤....................(y)Notes:d n L L 1''−=,d n W 1''=and it’s obvious that max 1Z <5.3Thirdly,for Brownie Pans in theoretical optimal shapeFigure18The arrangement of Brownie Pans in theoretical optimal shape in the oven It’s obvious that in the oven that can arrange multi-row Brownie Pans the optimal solution keeps unchanged.Namely,Z f=Z=1.215From all above,we can know that the Brownie Pan in the theoretical optimal shape can get greater optimal solution than either rectangular pans or round pans.So it’s a smart idea to choose a Brownie Pan in the theoretical optimal shape to cook your Brownie cakes!6References[1]Lars Mönch Robert Unbehaun You In Choung,Minimizing Earliness–Tardiness on a Single Burn-in Oven with a Common due Date and Maximum Allowable Tardiness Constraint, /static/pdf/374/art%253A10.1007%252Fs00291-005-0013-4.pdf?aut h66=1360982955_54758b58d2754d71008329400df3c11e&ext=.pdf,10Dec.2005[2]K.Venkateshmurthy&K.S.M.S.Raghavarao,Analysis of Modes of Heat Transfer in Baking Indian Rice Pan Cake(Dosa,)a Breakfast Food, /static/pdf/517/art%253A10.1007%252Fs13197-010-0204-0.pdf?aut h66=1360983288_e6bf23e753296afd9fcd43eea80dad4a&ext=.pdf,06Dec.2010[3]Jiang Qiyuan and Xie Jinxing,Mathematical model Mathematical programming model,3rd edn, Mathematical Modeling,Accessed Feb.2003[4]D.Pitts et al,Schaum's Outline of Theory and Problems of Heat Transfer,2nd edn,Science Press,2002[5]M.N.Aoqi Sigg,Heat Conduction,Higher Education Press,1984A dvertising sheet:AppendixFigure3Distribution of temperature of rectangular pan199.5198.5Figure4Distribution of temperature of circular panFigure5Distribution of temperature of oval pan。
2013美国大学生数学建模竞赛论文
summaryOur solution paper mainly deals with the following problems:·How to measure the distribution of heat across the outer edge of pans in differentshapes and maximize even distribution of heat for the pan·How to design the shape of pans in order to make the best of space in an oven·How to optimize a combination of the former two conditions.When building the mathematic models, we make some assumptions to get themto be more reasonable. One of the major assumptions is that heat is evenly distributedwithin the oven. We also introduce some new variables to help describe the problem.To solve all of the problems, we design three models. Based on the equation ofheat conduction, we simulate the distribution of heat across the outer edge with thehelp of some mathematical softwares. In addition, taking the same area of all the pansinto consideration, we analyze the rate of space utilization ratio instead of thinkingabout maximal number of pans contained in the oven. What’s more, we optimize acombination of conditions (1) and (2) to find out the best shape and build a function toshow the relation between the weightiness of both conditions and the width to lengthratio, and to illustrate how the results vary with different values of W/L and p.To test our models, we compare the results obtained by stimulation and our models, tofind that our models fit the truth well. Yet, there are still small errors. For instance, inModel One, the error is within 1.2% .In our models, we introduce the rate of satisfaction to show how even thedistribution of heat across the outer edge of a pan is clearly. And with the help ofmathematical softwares such as Matlab, we add many pictures into our models,making them more intuitively clear. But our models are not perfect and there are someshortcomings such as lacking specific analysis of the distribution of heat across theouter edge of a pan of irregular shapes. In spite of these, our models can mainlypredict the actual conditions, within reasonable range of error.For office use onlyT1 ________________T2 ________________T3 ________________T4 ________________ Team Control Number18674 Problem Chosen AFor office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________2013 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to your solution paper.)Type a summary of your results on this page. Do not includethe name of your school, advisor, or team members on this page.The Ultimate Brownie PanAbstractWe introduce three models in the paper in order to find out the best shape for the Brownie Pan, which is beneficial to both heat conduction and space utility.The major assumption is that heat is evenly distributed within the oven. On the basis of this, we introduce three models to solve the problem.The first model deals with heat distribution. After simulative experiments and data processing, we achieve the connection between the outer shape of pans and heat distribution.The second model is mainly on the maximal number of pans contained in an oven. During the course, we use utility rate of space to describe the number. Finally, we find out the functional relation.Having combined both of the conditions, we find an equation relation. Through mathematical operation, we attain the final conclusion.IntroductionHeat usage has always been one of the most challenging issues in modern world. Not only does it has physic significance, but also it can influence each bit of our daily life. Likewise,space utilization, beyond any doubt, also contains its own strategic importance. We build three mathematic models based on underlying theory of thermal conduction and tip thermal effects.The first model describes the process and consequence of heat conduction, thus representing the temperature distribution. Given the condition that regular polygons gets overcooked at the corners, we introduced the concept of tip thermal effects into our prediction scheme. Besides, simulation technique is applied to both models for error correction to predict the final heat distribution.Assumption• Heat is distributed evenly in the oven.Obviously, an oven has its normal operating temperature, which is gradually reached actually. We neglect the distinction of temperature in the oven and the heating process, only to focus on the heat distribution of pans on the basis of their construction.Furthermore, this assumption guarantees the equivalency of the two racks.• Thermal conductivity is temperature-invariant.Thermal conductivity is a physical quantity, symbolizing the capacity of materials. Always, the thermal conductivity of metal material usually varies with different temperatures, in spite of tiny change in value. Simply, we suppose the value to be a constant.• Heat flux of boundaries keeps steady.Heat flux is among the important indexes of heat dispersion. In this transference, we give it a constant value.• Heat conduction dom inates the variation of temperature, while the effects ofheat radiation and heat convection can be neglected.Actually, the course of heat conduction, heat radiation and heat convectiondecide the variation of temperature collectively. Due to the tiny influence of other twofactors, we pay closer attention to heat conduction.• The area of ovens is a constant.I ntroduction of mathematic modelsModel 1: Heat conduction• Introduction of physical quantities:q: heat fluxλ: Thermal conductivityρ: densityc: specific heat capacityt: temperature τ: timeV q : inner heat sourceW q : thermal fluxn: the number of edges of the original polygonsM t : maximum temperaturem t : minimum temperatureΔt: change quantity of temperatureL: side length of regular polygon• Analysis:Firstly, we start with The Fourier Law:2(/)q gradt W m λ=- . (1) According to The Fourier Law, along the direction of heat conduction, positionsof a larger cross-sectional area are lower in temperature. Therefore, corners of panshave higher temperatures.Secondly, let’s analyze the course of heat conduction quantitatively.To achieve this, we need to figure out exact temperatures of each point across theouter edge of a pan and the variation law.Based on the two-dimension differential equation of heat conduction:()()V t t t c q x x y yρλλτ∂∂∂∂∂=++∂∂∂∂∂. (2) Under the assumption that heat distribution is time-independent, we get0t τ∂=∂. (3)And then the heat conduction equation (with no inner heat source)comes to:20t ∇=. (4)under the Neumann boundary condition: |W s q t n λ∂-=∂. (5)Then we get the heat conduction status of regular polygons and circles as follows:Fig 1In consideration of the actual circumstances that temperature is higher at cornersthan on edges, we simulate the temperature distribution in an oven and get resultsabove. Apparently, there is always higher temperature at corners than on edges.Comparatively speaking, temperature is quite more evenly distributed around circles.This can prove the validity of our model rudimentarily.From the figure above, we can get extreme values along edges, which we callM t and m t . Here, we introduce a new physical quantity k , describing the unevennessof heat distribution. For all the figures are the same in area, we suppose the area to be1. Obviously, we have22sin 2sin L n n n ππ= (6) Then we figure out the following results.n t M t m t ∆ L ksquare 4 214.6 203.3 11.3 1.0000 11.30pentagon 5 202.1 195.7 6.4 0.7624 8.395hexagon 6 195.7 191.3 4.4 0.6204 7.092heptagon 7 193.1 190.1 3.0 0.5246 5.719octagon 8 191.1 188.9 2.2 0.4551 4.834nonagon 9 188.9 187.1 1.8 0.4022 4.475decagon 10 189.0 187.4 1.6 0.3605 4.438Table 1It ’s obvious that there is negative correlation between the value of k and thenumber of edges of the original polygons. Therefore, we can use k to describe theunevenness of temperature distribution along the outer edge of a pan. That is to say, thesmaller k is, the more homogeneous the temperature distribution is.• Usability testing:We use regular hendecagon to test the availability of the model.Based on the existing figures, we get a fitting function to analyze the trend of thevalue of k. Again, we introduce a parameter to measure the value of k.Simply, we assume203v k =, (7) so that100v ≤. (8)n k v square 4 11.30 75.33pentagon 5 8.39 55.96hexagon 6 7.09 47.28heptagon 7 5.72 38.12octagon 8 4.83 32.23nonagon9 4.47 29.84 decagon 10 4.44 29.59Table 2Then, we get the functional image with two independent variables v and n.Fig 2According to the functional image above, we get the fitting function0.4631289.024.46n v e -=+.(9) When it comes to hendecagons, n=11. Then, v=26.85.As shown in the figure below, the heat conduction is within our easy access.Fig 3So, we can figure out the following result.vnActually,2026.523tvL∆==.n ∆t L k vhendecagons 11 187.1 185.8 1.3 0.3268 3.978 26.52Table 3Easily , the relative error is 1.24％.So, our model is quite well.• ConclusionHeat distribution varies with the shape of pans. To put it succinctly, heat is more evenly distributed along more edges of a single pan. That is to say, pans with more number of peripheries or more smooth peripheries are beneficial to even distribution of heat. And the difference in temperature contributes to overcooking. Through calculation, the value of k decreases with the increase of edges. With the help of the value of k, we can have a precise prediction of heat contribution.Model 2: The maximum number• Introduction of physical quantities:n: the number of edges of the original polygonsα: utility rate of space• Analysis:Due to the fact that the area of ovens and pans are constant, we can use the area occupied by pans to describe the number of pans. Further, the utility rate of space can be used to describe the number. In the following analysis, we will make use of the utility rate of space to pick out the best shape of pans. We begin with the best permutation devise of regular polygon. Having calculated each utility rate of space, we get the variation tendency.• Model Design:W e begin with the scheme which makes the best of space. Based on this knowledge, we get the following inlay scheme.Fig 4Fig 5According to the schemes, we get each utility rate of space which is showed below.n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 shape square pentagon hexagon heptagon octagon nonagon decagon hendecagon utility rate(％)100.00 85.41 100.00 84.22 82.84 80.11 84.25 86.21Table 4Using the ratio above, we get the variation tendency.Fig 6 nutility rate of space• I nstructions:·The interior angle degrees of triangles, squares, and regular hexagon can be divided by 360, so that they all can completely fill a plane. Here, we exclude them in the graph of function.·When n is no more than 9, there is obvious negative correlation between utility rate of space and the value of n. Otherwise, there is positive correlation.·The extremum value of utility rate of space is 90.69％,which is the value for circles.• Usability testing:We pick regular dodecagon for usability testing. Below is the inlay scheme.Fig 7The space utility for dodecagon is 89.88％, which is around the predicted value. So, we’ve got a rather ideal model.• Conclusion:n≥), the When the number of edges of the original polygons is more than 9(9 space utility is gradually increasing. Circles have the extreme value of the space utility. In other words, circles waste the least area. Besides, the rate of increase is in decrease. The situation of regular polygon with many sides tends to be that of circles. In a word, circles have the highest space utility.Model 3: Rounded rectangle• Introduction of physical quantities:A: the area of the rounded rectanglel: the length of the rounded rectangleα: space utilityβ: the width to length ratio• Analysis:Based on the combination of consideration on the highest space utility of quadrangle and the even heat distribution of circles, we invent a model using rounded rectangle device for pans. It can both optimize the cooking effect and minimize the waste of space.However, rounded rectangles are exactly not the same. Firstly, we give our rounded rectangle the same width to length ratio (W/L) as that of the oven, so that least area will be wasted. Secondly, the corner radius can not be neglected as well. It’ll give the distribution of heat across the outer edge a vital influence. In order to get the best pan in shape, we must balance how much the two of the conditions weigh in the scheme.• Model Design:To begin with, we investigate regular rounded rectangle.The area224r ar a A π++= (10) S imilarly , we suppose the value of A to be 1. Then we have a function between a and r :21(4)2a r r π=+--(11) Then, the space utility is()212a r α=+ (12) And, we obtain()2114rαπ=+- (13)N ext, we investigate the relation between k and r, referring to the method in the first model. Such are the simulative result.Fig 8Specific experimental results arer a ∆t L k 0.05 0.90 209.2 199.9 9.3 0.98 9.49 0.10 0.80 203.8 196.4 7.4 0.96 7.70 0.15 0.71 199.6 193.4 6.2 0.95 6.56 0.20 0.62 195.8 190.5 5.3 0.93 5.69 0.25 0.53 193.2 189.1 4.1 0.92 4.46Table 5According to the table above, we get the relation between k and r.Fig 9So, we get the function relation3.66511.190.1013r k e -=+. (14) After this, we continue with the connection between the width to length ratioW Lβ=and heat distribution. We get the following results.krFig 10From the condition of heat distribution, we get the relation between k and βFig 11And the function relation is4.248 2.463k β=+ (15)Now we have to combine the two patterns together:3.6654.248 2.463(11.190.1013)4.248 2.463r k e β-+=++ (16)Finally, we need to take the weightiness (p) into account,(,,)()(,)(1)f r p r p k r p βαβ=⋅+⋅- (17)To standard the assessment level, we take squares as criterion.()(,)(1)(,,)111.30r p k r p f r p αββ⋅⋅-=+ (18) Then, we get the final function3.6652(,,)(1)(0.37590.2180)(1.6670.0151)1(4)r p f r p p e rββπ-=+-⋅+⋅++- (19) So we get()()3.6652224(p 1)(2.259β 1.310)14r p f e r r ππ--∂=-+-+∂⎡⎤+-⎣⎦ (20) Let 0f r∂=∂,we can get the function (,)r p β. Easily,0r p∂<∂ and 0r β∂>∂ (21) So we can come to the conclusion that the value of r decreases with the increase of p. Similarly, the value of r increases with the increase of β.• Conclusion:Model 3 combines all of our former analysis, and gives the final result. According to the weightiness of either of the two conditions, we can confirm the final best shape for a pan.• References:[1] Xingming Qi. Matlab 7.0. Beijing: Posts & Telecom Press, 2009: 27-32[2] Jiancheng Chen, Xinsheng Pang. Statistical data analysis theory and method. Beijing: China's Forestry Press, 2006: 34-67[3] Zhengshen Fan. Mathematical modeling technology. Beijing: China Water Conservancy Press, 2003: 44-54Own It NowYahoo! Ladies and gentlemen, please just have a look at what a pan we have created-the Ultimate Brownie Pan.Can you imagine that just by means of this small invention, you can get away of annoying overcookedchocolate Brownie Cake? Pardon me, I don’t want to surprise you, but I must tell you , our potential customers, that we’ve made it! Believing that it’s nothing more than a common pan, some people may think that it’s not so difficult to create such a pan. To be honest, it’s not just a simple pan as usual, and it takes a lot of work. Now let me show you how great it is. Here we go!Believing that it’s nothing more than a common pan, some people may think that it’s not so difficult to create such a pan. To be honest, it’s not just a simple pan as usual, and it takes a lot of work. Now let me show you how great it is. Here we go!Maybe nobody will deny this: when baked in arectangular pan, cakes get easily overcooked at thecorners (and to a lesser extent at the edges).But neverwill this happen in a round pan. However, round pansare not the best in respects of saving finite space in anoven. How to solve this problem? This is the key pointthat our work focuses on.Up to now, as you know, there have been two factors determining the quality of apan -- the distribution of heat across the outer edge of and thespace occupied in an oven. Unfortunately, they cannot beachieved at the same time. Time calls for a perfect pan, andthen our Ultimate Brownie Pan comes into existence. TheUltimate Brownie Pan has an outstandingadvantage--optimizing a combination of the two conditions. As you can see, it’s so cute. And when you really begin to use it, you’ll find yourself really enjoy being with it. By using this kind of pan, you can use four pans in the meanwhile. That is to say you can bake more cakes at one time.So you can see that our Ultimate Brownie Pan will certainly be able to solve the two big problems disturbing so many people. And so it will! Feel good? So what are you waiting for? Own it now!。
2010 美赛 MCM 优秀论文
3 Center of Minimum Distance Model.................................. 5
2016年美国大学生数学建模E题英文版论文正稿
In this paper, a model is established to provide a measure of the ability of a region to provide clean water to meet the needs of its population, and find out the reason for the lack of water resources. Specific tasks are as follows:For Task 1: We establish a model. In the model, we think the supply of clean water depends on the amount of surface water, underground water and sewage purification. The water requirements are decided by the amount of life water, agricultural water and industrial water in the region. In water supply, surface water is affected by the annual average temperature, annual average precipitation and forest coverage rate. The groundwater is impacted by the annual average temperature, annual average precipitation. The agricultural water is affected by the population of the region and annual average precipitation. The GDP of the region influences the industrial water consumption. We use the principle of multivariate nonlinear regression to find out the regression coefficient. And then make sure its function. The ratio of water supply and water requirements is used as a measure of the ability of a region to provide clean water. We find that the ability of a region to provide clean water is good or not by comparing the ratio with 1.For Task 2: The model selects the Shandong Province of China as the testing region. We analyse the data of China's Shandong area between 2005 and 2014, and then crystallize the model through the thought of the function fitting and multivariate nonlinear regression. By the model, we think Shandong province's ability to provide clean water is weak. And then from two aspects which physical shortage and shortage of economic, this paper analyses the causes of water shortage in Shandong Province, and thus test the applicability of the model.For Task 3: We select several factors affecting water supply and water demand badly, which is annual precipitation, annual average temperature, the forest coverage rate and population forecast. We analyse the data of China's Shandong area between 2005 and 2014, according which to predict the changes of those factors in 15 years. After that this paper uses the model to analyse the situation of Shandong’s water in 15 years.For Task 4: According to the model in Task 1 and the analysis of the Task 2. We find the main factors influencing the ability to provide clean water in China's Shandong province. By these factors we make the intervention program. In view of the low average annual rainfall, increase the average annual rainfall by artificial rainfall. In view of the forest coverage rate, forest plantation and protect vegetation is came up with. For sewage purification capacity, putting forward to improve sewage treatment technology and improve the sewage conversion rate and increases daily sewage quantity. In view of the total population, we put forward the policy of family planning for water consumption per capita, putting forward to set the daily water consumption per person. And putting forward the industrial wastewater must reach the indexes of the rules, developing seawater desalination technology to increase the supply of clean water.Water has always been the hot spot in the world.The future is also not exceptional. Only finding out the problem, we can suit the remedy to the case.The model measure the ability of a region to provide clean water by analysing the cases which influence the supply and remand of water. Based on this, make a good intervention program. Offering helps to solve global water issues.1 Introduction (4)1.1 Problem Statement (4)1.2Problem Analysis (4)1.2.1Task 1 Analysis (4)1.2.2Task 2 Analysis (4)1.2.3Task 3 Analysis (5)1.2.4Task 4 Analysis (5)1.2.5Task 4 and 5 Analysis (5)2 Assumptions and Notations (6)2.1 Assumptions (6)2.2 Notations (6)3 Model Establishment and Solution (7)3.1 The effect of single factor on the water supply in a certain area (7)3.1.1Effects of annual average temperature, annual average precipitation andforest coverage on surface water resources in a certain area (7)3.1.2 Effects of annual average temperature and annual precipitation ongroundwater resources in a certain area (8)3.1.3 Influence of total population and per capita water consumption on dailywater consumption in a certain area (8)3.1.4 The influence of average annual rainfall and total population onagricultural water consumption in a certain area (9)3.1.5 Effect of average annual rainfall and population in an area of agriculturalwater use (9)3.2 Function Arrangement (9)3.2.1 Water supply function (9)3.2.2 Water demand function (10)3.2.3 The ability of a region to provide clean water (10)3.3 In order to test the accuracy and usability of the model, this model is selectedas a test area in Shandong Province, China. (10)3.3.1 Total surface water resources (11)3.3.2 Total groundwater resources (14)3.3.3 Total industrial water consumption function (16)3.3.4 Total agricultural water consumption function (17)3.3.5 Assessment of water supply capacity (18)3.4.2 Remediation Measures (19)3.5 Forecast for the next 15 years (20)3.5.1 Forecast of average annual rainfall (20)3.5.2 Prediction of annual temperature (21)3.5.3 Prediction of forest cover (22)3.5.4 Prediction of population (23)3.6 Intervention Program (24)3.6.1 Present ofthe Intervention Program (24)3.6.2 Implement ofthe Intervention Program (25)4 Advantages and Shortcoming of the model (26)4.1Advantages： (26)4.2 Shortcoming (26)5 Improvement of model (26)6 Reference (27)7 Appendices (28)7.1 Data used in task 2 (28)7.2 Matlab Source Code (30)1 Introduction1.1 Problem StatementOn the earth, the water that human beings can use the water directly or indirectly, is an important part of natural resources. At present, the total amount of the earth's water is about billion cubic meters, of which the ocean water is billion cubic meters, accounting for about 96.5% of the total global water. In the remaining water, the surface water accounts for 1.78%, 1.69% of the groundwater. The fresh water that human mainly use of is about billion cubic meters, accounting for only 2.53% in the global total water storage. Few of them is distributed in lakes, rivers, soil and underground water, and most of them are stored in the form of glaciers, permafrost and permafrost, The glacier water storage of about billion cubic meters, accounting for 69% of the world's total water, mostly stored in the Antarctic and Greenland, the available clean water in the dwindlingwith time going by.In order to assess the ability to provide clean water of an area, we set up an assessment model.1.2Problem Analysis1.2.1Task 1 AnalysisTask 1 requires establishing a model to measure the ability of a region to provide clean water. At the same time,we also need to provide a measure standard.This paper make the ratio of water supply and water requirements of a region as the measure standard, by which to measure the ability of a region to provide clean water.A region's main source of water is groundwater, surface water and sewagepurification.The model assumes the volume of groundwater in a region is mainly affected by average annual temperature, annual precipitation;Thevolume of surface water is mainly affected by the average annual temperature, annual precipitation, the forest coverage rate.These factors decide water supply of an area.The waterdemand of an area mainly includes living water, agriculture water and industrial water. We assume living water is affected by the population and per capita consumption decision;Agricultural waterdepends an annual precipitation and population decision;Industrial water is mainly decided by a gross regional product.The above factors decide the water demand in a certain area.1.2.2Task 2 AnalysisAccording to the information provided in the map, in Asia, China's Shandong Province is the region meeting the requirements.Through the data collection of Shandong Province, we can find the annual temperature, annual precipitation, the forest coverage rate, groundwater, surface water, sewage treatment capacity, water, agricultural water, industrial water, population,per capita consumption and GDP data.And then according to the model of Task 1,we analyze the ratio by using multivariate nonlinear regression to make sure that Shandong province is a water-deficient area.After proving that Shandon province is short of water through the model, we analyze the reasons for lack of water from two aspects: physical shortage and economic shortage.1.2.3Task 3 AnalysisBecause we already have the relevant data, we can function to fit the relationship between the variables and the year.Thus it is possible to predict in Shandong Provinc e’s data in the 15 years, then input the data into the model to achieve the purpose of prediction.In addition, it can be combined with the actual situation and the selected areas of the corresponding policy to analysis which factors will have a great change in the15 years.We still can analyze from two aspects of society and the environment.Socialaspects includes the promotion of water conservation, population growth;Environmental aspects includes policy changes to the environment, sewage purification capacity enhancement and so on.1.2.4Task 4 AnalysisFormulating plans for intervention mainly start from the perspective of the main model. According to the content of the model, we can still divide all of factors into two types: the social and environmental factors. The intervention programs can be developed based on two types of factors that affect the supply of water, reducing as much as possible the negative impact of the factors that control factors and intensifying the development of a positive impact. In addition, because Shandong Province is beside the sea, desalination and other measures can be developed to increase clean water supply sources.1.2.5Task 4 and 5 AnalysisTask 4 intervention programs indirectly impact the water supply and demand water through a direct impact on GDP model of forest cover, annual precipitation, annual temperature, water emissions, sewage treatment capacity, population growth and the region.2 Assumptions and Notations2.1 Assumptions●The water resources in a region are derived from the purification of surfacewater, groundwater and sewage, and the demand of water resources comes from domestic water, industrial water and agricultural water.●The surface water supply in a certain area is affected only by the average annualtemperature, annual precipitation and forest coverage. The groundwater supply is affected by the annual average temperature and annual precipitation.●The region's water consumption of a certain region depends on the populationand per capita water consumption; Agricultural water consumption is affected by the average annual precipitation and the numberof people. Industrial water is mainly determined by a regional GDP.● A certain region will not suddenly increase or decrease the population largely.●There will not be a serious natural disasters in a region in the next periodof time.2.2 Notations3 Model Establishment and SolutionThe model established here is a use of a region's water supply and water demand ratio to determine whether the water shortage in the region, the main variables involved.3.1 The effect of single factor on the water supply in a certain area3.1.1Effects of annual average temperature, annual average precipitation and forest coverage on surface water resources in a certain areaDue to the average annual temperature, annual precipitation, the forest coverage rate and surface water of linear or nonlinear relationship exists, so first in order to determine the average annual temperature, annual precipitation, the forest coverage rate and surface water, and then the nonlinear multiple regression analysis method to determine the functional relationship between the three factors and surface water.The surface water content is 1y , Average annual precipitation, annual average temperature and forest coverage rate are 1x ,2x ,3x , Using nonlinear regression statistical methods, the use of MATLAB fitting toolbox were identified 1x ,2x ,3x of the highest regression power(MATLAB fitting toolbox of the highest fitting function is the 9 power, greater than the 9 power function is too complex, not much research value), According to the decision coefficient R 2of the regression equation, the corresponding probability value of the statistic P, the regression coefficients β,0β,1n β,2n β, get the regression equation:35612412312345699999910123123111 (1)n n n n n n n n n n n n n n n y x x x x x x βββββ====++++∑∑∑∑∑∑3.1.2 Effects of annual average temperature and annual precipitation on groundwater resources in a certain areaThere is a linear or nonlinear relationship between the average annual temperature, average annual rainfall and the supply of groundwater, according to the idea of 5.1.1, the relationship between the average annual amount of groundwater supply, the average annual precipitation and the supply of groundwater is calculated. And the regression coefficient is determined, and the function relationship between the average annual temperature, average annual precipitation and the supply of groundwater is based on the regression coefficient:Design of underground water for 2y , with an average annual precipitation, annual average air temperature respectively 1x ,2x , using nonlinear regression statistical methods, according to the regression equation with coefficient of determination R2, F statistic corresponds to the probability value p, to determine the regression coefficients β,0β,1n β,2n β, got the regression equation:312412123499992012121111n n n n n n n n n n y x x x x ββββ=====+++∑∑∑∑ （2）3.1.3 Influence of total population and per capita water consumption on daily water consumption in a certain areaThe total population of a region and the amount of water consumption per capita and the daily use of the product of the relationship between the amount of water = total amount *Water usage per person consumption.Set daily water consumption is 5y , the total population, per capita water consumption were 5x , Q , 5y ,Q ,5x , the function of the relationship between the:55y Qx =（3）3.1.4 The influence of average annual rainfall and total population on agricultural water consumption in a certain areaDue to the annual precipitation and the total population and the area of agriculture of area of a water there is a linear or nonlinear relationship, according to the thought of multivariate nonlinear regression can be calculated average annual precipitation and the total population and the area of agriculture with the function relationship between water and to determine the regression coefficient and regression coefficient write GDP and industrial functional relationship between Gross domestic product GDP and industrial water consumption.Let industrial water consumption of 3y , gross production set 4x , using statistical nonlinear regression, regression equation based on the coefficient of determination 2R , F statistical probability value p corresponding to the amount determined regression coefficients 0β,1n β, the regression equation:11193011n n n y x ββ==+∑（4）3.1.5 Effect of average annual rainfall and population in an area of agricultural water useAgricultural water consumption of 4y , design with an average annual rainfall of 1x , 5x , using statistical nonlinear regression, regression equation based on the coefficient of determination 2R , F statistical probability value p corresponding to the amount determined regression coefficients 0β,1n β, the regression equation:3124121234999940151511n n n n n n n n n n y x x x x ββββ===+++∑∑∑∑（5） 3.2 Function Arrangement3.2.1 Water supply functionThe model takes into account a region's water supply from three aspects: surfacewater resources, groundwater resources and the amount of sewage treatment. The function relation between surface water resources, groundwater resources, sewage treatment and water supply is the function, that is, the amount of water supply = surface water resources + groundwater resources.The amount of water supply isX , and the sewage treatment capacity is *Q , by(2)(1): *12X y y Q =++（6）3.2.2 Water demand functionThe model takes into account the need for a region from three aspects: daily water consumption, industrial water consumption and agricultural water consumption. Daily water consumption, industrial water and agricultural water consumption and water demand is a function of the relationship between the function and the function, that is: water demand = daily water consumption + industrial water + agricultural water consumption.A demand for Y , by (3) (4) (5) to:345Y y y y =++（7）3.2.3 The ability of a region to provide clean waterA region to provide clean water and the area of water supply and water demand about, if water supply is greater than demand, the region provide clean water ability strong; on the contrary, the region provide clean water ability is weak. This model provides that a region to provide clean water capacity by the area of water supply and water demand ratio λ determined by (6) (7) available:(1) 1λ>: the region's ability to provide clean water;(2) 1λ=: the area provides a warning of the ability to provide clean water;(3) 1λ<: the ability of the region to provide clean water is weak;3.3 In order to test the accuracy and usability of the model, this model is selected as a test area in Shandong Province, China.Provide the capacity of water resources in China's Shandong Province, we collected in 2005 to 2014 this decade, Shandong Province, the total water supply, surface water resources amount, quantity of groundwater resources, sewage treatment capacity, agricultural water consumption, industrial water, living water, sewage emissions, forest coverage, total population, per capita water use, annual precipitation, GDP (see Appendix for the specific data).3.3.1 Total surface water resourcesSurface water resources amount 1y , groundwater resources quantity is 2y , industrial water use 3y , agricultural water use for 4y , with an average annual precipitation 1x , with an average annual temperature 2x , the forest coverage rate 3x , GDP for 4x , with a total population of 5x .The factors 1y that is affected by 1x , 2x , 3x , in order to determine the relationship between the 1y , and 1x , 2x , 3x ,first use the data in the appendix table to make 1y with 1x , 2x , 3x scatter plots, such as the figure:Figure 1 Surface water resources and average annual rainfallFigure 2 Surface water resources and annual temperatureFigure 3 Surface water resources and forest coverFigure 1 is obtained by MATLAB fitting curve, the fitting found that 1x and 1y is the 6 power function model (εfor random error).（9）Figure 2 is obtained by MATLAB fitting curve, the fitting found that 2x and 1y isthe 61011n n n y x ββε==++∑8 power function model（10）Figure 3 is obtained by MATLAB fitting curve, the fitting found that 3x and 1y is the 8 power function model（11）Combined with the above analysis, the model (9) (10) (11) established the following regression model（12）Directly using the MATLAB statistics toolbox in the command regress solution, the format is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b and r is the residual vector, rint is the confidence interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 1 Surface water regression coefficientCan get the regression coefficient from the figure, the estimated value of the regression coefficient into the model (12) forecast equation81021n n n y x ββε==++∑31031n n n y x ββε==++∑33121212312456683999103331231131111n n n n n n n n n n n n n n n y x x x x x x βββββε=======+++++∑∑∑∑∑∑（13）3.3.2 Total groundwater resourcesFactors that affect the 2y include 1x ,2x , in order to determine the relationship between 2y and 1x ,2x , first use the data in the appendix table to make the A3 and A4 and A5 of the scatter diagram, as shown in figure:Figure 4 the amount of groundwater resources Figure 5 the amount of groundwater resources and annual average temperatureand the average annual rainfallFigure 4 is obtained by MATLAB fitting curve, the fitting found that 1x and 2y is the 6 power function model (ε for random error),（14）Figure 5 is obtained by MATLAB fitting curve, the fitting found that 2x and 2y is the 8 power function model.（15）Combined with the above analysis, the model (9) (10) (11) established the following regression model.（16） Directly using the MATLAB statistics toolbox in the command regress solution, the format ^9665444111138273322223724 1.7610 1.0610 1.8910 1.18103.8410 2.610 3.2910 1.0110y x x x x x x x x x --------=+⨯-⨯+⨯+⨯-⨯+⨯-⨯+⨯62011n n n y x ββε==++∑82021n n n y x ββε==++∑121212123468992013121111n n n n n n n n n n y x x x x ββββε=====++++∑∑∑∑is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b ,and 2R is the residual vector, rint is the confidence interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 2 Regression coefficients of groundwater resourcesCan get the regression coefficient from the figure, the estimated value of the regression coefficient into the model (16) forecast equation（17）Its image is shown in Figure 6^9665445111123821000 1.8210 1.3410 3.37108.49101.3610y x x x x x -----=+⨯-⨯+⨯+⨯-⨯Figure 6 groundwater resources3.3.3 Total industrial water consumption functionFactors that affect 3y is 4x , in order to determine the relationship between 3y and 4x , the first use of the data in the appendix table to make the X and the scatter diagram, as shown in figure:Figure 7 industrial water consumption and GDPFigure 8 industrial water useFigure 7 is obtained by MATLAB fitting curve, the fitting found that 4x and 3y is a function model (εfor random errors),（18）The regression coefficient can be got from the following chartTable 3 Regression coefficient of industrial water consumptionAccording to the above analysis, combined with the model to establish the following regression model, regression coefficient estimation values are substituted into the model (18) to forecast equation.（19）Image as figure 8 3014y x ββε=++^443105.2888410y x -=+⨯3.3.4 Total agricultural water consumption functionFactors that affect the 4y are 1x , 5x , in order to determine the relationship between 4y and 1x , 5x , first using the data in the appendix table to make the 4y and 1x , 5x scatter diagram, as shown in figure:Figure 9 total agricultural water consumption Figure 10 the amount of agricultural water and the average annual rainfalland populationFigure 9 is obtained by MATLAB fitting curve, the fitting found that 4x and 4y is a function model (εfor random errors),（20）Figure 10 is obtained by MATLAB fitting curve, the fitting found that 5x and 4y is a function model (εfor random errors),（21）Combined with the above analysis, the model (20) (21) established the following regression model.（22）Directly using the MATLAB statistics toolbox in the command regress solution, the format is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b ,and 2Ris the residual vector, rint is the confidence 3014y x ββε=++74051n n n y x ββε==++∑121212123468994013121111n n n n n n n n n n y x x x x ββββε=====++++∑∑∑∑interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 4 regression coefficients of agricultural water useAccording to the above analysis, combined with the model to establish the following regression model, regression coefficient estimation values are substituted into the model (22) to forecast equation.（22）Its image is shown in Figure 11Figure 11 function of agricultural water3.3.5 Assessment of water supply capacityAccording to the data model obtained in 3.2.3, Shandong Province in China, therelevant ^737424155514 1.010 2.110510910 2.56810y x x x x ----=⨯-⨯+⨯-⨯+⨯data and the above function is brought into the model and calculated results:By the conclusion of the model, 1λ< shows that the ability to provide clean water in Shandong province is weak.3.4 Cause Analysis and Treatment Measures Water Shortage.3.4.1 the causes of water shortage in Shandong.（1） Water and soil erosion in hilly areas is serious, and water cannot be brought together into a river（2） Shandong is a temperate monsoon climate. Instability is one of the characters of the monsoon climate. Shandong is located in a part a Plain, and it is short of water. It is a big agricultural province. The water used in industry and agriculture is a lot.（3） Water shortage is the basic situation in the province of Shandong, the contradiction between water supply and demand have become increasingly prominent.（4） Total water resources shortage, average, low mu water resources, less water and more and more people, water resources and population, cultivated land resources serious imbalance, which is the main reason caused by a very prominent contradiction between water supply and demand in Shandong.（5） Have a great relationship with the natural geographical location. Shandong is located at the junction of the north and the south, which is a warm temperate monsoon climate. From the rainfall, the first is the uneven distribution of rainfall during the year.（6） As to rainfall distribution, in the southeast of Shandong Province annual rainfall average is up to 8.5 mm, and northwest region's annual average rainfall is only 550 millimeters, basically showing decreases from the southeast Shandong Province to the northwest of successive trend.（7） East Province is a coastal province, but the sea is not the water for drinking.A lot of rain in the coastal areas is typhoon. The available water in these areas is actually very little.（8） Groundwater levels continue to decline due to over exploitation of underground water in many places. The eastern provinces have formed a number of super mining areas.A series of environmental geological problems, such as groundwater pollution, are formed by the formation of the super mining area.（9）Water must not lack of water in the Yellow River in Shandong province. However, the amount of water in the Yellow River is declining year by year, and the available amount is decreasing.（10） Water conservancy project aging, degradation, water supply reduction3.4.2 Remediation Measures（1）With more rain and floods, water conservation, improvement of water cycle, reserve of groundwater resources, to achieve the use of abundant dry.（2）In strict accordance with the requirements of the state on the implementation of 216.030.741289.69X Y λ===<。
1987年美国大学生数学建模竞赛试题
MCM1987A 盐的储存问题
大约15年以来，美国中西部的一个州一直把用于冬天洒在马路上的盐储存在球形屋顶的仓库里，图A-5表示了过去盐是怎样储存的，在用盐铺成的坡道上通过驾驶平头铲车把盐运进、运出仓库，用平头铲车上的铲斗把盐堆成25~30英尺高。
最近一个小组认为这种做法是不安全的，如果铲车太靠近盐堆的顶端，盐就要滑动，铲车就会翻到为加固仓库而筑的拥壁上去，小组建议，如果盐堆是用铲车堆起来的，那么盐堆最高不要超过5英尺。
对这种情况建立一个数学模型，并求出仓库内盐堆的最大高度。
MCM1987B 停车场问题
在New England(新英格兰，美国东北部一地区)一个镇上位于街角处，有一个100英尺×200英尺的停车场，场主雇你来设计这个停车场，也就是如何在停车场的地上画线。
你可能认为要把尽可能多的车驶进停车场，一定应该一辆挨一辆地直角停放，但是缺乏经验的司机感到这样停放有困难，会引起昂贵的保险费要求，为了减少停放车辆时可能造成的损坏，场主就要雇用一些专门停放汽车的有经验的司机。
另一方面，如果汽车从通道进来有一个足够大的转弯半径，那么大多数司机都能轻而易举地一次停放成功。
当然，通道越宽，能够容纳的车辆越少，这会导致停车场场主收入的减少。
2016 美国大学生数学竞赛优秀论文AB
2016年美赛A题热水澡一个人用热水通过一个水龙头来注满一个浴缸，然后坐在在浴缸中，清洗和放松。
不幸的是，浴缸不是一个带有二次加热系统和循环喷流的温泉式浴缸，而是一个简单的水容器。
过一会儿，洗澡水就会明显地变凉，所以洗澡的人需要不停地将热水从水龙头注入，以加热洗浴水。
该浴缸的设计是以这样一种方式，当浴缸里的水达到容量极限，多余的水通过溢流口泄流。
考虑空间和时间等因素，建立一个浴缸的水温模型，以确定最佳的策略，使浴缸里的人可以用这个模型来让整个浴缸保持或尽可能接近初始的温度，而不浪费太多的水。
使用你的模型来确定你的策略对浴缸的形状和体积，浴缸里的人的形状、体积、温度，以及浴缸中的人的运动等因素的依赖程度。
如果这个人一开始用了一种泡泡浴剂加入浴缸，以协助清洗，这会怎样影响你的模型的结果？除了要求的一页MCM摘要提交之外，你的报告必须包括一页的为浴缸用户准备的非技术性的说明书来阐释你的策略，同时解释为什么洗澡水的温度得到均衡地保持是如此之难。
2016年美赛B题太空垃圾在地球轨道上的小碎片的数量已引起越来越多的关注。
据估计，目前有超过500，000块的空间碎片，也被称为轨道碎片，由于被认为对空间飞行器是潜在的威胁而正在被跟踪。
2009年2月10日，俄罗斯卫星kosmos-2251和美国卫星iridium-33相撞之后，该问题受到了新闻媒体更广泛的讨论。
一些消除碎片方法已经被提出。
这些方法包括使用微型的基于太空的喷水飞机和高能量的激光来针对一些特定的碎片和设计大型卫星来清扫碎片。
碎片按照大小和质量分步，从刷了油漆的薄片到废弃的卫星都有。
碎片在轨道上的高速度飞行使得捕捉十分困难。
建立一个以时间为考量的模型，以确定最佳的方法或系列方法，为一个私营企业提供商机，以解决空间碎片问题。
你的模型应该包括定量和定性的对成本，风险，收益的估计，并考虑其他的一些重要因素。
你的模型应该能够评估某种方法，以及组合的系列方法，并能够研究各种重要的假设情况。
2005年美国大学生数学建模竞赛优秀论文翻译
优化和评价的收费亭的数量景区简介由於公路出来的第一千九百三十，至今发展十分迅速在全世界逐渐成为骨架的运输系统，以其高速度，承载能力大，运输成本低，具有吸引力的旅游方便，减少交通堵塞。
以下的快速传播的公路，相应的管理收费站设置支付和公路条件的改善公路和收费广场。
然而，随着越来越多的人口密度和产业基地，公路如花园州公园大道的经验严重交通挤塞收费广场在高峰时间。
事实上，这是共同经历长时间的延误甚至在非赶这两小时收费广场。
在进入收费广场的车流量，球迷的较大的收费亭的数量，而当离开收费广场，川流不息的车辆需挤缩到的车道数的数量相等的车道收费广场前。
因此，当交通繁忙时，拥堵现象发生在从收费广场。
当交通非常拥挤，阻塞也会在进入收费广场因为所需要的时间为每个车辆付通行费。
因此，这是可取的，以尽量减少车辆烦恼限制数额收费广场引起的交通混乱。
良好的设计，这些系统可以产生重大影响的有效利用的基础设施，并有助于提高居民的生活水平。
通常，一个更大的收费亭的数量提供的数量比进入收费广场的道路。
事实上，高速公路收费广场和停车场出入口广场构成了一个独特的类型的运输系统，需要具体分析时，试图了解他们的工作和他们之间的互动与其他巷道组成部分。
一方面，这些设施是一个最有效的手段收集用户收费或者停车服务或对道路，桥梁，隧道。
另一方面，收费广场产生不利影响的吞吐量或设施的服务能力。
收费广场的不利影响是特别明显时，通常是重交通。
其目标模式是保证收费广场可以处理交通流没有任何问题。
车辆安全通行费广场也是一个重要的问题，如无障碍的收费广场。
封锁交通流应尽量避免。
模型的目标是确定最优的收费亭的数量的基础上进行合理的优化准则。
主要原因是拥挤的随着经济的发展，交通系统逐渐形成和完善自己。
不同种类的车辆已迅速改善的数量，质量，速度，和类型。
为了支付维修费用的高速公路，收费站系统的建立。
然而，费时费给我们带来的拥塞，高度增加烦恼的司机。
一般来说，在收费亭的数量大于数量的车道。
美国数学建模竞赛优秀论文阅读报告
2.优秀论文一具体要求：1月28日上午汇报1）论文主要内容、具体模型和求解算法（针对摘要和全文进行概括）；In the part1, we will design a schedule with fixed trip dates and types and also routes. In the part2, we design a schedule with fixed trip dates and types but unrestrained routes.In the part3, we design a schedule with fixed trip dates but unrestrained types and routes.In part 1, passengers have to travel along the rigid route set by river agency, so the problem should be to come up with the schedule to arrange for the maximum number of trips without occurrence of two different trips occupying the same campsite on the same day.In part 2, passengers have the freedom to choose which campsites to stop at, therefore the mathematical description of their actions inevitably involve randomness and probability, and we actually use a probability model. The next campsite passengers choose at a current given campsite is subject to a certain distribution, and we describe events of two trips occupying the same campsite y probability. Note in probability model it is no longer appropriate to say that two trips do not meet at a campsite with certainty; instead, we regard events as impossible if their probabilities are below an adequately small number. Then we try to find the optimal schedule.In part 3, passengers have the freedom to choose both the type and route of the trip; therefore a probability model is also necessary. We continue to adopt the probability description as in part 2 and then try to find the optimal schedule.In part 1, we find the schedule of trips with fixed dates, types (propulsion and duration) and routes (which campsites the trip stops at), and to achieve this we use a rather novel method. The key idea is to divide campsites into different “orbits”that only allows some certain trip types to travel in, therefore the problem turns into several separate small problem to allocate fewer trip types, and the discussion of orbits allowing one, two, three trip types lead to general result which can deal with any value of Y. Particularly, we let Y=150, a rather realistic number of campsites, to demonstrate a concrete schedule and the carrying capacity of the river is 2340 trips.In part 2, we find the schedule of trips with fixed dates, types but unrestrained routes. To better describe the behavior of tourists, we need to use a stochastic model（随机模型）. We assume a classical probability model and also use the upper limit value of small probability to define an event as not happening. Then we use Greedy algorithm to choose the trips added and recursive algorithm together with Jordan Formula to calculate the probability of two trips simultaneously occupying the same campsites. The carrying capacity of the river by this method is 500 trips. This method can easily find theoptimal schedule with X given trips, no matter these X trips are with fixed routes or not. In part 3, we find the optimal schedule of trips with fixed dates and unrestrained types and routes. This is based on the probability model developed in part 2 and we assign the choice of trip types of the tourists with a uniform distribution to describe their freedom to choose and obtain the results similar to part 2. The carrying capacity of the river by this method is 493 trips. Also this method can easily find the optimal schedule with X given trips, no matter these X trips are with fixed routes or not.2）论文结构概述（列出提纲，分析优缺点，自己安排的结构）；1 Introduction2 Definitions3 Specific formulation of problem4 Assumptions5 Part 1 Best schedule of trips with fixed dates, types and also routes.5.1 Method5.1.1 Motivation and justification5.1.2 Key ideas5.2 Development of the model5.2.1Every campsite set for every single trip type5.2.2 Every campsite set for every multiple trip types5.2.3One campsite set for all trip types6 Part 2 Best schedule of trips with fixed dates and types, but unrestrained routes.6.1 Method6.1.1 Motivation and justification6.1.2 Key ideas6.2 Development of the model6.2.1 Calculation of p(T,x,t)6.2.2 Best schedule using Greedy algorithm6.2.3 Application to situation where X trips are given7 Part 3 Best schedule of trips with fixed dates, but unrestrained types and routes.7.1 Method7.1.1 Motivation and justification7.1.2 Key ideas7.2 Development of the model8 Testing of the model----Sensitivity analysis8.1Stability with varying trip types chosen in 68.2The sensitivity analysis of the assumption 4④8.3 The sensitivity analysis of the assumption 4⑥9 Evaluation of the model9.1 Strengths and weaknesses9.1.1 Strengths9.1.2 Weakness9.2 Further discussion10 Conclusions11 References12 Letter to the river managers3）论文中出现的好词好句（做好记录）；用于问题的转化We regard the carrying capacity of the river as the maximum total number of trips available each year, hence turning the task of the river managers into looking for the best schedule itself.表明我们在文中所做的工作We have examined many policies for different river…..问题的分解We mainly divide the problem into three parts and come up with three different….对我们工作的要求：Given the above considerations, we want to find the optimal。
2014美国数学建模-B题paper-30680
A Networks and Machine Learning Approach toDetermine the Best College Coaches of the20th-21st CenturiesTian-Shun Allan Jiang,Zachary T Polizzi,Christopher Qian YuanMentor:Dr.Dan TeagueThe North Carolina School of Science and Mathematics∗February10,2014Team#30680Page2of18Contents1Problem Statement3 2Planned Approach3 3Assumptions3 4Data Sources and Collection44.1College Football (5)4.2Men’s College Basketball (5)4.3College Baseball (5)5Network-based Model for Team Ranking65.1Building the Network (6)5.2Analyzing the Network (6)5.2.1Degree Centrality (6)5.2.2Betweenness and Closeness Centrality (7)5.2.3Eigenvector Centrality (8)6Separating the Coach Effect106.1When is Coach Skill Important? (11)6.2Margin of Win Probability (12)6.3Optimizing the Probability Function (13)6.3.1Genetic Algorithm (13)6.3.2Nelder-Mead Method (14)6.3.3Powell’s Method (14)7Ranking Coaches157.1Top Coaches of the Last100Years (15)8Testing our Model158.1Sensitivity Analysis (15)8.2Strengths (16)8.3Weaknesses (16)9Conclusions17 10Acknowledgments172Team#30680Page3of181Problem StatementCollege sport coaches often achieve widespread recognition.Coaches like Nick Saban in football and Mike Krzyzewski in basketball repeatedly lead their schools to national championships.Because coaches influence both the per-formance and reputation of the teams they lead,a question of great concern to universities,players,and fans alike is:Who is the best coach in a given sport? Sports Illustrated,a magazine for sports enthusiasts,has asked us tofind the best all-time college coaches for the previous century.We are tasked with creat-ing a model that can be applied in general across both genders and all possible sports at the college-level.The solution proposed within this paper will offer an insight to these problems and will objectively determine the topfive coaches of all time in the sports of baseball,men’s basketball,and football.2Planned ApproachOur objective is to rank the top5coaches in each of3different college-level sports.We need to determine which metrics reflect most accurately the ranking of coaches within the last100years.To determine the most effective ranking system,we will proceed as follows:1.Create a network-based model to visualize all college sports teams,theteams won/lost against,and the margin of win/loss.Each network de-scribes the games of one sport over a single year.2.Analyze various properties of the network in order to calculate the skill ofeach team.3.Develop a means by which to decouple the effect of the coach from theteam performance.4.Create a model that,given the player and coach skills for every team,canpredict the probability of the occurrence of a specific network of a)wins and losses and b)the point margin with which a win or loss occurred.5.Utilize an optimization algorithm to maximize the probability that thecoach skill matrix,once plugged into our model,generates the network of wins/losses and margins described in(1).6.Analyze the results of the optimization algorithm for each year to deter-mine an overall ranking for all coaches across history.3AssumptionsDue to limited data about the coaching habits of all coaches at all teams over the last century in various collegiate sports,we use the following assumptions to3Team#30680Page4of18 complete our model.These simplifying assumptions will be used in our report and can be replaced with more reliable data when it becomes available.•The skill level of a coach is ultimately expressed through his/her team’s wins over another and the margin by which they win.This assumes thata team must win to a certain degree for their coach to be good.Even ifthe coach significantly amplifies the skills of his/her players,he/she still cannot be considered“good”if the team wins no games.•The skills of teams are constant throughout any given year(ex:No players are injured in the middle of a season).This assumption will allow us to compare a team’s games from any point in the season to any other point in the season.In reality,changing player skills throughout the season make it more difficult to determine the effect of the coach on a game.•Winning k games against a good team improves team skill more than winning k games against an average team.This assumption is intuitive and allows us to use the eigenvector centrality metric as a measure of total team skill.•The skill of a team is a function of the skill of the players and the skill of the coach.We assume that the skill of a coach is multiplicative over the skill of the players.That is:T s=C s·P s where T s is the skill of the team,C s is the skill of the coach,and P s is a measure of the skill of the players.Making coach skill multiplicative over player skill assumes that the coach has the same effect on each player.This assumption is important because it simplifies the relationship between player and coach skill to a point where we can easily optimize coach skill vectors.•The effect of coach skill is only large when the difference between player skill is small.For example,if team A has the best players in the conference and team B has the worst,it is likely that even the best coach would not be able to,in the short run,bring about wins over team A.However, if two teams are similarly matched in players,a more-skilled coach will make advantageous plays that lead to his/her team winning more often than not.•When player skills between two teams are similarly matched,coach skill is the only factor that determines the team that wins and the margin by which they win by.By making this assumption,we do not have to account for any other factors.4Data Sources and CollectionSince our model requires as an input the results of all the games played in a season of a particular sport,wefirst set out to collect this data.Since we were unable to identify a single resource that had all of the data that we required,we4Team#30680Page5of18 found a number of different websites,each with a portion of the requisite data. For each of these websites,we created a customized program to scrape the data from the relevant webpages.Once we gathered all the data from our sources,we processed it to standardize the formatting.We then aimed to merge the data gathered from each source into a useable format.For example,we gathered basketball game results from one source,and data identifying team coaches from another.To merge them and show the game data for a specific coach,we attempted to match on commonfields(ex.“Team Name”).Often,however,the data from each source did not match exactly(ex.“Florida State”vs“Florida St.”).In these situations,we had to manually create a matching table that would allow our program to merge the data sources.Although we are seeking to identify the best college coach for each sport of interest for the last century,it should be noted that many current college sports did not exist a century ago.The National Collegiate Athletic Association (NCAA),the current managing body for nearly all college athletics,was only officially established in1906and thefirst NCAA national championship took place in1921,7years short of a century ago.Although some college sports were independently managed before being brought into the NCAA,it is often difficult to gather accurate data for this time.4.1College FootballOne of the earliest college sports,College Football has been popular since its inception in the1800’s.The data that we collected ranges from1869to the present,and includes the results andfinal scores of every game played between Division1men’s college football teams(or the equivalent before the inception of NCAA)[2].Additionally,we have gathered data listing the coach of each team for every year we have collected game data[4],and combined the data in order to match the coach with his/her complete game record for every year that data was available.4.2Men’s College BasketballThe data that we gathered for Men’s College Basketball ranges from the sea-son of thefirst NCAA Men’s Basketball championship in1939to the present. Similarly to College Football,we gathered data on the result andfinal scores of each game in the season and infinals[2].Combining this with another source of coach names for each team and year generated the game record for each coach for each season[4].4.3College BaseballAlthough College Baseball has historically had limited popularity,interest in the sport has grown greatly in the past decades with improved media coverage and collegiate spending on the sport.The game result data that we collected5Team#30680Page6of18 ranges from1949to the present,and was merged with coach data for the same time period.5Network-based Model for Team Ranking Through examination of all games played for a specific year we can accurately rank teams for that year.By creating a network of teams and games played, we can not only analyze the number of wins and losses each team had,but can also break down each win/loss with regard to the opponent’s skill.5.1Building the NetworkWe made use of a weighted digraph to represent all games played in a single year.Each node in the graph represents a single college sports team.If team A wins over team B,a directed edge with a weight of1will be drawn from A pointing towards B.Each additional time A wins over B,the weight of the edge will be increased by1.If B beats A,an edge with the same information is drawn in the opposing direction.Additionally,a list containing the margin of win/loss for each game is associated with the edge.For example,if A beat B twice with score:64−60,55−40,an edge with weight two is constructed and the winning margin list4,15is associated with the edge.Since each graph represents a single season of a specific sport,and we are interested in analyzing a century of data about three different sports,we have created a program to automate the creation of the nearly300graphs used to model this system.The program Gephi was used to visualize and manipulate the generated graphs. 5.2Analyzing the NetworkWe are next interested in calculating the skill of each team based on the graphs generated in the previous section.To do this,we will use the concept of central-ity to investigate the properties of the nodes and their connections.Centrality is a measure of the relative importance of a specific node on a graph based on the connections to and from that node.There are a number of ways to calculate centrality,but the four main measures of centrality are degree,betweenness, closeness,and eigenvector centrality.5.2.1Degree CentralityDegree centrality is the simplest centrality measure,and is simply the total number of edges connecting to a specific node.For a directional graph,indegree is the number of edges directed into the node,while outdegree is the number of edges directed away from the node.Since in our network,edges directed inward are losses and edges directed outwards are wins,indegree represents the total number of losses and outdegree measures the total number of wins.Logically,therefore,outdegreeeindegreee represents the winlossratio of the team.This ratiois often used as a metric of the skill of a team;however,there are several6Team#30680Page7of18Figure1:A complete network for the2009-2010NCAA Div.I basketball season. Each node represents a team,and each edge represents a game between the two teams.Note that,since teams play other teams in their conference most often, many teams have clustered into one of the32NCAA Div.1Conferences. weaknesses to this metric.The most prominent of these weaknesses arises from the fact that,since not every team plays every other team over the course of the season,some teams will naturally play more difficult teams while others will play less difficult teams.This is exaggerated by the fact that many college sports are arranged into conferences,with some conferences containing mostly highly-ranked teams and others containing mostly low-ranked teams.Therefore, win/loss percentage often exaggerates the skill of teams in weaker conferences while failing to highlight teams in more difficult conferences.5.2.2Betweenness and Closeness CentralityBetweenness centrality is defined as a measure of how often a specific node acts as a bridge along the shortest path between two other nodes in the graph. Although a very useful metric in,for example,social networks,betweenness centrality is less relevant in our graphs as the distance between nodes is based on the game schedule and conference layout,and not on team skill.Similarly, closeness centrality is a measure of the average distance of a specific node to7Team#30680Page8of18 another node in the graph-also not particularly relevant in our graphs because distance between nodes is not related to team skills.5.2.3Eigenvector CentralityEigenvector centrality is a measure of the influence of a node in a network based on its connections to other nodes.However,instead of each connection to another node having afixed contribution to the centrality rating(e.g.de-gree centrality),the contribution of each connection in eigenvector centrality is proportional to the eigenvector centrality of the node being connected to. Therefore,connections to high-ranked nodes will have a greater influence on the ranking of a node than connections to low-ranking nodes.When applied to our graph,the metric of eigenvector centrality will assign a higher ranking to teams that win over other high-ranking teams,while winning over lower-ranking nodes has a lesser contribution.This is important because it addresses the main limitation over degree centrality or win/loss percentage,where winning over many low-ranked teams can give a team a high rank.If we let G represent a graph with nodes N,and let A=(a n,t)be an adjacency matrix where a n,t=1if node n is connected to node t and a n,t=0 otherwise.If we define x a as the eigenvector centrality score of node a,then the eigenvector centrality score of node n is given by:x n=1λt∈M(n)x t=1λt∈Ga n,t x t(1)whereλrepresents a constant and M(n)represents the set of neighbors of node n.If we convert this equation into vector notation,wefind that this equation is identical to the eigenvector equation:Ax=λx(2) If we place the restriction that the ranking of each node must be positive, wefind that there is a unique solution for the eigenvector x,where the n th component of x represents the ranking of node n.There are multiple different methods of calculating x;most of them are iterative methods that converge on a final value of x after numerous iterations.One interesting and intuitive method of calculating the eigenvector x is highlighted below.It has been shown that the eigenvector x is proportional to the row sums of a matrix S formed by the following equation[6,9]:S=A+λ−1A2+λ−2A3+...+λn−1A n+ (3)where A is the adjacency matrix of the network andλis a constant(the principle eigenvalue).We know that the powers of an adjacency matrix describe the number of walks of a certain length from node to node.The power of the eigenvalue(x)describes some function of length.Therefore,S and the8Team#30680Page9of18 eigenvector centrality matrix both describe the number of walks of all lengths weighted inversely by the length of the walk.This explanation is an intuitive way to describe the eigenvector centrality metric.We utilized NetworkX(a Python library)to calculate the eigenvector centrality measure for our sports game networks.We can apply eigenvector centrality in the context of this problem because it takes into account both the number of wins and losses and whether those wins and losses were against“good”or“bad”teams.If we have the following graph:A→B→C and know that C is a good team,it follows that A is also a good team because they beat a team who then went on to beat C.This is an example of the kind of interaction that the metric of eigenvector centrality takes into account.Calculating this metric over the entire yearly graph,we can create a list of teams ranked by eigenvector centrality that is quite accurate. Below is a table of top ranks from eigenvector centrality compared to the AP and USA Today polls for a random sample of our data,the2009-2010NCAA Division I Mens Basketball season.It shows that eigenvector centrality creates an accurate ranking of college basketball teams.The italicized entries are ones that appear in the top ten of both eigenvector centrality ranking and one of the AP and USA Today polls.Rank Eigenvector Centrality AP Poll USA Today Poll 1Duke Kansas Kansas2West Virginia Michigan St.Michigan St.3Kansas Texas Texas4Syracuse Kentucky North Carolina5Purdue Villanova Kentucky6Georgetown North Carolina Villanova7Ohio St.Purdue Purdue8Washington West Virginia Duke9Kentucky Duke West Virginia10Kansas St.Tennessee ButlerAs seen in the table above,six out of the top ten teams as determined by eigenvector centrality are also found on the top ten rankings list of popular polls such as AP and USA Today.We can see that the metric we have created using a networks-based model creates results that affirms the results of commonly-accepted rankings.Our team-ranking model has a clear,easy-to-understand basis in networks-based centrality measures and gives reasonably accurate re-sults.It should be noted that we chose this approach to ranking teams over a much simpler approach such as simply gathering the AP rankings for vari-ous reasons,one of which is that there are not reliable sources of college sport ranking data that cover the entire history of the sports we are interested in. Therefore,by calculating the rankings ourselves,we can analyze a wider range of historical data.Below is a graph that visualizes the eigenvector centrality values for all games played in the2010-2011NCAA Division I Mens Football tournament.9Team#30680Page10of18 Larger and darker nodes represent teams that have high eigenvector centrality values,while smaller and lighter nodes represent teams that have low eigenvector centrality values.The large nodes therefore represent the best teams in the 2010-2011season.Figure2:A complete network for the2012-2013NCAA Div.I Men’s Basketball season.The size and darkness of each nodes represents its relative eigenvector centrality value.Again,note the clustering of teams into NCAA conferences. 6Separating the Coach EffectThe model we created in the previous section works well forfinding the relative skills of teams for any given year.However,in order to rank the coaches,it is necessary to decouple the coach skill from the overall team skill.Let us assume that the overall team skill is a function of two main factors,coach skill and player skill.Specifically,if C s is the coach skill,P s is the player skill,and T s is10Team #30680Page 11of 18the team skill,we hypothesize thatT s =C s ·P s ,(4)as C s of any particular team could be thought of as a multiplier on the player skill P s ,which results in team skill T s .Although the relationship between these factors may be more complex in real life,this relationship gives us reasonable results and works well with our model.6.1When is Coach Skill Important?We will now make a key assumption regarding player skill and coach skill.In order to separate the effects of these two factors on the overall team skill,we must define some difference in effect between the two.That is,the player skill will influence the team skill in some fundamentally different way from the coach skill.Think again to a game played between two arbitrary teams A and B .There are two main cases to be considered:Case one:Player skills differ significantly:Without loss of generality,assume that P (A )>>P (B ),where P (x )is a function returning the player skills of any given team x .It is clear that A winning the game is a likely outcome.We can draw a plot approximating the probability of winning by a certain margin,which is shown in Figure 3.Margin of WinProbabilityFigure 3:A has a high chance of winning when its players are more skilled.Because the player skills are very imbalanced,the coach skill will likely not change the outcome of the game.Even if B has an excellent coach,the effect of the coach’s skill will not be enough to make B ’s win likely.Case two:Player skills approximately equal:If the player skills of the two teams are approximately evenly matched,the coach skill has a much higher likelihood of impacting the outcome of the game.When the player skills are11Team #30680Page 12of 18similar for both teams,the Gaussian curve looks like the one shown in Figure 4.In this situation,the coach has a much greater influene on the outcome of the game -crucial calls of time-outs,player substitutions,and strategies can make or break an otherwise evenly matched game.Therefore,if the coach skills are unequal,causing the Gaussian curve is shifted even slightly,one team will have a higher chance of winning (even if the margin of win will likely be small).Margin of WinProbabilityFigure 4:Neither A nor B are more likely to win when player skills are the same (if player skill is the only factor considered).With the assumptions regarding the effect of coach skill given a difference in player skills,we can say that the effect of a coach can be expressed as:(C A −C B )· 11+α|P A −P B |(5)Where C A is the coach skill of team A ,C B is the coach skill of team B ,P A is the player skill of team A ,P B is the player skill of team B ,and αis some scalar constant.With this expression,the coach effect is diminished if the difference in player skills is large,and coach effect is fully present when players have equal skill.6.2Margin of Win ProbabilityNow we wish to use the coach effect expression to create a function giving the probability that team A will beat team B by a margin of x points.A negative value of x means that team B beat team A .The probability that A beats B by x points is:K ·e −1E (C ·player effect +D ·coach effect −margin ) 2(6)where C,D,E are constant weights,player effect is P A −P B ,coach effect is given by Equation 5,and margin is x .12Team#30680Page13of18This probability is maximized whenC·player effect+D·coach effect=margin.This accurately models our situation,as it is more likely that team A wins by a margin equal to their combined coach and team effects over team B.Since team skill is comprised of player skill and coach skill,we may calculate a given team’s player skill using their team skill and coach skill.Thus,the probability that team A beats team B by margin x can be determined solely using the coach skills of the respective teams and their eigenvector centrality measures.6.3Optimizing the Probability FunctionWe want to assign all the coaches various skill levels to maximize the likelihood that the given historical game data occurred.To do this,we maximize the probability function described in Equation6over all games from historical data byfinding an optimal value for the coach skill vectors C A and C B.Formally, the probability that the historical data occurred in a given year isall games K·e−1E(C·player effect+D·coach effect−margin)2.(7)After some algebra,we notice that maximizing this value is equivalent to minimizing the value of the cost function J,whereJ(C s)=all games(C·player effect+D·coach effect−margin)2(8)Because P(A beats B by x)is a nonlinear function of four variables for each edge in our network,and because we must iterate over all edges,calculus and linear algebra techniques are not applicable.We will investigate three techniques (Genetic Algorithm,Nelder-Mead Search,and Powell Search)tofind the global maximum of our probability function.6.3.1Genetic AlgorithmAtfirst,our team set out to implement a Genetic Algorithm to create the coach skill and player skill vectors that would maximize the probability of the win/loss margins occurring.We created a program that would initialize1000random coach skill and player skill vectors.The probability function was calculated for each pair of vectors,and then the steps of the Genetic Algorithm were ran (carry over the“mostfit”solution to the next generation,cross random elements of the coach skill vectors with each other,and mutate a certain percentage of the data randomly).However,our genetic algorithm took a very long time to converge and did not produce the optimal values.Therefore,we decided to forgo optimization with genetic algorithm methods.13Team#30680Page14of186.3.2Nelder-Mead MethodWe wanted to attempt optimization with a technique that would iterate over the function instead of mutating and crossing over.The Nelder-Mead method starts with a randomly initialized coach skills vector C s and uses a simplex to tweak the values of C s to improve the value of a function for the next iteration[7]. However,running Nelder-Mead found local extrema which barely increased the probability of the historical data occurring,so we excluded it from this report.6.3.3Powell’s MethodA more efficient method offinding minima is Powell’s Method.This algorithm works by initializing a random coach skills vector C s,and uses bi-directional search methods along several search vectors tofind the optimal coach skills.A detailed explanation of the mathematical basis for Powell’s method can be found in Powell’s paper on the algorithm[8].We found that Powell’s method was several times faster than the Nelder-Mead Method and produced reasonable results for the minimization of our probability function.Therefore,our team decided to use Powell’s method as the main algorithm to determine the coach skills vector.We implemented this algorithm in Python and ran it across every edge in our network for each year that we had data.It significantly lowered our cost function J over several thousand iterations.Rank1962200020051John Wooden Lute Olson Jim Boeheim2Forrest Twogood John Wooden Roy Williams3LaDell Anderson Jerry Dunn Thad Matta The table above shows the results of running Powell’s method until the probability function shown in Equation6is optimized,for three widely separated arbitrary years.We have chosen to show the top three coaches per year for the purposes of conciseness.We will additionally highlight the performance of our top three three outstanding coaches.John Wooden-UCLA:John Wooden built one of the’greatest dynasties in all of sports at UCLA’,winning10NCAA Division I Basketball tournaments and leading an unmatched streak of seven tournaments in a row from1967to 1973[1].He won88straight games during one stretchJim Boeheim-Syracuse:Boeheim has led Syracuse to the NCAA Tour-nament28of the37years that he has been coaching the team[3].He is second only to Mike Krzyzewsky of Duke in total wins.He consistently performs even when his players vary-he is the only head coach in NCAA history to lead a school to fourfinal four appearances in four separate decades.Roy Williams-North Carolina:Williams is currently the head of the basketball program at North Carolina where he is sixth all-time in the NCAA for winning percentage[5].He performs impressively no matter who his players are-he is one of two coaches in history to have led two different teams to the Final Four at least three times each.14Team#30680Page15of187Ranking CoachesKnowing that we are only concerned withfinding the topfive coaches per sport, we decided to only consider thefive highest-ranked coaches for each year.To calculate the overall ranking of a coach over all possible years,we considered the number of years coached and the frequency which the coach appeared in the yearly topfive list.That is:C v=N aN c(9)Where C v is the overall value assigned to a certain coach,N a is the number of times a coach appears in yearly topfive coach lists,and N c is the number of years that the coach has been active.This method of measuring overall coach skill is especially strong because we can account for instances where coaches change teams.7.1Top Coaches of the Last100YearsAfter optimizing the coach skill vectors for each year,taking the topfive,and ranking the coaches based on the number of times they appeared in the topfive list,we arrived at the following table.This is our definitive ranking of the top five coaches for the last100years,and their associated career-history ranking: Rank Mens Basketball Mens Football Mens Baseball 1John Wooden-0.28Glenn Warner-0.24Mark Marquess-0.27 2Lute Olson-0.26Bobby Bowden-0.23Augie Garrido-0.24 3Jim Boeheim-0.24Jim Grobe-0.18Tom Chandler-0.22 4Gregg Marshall-.23Bob Stoops-0.17Richard Jones-0.19 5Jamie Dixon-.21Bill Peterson-0.16Bill Walkenbach-0.168Testing our Model8.1Sensitivity AnalysisA requirement of any good model is that it must be tolerant to a small amount of error in its inputs.In our model,possible sources of error could include im-properly recorded game results,incorrectfinal scores,or entirely missing games. These sources of error could cause a badly written algorithm to return incorrect results.To test the sensitivity of our model to these sources of error,we decided to create intentional small sources of error in the data and compare the results to the original,unmodified results.Thefirst intentional source of error that we incorporated into our model was the deletion of a game,specifically a regular-season win for Alabama(the team with the top-ranked coach in1975)over Providence with a score of67to 60.We expected that the skill value of the coach of the Alabama team would15。
美国大学生数学建模比赛的论文格式
ContentsⅠIntroduction (1)1.1Problem Background (1)1.2Previous Research (2)1.3Our Work (2)ⅡGeneral Assumptions (3)ⅢNotations and Symbol Description (3)3.1 Notations (4)3.2 Symbol Description (4)ⅣSpread of Ebola (5)4.1 Traditional Epidemic Model (5)4.1.1.The SEIR Model (5)4.1.2 (6)4.1.3 (6)4.2 Improved Model (7)4.2.1.The SEIHCR Model (8)4.2.2 (9)ⅤPharmaceutical Intervention (9)5.1 Total Quantity of the Medicine (10)5.1.1.Results from WHO Statistics (10)5.1.2.Results from SEIHCR Model (11)5.2 Delivery System (12)5.2.1.Locations of Delivery (13)5.2.2 (14)5.3 Speed of Manufacturing (15)ⅥOther Important Interventions (16)6.1 Safer Treatment of Corpses (17)6.2 Conclusion (18)ⅦControl and Eradication of Ebola (19)7.1 How Ebola Can Be Controlled (20)7.2 When Ebola Will Be Eradicated (21)ⅧSensitivity Analysis (22)8.1 Impact of Transmission Rate (23)8.2 Impact of the Incubation Priod (24)ⅨStrengths and Weaknesses (25)9.1 Strengths (26)9.2 Weaknesses (27)9.3 Future Work (28)Letter to the World Medical Association (30)References (31)ⅠIntroduction1.1.Promblem Background1.2.Previous Research1.3.Our WorkⅡGeneral Assumptions●●ⅢNotations and Symbol Description3.1. Notataions3.2. Symbol DescriptionSymbol DescriptionⅣSpread of Ebola4.1. Traditional Epidemic Model4.1.1. The SEIR Model4.1.2. Outbreak Data4.1.3. Reslts of the SEIR Model4.2. Improved Model4.2.1. The SEIHCR Model4.2.2. Choosing paametersⅤPharmaceutical Intervention 5.1. Total Quantity of the Medicine 5.1.1. Results from WHO Statistics5.2. Delivery System5.2.1. Locations of Delivery5.2.2. Amount of Delivery5.3. Speed of Manufacturong5.4. Medicine EfficacyⅥOther Important Interventions 6.1. Safer Treatment of Corpses6.2. ConclusionⅦControl and Eradication of Ebola 7.1. How Ebola Can Be Controlled7.2. When Ebola Will Be EradicatedⅧSensitivity Analysis8.1. Impact of Transmission Rate8.2. Impact of Incubation PeriodⅨStrengths and Weaknesses 9.1. Strengths●●●9.2. Weaknesses●●●9.3.Future WorkLetter to the World Medical AssociationTo whom it may concern,Best regards,Team #32150References ［1］［2］［3］［4］。
05年美国大学生数学建模竞赛A题特等奖论文翻译
在每一时刻流出水的体积等于裂口的面积乘以水的速率乘以时间：
h h Vwater leaeing = wbreach (
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其中：V 是体积， w 是宽度， h 是高度， s 是速度， t 是时间。
我们假设该湖是一个大的直边贮槽，所以当水的高度确定时，其面积不改变。 这意味着，湖的高度等于体积除以面积
在南卡罗莱那州的中央，一个湖被一个 75 年的土坝抑制。如果大坝被地震破坏 将会发生什么事？这个担心是基于1886年发生在查尔斯顿的一场地震,科学家们 相信它里氏7.3级[联邦能源管理委员会2002]。断层线的位置几乎直接在穆雷湖 底(SCIway 2000;1997，1998年南CarolinaGeological调查)和在这个地区小地震 的频率迫使当局考虑这样一个灾难的后果。
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（我们的模型转换以距离和时间做模型的单位，因此每个步长为 1）。第二个条 件的作用是受潮尖峰，因为它看起来不同并补偿于在每个点的任一边上的点。
我们发现该模型对粗糙度参数 n 高度敏感（注意，这是只在通道中的有效粗 糙度）。当 n 很大（即使在大河流的标准值 0.03）。对水流 和洪水堆积的倾向有 较高的抗拒，这将导致过量的陡水深资料，而且往往使模型崩溃。幸运的是，我
我们的任务是预测沿着Saluda河从湖穆雷大坝到哥伦比亚的水位变化，如果 发生了1886年相同规模的地震破坏了大坝。特别是支流罗尔斯溪会回流多远和哥 伦比亚南卡罗来纳州的州议会大厦附近的水位会多高。
美国大学生数学建模竞赛MCM写作模板（各个部分）
美国⼤学⽣数学建模竞赛MCM写作模板（各个部分）摘要：第⼀段：写论⽂解决什么问题1．问题的重述a. 介绍重点词开头：例1：“Hand move” irrigation, a cheap but labor-intensive system used on small farms, consists of a movable pipe with sprinkler on top that can be attached to a stationary main.例2:……is a real-life common phenomenon with many complexities.例3：An (effective plan) is crucial to………b. 直接指出问题：例1：We find the optimal number of tollbooths in a highway toll-plaza for a given number of highway lanes: the number of tollbooths that minimizes average delay experienced by cars.例2：A brand-new university needs to balance the cost of information technology security measures with the potential cost of attacks on its systems.例3：We determine the number of sprinklers to use by analyzing the energy and motion of water in the pipe and examining the engineering parameters of sprinklers available in the market.例4: After mathematically analyzing the ……problem, our modeling group would like to present our conclusions, strategies, (and recommendations )to the …….例5：Our goal is... that (minimizes the time )……….2．解决这个问题的伟⼤意义反⾯说明。
数学建模 美赛特等奖论文(中文版)分析溃坝：针对南卡罗来纳州大坝坍塌建立模型
分析溃坝：针对南卡罗来纳州大坝坍塌建立模型 摘要萨鲁达大坝建立在卡罗莱纳州的墨累湖与萨鲁达河之间，如果发生地震大坝就会坍塌。
本文通过建立模型来分析以下四种大坝决口时水的流量以及洪水泛滥时水的流量：● 大坝的绝大部分被瞬间侵蚀看成是大坝瞬间彻底坍塌；● 大坝的绝大部分被缓慢侵蚀看成是大坝延期彻底坍塌；● 管涌就是先形成一个小孔，最终形成一个裂口；● 溢出就是大坝被侵蚀后，形成一个梯形的裂口。
本文建立了两个模型来描述下游洪水的泛滥情况。
两个模型都采用离散网格的方法，将一个地区看成是一个网格，每个网格都包含洪水的深度和体积。
复力模型运用了网格的速度、重力以及邻近网格的压力来模拟水流。
下坡模型假定水流速度与邻近网格间水位高度的成正比例。
下坡模型是高效率的、直观的、灵活的，可以适用于已知海拔的任何地区。
它的两个参数稳定并限制了水流，但该模型的预测很少依赖于它们的静态值。
对于萨鲁达溃坝，洪水总面积为25.106km ；它还没有到达国会大厦。
罗威克里克的洪水向上游延伸了km 4.4，覆盖面积达24.26.1km -变量及假设表1说明了用来描述和模拟模型的变量，表2列出了模拟程序中的参数。
表 1模型中的变量.变量 定义溃坝时的水流量速率1TF Q 瞬间彻底坍塌2TF Q 延期彻底坍塌PIPE Q 管涌OT Q 溢出peak Q 最大流速溃坝时水流出到停止所用时间1TF t 瞬间彻底坍塌2TF t 延期彻底坍塌PIPE t 管涌OT t 溢出V ∆ 溃坝后从墨累湖里流出的水的总体积Lm Vol 墨累湖的原来体积LM Area 墨累湖的原来面积breach d 从裂口到坝顶距离breach t 从裂口开始到溃坝形成的时间 近似圆锥的墨累湖的侧面一般假设● 正常水位是在溃坝前的湖水位置。
● 河道中的水流不随季节变化而变动。
● 墨累湖里的水的容积可以看作为一个正圆锥（图1 ）。
表2 模拟程序中的参数 参数 所取值 意义BREACH_TYPE 变量 瞬间彻底坍塌，延期彻底坍，管涌，溢出模型中的一种 T ∆ 0.10 时间不长的长度(s)MIN_DEPTH 0001.0 网格空时的水的深度(m) FINAT T 100000 大坝彻底决口所用时间 b T 3600 溃坝达最大值的时间(s) peak Q 25000 溃坝的最大流速(m 3/s) breach d 30 蓄水池的最初深度(m) LM Volume 910714.2⨯ 墨累湖的总体积(m 3) LM Area 610202⨯ 墨累湖的总面积(m 2)k 504.0 扩散因素 (控制两网格间交换的水的数量) MAX_LOSS_FRAC 25.0 单位网格中水的最大流失量图 1. 水库近似一个正圆锥.大坝假设● 萨鲁达大坝在以下四种方式之一坍塌：-瞬间彻底坍塌，-延期彻底坍塌，-管涌，-溢出。
2015美国数学建模竞赛优秀论文
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Contents 1 Introduction and restatement
1.1 Background…………………………………………………………………… 2 1.2 Restatement of problems……………………………………………………..... 2 1.3 An overview of sustainable development…………………………………........ 3
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2015
Mathematical Contest in Modeling (MCM/ICM) Summary Sheet
In order to measure a country's level of sustainable development precisely, we establish a evaluation system based on AHP(Analytic Hierarchy Process). We classify quantities of influential factors into three parts, including economic development, social development and the situation of resources and environment. Then we select 6 to 8 significant indicators of every part. With regard to the practical situation of the country we are focusing on, we are able to build judgment matrixes and obtain the weight of every indicator. Via liner weighting method, we give the definition of comprehensive index of sustainable development. Referring to classifications given by experts, we can judge whether this country is sustainable or not precisely. In task 2, we choose Cambodia as our target nation. We obtain detailed data from the world bank. Using standardized data of this country, via the process above, we successfully get the comprehensive index is 0.48, which means its sustainable development ability is weak. We notice that industrial value added, enrollment rate in higher learning institutions and other five indicators contribute most to the sustainable development according to our model, so we make policies and plans which focus on the improvement of these aspects, including developing industry and improving social security. We also recommend ICM to assistant Cambodia in these aspects in order to optimize the development. To solve task 3, we consider several unpredictable factors that may influence sustainable development, such as politics, climate changes and so on. After taking all of these factors into consideration, we predict the value of every indicator in 2020, 2030 and 2034 according to our plans. After calculating, we are delighted that the comprehensive index has grown up to 0.91, meaning this country is quite sustainable. This also reflects that our model and plans are reasonable and correct.
美赛：27688---数模英文论文
Team Control NumberFor office use only27688For office use onlyT1 ________________F1 ________________T2 ________________F2 ________________T3 ________________Problem Chosen F3 ________________ T4 ________________C F4 ________________2014Mathematical Contest in Modeling (MCM/ICM) Summary SheetThe research of influence based on the characteristic of a network To find the influential nodes in the network, the key is the definition of “influential”and how to measure the influence. In this paper, we use two kinds of metrics to measure the influence of coauthor network and citation network. In coauthor network, both the Authority and Importance of the researchers are proposed to measure the influential of researcher. And the second one in citation network take the citation times, publication time and the position in the network into account.For the evaluation of coauthor, we first construct a coauthor network with 511 vertices and 18000 edges and it is an undirected graph. Next, we use software UCInet to analyze the degree centrality, eigenvector centrality, closeness centrality and betweenness centrality of the network. Since there is no evident transfer relationship in the coauthor network, we using Authority and Importance to measure the influence of a research. In detail, the Authority is correlated with the coauthoring times with Paul Erdös and the Importance is measured by eigenvector centrality. Finally, we rank the researchers whose authority is larger than 2 according to their importance. And the top 5 most influential researchers are: RODL, VOJTECH; LOVASZ, LASZLO; GRAHAM, RONALD LEWIS; PACH, JANOS; BOLLOBAS, BELA. Finally, we search for some data through websites and verify these people are really influential.For the evaluation of papers, we first compare the difference between the citation network and coauthor network. According to the characteristic of Directed Acyclic Graph(DAG), we define a contribution coefficient and self-contribution coefficient by making an analogy with the energy transfer in the food chain. Considering the less-effectiveness of PageRank Algorithm and Hits Algorithm, we design an algorithm, which is effective in solving the DAG problem, to calculate the contribution coefficient. We find 3 most influential papers: Paper 14, Paper 4 and Paper 2 in the NetSciFoundation.pdf.In the third part, we implement our model to analyze a corporation ownership network. We use the value of the company’s cash, stock, real estate, technical personnel, patent and relationships to define its value. And we use the proportion of stock to measure the control ability of parent company. Applying the model and algorithm of citation network, we find 15 influential companies. Then we find that 9 of them are in the top 20 of authoritative ranking, which verifies the rationality of our result.Finally, we describe how we can utilize these influential models to do some socialized service, to aid in making decision on company acquisition and to carry out strategic attack.Team #27688Page 1 of 18 1. IntroductionNowadays, coauthor network and citation network are built to determine influence of academic research. Paul Erdös, one of the most influential researchers who had over 500 coauthors and published over 1400 technical research papers. There exists a coauthor network among those who had coauthored with Erdös and those who had coauthored with Erdös’s directed coauthors.In this paper, we first analyze this coauthor network and find some researchers who have significant influence. Then, we analyze the citation network of some set of foundational papers in the emerging field of network science. Furthermore, we determine some measures to find some most influential papers. After that, we use the data of US Corporate Ownership to construct a new network and test the applicability of our model and algorithm. Finally, we describe some applications of using the analysis of different networks.In section 3, the coauthor network is an undirected graph. We first analyze four kinds of centrality: Degree Centrality, Eigenvector Centrality, Closeness Centrality and Betweenness Centrality. Additional, the Degree distribution and Clustering coefficient are also the important properties of the network. Then, we define Authority and Importance to measure the influence of a researcher. Authority can be measured by the coauthoring times with Erdös. It is clearly that the researcher who coauthors with more people is more important. Since this is not a problem about “information flow”, we only cons ider the influence of those directed coauthor and neglect the transitivity of influence. That is to say, Importance can be measure by Eigenvector Centrality. Finally, we choose some people with higher authority and rank them according to their Eigenvector Centrality.In section 4, the citation network is different from the coauthor network. As the citation relation is related to publication time, the citation network is a Directed Acyclic Graph(DAG). Traditionally, we calculate the nodes’ importance of a n etwork by using PageRankAlgorithm[17] and HITS Algorithm[18]. However, both of them involve matrix multiplicationand repeated iterative process, which is less-effective. Since the network satisfies theproperty of Directed Acyclic Graph(DAG), we draw on the thought of topological sorting to design a more effective algorithm. In this citation network, there exists transitive relation that does not exist in the coauthor network. We first use software UCInet to calculate thecentrality of each paper. And then we take these metrics, publication time and times cited count into account to develop a new model. In this model, we learn from the energy transfersin the food chain and define an initial contribution coefficient to measure its authority. In addition, we define a self-contribution coefficient to measure the influence from other papers. Finally, we design an algorithm to calculate each paper’s final contribution coefficientto measure the paper’s influence.In section 5, we use nearly 500 US Media Companies to construct an ownership network. Then we set the initial value of each company according to their case, stock, real estate, technical personnel, patent and relationships. And we set a control coefficient to measure the ownership between two companies. Then we can use the algorithm in citation network to find someTeam #27688Page 2 of 18influential companies.In the fourth part, we utilize these influential models to do some social service, aid in making decision on company acquisition and carry out strategic attack.In general, the article is written follows:(1)Build a coauthor network for question 1.(2)Build the evaluation index of the most influential coauthor to estimate the influenceof coauthors in the coauthor network.(3)Build citation network and define the influence criterion of papers to estimate themost influential paper.(4)Implement our model to the US Corporate Ownership network to analyze theimportance and the value of the company.(5)Finally, we discuss about the basic theory, the use and effectiveness of the science ofnetwork.2. Assumptions and Justification(1)We use number 1..16 to represent the paper given in the NetSciFoundation.pdf according totheir sequence. It is worth mentioning that the information of paper 7 given in the file seems to be wrong. Hence, we regard it as an isolated vertex in the network.(2)The researchers’ authority, it is correlated with the coauthoring times with Paul Erdös. Inthe coauthor network, we know that all of them have coauthored with Erdös and Erdös is such an excellent mathematician. So it is suitable for us to assume that more times coauthored with Erdös, more authority the researcher is.(3)We do not consider the influence of the paper’s content and field because the cited times indifferent fields have no comparability. In question 3 we know that 16 papers are in the emerging field of network science, so it is reasonable for us to simplify this problem.(4)When constructing the citation network, we only take those papers citing more than twopapers in 16 given papers and also having been cited by other papers. Absolutely, the citation network is infinite. In this paper, we aim to find influential papers. Hence, we give up those less important papers and restrict the scale of our network.(5)We assume that the citation relation is effective. If a paper cited other papers, we considerthat the author admitted the positively effect of the cited paper. Since the influence of a paper is related to the citation times, our assumption can improve the validity of the result.(6)The data in our paper is effective. Our dataset is searched in Web of Science and GoogleScholar, which are equipped with high authority.Team #27688Page 3 of 18 3. Coauthor Network3.1 Building the modelA coauthor network can be built to help analyze the influence of the researchers whose Erdös Number are 1. Obviously, this is a social network. In the network, each node represents a researcher who has coauthored with Paul Erdös and each link could represent the coauthoring relationship between two researchers. Since the coauthor matrix is symmetrical, we know that there is no different between A coauthors withB and B coauthors with A. Therefore, the coauthor network is an undirected network which has 511 vertices. We use software Gephi to draw the graph and the network diagram is shown in Figure 1.Figure 1: the co-author networkIn this graph, the vertex represents a researcher and the edge represents the coauthoring relation. The size of the vertex represents its coauthoring times with Erdös and the darker the color is, the more people he coauthored with. There are 511 vertices and 18000 edges.In this network, there are many basic measures and metrics, such as Degree, Centrality, Clustering coefficient, Density, Betweenness and so on. In this paper, we first choose several important measures for analyzing this network and show them as follows. [1]Of course, the common property is CENTRALITY. Centrality is a crucial metric to evaluate the influence of a vertex. In the following, we discuss several classic Centralities and analyze theirdifference.⏹DEGREE CENTRALITYThe degree of a vertex in a graph is the number of edges connected to it. We will denote thedegree of vertex i by d i. And the simplest centrality measure, which is called degree centrality ( C d ), is just the degree of a vertex. That means:C d(i ) d iTeam #27688 Page 4 of 18In a social network, for instance, it seems reasonable to suppose that individuals whohave connections to many others might have more influence, more access to information, or more prestige than those who have fewer connections.⏹ EIGENVECTOR CENTRALITYSometimes, all neighbors of a vertex are not equivalent. Hence, Bonacich [2] puts forwardEigenvector centrality to cope with this situation. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.λd i = ∑r ij d j jWhere:r ij represents the elements in the adjacency matrix; d irepresents the degree centrality of vertex iUsually, we choose the eigenvector corresponding to the maximal eigenvalue to be the eihenvector centrality( C e )[3]. ⏹ CLOSENESS CENTRALITYCloseness centrality measures the mean distance from a vertex to other vertices, which canused to analyze the position of a vertex in the network [1]. ∑ D ijC c (i ) = j ( ≠i )-n 1Where: D ij is the distance between vertex i and vertex j ;C c (i ) is the closeness centrality of vertex i ;n is the number of vertices. ⏹ BETWEENNESS CENTRALITYBetweenness centrality measures the extent to which a vertex lies on paths between othervertices [1]. That is to say, a vertex with a higher betweenness centrality plays a moreimportant role in the connection of the network.n nC b ( k ) = ∑∑[ g ij ( k ) / g ij ] i jWhere: g ij ( k ) represents the number of shortest path between i andj through k ; g ij represents the number of shortest path between i andjThen, we use the UCINET to calculate some basic metrics and show them in table 1.Table 1: the basic data of centralitytype Degree Closeness Betweenness EigenvectorAverage 1.292 2.115 0.461 3.055Minimum 0.000 0.196 0.0000.000 Maximum 10.392 2.201 7.508 36.515According to the above table and Figure 1, we can know that about 30 vertices have 3 times more than the average degree. That is to say, these researchers have many coauthors. In addition, since the average value of closeness is close to its maximum, we know that there are few vertices。
美国大学生数学建模竞赛二等奖论文
美国⼤学⽣数学建模竞赛⼆等奖论⽂The P roblem of R epeater C oordination SummaryThis paper mainly focuses on exploring an optimization scheme to serve all the users in a certain area with the least repeaters.The model is optimized better through changing the power of a repeater and distributing PL tones,frequency pairs /doc/d7df31738e9951e79b8927b4.html ing symmetry principle of Graph Theory and maximum coverage principle,we get the most reasonable scheme.This scheme can help us solve the problem that where we should put the repeaters in general cases.It can be suitable for the problem of irrigation,the location of lights in a square and so on.We construct two mathematical models(a basic model and an improve model)to get the scheme based on the relationship between variables.In the basic model,we set a function model to solve the problem under a condition that assumed.There are two variables:‘p’(standing for the power of the signals that a repeater transmits)and‘µ’(standing for the density of users of the area)in the function model.Assume‘p’fixed in the basic one.And in this situation,we change the function model to a geometric one to solve this problem.Based on the basic model,considering the two variables in the improve model is more reasonable to most situations.Then the conclusion can be drawn through calculation and MATLAB programming.We analysis and discuss what we can do if we build repeaters in mountainous areas further.Finally,we discuss strengths and weaknesses of our models and make necessary recommendations.Key words:repeater maximum coverage density PL tones MATLABContents1.Introduction (3)2.The Description of the Problem (3)2.1What problems we are confronting (3)2.2What we do to solve these problems (3)3.Models (4)3.1Basic model (4)3.1.1Terms,Definitions,and Symbols (4)3.1.2Assumptions (4)3.1.3The Foundation of Model (4)3.1.4Solution and Result (5)3.1.5Analysis of the Result (8)3.1.6Strength and Weakness (8)3.1.7Some Improvement (9)3.2Improve Model (9)3.2.1Extra Symbols (10)Assumptions (10)3.2.2AdditionalAdditionalAssumptions3.2.3The Foundation of Model (10)3.2.4Solution and Result (10)3.2.5Analysis of the Result (13)3.2.6Strength and Weakness (14)4.Conclusions (14)4.1Conclusions of the problem (14)4.2Methods used in our models (14)4.3Application of our models (14)5.Future Work (14)6.References (17)7.Appendix (17)Ⅰ.IntroductionIn order to indicate the origin of the repeater coordination problem,the following background is worth mentioning.With the development of technology and society,communications technology has become much more important,more and more people are involved in this.In order to ensure the quality of the signals of communication,we need to build repeaters which pick up weak signals,amplify them,and retransmit them on a different frequency.But the price of a repeater is very high.And the unnecessary repeaters will cause not only the waste of money and resources,but also the difficulty of maintenance.So there comes a problem that how to reduce the number of unnecessary repeaters in a region.We try to explore an optimized model in this paper.Ⅱ.The Description of the Problem2.1What problems we are confrontingThe signals transmit in the way of line-of-sight as a result of reducing the loss of the energy. As a result of the obstacles they meet and the natural attenuation itself,the signals will become unavailable.So a repeater which just picks up weak signals,amplifies them,and retransmits them on a different frequency is needed.However,repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.In addition to geographical separation,the“continuous tone-coded squelch system”(CTCSS),sometimes nicknamed“private line”(PL),technology can be used to mitigate interference.This system associates to each repeater a separate PL tone that is transmitted by all users who wish to communicate through that repeater. The PL tone is like a kind of password.Then determine a user according to the so called password and the specific frequency,in other words a user corresponds a PL tone(password)and a specific frequency.Defects in line-of-sight propagation caused by mountainous areas can also influence the radius.2.2What we do to solve these problemsConsidering the problem we are confronting,the spectrum available is145to148MHz,the transmitter frequency in a repeater is either600kHz above or600kHz below the receiver frequency.That is only5users can communicate with others without interferences when there’s noPL.The situation will be much better once we have PL.However the number of users that a repeater can serve is limited.In addition,in a flat area ,the obstacles such as mountains ,buildings don’t need to be taken into account.Taking the natural attenuation itself is reasonable.Now the most important is the radius that the signals transmit.Reducing the radius is a good way once there are more users.With MATLAB and the method of the coverage in Graph Theory,we solve this problem as follows in this paper.Ⅲ.Models3.1Basic model3.1.1Terms,Definitions,and Symbols3.1.2Assumptions●A user corresponds a PLz tone (password)and a specific frequency.●The users in the area are fixed and they are uniform distribution.●The area that a repeater covers is a regular hexagon.The repeater is in the center of the regular hexagon.●In a flat area ,the obstacles such as mountains ,buildings don’t need to be taken into account.We just take the natural attenuation itself into account.●The power of a repeater is fixed.3.1.3The Foundation of ModelAs the number of PLz tones (password)and frequencies is fixed,and a user corresponds a PLz tone (password)and a specific frequency,we can draw the conclusion that a repeater can serve the limited number of users.Thus it is clear that the number of repeaters we need relates to the density symboldescriptionLfsdfminrpµloss of transmission the distance of transmission operating frequency the number of repeaters that we need the power of the signals that a repeater transmits the density of users of the areaof users of the area.The radius of the area that a repeater covers is also related to the ratio of d and the radius of the circular area.And d is related to the power of a repeater.So we get the model of function()min ,r f p µ=If we ignore the density of users,we can get a Geometric model as follows:In a plane which is extended by regular hexagons whose side length are determined,we move a circle until it covers the least regular hexagons.3.1.4Solution and ResultCalculating the relationship between the radius of the circle and the side length of the regular hexagon.[]()()32.4420lg ()20lg Lfs dB d km f MHz =++In the above formula the unit of ’’is .Lfs dB The unit of ’’is .d Km The unit of ‘‘is .f MHz We can conclude that the loss of transmission of radio is decided by operating frequency and the distance of transmission.When or is as times as its former data,will increase f d 2[]Lfs .6dB Then we will solve the problem by using the formula mentioned above.We have already known the operating frequency is to .According to the 145MHz 148MHz actual situation and some authority material ,we assume a system whose transmit power is and receiver sensitivity is .Thus we can conclude that ()1010dBm mW +106.85dBm ?=.Substituting and to the above formula,we can get the Lfs 106.85dBm ?145MHz 148MHz average distance of transmission .()6.4d km =4mile We can learn the radius of the circle is 40mile .So we can conclude the relationship between the circle and the side length of regular hexagon isR=10d.1)The solution of the modelIn order to cover a certain plane with the least regular hexagons,we connect each regular hexagon as the honeycomb.We use A(standing for a figure)covers B(standing for another figure), only when As don’t overlap each other,the number of As we use is the smallest.Figure1According to the Principle of maximum flow of Graph Theory,the better of the symmetry ofthe honeycomb,the bigger area that it covers(Fig1).When the geometric centers of the circle andthe honeycomb which can extend are at one point,extend the honeycomb.Then we can get Fig2,Fig4:Figure2Fig3demos the evenly distribution of users.Figure4Now prove the circle covers the least regular hexagons.Look at Fig5.If we move the circle slightly as the picture,you can see three more regular hexagons are needed.Figure 52)ResultsThe average distance of transmission of the signals that a repeater transmit is 4miles.1000users can be satisfied with 37repeaters founded.3.1.5Analysis of the Result1)The largest number of users that a repeater can serveA user corresponds a PL and a specific frequency.There are 5wave bands and 54different PL tones available.If we call a code include a PL and a specific frequency,there are 54*5=270codes.However each code in two adjacent regular hexagons shouldn’t be the same in case of interfering with each other.In order to have more code available ,we can distribute every3adjacent regular hexagons 90codes each.And that’s the most optimized,because once any of the three regular hexagons have more codes,it will interfere another one in other regular hexagon.2)Identify the rationality of the basic modelNow we considering the influence of the density of users,according to 1),90*37=3330>1000,so here the number of users have no influence on our model.Our model is rationality.3.1.6Strength and Weakness●Strength:In this paper,we use the model of honeycomb-hexagon structure can maximize the use of resources,avoiding some unnecessary interference effectively.It is much more intuitive once we change the function model to the geometric model.●Weakness:Since each hexagon get too close to another one.Once there are somebuildingsor terrain fluctuations between two repeaters,it can lead to the phenomenon that certain areas will have no signals.In addition,users are distributed evenly is not reasonable.The users are moving,for example some people may get a party.3.1.7Some ImprovementAs we all know,the absolute evenly distribution is not exist.So it is necessary to say something about the normal distribution model.The maximum accommodate number of a repeater is 5*54=270.As for the first model,it is impossible that 270users are communicating in a same repeater.Look at Fig 6.If there are N people in the area 1,the maximum number of the area 2to area 7is 3*(270-N).As 37*90=3330is much larger than 1000,our solution is still reasonable to this model.Figure 63.2Improve Model3.2.1Extra SymbolsSigns and definitions indicated above are still valid.Here are some extra signs and definitions.symboldescription Ra the radius of the circular flat area the side length of a regular hexagon3.2.2Additional AdditionalAssumptionsAssumptions ●The radius that of a repeater covers is adjustable here.●In some limited situations,curved shape is equal to straight line.●Assumptions concerning the anterior process are the same as the Basic Model3.2.3The Foundation of ModelThe same as the Basic Model except that:We only consider one variable(p)in the function model of the basic model ;In this model,we consider two varibles(p and µ)of the function model.3.2.4Solution and Result1)SolutionIf there are 10,000users,the number of regular hexagons that we need is at least ,thus according to the the Principle of maximum flow of Graph Theory,the 10000111.1190=result that we draw needed to be extended further.When the side length of the figure is equal to 7Figure 7regular hexagons,there are 127regular hexagons (Fig 7).Assuming the side length of a regular hexagon is ,then the area of a regular hexagon is a .The area of regular hexagons is equal to a circlewhose radiusis 22a =1000090R.Then according to the formula below:.221000090a R π=We can get.9.5858R a =Mapping with MATLAB as below (Fig 8):Figure 82）Improve the model appropriatelyEnlarge two part of the figure above,we can get two figures below (Fig 9and Fig 10):Figure 9AREAFigure 10Look at the figure above,approximatingAREA a rectangle,then obtaining its area to getthe number of users..The length of the rectangle is approximately equal to the side length of the regular hexagon ,athe width of the rectangle is ,thus the area of AREA is ,then R ?*R awe can get the number of users in AREA is()，2**10000 2.06R a R π=????????9.5858R a =As 2.06<<10,000,2.06can be ignored ,so there is no need to set up a repeater in.There are 6suchareas(92,98,104,110,116,122)that can be ignored.At last,the number of repeaters we should set up is,1276121?=2)Get the side length of the regular hexagon of the improved modelThus we can getmile=km 40 4.1729.5858a == 1.6* 6.675a =3)Calculate the power of a repeaterAccording to the formula[]()()32.4420lg ()20lg Lfs dB d km f MHz =++We get32.4420lg 6.67520lg14592.156Los =++=32.4420lg 6.67520lg14892.334Los =++=So we get106.85-92.156=14.694106.85-92.334=14.516As the result in the basic model,we can get the conclusion the power of a repeater is from 14.694mW to 14.516mW.3.2.5Analysis of the ResultAs 10,000users are much more than 1000users,the distribution of the users is more close toevenly distribution.Thus the model is more reasonable than the basic one.More repeaters are built,the utilization of the outside regular hexagon are higher than the former one.3.2.6Strength and Weakness●Strength:The model is more reasonable than the basic one.●Weakness:Repeaters don’t cover all the area,some places may not receive signals.And thefoundation of this model is based on the evenly distribution of the users in the area,if the situation couldn’t be satisfied,the interference of signals will come out.Ⅳ.Conclusions4.1Conclusions of the problem●Generally speaking,the radius of the area that a repeater covers is4miles in our basic model.●Using the model of honeycomb-hexagon structure can maximize the use of resources,avoiding some unnecessary interference effectively.●The minimum number of repeaters necessary to accommodate1,000simultaneous users is37.The minimum number of repeaters necessary to accommodate10,000simultaneoususers is121.●A repeater's coverage radius relates to external environment such as the density of users andobstacles,and it is also determined by the power of the repeater.4.2Methods used in our models●Analysis the problem with MATLAB●the method of the coverage in Graph Theory4.3Application of our models●Choose the ideal address where we set repeater of the mobile phones.●How to irrigate reasonably in agriculture.●How to distribute the lights and the speakers in squares more reasonably.Ⅴ.Future WorkHow we will do if the area is mountainous?5.1The best position of a repeater is the top of the mountain.As the signals are line-of-sight transmission and reception.We must find a place where the signals can transmit from the repeater to users directly.So the top of the mountain is a good place.5.2In mountainous areas,we must increase the number of repeaters.There are three reasons for this problem.One reason is that there will be more obstacles in the mountainous areas. The signals will be attenuated much more quickly than they transmit in flat area.Another reason is that the signals are line-of-sight transmission and reception,we need more repeaters to satisfy this condition.Then look at Fig11and Fig12,and you will know the third reason.It can be clearly seen that hypotenuse is larger than right-angleFig11edge(R>r).Thus the radius will become smaller.In this case more repeaters are needed.Fig125.3In mountainous areas,people may mainly settle in the flat area,so the distribution of users isn’t uniform.5.4There are different altitudes in the mountainous areas.So in order to increase the rate of resources utilization,we can set up the repeaters in different altitudes.5.5However,if there are more repeaters,and some of them are on mountains,more money will be/doc/d7df31738e9951e79b8927b4.html munication companies will need a lot of money to build them,repair them when they don’t work well and so on.As a result,the communication costs will be high.What’s worse,there are places where there are many mountains but few persons. Communication companies reluctant to build repeaters there.But unexpected things often happen in these places.When people are in trouble,they couldn’t communicate well with the outside.So in my opinion,the government should take some measures to solve this problem.5.6Another new method is described as follows(Fig13):since the repeater on high mountains can beFig13Seen easily by people,so the tower which used to transmit and receive signals can be shorter.That is to say,the tower on flat areas can be a little taller..Ⅵ.References[1]YU Fei,YANG Lv-xi,"Effective cooperative scheme based on relay selection",SoutheastUniversity,Nanjing,210096,China[2]YANG Ming,ZHAO Xiao-bo,DI Wei-guo,NAN Bing-xin,"Call Admission Control Policy based on Microcellular",College of Electical and Electronic Engineering,Shijiazhuang Railway Institute,Shijiazhuang Heibei050043,China[3]TIAN Zhisheng,"Analysis of Mechanism of CTCSS Modulation",Shenzhen HYT Co,Shenzhen,518057,China[4]SHANGGUAN Shi-qing,XIN Hao-ran,"Mathematical Modeling in Bass Station Site Selectionwith Lingo Software",China University of Mining And Technology SRES,Xuzhou;Shandong Finance Institute,Jinan Shandon,250014[5]Leif J.Harcke,Kenneth S.Dueker,and David B.Leeson,"Frequency Coordination in the AmateurRadio Emergency ServiceⅦ.AppendixWe use MATLAB to get these pictures,the code is as follows:1-clc;clear all;2-r=1;3-rc=0.7;4-figure;5-axis square6-hold on;7-A=pi/3*[0:6];8-aa=linspace(0,pi*2,80);9-plot(r*exp(i*A),'k','linewidth',2);10-g1=fill(real(r*exp(i*A)),imag(r*exp(i*A)),'k');11-set(g1,'FaceColor',[1,0.5,0])12-g2=fill(real(rc*exp(i*aa)),imag(rc*exp(i*aa)),'k');13-set(g2,'FaceColor',[1,0.5,0],'edgecolor',[1,0.5,0],'EraseMode','x0r')14-text(0,0,'1','fontsize',10);15-Z=0;16-At=pi/6;17-RA=-pi/2;18-N=1;At=-pi/2-pi/3*[0:6];19-for k=1:2;20-Z=Z+sqrt(3)*r*exp(i*pi/6);21-for pp=1:6;22-for p=1:k;23-N=N+1;24-zp=Z+r*exp(i*A);25-zr=Z+rc*exp(i*aa);26-g1=fill(real(zp),imag(zp),'k');27-set(g1,'FaceColor',[1,0.5,0],'edgecolor',[1,0,0]);28-g2=fill(real(zr),imag(zr),'k');29-set(g2,'FaceColor',[1,0.5,0],'edgecolor',[1,0.5,0],'EraseMode',xor';30-text(real(Z),imag(Z),num2str(N),'fontsize',10);31-Z=Z+sqrt(3)*r*exp(i*At(pp));32-end33-end34-end35-ezplot('x^2+y^2=25',[-5,5]);%This is the circular flat area of radius40miles radius 36-xlim([-6,6]*r) 37-ylim([-6.1,6.1]*r)38-axis off;Then change number19”for k=1:2;”to“for k=1:3;”,then we get another picture:Change the original programme number19“for k=1:2;”to“for k=1:4;”,then we get another picture:。
2014美国大学生数学建模特等奖优秀论文
Page 1 of 25
Best all time college coach Summary
In order to select the “best all time college coach” in the last century fairly, We take selecting the best male basketball coach as an example, and establish the TOPSIS sort - Comprehensive Evaluation improved model based on entropy and Analytical Hierarchy Process. The model mainly analyzed such indicators as winning rate, coaching time, the time of winning the championship, the number of races and the ability to perceive .Firstly ， Analytical Hierarchy Process and Entropy are integratively utilized to determine the index weights of the selecting indicators Secondly，Standardized matrix and parameter matrix are combined to construct the weighted standardized decision matrix. Finally, we can get the college men's basketball composite score, namely the order of male basketball coaches, which is shown in Table 7. Adolph Rupp and Mark Few are the last century and this century's "best all time college coach" respectively. It is realistic. The rank of college coaches can be clearly determined through this method. Next, ANOVA shows that the scores of last century’s coaches and this century’s coaches have significant difference, which demonstrates that time line horizon exerts influence upon the evaluation and gender factor has no significant influence on coaches’ score. The assessment model, therefore, can be applied to both male and female coaches. Nevertheless, based on this, we have drawn coaches’ coaching ability distributing diagram under ideal situation and non-ideal situation according to the data we have found, through which we get that if time line horizon is chosen reasonably, it will not affect the selecting results. In this problem, the time line horizon of the year 2000 will not influence the selecting results. Furthermore, we put the data of the three types of sports, which have been found by us, into the above Model, and get the top 5 coaches of the three sports, which are illustrated in Table10, Table 11, Table12 and Table13 respectively. These results are compared with the results on the Internet[7], so as to examine the reasonableness of our results. We choose the sports randomly which undoubtedly shows that our model can be applied in general across both genders and all possible sports. At the same time, it also shows the practicality and effectiveness of our model. Finally, we have prepared a 1-2 page article for Sports Illustrated that explains our results and includes a non-technical explanation of our mathematical model that sports fans will understand. Key words: TOPSIS Improved Model; Entropy; Analytical Hierarchy Process; Comprehensive Evaluation Model; ANOVA
2012年美国大学生数学建模竞赛B题特等奖文章翻译要点
2012年美赛B题题目翻译：到Big Long River（225英里）游玩的游客可以享受那里的风景和振奋人心的急流。
远足者没法到达这条河，唯一去的办法是漂流过去。
这需要几天的露营。
河流旅行始于First Launch，在Final Exit结束，共225英里的顺流。
旅客可以选择依靠船桨来前进的橡皮筏，它的速度是4英里每小时，或者选择8英里每小时的摩托船。
旅行从开始到结束包括大约6到18个晚上的河中的露营。
负责管理这条河的政府部门希望让每次旅行都能尽情享受野外经历，同时能尽量少的与河中其他的船只相遇。
当前，每年经过Big Long河的游客有X组，这些漂流都在一个为期6个月时期内进行，一年中的其他月份非常冷，不会有漂流。
在Big Long上有Y处露营地点，平均分布于河廊。
随着漂流人数的增加，管理者被要求应该允许让更多的船只漂流。
他们要决定如何来安排最优的方案：包括旅行时间（以在河上的夜晚数计算）、选择哪种船（摩托还是桨船），从而能够最好地利用河中的露营地。
换句话说，Big Long River在漂流季节还能增加多少漂流旅行数？管理者希望你能给他们最好的建议，告诉他们如何决定河流的容纳量，记住任两组旅行队都不能同时占据河中的露营地。
此外，在你的摘要表一页，准备一页给管理者的备忘录，用来描述你的关键发现。
沿着大朗河露营摘要我们开发了一个模型来安排沿大河的行程。
我们的目标是为了优化乘船旅行的时间，从而使6个月的旅游旺季出游人数最大化。
我们模拟团体从营地到营地旅行的过程。
根据给定的约束条件，我们的算法输出了每组沿河旅行最佳的日程安排。
通过研究算法的长期反应，我们可以计算出旅行的最大数量，我们定义为河流的承载能力。
我们的算法适应于科罗多拉大峡谷的个案分析，该问题的性质与大长河问题有许多共同之处。
最后，我们考察当改变推进方法，旅程时间分布，河上的露营地数量时承载能力的变化的敏感性。
我们解决了使沿大朗河出游人数最大化的休闲旅行计划。
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3.3 Stacking Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3.1 Pyramid Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3.2 Sparse Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Crossed Stack . . . . . . . . . . . . . . . . . . .. . . 8
We could have had it all, Rolling in the Deep. Your had my heart. Inside of your hand. And you played it. To the beat. Baby I have no story to be told, But I’ve heard one of you and I’m gonna make your head burn. Think of me in the depths of your despair. Making a home down there, as mine sure won’t be shared. The scars of your love, remind me of us. They keep me thinking that we almost had it all The scars of your love, they leave me breathless I can’t help feeling.
1
Team # 00000
Liang Qian
Page 2 of 000
Contents
1 Introduction
3
1.1 assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Introduction
6
3 Problem Analysis
7
3.1 The Track of the Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Number of Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
We could have had it all. Rolling in the Deep Your had my heart inside of your hand, And you played it To the beat WE Could have had it all,Rolling in the deep. You had my heart inside of your hand, But you played it with your beat Throw yourself through ever open door (Whoa), Count your blessings to find what look for (Whoa-uh), Turn my sorrow into treasured gold (Whoa), And pay me back in kind- You reap just what you sow. But you played it, You played it. You played it to the beat.
Rolling in the Deep
Team # 00000 2013.1.17
Team # 00000
Liang Qian
Page 1 of 000
Abstract
There’s a fire starting in my heart. Reaching a fever pitch and it’s bringing me out the dark. Finally, I can see you crystal clear. Go ahead and sell me out and I’ll lay your ship bare. See how I leave, with every piece of you. Don’t underestimate the things that I will do. There’s a fire starting in my heart, Reaching a fever pitch and it’s bringing me out the dark. The scars of your love, remind me of us. They keep me thinking that we almost had it all. The scars of your love, they leave me breathless, I can’t help feeling.
We could have had it all, Rolling in the Deep. Your had my heart. Inside of your hand. And you played it. To the beat. Baby I have no story to be told, But I’ve heard one of you and I’m gonna make your head burn. Think of me in the depths of your despair. Making a home down there, as mine sure won’t be shared. The scars of your love, remind me of us. They keep me thinking that we almost had it all The scars of your love, they leave me breathless I can’t help feeling.
We could have had it all. Rolling in the Deep Your had my heart inside of your hand, And you played it To the beat WE Could have had it all,Rolling in the deep. You had my heart inside of your hand, But you played it with your beat Throw yourself through ever open door (Whoa), Count your blessings to find what look for (Whoa-uh), Turn my sorrow into treasured gold (Whoa), And pay me back in kind- You reap just what you sow. But you played it, You played it. You played it to the beat.
3.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2
Team # 00000
Liang Qian
Page 3 of 000
1 Introduction
There’s a fire starting in my heart. Reaching a fever pitch and it’s bringing me out the dark. Finally, I can see you crystal clear. Go ahead and sell me out and I’ll lay your ship bare. See how I leave, with every piece of you. Don’t underestimate the things that I will do. There’s a fire starting in my heart, Reaching a fever pitch and it’s bringing me out the dark. The scars of your love, remind me of us. They keep me thinking that we almost had it all. The scars of your love, they leave me breathless, I can’t help feeling.