美国数学建模竞赛论文tex模板
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。
美赛数学建模比赛论文模板
The Keep-Right-Except-To-Pass RuleSummaryAs for the first question, it provides a traffic rule of keep right except to pass, requiring us to verify its effectiveness. Firstly, we define one kind of traffic rule different from the rule of the keep right in order to solve the problem clearly; then, we build a Cellular automaton model and a Nasch model by collecting massive data; next, we make full use of the numerical simulation according to several influence factors of traffic flow; At last, by lots of analysis of graph we obtain, we indicate a conclusion as follow: when vehicle density is lower than 0.15, the rule of lane speed control is more effective in terms of the factor of safe in the light traffic; when vehicle density is greater than 0.15, so the rule of keep right except passing is more effective In the heavy traffic.As for the second question, it requires us to testify that whether the conclusion we obtain in the first question is the same apply to the keep left rule. First of all, we build a stochastic multi-lane traffic model; from the view of the vehicle flow stress, we propose that the probability of moving to the right is 0.7and to the left otherwise by making full use of the Bernoulli process from the view of the ping-pong effect, the conclusion is that the choice of the changing lane is random. On the whole, the fundamental reason is the formation of the driving habit, so the conclusion is effective under the rule of keep left.As for the third question, it requires us to demonstrate the effectiveness of the result advised in the first question under the intelligent vehicle control system. Firstly, taking the speed limits into consideration, we build a microscopic traffic simulator model for traffic simulation purposes. Then, we implement a METANET model for prediction state with the use of the MPC traffic controller. Afterwards, we certify that the dynamic speed control measure can improve the traffic flow .Lastly neglecting the safe factor, combining the rule of keep right with the rule of dynamical speed control is the best solution to accelerate the traffic flow overall.Key words：Cellular automaton model Bernoulli process Microscopic traffic simulator model The MPC traffic controlContentContent (2)1. Introduction (3)2. Analysis of the problem (3)3. Assumption (3)4. Symbol Definition (3)5. Models (4)5.1 Building of the Cellular automaton model (4)5.1.1 Verify the effectiveness of the keep right except to pass rule (4)5.1.2 Numerical simulation results and discussion (5)5.1.3 Conclusion (8)5.2 The solving of second question (8)5.2.1 The building of the stochastic multi-lane traffic model (9)5.2.2 Conclusion (9)5.3 Taking the an intelligent vehicle system into a account (9)5.3.1 Introduction of the Intelligent Vehicle Highway Systems (9)5.3.2 Control problem (9)5.3.3 Results and analysis (9)5.3.4 The comprehensive analysis of the result (10)6. Improvement of the model (11)6.1 strength and weakness (11)6.1.1 Strength (11)6.1.2 Weakness (11)6.2 Improvement of the model (11)7. Reference (13)1. IntroductionAs is known to all, it’s essential for us to drive automobiles, thus the driving rules is crucial important. In many countries like USA, China, drivers obey the rules which called “The Keep-Right-Except-To-Pass (that is, when driving automobiles, the rule requires drivers to drive in the right-most unless theyare passing another vehicle)”.2. Analysis of the problemFor the first question, we decide to use the Cellular automaton to build models,then analyze the performance of this rule in light and heavy traffic. Firstly,we mainly use the vehicle density to distinguish the light and heavy traffic; secondly, we consider the traffic flow and safe as the represent variable which denotes the light or heavy traffic; thirdly, we build and analyze a Cellular automaton model; finally, we judge the rule through two different driving rules,and then draw conclusions.3. AssumptionIn order to streamline our model we have made several key assumptions●The highway of double row three lanes that we study can representmulti-lane freeways.●The data that we refer to has certain representativeness and descriptive●Operation condition of the highway not be influenced by blizzard oraccidental factors●Ignore the driver's own abnormal factors, such as drunk driving andfatigue driving●The operation form of highway intelligent system that our analysis canreflect intelligent system●In the intelligent vehicle system, the result of the sampling data hashigh accuracy.4. Symbol Definitioni The number of vehiclest The time5. ModelsBy analyzing the problem, we decided to propose a solution with building a cellular automaton model.5.1 Building of the Cellular automaton modelThanks to its simple rules and convenience for computer simulation, cellular automaton model has been widely used in the study of traffic flow in recent years. Let )(t x i be the position of vehicle i at time t , )(t v i be the speed of vehicle i at time t , p be the random slowing down probability, and R be the proportion of trucks and buses, the distance between vehicle i and the front vehicle at time t is:1)()(1--=-t x t x gap i i i , if the front vehicle is a small vehicle.3)()(1--=-t x t x gap i i i , if the front vehicle is a truck or bus.5.1.1 Verify the effectiveness of the keep right except to pass ruleIn addition, according to the keep right except to pass rule, we define a new rule called: Control rules based on lane speed. The concrete explanation of the new rule as follow:There is no special passing lane under this rule. The speed of the first lane (the far left lane) is 120–100km/h (including 100 km/h)；the speed of the second lane (the middle lane) is 100–80km8/h (including80km/h)；the speed of the third lane (the far right lane) is below 80km/ h. The speeds of lanes decrease from left to right.● Lane changing rules based lane speed controlIf vehicle on the high-speed lane meets control v v <, ),1)(min()(max v t v t gap i f i +≥, safe b i gap t gap ≥)(, the vehicle will turn into the adjacent right lane, and the speed of the vehicle after lane changing remains unchanged, where control v is the minimum speed of the corresponding lane.● The application of the Nasch model evolutionLet d P be the lane changing probability (taking into account the actual situation that some drivers like driving in a certain lane, and will not takethe initiative to change lanes), )(t gap f i indicates the distance between the vehicle and the nearest front vehicle, )(t gap b i indicates the distance between the vehicle and the nearest following vehicle. In this article, we assume that the minimum safe distance gap safe of lane changing equals to the maximum speed of the following vehicle in the adjacent lanes.Lane changing rules based on keeping right except to passIn general, traffic flow going through a passing zone (Fig. 5.1.1) involves three processes: the diverging process (one traffic flow diverging into two flows), interacting process (interacting between the two flows), and merging process (the two flows merging into one) [4].Fig.5.1.1 Control plan of overtaking process(1) If vehicle on the first lane (passing lane) meets ),1)(min()(max v t v t gap i f i +≥ and safe b i gap t gap ≥)(, the vehicle will turn into the second lane, the speed of the vehicle after lane changing remains unchanged.5.1.2 Numerical simulation results and discussionIn order to facilitate the subsequent discussions, we define the space occupation rate as L N N p truck CAR ⨯⨯+=3/)3(, where CAR N indicates the number ofsmall vehicles on the driveway,truck N indicates the number of trucks and buses on the driveway, and L indicates the total length of the road. The vehicle flow volume Q is the number of vehicles passing a fixed point per unit time,T N Q T /=, where T N is the number of vehicles observed in time duration T .The average speed ∑∑⨯=T it i a v T N V 11)/1(, t i v is the speed of vehicle i at time t . Take overtaking ratio f p as the evaluation indicator of the safety of traffic flow, which is the ratio of the total number of overtaking and the number of vehicles observed. After 20,000 evolution steps, and averaging the last 2000 steps based on time, we have obtained the following experimental results. In order to eliminate the effect of randomicity, we take the systemic average of 20 samples [5].Overtaking ratio of different control rule conditionsBecause different control conditions of road will produce different overtaking ratio, so we first observe relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.(a) Based on passing lane control (b) Based on speed control Fig.5.1.3Fig.5.1.3 Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.It can be seen from Fig. 5.1.3:(1) when the vehicle density is less than 0.05, the overtaking ratio will continue to rise with the increase of vehicle density; when the vehicle density is larger than 0.05, the overtaking ratio will decrease with the increase of vehicle density; when density is greater than 0.12, due to the crowding, it willbecome difficult to overtake, so the overtaking ratio is almost 0.(2) when the proportion of large vehicles is less than 0.5, the overtaking ratio will rise with the increase of large vehicles; when the proportion of large vehicles is about 0.5, the overtaking ratio will reach its peak value; when the proportion of large vehicles is larger than 0.5, the overtaking ratio will decrease with the increase of large vehicles, especially under lane-based control condition s the decline is very clear.● Concrete impact of under different control rules on overtaking ratioFig.5.1.4Fig.5.1.4 Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions. (Figures in left-hand indicate the passing lane control, figures in right-hand indicate the speed control. 1f P is the overtaking ratio of small vehicles over large vehicles, 2f P is the overtaking ratio of small vehicles over small vehicles, 3f P is the overtaking ratio of large vehicles over small vehicles, 4f P is the overtaking ratio of large vehicles over large vehicles.). It can be seen from Fig. 5.1.4:(1) The overtaking ratio of small vehicles over large vehicles under passing lane control is much higher than that under speed control condition, which is because, under passing lane control condition, high-speed small vehicles have to surpass low-speed large vehicles by the passing lane, while under speed control condition, small vehicles are designed to travel on the high-speed lane, there is no low- speed vehicle in front, thus there is no need to overtake. ● Impact of different control rules on vehicle speedFig. 5.1.5 Relationships among vehicle density, proportion of large vehicles and average speed under different control conditions. (Figures in left-hand indicates passing lane control, figures in right-hand indicates speed control.a X is the average speed of all the vehicles, 1a X is the average speed of all the small vehicles, 2a X is the average speed of all the buses and trucks.).It can be seen from Fig. 5.1.5:(1) The average speed will reduce with the increase of vehicle density and proportion of large vehicles.(2) When vehicle density is less than 0.15,a X ,1a X and 2a X are almost the same under both control conditions.Effect of different control conditions on traffic flowFig.5.1.6Fig. 5.1.6 Relationships among vehicle density, proportion of large vehicles and traffic flow under different control conditions. (Figure a1 indicates passing lane control, figure a2 indicates speed control, and figure b indicates the traffic flow difference between the two conditions.It can be seen from Fig. 5.1.6：(1) When vehicle density is lower than 0.15 and the proportion of large vehicles is from 0.4 to 1, the traffic flow of the two control conditions are basically the same.(2) Except that, the traffic flow under passing lane control condition is slightly larger than that of speed control condition.5.1.3 ConclusionIn this paper, we have established three-lane model of different control conditions, studied the overtaking ratio, speed and traffic flow under different control conditions, vehicle density and proportion of large vehicles.5.2 The solving of second question5.2.1 The building of the stochastic multi-lane traffic model5.2.2 ConclusionOn one hand, from the analysis of the model, in the case the stress is positive, we also consider the jam situation while making the decision. More specifically, if a driver is in a jam situation, applying ))(,2(x P B R results with a tendency of moving to the right lane for this driver. However in reality, drivers tend to find an emptier lane in a jam situation. For this reason, we apply a Bernoulli process )7.0,2(B where the probability of moving to the right is 0.7and to the left otherwise, and the conclusion is under the rule of keep left except to pass, So, the fundamental reason is the formation of the driving habit.5.3 Taking the an intelligent vehicle system into a accountFor the third question, if vehicle transportation on the same roadway was fully under the control of an intelligent system, we make some improvements for the solution proposed by us to perfect the performance of the freeway by lots of analysis.5.3.1 Introduction of the Intelligent Vehicle Highway SystemsWe will use the microscopic traffic simulator model for traffic simulation purposes. The MPC traffic controller that is implemented in the Matlab needs a traffic model to predict the states when the speed limits are applied in Fig.5.3.1. We implement a METANET model for prediction purpose[14].5.3.2 Control problemAs a constraint, the dynamic speed limits are given a maximum and minimum allowed value. The upper bound for the speed limits is 120 km/h, and the lower bound value is 40 km/h. For the calculation of the optimal control values, all speed limits are constrained to this range. When the optimal values are found, they are rounded to a multiplicity of 10 km/h, since this is more clear for human drivers, and also technically feasible without large investments.5.3.3 Results and analysisWhen the density is high, it is more difficult to control the traffic, since the mean speed might already be below the control speed. Therefore, simulations are done using densities at which the shock wave can dissolve without using control, and at densities where the shock wave remains. For each scenario, five simulations for three different cases are done, each with a duration of one hour. The results of the simulations are reported in Table 5.1, 5.2, 5.3.●Enforced speed limits●Intelligent speed adaptationFor the ISA scenario, the desired free-flow speed is about 100% of the speed limit. The desired free-flow speed is modeled as a Gaussian distribution, with a mean value of 100% of the speed limit, and a standard deviation of 5% of the speed limit. Based on this percentage, the influence of the dynamic speed limits is expected to be good[19].5.3.4 The comprehensive analysis of the resultFrom the analysis above, we indicate that adopting the intelligent speed control system can effectively decrease the travel times under the control of an intelligent system, in other words, the measures of dynamic speed control can improve the traffic flow.Evidently, under the intelligent speed control system, the effect of the dynamic speed control measure is better than that under the lane speed control mentioned in the first problem. Because of the application of the intelligent speed control system, it can provide the optimal speed limit in time. In addition, it can guarantee the safe condition with all kinds of detection device and the sensor under the intelligent speed system.On the whole, taking all the analysis from the first problem to the end into a account, when it is in light traffic, we can neglect the factor of safe with the help of the intelligent speed control system.Thus, under the state of the light traffic, we propose a new conclusion different from that in the first problem: the rule of keep right except to pass is more effective than that of lane speed control.And when it is in the heavy traffic, for sparing no effort to improve the operation efficiency of the freeway, we combine the dynamical speed control measure with the rule of keep right except to pass, drawing a conclusion that the application of the dynamical speed control can improve the performance of the freeway.What we should highlight is that we can make some different speed limit as for different section of road or different size of vehicle with the application of the Intelligent Vehicle Highway Systems.In fact, that how the freeway traffic operate is extremely complex, thereby,with the application of the Intelligent Vehicle Highway Systems, by adjusting our solution originally, we make it still effective to freeway traffic.6. Improvement of the model6.1 strength and weakness6.1.1 Strength●it is easy for computer simulating and can be modified flexibly to consideractual traffic conditions ,moreover a large number of images make the model more visual.●The result is effectively achieved all of the goals we set initially, meantimethe conclusion is more persuasive because of we used the Bernoulli equation.●We can get more accurate result as we apply Matlab.6.1.2 Weakness●The relationship between traffic flow and safety is not comprehensivelyanalysis.●Due to there are many traffic factors, we are only studied some of the factors,thus our model need further improved.6.2 Improvement of the modelWhile we compare models under two kinds of traffic rules, thereby we come to the efficiency of driving on the right to improve traffic flow in some circumstance. Due to the rules of comparing is too less, the conclusion is inadequate. In order to improve the accuracy, We further put forward a kinds of traffic rules: speed limit on different type of cars.The possibility of happening traffic accident for some vehicles is larger, and it also brings hidden safe troubles. So we need to consider separately about different or specific vehicle types from the angle of the speed limiting in order to reduce the occurrence of traffic accidents, the highway speed limit signs is in Fig.6.1.Fig .6.1Advantages of the improving model are that it is useful to improve the running condition safety of specific type of vehicle while considering the difference of different types of vehicles. However, we found that the rules may be reduce the road traffic flow through the analysis. In the implementation it should be at the 85V speed of each model as the main reference basis. In recent years, the85V of some researchers for the typical countries from Table 6.1[ 21]:Author Country ModelOttesen and Krammes2000 AmericaLC DC L DC V C ⨯---=01.0012.057.144.10285Andueza2000Venezuela ].[308.9486.7)/894()/2795(25.9885curve horizontal L DC Ra R V T++--=].[tan 819.27)/3032(69.10085gent L R V T +-= Jessen2001America][00239.0614.0279.080.86185LSD ADT G V V P --+=][00212.0432.010.7285NLSD ADT V V P -+=Donnell2001 America22)2(8500724.040.10140.04.78T L G R V --+=22)3(85008369.048.10176.01.75T L G R V --+=22)4(8500810.069.10176.05.74T L G R V --+=22)5(8500934.008.21.83T L G V --=BucchiA.BiasuzziK. And SimoneA.2005Italy DCV 124.0164.6685-= DCE V 4.046.3366.5585--=2855.035.1119.0745.65DC E DC V ---=FitzpatrickAmericaKV 98.17507.11185-= Meanwhile, there are other vehicles driving rules such as speed limit in adverseweather conditions. This rule can improve the safety factor of the vehicle to some extent. At the same time, it limits the speed at the different levels.7. Reference[1] M. Rickert, K. Nagel, M. Schreckenberg, A. Latour, Two lane trafficsimulations using cellular automata, Physica A 231 (1996) 534–550.[20] J.T. Fokkema, Lakshmi Dhevi, Tamil Nadu Traffi c Management and Control inIntelligent Vehicle Highway Systems,18(2009).[21] Yang Li, New Variable Speed Control Approach for Freeway. (2011) 1-66。
数学建模美赛写作各部分模板
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 small farms, 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：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 the engineering parameters of sprinklers available in the market.数学建模美赛论文例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。
2012美国数学建模大赛二等奖论文及格式——英文版
Dedicated Pipeline for Trip ArrangementSummaryIn the problem of camping, we should set reasonable schedule which can not only increase the utilization of campsites but also meet people's needs. Meanwhile, the carrying capacity of the river is also required. To solve the problem, this thesis will build optimization model with maximum campsite's utilization and river trips as the model's target function.The specific steps are as follows:step1: Determine the number of campsites Y. We use Computer Emulation Simulation to solve this problem by making full use of the given conditions that trips will spend6 to 18 nights on the river and the river is 225 miles long. We get 29 sets of data through programming, then curve fitting them by SPSS software. By comparing the value of sig. and adjusting R square and so on, the ideal number of the campsites is got .Step2: By using the number of the campsites 39 as well as the goal programming equation built in the first step, we get the number of river trips that are allowed to enter, namely the carrying capacity of the river.Step3: By using the campsites 39, we adjust the campsites of different camping program and then divide them into 4 kinds through clustering analysis using SPSS. Then we select representatives in various types of camping programs according to repetition rate and the average transfer rate. So we streamline the camping programs into the problem of goal programming for 6, 8, 11, 12, 16 nights.Step4: In those five camping programs, 39 campsites which will not repeat are distributed in 3 dedicated pipelines . The first line accounts for 12 campsites and can only be available for 6 or 12 nights trip. Each day, a couple of 6 nights trips are distributed, and the starting trip camps the campsites in turn according to the even number of the pipeline while the secondary trip camps in turn according to the odd number. The second pipeline accounting for 16 campsites is arranged just as the first one .Under the premise of guaranteeing the variety of camping project, trips start as a pipeline to make the total number of trips camping in this line the biggest and the utilization of the campsites maximum. There are 11 campsites in the third pipeline which are available for 6 to 11 nights trip.According to the above analysis, the carrying capacity of Dedicated Pipeline, namely C_line, is less than that of the river, namely C_river, within 180 days. the park managers need to grasp passenger flow(P) of the river in the following period(T) and calculate P/( C_line/T)The best distribution program: the best utirlization of campsites is P/( C_line/T) in one period.According to the best utilization of campsites, The best distribution program can be got.Key words: Cluster Analysis, Bus Rapid, Transit Pipeline System, Curve Fitting , Computer Emulation SimulationContentsI. Introduction (3)1.1Restatement of the problem (3)1.2 Theory knowledge introduction (3)II. Definitions and Key terms (4)1，The conditions given (4)2，Symbol definition (4)III. General Assumptions (4)IV Model Design (5)4.1Model Establishment (5)4.2 Model Solution (6)4.2.1.To determine Y (6)4.2.1 To Determine the Camping program (11)4.2.3 To find capacity of the river (15)4.2.3 Determine Dedicated Pipeline (15)4.3 Strength and Weakness............................................................. 错误！未定义书签。
美国数学建模论文格式
Journal Citation(to be inserted by the publisher)Copyright by Trans Tech PublicationsYour Paper's Title Starts Here:Please Centeruse Arial14First Author1,Second Author2and Others3(use Arial14)1Full address of first author,including country,email(use Arial11)2Full address of second author,including country,email3List all distinct addresses in the same wayKeywords::List the keywords covered in your paper.These keywords will also be used by the Keywordspublisher to produce a keyword index.(use Arial11)For the rest of the paper,please use Times New Roman12Abstract.This document explains and demonstrates how to prepare your camera-ready manuscript for Trans Tech Publications.The best is to read these instructions and follow the outline of this text. The text area for your manuscript must be17cm wide and25cm high(6.7and9.8inches,resp.). Do not place any text outside this e good quality,white paper of approximately21x29cm or8x11inches.Your manuscript will be reduced by approximately20%by the publisher.Please keep this in mind when designing your figures and tables etc.IntroductionAll manuscripts must be in English.Please keep a second copy of your manuscript in your office (just in case anything gets lost in the mail).When receiving the manuscript,we assume that the corresponding authors grant us the copyright to use the manuscript for the book or journal in question.Should authors use tables or figures from other Publications,they must ask the corresponding publishers to grant them the right to publish this material in their paper.Use italic for emphasizing a word or phrase.Do not use boldface typing or capital letters except for section headings(cf.remarks on section headings,below).Use a laser printer,not a matrix dot printer.Organization of the TextSection Headings.The section headings are in boldface capital and lowercase letters.Second level headings are typed as part of the succeeding paragraph(like the subsection heading of this paragraph).Page Numbers.Do not print page numbers:Please number each sheet slightly in the left corner near the bottom(outside the typing area)with a light blue pencil.Footnotes.Footnotes1should be single spaced and separated from the text.Ideally,footnotes appear on the page of their reference,and are placed at the foot of the text,separated from the text by a horizontal line.Tables.Tables(refer with:Table1,Table2,...)should be presented as part of the text,but in such a way as to avoid confusion with the text.A descriptive title should be placed above each table. The caption should be self-contained and placed below or beside the table.Units in tables should be given in square brackets[meV].If square brackets are not available,use curly{meV}or standard brackets(meV).1This is a footnote2Title of Publication(to be inserted by the publisher)Figures.Figures(refer with:Fig.1,Fig.2,...)also should be presented as part of the text, leaving enough space so that the capt-ion will not be confused with the text.The caption should be self-contained and placed below or beside the figure.Generally,only original drawings or photographic reproductions are acceptable.Only very good photocopies are acceptable.Utmost care must be taken to insert the figures in correct alignment with the text.Half-tone pictures should be in the form of glossy prints.If possible,please include your figures as graphic images in the electronic version.If TTP is required to scan and insert images,please keep the following points in mind:(a)the allotted space(for inserting illustrations)must exactly match the space made available inthe camera-ready version,so that the electronic version is identical to the hard copy with regard to page and line breaks.(b)the required positioning of any high-quality separate illustration must be clearly indicated onits reverse side.The size of the illustrations must exactly match the space left in the camera-ready manuscript.Equations.Equations(refer with:Eq.1,Eq.2,...)should be indented5mm(0.2").There should be one line of space above the equation and one line of space below it before the text continues.The equations have to be numbered sequentially,and the number put in parentheses at the right-hand edge of the text.Equations should be punctuated as if they were an ordinary part of the text.Punctuation appears after the equation but before the equation number,e.g.c2=a2+b2.(1)Literature ReferencesReferences are cited in the text just by square brackets[1].(If square brackets are not available, slashes may be used instead,e.g./2/.)Two or more references at a time may be put in one set of brackets[3,4].The references are to be numbered in the order in which they are cited in the text and are to be listed at the end of the contribution under a heading References,see our example below.SummaryOn your floppy disk,please indicate the format and word processor used.Please also provide your phone number,fax number and e_mail address for rapid communication with the publisher(will not be published).Please always send your disk along with a hard copy that must match the disk's content exactly.If you follow the foregoing,your paper will conform to the requirements of the publisher and facilitate a problem-free publication process.References[1]Dj.M.Maric,P.F.Meier and S.K.Estreicher:Mater.Sci.Forum Vol.83-87(1992),p.119[2]M.A.Green:High Efficiency Silicon Solar Cells(Trans Tech Publications,Switzerland1987).This document is available on the web at /download **Please submit your paper in hardcopy and also electronically to the conference editor.。
美赛论文LaTeX模板
\documentclass{icmmcm}\usepackage{url} % For formatting URLs and other web or% file references.\usepackage{mflogo} % Provides the METAFONT logo; you% won't need it for your report.\usepackage{graphicx} % For importing graphics.\usepackage{natbib}%%% Sample ICM/MCM Contest Submission%%%%%% Based on sample senior thesis document%%% Last modified by Jeremy Rouse%%% Summer 2000%%%%%% and on the LaTeX Hints document%%% created by C.M. Connelly <cmc@>%%% Copyright 2002-2012%%% ---------------%%% Local Command and Environment Definitions%%% If you have any local command or environment definitions, put them %%% here or in a separate style file that you load with \usepackage.% \newtheorem declarations\newtheorem{Theo1}{Theorem}\newtheorem{Theo2}{Theorem}[section]\newtheorem{Lemma}[Theo2]{Lemma}% Each of the above defines a new theorem environment.% Multiple theorems can be done in the same environment.% Theo2's number is defined by the subsection it's in.% Theo3 uses the same numbering counter and numbering system as% Theo2 (that's the meaning of [Theo2]).%%% You probably won't want any of the following commands, which are %%% here to allow various the names of commands, make examples typeset %%% properly, and so on. You can, of course, use them as examples for %%% your own user-defined commands.\newcommand{\bslash}{\symbol{'134}}%backslash\newcommand{\bsl}{{\texttt{\bslash}}}\newcommand{\com}[1]{\bsl\texttt{#1}\xspace}\newcommand{\file}[1]{\texttt{#1}\xspace}\newcommand{\pdftex}{PDF\tex}\newcommand{\pdflatex}{PDF\latex}\newcommand{\acronym}[1]{\textsc{#1}\xspace}\newcommand{\key}[1]{\textsf{\emph{#1}}\xspace}\newcommand{\class}[1]{\textsf{#1}\xspace}\newcommand{\package}[1]{\textsf{#1}\xspace}\newcommand{\env}[1]{\texttt{#1}\xspace}\newcommand{\prog}[1]{\texttt{#1}\xspace}\newcommand{\command}[1]{\texttt{\bsl{}#1}\xspace}\newcommand{\ctt}{\texttt{comp.text.tex}\xspace}\newcommand{\tex}{\TeX\xspace}\newcommand{\latex}{\LaTeX\xspace}%%% Note that the \xspace command comes from the xspace package. It %%% allows you type a command that inserts text without having to %%% worry about how you ``end'' that command.%%%%%% Without \xspace, you would need to end a command with a backslash %%% followed by a space or with an empty set of braces if you followed %%% the command with a space. For example,%%%%%% \foo is a very important algorithm.%%%%%% might produce%%%%%% The foobarbaz algorithmis a very important algorithm.%%%%%% whereas with the \xspace command, the same code would produce %%%%%% The foobarbaz algorithm is a very important algorithm.%%%%%% If you need to butt a command that produces text against a letter %%% of some sort -- say, to pluralize it -- you need to tell TeX%%% where your command name ends so that it expands the correct %%% macro. So you might do%%%%%% \bar{}s are very busy creatures.%%% TeX has an amazingly good hyphenation algorithm, but sometimes it %%% gets confused and needs some help.%%%%%% For words that only occur once or twice, you can insert hints%%% directly into your text, as in%%%%%% our data\-base system is one of the most complex ever devised %%%%%% For words that you use a lot, and that seem to keep ending up at %%% the end of a line, however, inserting the hints each time gets to %%% be a drag. You can use the \hyphenation command to globally tell %%% TeX where to hyphenate words it can't figure out on its own.\hyphenation{white-space}%%% End Local Command and Environment Definitions%%% ---------------%%% ---------------%%% Title Block\title{\latex Hints for ICM/MCM Contest Reports}%%% Which contest are you taking part in? (Just one!)\contest{ICM/MCM}%%% The question you answered. (Again, just the one.)\question{Report Sample}%%% Your Contest Team Control Number\team{21247}%%% A normal document would specify the author's name (and possibly %%% their affiliation or other information) in an \author command. %%% Because the ICM/MCM Contest rules specify that the names of the %%% team members, their advisor, and their institution should not%%% appear anywhere in the report, do *not* define an \author command.%%% Defining the \date command is optional. If you leave it blank, %%% your document will include the date that the file is typeset, in %%% the form ``Month dd, yyyy''.% \date{}%%% End Title Block%%% ---------------\begin{document}%%% ---------------%%% Summary\begin{summary}This document is meant to give you a quick introduction to \TeX\ and \LaTeX. It covers a lot of material, but still barely manages to scratch the surface. It should provide you with some inspiration and, I hope, with some useful code you can copy, modify, and use in your report.You should use the \file{blank-template.tex} file as a basis foryour report rather than this file. Be sure to change its name to something sensible (maybe your team control number), and to set the values of the \com{title}, \com{question}, and \com{team} commands to appropriate values.Good luck!\hfill{}-- Claire\end{summary}%%% End Summary%%% ---------------%%% ---------------%%% Print Title Block, Contents, et al.\maketitle\tableofcontents%%% Uncomment the following lines if you have figures or tables in %%% your report:\listoffigures\listoftables%%% End Print Title Block, Contents, et al.%%% ---------------\section{Introduction: What Is \latex?}%\label{sec:introduction}\latex is a tool that allows you to concentrate on your writing while taking advantage of the \tex typesetting system to producehigh-quality typeset documents.\latex's benefits include\begin{enumerate}\item Standardized document classes\item Structural frameworks for organizing documents\item Automatic numbering and cross-referencing of structural elements \item ``Floating'' figures and tables\item High-level programming interface for accessing \tex'stypesetting capabilities\item Access to \latex extensions through loading ``packages''\end{enumerate}\section{Structured Writing}%\label{sec:structured-writing}Like HTML,\footnote{HyperText Markup Language} \latex is a markup language rather than a \acronym{Wysiwyg}{}\footnote{What You See Is What You Get.} system. You write plain text files that use special \key{commands} and \key{environments} that govern the appearance and function of parts of your text in your final typeset document.\subsection{Document Classes}%\label{sec:document-classes}The general appearance of your document is determined by your choice of \key{document class}. Document classes also load \latex packagesto provide additional functionality.\latex provides a number of basic classes, including \class{article},\class{letter}, \class{report}, and \class{book}. There are also alarge number of other document classes available, including\class{amsart} and \class{amsbook}, created by the American Mathematical Society and providing some additional mathematically useful structures and commands; \class{foils}, \class{prosper}, and\class{seminar}, which allow you to create ``slides'' for presentations; the math department's \class{thesis} class, forformatting senior theses; and many journal- or company-specific classes that format your document to match the ``house style'' of a particular periodical or publisher.\subsection{Packages}%\label{sec:packages}%\label{sec:ctan}\latex packages, or \key{style files}, define additional commands and environments, or change the way that previously defined commands and environments work. By loading packages, you can change the fonts used in your document, write your document in a non-English language with a non-\acronym{Ascii} font encoding, include graphics, format program listings, add custom headers and footers to your document, and much more.A typical \tex installation includes hundreds of style files, andhundreds more are available from the Comprehensive \tex Archive Network (CTAN), at \url{/}.\subsection{Structural Commands}%\label{sec:structural-commands}\begin{table}\centering\begin{tabular}{ll}\topruleCommand & Notes \\ \midrule\com{part} & \class{book} \& \class{report} only \\\com{chapter} &\class{book} \& \class{report} only \\\com{section} \\\com{subsection} \\\com{subsubsection} \\\com{paragraph} \\\com{subparagraph} \\\bottomrule\end{tabular}\caption[Structural commands in \latex]{Structural commands in \latex.}% \label{tab:structural-commands}\latex provides a set of structural commands for defining sections ofyour document, as shown in Table~\ref{tab:structural-commands}.Note that the argument to structural commands are moving arguments (see Section~\ref{sec:fragile-commands}) because they can be reused in the table of contents or in page headers or footers. Structural commands can take an optional argument in which you specify nonfragile commands or a shorter version of the actual section title that fits.You'll generally know when you need to provide an optional argument by \TeX's behavior.\subsection{Labels and References}%\label{sec:labels-and-references}Sections are numbered automatically by \latex during typesetting. Ifyou change your mind and decide that a subsection should be promotedto a section, or moved to the end of your document, the sections willbe renumbered so that the numbers are consistent.Sections can also be \command{label}{}ed with a tag such as\begin{quote}\begin{verbatim}\section{Our Complicated Equations}%\label{sec:complicated-eqs}\end{verbatim}\end{quote}and referred to with a \command{ref} or \command{pageref} command, as in\begin{quote}\begin{verbatim}In Section~\ref{sec:complicated-eqs}, we pointed out...\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}On page~\pageref{fig:gordian-knot}, we illustrated...\end{verbatim}\end{quote}\latex substitutes the correct section number when typesetting yourThe same commands can be used with numbered environments such as\env{equation}, \env{theorem}, and so forth.Use \emph{meaningful} labels---labeling a section as \texttt{sec12}may seem useful, but it will be confusing if you end up moving it to a different place in the document and its number changes to Section~34.It's also easier to remember what reference you want if you use a meaningful name.You may also want to impose some additional organization through the use of \emph{namespaces}, as I've done in this document. Rather than give different types of objects undistinguished labels, I precedesection labels with \texttt{sec:}, equations with \texttt{eq:},figures with \texttt{fig:}, tables with \texttt{tab:}, and so on.Emacs with Aux\tex and Ref\tex gives you easy access to these labels,as do many other editors with \tex-specific features. It's mucheasier to find the particular label you're looking for if you havesome additional information to help you. Adding the prefixes also reminds you of what text should precede the \com{ref} command.\subsection{Commands}\latex uses commands for changes that are very limited in scope (a few words) or are unlimited in scope (the rest of a document). For example, the commands\begin{quote}\begin{verbatim}\textbf{bold}\emph{italic (emphasized)}\textsf{sans serif}\end{verbatim}\end{quote}produce the following output in a typeset document:\begin{quote}\textbf{bold} \emph{italic (emphasized)} \textsf{sans serif}\end{quote}These are ``commands with arguments''---the command itself starts witha backslash (\bsl), and its \key{argument} appears inside braces{\verb+{ }+). Some commands may also have \key{optional arguments},which are typed inside brackets (\verb+[ ]+).There are also commands that take no arguments, such as\command{noindent}, \command{raggedright}, and \command{pagebreak}.You can define your own commands, as discussed inSection~\ref{sec:customization}.\subsection{Environments}%\label{sec:environments}\latex provides a number of \key{environments} that affect the appearance of text, and are generally used for more structurally significant purposes. For example, the commands listed above are typeset inside a \env{verbatim} environment typed inside a \env{quote} environment. Their results were typeset inside a \env{quote} environment.Environments use special commands to start and close---\command{begin} and \command{end}, followed by the name of the environment in braces, as in\begin{quote}\begin{verbatim}\begin{quote}``This is disgusting---I can't eat this. That arugala is sobitter\ldots{} It's like my algebra teacher on bread.''\flushright -- Julia Roberts in \emph{Full Frontal}\end{quote}\end{verbatim}\end{quote}producing\begin{quote}``This is disgusting---I can't eat this. That arugala is sobitter\ldots{} It's like my algebra teacher on bread.''\flushright -- Julia Roberts in \emph{Full Frontal}\end{quote}Some environments may take additional arguments in braces (required)or brackets (optional).Note that the order in which environments nest is extremely important.If you type an environment inside another environment, the inner environment must be \command{end}{}ed \emph{before} the secondenvironment is closed. It's also vitally important that you have an\command{end} line for each \command{begin} line, or \latex will complain.\subsubsection{The \env{document} Environment and the Preamble}%\label{sec:document-environment}The most important environment is the \env{document} environment, which encloses the \key{body} of your document. The code before the \command{begin}\verb+{document}+ line is called the \key{preamble}, and includes the all-powerful \command{documentclass} command, which loads a particular document class (seeSection~\ref{sec:document-classes}); optional \command{usepackage} commands, which load in additional \latex packages (seeSection~\ref{sec:packages}); and other setup commands, such asuser-defined commands and environments, counter settings, and so forth.I generally also include the commands defining the title, author, anddate in my preambles, but other people include them just after\command{begin}\verb+{document}+, before the \command{maketitle} command, which creates the title block of your document.\subsubsection{Math Environments}%\label{sec:math-environments}One of the major hallmarks of \tex is its ability to typesetmathematical equations.The two primary ways of doing so are with the use of \key{inline} and\key{display math environments}. These environments are used so often that there are shorthands provided for typing them. Inline math environments, such as $a^2 + b^2 = c^2$, can be typed as\begin{quote}\begin{verbatim}\begin{math}a^{2} + b^{2} = c^{2}\end{math}\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}$a^{2} + b^{2} = c^{2}$.\end{verbatim}\end{quote}Display math environments set your equation apart from your running text. They're generally used for more complicated expressions, such as\[f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\]which can be typed as\begin{quote}\begin{verbatim}\begin{displaymath}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{displaymath}\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}\[f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\]\end{verbatim}\end{quote}Generally, you'll want to use the \verb+$+ %$ <- fool font-lock-modedelimited form for inline math, and the \com{[} \com{]} form for display math environments. [Besides being easy to type, these forms are \key{robust}, which means that they can be used in \key{moving arguments}, elements that \tex may need to typeset in more than one place (such as a table of contents) or adjust (such as footnotes).]\paragraph{The \env{equation} Environment}%\label{sec:equation-environment}You'll probably want to use the \env{equation} environment for any formula you plan to refer to. \latex not only typesets the contentsof an \env{equation} environment in display mode, it also numbers it, as in\begin{equation}\label{eq:myequation}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{equation}written as\begin{quote}\begin{verbatim}\begin{equation}\label{eq:myequation}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{equation}\end{verbatim}\end{quote}Note that you can refer to this formula asEquation~\ref{eq:myequation} with\begin{verbatim}\ref{eq:myequation}.\end{verbatim}\subsection{Fonts}%\label{sec:fonts}Generally you'll want to let \latex handle the fonts for you---Knuth's Computer Modern fonts are used by default, and include a wide range of variations that can cover most any use you can think of.If you want to get fancy (and portable; seeSection~\ref{sec:fuzzy-fonts}), you can use Type~1 PostScript fonts, such as Times, Palatino, Utopia, and so forth. These font sets are accessible with packages with names like \package{times},\package{palatino}, and \package{utopia}. There are others, aswell---a command such as \com{locate psnfss | grep sty} will find mostof them.You can also get fonts from CTAN (see Section~\ref{sec:ctan}), both bitmap and Type 1. There's even support for TrueType fonts in some\TeX\ systems.\subsubsection{Font Commands}%\label{sec:font-commands}Most of your concern about fonts is probably related to what you're writing. You might want some \emph{emphasized} or \textbf{bold} textto stress a point or highlight a key term. Filenames might be set in\texttt{typewriter text} (although you should consider using the\package{url} package to help you out---by default, text set intypewriter text isn't hyphenated, which can lead to some unattractiveline breaks).You can also set text in \textsf{sans serif} or \textsc{small caps}.Table~\ref{tab:font-commands} shows you some of the most commonly used font commands provided by \latex.\begin{table}[htbp]\centering\begin{tabular}{ll}\topruleCommand & Result\\\midrule\com{emph} & \emph{emphasized text}\\\com{textsf} & \textsf{sans-serif text}\\\com{texttt} & \texttt{typewriter text}\\\com{textbf} & \textbf{bold text}\\\com{textsc} & \textsc{small caps text}\\\com{textsl} & \textsl{slanted text}\\\com{textit} & \textit{italic text}\\\bottomrule\end{tabular}\caption[Commonly used font commands]{Commonly used font commands.} \label{tab:font-commands}\end{table}I recommend that you use \com{emph} in preference to \com{textit}, anduse \com{textbf} sparingly. \com{emph} is a smarter command than\com{textit}---it switches back to the roman font when necessary. For example, \emph{She loved \emph{Scooby Doo}.} versus \textit{He loved\textit{Titanic}.}For complicated font changes, or for special font usages that you'retyping a lot, creating a macro (Section~\ref{sec:customization}) isthe way to go. I often just write, tossing in custom commands as Igo, and waiting to define them until just before I compile thedocument.\subsection{Customization}%\label{sec:customization}The main advantage of using commands and environments is that they allow you to organize your writing. A useful side-effect is that youcan change your mind about the way an element is typeset, and change all the appearances of that element in document by editing one pieceof code. For example, in this document the names of environments have been set in ``typewriter text'', using a command I created called\command{env}, which is defined as\begin{quote}\begin{verbatim}\newcommand{\env}[1]{\texttt{#1}\xspace}\end{verbatim}\end{quote}All I have to do to make the names of all the environments in the document appear in sans-serif type instead is to change that one lineto\begin{quote}\begin{verbatim}\newcommand{\env}[1]{\textsf{#1}\xspace}\end{verbatim}\end{quote}You can do the same with almost anything you can conceptualize---key terms, people's names (especially names of people fromnon-English-speaking countries), files, functions, and so on.\section{Mathematical Notation}%\label{sec:mathematical-notation}As we saw in Section~\ref{sec:math-environments}, math is typed into one of several kinds of math environments. Choose your environment based on the context and importance of the content. Any formula you plan to refer to should be typed in an \env{equation} environment (ora similar environment that supports labels).You should punctuate your mathematics as if the formulae were normal parts of English sentences. Reading them aloud is often a useful method for ensuring that you have all the commas in the right places. Where appropriate, you should also follow a displayed formula at the end of a sentence with a period.\subsection{Sums and Products}%\label{sec:sums-n-products}It's easy to typeset sums and products. For example,\begin{equation}f(n) = \sqrt[n]{\sum_{k=1}^{n} {n \choose k} f \left( n - k \right)},~\prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1} = \frac{2}{3}.\end{equation}%%% The ~ in the equation puts a nonbreaking space (equivalent to an%%% interword space in text mode) between the two halves of the equation. %%%%%% Also, note that the use of the \choose command here causes the%%% amsmath package to issue the warning%%%%%% Package amsmath Warning: Foreign command \atopwithdelims; %%% (amsmath) \frac or \genfrac should be used instead %%% (amsmath) on input line 557.%%%%%% amsmath would prefer the use of the \binom command it supplies.\subsection{Matrices}%\label{sec:matrices}It's a little more difficult to create matrices, but not too bad:%%% In LaTeX, & is the alignment tab, and separates columns. \\ is the end of %%% line marker, and separates rows. The ccc denotes that there are three %%% columns. The array environment and the tabular environment are %%% more or less identical, so what goes here also applies to a table.%%%\begin{equation}\left[ \begin{array}{ccc}2 & 1 & 2\\1 & 0 & 2\\2 & 1 & 1\end{array} \right]\left[ \begin{array}{ccc}-2 & 1 & 2\\3 & -2 & -2\\1 & 0 & -1\end{array} \right] =\left[ \begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array} \right].\end{equation}\subsection{Symbols}%\label{sec:symbols}\LaTeX provides an enormous number of symbols. Additional packages (loaded with \com{usepackage}) may provide additional symbols and fonts.For example, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ require you to load the \package{amsfonts} package (which is automatically loaded by the \texttt{icmmcm} class). These symbols are generated by \com{mathbb}, which only works in math mode.Subscripts and superscripts are easy---\verb!$a_n$! produces $a_n$,and \verb!$x^2$! produces $x^2$. Ordinal numbers, such as$3^{\textrm{rd}}$, $n^{\textrm{th}}$, and so forth,\footnote{Somefonts may include their own ordinals that can be accessed withspecial commands.} can be produced with code like\verb!$3^{\textrm{rd}}$!, \verb!$n^{\textrm{th}}$!.Equation~\ref{eq:superscript} shows a formula with a superscript.\begin{equation}\label{eq:superscript}\int_{0}^{\pi} \, \cos^{2n+1} x \, {\rm d} x = 0 \qquad\forall \, n \in \mathbb{N}.\end{equation}Notice that \com{cos} produces a nice roman ``$\cos$'' within math mode. There are similar commands for common functions like \com{log}, \com{exp}, and so forth. More can be defined with the\com{DeclareMathOperator} command provided by the \package{amsmath} package.You can stack symbols over other symbols. In math formulas,\begin{equation}m\ddot{x} + \gamma\dot{x} + kx = 0,\end{equation}or to produce diacritical accents, as in\begin{quote}Paul Erd\H{o}s s'est reveill\'{e} t\^{o}t pour enseigner lefran\c{c}ais \`{a} son fr\`{e}re et sa s\oe{}ur.\end{quote}\LaTeX{} has lots of Greek letters and ellipses too, some of which areshown in Figure~\ref{fig:greek-symbols}.\begin{figure}\begin{center}\begin{equation}\sqrt{\left[\begin{array}{cccccc}\alpha & \beta & \gamma & \delta & \epsilon & \zeta \\\eta & \theta & \iota & \kappa & \lambda & \mu \\\nu & \xi & o & \rho & \pi & \sigma \\\tau & \upsilon & \phi & \chi & \psi & \omega \\\Gamma & \Delta & \Theta & \Lambda & \Xi & \Pi \\\Sigma & \Upsilon & \Phi & \Psi & \Omega & \varphi\\\cdots & \ldots & \vdots & \ddots & : & \cdot\end{array}\right ] }.\end{equation}\end{center}\caption[Greek letters and some symbols]{Greek letters and some symbols.}% \label{fig:greek-symbols}\end{figure}See \cite{gratzer-mil}, pp.~455--474, or \cite{kopka-daly-guide},pp.~123--127, for lists of the symbols available. Intext, you mightsee some of these symbols used as\begin{quote}The Strong Induction Principle asserts that if a statement holds forthe integers $1$,~$2$,\dots,~$n$, and if whenever it holds for $n =1$, \dots,~$k$ then it also holds for $n = k+1$, then the statementholds for the integers $1$,~$2$,~$3$, $\ldots\,$ Using thisPrinciple, it can be shown that $1+2+\cdots+n = n(n+1)/2$ for allpositive integers~$n$.\end{quote}Notice that in the lists of integers, the ellipsis was made using the\com{ldots} command, and that the periods were nicely spaced betweenthe commas. In the sum, the dots were made with \com{cdots} and were centered on the line. The \package{amsmath} package provides a``smart'' \com{dots} command that can generally get things right basedon the context.So, with \com{dots} alone, the previous examples come out as\begin{quote}$1$,~$2$,~\dots,~$n$\\$n = 1$, \dots,~$k$\\$1$,~$2$,~$3$, $\dots\,$\\$1+2+\dots+n = n(n+1)/2$\end{quote}The general $n \times n$ matrix can be typeset as follows:\begin{equation}\left[\begin{array}{cccc}a_{11} & a_{12} & \ldots & a_{1n}\\a_{21} & a_{22} & \ldots & a_{2n}\\\vdots & \vdots & \ddots & \vdots\\a_{n1} & a_{n2} & \ldots & a_{nn}\\\end{array}\right].\end{equation}A fine point: lists of numbers that you're using in a mathematicalsense (as opposed to dates, numbers of objects, etc.) should be typedin math mode. For example, $341$, $541$, $561$, and $641$. The same numbers without math mode are 341, 541, 561, and 641. Depending on the fonts and packages that you're using, you may notice a little bitmore space around the first set than the second. With some packages, numbers intext may be set using old-style figures by default, as in\oldstylenums{341}, \oldstylenums{541}, \oldstylenums{561}, and\oldstylenums{641}. %%% But without the \oldstylenums commands!\subsection{More Math}In Fourier analysis, we talk about the $z$-domain.If $a$ is an even number, then\[ a + \phi(a) < \frac{3 a}{2}, \]and\[ \sigma(a) > \frac{2^{\alpha+1}-1}{2^{\alpha}} \, a \geq \frac{3a}{2}, \]where $\alpha$ is the greatest power of 2 that divides $a$, $\phi(a)$is the number of integers less than $a$ and relatively primeto $a$, and $\sigma(a)$ is the sum of the divisors of $a$ (including$1$ and $a$).。
美赛论文LaTeX模板
\documentclass{icmmcm}\usepackage{url} % For formatting URLs and other web or% file references.\usepackage{mflogo} % Provides the METAFONT logo; you% won't need it for your report.\usepackage{graphicx} % For importing graphics.\usepackage{natbib}%%% Sample ICM/MCM Contest Submission%%%%%% Based on sample senior thesis document%%% Last modified by Jeremy Rouse%%% Summer 2000%%%%%% and on the LaTeX Hints document%%% created by C.M. Connelly <cmc@>%%% Copyright 2002-2012%%% ---------------%%% Local Command and Environment Definitions%%% If you have any local command or environment definitions, put them %%% here or in a separate style file that you load with \usepackage.% \newtheorem declarations\newtheorem{Theo1}{Theorem}\newtheorem{Theo2}{Theorem}[section]\newtheorem{Lemma}[Theo2]{Lemma}% Each of the above defines a new theorem environment.% Multiple theorems can be done in the same environment.% Theo2's number is defined by the subsection it's in.% Theo3 uses the same numbering counter and numbering system as% Theo2 (that's the meaning of [Theo2]).%%% Y ou probably won't want any of the following commands, which are %%% here to allow various the names of commands, make examples typeset %%% properly, and so on. Y ou can, of course, use them as examples for %%% your own user-defined commands.\newcommand{\bslash}{\symbol{'134}}%backslash\newcommand{\bsl}{{\texttt{\bslash}}}\newcommand{\com}[1]{\bsl\texttt{#1}\xspace}\newcommand{\file}[1]{\texttt{#1}\xspace}\newcommand{\pdftex}{PDF\tex}\newcommand{\pdflatex}{PDF\latex}\newcommand{\acronym}[1]{\textsc{#1}\xspace}\newcommand{\key}[1]{\textsf{\emph{#1}}\xspace}\newcommand{\class}[1]{\textsf{#1}\xspace}\newcommand{\package}[1]{\textsf{#1}\xspace}\newcommand{\env}[1]{\texttt{#1}\xspace}\newcommand{\prog}[1]{\texttt{#1}\xspace}\newcommand{\command}[1]{\texttt{\bsl{}#1}\xspace}\newcommand{\ctt}{\texttt{comp.text.tex}\xspace}\newcommand{\tex}{\TeX\xspace}\newcommand{\latex}{\LaTeX\xspace}%%% Note that the \xspace command comes from the xspace package. It %%% allows you type a command that inserts text without having to %%% worry about how you ``end'' that command.%%%%%% Without \xspace, you would need to end a command with a backslash %%% followed by a space or with an empty set of braces if you followed %%% the command with a space. For example,%%%%%% \foo is a very important algorithm.%%%%%% might produce%%%%%% The foobarbaz algorithmis a very important algorithm.%%%%%% whereas with the \xspace command, the same code would produce %%%%%% The foobarbaz algorithm is a very important algorithm.%%%%%% If you need to butt a command that produces text against a letter %%% of some sort -- say, to pluralize it -- you need to tell TeX%%% where your command name ends so that it expands the correct %%% macro. So you might do%%%%%% \bar{}s are very busy creatures.%%% TeX has an amazingly good hyphenation algorithm, but sometimes it %%% gets confused and needs some help.%%%%%% For words that only occur once or twice, you can insert hints%%% directly into your text, as in%%%%%% our data\-base system is one of the most complex ever devised %%%%%% For words that you use a lot, and that seem to keep ending up at %%% the end of a line, however, inserting the hints each time gets to %%% be a drag. Y ou can use the \hyphenation command to globally tell %%% TeX where to hyphenate words it can't figure out on its own.\hyphenation{white-space}%%% End Local Command and Environment Definitions%%% ---------------%%% ---------------%%% Title Block\title{\latex Hints for ICM/MCM Contest Reports}%%% Which contest are you taking part in? (Just one!)\contest{ICM/MCM}%%% The question you answered. (Again, just the one.)\question{Report Sample}%%% Y our Contest Team Control Number\team{21247}%%% A normal document would specify the author's name (and possibly %%% their affiliation or other information) in an \author command. %%% Because the ICM/MCM Contest rules specify that the names of the %%% team members, their advisor, and their institution should not%%% appear anywhere in the report, do *not* define an \author command.%%% Defining the \date command is optional. If you leave it blank, %%% your document will include the date that the file is typeset, in %%% the form ``Month dd, yyyy''.% \date{}%%% End Title Block%%% ---------------\begin{document}%%% ---------------%%% Summary\begin{summary}This document is meant to give you a quick introduction to \TeX\ and \LaTeX. It covers a lot of material, but still barely manages to scratch the surface. It should provide you with some inspiration and, I hope, with some useful code you can copy, modify, and use in your report.Y ou should use the \file{blank-template.tex} file as a basis foryour report rather than this file. Be sure to change its name to something sensible (maybe your team control number), and to set the values of the \com{title}, \com{question}, and \com{team} commands to appropriate values.Good luck!\hfill{}-- Claire\end{summary}%%% End Summary%%% ---------------%%% ---------------%%% Print Title Block, Contents, et al.\maketitle\tableofcontents%%% Uncomment the following lines if you have figures or tables in %%% your report:\listoffigures\listoftables%%% End Print Title Block, Contents, et al.%%% ---------------\section{Introduction: What Is \latex?}%\label{sec:introduction}\latex is a tool that allows you to concentrate on your writing while taking advantage of the \tex typesetting system to producehigh-quality typeset documents.\latex's benefits include\begin{enumerate}\item Standardized document classes\item Structural frameworks for organizing documents\item Automatic numbering and cross-referencing of structural elements \item ``Floating'' figures and tables\item High-level programming interface for accessing \tex'stypesetting capabilities\item Access to \latex extensions through loading ``packages''\end{enumerate}\section{Structured Writing}%\label{sec:structured-writing}Like HTML,\footnote{HyperText Markup Language} \latex is a markup language rather than a \acronym{Wysiwyg}{}\footnote{What Y ou See Is What Y ou Get.} system. Y ou write plain text files that use special \key{commands} and \key{environments} that govern the appearance and function of parts of your text in your final typeset document.\subsection{Document Classes}%\label{sec:document-classes}The general appearance of your document is determined by your choice of \key{document class}. Document classes also load \latex packagesto provide additional functionality.\latex provides a number of basic classes, including \class{article},\class{letter}, \class{report}, and \class{book}. There are also alarge number of other document classes available, including\class{amsart} and \class{amsbook}, created by the American Mathematical Society and providing some additional mathematically useful structures and commands; \class{foils}, \class{prosper}, and\class{seminar}, which allow you to create ``slides'' for presentations; the math department's \class{thesis} class, forformatting senior theses; and many journal- or company-specific classes that format your document to match the ``house style'' of a particular periodical or publisher.\subsection{Packages}%\label{sec:packages}%\label{sec:ctan}\latex packages, or \key{style files}, define additional commands and environments, or change the way that previously defined commands and environments work. By loading packages, you can change the fonts used in your document, write your document in a non-English language with a non-\acronym{Ascii} font encoding, include graphics, format program listings, add custom headers and footers to your document, and much more.A typical \tex installation includes hundreds of style files, andhundreds more are available from the Comprehensive \tex Archive Network (CTAN), at \url{/}.\subsection{Structural Commands}%\label{sec:structural-commands}\begin{table}\centering\begin{tabular}{ll}\topruleCommand & Notes \\ \midrule\com{part} & \class{book} \& \class{report} only \\\com{chapter} &\class{book} \& \class{report} only \\\com{section} \\\com{subsection} \\\com{subsubsection} \\\com{paragraph} \\\com{subparagraph} \\\bottomrule\end{tabular}\caption[Structural commands in \latex]{Structural commands in \latex.}% \label{tab:structural-commands}\latex provides a set of structural commands for defining sections ofyour document, as shown in Table~\ref{tab:structural-commands}.Note that the argument to structural commands are moving arguments (see Section~\ref{sec:fragile-commands}) because they can be reused in the table of contents or in page headers or footers. Structural commands can take an optional argument in which you specify nonfragile commands or a shorter version of the actual section title that fits.Y ou'll generally know when you need to provide an optional argument by \TeX's behavior.\subsection{Labels and References}%\label{sec:labels-and-references}Sections are numbered automatically by \latex during typesetting. Ifyou change your mind and decide that a subsection should be promotedto a section, or moved to the end of your document, the sections willbe renumbered so that the numbers are consistent.Sections can also be \command{label}{}ed with a tag such as\begin{quote}\begin{verbatim}\section{Our Complicated Equations}%\label{sec:complicated-eqs}\end{verbatim}\end{quote}and referred to with a \command{ref} or \command{pageref} command, as in\begin{quote}\begin{verbatim}In Section~\ref{sec:complicated-eqs}, we pointed out...\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}On page~\pageref{fig:gordian-knot}, we illustrated...\end{verbatim}\end{quote}\latex substitutes the correct section number when typesetting yourThe same commands can be used with numbered environments such as\env{equation}, \env{theorem}, and so forth.Use \emph{meaningful} labels---labeling a section as \texttt{sec12}may seem useful, but it will be confusing if you end up moving it to a different place in the document and its number changes to Section~34.It's also easier to remember what reference you want if you use a meaningful name.Y ou may also want to impose some additional organization through the use of \emph{namespaces}, as I've done in this document. Rather than give different types of objects undistinguished labels, I precedesection labels with \texttt{sec:}, equations with \texttt{eq:},figures with \texttt{fig:}, tables with \texttt{tab:}, and so on.Emacs with Aux\tex and Ref\tex gives you easy access to these labels,as do many other editors with \tex-specific features. It's mucheasier to find the particular label you're looking for if you havesome additional information to help you. Adding the prefixes also reminds you of what text should precede the \com{ref} command.\subsection{Commands}\latex uses commands for changes that are very limited in scope (a few words) or are unlimited in scope (the rest of a document). For example, the commands\begin{quote}\begin{verbatim}\textbf{bold}\emph{italic (emphasized)}\textsf{sans serif}\end{verbatim}\end{quote}produce the following output in a typeset document:\begin{quote}\textbf{bold} \emph{italic (emphasized)} \textsf{sans serif}\end{quote}These are ``commands with arguments''---the command itself starts witha backslash (\bsl), and its \key{argument} appears inside braces{\verb+{ }+). Some commands may also have \key{optional arguments},which are typed inside brackets (\verb+[ ]+).There are also commands that take no arguments, such as\command{noindent}, \command{raggedright}, and \command{pagebreak}.Y ou can define your own commands, as discussed inSection~\ref{sec:customization}.\subsection{Environments}%\label{sec:environments}\latex provides a number of \key{environments} that affect the appearance of text, and are generally used for more structurally significant purposes. For example, the commands listed above are typeset inside a \env{verbatim} environment typed inside a \env{quote} environment. Their results were typeset inside a \env{quote} environment.Environments use special commands to start and close---\command{begin} and \command{end}, followed by the name of the environment in braces, as in\begin{quote}\begin{verbatim}\begin{quote}``This is disgusting---I can't eat this. That arugala is sobitter\ldots{} It's like my algebra teacher on bread.''\flushright -- Julia Roberts in \emph{Full Frontal}\end{quote}\end{verbatim}\end{quote}producing\begin{quote}``This is disgusting---I can't eat this. That arugala is sobitter\ldots{} It's like my algebra teacher on bread.''\flushright -- Julia Roberts in \emph{Full Frontal}\end{quote}Some environments may take additional arguments in braces (required)or brackets (optional).Note that the order in which environments nest is extremely important.If you type an environment inside another environment, the inner environment must be \command{end}{}ed \emph{before} the secondenvironment is closed. It's also vitally important that you have an\command{end} line for each \command{begin} line, or \latex will complain.\subsubsection{The \env{document} Environment and the Preamble}%\label{sec:document-environment}The most important environment is the \env{document} environment, which encloses the \key{body} of your document. The code before the \command{begin}\verb+{document}+ line is called the \key{preamble}, and includes the all-powerful \command{documentclass} command, which loads a particular document class (seeSection~\ref{sec:document-classes}); optional \command{usepackage} commands, which load in additional \latex packages (seeSection~\ref{sec:packages}); and other setup commands, such asuser-defined commands and environments, counter settings, and so forth.I generally also include the commands defining the title, author, anddate in my preambles, but other people include them just after\command{begin}\verb+{document}+, before the \command{maketitle} command, which creates the title block of your document.\subsubsection{Math Environments}%\label{sec:math-environments}One of the major hallmarks of \tex is its ability to typesetmathematical equations.The two primary ways of doing so are with the use of \key{inline} and\key{display math environments}. These environments are used so often that there are shorthands provided for typing them. Inline math environments, such as $a^2 + b^2 = c^2$, can be typed as\begin{quote}\begin{verbatim}\begin{math}a^{2} + b^{2} = c^{2}\end{math}\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}$a^{2} + b^{2} = c^{2}$.\end{verbatim}\end{quote}Display math environments set your equation apart from your running text. They're generally used for more complicated expressions, such as\[f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\]which can be typed as\begin{quote}\begin{verbatim}\begin{displaymath}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{displaymath}\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}\[f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\]\end{verbatim}\end{quote}Generally, you'll want to use the \verb+$+ %$ <- fool font-lock-modedelimited form for inline math, and the \com{[} \com{]} form for display math environments. [Besides being easy to type, these forms are \key{robust}, which means that they can be used in \key{moving arguments}, elements that \tex may need to typeset in more than one place (such as a table of contents) or adjust (such as footnotes).]\paragraph{The \env{equation} Environment}%\label{sec:equation-environment}Y ou'll probably want to use the \env{equation} environment for any formula you plan to refer to. \latex not only typesets the contentsof an \env{equation} environment in display mode, it also numbers it, as in\begin{equation}\label{eq:myequation}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{equation}written as\begin{quote}\begin{verbatim}\begin{equation}\label{eq:myequation}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{equation}\end{verbatim}\end{quote}Note that you can refer to this formula asEquation~\ref{eq:myequation} with\begin{verbatim}\ref{eq:myequation}.\end{verbatim}\subsection{Fonts}%\label{sec:fonts}Generally you'll want to let \latex handle the fonts for you---Knuth's Computer Modern fonts are used by default, and include a wide range of variations that can cover most any use you can think of.If you want to get fancy (and portable; seeSection~\ref{sec:fuzzy-fonts}), you can use Type~1 PostScript fonts, such as Times, Palatino, Utopia, and so forth. These font sets are accessible with packages with names like \package{times},\package{palatino}, and \package{utopia}. There are others, aswell---a command such as \com{locate psnfss | grep sty} will find mostof them.Y ou can also get fonts from CTAN (see Section~\ref{sec:ctan}), both bitmap and Type 1. There's even support for TrueType fonts in some\TeX\ systems.\subsubsection{Font Commands}%\label{sec:font-commands}Most of your concern about fonts is probably related to what you're writing. Y ou might want some \emph{emphasized} or \textbf{bold} textto stress a point or highlight a key term. Filenames might be set in\texttt{typewriter text} (although you should consider using the\package{url} package to help you out---by default, text set intypewriter text isn't hyphenated, which can lead to some unattractiveline breaks).Y ou can also set text in \textsf{sans serif} or \textsc{small caps}.Table~\ref{tab:font-commands} shows you some of the most commonly used font commands provided by \latex.\begin{table}[htbp]\centering\begin{tabular}{ll}\topruleCommand & Result\\\midrule\com{emph} & \emph{emphasized text}\\\com{textsf} & \textsf{sans-serif text}\\\com{texttt} & \texttt{typewriter text}\\\com{textbf} & \textbf{bold text}\\\com{textsc} & \textsc{small caps text}\\\com{textsl} & \textsl{slanted text}\\\com{textit} & \textit{italic text}\\\bottomrule\end{tabular}\caption[Commonly used font commands]{Commonly used font commands.} \label{tab:font-commands}\end{table}I recommend that you use \com{emph} in preference to \com{textit}, anduse \com{textbf} sparingly. \com{emph} is a smarter command than\com{textit}---it switches back to the roman font when necessary. For example, \emph{She loved \emph{Scooby Doo}.} versus \textit{He loved\textit{Titanic}.}For complicated font changes, or for special font usages that you'retyping a lot, creating a macro (Section~\ref{sec:customization}) isthe way to go. I often just write, tossing in custom commands as Igo, and waiting to define them until just before I compile thedocument.\subsection{Customization}%\label{sec:customization}The main advantage of using commands and environments is that they allow you to organize your writing. A useful side-effect is that youcan change your mind about the way an element is typeset, and change all the appearances of that element in document by editing one pieceof code. For example, in this document the names of environments have been set in ``typewriter text'', using a command I created called\command{env}, which is defined as\begin{quote}\begin{verbatim}\newcommand{\env}[1]{\texttt{#1}\xspace}\end{verbatim}\end{quote}All I have to do to make the names of all the environments in the document appear in sans-serif type instead is to change that one lineto\begin{quote}\begin{verbatim}\newcommand{\env}[1]{\textsf{#1}\xspace}\end{verbatim}\end{quote}Y ou can do the same with almost anything you can conceptualize---key terms, people's names (especially names of people fromnon-English-speaking countries), files, functions, and so on.\section{Mathematical Notation}%\label{sec:mathematical-notation}As we saw in Section~\ref{sec:math-environments}, math is typed into one of several kinds of math environments. Choose your environment based on the context and importance of the content. Any formula you plan to refer to should be typed in an \env{equation} environment (ora similar environment that supports labels).Y ou should punctuate your mathematics as if the formulae were normal parts of English sentences. Reading them aloud is often a useful method for ensuring that you have all the commas in the right places. Where appropriate, you should also follow a displayed formula at the end of a sentence with a period.\subsection{Sums and Products}%\label{sec:sums-n-products}It's easy to typeset sums and products. For example,\begin{equation}f(n) = \sqrt[n]{\sum_{k=1}^{n} {n \choose k} f \left( n - k \right)},~\prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1} = \frac{2}{3}.\end{equation}%%% The ~ in the equation puts a nonbreaking space (equivalent to an%%% interword space in text mode) between the two halves of the equation. %%%%%% Also, note that the use of the \choose command here causes the%%% amsmath package to issue the warning%%%%%% Package amsmath Warning: Foreign command \atopwithdelims; %%% (amsmath) \frac or \genfrac should be used instead %%% (amsmath) on input line 557.%%%%%% amsmath would prefer the use of the \binom command it supplies.\subsection{Matrices}%\label{sec:matrices}It's a little more difficult to create matrices, but not too bad:%%% In LaTeX, & is the alignment tab, and separates columns. \\ is the end of %%% line marker, and separates rows. The ccc denotes that there are three %%% columns. The array environment and the tabular environment are %%% more or less identical, so what goes here also applies to a table.%%%\begin{equation}\left[ \begin{array}{ccc}2 & 1 & 2\\1 & 0 & 2\\2 & 1 & 1\end{array} \right]\left[ \begin{array}{ccc}-2 & 1 & 2\\3 & -2 & -2\\1 & 0 & -1\end{array} \right] =\left[ \begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array} \right].\end{equation}\subsection{Symbols}%\label{sec:symbols}\LaTeX provides an enormous number of symbols. Additional packages (loaded with \com{usepackage}) may provide additional symbols and fonts.For example, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ require you to load the \package{amsfonts} package (which is automatically loaded by the \texttt{icmmcm} class). These symbols are generated by \com{mathbb}, which only works in math mode.Subscripts and superscripts are easy---\verb!$a_n$! produces $a_n$,and \verb!$x^2$! produces $x^2$. Ordinal numbers, such as$3^{\textrm{rd}}$, $n^{\textrm{th}}$, and so forth,\footnote{Somefonts may include their own ordinals that can be accessed withspecial commands.} can be produced with code like\verb!$3^{\textrm{rd}}$!, \verb!$n^{\textrm{th}}$!.Equation~\ref{eq:superscript} shows a formula with a superscript.\begin{equation}\label{eq:superscript}\int_{0}^{\pi} \, \cos^{2n+1} x \, {\rm d} x = 0 \qquad\forall \, n \in \mathbb{N}.\end{equation}Notice that \com{cos} produces a nice roman ``$\cos$'' within math mode. There are similar commands for common functions like \com{log}, \com{exp}, and so forth. More can be defined with the\com{DeclareMathOperator} command provided by the \package{amsmath} package.Y ou can stack symbols over other symbols. In math formulas,\begin{equation}m\ddot{x} + \gamma\dot{x} + kx = 0,\end{equation}or to produce diacritical accents, as in\begin{quote}Paul Erd\H{o}s s'est reveill\'{e} t\^{o}t pour enseigner lefran\c{c}ais \`{a} son fr\`{e}re et sa s\oe{}ur.\end{quote}\LaTeX{} has lots of Greek letters and ellipses too, some of which areshown in Figure~\ref{fig:greek-symbols}.\begin{figure}\begin{center}\begin{equation}\sqrt{\left[\begin{array}{cccccc}\alpha & \beta & \gamma & \delta & \epsilon & \zeta \\\eta & \theta & \iota & \kappa & \lambda & \mu \\\nu & \xi & o & \rho & \pi & \sigma \\\tau & \upsilon & \phi & \chi & \psi & \omega \\\Gamma & \Delta & \Theta & \Lambda & \Xi & \Pi \\\Sigma & \Upsilon & \Phi & \Psi & \Omega & \varphi\\\cdots & \ldots & \vdots & \ddots & : & \cdot\end{array}\right ] }.\end{equation}\end{center}\caption[Greek letters and some symbols]{Greek letters and some symbols.}% \label{fig:greek-symbols}\end{figure}See \cite{gratzer-mil}, pp.~455--474, or \cite{kopka-daly-guide},pp.~123--127, for lists of the symbols available. Intext, you mightsee some of these symbols used as\begin{quote}The Strong Induction Principle asserts that if a statement holds forthe integers $1$,~$2$,\dots,~$n$, and if whenever it holds for $n =1$, \dots,~$k$ then it also holds for $n = k+1$, then the statementholds for the integers $1$,~$2$,~$3$, $\ldots\,$ Using thisPrinciple, it can be shown that $1+2+\cdots+n = n(n+1)/2$ for allpositive integers~$n$.\end{quote}Notice that in the lists of integers, the ellipsis was made using the\com{ldots} command, and that the periods were nicely spaced betweenthe commas. In the sum, the dots were made with \com{cdots} and were centered on the line. The \package{amsmath} package provides a``smart'' \com{dots} command that can generally get things right basedon the context.So, with \com{dots} alone, the previous examples come out as\begin{quote}$1$,~$2$,~\dots,~$n$\\$n = 1$, \dots,~$k$\\$1$,~$2$,~$3$, $\dots\,$\\$1+2+\dots+n = n(n+1)/2$\end{quote}The general $n \times n$ matrix can be typeset as follows:\begin{equation}\left[\begin{array}{cccc}a_{11} & a_{12} & \ldots & a_{1n}\\a_{21} & a_{22} & \ldots & a_{2n}\\\vdots & \vdots & \ddots & \vdots\\a_{n1} & a_{n2} & \ldots & a_{nn}\\\end{array}\right].\end{equation}A fine point: lists of numbers that you're using in a mathematicalsense (as opposed to dates, numbers of objects, etc.) should be typedin math mode. For example, $341$, $541$, $561$, and $641$. The same numbers without math mode are 341, 541, 561, and 641. Depending on the fonts and packages that you're using, you may notice a little bitmore space around the first set than the second. With some packages, numbers intext may be set using old-style figures by default, as in\oldstylenums{341}, \oldstylenums{541}, \oldstylenums{561}, and\oldstylenums{641}. %%% But without the \oldstylenums commands!\subsection{More Math}In Fourier analysis, we talk about the $z$-domain.If $a$ is an even number, then\[ a + \phi(a) < \frac{3 a}{2}, \]and\[ \sigma(a) > \frac{2^{\alpha+1}-1}{2^{\alpha}} \, a \geq \frac{3a}{2}, \]where $\alpha$ is the greatest power of 2 that divides $a$, $\phi(a)$is the number of integers less than $a$ and relatively primeto $a$, and $\sigma(a)$ is the sum of the divisors of $a$ (including$1$ and $a$).。
MCM美赛论文集
高教社杯全国大学生数学建模竞赛承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则。
我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。
我们知道，抄袭别人的成果是违反竞赛规则的,如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。
我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。
如有违反竞赛规则的行为，我们将受到严肃处理。
我们参赛选择的题号是（从A/B/C/D中选择一项填写）：A我们的参赛报名号为（如果赛区设置报名号的话）：99999所属学校（请填写完整的全名）：西安交通大学参赛队员(打印并签名）：1.一作者2.二作者3.三作者指导教师或指导教师组负责人(打印并签名)：导师日期：2011年8月1日赛区评阅编号（由赛区组委会评阅前进行编号）：2011高教社杯全国大学生数学建模竞赛编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：赛区评阅记录（可供赛区评阅时使用）：评阅人评分备注全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）:全国大学生数学建模竞赛L A T E X2ε模板摘要这是数学建模论文模板mcmthesis的示例文件。
特别地，这篇文档是“全国大学生数学建模竞赛（CUMCM）”模板的示例文件。
这个模板使用于参加高教社杯全国大学生数学竞赛的同学准备他们的建模论文，帮助他们更多的关注于论文内容而非论文的排版。
这个模板的设计是根据2010年修订的《全国大学生数学建模竞赛论文格式规范》[1]制作，完全符合该论文格式规范，但是该模板未得到官方认可，请使用者自己斟酌使用。
这个示例文档逐条展示其对[1]的实现效果，并对所有自定义命令进行说明。
这个示例文件还包含了一些对公示、插图、表格、交叉引用、参考文献、代码等的测试部分，以展示其效果，并作简要的使用说明。