2011年美国数学建模竞赛题目(2011-02-11 11:56:05)PROBLEM A: Snowboard CourseDetermine the shape of a snowboard course (currently known as a “halfpipe”) to maximize the production of “vertical air” by a skilled snowboarder. "Vertical air" is the maximum vertical distance above the edge of the halfpipe. Tailor the shape to optimize other possible requirements, such as maximum twist in the air.What tradeoffs may be required to develop a “practical” course?PROBLEM B: Repeater CoordinationThe VHF radio spectrum involves line-of-sight transmission and reception. This limitation can be overcome by “repeaters,” which pick up weak signals, amplify them, and retransmit them on a different frequency. Thus, using a repeater, low-power users (such as mobile stations) can communicate with one another in situations where direct user-to-user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.In addition to geographical separation, the “continuous tone-coded squelch system” (CTCSS), sometimes nicknamed “private line” (PL), technology can be used to mitigate interference problems. This system associates to each repeater a separate subaudible tone that is transmitted by all users who wish to communicate through that repeater. The repeater responds only to received signals with its specific PL tone. With this system, two nearby repeaters can share the same frequency pair (for receive and transmit); so more repeaters (and hence more users) can be accommodated in a particular area.For a circular flat area of radius 40 miles radius, determine the minimum number of repeaters necessary to accommodate 1,000 simultaneous users. Assume that the spectrum available is 145 to 148 MHz, the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.How does your solution change if there are 10,000 users?Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas.ICM PROBLEMPROBLEM C: How environmentally and economically sound are electric vehicles? Is their widespread use feasible and practical?Click the title below to download a PDF of the 2011 ICM Problem.Your ICM submission should consist of a 1 page Summary Sheet and a 20 page solution for a total of 21 pages.A题：滑雪场问题请设计一个单板滑雪场（现为“半管”或“U型池”）的形状，以便能使熟练的单板滑雪选手最大限度地产生垂直腾空。
“垂直腾空“是超出“半管”边缘以上的最大的垂直距离。
定制形状时要优化其他可能的要求，如：在空中产生最大的身体扭曲。
在制定一个“实用”的场地时哪些权衡因素可能需要？B题中继站的协调甚高频无线电频谱包含信号的发送和接受。
这种限制可以被中继站所克服。
中继站可以捕捉到微弱的信号，然后把它放大，再用不同的频率重新发送。
这样，低功耗的用户，例如移动电话用户，在不能直接与其他用户联系的地方可以通过中继站来保持联系。
然而,中继站之间会互相影响,除非彼此之间有足够远的距离或通过充分分离的频率来传送。
除了地理的分离、“连续编码音调控制系统”(CTCSS),有时被称为“私人专线”(PL)、通过这项技术可以减轻干扰问题。
该系统连接每个中继站，靠的是所有通过同一个中继站连接的用户发送的独立的亚音频音调来连接。
中继站只回应接收到的具有特殊PL的语调的信号。
通过这个系统,两个附近的中继站可以共享相同的频率对(包括接收和发送);对于更多的中继站(并且更多的用户)可以提供在一个特定的区域。
在一个半径40英里的圆形区域，请你设计一个方案，用最少量的中继站来容纳1000同时在线用户。
假设频谱范围是145到148兆赫,在中继站中的发射机的频率要么是600千赫以上，要么低于接收机频率600千赫、并且这里有54个不同的PL可用。
如果这里有10,000个用户，如何改变你的解决方案。
在由于山区引起信号传播的阻碍的地区，讨论这样的情形。
2011-2-11 14:57 潍戁|二级问题A：单板滑雪课程确定一个滑雪场（现为“半管”）的形状，最大限度地利用熟练的滑雪板“垂直空气”的生产。
“垂直的空气”是最大的半管以上的边缘的垂直距离。
定制优化其他可能的要求，如空气中的最大扭曲，形状。
什么权衡可能需要制定一个“实用”的课程？B题：直放站协调涉及的甚高频无线电频谱线的视线传输和接收。
这种限制是可以克服的“中继器”，它拿起微弱信号，放大后发他们在不同的频率。
因此，使用一个中继器，低功耗的用户（如移动站）与一个可以沟通的情况下直接用户到用户的联系将是不可能的另一个。
然而，中继器可以互相干扰，除非他们足够远或传送在充分分离频率。
除了地理上的分隔，“连续音频编码静噪系统”（亚音频），有时也称为“专用线路”（特等），技术可用于减轻干扰的问题。
该系统为每个员工一个单独的中继subaudible音，其所有用户谁希望通过该通信中继传输。
中继器只响应其具体的PL接收到的信号音。
有了这个系统，附近两个中继器可以共享相同的频率对（用于接收和发送）;让更多中继器（因此更多的用户）可以在一个特定区域安置。
对于一个圆半径40海里为半径的平坦区域，确定必要的留级容纳1000个并发用户的最小数目。
假设频谱提供145至148兆赫，在一个中继器发射频率为600 kHz的任一高于或低于600千赫频率接收器，并有54个不同的发光色调可用。
您的解决方案如何改变，如果有10,000个用户？在由于山区引起信号传播的阻碍的地区，讨论这样的情形。
2010PROBLEM A:The Sweet SpotExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when h it. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”? Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?PROBLEM B: CriminologyIn 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a “center of mass” of the locations of the attacks. In the end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the “geographical profile” of a suspected serial criminal based on the locations of the crimes.Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for law enforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings.In addition to the required one-page summary, your report should include an additional two-page executive summary. The executive summary should provide a broad overview of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool. The executive summary will be read by a chief of police and should include technical details appropriate to the intended audience问题一：甜蜜点解释“甜蜜”对棒球球棒。