【正文】 nnel Assignments We seek to model the assignment of radio channels to a symmetric work of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grid (honeybstyle), as shown in Figure 1, where a transmitter is located at the center of each hexagon. Figure 1 An interval of the frequency spectrum is to be allotted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1, 2, 3, ... . Each transmitter will be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided. Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assign channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span. Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interference. Requirement A: There are several constraints on frequency assignments. First, no two transmitters within distance of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these constraints, what can we say about the span in, Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions. Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance differ by at least some given integer k, while those at distance at most must still differ by at least one. What can we say about the span and about efficient strategies for designing assignments, as a function of k? Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider? Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings. 2022 Mathematical Contest in Modeling (MCM) Problem A: Choosing a Bicycle Wheel Cyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the petition,and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course. Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right. The director sportif needs specific information to help make a decision and has asked your team to acplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front butthere is a choice of wheels for the rear. Task 1. Provide a table iving the wind peed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the wind speeds for different road grades starting from zero percent to ten percent in one percent increments. (Road grade is defined to be the ratio of the total rise of a hill divided by the length of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportional to the road rider will lose about 8 kph for a five percent grade over 100 meters. Task 2. Provide an example of how the table could be used for a specific time trial course Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision. Problem B: Escaping a Hurricane39。 and tend to blur and feather out sharp boundaries between the original pixels. A more faithful, flexible algorithm implemented on a personal puter would be useful. (1)for planning minimally invasive treatments, (2)for calibrating the MRI machines, (3)for investigating structures oriented obliquely in space, such as postmortem tissue sections in a animal research, (4)for enabling crosssections at any angle through a brain atlas consisting (4)for enabling crosssections at any angle through a brain atlas consisting of blackandwhite line drawing To design such an algorithm, one can access the value and locations of the pixels, but not the initial data gathered by the scanners. Problem Design and test an algorithm that produces sections of threedimensional arrays by planes in any orientation in space, preserving the original grayscale value as closely as possible. Data Sets The typical data set consists of a threedimensional array A of numbers A(i,j,k) which indicates the density A(i,j,k) of the object at the location (x,y,z)i