Mathematics Colloquia and Seminars

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Introducing Eulerian Cycles to Minimize Snowplow Work

Given a map and two snowplows starting from two different locations, we are asked to find the most efficient way to plow a subset of the roads. We explore the problem as a variant of the Chinese Postman Problem, which looks for most efficient path through every edge of a graph. Using a minimum perfect matching of our map to turn it into a graph that has Euler cycles, we find approximations to an optimal solution. The problem in general is NP-hard, and we investigate various optimality conditions when multiple snowplows are used.

Bicycle Problem

For professional cyclists, choosing a right type of bicycle wheels is critical for them to win a race. There are two basic types of wheels they can choose: wire spoked wheel and solid wheel. The spoked wheels are lighter and can be used on the front and rear of the bike while the solid wheels are more aerodynamic and can only be used on the rear of the bike. After examining the various conditions of a racecourse, such as the number and steepness of the hills, the weather, wind speed, and the competition, cyclists will then pick an appropriate wheel to use. This presentation will focus on two most significant conditions, steepness of the hills and wind speed, in order to show a simplified analysis on how to choose a bicycle wheel. The analysis involves some physics concepts so as to calculate when the power required for a solid wheel is less than for a spoked wheel at different wind speeds and road grades. The relevant mathematical model is implemented in MATLAB and generates a table of data which can be used as a reference for specific time trial courses.