Now almost all engineering courses, as well as some humanities and social sciences, if you read a doctorate, you will find various mathematical models in them. take for example
1. Population growth model. At first, we just observed a village. Without external influence, people will gradually increase. That's just the most cursory observation. Later, it was found that the growth rate of people was roughly related to the number of people, so it could be described as a dynamic system by ordinary differential equations. We can know that the population will multiply. It turned out that it was not completely right. When the population reaches a certain level and resources are insufficient, the growth of people will be limited. So we added a modification to our model and studied a new model. Oh, it turns out that if the population is limited by resources, the population will eventually stop at a certain level. As we observe more and more, we can "add" the observed mathematical language to the old model, and we can get more mathematical results. After translation, we can know more about the problem of population growth.
2. Texas Hold 'em (or other poker games). This involves multiple players, and each player should maximize his own profits, so it can be simulated as a game. Because the flop is uncertain (I don't know what the next card will turn), it is a random process. Now everyone uses Markov game to model. What can I do after building the mold? Is it used to make money? There are many Texas Hold 'em softwares now. There is software that can guarantee to defeat humans one-on-one, but there are not enough people and the calculation takes too long.
3. Pregnancy prediction. Target is a large supermarket in America. They have records of all consumers. Through some statistical analysis, they found that a girl was likely to get pregnant recently, so they promoted related products to her. Where is the mathematical model? The model here is the quantitative relationship between girls' pregnancy probability and various girls' consumption behaviors.
4. Some magic related to playing cards. People often perform magic tricks through playing cards. Some magic tricks do not need quick hands, no cover-ups, no props, only mathematics (or probability). Through some steps, some people can make the probability that the next card turned out is the card you want is extremely high. Berkeley has a professor of mathematics who specializes in this. Cool is awesome!
5. Audio processing. Didn't you keep talking about the problem of tuning "I am a singer" and "The Voice of China" a while ago? Tuning is also related to mathematical modeling. A piece of music can be regarded as a signal with frequency and amplitude. We can model it as the superposition of some waves In this way, after modeling, we can easily make some music modifications. For example, the bass is ugly. If you want to get rid of it, get rid of some low frequency waves. To add another accompaniment, add a new wave representing accompaniment to the original wave.
I wrote a few here. Actually, there are quite a few interesting things. Just open a column. By the way, many people use mathematics to study theology and philosophy. You can find it on coursera online course.
Mathematical modeling is actually to "translate" realistic problems into mathematical problems with mathematical language. If the following steps are completed, we can "translate" the analysis results back and let us have a deeper understanding of the real world. Welcome to discuss!