Personal views on domain change of several theorems and applications

电子科技大学格拉斯哥学院 2017级吴博凡同组同学：无

Basically, this paper talks about domain changes from a basic substitution to the advanced domain change, those who are interested in this area with limited knowledge background can have a look at it.

Let’s start from the very beginning of substitution. During your high school life, one of the most important tricks in solving a complex problem is to substitute a complex part as a single variable. To some problems, it will be easier to do such a substitution as you have divided the problem into two parts. Solve the single variable and complex part next. However, there is an important thought hidden in the substitution which is the domain change. In this example, it is abstract to find the exact domain change, the next few parts can show the domain change more clearly.

The Bayes’ theorem is a key theorem we can learn from Probability. This theorem tells us if we want to calculate the conditional probability of B with A known in advance, we can calculate the union probability of AB and probability of A first, total probability can be applied and the formula can be proved easily. If we suppose A is the result and B is the cause, we can induce the cause from the result, vice versa. In this condition, we have already changed domain from A to B.

Convolution is also an important formula we can use during solving question with different variables. If we have obtained relationship of height which is determined by age, Weight with given probability density function (pdf), by using convolution formula we can solve pdf of height by finding the integral area and the equation. In this way, we can change the domain of multivariable to only one variable in the other domain. The key point of convolution is that we are able to change complex domain. Alternatively, we can apply a matrix called Jacobian matrix. If wanting to change variables with 3 or more variables, it’s recommended to substitute and multiply the absolute value of the determinant of Jacobian matrix. It’s practical and useful in 3 or more variables’ calculation. Lastly, convolution have a tight connection with Fourier transformation which we will talk in the following paragraph.

Finally, we have arrived at Fourier transformation! Firstly, it’s known to us all that all the waves can be the combination of different types of sine waves. Secondly, Fourier transformation is especially useful in changing domain from frequency domain to time domain, based on the methods I have mentioned before, the domain transformation is much easier to be understood and can be especially useful in processing waves or images. Here we introduce two applications. One is Fourier filter, which do Fourier transformation to wave signals and multiply the function by a selecting function which can modify (enlarge or decrease) some certain wave signals. After filter function, we can do inverse operation to change it back to initial modified wave function. Secondly, when we want to help color blind people to play game, it’s important to help them distinguish different colors. The resolution has already divided an image into a n by m matrix and we can operate on the matrix to change the certain color. As all the color can be consisted of three basic colors, we have a certain measurement called gray lever to represent the graph. By the Fourier transformation, we can change graph into grey lever and change the color which can’t be distinguished, a inverse operation can get back to initial graph. As a result, the color can be distinguished and people who are color blind can play games successfully.

If possible, we can apply domain transformation into complex number by Euler’s law. It’s much harder to extend it into complex number area and deserves to be studied furtherly.

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Personal views on domain change of several theorems and applications

Personal views on domain change of several theorems and applications相关推荐

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