In the last article, we saw the number of inhabitants became one specific value. I raised a question, does this result depend on the initial state? In other words, are these conversing numbers the property of the matrix only or the property of both the matrix and the initial population vector?
We can compute this by octave again. Set the initial state Berlin 0, Potsdam 1000.
octave:10> p = [0 1000]';
octave:11> M * p
300
700
octave:12> M^2 * p
450.00
550.00
octave:13> M^3 * p
525.00
475.00
octave:14> M^10 * p
599.41
400.59
octave:15> M^100 * p
600.00
400.00
Surprisingly, the results are the same even if we changed the initial state. Figure 12 shows many different initial states and how they are changed after many years. I believe you see a pattern. And finding a pattern is mathematics.
By the way, do you realize the inhabitants Berlin 600 and Potsdam 400 are the special numbers? Let's compute how many people are moving in this number.
\begin{eqnarray*} \mbox{Berlin} \rightarrow \mbox{Potsdam} &=& 600 * 0.2 \\ &=& 120 \\ \mbox{Potsdam} \rightarrow \mbox{Berlin} &=& 400 * 0.3\\ &=& 120 \end{eqnarray*}
When once the number of inhabitants becomes this number, the number of moving people from Berlin to Potsdam and from Potsdam to Berlin become the same. Therefore, once the inhabitants of the cities became this number, the number of inhabitants will not change anymore. As you see in Figure 11, there is also something that doesn't depend on the initial number of total population. This suggests there is some kind of property in the matrix, not in the population vector. Mathematics first thinks the mathematical object, namely, numbers. Initially, Mathematics's main concern was how we could operate the numbers, i.e., addition, subtraction, and so on. But one point, the concern of mathematics was moved to the operation. The mathematicians stopped thinking about numbers. Here, how many inhabitants was the first concern, however, the next level becomes what the matrix is.
I would like to talk about that mathematicians stopped thinking about numbers in the next article. It would be related with mathematics, but not so much.
- Hypothesis 3: If the initial state is different, the result may change. For example, we started Berlin 900, Potsdam 100. If we started Berlin 0, Potsdam 1000, the result maybe not the same.
We can compute this by octave again. Set the initial state Berlin 0, Potsdam 1000.
octave:10> p = [0 1000]';
octave:11> M * p
300
700
octave:12> M^2 * p
450.00
550.00
octave:13> M^3 * p
525.00
475.00
octave:14> M^10 * p
599.41
400.59
octave:15> M^100 * p
600.00
400.00
Surprisingly, the results are the same even if we changed the initial state. Figure 12 shows many different initial states and how they are changed after many years. I believe you see a pattern. And finding a pattern is mathematics.
 |
| Figure 11: Population history with various initial conditions. |
\begin{eqnarray*} \mbox{Berlin} \rightarrow \mbox{Potsdam} &=& 600 * 0.2 \\ &=& 120 \\ \mbox{Potsdam} \rightarrow \mbox{Berlin} &=& 400 * 0.3\\ &=& 120 \end{eqnarray*}
When once the number of inhabitants becomes this number, the number of moving people from Berlin to Potsdam and from Potsdam to Berlin become the same. Therefore, once the inhabitants of the cities became this number, the number of inhabitants will not change anymore. As you see in Figure 11, there is also something that doesn't depend on the initial number of total population. This suggests there is some kind of property in the matrix, not in the population vector. Mathematics first thinks the mathematical object, namely, numbers. Initially, Mathematics's main concern was how we could operate the numbers, i.e., addition, subtraction, and so on. But one point, the concern of mathematics was moved to the operation. The mathematicians stopped thinking about numbers. Here, how many inhabitants was the first concern, however, the next level becomes what the matrix is.
I would like to talk about that mathematicians stopped thinking about numbers in the next article. It would be related with mathematics, but not so much.
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