Volume 1: Student Life Chapter 104: Finding Connections from the Title
Shen Jing stood at the door of the library, wondering what he was thinking.
He didn't even notice Chen Zhou approaching.
Chen Zhou patted him lightly: "What are you thinking about? Are you thinking so deeply?"
Shen Jing came back to his senses and turned to look at Chen Zhou: "Uh... nothing..."
Yang Yiyi also took the initiative to greet him: "Hello, senior."
Shen Jing then noticed the girl next to Chen Zhou, and sighed in his heart again. He now fully understood what it felt like to have someone else's child around him...
Although he thought so in his heart, Shen Jing still replied politely: "Hello, junior sister, hello..."
On the way back to the dormitory, Yang Yiyi raised some questions about learning mathematics.
Although they were questions, they were actually just Yang Yiyi's own curiosity. Most of the questions she asked were about the origins of this theorem or that proof.
Chen Zhou patiently answered her questions one by one.
There are some things Chen Zhou is not sure about, and coincidentally there is an excellent senior from the Department of Mathematics nearby who he can consult.
However, Shen Jing only had a vague idea of most of the things that Chen Zhou couldn't answer.
Shen Jing felt a special feeling when he heard this, and when he saw the look Yang Yiyi gave Chen Zhou, he felt even more sour ...
Shen Jing just wanted to say, I also want a girlfriend like this...
After finally reaching the dormitory area, Shen Jing hurriedly said hello and quickly walked towards his dormitory.
After sending Yang Yiyi to the downstairs of the dormitory, Chen Zhou also returned to the dormitory.
In the dormitory.
Li Li and the other two were reading at the desk.
Seeing Chen Zhou coming back, Zhao Qiqi immediately came up to him and asked, "Brother, how are you?"
Chen Zhou looked at him and said, "I asked for you guys. If there is time, we can have a social gathering together. You can decide for yourself when the time comes."
"Nice!" Zhao Qiqi hurriedly helped Chen Zhou take off his backpack, let Chen Zhou sit on a chair to rest, and gave him a shoulder massage.
Then he said to Li Li and Zhu Mingli, "Well, haven't you heard yet? Dormitory party, this is not just for me, everyone has a share! Li, quickly pour a glass of water for Brother Chen. Old Zhu, quickly take out the snacks to entertain him."
Hearing this, Li Li took Chen Zhou's cup and poured him a glass of water.
Zhu Mingli took out a large bag of delicious food that he had treasured.
"You are too polite..." Chen Zhou ate a piece of beef and drank some water with embarrassment.
" You're welcome . You're welcome. Then, Brother Chen, when are we going to have the gathering?" Zhao Qiqi asked.
The three of them looked at Chen Zhou expectantly.
Chen Zhou thought for a moment and said, "It's almost midterm exam season. How about waiting until we're done with the exams?"
Zhao Qiqi: "Yes, yes."
Zhu Mingli: "This is a good time. If you do well in the exam, everyone will have fun. If you don't do well in the exam, everyone will cheer you up. Chen Zhou will make the arrangements."
Li Li: "So, let's prepare for the midterm exam first."
Chen Zhou: “That’s right.”
…
When the dormitory finally quieted down, Chen Zhou borrowed Li Li's computer and began to download the task information sent by Wu Xiping.
After the download was complete, Chen Zhou unzipped the file and looked through the information.
He received two questions, one of which was a zero-point question given to him by Wu Xiping based on his own example.
This is a problem of pure mathematical analysis solved using advanced algebraic methods.
Correspondingly, another question is to use mathematical analysis methods to solve pure advanced algebra problems.
It can be said that Wu Xiping allocated the tasks for this project very reasonably.
If we look at each person's task individually, it can be carried out independently as a small project.
This is also what Wu Xiping deliberately did to cultivate Chen Zhou and Shen Jing's research ability.
Chen Zhou copied the two questions onto a piece of draft paper and prepared to study them.
Both questions are very simple, with a concise and beautiful appearance.
But it is quite difficult to solve.
After all, saying it is one thing, but actually doing it and researching it is another.
Chen Zhou twirled his pen, thinking about the corresponding solution.
After thinking for a while, Chen Zhou picked up his pen and started to solve the problem.
“If f(x)≠0, then the conclusion is true…”
“…it can be proved that there exist at least N+1 x1, x2, x3, …, xn+1∈(a, b), with x1＜x2＜x3……＜xn+1, such that f(xi)=0, (i=1, 2, …, n+1)…”
Having written this, Chen Zhou paused for a moment. He had a very strange feeling.
But Chen Zhou couldn't describe what this feeling was.
Shaking his head, Chen Zhou continued to write: " Assume that there are only m such points... Then x0→x1∫C'Xf(x)dx＋x1→x2∫C'Xf(x)dx＋＋…＋xm→xm＋1∫C'Xf(x)dx=0"
"According to the mean value theorem of integrals, there exists ξi (i=1, 2, ..., m) such that ..."
"According to the arbitrariness of C and the fact that the Vandermonde determinant is not equal to zero, we get..."
“Thus f(x) = 0, which contradicts f(x) ≠ 0.”
To solve this problem, Chen Zhou followed his own ideas, making horizontal connections between mathematical analysis and advanced algebra knowledge and applying them to solving the problem.
Chen Zhou looked at the steps he had written down and used the methods of advanced algebra to solve the problem of pure mathematical analysis.
After sorting it out again, Chen Zhou had that strange feeling again.
Could it be because of the weird feeling of integrating different courses to solve the problem for the first time?
After thinking for a while, Chen Zhou did not get a definite answer.
He raised his hand to look at his watch. It was almost 12 o'clock, and Li Li and the other two were still reading.
Chen Zhou got up and washed his face, then returned to his desk to work on the next question.
The next question is to use mathematical analysis methods to solve pure advanced algebra problems.
This is a very typical question, with only one sentence in the stem.
"Let ai>0, and let ai be all different, i=1, 2, ..., n, prove that the square matrix A (1/(ai+aj )) is positive definite."
After Chen Zhou finished reading it and thought about it for a while, he already had an idea.
The reason why this question is typical is that it requires the use of typical mathematical analysis methods, the generalized integral ∫＋∞e^（－ax）dx=1/a（a≠0）.
"First, it is a real symmetric matrix. For any x..., we can introduce integrals to perform calculations."
His thoughts flow continuously and he writes like a god.
Chen Zhou's hand holding the pen kept moving as he wrote down his problem-solving process on the draft paper.
"…because a1, …, an are different from each other, if x1e^(-a1t)+…+xne^(-ant)=0, then x1=…=xn=0, so they contradict each other."
Having written this, the answer is basically out.
Chen Zhou's strange feeling came up again.
Chen Zhou ignored this feeling for the time being and solved the whole problem according to his thinking.
"...Using the above conclusion, we can prove that the matrix... is positive definite."
The problem in the question itself was solved, but Chen Zhou's strange feeling did not find the answer.
Chen Zhou looked at the time. Only half an hour had passed, so it was still early.
He put the draft paper aside and planned to do the two questions again.
For math problems, use the method of mathematical analysis; for advanced algebra problems, use the method of advanced algebra.
Chen Zhou wanted to find the connection in the question, he felt that the question would tell him the answer.
