The Science Fiction World of Xueba

"What are you talking about?"

Pang Shao'an was a little surprised. He knew his grandson's level. At the same age, Pang Xuelin was the best. Even if he won the IMO gold medal, Pang Shao'an would not feel much surprise.

But now Pang Xuelin tells himself that he has not only integrated mathematics knowledge in high school, but even taught himself some college-level courses such as calculus, mathematical analysis, and abstract algebra, which surprised him a little.

He did not see the content of his self-study university on weekdays, did he secretly learn it in private?

He Pang Xuelin laughed and said, "Grandpa, it is true or false, I don't know if you test it!"

"Okay, then I'll make this bet with you, let's go to the study now, I will ask the question on the spot, and you will answer it on the spot!"

Xuan Pang Xuelin was incomparable, and Yao Jianzhong, who was on the side, followed with interest to watch the excitement.

When I came to the study, Pang Shaoan found the manuscript paper, and quickly wrote the first question on it:

Let R be a set of all real numbers, find all functions f: R → R, so that for any real number x and y,

F (f (x) f (y)) + f (x + y) = f (xy).

This question seems to be simple and does not exceed the scope of high school mathematics. However, the problem itself is very difficult for high school students. At least, it was taken as an IMO test with no problems at all.

Xi Pang Xuelin took the test, only glanced at it, and then didn't think about it. He picked up the pen and wrote the answer in the blank part below the manuscript paper.

Solution: Record the equation of the problem as P (x, y), then we can know from P (0,0)

F (f (0) ^ 2) = 0 (1)

For any real number x ≠ 1, there exists a real number y that satisfies x + y = xy, then y = x / x-1, which can be known from P (x, x / x-1),

F (f (x) f (x / x-1)) = 0, x ≠ 1 (2)

The f (0) can be discussed in two cases.

Uh ...

Pang Shaoan thought that Pang Xuelin had to think for more than ten minutes. After finding the idea, he spent ten minutes solving the problem.

I did not expect that the other party only glanced at the question, and then directly solved the problem. It didn't take five minutes.

"Grandpa, look at me, am I right?"

Xi Pang Xuelin confidently handed the answer to Pang Shao'an.

The Philippine Prize winners do IMO questions even more easily than college students do elementary school questions. At the first glance of the question, Pang Xuelin has seen the intention of the questioner.

If you really want to say the level of mathematics, he is already more than Pang Shao'an.

Pang Shao'an took a look and nodded: "Yes, I didn't expect that your level has risen recently, and the problem solving speed is much faster. But this is only the content of high school competitions. Since you have already taught yourself in college courses, then My next topic is not limited to high school knowledge. "

Pang Xuelin smiled and said, "Grandpa, you can't answer, but I won't count!"

"Shit, when is this crazy ?!"

虽然 Although Pang Shaoan scolded, he was very happy. If Pang Xuelin really had such a level as he said, wouldn't it prove the success of his education?

"You said before that you learned linear algebra, then the second question is related to matrices.

Soon, Pang Shaoan wrote the second question on the manuscript:

Prove that any real n-th order square matrix A can be decomposed into A = A0 + A1 + A2, where A0 = aIn, a is a real number, and A1 and A2 are both nilpotent square matrices.

Like the first question, Pang Xuelin didn't think long, and wrote the answer directly on the manuscript paper:

Proof: We first prove a lemma.

Lemma: Let A be a real square matrix of order n and satisfy tr (A) = 0, then there is a reversible real square matrix P such that the diagonal elements of P ^ -1AP are all 0.

Inductive n. When n = 1, A = (0). The conclusion is obviously true. Let n ≥ 2 below. We consider two cases.

Case 1: All non-zero vectors in Rn are eigenvectors of A ...

Uh ...

Similarly, in only five minutes, Pang Xuelin completed the comprehension question.

This time, not only Pang Shaoan was surprised, but even Yao Jianzhong was surprised: "Lao Pang, your little forest can't stand it. This problem is not necessarily solved by ordinary undergraduates!"

"Come again!"

Pang Shao'an Road.

Next, the third, fourth, fifth ...

From calculus and differential equations to derivation of conjugate gradients of linear equations, then to group theory, differential geometry, and functional analysis.

Difficulty also started to rise straight, but Pang Xuelin's problem-solving speed was moderate. No matter how difficult the problem was, in his hands, it was almost three or five divided by two, and it took at most ten minutes to get it done.

Unconsciously, more than an hour passed.

Pang Shaoan and Yao Jianzhong were surprised, amazed, and shocked and numb from the beginning.

"When did you start learning these things, Kobayashi?"

Pang Shao'an looked at Pang Xuelin with some incredible words.

Pang Xuelin was a little embarrassed, "Isn't there a lot of college-level math books in your study? I secretly studied two years ago, grandpa, what do you think of my current level, can I stop going to high school?

Pang Shao'an said: "That is to say, you have spent two years not only learning the mathematics system of the undergraduate level, but also basically understanding it?"

Pang Xuelin nodded.

Pang Shao'an and Yao Jianzhong face each other, what else can he say?

大部分 Most of the questions he just gave were the contents of undergraduate mathematics competitions, and some of them were difficult for graduate students, which were easily solved by Pang Xuelin.

Yao Jianzhong raised his thumbs aside and said, "Lao Pang, your kid is really a monster. According to his level, it doesn't make much sense to go to a high school course. So, Xiaolin, let me help you go to Jiangda The College of Physics said that you can go directly to the Department of Physics and go through the admission procedures for the spring semester early next year! "

Pang Shao'an glared at him and said: "Lao Yao, this is my grandson, don't think about digging the wall! Xiaolin, so, I will go to your school tomorrow to talk to your teacher, and then I will go to Liu Tingbo. Go to the Mathematics Department of Jiangda University! "

She Pang Xuelin couldn't help crying, "Grandpa, but I want to study material science!"

Pang Shao'an stared and said, "How is this possible? With your mathematical talent, it is a waste of learning materials!"

Pang Xuelin said: "Grandpa, Grandpa Yao, your kindness is in my heart. But I still want to learn materials engineering. In this era, doing pure number research is not very meaningful. On the contrary, materials have great applications in engineering. Prospects are more suitable for me. "

"But ..."

"Grandpa, don't persuade me ~ www.readwn.com ~ If I study other majors, I might as well wait for high school graduation to take the college entrance examination and choose my own major!"

Pang Shao'an frowned, seeing Pang Xuelin's firm face, sighing: "That line, I won't force you anymore, the material is the material!"

Pang Xuelin said: "Grandpa, would you like to tell my parents about what happened to Jiang Da this time?"

This is the first time Pang Xuelin has mentioned his parents after entering the standard.

He wanted to ask before, but he couldn't find a suitable entry point.

In the real world middle school, their parents were always busy with business and did n’t have much time to take care of themselves. But on the wandering earth plane, the whole world line changed greatly, and the construction of the earth engine required a lot of resources. At present, the world economy is basically a planned economy. In a state supplemented by the market economy, Pang Xuelin didn't know if his father's company could still run.

Pang Shao'an said: "Your parents and Xiaoxia's mother both work on Shanghai No. 2 engine. How can I have time to control you? I will just talk to them!"

"Shanghai No. 2 Engine?"

Pang Xuelin frowned slightly and could not help frowning.

There was a scene in the movie. Before the earth set sail, Shanghai was swallowed up by the tsunami. The seawater even reached the spire of the Oriental Pearl Tower, causing heavy casualties.

It is logical to say that if we acted according to the normal plan of humankind, everyone on the surface had already evacuated into the dungeon before the earth set sail.

In this case, such a disaster will only occur if it is possible to encounter an uncontrollable disaster in the coastal area, triggering a large-scale tsunami, and seawater pouring into Shanghai.

Qijiang City is not far from Shanghai. I am afraid that this level of disaster will also be affected.

Pang Xuelin was shocked!