The Science Fiction World of Xueba

"What you said is true?"

Pang Shaoan was slightly taken aback. He knew his grandson's level. In his age group, Pang Xuelin was the best. Even if he won the IMO gold medal, Pang Shaoan would not be surprised.

But now Pang Xuelin told himself that not only had he mastered the mathematics knowledge in high school, he had even taught himself part of college courses such as calculus, mathematical analysis, and abstract algebra, which surprised him a bit.

I haven't seen him studying the content of the university by himself on weekdays. Could it be that he learned it secretly?

Pang Xuelin laughed and said, "Grandpa, if it's true or not, I'll know if you test it!"

"Okay, then I'll make a bet with you. Let's go to the study now. I will ask the questions on the spot, and you will answer them on the spot!"

Pang Xuelin had nothing to do, and Yao Jianzhong on the side also followed with great interest to watch the excitement.

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

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

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

This question seems simple, and it is not beyond the scope of high school mathematics, but the question itself is very difficult for high school students. At least it is taken as an exam question by IMO, and there is no problem at all.

Pang Xuelin took the test questions, just glanced at them, then without thinking, picked up a pen and wrote the answers in the blank part of the manuscript paper.

Solution: Record the equation in the question as P(x, y), then it can be known from P(0, 0)

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

For any real number x≠1, there is a real number y, satisfying x+y=xy, then y=x/x-1, from P(x, x/x-1),

f(f(x)f(x/x-1))=0, x≠1 (2)

There are two cases for f(0) to be discussed.

...

Pang Shaoan originally thought that Pang Xuelin would have to think for more than ten minutes, and after finding the idea, he would spend another ten minutes solving the problem.

Unexpectedly, the other party only glanced at the question, and then directly solved the problem. It didn't take five minutes.

"Grandpa, look, am I right?"

Pang Xuelin confidently handed the answer to Pang Shaoan.

It is even easier for the Philippine award boss to do the IMO test questions than for the college students to do the primary school test questions. The first time he saw the questions, Pang Xuelin had already seen through the intention of the question maker.

If you really want to talk about the level of mathematics, he is already more than a notch higher than Pang Shaoan.

Pang Shaoan took a look at it, nodded and said: "Yes, I didn't expect that your level has improved recently, and your problem-solving speed is much faster. But this is only the content of high school competitions. Since you have already studied the courses at the university level, then My next topic is not limited to high school knowledge."

Pang Xuelin said with a smile: "Grandpa, you can come out, if you can't answer, I will lose!"

"Stinky boy, when did you become so crazy?!"

Although Pang Shaoan cursed, he was very happy in his heart. If Pang Xuelin really has the level he said, wouldn't it prove the success of his education?

"You said before that you have studied linear algebra, so the second question is related to matrices.

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

Proof: Any real square matrix A of order n can be decomposed into A=A0+A1+A2, where A0=aIn, a is a real number, and A1 and A2 are nilpotent square matrices.

Like the first question, Pang Xuelin also did not think for long, and wrote down the answer directly on the manuscript paper:

Proof: We first prove a lemma.

Lemma: Suppose A is a real square matrix of order n and satisfies tr(A)=0, then there is a reversible real square matrix P,

Make the diagonal elements of P^-1AP all 0.

Induction to n, when n=1, A=(0), the conclusion is obviously established, let n≥2, we consider two situations.

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

...

Similarly, it only took five minutes for Pang Xuelin to solve the problem.

This time, not only Pang Shaoan was surprised, but even Yao Jianzhong was amazed: "Old Pang, your Xiaolin is amazing. This question may not be solved by ordinary undergraduates!"

"Come again!"

Pang Shaoan said.

Next, the third question, the fourth question, the fifth question...

From calculus and differential equations to deriving conjugate gradients for solving linear equations, to group theory, differential geometry, and functional analysis.

The difficulty also began to rise in a straight line, but Pang Xuelin's speed of solving the problems was moderate, and there was no change. No matter how difficult the problem was, in his hands, it was almost three times, five times and two times, and it took at most ten minutes to solve it.

Before I knew it, more than an hour passed.

And Pang Shaoan and Yao Jianzhong were also surprised and admired at the beginning, and then shocked and numb later.

"Xiaolin, when did you start learning these contents?"

Pang Shaoan looked at Pang Xuelin and said in disbelief.

Pang Xuelin was a little embarrassed and said: "Don't you have a lot of mathematics books for the college level in your study? I secretly learned it two years ago. Grandpa, what do you think of my current level? Can I not go to high school?"

Pang Shaoan said: "In other words, you spent two years not only learning the mathematics system at the undergraduate level by yourself, but even mastering it basically?"

Pang Xuelin nodded.

Pang Shaoan and Yao Jianzhong looked at each other, what else could he say?

Most of the exam questions he posed just now were the contents of the mathematics competition for undergraduates, and some of them were difficult for graduate students. Pang Xuelin easily solved them.

Yao Jianzhong gave a thumbs up to the side and said, "Old Pang, your kid is really a monster. According to his level, it really doesn't make much sense to go to high school. How about this, Xiao Lin, I will help you go to Jiang University." Let me tell you from the School of Physics, you can go directly to the Physics Department to audit, and you can go through the admission procedures for the spring semester early next year!"

Pang Shaoan glared at him, and said: "Old Yao, this is my grandson, don't try to poach someone! Xiao Lin, how about this, I will go to your school to talk to your teacher tomorrow, and then I will go to Liu Tingbo. Go to the Mathematics Department of Jiangda University to take classes!"

Pang Xuelin didn't know whether to laugh or cry: "Grandpa, but what I want to study is materials!"

Pang Shaoan stared and said: "How could this be possible? With your mathematical talent, learning materials is a waste!"

Pang Xuelin said: "Grandpa, Grandpa Yao, I appreciate your kindness. But I still want to study materials engineering. In this era, it is meaningless to study pure numbers. On the contrary, materials have strong applications in the field of engineering. Foreground disciplines are more suitable for me.”

"But……"

"Grandpa, don't try to persuade me. If I study other majors, I might as well wait until I graduate from high school to take the college entrance examination and choose my own major!"

Pang Shaoan frowned, and seeing Pang Xuelin's determined face, sighed: "That's fine, I won't force you anymore, the materials are the materials!"

Pang Xuelin said: "Grandpa, why don't you tell my parents about going to Jiang University to sit in this time?"

This is the first time that Pang Xuelin mentioned his parents after entering the original plane.

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

When I was in middle school in the real world, my parents were always busy with business and didn’t have much time to take care of myself. But on the plane of the Wandering Earth, the entire world has changed drastically. Building an earth engine requires a lot of resources. The current world economy is basically dominated by a planned economy. In a state supplemented by the market economy, Pang Xuelin didn't know whether his father's company could continue to operate.

Pang Shao'an said: "Your parents and Xiaoxia's mother both work in Shanghai No. 2 Engine, so I don't have time to take care of you. I'll just talk to them when the time comes!"

"Shanghai No. 2 engine?"

Pang Xuelin was taken aback for a moment, and could not help but frown.

There was a scene in the movie where Shanghai was engulfed by a tsunami before the earth set sail, and the sea level even reached the top of the Oriental Pearl Tower, causing heavy casualties.

It stands to reason that if humans acted according to the normal plan, everyone on the surface of the earth would have been evacuated into the underground city before the earth set sail, and such casualties are unlikely to occur.

In this case, such a disaster will only happen if there is an uncontrollable disaster in the coastal area, causing a large-scale tsunami, and seawater flooding into the urban area of ​​Shanghai.

Jiangcheng is not far from Shanghai, and a disaster of this level may also be affected.

Pang Xuelin was shocked!