The top student must be diligent.

Chapter 22 Expectations for Xiao Yi
"paper?"

When Xiao Yi heard the request from Academician Hu, his eyes lit up. As a high school student, he had never come across such a thing as a thesis in the past.

Well...he has only been exposed to "argumentative essays" in Chinese composition.

And now Academician Hu actually sent him a paper directly and asked him to take a look at it, which immediately aroused his interest.

"Okay, I'll take a look."

Then I downloaded the paper sent by Academician Hu and started reading it.

First, he saw the relevant information about the author named Zeng Ang, who turned out to be a graduate student at the School of Mathematics of USTC, and his supervisor was not Hu Guangde, but another name.

Of course, Xiao Yi didn't read much of this content. After all, studying other people's information is not a good habit. Then he turned to the next page.

"The convergence of the Riemann integral is closely related to the Lebesgue integrability. First, the convergence theorem of the Riemann integral is proved in detail, then it is explained... Finally, an application is given."

This part is the abstract, which mainly explains what the paper is about.

Then we move on to the main text.

Although he had only read the book for a few days, Xiao Yi did not find any difficulty in reading the main text. Basically, he had seen everything mentioned in the paper in the book, and even if he had not seen it, he could understand it with a little deduction in his mind.

This paper was probably written by a graduate student named Zeng Ang as a practice paper. It is not very deep, but at least the theoretical derivation in the first half is completely fine.

Soon, Xiao Yi saw the last part of the paper, which was the application of the conclusions derived previously.

But after looking at it for a few times, his brows frowned slightly.

Because he found an error in the application process.

"Using the Dirichlet criterion, under the existing conditions, it is not possible to directly determine whether the Riemann integral (R) has converged."

He couldn't help but wonder if he was wrong. After all, this paper was written by a graduate student who would become his senior in the future, so how could he be wrong?

There must be an explanation or a patch later.

So he continued reading, but when he turned the page again, he found that the graduate student had directly come to the conclusion that "In summary, the conclusion is valid."

The following is the summary.

"This……"

Xiao Yi was in a dilemma.

He went back and read it again, and finally came to the same conclusion: it was wrong!
However, after all, it was Academician Hu who asked him to read the paper. Although the supervisor of this student named Zeng Ang was not Academician Hu, there must be a certain relationship between them.

Wouldn't it be a bit bad to point out the error directly?

But soon, he stopped worrying about it, because mathematics is a science where "right is right, wrong is wrong".

So, he finally spoke: "Academician Hu, I have finished reading it."

Hu Guangde smiled and said, "Okay, tell me what you think."

"Okay." Xiao Yi nodded: "The first three parts of this paper are correct, without any mistakes, and the inferences in it should also belong to the relatively basic content of "Real Variable Function Theory", but it has also been studied to a certain extent."

"However, there is an error in the fourth part, the application of the inference." "There is an error in the part of using the Drichlet criterion to judge the convergence of the Riemann integral (R). Under the current known conditions, this judgment cannot be made directly."

Hearing Xiao Yi's words, Hu Guangde raised his eyebrows, and then said: "So, how do you think we should make up for this mistake?"

Xiao Yi was stunned. He didn't expect that Hu Guangde didn't ask him how he discovered it, but instead asked him how to correct the mistake.

But he said directly: "We need to start from the beginning."

"Let α∈R, and f(x)=sin1/x/x^α, x∈(0], then the following conclusions hold: (i) when α≤1, f∈R[0] and f belongs to L[0]..."

Xiao Yi began his story, and on the other side of the video, apart from Hu Guangde's increasingly bright smile, the other professors, including Liu Bin, opened their mouths wider and wider.

This paper is indeed nothing to these professors at the master's and doctoral supervisor level, but the problem is that the person sitting opposite is a high school sophomore who has just read through the Theory of Real Analysis and it has not even been two days!
After just less than two days of study, you have been able to master the real analysis function that you need to study ten times to this level, and you can even point out the errors in a paper written by a graduate student after reading it?
Remember this graduate student named Zeng Ang, he seems to be a student of Academician Hu’s student?

In other words, he is Hu Guangde's disciple, so he must have the ability.

After being amazed, several professors looked at each other, and finally, all of them focused their eyes on Liu Bin's face.

Lao Liu, Lao Liu, where on earth did you find such students?
Liu Bin's face was filled with sadness.

Before, he was thinking of bringing Xiao Yi to their school to calm down the other geniuses in the junior class, but he never thought of calming down the mathematics professors as well.

My self-confidence is gone.

He thought of the 40 books he gave to Xiao Yi. According to this guy's reading efficiency, so many... Can he last a month with such a small number of books?
At most, the particularly difficult books such as algebraic geometry, algebraic topology, etc., as well as those English textbooks would require Xiao Yi to spend a little more time, right?

Really...

A despairing gap.

"…by substituting the variable x = for the integral ∫2πk+π, 2πk |sint/t|dt…we can easily prove (16) to be correct, and thus fL[0], and finally, according to the Cauchy criterion, (iv) is proved."

"Proof completed."

Xiao Yi's voice came from the phone, which also meant that he had corrected the mistakes in the paper.

"very good."

Finally, Hu Guangde nodded and even gave a thumbs up. If Xiao Yi was right in front of him, he would probably applaud.

"That's very well said, and that's exactly why I asked you to read this paper."

"This error is relatively not obvious in this paper, because in Section 3.2, the discussion of Riemann integrals and Lebesgue integrability gives the illusion that the missing condition is included, but it is not."

"This kind of mistake may be hard to prevent even for some experienced people."

“Only those who can truly master this part of real variable function theory can avoid such problems.”

“Also, the Cauchy Criterion you used at the end made this section even more exciting.”

The mathematics scholar made a final comment: "In short, I am now looking forward to seeing you in the School of Mathematics at HKUST."

(End of this chapter)