The top student must be diligent

"Sorry to bother you." Zhang Yitang said helplessly: "You are right. I am in the lecture hall of Princeton University now. Professor Manjul Bhargava asked a very important question about the expansion of the circle method. The key question I can’t answer now.”

Xiao Yi said: "Okay, let's make a video."

"Then I'll trouble you."

No nonsense, they quickly started the video, and Xiao Yi's avatar also appeared on the multimedia screen at the back.

Xiao Yi looked at the scene through the camera. There were probably hundreds of people.

Then he said: "Professor Manjul Bhargava, please explain your question again."

"OK."

In the first row, Manjul Bhargava stood up and said with a smile: "In the circle method expansion, how do you deal with the high-order terms in the polynomial expansion? These high-order terms are in the traditional Fourier method Serious error accumulation will occur in the analysis, thus affecting the accuracy of the results.”

"For example, from your and Professor Zhang's paper, on page 46, there is a formula of the form f(x)=∞∑_(n=∞)Cne^(inx). In this formula, when n is smaller How do you control the error accumulation of the higher-order term Cne^(inx) when it is large?"

Hearing this question, Xiao Yi raised his eyebrows. This was indeed a very critical question. No wonder Zhang Yitang couldn't answer it.

The problem of higher-order terms exists in the traditional Fourier expansion, and his Shaw polynomial expansion is a technology derived from Fourier expansion, so it was also brought into the circle method expansion.

Professor Manjul Bhargava is the winner of the Fields Medal in 2014. He has conducted in-depth research on analytic number theory. It is no wonder that he was able to find this problem.

but……

Xiao Yi just smiled lightly.

"This is a good question. Of course, to solve it, we only need to use the adaptive weighting function in the sieve method to filter these high-order terms layer by layer."

"The specific operation...well, I'll ask Professor Zhang to write it down for me."

"Of course, no problem." Zhang Yitang walked to the blackboard, "Just tell me."

"Okay." Xiao Yi said: "We first perform a transformation on the polynomial expansion to map f(x) to a new function g(x), where n(x) is a special basis function that can significantly Reduce the influence of higher-order terms.”

【g(x)=∞∑n=∞dnn(x)】

"Then we introduced the adaptive weighting function w(n), which is of the form: w(n)=1/(1+|n|^k), where k is an appropriately chosen positive number to control high-order The weight of the item.”

"Next, we only need to further combine the sieving method with weighting technology. Through this transformation, we can control the error of higher-order terms within an acceptable range while ensuring the retention of the main number theory information."

"That's it."

As Xiao Yi said, Zhang Yitang also wrote these steps on the blackboard.

Manjul Bhargava looked at these steps and thought about Xiao Yi's explanation just now. After pondering for a long time, he finally nodded and clapped his hands and said: "A good solution. I have no problem."

Chapter 147 Absolute Electronicity Calculation

Everyone in the audience fell silent, staring dumbfounded at the solution steps given by Xiao Yi on the blackboard.

That's it...is it done?

Is it a little too fast?

Zhang Yitang also carefully looked at the steps he had written out himself, and seriously thought about the meaning of each step.

However, no matter how he thought about it, there was only one thought in his mind: can the problem of higher-order terms be solved in this way?

How was he thinking before...

Well, he couldn't figure it out before.

In the first row below, Enrique Bombieri, Pierre Deligne and many other big guys are also there.

Looking at the solution given by Xiao Yi, they squinted their eyes, and after thinking for a moment, they could only sigh.

"Sure enough, this kid is still needed to solve a problem like this." Deligne shook his head and sighed.

Bombieri also nodded: "Who says it's not?"

He thought carefully about Xiao Yi's method and suddenly said: "Do you think that his method, even if it is applied to the Fourier expansion itself, can solve the problem of error accumulation?"

"Really?" Deligne took another look. After thinking for a moment, his eyes lit up: "Yes, it is indeed true!"

At the same time, other people discovered this, and someone immediately stood up and asked: "Mr. Xiao, if your method can be used to solve the problem of circle expansion, can we also use it? Going to Fourier expansion, can it also be used to solve the error accumulation problem? "

Regarding this question, Xiao Yi did not have any surprises. He nodded with a smile and said: "Yes, you discovered it very quickly. The two are closely related. Of course, some small adjustments are needed in the process. Variety."

"Can you please give us a demonstration?" The questioner said excitedly: "This is very important to me. I feel that it can help me complete my project."

"Of course it's no problem." Xiao Yi smiled and nodded: "Professor Zhang, I still have to trouble you."

Zhang Yitang also came back from his thoughts and said, "No trouble, this also gave me inspiration, you tell me."

"Well." Xiao Yi continued.

"First of all, from the combination of sieving method and weighting technology mentioned just now, we can get: [g(x)=n=N∑Ndnn(x)+|n|\u003eN∑1+|n|kcnn(x)]"

"By calculating EN′(x) to verify the control effect of the error, due to the introduction of w(n), the influence of high-order terms is significantly weakened, thus ensuring the accuracy of the overall result."

[EN′(x)=|n|\u003eN∑dnn(x)=|n|\u003eN∑1+|n|kcnn(x)]

"..."

Following Xiao Yi's narration, Zhang Yitang continued to write on the blackboard.

Everyone in the audience started to think.

Although they may not understand the circular expansion, there are quite a lot of people who understand the Fourier expansion.

As a very classic mathematical method, it can express a periodic function as an infinite series of sine and cosine functions, and has been used to solve various problems.

As for the high-order term problem of Fourier expansion, it mainly involves the influence of high-frequency components in Fourier series on the accuracy of function reconstruction. Those who understand it should have heard of the Gibbs phenomenon, which is a manifestation of the high-order term problem.

It can be said that people in the mathematical community often have headaches because of this problem when using this method.

And now this problem seems to have been greatly solved, so how can they not listen carefully?

Maybe, like the questioner, it can help them solve a problem in the topic in the future.

"... In short, according to the above method, we can weaken the high-frequency term and stabilize the accumulated error."

"Thereby achieving the control of the high-order term problem in Fourier expansion."

About ten minutes later, Xiao Yi's narration ended.

He smiled and said, "Of course, I think there are still many places to explore, which can further reduce the impact of the high-order term problem, but time is limited. I will stop here for the time being. I believe you can find it."

"Are there any other questions?"

He glanced at the mathematicians present.

Time passed quickly, and no one continued to ask questions. At the same time, most people were recording what Xiao Yi had just said, without any distraction.

After a while, Zhang Yitang smiled and said, "It seems that there is no more."

"That's fine." Xiao Yi nodded, and then dozed off deeply: "It seems that I can go to rest."

"Haha, then you should rest early." Zhang Yitang smiled and said, "Oh, by the way, I will also ask you on behalf of the mathematics community at the end, what problems are you studying recently? Now we are a little uncomfortable with not seeing your results for so long."

"Uh..." Xiao Yi touched his nose slightly embarrassedly and said, "In short, it should be a very important thing, but it is more application-oriented."

"It should be soon that you will see it."

"Haha, okay, then we look forward to it."

Zhang Yitang nodded, then looked at the scholars on the scene and said, "By the way, don't forget to give applause to our Xiao Yi, thank him for taking the time to answer our questions!"

After that, he took the lead in applauding.

Hundreds of mathematicians present also reacted and applauded.

Listening to the applause, Xiao Yi smiled and said, "Thank you, everyone."

As the applause gradually died down, they finally said goodbye and hung up the phone.

"Well... I'm really a little uncomfortable."

Xiao Yi turned off the computer and sighed.

If he was on the scene, he would be able to answer the questions on the blackboard himself, but unfortunately, it was a live answer, so he could only let Zhang Yitang write it for him.

Turning back, he saw Ye Cheng and the other two staring at him in amazement.

He waved his hand: "Wake up, what are you thinking about?"

The three people finally came back to their senses, and Ye Cheng said blankly: "Is this... the feeling of giving a report on the stage?"

Song Ziyang also swallowed his saliva: "Hundreds of people are looking at me... I feel so stressed."

Luo Qiao: "Brother Xiao, don't you feel nervous when there are so many people?"

"Uh..." Xiao Yi shrugged and spread his hands: "It's just a few hundred people, probably less than 400, so it's just like that."

Three people: "..."

Then they all gave Xiao Yi a middle finger.

Pooh!

...

It was already late, and they didn't waste time anymore. They went to rest early. After all, Ye Cheng had a damn early morning tomorrow.

However, although Xiao Yi didn't have classes, they didn't envy him. After all... Xiao Yi might get up earlier than them.