Great Country Academician

Chapter 732: A world-class mathematical problem that was killed casually

The weak Riemann hypothesis is proved. By shrinking the Riemann function ζ back to Jensen's inequality, the critical band Re(s))=1/2 can be compressed in another way.

For the mathematical community, this is a gluttonous feast.

It can be said that most scholars in analytic number theory, algebraic number theory, function theory, and even algebraic geometry have more or less studied this idea and tried to advance it.

The new generation of Fields Medal winners, including Terence Tao and Schultz, are all interested in trying further research on this basis.

Even Perelman, who lives in seclusion in St. Petersburg, downloaded Xu Chuan's paper posted on arxiv and followed this idea to push its limits on a simple blackboard.

This is a storm that has swept through most of the mathematics community. Leaving aside those who are just trying to mess up their papers, it seems to have gradually evolved into a competition game for others.

A game in which most scholars in the field of number theory participate; a competition to use this tool to advance the offset of Jensen's inequality and study the value of Riemann's conjecture Re(s).

In such an atmosphere, in just three months, the offset of Jensen's inequality, that is, the critical band of the Riemann function Re(s) has been advanced to No(T)\u003e0.731N(T)

For the Riemann Hypothesis, which has been stuck on the value No(T)\u003e0.35N(T) for 44 years, nearly half a century, these three months are like dew in the Sahara Desert, nourishing the world. brought countless lives.

For the mathematics community, what makes many mathematicians even more curious is how deeply Professor Xu Chuan, who created this tool, has conducted research in this area.

After all, anyone with a discerning eye can see that in the paper he published, No(T)\u003e0.50N(T) was far from his limit.

Even some doctoral students in the field of analytic number theory can further expand on this basis. There is no reason why he, the creator, can only stop here.

Everyone is curious about how far the Jensen inequality offset and the critical zone of the Riemann function Re(s) can be pushed if Professor Xu takes action.

Jinling, Nanda.

After the Lantern Festival, students returned to school and the deserted campus gradually became lively.

As the "Annals of Mathematics" published the papers "The Weak Riemann Hypothesis and "Polynomial Algorithm for Positive Integer Factorization", the banner at the gate of Nanjing University also changed accordingly.

"Warmly congratulate Professor Xu Chuan from our school for conquering the weak Riemann Hypothesis! 》

"Warmly congratulate Liu Jiaxin, honorary professor of our school, for conquering the conjecture of "whether the factorization of positive integers has a polynomial algorithm"! 》

Two big red banners were hung in the most conspicuous position, "showing off" the "strength" of the NTU Mathematics Department to the outside world, and also attracted the admiring and respectful eyes of countless students and parents who came to report.

In the office, on the first day back to school in the new year, Xu Chuan called five students over and collected their ‘winter vacation’ homework.

In what sense does the "high-precision" format still maintain high-precision values ​​when the partial differential equation is nonlinear and has discontinuities in the solution?

This problem was a winter vacation assignment he left to several students at the end of last year. He did not require them to solve the problem, but he required them to think about it when facing a certain problem.

After receiving the homework from several people, Xu Chuan sat behind his desk and read through them one by one.

After briefly flipping through the homework of several students, he sighed, placed the manuscript in his hand on the table, and looked at the students.

"It has been about twenty days since the New Year celebrations. Have you all been busy setting off fireworks? What are the things you have handed in?"

His eyes fell on several students. One winter vacation, regarding the questions he left, judging from the answers of several students, their thinking was almost superficial.

Obviously, the answers of these five students fell far short of his requirements.

The best answer was given by the oldest Ding Rui, but even so, this answer did not touch the boundary of this question.

This result inevitably made him a little disappointed.

After all, when he raised this question, he already had some certainty and answers in his mind.

Give them twenty days. Not to mention solving this problem, at least there should be some research progress on it, right?

"But this question is really difficult, Professor."

In the office, Yin Shi lowered his head and muttered quietly.

Xu Chuan stared at her, not caring that she was a girl, and said coldly: "It is difficult to think about it. If it is easy, what will you do?"

"Only through actual actions can we truly understand the difficulty of the problem and its solution. But obviously, I didn't see much research on your homework."

"I don't want to see you retreating because of difficulties, but I also want to see your courage in mathematics."

After being criticized by Xu Chuan, Rong Xinji looked over with an innocent face, mustered up the courage to respond in a low voice: "But this is a world-class conjecture, professor!"

“How can progress be so easy?”

Leaving a world-class mathematical conjecture as a winter vacation homework, and requiring research progress within twenty days, is this something that humans can do?

Rong Xinji just wants to cry for him, this is too difficult!

Hearing this, Xu Chuan was stunned for a moment: "World-class guess?"

"Yes, Professor, if you don't believe it, search this question. It is the core part of the conjecture of 'High-precision format for discontinuous solutions of nonlinear partial differential equations'"

Hearing this, Xu Chuan glanced at him suspiciously, turned around, turned on the computer, cut a section of the question and put it on the search engine.

Soon, search results popped up one after another. He clicked on one at random and entered to take a look.

"A very important area of ​​research for numerical analysis and scientific computing is the design and analysis of numerical formats for solving partial differential equations."

"But so far, except for some special circumstances, such as knowing the number of discontinuities in advance and being able to accurately calculate the location of discontinuities, there is no mathematical theory to prove that the numerical solution of the "high-precision" format of the discontinuous solution of nonlinear equations is in a certain It still maintains high accuracy in this norm sense.”

"This is still a world-class problem that has not been solved since it was proposed in the 1930s."

"In the field of partial differential equations, its importance is no less than Zhou's conjecture in Mersenne prime numbers. Research on the rigorous mathematical theory of the true meaning of high precision in nonlinear equations will be very meaningful work."

Xu Chuan briefly looked through the searched information and was a little confused.

A question he asked casually before the Chinese New Year could be said to be an expanded question after being explained in a small class. How did it become a world-class mathematical conjecture?

Comparing this conjecture to Zhou's conjecture in Mersenne's prime numbers made him even more confused.

Zhou's guess is a guess put forward by Chinese mathematician and linguist Professor Zhou Haizhong in the article "Distribution Law of Mersenne Prime Numbers" in an accurate expression.

This guess is an accurate expression of Mersenne prime numbers and is quite mathematically beautiful.

However, so far, it has not been proven or disproven, and it has become a famous mathematical problem.

Although importance and difficulty are not the same thing, the comparison is enough to show that this problem is not that simple.

At least, it is enough to say that it is a world-class mathematical problem.

Looking away from the computer screen, Xu Chuan looked at several students, with expressions on his face that were a little embarrassed and confused.

It is indeed a bit outrageous to assign a world-class mathematical problem to students as winter vacation homework.

However, is this problem really that difficult?

After thinking for a while, Xu Chuan stood up and walked to the other side of the office. He dragged out a folded blackboard from the corner, opened the two sides, and stood in front of him.

He picked up the marker from the pen basket nearby and stared at the blackboard in front of him and thought for a while.

In general, the development of solutions to the Fisher equation is mainly carried out using analytical methods. Among various numerical iteration methods for solving differential equations, the non-standard finite difference method is one of the most effective methods.

However, when constructing a non-standard difference scheme, considering the concept of "exact" scheme makes constructing an exact finite difference scheme very important in the research of constructing new numerical algorithms.

That is, μt=μμxx+μ(1-μ), and its waveform solution form is as follows: μ(x, t)=1/(1+e^b·e·x-5t/6/6), and where t ≥0, b is a constant. And satisfy: 0≤μ(x,t)≤1, when xb u(x,0)= 0

Standing in front of the blackboard, Xu Chuan was thinking about the high-precision format for discontinuous solutions to nonlinear partial differential equations.

Behind him, several students looked at their professor in confusion, not understanding what he wanted to do for a while.

"Professor, what are you doing?" Standing behind him, Rong Xinji poked Ding Rui beside him with his finger and asked curiously in a low voice.

Ding Rui thought for a moment, shook his head and said, "I don't know, maybe you want to explain the problem to us?"

In front of the blackboard, Xu Chuan ignored the whispers of several students.

After thinking in his mind for a while, he raised the marker in his hand, and the white pink line fell on the black panel, outlining mathematical symbols one by one.

【u t +αuδ u x u xx =βu(1 uδ)(uδγ)】

[The waveform is: u(x, t)=(γ/2+γ/2tanh[A(x A t)])1/δ】

Looking at the calculations on the blackboard, Xu Chuan's eyes flashed with thought and he muttered softly.

"Using the idea of ​​the exact traveling wave solution of the equation and the construction of exact differences, a special Fisher equation can be given. However, even for the special Fisher equation, the exact difference format is very complex in form."

"For the convenience of use, two simple non-standard finite difference schemes can be obtained from the exact finite difference scheme. That is, a step function of the non-standard finite difference method is constructed so that it continuously changes during the calculation process."

Lines of calculations were written on the whiteboard in front of him, accompanied by mutterings.

Standing behind Xu Chuan, several students were completely confused.

It wasn't until the calculations in front of them covered the entire blackboard that anyone reacted.

Looking at the calculations on the blackboard in shock, Tong Yang, who had always been a little transparent, suddenly took a deep breath and couldn't help but said: "Damn it. Professor, he must be solving that problem, right?"

"Problem? Which problem?" someone asked subconsciously.

"The one for our winter vacation homework!"

"Huh? No way?! That's a world-class problem!"

Among the students, the small discussion was full of shock. Even though they knew that their tutor had strong mathematical abilities, no one dared to imagine that a world-class mathematical problem would be solved in this way.

Standing behind Xu Chuan, looking at the back in front of him and his arms writing quickly, he felt his scalp numb.

Just when Xu Chuan filled a panel and moved to the other side, she remembered something, quickly took out her mobile phone from her pocket, and turned on the camera.

Video recording is turned on, and the layout in front of you is quickly included in the photo.

Although she can no longer understand the calculations on the blackboard, don’t panic when something happens. Use your mobile phone to take a photo first, so you can post it on your WeChat Moments later. Ahem, it’s just a show off!

After all, if this is really as expected, then this is definitely a historic moment that cannot be missed!

She couldn't even imagine how powerful her tutor was in mathematics if this was true!

Can a world-class mathematical problem that was easily solved prove this?

terrible!

In front of the blackboard, Xu Chuan did not pay attention to the whispers of several students behind him. At this moment, he was completely immersed in his own world.

The chalk in his hand kept drawing across the blackboard, and line after line of calculations quietly jumped into his pupils.

Perform local construction and truncation errors on the non-standard finite difference scheme of the Fisher equation, and then conduct numerical examples through the Burgers equation, and use the coupled constant coefficient convection-diffusion equation to perform the non-standard θ-algorithm

Lines of mathematical formulas emerged in his hands, flowing on the blackboard in front of him like the beautiful notes played by the most perfect pianist.

At this moment, the numbers in front of him seemed to be the most wonderful thing, and all the calculations were like the most tacit partners, following his progress as he pleased.

The calculations on the blackboard in front of me were filled up and then erased, and clean characters and formulas quickly filled the entire page.

One problem after another was solved by formulas that were constantly written, and one difficulty after another was constantly crossed.

For him, the problem of high-precision format for discontinuous solutions of nonlinear partial differential equations does have some difficulties.

But in front of the inspiration pouring out of my mind at this moment, it was like a sand dam facing the impact of a flood. It fell into the rolling water in an instant, and finally gathered together and flowed towards the ocean of knowledge at the farthest reaches.