Chapter 732: A World-Class Math Problem That Was Killed Casually

The weak Riemann conjecture proves that the critical band Re(s))=1/2 can be compressed in another way by shrinking the Riemann function ζ back to the Jensen inequality.

It's a feast for the mathematical community.

It can be said that the vast majority of scholars in the fields of analytic number theory, algebraic number theory, function theory, and even algebraic geometry have studied this idea to a greater or lesser extent and tried to promote it.

The new generation of Fields Medal winners, including Tao Zhexuan and Schultz, have tried further research on this basis with great interest.

Even Perelman, who lived in seclusion in St. Petersburg, downloaded Xu Chuan's paper hanging on Arxiv, pushing its limits on a rudimentary blackboard along this line of thought.

It's a storm that has swept through much of the math world, and aside from those who are trying to mix papers, it seems to have evolved into a competitive game for others.

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

In such an atmosphere, in just three months, the Jensen inequality shift, i.e., the critical band of the Riemann function Re(s), has been advanced to No(T)>0.731N(T)

For the Riemann conjecture, which has been stuck at the No(T)>0.35N(T) value for 44 years and nearly half a century, these three months are like a nectar in the Sahara Desert, nourishing countless vitality.

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

After all, anyone with a discerning eye can see that in the paper he made public at the beginning, No(T)>0.50N(T) is far from his limit.

Even some PhD students in the field of analytic number theory can further expand on this basis, and it makes no sense that he, as a creator, can only stop there.

Everyone is curious about how far the Jensen inequality offset and the Riemann function Re(s) critical band can be advanced if Professor Xu makes a move.

Jinling, Nanda.

After the Lantern Festival, the return of students to school gradually livened up the deserted campus.

With the publication of the weak Riemann conjecture and the paper "Proof of Solving Polynomial Algorithm for Factorization of Large Positive Integer Factorization" in the Annals of Mathematics, the banner at the entrance of NTU also changed.

"Warmly celebrate Professor Xu Chuan's overcoming of the weak Riemann conjecture!" 》

"Warmly celebrate Professor Liu Jiaxin's overcoming of the conjecture of 'whether the factorization of large positive integers has polynomial algorithms'! 》

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

In the office, on the first day of returning to school for the new year, Xu Chuan called five students over and collected their 'winter vacation' homework from them.

When a partial differential equation is nonlinear and the solution is interrupted, in what sense does the "high precision" format still maintain a high accuracy value?

This problem is the winter vacation homework he left for several students at the end of last year, and he does not require a few people to be able to solve this problem, but requires them to have their own thinking when facing a certain problem.

Receiving the homework from the hands of several people, Xu Chuan sat behind the desk and flipped through it one by one.

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

"From the New Year to the present, has there been more than 20 days, are you all busy setting off fireworks? What did you hand in? ”

Eyes fell on a few students, a winter vacation, for the questions he left, judging from the answers of several students, their thinking was almost superficial.

Clearly, the five students' answers fell far short of what he was looking for.

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

Such a result inevitably made him a little disappointed.

After all, when he asked this question, he already had some certainty and answer in his heart.

Leave them twenty days, not to mention solving this problem, at least there is some research progress on it, right?

"But that's a really hard question, Professor."

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

Xu Chuan stared at her, didn't care that she was a girl, and said coldly: "Difficult is the point of thinking, if it is easy, what will you leave it for?" ”

"It is only through practical action that we can truly understand how difficult the problem is and how to solve it. But obviously, I don't see much research on your assignments. ”

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

was criticized by Xu Chuan, Rong Xinji on the side looked at him innocently, and mustered up the courage to respond in a low voice: "But this is a world-class conjecture, professor!" ”

"How can it be so easy to make progress?"

Is it something that a person can do to leave a world-class mathematical conjecture as a winter vacation homework, and also require research progress within 20 days?

This will make Rong Xinji just want to cry to him, this is too difficult!

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

"Yes, Professor, if you don't believe me, you can search for this problem, which is the core part of the conjecture of the 'high-precision format for the discontinuous solution of nonlinear partial differential equations'"

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

Soon, one search result after another popped up, and 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, with the exception of some special cases, such as knowing the number of discontinuities in advance and being able to accurately calculate the position of the discontinuities, there is no mathematical theory to prove that the numerical solution of the problem in the form of "high precision" for the discontinuous solution of nonlinear equations still maintains high accuracy in some normative sense. ”

"This is still a world-class problem that has been raised since the thirties of the last century and is still unsolved."

"In the field of partial differential equations, which are no less important than Zhou's guesses in Mersenne primes, the study of rigorous mathematical theories about the true meaning of high precision in nonlinear equations will be very interesting work."

After briefly flipping through the searched information, Xu Chuan was a little confused.

A question he casually asked before the Chinese New Year, which can be said to be a problem that was expanded after being explained in a small class, how did it become a world-class mathematical conjecture?

And comparing this conjecture to Zhou's guess in the Mersenne prime number made him a little dazed.

Zhou's guess is a conjecture made by Professor Zhou Haizhong, a Chinese mathematician and linguist, in the article "The Distribution Law of Mersenne Prime Numbers" with an exact expression.

This guess is an exact expression of a Mersenne prime, and it is mathematically beautiful.

To date, however, it has not been proven or disproved, and has become a well-known mathematical puzzle.

Although importance and difficulty are not the same thing, it is enough to see that the problem is not so simple.

At the very least, it's enough to say it's a world-class math puzzle.

Retracting his gaze from the computer screen, Xu Chuan looked at several students, the expression on his face a little embarrassed and a little dazed.

It's a bit outrageous to give students a world-class math problem as a winter vacation homework.

However, this question. Is it really that hard?

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

Picking up the marker from the pencil basket beside him, he stared at the blackboard in front of him for a moment.

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

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

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

Standing in front of the blackboard, Xu Chuan pondered the question of the high-precision format for the discontinuous solution of nonlinear partial differential equations.

Behind him, several students looked at their professor with bewildered expressions, and for a while they didn't understand what he wanted to do.

"What does this do, Professor?" Standing behind him, Rong Xinji poked Ding Rui beside him with his finger and asked in a curious whisper.

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

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

After thinking about it in his mind for a while, he raised the marker pen in his hand, and the white pink lines 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 equation on the blackboard, Xu Chuan's eyes flashed with thought, and his mouth was whispering.

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

"For the convenience of use, two simple non-standard finite difference formats can be obtained in the precise finite difference format. That is, the step size function of a non-standard finite difference method is constructed, so that it changes continuously during the calculation process"

Lines of calculations, accompanied by muttering, were constantly written on the whiteboard in front of them.

Standing behind Xu Chuan, several students were completely confused.

It wasn't until the equation in front of me covered the entire blackboard that someone reacted.

Looking at the equations on the blackboard in shock, Tong Yang, who had always been a little transparent, suddenly took a deep breath and couldn't help but speak: ". Teaching him, isn't he solving that problem? ”

"The problem? Which conundrum? Someone asked subconsciously.

"The one from our winter vacation homework!"

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

Among the students, the small discussion was full of shocking emotions, even if they knew that their tutor was very strong in mathematics, but no one dared to think 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 the arm that was writing quickly, he only felt that his scalp was 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 quickly earns in the photo.

Although she can't understand the equation on the blackboard, don't panic in case of trouble, take a picture with your mobile phone first, and then you can send a circle of friends. Ahem, it's a wave of showing off!

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

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

Can a world-class math problem that has been killed casually prove it?

It's horrible!

In front of the blackboard, Xu Chuan didn't 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 slicing across the blackboard, and line by line of calculations kept quietly jumping into his pupils.

The non-standard finite difference format of the Fisher equation is locally constructed and the truncation error is carried out, and then the numerical example is carried out by the Burgers equation, and the non-standard θ-algorithm is carried out on the flow-diffusion equation by using the coupling advance coefficient

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 footsteps as they wished.

The equations on the blackboard in front of him were filled and then erased, and the neat characters and formulas quickly spread all over the page.

One problem after another is solved by the constantly written formula, and one difficulty after another is constantly crossed.

For him, the difficulty of a high-precision format for discontinuous solution of nonlinear partial differential equations is difficult.

But in the face of the inspiration that gushes out of my mind at this moment, it is like a mud dam facing the impact of a flood, and it suddenly falls into the boiling water, and finally converges and flows towards the ocean of knowledge at the bottom.