Omnipotent Data

Chapter 385

"Fritzjohn Necessary Optimality Conditions on Riemannian Manifolds"!

This is the proposed subject of Cheng Nuo's research project in the next two months.

Professor Fresnel briefly told Cheng Nuo and Hull some things to pay attention to in the cubicle, and then asked the two to go back with the documents to make preparations, and then officially start the research project the next day.

Cheng Nuo naturally had no objection.

He also wanted to take advantage of this time to learn some knowledge about the subject.

Although his mission may only be to help Professor Fresnel, it is always right to make adequate preparations.

Cheng Nuo was sitting on the desk, propping his chin with one hand, and flipping through the documents Fresnel gave him with the other.

The subject of Riemannian manifolds is one of the 50 national key mathematical research projects in 2022 directly approved by the Clay Institute of Mathematics in the United States.

These 50 mathematical scientific research projects are among the top in the world in terms of project difficulty and importance.

In fact, as one of the most developed countries in the field of mathematics in the world today, the Clay Institute of Mathematics in the United States is responsible for leading the forefront of world mathematics.

In addition to the characteristics of the Clay Mathematics Institute's deep pockets, each of these fifty national key mathematics research projects has provided financial support of US$100,000.

Moreover, the mathematicians responsible for the research work of these 50 scientific research projects are all top mathematicians in the world.

Just like Cheng Nuo's current boss, Professor Fresnel, who is a super expert in the field of geometry, among the 50 projects related to three topics in the field of geometry, the Clay Institute assigned the most difficult one to him. .

That is the subject of the Riemannian manifold that Cheng Nuo got.

During the whole morning, Cheng Nuo read the documents while looking for related papers on the Internet.

Disaster! Really difficult!

This is the result of Cheng Nuo's research all morning.

He finally knew why the Cray Mathematics Institute gave this subject to Professor Fresnel, because in today's mathematics world, there are probably no more than five mathematicians who can guarantee to solve this subject within two months .

And Professor Fresnel is obviously the safest one.

Not to mention the short research time given, there are too few papers and materials on the Internet in this area, which means that they are almost starting from scratch.

The Riemannian manifold is originally a super difficult point in the field of geometry research. Coupled with the knowledge of function theory and differentiation, it is enough to drive most mathematicians in the world crazy.

Ask yourself, if you leave this project to Cheng Nuo alone, it will take at least three years to start.

"It seems that for the time being, we still have to hold on to Professor Fresnel's thigh!" Cheng Nuo sighed and continued to collect information.

………………

The next day, Cheng Nuo came to the office early.

As soon as Professor Fresnel arrived, Cheng Nuo and Hull were called to the cubicle again.

"How are you preparing?" Professor Fresnel asked as soon as he came up.

Hull smiled wryly, "Teacher, there are indeed too few information about this on the Internet, and there are no books with high relevance in the library, so..."

Professor Fresnel waved his hand, as if anticipating this situation.

"The current mathematical research in this direction is indeed a blank space, so we need to study and fill in it!" Professor Fresnel's eyes slowly swept across the faces of the two, "So I said yesterday, you have to do Be mentally prepared. This is a tough fight!"

"Starting from scratch, there is no reference material, and the time limit... is only two months!"

Professor Fresnel continued, "I won't say anything to encourage you. I just hope you two don't forget the purpose of coming here. If you want to quit, I welcome it at any time."

"The extra words are here, now let's talk about the subject matter."

Professor Fresnel asked the two to find a place to sit down, moved over a laptop, opened a ppt, and pointed, "This is a brief research process I did."

"I will lead this project, and the task of the two of you is to assist me and solve some not-so-difficult links."

Cheng Nuo and Hull nodded to show they knew.

With the abilities of the two of them, it is not enough to support the framework of this project.

Professor Fresnel continued to explain, "The proposed name of this project is called Fritzjohn Necessary Optimality Conditions on Riemannian Manifolds. Then we must first understand what is Riemannian Manifold and what is Fritzjohn Necessary Optimality Conditions!"

"It goes without saying that the concept of Riemannian manifold, but the Fritzjohn necessary optimality condition should be relatively unfamiliar to you." He first looked at Cheng Nuo, "Cheng Nuo, do you understand this concept?"

Cheng Nuo replied without thinking, "The so-called fritzjohn necessary optimality condition refers to the necessary optimality condition of minf(x), st.{g(x)≤0, h(x)=0, x∈m .”

"That's right, this is the Fritzjohn's necessary optimality condition. As you can see, if this Fritzjohn's necessary optimality condition is studied directly, not only are there too many variables, the function equation is not easy to define, but there are also formulas in the derivation process complicated question."

"Because of this, we need to change our thinking."

Professor Fresnel turned to the next page of ppt, and there was only one line of formula written on it:

f:m→r, g:m→r^l, h:m→r^n

Cheng Nuo glanced at it and suddenly realized, "lipshitz function?!"

Professor Fresnel glanced at Cheng Nuo with a hint of admiration, "To be precise, it's a local lipshitz function!"

The lipshitz function means that if f(x) satisfies any two different real numbers x1 and x2 of the domain d on the interval i: ∥f(x1)-f(x2)∥=k∥x1-x2∥ If it is established, there must be f(x) consistent and continuous on the interval i.

In Cheng Nuo's heart, he already roughly understood what Professor Fresnel's problem-solving point was for this project.

Professor Fresnel continued his theoretical explanation, "In this formula, we can regard m as an m-dimensional Riemannian manifold."

"Aytonke's paper on the mp problem in Hilbert space, you two should have read it?"

The two nodded at the same time.

"That's good. By analogy, we can extend the mp problem from a linear space to a differential manifold, and the differential manifold is non-smooth. Then we can have the following framework."

The next ppt was displayed in front of the two of them.

"The first step is to establish non-smooth analysis tools on the Riemannian manifold, that is, to define generalized directional derivatives and generalized gradients on the manifold."

"In the second step, discuss the properties of generalized gradients."

"The third step, on the basis of the first two steps, discusses the Fritzjohn-type optimality condition for the problem (mp) on the Riemannian manifold."

"the fourth step,……"

The framework has already been set up by Professor Fresnel.

And Cheng Nuo felt enlightened when he saw the orderly steps of the process.

It turns out that this project should be done in this way!

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