universal data

Chapter 382 The Lipschitz Function
385 chapter

"Fritz John's Necessary Optimality Condition 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 be fully prepared.

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 2022 national key mathematical research projects in 50 directly approved by the Clay Institute of Mathematics in the United States.

These fifty mathematics research projects are among the best 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 plays a leading role in the forefront of mathematics in the world.

Coupled with the deep financial characteristics of the Clay Mathematics Institute, each of these fifty national key mathematics research projects has received financial support of US$[-].

Moreover, the mathematicians responsible for the research work of these [-] 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 [-] 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.

Throughout the morning, Cheng Nuo read documents and found relevant 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 small cubicle again.

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

Hull smiled bitterly, "Teacher, there is indeed too little information about this on the Internet, and there are no highly relevant books in the library, so..."

Professor Fresnel waved his hand, as if he expected 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 of them to find a place to sit down, brought over a laptop, opened a PPT, and pointed, "This is a brief project 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 Fritz John Necessary Optimality Conditions on Riemannian Manifolds. Then we must first understand what is Riemannian Manifold and what is Fritz John Necessary Optimality Conditions!"

"It goes without saying that the concept of Riemannian manifolds, but Fritz John's necessary optimality conditions should be relatively unfamiliar to you." He first looked at Cheng Nuo, "Cheng Nuo, do you understand this concept?"

Cheng Nuo answered without thinking, "The so-called Fritz John necessary optimality condition refers to the necessary optimality of minf(x), st.{g(x)≤0, h(x)=0, x∈M condition."

"Yes, this is the necessary optimality condition of Fritz John. As you can see, if we study the necessary optimality condition of Fritz John directly, not only are there many variables, and the functional equation is difficult to define, but there is also a derivation process. Problems with complex formulas."

"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, "Lipschitz function?!"

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

Lipschitz函数，是指若f(x)在区间I上满足对定义域D的任意两个不同的实数x1、x2均有：∥f(x1)-f(x2)∥<=K∥x1-x2∥成立，必定有f(x)在区间I上一致连续.
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."

"You two must have read Aytonko's paper on the MP problem in Hilbert spaces, right?"

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 a non-smooth analysis tool on the Riemannian manifold, that is, define generalized directional derivatives and generalized gradients on the manifold."

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

"The third step is to discuss the Fritz John type optimality conditions of the problem (MP) on Riemannian manifolds based on the first two steps."

"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!

(End of this chapter)