Chapter 382

Chapter 385

fritzjohn necessary optimality conditions on Riemannian manifolds!

This is the project to be studied by Cheng Nuo in the next two months.

Professor Fresnel briefly told Cheng Nuo and hull some things to pay attention to in the cubicle, and asked them to go back with the documents for preparation, and then formally started the research next day.

Cheng Nuo naturally has no opinion.

He also wants to take advantage of this time to learn some knowledge related to the subject.

Although his task may only be to help Professor Fresnel, it is always right to be well prepared.

Cheng Nuo sits on his desk, one hand on his chin, the other hand over the papers Fresnel gave him.

The subject of Riemannian manifold is one of the 50 national key mathematical research projects in 2022 directly approved by clay Institute of mathematics in USA.

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

In fact, as one of the most developed countries in the field of mathematics in the world, clay Institute of mathematics in America plays a leading role in leading the world's mathematics.

In addition to the characteristics of Cray Institute of mathematics, each of the 50 national key mathematical research projects has given us $100000 in financial support.

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

For example, Professor Fresnel, the current boss of Cheng Nuo, is a giant in the field of geometry. Among the 50 projects, the Cray Institute has entrusted him with the most difficult one.

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

In the morning, Cheng Nuo was reading the documents while searching for relevant papers on the Internet.

Difficult! It's really hard!

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

He finally knew why the Cray Institute of mathematics had to hand over the project to Professor Fresnel, because in today's mathematical world, the number of mathematicians who can guarantee to complete the project in two months is not more than five fingers.

And Professor Fresnel, obviously, is the safest one.

The time given for scientific research is too short. Besides, there are too few papers and materials on the Internet, which means that they start almost from scratch.

Riemannian manifold is the super difficult point in the field of geometry. In addition to the knowledge of function theory and differential, most mathematicians in the world are crazy.

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

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

………………

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

Once professor Fresnel arrived, Chennault and hull were called into the cubicle again.

"How are you getting ready?" Professor Fresnel came up and asked.

Hull laughed bitterly. "Teacher, there is really too little information on this field on the Internet, and there are no books with too high relevance in the library, so..."

Professor Fresnel waved his hand as if expecting this.

"At present, mathematical research in this direction is indeed a blank, so we need to study and fill it in!" Professor Fresnel's eyes swept slowly across their faces. "So I said yesterday, you should be prepared. This is a tough battle

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

Professor Fresnel continued, "I won't say anything about encouragement. I just hope you two don't forget the purpose of coming here. If you want to quit, I'm always welcome."

"Here's the extra words. Now let's talk about the subject."

Professor Fresnel asked them to find a place to sit down, moved over to a laptop, opened a PPT, and pointed out, "this is a short research process I did."

"I am the leader of this project, and your two tasks are to assist me in solving some links that are not too difficult."

Cheng Nuo and hull nodded to show that they knew.

The ability of the two of them is not enough to support the framework of the project.

Professor Fresnel went on to explain, "the proposed name of this project is called Fritz John necessary optimality conditions on Riemannian manifolds. First of all, we must understand what Riemannian manifolds are and what Fritz John's necessary optimality conditions are! "

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

Cheng Nuo replied without thinking, "the so-called Fritz John necessary optimality condition is the necessary optimality condition of minf (x), St. {g (x) ≤ 0, H (x) = 0, X ∈ M"Yes, that's Fritz John's necessary optimality condition. You can also see that this Fritz John necessary optimality condition, if we study it directly, will not only have a lot of variables and the function equation is not easy to define, but also has the problem of complex formula in the process of derivation. "

"So we need to change our thinking."

Professor Fresnel turned to the next page of PPT, which only wrote a line of formula:

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 trace of appreciation. "To be exact, it's a local Lipschitz function!"

Lipschitz function means that if f (x) satisfies on the interval I for any two different real numbers X1 and X2 of the definition domain D: ‖ f (x1) - f (x2) ‖ = k ‖ x1-x2 ‖ holds, there must be f (x) uniformly continuous on the interval I.

in Cheng Nuo's mind, he has roughly understood what the breakthrough point of Professor Fresnel in this project is.

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

"You two should have read the paper by Aton on MP in Hilbert space?"

They both nodded at the same time.

"That's good. By analogy, we can extend the MP problem from linear space to Differential Manifolds, and differential manifolds are nonsmooth. Then we can construct the following framework."

The next PPT is shown in front of the two.

"The first step is to establish a nonsmooth analysis tool on Riemannian manifolds, that is, to define generalized directional derivatives and generalized gradients on the manifolds."

Second, we discuss the properties of generalized gradient

In the third step, on the basis of the first two steps, the fritz John type optimality conditions for the problem (MP) on Riemannian manifolds are discussed

The framework has already been set up by Professor Fresnel.

And Cheng Nuo in the see that orderly process steps, there is a sense of urgency.

Originally, this project should be done like this!

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