Chapter 385: Lipschitz function

Chapter 385
"Fritzjohn Necessary Optimality Conditions on Riemannian Manifolds"!
This is the proposed topic for Cheng Nuo to study the project in the next two months.
In the cubicle, Professor Fresnel briefly told Cheng Nuo and Hull about some matters needing attention, and asked them to take the documents back to make preparations, and then formally start the research project the next day.
Cheng Nuo naturally had no opinion.
He also wants to take advantage of this little time to learn some knowledge about the subject.
Although his task may be to lay hands on Professor Fresnel, it is always right to be adequately prepared.
Cheng Nuo sat on the desk with one hand propped on his chin and the other hand flipping through the documents Fresnel gave him.
The subject of Riemannian manifolds is one of the 50 national key mathematical scientific research projects approved by the Clay Institute of Mathematics in the United States in 2022.
These fifty research projects in mathematics are among the best 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 today, the Clay Institute of Mathematics in the United States plays a leading role in leading the forefront of mathematics in the world.
In addition to the rich wealth of the Cray Institute of Mathematics, these fifty national key mathematics research projects are each given a funding of 100,000 U.S. dollars.
Moreover, the mathematicians who are responsible for the research of these fifty scientific research projects are all the world's top mathematicians.
Just like Cheng Nuo’s current boss, Professor Fresnel, as a super man in the field of geometry, the Cray Institute gave him the most difficult one among the fifty projects on three topics in the field of geometry. .
That is, the subject of the Riemannian manifold that Cheng Nuo took.
In the morning, Cheng Nuo read the documents while looking for related papers on the Internet.
difficult! It's really hard!
This is the result of Cheng Nuo's research all morning.
He finally knows why the Cray Institute of Mathematics should entrust this subject to Professor Fresnel, because today’s mathematics community can guarantee that the mathematicians who can handle this subject within two months will probably not exceed five fingers. .
Professor Fresnel is obviously the safest one.
The research time given is too short. There are too few papers and materials related to this aspect on the Internet, which means that they almost started from scratch.
The Riemannian manifold is originally a super difficult point in the field of geometry, coupled with the relevant knowledge of function theory and differentiation, enough to drive most mathematicians in the world crazy.
Ask yourself, if you leave this project to Cheng Nuo alone to complete, it will start at least three years.
"It seems that for the time being, we still have to hold Professor Fresnel's thigh firmly!" Cheng Nuo sighed, and continued to collect information.
………………
The next day, Cheng Nuo came to the office early.
When Professor Fresnel arrived, Cheng Nuo and Hull were called into the cubicle again.
"How are your preparations?" Professor Fresnel asked when he came up.
Hull smiled bitterly, "Teacher, there is really too little information on this topic on the Internet, and there are no books with high relevance in the library, so..."
Professor Fresnel waved his hand, seeming to anticipate this situation.
"The current mathematics research in this direction is really blank, so we need to study and fill it up!" Professor Fresnel's gaze slowly swept across the two faces, "So I said yesterday, you do Be mentally prepared. This is a tough battle!"
"Starting from scratch, there is no information to learn from, and the time limit... only two months!"
Professor Fresnel continued, "I won't say anything to cheer and encourage. I just hope you two will not forget the purpose of coming here. If you want to quit, I'm always welcome."
"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."
"This project, I take the lead, and the task of the two of you is to assist me and solve some links that are not too difficult."
Cheng Nuo and Hull nodded, indicating that they knew.
With their abilities, they are not enough to support the framework of this project.
Professor Fresnel continued to explain, "The proposed name of this project is called the necessary optimality condition of fritzjohn on the Riemannian manifold. Then we must first understand what is the Riemannian manifold and what is the necessary optimality condition of fritzjohn!"
"It goes without saying that the concept of Riemannian manifolds, and the necessary optimality conditions of fritzjohn should be relatively unfamiliar to you." He first looked at Cheng Nuo, "Cheng Nuo, do you understand this concept?"
Cheng Nuo answered without hesitation, "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 ."
"Yes, this is the necessary optimality condition of fritzjohn. You can also see that this necessary optimality condition of fritzjohn, if you directly study it, not only has a lot of variables, and the function equation is not well defined, there are also formulas in the derivation process. complicated question."
"For this reason, we need to change our thinking."
Professor Fresnel turned to the next page of ppt, where only one line of formula was written:
f:m→r, g:m→r^l, h:m→r^n
Cheng Nuo glanced, and suddenly realized, "lipshitz function?!"
Professor Fresnel glanced at Cheng Nuo with a hint of appreciation, "To be precise, it is a partial lipshitz function!"
The lipshitz function means that if f(x) satisfies any two different real numbers x1 and x2 of the domain d in the interval i: ∥f(x1)-f(x2)∥=k∥x1-x2∥ Yes, there must be f(x) uniformly continuous on the interval i.
In Cheng Nuo's mind, he probably understood what Professor Fresnel's problem was.
Professor Fresnel continued his theoretical explanation, "In this formula, we can regard m as an m-dimensional Riemannian manifold."
"Aidunke's paper on mp in hilbert space, you two should have read it?"
The two nodded at the same time.
"That would be great~EbookFREE.me~ 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 frame construction ."
The next ppt is shown in front of the two.
"The first step is to establish a non-smooth analysis tool on the Riemannian manifold, that is, to define generalized directional derivatives and generalized gradients on the manifold."
"The second step is to discuss the nature of the generalized gradient."
"The third step, on the basis of the first two steps, discuss the fritzjohn-type optimality condition of the problem (mp) on the Riemannian manifold."
"the fourth step,……"
The framework has already been built by Professor Fresnel.
When Cheng Nuo saw the orderly steps of the process, he felt a kind of enlightenment.
It turns out that this project should be done like this!
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