Chapter 383

Chapter 386

Professor Fresnel explained the idea thoroughly.

In brief, the concepts of generalized directional derivative and generalized gradient of Lipschitz function are given on Riemannian manifold.

Then, the properties and operation rules of generalized gradient are derived by using the properties of local differential homeomorphism between Riemannian manifold and open set of Euclidean space, tangent mapping and cotangent mapping.

At the same time, it is proved that the necessary condition for a function defined on the Riemannian plague form to obtain a small value is that the generalized gradient contains zero elements. The fritz John type optimality conditions for mathematical programming problems on Riemannian manifolds are given by using these properties.

At present, the framework built by Professor Fresnel is not perfect, because the following specific context drawing still needs to be judged according to the specific situation of the study at that time. However, this imperfect framework is beyond the reach of Cheng Nuo at present.

After a glimpse, Cheng Nuo had to admire Professor Fresnel's attainments in the field of geometry, which was not only one level higher than himself.

"There's a long way to go." Cheng Nuo sighed.

There is a general framework, and Cheng Nuo is not a major researcher, so there is not much work left on him.

This time, with the purpose of learning, he participated in the research of this national key mathematical project.

After Professor Fresnel took the task here, Cheng Nuo and hull left the cubicle together.

Outside the compartment, hull breathed a little and patted Cheng Nuo on the shoulder. "Cheng, I'm afraid we'll be busy in the next two months."

He has been an assistant to Professor Fresnel for nearly two years, and has participated in several "big projects" like this one.

The days of that time can only be described as being very busy, even if a group of them were fighting for Professor Fresnel.

Although the tasks assigned to them by Professor Fresnel are certainly those with less difficulty and lower complexity coefficient. But in the eyes of Professor Fresnel.

For these assistants, Professor Fresnel thinks that a simple task is as difficult as writing a SCI paper in one district.

Of course, during that time, under all kinds of pressure, they really grew faster than usual.

Now that the same situation fell on him again, hull was in pain and happy at the thought of staying up late in the office for the next two months.

Fortunately, this time at least with Cheng Nuo, his heart pain can be slightly reduced.

Just, look at the expression on Cheng Nuo's face, how can you not have the slightest worry?

Cheng Nuo grinned. "Hal, don't worry, Professor Fresnel and I are here, you won't have to live a whole two months of hard life!"

Professor Fresnel's level of mathematics is naturally needless to say.

Cheng believes that even without him and hull, Professor Fresnel alone can complete the project in two months.

Now the horses leading the carriage have changed from one horse to three. Although the other two horses are very small, as long as they don't drag their legs, it doesn't take two months.

But Cheng Nuo, only others drag his hind legs, how can he drag others' hind legs?

"Come on Cheng Nuo patted hull, who didn't understand what he said, and walked back to his desk with a smile.

Cheng Nuo moved his fingers and started the day's work.

The research process of the project is tentatively divided into three stages.

In the first stage, his task is simply to construct a few concepts.

Cheng Nuo, who has experienced numerous problems for a long time, is already familiar with it. He crackles on the computer:

[let (m, g) be a m-dimensional Riemannian manifold, and give a quantity product g (U, V) = & & u, V &;,? U, V ∈ TXM; where | V | x = & V, V &; denotes the norm of vector V ∈ TXM at point X. In this way, an equivalent norm is defined in tangent space 】

【…… Therefore, for each point x, the Riemannian metric g induces an isometric isomorphism between TXM and t (n, x) M: V * = GX (V, n) (V *, U) = GX (V, U),? U ∈ TXM.]

[define the norm of vector on the tangent space t * XM as | V * | = | V | = GX (V, V), so there is the Cauchy inequality & & V *, u &;

a single operation is as fierce as a tiger, and Cheng Nuo's fingers almost fly on the keyboard.

Cheng Nuo, who has a clear mind, hardly meets any stuck. A formula, as smooth as should appear on the computer screen.

The progress bar of the task is also rising slowly.

3%，8%，15%，23%……

On the other side, our classmate hull, is looking at the task list in his hand, frowning and thinking hard.

At this stage, his task is to study some properties of Lipschitz function and the transformation relation of Fritz John type optimality conditions. The difficulty is similar to that of Cheng Nuo.But at this time, he was in a state of melancholy, writing and drawing with a pen on the paper.

f(y)-f(x)|≤L|φ(y)-φ(x)|

f(x，v)=lim… lim…

No, it's not like this. It's not like this!

Hull scratched it out.

What about this? It seems that it is not right!

Ah, ah, ah! How to do it!

Hal rubbed his short blond hair, his blue eyes full of loveless eyes.

Riemannian manifold is one of hull's strong points, but Lipschitz function obviously belongs to the field of function theory, and he doesn't know much about it. Therefore, starting this project is much more difficult than the previous several major projects.

Looking at the line on the paper that had been crossed out, hull just wanted to shout:

God, who's going to help me!

Falling into melancholy, hull's eyes wandered around the office and finally landed on Cheng Nuo, not far away.

Over there, Cheng Nuo is happily slapping the keyboard, appears extremely excited.

Hull's eyes flashed.

…………

"Cheng!" Cheng Nuo, who is beating the keyboard happily, suddenly sees a head coming out of his side. He is so scared that he almost doesn't throw the computer out directly.

He took a look at Hull squatting beside him, covered his forehead and said, "Herr, you don't go to work. What are you doing here?"

Not to say that it's OK. As soon as he mentioned this, hull showed an extremely helpless expression, "I also want to work well, but the task assigned to me by the teacher is that I don't understand some places, so I came to see you."

"What do you mean?" Cheng Nuo Hu asked.

"Shall we exchange missions?" Hull suggested.

Cheng Nuo shook his head, which was a refusal, "no way."

"Why?" Hull didn't understand.

Cheng Nuo said with a smile, "because my task has been completed one third."

"What, one third?" Helton's eyes widened.

He looked directly at Cheng Nuo's eyes. "Cheng, you're not kidding, are you?"

"Well, I don't believe it." Cheng Nuo moved to the side, revealing the computer screen.

Hull looked up at the computer screen and landed on the pile of two page formulas.

Hiss!

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