Omnipotent Data

Chapter 386

Professor Fresnel explained the ideas very thoroughly.

To put it simply, the concepts of generalized directional derivatives and generalized gradients of the lipsehitz function are first given on the Riemannian manifold.

Then, the property and algorithm of the generalized gradient are deduced by using the properties of Riemannian manifold locally and open set diffeomorphism in Euclidean space, tangent map and cotangent shot.

At the same time, it is proved that the necessary condition for the small value of the correspondence defined on the Riemannian form is that the generalized gradient contains zero elements. And using these properties, the fritzjohn-type optimality conditions for mathematical programming problems on Riemannian manifolds are given.

The framework built by Professor Fresnel is not perfect at present, because the subsequent drawing of the specific context still has to be judged according to the specific situation of the research at that time. But this imperfect framework alone is beyond Cheng Nuo's reach.

Taking a closer look, Cheng Nuo couldn't help but admire Professor Fresnel's accomplishments in the field of geometry, which was more than one grade above his own.

"There is still a long way to go!" Cheng Nuo sighed softly.

The general framework is already there, and Cheng Nuo is not the main researcher, so there is not much work for him.

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

After accepting the task from Professor Fresnel, Cheng Nuo and Hull exited the cubicle together.

Outside the cubicle, Hull breathed a sigh of relief and patted Cheng Nuo on the shoulder, "Cheng, I'm afraid we will be busy in the next two months!"

He has been Professor Fresnel's assistant for almost two years, and has participated in several "big projects" like this one.

The days during that time can only be described as being very busy, even though they were a group of helpers for Professor Fresnel.

Although the tasks assigned to them by Professor Fresnel must be less difficult and less complex. However, that is according to Professor Fresnel.

For these little assistants, what Professor Fresnel thought was a simple task, in their eyes, was no less difficult than writing an SCI paper in the first district.

Of course, during that time, under various pressures, they did grow much faster than usual.

Now the same situation befalls him again,

When Hull thought about the days of staying up late and working overtime in the office for the next two months, he felt pain and happiness in his heart.

Fortunately, at least Cheng Nuo was with him this time, so the pain in his heart could be relieved a little.

However, looking at the expression on Cheng Nuo's face next to him, why didn't he look worried at all?

Cheng Nuo grinned, "Hull, don't worry, with Professor Fresnel and I here, you won't have to live a miserable life for two whole months!"

Needless to say, Professor Fresnel's mathematics level.

Cheng Nuo believed that even without him and Hull, Professor Fresnel could finish the project within two months by himself.

Now the number of horses leading the carriage has changed from one to three. Although the other two horses are very thin, as long as they don't hold back, naturally it doesn't take two months.

As for Cheng Nuo, only others held him back, how could he hold others back?

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

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

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

And his task in the first stage was simply to construct a few concepts.

Cheng Nuo, who has experienced countless subjects for a long time, is already familiar with it. He typed on the computer:

[Assuming (m, g) is an m-dimensional Riemannian manifold, a quantity product g(u, v)=amp;lt;u,vamp;gt;,?u, v∈txm; where vx=amp;lt;v, vamp;gt; represents the norm of the vector v∈txm at the point x. In this way, an equivalent norm is defined in the 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.]

[Defining the norm of the vector on the tangent space t*xm is v*=v=gx(v, v), so there is the Gosi inequality \u0026amp;lt;v*,uamp;gt;≤v*u,?u ∈ txm.]

The operation was so fierce that Cheng Nuo's fingers almost flew out of phantoms on the keyboard.

Cheng Nuo, who had a clear mind, hardly encountered any lag. One by one, the formulas appeared on the computer screen as if they were going smoothly.

The progress bar of the task is also rising steadily.

3%, 8%, 15%, 23%...

On the other side, our classmate Hull looked at the task list in his hand, frowning and thinking hard.

His task at this stage is to study the transformation relationship between some properties of the Lipshitz function and the Fritzjohn-type optimality conditions, and the difficulty is similar to the task given by Cheng Nuo.

But at this time, he was in a sad state, writing and drawing on the paper with a pen.

f(y)-f(x)≤lφ(y)-φ(x)

f(x,v)=lim...lim...

No, no, no, no!

Hull scratched it out hard.

What about this? Hiss, it doesn't seem right!

Ahhhhh! How to do it!

Hull rubbed his short blond hair vigorously, and his blue eyes were full of loveless eyes.

Riemannian manifolds can be regarded as Hull's strong point, but the Lipshitz function obviously belongs to the field of function theory, and he doesn't know much about it. Therefore, starting this project is much more troublesome than the previous big projects.

Looking at the crossed-out notice on the paper, Hull just wanted to shout:

God, who will save me!

Student Hull, who was in a state of melancholy, wandered around the office, and finally fell on Cheng Nuo who was not far away.

Over there, Cheng Nuo was happily tapping the keyboard, looking extremely excited.

Hull's eyes suddenly lit up.

…………

"Cheng!" Cheng Nuo, who was happily typing on the keyboard, suddenly saw a head protruding beside him, and was so frightened that he almost threw the computer out.

He glanced at Hull, who was squatting beside him, and said, covering his forehead, "Student Hull, what are you doing here if you don't go to work?"

It’s okay not to mention this, but as soon as this is mentioned, Hull showed an extremely helpless expression, “I also want to work hard, but there are some tasks that the teacher assigned to me, I don’t understand some things, so I come here to find is you."

"Then what do you mean?" Cheng Nuohu asked.

"How about we exchange tasks?" Hull suggested.

Cheng Nuo shook his head as a refusal, "No."

"Why?" Hull was puzzled.

Cheng Nuo said with a smile, "Because of my mission, I have already completed one-third of it."

"What, one third?!" Holden's eyes widened.

He looked directly at Cheng Nuo, "Cheng, you're not joking, are you?"

"Hey, look at it if you don't believe me?" Cheng Nuo moved to the side, exposing the computer screen.

Hull looked at the computer screen, and then fell on the piles of formulas that took up two pages.

hiss!

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