Volume 1: Student Life Chapter 157: Advancing the Project
In the afternoon, Chen Zhou's cousin Chen Yong came over with his schoolbag.
Chen Zhou arranged him and Chen Xiao together, letting them do their homework by themselves and ask him if they didn't understand something.
Chen Zhou easily handed Chen Yong's mathematics textbook to Chen Xiao.
Chen Xiao took it silently. He knew that this textbook would always accompany him during this winter vacation.
Chen Zhou looked at the two of them for a while, then went back to the house and took out his notebook, draft paper and other necessary equipment.
Open the file on your notebook that deals with topics related to Clifford analysis.
He is currently studying the relevant parts of the Cauchy-Pompeiu formula in complex Clifford analysis.
After briefly sorting out his thoughts, Chen Zhou began to write on the draft paper:
【w1＊Dξ＋w2＊Dξ=∑j=0→n[(aw1＊/aξj＋aw2＊/aξj)ej]=0...(1)】
[Dξw1＊＋Dξw2＊=∑j=0→n［e j（aw1＊/aξj＋aw2＊/aξj）]=0……(2)】
These two are very important equations and need to be proved first.
Chen Zhou thought for a while, made some changes to the above two equations, and then began to prove them.
[∑j=0→n[(aw1＊/aξj＋aw2＊/aξj)ej]=……]
[Obviously, the sum of these two corresponding terms is zero, and the same applies to the remaining terms...so the above formula holds.]
[Similarly, it can be proved that Dξw1＊＋Dξw2＊=0]
After completing the proof, Chen Zhou wrote down the next content that needed to be proved.
【Let ΩcC^(n+1) be a bounded region, let f, g∈C1(Ω, Cl0, n(C)), define df=af+▔af, ..., then d［f· (w1+w2)］=df∧(w1+w2). 】
After a little thought, Chen Zhou began to prove it.
【Because d(f·g)=df·g＋f·dg, so d［f·(w1＋w2)］=df∧(w1＋w2)＋f·d(w1＋w2)=df∧(w1＋w2)＋f［a(w1＋w2)＋▔a(w1＋w2)］】
[Because ▔aw2=0, aw1=0, so…]
Chen Zhou had just finished writing when Chen Yong next to him poked him and said, "Brother, help me with this question. I don't know how to do it and I don't understand the answer even after looking at it."
Chen Zhou took the reference book from his hand and took a look at it. It was a question about a function. He raised his hand, wrote an a symbol, and then crossed it out immediately.
Shaking his head slightly, Chen Zhou muttered to himself, this really is a matter of seeing what you see.
After looking over the question again and organizing his thoughts, Chen Zhou began to write down the steps to solve the problem on a piece of draft paper while explaining them to Chen Yong.
After stopping writing, Chen Zhou glanced at Chen Yong, who was still staring at the draft paper.
This question is indeed beyond the scope of the curriculum for high school students.
Chen Zhou was not in a hurry. He just thought about his own topic while waiting for Chen Yong.
After a while, Chen Yong withdrew his gaze from the draft paper and turned to look at Chen Zhou.
Chen Zhou smiled and asked, "Do you understand everything?"
Chen Yong nodded: "Yes, thank you brother."
Chen Zhou: "You're welcome. Continue with the questions."
After saying that, Chen Zhou returned to his own topic.
The two preparatory theorems have been solved, and the following is the proof of the Cauchy-Pompieu formula.
The Cauchy-Pompieu formula is:
【Let ΩcC^(n+1) be a bounded region, let f∈C1(Ω,Cl0,n(C)), and f∈H(Ω,α)(0＜α＜1), then for any n+1-dimensional chain Γ, ▔ΓcΩ, f(z)=∫aΓf(ξ)·(w1+w2)-∫Γd［f(ξ)·(w1+w2)］. 】
Chen Zhou took the pen, tapped twice on the draft paper out of habit, and then began to prove.
【With z∈Ω as the center and a sufficiently small ε as the radius, make a small ball Bε=｛ξ||ξ－z|＜ε｝, then...】
Based on the Stokes formula in complex analysis, we can continue the proof.
【……, when ε→0, ∫aBε［f（ξ）－f（z）］（w1＋w2）→0,……】
After writing it, Chen Zhou reviewed it again. He mainly used the definition of limit and separated the part containing the singularity by digging out points.
Among them, the part containing singular points can be proved to have a limit of zero by using the definition of Herder continuity of the function.
For the part without singularities, we use Stokes' formula to prove that the result is a definite constant, thus solving the problem.
That afternoon, Chen Zhou spent the time alternating between research and lectures.
At night, I would video chat with Yang Yiyi to supervise and learn from each other.
It was not until Yang Yiyi urged Chen Zhou to go to bed quickly that he put down the pen and cleared his mind.
The next day, Chen Zhou spent the same time.
Except for the occasional questions asked by Chen Xiao and Chen Yong, Chen Zhou would take a short break and spend the rest of the time immersed in the subject.
Chen Zhou has advanced the progress of the project to the study of the properties of the T operator with B-M kernel in complex Clifford analysis.
Chen Zhou had already sorted out the relevant preliminary knowledge and definitions.
He was already familiar with Hadamard's lemma, Held's inequality, Minkowski's inequality, etc.
The T operator, whose full name is Teodorescu operator, is a singular integral operator. This singular integral operator has many excellent properties and can be applied to the study of partial differential equations theory, integral equations theory and generalized function theory.
Looking at the conclusion he obtained, Chen Zhou thought of the conclusion of the classic Hile's lemma, which is very similar.
However, because Hile's lemma cannot be directly used in complex Clifford analysis, Chen Zhou inserted appropriate terms according to different situations and proved the relevant conclusions.
This conclusion is an important tool to prove the Herder continuity of operators in complex Clifford analysis.
Chen Zhou, who was concentrating on his research project, felt that time passed very quickly.
It felt like he hadn't finished much content yet, and Yang Yiyi reminded him that it was time to go to bed...
February 14th, Valentine's Day.
According to the discussion between Chen Zhou and Yang Yiyi, neither of them planned to go out to meet, eat, watch movies, etc.
After all, we just broke up, and we were together in school and saw each other every day. There was no need to run out alone for the so-called Valentine's Day.
In general, both of them feel that as long as they are together, every day is Valentine's Day.
So, that day, Chen Zhou spent the morning reading books and doing research with Yang Yiyi as usual.
Tutor Chen Xiao and Chen Yong in the afternoon.
Chen Xiao and Chen Yong looked at each other, and Chen Xiao spoke first: "Brother, did you break up with your sister-in-law?"
Chen Zhou asked curiously: "Why do you say that?"
Chen Xiao explained: "I saw other people going out on dates on Valentine's Day, there were couples everywhere on the streets, but you just stayed at home."
Chen Yong also said: "When I came here, I also saw that there were people selling flowers on the street."
Chen Zhou glanced at the two boys and said helplessly, "You two are really... I'm not breaking up, you two should hurry up and do your homework well."
Chen Xiao said, "Brother, don't blame me for not reminding you. You still have to celebrate this necessary festival. If you really haven't broken up, even if you don't meet, you should prepare a gift for your sister-in-law, right?"
Chen Zhou glared at Chen Xiao, and Chen Xiao immediately lowered his head without saying a word.
However, after Chen Xiao's reminder, Chen Zhou felt that there was some truth in it.
But where can he prepare a gift now? It’s too late to prepare a gift now…
