I kidnapped an alien civilization

Li Mo found that no matter how early he went, the library was always full of people. He quietly came to a small corner, afraid of encountering the same thing as last time again.

Take out the manuscript paper, but can't write. Perhaps it is precisely because the definition of the four-color conjecture is very simple. Simple means that there are few starting points, and it is difficult to interpret it with a mature theorem system.

The four-color conjecture is like a hedgehog.

hedgehog! Li Mo remembered the story told by the old man in the basement of the library, "How did I answer that?"

"If I were this eagle, I would catch this hedgehog high into the sky and throw it down hard." Li Mo clearly remembered his answer.

"Four-color conjecture is equal to a hedgehog, what is it equal to catching a high altitude?" He felt that he was about to grasp the key to the problem, and he was just a little bit short.

"Four-color conjectures are equal to hedgehogs, four-color conjectures are equal to hedgehogs, four-color conjectures are equal to hedgehogs." Li Mo kept chanting silently in his heart, and suddenly his mind flashed.

"The four-color conjecture is equal to a hedgehog, so I can put this hedgehog in the three-dimensional coordinate system, so that I can use it to carry out precise strikes."

Li Mo felt that he had reached the threshold. He took out a piece of paper and wrote on it: We can convert the four-color conjecture, or the four-color theorem, from the "map" to the "three-dimensional coordinate system" equivalently. . A graph, loosely speaking, is a graph formed by connecting points and edges. In graph theory, there is a definition called a planar graph, which means that a graph can be drawn on a three-dimensional coordinate system, and the edges do not intersect each other. We regard each country on the map as a point, and if two countries are adjacent, it means that there is an edge between the two points. In this way, we get a three-dimensional coordinate system, and coloring the country becomes coloring the points in the coordinate system, so that adjacent points have different colors. The four-color theorem says that for any three-dimensional coordinate system, four colors are enough to satisfy the above conditions.

What we have to do now is to find out the mysterious function, the graph with five or more points connected in pairs cannot indeed be drawn in the coordinate system. First consider a given graph G and color its vertices so that the two vertices of any edge have different colors. We call the minimum required number of colors that satisfies the condition chromatic.

At the same time, we call the number of points of the largest complete graph subgraph contained in graph f as clique number, denoted as x. It is easy to find that a complete graph of n points requires at least n different colors because the points are adjacent to each other.

Let x(n) be a sequence of M items, which can represent any lattice in graph theory. By DFT transformation, the calculation of any X(m) requires M times of complex multiplication and N-1 times of complex addition, then find NM items The X(m) of the complex number sequence, that is, the N-point DFT transformation requires about M^2 operations. When N1=10 points or more, N3=10486 operations are required.

From the above, it is obvious that arbitrarily partitioning a figure and coloring each of its parts makes any part with a common edge have a different color, and only four colors can be used, and no more. This proposition is established.

Certificate completed.

Li Mo, who broke through the thinking barrier, wrote down all the ideas of the proof in one breath. No wonder so many mathematicians have fallen in front of the four-color conjecture over the past century. It looks very weak like a hedgehog, but it is actually difficult to find a place to put its mouth down. If a weakness is found, it is nothing more than a difficult proof problem.

Looking at the complete proof ideas on the paper, Li Mo's heart was filled with joy. He felt that he was working hard for a small step forward of human civilization. Human beings are curious creatures. Exploring the unknown is the innate instinct of human beings. It is precisely because of this instinct that human beings can stand out from many biological clocks and establish the current earth civilization.

The next thing he has to do is to sort out the papers. For Li Mo, who has the ability to write academic papers, this has become the easiest thing.

"Buzzing. Buzzing" the phone vibrated, Li Mo picked it up and took a look, Ying Sasha said on WeChat: "Li Mo, why didn't you come to the linear algebra class, the teacher is going to call everyone, hurry up. "

"Oops", Li Mo looked at the time on the phone and said something bad in his heart. It's just that he was so obsessed with solving problems that he forgot that there was another linear algebra class in the morning.

He didn't have time to tidy up, he put the papyrus in his schoolbag randomly, and went straight to the auditorium.

There were very few students on the road, Li Mo checked the time on his phone while running, "No, I can't catch up."

Sure enough, when we came outside the lecture theater, Teacher Guo on the podium had already started to roll the roll.

"Zhang Yu!", "Here!"

"Wang Chunyan!", "Here!"

"Su Yuhang!", "Here!"

Li Mo tiptoed to the back door, poked his head, and found that Mr. Guo was concentrating on checking the name list. He was about to slip quietly and slowly to his seat.

Teacher Guo on the podium: "Li Mo!"

Li Mo, who was sneaking in through the back door, replied subconsciously: "Here!".

"Oh no!"

Realizing something was wrong, Li Mo raised his head and looked up to the podium. On the podium, teacher Guo stared at him with wide eyes, waved to him and said, "Student, are you just here? Come on, please come to the podium first."

Li Mo had no choice but to walk slowly towards the podium under the gaze of his classmates.

"You dare to be late for my class. It seems that my prestige has dropped a lot." Teacher Guo said with a sly smile, "Isn't Li Mo in the high math class, I don't make it difficult for you. I will give you a problem if you can do it." If you can figure it out, let the past go. If you can’t answer it, you don’t want to get the usual final grades.”

As he spoke, he angrily wrote on the blackboard: Let the vector α=(a1, a2, a3)β=(b1, b2, b3) a1!=0 ​​b1!=0 α^Tβ=0 A=αβ^ T

(1) Find A^2

(2) Eigenvalues ​​and eigenvectors of matrix A

After finishing writing, he handed over the chalk in his hand, and said with a smile: "Please, Li Mo."

Li Mo took the chalk and pondered for a moment, then nodded to Teacher Guo, and then wrote on the blackboard: ^1) A^bai2 = ab^T ab^T

Because a^Tb=a1b1+a2b2+a3b3 = b^Ta =0

So duA^2=a 0 b^T

So A^2 is a 0 vector

2)A

a1b1 a1b2 a1b3

a2b1 a2b2 a2b3

a3b1 a3b2 a3b3

|A-λE|=0

Find the determinant directly, and the constant term and λ first-order term dao are all eliminated;

Using a1b1+a2b2+a3b3=0, the λ quadratic term is also eliminated;

Finally, λ^3=0, the eigenvalues ​​are all 0

Ax = 0

Since the rows of A are proportional, the rank is 1

The final eigenvector expression: x1=-b2/b1x2-b3/b1x3 (b1!=0)

In one go, Li Mo handed the chalk back to Mr. Guo who was staring at the blackboard in a daze, his face gradually turning blue, and went straight back to his seat.

After a long time, Teacher Guo on the podium came to his senses, smiled awkwardly and said, "This student named Li Mo answered very well, this time the roll call is over, and the class will start next."