Reading as a god

Chapter 170 This is outrageous!
Pro-Yuan Xian fish, as retreat webs!
Looking at Zhang Shan's mathematics and language skills, Ye Weiyang was very envious, but she didn't hesitate, and continued with her original study plan~
Zhang Shan is now a mute who eats Coptis chinensis, and can't tell the pain.

Although he wrote a lot eloquently, he knew that it was far from reaching the standard of an SCI paper.

It's not that the gold content of knowledge is not enough, the key is that the language in Zhang Shan's article is not concise enough.

In this case, no amount of insight can do it!

Under formal circumstances, an SCI paper has three to four thousand characters, and there are also tens of thousands of characters.

Although the research directions are different and the abilities of the authors are different, all of their articles strive to be capable without exception.

Having said that, the flaws are not concealed. Although Zhang Shan’s current writing is a bit lengthy, at this time he fully appreciates the accumulation of various skills he has learned before~
Because SCI is basically written in English, it is necessary to have a certain English foundation, and the grammar must not be wrong. In addition to knowing some regular English words, you must also know the professional terms related to your direction to avoid mistakes.

These are all a piece of cake for Zhang Shan~
Furthermore, if you write a thesis, you should choose the currently popular research direction, and then find out the existing problems according to your own research direction, and propose a solution of your own. The method needs to be innovative.

And Zhang Shan doesn't need to worry about this at all. The fragments of the system not only provide the book list to a certain extent, but also frame the direction of topic selection.

In the normal process of writing a paper, after proposing a solution, sometimes you need to use a computer to download some simulation software related to research needs for simulation, and verify whether your method is feasible and effective in the computer simulation software.

Although he has not yet reached this step, Zhang Shan feels that this is not a problem.

Although I have never used simulation software, for Zhang Shan who can use a lot of programming software, this is really handy~
~
Speaking of the proof of the four-color conjecture:

"Only four colors are needed to color a map" was originally conjectured by Francis Goodrey in 1852.

However, this conjecture was not taken seriously at first.

An early "proof" of the theorem was the London lawyer and mathematician Alfred Bray Kemp.

Kemp's proof is based on induction on the number of countries.

First of all, it is easy to prove that the four-color theorem holds when the number of countries is not more than 4.Kemp assumes that the four-color theorem holds true when the number of countries is not more than n. His purpose is to prove that a map composed of n+1 countries can be reduced to a map composed of no more than n countries, thus proving that the four-color theorem holds true.

Kemp first proved a conclusion about planar graphs: there must be a country in any map whose number of neighbors is less than or equal to 5.The proof is simple, in the graph theory version the map is transformed into a simple planar graph.

In a simple planar graph, if V is the number of vertices, E is the number of edges, and F is the number of regions, since each region is surrounded by at least three sides, and each side is exactly separated by two regions, the number of regions and the number of edges Meet: 2E ≥ 3F.Assuming that each country has at least 6 neighboring countries, that is to say, each vertex has no less than 6 edges connected, then since each edge corresponds to two vertices, the number of vertices and edges satisfies: 2E ≥ 6V.Together we have:

V+F≤E
But this contradicts the famous Euler's formula in graph theory: V + F = E + 2.

Therefore, it is impossible for every country to have no less than six neighbors, and there must be a country with no more than five neighbors.

Next, Kemp examines the country with the smallest number of neighbors among the n+1 countries, which is called country A. The number of neighboring countries of country A shall not exceed 5.If the number of neighboring countries of country A does not exceed 3, country A can be "removed" (for example, integrated with one of the neighboring countries) to form a map of n countries. This map can be colored with 4 colors, and The original 3 neighboring countries used at most 3 colors.At this time, "putting country A back" and dyeing it with the fourth color is equivalent to finding a way to color the original map 4-[4].

This kind of part that can "remove" a country and reduce the number of countries is called "reducible configuration" (reducible configuration).

Next, Kemp proves that the situation that country A has 4 neighbors and 5 neighbors is still a reducible configuration, so they can be reduced to the situation of no more than n countries.Therefore, the map of any n+1 countries can still be colored with four colors, so it can be known by induction that the four-color theorem holds.

Kempe's method was later called the "Kempe chain" method (Kempe chain) to prove reducibility~
Although Kemp's approach was later found to be wrong, Kemp's thinking has continued.

Since the 20th century, research on the four-color theorem has stagnated in European mathematics circles.On the contrary, this issue has received more attention in the United States.

Many outstanding mathematicians have studied this problem and made great contributions.Part of the effort was to revise Kemp's proof;
Another effort is to transform the four-color problem to use more powerful mathematical tools.

The transformation of the four-color problem has never stopped.

After converting from the topology version to the graph coloring version, a new transformation was proposed in 1898.

Kemp and other scientists have noticed that proving the four-color problem only needs to consider the case where three countries have a common "intersection point", and the case where more countries have a common intersection point can be transformed into the former.

Therefore, in such a corresponding coloring graph, each vertex will have exactly three edges.Such a map is called a "trivalent map".

Some mathematicians have observed that if any area surrounded by edges in a three-dimensional graph has the number of edges that is a multiple of 3, then the graph can be 4-colored.He further found that a graph can be 1-colored as long as there exists a way to assign a value of +1 or -3 to the vertices of the graph such that the sum of the vertex numbers in each region is divisible by 4.It can be shown that the 4-color and existential assignment methods are equivalent.

In the United States, mathematicians have never stopped studying the four-color theorem.

In addition to Johns Hopkins University's Pierce and Stoley et al., another researcher is Paul Winick.After graduating from the University of Göttingen, an academic mecca at the time, Winnick came to the United States and taught at the University of Kentucky.The rudiments of reducibility already appeared in his 1904 paper.However, the first substantial progress in the American mathematics community on the four-color problem appeared after 1912.Oswald Veblen of Princeton University (nephew of the economist Thorstein Veblen) was at the vanguard of this wave.His work focused on topology, and in 1905 he proved the Jordan curve theorem.With a deep understanding of the new algebraic tools developed by Poincaré, he naturally began to study the four-color theorem.He used the concept of finite geometry and the correlation matrix on the finite field as a tool to transform the four-color problem into an equation problem on the coefficient space of the finite field.This direction was called "the quantitative method" by the later cryptographer and mathematician William Thomas Tutt.That same year, his Princeton colleague George David Birkhoff also began exploring this direction, but a year later he turned to Kemp's method, which Tate calls "the qualitative method." , and put forward the concept of reducible ring. In 1913, Berkhoff published a paper called "The Reducibility of Maps", using the reducibility ring to prove that a map composed of no more than 12 countries can be colored with four colors. In 1922, Birkhoff's student Philip Franklin used the same method to strengthen the conclusion to the point that no more than 25 countries can be colored with four colors on the map.Since Bekhov first proved that the four-color theorem is true for maps with no more than 12 countries, the historically proven upper limit of the number of countries for a colorable map is called the Bekhov number.

The proof by Birkhoff et al. is a continuation and systematization of Kemp's method, which is summarized as finding an unavoidable set of reducible configurations.

This idea is already embodied in Kemp's proof.

He first explained that the following four configurations must exist in any map: 2-neighboring countries, 3-neighboring countries, 4-neighboring countries and 5-neighboring countries; then he proved that each configuration is a reducible configuration.Hill later called this classification method "inevitable set".

Berkhoff's idea is to use the method of proof by contradiction: if there is a map that needs to be dyed with at least five colors, then there must be a "five-chromatic map" (five-chromatic map) with the smallest number of countries.This map must be "irreducible".And as long as a set of configurations is found, one of the configurations will inevitably appear in the minimal five-color map, and each configuration is reducible, then the number of countries on the map can be reduced through reduction, thus leading to contradictions.

The inevitable set Kemp found consisted of four configurations, but he could not prove the reducibility of the last one (5-neighboring countries), so Birkhoff began to look for a new way to characterize the inevitable set.

He proposed to divide the entire map M into three parts with the ring formed by adjacent countries: the inner part A of the ring, the outer part B and the ring itself R.If the number of countries on a ring is n, it is called an n-ring.If any coloring of R does not prevent A from being colored, then A can be "ignored" and the coloring problem of M can be reduced to the coloring problem of B+R.At this time, A+R is said to be a reducible configuration, and R is called a reducible ring.Berkhoff proved: when R is 4-ring, or R is 5-ring and there is more than one country in A, or A+R is "Birkhoff rhombus", A+R are all reducible configurations shape.Therefore, it is impossible for the minimal five-color map to contain these configurations.

Franklin further proved that: the minimal five-color map must contain three adjacent pentagonal countries (countries with 5 neighboring countries), or two adjacent pentagonal countries and one hexagonal country, or one adjacent pentagonal country and two adjacent countries hexa.He thus obtained a series of reducible configurations, forming an inevitable set of reducible configurations for maps of less than 25 countries.Therefore, the extremely small five-color map must contain at least 26 countries.

Franklin found that the minimal five-color map must include one of the above six situations.

The ultimate goal of this approach is to find the set of unavoidable reducible configurations of all maps.However, as the number of countries increases, it becomes more difficult to find the inevitable set and prove its reducibility.This is mainly because the number of coloring methods increases rapidly as the ring size increases. There are 6 4-staining methods for 31-rings and 12 for 22144-rings.Therefore, it is very complicated work to verify the reducibility of the configurations surrounded by macrocycles.

1926年，C.N.Reynolds将别克霍夫数从25提高到27。1938年，富兰克林将其推进到31。1941年，C.E.Winn将之提高到35。而直到1968年，别克霍夫数才更新为40。

The next breakthrough in the study of the four-color problem was not in the United States, but by Henry Hill, a German mathematician from Gottington University.

He proposed the existence of inevitable sets in 1948, but the inevitable sets he proposed may contain 10000 configurations, including the huge configuration of 18-rings.Another achievement of Hill is the "discharging method" (discharging method) proposed in 1969, which provides a systematic method for finding inevitable sets.

It is too slow to manually search for the inevitable configuration set and verify the reducibility of configurations. Mathematicians began to consider using the new computer as an aid to improve the verification efficiency.While constructing the discharge method, the work of verifying the reducibility of the configuration by means of a computer is also progressing rapidly.

Hill, with the help of Karl Dürre, devised the first algorithm in 1965 to verify the reducibility of configurations.They used the Algol 60 language and it ran for the first time on a CDC 1504A computer in the computer center of the Hannover Technical Institute in Germany. Before 1967, due to insufficient memory, only configurations below 12-rings could be verified.However, the unavoidable sets found by Hill can contain large configurations up to 14-rings or even more, and the computer's ability is not enough to quickly complete the verification of reducibility.

At that time, the computer technology of the United States was ahead of Europe, so Hill hoped to prove the four-color theorem with the help of large-scale computers in the United States. In 1967, the dean of the School of Applied Mathematics at Brookhaven National Laboratory (BNL) in New York invited Hill to visit the United States and allowed him to use the CDC 6600, the fastest computer in the world at the time.In the following years, Hill went to the United States twice to seek opportunities to use large-scale computers.During this time, Dürre rewrote the program in FORTRAN.With the hope of finally solving the four-color problem in Germany, Hill returned to Germany, but to his disappointment, the German academic community was negative about his plan and was unwilling to allocate computing time to his program.

During several visits to the United States, Hill began collaborating with Wolfgang Hacken.

Haken attended Hill's lecture on inevitable sets in 1948 and became interested in the four-color theorem afterwards.The two men worked together to make a lot of progress through the exchange of letters, paving the way for the final solution of the four-color problem. In 1971, Appel also began to study the four-color problem under the introduction of Haken.However, at that time Haken was pessimistic about the future of solving the four-color problem, because it was too complicated to find and verify a suitable set of unavoidable reducible configurations, and it would take too much time even with the help of a computer.Tate also believed at the time that even in the most optimistic estimate, the inevitable set should contain at least 8000 configurations.However, Tate and others also introduced Hill's work to the United States (at that time, Hill's work was only published in Germany), which aroused the enthusiasm of many people.Including Franco Allaire, Edward Rainier Swart, Franco R. Bernhardt and others began to find inevitable sets and test reducibility.Relying on the computer's ability to work, Haken and Appel continued to refine the discharge process.They combined an algorithm for finding inevitable sets through the discharge process with verification of reducibility, when the configuration of an unavoidable set is not C-reducible (a type of reducibility) or difficult to be verified as C-reducible When , give up this inevitable set to improve efficiency.The two set a lot of empirical correction rules, such as setting three empirical "obstacles" (three specific configurations), when a configuration contains such obstacles, it is directly considered irreducible; Another example is that the size of the configuration cannot exceed 14-rings, and so on.

In 1975, Haken found a good discharge process, but it was difficult to turn it into an algorithm program.So the two temporarily began to return to paper and pen calculations.At this time they were supported by John Koch, who was still a doctoral student at the time, who provided them with help in the work of the reducibility verification algorithm. In March 1976, they finally obtained an inevitable set composed of 3 configurations, and the corresponding discharge process consisted of 1936 rules.At the same time, the main computer of the University of Illinois was also replaced with an IBM 487 with higher computing speed, which saved a lot of time for calculation.After 360 hours of computer verification, they finally concluded in June that all 1200 configurations are reducible configurations.This represents the final solution of the four-color theorem [6]:1936.At this time, the work of several of their competitors such as Alaire and Swat is coming to an end.

On June 1976, 6, Haken and Appel first announced their results at the Summer Meeting of the American Mathematical Society (AMS) held at the University of Toronto.Soon, the postmark of the University of Illinois Mathematics Department added the sentence "FOUR COLORS SUFFICE" to celebrate the solution of the four-color conjecture. In September, the announcement column of the American Mathematical Society published the news that two people had proved the four-color theorem.

In 1977, Haken and Appel wrote the results into a paper called "Every planar map is four colorable" (Every planar map is four colorable), which was divided into two parts and published in the "Illinois Journal of Mathematics".

So far, the four-color problem that has plagued people for a long time has finally been solved.

It can be seen that for a long time people have mainly worked on the four-color conjecture around the reducibility verification.

In this process, new tools such as computers have greatly helped to simplify calculations~
Good tools provide good assistance to scientific research~
However, it is not a good thing that goose tools are too advanced!

Zhang Shan saw a total of 9 new proof methods from the system library.

However, six of them cannot be used!

What the hell is proof using a quantum computer?

There is no suitable quantum computer at all, is it possible to invent some new tools for this proof.

There is also the use of special sub-computers to prove what the hell it is!This is beyond Zhang Shan's imagination.

I can't see any more~
This is outrageous!
Fortunately, there are still three methods that can be used~
Just use existing tools, and prove that the idea is also very clever.

This is very nice~
(End of this chapter)