When Ranger returned to Aries, he already had many titles.

National men's long jump champion for middle school students, national record holder for male middle school students, athletic athlete, the first national first prize player in the biology competition, a member of the national training team for students, and a student from Shuimu University... These titles were obtained by Langjie in the past two months, and he entered the explosive period.

Lan Jinhui and He Yuyi expressed satisfaction and happiness. He Yuyi said, Ajie, I will give you the house in the Sports Commission compound, and that house can only be lived in but not sold. You can keep it for marriage. Of course, Ajie, you will have a long time to get married, not to mention that you will not know whether you will develop in Aries City in the future. In short, I will give you a key to the Sports Commission compound, and you will go there from time to time to clean and maintain the tidyness of your property.

Ranger took the key and said thank you parents.

Handing over the house keys is an important family ceremony.

Although Ranger is legally underage, he already has the ability to behave, judge and think in his parents' hearts.

Turnkey means that from this point on, Ranger has the right to have a high degree of independence and freedom.

From childhood to adulthood, Ranjie's creditor was his sister and his financial support was his aunt.

My aunt Lan Yunhui has been sponsoring Lan Jie. Lan Yunhui gave Lan Jie 10,000 yuan and said, "You can keep this money for college."

With a house and a deposit, Ranger reached the peak of his life overnight.

Lan Ying, who was an exchange student in Toronto, had a video call with Ranger after learning about Ranger's latest record.

"Ajie, you actually won the first prize in the biology competition, and you are still the first in the total score."

"My performance is actually average, and it depends on the opponent's chance."

"I don't have to pay back the 30,000 yuan I owe."

"Sister, have our debts been cleared?"

"Clear."

"Sister, you are quite expensive in North America, why don't I give you a symbolic 1,200,000?"

"Langjie, don't be so fake."

"Okay, okay, thank you sister! Huh? Are you outdoor? You are in the snow and ice, you seem to be wearing a swimsuit?"

"Hey, Annie, help me hold my phone, I want to show it live for my brother." Lan Ying said in English.

The next second, in the snowy environment, Lan Ying jumped into a river without freezing.

"Wow! Sister, you're swimming in winter! The snow in Canada is so heavy! I've never seen such heavy snow!"

Lan Ying in winter swimming yelled in the river: "This is much more enjoyable than ****!"

"Wow! Is this your life for college students?" Lanjie opened his eyes. As expected, college students can still play.

Langjie achieved certain results, and his family rewarded and encouraged him in various ways.

After a few days of rest in Aries City, Ranjie embarked on his journey again. He followed the Sheep China Football Team to the capital and moved into the dormitory of Shuimu University.

This year's Cpho National Finals, the training of the national team of football teams was hosted by Shuimu University. Langjie is already a Shuimu member. He plays at home and feels that he has a buff bonus.

Ding Ding Ding!

The national finals of the game begin.

There are three theoretical questions, each with 10 points and five hours of answering.

The first question is to find the equivalent resistance between any two nodes of the plane square infinite resistor network. It is known that all small resistances in the network are r, and it is extended to other types of infinite resistor networks, such as plane rectangular networks, plane positive triangle networks, plane positive hexagonal networks, three-dimensional three-dimensional networks, etc. Try to find the equivalent resistance between any two nodes of these infinite resistor networks.

‘The first question is very difficult!’

Lanjie couldn't help but sigh that the national finals are the national finals after all. I wonder how many players will stump them?

The first question should make most contestants explode.

But not including me Ajie!

For a flat square infinite resistor network with r in each segment of resistance, it is easy to find the equivalent resistance between two adjacent sections.

However, the equivalent resistance between any two non-adjacent nodes is difficult to calculate.

‘So Fourier expansion of a two-dimensional plane must be used to derive the analytical solution of the equivalent resistance between any two nodes of the square infinite resistor network in the plane!’

Ranger drew an electrogram, and he established εη coordinates on the network plane, so that the coordinates of any two nodes a and b are a(0,0), b(m,n), and the coordinates of all other nodes can be expressed as (k,l). Where m, n, k,l are all integers. Suppose the current i(k,l) is the current flowing into the node (k,l). Considering that the current i flows from point a, and after stabilization, the current i flows out from point b, then:

i(k,l)=1/r[v(k,l)-v(k-1,l)]1/r[v(k,l)-v(k1,l)]     1/r    [v(k,l)-v(k,l-1)]     1/r    [v(k,l)-v(k,l1)]

Simplified to: v(k,l)-￡v(k,l)=r/4i(k,l)

Find the homogeneous equation v(k,l)-￡v(k,l)=0

Find the special solution of the non-homogeneous equation!

Find the function f(x,y), and let it expand into a two-dimensional Fourier series on the interval [-π,π; -π,π]!

After complex mathematical calculations, Ranger found rmn=r/4π^2∫-ππ∫-ππ∫-ππ1-cos(mxny)/2-cosx-cosydxdy.

This is the analytical solution to the equivalent resistance between any two nodes in a flat square infinite network!

Continue to calculate and find the equivalent resistance between any two nodes of the plane rectangular network, plane positive triangle network, plane positive hexagonal network, and three-dimensional three-dimensional network!

‘Only 20% of this question belongs to physics, and the rest is all mathematics.’

This is Ranger's evaluation of the first question, and he immediately overcomes the next two questions.

The last two questions are very physical, and Ranger finished all the theoretical questions half an hour in advance.

Ding Ding Ding!

carry out an assignment!

The contestants have half a day off and will hold the experimental competition tomorrow morning.

Li Zihan is also a national finalist of the Yang Zhong Football Team. His first time to come to the capital, and of course, it is also his first time to come to Shuimu University.

Accompanied by Lanjie, Li Zihan was quite interested in visiting Shuimu University.

"Langjie, are you quite familiar with Mizuki University?"

"I stayed here for a few days in early October, but I wasn't very familiar with each other, so it's okay."

"As far as I know, Yan University's football competition training class is very powerful. The previous Cpho national team was hosted by Yan University, and the training base of the national team of the football competition has always been located in Yan University. Why has Yan University and Shuimu taken turns to hold the Cpho national team of the football competition and took turns to undertake the training of the national team of the football competition?"

"What I know is that a few years ago, Shuimu said that Yan Da monopolized the holding of competitions and national team training. This is not suitable. Shuimu and Yan Da should take turns. Then the Physics Association agreed with Shuimu's suggestion, and then Yan Da and Shuimu come in turn."

"Langjie, as far as I know, the venue of the National Team of the Summit and the training base of the Summit National Team have always been Shuimu Affiliated Middle School. Didn't Yan Da put forward any suggestions or opinions?"

"Shuimu Affiliated Middle School is a middle school, and the middle school's business has nothing to do with Yanda."

"That's it, but Shuimu Affiliated Middle School is in Shuimu Garden. Look, this is Shuimu Affiliated Middle School."

While walking around, the two of them reached the door of Shuimu Affiliated Middle School. Li Zihan said: "I always feel that Shuimu is tougher than Yan Da. I really want Shuimu."
Chapter completed!