From child prodigy to chief scientist

Chapter 98 Precision Breakthrough

It took them another day to complete the splitting work together. They came to the computer room again to verify it. This time, the verification results did not disappoint them. After the decoupling and splitting of the degree constraint, the accuracy of the upper bound was improved by a full 40%.

This proves that their newly constructed framework is very effective, and this idea of ​​neighborhood hierarchical control is indeed very suitable for the problem they are studying.

"Great! With this level of precision, it should be considered a small achievement, right?"

"Well, it certainly has some academic value, but that alone is not enough."

In academic research, Xu Xiao is slightly more experienced than Pan Hong.

He knew that further research was needed to truly reach a level suitable for publication in a journal.

"The bound after the split has indeed converged, but during the scaling process, we still need to deal with the lower bound estimation of the integral function fm() —"

Listening to Xu Xiao's thoughts, Pan Hong quickly realized the problem.

"If we use a traditional linear lower bound, wouldn't we also lose precision in the logarithmic terms?"

"That's right, this will cause the principal coefficients of the final bound to be too large, making it impossible to obtain sufficiently tight explicit constants. We still need to find a more optimized scaling boundary."

After thinking for a while, Pan Hong put forward her idea.

"I think we can classify them based on the parity of m and optimize the precise lower bound of the integral function accordingly. The parity of the Ramsey number in the underlying cycle is not symmetrical, and their properties differ to some extent."

Xu Xiao also thought of this point, because Ramsey numbers in even cycles do not have any minor principal terms diverging, making them relatively simple to handle.

Ramsey numbers in odd cycles naturally have logarithmic divergence factors of secondary principal terms, which cannot be eliminated by ordinary scaling, making them much more difficult to handle.

"Okay, then let's deal with the even numbers first, and then we'll study the divergence problem of odd numbers later."

After spending another two days, Xu Xiao and Pan Hong successfully solved the even-number boundary scaling optimization problem.

Together they derived an exact lower bound for the integral function, replacing the original coarser linear lower bound with an exact lower bound containing logarithmic terms.

This new lower bound perfectly captures the logarithmic decay property of the function, offsetting the logarithmic terms lost during the scaling process.

This further improves the accuracy of the bounds for even numbers m by 25%, and the resulting explicit compact constants are in high agreement with the numerical enumeration results.

After completing the work on the even-numbered parts, they immediately moved on to the odd-numbered parts.

However, this time, they clearly felt that the problem was much more complicated than before.

As they had anticipated, the Ramsey number of an odd number m inherently carries a secondary subject divergence factor.

If the scaling of the wheel graph is directly substituted, the constant term of the global bound will diverge as n increases, making it impossible to obtain a fixed explicit constant.

To solve this problem, the two of them used dozens of sheets of draft paper to derive and calculate, but they still couldn't eliminate the divergence factor.

Just then, Xu Xiao's phone vibrated.

Although the caller ID showed an unfamiliar number, Xu Xiao still went outside to answer the call, just in case someone really needed to contact her.

"Hello."

"Xiaoxiao, it's me."

The voice on the other end of the phone sounded very familiar; it was Xu Xiao's father, Xu Lang.

"Dad, you changed your phone number?"

"Well, that mobile phone could only be used in Steel City, so I got a new phone and bought a SIM card from Beijing."

"

Because of the limitations of PHS (Personal Handyphone System), it can only make local calls and does not support nationwide roaming. Since I have chosen to work in Yanjing, I definitely have to replace my old PHS phone.

"Yeah, it's good to replace it. Anyway, those old-fashioned mobile phones will be obsolete in a few years, right?"

"Huh? Xiaoling will be eliminated in a few years? How is that possible?"

Xu Lang couldn't understand this at all, because this was the peak of PHS's popularity, with its price and tariffs being very cheap, occupying one-third of the current mobile communication market.

Some people even hoarded PHS phones to resell, but ultimately, due to the rapid decline of PHS phones, many people's goods ended up rotting in their hands.

Xu Xiao couldn't explain these things in too much detail to Xu Lang, so she changed the subject and asked, "Dad, what's the matter that you called me about?"

"It's nothing, we're already at our new home in Yanjing. When do you have time, let's have a meal together at home. If you're busy, we're not in a hurry."

"Oh? You've already arrived in Yanjing? Why didn't you tell me beforehand so I could pick you up at the station?"

"Hehe, do we two adults need a kid like you to pick us up? Actually, we arrived a couple of days ago and have been busy settling into our new home and starting our jobs, so we haven't had a chance to contact you."

Actually, there was another reason that Xu Lang didn't explicitly say: they were worried that it would affect Xu Xiao's normal study and life.

Two days ago, when they arrived in Yanjing, they had already inquired about Xu Xiao's situation with his homeroom teacher and confirmed that he was doing well recently, so they were not too worried.

It wasn't until today, the weekend, that Xu Lang thought Xu Xiao shouldn't be so busy, so he called him.

Xu Xiao naturally understood her parents' good intentions, and immediately replied, "I don't have anything to do right now, so I'll come find you in a bit."

"Okay, we'll wait for you at home."

After hanging up the phone, Xu Xiao returned to the study room and told Pan Hong about her parents' visit.

"Then you should go quickly, our project isn't in a rush."

"Pan Hong, why don't you come and have a meal with us?"

"Your family can get together. I'll continue to study the divergence problem of odd numbers m."

After exchanging pleasantries with Pan Hong, Xu Xiao packed her things and headed to Yandong Garden.

"Xiaoxiao, you're back."

"Dad, Mom, you didn't tell me you were coming to Yanjing sooner."

"It won't make a difference if we stay a couple more days. We can also tidy up the house properly. Why don't you go to the study for a while? The food will be ready soon."

Returning to this new home, Xu Xiao could indeed feel the changes.

Although Xu Lang and Yu Guilan didn't buy any big items, they kept the house in good order, and it already felt like a home.

Regarding the allocation of these three rooms, the two of them and Xu Xiao each have a bedroom, and the remaining room is reserved for Xu Xiao to use as a study.

Even though Xu Xiao doesn't usually stay here, they still make sure to give their son plenty of space.

After wandering around the various rooms, Xu Xiao didn't actually go to the study by herself. Instead, she went to the kitchen and started preparing meals with her parents.

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