It was proven that if the modal path corresponding to ρM(p)>0 coincides with a part of the algebraic variety V, then the geotric properties of V can be determined by the analytical behavior of the modal density distribution, such as curvature and singularity distribution.
Yu Wei responded quickly, clearly rembering the content of that class very well.
"Oh, then it should be analyzing the embedding behavior of algebraic varieties in Modal Space. What else was there?" Qiao Yu continued to ask.
"There was, like the isomorphism between modal paths and torus mapping, and the intersection properties of modal paths and algebraic curves. But he only ntioned these during class and didn’t go in-depth, and these topics weren’t included in the exams. He said if we want to study this, it should be at the graduate level."
Qiao Yu nodded, and in just a few conversations, he had already finished the review and abstract of the paper and started reading the main text.
Having understood the situation, he grew serious. After all, Old Yu is actually quite decent, seemingly cold on the outside but warm on the inside.
Even though satisfying his curiosity wouldn’t slow down his reading speed, Yu Wei is extrely ticulous, and might think he isn’t reading the paper seriously.
Qiao Yu stopped asking questions, and since Yu Wei is naturally a reserved person, he didn’t initiate any conversation either.
His good upbringing prevented him from moving Qiao Yu’s things, so he simply sat there watching Qiao Yu flip through the paper...
Half an hour later, Qiao Yu had already read through the entire paper.
The short ti spent was because Qiao Yu had been thinking about this problem for a long ti.
When he was reducing the upper bound interval of Pri Numbers, he didn’t reduce it to 4 because he was reluctant to prove so related tools.
But now, the other party’s ideas were quite similar to his at that ti.
Simply put, they used the theorem he derived before to construct a modal spectrum manifold, embedding the point set of the modal path for this problem, i.e., the Pri Number distribution, into a space with higher-dinsional geotric symtry, to capture local patterns in sparse distributions.
The modal spectrum manifold M (moda) here is a high-dinsional topological space, with its coordinates generated by the correlation values between pri numbers in the Modal Space.
The geotric properties of the manifold are directly used to represent the microstructure of the pri distribution. Then, a sparsity mapping S operator is introduced to describe the symtry of local sparse distribution on the modal spectrum manifold.
An expression is then derived:
Then cos the proof process.
"The paper indeed has no major issues and must have been revised several tis, right? But why did you only prove up to 4? Since you’ve developed the tools, why not go all the way and prove up to 2? You’re not lacking money, do you want to pump out more papers?"
Having read the paper, Qiao Yu looked at Yu Wei and asked.
Yu Wei snapped back from his daze, looked at Qiao Yu with speechless frustration and then said indignantly, "Who wouldn’t want to go all the way? But the distribution resolution of the density function is insufficient, limited by smoothness constraints.
This results in too small gradients in high-density regions, making smaller intervals unfeasible to handle, and besides, the flexibility isn’t enough... Moreover, I wrote about this in the conclusion!"
"How would I know if what you wrote in the conclusion is sincere? Maybe you exaggerated the difficulties on purpose in this paper and then solve the problem in your next paper, that way you appear competent. To be honest, I often think about doing that."
Qiao Yu retorted confidently.
Yu Wei opened his mouth but realized he couldn’t reason with Qiao Yu on this, so he straightforwardly asked, "So do you think this paper can be directly published?"
Qiao Yu didn’t answer directly but asked, "Why are you in such a hurry? By the way, how are you planning to apportion the first and second authorship of this paper? Or what are your and Professor Chen’s contributions?"
"I proposed the local isotric properties of the modal spectrum manifold, but I couldn’t prove its continuity under embedding mapping. Then I started discussing with Professor Chen.
The reconstruction of the modal density function and verifying the convergence of the isotric relation using the kernel integral characteristics of the modal path were all Professor Chen’s ideas.
So Professor Chen suggested, we could very well share the first authorship for this paper. But I think Professor Chen suggests this because he knows we’re on good terms. Strictly speaking, my contribution isn’t that big."
Qiao Yu nodded and then curiously asked, "So why didn’t Professor Chen co with you?"
Yu Wei answered honestly, "Professor Chen said he has a conjecture and told not to tell you."
Qiao Yu directly rolled his eyes at Yu Wei...
Yu Wei thought for a mont and said, "Well, he said he discussed so things with you before. A lot of inspiration for this paper ca from discussions with you. He felt you could complete this proof independently, so he was embarrassed to co."
As expected, Qiao Yu thought, no one in Mathematics is genuinely good-hearted; they’re all scheming.
But honestly, being able to polish the paper to this extent with the professor, Yu Wei was quite impressive to him. This guy didn’t seem as idiotic as he appeared during the IMO period.
"Initiating shared first authorship proves Old Chen still has a conscience. Alright, but since the paper has gotten to this point, instead of just publishing directly, you might as well push it to 2, then submit it to Ann.Math. To be honest, I’m quite familiar with Lott Degen, the chief editor of Ann.Math."
Qiao Yu gave his suggestion straightforwardly.
Yu Wei took a deep breath and said, "It’s very difficult; we’ve tried many ways, spent the entire sumr discussing. If it weren’t because we wanted to achieve 2, we’d have published last sester."
"That’s because you didn’t co to sooner. The problem is actually twofold, the one you just ntioned and the insufficient local analytical properties of your thod’s modal spectrum manifold.
The curvature of the manifold is limited, so just tweak it a bit. Redefine the regularity conditions of the modal density function, then introduce an embedding mapping with adjustable curvature, which enhances local analytical capability.
Then you’ll find it’s possible to optimize the modal path kernel function, and then you’ll find the problem becos simpler. Hold on for a bit..."
After speaking, Qiao Yu picked up a pen and began rapidly modifying the paper.
Indeed, the two’s proof wasn’t wrong directionally. It was just that so detailed aspects hadn’t been thought through deeply.
Qiao Yu believed it would be unprofitable to publish the paper directly like this. For if the paper were published, so keen individuals would soon be able to reduce the upper bound interval to 2.
And perhaps they could conveniently prove the Twin Pri Conjecture. Because proving the upper bound interval as 2 would allow further proof through mathematical induction or prove that the path generated by the modal density function under specific conditions always contains point pairs with an interval of 2, thus proving the Twin Pri Conjecture.
He couldn’t let outsiders reap this benefit.
Already having a good grasp of the matter, Qiao Yu spent about half an hour and then put down the pen.
The freshly printed paper, once clean and tidy, was now filled with modifications.
"Go back and research this with Professor Chen; follow my suggestions, and you should be able to resolve it. You’re so lucky to have a friend like .
Oh, and by the way, let Professor Chen know. Ask if he has signed up for my project at the Academy of Sciences. If not, he should do it quickly, or he’ll miss the opportunity.
As for you, keep working hard. After you graduate, you can directly apply for my graduate program. Oh, and don’t tell Gong Jiatao and Yu Yongjun about this, they’re too dumb!
If they find out about this, they’ll keep pestering . If I beco soft-hearted and actually take them in, then the hundred-year reputation of our Sect would truly be ruined by those two!"
Not paying attention to Qiao Yu’s bizarre words, Yu Wei just blankly received the modified paper, only asking after a while, "So... shall we give you the first authorship?"
"Heh... who do you think you’re belittling? Do you think I helped revise your paper just to snatch your first authorship? Do you think I’m the kind of person who cares about a garbage paper like this?!"
Qiao Yu sneered as he said.
Seeing Yu Wei, who rarely appeared flustered, Qiao Yu softened his tone and said, "Hey, if you really feel uncomfortable about it, once the paper is published, just take whatever reward the school gives you, double it, and send the money to as a manuscript fee for revising the paper!"
Although Qiao Yu didn’t lack money, knowing he had a second-generation wealthy person by his side while feeling he lacked money all the ti yet never making a di off this guy made him constantly disgruntled.
After all, squeezing money out of soone who is extrely stingy is, in itself, a joyful thing.
Happiness prevails!
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