In the vicinity of a singular point, define a new homology layer, for example Fxα, where α represents the fractional order. In this way, the growth rate can be controlled through fractional order values, reflecting the complex geotric structure at the singular point.
However, this thod eventually failed because it primarily relied on approximation. Such approximation at the complex singular point cannot provide sufficient convergence proof. This problem troubled us for a long ti.
It was not until Sam and Frank, using their thod, connected the feature layer with white noise that we finally had an answer. Are you sure the thod you provided can effectively avoid this situation?"
"Oh... the third floor, thank you."
At this mont, the four of them had already gotten into the elevator, and Danis, after telling the bellboy in the hotel elevator the floor, turned to look at Qiao Yu.
"Well, my thod is quite different from yours. But your idea definitely won’t work, because fractional homology layers aren’t truly integer-order homology, there will certainly be errors in details, and the more complex the singular point is, the larger the error will be, ultimately still leading to a mismatch between local and global."
My approach is more direct, to imdiately identify and filter out the white noise component introduced by the feature layer, allowing the structure of the local system to always maintain a very clean mapping on the global level. Besides, this thod isn’t really complex, it mainly involves introducing a denoising fuzzy operator, which is too complicated to explain. I’ll show you an example later, and you’ll understand."
Danis nodded.
The collision of mathematical ideas, indeed sotis cannot be clearly explained with just words.
Fortunately, the eting room was right ahead.
...
"Lemma 1: Local Reconstruction of Conjugate Ridge Class Complex Singular Points."
"Let X be an algebraic variety, x∈X representing a conjugate ridge singular point."
"At the conjugate ridge singular point x, one can construct a local reconstruction object Fx, LGIR, and a diminishing power staircase operator DLGIRD to address the non-convergence of traditional local systems at singular points, defined as follows:"
"Under the LGIR frawork, define a diminishing power staircase operator DLGIRD whose effect on Fx, LGIR is:"
In the eting room, Qiao Yu wrote down so recent research findings on the hotel-provided notepaper and handed it to Danis.
This wasn’t all of Qiao Yu’s ideas; Old Xue had taught him that academia isn’t as pure as it seems from the outside, so it’s good to keep a part of it to himself. Hence, he decided to disclose the remaining key solution process only at the conference.
However, Qiao Yu believes that mathematicians who have studied this proposition for over a decade can see the value of his idea.
After all, the limit structure he designed ensures that the local reconstruction object remains well-defined under high-order homology groups. Traditional thods might diverge due to high-order disturbances, but this limit construction guarantees stable local behavior for local reconstruction objects.
This is currently the best solution for this issue, Qiao Yu even feels it’s unparalleled.
In fact, not only Danis, but the two professors from Yanbei were also contemplating the limit structure and the newly introduced operator that Qiao Yu constructed.
After reading the process written by Qiao Yu, Tan Zhiqing subconsciously looked up at Qiao Yu and said, "Filtering?"
Qiao Yu thought for a mont, then nodded and said, "Yes, sowhat like that."
Professor Su from the School of Mathematics also remarked, "Indeed a good idea, using alternating symbols and stratified weighting to gradually filter out self-noise at each level."
Danis frowned at the formula and nodded after a while: "Yes, the stratified approach can indeed ensure effective noise elimination at different homological levels without disrupting the consistency of the overall structure."
Having said that, Danis looked up and asked: "And then?"
Qiao Yu helplessly spread his hands and said, "Next, we have to make the global consistency correction. But I haven’t figured out how to handle this part yet, although I have so thoughts, it still requires ti to perfect. After all, we need to smoothly embed the local system characteristics into the global system.
This even involves the technical analysis of the homomorphic space mapping of the local system to the global system. So I still need ti to think, maybe a few days, maybe a few months.
Oh... Professor Danis, honestly, my research on this type of issue heavily relies on inspiration that bursts out suddenly, and during this period, I may need to read so literature, so I really can’t promise when I will fully resolve this issue.
)
I made that call to Professor Pan simply because I had promised him that once I found a solution, I would notify him at the first opportunity. Well, I think I have thought of this thod. Of course, there are still so key steps, but please trust that I will certainly help you and Professor Pan completely resolve this problem."
Danis could see sincerity in Qiao Yu’s face.
Alright, there really isn’t another way. In fact, Qiao Yu proposing this idea already helped him trendously, providing an excellent problem-solving approach. It just made the problem beco complex again.
Danis glanced at the two professors from Yanbei, then looked at Qiao Yu and asked: "Qiao Yu, would you be willing to join our team and share this idea with us? You should know who are in our team, don’t you? For instance, the main collaborators include Professor Sam, Professor Frank, Professor Pan, and..."
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