Expand the Product Line and Cultivate Niche Markets

"I simply cannot find a weakly closed convex set that is a closely approximating set, thus obtaining Hustav Topology. I believe no mathematician in the world can prove from this approach that every ω* closed convex set of X* is a closely approximating Chebyshev set." Shen Qi was not too frustrated, but rather enthusiastic. "At first, I didn't realize it was a dead end, because I didn't reach the end of the dead end."

"Do you have a good solution?" Muller asked.

Shen Qi already has an answer, "My solution is to establish a new topic for argument (3) separately. I will temporarily name it 'Muller -Shen Approximation Theorem'. This theorem needs further verification. It can be regarded as a corollary of 'Muller -Shen Theorem'."

"Then the new 'Muller -Shen Theorem' contains two statements (1) and (2), which is sufficient. We will complete the revision work in a few days and publicly release the research results of the 'Muller -Shen Theorem'." Shen Qi provided the solution.

"In fact... we've essentially launched a 'downgraded' version of the 'Muller -Shen Theorem'." Professor Muller quickly understood Shen Qi's new plan.

Shen Qi nodded. "Professor Muller's analogy is very apt, yes, a downgraded version. In fact, the so-called downgraded version is the standard version, because assertion (3) can be regarded as a new product."

"Actually, they're just two car bodies with different lengths but roughly the same exterior design, equipped with the same powertrain system, and then sold separately. Well, Volkswagen does this all the time." Professor Muller, true to his German nature, hit the nail on the head.

"Haha!" Shen Qi laughed, "That's why I bought a Chevrolet."

After discussing it, Muller and Shen Qi decided to do just that: first, release a low-end version of the "Muller -Shen Theorem" to meet basic market demand and seize this niche market.

A high-end version of the "Muller -Shen Theorem" will be launched soon. It will be given a new name, "Muller -Shen Approximation Theorem," which would be a good choice to further enrich the product line and consolidate the market share and brand effect of "Muller -Shen" in this niche market.

The revision of the "Muller -Shen theorem" was completed quickly.

"... After proving, we obtain the following theorem:"

Let X be Banach space, then the following argument is equivalent to:

(1) If x*∈S(X*) attains its norm on S(X), then x* is an ω* concave point of the unit ball B(X*);

(2) X is a strongly smooth space.

After a final check of the paper, Professor Muller personally uploaded it to arVix for pre-recording.

The task of formally submitting the paper was assigned to Shen Qi.

The paper is titled "Problems with concave points in Banach space and strongly smooth spaces," with Muller and Shen Qi as the first authors.

The paper is 24 pages long and contains a complete proof of the "Muller -Shen theorem".

Of course, the "Muller -Shen theorem" is a claim unilaterally made by Muller and Shen Qi. Whether it will ultimately be included in the IMU's encyclopedia of mathematical theorems is up to the IMU to decide.

Shen Qi believed that this paper was of very high quality, belonging to the level of the Top Four Journals, so he submitted it to the Annals of Mathematics.

At the same time, the proof of the "Muller -Shen approximation theorem" was also being carried out concurrently. This work was mainly completed by Shen Qi, because it was Shen Qi proposed it.

It's Wednesday again in the blink of an eye.

This morning, Faltings, as usual, browsed arVix and found a paper titled "Problems of Concave Points in Banach space and Strongly Smooth Spaces," authored by his colleague and a PhD student in the Mathematics Department at Princeton University.

"Oh, so that's how it's proven... 'The Muller -Shen theorem' utilizes approximating compact closed convex subsets, as well as the RNP property of Banach space and Martingale Theory. Looks like old Allen can finally publish a paper this year." Faltings read through the entire paper and found no problems.

Grader Faltings is German. He became famous at a young age, proving the groundbreaking Mordell Conjecture in 1983 when Faltings was only 29 years old.

When Faltings received the Fields Medal in 1986, he was only 32 years old. His wife was also a mathematician, and his teacher was Grothendieck, a godfather figure in 20th-century Mathematics Community.

A student of a renowned teacher, he married a mathematician, and everyone around Faltings in his life was related to mathematics.

As one of the top masters in the Mathematics Community today, Faltings is the editor-in-chief of the most authoritative mathematics journal, the Annals of Mathematics.

In the afternoon, at the coffee shop on the third floor of the Mathematics Department building, it was our usual coffee time.

Faltings, carrying his coffee cup, walked over to the small table where Muller and Shen Qi sat down.

",helü@ ful." Faltings said something in German to Muller, his expression seemingly teasing.

"Uh,@@l!" Muller retorted unhappily to Faltings.

Shen Qi was dumbfounded. "Can the two professors speak English?"

"Old Allen, you've finally published a paper," Faltings said.

"You're a year older than me, old Grader!" Muller repeated his earlier expression.

These two German old men are quite adorable... Shen Qi chuckled to himself. Scholars who reach their sixties or seventies sometimes act as mischievously as six- or seven-year-old children.

"Allen, it seems like you haven't published a single paper or any academic book all year," Faltings said.

Princeton has 1,200 professors or adjunct professors who publish more than 2,300 papers or academic monographs each year, averaging nearly two papers or monographs per person.

It's already October, Muller hasn't published a single paper or a single monograph.

At Princeton, while the quantity and quality of papers or academic monographs are not linked to housing or academic titles, it is still a place where academics are paramount, and papers or academic monographs hold an important place in the minds of most professors.

"Wait, Grader, you just said I finally published a paper, is it that fast?" Muller asked.

"It should be pre-recorded, Allen. Didn't you upload the paper 'Problems of Concave Points in Banach space and Strongly Smooth Spaces' to arVix?" Faltings was puzzled, then looked at Shen Qi with sudden realization. "Was it you, Shen Qi?"

Shen Qi said, "Professor Muller handled the pre-recording, while I handled the formal submission of the paper."

Professor Faltings sent a congratulatory message: "Then I should congratulate you, Allen and Shen Qi. In my opinion, the proof of the 'Muller -Shen Theorem' is without any problems. I believe that in the near future, this theorem, named after your two surnames, will be widely cited in research on Bachmann Space -related topics."

"Grader, it seems you're not so bad after all," Muller said.

"If Professor Faltings says there's no problem, then there shouldn't be any problem." Shen Qi felt that there shouldn't be any major issues. The review experts invited by IMU couldn't possibly be more professional than Professor Faltings; at most, they would be on par.

"Which journal did you submit it to?" Faltings asked.

Shen Qi smiled, "The * Annals of Mathematics *."

"Really?" Faltings was somewhat surprised.

Faltings is the editor-in-chief of this journal, and the manuscript is probably still in the process of being accepted; it hasn't reached the stage where he makes the final approval.

"Then it needs to be formally reviewed according to the rules. I need to go back and study this paper again," Faltings said seriously.

"Old Grader, didn't you just say there was no problem?" Professor Muller was displeased again.

"My opinion doesn't count; the reviewers' opinion matters. I'm not a reviewer." Faltings walked away with his coffee cup.

The Annals of Mathematics is a mathematics journal jointly sponsored by the Princeton Mathematics Department and the Princeton Institute for Advanced Study. It is the leading journal among the Top Four Journals.

When the Princeton Institute for Advanced Study, the research building was under construction. Einstein, von Neumann, and other early researchers had nowhere to conduct their research, so they stayed in the Princeton Mathematics Department building for several years.

The Mathematics Department and Physics Department departments at Princeton University have a good relationship with the Princeton Institute for Advanced Study, with researchers from both institutions frequently visiting each other. Faltings also spent a few years in the mathematics department at the Princeton Institute for Advanced Study.

The editorial office of the "Annals of Mathematics" was located in the Mathematics Department building of Princeton University, just above the floor where Shen Qi drank coffee.