Scholar's Cultivation System Panel

Because of the fact that Delnewitten and others were heading to Jingzhou City on the way, Smith's face turned extremely ugly, and he was extremely angry when he thought about what this International Congress of Mathematicians had become like in his own hands.

It is unimaginable how deserted tomorrow’s conference will be, which will damage the reputation of the International Mathematical Union.

Just thinking about this.

I saw a middle-aged man wearing a navy blue suit walking over.

The other person is short and wears glasses.

After he came to a stop in front of Smith, he hurriedly said, "Secretary-General, don't worry so much."

"I have also done a lot of research on the Riemann Hypothesis. This is not something that can be truly proven at this stage."

Although the conversation was in English, it was not difficult to tell that this was a mathematician from Neon.

Smith was obviously familiar with the person who came. After hearing this, he raised his eyes and immediately said hello.

"Professor Aono, you came just in time. I am counting on you for the one-hour report at the conference tomorrow."

In the past, Smith didn't care about the mathematicians from Neon, but this time most of the well-known mathematicians went to Jingzhou City. Only most of the mathematicians from Neon were willing to participate in the International Congress of Mathematicians. As the secretary of the alliance, he Chang is naturally very happy.

Especially Professor Aono Daisuke from Tokyo University, who was arranged by him to be in charge of tomorrow's mathematics report.

"Don't worry, Secretary General, I'm ready." Aono Daisuke nodded to Smith.

It can be said to be full of confidence.

Judging from the joy in his eyes, he was in a good mood at the moment.

As a professor of mathematics at Dongjing University, although his status and level are not low in the school, he is somewhat unpopular in the entire mathematics community.

The mathematical papers he published were of limited value.

It is no exaggeration to say that if he were to participate in the International Congress of Mathematicians normally, he would hardly have the opportunity to give a report on stage.

That is to say, this year's International Congress of Mathematicians is special. All the slightly famous mathematicians went to Jingzhou City.

Witness the proof of the Riemann hypothesis.

Otherwise, it would not be his turn to go on stage to report such a thing.

But he didn't care about it. Anyway, it was an honor for him as long as he could go on stage to report.

In fact, Smith has been feeling anxious these days, very worried that Xu Yun will really prove the Riemann hypothesis.

Because this would have a huge impact on the alliance.

Suddenly hearing Aono Daisuke say this, it was exactly in line with his thoughts.

Then it was inevitable to mention two more sentences.

"Professor Aono also thinks that Xu cannot prove the Riemann hypothesis?"

"Of course."

Daisuke Aono nodded and answered affirmatively: "The Riemann hypothesis can be hailed as the Everest problem of the century in mathematics. It is not only a problem in the field of number theory, but also involves new fields such as quantum mechanics."

"Although the genius mathematician Xu Yun solved the Hodge conjecture and Goldbach's conjecture at a very young age, it was not enough to prove the Riemann hypothesis in mathematics."

"Your analysis makes sense." After hearing what Smith said, he gave a rare nod of approval.

And this aroused Aono Daisuke's nerves even more. He just finished what he said and added it before he paused for two seconds.

"As far as I know, Xu Yun has been studying battery and lithography machine projects in recent years, and there is simply not enough time to study the Riemann hypothesis."

"I'm afraid he used the Riemann Hypothesis as an excuse to block the International Congress of Mathematicians in order to deliberately retaliate against the alliance."

Smith was also thinking after listening to Aono Daisuke's words, thinking carefully about what happened after Xu Yun announced that he had proved the Riemann hypothesis and was indeed banned from participating in the conference.

Before this, there was no news that the Riemann hypothesis had been proven.

Judging from this time period, Aono Daisuke's guess does make sense.

After thinking about this clearly, Smith felt much better.

He was certain that Xu Yun could not prove the Riemann hypothesis.

Although the International Congress of Mathematicians was indeed abandoned because of this matter, the final winner can only be the International Mathematical Union.

As long as the academic community announces that the Riemann hypothesis has not been proven, he can contact the media to operate.

Completely erase Xu Yun's prestige in the mathematics community.

Let others know the authority and status of the International Mathematical Union.

If Xu Yun wants to reverse the situation, he must really prove the Riemann hypothesis.

But is this possible?

Obviously impossible.

After all, Xu Yun only had a few months to formally study the Riemann hypothesis.

"I want to see how he ends up tomorrow."

As his mood became more and more comfortable, Smith couldn't help but snorted coldly, because the shadow of Deliniwetten was instantly wiped away.

It seems that he has already imagined the scene in which Xu Yun was embarrassed and questioned in front of mathematicians around the world.

Aono Daisuke stood beside him and was equally happy, but he also made some suggestions.

"It is estimated that Xu Yun will prepare a proof process in advance. We only need to arrange for people to conduct questioning and verification on site. As long as errors are found, we can completely overturn and deny the other party's proof."

"This is a good method." Smith was somewhat moved.

In addition to wanting to see whether the Riemann Hypothesis was proven, many people who went to Jingzhou City to participate in the exchange event were impressed by Xu Yun's charisma and were willing to help.

If everyone does not question it in public out of dignity, then even if it is later announced that the Riemann hypothesis has been falsified, it will not have any impact on Xu Yun.

In this way, you must arrange for your own people to be on site, and you must also conduct corresponding research on the Riemann hypothesis of number theory.

Although Aono Daisuke has done research on the Riemann hypothesis, Smith is very aware of his opponent's level.

I'm afraid it's not enough to refute Xu Yun's proof process.

What's more, the International Congress of Mathematicians will continue to be held, and someone needs to come on stage to report and maintain the last dignity.

The thought stayed here, and soon Smith seemed to have made a decision.

He raised his gaze and looked at Aono Daisuke and arranged: "I plan to go to Jingzhou City in person to overturn his certificate. I will leave the report meeting to you tomorrow. I will let my assistant be responsible for other processes." From the tone of speaking It obviously doesn't look like a joke.

When Daisuke Aono heard Smith's words, he was startled at first, and then he couldn't help but get excited.

He naturally knew Smith's level. If Smith personally verified Xu Yun's proof process, then the overthrow would be certain.

Before Smith became the Secretary-General of the International Mathematical Union, his status in the mathematics community was not low.

In addition to winning the Fields Medal, he also won the Abel Prize.

The key is that he has in-depth research in the field of number theory.

Thinking of Xu Yun's outcome tomorrow, Aono Daisuke immediately expressed his stance: "Secretary-General, please rest assured, I will be responsible for tomorrow's report meeting."

Since it was a temporary idea, in order not to delay the exchange meeting held by the Institute for Advanced Mathematics at Peking University tomorrow, Smith will not waste any more time.

After talking to Aono Daisuke for a few more words, he asked his assistant to order the fastest flight to Jingzhou City.

Although he also knew that it was not appropriate to go to Jingzhou in his own capacity, he felt like the International Mathematical Union was being subdued.

However, in order to overturn Xu Yun's proof process and make him lose his authority in front of mathematicians around the world, this trip really needs to be done.

In this way, without notifying Jingzhou City in advance, Smith got on the plane to Jingzhou.

This makes things even more lively, and everyone is looking forward to the final result tomorrow.

At the same time, the online debate over whether the Riemann hypothesis has been proven has become heated. In addition to many netizens participating in the discussion, a large number of mathematicians also participated.

It is conceivable that no matter what the result is tomorrow, it will cause heated discussions around the world.

Time passed quickly, and at eight o'clock in the morning in Jingzhou City the next day, Xu Yun was greeted outside the largest lecture hall of Jingzhou University.

In order to suit this important occasion today, Xu Yun specially wore the clothes he had attended last time at the Science and Technology Awards Conference. He looked very energetic and energetic.

Compared to when I first proved Hodge's conjecture a few years ago, I feel a lot less green.

Become more stable and mature.

In addition to him, standing beside him were the president Wu Zhongping and the deans and professors of each college.

Although this is a matter between the Department of Mathematics and the Institute for Advanced Study in Mathematics, it is also related to the honor of Jingzhou University.

What's more, this time there are well-known mathematicians from all over the world. The lineup is much larger than the previous Hodge Conjecture report. Naturally, cooperation and help from various colleges are needed.

It just so happened that the students of Jingzhou University were on summer vacation, so it was quite quiet except for the students and volunteers who stayed on campus.

I just happened to be here for this exchange event in the mathematical community.

Two of his graduate students, Li Youzhi and Zhang Yang, and Xu Yun also arranged to go out and let them help with the reception.

After all, having more contact with well-known mathematicians around the world will only be beneficial to their future development.

Even though it was only eight o'clock, since most of the people had already arrived early and were staying in nearby hotels, it didn't take long to get here.

Looking around, there are already many people gathered in the Institute for Advanced Study. In addition to mathematics professors from various overseas university research institutes, there are also many domestic professors and academicians.

It can be said to be extremely lively.

It is no exaggeration to say that if you throw a brick at this time, you will hit anyone who is a full professor.

Not long after, Xu Yun saw two familiar figures, and immediately quickened his pace to meet them.

"Professor Deligne."

"Professor Witten."

"Welcome to come to Jingzhou University for exchange."

Looking at Deligne Witten, whom he had not seen for several years, Xu Yun took the initiative to say hello.

And he shook hands with the two of them one after another with a smile on his face.

To be honest, he was quite surprised that Deligne and Witten could come. After all, despite their age, even at the Institute for Advanced Study in Princeton, they were honorary professors and could not do much.

When the Hodge Conjecture Lecture was held at Kyongshu University, Deligne only participated online.

Later, Xu Yun attended the last International Congress of Mathematicians and went to Princeton University before actually meeting them.

In addition, he also holds a PhD certificate from Princeton University.

Ever since he was included in the security building's focus list, I thought I might not have the chance to communicate with him again.

I never thought that I would use this opportunity to attract Deligne and Witten to Jingzhou City.

Both Deligne and Witten have high hopes for Xu Yun, and naturally they like this student very much.

He shook hands briefly and then responded immediately with a voice.

“It’s really not easy to come here this time, but as long as we can see the Riemann Hypothesis, the century-old problem known as the Everest of Mathematics, being solved, then there will be no regrets in life.”

"I think you won't let us down." Witten took Deligne's words and asked.

Xu Yun already knew that Deligne and Witten were restricted from traveling in the security building. He also knew that they came from the International Congress of Mathematicians. He was very grateful and naturally he would not keep it secret.

The response was extremely decisive.

"Don't worry, professors. I believe today will be a memorable day for everyone who comes to the exchange meeting."

He always kept a confident smile on his face when he spoke.

Deligne and Witten were no stranger to Xu Yun's smile. It was almost exactly the same as when they proved Goldbach's conjecture.

Seeing that Xu Yun was so confident, they became more and more excited.

Understand that I am afraid that the Riemann hypothesis was really proved by Xu Yun.

"Then we'll wait and see." Deligne smiled, very satisfied with his choice.

Xu Yun did not forget the person standing next to him. When he came to his senses, he hurriedly introduced Delnie Witten.

"This is our president of Jingzhou University, Mr. Wu Zhongping, and Dean Qin Xiangxinqin of the Institute for Advanced Mathematics."

…

Everyone present, whether they were in the field of mathematics or not, were basically familiar with the names of Deligne and Witten. Naturally, they all greeted Xu Yun enthusiastically when they were introduced.

After a few brief exchanges of greetings, Xu Yun asked Delniewiten and others to lead them to the lecture hall lounge.

According to the arrangement of this exchange meeting, hundreds of the world's most famous mathematicians will participate in the verification process of the Riemann hypothesis in the lecture hall, giving it as much authority as possible.

The remaining participants in the exchange meeting can discuss issues in other mathematical fields.

Or witness the entire proof of the Riemann Hypothesis through the screen.

However, it is worth mentioning that the verification of this proof process will not be broadcast live online.

Must attend in person to view.

The main purpose of throwing out the gimmick of the Riemann Hypothesis was to compete with international mathematicians for participants. If you choose to launch an online live broadcast, many people will definitely choose not to participate.

In this way, as time goes by, the number of well-known mathematicians entering the lecture hall is increasing.

And everyone's face was filled with visible excitement.

I look forward to witnessing this milestone moment in mathematics.

I have long forgotten about the International Congress of Mathematicians.

At this time, Wu Zhongping, Qin Xiangxin and others went to work separately, while Xu Yun saw an old friend.

It was Hofer, the editor-in-chief of the editorial board of the Annals of Mathematics.

As for the reason why the other party came, it was obviously very simple. It was obviously for the proof of the Riemann hypothesis.

But what is surprising is that Hofer is accompanied by Li Youzhi and Zhang Yang at this moment.

As soon as he got close to his ear, he heard Hofer's voice.

"Xu."

"These two are your students, right? I've read the proof paper on the dot problem."

"It's a bit like you used to be."

Because the two have known each other for a long time and have a close relationship, Hofer used a more affectionate term.

By the way, I didn't forget to praise Li Youzhi and Zhang Yang next to me.

When they received the person, Li Youzhi and Zhang Yang learned about Hofer's identity and knew that the other person was the editor-in-chief of the editorial board of the Annual Journal of Mathematics.

As a newcomer in the field of mathematics, it is definitely a great honor to be praised by the editor-in-chief of the Annual Journal of Mathematics.

Li Youzhi and Zhang Yang were also very happy about this, at least they did not embarrass Xu Yun in front of outsiders.

But thinking of their usual teachings, the two of them replied humbly.

“I have to thank my supervisor for the guidance of that thesis, otherwise we wouldn’t have been able to complete it successfully just by ourselves.”

"Is such that."

Seeing the performance of the two of them, Xu Yun couldn't help but raise the corners of his mouth slightly with a smile.

Then he took over the conversation and added: "These two students of mine are indeed good."

He did not hesitate to praise Li Youzhi and others at all. After saying this in front of Hofer, he waved his hand and said, "You two go on to work."

"Good mentor."

Li Youzhi and Zhang Yang almost forcibly suppressed the joy in their hearts. When they heard Xu Yun's words, they immediately felt like they were being pardoned and quickly turned around and left.

Soon, when only Hofer was left, Xu Yun smiled and said, "Does Mr. Hofer have such confidence in me?"

"Of course, if the Riemann hypothesis can be proven, it will be a major event in the entire mathematical world."

Hodel answered truthfully.

Since he came all the way to Jingzhou City, he must believe that Xu Yun can prove the Riemann Hypothesis. Otherwise, if the proof process is overturned, the relevant paper will not be published in the Mathematics Annual.

Although this time Xu Yun publicly proved that the process was not the first paper through on-site verification.

It can be said that as long as the status of the Riemann Hypothesis in the mathematics community is recognized by most mathematicians, the impact factor of the annual mathematics journal will be greatly improved if a paper is published later.

If he didn't let other mathematics journals take over, he would really regret it.

After all, Xu Yun has been cooperating with their Mathematics Annual from the beginning to the present.

The Riemann Hypothesis is indeed very complex and difficult to imagine, otherwise why there has been no progress over the years.

He couldn't believe it when he first heard the news, and didn't think the Riemann hypothesis could be proven.

But when I thought carefully about my interactions with Xu Yun later, it seemed that the other person would never do anything he was unsure of.

Since I dare to publicly announce that I have proved the Riemann hypothesis and accept verification on-site by world-renowned mathematicians, I must be absolutely sure.

After thinking about this clearly, he resolutely overcame the resistance and came to Jingzhou on time to attend this exchange meeting.

Xu Yun naturally doesn't mind publishing the paper in the Annual Journal of Mathematics so that more mathematicians can watch the research. Anyway, he has already written the entire paper during this time.

Just thinking of the various troubles that Professors such as Delignewitten encountered when they came to participate in the exchange meeting, he needed to confirm one thing with Hofer.

"I have no objection to still cooperating with the Annals of Mathematics, but I wonder if your editorial board still dares to accept my paper?"

Hearing Xu Yun's words, Hofer's expression suddenly became awkward. He naturally understood the meaning of Xu Yun's words.

Since Xu Yun is no longer a simple mathematician and has been on the security building's focus list to conquer lithography machine technology, many things that were once easy have become difficult to do.

After all, mathematics knows no borders.

Mathematicians do.

At this moment, Xu Yun is also watching Hofer. If the other party is not sure about this, then he will not look for other mathematics annuals and will just publish it directly on the entire network.

After about three or four seconds, Hofer finally seemed to have made a decision.

He suddenly raised his eyes and his eyes became firm.

“As long as it is a valuable mathematical paper, our Mathematics Annual will accept and publish it. This is also the greatest significance of our existence.”

"Then I wish our cooperation goes smoothly." After hearing Hofer's words, Xu Yun suddenly smiled and reached out his hand.

Just as the two were secretly happy to be working together again, a slightly familiar shout caught Xu Yun's attention.

Looking in the direction of the sound, a handsome overseas man's face suddenly came into view.

Who else could it be if it wasn't Minter.

The last time the two met was at the International Congress of Mathematicians, four years apart.

Due to the other person's image that gave him the only sense of crisis, he had quite an impression of Griffith's favorite student.

As the host, he ignored Hofer and hurriedly turned around and took a few steps to say hello.

"Dr. Minter, we meet again."

Hofer, who was behind him, left temporarily with a wink and did not stay to disturb the conversation between the two.

"yes."

Mingte held Xu Yun's palm and answered in standard Mandarin.

Although he seemed calm on the outside, he was actually feeling mixed emotions inside.

As a person who has studied the Riemann hypothesis for many years without any results, you can imagine how he feels when the prover of the Riemann hypothesis stands in front of him.

Even if Minter thought he had given up his obsession, he was embarrassed to mention that he was also studying the Riemann hypothesis at this time.

Xu Yun didn't deliberately think about what Mingte was thinking. When he saw that there was only one person on the other side, there was some confusion in his eyes.

"Dr. Minter, where is Professor Griffith?"

"Unfortunately."

"My teacher was unable to participate in this exchange event. He asked me to apologize to you on his behalf." When Xu Yun mentioned his teacher, Mingte immediately answered truthfully.

After hearing these words, Xu Yun immediately guessed what was going on.

After all, Griffith's previous email stated that he wanted to participate.

Although it was a bit regretful, the fact that the other party allowed Minter to attend and apologize on his behalf already explained the problem.

I can only say that many times you need to bow to reality.

"It doesn't matter."

"I believe there will be opportunities in the future."

Xu Yun didn't say much about this matter. He patted the other party's shoulder and said a word of comfort before walking towards the lecture hall.

Currently, hundreds of world-renowned mathematicians are basically present in the lecture hall.

Everyone has been waiting very anxiously.

Xu Yun didn't want to waste any more time, so he planned to officially start the proof and verification of the Riemann hypothesis.

Soon as Xu Yun walked onto the stage, he instantly attracted the attention of everyone present.

Expectations were revealed in their eyes.

In addition to many drawing boards placed on the stage, there are also specially prepared computers.

It is convenient for everyone to verify.

"Thank you all for coming to participate in the first mathematics exchange event of the Institute for Advanced Mathematics at Peking University. I know why everyone is here, so I won't waste time by being pretentious."

"Next, I will complete the entire proof process of the Riemann Hypothesis and invite everyone to verify it."

Xu Yun faced hundreds of world-renowned mathematicians in the lecture hall. After briefly making these opening remarks, he planned to officially start the highlight of this exchange event.

Such an exciting moment also made many people couldn't help but murmur.

"We have finally waited for this moment. Will we be the ones to witness the proof of the Riemann hypothesis?"

“It’s really exciting!”

“May God bless the Riemann hypothesis to be solved.”

"This will go down in the history of mathematics as a landmark moment."

"I have a feeling that today will definitely not be an ordinary day."

…

But at this moment, a sound with a significantly higher decibel level attracted the attention of everyone in the lecture hall.

"How can the Riemann hypothesis, which has been pending in the mathematics community for so long, be so simple to prove?"

Because this voice came from outside the lecture hall.

Following everyone's gaze, the door to the lecture hall was pushed open from the outside.

I saw Smith, who was the Secretary-General of the International Mathematical Union, striding in.

It's as if this is his home court.

However, this scene was obviously beyond everyone's expectations. They didn't expect to see Smith here.

Normally, Smith should be presiding over the International Congress of Mathematicians right now.

Suddenly appearing here makes people think a lot.

Smith is undoubtedly very confident in his judgment. He never believes that Xu Yun can complete the real proof in the past few months without studying the Riemann hypothesis.

So in his opinion, Xu Yun's failure was doomed.

In this way, it doesn't matter who is playing at home.

Ignoring the gazes of the people around him, he walked directly towards Xu Yun.

"I have something I would like to ask Professor Xu to answer. How did you become so confident in your proof without formally studying the Riemann hypothesis? Isn't this just deceiving the mathematical community again?"

No one expected that Smith would take such an approach. As his voice fell, everyone else's expressions changed slightly.

It is inevitable that there will be some uncertainty as to where things are going.

Among them, Witten and Deligne, who were sitting in the front row, were also a little worried, fearing that Xu Yun's certification process would really be overturned.

"Things seem to be getting a little tricky. What should we do with this situation now?"

"We can't do anything to help. We can only put our hope in Xu Yun and pray that his proof process can withstand verification." Facing Witten, Deligne shook his head and said.

After all, the situation has developed into what it is now, and the final direction can only depend on Xu Yun's proof process.

Others have no solution.

In fact, Xu Yun was a little surprised when he saw Smith running here from the International Congress of Mathematicians, but soon his heart turned into joy.

With the Secretary-General of the Mathematics Union witnessing the Riemann Hypothesis being proven, the authority will be amplified invisibly.

You must know that Smith represents the International Mathematical Union.

It can only be said that the other party still underestimated him, so he can only bear this failure by himself.

The thought stayed here, Xu Yun did not answer the question Smith had asked in front of everyone, but answered calmly and at his own pace.

"Secretary-General Smith is willing to give up his work at the International Congress of Mathematicians to participate in the exchange activities of our high school. Please allow me to express my gratitude to you on behalf of Jingzhou University."

"In addition, Secretary-General Smith is welcome to help me verify the process of proving the Riemann hypothesis."

After hearing all Xu Yun's words, Smith was naturally very angry.

"I will definitely verify it carefully."

But in the end, he had no choice but to reply with this sentence.

After temporarily settling Smith's small episode, Xu Yun arranged for him to sit down in the front row seat, then turned around and officially started his proof.

The Riemann hypothesis is a conjecture about the zero-point distribution of the Riemann zeta function ζ(s).

In the study of the Riemann Hypothesis, the straight line with Re(s)=1/2 on the complex plane is called the critical line. In this way, the Riemann Hypothesis is expressed as all non-trivial zero points of the Mann zeta function are located on the critical line.

That is, all non-trivial zero points of the Riemann zeta function are located on the straight line Re(s)=1/2 on the complex plane.

There are already more than a thousand mathematical propositions in today's mathematics literature based on the premise that the Riemann hypothesis is established.

In other words, if the Riemann hypothesis is proven, then more than a thousand mathematical propositions may be true, which is definitely a very critical node for the development of mathematics.

However, with the development of mathematics and research, the Riemann hypothesis has now become a theoretical geometric topological carrier for the integration of general relativity and quantum mechanics.

This is also the reason why the Riemann hypothesis has not been proven yet.

Of course, this current mathematical problem, which can be called Mount Everest, is nothing to Xu Yun.

In addition to the entire proof process in his mind, he has also thoroughly understood and mastered it through the past few months of research, and there will be no pause or resistance at all when writing it now.

Soon one of the writing boards was filled with mathematical formulas and complicated steps.

ζ(s)=……(Re(s)＞1, n∈N+)

…

As for everyone in the audience, their eyes were fixed on the contents of the writing board, and they were completely immersed in it.

During this period, I didn’t forget to take out the scratch paper I carried with me to check the calculations.

I have to say that the charm of mathematics is indeed very great. Even Smith, who had already determined that Xu Yun could not prove the Riemann hypothesis and wanted to make it difficult for him to step down later, was unknowingly completely convinced after seeing these mathematical formulas and steps. Immersed in it.

I was attracted by the proof process completed by Xu Yun.

As time passed by, while the expressions on other people's faces became more and more excited and joyful, Smith became more and more ugly.

Because he suddenly discovered that he couldn't find the so-called loophole at all.

After trying the calculation in my own way, it seems to be completely reasonable and there are no errors.

This caused a bad thought to pop up in his mind.

It seems that Xu Yun really proved the Riemann hypothesis.

But as soon as this idea took shape in his mind, he immediately cleared it out.

Unwilling to accept this result.

After all, Xu Yun's proof process has not been completed yet, and even if he can find a loophole to overturn the proof, there may be other mathematicians present who can do it.

In short, he still has hope of making a comeback.

Although this hope is rapidly decreasing as Xu Yun's proof process continues to be completed.

I don't know how much time passed. When all the writing boards were filled with mathematical formulas and proof processes, Xu Yun finally wrote the two words "certificate" and then stopped writing and turned around.

He still had that calm and confident expression.

"From the above formulas, the Riemann hypothesis can be established. This is my entire proof process." (End of this chapter)