Chapter 156 Learn, Learning Has No End

Early in the morning, Chen Zhuo went to the Department of Mathematics with his schoolbag on his back.

The building housing the School of Mathematics is quite old, and the hallways are usually dimly lit.

Chen Zhuo went up the stairs to the third floor and pushed open the door to the large conference room at the end of the corridor.

The heating in the old building hadn't been turned on yet, and the large space meant that cold air was blowing straight into our collars.

Wu Tao was standing in front of the huge blackboard that took up the entire wall.

He was wrapped in a thick, dark blue down jacket, the zipper pulled all the way up to his chin. A cigarette dangled from his mouth, and as he gritted his teeth, the flame flickered up and down at the corner of his mouth.

The blackboard was covered with densely packed derivations of formulas, erased and rewritten on the left, leaving layers of white chalk dust. A pile of messy draft papers lay scattered on the conference table, with the five-page outline on homology mapping that Chen Zhuo had brought over a few days ago lying in the center.

Hearing the door open, Wu Tao turned around.

His eyes were bloodshot, and his eye bags were so heavy it looked like he hadn't had a good night's sleep in days.

Seeing Chen Zhuo walk in, Wu Tao threw the half-burnt piece of chalk in his hand into the blackboard tray, patted the dust off his hands, and casually took the cigarette out of his mouth and held it in his hand.

"They're here."

Wu Tao's voice was a little hoarse as he pulled out a chair next to the conference table and sat down.

Chen Zhuo walked over to him, put his schoolbag on the table, and pulled out another chair to sit down.

He glanced at the blackboard full of formulas, then at the scratch paper on the table covered in crosses, and asked casually, "Where are you stuck?"

Wu Tao gave a wry smile and tapped on the third page of Chen Zhuo's five-page outline.

"From this lemma, we can derive the next step: isomorphism groups."

Wu Tao drew a line between the two lines of formulas with his fingernail.

"The gap between these two is too large."

Chen Zhuo looked in the direction he was pointing.

"Logically, it is possible to directly substitute the boundary conditions of the continuous domain," Chen Zhuo said.

"It makes sense logically, but I can't calculate it."

Wu Tao stood up, walked back to the blackboard, picked up a new piece of chalk, and quickly wrote a long string of formulas with integral symbols in the blank space in the lower right corner of the blackboard.

The chalk made a crisp sound as it tapped on the blackboard.

"You see, when we substitute the boundary conditions and try to perform an integral expansion in the third dimension, the result diverges." Wu Tao stopped writing and turned to look at Chen Zhuo.

"In a topological space, once the integral diverges, the subsequent isomorphism group cannot stand."

Chen Zhuo did not respond immediately.

He sat in his chair, his eyes fixed on the divergent integral on the blackboard, his mind rapidly reviewing his original derivation. In his original conception, the continuity here was naturally established.

Wu Tao sighed, threw his cigarette into the trash can, took a lollipop from his down jacket pocket, unwrapped it, and popped it into his mouth. "These past few days, Liu Ming and I have tried four different integration paths, and they all diverged near the singularity."

Wu Tao spoke indistinctly, his mouth full of candy.

"If we can't get over this hurdle, everything after this will have to be scrapped and started all over again."

Chen Zhuo took a pen out of his bag and tore off a blank sheet of draft paper.

"Since the integral over a continuous field diverges at a singularity, we should not use the continuous field."

As Chen Zhuo spoke, he quickly wrote down a set of matrix equations on a piece of paper.

Wu Tao leaned over and took a look.

"On this fault line, we introduce a discrete algebraic matrix."

Chen Zhuo's pen tip swept across the paper, writing down line after line of clear algebraic expressions.

"Forcibly cut this continuous singularity, use discrete eigenvalues ​​to frame the divergent trend, and then remap it back to the continuous domain at the next safe node."

Wu Tao stared at the matrix on the draft paper for more than ten seconds.

He didn't immediately refute, but instead pulled up a chair, took the pen from Chen Zhuo, and began to quickly perform calculations next to the matrix. The conference room fell silent, with only the scratching sound of the pen tip on the paper.

After about five or six minutes, Wu Tao stopped writing and shook his head.

"It's a dead end."

Wu Tao pushed the draft paper back in front of Chen Zhuo and drew a circle on the last line of the equation with the tip of his pen.

"Using discrete matrices to cut singularities is a very unconventional approach, and it does bypass divergence, but you missed a step in the calculation."

Wu Tao spoke with great certainty, his voice carrying the intuition of someone who had long been engaged in fundamental reasoning.

"After a matrix is ​​closed, its Jacobian determinant is not equal to zero."

Chen Zhuo glanced down at Wu Tao's calculation process.

"In a system of topological invariants, if the Jacobian determinant is not zero, it means that the space has been distorted."

Wu Tao leaned back in his chair and pointed to the blackboard.

"Then the subsequent isomorphism group is invalid, which is incorrect."

Chen Zhuo looked at the non-zero result and tapped the table lightly with his fingers.

Wu Tao's calculations were flawless.

In practical calculations, this doctoral student has an extremely keen and solid foundation, and can immediately see the rejection reaction of algebraic matrices in topological space.

If the Jacobian determinant is not zero, the space will be distorted.

Chen Zhuo's mind raced. Continuous domain was not an option, and discrete cutting would lead to distortion.

Then the only option is to tamper with the cutting process.

"Senior Brother Wu."

Chen Zhuo suddenly stopped tapping his fingers on the table and looked up at Wu Tao.

"The Jacobian determinant is not zero because the matrix is ​​directly closed in the current dimension, right?"

Wu Tao nodded.

"Yes, that's why space is distorted."

"What if we don't let it close immediately?"

Wu Tao paused for a moment, and the lollipop in his mouth stopped spinning.

"Not closed? If it's not closed, the mapping process is incomplete."

Chen Zhuo stood up and walked to the blackboard.

He picked up the half piece of chalk that Wu Tao had just placed in the chalk tray, and next to the divergent integral that Wu Tao had written, he rewrote the discrete matrix that he had just conceived.

Then, he added a small variable t to the upper right corner of the matrix.

"Introduce a virtual time variable."

Chen Zhuo's voice was calm as the chalk continued to move across the blackboard.

"We don't require the matrix to close immediately in the spatial dimension. Instead, we allow it to be dynamically compensated during the mapping process, with the imaginary time t." Wu Tao suddenly stood up and stared intently at the blackboard.

"As long as the limit of compensation is zero when the imaginary time approaches infinity, the Jacobian determinant will still be zero on the final observation surface."

Chen Zhuo turned his head and looked at Wu Tao.

"There is no distortion in space."

The meeting room was so quiet you could hear someone breathing.

Wu Tao stood still, his eyes fixed on the matrix with imaginary time variables on the blackboard, his brain frantically verifying the feasibility of this new path.

Suddenly, Wu Tao yanked the lollipop out of his mouth.

"Damn! Dynamic time compensation!"

Wu Tao took two steps to the blackboard and snatched the chalk from Chen Zhuo's hand.

"By substituting t into the partial differential equation, the divergence term at the singularity can be directly canceled out by the imaginary time term!"

Wu Tao's voice trembled with excitement. He didn't even bother to wipe the blackboard. He found a slightly cleaner spot and began to frantically deduce the solution following Chen Zhuo's train of thought.

A string of brand-new formulas poured out like a flowing stream.

Chen Zhuo took two steps back to make room for Wu Tao.

He watched as Wu Tao was practically glued to the blackboard, not even bothering to replace the broken piece of chalk, just gripping the end and continuing to write. "No problem! Absolutely no problem!"

Wu Tao muttered to himself as he wrote.

"Damn it, they're stuffing the time dimension into the topological space as a compensation term..."

Chen Zhuo pulled out a chair and sat down again.

He didn't interrupt. Although he had a direction, proving the entire virtual time compensation process in a complete and rigorous manner still required a huge amount of computation.

After about half an hour, Wu Tao finally put down his pen.

There wasn't a single blank space left on the blackboard.

Wu Tao turned around, let out a long breath, and his face, which was originally full of fatigue, was now flushed with excitement.

"The first phase of the singularity has been crossed."

Wu Tao threw the last bit of chalk dust in his hand into the trash can, patted the dust off his hands, walked to the table, and gulped down a large mouthful of cold water.

He turned to look at Chen Zhuo and couldn't help but laugh and curse.

"What's wrong with your brain? How could you come up with such a shady idea as building a bridge on a cliff?"

Chen Zhuo took a new sheet of draft paper and drew a few auxiliary lines on it.

"The bridge is up, but you still need to build the piers."

Chen Zhuo didn't even raise his head as he spoke.

"Proving the convergence of virtual time compensation is quite a task."

"That's physical labor. As long as you're on the right track, it's just a matter of pulling a few all-nighters."

Wu Tao pulled out a chair and sat down, looking refreshed.

"Today, let's derive the characteristic equation of this compensation term."

Chen Zhuo pushed the draft paper in his hand over.

"You derive the partial differential expansion on the left side, and I'll calculate the convergence limit on the right side," Chen Zhuo said.

Wu Tao took the paper and pen without any hesitation.

"Okay, after you finish calculating this part, I'll treat you to lunch at the cafeteria."

The two stopped wasting words and began to perform tedious calculations on paper.

The meeting room fell silent again.

In mathematics, in front of the blackboard and scratch paper, there is only right and wrong, only what can be calculated and what cannot be calculated.

A few days later, one afternoon.

USTC Institute of Applied Physics, Underground Wind Tunnel Laboratory.

This place lacks the quiet, old-fashioned atmosphere of the School of Mathematical Sciences.

The spacious underground space was brightly lit, with rows of tall server racks lined up against the walls, their indicator lights flashing.

Chen Zhuo took off his coat and put on a white work uniform issued by the laboratory.

The clothes were a little too big for him, and the cuffs were rolled up twice.

He stood behind the explosion-proof glass of the main control unit, and through the glass, he could clearly see the small test wind tunnel a few meters long not far away. Zhang Yuan, wearing dark gray overalls and a pair of soundproof earmuffs around his neck, was adjusting parameters in front of the main control unit.

"The matrix algorithm has been burned into the underlying test board."

Zhang Yuan stared at the monitor in front of him and tapped a few times on the keyboard.

"The preset broad error compensation is also on. Today we're mainly measuring the wind resistance and pressure distribution of the high-speed rail model train head at a speed of 200 kilometers per hour." Chen Zhuo nodded, his gaze falling on another monitor next to him.

He was confident in his algorithm.

Once the fan starts spinning and the sensors capture data, the pressure curves of each cross-section of the front of the car should be smoothly rendered on the screen after matrix processing.

"Attention all workstations, prepare to power on."

Zhang Yuan picked up the walkie-talkie, said something, and then pressed the green start button on the control panel.

A deep rumble instantly came from outside the bulletproof glass.

The turbofan motors inside the wind tunnel duct began to accelerate, the muffled sound causing a subtle, high-frequency vibration to appear on the anti-static floor beneath their feet. Chen Zhuo's gaze was fixed intently on the monitor.

As the wind speed increases, the data stream on the left side of the screen begins to scroll rapidly, and curves begin to be drawn on the coordinate axis on the right side.

However, just a few seconds later, Chen Zhuo frowned.

What appeared on the screen was not the smooth and elegant aerodynamic curve that I had expected.

It was a complete mess.

The red and green lines representing the data from each pressure measurement point look like a group of frightened snakes jumping wildly up and down on the coordinate axis.

The edges of the lines are full of fine burrs, and even in the nose cone of the car, where the force should be evenly distributed, there are several abrupt sharp peaks and cliffs.

The roar of the motor lasted for three minutes.

Zhang Yuan pressed the stop button.

The wind tunnel's rotation speed gradually decreased, and the roaring sound gradually turned into a low-frequency hum.

No one spoke in the control room.

Zhang Yuan stared at the jagged, chaotic data on the screen and rubbed his stiff neck.

"The compensation measures didn't cover everything."

Zhang Yuan turned his head, looked at Chen Zhuo, and said something very true.

Chen Zhuo did not refute.

He approached the monitor and examined the jagged lines closely. The screen's refresh rate was low, and the fine scan lines were visible upon close inspection. "It's not a problem with the algorithm itself."

Chen Zhuo pointed to several huge spikes that appeared periodically on the screen.

"This is a recurring disturbance. The conventional broad error compensation term filters it out as normal fluid disturbance, but it doesn't filter it completely."

Zhang Yuan pulled out a chair and sat down, casually picking up a water glass from the table.

"This small wind tunnel is an old piece of equipment in the institute."

Zhang Yuan took a sip of water, his tone carrying a hint of the engineers' helplessness towards outdated equipment.

"It's fine for rough tests, but once you use a high-precision matrix algorithm, all its hardware flaws are exposed."

Chen Zhuo turned around and looked at Zhang Yuan.

"The main turbofan's drive bearings are aging. When they rotate at tens of thousands of revolutions per minute, they generate mechanical vibrations in a specific frequency range. These vibrations are transmitted to the test model through the pipeline."

Zhang Yuan pointed to the spike on the screen.

"The periodic fluctuations you see are not wind pressure, but the motor shaking."

Zhang Yuan then pointed to the data stream.

"There are also sensors No. 2 and No. 4. These two batches of goods were not purchased in the same batch. The response time of sensor No. 2 is about three milliseconds slower than that of sensor No. 4. This causes the data received by your matrix in the same time slice to be misaligned. Once the time is misaligned, your equation will result in a bunch of garbled characters."

Chen Zhuo listened quietly.

Conventional algorithms usually reserve a universal constant as compensation, but when faced with this erratic old wind tunnel, that universal solution is obviously useless.

"Let's eat first."

Zhang Yuan glanced at the digital clock on the wall; it was almost one o'clock.

He took out two white foam lunch boxes from the cardboard box next to him. Because they had been there for a long time, the surface of the lunch boxes had lost its heat.

Zhang Yuan handed Chen Zhuo a lunchbox and a pair of disposable chopsticks, then took another one for himself and skillfully walked to the corridor outside the control room. Along the wall of the corridor was a row of old-fashioned cast-iron radiators.

Zhang Yuan squatted down next to the radiator, opened his lunchbox, and found braised beef with potatoes and stir-fried cabbage from the cafeteria. The food was cold, and the potatoes had a thin layer of oil on them.

He broke off his chopsticks and shoveled large mouthfuls of rice into his mouth, completely unconcerned.

Chen Zhuo came out with his lunchbox and, imitating Zhang Yuan, squatted down next to the radiator.

He opened his lunchbox, picked up a slightly hard piece of cabbage, and put it in his mouth.

Occasionally, a few lab technicians wearing the same white lab coats would walk by in the corridor, and they would nod to each other as a greeting.

Chen Zhuo chewed on rice while looking at the fire hydrant on the opposite wall.

The jagged curves on the screen were still replaying in his mind.

"Senior Brother Zhang."

Chen Zhu swallowed a mouthful of rice and spoke.

"Um?"

Zhang Yuan, with potatoes stuffed in his mouth, mumbled a reply.

"Are the wind tunnel, the sensor manuals, and the maintenance records from all the major overhauls all in the machine room?" Chen Zhuo asked. Zhang Yuan put down his chopsticks and turned to look at him.

"They're all there, locked in the filing cabinet. What do you need that for?"

"I need to know the specific tolerance data for this equipment."

Chen Zhuo used his chopsticks to stir the rice in the lunchbox.

"What is the specific vibration frequency of the main bearing? What is the actual response delay of each sensor in milliseconds? I need the precise values." Zhang Yuan swallowed the food in his mouth.

"You need precise values? You want to re-derive the compensation constant?"

"We can't use broad constants to apply this."

Chen Zhuo said.

"I need to write the device's inherent frequency band and hardware delay as characteristic values ​​directly into the underlying logic of the matrix, and create a customized filter for it."

Zhang Yuan was stunned for a moment.

Customized underlying filtering?

This means translating all the hardware parameters of mechanical engineering into constraints in algebraic matrices, a task that demands an absurdly high level of understanding across multiple disciplines.

"The data in the computer room is all very large," Zhang Yuan said.

"The mechanical drawings and parameter tables for this wind tunnel alone are as thick as a dozen telephone directories, and they're all pure data and structural diagrams." "It's okay."

Chen Zhuo picked up the last piece of beef, put it in his mouth, chewed it twice, swallowed it, and closed the empty lunchbox.

"Go ahead and do other things this afternoon, and give me your computer room key."

Zhang Yuan looked at the calm-looking boy squatting beside him.

He found that Chen Zhuo's attitude when faced with such experimental failures was surprisingly calm.

They didn't complain about the bad equipment or the dirty data; they simply took it for granted that they would find solutions to the real-world problems.

"OK."

Zhang Yuan quickly finished the rest of his rice, stood up, and patted his legs.

"After we eat, I'll take you to the server room. Just so you know, it's quite dusty in there."

Zhang Yuan took out a bunch of keys from his pocket and tossed them in his hand twice.

"If you get sleepy while watching, there are cots in the corner of the computer room."

Thanks.

Chen Zhuo stood up and threw the empty lunchbox into the trash can next to him.

1:30 PM.

Chen Zhuo entered the research institute's data processing room.

In the computer room, rows of metal cabinets stood in the shadows. Zhang Yuan helped him find all the data cards and thick bound books about the No. 1 wind tunnel and sensors.

Stacked on the table, they were a full half meter high.

Zhang Yuan gave a few instructions and then went back to continue inspecting the equipment.

Chen Zhuo was the only one left in the computer room.

He pulled out a folding chair, sat down, and opened the top book, "Turbine Fan Transmission Assembly Tolerance Table".

The paper was slightly yellowed, and it was covered with densely packed tables and mechanical drawings.

Chen Zhuo took out a pen and a stack of blank draft paper.

I laid out the draft paper, my gaze falling on the first line of data, and began to translate the cold, hard mechanical tolerances into lines of dynamic algebraic equations in my mind.