stjorthotic Chapter 49: Learning points skyrocketed. Isn't this just a mental arithmetic problem?
Chapter 49: Learning points skyrocketed. Isn't this just a mental arithmetic problem?
Lu Feng closed his laptop and casually unplugged the USB drive, putting it in his pocket.
As soon as I turned around and went out with my backpack on, five senior students swarmed around me.
Li Hao walked at the front, took out his phone, and the screen was on the QQ friend adding interface.
"Hey Lu Feng, can I leave you my contact information? We can create a group chat for easier communication later."
Lu Feng took out his phone and read out a string of numbers.
Back in 2012, QR code scanning wasn't widespread yet; everyone manually entered their QQ numbers.
This made things difficult for Lu Feng, as he had to give out his QQ number several times.
A series of "beep" notification sounds rang out, and five friend requests popped up one after another.
Lu Feng clicked to approve each request and then put his phone back in his pocket.
Chen Jing put away her phone and walked over to Lu Feng, her height coming up to his shoulder.
"Junior, this time we're joining forces, we must aim for the national scholarship!"
She clenched her fist, full of enthusiasm.
Lu Feng casually chimed in, "Did you participate in the national competition before, senior?"
Chen Jing's fist-waving motion froze in mid-air before she slowly lowered it.
The people next to them paused for a moment, and the atmosphere suddenly became a little awkward.
"I have participated before..." Chen Jing scratched her head, lacking confidence. "Last year, I teamed up with Li Hao and the others to compete in the provincial competition, but we didn't get a ranking, and we didn't even get a ticket to the national competition."
The situation is clear.
These third-year students, who are the main force, are actually a group of seasoned veterans who have experienced defeats.
Lu Feng sighed inwardly.
I had originally hoped to learn some practical experience from them, and understand the judges' preferences and the unspoken rules of paper format.
In hindsight, I was just overthinking it.
This isn't a powerful alliance; it's like letting him solo queue and carry five noobs.
"It's alright, we'll proceed cautiously this year," Lu Feng casually reassured him.
Let's offer some words of comfort, otherwise what if the strike breaks out?
The group walked side by side toward the cafeteria.
Along the way, Wang Zhe and Liu Yu kept asking various professional questions about reinforcement learning.
"Junior, what's the best way to set the update frequency for the Q table? Won't real-time updates cause the computing power to crash?" Wang Zhe pressed.
"No need for real-time updates," Lu Feng replied as he walked. "Set up an experience replay pool, store the state transition data of each vehicle in it, and batch extract a small batch for training. This will break the data correlation and reduce the computing load."
Liu Yu adjusted his glasses and followed up, "Then how do we balance exploration and exploitation? Won't the ε-greedy strategy lead to excessively slow convergence in the later stages?"
"Introduce a dynamically decaying ε value." Lu Feng stepped onto the cafeteria steps. "In the early stages, set a larger value to fully explore the model. As the number of iterations increases, the ε value decays exponentially, allowing the model to gradually favor the use of known optimal strategies."
Two sentences that hit the nail on the head.
The three male students from the Computer Science Department nodded repeatedly, completely abandoning their previous airs of seniority.
In the face of absolute technological superiority, age is utterly insignificant.
It's because they're better than you.
After lunch, Lu Feng parted ways with them and headed straight for the library.
Push open the glass door of the second-floor reading room.
A wave of cold air hit me.
Lin Wan was sitting behind the management desk, looking down at a thick book of postgraduate entrance exam math review.
She looked up when she heard the door open.
Upon seeing Lu Feng, she closed her book and pointed to the second-to-last row by the window.
"I saved it for you."
Lu Feng walked over and tapped on the table.
"Thanks, senior."
Lin Wan waved her hand, reopened the documents, and pricked the draft paper twice with her pen, looking somewhat irritated. This question was still difficult.
Lu Feng walked to his designated seat.
The tabletop was spotless, without a speck of dust, clearly having been wiped clean on purpose.
He pulled out a chair and sat down, thinking he should first figure out those mathematical conjectures.
Before Lu Feng could even produce the blueprints for the cutting-edge technology...
"Beep beep~"
My phone vibrated twice in my pocket.
Lu Feng took out his phone and opened QQ.
It was a message from Lin Wan.
Two pictures.
The first picture shows a handwritten postgraduate entrance exam math problem. The handwriting is neat and beautiful, but the side is covered with messy auxiliary lines.
"Calculate the double integral: ∬(D) e^(-x^2-y^2) dxdy, where D is the closed region bounded by the circle x^2+y^2=R^2."
A variant of the classic Poisson integral.
The second image is an ordinary differential equation.
"Find the general solution: y'' - 2y' + y = x*e^x."
Lu Feng glanced at the question and his fingers flew across the keyboard.
"Question 1: π(1 - e^(-R^2))"
"Question 2: y = (C1 + C2*x)*e^x + (1/6)*x^3*e^x"
Click send.
Less than ten seconds later, the other end replied with a meme full of question marks.
Lin Wan: "???"
Lin Wan: "That's it?"
Lu Feng: "Yes, that's the answer."
Lin Wan: "The process! I need the solution process, bro!"
Lu Feng leaned back in his chair, looking at the text on the screen, and could even imagine the frantic look on Lin Wan's face at the front desk.
I was internally thinking, "Does this even need to be solved?"
Lu Feng: "Does this question require a process? Isn't this something you can solve mentally?"
A muffled "bang" came from the control panel.
Lu Feng turned his head and saw Lin Wan banging her forehead on the table, her shoulders twitching.
This really broke through their defenses.
For a senior student preparing for postgraduate entrance exams, this kind of "dimensional reduction attack" is the most devastating.
She spent half an hour biting her pen and calculating, but she still couldn't figure out the upper and lower limits of the integration range.
As a result, they glanced at it and gave the answer directly.
It also included a sarcastic remark about "mental math problems".
Lu Feng looked away and pulled over a blank sheet of draft paper.
Well, fine, I'll just list them out.
Let's start with the polar coordinate transformation for the first question.
Let x = r*cosθ, y = r*sinθ.
Jacobian determinant J = r.
The integration region D is transformed into 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π.
The original expression is: ∫(0 to 2π) dθ∫(0 to R) e^(-r^2)* r dr.
= 2π*[-1/2 * e^(-r^2)] (from 0 to R).
=π(1 - e^(-R^2)).
The five characters are clear and easy to understand.
Next, write the characteristic equation method for the second question.
The characteristic equation r^2 - 2r + 1 = 0 gives a double root r = 1.
The homogeneous general solution Y = (C1 + C2*x)*e^x.
Let the particular solution be y* = x^2 *(A*x + B)* e^x.
Substituting into the original equation, we get A = 1/6 and B = 0.
The general solution is y = Y + y*.
There was no unnecessary chatter; every step went straight to the core.
Lu Feng took out his phone, snapped a photo, and sent it over in its original form.
After sending the message, he put his phone on silent and tossed it back onto the table.
Five minutes passed.
The phone screen lit up.
Lu Feng glanced at it.
Lin Wan: "I get it! Polar coordinate substitution is brilliant!"
Lin Wan: "I always forget to multiply the Jacobian determinant, no wonder my calculations are always wrong."
Lin Wan: "I also understand the special solution method. In the case of repeated roots, it is indeed necessary to multiply by the square of x."
Lin Wan: "Thank you so much, junior, for saving my life! [clasped hands]"
After ending the chat, Lu Feng composed himself and pulled a stack of brand-new A4 draft paper from his bag.
In my mind, the black technology blueprint, exuding pure mathematical aura, slowly unfolded.
Part 1: Proof of Gödel's incompleteness theorems by transfinite induction.
The mathematical conjecture is on a completely different level from the previous engineering drawings.
Engineering problems have physical references, while mathematical conjectures rely solely on pure logic.
This thing is like walking a tightrope on the edge of human rationality.
Lu Feng removed the cap from his pen and wrote down the first set of axioms for a formal system on the paper.
Peano's axiomatic system of arithmetic is the cornerstone of all theories of natural numbers.
He attempted to construct a transfinite ordinal system that includes all computable functions.
Each mathematical proposition is mapped to a unique Gödel number.
The pen tip moved quickly across the paper, and rows of logic symbols appeared on the page.
Half an hour later, the pen stopped.
It's a dead end.
Transfinite induction encounters an insurmountable cardinal barrier when dealing with self-referential propositions.
When the proposition states "This proposition is unprovable in this axiomatic system".
Once such an undecidable proposition is introduced, the entire derivation process will fall into an infinite logical loop of regression.
If it is proven to be true, then the system is inconsistent.
If it is proven false, then the system is incomplete.
Lu Feng crumpled the full draft paper into a ball and threw it aside.
Let's try a different approach.
An attempt was made to extend the axiomatic system using the compulsion method of model theory.
I'll pick up my pen again and construct a new Boolean value model.
By adding generic sets, we are trying to find consistency in a larger universe.
Time passed by, second by second.
Sheets of draft paper covered the table.
Two hours later, Lu Feng put down his pen and rubbed his sore wrist.
Or not.
While the forced method can extend the model, it cannot guarantee the continuity of the transfinite induction steps at the limit ordinal.
After infinite iterations, Gödel numbers will produce singularities that cannot be eliminated.
This is a bottomless pit.
Lu Feng leaned back in his chair and closed his eyes.
Bring up the system panel.
Current learning value: 7450
Lu Feng suddenly opened his eyes and stared at the number.
After buying the blueprints, I only had 4050 left. After studying the draft for two and a half hours, the price increased by more than 3000!
Such purely theoretical mathematical conjectures are several orders of magnitude more difficult than their engineering applications.
The more difficult the question, the higher the brain activity level, and the more aggressive the reward given by the system.
This rate of return is faster than robbing money.
The high level of difficulty brought high rewards.
Just do it.
Lu Feng sat up straight again and pulled over a brand new sheet of draft paper.
Since traditional set theory methods are not feasible, we should introduce category theory.
Treat formal systems as a topological category and use the tools of layer theory to deconstruct Gödel numbers.
The pen tip fell again, this time much slower than before.
The derivation of each symbol requires a great deal of mental effort.
He transformed undecidable propositions into obstacle classes in homological algebra.
Instead of focusing on the truth or falsity of the proposition itself, we calculate the non-zero elements of this obstacle class in the homology group.
By using spectral sequences to perform step-by-step calculations, we can forcibly bypass that logical dead loop.
The sky outside the window gradually darkened.
The library's motion-sensor lights came on, casting incandescent light onto the draft paper.
Lu Feng wrote down the last equation for isomorphic mapping.
The logical loop is complete.
Gödel's incompleteness theorems have a near-perfect constructive proof within the framework of transfinite induction.
[Ding! Detected that the host has completed the derivation of the first part of the high-tech blueprint!]
[Reward: 2600 learning points!]
Current learning value: 10050
It's over 10,000.
Lu Feng let out a long breath.
This feeling is even more comfortable than drinking a whole bottle of ice-cold cola on a sweltering summer day.
He folded up the scattered draft papers on the table and tucked them into his textbook.
I glanced at my watch; it was 7:10 PM.
It's time to go back.