Blog

This post is revised by Chat GPT. This post presents a detailed walkthrough of the continuity equation from physical intuition to mathematical formalism. We aim to derive the equation step by step and explain all quantities involved. 1. Motivation: Conservation in Physical Systems Many physical quantities are conserved in nature. For instance: Mass in a closed system Charge in electromagnetic systems Energy in isolated systems Number of particles in fluid dynamics ...

This is a brief review for some topics of ECE220. Since this website cannot render asm code blocks properly, I used nasm instead — though this is actually LC-3 assembly code. I. Purpose and Description 1. Purpose This program serves as a review for: LC3 basics (including I/O) Run-time stack Recursion (LC3 implementation) Linked list (LC3 implementation) Binary tree (LC3 implementation) 2. Descriptions This file primarily serves as a direct LC-3 translation of a C program that prints a file directory structure. The content to be printed is pre-stored in specific data structures. The program prints this content to the console in a pre-order traversal with a specified output format (see desired output for details). To fully cover the previous purposes and simulate how C programs are translated into lower-level assembly language, all user-defined C functions are implemented on a run-time stack. Additionally, we’ve implemented our own “system call” equivalent to “OUT” and “PUTS” (corresponding to C’s printf() function). We name them “PRINT_A_CHAR” and “PRINT_A_STRING” respectively. These subroutines don’t utilize a run-time stack, just as typical system calls in real-world machines. Their implementation is exactly the same as how LC-3 machine implements “OUT” and “PUTS” in its System Space (from x0450 to x0462). The provided test case is absolutely not exhaustive. It was primarily generated by Gemini, and I’ve only revised some grammar mistakes to ensure it works properly for LC-3 syntax. Feel free to add any additional test cases. 3. Data Structure Design (1) Data Structure for Folders and Files Data Structure of Files in a Folder (linked list): ...

This post introduces the Gaussian integral, starting from basic integration and gradually building towards a probabilistic and linear algebraic perspective. The exposition proceeds from a purely mathematical derivation to a more intuitive understanding grounded in probability and linear algebra. I. The Fundamental Theorem The most fundamental Gaussian Integral is as follows: $$ \int_{-\infty}^{\infty} \exp \left(-\frac{1}{2}x^2 \right) dx = \sqrt{2\pi} $$ Equivalently, in the form of the standard Gaussian distribution: ...

Thanks to 子预. His article helped me a lot in understanding this topic. This is a brief (but not rigorous) proof of the Central Limit Theorem (CLT). The CLT is one of the most fundamental results in probability theory, showing that the sum of many independent random variables tends toward a normal distribution, regardless of their original distribution. Intuitively, the CLT explains why many natural phenomena follow the bell curve: individual randomness averages out into a predictable pattern. ...

This is a very basic summary of simple pendulum. I wrote this only for a brief review of LaTeX syntax. For more serious one, refer to this article. Torsion Pendulum We hope to find the moving function. i.e. $\theta$ and $t$. \begin{align*} \vec{\tau} &= -\kappa \vec{\theta} \ \vec{\tau} &= I \vec{\alpha} \end{align*} From definition, we know that: $$ \ddot{\theta} = \alpha $$ So, we obtain the final function: $$ \ddot{\theta} = - \frac{\kappa}{I} \theta $$ ...

Abstract This is my first blog, just for test whether everything gets in the way. Markdown Basics bold words and Italic and delete line list 1 list 2 The list 1 The list 2 a link to zjui website The below is an centaur to test the image display. Table: title1 title 2 title 3 data1 data2 data3 LaTeX Mathematics Formula $$ \int_0^{\infty} x^2 dx $$ ...