First Post in this Blog

This first semester of AY 2018-19, I am taking App Physics 186 which focuses (I think) on Image and Video Processing. Finally, a course that will help me understand how the popular freeware Tracker works when I use it to analyze footage taken in our Physics 191/2 courses.

As a start to this new blog, I'd like to test whether it is compatible with $\LaTeX$, and so, I will be adding a few equations. Please don't mind them as it this only assures me that if it works, I know it is working.
$$\begin{align}\label{euler} \mathrm{e}^{\mathrm{i}\,x}&=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}x^{2n}+\mathrm{i}\,\sum_{n=0}^\infty\frac{(-1)^{n}}{(2n+1)!}x^{2n+1} \\ \label{fourier} \mathscr{F}\{g(\omega)\}&=\int_{-\infty}^{\infty}f(t)\mathrm{e}^{2\pi\mathrm{i}\,\omega t}\,\mathrm{d}t\end{align}$$
Here, $\eqref{euler}$ refers to the summation form of Euler's Formula, while $\eqref{fourier}$ refers to a simple Fourier Transform. Another nice equation to note is that to get the $n$-th Laguerre polynomial $\mathcal{L}_n(x)$, given by:
$$\begin{align}\label{laguerre}\mathcal{L}_n(x)=\sum_{m=0}^n\dfrac{(-1)^m\,n!}{(n-m)!\,(m!)^2}\,x^m\end{align}$$
Lastly, I'd like to test if this blog can also handle showing codes, thus I show one to implement $\eqref{laguerre}$ and plot it for $x\in[0,5]$.

import matplotlib.pyplot as plt
import numpy as np
from scipy.special import factorial
 
# nth Laguerre Polynomial
n = 9
 
# Initialize variables
x = np.arange(0, 5, 0.01)
Ln, y = [], []
 
# Create the coefficients of the terms in the polynomial
for m in range(n+1):
    Ln.append((-1)**m * factorial(n) / (factorial(n-m) * factorial(m)**2))
for f in x:
    s = 0
    for m in range(n+1):
        s += Ln[m] * f**m
    y.append(s)
    
plt.xlabel("$x$")
plt.ylabel("$L_m(x)$")
plt.plot(x, y)

This code enables one to plot the $n$-th Laguerre Polynomial.

Another nice code is one to obtain the prime factorization of some number, which follows.

import java.util.*;
public class primefac {
   public static void main(String[] args) {
       nprime j=new nprime();
       Scanner m=new Scanner(System.in);
       long a, b=2, c=m.nextLong();
       System.out.println("The prime factors of "+c+" are:");
       for(a=1;a<=c;a++) {
           if(c%a==0) {
               System.out.print(a+" ");
               c/=a;
               a=1;
           }
       }
   }
}

Hopefully this course will help me understand the techniques of processing images and videos so that I may find ways to implement my own version of Tracker.