Advanced Markdown Features with LaTeX Support

Mathematical Expressions

Inline Math

You can include inline mathematical expressions like $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ within text.

The area of a circle is $A = \pi r^2$ where $r$ is the radius.

Display Math

For larger expressions, use display mode:

$$ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} $$

$$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} $$

Matrix Examples

$$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix} $$

Algorithms and Equations

The Big-O notation for time complexity:

$$ O(f(n)) = \{g(n) : \exists c > 0, n_0 \geq 0 \text{ such that } 0 \leq g(n) \leq cf(n) \text{ for all } n \geq n_0\} $$

Advanced Formatting

Tables

Algorithm Best Case Average Case Worst Case Space Complexity
Quick Sort $O(n \log n)$ $O(n \log n)$ $O(n^2)$ $O(\log n)$
Merge Sort $O(n \log n)$ $O(n \log n)$ $O(n \log n)$ $O(n)$
Heap Sort $O(n \log n)$ $O(n \log n)$ $O(n \log n)$ $O(1)$

Code Blocks with Syntax Highlighting

#include <iostream>
#include <vector>
#include <algorithm>

class BinarySearch {
public:
    static int search(const std::vector<int>& arr, int target) {
        int left = 0, right = arr.size() - 1;

        while (left <= right) {
            int mid = left + (right - left) / 2;

            if (arr[mid] == target) {
                return mid;
            } else if (arr[mid] < target) {
                left = mid + 1;
            } else {
                right = mid - 1;
            }
        }

        return -1; // Element not found
    }
};
def fibonacci(n):
    """Calculate the nth Fibonacci number using dynamic programming"""
    if n <= 1:
        return n

    dp = [0] * (n + 1)
    dp[1] = 1

    for i in range(2, n + 1):
        dp[i] = dp[i-1] + dp[i-2]

    return dp[n]

# Time complexity: O(n), Space complexity: O(n)

Blockquotes

Important Note: Always consider the time and space complexity of your algorithms.

A well-optimized algorithm can make the difference between a solution that works for small inputs and one that scales to handle large datasets efficiently.

Definition Lists

Algorithm : A step-by-step procedure for solving a problem or completing a task.

Data Structure : A way of organizing and storing data so that it can be accessed and modified efficiently.

Time Complexity : A computational measure that describes the amount of time an algorithm takes to run as a function of the input size.

Nested Lists with Math

  1. Sorting Algorithms

    • Comparison-based sorts:
      • Bubble Sort: $O(n^2)$ average and worst case
      • Quick Sort: $O(n \log n)$ average, $O(n^2)$ worst case
      • Merge Sort: $O(n \log n)$ guaranteed
    • Non-comparison sorts:
      • Counting Sort: $O(n + k)$ where $k$ is the range
      • Radix Sort: $O(d \cdot (n + k))$ where $d$ is digits

  2. Graph Algorithms

    • Shortest Path:
      • Dijkstra's Algorithm: $O((V + E) \log V)$ with binary heap
      • Bellman-Ford: $O(VE)$ - handles negative weights
    • Minimum Spanning Tree:
      • Kruskal's: $O(E \log E)$
      • Prim's: $O(E \log V)$

Mathematical Proofs

Theorem: The time complexity of binary search is $O(\log n)$.

Proof: Let $T(n)$ be the time complexity for searching in an array of size $n$.

In each step, we eliminate half of the remaining elements:

  • After 1 comparison: $\frac{n}{2}$ elements remain
  • After 2 comparisons: $\frac{n}{4}$ elements remain
  • After $k$ comparisons: $\frac{n}{2^k}$ elements remain

We stop when $\frac{n}{2^k} = 1$, which gives us $k = \log_2 n$.

Therefore, $T(n) = O(\log n)$. â–¡

Chemical Formulas (Using LaTeX)

Water molecule: $\text{H}_2\text{O}$

Photosynthesis reaction: $$6\text{CO}_2 + 6\text{H}_2\text{O} + \text{light energy} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + 6\text{O}_2$$

Complex Mathematical Expressions

Fourier Transform: $$\mathcal{F}\{f(t)\} = \int_{-\infty}^{\infty} f(t) e^{-2\pi i \xi t} dt$$

Taylor Series: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$

Probability Density Function (Normal Distribution): $$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

Links and Images

For more information, visit GitHub or check out the official documentation.

You can also include images (this would work if you had an image file):

This demonstrates the enhanced markdown capabilities with full LaTeX support, advanced formatting, and syntax highlighting.