Mathematical Expressions and Visual Content Test

This post demonstrates the rendering capabilities for various mathematical expressions, images, and visual content on the blog.

Basic Mathematical Expressions

Inline Math

Here’s some inline math: $E = mc^2$, and the Pythagorean theorem:

$$\begin{align} a^2 + b^2 &= c^2 \\ b^2 + c^2 &= d^2 + e^2 \end{align}$$

Block Math

Here are some more complex mathematical expressions:

$$\begin{align} \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \\ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \end{align}$$

Advanced Mathematical Concepts

Matrix Operations

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

Probability and Statistics

The probability density function of a normal distribution:

$$f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Bayes’ theorem:

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Calculus

Fundamental theorem of calculus:

$$\int_a^b f'(x) dx = f(b) - f(a)$$

Chain rule for derivatives:

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Machine Learning Mathematics

Neural Network Forward Pass

$$z^{[l]} = W^{[l]} a^{[l-1]} + b^{[l]}$$ $$a^{[l]} = \sigma(z^{[l]})$$

Loss Functions

Mean squared error:

$$\mathcal{L}_{MSE} = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$

Cross-entropy loss:

$$\mathcal{L}_{CE} = -\frac{1}{n} \sum_{i=1}^{n} \sum_{c=1}^{C} y_{i,c} \log(\hat{y}_{i,c})$$

Optimization

Gradient descent update rule: $\theta_{t+1} = \theta_t - \alpha \nabla_\theta \mathcal{L}(\theta_t)$

Adam optimizer:

$$m_t = \beta_1 m_{t-1} + (1-\beta_1) g_t$$ $$v_t = \beta_2 v_{t-1} + (1-\beta_2) g_t^2$$ $$\hat{m}_t = \frac{m_t}{1-\beta_1^t}, \quad \hat{v}_t = \frac{v_t}{1-\beta_2^t}$$ $$\theta_{t+1} = \theta_t - \frac{\alpha}{\sqrt{\hat{v}_t} + \epsilon} \hat{m}_t$$

Set Theory and Logic

Set Operations

$A \cup B = \{x : x \in A \text{ or } x \in B\}$ $A \cap B = \{x : x \in A \text{ and } x \in B\}$ $A \setminus B = \{x : x \in A \text{ and } x \notin B\}$

Logical Expressions

$\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : y > x$ $\neg (P \land Q) \equiv (\neg P) \lor (\neg Q)$

Complex Mathematical Structures

Fourier Transform

The Fourier Transform reveals the frequency content of a signal by decomposing it into its constituent sinusoidal components:

$$\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$$

This animated visualization shows the intuitive understanding of the Fourier Transform:

• Top Left: The combined time-domain signal being built up
• Top Right: Rotating phasors representing each frequency component
• Bottom Left: Individual sinusoidal components with different frequencies
• Bottom Right: The frequency spectrum showing the magnitude of each component

The rotating phasors demonstrate how each frequency component contributes to the overall signal, with the rotation speed corresponding to the frequency and the radius corresponding to the amplitude.

Taylor Series

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

Eigenvalue Decomposition

$A\mathbf{v} = \lambda\mathbf{v}$ where $\mathbf{v}$ is an eigenvector and $\lambda$ is the corresponding eigenvalue.

Citations Test

Academic References

Mathematical foundations are built upon rigorous proofs \cite{louDiscreteDiffusionModeling2024}. The understanding of discrete structures has evolved significantly \cite{gatDiscreteFlowMatching2024a}.

Visual Content Placeholders

Static Images

Note: In a real blog post, you would include images like:

• Mathematical diagrams and plots
• Algorithm flowcharts
• Neural network architectures
• Data visualizations

Animated Content

For animated content, you could include:

• GIFs showing mathematical transformations
• Interactive plots and graphs
• Algorithm step-by-step animations
• Mathematical concept demonstrations

Code Blocks with Math Comments

import numpy as np
import matplotlib.pyplot as plt

# Generate data for f(x) = x^2
x = np.linspace(-10, 10, 100)
y = x**2  # This represents the function f(x) = x²

# Plot the quadratic function
plt.figure(figsize=(8, 6))
plt.plot(x, y, 'b-', linewidth=2, label='$f(x) = x^2$')
plt.xlabel('$x$')
plt.ylabel('$f(x)$')
plt.title('Quadratic Function: $f(x) = x^2$')
plt.grid(True, alpha=0.3)
plt.legend()
plt.show()

Mathematical Tables

Function	Derivative	Integral
$x^n$	$nx^{n-1}$	$\frac{x^{n+1}}{n+1} + C$
$e^x$	$e^x$	$e^x + C$
$\ln(x)$	$\frac{1}{x}$	$x\ln(x) - x + C$
$\sin(x)$	$\cos(x)$	$-\cos(x) + C$
$\cos(x)$	$-\sin(x)$	$\sin(x) + C$

Conclusion

This test post demonstrates the blog’s capability to render:

1. Inline and block mathematical expressions using KaTeX

2. Complex mathematical notation including matrices, integrals, and summations

3. Academic citations with proper bibliography generation

4. Code blocks with mathematical comments

5. Tables with mathematical content

The rendering system successfully handles both simple expressions like $E = mc^2$ and complex multi-line equations with proper formatting and spacing.