KaTeX renders mathematical notation beautifully in web pages. This guide covers common patterns, syntax, and tips for writing mathematical expressions. Use Case Use KaTeX when you need to: Display mathematical formulas and equations Show scientific notation Write technical documentation with math Create educational …

Asymmetric (public-key) cryptography with mathematical foundations, including RSA, ECC, and key exchange protocols. Mathematical Foundation Public-Key Cryptosystem A public-key cryptosystem consists of: $$ \begin{aligned} (pk, sk) &\leftarrow KeyGen() \ C &= Encrypt(pk, M) \ M &= Decrypt(sk, C) …

Cryptographic hash functions with mathematical properties and practical implementations. Mathematical Definition A cryptographic hash function $H: {0,1}^* \to {0,1}^n$ maps arbitrary-length input to fixed-length output. $$ H(m) = h $$ Where: $m$: Message (arbitrary length) $h$: Hash digest (fixed length, e.g., 256 …

Digital signature algorithms with mathematical foundations. Mathematical Definition A digital signature scheme consists of: $$ \begin{aligned} (pk, sk) &\leftarrow KeyGen() \ \sigma &= Sign(sk, M) \ {0,1} &= Verify(pk, M, \sigma) \end{aligned} $$ Properties: Authentication: Proves message from holder of …

Deep Neural Network policy learning with mathematical foundations. Policy Gradient Methods Policy Parameterization Policy $\pi_\theta(a|s)$ parameterized by neural network with weights $\theta$. Objective Function Maximize expected return: $$ J(\theta) = \mathbb{E}{\tau \sim \pi\theta}\left[\sum_{t=0}^{T} \gamma^t …

Key Derivation Functions (KDFs) for password hashing and key derivation. Password-Based KDFs PBKDF2 (Password-Based Key Derivation Function 2) $$ DK = PBKDF2(password, salt, iterations, dkLen) $$ Algorithm: $$ \begin{aligned} T_i &= F(password, salt, iterations, i) \ F(password, salt, c, i) &= U_1 \oplus U_2 …

Numerical methods, regularization techniques, and useful concise algorithms.

Q-Learning algorithm theory with mathematical foundations. Markov Decision Process (MDP) An MDP is defined by the tuple $(S, A, P, R, \gamma)$: $S$: Set of states $A$: Set of actions $P$: Transition probability $P(s'|s,a)$ $R$: Reward function $R(s,a,s')$ $\gamma \in [0,1]$: Discount factor Value Functions State Value …

Quantum mechanics basics, common mathematics, and most useful methods.

Fundamental concepts and mathematical tools for signal processing, including Fourier analysis, convolutions, and correlations.

Symmetric encryption algorithms with mathematical foundations and practical implementations. Mathematical Definition Symmetric encryption uses the same key for encryption and decryption: $$ \begin{aligned} C &= E_K(P) \quad \text{(Encryption)} \ P &= D_K(C) \quad \text{(Decryption)} \end{aligned} $$ Where: $P$: …

Tensor mathematics fundamentals and backpropagation theory with detailed mathematical derivations. Tensor Basics What is a Tensor? A tensor is a generalization of scalars, vectors, and matrices to higher dimensions: Scalar (0D tensor): $x \in \mathbb{R}$ Vector (1D tensor): $\mathbf{x} \in \mathbb{R}^n$ Matrix (2D …