For dynamical system $f: M \to M$, observation $\phi: M \to \mathbb{R}$, define delay map: $$F(x) = (\phi(x), \phi(f(x)), \dots, \phi(f^{2m}(x)))$$ where $m = \dim(M)$. Generically, $F$ is embedding. Implementation: $d \geq 2m+1$, $\tau$ optimal: $$\text{MI}(\tau) = \sum p(x_t, x_{t+\tau}) \log\frac{p(x_t, x_{t+\tau})}{p(x_t)p(x_{t+\tau})}$$ Vectors: $X_i = [x_i, x_{i+\tau}, \dots, x_{i+(d-1)\tau}]^T$ . For point clouds $X, Y$, diagrams $D_X, D_Y$: $$W_\infty(D_X, D_Y) \leq d_H(X, Y)$$ where $W_\infty(D_1, D_2) = \inf_{\eta} \sup_{x \in D_1} ||x - \eta(x)||_\infty$ with $\eta: D_1 \to D_2$ bijection. $p$-Wasserstein between diagrams $D_1, D_2$: $$W_p(D_1, D_2) = \left( \inf_{\eta} \sum_{x \in D_1} ||x - \eta(x)||^p \right)^{1/p}$$ Kernel via heat diffusion: $$k(D_1, D_2) = \frac{1}{8\pi t} \sum_{p \in D_1} \sum_{q \in D_2} e^{-\frac{||p-q||^2}{8t}}$$ Sliced-Wasserstein: Project to lines $\theta \in S^1$: $$SW(D_1, D_2) = \int_{S^1} W_1(\pi_\theta(D_1), \pi_\theta(D_2)) d\theta$$ Define persistence: $p_i = d_i - b_i$ Total persistence: $L = \sum_{i=1}^n p_i$ Normalized persistence: $P_i = \frac{p_i}{L}$ Persistent entropy: $$H = -\sum_{i=1}^n P_i \log_2 P_i$$ Persistence intervals ${(b_i, d_i)}_{i=1}^n$ Properties: Max: $H_{\max} = \log_2 n$ when $P_i = \frac{1}{n} \ \forall i$ , Min: $H_{\min} = 0$ when $\exists k$ such that $P_k = 1$, $P_i = 0 \ \forall i \neq k$ , Monotonic: Adding zero-length intervals leaves $H$ unchanged, Scale-invariant: $H(\alpha p_i) = H(p_i)$ for $\alpha > 0$

Partition function: $Z(q, \epsilon) = \sum_i \mu(B_i(\epsilon))^q$, Scaling: $Z(q, \epsilon) \sim \epsilon^{\tau(q)}$ as $\epsilon \to 0$, Legendre transform: $$\alpha(q) = \frac{d\tau}{dq}, \quad f(\alpha) = q\alpha - \tau(q)$$, Proof: From large deviations: $\Pr(\alpha_\epsilon \approx \alpha) \sim \epsilon^{-f(\alpha)}$, Recurrence matrix: $$R_{ij} = \Theta(\varepsilon - \lVert X_i - X_j \rVert)$$ where: $\Theta$: Heaviside function, $\varepsilon$: recurrence threshold, $X_i \in \mathbb{R}^d$: embedded vectors, Recurrence rate (RR): $$RR = \frac{1}{N^2} \sum_{i,j=1}^N R_{ij}$$, Determinism (DET): Let $P_{\ell} = \text{number of diagonal lines of length } \ell \text{ in } R$ $$DET = \frac{\sum_{\ell=\ell_{\min}}^{N} \ell \cdot P_{\ell}}{\sum_{\ell=1}^{N} \ell \cdot P_{\ell}}$$, Laminarity (LAM): Let $P_{v} = \text{number of vertical lines of length } v \text{ in } R$, $$LAM = \frac{\sum_{v=v_{\min}}^{N} v \cdot P_{v}}{\sum_{v=1}^{N} v \cdot P_{v}}$$, Typical parameters: $\ell_{\min} = 2$ (minimal diagonal line length), $v_{\min} = 2$ (minimal vertical line length), Interpretation: $RR \in [0,1]$: density of recurrence points, $DET \in [0,1]$: predictability of system, $LAM \in [0,1]$: presence of laminar states For graph $G$, normalized Laplacian $L = I - D^{-1/2}AD^{-1/2}$, eigenvalues $0 = \lambda_0 \leq \lambda_1 \leq \dots$, Cheeger constant: $h(G) = \min_{S \subset V} \frac{|\partial S|}{\min(\text{vol}(S), \text{vol}(V\setminus S))}$, Inequality: $\frac{\lambda_1}{2} \leq h(G) \leq \sqrt{2\lambda_1}$, Proof: Rayleigh quotient minimax. Cost matrix $D_{ij} = ||x_i - y_j||$, recurrence: $$C(i,j) = D_{ij} + \min{C(i-1,j), C(i,j-1), C(i-1,j-1)}$$, with $C(0,0) = 0$, $C(i,0) = C(0,j) = \infty$, Sakoe-Chiba band: $|i-j| \leq w$, complexity $O(w \cdot \min(m,n))$

Based on Kolmogorov complexity $K(x)$: $$d(x,y) = \frac{\max{K(x|y), K(y|x)}}{\max{K(x), K(y)}}$$, NCD approximation: $C$ compressor, $$NCD(x,y) = \frac{C(xy) - \min{C(x), C(y)}}{\max{C(x), C(y)}}$$, Properties: $0 \leq NCD \leq 1 + \epsilon$, $NCD(x,x) \approx 0$, First digit law: $P(d) = \log_{10}\left(1 + \frac{1}{d}\right)$ for $d \in {1,\dots,9}$, Derivation: Scale invariance → unique solution: $P(S) = \int_S \frac{1}{x \ln 10} dx$, Test statistic: $\chi^2 = \sum_{d=1}^9 \frac{(n_d - nP(d))^2}{nP(d)} \sim \chi^2_8$, For sets $A,B$, permutations $\pi_1,\dots,\pi_k$: $$\hat{J}(A,B) = \frac{1}{k}\sum_{i=1}^k \mathbb{I}[\min(\pi_i(A)) = \min(\pi_i(B))]$$, Variance: $\text{Var}(\hat{J}) = \frac{J(1-J)}{k}$, Proof: $\Pr(\min(\pi(A)) = \min(\pi(B))) = \frac{|A \cap B|}{|A \cup B|} = J(A,B)$, Joint probability: $$p(\mathbf{w}, \mathbf{z}, \theta|\alpha, \beta) = \prod_d p(\theta_d|\alpha) \prod_n p(z_{dn}|\theta_d) p(w_{dn}|z_{dn},\beta)$$, M-step for $\beta$: $\beta_{kw} \propto \sum_{d,n} \phi_{dnk} \mathbb{I}[w_{dn} = w]$, Topic entropy: $H_d = -\sum_k \theta_{dk} \log \theta_{dk}$

Lyapunov exponent: $\lambda = \lim_{t \to \infty} \frac{1}{t} \ln \frac{|\delta(t)|}{|\delta(0)|}$; Wolf's algorithm: $\lambda \approx \frac{1}{M \Delta t} \sum_{k=1}^{M} \ln \frac{L'(t_k)}{L(t_{k-1})}$; Hurst exponent: $\frac{R(n)}{S(n)} \sim c n^{H}$, where $R(n) = \max_{1 \le k \le n} \sum_{j=1}^{k} (X_j - \langle X \rangle_n) - \min_{1 \le k \le n} \sum_{j=1}^{k} (X_j - \langle X \rangle_n)$, Isomap: Geodesic distance $d_G(i,j) = \min_{P} \sum_{(u,v)\in P} ||x_u - x_v||$, MDS: minimize $\sum_{ij} (d_G(i,j) - ||y_i - y_j||)^2$, LLE: Reconstruct weights $w_{ij}$ minimizing: $$\Phi(W) = \sum_i ||x_i - \sum_{j \in N(i)} w_{ij}x_j||^2, \quad \sum_j w_{ij} = 1$$, Koksma-Hlawka: $|\hat{I} - I| \leq V(f) D_N^*$ For Sobol sequence: $\displaystyle D_N^* = O!\left(\frac{(\log N)^d}{N}\right)$, Monte Carlo error: $\displaystyle O!\left(\frac{\sigma_f}{\sqrt{N}}\right)$ where $\displaystyle \sigma_f^2 = \mathbb{E}[(f-\mu_f)^2]$, Coverage: $\displaystyle \lim_{N \to \infty} D_N^* \to 0$, QMC achieves $\displaystyle D_N^* = o(N^{-1/2})$, Ensemble: Let $\displaystyle \mathcal{H} = {h_t: \mathcal{X} \to \mathcal{Y}}_{t=1}^T$, predictions $\displaystyle \hat{y}_i^t = h_t(x_i)$, true labels $y_i \in \mathcal{Y}$. Bayesian fusion: $\displaystyle p(y|x, \mathcal{D}) \propto p(y) \prod_{t=1}^T p(h_t(x)|y, \mathcal{D})$.