Math Tests

$$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}$$

The equation $(x_i \cdot x_j)^2$ is called kernel function and is often written as $k(x_i, x_j)$.

$$\arg\max_\alpha \sum_j \alpha_j - \frac{1}{2} \sum_{j,k} \alpha_j, \alpha_k y_j y_k (x_j \cdot x_k)$$ $$f(X) = \frac{1}{(2\pi)^{\frac{n}{2} |\Sigma|^{\frac{1}{2}}}} e^{ - \frac{1}{2} (X - \mu)^T \Sigma^{-1} (X - \mu)}$$ $$\mu_i = \sum_{j=1}^N \frac{p_{ij} x}{n_i} \\ \Sigma_i = \sum_{j=1}^N \frac{p_{ij} (x_j - \mu_i) (x_j - \mu_i)^T}{n_i}\\ w_i = \frac{n_i}{N}$$ $$S_i^{(t)} = \big \{ x_p : \big \| x_p - \mu^{(t)}_i \big \|^2 \le \big \| x_p - \mu^{(t)}_j \big \|^2 \ \forall j, 1 \le j \le k \big\}$$