Discrete Math

$$\overline{\bar{A}} = A$$ $$\overline{A ∪ B} = \bar{A} ∩ \bar{B}$$ $$\overline{A ∩ B} = \bar{A} ∪ \bar{B}$$ Let $f(x) = a_nx^n + a_{(n-1)}x^{(n-1)} + ... + a_1x + a_0.$ Then $f(x) = O(x^n).$ $${n \choose r}$$ $$\lceil N/k \rceil$$ $$P(n,r) = \frac{n!}{(n - r)!}$$ $$C(n,r) = \frac{n!}{r!(n - r)!}$$ $$(x + y)^n = \sum_{k=0}^{n} C(n,k)x^{n-k}y^k$$ $$n^r$$ $$\sum_{k=0}^{n} C(n,k) = 2^n$$ $$C(m + n,r) = \sum_{k=0}^{n} C(m,r - k)C(n,k)$$ $$\sum_{k=0}^{n} (-1)^k C(n,k) = 0$$ $$a_n = c_1a_{n-1} + c_2a_{n-2} + \cdots + c_k a_{n-k}$$ $$a_n = r^n$$ $$a_n = c_1r^{n-1} + c_2r^{n-2} + \cdots + c_k r^{n-k}$$ $$r^k - c_1r^{k-1} - c_2r^{k-2} - \cdots - c_{k-1}r - c_k = 0$$ $$a_n = a_{n-1} + a_{n-2}^2 \textrm{ is not an LHRR because}$$ $$H_n = 2H_{n-1} + 1 \textrm{ is not an LHRR because}$$ $$B_n = nB_{n-1} \textrm{ is not an LHRR because}$$ $$\{a_n\} \textrm{ is a solution to an LHRR of degree } 2$$ $$a_n = c_1 a_{n-1} + c_2 a_{n-2}$$ $$r^2 - c_1r - c_2 = 0$$ $$a_n = \alpha_1 r_1^n + \alpha_2 r_2^2$$ $$r^2 - c_1r - c_2 = 0$$ $$a_n = \alpha_1 r_0^n + \alpha_2 nr_0^2$$ $$\{a_n\} \textrm{ is a solution to an LHRR of degree } k$$ $$a_n = c_1 a_{n-1} + c_2 a_{n-2} + \cdots + c_k a_{n-k}$$ $$r^k - c_1r^{k-1} - \cdots - c_k = 0$$ $$a_n = \alpha_1 r_1^n + \alpha_2 r_2^2 + \cdots + \alpha_k r_k^n$$ $$n^m - C(n,1)(n - 1)^m + C(n,2)(n - 2)^m - \cdots + (-1)^{n-1}C(n,n - 1)^m$$ $$D_n = n!\left[1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + (-1)^n\frac{1}{n!}\right]$$ $$\hat{y}=\theta_0+\theta_1x_1+\theta_2x_2+\cdots +\theta_nx_n$$ $$\hat{y}=h_\theta(\textbf{x})=\theta^T\cdot \textbf{x}$$ $$MSE(\textbf{X},h_\theta)=\frac{1}{m}\sum_{i=1}^{m}\left(\theta^T\cdot \textbf{x}^{(i)}-y^{(i)}\right)^2$$ $$\hat{\theta}=\left(\textbf{X}^T\cdot\textbf{X}\right)^{-1}\cdot\textbf{X}^T\cdot\textbf{y}$$ $$\frac{\delta}{\delta\theta_j}=\frac{2}{m}\sum_{i=1}^{m}\left(\theta^T\cdot\textbf{x}^{(i)}-y^{(i)}\right)x_j^{(i)}$$ $$\nabla_\theta MSE(\theta)=\begin{pmatrix} \frac{\delta}{\delta\theta_0}MSE(\theta) \\ \frac{\delta}{\delta\theta_1}MSE(\theta) \\ \vdots \\ \frac{\delta}{\delta\theta_n}MSE(\theta) \end{pmatrix} =\frac{2}{m}\textbf{X}^T\cdot(\textbf{X}\cdot\theta-\textbf{y})$$ $$\theta^{(\text{next step})}=\theta-\eta\nabla_\theta MSE(\theta)$$ $$J(\theta)=MSE(\theta)+\alpha\frac{1}{2}\sum_{i=1}^{n}\theta_i^2$$ $$\hat{\theta}=\left(\textbf{X}^T\cdot\textbf{X}+\alpha\textbf{A}\right)^{-1}\cdot\textbf{X}^T\cdot \textbf{y}$$ $$J(\theta)=MSE(\theta)+r\alpha\sum_{i=1}^{n}\left|\theta_i\right|+\frac{1-r}{2}\alpha\frac{1}{2}\sum_{i=1}^{n}\theta_i^2$$ $$\hat{p}=h_\theta(\textbf{x})=\sigma\left(\theta^T\cdot\textbf{x}\right)$$ $$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}\left[y^{(i)}\text{log}\left(\hat{p}^{(I)}\right)+\left(1-y^{(i)}\right)\text{log}\left(1-\hat{p}^{(I)}\right)\right]$$ $$\frac{\delta}{\delta\theta_j}=\frac{1}{m}\sum_{i=1}^{m}\left(\sigma\left( \theta^T\cdot \textbf{x}^{(i)}-y^{(I)} \right) \right)x_j^{(I)}$$ $$s_k(\textbf{x})=\left(\theta^{(k)}\right)^T\cdot\textbf{x}$$ $$\hat{p}_k=\sigma(\textbf{s}(\textbf{x}))_k=\frac{exp(s_k(\textbf{x}))}{\sum_{j=1}^{K}exp(s_j(\textbf{x}))}$$ $$\hat{y}=\underset{k}{\text{argmax }}\sigma(\textbf{s}(\textbf{x}))_k=\underset{k}{\text{argmax }}s_k(\textbf{x})=\underset{k}{\text{argmax }}\left(\left(\theta^{(k)}\right)^T\cdot\textbf{x}\right)$$ $$J(\Theta)=-\frac{1}{m}\sum_{i=1}^{m}\sum_{k=1}^{K}y_k^{(i)}\text{log}\left(\hat{p}_k^{(i)}\right)$$ $$\nabla_{\theta^{(k)}}J(\Theta)=\frac{1}{m}\sum_{i=1}^{m}\left(\hat{p}_k^{(i)}-y_k^{(i)}\right)\textbf{x}^{(i)}$$ $$\phi_\gamma(\textbf{x},l)=\text{exp}\left(-\gamma\left\|\textbf{x}-l\right\|^2\right)$$