\batchmode
\documentclass{article}
\usepackage{epsfig}
\pagestyle{empty}
\begin{document}
$ \phi $
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\[ u_{i,j,k}^{n+1} = \left\{ \begin{array}{ll} \mbox{max} (u_{i,j,k}^{n} + \Delta t H_{i,j,k}^{n}, 0) & \mbox{$B_{i,j,k} = 1$} \\ \mbox{min} (u_{i,j,k}^{n} + \Delta t H_{i,j,k}^{n}, 0) & \mbox{$B_{i,j,k} = -1$} \end{array}\right. \]
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$ u_{i,j,k}^{n} $
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$ (i,j,k) $
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$ n $
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$ H $
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$ B $
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$ I_k $
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$ c $
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\[ I_t = F_{\mbox{minmax}} |\nabla I| \]
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$ F_{\mbox{minmax}} = \min(\kappa,0) $
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$ \mbox{Avg}_{\mbox{stencil}}(x) $
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$ T_{thresold} $
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$ \max(\kappa,0) $
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$ \kappa $
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$ x $
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$ R $
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$ T_{threshold} $
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$ \mbox{min} \int D^2 \Rightarrow D \nabla D $
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\[ I_t = \kappa |\nabla I| \]
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\[ g(I) = 1 / ( 1 + | (\nabla * G)(I)| ) \]
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\[ g(I) = \exp^{-|(\nabla * G)(I)|} \]
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$ I $
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$ (\nabla * G) $
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$ P(\mathbf{x}) $
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$ Z(\mathbf{x}) $
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$ A(\mathbf{x}) $
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$ g(I) $
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\[ P(\mathbf{x}) = g(\mathbf{x}) \]
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\[ Z(\mathbf{x}) = g(\mathbf{x}) \]
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$ \mathbf{A}(\mathbf{x}) $
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\[ \mathbf{A}(\mathbf{x}) = -\nabla g(\mathbf{x}) \]
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$ \zeta( \phi^{*} - \phi) $
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$ \phi^{*} $
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$ \zeta $
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$ \lambda_1 \times c $
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$ \lambda_1 $
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$ \lambda_2 $
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$ \frac{1}{1+x} $
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$ O_i, i=1,2,\ldots,N $
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$ N $
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$ E(u,K)=\int_{\Omega-K}||u(r,c)-g(r,c)||^2{d{\Omega}}+\lambda\cdot{L(K)} $
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$ \Omega $
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$ g(r,c) $
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$ u(r,c) $
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$ O_i $
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$ \partial O_i $
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$ K=\bigcup_{i=1}^N\partial{O_i} $
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$ L(K) $
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$ \lambda $
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$ O_j $
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$ E(u,K) $
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$ E(\hat{u},K-\partial(O_i,O_j)) $
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$ \hat{u} $
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$ g $
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$ \partial(O_i,O_j) $
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$ E(u,K)-E(\hat{u},K-\partial(O_i,O_j))= \lambda\cdot{L(\partial(O_i,O_j))}- {(|O_i| \cdot |O_j|)\over (|O_i|+|O_j|)} ||c_i-c_j||^2 $
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$ c_{i,j} = (c_i |O_i| + c_j |O_j|) \over (|O_i| + |O_j|) $
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$ \frac{1}{1+ \frac{ difference^2 }{ \lambda^2 } }$
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$ F_{\mbox{minmax}} = \max(\kappa,0) $
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$ \min(\kappa,0) $
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\[\begin{tabular}{ccc} \begin{tabular}{|c|c|c|} 1.3 & 1.3 & 1.3 \\ 1.3 & 1.5 & 1.3 \\ 1.3 & 1.3 & 1.3 \\ \end{tabular} & \begin{tabular}{|c|c|c|} 1.7 & 1.7 & 1.7 \\ 1.7 & 0 & 1.7 \\ 1.7 & 1.7 & 1.7 \\ \end{tabular} & \begin{tabular}{|c|c|c|} 1.3 & 1.3 & 1.3 \\ 1.5 & 1.5 & 1.3 \\ 1.3 & 1.3 & 1.3 \\ \end{tabular} \\ \end{tabular}\]
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\[ \frac{ H(A) + H(B) }{ H(A,B) } \]
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$\pm 1.0$
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$ \epsilon $
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$ f $
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$ U $
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$ L $
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$ f(x) = \left\{ \begin{array}{ll} g(x) - L & \mbox{if $(g)x < (U-L)/2 + L$} \\ U - g(x) & \mbox{otherwise} \end{array} \right. $
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$ f(x) $
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$T$
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\[ f(x) = T - MahalanobisDistance(x) \]
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$ 2N $
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$ N-1 $
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$ (low, high) $
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$ L = min + T * (max - min) $
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$ max, min $
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$ T $
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$U(\mathbf{x})$
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$U(\mathbf{x}, t)$
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\[\frac{d U(\mathbf{x})}{d t} = \nabla \cdot c \nabla U(\mathbf{x})\]
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$c$
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$U(\mathbf{x}, 0) = U_0(\mathbf{x})$
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$\mathbf{x}$
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\[\frac{d U(\mathbf{x})}{d t} = C(\mathbf{x})\Delta U(\mathbf{x}) + \nabla C(\mathbf{x}) \nabla U(\mathbf{x})\]
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$C$
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\[C(\mathbf{x}) = e^{-(\frac{\parallel \nabla U(\mathbf{x}) \parallel}{K})^2}\]
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$ \Delta t $
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\[ Output(x_i) = \begin{cases} InsideValue & \text{if $LowerThreshold \leq x_i \leq UpperThreshold$} \\ OutsideValue & \text{otherwise} \end{cases} \]
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\[ h(A,B) = \mathrm{mean}_{a \in A} \min_{b \in B} \| a - b\| \]
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$A$
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$B$
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$ a \in A $
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$a$
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\[ H(A,B) = \max(h(A,B),h(B,A)) \]
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\[ h(A,B) = \max_{a \in A} \min_{b \in B} \| a - b\| \]
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\[ f_t = \mid \nabla f \mid \nabla \cdot c( \mid \nabla f \mid ) \frac{ \nabla f }{ \mid \nabla f \mid } \]
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\[ \nabla \cdot \frac{\nabla f}{\mid \nabla f \mid} \]
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$ (s/mm^2) $
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$ \lambda_3 $
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$\lambda_2$
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$\lambda_1$
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\[ \lambda_1 < \lambda_2 < \lambda_3 \]
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$\lambda_3$
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$ d \in [0,1] $
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$k$
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$x_j$
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$u_{x_j}$
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\[ \frac{ 1 }{ \sigma \sqrt{ 2 \pi } } \exp{ \left( - \frac{x^2}{ 2 \sigma^2 } \right) } \]
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\[ \frac{ 1 }{ \sqrt{ 2 \pi } }; \]
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\[ \frac{ 1 }{ \sigma \sqrt{ 2 \pi } }; \]
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\[ f(x) = (Max-Min) \cdot \frac{1}{\left(1+e^{- \frac{ x - \beta }{\alpha}}\right)} + Min \]
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\[ S = \frac{2 | A \cap B |}{|A| + |B|} \]
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$|\cdot|$
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$\cap$
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$S$
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$ \mathbf{magnitude} = \left( \sum_{i=0}^n \sum_{j=0}^m \frac{\delta * \phi_j}{\delta \mathbf{x}_{i}}^2 \right)^{\frac{1}{2}} $
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$\phi_j$
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$j^{\mathbf{th}}$
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$\phi$
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$n$
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\[ p_{in} = p_{out} + d \]
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$ \vec{\lambda}_j $
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$ \vec{g}(\vec{\lambda}_j) $
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$ D(\vec{x}) $
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$ \vec{x} $
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$ \vec{\delta}_j $
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\[ \vec{y} = B(\vec{x}) + D(\vec{x}) \]
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$u_{\mathbf{i}}^{n+1}=u^n_{\mathbf{i}}+\Delta u^n_{\mathbf{i}}\Delta t$
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$ \Delta $
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$\nu$
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$\alpha = 8 ( 1 - \nu ) - 1$
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$ G(x) = [\alpha*r(x)*I - 3*x*x'/r(x) ]$
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\[ r(x) = \sqrt{ x_1^2 + x_2^2 + x_3^2 } \]
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$\alpha = 12 ( 1 - \nu ) - 1$
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$ G(x) = [\alpha*r(x)^2*I - 3*x*x']*r(x) $
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$ \Delta u^n_{\mathbf{i}} $
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$ i $
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$ u $
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$n+1$
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$i$
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\[ J=\left[ \begin{array}{cccc} \frac{\partial x_{1}}{\partial p_{1}} & \frac{\partial x_{2}}{\partial p_{1}} & \cdots & \frac{\partial x_{n}}{\partial p_{1}}\\ \frac{\partial x_{1}}{\partial p_{2}} & \frac{\partial x_{2}}{\partial p_{2}} & \cdots & \frac{\partial x_{n}}{\partial p_{2}}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial x_{1}}{\partial p_{m}} & \frac{\partial x_{2}}{\partial p_{m}} & \cdots & \frac{\partial x_{n}}{\partial p_{m}} \end{array}\right] \]
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$ p $
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$ q $
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$ d $
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$ d_i = q_i - p_i $
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$ q_i - p_i $
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\[\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 \\ \hline 1 & 1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 \\ \hline 4 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 6 \\ \hline 4 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 6 \\ \hline 7 & 7 & 7 & 8 & 8 & 8 & 9 & 9 & 9 \\ \hline 7 & 7 & 7 & 8 & 8 & 8 & 9 & 9 & 9 \\ \hline a & a & a & b & b & b & c & c & c \\ \hline a & a & a & b & b & b & c & c & c \\ \hline \end{tabular}\]
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\[\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline C & C & C & & C & C & C & & C & C & C \\ \hline C & C & C & E & C & C & C & E & C & C & C \\ \hline & E & & & & E & & & & E & \\ \hline C & C & C & & C & C & C & & C & C & C \\ \hline C & C & C & E & C & C & C & E & C & C & C \\ \hline & E & & & & E & & & & E & \\ \hline C & C & C & & C & C & C & & C & C & C \\ \hline C & C & C & E & C & C & C & E & C & C & C \\ \hline & E & & & & E & & & & E & \\ \hline C & C & C & & C & C & C & & C & C & C \\ \hline C & C & C & E & C & C & C & E & C & C & C \\ \hline \end{tabular}\]
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\[\begin{tabular}{|c|c|} \hline A & A \\ \hline B & C \\ \hline \end{tabular}\]
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$\phi_{t} + \alpha \stackrel{\rightharpoonup}{A}(\mathbf{x})\cdot\nabla\phi + \beta P(\mathbf{x})\mid\nabla\phi\mid = \gamma Z(\mathbf{x})\kappa\mid\nabla\phi\mid$
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$ \stackrel{\rightharpoonup}{A} $
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$ P $
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$ Z $
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$ \alpha $
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$ \beta $
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$ \gamma $
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$\frac{f_{i}(\overrightarrow{x})}{f_{j}(\overrightarrow{x})} > \frac{K_{j}}{K_{i}}$
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$j \not= i$
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$f_{i}$
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$K_{i}$
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$ M(x) $
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$ P_i(x) $
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$ T(x) $
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\[ s = M(T(x)) + \sum_i^{q} p[i] * \sigma[i] * P_i(T(x)) \]
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\[ \overrightarrow{P}= \sum_{i=1}^{N-1} w_i * \overrightarrow{P}_i + \left(1- \sum_{i=1}^{N-1} w_i\right) * \overrightarrow{P}_N \]
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\[ \overrightarrow{P}=\frac{(\overrightarrow{A}+\overrightarrow{B})}{2} \]
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$ (1-\alpha) $
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\[ \overrightarrow{P}=\alpha * \overrightarrow{A}+ (1-\alpha)*\overrightarrow{B} \]
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$ \alpha \in [0,1] $
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$ \overline{AB} $
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$ \overrightarrow{A} $
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$ \alpha < 0 $
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$ \alpha > 1 $
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$ \overrightarrow{B} $
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\[ \overrightarrow{P}= w_1 * \overrightarrow{P}_1 + w_2 * \overrightarrow{P}_2 + (1-w_1-w_2 ) * \overrightarrow{P}_3 \]
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$ \in [0,1] $
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$ R^2 log(R) $
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$ G(x) = r(x)^2 log(r(x) ) *I $
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$ G(x) = r(x)*I $
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\[ J=\left[ \begin{array}{cccc} \frac{\partial x_{1}}{\partial p_{1}} & \frac{\partial x_{1}}{\partial p_{2}} & \cdots & \frac{\partial x_{1}}{\partial p_{m}}\\ \frac{\partial x_{2}}{\partial p_{1}} & \frac{\partial x_{2}}{\partial p_{2}} & \cdots & \frac{\partial x_{2}}{\partial p_{m}}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial x_{n}}{\partial p_{1}} & \frac{\partial x_{n}}{\partial p_{2}} & \cdots & \frac{\partial x_{n}}{\partial p_{m}} \end{array}\right] \]
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$ G(x) = r(x)^3*I $
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\[ I(x,y) = \sum_{i = \lfloor x \rfloor + 1 - m}^{\lfloor x \rfloor + m} \sum_{j = \lfloor y \rfloor + 1 - m}^{\lfloor y \rfloor + m} I_{i,j} K(x-i) K(y-j), \]
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\[ K(t) = w(t) \textrm{sinc}(t) = w(t) \frac{\sin(\pi t)}{\pi t} \]
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$ m $
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$ 2 m d $
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$ I(i,j,k) $
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$ K(x-i), K(y-j), K(z-k) $
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$ d (2m)^d $
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$ O ( (2m)^d ) $
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\[ w(x) = cos(\frac{\pi x}{2 m} ) \]
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\[ w(x) = 0.54 + 0.46 cos(\frac{\pi x}{m} ) \]
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\[ w(x) = 1 - ( \frac{x^2}{m^2} ) \]
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\[ w(x) = \textrm{sinc} ( \frac{x}{m} ) \]
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\[ w(x) = 0.42 + 0.5 cos(\frac{\pi x}{m}) + 0.08 cos(\frac{2 \pi x}{m}) \]
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$ \frac{\pi}{2 m} $
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$ \frac{\pi}{m} $
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$ \frac{1}{m^2} $
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$ \frac{2 \pi}{m} $
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$ [-\infty, \infty] $
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$ G(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{ - \frac{{\left( {x - \mu } \right)^2 }}{{2\sigma ^2 }}} $
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$ I_1 $
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$ I_2 $
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$ C\left( x \right) = I_1 + \frac{{I_2 - I_1 }}{2}\left( {1 + erf\left( {\frac{{x - \mu }}{{\sigma \sqrt 2 }}} \right)} \right) $
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$ C\left( { - \infty } \right) = I_1 $
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$ C\left( \infty \right) = I_2 $
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$ {\frac{{x - \mu }}{{\sigma \sqrt 2 }}} $
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\[ p_{n+1} = p_n + \mbox{learningRate} \, \frac{\partial f(p_n) }{\partial p_n} \]
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\[ f(x_{vector}, parameter_{vector}) = \sum_i^l \left( \sum_j^{l-i} \left( parameter_ {ij} * P_i(x) *P_j(y)) \right) \right) \]
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\[ f(x_{vector}, parameter_{vector}) = \sum_i^l \left( \sum_j^{l-i} \left( \sum_k^{l-i-j} \left( parameter_{ijk} * P_i(x) * P_j(y) * P_k(z) \right) \right) \right) \]
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$\frac{(l+1)\cdot(1+2)}{2}$
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$\frac{(l+1)*(l+2)*(l+3){3!}$
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$\Sigma$
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$x_{i}$
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$\vec x$
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$\mu_{i}$
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$\vec\mu$
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$\sigma_{ij}$
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$ij$
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$\sigma_{ij} = (x_{i} - \mu_{i})(x_{j} - \mu_{j})$
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$ g(i, j) $
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$ = f_1 = \sum_{i,j}g(i, j)^2 $
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$ = f_2 = -\sum_{i,j}g(i, j) \log_2 g(i, j)$
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$g(i, j) = 0$
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$ = f_3 = \sum_{i,j}\frac{(i - \mu)(j - \mu)g(i, j)}{\sigma^2} $
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$= f_4 = \sum_{i,j}\frac{1}{1 + (i - j)^2}g(i, j) $
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$ = f_5 = \sum_{i,j}(i - j)^2g(i, j) $
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$ = f_6 = \sum_{i,j}((i - \mu) + (j - \mu))^3 g(i, j) $
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$ = f_7 = \sum_{i,j}((i - \mu) + (j - \mu))^4 g(i, j) $
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$ = f_8 = \frac{\sum_{i,j}(i, j) g(i, j) -\mu_t^2}{\sigma_t^2} $
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$\mu_t$
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$\sigma_t$
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$ \mu = $
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$ = \sum_{i,j}i \cdot g(i, j) = \sum_{i,j}j \cdot g(i, j) $
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$ \sigma = $
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$ = \sum_{i,j}(i - \mu)^2 \cdot g(i, j) = \sum_{i,j}(j - \mu)^2 \cdot g(i, j) $
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\[ f(I) = -p \log_2 p \]
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\[ p = \frac{q_I}{\sum_{i \in I} q_I} \]
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$q_I$
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$p$
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\[ f(I) = q_I \]
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\[ f(I) = \log_2( \frac{q_I}{\sum_{i \in I} q_I} ) \]
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\[ f(I) = \frac{q_I}{\sum_{i \in I} q_I} \]
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$ \sum^{k}_{i=1}x_{i}\log(x_{i}/n\pi_{0i})$
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$ x_{i} $
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$n\pi_{0i}$
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$ = \frac{1}{n}\sum^{n}_{i=1}x_{i}$
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$ \boldsymbol{iU} \cdot \boldsymbol{iV} $
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\[ exp(\Phi) = exp( \frac{\Phi}{2^N} )^{2^N} \]
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$ iM \cdot oX = iBx $
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$ iM \cdot oY = iBy $
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$ 1 $
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$ \frac{1}{\|\boldsymbol{p1} - \boldsymbol{p2} \|} $
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$ \text{cot} \alpha_{ij} + \text{cot} \beta_{ij} $
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$ \frac{\text{cot} \gamma_{ij} + \text{cot} \delta_{ij}}{\|\boldsymbol{p1} - \boldsymbol{p2} \|} $
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\[ G(\vec{x}) = m e^{-\|\Sigma^{-1} \vec{x}\|^2 / 2}, \]
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$ \frac{\sum_i^n{a_i * b_i }}{\sum_i^n{a_i^2}\sum_i^n{b_i^2}} $
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$ A=\{a_i\} $
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$ H(A) $
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$ H(A) = \sum_i^n{- p(a_i) \log({p(a_i)})} $
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$ p(a_i) $
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$ B=\{b_i\} $
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$ p_(a_i,b_i) $
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$ H(A,B) = \sum_i^n{-p(a_i,b_i) * log( p(a_i, b_i) )} $
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$ H(A,B)-H(A)-H(B) $
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\end{document}