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\[ d = \sqrt{\frac{1}{N} \sum_{i=0}^N |\frac{x_i}{\mu_x} - \frac{y_i}{\mu_y}|) } \]
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\[ RMSD = \sqrt{\frac{1}{N} \sum_{i=0}^N (x_i - y_i)^2 } \]
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\[ \theta = acos \left( \frac{\sum_{i=0}^N (x_i * y_i))}{\sqrt{\sum_{i=0}^N (x_i * x_i) \sum_{i=0}^N (y_i * y_i)} } \right) \]
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\[ w_{RT} \cdot \left( \frac{\left| RT_1 - RT_2 \right|}{\Delta RT_{max}} \right)^{p_{RT}} + w_{MZ} \cdot \left( \frac{\left| MZ_1 - MZ_2 \right|}{\Delta MZ_{max}} \right)^{p_{MZ}} + w_{int} \cdot \left( \frac{\left| int_1 - int_2 \right|}{int_{max}} \right)^{p_{int}} \]
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$ RT_i $
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$ MZ_i $
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$ int_i $
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$ {\Delta RT_{max}}, {\Delta MZ_{max}} $
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$ int_{max} $
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$ {\Delta RT_{max}} $
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$ {\Delta MZ_{max}} $
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$ p_X $
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$ w_X $
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$\texttt{PearsonValue(map1}\cap \texttt{map2)}*\Bigl(\frac{\texttt{N(map1 }\cap\texttt{ map2})}{\texttt{N(map1 }\cup\texttt{ map2})}\Bigr)$
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$\Delta_\textit{RT}$
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$\Delta_\textit{MZ}$
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\[ \frac{1}{ \big( 1 + \Delta_\textit{RT} \cdot \textit{diff\_intercept\_RT} \big)^\textit{diff\_exponent\_RT} \cdot \big( 1 + \Delta_\textit{MZ} \cdot \textit{diff\_intercept\_MZ} \big)^\textit{diff\_exponent\_MZ} } \]
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$\Delta=0$
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\[ q_{i,j} = \big( 1 - d_{i,j} \big) \cdot \big( 1 - \frac{g \cdot d_{i,j}}{d_{2,i}} \big) \cdot \big( 1 - \frac{g \cdot d_{i,j}}{d_{2,j}} \big) \cdot \]
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$ q_{i,j} $
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$ d_{i,j} $
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$ d_{2,i} $
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$d_{2,j} $
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$ 0 \leq d_{i,j} \leq 1 $
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$ g \cdot d_{i,j} \leq d_{2,i} $
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$ g \cdot d_{i,j} \leq d_{2,j} $
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$ q_i $
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$ q_j $
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\[ q = \frac{q_{i,j} + (s_i - 1) \cdot q_i + (s_j - 1) \cdot q_j}{s_i + s_j - 1} \]
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$ s_i $
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$ s_j $
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$ f(x) = x $
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$ f(x) = slope * x + intercept $
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$ y - x $
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$ y + x $
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$ \longrightarrow $
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$ 1 + 10^{-5} $
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$ 10^{-5} $
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$1.60217738 \cdot 10^{-19} C$
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$9.1093897 \cdot 10^{-31}$
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$1,822.88850204(77)^{-1}$
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$1.6726230 \cdot 10^{-27}$
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$1.00727646677(10)$
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$1.0033548$
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$1.6749286 \cdot 10^{-27}$
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$1.0086649156(6)$
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$mol^{-1}$
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$6.0221367 \cdot 10^{23} mol^{-1}$
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$1.380657 \cdot 10^{-23}$
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$6.6260754 \cdot 10^{-34}$
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$5.29177249 \cdot 10^{-11}$
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$C^2J^{-1}m^{-1}$
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$8.85419 \cdot 10^{-12} C^2J^{-1}m^{-1}$
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$Js^2C^{-2}m^{-1}$
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$4\pi \cdot 10^{-7} Js^2C^{-2}m^{-1}$
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$2.99792458 \cdot 10^8 ms^{-1}$
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$Nm^2kg^{-2}$
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$6.67259 \cdot 10^{-11} Nm^2kg^{-2}$
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$7.29735 \cdot 10^{-3}$
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$2^{24}/10^{6}=16.777216$
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$2^{53}/10^{15}=9.007199254740992$
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$x_0, x_1, x_2, ...$
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$x$
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$ i $
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\[ \mathrm{erosion}_i = \min\{x_{i-\mathrm{struc\_size}/2}, \ldots, x_{i+\mathrm{struc\_size}/2}\} \]
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\[ \mathrm{dilation}_i = \max\{x_{i-\mathrm{struc\_size}/2}, \ldots, x_{i+\mathrm{struc\_size}/2}\} \]
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$ \emph{coeffs} $
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$ \emph{frameSize} $
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\[ \emph{coeff}(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{-x^2}{2\sigma^2}} \]
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$ x=[-\frac{frameSize}{2},...,\frac{frameSize}{2}] $
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$ \sigma $
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$ f_i $
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$g_i$
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$ c_n $
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$C \in R^{frameSize \times frameSize}$
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\[ C= \left[ \begin{array}{cccc} c_{0,0} & c_{0,1} & \cdots & c_{0,frameSize-1} \\ \vdots & & & \vdots \\ c_{frameSize-1,0} & c_{frameSize-1,2} & \ldots & c_{frameSize-1,frameSize-1} \end{array} \right]\]
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$ \frac{frameSize}{2} $
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$ C $
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$ frameSize $
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$ frameSize-1 $
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$ frameSize-2 $
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\[ A=\left[ \begin{array}{ccc} x_0^0 & x_0^1 & x_0^2 \\ \\ x_1^0 & x_1^1 & x_1^2 \\ \\ x_2^0 & x_2^1 & x_2^2 \\ \\ x_3^0 & x_3^1 & x_3^2 \\ \\ x_4^0 & x_4^1 & x_4^2 \end{array} \right]. \]
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$ x=[0,\ldots, frameSize-1] $
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\[ Ac=b \]
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$ b=[1,0,\ldots,0] $
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\[ A^TAc=A^Tb. \]
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\[ c_n=\sum_{m=0}^8 \{(A^TA)^{-1}\}_{0,m} n^m, \]
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$ 0\le n \le 8$
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$c=[c_0,\ldots,c_8] $
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$ Y(c,x) = c_0 + c_1 x $
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$ (c_0,c_1) $
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$ Y = c_0 + c_1 X $
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$ (x, y) $
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$ (x,y) $
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\[ S = \frac{1}{1 + \exp(-(a + b \cdot \Delta_{RT}^2 + c \cdot d_{int}))} \]
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$ \Delta_{RT} $
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$ d_{int} $
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$ a, b, c $
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$y_i = a + bx_i + cx_i^2$
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$x_i$
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$a$
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$b$
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$c$
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\end{document}
