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\begin{document}
\[ x_{i+1}\leftarrow(ax_{i} + c) \bmod m \]
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$1$
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\[ x_{i}\leftarrow(x_{i - s} - x_{i - r} - carry_{i-1}) \bmod m \]
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$r$
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$(m^r - m^s - 1)$
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\[ p(x|\mu,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{{x - \mu}^ {2}}{2 \sigma ^ {2}} } \]
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\[ p(x|m,s) = \frac{1}{sx\sqrt{2\pi}} \exp{-\frac{(\ln{x} - m)^2}{2s^2}} \]
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\[ p(x|\alpha,\beta) = \frac{1}{\beta\Gamma(\alpha)} (x/\beta)^{\alpha - 1} e^{-x/\beta} \]
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$p(x|n) = \frac{x^{(n/2) - 1}e^{-x/2}}{\Gamma(n/2) 2^{n/2}}$
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$p(x|a,b) = (\pi b (1 + (\frac{x-a}{b})^2))^{-1}$
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\[ p(x|m,n) = \frac{\Gamma((m+n)/2)}{\Gamma(m/2)\Gamma(n/2)} (\frac{m}{n})^{m/2} x^{(m/2)-1} (1 + \frac{mx}{n})^{-(m+n)/2} \]
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\[ p(x|n) = \frac{1}{\sqrt(n\pi)} \frac{\Gamma((n+1)/2)}{\Gamma(n/2)} (1 + \frac{x^2}{n}) ^{-(n+1)/2} \]
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$p$
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$(1 - p)$
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$p(i|t,p) = \binom{t}{i} p^i (1 - p)^{t - i}$
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$t$
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$p(i|p) = p(1 - p)^{i}$
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$p(i) = \binom{n}{i} p^i (1 - p)^{t - i}$
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$p(i|\mu) = \frac{\mu^i}{i!} e^{-\mu}$
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$\mu$
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$p(x|\lambda) = \lambda e^{-\lambda x}$
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$\frac{1}{\lambda}$
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$\frac{\ln 2}{\lambda}$
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$zero$
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$[0, \infty]$
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\[ p(x|\alpha,\beta) = \frac{\alpha}{\beta} (\frac{x}{\beta})^{\alpha-1} \exp{(-(\frac{x}{\beta})^\alpha)} \]
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\[ p(x|a,b) = \frac{1}{b} \exp( \frac{a-x}{b} - \exp(\frac{a-x}{b})) \]
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$x_0$
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$ m_{lcg} = 2147483563, a_{lcg} = 40014, c_{lcg} = 0, and lcg(0) = value $
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$ x_{-r} \dots x_{-1} $
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$ lcg(1) \bmod m \dots lcg(r) \bmod m $
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$ x_{-1} = 0 $
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$mean$
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$\alpha$
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$\beta$
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$[0, 1]$
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$k$
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$\lambda$
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$a$
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$b$
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\[ \frac{|f - f_{test}|}{|f_{test}|} \]
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$ L_n^m(x) $
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$ \alpha $
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$ L_n^\alpha(x) $
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\[ L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} {}_1F_1(-n; \alpha + 1; x) \]
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$ (\alpha)_n $
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$ {}_1F_1(a; c; x) $
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$ \alpha = m $
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\[ L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) \]
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\[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \]
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$ x >= 0 $
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$ P_l(x) $
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\[ P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) \]
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$ B(a,b) $
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$B(a,b)$
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\[ B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} \]
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$ a > 0 $
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$ b > 0 $
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$ E(k) $
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$ K(k) $
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\[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \]
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$ F(k,\phi) $
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$ |k| <= 1 $
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\[ E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \]
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$ E(k,\phi) $
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$ \Pi(k,\nu) $
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$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) $
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\[ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2} \frac{d\theta} {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} \]
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$ \Pi(k,\nu,\phi) $
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$ I_{\nu}(x) $
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$ \nu $
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\[ I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty} \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \]
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$ J_{\nu}(x) $
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\[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \]
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$ K_{\nu}(x) $
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$ x $
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\[ K_{\nu}(x) = \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} \]
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$ \nu = n $
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$ lim_{\nu \to n} $
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\[ K_{-\nu}(x) = K_{\nu}(x) \]
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$ N_{\nu}(x) $
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\[ N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} {\sin \nu\pi} \]
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$ k $
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$ \phi $
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\[ F(k,\phi) = \int_0^{\phi}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \]
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$ \phi= \pi/2 $
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\[ E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} \]
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\[ \Pi(k,\nu,\phi) = \int_0^{\phi} \frac{d\theta} {(1 - \nu \sin^2\theta) \sqrt{1 - k^2 \sin^2\theta}} \]
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$ Ei(x) $
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\[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \]
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$ H_n(x) $
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\[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \]
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\[ H_n(-x) = (-1)^n H_n(x) \]
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$ L_n(x) $
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$ l $
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$ |x| <= 0 $
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\[ P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} \]
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$ l >= 0 $
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$ \zeta(s) $
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$ s $
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\[ \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1 \]
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\[ \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s} \hbox{ for } 0 <= s <= 1 \]
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\[ \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) \]
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$ j_n(x) $
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\[ j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) \]
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$ \theta $
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\[ Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}] P_l^m(\cos\theta) \exp^{im\phi} \]
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$ n >= 0 $
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\[ n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) \]
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$ Ai(x) $
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$ Bi(x) $
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$ {}_1F_1(a;c;x) $
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\[ {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!} \]
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$ (x)_k = (x)(x+1)...(x+k-1) $
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$ (x)_0 = 1 $
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$ {}_2F_1(a,b;c;x) $
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\[ {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!} \]
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$[min, max]$
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$\sqrt{\mathrm{end} - \mathrm{begin}}$
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$1/(\mathrm{oversampling} \cdot \mathrm{num\_parts})$
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\end{document}
