harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L(b) = \beta_b^2 - \alpha_{b-1}\alpha_{b+1}, M(b) = \left(\frac{\alpha_b + \beta_b}{2}\right)^2 - \left(\frac{\alpha_{b-1}+ \beta_{b-1}}{2}\right)\left(\frac{\alpha_{b+1}+ \beta_{b+1}}{2}\right), U(b) = \alpha_b^2 - \beta_{b-1}\beta_{b+1}.
t(\{2m\}^n)=\frac{(-1)^n}{2^{m-1}}\sum_{\varepsilon_j=\pm 1,\; 1\leq j\leq m-1}\frac{(-1)^{mn}\pi^{2mn}}{(2mn)!2^{2mn}}(w^{m-1}+\varepsilon_1w^{m-2}+\cdots+\varepsilon_{m-1})^{2mn}.
\begin{array}{l}{{e_{a}\circ_{X}( e_{b}\circ_{X}( e_{c}\circ_{X}e_{d}) ) = \pm e_{7}}}\\{{\mathrm{~ i f ~ e x a c t l y ~ o n e ~ o f ~}a , b , c , d = 0 \mathrm{~ o r ~}h = 7 ,}}\\{{e_{a}( e_{b}( e_{c}e_{d}) ) = \pm 1 \mathrm{~ o t h e r w i s e . ~}}}\\ \end{array}
u_{i j}= x_{i}x_{j}- q^{i - j - n}x_{j}x_{i}- ( q^{n}- q^{- n}) q^{i - j - r - r n}x_{i + r}x_{i - r}+ p ( 1 - q^{n}) x_{\frac{i + j}{2}}x_{\frac{i + j}{2}}
\Phi_{\mathcal{S}_{\alpha ,\theta}(t)}(\xi ):=\mathbb{E}e^{i\xi \mathcal{S}_{\alpha ,\theta}(t)}=\exp \{-t|\xi |^{\alpha}\sigma^{\alpha}\omega_{\alpha ,\theta}(\xi )\},\xi \in \mathbb{R},\;\alpha \in(0,2],\;\sigma >0,
\cdot \>{_3F}_2 \left\{\begin{array}{l}-n_1, -l+|m|, l+|m|+\delta_1+\delta_2+1 |m|+\delta_1+1, -n+|m|+1 \end{array}\biggr| 1 \right\}E_{n_1 n_2}^{lm}
d s^{2}\sim \, d x_{1 , 3}^{2}+ \alpha^{\prime}g_{s}N \, \left[ d \rho^{2}+ \rho^{2}\left( d \theta_{1}^{2}+ \sin^{2}\theta_{1}\, d \phi_{1}^{2}\right) + \sum_{a = 1}^{3}\left( \sigma^{a}- A^{a}\right)^{2}\right]
\lambda^{(2)}(x) = 0.0004 + 0.03x + 0.47x^2 + 0.06 x^3 + 0.44 x^4, \lambda^{(4)}(x) = 0.001 + 0.05x + 0.46x^2 + 0.09 x^3 + 0.41 x^4.
U_{\mu}^{+}(x) =\min\{\frac{a}{b+\chi_2\mu_2-\chi_1\mu_1-M}, \varphi_{\mu}(x)\}&=\begin{cases}\frac{a}{b+\chi_2\mu_2-\chi_1\mu_1-M}\ \x\leq \frac{-\ln(\frac{a}{b+\chi_2\mu_2-\chi_1\mu_1-M})}{\mu}e^{- \sqrt a \mu x}\qquad\qquad\x\geq \frac{-\ln(\frac{a}{b+\chi_2\mu_2-\chi_1\mu_1-M})}{\mu}.\end{cases}
\int \frac{\delta \hat{\Gamma}_{c l}}{\delta \hat{\varphi}_{0}}, \int \left\{A \frac{\delta}{\delta A}+ c \frac{\delta}{\delta c}\right\}\hat{\Gamma}_{c l}, m_{H}\partial_{m_{H}}\hat{\Gamma}_{c l}, e \partial_{e}\hat{\Gamma}_{c l},
\prod_{k=1}^{M-1}\prod_{l=1}^{\infty}(1-e^{2\pi i k/M}Y q^lq^{-\{ks/M\}}) = \prod_{k'}\prod_{l=1}^{\infty}(1-e^{2\pi i k't'/m}Y^r q^{rl}q^{-k'r/m}) = \prod_{l=0}^{\infty}(1-Y^r q^{lr/m}\eta^{-l})/ \prod_{l=0}^{\infty}(1-Y q^l).
a(z) =\sum_{k=1}^{\infty}\sum\limits_{n=k}^\infty (-1)^{k-1}{n-1 \choose k-1}{\Gamma(k/q+1)\over k!\,\Gamma(k/q -k+2)}\,q^k z^n &=\sum\limits_{n=1}^\infty z^n \sum_{k=1}^n (-1)^{k-1}{n-1 \choose k-1}{\Gamma(k/q+1)\over k!\,\Gamma(k/q -k+2)}\,q^k.
{H(t_f) a(t_f) \over H(t_i)a(t_i)}= \left({t_{\rm sing}-t_i \over t_{\rm sing}-t_f}\right)^{(1+\sqrt{3})/\sqrt{3}}= \left({\tilde t_{\rm sing}-\tilde t_i \over \tilde t_{\rm sing}-\tilde t_f}\right)^{2/3}={\tilde H(t_f) \tilde a(t_f) \over \tilde H(t_i) \tilde a(t_i)}\,.
\leq C_{\varepsilon}\left({\displaystyle\sum\limits_{i=1}^{N}}\left\langle x_{N}^{n}D_{x_{i}}^{m}u\right\rangle_{\omega\gamma,x_{i},\overline{Q}}^{(\gamma)}+\left\langle D_{t}u\right\rangle_{t,\overline{Q}}^{(\gamma/m)}\right)+C\varepsilon^{\gamma}\left\langle\left\langle u\right\rangle \right\rangle_{n,\omega\gamma,\overline{Q}}^{(m+\gamma)(2s)}.
\varphi^h_A(x) = \left( \begin{array}{c}\Phi_{23}v_3' + \Phi_{13}w \Phi_{12}v_2' + \Phi_{23}w \Phi_{13}v_2' + \Phi_{12}v_3' \end{array}\right) + hx_3 \left( \begin{array}{c}- \Phi_{23}v_2' - \Phi_{12}w \Phi_{13}v_2' + \Phi_{12}v_3' \Phi_{13}v_3' + \Phi_{23}w \end{array}\right) \,,
\psi_{\mathrm{( w a l l - a n t i w a l l )}}( y ) = \mathrm{a r c s i n}\left({\frac{m}{g}}\frac{k \sqrt 2}{\sqrt{1 + k^{2}}}\mathrm{s n}\left({\frac{\sqrt 2}{\sqrt{1 + k^{2}}}}m y , k \right) \right) ,
\frac{P^+[\hat{H}_B+F]}{1- \hat{\phi}(k) P^+[\partial_u F]}-\frac{P^-[\hat{H}_B+F]}{1-\hat{\phi}(k) P^-[\partial_u F]}= \frac{2\pi iF(u)}{(1-\hat{\phi}(k) P^+[\partial_u F])(1-\hat{\phi}(k) P^-[\partial_u F])}.
S_{\mathrm{K K 1 1}}^{\mathrm{g r a v .}}= -{\frac{T_{\mathrm{K K 1 1}}}{2}}\int d^{7}\xi \sqrt{| \gamma |}\left[ k^{4 / 7}\gamma^{i j}D_{i}X^{\mu}D_{j}X^{\nu}g_{\mu \nu}- 5 \right] \, ,
\left|\partial^{\alpha}(\psi-\chi)(\xi)\right| = \left|\int_{t=0}^1 \frac{d}{dt}(\partial^{\alpha}(\psi-\chi)(t\xi))\,dt \right| \leq \sum_{i=1}^d |\xi_i| \left|\int_{t=0}^1 \frac{\partial}{\partial \alpha_i}(\partial^{\alpha}(\psi-\chi)(t\xi))\,dt\right|,
\partial^2_{x_i x_j}\phi = \partial^2_{x_ix_j}\psi + \partial^2_{x_iy}\psi \Gamma_j +\partial^2_{x_jy}\psi \Gamma_i+ \partial^2_y \psi \Gamma_j\Gamma_i + \partial_y \psi \partial^2_{x_jx_i}T - \partial_y T \partial^3_{x_jx_iy}T +\partial_{y}\Gamma_i \partial_{y}\Gamma_j
S \prime \prime_{0}( x ) + \frac{2}{x}S_{0}^{\prime}( x ) - \frac{e^{2 S_{0}( x ) / \alpha}}{x^{4}}\{( 1 - K_{0}( x )^{2})^{2}+ 2 x^{2}K_{0}^{\prime}( x )^{2}\}= 0 ,
0= \Big\{a+\frac{1}{\lambda_2}\left[(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{+}+\chi_{1}\mu_{1}(\lambda_1-\lambda_2)_{+}\right]\overline{u}&-\frac{1}{\lambda_2}\left[(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{-}+\chi_1\mu_1(\lambda_1-\lambda_2)_{-}\right]\underline{u}\Big\}_+=a,
c_a =\frac 1 2\int_0^{\frac 1 2}[s_1(1-s_1)]^4\left(\int_0^{s_1}[s_2(1-s_2)]^2 ds_2\right)ds_1=\frac{1199}{309657600}c_b =\frac 1 2\int_{\frac 1 2}^1[s_1(1-s_1)]^4\left(\int_{\frac 1 2}^{s_1}[s_2(1-s_2)]^2 ds_2\right)ds_1=\frac{283}{103219200}
G ( x , 0 ) = \frac{{\displaystyle \int}{\cal D}S{\cal D}{\bar{\psi}}{\cal D}\psi ~{\bar{\psi}}( x ) ~ e^{- 2 i \gamma^{5}S ( x )}~ \psi ( x ) ~{\bar{\psi}}( 0 ) ~{\displaystyle e}^{- 2 i \gamma^{5}S ( 0 )}~ \psi ( 0 ) \, \, \, e^{i S_{e f f}}}{{\displaystyle \int}{\cal D}S{\cal D}{\bar{\psi}}{\cal D}\psi \, \, \,{\displaystyle e}^{i S_{e f f}}},
\delta R_{M N}( h ) +{\frac{1}{6}}\bar{F}_{\theta \phi}\bar{F}^{\theta \phi}( h_{M N}+ 2 \bar{g}_{M N}{\cal F}) - ( \bar{F}_{M \theta}\bar{F}_{N}^{~ \theta}+ \bar{F}_{M \phi}\bar{F}_{N}^{~ \phi}){\cal F}= 0 ,
\frac{\lambda_{1}+Q_{n}^{(N,M)}(z) -\psi_{n}(x;N,M) \psi_{n-1}(x;N+1,M+1) /4n}{\psi_{n-1}(x;N+1,M+1)}=C\Psi_{n-1}^{(n-1)}(x;N+1,M+1,\lambda_{1}^{\prime})
\int_{S^2}\tilde X^i |\beta|^2 dS^2= \int_{S^2}\tilde X^i \bar W_{Ljr0}\bar W_{Lkr0}(\delta^{jk}- X^kX^j) dS^2 = \int_{S^2}\tilde X^i \bar W_{Ljr0}\bar W_{Ljr0}dS^2 - \int_{S^2}\tilde X^i \bar W_{0r r0}\bar W_{0rr0}dS^2 = \frac{8 \pi}{15}D_{jm}\bar W_{0mij}.
{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\; i}\left[\frac{1}{2}\frac{\partial\Gamma^{(2)}}{\partial w^{i}}\right]=-{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\; i}\left[B^{w(2)}_{i}\right] +{e}_{\alpha}^{\;\;\mu}{e}_{\hat{0}}^{\;\;\nu}K^{b}_{\mu\nu},
G_0(f) =\{p_0, p_1, p_3, p_5, p_7\}, G_n(f) =\begin{cases}\cup_{j\in\{0,1,3,5,7\}}\varphi_j(G_{n-1}(f)) f(n)=0 \cup_{j=1}^8\varphi_j(G_{n-1}(f)) f(n)=1\end{cases}
\int^{\infty}_{0}\frac{1}{x^2}f(x)\mathrm{d}x= \frac{2\theta}{\operatorname{B}(a,b)}\sum_{n=0}^{\infty}\frac{(-1)^n\Gamma(b)}{\Gamma(b-n)n!}\int^{\infty}_{0}\frac{\exp(-t)}{[2(a+n)\theta]^2}\mathrm{d}t = \frac{1}{\theta\operatorname{B}(a,b)}\sum_{n=0}^{\infty}\frac{(-1)^n\Gamma(b)}{\Gamma(b-n)n!}\frac{1}{(a+n)^2}.
\tilde{N}= N^1 \cup N^2,\tilde{E}= E^1 \cup E^2, \tilde{h}_0 = h_0^1 \cup h_0^2 \tilde{H}= H^1 \cup H^2, &\tilde{L}: \tilde{N}\to \{1, ..., d\}\mbox{such that}\tilde{L}|_{N^i}= L^i.
\sum_{\substack{d\leq y (d,q)=1}}\mu(d)N_d(\ell) =\sum_{\substack{d\leq y (d,q)=1}}\mu(d)\frac{\varphi(q)}{q}U_d(\ell)\dfrac{\left|I(\ell)\right|}{d^2}+O_m\left(d(q)y\log y\right) &=\frac{\varphi(q)}{q}\prod_{p\nmid q}\left(1-\frac{u_p(\ell)}{p^2}\right)\left|I(\ell)\right|+O_m\left(\tau(q)y\log y+Xy^{-1}\log y\right),
u(\mathrm{d}x) = \frac{1}{\sqrt{f(x)}}v_F(\mathrm{d}x) - \frac{1- \sqrt{f(x)}}{1-\int_{[0,1]^d}\sqrt{f(y)}\,\mathrm{d}y}\int_{[0,1]^d}\frac{1}{\sqrt{f(y)}}v_F(\mathrm{d}y) \,\mathrm{d}x
\alpha = (\underbrace{0,\ldots,0}_{n},\underbrace{r_n}_{\neq 0},r_{n+1},\ldots)\mapsto \chi(\alpha) = (\underbrace{0,\ldots,0}_{n+1},-1,0,0,\ldots):\Gamma_{\log}\to\Gamma^{\leq}_{\log}
u={2c_{_{\rm P}}^2 v\cos\theta \pm c_{_{\rm P}}\sqrt{\big(c_{_{\rm P}}^2-v^2\big)\big(3c_{_{\rm P}}^2-(1+2\cos^2\theta)v^2\big)}\over 3c_{_{\rm P}}^2-v^2}+{\cal O}\{\Theta^4\}\ .
\int \frac{d k^{\prime \prime}}{2 \pi}\frac{1}{\sqrt{k^{2}+ 4}}\frac{2 i k^{\prime \prime}\epsilon}{( i k^{\prime \prime}- 2 \sigma_{1})^{2}}\frac{1}{( k + k^{\prime}- 2 k^{\prime \prime}- i \tau )}.
\begin{cases}A(0)-\sqrt{\varepsilon}\,\frac{\partial}{\partial x}A(0)=F_A(0),A(1)+\sqrt{\varepsilon}\,\frac{\partial}{\partial x}A(1)=0, begin{align*}5pt]\mathcal B(0)-\sqrt{\varepsilon}\,\frac{\partial}{\partial x}\mathcal B(0)=F_B(1),\mathcal B(1)+\sqrt{\varepsilon}\,\frac{\partial}{\partial x}\mathcal B(1)=0.\end{cases}
\log^{+}(-\Delta u(x)) =\log^{+}(-\Delta U(x))\leq \log^{+}(|x|^{-a}e^{|V|^\lambda}) =a\log\frac{1}{|x|}+|V|^\lambda \leq a\log\frac{1}{|x|}+C\left(\log\frac{1}{|x|}\right)^\lambda .
\| D\|_{\mathcal{H}_{p}(X)}= \Big(\int_{\mathbb{T}^{\mathbb{N}}}\Big\|\sum_{\alpha:1 \leq p^\alpha \leq N}a_{p^\alpha}\omega^\alpha \Big\|_X^p d \omega\Big)^{1/p}begin{align*}1ex] =\Big(\lim_{R\rightarrow \infty}{\frac{1}{2R}{\int_{-R}^{R}{\Big\| \sum_{n=1}^{N}{a_{n}\frac{1}{n^{it}}}\Big\|_X^{p}\: d t}}}\Big)^{\frac{1}{p}},
p\left({d_{p2}\leq \delta}\right) = p\left({d_{p2}^2 \leq \delta^2}\right) = M\iint\limits_\Omega{\sin^{2\left({n - 4}\right)}\theta_1 \sin^{2\left({n - 4}\right)}\theta_2 \left({\sin^2 \theta_1 - \sin^2 \theta_2}\right)^2 d\sin^2 \theta_1 d\sin^2 \theta_2}
\setlength{\arraycolsep}{2pt}\def\arraystretch{1.1}\begin{array}{lcllcl}p_{6}= x^3(x-1)^3, p_{5}= (p_{50}+p_{51}x)x^2(x-1)^2, p_{4}= (p_{40}+p_{41}x+p_{42}x^2)x(x-1),\ p_{3}= p_{30}+p_{31}x+p_{32}x^2+p_{33}x^3, p_{2}= p_{20}+p_{21}x+p_{22}x^2, p_{1}= p_{10}+p_{11}x, \end{array}\setlength{\arraycolsep}{3pt}
x(yz)=\lambda_1(xy)z +\lambda_2(yx)z+\lambda_3z(xy) + \lambda_4 z(yx) & + \lambda_5 (xz)y + \lambda_6 (zx)y + \lambda_7 y(xz) + \lambda_8 y(zx)\intertext{and}(yz)x=\mu_1(xy)z +\mu_2(yx)z+\mu_3z(xy) + \mu_4 z(yx) & + \mu_5 (xz)y + \mu_6 (zx)y + \mu_7 y(xz) + \mu_8 y(zx)
\sum_{x\in \mathbb{F}_p^{\times}}\chi(x) \exp\left( 2\pi i \frac{ax+bx^{-1}}{p}\right) \leftrightsquigarrow K_{\nu}(x) = \frac{1}{2}\int_{0}^{\infty}t^{-\nu}\exp(-\frac{x}{2}(t+t^{-1}))\frac{dt}{t}.
\ \ \ \quad\ \ :\ \lim_{n\rightarrow\infty}\left( n[Y_{l_{1}}^{m_{1}}\star_{\vec{c}}^{n}Y_{l_{2}}^{m_{2}}+Y_{l_{2}}^{m_{2}}\star_{\vec{c}}^{n}Y_{l_{1}}^{m_{1}}-2Y_{l_{1}}^{m_{1}}Y_{l_{2}}^{m_{2}}]\right) =0
\begin{array}{l c l}{{w_{3}}}&{{=}}&{{u_{3}- 2 \theta u_{2}^{\prime}}}\\{{w_{4}}}&{{=}}&{{u_{4}- \frac{9}{1 0 0}u_{2}^{2}- \theta u_{3}^{\prime}+ \frac{{4 \theta^{2}}}{5}u_{2}^{\prime \prime}}}\\ \end{array}
\sum_{\vec{k}}\left\{\delta \Pi \ \dot{\varphi}( \vec{k}, \, t) + \delta \dot{\varphi}\ \Pi (\vec{k}, \, t) - \left[ \delta \varphi \{\partial H\over \partial \varphi ( \vec{k}, \, t)}+ \delta\Pi \{\partial H \over \partial \Pi ( \vec{k}, \,t)}\right] \right\}= \dot{\Omega}\ .
\lim_{N\to\infty}\bigg|\Big\langle\Phi,\Big({U}_N( H_N-H_{2\mathrm{-mode}}){U}_N^*- \mathbb{H}- \mu_+ \mathcal{N}_\perp\Big)\Phi\Big\rangle &-\frac{\lambda}{\sqrt{2(N-1)}}\sum_{m\ge3}\Big\langle \Phi, \Big(\big(w_{+1-m}\,\Theta+w_{+2-m}\, \Theta^{-1}\big) a_m\mathfrak{D}+\mathrm{h.c.}\Big)\Phi\big\rangle\bigg|=0.
\left\|\sum_{i=r}^{s}\frac{1}{\mu(B_i)}\Big(\int_{B_i}f d\mu\Big)\chi_{B_i}\right\|_p^p =\int_\Omega\Big( \sum_{i=r}^{s}\frac{1}{\mu(B_i)^p}\left\|\int_{B_i}f d\mu\right\|^p\chi_{B_i}\Big)d\mu =\sum_{i=r}^{s}\frac{1}{\mu(B_i)^p}\left\|\int_{B_i}f d\mu\right\|^p\int_{\Omega}\chi_{B_i}d\mu.
\frac{1}{2}( \partial^{\mu}\partial^{\nu}h_{\mu \nu}- \partial_{\mu}\partial^{\mu}\tilde{h}) - \frac{3}{2}\partial_{4}\sigma \partial_{4}\tilde{h}+ 6 k^{2}h_{4 4}+ 3 \partial_{4}\sigma \partial^{\mu}h_{\mu 4}= 0 ,
P_{v}= \exp \left\{- \mu \tilde{n}\int d{\bf x}\right\}, \mu = \sum_{l = 0}^{\infty}\frac{( - 1 )^{l}\beta ( l + 1 )}{( l + 1 )^{3 / 2}\beta ( 1 )}\exp \left\{- l \pi \frac{( a M )^{2}}{\rho}\right\}.
L=\exp \left( \begin{array}{cc}\theta_{N}\overline{\theta}^{N}\end{array}\right) \exp \left( \begin{array}{cc}i\frac{1+i\gamma^{4}}{2}\gamma_{p}x^{p}0_{4}\end{array}\right) \exp \left( \begin{array}{cc}i\frac{\gamma^{4}}{2}\rho 0_{4}\end{array}\right)
d s_{d e S}^{2}= -{\frac{\ell^{2}}{4}}\left({\frac{d \rho}{\rho}}\right)^{2}+{\frac{1}{\rho}}\, d \vec{x}\cdot d \vec{x}= -{\frac{\ell^{2}}{4}}{\frac{d \rho^{2}}{\rho^{2}}}+{\frac{1}{\rho}}\, \delta_{a b}d x^{a}d x^{b}\, .
f= \sum_{D\in F}\left( \sum_{\gamma \in I_D}(f_D)|_{D_{\gamma,i_{\gamma}}}\right) = \sum_{D \in F}\left( \sum_{i=1}^k \sum_{\gamma \in I_D}(f_D)|_{D_{\gamma,i}}\right)= \sum_{i=1}^k \left( \sum_{D\in F}\sum_{\gamma \in I_D}(f_D)|_{D_{\gamma,i}}\right).
y=\arccos_p(x)=\!\!\!\!\!\!\int\limits_0^{(1-x^p)^{\frac{1}{p}}}\!\!\!\!\!\frac{dt}{(1-t^p)^{1/p}}=\int\limits_{x}^{1}\frac{s^{p-2}ds}{(1-s^p)^{1-\frac{1}{p}}},~~0\le{x}\le1.
\begin{array}{l c l l}{{\mathrm{f l o o r}}}&{{\mathrm{S U ( 2 / 1 )}}}&{{\mathrm{f i e l d}}}&{{\mathrm{h e l i c i t y}}}\\{{\mid \mathrm{g n d}>}}&{{( 0 ~ - \frac{7}{2})}}&{{e_{\mu}^{a}}}&{{+ 2}}\\{{\mid \mathrm{1 s t}>}}&{{( 1 ~ - \frac{7}{2})}}&{{2 \Psi_{\mu}}}&{{+ \frac{3}{2}}}\\{{\mid \mathrm{2 n d}>}}&{{( 0 ~ - \frac{5}{2})}}&{{A_{\mu}}}&{{+ 1}}\\ \end{array}
\begin{aligned}\left( \operatorname{proj}_{K_N \cap Q}\circ{\operatorname{proj}_{K_N}}^{-1}\right) \# \mu_N = \left( \operatorname{proj}_{K_N \cap Q}\circ{\operatorname{proj}_{K_\ell}}^{-1}\right) \# \mu_\ell.\end{aligned}
\frac{d}{dt}\int_{-\infty}^0|x| |u|^2dx = 2\,\int_{-\infty}^0u\bar{u}_xdx =\mathcal{Q}_0^- + 2\int_0^t|u_x(0,s)|^2ds - \frac{\alpha}{\gamma}\int_{-\infty}^0v^2dx.
\log \{Y_{n}^{3}f_{n}( S ) H_{n}( T ) \}= - 1 \Longrightarrow Y_{n}^{3}= e^{- 1}e^{- \frac{3 2 \pi^{2}k_{n}S}{c_{n}}}\frac{1}{H_{n}^{i}( T )}, i = 1 , 2 .
\frac{|A\oplus_2 B|^2}{|P_u(A\oplus_2 B)|^2}=\frac{|B_2^n|^2}{|B_2^{n-1}|^2}\rho_{A\oplus_2B}(u)^2 &\ge\frac{|B_2^n|^2}{|B_2^{n-1}|^2}(\rho_{A}(u)^2+\rho_{B}(u)^2)=\frac{|A|^2}{|P_uA|^2}+\frac{|B|^2}{|P_uB|^2}.
\begin{array}{r c l}{{S}}&{{=}}&{{\displaystyle \int d^{8}z E^{- 1}\left[ \bar{Q}Q + \bar{Q}^{c}Q^{c}+ G ( \bar{Q}^{c}Q ) ( \bar{Q}Q^{c}) + \bar{\xi}_{1}( \bar{Q}Q ) + \bar{\xi}_{2}( \bar{Q}^{c}Q^{c}) \right] ,}}\\ \end{array}
f_i^{\left({eq}\right)}\left({{\bf{x}},t}\right) ={{\hat w}_i}\rho \left[{1 + \frac{{{{\bf{u}}^{eq}}\cdot{{\bf{c}}_i}}}{{\hat c_s^2}}+ \frac{{{{\left({{{\bf{u}}^{eq}}\cdot{{\bf{c}}_i}}\right)}^2}}}{{2\hat c_s^4}}- \frac{{\left|{{{\bf{u}}^{eq}}}\right|}}{{2\hat c_s^2}}}\right],
(x_{1}^{+}\mid x_{2}^{+})[d,\langle a_{1}^{-},a_{2}^{-}\rangle]=(x_{1}^{+}\mid x_{2}^{+})(\langle da_{1}^{-},a_{2}^{-}\rangle+\langle a_{1}^{-},da_{2}^{-}\rangle).
A_4 := \begin{pmatrix}8 4 -2 6 4 4 2 4 -2 2 5 0 6 4 0 5\end{pmatrix}, A_6 := \begin{pmatrix}9 7 7 1 1 -3 7 6 5 3 2 -2 7 5 6 -1 0 2 1 3 -1 9 5 1 1 2 0 5 3 1 -3 -2 -2 1 1 3\end{pmatrix}.
\#( Q(M_n) \setminus M_n ) = (\ell_1 + 1) (\ell_2 + 1) (\ell_3 + 1) - n \le (n^{1/3}+ K n^{1/12}+{\rm O}(1))^3 - n = 3K n^{3/4}+{\rm o}(n^{3/4}),
{\cal C}_{k}^{\prime \prime}+ \biggl \{k_{L}^{2}+ k_{T}^{2}\biggl [ - \frac{\eta}{\eta_{1}}\biggr ]^{\gamma}- \frac{\mu^{2}- \frac{1}{4}}{\eta^{2}}\biggr \}{\cal C}_{k}= 0 ,
b a^{\dagger}b^{\dagger}= b^{\dagger}a^{\dagger}b , a b^{\dagger}a^{\dagger}= a^{\dagger}b^{\dagger}a , b b^{\dagger}a^{\dagger}= a^{\dagger}b^{\dagger}b + 2 a^{\dagger}, a a^{\dagger}b^{\dagger}= b^{\dagger}a^{\dagger}a + 2 b^{\dagger}
\mathcal{T}^{(1)}_n := |v_{13}|^{1/2}\bigl[ H_0 + k_n^2\bigr]^{-1}|v_{13}|^{1/2}, Q_n := \Bigl\{1 - \lambda_n |v_{13}|^{1/2}\bigl[H_0 + k_n^2\bigr]^{-1}|v_{13}|^{1/2}\Bigr\}^{-1}.
\left( \begin{array}{c c}{\partial_{1}}&{\partial_{2}}\\{\partial_{4}}&{\partial_{3}}\\ \end{array}\right) \mapsto \left( \begin{array}{c c}{\partial_{1}q^{\alpha L_{6}}}&{\partial_{2}}\\{\partial_{4}}&{\partial_{3}q^{- \alpha L_{6}}}\\ \end{array}\right) \equiv \tilde{\partial}\quad ,
\alpha_j =n(p-1)+pa_{j-1}=n(pq-1)+pq\alpha_{j-2}=n(pq-1)\sum\limits_{k=0}^{(j-3)/2}(pq)^k+(pq)^{\frac{j-1}{2}}\alpha_1 &=(n+\alpha_1)(pq)^{\frac{j-1}{2}}-n=\left(n+\frac{(n-1)p}{2}\right)(pq)^{\frac{j-1}{2}}-n,
\sum_e \mathbb{E}_\pi \sum_{j=1}^\infty (\Delta_{e,j}G)^2 \leq \frac{8}{\# B_m}\sum_{z \in B_m}\sum_e \mathbb{E}_{e^c}\left[ \int_{[I,D_{z,e})}(1-\log_2 F(t_e))\mathbb{E}_{\pi_e}\left[ L_z (1+N)\right]~\mu(t_e) \right]
\hat{U}^{\dagger}( t_{2}, t_{1}) = \hat{U}^{- 1}( t_{2}, t_{1}) = \hat{U}( t_{1}, t_{2}) , \hat{U}( t_{3}, t_{2}) \hat{U}( t_{2}, t_{1}) = \hat{U}( t_{3}, t_{1}) \ ,
P_{Y_{j}| Y^{j-1}, \underline S}( 1| y^{j-1}\in \beta_k^{j-1}, \underline S) &=P_{Y_{j}| Y^{j-1}, \underline S}( 0| y^{j-1}\in \beta_k^{j-1}, \underline S) = \frac{1}{2},
A_4^2 =kA_0+\frac{(k-m) k}{2m (n-1)}A_2+\frac{k^2}{2m n}A_3, A_4A_5 =A_5A_4=kA_1+\frac{(k-m) k}{2m (n-1)}A_2+\frac{k^2}{2m n}A_3, A_5^2 =kA_0+\frac{(k-m) k}{2m (n-1)}A_2+\frac{k^2}{2m n}A_3.
\kern-1em 3R_{ijkl}h_{il}h_{jk}+R_{jl}h_{lk}h_{jk}&=2(h_{11}^2+h_{22}^2+h_{33}^2+3h_{11}h_{22}+3h_{11}h_{33}+3h_{22}h_{33})-2(h_{12}^2+h_{13}^2+h_{23}^2) &=-(h_{11}^2+h_{22}^2+h_{33}^2)-2(h_{12}^2+h_{13}^2+h_{23}^2)=-|h|^2.
\Psi_1(0,y)={}_2F_1\bigg[\begin{matrix}-a \frac12(1-y) &1\end{matrix}\bigg|\,z\bigg]=(1-z)^a{}_2F_1\bigg[\begin{matrix}-a \frac12(1+y) &1\end{matrix}\bigg|\,\frac z{z-1}\bigg]=\Psi_2(0,-y),
\displaystyle\prod_{k}d\tau_{k}\prod_{i}ds_{i}\left[ \det(J^{\sigma}_{ij}) \det \left( k^{i}, ( 1 - P \frac{1}{P^{\dag}P}P^{\dag}) k^{j}\right){\cal D}\mbox{et}(P^{\dag}P ) \right]^{\frac{1}{2}}\; ,
\rho \ddot{u}_i = \big(A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L}- B_{iJKj}\dot{u}_{j})_{,K}\big)_{,J}+E_{ij}\dot{u}_{j}
\frac{\lambda_{\ell}^{2}}{8}A_{0,\ell}\, q(\mathbf{0})-(\mathbb{E}\left[{\cal N}^c(f_{\ell}) \right])^{2}= \frac{1}{8}\left[ 2 \ell^3 +\frac{2 \cdot 3^2}{\pi^2}\ell^2 \log \ell \right] \, q(\mathbf{0}) +O(\ell^2).
1-\alpha\lambda_\mathrm{n_0}^{(3)}= \mathcal{O}(\delta^\mathrm{h}) \iff \omega^2 = \omega_0^2 - \omega_\mathrm{p}^2\dfrac{\varepsilon_\mathrm{m}+\lambda_\mathrm{n_0}^{(3)}}{\lambda_\mathrm{n_0}^{(3)}(\varepsilon_\infty^{-1}\varepsilon_\mathrm{m}-1)-\varepsilon_\mathrm{m}}+ \mathcal{O}(\delta^\mathrm{h})
\sum_{|j| \leq m_0}q_{\delta,j,m}= F(z+\pi i \theta_1,0) - \int_{-1/2}^{1/2}F(z+\pi i \theta_1 + 2 \pi i u) + O\left( \frac{1}{m_0 - |z+u|}\right) \mathrm{d}u + O(m_0^2/m) &= F(z+\pi i \theta_1,0) - \int_{-1/2}^{1/2}F(z+\pi i \theta_1 + 2 \pi i u) \mathrm{d}u + O(1/m_0) + O(m_0^2/m),
dV^q_t=qV^{q-1}_{t-}dV_{t,(c)}+V^{q}_t-V^{q}_{t-}= qV^{q-1}_{t-}u_t dt+\left(V_{t-}+\Delta V_t\right)^q-V^{q}_{t-}= qV^{q}_{t-}u_t/V_{t-1}dt+ \sum_{k=1}^{q}{q \choose k}\left(\Delta V_t\right)^{k}V^{q-k}_{t-},
a u_{xx}+ b u_{x}+ c u - u_{t}=f~~Q_{T}a(0,t)u_{x}(0,t)=\chi_{1}(t),~0 \leq t \leq T a(1,t) u_{x}(1,t) = \chi_{2}(t),~0 \leq t \leq T u(x,0)=\phi(x),~0 \leq x \leq 1.
\begin{array}{l}{{\alpha_{i}\alpha_{j}^{*}( b_{i}b_{j}^{\dagger}- q_{i j}b_{j}^{\dagger}b_{i}) + \beta_{i}^{*}\alpha_{j}^{*}( b_{i}^{\dagger}b_{j}^{\dagger}- q_{i j}b_{j}^{\dagger}b_{i}^{\dagger}) + \alpha_{i}\beta_{j}( b_{i}b_{j}- q_{i j}b_{j}b_{i})}}\\{{+ \beta_{i}^{*}\beta_{j}( b_{i}^{\dagger}b_{j}- q_{i j}b_{j}b_{i}^{\dagger}) = \delta_{i j}.}}\\ \end{array}
g_i(x^*)\not=0\ \Rightarrow\g_i(x^k)\not=0\ \Rightarrow\ \lambda_i^k=0, x_i^*\not=0\ \Rightarrow\x_i^k\not=0\ \Rightarrow\ \gamma_i^k=0, \lambda_i^*\not=0\ \Rightarrow\ \lambda_i^k\not=0\ \Rightarrow\g_i(x^k)=0, \gamma_i^*\not=0,\ \Rightarrow\ \gamma_i^k\not=0\ \Rightarrow\x_i^k=0,
\frac{\delta < H_{A}^{\theta}>}{\delta G ( k )}= \frac{1}{4}[ k^{2}- G^{- 2}( k ) + \frac{4 \pi^{2}}{g^{2}}( 2 z k^{- 2}G^{- 3}( k ) - \frac{\delta z}{\delta G ( k )}4 \pi^{2}\int \frac{d^{2}p}{( 2 \pi )^{2}}p^{- 2}G^{- 2}( p ) ) ]
\langle \psi^{I_{1}}_{1}(x_{1})\cdots\psi^{I_{n}}_{n}(x_{n})\rangle=e^{i(\kappa_{1}+\cdots+\kappa_{n})\Omega}\langle \psi^{I_{1}}_{1}(x_{1})\cdots\psi^{I_{n}}_{n}(x_{n})\rangle\,,
S_{E}\simeq \left( - \frac{2{\cal V}( \phi_{0})}{n - 1}+ \left( \frac{n}{( n - 1 )}\right)^{1 / 2}{\cal V},_{\phi_{0}}\right) \frac{I_{n}}{H^{( n + 1 )}}V o l ( \mathrm{S}^{n}) ,
\partial_{s}\int_{\partial\Omega_{s}}|f|^{2}=-\int_{\partial\Omega_{s}}(\partial_{n}f\overline{f}+f\overline{\partial_{n}f})-\int_{\partial\Omega_{s}}\kappa(1-\kappa s)^{-1}|f|^{2}=-2\iint_{\Omega_{s}}|\nabla f|^{2}-\int_{\partial\Omega_{s}}\kappa(1-\kappa s)^{-1}|f|^{2},
\langle H_y \rangle_{u_{\alpha}\otimes 2\alpha^{3/2}\eta\Phi_*^1}= 4 \alpha^3|\eta|^2 \|\tilde{\Phi}_*^1\|^2_* + O(\alpha^4) = 4 \alpha^3|\eta|^2 \|\Phi_*^1\|^2_* + O(\alpha^4),
d_{TV}(\mathbb{P}_{\mathcal{P}_n}, \mathbb{P}_{Gen(n)}) \le d_{TV}(\mathbb{P}_{\mathcal{P}_n}, \mathbb{P}_{\mathcal{GS}_n}) + d_{TV}(\mathbb{P}_{\mathcal{GS}_n}, \mathbb{P}_{Gen(n)}) = e^{-\Theta(n)}.
C_1(q, N) =\lim_{z\to1}z\sum_{n=1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(-1)^{n+1}(q)_nq^{n(n+1)/2}}{(q)_{n+N}}\frac{1-q^n}{(1-zq^n)^2}&=\sum_{n=1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(-1)^{n+1}(q)_nq^{n(n+1)/2}}{(q)_{n+N}(1-q^n)}.
A(t)+B(t) = \varphi(-t)^2(G(-t)H(t)+G(t)H(-t)) = \varphi(-t)^2\frac{2\psi(t^2)}{\varphi(t^2)}= \varphi(t^2)\phi(t)\frac{2\varphi(t^4)^2/\varphi(t^2)}{\varphi(t^2)}= \frac{\phi(t)}{\varphi(t^2)}2\varphi(t^4)^2 = 2\varphi(t^4)^2(G(t)G(t^4)+tH(t)H(t^4))
( m , \tilde{m}= m + \frac{i k}{( k , 2 k^{\prime})}, n = 1 , p = k^{\prime}- \frac{i k^{\prime}}{( k , 2 k^{\prime})}\mathrm{~ o r ~}2 k^{\prime}- \frac{i k^{\prime}}{( k , 2 k^{\prime})}) .
f ( x ) \to{\hat{\mathcal O}}_{r}( f ) = \int \frac{d p_{1}d p_{2}}{2 \pi \hbar}{\tilde{f}}_{r}( p ) \exp [ \frac{i}{\hbar}( p_{1}{\hat{x}}_{0}^{1}+ p_{2}{\hat{x}}_{0}^{2}) ]
R^{(0)}=-\sum_{i,j}\frac{c_ic_j\prod_{k\ne i}(1-\xi^k\bar{\xi}^j)\prod_{k\ne j}(1-\xi^i\bar{\xi}^k)}{(1-\xi^i\bar{\xi}^j)\prod_{k\ne i}(\xi^k-\xi^i)\prod_{k\ne j}(\bar{\xi}^k-\bar{\xi}^j)}
\varepsilon_{a j k}\left\{L_{k}R_{b}, L_{j}, R_{3}\right\}= i \hbar \varepsilon_{a j k}\varepsilon_{k j m}L_{m}\left( R_{b}R_{3}- \frac{1}{2}\left\{R_{b}, R_{3}\right\}\right) = \frac{1}{2}\left( i \hbar \right)^{2}\varepsilon_{a j k}\varepsilon_{k j m}L_{m}\varepsilon_{b 3 c}R_{c}= \hbar^{2}L_{a}R_{c}\varepsilon_{b 3 c}\; .
\sum_{k = 0}^{n}\frac{x^k y^{n - k}}{\binom{n}{k}}= \frac{n + 1}{(x + y) \left(\frac{1}{x}+ \frac{1}{y}\right)^{n + 1}}\sum_{k = 1}^{n + 1}\frac{\left(x^k + y^k\right) \left(\frac{1}{x}+ \frac{1}{y}\right)^k}{k}
\begin{aligned}\mathfrak{F}_{U(3)}=4\frac{(k_1-k_2)^2}{(k_1+k_2)^2}\{(k_1+k_2)dz_1\odot dz_1^* -i(k_1-k_2)dz_1\wedge dz_1^*\}+4\frac{(k_1-k_3)^2}{(k_1+k_3)^2}\{(k_1+k_3)dz_2\odot dz_2^* -i(k_1-k_3)dz_2\wedge dz_2^*\}+4\frac{(k_2-k_3)^2}{(k_2+k_3)^2}\{(k_2+k_3)dz_3\odot dz_3^* -i(k_2-k_3)dz_3\wedge dz_3^*\}\end{aligned}
u = q^3 \prod_{\substack{n \geq 1 n \equiv \pm 1,\pm 2 (18)}}(1-q^n) / \prod_{\substack{n \geq 1 n \equiv \pm 7,\pm 8 (18)}}(1-q^n) v = q^2 \prod_{\substack{n \geq 1 n \equiv \pm 1,\pm 4 (18)}}(1-q^n) / \prod_{\substack{n \geq 1 n \equiv \pm 5, \pm 8 (18)}}(1-q^n).
\nabla\cdot\mathbf{A}=\frac{1}{r^{2}\sin\theta}\left[\sin\theta\frac{\partial\left(r^{2}A_{r}\right)}{\partial r}+r\frac{\partial\left(\sin\theta A_{\theta}\right)}{\partial\theta}+r\frac{\partial A_{\phi}}{\partial\phi}\right]
\psi_{0}= \varphi - \intop_{0}^{\infty}\left[ \bar{\omega}(\zeta_{s}, \varepsilon) - \bar{\omega}(\eta_{s}, \varepsilon) + \varepsilon^{5/2}\left( H (\zeta_{s}, \psi_{s}, \varepsilon) - H (\eta_{s}, \phi_{s}, \varepsilon) \right) \right] \mathrm{d}s,
\liminf \frac{r_{n}+1}{\log(h_{n})}\geq \liminf \frac{r_{n}+1}{n\log(r_{n}+1)}\geq \liminf \frac{r_{n}}{nr_{n}^{1/3}}= \liminf \frac{r_{n}^{2/3}}{n}\geq \liminf \frac{n^{4/3}}{n}= \infty.
\int_{S^{n-1}}(-L_K z) z dV_K = \frac{1}{n-1}\int_{S^{n-1}}h_K ((D^2 h_K)^{-1})^{i,j}z_i z_j dV_K & = \frac{1}{n (n-1)}\int_{S^{n-1}}((D^2 h_K)^{-1})^{i,j}(h_K z_i) (h_K z_j) dS_K ,
{\frac{C_{X}^{2}}{( 6 4 \pi^{2})^{2}}}\r ( C_{X}, C_{Y})^{2}= \lambda_{X}, \quad{\frac{C_{Y}^{2}}{( 6 4 \pi^{2})^{2}}}\r ( C_{X}, C_{Y})^{2}= \lambda_{Y},
\int_{\mathbb{R}^{n-1}}\int_{\mathbb{R}^{n-1}}a(x-y)\:b((1-t)y+tx)\:\textup{d}x\textup{d}y = \|a\|_{L^1(\mathbb{R}^{n-1})}\|b\|_{L^1(\mathbb{R}^{n-1})}
A_{2,\ell}=-\frac{16}{\lambda_{\ell}^3}\int_{C/\ell}^{\pi -C/\ell}\frac{\beta^2_{1,\ell}(\phi)}{(1-4 \alpha_2^2/\lambda_{\ell}^2)^{1/2}(1-4 \alpha_1^2/\lambda_{\ell}^2)^{3/2}}\sin \phi d \phi=-16 \int_{C/\ell}^{\pi -C/\ell}\frac{\beta^2_{1,\ell}(\phi)}{\ell^6}\sin \phi d \phi+O(\ell^{-2})=O(\ell^{-2}).
P\left(\frac{n-A_n^*}{f_1(n)}\leq\varepsilon\right)=P(A_n^*\geq n-\varepsilon f_1(n))=P(A_n(t)>t,\quad\forall t=a_n,\ldots,\lfloor n-\varepsilon f_1(n)\rfloor).