harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[\frac{k_4^2-k_5^2-\left[\frac{1-2n}{2}-\frac{1}{2k_3}\left(k_2-\sqrt{(k_3-k_2)^2-4A}\right)\right]^2}{2\left[\frac{1-2n}{2}-\frac{1}{2k_3}\left(k_2-\sqrt{(k_3-k_2)^2-4A}\right)\right]}\right]^2-k_5^2=0.
c_{\frac{n+1}{2}}= \left(x_{\frac{n-1}{2}}x_{\frac{n+3}{2}}\right)x_{\frac{n+1}{2}}= x_{\frac{n-1}{2}}\left(x_{\frac{n+3}{2}}x_{\frac{n+1}{2}}\right) = (-1)^{\frac{n+1}{2}}x_{\frac{n-1}{2}}z = (-1)^{\frac{n+1}{2}}b_{\frac{n-1}{2}}= 0.
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(6n+4)q^n =\dfrac{f_3^3}{f_1^3}\Bigg(2\dfrac{f_2^2f_4^4}{f_1^{10}}\left(4\dfrac{f_2^6f_3}{f_1^3f_{6}^2}a(q^2)-a(q^2)-a^2(q^2)+36q\dfrac{f_2^4f_{6}^4}{f_1^2f_3^2}\right) &\quad+\dfrac{f_2^{14}}{f_1^{14}f_4^4}\left(12\dfrac{f_2^8}{f_1^4}-3\dfrac{f_2^2f_{6}^2}{f_1f_3}\right)\Bigg).
\mathbb{E}(X_iX_j)= f_k(j)\bigg(f_k(i)+O_{\alpha}\Big(\frac{\tau_3(i)}{\sqrt{i}}\Big)\bigg)+O_{\alpha}\Big(\frac{\tau_3(j)}{\sqrt{j-i}}\Big) = f_k(i)f_k(j)+O_{\alpha}\Big(\frac{\tau_3(j)\tau_3(i)}{\sqrt{i}}\Big)+O_{\alpha}\Big(\frac{\tau_3(j)}{\sqrt{j-i}}\Big),
\overline{g}_i := in_{(0,\ldots,0,-e,1,\ldots,1,1+e)}(g_i) = g_i i=1,\ldots , \ell-1 \overline{g}_\ell := in_{(0,\ldots,0,-e,1,\ldots,1,1+e)}(g_\ell) = w_1\cdot \ldots \cdot w_n
\det(B_{ij})^{n+1}_{i,j=1}= 2\det(x^{n-j+1}_i+x^{-(n-j+1)}_i-x^{n-j}_i-x^{-(n-j)}_i)^{n}_{i,j=1}= 2(x^{\frac{1}{2}}_i-x^{-\frac{1}{2}}_i)\det(x^{n-j+\frac{1}{2}}_i-x^{-(n-j+\frac{1}{2})}_i)^{n}_{i,j=1}.
\frac{\lambda_1}{\mu_1}\sum_{n=0}^\infty\sum_{h=1}^{k_1+k_2}\phi^{(1)}_{n,h}(\xi) - \sum_{n=0}^\infty\sum_{h_1=1}^{k_1}\phi^{(1)}_{n,h_1}(\xi)= \frac{\lambda_1\mu_2}{\mu_1^2}P_0'(\xi),
\big\langle \delta\mathbf{N}, \mathbf{r}_{ij}\big\rangle = -\big\langle u^l\mathbf{r}_l, (h_{ij}\mathbf{N}+ \Gamma_{ij}^k \mathbf{r}_k - g_{ij}k_0\mathbf{r}) \big\rangle = g_{ij}k_0u.
\begin{aligned}G_k(\theta,z_1,\ldots,z_k) &=\int Q_{z_k}(dz)G_{k-1}(\theta,z_1,\ldots,z_{k-2},z_{k-1}+e^{s_{k-1}-s_{k}}z)\,F(\theta,(1-e^{s_{k-1}-s_k})z).\end{aligned}
\eta(\frac{3\xi_{1}^{2}+\xi_{2}^{2}}{2\xi_{1}^{2}(\xi_{1}^{2}-\xi_{2}^{2})}c_{\eta}(0,\xi_{1})-\frac{n_{\eta}^{\prime}(\xi_{1})}{2\xi_{1}^{2}})+(1\leftrightarrow2)
\mathbf{d}\int\omega_{1}...\omega_{k}& =\sum_{i=1}^{k}\left( -1\right)^{p_{1}+...+p_{i-1}+i}\int\omega_{1}...\omega_{i-1}\left( \mathbf{d}\omega_{i}\right) \omega_{i+1}...\omega_{k}& +\sum_{i=1}^{k-1}\left( -1\right)^{p_{1}+...+p_{i}+i+1}\int\omega_{1}...\omega_{i-1}\left( \omega_{i}\wedge\omega_{i+1}\right) \omega_{i+2}...\omega_{k}
\sigma_{\widetilde{i+1}}=L_{\widetilde{i+1}}\pi\stackrel{\eqref{piLx}}{=}L_{\widetilde 1}\pi L_{\widetilde i}=\sigma_{\widetilde 1}\sigma_{\widetilde i}\pi^{-1}= \sigma_{\widetilde 1}\sigma^i_{\widetilde 1}\sigma_{\widetilde 0}^{1-i}\sigma_{\widetilde 0}^{-1}=\sigma^{i+1}_{\widetilde 1}\sigma_{\widetilde 0}^{-i},
\left(f_{M,M'}\circ (E_i\otimes L_i^{-1})\right)_{|_{M_{\lambda,\mu}\otimes M'_{\lambda',\mu'}}}= q^{\langle\lambda+\alpha_i,\mu'\rangle-\langle\alpha_i,\mu'\rangle}(E_i\otimes 1)_{|_{M_{\lambda,\mu}\otimes M'_{\lambda',\mu'}}}.
C_\pm K \gamma^{(2n+1)}\varphi =C_\pm\overline{\gamma^{(2n+1)}}\overline{\varphi}=C_\pm \gamma^{(2n+1)}C_\pm C_\pm \overline{\varphi}=(-1)^n\gamma^{(2n+1)}C_\pm\overline{\varphi}=(-1)^n\gamma^{(2n+1)}C_\pm K\varphi,
S_{j}( \lambda ) t_{j}S_{i}( \lambda^{\prime}) - S_{j}( \lambda^{\prime}) t_{j}S_{i}( \lambda ) ={\frac{\prod_{k \neq i , j}( \lambda - \lambda_{k}) \prod_{k \neq i , j}( \lambda^{\prime}- \lambda_{k})}{\prod_{k \neq j}( \lambda_{j}- \lambda_{k}) \prod_{k \neq i}( \lambda_{i}- \lambda_{k})}}( \lambda_{i}- \lambda_{j}) ( \lambda - \lambda^{\prime})
\begin{aligned}F(\{x_{i,r}\}_{1\leq i\leq n}^{1\leq r\leq k_{i}})=0 x_{i,1}=v^{4}x_{i,2}=v^{2}x_{i+1,1}1\leq i\leq n-1, x_{i,1}=v^{4}x_{i,2}=v^{2}x_{i-1,1}2\leq i\leq n-1, x_{n,1}=v^{2}x_{n,2}=v^{4}x_{n,3}=v^{2}x_{n-1,1}.\end{aligned}
L = E^{- 1}\left( \begin{array}{c c}{{\frac{\sqrt{1 - 3 V_{o o}}}{\sqrt{1 + V_{o o}}}}}&{{0}}\\{{0}}&{{\frac{\sqrt{1 - V_{e e}}}{\sqrt{1 + 3 V_{e e}}}}}\\ \end{array}\right) E^{- 1}\, .
p_0 = (3 t^8 + 28 t^6 - 14 t^4 + 28 t^2 + 3)^3 p_{\infty}= 3^3 (t^6 - 5 t^4 - 5 t^2 + 1)^4 p_1 = 2^4 (9 t^{11}+ 11 t^9 + 66 t^7 - 66 t^5 - 11 t^3 - 9 t)^2 .
\begin{aligned}f(x) \,\to\,\min\limits_{x,y,z}g_i(x) \,\leq\,0 \qquad i\in\mathcal M h_j(x) \,=\,0 j\in\mathcal P y_l,z_l \,\leq\,0 l\in\mathcal Q (G_l(x)-y_l)(H_l(x)-z_l) \,=\,0 l\in\mathcal Q \end{aligned}
\langle \rho_G, \mu_{S_2}\rangle - \langle \rho_G, \mu \rangle = (\langle \rho_G, \mu_{S_1}\rangle - \langle \rho_G, \mu \rangle ) + (\langle \rho_{M_{S_1}}, \mu_{S_2}\rangle - \langle \rho_{M_{S_1}}, \mu_{S_1}\rangle).
\sum_{\sigma=0}^{p^k-1}\mathcal{C}(\mathbf{a}_\sigma^{t_1},\mathbf{a}_\sigma^{t_2})(\tau)=\sum_{r=0}^{p^{m}-\tau-1}\sum_{\sigma=0}^{p^{k}-1}\omega_q^{\left(a^{t_1}_{\sigma,r}-a^{t_2}_{\sigma,r+\tau}\right)}=0.
\vec{x}_i = \begin{cases}(x_{1,1}, \dots, x_{1,i-2}, 1.05x_{1,i-1}, x_{1,i}, \dots, x_{1,n})^\top x_{1,i-1}\neq 0\,, (x_{1,1}, \dots, x_{1,i-2}, 0.00025, x_{1,i}, \dots, x_{1,n})^\top \end{cases}
\triangle ( \tau^{\prime}, \tau )^{a b}= \theta^{a b}\frac{1}{2 \pi}\sum_{n}n e^{i n ( \tau^{\prime}- \tau )}\Rightarrow \triangle ( \tau , \tau^{\prime})_{a b}^{- 1}= \theta_{a b}\frac{1}{2 \pi}\sum_{n}\frac{1}{n}e^{i n ( \tau - \tau^{\prime})}\ .
\Psi^{u,w}(v) = \Psi^{u_1u_2,w}(v) &= \Psi^{u_1,w}(u_2v) + u_1\Psi^{u_2,w}(v) &= \Psi^{u_1,w_2w_1}(u_2v) + u_1\Psi^{u_2,w_2w_1}(v) &= \left(\Psi^{u_1,w_1}(u_2vw_2) +\Psi^{u_1,w_2}(u_2v)w_1\right) +u_1\left(\Psi^{u_2,w_1}(vw_2) + \Psi^{u_2,w_2}(v)w_1\right).
y \in \mathcal{L}(Y)\mkern33mu\iff\mkern35mu\hat{y}_-\mkern-2mu = \hat{y}_+\mkern-3mu = y_1^{\textsc{g}}\mkern20mu\mkern20muy_1^{\textsc{d}}\mkern-1.2mu\in\mkern-1.2mu{\mathbb{Z}}\mkern9mu\mkern8.5muy \mkern-1.2mu\in\mkern-1.2mu{\mathbb{Z}}\mkern-1.5mu\cup\mkern-2.5mu \{\infty\}.
\widetilde{\mathcal{P}}_{N}(z) =\frac{1}{(\pi\sqrt{1-\tau^2})^{N}\prod_{j=1}^{N}j!}\exp\!\Big\{-\frac{1}{1-\tau^2}\sum_{j=1}^{N}\left(|z_{j}|^2-\frac{\tau}{2}\big(z_{j}^{2}+\overline{z}_{j}^{2}\big)\right)\Big\}\, \prod_{j<k}|z_k-z_j|^2,
g((\overline{\nabla}^{\ast}_{X}A^{\ast})_{N}Y,Z)-g((\overline{\nabla}^{\ast}_{Y}A^{\ast})_{N}X,Z) =\widetilde{c}[\eta^{\ast}(X)g(Y,\varphi Z)-\eta^{\ast}(Y)g(X,\varphi Z) &\quad+\eta^{\ast}(Z)\{g(\varphi X,Y)-g(X,\varphi Y)\}],
Var(\hat{\pi}_{h,n}(x)) \le \frac{c}{T_n}\big[ \frac{1}{\prod_{l =1}^d h_l}(\delta_1 + \Delta_n) + \frac{1}{\prod_{l \ge 3}h_l}\log (\frac{\delta_2}{\delta_1}) + \delta_2^{1 - \frac{d}{2}}+ D + \frac{1}{(\prod_{l = 1}^d h_l)^2}e^{- \rho D}\big].
\int_0^{(1+t)^{\frac{1}{2}-\theta_*}}u_*(r,t) \nu_d(r)r^{d-1}\,dr =O(t^{-\frac{d}{2}})\int_0^{(1+t)^{\frac{1}{2}-\theta_*}}\nu_d(r)r^{d-1}\,dr =O(t^{-\frac{d}{2}})O(t^{\frac{d}{2}-d\theta_*})=O(t^{-d\theta_*})
\Delta^{ab}_{\mu\nu}=g^2\delta^{ab}\frac{\delta_{\mu\nu}}{p^2}-g^4N\frac{\Gamma(2-\omega) \Gamma(\omega)\Gamma(\omega-1)}{(4\pi)^{\omega}\Gamma(2\omega)}\cdot 4(2\omega-1)\delta^{ab}\frac{\delta_{\mu\nu}-p_\mu p_\nu/p^2}{p^{6-2\omega}}.
\sum_{t=0}^{m-k-1}\binom{m}{t+k+1}t!S_{a_{1},b_{1}+a_{1}m}^{a_{2},b_{2}+a_{2}m,p_{2}}\left( p_{1},t\right)=\sum_{t=k+1}^{m}\binom{t-1}{k}\left( b_{1}+a_{1}t\right)^{p_{1}}\left(b_{2}+a_{2}t\right)^{p_{2}}.
D^+_zF^2 =\bigl(D^+_zF\bigr)^2+2F D^+_zF D^+_zF^3 =\bigl(D^+_zF\bigr)^3+3F^2 D^+_zF+3F\bigl(D^+_zF\bigr)^2 D^-_zF^2 =-\bigl(D^-_zF\bigr)^2+2F D_z^- F D^-_zF^3 =\bigl(D^-_zF\bigr)^3+3F^2D^-_zF-3F\bigl(D^-_z F\bigr)^2\,.
H\left( \theta,\theta^{\ast}\right) =\theta^{4}+\theta^{\ast4}-\frac{7}{8}\left( \theta-\theta^{\ast}\right) \left( \theta+\theta^{\ast}\right)^{2}\left( \theta-\theta^{\ast}\right) \in\mathbb{R}\left\langle\theta,\theta^{\ast}\right\rangle .
|M_{p}^{n+1}(\varphi )(z)-M_{p}^{n}(\varphi )(z)| =\left| \sum_{\omega \in B_{n+1,z}, \omega |_{n}\in C_{n,z}}p_{\omega}+\sum_{\omega \in C_{n+1,z}}p_{\omega}\varphi (h_{\omega}(z))-\sum_{\gamma \in C_{n,z}}p_{\gamma}\varphi (h_{\gamma}(z))\right| .
\Delta_{0}\phi = g^{ij}(0,x)(\partial_{i}\partial_{j}\phi - \Gamma_{ij}^{k}(0,x)\partial_{k}\phi) Q(\phi) = Q^{ij}s^2\partial_{i}\partial_{j}\phi + Q^{A}s\partial_A \phi Q^{i,j}, Q^{A}\in C^{\infty}([0,\epsilon)\times \partial M).
-g'(x) - g^2(x) = -\left(g_1'(x) - \frac{C(1 + \epsilon)}{x^{2 + \epsilon}}\right) - \left(g_1^2(x) + \frac{2Cg_1(x)}{x^{1 + \epsilon}}+ \frac{C^2}{x^{2 + 2\epsilon}}\right) &= f_1(x) + \frac{C}{x^{2 + \epsilon}}\left[1 + \epsilon - 2x g_1(x) - \frac{C}{x^\epsilon}\right].
\mathcal C_\bullet(K\cap M,\Bbbk)=\mathcal C_\bullet(K,\Bbbk) \cap \mathcal C_\bullet(M,\Bbbk), \mathcal C_\bullet(K\cup M,\Bbbk)=\mathcal C_\bullet(K,\Bbbk) + \mathcal C_\bullet(M,\Bbbk)
\hat{g_j}^*(y) + \varepsilon/2 \ge \left(3\pi y\sqrt{16a_j/b_j}+9\pi\left(\sqrt{b_j/(16a_j)}+1\right)(16a_j/3)\right)^{-1}&=\left(12\pi(y/b_j)+12\pi+24\sqrt2\pi(2a_j/b_j)^{1/2}\right)^{-1}(a_jb_j)^{-1/2}.
\frac{d^{l-m}}{dx^{l-m}}(x^{2}-1)^{l}=\sum_{j=0}^{l-m}(\begin{matrix}l-m\\j\end{matrix})\frac{d^{l-m-j}}{dx^{l-m-j}}(x+1)^{l}\frac{d^{j}}{dx^{j}}(x-1)^{l}
i + j \geq \frac{(p^k - 1)r - b}{p^k}= \frac{p^k r - r - b}{p^k}\geq \frac{p^k r - r - r}{p^k}= \frac{(p^k - 2) r}{p^k}, &\geq \frac{(p^k - 2) (p^k - 1)p^{\ell - k + 1}}{p^k}= \frac{p^{\ell + 1}(p^{2k}- 3p^k + 2)}{p^{2k}}\geq \frac{p^{\ell + 1}(p^{2k}- 3p^k)}{p^{2k}}&= p^{\ell + 1}\left(1 - \frac{3}{p^k}\right).
\frac{x_n}{\lambda_2^n}=x_1\left(\frac{\lambda_1}{\lambda_2}\right)^n+\frac{x_2-\lambda_1x_1}{\lambda_2}\left[\frac{1-\left(\frac{\lambda_1}{\lambda_2}\right)^n}{1-\frac{\lambda_1}{\lambda_2}}\right].
\min_{\beta_{\ell}\in\mathbb{R}}~~ \left\{\sum_{j=0}^N\omega_j\left(\sum_{\ell=0}^{L}\beta_{\ell}\tilde{\Phi}_{\ell}(x_j)-f(x_j)\right)^2+\lambda\sum_{\ell=0}^L|\mu_{\ell}\beta_{\ell}|\right\},\quad\lambda>0,
\begin{cases}\qquad\quad\:\ \ \ \phi\ =\ \lambda^3f^{123}\mp\lambda\mu^2(f^1\wedge e^1+f^2\wedge e^2+f^3\wedge e^3) \ \psi\ :=\ *_\phi\phi\ =\ \mu^4e^{4567}-\lambda^2\mu^2(e^1\wedge f^{23}+e^2\wedge f^{31}+e^3\wedge f^{12})\end{cases}\ .
\tilde{Y}_{1}\hat{R}dK_{1}\hat{R}= \hat{R}dK_{1}\hat{R}\tilde{Y}_{1}\quad \Longleftrightarrow \quad \hat{R}dK^{\dagger}_1 \hat{R}Y_1= Y_1 \hat{R}dK^{\dagger}_1 \hat{R}\;;
Q(f_{\alpha}^{+}, f_{\alpha}^{-}, \chi) ={L(1,\eta)L(1, \pi,{\rm ad})\over \zeta_{F}(2) L(1/2, \pi_{E}, \chi)}\int_{E^{\times}/F^{\times}}\chi(t) (\pi(t)f_{\alpha}^{+}, f_{\alpha}^{-})\,{dt\over |d|^{1/2}}.
B_0 : = \bigg( \int_0^{\infty}V^{- \frac{r}{p}}(t)\bigg( \int_0^t U^q(\tau) w(\tau)\,d\tau \bigg)^{\frac{r}{p}}U^q (t) w(t)\,dt \bigg)^{\frac{1}{r}}, B_1 : = \bigg( \int_0^{\infty}W_*^{\frac{r}{p}}(t) \bigg( \int_0^t \bigg( \frac{U(\tau)}{V(\tau)}\bigg)^{p'}v(\tau)\,d\tau \bigg)^{\frac{r}{p'}}w(t)\,dt \bigg)^{\frac{1}{r}};
\psi_{\Omega^1}(\frac{d\hat{u}}{\hat{u}}) = \psi_{\Omega^1}(\Delta_{LT}(\hat{u})d\log_{LT}) = \psi_{\Omega^1}(\Delta_{LT}(\hat{u}) \varphi_{\Omega^1}(d\log_{LT})) = \psi_L(\Delta_{LT}(\hat{u}))d\log_{LT}= \pi_L^{-1}\psi_{Col}(\Delta_{LT}(\hat{u}))d\log_{LT}= \Delta_{LT}(\hat{u})d\log_{LT}= \frac{d\hat{u}}{\hat{u}}\ .
\tau_{+}(\pi)= x_{1}\pi, \tau_{+}(T_{i}) = T_{i}, \tau_{+}(x_i) = x_{i},\forall\, i, \tau_{-}(\pi)=\pi,\tau_{-}(T_{i})=T_{i}, \tau_{-}(x_i)=\left( \prod_{a=i-1}^{1}T_{a}\right) \pi \left( \prod_{a=n}^{i}T_{a}\right) x_{i}.
f_1 =1728^{1/6}\eta^{4}K^{-1/6}\_{3}F_2(-1/6, 1/6, 1/2; 4/5, 1/5; K), f_2 = 1728^{-1/30}\eta^{4}K^{1/30}\ \_{3}F_2(1/30, 11/30, 7/10; 6/5, 2/5; K), f_3 =1728^{-19/30}\eta^{4}K^{19/30}\_{3}F_2(19/30, 29/30, 13/10; 9/5, 8/5; K).
( 3 d ) = - \frac{\lambda^{2}}{2^{2}. 3}i^{5}\int{\cal D}k_{1}{\cal D}k_{2}{\cal D}k_{3}{\cal D}k_{4}\frac{1}{k_{1}^{2}k_{2}^{2}k_{3}^{2}k_{4}^{2}( p + k_{1}+ k_{2}- k_{3}- k_{4})^{2}}\; .
X \Psi_{n_{1}n_{2}m}( \mu , \nu , \varphi ; \delta_{1}, \delta_{2}) = - \alpha \frac{n_{1}- n_{2}+ \frac{\delta_{1}- \delta_{2}}{2}}{n + \frac{\delta_{1}+ \delta_{2}}{2}}\Psi_{n_{1}n_{2}m}( \mu , \nu , \varphi ; \delta_{1}, \delta_{2}) ,
\Gamma(L\mid_{H_G},0)[u_\ast] =\int^\infty_{0}{\left\langle\dot{\widetilde{A}}(0,t)u_\ast(t),u_{\ast}(t)\right\rangle\,dt}+\int^0_{-\infty}{\left\langle\dot{\widetilde{A}}(0,t)u_\ast(t),u_{\ast}(t)\right\rangle\,dt}&=\int^\infty_{0}{\arctan(t)\langle J\dot{S}_0 u_{\ast}(t),u_{\ast}(t)\rangle\,dt}&=-\int^\infty_{0}{\arctan(t)(t^2+1)e^{-2t\arctan(t)}\,dt}<0,
G^{(\psi)}(z_1,z_2) = \frac{1}{16}\left[ \begin{array}{ll}\bar{D}^2_1\bar{D}^2_2 G_{++}(z_1,z_2) \bar{D}^2_1 D^2_2 G_{+-}(z_1,z_2) D^2_1\bar{D}^2_2 G_{-+}(z_1,z_2) D^2_1 D^2_2 G_{--}(z_1,z_2) \end{array}\right]
v_1(t) =1-\frac{e^{2\sigma_1(t-t_L)}}{e^{2\sigma_1(t-t_L)}+e^{-2\sigma_1(t-t_L)}}\left(\frac{2j\sigma_1}{\lambda-\omega_1+j\sigma_1}\right) v_2(t) =\left(\frac{2j\sigma_1}{\lambda-\omega_1+j\sigma_1}\right)\frac{e^{j\phi_L+2j\omega_1 t}}{e^{2\sigma_1(t-t_L)}+e^{-2\sigma_1(t-t_L)}}e^{-2j\lambda t}
\Lambda_{\Phi_0}(s) = s^{1/q_0}R_{\Psi_0}(s) = (4{p_0})^{\frac1{p_0}}s^{\frac{{p_0}-1}{{p_0}}}=(4{p_0})^{\frac1{p_0}}s^{\frac{1}{{q_0}}},
\psi_2'(b) = \xi''(b)q-b\xi''(b)+\frac{\xi''(b)(1-q)}{z_2}+\xi''(b)(1-q)-\frac{\xi''(b)(1-q)(1+z_2)[\xi'(1)-\xi'(q)]}{z_2[\xi'(1)-\xi'(q)+(\xi'(b)-\xi'(q))z_2]}&=\frac{\xi''(b)[(q-b)[\xi'(1)-\xi'(q)]+[\xi'(b)-\xi'(q)](1-q+(1-b)z_2 ) ]}{\xi'(1)-\xi'(q)+[\xi'(b)-\xi'(q)]z_2}.
\tilde x_{j+1}=\tilde x_j+P_j \tilde C^* \left( \tilde C P_j \tilde C^*+\frac{(t_{j+1}-t_a)(t_b-t_{j+1})}{t_b-t_a}R \right)^{-1}\left( \tilde y - \tilde C \tilde x_j \right)
l ({\bf v}) , \quad - i v_{0}\frac{\partial}{\partial{\bf v}}, \quad - i [ \frac{\partial}{\partial{\bf v}}+{\bf v}({\bf v}\frac{\partial}{\partial{\bf v}}) + \frac{3}{2}{\bf v}] , \quad - i v_{0}({\bf v}\frac{\partial}{\partial{\bf v}}+ \frac{3}{2})
\beta_{6 0}={\frac{2 8 8}{N \pi^{2}}}\lambda_{6 0}^{2}( 1 - \lambda_{6 0}) , \qquad \beta_{0 6}={\frac{2 8 8}{N r \pi^{2}}}\lambda_{0 6}^{2}( 1 - \lambda_{0 6}) .
\zeta_{\mathcal P_X^*}(s)\zeta_{\mathcal P_X^*}(2s)\zeta_{\mathcal P_X^*}(4s)\zeta_{\mathcal P_X^*}(8s)\cdots = \zeta_{\mathcal P_X}(s) \eta_{\mathcal P_X}(s)\eta_{\mathcal P_X}(2s)\eta_{\mathcal P_X}(4s)\eta_{\mathcal P_X}(8s)\cdots = \eta_{\mathcal P_X^*}(s)
N_{1}( t ) = \frac{N}{2}\left( \frac{\sigma_{1 2}- \sigma_{2 1}}{\sigma_{1 2}+ \sigma_{2 1}}\right) e^{- ( \sigma_{1 2}+ \sigma_{2 1}) t}+ N \left( \frac{\sigma_{2 1}}{\sigma_{1 2}+ \sigma_{2 1}}\right)
a_{1,n}(k) =-\bigl(a_n(k-1)+a_n(k)\bigr)\,\delta a_n(k-1), \delta a_{1,n}(k) =-\bigl(\delta a_n(k-1)+\delta a_n(k)\bigr)\,\delta a_n(k)-\bigl(a_n(k-1)+a_n(k)\bigr)\,\delta^2a_n(k-1).
t_{j}=\frac{\tan (\frac{\pi}{4}-\frac{\pi j}{2H})}{2\cos^{2}(\frac{\pi j}{2H})}\prod_{i=0}^{j-1}\frac{\tan (\frac{\pi (j-1-2i+w_{+})}{2H})\tan (\frac{\pi (j-1-2i+w_{-})}{2H})}{\tan^{2}(\frac{\pi (i+1)}{2H})}.
\mathrm{b o s o n s :}\; \lambda = \left( \begin{array}{c c}{{0}}&{{B}}\\{{B^{T}}}&{{0}}\\ \end{array}\right) , \mathrm{f e r m i o n s :}\; \lambda = \left( \begin{array}{c c}{{A_{1}}}&{{0}}\\{{0}}&{{A_{2}}}\\ \end{array}\right) ,
\eta(\tau_{n-i+j}, \tau_n^{q-1}\tau_{n-i+\ell+1})=(\tau_n \tau_1)^{i-j}\tau_n^{q-2+D(n-i+\ell+2, n-i+j+1)+D(n-i+\ell+1)}\tau_{n-j+\ell+1}.
\mathrm{Tr}_{d,D}(f) := \frac{1}{2\pi}\sum_{Q \in \mathcal{Q}_{dD}/\mathrm{SL}_2(\mathbb{Z})}\chi_D^{}(Q)\int_{\mathrm{SL}_2(\mathbb{Z})_Q \backslash S_Q}f(z)\frac{\sqrt{dD}dz}{Q(z,1)},
G_{1}(-rx_{1};r\vec{x},\vec{s}) = r^{-\frac{\sigma_{3}}{4}}\begin{pmatrix}1 0 -i\frac{x_{1}}{2}\sqrt{r}1\end{pmatrix}R(-x_{1})E_{-x_{1}}(-x_{1})\begin{pmatrix}\Psi_{1,11}\Psi_{1,12}\Psi_{1,21}\Psi_{1,22}\end{pmatrix},
S_2 = (-1)^{k-2+\lfloor\frac{k-2}{p}\rfloor}q(k-2)! \left( (\lfloor k/p\rfloor+\varepsilon)\binom{q/p-1}{\lfloor (k-2)/p\rfloor}-q/p\binom{q/p-2}{\lfloor (k-2)/p\rfloor-1}\right),
\liminf_{n\to\infty}\int_\Omega V\, u_n^2\, dx=\liminf_{n\to\infty}\left[\int_\Omega V_+\, u_n^2\, dx-\int_{\Omega}V_-\, u_n^2\, dx\right]\ge \int_\Omega V\, u^2\, dx.
\int_{x+[0,2]\times[-2,2]}y_1\left( \frac{1}{|x+he_2-y|^{2+2\alpha}}- \frac{1}{|x-y|^{2+2\alpha}}\right){\rm sgn}\left(y_2-x_2-\frac h2 \right) 2M(y_1) dy & = \int_{x+[-2,2]^2}y_1\left( \frac{1}{|x+he_2-y|^{2+2\alpha}}- \frac{1}{|x-y|^{2+2\alpha}}\right){\rm sgn}\left(y_1 \left(y_2-x_2-\frac h2 \right) \right) M(y_1) dy,
D_l = \mathfrak{H}R(z)=\bigg\{\xi\in \mathbb{C}\oplus \mathcal{L}:\xi = c(1,(E|R(z))+(0,(f|)\bigg\}D_r = R(z)\mathfrak{H}=\bigg\{\xi\in\mathbb{C}\oplus \mathcal{L}:\xi = c\binom{1}{-R(z)|E)}+\binom{0}{R(z)f}\bigg\}
\langle 0 | T_{i k}( x ) | 0 \rangle = \lim_{x^{\prime}\rightarrow x}\partial_{i}\partial_{k}^{\prime}\langle 0 | \varphi ( x ) \varphi ( x^{\prime}) | 0 \rangle + \left[ \left( \xi - \frac{1}{4}\right) g_{i k}\nabla_{l}\nabla^{l}- \xi \nabla_{i}\nabla_{k}- \xi R_{i k}\right] \langle 0 | \varphi^{2}( x ) | 0 \rangle ,
\delta m^{2}= \frac{1}{4 V}\int_{V}d^{3}x \sum_{i,j=1}^{N}\frac{n_{i}n_{j}}{|{\bf x}-{\bf x}_{i}||{\bf x}-{\bf x}_{j}|}= -\frac{\pi}{V}\sum_{{\scriptstyle i,j=1\atop\scriptstyle i<j}}^{N}n_{i}n_{j}|{\bf x}_{i}-{\bf x}_{j}| .
\small\left(\begin{array}{c}\binom{s_{i,1}}{1}+t_{i,2}\binom{s_{i,2}}{1}+\cdots+t_{i,k_{i}}\binom{s_{i,k_{i}}}{1},\ldots,\binom{s_{i,1}}{n}+t_{i,2}\binom{s_{i,2}}{n}+\cdots+t_{i,k_{i}}\binom{s_{i,k_{i}}}{n}\end{array}\right).
\begin{array}{l}{{\displaystyle{H^{B}= \int d \sigma \left( \Pi_{i}\dot{x}_{i}- \mathcal{L}_{B}\right) =}}}\\{{\displaystyle{\qquad \frac{1}{4 \pi l_{s}^{2}}\int_{0}^{2 \pi l_{s}^{2}p^{+}}d \sigma \left[ \sum_{i = 1}^{8}({\dot{x}_{i}}^{2}+{{x}_{i}^{\prime}}^{2}) + \mu^{2}\left( \sum_{I = 3}^{8}x_{I}^{2}+ 4 x_{2}^{2}\right) + 4 \mu x_{2}\partial_{\sigma}x_{1}\right]}}}\\ \end{array}
\frac{\partial^\nu}{\partial |x|^\nu}g(x) = -\frac{1}{\Gamma(m-\nu)}\frac{1}{2\cos\frac{\pi\nu}{2}}\frac{\partial^m}{\partial x^m}\int_{-\infty}^{+\infty}\frac{1}{\left|x-s\right|^{1+\nu-m}}g(s) \mathrm ds, m-1<\nu<m, \x>0
(f+g)+h = \mu_B((\mu_B(f \otimes g) \Delta_A) \otimes h ) \Delta_A = \mu_B (\mu_B \otimes 1)(f \otimes g \otimes h) (\Delta_A \otimes 1) \Delta_A = \mu_B (1 \otimes \mu_B)(f \otimes g \otimes h)(1 \otimes \Delta_A)\Delta_A = \mu_B (f \otimes (\mu_B (g \otimes h) \Delta_A) \Delta_A = f+(g+h),
I_{I R}( \rho ) = \frac{1}{2 \pi i}\int_{\tilde{C}}\frac{d s}{s}e^{\rho / s}F ( s ) = \frac{1}{2 \pi i}\int_{\tilde{C}}\frac{d s}{s}e^{\rho / s}\tilde{F}( s^{a}) = \frac{k}{2 \pi i}\int_{\tilde{C}^{\prime}}\frac{d s}{s}e^{( \frac{\tilde{\rho}}{s})^{k}}\tilde{F}( s ) ,
q(x|z) = \prod_{j=1}^N \left(\sum_{k=1}^{H_0}b^0_{kj}\prod_{i=1}^M (a^0_{ik})^{x_{i}}\right)^{z_j}, p(x|z,A,B) = \prod_{j=1}^N \left(\sum_{k=1}^{H}b_{kj}\prod_{i=1}^M (a_{ik})^{x_{i}}\right)^{z_j}.
L(x_{1},\cdot\cdot\cdot,x_{N},t)=\int_{u_{1}=-\infty}^{\infty}\cdot\cdot\cdot\int_{u_{N}=-\infty}^{\infty}f_{C}(x_{1}-u_{1},\cdot\cdot\cdot,x_{N}-u_{N},t)G(u_{1},t)du_{1}\cdot\cdot\cdot G(u_{N},t)du_{N},
(\mathrm{I}) ~=~ \sum_{x=4}^\infty \sum_{y=4}^\infty \underbrace{\Bigl( x \sqrt{(x \wedge a)(x \wedge b)}- y \sqrt{(y \wedge a)(y \wedge b)}\Bigr)^2 p(x) p(y)}_{:=u_{x,y,a,b}}.
\#\mathcal G'\gtrsim |\log\delta|^{-1}\#\mathcal G=|\log\delta|^{-1}\sum_{\tilde R\in \tilde{\mathcal R}}\sum_{R\in\mathcal{R}(\tilde R)}\#\mathcal{G}(R)\sim |\log\delta|^{-1}\mu_2 M\# \tilde{\mathcal{R}},
\mathcal{U}_{wp}^{t \, *}=\mathcal{U}_{wp}^{-t}\ , \ \ \mathcal{U}_{wp}^t\circ\mathcal{U}^{t'}=\mathcal{U}_{wp}^{t+t'}\ , \ \ \mathcal{U}^0_{wp}={\rm Id}_{\mathfrak{F}^2}\ , \ \t,t'\in\mathbb{R}\ .
\Omega = ( \Gamma_{11}^2 (p) (\dot x(\phi^{-1}(p)))^2 + ( 2\Gamma_{12}^2 (p) - \Gamma_{11}^1 (p) ) (\dot x(\phi^{-1}(p)))^2 (\dot y(\phi^{-1}(p))) + (\Gamma_{22}^2 (p) - 2 \Gamma_{12}^1 (p)) \dot x(\phi^{-1}(p)) (\dot y(\phi^{-1}(p)))^2 - \Gamma_{22}^1 (p) (\dot y(\phi^{-1}(p)))^2) t^3 \sqrt{\Delta(\phi (p))}.
R ( s ; 1 ) \sim{\frac{1}{4}}s^{2}-{\frac{1}{8 s}}+{\frac{9}{6 4 s^{4}}}-{\frac{1 8 9}{1 2 8 s^{7}}}+{\frac{2 1 6 6 3}{5 1 2 s^{1 0}}}+ \cdots \ ( s \rightarrow - \infty ) .
|\int_{G}\int_{G}f(x)g(y^{-1}x)h(y)d\mu(x)d\mu(y)|\le(\int_{G}|f|^{p})^{\frac{1}{p}}(\int_{G}|g|^{q})^{\frac{1}{q}}(\int_{G}|h|^{r})^{\frac{1}{r}}.
g^{i\bar{j}}=c_ic_j\frac{(1-\xi^i\bar{\xi}^j)\prod_{k\ne i}(1-\xi^k\bar{\xi}^j)\prod_{k\ne j}(1-\xi^i\bar{\xi}^k)}{\prod_{k\ne i}(\xi^k-\xi^i)\prod_{k\ne j}(\bar{\xi}^k-\bar{\xi}^j)}
{1\over 2}({\,\lower0.9pt\vbox{\hrule\hbox{\vrule height 0.2 cm \hskip 0.2 cm \vrule height0.2cm}\hrule}\,}+m^2) F_{\mu\nu}+(\partial_{\mu}F_{\alpha\nu}^{\quad,\alpha}-\partial_{\nu}F_{\alpha\mu}^{\quad,\alpha}) = 0 \quad,
\mathbb{P}_{couple}\left(\overline{S}_{\left\{1,...,M_{2}\left(n\right)\right\}}>n^{-c}\right) \leq M_{2}\left(n\right)n^{2c}n^{-1}\mathbb{E}\left(J_{1}^{2}\right) \leq n^{-\alpha c'+2c}c\log n\mathbb{E}\left(J_{1}^{2}\right) =n^{-c}c\log n\mathbb{E}\left(J_{1}^{2}\right).
G_{0 0}= g_{a b}\dot{x}^{a}\dot{x}^{b}, \hspace *{5 m m}G_{1 1}= g_{3 3}\equiv g_{\phi \phi}, \hspace *{5 m m}G_{0 1}= G_{1 0}= g_{a 3}\dot{x}^{a}\equiv g_{a \phi}\dot{x}^{a},
\beta_1 = \sqrt{1 - r^2}\left(\pi_t\left(X_t e^{i\theta}L_t + e^{-i\theta}L^{\dagger}_t X_t\right) - \pi_t\left(X\right)\pi_t\left(e^{i\theta}L_t + e^{-i\theta}L^{\dagger}_t\right)\right), \beta_2 = \dfrac{\pi_t\left( L^{\dagger}_t X_t L_t \right)}{\pi_t\left(L^{\dagger}_t L_t\right)}- \pi_t\left(X\right).
C_2(U_a):\,\, U=(u_ae^{\sigma_2},v_ae^{\sigma_2},p_a,\rho_a,Z_a)^\top, &C_3(U_a):\,\, U=(u_a,v_a,p_a,\rho_ae^{\sigma_3},Z_a)^\top, &C_4(U_a):\,\, U=(u_a,v_a,p_a,\rho_a,Z_a+\sigma_4)^\top.
\tilde{J}= \{\ (i,j) \ | \ 0 \leq i < \ell_1, \ 0 < j \leq \ell_2, \ \}, \tilde{J}' =\{\ (i,j) \ | \ 0 < i \leq \ell_1, \ 0 \leq j < \ell_2, \ \}, \tilde{\tilde{J}}= \{\ (i,j) \ | \ 0 \leq i \leq \ell_1, \ 0 \leq j \leq \ell_2, \ \}.
L_{B}= -{\frac{a}{4}}C_{\mu \nu}^{a}C^{\mu \nu a}-{\frac{1}{2 g}}B_{\mu}^{a}B^{\mu a}+ \alpha \epsilon^{\mu \nu \lambda}C_{\mu}^{a}( \partial_{\nu}C_{\lambda}^{a}+{\frac{1}{3}}f^{a b c}C_{\nu}^{b}C_{\lambda}^{c}) .
\lambda_{( 0 )}^{\alpha a A}{\lambda_{( 0 ) \alpha a}}^{B}= \frac{3 \cdot 2^{6}}{g^{2}}\frac{\rho_{0}^{4}}{\left( \rho_{0}^{2}+ ( x - x_{0})^{2}\right)^{4}}\: \zeta^{\alpha A}{\zeta_{\alpha}}^{B}.
\int_{\mathcal S^n}Y_l(y)\,f(x\cdot y)\,d\sigma_n(y) &=Y_l(x)\cdot\frac{(4\pi)^\lambda\,l!\,\Gamma(\lambda)}{(2\lambda+l-1)!}\int_{-1}^1 f(t)\,C_l^\lambda(t)\left(1-t^2\right)^{\lambda-1/2}dt.
f_{4,4,3}(-q^3,-q^2;q) =\frac{1}{4}\overline{J}_{0,3}\mu(q^3) -\frac{1}{2}\overline{J}_{1,3}\phi(q) +\overline{J}_{1,3}+\overline{J}_{1,3}\Theta_{2}(q) +\frac{1}{4}\overline{J}_{0,3}\frac{J_{6,12}^2}{J_{3}^3}+\frac{1}{\overline{J}_{0,16}\overline{J}_{0,12}}\theta_{4,4,3}(-q^3,-q^2;q).
{\mathcal K}^{\bullet}_D(V)=[0\longrightarrow{\mathcal K}^{0}_D(V)\longrightarrow{\mathcal K}^{1}_D(V)\longrightarrow{\mathcal K}^{2}_D(V)\longrightarrow\ldots\longrightarrow{\mathcal K}^{|D|}_D(V)].
[\frac{e}{n+ \frac{1}{2}}]^{n+ \frac{1}{2}}\approx\frac{\sqrt{2\pi}}{n!}\frac{e^{\frac{1}{2}}}{(1+\frac{1}{2n})^{n+\frac{1}{2}}}\approx \frac{\sqrt{2\pi}}{n!}
\varOmega_3(u):={}W[\mu^1_{2,1}(u),\mu^1_{2,2}(u),\mu^1_{2,3}(u)] ={}\det \begin{pmatrix}D^{0}\mu^1_{2,1}(u) D^{0}\mu^1_{2,2}(u) D^{0}\mu^1_{2,3}(u) D^1\mu^1_{2,1}(u) D^1\mu^1_{2,2}(u) D^1\mu^1_{2,3}(u) D^2\mu^1_{2,1}(u) D^2\mu^1_{2,2}(u) D^2\mu^1_{2,3}(u) \end{pmatrix}
\frac{\| B_k^\frac{1}{2}(B_k - I) w_k \|}{\|B_k^\frac{1}{2}w_k\|}= \sqrt{\frac{w_k^TB_k^3w_k}{w_k^TB_kw_k}- 2\frac{w_k^TB_k^2w_k}{w_k^TB_kw_k}+ 1}&= \sqrt{\frac{w_k^TB_k^3w_k}{w_k^TB_kw_k}- \left( \frac{w_k^TB_k^2w_k}{w_k^TB_kw_k}\right)^2 + \left( 1 - \frac{w_k^TB_k^2w_k}{w_k^TB_kw_k}\right)^2}&\leq \sqrt{2\varphi_k + \psi_k}
\begin{array}{r c l}{{{\cal N}_{\mathrm{b u l k}}}}&{{\approx}}&{{\displaystyle \sum_{i = 1}^{N}\frac{( q^{2 i}l_{p})^{2}}{( q^{2 i}l_{p}- q^{2 ( i - 1 )}l_{p})^{2}}~ , ~ ~ q^{2 N}= R ~ ,}}\\{{}}&{{=}}&{{\displaystyle \frac{q^{4}\ln \left( \frac{R}{l_{p}}\right)}{2 ( q^{2}- 1 )^{2}\ln q}~ .}}\\ \end{array}
\left(D^0 F^{[i]}(x_i),D^1 F^{[i]}(x_i),\dots,D^{(m-1)}F^{[i]}(x_i)\right)^T= R_i \left(D^0 F^{[i-1]}(x_i),D^1 F^{[i-1]}(x_i),\dots,D^{(m-1)}F^{[i-1]}(x_i)\right)^T,
\sum\limits_{r=0}^{k-1}\frac{(-1)^r\Gamma(\mu)}{2^{r+\lambda}r!\Gamma(\mu-r)(r+\lambda)}+\sum\limits_{r=0}^{s-1}\frac{(-1)^r\Gamma(\lambda)}{2^{r+\mu}r!\Gamma(\lambda-r)(r+\mu)}
w(S^j_k) = \frac{1}{q^{j^2}{k \choose k-j}_q}\sum\limits_{i=0}^j q^{{j-i \choose 2}}{j \choose i}_q{k+i \choose i}_q &\leq \frac{q^{jk}}{q^{j^2}{k \choose k-j}_q}\sum\limits_{i=0}^j q^{-i}&\leq \left(\frac{q}{q-1}\right)^{\min(j,k-j)+1}
s_{2j}=s_1^{2j}+\sum_{k=0}^{j-1}(-1)^{j-k}\left(C_{j+k}^{j-k}+C_{j+k-1}^{j-k-1}\right)s_1^{2k}, s_{2j+1}=s_1^{2j+1}+\sum_{k=0}^{j-1}(-1)^{j-k}\left(C_{j+1+k}^{j-k}+C_{j+k}^{j-n-1}\right)s_1^{2k+1}, s_1^{2j}=C_{2j}^j +s_{2j}+\sum_{k=1}^{j-1}C_{2j}^k s_{2(j-k)}, s_1^{2j+1}=s_{2j+1}+\sum_{k=1}^{j-1}C_{2j+1}^ns_{2(j-k)+1}.