harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Y_{i_{i'}, 2}= \mu'_{i'}+ \ell' - 2i' + 1 = \mathcal{C}_1(\alpha', \nu', \tau^{-1}(i'), I'_b \setminus \lbrace \tau^{-1}(i') \rbrace, I_b) \geq \mu^{\circ}+ \ell - 2i_{i'}= X_{i_{i'}, 1}+ \ell - 2i_{i'}= Y_{i_{i'}, 1}- 1.
S = S_{a}+S_{vis}= \int\frac{d^4x}{z_v^4}\left( \left(\lambda_{v1}+\lambda_{h1}\left(\frac{z_v}{z_h}\right)^{2-\nu}\right) \chi + \left( \lambda_{v2}^{(0)}+ \lambda_{h2}^{*(0)}+ (\lambda^{(0)}_{h2}-\lambda_{2}^{*(0)}) \left(\frac{z_v}{z_h}\right)^{-2\nu}\right) \frac{\chi^2}{2}\right).
\begin{array}{ll}(\overline \nabla_XJ_1)Y=\frac{1}{2(2n-1)}\left[g(X,Y)p_1-\overline \theta_1(Y)X+g(X,J_1Y)J_1p_1\right. \qquad\qquad\quad\left.-\overline \theta_1(J_1Y)J_1X\right] ,\end{array}
P\bigl(E_k^c\bigr)\leq\sum_{l\geq\underline{l}}P\bigl(E^c_k(l)\bigr)\leq\sum_{l\geq\underline{l}}\mathrm{e}^{-c_2 l}\leq K \mathrm{e}^{-c_2 \underline{l}}=Kn^{-2c_2A/ p},
\overline{\mbox{\Large$v$}}_{4}(x)={4\pi^{2}\over g^{2}\Lambda^{2}}x-\left({x^{2}\over2}-{{\cal E}^{2}\over3}\right)E_{i}(-x)+{1\over2}(1-x){\mathrm e}^{-x}-{1\over2}\ ,
\|\partial_{k}P_{3}(x) \|_{L^{1,3/2}(Q(r))}\lesssim \left(\frac{r}{\rho}\right)^{2}\| \Pi-\overline{\Pi}_{\frac{\rho}{2}}\|_{L^{5/4,15/7}(Q(\rho/2))}&\lesssim \left(\frac{r}{\rho}\right)^{2}\| \nabla \Pi \|_{L^{5/4}(Q(\rho/2))}.
S_2 \ll \sum_{l\le l_0}\frac{g(l)}{\phi(l)}\sum_{r+s+t+u\le w}\sum_{j=0}^t \frac{(\log_2 y)^{r+s+u+j}}{r!s!u!j!}\ll \sum_{r+s+u+j \le w}\frac{(\log_2 y)^{r+s+u+j}}{r!s!u!j!}(w-r-s-u-j+1) = \sum_{v\le w}(w-v+1) \frac{(4\log_2 y)^v}{v!}.
V(x) = x^6 + 6a |x|^5 + (9a^2 - 4b) x^4 - (12ab - 2c) |x|^3 + (4b^2 + 6ac - 7) x^2 - (18a + 4bc - 2x_1 - 2x_2) |x|, E_2^{(+)}= - 10b - c^2 - 6a(x_1+x_2) + 2(x_1^2+x_2^2), \psi_2^{(-)}(x) = e^{- \frac{1}{4}x^4 - a |x|^3 + b x^2 - c |x|}(|x|+x_1) (|x|+x_2),
\widehat{T}_{jk}\widehat{w}_k= \widehat{w}_j+\sum_{|\alpha|\geq n+1}M_j\cdot f_{kj, \alpha}\cdot \prod_{\lambda=1}^r\left(\sum_{\mu=1}^r(M_j^{-1})^\lambda_\mu \cdot \widehat{w}_j^\mu\right)^{\alpha_\nu}= \widehat{w}_j+O(|\widehat{w}_j|^{n+1}),
\ell_{t,Y_{n+1}}(x) = \prod_{s \in T \setminus \{t\}}{m(x) - m(s) \over m(t) - m(s)}- k(x) \sum_{y \in Y_{n}}{\ell_{y,Y_{n}}(x) \over k(y)}\prod_{s \in T \setminus \{t\}}{m(y) - m(s) \over m(t) - m(s)},
(\mu_2\otimes\mathbb{I}_N)\circ(\mathbb{I}_A\otimes\delta^1_N)\circ\phi^1+ (\mu_2\otimes\mathbb{I}_N)\circ(\mathbb{I}_A\otimes\phi^1)\circ\delta^1_M+ (\mu_1\otimes\mathbb{I}_N)\circ\phi^1 = 0.
E_{p,q}(r) = \frac{\pi_{p,q}}{2}\sum\limits_{n = 0}^\infty{\frac{1}{p}\choose n}\frac{1}{{(1 - \lambda )^{n - \frac{1}{p}}}}\sum\limits_{j = 0}^n{( - 1)^j}{n \choose j}{- \frac{1}{q}\choose j}{\frac{1}{p}- 1 - \frac{1}{q}\choose j}^{-1}\lambda^{n - j}r^{qj}.
H_{c}= \int d^{3}x \left[ \frac{1}{2}( \pi_{i}^{a})^{2}+ \frac{1}{4}F_{i j}^{a}F^{a , i j}+ \frac{1}{2}m^{2}\{( A^{a , 0})^{2}+ ( A^{a , i})^{2}\}- A_{0}^{a}\Theta_{2}^{a}\right] ,
\Phi_{\alpha}(A_{\alpha}^{1},\cdot\cdot\cdot,A_{\alpha}^{n})_{+}=max_{W_{\alpha-}^{i}\le w_{i}\le W_{\alpha+}^{i}A_{\alpha-}^{i}\le a_{i}\le A_{\alpha+}^{i}}\sum_{i=1}^{n}w_{i}a_{\sigma(i)}/\sum_{i=1}^{n}w_{i}
\textbf{L}(f) &=\frac{1}{\pi}\int_{\mathbb{D}}f'(z)(1-|z|^{2})\bar{h}(z)dm(z) &=\frac{1}{\pi}\int_{\mathbb{D}}f'(w)\left(\frac{2}{\pi}\int_{\mathbb{D}}\frac{\bar{h}(z)(1-|z|^{2})}{(1-\bar{w}z)^{3}}dm(z)\right)(1-|w|^{2})dm(w) &=\frac{1}{\pi}\int_{\mathbb{D}}f'(w)\bar{g}'(w)(1-|w|^{2})dm(w),
A_\mu^{(I)}:= \sum_{b=1}^N U_{I b}a_\mu^{(b)}\; \; \stackrel{U^T = U^{-1}}{\Longleftarrow \!\!=\!\!=\!\!=\!\!=\!\!= \!\!\! \Longrightarrow}\; \;a_\mu^{(b)}= \sum_{I=1}^N U_{I b}A_\mu^{(I)}\; .
\begin{aligned}\frac{1}{2}\frac{d^{2}}{dt^{2}}F(A^{(t)})|_{t=0}=\int_{\partial A}\int_{\partial A}G(x,y)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy &\quad+\int_{\partial A}\langle\nabla T_{\rho}1_{A}(x),X(x)\rangle\langle X(x),N(x)\rangle dx.\end{aligned}
\widetilde{a}_+( (\tau_{1,s}(\vartheta \widetilde{\varphi}), \tau_{1,s}(\vartheta \widetilde{w})), (\widetilde{\psi}, \widetilde{v})) = - \widetilde{a}_+( (\widetilde{\varphi}, \widetilde{w}), (\vartheta \tau_{1,-s}\widetilde{\psi}, \vartheta \tau_{1,-s}\widetilde{v})) + \widetilde{r},
\displaystyle{\frac{m_{\pi}^2 v j_{\pi}(q)}{m_{\pi}^2-q^2}(=\partial^{\mu}J_{\mu}(q))=\frac{q^2 v j_{\pi}(q)}{m_{\pi}^2-q^2}+\partial^{\mu}\widehat J_{\mu}(q).}
W = f(X) := \sum_{i=1}^{n-1}1_{\{\pi(i)> \pi(i+1)\}}+ \sum_{i=1}^{n-1}1_{\{\sigma(i)>\sigma(i+1)\}}&= \sum_{i=1}^{n-1}1_{\{V_{(i)}> V_{(i+1)}\}}+ \sum_{i=1}^{n-1}1_{\{U^{(i)}> U^{(i+1)}\}}\,.
\left(\ddot{\phi}_t-|\nabla \dot{\phi}_t|^2_{\omega_{\phi_t}}\right)\omega_\phi^n \wedge \eta=\epsilon (\omega^T)^n\wedge \eta=\left(\ddot{\varphi}_t-|\nabla \dot{\varphi}_t|^2_{\omega_{\varphi_t}}\right)\omega_\varphi^n \wedge \eta,
B_{1}^{in}=(\int_{M}dv a_{1}+\int_{S^{2}}dsc_{1})V''^{1/2-s}=\int_{M}(\frac{\lambda}{2}\hat{\phi}^{2})^{3/2-s}dv+\frac{2}{3R}\int_{S^{2}}(\frac{\lambda}{2}\hat{\phi}^{2})^{1/2-s}ds
~w(f)^{p-r}\prod_{(u,v)\in f}B((u,v),f) = ~\frac{[w_1(\pi_1(f))w_2(\pi_2(f))]^{p-r}}{(r!)^{p-r}}\cdot\prod_{(u,v)\in f}\frac{B_1(u,\pi_1(f))B_2(v,\pi_2(f))}{(r-1)!}= ~\frac{r^r\alpha_1\alpha_2}{(r!)^p}.
ds^2 = 2 e^{2\phi}d\tilde u (d \hat v +\omega_i d\tilde x^i) + \xi^2 d\tilde u^2 -\sum_{i=1}^{i=3}d\tilde x^id \tilde x^i -(d\hat x)^2 -\sum_{i=5}^{i=8}d\tilde x^id \tilde x^i \ ,
\begin{array}{lll}\Lambda_{24}^{2}= \{<A,B,0>,\; <0,A,B>,\; <B,0,A>: \; \; A,B \in \frac{1}{\sqrt{2}}{\cal E}_{8}^{odd}, & AB^{\dagger}=\pm\frac{1}{2}e_{a}, \; \; a\in \{0,...,7\}\},\end{array}
M_{--}^2 = g^2 v^2\left(\begin{array}{ccccccc}2 -1 0 \cdots 0 0 0 -1 2 -1 \cdots 0 0 0 0 -1 2 \cdots 0 0 0 & \ddots 0 0 0 \cdots -1 2 -1 0 0 0 \cdots 0 -1 2\end{array}\right).
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}_{VII}^+}d\omega_{k+1}dv_{s+k+1}d\tau \leq C_{d,s,k}C_{d,\alpha}T R^d \theta^{(d-1)/2}\end{aligned}
\frac{m ( \kappa^{2})}{\pi}\int_{\alpha}^{\infty}k \frac{d}{d k}\left[ \ln \left( 1 - e^{- a \sqrt{1 + k^{2}}}\right) \right] d k + \frac{b ( \kappa^{2})}{\pi}\int_{\alpha}^{\infty}\frac{d}{d k}\left[ \ln \left( 1 - e^{- a \sqrt{1 + k^{2}}}\right) \right] d k
\omega^{(1)}_{(\bar \ast)}+\omega^{(1)}_{(\ast)}+D_y\frac{\omega^{(0),{\rm reg}}_{(\ast,\ast)}}{2\,dx}+D_y^2\frac{D_x^2\omega^{(0)}_{(\ast)}}{24}=0, \omega^{(1)}_{(\bar\ast)}+\omega^{(1)}_{(\ast)}+D_x\frac{\omega^{(0),{\rm reg}}_{(\bar\ast,\bar\ast)}}{2\,dy}+D_x^2\frac{D_y^2\omega^{(0)}_{(\bar\ast)}}{24}=0
\mathbb{E}^V \theta(N \zeta_{\alpha^{\prime}}(\mu)\zeta_{\alpha^{\prime}}(\nu))= \mathbb{E}^V \theta(\frac{N \eta}{\pi}\int_{\lambda^{+}_{\alpha^{\prime}+1}}^{\lambda_{\alpha^{\prime}}^{+}}\frac{\zeta_{\alpha^{\prime}}(\mu)\zeta_{\alpha^{\prime}}(\nu)}{(E-\lambda_{\alpha^{\prime}})^2+\eta^2}dE)+o(1),
U = [U_k \ \, \widehat{U}]~, \ \, V = [V_k \ \, \widehat{V}]~,\ \, W = [W_k \ \, \widehat{W}]~, \ \, Z = [Z_k \ \, \widehat{Z}]~, begin{align*}1mm] D_A ={\sf diag}( D_{A_k},\,\widehat D_A) ~,\ \, D_B ={\sf diag}(D_{B_k},\, \widehat D_B)~, \ \, D_G ={\sf diag}( D_{G_k}, \, \widehat D_G),
Y_{v}^{\left( k\right)}\left( \lambda \right) =\left( -1\right)^{k}2^{k}\lambda^{v}\sum_{m=0}^{\infty}S_{1}\left( v,m\right)\sum_{n=0}^{\infty}\left( \begin{array}{c}n+k-1 n\end{array}\right) \lambda^{n}n^{m}.
ham(\begin{matrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{matrix})=det^{2}(A_{11})ham(\begin{matrix}A_{11}^{-1}&A_{11}^{-1}A_{12}\\A_{21}A_{11}^{-1}&A_{22}+A_{21}A_{11}^{-1}A_{12}\end{matrix})
|\mathsf E f(X_n^\varepsilon)-\mathsf E f(X_n)|\le C\mathsf E |X_n^\varepsilon-X_n|\le \frac{C}{s_n}\left(\mathsf E \left(\sum_{j=0}^{n-1}\psi_\varepsilon(\lambda_j)\xi_{j+1}\right)^2\right)^{1/2}= C\left(\mathsf E\sum_{j=0}^{n-1}\frac{\sigma_{j+1}^2}{s_n^2}\psi_\varepsilon^2 (\lambda_j)\right)^{1/2}\le C\varepsilon,
p_n(x;Q_1,...,Q_n) = (x-Q_n)\cdot p_{n-1}(x;Q_1,...,Q_{n-1}) - p_{n-2}(x;Q_1,...,Q_{n-2}) L_{2n}= (p_n + p_{n-1})(p_n - p_{n-1}) L_{2n+1}= p_n(p_{n+1}- p_{n-1})
M_x^\varepsilon-L_y^\varepsilon=\Gamma^2_{22}L^\varepsilon-2\Gamma^2_{12}M^\varepsilon +\Gamma^2_{11}N^\varepsilon+o_1^\varepsilon(1), N_x^\varepsilon-M_y^\varepsilon=-\Gamma^1_{22}L^\varepsilon+2\Gamma^1_{12}M^\varepsilon -\Gamma^1_{11}N^\varepsilon+o_2^\varepsilon(1),
\begin{array}{l}{\displaystyle x_{j}^{2}= \frac{\prod_{i = 1}^{D}( \rho_{i}- e_{j})}{\prod_{i \not = j}^{D}( e_{i}- e_{j})}\enspace , \qquad ( j = 1 , \dots , D )}\\{\displaystyle e_{1}< \rho_{1}< e_{2}< \dots < e_{D}< \rho_{D}\enspace .}\\ \end{array}
\frac{1}{ct}x=\frac{w}{c}=\beta=(\begin{matrix}\frac{x_{1}}{ct}\\ \frac{x_{2}}{ct}\\ \frac{x_{3}}{ct}\end{matrix})=(\begin{matrix}\beta_{x}\\ \beta_{y}\\ \beta_{z}\end{matrix})=(\begin{matrix}{\Lambda_{0}}^{1}/{\Lambda_{0}}^{0}\\{\Lambda_{0}}^{2}/{\Lambda_{0}}^{0}\\{\Lambda_{0}}^{3}/{\Lambda_{0}}^{0}\end{matrix}).
+{\tilde{d}}( q - 1 ) ( a_{1}b_{2}+ a_{2}b_{1}) + r ( q - 1 ) ( a_{2}f_{1}+ a_{1}f_{2}) + r{\tilde{d}}( f_{1}b_{2}+ f_{2}b_{1}) + \frac{\phi_{1}\phi_{2}}{2}= 0 ;
\|v\|_{C^{k,\alpha}(\Omega^0)}\le C(\boldsymbol{\mathcal C^0}) \big(\|\Delta \dot h^0\|_{C^{k-2,\alpha}(\Omega^0)}+ |\dot c^0|\big)\le C(\boldsymbol{\mathcal C^0}) \big(\| \dot h^0\|_{C^{k,\alpha}(\Omega^0)}+ |\dot c^0|\big).
\begin{aligned}f_3(t) =(t^3+6 t^2+18 t+24)e^{8 t}-4t(4 t^3+6t^2+6 t+3) e^{7 t}&\quad-4 (t^3+6 t^2+18 t+24)e^{6 t}-4 t (16 t^3-12 t-9)e^{5 t}&\quad+6 (t^3+6 t^2+18 t+24) e^{4 t}-4t (4 t^3-6 t^2+6 t+9) e^{3 t}&\quad-4 (t^3+6 t^2+18t+24)e^{2 t}+12 te^t +t^3+6 t^2+18 t+24\end{aligned}
\begin{aligned}I_1 =\iint_{[\alpha_1, \beta_1]\times[\alpha_{2},\beta_{2}]}\biggl[\Phi(x,y) +\frac{\partial \Phi(x,y)}{\partial x}P_1(x) &+\frac{\partial \Phi(x,y)}{\partial y}P_1(y) +\frac{\partial^{2}\Phi(x,y)}{\partial x\partial y}P_1(x)P_1(y)\biggl] dxdy,\end{aligned}
\Delta T_a(p+\frac{q}{2}, p-\frac{q}{2}) \;=\; -\frac{\mbox{PP}}{[(p-\frac{q}{2})^2-a] + 2pq}\: T_a(p-\frac{q}{2}) \:-\: T_a(p+\frac{q}{2}) \: \frac{\mbox{PP}}{[(p+\frac{q}{2})^2-a] - 2pq}
\theta_*(h_k\cdot \pi^*(\gamma)) = (pr_1)_*(\theta_1)_*(\theta_1^*(\tilde{h}_k)\cdot \pi^*(\gamma)) &= (pr_1)_* (\tilde{h}_k\cdot (\theta_1)_* \pi^*(\gamma)) = (pr_1)_* (\tilde{h}_k\cdot pr_2^* (\theta')_*(\gamma)) &= (pr_1)_* (\tilde{h}_k\cdot pr_2^*(\underline\gamma)) \/.
\begin{array}{r c l}{d A + A^{2}+ \hat{\psi}^{2}- \psi^{2}+ \hat{A}}&{=}&{0 ,}\\{d \hat{\psi}+ [ A , \hat{\psi}] - \{\psi , \hat{A}\}}&{=}&{0 ,}\\{d \psi + \{A , \psi \}+ [ \hat{A}, \hat{\psi}]}&{=}&{0 ,}\\{d \hat{A}+ [ A , \hat{A}] + \{\hat{\psi}, \psi \}}&{=}&{0 .}\\ \end{array}
\tilde\psi((x_0,(s_0,t_0)),\ldots,(x_{q-1},(s_{q-1},t_{q-1})))=\sum_{n=0}^{\lfloor\max(t_j)\rfloor}(h_n^*\psi_1)((x_0,(s_0,t_0)),\ldots,(x_{q-1},(s_{q-1},t_{q-1}))).
\begin{cases}_{0}D_{t}^{\alpha}S(t) = \Lambda^{\alpha}- d^{\alpha}S(t)- F\left(S(t),I(t)\right), begin{align*}0.2 cm]_{0}D_{t}^{\alpha}E(t) = F\left(S(t),I(t)\right) - m_{1}E(t), begin{align*}0.2 cm]_{0}D_{t}^{\alpha}I(t) = \sigma^{\alpha}E(t) - m_{2}I(t),\end{cases}
\Psi_{0}( \phi ( \vec{x}) ) \propto \int{\cal D}{\tilde{\phi}}e^{-{\frac{1}{2}}\int ({\dot{\tilde{\phi}}}^{2}+ ( \bigtriangledown{\tilde{\phi}})^{2}+ \mu^{2}{\tilde{\phi}}^{2}) d \vec{x}d \tau}.
\mathcal{B}_{a2n-1-a}^{b2n-1-b}-(\mathcal{B}^{-1})_{a2n-1-a}^{b2n-1-b}=(q^{\frac{1}{4}}-q^{-\frac{1}{4}})(\mathcal{M}_{a 2n-1-a}\mathcal{M}^{b2n-1-b}-\delta_{a}^{b}\delta_{2n-1-a}^{2n-1-b})\hbox{on}V_{2n-1}.
R_{a,b}(x_a/x_b)R_{a,c}(x_a/y_c)R_{b,c}(x_b/y_c)=R_{b,c}(x_b/y_c)R_{a,c}(x_a/y_c)R_{a,b}(x_a/x_b), R_{b,a}(y_b/y_a)R_{c,a}(x_c/y_a)R_{c,b}(x_c/y_b)=R_{c,b}(x_c/y_b)R_{c,a}(x_c/y_a)R_{b,a}(y_b/y_a).
\sum_{n = - \infty}^{+ \infty}\exp \left( - \gamma \, ( n + 1 / 2 )^{2}\right) \, \left[ \gamma \, ( n + 1 / 2 )^{2}\right]^{p}= \left({\frac{\pi}{\gamma}}\right)^{1 / 2}\,{\frac{( 2 p - 1 ) ! !}{2^{p}}}\left[ 1 + 2 \sum_{n = 1}^{\infty}( - 1 )^{n}\, \exp ( - \pi^{2}n^{2}/ \gamma ) \sum_{l = 0}^{p}\left({\frac{- \pi^{2}n^{2}}{\gamma}}\right)^{l}C_{l}^{p}\right]
r_2(z)=4 z r_1(z), z \in \mathbb{R}, &\bar{r}_1(z) r_2(z)=|r(k)|^2, z \in \mathbb{R}^{+}, k \in \mathbb{R}, & \bar{r}_1(z)r_2(z)=-|r(k)|^{2}, z \in \mathbb{R}^{-}, k \in i \mathbb{R}.
[\Theta_{g_k^*},\Theta_{f_i^*}]=-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}\bigg(\Theta_{[g_k^*,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,f_i^*]^{\sharp}}-\Theta_{[f_i^*,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,g_k^*]^{\sharp}}\bigg)\in\sum_{\alpha\in\Phi^+_{e,0}}U(\ggg,e)\cdot\Theta_{e_\alpha}.
F_3(k) = \sum_{h=1}^{+\infty}\frac{\big ( k^{2s-1}+ h^{2s-1}\big )^2}{(1+k+h)^{2s}(1+h)^{2r}}&\leq 2 \sum_{h=1}^{+\infty}\frac{k^{2(2s-1)}+ h^{2(2s-1)}}{(k+h)^{2s}h^{2r}}&= 2 k^{2(2s-1)}\sum_{h=1}^{+\infty}\frac{1}{(k+h)^{2s}h^{2r}}+ 2 \sum_{h=1}^{+\infty}\frac{1}{(k+h)^{2s}h^{2(r+1-2s)}}.
\left|P\cap \{0,1\}^n \right| = \frac{e^{g(z)}}{(2\pi)^{k_1+\ldots+k_{\nu}-\nu+1}}\int_{\Pi}e^{-i\left<t,b\right>}\prod_{j=1}^{n}\left(1-\zeta_j+\zeta_j e^{i\left<a_j,t\right>}\right)dt.
-\langle \nabla_{X_{2}}\nabla_{X_{1}}X_{2},X_{1}\rangle=\displaystyle\sum^{n}_{j=3}\langle\nabla_{X_{2}}X_{1},X_{j}\rangle^{2}\displaystyle\frac{(\lambda-\gamma)}{\mu-\gamma}- \displaystyle\frac{c\mu(\nu^{2}-b_{2}^{2})}{\mu-\lambda}-\displaystyle\frac{\nu^{2}b_{1}^{2}}{(\mu-\lambda)^{2}}.
M_n^{(\alpha,\beta)}= \frac{C_{n+1}^{(a,b)}}{a}=a^{n}\sum_{k=0}^{n+1}\frac1{n+1}\binom{n+1}{k}\binom{n+1}{k+1}\cdot\bigg(\frac ba\bigg)^k = \sum_{k=0}^{n+1}\bigg(\frac{\alpha+\sqrt{\alpha^2-4\beta}}2\bigg)^{n-2k}\cdot\frac{\beta^k}{n+1}\binom{n+1}{k}\binom{n+1}{k+1}.
\psi_{m}( r ) = r^{| m |}_{1}F_{1}\left({\frac{2 - g}{4}}+{\frac{| m | - m}{2}}-{\frac{\epsilon R^{2}}{2 \delta}}, 1 + | m | ,{\frac{\delta r^{2}}{R^{2}}}\right) e^{-{\frac{\delta r^{2}}{2 R^{2}}}}.
c_3 = d_3 = \tanh^{-1}-\varepsilon_1 = 3.8 c_2 = d_2 = 2 c_3/\tau_{f_n}= 7.6/\tau_{f_n}c_1 = c_4 = 0.5 \Delta \phi_n d_1 = d_4 = 0.5 \Delta v_n
y''=\underbrace{-\Gamma^2_{11}}_{K_0}+ \underbrace{(\Gamma^1_{11}-2\Gamma^2_{12})}_{K_1}y'+\underbrace{(2\Gamma^1_{12}-\Gamma^2_{22})}_{K_2}y'^2 +\underbrace{\Gamma^1_{22}}_{K_3}y'^3.
\frac{1}{\lambda ( k )}- \frac{9 N + 4 2}{1 6 \pi^{2}( N + 8 )}\ln \left( \frac{N + 8}{1 6 \pi^{2}}\lambda ( k ) \right) = \frac{N + 8}{1 6 \pi^{2}}\left( \ln \frac{\Lambda}{k}+ C \right) ,
\begin{array}{cl}\sigma^j(P_i) := (\alpha_{i\ell + j}:\beta_{i\ell + j}), \textrm{for}i \in \{0,\dots,\frac{n}{\ell}-1\}, j \in \{0,\dots,\ell-1\}\sigma^j(Q) := (\gamma_{j}: \delta_{j}), \textrm{for}j \in \{0,\dots,\ell-1\}.\end{array}
h_{\hat{D}_l(k)}(\theta)=h_{{j}_l(k\hat{r})}(\theta)+h_{\hat{y}_l(\hat{r};k)}(\theta)=(\hat{r}+\hat{B}(\hat{r}))|\sin\theta|,\,\theta\in[0,2\pi],\,\hat{r}_0\leq \hat{r}\leq\hat{r}_1,
\frac{Z_{\epsilon}}{QR}=[\frac{1}{Q}+(\frac{\epsilon^{\prime\prime}}{(\epsilon^{\prime})^{2}+(\epsilon^{\prime\prime})^{2}})\frac{\omega_{0}}{\omega}]+j[\frac{\omega}{\omega_{0}}-(\frac{\epsilon^{\prime}}{(\epsilon^{\prime})^{2}+(\epsilon^{\prime\prime})^{2}})\frac{\omega_{0}}{\omega}].
{f_X}\!\left( x \right) = \frac{{{m^m}}}{{\Gamma \left( m \right)}}\sum\limits_{j = 0}^M{\frac{{{K^j}{\alpha_j}}}{{j!}}}\frac{{{x^j}}}{{\Gamma \left({j \!+ \!1}\right){{\left({2{\sigma^2}}\right)}^{j + 1}}}}\exp\!\left(\!{- \frac{x}{{2{\sigma^2}}}}\!\right),
U(\vec{a},\Lambda) \phi^{\alpha,n}(f) U(\vec{a},\Lambda)^{-1}\left( S, g_\beta \otimes \rho_n(E^\beta)\right) &=\phi^{\alpha,n}[f(\Lambda^{-1}(\cdot - \vec{a}) )]\left( S, g_\beta \otimes \rho_n(E^\beta)\right) .
\vec s (t) = \sum_{k = -\infty}^{\infty}\left( \vphantom{\sum_{l = 1}^{f_\mathrm{IP}-1}}\vec x [k] \, h(t - kT) \right. &\left. + \sum_{l = 1}^{f_\mathrm{IP}-1}\vec{SI}\left( \vec{x}[k], \vec{x}[k+1], \frac{l}{f_\mathrm{IP}}\right) \, h\left(t - \left(k + \frac{l}{f_\mathrm{IP}}\right) T\right) \right).
\sup_{\mu(E)=\kappa}\int_{E}(M_{\mathcal{T}}\phi)^{p}d\mu=\int_{{\textstyle\bigcup\limits_{j}}I_{j}}(M_{\mathcal{T}}\phi)^{p}d\mu= =\sum_{j}\mu(I_{j})[u_{0}^{p}+\frac{N^{p}-1}{N^{p}-N}(\frac{\alpha_{j}}{\mu(I_{j})}-u_{0}^{p})]= =\kappa u_{0}^{p}+\frac{N^{p}-1}{N^{p}-N}(F-u_{0}^{p-1}f)
[-\frac{\hbar^{2}}{2m_{1}}\frac{\partial^{2}}{\partial x_{1}^{2}}-\frac{\hbar^{2}}{2m_{2}}\frac{\partial^{2}}{\partial x_{2}^{2}}+(\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{|x_{1}-x_{2}|})]\Psi(x_{1},x_{2},t)=i\hbar\frac{\partial}{\partial t}\Psi(x_{1},x_{2},t)
\chi=\frac{C_{F}g^{2}}{8\pi}\Biggl( 1-\frac{g^{2}}{8\pi^{2}}\biggl[ \frac{11C_{A}-2n_{f}}{3}\Biggl( \frac{1}{n-4}+ln\;\frac{m^{2}}{4\pi\nu^{2}}\Biggr)+\Phi+O(g^{2}_{0};n-4) \biggr] \Biggr),
\langle T_{\mu \nu}\rangle = \lim_{x^{\prime}\rightarrow x}\biggl ({\frac{2}{3}}D_{; \nu^{\prime}\mu}-{\frac{1}{3}}D_{; \nu \mu}-{\frac{1}{6}}g_{\mu \nu}D^{; \sigma^{\prime}}{}_{\sigma}+{\frac{1}{3}}g_{\mu \nu}D^{; \sigma^{\prime}}{}_{\sigma^{\prime}}+{\frac{1}{6}}( R_{\mu \nu}-{\frac{1}{2}}R g_{\mu \nu}) D \biggr ) .
\limsup_{n\to\infty}\frac{1}{b_c^{(n)}}\log P\left(\frac{n-A_n^*}{f_2(n)}\in C\right) \leq (-H(\ell_2 M))\vee(-H(\ell_2 m)) &=-\inf_{y\in C\cap [0,\ell_2^{-1})}I_2(y)\vee -\inf_{y\in C\cap (\ell_2^{-1},\infty)}I_2(y) &=-\inf_{y\in C}I_2(y),
\int_{D}|{v_k^{+}}|^p\,dx\, \le \, \int_{D}|v_k|^p\,dx \int_{D}|\nabla{v_k^{+}}|^p\,dx =\int_{D}|\nabla v_k|^p\chi_{\{v_k>0\}}\,dx \,\le \, \int_{D}|\nabla v_k|^p\,dx.
i_*(\Pi \times ev)_* \left[ \O_{\overline{{\mathcal{M}}_{0,r}}(X, \d)}\right] = \sum_{e_1, \dots, e_r \in I}\left((ev_1^* \phi_{e_1}) \otimes \dots \otimes (ev_r^* \phi_{e_r})\right) \boxtimes \left(\pi^*(\phi_{e_1}^{\vee}) \boxtimes \dots \boxtimes \pi^*(\phi_{e_r}^{\vee}) \right)
{\cal L}_{0}= s \Psi_{0}= \sum_{i}\bar{\theta}_{i}C^{a}( x_{i}) + \sum_{i}\bar{\gamma}_{i}\left( \phi^{a}( x_{i}) +{\frac{1}{2}}f_{\phantom{a}b c}^{a}C^{b}( x_{i}) C^{c}( x_{i}) \right) .
=\frac{y\rho}{a(t)r}\left[ \left( \ddot{a}(t)-\frac{\xi^{2}}{a(t)^{3}}\right) r+K\dot{f}(s)+\frac{2\pi}{s}{\displaystyle\int\limits_{0}^{s}}e^{f(\tau)}\tau d\tau\right]
G^{(10)}_{1,1,{1\over3}}\,\left| \,{p\;\,q\;\; i \atop p\;p\;(p + 2q)/3}\, \right\rangle=\sqrt{{(p + 1)\, (p + q + 2)}\over{p + 2}}\;\sum_{i'}\;\left| \,{p+1\;\,q\;\; i' \atop p\;p\; (p + 2q - 2)/3}\,\right\rangle \;A_{i'\;, i}\; .
\int_0^t Mf'(x(u))\,du = K_1(\epsilon) + \int_{T_1}^t \frac{f'(x(u))}{f(x(u))}\cdot\frac{Mf(x(u))}{x'(u)}x'(u)\,du &\leq K_1(\epsilon) + \frac{1}{1-\epsilon}\int_{T_1}^t \frac{f'(x(u))}{f(x(u))}x'(u)\,du &= K_1(\epsilon) + \frac{1}{1-\epsilon}\left( \log f(x(t)) - \log f(x(T_1))\right).
S_{wv}=-\frac{T_p}{\sqrt{2{\cal V}}\kappa_{\rm orb}}\left\{\int d^{p+1}\xi \sqrt{-|g_{\alpha\beta}|}e^{-\kappa_{\rm orb}\phi\frac{(1-p)}{2}}\prod_a e^{-\kappa_{\rm orb}\frac{\eta_a}{2}}-\sqrt{2}\kappa_{\rm orb}\int C_{p+1}\right\}
E_B = \dot q^r \frac{\partial L_B}{\partial \dot q^r}+ \dot{\bar{c}}_\alpha\frac{\partial L_B}{\partial \dot{\bar{c}}_\alpha}+ \sum_{k=0}^{N-1}\c{N-k}^\alpha \sum_{l=1}^{k+1}(-1)^{k+1-l}\frac{d^{k+1-l}}{dt^{k+1-l}}\left( \frac{\partial L_B}{\partial \c{N-l+1}^\alpha}\right) - L_B .
\bar{I}= \int (h (x,t) - h (x+y,t)) w(x+y,t) \overline{K}(x,y) dy &= e^{-|\lambda_1| t}\int (|x|^{\xi(d+\alpha)}-|x+y|^{\xi(d+\alpha)})w(x+y,t) \overline{K}(x,y) dy .
\begin{array}{c}r^{12}=-r^{21},~~~r^{23}r^{12}+r^{31}r^{23}+r^{12}r^{31}=0, begin{align*}5mm]a^{12}a^{31}=a^{31}a^{12}, begin{align*}5mm]\sigma^{23}a^{13}a^{12}=a^{12}r^{23}-r^{23}a^{12}, begin{align*}5mm]a^{32}a^{12}=r^{13}a^{12}-a^{32}r^{13}.\end{array}
\left( \partial_{n}r \right)_{\rho_{0}= R}={\frac{R^{2}+ r^{2}- \rho^{2}}{2 R r}}~ ~ , ~ ~ \left( \partial_{n}r_{1}\right)_{\rho_{0}= R}={\frac{\rho^{2}+ r^{2}- R^{2}}{2 \rho r}}
\sum_{\substack{u_1,\ldots,u_k (u_i,VP_f)=1\forall i}}\biggl( \prod_{i=1}^k\frac{\mu(u_i)^2}{\varphi(u_i)}\biggr)F\bigl(\frac{\log u_1}{\log R},\ldots,\frac{\log u_k}{\log R}\bigr)^2 &=\frac{\varphi(VP_f)^k(\log R)^k}{(VP_f)^k}I_k(F)+O\Bigl(\frac{F_{max}^2\varphi(VP_f)^k(\log X)^{k-1}\log\log X}{(VP_f)^kD_0}\Bigr)
\frac{\partial V_{F}}{\partial z^{a}}( S , \bar{S}_{0}, z_{0}^{a}, \bar{z}_{0}^{\bar{a}}) = 0 , \qquad \qquad V_{F}( S_{0}, \bar{S}_{0}, z_{0}^{a}, \bar{z}_{0}^{\bar{a}}) = 0 , \qquad \qquad F_{S}( S_{0}, z_{0}^{a}) \ne 0 ,
\mathcal F := \eta L_1 \frac{\gamma^3}{L_2^2}\cdot \log^{-3}(d\kappa /\delta), &\mathcal P :=\sqrt{\eta L_1}\frac{\gamma}{L_2}\cdot \log^{-1}(d\kappa /\delta), &\mathcal J := \frac{\log (d\kappa /\delta)}{\eta \gamma},
A ( X , f ; X^{\prime}, f^{\prime}) = N \int d [ \rho ] \int_{0}^{\infty}\left[ e^{- \frac{m^{2}c}{2}}\int \, d [ X ] \exp \left( - \int_{0}^{c}g_{\mu \nu}\frac{d X^{\mu}}{d f}\frac{d X^{\nu}}{d f}\, d f \right) \right] \, d c \: . \nonumber
\partial_i(a^{ij}\partial_j \tilde{u})-(\partial_i \phi) a^{ij}\partial_j \tilde{u}-\partial_i(a^{ij}\tilde{u}\partial_j \phi)+(\partial_i \phi) (\partial_j \phi) a^{ij}\tilde{u}-Kd^{-2}\tilde{u}=\partial_i(e^\phi F^i)-(\partial_i \phi) e^\phi F^i+e^\phi g.
x = \sum_{i=1}^n \lambda_i \alpha_i \beta_i^* = \sum_{i=1}^n \lambda_i \alpha_i r(\alpha_i) \beta_i^* = \sum_{i=1}^n \lambda_i \alpha_i \left( \sum_{\gamma \in X_{r(\alpha_i),k-|\beta_i|}}\gamma \gamma^* \right) \beta_i^* = \sum_{i=1}^n \sum_{\gamma \in X_{r(\alpha_i),k-|\beta_i|}}\lambda_i \alpha_i \gamma (\beta_i \gamma)^*.
\vec \lambda^{(2N)}_j= \sum_{k=1}^j \vec e_k,\quad j\leq N-2, \quad\vec \lambda^{(2N)}_{N-1}={1\over 2}\left (\sum_1^{N-1}\vec e_k-\vec e_N\right ),\quad \vec \lambda^{(2N)}_{N}= \sum_1^{N}{\vec e_k\over 2}.
\widetilde{S}_{0}^{( D_{0})}( k ) = - \frac{1 + i \left( 2 \omega_{_\mathrm{( g s )}}^{- 1}/ \pi - Q_{_\mathrm{( g s )}}\right)}{1 - i \left( 2 \omega_{_\mathrm{( g s )}}^{- 1}/ \pi - Q_{_\mathrm{( g s )}}\right)}\; ,
\left\| w_{r_k}\right\|_{L^2(B_1)}= \frac{\left\| w(r_k \cdot) \right\|_{L^2(B_1)}}{r^{- \frac{n+1}{2}}_k r^{\frac{2s-1}{2}}_k\left\| y_{n+1}^{\frac{1-2s}{2}}w \right\|_{L^2(B_{r_k}^+)}}& \lesssim \frac{r^{- \frac{n}{2}}_k \left\| w \right\|_{L^2(B_{r_k})}}{r^{- \frac{n+1}{2}}_k r^{\frac{2s-1}{2}}_k k \left\| w \right\|_{L^2(B_{r_k})}}= r_k^{1-s}k^{-1}.
\mu\sqrt{gh}(2\frac{\Theta^{\prime}}{\Theta}+\frac{b^{\prime}}{b}+\frac{1}{2}\frac{h^{\prime}}{h})\Theta\frac{dF}{d\tau}+\mu^{2}gh(\frac{\Theta^{\prime\prime}}{\Theta^{\prime}}+\frac{b^{\prime}}{b}+\frac{1}{2}\frac{h^{\prime}}{h})\Theta^{\prime}F.
\begin{aligned}\nabla_{ijkl}v - \nabla_{klij}v= R^m_{ljk}\nabla_{im}v + \nabla_i R^m_{ljk}\nabla_m v + R^m_{lik}\nabla_{jm}v + R^m_{jik}\nabla_{lm}v + R^m_{jil}\nabla_{km}v + \nabla_k R^m_{jil}\nabla_m v.\end{aligned}
\frac{4\alpha_{1}^2 a_{1}^{2}}{(-1 + \frac{1+\lambda}{2})^{2}+ a_{1}^{2}}= \frac{4\alpha_{2}^2 a_{2}^{2}}{(-1 + \frac{1+\lambda}{2})^{2}+ a_{2}^{2}}.
{\cal O}_{1 / 4}^{[ p , 0 , q ]}= [ W^{1}( \theta_{2 , 3 , 4}, \bar{\theta}^{1}) ]^{p}\ [ \bar{W}_{4}( \theta_{4}, \bar{\theta}^{1 , 2 , 3}) ]^{q}\, , \qquad \ell = p + q
n_{i}=|(C_{0}\bigcup C_{1})\bigcap((C_{0}\bigcup C_{1})+i)| =|C_{0}\bigcap(C_{0}+i)|+|C_{0}\bigcap(C_{1}+i)|+|C_{1}\bigcap(C_{0}+i)|+|C_{1}\bigcap(C_{1}+i)| =|i^{-1}C_{0}\bigcap(i^{-1}C_{0}+1)|+|i^{-1}C_{0}\bigcap(i^{-1}C_{1}+1)|+|i^{-1}C_{1}\bigcap(i^{-1}C_{0}+1)|+|i^{-1}C_{1}\bigcap(i^{-1}C_{1}+1)|.
\mathcal{M}_\ell(\hat{H}_s,\hat{S}) = \mathcal{M}_2(\hat{S}) \mathcal{M}_1(\hat{H}^\top) \mathcal{M}_1(\hat{H}), \mathcal{M}_s(\hat{H}^\top,\hat{S}) = \mathcal{M}_1(\hat{H})\mathcal{M}_2(\hat{S}) \mathcal{M}_1(\hat{H}^\top).
Q_{Z}= Q_{1}+ \frac{1 2 1}{1 2 \sqrt{1 0 5}}S^{\prime \prime}- \frac{2}{3}\sqrt{\frac{2}{3}}( J \; S^{\prime}) - \sqrt{\frac{2}{3}}( J^{\prime}\; S ) + \frac{1}{3}\sqrt{\frac{3 5}{3}}( J \; J \; S ) - \frac{1}{3}\sqrt{\frac{3 5}{3}}( T \; S )
L = \bar{\psi}i \gamma^{\mu}\partial_{\mu}\psi +{\bar{\psi}}\gamma^{\mu}( B_{\mu}+ A_{\mu}+ \epsilon_{\mu \nu}\partial^{\nu}\theta ) \psi + \partial^{\mu}\eta B_{\mu}+ \frac{1}{2 \pi}\partial^{\mu}\theta \partial_{\mu}\theta + \frac{1}{\pi}\epsilon^{\mu \nu}A_{\mu}\partial_{\nu}\theta \quad .
H_{LC}= \int_0^l \mathcal{H}_{LC}= \int_0^l d\sigma \, \frac{G_{+-}}{2 \,\pi_-}\left( G^{ij}\pi_i \pi_j + \frac{1}{(2\pi\alpha')^2}G_{ij}Z^{i\,\prime}Z^{j\,\prime}\right)
\begin{array}{l c r}{{L_{H}= \int d^{2}x \sum_{h y p e r m u l t}[ | D_{\mu}\varphi_{H}|^{2}+ | V \varphi_{H}|^{2}+ \overline{{{\psi}}}_{H}\gamma^{\mu}D_{\mu}\psi_{H}+ \overline{{{\psi}}}_{H}V{\psi}_{H}+ \overline{{{\psi}}}_{V}V{\psi}_{H}}}\\{{+ \overline{{\varphi}}_{H}D \varphi_{H}]}}\\ \end{array}
\prod_{i=1}^{m^\prime_a}\psi^{(a)}_2({y^\prime}_i^{(a)})\prod_{i=1}^{n_a}\overline{\psi}^{(a)}_1(x_i^{(a)}\!+ \hat{\tau})\prod_{i=1}^{m_a}\psi^{(a)}_2(y_i^{(a)}\!+ \hat{\tau})\prod_{i=1}^{n^\prime_a}\overline{\psi}^{(a)}_1({x^\prime}^{(a)}_i)\Big \rangle_0 \; .
y = c_1 + \int_\gamma \frac{1}{A\Re\hat g(x,A)}dA-\frac{\Re\hat h(x,A)}{\Re\hat g(x,A)}dx, &\Phi = c_2 + \int_\gamma \frac{\Im\hat g(x,A)}{A\Re\hat g(x,A)}dA+ \Big( \Im\hat h(x,A) - \frac{\Re\hat h(x,A)\Im\hat g(x,A)}{\Re\hat g(x,A)}\Big) dx