harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Pi_{\mathsf{V},\alpha}(t^n, x^n) V_{{t^n}}(x^n) \Pi_{\mathsf{V},\alpha}(t^n,x^n)\allowdisplaybreaks &\leq 2^{-n(S(\mathsf{V}|r) + \delta(\alpha)')}\Pi_{\mathsf{V},\alpha}(t^n,x^n)\allowdisplaybreaks &= 2^{-n(\sum_{t,x}r(t,x)S(\mathsf{V}(t,x)) + \delta(\alpha)')}\Pi_{\mathsf{V},\alpha}(t^n,x^n)
S=\begin{cases}\sum z_{1,2}^{(i_1)}z_{3,4}^{(i_3)}\cdots z_{K-1,K}^{(i_{K-1})}\sum z_{1,2}^{(i_1)}z_{3,4}^{(i_3)}\cdots z_{K-2,K-1}^{(i_{K-2})}\end{cases}
\frac{1}{2}\sum_{\alpha \in \Phi_{P^u}}[z_{\alpha}e_{-\alpha},\theta(z_{\alpha}e_{-\alpha})] = \frac{1}{2}\sum_{\alpha \in \Phi_{P^u}}z_{\alpha}\bar{z}_{\alpha}[e_{-\alpha},\theta(e_{-\alpha})] & = \frac{1}{2}\sum_{\alpha \in \Phi_{P^u}}z_{\alpha}\bar{z}_{\alpha}t_{\alpha}
b^{*}\phi |_{\square}(a^{0}, a^{1}, a^{2}, a^{3}) = a^{0}a^{1}[F,a^{2}][F, a^{3}] - a^{0}[F,a^{1}a^{2}][F,a^{3}] + a^{0}[F,a^{1}][F,a^{2}a^{3}] - a^{3}a^{0}[F,a^{1}][F,a^{2}] = a^{0}[F,a^{1}][F,a^{2}]a^{3}- a^{3}a^{0}[F,a^{1}][F,a^{2}].
\begin{array}{rcl}\displaystyle \left[a_q,a_q^\dagger\right] = \left[N+1+b\gamma\right]_q-\left[N+b\gamma\right]_q~, \displaystyle \left[N, a_q\right] = -a_q~,~~~\displaystyle \left[N, a_q^\dagger\right]=a_q^\dagger~.\end{array}
\frac{1}{5}\Pr(\tau=k) \leq \Pr(\tau=k,|S_{k+1,n}|\leq 4t) &\leq \Pr(\tau=k,||S_{n}|-a_{n}|>11t-4t-|a_{k}-a_{n}|) &\leq\Pr(\tau=k,||S_{n}|-a_{n}|>t)
\phi((f_1)_{N},h_{2}) &= (\mathcal{F}((f_{1})_{N})|h_{2})-(f_{1}|\mathcal{F}^{-1}(h_{2})) &= (\mathcal{F}(f_{1})|\mathcal{F}(F_{2}))-((f_{1})_{N}|(\tilde{f}_{2})_{N}).
\xi_{k,\ell}( \beta, \varrho, \mathbf{b})= \beta_0 \eta_{k,\ell}-\frac{\beta^2}{2\left( 1-\varrho \right)}+ c^{(1)}_{\ell}(\beta, \varrho, \pmb{b}) + ( \varrho_0 - \varrho ) c^{(2)}_{k,\ell}+ c^{(3)}_{\ell},
\int_{(\Omega_{q})_{\delta/2}}F(x,u)dx \leq c_{2}(\|u\|_{L^{q_{1}^{-}}(\Omega_{q})_{\delta/2}}^{q_{1}^{-}}+\|u\|^{\beta}+\|u\|_{L^{(p^{*})^{-}}(\Omega_{q})_{\delta/2}}^{(p^{*})^{-}}) &\leq c_{3}(\|u\|^{q_{1}^{-}}+\|u\|^{\beta}+\|u\|^{(p^{*})^{-}})
P\{T_{\ell}= T_{\min}\}= & (-1)^g \sum_{k_g = 1}^{M_g}\cdots \sum_{k_1 = 1}^{M_1}(-1)^{k_1 + \cdots + k_g}\binom{M_1}{k_1}\cdots \binom{M_g}{k_g}\frac{k_{\ell}p_{\ell}}{k_1 p_1 + \cdots + k_g p_g}.
\hat{{\cal H}}( s , \chi ) = s \left( m^{2}- \Pi^{2}+ \frac{i g}{2}F_{\alpha \beta}\tilde{\gamma}^{\alpha}\tilde{\gamma}^{\beta}\right) + \left( \Pi_{\nu}\tilde{\gamma}^{\nu}- m \tilde{\gamma}^{5}\right) \chi .
\binom{N}{k}\left(\frac{-\alpha}{(\theta+\alpha)_{n\uparrow}}\right)^{k}\left( \frac{(\theta+\alpha)_{n\uparrow}}{(\theta)_{n\uparrow}}\right)^{N}\prod_{i=1}^{k}(1-\alpha)_{m_{i}-1\uparrow}(\theta+\alpha)_{n-m_{i}\uparrow},
\zeta_{0,\infty}(s,c)=\frac{\pi}{\sqrt{c}}\,\frac{1}{s-1}+\pi^{2}+\frac{\pi}{\sqrt{c}}\left(2\gamma-\log\left(4c\right)-4\,\sum_{n=1}^{\infty}\frac{1}{n}\cdot\frac{1}{e^{2\pi\sqrt{c}n}+1}\right)+O(s-1)
I_1=\mathbb{E}[\int_0^T\{f(t,y)-\widehat{f}(t,y)\}\mathbb{E}[\delta_Y(y)|\mathcal{F}_t]dt], I_2=\mathbb{E}[\{g(x(T,y))-g(\widehat{x}(T,y))\}\mathbb{E}[\delta_Y(y)|\mathcal{F}_T]].
q(t_0,t) =\Big(1+\int_{t_0}^te^{\rho(t_0,u)}\mu(u)du\Big)^{-1},p(t_0,t)=e^{\rho(t_0,t)}q(t_0,t),\rho(t_0,t)=\int_{t_0}^t(\mu(u)-\lambda(u))du.
\gamma_{k+1}\leq -K_8\gamma_k^{\frac{2}{1+\epsilon}}+ \alpha_k \gamma_k + K_{11}\beta_k \leq- K_8\gamma_k^{\frac{2}{1+\epsilon}}+ d_k \gamma_k = d_k \gamma_k\bigg( 1- e_k\gamma_k^{\frac{1-\epsilon}{1+\epsilon}}\bigg),
\bigg|\sum_{A_{0}t^{\frac{1}{3}}< n \leq \sqrt{\frac{t}{2\pi}}}n^{-\frac{1}{2}- it}\bigg| \leq \sum_{j=1}^{J}\frac{1}{X_{j-1}^{\frac{1}{2}}}\max_{L \leq N_{j}- N_{j-1}}\bigg| \sum_{n= N_{j-1}+1}^{N_{j-1}+ L}e^{-it\log n}\bigg|.
{\bf{\hat g}}= \frac{1}{(N-1)^2}\Bigg[ (2-N) \sum_{i=1}^N\Big( \frac{d{\hat u}_i}{{\hat u}_i}\Big)^2 + \sum_{r \neq s}\frac{d{\hat u}_r}{{\hat u}_r}\,\frac{d{\hat u}_s}{{\hat u}_s}\Bigg]
\langle \Psi_A(\boldsymbol{x}')|T_{z_0}|\Psi_A(\boldsymbol{x})\rangle=\widetilde \Lambda^\dagger_{1}(x'_{N}) \widetilde \Lambda_{2}^\dagger(x'_{N-1}) \cdots\widetilde \Lambda_{N}^\dagger(x'_1)\,\widetilde \Lambda_{N}^{(z_0)}\left(x_1\right)\cdots \widetilde \Lambda_{2}^{(z_0)}(x_{N-1})\,\widetilde \Lambda_{1}^{(z_0)}(x_{N})\,.
h(s,r,z) =(1-s) \underbrace{z\varphi_0( \frac{r -\frac 2{\sqrt 3}}{\delta}) \varphi_0( \frac{|z|-(1-\epsilon)}{\epsilon^2})}_{h_0(r,z)}+s \underbrace{z\varphi_0( \frac{r -\frac 2{\sqrt 3}}{\delta}) \varphi_0( \frac{|z|-(1+\epsilon)}{\epsilon^2})}_{h_1(r,z)}.
A = K[x,y]/(x^{\frac{n}{2}+1}, y^{\frac{n}{2}}, xy), \textrm{for}n\textrm{even}A = K[x,y]/(x^{\frac{n+1}{2}}, y^{\frac{n+1}{2}}, xy), \textrm{for}n\textrm{odd.}
\begin{aligned}2g-2 = -2 d + \sum_{i=0}^{r_0}\left(\lambda_{0,i}- 1\right) + \sum_{i=0}^{r_1}\left(\lambda_{1,i}- 1\right)+ \sum_{i=0}^{r_\infty}\left(\lambda_{\infty,i}- 1\right) + \rho &= d-r_0-r_1-r_\infty+\rho.\end{aligned}
\sum_{v \in \mathcal{M}}(g+h)(v) = \sum_{v \in \mathcal{M}}g(v) + \sum_{v \in \mathcal{M}}h(v) = \bigg( \sum_{v \in \mathcal{M}-\{s\}}g(v) + g(s) \bigg) + \bigg(\sum_{v \in \mathcal{M}-\{s\}}h(v) + h(s) \bigg) = \sum_{v \in \mathcal{M}-\{s\}}g(v) + \sum_{v \in \mathcal{M}-\{s\}}h(v) + f(s) = \sum_{v \in \mathcal{M}}f(v).
\begin{cases}(\xi_G\cdot e_{4g-2}) =-1+(e_n\cdot e_{4g-2})-(f_1\cdot e_{4g-2})-2(e_{n+1}\cdot e_{4g-2}), (\xi_G\cdot f_1) = (f_1\cdot e_{4g-2})-2, (\xi_G\cdot f_2) = (f_2\cdot e_{4g-2})-1.\end{cases}
V ( M_{1}, M_{2}) = \frac{1}{N}\mathrm{T r}M_{1}^{2}+ \frac{1}{N}\mathrm{T r}M_{2}^{2}+ \frac{1}{N^{2}}\mathrm{T r}M_{1}^{2}\mathrm{T r}M_{2}^{2}.
\inf_{\gamma\in \Pi(\mu_1,\mu_2)}\mathbb{E}_{\gamma}[g(S_1, S_2)] = \sup_{x\in \mathbb{R}}\max \left\{\mu_1(x)+\mu_2(z-x)-1,0 \right\}\ \sup_{\gamma\in \Pi(\mu_1,\mu_2)}\mathbb{E}_{\gamma}[g(S_1, S_2)] = 1+\inf_{x\in \mathbb{R}}\min \left\{\mu_1(x)+\mu_2(z-x)-1,0 \right\}.
\begin{cases}\rho_t + \left(\rho \, v\right)_x=0, t>0,~ x\in{\mathbb{R}}, (\rho\,w)_t + \left(\rho\,v\,w\right)_x=0, t>0,~ x\in{\mathbb{R}}, v(0,x)= \bar{v}(x), x\in{\mathbb{R}}, w(0,x)= \bar{w}(x), x\in{\mathbb{R}},\end{cases}
\beta_{c}(I) =\sum_{w\in\mathcal{L}_t^f(u)}\binom{n-\min(w)-(d-1)t}{n-2-(d-1)t}-\sum_{\substack{w\in\mathcal{L}_t^f(v) w\ne v}}\binom{\max(w)-(d-1)t-1}{n-2-(d-1)t}&=\binom{a}{a-1}+m\binom{a-1}{a-1}-\binom{a}{a-1}=m>0.
\int \mathbf{1}_A(z)\prod_{v\in V}g_v(z_v,y_v)\,\gamma_V(x,dz) &=\int \gamma_W^y(z',A)\int\prod_{w\in W}g_w(z_w,y_w)\,\gamma_W(z',dz)\prod_{v\in V\backslash W}g_v(z_v',y_v)\,\gamma_V(x,dz') &=\int \gamma_W^y(z,A)\prod_{v\in V}g_v(z_v,y_v)\,\gamma_V(x,dz).
h(x) \le \Bigl(g_{10-}(x)+\frac{2}{6^{11}}(2^{11}x - 853)(73789-3^{11}x)\Bigr)\Bigr|_{x=\frac{73789}{3^{11}}}& + \frac1{10\cdot 6^{11}}- \eta &=-\frac{29347546710908416454761350780039732607}{65143838749669376004339536681901474208962048000}<0,
\rho^{\rm{NA}}_{u,\omega_1}(\lambda) = \rho^{\rm{NA}}_{u,\omega_1}(\lambda_{C}^2) = \begin{pmatrix}e^{-4(2a+1)u}-\frac{\sinh(4(2a+1)u)}{\sinh(u)}0 e^{4(2a+1)u}\end{pmatrix}.
(a_1,a_2,\ldots,a_k;q)_n= \prod_{j=1}^k \prod_{i=0}^{n-1}(1-q^i a_j), &{}_{r+1}\phi_r\left[{a_1,a_2,\ldots,a_{r+1}\atop b_1,b_2,\ldots,b_{r}};q,x \right]=\sum_{m\geq 0}{(a_1,a_2,\ldots,a_{r+1};q)_m \over (q,b_1,b_2,\ldots,b_{r};q)_m}x^m.
\begin{aligned}x^{k+1}(\ell) - [\bar x^{k+1}]_\ell{\bf 1}=(I+\gamma_1^k R)\left(x^k(\ell) -[\bar x^k]_\ell{\bf 1}\right) &\quad+\gamma_1^k \left(\zeta_w^k(\ell)- [\bar\zeta_w^k]_\ell{\bf 1}\right) -\lambda^k\left(I -\frac{{\bf 1}u^T}{m}\right)y^k(\ell)\end{aligned}
\left| \log \frac{(T^q)' (x)}{(T^q)' (y)}\right| \le \sum_{i=0}^{q-1}\frac{T''(\xi_i)}{T'(\xi_i)}\left| T^i(x) - T^i(y) \right| \le C_1 \sum_{i=0}^{q-1}b_{j-i}^{2\beta+2}\le C_2 \sum_{i=0}^{q-1}n_{j-i}^{-\alpha (2\beta+2)}\le C_2 \sum_{n=1}^\infty n^{-2}:= C_3,
\frac{|E(X,V\setminus X)|}{\frac{1}{2}\big((n-i)d - |E(X,V\setminus X)|\big) + |E(X,V\setminus X)|}= \frac{2|E(X,V\setminus X)|}{(n-i)d + |E(X,V\setminus X)|}.
\Omega\equiv \sum_A r_A\,(d\psi_A+\cos\theta_A\,d\psi_A)=\frac{i}{4}\sum_A \left(\xi_A^*d\xi_A-\xi_Ad\xi_A^*-\zeta_A^*d\zeta_A+\zeta_Ad\zeta_A^*\right),
Q \bar{Q}Q = \left( \begin{array}{c c}{- u_{1}d_{1}d_{2}+ u_{2}d_{1}^{2}}&{- u_{1}u_{2}d_{1}+ u_{1}^{2}d_{2}}\\{u_{2}d_{1}d_{2}- u_{1}d_{2}^{2}}&{u_{1}u_{2}d_{2}- u_{2}^{2}d_{1}}\\ \end{array}\right)
\varphi^{'^{\prime}}( x ) + ( \gamma + 1 - 2 \alpha ) K s g n ( x ) A^{- 1}\varphi^{'}+ [ \alpha ( \alpha - \gamma ) K^{2}A^{- 2}+ k_{x}^{2}A^{- 2 ( \gamma + 1 )}] \varphi = 0
u_i^B = \frac{1}{2\pi}\int_0^\infty{e^{\sigma (x_3-S_3)}}\int_0^{2 \pi}e^{i \sigma ((x_1-S_1) \cos \alpha + (x_2-S_2) \sin \alpha)}M_i(\alpha) W(\sigma,\alpha) \, d\alpha \, d\sigma,
\left\langle \Theta_{22}(\vec{x})\right\rangle =-\frac{1}{16\pi^{2}}D_{2}(\vec{x})+\frac{1}{48\pi^{2}}D_{1}(\vec{x})-\frac{1}{24\pi^{2}}D_{22}(\vec{x})+\frac{1}{64\pi^{2}}F_{2}(\vec{x})-\frac{1}{192\pi^{2}}F_{1}(\vec{x})
L^{\frak{g}}\Bigl(\Bigl(n-\frac{1}{2}\Bigr)\omega_1\Bigr)\Bigl|_{\frak{g}^\sigma}\simeq\bigoplus_{k=0}^{\infty}L^{\frak{so}(2,2l+1)}\Bigl(\Bigl(n+k-\frac{1}{2}\Bigr)\mu_1\Bigr)\boxtimes F^{\frak{so}(2n-2l)}(k\nu_1)
\delta S_{g}= \frac{i}{4 8}[ \int ( H -{\bar{H}}) F^{{\mathbf i}{\mathbf j}}F_{{\mathbf i}{\mathbf j}}+ \frac{3}{1 0}( H -{\bar{H}})_{{\mathbf i}{\mathbf j}{\mathbf k}{\mathbf l}}F^{{\mathbf i}{\mathbf j}}F^{{\mathbf k}{\mathbf l}}\; ]
D_{nm}\left( \underline{\theta}\right) =\delta_{nm}~\sum_{\ell =1,~\ell\neq n}^{N}\cot \left( \theta_{n}-\theta_{\ell}\right) +\left( 1-\delta_{nm}\right) ~\frac{\sigma_{n}\left( \underline{\theta}\right)}{\sigma_{m}\left( \underline{\theta}\right)}~\frac{1}{\sin \left( \theta_{n}-\theta_{m}\right)}~,
s(g)\sum_{k=1}^{N}a_{ik}(h) = \sum_{k, k' = 1}^{N}a_{ik'}(g)a_{ik}(h) & = \sum_{k, k'=1}^{N}\delta_{k, k'}a_{ik}(gh) & = \sum_{k=1}^{N}a_{ik}(gh)
|A_{n,k}| = |\psi(A_{n,k})| \leq 2^{n-1}\cdot 2^{n-1}k^{\left\lfloor \frac{n-1}{2}\right\rfloor}n^{\left\lceil\frac{n-1}{2}\right\rceil+1}& \leq 4^{n-1}k^{\frac{n}{2}}n^{\frac{n}{2}+1}.
y_n = \sum \limits_{i=0}^{n-1}C_0A_0^{n-1-i}B_0u_i + \sum \limits_{i=1 0}^{n-1}C_0A_0^{n-1-i}w_i + v_{n}= \sum \limits_{l=t-2T+1}^{t-1}C_0A_0^{t-1-l}Bu_l + \sum \limits_{l=0}^{t-2T}C_0A_0^{t-1-l}B_0u_l +\sum \limits_{l=0}^{t-1}C_0A_0^{t-1-l}w_l + v_{t}:= g_0X_t + \bar{g}_0 \bar{X}_t + hW_t + v_t,
T^j \varphi (T^i \varphi^n (x)) = T^j T_U^i \varphi^{n+1}(x) \hbox{and}T^{r_U (T_U^i ( \varphi^{n+1}(x))) -1}\varphi (T^i \varphi^n (x)) = T^{-1}T_U^{i+1}\varphi^{n+1}(x).
\begin{array}{c}\Gamma ( Th\circ \rho ,h\circ \eta ) ( z^{\alpha}t_{\alpha}) ( f_{\mid V})=z^{\alpha}\rho_{\alpha}^{i}( \frac{\partial ( f\circ h_{\mid U})}{\partial x^{i}}\circ h^{-1})_{\mid V},\end{array}
{\delta}B_{\mu \nu}^{p}={\cal D}_{\mu}^{p q}{\xi}_{\nu}^{q}-{\cal D}_{\nu}^{p q}{\xi}_{\mu}^{q}\equiv R_{\mu \nu \alpha}^{p q}{\xi}^{q \alpha}, \, \, \, \, \, \, \, \,{\delta}A_{\mu}^{p}= 0 ,
D_i^{(N)}(F(x_i,\{p_n\}))=x_i\frac{\partial}{\partial x_i}F(x_i,\{p_n\})+\beta x_i\oint\frac{d\xi}{\xi^2}\frac{\phi^-(\xi)-1}{1-\frac{x_i}{\xi}}\left(\tilde{V}'(\xi)\tilde{V}(x_i)F\right)(\xi,\{p_n\}),
h(Y_A^n|W_C)-h(Y_B^n|W_C) &\leq I(X_B^n;h_2 X_B^n+Z_A^n|W_C) - I(X_B^n; X_B^n+\frac{1}{h_2}Z_B^n|W_C) & + I (\tilde{X}_A^{n}; \tilde{X}_A^{n}+ Z_B^n|W_C) &= h (\tilde{X}_A^{n}+ Z_B^n|W_C) - h(Z_B^n|W_C) &\leq h ( \tilde{X}_A^{n}+ Z_B^n) - h(Z_B^n),
\tau_{\widehat{\alpha}}(\gamma)= \sum_{j_1,\ldots,j_{\hat{\ell}}}c^\gamma_{j_1,\ldots, j_{\hat{\ell}}}\,\tau_{\widehat{\alpha}_1-1}(\theta_{j_1})\cdots\tau_{\widehat{\alpha}_{\hat{\ell}}-1}(\theta_{j_{\hat{\ell}}})\ .
S = S_{0}+ \int d t \Big \{{\frac{1}{2}}h_{M N}T^{M N}+ J^{a_{1}a_{2}a_{3}; a_{4}}\partial_{a_{4}}C_{a_{1}a_{2}a_{3}}^{( 3 )}+ 3 0 M^{+ - m_{1}\ldots m_{4}; a}\partial_{a}C_{+ - m_{1}\ldots m_{4}}^{( 6 )}\Big \}
\int_{0}^{T}\left( \partial_{i}v_{n},\mu_{\varepsilon}(u_{n})\psi\right) =\int_{0}^{T}\left( \partial_{i}v,\mu_{\varepsilon}(u)\psi\right) +\int_{0}^{T}\left( \partial_{i}v_{n}-\partial_{i}v,\mu_{\varepsilon}(u)\psi\right) +\int_{0}^{T}\left( \partial_{i}v_{n},\left( \mu_{\varepsilon}(u_{n})-\mu_{\varepsilon}(u)\right) \psi\right) .
Z_{NR}(t_{k-1}) = Z_{R}(t_{k}) \int \epsilon (1+t^2)^{1/4}\frac{\xi}{k^2}((\frac{\xi}{k^2})^2)^{-1/4}\frac{1}{(1+(t-\frac{\xi}{k})^2)^{3/4}}dt \approx C \epsilon \sqrt{\frac{\xi}{k}}\sqrt{\frac{\xi}{k^2}}Z_{R}(t_{k}),
\varphi(\Gamma a_2a_0[F,\delta(a_1)]|D|^{-1}) =\varphi(\Gamma a_0[F,\delta(a_1)]|D|^{-1}a_2) &=\varphi(\Gamma a_0[F,\delta(a_1)]a_2|D|^{-1})+\varphi(\Gamma a_0[F,\delta(a_1)][|D|^{-1},a_2]).
\left( \begin{array}{c c}{{R^{\alpha \beta}}}&{{0}}\\{{0}}&{{R^{\alpha \beta}}}\\ \end{array}\right) - \frac{1}{2}\left( \begin{array}{c c}{{g^{\alpha \beta}}}&{{0}}\\{{0}}&{{g^{\alpha \beta}}}\\ \end{array}\right) R = \left( \begin{array}{c c}{{T^{\alpha \beta}}}&{{0}}\\{{0}}&{{T^{\alpha \beta}}}\\ \end{array}\right)
d s^{2}- d s_{0}^{2}={\frac{r_{0}^{2}W^{2}}{2 \sigma U}}\left( - r_{0}^{2}\sigma^{2}\Delta + k + \Delta \dot{f}^{2}- 2 a_{i}\dot{f}^{i}\right) + h i g h e r \o r d e r \i n \U .
\begin{array}{lll}(I-J^{(r)}\hat{C}C\hat{C}^{-1}J^{(r)}) X^{*}= J^{(r)}\Big[(\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}\Big(Dp - U(J^{(r)}X^{*}) - \mathbf{1}\frac{\beta}{2}\Big)\Big]. \end{array}
\mu(E^0_{\bar{a}}) = \mu'\left( \left\{x \in V' : f^0_+ \left( \zeta_{\bar{a}_I}(x) : I \subseteq [k] \right) = 1 \right\}\right) = \mu'\left( \left\{x \in V' : \left( \zeta_{\bar{a}_I}(x) : I \subseteq [k] \right) \in S_{+}\right\}\right) = \nu_{2^k}\left( S_+ \right).
E_2(\frac{B_{13,1}}{b_1-b_3})-\frac{B_{12,1}}{b_1-b_2}[\frac{B_{13,1}}{b_1-b_3}-\frac{B_{23,2}}{b_2-b_3}]=0, &E_3(\frac{B_{13,1}}{b_1-b_3})-[(\frac{B_{13,1}}{b_1-b_3})^2-\frac{B_{12,1}B_{23,3}}{(b_1-b_2)(b_2-b_3)}]=R_{1313}=b_1b_3+a_1+a_3.
\tilde{E}\big[e^{\sigma B_T^H}|\mathcal{F}_{r}^{H}\big]=e^{\sigma B_T^H}\sum\limits_{l=0}^{+\infty}\frac{(-\sigma(T-r)^H)^l}{l!}\sum\limits_{k=0}^{l}{l\choose k}\left(-\frac{\sigma(T^{2H}-r^{2H})}{(T-r)^{H}}\right)^kh_{l-k}\left( \frac{B_{T}^{H}-B_{r}^{H}}{(T-r)^{H}}\right).
\lim_{\sigma \rightarrow \infty}F_{n}^{\mathcal{O}|\mu_{1}\ldots \mu_{n}}(\theta_{1}+\eta_{1}\sigma ,\ldots ,\theta_{n}+\eta_{n}\sigma)=F_{\nu_{1}\cdots \nu_{n}}^{\mathcal{O}|\mu_{1}\ldots \mu_{n}}(\theta_{1},\ldots ,\theta_{n}),
\Vert (\mathbf{D}\widetilde{\Lambda})^\varepsilon (\Pi_\varepsilon -S_\varepsilon)\boldsymbol{\Phi}\Vert_{L_2(\mathbb{R}^d)}\leqslant \varepsilon \left(\widetilde{\beta}_1(r_0^{-1}+r_1 )^2+4\widetilde{\beta}_2\widetilde{M}_1^2\right)^{1/2}\Vert \boldsymbol{\Phi}\Vert_{H^2(\mathbb{R}^d)}.
(u_m(T),\psi(T)v)+(u_0,\psi(0)v)=\int_0^T\langle \partial_t u_m(t),\psi(t)v\rangle+\langle \psi'(t)v,u_m(t)\rangle\mathrm{dt}= =-\int_{Q_T}\psi(t)( \mathsf{A}\nabla u_m(t)+T_m(u_m(t)){\bf E}) \cdot \nabla v\mathrm{dx}\mathrm{dt}+ +\int_{\Sigma_T}\psi(t)(h(t)-b(u_m(t))u_m(t)) v\mathrm{ds}\mathrm{dt}+\int_0^T \psi'(t)(v ,u_m(t))\mathrm{dt},
\int_{-\infty}^\infty|\eta(t)|\,\|\partial_t\varphi\|_{2,q}\,dt\leq\,\omega_{p-q}\cdot 2^{p-q}\cdot\|\varphi\|_{2,p}\cdot|\eta|_0\cdot\left\{\sum_{n=0}^\infty(2n+2)^{-2p}\right\}^{\frac{1}{2}},
\int_0^1 \left(A(n_{2j},n_{2jx})-A(n_{1j},n_{1jx})\right)\varphi_xdx+\int_0^1 \left(n_{2j}-n_{1j}\right)\varphi dx=\int_0^1 \left(b_2-b_1\right)\varphi dx\le0,
-\frac{c^{-2}}{4}r^{2(1-\alpha)}\left[ \frac{d^{2}}{dr^{2}}+ (3 - 2\alpha) \frac{1}{r}\frac{d}{dr}-\frac{1}{r^{2}}(m^{2}+2(\alpha -1)m) \right] f_{m}= \lambda^{2}f_{m}
\mu(\sigma,\tau; \rho) := \frac{\omega(\sigma\tau\sigma^{-1}, \tau,\rho)}{\omega(\sigma\tau\sigma^{-1}, \sigma\rho\sigma^{-1}, \sigma)\omega(\sigma, \tau, \rho)}, \gamma(\sigma;\tau,\rho) := \frac{\omega(\sigma, \tau, \rho) \omega(\sigma\tau\rho(\sigma\tau)^{-1},\sigma,\tau)}{\omega(\sigma, \tau\rho\tau^{-1}, \tau)},
c_{(1), \lambda}^\mu = \begin{cases}\tilde{c}_{(1), \lambda}^\mu \textrm{if}\lambda \neq \mu \/; \tilde{c}_{(1), \lambda}^\lambda -C(t) = w_0 (\omega_k) - w_\lambda (\omega_k) \textrm{if}\lambda= \mu \/, \end{cases}
dX_t = (r_t (X_t- \varphi_t^1 - \varphi_t^2)+\varphi_t^1 \mu^1_t +\varphi_t^2 \mu^2_t ) dt +(\varphi_t^1 \sigma^1_t + \varphi_t^2 \sigma^2_t) dW_t +(\varphi_t^1 \beta^1_t + \varphi_t^2 \beta^2_t) d\tilde N_t & = (r_t X_t+\varphi_t' (\mu_t - r_t{\bf 1}) ) dt+ \varphi_t' \sigma_t dW_t + \varphi_t' \beta_t d\tilde N_t,
p(n-1)+p(n-2)-p(n) =\widetilde P_1(n) p\left(n-\frac{k(3k-1)}{2}\right)+p\left(n-\frac{k(3k+1)}{2}\right) = \widetilde P_{k-1}(n)+\widetilde P_k(n), \k\geqslant 2.
A_{p,q}= \partial_p\partial_q - \partial_{p+1}\partial_{q-1}{\rm where}(p,q) \in [1,k-1]\times [2,k] & \mathcal{T}^m := \partial_1\partial_{m-1}+ \partial_mE{\rm where}E := \sum_{h=1}^k \sigma_h\partial_h{\rm and}m \in [2, k]
tr(ab) = \sum_{n,m,i,j\geqslant 0}tr(a_{n,m}b_{i,j}) = \sum_{n,m,i,j\geqslant 0}\mu_V(v)^{-2}Tr(a_{n,m}b_{i,j}p_v) = \sum_{n,m\geqslant 0}\mu_V(v)^{-2}Tr(a_{n,m}b_{n,m}p_v) a_{n,m}\perp b_{i,j}p_v (n,m)\neq (i,j) & = \sum_{n,m\geqslant 0}tr(a_{n,m}b_{n,m}).
x^{i}{'}x^{j}{'}\tilde{R}_{iklj}n^{k}_R n^{l}_S=F\delta_{RS}(\frac{1}{2}\tilde{R}-\tilde{R}_{ij}x^{i}{'}x^{j}{'})-\tilde{R}_{ij}n^{i}_R n^{j}_S,
{\cal L}= -{1\over 4}F_{\mu\nu}F^{\mu\nu}+ (D_{\mu}\phi_{a})^{*}D^{\mu}\phi_{a}- m^{2}\phi_{a}^{*}\phi_{a}-{ie\over 2}\epsilon^{\mu\nu}F_{\mu\nu}\phi_{a}^{*}(\sigma_{3})_{ab}\phi_{b},
\alpha :=\alpha(W,\lambda,\eta)= \frac{1}{\eta}+\frac{64\lambda}{75}\sqrt{\eta+W^{-1/3}}+\frac{\lambda \eta}{W^{1/3}}+ \frac{\lambda}{W^{2/3}}, \beta :=\beta(W,\eta)= \frac{64}{15}\frac{1}{\sqrt{\eta}}+ \frac{3}{W^{1/3}}.
\mathfrak{D}_I \mathfrak{D}_I^*= \nabla^* \nabla +2r +\mathcal{R}_0 + \sqrt{r}\mathcal{R}_1 &\mathfrak{D}_I^* \mathfrak{D}_I = \nabla^* \nabla +2r +\tilde{\mathcal{R}_0}+ \sqrt{r}\tilde{\mathcal{R}_1}.
\begin{cases}\|f_1,\ldots, f_r\|_v:=\max\left(|c_{f_i}(m)|_v\ :\ 1\le i \le r,m\in M_{f_i}\right)\ (v\in M_K), &\|f_1,\ldots, f_r\|_{v,1}:=\|f_1,\ldots ,f_r\|_v\ (v\in M^0_K), &\|f_1,\ldots , f_r\|_{v,1}:= \|\sigma_v(f_1),\ldots ,\sigma_v(f_r)\|_1^{[K_v:R]/[K:Q]}\ (v\in M^\infty_K),\end{cases}
w_{n1}\big|_{\gamma_0'\times(t_1+\delta,t_2-\delta)}=w_{n3}\big|_{\gamma_0'\times(t_1+\delta,t_2-\delta)}, \frac{\partial w_{n1}}{\partial \nu}\Big|_{\gamma_0'\times(t_1+\delta,t_2-\delta)}= \frac{\partial w_{n3}}{\partial \nu}\Big|_{\gamma_0'\times(t_1+\delta,t_2-\delta)}.
\begin{pmatrix}u_1 u_2 u_3 \end{pmatrix}= \begin{pmatrix}e^{-\frac53r}\,\sin(\theta)\,\cos(\varphi) e^{-\frac23r}\,\sin(\theta)\,\sin(\varphi) e^{\frac43r}\,\cos(\theta)\end{pmatrix}\,,
\frac{d^2}{dz^2}\left( \sum_{j=1}^N \left[\begin{matrix}N j \end{matrix}\right] \frac{q^{j^2}(q)_j}{(z q)_j (z^{-1}q)_j}\right)_{z=1}=\sum_{n=1}^{\infty}N_{2,N}(n)q^n.
\frac{1}{[(k+k_{i})^{2}-m^{2}]}=\sum_{j=0}^{N}\frac{\left( -1\right)^{j}\left( k_{i}^{2}+2k_{i}\cdot k\right)^{j}}{\left( k^{2}-m^{2}\right)^{j+1}}+\frac{\left( -1\right)^{N+1}\left( k_{i}^{2}+2k_{i}\cdot k\right)^{N+1}}{\left( k^{2}-m^{2}\right)^{N+1}\left[ \left( k+k_{i}\right)^{2}-m^{2}\right]},
\begin{aligned}\Vert \partial_x v^1_p\Vert_{L^2}^2 \leq \iint_{\{\sqrt{\epsilon}y\leq 1\}}| \partial_x v^1_p|^2\mathrm{d}x\mathrm{d}y &\lesssim\epsilon^{-1/2}\iint \langle y\rangle^l |\partial_{xy}v^1_p|^2\mathrm{d}x\mathrm{d}y \leq C(L,\zeta)\epsilon^{-1/2-2\zeta}\end{aligned}
\varOmega_{i_{n}}\varOmega_{i_{n-1}}\cdots \varOmega_{i_{1}}=\boldsymbol{\omega}_{i_{n}}^{T}\boldsymbol{\delta}_{i_{n}}\boldsymbol{\omega}_{i_{n-1}}^{T}\boldsymbol{\delta}_{i_{n-1}}\cdots \boldsymbol{\omega}_{i_{1}}^{T}\boldsymbol{\delta}_{i_{1}},
y_{p,2}^c =\mathbf{h}_{sp}^H\mathbf{F}\mathbf{y}_r+\mathbf{h}_{sp}^H\mathbf{w}_2{s}_2+n_p &=\mathbf{h}_{sp}^H\mathbf{F}(\mathbf{H}_0\mathbf{w}_1{s}_1+\mathbf{n}_r)+\mathbf{h}_{sp}^H\mathbf{w}_2{s}_2+n_p, y_s^c =\mathbf{h}_{ss}^H\mathbf{w}_2{s}_2+\mathbf{h}_{ss}^H\mathbf{F}\mathbf{H}_0\mathbf{w}_1{s}_1+\mathbf{h}_{ss}^H\mathbf{F}\mathbf{n}_r+n_s.
c+o(1) =I[u_n]=\frac{1}{p}[u_n]_{s,p}^p-\frac{b}{2(p^\sharp)^2}\int_{\mathbb{R}^N}(K\ast |u_n|^{p^\sharp}) |u_n|^{p^\sharp}dx+\frac{\varepsilon_g}{p_g}\|u_n\|_{p_g}^{p_g}+o(1), o(1) =I'[u_n]u_n= [u_n]_{s,p}^p-\frac{b}{p^\sharp}\int_{\mathbb{R}^N}(K\ast |u_n|^{p^\sharp}) |u_n|^{p^\sharp}dx-\varepsilon_g\|u_n\|_{p_g}^{p_g}+o(1).
{\sf L}f(\xi) =\left[\log\left((a+1)^2+\frac12\right)+\log\left((a-1)^2+\frac12\right)-2\log\left(a^2+\frac12\right)\right] &+m\left[\log\left(a^2-\frac12\right)-\log\left(a^2+\frac12\right)\right] &=\log\left(\frac{4a^4-4a^2+9}{4a^4+4a^2+1}\right)+m\left[\log\left(\frac{2a^2-1}{2a^2+1}\right)\right]\leq 0.
\begin{cases}\rho(x) w_{tt}- \operatorname{div}(K(x) \nabla w) + \alpha^2 z^2w =0 ~\hbox{in}~ \Omega \times (0,T), \rho(x) z_{tt}- \operatorname{div}(K(x) \nabla z) + \alpha^2w^2z=0 ~\hbox{in}~ \Omega\times (0,T), w_t=z_t=0 ~\hbox{in}~ \omega\times(0,T).\end{cases}
S_{L_i}(x)\cap F^1 = S_{\widehat{L}_i}(x) \cap F^1 \enspace \enspace\frac{1}{3}|S_{\widehat{L}_i}(x) \cap F^2| \leq |S_{L_i}(x)\cap F^2| \leq \frac{1}{2}|S_{\widehat{L}_i}(x) \cap F^2|.
a_{k}(\sum_{i=1, i \ne k}^{n-1}\zeta^{pi+r(n+k-i)}+ \zeta^{ri+p(n+k-i)}) =a_k\left( (-1-\zeta^{k(p-r)})\zeta^{rk}+(-1-\zeta^{k(r-p)})\zeta^{pk}\right) &=2(\zeta^{rk}+\zeta^{pk})a_k,
\delta \langle \partial_x z^{\otimes 2},\mathbf{1}\rangle_{s,t}=\langle (\mathcal D +I+\mathbb{I}+\tilde{\mathbb{I}})_{s,t}(1,1),\mathbf{1}\rangle +\langle (\partial_x z)_{s,t}^{\natural,2},\mathbf{1}\rangle\, .
|Z_A^{(n)}-Z_A^{(n-1)}| =|Z_{A,0}(w^{(n)}-w^{(n-1)})\int_{0}^{1}\mathrm{exp}(sw^{(n)}+(1-s)w^{(n-1)})ds| &\leq\frac{Z_{A,0}e^{-C_*x}}{\rho_{A,0}u_{A,0}A(0)}\int_0^x A(\tau)|\rho_A^{(n)}\phi(T_A^{(n)})-\rho_A^{(n-1)}\phi(T_A^{(n-1)})|d\tau &\leq O(1)\delta_0\max\limits_{0\leq\tau\leq x}(|\rho_A^{(n)}-\rho_A^{(n-1)}|+|T_A^{(n)}-T_A^{(n-1)}|).
\mathrm{d}\varepsilon = \frac{1}{80\pi^2}|J| \, \left( \mathrm{d}^{(1/20,2/20)}+ \mathrm{d}^{(1/20,2/5)}+ \mathrm{d}^{(3/20,3/10)}+ \mathrm{d}^{(3/20,1/5)}+ 2\mathrm{d}^{(1/10,3/10)}\right),
F_0= \frac{g_0p_*\left(f_{\tilde u}\left(\prod_{1\le j\le m}f_{z_j}\right)\left(\sum_{1\le j\le m}p_j\frac{df_{z_j}}{f_{z_j}}\right)\right)}{c_0dz}= \frac{p_*\left(f_{u}\left(\prod_{1\le j\le m}f^{k_j+1}_{z_j}\right)\left(\sum_{1\le j\le m}p_j\frac{df_{z_j}}{f_{z_j}}\right)\right)}{c_0dz},
\mathbb{E}[\int_{\mathcal{D}}|U(x)\partial^{\alpha}\varphi(x)|dx] = \mathbb{E}[\int_{\alpha}|U(x)\partial^{\alpha}\varphi(x)|dx] = \int_{\alpha}|\partial^{\alpha}\varphi(x)|\mathbb{E}[|U(x)|]dx &\leq \int_{\alpha}|\partial^{\alpha}\varphi(x)|\sigma(x)dx < +\infty
u_\gamma(x) = \begin{cases}u_i p_\gamma(x) \in Q_i^\gamma 1 \leq i \leq d, \frac1\gamma \left(p_\gamma(x) -\tfrac\alpha2(u_i+u_{i+1})\right) p_\gamma(x) \in Q_{i,i+1}1 \leq i < d. \end{cases}
{}_{p}F_{q}\left(\ \begin{array}{lll}\alpha_{1},...,\alpha_{p}~;~ \beta_{1}, ..., \beta_{q}~;~\end{array}z\right)=\sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n}... (\alpha_{p})_{n}}{(\beta_{1})_{n}... (\beta_{q})_{n}}\frac{z^{n}}{n!}~,
\delta_{\bf a}{\cal L}_{0}=\mu\dot{\bf x}\cdot\delta_{\bf a}\dot{\bf x}+\frac{\mu}{R^{2}}\,{\bf x}\cdot\delta_{\bf a}{\bf x}=\frac{\mu}{R}\,{\bf a}\cdot\frac{d}{dt}\left[{\bf x}\sinh\left(\frac{t}{R}\right)\right].
\sigma_1(\Lambda_{g,F}^\omega) = -\sqrt{\widetilde{q_2}}, \sigma_0(\Lambda_{g,F}^\omega) = \frac{1}{2\sqrt{\widetilde{q_2}}}(\nabla_{\xi'}\sqrt{\widetilde{q_2}}\cdot D_{t'}\sqrt{\widetilde{q_2}}-\widetilde{q_1}-\partial_{t_3}\sqrt{\widetilde{q_2}}+\widetilde{E}\sqrt{\widetilde{q_2}}),
\dot{x}= y, \dot{y}= \begin{cases}-\frac{k}{m}\,x - \frac{b}{m}\,y + \frac{F}{m},{\rm until~}x = \mu, -\frac{k}{m}\,x - \frac{b}{m}\,y - \frac{F}{m},{\rm until~}x = -\mu,\end{cases}
\int_{0}^{1}\left[-\frac{1}{2e}\left(\dot{x}^{\mu}-i{\psi}_{a}^{\mu}{\chi}_{a}\right)^{2}+\frac{i}{2}f_{ab}\left[\psi_{a\mu},~\psi^{\mu}_b\right]_{-}-i\psi_{a\mu}\dot{\psi}_a^\mu\right]d\tau\;.
\{\,\,(t_{1},t_{2})\,\,|\,\,A(\epsilon_{(t_{1},t_{2},n)})<M\,\,\}=\{\,\,(t_{1},t_{2})\in \mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}\,\,|\,\,t_{1}R_{1}+t_{2}R_{2}<M-T_{n}\,\,\}
|B_n(\beta)| = \sum_{j=0}^{k-1}\binom{n-1}{j}+ \sum_{j=0}^{k-2}\binom{n-1}{j}&= 1 + \sum_{j=1}^{k-1}\left( \binom{n-1}{j}+ \binom{n-1}{j-1}\right) &= \sum_{j=0}^{k-1}\binom{n}{j},
\frac{1}{\sqrt{2}k}\left[ 1-\kappa\left( \frac{d}{dr}+\frac{\kappa+1}{r}\right)\right]R_{E\ell(\kappa)}(r)=\left(E+\frac{1}{2k^{2}}\right)^{\mbox{$\frac{1}{2}$}}R_{E\ell(-\kappa)}(r).
{X^2_1\over X^2_3}={1\over s}\;,\quad{X^2_2\over X^2_3}={t\over s}\;, \quad{2X_1\cdot X_3\over X^2_3}={1+s-t\over s}\;, \quad{2X_2\cdot X_3\over X^2_3}={1-s-t\over s}\;.
\hat{\Theta}^{\hat{a}\hat{b}\hat{c}}\equiv - \, \ell_{p}^{3}\, \hat{W}^{\hat{a}\hat{b}\hat{c}}= - \, \ell_{p}^{3}\, z^{- 2}\, \hat{g}^{\hat{a}\hat{k}}\hat{\cal H}_{\hat{k}\hat{l}\hat{m}}\, ( \hat{G}_{O M})^{\hat{b}\hat{l}}\, ( \hat{G}_{O M})^{\hat{c}\hat{m}}\, .