harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_{{\mathbf i}{\mathbf j}}{\bar{D}}^{{\mathbf k}{\mathbf l}}F_{{\mathbf k}{\mathbf l}}={\bar{D}}_{{\mathbf i}{\mathbf j}}( F^{{\mathbf k}{\mathbf l}}F_{{\mathbf k}{\mathbf l}}) + \frac{3}{1 0}{\bar{D}}^{{\mathbf k}{\mathbf l}}( F_{({\mathbf i}{\mathbf j}}F_{{\mathbf k}{\mathbf l})}) \ ,
\sum_{i\in I}2d_i+\frac{2\sum_{i\in I(\Gamma)}a_i+(k_1(\Gamma)-1)(r-2)}{r}+\frac{2\sum_{i\in I(\Gamma)}b_i+(k_2(\Gamma)-1)(s-2)}{s}&=2|I|+k_1(\Gamma)+k_2(\Gamma)-2.
T_{1}\oplus N_{1}+T_{2}\oplus N_{2}=(T_{1}+T_{2})\oplus (N_{1}+N_{2})\in\Pi(\mathcal{M})'; \zeta(T_{1}\oplus N_{1})=\zeta T_{1}\oplus \zeta N_{1}\in\Pi(\mathcal{M})';\,T_{1}^{*}\oplus N_{1}^{*}\in\Pi(\mathcal{M})'; (T_{1}\circ T_{2})\oplus (N_{1}\circ N_{2})\in\Pi(\mathcal{M})'.
( \delta g^{( 1 )}, \delta g^{( 2 )}) = \int \! \sqrt{g}\delta g_{\mu \nu}^{( 1 )}\left( g^{\mu \alpha}g^{\nu \beta}+ g^{\mu \beta}g^{\nu \alpha}+ C g^{\mu \nu}g^{\alpha \beta}\right) \delta g_{\alpha \beta}^{( 2 )},
\Bigg [ : e^{- i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi \! + \! g N}}U_{1 b}\Phi^{( 1 )}( x )}:_{M^{( 1 )}}\prod_{I = 2}^{N}: e^{- i 2 \sqrt{\pi}U_{I b}\Phi^{( I )}( x )}:_{M^{( I )}}e^{+ i \frac{\theta}{N}}
\frac{\lambda_s}{\lambda}+b=O(b_c^{\frac{5}{2}}+\|\varepsilon\|_{H^1_{}}), &\frac{x_s}{\lambda}-1=O(b_c^{\frac{5}{2}}+\|\varepsilon\|_{H^1_{}}), &b_s+c_p\tilde{b}b_c=O(b_c^3+b_c\|\varepsilon\|_{H^1_{}}).
\begin{aligned}d u_m = i2^m\left(\overline{u_{m+1}}u_{m+2}- \frac{\delta}{2}\overline{u_{m-1}}u_{m+1}- \frac{\delta -1}{4}u_{m-2}u_{m-1}\right) - \delta 2^{2m}u_m + q_m dW_t^{(m;R)}+ ip_m dW_t^{(m;I)},\end{aligned}
V ( z ,{\overline{{z}}}) ={\tilde{V}}( z ,{\overline{{z}}}) \exp ( i \sum_{a}H_{a}\rho_{a}({\overline{{z}}}) + i \sum_{I}Q^{I}\phi^{I}({\overline{{z}}}) + i \sum_{A}Q^{A}\varphi^{A}( z ) ) \exp ( i k_{\mu}{\cal X}^{\mu}( z ,{\overline{{z}}}) ) ~ ,
\frac{1}{A ( q^{2})}= 1 - \frac{e^{2}( \xi - \xi_{0})}{1 6 q}+ \left( \frac{e^{2}( \xi - \xi_{0})}{8 \pi q}\right)^{2}- \frac{1}{3}\left( \frac{e^{2}( \xi - \xi_{o})}{8 \pi q}\right)^{4}+ \ldots
\int_{\mathbb{R}^{n}}|K_{m}^{(j)}(x,y)|^{p}dx =\int_{|x|\leq 2|y|}|K_{m}^{(j)}(x,y)|^{p}dx+\int_{|x|> 2|y|}|K_{m}^{(j)}(x,y)|^{p}dx &=\mathcal{I}_{1}(y)+\mathcal{I}_{2}(y).
P_{{\bf X_2}|Y_1}({\bf x}_2 | 0) = \begin{cases}r_0^{y_1 =0}\{\bf x}_2 = 0^M r_b^{y_1 =0}\alpha_{b, \ell}^{y_1 =0}{\bf x}_2 ={\bf b}_{\ell}\end{cases}, & \forall \ell \in \left[{M \choose b}\right], b \in[ B_].
\eta_{n+q+1}\int_0^T || \mathcal{\mathbf{B}}_{n+q+1}^{\ast}W_{n+q+1}||_{G_{n+q+1}}^2 & \ge\int_0^T e_1(W_{n+q+1})\,dt - \delta_{n+q+1}\int_0^T ( |w_{n-1}|^2 + |w_n|^2 +\sum_{k=1}^q |w_{n+k}|^2)\,dt\,.
e_1^{(i-1)}-\mu^{(i)}f_1^{(i-1)}, e_2^{(i-1)}-\mu^{(i)}f_2^{(i-1)}, \ldots , e_{m-(i-1)}^{(i-1)}-\mu^{(i)}f_{m-(i-1)}^{(i-1)}&e_1^{(i-1)}+\mu^{(i)}f_1^{(i-1)}, e_2^{(i-1)}+\mu^{(i)}f_2^{(i-1)}, \ldots , e_{m-(i-1)}^{(i-1)}+\mu^{(i)}f_{m-(i-1)}^{(i-1)}
\tilde{m}_{2}={\frac{2 \int d \tilde{p}( \Sigma ( \tilde{p}) )^{2}f ( \tilde{p}) -{\frac{\beta^{2}}{1 6 \pi}}( \tilde{\mu}^{2}( \varphi_{0}) - \tilde{m}_{0}^{2}) ( \int d \tilde{p}{\frac{\Sigma ( \tilde{p})}{f ( \tilde{p})}})^{2}}{\int d \tilde{p}( \Sigma ( \tilde{p}) )^{2}}}\; ,
L_{1}(l)=(-1)^{l+1}\frac{\sqrt{5}}{25}\sum_{k=0}^{l-1}(-1)^{k}5^{l-k}\frac{2l}{2l-k}\binom{2l-k}{k}=(-1)^{l+1}\frac{\sqrt{5}}{25}\Bigg\lbrack C_{2l}(\sqrt{5}) -(-1)^{l}2\Bigg\rbrack ,
\begin{array}{l}{\partial_{-}\partial_{0}= q^{- 1}\partial_{0}\partial_{-}~ , ~ ~ ~ ~ ~ \partial_{0}\partial_{+}= q^{- 1}\partial_{+}\partial_{0}~ ,}{\partial_{-}\partial_{+}- \partial_{+}\partial_{-}= ( q^{1 / 2}- q^{- 1 / 2}) \partial_{0}\partial_{0}~ .}\end{array}
B_3=\begin{pmatrix}a_{1,4}a_{1,5}a_{1,7}a_{1,8}\cdots a_{1,n}a_{2,4}a_{2,5}a_{2,7}a_{2,8}\cdots a_{2,n}a_{3,4}a_{3,5}a_{3,7}a_{3,8}\cdots a_{3,n}\end{pmatrix}.
f_n(\lambda_n) =a\lambda_n^\beta-\lambda_n-b\lambda_n(1-q)^{-\beta/(\beta-1)}=\lambda_n\left\{\left(a-b(1-q)^{-\beta/(\beta-1)}\right)\lambda_n^{\beta-1}-1\right\}&=\lambda_n((\lambda_n/\Lambda)^{\beta-1}-1)<0.
\widetilde{c}_{j l}\equiv \cos \widetilde{\tau}_{j l}= \frac{( n_{j}\cdot n_{l})}{\sqrt{n_{j}^{2}n_{l}^{2}}}= \frac{( - 1 )^{j + l}\; D_{j l}^{( N - 1 )}}{\sqrt{D_{j j}^{( N - 1 )}\; D_{l l}^{( N - 1 )}}}.
\omega^{i}= \phi^{j}( \mathrm{1}+ \frac{1}{2 !}A^{2}+ \frac{1}{4 !}A^{4}+ \cdots )_{j}^{i}- d \xi^{\beta}( \frac{1}{2 !}A + \frac{1}{4 !}A^{3}+ \cdots )_{\beta}^{i}
{\mathfrak K}= A (y^0\frac{\partial}{\partial y^4}-y^4\frac{\partial}{\partial y^0})- B_i( y^0\frac{\partial}{\partial y^i}+y^i\frac{\partial}{\partial y^0})- C_j (y^4\frac{\partial}{\partial y^j}+y^j\frac{\partial}{\partial y^4})+ D_p \epsilon_{pqr}y^q\frac{\partial}{\partial y^r}.
\limsup_{L \to \infty}\max_{0\leq |\alpha| \leq 3}\sup_{x \in Q}L^{-2|\alpha|}| D_x^{\alpha}D_y^{\alpha}(K_L)_{i_1 i_2}(x,y) |_{y=x}| \leq M, \i_1,i_2 = 1,2,\dots, m.
X^2_+(t)=\sum_{k+\ell\leq s}\left(\sum_{i\in I_1}\left( |\xi^{(k,\ell)}_{i,1}(t)|^2+|\xi^{(k,\ell)}_{i,2}(t)|^2 \right)+\sum_{i\in\mathbb{N}}|\xi^{(k,\ell)}_i(t)|^2+\sum_{1\leq j\leq J}|\bar{z}^{(k,\ell)}_j(t)|^2 \right).
\begin{aligned}\int_{Q_T}\left(s\Theta (v_x)^2 + s^3 \Theta^3\left(\frac{x-x_0}{a}\right)^2v^2\right)e^{2s\varphi}dxdt &\le C\left(\int_{Q_T}h^{2}\frac{e^{2s\varphi}}{a}dxdt +sd_1\int_0^T\left[\Theta e^{2s \varphi}(x-x_0)(v_x)^2dt\right]_{x=0}^{x=1}\right),\end{aligned}
\|\phi\|^2=\frac{\mu_2!\nu!}{\pi^{\mu_1+\mu_2+\nu}}\sum^{\mu_2}_{j=0}{\mu_2\choose j}(j+r)!(\mu_2-j)!=\frac{\mu_2!\nu!}{\pi^{\mu_1+\mu_2+\nu}}\frac{(\mu_1+1)!}{r+1}.
\tilde{\Theta}^{Q,j}(x) = \Psi(x_{\bar{j}}, x_1, \dots,x_{k-1},x_{k+1},\dots,x_{\bar{j}-1}, x_{\bar{j}+1},\dots,x_d), \tilde{H}^{Q,j}(x) = \pm\tilde{\Psi}(x_{\bar{j}}, x_1, \dots,x_{k-1},x_{k+1},\dots,x_{\bar{j}-1}, x_{\bar{j}+1},\dots,x_d).
\pi ( u ) = \frac{2 k - 1}{4 \pi}\int_{\tau = 0}^{\infty}d \tau \sinh \tau \int_{0}^{2 \pi}d \theta \left[ \cosh \frac{\sigma}{2}- i \hat{n}. \hat{r}\sinh \frac{\sigma}{2}\right]^{- 2 k}
\mathit{CWE}(\mathcal{C}_{D}) =w_{c}^{p^{e-1}-(p-1)p^{m-1}-1}+\bigl(p^{e-1}-(p-1)p^{m-1}-1\bigr)w_{c}^{p^{e-2}-(p-1)p^{m-1}-1}\prod_{\substack{0\le i\leq p-1 i \ne c}}w_{i}^{p^{e-2}}+(p-1)(p^{e-1}+p^{m-1})w_{c}^{p^{e-2}-1}\prod_{\substack{0\le i\leq p-1 i \ne c}}w_{i}^{p^{e-2}-p^{m-1}}.
\langle \rho(\cdot,t),\varphi\rangle=\int_{\Omega}\rho_0(x,t)\,\varphi(x)\,dx+\int_{\Omega}\varphi(x)\,d\mu_t(x)=\int_{\Omega}\rho_0(x,t)\,\varphi(x)dx+\int_{\Gamma}\varphi(x)\,d\mu_t(x),
W^s_{loc}(\underline{a}) =\{\underline{b}\in\Sigma_{A^1}:d(\sigma^n\underline{a},\sigma^n\underline{b})\leq 1/3\ \forall\n\geq 0\}W^u_{loc}(\underline{a}) =\{\underline{b}\in\Sigma_{A^1}:d(\sigma^n\underline{a},\sigma^n\underline{b})\leq 1/3\ \forall\n\leq 0\}.
KP=K(K+\bar K-I)=K+K\bar K-K=K\bar K, & B=I-S\bar S=I-\partial (T\bar\partial)\bar T=I-\partial(I-K)\bar T=\partial K\bar T, & B^2=\partial K(\bar T \partial) K\bar T=\partial K(I-\bar K) K\bar T=B-\partial KPK\bar T.
G_{2}= \frac{1}{2 \sqrt{3}g}[ ( 6 \sqrt{3}g + \sqrt{1 0 8 g^{2}- 1})^{\frac{1}{3}}+ ( 6 \sqrt{3}g - \sqrt{1 0 8 g^{2}- 1})^{\frac{1}{3}}- \sqrt{3}]
-\langle\nabla_{[X_{1},X_{2}]}X_{2},X_{1}\rangle=-\displaystyle\frac{\nu^{2}(b_{1}^{2}+b_{2}^{2})}{(\mu-\lambda)^{2}}-\sum_{j=3}^{n}\langle\nabla_{X_{2}}X_{j},X_{1}\rangle^{2}\displaystyle\frac{(\gamma-\lambda)}{\mu-\lambda}\left(1-\displaystyle\frac{(\lambda-\gamma)}{\mu-\gamma}\right).
\begin{array}{l}g_n(x_n,x_{n+1})=x_n h_n(x_n,x_{n+1}), g_{k-1}(x_{k-1},x_k)=x_{k-1}h_{k-1}(x_{k-1},x_k)+g_{k-1}(0,x_k).\;\;( \;2 \leq k \leq n.)\end{array}
\bar{\cal K}[ A ] ={\cal K}[ A ] - \frac{1}{2 i}\mathrm{T r}\ln A^{- 1}= \int{\cal D}\varphi \left. e^{- \frac{1}{2}\varphi A^{- 1}\varphi - \frac{i \lambda}{4 !}\varphi^{4}}\right|_{\mathrm{e x c l . 1 V R}}
U_{k}=\{m \in \bar{I}_{k-1}\mid j_{2p+2}<m<j_{2p+1}\mbox{for some}p\geq 0\}, begin{align*}3pt]V_{k}=\{m \in \bar{I}_{k-1}\mid j_{2p+1}<m<j_{2p}\mbox{for some}p\geq 0\},
\gamma_{c,k}=\frac{|{\vec{h}}_{k}^{H}\vec{p}_{c}|^2}{\sum\limits_{j\in\mathcal{K}}|\vec{h}_{k}^{H}\vec{p}_{j}|^2+1}, \gamma_{p,k}=\frac{|{\vec{h}}_{k}^{H}\vec{p}_{k}|^2}{\sum\limits_{j\in\mathcal{K}\setminus k}|\vec{h}_{k}^{H}\vec{p}_{j}|^2+1}.
\mathbb{P}\left( \lim_{n\rightarrow \infty}\textrm{diam}( \varphi_{t_n}(\cdot,A)) =0 \right) \geq \mathbb{P}\left( A \subset U \right) \cdot \mathbb{P}\left( \lim_{n\rightarrow \infty}\textrm{diam}( \varphi_{t_n}(\cdot,U)) =0 \right) > 0.
\langle v(t),z \rangle =\langle v_0,z\rangle + \int_0^t \langle \Delta v(s) , z\rangle \,{\rm d}s+\int_0^t \langle (v(s) \cdot \nabla)v(s), z\rangle\,{\rm d}s+\langle \int_0^tG(v(s))\,{\rm d}W(s),z\rangle.
\mathbb E[Z^*_{\lfloor\varepsilon n\rfloor+1,n+\lfloor \varepsilon n\rfloor}]=\mathbb E[Z_{\lfloor\varepsilon n\rfloor}]+\mathbb E[Z^*_{\lfloor\varepsilon n\rfloor+1,n+\lfloor \varepsilon n\rfloor}-Z_{\lfloor\varepsilon n\rfloor}]=0+\mathbb E[Z_n^*]=\mathbb E[Z_n^*]\, .
\lim_{r\to\infty}r^{\rho-\rho(r)}\left|\sum_{r<|z_k|}\frac 1{z_k^\rho}\right| \le\lim_{r\to\infty}r^{\rho-\rho(r)}\left(\left|\sum_{R_n<|z_k|}\frac 1{z_k^\rho}\right|+\left|\sum_{R_n<|z_k|\le r}\frac 1{z_k^\rho}\right|\right)= 0.
A_{0}( \vec{x}, t ) \rightarrow A_{0}^{\prime}={\frac{1}{\beta}}\int_{0}^{\beta}d t A_{0}( \vec{x}, t ) , \qquad A_{i}( \vec{x}, t ) \rightarrow A_{i}^{\prime}= \bar{A}_{i}( \vec{x})
\sum_{u_1, \ldots, u_k}\Big(\prod_{i = 1}^k \frac{\mu(u_i)^2}{g(u_i)}\Big) \sideset{}{''}\sum_{s_{1, 2}, \ldots, s_{k-1, k}}\Big(\prod_{\substack{1 \leq i, j \leq k i \neq j}}\frac{\mu(s_{i, j})}{g(s_{i, j})^2}\Big) y^{(m)}_{a_1, \ldots, a_k}y^{(m)}_{b_1, \ldots, b_k}.
\widetilde E(t) = \left( U(t) - \mathcal R_h R_\tau^{k+1}U(t) \right) + \left( \mathcal R_h R_\tau^{k+1}U(t)- L_\tau U_{\tau,h}\right) =: \Theta(t) + \widetilde E_{\tau,h}(t)
\langle\{a_j\}, C\{a_j\}\rangle_{\mathcal A}=\sum\limits_{j,k}a^*_j\langle\omega_j,\omega_k\rangle_{\mathcal A}a_k=\Big\langle\sum\limits_j\omega_j a_j, \sum\limits_k\omega_k a_k\Big\rangle_{\mathcal A}\geq0.
\int_{0}^\infty P_r^{\bf V}(y)P_s^{\bf V}(y)\,y^{-1}e^{-y}\,dy\, \propto \int_{0}^\infty L_{r-1}^{(1)}(y)L_{s-1}^{(1)}(y) y\, e^{-y}\,dy\, \propto\delta_{r\,s},
\left\|u\right\|_{\mathcal{C}^{k,\alpha'}_{\beta}}:=\sum_{j=0}^{k}\sup_{\tilde{M}_{\eta_1,\eta_2}}\rho^{-\beta+j}|\nabla^{j}u| + \sup_{x\neq y \in \tilde{M}_{\eta_1,\eta_2}}\frac{|\rho^{-\beta+k}(x)\nabla^{k}u(x)-\rho^{-\beta+k}(y)\nabla^{k}u(y)|}{d(x,y)^{\alpha'}}.
\sum_{j = 0}^k{j^6 G_j}= - 33305G_0 - 53888G_1 \qquad+ (k^6 - 6k^5 + 75k^4 - 620k^3 + 3855k^2 - 16026k + 33305)G_k &+ (k^6 - 12k^5 + 120k^4 - 1000k^3 + 6240k^2 - 25932k + 53888)G_{k + 1}\,.
S = \int d^{4}x \frac{1}{2}\epsilon^{\mu \nu \lambda \rho}\left( B_{\mu \nu}^{a}( x ) R_{\lambda \rho}^{a}( x ) + \phi^{a b}( x ) ( B_{\mu \nu}^{a}B_{\lambda \rho}^{b}- \frac{1}{3}\delta^{a b}B_{\mu \nu}^{c}B_{\lambda \rho}^{c}) \right) \ .
K(L,r) =2^{2n-2L-2}\left[\sum_{j=0}^{n-2}\binom{n-2}{j}\frac{(-1)^j}{n-2L+j}\right]\left[\int_0^{\pi/2}(\cos\theta)^{2n-2L-2}d\theta\right] (1-r^2)^{2L-n}+\cdots
{K}^{(2)}_i(Y,Z)= \sum_{j \in I_p}\sum_{l \in J_p}K_{ijl}^{(2)}Y_{j}Z_{l}, \tilde{K}_k^{(2)}(Y,Z)= \sum_{j \in I_p}\sum_{l \in J_p}\tilde K_{kjl}^{(2)}Y_{j}Z_{l},
X_{\ell}+ [u,v] = \bigcup_{i=0}^{\ell-1}(ru_0 - u +i(v-u+1)) + [u,v] & = \bigcup_{i=0}^{\ell-1}[ru_0+i(v-u+1) , ru_0 + v - u +i(v-u+1) ] & = \bigcup_{i=0}^{\ell-1}[ru_0+i(v-u+1) , ru_0+ (i+1)(v-u+1) - 1 ] & = [ ru_0 , ru_0+ \ell (v-u+1) -1].
S^{-1}\left(p\right) =Z_{2}\left( p\hspace{-0.2cm}/ -m_{0}\right)-iZ_{1}e^{2}\int^{\Lambda}\frac{d^{4}k}{\left( 2\pi \right)^{4}}\gamma^{\mu}S\left( k\right) \gamma^{\nu}D_{\mu \nu}\left( k-p\right)
\mathbf{u}^{\mathcal{L}}(x_j^{(n)}) \mathcal{C}_n\mathcal{C}_n^T \mathbf{u}^{\mathcal{R}}(x_k^{(n)})=\left[\mathcal{C}_n^T[\mathbf{u}^{\mathcal{L}}(x_j^{(n)})]^T\right]^T [\mathcal{C}_n^T \mathbf{u}^{\mathcal{R}}(x_k^{(n)})]= \omega_{n,j,k}^{-1}\delta_{j,k}.
A = A^{a}\tau_{a}= \left( \omega^{a}+ \frac i l e^{a}\right) \tau_{a}\, , \qquad \bar{A}={\bar{A}}^{a}\tau_{a}= \left( \omega^{a}- \frac i l e^{a}\right) \tau_{a}\, , \quad a = 0 , 1 , 2 \, ,
\frac{d}{dt}\Bigl(L-p\dot{q}-\frac{1}{2}p_{\frac{1}{2}}\cdot{_a^CD_t}^{\frac{1}{2}}q\Bigr)= \frac{d}{dt}\Bigl(\frac{1}{2}p_{\frac{1}{2}}\cdot{_a^CD_t}^{\frac{1}{2}}q-H\Bigr) = \frac{d}{dt}\Bigl(\frac{\gamma}{2}\left({_a^CD_t}^{\frac{1}{2}}q\right)^2-H\Bigr) = 0.
\mathop{Sym}_{z_{1},\dots,z_{1-a_{ij}}}\sum_{k=0}^{1-a_{ij}}(-1)^{k}\left[\begin{matrix}1-a_{ij}k\end{matrix}\right]_{v_{i}}e_{i}(z_{1})\cdots e_{i}(z_{k})e_{j}(w)e_{i}(z_{k+1})\cdots e_{i}(z_{1-a_{ij}})=0\forall\i\neq j.
\int_{D_0}|F_j-(1-b_{t_j,1}(k\Psi))f_jF^{2k}|^2e^{-k\varphi+v_{t_j,1}(k\Psi)-k\Psi}\le (2-e^{-t_j-1})\int_{D_0}\mathbb{I}_{\{-t_j-1<k\Psi<-t_j\}}|f_j|^2e^{-k\Psi}\le (2e^{t_j+1}-1)\int_{\{k\Psi<-t_j\}\cap D_0}|f_j|^2,
\mathfrak{G}_{s}:=\{g_{s}(s_{1},s_{2},s_{3})(t_{1},t_{2},t_{3}) \{(t_{1},t_{2},t_{3})^{\pm1},(t_{1},t_{1}/t_{3},t_{1}/t_{2})^{\pm1},(t_{2}/t_{3},t_{2},t_{2}/t_{1})^{\pm1},(t_{3}/t_{2},t_{3}/t_{1},t_{3})^{\pm1}\}\}\simeq C_{2}^{3},
\begin{aligned}\sum_{y\in\mathcal{Y}}\sum_{\ell=1}^{J_y}(\chi(\Sigma^y_{\ell})-b^y_{\ell}) =\sum_{y\in\mathcal{Y}}\sum_{\ell=1}^{J_y}(2-2 genus(\Sigma^y_{\ell})-2b^y_{\ell}) \geq \sum_{y\in\mathcal{Y}}\sum_{\ell=1}^{J_y}(2b^y_{\ell}-3) - 3\sum_{y\in\mathcal{Y}}\sum_{\ell=1}^{J_y}index(\Sigma^y_{\ell}) \geq 3-3j.\end{aligned}
H^{\prime}\equiv - i \frac{3 \hbar c}{2}\gamma^{0}\gamma^{a}( - \sigma^{- 1})_{, a}= - i \frac{3 \hbar c}{2}\left( \begin{matrix}{\epsilon_{0}}&{\vec{\sigma}\cdot \vec{\epsilon}}\\{\vec{\sigma}\cdot \vec{\epsilon}}&{\epsilon_{0}}\\ \end{matrix}\right){,}
|\Psi\rangle=\sum_{s_{z1}}\sum_{s_{z2}}\cdot\cdot\cdot\sum_{s_{zN}}\int_{V_{1}}\int_{V_{2}}\cdot\cdot\cdot\int_{V_{N}}dr_{1}dr_{2}\cdot\cdot\cdot dr_{N}\Psi|r,s_{z}\rangle
1=\lim_{\theta\rightarrow 0}\frac{\int_0^{1/\theta}f_n(x) \d x}{\int_0^{\infty}e^{-\theta x}f_n(x) \d x}=\lim_{\theta\rightarrow 0}\frac{\int^{1/\theta}_0 \sum_{k=1}^\infty f_{\xi_k I_k}(x) \d x}{\int_0^{\infty}e^{-\theta x}\sum_{k=1}^{\infty}f_{\xi_k I_k}(x) \d x},~\forall n\in\mathbb{N}
\Sigma_{s}=[\begin{matrix}\langle x_{s}\cdot x_{s}\rangle&\langle x_{s}\cdot x_{s}^{\prime}\rangle\\ \langle x_{s}\cdot x_{s}^{\prime}\rangle&\langle x_{s}^{\prime}\cdot x_{s}^{\prime}\rangle\end{matrix}]=[\begin{matrix}\sigma_{s,11}&\sigma_{s,12}\\ \sigma_{s,21}&\sigma_{s,22}\end{matrix}]
E_{n\ell}=\alpha-\frac{1}{8a^2}\left[\frac{2\alpha a^2+\left[\left(n+\frac{1}{2}\right)+\frac{1}{2}\sqrt{(2\ell+1)^2+8\beta a^2}\right]^2}{\left[\left(n+\frac{1}{2}\right)+\frac{1}{2}\sqrt{(2\ell+1)^2+8\beta a^2}\right]}\right]^2,
A = (2a+2n)(2a+2n+1), B_1 = - 2(4a+4n-1) \sum_i z_i^2 - 8 \sum_{i<j}z_i z_j + 2(4a+4n-4b_1-2l-d-1) \sum_i z_i{}+ 2b_1(4a+4n-2b_1-2l-d-1) - 4b_2(4a+4n-3), E_{n,l}= \lambda [2(4a+4n-1) \sum_i z_i + 8(a+n)b_1 + 2(a+n)(2l+d) - 2b_1{}- l(l+d-1)],
(a^+_i)^mP^{(a,b)}_{\alpha}=\prod_{p=0}^{m-1}\frac{2(\alpha_i+p+1)(\alpha_i+b_i+a_i+p+1)}{(2\alpha_i+2p+b_i+a_i+1)(2\alpha_i+2p+b_i+a_i+2)}P^{(a,b)}_{\alpha_{0,\dots,0,m,0,\dots,0}}
\left| \dfrac{d^{n+m}}{dx^{n+m}}\big(\phi \ast f \big)(x) \right| \leqslant \int_{B(0,R)}\left| \dfrac{d^{n+m}}{dy^{n+m}}\phi(y) \right| |f(x-y)| \, dy = c(\phi, f, m, j) \, M(\phi, f, m, j)^n,
\sum\limits_{i=1}^{n-1}g(R(\phi E_i,\gamma^{\prime})\gamma^{\prime},\phi E_i)=\sum\limits_{i=1}^{n}g(R(\phi E_i,\gamma^{\prime})\gamma^{\prime},\phi E_i)=\phi^2Ric(\gamma^{\prime},\gamma^{\prime}).
\left.\frac{\partial}{\partial\Theta}Y_l^{(k_1,0,\dots,0)}(\Upsilon_{\!\Theta}x)\right|_{\Theta=0}=\beta_{l,k_1-1}\,Y_l^{(k_1-1,0,\dots,0)}(x)-\beta_{l,k_1}\,Y_l^{(k_1+1,0,\dots,0)}(x)
\pi^G_k(X) = \begin{cases}\displaystyle\operatorname*{colim}_{V \in s(\mathcal{U}_G)}\;\; [S^{k+V}, X(V)]_*^G \mathrm{for}\;\; k \geq 0 \displaystyle\operatorname*{colim}_{{V \in s(\mathcal{U}_G)}}\;\; [S^{k+V}, X(\mathbb{R}^{-k}\oplus V)]_*^G \mathrm{for}\;\; k \leq 0\end{cases}
\phi^{\prime}(\mathbf{z}_{B})\phi_{E}(\mathbf{z}_{E})=\phi (\mathbf{K}_{1}\mathbf{z}_{B}+\tilde{\mathbf{K}}\mathbf{z}_{E})\exp \left[ -\frac{1}{2}|\sqrt{\mathbf{I}_{B}-\mathbf{K}_{1}^{\ast}\mathbf{K}_{1}}\mathbf{z}_{B}-\mathbf{K}_{1}^{\ast}\mathbf{z}_{E}|^{2}\right] .
\left(\begin{matrix}x=0 x=\infty \, \left( \frac13 \right) \overbrace{\begin{matrix}1 \theta^0_2 0 \theta^0_1 0 0 \end{matrix}}\overbrace{\begin{matrix}-t^{\frac13}\theta^\infty_1/3 -\omega t^{\frac13}\theta^\infty_1/3 -\omega^2 t^{\frac13}\theta^\infty_1/3 \end{matrix}}\end{matrix}\right),
\int_\Omega\sigma(u) \nabla\phi\cdot\nabla w\mathrm{dx}=\int_\Omega \sigma(u) \alpha_\mathrm{S}(u)\nabla u\cdot \nabla w \mathrm{dx}+ \int_{\Gamma_\mathrm{N}}gw \mathrm{ds},\qquad\forall w\in V,
W^{(1)}-W^{(3)}= b_0 y+( 36 b_0 + b_1 )y^5+(- 415 b_0 - 10 b_1 )y^9 &+(- 39056 b_0 - 724 b_1 )y^{13}+(- 742131 b_0 - 3694 b_1 )y^{17}+ \cdots.
L_{+}= \left( \begin{matrix}{D}&{( 1 - q^{2}) B}\\{0}&{D^{- 1}}\\ \end{matrix}\right) , \ \L_{-}= \left( \begin{matrix}{D^{- 1}}&{0}\\{( q^{2}- 1 ) C}&{D}\\ \end{matrix}\right) \ ,
\gamma' \mathbf{z}= \begin{pmatrix}\frac{y_2}{b}+ b' \Lambda_1 - a' \Lambda_2 -b \Lambda_2 + a \Lambda_2 \end{pmatrix}= g M_k \mathbf{z}= z_2 \begin{pmatrix}m \epsilon_k + (-1)^{k - 1}\zeta \epsilon_{k - 1}- \zeta^{-1}\epsilon_k \end{pmatrix}.
J_{3}= \frac{\beta-1}{\beta}\lambda_{1}^{-1}\Big(\lambda_{1}^{-1}-\frac{2}{\beta}F^{-1}\frac{\partial F}{\partial \lambda_{1}}\Big)(\nabla_{1}F)^{2}+2F\lambda_{1}^{-2}\frac{\partial F}{\partial \lambda_{i}}\sum_{i>\mu}(\lambda_{i}-\lambda_{1})^{-1}h_{1ii}^{2}&~+F\lambda_{1}^{-2}\frac{\partial^{2}F}{\partial \lambda_{i}\partial \lambda_{j}}h_{ii1}h_{jj1}.
G_{\beta}( t , t^{\prime}) = \frac{1}{\beta}\Sigma_{m = - \infty}^{\infty}\exp [ - i \omega_{m}( t - t^{\prime}) ] \frac{1}{\omega_{m}^{2}+ \Omega^{2}}= \frac{\cosh [ \frac{\Omega}{2}( | t - t^{\prime}| - \beta ) ]}{\Omega \sinh [ \frac{\beta \Omega}{2}]},
\left ( \begin{array}{l}\nabla f(x^*)+\displaystyle \sum_{j=1}^p[y^I]^*_j\nabla^2c_j(x^*)\eta_1^*+ -A^TA\eta_1^*+{\cal J}c(x^*)^T[\xi_b]^*{\cal J}c(x^*)\eta_1^*+[\xi_a]^*+[\xi_b]^* \end{array}\right )=0.
u^{\prime}=\frac{dx^{\prime}}{dt^{\prime}}=\frac{\gamma(dx-vdt)}{\gamma(dt-\frac{vdx}{c^{2}})}=\frac{\frac{dx}{dt}-v}{1-(\frac{v}{c^{2}})(\frac{dx}{dt})}=\frac{u-v}{1-uv/c^{2}}.
M^{( 2 )}\equiv \frac{d^{2}V}{d \rho^{2}}\left. \right|_{\rho = 0}\simeq \frac{4 N_{c}\Lambda^{D - 2}}{( 4 \pi )^{D / 2}\Gamma ( D / 2 )}\frac{g_{c r}- g}{g_{c r}g}.
\|(N-T)x\| = \|\eta \Psi_2(x)+(1-\epsilon)(1-\eta)Tx-Tx\| &=\|\eta \Psi_2(x)-\epsilon(1-\eta)Tx+Tx-\eta Tx-Tx\| &=\|\eta(\Psi_2(x)-Tx)-\epsilon(1-\eta)Tx\| & \leq \|\eta(\Psi_2(x)-Tx)\|+\epsilon \|1-\eta\| \|Tx\| & \leq \|\eta(\Psi_2(x)-Tx)\|+2\epsilon.
\left\lbrace \begin{array}{l}\dot{x}^\pm(t)=\xi^\pm(t) \dot{\xi}^\pm(t)=-\partial_x \lambda_\delta^\pm(x^\pm(t)) \end{array}\right. \textrm{with}\left\lbrace \begin{array}{l}x^\pm(0)=x^\pm_0 \xi^\pm(0)=\xi^\pm_0 \end{array}\right.
C_{\mu \alpha}= \overline{{{\Psi}}}( x ) \Gamma_{\mu}\overline{{{\Gamma}}}_{\alpha}\Psi ( x ) \hspace{0 . 3 i n}\Theta_{\mu [ \alpha \beta ]}= \overline{{{\Psi}}}( x ) \Gamma_{\mu}\overline{{{\Gamma}}}_{[ \alpha}\overline{{{\Gamma}}}_{\beta ]}\Psi ( x )
III''_1 =\frac16(p+s-n-3)(p+s-n-4)(-2p+4s-n-5) III''_2 = \sum_{j=1}^{s-2}[2j^2+(2p-3n-7)j -(n+2)(p-n-3)] -(n+3-s)(p+s-n-4) =(p-\frac32n-\frac72)(s-1)(s-2)+(s-2)(s-1)(\frac23s-1) -(n+2)(p-3-n)(s-2) -(n+3-s)(p+s-4-n) = (s-1)n^2-[ps- p+\frac32 s^2-\frac{15}2s+6] n +(p-\frac72)(s-1)(s-2) +(s-2)(s-1)(\frac23ps-1)-2(s-2)(p-3)+(p+s-4)(s-3).
h_{SU(2)}(e^{\phi_1\lambda_3}e^{\psi_1\lambda_1}e^{\omega_1\lambda_3})=\sum_{j=1}^M c_je^{ik_j\phi_1}\sin^{m_j}(\psi_1)e^{il_j\omega_1}+c_j'e^{ik_j'\phi_1}\sin^{m_j'}(\psi_1)\cos(\psi_1)e^{il_j'\omega_1}.
\inf_{\mathcal{T},\left(y_{\mathcal{T}}(\cdot),u_{\mathcal{T}}(\cdot)\right)}\left\{\frac{1}{\mathcal{T}}\int_0^{\mathcal{T}}k(y_{\mathcal{T}}(t),u_{\mathcal{T}}(t))dt \right\}:=v_{per}(y_0),
g l ( 3 n ) \downarrow g l ( 3 ) + g l ( n ) \quad{\{}\overline{{1^{K}}}{\}}\cdot{\{}1^{L}{\}}\downarrow \sum_{\rho \vdash K , \sigma \vdash L}{\{}\overline{{\rho^{\prime}}}{\}}\cdot{\{}\sigma^{\prime}{\}}\times{\{}\overline{{\rho}}{\}}\cdot{\{}\sigma{\}},
x(E):=\frac{N \eta}{\pi}\sum_{k=M+1, \ \ \neq \mu, \nu}^{M+N}X_{\mu\nu,k}(E+i\eta), \y(E):=\frac{\tilde{\eta}}{\pi}\int_{E^{-}}^{E_U}\sum_k \sum_{\beta \neq k}X_{\beta \beta,k}(E+i\tilde{\eta})dE.
{\cal F}(x^a,\varphi ,\tilde{p}_a, \tilde{p}_\varphi )=\varrho\int\limits_{\varphi_0}^{\varphi}\frac{(\tilde{p},\partial_\varphi n)}{(\tilde{p},n)}d\varphi+ \varphi\tilde{p}_\varphi+x^a\tilde{p}_a\ ,
\Gamma_{jk}^{(v)}(z_j) =\sum_{u=1}^m h_u^{(v)}\theta_{ujk}(z_j)+B_{jk}(z_k)g_k^{(v)}(z_k)-g_j^{(v)}(z_j), \lambda_j^{(v)}(z_j) =\sum_{u=1}^mh_u^{(v)}\Lambda_{uj}'(z_j)-[\sum_{r,s=1}^n \Lambda_{M_{0j}}^{r,s}(z_j)\frac{\partial}{\partial z_j^r}\wedge \frac{\partial}{\partial z_j^s},\sum_{\alpha=1}^n g_j^{\alpha(v)}(z_j)\frac{\partial}{\partial z_j^{\alpha}}]
S ( g ) = -{\frac{k}{8 \pi}}\int_{S^{2}}d^{2}\rho \eta^{\mu \nu}T r ( g^{- 1}\partial_{\mu}g ) ( g^{- 1}\partial_{\nu}g ) +{\frac{k}{1 2 \pi}}\int_{B}T r ( g^{- 1}d g )^{3}
\Phi (\xi_1, \eta_1, \xi ,\eta) = \xi_1 \xi(\xi_1 + \xi) + \eta_1 \eta (\eta_1 + \eta), F (\xi_1, \eta_1, \xi ,\eta) = \xi_1 \eta + \xi \eta_1 + 2 (\xi_1 \eta_1 + \xi \eta).
\P\left( \Big| \sum_{i=1}^n u_i(f,f') \Big| \geq \frac{C_1 \|w-w'\|_2}{\sqrt n h^{m/2+1}}\left( \sqrt{t}+ \sqrt{\frac{(\log n)^2}{n h^m}}\sqrt t + \sqrt{\frac{(\log n)^6}{n h^m}}t \right) \right) \leq C_1 e^{-t}.
E(z)=\sqrt{(exp[-4(\frac{(z-z_{0})}{L_{c}})^{2}]cos(4\pi\frac{z-z_{0}}{\lambda_{0}}))^{2}+(exp[-4(\frac{(z-z_{0})}{L_{c}})^{2}]sin(4\pi\frac{z-z_{0}}{\lambda_{0}}))^{2}}
E_{p , N_{s}}^{( q )}( F ( t ) ) = \frac{q}{2}F ( t ) \partial_{z}Y_{p , N_{s}}( t ) + \frac{\epsilon ( F )}{2}[ D F ( t ) ] D Y_{p , N_{s}}( t ) + [ \partial_{z}F ( t ) ] Y_{p , N_{s}}( t ) .
E = \int d^{4}x \left\{\frac 1 2 \left( F_{1 2}+ s ( | T |^{2}- \zeta ) \right)^{2}+ | D_{1}T - i s D_{2}T |^{2}+ s \zeta F_{1 2}+ V ( T ) - \frac 1 2 ( | T |^{2}- \zeta )^{2}\right\}
+\sum_n\pi_{n,1}\frac{\partial\overline{H}_0}{\partial q_{n,0}}+\sum_n\sum_{j_n=2}^{m_n-1}\pi_{n,j_n}\left[\frac{\partial\overline{H}_0}{\partial q_{n,j_n-1}}-\mu_{n,j_n-1}\right]\ .
(\Lambda Af)(x)= \frac{1}{\Pi(x)}\frac{1}{2C}\int^x_0\!\left( \frac{2a(y)}{b(y)}\frac{f'(y)}{s(y)}+\frac{f''(y)}{s(y)}\right)\,dy = \frac{1}{\Pi(x)}\frac{1}{2C}\int^x_0 \frac{d}{dy}\frac{f'(y)}{s(y)}\,dy = \frac{1}{\Pi(x)}\frac{1}{2C}\left[ \frac{df}{dS}(x)-\frac{df}{dS}^+(0) \right] = \frac{1}{2}\frac{b(x)\pi(x)}{\Pi(x)}f'(x),
W^{\left({\rm i}\right)}{\left( x_{1},x_{2},x_{3},x_{4}\right)}=\left\langle \Omega ,\phi_{{i}_{1}}\left( x_{{i}_{1}}\right) \phi_{{i}_{2}}\left( x_{{i}_{2}}\right) \phi_{{i}_{3}}\left( x_{{i}_{3}}\right)\phi_{{i}_{4}}\left( x_{{i}_{4}}\right) \Omega \right\rangle ,
\left[1 +\frac{\,g^2}{2m^2}\,V^{(2)}(\phi)\right]\frac{dp^\mu}{d\tau}=-\frac{\,g^2}{2m}\left[\partial^\mu V^{(2)}(\phi) - \frac{p^\mu}{p^2}\,p\cdot\partial\,V^{(2)}(\phi)\right]\,,
\tilde{\mathbf{P}}_0^{i+1}:=\breve{\mathbf{P}}_0^i-\max\left\{1-\frac{\bar{S}_f}{\Sigma},0 \right\}\frac{\mathbf{1}\mathbf{1}^{\top}\breve{\mathbf{P}}_0^i}{3}\tilde{\mathbf{Q}}_0^{i+1}:=\breve{\mathbf{Q}}_0^i-\max\left\{1-\frac{\bar{S}_f}{\Sigma},0 \right\}\frac{\mathbf{1}\mathbf{1}^{\top}\breve{\mathbf{Q}}_0^i}{3}
d_2(\mathbf{Geo}_{\tau}(\{\mu_{k}^{\tau}\})(t),\mathbf{Geo}_{\tau}(\{\mu_{k}^{\tau}\})(s)) = \int_s^t \lvert \mathbf{Geo}_{\tau}(\{\mu_{k}^{\tau}\}) '(r) \rvert dr \leq (t-s)^{1/2}\left( \sum_{k = 1}^{\infty}\frac{d_2(\mu_{k}^{\tau},\mu_{k+1}^{\tau})^2}{\tau}\right)^{1/2}\leq C (t-s)^{1/2}
\langle \psi({\bf{x}}^{'},t^{'})\psi^{\dagger}({\bf{x}},t)\rangle = \rho_{0}e^{-\sum_{{\bf{k}},{\bf{q}}\neq 0}f^{*}_{{\bf{k}},{\bf{q}}}({\bf{x}}^{'})f_{{\bf{k}},{\bf{q}}}({\bf{x}}^{'})}
\overline{\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_{\ell}-1}(\gamma_{\ell})}=\sum_{P \{1,\ldots,l\}}\ \prod_{S\in P}\ \sum_{\widehat{\alpha}}\tau_{\widehat{\alpha}}(\widetilde{\mathsf{K}}_{\alpha_S,\widehat{\alpha}}\cdot\gamma_S)
({\widehat{L}}^* k)(\eta)= -\vert \eta \vert k(\eta) &+z\sum_{x\in \eta}e^{-E^\phi (x,\eta\setminus x)}\int_{\Gamma_0}e_\lambda(e^{-\phi (x - \cdot)}-1,\xi) k((\eta\setminus x)\cup\xi)\,d\lambda (\xi).
F(w_{1},w_{2},...,w_{m})=\sum_{n_{1}=-\infty}^{\infty}\sum_{n_{2}=-\infty}^{\infty}\cdot\cdot\cdot\sum_{n_{m}=-\infty}^{\infty}f(n_{1},n_{2},...,n_{m})e^{-iw_{1}n_{1}-iw_{2}n_{2}\cdot\cdot\cdot-iw_{m}n_{m}}
< A_{p}^{( n )}( 0 , 0 ) A_{p}^{( n )}( L , \tau ) > = \int \frac{d^{4}p}{( 2 \pi )^{4}}\frac{e^{- i p_{0}\tau + i \vec{p}\cdot \vec{L}}}{p_{0}^{2}+ p^{2}+ m_{n}^{2}}