harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma^{n+m-k-1}_{i_0i_1\cdots i_k}= (-1)^m\sum^k_{s=0}(-1)^{m(k-s)+s(k-1)}{\omega^{m-s}_{i_0\cdots i_s}}|_Y \wedge \beta^{n-k-1+s}_{i_s\cdots i_k}
\Psi [ \phi ] = \frac{{1}}{[ \mathrm{\det}( 2 \pi \hbar G ) ]^{1 / 4}}\mathrm{\exp}\left[ - \int_{x , y}( \phi_{a}( x ) -{\hat{\phi}}_{a}( x ) ) \frac{{G_{a b}^{- 1}( x , y )}}{4 \hbar}( \phi_{b}( y ) -{\hat{\phi}}_{b}( y ) ) \right] ,
\begin{array}{rcl}m_3 = \min\big\{{2 \over 3}- \lambda > 0\;|\; \lambda \in \Lambda_n \big\}= \min\{{2 \over 3}-{n + a \over 2n}> 0\;|\; a = 0, \ldots, n \}\geq{1 \over 6n}.\end{array}
W_{\psi_{0}}(\psi_{1}\otimes\psi_{2}\otimes\psi_{3}; z_{i})\sim \left(z_{2}-z_{3}\right)^{-\bar{h}t_{23}}\left(z_{1}-z_{3}\right)^{-\bar{h}(t_{12}+t_{13})}W_{\psi_{0}}^{2}(\psi_{1}\otimes\psi_{2}\otimes\psi_{3}).
{\bigl < t_{\eta , v}^{( \rho )}; t_{\theta , w}^{( \sigma )}\bigr >}_{L^{2}}= \int_{G}\overline{{{t_{\eta , v}^{( \rho )}( g )}}}\cdot t_{\theta , w}^{( \sigma )}( g ) \, d g = \frac{1}{\dim V_{\rho}}\delta_{\rho \sigma}\overline{{{\left< \eta ; \theta \right>}}}\left< v ; w \right> ,
T_{~ \mu}^{\mu}= - \frac{T_{3}V ( T )}{\sqrt{- X}}\left[ ( 4 + 3 \partial_{\rho}T \partial^{\rho}T ) + \left( ( 1 + \frac 1 2 \partial_{\rho}T \partial^{\rho}T ) \delta_{~ \sigma}^{\mu}- \partial^{\mu}T \partial_{\sigma}T \right) F_{\mu \nu}F^{\sigma \nu}\right] .
\min_{\mathbf{y}\in Y}\Big\{\Big(\frac{1}{K}\sum_{k = 1}^K \max_{\hat{\mathbf{c}}^{\mbox{\tiny \upshape (\itshape k\upshape)}}\in \mathcal{S}^{(bf)}_k}\hat{\mathbf{c}}^{\mbox{\tiny \upshape (\itshape k\upshape)}\top}\mathbf{x}+ \varepsilon_K\Big)y_1 + hy_2\Big\}.
B = - \frac{\vert e \vert H}{( 2 \pi )^{2}}\frac{2 \sqrt{\pi}\vert e \vert H}{\Gamma ( 1 + \delta )}\int_{0^{+}}^{\infty}d \eta \; \eta^{- 3 / 2 + 1 + \delta}\mathrm{e}^{- ( 4 + \beta^{2}m^{2}) \eta}\mathrm{c s c h}^{2}( \beta^{2}\vert e \vert H \eta )
h_v (z-b,D(b,z)) = h_{E_1 ,v}(f_1 (z)-f_1 (b),w(f_1 (z))+w(f_1 (b))-2\infty ) &- h_{E_2 ,v}(f_2 (z)-f_2 (b),w(f_2 (z))+w(f_2 (b))-2\infty ).
= \frac{1}{3}( 1 \times 2 \times 3 - 0 \times 1 \times 2 + 2 \times 3 \times 4 - 1 \times 2 \times 3 + 3 \times 4 \times 5 - 2 \times 3 \times 4 + \cdots + 2 0 1 6 \times 2 0 1 7
\langle Q_{\lambda}, P_{\mu}\rangle=\delta_{\lambda,\mu},Q_{\lambda}(x_1,\dots,x_n;t)=b_{\lambda}(t)P_{\lambda}(x_1,\dots,x_n;t),b_{\lambda}(t)=\prod_{i \geq 1}\prod_{j=1}^{m_i(\lambda)}(1-t^j).
d s^{2}= - g^{2}r^{2}\d t^{2}- \Delta^{2}( r ) \left( d t + \frac{a}{2 r^{2}\Delta ( r )}\ \sigma_{3}\right)^{2}+ \frac{d r^{2}}{V ( r )}+ \frac{r^{2}}{4}\left( \sigma_{1}^{2}+ \sigma_{2}^{2}+ \sigma_{3}^{2}\right) ,
\mathcal{L}_{X_i}\Delta_{g_{F}}\vert_{\lbrace x_0 \rbrace \times U_F}=\sum_{j=1}^n [X_i,Y_j]Y_j + Y_j [X_i,Y_j] - [X_i,\nabla_{Y_j}Y_j] =-\sum_{j=1}^n \big(\tilde\nabla_{Y_j}X_i\big) Y_j + Y_j \big(\tilde\nabla_{Y_j}X_i\big) + [X_i,\nabla_{Y_j}Y_j]\,.
\left. + \, \Lambda_{2 3}^{+ +}\, V_{2 3}\, \Lambda_{2 3}^{+ +}\, \psi_{2 3}( h_{1}) \, + \, \Lambda_{3 1}^{+ +}\, V_{3 1}\, \Lambda_{3 1}^{+ +}\, \psi_{3 1}( h_{2}) \, \right]
\frac{1+\int_{0}^{B_{max}}(1-F_n(y)) \mathrm{d}y}{1+{\displaystyle\sup\limits_{0\leq x \leq B_{max}}{x (1-F_n(x))}}}\stackrel{(a)}{=}\frac{1+\int_{0}^{1}1 \mathrm{d}y +\int_{1}^{n}\frac{1}{y}\mathrm{d}y}{1+1}= 1+\frac{\ln(n)}{2}
\left\langle \left(-\frac{i}{n}S+1\right)^{-1}\xi, T^*\eta\right\rangle= \left\langle \left(\frac{i}{n}S+1\right)^{-1}(S+T^*)^*\xi,\eta\right\rangle &-\left\langle Sc\left(\frac{S}{n}\right)\left(-\frac{i}{n}S+1\right)^{-1}\xi, \eta\right\rangle+\left\langle R_{\frac{1}{n}}^*\xi,\eta\right\rangle,
\kappa_m= \frac{\left(-m (m + 1) - \eta+ \eta^2\right)^2}{4 (2 m + 1)^2}=\frac{(m+\eta)^2(m-\eta+1)^2}{4 (2 m + 1)^2},m\ge 1 \lambda_m= \frac{(1 + m)^2 - \eta + \eta^2}{(2 m + 1) (2 m + 3)},m\ge 0.
\mid{P}_{0}\rangle \: = \: f ( p ) \exp \left[ 2{({\cal W}^{- 1})}_{\nu \rho}^{\mu \sigma}{\mathbf a}^{\dagger \nu}{\mathbf b}_{\mu}^{\dagger}{\mathbf a}^{\dagger \rho}{\mathbf b}_{\sigma}^{\dagger}\right] \mid 0 \rangle .
H_{a}\left( \frac{d S_{a}}{d \lambda_{a}}, \lambda_{a}\right) = - 2 A ( \lambda_{a}) \left( \frac{d S_{a}}{d \lambda_{a}}\right)^{2}- \frac{1}{2}\left( \lambda_{a}^{N + 1}- ( \alpha - 1 ) \lambda_{a}^{N}+ ( 1 - \alpha + \beta ) \lambda_{a}^{N - 1}\right) ,
\lambda_{k}^{\epsilon, \hat{v}_k}= -\limsup_{t \rightarrow \infty}\frac{1}{t}\log \mathbb{P}_{x, k}^{\epsilon, \hat{v}_k}\bigl \{\tau_{k}^{\epsilon, \hat{v}_k}> t \bigr\}, \,\,\, x \in D, \,\,\, k \in \{1,2, \ldots, n\}.
\Psi = \exp \left\{\frac{\theta}{2}\Gamma_{\underline{{r}}\underline{{\theta}}}- A -{\frac{1}{4}}B + \int^{r}\mathrm{d}\rho \mathrm{e}^{- B ( \rho ) / 2}\left({\frac{1}{2 a}}- \mathrm{i}\Gamma_{7}{\frac{e}{2}}\Gamma_{\underline{{r}}\underline{{\theta}}}A_{\theta}( \rho ) \right) \right\}\psi ,
f_{X}(x)=\int_{-\sqrt{1-y^{2}-x^{2}}}^{+\sqrt{1-y^{2}-x^{2}}}\int_{-\sqrt{1-x^{2}}}^{+\sqrt{1-x^{2}}}\frac{3dydz}{4\pi}=3\sqrt{1-x^{2}}/4;
\sum_{\substack{m=n_0 m\not=n}}^\infty\ |K( \sigma_{n}-\overline{\sigma_m})|+\sum_{m=n_0}^\infty|K( \sigma_n+\sigma_m)|\le\frac{4\pi}{T\gamma^2(1-\varepsilon)}\bigg(1+\sum_{m=n_0}^\infty\frac{1}{4m^{2}-1}\bigg)\,,
[-\xi_{n+1}(t)]=[\xi_{1}(t)-\xi_{n+1}(t)...\xi_{k}(t)-\xi_{n+1}(t)...\xi_{n}(t)-\xi_{n+1}(t)]^{T}[\begin{matrix}\alpha_{1}(t)\\ \vdots\\ \alpha_{k}(t)\\ \vdots\\ \alpha_{n}(t)\end{matrix}]
\zeta_{n+1}(x)=x \zeta_{n}(x)-\alpha \zeta_{n-1}(x)-\gamma \zeta_{n-2}(x)\, , n \geq 2, & \zeta_{0}(x)=1,\zeta_{1}(x)=x, \zeta_{2}(x)=x^{2}-\alpha\, ,
\sum_{x\in B_r(x_0)}\!\!\!V(x)\,v^2_+(x,T)\mu(x) \leq \frac{c(T,\lambda,r)}{(r_1)^2}\sum_{x\in G}e^{-\Lambda \mathbf{d}^\alpha(x)}\chi_{\{r< \mathbf{d}(x)\le r_1\}}(x)\mu(x) = \frac{c(T,\lambda,r)}{(r_1)^2}\sum_{x\in G}e^{-\Lambda d^\alpha(x,x_0)}\chi_{\{r< \mathbf{d}(x)\le r_1\}}(x)\mu(x).
\int_{m-x}^\infty (z+x) \frac{2}{\sqrt{T-t}}\varphi\Big(\frac{z}{\sqrt{T-t}}\Big) dz \ = \ 2\sqrt{T-t}\int_{\frac{m-x}{\sqrt{T-t}}}^\infty z\varphi(z) dz + 2x \int_{\frac{m-x}{\sqrt{T-t}}}^\infty \varphi(z) dz &= \ \sqrt{\frac{2(T-t)}{\pi}}e^{-\frac{(m-x)^2}{2(T-t)}}+ 2x \Big(1 - \Phi\Big(\frac{m-x}{\sqrt{T-t}}\Big)\Big).
Spec~(A_\alpha(G))=\left\{\frac{2m}{n}, \left[\frac{2m}{n}(2\alpha-1)\right]^{l-1}, \left[\frac{2m\alpha}{n}+(1-\alpha)\right]^a, \left[\frac{2m\alpha}{n}-(1-\alpha)\right]^b\right\}.
Cap(z,U_{+}\cup U_{-}) =h(z_{+})=h(\zeta)=h(\zeta_{+})+h(\zeta_{-}) & =H(\zeta)[h_{+}(\zeta_{+})+h_{-}(\zeta_{-})] & =(1-d(z,\zeta)h(\zeta_{+}))[Cap(\zeta,U_{+})+Cap(\zeta,U_{-})] & =(1-d(z,\zeta)\cdot Cap(z,U_{+}\cup U_{-}))[Cap(\zeta,U_{+})+Cap(\zeta,U_{-})].
= \; C_{m_{d}}( w \! - \! z ) \; - \; \frac{m c}{2 \pi}\frac{4 \pi^{2}}{\pi \! + \! g}\; \int_{\Lambda}d^{2}x \; C_{m_{d}}( x \! - \! w ) C_{m_{d}}( x \! - \! z ) \; 2 \cos ( \theta )
\ell_{j}\left( s\right) \left( u,v\right) :=\left\langle \mathbb{A}_{j}^{\operatorname*{ext}}\nabla u,\overline{\nabla v}\right\rangle_{\mathbb{R}^{3}}+s^{2}\left\langle p_{j}^{\operatorname*{ext}}u,\overline{v}\right\rangle_{\mathbb{R}^{3}}\qquad\forall u,v\in H^{1}\left(\mathbb{R}^{3}\right) ,
\textrm{LHS(\ref{e.toprovelhs})}= \frac{(\nu/\mu;q)_{\ell}}{(\nu;q)_y}\sum_{r=0}^{\ell}\frac{(q^{-\ell};q)_r \, (\nu^{-1}q^{1-y};q)_{r}}{(q;q)_r\, (\mu \nu^{-1}q^{1-\ell};q)_r}(\mu q^y)^r &=\frac{(\nu/\mu;q)_{\ell}}{(\nu;q)_y}{_2\phi_1}(q^{-\ell},\nu^{-1}q^{1-y};\mu\nu^{-1}q^{1-\ell};q;\mu q^y).
{\frac{g_{\tiny \mathrm{U V}}^{( a )}}{g_{\tiny \mathrm{I R}}^{( a )}}}={\frac{\operatorname{s i n}{\frac{\pi a}{k + l + 2}}}{\operatorname{s i n}{\frac{\pi}{k + l + 2}}}}{\frac{\operatorname{s i n}{\frac{\pi}{l + 2}}}{\operatorname{s i n}{\frac{\pi ( a - k )}{l + 2}}}}.
r^{2}\frac{d^{2}}{d r^{2}}{\delta}\overline{{\cal{N}}}( r ) + 2 r \frac{d}{d r}{\delta}\overline{{\cal{N}}}( r ) - 2{\delta}\overline{{\cal{N}}}( r ) = 2 g^{2}{\Delta}( r ) r{\cal T}_{C}( r ) .
b_1(2n)^2-b_1(2n-1)b_1(2n+1) =b_1(2n)b_1(n), b_1(2n-1)^2-b_1(2n-2)b_1(2n) =-b_1(2n-2)b_1(n), b_2(2n)^2-b_2(2n-1)b_2(2n+1) =\left(\sum_{j=0}^n b_2(j)\right)^2, b_2(2n-1)^2-b_2(2n-2)b_2(2n) =\left(\sum_{j=0}^n b_2(j)\right)^2-b_2(2n-2)b_2(n).
\left(\sum_{i=0}^{p-1}\frac{p(-1)^{i}}{2i-1}\right)^2\equiv 1+4p(-1)^{\frac{p-1}{2}}+4p^2-p^2\sum_{k=1}^{\frac{p-3}{2}}\frac{(-1)^k}{k^2}\pmod{p^3}.
\begin{aligned}\limsup_{N\rightarrow \infty}\left\Vert \left( f_N^{(s)}(t) - \int_{\mathcal{P}(\mathbb{R}^{2d})}h_N^{(s)}(t;h_0) d\pi (h_0)\right)\mathbf{1}_{Z_s \in \mathcal{G}_s \cap\hat{\mathcal{U}}_s^{\eta (\varepsilon)}}\mathbf{1}_{E_s (Z_s) \leq R^2}\right\Vert_{L^\infty_{Z_s}}& = 0\end{aligned}
\Delta_{MAC}= R_{MAC}-R_{l,s}= \frac{1}{2}\log\left(\frac{1 + (\sum_{i=1}^m \sqrt{\gamma_i})^2}{1 + \sum_{i=1}^m \gamma_i}\right) \leq \frac{1}{2}\log\left(\frac{1 + m \sum_{i=1}^m \gamma_i}{1 + \sum_{i=1}^m \gamma_i}\right) < \frac{1}{2}\log(m),
cov\{\hat{m}_{2y}(-\lambda),\hat{m}_{2y}(\beta)\}= \frac{1}{N}\sum_{\tau = -K_N}^{K_N}[\hat{m}_{4y}(-\lambda,\tau,\tau+\beta) - \hat{m}_{2y}(-\lambda)\hat{m}_{2y}(\beta)],
\Bigl( \theta_q(\frac{n-5}{2})(d-1)q^{\frac{n-1}{2}}+ \theta_q(\frac{n-1}{2}) \Bigr) - \Bigl( \theta_q(\frac{n-3}{2}) \bigl( (d-1)q^{\frac{n-3}{2}}+1 \bigr) \Bigr) = q^{\frac{n-3}{2}}(q+1 -d),
E_V \log \sum_{k=1}^N p_k q_k^\gamma \leq ( 1 - \gamma ) \mathbb{E}\log \sum_{k=1}^N p_k + \mathbb{E}\log \left( \sum_{k=1}^N p_k q_k \right)^\gamma = \mathbb{E}\log \left( \sum_{k=1}^N p_k q_k \right)^\gamma.
M(x_i) = \{(\sigma, \sigma \cup \{x_i\}) \mid x_i \notin \sigma, \sigma, \sigma \cup \{x_i\}\in K_{i-1}\}, N(x_i) = \{\sigma \in K_{i-1}\mid \sigma \in \eta \eta \in M(x_i)\}, K_i = K_{i-1}\setminus N(x_i).
E_{(r,A,r',\mathcal{U},p,q)}=(E_{(r,A,r',\mathcal{U},p,q)},\nabla_{(r,A,r',\mathcal{U},p,q)}), &E_{(s,B,s',\mathcal{V},u,v)}=(E_{(s,B,s',\mathcal{V},u,v)},\nabla_{(s,B,s',\mathcal{V},u,v)}),
\ell(\hat y_t,y_t) \leq \sum_{d=1}^{D_t}\hat p_{d,t}\ell(f_{d,t},y_t) \leq -\frac{1}{\eta_{t}}\log \sum_{d=1}^{D_t}\hat p_{d,t}e^{-\eta_{t}\ell(f_{d,t},y_t)}+ \frac{\eta_{t}}{8}
E_{\mu}^{[\alpha,\gamma],\varepsilon}=\bigcup_{n\geq 1}E_{\mu}^{[\alpha,\gamma],\frac{1}{n},\varepsilon}\ \ \ \mbox{and}\ \ \ \F_{\mu}^{[\alpha,\gamma],\varepsilon}=\bigcup_{n\geq 1}F_{\mu}^{[\alpha,\gamma],\frac{1}{n},\varepsilon}.
\langle \tau_t(c_j^*) \tau_t(c_j) \rangle_{\rho}=\sum_{k,\ell=1}^n \overline{\left(e^{-2itA}\right)_{j\ell}}\left(e^{-2itA}\right)_{jk}\langle c_\ell^* c_k\rangle_{\rho}.
\frac{d^{2}f}{d x^{2}}+ \frac{a^{2}- 6 a + 2 - ( a^{2}- 8 a + 2 ) x^{a}}{2 x ( 1 - x^{a})}\frac{d f}{d x}+ \frac{\lambda^{2}}{x^{4 - a}( 1 - x^{a})^{2}}f - \frac{l ( l + a )}{x^{2}( 1 - x^{a})}f = 0 \, .
\int_{\mathbb{R}^{N}}\left( 1+ 2w^2 \right) \nabla w\cdot \nabla h\, dx+ \int_{\mathbb{R}^{N}}2w |\nabla w|^2 \, h \, dx + \int_{\mathbb{R}^{N}}V\left( \left| x\right| \right) wh\,dx=\int_{\mathbb{R}^{N}}K(|x|) g(w) h\,dx
P_A \left( \sigma(x_1) = y_1,\ldots,\sigma(x_p) = y_p \right) = \frac{1}{2}\frac{\prod_{i=1}^p w_A(x^i,y^i)}{Y_{m_1,m_2}(A)}\sum_{\theta \in \{0,1\}^2}(-1)^{(\theta_1+m_1+1)(\theta_2+m_2+1)}\frac{\partial}{\partial W_{x^1,y^1}}\ldots \frac{\partial}{\partial W_{x^p,y^p}}\det \limits_{x,y \in \mathbb{T}_{m_1,m_2}}K_{A,\theta}(x,y) .
\sum_{i,j=1}^n a_{ij}\frac{\partial^2\delta^\gamma}{\partial x_i\partial x_j}=\gamma(\gamma-2)\delta^{\gamma-4}\sum_{i,j=1}^n a_{ij}\sum_{k=1}^m b_{ki}y_k\sum_{l=1}^m b_{lj}y_l+\gamma\delta^{\gamma-2}\sum_{i,j=1}^n a_{ij}\sum_{k=1}^mb_{ki}b_{kj}.
\tilde{\lambda}_i^2 = (1+\rho t_i) \lambda_i^2, G_i = 1 - \frac{\frac{1}{2}t_i}{1 + \frac{\theta}{2}t_i}, \tilde{g}_i = \frac{\frac{1}{2}t_i}{1 + \frac{\theta}{2}t_i}, g_i = \frac{t_i (1 + \rho t_i)}{1 + \frac{\theta}{2}t_i}, r_i = -\rho t_i, 1 + \tilde{r}_i = \frac{1}{1 + \rho t_i}, \hat{r}_i = 0,
\begin{aligned}\Phi(x,y) = \langle \tilde{B}_1x - a, y \rangle +y^TB_1y +h(y) - x^TB_1x + a^T x - h(x) & \leq \langle \tilde{B}_1u^I - a, y \rangle +y^TB_1y +h(y) -(l^I)^TB_1l^I + a^Tu^I - h(l^I) .\end{aligned}
|\mathbf{1}_{Z\le y}- \mathbf{1}_{Z'\le y}| =\mathbf{1}_{Z\le y< Z'}+\mathbf{1}_{Z' \le y <Z}\le2 \mathbf{1}_{|Z-Z'| \ge\alpha}+ \mathbf{1}_{y-\alpha< Z' < y+\alpha}.
\left( \prod_{s=1}^{l}\prod_{p=1}^{i}(u - \hbar'(p - 1) + a_s) \right) C_i(u) \xi_{\Lambda}= \sum_{(s,p) \in A_{\Lambda,i}}\beta_{i,p}^{(s)}\left( \prod_{(s',p') \neq (s,p)}\dfrac{u - \nu_{i,p'}^{(s')}}{\nu_{i,p}^{(s)}- \nu_{i,p'}^{(s')}}\right) \xi_{\Lambda - \delta_{i,p}^{(s)}}.
\begin{aligned}\left(\frac{\boldsymbol{u}^1_h-\widetilde{\boldsymbol{u}}^1_h}{\Delta t}+\nabla (p_h^1-p_h^0), \boldsymbol{y}_h\right) =0, (\nabla \cdot \boldsymbol{u}_h^1, q_h) =0 \boldsymbol{u}_h^1|_{\partial\Omega}=0 \end{aligned}
\begin{cases}dx_i =( Ax_i +BR^{-1}B^Ty_i +G m_i+\gamma p_i) dt +D dW_i, \dot{m_i}= (A + G) m_i +BR^{-1}B^T{\mathbb E}y_i +\gamma p_i, \dot{p_i}= -(A+G)^T p_i -(I-\Gamma)^T Q [\mathbb{E}x_i- (\Gamma m_i+\eta)] , dy_i=\left\{-A^T y_i +Q[x_i-(\Gamma m_i+\eta)]\right\}dt +\zeta_i dW_i,\end{cases}
\begin{cases}\varphi_{+}\sim \pi^{\frac{1}{2}}t^{-\frac{1}{2}}e^{\frac{t}{4}}e^{-\frac{t\lambda}{2}}, begin{align*}0.2cm]\varphi_{-}\sim \pi^{-\frac{1}{2}}t^{-\frac{1}{2}}e^{-\frac{t}{4}}e^{\frac{t\lambda}{2}}-i\pi^{-\frac{1}{2}}t^{-\frac{1}{2}}e^{\frac{t}{4}}e^{-\frac{t\lambda}{2}}.\end{cases}
|X(-J,-1)| + k - 2 = \sum_{j \neq i}\mathcal{D}^{ij}(-J,-1) = - \sum_{j \neq i}\mathcal{D}^{ij}(0) \mathcal{D}^{ij}(-J,-1) = -\frac 12 \sum_{j \neq l}\mathcal{D}^{jl}(0) \mathcal{D}^{jl}(-J,-1).
1-v^{cd}\rho_{e}(\sigma+\phi^{cd})=-v^{cd}\rho_{e}'(\phi^{cd})\sigma-\frac{v^{cd}\rho_{e}''(\phi^{cd})}{2}\sigma^{2}\underbrace{-v^{cd}\int_{\phi^{cd}}^{\phi}\rho_{e}'''(\varrho)\frac{(\phi-\varrho)^{2}}{2}d\varrho}_{I_0}.
r^{S}= \left( \begin{array}{c c c c}{{r_{+}^{S}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{r_{+}^{S}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{r_{\chi}^{S}}}\\ \end{array}\right) \qquad r^{\delta_{i}}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{r_{\delta_{i}}}}\\ \end{array}\right)
\begin{aligned}\overline{\alpha}(A(\varphi(\theta_0+\pi/2))X(\theta_0+\pi/2)K) =\overline{\alpha}(X(\theta_0+\pi/2)K)=\overline{a}(A(\varphi(\theta_0))X(\theta_0)K), \overline{a}(A(\varphi(\theta_0+\pi/2))X(\theta_0+\pi/2)K) =\overline{a}(X(\theta_0+\pi/2)K)=\overline{\alpha}(A(\varphi(\theta_0))X(\theta_0)K).\end{aligned}
( A_{1^{N}}^{\phi})^{d}= ( A_{1^{N}}^{d})^{\phi}= D_{1^{N}}^{\phi}\, A_{1^{N}}^{d}\, D_{1^{N}}^{- \phi}= \sum_{\pi \in S_{N}}f^{d}( \pi ) R^{\phi}( \pi ) \, .
\frac{dR}{dt}=\frac{d\overline{x}}{dt}\cdot t+(\sum\pi_{z}^{f}\frac{dz^{f}}{dt}-\sum E_{z}^{h}\frac{dz^{h}}{dt})=\frac{d\overline{x}}{dt}\cdot t+(\Pi^{t}-B^{t})
V :=\big\{h\in H_c\ \big|\ \alpha(h)\leq a,\ \beta(h)\leq b \big\}, W :=\big\{h\in H_c\ \big|\ \alpha(h)< a,\ \beta(h)\leq b \big\}, Z :=\big\{h\in H_c\ \big|\ \alpha(h)\leq a,\ \beta(h)< b \big\},
0<\lambda_{1n}\ll \lambda_{2n}\ll\cdots\ll\lambda_{J_0n}, \footnote{\, For two sequences of positive numbers $a_n,\,b_n$, we write $a_n\ll b_n$ if $\lim\limits_{n\to\infty}\frac{a_n}{b_n}=0$.}\,\,{\rm as}\,\,t_n\to T_+,
\begin{cases}2 \kappa_0 - \eta^2 + 2 \eta^3 - \eta^4=0 &-3 \kappa_0 - 3 \kappa_0 \lambda_0 - 3 \kappa_0 \eta + 2 \eta^2 + 3 \kappa_0 \eta^2 - 3 \eta^3 - \eta^4 + 3 \eta^5 - \eta^6=0 &g(x)=0,\end{cases}
\mbox{\boldmath$I$}_{21}(F_1)\mbox{\boldmath$I$}_{21}(F_2) =\mbox{\boldmath$I$}_2(\mbox{\boldmath$I$}_1(F_1)\mbox{\boldmath$I$}_1(F_2)) &=\mbox{\boldmath$I$}_2(\mbox{\boldmath$I$}_1(F_1 F_2)) &=\mbox{\boldmath$I$}_{21}(F_1 F_2).
\varphi(C_1) =x_1+x_2, \varphi(C_2) =(x_1+x_2)(x_1-x_2+1), \varphi(C_3) =(x_1+x_2)(x_1^2-x_1x_2+x_2^2+x_1-2x_2+1) =\varphi(C_1)(\frac{3\varphi(C_2)^2}{4\varphi(C_1)^2}+\frac{\varphi(C_1)^2}{4}-\frac{\varphi(C_1)}{2}+\frac{1}{4}), \dots
\underset{_{\substack{(\rho )=\frac{1}{2}T<\left\vert \gamma\right\vert \leq eT}}}{\sum}\frac{(x+h)^{\rho +1}}{\rho \left( \rho +1\right)}=O\left( x^{\alpha}(\log x)^{\beta}(\log \log x)^{\beta}\right)
\sqrt{a_{11}^{2}+a_{21}^{2}+2a_{11}a_{21}\frac{a_{12}^{2}+a_{22}^{2}-(a_{11}^{2}+a_{21}^{2})}{4a_{11}a_{21}}}+\sqrt{a_{12}^{2}+a_{22}^{2}+2a_{12}a_{22}\frac{a_{12}^{2}+a_{22}^{2}-(a_{11}^{2}+a_{21}^{2})}{4a_{11}a_{21}}}=2^{\frac{1}{2}}\sqrt{a_{11}^{2}+a_{21}^{2}+a_{12}^{2}+a_{22}^{2}}.
P = P_{g}+ P_{b}= \frac{- 1}{9 \times 9 6 0 \pi^{2}}( \Lambda_{b e t w}^{2}- \Lambda_{o u t}^{2}) - \frac{\eta^{4}\Lambda_{b e t w}^{2}J_{3}( B_{1}, B_{2})}{3 6 \pi^{3 / 2}a^{4}\Gamma ( 3 / 2 )}.
{K}_n(x,y) = \frac{\mu}{2\pi i}\oint_{\Sigma}\frac{ds}{s}\int_0^1 \frac{dt}{t}\left(\frac{t}{s}\right)^{\nu}\, k_n \left( \frac{x}{s}, \frac{y}{t}\right) \left(1- s \right)^{-\mu-1}\left(1- t \right)^{\mu-1},
\mu(Q_2)\big(\psi_3(w) +\psi_3(v)\big) + \mu(Q_3)\big(\psi_2(w)+\psi_2(v)\big) &=\int_{Q_2}\int_{Q_3}g(p-v)+g(p-w)+g(q-v)+g(q-w) \, d\mu(p)\,d\mu(q) &\geq -(3+\eta) \mu(Q_2)\mu(Q_3)\geq -(3+\eta) \bigg(\frac 13+\eta\bigg)^2\,.
\omega_0(z;q)= \frac{1}{\xi_0}, \omega_{n+1}(z;q)= \frac{1}{\bar{\pi}_{n(n+1)}}\Big[z^{2}\omega_{n}(z;q)-\sum_{k=0}^{n}\bar{\pi}_{nk}\omega_k(z;q)\Big], \, n\in \mathbb{N}_0.
\partial_{\varphi}\left\{\partial_{z}S^{z}\partial_{\bar{z}}S^{\bar{z}}- \partial_{z}S^{\bar{z}}\partial_{\bar{z}}S^{z}+{\cal{H}}\right\}= \partial_{z}p + \partial_{\bar{z}}\bar{p}+ \partial_{\varphi}{\cal H}= \partial_{\varphi}\left\{\partial_{z}S + \partial_{\bar{z}}\bar{S}+{\cal H}\right\}= 0 \, .
1\otimes T_{n_{s_1\dotsm s_l}}^*T_{n_{w}}=1\otimes T_{n_{s_1\dotsm s_{l - 1}}}^*(T_{n_{s_l}}- c_{n_{s_l}})T_{n_{w}}& = 1\otimes T_{n_{s_1\dotsm s_{l - 1}}}^*T_{n_{s_lw}}- 1\otimes T_{n_{s_1\dotsm s_{l - 1}}}^*c_{n_{s_l}}T_{w}& = 1\otimes T_{n_{s_1\dotsm s_{l - 1}}}^*T_{n_{s_lw}}.
-\Xi'_{\imath}+\Xi'_{\jmath}\begin{cases}=r^{(\jmath)}\ \imath\in{{}^{\jmath}B}, \in r^{(\jmath)}+4\mathbb{Z}_{>0}\ \imath\notin{{}^{\jmath}B}\ \ \jmath\neq n, \in r^{(n)}+2\mathbb{Z}_{>0}\ \imath\notin{{}^{\jmath}B}\ \ \jmath=n.\end{cases}
\rho =\frac{\Pi_{1}(B^{2}E_{2}^{2}-E_{1}^{2})}{E_{1}(E_{2}^{2}-1)}-\frac{E_{1}T_2^2}{\Pi_{1}(E_{2}^{2}-1)}\frac{1}{1+(1-u^{2})\sinh^{2}(x/\zeta_B)}.
0=\psi(x)= \sum_ie_{11}([a_i,b_i]-\overline{[a_i,b_i]})+e_{11}(c_0-\bar{c}_0)+e_{22}(\rho(c_0))-e_{33}(\rho(\bar{c}_0)) &+{f}(c_1)+{g}(c_2)+{v}(c_3)+{w}(c_4)
\begin{aligned}D^d_{2^{\mu},M}(x_{n,i})=D^d_{2^{\mu},M}(q_{n,i}) \:\:\: \leq \:\:\: \frac{2^{\mu}D^d_{2^{\mu}}(q_{n,i})+MD^d_M(q_{n,i})}{M-2^{\mu}}\leq \:\:\: \frac{2^{\mu}\sqrt{d}/\sqrt{2^{\mu}}+M\sqrt{d}/\sqrt{M}}{M-2^{\mu}}< \:\:\: \frac{2\sqrt{d}\sqrt{2^{m+1}}}{M-2^{\mu}}. \end{aligned}
\Gamma = \pi A_{h}\frac{( \omega^{2}- m^{2})^{3 / 2}}{m ( \omega - m \phi_{4})}\sqrt{\frac{r_{1}r_{2}r_{3}}{r_{4}}}\frac{1}{( e^{2 \pi m \sqrt{\frac{r_{1}r_{2}r_{3}}{r_{4}}}}- 1 )}\frac{1}{( e^{\frac{\omega - m}{T_{H}}}- 1 )}\frac{d^{3}k}{( 2 \pi )^{3}},
f^{\prime \prime}={\frac{{f^{\prime}}^{2}}{2 f}}+{\frac{3}{2}}f^{3}+ 2 \omega x f^{2}+ ( \mathrm{\frac{1}{2}}\omega^{2}x^{2}+ k_{3}- k_{1}) f -{\frac{k_{2}^{2}}{2 f}}.
c^{\mathcal{J}, T, (A)}_t(\sigma ) = \frac{e^{- \Delta_A\mathcal{H}_{\mathcal{J}}(\sigma ) / T(t)}}{e^{\Delta_A\mathcal{H}_{\mathcal{J}}(\sigma ) / T(t)}+e^{- \Delta_A\mathcal{H}_{\mathcal{J}}(\sigma )/ T(t)}}= \frac{1}{1+ e^{2\Delta_A \mathcal{H}_{\mathcal{J}}(\sigma ) / T(t)}}.
c^{2}d\tau^{2}=(g_{tt}-\frac{g_{t\phi}^{2}}{g_{\phi\phi}})dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}(d\phi+\frac{g_{t\phi}}{g_{\phi\phi}}dt)^{2}
\left\vert x\left(\zeta^{u}-\frac{y_{0}^{u}}{x_{0}^{u}}\right)\right\vert\leq X\left\vert\zeta^{u}-\frac{y_{0}^{u}}{x_{0}^{u}}\right\vert\leq \frac{x_{0}^{2}}{2c_{1}}\cdot c_{1}x_{0}^{-3}= \frac{1}{2}x_{0}^{-1}.
\begin{array}{llllllll}K^{1}_2 = \langle x_1+x_2, h_1+h_2 \rangle, K^{2,a}_2 = \langle x_1, h_1 + a h_2 \rangle, K^{3}_2 = \langle x_1, h_1 + x_2 \rangle, K^{4,a}_2 = \langle x_2, h_2 + a h_1 \rangle, K^{5}_2 = \langle x_2, h_2 + x_1 \rangle,\end{array}
\Upsilon_{\omega}^{+}:=\left\{Y_{\alpha\omega}^{mi},Y_{\beta\omega}^{mj}\right\},\quad\Upsilon_{\omega}^{-}:=\left\{Y_{\alpha\omega}^{mj},Y_{\beta\omega}^{mi}\right\},\quad\Upsilon_{\omega}:=\Upsilon_{\omega}^{+}\cup\Upsilon_{\omega}^{-}
\nu_{k}\ = \ \Bigl [ \frac{4 \i}{\mu_{k}- \bar{\mu}_{k}- \mu_{k}^{- 1}+ \bar{\mu}_{k}^{- 1}}\cdot \frac{\mu_{k}- \mu_{k}^{- 1}- 2 \i}{\bar{\mu}_{k}- \bar{\mu}_{k}^{- 1}+ 2 \i}\Bigr ]^{1 / 2}\quad .
\left\langle \frac{1}{2 N}\mathrm{T r}\, ( \vec{\sigma}( \vec{A}\times \vec{A}) \vec{\sigma}( \vec{A}\times \vec{A}) ) \right\rangle = \left\langle \frac{6}{N}\mathrm{T r}\, ( A_{1}A_{2}A_{1}A_{2}- A_{1}^{2}A_{2}^{2}) \right\rangle = - 6 \left( \frac{a^{2}}{4}\right)^{2}
{\displaystyle\sum\limits_{n=1}^{\infty}}(-1)^{n+1}\,(2n-1)^{2n}=\frac{i}{2}{\displaystyle\int\limits_{0}^{\infty}}\left( \frac{W(-i\,x)}{\left[ 1+W(-i\,x)\right]^{3}}-\frac{W(i\,x)}{\left[ 1+W(i\,x)\right]^{3}}\right) e^{-x}dx & =0.0111203007....
p(n) =p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots+ \ \ \ \ (-1)^{k-1}p\left(n-\dfrac{3k^{2}\pm k}{2}\right)+\cdots\cdots =\sum\limits_{k=1}^{k_{1}}(-1)^{k-1}p\left(n-\dfrac{3k^{2}+k}{2}\right)+\sum\limits_{k=1}^{k_{2}}(-1)^{k-1}p\left(n-\dfrac{3k^{2}-k}{2}\right),
\sqrt{\max\left\{_0\widetilde{\tau}^{(n)}_{M_n},_n\widetilde{\tau}^{(n)}_{M_n}\right\}}=o\left(\max\left\{\mathbb{E}_0\widetilde{\tau}^{(n)}_{M_n},\mathbb{E}_n\widetilde{\tau}^{(n)}_{M_n}\right\}\right).
W^{1,p}(\Omega,w)\hookrightarrow W^{1,p_s}(\Omega)\hookrightarrow \begin{cases}L^q(\Omega), q\in[p_s,p_s^{*}], 1\leq p_s<N L^q(\Omega), q\in[p_s,\infty), p_s=N C^{0,\alpha}(\overline{\Omega}), p_s>N.\end{cases}
S = \displaystyle \int d^{8}z E^{- 1}\left[ \bar{Q}Q + \bar{Q}^{c}Q^{c}+ \lambda_{0}( \bar{Q}^{c}Q ) ( \bar{Q}Q^{c}) + \bar{\xi}_{1}( \bar{Q}Q ) + \bar{\xi}_{2}( \bar{Q}^{c}Q^{c}) \right] ,
\rho_{n-w-2}(u_2,\ldots,u_{2n-1})-\rho_{n-w-1}(u_1,\ldots,u_{2n}) = \Delta^n_{2w+2}-\nabla^n_{2w+2}= 2\Delta^n_{2w+2}+\sum_{s=1}^{2w+1}(-1)^{s}\nabla^n_{s}\Delta^n_{2w+2-s}=0.
\sum_{j'=1}^J \frac{n_0^{(j')}}{2}\log ( k_n^{(j)}) \leq_n \frac{\epsilon'}{2}\frac{n}{2}\log k_n^{(j)}+ \sum_{j'=1}^J (1+\epsilon') \beta^{(j')}\ell_n H_2 \left( \alpha_n^{(j')}\right) .
\begin{aligned}\inf_X\left[ (1-t)\Theta_0 +t\Theta_1 +b_t\right] = (1-t)\Theta_0(p) +t\Theta_1(p) +b_{t}&= \Theta_0(p) +t(\Theta_1(p) -\Theta_0(p)) +b_t &= \Theta_0(p) +t\sup_{X}(\Theta_1-\Theta_0) +b_t &\geq \Theta_0(p) &> (n-2)\frac{\pi}{2}.\end{aligned}
D_{\theta'}X_{\theta}^{n}- K_H(\theta,\theta')I_d = \sum_{m=1}^{\infty}\int_{\Delta_{\theta',\theta}^m}\prod_{j=1}^m b_n'(s_j,X_{s_j}^{n}) (K_H(s_m,\theta') I_d) \, ds_m \cdots ds_1.
\mathbb{P}(S_{(\eta)}>x) =\sum\limits_{n=1}^{\infty}\mathbb{P}(S_{(n)}>x)\mathbb{P}(\eta=n) &\geqslant\mathbb{P}\bigl(\max\{S_1,\dots,S_a\}>x\bigr)\mathbb{P}(\eta=a) &\geqslant\mathbb{P}(S_a>x)\mathbb{P}(\eta=a).
f(t) =\langle z|e_{\lambda}^{a^{\dagger}a}(t)|z\rangle=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\langle z|(a^{\dagger}a)_{k,\lambda}|z\rangle &=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\sum_{l=0}^{k}S_{2,\lambda}(k,l)|z|^{2l}=\sum_{k=0}^{\infty}\phi_{k,\lambda}(|z|^{2})\frac{t^{k}}{k!}.
\sum_{i=-1}^{r}\tbinom{(p-w)-b}{2(p-w)-2b+i-r}\tbinom{a-(p-w)}{2a-2(p-w)-i-1}= \sum_{i=c}^{e}\tbinom{(p-w)-b}{2(p-w)-2b+i-r}\tbinom{a-(p-w)}{2a-2(p-w)-i-1}.
{{\cal G}^{- 1}}_{\mu a}^{\nu b}\eta_{\nu b}^{\mathrm{T , L}}= \frac{1}{2}\left( k^{2}- \Lambda_{l}+ \frac{1}{2 r^{2}}( N - 1 ) ( N - 2 ) - \frac{N + d}{N + d - 2}\Lambda \right) \eta_{\mu a}^{\mathrm{T , L}},
y''(z) + \left(2 \frac{\phi'(z)}{\phi(z)}+ \frac{\tilde{\tau}(z)}{\sigma(z)}\right) y'(z) + \left(\frac{\phi''(z)}{\phi(z)}+ \frac{\phi'(z)}{\phi(z)}\frac{\tilde{\tau}(z)}{\sigma(z)}+ \frac{\tilde{\sigma}(z)}{\sigma^2(z)}\right) y(z) = 0.
\left|\sum_{|\alpha|>CL}a_{\alpha}\left(\frac{\Gamma(L+|\alpha|)}{\alpha!\Gamma(L)}\right)^{1/2}z^{\alpha}\right| \leq \sum_{m>CL}\left[h(C)r^2\right]^{m/2}\left(\sum_{|\alpha|=m}|a_{\alpha}|^2\right)^{1/2}& \leq \sum_{m>CL}(1-\delta)^m\left(\sum_{|\alpha|=m}|a_{\alpha}|^2\right)^{1/2}.
\phi_{1}={\frac{\phi_{0}\; ( \eta_{a b}\; \zeta^{a}\; \zeta^{b}) \; \left\{\eta_{a b}\; [ x^{a}- x_{0}^{a}] \; [ x^{b}- x_{0}^{b}] - \eta_{a b}\; \zeta^{a}\; \zeta^{b}\right\}}{( \eta_{a b}\; [ x^{a}- x_{0}^{a}] \; [ x^{b}- x_{0}^{b}] - \eta_{a b}\; \zeta^{a}\; \zeta^{b})^{2}+ 4 ( \eta_{a b}\; [ x^{a}- x_{0}^{a}] \; \zeta^{b})^{2}}};
^{H}\mathbb{D}_{b-}^{\alpha ,\beta ;\psi}f\left( x\right) =I_{b-}^{\beta\left( n-\alpha \right) ;\psi}\left( -\frac{1}{\psi^{\prime}\left(x\right)}\frac{d}{dx}\right)^{n}I_{b-}^{\left( 1-\beta \right) \left(n-\alpha \right) ;\psi}f\left( x\right).
wt(C_{5})=\begin{cases}(p-1)p^{e-2}, \qquad\ \biggl(\frac{c^{2}-4ai}{p}\biggr)=-1, (p-1)p^{e-2}+2p^{m+d-1}, \qquad\ \biggl(\frac{c^{2}-4ai}{p}\biggr)=1.\end{cases}
\gamma_{\Psi}A^{* \mu}= 0 , \gamma_{\Psi}\phi^{*}= 0 , \gamma_{\Psi}C^{*}= - \partial_{\mu}A^{* \mu}, \gamma_{\Psi}b^{*}= - \overline{{C}}^{*}, \gamma_{\Psi}\overline{{C}}^{*}= 0 .