harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(E_r, AU) = (A^TE_r, U) = ( \sum_{s=1}^p \alpha_{rs}E_s + A^T E_r -\sum_{s=1}^p \alpha_{rs}E_s, U) = \sum_{s=1}^p \alpha_{rs}(E_s, U)+ ( A^T E_r - \sum_{s=1}^p \alpha_{rs}E_s, U) = \sum_{s=1}^p \alpha_{rs}\psi_s+ ( A^T E_r - \sum_{s=1}^p \alpha_{rs}E_s, U).
\alpha_{\phi}^{i j}( x ; y ) = \alpha^{i j}( x ) + \nabla_{r}\alpha^{i j}( x ) \, y^{r}+ \left( \frac 1 2 \, \nabla_{r}\nabla_{s}\alpha^{i j}( x ) - \frac 1 6 \, R_{r k s}^{[ i}( x ) \alpha^{j ] k}( x ) \right) \, y^{r}y^{s}+ \cdots ,
F ( x^{-}, x^{+}) = \sum_{i}\left|{\frac{L}{\pi}}\langle 0 | T^{+ +}( K ) | i \rangle \right|^{2}\left({\frac{x^{+}}{x^{-}}}\right)^{2}{\frac{M_{i}^{4}}{8 \pi^{2}K^{3}}}K_{4}\left( M_{i}\sqrt{2 x^{+}x^{-}}\right) ,
\log\left( \frac{Z_{n,\alpha}(\tfrac{s}{4n^{2}};r)}{Z_{n,\alpha}(\tfrac{\epsilon}{4n^{2}};r)}\right) = \int_{\epsilon}^{s}\partial_{x}\log Z_{n,\alpha}\left( \frac{x}{4n^{2}};r \right)dx.
f =\Delta_t \phi=(\partial_z^2+a^2\partial^L_{vv}+b\partial^L_{v})\phi =(\partial_z^2+a^2(\partial_{v}-t\partial_z)^2+b(\partial_{v}-t\partial_z))\phi & =\Delta_L \phi + ((a^2-1)\partial^L_{vv}+b\partial^L_{v})\phi.
k_q \geq \sum_{I\in S_\phi}y_I^{p-q}\Biggl\{\frac{(\mu(I) y_I)^q}{(\mu(I)\tau_I)^{q-1}}- \sum_{\substack{J\in S_\phi J^\star=I}}\frac{(\mu(J)y_J)^q}{((\beta+1)\mu(J))^{q-1}}\Biggr\}= = \sum_{I\in S_\phi}\mu(I)\frac{y_I^p}{\tau_I^{q-1}}- \sum_{I\in S_\phi}y_I^{p-q}\sum_{\substack{J\in S_\phi J^\star=I}}\frac{y_J^q}{(\beta+1)^{q-1}}\mu(J).
\int_{-\infty}^{\infty}p_{\beta=1}(y)\,e^{ys}\, dy=K^{s}\Gamma(1+s) \tilde{M}_{\beta=1}(1+s) = \frac{1}{C}K^{s}\Gamma^2(1+s) \frac{G(s+7/2)^{2}G(s+3)G(s+4)}{G^2(s+3)G(s+6)G(s+2)}=\frac{1}{C}K^{s}\frac{\Gamma(1+s)G(s+7/2)^{2}\Gamma(s+3)}{G(s+1)G(s+6)}.
\ll \sum_{\substack{n \le x: \, p\parallel n \lambda \le R_p(n)<\lambda+1/\Omega(n)}}\frac1{\Omega(n)}\ll \sum_{\substack{n \le x: \, p|n \Omega(n) \le \frac13\log_2 x}}1 + \frac1{\log_2 x}\sum_{\substack{n \le x: \, p\parallel n \lambda \le R_p(n)<\lambda+1/\Omega(n)}}1.
b_{11}^{(k+1)}-b_{33}^{(k+1)}= b_{11}^{(k)}-b_{44}^{(k)}+ b_{13}^{(k)}\tan\phi_{k+1}+b_{24}^{(k)}\tan\psi_{k+1}& = b_{11}^{(k)}-b_{44}^{(k)}+ (b_{13}^{(k)}-b_{24}^{(k)})\tan\phi_{k+1}+b_{24}^{(k)}(\tan\phi_{k+1}+\tan\psi_{k+1}).
n_p\bigl(a,a+b,a j_{k-1}(v)+b j_k(v)\bigr) &=\frac{1}{2}\biggl(-v(a-r^2)+(a-1)(b-1)+\frac{v(a+r)(a-r)j_{k-1}(v)}{j_k(v)}&+v(a-r)\bigl(j_k(v)-j_{k-1}(v)\bigr)\biggr) &+\frac{p}{2}a j_k(v)\bigl(2(v a+b)-v(j_k(v)-j_{k-1}(v)\bigr)-\frac{p^2}{2}a v j_k(v)\bigl(j_k(v)-j_{k-1}(v)\bigr)\,,
B_{m n}^{( + )}= -{\frac{i^{k}}{2 \pi}}\sqrt{{\frac{n}{m}}}( m + n )^{- k}e^{i n T^{+}( 0 )}\int_{0}^{2 \pi}e^{i ( m + n ) y}{\frac{d^{k + 1}\varphi_{t}}{d y^{k + 1}}}\, d y \, ,
\left[ \, \left( T_{x}^{g}\right)^{\vee}\otimes \left( h^{*}T_{x}^{g}\right) \, \right] \: \otimes \: \left[ \, \left( T_{x}^{h}\right)^{\vee}\otimes \left( g^{*}T_{x}^{h}\right) \, \right]^{\vee}
P^{y^{\prime}z^{\prime}}~ T^{A , b}~ ( P^{y^{\prime}z^{\prime}})^{- 1}= T^{A , b}~ , ~ P^{y^{\prime}z^{\prime}}~ T^{A , \hat{b}}~ ( P^{y^{\prime}z^{\prime}})^{- 1}= - T^{A , \hat{b}}~ . ~ \,
\cos[\pi(\gamma-\overline{\gamma})]=\frac{e^{2\pi\Im\gamma}+e^{-2\pi\Im\gamma}}{2},e^{i\pi(\gamma-\overline{\gamma})}=e^{-2\pi\Im\gamma},e^{i\pi(\overline{\gamma}-\gamma)}=e^{2\pi\Im\gamma}.
S_{I I}= k^{2}~ \frac{\sigma_{0}}{\pi}\int_{\tau_{0}}^{\tau_{f}}d \tau ~ \sum_{n \neq 0}^{}~ | F_{\nu}\left( \frac{2 \pi}{\sigma_{0}}| n | \tau , \frac{2 \pi}{\sigma_{0}}| n | \tau_{0}\right) |^{2}~ | \phi_{n}( \sigma_{0}) |^{2}
\begin{array}{l r l l}{{}}&{{( f_{0}+ Q^{- 1 / 2}f_{1}) ( Q - 1 )}}&{{>}}&{{- 1}}\\{{\mathrm{~ a n d ~}}}&{{Q^{- 1 / 2}f_{1}( Q - 2 )}}&{{>}}&{{- 1 \; ,}}\\ \end{array}
A = \int \frac{d^{3}p}{( 2 \pi )^{3}}\ \overline{{{\Psi}}}( p ) S^{- 1}( p ) \Psi ( p ) \ + \int \frac{d^{3}p \d^{3}q}{( 2 \pi )^{3}}\ \overline{{{\Psi}}}( p + \frac{\ 1}{\ 2}\q ) \Gamma ( p , q ) \Psi ( p - \frac{\ 1}{\ 2}\q ) ,
\vec{v}_{B|A}=\frac{1}{\gamma_{A}(1-\frac{\vec{v}_{A}\vec{v}_{B}}{c^{2}})}[\vec{v}_{B}-\vec{v}_{A}+\vec{v}_{A}(\gamma_{A}-1)(\frac{\vec{v}_{A}\cdot\vec{v}_{B}}{v_{A}^{2}}-1)]
\left(\frac{n+a_{p+1}+p+1}{n+1}S_n-1\right)R_{p,b}(n) &\quad=\frac{n+a_{p+1}+p+1}{n+1}R_{p,b}(n+1)S_n-R_{p,b}(n) &\quad=\frac{n+a_{p+1}+p+1}{n+1}R_{p,b}(n+1)L_b+\left(\frac{n+a_{p+1}+p+1}{n+b+1}R_{p,b}(n+1)-R_{p,b}(n)\right).
\big|{y}_{0}(x)-\hat{y}_{0}(x)\big| =\bigg|\frac{y_a}{\Gamma(\gamma)}(x-a)^{\gamma-1}-\frac{(y_a+{\epsilon})}{\Gamma(\gamma)}(x-a)^{\gamma-1}\bigg| \big|{y}_{0}(x)-\hat{y}_{0}(x)\big| \leq \big|{\epsilon}\big|\frac{(x-a)^{\gamma-1}}{\Gamma(\gamma)}.
g_{1}(\hat z^{\pm}_{i,j-1};a) = h\left(\hat z_{i,j-1}^{-}\,|\,\gamma^{-1}|I| k_-^{i,j-1}(a_{i,j-1}) c_M\right)+h(\hat z_{i,j-1}^{+}\,|\,\gamma^{-1}|I| k_+^{i,j-1}(a_{i,j-1})c_M) g_{2}(\hat z^{\pm}_{i,j-1};a) = h\left(\hat z_{i,j-1}^{-}\,|\,\gamma^{-1}|I| k_-^{i,j-1}(a_{i,j-1})c_m)+h(\hat z_{i,j-1}^{+}\,|\,\gamma^{-1}|I| k_+^{i,j-1}(a_{i,j-1})c_m\right).
(\nabla_{p}\log F)^{2}=(\sum_{i}\frac{\partial\log F}{\partial \lambda_{i}}h_{iip})^{2}\leq(\sum_{i}\lambda_{i}\frac{\partial\log F}{\partial \lambda_{i}})(\sum_{i}\lambda_{i}^{-1}\frac{\partial\log F}{\partial \lambda_{i}}h_{iip}^{2}) =\beta(\sum_{i}\lambda_{i}^{-1}\frac{\partial\log F}{\partial \lambda_{i}}h_{iip}^{2}).
s_\mathcal{A}(r,m) =\sum_{n=m2^r}^{m2^{r+1}-1}f_{\mathcal{A}}(n) =\sum_{n=m2^{r-1}}^{m2^r-1}f_{\mathcal{A}}(2n)+f_{\mathcal{A}}(2n+1) =\sum_{n=m2^{r-1}}^{m2^r-1}f_{\mathcal{A}}(n)+f_{\mathcal{A}}(n-b_2)+\cdots+f_{\mathcal{A}}(n-b_s)+f_{\mathcal{A}}(n-c_1)+\cdots+f_{\mathcal{A}}(n-c_t).
|{\mathcal B}_{y^j}( \underline x^j) | \geq y_j \min(|{\mathcal B}_{y^{j-1}}( \underline x^{j-1}) |, b(\underline x_j) ) &+ (1-y_j) [ |{\mathcal B}_{y^{j-1}}( \underline x^{j-1}) | - b(\underline x_j) ]^+, \forall j \leq L, |{\mathcal B}_{y^0}( \underline x^0) | = M.
N(\star_{i=1}^{k+l}G,\alpha^{k+1}\star \alpha^{l+1})= N(\star_{i=1}^{k+l}G, \alpha^{k+l+1}) = N(\star_{i=1}^{k+l}G, \star_{i=1}^{k+l}\alpha^{2}) \le N( \star_{i=1}^{k}G,\alpha^{k+1}) N(\star_{i=1}^{l}G,\alpha^{l+1}) = N(\star_{i=1}^{k}G, \star_{i=1}^{k}\alpha^{2}) N(\star_{i=1}^{l}G, \star_{i=1}^{l}\alpha^{2}).
m_{[i_{2}-1,i_{2}-1]}m_{[i_{2},i_{2}]}+\sum_{s=i_{2}}^{j_{2}-1}m_{[s,s]}m_{[s+1,s+1]}+m_{[j_{2},j_{2}]}m_{[j_{2}+1,j_{2}+1]}&=(2(j_{2}-i_{2})+2)m_{[i_{2},j_{2}]}m_{[i_{1},n]}+\cdots,
R =\frac{1}{NB}\left(|\mathcal{M}_2|\times|\mathcal{A}| +|\mathcal{M}_1|\times|\mathcal{A}^{*c}|+|\mathcal{I}|\times B\right) =[H(p_2)-H(p_1)]\times q_1 +[H(p_2^*)-H(p_1^*)]\times q_2^*+[H(p_1^*)-H(p_2)] =q_1^* H(p_1^*)+q_2^* H(p_2^*)-q_1H(p_1)-q_2H(p_2).
E_1(x,y) =|(Id-L)(F_y;x)|\leq\displaystyle\sum_{\rho=0}^r\Gamma_{\rho,L}(x)\cdot\omega_{\rho}\left(F_y;\Lambda_{\rho,L}(x)\right) &\leq\displaystyle\sum_{\rho=0}^r\Gamma_{\rho,L}(x)\cdot\omega_{\rho}\left(F;\Lambda_{\rho,L}(x),0\right).
\mu_3^2 \geq \nu_1 \mu_2^2 \nu_1 := \left(\sqrt{1 + \frac{c_{1,\infty}}{2c_{3,\infty}}}- 1\right)^2\left(-1 + \sqrt{1 + \frac{c_{3,\infty}+ c_{4,\infty}}{c_{2,\infty}}}\right)^{-2}
\left\|h(t)-\frac1n\sum_{j=1}^n g_j(t)\right\|=\left\|h(t)-\frac{n-1}{n}h(t)-\frac{1}{n}g_k(t)\right\|=\frac1n\|h(t)-g_k(t)\|\leqslant \frac{2}{n};
S = \frac{V}{4 g^{2}}\mathrm{T r}\left( \sum_{\rho \sigma}( F_{\rho \sigma}+ \phi_{\rho \sigma}\cdot{\bf 1}) g^{\rho \mu}g^{\sigma \nu}( F_{\mu \nu}+ \phi_{\mu \nu}\cdot{\bf 1}) \right) \, .
(II)\leq (A)+(B) \leq \frac{1}{(j-1)!}\left(\frac{2M}{\mu}\right)^{j-1}+ \frac{1}{j!}\left(\frac{2M}{\mu}\right)^{j}= \frac{1}{j!}\left(1 + \frac{j\mu}{2M}\right)\left( \frac{2M}{\mu}\right)^j.
D^2 =-\sum_{i=1}^n \epsilon_i^2 \otimes 1 -\frac{1}{4}\mathbf r^2\sum_{\alpha>0, \beta>0, s_{\alpha}(\beta)<0}c_{\alpha}c_{\beta}|\alpha||\beta|s_{\alpha}s_{\beta}\otimes \widetilde{s}_{\alpha}\widetilde{s}_{\beta}
\int_{\Omega}\mathrm{div}\left( (\nabla u_{0})q_{0}u_{0}^{q_{0}-1}\phi_{1}\right) =\int_{\Omega}\mathrm{div}\left( (\nabla u_{0})q_{0}\left(\frac{\phi_{1}}{u_{0}}\right)^{1-q_{0}}\phi_{1}^{q_{0}}\right) & =\int_{\partial\Omega}\frac{\partial u_{0}}{\partial\nu}q_{0}\left(\frac{\phi_{1}}{u_{0}}\right)^{1-q_{0}}\phi_{1}^{q_{0}}& =0,
\varphi^{(16)}_1(\tau,z) :=8\varphi_1^{(4)}(\tau,z)\varphi_1^{(13)}(\tau,z)-\varphi_1^{(5)}(\tau,z)\varphi_1^{(12)}(\tau,z)+\varphi_1^{(7)}(\tau,z)\varphi_1^{(10)}(\tau,z), \varphi^{(16)}_2(\tau,z) :=12\varphi_1^{(4)}(\tau,z)\varphi_1^{(13)}(\tau,z)-\varphi_1^{(5)}(\tau,z)\varphi_1^{(12)}(\tau,z)-3\varphi_1^{(16)}(\tau,z).
\frac{\mathcal{B}_{n+m+l}^{\left( l\right)}\left( \lambda\right)}{\binom{n+m+l}{l}l}=\sum_{k=0}^{m}\sum_{j=0}^{n}\genfrac{\{}{\}}{0pt}{}{m}{k}\binom{n}{j}\binom{l+k+j}{j}^{-1}\frac{\left( -\lambda\right)^{k}k^{n-j}}{l+k}\mathcal{B}_{l+k+j}^{\left( l+k\right)}\left( \lambda\right) .
\epsilon^{\lambda \mu \nu \rho}\, \mathbf{F}_{\lambda \mu}\, \mathbf{F}_{\nu \rho}\longrightarrow \epsilon^{\lambda \mu \nu \rho}\, \left\{\, X_{\lambda}( \sigma ) \ , X_{\mu}( \sigma ) \, \right\}_{\mathrm{P B}}\left\{\, X_{\nu}( \sigma ) \ , X_{\rho}( \sigma ) \, \right\}_{\mathrm{P B}}\equiv 0
\sum_{i_1,\cdots,i_{k}\in\Lambda(I)\}\prod_{j=1}^k({\tau}^*_{i_j}-x_j)_+ = (nS(I)/4)^k(2\ln n)^{\frac{k}{2}}\sum_{i_1,\cdots,i_{k}\in\Lambda(I)\}\prod_{j=1}^k({m}^*_{i_j}-G_n(x_j)/S(I))_+ = (nS(I)/4)^k(2\ln n)^{\frac{k}{2}}|\Sigma_k(G_n(x_1)/S(I),\cdots,G_n(x_k)/S(I))|.
\displaystyle\sum_{0\leq \ell\leq k}f^{(n-\ell,\ell)}=\displaystyle\sum_{0\leq \ell\leq k-1}f^{(n-\ell,\ell)}+f^{(n-k,k)}&={\begin{pmatrix}n k-1 \end{pmatrix}}+\frac{n!(n-2k+1)}{(n-k+1)!k!}&={\begin{pmatrix}n k \end{pmatrix}}.
\eta_1 = \frac{\alpha_0}{r}- \frac{3T+1}{2r}+ \frac{t+x}{2r}, \eta_2 = \frac{\beta_0}{r}- \frac{3T+1}{2r}+ \frac{3t+x}{2r},\eta_3 = \eta_2 - \frac{1}{r}, \eta_4 = \frac{\alpha_0-\beta_0}{r}- \frac{x}{r},\eta_5 = \eta_4+\frac{1}{r}.
q_d(k) =\left[\prod_{j=1}^{d-1}\left( 1 + \gamma_j \, \omega\left( \left\{\frac{k Y_j z_j}{N}\right\}\right) \right)\right]\left[\prod_{j=d+1}^{s}\left( 1 + \gamma_j \, \omega\left( \left\{\frac{k z_j^0}{N}\right\}\right) \right)\right] .
([M, S])_j^{j-\pi(\ell)}(\ell)= \sum_{\ell_1+\ell_2=\ell}M_j^{j-\pi(\ell_1)}(\ell_1) S_{j-\pi(\ell_1)}^{j-\pi(\ell)}(\ell_2)- S_j^{j-\pi(\ell_1)}(\ell_1) M_{j-\pi(\ell_1)}^{j-\pi(\ell)}(\ell_2)\, .
Q_\mathfrak{g}(x_1\wedge\cdots\wedge x_k) :=\sum_{1\leq i<j\leq k}(-1)^{i+j-1}[x_i,x_j]_\mathfrak{g}\wedge x_1\wedge\cdots \widehat{x}_i\cdots \widehat{x}_j\cdots\wedge x_k,
P\lbrace \max_{0\le k \le n}S_k > \beta, S_n = x\ |\S_1 = 1\rbrace = P\lbrace \max_{0\le k \le n-1}S_k > \beta-1, S_{n-1}= x-1 \rbrace & = P\lbrace S_{n-1}= 2\beta-x+1\rbrace = P\lbrace S_n = 2\beta-x\ |\S_1 = -1\rbrace,
\begin{aligned}\, 2 K_1^{\frac{p-1}{p-2+m}}\frac{C^{m-1}m (N-1) k \coth(k)}{a(m-1)}(\tau+t)^{\alpha m-\beta}\leq \, C^{\frac{(p-1)(m-1)}{p-2+m}}\left(\frac{C^{m-1}m}{2 a^2(m-1)^2}\right)^{\frac{p-1}{p-2+m}}(\tau+t)^{\frac{\alpha p(m-1)+ (\alpha m - 2\beta )(p-1)}{p-2+m}}\forall t > 0 \, .\end{aligned}
h_f(f_0,g_0)=\frac{\left|A(e^{i\theta})g_0(\theta)+\sum\limits_{j=0}^{\infty}((\bold{B}^0)^{-1}\bold{R}^0\bold{a})_je^{ij\theta}\right|^2}{(f_0(\theta)+g_0(\theta))^2},
\mathrm{g h}( \varphi_{u}) = \mathrm{g h}( \varphi_{\xi}) = 0 , \quad \mathrm{g h}( c_{\xi}) = \mathrm{g h}( c_{u}) = 1 , \quad \mathrm{g h}( \Phi^{*}) = - \mathrm{g h}( \Phi ) - 1 .
\begin{array}{rcl}\phi = \displaystyle\phi_1+i\phi_2 =\sqrt{2}\frac{dg/dz}{1-\bar{g}g}, \omega = \displaystyle\omega_\mu dx^\mu = \frac{i}{2}\Big( \frac{g d\bar{g}- \bar{g}dg}{1-\bar{g}g}\Big), \end{array}
\Psi_{k}({\bf x}_{1}, i_{1}, M_{1}; \cdots ;{\bf x}_{k}, i_{k}, M_{k}) = \Psi_{k}({\bf x}_{k}, i_{k}, M_{k};{\bf x}_{1}, i_{1}, M_{1}; \cdots ;{\bf x}_{k - 1}, i_{k - 1}, M_{k - 1}) ,
M \cos(\tau (t^2 S)^{1/2}P) P M^* = M_0 \cos (\tau (t^2 M_0 \widehat{S}M_0)^{1/2}) M_0 \widehat{P}, & M (t^2 S)^{-1/2}\sin(\tau (t^2 S)^{1/2}P) P M^* = M_0 (t^2 M_0 \widehat{S}M_0)^{-1/2}\sin (\tau (t^2 M_0 \widehat{S}M_0)^{1/2}) M_0 \widehat{P}.
\begin{array}{l l l}{{X_{3 7_{1}}= \gamma_{\omega_{2}}^{\frac{1}{2 n}},}}&{{X_{3 7_{2}}= \gamma_{\omega_{1}}^{- \frac{1}{2 n}}\gamma_{\omega_{2}}^{- \frac{1}{2 n}},}}&{{X_{3 7_{3}}= \gamma_{\omega_{1}}^{\frac{1}{2 n}}.}}\\ \end{array}
F_{x}\left( f^{0},f\right) +F_{y}\left( f^{0},f\right) f_{x}\left(f^{0}\right) =0, F_{x}\left( f,f^{2}\right) +F_{y}\left( f,f^{2}\right) f_{x}\left(f\right) =0, & \cdots F_{x}\left( f^{j-1},f^{j}\right) +F_{y}\left( f^{j-1},f^{j}\right)f_{x}\left( f^{j-1}\right) =0, & \cdots
\forall r\in [s,t], \widetilde{I}_{r}^N := \frac{1}{N}\sum_{i=1}^N \int_s^r \nabla e^{(t-u)\Delta}V^N(X_u^{i,N}-x) \cdot dW^i_u \, , \forall r\in [0,s],\widetilde{II}_{r}^N := \frac{1}{N}\sum_{i=1}^N \int_0^r \nabla e^{(s-u)\Delta}\left[e^{(t-s)\Delta}V^N(X_u^{i,N}-x)- V^N(X_u^{i,N}-x)\right] \cdot dW^i_u \, .
F^{-}(x) =\left(\begin{array}{cc}1 - 1 & - 1 2\end{array}\right),\; F^{+}(x) =\left(\begin{array}{cc}2 1 & 1 1\end{array}\right),\; \Lambda(x) ={\Lambda_{0}(x) =}\left(\begin{array}{cc}\frac{x - i}{x + i}0 & 0 1\end{array}\right).
\frac{p_{02}}{p_{01}}=(\frac{\frac{\gamma+1}{2}M_{1}^{2}}{1+\frac{\gamma-1}{2}M_{1}^{2}})^{\frac{\gamma}{\gamma-1}}(\frac{1}{\frac{2\gamma}{\gamma+1}M_{1}^{2}-\frac{\gamma-1}{\gamma+1}})^{\frac{1}{\gamma-1}}
\left\langle \alpha_{1}, \alpha_{2}, . . . , \alpha_{2 N}, t \right| U_{L}\left| \alpha_{1}^{\prime}, \alpha_{2}^{\prime}, . . . , \alpha_{2 N}^{\prime}, t \right\rangle = R_{\alpha_{1}\alpha_{2}}^{\alpha_{2}^{\prime}\alpha_{3}^{\prime}}\: R_{\alpha_{3}\alpha_{4}}^{\alpha_{4}^{\prime}\alpha_{5}^{\prime}}\: . . . \: R_{\alpha_{2 N - 1}\alpha_{2 N}}^{\alpha_{2 N}^{\prime}\alpha_{1}^{\prime}}
\lim_{t \to 0}u(t,x) = \lim_{t \to 0}t^{-\frac{1}{\alpha}}f \Bigl( \frac{x}{\sqrt t}\Bigr)= |x|^{-\frac{2}{\alpha}}\lim_{r\to\infty}r^{\frac{2}{\alpha}}f \Bigl( \frac{x}{|x|}r \Bigr)= \omega (x) |x|^{-\frac{2}{\alpha}},
g_0 = \begin{bmatrix}\sqrt{x_0}0 \sqrt{-y_0}\frac{1}{\sqrt{x_0}}\end{bmatrix}= \begin{bmatrix}\sqrt{x_0}0 0 \frac{1}{\sqrt{x_0}}\end{bmatrix}\cdot \begin{bmatrix}1 0 \sqrt{-{x_0}{y_0}}1 \end{bmatrix}.
(M_r{V})_S= &\sum_{a,b}\left [\binom{a}{2}+ \binom{b}{2}\right ] \left [ \binom{|S|-2}{a-2}\binom{r|S|-2}{b}+ \binom{|S|-2}{a}\binom{r-|S|-2}{b-2}\right . &\left . -\binom{|S|-2}{a-1}\binom{r-|S|-2}{b-1}-\binom{|S|-2}{a-1}\binom{r-|S|-2}{b-1}\right ] &=P_1+P_2-2P_3
X_{M_{\mathbf i'}}(x_1, \dots,x_{i_1}',\dots, x_n) = x_{\mathbf i'}(x_1, \dots,x_{i_1}',\dots x_n) = x_\mathbf i(x_1, \dots, x_{i_1}, \dots, x_n)\in \mathcal A(B).
\begin{cases}D_{r_i r_i}g= 8 \sin^2 \frac{\pi}{n}+4n \cos^2 \frac{\pi}{n}, D_{r_i r_j}g= (8-4n) \sin^2 \frac{\pi}{n}-2n \cos^2 \frac{\pi}{n},\ \mbox{for}\ |j - i|=1, D_{r_i r_j}g= 8 \sin^2 \frac{\pi}{n}, \ \mbox{otherwise}.\end{cases}
w^{S_0 \oplus S_1}_{(a^-, i)}= 1 + \beta \sum_{j \in N}p_{ij}g^{S_0 \oplus S_1}(1, j) - \beta g^{S_0 \oplus S_1}_{(0, i)}= 1 + \beta \sum_{j \in S_1}p_{ij}g^{S_1}_j = w^{S_1}_i + \beta g^{S_1}_i = w^{S_1}_i/(1-\beta),
\#\xi(r_i)=\sum_{\omega \in \Omega_i}\#\xi(\gamma(v_{i,\omega};y_{i,\omega,1},y_{i,\omega,2}))= \sum_{\omega\in \Omega_i}\gamma(v_{i,\omega};\#\xi (y_{i,\omega,1}),\# \xi( y_{i,\omega,2})),
\partial_t P_N\partial_xP_N \le a'(t)\,N^{5/4}\times \;\;\Big\{\frac54 aN^{1/4}+\frac{5^2}{4^2}a^2N^{2/4}(x-N)+\frac{5^3}{4^3}a^3\frac{N^{3/4}}{2}(x-N)^2\Big\}\,e^{2aN^{5/4}}.
\hat{\sigma}_{ABM|Z=\mathbf{z}}= \hat{D}_A\left(\frac{\lambda\mathbf{z}}{\sqrt{\eta}}\right)\hat{D}_B\left(\frac{(1-\lambda)\mathbf{z}}{\sqrt{1-\eta}}\right)\hat{\rho}_{ABM}{\hat{D}_A\left(\frac{\lambda\mathbf{z}}{\sqrt{\eta}}\right)}^\dag{\hat{D}_B\left(\frac{(1-\lambda)\mathbf{z}}{\sqrt{1-\eta}}\right)}^\dag
\lim_{\varepsilon \downarrow 0}\big\| P_\varepsilon\big[e^{- \frac{\phi}{2\varepsilon}}\big]^{-2\varepsilon}\big\|_{L^{-\frac{q_\varepsilon}{2\varepsilon}}(\gamma)}=\int_{\mathbb{R}^n}e^{\lim_{\varepsilon \downarrow 0}u^\varepsilon}\, d\gamma\le \int_{\mathbb{R}^n}e^{Q_1\phi(x)}\, d\gamma.
\left\{\frac{d^{2}}{d \varrho^{2}}+ \alpha^{2}\varrho^{2 ( \alpha - 1 )}\left[ E - V ( \varrho^{\alpha}) \right] - \frac{\alpha^{2}( l + \nu )^{2}- 1 / 4}{\varrho^{2}}\right\}\widetilde{u}( \varrho ) = 0 \; ,
\frac{1}{4 \sin^2[(x_i - x_j)/2]}= \frac{z_i \, z_j}{z_{ij}\,z_{ji}}= \frac{-z_i\,z_j}{(z_i-z_j)^2}= \frac{1}{|z_i - z_j|^2}\, , z_j = \mathrm{e}^{\mathrm{i}\mspace{1mu}x_j}\, ,
\hat{y}_- \mkern-1mu=\mkern1.5mu \hat{y}_+ \mkern-1mu=\mkern1.5mu -y_1^{\textsc{d}}-y,\;\;\;\;\;\;\;\;\;y_- \mkern-1mu=\mkern1mu y_+ \mkern-1mu=\mkern1mu -y_1^{\textsc{d}}-y_1^{\textsc{g}}.
\frac{1}{1-c^2b_1^hA_j^v}+c^2b_1^h\sum_{i=j}^{m_1+m_2}\frac{(a_i^v)^2}{(1-c^2b_1^hA_i^v)(1-b_1^hA_{i+1}^v)}=\frac{1}{1-c^2b_1^hb_2^v},
\left[G_n(v)\psi \right]= n^{-n}P^{(n)}\int_{\mathbb{R}^n}\left[G\left(\sum_{i=1}^nx_ie_i \right)\psi \right] \pmb{1}_n\left(\left|\sum_{i=1}^n x_ie_i \right|_{H^{1,2}}\right) &\times \rho\left(n (\hat{P}^{(1)}v-x_1)\right)\cdot \cdot \cdot \rho\left(n (\hat{P}^{(n)}v-x_n)\right)\,{\rm d}x_1\ldots{\rm d}x_n.
E_{\tilde \varphi_c}^Q \left ( \frac{f_{j_n}}{\varphi_c (Z_n)}\right) \ = \E_{\tilde\varphi_c}^Q \left ( f_{j_n}E_{Z_{j_n}}\left ( \frac{1}{\varphi_c (Z_{n-j_n})}\right)\right).
\begin{array}{l c l}{{{\widehat \Sigma}}}&{{=}}&{{\oplus_{p \leq q}{\widehat \Sigma}^{( p , q )}}}\\{{}}&{{=}}&{{\oplus_{p \leq q}\oplus_{m \in Z}{\widehat \Sigma}_{m}^{( p , q )}.}}\\ \end{array}
H^s(\mathcal{E}) := \{v \in L_2(\Omega): \forall i \in \{1,\ldots,N\}\; \; v_i := v|_{\Omega_i}\in H^s(\Omega_i)\}\subset L_2(\Omega), H^s(\mathcal{T}_h) := \{v \in L_2(\Omega): \forall \tau \in \mathcal{T}_h \; \; v|_{\tau}\in H^s(\tau)\}\subset L_2(\Omega).
\mathcal{V}_{j}=\{\overline{a}_{j}^{0}+\sum_{l=1}^{q_{j}}v_{j}^{l}\overline{a}_{j}^{l}:\Vert (v_{j}^{1},\dots ,v_{j}^{q_{j}})\Vert \leq 1\}\times \lbrack\underline{b}_{j},\overline{b}_{j}],
Q_g =-\frac{1}{2(n-1)}\Delta_gR_g - \frac{2}{(n-2)^2}|\mathrm{Ric}_g|^2_g + \frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}R_g^2, P_gu =\Delta^2u + \mathrm{div}_g\left(\left(\frac{4}{n-2}\mathrm{Ric}_g - \frac{n^2-4n+8}{2(n-1)(n-2)}R_g\: g \right)(\nabla u,\cdot) \right) + \frac{n-4}{2}Q_gu.
\sup_{x\in G\setminus \{e\}}\sup_{r>0}\frac{\| \delta_{r}^{A}x\|}{\| \delta_{r}^{B}x\|}= \sup_{x\in G\setminus \{e\}}\sup_{r>0}\frac{\| \delta_{1/r}^{A}x\|}{\| \delta_{1/r}^{B}x\|}\leq R,
|\sin \angle \left( (\xi_1, \, \eta_1), \, (\xi_2, \eta_2) \right)| =\frac{|\xi_1 \eta_2 - \xi_2 \eta_1|}{|(\xi_1,\eta_1)| | (\xi_2, \eta_2)|}\geq \frac{|\xi_1 \eta - \xi \eta_1|}{4 |(\xi_1,\eta_1)| |(\xi, \eta)|}= 2^{-2}|\sin \angle \left( (\xi, \, \eta), \, (\xi_1, \eta_1) \right)| > 2^{-22},
\forall J \geq 1, \lim_{n \rightarrow \infty}\left [ \left \| \sum_{j = 1}^J U^j_n\right \|_{S(0,\theta_n)}^{2(N+1)/(N-2)}- \sum_{j = 1}^J \left \| U^j_n\right \|_{S(0,\theta_n)}^{2(N+1)/(N-2)}\right ] = 0.
R_{i j}^{\pm}= \frac{1}{\sqrt{2}}[ 1 \pm ( i j ) ] ~ , ~ ~ R_{i l}^{\pm}= \frac{1}{\sqrt{2}}[ 1 \pm ( i l ) ] ~ , ~ ~ R_{j l}^{\pm}= \frac{1}{\sqrt{2}}[ 1 \pm ( j l ) ] ~ ,
\psi_{n,k}(z) =O(z^{-N(n,k)})=O(z^{-Z(n,k-1)+Z(n,k)}), \psi_{n+1,k}(z) =O(z^{-N(n+1,k)})=O(z^{-Z(n+1,k-1)+Z(n+1,k)-1}),
-H(\rho^\varepsilon_0)(z_l) \geq \frac{1}{2R}\phantom{xx}\phantom{xx}H(\rho^\varepsilon_0)(z_l) + H(\rho^\varepsilon_0)(z_r)= \frac{2\varepsilon\rho_0}{R^2-\varepsilon^2\rho_0^2}\leq \frac{4\varepsilon\rho_0}{R^2}
\frac{\partial \log{f_i}}{\partial t}= \frac{1}{V^{2}}\Big\{(\log f_i)^{\prime\prime}+\Big( (\log f_i)^{\prime}\Big)^{2}-\frac{1}{2}(\log f_i)^{\prime}(\log V^{2})^{\prime}\Big\}-\frac{1}{3}\mathcal{T}
\delta = \epsilon^{0}\, \underline{{\epsilon}}_{0}+ \epsilon^{+}\, \underline{{\epsilon}}_{+}+ \epsilon^{-}\, \underline{{\epsilon}}_{-}+ \alpha^{I}\, \underline{{\alpha}}_{I}+ \beta_{I}\, \underline{{\beta}}^{I}+ \hat{\alpha}{}^{I}\, \underline{{{\hat{\alpha}}}}_{I}+ \hat{\beta}_{I}\, \underline{{{\hat{\beta}}}}{}^{I}+ \omega^{i}\, \underline{{\omega}}_{i}\ .
|[ Q_{|\textup{\textbf{m}}|}^{\textup{\textbf{F}}}\Phi_k]_{\nu}|=\left|\frac{1}{2\pi i}\int_{\Gamma_{\rho_2-2\delta}}\frac{Q_{|\textup{\textbf{m}}|}^{\textup{\textbf{F}}}(t) \Phi_k(t) \Phi'(t)}{\Phi^{\nu+1}(t)}dt \right|\leq c_{12}\frac{(\rho_2-\delta)^k}{(\rho_2-3\delta)^{\nu}},
\left({}^{c}D_{0+}^{0,\alpha',\beta,\beta',\gamma}f\right)(x)=\left({}^{c}D_{0+}^{\gamma,\alpha'-\gamma,\beta'-\gamma}f\right)(x),\ \ \ \ \left({}^{c}D_{-}^{0,\alpha',\beta,\beta',\gamma}f\right)(x)=\left({}^{c}D_{-}^{\gamma,\alpha'-\gamma,\beta'-\gamma}f\right)(x).
E_iG^{\mathrm{low}}(b).v_{r\rho}= [ \varphi_i^{r\rho}(b)+1]_iG_{r\rho}^{\mathrm{low}}(\tilde{e}_i\pi_{r\rho}(b)) &+\sum_{\substack{\tilde{b}\in B(\infty) \varphi_i^{r\rho}(b)+1<\varphi_i^{r\rho}(\tilde{b})}}\widehat{E}_{\pi_{r\rho}(b), \pi_{r\rho}(\tilde{b})}^{(1), i}G_{r\rho}^{\mathrm{low}}(\pi_{r\rho}(\tilde{b})),
\frac{\nu}{2}\int_\Omega|\mathbf{b}_{x}|^2dx+ \iint_{Q_t}|\mathbf{b}_t|^2 dxdt =\frac{\nu}{2}\int_\Omega|\mathbf{b}_{0x}|^2dx+\iint_{Q_t}\big[\mathbf{w}_x - (u\mathbf{b})_{x}\big]\cdot\mathbf{b}_t dxdt.
T_1 \, \Theta(z+\iota(P-A_r)) \, \Theta(z+\iota(A_p-O)) + T_2 \, \Theta(z+\iota(P-O)) \, \Theta(z+\iota(A_p-A_r))\, = T_3 \, \Theta(z+\iota(P+A_p-A_r-O)) \, \Theta(z) \,
[G^{ij}]=\left[\begin{array}{ccc}\frac{1}{R^2}0 0 0 \frac{1}{(L+R\cos\theta)^2}-\frac{\sin\theta}{(L+R\cos\theta)^2}0 -\frac{\sin\theta}{(L+R\cos\theta)^2}\frac{M}{I}+\frac{\sin^2\theta}{(L+R\cos\theta)^2}.\end{array}\right]
\begin{pmatrix}x_{*}^{H}X_{\bot}^{H}\end{pmatrix}T(\lambda_{*})\begin{pmatrix}x_{*}X_{\bot}\end{pmatrix}=\begin{pmatrix}0 x_{*}^{H}T(\lambda_{*})X_{\bot}0 L(\lambda_{*}) \end{pmatrix}.
\left\{\varphi (x),\pi (x^{\prime})\right\}=(1-{\cal P})(x,x^{\prime})\,,\qquad \left\{P,\varphi \right\}=\left\{P,\pi \right\}=0\,,\qquad\left\{\varphi ,\varphi \right\}=\left\{\pi ,\pi \right\}=0\,.
\frac{4 m}{\hbar^{2}}U = \sigma_{1}\bar{\beta}_{1 1}| \phi |^{4}+ ( \sigma_{1}+ \sigma_{2}) \bar{\beta}_{1 2}| \phi |^{2}| \chi |^{2}+ \sigma_{2}\bar{\beta}_{2 2}| \chi |^{4}.
\psi_{\alpha}(x^{\prime})=(\frac{m\omega}{\pi\hbar})^{\frac{1}{4}}e^{\frac{i}{\hbar}\langle\hat{p}\rangle_{\alpha}(x^{\prime}-\frac{\langle\hat{x}\rangle_{\alpha}}{2})-\frac{m\omega}{2\hbar}(x^{\prime}-\langle\hat{x}\rangle_{\alpha})^{2}}.
\begin{array}{l l}{{d s_{6}^{2}=}}&{{e^{2 \lambda}\left( d y^{2}+ d z^{2}\right) + e^{2 \varphi}\left( - d \tau^{2}+ \tau^{2}d \Omega_{3 , - 1}^{2}\right) ,}}\\{{H_{e}=}}&{{i d \left( e^{2 \lambda}d y \wedge d z \right) .}}\\ \end{array}
\Delta_q(u\cdot\nabla v)= \sum_{|q-p|\leq 2}\Delta_q(u_{\leq{p-2}}\cdot\nabla v_p)+\sum_{|q-p|\leq 2}\Delta_q(u_{p}\cdot\nabla v_{\leq{p-2}}) &+\sum_{p\geq q-2}\Delta_q(\tilde u_p \cdot\nabla v_p),
\int \frac{\mathrm{d}U}{\left[C_2 + C_1 U + U^2\left(\kappa - \gamma U^{n}\right)\right]^{1/(m+1)}}= \pm\xi + \xi_0,\kappa\equiv \frac{(m+1)}{2m}c,\gamma\equiv \frac{(m+1)}{(n+1)(n+2)m},
\omega_{k,l}(F;\delta_1,\delta_2) :=\displaystyle\sup\left\{\left|\displaystyle\sum_{\nu=0}^k\sum_{r=0}^{l}(-1)^{\nu+\mu}{k\choose \nu}{l\choose \mu}F(x+\nu\cdot h_1, y+\mu\cdot h_2)\right|: \right. \left. (x,y),(x+kh_1,y+lh_2)\in[0,1]^2, |h_i|\leq\delta_i,i=1,2 \right\},
^{+}S_2=\frac{i}{2}{^+\tau}\int_M e F^{\mu \nu i}F^{\alpha \beta j}g_{ab}\,{\cal R}_{\mu \nu}^{ab}\,{\cal R}_{\alpha \beta}^{cd}\,\epsilon_{abcd}-\frac{1}{2}{^+\tau}\int_M e F^{\mu \nu i}F^{\alpha \beta j}g_{ij}\,{\cal R}_{\mu\nu}^{ab}\,{\cal R}_{\alpha \beta}^{cd}\delta_{ab,cd}\, .
\mathfrak{E}_h^k(s) = \frac{(2\pi)^{\frac{k+1}{2}}(\overline{\rho_\infty^k(h, \frac{k+1}{2}) + \rho_0^k(h, \frac{k+1}{2})})}{(2 h)^{s-1}}\frac{\Gamma(s - \frac{k}{2}- \frac{1}{2})}{\Gamma(s - \frac{k}{2})}.
m_{(1,0;1,1)}(x;\theta)=\theta_1\theta_2(x_{1}-x_2)x_3x_4+\theta_1\theta_3(x_{1}-x_3)x_2x_4+\theta_1\theta_4(x_{1}-x_4)x_2x_3 \phantom{m_{(1,0;1,1)}(x;\theta)=}{}+\theta_2\theta_3(x_{2}-x_3)x_1x_4+\theta_2\theta_4(x_{2}-x_4)x_1x_3+\theta_3\theta_4(x_{3}-x_4)x_1x_2.
\frac{{{\partial^2}{{\cal F}_i}\left({\alpha ,{\pmb \rho}}\right)}}{{\partial{\alpha^2}}}= \frac{{2{Z_i}{{\left({{X_i}{W_i}}\right)}^2}}}{\left[{{{\left({\alpha{X_i}{W_i}+{X_i}-{Y_i}{Z_i}+{Z_i}}\right)}^2}+ 4{Y_i}Z_i^2}\right]^{3/2}}
\mathcal{C}_{X_1}[\Psi(P_{X_1},P_{X_2})]:= \sup_{P_{X_1U}}\left\{h(X_1+X_2+Z_1+Z_2|U)\right. +h(X_1+Z_1|U) \left.-(u+1)h(X_1+Z_1+Z_2|U)\right\}
\displaystyle\sum\limits_{k=0}^{N-1}A_{0}\left( x,v_{k}\right) \psi_{k}^{\prime}\left( x_{0}\right) +F_{2}\left( x,\displaystyle\sum\limits_{m=0}^{N-1}\nabla v_{m}\left( x\right) \psi_{m}\left(x_{0}\right) ,\displaystyle\sum\limits_{k=0}^{N-1}\nabla v_{k}\left(x\right) \psi_{k}^{\prime}\left( x_{0}\right) \right) =0,
\Phi_{1}= 2 \sum_{n / 1}^{\infty}D_{n}\sin ( \omega_{n}\tau + \varphi_{n}) \left[ \tan \lambda \sigma \cos \left( \omega_{n}\sigma + \frac{n \pi}{2}\right) - \, \frac{\omega_{n}}{\lambda}\sin \left( \omega_{n}\sigma + \frac{n \pi}{2}\right) \right] ,