harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
r_{\ell}\leq \frac{e^{-( 2L-55 \varepsilon) q_{n}}}{\max (|\ell|, 1)}\max \left(r_{\ell-1}, r_{\ell+1}\right) \times \begin{cases}\max \left(|\ell|, e^{\delta_{n}q_{n}}\right), \beta_{n}\geq \delta_{n}+200 \varepsilon e^{\beta_{n}q_{n}}, \beta_{n}<\delta_{n}+200 \varepsilon\end{cases}.
R_k =\mathtt{I}(x_k;y_{_k}) &=\log\bigg(1+\frac{\eta_k p_k|h_{kk}|^{2}}{\sigma^{2}_k+\eta_k(\varrho^{2}_k+\sum_{\substack{j=1 j\neq k}}^{K}p_j|h_{kj}|^{2})}\bigg),\ \forall k\in\mathcal{K}, E_k =(1-\eta_k)\bigg(\sum_{j=1}^{K}p_j|h_{kj}|^{2}+\varrho^{2}_k\bigg),\forall k\in\mathcal{K},
\left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right)_{\xi \eta}= \left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right)_{\xi}\otimes \left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right)_{\eta}
V(\hat{q}) = \sum_{A=1}^{\frac{N}{2}}\exp \left \{\sqrt{\frac{2}{N}}K_{1}+\frac{4\lambda}{\sqrt{N}}\sum_{\alpha=1}^{\frac{1}{2}N-1}C_{\alpha}\hat{q}_{\alpha}\sin \varphi_{\alpha}\cos [(2A-1)\varphi_{\alpha}+ \frac{\pi}{4}]\right \}
\mathbf{\Sigma}_{l}=\frac{w_{l}}{\hat{\mu}}\left(\hat{\mathbf{\Omega}}_{l}^{-1}-\left(\hat{\mathbf{\Omega}}_{l}+\mathbf{H}_{l,l}^{\dagger}\hat{\mathbf{\Sigma}}_{l}\mathbf{H}_{l,l}\right)^{-1}\right)l=1,\ldots,L,
L_{i j}= \left( \begin{array}{c c c c}{{0}}&{{k_{1}- k_{3}}}&{{- k_{1}+ k_{2}}}&{{k_{3}- k_{2}}}\\{{- k_{1}+ k_{3}}}&{{0}}&{{k_{1}}}&{{- k_{3}}}\\{{k_{1}- k_{2}}}&{{- k_{1}}}&{{0}}&{{k_{2}}}\\{{- k_{3}+ k_{2}}}&{{k_{3}}}&{{- k_{2}}}&{{0}}\\ \end{array}\right)
O_d =\bigl\{\bigl((x,d),(y,d)\bigr):(x,y)\in V_{n_d}\bigr\}\cup\bigl\{\bigl(-(x,d),-(y,d)\bigr):(x,y)\in V_{n_d}\bigr\}O_d =\bigl\{\bigl((x,d),(y,d)\bigr):(x,y)\in Z\times Z\bigr\}\cup\bigl\{\bigl(-(x,d),-(y,d)\bigr):(x,y)\in Z\times Z\bigr\}
\sum_{k \geq l}\binom{k}{l}\,\frac{I(q^{-k})}{k}=\frac{1}{l}\sum_{k \geq l}\sum_{m \geq 1}\frac{\mu(m)}{m}\,\binom{k-1}{l-1}\,\log\left(\frac{1}{1-q^{1-km}}\right) &= \frac{1}{l}\sum_{m,n\geq 1}\frac{\mu(m)}{nm}\,q^n\,\sum_{k\geq l}\binom{k-1}{l-1}q^{-k(mn)}&=\frac{1}{l}\sum_{D \geq 1}\frac{|\mathfrak{I}_D|}{(q^D-1)^l}
Z_{\mathrm{U E}}= \int_{N \times N \, \mathrm{\scriptsize ~ h e r m i t e}}\! \! \! \! \! \! \! \! \! \! d M \, \mathrm{e}^{- N \, \mathrm{t r}\, V ( M^{2})}\, | \mathrm{d e t}\, M |^{\alpha}
{\bf v}_{0}= \frac{\sqrt 2}l \left[ \begin{array}{c}{\, 1}{\, \, a^{2}\,}\end{array}\right] , \, \, \,{\bf v}_{i}= \frac{{\sqrt 2}a}l \left[ \begin{array}{c}{1 / J_{2}( j_{i})}{- 1 \,}\end{array}\right] \ .
[ \Delta T ]_{1}={\frac{2}{\mu_{1}V_{1}^{2}}}\ln \Big [ 1 -{\frac{4 \mu_{1}\mu_{2}V_{1}V_{2}}{( V_{1}- V_{2})^{2}+ ( \mu_{1}V_{1}+ \mu_{2}V_{2})^{2}}}\Big ] .
p|_2 = \rho_2 g \left(\sum_{j=2}^N D_j + b - z + \frac{\rho_1}{\rho_2}\D_1\right)= \rho_2 g \left(\sum_{j=1}^N D_j + b - z + \frac{\rho_1 - \rho_2}{\rho_2}\D_1\right).
\frac{\partial}{\partial \alpha}\int{\cal D}v \; \delta^{n}\left( \int v d \tau \right) F_{q G}[ v , I , \alpha ] = \int{\cal D}v \; \delta^{n}\left( \int v d \tau \right) \frac{\partial}{\partial \alpha}F_{q G}[ v , I , \alpha ] \; .
S = \int d^{1 1}x \sqrt{- g_{1 1}}[{\frac{1}{4}}R -{\frac{1}{2^{.}4 !}}\hat{F}_{M_{1}. . . M_{4}}^{( 4 )}\hat{F}^{( 4 ) M_{1}. . . M_{4}}] + \int{\frac{1}{3}}A^{( 3 )}\wedge d A^{( 3 )}\wedge d A^{( 3 )}
| \Gamma_{\alpha}^0(x,y)-L^0(x,y) | \le C \bigg ( \min \big\{\frac{1}{|\alpha| |x|^2}, \frac{1}{|\alpha|^{\frac{1}{2}}|x|}\big\}+ |y| \min \big\{\frac{1}{|\alpha| |x|^3}, \frac{1}{|x|}\big\}+ \frac{|y|^2}{|x|^2}\bigg )\,, |x| > 2 |y|\,.
\mathbb H_{2,\mathbb K}^2=\mathbb H_{2,\mathbb K}^{2,\Theta}=\mathbb K[\Theta_{2,0,0},\Theta_{1,1,0},\Theta_{1,0,1},\Theta_{0,2,0},\Theta_{0,1,1},\Theta_{0,0,2}].
\big(I\otimes |k\rangle\langle k|\big)\mathrm{G}\big(P_\psi\otimes|0\rangle\langle 0|\big)\mathrm{G}^\ast \big(I\otimes |k\rangle\langle k|\big)=E_kP_\psi E_k^\dag\otimes |k\rangle\langle k|,
{\frac{\partial W}{\partial \theta}}= \int{\cal D}\bar{K}^{4 i}{\cal D}\bar{K}\raisebox{0 . 9 7 e x}{\mathrm{\scriptsize{4 i}}}\raisebox{- 0 . 3 6 e x}{\mathrm{\scriptsize{, \! \theta}}}{\cal D}\bar{t}_{4}{\cal D}\bar{t}_{4 , \theta}{\frac{\partial \bar{W}}{\partial \theta}}
\int_0^\omega \Big(\dot x_*(t)\dot x(t)dt +\frac{\partial L}{\partial x}\big(x_*(t),\phi_*(t),\dot x_*(t),\dot \phi_*(t)\big)x(t)\Big)dt=0, \int_0^\omega\Big( (1+x_*^2(t))\dot\phi_*(t)\dot\phi(t)dt +\frac{\partial L}{\partial \phi}\big(x_*(t),\phi_*(t),\dot x_*(t),\dot \phi_*(t)\big)\phi(t)\Big)dt=0.
\mathcal{F}^{\omega,\{U_i\}_{i=0,\dots,K}}=\left\{\psi:=(\psi_0,\psi_1,\dots,\psi_K)|\;\psi_i:\Phi_i(U_i)\rightarrow\psi_i(\Phi_i(U_i)) \right. &\qquad\qquad\qquad\qquad\left.i=0,\dots,KDF^{\omega}(\psi,T\psi)\right.\}.
\left\{{{Det^{\prime}\left[- (d^2/d\tau^2) + V^{\prime \prime}[x_c(\tau)] \right]}\over{Det \left[- (d^2/d\tau^2) + \omega^2 \right]}}\right\}^{- 1/2}sqrt{{{S_{eo}}\over{2 \pi}}}\d\tau_{c}
\int d x^{4}\phi_{[ \omega , l , m_{1}, m_{2}]}^{\ast}( x ) \phi_{[ \omega^{\prime}, l^{\prime}, m_{1}^{\prime}, m_{2}^{\prime}]}( x ) = ( 2 \pi ) \delta ( \omega - \omega^{\prime}) \delta_{l l^{\prime}}\delta_{m_{1}m_{1}^{\prime}}\delta_{m_{2}m_{2}^{\prime}}
( A_{\mu}, \psi_{\mu}, \bar{\psi}_{\mu}, \phi , V_{\mu}, \bar{\phi}, \eta , \bar{\eta}, Y , \chi_{\mu \nu}, B_{\mu \nu}) \rightarrow ( A_{\mu}, \bar{\psi}_{\mu}, \psi_{\mu}, \phi , - V_{\mu}, \bar{\phi}, \bar{\eta}, \eta , - Y , \tilde{\chi}_{\mu \nu}, B_{\mu \nu}) .
\begin{array}{rcl}\displaystyle\mu_{g_s}(\hat x,t) = \displaystyle\frac{1}{(2t)^{k-1}}\,\int_{x_{k-1}-t}^{x_{k-1}+t}\cdots\int_{x_1-t}^{x_1+t}g_s(\tau_1,\ldots,\tau_{k-1})\,d\tau_1\cdots d\tau_{k-1}& &= \displaystyle \frac{1}{(2t)^{k-1}}\,\int_{x_{k-1}-t}^{x_{k-1}+t}\cdots\int_{x_1-t}^{x_1+t}f(\tau_1,\ldots,\tau_{k-1},s)\,d\tau_1\cdots d\tau_{k-1}\end{array}
d_{i,i}(x_{i,1}^3 - x_{i,2}^3) + \sum_{j=R+\omega+1}^S d_{i,j}(x_{j,1}^3+x_{j,2}^3-x_{j,3}^3-x_{j,4}^3) = 0 (\omega+1 \le i \le R)
e_1(W_{n+q+1})(0) &\le C(T) \int_0^T\big( \sum_{l=0}^{q}|| \mathcal{\mathbf{B}}_{n+l}^{\ast}U_{n+l}||_{G_{n+l}}^2 + ||\mathcal{\mathbf{B}}_{n+q+1}^{\ast}W_{n+q+1}||_{G_{n+q+1}}^2\big) dt .
S[x^\mu,p_\mu;\lambda]=\int_{\tau_i}^{\tau_f}d\tau\left[\dot{x}^\mu P_\mu-\lambda\phi\right]=\int_{\tau_i}^{\tau_f}d\tau\left[\dot{x}^\mu P_\mu-\frac{1}{2}\lambda\left[P^2+m^2\right]\right]\ .
E_1 =\left\langle \frac{\partial}{\partial x_1}+p_1\frac{\partial}{\partial z},\dots,\frac{\partial}{\partial x_n}+p_n\frac{\partial}{\partial z}\right\rangle,E_2 =\left\langle \frac{\partial}{\partial p_1},\dots,\frac{\partial}{\partial p_n}\right\rangle,
h_{2i+1}^j =\frac{(-1)^{j+i}(2j+1)!(-1)^{j-i}(2i)!!(4i+2)}{(2i+1)(2i)!!(2j-2i)!!(2i+2j+2)!!}=2^{1-2j}\binom{2j+1}{j-i},
||\tilde{\mathcal{A}}\mathcal{A}_3{\mathbf V}||^2_{s-1,\ast,t}\leq C(K)\Big( ||{\mathbf V}||^2_{s-1,\ast,t}+||\dot{{\mathbf U}}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s+1,\ast,t}\Big),
\frac{\det D(A)}{\det D (0)}\,=\, \prod_k\left \{\,\frac{v^{\dagger}_+(k\mid 0) v_+(k \mid A)}{\mid v^{\dagger}_+ (k\mid 0) v_+(k \mid A) \mid}\frac{v^{\dagger}_+(k\mid A) v_-(k \mid A)}{v^{\dagger}_+(k\mid 0) v_-(k \mid 0)}\frac{v^{\dagger}_-(k\mid A) v_-(k \mid 0)}{\mid v^{\dagger}_-(k\mid A) v_-(k \mid 0)\mid}\,\right \}\;,
{\rm Tr}=F_{c}+G_h=F_c-1=\frac{\Gamma}{(K+c)}\left(-\frac{K}{K+c}+\frac{(1-b)\Gamma c}{(1+c)(b+c)}\right)-1 &{\rm Det}=F_cG_h-F_hG_c=-\frac{\Gamma}{(K+c)}\left(-\frac{K}{K+c}+\frac{(1-b)\Gamma c}{(1+c)(b+c)}\right)+\frac{2\Gamma c^2}{(1+c^2)(K+c)}&{\rm Discr}=({\rm Tr})^2-{\rm 4Det}.
\hat{Q}_k(\mu)=\min_x \biggl\{\sum_{i\neq j_{k+1}}\tilde{f}^{k+1}_i(x) + (1-\mu)\left(f_{j_{k+1}}(z^{k}_{j_{k+1}}) + \langle g^{k}_{j_{k+1}}, x - z^{k}_{j_{k+1}}\rangle\right) + \mu\left(f_{j_{k+1}}(z^k_{j_k}) + \langle s_{j_{k+1}}^k, x-z^k_{j_k}\rangle\right) + \frac{1}{2}\big\|x-x^k\big\|_D^2\biggr\},
\gamma_n(\pi, \sigma,\tau^*)=\gamma_n(\pi, \sigma,\hat{\tau}^*) \leq \gamma_n(\pi, \hat{\sigma}_n,\hat{\tau}^*)=\hat{\gamma}_n\left(\Phi(\pi), \hat{\sigma}_n,\hat{\tau}^*\right) \leq w^*_\infty\left(\Phi(\pi)\right)+\varepsilon
\|f^{\star}- (1-\alpha_m)f_{m-1}- \alpha_mcv_fh_m \|^2 = (1-\alpha_m)^2\|f^{\star}- f_{m-1}\|^2 & - 2\alpha_m(1-\alpha_m)\langle f^{\star}- f_{m-1}, ch_mv_f - f^{\star}\rangle + \alpha^2_m\|f^{\star}- ch_mv_f\|^2.
\|\mathcal{K}(t,s)-\mathcal{K}(t,r)\|_{L(L_{p_0})}=\sup_{\|f\|_{L_{p_0}}=1}\left\|\int_{\mathbf{R}^{d}}(K(t,s,x-y)-K(t,r,x-y))f(y)dy\right\|_{L_{p_0}}\leq \sup_{\|f\|_{L_{p_0}}=1}\|f\|_{L_{p_0}}\int_{\mathbf{R}^{d}}|K(t,s,x)-K(t,r,x)|dx = \int_{\mathbf{R}^{d}}|K(t,s,x)-K(t,r,x)|dx.
\Theta_{\rm eb}(t)=\Theta_{\rm eb}(t,\tau_0)=\Big\{\theta\in\mathbb{R}^n : \textstyle{\sum_{i\in I_*^c}}\theta_i^2 \le t \sigma^2\big(1+|I_*|\log(\tfrac{en}{|I_*|})\big)\Big\},
U_{(0,\,r-1)}\equiv \frac{1}{2}(Q_{0})_{\alpha_{0}\beta_{0}}(Q_{1})_{\alpha_{1}\beta_{1}}(Q_{2})_{\alpha_{2}\beta_{2}}\cdots (Q_{r-1})_{\alpha_{r-1}\beta_{r-1}}\epsilon^{\beta_{0}\alpha_{1}}\epsilon^{\beta_{1}\alpha_{2}}\cdots\epsilon^{\beta_{r-1}\alpha_{0}}.
\mathrm{d}s^{2}= U_{I J}( x ) \, \mathrm{d}\vec{x}^{I}\cdot \mathrm{d}\vec{x}^{J}+ ( U^{- 1}( x ) )^{I J}\Big ( \mathrm{d}\varphi_{I}+ \vec{W}_{I K}( x ) \cdot \mathrm{d}\vec{x}^{K}\Big ) \Big ( \mathrm{d}\varphi_{J}+ \vec{W}_{J L}( x ) \cdot \mathrm{d}\vec{x}^{L}\Big ) \ .
\begin{aligned}F(r)=r^{\kappa}e^{-\sqrt{-E}r}L^{(2\kappa-1)}_{N_r}(2\sqrt{-E}r),~~~\kappa=2\sum_{s=1}^{N-1}J_s+N-\frac{1}{2}+\frac{1}{2}\sum_{s=1}^{N}\sqrt{1+4\lambda_s}, \end{aligned}
x^{\underline{n-j}}\, x^{\underline{n-k}}= \sum_{s=0}^{\min(n-j,n-k)}\binom{n-j}{n-j-s}\binom{n-k}{n-k-s}s!\, x^{\underline{n-j+n-k-s}}= \sum_{l = j+k-n}^{\min(j,k)}\binom{n-j}{k-l}\binom{n-k}{j-l}(n+l-j-k)!\, x^{\underline{n-l}}.
\frac{1}{a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}}= \frac{\Gamma [ \alpha_{1}+ \alpha_{2}]}{\Gamma [ \alpha_{1}] \Gamma [ \alpha_{2}]}\int_{0}^{1}d x \frac{x^{\alpha_{1}- 1}( 1 - x )^{\alpha_{2}- 1}}{[ x a_{1}+ ( 1 - x ) a_{2}]^{\alpha_{1}+ \alpha_{2}}}
\langle A^{1/2}x,A^{1/2}y\rangle_{\mathbf{R}(A^{1/2})}:=\langle P_{\overline{\mathcal{R}(A)}}x\mid P_{\overline{\mathcal{R}(A)}}y\rangle,\;\forall\, x,y \in \mathcal{H}.
\sum_i x_i .m \otimes Y_i \otimes_{k[U_\ell]}1 = \sum_i x_i .m \otimes 1 \otimes_{k[U_\ell]}\lambda_i = \sum_i \lambda_i x_i.m \otimes 1 \otimes_{k[U_\ell]}1 = x_{\alpha}.m \otimes 1 \otimes_{k[U_\ell]}1
I_2 = \delta^{-\frac{n-4}{2}}\int_{B}\varphi_0\Big(\frac{e_n-x}{\delta}\Big)U_{\delta,e_n}^p -\varphi_0\Big(\frac{e_n+x}{\delta}\Big)U_{\delta,e_n}^p -\varphi_0\Big(\frac{e_n-x}{\delta}\Big)U_{\delta,-e_n}^p +\varphi_0\Big(\frac{e_n+x}{\delta}\Big)U_{\delta,-e_n}^p.
\left\|\Box_k^{\alpha_1}|~M_1\rightarrow M_2\right\| \gtrsim \frac{\|\Box_k^{\alpha_1}f_l^{\alpha_2}\|_{M_2}}{\|f_l^{\alpha_2}\|_{M_1}}\sim2^{j\widetilde{A_1}(\mathbf{p},q,\alpha_1,\alpha_2)}
|\int_{\epsilon}^{1}\frac{1}{((z+1)^{2}z^{2}+\epsilon^{2})^{3/2}}-\frac{1}{(z^{2}+\epsilon^{2})^{3/2}}dz|\le\int_{\epsilon}^{1}\frac{1}{|(z+1)^{2}z^{2}+\epsilon^{2}|^{3/2}}+\frac{1}{|z^{2}+\epsilon^{2}|^{3/2}}dz
K_{X' / X}= \sum_{j=1}^u (\vartheta_{X'}-\vartheta_X) (y,\kappa(p_j)p_j,\kappa(p_j)) D_{(y_j,p_j)}' + \sum_{i=1}^e (\vartheta_{X'}-\vartheta_X) (C,\rho_i,0) D_{\rho_i}'.
\zeta_1+\zeta_2+\zeta_3+\zeta_4+\zeta_5+\zeta_6=\delta_1\alpha+\delta_2, \ \ \zeta_1, ..., \zeta_6>0 \frac1{\theta_1}+\frac1{\theta_2}+\frac1{\theta_3}+\frac1{\theta_4}+\frac1{\theta_5}+\frac1{\theta_6}=1, \theta_1=\frac2{\delta_1}, \ \theta_3=\frac2{\delta_2}, \ \theta_5=\frac2{\delta_3}, \ 1<\theta_2,\theta_4,\theta_6<\infty
\left\langle v_{l}\, ,\, e^{tL_{\delta}^{\ast}}h\right\rangle_{d}\, =\, \left\langle v_{l}\, ,\, e^{tL_{\delta}^{\ast}}h\right\rangle_{2, d}\, =\, \left\langle e^{tL_{\delta}}v_{l}\, ,\, h\right\rangle_{2, d}\, =\, \left\langle e^{tL_{\delta}}v_{l}\, ,\, h\right\rangle_{d}.
p_0(\omega)=\sum\limits_{i=1}^{n}\sum\limits_{\alpha \in \mathbb{N}^n}c_\alpha^i (b-\mu_1^{\alpha_1+1}\dots \mu_n^{\alpha_n+1}\mu_i^{-1}) z_1^{\alpha_1}\dots z_n^{\alpha_n}\widehat{dz_i},
\phi^{(\epsilon)}_{ij}(\xi) =-\phi_{ji}(-\epsilon)+\int_{\epsilon}^\xi\phi_{ji}'(-\rho_\epsilon(\eta))\,d\eta &=-\phi_{ji}(-\epsilon)-\int_{-\epsilon}^{-\xi}\phi_{ji}'(-\rho_\epsilon(-\eta))\,d\eta &=-\phi^{(\epsilon)}_{ji}(-\xi).
\hat{Z}_{8 8 8}^{1 0 , 1 0}( \Omega ) = \frac{( d e t \I m \ \Omega )^{2}}{2^{3}}\left| \Theta \left[ \begin{array}{l l l}{1 / 2}&{1 / 2}&{1 / 2}\\{1 / 2}&{0}&{1 / 2}\\ \end{array}\right] ( 0 \vert \Omega ) \right|^{8}
\begin{cases}\big(-\alpha_1{\eta^+}'\big)'\,+\,a_1 \eta^+\,+\,b_1\zeta^+\,=\,0, begin{align*}2mm] \big(-\alpha_2{\eta^-}'\big)'\,+\,a_1 \eta^-\,+\,b_2\zeta^-\,=\,0, begin{align*}2mm] \big(-\beta_1{\zeta^+}'\big)'\,+\,d_1 \zeta^+\,+\,b_1\eta^+\,+\,b_3\zeta^-\,=\,0, begin{align*}2mm] \big(-\beta_2{\zeta^-}'\big)'\,+\,d_2 \zeta^-\,+\,b_2\eta^-\,+\,b_3\zeta^+\,=\,0. \end{cases}
_0 \left< \right| T \left( \Psi^{R}( x_{2}) \bar{\Psi}^{R}( x_{1}) \right) \left| \right>_{0}= - \frac{1}{( 2 \pi )^{3}}\int{\mathrm d}^{3}p \frac{- i \hat{p}+ m}{2 E}e^{i p ( x_{2}- x_{1})}
\int_{\mathbb R}\tilde\phi_1(x_1,x_2,x_3) dx_1=0\quad{\rm for\each\fixed\}(x_2,x_3)\{\rm in\}\mathbb R^2; &\int_{\mathbb R^2}\tilde\phi_1(x_1,x_2,x_3) dx_2dx_3=0 \quad{\rm for\each\fixed\}x_1\{\rm in\}\mathbb R
\beta_m = \left(\begin{array}{cc}0 p^{-m}-p^m 0\end{array}\right) = \left(\begin{array}{cc}0 p^{-m}p^m 0\end{array}\right)\left(\begin{array}{cc}-1 0 0 1\end{array}\right) = p^{-m}\left(\begin{array}{cc}0 1 p^{2m}0\end{array}\right)\left(\begin{array}{cc}-1 0 0 1\end{array}\right).
\kappa_{+}\neq 0,\kappa_{-}\neq 0,\exists i\in \left\{0,1\right\},\mathsf{M}\in \left\{0,...,\mathsf{N}-1\right\}:Y^{(i,2\mathsf{M})}(\tau_{\pm},\alpha_{\pm},\beta_{\pm})=0,
\begin{matrix}10x_{1}&-x_{2}&+2x_{3}&&=6,\\ -x_{1}&+11x_{2}&-x_{3}&+3x_{4}&=25,\\ 2x_{1}&-x_{2}&+10x_{3}&-x_{4}&=-11,\\ &3x_{2}&-x_{3}&+8x_{4}&=15.\end{matrix}
S(x,x)=\frac{1}{B(L/2,1/2)}\frac{I_{1-x^2}(L+1,N-1)}{1-x^2}+\frac{(1-x^2)^{(L-2)/2}|x|^{N-1}}{B(N/2,L/2)}I_{x^2}\left(\frac{N-1}2,\frac{L+2}2\right),
y_{\mathbf{r},r_0}=\mu(r_0r)\phi_{w*}(r_0r)\sum_{\substack{\mathbf{r}|\mathbf{d}r_0|d_0 (d_0,d)=1}}\frac{\lambda_{\mathbf{d}}\tilde{\lambda}_{d_0}}{dd_0},\tilde{y}_{r_0}=\mu(r_0)\phi(r_0)\sum_{r_0|d_0}\frac{\tilde{\lambda}_{d_0}}{d_0}.
[a_c(\mathbf{p}), a_c^{*}(\mathbf{q})]=2\omega_{\mathbf{p}}\delta^n(\mathbf{p}-\mathbf{q}), [a_c(\mathbf{p}), a_c(\mathbf{q})]=0=[a_c^{*}(\mathbf{p}), a_c^{*}(\mathbf{q})].
(z) = \zeta \mathcal{A}h (z) - \mathcal{A}(\zeta h) (z) = \int h (t, x - y, v) \big(\zeta (t, x, v) - \zeta (t, x-y, v)\big) \, \frac{y}{|y|^{d+5/3}}\, dy & = \bigg(\int_{|y| \le 1}\ldots + \int_{|y| > 1}\ldots\bigg) =: I_1 (z) + I_2 (z).
\Gamma(f)\Gamma(g) =[\gamma(f)]\ [\gamma(g)] &=[f_-(-M_x)+f_+(H)-f(0)]\ [g_-(-M_x)+g_+(H)-g(0)] &=\Big([f_-(-M_x)][g_-(-M_x)]+[f_+(H)][g_+(H)]\Big) &\ \ +\Big([f_-(-M_x)][g_+(H)]+[g_-(-M_x)][f_+(H)]\Big) &\ \ -\Big(f(0)[g_-(-M_x)+g_+(H)]+g(0)[f_-(-M_x)+f_+(H)]\Big) &\ \ +f(0)g(0) &=\ \A+B-C+f(0)g(0).
-2tf(1-f)^{-1}\left\|\overline{\nabla}f\right\|\left\|\overline{\nabla}\varphi\right\|\phi \leq t\phi\left(\frac{\left\|\overline{\nabla}f\right\|^{2}}{1-f}\varphi +\frac{\left\|\overline{\nabla}\varphi\right\|^{2}}{\varphi}\frac{f^{2}}{1-f}\right) &=t(1-f)F^{2}+tF\frac{c_{2}}{\rho^{2}}\frac{f^{2}}{1-f}.
\boldsymbol{x}=\sum\limits_{i\in I_{\boldsymbol{x}}}\boldsymbol{\lambda}(a_i,b_i)+\sum\limits_{j=1}^m\mathbf{h}_{j1}(1,c_j)+\sum\limits_{k=2}^n\mathbf{h}_{1k}(1,d_k),
\delta{\hat{\xi}}^{{\hat{\imath}}}= \delta^{{\hat{\imath}}6}[ - \Lambda^{( 0 )}+ \frac{m}{2}( 2 \pi \alpha^{\prime}){\rho}^{( 0 )}- \frac{m}{2}( 2 \pi \alpha^{\prime}) \sigma^{( 0 )}\omega^{( 0 )}] \, ,
\sum_{n\neq0}n \left[ N_n + \widetilde{N}_n + \sum_{I=3}^8 N^{(I)}_n + \sum_{b=1}^8 F^{(b)}_n \right] = \frac{\sqrt{f}}{l_s}mR_T ( \tilde{a}^\dagger_0 + \tilde{a}_0 ) ~.
\begin{array}{l c l}{{\partial{\bar{\partial}}\phi_{1}}}&{{=}}&{{A e^{- \frac{1}{2 \theta}( \phi_{1}+ \frac{1}{2}\phi_{2})}}}\\{{\partial{\bar{\partial}}\phi_{2}}}&{{=}}&{{B e^{- \frac{1}{2 \theta}( \phi_{1}+ 2 \phi_{2})}}}\\ \end{array}
e^{(0)M}_{\mu}e^{(0)N}_{\nu}\left[{\cal L}_V{\cal L}_{n^{(0)}}g^{(0)}_{MN}-{\cal L}_{W[V]}g^{(0)}_{MN}+ n^{(0)L}(V_{L;MN}+V_{L;NM}+R_{L'MLN}V^{L'}+R_{L'NLM}V^{L'})\right]=0,
g_{\mu\mu}^\prime(\mu,\sigma) = \frac{2\mu^2}{\sigma_0^4}+ \frac{1}{\mu_0^2}, g_{\mu\sigma}^\prime(\mu,\sigma) = \frac{2\mu\sigma}{\sigma_0^4},g_{\sigma\sigma}^\prime(\mu,\sigma) = \frac{2\sigma^2}{\sigma_0^4}.
(-i(\sum_{i=1}^{n-1}(y_i-x_i)e_i+y_ne_n)e_n)^\mu=1^\mu r^\mu\Pi_++(-1)^\mu r^\mu\Pi_- = \frac{1}{2}(1^\mu +(-1)^\mu)r^\mu +\frac{1}{2}(1^\mu -(-1)^\mu)(-i{(\sum_{i=1}^{n-1}(y_i-x_i)e_i+y_ne_n)e_n})r^{\mu-1}.
\alpha(\xi,\eta)\frac{\partial^{2}u}{\partial\xi^{2}}+2\beta(\xi,\eta)\frac{\partial^{2}u}{\partial\xi\partial\eta}+\gamma(\xi,\eta)\frac{\partial^{2}u}{\partial\eta^{2}}+\delta(\xi,\eta,u,\frac{\partial u}{\partial\eta},\frac{\partial u}{\partial\eta})=0
\operatorname*{l i m}_{\{\alpha_{f}\}\to \infty}\frac{\operatorname*{d e t}[ C ( \alpha_{f}, i \mu_{g}) ]_{f , g = 1 , \dots , N_{f}}}{\Delta_{N_{f}}( \alpha )}= \mathcal{F}\cdot \operatorname*{d e t}\mathcal{B}
\bigg[\int_0^x\, dx' \, U_{\infty}(E,x')^{-1}\big[U_{\infty}(E,x')^{-1}\big]^*\bigg]^{-1}& =\begin{pmatrix}4\delta_- (E)/(e^{2\delta_- (E) x}-1) 0 0 4\delta_+ (E)/(e^{2\delta_+ (E) x}-1) \end{pmatrix}\underset{x\uparrow \infty}{\rightarrow}0.
P :=r^3\epsilon^{AB}\tilde{\nabla}_B (r^{-2}h_{A\alpha})dx^\alpha, Q :=\Big(\tilde{\nabla}^A r\Big)\left(\mathring{\nabla}_\alpha h_{A\beta}+\mathring{\nabla}_\beta h_{A\alpha}-r^2\tilde{\nabla}_A (r^{-2}h_{\alpha\beta})\right) dx^\alpha dx^\beta.
\eta'_2 = \theta' \circ \eta'_1 \circ \theta', \eta'_1 \circ \theta' = \sigma_{12}^{-1}\circ \theta^{-1}\sigma_{21}= \sigma_{11}^{-1}\circ \theta^{-1}\circ \sigma_{22}, \theta' \circ \eta'_1 = \eta'_2 \circ \theta' = \sigma_{22}^{-1}\circ \theta \circ \sigma_{11}= \sigma_{21}^{-1}\circ \theta \circ \sigma_{12}.
\frac{\partial}{\partial y_l}\left( \frac{1}{2}\langle y , (Id+ P(M_{ij})NP^* )y \rangle_Y \right) = \frac{1}{M}y_l + \frac{1}{N}\sum_{n=1}^M \sum_{i \in B(l), j \in B(n)}M_{ij}y_n.
W^-=\begin{bmatrix}U_1^2+U_n^2 U_n U_{\tau}\end{bmatrix},\Lambda^- =\begin{bmatrix}\frac{1}{2 U_n \sqrt{U_1}}0 0 \frac{1}{2 U_n \sqrt{U_1}}\end{bmatrix},W^+= U_1^2,\Lambda^+ = - \frac{1}{2 U_n \sqrt{U_1}}.
|A(t,x)|\cdot|\xi| \leq \frac{1}{p}|A|^p + \frac{1}{q}|\xi|^q \leq \frac{1}{p}<x>^{p(M+1-\delta)}+ \frac{1}{q}|\xi|^q &= \frac{1}{p}<x>^{2(M+1)\delta_1}+ \frac{1}{q}|\xi|^{2\delta_2}.
\frac{1}{2}C_1|\alpha-\beta|\leq |z(\alpha,t)-z(\beta,t)|\leq 2|\alpha-\beta|, \|u_t(\cdot,t)\|_{H^1}^2\leq 2\|u_t(\cdot,0)\|_{H^1}^2, \inf_{\alpha\in \mathbb{R}}a|z_{\alpha}|(\alpha,t)\geq \frac{1}{2}\alpha_0.
D_{\alpha}^{\mathbf i}\left[ - 3 i{\bar{\sigma}}^{0 \dot{\alpha}\gamma}{\bar{D}}_{\dot{\alpha}{\mathbf i}}D_{\beta \gamma}T \right] = - 2 4 \left[ \sigma_{\alpha}^{\mu 0 \gamma}\partial_{\mu}D_{\beta \gamma}T - \frac{1}{2}\partial^{0}D_{\alpha \beta}T \right] .
B_t^E\left[B_{2/3}^E\right]^{-1}= \begin{bmatrix}E - q(\omega) -[(1-\eta(t))+\eta(t)p(T^{-1}\omega)] (1-\eta(t))+\eta(t)p(\omega) 0 \end{bmatrix}\begin{bmatrix}0 1 -1 E - q(\omega) \end{bmatrix}& = \begin{bmatrix}(1-\eta(t))+\eta(t)p(T^{-1}\omega) * 0 [1-\eta(t)]+\eta(t)p(\omega) \end{bmatrix},
{J_{tv}}(s) =\!\frac{\Gamma(\nu_{tv})}{(\frac{Z_2}{Z_1})^{\nu_{tv}}(1\!+\!\frac{a^2s\alpha_j\bar{\gamma}_j}{Z_2})^{\nu_{tv}+N_j}}\sum^{N_{j}}_{k=0}\binom{N_j}{k}\left(\frac{a^2s\alpha_j\bar{\gamma}_j}{Z_2}\right)^{k}&\times U\left(\nu_{tv},\nu_{tv}+1-k,\frac{Z_1}{Z_2\left(1+\frac{a^2s\alpha_j\bar{\gamma}_j}{Z_2}\right)}\right),
A_{f}^{\dagger}( \vec{p}) = ({\frac{1}{2 ( 2 \pi )^{D}f ( \vec{p})}})^{1 / 2}\int_{x}e^{i \vec{p}\cdot \vec{x}}[ f ( \vec{p}) ( \phi_{x}- \varphi_{x}) - i \Pi_{x}]
\iota (\mathbf{h}\bullet \mathbf{h}') = \iota\left(x+x',\omega+\omega',\tau+\tau'+ \tfrac{1}{2}(x' \cdot \omega - x \cdot \omega') \right) & = \left(x+x',\omega+\omega',\tau+\tau'+\tfrac{1}{2}(x' \cdot \omega - x \cdot \omega')+\tfrac{1}{2}(x+x')\cdot(\omega+\omega')\right) .
\mathbf{A}_{f}:=\left(\begin{array}{cccccc}0 0 0 \cdots 0 0 f\left(1\right) 0 0 \cdots 0 0 f\left(2\right) f\left(2\right) 0 \cdots 0 0 \vdots \vdots \vdots \ddots \vdots \vdots f\left(N-1\right) f\left(N-1\right) f\left(N-1\right) \cdots f\left(N-1\right) 0\end{array}\right).
\Delta_1(X_1) =0\,, \Delta_1([X_1,X_2])=\Delta_1([X_1,X_2]) =1\,, \Delta_1([[X_1,X_2],X_3]) =2\,, \Delta_1([X_1,[X_2,X_3]]) =1\,, \Delta_2([X_1,[X_2,X_3]])=\Delta_3([X_1,[X_2,X_3]]) =2\,.
\begin{cases}\underline{R}^{I,b}_t=R^I_0\wedge b+2\int_0^t\sqrt{\underline{R}^{I,b}_s}d\beta^I_s+(|I|-1)\Big(2-\frac{\chi|I|}{4\pi N}\Big)t-c\int_0^t\sqrt{\underline{R}^{I,b}_s}ds-L_t (L_s)_{s\geq 0}\mbox{is an adapted increasing process such that}L_0=0\mbox{and}\int_0^t(b-\underline{R}^{I,b}_s)dL_s=0 \end{cases}
C'_{q,x}(s)=q\big(\int_{y<s}(s-y)^{q-1}d\rho_x(y)-\int_{y>s}(y-s)^{q-1}d\rho_x(y)\big) &=q\Big(\int_{y<0}(s-y)^{q-1}d\rho_x(y)+\int_{0\leq y<s}(s-y)^{q-1}d\rho_x(y)-\int_{y>s}(y-s)^{q-1}d\rho_x(y)\Big) &\geq q\Big(\int_{y<0}(-y)^{q-1}d\rho_x(y)+\int_{0\leq y<s}(s-y)^{q-1}d\rho_x(y)-\int_{y>s}(y-s)^{q-1}d\rho_x(y)\Big).
P_{ED}^{LMP>0}:= \sum_{t\in\mathcal{T}}\sum_{n \in \mathcal{N}_t^+}\left[ \sum_{i\in\mathcal{G}_n}p_{it}+ \sum_{i\in\mathcal{I}_n}\left( M_{it}- m_{it}\right) \right. \left .+ \sum_{i\in\mathcal{W}_n}\left(W_{it}- w_{it}\right) + \sum_{i\in\mathcal{R}_n}\left(R_{it}-r_{it}\right)\right]
{\sf t}_9 = \{f({\sf U}_\infty+\Theta_{\sf a}+\theta) - f_2({\sf U}_\infty+\Theta_{\sf a}+\theta) + f_2({\sf U}_\infty) +{\sf U}_\infty^{q-1}(\Theta_{\sf a}+\theta) + P_1\theta \}\times (1-\chi_{2,{\sf a}}) \chi_{3,{\sf e}}.
\int_{2^{-i}}^{c_i}|x-x_i|^{-b_i}|x-x_{i-1}|^{-b_{i-1}}\, dx \leq \left( 2^{-i-1}\right)^{-b_{i-1}}\cdot \int_{0}^{2^{-i-1}}t^{-b_i}\, dx & = \left( 2^{-i-1}\right)^{-b_{i-1}}\cdot \frac{(2^{-i-1})^{1-b_i}}{1-b_i}.
\operatorname E\|X_0\|^p\le2^{p-1}\Bigl(C\frac p{\log p}\Bigr)^p\Bigl[(\operatorname E\|X_0\|^2)^{p/2}+\sum_{j=0}^\infty\operatorname E\|u_j\varepsilon_{k-j}\|^p\Bigr]
\hat{F}(s,\xi) = |\xi|^{-n/2-\delta}\hat{f}(s,\xi),\hat{G}(s,\xi) = |\xi|^{-n/2-\delta}\hat{g}(s,\xi),\hat{H}(s,\xi) = |\xi|^{-n/2-\delta}\hat{h}(s,\xi),
G(\nu_0, \boldsymbol{\nu}) = \mathbb{P}(\max \{Y_1, \dots, Y_k \}< Y_0) &= \int_{-\infty}^{\infty}f_{Y_0}(t) \mathbb{P}(\max \{Y_1, \dots, Y_k \}< t) dt &= \frac{1}{\sqrt{\rho}}\int_{-\infty}^{\infty}\phi \left ( \frac{t-\nu_0}{\sqrt{\rho}}\right ) \prod_{j=1}^k \Phi \left ( \frac{t-\nu_j}{\sqrt{1-\rho}}\right ) dt.
\big{\langle}M(Z,W)^2X,X \big{\rangle}= - \big{\langle}J_WJ_Z(\nabla_Z J)_WX, J_ZJ_W(\nabla_ZJ)_WX \big{\rangle}&= \big{\|}J_W J_Z( \nabla_ZJ)_WX\big{\|}^2+2\langle Z,W \rangle \big{\langle}J_W J_Z (\nabla_ZJ)_W, (\nabla_ZJ)_WX \big{\rangle}&= \Big{(}\|Z\|^2\|W\|^2-2\langle Z,W \rangle^2 \Big{)}\big{\|}(\nabla_ZJ)_WX\big{\|}^2.
\frac{\partial \Gamma_{\Lambda}[\phi]}{\partial \Lambda}=- \frac{\Omega}{2}\mathrm{tr}\int^{\infty}_0\!\!\!\!\! dq \,\frac{q^{D-1}}{C(q,\Lambda)}\frac{\partial C(q,\Lambda)}{\partial \Lambda}\left \langle \left [1 + C.\frac{\delta^2 \Gamma_{\Lambda}}{\delta \phi\delta \phi}\right ]^{-1}\!\!\! (\mathbf{q},\mathbf{-q}) \right \rangle
\operatorname{dim}_{\mathbb{F}}( \operatorname{Ext}^1_{\mathcal{O}}(M(T_2),M(T_1))_{\operatorname{tors}}\otimes_{\mathcal{O}}\mathbb{F}) = \operatorname{dim}_{\mathbb{F}}\operatorname{Hom}_{\operatorname{BK}}(\overline{M}_2,\overline{M}_1) - \operatorname{dim}_{E}\operatorname{Hom}_{E[G_K]}(V_2,V_1)
\exp(t \alpha ) \exp(-t^2 \beta) \exp( (t^3-p_\Gamma t)\Gamma)x = \tilde U^* \mbox{diag}(e^{-a_j t - b_j t^2 - \gamma_j (t^3-p_\Gamma t)}) \tilde Ux &= \sum_{j=1}^d e^{-a_j t - b_j t^2 - \gamma_j (t^3-p_\Gamma t)}\langle x, v_{j}\rangle v_{j},
\Omega^{(2m-1)}_{\rho i_2 \ldots i_{2m-2}\sigma}={1\over (2m-3)!}k_{\rho l_1 \ldots l_{m-1}}{C_{j_2 j_3}}^{l_1}\ldots{C_{j_{2m-2}\sigma}}^{l_{m-1}}\epsilon^{j_2 \ldots j_{2m-2}}_{i_2 \ldots i_{2m-2}}\, \, ,
= \ell \int_{\Omega}\nabla \varphi_{\varepsilon}(t) \cdot \nabla \kappa_{\varepsilon}(t)- \ell \int_{\Omega}\nabla \varphi_0 \cdot \nabla \kappa_0- \ell \int_{Q_t}\nabla(\partial_t \varphi_{\varepsilon}) \cdot \nabla \kappa_{\varepsilon}.
\left( \frac{i \epsilon_{\mu \lambda \alpha}p_{\lambda}}{p^{2}}\right) \left( \frac{i \alpha \epsilon_{\alpha \sigma \beta}p_{\sigma}}{\sqrt{p^{2}}}\right) \left( \frac{i \epsilon_{\beta \kappa \nu}p_{\kappa}}{p^{2}}\right) = \frac{i \alpha \epsilon_{\mu \lambda \nu}p_{\lambda}}{\left( p^{2}\right)^{3 / 2}}\, .
I ( Q^{2}) = v_{4}\frac{k^{2}}{Q^{2}}\int_{0}^{\frac{1}{2}}d z \left( z^{- 2}- \frac{Q^{2}}{k^{2}}z^{- 1}- z^{- 2}e^{- \frac{Q^{2}}{k^{2}}z}\right) .