harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Phi^{\alpha}_{ij}=F_1^{ij}\biggl [F_2^{ij}(F_1^{ji}-F_{2}^{ji})F_3^{ij}\mathcal{D}^{\alpha}(F_1^{ji}-F_{3}^{ji})-F_3^{ij}(F_1^{ji}-F_{2}^{ji})F_2^{ij}\mathcal{D}^{\alpha}(F_1^{ji}-F_{2}^{ji})\biggl ]
- \epsilon_{b k}^{q}\hat{R}_{i b , a m}\epsilon_{c m}^{q}K_{a c}= ( \epsilon^{q})_{k b}^{t}\hat{R}_{i b , a m}( \epsilon^{q})_{m c}^{t - 1}K_{a c}= \hat{R}^{\epsilon}\,_{i k , a c}K_{a c}= K_{i k}^{\epsilon}
\sum_{n}H ( x_{1}+ \omega_{n}, x_{2}+ \omega_{n}) \simeq \left( \frac{z}{s + 1}\frac{1}{x_{1}- x_{2}}+ \frac{1}{2}\right) \frac{\tanh ( \mu - x_{1})}{2 \mu - x_{1}+ x_{2}}+ X_{1 , 2}+ \mathrm{p . p . c .}\, .
1=\int DS_{\mu\nu}\delta\left(S_{\mu\nu}-\Sigma_{\mu\nu}^{\rm gas}\right)=\int DS_{\mu\nu}Dl_{\mu\nu}\exp\left[-i\pi\int d^4x l_{\mu\nu}\left(S_{\mu\nu}-\Sigma_{\mu\nu}^{\rm gas}\right)\right]
\sum_{j=1}^{m_0}\big(\omega^p_{\mathbb{B}^n}- (F_j^*\omega_{\mathbb{B}^{N_j}})^p \big) + (\lambda- m_0) \omega^p_{\mathbb{B}^n}= \sum_{j=m_0+1}^m F_j^*(\omega_{\mathbb{B}^{N_j}}^p)
u\left(X\cdot Y\cdot Z\cdot U\cdot \bar{\psi},\bar{\psi}\right)+u\left(X\cdot Y\cdot Z\cdot \bar{\psi},U\cdot\bar{\psi}\right)= &2u\left(X\cdot Y\cdot Z\cdot U\cdot \bar{\psi},\bar{\psi}\right)-2u g(U,X)g(Y,Z)+2ug(U,Y)g(X,Z)-2ug(U,Z)g(X,Y).
[ a_{0}, a_{n}] ={\frac{1}{A}}{\frac{1}{n}}a_{0}a_{n}\hbar +{\frac{1}{2 A}}\sum_{k \ne 0 , n}{\frac{\epsilon ( n )}{\sqrt{| k n ( n - k ) |}}}a_{k}a_{n - k}\hbar + O ( \hbar^{2}) , n \ne 0 ,
\tilde s^{(2)}(p^{2};m^{2})={1\over 2\pi}\,{1\over p^{2}\sqrt{-1-4m^{2}/p^{2}}}\, \arctan{1\over\sqrt{-1-4m^{2}/p^{2}}}\, \raise 2pt\hbox{,}\quad{\rm for}\ 0< -p^{2}< 4m^{2}.
\Big(\sum_{i=1}^d F^\ast_i(x_i) - (d-1)\Big)^+ \vee \max\Big\{\pi_s - \sum_{i=1}^d \big(F_i^\ast(s_i) - F_i^\ast(x_i)\big)^+ : s\in S\Big\}&\le F(x_1,\dots,x_d) \le \min_{i=1,\dots,d}F^\ast_i(x_i) \wedge \min\Big\{\pi_s + \sum_{i=1}^d \big(F_i^\ast(x_i) - F_i^\ast(s_i)\big)^+ : s\in S\Big\}.
\begin{array}{llll}V_{a,a}= \{a\}, V_{a,b}= \{b\}, V_{a,c}= \emptyset, V_{a,d}= \{c,d\}, V_{b,a}= \{a,d\}, V_{b,b}= \{b\}, V_{b,c}= \{c\}, V_{b,d}= \emptyset, V_{c,a}= \emptyset, V_{c,b}= \{a,b\}, V_{c,c}= \{c\}, V_{c,d}= \{d\}, V_{d,a}= \{a\}, V_{d,b}= \emptyset, V_{d,c}= \{b,c\}, V_{d,d}= \{d\}. \end{array}
\tau \rightarrow \frac{a^{\prime}\tau + b^{\prime}}{c^{\prime}\tau + d^{\prime}}\; \; , \; \; \Gamma_{\gamma}^{\prime}= \left( \begin{array}{c c}{{a^{\prime}}}&{{b^{\prime}}}\\{{c^{\prime}}}&{{d^{\prime}}}\\ \end{array}\right)
T_n^{*}=\frac{(\hat{C}_0(2u_n)-\hat{C}_1(u_n))-(\hat{C}_0(2u_n)-\hat{C}_0(u_n))}{\hat{C}_1(u_n)}=\frac{\hat{C}_0(u_n)-\hat{C}_1(u_n)}{\hat{C}_1(u_n)},
(x+m+1)\sum_{i=0}^{m}(-1)^{i}(\begin{matrix}x+y+i\\m-i\end{matrix})(\begin{matrix}y+2i\\i\end{matrix})-\sum_{i=0}^{m}(\begin{matrix}x+i\\m-i\end{matrix})(-4)^{i}=(x-m)(\begin{matrix}x\\m\end{matrix}).
c(v_{t}D(\alpha,\alpha) w_{-\theta,\theta - \frac{\pi}{2}}D(\alpha,\alpha)) = c(v_{t+\phi+\phi'}D(\beta,\gamma)), c(v_t D(\alpha,0) \iota(u_\theta) D(\alpha,0)) = c(v_{t+\omega_1+\omega_2}D(\beta,\gamma)).
\begin{aligned}\frac{{\rm d}h(t)}{{\rm d}t}= \frac{-2}{t^{2}}+ e^{t}(\frac{2}{t^2}-\frac{2}{t}+1) &= \frac{-2}{t^{2}}+ (1+t+\frac{t^2}{2}+\mathcal{O}(t^2))(\frac{2}{t^2}-\frac{2}{t}+1) &\geq \frac{-2}{t^{2}}+ (1+t+\frac{t^2}{2})(\frac{2}{t^2}-\frac{2}{t}+1) &= \frac{t^{2}}{2}> 0,\end{aligned}
ii)\hbox{}\bigcup_{p+q=k}\Sigma_{\pi ,p}(\rho_1)\times\Sigma_{\pi ,q}(\rho_2)\subseteq\Sigma_{\pi ,k}(\rho)\subseteq Sp_{\pi ,k}(\rho)\subseteq &\bigcup_{p+q=k}Sp_{\pi ,p}(\rho_1)\times Sp_{\pi,q}(\rho_2).
\binom{2n}{2n-j-u}\binom{2n-j-u}{n}(-1)^{j+u+r-1}\binom{j+u+r-1}{j+u}\frac{1}{2}S(n,r)\cdot \cdot\sum_{l=0}^{r-1}(-1)^l\binom{2n-j-u+l}{l}\frac{\binom{2(j+u+r-1-l)}{j+u+r-1-l}\binom{2(n-j-u+l+1)}{n-j-u+l+1}\binom{n-j-u}{j+u+r-l-1}}{2\binom{2n-j-u+l+1}{n}}
\frac{d}{dt}G(t) =\zeta_{2}(T)\| \omega(t)\|_{L^{\infty}}&\leq\zeta_{2}(T)\big(\|\omega(0)\|_{L^{\infty}}+C(T)\int_{0}^{t}\|\nabla j(\tau)\|_{L^{\infty}}\,d\tau\big) &\leq\zeta_{2}(T)\big(\|\omega(0)\|_{L^{\infty}}+C(T)G(t)\big).
Y_0(x) = \frac{2}{\pi}J_0(x)\left(\ln \frac{x}{2}+ C\right) - \frac{2}{\pi}\sum_{m =1}^{\infty}\left[\frac{(-1)^m}{(m!)^2}\left( \frac{x}{2}\right)^{2m}\sum_{k = 0}^m \frac{1}{m}\right],
E_{0}(\tilde{g};[0,1]) \le \max_{x\in [m^{-2},1]}\left|x^{-r}\ln(m^2x) + \sum_{l=1}^r a_lx^{-l}\right| &\le m^{2r}\cdot \max_{z\in [1,\infty)}z^{-r}\ln z + \sum_{l=1}^r |a_l|m^{2l}&\le m^{2r}+ \sum_{l=1}^r |a_l|m^{2l}&\le \frac{1}{2}E_{0}(\tilde{g};[0,1]) + r\cdot \max_{1\le l\le r}|a_l|m^{2l}.
-\beta{\cal F}(\beta)\simeq-\frac{2^{-15}\pi^{-26}}{(2\alpha')^{13}}\int_{0}^{\varepsilon}d\tau_2\tau_2^{-14}\left[\theta_3\left(0,\frac{i\beta^2}{4\pi^2\tau_2\alpha'}\right)-1\right]e^{\frac{4\pi}{\tau_2}}\tau_2^{51/2}
S_\lambda S_\lambda^* \xi(x) = \begin{cases}(\Phi_{\tau_\lambda}(\tau^{d(\lambda)}(x)))^{-1/2}( S_\lambda^* \xi)(\tau^{d(\lambda)}x) 0\quad\quad\quad\end{cases}= \begin{cases}\xi(x)\quad 0\quad\quad\quad\end{cases}
\partial_x\int_B G(x,z)P_{B^c}(y,z)dz = \partial_x\int_B G(x,z)P_1(y,z)dz + \partial_x\int_B G(x,z)P_2(y,z)dz & = \int_B \partial_x G(x,z) P_1(y,z)dz + \int_B \partial_x G(x,z)P_2(y,z)dz & = \int_B \partial_xG(x,z)P_{A}(y,z)dz,
\Gamma (x) =\sum_{k = 1}^d e_{kk}\left(\kappa'(x_{k-1}+x_{k+1})x_{k}+ \lambda' + \delta'x_k\right) -\sum_{k = 1}^d \kappa'x_kx_{k+1}(e_{k,k+1}+e_{k+1,k}),
\frac{\partial}{\partial x_1}[z^n] P_1 \Big|_{x_1 = x_2 = x_3 = 1}= \frac{1}{n}[t^{n-1}] (1+(2n-1)t) (1+t)^{4n-3}&= \frac{1}{n}\Big( \binom{4n-3}{n-1}+ (2n-1) \binom{4n-3}{n-2}\Big).
\begin{array}{l}\lim_{t\to 0}\mu_f(x,t) = f(x)\,,\ \ \lim_{t\to 0}\nabla_x \mu_f(x,t) = \nabla f(x)\,,\ \ \lim_{t\to 0}\,\nabla_{xx}\mu_f(x,t) = \nabla^2 f(x)\,, \lim_{t\to 0}\,\nabla_{tx}\mu_f(x,t) = 0\,.\end{array}
\frac 1 q =\frac 1{2n \kappa-(n+1)}\left(\frac{n-1}2 +\frac 6{m+2}+\frac m{\mu_\ast(m+2)}\right) \intertext{and}\frac 1 s = \frac{(m+2)(n-1)}4 \left( \frac 1 2 -\frac 1 q\right)+\frac m{4 \mu_\ast}.
\begin{array}{r c l}{{{\cal A}( \omega ; \omega_{\bot})}}&{{=}}&{{\displaystyle \int d x_{0}d x_{3}~ e^{i ( \omega x_{0}- \sqrt{\omega^{2}- \omega_{\bot}^{2}}x_{3})}\tau ( x_{0}) g ( x_{0}, x_{3}^{2}; \omega_{\bot}) .}}\\ \end{array}
\frac{\Gamma_{p}(\frac{\nu+\delta+p-1}{2})}{\Gamma_{p}(\frac{\nu}{2})\Gamma_{k}(\frac{\delta+p-1}{2})|\Psi|^{\frac{\nu}{2}}}|X|^{\frac{\nu-p-1}{2}}|I_{p}+X\Psi^{-1}|^{-\frac{\nu+\delta+p-1}{2}}
(\mathbb{A}^{-1}_R)_{21}=-\left(c+(d-ca^{-1}b)^{-1}[c,d]\right)(\Delta')^{-1}&=-(d-ca^{-1}b)^{-1}\left((d-ca^{-1}b)c+[c,d]\right)(\Delta')^{-1}=-(d-ca^{-1}b)^{-1}c\left(d-a^{-1}bc\right)(\Delta')^{-1}&=-(d-ca^{-1}b)^{-1}ca^{-1}&=-(ac^{-1}d-b)^{-1}.
\Bigl|\mathrm{div}\,\bar{\nu}+\overline{H}\Bigr| = \Bigl|\mathrm{div}\,\bar{\nu}-\overline{\mathrm{div}_\Gamma\nu}\Bigr| \leq c|d|\Bigl|\overline{W}\Bigr| \leq c|d| \quad\overline{N}
\begin{array}{c c c}{{x_{1 2}^{A}= x_{1}^{A}- x_{2}^{A}- i \bar{\theta}_{2 i}\gamma^{A}\theta_{1}^{i}- i \bar{\tilde{\theta}}_{2 \tilde{\imath}}\tilde{\gamma}^{A}\tilde{\theta}_{1}^{\tilde{\imath}}~}}&{{\theta_{1 2}^{i}= \theta_{1}^{i}- \theta_{2}^{i}~}}&{{\tilde{\theta}_{1 2}^{\tilde{\imath}}= \tilde{\theta}_{1}^{\tilde{\imath}}- \tilde{\theta}_{2}^{\tilde{\imath}}}}\\ \end{array}
\mathfrak{E}_{W}({v})= \mathfrak{E}_{W}({v},(-\infty,s_0])+\mathfrak{E}_{W}({v},[s_0,+\infty))\geq \mathfrak{E}_{W}({v},(-\infty,s_0])+d_K(B(a^-,\varepsilon_1),a^+).
\Psi ( a , \psi , \theta ) = \psi - \frac{1}{2}\theta^{\alpha \beta}a_{\alpha}\mathrm{P}_{-}\partial_{\beta}\psi + \frac{i}{8}\theta^{\alpha \beta}[ a_{\alpha}, a_{\beta}] \mathrm{P}_{-}\psi + o ( \theta^{2}) .
{\cal L}= \frac{1}{2 g}[ \sum_{i = 1}^{9}\mathrm{t r}\dot{X}_{n}^{i}\dot{X}_{- n}^{i}- \sum_{j = 2}^{9}\mathrm{t r}S_{n}^{j}( S_{n}^{j})^{+}- \frac{1}{2}\sum_{j , k = 2}^{9}\mathrm{t r}T_{n}^{j k}( T_{n}^{j k})^{+}] ,
\begin{cases}(\phi,\psi)|_{y=0}=(\phi,\psi)|_{y\rightarrow\infty}=(\phi,\psi)|_{x=1}=(\phi,\psi)|_{x\rightarrow\infty}=(0,0), (\phi_x,\psi_x)|_{x=1}=(\phi_x,\psi_x)|_{x\rightarrow\infty}=(0,0).\end{cases}
G^{{\mu}{\nu}{\sigma}}( B ) = -{\frac{1}{3 !}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\partial}_{[{\rho}}B_{{\eta}{\delta}]}\equiv -{\frac{1}{3 !}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}F_{{\rho}{\eta}{\delta}}( B ) ,
\mu(R_\lambda) =\int_{D_\lambda}\frac{d(\mu\circ \tau_\lambda)}{d\mu}\, d\mu=\int_{D_\lambda}\Phi_{\tau_\lambda}\, d\mu =\Phi_{\tau_\lambda}\mu(D_\lambda)=C^{-\omega d(\lambda)}\mu(R_{s(\lambda)})
g_3(x) = a\mu (x - x_0)^2 + a(c_A - \mu) -\delta(1 - \mu) - a\mu x_0^2 &\geq a(c_A - \mu) -\delta(1 - \mu) - a\mu \left(\frac{2\mu}{2 a\mu}\right)^2 &\geq a(c_A - \mu) -(1 - \mu) - \mu/a &\geq \left[(c_A - 1)a - 1\right] + \mu (1 - 1/a).
\epsilon:= ||X_P||_{s,r,p-1,\mathbf{a};\mathcal{O}_r}+ \frac{\alpha}{M}||X_P||^{lip}_{s,r,p-1,\mathbf{a}; \mathcal{O}_r}=O(r^{\frac74})\leq (\alpha\gamma)^{1+\beta},
\sum_{n=0}^\infty \sum_{m=0}^{n}\sum_{k + j = m}Z_k^{\mu}(t)Z_j^{\l}(s) r^n = \frac{1}{1-r}\sum_{n=0}^\infty \sum_{k + j = n}Z_k^{\mu}(t)Z_j^{\l}(s) r^n = \frac{1}{1-r}\sum_{k=0}^\infty Z_k^{\mu}(t) r^k \sum_{j=0}^\infty Z_j^{\l}(s) r^j = \frac{(1+r) (1-r^2)}{(1-2 r s + r^2)^{\l+1}(1-2 r t + r^2)^{\mu+1}}
\cal{M}(\phi_{1, \Delta}) = P \circ \begin{bmatrix}\lambda_1 0 \lambda_1+1 -1\end{bmatrix}\circ P^{-1}, \cal{M}(\phi_{2, \Delta}) = P \circ \begin{bmatrix}\lambda_2 0 -(\lambda_2+1) -1\end{bmatrix}\circ P^{-1}, \cal{M}(\phi_{3, \Delta}) = P \circ \begin{bmatrix}-1 0 0 \lambda_3\end{bmatrix}\circ P^{-1}
\P_x(\rho>n) = \int_{W^d}\P_x(\rho>[n/2], S_{[n/2]}\in dy ) \P_y(\rho>n-[n/2]) &\le \int_{W^d\cap \{\max_j (y_j-y_{j-1})\le \sqrt n\}}\P_x(\rho>[n/2], S_{[n/2]}\in dy ) \P_y(\rho>n-[n/2]) &+\sum_{j=2}^d \int_{W^d\cap \{(y_j-y_{j-1})> \sqrt n\}}\P_x(\rho>[n/2], S_{[n/2]}\in dy ) \P_y(\rho>n-[n/2]) &=: P_1+\sum_{j=2}^d P_j.
n^{-1/2}\big \Vert \sum_{i=1}^n d_{i,N}\big \Vert_2 \leq n^{-1/2}\Vert S_n \Vert_2 + 2 n^{-1/2}\Vert g_{0,N}\Vert_2 + n^{-1/2}\big \Vert \sum_{i=1}^n Y_{i,N}\big \Vert_2 \, .
L_{2,\phi}^m = -\Delta_\phi + \frac{m+n-1}{2}J_\phi^m , L_{4,\phi}^m = \Delta_\phi^2 + \delta_\phi\left(4P_\phi^m - (m+n-1)J_\phi^mg\right)d + \frac{m+n-3}{2}Q_\phi^m
F ( q^{2}) = 1 + a \alpha{\frac{q^{2}}{m^{2}}}\sqrt{\frac{m^{2}}{- q^{2}}}+ b \alpha{\frac{q^{2}}{m^{2}}}\log ( - q^{2}) + c \alpha{\frac{q^{2}}{m^{2}}}+ \ldots
+ \sum_{l = \mathrm{m a x}( n , 0 )}^{\infty}E_{l}( x | A ) \frac{\ln ( t m^{2})^{- 1}- \gamma + \psi ( l + 1 - n ) - \psi ( \frac{d}{2}- l )}{\Gamma ( \frac{d}{2}- l ) \Gamma ( l + 1 - n )}( - t )^{l - n}+
\Omega = a_{12}e^1\wedge e^2+a_{13}(e^1\wedge e^3-e^2\wedge e^6)+a_{14}(e^1\wedge e^4+e^2\wedge e^5-2e^3\wedge e^6) & +a_{15}(e^1\wedge e^5-e^2\wedge e^4) + a_{16}(e^1\wedge e^6+e^2\wedge e^3).
M^\nu_{\alpha,\mu}(1\vert \lambda) = \int_0^\infty e^{- \lambda t}\, dR^\nu_{\alpha,\mu}(t) (\lambda\ge0){\rm where}dR^\nu_{\alpha,\mu}(t) = \frac{1}{\Gamma(\mu)}\, \{I_+^\nu\, f_\alpha(\cdot\vert t)\}(1)\, t^{\mu-1}\, dt = \frac{1}{\Gamma(\mu)}\, \{I_+^\nu\, f_\alpha\}(t^{-1/\alpha})\, t^{\mu+(\nu-1)/\alpha-1}\, dt
\dot{\mathfrak{q}}_1(\epsilon,\partial_y\mathcal{Z}_1)-\big(\partial_y(\mathcal{L}_1\epsilon),\mathcal{Z}_1\big)-p\big([|V_0|^{p-1}-R_1^{p-1}]\epsilon,\partial_y\mathcal{Z}_1\big) &=\Big(\big[|V_0+\epsilon|^{p-1}(V_0+\epsilon)-|V_0|^{p-1}V_0-p|V_0|^{p-1}\epsilon\big],\partial_y\mathcal{Z}_1\Big)-(\Psi_{V_0},\mathcal{Z}_1),
C(a^{ij}) = \sup\limits_{A_{\rho,r}^+}\left| \frac{a^{ij}(x)-\delta^{ij}}{|x|^{\gamma}}\right| + \sup\limits_{\rho\leq \tilde{r}\leq r/2}\left\| |x|^{-\gamma}\nabla a^{ij}\right\|_{L^{n+1}(A_{\tilde{r},2\tilde{r}}^+)}.
\widetilde{\psi}_{k}^{( n^{2})}(x;N,M,\lambda_{1})=[ E_{n}(N,M) -E_{k}(N,M) ] \psi_{k}( x) -\frac{W(\psi_{n},\psi_{k}\,|\, x)}{\Psi_{n}^{(n)}(x;N,M,\lambda_{1})},
\tilde{u}(r) \geq \frac{Ca_{0}^{p}r^{2k + p\beta_0}}{(n + 2k + p\beta_0)^{2k}}\geq \frac{Ca_{0}^{p}r^{m + p\beta_0}}{(m + p\beta_0)^{m}},~ m = n+2k.
\delta v' \cdot\tau= \delta u_0' \cdot\tau_0\ \exp\left(-\int_0^t G{v^+}' \cdot \tau \right) - \exp\left(-\int_0^t G{v^+}'\cdot\tau \right) \int_0^t G^{-1}\mathcal{A}\exp\left(\int_0^s G{v^+}'\cdot\tau\right)\ds\,.
I=(\sigma^2i\partial_\mu\gamma^\mu)_{\alpha_j\alpha}(i\partial_\nu\gamma^\nu)_{\alpha\alpha_i}\Psi^{(\pm)}_{\alpha_i\alpha_j}=m^2\sigma^2_{\alpha_j\alpha_i}\Psi^{(\pm)}_{\alpha_i\alpha_j}=-I\;.
D^r_{sh}(U,F) = \left( H^r(U,F) \rightarrow \prod_{v \in X^{(1)}_0}H^r(K_v,F)\right), \mathcal{D}^{d+4-r}(U,F'):= \left( H^{d+4-r}_c(U,F') \rightarrow H^{d+4-r}(K_0, F')\right) .
A^{T}= \Gamma_{t}^{T}\cdots \Gamma_{1}^{T}= ( - \eta )^{t}{\cal C}\Gamma_{t}\cdots \Gamma_{1}= \eta^{t}{\cal C}A^{- 1}{\cal C}^{- 1}= ( - )^{t ( t + 1 ) / 2}\eta^{t}{\cal C}A{\cal C}^{- 1}.
E(V)=E_{0}+K_{0}V_{0}[\frac{1}{K_{0}^{\prime}(K_{0}^{\prime}-1)}(\frac{V}{V_{0}})^{1-K_{0}^{\prime}}+\frac{1}{K_{0}^{\prime}}\frac{V}{V_{0}}-\frac{1}{K_{0}^{\prime}-1}].
l_1(W,F,G,H)= W_{T,1}(F_T)+W_{T,2}(G_T)+W_{T,3}(H_T) &-W_{0,1}(F_0)-W_{0,2}(G_0)-W_{0,3}(H_0) &-\int_0^TW_{s,1}(\partial_sF_s)+W_{s,2}(\partial_sG_s)+W_{s,3}(\partial_sH_s)ds.
\begin{array}{c}B'_{\mu}(x)=B_{\mu}(x)-\frac{\mbox{$1$}}{\mbox{$h_{i}$}}\partial_{\mu,x}\alpha(x) \Delta'_{\mu}(x)=\Delta_{\mu}(x)-\frac{\mbox{$1$}}{\mbox{$g_{i}$}}\partial_{\mu,x}\gamma(x). \end{array}
\begin{aligned}\boldsymbol{W}\boldsymbol{1}\boldsymbol{1}^\top = \boldsymbol{1}\boldsymbol{1}^\top \boldsymbol{1}\boldsymbol{1}^\top\boldsymbol{W}^\top = \boldsymbol{1}\boldsymbol{1}^\top \boldsymbol{W}\boldsymbol{1}\boldsymbol{1}^\top\boldsymbol{W}^\top = \boldsymbol{1}\boldsymbol{1}^\top\end{aligned}
\left. \begin{array}{l l}\Lambda \!\!\!\! = x_E(y_3z_1\!-\!y_1z_3) \!+\! y_E(x_1z_3\!-\!x_3z_1) \!+\!z_E(x_3y_1\!-\!x_1y_3) ~~ - x_2[y_3(z_1\!-\!z_E)+y_1(z_E\!-\!z_3)+y_E(z_3\!-\!z_1)] ~~ - y_2[x_3(z_E\!-\!z_1)+x_1(z_3\!-\!z_E)+x_E(z_1\!-\!z_3)] ~~ - z_2[x_3(y_1\!-\!y_E)+x_1(y_E\!-\!y_3)+x_E(y_3\!-\!y_1)]. \end{array}\right.
\frac{c}{2a(1-a)}|(y-l_\alpha)^{1-a}-(x-l_\alpha)^{1-a}|=\int_x^{y}\frac{c}{2a(s-l_\alpha)^a}ds\leq \Big|\int_x^{y}D^{n}\varphi(s)ds\Big| &\leq |D^{n-1}\varphi(x)-D^{n-1}\varphi(y)|\leq \|D^{n-1}\varphi\|_{C^{1-a+\tau}}|(y-l_\alpha)-(x-l_\alpha)|^{1-a+\tau}.
\lim_{N \rightarrow \infty}\max_{\delta N_k \leq l \leq (1-\delta)N_k}\max_{\mu, \nu}\left \vert \mathbb{E}^V \theta(N \zeta_{\alpha^{\prime}}(\mu)\zeta_{\alpha^{\prime}}(\nu))-\mathbb{E}^V \theta[\frac{N}{\pi}\int_{I}\tilde{G}_{\mu \nu}(z) \mathcal{X}(E) dE] \right\vert =0,
I_{2\omega}=\frac{2\omega^{2}}{\pi c^{2}\epsilon_{0}w_{0}^{2}n_{2\omega}n_{\omega}^{2}}I_{\omega}^{2}(\chi^{(2)})^{2}(\int_{z_{0}}^{z_{0}+L}\frac{e^{i\Delta kz}}{1+iz/z_{R}})^{2}dz.
\sum_{k=0}^\infty u^kP\{\max_{0\le s\le t}\mathcal{T}(s)< \beta\ |\V(0) = c,\N(t) = 2k+1\}= \frac{\beta}{ct}\sum_{k=0}^\infty u^k\sum_{j=0}^{k}\binom{2j}{j}\Bigl(\frac{\sqrt{c^2t^2-\beta^2}}{2ct}\Bigr)^{2j}=
T_{1,k}=\Delta_{\tilde{n}}\left( \frac{xe^x}{e^x-1}\cdot\left( \frac{e^x-1}{e^{\frac{x}{M}}-1}\right)^{\tilde{n}-k}\cdot\left( \frac{e^x-1}{e^{\frac{x}{N}}-1}+\frac{e^x-1}{e^{\frac{x}{M-N}}-1}\right)^k\right),
\begin{aligned}Q(r_1,r_2;s_1,s_2) = \, Q(r_1+r_2,s_1+s_2)\oplus\left(\bigoplus_{k=1}^{\min(r_1,r_2)}Q(r_1+r_2-2k,s_1+s_2+k)\right) \ \ \oplus\left(\bigoplus_{k=1}^{\min(s_1,s_2)}Q(r_1+r_2+k,s_1+s_2-2k)\right) \ .\end{aligned}
\mathcal{L}_1^x R_2({\bf n; x}; \beta;N)=-n_1(n_1+\beta_2-\beta_0-1) R_2({\bf n; x}; \beta;N), \mathcal{L}_2^x R_2({\bf n; x}; \beta;N)=-(n_1+n_2)(n_1+n_2+\beta_2-\beta_0-1) R_2({\bf n; x}; \beta;N),
\Omega_{i}= \kappa_{i}( \flat_{i}^{+}\flat_{i}- \flat_{i + 1}^{+}\flat_{i + 1}) + \sum_{1 \leq q \leq i , \; i + 1 \leq p \leq n + 1}x_{p q}^{[ i ]}\flat_{p}^{+}\flat_{q}.
\Gamma_{\varepsilon}: \,\zeta_1=ib_0(\zeta_2)+\varepsilon e^{i\phi}, &\Gamma_{+, \varepsilon}: \,\zeta_1=e^{i\pi/2}w+\varepsilon , &\Gamma_{-, \varepsilon}: \,\zeta_1=e^{i\pi/2}w-\varepsilon .
r_{ij}=\frac{\sum_{m}\sum_{n}[f(m+i,n+j)-\overline{f}][g(m,n)-\overline{g}]}{\sqrt{\sum_{m}\sum_{n}[f(m,n)-\overline{f}]^{2}\sum_{m}\sum_{n}[g(m,n)-\overline{g}]^{2}}}.
u_1\circ \phi_{x_1}= \exp (u_0\circ \phi_{x_1}, \mathfrak{a}\circ \phi_{x_1}+ \xi\circ \phi_{x_1}) = \exp (u_0\circ \phi_{x_0}, \mathfrak{a}\circ \phi_{x_0}+ \xi\circ \phi_{x_0}+ du_0(\phi_{x_0})\eta +Q_{u_0}),
{\delta}B_{\mu\nu}^p={\cal D}_{\mu}^{pq}{\xi}_{\nu}^q -{\cal D}_{\nu}^{pq}{\xi}_{\mu}^q\equiv R_{\mu\nu\alpha}^{pq}{\xi}^{q\alpha},\,\,\,\,\,\,\,\,{\delta}A_{\mu}^p = 0,
I_k^1 = \int_{E\cap B_k}\int_{B_k\setminus E}p(t,x,y)\, d\mu(y)d\mu(x) \leq \int_{E\cap B_k}\int_{B_k\setminus E}\frac{C}{\mu(B_{\sqrt{t}}(x))}\,d\mu(y)d\mu(x) & \leq C\frac{\mu(B_k\setminus E)}{\mu(B_k)}\mu(E\cap B_k) \leq C\sqrt{t}P(E,2\lambda B_k).
\begin{aligned}\sum_{n\leq x}g_F(n) \ll x\exp\bigg\{-K\sum_{p\leq x}\frac{1-g_F(p)}{p}\bigg\}& \ll x\exp\bigg\{-2K\sum_{\substack{p\leq x g_F(p)=-1}}\frac{1}{p}\bigg\}& \ll \frac{x}{(\log x)^{K/16}}. \end{aligned}
\varphi=\frac{\det d\Phi}{\left\Vert \gamma_{t}^{\prime}\right\Vert}\mathbf{N}=\frac{\left( -\partial y/\partial\sigma\right) \mathbf{e}_{1}+\left( \partial x/\partial\sigma\right)\mathbf{e}_{2}}{\left\Vert \gamma_{t}^{\prime}\right\Vert}
\mathbb{P}[\exists \mathbf{x}\in \mathcal{F}^l ~\textrm{s.t.}~ \mathbf{D}\mathbf{x}= \mathbf{e}_i] = \mathbb{P}[Y_i > 0] \ge \frac{\mathbb{E}^2[Y_i]}{\mathbb{E}[Y_i^2]}\ge \frac{1}{1+\frac{1}{\mathbb{E}[Y_i]}}
\varphi(\bar{a})=\bar{a}^{i_1}z_1, \hskip3mm \varphi(\bar{b}) = \bar{a}^{i_2}\bar{b}^{j_2}z_2,\hskip3mm \varphi(\bar{c}) = \bar{a}^{i_3}\bar{b}^{j_3}\bar{c}^{k_3}z_3 \hskip3mm \varphi(\bar{d}) = \bar{a}^{i_4}\bar{b}^{j_4}\bar{c}^{k_4}\bar{d}^{l_4}z_4,
{\cal V}[ \phi_{0}] = V [ \phi_{0}, m ( \mu_{0}) , \lambda ( \mu_{0}) ] +{\frac{\hbar}{6 4 \pi^{2}}}\left\{{\cal M}_{0}^{4}( \mu_{0}) - m_{0}^{4}\right\}\ln{\frac{\mu^{2}}{\mu_{0}^{2}}}+ \hbar X [ \phi_{0}, m ( \mu ) , \lambda ( \mu ) , \mu ] + O ( \hbar^{2}) .
R_{l}( r ) = \frac{u_{l}( r )}{r^{( D_{0}- 1 ) / 2}}\stackrel{( r \rightarrow 0 )}{\sim}\mathrm{\boldmath \large \left\{\right.}\! \! \! r^{\alpha_{+ , l}}\mathrm{\boldmath \large ,}\! r^{\alpha_{- , l}}\mathrm{\boldmath \large \left. \right\}}\! \! ,
\frac{1}{2}\int_0^ku^2dx+\frac{Ca}{2}\frac{1}{\gamma-1}\int_0^kH(\rho)dx+P(R)+\mu\int_0^t\int_0^k\rho(r^2u_x)^2dxd\tau+2\mu\int_0^t\int_0^k\rho^{-1}\frac{u^2}{r^2}dxd\tau=E_0.
\frac{1}{S_k}\sum_{i=0}^k\frac{S_i}{(-\alpha_i)^{\frac{\rho}{2-\rho}}}\leq\frac{p+1}{(\sigma_d\bar\sigma_f+pL+(p+1)\sigma_d\beta)^{\frac{\rho}{2-\rho}}}\frac{1}{(k+1)^{p+1}}+ 3^{p+1}\left(\frac{2^{p-1}(p+1)^2}{\sigma_d\sigma_f}\right)^{\frac{\rho}{2-\rho}}P(k),
S ( B ) \approx S ( A ) +{\frac{1}{2}}\int \! d^{4}x d^{4}y b_{\mu}^{a}( x ){\frac{\delta^{2}S ( A )}{\delta A_{\mu}^{a}( x ) \delta A_{\nu}^{b}( y )}}b_{\nu}^{b}( y ) .
u(z)= t(z) \phi^{n\sigma_3}(z) = O_n\left(\frac{1}{z}\right) \left(\begin{matrix}z^n + O_n(z^{n-1}) 0 0 z^{-n}+ O_n\big(z^{-n-1}\big)\end{matrix}\right) \hphantom{u(z)}{}=O_n\left(\begin{matrix}z^{n-1}z^{-n-1}z^{n-1}z^{-n-1}\end{matrix}\right).
u_1(z) = \frac 13 z u_2(z) = \frac 13 e^{\pi i/3}z + \frac 13 u_3(z) = \frac 13 e^{-\pi i/3}z + \frac 13 + \frac 13 e^{2\pi i/3}u_4(z) = \frac 13 z + \frac 23.
\Theta \left[^a_b\right]\left(z+n+\Omega\cdot m|\Omega\right)=e^{-\pi i m \cdot\Omega\cdot m -2\pi i m \cdot z+2\pi i (a\cdot n-b\cdot m)}\Theta\left[^a_b\right]\left(z|\Omega\right).
\overline{{{V_{A}^{a}}}}= V_{\dot{A}a}, \overline{{{V^{A a}}}}= - V_{a}^{\dot{A}}, \overline{{{V_{A a}}}}= - V_{\dot{A}}^{a}, \overline{{{V_{a}^{A}}}}= V^{\dot{A}a}, \overline{{{\varepsilon_{a b}}}}= \varepsilon^{a b}, \overline{{{\chi^{A a}\psi^{B b}}}}= - \overline{{{\chi^{A a}}}}\ \overline{{{\psi^{B b}}}}
q^{r-2a}[2\ell-r] +q^{2\ell-a}[a]+q^{2c+r-2\ell-a}[r-2c-a]\frac{q^{-2r}K^{-2}-1}{q^{4c-2r}K^{-2}-1}&+q^{2c-2\ell}[2c] \frac{q^{-2a}K^{-2}-1}{q^{4c-2r}K^{-2}-1}= [2\ell].
\gamma_1 = \xi(1,-5)-\xi(2,5)-\xi(1,-2) = \left\{\frac15,\frac25\right\}\gamma_2 = \langle 2 \rangle_* \gamma_1 = \xi(2,3)-\xi(4,-3)-\xi(2,-4) \gamma_3 = \xi(1,-3)-\xi(1,3) = \left\{\frac13,-\frac13\right\}\gamma_4 = \langle 2 \rangle_* \gamma_3 = \xi(2,-6)-\xi(2,6).
\sigma_0^{B=0}={{\pi^4 (\tilde \omega R_0)^8}\over{8 \omega^5}}\left \{1 -{1\over 6}\left ( \tilde \omega R_0 \right )^4 \log (\tilde \omega \bar R_0 ) +{7 \over 72}(\tilde \omega R_0)^4 + \cdots \right \}
H=\begin{pmatrix}\cosh(s) 0 0 \sinh(s)\cos(d) \sinh(s)\sin(d) 0 \cosh(s) 0 \sinh(s)\sin(d) -\sinh(s)\cos(d) 0 0 1 0 0 \sinh(s)\cos(d) \sinh(s)\sin(d) 0 \cosh(s) 0 \sinh(s)\sin(d) -\sinh(s)\cos(d) 0 0 \cosh(s)\end{pmatrix}\, ,
\Pi^{\mathrm{o n e - l o o p}}( p^{2}) = \frac{N g^{2}m}{8 \pi}\left\{\left( \frac{1}{1 6}\frac{p^{6}}{m^{6}}- \frac{1}{2}\frac{p^{4}}{m^{4}}- \frac{5}{2}\frac{p^{2}}{m^{2}}+ 2 \right) \frac{2 m}{p}\tan^{- 1}\frac{p}{2 m}+ \frac{1}{4}\frac{p^{4}}{m^{4}}- \frac{p^{2}}{m^{2}}- 2 \right\}
{\ \chi_{n}^{\mu \upsilon}=\left( \eta_{n_{1}}^{\mu}\sin n\tau -\eta_{n_{2}}^{\mu}\cos n\tau \right) \left( \eta_{n_{1}}^{v}\sin n\tau -\eta_{n_{2}}^{v}\cos n\tau \right) n^{2}\cos^{2}n\sigma -}
\frac{(\#\mathcal{E}')^{5/4}}{B^2}\frac{X}{N K}\ll X^{115/160}=X^{23/32}, \frac{(\#\mathcal{E}')^{3/2}}{B^2 X^{1/2}}\Bigl(\frac{X}{N K}\Bigr)^2 \ll X^{75/80}=X^{15/16}.
\sum_{\nu}\,^{\prime}\,\sum_{\mu}\sum_{\kappa}\int d^2 x \int d^2 y \int d^2 z(\Psi_{\nu}^{*}(x) \delta\lambda_{\mu}\Psi_{\mu}(x))(\Psi_{\mu}^{*}(y) e^{-\epsilon\lambda_{\kappa}}\Psi_{\kappa}(y))(\Psi_{\kappa}^{*}(z){1\over{\lambda_{\nu}}}\Psi_{\nu}(z))\, ,
\||x|^\gamma u_2\|^p_p\leq C \int_{E_2}\int_{E_2}|x|^\gamma\Gamma(x,y)|f(y)|^p dydx &= C\int_{E_2}|f(y)|^p \left(\int_{E_2}|x|^\gamma \Gamma(x,y)dx\right)dy\leq C\|f\|_p^p\;.
F(x_1 , \cdots , x_n | z_1 , \cdots ,z_N )=\frac{\Delta^{(n l)}(x_1 , \cdots , x_n | z_1 , \cdots , z_n | z_{n+1}, \cdots , z_N )}{\displaystyle\prod_{j=1}^{n}\prod_{i=n+1}^{N}(z_i -z_j \tau^2 )},
{\cal Z}_{open}^{9-9}=\int_0^\infty\frac{ds}{s}\left[{\rm Tr}_{\rm NS}\left({{1+(-1)^{F}\over 2}}{1+ \Omega \over 2}\,q^{2L_0-1}\right)-{\rm Tr}_{\rm R}\left({{1+(-1)^{F}\over 2}}{1+ \Omega \over 2}\,q^{2L_0}\right)\right]~~,
\langle \zeta^{\prime}| p_{\mu}| \zeta^{\prime \prime}\rangle = - i \hbar \frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}\langle \zeta^{\prime}| \zeta^{\prime \prime}\rangle + \frac{\partial F ( \zeta^{\prime \prime})}{\partial \eta^{\prime \prime}{}^{\mu}}\langle \zeta^{\prime}| \zeta^{\prime \prime}\rangle ,
N^{(1)}_{(f,H_1),\leqslant k_1}(r)\geqslant\sum_{s\ne 1,i}N^{(1)}_{(f,H_s),\leqslant k_s}(r)-N^{(1)}_{(f,H_i),\leqslant k_i}(r)-2\sum_{u=1}^3\sum_{s=1,i}N^{(1)}_{(f^u,H_s),>k_s}(r)+o(T(r)).
A(\beta, \varrho) = -\frac{KL}{2}\log \left(2\pi \beta^2\right) - \frac{KL \left( \sigma^2(\varrho)+m^2(\beta , \varrho) \right)}{2\beta^2}- \frac{K^2 m^2(\beta , \varrho)}{4 \beta^2},
\begin{aligned}\varkappa_d(F)_i=3 \Longleftrightarrow \quad\mathcal{G}_i\simeq\langle 243,27\rangle \varkappa_s(\mathcal{G}_i)=(1,0,0,0), \varkappa_d(F)_i=9 \Longleftrightarrow \quad\mathcal{G}_i\simeq\langle 243,26\rangle \varkappa_s(\mathcal{G}_i)=(0,0,0,0).\end{aligned}
g_{11}^{\ast}=\left \vert \delta \right \vert^{-5}\left[ \frac{2\delta \delta \,\acute{}\,\acute{}-3\delta \,\acute{}\,^{2}-4\delta^{2}}{4}+\frac{\left( 2\delta \kappa \,\acute{}-3\delta \,\acute{}\, \kappa \right) v}{2}\right]^{2}\!+\kappa^{2}v^{2}\left \vert\delta \right \vert^{-3}+\kappa^{2}\left \vert \delta \right \vert^{-1},
\left( \frac{\partial}{\partial t}-{\mathcal L}\right) U(t, x,\omega) +{Q (x,\omega)}\lozenge U(t, x,\omega) = F(t, x ,\omega), t\in (0,T],\, x\in \mathbb R^d,\, \omega \in \Omega U(0, x,\omega) = G(x,\omega)x\in \mathbb R^d,\, \omega \in \Omega .
G^j(z)=[G^j(z)]_+ + \sum\limits_{\chi}\left( \sum\limits_{i=0}^{\infty}\displaystyle\frac{\bar P_{i,\chi}^j}{(z+\chi)^i}\sum\limits_{l=0}^{i-1}\displaystyle\frac{\partial^i}{\partial z^l}S_{u^j}(\chi)(-z-\chi)^l \right) .
A_1=1-Im\Big\{(I-\mathbb{H})(F_z\circ Z Z_{\alpha}D_t Z)+(I-\mathbb{H})\sum_{j=1}^N \frac{\lambda_j i (D_t Z Z_{\alpha}-\dot{z}_j(t)Z_{\alpha})}{2\pi(Z(\alpha,t)-z_j(t))^2}\Big\}
\frac{\sqrt{\pi}}{8}\frac{L}{\sqrt{\tau}}\int_{\left\vert y\right\vert \geq 1}e^{-\left\vert y\right\vert^{2}L^{2}/(4\tau)}dy \geq \frac{\sqrt{\pi}}{8}\frac{L}{\sqrt{\tau}}2e^{-L^{2}/\tau}& = \frac{\sqrt{\pi}}{4}\frac{L}{\sqrt{\tau}}e^{-L^{2}/\tau},
(FG)_{lm}\ =\ (-1)^{n+m}\displaystyle{\sum_{l_{1},l_{2}=0}^{n}\sum_{m_{1}=-l_{1}}^{l_{1}}}\left[\begin{array}[c]{ccc}l_{1}l_{2}l_{{}}m_{1}m_{2}-m_{{}}\end{array}\right] \!\!{[j]}F_{l_{1}m_{1}}G_{l_{2}m_{2}}
\xi^*_n=\begin{cases}\beta-\left(X(\tau^*_n-)+\Delta Y(\tau^*_n)\right), \textrm{if}~X(\tau^*_n-)+\Delta Y(\tau^*_n)\ge b , \alpha-\left(X(\tau^*_n-)+\Delta Y(\tau^*_n)\right), \textrm{if}~X(\tau^*_n-)+\Delta Y(\tau^*_n)\le a. \end{cases}