harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbf{M}=\Big(M_{T_1,T_2}^{1,2,5-\alpha}(\omega),M_{T_1,T_2}^{3,N,2-\alpha}(\omega),\dots,M_{T_1,T_2}^{\frac{N}{2},\frac{N}{2}+3,2-\alpha}(\omega),M_{T_1,T_2}^{\frac{N}{2}+1,\frac{N}{2}+2,5-\alpha}(\omega)\Big)^{\top}
\frac{1}{8 \pi}\int_{\Sigma_r}X_i ( div_{\sigma}\alpha_{H}- div_{\sigma}\alpha_{H_0})d \Sigma_r = \frac{1}{8 \pi}\int_{\Sigma_r}r \tilde X^i ( div_{\sigma}\alpha_{H}-\frac{1}{2}\tilde \Delta ( \tilde \Delta +2 ) X_0^{(3)}) d \Sigma_r +O(r^4). = \frac{1}{8 \pi}\int_{\Sigma_r}X^i ( div_{\sigma}\alpha_{H})d \Sigma_r +O(r^4).
d s^{2}= \left( \frac{r}{R_{0}}\right)^{2 ( d_{\perp}- 2 ) / ( p + 1 )}d x^{\mu}\otimes d x^{\nu}\, \eta_{\mu \nu}+ \left( \frac{R_{0}}{r}\right)^{2}d y^{I}\otimes d y^{J}\, \delta_{I J}\ .
x^n_{t\wedge \tau_n}-x_{t\wedge \tau_n}=\int_0^{t\wedge \tau_n}\Big\{\tilde{f}_n(s,x_s^n)-\tilde{f}(s,x_s)\Big\}ds &=\int_0^t \chi_{_{[0,\tau_n)}}(s)\Big\{\tilde{f}_n(s,x_{s\wedge\tau_n}^n)-\tilde{f}_n(s,x_{s\wedge\tau_n})\Big\}ds.
\nabla_k R^k_{\;j}= - \nabla_k \nabla^k\nabla_j f &=(\nabla_j \nabla_k -\nabla_k \nabla_j) \nabla^k f - \nabla_j \nabla_k \nabla^k f &=R^k_{\; ajk}\nabla^a f + \nabla_j R &=- R_{aj}\nabla^a f + 2\nabla_k R^k_{\;j}
\mathcal{J}^{\pm}_{1}(y; \mu) = \Bigl| \frac{y_1}{y_2}\Bigr|^{\frac{1}{2}\mu_2}\int_0^\infty J^{\pm}_{3\nu_3}\left(2\pi |y_1|^{1/2}\sqrt{1+u^2}\right) J^{\pm}_{3\nu_3}\left(2\pi |y_2|^{1/2}\sqrt{1+u^{-2}}\right) u^{3\mu_2}\frac{du}{u},
\frac{t_r}{t_s}= \sum_{i=0}^{s-r}\frac{a^{-i}q^{i(2s-d)}(\theta_r - \theta_s)(\theta_r - \theta_{s-1})\cdots (\theta_r - \theta_{s-i+1})}{(q^2;q^2)_i}, \frac{t_s}{t_r}= \sum_{i=0}^{s-r}\frac{a^{i}q^{i(d-2s)}(\theta_r - \theta_s)(\theta_r - \theta_{s-1})\cdots (\theta_r - \theta_{s-i+1})}{(q^{-2};q^{-2})_i}.
\frac{d\Phi^1_t(x_1^0,x_2)}{dt}=u_1(\Phi_t^1(x_1^0,x_2),x_2,t),\frac{d}{dt}\Phi_t^2(x_1^0,x_2)\equiv 0,\ \Phi_{t=0}(x_1^0,x_2)=(x_1^0,x_2).
\sum_{1-n \leq 2r \leq n}\frac{1}{4r^2 - 1}\left(1 +\frac{r}{j n}\right)^{-s-1}& = -1 + \sum_{k=1}^{\frac{n-1}{2}}\frac{1}{4k^2 - 1}\left[ \left(1 + \frac{k}{jn}\right)^{-s-1}+ \left( 1 - \frac{k}{jn}\right)^{-s-1}\right].
A_{ij}y_{j}=\begin{pmatrix}a_{ij}b_{ij}b_{ij}a_{ij}\end{pmatrix}x_{j}\mathbf{f=}\begin{pmatrix}a_{ij}-b_{ij}b_{ij}-a_{ij}\end{pmatrix}x_{j}=c_{ij}x_{j}\mathbf{f}
\begin{cases}3\,\lambda + 2\,\mu + \kappa > 0, 2\,\mu + \kappa > 0,\kappa > 0, & 3\,\alpha + \beta + \gamma > 0, \gamma + \beta > 0, \gamma - \beta > 0, & d >0, k > 0,T_0 >0.\end{cases}
\sum_{k=1}^{j_2}C_k = \left(C_1+C_{j_2}\right)+\sum_{k=2}^{j_2-1}C_j &\leq \left(2p_1+L+\sum_{k=1}^{j_2}p_k+p_{j_2-1}+p_{j_2}+2L\right)+\sum_{k=2}^{j_2-1}C_j^{OPT_f}&\leq 1.5\left(C_1^{OPT_f}+C_{j_2}^{OPT_f}\right)+\sum_{k=2}^{j_2-1}C_j^{OPT_f}\leq 1.5\sum_{k=1}^{j_2}C_j^{OPT_f}
\sum_{j=0}^{\infty}\mu^{j/2}\exp\left( -\frac{2 \pi M \mu^j}{\tau_{0}^{\alpha}}\right) \geq\frac{2}{\ln\mu}\left( \frac{\pi e^{-4 (\frac{2 \pi M}{\tau_{0}^{\alpha}})}}{12\left( \frac{2\pi M}{\tau_{0}^{\alpha}}\right)}\right)^{1/2}& =\frac{1}{\ln\mu}\sqrt{\frac{\tau_{0}^{\alpha}}{6M}}\, e^{-4 \pi M / \tau_{0}^{\alpha}}.
\left[ \delta_{\zeta_{1}}, \delta_{\zeta_{2}}\right] f = \{f , \{\zeta_{1}^{\alpha}\phi_{\alpha}, \zeta_{2}^{\beta}\phi_{\beta}\}\}= \{f , \zeta_{1}^{\alpha}\zeta_{2}^{\beta}{C_{\alpha \beta}}^{\gamma}\phi_{\gamma}\}\approx \zeta_{1}^{\alpha}\zeta_{2}^{\beta}{C_{\alpha \beta}}^{\gamma}\{f , \phi_{\gamma}\}\ .
\mathbf{a}(\theta_{0})^{T}\mathbf{v}^{0}=e^{-jm\phi}+e^{-jn\phi}+e^{-jk\phi}+e^{-jl\phi}&=e^{-jm\phi}. &(1+e^{-j(n-m)\phi}(1+e^{-j(k-n)\phi}(1+e^{-j(l-k)\phi})))
V(t,y) =\Theta(\vec{x}(t),\vec{\mu}(t),y)=\sum_{i=1}^n\sigma_iQ_{1+\mu_{i}(t)}(y-x_{i}(t))+\sum_{\substack{i,j=1 j\not=i}}^n\frac{A_{ij}(t,y-x_{i}(t))}{x^2_{ij}(t)}&+\sum_{\substack{i,j=1 j\not=i}}^n\frac{B_{ij}(t,y-x_{i}(t))}{x_{ij}^3(t)}\varphi_{ij}(t,y),
\begin{aligned}\lim_k \bigl( \|X_1J_{\theta,1}^{-1}U_\mu^{n_k}p\|^2 -\|pX_1J_{\theta,1}^{-1}\chi^{-n_k}\|^2\bigr)= 0 \ \ \lim_k \bigl(\|X_1J_{\theta,1}^{-1}U_\mu^{-n_k}p\|^2-\|pX_1J_{\theta,1}^{-1}\chi^{-n_k}\|^2\bigr)=0\end{aligned}
lim_{N\rightarrow\infty}\int_{0}^{2\pi}\int_{0}^{\pi}|f(\theta,\varphi)-\sum_{l=0}^{N}\sum_{m=-l}^{l}f_{l}^{m}Y_{l}^{m}(\theta,\varphi)|^{2}sin\theta d\theta d\varphi=0.
\biguplus_{n\in\omega}K_n \uplus \{x\}= \biguplus_{n\in\omega}(K\cap F_n) \uplus \{x\}= \left(K\cap\biguplus_{n\in\omega}F_n\right) \uplus \{x\}= K\cap\left(\biguplus_{n\in\omega}F_n\uplus \{x\}\right) = K \cap E = K.
\mathcal{L}_{_{5},\mathfrak{C}_{5}}=\alpha_{0}\,\varepsilon_{abcde}\left( R^{ab}e^{c}e^{d}e^{e}+\frac{3}{10\ell^{2}}e^{a}e^{b}e^{c}e^{d}e^{e}+\frac{3}{4}\ell^{2}R^{ab}R^{cd}h^{e}+\frac{1}{\ell^{2}}e^{a}h^{b}e^{c}h^{d}h^{e}+\frac{3}{2}R^{ab}e^{c}h^{d}h^{e}\right) ,
c^{(1)}_j = \frac{\theta^j}{j!}+\eta_1\theta^{j-1}\left( \frac{g\delta^2+\delta(d-g+1-n)}{(j-1)!}+\frac{\delta-\delta^2}{(j-2)!}\right) +\frac{\delta}{(j-1)!}\gamma_{1,3}\theta^{j-1}
\partial_k v_k = \int_{|(y,s)|<1}\sum_{m=1}^d \sum_{\substack{|\mu|+ 2l =m \mu_k >0}}D_x^{\mu}D_t^l K_{jk}(-y, -s) \frac{x^{\mu-e_k}t^l}{(\mu-e_k)! l!}f_j (y,s) \,dy \,ds,
\begin{aligned}\log \hat Z_{N,2,1}\geq -( I_{1,2,1}+I_{2,2,1}) \geq -C \left[\gamma N^{2}\big( (\beta^{-1/2}a^{-1}N \log N) \vee 1 \big) +\beta N^2 a^2\right].\end{aligned}
\Pi_{_h\,\rho ,}^\rho \,^{\mu \nu}\left( k\right) =-\kappa\;\Gamma_{_h\,}^{\mu \nu}=-\kappa^2\omega\frac{\pi^2T^4}{180}\left( 4\,\eta^{\mu 0}\eta^{\nu 0}-\eta^{\mu \nu}\right) ,
{\cal W}(x,x_0) =\exp\left\{-i\biggl[ \int_{x_0}^x dx_j A_j + \frac{\theta}{2}\, \int_{x_0}^x dx_j \,\left[ m \, ({\bf v}\times \nabla A^{(0)}_j)_3 - ({\bf A^{(0)}}\times \nabla A^{(0)}_j)_3 \right]\biggr]\right\}.
\Phi(\alpha, e_i) = \frac{\theta(a, \alpha(a), \xi_{c'}^{(i)})}{\theta(a, \alpha(a), \xi_{c}^{(i)})}=\frac{u_{\alpha}(\xi_{c'}^{(i)})}{u_{\alpha}(\xi_{c}^{(i)})}=\frac{u_{\alpha}(\xi_{c'}^{(i)})}{u_{\alpha}(z_0)}
a_{63E}(t)= \, 16 + 576 t - 1408 t^2 - 1440 t^3 + 1608 t^4 + 720 t^5 - 352 t^6 - 72 t^7 + t^8, b_{63E}(t)= \, 27648 (-2 + t)^3 t^3 (1 + t)^3 (2 + t^2)^2,
\sum_{k=1}^{\ell}\xi^{-a_{i+k}}\xi^{a_{j+k}}=\sum_{k=1}\xi^{a_{j+k}-a_{i+k}}=\xi^{a_j-a_i}\sum_{k=1}^\ell \xi^{k(j-i)}=\delta_{ij}\ell \xi^{a_j-a_i}=\delta_{ij}\ell
U(\gamma_1,\gamma_2;m,r):=\frac{(-2\gamma_1-1)^{-r}(-2\gamma_2-1)^{-r}(-\gamma_1-\gamma_2-1)^{-(m-2r)}}{\left[(-\gamma_1-\gamma_2-1)^{-2}+ (-2\gamma_1-1)^{-1}(-2\gamma_2-1)^{-1}\right]^{m/2}}.
\Delta{C}_t= \big(\mu_1(\beta-1)-\mu_0(\alpha-1)\big)+H(\alpha)-H(\beta)+\log\Big(\frac{1+2^{\mu_1+\Delta{C}_{t+1}}}{1+2^{\mu_0+\Delta{C}_{t+1}}}\Big),~\Delta{C}_{n+1}=0, = f(\alpha,\beta,\mu_0,\mu_1,\Delta{C}_{t+1}),~t\in\{n,\ldots,0\}
|\partial_{t}^{n}D_{x}^{m}q(t,x)| \leq N|x|^{-d-m}t^{-n+\alpha-1}(R^2+R^2 \ln R \cdot 1_{d=2}) + N|x|^{-d}t^{-n+\alpha-1}\left( R^{1/2}\cdot 1_{d=1}+ R \cdot 1_{d=2}+R^2 \ln R \cdot 1_{d=4}\right) 1_{m = 0}.
\Delta \phi \leq \int_{t_{0}}^{\infty}\dot{\phi}d t = \dot{\phi}_{0}\int_{t_{0}}^{\infty}\left({\frac{t_{0}}{t}}\right)^{\frac{2}{1 + \alpha}}d t ={\frac{1 + \alpha}{1 - \alpha}}\dot{\phi}_{0}t_{0}.
G_{\boldsymbol{Y}}(\boldsymbol{s}) = \sum_{\boldsymbol{y}\in \mathbb{N}^J}\Gamma\left( \vert \boldsymbol{y}\vert +1 \right) P\left( \vert \boldsymbol{Y}\vert = \vert \boldsymbol{y}\vert \right) \frac{\prod_{j=1}^J(\alpha_j)_{y_j}}{(\vert\boldsymbol{\alpha}\vert)_{\vert\boldsymbol{y}\vert}}\prod_{j=1}^J \frac{s_j^{y_j}}{y_j!}.
\Gamma_3=\frac{\lambda^3}{8\left(2\pi\right)^5}\int^\infty \frac{{\rm d}|{\bf Q}|}{|{\bf Q}|}N\left(|{\bf Q}|\right)\int{\rm d}\Omega\left\{\frac{k_1^2\,k_2\cdot k_3}{\left(\hat Q\cdot k_1\right)^2 \hat Q\cdot k_2 \hat Q\cdot k_3}+ permutations\right\},
||a_j - u \varphi(a_j)u^*||_2 \quad\leq ||a_j - b_j||_2 + ||b_j - u\varphi_K(b_j)u^*||_2 \;+ & ||u\varphi_K(b_j)u^* - u \varphi(a_j)u^*||_2 = ||a_j - b_j||_2 + ||b_j - u\varphi_K(b_j)u^*||_2 + ||\varphi(b_j) - \varphi(a_j)||_2 < \frac{\varepsilon}{3}+ \frac{\varepsilon}{3}+ \frac{\varepsilon}{3}= \varepsilon.
\hat{\mathrm{T}}_{b}\equiv \hat{\pi}_{[ \frac{e_{+}b \mathrm{L}}{2{\pi}{\hbar}}]}- \frac{2 \pi}{2 \pi}{L}\int_{- \mathrm{L}/ 2}^{\mathrm{L}/ 2}d x x \cdot ( \hat{j}_{+}( x ) + \mathrm{N}\hat{j}_{-}( x ) ) \equiv - i \hbar \frac{d}{d [ \frac{e_{+}b \mathrm{L}}{2{\pi}{\hbar}}]}+ i{\hbar}\sum_{\stackrel{n \in \cal Z}{n \neq 0}}\frac{( - 1 )^{n}}{n}\rho_{\mathrm{N}}( n )
\begin{aligned}\int_{\Gamma_h}e^{\rho \cdot \mathbf x}\mathrm d\sigma =\sin\theta_0\frac{2\pi\Gamma(2)}{\tau^2}\frac{1}{((\mathbf d+\mathrm i\mathbf d^{\perp})\cdot \mathbf{\hat{x}}(\theta_0,\varphi_\xi))^2}-\sin\theta_0\int_{0}^{2\pi}I_{R}\mathrm d\varphi, \end{aligned}
f_1(x) =1+x^e, f_2(x) =1-x^e, f_3(x) =1+x^{2e}, f_4(x) =1-x^{2e}, f_5(x) =1+x^e+x^{2e}, f_6(x) =1-x^e+x^{2e}, f_7(x) =1+x^e-x^{2e}, f_8(x) =1-x^e-x^{2e}.
\kappa_p^{(N)}(k_1, \ldots, k_p)=\sum_{m=1}^p \frac{(-1)^{m}}{m}\*\sum_{\substack{(p_1, \ldots, p_m): p_1+\ldots +p_m=p, \p_1, \ldots p_m\geq 1}}\frac{1}{p_1!\cdots p_m!}\* \sum_{\sigma \in S_p}\* J_N(p_1, \ldots, p_m; k_{\sigma(1)},\ldots, k_{\sigma(p)}),
|\eta''^{\!}(\theta)| =\Bigl|\tfrac{d^2}{d\theta^2}u\bigl(3r+\theta-a-cu(3r+\theta)\bigr)\Bigr| & =\Bigl|\bigl(1-c\dot{u}(3r+\theta)\bigr)^2\ddot{u}\bigl(3r+\theta-a-cu(3r+\theta)\bigr)-c\ddot{u}(3r+\theta)\bigr)\dot{u}\bigl(3r+\theta-a-cu(3r+\theta)\bigr)\Bigr| & \leq\bigl(D_1^2 +(|\mu|+|\sigma|)^3|c|\delta\bigr)\bigl(1+|\sigma c|\delta\bigr)\delta=D_2 \delta.
z_{\gamma_m}= \frac{1}{2\pi}[-\ln|c| \cdot c_1 + \arg c \cdot c_2 + c_1 + S], z_{\gamma_e}= c_2 \theta_{\gamma_e}= \frac{2\pi \cdot t_1}{S_1 - \ln|c|},\theta_{\gamma_m}= t_2 -\frac{S_2+ \arg c}{S_1 - \ln |c|}\cdot t_1
N_m=\left(\frac{m!(2m+2A)\Gamma(m+2A)}{2^{2A-2}(m+A-B+\frac{1}{2})(m+A+B+\frac{1}{2})\Gamma(m+A-B-\frac{1}{2})\Gamma(m+A+B-\frac{1}{2})}\right)^{\frac{1}{2}}
-\frac{1}{\pi}\int_{\rho}^{\infty}{\frac{\bar K_{\nu}(x)}{\bar I_{\nu}(x)}\left[ e^{-\nu \pi i}f(xe^{\pi i/2})+e^{\nu \pi i}f(xe^{-\pi i/2})\right]dx}+ \int_{\gamma_{\rho}^{+}}{f(z) \frac{\bar H^{(1)}_{\nu}(z)}{\bar J_{\nu}(z)}dz}-\int_{\gamma_{\rho}^{-}}{f(z) \frac{\bar H^{(2)}_{\nu}(z)}{\bar J_{\nu}(z)}dz},
\|f\|_{\mathbb{M}^s_{\mathrm{e}^2}}= \sum_{k=0}^\infty |\hat{f}_k|^2 \Big(\frac{\pi}{\pi - s}\Big)^{k+1}= \frac{\pi}{\pi - s}\sum_{k=0}^\infty |\hat{f}_k|^2 \exp \Big( \ln (\frac{\pi}{\pi - s}) k \Big) = \frac{\pi}{\pi - s}\|f\|_{\mathbb{E}^1_q}\, .
\frac{\Delta_{i,i+1}(u^{(n)}w_0^{(n)})}{\Delta_i(u^{(n)}w_0^{(n)})}=\frac{1}{y_{n-1,n-i}}+\frac{y_{n-1,n-i-1}}{y_{n-1,n-i}}\frac{\Delta_{i,i+1}(u^{(n-1)}w_0^{(n-1)})}{\Delta_i(u^{(n-1)}w_0^{(n-1)})},
\mathfrak{P}_\nu^{1}(P,Q):= \biggl\{f\in\mathcal{H}_k^*(N): \biggl|\sum_{\substack{P<p\leq Q p\nmid N}}\frac{\lambda_f(p^\nu)}{p}\biggr|>\frac{10(\nu+1)}{(\log (kN)) (\log P)}\biggr\}.
g_{2}((\nabla\pi_*)(U,X),\pi_*Y) =g_{1}(\nabla^{^{M_1}}_{U}\phi \mathcal{B}X,Y)+g_{1}(\nabla^{^{M_1}}_{U}\omega\mathcal{B}X,Y)-g_{1}(\mathcal{T}_{U}\mathcal{C}X,\mathcal{B}Y)-g_{2}(\pi_*(\mathcal{H}\nabla^{^{M_1}}_{U}\mathcal{C}X),\pi_*\mathcal{C}Y) &+\eta(X)g_{1}(QU,\varphi Y)-U\eta(X)\eta(Y)-g_{1}(X,\omega U)\eta(Y).
\sum_{i=0}^{n-1}a_{k,i}a_{l,i}=\sum_{i=0}^{n-1}h_{0,j}h_{k,i}h_{0,j}h_{l,i}=\sum_{i=0}^{n-1}h_{0,j}^{2}h_{k,i}h_{l,i}=\sum_{i=0}^{n-1}h_{k,i}h_{l,i}=0
r_k \leq I(y_k;{\bf x}_{B_k})=h({y}_k)-h(y_k|{\bf x}_{B_k}),\quad\forall k\in\mathcal{C}, r^{'}_j \leq I(\hat{z}_j;{\bf x}_l)=h(\hat{z}_j)-h(\hat{z}_j|{\bf x}_{l}),\quad\forall j\in\mathcal{D},
\dfrac{{\large (}rf^{(r-1)}(b)-rbf^{(r)}(b){\large )}-{\large (}rf^{(r-1)}(a)-raf^{(r)}(a){\large )}}{f^{(r)}(b)-f^{(r)}(a)}= r\dfrac{{\large (}f^{(r-1)}(b)-bf^{(r)}(b){\large )}-{\large (}f^{(r-1)}(a)-af^{(r)}(a){\large )}}{f^{(r)}(b)-f^{(r)}(a)}
\int_0^t \int_{\Omega}\lvert \nabla d(s) \rvert^2 \varphi (s)^2 \; \d x \; \d s = \int_0^{t_0}\int_{\Omega}\lvert \nabla d(s) \rvert^2 \varphi (s)^2 \; \d x \; \d s + \int_{t_0}^t \int_{\Omega}\lvert \nabla d(s) \rvert^2 \varphi (s)^2 \; \d x \; \d s =: \mathrm{I}+ \mathrm{II}.
\partial_{m}\hat{X}^{\underline{{m}}}( \xi ) \partial_{n}\hat{X}_{\underline{{n}}}( \xi ) ={\frac{1}{2}}g_{m n}( \xi ) g^{p q}( \xi ) \partial_{p}\hat{X}^{\underline{{m}}}( \xi ) \partial_{q}\hat{X}_{\underline{{n}}}( \xi ) .
\tilde{w}_y+\lambda_1\tilde{w}_x=\varepsilon \tilde{w}_{xx}-\frac{2\varepsilon v_x\tilde{w}_x}{v}+(f(u,v)+g(u,v))B+\frac{B^\prime}{B}\tilde{w}, &\tilde{z}_y+\lambda_2\tilde{z}_x=\varepsilon \tilde{z}_{xx}-\frac{2\varepsilon v_x\tilde{z}_x}{v}+(f(u,v)-g(u,v))B+\frac{B^\prime}{B}\tilde{z}.
\operatorname{fix}(y,M^G)<\sum_{\sum il_i=m}\frac{\Delta(l)}{|C_{GL_m(q)}([y^U]_\beta)|}<\sum_{\sum il_i=m}cq^{\frac{nm}{4}+(\sum l_i-\frac{m}{4})s-\sum il_i^2 -2\sum_{i<j}il_il_j}<q^{\frac{1}{4}m(n-m)+cm}
\int_{0}^{L}\rho_s|\partial_{tt}h|^2 -\int_{0}^{L}\phi(u,p,h) \partial_{tt}h =-\alpha \int_0^L \partial_{xxxx}h \partial_{tt}h + \beta\int_0^L \partial_{xx}h \partial_{tt}h +\gamma\int_0^L \partial_{xxt}h \partial_{tt}h
C \le C_{\infty}= \frac{1}{2}\log(1+\mathbb{E}(E_t)) \stackrel{(a)}{=}\frac{1}{2}\log\left(1+\int_{0}^{\infty}(1-\tilde{F}(y)) \mathrm{d}y \right) \stackrel{(b)}{=}\frac{1}{2}\log\left(1+\int_{0}^{B_{max}}(1-F(y)) \mathrm{d}y \right)
= \sum_{\substack{\mu\subset n^m 0\leq r\leq m}}(-1)^{m(n+1)-|\mu |-r}\Bigl( P_{m^n-\mu^\prime}(z_1,\ldots ,z_n;t,q,\mathbf{t}) \langle z;t_0\rangle_{t,m-r}& \times E_{r}(x_1,\ldots,x_m;t,t_0) P_{\mu}(x_1,\ldots ,x_m;q,t,\mathbf{t}) \Bigr) .
\Gamma[A,V]=\int_0^\infty{{\rm d}T\over T}{\rm e}^{-m^2T}{\rm tr}\;\;{\cal P}\int \! \mbox{d}^d x_0 \int \!{\cal D}y \exp \Bigl[ - \int_0^T \!\!\!{\rm d}\tau \bigl({\dot{y}^2\over 4}\!+\! i g \dot{y}^\mu A_\mu \!+\! V \bigr)\Bigr]\,. \,\,
{^Td}_{5}^{3,6}(v_i'v_j'v_k' x_1)={^Td}_{5}^{3,6}((v_i'v_j'+v_j'v_k'+v_k'v_i')v_nx_1)={^Td}_{5}^{3,6}(v_n^3x_1)=0.
\nabla^{B}\nabla_{\ \ \dot{A}}^{C}W_{A B C}= - 2 i \nabla_{\ \ \dot{A}}^{B}\nabla_{\underline{{{A}}}}^{\ \ \dot{B}}G_{\underline{{{B}}}\dot{B}}+ G_{\ \ \dot{A}}^{B}\nabla_{\underline{{{A}}}}^{\ \ \dot{B}}G_{\underline{{{B}}}\dot{B}}+ \frac{1 1}{2}i \left( \nabla^{B}G_{\ \ \dot{A}}^{C}\right) W_{A B C}
H_NS_N(i,j) =\frac{1}{J_N}(-J_{i-2}J_{N-j}(J_{j-i+1}+J_{j-i+2})-J_{i-1}J_{N-j}(J_{j-i}+J_{j-i+1}) ~+2J_iJ_{N-i-1}) =\frac{1}{J_N}(-J_{j-3}J_{N-j}(J_2+J_3)-J_{j-2}J_{N-j}(J_1+J_2)+2J_{j-1}J_{N-j}) =\frac{1}{J_N}(-4J_{j-3}J_{N-j}-2J_{j-2}J_{N-j}+2J_{j-1}J_{N-j}) =\frac{2J_{N-j}}{J_N}(-2J_{j-3}-J_{j-2}+J_{j-1}) =0.
\sharp T(\varepsilon,K,\Lambda_d,b) &=\sum_{1\leqslant k\leqslant N}\left(\frac{k\varepsilon K^2}{N^2\det(\Lambda_d)}B^{2-\frac{1}{r}}+O\left(\frac{K^2}{N^2 \det(\Lambda_d)}B^{2-\frac{1}{r}}\right)+O\left(b\det(\Lambda_d) \log\left(\frac{kK(\lambda_2-\alpha\mu_2)}{N\det(\Lambda_d)}B\right)\right)\right).
\left.{\frac{1 6 \, \pi^{2}}{g_{i , \, \mathrm{e f f}}^{2}}}\right\vert_{\mathrm{\f i e l d \t h e o r y}}^{\ \overline{{{D R}}}}= \frac{1 6 \, \pi^{2}}{g_{i}^{2}( \mu )}+ b_{i}\, ( 2 \gamma + 2 ) \, ;
\lim_{K \to \infty}\limsup_{N \to \infty}\mathbb{P}\left(\int_{\mathbb{H}}(|\phi(z,\xi_N(z))|-K)^+\,dz \geq \epsilon \right) = 0, \inf_{B\subset \mathbb{H}, \lambda(B)<\infty}\limsup_{N\to\infty}\mathbb{P}\left(\int_{\mathbb{H}- B}|\phi(z,\xi_N(z))|\,dz \geq \epsilon \right) = 0,
\langle J_{\lambda,\mu}'(u,v),(\phi,\psi)\rangle = \mathcal{A}(u,\phi)+\mathcal{A}(v,\psi)-\int_\Omega \left(\lambda |u|^{q-2}u\phi+ \mu |v|^{q-2}v\psi \right)dx &-\frac{2\alpha}{\alpha+\beta}\int_\Omega|u|^{\alpha-2}u|v|^\beta\phi dx-\frac{2\beta}{\alpha+\beta}\int_\Omega|u|^{\alpha}|v|^{\beta-2}v\psi dx
\omega((a_1,k_1),(a_2,k_2),(a_3,k_3)) := \widehat{F}(k_1,k_2)(a_3) \ \epsilon(k_1,k_2,k_3) \widehat{\omega}((k_1, \rho_1), (k_2,\rho_2),(k_3 ,\rho_3)) := \epsilon(k_1,k_2,k_3) \ \rho_1(F(k_2,k_3)).
\phi(x,y)=\sum_{k\in\Bbb Z}a_k(y){\bf e}^{ikx}, \psi(x,y)=\sum_{k\in\Bbb Z}b_k(y){\bf e}^{ikx}, \xi(x)=\displaystyle \sum_{k\in\Bbb Z}d_k{\bf e}^{ikx},
\mathcal A \mathcal A+ \mathcal F=\mathcal A \mathcal A, \ \ \mathcal P\mathcal A \mathcal A + \mathcal F = \mathcal P\mathcal A \mathcal A \ \ \ \ \mathcal A \mathcal A+ \mathcal K \not\subset \mathcal P \mathcal A \mathcal A
P\left(\sup_{t\leq T}|\int_0^t\sigma(X_s,Y_s)dW_s|\geq K\right) =P\left(\sup_{t\leq T}|W'_{\beta(t)}|\geq K\right)=P\left(\sup_{t\leq \beta(T)}|W'_{t}|\geq K\right) &\leq P\left(\beta(T)\geq \gamma\right)+P\left(\sup_{t\leq \gamma}|W'_{t}|\geq K\right).
u^{(1)}(\vec{0})=[\begin{matrix}1\\ 0\\ 0\\ 0\end{matrix}]u^{(2)}(\vec{0})=[\begin{matrix}0\\ 1\\ 0\\ 0\end{matrix}]v^{(1)}(\vec{0})=[\begin{matrix}0\\ 0\\ 1\\ 0\end{matrix}]v^{(2)}(\vec{0})=[\begin{matrix}0\\ 0\\ 0\\ 1\end{matrix}]
\frac{d\theta_j(\gamma_k(t))}{dt}=X_T(\gamma_k(t))\cdot \nabla\theta_j(\gamma_k(t)) &=2\tau_k B(\gamma_k(t))\mu_k(\gamma_k(t))\,\frac{(x-x_k)(x-x_j)+(y-y_k)(y-y_j)}{(x-x_j)^2+(y-y_j)^2}\Big|_{\gamma_k(t)}\,,
{\cal E}:={\cal E}_0 +{\cal E}_1 + D G_\delta ({\mathtt u}_\delta)R_Z D{\widetilde G}_\delta ({\mathtt u}_\delta)^{-1}\,,{\cal E}_\omega := D G_\delta ({\mathtt u}_\delta){\mathbb R}_\omega D{\widetilde G}_\delta ({\mathtt u}_\delta)^{-1}\,,
\Vert u-\omega\Vert_{L^{\frac{2n}{n-2}}}^{\frac{n}{n-2}}\leq C\Vert u^{\frac{n}{n-2}}-\omega^{\frac{n}{n-2}}\Vert_{L^{2}}\leq C(J(u)-J(\infty))^{\frac{\gamma}{1+\gamma}}\leq \frac{C}{(1+t)^{\frac{\gamma}{1-\gamma}}}.
\sum_{n=0}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(q)_nq^{n^2}}{(zq)_n(z^{-1}q)_n}=\frac{1}{(q)_N}+(1-z)\sum_{n=1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(-1)^n(q)_nq^{n(3n+1)/2}}{(q)_{n+N}}\left(\frac{1}{1-zq^n}-\frac{1}{z-q^n}\right),
x_{m,n}=y_{m,n}=u_{m,n}=u_{x}^{m,n}=u_{y}^{m,n}=u_{xx}^{m,n}=u_{xy}^{m,n}=0, u_{yy}^{m,n}=1,u_{x^{k+3}}^{m,n}=u_{x^{k+2}y}^{m,n}=\cdots=0,k\geq 0.
\rho_{\mu}\left( x,\xi\right) =\sum_{k=0}^{2mn}A_{k}\left( x\right)\left[ \left( \partial_{x}+i\xi\right)^{k}-\left( i\xi\right)^{k}\right] \Gamma\left( x,\xi\right) & =\sum_{k=1}^{2mn}\sum_{j=1}^{k}\binom{k}{j}A_{k}\left( x\right) \left(i\xi\right)^{k-j}\partial_{x}^{j}\Gamma\left( x,\xi\right) .
\Gamma_{A^\mu A^\nu}(p,-p) \equiv \Gamma_{\mu \nu}(p,-p) = (\eta_{\mu \nu}- \frac{p_{\mu}p_{\nu}}{p^2}) \Gamma^T (p^2) + \frac{p_{\mu}p_{\nu}}{p^2}\Gamma^L(p^2) \; .
\Bigr\{\bigr[ \, 0:1:0 \, \bigr| \, \Delta_{456}:0:\Delta_{467}:0 \, \bigr] \Bigr\}\Bigr\{\bigr[ \, \Delta_1:\Delta_2:0 \, \bigr| \, 0:\Delta_{457}:\Delta_{467}:0 \, \bigr] \, \Bigr| \, \Delta_2 \Delta_{457}= \Delta_1 \Delta_{467}\, \Bigr\}.
{(-i)^d\over N}\sum_m{}^\prime e^{2\pi i{2\mu\over N}m}{1\over 2^d\prod\limits_j\sin{\alpha_j m\over 2}}=\sum_\nu w_\nu{(-i)^d\over N}\sum_m{}^\prime e^{2\pi i{\mu-\nu\over N}m}{1\over 2^d\prod\limits_j\sin{\alpha_j m\over 2}}
\lim_{r\to 0}\begin{vmatrix}f^+(r) \overline{\widetilde{f}^+(r)}f^-(r) \overline{\widetilde{f}^-(r)}\end{vmatrix}= \begin{vmatrix}D\begin{pmatrix}A^+ A^-\end{pmatrix}\overline{D\begin{pmatrix}\widetilde A^+ \widetilde A^-\end{pmatrix}}\end{vmatrix},
\omega ( z ) = \omega_{0}{\frac{\cal A}{\cal B}}\ , \quad \Delta \omega = \omega^{\prime}( z ) \, \chi ( z ) = \omega_{0}{\frac{\cal A}{\cal B}}\sum_{i = 1}^{\cal N}\left({\frac{1}{z - a_{i}}}-{\frac{1}{z - b_{i}}}\right) \chi ( z ) \ .
\overline{\mathcal{Z}}_\mathcal{S}=\big\{D+\sum_{p\in\mathcal{B}(\mathcal{S})}a_pe_p\,\vert\, D\in\mathrm{N}^1(\mathcal{S}),\,a_p\in\mathbb{R},\,\sum a_p^2<+\infty\big\}.
T_{a}N =dl_{a}( \ \ \mathfrak{g}_0 +\sum_{\substack{\alpha \in \Delta^+ \langle \alpha, w \rangle \notin 2 \pi \mathbb{Z}}}\mathfrak{g}_{\alpha}\ \ ), T^\perp_{a}N =dl_{a}(\ \mathfrak{t}+ \sum_{\substack{\alpha \in \Delta^+ \langle \alpha, w \rangle \in 2 \pi \mathbb{Z}}}\mathfrak{g}_{\alpha}\ \ ).
\sum \mathcal{D}_{1}^{\mathrm{r e t}}( x - x_{2}) \mathcal{D}_{2}^{\mathrm{r e t}}( x - x_{3}) : B_{0}( x ) B_{1}( x_{2}) B_{2}( x_{2}) B_{3}( x_{3}) B_{4}( x_{3}) : + ( x_{2}\leftrightarrow x_{3})
K_m^{\rm qtm}(1+(1-q^{-1})qx;p^{-1},N;q^{-1})=((pq)^{-1};q^{-1})_n\,K_n^{\rm Aff}\Big(1+(1-q)q^{-N}\big(\tfrac{1-q^N}{1-q}-x\big);p,N;q\Big).
\nu(A_{R,\eta})\leq \liminf_{k \to \infty}\nu_{n(k)}(A_{R,\eta})= \liminf_{k \to \infty}\int_{A_{R,\eta}}f_{n(k)}(x)\ \d{x}\leq \liminf_{k \to \infty}\frac{\lambda_d(A_{R,\eta})}{k}=0.
d_i\left(x_i,h_i^k(x_i)\right) \le d_i\left(x_i,h_i^\ell(x_i)\right)+d_i\left(h_i^\ell(x_i),h_i^k(x_i)\right) =d_i\left(x_i,h_i^\ell(x_i)\right)+d_i\left(x_i,h_i^{k-\ell}(x_i)\right)\le r\;.
\frac{1}{2}\frac{\partial^2 C^2}{\partial z^i_{\epsilon}\partial z^j_{\eta}}\frac{\partial^2\varphi^j}{\partial x^{\epsilon}\partial x^{\eta}}v^i= f(x^1)\left[ -f(x^1)f''(x^1)+1+f'^2(x^1)\right],
\sum_{i\in\mathcal{I}}\left( (A_i-X)^T\otimes( \partial_{x_jx_k}\Phi^{s}_i(X)) \right)+ \sum_{i\in\mathcal{I}^{s'}}\left( (A_i-\overline{X}_i)^T\otimes( \partial_{x_jx_k}\Phi^{s}_i(X)) \right) (\overline{M}^s -I_4) = 0_{2\times4}.
G^{R}( k ) ={\frac{N^{2}\epsilon^{2 ( \Delta - 4 )}}{8 \pi^{2}( \Delta - 3 ) !^{2}2^{2 \Delta - 5}}}( q^{2})^{\Delta - 2}\left[ \ln{q^{2}}- i \pi \, \mathrm{s g n}\, \omega \right] \, , \; \Delta = 2 , 3 , 4 , . . .
f_{n}:= \coprod_{D \in \mathcal{D}_n}G~\times_{\bar{D}}~ G \times~ X_{n,D}\to \coprod_{D \in \mathcal{D}_n}G~\times_{\bar{D}}~ E_{n,D}\times G ~\colon~ ~ (g,h,x)\to (g,f_{n,D}(h,x),gh)
[\lambda'\otimes e',x^{+}\otimes a^{-}] =\lambda'\diamond x^{+}\otimes\frac{[e',a^{-}]_{E'}}{2}+[\lambda',x^{+}]\otimes\frac{(e'\circ a^{-})_{C'}}{2},{}[\lambda'\otimes e',x^{-}\otimes a^{+}] =\lambda'\diamond x^{-}\otimes\frac{[e',a^{+}]_{C'}}{2}+[\lambda',x^{-}]\otimes\frac{(e'\circ a^{+})_{E'}}{2}.
\tilde{\mathcal{L}}f =\sum_{i=1}^{3}\tilde{D}_{i}\frac{\partial f}{\partial x_{i}}+\frac{\tilde{\sigma}^{2}}{2}\sum_{i,j=1}^{3}(\tilde{\Lambda})_{i,j}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}.
[\begin{matrix}d_{0}\\d_{1}\\d_{2}\\d_{3}\end{matrix}]=[\begin{matrix}2&3&1&1\\ 1&2&3&1\\ 1&1&2&3\\ 3&1&1&2\end{matrix}][\begin{matrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{matrix}]
a(0)=a_{0}\; ; \;\dot{a}(0)=\dot{a}_{0}\; ; \;b(0)=b_{0}\; ; \;\dot{b}(0)=\dot{b}_{0}\; ; \; \phi(0)=\phi_{0}\; ; \;\dot{\phi}(0)=\dot{\phi}_{0}\; ; \; \rho(0)=\rho_{0}.
Z_{K_{+}}: = \frac{5 9^{3 / 2}}{( 2 \pi )^{2}}\, \frac{\zeta_{K_{+}}( 2 )}{2 \zeta ( 2 )}= 5 . 1 3 7 9 4 1 2 0 1 8 7 3 4 1 7 7 6 9 8 4 1 3 4 8 3 3 9 \ldots
\beta_{n,k}^{1}=2-\frac{1}{n}-\frac{k-1}{2n^2}+{\rm o}\left(\frac{1}{n^2}\right) \beta_{n,k}^{l}\sim n\left(1-\exp(i2\pi(l-1)/k)\right), l \in \{2, \ldots, k\}.
\frac{\partial\zeta}{\partial y}(t,s,\tilde{x},y)=-\Big[\frac{\partial \tilde{x}}{\partial\zeta}(t,s,y,\zeta)\Big]^{-1}\frac{\partial \tilde{x}}{\partial y}(t,s,y,\zeta) ,{\rm with}\ \zeta=\zeta(t,s,\tilde{x},y).
\mathcal{Q}_N = \frac{T}{2}\sum_{k=N+1}^\infty \|\boldsymbol{y_d}_k\|_{\Omega}^2 = \frac{T}{2}\sum_{k=N+1}^\infty \left(\|\boldsymbol{y_d}_k^c\|_{\Omega}^2 + \|\boldsymbol{y_d}_k^s\|_{\Omega}^2\right) = \|\boldsymbol{y_d}- \boldsymbol{y_d}_N\|
\begin{array}{l}{s_{2}= ( 1 / 3 ) ( C - 2 B )}\\{s_{1}= ( 1 / 3 ) ( C + B ) - ( 1 / 2 ) A}\\{s_{0}= ( 1 / 3 ) ( C + B ) + ( 1 / 2 ) A .}\\ \end{array}
C^{\mathrm{dl}}_{1}\left( N_{\mathrm{sc}}\right) = \sum_{k=1}^K \min \left\lbrace C^k_{\mathrm{b2s},1}\left( N_{\mathrm{sc}}\right), C^k_{\mathrm{s2u},1}\left( N_{\mathrm{sc}}\right) \right\rbrace.
\begin{aligned}\limsup_{N\to\infty}\max_{\beta_L\in B_L}\frac{\left|\sum_{n=1}^Nf_{\beta_{L}}(M_nx)\right|}{\sqrt{2N\log(\log(N))}}\:\:\: = \:\:\: \max_{\beta_L\in B_L}\limsup_{N\to\infty}\frac{\left|\sum_{n=1}^Nf_{\beta_{L}}(M_nx)\right|}{\sqrt{2N\log(\log(N))}}= \:\:\: \max_{\beta_L\in B_L}||f_{\beta_L}||_2=\frac{1}{2}\textnormal{a.e.}\end{aligned}
|h(x) - \Psi_v \star T_{\alpha,v}(\omega_0)(x)| &\leq \Bigl|h(x) - \int_{-2c_x \log n}^{2c_x \log n}T_{\alpha,v}(\omega_0)(y) \Psi_v(x-y) dy \Bigr| + \Bigl| \int_{-\infty}^{-2c_x \log n}T_{\alpha,v}(\omega_0)(y) \Psi_v(x-y) dy \Bigr| + \Bigl|\int_{2c_x \log n}^\infty T_{\alpha,v}(\omega_0)(y) \Psi_v(x-y) dy\Bigr|
\varphi (\gamma) = \begin{pmatrix*}[r]\gamma_{1}+\gamma_{2}a+\gamma_{4}b \gamma_{3}+ \gamma_{4}a-\gamma_{2}b -\gamma_{3}+ \gamma_{4}a-\gamma_{2}b \gamma_{1}-\gamma_{2}a-\gamma_{4}b \end{pmatrix*},
V_{\mathrm{d i p o l e}}= -{\frac{1 5}{3 2}}\pi^{3}i \bar{\Theta}_{1}\Gamma^{0 i j}\Theta_{1}\bar{\Theta}_{2}\Gamma^{0 i k}\Theta_{2}\partial_{j}\partial_{k}\left\{{\frac{1}{r^{7}}}\right\},
V(n)= \frac{1}{(n-1)!}\sum_{k=1}^{n-1}\sum_{i=0}^{n-1}(-1)^i{n-1\choose i}\{M(k-ia)^{n-1}-M(k-(i+1)a)^{n-1}\}= \frac{1}{(n-1)!}\sum_{k=1}^{n-1}\sum_{i=0}^{n-1}(-1)^i{n-1\choose i}M(k-ia)^{n-1}+ \frac{1}{(n-1)!}\sum_{k=1}^{n-1}\sum_{i=1}^{n}(-1)^i{n-1\choose i-1}M(k-ia)^{n-1}= \frac{1}{(n-1)!}\sum_{k=1}^{n-1}\sum_{i=0}^{n}(-1)^i{n\choose i}M(k-ia)^{n-1}.
C(d,\delta)\ge \begin{cases}h(w)-{\displaystyle\max_{0\le x\le\delta/2}}\Big[w\,h\Big(\frac{x}{w}\Big) +(1-w)h\Big(\frac{x}{1-w}\Big)\Big] \textup{if}w<\frac{1}{2}1-h(\delta) \textup{otherwise}. \end{cases}