harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(1-x+\frac{xv}{1-v}\right)C(x,v) =\frac{xv^3}{1-v}C(x,1)+\frac{x^3v^3(2-x-xv)}{(1-x)^2(1-xv)^2}, \left(1-x+\frac{xv}{1-v}\right)D(x,v) =xB(x,v)+\frac{x}{v}C(x,v)+\frac{xv^2}{1-v}D(x,1).
\lim_{n\to\infty}\Bigl( \pi(n^{-\frac12}u) \cdot \pi(n^{-\frac12}v) ) \Bigr)^n = \lim_{n\to\infty}\Bigl( 1-(2n)^{-1}\sum_{j=1}^m(u_j-v_j)^2+n^{-2}O\bigl(|u|^4+|v|^4\bigr)\Bigr)^n &= \exp\Bigl\{-\frac12 \sum_{j=1}^m(u_j-v_j)^2\Bigr\}\,.
\mathcal{T}_{n+1}(x) = x\mathcal{T}_n(x)-\dfrac{1}{4}(x^2+1)\mathcal{T}_{n-1}(x), n\geq 1, & \mathcal{T}_{1}(x)=x+1/2, \mathcal{T}_{0}(x)=1.
d(\theta^1\wedge\dots\wedge\theta^k)=\sum \epsilon(i_1\dots i_k)\,d\theta^{i_1}\wedge\theta^{i_2}\wedge\dots\wedge \theta^{i_k}=(-1)^{k-1}\sum \epsilon(i_1\dots i_k)\,\theta^{i_1}\wedge\dots \wedge\theta^{i_{k-1}}\wedge d\theta^{i_k},
|g-\tilde g| = |h(F_*, F_*)-h (\widetilde F_* , \widetilde F_*)| &= |h( F_* ,F_*) -h(P\widetilde F_* , P\widetilde F_*)| &= |h( P\widetilde F_*-F_* , P\widetilde F_*) + h( F_* ,P\widetilde F_*-F_*)| &\le C(|F_*|+|P\widetilde F_*|) |P\widetilde F_* - F_*|.
\left|\mathbb{E}\left(\widetilde{\alpha}_{t,\varepsilon}^{(k)}\right)^n\right| \leq c_n\left(\int_{[0,t]^n}\prod_{j=1}^n|r_j-r_{j-1}|^{-\frac{H_1H_2}{H_1+H_2}}dr\right)^2,
p_{sr}=p_{rd}=\frac{1}{2}\sum_{k=1}^{n-2}{\binom{n-2}{k}(\frac{1}{m^2})^k (1-\frac{1}{m^2})^{n-1-k}\cdot \frac{1}{k+1}}=\frac{1}{2}\left \{\frac{m^2-1}{n-1}-\frac{m^2}{n-1}(1-\frac{1}{m^2})^n-(1-\frac{1}{m^2})^{n-1}\right \}
{h}_{2 n}^{( \alpha )}= \frac{1}{N^{2 \alpha}}{\Gamma ( \alpha + \frac 1 2 )^{2}}\left( \frac{2}{u ( t )}\right)^{2 \alpha}\sqrt{r ( t )}+ \mathrm{h i g h e r ~ o r d e r s ~ i n ~}\frac 1 N .
\Delta ( x | z_{1}| z_{2}, \cdots , z_{N}) = ( x - z_{1}\tau ) \sum_{\kappa = 0}^{N - 3}( 1 - \tau^{2 ( N - 2 - \kappa )}) ( - \tau )^{\kappa}x^{N - 3 - \kappa}\sum_{\lambda = 0}^{\kappa}( - z_{1}\tau^{2})^{\lambda}\sigma_{\kappa - \lambda}( z_{2}\cdots z_{N}) ,
(\psi\otimes\operatorname{id})\bigl(\Delta((\operatorname{id}\otimes\omega)[(1\otimes\tilde{y}c)(\Delta a)])\bigr) &=(\psi\otimes\operatorname{id})\bigl((\operatorname{id}\otimes\operatorname{id}\otimes\omega)[(1\otimes1\otimes\tilde{y})(1\otimes1\otimes c)(1\otimes E)\Delta_{13}(a)]\bigr).
S_{w_{G_n}}(\theta_{\{b_{i,j}\}}^{\{c_{i,j}\}};\ell) = S_2( u_1p^{\ell}, c_{n-1,n}p^{\ell-b_{n-1,n}};p^{\ell}) \hskip 10pt \cdot S_2( u_2p^{\ell}, c_{n-2,n-1}p^{\ell-b_{n-2,n-1}};p^{\ell}) \cdots \hskip 10pt \cdot S_2( u_{n-1}p^{\ell}, c_{12}p^{\ell-b_{12}};p^{\ell}),
{\mathcal L}^{( < )}( \Theta_{l}) = - p \, \frac{\Theta_{l}}{\left( \lambda_{l}^{( \ast )}\right)^{1 / 2}}\tan \left\{p \left[ \frac{\Theta_{l}}{\left( \lambda_{l}^{( \ast )}\right)^{1 / 2}}- \tan^{- 1}\left( \frac{\Theta_{l}}{\left( \lambda_{l}^{( \ast )}\right)^{1 / 2}}\right) \right] - \frac{\pi}{4}\right\}+ O ( 1 ) \; .
\psi_{\varepsilon}^{(k)}(a)+\chi_{\varepsilon}^{(k)}(a)=(-1)^{k-1}\sum_{\ell=0}^{k-1}(-1)^{\ell}p^{(k-\ell-1)}(0)\psi^{(\ell)}(a)\,\varepsilon^{(\ell-k)/(2N)}+\frac12\,\psi^{(k)}(a)+\chi^{(k)}(a)+R_{k,\varepsilon}
A_{D}\left( x_{1}, \ldots , x_{n}, f_{1}, \ldots , f_{n - 1}\right) \equiv \frac{1}{4 \pi}\sum_{\begin{array}{c}{{1 \leq s \leq n}}\\{{1 \leq t \leq n}}\\{{s \neq t}}\\ \end{array}}\frac{\left( x_{s}- x_{t}\right)^{2}\left( f_{s}- f_{s - 1}\right) \left( f_{t}- f_{t - 1}\right)}{4 \sin^{2}\left( \frac{f_{s - 1}+ f_{s}- f_{t - 1}- f_{t}}{4}\right)}
f_{x_1,x_2}(P)=\Pr_{P}\{X_1=x_1,X_2=x_2\},\forall (x_1,x_2)\in\mathcal{X}_1\times \mathcal{X}_2 &f_I(P)=I_P(X_2;Y|X_1), &f_{II}(P)= I_P(X_1;Y|X_2), &f_{III}(P)= I_P(X_1X_2;Y), &f_{IV}(P)= -I_P(X_1X_2;U)+I_P(X_2;U|X_1)+I_P(X_1;U|X_2)
\mathcal{E}_1 = \| \sqrt{-\dot{M}M}m^\alpha G\partial_{Y}^L\partial_{Y}^LP \|_{H^s}, \mathcal{E}_2 = \| \sqrt{-\dot{M}M}m^\alpha \psi_z \partial_{Y}^L\partial_{Z}P \|_{H^s}, \mathcal{E}_3 = \| \sqrt{-\dot{M}M}m^\alpha \Delta_t C \partial_Y^L P \|_{H^s}.
m\ddot{\boldsymbol{q}}_a=-\nabla V(\boldsymbol{q}_a)-\frac{\hbar^2}{8m}{\frac{\partial}{\partial \boldsymbol{q}_a}\int \frac{\sum_{b}w_b \nabla K(\boldsymbol{y}-\boldsymbol{q}_a)\cdot\nabla K(\boldsymbol{y}-\boldsymbol{q}_b)}{\sum_{c}w_c K(\boldsymbol{y}-\boldsymbol{q}_c)}\,d^3y}\,.
I_{D_{Z,+},d}=\frac{\prod_{0<a< v_2\cdot d}(v_2+az)}{\prod_{i\in I_+, D_i\cdot d>0}(\bar D_i+(D_i\cdot d)z)}[\textbf{1}]_{(-D_i\cdot d)_{i\in I_+},v_2\cdot d},
\mathrm{Vol}_{\omega_0}(F_b)=\int_{F_b}\Omega_0^m = \int_{{F_{\tilde{\pi}^{-1}(\gamma(b))}}}\gamma^* (\Omega_0)^m = \int_{{F_{\tilde{\pi}^{-1}(\gamma(b))}}}{{c_\gamma^m}}\cdot \Omega_0^m ={{c_\gamma^m}}\int_{F_b}\Omega_0^m,
\Delta{C}^{1,\infty}= (\alpha-\beta)\Big(\Delta{C}^{2,\infty}+\log\big(1+2^{\Delta{C}^{1,\infty}}\big)\Big) \Delta{C}^{2,\infty}= (\beta-\gamma)\Big(\Delta{C}^{2,\infty}+\log\big(1+2^{\Delta{C}^{1,\infty}}\big)\Big).
{\bf ( p_{L}^{D})}_{i}={\cal R}_{i}^{\, \, j}{\bf ( p_{L})}_{j}\, \, \mathrm{w i t h}\, \,{\cal R}_{i}^{\, \, j}= -{\bf e^{a}}_{i}( e^{t}. e^{- 1})_{a}^{\, \, b}{\bf e_{b}}^{j}\, .
\left\langle \int d^{2}x : \Psi^{\dagger}( x ) M \Psi ( x ) : \right\rangle = - \frac{1}{2}\left( \sum_{\mathrm{o c c u p i e d}}\int d^{2}x \psi_{E}^{\dagger}( x ) M \psi_{E}( x ) - \sum_{\mathrm{u n o c c u p i e d}}\int d^{2}x \psi_{E}^{\dagger}( x ) M \psi_{E}( x ) \right) .
2^{3}u^{3/2}\varOmega_3(u)=\det\begin{pmatrix}\mu^1_{2,1}(u) \mu^1_{2,2}(u) \mu^1_{2,3}(u) \acute\mu^1_{2,1}(u) \acute\mu^1_{2,2}(u) \acute\mu^1_{2,3}(u) \mu^2_{2,1}(u) \mu^2_{2,2}(u) \mu^2_{2,3}(u) \end{pmatrix}
d s^{2}= - \left[ \frac{2 m_{0}}{t}+ \frac{1}{3}\Lambda t^{2}- 1 \right]^{- 1}d t^{2}+ \left[ \frac{2 m_{0}}{t}+ \frac{1}{3}\Lambda t^{2}- 1 \right] d x^{2}+ t^{2}d \Omega^{2}.
f_*\mu_X(\,-\infty,a_k+r\,) \le \sum_{i=0}^k f_*\mu_X[\,a_i,a_i+r\,] + f_*\mu_X(\,-\infty,a_0\,) + \sum_{i=1}^k f_*\mu_X[\,a_{i-1}+r,a_i\,) < N(1-\kappa) + \sum_{i=0}^k \kappa_i \le 1-\sum_{i=k+1}^N \kappa_i,
K_{\zeta}^{\rm ASEP}(w,w') = \frac{1}{2\pi \mathrm{i}}\int_{\mathcal{D}}\Gamma(-s)\Gamma(1+s)(-\zeta)^s \frac{g_{j,t}(w)}{g_{j,t}(\tau^s w)}\frac{(-1)}{\tau^s w-w'}ds
B_{1}= - \frac{\tilde{n}_{1}}{a}\sinh{a \chi_{1}}\left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{0}}&{{\sqrt{\frac{n_{2}}{n_{3}}}\sin{\eta_{1}}}}&{{\cos{\eta_{1}}}}\\{{0}}&{{- \cos{\eta_{1}}}}&{{\sqrt{\frac{n_{3}}{n_{2}}}\sin{\eta_{1}}}}\\ \end{array}\right) .
\mathrm{WCE}(n, \mathbb{M}^2_{\mathrm{e}}) := \sup_{\substack{f\in \mathbb{M}^s_{\mathrm{e}}, \|f\|_{\mathbb{M}^s_{\mathrm{e}}}\leq 1}}\left| \int\limits_{\,-\infty}^\infty f(x) W(x) \mathrm{d}x- \sum_{x\in X_n}\omega(x) f(x)\right|^2 & \asymp \sum_{x,y\in X_n}\omega(x)\omega(y) \sum_{k=2n}^\infty \mathrm{e}^{-q\sqrt{k}}h_{k}(x)h_{k}(y).
M_k (t) = \int_0^t \sigma(u, X(u)) e_k(u) du + \int_0^t \Big( \int_0^r U(u, r) e_k(u) du\Big) dr + \int_0^t \Big( \int_0^r V(u, r) e_k(u) du \Big) dW(r) &+\int_0^t \nabla_x b(r, X(r)) M_k(r) dr + \int_0^t \nabla_x \sigma (r, X(r)) M_k(r) dW(r).
(-iu)^{\gamma\cdot \mathrm c_1(S)}\cdot \left\langle \gamma_1, \dots , \gamma_N \right\rangle^{- \infty}_{g, \gamma}=(e^{iu/2}-e^{-iu/2})^{\gamma \cdot \mathrm c_1(S)}\cdot \left\langle \gamma_1, \dots, \gamma_N \right\rangle^{0}_{g, \gamma}.
\mu \int_{\Omega}\varepsilon S'(\overline{u},h) : \varepsilon p +\nu \int_{\Omega}\mathsf{m}'\left( \varepsilon \overline{y};\varepsilon S'(\overline{u},h) \right) : \varepsilon p = -\alpha \int_{\Omega}h \cdot \overline{u}= - J_u (\overline{y}, \overline{u})h.
\sum_{k=0}^{\lfloor n/2 \rfloor}\frac{2^k n!}{k!(n-2k)!}\frac{\prod_{i<j}|z_i-z_j|}{(\prod_{i=1}^n |1-z_i|)^{(n+1)}}d\ell_{n,k}(z_1,\dots,z_n) & = \sum_{k=0}^{\lfloor n/2 \rfloor}\frac{2^k n!}{k!(n-2k)!}\frac{\prod_{i<j}|z_i-z_j|}{\|\tilde{a}\|_1^{n+1}}d\ell_{n,k}(z_1,\dots,z_n).
F(z,s) := \sum_{j \in \mathbb{Z}}\frac{e^{2 \pi i j s}}{z + 2 \pi i j}= \frac{e^{- zs}}{1 - e^{- z}}\sum_{j \in \mathbb{Z}+1/2}\frac{e^{2 \pi i j s}}{z + 2 \pi i j}= \frac{e^{- zs}}{1 + e^{- z}}
\Sigma^{A B}={\frac{i}{4}}\left[ \Gamma^{A}, \Gamma^{B}\right] \to{\frac{i}{4}}\left[ \omega \left( \Gamma^{A}\right) , \omega \left( \Gamma^{B}\right) \right] \equiv \Omega \Sigma^{A B}\Omega^{- 1}.
2f_1(x)-f_2(x) =-2x_1^2-x_2^2+5x_2 &=-2(x_1+0.25)^2-(x_2-3.5)^2+x_1-2x_2+12.375 &\leq x_1-2x_2+12.375\leq -20+12.375\leq 10.
\partial_{k\ell}(\partial_i u^j +\partial_j u^i) +\partial_{ij}(\partial_k u^\ell +\partial_\ell u^k) = \partial_{kj}(\partial_i u^\ell +\partial_\ell u^i) +\partial_{i\ell}(\partial_k u^j +\partial_j u^k)\;,
e^{-\sum_{i,j}R_{ij}\alpha_ix_j}\prod_{k=1}^{d}\left(1+\sum_{\ell=1}^{d}R_{k\ell}\frac{x_\ell}{\alpha_k}\right)^{i_k}=\sum_{n_1,\ldots,n_{d}=0}^{\infty}C_{n_1,\ldots,n_{d}}(i_1,\ldots,i_{k})\,\frac{x_1^{n_1}x_{2}^{n_2}\cdots x_{k}^{n_k}}{\sqrt{n_1!n_2!\cdots n_{k}!}}
L=(\begin{matrix}0&0&0\\ 0&0&1\\ 0&-1&0\end{matrix})i+(\begin{matrix}0&0&-1\\ 0&0&0\\ 1&0&0\end{matrix})j+(\begin{matrix}0&1&0\\ -1&0&0\\ 0&0&0\end{matrix})k.
\left(f_{\mathcal{S}}\right)_\lambda(s_1, s_2) = \sup_{(y_1^\prime, y_2^\prime, x^\prime ) \in \mathcal{S}}\left\{f_1(y_1^\prime) + f_2(y_2^\prime) - \sum_{1 \leq \ell \leq 2}\lambda_\ell c_\ell\left( (y_\ell, x_\ell), (y^\prime_\ell, x^\prime) \right) \right\}.
R_3=\frac{1}{2}\log\frac{|\sigma^{2}{\bf I}_2+\sum_{j=1}^{3}{\bf G}_{2j}{\bf Q}_j{\bf G}_{2j}^{T}|}{|\sigma^{2}{\bf I}_2+\sum_{j=1}^{2}{\bf G}_{2j}{\bf Q}_j{\bf G}_{2j}^{T}|}.
\bigl(\iota_*X)_{g^{-1}}f =X_g(f\circ \iota) =\left.\frac{d\}{dt}\right|_{t=0}(f\circ\iota)(g\exp(tX)) =\left.\frac{d\}{dt}\right|_{t=0}f\bigl((g\exp(tX))^{-1}\bigr) =\left.\frac{d\}{dt}\right|_{t=0}f(\exp(-tX)g^{-1})=-\hat X_{g^{-1}}(f).
H_{o}= - \alpha^{\prime}E^{2}+ \alpha^{\prime}\frac{n^{2}}{R^{2}N^{2}}+ \frac{\rho^{2}}{\pi^{2}\alpha^{\prime}}\sin^{2}\Big [ \frac{( \alpha - \beta )}{2}\pi \alpha^{\prime}E + \pi \frac{( \alpha + \beta )}{2 R}\alpha^{\prime}\frac{n}{N}+ \frac{\pi m}{N}\Big ] + \hat{N},
F_T(x) =1+xF_T(x)+\frac{x^3-7x^2+5x-1}{(1-x)(1-2x)}-\frac{3x^3-9x^2+6x-1}{(1-x)^2}C(x)+xC(x)F_T(x) &+\frac{x}{1-x}\big(C(x)-1-xC(x)\big)+\frac{x^4}{(1-x)^3}\frac{1-x}{1-2x}C(x)\,,
\begin{array}{rl}\tan\left(\frac{\pi}{4}+\frac{\Theta}{2}\right) = (-1)^n \left(\frac{E+m}{E-m}\right)^{1/2}\left(\frac{k}{2m}\right)^{2\nu+1}& \times\frac{\Gamma(-\nu)}{\Gamma(\nu+1)}\tan\mu .\end{array}
B = \frac{4 \, \tilde{e}^{2}\, e^{\beta \mu}}{3 ( 2 \pi )^{3 / 2}\tilde{m}^{2}}\, \sqrt{\frac{\tilde{m}}{\beta}}\Bigl ( \cos \frac{\theta}{3}+ \frac{1}{\sqrt{3}}\sin \frac{\theta}{3}\Bigr )^{- 1}\, ,
\textrm{i)}\sup_{N\in\mathbb{N}}\mathbb{E}\left\vert H_t^N - H_u^N \right\vert^p \leq C \left\vert t- u\right\vert^{\alpha}\mbox{for some $C>0$ and $p, \alpha>1$;}& \textrm{ii)}\lim_{K \to \infty}\sup_{N\in\mathbb{N}}\mathbb{P}\left(\left\vert H_0^N-{a}\right\vert>K \right) =0 \mbox{for some $a \in \mathbb{R}$.}
\bigl\lvert\lvert\varphi\rvert(x)\bigl\rvert \leq \lvert\varphi\rvert\bigl(\lvert x\rvert\bigl)=\Bigl\{\bigl\lvert\varphi(y)\bigl\rvert\ : \lvert y\rvert \leq \lvert x\rvert\Bigl\}\leq \varepsilon.
g(A_XY, A_Z W) = g(A_XY, \frac{e_0}{\sqrt{a}}) +g(\frac{e_0}{\sqrt{a}}, A_Z W) = \,\frac{1}{4a}\,g(F(X,Y),e_0) \cdot g(F(Z, W),e_0) = \,\frac{a(\tau)}{4}\left (\sum_i q_i \omega_i(X,Y) \right ) \cdot \left (\sum_j q_j\omega_j (Z,W) \right ),
R_\mu(P_{XY}) \le I(\tilde{W}\wedge \tilde{X}) + \mu H(\tilde{Y}|\tilde{W}) &= I(\tilde{W}\wedge \tilde{X},\tilde{Y}) + \mu H(\tilde{Y}|\tilde{W}) &\le I(W \wedge X,Y) + \mu H(Y|W) + \delta^\prime &= R_\mu(\delta | P_{XY}) + \delta^\prime,
D_{S L ( 3 , R )}^{\mathrm{d i s c}}({\frac{3}{2}}; R ) : \quad \{J \}= \{{\frac{3}{2}},{\frac{5}{2}},{\frac{7^{2}}{2}},{\frac{9^{2}}{2}},{\frac{1 1^{2}}{2}}\ldots \}.
k=-\frac{1}{2}(\frac{1}{(y^{\prime\prime})^{\frac{2}{3}}})^{\prime\prime}=\frac{y^{\prime\prime\prime\prime}}{3(y^{\prime\prime})^{\frac{5}{3}}}-\frac{5(y^{\prime\prime\prime})^{2}}{9(y^{\prime\prime})^{\frac{8}{3}}}
(C_m,S_m) =\frac{1}{\sqrt{p_{r_{m+1}-1}}}\sum_{j=1}^{r_{m+1}-r_m}H_{j,m}=\frac{b_m}{\sqrt{\sum_{j=1}^{r_{m+1}-r_m}c_{j,m}}}\sum_{j=1}^{r_{m+1}-r_m}H_{j,m},
\dot{x}_{\mu}^{c l}( \sigma ) = u_{\mu}-{\frac{1}{2}}u_{\mu}( \tau_{1}- \tau_{2}) \frac{\dot{G}_{B}( \sigma , \tau_{1}) - \dot{G}_{B}( \sigma , \tau_{2})}{L + G_{B}( \tau_{1}, \tau_{2})}
\left(\left(I_n\otimes T_2^{j_2}\right)\left(I_n\otimes T_1^{j_1}\right)p(T^*,T)\left(I_n\otimes T_1^{k_1}\right)\left(I_n\otimes T_2^{k_2}\right)\right)_{j_1,k_1=-N}^N.
G_{\alpha_{1}, \alpha_{2}, \alpha_{3}}( x_{1}, x_{2}, x_{3}) = \left| x_{1 2}\right|^{2 \gamma_{3}}\left| x_{2 3}\right|^{2 \gamma_{1}}\left| x_{3 1}\right|^{2 \gamma_{2}}C ( \alpha_{1}, \alpha_{2}, \alpha_{3})
\Theta_{\mu}^{\mu}= \frac{2 g^{\mu \nu}}{\alpha^{2}}\frac{\partial \alpha}{\partial g^{\mu \nu}}\frac{1}{1 6 \pi}\mathrm{t r}F^{2}= \frac{1}{8 \pi}\frac{\beta ( \alpha )}{\alpha^{2}}\mathrm{t r}F^{2}
tH_{\mathrm{KFS}}^{\frac32+\frac32}\left(\theta^\infty_1-\theta^\infty_2,\, 1-\theta^\infty_1 ;t;{q_1,p_1 \atop q_2,p_2}\right) =tH_{\mathrm{III}(D_7)}\big( \theta^\infty_1-\theta^\infty_2; t; q_1,p_1 \big) \qquad{}+tH_{\mathrm{III}(D_7)}\big( 1-\theta^\infty_1; t; q_2,p_2\big)-p_1q_1p_2q_2-t(p_1p_2+p_1+p_2).
A^{(1)}=\frac{a}{2}\left( \beta_\lambda^{(1)}-4\lambda\gamma^{(1)}\right), \ \B^{(1)}=\frac{b}{2}\left(\beta_\xi^{(1)}-2\xi \gamma^{(1)}\right), \ \D^{(1)}=\frac{d}{2}\left( \beta_\zeta^{(1)}-2\zeta \gamma^{(1)}\right).
\Upsilon_{k}(x) = \Upsilon_{k, a_1, \ldots, a_{m_1}, b_1, \ldots, b_{m_2}}(x) = \frac{x}{\log x - 1 - A(x)}\left( 1 - \frac{1}{k}- \frac{1}{k}\,\frac{\log k - A(x) + B(x/k)}{\log (x/k) - 1 - B(x/k)}\right).
-\sum_{A,B}\sum_{\vert J \vert =1}\sum_{\vert K \vert =1}\int_{\Omega}D_J\biggl( E^J_A(h)I^{(K)}_{AB}E^K_B(f)\biggr) -\sum_{A,B}\sum_{\vert K \vert =2}\int_{\Omega}E^0_A(h)I^{(K)}_{AB}E^K_B(f) .
\begin{aligned}X[x,y] = -(1-2n^2+m^2(1+n)^2) + \bigg(\frac{y-m(1+n)}{x-n^2+1}\bigg)^2-n^2+1-x Y[x,y] =-m(n+1)-(X[x,y]-n^2+1)\bigg(\frac{y-m(1+n)}{x-n^2+1}\bigg).\end{aligned}
\begin{aligned}\deg H_0(W) \leq \phantom{\max\big(\,\deg H_0(V),\ \}\deg H_0(M) \deg H_1(W) \leq \max\big(\,\deg H_0(V),\ \ \deg H_1(M)\phantom{,\ \ \deg H_2(W)}\,\big) \deg H_0(K) \leq \max\big(\,\deg H_0(V),\ \ \deg H_1(M),\ \ \deg H_2(W)\,\big) \end{aligned}
P(\phi_1(u,v) w) =P(\phi_1(u,v) w_1 w_2\cdots w_k) = P( \phi_1(u,v) P(\tilde{w})w_2\cdots w_k) &= \sum_i c_i P( p_i|_{s_i}w_2 \cdots w_k) = \sum_i c_i q_i|_{s_i}
\begin{aligned}\lVert \widetilde{M}_{k}(\alpha,\theta,E,n)\rVert\leq \lVert M_k(\alpha,\theta,E,n)\rVert +\sum_{i=0}^{k-1}|g(n+i)|\lVert M_{k-i-1}(\alpha,\theta,E,n+i+1)\rVert\lVert\widetilde{M}_{i}(\alpha,\theta,E,n)\rVert. \end{aligned}
\frac{q_1 + p_2q_2}{q_1+q_2}- \frac{p_2 q_2}{q_2+q_3}=\frac{q_1}{q_1+q_2}\leq \frac{q_1+q_3}{q_1+q_2+q_3}= \P( (G\cap H^c) \cup (H\cap G^c)\mid G\cup H).
\partial \gamma_f = \sum_{x \in E_N}\xi_f(x) ([g_x \rho^2]-[g_x \rho]) & = \sum_{x \in E_N}\xi_f(x) ([g_{x\sigma}\rho]-[g_x \rho]) & = -2 \sum_{x \in E_N}\xi_f(x) [g_x \rho].
\frac{1}{2}\partial_{t}\left(\int_{0}^{\infty}g(s)\left\|\partial_{x}^{k}\eta^{t}\right\|^{2}d s\right)=\frac{1}{2}\int_{0}^{\infty}g^{\prime}(s)\left\|\partial_{x}^{k}\eta^{t}\right\|^{2}d s+(-1)^{k}\int_{0}^{L}u \int_{0}^{\infty}g(s) \partial_{x}^{2 k}\eta^{t}d s d x,
\|f(t,\cdot)\|_{H^{\beta t-d}}^2 = \|M_\infty(t,D_v)f(t,\cdot)\|_{L^2}^2\le\|f_0\|_{H^{-d}}^2 \exp\left( \int_0^t A(\beta,\tau)\, \mathrm{d}\tau \right) .
\frac{|C_1|+|C_2|+\cdots +|C_N|}{|H|}=\frac{N|C|}{|H|}&\leq \frac{N|C|}{|R_1|+|R_2|+\cdots+|R_N|}&\leq \frac{N|C|}{A(s_1)+A(s_2)+\cdots+A(s_N)}&\leq \frac{|C|}{A(6)},
\int_{[-1,1]}m_\omega(\xi) \cdot \hat g(\ell \xi) \; \hat h^*(\ell \xi) \ \mathrm{d}\xi = \sum_{k=0}^\infty \ell^k e_k[g,h] \int_{[-1,1]}\xi^k m_\omega(\xi) \ \mathrm{d}\xi
\lim_{\delta\rightarrow 0}\limsup_{n\rightarrow+\infty}a_n^{-2}\sum_{x,y\in\mathbb Z^{d_0}}\mathbf E\Big[(h(\xi_1,\xi_2)\zeta_{n,x,y})^2\mathbf 1_{\{|h(\xi_1,\xi_2)\zeta_{n,x,y}|\le a_n\delta\}}\Big|\tilde{\mathcal F}\Big]=0.
P =P_{QU_1V_1V_2U_2X_2}(q,u_1,v_1,v_2,u_2,x_2) &=P_{Q}(q)P_{U_1|Q}(u_1|q)P_{V_1V_2|U_1Q}(v_1,v_2|u_1,q)P_{U_2|Q}(u_2|q)P_{X_2|U_2Q}(x_2|u_2,q).
V^\pm_{\partial X}(x,y) := \frac{V^\pm \cdot \hat n_{X_=}}{\sqrt{1-(\hat n_X \cdot \hat n_{X_=})^2}}\hat n_\partial, \hat n_\partial(x,y) :=\frac{\hat n_{X_=}- (\hat n_{X_=}\cdot \hat n_X) \hat n_X}{\sqrt{1 - (\hat n_X \cdot \hat n_{X_=})^2}},
(P^*S|_{\widetilde F}- S|_{F}) (a_1, \cdots, a_p, b^1 , \cdots, b^q) = \int_0^1 \partial_s \big( S(a_1(s), \cdots, a_p(s), b^1(s), \cdots, b^q(s)\big) ds &=-\int_0^1 (\bar\nabla_{\dot\gamma}S)( a_1(s), \cdots, a_p(s), b^1(s), \cdots, b^q(s)) ds
U_n^\pm\geq 0 r\geq 0,U_n^\pm\rightarrow t^{\pm}\mbox{as}r\rightarrow \infty, begin{align*}2mm]U_n^\pm(0)=0\mbox{if}n_{\pm}\neq 0,{U_n^\pm}'(0)=0\mbox{if}n_{\pm}= 0.
E_{C}^{{\mathrm{f}i x e d}}= \frac{1}{2 \pi}\int_{0}^{\infty}d \omega \ln \left( 1 - \exp{( - 2 \omega R )}\right) = - \frac{R}{\pi}\int_{0}^{\infty}\frac{\omega d \omega}{\exp{( 2 \omega R )}- 1}{.}
\beta_2(x,v,x',v')=\int_0^{\tau_-(x,v)}\!\!\!\!\!\!\! E(x,x-tv,x'){k(x-tv,v_1,v)k(x',v',v_1)_{|v_1= \widehat{x-tv-x'}}\over |x-tv-x'|^{n-1}}dt,
\mathcal{R}(z) = \sum_{p=0}^\infty \kappa_{p+1}(\mu_{k,c})z^p = \frac{c(k^2-1)}{1-z}- \frac{c(k-1)}{1+(k-1)z}K(z) = \mathcal{R}(z) + \frac{1}{z}= \frac{1}{z}+ \frac{c(k^2-1)}{1-z}- \frac{c(k-1)}{1+(k-1)z}.
\int \frac{d^{3}q}{( 2 \pi )^{3}}\, \frac{1}{( q^{2}+ m^{2}) [ ( p - q )^{2}+ m^{2}] [ ( q - p_{1})^{2}+ m^{2}]}= \frac{1}{1 6 \pi \mu^{3}}\, \ln \left( \frac{4}{9 \Delta}\right) .
g(t_1,\ldots,t_{p+1}) =f(t_1,\ldots,t_{p+1}) \prod_{i=1}^p \frac{1-t_i}{1-{t_i}/{t_{p+1}}}, \widetilde{g}_N(t_1,\ldots,t_{p+1}) =f(t_1,\ldots,t_{p+1}) \prod_{i=1}^p Q_{i}^N.
\lambda_{n,j}=\frac{\eta_{n,j}}{\eta_{n,j}+i}= \frac{t_p J_n^{\prime}(t_p) + in J_n(t_s)\sigma_j^{(n)}}{t_p H_n^{(1)\prime}(t_p) + in H_n^{(1)}(t_s)\sigma_j^{(n)}}, \j=1,2.
f_0(x,p)={1\over 2\pi}\int\! \! dy ~{e^{-iyp}\over 2 \cosh(x/\hbar - y / 2 )\cosh(x/\hbar + y/2)}={1\over \pi}\! \int^{\infty}_0\! \! \! \! dy ~{\cos(yp) \over \cosh(2x/\hbar )+\cosh(y)}
P_{K}( V ;{\bf a}_{1}, \ldots ,{\bf a}_{K}, \rho_{1}, \ldots , \rho_{K}) = \frac{1}{Z ( V )}\frac{C^{K}}{K !}\prod_{j = 1}^{K}\rho_{j}^{\alpha}\ \Theta ( \{{\bf a}_{j}\}, \{\rho_{j}\}) .
\begin{cases}l_1(x, m) = m^E_1 (s(x), i(m)) &r_1(m,x)= m^E_1 ( i(m), s(x))\end{cases};\begin{cases}l_2(x, m) = m^E_2 (s(x), i(m)) &r_2(m,x)= m^E_2 ( i(m), s(x)).\end{cases}
H_{1,1}=\int_0^{2\pi}\int_0^{\frac{3}{2}}\frac{r dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^q|re^{i\theta}-1|^q}, H_{1,2}=\int_0^{2\pi}\int_{\frac{3}{2}}^\gamma\frac{r dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^q|re^{i\theta}-1|^q}.
\begin{array}{lll}d:=vec(D)=[D_{\bullet 1}^T \ldots D_{\bullet m}^T]^T, \mathbf{\hat e_i}:= e_i\otimes \mathbf 1_m=[\mathbf 0^T\ldots \mathbf 0^T \, \mathbf 1^T_m \, \mathbf 0^T \ldots \mathbf 0^T]^T, \mathbf{\tilde e_i}:= \mathbf 1_m \otimes e_i =[e_i^T\ldots e_i^T]^T, \mathbf{\theta_i}:=[p_1e_i^T\ldots p_m e_i^T]^T.\end{array}
\mathbb{P}_{\ell}^{\lambda,k}:= \mathbb{P}\bigg[ \sup_{X_{\ell -1}< x_i \leqslant X_{\ell}}\sup_{\substack{0 \leqslant j\leqslant J}}\lambda_{\ell}^{(k)}(x_i,y_j,f)> \frac{T(\ell)}{\ell^{K/2}\ell \log \ell}\bigg]
\phi^{A}\psi^{B}- \phi^{B}\psi^{A}= \frac{1}{8}\left( \frac{1}{2 !}( \phi^{T}\Gamma^{\mu \nu}\psi ){\Gamma_{\mu \nu}}^{A B}+ \frac{1}{6 !}( \phi^{T}\Gamma^{\mu \nu \alpha \beta \lambda \rho}\psi ){\Gamma_{\mu \nu \alpha \beta \lambda \rho}}^{A B}\right) .
\begin{array}{l}\tilde{R}_{i}\mathop{=}\limits^{\Delta}\frac{1}{2}\log_{2}(1{{\rm \; \; \; \;\; \; \; \;\; \; \; \;\; \; \; \;}+\gamma_{i,i}[\frac{e-1}{\gamma_{i,i}^{}},\frac{1}{2\eta \sum_{l=1}^{L}\Phi_{i,i}^{s,(l)}+ 2\kappa \sum_{l=1}^{L}\Phi_{i,i}^{r,(l)}}-\frac{1}{\gamma_{i,i}^{}}]^{+}).}\end{array}
\small\setlength{\abovedisplayskip}{5 pt}\setlength{\belowdisplayskip}{5 pt}\boldsymbol{\Gamma}=\left[\!\!{\begin{array}{*{20}{c}}{\sigma_t^2\sum\limits_{j = 0}^\infty{{\mathbb{E}\left[w^{2Wj}\right]}}}0 0 0 \ddots 0 0 0{\sigma_{t - (W - 1)}^2\sum\limits_{j = 0}^\infty{{\mathbb{E}[w^{2Wj + 2(W - 1)}]}}}\end{array}}\!\right],
\sum_{n=1}^{\infty}\left\{\frac{1}{\sqrt{\left(2n-1\right)^{2}+x^{2}}}-\frac{1}{2n-1}\right\}+\frac{\gamma}{2}+\frac{\log\left(x\right)}{2}=\sum_{n=1}^{\infty}(-1)^{n}\,K_{0}(\pi nx).
\begin{array}{l l}{{L^{- 1}( T , X , Y , Z ) = ( \cosh t \cos \frac{r_{+}- r_{-}}{2},}}&{{\sinh t \cos \frac{r_{+}+ r_{-}}{2},}}\\{{}}&{{\sinh t \sin \frac{r_{+}+ r_{-}}{2}, - \cosh t \sin \frac{r_{+}- r_{-}}{2}) .}}\\ \end{array}
\liminf_{N \to \infty}\; \mathbb{E}\left[ \frac{1}{N}\displaystyle\sum\limits_{t = 1}^N \frac{1}{2}\log\left( 1 + g^\prime(t)\right) \right] & \geq \mathbb{E}\left[ \displaystyle\sum\limits_{j = 0}^{T_1-1}\frac{1}{2}\log\left( 1 + \tilde{g}(j)\right)\right] \bigg / \mathbb{E}[T_1].
\frac{1}{k^2(k.n)^m}\rightarrow \frac{1}{k^2+i\varepsilon}\left\{\frac{\Theta(-k^0)}{(k.n-i\xi)^m}+\frac{\Theta(k^0)}{(k.n+i\xi)^m}\right\},\,\,\begin{array}{c}\varepsilon \equiv2\epsilon k_0^2 \rightarrow 0^{+}\xi \equiv \epsilon|k^0|n^0 \rightarrow 0^{+}\end{array},
K_{0}={\frac{x_{0}^{d - 4}}{( x_{0}^{2}+{\bf x}^{2})^{d - 2}}}d x^{i}d x^{j}-{\frac{x_{0}^{d - 5}x^{i}}{( x_{0}^{2}+{\bf x}^{2})^{d - 2}}}d x^{0}d x^{j}+{\frac{x_{0}^{d - 5}x^{j}}{( x_{0}^{2}+{\bf x}^{2})^{d - 2}}}d x^{0}d x^{i},
\sigma_n =\frac{\alpha^2}{4}\cdot\frac{R_n}{1-R_n}-\frac{1}{4}(4n+2\alpha-t) tR_n-\frac{1}{4}t^2 R_n^2-\frac{1}{4}\frac{t^2(R_n')^2}{R_n(1-R_n)}&=-\frac{\alpha^2}{4}\cdot\frac{1}{S_n}+\frac{t}{4}\cdot\frac{4n+2\alpha-t}{S_n-1}-\frac{1}{4}\cdot\frac{t^2}{(S_n-1)^2}+\frac{1}{4}\cdot\frac{t^2 (S_n')^2}{(S_n-1)^2S_n}.
\partial_{\mu}\Omega ( \Sigma , x ) = - \int_{\Sigma}d^{D - 1}\tilde{\sigma}_{\mu}\delta^{D}( x - \bar{x}( \sigma ) ) - \int_{\Sigma}d^{D - 1}\tilde{\sigma}_{\nu}( \delta_{\mu \nu}\partial^{2}- \partial_{\mu}\partial_{\nu}) D ( x - \bar{x}( \sigma ) ) \ .
{\cal L}_{\mathrm{r e l}}^{( 2 )}= \frac{1}{2}\biggl ( \mu + \frac{g^{2}\lambda}{8 \pi r}\biggr ) \dot{{\bf r}}^{2}+ \frac{1}{2}\left( \frac{g^{2}\lambda}{8 \pi}\right)^{2}\biggl ( \mu + \frac{g^{2}\lambda}{8 \pi r}\biggr )^{- 1}( \dot{\psi}+{\bf w}({\bf r}) \cdot \dot{{\bf r}})^{2}.
p_G=(\nu-1)m_{\nu}^{\nu-1}\!\left(\frac{\ddot{a}}{n^2a}-\frac{\dot{n}\dot{a}}{n^3a}\right)+\frac{1}{2}(\nu-1)(\nu-2)m_{\nu}^{\nu-1}\!\left(\frac{\dot{a}^2}{n^2a^2}+\frac{k}{a^2}\right).
V(h) = p \left( |\cdot|^{-(N-\gamma)}* (Q^{p-1}\Re h) \right) Q^{p-1}&+ \left( |\cdot|^{-(N-\gamma)}* Q^p \right) \left( \left( \frac{p}{2}-1 \right) Q^{p-2}\bar{h}+ \frac{p}{2}Q^{p-2}h \right)
D_{i \alpha}\lambda^{j \beta}( z ) = \delta_{i}^{j}( \textstyle{\frac{1}{2}}\hat{\lambda}( z ) \delta_{\alpha}^{\beta}- \tilde{\hat{\omega}}_{\alpha}{}^{\beta}( z ) ) - \hat{T}_{i}^{j}( z ) \delta_{\alpha}^{\beta}
\dfrac{\pi}{2}\Bigl(\dfrac{1}{2}\Bigr)_{r-1}\dfrac{1}{(n+r)^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{n^r}< H_r \leq \dfrac{\pi}{2}\Bigl(\dfrac{1}{2}\Bigr)_{r-1}\dfrac{1}{n^{r-\frac{1}{2}}}-\dfrac{(r-1)!}{(n+r)^r}+\sum_{k=1}^{\infty}\dfrac{1}{k\pi^k}\Bigl(\dfrac{k}{2}\Bigr)_r \dfrac{1}{n^{r+\frac{k}{2}}}.
\sum_{k=1}^\infty\|\varepsilon_0\|_p^{2k}\sum_{j_k\ge m,\ 0 \le j_1 < \cdots< j_k}g^2_k(j_1, \ldots,j_k) = O\bigl(m^{-1-2\tau}(\log m)^{-2A}\bigr)
S(a_1\circ b)a_2 = S(a_1\circ b_1\epsilon(b_2))a_2 = S(a_1\circ b_1)\epsilon(b_2)a_2 &=S(a_1\circ b_1)(a_2\circ b_2)S(a_3)(a_4\circ S(b_3)) &=\epsilon(a_1\circ b_1)S(a_2)(a_3\circ S(b_2))=S(a_1)(a_2\circ S(b)).
ik \sum_{k_1+k_2+k_3 = k}^* (k_1 + k_2)\frac{e^{-it(k^3-k_1^3-\alpha k_2^3-\alpha k_3^3)}m_{k_1}n_{k_2}n_{k_3}}{k^3-\alpha(k_1+k_2)^3 - \alpha k_3^3}= -\frac{i}{3\alpha}\sum_{k_1 + k_2 + k_3 = k}^* (k_1 + k_2)\frac{e^{-it(k^3-k_1^3-\alpha k_2^3-\alpha k_3^3)}m_{k_1}n_{k_2}n_{k_3}}{(k_1+k_2 - c_1k)(k_1+k_2-c_2 k)}.
\mathbb{P}\left(\exists z\in\mathbb{R}^2\colon\sup_{t\leq T}\phi_{t}^{(1)}(z)=\infty\right)\geq\lim_{x_0\to\infty}\mathbb{P}\left(\sup_{z\in \mathfrak{I}}\sup_{t\leq T}\phi_{t}^{(1)}(z)=\infty\right)=1
\tau\sum_{k=1}^{m}U_4^{k}(\overline{\partial_{\tau}\theta}_{\mathbf{A}}^{k}) \leq C\big(h^{2r}+ \tau^{4}\big) + C\tau \sum_{k=0}^{m}\big(\Vert\nabla \theta_{\Psi}^{k}\Vert_{\mathbf{L}^{2}}^{2}+ \Vert \partial_{\tau}\theta_{\mathbf{A}}^{k}\Vert_{\mathbf{L}^{2}}^{2}\big).
\Psi_{k,\ell,m,n}(r,\alpha,\theta ,\varphi )=\frac{F_{k,\ell,m,n}(r)}{r \sqrt{r}}\frac{G_{\ell,m,n}(\alpha)}{\sin \alpha}\frac{\Theta_{m,n}(\theta )}{\sqrt{\sin\theta}}\Phi_n (\varphi ).
\phi_{1,p}(\kappa, \pi;u)= \int_{\mathcal{C}_{<}}\frac{dw}{2\pi i}e^{\frac{1}{3}(w-\kappa)^3-u(w-\kappa)}\frac{1}{\sqrt{2w}}\prod_{k = 1}^{p}\frac{w+\pi_{k}}{w- \pi_{k}}.
\sum_{0\le m_1{\le}\cdots{\le}m_{n}<\infty}\frac{(\alpha)_{m_1}}{{m_1}!}\frac{{m_n}!}{(\alpha)_{m_n}}\left\{\prod_{i=1}^{n}\frac{1}{(m_i+\alpha)^{a_i}(m_i+\beta)^{b_i}}\right\},