harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_0 =\bar{E}[H_0]^{\leq 2+m}+\bar{E}[P]^{\leq 5+m}, I_1 =\bar{E}[H_1,H_2]^{\leq 2}+\bar{E}[Q]^{\leq 5}+\bar{E}[H_0]^{\leq 5+m}+\bar{E}[P]^{\leq 8+m}.
{\cal F}_{g a u g e}= \frac{i ( 4 - N_{f})}{4 \pi}a^{2}\Big [ \log \frac{a^{2}}{\Lambda^{2}}- \sum_{n = 1}^{\infty}{\cal F}_{n}\Big ( \frac{\Lambda}{a}\Big )^{\frac{n}{\alpha}}\Big ] .
P_{m^{n+1}-\lambda^\prime}(z_1,\ldots ,z_n,z;t,q,\mathbf{t}) = & \sum_{\substack{\mu\subset n^m, \, 0\leq r\leq m \mu\sim_{r}\lambda}}(-1)^{r+|\lambda|-|\mu|}C^{\lambda,m}_{\mu , r}(q,t,\mathbf{t}) P_{m^n-\mu^\prime}(z_1,\ldots ,z_n;t,q,\mathbf{t}) \langle z;t_0\rangle_{t,m-r},
L_1=\left( \begin{array}{ccccc}0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0\end{array}\right) ,\,\,\,L_2=\left( \begin{array}{ccccc}0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0\end{array}\right)
S_{FT}=\int{\cal L}dx\;,\;\;{\cal L}=\sqrt{-g}\left[ \left( P_{\mu}\varphi\right)^{\ast}g^{\mu \nu}P_{\nu}\varphi -m^{2}\varphi^{\ast}\varphi \right] ,\;\;P_{\mu}=i\partial_{\mu}-qA_{\mu}\;.
z\,{}_2 F_1\bigg (1, \frac{d-1}{2}+1;\frac{1}{2}+1; z \bigg) = \frac{1}{d}\bigg\{(1-z)\,{}_2 F_1\bigg (1, \frac{d-1}{2}+1;\frac{1}{2}; z \bigg) - 1 \bigg\}.
\frac{1}{n}\sum_{k\in N}e_{k,g}^n +\frac{1}{n}e^{n-1}_{i,g+1}+\frac{1}{n}e_{i,g+2}^{n-2}&+\left(1-\frac{1}{n}\right)\left[\frac{1}{n^2}e_{i,g+1}^{n-3}+\frac{1}{n}e_{i,g+2}^{n-3}\right]+\frac{1}{n}\sum_{\tau=1}^{n-4}e_{i,g+\tau+2}^{n-\tau-3}.
v_k(r) = v_k(r; c_k^+, c_k^-) := \begin{cases}c_k^+ r^{\gamma^+_k}+ c_k^- r^{\gamma^-_k}= r^{-\frac{n-2}{2}}(c_k^+ r^{b_k}+c_k^- r^{-b_k}), b_k\neq 0; r^{-\frac{n-2}{2}}(c_k^+ + c_k^- \log r), b_k = 0. \end{cases}
III' = 4\pi c - \frac12 c^{-1}W(ha) \int_{|x-a|<h^{-1}}Q_{a, c}(x)^2 \, dx & +\frac14 c^{-1}h^2W''(ha) \int_{|x-a|<h^{-1}}(x-a)^2 Q_{a, c}(x)^2 \, dx + O(h^3)
\left( \begin{array}{c c}{{A}}&{{B}}\\{{C}}&{{D}}\\ \end{array}\right) = \left( \begin{array}{c c}{{A}}&{{0}}\\{{C}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c c}{{1}}&{{A^{- 1}B}}\\{{0}}&{{D - C A^{- 1}B}}\\ \end{array}\right) ~ .
0\leq u\leq \nu<1 \quad\hbox{on}\partial^T D_2\backslash (F_1\cap F_2),\quad\hbox{(by \eqref{eq:A_u_boundary}with $i=1$ and $j=2$)}0\leq u\leq 1 \quad\hbox{on}D_2\backslash (F_1\cap F_2),\quad\hbox{(by \eqref{eq:A_u_bounds})}Au = 0 \quad\hbox{on}D_2\backslash \partial^T D_2,\quad\hbox{(by \eqref{eq:A_u})},
c_2 =3 \nu^{7}-123 \nu^{6}+1330 \nu^5-1918 \nu^4-28897{\nu}^3+65177 \nu^2+93100 \nu+120544 + \left(36 \nu^{7}-624 \nu^{6}+348 \nu^5+25616 \nu^4-7332 \nu^3-272368 \nu^2-134556 \nu+688784 \right) + \left(108 \nu^{7}-72 \nu^{6}-4848 \nu^5-35916 \nu^4+247548 \nu^3-252720 \nu^2+144456 \nu-647676 \right){\mu}^2
\sum_{k=1}^{n}\frac{1}{k^{2}}\sim\frac{\pi^{2}}{6}-\frac{1}{n}+\frac{1}{2n^{2}}-\frac{1}{6n^{3}}+\frac{1}{30n^{5}}-\frac{1}{42n^{7}}+\cdot\cdot\cdot.
\langle 1 \cdots M|\mathcal{B}^\prime(z_1)\cdots \mathcal{B}^\prime(z_N)|\overline{x_1}\cdots\overline{x_N}\rangle=\sum_{\overline{y}}\langle 1 \cdots M|\mathcal{B}^\prime(z_1)|\overline{y}\rangle\langle \overline{y}|\mathcal{B}^\prime(z_2) \cdots \mathcal{B}^\prime(z_N)|\overline{x_1}\cdots\overline{x_N}\rangle,
\sum_{i=0}^{2\ell}\binom{2\ell}{i}\,K_n(i)K_m(i)=\delta_{n,m}\, 2^{2\ell}\, \binom{2\ell}{n}^{-1}, -i\, K_n(i) = \frac{N-n}{2}\, K_{n+1}(i) - \frac{N}{2}\, K_n(i) +\frac{n}{2}\, K_{n-1}(i).
\Gamma(Q_\infty,Q)=\Bigg\{(mj_1-i(q^2-q),i+mj_2) \mid 1\leq i\leq m-1,\; j_1\geq \left\lceil\frac{i(q^2-q)}{m}\right\rceil,\; j_2\geq 0, &\mbox{and}j_1+j_2=(q-2)\left\lceil \frac{i}{s}\right\rceil+1 \Bigg\},
\mathbb{P}[ (w_n x_0 \cdot w_m x_0)_{x_0}\ge l(n) ] = \mathbb{P}[ (u_m x_0 \cdot x_0)_{w_m^{-1}x_0}\ge l(n) ] & = \mathbb{P}[u_m x_0 \in S_{w_m^{-1}x_0}(x_0,R) ]
(1+ \lambda t)^{\frac{x+y}{\lambda}}= (1+\lambda t)^{\frac{x}{\lambda}}(1+\lambda t)^{\frac{y}{\lambda}}= \left( \sum_{l=0}^\infty{x \choose l}_\lambda t^l \right) \left( \sum_{m=0}^\infty{y \choose m}_\lambda t^m \right) &= \sum_{n=0}^\infty \left( \sum_{m=0}^n{y \choose m}_\lambda{x \choose n-m}_\lambda \right) t^n.
\sum_{0\leq k\leq s/3}\left(C_{0,s-3k,2k}(n_{1})-\frac{C_{0,s-3k,2k}(n_{2})C_{0,1,0}^{s-3k}(n_{1})C_{0,0,1}^{2k}(n_{1})}{C_{0,1,0}^{s-3k}(n_{2})C_{0,0,1}^{2k}(n_{2})}\right)\left(\frac{b_{6}^{2}(E_{1})}{b_{4}^{3}(E_{1})}\right)^{k}=0.
C_\ell:=\inf_{\substack{m\in\mathbb{N}m\geq 2}}\left(\frac{C_{\geq m}}{\ell}+C_{\leq m,\ell}\right),C_\ell':=\inf_{\substack{m\in\mathbb{N}m\geq 2}}\left(\frac{C_{\geq m}}{\ell}+C'_{\leq m,\ell}\right).
d\theta^1_q = 0, d\theta^2_q = 0, d\theta_{x}= 0, d\theta_p^1 = \theta_{x}\wedge \theta_{q}^1, d\theta_p^2 = \theta_{x}\wedge \theta_{q}^2, d\theta^1_y = \theta_x \wedge \theta_p^1 , d\theta^2_y = \theta_x \wedge \theta_p^2 , d\theta_z = - \theta^1_y\wedge \theta^2_q + \theta^2_y\wedge \theta^1_q + \theta^1_p \wedge \theta^2_p,
\langle m_{1}; m_{2}; . . . ; m_{n}; A ; \pm | \overline{{m}}_{1}; \overline{{m}}_{2}; . . . ; \overline{{m}}_{n}; A ; \pm \rangle ={\delta}_{m_{1}\overline{{m}}_{1}}{\delta}_{m_{2}\overline{{m}}_{2}}. . .{\delta}_{m_{n}\overline{{m}}_{n}}.
v_{0}= u -w_1 a^{-\gamma}-(\mbox{$\frac{1}{12}$}u'' + u^2) a^{-2 \gamma}+2 (\mbox{$\frac{1}{6}$}w_1'' +u w_1) a^{-3 \gamma}+ \mbox{O}(a^{-4 \gamma}).
\sum_{k=0}^{j-2}(-1)^kb_{j-1-k}=\sum_{k=0}^{j-2}(-1)^k(|\mathcal{C}_{j-1-k}|+|\mathcal{D}_{j-1-k}|) &=|\mathcal{C}_{j-1}|+(-1)^{j-2}|\mathcal{D}_1|=|\mathcal{C}_{j-1}|.
\{\tilde{w},w\}_0 =\lim_{z\to w}\log\frac{\tilde{z}-\tilde{w}}{z-w}, \{\tilde{w},w\}_1 =2\lim_{z\to w}\frac{\partial}{\partial z}\log\frac{\tilde{z}-\tilde{w}}{z-w}, \{\tilde{w},w\}_2 =6\lim_{z\to w}\frac{\partial^2}{\partial z\partial w}\log\frac{\tilde{z}-\tilde{w}}{z-w}.
A(x,y) = \frac{(\cos(x)-\cos(x-y) + R(x)\cos(x) - R(x-y)\cos(x-y))^{2}}{4\left|\sin\left(\frac{y}{2}\right)\right|^{2}}& + \frac{(r(\sin(x)-\sin(x-y))+ R(x)\sin(x) - R(x-y)\sin(x-y))^{2})}{4\left|\sin\left(\frac{y}{2}\right)\right|^{2}}.
W_{\bar{i}^0}' := \left\{\bar{i}\in [m]^k : i_1 \in [m] \setminus \{i_1^0 \}, \ldots, i_k \in [m] \setminus \{i^0_k \}\right\}, &W^*_{\bar{i}^0}:= \left\{\bar{i}\in [m]^k : \bar{i}\neq \bar{i}^0 \land \left( \bigvee_{s \in [k]}i_s = i_s^0 \right)\right\}.
|\partial_{r_n}D' h_k (p) - \partial_{r_n}D' h_k(q) | \le | \partial_{r_n}D' h_k (p) - \partial_{r_n}D' h_k (p')| + | \partial_{r_n}D' h_k (p') - \partial_{r_n}D' h_k (q')| & + | \partial_{r_n}D' h_k(q') - \partial_{r_n}D' h_k (q)|.
\int_{\Omega}C: \nabla H|\nabla H|^{q-2}x=\int_{\Omega}\big(u|\nabla H|^q+(H\otimes \nabla u) :\nabla H|\nabla H|^{q-2}\big) x \leq C|D(u)|_\infty |\nabla H|^q_q+C|H|_\infty|\nabla H|^{q-1}_q |u|_{D^{2,q}},
\Big\|\sum_{k\leq i}\sum_{l\leq i}\frac{1}{\sqrt{\lambda_k+y-\lambda_i}}\frac{1}{\sqrt{\lambda_l+y-\lambda_i}}P_kEP_l\Big\|_\infty^2 &=\Big\|\sum_{r\geq 1}\sum_{s\geq 1}T_rET_s\Big\|_\infty^2\leq \sum_{r\geq 1}\sum_{s\geq 1}\|T_rET_s\|_\infty^2.
\mathcal{N}[m](J,K)= \sum_{\substack{(jk,q)=1 j\sim J, k\sim K}}N[m](X,q;j,k) = \#\{(j,k,u,v); j\sim J,k\sim K, 0<j^2u,k^2v<X,mj^2u\equiv k^2v\!\!\!\pmod{q}\}
\frac{1}{2t_n}\left(u_n \Phi(x) u_n^{-1}- \Phi(x)\right) = -\Phi(x_n)+\frac{1}{2t_n}\sum\limits_{k=1}^{n-1}\left(u_n \Phi(t_k x_k)u_n^{-1}- \Phi(t_k x_k)\right).
\lim_{t\to 0}\frac{1}{t}(K(u+tv)-K(u)) =\lim_{t\to 0}\int_\Omega f(x,u+z tv)vd\mu_g(x) &=\int_\Omega\lim_{t\to 0}f(x,u+z tv)vd\mu_g(x) &=\int_\Omega f(x,u)vd\mu_g(x).
L_{\alpha}^{\prime ab}=X^{(1)}\left( L_{\alpha}^{ab}\right)+\tfrac{\partial v^{\beta}}{\partial y^{\alpha}}L_{\beta}^{ab}-2\tfrac{\partial u^{a}}{\partial x^r}L_{\alpha}^{br}+\operatorname{div}(X^{\prime})L_{\alpha}^{ab},
\det \begin{pmatrix}\langle e_{\xi_0}, d\Phi_0(\partial_x)\rangle \langle e_{\xi_0}, d\Phi_0(\partial_y)\rangle \langle e_{\zeta_0}, d\Phi_0(\partial_x)\rangle \langle e_{\zeta_0}, d\Phi_0(\partial_y)\rangle \end{pmatrix}(p)\ne 0.
( c^{2}M_{3}^{2}- m_{2}^{2}G^{2})^{1 / 2}( c | M_{3}| )^{- 1}= ( 1 + 4 a^{2}( \omega c^{- 1})^{2}( 1 - e^{2})^{- 1}( 1 + ( 1 - ( 2 a \omega c^{- 1})^{2})^{1 / 2})^{- 2})^{- 1 / 2}.
\left( \begin{matrix}{X^{\Lambda}}\\{-{\frac{i}{2}}F_{\Lambda}}\\ \end{matrix}\right) \Longrightarrow \left( \begin{matrix}{X^{0}}\\{- i X^{1}}\\ \end{matrix}\right) , \left( \begin{matrix}{X^{1}}\\{- i X^{0}}\\ \end{matrix}\right) .
\mathbb{P}(E_0) = \mathbb{P}\left( \displaystyle\sum\limits_{i = 1}^{L}T_i > \frac{L}{p}(1 + \delta_2) \right) & = \mathbb{P}\left( \frac{1}{L}\displaystyle\sum\limits_{i = 1}^{L}T_i > \frac{1}{p}(1 + \delta_2) \right) \le \epsilon_2 ( L) \rightarrow 0
\begin{aligned}w^2 \, \partial_w^2 \, f(w) = \sum_{i=1}^N f(z_i) \sum_{\substack{j\neq j' (\neq i)}}^N \frac{w}{z_{ij}}\, \frac{w}{z_{ij'}}\prod_{k(\neq i,j,j')}^N \frac{w-z_k}{z_i - z_k}\, .\end{aligned}
\frac{\sqrt{-\hat{g}}}{\hat{g}_{str}}\frac{\hat{g}^{00}\hat{g}^{11}F_{01}}{\sqrt{1+(2\pi\alpha^\prime)^2 \hat{g}^{00}\hat{g}^{11}F_{01}^2}}= B_{23}=\frac{s}{2\pi R^2},
A_{k}\left( \mathbf{v},\mathcal{N}_{1}^{\mathcal A}\mathbf{w}\right) =\left(\kern-.1em\left(\kern-.1em\mathbf{v},L_{\Omega}\mathbf{w}\kern-.1em\right)\kern-.1em\right) -\operatorname*{i}kb_{k}\left( \mathbf{v}^{\operatorname*{curl}},\left( L_{\Omega}\mathbf{w}\right)^{\operatorname*{curl}}\right)\qquad\forall\mathbf{v}\in\mathbf{X}.
0 \leq\lim_{n\to\infty}\left( f^{i,i}\circ c^i(s_n) +\sum_{j=1,j\not=i}^mf^{j,i}\circ c^j(s_n)+g^j(s_n)\right) &\leq f^{i,i}\circ c^i(s) +\sum_{j=1,j\not=i}^mf^{j,i}\circ c^j(s)+g^j(s)=0.
(\varphi^{*}C^{\infty}(N))\widehat{}:= \left\{f\in\mathcal{C}^{\infty}(M):\,\forall_{b\in\varphi(M)}\,\exists_{G_{b}\in\widehat{\mathcal{Q}}_{b}}: \left(\hat{f}_{a}=\hat{\varphi}_{a}^{*}(G_{b})\,,\forall_{a\in\varphi^{-1}(b)}\right)\right\}.
_2F_1(a,b;c-1;x) = \left(1-\frac{a b x}{(c-1) c (x-1)}\right) \,_2F_1(a,b;c+1;x) &+\frac{a b x (x (-(a+b+1))+c (2 x-1)+1)}{(c-1) c (c+1) (x-1)}\,_2F_1(a+1,b+1;c+2;x)
A^{11}= \frac{b^{\prime}r - b}{2r^{3}}+ \frac{1}{r \left (e^{2\psi}\alpha^{2}- \beta^{2}r^{2}\right )}\left [ e^{2\psi}\psi^{\prime}\left ( 1 - \frac{b}{r}\right ) \alpha^{2}- \beta^{2}b \right ]
\partial_{i}V ={\frac{1}{4}}\bar{Z}^{C D}\bar{Z}^{A B}P_{A B C D , i}+{\frac{1}{4}}Z_{A B}Z_{C D}P^{A B C D}{}_{, i}+{\frac{1}{2}}\bar{Z}_{I}P_{A B , i}^{I}\bar{Z}^{A B}+{\frac{1}{2}}Z^{I}P_{I , i}^{A B}Z_{A B}= 0 \ .
\partial_{t}v_{4}(t,r) = -\int_{t}^{\infty}ds \frac{v_{4,s}(t,r)}{(s-t)}-\frac{1}{2\pi}\int_{t}^{\infty}ds \int_{B_{s-t}(0)}\frac{dA(y)}{\sqrt{(s-t)^{2}-|y|^{2}}}_{v_{4,1}}
\frac{1}{m}\sum_{j = -m/2+1}^{m/2}\frac{e^{2 \pi i \frac{j + \theta_1/2}{m}\lfloor sm \rfloor}}{1 - \exp \left\{- 2 \pi i \frac{j + \theta_1/2}{m}\right\}+ \frac{z}{m}}= e^{\pi i \theta_1 \lfloor sm \rfloor/m}\sum_{j = - m/2 +1}^{m/2}a_{j,m}b_m^j
\int{\cal D}[g]{\cal F}= \int{\cal D}[f] \: \int{\cal D}[\sigma]{d\tau_i\over v(\tau)}\left[{\det'(P^\dagger P)\over \det(\phi_a,\phi_b)\det(\psi_k,\psi_l)}\right]^{1\over 2}\det (\psi_m,{\partial g\over \partial\tau_n}) \, .
\sum_{k=0}^n \binom{a}{k}\binom{b}{n-k}\binom{k}{c}\binom{n-k}{d}H_{a-k}={}\binom{a+b-c-d}{n-c-d}\binom{a}{c}\binom{b}{d}(H_{a+b-n}-H_{a+b-c-d}+H_{a-c})
\sum_{k=0}^{\infty}\binom{2k+2n+1}{k+n}x^k = \frac{1}{\sqrt{1-4x}}\left( \frac{1 - \sqrt{1-4x}}{2x}\right)^{2n+1}= \frac{\mu_0}{\displaystyle 1 + a_0 x - \frac{b_1 x^2}{\displaystyle 1 + a_1 x -\frac{b_2 x^2}{\displaystyle 1 + a_2 x - \cdots}}}
C(\Delta(r+2)) \geq c_r+d_r \geq 4(c_{r-2}+d_{r-2})+1 = 4\cdot C(\Delta(r)) + 1 &\geq 4\cdot\frac{2^r-2^{r-3}-1}{3}+1 = \frac{2^{r+2}-2^{r-1}-1}{3}
\mathcal{E}(d)=\sum_{i \in Q_0^{\sigma}}\frac{d_i(d_i -s_i)}{2}+ \sum_{i \in Q_0^+}d_{\sigma(i)}d_i - \sum_{\sigma(i) \xrightarrow[]{\alpha}i \in Q_1^{\sigma}}\frac{d_i(d_i + \tau_{\alpha}s_i)}{2}-\sum_{i \xrightarrow[]{\alpha}j \in Q_1^+}d_{\sigma(i)}d_j.
\nu = \frac{1}{4}\sum_{p=1}^{n}{\sum_{q > p}{n^{-2}(n-1)^{-2}\lambda^2_n(x_p,x_q)}}=\frac{1}{4n^2(n-1)^2}\sum_{p=1}^{n}\sum_{q > p}{\lambda^2_n(x_p, x_q)}
\sup\left\{\int_{0}^{k}G[(M_{\mathcal{T}}\phi)^{\ast}(t)]h(t)dt:\phi\text{measurable on}X\phi^{\ast}=g\right\}= =\int_{0}^{k}G\left( \frac{1}{t}\int_{0}^{t}g(u)du\right) h(t)dt.
A_1 = \begin{pmatrix}0 1 0 1 \end{pmatrix}, A_2 = \begin{pmatrix}1 0 0 0 \end{pmatrix}, A_3 = \begin{pmatrix}1 0 1 0 \end{pmatrix}, A_4 = \begin{pmatrix}0 1 -1 1 \end{pmatrix}, A_5 = \begin{pmatrix}1 -1 1 -1 \end{pmatrix}, A_6 = \begin{pmatrix}0 0 1 0 \end{pmatrix}.
(\rho u)_{\kappa}(x,t) \cdot \varphi_{\kappa}(x,t)=m_{0,\kappa}(x) \cdot \varphi_{\kappa}(0) + \int_{0}^{t}\big [\varphi_{\kappa}(x,s) \cdot g_{\kappa}(x,s)+ \partial_{t}\varphi_{\kappa}\cdot (\rho u)_{\kappa}(x,s) \big ]ds + \int_{0}^{t}\varphi_{\kappa}(x,s)dM_{s}(\xi_{\kappa,x}).
A(\xi) = \min\left( \frac{|\xi - \xi'|}{1+|\xi'|},\frac{1}{1+|\xi|}\right)~,~A(h^T \xi) = \min\left( \frac{|h^T \xi - \xi''|}{1+|\xi''|}, \frac{1}{1+|h^T \xi|}\right)~.
\| X_t - Y_t^{N,I,R}\|_{L^p(\P;H)}\leq \| X_t - X^{\tilde I, \operatorname{Id}_U}_t \|_{L^p(\P;H)}+ \| X_t^{\tilde I, \operatorname{Id}_U}- X^{I, \operatorname{Id}_U}_t \|_{L^p(\P;H)}&+\| X_t^{I, \operatorname{Id}_U}- X^{I, R}_t\|_{L^p(\P;H)}+\| X^{I,R}_t - Y_t^{N,I,R}\|_{L^p(\P;H)}.
\sum_{i=1}^N\beta_i\log_2\left(1+\frac{\left|g^i_{ss}\right|p_s^i}{\sigma_n^2+\left(\left|\hat{g}^i_{ps}\right|^2+2\varepsilon\left|\hat{g}^i_{ps}\right|+\varepsilon^2\right)p_p}\right)
N ={\cal N}|_{\pi^{*}{\cal S}}={\cal O}_{\pi^{*}{\cal S}}( ( n ( \frac{1}{2}+ \lambda ) \sigma + ( \frac{1}{2}- \lambda ) \pi^{*}\eta + ( \frac{1}{2}+ n \lambda ) \pi^{*}c_{1}( B ) ) \cdot \pi^{*}{\cal S}) .
[b_0(t,y)-\pi(t,y)\sigma_0^2(t,y)-\frac{\lambda\pi(t,y) \gamma_0^2(t,y)}{1+\pi(t,y)\gamma_0(t,y)}]\mathbb{E}[\delta_Y(y)|\mathcal{F}_t]+\sigma_0(t,y)\mathbb{E}[D_{t}\delta_Y(y)|\mathcal{F}_t] &+\frac{\lambda\gamma_0(t,y)}{1+\pi(t,y)\gamma_0(t,y)}\mathbb{E}[D_{t,1}\delta_Y(y)|\mathcal{F}_t]= 0.
\tilde{g}_{\mu\nu}=\left(\begin{array}{lc}e^{-U}e^{-U}b_j begin{align*}9pt]e^{-U}b_i e^{-U}b_i b_j + e^{-U}\tilde{\gamma}_{ij}\end{array}\right)\ ,\hspace{30pt}B_{\mu\nu}=\left(\begin{array}{lc}0 \omega_j begin{align*}9pt] -\omega_i b_{ij}\end{array}\right) \ .
(\frac{1/\sigma_{x}^{2}}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}})^{2}\sigma_{x}^{2}+(\frac{1/\sigma_{d}^{2}}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}})^{2}\sigma_{d}^{2}=\frac{1}{1/\sigma_{x}^{2}+1/\sigma_{d}^{2}}
\alpha(n;k_1,\ldots,k_n)= \sum_{\substack{(l_1,\ldots,l_{n-1})\in\mathbb{Z}^{n-1}, k_1 \leq l_1 \leq k_2 \leq l_2 \leq \cdots \leq k_{n-1}\leq l_{n-1}\leq k_n, l_i < l_{i+1}}}\alpha(n-1; l_1,\ldots,l_{n-1}),
\frac{x}{(1-x\rho_j^2)^2(1-y\rho_j^2)}-\frac{y}{(1-x\rho_j^2)(1-y\rho_j^2)^2}=\frac{x-y}{(1-x\rho_j^2)^2(1-y\rho_j^2)^2}, \frac{1}{(1-x\rho_l^2)(1-y\rho_j^2)}-\frac{1}{(1-x\rho_j^2)(1-y\rho_l^2)}=\frac{(\rho_l^2-\rho_j^2)(x-y)}{(1-x\rho_j^2)(1-x\rho_l^2)(1-y\rho_j^2)(1-y\rho_l^2)},
\tilde{t}=t^{\ast}+\frac{1}{f(u_{0})}\bigl(\Delta^{-1}\otimes\Phi^{-1}(u_{0})\bigr)\sum_{\mathbf{i}\in G_{N}}Z_{\mathbf{i}}\psi(\varepsilon_{\mathbf{i}})K\biggl(\frac{U_{\mathbf{i}}-u_{0}}{h_{\mathbf{n}}}\biggr).
\mathcal{A}_{t h}( 0 , \infty ) = \mathcal{N}= \mathcal{A}_{0}( 1 , \infty ) , \mathcal{A}_{t h}( - \infty , 0 ) = \mathcal{N}^{\prime}\cap \mathcal{M = A}_{0}( 0 , 1 )
\int \delta^{1 0}u \sqrt{| g_{\alpha \beta}|}e^{- \Phi^{\prime}}\sum_{I = 1}^{1 6}\widehat{F}^{I , \mu \nu}\widehat{F}_{\mu \nu}^{I}= \int \delta^{n^{\prime}+ m^{\prime}}u \sqrt{| g_{\alpha^{\prime}\beta^{\prime}}|}e^{- \phi}~ \sum_{I = 1}^{1 6}[ \widehat{F}^{I , \mu^{\prime}\nu^{\prime}}\widehat{F}_{\mu^{\prime}\nu^{\prime}}^{I}+ 2
u_t-d_1\Delta u+\chi_1 \nabla u \cdot \nabla w & \leq u\left\{a_0-(a_1-\frac{\chi_1}{d_3}k)u+|\Omega|(a_3)_-\overline{u}-\left(a_2 +|\Omega|a_4\right)\underline{v}\right\}&+u\left\{\left(a_2+|\Omega|((a_3)_-+a_4)+k\frac{\chi_1}{d_3}+2l\frac{\chi_{1}}{d_{3}}\right)\epsilon\right\},
K^0_{ij}(\,p'_{ij0},\,p_{ij0},\,P_{ij0})\,=\beta_i\beta_j\,\Lambda^{++}_{ij}\,\beta_i\beta_j\,K^T_{ij}(\,s_{ij},\,s_{ij},\,P_0-h_k\,)\,\Lambda^{++}_{ij}
K_{V^{\gamma}}=\frac{2}{\pi^{3}}\frac{e^{w\bar{\eta}}(1-z^{\prime}\bar{\zeta^{\prime}}+2e^{w\bar{\eta}}z\bar{\zeta})}{(1-e^{w\bar \eta}z\bar{\zeta}-z^{\prime}\bar{\zeta^{\prime}})^{4}}.
|D_V(t_1,t_2)|M_{t_1}M_{t_2}\leq Cn^{4i-2i_1-2i_2}M_{2i_1}M_{2i_2}= C(1+O(n^{-1}))^2n^{4i-2i_1-2i_2}n^{i_1(i_1 + 3) + i_2(i_2+3)}p^{i_1+1+i_2+1}\leq Cn^{-5i +k -1 + i_1^2+i_2^2}p^{i+1}\leq Cn^{i(i+3)}p^{i+1}n^{k(k-1) - 2i_1i_2}.
\int_{y>f_q(x)}(y-f_q(x))^{q-1}d\rho_x(y)\geq \int_{y>f_q^\epsilon(x)+\epsilon}(y-f_q(x))^{q-1}d\rho_x(y) &\geq\int_{y>f_q^\epsilon(x)+\epsilon}(y-f_q^\epsilon(x)-\epsilon)^{q-1}d\rho_x(y)=\int_{y<f_q^\epsilon(x)-\epsilon}(f_q^\epsilon(x)-\epsilon-y)^{q-1}d\rho_x(y).
P\big(\sup_{0\leq t\leq T}[\frac{|G_t^{d,p}|}{pe^{-t}}-r^d]^2>\frac{\epsilon^2}{p^2}r^{2d}\big) \leq \frac{p^2}{\epsilon^2r^{2d}}{\rm E}[\frac{|G_T^{d,p}|}{pe^{-T}}-r^d]^2 &=\frac{e^{2T}}{\epsilon^2r^{2d}}{\rm E}[|G_T^{d,p}|-r^dpe^{-T}]^2.
\mathbb{P}\left(\tau \geq t\right) \leq \mathbb{P}\left(\#{\cal N}_t \geq 1\right) \leq \mathbb{E}\#{\cal N}_t \leq C_u^{t}= \exp\left(-2\frac{\log{C_u}}{\log{C_1}}\log{n}\right) \leq \frac{1}{n^2}
\begin{array}{ll}d\geq 5 \sqrt{\delta}\frac{d \sqrt{\delta}}{2}\bigg( \sqrt{\frac{d^2 \delta}{4}+ (d-2)^2}+ d-2 \bigg) < d-1, d=4 \sqrt{\delta}< \frac{2(\sqrt{3}-1)}{d}\approx 0.36602, begin{align*}2mm]d=3 \sqrt{\delta}< \frac{2\sqrt{2}-1}{d}\approx 0.60947,\end{array}
\int_{\omega_+}|W_{\delta}(x)-\psi(x)|^{p+1}=\int_{\omega_-}|W_{\delta}(x)-\psi(x)|^{p+1}=\frac{\mathfrak{A}}{2}+(-\frac{\mathfrak{B}}{2}+n \mathfrak{C})\delta+o(\delta),
\sum_{0\neq z\in Z^d}e^{-i(z,k)}u(\dot{z})|z|^{-d-\gamma}=c(\dot{k})|k|^{\gamma}-\sum_{0\neq z\in Z^d}e^{-i(z,k)}\sum_{s=1}^{n-1}v_s(\dot{z})|z|^{-d-\gamma-s}+h(k), ~~h\in C^m,
\begin{array}{c}{{A^{b a}= ( - )^{\epsilon_{a}+ \epsilon_{b}+ \epsilon_{a}\epsilon_{b}}A^{a b}, \hspace{3 m m}( A^{a b})^{*}= A^{b a},}}\\{{( B^{a})^{*}= ( - )^{\epsilon_{a}}B^{a}, \hspace{3 m m}C^{*}= C .}}\\ \end{array}
\psi_{N, \ell=(N-1), m}(\chi, \theta, \varphi)={\mathbf F}^{-1}(\chi)\,Y_{K=(N-1), \ell=(N-1), m}(\chi ,\theta, \varphi)\equiv{\widetilde Y}_{K=(N-1), \ell=(N-1), m}(\chi,\theta,\varphi),
[\dots,(\alpha_{n-1},\epsilon_{n-1}),(\alpha_n,\epsilon_n),(\alpha_{n+1},\epsilon_{n+1}),\dots] &\qquad=[\dots,(\alpha_{n-1},\epsilon_{n-1}),(\alpha_n+\epsilon_n,-\epsilon_n),(1,1),(\alpha_{n+1}-1,\epsilon_{n+1}),\dots].
\Bigl(\int_a^b\int_{B(x,\sqrt{b})}|u(t,y)|^2\,{\rm d}y\,{\rm d}t\Bigr)^{\frac{1}{2}}\leq \sum_{j=0}^M \sum_{k=1}^N \Bigl(\int_{2^ja}^{2^{j+1}a}\int_{B(x+z_k,\sqrt{a})}|u(t,y)|^2\,{\rm d}y\,{\rm d}t\Bigr)^{\frac{1}{2}}&\lesssim \sum_{k=1}^N \tilde{N}(u)(x+z_k).
\begin{gathered}y_1=\cdots=y_{p-l}= d-1,\y_{p-l+1}= p-l-1,\y_{p-l+2}=\cdots=y_{n-l}=0, u_1=\cdots=u_{p-l}=l,\u_{p-l+1}= \cdots=u_{n-l}= 0, \end{gathered}
\|u_{\mathbf{0}}(t,\cdot)\|_{L^2}\leq M(t) \left( \|g_{\mathbf{0}}\|_{L^2}+ \int_0^t \| f_{\mathbf{0}}(s,\cdot)\|_{L^2}\,ds \right) \leq M(t) \left( \|g_{\mathbf{0}}\|_{L^2}+ t \| f_{\mathbf{0}}\| \right)=M(t)a_{\mathbf{0}}(t).
|-(y_2-y_1)^2+(y_2^0-y_1^0)^2| =|(y_2^0-y_2)+(y_1-y_1^0)|\,|(y_2^0-y_1^0)+(y_2-y_1)| &\le 2\rho \cdot 2 C_0\rho=4 C_0\rho^2.
A_0(\ell) = \frac{(-1)^{\ell+1}(\ell+1)!}{2^{\ell}}\sum_{n=0}^{r}\frac{(-1)^n(2\, \ell-2\, n)!(2\, n)!}{(2\, n+1)! (\ell-2\, n)! (\ell-n)!\, n!},
B_{h}^{n}(x) = \sum_{i=k+l}^{n}\binom{n}{h}\binom{n-k-l}{h-k}^{-1}b_{h-k,i-k-l}^{(n-k-l,\alpha+2l,\beta+2k)}(1-x)^lx^k &\times R_{i-k-l}^{(\alpha+2l,\beta+2k)}(x) (h=k,k+1,\ldots,n-l),
g^{( 4 )}\simeq \Lambda^{2}( \zeta^{2}d T^{2}+ d \zeta^{2}+ d \rho^{2}) + \Lambda^{- 2}\rho^{2}d \phi^{2}, \qquad \Lambda \simeq 1 + \frac{B^{2}\rho^{2}}{4},
\mathbb{E}(S_{0,0}) = \frac{\gamma}{\lambda + \mu + \xi}\mathbb{E}(S\mid X_{0}=(0,1), X_{1}=(1,1)) + \frac{\lambda}{\lambda + \mu + \xi}\mathbb{E}(S\mid X_{0}=(0,1), X_{1}=(0,2)) \qquad+ \frac{\xi}{\lambda + \mu + \xi}\mathbb{E}(S\mid X_{0}=(0,1), X_{1}=(0,0)),
H_{n+1,bj}:= H_{n,bj}\frac{((W_n)^\top V)_{bj}}{(W_n^\top W_n H_n)_{bj}}(b = 1,2,\ldots,R, j= 1,2,\ldots,N), &W_{n+1,ia}:= W_{n,ia}\frac{(VH_{n+1}^\top)_{ia}}{(W_n H_{n+1}H_{n+1}^\top)_{ia}}(i = 1,2,\ldots,M, a= 1,2,\ldots,R),
\mathcal{S}_{d}(e^{(3)}_{n})(x) = \frac{a^{2}\, k^{2}}{2}\, N(e^{(3)}_{n})(x) + \frac{a^{2}k^{2}}{2}\int_{B}\Phi_{0}(x,y) \frac{A(x,y)\cdot e^{(3)}_{n}(y)}{\Vert x -y \Vert^{2}}dy
( 1 + \frac{1}{2}) ( 1 - \frac{1}{2}) ( 1 + \frac{1}{3}) ( 1 - \frac{1}{3}) ( 1 + \frac{1}{4}) \cdots ( 1 + \frac{1}{2 0 1 3}) ( 1 - \frac{1}{2 0 1 3})
\m^{2}( \beta ) = m_{0}^{2}+ \lambda_{0}\beta^{2 - D}\sum_{k = 0}^{\infty}f ( D , k ) \left( m ( \beta ) \beta \right)^{2 k}\int_{m ( \beta ) \beta}^{\infty}d \tau \frac{\tau^{D - 3 - 2 k}}{e^{\tau}- 1},
n_1(\cdot,t) =e^{(t-t_0)\Delta}n_1(t_0) - \int_{t_0}^t e^{(t-s)\Delta}\nabla\cdot \left(\chi n_1(\cdot,s)\nabla c(\cdot,s) + n_1(\cdot,s)u(\cdot,s) \right)ds &\quad\,+\mu_1\int_{t_0}^t e^{(t-s)\Delta}n_1(1- n_1 - a_1n_2)ds =:I_1(\cdot,t)+I_2(\cdot,t)+I_3(\cdot,t).
\theta \! \left(\partial_K \! \left(\bigcup\nolimits_{i\in F}A_i\right)\right)\! \, \leq \, \theta\!\left( \bigcup\nolimits_{i\in F}\partial_K A_i\right) & \leq \, \sum\nolimits_{i\in F}\theta(\partial_K A_i)\, = \, \sum\nolimits_{i\in F}\theta(A_i)\frac{\theta(\partial_K A_i)}{\theta(A_i)}\, \leq \, M \sum\nolimits_{i\in F}\theta(A_i).
{\cal Z}={\frac{y^{2}( q ) \prod_{i < j}A_{i j}B_{i j}}{c^{2}( \mathrm{d e t}K )^{5}\prod_{i < j}( a_{i}- a_{j})}}={\frac{y^{2}( q ) Q_{\delta}}{c^{2}( \mathrm{d e t}K )^{5}P ( a )}},
\mathcal{N}(x)=\begin{cases}\dfrac{1}{2\pi}\log\dfrac{1}{|x|}\N=2, begin{align*}10pt]\dfrac{\Gamma(\frac{N}{2}+1)}{N(N-2)\pi^{\frac{N}{2}}}|x|^{2-N}\N\geq 3.\end{cases}
\hat{T}= \exp \{- a_{0}[ \frac{g^{2}}{2 a}\sum_{\ell}\hat{E}_{\ell}^{2}+ \frac{1}{2 a g^{2}}\sum_{p}( \hat{W}_{p}+ \hat{W}_{p}^{\dagger}) ] + O ( a_{0}^{2}) \},
\left\|\sum_{\boldsymbol{m}}|D^{\boldsymbol{r}}\varphi_{\boldsymbol{N},\boldsymbol{m}}|\right\|_\infty=O(1),\quad\left\|\sum_{\boldsymbol{k}}|D^{\boldsymbol{r}}\psi_{\boldsymbol{j},\boldsymbol{k}}|\right\|_\infty\lesssim\prod_{l=1}^d2^{(1/2+r_l)j_l}.
\mu_{\beta,h}^{\Lambda,\sigma}(\{\omega_{\Lambda}\}) := \frac{1}{Z_{\beta,h}^{\Lambda,\sigma}}e^{-H_{\beta,h}^{\Lambda,\sigma}(\omega_{\Lambda})}, Z_{\beta,h}^{\Lambda,\sigma}:= \sum_{\omega_{\Lambda}\in \tilde{\Omega}_{\Lambda}}e^{-H_{\beta,h}^{\Lambda,\sigma}(\omega_{\Lambda})},
g_0(\frac{\partial\Psi}{\partial x_2},\frac{\partial\Psi}{\partial x_2}) = \frac{1}{4x_1^2}g_0(\xi_3 - d\psi_{x_1,0,0}(\xi_3(m)),\xi_3 - d\psi_{x_1,0,0}(\xi_3(m))) &=\frac{1}{4x_1^2}[\rho^2 -2\rho^2(m) + g_{x_1,0,0}(\xi_3(m),\xi_3(m))].
\inf_{\substack{\eta,{\zeta}\in V, \boldsymbol{\tau},\boldsymbol{\rho}\in H(\emph{div},\Omega), v_{\zeta}= \mathbb{P}_{[u_a,u_b]}\{u_d - \frac{1}{\lambda}{\zeta}\}, \alpha,\beta > 0}}\mathcal{M}_1^\oplus(\alpha,\beta;\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho},v_{\zeta}) = 0.
\nabla(\varphi.\omega) = \frac{\partial \varphi}{\partial \lambda}.b.\omega + \varphi.\nabla(\omega){\rm for}\varphi \in \mathcal{O}_{\lambda}{\rm and}\omega \in \mathcal{H}^{\bullet}b.\nabla = \nabla.b \quad{\rm and}\nabla.a = (a - b).\nabla.
b_{\nu,n}^{(\alpha)}:= [Q_{|\textup{\textbf{m}}|}^{\textup{\textbf{F}}}Q_{n,\textup{\textbf{m}}}F_\alpha]_{\nu}=\frac{1}{2\pi i}\int_{\Gamma_\rho}\frac{Q_{|\textup{\textbf{m}}|}^{\textup{\textbf{F}}}(t) Q_{n,\textup{\textbf{m}}}(t) F_\alpha(t) \Phi'(t)}{\Phi^{\nu+1}(t)}dt, \nu \geq n+|\textup{\textbf{m}}|-m_\alpha+1,
-\frac{A}{\epsilon^{2}}\frac{\partial S}{\partial x_{i}}\frac{\partial S}{\partial x^{i}}+\frac{2i}{\epsilon}\frac{\partial A}{\partial x_{i}}\frac{\partial S}{\partial x^{i}}+\frac{iA}{\epsilon}\frac{\partial^{2}S}{\partial x_{i}\partial x^{i}}+\frac{\partial^{2}A}{\partial x_{i}\partial x^{i}}=0.
\mathfrak{t}_i=1^{\otimes{i-1}}\otimes \mathfrak t\otimes 1^{l-i-1}: V_q^{\otimes l}\rightarrow V_q^{\otimes l}\ \ \ \ \mathfrak{c}_i=1^{\otimes{i-1}}\otimes \mathfrak c\otimes 1^{l-i-1}: V_q^{\otimes l}\rightarrow V_q^{\otimes l}.
\psi_{m}( z ) = c_{1}{\frac{\sqrt{{\cal Z}- z}}{z}}J_{{\frac{1}{2}}\sqrt{1 + 8 K}}[ \sqrt{2}m ({\cal Z}- z ) ] + c_{2}{\frac{\sqrt{{\cal Z}- z}}{z}}Y_{{\frac{1}{2}}\sqrt{1 + 8 K}}[ \sqrt{2}m ({\cal Z}- z ) ] \, ,
tx_{\alpha+\beta}t^{-1}= \begin{bmatrix}1 0 0 0 1 0 a^{-1}c 0 1\end{bmatrix}, \tx_{\alpha}t^{-1}= \begin{bmatrix}1 0 0 a^{-1}b 1 0 0 0 1\end{bmatrix}, \tx_{\beta}t^{-1}= \begin{bmatrix}1 0 0 0 1 0 0 b^{-1}c 1\end{bmatrix},
\left[c_{2,1}\left(\bigcap_{n=l}^{\infty}G_{n}\right)\right]^{2}\leq\left[c_{2,1}\left(\bigcap_{n=l}^{N}G_{n}\right)\right]^{2}\leq C'(N-l)^{2}C_{H}e^{-C_{3}(N^{1-\gamma}-l^{1-\gamma})},
\begin{array}{l c l c l}{{z = t + x}}&{{,}}&{{w = \tau + \theta}}&{{,}}&{{\phi_{+}= \tilde{t}+ \phi}}\\{{\bar{z}= t - x}}&{{,}}&{{\bar{w}= \tau - \theta}}&{{,}}&{{\phi_{-}= \tilde{t}- \phi}}\\ \end{array}