harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E(a,x+y)\sqcup E(x+y-a,y)\sqcup E(x+y-b,x)\hookrightarrow E(a,x+y)\sqcup \sqrt{\alpha}E(x+y-a,x+y) &\subset \sqrt\alpha\left(E(a,x+y)\sqcup E(x+y-a,x+y)\right) \hookrightarrow \alpha E(x+y,x+y)=\alpha B^4(x+y).
d(x^\alpha dx_\sigma) := \sum_{i=1}^n \left(\frac{\partial x^\alpha}{\partial x_i}dx_i\right)\wedge dx_{\sigma(1)}\wedge\dots\wedge dx_{\sigma(k)}, \kappa(x^\alpha dx_\sigma) := \sum_{i=1}^k \left((-1)^{i+1}x^\alpha x_{\sigma(i)}\right)dx_{\sigma(1)}\wedge\cdots\wedge\widehat{dx_{\sigma(i)}}\wedge\cdots\wedge dx_{\sigma(k)}.
\begin{array}{ccl}0 = \mathrm{Cap}_t(v_1\otimes 1,\dots,v_{t-1}\otimes 1,v_t\otimes b_j,w_1\otimes 1,\dots,w_{t+1}\otimes 1) = \mathrm{Cap}_t(v_1,\dots,v_{t-1},v_t,w_1,\dots,w_{t+1})\otimes b_j.\end{array}
\theta'=\operatorname{tr}(B')=-\operatorname{tr}(\overset{\mathrm{sf}}R)-\operatorname{tr}(B^2)= -\operatorname{tr}(\overset{\mathrm{sf}}R)-\operatorname{tr}\left[\left(\omega+\sigma+ \frac{\theta}{n-1}\mathrm{Id}\right)^2\right].
\|f\|_{C^{k,\alpha}_{\gamma,\delta}}= \sum_{i=0}^k \sup \sigma(x)^i w_0(x) |\nabla^i f| + \sup \sigma(x)^{k+\alpha}w_0(x) \sup_{y\in B_x(\sigma(x))}\frac{|P_{y,x}\nabla^k f(y) - \nabla^kf(x)|}{d(x,y)^\alpha},
y\bigl(\bigl\{X_{ni}^{(1)}\bigr\}\bigr)\cdot y\bigl(\bigl\{X_{ni}^{(2)}\bigr\}\bigr) =Z_1+Z_2+\sum_{l=3}^kB_l^{(2)}\bigl(\bigl\{X_{ni}^{(1)}\bigr\},\bigl\{X_{ni}^{(2)}\bigr\}\bigr).
0 \le n_1n_2/k^2 -\lfloor n_1/k\rfloor\lfloor n_2/k\rfloor = (n_2/k-\lfloor n_2/k\rfloor) (n_1/k)+(n_1/k-\lfloor n_1/k\rfloor)\lfloor n_2/k\rfloor \le (n_1+n_2)/k\,,
\stackrel{1}{P^{i}_j}=\delta^{i}_j-\frac{y_jy^{i}}{F^2+K^2}\,,\,\stackrel{2}{P^{i}_j}=\delta^{i}_j-\frac{p^ip_j}{F^2+K^2}\,,\,\stackrel{3}{P_{ij}}=-\frac{y_jp_i}{F^2+K^2}\,,\,\stackrel{4}{P^{ij}}=-\frac{p^jy^{i}}{F^2+K^2}.
\frac{\sin \gamma_{01}}{\sigma_{01}}= \frac{X_0 Y_1 - X_1 Y_0}{\zeta_2 \beta_0 \beta_1}\ \ \ \ \ \ \frac{\sin \gamma_{12}}{\sigma_{12}}= \frac{X_1 Y_2 - X_2 Y_1}{\zeta_0 \beta_1 \beta_2}\;.
\frac{1}{2i\pi}\int\limits_{c+i\infty}^{c-i\infty}R(t) \frac{\pi^3}{\sin(\pi t)^3}e^{u t}dt =\sum_{k=n+1}^\infty \left(\frac{\pi^2+u^2}{2}R(k)+ uR'(k)+ \frac12R''(k)\right)(-e^u)^k.
\begin{array}{ll}w^1(t,y) = \frac{1}{2\sqrt{\pi t}}\int\limits_0^{+\infty}\Big( e^{-\frac{(y-\tilde{y})^2}{4t}}- e^{-\frac{(y+\tilde{y})^2}{4t}}\Big)w_0^1(\tilde{y}) \,\mathrm{d}\tilde{y}<0,\end{array}
\langle \pi_\lambda(h) \circ T^{(\lambda)}_\varphi f,g \rangle = \langle B_\lambda \circ M_\varphi f, \pi_\lambda(h^{-1}) g \rangle = \langle M_\varphi f, \pi_\lambda(h^{-1}) g \rangle = \langle M_{\varphi_h}\circ \pi_\lambda(h) f, g \rangle = \langle T^{(\lambda)}_{\varphi_h}\circ \pi_\lambda(h) f, g \rangle.
\eta_{2,2}(q\,|\,1, b) = \prod\limits_{k=0}^\infty \Bigl[\frac{b_0+k}{q+b_0+k}\frac{q+b_0+b_1+k}{b_0+b_1+k}\frac{q+b_0+b_2+k}{b_0+b_2+k}\frac{b_0+b_1+b_2+k}{q+b_0+b_1+b_2+k}\Bigr]^{k+1}.
h(t) = \begin{bmatrix}a_3 \Phi^{(0)^{\scriptstyle 2}}_t + a_4 \Phi^{(0)}_t \Psi^{(0)}_t + a_5 \Psi^{(0)^{\scriptstyle 2}}_t b_3 \Phi^{(0)^{\scriptstyle 2}}_t + b_4 \Phi^{(0)}_t \Psi^{(0)}_t + b_5 \Psi^{(0)^{\scriptstyle 2}}_t\end{bmatrix}.
\int_{0}^{\infty}P_{r}^{\bf V}( y ) P_{s}^{\bf V}( y ) y^{- 1}e^{- y}d y \propto \int_{0}^{\infty}L_{r - 1}^{( 1 )}( y ) L_{s - 1}^{( 1 )}( y ) y e^{- y}d y \propto \delta_{r s},
\langle T_{z z}( z , \bar{z}) T_{z z}( 0 , 0 ) \rangle = \frac{\pi}{3 \cdot 1 6}\int_{0}^{\infty}d \mu^{2}\, \tilde{c}( \mu^{2}) \int \frac{d^{2}p}{( 2 \pi )^{2}}\frac{e^{\frac{i}{2}( p \bar{z}+ \bar{p}z )}}{p \bar{p}+ \mu^{2}}\, \bar{p}^{4},
u_{i,j}=-2(3c_{i}+4s_{i}+4z_i)-2(3c_{j}+4s_{j}+4z_j) &\phantom{{}={}}+(3c_{|i-j|}+4s_{|i-j|}+4z_{|i-j|}) +(3c_{i+j}+4s_{i+j}+4z_{i+j}) &= 3c_{i,j}+4t_{i,j}+4z_{i,j}.
\tilde{\mu}\big(\tilde{\mathcal{C}}_V^{(n)}\big)=\prod_{\mathbf{u}\in\mathcal{U}^n}\prod_{j\in\mathcal{J}_n}p^n_{V|U}\big(\tilde{\mathbf{v}}(\mathbf{u},j)\big|\mathbf{u}\big),
\varepsilon_{\mu_{1}\ldots \mu_{d}}= e^{- 1}e_{\mu_{1}}^{a_{1}}\ldots e_{\mu_{d}}^{a_{d}}\varepsilon_{a_{1}\ldots a_{d}}, \qquad \varepsilon^{\mu_{1}\ldots \mu_{d}}= e e_{a_{1}}^{\mu_{1}}\ldots e_{a_{d}}^{\mu_{d}}\varepsilon^{a_{1}\ldots a_{d}}.
\begin{aligned}\left.\frac{\partial}{\partial s}\right|_{s=0}H(\Psi_s \circ F \circ \Phi_s) = \frac{1}{2}\langle f_R \nu, \,\nu\rangle + \frac{1}{2}\langle F, -\nabla f_R\rangle = \frac{1}{2}f_R - \frac{1}{2}\langle F, \nabla f_R\rangle,\end{aligned}
\left|\int_{B_1^c\times B_{1/4}}\frac{|u(x)-u(y)|^p-|u(x)-\xi(y)\, u(y)|^p}{|x-y|^{N+ps}}\, dx\, dy\right| \leq 2^p\int_{B_1^c\times B_{1/4}}\frac{|u(x)|^p+|u(y)|^p}{|x-y|^{N+ps}}\, dx\, dy &\leq C\int_{B_1^c}\frac{|u|^p}{|x|^{N+ps}}\, dx +C\int_{B_{1/4}}|u(y)|^p\, dy,
D_{\mu\nu}= \overline{P}_{\mu\nu}\,\frac{1}{p^{2}+\Pi_{\rm T}}+Q_{\mu\nu}\,\frac{1}{p^{2}+\Pi_{\rm L}}+R_{\mu\nu}\,\frac{1}{p^{2}+\tilde{\Pi}_{\rm T}}+\xi\,\frac{p_{\mu}p_{\nu}}{p^{2}}
\frac{2}{\nu}\dot{a}a^{\nu}G_{\perp \perp}= - \frac{\partial}{\partial t}\! \left( \frac{\dot{a}^{2}}{n^{2}}a^{\nu - 1}-{a^{\prime}}^{2}a^{\nu - 1}+ k a^{\nu - 1}\right) = 0 ,
\left|\ln \frac{g^\theta (x;y)}{g^\theta (x';y)}\right| =\left|(r+y)\left[\ln(1+x')-\ln(1+x)\right]+y\left[\ln x-\ln x'\right]\right| &\leq \left[(r+y)(1+\underline \omega)^{-1}+y\,\underline \omega^{-1}\right]\;|x-x'| &\leq \left[\overline r +y\,(1+\underline \omega^{-1})\right]\;|x-x'| \;.
\{b ( p , s ) , b^{\dagger}( p^{\prime}, s^{\prime}) \}= \delta ( p^{+}- p^{\prime + )}\delta^{2}( p_{\perp}- p_{\perp}^{\prime}) \delta_{s , s^{\prime}}= \delta^{3}( p - p^{\prime}) \delta_{s , s^{\prime}}\; ,
\hspace{1 5 p t}i \mathrm{t r}( e_{i}[ e_{j}, e_{k}] ) = - i \mathrm{t r}( \bar{e}_{i}[ \bar{e}_{j}, \bar{e}_{k}] ) = \partial_{i}B_{j k}+ \partial_{j}B_{k i}+ \partial_{k}B_{i j}.
|[P_t\psi]_v - [P_t\psi]_{v'}| \leq \sum_{v'' \in V}|p_t(v,v'') - p_t(v',v'')| |\psi_{v''}| &\leq C(1 + t)^{-\frac{\eta}{2}}\sum_{v'' \in V}p_{2t}(v,v'') |\psi_{v''}| &= C(1 + t)^{-\frac{\eta}{2}}[P_{2t}|\psi|]_v.
2\nu \left\|Dr_\rho\right\|_2^2+\left(\mathbf{curl}\,\sigma\left(\mathbf r_\rho\right)\times \mathbf y+ \mathbf{curl}\,\sigma\left(\mathbf y_\rho\right)\times \mathbf r_\rho+\mathbf{curl}\,\sigma\left(\mathbf y_\rho-\mathbf y\right)\times \mathbf z,\mathbf r_\rho\right)=0.
U(\beta, t)= 1+\sum_{k=1}^{\infty}(-H\beta^2t)^k\prod_{j=0}^{k-1}\frac{\sqrt{\pi}}{2^{2j+1}}\binom{2j+1}{j+1}= 1+\sum_{k=1}^{\infty}\left(-H\sqrt{\pi}\beta^2 t\right)^k\prod_{j=1}^{k}\frac{1}{2^j}\binom{2j}{j}.
\tilde{\Psi}_{K_Y}(Y,T)=\frac{1}{L}\sum_{n=-\infty}^{\infty}e^{\pi^2 n\over H_5 L}H^{(2)}_{2 i\pi n / H_5 L}(mt)e^{i{2 \pi n\over L}(y-H_5^{-1}\sinh^{-1}(\frac{K_Y}{m}))}.
\mathbb E\left[Y_{n,i}^{(t)}\right]^\alpha\leq Cn^{1+\frac{2^m-1}{2^m}\alpha}\left[\max\{1,\; p^{\alpha-1}\frac{\rho(\alpha t)}{\rho(t)^\alpha},\;p^{\frac{2^l-1}{2^l}\alpha}\frac{\rho(2^lt)^{\alpha/2^l}}{\rho(t)^\alpha},l=1,2,\cdots,m\}\right]^n
b_{q,m}^{(l)}=\frac{2(q+m)-k^2+2(l-1)(2k-l+1)}{2l}b_{q,m}^{(l-1)}+ \frac{1}{l}b_{q,m-1}^{(l-1)}+\frac{l-k-2}{l}b_{q-1,m}^{(l-2)}.
D_{L}\equiv D_{L}( \Omega_{B}) = \frac{V_{0}^{\prime}( 1 + z )}{2 H_{0}( \omega + 1 ) V_{0}^{\frac{1}{2}+ \frac{1}{3 ( \omega + 1 )}}}\int{\phi ( 0 )}^{\phi ( z )}\frac{d \phi}{V^{\frac{3 \omega + 1}{6 ( \omega + 1 )}}},
\bigl[\mathcal{H}^{\,(+)}, [\mathcal{H}^{\,(+)},\mathcal{E}(n)]\bigr] =\mathcal{E}(n)R_0\bigl(\mathcal{H}^{\,(+)}\bigr) +[\mathcal{H}^{\,(+)},\mathcal{E}(n)] R_1\bigl(\mathcal{H}^{\,(+)}\bigr) +R_{-1}\bigl(\mathcal{H}^{\,(+)}\bigr).
\theta^{r-1}a_{i}=\sum_{j=1}^{n}k_{ij}\theta^{r-1}\theta^{j-1}=\sum_{j=1}^{n}k_{ij}\sum_{k=1}^{n}c_{rjk}\theta^{k-1}=\sum_{k=1}^{n}\Big(\sum_{j=1}^{n}k_{ij}c_{rjk}\Big)\theta^{k-1}.
\phi(g_1,\ldots,g_4) = \sum_{N_1\ldots N_4}\sum_\Lambda \left(\phi^{N_1\ldots N_4}_{\alpha_1\ldots\alpha_4}\C^{N_1\ldots N_4\,\Lambda}_{\alpha_1\ldots\alpha_4}\right) \left({C}^{N_1\ldots N_4\,\Lambda}_{\beta_1\ldots\beta_4}D^{(N_1)}_{\beta_1\gamma_1}(g_1) \ldots D^{(N_4)}_{\beta_4\gamma_4}(g_4)\w_{\gamma_1}\ldots w_{\gamma_4}\ \right),
u_{0}=\frac{1}{\sqrt{N}}(1,1,1,\cdots,1,1,1)^\top\, u_{n}=\frac{1}{\sqrt{n(n+1)}}(0,\underbrace{-1,\cdots,-1}_,n,0,0,\cdots,0 )^\top, &n=1,2,\cdots,N-2\,, u_{N-1}=\frac{1}{\sqrt{N(N-1)}}(1-N,1,\cdots,1,1)^\top\,.
\sigma_{1,1}:= \frac{2KL}{\beta_0^2}+ \frac{K(K+2L)}{2\left(1-\varrho_0\right)^2}, \sigma_{2,2}:= \frac{KL}{1-\varrho_0^2}+ \frac{K(K+2L)\beta_0^2}{2\left(1-\varrho_0\right)^4}, \sigma_{1,2}= \sigma_{2,1}:= \frac{K(K+2L)\beta_0}{2\left(1-\varrho_0\right)^3},
[P,a^1]\cdots [P,a^{2k+1}]= (-1)^kPa^1(1-P)a^2P\cdots (1-P)a^{2k}Pa^{2k+1}(1-P) &\quad-(-1)^k(1-P)a^1Pa^2(1-P)\cdots Pa^{2k}(1-P)a^{2k+1}P = (-1)^kPa_+^1(1-P)a^2P\cdots (1-P)a_-^{2k}Pa_+^{2k+1}(1-P) &\quad-(-1)^k(1-P)a_-^1Pa_+^2(1-P)\cdots Pa^{2k}(1-P)a_-^{2k+1}P.
- \overline{G}^{(d)}_n(a_0, b_0) = \int_0^1 \nabla_{(a, b)}\overline{G}^{(d)}_n(a_0 + t (\hat{a}_n - a_0), b_0 + t (\hat{b}_n - b_0)) dt \begin{pmatrix}\hat{a}_n - a_0 \hat{b}_n - b_0\end{pmatrix}.
\psi^*(j) = \psi^*(j)^{1/2}\sum_{i=1}^{\infty}\Theta_{x_i,x_i}\psi^*(j)^{1/2}= \sum_{i = 1}^{\infty}\Theta_{\psi^*(j)^{1/2}x_i, \psi^*(j)^{1/2}x_i}.
\begin{cases}\partial_th=|\partial_x|\omega- |\partial_x|T_{|\partial_x|\omega}P_{\leq 0}h-\partial_xT_Vh + G_2 + G_{\geq 3}, &\partial_t\omega=\partial_x^2h+\partial_xT_\sigma h_x-T_V\omega_x + H_2 + T_{|\partial_x|^3 h}P_{\geq 1}h + \Omega_{\geq 3},\end{cases}
\frac{[z^n] \frac{\partial^2}{\partial x_h^2}(P_1+P_2+P_3) |_{x_1=x_2=x_3 = 1}+ [z^n] \frac{\partial}{\partial x_h}(P_1+P_2+P_3) |_{x_1=x_2=x_3 = 1}}{[z^n] (P_1+P_2+P_3) |_{x_1=x_2=x_3 = 1}},
{\frac{B}{4 \pi}}d \left({\frac{1}{4}}4 \pi a^{2}\right) -{\frac{1}{2}}d a = - T_{r}^{r}( a ) d \left({\frac{4 \pi}{3}}a^{3}\right) = - T_{r}^{r}( a ) [ 4 \pi a^{2}] d a
\phi_{\mathbf g, \infty}\left(\left(\begin{smallmatrix}x_1 x_2 x_3 x_4 \end{smallmatrix}\right) \right) = (x_1 + \sqrt{-1}x_2 + \sqrt{-1}x_3 - x_4)^{k+1}\mathrm{exp}(-\pi \mathrm{tr}(x^t x)).
\epsilon^{ijk}\partial_{j}[ve^{\phi}(\delta_{kl}(1 + e^{-\phi}B_{i}^1B_{i}^1 ) - e^{-\phi}B_{k}^1 B_{l}^1 )B_{l}^2 - E_{k}^1 - ve^{\phi}\psi B_{k}^1 ]= 0.
(x_k+y_k-1)(x_{k+1}+y_{k+1}-1) &=x_kx_{k+1}+x_ky_{k+1}-x_k+y_kx_{k+1}+y_ky_{k+1}-y_k-x_{k+1}-y_{k+1}+1 &=(x_ky_{k+1}-x_k-y_{k+1}+1)+(y_kx_{k+1}-x_{k+1}-y_k+1)+(x_kx_{k+1}+y_ky_{k+1}-1) &=(x_k-1)(y_{k+1}-1)+(y_k-1)(x_{k+1}-1)+(x_kx_{k+1}+y_ky_{k+1}-1).
E ( x ) = \frac{1}{{\partial}_{-}^{2}}D ( x ) - \frac{1}{{\partial}_{\bot}^{2}+ n^{2}{\partial}_{+}^{2}}D_{s}( x ) - \frac{\mathrm{s i n}2{\theta}{\partial}_{+}+{\partial}^{-}}{({\partial}_{\bot}^{2}+ n^{2}{\partial}_{+}^{2})^{2}}{\partial}_{-}D_{s}( x )
(1-e_1) \cdot \vartheta (\delta_n^{(t)}) = \omega' \cdot [\mathbb{Q}(\zeta_n): k^n \mathbb{Q}(\zeta_{n/t})] \cdot s\big( G_{n/t}\big) \cdot \prod_{\ell \mid n}\left( 1-\sigma_{\ell}^{-1}\cdot e_{\ell}\right).
\|\widehat{g}\|^2_{L^2(P, d\sigma)}= \frac{1}{|P|}\sum_{x\in P}|\widehat{g}(x)|^2 \sim \frac{1}{|C|}\sum_{s\in \mathbb F_q^*}\sum_{x\in P}|\widehat{g}(x)|^2 = \frac{1}{|C|}\sum_{s\in\mathbb F_q^*}\sum_{\substack{(x^\prime, x_d)\in \mathbb F_q^d : x_1^2+\cdots+x_{d-1}^2=x_d s}}|\widehat{g}(x^\prime, x_d s)|^2 = \|\widehat{G_g}\|^2_{L^2(C, d\sigma_c)},
t r \left[ \ \gamma^{5}\sum_{n}\varphi_{n}( x ) \varphi_{n}^{\dag}( y ) \right]{\cal A}_{\alpha}( y ) = \frac{\partial}{\partial y^{\alpha}}t r \left[ \ \gamma^{5}\sum_{n}\varphi_{n}( x ) \varphi_{n}^{\dag}( y ) \right] .
I = \sum_{j = 0}^{K-1}\Big[ r(z^{u, N}(t_j), \nu_j) - r(\tilde{z}^u(t_j), \nu_j) + \int_{t_j}^{t_{j+1}}\Big(\phi(z^{u, N}(s), \nu_{j}) - \phi(\tilde{z}^{u}(s), \nu_{j})\Big) \,ds\Big] + \psi(z^{u,N}(T)) - \psi\big(\tilde{z}^u(T)\big)\,.
M_{0,t}(z) = (1-z) \frac{1}{2i\pi}\int_{\gamma_0}\frac{K_{2t}(y))^2-1}{[t(K_{2t}(y))^2 + (2-t)]}\frac{dw}{w R(z,w)}& = \frac{(1-z)(K_{2t}(y(z,0))^2-1)}{[t(K_{2t}(y(z,0)))^2 + (2-t)]R(z,0)}.
\sum_{i,j\geq 1}\frac{\omega_{i,j}^{[0]}}{\lambda^{i+1}\mu^{j+1}}=\frac{B(\lambda) B(\mu)((\lambda-v)(\mu-v)-4w)-1}{2 \, (\lambda-\mu)^2}, &\frac1\lambda + \sum\limits_{i\geq 1}\frac{\varphi_i^{[0]}}{\lambda^{i+1}}= B(\lambda),
\hat{\phi}=(c_{1,0}\bar{\rho}_1+c_{0,1}\bar{\rho}_2)\xi+\Biggl({\lambda(c_{1,0}^3\bar{\rho}_1^3+c_{0,1}^3\bar{\rho}_2^3)\over 8m^2}-{3\lambda(c_{1,0}^2c_{0,1}\bar{\rho}_1^2\bar{\rho}_2+c_{1,0}c_{0,1}^2\bar{\rho}_1\bar{\rho}_2^2)\over 4(k_{1,0}k_{2,0}-{\bf k_1}.{\bf k_2}-m^2)}\Biggr)\xi^3+...,
\|(g_c,H_c)-(\widetilde{g}^s_i(T_i),\widetilde{H}_i^s(T_i))\|_{C^{k-3}}= \|(g_c,H_c)-(\widetilde{g}_i(1),\widetilde{H}_i(1))\|_{C^{k-3}}\leq\|(g_c,0)-(\widetilde{g}_i(1),\widetilde{b}_i(1))\|_{C^{k-2}}\rightarrow 0
S ={\frac{N}{g}}t r \left({\frac{1}{2}}A ( \beta ) S^{2}+{\frac{1}{2}}B ( \beta ) D^{2}+{\frac{1}{4}}S^{4}+{\frac{1}{4}}D^{4}+{\frac{1}{2}}( S D S D + 2 S^{2}D^{2}) \right) .
W[u,u^*]^2 = \int_0^M(G(\omega)-G^*(\omega))^2\, d\omega= \int_0^M\bigg(\int_0^\omega(g(\eta)-g^*(\eta))d\eta\bigg)^2 \, d\omega &= \int_0^M\int_0^M(M-\max\{\eta,\eta'\})(g(\eta)-g^*(\eta))(g(\eta')-g^*(\eta'))\, d\eta\,d\eta'.
[q^{n}](\sum_{i\ge t}\frac{a_{i}q^{mi}}{(1-q^{i})^{k+1}})=\sum_{\begin{matrix}d|n\\t\le d\le\lfloor\frac{n}{m}\rfloor\end{matrix}}(\begin{matrix}\frac{n}{d}-m+k\\k\end{matrix})a_{d}.
m_{n}(t)=\allowbreak \sqrt{\frac{n!}{\left( \beta \right)^{\left( n\right)}}}\frac{1}{(1-t)^{\beta}}\frac{1}{\Gamma (\beta )}\times \int_{0}^{\infty}\left( \sum_{j=0}^{n}\left( \frac{-ty}{1-t}\right)^{j}L_{n-j}(x|(\beta +j))/j!\right) y^{\beta -1}\exp (-y)dy.
\underset{j}{\lim}\ (\Gamma^{B,C}(v_j)(Y)Z\Gamma^{A,B}(u)(X)) =(\Gamma^{B,C}(v)(Y)Z\Gamma^{A,B}(u)(X)) & = (\Gamma^{A,B}(u)(X)\Gamma^{B,C}(v)(Y)Z).
\lim_{r \to0}\mathop{\sup_{s,t \in I}}_{\tau_{E}(s-t) \le r}\frac{|X(s) - X(t)|}{\tau_{E}(s-t)\sqrt{\log( 1 +\tau_{E}(s-t)^{-1})}}= C_{6}\qquad\mbox{a.s.},
d_{\overrightarrow{\boldsymbol{N}}}:=\mu^{V;\boldsymbol{A}}_{N-M, \frac{\overrightarrow{\boldsymbol{N}}}{N-M}\beta}(\prod_{i=1}^{M}e^{\sum_{j=1}^{N-M}(-\frac{\beta}{2}V(\eta_{i})+\beta \ln|\lambda_{j}-\eta_{i}|-\frac{\beta}{2}V(\lambda_{j}))}).
\begin{array}{ll}\sum\limits_{i=n+1}^{m}g (\nabla_{U_i}U_i, U_i) + \sum\limits_{j=1}^{n}g (\nabla_{X_j}X_j, X_j) + s^{KerF_\ast}+ \frac{1}{\lambda^2}s^{N}+ \mu (m-n+n) = 0,\end{array}
\sum_{i=1}^nY_{i,n}(u)^2-\frac{u^2\pi_{m_0}^2}{n}\sum_{i=1}^nZ_{i,n}^2=\pi_{m_0}^2 \sum_{i=1}^n(R_{i,n}(u)-R_{i,n}(0))^2 &+\frac{2u \pi_{m_0}^2}{\sqrt{n}}\sum_{i=1}^n(R_{i,n}(u)-R_{i,n}(0))Z_{i,n}
v_f^i (\sigma,t) = \sum_{n=1}^\infty W(n+\frac{\epsilon}{\tau})\Biggl[ \frac{c_n^i}{\sqrt{n}}\, \dot{x}_n^++ \frac{\tilde{c}_n^i}{\sqrt{n}}\, \dot{x}_n^- + h.c. \Biggr] \, .
\frac{\overline{F}_{\xi_{(\eta)}}(xy)}{\overline{F}_{\xi_{(\eta)}}(x)}=\frac{\sum_{n=1}^{K}\mathbb{P}\big(\bigcup_{k=1}^{n}\{\xi_k>xy\}\big)\mathbb{P}(\eta=n)}{\mathbb{P}(\xi_{(\eta)}>x)}&+\frac{\sum_{n=K+1}^{\infty}\mathbb{P}\big(\bigcup_{k=1}^{n}\{\xi_k>xy\}\big)\mathbb{P}(\eta=n)}{\mathbb{P}(\xi_{(\eta)}>x)}&=:\mathcal{J}_1+\mathcal{J}_2,
\begin{array}{rl}\overline{z}h(\overline{z}) = \phi(\overline{z}) = \rho_l(a)^{-1}\phi(\rho(a)\overline{z}) = \rho_l(a)^{-1}(\rho(a)\overline{z}h(\rho(a)\overline{z})) & = \overline{z}Q_a(\overline{z}) \rho_l(a)^{-1}(h (\rho(a)\overline{z})).\end{array}
\kappa_2 (\mathbf{C}^{(\ell)}\mathbf{A}) = \begin{cases}[1 - \lambda_{\max}(\mathbf{H})] / [1 - \lambda_{\min}(\mathbf{H})] \mbox{for}\ell \mbox{odd}, (1 - \delta^\ell) / [1 - \nu(\mathbf{H})^\ell] \mbox{for}\ell \mbox{even},\end{cases}
d s^{2}= \frac{X^{1 / 2}}{\rho}e^{2 A ( r )}d x_{/ /}^{2}- \frac{X^{1 / 2}}{\rho}\left( d r^{2}+ \frac{L^{2}}{\rho^{2}}\left[ d \theta^{2}+ \frac{\sin^{2}\theta}{X}d \phi^{2}+ \frac{\rho^{6}\cos^{2}\theta}{X}d \Omega_{3}^{2}\right] \right) ,
\tilde r_i(\nu) \equiv \int dt e^{-i\nu t}r_i(t) \,,\kern 0.35 in\tilde\rho_\theta^i(\nu) \equiv \int dt e^{-i\nu t}\rho_\theta^i(t) \,,\kern 0.35 in\tilde\rho_\phi^i(\nu) \equiv \int dt e^{-i\nu t - i\delta_i(t)}\rho_\phi^i(t) \,.
\begin{aligned}{\rm div}(\varphi_{\Lambda}) =\ell\,\{(a_{g}-2^{2g-1})\,{\mathcal D}_{\Lambda}^{-}+(b_{g}-2^{4})\,{\mathcal H}_{\Lambda}\}+\ell(c_{g}-2^{4}\cdot7)\,{\mathcal D}_{\Lambda}^{+}.\end{aligned}
G^{\prime}( \sigma_{i}^{a}, \sigma_{j}^{a}) = \sum_{n_{1}= - \infty}^{\infty}\sum_{n_{2}= 0}^{\infty}{\frac{4 \pi \alpha^{\prime}}{\omega_{n_{1}n_{2}}C_{n_{1}n_{2}}}}{\bar{\Psi}}_{- n_{1}n_{2}}( \sigma_{i}^{a}) \Psi_{n_{1}n_{2}}( \sigma_{j}^{a}) \quad .
{Q}^{( 2 )}\, = \frac{1}{8 \pi^{2}}\sum_{a = 1}^{3}\left( \dot{\phi}_{a}\beta_{a}+ \sum_{c y c l i c}^{i , j , k}\dot{\chi}_{a}M_{a i}^{T}( \phi_{j}- \phi_{k}) \, b_{i}\right) \, .
\begin{array}{c}{{U n k n o w n}}\\{{( 6 , 2 , 1 )}}\\ \end{array}\, \, \, d \hat{s}_{I I B}^{2}= H \left[ H^{- 1 / 2}\eta_{i j}d y^{i}d y^{j}- H^{1 / 2}d x^{2}\right] - H^{- 1 / 2}d y^{2}- H^{- 3 / 2}d z^{2}\, .
||D\Phi^{\nu}||_{s_0,r_0,s_{\nu},r_{\nu};D_{\nu}\times\mathcal{O}_{\nu}}\leq\prod_{\mu=1}^{\nu}||D\Phi_{\mu}||_{s_{\mu},r_{\mu},s_{\mu},r_{\mu};D_{\mu}\times\mathcal{O}_{\mu}}\leq\prod_{\mu=1}^{\infty}(1+\frac{\gamma_0}{2^{\mu+6}})\leq2,
\acute{\rho}_\sigma(\hat{s}) = \sum_{0\leq a < b \leq 3}\acute{\rho}_\sigma^{ab}(\hat{s})[\det \acute{J}_{ab}^\sigma](\hat{s}), |\acute{\rho}_\sigma|(\hat{s}) = \left| \sum_{0\leq a < b \leq 3}\acute{\rho}_\sigma^{ab}(\hat{s})[\det \acute{J}_{ab}^\sigma](\hat{s})\right|.
\tilde{\Phi}_5(\tilde{\gamma}_5,\mathcal{F}(\tilde{\sigma}_3,\tilde{\sigma}_2;\tilde{\Phi}_1(\tilde{\gamma}_1;V_a+\widetilde{\mathcal{V}}(V_a,Z_{a}h)Z_{a}h))) &=\tilde{\Phi}_5(\gamma_5,\mathcal{F}(\sigma_3,\sigma_2;\tilde{\Phi}_1(\gamma_1;V_a)))+\widetilde{\mathcal{V}}(V_b,Z_{b}h)Z_{b}h.
\lambda(O_1) = (2n-k \geq 2n- k- 1 \geq 2n-k -2 \geq ...\geq 2n-2k+ 2 \geq 0) \/; \lambda(O_2) = (2n-k +1 \geq 2n- k -1 \geq ... \geq 2n-2k+2 \geq -1) \/ .
F_{A_{/ 1}}= 0 \Longleftrightarrow q^{a_{i / 1}\left[ i \right]}= \phi^{a_{i / 1}}\left( q^{b_{i / 1}\left[ i - 1 \right]}, q^{b_{i / 2}\left[ i - 1 \right]}, q^{b_{i / 2}\left[ i \right]}\right) .
w'=\left({w'}_{1},\dots,{w'}_{N},\frac{{w'}_{N+1}}{G^{\left(N+1\right)}_{0;1,0, \dots,0,\dots, 0}(z)},{w'}_{N+2},\dots,{w'}_{N'}\right), &{z'}=\left({z'}_{1},\dots,{z'}_{N},\frac{{z'}_{N+1}}{\gamma_{N+1,1}},{z'}_{N+2},\dots,{z'}_{N'}\right) ,
g_{M N}= \left( \begin{array}{c c c}{{- 1}}&{{0}}&{{0}}\\{{0}}&{{a_{4}^{2}( t ){\tilde{g}}_{r s}}}&{{0}}\\{{0}}&{{0}}&{{a_{7}^{2}( t ){\tilde{g}}_{\Delta \Sigma}}}\\ \end{array}\right) .
\Vert \sigma(D_x) f \Vert_{B^s_{\infty,\infty}(G)}= \sup_{l\geq 0}2^{ls}\Vert \psi_l(\mathcal{R})\sigma(D_x)f \Vert_{L^\infty(G)}& \lesssim \sup\{C_{\alpha}:{|\alpha|\leq k}\}\sup_{l\geq 0}2^{ls}\Vert \psi_l(\mathcal{R})f \Vert_{L^\infty(G)}&\asymp \sup\{C_{\alpha}:{|\alpha|\leq k}\}\Vert f\Vert_{B^s_{\infty,\infty}(G)}.
\Bigl[4 J_n^+ J_n^0 - 4 J_n^+ J_n^- + 2(4a+1+3n) J_n^+ - 2(4a-4b_1-2l-d+3+2n) J_n^0 - 8b_1 J_n^- - 4a^2 + 8ab_1 + 2a(2l+d-1) - 2b_1 - n(4a-4b_1-2l-d+3+2n)\Bigr] \phi_n(z) = \epsilon_n \phi_n(z),
\sum_{s=0}^{T_1}(1-a)^{s}(T_1-s+1)^{-\frac{2\beta-1}{\beta}}=\sum_{m=1}^{T_{1}}\rho^{T_{1}-m+1}m^{-\frac{2\beta-1}{\beta}}+(T_{1}+1)^{-\frac{2\beta-1}{\beta}}&\leq (24\frac{2\beta-1}{\beta}+3(2^{\frac{\beta-1}{\beta}}))\frac{T_{1}^{-\frac{2\beta-1}{\beta}}}{a}+(T_{1}+1)^{-\frac{2\beta-1}{\beta}}&\leq\mathcal{A}_2T_{1}^{-\frac{2\beta-1}{\beta}}(1+\frac{1}{a}).
\overline{F(P)}^T\cdot h_P \cdot F(P)= Id\left(1+ O\left(t^{-\frac{2}{b}}\right)\right)\ \ \ \ \ \ \overline{F(\gamma(s))}^T\cdot h_{\gamma(s)}\cdot F(\gamma(s))= Id\left(1+ O\left(t^{-\frac{2}{b}}\right)\right).
\widetilde{\psi}_{n,m}^{(-)}(r) = (\omega r)\left(\frac{L_m^{\alpha+1}(-y)}{L_m^\alpha(-y)}- \frac{\partial_yL_n^{\alpha}(y)}{L_n^\alpha(y)}\right)_{y=\frac{1}{2}\omega r^2}\widetilde{\psi}_n(r).
d_1(\xi):= \frac{1}{1152}\left(846168+964186e^{i(\xi_1+\xi_2)}-581936e^{i\xi_1}+648973e^{i(\xi_1-\xi_2)}-65e^{2i\xi_2}-1876500e^{i\xi_2}\right. &\left.-1253380e^{-i\xi_2}+912657e^{i(\xi_2-\xi_1)}+576e^{-i\xi_1}+604428e^{-i(\xi_1+\xi_2)}+263955e^{-2i\xi_1}\right), d_2(\xi):= (1-d_1(\xi)/d_1(0))^2,d_3(\xi):=\frac{d_2(\xi)^2-1}{d_1(\xi)}.
f ( \vec{x}, \vec{p}, t ) = \int_{\Delta V}{\frac{d^{3}x^{\prime}d^{3}p^{\prime}}{( 2 \pi )^{3}}}W ( \vec{x}- \vec{x}^{\prime}, \vec{p}- \vec{p}^{\prime}, t ) \cdot g ( \vec{x}^{\prime}, \vec{p}^{\prime}) ,
\mathcal{E}_{\beta_\chi}^{\ell'+1,k'+1}\left(\tau,\begin{pmatrix}0 -1 a 0 \end{pmatrix}\right) = \frac{\pi^{k'+\ell'+1}}{\ell'! N^{k'+\ell'}\cdot \mathrm{Im}(\tau)^{k'+\ell'+1}}\lim_{s \to -\ell'}\Gamma(s+\ell-k) E_{\ell-k,N}(\tau,s,\overline{\chi}).
L_v\Omega_{2+1}=(i_vd+di_v)(\tilde{w}_{1/2}dV\wedge dt)=\left(\frac{\partial}{\partial t}\tilde{w}_{1/2}+\frac{\partial}{\partial x^i}(\dot{\tilde x}^i\tilde{w}_{1/2})+\frac{\partial}{\partial p_i}(\tilde{w}_{1/2}\dot{\tilde p}_i)\right)dV\wedge dt.
\tilde{L}_y=-\frac{v_y}{v^2}, \tilde{L}_{xx}=\frac{2v_x^2-vv_{xx}}{v^3}, &\tilde{N}_x=\frac{(2uu_x-2vv_x)v-(u^2-v^2)v_x}{v^2}=\frac{2uvu_x-v^2v_x-u^2v_x}{v^2},
\big( (1-Q')+ e^{(2t-1)\pi i}Q'\big) u P_0 u^* \big((1-Q')+ e^{(1-2t)\pi i}Q' \big) = \big((1-Q')+ e^{(2t-1)\pi i}Q' \big) v P_0 v^* \big((1-Q')+ e^{(1-2t)\pi i}Q' \big) = \big((1-Q')+ e^{-(2-2t)\pi i}Q' \big) P_0 \big((1-Q')+ e^{(2-2t)\pi i}Q' \big) = v_{2-2t}P_0 v_{2-2t}^* = g_2(2-2t).
r_2^{-((n+1)/2 + \kappa_0 + \frac{3\epsilon}{4})}\left\| w \right\|_{L^2(A^+_{r_2/2,r_2}(0))}\lesssim r_1^{-((n+1)/2 + \kappa_0 + \epsilon)}\left\| w\right\|_{L^2(A^+_{r_1,2r_1}(x))}& + r_3^{-((n+1)/2 + \kappa_0 + \epsilon)}\left\| w\right\|_{L^2(A_{r_3,2r_3}^+(x))}.
D(r)- \int_{\partial B_r}|y|^{1-2s}\mu(z)UU_\nu dH^n(z) &=\int_{\partial B_r}|y|^{1-2s}U\nabla U\cdot\kappa dH^n(z) &=-\frac{1}{2}\int_{\partial B_r}|y|^{1-2s}U^2div(\kappa) dH^n(z)-\frac{1}{2}\int_{\partial B_r}U^2\nabla |y|^{1-2s}\cdot\kappa dH^n(z).
A_{\sigma(UV)}A_{\sigma(UW)}^* =f(UV)\overline{f(UW)}A_VA_W^* ,A_{\sigma(UV)}\Psi_{\sigma(UW)}^*=f(UV)\overline{f(UW)}A_V\Psi_W^*, \Psi_{\sigma(UV)}\Psi_{\sigma(UW)}^* =f(UV)\overline{f(UW)}\Psi_V\Psi_W^*,~ \forall U,V,W \in \mathcal{U}.
Q = Q (k, l, m, n) = - \mu \frac{(1 - a) (1 - b) c}{(q - c) (1 - c)}\frac{x^{1 - \max\left\{m, 0 \right\}}(x)_{\min\left\{n, 0 \right\}}}{(a b q x / c)_{\max\left\{k + l - m + n, 0 \right\}- 1}}\widetilde{P}\c{k}{l}{m}{n}{a}{b}{c}{x},
\tilde{\nabla}_0(\varphi v_i dr\otimes dr)_{00}=\varphi'v_i,\tilde{\nabla}_k(\varphi v_i dr\otimes dr)_{00}=\varphi\nabla_k v_i, \tilde{\nabla}_k(\varphi v_i dr\otimes dr)_{l0}=\tilde{\nabla}_k(\varphi v_i dr\otimes dr)_{0l}=\varphi v_ig_{kl}f'f,
\frac{\partial g_{ij}}{\partial t}=-2R_{ij}+2h_{ik}h_{jl}g^{kl}, \frac{\partial h_{ij}}{\partial t}= \triangle h_{ij}-R_{im}h_{nj}g^{mn}-R_{jm}h_{ni}g^{mn}\ \ \ \ +2h_{im}h_{jn}h_{kl}g^{mk}g^{ln}-h_{mn}h_{kl}g^{mk}g^{nl}h_{ij}.
\partial_{\nu}\left\{\begin{array}{c}{{L \left( \begin{array}{c}{{G_{\mu}}}\\{{H_{\mu}}}\\ \end{array}\right)}}\\ \end{array}\right\}= \{\partial_{\nu}( L A_{\mu}^{a}) \}\left( \begin{array}{c}{{\frac{\partial n^{a}}{\partial \theta}}}\\{{\frac{1}{\sin \theta}\frac{\partial n^{a}}{\partial \varphi}}}\\ \end{array}\right) + D_{\nu \mu}
j \overline{{{\Psi}}}_{L}\Psi_{R}= j \overline{{{\psi}}}_{L}j \psi_{L}^{\prime}- j \overline{{{\psi}}}_{R}j \psi_{R}^{\prime}+ j ( \overline{{{\psi}}}_{L}\psi_{R}+ \overline{{{\psi}}}_{R}^{\prime}\psi_{L}^{\prime}) .
D_{t,q}R_{O,{\rm approx}}= \sum_j \chi_j^2 D_{t,q}\Big( \mathring R_q - \mathring R_{q,j}\Big) + 2 \sum_j \chi_j \chi_j' \Big( \mathring R_q - \mathring R_{q,j}\Big) = \sum_j \chi_j^2 \Big(D_{t,q}\mathring R_q\Big) + 2 \sum_j \chi_j \chi_j' \Big( \mathring R_q - \mathring R_{q,j}\Big).
d\|X-\widetilde X\|^2 = \left( -2\kappa \|X-\widetilde X\|^2 + d\,\nu^2(Y^{1/2}-\widetilde Y^{1/2})^2\right)dt + 2\nu (Y^{1/2}-\widetilde Y^{1/2})(X-\widetilde X)^\top dW.
\mathcal{L}=-\overline{\Psi}(x)\left[ \Gamma_\mu \left( \partial_\mu-ieA_\mu -gA_\mu^BI^B\right) +m\right] \Psi (x)-\frac 14\mathcal{F}_{\mu\nu}^2-\frac 14\left( F_{\mu \nu}^B\right)^2
\mathcal{K}_n\mathbf{u}^{\mathcal{R}}(x)=x\mathcal{J}_n\mathbf{u}^{\mathcal{R}}(x)+\sqrt{d_{n}}(x-i\omega)u_{n}^{\mathcal{R}}(x)\mathbf{e}_n, &\mathbf{u}^{\mathcal{L}}(x)\mathcal{K}_n=x\mathbf{u}^{\mathcal{L}}(x)\mathcal{J}_n+\sqrt{d_{n}}(x+i\omega)u_{n}^{\mathcal{L}}(x)\mathbf{e}_n^{T},
\tau =\lim_{P\to\infty}\lim_{F\to\infty}\frac{T\log P}{F}&\ge\frac{1}{l}\left\{(s_1+s_2)-(N_T-l)s_2\mu_T-\left(\frac{2s_2+s_1+1}{2}\cdot s_1+s_2^2\right)\mu_R\right. &\qquad\quad\left.+\left(\frac{2s_2+s_1}{2}(s_1-1)+s_2^2\right)(1-N_T\mu_T)^+\right\}.
\partial_t\big(x^2 w(t,x)\big) =\partial_x\,\Big(\frac{1}{2}\partial_x\big(x^2 w(t,x)\big)-\frac{1}{x}\big(x^2 w(t,x)\big)\!\Big)=\Big(\frac{1}{2}\partial_{xx}+\frac{1}{x}\partial_x\Big)^*\big(x^2 w(t,x)\big),