harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\overline u_{0n}\in C^2([\overline g_{0n}, \overline h_{0n}]),\; \overline u_{0n}(\overline g_{0n})=\overline u_{0n}(\overline h_{0n})=0,\; \overline u_{0n}(x)>0 \mbox{for}x\in (\overline g_{0n}, \overline h_{0n}),
[\partial x_\lambda y] =- \lambda[x_\lambda y], [x_\lambda \partial y]= (\partial+\lambda)[x_\lambda y] (conformal \sesquilinearity),{}[x_\lambda y] = -(-1)^{|x||y|}[y_{-\lambda- \partial}x](skewsymmetry),{}[x_\lambda [y_\mu z]] = [[x_\lambda y]_{\lambda+ \mu}z]+ (-1)^{|x||y|}[ y_\mu [x_\lambda z]](Jacobi\identity),
D_{x_{N}}^{m-n-1}u={\displaystyle\int\limits_{x_{N}}^{\infty}}d\xi{\displaystyle\int\limits_{\xi}^{\infty}}D_{x_{N}}^{m-n+1}u(x^{\prime},\eta,t)d\eta={\displaystyle\int\limits_{x_{N}}^{\infty}}d\xi{\displaystyle\int\limits_{\xi}^{\infty}}\frac{1}{\eta}a(x^{\prime},\eta,t)d\eta.
\delta T_{N}^{M}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{0}}&{{- \delta \rho}}&{{- ( \rho + p ) e^{\alpha_{0}}v_{, i}}}\\{{0}}&{{( \rho + p ) e^{- \alpha_{0}}v_{, i}}}&{{\delta p \: \delta_{i j}}}\\ \end{array}\right) \: \delta ( y ) ,
-\tau\langle u,(\partial_{n+1}^{4}\phi)u\rangle+(2s+1)(2s-1)\tau\langle u,x_{n+1}^{-3}(\partial_{n+1}\phi)u\rangle = 2\tau(1-2s)(1+2s)^{2}\|x_{n+1}^{-1-s}u\|^{2}-8 \tau \mathring{c}_{s}\|x_{n+1}^{-1}u\|^{2}.
\mathbb{E}\left[ 1_{\mathcal{E}_\delta}\left(\frac{1}{n}\sum_{i=1}^n(\hat{f}_{V})_1(X_1^i)\right)^2\right] \leq 2\mathbb{E}\left[1_{\mathcal{E}_\delta}\left(\frac{1}{n}\sum_{i=1}^n(\hat{f}_{V})_1(X_1^i)-f_1(X_1^i)\right)^2\right] +2\mathbb{E}\left[1_{\mathcal{E}_\delta}\left(\frac{1}{n}\sum_{i=1}^nf_1(X_1^i)\right)^2\right].
\Im E ( R = 2 \pi L k + c ) = - 1 6 \pi^{2}G_{\mathrm{N}}D^{2}L \int_{0}^{c}d r \int_{0}^{r}d \ell \ \ln{\frac{\Bigl | \sin{\frac{r + \ell}{2 L}}\Bigr |}{\sin{\frac{r - \ell}{2 L}}}}, \quad 0 \le c < 2 \pi L
\int_{\mathbb{R}^{3}}v(-\Delta)^{t}\phi_{u}^{t}dx=\int_{\mathbb{R}^{3}}(-\Delta)^{\frac{t}{2}}\phi_{u}^{t}(-\Delta)^{\frac{t}{2}}vdx=\int_{\mathbb{R}^{3}}u^{2}vdx,v\in D^{t,2}(\mathbb{R}^{3}).
\int\limits_0^\infty{{x^{s - 1}}\sum\limits_{n = 0}^\infty{\phi \left({2n + 1}\right)\frac{{{{\left({- 1}\right)}^{n}}}}{{\left({2n + 1}\right)!}}{x^{2n + 1}}dx = \phi \left({- s}\right)\Gamma \left( s \right)\sin \frac{{\pi s}}{2}}},
|c_{l}^{\pm}| \leq |f(t)| \begin{cases}(\xi/k)^{1/2}(1+(t-\frac{\xi}{k})^2)^{-3/4}c_{l}^{\pm}=c_{k\pm 1}^{\mp}, (\xi/k)^{-1/2}(1+(t-\frac{\xi}{k})^2)^{-1/4}. \end{cases}
\inf_{\textbf{v}\in \partial \mathcal{B}(x), t \ge 0}\!\left\| \textbf{v}\!-\! \tilde{\textbf{x}}^0(t) \right\|_2 \!=\! \inf_{\textbf{v}\in \partial \mathcal{B}(x)}\left\| \textbf{v}- \hat{\textbf{x}}_{}\right\|_2 > 0.
\frac{d^\ell}{dy^\ell}G(x,y)\bigg|_{y=1}= G(x,y)\bigg( A_{0,1}(x)^\ell + \sum_{0\le k_1\le \cdots\le k_{\ell-1}\atop k_1+\cdots+k_{\ell-1}\le \ell}d'_{k_1,\dots,k_{\ell-1}}\prod_{1\le i\le \ell-1}A_{0,k_i}(x)\bigg).
\frac{1}{\prod_{j=1}^{n}\tau_{j}^{2}\left(\zeta,\delta_{\Omega}(\zeta)\right)}\tau_{m}\left(\zeta,\delta_{\Omega}(\zeta)\right)\tau_{l}\left(\zeta,\delta_{\Omega}(\zeta)\right)\frac{1}{\left|z-\zeta\right|}.
\rho_1= \,\rho(p_\ell;f_\ell(\mathbf{a}_m),f_\ell(\mathbf{b}_m)), \rho_2= \,\rho(q_\ell^+;g_\ell^+(\delta^+_m\mathbf{a}_m),g_\ell^+(\delta^+_m\mathbf{b}_m)), \rho_3= \,\rho(q_\ell^-;g_\ell^-(\delta^-_m\mathbf{a}_m),g_\ell^-(\delta^-_m\mathbf{b}_m)),
\frac{r}{V}=n_{A}n_{B}\int_{0}^{\infty}\frac{S(E)}{E}e^{-\sqrt{\frac{E_{G}}{E}}}2\sqrt{\frac{E}{\pi(kT)^{3}}}e^{-\frac{E}{kT}}\sqrt{\frac{2E}{m_{R}}}dE
\sum_{N=1}^\infty |h_N||z|^N = \bigg(\sum_{N=1}^{r-1}+\sum_{N=r}^\infty\bigg)\frac{N+r}{N}|b_{N+r}||z|^N \ll \sum_{N=1}^{r-1}\frac{N+r}{r}|z|^N +\sum_{N=r}^\infty\frac{r}{N}\big(r^{1/r}|z|\big)^N \ll|z|
H:=\begin{pmatrix}\sqrt{2}x_1 y_{12}0 \cdots \cdots 0 y_{12}\sqrt{2}x_2 y_{23}\vdots 0 y_{23}\ddots \vdots \vdots \ddots \ \ddots y_{N-2N-1}0 \vdots y_{N-2N-1}\sqrt{2}x_{N-1}y_{N-1N}0 \cdots \cdots 0 y_{N-1N}\sqrt{2}x_N\end{pmatrix},
\frac{1}{\rho}\|\eta_{\rho}- u_k\| =\|\frac{(\xi_k(w_\rho)-1)}{\rho}u_k - \xi_k(w_\rho) f\| \rightarrow \|u_k \langle \xi^{'}_k(0),f \rangle -f \| \ \ \ \ \rho \rightarrow 0.
\mathcal{E}(q(t))+\int_0^t\int_0^1{q^2 (q_{xx}- \nu^{-\frac{1}{2}}f)^2 \,dx}+\nu^{-1/2}\int_0^t\int_0^1q_x\left[\int_0^xf_\tau\,ds\right]\,dxd\tau=\mathcal{E}(q_0),
J_{\mu}^{( 1 )}= \partial_{\mu}\phi +{\textstyle \frac 1 2}\varepsilon_{\mu \nu}[ \partial^{\nu}\phi , \phi ] -{\textstyle \frac 1 2}[ J_{\mu}, \phi ( - \infty ) ] = Z_{\mu}-{\textstyle \frac 1 2}[ J_{\mu}, \phi ( - \infty ) ] .
\langle M_{i j}M_{k l}\rangle = \frac{1}{N}[ \delta_{i l}\delta_{k j}+ \mathrm{e}^{- \frac{3}{2}\beta}L_{i l}( L^{2})_{k j}+ \mathrm{e}^{- \frac{3}{2}\beta}( L^{2})_{i l}L_{k j}] ,
M^{(2m-1)}_n := E[(S_n)^{2m-1}], f^{(2m-1)}_n :={\displaystyle \sum_{\ell =1}^{m-1}\left\{\binom{2m -1}{2\ell-1}+\frac{\alpha}{n}\binom{2 m-1}{2 \ell -2}\right\}M^{(2\ell-1)}_n}, g^{(2m-1)}_n := 1+\frac{(2m-1)\alpha}{n}.
\varphi(H(\{x_{n}\},\{x_{n+p}\})) \leq \varphi \left( \sum_{k=0}^{p-1}H(\{x_{n+k}\},\{x_{n+k+1}\}) \right) \leq \sum_{k=0}^{p-1}\varphi(H(\{x_{n+k}\},\{x_{n+k+1}\}) ) \leq \sum_{k=0}^{p-1}q^{n-n_0+k}f(x_{n_0}) \leq \frac{q^{n-n_0}}{1-q}f(x_{n_0}).
\frac{(\alpha_1-\lambda_i)}{(\alpha_1-\lambda_1)}\cdot \mathbf M_{ij}=\frac{(\lambda_{n+2}-\lambda_i)(\lambda_{n+3}-\lambda_i)}{(\lambda_{n+2}-\lambda_1)(\lambda_{n+3}-\lambda_1)}\cdot\mathbf M_{1j}.
|(\partial_x^{\,\delta'}|x|^N)|\leq a_N\,|x|^{N-|\delta'|}\leq b_N\,\sup_{\{\nu:|\nu|=N-|\delta'|\}}|x^\nu|\leq b_N\,\sup_{\{\nu:|\nu|=N\}}|(\partial_x^{\,\delta'}x^\nu)|\;.
-\sum\limits_{i = 1}^{n}g(X_{i}, Y_{i}) = -\sum\limits_{i=1}^{n}\frac{h'_i}{2}|f(X_i) - f^*(X_i)| + \sum\limits_{i=1}^{n}\frac{\xi^{(h')}_{i}}{2}|f(X_i) - f^*(X_i)| &\le -(\min\limits_{i}h'_{i})\sum\limits_{i = 1}^{n}\frac{1}{2}|f(X_i) - f^*(X_i)| + \sum\limits_{i = 1}^{n}\frac{\xi^{(h')}_{i}}{2}|f(X_i) - f^*(X_i)|.
(\tilde{l_p})_{\sigma_j\sigma_k}(0,0)=-\vert\nabla_x l_p(x_0,y_0)\vert\left\{\nu_{x_0}\cdot\frac{\partial^2 x}{\partial\sigma_j\partial\sigma_k}(0)+\hat{\nu}_{x_0}\cdot\frac{\partial^2 x}{\partial\varphi_j\partial\varphi_k}(0)\right\}.
E_{\mathrm{c a s}}[ \sigma ] = \sum_{\ell}N_{\ell}\int_{m}^{\infty}\frac{d t}{2 \pi}\frac{t}{\sqrt{t^{2}- m^{2}}}\left[ \beta_{\ell}( t ) \right]_{N}+ E_{\mathrm{F D}}^{N}+ E_{\mathrm{C T}},
p_t^{\theta^*\vee l}:= E_x\left[\left.\int_t^{\infty}c_{\theta}(u,X_u,\theta_u^*\vee l)e^{-ru}\,du\,\right\vert\,\mathcal{F}_t \right] &= E_x\left[\left.\int_{\sigma(t,l)}^{\infty}c_{\theta}(u,X_u,\theta_u^*\vee l)e^{-ru}\,du + \int_t^{\sigma(t,l)}c_{\theta}(u,X_u,\theta_t^*\vee l)e^{-ru}\,du\,\right\vert\,\mathcal{F}_t \right]
p^n= \big|Z(G)\big| \bigg|\frac{Z_3(G)}{Z(G)}\bigg| \bigg|\frac{Z_{n-r-1}(G)}{Z_3(G)}\bigg| \bigg|\frac{G}{Z_{n-r-1}(G)}\bigg| \ge p\p^{{{d(G)+1}\choose{2}}+1}\p^{n-r-4}\p^2,
\mathcal{E}(U)_{(s,t]}= e^{U_t - U_s - \sigma_U^2 (t-s)/2}\prod_{s< u \leq t}(1 + \Delta U_u) e^{-\Delta U_u}, \mathcal{E}(U)_{(s,t)}= e^{U_{t-}- U_s - \sigma_U^2 (t-s)/2}\prod_{s< u < t}(1 + \Delta U_u) e^{-\Delta U_u}.
u(t,x) =N\Big(1+\rho-\frac{2\rho x}{\ell}\Big) -\frac{1}{\ell}\Lambda(t)+ 4\mu N \sum_{n=1}^\infty \cos(\frac{n\pi x}{\ell}) \frac{((-1)^n-1)}{n^2\pi^2}e^{-2\kappa n^2\pi^2\, t}&-\frac{2}{\ell}\sum_{n=1}^\infty \cos(\frac{n\pi x}{\ell})\int_0^te^{-2\kappa n^2\pi^2(t-s)}\,\Lambda(ds)
+ \; \frac{1}{2}\frac{1}{(2\pi)^2}\Big( m c \Big)^2 \int_{\Lambda}d^2x \; d^2y\bigg[2e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi+g}}\big( \delta(x)-\delta(y), C_{m_d}\alpha \big)}
T_{n}(R)_{k^{\prime}k}\equiv\langle\chi_{k^{\prime}}|T_{n}|\chi_{k}\rangle_{(r)}=\delta_{k^{\prime}k}T_{n}+\sum_{A,\alpha}\frac{1}{M_{A}}\langle\chi_{k^{\prime}}|(P_{A\alpha}\chi_{k})\rangle_{(r)}P_{A\alpha}+\langle\chi_{k^{\prime}}|(T_{n}\chi_{k})\rangle_{(r)}.
F(0) = \left( 1+ \sum_{j=0}^{\infty}\left( 1 + \dfrac{V^{*}(\theta + j\omega_2)}{V^{*}(\lambda) B^{*}(\theta+j\omega_2)}\right)\prod_{m=0}^{j}\lambda B^{*}(\theta + m\omega_2) \right)^{-1}.
\|z\|_{L^{\infty}_x(\Omega_{T})}=\sup_{t\le T}\|z(\cdot,t)\|_{L^{\infty}_x}\le\left(\bigl\|z_0\bigr\|_{L^{\infty}_x}+ \int_0^T \bigl\|f(\cdot,s)\bigr\|_{L^{\infty}_x}ds\right) e^{LT},
d s^{2}= - ( 1 -{\frac{\lambda}{r^{d - 2}}}+{\frac{r^{2}}{l^{2}}}) d t^{2}+ ( 1 -{\frac{\lambda}{r^{d - 2}}}+{\frac{r^{2}}{l^{2}}})^{- 1}d r^{2}+ r^{2}d \Omega_{d - 1}^{2}\, ,
\Psi_a(x)=\varphi_a(x_a)\left(1-\sum_{\substack{|a_{k-1}|=k-1 a<a_{k-1}}}\varphi_{a_{k-1}}(x_{a_{k-1}})\right)\dots\left(1-\sum_{\substack{|a_{2}|=2{a<a_2}}}\varphi_{a_{2}}(x_{a_{2}})\right).
i \hbar \left\langle \frac{\partial A}{\partial t}\right\rangle_{\scriptstyle \! \! \Psi}= \left\langle [ A , H ]_{\mathrm{e x t r a}}\right\rangle_{\scriptstyle \! \Psi}= - \left\langle [ A , H ]_{\mathrm{r e g u l a r}}\right\rangle_{\scriptstyle \! \Psi},
\Upsilon_\eta(A) = \bigcup_{i=-1}^{k-1}\bigcup_{\sigma \in X(i) \setminus S_\eta^i(A)}\Gamma^{k+1}(I^\sigma( E(I_\sigma(S_\eta^{i+1}(A)),I_\sigma(S_\eta^{i+1}(A)))))
\aligned (\hat d_{2}^{i+1}\circ(\hat d_{2}^{i+1}\circ \frak H_{ab}^{i+1 i}-\frak H_{ab}^{i+1 i}\circ \hat d_{1}^{i}) &+(\hat d_{2}^{i+1}\circ \frak H_{ab}^{i i+1}-\frak H_{ab}^{i+1 i}\circ \hat d_{1}^{i})\circ \hat d_{1}^{i})_{\alpha'_2\alpha_1}= 0.\endaligned
\frac{\partial l}{\partial\sigma^{2}}=-\frac{n}{2\sigma^{2}}+\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{2\sigma^{4}}+\frac{2\mu}{\sigma^{4}}\sum_{i=1}^{n}\frac{x_{i}}{1+e^{\frac{2\mu x_{i}}{\sigma^{2}}}}
\begin{array}{c}{- a + \alpha = \frac \lambda{\lambda - \mu}2 \alpha_{+}- \frac \mu{\lambda - \mu}2 \alpha_{-},}\\{- a - \alpha = \frac \mu{\lambda - \mu}2 \alpha_{+}- \frac \lambda{\lambda - \mu}2 \alpha_{-}.}\\ \end{array}
x_{\sigma(\iota)}= w_{j_k,i_{\iota}}\bigl(1+\frac{\ell(i_{\iota})}{\mu_{i_{\iota}}}\bigr)\Biggl(\frac{y_{\sigma(\iota_{i^*_k})}}{w_{j_k,i^*_k}\bigl(1+\frac{\lambda_{\ell(i^*_k)}}{\mu_{i^*_k}}\bigr)}-\frac{y_{\sigma(\iota)}}{w_{j_k,i_{\iota}}\bigl(1+\frac{\lambda_{\ell(i_{\iota})}}{\mu_{i_{\iota}}}\bigr)}\Biggr);
\lim_{\varepsilon \to 0}w_0^\varepsilon (x) = \gamma - a \ \ \x \in \overline{\Omega (\delta )}, \lim_{\varepsilon \to 0}w_i^\varepsilon (x) = v_i \circ H(x) - a \ \ \x \in \overline{\Omega}_i \setminus \Omega_i (\delta_i/2 ),
\underbrace{0 = d_{j_1}= \cdots = d_{j_{u_1}}}_{=\sigma_0}< \underbrace{d_{j_{u_1 +1}}= \cdots =d_{j_{u_2}}}_{=\sigma_1}< \cdots <\underbrace{d_{j_{u_{s-1}+1}}= \cdots =d_{j_r}}_{=\sigma_{s-1}}<1 = \sigma_{s};
K^t_{j\bar k \alpha\bar\beta}(\zeta,\eta) =( ({\cdot_\beta\eta})_{\bar k}, ({\cdot_\alpha\zeta})_{\bar j}) +\int_{\partial D_t}\theta_{j\bar k}(\rho) \langle{\cdot_\beta\eta},{\cdot_\alpha\zeta}\rangle d \sigma + (\mathbb H b,\mathbb H b'),
\int\chi(x)e^{-ix\cdot \xi_0/h\Tilde{h}}(-ihX-z-i(\tilde{h}^{-1}Q_{\infty,\tilde{h}}+\tilde{h}^{-1}q_{1}+W))^{-1}(\psi(x)e^{ix\cdot r\eta_0/h\Tilde{h}})dx=\mathcal{O}(h^\infty\Tilde{h}^\infty).
M_{Y_{7}Y_{6}}^{2}= 2 \left( \begin{array}{c c}{H}&{- 2 i c}\\{2 i c}&{H}\\ \end{array}\right) . M_{Y_{5}Y_{4}}^{2}= 2 \left( \begin{array}{c c}{H}&{- 2 i c}\\{2 i c}&{H}\\ \end{array}\right) .
r_I^{-1}(\overline{V}) = \psi^{-1}(\overline{V}) \times_{s,\alpha}\alpha^{-1}(s(\psi^{-1}(\overline{V})))= (\psi^{-1}(\overline{V}) \times \alpha^{-1}(s(\psi^{-1}(\overline{V})))) \cap E^1_I
L_0 =\sum_{i=1}^{2\ell-1}\sum_{m= 1}^\infty \xi_{i,m}\Big(m\partial_{\xi_{i,m}}-(-1)^i(1-\delta_{i,1})\partial_{\xi_{i-1,m}}\Big) &+\sum_{i=\ell+2}^{2\ell-1}(-1)^{i+1}\xi_{i,0}\partial_{\xi_{i-1,0}}+(-1)^\ell \xi_{\ell+1,0}\xi_{\ell,0}+\frac{2\ell-1}{16}I.
G_{\mu\nu}^{ab}(p) = \frac{-i\delta^{ab}}{(2\pi)^{2\omega}(p^2+i\epsilon)}\Big [ g_{\mu\nu}- \frac{p_{\mu}n_{\nu}+p_{\nu}n_{\mu}}{p\cdot n}+\frac{n^2p_{\mu}p_{\nu}}{(p\cdot n)^2}\Big ], \ \n^2>0, epsilon > 0,
\langle \hat u_j-u_j, Eu_k\rangle =\sum_{l\geq 1}\langle \hat u_j-u_j,u_l\rangle\langle u_l, Eu_k\rangle &=\sqrt{\lambda_j\lambda_k}\langle \hat u_j-u_j,u_j\rangle\bar\eta_{jk}+\sum_{l\neq j}\sqrt{\lambda_k\lambda_l}\langle \hat u_j,u_l\rangle\bar\eta_{kl}
|\mathfrak m_N^{B^{q}}(\xi)-1| \le C_q\kappa_q(d, N)|\xi|, |\mathfrak m_N^{B^{q}}(\xi)| \le C_q(\kappa_q(d, N)|\xi|)^{-1}, |\mathfrak m_{N+1}^{B^{q}}(\xi)-\mathfrak m_{N}^{B^{q}}(\xi)| \le C_qN^{-1},
-F^{s-1}df_{1}^{*}a=\frac{t_{2}^{-1}f_{1}^{*}\alpha_{1}d\log t_{1}+(t_{2}^{-1}f_{1}^{*}\alpha_{1}+f_{1}^{*}\beta_{2})d\log t_{2}+t_{1}t_{2}f_{1}^{*}\gamma}{t_{1}^{n_{1}}t_{2}^{n_{1}+n_{2}-1}}
\Gamma (s_1) \Gamma (s_2) \zeta_2 (s_1,s_2 \,; a) = &\int_0^\infty \frac{y^{s_2-1}}{e^y-1}\int_0^\infty \!\!\! H(a,x+y)x^{s_1-1}dx dy +\Gamma (s_1) \Gamma (1-s_1) \int_0^\infty \!\!\! H(1,y)y^{s_1+s_2-2}dy,
\frac{\partial P_n}{\partial x}(-\bar\beta,\bar\alpha)=\left((n+\lambda)(\bar\beta y+\bar \alpha x)\beta-(n-\lambda)(\alpha y-\beta x)\bar\alpha\right)P_{n-1}(-\bar\beta,\bar\alpha)=\left((n+\lambda)y-2n(\alpha y-\beta x)\bar\alpha\right)P_{n-1}(-\bar\beta,\bar\alpha),
\begin{aligned}|\nabla F(f)(t)|_p \leq t^{\frac{1}{p}-\frac{3}{2}+\gamma}c\mathcal{B}^2\eta_t \max\left\{1,\frac{4}{\theta_i^2},\i=1,2,...,N\right\}B\left(\frac{2}{p}-\frac{3}{2}+\gamma,1-\frac{1}{p}\right)\|f\|^2.\end{aligned}
\langle f, \rho(x, \omega, \tau) g \rangle = e^{-2 \pi i \tau}\langle f, \rho(x,\omega) g \rangle = e^{-2 \pi i \tau}A(f,g)(x,\omega) = e^{-2 \pi i \tau}e^{\pi i x \cdot \omega}V_g f(x, \omega).
\langle{\cal B}({\{x \}}) \rangle_{0}^{\theta}: = e^{i \theta \frac{Q_{5}({\scriptstyle{\cal B}})}{2 N}}\lim_{\tau \rightarrow \infty}\langle{\cal U}_{\tau}({\scriptstyle{\cal B}}){\cal B}({\scriptstyle \{x \}}) \rangle_{0}.
\tilde{\mathcal{W}}\left(Z',Z''\right) = \left( \frac{\left(I_{q}-A{Z'''}\right){Z'}^{t}}{\sqrt{I_{q}+A}\left(I_{q}+A^{2}-A{Z'''}-A{Z''}^{t}\right)},\frac{\left(A-Z'''\right){Z'}^{t}}{\sqrt{I_{q}+A}\left(I_{q}+A^{2}-A{Z'''}-A{Z''}^{t}\right)},\frac{A{Z''}^{t}+A{Z'''}-\left(I_{q}+A^{2}\right){Z'''}{Z''}^{t}}{I_{q}+A^{2}-A{Z'''}-A{Z''}^{t}}\right).
\left( \begin{array}{c}{x^{1}}\\{x^{0}}\\ \end{array}\right) \simeq \left( \begin{array}{c}{x^{1}}\\{x^{0}}\\ \end{array}\right) + \left( \begin{array}{c}{2 \pi R_{s}}\\{0}\\ \end{array}\right) , x^{i}\simeq x^{i}, p^{1}= \frac{N}{R_{s}}.
P\left( a_{\hbar},a_{\hbar}^{\dag}\right) =\sum_{k=0}^{d}\sum_{\mathbf{b}=\left( b_{1},\dots,b_{k}\right) \in\Lambda_{1}^{k}}\hbar^{k/2}c_{k}\left(\mathbf{b}\right) u_{\mathbf{b}}\left( a,a^{\dag}\right)
t :=\frac{2^*}{2_{N/r}^*-1}\left(1-\frac{q}{r}\right)<q &\iff \frac1{q}-\frac1{r}<\frac{2_{N/r}^*-1}{2^*}=1-\frac1{r}-\frac12+\frac1N &\iff \frac1{q}<\frac12+\frac1N\iff q>\frac{2N}{N+2}\ \checkmark
\bar{f}(r,t,x)dr+ \mathbb{P}(\tau_x =\infty |\mathcal{G}_t)\delta_\infty (dr) \mbox{and}\mathbb{P}(\tau_x =\infty |\mathcal{G}_t)={\bf 1}_{\tau_x>t}\mathbb{E}(G(\infty, x-X_t)|\mathcal{G}_t ),
q_{\infty}\left(\mathcal F_1^{c'},\mathcal F_2^{c'},\mathbf T^{'}\right)-q_{\infty}^{*}=\left(a_{n_1}-a_{n_3}\right)\left(f_{2,K_2^c,\infty}(T_{n_3}^{*})-f_{1,K_1^c+F_1^{b*},\infty}\right)= 0,
\prod_{i=1}^{\ell(t)-1}\left(\frac{z_{i}}{1+z_{i}}\right)^{x_{\tau'(i)}}\prod_{i=\ell(t)+1}^{k}\left(\frac{z_{i}}{1+z_{i}}\right)^{x_{\tau'(i-1)}}=\prod_{i=1}^{k-1}\left(\frac{w_{i}}{1+w_{i}}\right)^{x_{\tilde{\tau}(i)}}.
\mathbb{V}\left( \mathbb{G}^{(N)}(f)\right) -\mathbb{V}\left(\mathbb{G}(f)\right) =\sum_{k=1}^{N}\Phi_{k}^{(N)}(f)^{t}\cdot\mathbb{V}\left( \mathbb{G}[\mathcal{A}^{(k)}]\right) \cdot\Psi_{k}^{(N)}(f),
S_2 \leq \sum_{i= 1}^I \sum_{k=1}^{i-1}\frac{x_k}{x_i}\beta_{\alpha_{i,k}-1,k}\Delta x_k (x_{\alpha_{i,k}-1/2}- (x_{i+1/2}-x_k))\, |n_k n_{\alpha_{i,k}-1}-\hat{n}_k \hat{n}_{\alpha_{i,k}-1}|.
\sum_{j = 0}^k{\frac{{G_j^2}}{{2^j}}}= \frac{4}{3}G_0^2 - \frac{2}{3}G_2^2 - \frac{1}{3}\frac{{G_{k + 1}^2}}{{2^{k - 1}}}+ \frac{1}{3}\frac{{G_{k + 3}^2}}{{2^k}}
S^{(n+3)}(0)=-K^{(n+3)}(0)-5(n+3)K^{(n+2)}(0)-15(n+3)(n+2)K^{(n+1)}(0) -35(n+3)(n+2)(n+1)K^{(n)}(0)-(n+3)!\dbinom{n+7}{n+3}K(0)= -(n+3)!(n+2)(n-1)\left( n^{2}+13n+6\right) +15(n+3)(n+3)!n\left(n^{2}+9n+2\right) -45(n+3)(n+2)(n+3)!n\left( n+5\right) +35(n+3)(n+2)(n+1)(n+3)!(n+2) +12\dbinom{n+7}{n+3}(n-1)
\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}g^{j,t}\|_{L^2(B^g(x^t_j,2 \sqrt{t}), \mu_2)}\leq C \left( \|g_0^{j,t}\|_{L^2(C^{j,t}_0, \mu_2)}+\sum_{k\geq 1}e^{-c 2^k}\|g_k^{j,t}\|_{L^2(C^{j,t}_k, \mu_2)}\right ).
\tau (r_q+\ell') < 5\tau\rho_0/2\leqslant \frac{10~C_0\ell'\rho_0^2}{((r_q-\rho_0)/2)^{5/4}}\leqslant \frac{10~ C_0\ell' \rho_0^2}{((r_q-\rho_0)/2)^{5/4}}=\frac{10~C_0 \rho_0^2}{(r_0/2)^{5/4}}\delta\leqslant \frac{\delta}{4},
= \frac{\beta}{2 \pi}( k^{\mu}k^{\nu}- k^{2}g^{\mu \nu}) \left[ \frac{1}{\sqrt{k^{2}}}\left( 1 + \frac{4 m^{2}}{k^{2}}\right) \log \left( \frac{1 - \sqrt{\frac{4 m^{2}}{k^{2}}}}{1 + \sqrt{\frac{4 m^{2}}{k^{2}}}}\right) + \frac{4 m}{k^{2}}\right] \, \, ,
\mathcal{A}\left( C,u\right) :=\arg \min \sum\limits_{i=1}^{m}\left(u_{i}\right)^{p}d_{S_{i}}^{p}\left( x\right) +I_{C}\left( x\right) ,\bigskip \mathcal{A}\left( l\right) :=\arg \min \sum\limits_{i=1}^{m}\left(l_{i}\right)^{p}d_{S_{i}}^{p}\left( x\right) .
\|(I-T)a\|^2 = \|\sum_{i=1}^{\infty}a_i(e_i-x_i)\|^2 &\le \sum_{i=1}^{\infty}|a_i|\|e_i-x_i\| &\le \left ( \sum_{i=1}^{\infty}|a_i|^2\right ) \left (\sum_{i=1}^{\infty}\|x_i-e_i\|^2 \right ) &\le (1-\epsilon)\|a\|^2.
S_{J,n}= \sum_{j=1}^J U_{j,n}; (\tilde{w}_{0,n}^J, \tilde{w}_{1,n}^J) = (w_{0,n}^J, w_{1,n}^J) + \sum_{j=1}^J \left(V_{j,n}(\cdot, 0) - U_{j,n}(\cdot, 0), \partial_t V_{j,n}(\cdot, 0) - \partial_t U_{j,n}(\cdot, 0)\right).
G_{\omega}^{\theta}( t - t^{\prime}) ={\frac{e^{- i \theta / 2}}{4 \omega}}\biggl [{\frac{e^{i \omega ( t - t^{\prime}- \tau / 2 )}}{\sin ({\frac{\omega \tau + \theta}{2}})}}+{\frac{e^{- i \omega ( t - t^{\prime}- \tau / 2 )}}{\sin ({\frac{\omega \tau - \theta}{2}})}}\biggr ] \; \; ; \; \; t - t^{\prime}\, e p s i l o n \; [ 0 , \tau )
\int\limits_M^{M+\Delta}J_{\kappa-1}\!\left(\frac{4\pi\sqrt{nx}}k\right)w(x)\,\mathrm dx &\quad=\frac{k^2}{4\pi^2 n}\int\limits_M^{M+\Delta}J_{\kappa+1}\!\left(\frac{4\pi\sqrt{nx}}k\right)\left(x\,w''(x)+\left(1-\kappa\right)w'(x)+\frac{\kappa^2-1}{4x}\cdot w(x)\right)\mathrm dx &\quad\ll k^{5/2}\,n^{-5/4}\,M^{3/4}\,U^{-1}.
(S_{I_{k;q}})_j:=ad^{-1}\,G_{I_{k;q}}\,((V^{(k,q-1)}_{I_{k;q}})^{od}_j)=\frac{1}{G_{I_{k;q}}-E_{I_{k;q}}}P^{(+)}_{I_{k;q}}\,(V^{(k,q-1)}_{I_{k;q}})_j\,P^{(-)}_{I_{k;q}}-h.c.\,.
\begin{aligned}Hom_{S_n}(\Pi \otimes \mathcal{C}_c^\infty(F^m),|.|^{-s/2}) \simeq Hom_{S_n}(\Pi \otimes ind_{P_n\cap S_n}^{S_n}(\Theta^{-1}),|.|^{-s/2}) &\simeq Hom_{S_n}(\Pi, Ind_{S_n\cap P_n}^{S_n}(|.|^{-s/2+1/2}\Theta)) &\simeq Hom_{S_n\cap P_n}(\Pi, |.|^{-(s-1/2)}\Theta)),\end{aligned}
\displaystyle \mbox{$_4$F$_3$}\left( -n,a,b,c;c-1,e,a+b+2-n-e;1 \right) =A{\frac{\Gamma \left( a+b+2-n-e \right) \mbox{}\Gamma \left( n+e-a-1 \right) \Gamma \left( b+1-e \right) \Gamma \left( e \right)}{\Gamma \left( b+2-n-e \right) \Gamma \left( a+b+2-e \right) \Gamma \left( n+e \right) \mbox{}\Gamma \left( e-a \right)}}
C^n_\mathrm{RBA}((A, P), (M, Q)) = \begin{cases}M n \geq 0, C^n (A, M) \oplus C^{n-1}(A_P, \overline{M}) = \mathrm{Hom}(A^{\otimes n}, M) \oplus \mathrm{Hom}(A^{\otimes n-1}, M) n \geq 1.\end{cases}
d \star \widehat{\mathcal{H}}_{+}- \mathrm{i}\, Q \wedge \, * \widehat{\mathcal{H}}_{+}= \mathrm{i}\, \mathcal{F}_{[ 5 ]}\, \wedge \, \widehat{\mathcal{H}}_{+}- P \wedge \star \widehat{\mathcal{H}}_{-}
\frac{Z(W,s,\mu_p)}{L(s,\mu_p\tilde{\pi}_{p})}\epsilon(s,\mu_p\tilde{\pi}_{p})=\frac{Z(\tilde{\pi}_p(w)W,1-s,\mu_p^{-1}\omega_{\tilde{\pi},p}^{-1})}{L(1-s,\mu_p^{-1}\pi_{p})}.
{\mbox{}^*\mathcal{L}}_{eff}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \beta^2\sqrt{1+\frac{1}{6 m^2\beta^2}H_{\mu \nu \rho}H^{\mu \nu \rho}}-\frac{\chi}{2\theta}\varepsilon^{\mu \nu \rho \lambda}B_{\nu \rho}\partial_\mu A_\lambda\; .
r(p^{\alpha}) = \Big( f_1(p^{\alpha})f_2(p^{\alpha}) - \frac{f_1(p^{\alpha+1})f_2(p^{\alpha+1})}{p^{\tau_2}}\Big) \Big ( 1 - \frac{f_1(p) f_2(p)}{p^{\tau_2}}\Big )^{-1}.
\int b(x)|u|^{\tilde{\delta}(x)}|u-v| = \int b(x)|u|^{\tilde{r}(x) - 1}|u-v| \leq \int b^{\frac{\tilde{r}(x)}{\tilde{r}(x)-1}}|u|^{\tilde{r}(x)}+ \int |u-v|^{\tilde{r}(x)}, &\leq \int \epsilon_4^{-\frac{r(x)}{r(x)-\tilde{r}(x)}}+ \int \epsilon_4 b(x)^{\frac{\tilde{r}(x)}{\tilde{r}(x)-1}\frac{r(x)}{\tilde{r}(x)}}|u|^{r(x)}+\int |u-v|^{\tilde{r}(x)}.
F(z_{1},z_{2},...,z_{m})=\sum_{n_{1}=-\infty}^{\infty}\cdot\cdot\cdot\sum_{n_{m}=-\infty}^{\infty}f(n_{1},n_{2},...,n_{m})z_{1}^{-n_{1}}z_{2}^{-n_{2}}...z_{m}^{-n_{m}}
( - g ) \,{\epsilon}^{{\delta}_{( 1 )}b a}\langle{A_{{\sf G I}\, i}^{{\delta}_{( 1 )}}({\bf{r}}) \rangle \partial_{i}\int \frac{d{\bf x}_{1}}{4{\pi}|{\bf r}-{\bf x}_{1}|}\, j_{0}^{a}({{\bf x}_{1}}}) = i g \,{\epsilon}^{{\delta}_{( 1 )}b a}{\phi}( r ) L^{{\delta}_{( 1 )}}{\int \frac{d{\bf x}_{1}}{4{\pi}|{\bf r}-{\bf x}_{1}|}\, j_{0}^{a}({{\bf x}_{1}}}) ,
\begin{bmatrix}\xi+\eta -2\eta -2\tanh\xi -\xi 0 \tanh(\xi+\eta) -\tanh\eta \tanh(\xi+\eta) 0\end{bmatrix}\begin{bmatrix}A^h(\xi,\eta) B^h(\xi,\eta) C^h(\xi,\eta)\end{bmatrix}= \begin{bmatrix}0 i\xi\eta 0\end{bmatrix}.
A(w) :=\frac{r(d-1)(2R-rd+r)}{R}\frac{1}{c-rw}, B(w) :=\frac{-\lambda r^2 (d-1)^2}{R}\frac{1}{(c-rw)^2}, C(w) :=\frac{r^2 d^2}{(c-rb)^{2d}}(c-rw)^{2d-2}.
\Upsilon_{\omega}^{+}=\left\{\tau^{-1}-1,\tau\theta^{-1}\cdot\Omega\right\},\quad\Upsilon_{\omega}^{-}=\left\{\tau-1,\Omega\right\},\Omega\in\{-\vartheta\tau^{-1}-1\}\cup\mathbb{C}.
\begin{aligned}\phi''(z^*) +\left(\frac{1}{z^*-s}-\frac{1}{z^*-s+r(s)}\right)\phi'(z^*) &+\left(\frac{\frac{r'(s)^2}{4 r(s)}-\frac{r'(s)}{2 r(s)}-s r(s)+r^2(s)}{z^*-s}+\frac{\frac{r'(s)}{2 r(s)}}{z^*-s+r(s)}-z^*\right)\phi(z^*)=0.\end{aligned}
N = - \frac{\dot{U}V^{\prime}- \dot{V}U^{\prime}}{2 U^{\prime}V^{\prime}}\sqrt{- A U^{\prime}V^{\prime}}, \quad N^{\rho}= \frac{\dot{U}V^{\prime}+ \dot{V}U^{\prime}}{2 U^{\prime}V^{\prime}}\ .
M_x^2 =\frac{\omega}{3L_{xy}}\left(\Psi^0 + \frac{4}{(1-\theta)^2}(\delta_x+\delta_y)+\frac{\Sigma^2}{2(1-\theta)}\right), M_y^2 =\frac{1}{4L_{xy}\omega}\left(\Psi^0 + \frac{4}{(1-\theta)^2}(\delta_x+\delta_y)+\frac{\Sigma^2}{2(1-\theta)}\right),
0 \leq R_g- R_{g_0}= \frac{1}{2}\sum_{i}\frac{b_id_i}{\lambda_i}(1-\lambda_i) -\frac{1}{4}\sum_{i, j, k}A^k_{ij}\left(\frac{\lambda_{k}}{\lambda_i\lambda_{j}}-1\right) &= \sum_{i}\frac{c_id_i}{\lambda_i}(1-\lambda_i) -\frac{1}{4}\sum_{i, j, k}A^k_{ij}\left[ \frac{\lambda_{k}}{\lambda_i\lambda_{j}}+1 - \frac{2}{\lambda_i}\right].
\begin{array}{ll}9 \; y_{11}+ 18 \; y_{12}+ 9 \; y_{22}-30 \; y_{13}- 30 \; y_{23}+ 25 \; y_{33}= 18 y_{11}+ 2 \; y_{12}+ y_{22}+10 \; y_{13}+10 \; y_{23}+ 25 \; y_{33}= 2 4 \; y_{22}+20 \; y_{23}+ 25 \; y_{33}= 4\end{array}
\frac{(-1)^l}{w\cdots(w+l)}\frac{\chi_f^{(m-r)}}{\chi_f}(1-s-w)\ll\begin{cases}|v|^{-(l+1)}(\log|v|)^{m-r}, w\in L_{\pm1,\pm2}, |t|^{-\frac{l+1}{2}}(\log|t|)^{m-r}, w\in\mathcal{F}. \end{cases}
\min_{\mathcal{A}\epsilon-}\max_{m \in [\frac{1}{3}\binom{n}{2},\frac{2}{3}\binom{n}{2}]}\mathbb{E}_{G \sim G(n,m)}[ ( A(G) - \frac{m}{\binom{n}{2}})^2]=\Omega\left(\frac{1}{n^3\epsilon^2}\right).
\gamma_{1j}\sigma_1^{-1}\gamma_{1k}- \gamma_{1k}\sigma_1^{-1}\gamma_{1j}= \Phi A_1 \Phi^* \left( \sigma_j \sigma_1^{-1}\sigma_k - \sigma_k \sigma_1^{-1}\sigma_j \right) \Phi A_1^* \Phi^*.
{\L_{Y5}}= \sum_{n_1,\, n_2}\,\!\!\!^\prime\g H_5 \left[ i e^{- i \theta}{\bar{\phi}}^{(n_1,n_2)}_e{\bar{\chi}}^{(n_1,n_2)}_e +i e^{i \theta}{\bar{\phi}}^{(n_1,n_2)}_o{\bar{\chi}}^{(n_1,n_2)}_o\right]+ h.c.
\begin{array}{c}D_{TM}^{(-1)}(t) = \displaystyle{\frac{-\left({n_{0}}^2\,R^2\,t\, \left( 8\,a^2 +{n_{0}}^2\,R^2\,t^2 \right) \right)}{8\,{\left( a^2 +{n_{0}}^2\,R^2\,t^2 \right)}^2\,{\sqrt{1 + \frac{{n_{0}}^2\,R^2\,t^2}{a^2}}}}- \frac{\left( 2 +{n_{0}}^2\,t^2 \right) \,Z^2}{2\,t^3\,{\sqrt{1 +{n_{0}}^2\,t^2}}}},\end{array}
\begin{aligned}(iv)\u-h \geq (u_0-h_0) &-Cx\vartheta_0^{-4}\left(P(C(u_e,u_b,h_e)+\Vert (u_0,h_0)\Vert_{H^m_l})+Cx\right)^{1/2}&\cdot\left\{1-C\vartheta_0^{-24}\left(P(C(u_e,u_b,h_e)+\Vert (u_0,h_0)\Vert_{H^m_l})+Cx\right)^2x\right\}^{-1/4}.\end{aligned}
\frac{SU(1,1)}{U(1)}\times\frac{SO(2,\frac{n_A(n_A-1)}{2}+\frac{n_B(n_B-1)}{2})}{SO(2)\times SO(\frac{n_A(n_A-1)}{2}+\frac{n_B(n_B-1)}{2})}\to\frac{SU(1,1)}{U(1)}\times \frac{SO(2,n_1 n_2 + n_3 n_4)}{SO(2)\times SO(n_1 n_2 + n_3 n_4)}.
f(z_0) = \dfrac{1}{2\pi i}\int_{\partial D_r}\dfrac{f(z)}{z-z_0}\, dz - \frac{1}{\pi}\int_{D_r}\frac{f_{\overline{z}}(z)}{z-z_0}\, dx\, dy\, , \frac{1}{\pi}\int_{D_r}\frac{f(z)}{(z-z_0)^2}\, dx\, dy =\frac{1}{\pi}\int_{D_r}\frac{f_z(z)}{z-z_0}\, dx\, dy+\frac{1}{2\pi i}\int_{\partial D_r}\frac{f(z)}{z-z_0}\, d\overline{z}\, .
W(\rho,\psi) =R(\psi,c)\cdot\det(\rho)(c)q_F^{-\frac{1}{2}}\cdot\sum_{x\in A_E}\chi_E^{-1}(x)(c^{-1}\psi)(\rm{Tr}_{E/F}(x)) =R(\psi,c)\cdot\det(\rho)(c)q_F^{-\frac{1}{2}}\cdot q_F\psi(mc^{-1}) =R(\psi,c)\cdot\det(\rho)(c)q_F^{\frac{1}{2}}\psi(mc^{-1}).