harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
+\cos\psi\csc\eta\cdot\left(\frac{F[p,q](i\mu_{0}+1)w_{2}[p,q](i\mu_{0}+1)w_{3}[p,q](i\mu_{0}+1)}{w_{1}[p,q](i\mu_{0}+1)}\right)^{-\frac{1}{2}}\gamma^{1}\partial_{\phi}u_{n}(\mu)\mid_{\mu=\mu_{0}-i}
\delta \left( Z_{\beta_{2}}^{\alpha_{1}}\eta_{\alpha_{1}}^{*}+ \frac 1 2 C_{\beta_{2}}^{\alpha_{0}\beta_{0}}\Phi_{\alpha_{0}}^{*}\Phi_{\beta_{0}}^{*}\right) = D_{\beta_{2}}^{\alpha_{2}}\Phi_{\alpha_{2}}^{*},
\mathrm{d}p_{X}(t)(\mathbf{x}) = \mathrm{e}^{-\frac{|\mathbf{x}|^2}{2t}}\frac{\mathrm{d}^{2n}x}{(2\pi\,t)^n}\;,\mathbf{x}\in\mathbb{R}^{2n}\;,
\mathcal{Q}_w(z)=\left( \begin{array}{cc}\Lambda^{-1}E_{w,-+}(z) R_{w,-}(z) R_{w,+}(z) 0 \end{array}\right): L^2(\partial\mathcal{O};\mathbb{C}^N)\times\mathbb{C}^M \to L^2(\partial\mathcal{O};\mathbb{C}^N)\times\mathbb{C}^M,
\int_{\mathbb{S}^n}\ln \tilde{K}dg_0\leq C\int_{B_1}\ln \left(\frac{1}{|x|}\right)dx+C\int_{\mathbb{R}^n\backslash B_1}\frac{\ln (1+|x|)}{|x|^{2(n-1)}}dx\leq C.
\langle\|7M(1+7\frac{S}{M}(1+7\frac{B}{S}(1+7\frac{C}{B}(1+17\frac{F}{C}(1+7\frac{E}{F})))))\|_{\rightarrow A}(1+6\frac{B}{A}(1+7\frac{F}{B}))|F\rangle
Z = \int \prod\limits_{x,i,a}d\bar E_{x,i}^a \, d\mu(U_{x,i}) \, \exp\left[-{\bar H_{\rm el}+ \bar H_{\rm mag}\over T_{\rm G}}\right] \, \delta(D_{x,i}^{ab}\bar E_{x,i}^b) \, \delta(F(U_{x,i})) \,,
\sideset{}{^\star}\sum_{\chi\bmod{q}}\chi(n)(1+\chi(-1))\bar\chi(\ell) =\sum_\pm\sideset{}{^\star}\sum_{\chi\bmod{q}}\chi(n)\bar\chi(\pm\ell) =\sum_{\pm}\prod_{i=1}^2\left(\sideset{}{^\star}\sum_{\chi_i\bmod{q_i}}\chi_i(n)\bar\chi_i(\pm\ell)\right) =\sum_{\pm}\prod_{i=1}^2\left(\phi(q_i)1_{n\equiv \pm\ell\bmod{q_i}}-1\right).
\int_{\sum_{i=1}^{d}i^{-2\alpha-1}x_{i}^2 \leq d^{-2\alpha}}\frac{\exp\{-\sum_{i=1}^{d}x_{i}^2 /2 \}}{(2\pi)^{d/2}}\mathrm{d}x &\geq [ \{\Gamma(d+1)\}^{\alpha+1/2}/ \{d^{d\alpha}\Gamma(d/2+1) \}]\exp(-\tilde{c}_{1}d).
\partial_{\nu}G_{a}^{\nu \mu}+ g \varepsilon^{a b c}G_{b}^{\mu \nu}A_{\nu}^{c}+ \frac{m}{2}\varepsilon^{\mu \alpha \beta}G_{\alpha \beta}^{a}+ g^{\mu 0}\Pi_{0 0}A_{0}^{a}= 0 .
\rho_{1}^{*}\psi_{1}^{*}(i)=\inf_{v_1\in V_1(i)}\biggl[\sum_{j\in S}\psi_{1}^{*}(j)\pi_{ij}(v_1,\tilde{v}_{2}(i))+c_1(i,v_1,\tilde{v}_{2}(i))\psi_{1}^{*}(i)\biggr].
H_3=\sum_{n=M_+}^\infty E_M(t,n)|C_n| \leq c\sum_{n=M_+}^\infty p_M(t,\hat x_n)\,n^{d-1}= c \sum_{n=M_+}^\infty\frac{n^{d-1}}{M^d}P_{\hat x_n}(Y_t\in B_M),
\sup_{\lambda/n\leq s\leq1-\lambda/n}\frac{n^{\eta}\left\vert \beta_{n}\left(s\right) -B_{n}\left( s\right) \right\vert}{\left[ s\left( 1-s\right)\right]^{1/2-\eta}}=O_{\mathbb{P}}\left( 1\right) ,
\langle P M_l T_k g, g \rangle = \langle X T_l M_{-k}\widehat{g}, \widehat{g}\rangle & = l \langle T_l M_{-k}\widehat{g}, \widehat{g}\rangle + \langle T_l M_{-k}X \widehat{g}, \widehat{g}\rangle & =\langle M_l T_k P g, g \rangle.
V_r(z)=\binom{\phi_r(x_1)}{\phi_r(x_2)}\;,\phi_r(x)=\frac1{\sqrt{2}}\frac{\eta(x+r/\sqrt{2})-\eta(x-r/\sqrt{2})}{\eta(x+r/\sqrt{2})+\eta(x-r/\sqrt{2})}\;.
\frac{\partial^n}{\partial \hat{t}^n}= \left(\frac{\partial \hat{z}_i}{\partial \hat{t}}\right)^n \frac{\partial^n}{\partial \hat{z}_i^n}+ \sum_{j=1}^{n-1}O_{d_0}(\rho^{-1}) \frac{\partial^j}{\partial \hat{z}_i^j}.
\| [\langle x \rangle^a, D^{-\frac12}\nabla] f \|_{L^q(\mathbb R^n)}\le\| [\langle x \rangle^a, R] D^\frac12 f \|_{L^q(\mathbb R^n)}+\| R [\langle x \rangle^a, D^\frac12] f \|_{L^q(\mathbb R^n)}.
h_D({\bf r}_1, \cdots,{\bf r}_5) = \frac{1}{5!}\sum_{\pi \in \mathfrak{S}_5}\frac{\phi \left(r^{(1)}_{\pi_1}, \dots, r^{(1)}_{\pi_5}\right) \phi\left(r^{(2)}_{\pi_1}, \dots, r^{(2)}_{\pi_5}\right)}{4},
\max_{p(x): |X|^2 \le \mathcal{E}_j}I( X; Y) \ge \frac{1}{2}\log \left(1 + \frac{\mathcal{E}_j}{3}\right) - \frac{1}{2}\log\left(\frac{\pi e}{6}\right) \ge \frac{1}{2}\log(1 + \mathcal{E}_j) - \left(\frac{1}{2}\log(3) + \frac{1}{2}\log\left(\frac{\pi e}{6}\right) \right) = \frac{1}{2}\log(1 + \mathcal{E}_j) - 1.04
\mathbf{C}_{\mathbf{s}_\mathcal{Q}}=\frac{2}{\pi}\left(^{-1}\left(\mathbf{K}\mathfrak{R}\{\mathbf{C}_{\mathbf{s}}\}\mathbf{K}\right)+j^{-1}\left(\mathbf{K}\mathfrak{I}\{\mathbf{C}_{\mathbf{s}}\}\mathbf{K}\right)\right).
\dot{Z}^{2}= - \overline{{{h}}}+ \left[ \frac{4 \pi G_{n}}{n - 2}( \sigma + \rho ) Z \right]^{2}+ \left[ \frac{( n - 2 ) \Delta h}{1 6 \pi G_{n}( \sigma + \rho ) Z}\right]^{2}
\int_{A_\delta}\frac{1}{( |y_1 - \frac1{\sqrt2}e_1|^2 + |y_2|^2 + |z-\frac1{\sqrt2}e_1|^2)^{\frac{N+s}{2}}}& =\int_{\tilde A_\delta}\frac{1}{( |y_1 - \frac1{\sqrt2}e_1|^2 + |y_2|^2 + |z-\frac1{\sqrt2}e_1|^2)^{\frac{N+s}{2}}}
\left|\frac{\Psi'(z(t))}{\Psi(z(t))}\right| =\left|\frac{1}{R(t)t}+\beta\int_{\mathbb{T}}\frac{2\xi}{(\xi-R(t)t)^{2}}d\mu_{1}(\overline{\xi})\right| =\frac{1}{R(t)}\left|1+\beta\int_{\mathbb{T}}\frac{2\xi R(t)t}{(\xi-R(t)t)^{2}}d\mu_{1}(\overline{\xi})\right| \ge\frac{1}{R(t)}\left[1-\frac{2R(t)\log R(t)}{R(t)^{2}-1}\right].
\Delta m^{2}\sim - \frac{\lambda}{2 \pi L}\, \int_{0}^{\infty}d x \, \ln \Bigl ( 1 - e^{- 2 \pi L \sqrt{x^{2}+ m^{2}/ 4 \pi^{2}}}\Bigr ) = \frac{\lambda}{4 \pi^{2}}m^{2}\int_{1}^{\infty}{\frac{( t^{2}- 1 )^{{\frac{1}{2}}}d t}{\mathrm{e}^{m L t}- 1}}\, .
f_1(t)f_2(t)=\frac{\sqrt{\pi}}{\sqrt{32}N^{3/2}\Sigma^4tt_c}I_0\left(\Sigma u_p\sqrt{1/t_c-1/t}\right) \frac{e^{-N\left(\frac{t}{t_c}-1-\log\frac{t}{t_c}\right)}}{t/t_c-1}
\begin{matrix}{\rm Chow}(\kappa) (L_f) \,\,= \,\,(\Delta_{\{2^{k-2},4\}})^\vee \hbox{and}{\rm Chow}(\nu) (L_f) \,\,= \,\, (\Delta_{\{2^{k-3},3^2\}})^\vee.\end{matrix}
\left.\begin{array}{ll}\lambda_{n+1}\left(a_{n+1}p_n+p_{n-1}\right) = a_{n+1}\hat{q}_{n-1}+\hat{q}_{n-2}\lambda_{n+1}\left( a_{n+1}q_n+q_{n-1}\right) =a_1\left(a_{n+1}\hat{q}_{n-1}+\hat{q}_{n-2}\right)+\left(a_{n+1}\hat{p}_{n-1}+\hat{p}_{n-2}\right)\end{array}\right.
V_{\mathrm{Z}}( r ) = \left( 1 -{\frac{2 M}{r}}\right) \left\{\frac{2 \lambda^{2}( \lambda + 1 ) r^{3}+ 6 \lambda^{2}M r^{2}+ 1 8 \lambda M^{2}r + 1 8 M^{3}}{r^{3}( \lambda r + 3 M )^{2}}\right\}
E\left[S^{tot,m}_{s,t}\right]-E\left[S^{tot,0}_{s,t}\right]=E\left[S^{env,m}_{s,t}\right]-E\left[S^{env,0}_{s,t}\right]-\frac{n}{2}E\left[ \ln(\beta(t,q_t)/\beta(s,q_s))\right].
\Sigma_{{\rm BFYM}}=\int d^nx\,\,\left\{\varepsilon^{\mu_1\ldots \mu_n}B_{\mu_1\ldots \mu_{n-2}}^aF_{\mu_{n-1}\mu_n}^a+\left( B_{\mu_1\ldots \mu_{n-2}}^a+D_{[\mu_1}\eta_{\mu_2\ldots \mu_{n-2}]}^a\right)^2\right\}\,\,,
\int_{- \infty}^{+ \infty}d x \, \, \Psi_{k}( x ) \Psi_{k \,^{\prime}}( x ) = \delta ( k - k \,^{\prime}) , ~ ~ \int_{0}^{+ \infty}d k \, \, \Psi_{k}( x ) \Psi_{k}( x^{\prime}) = \delta ( x - x \,^{\prime})
\{s_{m}^{a}, s_{n}^{b}{}^{\dagger}\}= \delta^{a , b}\delta_{m , n}, \{\tilde{s}_{m}^{a}, \tilde{s}_{n}^{b}{}^{\dagger}\}= \delta^{a , b}\delta_{m , n}, \{s_{0}^{a}, s_{0}^{b}{}^{\dagger}\}= \delta^{a , b}
A_{i}= \begin{pmatrix}0 \cdots 0 1 1 \ddots \ddots 0 \vdots \ddots \ddots \vdots 0 \cdots 1 0 \end{pmatrix}_{p^{a}\times p^{a}},~ C_{i}= \begin{pmatrix}\lambda^{j_i}\lambda^{rj_i}\ddots \lambda^{r^{p^a-1}j_i}\end{pmatrix}_{p^{a}\times p^{a}}.
{\cal V}( x , y , z ) = \int d^{2 \omega}\zeta{\frac{\partial}{\partial z^{\alpha}}}\left[{\frac{\partial}{\partial x^{\alpha}}}{\frac{\partial}{\partial y^{+}}}-{\frac{\partial}{\partial y^{\alpha}}}{\frac{\partial}{\partial x^{+}}}\right] F ( x - \zeta ) F ( y - \zeta ) G ( z - \zeta )
\int_q^z \frac{S(t)}{t^2}\, dt = \int_q^z \bigg( \frac{\log\log t}{\phi(q)\, t \log t}+ O\bigg(\frac{1}{t \log t}\bigg) \bigg) \, dt &= \frac{(\log\log t)^2}{2 \phi(q)}\bigg|_q^z + O\big(\log\log t \big|_q^z\big) = \frac{(\log\log z)^2}{2 \phi(q)}+ O(\log\log z).
\beta \circ \alpha (a) \approx_\varepsilon \sum_{k=1}^m \beta ( \Omega_{\xi^{(k)}}(a) )=\sum_{k=1}^m \beta \left( \left[ \l< \xi_i^{(k)}, \pi_H (a) \xi_j^{(k)}>\right]_{i,j=1}^n \right) &=\sum_{k=1}^m \sum_{l=1}^n \sum_{i,j=1}^n \l< \xi_i^{(k)}, \pi_H (a) \xi_j^{(k)}> x_{li}^* x_{lj}=\sum_{k=1}^m \sum_{l=1}^n \Omega_{\eta_l^{(k)}}(a)
f'(x)-\Phi(x) = \sum_k (f_k'(x)-\Phi(x))\phi_k(x)+ \sum_k (f_k(x)-g(x))\phi_k'(x), \shortintertext{hence}\|f'(x)-\Phi(x)\| \le \sum_k \|f_k'(x)-\Phi(x)\|\phi_k(x)+ \sum_k |f_k(x)-g(x)|\|\phi_k'(x)\| &\le \sum_k (\xi(x)+\omega_0(x))\phi_k(x)+\sum_k 2^{-k-1}\omega_0(x) &\le \xi(x)+2\omega_0(x).
F\{\sum_{n=0}^{\infty}e^{-\frac{1}{2}x^{2}}H_{n}(x)\frac{t^{n}}{n!}\}=\sum_{n=0}^{\infty}F\{e^{-\frac{1}{2}x^{2}}H_{n}(x)\}\frac{t^{n}}{n!}.
S_{s L}= \mathrm{c o n s t}\left[ \sum_{i , j \neq i}\frac{( 1 - \alpha_{i}) ( 1 - \alpha_{j})}{( 2 \alpha_{i}- 1 )}\ln ({\bf z}_{i j}{\bar{\bf z}}_{i j}) + \frac{\lambda_{0}}{2}\sum_{i}\left( 2 \alpha_{i}- 1 - \frac{1}{2 \alpha_{i}- 1}\right) + F [ \alpha ] \right] .
t\int_t^1 u_0(1-\tau^2,\varphi) \frac{d\tau}{\tau}-\frac{1}{2\pi}\int_0^{2\pi}\frac{\partial}{\partial t}\sum_{j=1}^{\infty}s_j F_j(1, \varphi)\cdot \int_t^1 u_0\Bigl(\frac{\rho^2-t^2}{\rho}, \theta\Bigr) \overline{Q_j(\rho,\theta)}\rho d\rho d\theta=0.
[T(a),T(b)]-T\big(\rho(T(a))b-\rho(T(b))a+\lambda[a,b]_V\big) =T(a)\cdot T(b)-T(b)\cdot T(a)-T\big((\mathfrak{l}-\mathfrak{r})(T(a))b-(\mathfrak{l}-\mathfrak{r})(T(b))a+\lambda(a\cdot_V b-b\cdot_V a\big)=0.
Q_{\hat{\beta}}^{( 0 )}= - i \Theta_{\hat{\alpha}A}\gamma_{\hat{\beta}\hat{\alpha}}^{t}P_{A B}p_{s t}\partial_{y_{s B}}+ r ( \vec{e}\times \vec{y}_{t}) E_{s}\gamma_{\hat{\beta}\hat{\alpha}}^{s t}\vec{\Theta}_{\hat{\alpha}}
\lim_{x\rightarrow 0^{+}}\frac{F\left( x\right)}{G\left( x\right)}=\lim_{t\rightarrow \infty}\frac{f\left( t\right)}{g\left( t\right)}\text{\and \}\lim_{x\rightarrow \infty}\frac{F\left( x\right)}{G\left(x\right)}=\lim_{t\rightarrow 0^{+}}\frac{f\left( t\right)}{g\left(t\right)}
\prod_{\substack{l,m,n \geq 1 l,m \leq n ; \, \gcd(l,m,n)=1}}\left( \frac{1}{1-x^l y^m z^n}\right)^{\frac{1}{l^a m^b n^c}}= \exp\left\{\sum_{n=1}^{\infty}\left( \sum_{l=1}^{n}\frac{x^l}{l^a}\right) \left( \sum_{m=1}^{n}\frac{y^m}{m^b}\right) \frac{z^n}{n^c}\right\}.
\left( J(h_{+}\allowbreak )+\sqrt{M\left( h_{+}\right)}\right)^{2}=\dfrac{4p_{1}\left( s-v\right)^{2}(s-2h_{+})^{2}}{s^{4}}\times \left( \sqrt{p_{1}}+\dfrac{\allowbreak \left\vert2wst-(t^{2}-s^{2})v\right\vert}{\sqrt{s^{2}+t^{2}}}\right)^{2}
A((v_{i,j})_{1\leq i<j\leq 2d})=\begin{pmatrix}M_1((v_{i,j})_{1\leq i<j\leq 2d}) M_2((v_{i,j})_{1\leq i<j\leq 2d}) \vdots M_{2d}((v_{i,j})_{1\leq i<j\leq 2d}) \end{pmatrix}.
\mathcal{A}^u_{k^1,k^2}a_{k^1,k^2}(\omega^1,\omega^2)=I^{p^1(\omega^2)}_{k^1}(\mathcal{J}^{u,\mathcal{S}^2}_{k^2}(a_{k^1,k^2}(t^1,\cdot))(\omega^2))(\omega^1),
\Pi = \frac{1}{2}\left[1 + \frac{\theta_-}{\theta_+}+ \frac{(u_- - u_+)^2}{a^2\theta_+}\right] \pm \sqrt{\frac{1}{4}\left[1 + \frac{\theta_-}{\theta_+}+ \frac{(u_- - u_+)^2}{a^2\theta_+}\right]^{\! 2}- \frac{\theta_-}{\theta_+}}.
\frac{\prod_{m=0}^{\frac{i-3}{2}}(2n-i+2l-2m)}{(n-\frac{i}{2}+1)_l}=2^{\frac{i-1}{2}}\frac{(n-i+\frac{3}{2})_{\frac{i-1}{2}}}{(n-i+\frac{3}{2})_l}
\frac{\ddot{b}}{b}+ (D - 1) \frac{\dot{b}^2}{b^2}+ 3 \frac{\dot{a}}{a}\frac{\dot{b}}{b}= \frac{8 \pi \overline{G}}{D+2}\left[\rho - 3 p_a + 2 p_b\right] .
\widetilde{c}_{j l}\equiv \operatorname{c o s}\widetilde{\tau}_{j l}= \frac{( n_{j}\cdot n_{l})}{\sqrt{n_{j}^{2}n_{l}^{2}}}= \frac{( - 1 )^{j + l}\; D_{j l}^{( N - 1 )}}{\sqrt{D_{j j}^{( N - 1 )}\; D_{l l}^{( N - 1 )}}}.
\langle \overline{Z |b|^2(H) \Gamma_H}F , \widetilde{G}\rangle_T = \langle \Gamma_H F , |b|^2(H) Z^* \widetilde{G}\rangle_T = \langle \Gamma_{H_0}F , |b|^2(H) Z^* \widetilde{G}\rangle_T -i \langle \overline{Y \Gamma_{H_0}}F , \overline{Z b (H) \Gamma_H^*}b(H)^* Z^* \widetilde{G}\rangle_T.
Z_m(x_i) = a_0 +\sum_{0<j\leq i}S(x_j,y_j)y_j^{-1/\alpha (Z_m(x_{j-1}))},Z_n(x_i) = a_0 +\sum_{k: 0<i_k\leq i}S(x_{i_k},y_{i_k})y_{i_k}^{-1/\alpha (Z_n(x_{{i_k}-1}))}.
\int \, 2 \, R [ g ] \, \sqrt{- g}\, d^{D}x = \int \, \exp \left[ ( D - 2 ) \alpha \, \phi \right] \, \left( 2 R [ \overline{{{g}}}] - \alpha^{2}\left( D - 1 \right) \left( D - 2 \right) \partial^{\mu}\phi \partial_{\mu}\phi \right) \sqrt{- \overline{{{g}}}}\, d^{D}x
((T^*)^n (x_1\oplus h_1), (T^*)^n(x_2\oplus h_2)) = (h_1,h_2)+(P_{H^2_-}\overline\chi^n Y_0^*x_1,P_{H^2_-}\overline\chi^n Y_0^*x_2)+((T_0^*)^nx_1,(T_0^*)^nx_2).
\int\,P(D^{(k)}+T)(\mathrm{d}T)\int\prod_{s=1}^{(N-k)/2}\delta\left(G_s-\mathrm{diag}(x_s\pm\mathrm{i}y_s)\right)(\mathrm{d}G_s)=\prod_{l=1}^kw_r(\lambda_l)\prod_{j=1}^{(N-k)/2}w_c(x_j,y_j)
\varkappa(m,p_1,p_2):=\sum_{s=0}^n\sum_{\nu\in I_s}k^{\frac{sr}{2}(\frac{1}{p_2}+\frac{1}{3p_1}-1)}\cdot (\prod_{\nu_j>k}\nu_j)^{\frac{r}{2}(\frac{1}{p_2}+\frac{1}{3p_1}-1)}|m(\nu)|^r<\infty,
s' = (1+b + a\circ (1+b))\circ (1+c) - (1+a) \circ (1+c + b \circ (1+c)) &= (1+c + b \circ (1+c) +(a \circ (1+b))\circ (1+c)) - (1+ c + b \circ(1+c) +a \circ (1+c+b \circ(1+c))) &= (a \circ (1+b))\circ (1+c) - a \circ (1+c+b \circ(1+c)).
(\nabla_{j}f)(y_{1},y_{2})=\frac{2\pi i}{\det\mathcal{T}}(\mathcal{C}_{1j}y_{1}+\mathcal{C}_{2j}y_{2})f(y_{1},y_{2})+\frac{\mathcal{C}_{3j}}{\det\mathcal{T}}\frac{\partial f}{\partial y_{1}}(y_{1},y_{2})+\frac{\mathcal{C}_{4j}}{\det\mathcal{T}}\frac{\partial f}{\partial y_{2}}(y_{1},y_{2}),
A_{i}^{\prime}( \vec{x}, t ) = e^{- i t \tilde{A}_{0}}\left[ A_{i}^{\prime}( \vec{x}, 0 ) + \int_{0}^{t}d t^{\prime}\, e^{i t^{\prime}\tilde{A}_{0}}( \partial_{i}\tilde{A}_{0}) e^{- i t^{\prime}\tilde{A}_{0}}\right] e^{i t \tilde{A}_{0}}
\left( -{\mathrm{e x p}}( - \frac{\epsilon P_{0}}{2}) \gamma_{i}P_{i}+ \gamma_{4}\frac{1}{\epsilon}\sinh ( \epsilon P_{0}) - \frac{i \epsilon}{2}\gamma_{4}P_{i}P_{i}\right) \psi = m ( 1 + \frac{\epsilon^{2}m^{2}}{4})^{\frac{1}{2}}\psi ,
g(x):= (1-4p^2x^2)^{-1/2}\Big[(1-2p^2)+ 4p(1-2p)e^{-\frac{px^2}{1-4px^2}}+ 2p^2e^{-\frac{x}{(1-2px)}}+ 2p^2e^{\frac{x}{(1+2px)}}\Big]
S \; = \; \int d t d^{2}x \, \left[ - h b_{0}{\tilde{J}}_{0}- h b_{k}{\tilde{J}}_{k}+ \frac{m}{2 \rho_{b}}{\vec{J}}_{b}^{2}- \frac{\hbar^{2}}{8 m \rho_{b}}\partial_{j}\rho_{b}\partial_{j}\rho_{b}\, - \, V ( \rho_{b}) \right] \; ,
\overline{\Theta}_{\mu \nu}^{a b}= -g_{\mu\nu}\left[\left( \nabla_{\lambda}-i g(n^c T^c)\overline{A}_{\lambda}\right)^2\right]^{a b}+ig (n^c T^c)^{a b}\overline{F}_{\mu \nu}-\delta^{a b}R_{\mu \nu}.
\phi = A ( x - y ) \Sigma_{l m}\int d \omega ( f_{\omega l m}^{b}b_{\omega l m}^{b}+ \bar{f}_{\omega l m}^{b}b_{\omega l m}^{b \dagger}+ f_{\omega l m}^{a}b_{\omega l m}^{a}+ \bar{f}_{\omega l m}^{a}b_{\omega l m}^{a \dagger}) ,
b_{s}= b_{r}\prod_{s \le t \le r - 1}{\frac{q^{{\frac{N}{2}}+ p}[ \! [ 2 t + 2 ] \! ]}{[ \! [{\frac{N}{2}}+ p + 2 t ] \! ] - [ \! [{\frac{N}{2}}+ p + 2 r ] \! ]}}\ .
p^*\left(\frac{\prod\limits_{i=1}^k L_{i,b-k+i}}{\prod\limits_{i=1}^{k-1}L_{i,b-k+i+1}}\right)= \frac{\Delta_{\{k-a,\dots,k-1\}}}{\Delta_{\{k-a+1,\dots, k\}}}\prod_{i=k-a}^{k-1}\lambda_i=p^* (M_k ),
{\rm Vol(Q^{1,1,1})}=\oint_{Q^{1,1,1}}\left(\frac{3}{8\Lambda}\right)^{7/2}8\,\prod_{i=1}^{3}\sin\theta_i\,d\theta_i d\phi_i\,d\psi=\frac{\pi^4}{8}\left(\frac{6}{\Lambda}\right)^{7/2}\,.
\left<\tau^M_\varphi, [V(D)]-[e_1]\right>=\lim_{t\downarrow 0}\left<\tau^M_\varphi,[V(tD)]-[e_1]\right>=\frac{p!}{2p !}\frac{(-1)^p}{(2\pi i)^p}\,\int_M \chi\hat{A}(M)\wedge\omega_\varphi.
\|m_{u_{n+1}v_{n+1}}-m_{p_iq_i}\| = \frac{d(u_{n+1},p_i)+d(q_i,v_{n+1})+|d(u_{n+1},v_{n+1})-d(p_i,q_i)|}{\max\big\{d(u_{n+1},v_{n+1}),d(p_i,q_i)\big\}}=\frac{1+1+|2-2|}{\max\big\{2,2\big\}}=1
\widehat{R}_{\nu}\left(\boldsymbol{\ell}_{1}\right)=Q\left(\boldsymbol{\ell}_{1}^{-1}\right)+a\left(\boldsymbol{\ell}_{1}\right),a\left(\boldsymbol{\ell}_{1}\right)=\sum_{n\ge n_{0}}a_{n}\boldsymbol{\ell}_{1}^{n},\a_n\in\mathbb C,
\Sigma = \frac{\lambda}{3 2 \pi^{2}}\biggl \{\Lambda^{2}- m^{2}\biggl [ \gamma + \ln \biggl ( \frac{m^{2}}{\Lambda^{2}}\biggr ) \biggr ] + O \biggl ( \frac{m^{2}}{\Lambda^{2}}\biggr ) \biggr \}+ O ( \lambda^{2}) .
K_{N,L}=\frac{\mathrm{vol}\left(O(L)\right)\mathrm{vol}\left(O(N)\right)}{\mathrm{vol}\left(O(L+N)\right)}\left(\frac{(2\pi)^L}{L!}\right)^{N/2}=\bigg(\frac{2^L}{L!}\bigg)^{\!N/2}\prod_{j=1}^N\frac{\Gamma(\frac{L+j}2)}{\Gamma(\frac j2)}.
K_{0}=2^{-8}(n-6)^2(n-2)^2(n+2)^2 K_{2}=2^{-4}(3n^4-24n^3+72n^2-96n+304) &K_{4}=2^{-2}(3n^2-12n+44) &J_{0}=2^{-3}(3n^4-18n^3-192n^2+1864n-3952) &J_{1}=2^{-1}(3n^3+3n^2-244n+620) &J_{2}=2 n^2+13n-68 &J_{3}=2 (n+1) &L_{0}=2^{-2}(3 n^2-12n-20)
H_{2}\leq 2^{\tau+q}\int_{0}^{2\pi}\int_{\frac{1}{2}}^{\frac{3}{2}}\frac{dr d\theta}{|\sin(\theta+\varphi)|^{\tau}|re^{i\theta}-1|^{q}}&=2^{\tau+q}\sum_{j=1}^4\int_{\frac{(j-1)}{2}}^{\frac{j\pi}{2}}\int_{\frac{1}{2}}^{\frac{3}{2}}\frac{dr d\theta}{|\sin(\theta+\varphi)|^{\tau}|re^{i\theta}-1|^{q}}\doteq2^{\tau+q}\sum_{j=1}^4H_{2,j}.
Q^{\mu\nu}=R^{\mu\nu}+iS^{\mu\nu}=R^{\mu\nu}-\frac{i}{2}e^{\mu\nu\sigma\tau}F_{\sigma\tau}, P_{\mu\nu}=F_{\mu\nu}+iG_{\mu\nu}=F_{\mu\nu}-\frac{i}{2}e_{\mu\nu\sigma\tau}R^{\sigma\tau}.
R_{ijkt, lr}-R_{ijkt, rl}= R_{sjkt}R_{silr}+R_{iskt}R_{sjlr}+R_{ijst}R_{sklr}+R_{ijks}R_{stlr}; R_{ijkt, lrs}-R_{ijkt, lsr}= R_{vjkt, l}R_{virs}+R_{ivkt, l}R_{vjrs}+R_{ijvt, l}R_{vkrs}+R_{ijkv, l}R_{vtrs}+R_{ijkt, v}R_{vlrs}.
0=e^{-\delta s}\epsilon\ln{x}+\frac{\partial \phi}{\partial s}(s,x) +\Gamma x\frac{\partial \phi}{\partial x}(s,x) +\frac{1}{2}\sigma^2x^2\frac{\partial^2 \phi}{\partial x^2}(s,x) &+ \int_\mathbb{R}\Big\{\phi(s,x(1+\gamma(z))-\phi(s,x)-x\gamma(z)\frac{\partial\phi}{\partial x}\Big\}\nu(dz).
{\frac{S_{{2\,n-6}}S_{{2\,n}}-{\tilde\alpha}^{2}S_{{2\,n-4}}S_{{2\,n-2}}}{\tilde\beta\, \left( S_{{2\,n-6}}S_{{2\,n-2}}+\tilde\alpha\,{S_{{2\,n-4}}}^{2}\right)}}={\frac{\tilde\beta\, \left( S_{{2\,n-6}}S_{{2\,n-2}}+\tilde\alpha\,{S_{{2\,n-4}}}^{2}\right)}{S_{{2\,n-8}}S_{{2\,n-2}}-{\tilde\alpha}^{2}S_{{2\,n-6}}S_{{2\,n-4}}}},
|H|_{D^{2,q}}\leq C(\sigma)(|H_t-(u\times H)|_q+|\nabla H|_q)\leq C(\sigma)(|\nabla H_t|_2+1), |\theta|_{D^{2,q}}\leq C(\big|\rho \theta_t+\rho u\cdot \nabla \theta+Pu|_q+||\nabla u|^2|_q+|\nabla |_q)+C\frac{1}{\sigma}|H|^2|_q \leq C(\sigma)(1+|\theta_t|_{D^1}+|u|_{D^{2,q}}+|H|_{D^{2,q}}),
d(\frac{[a_{1}^{-},a_{2}^{-}]_{A^{-}}}{2}+\frac{(a_{1}^{-}\circ a_{2}^{-})_{A^{+}}}{2}) =\frac{[da_{1}^{-},a_{2}^{-}]_{A^{-}}}{2}+\frac{(da_{1}^{-}\circ a_{2}^{-})_{A^{+}}}{2}+\frac{[a_{1}^{-},da_{2}^{-}]_{A^{-}}}{2}+\frac{(a_{1}^{-}\circ da_{2}^{-})_{A^{+}}}{2},
S \simeq \int d^{3}x \left[ \frac 1 2 ( \partial_{i}\chi )^{2}- ( g_{m}\zeta )^{2}\cos ( 2 g_{m}\chi ) - \frac{3 2 \sqrt{\pi}}{3}( g_{m}^{2}\zeta )^{3}\cos^{3}( g_{m}\chi ) \right] .
2B\delta(B)= \delta(B^2)=\delta(B(BA)B) = B^2\delta(BA)+3B(BA)\delta(B)-(BA)B\delta(B) = B^2(2A\delta(B)+2B\delta(A))+3B^2A\delta(B)-B\delta(B) = 5B^2A\delta(B)+2B^3\delta(A)-B\delta(B),
\frac{3}{2}\int_{\mathcal{N}}|W^{+}|_{h}^2d\mu_h \geq 8\pi^2\sum_{i=1}^{r}k_i + \frac{1}{4}\int_{M}|\mathring{\widetilde{Ric}}|_{\tilde{h}}^2 d\mu_{\tilde{h}}\geq 8\pi^2\sum_{i=1}^{r}k_i.
\big(1-(1+v)x\big)A^+(x;v) =\big(1-(1+v)x\big)G(x;v)+\frac{1-vx}{v^2}B'(vx;1/v)-x^3v-\frac{v^2x^4}{1-2vx}&+\frac{1}{1-v}(B(x;1)-\frac{1}{v^3}B(vx;1/v))+\frac{x^2}{v}C(x).
\zeta ( s ) = 2 \left( \frac{\tau_{2}}{2 \pi}\right)^{2 s}\sum_{Z^{2}}\left[ \left( n_{1}- g h_{1}+ \frac{1}{2}- \tau_{1}\left( n_{2}- g h_{2}+ \frac{1}{2}\right) \right)^{2}+ \tau_{2}^{2}\left( n_{2}- g h_{2}+ \frac{1}{2}\right)^{2}\right]^{- s},
P_{\pm}(n)\equiv |\langle n|\widetilde{\alpha_{\pm}}\rangle|^2 =2|\widetilde{N_{\pm}}(|\alpha|^2\mbox{,}\varphi)|^2 e^{-|\alpha|^2}\frac{|\alpha|^{2n}}{n!}(1\pm\cos(2n\varphi)).
{\cal Q}_{N}( \vartheta ) = \mathop{\lim}_{\eta \rightarrow 0}\frac{1}{i}\log \frac{1 + ( - 1 )^{\delta}e^{i Z_{N}( \vartheta + i \eta )}}{1 + ( - 1 )^{\delta}e^{- i Z_{N}( \vartheta - i \eta )}}\: .
I_1 := \int_\mathbb{R}\bigg(\sum_{k=k_0}^N 2^{-k\alpha}\|\mathcal{C}_{k}g(\cdot,t)\|_{L^2_x}\bigg)^{q_1}\, \mathrm{d}t I_2 := \int_\mathbb{R}\bigg(\sum_{k=N+1}^\infty 2^{-k\alpha}\|\mathcal{C}_{k}g(\cdot,t)\|_{L^2_x}\bigg)^{q_2}\, \mathrm{d}t,
\Sigma(W)\ll \widehat{Y}^{(n+5)/2+\varepsilon}\sum_{|d|\leq \widehat{K}}\sum_{d=e_1f_1f_2^2d_3}\sum_{d_3=d_3'g_1g_2}|e_1f_1|^{-1}|f_2|^{-1/2}|g_2||m(g_2)|^{-1}|d_3'|^{-1/2}\left(\widehat{W}+\widehat{Y}^{n/3}\right).
- \sum_{t \ge 0}\sum_{i \ge 0}\frac{(-1)^{t}}{(t+i+1)!}\sum_{i \le p \le t+i}\binom{p}{i}\sum_{\bold{q}= (q_1,\ldots,q_{i+1})}\tau_{\bold{q}}b_{\bold{q}}D^{t}\,.
\phi_h(x,y) = \begin{pmatrix}x y \end{pmatrix}+ hb_1\begin{pmatrix}u(k_1(x)) v(k_1(x),l_1(k_1(x),y)) \end{pmatrix}, \intertext{with the internal stages}\begin{pmatrix}k_1(x) l_1(k_1(x),y) \end{pmatrix}= \begin{pmatrix}x y \end{pmatrix}+ ha_{11}\begin{pmatrix}u(k_1(x)) v(k_1(x),l_1(k_1(x),y)) \end{pmatrix}.
\langle Op_\mathcal{A}(a)f,g\rangle =\langle a,W_\mathcal{A}(g,f)\rangle=\langle a, \mu(V_C^T\mathcal{A}_{FT2}\mathcal{D}_L)(g\otimes \bar f)\rangle=\langle \mathfrak{T}_{L^{-1}}\mathcal{F}_2^{-1}\mu(V_C^{-T})a,g\otimes \bar f\rangle,
\begin{cases}\Delta^{2}H_{\tilde{h}\tilde{k}}^{I}=0 w\in B_{1}H_{\tilde{h}\tilde{k}}^{I}=\tilde{h}w\in \partial B_{1}\Delta H_{\tilde{h}\tilde{k}}^{I}=\tilde{k}w\in \partial B_{1}\end{cases}\,.
\begin{aligned}\big|\big\langle x-\xi, \nu(x_{0})\big\rangle-\big\langle x-\xi^{'}, \nu(x_{0})\big\rangle\big|= |\xi-\xi^{'}|\leq C|x_{0}-\xi|^{1+\lambda}& \leq C |x-\xi|^{1+\lambda}.\end{aligned}
G^{(N)}_{\varepsilon}(\zeta_1,\cdots,-\zeta_j,\cdots,\zeta_N)^{-\cdots-+\cdots+}=(-1)^{N-j+H(n\ge j)}G^{(N)}_{-\varepsilon}(\zeta_1,\cdots,\zeta_j,\cdots,\zeta_N)^{-\cdots-+\cdots+},
\delta^1_\Pi (X) (\alpha_1, \ldots, \alpha_n) = \sum_{k=1}^{n}(-1)^{n-k}\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \hat{\alpha}_k \wedge \cdots \wedge \alpha_n) (X(\alpha_k))- X([\alpha_1,\ldots, \alpha_n])
\mathcal Z (\mathcal{H}):= \left\{f \in L^1 (\Omega) \cap \mathcal D (\mathcal{H}) : \sup_{j \in\mathbb Z}2^{M |j|}\big\| \phi_j \big(\sqrt{\mathcal{H}}\big ) f \big\|_{L^1(\Omega)}< \infty M \in \mathbb N \right\}.
u_a(x,t) = \frac{1}{2 \pi}\int\limits_0^\infty \left[ \cos \left(\xi \left(x+\frac{t}{\sqrt{1+ a_\alpha\xi^{\alpha}}}\right)\right)+\cos \left(\xi \left(x-\frac{t}{\sqrt{1+ a_\alpha\xi^{\alpha}}}\right)\right)\right]e^{-\frac{a^2 \xi^2}{4}}d\xi.
\|v_nw_i^*v_n^* - w_i^*\|_2 = \|[1-2(v_np_{k_i}v_n^*)q_{k_i}] - [1 - 2p_{k_i}q_{k_i}]\|_2 \le 2\|v_np_{k_i}v_n^* - p_{k_i}\|_2 \le 2\cdot \frac{1}{2^{n+1}}= \frac{1}{2^n},
\Delta(x^i\otimes a_1\otimes \cdots \otimes a_n)=\sum_{k=1}^{n-1}\sum_{r+s=i}(-1)^\epsilon (x^r\otimes a_1\otimes \cdots \otimes a_k)\otimes (x^s\otimes a_{k+1}\otimes \cdots \otimes a_n),
\Gamma_{4}= \frac{i \lambda^{2}}{2}\cos \frac{{\tilde{p}}_{1}p_{2}}{2}\cos \frac{{\tilde{q}}_{1}q_{2}}{2}\int \frac{d^{2}k d \omega}{( 2 \pi )^{3}}\frac{\mathrm{c o s}^{2}\frac{{\tilde{P}}k}{2}}{( \omega - \frac{k^{2}}{2}+ i \epsilon ) ( E - \omega - \frac{( k - P )^{2}}{2}+ i \epsilon )},
\left( \begin{array}{c c}{l_{1 1}}&{l_{1 2}}\\{l_{2 1}}&{l_{2 2}}\\ \end{array}\right) = \left( \begin{array}{c c}{1}&{0}\\{1}&{0}\\ \end{array}\right) , \left( \begin{array}{c c}{1}&{1}\\{0}&{0}\\ \end{array}\right) \mathrm{e t c .}
\sum_{|\gamma|>1}|k_2(\rho)| \leq x^2 \sum_{m=1}^{\infty}\{n_L(2m)+n_L(-2m)\}x^{-(2m-1)^2}\leq 5.44 x^2 \sum_{m=1}^{\infty}\{\log d_L + n_L \log (2m+2)\}x^{-(2m-1)^2}{\mbox{by (\ref{nLt-WithoutGRHShort})}}\leq c_{20}x \log d_L,
\mathbb{P}_{\beta ,M}(X_{1}=\xi_{1},X_{2}=\xi_{2},\ldots ,X_{M\,}=\xi_{M})~=~ Z_{\beta ,M}^{-1}\,\,\frac{1}{2^{M}}\,e^{\frac{\beta}{2M}(\sum \xi_{j})^{2}}
g_0(\nabla_{H}Y_j, Y_{\ell}) = \frac{1}{2}\left(g_0([H,Y_j],Y_{\ell}) - g_0([H, Y_{\ell}], Y_j) - g_0([Y_j,Y_{\ell}], H) \right) &= \frac{1}{2}\left(\alpha_j(H) \delta_{j\ell}- \alpha_{\ell}(H) \delta_{j\ell}- 0 \right) &= 0,
p_{\mathrm{W}}(\lambda) =\frac{\Gamma(n)}{\Gamma(s+n)}\lambda^{s}e^{-\lambda}&\times [L_{n-1}^{(s)}(\lambda)L_{n}^{(s+1)}(\lambda)-L_{n}^{(s)}(\lambda)L_{n-1}^{(s+1)}(\lambda)].
b_{\alpha}^{\dagger}|n_{\alpha}\rangle=\frac{1}{\sqrt{n_{\alpha}+1}}\psi_{\alpha}\otimes_{+}\psi_{\alpha}^{\otimes n_{\alpha}}=\sqrt{n_{\alpha}+1}\psi_{\alpha}^{\otimes(n_{\alpha}+1)}=\sqrt{n_{\alpha}+1}|n_{\alpha}+1\rangle.
\sum_{n=0}^{\infty}\frac{(-1)^n (-q;q)_n (- \frac{aq}{b}; q)_n \, b^n}{(aq; q^2)_{n+1}}= \sum_{n=0}^{\infty}\frac{(-1)^n (- \frac{aq}{b}; q)_n \, b^n q^{\frac{n(n+1)}{2}}}{(-b; q)_{n+1}}.
{\frac{1}{2}}\gamma_{b}^{a}\left( \xi_{i j}^{b}+ \xi_{j i}^{b}- \partial_{i}\rho_{j}^{b}- \partial_{j}\rho_{i}^{b}- \nu_{i c}^{b}\rho_{j}^{c}- \nu_{j c}^{b}\rho_{i}^{c}\right) = 0 \, \, \, .
( e ) = -{\frac{\lambda^{2}}{3 2 ( 2 \pi )}}{\tilde{G}}_{\beta}^{2}( 0 ) \int_{0}^{\frac{\beta}{2}}d \tau \, \, \int_{- \infty}^{+ \infty}{\frac{d k}{g^{2}( k )}}\left[ e^{- g ( k ) \tau}+{\frac{e^{g ( k ) \tau}+ e^{- g ( k ) \tau}}{e^{\beta g ( k )}- 1}}\right]^{2}\; ,
\begin{array}[c]{c}\omega_{M}:=\psi^{\ast}\omega_{N}+\frac{\sqrt{-1}}{2\pi}\frac{\varepsilon_{0}}{r_{S^{1}}^{2}}\sum_{i,j=1}^{2}(\delta_{ij}-\frac{\overline{z}_{i}z_{j}}{r_{S^{1}}^{2}})dz_{i}\wedge d\overline{z}_{j}\end{array}