harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
s_{b_1}^{\beta_1}s_{b_2}^{\beta_2}\cdots s_{b_q}^{\beta_q}f_1(t, s)^{\mu_1}f_2(t, s)^{\mu_2}\cdots f_n(t, s)^{\mu_n}s_{w_1}^{\partial_1}s_{w_2}^{\partial_2}\cdots s_{w_l}^{\partial_l}= s_{a_1}^{\alpha_1}s_{a_2}^{\alpha_2}\cdots s_{a_p}^{\alpha_p}.
F ( y ) = \int_{1}^{y}\frac{d u}{u^{3 / 2}\sqrt{u^{3}- 1}}\equiv \frac 2 3 ( y^{3}\! - \! 1 )^{\frac 1 2}\y^{\frac 7 2}\_{2}F_{1}( 7 / 6 , 1 ; 3 / 2 | 1 \! - \! y^{- 3}) \ .
\frac{|1 + \big(\frac{i+ip}{1-p}\big)^2 - \big(-\frac{is}{1-p}\big)^2|}{1 + |\frac{i+ip}{1-p}|^2 - |-\frac{is}{1-p}|^2}= \frac{|\varphi_{z_1}(z_2)|^2}{2- |\varphi_{z_1}(z_2)|^2}\in [0,1).
\begin{array}{c c}{{\displaystyle{\langle \psi^{i}( z_{1}) \psi^{j}( z_{2}) \rangle = C_{\psi}\frac{V^{i j}( z_{1 2})}{( \det{\cal X}_{1 2})^{\frac{1}{2}\eta}}}~}}&{{~ V^{i j}( z_{1 2}) = V_{k}^{~ j}( z_{1 2}){\cal E}^{k i}}}\\ \end{array}
\left\langle J_{(ab,i)},J_{(cd,j)},P_{(e,k)}\right\rangle =\alpha_{\gamma}K_{ijk}^{\gamma}\left\langle \tilde{J}_{ab},\tilde{J}_{cd},\tilde{P}_{e}\right\rangle , & =\frac{1}{8}\left( \alpha_{0}K_{ijk}^{0}+\alpha_{1}K_{ijk}^{1}+\alpha_{2}K_{ijk}^{2}+\alpha_{3}K_{ijk}^{3}\right) \varepsilon_{abcde}.
\delta x(t) = \sum_{k = 1}^{K}\alpha_k \Phi^{\bar{u}}(t,\tau_{k}) \left[f(x^{(\bar{u},x_0)}(\tau_k),v_k) - f(x^{(\bar{u},x_0)}(\tau_k), \bar{u}(\tau_k)) \right]l_k
p_{n+1}(A)\big(p_n(A)\big)^{-1}= M\begin{bmatrix}\frac{p_{n+1}(x_1)}{p_n(x_1)}\frac{W(p_n,p_{n+1})(x_1)}{(p_n(x_1))^2}0 \frac{p_{n+1}(x_1)}{p_n(x_1)}\end{bmatrix}M^{-1}, (I_2x-A)^{-1}=M\begin{bmatrix}\frac{1}{x-x_1}\frac{1}{(x-x_1)^2}0 \frac{1}{x-x_1}\end{bmatrix}M^{-1},
\|T_{\mu}^{\beta}f_k\|_{p_2,\alpha_2}^{p_2}\lesssim \sum_{j=1}^{\infty}\left (\frac{\mu(D_j)}{(1-|a_j|^2)^{(n+1+\gamma )\l}}\right )^{p_2}\left(\int_{\tilde D_j}\!\! |f_k(z)|^{p_1}\,dv_{\alpha_1}(z)\right)^{p_2/p_1}.
\partial_t^k f =\partial_t (\partial_t^{k-1}f)=-\mathcal L(\partial_t^{k-1}f)+\partial_t^{k-1}\Gamma(f, f) &=-\mathcal L(\partial_t^{k-1}f)+\sum_{0\le j\le k-1}C^j_{k-1}\Gamma(\partial^j_tf,\partial^{k-1-j}_tf).
Q^{( 1 )}={\frac{l}{a^{2}}}\left[{\frac{1}{6}}R ( h ) +{\frac{y e^{\phi}}{l}}\left( \phi_{\ | \alpha}^{| \alpha}+ \phi^{| \alpha}\phi_{| \alpha}\right) -{\frac{y^{2}e^{2 \phi}}{l^{2}}}\phi^{| \alpha}\phi_{| \alpha}\right] \ .
\left(\begin{matrix}x=0 x=\infty \, \left( \frac13 \right) \overbrace{\begin{matrix}1 \theta^0_2 0 \theta^0_1 0 0 \end{matrix}}\overbrace{\begin{matrix}-t^{\frac13}\theta^\infty_1/3 -\omega t^{\frac13}\theta^\infty_1/3 -\omega^2 t^{\frac13}\theta^\infty_1/3 \end{matrix}}\end{matrix}\right).
\frac{1}{4 d}\left[ \mathrm{T r}\left( G^{- 1}\dot{G}\right) \right]^{2}= \frac{1}{4 d}\left( \frac{d}{d u}{\log \det G}\right)^{2}= \frac{1}{d}\left( \dot{\phi}- \dot{\overline{{{\phi}}}}\right)^{2}.
\widetilde{(\eta^{n})}_l =e^{i\beta_l^+\tau}\widetilde{(\eta^{n,+})}_l - e^{i\beta_l^-\tau}\widetilde{(\eta^{n,-})}_l - e^{\frac{i\tau}{\alpha}}(\widetilde{(\eta^{n-1,+})}_l - \widetilde{(\eta^{n-1,-})}_l), n\ge 1,
J \ =\ \left\{\left[-\frac{1}{4}, 2+\frac{1}{4}\right], \left[4-\frac{1}{4}, 5+\frac{1}{4}\right], \left[9-\frac{1}{4}, 9+\frac{1}{4}\right], \left[12-\frac{1}{4}, 14+\frac{1}{4}\right]\right\}.
K_{\theta}( x_{1}, x_{2}, x_{3}) = \frac{1}{\pi^{2}\theta^{2}}\exp \left[ - \frac{2 i}{\theta}\epsilon_{\alpha \beta}\Bigl ( x_{1}^{\alpha}x_{2}^{\beta}+ x_{2}^{\alpha}x_{3}^{\beta}+ x_{3}^{\alpha}x_{1}^{\beta}\Bigr ) \right]
~\bigg(\frac{z_1-1}{z_1+1}\bigg)^{1+(i-s)/2}F\bigg(1+i, 1-s, r+2, \frac{2z_1}{z_1+1}\bigg) = ~(-z)^{1+(i-s)/2}F\bigg(1+i, 1-s, r+2, 1-z\bigg) = ~(-1)^{1+(i-s)/2}(\cosh t)^{-2-i+s}F(1+i, 1-s, r+2, \tanh^2 t).
|I_{1,\rho}(x)| \leq C \sum_{k=-\infty}^{-1}\frac{\rho(2^{k}R)}{(2^{k}R)^{\gamma}}\int_{|xy^{-1}|<2^{k+1}R}|f(y)|dy \leq C Mf(x) \sum_{k=-\infty}^{-1}\frac{\rho(2^{k}R)}{(2^{k}R)^{\gamma-Q}}\leq C Mf(x) \sum_{k=-\infty}^{-1}\int_{2^{k}R}^{2^{k+1}R}\frac{\rho(t)}{t^{\gamma-Q+1}}dt = C Mf(x) \int_{0}^{R}\frac{\rho(t)}{t^{\gamma-Q+1}}dt,
\mu_{n}:= \frac{Q_{r}(\lambda_{n})}{P_{r}^{\, \prime}(\lambda_{n})}= \left(\sum\limits_{k=0}^{r-1}\dfrac{P_{k}(\lambda_{n})^{2}}{D_{k}D_{k-1}}\right)^{-1}\in (0, +\infty) \ , \ \ \ \ 1 \leq n \leq r \ ,
\eta^{I J}P_{I}^{x}P_{J}^{x}= \eta^{I J}\big ( h_{u v}k_{I}^{u}k_{J}^{v}- m_{I}^{A}m_{J}^{B}\rho_{A B}\big ) = \eta^{I J}h_{u v}k_{I}^{u}k_{J}^{v},
[M']_t = \sum_{n\geq 1}[\langle M', h_n\rangle]_t = \sum_{n\geq 1}\sum_{0\leq s\leq t}\Delta[\langle M', h_n\rangle]_s = \sum_{0\leq s\leq t}\sum_{n\geq 1}\Delta[\langle M', h_n\rangle]_s =\sum_{0\leq s\leq t}\Delta[M']_s.
A_r =a_{2r,1}+a_{2r,-1}+a_{2r-1}, A'_s =a'_{2s,1}+a'_{2s,-1}+a'_{2s-1}, B_r =a_{2r,1}+a_{2r,-1}+a_{2r+1}, B'_s =a'_{2s,1}+a'_{2s,-1}+a'_{2s+1}, C_r =\frac{\alpha_{-1}}{\beta_r}a_{2r,1}-\frac{\alpha_{1}}{\beta_r}a_{2r,-1}, C'_s =q^{-s}(a'_{2s,1}-a'_{2s,-1}),
\max_{a,b}f(a,b)=f\left(\left(\frac{n}{p}-2+\gamma\right)^{p-1}\left(\frac{n(p-1)}{p}-\gamma\right)^{p-1},\frac{n}{p}-2+\gamma\right)=\left(\frac{n}{p}-2+\gamma\right)^p\left(\frac{n(p-1)}{p}-\gamma\right)^p,
\Psi^{\alpha_{1}^{1}. . . . \alpha_{1}^{m}, \alpha_{2}^{1}. . . . \alpha_{2}^{m}}( X_{1}, X_{2}) = - \Psi^{\alpha_{2}^{1}. . . . \alpha_{2}^{m}, \alpha_{1}^{1}. . . . \alpha_{1}^{m}}( X_{2}, X_{1}) ,
\Big\|\big(\sum_{n\in\mathbb Z}\sum_{m = 0}^{2^{l}-1}|(M^G_{2^n+{2^{n-l}(m+1)}}- M^G_{2^n+{2^{n-l}m}})S_{j+n}f|^2 \big)^{1/2}\Big\|_{L^p}&\lesssim 2^{-\theta l/2+(1-\theta)l}\min\big\{1, 2^{l}2^{-|j|/2}\big\}^{\theta}\|f\|_{L^p},
\hat{g}= \left({\frac{L}{u}}\right)^{2}\left( \begin{array}{l l l}{}&{{\dot{u}}^{2}- u^{4}}&{{\dot{u}}{u^{\prime}}}\\{}&{{\dot{u}}{u^{\prime}}}&{{u^{\prime}}^{2}+ u^{4}}\\ \end{array}\right) ,
\int_0^1 \left|\int^s_0\nabla^i_x \beta(t) dt\right|^2 ds \leq \int_0^1 s \int^s_0 |\nabla^i_x \beta(t)|^2 dt ds & \leq \frac{1}{2}\left( \int^1_0 |\nabla^i_x \beta(t)|^2 dt - \int^1_0 s |\nabla^i_x \beta(s)|^2 ds \right) & \leq \int^1_0 |\nabla^i_x \beta(t)|^2 dt.
\frac{\partial G_{k}}{\partial t}(x,t)=\sum_{i=0}^{l-1}(-1)^{i+1}(\frac{1}{\sqrt{x}}S_{i+1}G_{k}-T_{i}G_{k})(\frac{2t}{\sqrt{x}})^{i}+(-1)^{l}T_{l+2}G_{k}(\frac{2t}{\sqrt{x}})^{l}.
-\frac{3\sqrt{c}\, g}{4R^4}\, (1+\rho_*\kappa_1^*)^{3}(1+\rho_*\kappa_2^*)^{3}=(1+\rho_{*}\kappa^{*}_{1})^{2}\left(\frac{\partial \kappa^{*}_{1}}{\partial z_{1}}\right)^{2}+ (1+\rho_{*}\kappa^{*}_{2})^{2}\left(\frac{\partial \kappa^{*}_{1}}{\partial z_{2}}\right)^{2}.
{\Gamma^{\prime}}_{\mathrm{e f f}}^{{( 2 )}}= \int \frac{d^{4}\! p}{( 2 \pi )^{4}}\phi ( p ) \phi ( - p ) \left( p^{2}- \frac{\hbar g^{2}}{9 6 \pi^{2}}\left( \frac{1}{\tilde{p}^{2}}- \frac{1}{\tilde{p}^{2}+ \frac{1}{\Lambda^{2}}}\right) + O ( g^{4}) \right)
A = 2a(2a+1) +2b_1(4a-2l-d+3), B_1 = 2b_1(4a+2b_1+3), E_{0,l}= \lambda [2a(2l+d) + 2b_1(4a-2l-d+3) - l(l+d-1)], \psi_{0,l}(r) \propto r^l (1+\lambda r^2)^{-a}\exp\left(- \frac{b_1}{1+\lambda r^2}\right),
C_1(T)\Big(\displaystyle{\sum_{i=1}^n e_{1+i-n}(U_i)(0)}+ \sum_{j=1}^q e_1(U_{n+j})(0)\Big) \le \int_0^T \sum_{l=0}^q || \mathcal{\mathbf{B}}_{n+l}^{\ast}U_{n+l}||_{G_{n+l}}^2 \, dt \,.
W_{\theta}\times \overline{P}^1_{\theta}(x,x'):= W_{\theta}(x|x')\overline{P}^1_{\theta}(x') = \frac{\overline{P}^3_{\theta}(x) \overline{W}_{\theta}(x|x') \overline{P}^2_{\theta}(x')}{\lambda_{\theta}\sum_{x''}\overline{P}^3_{\vec{\theta}}(x'')\overline{P}^2_{\vec{\theta}}(x'')}
\Delta_{2}(t,\varepsilon)=\int_{0}^{+\infty}\Big(\Psi_{3}(\sigma,\varepsilon)\Gamma_{1}(\sigma+t,\varepsilon) +(1/\varepsilon)\Psi_{4}(\sigma,\varepsilon)\Gamma_{2}(\sigma+t,\varepsilon)\Big)d\sigma,\t\ge 0.
\frac{\partial}{\partial \tau}[ \Delta ( \tau , \tau_{2}) - \Delta ( \tau , \tau_{1}) ] ={\frac{1}{2}}\frac{\partial}{\partial \tau}[ G_{B}( \tau , \tau_{1}) - G_{B}( \tau , \tau_{2}) ] , \quad \frac{\partial^{2}}{\partial \tau \partial \tau^{\prime}}\Delta ( \tau , \tau^{\prime}) ={\frac{1}{2}}\ddot{G}_{B}( \tau , \tau^{\prime}) ,
|E_1| = C \Biggl|\int_{\mathbb R}e^{-v^2}\sigma(s+h,x_1-2av\sqrt{t-s-h})\, dv &\quad{}-\int_{\mathbb R}e^{-v^2}\sigma(s+h,x_2-2av\sqrt{t-s-h})\,dv \Biggr| &\stackrel{\rm A6}{\le}C \int_{\mathbb R}e^{-v^2}|x_1-x_2|^{\beta(\sigma)}\,dv =C|x_1-x_2|^{\beta(\sigma)}.
H_{R}= \prod_{u}H_{R u}( x_{a_{1}^{u}}, \dots , x_{a_{n_{u}}^{u}}) , ~ ~ g_{a_{i}^{u}a_{i}^{u}}= g_{a_{j}^{u}a_{j}^{u}}, ~ ~ \eta_{M N}\partial_{M}\partial_{N}H_{R u}= 0 ,
\|\langle x\rangle I_1^{\prime}(x)\|_{L^2(\mathbb{R}^+)}= 2\pi^{-1}\|\langle x\rangle \widehat{\bar{r}_1(z)}(2x)\|_{L^2(\mathbb{R})}\leq c \| r_1(z) \|_{H^1(\mathbb{R})},
\vert t_{N}^{a}\zeta^{a}-s_{N}^{a}\vert=\vert t_{N}\zeta-s_{N}\vert\cdot\vert t_{N}^{a-1}\zeta^{a-1}+\cdots+s_{N}^{a-1}\vert \leq D(a,\zeta)t_{N}^{-\nu_{N}+a-1}\vert t_{N+1}^{a}\zeta^{a}-s_{N+1}^{a}\vert=\vert t_{N+1}\zeta-s_{N+1}\vert\cdot\vert t_{N+1}^{a-1}\zeta^{a-1}+\cdots+s_{N+1}^{a-1}\vert \leq D(a,\zeta)t_{N+1}^{-\nu_{N}+a-1}.
\mathcal{G}^{k}_1(\vec{a};\vec{b}) =\int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}dx_1dx_2\cdots dx_n &=\prod_{i=1}^{n}|b_i|-|a_i|.
\sum_{r,s\in\mathbb Z}(-1)^{n+k+s}{k\choose n+r-s-m}\!\!\left\{\!{n-1-k\choose m-r}\!+\!{n-1-k\choose m-r-1}\right\}\!M(r-sa)^{n-1}&=\!\!\sum_{r,s\in\mathbb{Z}}(-1)^{n+k+s}{k\choose n+r-s-m}{n-k\choose m-r}M(r-sa)^{n-1},
R_{\gamma \alpha \beta}^{\mu}= \partial_{\alpha}\Gamma_{\beta \gamma}^{\mu}- \partial_{\beta}\Gamma_{\alpha \gamma}^{\mu}+ \Gamma_{\alpha \lambda}^{\mu}\Gamma_{\beta \gamma}^{\lambda}- \Gamma_{\beta \lambda}^{\mu}\Gamma_{\alpha \gamma}^{\lambda}
m^{2}\alpha^{2}= - 2 \sum_{\alpha}\sum_{n , l}l ( n - l ) [ A_{n - l}^{\alpha}A_{l}^{\alpha}e^{- i n ( \sigma + \tau )}+ \tilde{A}_{n - l}^{\alpha}\tilde{A}_{l}^{\alpha}e^{- i n ( \sigma - \tau )}] ,
0=\int\limits_{\Sigma_{\varepsilon}}d(\frac{dJ}{J}\wedge \Phi^*\omega)=\int\limits_{\partial \Sigma_{\varepsilon}}\frac{dJ}{J}\wedge \Phi^*\omega= \int\limits_{S^1 \times M^k}\frac{dJ}{J}\wedge\Phi^*\omega - \sum_j \int\limits_{\partial N_{j,\varepsilon}}\frac{dJ}{J}\wedge\Phi^*\omega.
\Omega(s)=\int_0^\infty\Bigl(\bigl((n+1-q)t^{s-1}-2t^{s-2}\bigr)\bigl(e^{(n+1-q)t}-\sum_{j=0}^{2n-1}(n+1-q)^j\frac{t^j}{j!}\bigr)+2(n+1-q)^{2n}t^{s+2n-2}\Bigr) S_{n,q}(e^{-t})e^{-(n+1-q)t}dt.
\int_{\Omega\setminus B(\xi_m,\rho)}\bigg|f_0^{'}\Big(U_{\mu^i_m,\xi^i_m}(x)\Big) \psi_{\mu^i_m,\xi^i_m}^h (x) u_m(x) \bigg|dx \leq C|\psi_{\mu^i_m,\xi^i_m}^h|_{\frac{2n}{n-2}}|u_m |_{\frac{2n}{n-2}}\Big(\int_{\Omega\setminus B(\xi_m,\rho)}U_{\mu^i_m,\xi^i_m}^{\frac{2n}{n-2}}dx\Big)^{\frac{2}{n}}= O(\mu^i_m).
\mathrm{Res}_{\xi =\xi_{1 +}(\omega)}\frac{1}{\zeta_1(\xi) - \omega}= \mathrm{Res}_{\xi =\xi_{2 +}(\omega)}\frac{1}{\zeta_1(\xi) - \omega}= \left. \frac{1}{\zeta_1'(\xi)}\right|_{\xi = \xi_{2+}(\omega)}&= \frac{1}{\zeta_1'( H( \mathrm{i}\sqrt{\Lambda - \omega}))}= \sum_{k=-1}^\infty c_k (\Lambda - \omega)^{k/2}
\langle( I_{2})'(u),v\rangle = \lim_{t\to0}\frac{1}{t}( I_{2}(u+tv)- I_{2}(u)) = \frac{1}{p}\lim_{t\to 0}\frac{1}{t}\int_{M}(|u(x)+tv(x)|^{p}-|u(x)|^{p})d\mu_g(x).
\underline{{\Omega}}_{I J}= \left( \begin{array}{c c c c}{0}&{0}&{0}&{1}\\{0}&{0}&{\frac{1}{V}}&{- \underline{{\Omega}}_{C + +}}\\{0}&{- \frac{1}{V}}&{0}&{- \underline{{\Omega}}_{C --}}\\{- 1}&{\underline{{\Omega}}_{C + +}}&{\underline{{\Omega}}_{C --}}&{0}\\ \end{array}\right) .
\frac{\partial}{\partial t}\rho + \div(\rho u) =0 \frac{\partial}{\partial t}(\rho u) + \div (\rho u \otimes u) + \nabla (p(\rho) + C_\kappa \frac{\alpha}{2}\rho^2) = \div(\sigma[ u]) + C_\kappa\alpha \rho \nabla c c - \frac{1}{\alpha}\triangle c = \rho\, ,
\hat{C}_{( 2 n )}^{+}= C_{( 2 n )}^{+}- C_{( 2 n - 1 )}^{+}\left( d \underline{{y}}+ A^{( 2 )}\right) + \underline{{y}}\G_{( 0 )}^{+}\textstyle{\frac{1}{n !}}\hat{B}^{n},
(v) =\sum^{k+1}_{i=1}(-1)^{i}\left[\sum^{k}_{j=1}(-1)^{j}f \circ (\alpha^{j-1,1}_{k})^{-1}\circ \mu^{j-1}_{k+1}\circ \alpha^{j-1,2}_{k+1}\right] \circ (\alpha^{i-1,1}_{k+1})^{-1}\circ \mu^{i-1}_{k+2}\circ \alpha^{i-1,2}_{k+2}=\sum^{k+1}_{i=1}\left[\sum^{k}_{j=1}b_j\right]a_i=\sum^{k+1}_{i=1}\sum^{k}_{j=1}b_ja_i=0,
\begin{array}{cccc}\Gamma_{1,\epsilon}\Gamma_{2,\epsilon}\epsilon=1,\;2 \Gamma_{1,\epsilon}n_1 \in Z\ +\;\frac{1}{2}n_2\;odd \Gamma_{2,\epsilon}n_1 \in Z n_2\;even \Gamma_{3,\epsilon}n_1 \in Z + \frac{n_2 +1}{2}n_2\;\in\; Z,\end{array}
\big\{c\left((x_1,\ldots,x_{a-1}, x_{a}= u^{\ast}, x_{a+1},\ldots, x_t); (u, v)\right)~:~(x_1,\ldots, x_{a-1}, x_{a+1},\ldots, x_t) \in [r]^{t-1}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({u},{v}) \in [r] \times [s]\big\}.
\partial_{y}\Big [ \big ( \partial_{y}+ 2 \ell^{- 1}\big ) h + \frac{( 1 - \gamma )^{- 1}\ell}{3 M_{*}^{3}}e^{- 2 y / \ell}T_{5 5}( y ) \Big ] = \frac{( 1 - \gamma )^{- 1}\ell}{3 M_{*}^{3}}\partial_{\mu}T^{\mu 5}.
\int_{t-d}^{t+d}\int_{\lambda> 1}e^{is\lambda}\hat{g}(\lambda) d\lambda ds = \int_{\lambda> 1}\hat{g}(\lambda) \int_{t-d}^{t+d}e^{is\lambda}ds d\lambda = -i \int_{\lambda> 1}\hat{g}(\lambda) \left(\frac{e^{i(t+d)\lambda}}{\lambda}- \frac{e^{i(t-d)\lambda}}{\lambda}\right) d\lambda.
E\{(\|X_{1}-X_{2}\|_{2}^{2}-p\tau)^{l}\}+E\{(\|Y_{1}-Y_{2}\|_{2}^{2}-p\tau)^{l}\}-2E\{(\|X_{1}-Y_{1}\|_{2}^{2}-p\tau)^{l}\}&=\sum^{l}_{a=0}\binom{l}{a}(-p\tau)^{l-a}\left(E\|X_1- X_2\|^{2a}_2+E\|Y_1-Y_2\|^{2a}_2-2E\|X_1- Y_1\|^{2a}_2\right).
\overline{C}_{X^{\ast}}(h,yI_{2m}) =\overline{C}_{X^{\ast}}([\lambda_h, g_h],[y, g_y]) &=m_{X^{\ast}}(g_h^{y}g_y)^{-1}m_{X^{\ast}}(g_h)m_{X^{\ast}}(g_y)\widetilde{C}_{X^{\ast}}([\lambda_h, g_h], [y, g_y]) &=m_{X^{\ast}}(g_h^{y}g_y)^{-1}m_{X^{\ast}}(g_h)m_{X^{\ast}}(g_y) &=(x(g_h^y), x(g_y))_F=(x(g_h^y),y^{m})_F=1.
t_{k}=\omega_{k}\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\omega_{k}^{2}\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}
\begin{aligned}f(x) \,\rightarrow\,\min g_i(x) \,\leq\,0 \qquad i\in\mathcal M h_j(x) \,=\,0 \qquad j\in\mathcal P \varphi^t(G_l(x),H_l(x)) \,\leq\,0 \qquad l\in\mathcal Q \end{aligned}
\left|\frac{zf_2'(z)}{f_2(z)}\right| =\left|1+\frac{\rho_2}{(1-\rho_2^2)}\left( \frac{u\rho_2^2+4\rho_2+u}{\rho_2^2+u\rho_2+1}+\frac{v\rho_2^2+2\rho_2+v}{v\rho_2+1}+\frac{q \rho_2^2+4\rho_2+q}{\rho_2^2+q \rho_2+1}\right)\right| = \frac{5}{3}
G\left(x,t\right) =\sum\limits_{n= 0}^{\infty}P_n(x)\frac{t^n}{n!}&=\left(1-at\right)^\beta \exp\left\{\frac{xt+\theta}{1-at}+b_{0}+b_{1}t+...+b_{d-1}\frac{t^{d-1}}{(d-1)!}\right\}.
\{\dots \}_{2}= + \frac{1}{2}( 1 + \beta ) \vert \eta_{\mathrm{j}}\vert^{- 1}+ i n \biggl ( 1 \mp \biggl \vert \frac{\eta_{\mathrm{j}}}{\eta_{1}^{\prime}}\biggr \vert \biggr ) .
\langle ({\cal O}^{r_1~+}_{j_1m_1}{\cal O}^{r_2~+}_{j_2m_2}){\cal O}^{r_3~-}_{j_3~{-m_3}}\rangle = \langle{\cal O}^{r_1~+}_{j_1m_1}({\cal O}^{r_2~+}_{j_2m_2}{\cal O}^{r_3~-}_{j_3~{-m_3}}) \rangle
u\left( \theta,t\right) =a\left( t\right) \cos\theta+b\left( t\right)\sin\theta+\int_{0}^{\theta}\frac{1}{k\left( \sigma,t\right)}\sin\left(\theta-\sigma\right) d\sigma,\ \ \left( \theta,t\right) \in S^{1}\times\lbrack0,\infty),
\begin{aligned}fmac(\tau) = \alpha(\tau) \circ K_0(\psi(\tau)) &= \alpha({\tilde \tau}) \circ K_0( W \circ \psi(\tau)) &= \alpha({\tilde \tau}) \circ K_0(\psi({\tilde \tau}) \circ W) &= \alpha({\tilde \tau}) \circ K_0(\psi({\tilde \tau}) \circ K_0( W) &= \alpha({\tilde \tau}) \circ K_0(\psi({\tilde \tau}) = fmac({\tilde \tau}).\end{aligned}
\psi^{\prime \prime}+ [ 3{\cal H}- 2 \frac{\varphi^{\prime \prime}}{\varphi^{\prime}}] \psi^{\prime}+ [ 4{\cal H}^{\prime}- 4{\cal H}\frac{\varphi^{\prime \prime}}{\varphi^{\prime}}] \psi - \partial_{\alpha}\partial^{\alpha}\psi = 0 .
Q_RQ_R(p\otimes b) =Q_R\bigl((\operatorname{id}\otimes\operatorname{id}\otimes\varphi)(\Delta_{13}(a^*)(1\otimes E)(1\otimes b\otimes x))\bigr) &=Q_R\bigl((\operatorname{id}\otimes\operatorname{id}\otimes\varphi)(\Delta_{13}(a^*)(1\otimes E)(1\otimes E)(1\otimes b\otimes x))\bigr) &=Q_R(p\otimes b),
\frac{R(t)}{R_0}=\left\{\frac{\Omega_M}{1-\Omega_{\Lambda}^2}\left[\Omega_{\Lambda}+\sin\left(\frac{3}{2}h_0^2\sqrt{1-\Omega_{\Lambda}^2}t\right)\right]\right\}^{\frac{1}{3}}-\left(\frac{\Omega_M\Omega_{\Lambda}}{1-\Omega_{\Lambda}^2}\right)^{\frac{1}{3}},
\lVert (H_{\eta}(v_g,\omega)+C_g+i)^{-1}- (H_{0}(v_g,\omega)+C_g+i)^{-1}\lVert &=\lVert (V_gH_{\eta}(v_g,\omega)V_g^*+C_g+i)^{-1}-(V_gH_{0}(v_g,\omega)V_g^*+C_g+i)^{-1}\lVert
\Gamma [ A , V ] = \int_{0}^{\infty}{\frac{\mathrm{d}T}{T}}\mathrm{e}^{- m^{2}T}\mathrm{t r}\; \;{\cal P}\int \! \mathrm{d}^{d}x_{0}\int \!{\cal D}y \exp \Bigl [ - \int_{0}^{T}\! \! \! \mathrm{d}\tau \bigl ({\frac{\dot{y}^{2}}{4}}\! + \! i g \dot{y}^{\mu}A_{\mu}\! + \! V \bigr ) \Bigr ] \, . \, \,
\|f\|_{M_{p,q}^{s,\alpha_1}}\sim\begin{cases}\big\|\{\|\Box_k^{\alpha_2}f\|_{M_{p,q}^{0,\alpha_1}}\}|~{l_{q}^{s,\alpha_2}}\big\|,~ ~\alpha_2<1 \big\|\{\|\Delta_jf\|_{M_{p,q}^{0,\alpha_1}}\}|~{l_{q}^{s,1}}\big\|,~ ~\alpha_2=1.\end{cases}
\begin{array}{r c l}{{L_{3}v_{. . , \lambda , l , m}}}&{{=}}&{{m v_{. . , \lambda , l , m},}}\\{{L_{\pm}v_{. . , \lambda , l , m}}}&{{=}}&{{\sqrt{l ( l + 1 ) - m ( m \pm 1 )}v_{. . , \lambda , l , m \pm 1}.}}\\ \end{array}
d \hat{s}^{2}= \epsilon^{2}d s^{2}= - \hat{n}^{2}\hat{c}^{2}d \hat{t}^{2}+ \hat{a}^{2}\hat{\gamma}_{i j}d \hat{x}^{i}d \hat{x}^{j}+ \hat{b}^{2}d \hat{y}^{2},
\Gamma(s,s^\prime,t) = \frac{1}{k_1!k_2!}\, \frac{\det\left[\gamma(s_{p1},s_{p'1}^\prime,t)\right]_{p,p'=1,\ldots,k_1}\det\left[\gamma(s_{q2},s_{q'2}^\prime,t)\right]_{q,q'=1,\ldots,k_2}}{B_{k_1k_2}(s^2)B_{k_1k_2}(s^{\prime 2})}\ ,
P\{\max_{0\le s\le t}\mathcal{T}(s)< \beta\ |\V(0) = c_1,\N(t) = 2\}= \frac{(c_2t+\beta)^2-\bigl(\frac{c_2}{c_1}\bigr)^2(c_1t-\beta)^2}{(c_1+c_2)^2t^2}=
K=\left(\begin{array}{ll}-\sigma_1\bar\beta_{11}\ \ -\sigma_1\bar\beta_{12}-\sigma_2\bar\beta_{12}\ \ -\sigma_2\bar\beta_{22}\end{array}\right) \equiv\left(\begin{array}{ll}K_{11}\ \K_{12}K_{21}\ \K_{22}\end{array}\right).
{\Pr}_r(S,K) = \frac{1}{|S||K|}\underset{r\in [s, K]}{\underset{s\in S}{\sum}}|C_K(s)| = \frac{1}{|S|}\underset{r\in [s, K]}{\underset{s\in S}{\sum}}\frac{1}{|[s, K]|}.
\lambda^{-\frac{n-2}{2}}\sigma{\sf A}_1 = - \eta^\frac{2}{1-q}|y|\to\infty,\ |\xi|\to0, - \eta^\frac{2}{1-q}{\sf B}_1 = K{\sf D}_J (T-t)^J \eta^\gamma |\xi|\to\infty,\ |z|\to0.
II^L=\left( \begin{array}{cc}\langle \nabla^{L}_{e_1}v_L,e_1)\rangle_{L}, \langle \nabla^{L}_{e_1}v_L,e_2)\rangle_{L}\langle \nabla^{L}_{e_2}v_L,e_1)\rangle_{L}, \langle \nabla^{L}_{e_2}v_L,e_2)\rangle_{L}\end{array}\right).
t(s_{k+1})-t(s_k)=\int_{s_k}^{s_{k+1}}t'(s) ds = -2 \int_{s_k}^{s_{k+1}}r(s)\cos(\theta(s)) ds &=\frac{3}{c}\int_{s_k}^{s_{k+1}}\theta'(s) \cos(\theta(s)) ds =\frac{-6}{c}.
\frac{\delta\ell}{\delta\xi}(\operatorname{Ad}_h\xi+\eta,\rho^*_h(a))=\operatorname{Ad}^*_{h^{-1}}\frac{\delta\ell}{\delta \xi}(\xi,a)\frac{\delta\ell}{\delta a}(\operatorname{Ad}_h\xi+\eta,\rho^*_h(a))=\rho_{h^{-1}}\frac{\delta \ell}{\delta a}(\xi,a).
(S(c_{f,v}^{\Lambda})\triangleleft Y)(X) = S(c_{f,v}^{\Lambda})(YX) = c_{f,v}^{\Lambda}(S(YX)) = f(S(X) \triangleright S(Y)\triangleright v) & = f(S(X) \triangleright (v \triangleleft Y)) = c_{f, v \triangleleft Y}^\Lambda (S(X)) = S(c_{f, v \triangleleft Y}^\Lambda) (X).
\mathcal{E}_{[2,5]}(t) :=\sum_{k'=2}^5\int_{\Sigma_t}|\partial_t \nabla^{k'}\Psi|^2+(1 + \Psi)^P |\nabla \nabla^{k'}\Psi|^2+ |\nabla^{k'}\Psi|^2\, d \underline{x}.
D_{s}\varphi (s, x) \, :=\, \frac{\,\partial \varphi \,}{\, \partial s\,}(s, x) \, , D_{i}\varphi (s, x)\, :=\, \frac{\partial \varphi}{\partial x_{i}}(s, x) \, , D_{i}^{2}\varphi (s, x) \, :=\, \frac{\,\partial^{2}\varphi \,}{\,\partial x_{i}^{2}\,}(s, x) \, .
R_{i j}\! \! - \! \frac{1}{2}R g_{i j}\! = \! 8 \pi G T_{i j}^{m}\! + \! ({\cal R}_{\mu \nu}\! - \! \frac{1}{2}{\cal R}{\cal G}_{\mu \nu}) Z_{, i}^{\mu}Z_{, j}^{\nu}\! + \! Q_{i j}+ S_{i j}
\sum_{i=0}^{p-1}\frac{y_{p-i}^{N}}{y_{p-i}\prod_{j\neq p-i}(y_{p-i}-y_j)}=\begin{cases}\frac{(-1)^{p-1}}{\prod_{j=1}^{p}y_j}, \mbox{if}\N=0; 0, \mbox{if}\ 1\leq N\leq p-1. \end{cases}
N \ge \lim_k \bigg| \frac{\partial_{x_i}u_k(d_k x+(x^k)')}{d_k x_n}\bigg| = \lim_k \bigg| \frac{\partial_{x_i}\tilde u_k(x)}{x_n}\bigg| =\bigg|\frac{\partial_{x_i}u_0(x)}{x_n}\bigg|
\int_{\{\Psi<-t_1\}}|\tilde{F}_1-(1-b_{t_0,B}(\Psi))F_{t_0}|^2_he^{v_{t_0,B}(\Psi)-\Psi}c(-v_{t_0,B}(\Psi)) \le (\int_{t_1}^{t_0+B}c(s)e^{-s}ds) \int_{M}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|F_{t_0}|^2_he^{-\Psi}<+\infty.
\lambda_1(D(T(a,b)))\geq\lambda_1(D(T(1,1))=8.2882, \lambda_2(D(T(a,b)))\geq\lambda_2(D(T(1,1))=-0.5578, \lambda_3(D(T(a,b)))\geq\lambda_3(D(T(1,1))=-0.7639, \lambda_4(D(T(a,b)))\geq\lambda_4(D(T(1,1))=-1.7304, \lambda_n(D(T(a,b)))\leq\lambda_5(D(T(1,1))=-5.2361.
-(N-2)S_{N-1}\alpha =d\left[ \omega^{A}\int_{R_{n}^{\prime}}\frac{(\delta^{\prime}\alpha^{\prime})\wedge \omega^{\prime \bar{A}}}{r_{12}^{N-2}}\right] +\delta \left[ \omega^{A}\int_{R_{n}^{\prime}}\frac{(d^{\prime}\alpha^{\prime})\wedge \omega^{\prime \bar{A}}}{r_{12}^{N-2}}\right] +\mathcal{F},
0 = \frac{1}{2}\frac{dG(t, \beta)}{dt}\Big|_{t=0}= \int \left(\langle \nabla u, \nabla \phi \rangle + \alpha_1 u \phi - \frac{\alpha_2}{2}u\phi \log u^2 - \alpha_2 u\phi + \beta u \phi \right) dm.
\sqrt n \left( \hat{\Sigma}_t^{n,m}-tQ\right) = \sqrt n \left( RV_t^n -tQ\right)+\left( \sqrt n \left( \hat{\Sigma}_t^{n,m}-\Pi_mtQ\Pi_m\right)-\sqrt n \left( RV_t^n -tQ\right)\right)+\sqrt n \left( \Pi_m tQ\Pi_m-tQ\right).
d_{n+1}^{\otimes}(H,G)= &\frac{1}{\vert H\vert^{n+1}\vert G \vert}\vert \lbrace{(x_{1},\ldots ,x_{n+1},y) :[x_{1},\ldots ,x_{n+1}] \otimes y=1_{H\otimes G}, x_{i}\in H, y \in G}\rbrace \vert.
\displaystyle{\frac{2}{\kappa}}V_{\mu \nu}^{^1{s c a l a r}}( k_{1}; p_{1}, p_{2}) = \, p_{_1 \mu}\, p_{_2 \nu}+ \, p_{_1 \nu}\, p_{_2 \mu}- p_{1}\cdot p_{2}\, \eta_{\mu \nu}-{m^{2}}\, \eta_{\mu \nu}+ 2 \, \xi \, \left( \, k_{_1 \mu}\, k_{_1 \nu}- \, k_{1}^{2}\, \eta_{\mu \nu}\right)
z_1(A) =_H\begin{cases}b_2(A)^{-1}s_2(A)a_1(A), \textrm{if}o(A)=1, a_2(A)^{-1}a_1(A), \textrm{if}o(A)=-1,\end{cases}\textrm{and}begin{align*}.5em] z_2(A) =_H \begin{cases}b_1(A)^{-1}s_1(A)a_2(A), \textrm{if}o(A)=1, b_1(A)^{-1}s_1(A)s_2(A)^{-1}b_2(A), \textrm{if}o(A)=-1.\end{cases}
\sum_{k=0}^\infty[4k+1]_{q^2}\frac{(aq^2;q^4)_k(q^2/a;q^4)_k(q^2;q^4)_k q^{-k}}{(aq^4;q^4)_k(q^4/a;q^4)_k(q^4;q^4)_k}=\frac{(q^2;q^4)_\infty(q^6;q^4)_\infty (aq;q^4)_\infty(q/a;q^4)_\infty}{(1-q^{-1})(q^3;q^4)_\infty^2 (aq^4;q^4)_\infty(q^4/a;q^4)_\infty}.
\lambda_{\deg(P_n)}(\mu_{[-1,1]},0) \leq \int_{-1}^{1}|P_n(x)|^2 w_{[-1,1]}(x) dx = \frac{1}{2}\int_{-\pi}^{\pi}|P_n(\cos t)|^2 w_\mathbb{T}(e^{it}) dt.
\Phi\big(t(\varepsilon)u, t(\varepsilon)\varepsilon v\big)-\Phi(u,0) = \big(\frac{1}{2}-\frac{1}{2^*(s_1)}\big)\lambda|u|_{2^*(s_1),s_1}^{2^*(s_1)}\big[2^*(s_1)\tilde{M}(v)\big]|\varepsilon|^2 \big(1+o(1)\big)+o(|\varepsilon|^2) = \frac{1}{2}\|v\|^2|\varepsilon|^2\big(1+o(1)\big) > 0\;\hbox{when $\varepsilon$ is small enough}.
\frac{R_s - n_s}{q_s}= \frac{\sum_{k=0}^s r\bigl( \frac{k-s}{\sqrt{s}}\bigr) \frac{(s\rho)^k}{k!}+ \frac{(s\rho)^s}{s!}\sum_{k=s+1}^\infty r\bigl( \frac{k-s}{\sqrt{s}}\bigr) \rho^{k-s}f\bigl( \frac{k-s}{\sqrt{s}}\bigr)}{\sum_{k=0}^s \frac{(s\rho)^k}{k!}+ \frac{(s\rho)^s}{s!}\sum_{k=s+1}^\infty \rho^{k-s}f\bigl( \frac{k-s}{\sqrt{s}}\bigr)}.
\frac{1}{\sqrt{nh_n}}\sum_{i=1}^n [w_n(x_i/h_n)-w_\infty(x_i/h_n)]u_i-\frac{1}{\sqrt{nh_n}}\sum_{i=1}^n [w_n^*(x_i/h_n)-w_\infty(x_i/h_n)]u_i.
\bigg(L-\frac{\partial}{\partial t}\bigg)\big[\Phi''(P_t f)\Gamma(P_t f)\big] =\frac{2\,\Gamma_2(\Phi'(P_t f))}{\Phi''(P_t f)}-\bigg(\frac1{\Phi''}\bigg)''(P_t f)\big[\Phi''(P_t f)\Gamma(P_t f)\big]^2.
D(\alpha,{\alpha '})={\sum\limits_{i,j}p_i''\langle\Phi_i|C_\alpha|\Psi_j\rangle\langle\Phi_i|C_{\alpha '}|\Psi_j\rangle^* p_j'\over\sum\limits_{i,j}p_i''|\langle\Phi_i|C_u|\Psi_j\rangle|^2 p_j'}.
{\cal C}={\cal C}^{- 1}= \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- 1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- 1}}\\ \end{matrix}\right) \quad , \quad{\cal C}\alpha^{\mu}{\cal C}^{- 1}= \overline{{{\alpha}}}^{\mu \, \, \ast}
\left(2(1-\alpha)+ \sum_i \gamma_i\right)^2 = (\sum_i \sqrt{\gamma_i(\gamma_i+4\mu)})^2 \leq \left(\sum_i \gamma_i\right) (4m\mu+\sum_i \gamma_i ) = 4m\mu \sum_i\gamma_i + (\sum_i\gamma_i )^2,
D(A_k)=\left\{x= (\mathfrak{f})\sum\limits_{n=1}^{\infty}c_{n}e_n \in \ell_{p,k}\left(\{e_n\}\right):\: \{f(n) \cdot c_{n}\}_{n=1}^{\infty}\in \ell_p(\Delta^k)\right\},
\mathrm{I )}\quad{\cal L}= \partial_{z}\varphi \partial_{\bar{z}}\varphi - 4 \varphi^{2}+ 2 \varphi^{4}\quad \mathrm{a n d}\quad \mathrm{I I )}\quad{\cal L}= \partial_{z}\varphi \partial_{\bar{z}}\varphi + 2 \varphi^{2}- 2 \varphi^{4}\, .