harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{4\, \pi^{(d-2)/2}}{\Gamma((d-2)/2)}\, \int_0^\alpha f(\eta)\, \sin^{d-2}\eta \, d\eta \, \frac{1}{a^{d-3}\, b^{d-3}}\int_0^{\min{(a,b)}}\frac{t^{d-3}\,dt}{\sqrt{a^2-t^2}\sqrt{b^2-t^2}}= F_Q - Q(\theta), 0 \leq \theta \leq \alpha,
R_{m n}{}^{p q}\rightarrow W_{m n}{}^{p q}- \frac{1 6}{d - 2}\delta_{[ m}{}^{[ p}( \bar{\psi}_{| r |}\Gamma^{| r |}\psi_{n ]}{}^{q ]}- \bar{\psi}^{| r |}\Gamma^{q ]}\psi_{n ] r}) \, .
\partial_{w_{0,a,b}}=\left(\frac{1}{x_{k_1}-x_{k_1+1}}-\frac{s_{k_1}}{x_{k_1}-x_{k_1+1}}\right)\ldots \left(\frac{1}{x_{k_{ab}}-x_{k_{ab}+1}}-\frac{s_{k_{ab}}}{x_{k_{ab}}-x_{k_{ab}+1}}\right).
\begin{cases}k(x-1)(x-\lambda)\prod_{i=1}^{\frac{D}{2}-1}(x-\alpha_i)^2-x\prod_{i=1}^{\frac{D}{2}-1}(x-\beta_i)^2=k\prod_{i=1}^{\frac{D}{2}}(x-\gamma_i)^2 k(x-1)(x-\lambda)\prod_{i=1}^{\frac{D}{2}-1}(x-\alpha_i)^2-\lambda x\prod_{i=1}^{\frac{D}{2}-1}(x-\beta_i)^2=k\prod_{i=1}^{\frac{D}{2}}(x-\delta_i)^2\end{cases}
R_0=\langle e_{11}, e_{22}, e_{33}, e_{44}\rangle, R_a=\langle e_{12},e_{34}\rangle, R_b=\langle e_{23}\rangle, &R_{a+b}=\langle e_{13}, e_{24}\rangle, R_{2a+b}=\langle e_{14}\rangle,R_{-a}=\langle e_{21}, e_{43}\rangle, R_{-b}=\langle e_{32}\rangle, R_{-a-b}=\langle e_{31}, e_{42}\rangle, R_{-2a-b}=\langle e_{41}\rangle.
\begin{array}{cc}\displaystyle{\{D_{i\alpha},D_{j\beta}\}=-2i\bar{\cal E}_{ij}\gamma^{A}_{\alpha\beta}\partial_{A}}~~~ ~~~\displaystyle{\{\tilde{D}_{\tilde{\imath}}^{\alpha},\tilde{D}_{\tilde{\jmath}}^{\beta}\}=-2i\bar{\tilde{\cal E}}_{\tilde{\imath}\tilde{\jmath}}\tilde{\gamma}^{A\alpha\beta}\partial_{A}}\end{array}
\begin{aligned}\sum_{s_1\ge s_2\ge\cdots\ge s_{k}\ge 0}\frac{q^{s_{1}^2+\cdots +s_k^2+s_{k-r+1}+\cdots +s_k}}{(q)_{s_1-s_2}\cdots (q)_{s_{k-1}-s_{k}}(q^2;q^2)_{s_{k}}}&=\frac{(q^{k+1-r},q^{k+1+r},q^{2k+2};q^{2k+2})_\infty}{(q;q)_\infty},0\le r\le k.\end{aligned}
L={1 \over 4}\int d^2 \theta\left({1 \over g^2(M)}+{b_0 \over 8 \pi^2}\log{M \over \mu}\right) W^a W^a + h.c. + \int d^4\theta \sum_i Z_i(\mu, M) \Phi_i^{\dagger}e^{2V_i}\Phi_i \, ,
K_f^2= \;\frac{n-1}{r-1}\left(\frac{(n-1)r-2n}{n}(\alpha_0-2\varepsilon)+(n+1)\sum_{k\ge 1}k(-nk+r)\alpha_k\right)-n\sum_{k\ge 1}\alpha_k+\varepsilon, \chi_f= \;\frac{n-1}{12(r-1)}\left(\frac{(2n-1)r-3n}{n}(\alpha_0-2\varepsilon)+(n+1)\sum_{k\ge 1}k(-nk+r)\alpha_k\right).
n\pi_n(x a_c^{(n)}) \sim_e n\frac{x^{r}}{r!}(p_n a_c^{(n)})^{r}&=x^{r}(a_c^{(n)})^{r-1}\frac{n p_n^{r}}{r!}a_c^{(n)}&=x^{r}\left(1-\frac{1}{r}\right)^{r-1}\frac{(r-1)!}{np_n^{r}}\frac{np_n^{r}}{r!}a_c^{(n)}&=\frac{1}{r}\left(1-\frac{1}{r}\right)^{r-1}x^{r}a_c^{(n)}.
\Lambda_1 (\L, I) := \inf \{\lambda , \exists \phi \in\mathcal{C}^2 (I)\cap \mathcal{C}^0 (\overline{I}), \phi >0 \hbox{in}I, \phi=0 \hbox{in}\partial I, \L \phi \leq \lambda \phi \hbox{in}I\}.
\begin{aligned}Z_{k,\,\Gamma}:= \left\{x \in X;\, x^{(k)}\in \Gamma_k\right\}Z_{k,\,\epsilon}:= \left\{x \in X;\, \epsilon|x^{(k)}| \le \sum_{j \in J_{\prec k}}|x^{(j)}|\right\}.\end{aligned}
r(1,1) =p(1)\cdot q(1)=1=r(\mathbf{1}) r(a,0) =p(a)\cdot q(0)=0=r(\mathbf{0}) r(a,b_1\vee b_2) =p(a)\cdot q(b_1\vee b_2)=p(a)\cdot (q(b_1)\vee q(b_2)) &=(p(a)\cdot q(b_1))\vee (q(a)\cdot p(b_2)) &=r(a,b_1)\vee r(a, b_2)=r((a,b_1)\vee(a, b_2)).
P\{T_1 < T_2\}\sim \frac{\nu_2 \lambda \Gamma(\lambda)}{\nu_1^{\lambda}}\cdot \frac{1}{M^{\lambda - 1}}= \frac{\nu_2 \Gamma(\lambda + 1)}{\nu_1^{\lambda}}\cdot \frac{1}{M^{\lambda - 1}}= \frac{\nu_2}{\nu_1}\cdot \frac{\Gamma(\lambda + 1)}{M_1^{\lambda - 1}}
\frac{1}{2 < \psi_{A B}| \psi_{A B}>}\left[ < \psi_{A B}| z_{A}\frac{\partial}{\partial z_{A}}\psi_{A B}> + < z_{A}\frac{\partial}{\partial z_{A}}\psi_{A B}| \psi_{A B}> \right] = \frac{1}{2}( 1 - \frac{N}{2})
\tilde{\Gamma}_{t}- \int_{0}^{t}h(t-z) \left< \tilde{\eta}_{z}, \Psi(\cdot,\overline{\gamma}(z)\right> dz - \int_{0}^{t}h(t-z) \left< u_{z},\frac{\partial \Psi}{\partial y}(\cdot,\overline{\gamma}(z))\right> \tilde{\Gamma}_{z}dz=0.
L_2^{\mu}(\beta b(X),\xi) = b(X) \big{(}\mathrm{sech}(\sqrt{\mu}|\xi|) -1 \big{)}\frac{1}{\mu |\xi|^2}+ \frac{1}{6\beta}\Big(\beta^3 b(X)^3 \int_0^1 \cosh(t\beta b(X) \sqrt{\mu}|\xi|) (1-t)^2\mathrm{d}t\Big) \mathrm{sech}(\sqrt{\mu}|\xi|),
\|\alpha\|_p^2=\int_{\Delta^n(0,2r_p)}|\overline\partial\chi_p|^2e^{-2p(\psi_x^\prime+ \widetilde\psi_x)}\,\frac{\omega^n}{n!}\leq\frac{\|\overline\partial\chi\|^2}{r_p^2}\,\left(\frac{\pi}{2}\right)^n\,\frac{(1+4C_1r_p^2)\exp\!\big(16C_2p\,r_p^3\big)}{p^n\lambda_1\ldots\lambda_n}\,\cdot
\Delta^\perp (\mathbf{m},\mathbf{a};k)=2^q \cdot [ 2^{k-1}+ 1] -t \,\,\, \,\,\, k\Delta^\perp(\mathbf{m},\mathbf{a};k) = C(\mathbf{m},\mathbf{a},\mathcal{O};k)),
L = \int d^{p}x{\cal L}, \; \;{\cal L}= 1 - \sqrt{1 - ( \partial_{i}\phi )^{2}}- \Sigma_{p}e \phi \delta ({\bf r}) , \; \; \Sigma_{p}= \frac{p \pi^{\frac{p}{2}}}{( \frac{p}{2}) !}
\mathbf{1}_{\alpha,\beta}(x)= \textup{Im}\int_0^L G\left(\frac{u}{L}\right)e^{2\pi i ux}f_{\alpha,\beta}(u)\frac{du}{u}+ O\left(\frac{\sin^2(\pi L(x-\alpha))}{(\pi L (x-\alpha))^2}+ \frac{\sin^2(\pi L(x-\beta))}{(\pi L (x-\beta))^2}\right).
\begin{aligned}\rho_1(\bar{\mathcal M}_1) \geq\rho\left(\frac{1}{N^2}\sum_{i, j =1}^N W_{ij}\otimes B_{ij}^{(1)}\right) &=\rho\left(\sum_{i, j = 1}^N q_{ij}W_{ij}\otimes A_i \otimes (e_j e_i^\top)\right). \end{aligned}
\Psi_{\omega}=\sum_{\alpha_{1},...,\alpha_{G}}A_{\omega,\alpha_{G}}^{G}A_{\alpha_{G},\alpha_{G-1}}^{G-1}\cdot\cdot\cdot A_{\alpha_{3},\alpha_{2}}^{2}A_{\alpha_{2},\alpha_{1}}^{1}x_{\alpha_{1}}
- \mathrm{T r}_{2 + 1}\ln ( -{\hat{\partial}}^{2}- \partial_{2}^{2}+ \kappa^{2}) \, + \, \frac{1}{2}\mathrm{T r}_{2 + 1}\ln ( -{\hat{\partial}}^{2}- \partial_{2}^{2}+ 4 \kappa^{2}) \; ,
\delta W_{\mu}^a = -i\overline{\eta}\gamma_{\mu}\Lambda^a, \;\;\;\delta \Lambda^a = -i{}^{\star}{W^{\lambda}}^a\gamma_{\lambda}\eta, \;\;\;\delta W^a = \overline{\eta}\chi_{\vec{W}}^a, \;\;\;\delta A = \overline{\eta}\chi_A, \;\;\; \delta B = \overline{\eta}\chi_B,
\mathrm{P}_o = \mathrm{P}\left(\min\{|g_{n,2}|^2, |g_{n,1}|^2, |h_n|^2\}<\xi_1, \forall n\in\{1, \cdots, N\}\right) =\left[1-\mathrm{P}\left(\min\{|g_{\pi(1),2}|^2, |g_{\pi(1),1}|^2, |h_{\pi(1)}|^2\}>\xi_1\right)\right]^N = [1-e^{-3\xi_1}]^N,
T_{\mu \nu}( k ) = i \int \frac{d^{2}p}{( 2 \pi )^{2}}\frac{\mathrm{t r}\{\gamma_{\mu}\left[ ( p \! \! \! \slash + k \! \! \! \slash ) + m \right] \gamma_{\nu}( p \! \! \! \slash + m ) \}}{\left[ ( p + k )^{2}- m^{2}\right] ( p^{2}- m^{2})}.
J_3 =\int_1^{m/2}\int_1^{n/2}\int_{1}^{\infty}\frac{dy_1dsdy_3}{(y_1^{\alpha_1}+(n-s)^{\alpha_2}+y_3^{\alpha_3})((y_1+k)^{\alpha_1}+s^{\alpha_2}+(m-y_3)^{\alpha_3})}&\leq C\int_1^{m/2}\int_1^{n/2}\int_{1}^{\infty}\frac{dy_1dsdy_3}{(y_1^{\alpha_1}+y_3^{\alpha_3})(s^{\alpha_2}+m^{\alpha_3})},
1)\,[[X,Y],m]_{L}=[[X,m]_{L},Y]_{R}+[X,[Y,m]_{L}]_{L}, & 2)\,[[m,Y]_{R},X]_{R}=[[m,X]_{R},Y]_{R}+[m,[Y,X]]_{R}, & 3)\,[[Y,m]_{L},X]_{R}=[[Y,X],m]_{L}+[Y,[m,X]_{R}]_{L},\quad\forall X,Y\in \mathfrak{g},\quad\forall m\in M,
\int\limits_{0}^{\infty}f_1(x \mid b_1, q_1) \, \ln \frac{f_1(x \mid b_1, q_1)}{f_2(x \mid b_2, q_2)}dx = \ln \frac{e^{q_1}\, b_1 \, q_1}{e^{q_2}\, b_2 \, q_2}+ e^{q_1}\left[ \left(\frac{b_2}{b_1}- 1 \right) \, \operatorname{Ei}(- q_1) + \frac{q_2}{q_1^{\frac{b_2}{b_1}}}\, \Gamma \left(\frac{b_2}{b_1}+1, q_1 \right) \right]- (q_1 + 1).
h_1'+\left\{\frac{n-2-(m/\tilde k)}{r}-(1-m)\frac{g'}{g}+\frac{\tilde\beta}{m}r^{\frac{n-2-nm}{m}-1}g^{1-m}\right\}h_1=\left(n-2-\frac{m}{\tilde k}\right)\frac{g}{r}\geq0
\tilde{f}(\mathbf{n}_{1}) =\int_{S^{2}}\mathbb{U}_{\vec{c},\frac{1}{\vec{c}}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2})f(\mathbf{n}_{2})d\mathbf{n}_{2}\ ,{f}(\mathbf{n}_{1}) =\int_{S^{2}}\mathbb{U}_{\frac{1}{\vec{c}},\vec{c}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2})\tilde{f}(\mathbf{n}_{2})d\mathbf{n}_{2}
=\frac{(a_{1}-a_{2})pq}{a_{1}pq-Q(q-p)}\left(\frac{Q(q-p)-a_{2}pq}{a_{1}pq-Q(q-p)}\right)^{\frac{a_{2}pq-Q(q-p)}{(a_{1}-a_{2})p^{2}}}\|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}\phi\|_{L^{p}(\mathbb{G})}^{p-q}
\mid{\bf \cdot}\, \rangle = \; \mid p\, \rangle^0, \; \; \; \; \; \;P = \; \mid p \, \rangle^{0 \; 0}\langle \, p \mid, \; \; \; \; \; \; \mid \Psi_P \, \rangle = \; \mid p \, \rangle^{0 \; 0}\langle \, p \mid p \, \rangle.
\cos ( \beta ({\frac{\delta}{\delta J_{r}}}+ \Phi ) ) \exp \{{\frac{1}{2}}J f^{- 1}J \}= \exp \{-{\frac{\beta^{2}}{2}}f_{r r}^{- 1}\}\cos ( \beta ( \int d^{2}r^{\prime}f_{r^{\prime}r}^{- 1}J_{r^{\prime}}+ \Phi ) ) \exp \{{\frac{1}{2}}J f^{- 1}J \}
G_1(t) =\idotsint\limits_{t_1,\dots,t_{j}\ge 0}G(t)\Phi\Bigl(\sum_{i=1}^{j}t_i+t+\sum_{i=j+2}^ru_i\Bigr)\Bigl(\prod_{i=1}^{j}G(t_i)dt_i\Bigr)\Bigl(\prod_{i=j+2}^r G(u_i)\Bigr), G_2(t) =G(t)\Bigl(\int_{t\ge 0}G(t')dt'\Bigr)^j\Bigl(\prod_{i=j+2}^r G(u_i)\Bigr),
\frac 1 8 \theta ( A - B ) \left\{\Delta^{2}( 0 ) ( 3 + z^{2}) + \Delta ( A , B ) \Delta ( 0 ) ( 1 - z^{2}) \right\}U^{- 1}\partial_{\mu}^{2}\partial_{\rho}U U^{- 1}\partial_{\rho}U U^{- 1}.
\mathcal{C}\sim a^{4-h}\, \sum_{n}\, \frac{1}{\left( 1 + \eta \, \lambda_{n}^{(3)}\right)}\, \langle \overset{1}{\mathbb{P}}\left(N\left(e_{n}^{(3)}\right)\right) , e^{(1)}_{n_{0}}\rangle \langle \mathcal{P}(\cdot,0), e^{(1)}_{n_0}\rangle \, \int_{B}e^{(3)}_{n}(x) \, dx = \mathcal{O}\left( a^{6-h}\right).
\int \left( \sum_{n \in I}\frac{w_n \mathbf{1}_{U(x_{n}, t_n r_{n})}}{\mu U(x_{n}, t_n r_{n})}\right) h \d\mu =\sum_{n \in I}\frac{w_n}{\mu U(x_{n}, t_n r_{n})}\int h \mathbf{1}_{U(x_{n}, t_n r_{n})}\d\mu
\left. \left( \overline{{g}}^{a 1}\partial_{a}X_{\mu}\mathcal{P}^{2}+ \frac{1}{3}\Pi^{( U ) a 1}\mathcal{P}^{\nu}\partial_{a}X^{\lambda}A_{\mu \nu \lambda}\right) \right|_{\sigma_{1}= 0 , \pi}\approx 0
\int_{\tau_{j-\ell+1}}^{\tau_j}G_{j - \ell}(v) \: dv = \sum_{m = 1}^{\ell - 1}\bigg( \int_{\tau_{j-m}}^{\tau_{j-m+1}}G_{j-m}(v) \: dv - s (\ell - m)(\tau_{j-m+1}- \tau_{j-m}) \bigg) &\leq \bigg( \sum_{m=1}^{\ell - 1}\int_{\tau_{j-m}}^{\tau_{j-m+1}}G_{j-m}(v) \: dv \bigg) - \frac{s \ell (\ell - 1) a_N}{6k_N},
\begin{array}{rcl}\overline{D_m(k_1+p)}= \displaystyle D_m(k_1)\left[-2k_1\cdot pD_m(k_1)\right. & \displaystyle \left.-p^2D_m(k_1)+4(k_1\cdot p)^2D_m^2(k_1)+O(p^3)\right].\end{array}
\begin{aligned}R := \left(\frac{2}{\gamma\alpha}\frac{D}{u(x)+ m}\right)^{1/\alpha}D =e^{rt}\| u_0 \|_{L^{s+1}}L^\frac{s}{2(s+1)}:= \min \left\{e^{rt}\|u_0\|_{L^{s+1}}, c_{\alpha}\right\}.\end{aligned}
Q(z,\overline{z})=\left(q_{1}(z,\overline{z}),q_{2}(z,\overline{z}),\dots,q_{N}(z,\overline{z})\right), \mbox{for:}q_{l}(z,\overline{z})=z_{l}\overline{z}_{\tau\left(l\right)}+\lambda_{l}\left(z_{l}z_{\sigma\left(l\right)}+\overline{z}_{l}\overline{z}_{\sigma\left(l\right)}\right),
\left[ \widehat{{\mathcal J}}_{\ell/r}^0 \left( t, \underline{x}\right),~\widehat{{\mathcal J}}_{\ell/r}^0 \left( t, \underline{y}\right) \right] \;=\;\pm~i~\frac{q^2}{4\pi^2}~\left( \underline{B}\left( \underline{x}, t\right) \,\cdot\, \underline{\nabla}\right)\; \delta \left( \underline{x}-\underline{y}\right)~,
\psi_k(z) = \rho\sum_{m=1}^N \sum_{m_1,m_2 \in \mathbb Z}\nolimits^* \left(W^{(m)}_{m_1,m_2}\psi_m\right)(z)+1,\;|z -a_k| \leq r\;(k=1,2,\ldots, N),
x=z\,\mbox{e}^{\frac{\theta}{M+1}}\,, \; \bar{x}= \bar{z}\,\mbox{e}^{-\frac{\theta}{M+1}}\,, \; E= s^{nM}\mbox{e}^{\frac{n\theta M}{M+1}}\,, \; \bar{E}= s^{nM}\mbox{e}^{-\frac{n\theta M}{M+1}}\,.
\hat{s}ec(U, V) - \lambda^2 \mid \nabla (f + \log \lambda )\mid^2 = sec(U, V) - \frac{\lambda^4}{4}\mid \nabla_{\mathcal{H}}\frac{1}{\lambda^2}\mid^2 & - \frac{\lambda^2}{2}\left( g(T_U U, \nabla \frac{1}{\lambda^2}) + g(T_V V, \nabla \frac{1}{\lambda^2}) \right).
\begin{aligned}g_{1}=(v^{6}+1)x_{1,1}^{2}x_{1,2}^{2}+(v^{6}+1)x_{1,1}x_{1,2}(x_{2,1}x_{2,2}+x_{2,1}x_{2,3}+x_{2,2}x_{2,3}) \ \ \ \ \ \ -v^{3}(x_{1,1}+x_{1,2})(x_{1,1}x_{1,2}x_{2,1}+x_{1,1}x_{1,2}x_{2,2}+x_{1,1}x_{1,2}x_{2,3}+x_{2,1}x_{2,2}x_{2,3}).\end{aligned}
I_\Gamma(u)=\frac 12 \int_{\Omega_\Gamma}(|\nabla u|^{2}+|u|^{2})dx -\frac{1}{2p}\int_{\Omega_\Gamma}\Big(\int_{\Omega_\Gamma}\frac{|u|^p}{|x-y|^{\mu}}dy\Big)|u|^{p}dx
\langle f, \pi(g) K_z\rangle = \langle \pi(g^{-1})f, K_z \rangle = (\pi(g^{-1})f)(z) = j_\lambda(g^{-1},z)f(g^{-1}\cdot z) = \langle f, \overline{j_\lambda(g^{-1},z)}K_{g^{-1}\cdot z}\rangle.
\mathcal{L}_{\star}^{full}\left( t \right) \, \varrho= \mathcal{L}_{\star}^h \, \varrho+\left[ \alpha \left( t \right) a^\dagger - \overline{\alpha \left( t \right)}a + \overline{\beta \left( t \right)}\sigma^{-}- \beta \left( t \right) \sigma^+ ,\varrho \right]
G_{\parallel}= \frac{V_3N u^4}{2\pi^2 \lambda^2}\left ( \sqrt{1-\frac{2m}{u^4}-\frac{\lambda \Omega^2(u^2+l^2)}{u^4}+\frac{2\sqrt{2m\lambda}l\Omega}{u^4}}-1 -\frac{\sqrt{2m\lambda}l\Omega}{u^4}\right).
D=\int_{-Q_1}^{Q_1}{\rho_1(k)dk},\;\;\;D_m=\int_{-B_1}^{B_1}{\sigma_1(\Lambda)d\Lambda},\;\;\;E_1/N=D^{-1}\int_{-Q_1}^{Q_1}{k^2\rho_1(k)dk}.
\left(s^{\prime}z_{1}s^{\prime}z_{2}\right)^{-1}z_{2}\left(s^{\prime}z_{1}s^{\prime}\right)=\left[z_{2}^{-1},s^{\prime}z_{1}s^{\prime}\right]\mbox{and}z_{2}\left(s^{\prime}z_{1}s^{\prime}z_{2}\right)^{-1}\left(s^{\prime}z_{1}s^{\prime}\right)=e.
\sum_{n=1}^{\infty}n^2\Vert f_n^d(t) \Vert_{L^1(S_2)}\Vert I_{1n}\Vert_{L^1(S_2)}\leq \sum_{n=1}^{\infty}\sum_{q=1}^{\infty}n^2\vert a_{qn}\vert\Vert f_n^d \Vert_{L^1(S_2)}\Vert f_q^d \Vert_{L^1(S_2)}\leq \sum_{n=1}^{\infty}\sum_{q=1}^{\infty}\dfrac{\vert a_{qn}\vert}{q^2}n^2\Vert f_n^d \Vert_{L^1(S_2)}q^2 \Vert f_q^d \Vert_{L^1(S_2)}
{\cal L}^{( 1 )}= - \partial_{\mu}\overline{\psi}_{1}\gamma_{\mu \nu}\partial_{\nu}\psi_{1}- \partial_{\mu}\overline{\psi}_{2}\gamma_{\mu \nu}\partial_{\nu}\psi_{2}- m^{2}\overline{\psi}_{1}\psi_{1}+ m^{2}\overline{\psi}_{2}\psi_{2}\quad .
|\frac{a_{{\eta}'}}{a_B}{\eta}-\frac{a_B}{a_{{\eta}'}}{\eta}'|^{2-n}= \sum_{N=0}^{N_0}\sum_{\alpha=0}^{N}\frac{1}{\lambda_N}Y_{\alpha}^N(\frac{a_{{\eta}'}}{a_B}{\eta})Y_{\alpha}^N(\frac{a_B}{a_{{\eta}'}}{\eta}')
G_{\alpha \beta}= \left( \begin{array}{c c c c c}{- ( 1 - \tilde{v}^{2})}&{0}&{0}&{0}&{- \tilde{v}}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{- \tilde{v}}&{0}&{0}&{0}&{1}\\ \end{array}\right) ,
\Delta_{l}^{(m+n)}:=\Delta^{(m+n-1,1)}_{\texttt{b}; l}= \left( 2(m+n)+ u_{q_0}\bigl( \xi^{(m+n)}_l \bigr)+ u_{q_1}\bigl( \xi^{(m+n)}_l \bigr) \right)^{-1}
S_n^k(x,b_1,b_2,h):=\left\{\forall i\in I_n:\omega_i^0\leq\frac{1}{k}\right\}\cap\left\{\begin{matrix}V(x)=\min_{y\in I_n}V(y) begin{align*}1mm]V(x)-V(x-b_1\ln n)\geq h\ln n begin{align*}1mm]V(x-b_2\ln n)-V(x)\geq h\ln n \end{matrix}\right\}
\hat{h}(\frac{w}{2\pi},u) = \begin{cases}0 \textrm{if}u \le \gamma \frac{1}{jw}\Big(e^{-jwu}- e^{jw \big(1 \!-\! \frac{1+\gamma}{\gamma}u \big)}\Big) \textrm{if}u > \gamma \end{cases}.
\widetilde z = (r^2 t + t_0, r^3 x + x_0 - r^2 t v_0, r v + v_0),\widetilde h (z) = h (\widetilde z), Y = \partial_t - v \cdot D_x, \widetilde P = \partial_t - v \cdot D_x - a^{ij}(\widetilde z) D_{v_i v_j}.
\lim\phi_{\sigma(n)}(u)=\phi_0(u), \lim\psi_{\sigma(n)}(u)=\psi_0(u), \lim\xi_{\sigma(n)}(u)=\xi_0(u), &\lim\phi_{\sigma(n)}(z)=\phi_0(z), \lim\psi_{\sigma(n)}(z)=\psi_0(z), \lim\xi_{\sigma(n)}(z)=\xi_0(z).
\|H(\xi_{1})-H(\xi_{0})\|_{p}\leq \int_{0}^{\xi_{0}-2\delta}\big\|(F(\xi_{1}-s)-F(\xi_{0}-s))G(s)\big\|_{p}\, \textnormal{d}s + \sum_{j=0}^{1}\int_{\xi_{0}-2\delta}^{\xi_{j}}\|F(\xi_{j}-s)G(s)\|_{p}\, \textnormal{d}s.
\tilde{A}\left(W',\overline{W'}\right)\otimes\left(\left(\overline{I_{q'^{2}}+R\left(W'\right)}\right)^{t}\otimes W'-\left(\overline{\left(\overline{I_{q'^{2}}+R\left(W'\right)}\right)^{t}\otimes W'}\right)^{t}\right)=2\sqrt{-1}\left(V\otimes Z'\right)\overline{\left(V\otimes Z'\right)}^{t},
\Psi(\varphi)=\Psi(L,D,R)=\sum\limits_{\alpha,\beta\in\Omega}\sum\limits^{N(\alpha)}_{m,n=1}\sum\limits^{N(\beta)}_{k,l=1}D^{\alpha}_{mn}(L)f^{\alpha\beta}_{nk|ml}(D)D^{\beta}_{kl}\left(R^{-1}\right).
q^{\frac{1}{2}}X_1X_4= q^2X(-1,3)+(q+q^2)hX(-1,2)+(1+qh^2+q^2)X(-1,1)+(1+q)hX(-1,0) &+X(-1,-1) +qhX(0,1)+q^{\frac{1}{2}}h^2+hX(0,-1)+X(1,-1)
|E[g(X(t))] - E[g(X(s))]| &\leq \int_s^t \int_U \int_AE |g(X(r) + f(r,X(r), u, a, m(r))) -g(X(r))| \rho_r(da)\nu(du)dr &\leq \int_s^t \int_U \int_A 2|g|_{\infty}\rho_r(da)\nu(du)dr = 2\nu(U) |g|_{\infty}(t-s)
\Phi^{a}= \hat{x}^{a}\, \phi ( r ) , \qquad B^{a}= - \frac{\epsilon_{a b c}\, x^{b}\, d x^{c}}{e \, r^{2}}\, [ \, 1 - u ( r ) \, ] - \hat{x}^{a}v ( r ) \, d t .
\Psi(2,\{\alpha_i\}_{i=1}^2,K,\delta) :=\Phi(\alpha_1,\alpha_2,K,\delta) \Psi(m+1,\{\alpha_i\}_{i=1}^{m+1},K,\delta) :=\Phi(\alpha_1\star\ldots\star\alpha_m,\alpha_{m+1},\rho\mapsto\max(\Psi(m,\{\alpha_i\}_{i=1}^m,K,\rho),K(\rho)) ,\delta)
\begin{tabular}{lll}$ v_1=(4,2,0,1) $ $ v_2=(0,0,-2,1)$ $v_3=(0,0,1,-1)$ $ v_4=(6,2,1,1) $ $ v_5=(6,-6,1,1) $ $v_6=(-6-6n,-6,1,1)$ $ v_7=(2n-6,2,1,1) $ $ v_8=(2n-4,2,0,1) $. \end{tabular}
g_n =\left(\frac{d}{dt}P_Nu_n+\nu AP_Nu_n+B_N(u_n,u_n)\right)-\left(\frac{d}{dt}P_Nu+\nu AP_Nu+B_N(u,u)\right) =B_N(P_Nu+Q_Nv_n,P_Nu+Q_Nv_n)-B_N(P_Nu+Q_Nu,P_Nu+Q_Nu) =B_N(Q_Nw_n,Q_Nw_N)+DB_N(u)Q_Nw_n,
F(\underline{k}, t,\mathbf{p}_{n+1}) = \left( \prod_{j=0}^{k}p_j^{}\right) \sum_{j=0}^{k}\frac{q_j^{t-k}}{1-q_j}\prod_{\substack{i=0 i \neq j}}^{k}\frac{1}{q_j^{}- q_i^{}}\,.
G_{u}:=c \left( R^{\frac{6}{r}}\left( \frac{1}{R^3}\int_{\mathcal{C}(R/2,R)}\vert \nabla u\vert^{\frac{p}{2}}dx\right)^{\frac{2}{p}},S_{u}:=R^{\frac{6}{r}}\left(\frac{1}{R^3}\int_{\mathcal{C}(R/2,R)}\vert u\vert^{p}dx\right)^{\frac{2}{p}}\right),
I_{\mu_1\cdots\mu_n}(0)\sim\mathcal{C}_{\left[1+\frac{n}{2}\right]}+p^2\mathcal{C}_{\left[\frac{n}{2}\right]}+\cdots+\frac{p^{2\left[\frac{n}{2}\right]}}{\left[\frac{n}{2}\right]!}\mathcal{C}_{1}+\mathrm{const.}
\nu_k\ =\ \Bigl[\,\frac{4\i}{\mu_k-\bar\mu_k-\mu_k^{-1}+\bar\mu_k^{-1}}\,\cdot\,\frac{\mu_k-\mu_k^{-1}-2\i}{\bar\mu_k-\bar\mu_k^{-1}+2\i}\,\Bigr]^{1/2}\quad.
\mathrm{\boldmath ~ \Psi ~ \u n b o l d m a t h}= \left( \begin{array}{c}{{\Psi_{+ 1}({\bf x})}}\\{{\Psi_{- 1}({\bf x})}}\\ \end{array}\right) \; , \; \; \Psi_{\zeta}({\bf x}) = \left( \begin{array}{c}{{\chi_{\zeta}({\bf x})}}\\{{\varphi_{\zeta}({\bf x})}}\\ \end{array}\right) \; ,
C_{11}=\frac{1}{2}(R_{1212}-2R_{1234}+R_{3434}), C_{12}=C_{21}=\frac{1}{2}(R_{1213}-R_{1334}-R_{1242}+R_{3442}), C_{13}=C_{31}=\frac{1}{2}(R_{1214}-R_{1434}-R_{1223}+R_{2334}), C_{22}=\frac{1}{2}(R_{1313}-2R_{1342}+R_{4242}), C_{23}=C_{32}=\frac{1}{2}(R_{1314}-R_{1442}-R_{1323}+R_{2342}), C_{33}=\frac{1}{2}(R_{1414}-2R_{1423}+R_{2323}).
f(b) = \begin{pmatrix}a_{11}a_{12}\cdots a_{1n}a_{21}a_{22}\cdots a_{2n}\vdots \vdots \ddots \vdots a_{n1}a_{m2}\cdots a_{nn}\end{pmatrix}\begin{pmatrix}b_1^{p^e}\vdots b_n^{p^e}\end{pmatrix}.
\|I_{\alpha}(|f|^{l})^{1/l}\|_{\mathcal{M}_{v_{1}}^{u_{1}}}\lesssim\|f^{l}\|_{\mathcal{M}_{q_{1}/l}^{p_{1}/l}}^{\frac{1}{l}}=\|f\|_{\mathcal{M}_{q_{1}}^{p_{1}}}.
a_{i}^{I}({\bf r}, t ) = \epsilon^{i j}\nabla_{j}\frac{1}{\kappa_{I}}\sum_{p}q_{p}^{I}\int \! d^{2}\!{\bf r}^{\prime}\, G ({\bf r}-{\bf r}^{\prime}) \rho_{p}({\bf r}^{\prime}, t ) ,
\mathbb E_x\left[e^{-p \int_{0}^{e(q)}\rm{\bf{1}}_{\{X_s<b\}}ds}\rm{\bf{1}}_{\{X_{e(q)}\in dy\}}\right]=-qW_{x-b}^{(q,p)}(x-y)dy &+\frac{q}{p}(\Phi(p+q)-\Phi(q))H^{(p+q,-p)}(x-b)H^{(q,p)}(b-y)dy,
t = (t_1, t_2, t_3, t_4, t_{12}, \dotsc, t_{1234}) = \left(0,0,0,0,0,-1,0,-1,\frac{2}{3},-1,\eta_4,\frac{4}{3}\eta_4,\eta_4, -\frac{2}{3}\eta_4,\frac{1}{3}\right),
q_{V}( T_{\mu}x - T_{\mu}y) = \langle T_{\mu}x - T_{\mu}y , x^{*}_{V}\rangle= \mu_{t}\langle T_{t}x - T_{t}y , x^{*}_{V}\rangle \leq \| \mu\| \sup_{t}| \langle T_{t}x - T_{t}y , x^{*}_{V}\rangle | \leq \sup_{t}q_{V}( T_{t}x - T_{t}y) \leq q_{V}(x-y),
\frac{n_{1}\log (2)\, (1+2^{n_{2}-n_{1}}+\cdots+2^{n_{\tau(N)}-n_{1}})}{\log (2)\, (n_{1}+n_{2}2^{n_{2}-n_{1}}+\cdots+n_{\tau(N)}2^{n_{\tau(N)}-n_{1}})}=\frac{1+2^{n_{2}-n_{1}}+\cdots+2^{n_{\tau(N)}-n_{1}}}{1+\big(\frac{n_{2}}{n_{1}}\big)\,2^{n_{2}-n_{1}}+\cdots+\big(\frac{n_{\tau(N)}}{n_{1}}\big)\,2^{n_{\tau(N)}-n_{1}}}=:\frac{c_{N}}{d_{N}}.
W^C_{58}= 1 +(319 - 24 \beta - 2 \gamma)y^{10}+(3132 + 152 \beta + 2\gamma)y^{12}+ \cdots, W^S_{58}= \beta y^5+ \gamma y^9+ (24128 - 54 \beta - 10 \gamma )y^{13}&+ (1469952 + 320 \beta + 45 \gamma )y^{17}+ \cdots,
\mathrm{M}= \begin{bmatrix}2k\nabla^2 f(x) - 2A^TA -q_1\mathrm{I}(A\nabla^2f(x))^T A\nabla^2f(x) 2AA^T-q_1\mathrm{I}\end{bmatrix}\geq \begin{bmatrix}2k\nabla^2 f(x) - 2A^TA -q_1\mathrm{I}(A\nabla^2f(x))^T A\nabla^2f(x) AA^T \end{bmatrix}.
\sum_{1\leq i \leq 3}\eta_1^{ij}(x,y)y_i \, =\, 0\quad 1\leq j\leq 3, \sum_{1\leq j \leq 3}\eta_1^{ij}(x,y)y_j \, =\, 0\quad 1\leq i\leq 3.
\delta(k^2-m^2) \, \delta(q^2-m^2) \big|_{q = (\lambda, \vec{k})}= \delta \big( \omega^2 - |\vec{k}|^2 -m^2 \big) \:\delta \big( \lambda^2 - |\vec{k}|^2 -m^2 \big) &= \delta(\lambda^2 - \omega^2)\: \delta \big( \omega^2 - |\vec{k}|^2 -m^2 \big) \:.
\int_{0}^{\infty}\frac{d \rho}{\rho^{n + 1}}\exp \left( - \rho - \frac{1}{\rho}a^{2}\right) = \left( - \frac{1}{2 a}\frac{d}{d a}\right)^{n}[ 2 K_{0}( 2 a ) ] \, .
\sum_{k=0}^{\infty}\frac{1}{(3k)!}=\frac{1}{0!}+\frac{1}{3!}+\frac{1}{6!}+\frac{1}{9!}+\frac{1}{12!}+\cdot\cdot\cdot=\frac{1}{3}(e+\frac{2}{\sqrt{e}}cos\frac{\sqrt{3}}{2})
[\begin{matrix}X_{B}\\Y_{B}\\Z_{B}\end{matrix}]=[\begin{matrix}c_{x}\\c_{y}\\c_{z}\end{matrix}]+(1+s\times10^{-6})[\begin{matrix}1&-r_{z}&r_{y}\\r_{z}&1&-r_{x}\\ -r_{y}&r_{x}&1\end{matrix}][\begin{matrix}X_{A}\\Y_{A}\\Z_{A}\end{matrix}].
Pr(f,\sigma)=Pr(f,\sigma\nu), Pr_D(f,\sigma)=Pr_D(f,\sigma\nu), Pr(f,\sigma,\{k_j\})=Pr(f,\sigma\nu,\{k_j'\}), Pr(f,\sigma,\{k_j\},L,\{R_i\})=Pr(f,\sigma\nu,\{k_j'\},L,\{R_i\}).
\partial_x \Gamma_i^{*}(t,\lambda) = \Phi_i (t) \psi_{BA}^{*}(t,\lambda)\;\; ; \; \; -\partial_x \psi_{BA}^{*}(t,\lambda) - \sum_{i=1}^{m}\Psi_i \Gamma_i^{*}(t,\lambda) = \lambda \psi_{BA}^{*}(t,\lambda) \, .
\frac{\mu^{s + 1}}{2}\left( \frac{2 \pi}{L}\right)^{- s}\frac{1}{\Gamma \left( \frac{s}{2}\right)}\int_{0}^{\infty}d t \, t^{( \frac{s}{2}- 1 )}e^{- ( \frac{m L}{2 \pi})^{2}t}\Theta ( 0 , \frac{t}{\pi}\rfloor_{s = - 1}\, ,
\max_{1\leq k\leq n}\left\vert S_{k}\right\vert \leq\max_{1\leq k\leq n}|E^{k}S_{k}|+\max_{2\leq k\leq n}\left\vert \sum\nolimits_{i=1}^{k-1}P^{i}(S_{i})\right\vert ,
\begin{alignedat}{1}Y_{0}= X_{0}+X_{1}+X_{2}-X_{3}-X_{4}-X_{5}, Y_{1}= X_{1}-X_{5}, Y_{2}= X_{3}-X_{2}, Y_{3}= X_{4}-X_{2}, Y_{4}= X_{0}-X_{5}, Y_{5}= X_{0}X_{1}+X_{0}X_{2}+X_{1}X_{2}-X_{3}X_{4}-X_{3}X_{5}-X_{4}X_{5},\end{alignedat}
\max_{\mathbf{i}}|\hat{Y}_{\mathbf{i}}-Y_{\mathbf{i}}|=\mathrm{O}_{p}(\epsilon_{\mathbf{n}}),\max_{\mathbf{i}}\|\hat{X}_{\mathbf{i}}-X_{\mathbf{i}}\|=\mathrm{O}_{p}(\epsilon_{\mathbf{n}}),\max_{\mathbf{i}}\|\hat{U}_{\mathbf{i}}-U_{\mathbf{i}}\|=\mathrm{O}_{p}(\epsilon_{\mathbf{n}}).
\tilde{\rho}( \tilde{\theta}_{1}^{\alpha}, x) = P_{\tilde{\pi}^{GEL}}(\theta_1 < \tilde{\theta}^{\alpha}_1| x) = \frac{\int^{\tilde{\theta}^{\alpha}_1}\ldots \int e^{- \frac{1}{2}\tilde{l}_\gamma^{GEL}(\theta) + \xi(\theta)}d\theta_d \ldots d\theta_1}{\int \ldots \int e^{- \frac{1}{2}\tilde{l}_\gamma^{GEL}(\theta) + \xi(\theta)}d\theta_d \ldots d\theta_1}= \alpha \,.
\hat{\tau}= \left( -\frac{\rho_N^{\frac{1}{4}}}{8\sqrt{2}C_{\beta}n}\ln \left(\frac{\sqrt{\rho_N}}{D_{\beta}}\right) + \sqrt{\left(\frac{\rho_N^{\frac{1}{4}}}{8 \sqrt{2}C_{\beta}n}\ln \left(\frac{\sqrt{\rho_N}}{D_{\beta}}\right) \right)^2+\frac{G_{\beta}}{C_{\beta}}\rho_N^{\frac{1}{4}}}\right)^4.
{\cal L'}^{M'}_h(V_h,\phi) = \frac{1}{16}\int d^2 \theta \frac{1}{g'^2_h}W^a(V_h)W^a(V_h) + \mbox{h.c.}+ \int d^4 \theta \sum_i \phi^{\dagger}e^{2 V_h^i}\phi
\eta \delta_n(\beta) = \sum_{w \in W_{\rho}^{m_0}}\dfrac{(f^{\tilde{\phi}^{-(n+1)}})'(v_{n+1}+w)}{f^{\tilde{\phi}^{-(n+1)}}(v_{n+1}+w)}= \operatorname{T}_{n+1,n}(\delta_{n+1}(\beta)).
\Phi(r)=\frac{q}{4\pi\epsilon r}\sum_{k=0}^{\infty}(\frac{a}{r})^{k}P_{k}(cos\theta)\equiv\frac{1}{4\pi\epsilon}\sum_{k=0}^{\infty}M_{k}(\frac{1}{r^{k+1}})P_{k}(cos\theta)
\frac{C_{3,n}}{(k-5)!}= - [2(n-1) a_k + b_{k-1}] m_{(3,\dot{0})}- [2(n-1) a_{k-1}+ b_{k-2}] m_{(2,\dot{0})}- [2(n-1) a_{k-2}+ b_{k-3}] m_{(1,\dot{0})}- 2a_k m_{(2,1,\dot{0})}- 2a_{k-1}m_{(1^2,\dot{0})}- \frac{6n(n-1)}{k(k-1)}(k-2) a_{k-3}- \frac{3n}{k-1}b_{k-4},
\hom(G\cup G,\ell(F^B)) =\hom((G\cup G)^{\circ},F^B)= &=\hom((G\cup G)^{\circ}\times B,F)= &=\hom(G^{\circ}\times B\cup G^{\circ}\times B,F)\geq &\geq \hom(H^{\circ}\times B,F)= &=\hom(H^{\circ},F^B)= &=\hom(H,\ell(F^B)).
\tau \rightarrow \frac{a^{\prime}\tau + b^{\prime}}{c^{\prime}\tau + d^{\prime}}\; \; , \; \; \; \; \; \Gamma_{\gamma}^{\prime}= \left( \begin{array}{c c}{a^{\prime}}{b^{\prime}}{c^{\prime}}{d^{\prime}}\end{array}\right)