harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Omega_1 = \Big\{\min_{[0,s_0]}\tilde Y_s =0,\;\max_{[0,s_0]}\tilde Y_s < \frac{(2a+1)^2d^2}{16}\Big\},\Omega_2 = \Big\{\max_{[0,s_0]}|\gamma_s| < \frac{(2a-1)d}{4}\Big\}, \Omega_3 = \Big\{\sigma > s_0, \; \min_{s\in[0,s_0],j\geq 3}|\tilde{X}^{j,N}_s-\bar{x}|> b d \Big\}.
{\cal L}_{\mathrm{e f f}}( \vec{E}= 0 , \vec{n}\parallel \vec{B}) = - [ \Pi ( m a ) + F_{2}( m a ) ] \frac{\vert B \vert^{\, 2}}{2}\equiv \frac{- 1}{\mu ( a m )}\frac{\vert B \vert^{\, 2}}{2}\, .
\begin{aligned}\int_K\int_{G_1}|x-y|^{-(d+\alpha)}|x|^{\alpha-d}|y|^{\alpha-d}dydx \leq \int_K\int_{G_1}\left(\frac{|y|}{|x-y|}\right )^{\alpha+d}|x|^{\alpha-d}|y|^{-2d}dydx.\end{aligned}
\tilde{S}_{Y_p}\cap[N_{Y_p}/2, N_{Y_p}-K]= \{\xi r_-+\xi r_+,\; (\xi-1)r_-+(\xi+1)r_+,\dots,\;r_-+(2\xi-1)r_+, (2\xi+1)r_-,2\xi r_-+r_+,\dots,\; (\xi+3)r_-+(\xi-2)r_+, \; (\xi+2)r_-+(\xi-1)r_+ \}.
\Phi_{\xi}^{- 1}( s , 0 ) \delta_{\mu}( s^{\prime}) \Phi_{\xi}( s , 0 ) = - i g \theta ( s - s^{\prime}) \Phi_{\xi}^{- 1}( s , 0 ) E_{\mu}[ \xi | s ] \Phi_{\xi}( s , 0 ) .
B_L =\frac{1}{\beta_{L}}\sum_{p}\sum_{n\in int(C_{p})}\sum_{(m,k)\in \partial C_{p}}|G^{C_{p}}(z_L;n,m)||G^{\Lambda_L}(z_L;k,n)| =\frac{1}{\beta_{L}}\sum_{p}\sum_{n\in int(C_{p})}\sum_{(m,k)\in \partial C_{p}}|G^{\Lambda_L}(z_L;k,n)|^s|G^{\Lambda_L}(z_L;k,n)|^{1-s}|G^{C_{p}}(z_L;n,m)|
{\frac{\delta}{\delta \phi^{a}( x )}}= \int d^{d}z \int d^{d}y{\frac{\delta \sigma ( z , y )}{\delta \phi^{a}( x )}}{\frac{\delta}{\delta \sigma ( z , y )}}= \int d^{d}z \phi^{* a}( z ){\frac{\delta}{\delta \sigma ( z , x )}}
\# X \equiv \sum_{(x_1,\dots,x_n) \in T_q^n}\prod_{j=1}^r (1-P_j(f_1(x_1),\dots,f_n(x_n))^{(q-1)q^n}) &\equiv \sum_{(i_1,\dots,i_r) \in \{0,1\}^r}\sum_{(x_1,\dots,x_n) \in T_q^n}\prod_{j=1}^r \left(-P_j(f_1(x_1),\dots,f_n(x_n))^{(q-1)q^n}\right)^{i_j}\!\!\!\!\!\!\!\pmod{q^n},
\|\xi_1 \xi_2\|_{L^p(\mathcal M)}\leq \|\xi_1 \|_{L^{p_1}(\mathcal M)}\| \xi_2\|_{L^{p_2}(\mathcal M)}, \textstyle \frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}
\mathbb{E}_{f_0,T}\|f_0-\hat{f}\|_2= \mathbb{E}_{f_0,T}\|f_0-\tilde{f}(\hat{j})\|_2 1_{\hat{j}>j^*}+ \mathbb{E}_{f_0,T}\|f_0-\tilde{f}(\hat{j})\|_2 1_{\hat{j}\leq j^*},
\frac{(q;q^2)_{n-1}}{(q^2;q^2)_{n-1}}=\frac{(q;q^2)_{n-1}}{(q;q)_{n-1}(-q;q)_{n-1}}\equiv \frac{-q[n]}{(-q;q)_{n-1}}\equiv -q[n]\pmod{\Phi_n(q)^2}.
\mathfrak{s}_1\in \begin{cases}\{1\}, \mathfrak{d}_1=2; [2^{-\frac{\mathfrak{k}_0}{\mathfrak{d}_1}}, \frac{1}{2}] \mathfrak{d}_1>2, \end{cases}\ \ 2^{\mathfrak{k}_1}=2^{\mathfrak{k}_0}(\mathfrak{s}_1)^{\mathfrak{d}_1}.
\overline{\overline{N}}(b)\overline{J}_{1}=\overline{\overline{N}}(b)\overline{J}_{4}=0 \overline{\overline{N}}(b)\overline{J}_{2}=c_{k}^{-1}\pi_{be_{k}}\overline{i}_{3}\theta_{2}\overline{\overline{N}}(b)\overline{J}_{3}=c_{k}^{-1}\pi_{be_{k}}\overline{j}\overline{\sigma}\theta_{3}
p^{\alpha_{m+1}}_{1}(m+1,t) =-\lambda I_t^{\alpha_{m+1}}\left(p^{\alpha_{m+1}}_{0}(m+1,t)-p^{\alpha_{m}}_{0}(m,t)\right)=0, p^{\alpha_{m+1}}_{2}(m+1,t) =-\lambda I_t^{\alpha_{m+1}}\left(p^{\alpha_{m+1}}_{1}(m+1,t)-p^{\alpha_{m}}_{1}(m,t)\right)=0.
w^1_1+w^1w^1_2-w^1_{22}-\frac{(w^2)^2}{z_2}-\frac1{z_2^3}=0, w^2_1+w^1w^2_2-w^2_{22}+\frac{w^1w^2}{z_2}+2\frac{w^2}{z_2^2}=0.
\widehat{\mathbb E}|Y_i|^{2+\delta}=\sup_{\lambda_{i-1}\in [\underline\lambda_{i-1},\overline\lambda_{i-1}]}\mathsf E\left|\lambda_{i-1}\xi_i/\sigma_i\right|^{2+\delta}=\overline\lambda_{i-1}^{2+\delta}\mathsf E|\xi_i/\sigma_i|^{2+\delta}\le M,
{\bf 70}\, \stackrel{Usp(4) \, \times \, SU(4) \, \times\, U(1)}{\longrightarrow}\, \left({\bf 1},{\bf 1},{\bf 1}+{\bar{\bf 1}}\right) \, \oplus\, \left({\bf 5},{\bf 6},{\bf 1}\right) \, \oplus \,\left({\bf 1},{\bf 6},{\bf 1}\right) \, \oplus \,\left({\bf 4},{\bf 4},{\bf 1}\right) \, \oplus \,\left({\bf 4},{\bf 4},{\bf 1}\right)
U_{n2}= u_n + \tfrac{1}{2}\Delta t \varphi_{1}( \tfrac{1}{2}\Delta t J_n)F(u_n), U_{n3}= u_n + \Delta t \varphi_{1}(\Delta t J_n)F(u_n), u_{n+1}= u_n + \Delta t \varphi_{1}(\Delta t J_n)F(u_n) + \Delta t \varphi_3 (\Delta t J_n) (16 D_{n2}-2 D_{n3}) + \Delta t \varphi_4 (\Delta t J_n) (-48 D_{n2}+12 D_{n3}).
\alpha_p^{\sharp}(\mathbf h, \breve{\mathbf g}, \pmb{\phi}) = \frac{1-p^{-1}}{p^2+w_p}(p^{-1}(p^2-w_p) + 1 - w_p) = \frac{p-1}{p^2}\cdot \frac{p^2+p(1-w_p)-w_p}{p^2+w_p}= \frac{p-w_p}{p^2+w_p}\zeta_p(2)^{-1}.
Z=[Q_a\,Q_{b_U}\,Q_{b_S}\,Q_{b_L}\,Q_{c}]\left[\begin{array}{ccccc}\Lambda_a+I_{|a|}begin{align*}4pt] I_{|b_U|}begin{align*}4pt]{\rm Diag}\,(w_{b_S}) begin{align*}4pt] -I_{|b_L|}begin{align*}4pt] \Lambda_c-I_{|c|}\end{array}\right]\left[\begin{array}{c}Q_a^T begin{align*}4pt]Q_{b_U}^T begin{align*}4pt]Q_{b_S}^T begin{align*}4pt]Q_{b_L}^T begin{align*}4pt]Q_{c}^T\end{array}\right].
d_{G - \{v_{1}, v_{2}\}}(u) \ge \sigma_{2}(G) - d_{G}(v_{1}) - d_{G[\{v_{1}, v_{2}\}]}(u) \ge \big( 2(s_{1}+ s_{2}) - 1\big) - (s_{1}+ s_{2}- 2) - 1 = s_{1}+ s_{2}.
\chi = \frac{\partial}{\partial \eta^\sigma_{k+1}}+ \sum_j \bigg( \gamma^k_j + \frac{\partial a^k_j}{\partial \eta^\sigma_k}\bigg) \frac{\partial}{\partial y^j_{k+1}}+ \sum_\rho \bigg(g^k_\rho + \frac{\partial \beta^k_\rho}{\partial \eta^\sigma_k}\bigg) \frac{\partial}{\partial \eta^\rho_{k+1}}+ Z,
J_{i}\boldsymbol{I}_{jk}=0,\;\ \ \J_{j}\boldsymbol{I}_{jk}^{\pm}=\pm \frac{1}{2}w^{k}\boldsymbol{x}_{jk},\;\ \ \J_{k}\boldsymbol{I}_{jk}^{\pm}w^{j}\boldsymbol{a}_{jk}.
\begin{array}{rcl}\left({\partial\over\partial x}Q\right)\left({\partial\over\partial x}F^{n}\right) = nF^{n-1}\left({\partial \over\partial x}F \right) \left({\partial\over\partial x}Q\right) &= nF^{n-1}(\Delta F^{2}-4F).\end{array}
\P \left[(X + \Delta X) \in dx \, | \, Y =-i \right] = \sum_{j,i \notin j}\P\left[X\in d(x-e_{j}), \Delta X= e_{j}\, | \, Y=-i \right] &=\sum_{j,i \notin j}\frac{a(j)m(d(x-e_{j}))}{\hat a(-i)}&=\sum_{j,i \notin j}\left(\frac{a(j)}{\hat a(-i)}m\right)_{\sharp e_j}(dx).
eT_{\mu\nu}={\tilde{\overline\Psi}_{L_0}}{\gamma_\mu}i\left(D_\nu -\frac{1}{2}\Gamma^\alpha_{\alpha\nu}\right){\tilde\Psi}_{L_0}-\frac{i}{2}g_{\mu\nu}\partial_\alpha({\tilde{\overline\Psi}}_{L_0}\gamma^\alpha{\tilde\Psi_{L_0}}).
\Psi_{\nu}^{( \nu^{k})}(x;\Lambda_{k})=\frac{\lambda_{k}+{\displaystyle \int}_{x_{0}}^{x}dt\big( \Psi_{\nu}^{( \nu^{k-1})}(t;\Lambda_{k-1})\big)^{2}}{\Psi_{\nu}^{( \nu^{k-1})}(x;\Lambda_{k-1})},
\|K-\psi(s_v^{\Lambda^i})\|_{\mathcal{L}_{C^*(\Lambda^i)}(X)}=\sup_{\|x\|_X\leq 1}\|(K-\psi(s_v^{\Lambda^i}))(x)\|_X &\geq \|(K-\psi(s_v^{\Lambda^i}))(s_\nu^\Lambda)\|_X=\|s_\nu^\Lambda\|_X=1,
{\frac{d s^{2}}{l_{s}^{2}\sqrt{4 \pi g_{s}N}}}= \left( \rho^{2}-{\frac{b^{4}}{\rho^{2}}}\right)^{- 1}d \rho^{2}+ \left( \rho^{2}-{\frac{b^{4}}{\rho^{2}}}\right) d \tau^{2}+ \rho^{2}\sum_{i = 1}^{3}d x_{i}^{2}+ d \Omega_{5}^{2},
h^*_i =\begin{cases}2, \mbox{if}\begin{cases}2(k+1)-\alpha \leq i \leq 2(k+1)-1, \, 3(k+1)-\alpha -\beta \leq i \leq 3(k+1)-\alpha-1,\, 4(k+1)-2\alpha -\beta \leq i \leq 4(k+1)-\alpha-\beta-1; \, \end{cases}1, \mbox{otherwise.}\end{cases}
Y_{\tau}^{t,x}=\int_\tau^Te^{-(s-\tau){A}}G B(s,\Xi^{t,x}_s)\,ds+\int_\tau^Te^{-(s-\tau){A}}Z_s^{t,x}\, B(s,\Xi^{t,x}_s)\,ds- \int_\tau^Te^{-(s-\tau){A}}Z^{t,x}_{s}\;d W_s,
\frac{\partial}{\partial a}\mathrm{B}_{z}\left( a,b\right) =\ln z\,\mathrm{B}_{z}\left( a,b\right) -\frac{z^{a}}{a^{2}}\,_{3}F_{2}\left( \left.\begin{array}{c}1-b,a,a a+1,a+1\end{array}\right\vert z\right) .
+ \gamma^{4}[ \partial_{1}a ( \partial_{2}b_{4}+ \partial_{1}a_{4}) - \partial_{2}a ( \partial_{1}b_{4}- \partial_{2}a_{4}) + 4 c ( - \partial_{1}a_{2}+ \partial_{2}b_{2}) + 4 d ( \partial_{2}a_{2}+ \partial_{1}b_{2})
\frac{3}{10}u_{j-\frac{1}{2}}^{-}+\frac{6}{10}u_{j+\frac{1}{2}}^{-}+\frac{1}{10}u_{j+\frac{3}{2}}^{-}=\frac{1}{30}\bar{u}_{j-1}+\frac{19}{30}\bar{u}_{j}+\frac{10}{30}\bar{u}_{j+1}.
\mathcal P\exp\left(\int_{\frac 1 2}^t\left(-\frac{1}{i\pi}\log(\varepsilon)\t_{23}+\left(\tau^s\right)^{2,3}+\left(\tau^s\right)^{1,23}\right)ds\right) = \exp\left(\frac{2t-1}{2\pi i}\log(\varepsilon)\t_{23}\right)\cdot \mathcal P\exp\left(\int_{\frac 1 2}^t\left(\left(\tau^s\right)^{2,3}+\left(\tau^s\right)^{1,23}\right)ds\right).
\rho_{0}^{\mathrm{\tiny{b o s}}}( \omega , k ) = \frac{\pi k ( \omega^{2}- k^{2})^{2}}{3 2}\left[ 1 + \frac{2 T}{k}\log \left( \frac{1 - e^{- \frac{\omega + k}{2 T}}}{1 - e^{- \frac{\omega + k}{2 T}}}\right) \right] \, .
\begin{array}{l}{\mu_{+}x^{+}= q^{2}x^{+}\mu_{+}, \mu_{+}\partial_{+}= q^{- 2}\partial_{+}\mu_{+},}\\{\Lambda x^{k}= q^{2}x^{k}\Lambda , \Lambda \partial_{k}= q^{- 2}\partial_{k}\Lambda , \mathrm{f o r}k = 0 , \pm .}\\ \end{array}
\zeta(2n)=\sum_{k=1}^{\infty}\frac{1}{k^{2n}}=\frac{1}{1^{2n}}+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\frac{1}{4^{2n}}+\cdot\cdot\cdot=(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}
\int_{r_{1}}^{r_{2}}dr \sqrt{2\mu ( E - V_{\it eff}(r) )}=\left( n + \frac{1}{2}+ \frac{1}{2}|l-\alpha| - \frac{1}{2}\sqrt{\eta}\right) \pi, \;\;\;\;\; ( n = 0, 1, 2, \cdots )
\{\theta_1,\theta_2\}_{\mbox{c.b.}}=W^{0}_{~12}+ \left\{\theta_1[\theta_2,r]\right\}+\mbox{Tr}_{34}(W^2_{~1234}\theta_3\theta_4)+\mbox{Tr}_{3456}(W^4_{~123456}\theta_3\theta_4\theta_5\theta_6)\ldots.
{\cal W}_{F}^{( 2 ; s e )}={\frac{C_{F}C_{A}}{6 4{\pi}^{4}}}{\frac{\Gamma^{2}( D / 2 - 1 ) ( 3 D - 2 )}{( D - 4 )^{3}( D - 3 ) ( D - 1 )}}{\cal E},
x w t_\ell \zeta_k(a_1) = x t_\ell^{-1}w \zeta_k(a_1) & = (-1)^{n-k}x t_\ell^{-1}\zeta_{n+1-k}(a_1) & = (-1)^{n-k}t_\ell^{-1}x_\lambda \zeta_{k+1}(a_1) = x w \zeta_k(b_1) = (-1)^{n-k}x \zeta_{k+1}(b_1) & = (-1)^{n-k}(n-k) \zeta_k(b_1) & = (-1)^{n-k}(n-k) t_\ell \zeta_k(a_1)
\Bigl ( X_{A B}^{a b}( x , y ) \Bigr ) = \left( \begin{array}{c c}{0}&{- m^{2}\delta^{a b}}\\{\delta^{a b}}&{\frac{1}{2}g f^{a b c}( D_{i}\pi^{i})^{c}}\\ \end{array}\right) \delta ( x - y ) .
|\sum_{i,j}M_{ij}\sigma_i^* \sigma_j^* - \sum_{i,j}M_{i,j}\sigma_i^{\star}\sigma_j^{\star}| =2\, | \sum_{i \in W}\sum_{j \in W^c}M_{i,j}\sigma_i^* \sigma_j^*| \leq 20 C(\xi) |W| + 2\, \sum_{i,j \in W}|M_{i,j}|.
\ddot{x}_n = (1 + g\dot{x}_{n+1}) (1 + g \dot{x}_{n}) \frac{e^{x_{n+1}- x_{n}}}{1+ g^2 e^{x_{n+1}- x_{n}}}- (1 + g \dot{x}_{n}) (1 + g \dot{x}_{n-1}) \frac{e^{x_{n}- x_{n-1}}}{1+ g^2 e^{x_{n}- x_{n-1}}},
\left[B(u,v)\right]_n=-i\left( a k_{n+1}u_{n+1}^* v_{n+2}^*+b k_{n}u_{n-1}^* v_{n+1}^* -a k_{n-1}u_{n-1}^* v_{n-2}^*-b k_{n-1}u_{n-2}^* v_{n-1}^*\right)
\mathbf{a}_{\mathrm{v}}\left(\theta\right) = \frac{1}{\sqrt{M_{\mathrm{v}}}}\left[1, \,\, e^{j2\pi \frac{d_{\mathrm{v}}}{\lambda}\cos\theta}, \,\, \cdots, \,\, e^{j2\pi \left(M_{\mathrm{v}}-1\right)\frac{d_{\mathrm{v}}}{\lambda}\cos\theta}\right]^{\mathrm{T}},
i e ( \varphi \partial^{\mu}\varphi^{\ast}- \varphi^{\ast}\partial^{\mu}\varphi ) + 2 e^{2}A^{\mu}| \varphi |^{2}+ \mu \varepsilon^{\mu \nu \kappa \lambda}v_{\nu}\partial_{\kappa}A_{\lambda}= \partial_{\nu}F^{\mu \nu},
F({\cal A},G,q) = \sum_{k=1}^{\infty}F_k^{(\rm cov)}(G) \sum_{n=-\infty}^{+\infty}{\rm e}^{-k{\cal A}v(n)}q^k = \sum_{n=-\infty}^{+\infty}F^{(\rm cov)}(G,{\rm e}^{-{\cal A}v(n)}q)
f_{\boldsymbol{R}}\left(\boldsymbol r\right)=\frac{{\beta}^{n+1\}}{{\alpha}^{\beta (n+1)}}e^{-{\left(\frac{r_n}{\alpha}\right)}^\beta}\prod^n_{i=0}{{r_i}^{\beta-1}}\ \ \ \ \ 0<r_0<r_1<\dots <r_n.
A = \begin{pmatrix}a 1 0 0 b 0 0 1 a\end{pmatrix},B = \begin{pmatrix}a 0 0 1 b 1 0 0 a\end{pmatrix},C=\begin{pmatrix}a-1 1 0 1 b 1 0 1 a-1\end{pmatrix},D=\begin{pmatrix}a 0 0 0 b+2 0 0 0 a\end{pmatrix}.
p=\frac{\partial L}{\partial\dot{q}}=(\frac{\partial L}{\partial\dot{q}_{1}},\frac{\partial L}{\partial\dot{q}_{2}},\cdot\cdot\cdot\frac{\partial L}{\partial\dot{q}_{N}})=(p_{1},p_{2}\cdot\cdot\cdot p_{N}),
(\prod_{i=1}^{r-2}T_{a_i}^{x_{i,1}\ldots x_{i,s}})(\prod_{j=1}^{s}T_{x_{1,j}\ldots x_{r-2,j}x_{w,j}}^{b_j})T_{a_{r-1}a_r}^{x_{w,1}\ldots x_{w,s}}
s(x) :=\sup_{a\in E}\left( f(a)+ \langle G(a), x-a \rangle + \frac{1}{2M}\|G(x)-G(a)\|^2 \right) &\leq I(x):= \inf_{b\in E}\left( f(b)- \langle G(x), b-x \rangle - \frac{1}{2M}\| G(x)-G(b)\|^2 \right),
\|\bar{Z}\bar{Z}^\top\| = \left\| \sum_{i \in \widehat{\mathcal{C}}_1 \cap \overline{\mathcal{C}_1}}\sum_{l=1}^{N_i}\sum_{t=0}^{T-1}z^{(i)}_{l,t}z^{(i),\top}_{l,t}\right\| \leq \sum_{i \in \widehat{\mathcal{C}}_1 \cap \overline{\mathcal{C}_1}}\left\| \sum_{l=1}^{N_i}\sum_{t=0}^{T-1}z^{(i)}_{l,t}z^{(i),\top}_{l,t}\right\|
\int_{C_h^-}{f(z)dz}-\int_{C_h^+}{f(z)dz}=2\int_{a}^{b}{f(x)dx}+2\pi i\left[ \sum_{k}{\mathrm{Res}}_{{\mathrm{Im}}z_{f,k}<0}f(z)-\sum_{k}{\mathrm{Res}}_{{\mathrm{Im}}z_{f,k}>0}f(z)\right] .
1 = \sum_{k=0}^{n}P(S_{2k}=0)P^{0}(\{S_m\}_{m=1}^{2n-2k}\cap \{0,b_{i}\}= \emptyset ) + \sum_{k=0}^{n-1}P(S_{2k+1}=b_i)P^{b_i}(\{S_m\}_{m=1}^{2n-2k-1}\cap \{0,b_{i}\}= \emptyset).
r_n^{(k+1)}=\frac{\iint_{\, \mathbb{C}_n^{(k)}}g(\phi_n,x,y)f\left(\sqrt{x^2+y^2}\right) \, \mathrm{d}x \, \mathrm{d}y}{\iint_{\, \mathbb{C}_n^{(k)}}\frac{1}{\sqrt{x^2+y^2}}f\left(\sqrt{x^2+y^2}\right) \, \mathrm{d}x \, \mathrm{d}y},
u_{\underline{{{m}}}}^{\underline{{{a}}}}\equiv ( 1 / 2 ( u_{\underline{{{m}}}}^{+}+ u_{\underline{{{m}}}}^{-}) , u_{\underline{{{m}}}}^{i}, 1 / 2 ( u_{\underline{{{m}}}}^{+}- u_{\underline{{{m}}}}^{-}) ) \; .
D=(\begin{matrix}6&0&0\\ 0&4&0\\ 0&0&5\end{matrix}),U=(\begin{matrix}0&2&3\\ 0&0&2\\ 0&0&0\end{matrix}),L=(\begin{matrix}0&0&0\\ 1&0&0\\ 3&1&0\end{matrix}).
\begin{array}{rll}P_j\, =\,\frac 1{\phi_j(b(j))}\phi_j(X)\,-\,\frac{\phi_j'(b(j))}{\phi_j(b(j))^2}\bigl(X-b(j)\bigr) \phi_j(X)\qquad j=0,\ldots,m & N_j\, =\,\frac 1{\phi_j(b(j))}\,\bigl(X-b(j)\bigr) \phi_j(X)\qquad j=0,\ldots\,(m-1) & N_j^+\, =\,T_j\,N_j\qquad N_j^-\,=(1-T_j)\,N_j \end{array}
\delta_{tot}|\phi\rangle=\Bigl(\xi\partial+\frac{d-2}{2z}\xi^z+\frac{1}{2}\partial^I\xi^JM^{IJ}+M^{IJ}\partial^+\xi^I\frac{\partial^J}{\partial^+}-\frac{d-4}{2z\partial^+}\partial^+\xi^IR^{zI}-\frac{\partial^+\xi^z}{z\partial^+}\Bigr)|\phi\rangle
w_0 = \left( a_0 \prod_{i=1}^n a_1^{(i)}a_2^{(i)}\right)^{m-n}\,,~~~ w_1 = a_0^n \prod_{i=1}^n (a_1^{(i)})^{i-1}(a_2^{(i)})^{2n-i}
\|h\|_2 =\|D^*h\|_2 &=\sqrt{\|D^*_{S_0}h\|_2^2+\|D^*_{S_0^c}h\|_2^2}&\leq\sqrt{\|D^*_{S_0}h\|_2^2+(\|D^*_{S_0}h\|_2+R)^2}\leq\sqrt{2\|D^*_{S_0}h\|_2^2}+R\leq\sqrt{2}z+R &\leq\frac{\sqrt{2(1+\delta)}}{1-\sqrt{t/(t-1)}\delta}\epsilon+\left( \frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1 \right) R.
\mu\gamma_{k+1}^{{\scriptscriptstyle (\mu)}}=\frac{\mu\left(\gamma_{k}^{{\scriptscriptstyle (\mu)}}-\gamma_{k}\right)}{\mu\left(\gamma_{k}^{{\scriptscriptstyle (\mu)}}-\gamma_{k}\right)+\delta_{k+1}}<\frac{\mu\gamma_{k}^{{\scriptscriptstyle (\mu)}}}{\mu\gamma_{k}^{{\scriptscriptstyle (\mu)}}+\delta_{k+1}}\leq\frac{\phi_{k}}{\phi_{k}+\delta_{k+1}}=\phi_{k+1}.
\mu_*( B((t,h(t)),r))\geq \frac{\prod\limits_{i=1}^{q}p_{\sigma_i}}{\prod\limits_{i=1}^{q}C_{\sigma_i}}r C_{\min}{\beta}^{-1}= C_* \cdot r \frac{\prod\limits_{i=1}^{q}p_{\sigma_i}}{\prod\limits_{i=1}^{q}C_{\sigma_i}},
w(k) =\sum_{i=0}^{ak-bk}i(bk+i+1), = \left(\frac{1}{3}a^3-\frac{1}{2}a^2b+\frac{1}{6}b^3\right)k^3+\left(a^2-\frac{3}{2}ab+\frac{1}{2}b^2\right)k^2+\left(\frac{2}{3}a-\frac{2}{3}b\right)k.
K_{1}(w,\lambda(t)) = \int_{0}^{\infty}dR \frac{R}{(1+R^{2})^{3}}\frac{R^{2}\lambda(t)^{2}}{w}&- \int_{0}^{\infty}dR \frac{R}{(1+R^{2})^{3}}\frac{4 R^{2}\lambda(t)^{2}}{2 w \left(\sqrt{\left(-R^{2}\lambda(t)^{2}+w^2+1\right)^2+4 R^{2}\lambda(t)^{2}}-R^{2}\lambda(t)^{2}+w^2+1\right)}
P(L_{\sigma(k)}\ge 2^k N(x+4\delta)- c\sum_{i=0}^{k-1}2^{k-i}[\sigma(i)]^{\xi}) \ge c_3 \zeta^{d[\sigma(k-1)]^{\xi}}[P( L_{\sigma(k-1)}\ge 2^{k-1}N(x+4\delta)- c\sum_{i=0}^{k-2}2^{k-1-i}[\sigma(i)]^{\xi})]^2.
\left\langle \, \prod_{j}T^{\mathrm{\scriptscriptstyle L}}( w_{j}) \prod_{k}\widetilde T^{\mathrm{\scriptscriptstyle L}}( \bar{w}_{k}) \prod_{r = 1}^{N}P ( z_{r}, \bar{z}_{r}) \, \right\rangle^{\! \! \! \mathrm{\scriptscriptstyle L}}\ .
\mu \big( \big( \sum_{i=1}^n t_i a_i \big)^k \big) = \sum_{\substack{k_i\ge 0 k_1 + \dots + k_n = k}}\frac{k!}{k_1! \dots k_n!}\mu(a_1^{k_1}\dots a_n^{k_n}) t_1^{k_1}\dots t_n^{k_n}
h^q\Bigl(Y, \sum_{i =1}^r m_{i,k}D_i\Bigr) - \frac{m_k^n}{n!}\widehat{h}^q\Bigl(Y, \sum_{i =1}^r \frac{m_{i,k}}{m_k}D_i \Bigr) \geq \alpha m_k^n.
{\delta{\cal E}(h) \over h^2}= \alpha_{0}\beta_1\ln{h\over\Lambda}+\alpha_{0}{\beta_2\over\beta_1}\ln\ln{h\over\Lambda}+\alpha_1+{\cal O}{\hbox{\Large (}}\ln \ln{h \over \Lambda}{\hbox{\Large /}}\ln{h \over \Lambda}{\hbox{\Large )}}\, .
\mu^{\prime \prime}+ \mu \biggl [ n^{2}+ n^{2}\sum_{q = 1}^{m}\frac{b_{q}}{( 2 \pi )^{2 q}}\biggl ( \frac{\epsilon n}{\vert \eta \vert^{1 + \beta}}\biggr )^{2 q}- \frac{a^{\prime \prime}}{a}\biggr ] = 0 .
\left< R \left| \chi_{R_{1}}( U ) \chi_{R_{2}}( U ) \ldots \chi_{R_{k}}( U ) \right| R^{\prime}\right> ~ = ~ \int [ d U ] \chi_{R}^{*}( U ) \chi_{R_{1}}( U ) \chi_{R_{2}}( U ) \ldots \chi_{R_{k}}( U ) \chi_{R^{\prime}}( U )
S=\begin{cases}(\emptyset,\emptyset) (\{p_{i,j}\}, \{q_{i,j}\})=0, (\mathbf{k}) \mid_{\eta_=A}(\{p_{i,j}\}, \{q_{i,j}\})=+1, -(\mathbf{k}) \mid_{\eta_=A}(\{p_{i,j}\}, \{q_{i,j}\})=-1. \end{cases}
R^{\frac{1}{2}}(\theta )^{\left| \left. b_{k}\dots \frac{1}{2},b_{1},\frac{1}{2}\right| n_{k}\dots ,m_{1},n_{1}\right\rangle}_{\left| \left. a_{k}\dots \frac{1}{2},a_{1},\frac{1}{2}\right| n_{k}\dots ,m_{1},n_{1}\right\rangle}=R_{a_{1}b_{1}}^{\frac{1}{2}}(\theta )\prod^{k-1}_{i=1}f^{a_{i}a_{i+1}}_{b_{i}b_{i+1}}(w_{m_{i}},\nu_{n_{i+1}},\theta )
J(a,z)(t)=J_{\mathrm{sect}(t)}(a)(z)=J_{\mathrm{sect}(t)}(a^{(1)})(\mathrm{sect}(\delta/2))J_{\mathrm{sect}(\delta/2)}(a^{(2)})(z) =J(a^{(1)},\mathrm{sect}(\delta/2))(t)\cdot J_{\mathrm{sect}(\delta/2)}(a^{(2)})(z)
F_1(t): = t^{\nu + \kappa_1 + \kappa_2 + 2}F_{(y_1, y_2)}^{\prime}(t) + ( - \nu - \kappa_1 - \kappa_2 - 1)\int_{\Omega}\left[a_1\vert y_1 \vert^{1 - \nu}+ a_2\vert y_2 \vert^{1 - \nu}\right]dz.
\frac{\partial P_{i+n+1}}{\partial y_j}(y_{j+n+1},y_{j+2n+2}) +a_j\wp'(a_jf_j(x,y))\frac{\partial P_{i+n+1}}{\partial y_{j+n+1}}(y_{j+n+1},y_{j+2n+2}) &+a_j\wp''(a_jf_j(x,y))\frac{\partial P_{i+n+1}}{\partial y_{j+2n+2}}(y_{j+n+1},y_{j+2n+2})=0.
\zeta^\alpha_k := \langle \zeta_k,{\hat{\alpha}}\rangle \ \ \ , \S^\alpha_n := \langle S_n,{\hat{\alpha}}\rangle \ \ \ , \x^\alpha_0 := \langle x_0, \hat{\alpha}\rangle \,, \zeta^\perp_k := \langle \zeta_k,{\alpha}^\perp \rangle \ , \S_n^\perp := \langle S_n,{\alpha}^\perp \rangle \ ,\x^\perp_0 := \langle x_0, \alpha^\perp \rangle \,,
\zeta_{L N}^{\prime}( 0 ) ={\frac{7 4}{4 5}}\ln ( r_{+}/ r_{-}) - \ln ( \pi ) - 2 \sum_{n = 2}^{\infty}n^{2}\ln \Bigr [ 1 - ( r_{-}/ r_{+})^{2 n}\Bigr ] .
s^i(\hat{x}_t,\theta_t) = \left[\frac{(\theta_t^i)^{\frac{1}{\gamma}}}{\sum_j(\theta_t^j)^{\frac{1}{\gamma}}}\right]\,\hat{x}_t + \frac{\gamma\hat{\phi}}{\eta}\left[\frac{(\theta_t^i)^{\frac{1}{\gamma}}}{\sum_j(\theta_t^j)^{\frac{1}{\gamma}}}-\frac{\phi^i}{\hat{\phi}}\right],
B(t)=\begin{cases}2^{-1}|t|^{s+1}|b|^{-2s-1}(2^{2s+2}-1)(1-2^{-2s-1})^{-1}(|b|=1), |t|^{s+1}|b|^{-2s-1}\{(2^{2s+1}-2^{-1})(1-2^{-2s-1})^{-1}+\delta(|t|=|2b|)(2^s-2^{2s+1})\}(|b|>1).\end{cases}
k,k+1,\ldots,k+p,k' = k,k+1,\ldots,k+p,k < k,k+1,\ldots,k+p,k+p+1 \leq k,k+1,\ldots,k + d_1 < k,k+1,\ldots,k+p,k + d'_1 ={\bf i'}^{(1)}
M_{C M}= \left( \begin{array}{l l l}{\mathcal{A}_{C M}}&{\mathcal{B}_{C M}}&{\mathcal{E}_{C M}}\\{\mathcal{B}_{C M}}&{\mathcal{A}_{C M}}&{\mathcal{E}_{C M}}\\{( \mathcal{E}_{C M})^{t}}&{( \mathcal{E}_{C M})^{t}}&{\mathcal{G}_{C M}}\\ \end{array}\right) ,
\int_X\varphi d\beta_x =\lim_{n\to\infty}\frac{1}{|F_n|}\int_{hF_n}T_g\varphi(x)dg=\lim_{n\to\infty}\frac{1}{|F_n|}\int_{F_n}T_{hg}\varphi(x)dg &=\lim_{n\to\infty}\frac{1}{|F_n|}\int_{F_n}T_{g}T_h\varphi(x)dg &=\int_XT_h\varphi d\beta_x
p(s_{1},s_{2}) = \lim_{n\to\infty}\left( \frac{1}{4n^{2}}\right)^{2}p_{n}\left( \frac{s_{1}}{4n^{2}},\frac{s_{2}}{4n^{2}}\right) = \frac{e^{-\frac{s_{2}}{4}}}{16}\left( \frac{s_{2}}{s_{1}}\right)^{\alpha}D(s_{1},s_{2}),
I_{C}(k,q)=\frac{1}{|\vec k|q^2(q+k)^2}\left[8q\cdot u k\cdot q - k^2 k\cdot u - 6 k\cdot q k\cdot u -8(k\cdot q)^2\frac{k\cdot u}{k^2}+ 4 k^2 q\cdot u\right]