harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\int_{[b_i-2r_i,b_i]}|g|}{4r_i}\ge \frac{g(b_i-2r_i)}{2}=\frac{g(b_i)}{2}-\frac{\eta([b_i-2r_i,b_i])}{2}\ge \frac{g(b_i)}{2}- \frac{\mu[b_i,b_i+r_i]}{8}.
K(n_i, y; n_j, z) = - f_{n_j - n_i}(z - y) 1_{\{i < j\}}& + \sum_{k, l = 1}^d \int_{y}^{\infty}f_{n_m - n_i}(u-y)u^{k-1}du (B^{-1})_{lk}f_{n_j-1+l}(z - x_l)
( \partial^{2}- 3 f \partial_{z}) h_{\mu \nu}+ 4 \alpha \kappa e^{2 A ( z )}\Big \{\partial_{z}f \partial_{\lambda}^{2}- f^{2}\partial_{z}^{2}- f \big ( 2 \partial_{z}f - f^{2}\big ) \partial_{z}\Big \}h_{\mu \nu}= - \kappa T_{\mu \nu}^{( m )}\, ,
Q^{(0)}:=Q_{\rho}^{(\lambda)}(z_0),Q^{(1)}:=Q_{\rho_1}^{(\lambda)}(z_0),Q^{(2)}:=Q_{\rho_1^+}^{(\lambda)}(z_0),Q^{(3)}:=Q_{\rho_2^-}^{(\lambda)}(z_0), begin{align*}5pt] Q^{(4)}:=Q_{\rho_2}^{(\lambda)}(z_0),Q^{(5)}:=Q_{16\rho}^{(\lambda)}(z_0),Q^{(6)}:=Q_{32\rho}^{(\lambda)}(z_0),
Dg_{\ell}(t,s)=\frac{D\lambda^u_{f^{-\ell-1}(\Gamma_x(t,s))}Df^{-\ell-1}(\Gamma_x(t,s))D\Gamma_x(t,s)}{\lambda^u_{f^{-\ell-1}(\Gamma_x(t,s))}}.
{_{2}}G_2\left[\begin{array}{cc}\frac{1}{4}, \frac{3}{4}\frac{1}{3}, \frac{2}{3}\end{array}|\frac{27b^2}{4a^6}\right]_q={_{2}}G_2\left[\begin{array}{cc}\frac{1}{4}, \frac{3}{4}\frac{1}{6}, \frac{5}{6}\end{array}|\frac{27b^2}{4a^6}\right]_q.
( \tilde{\Delta}Z^{M}) \, \frac{\hat{\partial}L}{\hat{\partial}Z^{M}}+ ( \tilde{\Delta}A_{\mu}) \, \frac{\hat{\partial}L}{\hat{\partial}A_{\mu}}= \partial_{\mu}\left[ \lambda ( \varphi ) \, j_{\Delta}^{\mu}\right] \ .
\vartheta_x=\frac{x}{t}\, \frac{dt}{dx}\, \vartheta_t = \frac{2+2t}{1-t}\vartheta_t, \vartheta_x^2 = \frac{4(1+t)^2}{(1-t)^2}\vartheta_t^2 + \frac{8t(1+t)}{(1-t)^3}\vartheta_t.
I_{g h}^{\prime}= \int d^{3}x \mathrm{T r}( Z_{g h}^{- 1}\partial^{\mu}\overline{{c}}^{( r )}p a r t i a l_{\mu}c^{( r )}+ \tilde{Z}_{g}^{- 1}\partial^{\mu}\overline{{c}}^{( r )}[ A_{\mu}^{( r )}, c^{( r )}] ) .
L_{+}= \frac{\ell^{2}}2 \left[ \frac 3 2 \left( \frac{f_{+}^{\prime \prime}}{f_{+}^{\prime}}\right)^{2}- \frac{f_{+}^{\prime \prime \prime}}{f_{+}^{\prime}}\right] \qquad , \qquad L_{-}= \frac{\ell^{2}}2 \left[ \frac 3 2 \left( \frac{f_{-}^{\prime \prime}}{f_{-}^{\prime}}\right)^{2}- \frac{f_{-}^{\prime \prime \prime}}{f_{-}^{\prime}}\right] \qquad .
ev_{(X_1,...,X_n)}^{\sigma (e) h-an}(\partial^{|n|}_{n(X)}P):= \left([\partial^{|n|}_{n(X)}P](X)\oplus \bigoplus_{n\subset m\in M(Y_1,...,Y_n)}\iota_{(n\subset m)(Y,Z)}(ev_{(X_1,...,X_n)}^{\sigma (e) h\otimes I(m)}[\partial^{|m|}_{m(X)}(P)])\right) &\in A_{\sigma (e) h \otimes I(n)}\langle Z_1,...,Z_n:I,S,C\rangle
a_{2}=\sum_{k=1}^{\infty}\left(\frac{k}{2}\right)_{r}\left(\frac{1}{48r-2}\right)^{\frac{k-1}{2}}\frac{6^{k}}{k\pi^{k}24^{\frac{k}{2}}}, begin{align*}6pt] a_{3}=\sum_{k=1}^{\infty}\left(\frac{k}{2}\right)_{r+1}\left(\frac{1}{48r-2}\right)^{\frac{k+1}{2}}\left(\frac{48}{47}\right)^{\frac{k}{2}+r+1}\frac{25\cdot6^{k}(r+\frac{k}{2})}{k\pi^{k}24^{\frac{k}{2}+1}}.
\begin{array}{l}p(x,\xi)=p_2(x,\xi)+p_1(x,\xi)+p_0(x,\xi), p_2(x,\xi):=|\xi|_{g^*(x)}^2,p_1(x,\xi):=a_1^i(x)\xi_i,p_0(x,\xi):=a_0(x)\,.\vphantom{\vrule height 12pt}\end{array}
\Vert \Phi_{-T}\vert_{\{L_T\}\times (E^u_\sigma \cap L^{\perp}_T)}\Vert \leq \Vert \Phi^*_{-T}\vert_{\{L_T\}\times E^u_\sigma}\Vert = \Vert X_{-T}\vert_{E^u_\sigma}\Vert,
\Delta_{[w]}^{cyc}+\Delta_{[\pi]}^{cyc}=2\cdot 2^4M^{cyc}_{(1,1,1,1)}+8\cdot 2^3M^{cyc}_{(2,1,1)}+5\cdot 2^2M^{cyc}_{(3,1)}+4\cdot 2^2M^{cyc}_{(2,2)}+2\cdot 2M^{cyc}_{(4)},
S = \int d x^{D}\sqrt{g_{D}}\left[ - \frac{M_{*}^{D - 2}}{1 6 \pi}( R + 2 \Lambda ) - \frac 1 2 g^{A B}\nabla_{A}{\bf \Phi}\cdot \nabla_{B}{\bf \Phi}- \frac{\lambda}{4}({\bf \Phi}^{2}- v^{2})^{2}\right] ,
\mathfrak{M}_{1}(\mathbb{H}; t^{a/2}):= \{\eta \in \mathfrak{M}(\mathbb{H}), x \in \mathbb{H}\to d_{\mathbb{H}}(x,z)^{a/2}\in L^{1}(|\eta|) &z \in \mathbb{H}\eta(\mathbb{H})=0 \}.
\tau^{j,*}_{0}=0,\quad\tau_k^{j,*}=\inf\{t\ge\tau^{j,*}_{k-1}:u^{\xi^{j,*}_{k-1}}(X_t)=H^{\xi^{j,*}_{k-1}}(X_t,u(X_t))\}\wedge \zeta,k\ge1,
\begin{aligned}\nabla H_{3}^{(3)}(u)= -3\,c_1\,\partial_x(z_x^2)-6\,c_1\,\partial_x\Pi_S^{\perp}[v_x\,z_x]-c_2\partial_{xx}(z^2)+c_2 \pi_0[z_x^2]-2\,c_2\partial_{xx}\Pi_S^{\perp}[v\,z]+ &+2\,c_2\Pi_S^{\perp}[v_x\,z_x]+3\,c_3 \pi_0[z^2]+2\,c_3 \Pi_S^{\perp}[v\,z].\end{aligned}
\frac i2\lambda(I_q^{p;1}+I_q^{p^*;1}) = \sum_{n=0}^{\infty}g_n\lambda^n\,, \frac i2\lambda(I_q^{s;1}+I_q^{s^*;1}) = \sum_{n=0}^{\infty}g'_n\lambda^n
{\hat{\Pi}}= D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}{\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}= \partial{\hat{X}}^{\hat{\mu}}\partial{\hat{X}}^{\hat{\nu}}\left({\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}+ |{\hat{k}}|^{- 2}{\hat{k}}_{\hat{\mu}}{\hat{k}}_{\hat{\nu}}\right) \, ,
t_1[\pi_1] + t_2[\pi_1] + \cdots + t_n[\pi_n] = [\sigma_M(\pi_1 \otimes p_1) + \sigma_M(\pi_1 \otimes p_2) + \cdots + \pi_n \otimes p_n] & = [\sigma_M(\pi_1 \otimes (p_1 + p_2)) + \cdots + \pi_n \otimes p_n] & = (t_1 + t_2)[\pi_1] + \cdots + t_n[\pi_n].
[\begin{matrix}A_{11}&0&A_{1\Gamma}\\ 0&A_{22}&A_{2\Gamma}\\A_{\Gamma1}&A_{\Gamma2}&A_{\Gamma\Gamma}\end{matrix}][\begin{matrix}U_{1}\\U_{2}\\U_{\Gamma}\end{matrix}]=[\begin{matrix}F_{1}\\F_{2}\\F_{\Gamma}\end{matrix}],
\limsup_{M\to\infty}\limsup_{n\to\infty}P\left(\|\hat{K}_{\lambda_n}-K_Y\|_{\ell_2}>M\sqrt{s_n}\lambda_n\right)\leq\limsup_{L\to\infty}\limsup_{n\to\infty}P(\Omega_{n,L}^c)=0
\sigma_z^{-j-1}q_j\psi = (-1)^j\sigma_z^{-1}(L \sigma_z^{-1})^j\psi &= (-1)^j\sum_{a+b+c = j}\sigma_z^{-1-a-2b-3c}p_{a,b,c}(L,[L,c(\delta u)^2],[[L,c(\delta u)^2],c(\delta u)^2])\psi,
\Big\|K\Big(\frac{.}{\sqrt{t - \tau}}\Big)\Big\|_{\mathcal{L}^{r,1}}= (t - \tau)^{\frac{d}{2r}}\big\|\hat K\big\|_{L^{r', 1}}\simeq (t - \tau)^{\frac{d}{2}(1+ \frac{1}{p}- \frac{2}{\tilde p})}.
- \frac{1}{2}e^{- 2 A}\overline{{\nabla}}^{2}V + H^{2}V - \frac{1}{2}V^{'^{\prime}}+ \left( A^{'^{\prime}}+ 2 ( A^{\prime})^{2}\right) V = - \lambda \delta^{( 3 )}( x ) \delta ( y )
ds^2\approx\left[1-\left(\frac{4\pi G_D-2|\Lambda|}{d(p+d-1)}-\frac{\hat{R}}{d(p+1)}\right) r^2\right]\hat{g}_{\mu\nu}(x)dx^\mu dx^\nu+\frac{dr^2}{1+\bar{B}_2r^2}+r^2d\Omega_{d-1}^2,
\left|\frac{dP_n(\varphi+t\psi)}{dt}\right| \!=\!\frac{1}{C_n((G,G_1),t \varphi,\varepsilon)}\frac{1}{n}\Big(\frac{1}{m^n}\sum_{{\underline g \in G_n^*}}\displaystyle\inf_{\mathcal{U}}\Big\{\sum_{U\in\mathcal{U}}S_{\underline g}\psi(U) e^{S_{\underline g}(\varphi+t\psi) (U)}\Big\}\Big)
U_{t_k}=S_{\Delta t}^{k-\ell-1}h+\Delta t \sum_{m=\ell+1}^{k-1}S_{\Delta t}^{k-m}BF'(X_m).U_{t_m}+\sum_{m=\ell+1}^{k-1}S_{\Delta t}^{k-m}\bigl(\sigma'(X_m).U_{t_m}\bigr)\Delta W_m,
L_{\hbox{\tiny{WF}}}u = \frac{1}{2}\sum_{i,j=1}^n\left(\delta_{ij}x_i-x_ix_j\right)u_{x_ix_j}+ \sum_{i=1}^n\left(b_i-x_i\sum_{j=1}^{n+1}b_j\right)u_{x_i},
\begin{cases}1 = (\partial\Psi)\circ\Phi\cdot\partial\Phi+(\bar\partial \Psi)\circ\Phi\cdot\overline{\bar \partial\Phi}, 0 = (\partial\Psi)\circ\Phi\cdot\bar\partial\Phi+(\bar\partial\Psi)\circ\Phi\cdot \overline{\partial \Phi}. \end{cases}
\gamma_{\textrm{FHN}}^{\textrm{OFDMA}}=\frac{P_\textrm{N}{|h_{\textrm{BN}}|^2}}{{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^2}\left( 1-\theta\right)}+\frac{\beta_{\textrm{F}}\eta P_\textrm{B}|h_{\textrm{BF}}|^2 |h_{\textrm{NF}}|^2}{d_{\textrm{BF}}^{\alpha}d_{\textrm{NF}}^{\alpha}\sigma_{\textrm{N}}^2 \left( 1-\theta\right)}.
\varphi_h^{\mathcal{E}+ \mathcal{B}}\left( \begin{bmatrix}q_0 p_0 \end{bmatrix}\right)= \begin{bmatrix}q_0 \exp \big(h \Omega(q_0) \big) p_0 + h \phi_1 \big(h \Omega(q_0) \big) F(q_0) \end{bmatrix}.
\tau_{\alpha}(s_0) = \frac{\langle\alpha(s_0)-P_S(s_0),\alpha'(s_0)\times\alpha'''(s_0)\rangle}{1+J^2(s_0)}=\frac{J'(s_0)}{1+J^2(s_0)}+\frac{\kappa(s_0)\varsigma(s_0)}{1+J^2(s_0)}\,,
\sum_{j \in \mathcal{I}_n(t)}\psi_j^2 (t) \geq \sum_{j \in \mathcal{I}_{i_n(t)-1}(t)}\psi_j^2 (t) & \geq 2 |w_{i_n(t)-1}''(t)| + 1 + 2^{-((i_n(t)-1)-1)}& \geq 2|w_{i_n(t)}''(t)| - 2 \cdot 2^{-i_n(t)}+ 1 + 2^{-((i_n(t)-1)-1)}& = 2|w_{i_n(t)}''(t)| +1 + 2^{-(i_n(t)-1)}& \geq 2|w_n''(t)| + 1 + 2^{-(n-1)},
\mathop{\mathtt{Min.}}\limits_{\{\mathbf{w}^{}_i\}}~~ \sum_{i=1}^{K}\|\mathbf{w}^{}_i\|^2 \mathtt{s.t.}~~^{}_i \triangleq \frac{|\mathbf{g}_i^{H}\mathbf{w}^{}_i|^2}{\sum_{j \neq i}|\mathbf{g}_i^{H}\mathbf{w}^{}_j|^2 + \sigma^2}\geq \gamma^{}_i, \forall i \in \mathcal{K}
\int_0^1 \int_0^1 a(y_1,y_2) R_1'(y_1+y_2)\,\mathrm{d}y_1\,\mathrm{d}y_2 = \int_0^1 \int_0^1 a(y_1,y_2) R_2'(y_1-y_2)\,\mathrm{d}y_1\,\mathrm{d}y_2 = 0,
\begin{array}{lll}q = (+\frac{1}{2},\, \, \vartheta_{1}-\frac{1}{2},\vartheta_{2}-\frac{1}{2},\vartheta_{3}-\frac{1}{2}) q = (\pm \frac{1}{2},\pm \frac{1}{2},\, \, \vartheta_{2}-\frac{1}{2},\vartheta_{3}-\frac{1}{2}) q = (\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2},\, \, \vartheta_{3}-\frac{1}{2})\end{array}
m_1= \min\left\{d-\sqrt{(2-\frac{1}{r})k}-(2-\frac{1}{r})^{\frac{1}{4}}k^{\frac{1}{4}}-1,\c(k)\left(-c(k)+d-\sqrt{(2-\frac{1}{r})k}-(2-\frac{1}{r})^{\frac{1}{4}}k^{\frac{1}{4}}\right)\right\}, m_2= (\frac{1}{r}-1)k-c(k)\left(\sqrt{(\frac{1}{r}-1)k}+(\frac{1}{r}-1)^\frac{1}{4}k^\frac{1}{4}\right).
S_{E G H}^{b o u n d}= -{\frac{1}{3 2 \pi^{2}}}\int_{\partial M}\epsilon_{a b c d}( 2 \theta^{a b}\wedge R^{c d}-{\frac{4}{3}}\theta^{a b}\wedge \theta^{a}{}_{e}\wedge \theta^{e b})
\frac{1}{N}\sum_{i=1}^N \mathbb{E}\big[ \mathfrak{d}_{i,1}Q_i \mathfrak{l}^{(p-1,p)}\big]=\mathbb{E}\big[O_\prec(\hat{\Pi}^2) \mathfrak{l}^{(p-1,p)}\big]+\mathbb{E}\big[O_\prec(\Psi^2\hat{\Pi}^2) \mathfrak{l}^{(p-2,p)}\big]+\mathbb{E}\big[O_\prec(\Psi^2\hat{\Pi}^2) \mathfrak{l}^{(p-1,p-1)}\big],
\left.{D_{u,m}}\right|_{\alpha_m=0}= \displaystyle c_{u,m}\prod_{i=1}^{m-1}(-\alpha_i)^{(2n+1)r^2}\prod_{1\le i<j\le m-1}(\alpha_{j}-\alpha_{i})^{(2n+1)r^2}\enspace &=\Psi_{\boldsymbol{1}}\left(Q_{u,m-1}(\boldsymbol{t}){{\mathfrak{A}}}(\boldsymbol{t})\left.{{\mathfrak{B}}}(\boldsymbol{t})\right|_{\alpha_m=0}\right)\enspace.
c^*_e(A_L,q_L,\mu_L)=\min_{\lambda>0}\frac{-k_{\lambda e}(A_L,q_L,\mu_L)}{\lambda}\geq \min_{\lambda>0}\big(\frac{1}{\lambda}\overline{\mu}+\lambda D_e(A)\big)=2\sqrt{\overline{\mu}D_e(A)}.
= \left(I + \varepsilon N_{1,\varphi}^{-}(x) + \ldots + \varepsilon^k N_{k,\varphi}^{-}(x)\right) \left(I + \varepsilon N_{1,\varphi}^{+}(x) + \ldots + \varepsilon^k N_{k,\varphi}^{-}(x)\right),
\psi(u_i)(a_1,\ldots,a_r) = \psi(\pi(t_i))(p(x_1),\ldots,p(x_r)) = \psi(\pi(t_i))((p\circ\pi)(x_1),\ldots,(p\circ\pi)(x_r)) = (p \circ \pi)(t_i) = p(u_i),
\left(\Delta \tilde{u}_\varepsilon\right)(z)=\frac{(\Delta u_\varepsilon)(z_\varepsilon)}{(\Delta u_\varepsilon)(y_\varepsilon)}=\frac{\Psi'_{N_\varepsilon}(z_\varepsilon)}{\Psi'_{N_\varepsilon}(y_\varepsilon)}\,.
0<\varepsilon< \left(\frac{1}{16 a_2[w]_{A_\infty}}\right)^\alpha \cdot \frac{1}{a_2a_3^2}=\min \left\{\frac{1}{2}, \frac{1}{a_2a_3}, \left(\frac{1}{16 a_2[w]_{A_\infty}}\right)^\alpha \cdot \frac{1}{a_2a_3^2}\right\}
-30 \int_{S^2}P_i P_j dS^2 = 5 \int_{S^2}W_0 \tilde X^i P_j dS^2 = \frac{1}{3}\int_{S^2}W_0\bar W_{0j0n}\tilde X^i \tilde X^n dS^2 - \frac{5}{6}\int_{S^2}W_0^2 \tilde X^i \tilde X^j dS^2,
\#\{\beta\in \Phi^+\setminus I\mid \varphi(v'\beta)<0\}= \#\{\beta\in \Phi^+\setminus (I\cup\{\alpha\})\mid \varphi(v s_\alpha\beta)<0\}= \#\{\beta\in \Phi^+\setminus (I\cup\{\alpha\})\mid \varphi(v \beta)<0\}= \#\{\beta\in \Phi^+\setminus I\mid \varphi(v \beta)<0\}-1.
\eta_{w,q}(Y_{T_{\ell-n+i}}) =\eta_{w,q}([D_{w\varpi_{i},\varpi_{i}}]) =q^{-(\varpi_{i},w\varpi_{i}-\varpi_{i})}[D_{w\varpi_{i},\varpi_{i}}]^{-1}=q^{\dim_{\mathbb{C}}\epsilon_{i}.T_{\ell-n+i}}Y_{T_{\ell-n+i}}^{-1},
\Big\|M_F \|F_1^+(\cdot) - G_1^+(\cdot)\|^2\Big\|_{L_1}\leq \|M_F\|_{L_\infty}\left\|F_1^+ - G_1^+\right\|_{H_2}^2 = \max\left\{\|F\|_{L_{\infty}}, 1\right\}\left\|F_1^+ - G_1^+\right\|_{H_2}^2,
L^{(k)}(1)=-2\left( 2p-3\right) \left( 4p^{2}-12p-1\right) \left(\prod\limits_{j=0}^{k-1}(p-j)\right) +12p\left( 4p^{2}-16p+13\right) \left( \prod\limits_{j=1}^{k}(p-j)\right)-24p(p-1)(2p-5)\left( \prod\limits_{j=2}^{k+1}(p-j)\right) +16p(p-1)(p-2)\left( \prod\limits_{j=3}^{k+2}(p-j)\right)
\Big\|\delta^k(a)(D+i\lambda)^{-p-1}\Big\|_1 = O(\lambda^{-1}),\quad\lambda\to\infty, \Big\|\delta^k(\partial(a))(D+i\lambda)^{-p-1}\Big\|_1 = O(\lambda^{-1}),\quad\lambda\to\infty.
\mbox{LLR}_k = \mbox{ln}\frac{\sum_{(\tilde{m},\tilde{q})\in\tilde{B}_0^\upsilon}f_\mathbf{U}\left(\mathbf\Lambda_k^{(\tilde{m}, \tilde{q})}|\alpha, \beta\right)}{\sum_{(\tilde{m},\tilde{q})\in\tilde{B}_1^\upsilon}f_\mathbf{U}\left(\mathbf\Lambda_k^{(\tilde{m}, \tilde{q})}|\alpha, \beta\right)}
ab=\left(ab^{\frac{2-p}{p}}\right)b^{2\frac{p-1}{p}}\le \frac{1}{p}\left(ab^{\frac{2-p}{p}}\right)^p+\frac{1}{q}b^{2\frac{p-1}{p}q}=\frac{1}{p}a^pb^{2-p}+\left(1-\frac{1}{p}\right)b^2.
|Y(\mathbb{F}_q)| = \sum_{M \in \mathbb{G}}\left( |(M \cap X)(\mathbb{F}_q)|-|L_1(\mathbb{F}_q)| \right) +|L_1(\mathbb{F}_q)| =\theta_q(\frac{n-3}{2})q^{\frac{n-1}{2}}(d-1) + \theta_q(\frac{n-1}{2}),
\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{k_{1}}\cdot\cdot\cdot\sum_{k_{n}=0}^{k_{n-1}}|a_{1,k_{n}}a_{2,k_{n-1}-k_{n}}\cdot\cdot\cdot a_{n,k_{1}-k_{2}}|
\begin{aligned}f_N^{(s)}(0,Z_s) \geq\mathcal{Z}_N^{-1}\mathbf{1}_{Z_s \in \mathcal{D}_s}f_0^{\otimes s}(Z_s) \times & \times \left[ \mathcal{Z}_{N-s}- s (N-s) \mathcal{Z}_{N-s-1}\varepsilon^d|B_1^d| \left\Vert f_0 \right\Vert_{L^\infty_x L^1_v}\right]\end{aligned}
\begin{array}{ccc}\displaystyle\tilde{\omega}_{1}(\pi) = \prod_{i=1}^{\nu(\pi)-1}(\lambda_{i}-\lambda_{i+1}-1) \displaystyle\tilde{\omega}_{2}(\pi) = (\lambda_{\nu(\pi)}-2)\cdot\prod_{i=1}^{\nu(\pi)-1}(\lambda_{i}-\lambda_{i+1}-1).\end{array}
\mathrm{D e t}\left[{\cal D}\right] = \left( p_{0}^{2}- p_{1}^{2}- p_{2}^{2}- p_{3}^{2}- m^{2}\right)^{2}\frac{( p_{0}^{2}- p_{1}^{2}- p_{2}^{2}- p_{3}^{2}+ 3 m^{2}+ 4 m p_{0})^{2}}{1 6 m^{4}( p_{0}+ m )^{4}}\quad .
\frac1n\sum_{k\in J(j,n,\epsilon)}|f(k\theta)| =\frac1n\sum_{\nu=1}^L|f(k_\nu\theta)| \le \frac1n\sum_{\nu=1}^{[3\epsilon n]}h(\tfrac{C\nu}{2n}) \le \int_0^{\frac{[3\epsilon n]}n}h\bigl(\tfrac{Cx}2\bigr)\,dx \le \frac 2C\int_0^{\frac{6\epsilon}C}h(t)\,dt,
i_{2 \times 2}(H^F) < \sum_{a_{k,n}}|a_{k,n}^{H^F}| i_{2 \times 2}(C_{H^F}(a_{k,n})) + \sum_{b_{k,n}}|b_{k,n}^{H^F}| i_{2 \times 2}(C_{H^F}(b_{k,n})) + \sum_{c_{k,n}}|c_{k,n}^{H^F}| i_{2 \times 2}(C_{H^F}(c_{k,n})) &= \Sigma_a+\Sigma_b+\Sigma_c.
\left\|\frac{f}{x^{1-\sigma'}}\right\|_{L^{2}}\leq\left\|\left\| \frac{f}{(x-1)^{1-\sigma'}}\right\|_{L_{x}^{2}}\right\|_{L_{y}^{2}}\lesssim\|\| f_{x}(x-1)^{\sigma'}\left\|_{L_{x}^{2}}\right\|_{L_{y}^{2}}\lesssim\left\|f_{x}x^{\sigma'}\right\|_{L^{2}}
\mu\equiv m_H^2,~ C\equiv\sqrt{1+\frac{Ka_2}{2}(\xi A(0)B(0))^2\left(1+\frac{\Lambda'^2}{m_H^2}\right)^{\frac{K}{2\pi}}}\simeq\sqrt{1+\frac{Ka_2}{2}(\xi A(0)B(0))^2\left(1+\frac{\Lambda^2}{m_H^2}\frac{K}{2\pi}\right)}.
L_\rhd(\{y,z\})L_\rhd(x) t=\{y,z\}\rhd(x\rhd t)= \{\{x,y\},z\}\rhd t+y\rhd(\{x,z\}\rhd t)+x\rhd(y\rhd( z\rhd t))-z\rhd(x\rhd( y\rhd t)) &=L_\rhd (\{\{x,y\},z\}) t+L_\rhd (y)L_\rhd(\{x,z\}) t+L_\rhd (x)L_\rhd(y)L_\rhd(z)t-L_\rhd(z)L_\rhd(x)L_\rhd( y)t.
\vert u\vert_{C^{\alpha}(\overline{\Omega})}= \displaystyle\sup_{x,y \in \overline{\Omega},x\neq y}\frac{\vert u(x) - u(y) \vert}{\vert x-y\vert^{\alpha}}&C^{0,\alpha}(\overline{\Omega}) = \{u \in C(\overline{\Omega}): |u|_{C^{\alpha}(\overline{\Omega})}< \infty \}.
p(a, \theta)=\|f(t)-a(t)\cos\theta(t)\|_{L^2}\approx 72.4, &p(a_1, \theta_1)=\|a_2\cos\theta_2\|_{L^2}^2\approx 84, &p(a_2, \theta_2)=\|a_1\cos\theta_1\|_{L^2}^2\approx 84.
B(\overrightarrow{x},\overrightarrow{y}) =\frac12 \bigl[ Q(\overrightarrow{x}+\overrightarrow{y})- Q(\overrightarrow{x}) - Q(\overrightarrow{y}) \bigr] &=x_1y_1+\dots+x_py_p-x_{p+1}y_{p+1}-\dots -x_{p+q}y_{p+q},
\frac{g(z)dz_{1}\wedge\cdot\cdot\cdot\wedge dz_{n}}{f(z)}\mapsto(-1)^{i-1}\frac{g(z)dz_{1}\wedge\cdot\cdot\cdot\wedge\hat{dz_{i}}\wedge\cdot\cdot\cdot\wedge dz_{n}}{\partial f/\partial z_{i}}|_{f=0}.
s_{\lambda}=\sum_{\nu}\frac{\chi_{\nu}^{\lambda}}{z_{\nu}}p_{\nu}=\sum_{\rho=(1^{r_{1}},2^{r_{2}},3^{r_{3}},...)}\chi_{\rho}^{\lambda}\prod_{k}\frac{p_{k}^{r_{k}}}{r_{k}!k^{r_{k}}},
(v_1^{(i)},v_2^{(i)},v_0^{(i)})^T \mapsto \left (v_1^{(i)}, v_2^{(i)}- j v_1^{(i)}, v_0^{(i)}- k v_1^{(i)}\right)^T = (v_1^{(i+1)}, v_2^{(i+1)}, v_0^{(i+1)}),
\phi Y=AY, A=\begin{pmatrix}0 1 0 \dots 0 0 0 1 \ddots \vdots \vdots \vdots \ddots \ddots 0 0 0 \dots 0 1 -\frac{a_{0}}{a_{n}}-\frac{a_{1}}{a_{n}}\dots \dots -\frac{a_{n-1}}{a_{n}}\end{pmatrix}\in \operatorname{GL}_{n}(K).
\Psi_{\mathcal{H}_t}=\Psi_{\mathfrak{H}_t}=\{(\omega,Q)\in\Omega_t\times\mathfrak{P}(\Omega):\E_Q|\Delta S_t(\omega,\cdot)|<\infty,\ \sup_{y\in\mathfrak{H}_t(\omega)}yE_Q[\Delta S_t(\omega,\cdot)]\leq 0\}.
\begin{array}{c c c}{{e^{a}}}&{{\rightarrow}}&{{\frac{1}{\alpha}e^{a}}}\\{{\omega^{a b}}}&{{\rightarrow}}&{{\omega^{a b}}}\\{{b}}&{{\rightarrow}}&{{\frac{1}{3 \alpha}b}}\\{{\psi_{i}}}&{{\rightarrow}}&{{\frac{1}{\sqrt{\alpha}}\psi_{i}}}\\{{\overline{{{\psi}}}^{i}}}&{{\rightarrow}}&{{\frac{1}{\sqrt{\alpha}}\overline{{{\psi}}}^{i}}}\\{{a_{j}^{i}}}&{{\rightarrow}}&{{a_{j}^{i}.}}\\ \end{array}
\mathcal D(-\left. \Delta \right|_{\mathit{D}}) = H^2(\Omega) \cap H^1_0(\Omega),\mathcal D(-\left. \Delta \right|_{\mathit{N}}) = \left\{u \in H^2(\Omega) : \left. \partial_n u\right|_{\partial \Omega}= 0 \right\}
C_{k_1+k_2+1,1}^{k_1+k_2}(t)=\sum_{l_1=0}^{k_1}\sum_{l_2=0}^{k_2}\binom{k_1}{l_1}\binom{k_2}{l_2}(-1)^{l_1+l_2+k_1+k_2}(l_1+k_1t)^{k_1+k_2}=0.
\Lambda_{\geq 4}[\frac{d}{d t}E_{modi}^{S, low1}]=\sum_{\tau=p, N_1}\sum_{\mu, \nu, \kappa\in\{+,-\}}\mathfrak{G}_{\mu, \nu}^{\kappa,\tau}+\sum_{\mu, \nu, \mu',\nu' \in\{+,-\}}\mathfrak{G}_{\mu, \nu;\mu',\nu'}^{\tau},
\sum_{i=0}^{n-m}\frac{(-1)^ii}{i!(n-m-i)!}=\frac{1}{(n-m)!}\sum_{i=0}^{n-m}\binom{n-m}{i}(-1)^ii &=\frac{1}{(n-m)!}\frac{d}{dx}\left.\left\{\sum_{i=0}^{n-m}\binom{n-m}{i}(-x)^i\right\}\right|_{x=1}&=\frac{1}{(n-m)!}\frac{d}{dx}\left.(1-x)^{n-m}\right|_{x=1}&=0,
\big | \pi_{i = 1 \sim N_{c}}\big | = \Lambda_{T}^{2}, \ \ \big | \pi_{i = N_{c}+ 1 \sim N_{f}}\big | = \xi \big | \pi_{i = 1 \sim N_{c}}\big | , \ \| \pi_{B}| = | \pi_{\bar{B}}| \sim \Lambda_{T}^{N_{c}}, \ \, b i g | \pi_{\lambda}\big | \sim \Lambda^{3}\xi^{\frac{\rho + N_{f}- N_{c}}{N_{f}- N_{c}}},
\mu_{\sigma}=\delta_{b_{\sigma_{1}}^{-1}D_{\sigma_{1}}}* \delta_{\left(b_{\sigma_{1}}b_{\sigma_{2}}\right)^{-1}D_{\sigma_{2}}}* \delta_{\left(b_{\sigma_{1}}b_{\sigma_{2}}b_{\sigma_{3}}\right)^{-1}D_{\sigma_{3}}}* \cdots.
\lim_{u\nearrow u^*_t(r)}\rho_t(u,r) = \psi_0 \left(T^{-1}_{0,t}(u^*_t(r),r),r\right)\exp \left\{- \int_0^t \left[ \varphi\left( T^{-1}_{s,t}(u^*_t(r),r),r\right) - \alpha - \lambda(r) \right] ds \right\},
\left( \sum_{j_{1},\ldots,j_{=m}}^{n}|T(e_{j_{1}},\ldots,e_{j_{m}})|^{\frac{1}{1-\left|\frac{1}{\mathbf{p}}\right|}}\right)^{1-\left|\frac{1}{\mathbf{p}}\right|}\leq 2^{(s-1)\left[1-\left(\frac{1}{p_{k_1}}+\cdots+\frac{1}{p_{k_s}}\right)\right]}\Vert T\Vert,
{\cal L}= \frac{1}{2}\left\{\partial_{m}\phi_i \partial^{m}\phi_i + \imath\bar{\psi}_i \gamma^m \partial_m \psi_i - \frac{\partial{\cal W}}{\partial \phi_i}\frac{\partial{\cal W}}{\partial \phi_i}- \frac{\partial^2{\cal W}}{\partial \phi_i\partial \phi_j}\bar{\psi}_i \psi_j \right\}.
H_F\ = \ \left( \vec{q}_1 + i\,\vec{q}_2\right)\;\cdot\; \left( \vec{\lambda}\wedge \vec{\lambda}\right) - \left( \vec{q}_1 - i\,\vec{q}_2\right) \cdot\left(\frac{\partial}{\partial\vec{\lambda}}\wedge \frac{\partial}{\partial\vec{\lambda}}\right) \ .
\forall \eta_\Lambda \in \Omega_\Lambda: \sum_{\Delta \subset \Lambda}\sum_{\xi_\Delta}\left(c_\Delta(\eta_\Lambda, \xi_\Delta) - \frac{\mu(\xi_\Delta \eta_{\Lambda \setminus \Delta})}{\mu(\eta_\Lambda)}c_\Delta(\xi_\Delta\eta_{\Lambda \setminus \Delta}, \eta_\Delta)\right) = 0.
\frac{b+\lambda-c|x|^2}{b'}\exp\left(\frac{|x|^2}{16t}\right)= 4t\exp\left(-n-\frac{R^2}{8t}+1\right)\left(\frac{n}{8t}+\frac{R^2}{32t^2}+\frac{n}{8t}-\frac{|x|^2}{64t^2}\right)\exp\left(\frac{|x|^2}{16t}\right) = \left(n+\frac{R^2}{8t}-\frac{|x|^2}{16t}\right)\exp\left(-\left(n+\frac{R^2}{8t}-\frac{|x|^2}{16t}\right)+1\right).
\ln \frac{\det{}^{\prime}D}{\det{}^{\prime}D^{\left( N \right)}}= - \Gamma \left[ a_{\mu}\right] - \overline{{{\Gamma}}}\left[ A_{\mu}^{\left( N \right)}, a_{\mu}\right] + \ln \det \left| \left\langle \varphi_{0 i}^{\left( N \right)^{\dagger}}\exp \left( 2 \phi \gamma_{5}\right) \varphi_{0 j}^{\left( N \right)}\right\rangle \right| ,
{\times}\int{d^{3}x\:\mbox{tr}\Bigl[(\partial_{j}A_{k}+A_{j}A_{k})(\partial_{i}f\, g-\partial_{i}g\, f)+\partial_{i}(A_{j}A_{k})\Bigl(\frac{1}{1+\mu}\,fg-\frac{\mu}{1+\mu}\,gf\Bigr)\Bigr]}.
\mathbb{E}[ -\log( q_{V|X_A}(V|X_A) ) ]= \mathbb{E}\left[ \frac{(V-X_A^2 \Delta \tilde{G}- \sigma^2_N)^2}{4 X_A^2 \Delta^2 \sigma^2_N}\right] & + \log{\Delta}+ \frac{1}{2}\log(4 \pi\sigma^2_N)+ \frac{1}{2}\mathbb{E}[ \log(X_A^2) ]
\langle\, a_{\omega}^{R \dag}a_{\omega'}^{R}\, \rangle \,_{\rm{in}}= \frac{1}{2\pi}\,\int_{0}^{\infty}\,d\omega''\,\tilde{\beta}_{\omega''\omega}^{R *}\,\tilde{\beta}_{\omega''\omega'}^{R}\,.
\Xi_n(\tau_i) = \widetilde{\xi}_{(1,\gamma_{\sigma_i}/\gamma_{1^n}-1)}+ \widetilde{\xi}_{(s_i,\gamma_{\sigma_i}/\gamma_{1^n})}= \widetilde{\xi}_{1,\frac{\Delta_{i,i+1}}{x_{i+1}-x_i}}+\widetilde{\xi}_{s_i,1+\frac{\Delta_{i,i+1}}{x_{i+1}-x_i}}.
\left(\frac{(r^{2}-1-(s-t)^{2})}{\sqrt{\beta}(1+r^{2}+(s-t)^{2}+\sqrt{\beta})}\right) = \frac{-1}{2(s-t)^{2}}\left(1+O\left(\frac{1+r^{2}}{(s-t)^{2}}\right)\right)
G_{j}G_{j+1}M_{j}= \begin{bmatrix}c_{j}-s_{j}s_{j}\phantom{-}\overline{c}_{j}1 \end{bmatrix}\begin{bmatrix}1 c_{j+1}-s_{j+1}s_{j+1}\phantom{-}\overline{c}_{j+1}\end{bmatrix}\begin{bmatrix}\gamma_{j}-\sigma_{j}\sigma_{j}\phantom{-}\overline{\gamma}_{j}1 \end{bmatrix}.
\bigg ( \langle \langle F_{\mu \nu}( z_{j}) F_{\rho \sigma}( z_{j}^{\prime}) \rangle \rangle - \langle \langle F_{\mu \nu}( z_{j}) \rangle \rangle \langle \langle F_{\rho \sigma}( z_{j}^{\prime}) \rangle \rangle \bigg )^{\mathrm{S R}}= 0 .
\frac{\partial h^{(1)}(c,t)}{\partial t}= - \sum_{k=1}^K \left[ \sum_{l=1}^L R_t^{kl}(A_k^{OPR}(c,t), h^{(1)}(c,t) - h^{(0)}(c-e_l,t)) - \lambda_k(t) \mathbf{E}[P^k(0,A_k^{OPR}(c,t))] \Delta^{(1)}(c,t) \right] & = - \sum_{k=1}^K \left[ \sum_{l=1}^L R_t^{kl}(A_k^{OPR}(c,t), \Delta_l V^{PR}(c,t)) - \lambda_k(t) \Delta^{(1)}(c,t) \right],
\frac1X \int_X^{2X}\Big(\Delta_2\Big(x+\frac{X^{1/2}}{L}\Big)-\Delta_2(x) \Big)^2 \, dx=\frac{8}{\pi^2}\frac{X^{1/2}}{L}(\log L )^3+O\Big(\frac{X^{1/2}}{L}(\log L)^2 \Big).
\left\{\textbf{S}_k ( \textbf{x}) \right\}_{ii}= \textbf{S}_1( \textbf{x}_{i}) \left\{\textbf{S}_k ( \textbf{x}) \right\}_{ij}= \left\{\textbf{S}_2 ( \textbf{x}_{ij}) \right\}_{12},
\begin{pmatrix}\displaystyle\cos\frac{2\pi}{k}+ \sqrt{\cos^2\frac{2\pi}{k}- 1- \alpha_{2}\alpha_{3}}\alpha_2 0 \alpha_3 \displaystyle\cos\frac{2\pi}{k}- \sqrt{\cos^2\frac{2\pi}{k}- 1- \alpha_{2}\alpha_{3}}0 \alpha_5 \alpha_6 1\end{pmatrix}
\partial_{x_1}^{\sigma}\big\{\psi(x_1,x_1+x_2,\ldots,x_1+x_M,x_{M+1},\ldots,x_N)\big\}&=M^{|\sigma|/2}\big(\partial_{x_Q}^{\sigma}\psi\big)(x_1,x_1+x_2,\ldots,x_1+x_M,x_{M+1},\ldots,x_N)\,.
\langle \sigma^{2}(x)\,\sigma^{2}(y)\,\sigma^{2}(0)\rangle ={\frac{1}{\pi^{6}|x||y||x-y|}}\left( 4u(s)(u(s)+2)\right)^{3/4}\left({\frac{\alpha (\mu)}{(1-\alpha^{2}(\mu ))(1+\alpha (\mu ))^{2}}}\right)^{3/2},
h_{R,l}(2N,2N) \ =\ \|(\underbrace{1,1,\ldots, 1}_{\mbox{length}2N}, \ldots)\|\ =\ 1\mbox{and}h_{R,l}(N, 2N) \ \ge\ \|(\underbrace{0,1,0\ldots, 0, 1}_{\mbox{length}2N}, 0, \ldots)\| \ =\ 2N.
q_{c}( a , r ) = \frac{1}{4 \pi^{2}a^{4}}\sum_{m = 0}^{\infty}{'}\int_{0}^{\infty}{d z \chi_{\mu}( z / a ) F_{c m}^{( q )}( z , x )}, \quad r > a , \quad q = \varepsilon , \, p_{i},
N(Y,\sqrt c) = \begin{pmatrix}\beta \alpha -\frac32 \alpha^2 - \frac23d_3(0) \left( A^2 + B^2\right) -B\left( \frac1{\sqrt c}+ \omega_1(0)\sqrt c + m(0)\sqrt c \alpha\right) A\left( \frac1{\sqrt c}+ \omega_1(0)\sqrt c + m(0)\sqrt c \alpha\right)\end{pmatrix},
S_0 = \sum_l \frac{1}{2\Delta_{l}^{2}}{\rm Tr}\, \phi^{a}_{l}\phi^{a}_{-l}+ \sum_l \frac{1}{2\epsilon_{l}^{2}}{\rm Tr}\, f^{a}_{l}f^{a}_{-l}+ \sum_r \frac{1}{g_{r}}{\rm Tr}\, \bar{\psi}^a_r \psi^a_r \,.
(\sigma_T,v_T) = \sum_{k\geq K_0}a_k \Phi_{\lambda^p_{k}}+ \sum_{|k|\geq K_0}b_k \Phi_{\lambda^h_k}=\sum_{k\geq K_0}a_k(\xi_{\lambda^p_k}, \eta_{\lambda^p_k}) + \sum_{|k|\geq K_0}b_k (\xi_{\lambda^h_k}, \eta_{\lambda^h_k}) ,
\widehat{I}_\xi=\left(\begin{array}{ccc}\left<\psi_0,\phi_0\right> \left<\psi_0,v_1^{(1)}\right>+\left<\tilde v_0^{(1)},\phi_1\right> 0 0 \left<\psi_1,\phi_1\right> 0 0 \left<\psi_2,v_1^{(1)}\right>+\left<\tilde v_2^{(1)},\phi_1\right> \left<\psi_2,\phi_2\right>\end{array}\right) + o(1).