harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\psi (x) = Z^{a +\frac{1}{2}}\,\,\,r^{-(l + 1)}\,\,\,( y - 1)^{(\lambda - s)\over 2}\,\,\,( y + 1)^{-(\lambda + s)\over 2}\,\,\,P_n^{(\lambda - s - \frac{1}{2}) , (-\lambda - s -\frac{1}{2})}(y) \,\,\, P_k (x)
A ( x , p ) \star B ( x , p ) = e^{\kappa ( \partial_{x}\partial_{\widetilde{p}}{~}-{~}\partial_{p}\partial_{\widetilde{x}})}A ( x , p ) B ({\widetilde{x}},{\widetilde{p}}) \Big |_{{\widetilde{x}= x},{\widetilde{p}= p}}.
\min_{\{\mathbf{e}_m\}_{m=1}^M}~ ~C(\{\mathbf{e}_m\}):=\sum_{t=1}^T C_t\left(d(t)+\sum_{m=1}^M e_m(t)\right) \textrm{s.to}~ ~\mathbf{e}_m \in\mathcal{E}_m,~\forall~m=1,\ldots,M
\tilde{p}^{\epsilon,\Delta t}(t)=e^{-\frac{t-t_n}{\epsilon^2}}\tilde{p}^{\epsilon,\Delta t}(t_n)+\epsilon(1-e^{-\frac{t-t_n}{\epsilon^2}})f(\tilde{q}^{\epsilon,\Delta t}(t_n))+\frac{1}{\epsilon}\sigma(\tilde{q}^{\epsilon,\Delta t}(t_n))\int_{t_n}^{t}e^{-\frac{t-s}{\epsilon^2}}d\beta(s).
{\cal J}F(x,y,z)=\left[\begin{array}{ccccc}\displaystyle \sum_{j=1}^{p}y^I_j\nabla^2 c_j(x) A^T{\cal J}c(x)^T 0 0 begin{align*}4pt]A I_q 0 -I_q 0 begin{align*}4pt]{\cal J}c(x) 0 I_p 0 -I_p\end{array}\right].
\|f^{\Sigma}_{\Lambda}(t,\cdot)\|_{L^{\infty}}\le \frac{A'}{8 K}\varepsilon + CB'\varepsilon\int_1^t \tau^{-\frac{5}{4}+\sigma}\, d\tau & \le \frac{A'}{8K}\varepsilon + C'B'\varepsilon & \le \frac{A'}{4K}\varepsilon \, ,
(f^{\ast})^{2}X =X-t^{\ast}h^{\ast}X, h^{\ast}f^{\ast}X+s^{\ast}h^{\ast}X =0, f^{\ast}t^{\ast}V+t^{\ast}s^{\ast}V =0, (s^{\ast})^{2}V =V-h^{\ast}t^{\ast}V.
(c\tau+d)^{-k}F(\gamma\tau) = \sum_{m=0}^r f_m(\tau)\Big(\frac{c}{c\tau+d}\Big)^m - \frac{1}{12^r}f_r(\tau)\Big(E_2(\tau) + \frac{12c}{c\tau+d}\Big)^r &= \sum_{m=0}^r f_m(\tau)\Big(\frac{c}{c\tau+d}\Big)^m &-\frac{1}{12^r}f_r(\tau) \bigg(\sum_{m=0}^r{r \choose{r-m}}E_2(\tau)^{r-m}\Big(\frac{12c}{c\tau+d}\Big)^{m}\bigg).
\frac{1}{a_{1}^{\alpha_1}a_{2}^{\alpha_2}}= \frac{\Gamma[\alpha_1+\alpha_2]}{\Gamma[\alpha_1]\Gamma[\alpha_2]}\int_{0}^{1}d x \,\frac{x^{\alpha_1-1}(1-x)^{\alpha_2-1}}{[x a_1 + (1-x) a_2]^{\alpha_1+\alpha_2}}
\widehat{\Theta}^+_1 =\frac{1}{|\widehat{\mathcal{C}}_1|}\sum_{i \in \widehat{\mathcal{C}}_1}\tilde{\Theta}_i=\frac{1}{|\widehat{\mathcal{C}}_1|}\sum_{i \in \widehat{\mathcal{C}}_1 \cap \mathcal{S}_1}\tilde{\Theta}_i + \frac{1}{|\widehat{\mathcal{C}}_1|}\sum_{i \in \widehat{\mathcal{C}}_1 \cap \overline{\mathcal{S}_1}}\tilde{\Theta}_i
\big\Vert v_\Sigma^{(n)}- v_{\Sigma}^{(n+1)}\big\Vert_{L^\infty(\Sigma)}= O\left(\frac{1}{n}\right),\big\Vert C_{v^{(n)}_\Sigma}- C_{v^{(n+1)}_\Sigma}\big\Vert_{L^2 \to L^2}= O\left(\frac{1}{n}\right).
\beta_\# (1,0) = (p_1)_\# ( \varphi(1,0)) = (p_1)_\# ( a l_\sigma ( a) ) + (p_1)_\# ( B , 0 , 0) = (p_1)_\# ( a l_\sigma ( a) ), \; \beta_\# (0,1) = (p_1)_\# ( \varphi(0,1)) = (p_1)_\# ( b ),
T_{i_1 \cdots i_n}= \sum_{J=1}^{J_0^n}\alpha_J H_{i_1 \cdots i_n}^{J}+ \sum_{J=1}^{J_1^n}H_{i_1 \cdots i_n j}^{J}[v_j^J] + \sum_{s=2}^{J_s^n}\sum_{J=1}^{J_0^n}H_{i_1 \cdots i_n j_1 \cdots j_s}^{J}[D_{j_1 \cdots j_n}^{J}]
(\sigma_2)_\mathrm{I}=C_\mathrm{I}\,(\alpha-1) -\frac{1+v^2}{4v^2},(\sigma_2)_\mathrm{II}=C_\mathrm{II}\,\alpha +\frac{1+v^2}{4v^2}+\frac{(1-v^2)^2}{16C_\mathrm{II}\, v^4},
{\frac{1}{1 6}}\omega^{2}( 3 + 4 \omega )^{2}+ \omega^{2}( \omega + 1 )^{2}\left({\frac{z_{0}^{2}}{l^{2}}}\right)^{2}+{\frac{\omega^{2}}{2}}( 3 + 4 \omega ) ( \omega + 1 ){\frac{z_{0}^{2}}{l^{2}}}> 0 \, .
\partial_{\eta}L_n^{(\alpha)}(\eta)=-L_{n-1}^{(\alpha+1)}(\eta), L_n^{(\alpha)}(\eta)-L_n^{(\alpha-1)}(\eta)=L_{n-1}^{(\alpha)}(\eta), \eta L_{n-1}^{(\alpha+1)}(\eta)-\alpha L_{n-1}^{(\alpha)}(\eta) =-nL_n^{(\alpha-1)}(\eta).
\frac{\partial F}{\partial t}={\textstyle{\frac{1}{2}}}(1+\xi\bar{\xi})^2\bar{\partial}\left(\frac{\partial r}{\partial t}\right)=-{\textstyle{\frac{1}{2}}}(1+\xi\bar{\xi})^2\bar{\partial}{\cal K}=-{\textstyle{\frac{1}{2}}}(1+\xi\bar{\xi})^2({\cal K}_{10}\bar{\partial}\psi+{\cal K}_{01}\bar{\partial}|\sigma|).
\left( \sqrt{r_{k}^{2}-r_{k+1}^{2}}\right)^{n-2}h^{\ast}\left(x_{1},r_{k}\right) \left( r_{k}-r_{k+1}\right) =\left\vert \widetilde{E}\left( x,r_{k}\right) \right\vert \approx \left\vert B\left( x,r_{k}\right)\right\vert &\approx s_{r_{k}}\left( r_{k}-r_{k+1}\right) ,
G ({\bf x}_{n + 1},{\bf x}_{n}; \varepsilon \vert{\hat{\bf A}}) \simeq \biggl ({\frac{m}{2 \pi i \varepsilon}}\biggr )^{N}e x p \biggl [ i \sum_{a = 1}^{N}\biggl ({\frac{m ( \Delta{\bf x}_{n}^{a})^{2}}{2 \varepsilon}}+ e \Delta{\bf x}_{n}^{a}\cdot{\hat{\bf A}}({\bar{\bf x}}^{a}) \biggr ) \biggr ] .
\left(\int_{\mathbb R^{n}}|f|^{2}dx\right)^{2}\leq \left(\frac{2}{n}\right)^{2}\int_{\mathbb R^{n}}\left| \nabla f \right|^{2}dx \int_{\mathbb R^{n}}\|x\|^{2}|f|^{2}dx, n\geq2,
\tilde{\mathcal{J}}(\tilde{u}_{k}, \tilde{v}_{k}) = \tilde{\mathcal{J}}(u_{k}, v_{k}) \longrightarrow \tilde{c}_{\lambda}\,\,\,\,\,\, \,\,\,\,\, \tilde{\mathcal{J}}'(\tilde{u}_{k}, \tilde{v}_{k})\longrightarrow 0.
\omega^{i_{p}}( g , \lambda ) = \sum_{\alpha = 0}^{\infty}\lambda^{\alpha}\omega^{i_{p}, \alpha}( g ) = \omega^{i_{p}, 0}( g ) + \lambda \omega^{i_{p}, 1}( g ) + \lambda^{2}\omega^{i_{p}, 2}( g ) + \ldots \; ; \quad p = 0 , 1 \; ,
|||(\tilde{\mathcal{A}}\mathcal{A}_3\mathbf V)(t)|||^2_{s-1,\ast}\lesssim||\tilde{\mathcal{A}}\mathcal{A}_3\mathbf V||^2_{s,\ast,t}&\leq C(K)\Big(||\mathbf V||^2_{s,\ast,t}+||\dot{{\mathbf U}}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s+2,\ast,t}\Big) &\leq C(K)\mathcal{M}(t).
\sum_{r,s\in{\mathbb F}_p}\!\!a_{r,s}T_{1k}^rT_{1k+1}^s\!\!=\!\!\sum_{r,s\in{\mathbb F}_p}\!\!a_{r,0}T_{1k}^rT_{1k+1}^s\!\!=\!\!\sum_{r\in{\mathbb F}_p}\!\!a_{r,0}T_{1k}^r\sum_{s\in{\mathbb F}_p}T_{1k+1}^s\!\!=\!\!p^{-1}f_k(T_{1k})c(T_{1k+1}).
\frac{1}{\rho}\frac{d}{d\rho}\left(\rho\frac{d\psi_4}{d\rho}\right)-\frac{a_+^2}{\rho^2}\psi_4-g_1\eta^2\psi_4(\psi_4^2+\psi_6^2-1)-g_2\eta^2\psi_4(\psi_4^2-\psi_6^2-c)=0,
e^{- F ( \vec{x}_{1}, \ldots , \vec{x}_{m}, \vec{y}_{1}, \ldots , \vec{y}_{n}) / T}= \langle P ( \vec{x}_{1}) \ldots P ( \vec{x}_{m}) P^{\dagger}( \vec{y}_{1}) \ldots P^{\dagger}( \vec{y}_{n}) \rangle
\tau\bigg({\phi_1'}^{-1}\cdot{f_1'}U^\ast_{\phi_1}{\phi_0'}^{-1}\cdot{f_0\cdot \log(\phi_0')}U^\ast_{\phi_0}\bigg)=\tau\bigg(f_0 U^\ast_{\phi_0}{\phi_0'\circ\phi_1}^{-1}\cdot{\log(\phi_0'\circ\phi_1)}\cdot{(\phi_1')^{-2}}{f_1'}U^\ast_{\phi_1}\bigg) &=-\tau\bigg(f_0 U^\ast_{\phi_0}{\phi_1'}^{-2}\cdot{f_1' \cdot \log(\phi_1')}U^\ast_{\phi_1}\bigg)
{\sf a}= \sqrt{|{\kappa}| \left( \frac{\sqrt{4{\ell}({\ell}+ 1 ) + 1}- 1}{2}\right) \;}\; \; \mathrm{a n d}\;{\sf b}= \sqrt{|{\kappa}| \left( \frac{\sqrt{4{\ell}({\ell}+ 1 ) + 1}+ 1}{2}\right) \;}.
P_1(z) = -\sum_{j=3}^n b_jQ(C_j, \alpha_1), P_2(z) = \sum_{j\in J-\{1,2\}}b_j \Bigl(r(C_j(m_j)) - \Bigl(Q(C_j, \tau_1)-Q(C_j, \tau_3)\Bigr)r(C_1(m_1))\Bigr), P_3(z) = \sum_{j\in \{3,4,\dots, n\}-J}b_j r(C_j(m_j))\:- \: \sum_{j=3}^n b_j\Bigl(Q(C_j,\tau_2)-Q(C_j,\tau_3)\Bigr)r(C_2(m_2)).
\|d(\nabla\varphi,SO(n))^2\|_{L^{p(\cdot)}(Q')}\le 2\|d(\nabla v,SO(n))\|_{L^{2p(\cdot)}(Q')}^2+2\|\nabla \psi\|_{L^{2p(\cdot)}(Q')}^2 &\le C\|d(\nabla v,SO(n))\|_{L^{2p(\cdot)}(Q')}^2=C\|d(\nabla v,SO(n))^2\|_{L^{p(\cdot)}(Q')}&\le C(M+\sqrt{n})\|d(\nabla v,SO(n))\|_{L^{p(\cdot)}(Q)}
\big(T^* f\big)(x) =\int_a^b\overline{T (y,x)}f(y)\d y, \big(T^\# f\big)(x) =\int_a^b T (y,x)f(y)\d y, \big(\overline{T}f\big)(x) =\int_a^b \overline{T (x,y)}f(y)\d y.
~~~ \| G_{\alpha}(\beta, w)\|_{L^2 (\Omega)}& \leq C\bigg ( |\alpha| \, \| \nabla w\|_{L^2 (\Omega)}+ |\beta| \, \|w\|_{L^\infty_{1}(\Omega)}+ \| w\|_{L^\infty_{1}(\Omega)}\| \nabla w\|_{L^2 (\Omega)}| \log \| \nabla w \|_{L^2 (\Omega)}| \bigg ) \,.
\phi ( x ) = \frac{1}{2 \pi \sqrt{R_{2}\sin \theta}}\sum_{n_{+}= 0}^{\infty}\sum_{n_{2}= - \infty}^{\infty}[ e^{- i ( \frac{n_{+}}{R_{+}}Z_{+}+ \frac{n_{2}}{R_{2}}Z_{2})}A ( n_{+}, n_{2}) + e^{i ( \frac{n_{+}}{R_{+}}Z_{+}+ \frac{n_{2}}{R_{2}}Z_{2})}A^{\dagger}( n_{+}, n_{2}){]}\, ,
D_{0}= \frac{1}{2}\left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \; , D_{+}= \left( \begin{array}{c c}{{0}}&{{\sigma_{1}}}\\{{0}}&{{0}}\\ \end{array}\right) \; , D_{-}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{\sigma_{1}}}&{{0}}\\ \end{array}\right) ;
0 = - 3 s - 3ws - 3sw + 3cws + 3 swc + \frac{1}{2}x^3 wu + \frac{1}{2}u^{-1}wx^3 + \frac{1}{2}uwx^3w & + \frac{1}{2}wx^3wu^{-1}+ \frac{1}{4}x^4wu - \frac{1}{4}u^{-1}wx^4 + \frac{1}{4}uwx^4w - \frac{1}{4}wx^4wu^{-1}
|\psi\rangle=\sum_{\alpha,\beta,\gamma=1}^{\chi}\sum_{i,j=1}^{M}\lambda_{\alpha}^{[C-1]}\Gamma_{\alpha\beta}^{[C]i}\lambda_{\beta}^{[C]}\Gamma_{\beta\gamma}^{[D]j}\lambda_{\gamma}^{[D]}|\alpha ij\gamma\rangle
\Delta\vec{p}=\int_{-\infty}^{+\infty}\vec{F}dt=\int_{-\infty}^{+\infty}e\vec{E}dt=\int_{-\infty}^{+\infty}\frac{eg}{4\pi}\frac{\rho}{(\rho^{2}+z^{2})^{\frac{3}{2}}}\hat{e}_{\phi}dt
\inf_{\substack{t > 0 \eta_t\in H_\xi}}h_\xi(t)= \inf_{t > 0}h_\xi(t)= h_\xi(t_\xi) = \left[\left(t_\xi - \log\frac{1}{r_\xi}\right)(k_\xi - \delta) - \Psi(t_\xi)\right]
- i \overline{{{p}}}\Psi_{p}^{( 1 / 2 )}= \overline{{{\varepsilon}}}m \Psi_{p}^{( 1 / 2 )}, \hspace{0 . 5 i n}\overline{{{\sigma}}}_{p}^{( 1 / 2 )}\Psi_{p}^{( 1 / 2 )}= \overline{{{s}}}_{p}\Psi_{p}^{( 1 / 2 )},
\chi_{r , s}^{( p , p^{\prime})}( q ) = \frac{1}{( q )_{\infty}}\sum_{j = - \infty}^{\infty}\left\{q^{j ( p p^{\prime}j + p^{\prime}r - p s )}- q^{( j p + r ) ( j p^{\prime}+ s )}\right\},
g_k\circ\ell\left(1\right)-g_k\circ\ell\left(0\right) =\frac{dg_k\circ\ell}{ds}\left(s_k^*\right) &=\left({\nabla_x}g_k\left(\ell\left(s_k^*\right)\right)\right)^\mathrm{T}\ell^\prime\left(s_k^*\right).
D_{s\eta^{i+1}}=(1+\varrho^{i+1})D^{(i+1)}_{s\eta^i}+s((\chi_{i+1})_z\eta_{i+1}+(\chi_{i+1})_{\bar{z}}\bar{\eta}_{i+1})e_1\partial_t &+\Theta^{i}_s+\mathcal{R}^{i}_{s}+\hat{\mathcal{H}}^{i}_s+\mathcal{F}^{i}_s+\delta^{(i+1)}
l_\sigma(v^{2k+1};0,0) = l_\sigma(v^{2k};0,0) l_\sigma(v;0,0) = ((uv)^{-2k};0,2k) (v^{-1}B;0,1) & = ( (uv)^{-2k}\theta(0,2k)(v)^{-1}\theta(0,2k)(B); 0, 2k + 1) & = ( (uv)^{-2k}v^{-1}B; 0, s) = ( (uv)^{-s}(u B)^{\delta_s}, 0, s).
\begin{gathered}I_1 + I_6 = \frac{1}{\nu}\left ( \eta(t) \cdot \nabla \left ( g_{\nu t}* \left ( \tau \circ X^{-1}(t) \right ) \right ) - \nabla \left ( g_{\nu t}* \left (\eta(t) \tau \circ X^{-1}(t) \right ) \right ) \right ) - \frac{1}{\nu}g_{\nu t}* \left (\nabla \cdot \eta(t) \left ( \tau \circ X^{-1}(t) \right ) \right ) \end{gathered}
(-1)^k_\mu (\iota_\alpha P) + \iota_{d_A \alpha}P = (-1)^k_\mu (\iota_\alpha (X_1 \wedge \cdots \wedge X_k)) + d_A \alpha (X_1, \ldots, X_k) = (-1)^k \sum_{i=1}^k (-1)^{i+k}_\mu (\alpha(X_1, \ldots, \hat{X_i}, \ldots, X_k) X_i)
{1\over{\sqrt{g\left( x+ \pi (\xi ) \right)}}}={1\over{\sqrt g}}\left[ 1 -{\cal D}\phi.\xi +{{{\cal D}^2}\over 2}\left( \phi .\xi\right)^2 -{{\cal D}\over 2}\phi_{\lambda;\sigma}\xi^\lambda \xi^\sigma -{1\over 6}R_{\lambda \sigma}\xi^\lambda \xi^\sigma +...\right].
D_{\mu}^{i j}[ A ] \frac{\delta W}{\delta A_{\mu}^{j}}+ g f^{i j k}( J_{\mu}^{j}\frac{\delta W}{\delta J_{\mu}^{k}}+ \xi^{j}\frac{\delta W}{\delta \xi^{k}}+ \overline{{{\xi}}}^{j}\frac{\delta W}{\delta \overline{{{\xi}}}^{k}}) = 0 .
A^{\mu}\hat{D}_{\mu \nu}^{- 1}\tilde{\delta}A^{\nu}+ a^{\mu}\bar{D}_{\mu \nu}^{- 1}\tilde{\delta}a^{\nu}= - ( A_{\mu}+ a_{\mu}) D_{\mu \nu}^{- 1}\partial^{\nu}( C + c ) \zeta .
\mu \circ T^{-1}((r, \infty) \times B) = \mu( \{x : \rho(x) > r, \, \theta(x) \in B \}) = \mu( r \, \{x : \rho(x) > 1, \, \theta(x) \in B \}) = r^{-\alpha}\, \mu( \{x : \rho(x) > 1, \, \theta(x) \in B \}) = r^{-\alpha}\, H(B).
\frac{1}{3n^2}\sum_{k=0}^{n-1}(4k+3)g_k =\sum_{k=0}^{n-1}\frac{1}{k+1}{2k\choose k}{n-1\choose k}^2, begin{align*}5pt]\sum_{k=0}^{p-1}kg_k \equiv -\frac{3}{4}\pmod{p^2},
\begin{aligned}B_\mathbf{u}(c_h^{n-1};(\mathbf{u}_h^{n},p_h^{n},\lambda_h^{n});(0,1,0))=-\langle\mathbf{u}_h^{n}\cdot\mathbf{n},1\rangle_{\partial K}=-(q^{n},1)_{K}. \end{aligned}
i \int d^{2}\xi d^{2}\theta [ ( A_{+}^{a}) D_{-}\tilde{X}_{a}- ( D_{+}\tilde{X}_{a}) C_{-}^{a}] = i \int d^{2}\xi d^{2}\theta \tilde{X}_{a}F^{a}+ \mathrm{s u r f a c e ~ t e r m},
T^{\nabla^{E}}( \omega \otimes \tau \otimes s) = - 8 ({\mathcal H}\wedge \omega )\otimes\tau \otimes s \ + (-1)^{|\omega |}\omega \otimes \tau \otimes C s + 2 (-1)^{| \omega | +1}\sum_{i,j,k}(R(X_{i}, X_{j}), r_{k})^{\mathcal G}(\alpha_{i}\wedge\alpha_{j}\wedge \omega )\otimes\tau \otimes{\tilde{r}_{k}}s .
A_i = X^{-1}\begin{pmatrix}A^{(1)}_i D^{(1,2)}_i D^{(1,3)}_i \cdots D^{(1,k)}_i 0 A^{(2)}_i D^{(2,2)}_i \cdots D^{(2,k)}_i 0 0 A^{(3)}_i \cdots D^{(3,k)}_i \vdots \vdots \vdots \ddots \vdots 0 0 0 \cdots A_i^{(k)}\end{pmatrix}X
\frac{\mathrm{d}}{\mathrm{d}\ln s}\,\, \ln\left|\frac{F(s)}{s}\right| \,\, \rightarrow \,\,\frac{2}{\pi}\;\arctan\left( \frac{\mathrm{Re}\,F(s)}{\mathrm{Im}\,F(s)}\right) \,\,\, .
s^{(\alpha)}({\bf{x}};\theta) := \nabla\ln p^{(\alpha)}_\theta({\bf{x}}) = \nabla\ln \frac{p_\theta({\bf{x}})^\alpha}{\|p_\theta\|^\alpha}= \nabla \big[ \ln p_\theta ({\bf{x}})^\alpha -\ln \|p_\theta\|^\alpha\big]= \alpha \big[s({\bf{x}};\theta) - \nabla\ln \|p_\theta\| \big].
P_{m_{12}}^{l_{12}}(x + y) \displaystyle\sum\limits_{m_1,m_2}\langle m_1,l_1,m_2,l_2 | 0,l_{12}\rangle P_{m_1}^{l_1}(x) P_{m_2}^{l_2}(y) = \displaystyle\sum\limits_{m_1,m_2}\langle m_1,l_1,m_2,l_2 | m_{12}, l_{12}\rangle P_{m_1}^{l_1}(x) P_{m_2}^{l_2}(y)
\psi_n^{(m)}(q) =J_{(n,0)}^{\mathfrak{sl}_3}(T(2,2m))=q^{-\frac{2m}{3}(n^2+3n)}\sum_{i=0}^n\frac{\Delta(i,i)}{\Delta(n,0)}q^{m(i^2+2i)}&=q^{-\frac{2m}{3}(n^2+3n)+n}\sum_{i=0}^nq^{-2i}q^{m(i^2+2i)}\frac{(1-q^{i+1})^3(1+q^{i+1})}{(1-q)(1-q^{n+1})(1-q^{n+2})}.
F ( x_{1}, \cdots , x_{n}| z_{1}, \cdots , z_{N}) = \frac{\Delta^{( n l )}( x_{1}, \cdots , x_{n}| z_{1}, \cdots , z_{n}| z_{n + 1}, \cdots , z_{N})}{\displaystyle \prod_{j = 1}^{n}\prod_{i = n + 1}^{N}( z_{i}- z_{j}\tau^{2})},
g_{y \bar{y}}= \partial_{y}\partial_{\bar{y}}K = \frac n{n + 1}+{\frac{| y |^{2 ( n - 1 )}}{( n - 1 ) !}}{\frac{e^{- | y |^{2}}}{Q ( | y |^{2})}}\left( n - | y |^{2}-{\frac{| y |^{2 n}}{( n - 1 ) !}}{\frac{e^{- | y |^{2}}}{Q ( | y |^{2})}}\right) ,
\begin{aligned}|S(\cdot,t)| \leqslant C_S \biggl[ (|F^{j-1}|^{p-2}+|F^{j}|^{p-2})|\delta F^j|^2 + |{\delta\Theta}^j|^2 +(|F^{j-1}|^{p-2}+|F^{j}|^{p-2})|\delta F^j| + |\delta \Theta^j| + d(\Theta,\bar{\Theta}) \biggr]\end{aligned}
|r^{2}\partial_{r}^{2}v_{5}(t,r)| \leq \frac{C r}{t^{4}\log^{3b}(t)}+ \frac{C r^{2}}{t^{4}\log^{3N+b-2}(t)}+ \frac{C r}{t^{7/2}\log^{\frac{5N}{2}+3b-3}(t)}+ \frac{C r}{t^{7/2}\log^{\frac{5N}{2}+3b-3}(t)}&\leq \frac{C r}{t^{3}\log^{3N+b-2}(t)}, r \leq \frac{t}{2}
S_{e f f}= \frac{1}{2}E^{a}E^{b}\left( \delta^{a b}+ \frac{2 2}{3}\frac{g^{2}}{( 4 \pi )^{2}}\ln \left( \frac{1}{\mu^{2}}\sqrt{2 g^{2}E^{a}E^{b})}\right) - \frac{1 1 i}{4 8 \pi}g^{2}\delta^{a b}\right)
H^{2}( u ) = \left[ \frac{e^{4 u}+ b^{4}}{e^{4 u}- b^{4}}- \frac{1}{2 u}+ \frac{2 K}{u \Lambda^{2}}\left( \frac{e^{2 u}}{e^{4 u}- b^{4}}\right) \right] .
\sum_{\substack{v_2(k)=1 b_k\in B_2}}b_k=R_{n-1}-\sum_{\substack{v_2(k)=1 b_k\in B_1}}b_k=R_{n-1}-A, &\sum_{\substack{v_2(k)=0 b_k\in B_2}}b_k=R_{n}-\sum_{\substack{v_2(k)=1 b_k\in B_1}}b_k=2^{N-1}-B.
\tau_{1,0}=2a_{2,0},\left|\begin{array}{lll}-\tau_{1,1}a_{2,2}0 -2a_{2,0}a_{2,1}-\tau_{1,1}2a_{3,2}+2a_{2,2}0 -a_{2,0}2a_{3,1}+2a_{2,1}-\tau_{1,1}\end{array}\right|=0.
\dfrac{\Big( \hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'}) \Big) \Big( \hbar'(p - i -1) - (a_s - a_{s'}) \Big)}{\hbar'(p - l_{i}^{(s')}- 1) - (a_s - a_{s'})}\times \dfrac{1}{\hbar'(p - i - 1) - (a_s - a_{s'})}&= \dfrac{\hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'})}{\hbar'(p - l_{i}^{(s')}- 1) - (a_s - a_{s'})}.
H=-\frac{\mu}{2}\sum_{j=1}^{N}(1+i\gamma_{B,j}\gamma_{A,j})-\frac{i}{2}\sum_{j=1}^{N-1}\,\{(\Delta +\tau )\gamma_{B,j}\gamma_{A,j+1}+(\Delta-\tau)\gamma_{B,j}\gamma_{A,j+1}\}\,.
\tilde{y}_{1,i}=[\mathbf{R}_1]_{i,i}\alpha_i s_i + [\mathbf{R}_1]_{i,i}\beta_i w_i + \sum^{N}_{j=i+1}\left([\mathbf{R}_1]_{i,j}\alpha_j s_j + [\mathbf{R}_1]_{i,j}\beta_j w_j\right) +n_{1,i}.
b_n = \sum_{s_1,\ldots,s_n\in S_n}{n \choose{s_1,\ldots,s_k}}2^{\sum_{v_i,v_j\in E(D)}s_is_j}&\leq 2^{n^2d(1-1/n+1/n^2)}\sum_{s_1,\ldots,s_n\in S_n}{n \choose{s_1,\ldots,s_k}}\leq 2^{n^2d_n(1-1/n+1/n^2)}n^k
\begin{array}{l l}{\lefteqn{\! \! J^{P}= 5^{-}:}\vspace{1 m m}}{\! \! 0 \; \; =}{\{\{S_{2}, S_{1}\}_{3}, S_{2}\}_{5}\, ; \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad}\end{array}
\lim_{k \to + \infty}\int_{\Omega_r(z_0) \cap \left\{t<t_0-\varepsilon_k \right\}}\left( \tfrac{1}{r^N}\left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z)\right) dz = & \int_{\Omega_r(z_0)}\left( \tfrac{1}{r^N}\left( \div \, b(z) - c(z) \right) u(z) - v(z) f(z)\right) dz.
S_{0}= \int d^{4}x \left\{\int d^{2}\theta \frac{1}{4}W^{\alpha}W_{\alpha}+ \int d^{2}{\overline{\theta}}\frac{1}{4}{\overline{W}}_{\dot{\alpha}}{\overline{W}}^{\dot{\alpha}}+ \int d^{4}\theta m^{2}V^{2}\right\}.
g_{-1}=\{(\begin{matrix}0&^{t}p&0\\ 0&0&J^{-1}p\\ 0&0&0\end{matrix})|p\in\mathbb{R}^{n}\},g_{-1}=\{(\begin{matrix}0&0&0\\^{t}q&0&0\\ 0&qJ^{-1}&0\end{matrix})|q\in(\mathbb{R}^{n})^{*}\}
{}_2F_1\bigg[\begin{matrix}\frac14 \frac14 &1\end{matrix}\bigg|\,4\bigg]_{p-1}\equiv \begin{cases}a+b\sqrt{-3}\pmod{p^2}, p\equiv1\pmod{4}, a\sqrt{-3}-3b\pmod{p^2}, p\equiv3\pmod{4}.\end{cases}
v_{\Lambda \Omega}= \frac{1}{f_{\Lambda}}\mathrm{{s g n}}( \sigma_{R_{i j}^{( \ell )}\Lambda}^{a}) \frac{v_{\Lambda_{i j}\left( R_{i j}^{( \ell )}\Lambda \right)_{i j}}}{f_{\left( R_{i j}^{( \ell )}\Lambda \right)_{i j}}}f_{\Omega}\# ( \Omega , \Lambda ) .
\xi^*(r) \doteq x + \int_0^r w^*(s)\, ds\,,\hat\xi(r)\doteq x + \int_0^r w^*(s) + \delta\, \tilde w(s)\, ds = \xi^*(r) + \delta\, \tilde\xi(r)\,, \tilde\xi(r) \doteq \int_0^r \tilde w(s)\, ds\,,
\|(I-M_{2r})\l_{(q_N\chi_{A_n})}M_r\| \leq \sum_{n \geq N_r}\|\l_{(q_N\chi_{A_n})}\| &\leq \sum_{n \geq N_r}R^{-2n}C_4 J_D(f) = 2C_4 R^{-2N_r}J_D(f) .
\frac{1}{\Gamma(n)}\frac{d^{n-1}}{dz^{n-1}}(A-zId)_{ij}^{-1}=\frac{1}{\Gamma(n)}\int_0^{\pm\infty}\langle i|e^{-(A-zId)r}r^{n}|j\rangle\;d\log r=(A-zId)_{ij}^{-n}
\frac{f_{\mathcal{G}}(x)}{\int_0^\infty f_{\mathcal{G}}(x) dx}= \frac{f_{\mathcal{G}}(x)}{1 - F_{\mathcal{G}}(0)}= \frac{f_{\mathcal{G}}(x)}{1 - (1 - e^{-q})}= e^{q}f_{\mathcal{G}}(x) = e^q \, b \, q \, e^{bx}\, e^{-q \, e^{bx}}.
\vec{S}_{P}(c,z) \cdot\vec{\mu}(c ) =\frac{1}{z}\,{}_{p+1}F_{q+1}\left[\begin{matrix}-\frac{z}{\lambda+\tau c},\ \vec{\alpha}1-\frac{z}{\lambda+\tau c},\ \vec{\beta}+1\end{matrix};c\right] .
\Delta_{k}= \left| \begin{array}{c c c}{{D_{1}^{1}}}&{{\cdots}}&{{D_{1}^{k}}}\\{{\vdots}}&{{}}&{{\vdots}}\\{{D_{k}^{1}}}&{{\cdots}}&{{D_{k}^{k}}}\\ \end{array}\right| \quad , \qquad k = 1 , \ldots , r
\Vert \textbf{G}_{i}(0_{i})(p) \Vert = \Vert_{p}^{-1}\circ g_{i}\circ_{f_{i}^{-1}(p)}(0_{f_{i}^{-1}(p)})\Vert =\Vert_{p}^{-1}( g_{i}f_{i}^{-1}(p) )\Vert &= d ( g_{i}f_{i}^{-1}(p),p)= d ( g_{i}f_{i}^{-1}(p),f_{i}f_{i}^{-1}(p))<\xi^{\prime}
\sum_{p > \log Z/(2kd)}\frac{1}{p^2}= O_k \left(\sum_{j=0}^\infty \left(\frac{2^{j+1}\log Z}{(j+1) \log 2 + \log \log Z}\right) \frac{1}{2^{2j}(\log Z)^2}\right) & = O_k \left(\frac{1}{\log Z \log \log Z}\right).
M_{ij}=-\frac{1}{kf_k(B)^{k-1}}\sum_{\mbox{\tiny$\begin{array}{c}i\leq j_1<j_2<...<j_{k-2}\leq n, j_1,...,j_{k-2}\neq i,j \end{array}$}}\det{B_{(j,j_1,...,j_{k-2})(i,j_1,...,j_{k-2})}},
\begin{array}{c}B_{1j}={\partial (g_1(c_g(t^1_1)), \cdots, g_1(c_g(t^1_{3d+1})))\over \partial \theta_j}B_{2j}={\partial (g_2(c_g(t^2_1)), \cdots, g_2(c_g(t^2_{2d+1})))\over \partial \theta_j}B_{3j}={\partial (g_3(c_g(t^3_1)), \cdots, g_3(c_g(t^3_{2d+1})))\over \partial \theta_j}.\end{array}
\sum\limits_{i = 1}^k{A_i^\beta{B_i}}\sum\limits_{i = 1}^k{A_i^{2\alpha - \beta - 1}{C_i}}\le\sum\limits_{i = 1}^k{A_i^\beta}\sum\limits_{i = 1}^k{A_i^{2\alpha - \beta - 1}{B_i}{C_i}}.
P_{\mathcal{D},n}\bigl({-}\eta;(g,h)\bigr) =(-1)^nP_{\mathcal{D},n}\bigl(\eta;(h,g)\bigr),\Xi_{\mathcal{D}}\bigl({-}\eta;(g,h)\bigr) =\Xi_{\mathcal{D}}\bigl(\eta;(h,g)\bigr).
h_{1,j'-1}(t)~_{s+1}\mathcal{F}_{s}({\bf a};{\bf b})(\alpha_1)^{q^d}+\cdots+h_{r-j'+1,j'-1}(t)~_{s+1}\mathcal{F}_{s}({\bf a};{\bf b})(\alpha_{r-j'+1})^{q^d}+R_{j'-1}=0
W_{e f f}= ( N_{c}^{\prime}- N_{c}- 1 ) \left[ \frac{\Lambda^{\prime 6 ( N_{c}^{\prime}- 1 ) - 2 N_{c}}}{\det ( q_{a}q_{b})}\right]^{\frac{1}{2 ( N_{c}^{\prime}- 1 ) - 2 N_{c}}}.
\sigma_3(q(1,1)^2) =\sigma_3(d_2^{-1}d_{14})\equiv d_6^{-1}d_{10}\equiv q(1,3)^2\pmod{2}, \sigma_3(q(1,1)^2) \equiv\sigma_3(1+r_4+r_6)\equiv1+r_4+r_2\pmod{2}.
U(\xi) \sim \mu^{-\frac{1}{p}}\sim\begin{cases}\left(B_{1}e^{\omega_{1}\xi}+B_{2}e^{\omega_{2}\xi}+\dfrac{1}{\mu}\right)^{\frac{1}{p}}, (D>0), \left((B_{3}\xi+B_{4})e^{\omega\xi}+\dfrac{1}{\mu}\right)^{\frac{1}{p}},(D=0), \left(e^{-\frac{\mu c}{2}\xi}\bar{Z}(\xi) +\dfrac{1}{\mu}\right)^{\frac{1}{p}},(D<0),\end{cases}
|\frac{-c_{b}r}{2\pi}\int_{0}^{\pi}\sin^{2}(\theta) d\theta \int_{0}^{\infty}d\xi \psi(\xi)\left(\frac{\sin(\xi t_{+})}{t_{+}^{2}}+ \frac{\sin(\xi t_{-})}{t_{-}^{2}}\right)| \leq C r \int_{0}^{\pi}d\theta \int_{0}^{\infty}\frac{|\psi'(\xi)| d\xi}{t^{3}}&\leq \frac{C r}{t^{3}}, r \leq \frac{t}{2}
P^\omega((C_t\cap D_t)^c \mid S_t) = 1-P^\omega(C_t\cap D_t \mid S_t) &\leq 1-P^\omega(D_t\mid C_t, S_t)P^\omega(C_t \mid S_t) &=1-(1-P^\omega(D_t^c\mid C_t, S_t))(1-P^\omega(C_t^c \mid S_t)) &\leq P^\omega(D_t^c\mid C_t, S_t)+P^\omega(C_t^c \mid S_t).
\mathcal{H}= \left( a_{i}^{\dagger}, b_{j}\right) \left( \begin{matrix}{\mathcal{E}_{i l}}&{\mathcal{F}_{j l}^{\dagger}}\\{\mathcal{F}_{i m}}&{- \mathcal{E}_{j m}^{T}}\\ \end{matrix}\right) \left( \begin{matrix}{a_{l}}\\{b_{m}^{\dagger}}\\ \end{matrix}\right) .
L_{1}^{\pm}(v)^{-1}=\begin{pmatrix}* 0 -e_{1}^{\pm}(v)k_{2}^{\pm}(v)^{-1}0 0 * 0 -e_{1}^{\pm}(v)k_{2}^{\pm}(v)^{-1}-k_{2}^{\pm}(v)^{-1}f_{1}^{\pm}(v) 0 k_{2}^{\pm}(v)^{-1}0 0 -k_{2}^{\pm}(v)^{-1}f_{1}^{\pm}(v) 0 k_{2}^{\pm}(v)^{-1}\end{pmatrix}
\begin{cases}\|\sum_{l\le j\epsilon}T_a^{j, l}\|_{L^1\to L^{\infty}}\lesssim_{\epsilon}2^{jm+jn}, \|T_a^{j, l}\|_{L^1\to L^{\infty}}\lesssim_{\epsilon}2^{10n(m-n)(j+l)}, l>j\epsilon.\end{cases}
\tilde{a}=\frac{X_x^2}{\dot{T}}a, \tilde{b}=\frac{X_x}{\dot{T}}\left[b-2a \frac{\theta_x}{\theta}+a\frac{X_{xx}}{X_x}-\frac{X_t}{X_x}\right], \tilde{c}=\frac{1}{\dot{T}}\left[c-b \frac{\theta_x}{\theta}+2a \left(\frac{\theta_x}{\theta}\right)^2-a\frac{\theta_{xx}}{\theta}+\frac{\theta_t}{\theta}\right],
A - A_k = \widehat{Z}\, \widehat{D}_A \, \widehat{W}^T, B - B_k = \widehat{Z}\, \widehat{D}_B \, \widehat{U}^T, G - G_k = \widehat{V}\, \widehat{D}_G \, \widehat{W}^T.
\mu_1|u_1|_4^4\sum_{i=2}^d\mu_i|u_i|_4^4+2b\mu_1|u_1|_4^4\sum_{i<j}|u_iu_j|_2^2 \leq 2b\mu_1|u_1|_4^4\sum_{j=2}^d |u_1u_j|_2^2 &+b^2\Big(\sum_{j=2}^d |u_1u_j|_2^2\Big)^2,
{\cal L}_{0}= - \frac{1}{4}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}- \frac{1}{2}\partial_{\mu}\phi_{\alpha}\partial^{\mu}\phi^{\alpha}- D_{( 0 ) \mu}T \overline{{\: D_{( 0 )}^{\mu}T}}- m^{2}T \overline{{T}}
\mathrm{tr}\left(\mathcal{D}f(A)(vv^{\ast})\right)=\mathrm{tr}\left(U\left[f^{[1]}(\Lambda)\circ(U^*vv^{\ast}U)\right]U^*\right)=\mathrm{tr}\left(\left[f^{[1]}(\Lambda) \circ(U^*vv^{\ast}U)\right]\right),
\widetilde{V}(n):= \sum_{\substack{1 \leqslant j\leqslant J \frac{n}{y_{j}}>\ell^{100K}}}\sum_{y_{j-1}<k \leqslant y_j}\frac{1}{k}\bigg| \sum_{\substack{|\lambda|=n-k \lambda_1 < k}}a(\lambda) \bigg|^2
b_3 = -\frac{1}{2 X_{1+2+3}}\left( 3 b_2 \sum_{j=1}^2 a_{2-j}a_{j-1}(3-j)! j!\sum_{Q\in Q^{(3,j)}}X_{Q}+ \sum_{j=1}^3 a_{3-j}a_{j-1}(4-j)! j! \sum_{Q\in Q^{(4,j)}}X_{Q}\right) .
\frac{1}{2}\int_{\Omega_R}\rho |\overline{u}^R|^2(t)x+\int_0^t \int_{\Omega_R}\mu(\varphi^R)^2|\nabla \overline{u}|^2xs\leq C \int_0^t|\sqrt{\rho}\overline{u}_R|^2_2s+Q_R,
\int d\tau \, \delta M^2 = - \frac{2 \, G_N}{\pi \, \ell_s^2}\int\frac{d^D \, k}{(2\pi)^D}\, \frac{1}{k^2 - i \varepsilon}\int \int d^2 \sigma_1 \, d^2 \sigma_2 \, e^{i \, \ell_s^2 \, k \cdot p \, (\tau_1 -\tau_2)}\, W (k,1,2) \, ,
\mathcal{I}_{N/K}/\mathcal{P}_K\simeq\left(\bigoplus_{i=1}^{n}\,\mathbb{F}_5\,\mathfrak{Q}_i\right)\oplus\left(\bigoplus_{i=1}^{s_2}\,(\mathbb{F}_5\,\mathfrak{L}_i\oplus\mathbb{F}_5\,\mathfrak{L}_i^{\tau})\right)\oplus\left(\bigoplus_{i=s_2+1}^{s_2+s_4}\,(\mathbb{F}_5\,\mathfrak{L}_i\oplus\cdots\oplus\mathbb{F}_5\,\mathfrak{L}_i^{\tau^3})\right)\,.
\begin{aligned}\boldsymbol\Sigma_{\textbf{x}}\!=\! \varepsilon_S^2 \cdot\! \!\int_0^\infty \!e^{\mathbf{E}v}\mathbf{F}e^{\mathbf{E}^v}dv\!=\! \frac{\varepsilon_S^2}{2\varepsilon_S \!-\!1}\mathbf{I}_S(\boldsymbol{\psi}, \widetilde{\textbf{W}}^*)^{\!-1}\!.\! \end{aligned}