harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}kx(x-1)(x-\alpha)^2-(x-\lambda)(x-\beta)^2=k(x^2+ux+v)^2 kx(x-1)(x-\alpha)^2-\lambda(x-\lambda)(x-\beta)^2=k(x^2+u^{\prime}x+v^{\prime})^2\end{cases}
D_n(x+y|a_1,\dots,a_r) =\sum_{j=0}^n\binom{n}{j}D_j(x|a_1,\dots,a_r)(y)_{n-j}\,, \widehat D_n(x+y|a_1,\dots,a_r) =\sum_{j=0}^n\binom{n}{j}\widehat D_j(x|a_1,\dots,a_r)(y)_{n-j}\,.
G_t= \{(x,r):x\in \mathbb{R}^k, 0<r<d(x,K),z_1,z_2\in K &d(x,z_1),d(x,z_2)< d(x,K)+\epsilon r+t & |\theta|>\cos^{-1}\left(2\left(\frac{1-(2\delta +\epsilon)}{1+2\delta}\right)^2-1\right)\},
\hat{C}\;=\; C_{(3)}-\textstyle{\frac{1}{2}}\vec{A}^{T}_{(1)}\varepsilon\vec{A}_{(2)}+\vec{A}_{(2)}^{T}\vec{\theta}+\textstyle{\frac{1}{2}}A_{(1)}\vec{\theta}^{T}\varepsilon\vec{\theta}\; ,
(x-1)^{\kappa(n,{\boldsymbol{m}})}{\widetilde{T}}_n({\boldsymbol{m}},x,y){}={\rm Coef\/f}_{[t_1 \cdots t_n]}\left[ \left( \sum_{\ell_1,\ldots,\ell_n\atop \ell_i \in \{0,1 \}, \, \forall\, i}y^{\sum\limits_{1 \le i < j \le n}m_{ij}\ell_i \ell_j}(y-1)^{- (\sum_{j}\ell_{j})}t_1^{\ell_{1}}\cdots t_n^{\ell_{n}}\right)^{(x-1)(y-1)}\right] ,
c^{27}=a_4^{67}=:t+t_6,c^{16}=a_1^{67}=:t_1+t &z_3^{36}-z_4^{37}=a_1^{67}-a_4^{67}=-z_1^{56}+z_2^{57}, &z_3^{36}-z_1^{56}=c^{35}=z_2^{57}-z_4^{37},
\Big|\sum_{m=2}^\infty\sum_{n=2}^\infty \frac{a_m b_n}{\sqrt{mn}\log(mn)}\Big|\leq \pi\Big(\sum_{m=2}^\infty |a_m|^2\Big)^\frac{1}{2}\Big(\sum_{n=2}^\infty |b_n|^2\Big)^\frac{1}{2}.
-2\ln\left(\frac{2\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{1}{4}\right)}\right) = \frac{\gamma}{2}+ \frac{1}{2}\ln\left(\frac{\pi}{2}\right) - \frac{1}{\pi}\left(\Delta_1 + \frac{\pi\gamma}{2}\right).
\bar{g}_1 (x)= \lim_{K \to -\infty \atop{L \to +\infty}}\bar{g}_1^{\,[K,L]}(x) = (q-1) \sum_{n=-\infty}^{+\infty}(q^n\xi)^{\mu}\frac{\big(q^{\lambda+n+1}\xi /x, q^{n+1}\xi\alpha_1, q^n \xi\alpha_2; q\big)_{\infty}}{\big(q^{n+1}\xi /x, q^n \xi\beta_1, q^n \xi\beta_2 ;q\big)_{\infty}}.
\mathcal{I}_1\ll L^{8}\sum_{\substack{|\mathbf{x}|,|\mathbf{x}'|,|s|,|s'|,|z|,|z'|\ll X \mathbf{x}^{T}A_1\mathbf{x}'+2\mathbf{x}^{T}\gamma_3s'+2s\gamma_3^{T}\mathbf{x}'+hss'+d_izz'=0 \mathbf{x}^{T}C+ z \upsilon_i^{T}=0 s=\epsilon_i z}}1.
\partial_t v(p_1,p_2,t) =\frac{-v(p_1,p_2,t)}{2\left(1-t\right)}+\frac{p_1}{2\left(1-t\right)}\, \partial_1 u\left(\frac{p_1}{\sqrt{1-t}},\frac{p_2}{\sqrt{1-t}}\right) & +\frac{p_2}{2\left(1-t\right)}\, \partial_2 u\left(\frac{p_1}{\sqrt{1-t}},\frac{p_2}{\sqrt{1-t}}\right).
\frac{\partial w}{\partial t}+ Lw - (\lambda + \mu) w = - \frac{\partial u}{\partial y}\frac{\partial v}{\partial y}\: \mbox{in}\: D_T, w(\pm 1, y) = w(\pm 1, -e y) \: \mbox{in}\: D_T^\pm, w(T) = 0 \: \mbox{in}\: D
\overline{\tau}_{k+1}(\iota_k (s_{\alpha}p_{r(\beta \alpha)}s_{\alpha}^*)) =\overline{\tau}_{k+1}(\sum_{a \in \{0,1\}}s_{\alpha a}p_{r(\beta \alpha a)}s_{\alpha a}^*) & =\sum_{a \in \{0,1\}}\tau (p_{r(\beta \alpha a)}) = \tau (p_{r(\beta \alpha)}) & = \overline{\tau}_k (s_{\alpha}p_{r(\beta \alpha)}s_{\alpha}^*).
m_2 = \frac{1}{2}- \frac{96}{\mu \pi^7}\sum_{n}\frac{a_n}{n^3}= \frac{1}{2}- \frac{384}{\mu \pi^7}\sum_{n}\frac{\tanh \left( n\pi / 2 \right)}{n^7}+ \frac{96}{\mu \pi^6}\cdot \zeta(6) \cdot \frac{63}{64}.
P_{1}(v\cdot\nabla_{x}M)=P_{1}\big\{v\cdot(\frac{|v-u|^{2}\nabla_{x}\widetilde{\theta}}{2R\theta^{2}}+\frac{(v-u)\cdot\nabla_{x}\widetilde{u}}{R\theta})M\big\}+\frac{1}{\varepsilon}L_{M}\overline{G}.
(N-M)_M(N-M+\frac{1}{2})_{M}= = \sum_{l=0}^M(-1)^{l}{M\choose l}(j-l+1)_l(N-M-j+l)_{M-l}\times \times (N-M-j+l+\frac{1}{2})_{M-l}(j-l-2N+M+\frac{3}{2})_{l},
\Gamma^{( 1 )}= \int d^{3}x \left( f^{( 0 )}u^{4}+ f^{( 2 )}u^{3}\left[ \psi^{2}\right] + f^{( 4 )}u^{2}\left[ \psi^{4}\right] + f^{( 6 )}u \left[ \psi^{6}\right] + f^{( 8 )}\left[ \psi^{8}\right] \right) ,
\int_{t}^{\infty}|e_{1}''(s)-e_{2}''(s)| |\frac{1}{\log(\lambda_{0}(t))}-\frac{1}{\log(\lambda_{0}(s))}| \frac{ds}{(\lambda_{0}(t)^{1-\alpha}+s-t)(1+s-t)^{3}}&\leq \frac{C ||e_{1}-e_{2}||_{X}}{t^{3}\log^{b+2}(t) (\log(\log(t)))^{5/2}}
\vartheta_{1}(u\mid q^{2})\vartheta_{4}(v\mid q^{2})+\vartheta_{1}(v\mid q^{2})\vartheta_{4}(u\mid q^{2})= \vartheta_{1}\left(\frac{u+v}{2}\,\Big|\, q\right)\vartheta_{2}\left(\frac{u-v}{2}\,\Big|\, q\right)\!,
u\leftarrow\frac{\bar{s}}{\left\vert s\right\vert^{3}}v-\frac{\bar{s}}{\left\vert s\right\vert^{3}}\frac{1}{p_{j}^{\operatorname*{ext}}}\operatorname*{div}\mathbf{m\quad}\mathbf{j}=-\frac{s}{\left\vert s\right\vert}\left( 1+\frac{1}{\left\vert s\right\vert^{2}}\right) \mathbf{m}
\begin{aligned}Q(2\tau) =\frac{1}{8}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi}\frac{q(\rho,\theta,\phi)\sinh\rho \sin\phi\sqrt{\cosh^{2}\rho-\cos^{2}\phi}}{\sqrt{(4\tau^{2}-\cos^{2}\phi)(4\tau^{2}-1)}}d\theta d\phi,\end{aligned}
b_{\nu,n}(x)=(\begin{matrix}n\\ \nu\end{matrix})\sum_{k=0}^{n-\nu}(\begin{matrix}n-\nu\\k\end{matrix})(-1)^{n-\nu-k}x^{\nu+k}=\sum_{l=\nu}^{n}(\begin{matrix}n\\l\end{matrix})(\begin{matrix}l\\ \nu\end{matrix})(-1)^{l-\nu}x^{l},
\int_0^{r_0}r dr (\tilde T^{\theta}_{_{(0)}\theta}+\tilde T^r_{_{(0)}r})= r_0^2 \tilde T^r_{_{(0)}r}(r_0) = r_0^2\left[\frac{A'^2(r_0)}{2 e^2}+ \frac{\alpha}{2}\Lambda'^2(r_0)\right] .
\mathbb E \big[{}- \log Z_{T}\vert \mathcal F_{T}^{X}\big] \, =\, \mathbb E \Big[ \int^{T}_{0}b(s, X_{s}, F_{s}){\mathrm d}X_{s}- \frac{1}{2}\int^{T}_{0}\lvert b(s, X_{s}, F_{s}) \rvert^{2}{\mathrm d}s \Big \vert \mathcal F_{T}^{X}\Big] \, .
{\cal W}_{{\cal C}}^{U(N)}=1-\frac{1}{2}~N g^2{\cal A}_{{\cal C}}+\frac{1}{8}(\frac{2}{3}N^2+ \frac{1}{3})~g^4{\cal A}_{{\cal C}}^2-\frac{1}{48}(\frac{5}{15}N^3+ \frac{10}{15}N)~g^6{\cal A}_{{\cal C}}^3 +{\cal O}(g^8),
\frac{\partial E(sign(y_{1})sign(y_{2}))}{\partial\rho}=\frac{\partial I(\rho)}{\partial\rho}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}4\delta(y_{1})\delta(y_{2})p(y_{1},y_{2})dy_{1}dy_{2}=\frac{2}{\pi\sqrt{1-\rho^{2}}}
{\cal A}( \alpha ) = \frac{2^{( 1 + 2 \alpha - d ) / 2}}{\pi^{( 1 + d ) / 2}\Gamma ( - \alpha )}\left( \frac{m}{R}\right)^{( 1 + 2 \alpha + d ) / 2}F \left( \frac{1 + 2 \alpha + d}{2}; m R \right) ,
\mathcal{G}_{n}(\varphi v) := \left< D^{+}_{t}u^{n-1/2}, \varphi v \right> - \frac{1}{2}\left< \frac{\left(u^{n+1/2}\right)^{2}+ \left(u^{n-1/2}\right)^{2}}{2}, (\varphi v)_{x}\right> &+ \left< \mathcal{H}\left( \frac{u^{n+1/2}+ u^{n-1/2}}{2}\right)_{x}, (\varphi v)_{x}\right> = 0.
e_{ij}=\begin{cases}\delta_{ij}\j\neq k, -1 \i=j=k, \max(0,-b_{ik}) \i\neq j=k,\end{cases}f_{ij}=\begin{cases}\delta_{ij}\i\neq k, -1 \i=j=k, \max(0,b_{kj}) \i=k\neq j.\end{cases}
\begin{aligned}T_{f}\big(A^{C^{*}}, h^{C^{*}}\big)+\widetilde{f}\big(\nabla^{H(C^{*})}, \widetilde{h}^{H(C^{*})}, h^{H(C^{*})}\big) &=T_{f}\big(A^{E_{0}}_{0}, h^{E_{0}}\big)+\sum_{k=1}^{k_{0}}T_{f}\big(A^{E_{k}}_{k}, h^{E_{k}}\big)+\widetilde{f}\big(\nabla^{E_{\infty}},\widetilde{h}^{H(C^{*})}, h^{E_{\infty}}\big),\end{aligned}
g\left(L_1-\sum^{j}_{i=1}\frac{1}{2^{i+1}}d_2-\sum^{j-1}_{i=1}(T-u_i)\right) =g\left(L_1-\sum^{j}_{i=1}\frac{1}{2^{i+1}}d_2-\sum^{j}_{i=1}(T-u_i)\right) &= 1+2^{-j},~ j = 1,\dots, r.
\int (q^2x^2;q^2)_\infty \left( (cq+x)[n]_q-x\right) h_n(x;q) d_qx= &q^{n-1}(1-q) (x^2;q^2)_\infty\left( q h_n(\frac{x}{q};q)- [n]_q(cq+x) h_{n-1}(\frac{x}{q};q)\right),
1/\left( \frac{\partial w_{\nu}(x)}{\partial E_{\nu}}\right)_{\nu =\nu_{1}}=\left[ \log W\left( \psi_{\nu_{1}},\left( \frac{\partial \psi_{\nu}(x)}{\partial E_{\nu}}\right)_{\nu =\nu_{1}}\,|\, x\right) \right]^{\prime}
v_B^{-1}\overset{\mathcal{X}^+(2)}{W}\!\!\!\!\!_A \triangleright \chi^{\epsilon}_s = v_B^{-1}\triangleright \chi^{\epsilon}_s\big({-}\hat q^2C ?\big) & = \sum_{\substack{\ell=1 \sigma \in \{\pm\}}}^p -\epsilon \big(q^s+q^{-s}\big) \lambda^{\sigma}_{\ell}(\epsilon, s) \chi^{\sigma}_{\ell}+ \sum_{j=1}^{p-1}-\epsilon\big(q^s + q^{-s}\big)\delta_j(\epsilon, s)G_j.
(A_1)_j^{j'}(l)=\begin{cases}-\dfrac{(\mathfrak{B}_1)_j^{j'}(l)}{i(\omega\cdot l+m_3(j'^3-j^3))}\mbox{if}\,\,\overline{\omega}\cdot l+j'^3-j^3\neq 0, j, \,j'\in S^c, \,l\in\mathbb{Z}^{\nu}begin{align*}3mm]0 \mbox{otherwise}\end{cases}
\mathcal{L}= \frac{_1}{^2}i \hbar ( \Psi^{*}\partial_{t}\Psi - \partial_{t}\Psi^{*}\Psi ) - \frac{\hbar^{2}}{2 m}\Psi^{*}\overleftarrow{\nabla}\cdot \overrightarrow{\nabla}\Psi - V \left( \mathbf{x}\right) \Psi^{*}\Psi .
\sum_{p \mid m,n \Rightarrow T_{j-1}\leq p < T_j}\frac{(k-1)^{\Omega(n) + \Omega(m)}g(n) g(m)}{[n,m]}= \prod_{T_{j-1}\leq p < T_j}\left(1 + \frac{2(k-1) +(k-1)^2}{p}+ O\left(\frac{1}{p^2}\right) \right) &= \prod_{T_{j-1}\leq p < T_j}\left(1 + \frac{k^2 - 1}{p}+ O\left(\frac{1}{p^2}\right) \right)
\begin{bmatrix}I_m 0\end{bmatrix}Q V_k = U_{k+1}J_k, Q Q^T \begin{bmatrix}U_{k+1}0\end{bmatrix}= Q V_k J_k^T + \alpha_{k+1}Q v_{k+1}e_{k+1}^T, \begin{bmatrix}0 I_p\end{bmatrix}Q V_k = \widehat{U}_{k}\check{J}_k, Q Q^T \begin{bmatrix}0 \widehat{U}_{k}\end{bmatrix}= Q V_k \check{J}_k^T + \check{\beta}_{k}Q v_{k+1}e_{k}^T,
H_0=2\pi\left\{\frac{1}{4r}(f_A^{\prime 2}+f_B^{\prime 2}) +\frac{\eta^2}{2}\left[r(H^{\prime 2}+K^{\prime 2})+\frac{1}{r}(Kf_A-Hf_B)^2\right]+\frac{\lambda\eta^4}{4}r(1-H^2-K^2)^2\right\}
h(x) \coloneqq \begin{cases}0, x \in (-\infty,0), \sum_{k=1}^\infty \frac{a_k}{2^k}, x = \sum_{k=1}^\infty \frac{2a_k}{3^k}\in \mathcal C (a_k)_{k=1}^\infty \in \{0,1\}^{\mathbb N}, \sup_{y \in \mathcal C \cap [0,x]}h(y), x \in [0,1] \setminus \mathcal C, 1, x \in (1,\infty).\end{cases}
M_{\alpha}= e^{- d \tau}\frac{\partial j_{\beta}^{\prime}}{\partial j_{\alpha}}\frac{1}{V^{\prime}}\frac{\partial}{\partial j_{\beta}^{\prime}}\operatorname{l n}Z \equiv e^{- d \tau}\frac{\partial j_{\beta}}{\partial j_{\alpha}}M_{\beta}^{\prime}.
\mathbb P_-S R f(q) =\mathbb{P}_-S H_\varphi f(q)=\mathbb{P}_- S \mathbb P_-\varphi\star f(q) = \mathbb{P}_- q \star \varphi\star f (q) &=\mathbb{P}_- \varphi\star q \star f (q)=H_{\varphi}S f(q)=H_{\varphi}T f(q)=RTf(q).
\min_{r}\min_{\sum_{l=1}^{2N-2}|v_l|^2=2r^2}\frac{\lambda_0 + \sum\limits_{l=1}^{2N-2}\lambda_l |v_l|^2}{2N-1-\frac{1+2r^2}{K}}, = \min_{r}\frac{\lambda_0 + 2 \lambda_1 r^2}{2N-1-\frac{1}{K}-\frac{2r^2}{K}},
\begin{aligned}\|F(f) - T_{F'(f)}f \|_{H^{s}}\lesssim_{F,\|f\|_{W^{1,\infty}}}\|f\|_{W^{1,\infty}}\|\partial_x^{-1}f\|_{H^s}, \|F(f) - T_{F'(f)}f \|_{\dot H^{s}}\lesssim_{F,\|f\|_{W^{1,\infty}}}\|f\|_{W^{1,\infty}}\|\partial_x^{-1}f\|_{\dot H^s}.\end{aligned}
g_{2}((\nabla\pi_*)(U_{2},V_{2}),\pi_*Z) =g_{1}(\nabla^{^{M_1}}_{U_{2}}\phi^{2}V_{2},Z)+g_{1}(\pi_*(\mathcal{H}\nabla^{^{M_1}}_{U_{2}}\omega\phi V_{2}),\pi_*Z)-g_{1}(\mathcal{T}_{U_{2}}\omega V_{2},\mathcal{B}Z) &-g_{2}(\pi_*(\mathcal{H}\nabla^{^{M_1}}_{U_{2}}\omega V_{2}),\pi_*(\mathcal{C}Z))-g_{1}(V_{2},\phi U_{2})\eta(Z).
\begin{array}{rl}\displaystyle \sqrt{n}\sigma_j^{-1}(\check\beta_j-\beta_{0j}) = \frac{1}{\sqrt{n}}\sum_{i=1}^n\{\bar\psi_j(y_i,z_i)-\sigma_j^{-1}\Sigma_j^{-1}(e^j-\mu^j_0)^T\varphi(z_i)\beta_0\}\displaystyle + O_\P(\sigma_j^{-1}\Sigma_j^{-1}\delta_n)\end{array}
Y=(y_1+iy_2)/2=\sum_n\left[{1\over\sqrt{4p^+\omega_n}}\,\alpha_n^+e^{-i\omega_n\tau+in\sigma/\alpha^{\prime}p^+}+{1\over\sqrt{4p^+\omega_{-n}}}\left(\alpha_n^-\right)^{\dagger}e^{i\omega_{-n}\tau-in\sigma/\alpha^{\prime}p^+}\right] \ .
\omega=a(r(t))\left(\theta^{2}\otimes e_{3}-\theta^{3}\otimes e_{2}+\theta^{4}\otimes e_{1}\right)+\left(b(r(t))-1\right)\left(\theta^{5}\otimes e_{3}-\theta^{6}\otimes e_{2}+\theta^{7}\otimes e_{1}\right).
\mathbb{P}^*(\rho)=\frac{1}{K}e^{-\frac{\mathcal{F}(\rho)}{\beta}},\textrm{where $K=\int_{\mathcal{B}}\Pi(\rho)^{-\frac{1}{2}}e^{-\frac{\mathcal{F}(\rho)}{\beta}}d\textrm{vol}$}.
\Psi_p(x\otimes g)^* \Psi_p(y\otimes h) = U_g^* \psi_p^*(x) \psi_p(y) U_h = U_g^* \psi_{e}(\langle x,y\rangle) U_h = \psi_{e}(\beta_{g^{-1}}(\langle x,y\rangle)) U_{g^{-1}h}= \Psi_{e}(\beta_{g^{-1}}(\langle x,y\rangle)\otimes g^{-1}h) = \Psi_{e}(\langle x\otimes g, y\otimes h\rangle).
X^{\mu}(\sigma,\tau)= X^{\mu}+ P^{\mu}\tau + 2L^{\mu}\sigma + \frac{i}{2}\sum_{k\neq 0}\frac{1}{k}\left(\alpha^{\mu}_{k}\mbox{e}^{-2ik\sigma_{+}}+\tilde{\alpha}^{\mu}_{k}\mbox{e}^{-2ik\sigma_{-}}\right)
B^{a}= \left( \begin{array}{c c c c}{{c_{1}^{a}}}&{{\ldots}}&{{0}}&{{0}}\\{{\vdots}}&{{\ddots}}&{{\vdots}}&{{0}}\\{{0}}&{{\ldots}}&{{c_{l}^{a}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{Y_{n}^{a}}}\\ \end{array}\right) .
B_j=\sum_{a=0}^{l/s-1}\lambda_{j,a_j}e_a^{(l/s)}(e_{a(j,a_j\oplus 1)}^{(l/s)})^T j\in[i-1], B_j=\sum_{a=0}^{l/s-1}\lambda_{j+1,a_j}e_a^{(l/s)}(e_{a(j,a_j\oplus 1)}^{(l/s)})^T i\le j<n,
f(g\cdot x_{(i\,j)}^2g_{(j\,k)}^{-2}) = \chi_{i\,j}^2(g) f(x_{(i\,j)}^2g_{(j\,k)}^{-2}) = g\cdot f(x_{(i\,j)}^2g_{(j\,k)}^{-2}), f(x_k \cdot x_{(i\,j)}^2g_{(j\,k)}^{-2}) = 0 = x_k \cdot f(x_{(i\,j)}^2g_{(j\,k)}^{-2}),
\alpha = \frac{1}{\Delta}( \Delta - F_{1 1}) , \hspace{0 . 4 i n}\beta = \frac{F_{0 1}}{\Delta}, \hspace{0 . 4 i n}\gamma = \frac{1}{\Delta}( \Delta - F_{0 0}) , \hspace{0 . 4 i n}\Delta \equiv F_{0 0}F_{1 1}- F_{0 1}^{2}.
S = -{\frac{1}{2 \pi}}\int d \tau d \sigma \left( ( U^{'})^{2}+ \frac{U^{4}L^{2}}{d_{3}e^{2}}\right)^{1 / 2}+{\frac{1}{2 \pi \alpha^{\prime}}}\int d \tau A_{0}\big \vert_{\sigma = 0}^{\pi}\, .
|\nabla\psi''(\iota\varphi_1 + (1-\iota)\varphi_2)|=|\psi'''(\iota\varphi_1 + (1-\iota)\varphi_2)(\iota\nabla\varphi_1 + (1-\iota)\nabla\varphi_2)| &\lesssim(1+ |\varphi_1| + |\varphi_2|)(|\nabla\varphi_1| + |\nabla\varphi_2|) &\lesssim 1+ |\varphi_1|^2 + |\varphi_2|^2 + |\nabla\varphi_1|^2 + |\nabla\varphi_2|^2\,,
\epsilon^{\lambda\mu\nu\rho}\, \mathbf{F}_{\lambda\mu}\, \mathbf{F}_{\nu\rho}\longrightarrow \epsilon^{\lambda\mu\nu\rho}\,\left\{\, X_\lambda(\sigma)\ , X_\mu(\sigma)\,\right\}_{\mathrm{PB}}\left\{\, X_\nu(\sigma)\ , X_\rho(\sigma)\,\right\}_{\mathrm{PB}}\equiv 0
S=(-1)^s\int \left(-\nabla^{\mu}h^{\mu_1\ldots\mu_s}\nabla_{\mu}h_{\mu_1\ldots\mu_s}+[s^2+(d-6)s-2(d-3)]h^{\mu_1\ldots\mu_s}h_{\mu_1\ldots\mu_s}\right)
\mathcal{M}\left( \partial_{M}\partial^{M}A^{N}- M^{2}A^{N}\right) + \delta ( x^{\perp}) \eta^{N}{}_{\nu}\left( \partial_{\mu}\partial^{\mu}A^{\nu}+ \partial^{\nu}\partial_{\perp}A^{\perp}\right) = - \delta ( x^{\perp}) \eta^{N}{}_{\nu}j^{\nu}.
d s^{2}= ( 1 - \frac{r_{+}}{r}) ( 1 - \frac{r_{-}}{r})^{\sigma}d t^{2}- ( 1 - \frac{r_{+}}{r})^{- 1}( 1 - \frac{r_{-}}{r})^{- \sigma}d r^{2}- ( 1 - \frac{r_{-}}{r})^{1 - \sigma}r^{2}( d \theta^{2}+ s i n^{2}\theta d \phi^{2})
p(\beta \alpha^{2}\beta) \geq \phi\left(\lambda \alpha^{2}\lambda\right) \geq \phi\left(\lambda \beta^{2}\lambda\right)+\frac{m}{2}\phi \left(\lambda^{2}\right) &\geq \frac{m+2}{2}\phi\left(\lambda \beta^{2}\lambda\right)=\frac{m+2}{2}p(\lambda \beta^{2}\lambda)
\mathbf{e}^{j}(l_{1},m_{1})\mathbf{e}^{j}(l_{2},m_{2})=\sum_{l=0}^{2j}\mathcal{M}[j]_{m_{1},m_{2},m}^{\l_{1},\l_{2},\l}\mathbf{e}^{j}(l,m)
\int_Q |v_{2k+1}|^p dx \leq \int_{\Sigma}\int_{|x_N|\geq \frac{1}{2 (2k+1)}}|v|^p dx + \int_{\Sigma}\int_{|x_N|\leq \frac{1}{2(2k+1)}}(2k+1)|\varphi((2k+1)x)|^p dx &\leq C + \displaystyle{\int_{\Sigma}\int_{-\frac{1}{2}}^{\frac{1}{2}}|\varphi((2k+1)x',x_N)|^p dx = C + \int_{\Sigma}\int_{-\frac{1}{2}}^{\frac{1}{2}}|\varphi(x',x_N)|^pdx \leq C},
\Phi_{\alpha_{0}}^{* ( 1 )}= N_{\alpha_{1}\alpha_{0}}n_{1}^{\alpha_{1}}+ \cdots , \; \varphi_{\alpha_{1}}^{* ( 1 )}= \left( - \right)^{\epsilon_{\alpha_{0}}\left( \epsilon \left( f_{\alpha_{1}}\right) + 1 \right)}N_{\alpha_{1}\alpha_{0}}\nu_{1}^{\alpha_{0}}+ \cdots ,
[\nabla u_R]^p_{W^{\tau,p}(B_{1/4})}\le\frac{h_0^{-\tau\,p}}{(\Gamma-1-\tau)\,\tau}\, \frac{Ci_0^{2\,p}}{(2-\Gamma)^p\,(\Gamma-1)^p}\,\left[C\,i_0^{2\,p}\,\left(C_5\,\frac{i_0^{4(N+p)}}{(1-\gamma_{i_0-1})^p}\right)^{i_0+1}\,\frac{1-s}{(\gamma_{i_0}-1)^p}\,\mathcal{A}_1\right. &+\left.h_0^{-\Gamma\,p}\,\|u_R\|^p_{L^p(B_1)}\right],
I=\int_{0}^{2\pi}\int_{0}^{R}\frac{drd\theta}{|r\sin\theta|^{\tau}r^{q-1}}=\int_{0}^{2\pi}\frac{d\theta}{|\sin\theta|^{\tau}}\int_{0}^{R}\frac{dr}{r^{\tau+q-1}}= C(\tau)\displaystyle\frac{R^{2-\tau-q}}{2-\tau-q},
||f||^2_{I_F\otimes h_E\otimes\mathrm{det}\,h_E,\omega}=\int_U|f|^2_{I_F\otimes h_E\otimes\mathrm{det}\,h_E,\omega}dV_\omega=\sum_k \int_U|f_k|^2_{h_E\otimes\mathrm{det}\,h_E,\omega}dV_\omega<+\infty,
\sum_{j=1}^{\infty}\mu(S_{j}) \leq c\sum_{j=1}^{\infty}\mu(B(x_{j},r_{j}))=c\mu\left( \bigcup\limits_{j=1}^{\infty}B(x_{j},r_{j})\right)\leq & \leq c\mu(U)\leq c\frac{A}{\lambda_{0}}\mu(S)=\frac{1}{2}\mu\left(S\right) ,
e^{i \Gamma_{f}[ A_{+}, A_{-}]}= \int d \psi^{\dagger}d \psi e^{i \int d^{2}x ( \psi_{L}^{\dagger}( i \partial_{+}+ A_{+}) \psi_{L}+ \psi_{R}^{\dagger}( i \partial_{-}+ A_{-}) \psi_{R})},
m_\psi^*([\sum_{g\in G}a_gg,\sum_{g\in G}b_gg,\sum_{g\in G}c_gg,\sum_{g\in G}d_gg]) &=[\sum_{g\in G}a_g\psi^{-1}(g),\sum_{g\in G}b_g\psi^{-1}(g),\sum_{g\in G}c_g\psi^{-1}(g),\sum_{g\in G}d_g\psi^{-1}(g)]
V_F = \frac{\tau_p(g_s)}{4N(2\pi l_s^2)^2}\left( |\Phi_1\Phi_2 - \epsilon^{-1}\Phi_2 \Phi_1|^2 + |\Phi_2 \Phi_3 -\epsilon^{-1}\Phi_3 \Phi_2|^2 + |\Phi_3 \Phi_1 -\epsilon^{-1}\Phi_1 \Phi_3|^2 \right) ,
M(s) = \left( 1- \frac{(\overline{\gamma}_{S,D}/P_{L_{S,D}}+\overline{\gamma}_{R,D}/P_{L_{R,D}})}{m}s + \frac{(1-\rho)\overline{\gamma}_{S,D}\overline{\gamma}_{R,D}/(P_{L_{S,D}}P_{L_{R,D}})}{m^2}s^{2}\right)^{-m}, \;s < 0
\sim{1\over (4\pi)^{d/2}}\sum_{n=0}^\infty\left[a_{n-\alpha}~{\lambda^{d/2-n+\alpha-1}\over \Gamma\left(\frac d2 -n +\alpha\right)}+b_{n-\alpha}~{\lambda^{(d-1)/2-n+\alpha-1}\over \Gamma\left({d-1 \over 2}-n +\alpha\right)}\right],
z =\mathbf{g}\mathbf{x}+n_{E}&=\sqrt{\alpha P_A}\mathbf{g}\mathbf{v}_{I}s_{I}+\sqrt{\frac{(1-\alpha)P_A}{N_t-1}}\mathbf{g}\mathbf{V}_{N}\sqrt{\mathbf{\Omega}}\mathbf{s}_N+{n}_{E},
\theta_s(\sigma) =\tau^{r-s}d(v) \mbox{for $\sigma=\{v\}$, i.e.\ $|\sigma|=1$}\theta_s(\sigma) = 2^{r\choose2}\tau^{r-s}\sum_{\ell=0}^{r-s}2^{-{s+\ell\choose2}}\tau^{-\ell}d^{(|\sigma|+\ell)}(\sigma) \mbox{for $|\sigma|\ge 2$}
\Biggl\langle{0 \choose 1}, \Bigl(b_{-n_1}\dots b_{-n_N}\bar b_{-m_1}\dots \bar b_{-m_M}\vert 0 \!\!> \Bigr)_0{1 \choose 0}\Biggr\rangle=\pi ( b_{-n_1}\dots b_{-n_N}\bar b_{-m_1}\dots \bar b_{-m_M}\vert 0 \!\!>)
2 \Psi(v)=2 \sum_{j=1}^{N}\Psi(v^j)=\sum_{j=1}^{N}\left[2\psi(\lambda_1(v^j))+\psi(\lambda_2(v^j))+\psi(\lambda_2(v^j))\right]:= \sum_{j=1}^{4N}\psi(z_j),
\langle \psi^{\dagger}({\bf{0}},0)\psi({\bf{x}},t)\rangle = exp\mbox{}[ \mbox{}(\frac{1}{4N_{0}})\sum_{{\bf{q}}}f_{{\bf{q}}}({\bf{x}},t) \mbox{}]\langle \psi^{\dagger}({\bf{0}},0)\psi({\bf{x}},t)\rangle_{free}
\sum_{ij}N_{ij}t_{ijij}= \sqrt n \sum_{ij}X_{ij}t_{ijij}+ n \sum_{ij}\pi_i \pi_j t_{ijij}= O(n^{-1/2}) \sum_{ij}|X_{ij}| + \frac{1}{d}\sum_{ij}A_{ii}A_{jj}\pi_i \pi_j + O(n^{-1});
f_k \stackrel{\eqref{lo3}}\leq h_k \stackrel{\eqref{lo3}}{\leq}\left(-\Delta u_{k-1}\right) \vee f_k \stackrel{\eqref{lo1}}{\leq}h_{k-1}\vee f^* \leq (h_{k-2}\vee f^*) \vee f^* = h_{k-2}\vee f^* \leq \cdots \leq h_1 \vee f^* \stackrel{\eqref{lo3}}\leq \left(-\Delta u_0\right) \vee f^*\mbox{a.e.~in}\Omega,
U^{2,\nu}{{H\rightarrow 0,\nu \rightarrow\infty}\over H\nu=m}\rightarrow P^>(m,2)\qquad\mbox{or}\qquad U^{2,-\nu}{{H\rightarrow 0,\nu\rightarrow\infty}\over H\nu=m}\rightarrow P^<(m,2).
h_{S_\beta}(\eta) = \max_{t\in[-\frac{1}{2},\frac{1}{2}]}\bigl\lbrace t\bigl(-\cos (\beta)\sin(\eta)+\sin(\beta)\cos(\eta)\bigr)\bigr\rbrace & =\max_{t\in[-\frac{1}{2},\frac{1}{2}]}\bigl\lbrace t\sin(\beta-\eta )\bigr\rbrace &= \frac{1}{2}\bigl\vert\sin(\beta-\eta)\bigr\vert \Rightarrow H_{S_\beta}(\eta) =\bigl\vert\sin(\beta-\eta)\bigr\vert.
{\cal P}^* \triangleq \arg \max_{p_k(\cdot),k=1,2,\ldots,K}\Big\{\mathbb E_{\boldsymbol{H},\boldsymbol{E},\boldsymbol{A}}\Big[\sum_{k=1}^K \gamma^{k-1}r(p_k(\Gamma_k),h_k)\Big]\Big\}\ ,
Z_{k}(\lambda_{1}^{2},\lambda_{2}^{2},q^{2};\lambda^{2})=\int_{0}^{1}dzz^{k}ln\left( \frac{q^{2}z(1-z)+\left( \lambda_{1}^{2}-\lambda_{2}^{2}\right) z-\lambda_{1}^{2}}{-\lambda^{2}}\right) ,
E_1 v_{q_1, q_2}^{q_1 - q_2}= A_{1}^{\pi_p}(q) v_{q_1 + 1, q_2}^{(q_1 + 1) - q_2}F_2 v_{q_1, q_2}^{q_1 - q_2}= B_{2}^{\pi_p}(q) v_{q_1, q_2 - 1}^{q_1 - (q_2 - 1)},
\mu(K+\epsilon B)\ge(\mu(K)^{1/n}+\epsilon V(B)^{1/n})^{n}=\mu(K)(1+\epsilon(\frac{\mu(B)}{\mu(K)})^{1/n})^{n}\ge\mu(K)(1+n\epsilon(\frac{\mu(B)}{\mu(K)})^{1/n}),
\int_{\mathbb{R}^d}\vert a_j(\mathbf{x})^*\mathbf{e}_k\vert^2\vert u(\mathbf{x})\vert^2\,d\mathbf{x}\leqslant \int_{\mathbb{R}^d}\vert a_j (\mathbf{x})\vert^2\vert u(\mathbf{x})\vert^2\,d\mathbf{x}&\leqslant C_{\Omega ,\rho}^2\Vert a_j\Vert^2_{L_\rho (\Omega)}\Vert u\Vert^2_{H^1(\mathbb{R}^d)},
|\tilde{A}|^2_{\tilde{g}}+ \widetilde{Ric}(\tilde{\nu},\tilde{\nu}) = e^{\frac{|x|^2}{2n}}\biggl( |A|^2_g+1-\frac 1{4n}|x|^2+\frac{1}{4n^2}|x^{\top}|^2\biggr),
\tilde{\rho}_{\mathrm{s}}( x ) ={\frac{1}{8 \pi^{2}\beta}}\int_{a^{2}}^{b^{2}}{\frac{d y}{x + y}}\sqrt{{\frac{x ( b^{2}- x ) ( y + a^{2})}{y ( b^{2}+ y ) ( x - a^{2})}}}.
(\begin{matrix}{A^{\prime}}^{0}\\{A^{\prime}}^{1}\\{A^{\prime}}^{2}\\{A^{\prime}}^{3}\end{matrix})=(\begin{matrix}1&0&0&0\\ 0&cos\theta&-sin\theta&0\\ 0&sin\theta&cos\theta&0\\ 0&0&0&1\\ \end{matrix})(\begin{matrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{matrix}).
h_{(0,\dots, (k-1)_2),\ell}(z,\dots,z_{\ell}):=\frac{q_{(0,\dots, (k-1)_2)}(x_1,\dots,x_h,z_{1},\dots,z_{\ell})}{q_{(0,\dots,(h-1)_2)}(x_1,\cdots,x_h)\prod_{1\leq i\leq 1<j\leq \ell,\a_j=a_i}(z_j-x_i)}.
L^2_{\mathrm{sym}}(\mathbb{R}^6):=\left\{\psi\in L^2(\mathbb{R}^6)\,\big|\;\psi(\vec{x},\vec{y})=\psi(-\vec{x},\vec{y})=\psi\!\left(\tfrac{1}{2}\,\vec{x}+\vec{y},\tfrac{3}{4}\,\vec{x}-\tfrac{1}{2}\,\vec{y}\right)\right\}\!.
R_{1 \bar{1}1 \bar{1}}=-32p^3K, &R_{1 \bar{2}1\bar{2}}=(p-1)f^{(1)}(0), &R_{1 \bar{1}2 \bar{2}}=-pf^{(1)}(0), &R_{2 \bar{2}2 \bar{2}}=-\frac{f^{(3)}(0)}{16}+\frac{f^{(1)}(0)^2}{16pK}.
\|T+xS\|_{(p,k)}^p+\|T-xS\|_{(p,k)}^p \leq 2\|T\|_{(p,k)}^p+2\|xS\|_{(p,k)}^p &=2\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T)+2\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(x^2S^*S).
\int_{\mathbb{S}(\mathbb{C}_{n})}\sum_{m=1}^{n}\Big [f^{\prime}(\lambda_{m}^{(n)}(X_{s-}^{(n)})) \mathrm{tr}(D\lambda_{m}^{(n)}(X_{s-}^{(n)})vv^{\ast})\Big ]\pi(dv)=\int_{\mathbb{S}(\mathbb{C}_{n})}\mathrm{tr}\left(\mathcal{D}f(X_{s-})(vv^*)\right)\pi(dv).
\frac{1}{4\pi}\int_{S^2}\frac{1}{4}|\dot h|^2 = \frac{1}{4\pi}\int_{S^2}\frac{1}{18}R_{ij}X^iX^j R_{kl}X^kX^l + \frac{1}{4}|k|^2 &= \frac{1}{18}B + \frac{1}{4\pi}\int_{S^2}\frac{1}{8}\cdot \frac{1}{6}R_{ij}(X^iX^j - \frac{1}{3}\delta^{ij}) R_{kl}(X^kX^l - \frac{1}{3}\delta^{kl}) = \frac{1}{18}B + \frac{1}{48}B - \frac{R^2}{432}
V_\theta={1\over (4\pi)^2}\left[{{\cal I}_K \over z_<^4}+{{\cal I}_I\over z_>^4}+{{\cal V}(\tau)\over z_>^4}\right] +{\beta_4 (\beta\theta +1) \over (4\pi)^{2}}\left[{1\over z_+^4}\ln\left({z_+\over z_0}\right) +{1\over z_-^4}\ln\left({z_-\over z_0}\right)\right]
\begin{array}{r c l}{{{\bf{Q}}_{-{\frac{1}{k}}}^{+}\vert \Omega_{- s}^{-}\rangle}}&{{=}}&{{0}}\\{{\{{\frac{({\bf{Q}}_{-{\frac{1}{k}}}^{-})^{r}}{[ r ] !}}\vert \Omega_{- s}^{-}\rangle}}&{{;}}&{{o \le r \le k - 1 \}}}\\ \end{array}
T_iT_j =T_jT_i,\ \ \ \a_{ij}=0, T_iT_jT_i =T_jT_iT_j,\ \ \ \a_{ij}a_{ji}=1, (T_iT_j)^2 = (T_jT_i)^2,\ \ \ \a_{ij}a_{ji}=2, (T_iT_j)^3 = (T_jT_i)^3,\ \ \ \a_{ij}a_{ji}=3,
\partial_{\sigma}X^0|_{\sigma=0,\pi}=\partial_{\tau}X'^{5\cdots 9}|_{\sigma=0,\pi}=\partial_{\tau}X^{1\cdots4}|_{\sigma=0}=g_{ij}\partial_{\sigma}X^j+2\pi\alpha'B_{ij}\partial_{\tau}X^j|_{\sigma=\pi}=0
(a_1\circ b)S(\sigma(a_2))(a_3\circ c)= (B_1(a_1)bS(B_2(a_2)))S(\sigma(a_3))(B_1(a_4)cS(B_2(a_5))) = (B_1(a_1)bS(B_2(a_2)))B_2(a_4)S(B_1(a_3))(B_1(a_5)cS(B_2(a_6))) = (B_1(a_1)bS(B_2(a_2)))B_2(a_3)S(B_1(a_4))(B_1(a_5)cS(B_2(a_6))) = B_1(a_1)bcS(B_2(a_2))=a\circ(bc),
\begin{cases}\Delta_{X,z}^{\mu}\phi_1 = \mu \beta F \ \ \mathrm{in}\ \ \mathcal{S}_b, \phi_1|_{z=0}= 0, \ \ \partial_z \phi_1|_{z=-1 + \beta b}= - \nabla_X \cdot \big(\mathcal{L}_{1}^{\mu}[\beta b]\nabla_X\psi), \end{cases}
y := -wxw = -wsw = cws + swc & = wx + \frac{1}{2}x^2 wx + xw + \frac{1}{2}xwx^2; h := [x, y] = xwx + x^2 w + \frac{1}{2}x^2wx^2 - wx^2 - \frac{1}{2}x^2wx^2 - xwx & = x^2 w - wx^2
\sum_{H=1}^{6}(H-1)^{A+B}\cdot C(H-1)^{C-1}+\sum_{H=7}^{8}6^{A}\cdot(H-1)^{B}\cdot C(H-1)^{C-1}+\sum_{H=9}^{10}6^{A}\cdot8^{B}\cdot C(H-1)^{C-1}.
:{\hat \psi_{m}}^\dagger \hat \psi_n:~=~{\hat \psi_{m}}^\dagger \hat \psi_n{}~~{\rm if}~ n>0~~~~~~~~:{\hat \psi_{m}}^\dagger \hat \psi_n:~=~- \hat \psi_n{\hat \psi_{m}}^\dagger~~{\rm if}~ n\le 0
K_r (u)=\left( \begin{array}{ll}2y_{0}(qtu-q^{-1}t^{-1}u^{-1}) \; y_{+}(qu^2 -q^{-1}u^{-2}) y_{-}(qu^2 -q^{-1}u^{-2}) \; -2y_{0}(t^{-1}u-tu^{-1})\end{array}\right )\, ,
S_{SMaxwell}=\int d^{4}x[-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{\vartheta}{32\pi^{2}}F_{\mu\nu}*F^{\mu\nu}-\frac{i}{e^{2}}\lambda\sigma^{\mu}\partial_{\mu}\overline{\lambda}+\frac{1}{2e^{2}}D^{2}]