harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lambda_{1,j}:= \left| \frac{\partial^{l+j}}{\partial m^{l+j}}(qf(mn+mh) - qf(mn))\right| &= \frac{qh}{m^{l+j-1}}t_0^{l+j-1}\left| (l+j)f^{(l+j)}(t_0) + t_0 f^{(l+j+1)}(t_0) \right| &= \frac{qh}{m^{l+j-1}}t_0^{l+j-1}\left| f^{(l+j)}(t_0) \right| \left| l+j + \frac{t_0 f^{(l+j+1)}(t_0)}{f^{(l+j)}(t_0)}\right|
q^{{n}\choose{2}}C_n\left(\frac{1}{q},t\right)=\sum_{\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k),\atop \lambda_1+k\leq n}\prod_{i \geq 1}\left[\lambda_i-\lambda_{i+2}+1 \atop \lambda_{i+1}-\lambda_{i+2}\right]_t q^{{\lambda_1\choose 2}+|\lambda|}.
u(x,t) = b \left[ v_G(x) - t v_G'(x) \right] + a_0 v_G'(x) + \sum_{n \in \mathbb{N}}a_{+n}u_{+n}(x) e^{i \omega_n t}+\sum_{n \in \mathbb{N}}a_{-n}u_{-n}(x) e^{-i \omega_n t},
h(z) = \exp\left( + 4\eta(1) \bar z + \eta(1) - \frac{m}{2}\eta(\tau) - 4 \eta(1)\bar z + \eta(1) \sum_{k=1}^4 \Re(b_k) + \eta (1) \frac{m}{2}\tau\right) = \exp\left(\eta(1)\left( 1 + \sum_{k=1}^4 \Re(b_k)\right)\right) \exp\left( m \pi i \right),
[ P , R ]_{\scriptscriptstyle S N}\; \hat{\longrightarrow}\; \sum_{i = 1}^{p}( - 1 )^{i + 1}\widehat{X}_{( 1 )}\cdots{\widehat{\widehat X}}_{( i )}\cdots \widehat{X}_{( p )}\{- \{\widehat{X}_{( i )}, Q \}, \widehat{R}\}
\int_{0}^{1}| \tau_{E,\alpha}^{\prime}(h)|^{s}\mathrm{d}h \le \sum_{k=1}^{M}\int_{I(h_{k})}C_{h_{k}}^{s}(1 + |h-h_{0}|^{-\frac{1}{\sigma^{\prime}}})^{s}\mathrm{d}h \le 2^{s}\overline{C}^{s}M\int_{-1}^{1}(1 + |h|^{\frac{-s}{\sigma^{\prime}}}) \mathrm{d}h,
\int_{\mathbb R^{j}}g(x){\mathrm M}_{t}^{(j)}({\mathrm d}x) \, =\, \int_{\mathbb R^{j}}g(x){\mathrm M}_{0}^{(j)}({\mathrm d}x ) + \int^{t}_{0}[ \mathcal A_{s}^{(j)}({\mathrm M}^{(j+1)}) g]{\mathrm d}s \, ; 0 \le t \le T\, , \, \, j \, =\, 2, \ldots , k-1 \,
\Gamma_{a , s}( l_{1 s}) = \tilde{\Gamma}_{s}^{- 1}\Gamma_{a , s}^{( o )}( l_{1 s}) \tilde{\Gamma}_{s}, \qquad \Gamma_{b , s}( l_{2 s}) = \tilde{\Gamma}_{s}^{- 1}\Gamma_{b , s}^{( o )}( l_{2 s}) \tilde{\Gamma}_{s}
S_{e n / I}^{e^{\prime}n^{\prime}/ I}= i V \mathrm{e x p}- \frac{2 i \pi}{k}[ e^{\prime}( \widetilde{n}- 1 / 2 ) + e ( \widetilde{n^{\prime}}- 1 / 2 ) ] , \S_{e n / I}^{e^{\prime}n^{\prime}/ I I}= 0
c(Y)=(3f^* H+6f^* \pi^* L-2E_1)(1+E_1)\frac{(1+f^* Z_1-E_1)(1+f^* Z_2 -E_1)(1+f^* Z_3-E_1)}{(1+f^* Z_1)(1+f^* Z_2)(1+f^* Z_3)}f^*c(X_0).
\phi(x,y) = 1 \wedge (\alpha x + \beta y) \wedge (\alpha x + \beta y)^2 = 1 \wedge (\alpha x + \beta y) \wedge [(\ell - b\alpha + a\beta)x^2 + mxy + (n - d\alpha + c\beta)] = (ax^3 + bx^2y + cxy^2 + dy^3)(1 \wedge \alpha \wedge \beta).
\sum\limits_{s,\hat{s}\in\mathcal{S}:s\neq\hat{s}}p_{\hat{S}|S}(\hat{s}|s) \leq\sum\limits_{s,\hat{s}\in\mathcal{S}:s\neq\hat{s}}p_{\hat{S}'|S}(\hat{s}|s) &\leq\frac{1}{\rho}\sum\limits_{s,\hat{s}\in\mathcal{S}:s\neq\hat{s}}p_S(s)p_{\hat{S}'|S}(\hat{s}|s) &=\frac{\mathbb{P}(S\neq\hat{S}')}{\rho}.
\begin{array}{lll}n = 0 (\lambda + c\gamma )p_{0,0}= \xi p_{0,1}+ \mu p_{1,1}n\geq 1 (\lambda + c\gamma + n \xi)p_{0,n}= \lambda p_{0,n-1}+ (n+1)\xi p_{0,n+1}. \end{array}
{\cal L}_{c . t .}^{( 1 )}\, = \,{\frac{\sqrt{- g}}{( 4 \pi )^{2}\epsilon}}\lbrack{\frac{1}{1 2 0}}R^{2}+{\frac{7}{2 0}}R_{\mu \nu}R^{\mu \nu}-{\frac{5 3}{1 8 0}}\epsilon^{\alpha \beta \gamma \delta}\epsilon_{\mu \nu \rho \sigma}{R^{\mu \nu}}_{\alpha \beta}{R^{\rho \sigma}}_{\gamma \delta}\rbrack
\langle e_{a,1}(t,R \lambda(t)),\phi_{0}(R)\rangle_{L^{2}(R dR)}&= -\frac{4}{\lambda(t)}\int_{t}^{\infty}\frac{\lambda''(s)}{1+s-t}ds -2 \int_{0}^{\infty}\sin(t\xi) \widehat{v_{2,0}}(\xi) \xi^{2}K_{1}(\xi \lambda(t))d\xi +\frac{4 \lambda''(t) \log (\lambda(t))}{\lambda(t)}+f_{1}(\lambda(t),\lambda'(t),\lambda''(t))
T_1\ll \sum_{\stackrel{0\leq m_j\leq Q}{G(\mathbf{m})\neq0}}\sum_{h|G(\mathbf{m})}\sum_{d'\leq \frac{Q}{h}}(hd')^{-2+\varepsilon}h^{\frac{1}{2}}\prod_{j=1}^3\min\big\{M,\frac{hd'}{m_j}\big\}&\ll M^2Q^{\varepsilon}+MQ^{1+\varepsilon}+Q^{2+\varepsilon}.
\mathbb{P}(X_{l+1}=\gamma|X_{l}=\gamma^{\prime})=K_{\delta,\gamma-\gamma^{\prime}}\frac{\vec{t}^{\delta-w(\gamma-\gamma^{\prime})}}{S_{\delta}(\vec{t})},\mathbb{P}(X_{l+1}^{\prime}=\gamma|X_{l}^{\prime}=\gamma^{\prime})=K_{\delta,\gamma-\gamma^{\prime}}\frac{\vec{t}^{\prime\delta-w(\gamma-\gamma^{\prime})}}{S_{\delta}(\vec{t}^{\prime})}.
\frac{d\,h\left( t,\frac{{D_B}^{\frac12}}{\mu}\right)}{dt}= \sum\limits_{k=0}^d(-k)\frac{1}{2\mu}\frac{\Gamma \left( \frac{k+1}2\right)}{\Gamma \left(\frac 12\right)}a_{d-k}\left( \frac{t}{2\mu}\right)^{-k-1}+
\begin{aligned}\int_0^t(t-s)^{1-\gamma-\frac{1}{\alpha}}|M(y(s))|_\alpha ds \leq C\eta_\infty\int_0^t (t-s)^{1-\gamma-\frac{4-p}{2p}}\cdot s^{\frac{2}{p}-\frac{5}{2}+2\gamma-\gamma}ds \ \|y\|^2 &=C\eta_\infty B\left(\frac{2}{p}-\frac{5}{2}+\gamma+1,1-\gamma-\frac{4-p}{2p}+1\right)\|y\|^2,\ \forall t\geq0. \end{aligned}
\frac{{g\left({x_1}\right)f\left({x_2}\right) - g\left({x_2}\right)f\left({x_1}\right)}}{{g\left({x_2}\right) - g\left({x_1}\right)}}= f\left( \xi \right) -\frac{{g\left( \xi \right)}}{{g'\left( \xi \right)}}f'\left( \xi\right).
\left(f_{\omega}^{R*}\,\tilde{\beta}_{\omega''\omega}^{R*}+f_{\omega}^{R}\,\tilde{\alpha}_{\omega''\omega}^{R}\right)\left(\tilde{\beta}_{\omega''\omega'}^{R}\,f_{\omega'}^{R}+\tilde{\alpha}_{\omega''\omega'}^{R*}\,f_{\omega'}^{R*}\right) \,,
\widetilde{X}(T) :=\Vert \sigma\Vert_{\widetilde{L}^{\infty}_{T}(\widetilde{B}^{\frac{N}{2}+1-\alpha,\frac{N}{2}})}+\Vert \widetilde{u}\Vert_{\widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1})}+\Vert \sigma\Vert_{L^1_{T}(\widetilde{B}^{\frac{N}{2}+1,\frac{N}{2}+2-\alpha})}+ \Vert \widetilde{u}\Vert_{L^1_T (\dot{B}^{\frac{N}{2}+1}_{2,1})}.
H_{c}= \int \mathrm{d}^{2}x \left[ \Pi_{\alpha}^{*}\Pi_{\alpha}+ \partial_{i}Z_{\alpha}^{*}\partial_{i}Z_{\alpha}- ( Z_{\alpha}^{*}\partial_{i}Z_{\alpha}) ( Z_{\beta}\partial_{i}Z_{\beta}^{*}) \right]
y(s_l)=y(t_n) +\int_{t_n}^{t_n+\zeta h}[(1-\theta)f(t_n,y(t_n),y(qt_n))+\theta f(t_{n+1},y(t_{n+1}),y(qt_{n+1}))]{\rm d}t &+\int_{t_n}^{t_n+\zeta h}g(t_n,y(t_n),y(qt_n)){\rm d}B(t).
V(G[H;U]) = V(G) \times V(H) , E(G[H;U]) = \left\{(u_1,v_1)(u_2,v_2) \ \middle|\ \left.\begin{aligned}u_1 = u_2 v_1 v_2 \in E(H) , & &u_1 u_2 \in E(G) v_1, v_2 \in U\end{aligned}\right. \right\}.
\left[\frac{2\sqrt{2L}}{3}\right]^d-\left[\frac{2\sqrt{2L}}{3}-\frac{L^2}{12b^2}\right]^d \leq d\frac{L^2}{12b^2}\left[\frac{2\sqrt{2L}}{3}\right]^{d-1}=\frac{dL^{\frac{d+3}{2}}}{12b^2}\left[\frac{\sqrt{8}}{3}\right]^{d-1}.
A = 2\int_{0}^{1/2}\left( \frac{1}{N}\# \left\{1 \leq m \neq n \leq N: |x_m - x_n| \leq s \right\}- 2Ns \right)^2 ds =2\int_{0}^{1/2}\left( 4s\right)^2 ds = \frac{4}{3}.
\big\||x|^{-\tau-1}|u|^{p-1}|I_\alpha\ast|\cdot|^{-\tau}|u|^p|\big\|_{L_x^{\frac{2n}{n+2}}}\lesssim \||x|^{-\tau-1}|u|^{p-1}\|_{L_x^{a_1}}\||x|^{-\tau}|u|^p\|_{L_x^{b_1}}&=\||x|^{-\frac{\tau+1}{p-1}}u\|^{p-1}_{L_x^{(p-1)a_1}}\||x|^{-\frac{\tau}{p}}u\|^{p}_{L_x^{pb_1}}&\lesssim \|\nabla u\|_{L_x^r}^{2p-1},
I_{2}= ( - 1 ) \int \, \delta^{3}\! \left({\vec{r}_{1}+ \vec{r}_{2}- \vec{r}}\right) \, \delta ( r_{1}+ r_{2}- t ) \, \frac{d^{3}r_{1}}{4 \pi \, r_{1}}\, \frac{d^{3}r_{2}}{4 \pi \, r_{2}\,{\ell_{0}}^{2}}
G^{(\delta)}(t)=G_{1,k}^{(\delta)}(t)+G_{2,k}^{(\delta)}(t), G_{1,k}^{(\delta)}(t):={V^{\delta}_1(t)}\int_t^{\eta_k}\frac{g(x)}{V_1(x)}\Bigl(\int_{a(x)}^tv_1^{-p'}\Bigr)^{1-\delta}dx, G_{2,k}^{(\delta)}(t):={V^{\delta}_1(t)}\int_{\eta_k}^{a^{-1}(t)}\frac{g(x)}{V_1(x)}\Bigl(\int_{a(x)}^tv_1^{-p'}\Bigr)^{1-\delta}dx.
Uf(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-(x^{2}+y^{2})}e^{-2ixy}f(t+x)e^{-t^{2}/2}dt=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}B(z,t)f(t)dt,
u^{(N-1)}_{1N}=u^{(N-2)}_{1N}e^{-i\phi_{N-1}}\cos(\psi_{N-1})-u^{(N-2)}_{NN}\sin(\psi_{N-1}), u^{(N-1)}_{NN}=u^{(N-2)}_{1N}e^{-i\phi_{N-1}}\sin(\psi_{N-1})+u^{(N-2)}_{NN}\cos(\psi_{N-1}).
V_{{\rm CM}}(D)=\left( \begin{array}{cc}e^{-D^- D^+}e^{-\frac{1}{2}D^- D^+}\left( \frac{I- e^{-D^- D^+}}{D^- D^+}\right) D^- e^{-\frac{1}{2}D^+ D^-}D^+ I- e^{-D^+ D^-}\end{array}\right)
j_P^+(T_w^{P*}) = \sum_v c_vj_P^+(E^P_{o_{-,P}}(v)) & = \sum_v c_vE_{o_-}(v) & = \sum_v c_vj_P^{-*}(E^P_{o_{-,P}}(v)) & = j_P^{-*}(T_w^{P*}) = T_w^{*}.
n_s(I\cup J) =\max \{\lvert v\rvert_S\mid v\in I\cup J, \tau_s(v)\neq v\}&=\max (\max\{\lvert v\rvert_S\mid v\in I, \tau_s(v)\neq v\},\max \{\lvert v\rvert_S\mid v\in J, \tau_s(v)\neq v\}) &=\max (n_s(I),n_s(J))
\begin{aligned}\left|\sum_{i\ge n}u_i^{(r)}\right|\le\sum_{i\ge n}|u_i^{(r)}|\le M\cdot\sum_{i\ge n}\frac{2^{p_i}}{p_i!}&\le Mk\cdot\sum_{p\ge p_n}\frac{2^{p}}{p!}\le \frac{Mke^2\cdot 2^{p_n}}{p_n!},\end{aligned}
\sum_{\sigma \in X(j)}\|I^\sigma(I_\sigma(A))\| = \sum_{\sigma\in X(j)}\sum_{\sigma \subset \tau \in A}w(\tau)= \sum_{\tau \in A}\sum_{\sigma \subset \tau}w(\tau) = \sum_{\tau \in A}{k+1 \choose j+1}\cdot w(\tau) ={k+1 \choose j+1}\cdot \|A\|.
|Df^{n}(x)(v)| \geq \frac{1}{2}|(f_0^n)'(w_0)| &\geq \frac{C}{2}\lambda_0^n \frac{\min_{i=0}^{n}|f_0^i(w_0)-c(0)|}{|w_0-c(0)|}&\geq \frac{C}{8}\lambda_0^n \frac{\min_{i=0}^{n}|\xi_{i}(x)-c(\lambda^{i-1}z_0)|}{|w_0-c(z_{0}/\lambda)|}.
4\pi\int_{M}(\phi_{0}+\epsilon)f_{\epsilon}\omega\cdot\nabla(\phi_{0}+\epsilon) =2\pi\int_{M}f_{\epsilon}\omega\cdot\nabla(\phi_{0}+\epsilon)^{2}=-2\pi\int_{M}(\phi_{0}+\epsilon)^{2}\nabla f_{\epsilon}\cdot\omega \,.
{\cal H}_{c a n .}( x , t ) = \frac{1}{2}\pi^{2}+ \frac{1}{2}( \partial_{x}\varphi )^{2}+ ( 1 - \cos ( \varphi ) ) \quad \mathrm{w i t h}\quad \pi = \frac{\partial{\cal L}}{\partial ( \partial_{t}\varphi )}
0 \equiv A+B =\varphi(f+D^2u(\nu, \nu))+\bar{\nabla}f\cdot \bar{\nabla}\varphi- h(\bar{\nabla}f, \bar{\nabla}f) =\varphi^2(-H+(k-1)\cot \theta-(n-k) \tan \theta)+ \cot \theta |\bar{\nabla}f|^2 -h(\bar{\nabla}f, \bar{\nabla}f).
S^{f,N, i}_t:=f \big(Z^i_t, B_t \big)-\varphi(Z^i_0, B_0)-\int_0^t \frac{1}{2}\mathrm{Tr}\big[ \hat a_0(s, X^{\alpha,i}, \varphi^{N,X}_s) \nabla^2 \varphi(Z^i_s,B_s)\big]\mathrm{d}s,~ t \in [0,T].
\lim_{B\rightarrow0+0}\left(\left(\frac{1}{\delta}+1\right)e^{B}-1 \right)\frac{G(0)-G(B)}{B}= \lim_{B\rightarrow0+0}\left(\left(\frac{1}{\delta}+1\right)e^{B}-1 \right)\frac{G(-\log1)-G(-\log e^{-B})}{1-e^{-B}}\cdot\frac{1-e^{-B}}{B}\le \frac{1}{\delta}G(0).
\begin{gathered}\mathcal N_T =\left\lbrace Q\in\mathcal S\colon Q=s_*\left(n\otimes n -\frac 13 I\right),\: n\in\mathbb S^2 \right\rbrace, s_*= s_*(T) = \frac{b+\sqrt{b^2-24 \alpha (T-T_*)c}}{4c}.\end{gathered}
{{\mathcal B}_i^{AF}}\left(\pmb \rho \right) = \frac{1}{{1 + \sqrt{\frac{{{Z_i}}}{{\left({{X_i}+{Y_i}+ 1}\right)\left({{W_i}+ 1}\right)}}+ \frac{1}{{{X_i}+{Y_i}+ 1}}}}},
\mu_{0}J_{\mathrm{e m}\, \mu}= \frac{i \hbar}{2 q \lambda^{2}}\left[ \psi^{*}\left( \nabla_{\mu}\psi + i \frac{q}{\hbar}A_{\mu}\psi \right) - \left( \nabla_{\mu}\psi^{*}- i \frac{q}{\hbar}A_{\mu}\psi^{*}\right) \psi \right] .
\frac{\partial(g_{i}\circ f)}{\partial x_{j}}(x)=\frac{\partial g_{i}}{\partial y_{1}}(y)\frac{\partial f_{1}}{\partial x_{j}}(x)+\cdot\cdot\cdot+\frac{\partial g_{i}}{\partial y_{m}}(y)\frac{\partial f_{m}}{\partial x_{j}}(x),
\begin{cases}\sum_{\beta}\Phi_\beta(r) =1 , r>0 \operatorname{supp}\Phi_\beta \subset \{r: 2^{\beta-2}\leq r \leq 2^\beta\}, \ \beta =1,2,\dots,m-1 \operatorname{supp}\Phi_0 \subset \{r:0<r\leq 1\}\operatorname{supp}\Phi_m \subset \{r: r\geq \frac{s}{40}\}\end{cases}
C_{\mathcal{R}_{1}(i)\upharpoonright (T' \cap T)}(p)=|\mathcal{R}_{1}(i)\upharpoonright (T' \cap T)\cap g^{-1}(p)| = |\mathcal{R}_{1}(i)\upharpoonright (T' \cap T) \cap \pi_{T(i')}^{-1}(p)|\le{n-i' \choose i - i'}.
D_uX_t - D_v X_t = -\int_u^t \sum_{j=1}^l \theta_j \nabla f_j(X_r) (D_u X_r - D_v X_r) dr + \int_v^u \sum_{j=1}^l \theta_j \nabla f_j(X_r) D_vX_r dr \,.
\frac{\partial W(F)}{\partial F_{\alpha\beta}}=2\, F_{\alpha\beta}\, \psi^{\prime}(I)\,,\mathbb{C}_{\alpha\beta\gamma\delta}= 4\, F_{\alpha\beta}\, F_{\gamma\delta}\, \psi^{\prime\prime}(I)+2\, \delta_{\alpha\gamma}\,\delta_{\beta\delta}\,\psi^{\prime}(I)\,,
|H_p^n(\Delta)|_n=\frac{1}{n}\left(1+\Delta-\frac{1}{(1+\Delta)^{\frac{1}{p-1}}}\right)\left(1-\frac{1}{(1+\Delta)^{\frac{p}{p-1}}}\right)^{\frac{n-1}{p}}|B_p^{n-1}|_{n-1}- \left| C_p^n\left(1-\frac{1}{(1+\Delta)^{\frac{1}{p-1}}}\right)\right|_n.
\{\Pi_{i}( x ) , \theta ( y ) \}^{*}= 3 D{\frac{\partial_{i}}{\nabla =^{2}}}\delta ( x - y ) ; ~ ~ \{\Pi_{i}( x ) , \Pi_{0}( y ) \}^{*}= 3 D - m^{2}{\frac{\partial_{i}}{={\nabla^{2}}}}\delta ( x - y ) .
\nabla_p H_\epsilon(t,x,m,p) = \frac{(1+\frac{\epsilon}{r}H(t,x,\kappa_\epsilon(m),p)^{(r-1)/r})\nabla_p H(t,x,\kappa_\epsilon(m),p)}{(1+\epsilon H(t,x,\kappa_\epsilon(m),p)^{(r-1)/r})^2}
\left( p_{2}p_{3}- p_{1}p_{2}^{2}+ p_{1}q_{1}^{- 1}q_{2}^{- 2}\right) \partial_{p_{2}}+ \left( p_{3}^{2}- q_{2}p_{2}^{2}q_{1}^{- 1}- q_{1}^{- 2}q_{2}^{- 1}\right) \partial_{p_{3}},
\chi_{l l}^{R}\approx 0 , \quad \chi_{l k}^{R}\approx \chi_{k l}^{R}, \quad \chi_{l k}^{J}\approx \chi_{k l}^{J}, \quad ( l \neq k ) \quad \sin \chi_{1 1}^{J}+ \sin \chi_{2 2}^{J}+ \sin \chi_{3 3}^{J}\approx 0 .
\sum^{j^{*}}_{i=1}\Pr \{X \in A_{D}(y_i) \}- \sum^{i^{*}}_{i=1}\Pr \{X \in A_{D}(y_i) \}& \overset{(a)}{\leq}1- \gamma - \delta + \Pr \{X \in A_{D}(y_{j^*}) \}- (1-\epsilon-\delta) & = \Pr \{X \in A_{D}(y_{j^*}) \}+ \epsilon - \gamma & \overset{(b)}{\leq}\Pr \{X \in A_{D}(y_{j^*}) \}+ \Pr \{X \in A_{D}(y_{i^* + 1}) \},
(\sigma f)(x_1,x_2,x_3) = f(x_2,x_3,x_1) = x_2 x_3^2 x_1^3 = x_1^3 x_2 x_3^2 (\tau f)(x_1,x_2,x_3) = f(x_2,x_1,x_3) = x_2 x_1^2 x_3^3 = x_1^2 x_2 x_3^3 ( (\sigma \tau)f)(x_1,x_2,x_3) = f(x_3,x_2,x_1) = x_3 x_2^2 x_1^3 = x_1^3 x_2^2 x_3 (\tau\sigma f)(x_1,x_2,x_3) = f(x_1,x_3,x_2) = x_1 x_3^2 x_2^3 = x_1 x_2^3 x_3^2
\frac{1}{2}\int d^4x k_{ab}{\bf B}^a ({\bf \dot{A}}^b + \mbox{\boldmath $\alpha^b$})-\frac{1}{2}\sum_n k_{ab}q^a_n \int{\bf A}^b (z_n) \cdot d{\bf z}_n= \int d^4 x \left( \partial_\mu V^\mu +\frac{1}{2}\, k_{ab}\mbox{\boldmath $\alpha^a\beta^b$}\right) \ .
\iint_{\Omega_i}b uv \,dx\,dy = \iint_{\Omega_i^{need}}b u v\,dx\,dy = \iint_{S^{need}}\pi_1(x,y) \pi_2(x,y)\, \widehat b |\operatorname*{det}M_{A,i}^\prime| \,dx\, dy .
| \langle n_{1}, \ldots , n_{r}; \bar{n}_{1}, \ldots , \bar{n}_{r}| \mathrm{\boldmath ~ \e t a ~}; M ; C_{r}\rangle |^{2}={\frac{M !}{n_{1}! \cdots n_{r}! \bar{n}_{1}! \cdots \bar{n}_{r}!}}| \eta_{1}|^{2 n_{1}}\cdots | \eta_{r}|^{2 n_{r}}| \eta_{- 1}|^{2 \bar{n}_{1}}\cdots | \eta_{- r}|^{2 \bar{n}_{r}}.
s_{0,\gamma}:=\inf_{[0,\gamma]}\delta(f,\alpha) - 2\sup_{[0, \gamma]}\sqrt{\Delta(Dg,\alpha)}\quad\hbox{and}\Sigma_{\gamma, 1}:=\sup_{[\gamma, 1]}\delta(f,\beta) + 2\sup_{[\gamma, 1]}\sqrt{\Delta(Dg,\beta)}.
P_{DH,i,j}'(p)=P_{DH}'(p)\left(\sum_{\alpha,\beta}\frac{\alpha^{\vee,i}\beta^{\vee,j}}{(2\chi-p)(\alpha^{\vee})(2\chi-p)(\beta^{\vee})}+\sum_{\alpha}\frac{\alpha^{\vee,i}\alpha^{\vee,j}}{(2\chi-p)(\alpha^{\vee})^2}\right).
\left| \mathbb{E}_{\mu_N^{\sigma}}\left[ X_i \right] \right| = \left| \mathbb{E}_{\nu_{N, 1}^{\sigma}}\left[ X_i \right] \right| \lesssim \left| \mathbb{E}_{\nu_{N, 0}^{\sigma}}\left[ X_i \right] \right| +1
\prod_{k=1}^\infty (1+\alpha^{-1}X^{2k-1}) =\prod_{k=0}^\infty (1+\alpha^{-1}X^{2k+1})=\sum_{k=0}^\infty\frac{\alpha^{-k}X^{k^2}}{(1-X^2)\ldots(1-X^{2k})}&=\prod_{k=1}^\infty\frac{1}{1-X^{2k}}\sum_{k=0}^\infty\alpha^{-k}X^{k^2}\prod_{l=0}^\infty(1-X^{2l+2k+2}).
= \exp\left\{\sum_{n=1}^{\infty}\left(\frac{- n^2 y^{n+3}+ (2n^2+2n-1)y^{n+2}-(n^2+2n+1) y^{n+1}+ y^2 + y}{(1-y)^3}\right) \frac{z^n}{n^3}\right\}
f_{2,\left( 0,1\right)}\left( r\right) =\sqrt{2\pi}J_{0}\left(\sqrt{a^{2}+b^{2}}r\right) ,\qquad f_{2,\left( 1,1\right)}\left( r\right) =f_{2,\left( 1,2\right)}\left( r\right) =0,
\lambda_{2n-1}\left(\begin{bmatrix}A X X^* B\end{bmatrix}\right) \le \lambda_{n}\left(\begin{bmatrix}A 0 0 0\end{bmatrix}\right)+ \lambda_{n}\left(\begin{bmatrix}0 0 0 B\end{bmatrix}\right) &= \lambda_{n}(A) + \lambda_n(B) \le \lambda_n(A+B).
\psi ({\bf x}, 0 ) = \left( \frac{\epsilon_{p}+ m}{2 \epsilon_{p}}\right)^{\frac{1}{2}}\left( \begin{array}{c}{{\displaystyle - \frac{p}{\epsilon_{p}+ m}}\chi_{1}}\\{\chi_{1}}\\ \end{array}\right) e^{i p z}
{\cal L}_{\rm eff}=\sum_{i,j}{1\over 2}K_{\phi_i,{\bar\phi}_j}\nabla \phi_i\cdot \nabla{\bar\phi}_j -{\rm e}^K \left[\ \sum_{i,j}K^{\phi_i\{\bar\phi}_j}(D_{\phi_i}W)(D_{{\bar\phi}_j}W)^{\dag}- 3 |W|^2\ \right]
d \pi_{q}^{1 +}= \frac 1 2 \Omega^{( 0 )}\pi_{q}^{1 +}- \frac 1 4 \Omega_{q p}\pi_{p}^{1 +}- \frac 1 2 \gamma_{q \dot{p}}^{i}\Omega^{+ 2 i}\pi_{\dot{p}}^{1 -},
\sum_{i_q=i_{q-1}+x/2}^{\min(i_{q-1}+t,m)}u(m - i_{q}) u(i_q-i_{q-1})^2 \leq c_{7}c_5 u(m-i_{q-1}) \sum_{i_q= i_{q-1}+x/2}^{\min(i_{q-1}+t,m)}u(i_q-i_{q-1})^2 \leq c_{7}c_5 u(m-i_{q-1}) D(t).
\partial_i K(\xi-d\Gamma(m)) Q_0(k,\xi)=\partial_i K(\xi-d\Gamma(m))\frac{\omega(k)}{H_\mu(\xi)-\Sigma(\xi)+\omega(k)+1}&+\partial_i K(\xi-d\Gamma(m))\frac{1}{H_\mu(\xi)-\Sigma(\xi)+\omega(k)+1}Q_0(k,\xi)
F_{\alpha}({\bf x}) =\sum_{j\not\in Q_{\alpha}}\big(-\tfrac{Z}{2}|x_j|+\tfrac{Z}{2}\sqrt{|x_j|^2+1}\big) &\quad+\sum_{j,k\not\in Q_{\alpha}, j<k}\big(\tfrac{1}{4}|x_j-x_k|-\tfrac{1}{4}\sqrt{|x_j-x_k|^2+1}\big)\,.
\bar{\Psi}^{\prime\rho}{}_{r}(z^{\prime})=\Omega(z;g)^{-\eta}D^{\rho}{}_{\sigma}(L^{-1}(z;g))D_{r}{}^{s}(U^{-1}(z;g))\bar{\Psi}^{\sigma}{}_{s}(z)\,.
\tan^{- 1}\left( \frac{{\sqrt{4 \xi - 1}}}{3}\right) = \frac{{\sqrt{4 \xi - 1}}}{3}- \frac{( 4 \xi - 1 )^{\frac{3}{2}}}{2 7}+ O ( 4 \xi - 1 )^{\frac{5}{2}}.
\int_R^\infty Q(r\omega) |u(r)|^2\,dr = \int_0^\infty W(r)|\tilde u(r)|^2\,dr \leq 4 \left( \sup_{s>0}s \int_s^\infty W(t)\,dt \right) \int_0^\infty |\tilde u'(r)|^2\,dr & = 4 \left( \sup_{s>R}s \int_s^\infty Q(t\omega)\,dt \right) \left( \int_R^\infty |\tilde u'(r)|^2\,dr + c_R |u(R)|^2 \right).
[v]_{W^{s,p}(K_T)}:=\left(\,\,\iint_{K_T \times K_T}\frac{|v(x,t)-v(x^\prime,t^\prime)|^p}{\left(\sqrt{|x-x^\prime|^2+(t-t^\prime)^2}\right)^{n+1+sp}}\,dxdtdx'dt' \right)^\frac{1}{p},
d(x,y)\leq\|h\|_{\bar{\mathcal{H}}}=\left\|\left(\int_{0}^{\cdot}(V^*(VV^*)^{-1})(z_{s})ds\right)\cdot(y-x)\right\|_{\bar{\mathcal{H}}}&=\|\gamma\|_{\bar{\mathcal{H}}}|y-x|\leq C_{H,V}|y-x|.
d \Theta^{1 \underline{{\mu}}}= v_{( \underline{{\alpha}})}^{\underline{{\mu}}}\pi^{1 ( \underline{{\alpha}})}= v_{q}^{- \underline{{\mu}}}\pi_{q}^{1 +}+ v_{\dot{q}}^{+ \underline{{\mu}}}\pi_{\dot{q}}^{1 -},
\prod_{n = 1}^{\infty}( 1 - q^{n})^{3}= \lim_{\epsilon \rightarrow 0}\sum_{- \infty}^{\infty}\frac{1}{\epsilon}\left( \frac{- 1}{1 - \epsilon}\right)^{n}q^{\frac{1}{2}n^{2}i \frac{1}{2}n}
\hat{\phi}(t,r) =\frac{1}{2\pi}\int_{0}^{\infty}\frac{d\omega}{\sqrt{\omega}}\, \left[ b_{\omega}e^{-i\omega t}\psi_{\omega}(t,r) +b^{\dagger}_{\omega}e^{i\omega t}\psi_{\omega}^{\ast}(t,r)\right].
S_1(z, q, N)=\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]_{q^2}\left(\frac{(q;q)_{2n-2}z^{2n-1}q^{n(2n-1)}}{(zq;q)_{2n-1}}+\frac{(q;q)_{2n-1}z^{2n}q^{n(2n+1)}}{(zq;q)_{2n}}\right)\frac{(q^{2};q^2)_n}{(zq^{2N+1};q^2)_{n}}.
y^{00}= \phi\Omega((\phi-z\phi_z)^2+z^2\phi\phi_{zz}) + (r^2-s^2)\phi \left(\phi^2\phi_{ss}+2z\phi(\phi_s\phi_{sz}-\phi_z\phi_{ss}) + z^2\delta_3\right) y^{0i}= \phi\left[-(\Omega+s\phi_s)(\phi\Omega)_z+(r^2-s^2)(\phi(\phi_s\phi_{sz}-\phi_z\phi_{ss})+z\delta_3)\right]u^i + \phi^2[\phi_s\Omega_z-\phi_{sz}\Omega]x^i
f(s,x,t,y)=G(s,x,t,y)\left(-\phi'(t)y+\phi'(s)x-\frac{1}{2}\int_{s}^{t}(\phi'(u))^2du \right )\times \mathbb{E}\left[ \left(-\int_{s}^{t}\phi''(u)B(u)du\right) \vert B(s)=-x, B(t)=-y\right]
(m,L^{\mu,t}_nm')_\mu =(L^{\mu,s-t}_{-n}m,m')_\mu, (m,(J^{\{u\}})^{\mu,t}_nm')_\mu =-((J^{\{\phi(u)\}})^{\mu,s-t}_{-n}m,m')_\mu, (m, (G^{\{v \}})^{\mu,t}_nm')_\mu =((G^{\{\phi(v) \}})^{\mu,s-t}_{-n}m,m')_\mu.
R_{\mathcal{BS}(\alpha)}(\mathcal{S^{*}}[A,B]) =\begin{dcases}\min \left\{1,\dfrac{2\sqrt{\alpha}}{(1+\alpha)\sqrt{4\alpha (A-B)^2 +B^2}}\right\}, 4A \alpha \geq (5\alpha-1)B, \min \left\{1,\dfrac{1}{(1-\alpha)(A-B)-B}\right\}, 4A \alpha\leq (5\alpha-1)B.\end{dcases}
2 8{\cal T}_{4}( p_{1}^{\prime}, p_{1}; p_{2}^{\prime}, p_{2}) = 2 i E \int \frac{d^{2}z}{( 2 \pi )^{3}}e^{i q \cdot z}{\cal T}_{2}( p_{2}^{\prime}, p_{2}| h^{C}) \, ,
\hat{h}= \frac{\mu}{\mu ( q )}\hat{h}^{( 0 )}+ \frac{C_{2}- C_{1}}{\mu c^{2}}\frac{\alpha^{2}}{q^{2}}+ \frac{\hbar^{2}}{2}\frac{\alpha}{\mu^{2}c^{2}}C_{1}\left( \frac{{\bf q}}{q^{3}}\frac{\partial}{\partial{\bf q}}+ 4 \pi \lambda \delta ({\bf q}) \right) ,
\displaystyle\frac{1}{k}(w,\bar{u}_h)+ (\nabla w, \nabla \bar{u}_h)=-(\widetilde{u}{\boldsymbol \tau},\nabla\bar{u}_h)-(\nabla u^{n-1}_h, \nabla\bar{u}_h)-(\widetilde{u}{\boldsymbol \sigma}^{n-1}_h,\nabla\bar{u}_h), \ \forall \bar{u}_h\in U_h,
\int^t_0 \left ( |u_x ( 1, s) |^2 + |u_x ( 0, s) |^2 \right ) ds= D_0 + 2 \int^t_0\int^1_0\left ( 2 |u_x ( x, s) |^2 - \lambda ( 1- (2/p)) |u ( x, s)|^p \right ) dx ds & - 2i \int^1_0 ((x-(1/2))u (x, t) \bar u_x( x, t) ) dx -2\int^t_0 \Big ( u(1, s) \bar u_x(1, s) - u(0, s) \bar u_x(0,s)\Big )ds \, .
\Phi_b=-b\Lambda Q_b-(Q_b''-Q_b+Q_b^p)', R_b(\varepsilon)=p(Q_b^{p-1}-\mathcal{Q}_p^{p-1})\varepsilon, R_{}(\varepsilon)=(\varepsilon+Q_b)|\varepsilon+Q_b|^{p-1}-p\varepsilon Q_b^{p-1}-Q_b^p.
\|z^{s+1}-z^s\|\leq \|z^s-z^s_*\|+\|z^{s+1}-z_*^s\|\overset{\eqref{c3}, \eqref{ineq}}{\leq}\|z^s-x^*\|+\rho_s\tau_s \overset{\eqref{ppae2}}{\leq}\|z^0-x^*\|+\sum_{k=0}^{s}\rho_k\tau_k.
\mathcal{D}_{R}:=\big\{\overline{k}=\big(k_1m^{-R},\cdots,k_dm^{-R}\big)\, \big|\, k_i=0,\ldots,m^{R}-1, \i=1,\ldots,d\big\}, \Omega_n :=\bigg\{\big(\overline{k}+[0,m^{-j})^d\big)\cap[0,1)^d\, \bigg|\, j=0,\ldots,J-1,\, \overline{k}\in\mathcal{D}_{R}\bigg\};
\chi^{(k)}(X,G)=\frac{1}{\vert G\vert}\sum_{{{\bf g}\in G^{k+1}:}\atop{g_ig_j=g_jg_i}}\chi(X^{\langle{\bf g}\rangle})=\sum_{[g]\in G_*}\chi^{(k-1)}(X^{\langle g\rangle}, C_G(g))\,,
\langle T_{z z}( z , \bar{z}) T_{z z}( 0 , 0 ) \rangle = \frac{\pi}{3 \cdot 1 6}\int_{0}^{\infty}d \mu^{2}\tilde{c}( \mu^{2}) \int \frac{d^{2}p}{( 2 \pi )^{2}}\frac{e^{\frac{i}{2}( p \bar{z}+ \bar{p}z )}}{p \bar{p}+ \mu^{2}}\bar{p}^{4},
\begin{aligned}[b]T_2(Z_n, Z_n){}= T_2\Bigl( G_n + i[\tilde H_n,F_n],\,G_n + i[\tilde H_n,F_n] \Bigr) &{}= T_2\Bigl(G_n, \, G_n\Bigr){}+ T_2\Bigl( G_n, \, i[\tilde H_n,F_n] \Bigr) &{}+ T_2 \Bigl( i[\tilde H_n,F_n],\, G_n \Bigr) &{}+ T_2\Bigl( i[\tilde H_n,F_n],\,i[\tilde H_n,F_n] \Bigr).\end{aligned}
Q_{p,q}(\psi(t)) =2N(q-2)E_{p,q}(\psi(t))-(N(q-2)-8)\|\Delta \psi(t)\|_{2}^{2}-\frac{2N(p-q)}{p}\|\psi(t)\|_{p}^{p}&\leq\rho\frac{N(q-2)-8}{2}-1-\frac{N(q-2)-8}{2}\rho-\frac{N(q-2)-8}{2}\|\Delta \psi(t)\|_{2}^{2}.
\begin{aligned}\|u\|_{C^{k,2+\alpha}_{WF}([0,T]\times \bar B_r(z^0))}\leq C\left(\|u_t-Lu\|_{C^{k,\alpha}_{WF}([0,T]\times\bar B_{2r}(z^0))}\right. &\quad\left.+ \|u(0,\cdot)\|_{C^{k,2+\alpha}_{WF}(\bar B_{2r}(z^0))}+ \|u\|_{C([0,T]\times\bar B_{2r}(z^0))}\right).\end{aligned}
\tau_{k}=\frac{q_{k}}{2n-2}+q_{k}\cdot\frac{2n-2-\widehat{w}_{n}(\zeta)}{(2n-2)(1+\widehat{w}_{n}(\zeta))}=q_{k}\cdot \frac{2n-1}{(2n-2)(1+\widehat{w}_{n}(\zeta))}.
\sum_{(x_1,\ldots, x_n) \in (\mathbb{Z}/p\mathbb{Z})^N \setminus \{\textbf{0}\}}\left( \frac{1}{2N}\sum^{N}_{j=1}\left( \cos \left(\frac{2 \pi x_ja}{p}\right) + \cos \left(\frac{2 \pi x_j}{p}\right) \right) \right)^{2t}\leq \alpha e^{-\beta}.
\delta_{I}\mathcal{P}_{2 i}^{a}= - \frac{1}{2}\epsilon_{0 i j k}F^{j k a}+ \partial_{i}\pi^{a}, \; \delta_{I}\mathcal{P}_{1}^{a}= - \pi^{a}+ \pi^{( 1 ) a}, \; \delta_{I}\mathcal{P}_{2}^{a}= \pi^{( 2 ) a},
\vert \phi \rangle ={\frac{1}{\sqrt{2 \pi}}}\sum_{m = - \infty}^{\infty}e^{i m \phi}\vert m ; \alpha \rangle \left( \equiv \lim_{\varepsilon \to 0}{\frac{1}{\sqrt{2 \pi}}}\sum_{m = - \infty}^{\infty}e^{i m \phi - \vert m \vert \varepsilon}\vert m ; \alpha \rangle \right) \ ,
\frac{\partial q}{\partial t}=-\frac{\sqrt{3}e^{-\omega}}{4}q \ \alpha_2\times\alpha_3, &\frac{\partial q}{\partial u}=\frac{e^{-\omega}}{8}(4e^{\omega}q \alpha_2-4 q \alpha_3+\omega_v q\ \alpha_2\times\alpha_3), &\frac{\partial q}{\partial v}=-\frac{e^{-\omega}}{8}(4 q \alpha_2-4e^{\omega}q \alpha_3+\omega_u q\ \alpha_2\times\alpha_3).
g = Z_2 = Z_0^{-2}\Biggl (\frac{a_0 + Z_0}{\bar a_0 + Z_0^{-1}}\Biggr)\Biggl (\frac{\mu_+ + Z_0}{\bar \mu_+ + Z_0^{-1}}\Biggr)\Biggl (\frac{\mu_- + Z_0}{\bar \mu_- + Z_0^{-1}}\Biggr)
\phi_{\Lambda}(x_1, \ldots, x_k) := (x_1, \ldots, x_{\delta_1},{\bf y_{1}}, x_{\delta_1+1}, \ldots, x_{\delta_2},{\bf y_{2}}, \ldots,{\bf y_{\ell}}, x_{\delta_{\ell}+1}, \ldots, x_{k}) .
\begin{array}{rl}-{\frac{1}{b}(q^b-2q^{b/2})}\frac{bq^{-b}}{(1-q^{-b})^2}\omega(1,q^b)^b = -(1-2q^{-b/2})\frac{\omega(1,q^b)^b}{(1-q^{-b})^2}&< \displaystyle\frac{-(1-2q^{-b/2})(1-2bq^{-b})}{(1-q^{-b})^2}&= -1 + \displaystyle\frac{2q^{-b/2}+2(b-1)q^{-b}-4bq^{-3b/2}+q^{-2b}}{(1-q^{-b})^2}&< -1 + 2(2q^{-b/2}+2(b-1)q^{-b}-4bq^{-3b/2}+q^{-2b}).\end{array}