harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\partial}{\partial \theta}\Psi_j (\bar{x})\left |_{\Psi_j (\bar{x})}\equiv 0 = \int d^4x \frac{\partial}{\partial \theta}S_k(\bar{y})|_{\Psi_j}\frac{{\delta^\prime}^2 \bar{Z}[S]}{\delta S_k(\bar{y})\delta S_j (\bar{x})}\right .
c_{\mu}^{( s )}( \rho ) = \frac{i \sqrt{\sinh \pi \mu}}{2 \pi}\, \phi_{M}( 0 , \rho ) \left\{\Gamma ( - i \mu ) \left( \frac{m \rho}2 \right)^{i \mu}- \Gamma ( i \mu ) \left( \frac{m \rho}2 \right)^{- i \mu}\right\},
S_{W}\cong \int \mathcal{A}\star Q_{NS}\mathcal{A}+ \frac23\int\mathrm{X}\mathcal{A}\star \mathcal{A}\star \mathcal{A}+\int Y(i)\Psi \star Q_{R}\Psi +2\int \Psi \star \mathcal{A}\star \Psi .
\varphi_{\alpha}(t)=\frac{\sqrt{\pi}}{\Gamma(1/\alpha)}e^{it\mu}{\rm{H}}_{1,2}^{1,1}\left[\frac{\sigma^2\Gamma(1/\alpha)}{4\Gamma(3/\alpha)}t^2\left|\genfrac{}{}{0pt}{}{(1-\frac{1}{\alpha},\frac{2}{\alpha})}{(0,1),(\frac{1}{2},1)}\right.\right],
m(H,k-1)= m(H-v,k-1)+\sum\limits_{w'\in N_{H}(v)}m(H-v-w',k-2) = m(T(a,y-1,c-1)\cup P_{x-a-2},k-1)+m(T(a,y-1,c-1)\cup P_{x-a-3},k-2) +m(T(a,y-2,c-1)\cup P_{x-a-2},k-2)+m(T(a,y-1,c-2)\cup P_{x-a-2},k-2) = m(T(a,y-1,c-1)\cup P_{x-a-1},k-1)+m(T(a,y-2,c-1)\cup P_{x-a-2},k-2) +m(T(a,y-1,c-2)\cup P_{x-a-2},k-2).
|\langle T_jx',x'\rangle|=\bigl|\langle T_jx, x\rangle +\frac{1}{2^{k+1}}\langle T_ju,u\rangle\bigr|=\bigl|\xi_j+\frac{1}{2^{k+1}}\langle T_ju,u\rangle\bigr|\le\frac{r}{2^{k+2}}
{_{3}}F_{2}\left(\begin{array}{ccc}\chi_4, \overline{\chi_4}, \overline{\chi_4}\varphi, \varepsilon\end{array}| 1\right) =\frac{1}{q^2}J(\chi_8,\chi_8^2)\times 2 Re(J(\chi_8,\chi_8))\times \chi_8(-1) &=\frac{1}{q^2}[-2u(-p)^t],
\sum_{k=0}^n{n\choose k}\varphi_{k}(x_1)\varphi_{n-k}(x_2)= \sum_{k_1,k_2\geq 0, k_1+k_2=n}{n \choose k_1,k_2}\varphi_{k_1}(x_1)\varphi_{k_2}(x_2) \left(x_{1}, x_{2}\in G\right),
\begin{aligned}\bigg|\int_\Omega w_\varepsilon\Phi dx\bigg|\leq \bigg|\int_\Omega \tilde{f}\cdot\nabla\phi_\varepsilon dx\bigg|+ \bigg|\int_\Omega (\theta_\varepsilon-\sigma)\eta dx\bigg| =: I_1 + I_2,\end{aligned}
E = \bigl\{(q_0,g,\lambda') : (i,g,\lambda') \in A \bigr\}\cup \bigl\{(\lambda',p_{\lambda'i'}g,\lambda'') : \lambda' \in \Lambda_A, \:(i',g,\lambda'') \in A \bigr\}\cup \bigl\{(\lambda,p_{\lambda i},t) \bigr\}\subseteq Q \times G \times Q.
\mathcal E_{\alpha,g, T}(t):= \frac{2}{\alpha-1}t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x(t) - x^{*}+ \frac{t}{\alpha-1}\dot{x}(t) \|^2 + 2\int_t^{T}\tau \langle x(\tau) - x^{*}+ \frac{\tau}{\alpha-1}\dot{x}(\tau), g(\tau) \rangle d\tau.
\widetilde{\psi}_{\mu}^{( \nu ,\nu )}(x;\lambda_{1})=\widehat{A}\big(W_{\nu}^{(\nu)}\big)\psi_{\mu}^{(\nu)}(x)=( E_{\nu}-E_{\mu}) \psi_{\mu}(x) -\frac{W(\psi_{\nu},\psi_{\mu}\,|\, x)}{\Psi_{\nu}^{(\nu)}(x;\lambda_{1})}.
\begin{array}{l l}{Z = \frac{1}{2}(}&{\mathrm{T r \thinspace}e^{- \beta ( H^{( + )}- \mu \hat{N})}+ \mathrm{T r \thinspace}e^{- \beta ( H^{( - )}- \mu \hat{N})}+}\\{}&{\mathrm{T r \thinspace}\tau e^{- \beta ( H^{( + )}- \mu \hat{N})}- \mathrm{T r \thinspace}\tau e^{- \beta ( H^{( - )}- \mu \hat{N})})}\\ \end{array}
t_{x_1}t_{y_1}t_{\alpha_1}^{-1}(z_1) = t_{\beta_1}t_{\gamma_1}t_{\delta_1}t_{z_1}^{-1}(z_1) = z_1, t_{y_1^\prime}t_{z_1^\prime}t_{\gamma_1}^{-1}(x_1^\prime) = t_{x_1^\prime}^{-1}t_{\alpha_1}t_{\beta_1}t_{\delta_1^\prime}(x_1^\prime) = x_1^\prime.
\begin{aligned}w_{2s}=\frac{2(1-\alpha)\kappa_N (-j)_s(2a)_s(a+1)_s(a-c-j+1)_s(a+c)_s}{j! (a+c)_j(c-a)_j(2a+1)_j(a)_s s! (2a+1+j)_s(a-c+1)_s(a+c+j)_s}, w_{2s+1}=\frac{-2 \alpha \kappa_N (-j+1)_s(2c)_s(c+1)_s(c-a-j)_s(a+c)_s}{(j-1)! (a+c)_{j+1}(a-c)_{j+1}(2c+1)_{j-1}(c)_s s! (2c+j)_s(c-a+1)_s(a+c+j+1)_s}, \end{aligned}
u_j=\sum_{s_1,\dots,s_j=1}^{d_1}\alpha_{s_1\dots s_j}X_{s_1}\otimes\cdots\otimes X_{s_j}=\sum_{s_1,\dots,s_j=1}^{d_1}\big(\alpha_{s_1\dots s_j}^1X_{s_1}\big)\otimes\cdots\otimes \big(\alpha_{s_1\dots s_j}^j X_{s_j}\big) =:\sum_{n=1}^{d_1^j}X_{n1}\otimes \cdots\otimes X_{nj},
\theta_{p+1}\nabla^{\rm DS}\big(P_j^p{\nabla}_{\mathfrak{U}^2}^j(s_{U^2})\big) = P_j^p\theta_{p+1}\nabla^{\rm DS}\big({\nabla}_{\mathfrak{U}^2}^j(s_{U^2})\big)+{\nabla}_{\mathfrak{U}^2}^j(s_{U^2})dP_j^p(\xi_{p+1}^2) &= P_j^p\theta_{p+1}\beta_{j,p}{\nabla}_{\mathfrak{U}^2}^{j+1}(s_{U^2})+{\nabla}_{\mathfrak{U}^2}^j(s_{U^2})dP_j^p(\xi_{p+1}^2),
\varphi_{\alpha_{1}, \alpha_{2}, \alpha_{3}}( z_{1}, z_{2}, z_{3}) \equiv z_{1}^{\alpha_{1}- h}z_{2}^{\alpha_{2}- h}z_{3}^{\alpha_{3}- h}z_{1 2}^{\alpha_{3}}z_{2 3}^{\alpha_{1}}z_{3 1}^{\alpha_{2}}\, ,
\Pi_{\gamma}(m):=\prod_{j=1}^{m-1}\frac{\Gamma \left(\sum_{l=1}^{j}w_l + \left(H-\frac{1}{2}-\gamma \right)\sum_{l=1}^{j}\varepsilon_l + j\right)\Gamma \left( w_{j+1}+1\right)}{\Gamma \left( \sum_{l=1}^{j+1}w_l + \left(H-\frac{1}{2}-\gamma \right)\sum_{l=1}^{j}\varepsilon_l + j+1\right)}.
\varepsilon^+_u = \eta_{12}\gamma^{12}\varepsilon^+_l,\ \ \ \ varepsilon^+_u = -i \eta_{23}\gamma^{23}\varepsilon^+_l, \ \ \ \ varepsilon^+_u = i \eta_{1}\gamma^{1}\varepsilon^-_l, \ \ \ \ varepsilon^+_u = - \eta_{3}\gamma^{3}\varepsilon^-_l
A_{\alpha_{1}\alpha_{2}\cdot\cdot\cdot\alpha_{p},\alpha_{p+1}\cdot\cdot\cdot\alpha_{q}}=\frac{\partial}{\partial x^{\alpha_{q}}}\cdot\cdot\cdot\frac{\partial}{\partial x^{\alpha_{p+2}}}\frac{\partial}{\partial x^{\alpha_{p+1}}}A_{\alpha_{1}\alpha_{2}\cdot\cdot\cdot\alpha_{p}}.
\mathbb{P}\big[ W(t_1) \in K_1, \, \dots, \, W(t_n) \in K_n \big] = \int_{K_1 \times \cdots \times K_n}p(t_1, 0, y_1) \, \prod_{k=2}^n p(t_k -t_{k-1}, y_{k-1}, y_k) \, \mbox{d}y_1 \cdots \mbox{d}y_n
D_{r}F_{n,\lambda}(x) =\frac{1}{r!}\bigg(\frac{d}{dx}\bigg)^{r}\bigg[\sum_{k=0}^{n}S_{2,\lambda}(n,k)k!x^{k+r}\bigg] &=\frac{1}{r!}\bigg[\sum_{k=0}^{n}S_{2,\lambda}(n,k)k!(k+r)_{r}x^{k}\bigg] &=\sum_{k=0}^{n}S_{2,\lambda}(n,k)\binom{k+r}{k}k!x^{k}.
\alpha_{j,k}=\int_{-1}^{1}\mathrm{d}x\,w_r^{(m)}(x)\int_{-1}^{1}\mathrm{d}y\,w_r^{(m)}(y)p_{j-1}(x)p_{k-1}(y)\mathrm{sgn}(y-x), \beta_{j,k}=2\mathrm{i}\int_{D_{+}}\mathrm{d}x\mathrm{d}y\,w_c^{(m)}(x,y)\left(p_{j-1}(x+\mathrm{i}y)p_{k-1}(x-\mathrm{i}y)-p_{k-1}(x+\mathrm{i}y)p_{j-1}(x-\mathrm{i}y)\right),
\frac{d^l}{dx^l}p^{n,k}(x)\Big|_{x=a}=\frac{d^l}{dx^l}\theta_{p}^{n,k}(x)\Big|_{x=a}&=\frac{d^l}{dx^l}\left(\exp(x-a)(1-F_{n-1,k}(a;x))\right)\Big|_{x=a}&=1+\sum_{j=1}^{l}\binom{l}{j}\frac{d^j}{dx^j}(1-F_{n-1,k}(a;x))\Big|_{x=a}&=1-\sum_{j=1}^{l}\binom{l}{j}\frac{d^{j-1}}{dx^{j-1}}f_{n-1,k}(a;x)\Big|_{x=a}&=1.
\mathcal{T}_z\mathcal{G}^Z_\kappa(x;\hbar)=\mathcal{T}_z(\pi\hbar)^{-d/4}e^{\frac{i}{\hbar}\Big(\kappa\cdot x+\frac{1}{2}x\cdot Zx\Big)}=\mathcal{G}^Z_{\kappa-p}(x-q;\hbar) \ .
\langle V_3|(\varphi(0)|k_1\rangle_1)\otimes |k_2\rangle_2\otimes |k_3\rangle_3\sim \langle\varphi(z_1)e^{ik_1X(z_1)}e^{ik_2X(z_2)}e^{ik_3X(z_3)}\rangle\neq 0.
(-m_{\pi}^{2}+\partial^{2}_{\bot})\omega^{(1)}_{\pi}=\frac{1}{2L}\int^{L}_{-L}dx^{-}(\varphi^{3}_{\pi}+\varphi_{\pi}\varphi^{2}_{\sigma}+2v\varphi_{\pi}\varphi_{\sigma}).
\det \left(X_{\kappa_n,i_{(1,1)}'}f(p) \wedge \cdots \wedge X_{\kappa_n,i_{(n,1)}}\cdots X_{\kappa_n,i_{(n,\kappa_n)}'}f(p) \right) = \sum_{i}c_{ii'}\det \left(X_{1,i_{(1,1)}}f(p) \wedge \cdots \wedge X_{\kappa_n,i_{(n,1)}}\cdots X_{1,i_{(n,\kappa_n)}}f(p) \right)
\varphi_{\underline{d},\underline{d}''}(V_{\underline{d}})=(V_{\underline{d}''}^{X_{3}^{c},0}+V_{\underline{d}''}^{X_{2}^{c},0})+\left\langle u_1,\ldots,u_{\beta}\right\rangle=V_{\underline{d}''}^{X_{1},0}.
\sum_{t=0}^{p}r^t \begin{bmatrix}p+1 t+1 \end{bmatrix}= p! \left[ \binom{r+p+1}{p+1}- 1 \right] - \sum_{j=1}^{r-1}\frac{(p+j)!}{j!}& = p! \left[ \binom{r+p+1}{p+1}- 1 \right] - p! \sum_{j=1}^{r-1}\binom{p+j}{j}.
\|g_1(x)\|_{L^2}\geq e^{-\pi \alpha \left( \frac{n-1}{2}\right)^2}\left( \sum_{1 \leq |k| \leq \frac{n-1}{2}}\left| \ \sum_{j=1}^{n}\varepsilon_j e^{-2 \pi i k \frac{j}{n}}+ \mathcal{O}\left(k^2 \sum_{j=1}^{n}\varepsilon_j^2\right) \right|^2 \right)^{1/2}.
|U_i| =|U_i'|=\ell' +e_{C'}(U_{i+1}', U_i')+e_{C'}(U_i', U_{i-1}') &=\ell-1+e_{C'}(U_{i+1}', U_i')+e_{C'}(U_i', U_{i-1}')=\ell+e_C(U_{i+1}, U_i)+e_C(U_i, U_{i-1}).
\int_{({\mathbb R}^d)^{k-1}}\varphi_f(K_1\cap K_2^{x_2}\cap \dots \cap K_k^{x_k})\, \lambda^{k-1}(d(x_2,\dots ,x_k)) & = \sum_{{m_1,\dots,m_k=j}\atop{m_1+\dots +m_k=(k-1)d+j}}^d\varphi^{(j)}_{f,m_1,\dots,m_k}(K_1,\dots,K_k),
\hat{V}_{\mu \nu \rho}^{( 3 \, z ) \, a b}= \hat{V}_{\mu \nu \rho}^{( 3 \, x )}= \hat{V}_{\mu \nu \rho}^{( 3 \, y ) \, a b}= - Y_{L}^{( e ) \, 3}\, \mathrm{T r}\, \tau^{a}\tau^{b}\, V_{\mu \nu \rho}^{( 3 )}\, .
\begin{aligned}\Vert e^{-itH}f-\psi_a(H^a)e^{-itH}f\Vert^2 =(e^{-itH}f-\psi_a(H^a)e^{-itH}f,(I-\psi_a(H^a))e^{-itH}f) &\sim\sum_{b<a}(e^{-itH}f-\psi_a(H^a)e^{-itH}f,e^{-itH}g_b)\end{aligned}
s^{(1)}_m(y) =(\lambda-\beta+2m)s^{(-)}_{m}(y) -2m(\lambda+2m)(\lambda-\tfrac{\beta}{2}+m)s^{(-)}_{m-1}(y) +m(m-1)(\lambda+\beta+2m)(\lambda-\tfrac{\beta}{2}+m-1)_2s^{(-)}_{m-2}(y) \quad-(\lambda)_m(\lambda-\beta)(\tfrac{\beta}{2})_m s^{(-)}_0(y).
\mathcal{H}_{\Lambda}(\rho_{\mu})_n \geq \sum_{j=1}^n \frac{b}{\ell^2}Q_{\Lambda,j}+ \frac{1}{\lvert B \rvert}\int_{\Lambda}\, \mathcal{H}_{B_u}(\rho_{\mu})_n \mathrm{d}u.
\hat{S}= \int d^{1 1}\hat{x}\sqrt{| \hat{g}|}\left[ \hat{R}-{\textstyle \frac{1}{2 \cdot 4 !}}\hat{G}{}^{2}-{\textstyle \frac{1}{( 1 4 4 )^{2}}}{\textstyle \frac{1}{\sqrt{| \hat{g}|}}}\hat{\epsilon}\hat{G}\hat{G}\hat{C}\right] .
M:= \Big(\sup_{0\leq t\leq T}\big\|u^h(t)\big\|\Big)\vee \Big(\sup_{0\leq t\leq T}\big\|v^h(t)\big\| \Big)\vee \Big(\int_{0}^{T}\big\|u^h(s)\big\|_V^2ds \Big)\vee \Big(\int_{0}^{T}\big\|v^h(s)\big\|_V^2ds \Big).
\vert J_2\vert\leq \Vert f\Vert_p\bigg(\int_{\vert x-t\vert>R}\frac{dt}{\vert x-t\vert^q}\bigg)^{1/q}=\frac{\Vert f\Vert_p2^{1/q}}{(q-1)^{1/q}R^{(q-1)/q}},
T_{n k}( x | g ) = \left( \begin{array}{c c}{\Lambda_{\mu \nu}( x | g )}&{0}\\{0}&{- \varepsilon}\\ \end{array}\right) , \Lambda_{\mu \nu}( x | g ) = \eta_{\mu \nu}\varepsilon - \frac{e_{0}}{2}\varepsilon g F_{\mu \nu}( x ) \varepsilon .
\frac{D - 2}{2}\mathrm{T r}\, \ln ( - \partial_{\alpha}\alpha^{\alpha \beta}\partial_{\beta}) \, = \, \frac{D - 2}{2}T \, \sum_{n}^{}\int_{- \infty}^{+ \infty}\frac{d q_{0}}{2 \pi}\ln \left( \alpha^{0 0}q_{0}^{2}\, + \, \alpha^{1 1}{\omega_{n}^{2}}\right) \, = \,{T}\sqrt{\frac{\alpha^{1 1}}{\alpha^{0 0}}}\, E_{\mathrm{C}}{.}
\mathcal{Y}_{\vec{f}}^N(t, \xi^N)=\exp\left(\frac{\gamma_N^2}{N}\left(\eta_t^N(\vec{f})-\eta_0^N(\vec{f})-\int_0^t\eta_s^N(\mathcal{A}_s^N\vec{f})ds+\sum_{k=1}^4\varepsilon_{k,t}^N(\vec{f})\right)\right),
\sum_{i=1}^{N}\delta \alpha_{i}\, \lambda_{i}= \sum_{i=1}^{N}(\alpha_{i}-1) \delta \left[ \sum_{j \neq i}2\pi (\alpha_{j}-1)G(\omega_{i}-\omega_{j}| \tau) \right] + \frac{1}{2}\delta \left[\sum_{i=1}^{N}(\alpha_{i}-1)^{2}\log |2\pi \eta^{2}| \right]
\varphi^* \left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{2 \, p^{e_1}}a^{p^{e_1}}\, b^{p^{e_1}}b^{2 \, p^{e_1}}0 1 0 0 0 d^{2 \, p^{e_1}}\end{array}\right) ( \, e_1 \geq 0 \, ) .
\begin{aligned}\big|\big\{x\in(E^\ast)^c:\sum_{s>-3}\sup_j|\Omega_j\ast B_{j-s}|(x)>\frac\lambda2\big\}\big| &\le \big|\big\{x\in(E^\ast)^c:\sum_{s>-3}\sup_j|\Omega_j^s\ast B_{j-s}|(x)>\frac\lambda4\big\}\big| &+ \big|\big\{x\in(E^\ast)^c:\sum_{s>-3}\sup_j|K_j^s\ast B_{j-s}|(x)>\frac\lambda4\big\}\big|.\end{aligned}
G(t,s)=\frac{1}{\Gamma(\alpha)}\begin{cases}\frac{(t-a)^{\alpha -1}(b-s)^{\alpha -1}}{(b-a)^{\alpha -1}}-(t-s)^{\alpha-1},a\leq s\leq t\leq b, \frac{(t-a)^{\alpha -1}(b-s)^{\alpha -1}}{(b-a)^{\alpha -1}},a\leq s\leq t\leq b,\end{cases}
\pi_{\alpha}^{0}* \tilde{H}^{( 1 )}* \pi_{\alpha}^{0}= \pi_{\alpha}^{0}* H^{( 1 )}* \pi_{\alpha}^{0}+ i \hbar \pi_{\alpha}^{0}\underbrace{\Delta \Gamma ( X_{\lambda_{\alpha}})}_{\mathrm{` ` B e r r y p h a s e ' ' t e r m}}\pi_{\alpha}^{0}
\theta( m + 1 , 0)(u) = \theta(m,0) (\theta(1,0)(u)) = \theta(m,0)(B u B^{-1}) = B^{m+1}u B^{-(m+1)}, \theta( m + 1 , 0)(v) = \theta(m,0) (\theta(1,0)(v)) = \theta(m,0)( B v u^{-2}B^{-1}) = B^{m+1}v u^{- 2(m+1)}B^{- (m+1)},\; \theta(m+1,0)(B) = \theta(m,0)( \theta(1,0)(B)) = \theta(m,0)(B) = B,
\Theta = a_{n}\mathrm{G}_{n}= a_{n}( - 1 )^{\frac{n}{2}}\varepsilon_{\mu_{1}\nu_{1}\cdots \mu_{\frac{n}{2}}\nu_{\frac{n}{2}}}\varepsilon^{\alpha_{1}\beta_{1}\cdots \alpha_{\frac{n}{2}}\beta_{\frac{n}{2}}}\prod_{i = 1}^{\frac{n}{2}}R_{\alpha_{i}\beta_{i}}^{\mu_{i}\nu_{i}}
\begin{aligned}\cdots\rightarrow P_2(M) \rightarrow P_1(M)\rightarrow P_0(M)\rightarrow k\rightarrow 0 \, , \cdots \rightarrow P_2(N)\, \rightarrow P_1(N)\, \rightarrow P_0(N)\, \rightarrow k\rightarrow 0 \, .\end{aligned}
\int_{\varphi(a)}^{\varphi(b)}{f_2(b,x) \, dx}+ \int_{\varphi(b)}^{\psi(b)}f_2(b,x) \, dx -\int_{\psi(a)}^{\psi(b)}{f_2(b,x) \, dx}-\int_{\varphi(a)}^{\psi(a)}f_2(a,x) \, dx = \int_{\varphi(a)}^{\psi(a)}f_2(b,x) \, dx-\int_{\varphi(a)}^{\psi(a)}f_2(a,x) \, dx = \int_{\varphi(a)}^{\psi(a)}[f_2(b,x)-f_2(a,x)] \, dx
\ell^*_1 = \sup_{\substack{h>0 t\in (0,T-h]}}\left(\frac th\right)^\gamma \int_t^{t+h}|\ell(\tau)| d\tau<\infty, &16 B(1-\delta, 1-2\gamma) T^{1-\delta}({L_f^*}^2 +{K_f^*}^2{\ell^*_2}^2)<1,
\int\limits_{\psi_0}^{\pi-\psi_0}Cf(0, e_n, \psi)g(\psi_0,\psi)d\psi =\int\limits_{\psi_0}^{\pi/2}Cf(0, e_n, \psi)g(\psi_0,\psi)d\psi &+\int\limits_{\pi/2}^{\pi-\psi_0}Cf(0, e_n, \psi)g(\psi_0,\psi)d\psi.
\mathrm{\int_{0}^{1}d x \int_{0}^{x}d G \int_{G}^{1 - x + G}d y \int_{0}^{x - G}d v = - \int_{0}^{1}d x \int_{0}^{x}d t \int_{1 + t - x}^{t}d Y \int_{0}^{t}d v ,}
- \Bigl [ \left[ Q ( \xi ) , Q ( \eta ) \right] , b ( k , \sigma ) \Bigr ] \stackrel{!}{=}\Bigl [ \left[ Q ( \eta ) , b ( k , \sigma ) \right] , Q ( \xi ) \Bigr ] + \Bigl [ \left[ b ( k , \sigma ) , Q ( \xi ) \right] , Q ( \eta ) \Bigr ]
\L\big(\textstyle \sum_{x_k}f(y_i,x_k,\bar x_{k,i}) \mu_k^{x_i,\bar x_{k,i}}(x_k) , \sum_{x_k}f(x_i,x_k,\bar x_{k,i}) \mu_k^{x_i,\bar x_{k,i}}(x_k)\big) \geq\textstyle \sum_{x_k}\L\big(f(y_i,x_k,\bar x_{k,i}),f(x_i,x_k,\bar x_{k,i})\big) \mu_k^{x_i,\bar x_{k,i}}(x_k)=\bar \nu_{k,i}^{x_i,y_i,\bar x_{k,i}}.
\frac{dF}{dt}=\sum_{i}\frac{\partial F(T,V,N)}{\partial N_{i}}\frac{dN_{i}}{dt}=\sum_{i}\mu_{i}\frac{dN_{i}}{dt}=-VRT\sum_{r}(lnw_{r}^{+}-lnw_{r}^{-})(w_{r}^{+}-w_{r}^{-})\le0
\big(1-\frac{\varepsilon}{3}\big) |I\cap V| \leq \sum_{\ell\in \Gamma}|A_{\ell}\cap V| = \Big| \bigcup_{\ell\in \Gamma}A_{\ell}\cap V\Big| \leq \Big| \bigcup_{i}B_{\ell(i)}\cap V\Big| \leq \Big| \bigcup_{i}B_{\ell(i)}\Big|.
C^{0, \alpha}= \{h \in{C^{0,0}(\mathbb R^d; \mathbb R^m)}: \|h\|_{C^{0,\alpha}}<\infty \}&C^{1, \alpha}=\{h \in{C^{1,0}(\mathbb R^d; \mathbb R^m)}: \|h\|_{C^{1,\alpha}}<\infty \},
\dim(\overline{\mu}^{-1}(y))=\dim(\mu^{-1}(\mathcal{O}(y))) = \dim(X)-\dim(g_{})+\dim(\mathcal{O}(y)) =(\dim(G)+\mathrm{rk}(G))-\dim(G)+(\dim(G)-\mathrm{rk}(G)) & = \dim(G),
\begin{array}{rcl}\partial_j^{}M_k^\alpha(q,u) \! = \! \alpha_j^{}\, \Bigl( \, \zeta(u-v) \,+\, \zeta(v) \,+\, \zeta(\alpha(q)) \,-\, \zeta(\alpha(q)+u) \Bigr) begin{align*}2mm] \times \, \Bigl( M_k^\alpha(q,u) - M_k^\alpha(q,v) \Bigr)~, \end{array}
g_{2}((\nabla \pi_{\ast})(Y,\phi V),\pi_{\ast}(X))+g_{2}((\nabla \pi_{\ast})(X,\phi V),\pi_{\ast}(X)) =g_{1}(\phi V,\mathcal{V}(\nabla^{^{M_1}}_{X}\mathcal{B}Y+\nabla^{^{M_1}}_{Y}\mathcal{B}X))
U_{p,i,h}(J) = \frac{\gamma_{p,i,h}(1)^{j_1}}{j_1!}\ \frac{\gamma_{p,i,h}(2)^{j_2}}{j_2!}\ \frac{\gamma_{p,i,h}(3)^{j_3}}{j_3!}\cdots, \gamma_{p,i,h}(k) = i \alpha_p(k) + h \beta_p(k),
\begin{aligned}=\gamma_{0}(y,t)\left(\sum_{x\in T^{-1}(x')}\frac{h(x)}{\vert T'(x) \vert}\right)+\gamma_{1}(y,t)\sum_{x\in T^{-1}(x')}(4x^{3}-3x)\frac{h(x)}{\vert T'(x) \vert}= \sum_{x\in T^{-1}(x')}\frac{h(x)}{\vert T'(x) \vert}\left(\gamma_{0}(y,t)+4x^{3}\gamma_{1}(y,t)\right)\end{aligned}
(\text{the second term in the right-hand side of \eqref{eq_tree_fixed_difference_decomposed}}) &\le 2^{2\sum_{j=1}^n(m_j-k_j)}D_{et}^{\frac{1}{2}\sum_{j=1}^n(m_j-k_j)-n+1}&\quad\cdot\sum_{v=1}^n(1_{v=1}D(\beta_1,\beta_2)+1_{v\ge 2})\prod_{i=2}^nA(\widetilde{C_o}(\beta_1),\widetilde{C_o}(\beta_2))_l^{(v,i)}\prod_{k=1}^n\|J_{m_k}(\beta_2)\|_{l,0}.
RD_f = \{\omega\in \Omega : \left(\phi(f(\omega)^+,\omega)-\phi(f(\omega),\omega)\right) >0\}LD_f = \{\omega\in \Omega : \left(\phi(f(\omega),\omega)-\phi(f(\omega)^-,\omega)\right) >0\}D_f = \{\omega\in \Omega : \left(\phi(f(\omega)^+,\omega)-\phi(f(\omega)^-,\omega)\right) >0\}
{\cal I}_{\mathrm{d e g}}^{\mathrm{h e t}}= \frac{V^{( 8 )}T_{2}}{2^{8}\pi^{4}}t_{8}F_{+}^{4 - l}F_{-}^{l}\Delta_{F_{+}^{4 - l}F_{-}^{l}}^{\mathrm{d e g}}( T_{2}, U ) \leftrightarrow{\cal I}_{\mathrm{1 - l o o p}}^{\mathrm{I}}\, .
\begin{aligned}-h^2\Delta_{x}\Psi-{\partial^2\Psi\over\partial z^2}=f, -D(x)<z<0\cr{\partial\Psi\over\partial z}=\psi^+(x) \ @ \z=0\cr \Psi=\varphi^-(x) \ @ \z=-D(x)\cr\end{aligned}
\lim_{n\to \infty}\P \!\left[ \sup\nolimits_{t\in [0,T]}\| X_t^0 - X_t^{n}\|_{H_\gamma}\geq \varepsilon \right ]=\P\!\left[ \sup\nolimits_{t\in [0,T]}\|X_t^0 \|_{H_\gamma}\geq R \right].
\omega(\zeta,z)=\frac{(n-1)!}{(2\pi i)^{n}}\frac{1}{|z-\zeta|^{2n}}\sum_{1\le j\le n}(\overline{\zeta}_{j}-\overline{z}_{j})d\overline{\zeta}_{1}\wedge d\zeta_{1}\wedge\cdot\cdot\cdot\wedge d\zeta_{j}\wedge\cdot\cdot\cdot\wedge d\overline{\zeta}_{n}\wedge d\zeta_{n}
\begin{aligned}M_{\tau}''(t) =M_{\tau}(t)\left[\ln \left(\frac{Kk_2}{k_2-1}\right)\cdot\frac{1}{k_2\sigma}\cdot \frac{1}{\sqrt{(a')^{2}-2t}}\right]^{2}&+M_{\tau}(t)\left[\ln \left(\frac{Kk_2}{k_2-1}\right)\cdot\frac{1}{k_2\sigma}\cdot \frac{1}{(\sqrt{(a')^{2}-2t})^{3}}\right].\end{aligned}
I(T_1;x) =\!\begin{multlined}[t] x^{14}+ 51x^{13}+ 2979x^{12}+ 18683x^{11}+ 55499x^{10}+ 100144x^9 + 121376x^8 + 103736x^7 + 63933x^6 + 28551x^5 + 9142x^4 + 2040x^3 + 300x^2 + 26x + 1,\end{multlined}
{\cal T}_{2}( z_{1}^{\prime}, z_{2}) R_{2 1}( z_{1}+ z_{2}){\cal T}_{1}( z_{1}, z_{2}) R_{1 2}( z_{1}- z_{2}) = R_{2 1}( z_{1}^{\prime}- z_{1}){\cal T}_{1}( z_{1}, z_{2}) R_{1 2}( z_{1}+ z_{1}^{\prime}){\cal T}_{2}( z_{1}^{\prime}, z_{2}) .
M_{L}^{2}= - c_{L}+ 1 / 8 \, \alpha_{L}\cdot \alpha_{L}+ \sum_{L - m o v .}( \mathrm{f r e q u e n c i e s}) = - c_{R}+ 1 / 8 \, \alpha_{R}\cdot \alpha_{R}+ \sum_{R - m o v .}( \mathrm{f r e q .}) = M_{R}^{2}\, .
{\cal Z}=\int DB_\mu D\theta^{{\rm sing.}}D\theta^{{\rm reg.}}\exp\left\{-\int d^4x\left[\frac14\left(F_{\mu\nu}+F_{\mu\nu}^E\right)^2+\frac{\eta^2}{2}\left(\partial_\mu\theta-2g_mB_\mu\right)^2\right]\right\},
\nabla^{2}\sqrt{\rho}=\nabla\nabla\rho^{1/2}=\nabla(\frac{1}{2}\rho^{-1/2}\nabla\rho)=\frac{1}{2}[(\nabla\rho^{-1/2})\nabla\rho+\rho^{-1/2}\nabla^{2}\rho]
\frac{E_{s}(\alpha_{N})-I_{s}(\sigma) N^{2}}{N^{1+s}}= \sum_{k=1}^{t-1}\left(\frac{2^{n_{k}+1}}{N}\right)^{1+s}\left(\sum_{j=k+1}^{t}2^{n_{j}-n_{k}}\right)\mathcal{R}_{s}(2^{n_{k}+1}) + \sum_{k=1}^{t}\left(\frac{2^{n_{k}}}{N}\right)^{1+s}\left(1-\sum_{j=k+1}^{t}2^{n_{j}-n_{k}+1}\right)\mathcal{R}_{s}(2^{n_{k}}).
\int_{- 1}^1 (1 - t^2)^{\alpha}C^{(\alpha)}_{n + 2}(t) = \frac{(n - 1) (n + 2 \alpha)}{(n + 2) (n + 2 \alpha + 3)}\int_{- 1}^1 (1 - t^2)^{\alpha}C^{(\alpha)}_n (t) .
\big|g(u_i)-g(x)\big| = \big|(1-\varepsilon)f(v_i)+(1-\varepsilon)d(u_i,v_i)-(1-\varepsilon)f(x)\big| \le (1-\varepsilon)\big(d(u_i,v_i)+|f(x)-f(v_i)|\big) \le (1-\varepsilon)\big(d(u_i,v_i)+d(x,v_i)\big) \le d(x,u_i).
F_{\alpha}(\vec{x},\vec{s}) = \det (1-\mathcal{K}_{\vec{x},\vec{s}}), \mathcal{K}_{\vec{x},\vec{s}}:=\hat\chi_{(0,x_{m})}\sum_{j=1}^{m}(1-s_{j})\mathcal{K}_{\alpha}^{\mathrm{Be}}\hat\chi_{(x_{j-1},x_{j})},
\frac{\partial c_{0}}{\partial\tau}=(1+\frac{Pe^{2}}{48})\frac{\partial^{2}c_{0}}{\partial\xi^{2}}\Rightarrow\frac{\partial c_{0}}{\partial t}=(1+\frac{Pe^{2}}{48})\frac{\partial^{2}c_{0}}{\partial x^{2}}.
|\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{matrix}|_{11}=a_{11}-a_{12}{a_{22}}^{-1}a_{21}|\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{matrix}|_{12}=a_{12}-a_{11}{a_{21}}^{-1}a_{22}.
\det (t I + B'') = (t+\varepsilon_2) \det ((t+\varepsilon_2) I + (1+(k+1)\varepsilon_2)B), &= \sum_{i=0}^n (t+\varepsilon_2)^{n+1-i}I_i(B) (1+(k+1)\varepsilon_2)^i, &= \sum_{m=0}^{n+1}t^{n+1-m}\sum_{i=0}^{n}\varepsilon_2^{m-i}{n+1-i \choose n+1-m}I_i(B) (1+(k+1)\varepsilon_2)^i,
\sum_{j \in \mathbb{J}}\langle F_jx ,F_jx\rangle =\sum_{j \in \mathbb{J}}\left\langle \sum_{k \in \mathbb{L}_j}a_{j,k}e_{j,k},\sum_{m \in \mathbb{L}_j}a_{j,m}e_{j,m}\right \rangle =\sum_{j \in \mathbb{J}}\sum_{k \in \mathbb{L}_j}a_{j,k}a_{j,k}^* =\langle x, x \rangle.
I_{2}= -4r^2\int_{0}^{2\pi}\frac{\cos\left(\frac{x-y}{2}\right)\cos\left(\frac{x+y}{2}\right)(h(x)\sin(x)-h(y)\sin(y))}{((1+r^2)+(r^2-1)\cos(x+y))}dy & = -2r^2\int_{0}^{2\pi}\frac{(\cos(x)+\cos(y))(h(x)\sin(x)-h(y)\sin(y))}{((1+r^2)+(r^2-1)\cos(x+y))}dy
\zeta_{\bar{\cal V}}(0)=0 \hspace{0.2cm}, \hspace{0.2cm}\zeta_{P\bar{\cal K}}(0)=-1\hspace{0.2cm}, \hspace{0.2cm}\frac{d \zeta_{\bar{\cal V}}}{ds}(0)= -\sqrt{2}mL\hspace{0.2cm},\hspace{0.2cm}\frac{d \zeta_{P\bar{\cal K}}}{ds}(0)=-\sqrt{2}mL+\log 48\, .
p_{a}= \frac{1}{\sqrt{k}}\times \frac{1}{l_{s}}\left( \frac{n_{a}}{r_{a}}+ w^{a}r_{a}\right) ~ , \bar{p}_{a}= \frac{1}{\sqrt{k}}\times \frac{1}{l_{s}}\left( \frac{n_{a}}{r_{a}}- w^{a}r_{a}\right) ~ ,
v(0,x) =e^{-\tau A}\widetilde Y_\tau^{x}+\int_0^\tau e^{-sA}G B(s,X^{x}_s)\,ds -\int_0^\tau e^{-sA}\widetilde Z^{x}_{s}\;dW_s =e^{-\tau A}v(\tau,X_\tau^{x})+\int_0^\tau e^{-sA}G B(s,X^{x}_s)\,ds-\int_0^\tau e^{-sA}\nabla^G v(s,X_s^x)\;dW_s.
\mathbb{E}[|\hat{\pi}_{h,n}(x) - \pi (x)|^2] \le c(h_1^{2 \beta_1}+ h_2^{2 \beta_2}) + \frac{c}{T_n}\sum_{j = 1}^2 |\log h_j| + \frac{c}{T_n}\frac{\Delta_n}{h_1 h_2}& \le c(h_1^{2 \beta_1}+ h_2^{2 \beta_2}) + c (\frac{1}{n})^{1 - \frac{1}{\bar{\beta}+ 1}}\frac{\sum_{j = 1}^2 |\log h_j|}{\log (n \Delta_n)}+ \frac{c}{n h_1 h_2}.
[\chi_NZ+\bar \chi_NZ^{-1}]_{\alpha_1 \alpha_2}^{\beta_1 \beta_2}=z_{\alpha_1 \alpha_2}\, \delta_{\alpha_1}^{\beta_1}\delta_{\alpha_2}^{\beta_2}\quad , \quad [\chi_NZ^{-1}+\bar \chi_NZ]_{\alpha_1 \alpha_2}^{\beta_1 \beta_2}=z_{\alpha_1 \alpha_2}^{-1}\, \delta_{\alpha_1}^{\beta_1}\delta_{\alpha_2}^{\beta_2}\, ,
\left( g_{1}g_{2}\right) A_{W}=g_{2}\mathfrak{p}_{+}g_{1}A_{W}+g_{2}\mathfrak{p}_{-}g_{1}A_{W}& =g_{2}g_{1}R_{W}\left( g_{1}\right) +g_{2}A_{g_{1}W}g_{1}\left(I+R_{W}\left( g_{1}\right) \right)
\begin{array}{l c l}{{\mathrm{N S 5}}}&{{:}}&{{0 1 2 3 4 5}}\\{{\mathrm{N S^{\prime}5}}}&{{:}}&{{0 1 2 3 8 9}}\\{{\mathrm{D 6}}}&{{:}}&{{0 1 2 3 7 8 9}}\\{{\mathrm{D 4}}}&{{:}}&{{0 1 2 3 6}}\\{{\mathrm{O 4}}}&{{:}}&{{0 1 2 3 6}}\\ \end{array}
+ \frac{1}{2}\frac{1}{( 2 \pi )^{2}}\Big ( m c \Big )^{2}\int_{\Lambda}d^{2}x d^{2}y \bigg [ 2 e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) - \delta ( y ) , C_{m_{d}}\alpha \big )}
\varphi_{a_1}(\alpha) = \alpha_1 \alpha_2+ \alpha_1 \alpha_3 \varphi_{a_2}(\alpha) = \alpha_1 \alpha_2 + 2\alpha_2^2 \alpha_4 + \alpha_2\alpha_3\alpha_4 \varphi_{a_3}(\alpha) = \alpha_1 \alpha_3 + \alpha_2\alpha_3\alpha_4 + 2\alpha_3^2 \alpha_4 \varphi_{a_4}(\alpha) =\alpha_2^2 \alpha_4 + \alpha_2\alpha_3\alpha_4 + \alpha_3^2\alpha_4,
p\left({{d_{pF}}\leq \delta}\right) = \frac{M}{{k!}}\int_{{{\mathbf{\Omega}}_1}}{\prod\limits_{i = 1}^k{{{\left({{{\sin}^2}{\theta_i}}\right)}^{n - 2k}}}\prod\limits_{1 \leq i < j}^k{{{\left({{{\sin}^2}{\theta_i}-{{\sin}^2}{\theta_j}}\right)}^2}}d{{\sin}^2}{\theta_1}\cdots d{{\sin}^2}{\theta_k}}
D_{++}^{(ML)cd}(x) =D_{++}^{ML}(x) \delta^{cd}={i \delta^{cd}\over \pi^2}\int d^2k\, e^{ikx}{k_+^2\over (k^2 + i \epsilon)^2}={\delta^{cd}\over \pi}{(x^-)^2\over (-x^2 + i\epsilon)}
(\omega_r \omega_n^q)^{n-1}\omega_r =(\omega_r \omega_n^q)^{n-2}\omega_{2r}\omega_n^{m_1}=(\omega_r \omega_n^q)^{n-3}\omega_{3r}\omega_n^{m_2}=\dots=(\omega_r \omega_n^q)^{n-j-1}\omega_{(j+1)r}\omega_n^{m_j}=\dots=\omega_{nr}\omega_n^{m_{n-1}}.
\frac{1}{n}\mathbb{E}[X^n(X^n)^T]\leq P, &\frac{1}{n}\mathbb{E}[(\mathbf{S}^n_i-\hat{\mathbf{S}}^n_i)(\mathbf{S}^n_i-\hat{\mathbf{S}}^n_i)^T]\preceq\mathbf{D}_i,i=1,2,
\partial^2_\mu D(p||p_{\mu,\sigma}) = \frac{1}{\sigma^2}, \partial_\sigma \partial_\mu D(p||p_{\mu,\sigma}) = \frac{2}{\sigma^3}\mathbb{E}_p[x-\mu], \partial^2_\sigma D(p||p_{\mu,\sigma}) = - \frac{1}{\sigma^2}+ \frac{3}{\sigma^4}\mathbb{E}_p[(x-\mu)^2].
q_1 = z_1 z_2, q_2 = z_3^2, q_3 = z_1^2 - z_2z_4, q_4 = z_2^2 + z_4^2 - z_2z_3, \mu_{23}= 1 = -\mu_{34}, (\mu_{14})^2 = \mu_{24}= -1,\mu_{13}= -\mu_{14},
\|u\|_{W^{m,\frac{n}{m}}(\Omega)}= \left( \|u\|^{\frac{n}{m}}_{\frac{n}{m}}+ \sum_{|\alpha|=1}^m \|D^{\alpha}u\|^{\frac{n}{m}}_{\frac{n}{m}}\right)^{\frac{m}{n}}.
\mathcal{P}_d = \begin{bmatrix}\hat{A}0 0 -\hat{S}\end{bmatrix}, \mathcal{P}_\ell = \begin{bmatrix}\hat{A}0 B -\hat{S}\end{bmatrix}\mathcal{P}_u = \mathcal{P}_\ell^\top = \begin{bmatrix}\hat{A}^\top B^\top 0 -\hat{S}^\top \end{bmatrix}.