harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta \, S_{C S}^{\prime}= \frac \ell{8 \pi G}\int_{| r | = \bar{r}}T r [ \partial_{t}g_{1}\, g_{1}^{- 1}\, \delta \zeta - \partial_{t}g_{2}\, g_{2}^{- 1}\, \delta \tilde{\zeta}\, ] |_{\varphi = 0}\, d t \qquad ,
\gamma_F=\begin{bmatrix}1_2 \!\!-1_2 0_4 \end{bmatrix}\otimes\begin{bmatrix}1 \!\!-1_3 \end{bmatrix}+\begin{bmatrix}0_4 \!\!-1 1_3 \end{bmatrix}\otimes\begin{bmatrix}1_2 \!\!-1_2 \end{bmatrix}\;.
j_{A,p}=\min\left\{\frac{|A|}{|\partial A|^{\frac{p}{p-1}}}+\frac{1}{|\partial A|^{\frac{1}{p-1}}},\frac{\deg(\Gamma)|A|}{|\partial^EA|^{\frac{p}{p-1}}}+\frac{1}{|\partial^EA|^{\frac{1}{p-1}}}\right\}.
-W(x,t) = -\sum ca_k P_t \ast K\tilde{H}(x-x_k) & = -\sum_{k<k_0}ca_k P_t \ast K\tilde{H}(x-x_k) -\sum_{k \geq k_0}ca_k P_t \ast K\tilde{H}(x-x_k).
\int_{\Omega}\sum_{i=1}^m q_i \psi_h^n(c^{n+1}_{ih}- c^{n}_{ih})dx= \int_{\Omega}(\rho_h^{n+1}-\rho_h^n)\psi_h^n dx=A_1(\psi_{h}^{n+1}-\psi_h^n, \psi_h^n).
\alpha(t,x)=\ \Phi ({X}_T^{t,x})-\int^T_t \beta(r, X_r^{t,x}) \mathrm d{W}_r &+\int^T_t f(r,{X}_r^{t,x}, \alpha(r, X_r^{t,x}),\beta(r, X_r^{t,x}))\mathrm d r &-w(t,x)-\int^T_t\nabla w(r,{X}_r^{t,x})\mathrm d{W}_r.
\rho(t)=C_4 +C_3t +(\frac{C_1}{2}+\frac{k^2}{4})t^2 +\frac{C_2}{6}t^3+\frac{C_1}{6}t^4+\frac{C_2}{10}t^5+\frac{C_1}{30}t^6+\frac{C_2}{42}t^7,
\eta(\eta(\tau_i, \tau_j),\eta(\tau_i,\tau_\ell)) =\eta((\tau_n^q\tau_1)^{n-i}\tau_n^{q-1+D(n-i+j)}\tau_{n-i+j+1}, (\tau_n^q\tau_1)^{n-i}\tau_{n-i+\ell}) &=\eta(\tau_n^{q-1+D(n-i+j)}\tau_{n-i+j+1}, \tau_{n-i+\ell}).
\sum_{\ell=1}^m \det \! \left( \begin{array}{@{}l@{\;}l@{}}a_{i,\,j}\rangle^{i\leq m}_{i\neq \ell}c_j\,a_{i,\,j}\rangle_{i=\ell}\end{array}\right)=\Big( \sum_{j=1}^m c_j \Big) \det \! \Big( \begin{array}{@{}l@{\;}l@{}}a_{i,\,j}\rangle_{i=1}^{m}\end{array}\Big).
\ell_0\ge\frac{(\nu|\nu+2\rho^\natural)^\natural}{2(M_1+m-3)}+\frac{r (M_1-r-1)}{2 (m+M_1-3)}=-\frac{(\nu|\nu+2\rho^\natural)^\natural-r (2k+r+2)}{2 (2k-m+4)},
d\rho_t = \underbrace{\left[-i\left[H_t,\rho_t\right] + \mathbf{L}^{\top}\rho_t\mathbf{L}^{\ast}- \frac{1}{2}\mathbf{L}^{\dagger}\mathbf{L}\rho_t - \frac{1}{2}\rho_t\mathbf{L}^{\dagger}\mathbf{L}\right]dt}_{}+ \underbrace{\zeta^{\top}\Sigma^{-\top}d\mathbf{W}}_{}.
-L_{v_{-1}}= 2 \rho^{-2}\bigl(\textrm{log}\rho\bigl)^{-2}v_{-1}+ (N-k+1) \rho^{-2}\bigl(\textrm{log}\rho\bigl)^{-1}v_{-1}-\l \eta \rho^{-2}v_{-1}+O\bigl((\textrm{log}\rho) \rho^{-3/2}\bigl) v_{-1}.
S_{\mathrm{b o s o n}}^{\mathrm{r o o t}}= \int d^{4}x \left\{\frac 1{4 g^{2}}F_{\mu \nu}^{2}+ \bar{\nabla}_{\mu}\bar{q}_{A f}\nabla_{\mu}q^{f A}+ \frac{g^{2}}8 [ \mathrm{T r}( \bar{q}\tau^{m}q ) ]^{2}\right\},
T^*=\left[T\in\mathbb{R}:\sum_{n=1}^{N}\frac{\mu_n}{d_n}\left(\ln\left(1+\frac{c_n}{T}-c_n\right)\right.\right. \left.\left.-\frac{c_n}{T(1-c_n)+c_n}\right)=0\right]_{\epsilon}^{\varepsilon}.
\begin{array}{l}S^\omega(\mathsf{E}_\alpha)=-\left(\sum_{\lambda\in X}q^{-(\lambda,\alpha)}\omega(\lambda,\lambda)\omega(\lambda-\alpha,\lambda-\alpha)\right)K_\alpha^{-2}\mathsf{E}_\alpha, S^\omega(F_\alpha)=-\left(\sum_{\lambda\in X}q^{(\lambda,\alpha)}\omega(\lambda,\lambda)\omega^{-1}(\lambda+\alpha,\lambda+\alpha)\right) F_\alpha.\end{array}
L_{i}\ast_{M}^{}F = L_{i}F + \nu \sum_{1 \leq j , k \leq 3}\varepsilon_{i j k}L_{k}{\frac{\partial F}{\partial L_{j}}}+ \nu^{2}\Bigl ( 2{\frac{\partial F}{\partial L_{i}}}+ \sum_{1 \leq j \leq 3}L_{j}{\frac{\partial^{2}F}{\partial L_{i}\partial L_{j}}}\Bigr ) , \quad 1 \leq i \leq 3 .
W_{n}^{m}\approx \sqrt{\frac 2{\pi \nu_{n m}}}\frac{| \eta |^{| n - m |}}{2^{\mu_{n m}}\mu_{n m}! | \xi |^{| n - m | + 1}}\left[ H_{\mu_{n m}}\left( \sqrt{\frac{| n - m | + 1}{2 | \xi |^{2}}}\right) \right]^{2}.
X_{aa}={e^4 \over 16\, m^2 (pk)^2}{\rm Tr}\left[ \gamma^{\beta}(\gamma^{\mu}(p + k)_{\mu}+ m)\gamma^{\alpha}( \gamma^{\nu}p_{\nu}+ m) \gamma_{\alpha}(\gamma^{\rho}(p + k)_{\rho}+ m)\gamma_{\beta}( \gamma^{\lambda}{p'}_{\lambda}+ m) \right]
[x\ast[y,[z,u]],v]+[v\ast[y,[z,u]],x] = (x[y,[z,u]]+[y,[z,u]]x)v-v(x[y,[z,u]]+[y,[z,u]]x) &+(v[y,[z,u]]+[y,[z,u]]v)x-x(v[y,[z,u]]+[y,[z,u]]v) = x([y,[z,u]]v)+[y,[z,u]](xv)-(vx)[y,[z,u]]-v([y,[z,u]]x) &+v([y,[z,u]]x)+[y,[z,u]](vx)-(xv)[y,[z,u]]-x([y,[z,u]]v) = [y,[z,u]](x\ast v)-(x\ast v)[y,[z,u]] = -[x\ast v,[y,[z,u]]].
A =\Big([f_-(-M_x)][g_-(-M_x)]+[f_+(H)][g_+(H)]\Big) &=\Big([f_-g_-(-M_x)]+[f_+g_+(H)]-f(0)g(0)\Big)+f(0)g(0) &=[\gamma(fg)]+f(0)g(0) &=\Gamma(fg)+f(0)g(0),
p_g(X,\Lambda) =\dim\ker \mathcal{R}_{X_1\to X_1\cap X_2}+\dim \ker \mathcal{R}_{X_2\to X_1\cap X_2}=h^0(X_1,(K_X+\Lambda)|_{X_1})+h^0(X_2,(K_X+\Lambda)|_{X_2})-4 < (K_X+\Lambda)^2,
\sum_{i,j=1}^n a_{ij}(z)\xi_i\xi_j + \sum_{i=1}^n\sum_{l=1}^m \left(a_{i,n+l}(z)+a_{n+l,i}(z)\right)\xi_i\eta_l +\sum_{l,k=1}^m a_{n+l,n+k}\eta_l\eta_k &\quad\geq \lambda\left(|x\cdot\xi|^2+|\eta|^2\right)+(1-r_0)\lambda|\xi|^2 &\geq (1-r_0)\lambda \left(|\cdot\xi|^2+|\eta|^2\right).
A_k(n) = \{(\lambda = (\lambda_1^{m_1}, \lambda_2^{m_2}, \ldots, \lambda_\ell^{m_\ell}), \lambda_{i_0}, m)\mid & \qquad\lambda \in \mathcal{P}(n), \,m_i <k \, \forall i, \,1 \leq i_0 \leq \ell, \,k \mid \lambda_{i_0}, \, 0 \leq m < m_{i_0}\}.
d\xi_1=-\mu\omega_1Y_2-\mu\omega_2Y_1+\sum_{\alpha=1}\theta_{1\alpha}\xi_\alpha, d\xi_2=-\mu\omega_1Y_1+\mu\omega_2Y_2+\sum_{\alpha=1}\theta_{2\alpha}\xi_\alpha, d\xi_{\alpha}=-\theta_{1\alpha}\xi_1-\theta_{2\alpha}\xi_2+\sum_{\beta}\theta_{\alpha\beta}\xi_{\beta}.
\hat{\rm T}_{b}\equiv \hat{\pi}_{[\frac{e_{+}b{\rm L}}{2{\pi}{\hbar}}]}-\frac{2\pi}{\rm L}\int_{-{\rm L}/2}^{{\rm L}/2}dx x \cdot(\hat{j}_{+}(x) +{\rm N}\hat{j}_{-}(x)) \equiv-i \hbar \frac{d}{d[\frac{e_{+}b{\rm L}}{2{\pi}{\hbar}}]}+ i{\hbar}\sum_{\stackrel{n \in \cal Z}{n \neq 0}}\frac{(-1)^n}{n}\rho_{\rm N}(n)
\sqrt{{\frac{4 v}{m_{\mathrm{b}}^{2}}}- 1}\, \operatorname{l n}{\frac{v}{\Lambda^{2}}}= \pi - 2 \, \operatorname{a r c t a n}\sqrt{{\frac{4 v}{m_{\mathrm{b}}^{2}}}- 1}\cdotp
R_{\mathcal{BS}(\alpha)}(\mathcal{K}) =\begin{dcases}\dfrac{1}{2-\alpha}, 0\leq \alpha\leq \frac{1}{5}, \dfrac{2\sqrt{\alpha}}{(1+\alpha)\sqrt{1+4\alpha}}, \frac{1}{5}\leq\alpha \leq 1.\end{dcases}
\begin{aligned}\hat{F}(s,\phi,\pi) =\hat{F}(s,\pi-\phi,0), \hat{G}(s,\phi,\pi) =-\hat{G}(s,\pi-\phi,0), \hat{H}(s,\phi,\pi) =-\hat{H}(s,\pi-\phi,0)\end{aligned}
\phantom{=}\lim_{n\to\infty}I_{1,n}'=\int_{\mathbb{R}^N}\varphi_{x_i,\varepsilon}d\mu = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u_0(x)-u_0(y)|^{p}\varphi_{x_i,\varepsilon}(x)}{|x-y|^{N+ps}}dxdy +\mu_i,
M(f) = f(1) + m(f - f(1)) = f(1) + (\lim \limits_{i \in I}\beta_i) (f - f(1)) & = f(1) + \lim \limits_{i \in I}(f(\beta_i) - f(1)) & = \lim \limits_{i \in I}f(\beta_i) = (\lim \limits_{i \in I}\beta_i) (f).
\int \frac{d^3k}{(2 \pi)^3}\left\{D_{\mu \nu}^{\rm T}(k - \ell){\rm tr}\left[\gamma_\mu S(k) \Gamma_\nu^{\rm BC}(k,\ell) S(\ell)\right] + \frac{1}{(k - \ell)^2}F(k,\ell) \right\}= 0,
\begin{cases}\partial_t \omega^{(l)}+ (u^{(l)}\cdot \nabla) \omega^{(l)}=0, u^{(l)}= \Delta^{-1}\nabla^{\perp}\omega^{(l)}, \omega^{(l)}\Bigr|_{t=0}=\omega_0^{(l)}; \end{cases}
|D_{2}\rangle=[(\frac{1}{2}k_{\mu}k_{\nu}\theta_{\lambda}^{2}+2\eta_{\mu\nu}\theta_{\lambda}^{2})\alpha_{-1}^{\mu}\alpha_{-1}^{\nu}\alpha_{-1}^{\lambda}+9k_{\mu}\theta_{\nu}^{2}\alpha_{-2}^{[\mu}\alpha_{-1}^{\nu]}-6\theta_{\mu}^{2}\alpha_{-3}^{\mu}]|0,k\rangle,k\cdot\theta^{2}=0.
\overline{T}_{a,b}(\mathcal{C}) \leq \min_{t \in \mathbb{N}}\left\{t + \left\lfloor2^{-t}\sum_{u=1}^{a}\binom{n}{u}\sum_{i=0}^{b}i\binom{t}{b-i}\right.\right. &+\left.\left.2^{-t(n-k)}\sum_{r=0}^{n-k}(n-k-r){n-k \brack r}_2\prod_{i=0}^{r-1}(2^t-2^i)\right\rfloor\right\}.
\frac{1}{|{\bf x}|}\equiv\sum\limits_{n=-\infty}^{+\infty}\frac{1}{\sqrt{\vec x^2+(\beta n)^2}}=2T\sum\limits_{n=-\infty}^{+\infty}K_0(2\pi T|\vec x{\,}|n)\simeq-2T\ln(\mu|\vec x{\,}|).
\gamma^{-\frac{4}{n-4}}_nH^2_{ij}=\frac{1}{3}R_{imnj}(p)\frac{z^m z^n}{\rho^4}- \frac{2}{3}\underbrace{R_{kmnj}(p)z^kz^m}_{=0}\frac{z^n z^i}{\rho^6}-\frac{2}{3}\underbrace{R_{imnl}(p)z^nz^l}_{=0}\frac{z^m z^j}{\rho^6}&+\frac{4}{3}\underbrace{R_{kmnl}(p)z^kz^m}_{=0}\frac{z^nz^l z^iz^j}{\rho^8}, &=\frac{1}{3}R_{imnj}(p)\frac{z^m z^n}{\rho^4}
\big(\pi(T_xT_y^*)(e_{k,\sigma}\otimes h)\,|\,e_{k,\sigma}\otimes h\big) =\big(\pi(T_yT_{b^t}T_y^*)(e_{k,\sigma}\otimes h)\,|\,e_{k,\sigma}\otimes h\big) &=\big(\pi(T_{b^t})\pi(T_y^*)(e_{k,\sigma}\otimes h)\,|\,\pi(T_y^*)(e_{k,\sigma}\otimes h)\big).
\eta(p\cdot g) =|\det pg| \cdot h(e_r\cdot pg)^{-r}= |\det pg|\cdot h(de_r\cdot g)^{-r}&= |\det p|\cdot |\det g|\cdot |d|^{-r}\cdot h(e_r\cdot g)^{-r}= 1\cdot |\det g|\cdot h(e_r\cdot g)^{-r}= \eta (g).
\int_{\textstyle\prod_{t=1}^m [-b_t,b_t]}\ \frac{du_1du_2\ldots du_m}{\displaystyle\sum_{t=1}^m(u_t^2-a_t^2)^{2s}}\hskip2cm &\le2^m\int_{\textstyle\prod_{j=1}^m[-a_t,b_t-a_t]}\ \frac{dv_1dv_2\ldots dv_m}{\displaystyle\sum_{t=1}^mv_t^{4s}}\hskip2cm
P_{\gamma}^{\mathrm{in}}(M)=P_{\gamma}^{\mathrm{out}}(M):= \begin{cases}(2,...,\,2) \mathrm{if}\,\,M\,\,\mathrm{is\,\,even}, (2,...,\,2,\,1) \mathrm{if}\,\,M\,\,\mathrm{is\,\,odd}, \end{cases}
r \equiv \int d{\tau}_1 d{\tau}_2 d{\tau}_3 \delta ({\tau}_1 ) \delta ({\tau}_2 ) \delta ({\tau}_3 ) \frac{|{\tau}_3 -{\tau}_1 | + |{\tau}_3 -{\tau}_2 |}{|{\tau}_3 -{\tau}_1 | + |{\tau}_2 -{\tau}_1 | + |{\tau}_3 -{\tau}_2 |}\;
A_2^{\dagger}= \frac{1}{\sqrt{6}}\left(b_1\, E_1 + b_2\, E_2 + b_4\, E_4 + b_5 \, E_5 \right) = b_1 A_1 + b_2 A_2 + b_4 A_1 A_2 + b_5 A_2 A_1 ,
\Gamma_2^{(\kappa)}(y) =y(1-y)(\Psi_0(y)^*)^{-1}\,(T^{(\kappa)})^{-1}T^{(\kappa+1)}\, \Psi_0(y)^*, \Gamma_1^{(\kappa)}(y) =(W^{(\kappa)}(y))^{-1}(W^{(\kappa+1)}(y))'
\theta( |\mathbf{p}+ \mathbf{q}| - 1) = \theta(\mathbf{\hat{p}_.q}+ p/2)= \theta(\mathbf{q_.\hat{p}}) + \sum^{\infty}_{m=1}\frac{(p/2)^m}{m!}\delta^{(m-1)}(\mathbf{q_.\hat{p}})
\begin{aligned}|\lambda|^{-n}\Big\| \vec{\psi}(n)-\begin{pmatrix}\lambda^n 0\end{pmatrix}\Big\| \leq 2K\sum_{s=n_0}^{n-1}|\lambda|^{-(n-s)}|R(s)||\lambda|^{s-n}+2K\sum_{s=n}^\infty |\lambda|^{n-s}|R(s)||\lambda|^{s-n}\leq 2K\sum_{s=n_0}^{\infty}|R(s)|\leq \epsilon. \end{aligned}
\min \;\;\;\sum_{k\in M_{D_x}}a^2_k(x) \hbox{subject to}\;\;\; \sum_{k\in M_{D_x}}a_k(x) =0, \;\;\; \sum_{k\in M_{D_x}}a_k(x) (k\delta)^j = j x^{j-1},\;\;\;\;\;j=1,\ldots, \ell.
\frac{\left(v^{n+1}_{h,c}\right)^2 - \left(v^n_{h,c}\right)^2}{2 \tau}=v_{h,c}^{n+ \frac{1}{2}}\frac{v^{n+1}_{h,c}- v^{n}_{h,c}}{\tau}= & = v_{h,c}^{n+ \frac{1}{2}}\left( \Delta_h\, u_h^{n + \frac{1}{2}}\right)_c - v_{h,c}^{n+ \frac{1}{2}}f'\left( u_{h,c}^{n+\frac{1}{2}}\right).
\sum_{a\le q}\sum_{\substack{|\eta|\le E/Y (\eta+a/q)Y\in\mathbb{Z}}}F_Y\Bigl(\frac{a}{q}+\eta\Bigr) \ll (q E)^{27/77}+\frac{q E}{Y^{50/77}}, \sum_{\substack{q<Q d|q}}\sum_{\substack{a\le q (a,q)=1}}\sum_{\substack{|\eta|\le E/Y (\eta+a/q)Y\in\mathbb{Z}}}F_Y\Bigl(\frac{a}{q}+\eta\Bigr) \ll \Bigl(\frac{Q^2E}{d}\Bigr)^{27/77}+\frac{Q^2E}{d Y^{50/77}}.
E\big(Cov_N(\delta_i, y_i) | \delta_U\big) = \frac{1}{N}\sum\limits_{i\in U}\delta_i \mu_i - \big( \frac{1}{N}\sum\limits_{i\in U}\delta_i \big) \big( \frac{1}{N}\sum\limits_{i\in U}\mu_i \big) \rightarrow 0
g_1^{i_0}(x_1,x_2) + g_2^{i_0}(x_1,x_2) =\bigg(g_1^{i_0}(x_1,x_2) -{\chi_{\overline{R}}(x_1,x_2)\over |\overline{R}|}\iint_{\overline{R}}f_1(y_1,y_2)dy_1dy_2\bigg) &\quad+ \bigg( g_2^{i_0}(x_1,x_2) -{\chi_{\overline{R}}(x_1,x_2)\over |\overline{R}|}\iint_{\overline{R}}f_2(y_1,y_2)dy_1dy_2\bigg) =: f_1^{i_0+1}+f_2^{i_0+1}.
C:\quad\left(\gamma_{\mu\nu}\,D^\mu_{(+)}\,D^\nu_{(+)}\,+\,m^2\right)\,u(x)\,=\,0\,\,\,\longrightarrow\,\,\,\left(\gamma_{\mu\nu}\,D^\mu_{(-)}\,D^\nu_{(-)}\,-\,m^2\right)\,v(x)\,=\,0\quad,
\Lambda_0 = B^TP+ D^TPC + G^TPF + S, \Lambda_1 =(B^T+\bar{B}^T)\Pi+ (D^T+\bar{D}^T)P(C+\bar{C}) + (G^T+\bar{G}^T)\Pi(F + \bar{F}) + S + \bar S, \Sigma_0 = D^TPD + R, \Sigma_1 = (D^T+\bar{D}^T)P(D+\bar{D}) + (G^T+\bar{G}^T)\Pi(G + \bar{G}) + (R+\bar R).
0 = 2(k-\ell-1)f^{(\ell+1)}+ \left(\Delta + \ell(\ell-k+n+1-m) + \frac{(n+1-m)(m+n+1-2k)}{4}\right)f^{(\ell)}& + \frac{\ell}{2}\left(\Delta + (\ell-1)(\ell-k-1+2m) + \frac{m(m+n+1-2k)}{2}\right)f^{(\ell-1)}& + \frac{\ell(\ell-1)}{4}\left((\ell-2)(k-\ell+1-m-n) - \frac{(m+n+1)(m+n+1-2k)}{4}\right)f^{\ell-2}.
\zeta_i = \frac{1-\beta T}{2-\beta T}(1-\beta T)^{-n+i-q_i}c_{n-i+q_i}(D_{i-1}^\perp/D_{i-1}- D_i/D_{i-1}- F_{q_i}/D_{i-1}) & - \frac{1}{2}(-1)^{\dim(E/D_{i-1}\cap F/D_{i-1})}\gamma(E/D_{i-1},F/D_{i-1}) \,c_{n-i+q_i}(E/D_i+F/F_{q_i}).
\begin{aligned}\alpha_{\tau\sigma}(hM)=\Big(\frac{1}{6}+\frac{\tau-\sigma}{2}\Big)\varphi_{1}((\tau-\sigma) hM), & \beta_{\tau}( hM )=(1-\tau)\varphi_{1}((1-\tau)hM),\quad\gamma_{\tau}( hM )=\varphi_{0}((1-\tau)hM),\end{aligned}
\mathbf{r}\in\mathbf{X}Z\in H^{1}\left(\Omega\right) /\mathbb{R}\left(\kern-.1em\left(\kern-.1em\nabla Z,\nabla\xi\kern-.1em\right)\kern-.1em\right) =\left(\kern-.1em\left(\kern-.1emL_{\Omega}\mathbf{r},\nabla\xi\kern-.1em\right)\kern-.1em\right) \qquad\forall\xi\in H^{1}\left( \Omega\right) .
\hat{\Lambda}_k(t \mid a, \boldsymbol{z^{(k)}}) = \hat{\Lambda}_{0k}(t) \exp(\hat{\beta}_{kA}a + \skew{3.5}\hat{\boldsymbol{\beta}}_{\boldsymbol{kZ}}^T \boldsymbol{z^{(k)}}), \hat{\Lambda}_{0k}(t) = \int_0^t \frac{dN_k(s)}{\sum_{i=1}^n Y_i(s) \exp(\hat{\beta}_{kA}A_i + \skew{3.5}\hat{\boldsymbol{\beta}}_{\boldsymbol{kZ}}^T \boldsymbol{Z_i^{(k)}})}.
\epsilon \propto 4 \frac{v^{2}}{r^{2}}+ \frac{2}{3}\left( \frac{w^{\prime}}{r}- \frac{w}{r^{2}}\right) + 4 \frac{v^{2}w^{2}}{r^{4}}+ \frac{2}{r^{2}}\left( v^{2}- 1 \right) .
\sum_{a<n<b}g(n) \exp(2\pi if(n)) =\sum_{\alpha-\eta<m<\beta+\eta}\int_a^b g(x) \exp(2\pi i (f(x)-mx)) dx &+ O\left(G_1\bigg(\eta^{-1}+\log \Big(1+\frac{\beta-\alpha}{\eta}\Big)\bigg)+G_2(\beta-\alpha+\eta)\right),
L^n_p(x,r,y,s) = \left\{\begin{array}{c}\mbox{There exists a chain}((x_1,r_1),\dots , (x_{p-1}, r_{p-1})) \mbox{of length}p-1 \mbox{starting in}B(x,r) \mbox{and stopping in}B(y,s) \mbox{s.t. for all}i\in \{1,\dots, p-1 \}, \, x_i \in [-n,n]^d\end{array}\right\}.
\begin{bmatrix}- d_{n-1}^{\mathbb{H}}0 - f_{n-1}d_n^{\mathbb{G}}\end{bmatrix}:{\begin{array}{lcl}\mathbb{H}_{n-1}\longrightarrow \mathbb{H}_{n-2}\bigoplus \searrow \bigoplus \mathbb{G}_n \longrightarrow \mathbb{G}_{n-1}\end{array}}
\Xi(x,y)\!=\!\left(\!\frac{B^rx}{\mu_r}\!\right)^{y}\!\!{\rm G}_{3,3}^{3,3}\!\!\left[\!{\frac{\kappa m{\cal C}}{\kappa_I m_I\bar{\gamma}}}\!\!\left| \begin{array}{cccc}\!\!\!1\!-\kappa_I\!,1\!-\!Lm_I\!,1+y\! \kappa,Nm,0\end{array}\!\!\!\!\right.\right],
\sigma^{AB}(t)=\frac{\rho(v+\beta t)-\rho_{-}v_{-}}{\rho-\rho_{-}}=v+\frac{\rho_{-}(v-v_{-})}{\rho-\rho_{-}}+\beta t=v_{-}+\frac{\rho(v-v_{-})}{\rho-\rho_{-}}+\beta t.
\rho(\lambda) = \frac{p_0'(\lambda)}{2\pi}+ \sum_i\int_{\lambda_{iL}}^{\lambda_{iR}}\frac{d\nu}{2\pi}K(\lambda-\nu)\rho(\nu) + \frac{K(\lambda-\lambda_p)}{2\pi L}+\frac{1}{24 L^2}\sum_{ia}\frac{s_aK'(\lambda-\lambda_{ia})}{2\pi\rho(\lambda_{ia})}
- \frac{\,1\,}{\,2\,}\int^{t}_{0}f^{\prime\prime}( \langle \mathrm m_{s, n}, g \rangle ) \frac{1}{\, n^{2}\,}\sum_{i=1}^{n}\lvert g^{\prime}( X_{s, i}^{(u)})\rvert^{2}{\mathrm d}s \, =\, \int^{t}_{0}f^{\prime}( \langle \mathrm m_{s, n}, g \rangle) \frac{\,1\,}{\,n\,}\sum_{i=1}^{n}g^{\prime}( X_{s,i}^{(u)}){\mathrm d}W_{s, i}\, ,
l(v_i) =\begin{cases}(\underbrace{1,1,\ldots,1}_{\frac{n}{2}-i+2},\underbrace{2,2,\ldots,2}_{i-1}) 1\leq i \leq \frac{n}{2}(1,\underbrace{2,2,\ldots,2}_{\frac{n}{2}-1},1) i=\frac{n}{2}+1 (\underbrace{2,2,\ldots,2}_{n-i+1},\underbrace{1,1,\ldots,1}_{i-\frac{n}{2}}) \frac{n}{2}+2\leq i \leq n \end{cases}
0 = g([{f},{f}]({f}X,Y) - 2\sum\nolimits_{j}d\eta^j({f}X,Y)\,\xi_j,\ \xi_i) = g([{f}^2 X,{f}Y],\xi_i) - ({f}X)(\eta^i(Y)) + \eta^i([{f}X,Y]) ,1\le i\le p.
R \Bigl(t,x, v+w, \frac{\partial v}{\partial x}+ \frac{\partial w}{\partial x}\Bigr) - R \Bigl(t,x, v, \frac{\partial v}{\partial x}\Bigr) =c_1(t,x)w + c_2(t,x)\frac{\partial w}{\partial x}+ \sum_{j+\alpha \geq 2}c_{j,\alpha}(t,x) w^j \Bigl(\frac{\partial w}{\partial x}\Bigr)^{\alpha},
\lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2\overline{s}}\partial_{n+1}v\bigg|_{C_{\overline{s},1/2}'}=\lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2\overline{s}}(\partial_{n+1}x_{n+1}^{1-2s}\partial_{n+1}\tilde{w})\bigg|_{C_{\overline{s},1/2}'}=-\lim_{x_{n+1}\rightarrow0}\sum_{j,k=1}^{n}\partial_{j}a_{jk}\partial_{k}\tilde{w}\bigg|_{C_{\overline{s},1/2}'}=0.
S = - 2^{k}( 0 . 6 0 6 )^{2 k + 1}1 . 2 T_{p}\int_{R^{1 , 8 - 2 k}}d^{p + 1}x \left[ 1 + \frac{( 2 \pi \alpha^{\prime})^{2}}{4}( F_{\mu \nu}F^{\mu \nu}+ 2 \partial_{\mu}X^{i}\partial^{\mu}X^{i}+ 2 i \overline{{{\theta}}}\Gamma^{\mu}\partial_{\mu}\theta ) \right]
\mu(P,z)= \left\langle \sum z^{-n_k}e^{\nu_k}\prod_{q_i=0}{\eta_i \over e^{\eta_i\over2}-e^{-{\eta_i\over2}}}\prod_{q_j\not = 0}{1 \over z^{-{q_j\over2}}e^{\eta_j\over2}-z^{q_j\over2}e^{-{\eta_j\over2}}},[P]\right\rangle ,
\frac{d}{dt}L(t) =-I(u(t))\le \int_\Omega |u|^q\ln|u|dx<\frac{1}{\alpha}\|u\|_{q+\alpha}^{q+\alpha}&<\frac 1\alpha\Big[C_*^{\theta(q+\alpha)}\alpha^{-\frac{\theta(q+\alpha)}{p}}[(\Omega)]^{\frac{s(1-\theta)(q+\alpha)}{2}}\Big]^{\frac{p}{p-\theta(q+\alpha)}}2^\kappa L^\kappa(t).
n_{k}= \frac{1}{2}\, \langle \, \frac{p_{k}^{2}}{\omega_{k}}+ \omega_{k}q_{k}^{2}\, \rangle \approx \frac{{\cal B}_{k}}{\omega_{k}}+ \frac{\omega_{k}}{1 6{\cal A}_{k}}- \frac{1}{2}\, ,
\begin{cases}dp(t,y) = - \pi(t,y) [b_0(t,y)p(t,y) + \sigma_0(t,y)q(t,y)]dt+ q(t,y) dB(t); t\in[0, T] p(T,y) = U'(x(T,y))\mathbb{E}[\delta_Y(y)|\mathcal{F}_T]\end{cases}
v_{5}(t,r) =\frac{-r}{2\pi}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\int_{0}^{1}\partial_{2}G_{5}(s,r\beta,\rho) d\beta
\Pi_{\beta}( \beta \in B \mid Y,\sigma^{2}) =\frac{\int_{B^{\mathrm{c}}}\mathrm{e}^{-\langle P(\varepsilon+r),X(\beta-\beta_{0})\rangle/\sigma^{2}-\|X(\beta-\beta_{0})\|^{2}/(2\sigma^{2})}\pi(\beta)d\beta}{\int \mathrm{e}^{-\langle P(\varepsilon+r),X(\beta-\beta_{0})\rangle/\sigma^{2}-\|X(\beta-\beta_{0})\|^{2}/(2\sigma^{2})}\pi(\beta)d\beta}.
\left ( \begin{array}{c}(\mathcal{F}\tilde{f_{1}})(\xi) \vdots (\mathcal{F}\tilde{f}_{N})(\xi) \end{array}\right ) = \prod_{j = 1}^{\infty}\tilde{\mathcal{A}}( e^{- i 2^{-j}\xi}) \left ( \begin{array}{c}(\mathcal{F}\tilde{f_{1}})( 0 ) \vdots (\mathcal{F}\tilde{f}_{N})( 0 ) \end{array}\right ) .
\mathfrak{M}\bigl(\frac{q}{\tau}\,\Big|\,\frac{1}{\tau},\tau\lambda_1,\tau\lambda_2\bigr) (2\pi)^{-\frac{q}{\tau}}\,\Gamma^{\frac{q}{\tau}}(1-\tau) \Gamma(1-\frac{q}{\tau}) = \mathfrak{M}(q\,|\,\tau,\lambda_1,\lambda_2) (2\pi)^{-q}\times \times \Gamma^{q}(1-\frac{1}{\tau}) \Gamma(1-q).
C \Big [ i \pi \epsilon ( p_{0}) e^{-{\frac{m^{2}}{2 \mu^{2}}}}\Bigg ( \delta ( p^{2}) + \theta ( p^{2}){\frac{m}{2 \mu^{2}}}{\frac{1}{\sqrt{p^{2}}}}I_{1}\Big ({\frac{m \sqrt{p^{2}}}{\mu^{2}}}\Big ) \Bigg ) e^{- p^{2}/ 2 \mu^{2}}\Big ] .
{\frac{2 i ( j_{1 , c}+ j_{2 , c}+ 1 ) ( j_{1 , c}- j_{2 , c})}{i c}}= 2 c \left({\frac{j_{1 , c}^{2}+ j_{2 , c}^{2}+ j_{1 , c}+ j_{2 , c}}{c^{2}}}- 2{\frac{j_{2 , c}^{2}+ j_{2 , c}}{c^{2}}}\right)
[h_0,e_{r,r+1}+e_{n-r,n-r+1}] = (-1)^{r-1}\left( C(n-1,r)e_{r,r+1}+ C(n-1,n-r) e_{n-r,n-r+1}\right) = (-1)^{r-1}C(n-1,r) (e_{r,r+1}-e_{n-r,n-r+1}).
=\sum_{s=0}^{n}\binom{n}{s}y^{n-s}(1-\rho )^{s}\sum_{t=s}^{n}(-1)^{n-t}\frac{(n-s)!}{(t-s)!t!}\rho^{t-s}(\beta +t-s)^{(s)}=\sum_{s=0}^{n}\binom{n}{s}y^{n-s}(1-\rho )^{s}\sum_{m=0}^{n-s}(-1)^{n-m-s}\binom{n-s}{m}\rho^{m}(\beta +m)^{(s)}.
\Gamma_{j_{0}, j_{1}, \cdots, j_{\tau_{1}-1}}:=\bigcup_{i=0}^{\tau_{1}-1}\bigcup_{l=0}^{t_{1}-1}\left(\frac{1}{q}\left(i+\tau_{1}j_{i}+\tau_{1}p_{1}l \right)+\Lambda_{i+\tau_{1}j_{i}+\tau_{1}p_{1}l}\right)
\int_{\mathbb T^{d}_{\ell}}S^0_{\ell,\iota}|y|^2 = \int_{\mathbb T^d_{\ell}}|[\sqrt{R_0}\chi_\ell] \ast \zeta_{\iota}|^2 |y|^2 \leq \int_{\mathbb T^{d}_{\ell}}[|\sqrt{R_0}\chi_{\ell}|^2 * \zeta_{\iota}] |y|^2 & \leq \int_{\mathbb T^{d}_{\ell}}|\sqrt{R_0}\chi_{\ell}|^2 ((1+\iota)|y|^2 + C\iota),
0 = 2^{2m-2}\phi_1(4ai) \cos(4az) + \cosh(2ar)\phi_1(2ai) \cos(2az) & = 2^{2m-2}\phi_1(4ai) \big(2 \cos^2(2az)-1\big) + \cosh(2ar)\phi_1(2ai) \cos(2az).
T_{j}(pg_{L+1})(\mathbf{x}) =p(\mathbf{x})T_{j}g_{L+1}(\mathbf{x})+g_{L+1}(\mathbf{x})T_{j}p(\mathbf{x}) &=2(L+1)x_jp(\mathbf{x})g_{L}(\mathbf{x})+g_{L+1}(\mathbf{x})T_{j}p(\mathbf{x}).
\Bigl\langle \hat \theta_r-\sqrt{1+b_r}\theta_r,u\Bigr\rangle =\frac{\rho_r(u)}{\sqrt{1+b_r+\rho_r(\theta_r)}}&- \frac{\sqrt{1+b_r}}{\sqrt{1+b_r+\rho_r(\theta_r)}\Bigl(\sqrt{1+b_r+\rho_r(\theta_r)}+\sqrt{1+b_r}\Bigr)}\rho_r(\theta_r)\langle \theta_r,u\rangle.
\sum_{k \in \mathcal{P}_H}\mathcal T_{\rm{com}}(k) = - \frac{1}{\vert \Lambda \vert^2}\sum_{k \in \mathcal{P}_H, p \in \mathcal P_L}\frac{\vert z \vert^2 \widehat g(k)^2}{(1-\varepsilon_K) \mathcal D_k}a_p^\dagger a_p + \mathcal E,
\sup_{j\ge 0}\ \|P_{n}T^{\,j}\,e_{k}\|\le |v_{l}| Bigl(\prod_{s=k+1}^{b_{l+1}-1}|w_{s}|\Bigr)\ \sup_{j\ge 0}\ \|P_{n}T^{\,j}e_{b_{\varphi(l)}}\|
P ( \ell , \Phi ) = \left[ \ell^{3}- \frac{\ell^{2}\Phi}{\pi}+ \ell \left( \frac{{\Phi}^{2}}{2{\pi}^{2}}- 1 \right) + \frac{2{\pi}^{2}\Phi -{\Phi}^{3}}{8{\pi}^{3}}\right] ,
\gamma_0 := (x, m, y) \gamma_1 := (y, -d(\lambda_n), \lambda_n y) \gamma_2 := (\lambda_n y, -d(\lambda_{n-1}), \lambda_{n-1}\lambda_n y) \vdots \gamma_n := (\lambda_2\cdots\lambda_n y, -d(\lambda_1), \lambda_1\cdots\lambda_n y),
\lim_n \frac{\lvert T^{-n}[\phi w]\rvert_S}{n}=\lim_n \frac{\lvert (\phi w)_n\rvert_S}{n}&=\lim_n \frac{(n-N)sp(w)+\lvert (\phi w)_N\rvert_S}{n}&=sp(w)+\lim_n \frac{-Nsp(w)+\lvert (\phi w)_N\rvert_S}{n}&=sp(w).
\bar{A}_i^T Q \bar{A}_i + \bar{C}_i^T \bar{C}_i - Q + 2p(1-p) \lambda_i \big(A^{cT}Q A^c + C^{cT}C^c\big) \prec 0 \bar{B}_i^T Q \bar{B}_i + \bar{D}_i^T \bar{D}_i - Z_i + 2p(1-p) \lambda_i \big(B^{cT}Q B^c + D^{cT}D^c\big) \prec 0
\sup_{0\leq s\leq t}\|\partial_zu\|_2^2+\int_0^t\|\nabla\partial_zu\|_2^2ds \leq e^{C\int_0^t(\|u\|_6^4+\|v\|_6^4)ds}\left(\|\partial_zu_0\|_2^2+C\int_0^t\|\nabla_HT\|_2^2ds\right) \leq Ce^{K_1^2(t)t}(\|v_0\|_{H^2}^2+K_1(t))=:K_2'(t),
(P\cap H_0) \times \{1\}= ((P\times \mathbb{C}^*) \cap \tilde{H}_0) \cap (G\times \{1\}) = ((L\times \mathbb{C}^*) \cap \tilde{H}_0) \cap (G\times \{1\}) = (L\cap H_0) \times \{1\}
\begin{aligned}\scriptstyle \mathcal{I}_1^+=\{(0|1)\}, \mathcal{I}_2^+=\{(0,1|1,1),(0,1|1,2)\}, \mathcal{I}_3^+=\{(0,1,0|1,1,1),(0,1,2|1,2,2),(0,1,2|1,2,3)\}, \scriptstyle \mathcal{I}_4^+=\{(0,1,0,1|1,1,1,1),(0,1,0,1|1,1,1,2),(0,1,2,1|1,2,2,1), (0,1,2,1|1,2,2,2),(0,1,2,3|1,2,3,3),(0,1,2,3|1,2,3,4)\}. \end{aligned}
p\left(u, v, w, s, x, y_1, y_2, y_3 \right)= p\left(s \right) p\left(u\mid s \right) p\left( v\mid u,s \right)p\left(w\mid u,v,s\right)p\left(x\mid u,v,w,s \right) p\left(y_1,y_2,y_3\mid x,s\right).
L^1 = \frac{1}{N+1}\sum_{i=0}^N \left| u_i(T) - u(x_i, T) \right|, L^2 = \sqrt{\frac{1}{N+1}\sum_{i=0}^N \left( u_i(T) - u(x_i, T) \right)^2}, L^{\infty}= \max_{0 \leqslant i \leqslant N}\left| u_i(T) - u(x_i, T) \right|,
\frac{1+\int_{0}^{B_{max}}1-F_k(y) \mathrm{d}y}{1+{\displaystyle\max\limits_{1\leq i \leq k}{A_i (1-F_k(A_i^-))}}}= \frac{1+\sum_{i=1}^{k}{\frac{1}{i}}}{1+\displaystyle\max\limits_{1\leq i \leq k}{i \times \frac{1}{i}}}= \frac{1+\sum_{i=1}^{k}{\frac{1}{i}}}{2}\geq \frac{1+\ln(k+1)}{2}
\Sigma_{2}^{(t+1)}=\frac{\sum_{i=1}^{n}T_{2,i}^{(t)}(x_{i}-\mu_{2}^{(t+1)})(x_{i}-\mu_{2}^{(t+1)})^{\top}}{\sum_{i=1}^{n}T_{2,i}^{(t)}}.
a_k = \frac{1}{p_k^2\Big(x_k-x_{k-1}+\displaystyle\frac{\alpha}{p_{k-1}}\Big)\Big(x_{k+1}-x_k+\displaystyle\frac{\alpha}{p_k}\Big)},b_k =-\frac{1}{x_k-x_{k-1}+\displaystyle\frac{\alpha}{p_{k-1}}}\,\left(\frac{1}{p_{k-1}}+\frac{1}{p_k}\right)
\sigma^{1}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) , \sigma^{2}= \left( \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) , \sigma^{3}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) .
\delta{\hat{\xi}}^{{\hat{\imath}}}= \delta^{{\hat{\imath}}6}[ - \Lambda^{( 0 )}+ \frac{m}{2}( 2 \pi \alpha^{\prime}){\rho}^{( 0 )}- \frac{m}{2}( 2 \pi \alpha^{\prime}) \sigma^{( 0 )}\omega^{( 0 )}] ,
\log p(n)= \log \frac{\pi^{2}}{6\sqrt{3}}-3\log \mu(n) +\log (\mu(n)-1)+\mu(n) \quad+\log \left(1+\frac{\mu(n)+1}{\mu(n)-1}e^{-2\mu(n)}\right)+\log \left(1+\frac{f_{2}(n)}{f_{1}(n)}\right) \quad+\log \left(1+\frac{\sum_{k\geq 3}^{\infty}f_{k}(n)}{f_{1}(n)+f_{2}(n)}\right).
\mathbb{E}_t(f(\check{X}_t))\le \mathbb{E}[f(\check{X}_0)]+2\mathbb{E}\left[\int_{s=0}^{\hat T}\left\{(\alpha(0)L-\frac 12\bar\beta) f(\check{X}_s) +\frac 12\alpha(s)\sigma \right\}ds\right] -2\mathbb{E}\left[\int_{s=\hat T}^t \left\{\frac{1}{2}\bar\beta f(\check{X}_s) -\frac 12\alpha(s)\sigma \right\}ds\right]