harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(x_{1}c_{1}+x_{2}c_{2}).(y_{1}z_{1}+y_{2}z_{2})=(x_{1}y_{1}+x_{2}y_{2}).(c_{1}z_{1}+c_{2}z_{2})+(x_{1}z_{2}-x_{2}z_{1}).(c_{1}y_{2}-c_{2}y_{1})
(U_\theta(t)U_\theta(s)f)(r) =U_\theta(t)(e^{-\frac i2(s,\theta r)}f(r-s)) =e^{-\frac i2(t,\theta r)}e^{-\frac i2(s,\theta (r-t))}f(r-s-t) =(e^{\frac i2(s,\theta t)}U_\theta(t+s)f)(r).
\sum_{\nu=1}^N \beta_{\mu\nu}X_{\mu\nu}+ \sum_{\mu,\nu=1}^N\lambda_{\mu\nu}X_{\mu\nu}= \sum_{\nu=1}^N \beta_{\mu\nu}Y_{\mu\nu}+ \sum_{\mu,\nu=1}^N\lambda_{\mu\nu}Y_{\mu\nu}.
\mathbb{E}_{\gamma}\theta [\int_I x^S q(y^S)dE]-\mathbb{E}_{\gamma}\theta [\int_I x^R q(y^R)dE] & = \mathbb{E}_{\gamma}\theta^{\prime}[\int_I x^R q(y^R)dE][\int_I \Delta x_0^{(3)}q(y^R)+x^R q^{\prime}(y^R)\Delta y_0^{(3)}dE]+ A_4+A_5+o(N^{-2+t}).
W_{\mathrm{e f f}}= S \left\{\ln \left[ \frac{S^{N_{c}- N_{f}+ 1}\mathrm{d e t}({\tilde{T}}) f ({\tilde{Z}})}{{\tilde{\Lambda}}^{3 N_{c}- N_{f}+ 1}}\right] + N_{f}- N_{c}- 1 \right\},
\left|\Lambda_\lambda(0)\right| =\left|\left\{g\in\mathbb H^n: \delta_{d_K(g)^{-1}}( g)\in \Gamma,{|b(g)|\over d_K(g)^Q}>\lambda\right\}\right| &\leq{1\over{C_{K_j}\lambda}}\|[b,R_j]\|_{L^1(\mathbb H^n)\rightarrow L^{1,\infty}(\mathbb H^n)}.
z_{2}= \frac{3 \mu^{2}( \mu - 1 ) \Gamma ( \mu ) ( \eta_{1}^{\mathrm{o}})^{2}}{2 ( \mu - 2 )^{2}( 2 \mu - 1 )^{2}}\left[ \hat{\Theta}- \frac{1}{( \mu - 1 )^{2}}\right]
\begin{aligned}x_{n_i}=r_i \cos \phi^{i}_{d_i-1}x_{n_i-1}=r_i \sin \phi^{i}_{d_i-1}\cos \phi^{i}_{d_i-2}\cdots \cdots x_{n_{i-1}+1}=r_i \sin \phi^{i}_{d_i-1}\cdots \sin \phi^{i}_2\sin \phi^{i}_1. \end{aligned}
\begin{cases}(\partial_t^2 - c^2(x)\Delta) u(x,t) = 0, (x,t) \in \left(\mathbb{R}^n \setminus \Omega \right) \times (0, \infty), \, n \ge 2, u(x,0) = u_0(x), \partial_t u(x,0) = u_1(x), u(t,x) = 0, (x,t) \in \partial \Omega \times (0,\infty),\end{cases}
\sum_{k=0}^{n-1}\frac{(q;q^2)_k(-1;q^2)_k}{(q^2;q^2)_k}q^{(n-1)^2-k^2}begin{align*}7pt] \equiv -\frac{\left(1+(-1)^{\frac{n-3}{2}}\right)q(1-q^n)}{2}begin{align*}7pt] +(-1)^{\frac{n-1}{2}}\left(q^{-n+1}+\frac{(1-n)q(1-q^n)}{2}\right)\pmod{\Phi_n(q)^2}.
\frac{\partial}{\partial x^{\mu}}\psi^{q}= - i \, D_{x}( \hat{A}) \left( D_{x}^{\dagger}( \hat{A}) D_{x}( \hat{A}) \right)^{- 1}\sigma_{\mu}^{\dagger}\psi^{q}+ \psi^{r}~ R_{\mu}^{r q}( x ) ,
Q_{1, \pm}(x;k)=M_{\pm}(x;z) V_1(k)-V_1(k)=\left(\left(M_{\pm,1}(x;z)-e_1\right), (2ik)M_{\pm,2}(x;z)-e_2\right), &Q_{2, \pm}(x;k)=M_{\pm}(x;z) V_2(k)-V_2(k)=\left((2ik)^{-1}\left(M_{\pm,1}(x;z)-e_1\right), M_{\pm,2}(x;z)-e_2\right),
f(\zeta)=\frac{1}{(2\pi i)^{n}}\int\cdot\cdot\cdot\iint_{\partial D_{1}\times\cdot\cdot\cdot\times\partial D_{n}}\frac{f(z_{1},...,z_{n})}{(z_{1}-\zeta_{1})\cdot\cdot\cdot(z_{n}-\zeta_{n})}dz_{1}\cdot\cdot\cdot dz_{n}
\sum_{l=0}^k \binom{n+k-2}{k-l}\sigma^{(k-l)}v_n^{(l)}+ \sum_{l=0}^{k-1}\binom{n+k-2}{k-l-1}\tau^{(k-l-1)}v_n^{(l)}+ \sum_{l=0}^{k-2}\binom{n+k-2}{k-l-2}h^{(k-l-2)}v_n^{(l)}= 0.
\sup_{z\in S^1}\left|\left[D^L_{C^n Z}(\phi - \sum_{i=1}^n c_{\omega_i, v_i})\right]^{\widetilde{}}(z)\right| = \sup_{z\in S^1}\left|\left[\phi(C^n Z) - \sum_{i=1}^n c_{\omega_i, v_i}(C^n Z)\right](z)\right| & = \sup_{z\in S^1}\left|D^L_{C^n}\left(\phi( Z) - \sum_{i=1}^n c_{\omega_i, v_i}( Z)\right)(z)\right|
\tilde{\Psi}_{y}= \tilde{\Psi}_{x x}+ 2 ( \alpha - 1 ) ( \tilde{\Psi}\tilde{\Psi}^{*}) \Psi_{x}+ 2 \alpha \tilde{\Psi}^{2}\tilde{\Psi}_{x}^{*}- \alpha ( 1 + \alpha ) ( \tilde{\Psi}\tilde{\Psi}^{*})^{2}\tilde{\Psi}.
\begin{aligned}E_{P}\biggl[\varphi\biggl(x+\sum_{i=1}^{n}Z_i^\theta\biggr)\biggr] = \xi_0 E_P\biggl[\varphi\biggl(x+\sum_{i=1}^{n}Z_i^\theta\biggr)\vert H_1\biggr]+(1-\xi_0)E_P\biggl[\varphi\biggl(x+\sum_{i=1}^{n}Z_i^\theta\biggr)\vert H_2\biggr],\end{aligned}
\m^{2}(\beta)=m^{2}_{0}+\lambda_{0}\beta^{2-D}\sum^{\infty}_{k=0}f(D,k)\left(m(\beta)\beta\right)^{2k}\int^{\infty}_{m(\beta)\beta}d\tau\frac{\tau^{D-3-2k}}{e^{\tau}-1},
\Sigma_3(t) \lesssim||\partial_1\mathcal{A}_{(0)}\partial_1{\mathbf V}||^2_{s-4,\ast,t}+||\partial_1(\partial_1\mathcal{A}_{(0)}\partial_1{\mathbf V})||^2_{s-4,\ast,t}&\leq C(K)\Big(||{\mathbf V}||^2_{s,\ast,t}+||\dot{{\mathbf U}}||^2_{W^{2,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s,\ast,t}\Big).
J_{\lambda_k}(u) \ge \dfrac{\lambda_{k+1}-\lambda_k}{2}\int_{\Omega}|u(x)|^2 dx- \int_{\Omega}F(x,u(x)) dx & \ge - M \int_{\Omega}|u(x)|dx \ge - \tilde{M}\|u(x)\|_{\mathbb{X}(\Omega)}\ge - \tilde{M}R=:-C_{R},
C_{0,B}=\frac12\dot\phi^2+\frac{m_B^2}2\phi^2+\frac{\lambda_B}8\phi^4+\frac{\lambda_B}{32\pi^2}\int_0^\Lambda\,\coth\frac{\beta W_k}2 \Biggl[|\phi\dot\varphi_k-\dot\phi\varphi_k|^2 +k^2|\varphi_k|^2\phi^2\Biggr]\,dk\ ,
\left< v_{1,2}^t, v_{1,3}^t \ \middle| \t = 1, \dots, d \right> \cap \left< v_{1,4}^t \ \middle| \t = 1, \dots, d \right> = \{0 \}\ \ \left< v_{1,2}^t \ \middle| \t = 1, \dots, d \right> \cap \left< v_{1,3}^t \ \middle| \t = 1, \dots, d \right> = \{0 \}.
W^{-2}(V \log \log T/2e) (1.001 \log \log T) \leq W^{-2}V (\log \log T)^2/5.4 = V^{-1}(\log \log T)^2 (1-1.1\varepsilon)^{-2}/5.4 \leq 2 (1-1.1\varepsilon)^{-2}/5.4 \leq 1/2
\log \frac{\,{\mathrm d}\mathbb P^{(u)}\,}{\,{\mathrm d}\mathbb P_{0}\,}\Big \vert_{\mathcal F_{T}}\, =\, \int^{T}_{0}( X_{t}- u \widetilde{X}_{t}){\mathrm d}X_{t}+ \frac{\,1\,}{\,2\,}\int^{T}_{0}( X_{t}- u \widetilde{X}_{t})^{2}{\mathrm d}t \, .
\frac{1}{\rho}k ( \rho \mu )^{2}= \left. \frac{1}{\rho}\frac{a ( \rho ) c_{1}( \rho \mu )^{2}+ b ( \rho ) c_{2}( \rho \mu )^{- 2 \gamma}}{a ( \rho ) c_{3}+ c_{4}b ( \rho ) ( \rho \mu )^{- 2 \gamma}}\right.
-\operatorname{div}\left({\mathbf{A}}^{-1}\nabla\varphi\right)=\operatorname{div}\left({\mathbf{A}}^{-1}R^{\operatorname{curl}}(\mathbf{f})\right) \quad\mbox{in $\omega$},\qquad{\mathbf{n}}\cdot{\mathbf{A}}^{-1}\nabla\varphi=-{\mathbf{n}}\cdot{\mathbf{A}}^{-1}R^{\operatorname{curl}}({\mathbf{f}})\quad\mbox{on $\partial\omega$.}
\sum_{\nu}\sum_{j}(u_\nu^j)^2-\sum_{j}y_j^2-\sum_\gamma\sum_j(x_\gamma^j)^2=0, &\sum_{\nu}\sum_{j}\nu u_\nu^j-\sum_\gamma\sum_j\gamma\,x_\gamma^j=0,\sum_{\nu}\sum_{j}\nu^2-\sum_\gamma\sum_j\gamma^2=n.
\{\Phi_x,\Pi_y\}=\langle D_1\Phi_x,D_2 \Pi_y\rangle_{H^1}-\langle D_1\Pi_x,D_2 \Phi_y\rangle_{H^1}&=\langle\mathcal{E}_x,\mathcal{E}_y\rangle_{H^1}=\mathcal{E}_x(y)=\mathcal{E}_y(x)\,.
[\frac{\lambda}{\mu}]_{2}[\frac{\mu}{\lambda}]_{2}=(-1)^{\frac{N\lambda-1}{2}\frac{N\mu-1}{2}},[\frac{1-\omega}{\lambda}]_{2}=(\frac{a}{3}),[\frac{2}{\lambda}]_{2}=(\frac{2}{N\lambda}).
\lim_{\lambda\rightarrow\infty}\frac{\Psi_{\lambda p(p+1)+\rho+1,k}(z)}{\Psi_{\lambda p(p+1)+\rho,k}(z)}=\frac{\varepsilon_{k}^{(\rho)}\,g_{k}^{(\rho)}(z)}{(\kappa_0^{(\rho)}\cdots\kappa_{k-1}^{(\rho)})^2}\frac{\widetilde{F}_k^{(\rho)}(z^{p+1})}{\widetilde{F}_{k-1}^{(\rho)}(z^{p+1})}, z\in\mathbb{C}\setminus(\Gamma_{k-1}\cup\Gamma_{k}\cup\{0\}),
\stackrel{( 3 )}{\varphi}_{\mu}\, = \, p_{1 \mu}\, - \, m \, \frac{q_{2 \mu}}{\sqrt{q_{3}^{2}}}\, - \, \beta \, \frac{\varepsilon_{\mu \nu \lambda}\, q_{2}^{\nu}\, q_{3}^{\lambda}}{q_{2}^{2}\, \sqrt{q_{2}^{2}}}\, = \, 0 , \quad \mu \, = \, 0 , \, 1 , \, 2 \,{.}
|-\frac{1}{\omega}\int_{\frac{2}{\sqrt{\omega}\lambda(t)}}^{\infty}\phi(r,\omega \lambda(t)^{2}) \sqrt{r}F_{4}(t,r\lambda(t)) dr| \leq \frac{C |a(\omega \lambda(t)^{2})|}{\omega^{5/4}\lambda(t)^{1/2}}\int_{\frac{2}{\sqrt{\omega}\lambda(t)}}^{\infty}\sqrt{r}|F_{4}(t,r\lambda(t))| dr
\cos^2\psi_0-\cos^2\psi=\frac{\sec^2\psi-\sec^2\psi_0}{\sec^2\psi_0 \sec^2\psi}=\frac{\tan^2\psi-\tan^2\psi_0}{\sec^2\psi_0 \sec^2\psi}= \frac{r^2-z^2\tan^2\psi_0}{z^2\sec^2\psi_0 \sec^2\psi},
\begin{array}{l}\displaystyle v(s,y) = \alpha(s) \varphi_0(y) + f(s,y), begin{align*}5pt]\displaystyle w(s,y) = \frac{d\alpha}{ds}(s) \varphi_0(y) + \alpha(s)\psi_0(y) + g(s,y),\end{array}
\inf_{\omega \in O(\frac{n(n-1)}{2})}\sup_{B_{2^{j/20}}(x_2)}\max_\alpha \bigg | K_\alpha^{(x_1,t_1)}- \sum_{\beta=1}^{\frac{n(n-1)}{2}}\omega_{\alpha\beta}K_\beta^{(x_2,t_2)}\bigg | \leq C \, 2^{-j+\frac{j}{10}}.
\psi_{N}^{+}= \frac{1}{[ 2 r ( r + r_{3}) ]^{\frac{1}{2}}}\left( \begin{array}{c}{r + r_{3}}\\{r_{1}+ i r_{2}}\\ \end{array}\right) = \left( \begin{array}{c}{\cos \frac{\theta}{2}}\\{e^{i \phi}\sin \frac{\theta}{2}}\\ \end{array}\right)
\{Y_{l_{1},m_{1}}^{g},Y_{l_{2},m_{2}}^{g}\}={\displaystyle\sum\limits_{l}}K_{m_{1},m_{2},m}^{l_{1},\l_{2},\l}\Y_{l,m}^{g}& ={\displaystyle\sum\limits_{l,\mu}}D_{\mu,m}^{l}(g)K_{m_{1},m_{2},m}^{l_{1},\l_{2},\l}\Y_{l,\mu}\ .
w(e)^{p-r}\cdot\prod_{v\in e}B(v,e) =\Bigg(\frac{r}{\lambda^{(p)}(G[S])}\prod_{u\in e}x_u\Bigg)^{p-r}\cdot\prod_{v\in e}\frac{\prod_{u\in e}x_u}{\lambda^{(p)}(G[S])x_v^p}& =\frac{r^{p-r}}{(\lambda^{(p)}(G[S]))^p}=\alpha.
F [ L , \beta , \nu ] \propto \cos{( \pi \nu )}\frac{\left( \beta^{2}- 8 \pi^{2}\alpha^{\prime}\right)^{3 / 2}}{L^{4}}e^{- L \sqrt{\beta^{2}- 8 \pi^{2}\alpha^{\prime}}/ 2 \pi \alpha^{\prime}}.
\int\limits_{\mathbb{R}}f\left( y\right) d\mu \left( y\right) =\frac{1}{2\pi}\int\limits_{0}^{+\infty}\left| \frac{\Gamma \left( a+it\right) \Gamma\left( b+it\right) \Gamma \left( c+it\right)}{\Gamma \left( 2it\right)}\right| \frac{f\left( t^{2}\right)}{\Gamma \left( a+b\right) \Gamma \left(a+c\right) \Gamma \left( b+c\right)}dt
I_{\psi,{\widetilde \psi}}\equiv{1\over 2}\int_{M}\epsilon^{\mu \nu \rho \sigma}\left[{\widetilde \psi}_{\; \; \; \mu}^{A'}\;e_{AA' \nu}\; D_{\rho}\; \psi_{\; \; \sigma}^{A}-\psi_{\; \; \mu}^{A}\; e_{AA' \nu}\; D_{\rho}\;{\widetilde \psi}_{\; \; \; \sigma}^{A'}\right] d^{4}x \; ,
S_t = x-y + \int_s^t \{f(\phi(s,r,x)) - f(\phi(s,r,y))\}dr & = x-y + \int_s^t \int_0^1 f'(u \phi(s,r,x) + (1-u)\phi(s,r, y)) (\phi(s,r,x) - \phi(s,r,y))du dr &= x-y + \int_s^t \biggl\{\int_0^1 f'(Z_r^u)du\biggr\}S_rdr
\|x^{k+1}_i - x^*_i\|^2 \leq \|x_i^k-\alpha_{i}F_i(x_i^k,N\hat v^k_i) - x^k_i - x^*_i + \alpha_{i}F_i(x^*_i,\bar x^*)\|^2 \cr = \|x_i^k-x^*_i \|^2 + \alpha_{i}^2\| F_i(x_i^k,N\hat v^k_i) -F_i(x^*_i,\bar x^*) \|^2 \cr - 2\alpha_{i}(F_i(x_i^k,N\hat v^k_i) -F_i(x^*_i,\bar x^*))^T(x^k_i-x^*_i).
\int_x^{h^nx}\widetilde{F}+\int_x^{p x}\widetilde{F}= \int_{px}^{p h^nx}\widetilde{F}+\int_x^{p x}\widetilde{F}\ge -2L(r,B)-C_{F,\xi}(x,p h^nx) &\ge -3L(r,B)+\int_x^{p h^nx}\widetilde{F}.
\gamma_1 \frac{r_n}{\sqrt{k}}\leq D_l(k) \leq \gamma_2 \frac{r_n}{\sqrt{k}}\mathbb{E}\left(d^2(Y_{k},\{Y_u\}_{1 \leq u \leq k-1}) | N_l = k\right) \leq \gamma_3 \frac{r_n^2}{k}.
{(a_1+a_2)b_1\neq a_1b_2},\d_{1}\in C^{2}(\mathbb{R}_{\geq 0};\mathbb{R}), &u_0\in \mathbb{H}^2_{(0)}:=\{u\in H^2(0,1); a_1u(1)+a_2u_x(1)= 0, &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b_1u(0)+b_2u_x(0)=0\}.
u(x,x_0) = e^{i\tau\Phi}(1-\frac{1}{4}e^{-i\tau (\bar\Phi+\Phi)}\partial^{-1}(e^{i\tau (\bar\Phi+\Phi)}(\bar\partial^{-1}q-\bar\partial^{-1}q(x_0))) + r(x,x_0)),
[\hat{a}_i({\bf k},t),\hat{a}_j^\dag({\bf q},t)]=\delta_{ij}\delta({\bf k}-{\bf q})\;,~~[\hat{a}_i({\bf k},t),\hat{a}_j({\bf q},t)]=[\hat{a}_i^\dag({\bf k},t),\hat{a}_j^\dag({\bf q},t)]=0\;,
\theta(V, p, 1) = \frac V{p-1}+ O\bigg( \frac{V}{(\log V)^3}\frac{\log Q}Q \bigg) &\le \frac VQ + O\bigg( \frac{V}{(\log V)^3}\frac{\log Q}Q \bigg) = \log x - \frac{\log x}{\log\log x}+ O\bigg( \frac{\log x}{(\log\log x)^2}\bigg).
\delta^{N e w}B_{\mu_{1}\ldots \mu_{p_{a}}}^{a}= \partial_{[ \mu_{1}}\Lambda_{\mu_{2}\ldots \mu_{p}]}^{a}+ g_{b}p_{a}! p_{b}k_{b \mu_{1}\ldots \mu_{p_{a}}}^{a \nu_{2}\ldots \nu_{p_{b}}}\Lambda_{\nu_{2}\ldots \nu_{p_{b}}}^{b}.
\tilde{D}_{\alpha}= e^{i \partial_{i}^{p}D_{i}}e^{- i x_{\mu}p_{\mu}}D_{\alpha}e^{i x_{\nu}p_{\nu}}e^{- i \partial_{j}^{p}D_{j}}\, , \quad ( \partial_{i}^{p}= \frac{\partial}{\partial p_{i}}) .
\mathbb{P}\Bigg(\min\limits_{2 \leq l \leq \frac{N^{1+\delta}}{2}}\Gamma_{l}> N^{1+\delta}\Bigg) = 1 - \mathbb{P}\Bigg(\min\limits_{2 \leq l \leq \frac{N^{1+\delta}}{2}}\Gamma_{l}\leq N^{1+\delta}\Bigg) \geq 1 - N^{1+\delta}\Bigg(\frac{2}{N^{2}}\Bigg) - \sum\limits_{l=3}^{\frac{N^{1+\delta}}{2}}\frac{8N^{1+\delta}}{N^{3}}
R_4 =\{\iota^2\}\cup\{\iota\tilde\alpha_0=\tilde\alpha_0\iota\}\cup\{\tilde\alpha_i\iota=\varepsilon_i^2\iota\tilde\alpha_i\mid 1\le i\le 4\}\cup\{\varepsilon_i\iota\varepsilon_i\iota\mid 1\le i\le 4\}
\varLambda_j= \left\lceil\frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{S_j}\right\rceil,S_j=\sum_{\ell=1}^j \mathcal E_{\ell}, \varLambda'_j= \,\left\lceil\frac{n^{1/2}}{c}\log\frac{\tfrac{n^{1/2}}{c}}{S'_j}\right\rceil,S'_j=\sum_{\ell=1}^j\mathcal E'_{\ell},
\int_{\Gamma_2}g(\lambda- \lambda_h) (u_h-u)\, ds =\int_{\Gamma_2}g \ \lambda \u_h\, ds - \int_{\Gamma_2}g \ \lambda \u\,ds -\int_{\Gamma_2}g \ \lambda_h \u_h\,ds + \int_{\Gamma_2}g \ \lambda_h \u\,ds \leq \int_{\Gamma_2}g \ |u_h|\, ds - \int_{\Gamma_2}g \ |u|\,ds -\int_{\Gamma_2}g \ |u_h|\,ds + \int_{\Gamma_2}g \ |u|\,ds = 0.
J_3^n(\theta',\theta) := \int_{\Delta_{\theta',\theta}^m}\frac{\partial}{\partial x_{l_1}}b_n^{(i)}(s_1,x+B_{s_1}^H) \cdots \frac{\partial}{\partial x_j}b_n^{(l_{m-1})}(s_m,x+B_{s_m}^H) K_H(s_m,\theta') ds_m \cdots ds_1.
g(\tau +h) + (\Delta_h K * dZ)_\tau = g(\tau +h) + (\Delta_h K * L)(0)(X - g)(\tau) &+ (d(\Delta_h K * L) * X)_{\tau}- (d(\Delta_h K * L) *g) (\tau) &\geq g(\tau+h) - (d(\Delta_hK*L)*g)(\tau) - (\Delta_hK * L)(0) g(\tau),
M_{T_1,T_2}^{k,l,\beta}(\nu ,\omega ) =\sup\limits_{t\in [T_1,T_2]}\Bigg(\frac 4\beta e^{-2O_t^{(k)}}\|f^{(k)}(e^{O_t^{(k)}}x^{(k)}(t))\|^2+\frac 4\beta |O_t^{(k)}|^2\|x^{(k)}(t)\|^2\Bigg) & +\sup\limits_{t\in [T_1,T_2]}\Bigg(\frac 4\beta e^{-2O_t^{(l)}}\|f^{(l)}(e^{O_t^{(l)}}x^{(l)}(t))\|^2+\frac 4\beta|O_t^{(l)}|^2\|x^{(l)}(t)\|^2\Bigg)
\tilde{Z}_{G}(q,v)=q^{|V|-|E|-1}v^{|E|}\tilde{Z}_{G^{*}}(q,\frac{q}{v})withv=\frac{p}{1-p}and\tilde{Z}_{G}(q,v)=(1-p)^{-|E|}Z_{G}(q,v)
\displaystyle \chi(\mathbb{R}^4)-\frac{1}{4\pi^2}\int_{\mathbb{R}^4}Q_g dv_g= \sum_{j=1}^{k}\lim_{r\rightarrow\infty}\frac{vol_g(\partial B_{j}(r))^{4/3}}{4(2\pi^2)^{1/3}vol_g(B_{j}(r))},
[z\otimes\alpha,(E_{1,3}+E_{3,1})\otimes\frac{1}{2}[b_{1},b_{2}]_{C}+[z\otimes\alpha,(E_{1,3}-E_{3,1})\otimes\frac{1}{2}(b_{1}\circ b_{2})_{E}]=[\varepsilon e_{2}\otimes\alpha b_{1},u_{2}\otimes b_{2}].
A_{\overline{{{T}}}T \overline{{{T}}}T}( k_{1}; k_{2}; k_{3}; k_{4}; a_{i}) \sim ( 2 \pi )^{p + 1}\delta ( \sum k ) \: \frac{1}{\alpha^{\prime}}{\cal C}_{0}{\cal N}_{T}^{4}V_{\bot}\frac{s + t}{s t}\left( 1 + \alpha^{\prime}( s + t ) \right)
= i \int \frac{d^{4}x_{1}d^{4}p^{\prime}}{( 2 \pi )^{4}}e^{- i p_{\mu}^{\prime}x_{2}^{\mu}}\frac{p_{\nu}^{\prime}\gamma^{\nu}+ m}{p_{\tau}^{\prime}p^{\tau \, \prime}- m + i \epsilon}e^{i p_{\eta}^{\prime}x_{1}^{\eta}}N_{\sigma}\gamma^{\sigma}\psi ( 1 )
{\bf I}_1 = -2p\kappa Z + 2p\nu^2(1-p) Z + 2p^2\nu^2(Y^{p-1/2}-\widetilde Y^{p-1/2})^2 \le -2p\nu^2\left(\frac{\kappa}{\nu^2}- 1+p - \frac{(2p-1)^2}{4(1-p)}\right)Z.
K(H_{11}^fH_{22}^f-H_{12}^f{}^2) = cK^2\Delta - K^2 (f_1{}^2G-2f_1f_2F+f_2{}^2E) = \frac{b^4c}{E^3}-\frac{b^4}{E^4}(f_1{}^2+f_2{}^2E) = \frac{b^4}{E^3}\left( c-\frac{1}{E}f_1{}^2-f_2{}^2 \right).
\hat{S}=\int d^{D}x\sqrt{-\hat{G}}e^{-\hat{\phi}}\left( R_{\hat{G}}+\left(\hat{\partial}\hat{\phi}\right)^{2}-\frac{1}{12}\hat{H}_{\hat{\mu}\hat{\nu}\hat{\rho}}\hat{H}^{\hat{\mu}\hat{\nu}\hat{\rho}}\right) ,
F(a,b;t)= \large\left(\frac{1-b}{1-t}\large\right)\large\left(1-\frac{(b-atq)}{(b-aq)}aq\large\right) + \frac{(1-aq)(b-atq)(b-atq^2)}{(1-t)(b-aq)}F(aq,b;tq),
(\eta^i,\widetilde{\eta}_i,\kappa)\cdot (\zeta^i,\widetilde{\zeta}_i^{},\sigma^{})=(\zeta^i+\eta^i, \widetilde{\zeta}_i^{}+\widetilde{\eta}_i, \sigma^{}+\kappa +\widetilde{\zeta}_i^{}\eta^i-\zeta^i\widetilde{\eta}_i)\,.
D_{0}\, = \, \begin{pmatrix}1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 \end{pmatrix}D_{1}\, = \, \begin{pmatrix}0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 \end{pmatrix}.
\bigg\|E\left[\sum_{i=1}^{n}\vartheta_i|\xi^i|\bigg|\mathcal{F}_t\right]\bigg\|_{\infty}\leq \bigg\|E\left[\sqrt{\sum_{i=1}^{n}\vartheta_i^2}\sqrt{\sum_{i=1}^n|\xi^i|^2}\bigg|\mathcal{F}_t\right]\bigg\|_{\infty}\leq\sqrt{\sum_{i=1}^{n}\vartheta_i^2}\sqrt{\bigg\|\sum\limits_{i=1}^n|\xi^i|^2\bigg\|_{\infty}},
49(J_{n+r}^{(3)}j_{n+s}^{(3)}) =2^{2n+4+r+s}+3\cdot 2^{n+r+1}(a\omega_{1}^{n+s}+b\omega_{2}^{n+s})-2^{n+s+3}(a\omega_{1}^{n+r}+b\omega_{2}^{n+r}) &\ \ -3(a^{2}\omega_{1}^{2n+r+s}+b^{2}\omega_{2}^{2n+r+s})-7(\omega_{1}^{r}\omega_{2}^{s}+\omega_{1}^{s}\omega_{2}^{r}),
\mathbb E\left[Y_{1,i}^{(2t)}\right]^{\alpha/2}\leq Ck^{1+\frac{2^m-1}{2^{m+1}}\alpha}\left[\max\{1,\; p^{\alpha/2-1}\frac{\rho(\alpha t)}{\rho(2t)^{\alpha/2}},\;p^{\frac{2^l-1}{2^{l+1}}\alpha}\frac{\rho(2^{l+1}t)^{\alpha/2^{l+1}}}{\rho(2t)^{\alpha/2}},l=1,2,\cdots,m\}\right]^k.
\int_{1+c}^{\infty}\mu(A_{t,n}\cap B)\;t=\int_{B}\int_{1+c}^{\infty}1_{\{\varphi_n\geqslant t\}}\;t\;\mu=\int_{B}(\varphi_n-(1+c))1_{\{\varphi_n\geqslant (1+c)\}}\;\mu\geqslant\delta_0.
\int d^{3}x \phi_{f}^{4}({\bf x}, t ) = \frac{3}{( 2 \pi )^{3}}\sum_{\alpha \neq \beta}\phi_{\alpha}^{2}\phi_{\beta}^{2}+ \frac{1}{( 2 \pi )^{3}}\sum_{\alpha}\phi_{\alpha}^{4}.
\bar{P}_f(\bar{\gamma}) = \int_{0}^{\gamma_t}P_f^G(\gamma) f_{\gamma}(\gamma,\bar{\gamma}) \gamma+ \int_{\gamma_t}^{\infty}P_f^G(\gamma) f_{\gamma}(\gamma,\bar{\gamma}) \gamma &\approx \int_{0}^{\gamma_t}f_{\gamma}(\gamma,\bar{\gamma}) \gamma =F_{\gamma}(\gamma_t,\bar{\gamma}),
\sup\limits_{g' \in\mathcal{G}_{f^{*}}}\left(\sum\limits_{i = 1}^{n}\xi_{i}g'(X_{i}) - \frac{h}{3}g'(X_i)\right) &\le \gamma\left(1 + \frac{h}{12}\right)\!+\!\sup\limits_{g' \in\mathcal{G}_{f^{*}}}\left(\sum\limits_{i = 1}^{n}\xi_{i}p(g'(X_{i})) - \frac{h}{12}p(g'(X_i))\right).
\{O_{1},O_{2}\}=\frac{\partial O_{1}}{\partial x_{1}^{\mu}}\frac{\partial O_{2}}{\partial p_{1\mu}}-\frac{\partial O_{1}}{\partial p_{1}^{\mu}}\frac{\partial O_{2}}{\partial x_{1\mu}}+\frac{\partial O_{1}}{\partial x_{2}^{\mu}}\frac{\partial O_{2}}{\partial p_{2\mu}}-\frac{\partial O_{1}}{\partial p_{2}^{\mu}}\frac{\partial O_{2}}{\partial x_{2\mu}}.
\delta^1(i\otimes i') = \sum_{\substack{(\xi_k,\eta_k)\in C_{\tau}ia(\xi_k)=a(\xi_k)j i'\overline{a(\eta_k)}=\overline{a(\eta_k)}j' j\otimes j' \mathrm{is~a~generator}}}(a(\xi_k)\otimes\overline{a(\eta_k)}) \otimes (j\otimes j'),
x[e^{-\frac{1}{2}\frac{(x-\mu)^{2}}{\sigma^{2}}}+e^{-\frac{1}{2}\frac{(x+\mu)^{2}}{\sigma^{2}}}]-\mu[e^{-\frac{1}{2}\frac{(x-\mu)^{2}}{\sigma^{2}}}-e^{-\frac{1}{2}\frac{(x+\mu)^{2}}{\sigma^{2}}}]=0
\frac{1}{\varepsilon^{n}}\int_{M}\psi_{\varepsilon}Z_{\varepsilon,\xi}^{2}d\mu_{g}\le\frac{1}{\varepsilon^{n}}\|\psi_{\varepsilon}\|_{H_{g}^{1}}\left(\int_{M}Z_{\varepsilon,\xi}^{12/5}d\mu_{g}\right)^{5/6}\le ct_{\varepsilon}^{2}\frac{1}{\varepsilon^{n}}\varepsilon^{\frac{10n}{6}}=ct_{\varepsilon}^{2}\varepsilon^{\frac{2n}{3}},
M_p(f_I^c,r)^p\le \frac{2^{\frac{p}{2}-1}}{2\pi}\int_{-\pi}^{\pi}\left(\left|\overline{F(re^{-I\theta})}\right|^p+\left|G(re^{I\theta})\right|^p\right) d\theta =2^{\frac{p}{2}-1}\left(M_p(F,r)^p+M_p(G,r)^p\right).
E(v^l_n) = \frac{E(\tilde{v}^l_n)}{(\lambda^l_n)^2}+ \frac{(\lambda^l_n)^\alpha-1}{\alpha+2}\|v^l_n\|^{\alpha+2}_{L^{\alpha+2}}\geq \frac{E(\tilde{v}^l_n)}{(\lambda^l_n)^2}.
S = \frac{1}{4 \pi}\int d^{2}x \left( \frac{1}{2}\partial_{\mu}\phi ( x ) \partial^{\mu}\phi ( x ) - 4 \pi \frac{\tilde{\lambda}}{2}e^{- i \beta \phi ( x ) / \sqrt{4 \pi}}+ \sqrt{2}i \alpha_{0}\phi ( x ) R ( x ) \right)
\left[\overline{\mathcal{W}}_{3,1}\right] \cdot \left[\overline{\Theta}_{3,1}\right] |_{T_1}= 252, \left[\overline{\mathcal{H}yp}_{3,1}\right] |_{T_1}= 18, \left[\overline{\mathcal{W}}_2 \right] |_{T_1}= -6, \gamma_1 |_{T_1}=\gamma_2 |_{T_1}=0.
\aligned 0 =\delta^\nabla \big (\exp\, (\frac 12||R^\nabla||^2)R^\nabla \big) &= - \sum_{i=1}^m \big (\nabla_{e_i}\exp\, (\frac 12||R^\nabla ||^2)R^\nabla\big )(e_i,\cdot ) &=\exp\, (\frac 12||R^\nabla ||^2)\delta^\nabla R^\nabla -i_{\big (\exp\, (\frac 12||R^\nabla ||^2)\big )}R^\nabla\, .\endaligned
g_{i j}( x^{k}) \rightarrow g_{i j}( x^{k}- \varepsilon x^{l}\epsilon_{l}^{~ k}) - \varepsilon \epsilon_{i}^{~ i^{\prime}}g_{i^{\prime}j}( x ) - \varepsilon \epsilon_{j}^{~ j^{\prime}}g_{i j^{\prime}}( x )
\sum_i F^{ii}(\nu^{n+1})_{ii}= \ 2\sum_i F^{ii}\frac{u_i}{u}(\nu^{n+1})_i+(n-1)\sigma (1+(\nu^{n+1})^2) &-\nu^{n+1}\left(\sum_i F^{ii}+\sum_i F^{ii}\kappa_i^2\right).
\check{D}_{[0]}=x\frac{\partial}{\partial x},\check{D}_{[1]}=X\frac{\partial}{\partial X}\quad\mbox{with}X=1-x, & b_2^{[0]}=H_1\cdot (1-x), b_1^{[0]}=\frac{N}{2}H_1+(b_1+2H_1(D\omega))(1-x), &b_2^{[1]}=H_1\cdot x, b_1^{[1]}=\frac{5}{2}H_1-(b_1-2H_1(D\omega))x, &b_0^{[\mu]}=b_0-(-1)^{\mu}(H_1\Lambda -b_1\check{D})\omega.
J\left( \sigma\right) =1+\operatorname{Re}\int_{0}^{1}{\left[{\left({\frac{t^{2}-\sigma^{2}}{t^{2}-1}}\right)^{1/2}-1}\right] dt}=\int_{0}^{\sigma}{\left({\frac{\sigma^{2}-t^{2}}{1-t^{2}}}\right)^{1/2}dt};
d(t)= \frac{|\textbf{D}(i)|}{n^{1-s}},\f(t)= \frac{|\textbf{F}(i)|}{n},\x(t)= \frac{|\textbf{X}(i)|}{n^{1-2s}}, \y(t)= \frac{|\textbf{Y}(i)|}{n^{1-s}},\z(t)= \frac{|\textbf{Z}(i)|}{n}.
\mathcal{U}\cdot W=V_{\mathbb{Z}\beta+\frac{i}{4k}\beta}\otimes V_{\mathbb{Z}\beta+\frac{j}{4k}\beta}+V_{\mathbb{Z}\beta+\frac{2k+i}{4k}\beta}\otimes V_{\mathbb{Z}\beta+\frac{2k-j}{4k}\beta}.
\boldsymbol{v}_{k+1}=\boldsymbol{v}_{k}- \mu \boldsymbol{x}_k \boldsymbol{x}^T_k\boldsymbol{v}_k + \mu n_k \boldsymbol{x}_k - \rho_{\rm r}\frac{(\boldsymbol{w}_k)}{\epsilon_{\rm r}+ |\boldsymbol{w}_{k-1}|}.
\frac{(\beta)_{m_1}}{(\beta)_{m_q+1}}=\frac{1}{(m_1+\beta)^{c^{'}_1}}\left(\prod_{i=2}^{q}\frac{(\beta)_{m_{i-1}+c^{'}_{i-1}}}{(\beta)_{m_{i}+c^{'}_i}}\right),
D + \rho = \left( \begin{array}{l l}{\partial \kern - 0 . 5 e m / \otimes{1 \kern - . 4 e m 1}\otimes{1 \kern - . 4 e m 1}}{\gamma^{5}\otimes H \otimes K}{\gamma^{5}\otimes H^{\dag}\otimes K^{\dag}}{\partial \kern - 0 . 5 e m / \otimes{1 \kern - . 4 e m 1}\otimes{1 \kern - . 4 e m 1}}\end{array}\right)
\frac{\Phi_{t_0}(x_0e^{i A_t(x_0)})}{\Phi_s(y e^{i A_s(y)})}=\frac{\Phi_{t_0}(ye^{i A_s(y)})}{\Phi_s(y e^{i A_s(y)})}\frac{\Phi_{t_0}(x_0e^{i A_t(x_0)})}{\Phi_{t_0}(ye^{i A_s(y)})}.
p_L := \inf \left\{p\in(1,2];\ \textrm{for all $t>0$, $e^{-tL}$ satisfies $L^p$-$L^2$ off-diagonal estimates}\right\}, p^L := \sup \left\{p\in[2,\infty);\ \textrm{for all $t>0$, $e^{-tL}$ satisfies $L^2$-$L^p$ off-diagonal estimates}\right\}.
\mathbb{V}_{1}=V\left( \mathbb{V}_{1}si\right) \subseteq HSPIS\mathcal{T}_{i}\left( P\left({\mathbf{2}}\right) \right) \subseteq V\left(\mathcal{T}_{i}\left( P\left({\mathbf{2}}\right) \right) \right)=V\left( \left\{\mathcal{T}_{i}\left( \mathbf{2}^{\kappa}\right):i\in\left\{1,2\right\},\kappa\right\}\right) .
\|v \|^{2}_{DG}= \sum_{i=1}^{N}\alpha_{i}\|\nabla_{\Omega}v_{i}\|_{L^{2}(\Omega^{i})}^{2}+ \sum_{\gamma \in \mathcal{E}_{I}\cup \mathcal{E}_{D}}\alpha_{\gamma}\frac{\delta}{h_{\gamma}}\| [v] \|_{L^{2}(\gamma)}^{2},
T_{\mu}= \frac{( 2 \imath )^{\frac{D - 2}{2}}}{( D - 2 ) !}\epsilon_{\nu \mu \rho_{2}\ldots \rho_{D - 1}}\pi^{\mu}\psi^{\rho_{2}}\ldots \psi^{\rho_{D - 1}}+ \frac{\alpha}{2^{\frac{D - 2}{2}}}\pi_{\nu}= 0 .
L_{V , A}= -{\frac{1}{2 ( p + 1 ) !}}\, \left( \, \partial_{\, [ \, \mu_{1}}V_{\mu_{2}\dots \mu_{p + 1}\, ]}+ m \, A_{\mu_{1}\dots \mu_{p + 1}}\, \right)^{2}-{\frac{1}{2 ( p + 2 ) !}}F^{2}{}_{\mu_{1}\dots \mu_{p + 2}}( A )
\sigma(i,j,k)_i =i; \sigma(i,j,k)_{i+1}=j+1; \sigma(i,j,k)_{k-j+i}=k; \sigma(i,j,k)_{k-j+i+1}=i+1; \sigma(i,j,k)_{k}=j; \sigma(i,j,k)_{k+1}=k+1.
\sigma_{pair}=Z^{2}\alpha r_{e}^{2}\frac{2\pi}{3}(\frac{k-2}{k})^{3}(1+\frac{1}{2}\rho+\frac{23}{40}\rho^{2}+\frac{11}{60}\rho^{3}+\frac{29}{960}\rho^{4})
\mathcal I_p^{\sharp}(\mathbf h, \breve{\mathbf g}, \pmb{\phi}) = \frac{L(1,\pi_v,\mathrm{ad})L(1,\tau_v,\mathrm{ad})}{L(1/2,\pi_v\times \mathrm{ad}\tau_v)}\alpha_p^{\sharp}(\mathbf h, \breve{\mathbf g}, \pmb{\phi}),
t\frac{8\pi^{3}g^{2}}{h^{4}}\times|\int v_{m}^{*}u_{n}d\tau|^{2}\frac{p_{\sigma}^{2}}{v_{\sigma}}(\tilde{\psi}_{s}\psi_{s}-\frac{\mu c^{2}}{K_{\sigma}}\tilde{\psi}_{s}\beta\psi_{s}),
\Pi_{\beta}^{B}(d\beta\mid Y,\sigma^{2}):=1_{\beta\in B}\mathrm{e}^{-\|Y-X\beta\|^{2}/(2\sigma^{2})}\pi(\beta)d\beta \bigg{/}\int_{B}\mathrm{e}^{-\|Y-X\tilde{\beta}\|^{2}/(2\sigma^{2})}\pi(\tilde{\beta})d\tilde{\beta}.
\mu_l^{[12]}=\mu_{l-1}^{[12]}+\mu_{l-2}^{[12]}+\mu_{l-3}^{[12]}+\delta_{l,2}+\delta_{l,4},\mu_{l<2}^{[12]}=0, \mu_l^{[13]}=\mu_{l-1}^{[13]}+\mu_{l-2}^{[13]}+\mu_{l-3}^{[13]}+2\delta_{l,3}, \mu_{l<3}^{[13]}=0, \mu_l^{[23]}=\mu_{l-1}^{[23]}+\mu_{l-2}^{[23]}+\mu_{l-3}^{[23]}+\delta_{l,3}+\delta_{l,5}, \mu_{l<3}^{[23]}=0.
\langle f_i,f_j\rangle=\delta_{ij}, (f_1,f_1)=\frac{1}{\lambda_1^2},(f_2,f_2)=-\frac{1}{\lambda_2^2},(f_3,f_3)=\frac{1}{\lambda_3^2},(f_4,f_4)=-\frac{1}{\lambda_4^2},