harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
2=|f(\widetilde{x}+\widetilde{y})| =|f(x+y)+[T^*f](u+v)| \leq \|f\|_{X^*}\|x+y\| +\|T^*f\|_Y\|u+v\| &\leq 2(\|f\|_{X^*}+\|T^*f\|)=2,
\biggl(L\biggl(x-1,{\partial\over\partial x}\biggr)-z^p\biggr)\Psi(x,0,z)=0\,\ \hbox{and}\ \,\biggl(L\biggl(z,{\partial\over\partial z}-1\biggr)-(x-1)^p\biggr)\Psi(x,0,z)=0\,,
v_\infty=\frac{1}{h_\infty}\left(\frac{\sigma^2_+(0)\int_{0}^\infty k^+(u)^2\, du}{p_{X,+}(0)\left[\int_0^\infty k^+(u)\, du\right]^2}+\frac{\sigma^2_-(0)\int_{-\infty}^0 k^-(u)^2\, du}{p_{X,-}(0)\left[\int_{-\infty}^0 k^-(u)\, du\right]^2}\right).
H_t(u,\theta)=\,(1-\kappa(t,\tau))\,\sum^{8N}_{k=1}\,\left(\,\sum^{N}_{k=1}\sum^4_{\mu=1}\,\beta^\mu_{x k}(u)\theta_{x k \mu}+\,\sum^{N}_{k=1}\sum^4_{\mu=1}\,\beta^\mu_{y k}(u)\theta_{y k \mu}\right),
M_\lambda(\zeta) \mu^{\frac{1}{2}}=|\sigma_1|^{\frac{1}{2}}M_\lambda P \bar\Lambda=|\sigma_1|^{\frac{1}{2}}\Bigl( \sigma_1 v_1 \bar\Lambda_1 + \sum_{l=2}^k \bar v_l \bar \Lambda_l \Bigr) ,
\int_{X}^{2X}x^{\alpha}\bigg|\sum_{1 \le n \le N}a_n e(k \sqrt[k]{nx}) \bigg|^2 \, dx & = \sum_{1 \le n \le N}|a_n|^2 \cdot\Big(\frac{2^{1+\alpha}-1}{1+\alpha}X^{1+\alpha}+O\Big(X^{1+\alpha-\frac1k}N^{1-\frac1k}\log N \Big)\Big) .
\frac{\theta_r}{2^{n_{r+1}}}= \sum_{k=r+1}^{\infty}\frac{a_k}{2^{n_k}q_k}< \sum_{k=r+1}^{\infty}\frac{1}{2^{n_k}}= \frac{1}{2^{n_{r+1}}}\sum_{k=r+1}^{\infty}\frac{1}{2^{n_k - n_{r+1}}}\leq \frac{2}{2^{n_{r+1}}}.
\Omega_L=\sum_{i=1}^n\left(\sum_{b,a=1}^m\left(\frac{\partial^2{\rm L}}{\partial u^b\partial u^a}\vartheta^b\wedge\vartheta^a+\sum_{j=1}^n\frac{\partial^2{\rm L}}{\partial u^b_j\partial u^a_i}\vartheta_j^b\wedge\vartheta^a\right)\right)\wedge\nu_i.
H = \frac{R}{2 \hbar}\Pi_{\lambda}^{k}\Pi_{\lambda}^{k}- \frac{R}{1 6 \pi \alpha^{3}}F_{\rho}^{\mu \nu}F_{\rho}^{\tau \epsilon}Y_{\mu}^{i}Y_{\tau}^{i}Y_{\nu}^{j}Y_{\epsilon}^{j}- \frac{i R}{{2 \pi}\left( \hbar \alpha^{\prime}\right)^{3 / 2}}\pi^{\mu}Y_{\sigma}^{i}F_{\mu}^{\sigma \nu}\gamma_{i}\theta_{\nu}
S_{q}( y , M ) = \sum_{\alpha , \beta \gamma , Z , K K}g_{\alpha}^{2}( M ) g_{\beta}^{2}( M ){\frac{( v_{e}^{\alpha}v_{e}^{\beta}+ a_{e}^{\alpha}a_{e}^{\beta}) ( v_{l}^{\alpha}v_{l}^{\beta}+ a_{l}^{\alpha}a_{l}^{\beta})}{( s - m_{\alpha}^{2}+ i \Gamma{_\alpha}m_{\alpha}) ( s - m_{\beta}^{2}- i \Gamma_{\beta}m_{\beta})}}\ .
\widetilde{Y}^{m+1}_{\cdot\wedge \tau_m}= \widetilde{Y}^{m}_{\cdot\wedge \tau_m}\ \mathbb{P}, \mathbb{I}_{\{t\leq \tau_m\}}\lambda_t\widetilde{Z}^{m+1}_t = \lambda_t\widetilde{Z}^m_t\dA \otimes d\mathbb{P}, \widetilde{N}^{m+1}_{\cdot\wedge \tau_m}= \widetilde{N}^m_{\cdot\wedge \tau_m}\ \mathbb{P}
\langle A_{\mu}^{a}A_{\nu}^{b}\rangle \sim{\frac{\delta^{a b}}{k^{2}}}\left( g_{\mu \nu}-{\frac{k_{\mu}k_{\nu}}{k^{2}}}\right) + \varepsilon \alpha^{a}\alpha^{b}{\frac{k_{\mu}k_{\nu}}{( k^{2})^{2}}},
\widetilde L_4:={}u^2(u-25) (u-9) (u-1) D^{4}+2 u (5 u^3-140 u^2+777 u-450)D^3{}+(25 u^3-518 u^2+1839 u-450)D^{2}+(3 u-5) (5 u-57)D^{1}+(u-5)D^{0}
\overset{+\infty}{\underset{-\infty}{\int}}\left( \underset{{\scriptsize\begin{array}{c}t<\left\vert \rho \right\vert \leq t+1 \left\vert \gamma \right\vert \leq T\end{array}}}{{\sum}}\frac{1}{\left\vert \rho \right\vert}\right)^{2}dt=O\left(\overset{T+1}{\underset{\frac{1}{2}}{\int}}dt\right) =O\left( T\right)
\xi_{n\ell}=\sqrt{-\frac{2\mu b^2E_{n\ell}}{\hbar^2}+\ell(\ell+1)D_0}, \ \ \ \ \ \beta_1=\tilde{A}-\ell(\ell+1)D_1\ \ \ \mbox{and}\ \ \ \beta_2=\alpha(\alpha-1)+\ell(\ell+1)D_2,
(i_{[L,L]_{RN}}\alpha)(X,Y,Z) =\alpha([L,L]_{RN}(X,Y,Z)) &=2i_L^2\alpha(X,Y,Z) &=2\circlearrowright i_L\alpha(L(X,Y),Z) &=2\circlearrowright\alpha(L(L(X,Y),Z)) &=2\circlearrowright\alpha(T_N(T_N(X,Y),Z))=0,
\|b_{jk}(t,s)\|_{L^2(B)}=(t-s)^{-2}\|\frac{\partial \xi_j}{\partial \eta_k}(t,s,y,\zeta/(t-s))-\delta_{jk}\|_{L^2(B)}&\leq C(t-s)^{-2}\|\frac{\partial \xi_j}{\partial \eta_k}(t,s,y,\zeta/(t-s))-\delta_{jk}\|_{L^\infty(B)}\leq C'
M'(t, y)= \dfrac{1}{t^2}A_1 A_2 A_3\begin{pmatrix}1 1 1\end{pmatrix}+ \dfrac{1}{A_1 A_2 A_3}\begin{pmatrix}-A_1^2 A_2^2 A_3^2 A_1^2 -A_2^2 A_3^2 A_1^2 A_2^2 -A_3^2\end{pmatrix}y-\dfrac{1}{t}2\begin{pmatrix}-1 1 1 1 -1 1 1 1 -1\end{pmatrix}y
I_{\mathrm{s u g r a}}={\frac{1}{2}}\left( I_{3 / 2}( R ) - I_{1 / 2}( R ) \right)_{1 2}= -{\frac{1}{2}}\left( I_{A}( R ) \right)_{1 2}={\frac{1}{1 6}}\left( L ( R ) \right)_{1 2}.
\{\lambda \}\geq \{\mu \}\Longleftrightarrow | \lambda | = | \mu | \ \ \{\mathrm{a}n d}\ \ \, s u m_{i}^{r}\lambda_{i}\geq \sum_{i}^{r}\mu_{i}\ \ \{\mathrm{f}o r \ \a l l}\r \geq 1 .
\int_{\mathbb{T}^n}(A P)(x)\overline{Q(x)}dx =\int_{\mathbb{T}^n}\left(\sum_\xi e^{i2\pi(x,\xi)}{a(x,\xi)\delta_{m,\xi}}\right)\overline{Q(x)}dx &=\int_{\mathbb{T}^n}e^{i2\pi(x,m)}a(x,m)\overline{Q(x)}dx=\int_{\mathbb{T}^n}e^{i2\pi(x,m)-i2\pi kx}a(x,m)dx.
\left\langle f,g\right\rangle_{\widehat{S}_p}:=\frac{1}{p}\int\limits_S \sum_{j=0}^{p-1}f(e^{\frac{j\pi i}{p}}\zeta )\overline{g(e^{\frac{j\pi i}{p}}\zeta )}\,d\sigma (\zeta),
(\varphi(\widehat{F}) \delta_G \beta) (k_1,k_2)(x_1,(a_2,x_2)) = \widehat{F}(k_1,k_2)({}^{x_1}a_2) \ \widehat{F}(k_1,k_2)(F(x_1,x_2)) & \epsilon(k_1,k_2,x_1x_2) \ \epsilon(k_1,k_2,x_1)^{-1}.
\tilde{\cal E}_{q}[ \phi ( x ) ] \sim \frac{1}{2}\left( \sum_{\tilde{E}_{j}}| \tilde{E}_{j}| - \sum_{\tilde{E}_{j}^{0}}| \tilde{E}_{j}^{0}| \right) + \tilde{\cal E}_{c t}[ \phi ( x ) ]
\begin{array}{l l l}{{\left( E_{4 s}- 2 Q - 2 Q^{- 1}\right) A ( k_{1}, k_{2})}}&{{=}}&{{A ( k_{1}- 1 , k_{2}) + A ( k_{1}+ 1 , k_{2})}}\\{{}}&{{}}&{{}}\\{{}}&{{+}}&{{A ( k_{1}, k_{2}- 1 ) + A ( k_{1}, k_{2}+ 1 )}}\\ \end{array}
N(t) =\int_{0}^{\infty}gp(t,V_F,g)dg=\frac{1}{V_F}\int_{0}^{\infty}g \left(\sum_{k=-\infty}^{+\infty}p_k(t,g)\right)dg. =\sum_{k=-\infty}^{+\infty}\frac{1}{V_F}\int_{0}^{\infty}gp_k(t,g)dg=\sum_{k=-\infty}^{+\infty}N_k(t),
\begin{array}{lll}\tau(E_i) = F_i, \tau(F_i) = E_i, \tau(K_i) = K_i^{-1}, \tau(\bar{E}_i) = -\sqrt{-1}\bar{F}_i, \tau({\bar{F}_i}) = -\sqrt{-1}\bar{E}_i, \tau({\bar{K}_j}) = \sqrt{-1}\bar{K}_j.\end{array}
f_{N}\left[ \phi ( x ) \right] ={\frac{1}{Z_{0}}}\sum_{n = 0}^{N}{\frac{\left( - 1 \right)^{n}}{n !}}\left({\frac{\lambda}{4}}\int \mathrm{d^{d}}x \phi^{4}\right)^{n}e^{- \int \mathrm{d^{d}}x \left[{\frac{1}{2}}\left( \partial_{\mu}\phi \right)^{2}+{\frac{1}{2}}m^{2}\phi^{2}\right]}
\frak{aut}_{CR}(M_k):=\underbrace{\frak g_{-\rho}\oplus\cdots\frak g_{-1}}_{\frak g_-}\oplus\frak g_{0}\oplus\underbrace{\frak g_{1}\oplus\cdots\oplus\frak g_{\varrho}}_{\frak g_+}, \ \ \ \ \ \ \varrho,\,\rho\,\in\,\mathbb N,
\gamma_{+}({\bf p}) = - \frac{e^{2}}{2}\int_{- \infty}^{+ \infty}d t \left[ \frac{p_{3}}{{\bf p}^{4}}p_{k}\epsilon_{i j k}+ \pi \delta ( p_{1}, p_{2}) \Delta ( p_{3}) \epsilon_{i j 3}\right] a_{j}\dot{a}_{i}.
\Xi_{\mathcal{C}^c}:=\{X\in\Xi~|~X(\mathcal{C}^c)\subseteq\mathcal{C}^c\}~~~~~ ~~~~~ \Xi_{\mathcal{C}\mathcal{C}^c}:=\{X\in\Xi~|~X(\mathcal{X})\subseteq\mathcal{C}^c\},
K_{m}:=-m-\alpha\gamma\sum_{d=1}^{m-1}\frac{[d]}{1+\beta\gamma[d]}+\sum_{r=1}^{m}\frac{(-\beta\delta)^{r-1}[r-1]!\,q^{-r(r-1)}}{\prod_{p=0}^{r-1}(1+\beta\gamma[m-1-p])}e_{r}(1, q, \ldots , q^{m-1}).
\begin{aligned}A=A(a,b)= a\, \vartheta \big (1 +b\, \pi -a\,\vartheta (1 + \theta_r) \big)^{-1}\,, B=B(a,b)= b\, \pi \big (1+b\, \pi -a\,\vartheta (1+ \theta_r) \big)^{-1}\,. \end{aligned}
\begin{cases}dp(t,x,z)= -[A_{\pi(t,z)}p(t,x,z)+xq(t,x,z)] dt + q(t,x,z) dR(t); 0 \leq t \leq T, p(T,x,z)=U(x,z)\mathbb{E}_{Q}[\delta_{Z}(z)|\mathcal{R}_T].\end{cases}
Y_{q}= \begin{array}{cc}\begin{array}{cccc}u_{L}\; d_{L}\; u_{R}\; d_{R}\;\end{array}\; \begin{array}{c}u_{L}d_{L}u_{R}d_{R}\end{array}\left( \begin{array}{cccc}0 0 M_{u}0 0 0 0 M_{d}M_{u}^{*}0 0 0 0 M_{d}^{*}0 0\end{array}\right)\end{array}
\mathcal{K}_{1}:= -u_{i}^{p-2}(y)u_{i}(x), \mathcal{K}_{2}:= \left(u_{i}^{p-2}(y)-|u_{i}(x)-u_{i}(y)|^{p-2}\right)u_{i}(x), \mathcal{K}_{3}:= -\left(u_{i}^{p-2}(y)-|u_{i}(x)-u_{i}(y)|^{p-2}\right)u_{i}(y), \Theta:= -|u_{i}(x)-u_{i}(y)|^{p-2}(u_{i}(x)-u_{i}(y)).
\phi_{r_{1}r_{2}}=\frac{2\sqrt{1-\rho^{2}}}{\pi}\int_{0}^{\pi/2}\frac{d\theta}{1-\rho sin2\theta}-1=-\frac{2}{\pi}arctan(\frac{\rho-tan\theta}{\sqrt{1-\rho^{2}}})|_{0}^{\pi/2}-1=\frac{2}{\pi}arcsin(\rho)
\hat h(z) = \begin{cases}h(z), \;\;{\rm Re}\, z \geq 0, h(z^*) + \frac{(2{\rm Re}z) h'(z^*)}{1 - ({\rm Re}\, z) Ph(z^*)}(z^* := -\bar z), \;\; -\tau <{\rm Re}\, z < 0,\end{cases}
\begin{aligned}I_4^{\prime}(x) =-4\pi^{-1}\int_{-\infty}^{\infty}r_2(z) e^{-2 i z x}\left(M_{+,22}(x;z)-1\right) dz &-2i\pi^{-1}\int_{-\infty}^{\infty}s^{-1}r_2(z) e^{-2 i z x}\partial_{x}M_{+,22}(x ; z) dz. \end{aligned}
\frac{d}{dt}P(w) = -\int_{|v|< |u|}(v_x \log|v| - u_x \log|u|) w dx - \int_{|v|\geq |u|}(v_x \log|v| - u_x \log|u|) w dx &= -\int_{|v|< |u|}v_x (\log|v| - \log|u|) w dx -\int_{|v|< |u|}\log|u| w w_x dx &- \int_{|v|\geq |u|}u_x (\log|v| - \log|u|) w dx -\int_{|v|\geq |u|}\log|v| w w_x dx.
\langle f,g \rangle_{\alpha}: = 2 \langle f,g \rangle_{L^2 (\mathbb{R}^2)}+ \langle \widehat{u}f,\widehat{u}g \rangle_{L^2 (\mathbb{R}^2)}+ \langle \widehat{v}f,\widehat{v}g \rangle_{L^2 (\mathbb{R}^2)}
=\frac{\theta}{2}\left(\left[\frac{\partial \Phi}{\partial y^0}\right]^2-2\int_{\mathbb{R}^n}\textbf{f}\;|\theta|^3 \sqrt{|\Theta|}\;\widetilde{p}^1\;d\widetilde{p}'\right)-\frac{\theta^2}{2}(\widetilde{R}^{(n-1)}+2\frac{\Theta^{ab}\widetilde{T}_{ab}}{n-1}).
\begin{array}{rcl}\partial_aW = \partial_aW' + \partial_aW'' = \sum_{(x,y)\in \Sigma'}\varphi_{(x,y)}\alpha_{(x,y)}\psi_{(x,y)}+ \sum_{(x,y)\in \Sigma''}\varphi_{(x,y)}\alpha_{(x,y)}\psi_{(x,y)}\end{array}
2E_0x =\theta x+(-1)^{(p+\bar{p}+q+\bar{q})}x\theta, 2F_1x =\theta x-(-1)^{(p+\bar{p}+q+\bar{q})}x\theta 2F_2x =\bar{\theta}x+(-1)^{(p+\bar{p}+q+\bar{q})}x\bar{\theta}, 2E_3 =\bar{\theta}x-(-1)^{(p+\bar{p}+q+\bar{q})}x\bar{\theta}
\partial_1^*(e_1^*) = e_1^*, \partial_1^*(e_2^*) = e_2^*, \partial_1^*(e_3^*) = 2 e_3^*; \partial_2^*(e_1^*) = -e_2^*, \partial_2^*(e_2^*) = e_1^* + e_2^*, \partial_2^*(e_3^*) = e_3^*.
= 2 ( N_{c}^{2}- 1 ) V \int \! \frac{d^{3}k}{( 2 \pi )^{3}}\frac{| k |}{2}+ \begin{array}{c}{\mathrm{p e r t u r b a t i v e}}\\{\mathrm{c o r r e c t i o n s}}\\ \end{array}
{\cal A}_{1}^{( 0 )}= \int d^{2}x \left[ \frac{( \partial_{\mu}\Phi )^{2}}{8 \pi}+ \frac{( \partial_{\mu}\varphi_{1})^{2}}{8 \pi}+ 2 \mu^{\prime}\cos ( \gamma \Phi ) e^{b \cdot \varphi_{1}}\right] .
\rho(L_{-1}) \, =\, h(z)^{-1}\left(\rho(L_0)^2 + \rho(L_0) -c(c+1)\right)\, , \rho(L_{0}) \, =\, -\frac{h(z)}{h'(z)}\partial + b(z) \, , \rho(L_1) \, =\, h(z)\, .
\frac{qx^2 \nu_z\nu_x - q^{-1}x^2 \nu_x \nu_z}{q-q^{-1}}= q^{-5}\nu_z x^2 + (q^{-1}+q^{3}-q^{-5})\nu_z + q^{-4}\nu_z^2 + (q^{-5}\nu_z)(1-x^2) &= x^2 - (1-q^{-2}(q+q^{-1})\nu_z + q^{-4}\nu_z^2).
\mathcal{A}=\begin{pmatrix}-2\dfrac{v_1\sqrt{\alpha}-v_2}{\alpha(1-\sqrt{\alpha})}2\dfrac{1+\sqrt{\alpha}}{1-\sqrt{\alpha}}begin{align*}10pt]-2\dfrac{\big[v_1\alpha(2+\sqrt{\alpha})-v_2(1+2\sqrt{\alpha})\big](v_1\sqrt{\alpha}-v_2)}{\alpha^2(1-\alpha)(1+\sqrt{\alpha})^2}2\dfrac{v_1\sqrt{\alpha}-v_2}{\alpha(1-\sqrt{\alpha})}\end{pmatrix}.
\frac{1}{2^m}\sum_{j=1}^{r}\psi^{(m-1)}\left(\frac{n-r+j}{2}\right) \sim\frac{1}{2^m}\sum_{k=n-r+1}^n\,\frac{(-1)^{m-2}(m-2)!}{(k/2)^{m-1}}&=\frac{(-1)^m\,(m-2)!}{2}\bigg[\sum_{k=1}^n\,\frac{1}{k^{m-1}}-\sum_{k=1}^{n-r}\,\frac{1}{k^{m-1}}\bigg]\sim\frac{(-1)^{m}(m-3)!}{2\cdot n^{m-2}}\left(\frac 1{(1-\alpha)^{m-2}}- 1\right),
S = \frac{1}{2}\int d^{4}x \epsilon^{\mu \nu \alpha \beta}\epsilon_{a b c d}\bigg [ ({^+}\tau -{^-}\tau ) F_{\mu \nu}^{a b}F_{\alpha \beta}^{c d}+ ({^+}\tau +{^-}\tau ) F_{\mu \nu}^{a b}\tilde{F}_{\alpha \beta}^{c d}\bigg ] .
a_{[1:9]}=121211100, b_{[1:9]}=102232101, c_{[1:9]}=123332100 &d_{[1:9]}=100234543, e_{[1:9]}=121002343, f_{[1:9]}=102102323 &g_{[1:9]}=100234543, h_{[1:9]}=100002345, i_{[1:9]}=210101100
u(x,t)=\frac{2}{\pi}\int_{0}^{\infty}sin(\lambda x)e^{-\lambda^{2}t}[\int_{0}^{\infty}sin(\lambda y)u_{0}(y)dy+\lambda\int_{0}^{t}e^{\lambda^{2}\tau}g_{0}(\tau)d\tau]d\lambda.
\frac{\partial}{\partial \eta^\rho_k}= \frac{\partial}{\partial \eta^\rho_{k+1}}+ \sum_l \bigg(\frac{\partial a^k_l}{\partial \eta^\rho_k}\bigg) \frac{\partial}{\partial y^l_{k+1}}+ \sum_\tau \bigg(\frac{\partial \beta^k_\tau}{\partial \eta^\rho_k}\bigg) \frac{\partial}{\partial \eta^\tau_{k+1}}.
u(x) \geq u(x) - u(\delta_0) = - \int_x^{\delta_0}u^\prime(s) ds \geq \epsilon_0 \int_x^{\delta_0}\frac{u(s)}{s}ds & \geq \epsilon_0 u(\delta_0) \int_x^{\delta_0}s^{-1}ds = \epsilon_0 u(\delta_0) \left[ \ln ( \delta_0 ) - \ln (x) \right] \rightarrow \infty\ \x \to 0.
\frac{1}{( \lambda - \lambda_1^{(0)}) \prod_{n=1}^{N-1}( \lambda - \lambda_n^{(0)})}= = \frac{1}{( \lambda - \lambda_1^{(0)})}\sum_{n=1}^{N-1}\frac{1}{\lambda - \lambda_n^{(0)}}\frac{1}{\prod_{n'\neq n}(\lambda_{n}^{(0)}- \lambda_{n'}^{(0)})}
n^{-(\frac{3}{2}I-N)}S_{n}= U^{*}(n^{D-\frac{3}{2}}(U S_{n}))=U^{*}\left( n^{D-\frac{3}{2}}\Big( U \sum_{k=1}^{n}(U^{*}Z_{k})\Big) \right) = U^{*}\left( n^{-(\frac{3}{2}-D)}\sum_{k=1}^{n}Z_{k}\right)= U^{*}\left( n^{-(\frac{3}{2}-D)}\tilde{S}_{n}\right) .
\begin{aligned}\mathcal{R}_kq_0(a-t)=\mathcal{K}_0(\mathcal{R}_{k-1}q_0(a-t)), \mathcal{R}_k\tilde{q}_0(a-t) =\mathcal{K}_0(\mathcal{R}_{k-1}\tilde{q}_0(a-t)), \mathcal{R}_kp(a-t)=\mathcal{K}_1(\mathcal{R}_{k-1}p(a-t)), \mathcal{R}_k\tilde{p}(a-t) =\mathcal{K}_1(\mathcal{R}_{k-1}\tilde{p}(a-t))\end{aligned}
\mu^{2}( \varphi , \beta ) = \int_{- \infty}^{\infty}{\frac{d \alpha}{2 \sqrt{\pi}}}e^{-{\frac{\alpha^{2}}{4}}}V^{( 2 )}({\frac{\alpha}{2}}\sqrt{J_{1}( g ) - I_{1}( Q^{2})}+ \varphi )
|\nabla v|^2= \cos^2\theta\sin^2\theta|\nabla\theta|^2=\cos^2\sin^2\theta\sum_i|h_{1i}^4-h_{2i}^3|^2 \leq 2\sin^2\theta\sum_i((h_{1i}^4)^2+(h_{2i}^3)^2)\leq 2|h|^2, |\mathcal{L}v|= \frac{1}{2}|\mathcal{L}\cos^2\theta|=|\sin^2\theta|\nabla\theta|^2-\frac{1}{4}\cos^2\theta|\bar D \mathbf J_M|^2|\leq6|h|^2,
\alpha_{ij}^{(p)}\coloneqq A_{ij}\bullet \begin{pmatrix}I_d (U^{(p)})^\top\end{pmatrix}\begin{pmatrix}I_d (V^{(p)})^\top\end{pmatrix}^\top -d_{ij}^2 , \alpha_{ik}^{(p)}\coloneqq A_{ik}\bullet\begin{pmatrix}I_d (U^{(p)})^\top\end{pmatrix}\begin{pmatrix}I_d (V^{(p)})^\top\end{pmatrix}^\top -d_{ik}^2 .
\begin{array}{c}{{\displaystyle{\sum_{\nu = \nu_{<}+ 1}^{\nu_{>}}\left( f ( \nu ) - g ( \nu ) \right) = \int_{\nu_{<}+ 1}^{\nu_{>}}\left( f ( \nu ) - g ( \nu ) \right) \, d \nu +}}}\\{{\displaystyle{+ \left( f ( \nu_{<}+ 1 ) - g ( \nu_{<}+ 1 ) \right) + \int_{\nu_{<}+ 1}^{\nu_{>}}\left( x - [ x ] \right) \left( f^{\prime}( x ) - g^{\prime}( x ) \right) \, d x}.}}\\ \end{array}
\left( \begin{array}{c c}{\frac{\tilde{B}_{\lambda}}{( k^{2})^{3 / 2 - \beta_{\lambda}}}}{i \vartheta \frac{\tilde{B}_{\lambda \sigma}}{( k^{2})^{\beta_{\lambda \sigma}}}}{i \vartheta \frac{\tilde{B}_{\lambda \sigma}}{( k^{2})^{\beta_{\lambda \sigma}}}}{\frac{\tilde{B}_{\sigma}}{( k^{2})^{3 / 2 - \beta_{\sigma}}}}\end{array}\right)
\triangle^{( 3 / 2 )}( D_{\mu}\psi ) = D_{\mu}\left( ( \triangle^{( 1 / 2 )}- 1 ) \psi \right) , D^{\mu}\triangle^{( 3 / 2 )}\psi_{\mu}= ( \triangle^{( 1 / 2 )}- 1 ) D^{\mu}\psi_{\mu},
[z^n x_1^{\ell_1}\cdots x_k^{\ell_k}] P_h = [z^{n-1}x_1^{\ell_1}\cdots x_k^{\ell_k}] f(A) &= \frac{1}{(n-1)}[t^{n-2}x_1^{\ell_1}\cdots x_k^{\ell_k}] f'(t) \Phi(t)^n.
F V_1 = F v_1 \otimes K^{-1}v_0 + v_1 \otimes F v_0 - q^2 F v_0 \otimes K^{-1}v_1 - q^2 v_0 \otimes F v_1 = [2]^{1/2}(v_0 \otimes v_0 + v_1 \otimes v_{-1}- v_{-1}\otimes v_1 - q^2 v_0 \otimes v_0) = [2]^{1/2}V_0.
\int_K ||\tau_0||_{\omega_{\epsilon_1}}\le \liminf_{\epsilon\to 0}\int_K ||\tau_{\epsilon}||_{\omega_{\epsilon_1}}\le \liminf_{\epsilon\to 0}\int_K ||\tau_{\epsilon}||_{\omega_{\epsilon}}\le \liminf_{\epsilon\to 0}\int_M ||\tau_{\epsilon}||_{\omega_{\epsilon}}.
\int_{S^{n-1}_1}\partial_r R_g r^{n-1}d\omega_{n-1}=\frac{\partial}{\partial r}\left(r^{n-1}\int_{S^{n-1}_1}R_g d\omega_{n-1}\right) - (n-1)r^{n-2}\int_{S^{n-1}_1}R_g d\omega_{n-1}.
\Phi w\in C^{\mu}_{loc}((0,T_*]; L^q_\sigma(\Omega)), \forall q\in [2,\infty),\, \forall \mu\in (0,\mu_0), &\nabla\Phi w\in C^{\mu}_{loc}((0,T_*]; L^q(\Omega)), \forall q\in [2,6),\, \forall \mu\in (0,\mu_0-1/2),
\{x\in B_R{\mbox{s.t.}}\omega\cdot x\le -aR\}\subseteq\left\{x\in B_R{\mbox{s.t.}}u(x)\le -\frac{9}{10}\right\}{\mbox{and}}\; \left\{x\in B_R{\mbox{s.t.}}u(x)\le \frac{9}{10}\right\}\subseteq\{x\in B_R{\mbox{s.t.}}\omega\cdot x\le aR\}.
\alpha_t(\lambda) = \begin{pmatrix}1 0 0 1\end{pmatrix}, \beta_t(\lambda) = \begin{pmatrix}e^{-4(2a+1)u_t+2l\pi\sqrt{-1}}-\frac{\sinh\bigl(4(2a+1)u_t\bigr)}{\sinh(u_t)}0 e^{4(2a+1)u_t-2l\pi\sqrt{-1}}\end{pmatrix}
\widehat{X}(w_1,\dots,w_r) = \overline{(P_{w_1}\cap G_{w_1}) w_1 (P^{w_1}\cap G_{w_1})}\times^{(P^{w_1}\cap G_{w_1})}\widehat{X}(w_2,\dots,w_{r}) &= \overline{B_{I_1}w_1 L_{I_1}}\times^{B_{I_1}}\widehat{X}(w_2,\dots,w_{r}) &= L_{I_1}\times^{B_{I_1}}\widehat{X}(w_2,\dots,w_{r}) &\simeq P_{I_1}\times^B\widehat{X}(w_2,\dots,w_{r}).
\widetilde{\mathsf{V}}\cdot \big( \mathsf{D}_{t}^{p}\cdot P \big) \big\vert_{U_{x,k}}= H_{0}^{p}\,P + H_{1}^{p}\, \big(\mathsf{D}_{t}^{1}\cdot P\big) + \dotsb + H_{k}^{p}\, \big(\mathsf{D}_{t}^{k}\cdot P\big){\scriptstyle(p=0,1,\dotsc,k)}.
\Big| [-(- \Delta)^{\gamma/2}G ](u)\Big |= c_{\gamma}\Big| \int_{0}^{\infty}\frac{G(u-w)}{w^{\gamma+1}}dw \Big| = c_{\gamma}\Big| \int_{u-b_G}^{u+b_G}\frac{G(u-w)}{w^{\gamma+1}}dw \Big| \leq H^G(u).
R(z) = \left( \begin{array}{cc}0 0 0 R_{\Omega}(z)\end{array}\right) +\left( \begin{array}{c}1 -R_{\Omega}(z)|E )\end{array}\right)\frac{1}{z-\i \pi \sigma(z)}\left( 1, (E | R_{\Omega}(z) \right ) .
\xi^0 \ge |x_1| \eta^0 \ge |x_2| \xi^j = \ \ \ \ \cos (\theta_j) \xi^{j-1}+ \sin (\theta_j) \eta^{j-1}\j=1,\dots,\nu \eta^j \ge |-\sin (\theta_j) \xi^{j-1}+ \cos (\theta_j) \eta^{j-1}| j=1,\dots,\nu \xi^\nu \le x_3 \eta^\nu \le \tan (\theta_\nu) \xi^\nu
I'_{h,m,n}(x)=\frac{1}{\sqrt{2 \pi n^{2/3}}}\, \sqrt{\frac{L_m(x)}{3}}\, \exp\left( \tfrac{3}{2}n^{2/3}L_m(x)\right)\, \left( 1+ O_X( n^{- 2/3}) \right) .
\mu(c_{\textbf{j},\textbf{s}}(F_{p,q})x_{\textbf{s},\textbf{i}}1_{\chi}\otimes 1\otimes (F_{p,q}^*)^{(k)}v_{\textbf{j}})=\begin{cases}0, \ \ \mbox{if}\ \textbf{s}\neq \textbf{i}\ \mbox{or}\c_{\textbf{j},\textbf{s}}(F_{p,q})=0, c_{\textbf{j},\textbf{i}}(F_{p,q})(F_{p,q}^*)^{(k)}v_{\textbf{j}},\ \mbox{otherwise}.\end{cases}
\int_{A_{n,j}\cap A_n}\int_{\eta_j(K)}^{\tau_{n+1}(\omega) - \tau_n(\omega)}D^p(x_{\tau_n +t}(\omega))dt d\P \ge c \delta^p K^{2rp}\int_{A_{n,j}\cap A_n \cap \{\bar{\tau}\ge T\}}(\tau_{n+1}(\omega) - \tau_n(\omega))d\P - c\eta_j(K) \delta^p K^{2rp}\P(A_{n,j}\cap A_n).
(\Theta_{j\bar k}u_j,u_k)=\pi_*(c_n\{[ L_j, L_{\bar k}]\textbf u_j, \textbf u_k\}) +(D_{t^j}u_j,D_{t^k}u_k)-\pi_*(c_n\{L_j \textbf u_j,L_k\textbf u_k\}) .
\nabla_i^\perp(\partial_t H_j-\nabla_j V) =-\nabla_l^\perp V\nabla_l \nabla_i^\perp H_j +\nabla_l\nabla_i^\perp V \nabla_l^\perp H_j &=\nabla_l V\nabla_l^\perp \nabla_i^\perp H_j +\nabla_l\nabla_i^\perp V \nabla_l^\perp H_j &=\nabla_i^\perp (\nabla_l V\nabla_l^\perp H_j),
d\mu_t^{\epsilon,K}\,=\,\left(\frac{\epsilon^{1-\kappa}}2|Dv_{\epsilon,K}|^2\,+\,\frac 1{2\epsilon^{1-\kappa}}(v_{\epsilon,K}^2-1)^2\right)\, dx, & d\xi_t^{\epsilon,K}\,=\,\left(\frac 1{2\epsilon^{1-\kappa}}(v_{\epsilon,K}^2-1)^2\,-\,\frac{\epsilon^{1-\kappa}}2|Dv_{\epsilon,K}|^2\right)\, dx.
\langle A_{0}\otimes B_{0}\rangle=\frac{1}{\sqrt{2}},\langle A_{0}\otimes B_{1}\rangle=\frac{1}{\sqrt{2}},\langle A_{1}\otimes B_{0}\rangle=\frac{1}{\sqrt{2}},\langle A_{1}\otimes B_{1}\rangle=-\frac{1}{\sqrt{2}}.
\frac{d^{2}\overline{{\phi}}( x )}{d x^{2}}= V^{\prime}( \overline{{\phi}}) \quad \longrightarrow \quad \frac{d}{d x}\left[ \frac{1}{2}\left( \frac{d \overline{{\phi}}}{d x}\right)^{2}\right] = \frac{d}{d x}V ( \overline{{\phi}}) .
\left|\frac{r(r^2 + t_+^2)}{t_+}- 2r^2\right| = \frac{r}{t_+}\left|2r^2 + e^{2\tau}- 2r t_+\right| \leq \frac{e^{4\tau}}{2 r^2 + e^{2\tau}+ 2r t_+}\leq r^{-2};
I_{l}^{m}(r+a)=\sum_{\lambda=0}^{\infty}(\begin{matrix}2l+2\lambda+1\\ 2\lambda\end{matrix})^{1/2}\sum_{\mu=-\lambda}^{\lambda}R_{\lambda}^{\mu}(r)I_{l+\lambda}^{m-\mu}(a)\langle\lambda,\mu;l+\lambda,m-\mu|lm\rangle
(\l_{h}^{(i)}-\l )b_h(u_{h},u_{h}) =a(u-u_{h},u-u_{h})-\l b(u-u_{h},u-u_{h}) &\quad+(a_{h}(u_{h},u_{h})-a(u_{h},u_{h}))+\l\left[b(u_{h},u_{h})-b_{h}(u_{h},u_{h})\right].
\left< tr \frac{\delta}{\delta \gamma_\mu (s)}{\cal{F}}_\mu (\gamma(0))\right> = \frac{\partial^2 W(\gamma )}{\partial \gamma^2_\mu (0)}=-e^2\int\delta(\gamma (s) -\gamma (0))W(\gamma_{0s})W(\gamma_{s0})\dot{\gamma_\mu}(0)\dot{\gamma_\mu}(s)ds
E_{modi}^{S, low 1}(t) :=\sum_{\tau=p,N_1}\sum_{\mu, \nu, \kappa \in\{+,-\}}\mathfrak{Re}\big[ \int \overline{\widehat{(SU)_{\mu}}(\xi)}\widehat{(SU)_{\nu}}(\xi-\eta) \widehat{U_{\kappa}}(\eta) q_{\mu,\nu}^{\kappa,\tau}(t,\xi-\eta, \eta) d \eta d \xi\big],
\left[ \sum_{i=1}^{L}\bar{a}(x_i , y_i) - \frac{1}{(L-1)!}\sum_{i=1}^{L}\prod_{j=1}^{L}\bar{a}(x_i , y_j) \prod_{\stackrel{j=1}{j \neq i}}^{L}\frac{\bar{a}(x_j , x_i)}{\bar{b}(x_j , x_i)}\frac{\partial^{L-1}}{\partial x_i^{L-1}}\right] \bar{Z}(\bar{X}^{1,L}) = 0 \; ,
\frac{1}{2}\|u^{1}_{h}\|^{2}_{2}+\frac{\nu\Delta t}{4}\|\nabla u^{1}_{h}\|^{2}_{2}\leq \frac{1}{2}\|u_{0}\|^{2}_{2}+\frac{\nu\Delta t}{4}\|\nabla u^{0}_{h}\|^{2}_{2}\leq \big(\frac{1}{2}+\frac{\nu\Delta t}{4 h^{2}}\big) \|u_{0}\|^{2}_{2}:=K_{3},
\{\Psi_{2}^{2}, \Psi_{3}^{2}\}\stackrel{\Sigma}{=}0 , \{\Psi_{2}^{2}, \Psi_{2}^{4}\}\stackrel{\Sigma}{=}0 , \{\Psi_{2}^{3}, \Psi_{3}^{1}\}\stackrel{\Sigma}{=}0
\bigl \{\Phi(\sigma_1, \tau),{\partial\over \partial \tau}\Phi(\sigma_2, \tau) \bigr\}_{\hbox{P.B.}}=4\pi \, \delta(\sigma_1-\sigma_2),\quad\bigl \{\Phi(\sigma_1, \tau),\Phi(\sigma_2, \tau) \bigr\}_{\hbox{P.B.}}= 0
T_{m}^{\prime}= R_{m}{}^{n}T_{n}\, , \hspace{1 c m}[ T_{m}^{\prime}, T_{n}^{\prime}] = R_{m}{}^{r}R_{n}^{s}\epsilon_{r s p}T_{p}= \mathrm{d e t}R \, \epsilon_{m n q}\left( R^{- 1}\right)_{p}{}^{q}\left( R^{- 1}\right)_{p}{}^{r}T_{r}^{\prime}\, ,
S_p(\mathbf{r},\mathbf{s},r_0,s_0) =\begin{cases}p-1, p|(\mathbf{r},\mathbf{s})(r_0,s_0), -1, p|rr_0p|ss_0p\nmid (\mathbf{r},\mathbf{s})(r_0,s_0), 0, (p|rr_0p\nmid ss_0)(p|ss_0p\nmid rr_0).\end{cases}
K_{N}(z_{1},z_{2}) = N^{-2}\sum_{k=1}^{N}\mathbb{E}_{k-1}\sum_{i<k,j<k}\mathbb{E}_{k}(G^{(1)}_{k})_{ij}\mathbb{E}_{k}(G^{(2)}_{k})_{ji}&+N^{-2}\sum_{k=1}^{N}\mathbb{E}_{k-1}\sum_{i<k}\mathbb{E}_{k}(G^{(1)}_{k})_{ii}\mathbb{E}_{k}(G^{(2)}_{k})_{ii}\beta_{ik}
\begin{aligned}T(u,v)(t):= \left(\begin{array}{c}\gamma_{1}(t)H_{1}(\alpha_{1}[u])+\delta_1(t)L_{1}(\beta_{1}[v])+G_1(u)(t)+F_1(u,v)(t) \gamma_{2}(t)H_{2}(\alpha_{2}[v])+\delta_2(t)L_{2}(\beta_{2}[u])+G_2(v)(t)+F_2(u,v)(t)\end{array}\right) :=\left(\begin{array}{c}T_1(u,v)(t) T_2(u,v)(t)\end{array}\right) ,\end{aligned}
\partial_{tt}\tilde{u}+2\alpha \partial_t \tilde{u}-\Delta \tilde{u}+\tilde{u}-|\tilde u|^{p-1}\tilde{u}=e, \|e\|_{L^1_t(I;L^2_x)}+\|S_{1,\alpha}(t-t_0)(\tilde{u}-v)(t_0)\|_{L^{Q}_t(I;L^{2p}_x)}+\|S_{2,\alpha}(t-t_0)(\partial_t\tilde{u}-\partial_tv)(t_0)\|_{L^{Q}_t(I;L^{2p}_x)}\le \varepsilon.
(S_{I_{k+1;1}})_j:=ad^{-1}\,G_{I_{k+1;1}}\,((V^{(k, N-k)}_{I_{k+1;1}})^{od}_j):=\frac{1}{G_{I_{k+1;1}}-E_{I_{k+1;1}}}P^{(+)}_{I_{k+1;1}}\,(V^{(k, N-k)}_{I_{k+1;1}})_j\,P^{(-)}_{I_{k+1;1}}-h.c.\,,
+iq_{0}^{-1}\sin\psi\csc\eta\cdot\left(\frac{F[\frac{1}{q_{0}}](i\mu_{0})w_{1}[\frac{1}{q_{0}}](i\mu_{0})w_{3}[\frac{1}{q_{0}}](i\mu_{0})}{w_{2}[\frac{1}{q_{0}}](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{2}\partial_{\phi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\min_{\beta_{\ell}\in\mathbb{R}}~~ \left\{\sum_{j=0}^N\omega_j\left(\sum_{\ell=0}^{L}\beta_{\ell}\tilde{\Phi}_{\ell}(x_j)-f(x_j)\right)^2+\lambda\sum_{\ell=0}^L(\mu_{\ell}\beta_{\ell})^2\right\},\quad\lambda>0,
||f||_{M\dot{K}_{p,\vec{q}}^{\alpha_2,\lambda}}=2^{k(\alpha_2-\alpha_1)}\mathop{sup}\limits_{k_0\in \mathbb{Z}}2^{-k_{0}\lambda}\left(\sum_{k=-\infty}^{k_0}2^{k\alpha_{1}p}||f\widetilde{\chi}_k||_{\vec{q}}^p\right)^{\frac{1}{p}}\leq C ||f||_{M\dot{K}_{p,\vec{q}}^{\alpha_1,\lambda}}.
\delta_{n , j}^{( 1 )}( k ) = - \frac{\pi}{2}\int_{0}^{\infty}d r V ( r ) r \left( J_{\frac{n}{2}+ j - \frac{3}{2}}( k r )^{2}+ J_{\frac{n}{2}+ j - \frac{1}{2}}( k r )^{2}\right)
{}_2\!F_{1}(a,b;a+b;t) =1+\frac{ab}{a+b}\int_0^t(1-s)^{-1}{}_2\!F_{1}(a,b;a+b+1;s)dt &\asymp1+\ln\frac{1}{1-t}\asymp\ln\left(\frac{1}{1-t}+2\right),-1\le t<1.
\Sigma=\big\{[\epsilon_1:1],[\epsilon_2:1],\ldots,[\epsilon_{2r}:1],[\beta_1:1],[\bar{\beta}_1:1],\ldots,[\beta_s:1],[\bar{\beta}_s:1]\big\}, \Sigma_{1}=\big\{[a_1,1],[b_1,1]\big\},\ \ Sigma_{2}=\big\{[a_2,1],[b_2,1]\big\}.