harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
xG^*(1/x) \leq \psi^{-1}(\log_2 \Phi^{-1}((1/x))) \leq \psi^{-1}\left( \log_2 \left( \frac{1}{x\log (1/x)}\left[1 + \frac{1}{\log(1/x)}\right] \right) \right) & = \psi^{-1}\left( \log \left[y - \log y + \log\left( 1 + \frac{1}{y}\right) \right] \right) .
g_{2k}=\sum_{j=0}^{\lfloor k/8\rfloor}f_{k-8j}g_j, g_{2k+1}=\sum_{j=0}^{\lfloor k/2\rfloor}(-1)^jf_{k-2j}h_j, h_{2k}=\sum_{j=0}^{\lfloor k/2\rfloor}(-1)^jf_{k-2j}g_j, h_{2k+1}=\sum_{j=0}^{\lfloor (k-1)/8\rfloor}f_{k-8j-1}g_j.
\left(\int_{5c\widetilde Q}|\nabla Eu(x) |^s \,\d x \right)^\frac{p}{s}\ge \left( C (5c)^{n-s}\ell(\widetilde Q)^{n-s}\right)^\frac{p}{s}= C' \ell(\widetilde Q)^{n - p}\ell(\widetilde Q)^{(\frac{p}{s}- 1)n}
1 + \sum_{N>A}e^{-\beta N + N^\kappa}= 1 + \sum_{N> A}e^{-\beta N^\kappa(N^{1-\kappa}-1/\beta)}\le 1 + \sum_{N> A}e^{-B \beta N^\kappa}\le \sum_{N\in \mathbb{N}}e^{-B \beta N^\kappa}.
\frac{\partial}{\partial y}\left(y\beta(y,t)\right)+\frac{1}{4}\frac{\partial^{2}}{\partial y^{2}}\beta(y,t)- \frac{\partial}{\partial t}\beta(y,t)=\left(\frac{2}{\pi}\right)^{1/2}\left(-\frac{4}{3}y^{4}-2y^{2}+\frac{3}{4}\right)e^{-2y^{2}}
\widetilde{\chi}(\Delta(P(a,b))) = \sum_{i=0}^{\infty}(-1)^i 2 \sum_{t=0}^i \binom{a-3-i}{t-1}\bigg[ \binom{i}{t}\binom{b-2-i}{i-t}+ \binom{i}{t-1}\binom{b-3-i}{i-t}\bigg].
z_iR(z_i) =q(z_i+y)\prod_{p=1}^{r}(1+z_ix_p), \prod_{j=1}^{N}(z_j-\alpha_i) =(-1)^{N+1}\frac{P(\alpha_i)}{z_{N+1}-\alpha_i}=(-1)^Nq\frac{(\alpha_i+y)\prod_{p=1}^{r}(1+\alpha_ix_p)}{z_{N+1}-\alpha_i}.
H_S=\sum_n \frac{1}{2}\left[ e^{-3Ht}L^3\pi_{\phi}({\bf k}_n,t)\pi_{\phi}(-{\bf k}_n,t)+ \frac{e^{3Ht}}{L^3}(e^{-2Ht}{\bf k}_n^2+m^2)\phi({\bf k}_n,t)\phi(-{\bf k}_n,t)\right] +\frac{e^{3Ht}}{L^3}V_F,
\omega_1 \star a = (\iota \otimes c \cdot \omega_1') \Delta(a) = (\iota \otimes \omega_1')(\Delta(a) (1 \otimes c)), a\star \omega_2^\sharp = (d \cdot \omega_2' \otimes \iota)\Delta(a) = (d \cdot\omega_2' \otimes \iota)(\Delta(a)(d \otimes 1)).
(q-1)^{E(\Gamma_1)}q^{E(\Gamma_1)(\alpha-1)}(q-1)^{E(\Gamma_2)-E(\Gamma_1)}q^{\left(E(\Gamma_2)-E(\Gamma_1)\right)(\alpha-2)}\cdots \cdots (q-1)^{E(\Gamma_{\alpha-1})-E(\Gamma_{\alpha-2})}q^{E(\Gamma_{\alpha-1})-E(\Gamma_{\alpha-2})}(q-1)^{E(\Gamma_\alpha)-E(\Gamma_{\alpha-1})}&= (q-1)^{E(\Gamma_\alpha)}q^{\sum_{k=1}^{\alpha-1}E(\Gamma_k)},
\log\left(\frac{g(t)+T_0}{C_0+T_0}\right)=\int_{C_0}^{g(t)}\frac{1}{x +T_0}\,dx=\int_\mu^t\frac{S_0\,\dot{u}\,\frac{d}{dt}\Phi'(\dot{u})}{g(\tau) +T_0}\,d\tau\leq 2 S_0 r_0.
L=c_0s+c\sqrt{{\dot{\bf e}}_3^2-k^2_2}+{\bf p}({\dot{\bf x}}-s{\bf e}_1)+\sum_{i}{\bf p}_{i-1}({\dot{\bf e}}_{i-1}-k_{i-1}{\bf e}_{i}+k_{i-2}{\bf e}_{i-2})-\sum_{i,j}d_{ij}({\bf e}_i{\bf e}_j-\delta_{ij}),
\mathrm{Var}(\hat{p}^{n,k}(x))= \frac{P(x)^2}{n}\left(-\log(P(x))+\frac{\left[\left((\log (P(x))\right)^2-\log(P(x))\right](k-1)}{2n}+{\rm o}\left(\frac{1}{n}\right)\right).
\int_{\Omega_\rho^+(\eta_k)}\frac{H^{(1)}_{\mu}(\lambda z)}{H^{(1)}_{\nu}(z)}F(z)dz+\int_{\Omega_\rho^-(\eta_k)}\frac{H^{(2)}_{\mu}(\lambda z)}{H^{(2)}_{\nu}(z)}F(z)dz=\left( 2-\delta_{0\eta_k}\right) \pi i\,{\mathrm{Res}}_{z=\eta_k}\frac{H^{(1)}_{\mu}(\lambda z)}{H^{(1)}_{\nu}(z)}F(z),
S ( \tau_{21})= - (\tau_1^{+})^{-1}\tau_{21}(\tau_2^{+})^{-1}, \S ( \tau_{12})= - (\tau_2^{-})^{-1}\tau_{12}(\tau_1^{-})^{-1}, \S (\tau_i^{\pm}) = (\tau_i^{\pm})^{-1}
\mathrm{SG}_0^\pm:=\left\{{\left.\begin{pmatrix}\cos(q^3+q^4) \mp\sin(q^3+q^4) 0 0 q^1 \pm\sin(q^3+q^4) \cos(q^3+q^4) 0 0 q^2 0 0 1 0 q^3 0 0 0 1 q^4 0 0 0 0 1\end{pmatrix}\right|q\in\mathbb{R}^4}\right\}.
\begin{array}{c}{{\dot{\bf x}= \mathrm{i}\left({\bf z}_{1}-{\bf{\bar{z}}}_{1}\right) ,}}\\{{\dot{\bf z}_{i - 1}= - \mathrm{i}\delta_{1 . i - 1}{\bf p}+ k_{i - 1}{\bf z}_{i}- k_{i - 2}{\bf z}_{i - 2},}}\\{{\dot{\bf z}_{N}= - \mathrm{i}\delta_{1 . N}{\bf p}+ \mathrm{i}k_{N}{\bf z}_{N}- k_{N - 1}{\bf z}_{N - 1},}}\\{{{\dot{\bf p}}= 0 ,}}\\ \end{array}
\lambda+\lambda^{l-1}= \left( \cos \left( \frac{2\pi r}{l}\right) + i \sin \left( \frac{2\pi r}{l}\right) \right)+ \left(\cos \left( \frac{2\pi r}{l}\right) - i \sin \left( \frac{2\pi r}{l}\right) \right) &= 2\cos \left( \frac{2\pi r}{l}\right) \in \mathbb{R}.
\left|\frac{E_{n_{r}0}}{E_{_{\rm (gs)}}}\right|=\exp \left[-2^{4/3}\, \left( C_{n_{r}}- C_{1}\right)\,\left(\lambda^{(\ast)}_{_{\rm (gs)}}\right)^{-1/3}\,\epsilon^{-1/3}\right] \stackrel{(\epsilon \rightarrow 0, n_{r}>1 )}{\longrightarrow}0\; .
Q\circ \Pi \psi(\omega_0,\hat{v}) = \int \Pi\psi(\omega_1, \hat{A}(\omega_1,\omega_0)\hat{v}) \; dK_{\omega_0}(\omega_1) &=\int \int \psi(\omega_2,\omega_1, \hat{A}(\omega_1,\omega_0)\hat{v}) \; dK_{\omega_1}(\omega_2) \, dK_{\omega_0}(\omega_1).
\underline{Q}(t) = (t-a)\det\begin{bmatrix}\tilde{s}_0 \tilde{s}_1 \ldots \tilde{s}_{m-1}1 \tilde{s}_1 \tilde{s}_2 \ldots \tilde{s}_{m}t \vdots \vdots \vdots \vdots \vdots \tilde{s}_{m}\tilde{s}_{m+1}\ldots \tilde{s}_{2m-1}t^{m}\end{bmatrix},
\phi_1(z):= \frac{1}{2(1-m(k))}\left[(\partial_b c)\partial_a w-(\partial_a c)\partial_b w\right](k,A,b)(z) = \cos(z)+\frac{-1/2+\cos(2z)}{m(k)-m(2k)}~A-6\cos(z)b+\mathcal{O}(A^2+b^2), \phi_2(z):= -\frac{1}{a}\partial_zw(k,A,b)(z) = \sin(z)+\frac{\sin(2z)}{m(k)-m(2k)}~A+\mathcal{O}(A^2+b^2) \phi_3(z):= 1
\mathrm{}[ \hat{\Phi}( x^{0}, x^{1}) , \hat{\Phi}( x^{0}, y^{1}) ] = - \frac{i}{2}\left\{\frac{1}{i \pi}{\sum_{n = - \infty}^{+ \infty}}^{\prime}\left( \frac{\pi}{R}\right) \frac{e^{\frac{i n \pi}{R}( x^{1}- y^{1})}}{\frac{n \pi}{R}}\right\}.
C_{\mu \nu}^{- 1}= \frac{1}{\nabla^{2}}\left( \begin{array}{c c c c}{0}&{0}&{0}&{1}\\{0}&{0}&{- 1}&{0}\\{0}&{1}&{0}&{0}\\{- 1}&{0}&{0}&{0}\\ \end{array}\right) \delta^{2}( x - y ) .
\mathcal{M}_{|\Delta_n|^2}(s) \approx \left(\frac{\tilde{A}_n}{2}\right)^{\frac{\alpha_n-1}{2}}\frac{\left(\alpha_n/b_{0,n}\right)^{\frac{\alpha_n+1}{2}}}{s\Gamma(\alpha_n)}& \times K_{\alpha_n-1}\left(\sqrt{\frac{2\tilde{A}_n\alpha_n}{b_{0,n}}}\right).
T^{n}z =\sum_{k=1}^{r}a_{k}T^{n}z_{k}=\sum_{k=1}^{r}a_{k}T^{l_{1}+\cdots+l_{k}}z_{k}\qquad\hbox{by (iii)},\intertext{so that}\|T^{n}z-x\| \le\sum_{k=1}^{r}|a_{k}|\,\|T^{l_{1}+\cdots+l_{k}}z_{k}-x_{k}\|<\varepsilon\hbox{by (ii).}
NLP = \int_1^T \int \langle D \rangle^{N-1}M U_{\neq}^1 \langle D \rangle^{N-1}M \partial_X \Delta_t^{-1}(\partial_i^t U^j_{\neq}\partial_j^t U^i_{\neq}+ 2 \partial_i^t U^j_{0}\partial_j^t U^i_{\neq}) \,dV & = NLP_{\neq}+ NLP_0.
L ( \tilde{x}_{0}) - \frac{g}{1 + g}L \Bigl ( \tilde{x}_{0}^{1 + \frac 1 g}\Bigr ) > L \Bigl ( \frac{1}{1 + \tilde{y}_{0}}\Bigr ) , \qquad \mathrm{f o r}\quad 0 < g < 1 .
p_i\! =\! \frac{1}{2}p{_i}^{ab}e^{(2)}_{ab}\wedge \bar{e}^{(r)}- p{_i}^{ak}e^{(3)}_{a}\wedge \bar{e}^{(r-1)}_k + \frac{1}{2}p{_i}^{jk}e^{(4)}\wedge \bar{e}^{(r-2)}_{jk}
W_2(\{b^{-2}a,b^{-1}a,ab^{-1}a\})= W_2(b^{-2}a)\cup W_2(b^{-1}a)\cup W_2(ab^{-1}a) = \{b^{-2}a,ab^{-2}a,ab^{-3}a\}\cup \{b^{-1}a,ab^{-1}a,ab^{-2}a\}\cup \{ab^{-3}a\}= \{b^{-2}a,ab^{-2}a,ab^{-3}a,b^{-1}a,ab^{-1}a\}.
\forall (x,\theta)\in (T^*G\setminus \{0\})\cong G\times \mathfrak{g}^*\setminus \{0\},\,\, \tilde \sigma_{\textnormal{loc}}(x,\theta)=\sigma_{\textnormal{loc}}(x,\theta)\|\theta\|_g^{-m}= \sigma_{\textnormal{glob}}(x,\theta).
\widehat{L}_{\rm part}={\cal P}_i \dot{\eta}_i - \frac{1}{2}{\cal P}_i^{2}+ e( \widehat{A}_i \dot{X}_i + \widehat{A}_0 +\frac{e\theta}{2}\varepsilon_{ij}\widehat{A}_i \frac{d}{dt}\widehat{A}_j) - \frac{1}{2}\frac{d}{dt}(e \theta\varepsilon_{ij}{\cal P}_i \widehat{A}_j )\, .
\bar{P}_{V, t}(Y) = C_{V, t}\int_{W\in \mathbb{R}^{p_1\times r}: \sigma_{\min}(W) \geq 1/2}\frac{1}{(2\pi)^{p_1p_2/2}}\exp(-\|Y - 2t WV^{\intercal}\|_F^2/2 ) & \cdot (\frac{p_1}{2\pi})^{p_1r/2}\exp(-p_1\|W\|_F^2/2) dW.
J_1^n=\begin{pmatrix}U_1^n nc U_1^n 0 0 U_1^n 0 0 0 U_1^n\end{pmatrix}, \ \ \J_2^m=\begin{pmatrix}U_2^m 0 md U_2^m 0 U_2^m 0 0 0 U_2^m\end{pmatrix}
\begin{aligned}f(x) \,\to\,\min g_i(x) \,\leq\,0 \qquad i\in\mathcal M h_j(x) \,=\,0 j\in\mathcal P 0\,\leq\,\bar G_l(x)\,\perp\,\bar H_l(x) \,\geq\,0 l\in\mathcal Q. \end{aligned}
H_{\mu\iota}:=u_{\mu}(t,x)\nabla_{y}\overline{u_{\mu}(t,y)}+\nabla_{x}u_{\iota}(t,x)\overline{u_{\iota}(t,y)}, G_{\mu\iota}:=u_{\mu}(t,x)\nabla_{y}u_{\iota}(t,y)-\nabla_{x}u_{\mu}(t,x)u_{\iota}(t,y).
\frac{1}{\lambda}- \frac{1}{\lambda}\int_0^{\infty}dw \, e^{-w}(1-dt f(\lambda w)) & =dt \, \frac{1}{\lambda}\int_0^{\infty}dw \, e^{-w}\int_0^{\infty}\nu(ds) (1-e^{-\lambda w s}) & = dt \int_0^{\infty}\nu(ds) \frac{s}{1+\lambda s}
\text{LHS of (\ref{eqn:final})}= \sum_{m=2}^4 (-1)^m \binom{2m-1}{3}\binom{4}{m}= \binom{3}{3}\binom{4}{2}- \binom{5}{3}\binom{4}{3}+ \binom{7}{3}\binom{4}{4}= 1
\begin{aligned}\cos \theta\,\sqrt{i+1}\,\widetilde{W}_{i+1,k}+\sin \theta\,\sqrt{k+1}\,\widetilde{W}_{i,k+1}+\alpha\,\widetilde{W}_{i,k}=0, -\sin \theta\,\sqrt{i+1}\,\widetilde{W}_{i+1,k}+\cos \theta\,\sqrt{k+1}\,\widetilde{W}_{i,k+1}+\beta\,\widetilde{W}_{i,k}=0.\end{aligned}
\langle T_{\mu \nu}\rangle = \lim_{x^{\prime}\rightarrow x}\left[{\frac{2}{3}}{D}_{; \nu^{\prime}\mu}-{\frac{1}{3}}{D}_{; \nu \mu}-{\frac{1}{6}}g_{\mu \nu}{D}^{; \sigma^{\prime}}{}_{\sigma}\right]
\sigma(B)-\sigma(0)=-\frac{e^{2}}{2\pi^{2}\hbar}[\psi(\frac{1}{2}+\frac{H_{1}}{H})-\psi(\frac{1}{2}+\frac{H_{2}}{H})+\frac{1}{2}\psi(\frac{1}{2}+\frac{H_{3}}{H})-\frac{1}{2}\psi(\frac{1}{2}+\frac{H_{4}}{H})]
e^d=\left(\begin{array}{lllll}e^\alpha 0 \ldots 0 \beta' 0 e^{2\alpha}\ldots 0 (e^{2\alpha}-e^{2^{n-1}\alpha})a_{1n}\vdots \vdots \ddots \vdots \vdots 0 0 \ldots e^{2^{n-2}\alpha}(e^{2^{n-2}\alpha}-e^{2^{n-1}\alpha})a_{n-2,n}0 0 \ldots 0 e^{2^{n-1}\alpha}\end{array}\right),
T = \left( \begin{array}{c c}{{1}}&{{0}}\\{{\zeta}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c c}{{a}}&{{0}}\\{{0}}&{{\hat{d}}}\\ \end{array}\right) \left( \begin{array}{c c}{{1}}&{{\xi}}\\{{0}}&{{1}}\\ \end{array}\right)
|I_0| = \frac{n!}{2!2!(n-4)!}= \frac{n(n-1)(n-2)(n-3)}{4}, |I_1| = \frac{n!}{1!1!1!(n-3)!}= n(n-1)(n-2), |I_2| = \binom{n}{2}= \frac{n(n-1)}{2}.
<T_{\mu \nu}(x)\,T_{\rho \sigma}(0)>={\frac{1}{480\pi^{4}}\prod}_{\mu\nu ,\rho \sigma}^{(2)}\left({\frac{c(g(x))}{x^{4}}}\right) +\pi_{\mu \nu}\pi_{\rho \sigma}\left({\frac{f(\ln x\mu ,g))}{x^{4}}}\right) .
E^{{\mbox fixed}}_C= \frac{1}{2\pi}\int\limits_{0}^{\infty}d\omega \,\ln \left ( 1- \exp{(-2 \omega R)}\right )= - \frac{R}{\pi}\int\limits_{0}^{\infty}\frac{\omega\, d\omega}{\exp{(2\omega R)}-1}\,{.}
\begin{array}{rcl}v_\mu A_\nu^a =i\,\mbox{sgn}(k){\epsilon}_{\nu \mu \rho}{\partial}^\rho c^a , v_\mu c^a=0 \, , begin{align*}2mm] v_\mu{\bar{c}^a}=A_\mu^a , v_\mu B^a=-D_\mu c^a\, ,\end{array}
I_1 =\sum_{j\neq k}\sum_{v\in\Lambda}\int_1^\infty e^{-|x_j-x_k+v|^2t}\frac{t^{\frac{s}{2}-1}}{\Gamma(\frac{s}{2})}\mathrm{d}t, I_2 =\sum_{j\neq k}\sum_{w\in\Lambda^*\setminus\{0\}}e^{2\pi i w\cdot (x_j-x_k)}\int_0^1 \frac{\pi^{\frac{d}{2}}}{t^\frac{d}{2}}e^{-\frac{\pi^2|w|^2}{t}}\frac{t^{\frac{s}{2}-1}}{\Gamma(\frac{s}{2})}\mathrm{d}t.
\dfrac{d}{dt}u_x\big(t,x_\varepsilon(t)\big) =u_{tx}+u^2u_{xx}=-\dfrac{1}{2}uu_x^2+u^3-p*\Big(\dfrac{3}{2}uu_x^2+u^3\Big)-\dfrac{1}{2}p_x*u_x^3
{\mathbb E}(\|{\mathbb E}(U_{[n/q]+1}|{\mathcal F}_{({[n/q]-1})q})\|_H)\leq\sum_{j=q[n/q]+1}^{n}\|{\mathbb E}(Y_j|{\mathcal F}_{({[n/q]-1})q})\|_H\leq (n-q[n/q])\theta(q)\, .
c_{u_0,m,r}(u) =e^{-mu_0}\sum_{n=0}^N\int_0^1a_{m,n}(z)b_r(z,u)z^{n+r}e^{-u_0z}\,dz &=e^{-mu_0}\sum_{n=0}^N\Theta(I_{m,n,r})\int_0^1z^{n+r}e^{-u_0z}\,dz,
|E(\Gamma_N(QD_{2^\alpha}))| =|E(K_2+\Gamma_N(D_{2^{\alpha-2}}))|+|E(K_2+\Gamma_N(D_{2^{\alpha-3}}))| &=2|V(\Gamma_N(D_{2^{\alpha-2}}))|+1+|E(\Gamma_N(D_{2^{\alpha-2}}))| &\quad{}+2|V(\Gamma_N(D_{2^{\alpha-3}}))|+1+|E(\Gamma_N(D_{2^{\alpha-3}}))|.
f \star g = \frac{1}{( 2 \pi )^{2}}\sum_{m , n , P^{\prime}, P^{\prime \prime}}\int_{- \pi}^{+ \pi}d \theta^{\prime}d \theta^{\prime \prime}\delta_{P^{\prime}, P_{\theta}+ m}\delta_{P^{\prime \prime}, P_{\theta}- n}e^{[ - 2 i A ( \tilde{\bf r}, \tilde{\bf r}^{\prime}, \tilde{\bf r}^{\prime \prime}) ]}f ( \tilde{\bf r}^{\prime}) g ( \tilde{\bf r^{\prime \prime}})
\int_0^{1/2}u\varphi\,dx=\Big[u\Phi\Big]_0^{1/2}+\int_0^{1/2}(-u')\Phi\,dx=u\left(\frac{1}{2}\right)\Phi\left(\frac{1}{2}\right)+\int_0^{1/2}(-u')\Phi\,dx.
\left\{\begin{array}{cccll}A^{I}_\mu \null \null (\mathbf{I}=1,\dots,n_V) \mbox{vectors}\null \lambda^i_A \phi^i (i=1,\dots,n_V+n_T \equiv n) (A=1,2) B^\mathcal{M}_{\mu \nu}\null \null (\mathcal{M}=1,\dots,n_T) \mbox{tensors}\ \end{array}\right\}
s_n^2 := \sum_{i=1}^n \dfrac{\Gamma(i)^2}{\Gamma(i+\alpha)^2},\intertext{and}\sigma_n^2 := \begin{cases}\dfrac{\Gamma(n)^2}{\Gamma(n+\alpha)^2}\cdot \dfrac{n}{1-2\alpha}(-1<\alpha<1/2), begin{align*}4mm]\dfrac{\Gamma(n)^2}{\Gamma(n+\alpha)^2}\cdot n\log n (\alpha=1/2).\end{cases}
\textbf{c}'=\left[\begin{array}{cccc}\lambda c_{n-1,0}\lambda c_{n-1,1}\dots \lambda c_{n-1,\ell-1}c_{00}c_{01}\ldots c_{0,\ell-1}\vdots \vdots \ddots \vdots c_{n-2,0}c_{n-2,1}\ldots c_{n-2,\ell-1}\end{array}\right]
H_2(f, g) \ = \ - \sum_{\mu_1 = 1}^f \frac{1}{\mu_1 !}\binom{f - \mu_1}{g - \mu_1}\frac{(-1)^{f - \mu_1}}{(f-\mu_1)!}&= \ - \frac{(-1)^f}{(f-g)!}\sum_{\mu_1 = 1}^f \frac{(-1)^{\mu_1}}{\mu_1 ! (g-\mu_1)!}\ = \ \frac{(-1)^f}{g!(f-g)!},
\frac{1}{q_3^{n+t}}=\sum_{i=1}^n \eta_i\left( q_1^{-i}-q_3^{-i}\right)\le M\sum_{i=1}^{\infty}\left( q_1^{-i}-q_3^{-i}\right) &=M\left( \frac{q_1^{-1}}{1-q_1^{-1}}-\frac{q_3^{-1}}{1-q_3^{-1}}\right) =\frac{M(q_3-q_1)}{(q_1-1)(q_3-1)}.
0= m_{1}\left(\frac{3}{2}- \left(\cot \frac{\vartheta}{2}\right)^{2}\right)+m_{1}\left(\frac{3}{2}- \left( \tan \frac{\vartheta}{2}\right)^{2}\right) & +4n\left(\frac{3}{2}- (\tan \vartheta )^{2}\right)+4m_{2}\left(\frac{3}{2}- (\cot \vartheta )^{2}\right)
\begin{cases}\eta_t + w_x+w_{xxx}= 0, \,\, (0,L)\times (0,+\infty), w_t +\eta_x +\eta_{xxx}=0, \,\, (0,L)\times (0,+\infty), \eta(x,0)= \eta_0(x), w(x,0)= w_0(x), \,\, (0,L),\end{cases}
\sigma_{0}:= \dfrac{2(3-2\sqrt{2})}{(2-2\alpha+\beta)+\sqrt{(-2+2\alpha-\beta)^2-4(3-2\sqrt{2})(2\sqrt{2}-1-2\alpha-\beta)}},\intertext{and}\tilde{\sigma_{0}}:= \dfrac{2}{(2-2\alpha+\beta)+\sqrt{(-2+2\alpha-\beta)^2-4(2\alpha-3+\beta)}}.
\bigg|f(x)-\frac{\nu(J_j)}{\nu_0(J_j)}\bigg| =|f(x)-\hat f_m(x_i^*)|\leq |f(x)-\hat f_m(x)|+|\hat f_m(x)-\hat f_m(x_i^*)| \leq 3K\ell_m^{1+\gamma}+ |\hat f_m'(x_i^*)(x-x_i^*)| \leq 3K\ell_m^{1+\gamma}+ M \ell_m.
\varrho(\mathsf{A}) :=\lim_{n \to \infty}\sup\left\{\left\|A_n\cdots A_1 \right\|^{\frac{1}{n}}\colon A_i \in \mathsf{A}\right\}&=\sup_{(A_i)_{i=1}^\infty \in \mathsf{A}^{\mathbb{N}}}\limsup_{n \to \infty}\left\|A_n \cdots A_1\right\|^{\frac{1}{n}},
L_i\ast^{}_M F= L_i F +\nu\sum_{1\leq j,k \leq 3}\varepsilon_{ijk}L_k{\partial F \over\partial L_j}+ \nu^2 \Bigl( 2{\partial F \over\partial L_i}+\sum_{1\leq j \leq 3}L_j{\partial^2 F \over\partial L_i \partial L_j}\Bigr), \quad 1\leq i\leq 3.
H\backslash A =\{(a,b,c,-a):a,b,c>0b\geq a,c\geq a\}& \cup\{(a,b,c,-b):a,b,c>0a\geq b,c\geq b\}& \cup\{(a,b,c,-c):a,b,c>0a\geq c,b\geq c\}.
D_{\psi_{i}}= \sum_{N = 0}^{+ \infty}{\left( \frac{\Gamma_{i}e^{-{\frac{\hbar \omega_{i}}{k T}}}}{1 - \left( 1 - \Gamma_{i}\right) e^{-{\frac{\hbar \omega_{i}}{k T}}}}\right)}^{N}\left| \rho_{i}^{N}\right> \left< \rho_{i}^{N}\right| \; .
\delta s^{2}= \Omega^{2}( x^{i}, v ) [ g_{1}( d x^{1})^{2}+ g_{2}( d x^{2})^{2}+ g_{3}( d x^{3})^{2}+ h_{4}( \hat{{\delta}}v )^{2}+ h_{5}( \delta p )^{2}] ,
\begin{cases}\| \sum_{l \leq j \epsilon}T_a^{j,l}\|_{L^s \to L^s}\lesssim_{\epsilon}2^{jm + j(n(1-\rho)/2 + \varepsilon)(\frac{2}{s}-1)}\|T_a^{j, l}\|_{L^s\to L^s}\lesssim_{\epsilon}2^{10n(m-n)(j+l)}, l>j\epsilon\end{cases}
(f,g) =S_{\chi\oplus \xi}S_{\chi\oplus \xi}^{-1}(f,g) &=\int_{\Theta}v^{2}(w)\pi_{F(w)\oplus G(w)}(\chi_{w}\oplus\xi_{w})^{\ast}(\chi_{w}\oplus \xi_{w})\pi_{F(w)\oplus G(w)}S_{\chi\oplus \xi}^{-1}(f,g)d\mu(w)
r_{00}=c\alpha^2, \ \r_{0j}=cy_j, \ \r_i=cb_i, \ \r=cb^2, \ \r^i_j=c\delta^i_j, r_{0k}s^k_{~0}=0, \ \r_{0k}s^k=cs_0, \ \s^k_{~0}r_k=cs_0, \ \r_{00|0}=c_0\alpha^2.
\begin{array}{c}{{- a + \alpha = \frac \lambda{\lambda - \mu}2 \alpha_{+}- \frac \mu{\lambda - \mu}2 \alpha_{-},}}\\{{- a - \alpha = \frac \mu{\lambda - \mu}2 \alpha_{+}- \frac \lambda{\lambda - \mu}2 \alpha_{-}.}}\\ \end{array}
|\Omega^{(n)\, \alpha_{1}\ldots \alpha_{n}}_{i_{1}\ldots i_{n}}\rangle = \frac{1}{\sqrt{n!}}\left( a_{\alpha_{n}}^{i_{n}}\right)^{\dagger}\ldots \left( a_{\alpha_{1}}^{i_{1}}\right)^{\dagger}|\Omega \rangle \, .
Z_{g h}= \int D [ C , \tilde{\cal P};{\cal P}, \tilde{C}] \cdot \exp \{i \int_{\tau_{i}}^{\tau_{f}}( \tilde{\cal P}_{a}\dot{C}^{a}- \dot{\tilde{C}}_{a}{\cal P}^{a}- \tilde{\cal P}_{a}{\cal P}^{a}) d \tau \}\, .
\int d^{4}x ( \epsilon_{\mu \nu \rho \sigma}\partial^{\nu}B^{\rho \sigma})^{2}= \int d^{4}x 2 ( - B_{\mu \nu}\partial^{2}B_{\mu \nu}- B_{\mu \nu}\partial_{\nu}\partial_{\lambda}B_{\lambda \mu}+ B_{\nu \mu}\partial_{\mu}\partial_{\lambda}B_{\lambda \nu})
0<\beta\le t_{\epsilon,\alpha}:=\frac{(1-\sqrt{\alpha})\epsilon}{16\ln \tfrac{4}{(1-\sqrt{\alpha})\epsilon}}\le \frac{(1-\sqrt{\alpha})\epsilon}{16\ln 4}< \frac{(1-\sqrt{\alpha})\epsilon}{2}\le \frac{(1-\sqrt{\alpha})\epsilon}{1+\epsilon}.
\Delta w_1 = \frac{4K_{11}}{\sqrt{|K|}}\bigg(e^{u_0^{'}+\sqrt{|K|}w_1}-1\bigg) &+\frac{4K_{12}}{\sqrt{|K|}}\bigg(e^{u_0^{''}+\frac{1}{K_{11}}(|K|w_2+K_{21}\sqrt{|K|}w_1)}-1\bigg)+h_1, \Delta w_2 =4 \bigg(e^{u_0^{''}+\frac{1}{K_{11}}(|K|w_2+K_{21}\sqrt{|K|}w_1)}-1\bigg)+h_2,
\rho^{[k+2..N]}=\sum_{\gamma}(\lambda_{\gamma}^{[k+1]})^{2}|\Phi_{\gamma}^{[k+2..N]}\rangle\langle\Phi_{\gamma}^{[k+2..N]}|=\sum_{\gamma}(\lambda_{\gamma}^{[k+1]})^{2}|\gamma\rangle\langle\gamma|.
(\widetilde{X}_1 \widetilde{X}_2 \widetilde{X}_c)^2 \widetilde{B}\widetilde{C}\widetilde{A}\widetilde{W}_1 \widetilde{W}_2 \widetilde{W}_c \widetilde{W_1^\prime}\widetilde{W_2^\prime}\widetilde{W_c^\prime}= \partial^2.
u_{\vec{\xi}}(\vec{x}) = \sum_{\sigma \in S_N}A_{\sigma}\prod_{i =1}^{N}z_{\sigma(i)}^{x_i}= \sum_{j=1}^{N}\left ( \underset{\sigma^{-1}(j)=1}{\sum_{\sigma \in S_N}}A_{\sigma}\prod_{i =2}^{N}z_{\sigma(i)}^{x_i}\right) z_j^{x_1}=0.
\left| \int \hat{B}(q) \left( q^2 - E - i \eta \right)^{-1}d q \right| \leq \int \left| \hat{B}(q)\right| \left| \left( q^2 - E - i \eta \right)^{-1}\right| d q \leq \int \left| \hat{B}(q) \right| \left| \eta^{-1}\right| dq &= \eta^{-1}\int \left| \hat{B}(q) \right| dq &= \| \hat{B}\|_1 \eta^{-1}
\big\| e^{Q_1 \big[ -\frac12(1-\frac1\beta)|\cdot|^2 \big]}\big\|_{L^a(\gamma)}\big\| e^{\frac12(1-\frac1\beta)|\cdot|^2}\big\|_{L^{1-a}(\gamma)}=\big( \beta^{\frac1{1-a}}(\beta a - a +1)^{- \frac{1}{a(1-a)}}\big)^\frac{n}{2}.
\eth_1^{\pm}(n^2_1n_2,u,q,p_1) = d{\pi}^{3}\int_{0}^\infty V_1^{\pm}(z) z^{-1/3+i\nu}\, e\left(\frac{Nuz}{p_1qQ}\pm \frac{3(Nz n_1^2 n_2)^{1/3}}{p_1q r^{1/3}}\right)dz ,
\sum_{n=2^{k-1}}^{2^k-1}1_{[-B,B]}\left( t + \sum_{i=0}^{n-1}f(T^i x) \right)\ge\beta \sum_{n=0}^{2^k - 1}1_{[-B,B]}\left( t + \sum_{i=0}^{n-1}f(T^i x) \right)
p_{0 \mu}= p_{\mu}- \frac{1}{2}\tilde{S}^{a b}e_{a \mu b}= p_{\mu}- \frac{1}{2}\tilde{S}^{a b}\omega_{a b \mu}\;,\;\omega_{a b \mu}=\frac{1}{2}(e_{a \mu b}- e_{b \mu a}),
\psi(g,1/2+it)= \sum_{\mathfrak{a}\in \mathcal{E}_k}\frac{2\zeta(1/2+2it)}{(\vec{h}_{\mathfrak{a}}^{\intercal}P_0\vec{h}_{\mathfrak{a}})^{1/4+it}y_{\mathfrak{a}}(\vec{x}_0)^{1/2+2it}}E_{\mathfrak{a}}(g,1/2+2it).
{\cal D}^{(n)}_{q,t:*}(q^{-1/2}) = \sum_{\sigma\in \{\pm 1\}^n}\prod_{1\le i\le n}\frac{z_i^{\sigma_i}}{1-z_i^{2\sigma_i}}\prod_{1\le i<j\le n}\frac{1-t z_i^{\sigma_i}z_j^{\sigma_j}}{1-z_i^{\sigma_i}z_j^{\sigma_j}}\prod_{1\le i\le n}T_i^{\sigma_i/2}.
(\chi + t_i \chi + \tilde \chi + \sqrt{-1}\partial\bar\partial \varphi_{t_i})^n = c (\chi + t_i \chi + \tilde \chi + \sqrt{-1}\partial\bar\partial \varphi_{t_i})^m \wedge \omega^{n - m}+ b_{t_i}f_1 \omega^n
\max \sum_{l=1}^L \sum_{k=1}^{K_l}\|\mathbf{V}^\mathrm{T}_{k_l}\mathbf{U}_{\mathrm{p}}\|_\mathrm{F}^2 = \max \mathrm{tr}\left[ \left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{R}\mathbf{W}^{\mathrm{RF}}\right],
\forall \eta_\Lambda \in \Omega_\Lambda: \sum_{\Delta \subset \Lambda}\sum_{\xi_\Delta}\mu(\eta_\Lambda) c_\Delta(\eta_\Lambda, \xi_\Delta) = \sum_{\Delta \subset \Lambda}\sum_{\xi_\Delta}\mu(\xi_\Delta \eta_{\Lambda \setminus \Delta})c_\Delta(\xi_\Delta\eta_{\Lambda \setminus \Delta}, \eta_\Delta),
\int_M^N t^\nu j_\alpha(rt)\, dt = \sum_{i=1}^kC_{i,\nu,\alpha}\big( N^{\nu+1}j_{\alpha+i}(rN)-M^{\nu+1}j_{\alpha+i}(rM)\big) &\phantom{=}+ C'_{k,\nu,\alpha}\int_M^N t^\nu j_{\alpha+k}(rt)\, dt,
|\tau(\Phi_q(y)w-y\Phi_pw)| = |\tau\left((\Phi_qy- \Phi_qy_1-i\Phi_qy_2 )w+(\Phi_qy_1+i\Phi_qy_2)w-(y_1+iy_2)\Phi_p(w)\right)| &\leq 2Q[\Phi_q]\|y\|_q\|w\|_p+ 2 L_{\varphi}(\|y_1\|_q+\|y_2\|_q)\|w\|_p &\leq \left(2Q[\Phi_q]+ 4 L_{\varphi}\right)\|y\|_q\|w\|_p.
\|x_i^{k+1}-x_i^*\|^2 = \|\Pi_{K_i}[x_i^k-\alpha_k F_i(x_i^k,N{\hat v_i^k})] - x^*_i\|^2\cr = \|\Pi_{K_i}[x_i^k-\alpha_k F_i(x_i^k,N{\hat v_i^k})] - \Pi_{K_i}[x^*_i-\alpha_k F_i(x^*_i,\bar{x}^*)]\|^2\cr \leq \|x_i^k-x^*_i-\alpha_k (F_i(x_i^k,N{\hat v_i^k}) - F_i(x^*_i,\bar{x}^*))\|^2.
a_1(n) =\begin{cases}1 p\mid n\Rightarrow p\leq Y, & \Omega(n)\leq 100\log\log Tp\leq Y, 0 .\end{cases}\tilde{a}_1(n) =\begin{cases}1 p\mid n\Rightarrow p\leq Y, & \Omega(n)> 100\log\log Tp\leq Y, 0 .\end{cases}
\lVert\hat{\lambda}^\infty_{g_1g_2}- \hat{\lambda}^\infty_{g_1}\circ \hat{\lambda}^\infty_{g_2}\rVert= \lim{}\lVert\hat{\lambda}^i_{g_1g_2}- \hat{\lambda}^i_{g_1}\circ \hat{\lambda}^i_{g_2}\rVert\leq \lim r_i\leq \lim 2^{-i}r_0= 0.
D = \det \left( \begin{array}{c c c}{{D_{t t}- D_{t b}( Q^{- 1})_{b c}D_{c t}}}&{{D_{t j}- D_{t b}( Q^{- 1})_{b c}D_{c j}}}&{{D_{t b}}}\\{{D_{i t}- D_{i b}( Q^{- 1})_{b c}D_{c t}}}&{{D_{i j}- D_{i b}( Q^{- 1})_{b c}D_{c j}}}&{{D_{i b}}}\\{{0}}&{{0}}&{{Q^{a b}}}\\ \end{array}\right)
\begin{pmatrix}k_1 k_2\end{pmatrix}_t = \begin{pmatrix}-D^3+k_1D+Dk_1 Dk_2+k_2D Dk_2+k_2D D^3-Dk_1-k_1D\end{pmatrix}\begin{pmatrix}r_1 r_2\end{pmatrix}= P\begin{pmatrix}r_1 r_2\end{pmatrix}.
S_1 := C_R(a + b) &= Z \cup (Z + (a + b)) \cup (Z + 2(a + b)) \cup (Z + 3(a + b)), S_2 := C_R(a + 2b) = Z \cup (Z + (a + 2b)), S_3 := C_R(a) = Z \cup (Z + a) S_4 := C_R(b)= Z \cup (Z + b) \cup (Z + 2b) \cup (Z + 3b).
j(k)(j(h\upharpoonright Z_s))=(j(k)\upharpoonright Z_t)(j(h\upharpoonright Z_s))=j(k\upharpoonright Z_t)(j(h\upharpoonright Z_s))= =j((k\upharpoonright Z_t)(h\upharpoonright Z_s))=j(k(h\upharpoonright Z_s)).
-T^{2q}+2T^{q+1}+ T^{q(q+1)}\sum_{c}(c+T)^{1-q}= \frac{(-T^{2q}+2T^{q+1})\cdot \prod_{c}(c+T^{q})+T^{q(q+1)}\cdot (T^{q}-T)}{\prod_{c}(c+T^{q})}= \frac{T^{q(q+1)+1}+}{\prod_{c}(c+T^{q})}.
T^{m+m'}z=T^{m'}x-\sum_{l<2^{k_{0}}}\ \sum_{j=b_{l}}^{b_{l+1}-1}\Biggl(v^{(k)}\prod_{i=j-b_l+m'+1}^{\Delta^{(k)}-1}w_{i}^{(k)}\Biggr)^{-1}\cdot &\Bigl(\,\,\prod_{i=1}^{j-b_l}w_{b_{l}+i}\Bigr)^{-1}\,x_{j}e_{b_{2^{k-1}+l}+j-b_{l}+m'}.
\epsilon^k_0(g_1, \ldots, g_k) = (g_2, \ldots, g_k) \epsilon^k_i(g_1, \ldots, g_k) = (g_1, \ldots, g_{i}g_{i+1}, \ldots, g_k) 1 \leq i \leq k-1 \epsilon^k_k (g_1, \ldots, g_k) = (g_1, \ldots, g_{k-1})
\sup_{\pi}\Bigg\{\Bigg[\sum_{j=1}^{N'}\Big[\sum_{i=1}^N|\Delta_i\Delta_j (\nabla^w_x F_{t_i})(^{x_j}c_{t_i})|^a \Big]^{\frac{b}{a}}\Bigg]^{\frac{1}{b}}\Bigg\}< \infty;c\in C([0,T];\mathbb{R})
\textrm{Hess}_W\mathcal{H}(\rho|\rho^*)(V_{\Phi_1}, V_{\Phi_2})= \int_{M}\frac{\Delta_\rho\Phi_1\Delta_\rho\Phi_2}{\rho}+\big(\nabla\nabla\log{\rho}\nabla\Phi_1,\nabla\Phi_2\big)\rho+\big(\nabla\nabla h\nabla\Phi_1,\nabla\Phi_2\big)\rho dx.
\delta \phi^i \left( \frac{\partial}{\partial \phi^i}f^{(p)}\right) \left[ \psi^p \right]_{\rm max}= \delta \phi^i \phi^i \left( \frac{d}{\phi d \phi}f^{(p)}\right) \left[ \psi^p \right]_{\rm max}= - f^{(p+2)}\delta_{\psi}\left[ \psi^{p+2}\right]_{\rm max},
\Pi_{\mu \nu}^{a b}( 4 ) = \frac i{1 4 4 \pi^{2}}g^{2}\frac{N_{f}}{2}\delta^{a b}( p_{\mu}p_{\nu}- p^{2}g_{\mu \nu}) [ 1 2 \mathrm{l n}( \bar{\lambda}^{2}/ p^{2}) - 4 ] .
\sum_{\ell=K}^{K+k-1}\left[(f+g)(x_*)-(f+g)(x^{\ell+1})\right]\ge \frac{1}{2\beta}\left(\|x^{K+k}-x_*\|^2-\|x^K-x^*\|^2+2\left[(f+g)(x^{K+k})-(f+g)(x^{K})\right]\right).
||\partial_x^k(\Phi_r-\Phi_0)||_{s,r,p;\mathbb{T}^n\times\mathcal{O}_r^-}+\frac{\alpha}{M}||\partial_x^k(\Phi_r-\Phi_0)||^{lip}_{s,r,p;\mathbb{T}^n\times\mathcal{O}_r^-}=O(\epsilon^{\frac{1}{1+\beta}}/\alpha)=O(r^{1/40}),
\omega_{(-)}\partial_-\omega_{(-)}^{-1}=\left (g_{(0)}^{-1}\, y_{(0)+}\right )^{-1}\, L_{(-1)}\, g_{(0)}^{-1}y_{(0)+}=y_{(0)+}^{-1}\, N_{(-1)}^{-1}\, \partial_-N_{(-1)}\,y_{(0)+}.
\widetilde{J}_i^{(a)}= \begin{cases}J_i^{(a)}i \neq i^{(a)}, p_{\ell^{(a)}}^{(a)}(\widetilde{\nu}_{i^{(a)}}^{(a)}, B^{1,s_{m-1}}\otimes \cdots \otimes B^{1,s_1}) i = i^{(a)},\end{cases}