harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\chi ( r ) \to a_{0}e^{- r / L}+ \ldots \ ; \qquad \alpha ( r ) \to{\frac{2}{3}}a_{0}^{2}{\frac{r}{L}}e^{- 2 r / L}+{\frac{a_{1}}{\sqrt{6}}}e^{- 2 r / L}+ \ldots
P_{\perp}\begin{pmatrix}Mz begin{align*}.1cm] \overline{N}z \end{pmatrix}+ P_{\perp}\begin{pmatrix}N \bar{z}begin{align*}.1cm] \overline{M}\bar{z}\end{pmatrix}= P_{\perp}\begin{pmatrix}p begin{align*}.15cm] \overline{p}\end{pmatrix}
ic(n-1) = 2\sum_{i, j \in I}w(ij) + \sum_{i \in I, j \in I^{c}}w(ij)= 2\sum_{i, j \in I}w(ij) - 2\sum_{i \in I, j \in I^{c}}w(ij)\le 2\sum_{i, j \in I}w(ij) - 2f(i) &= 2\sum_{i,j\in I}w(ij) + ci(n-1) - a_{i},
\partial_{x_1}u(x_1,x_2) = 4^{-\frac{1}{3}}(\partial_x + \partial_y) v(x,y), & \partial_{x_2}u(x_1,x_2) = \sqrt{3}4^{-\frac{1}{3}}(\partial_x - \partial_y) v(x,y), & \partial_{x_1}(\partial_{x_1}^2+ \partial_{x_2}^2) u (x_1,x_2) =(\partial_x^3 + \partial_y^3) v(x,y).
\begin{aligned}d x(t) = v(t)\, d t, m\, d v(t) =\big(-\gamma v(t)-\Phi'(x(t))-\sum_{k\geq 1}\sqrt{c_k}z_k(t)\big)\,dt+\sqrt{2\gamma}\, dW_0(t) , d z_k(t) =\left(-\lambda_k z_k(t)+ \sqrt{c_k}v(t)\right) \, dt+\sqrt{2\lambda_k}\, dW_k(t),k\geq 1, \end{aligned}
\begin{aligned}\tau^{(2)}{G}= \lbrack\ 1^2;\ (\ \mathrm{A}(3,c-k-t);\ \mathrm{A}(3,c-1-k-t)\times C_3,\ (\mathrm{B}(3,c-1-k-t)\times C_3)^3\ ), (\ 21;\ \mathrm{A}(3,c-1-k-t)\times C_3,\ (21)^3\ )^3\ \rbrack,\end{aligned}
\max (L_1, L_2) \gtrsim |(\tau_1 - \xi_1^3 - \eta_1^3) + (\tau- \tau_1) - (\xi- \xi_1)^3 - (\eta- \eta_1)^3| & = |(\tau- \xi^3 -\eta^3) + 3(\xi \xi_1 (\xi-\xi_1) + \eta \eta_1 (\eta- \eta_1))|
\nabla_y \times \widehat{\widetilde{\Upsilon}}(s,{\scriptstyle X}, y)= \widehat{\widetilde{\mathcal{F}}}(s,{\scriptstyle X}, y)= \widehat{\widetilde{\bar{\bf b}}}(s,{\scriptstyle X})- \widehat{\widetilde{\bf b}}(s,{\scriptstyle X}, y).
|\log \gamma_2| <\frac{B_2+|\log \gamma_1|}{2h}<\frac{G(B_2)\log \alpha_2 +G(|\log \gamma_1|)\cdot \log \alpha_2 \log c}{2h}&<\frac{\left(\frac{G(B_2)}{\log 10^5}+G(|\log \gamma_1|)\right) \log \alpha_2 \log c}{2G(h)\cdot \log \alpha_2 \log c}=\frac{\frac{G(B_2)}{\log 10^5}+G(|\log \gamma_1|)}{2G(h)}=:G(|\log \gamma_2|).
\langle x_{\mu}( \tau ) x_{\nu}( \tau^{\prime}) \rangle = \delta_{\mu \nu}\Delta ( \tau - \tau^{\prime}) , \quad \Delta ( \tau \rightarrow \tau^{\prime}) = \Delta ( 0 ) \sim \frac{\alpha^{\prime}}{\epsilon}, \quad \partial_{\tau}\Delta ( \tau \rightarrow \tau^{\prime}) = 0 ,
i \partial_{t}\Psi [ \xi , t ) = \int d^{2}x \biggl [ i \biggl ( A_{0}+{\frac{1}{\theta}}J_{i}\epsilon_{i j}{\frac{\partial_{j}}{\nabla^{2}}}\biggr ){\frac{\delta}{\delta \xi}}+ A_{0}\rho - \xi{\dot{\rho}}\biggr ] \Psi [ \xi , t ) \, ,
R|_V=\left(\begin{array}{cc}0 I_2 I_2 0 \end{array}\right), J|_V=\left(\begin{array}{cc}J_2 0 0 J_2 \end{array}\right), \quad\hbox{and}L|_V=\left(\begin{array}{cc}J_2 0 0 -J_2 \end{array}\right).
\mathcal{A}^{\prime}(\mu) := \sum_{l=1}^{Q}\frac{\alpha_{l}}{|\Omega_{1D}|}A^{\prime}(\mathcal{P}(\mu) ;\mu_{l}), (\mathcal{A}^{h})^{\prime}(\mu) := \sum_{l=1}^{Q}\frac{\alpha_{l}}{|\Omega_{1D}|}A^{\prime}(\mathcal{P}^{h}(\mu) ;\mu_{l}).
y_{1,k}(\theta+\frac{\pi}{2}i) y_{1,k}(\theta-\frac{\pi}{2}i) =( t_{0,k-1})^2 (y_{1,t}(\theta)^2+1), y_{1,t}(\theta+\frac{\pi}{2}i) + y_{1,t}(\theta-\frac{\pi}{2}i) = y_{1,k}(\theta).
\Phi\left(Z^{\chi(1), \epsilon(1)}_{k_1}\cdots Z^{\chi(n), \epsilon(n)}_{k_n}\right) = \sum_{\pi \in BNC(\chi)}\left[ \sum_{\substack{\sigma \in BNC(\chi) \pi \leq \sigma \leq \epsilon}}\mu_{BNC}(\pi, \sigma) \right] \Phi_\pi\left(Z^{\chi(1), \epsilon(1)}_{k_1}, \ldots, Z^{\chi(n), \epsilon(n)}_{k_n}\right)
\begin{array}{l}\displaystyle{H^B = \int d \sigma \left( \Pi_i \dot{x}_i - \mathcal{L}_B \right) =}\displaystyle{\qquad \frac{1}{4 \pi l_s^2}\int_{0}^{2 \pi l_s^2 p^+}d \sigma \left[ \sum_{i=1}^8 ({\dot{x}_i}^2 +{{x}_i^{\prime}}^2 ) + \mu^2 \left( \sum_{I=3}^8 x_I^2 + 4 x_2^2 \right) + 4 \mu x_2 \partial_\sigma x_1 \right]}\end{array}
\int_{M}\Delta_{g}\left(W_{\delta,q}+\delta V_{\delta,q}\right)\Phi d\mu_{g}\le\|\Delta_{g}(W_{\delta,q}+\delta V_{\delta,q})\|_{L^{\frac{2n}{n+2}}(M)}\|\Phi\|_{H}=O(\delta^{2})\|\Phi\|_{H}=o(\varepsilon^{2})
\int_0^1 \varphi(t)\varrho(t)\,dt =\int_0^{x_0}\varphi(t)\varrho(t)\,dt+\int_{x_0}^1 \varphi(t)\varrho(t)\,dt &\ge \int_0^{x_0}\varphi(x_0)\varrho(t)\,dt+\int_{x_0}^1 \varphi(x_0)\varrho(t)\,dt &=\varphi(x_0)\int_0^{1}\varrho(t)\,dt &> 0 ( \ge 0)\,.
\Phi^{\alpha}( z , \bar{z}) = \Phi_{0}^{\alpha}- i \Psi_{0}^{\alpha}\ln | z |^{2}+ i \sum_{n \neq 0}\frac{\psi_{n}^{\alpha}}n z^{- n}+ \frac{\bar{\psi}_{n}^{\alpha}}n \bar{z}^{- n}.
M_{1,1}= \frac{2P_3(1)}{L q^{1/2}}\mathop{\sum \sum}_{\substack{vw = q (v,w)=1}}\mu^2(v) \varphi(w) \mathop{\sum \sum}_{\substack{b \leq y_2 (bn,q)=1}}\frac{\Lambda(b) P_2[b]}{(bn)^{\frac{1}{2}}}V_1 \left( \frac{n}{q^{\frac{1}{2}}}\right)\cos\left(\frac{2\pi bn\overline{v}}{w}\right),
J_{\epsilon}(u) = \int_{\mathbb{R}^{N}}\phi(\vert \nabla u\vert)\vert \nabla u\vert^{2}dx + \int_{\mathbb{R}^{N}}V_{\epsilon}(x)\phi(\vert u\vert)\vert u\vert^{2}dx - \int_{\mathbb{R}^{N}}f(u)udx.
S\left(\frac{1}{x^{i_1}},\frac{1}{x^{i_2}},\frac{1}{x^{i_3}};\infty,1\right)= A\left(\frac{1}{x^{i_1}},\frac{1}{x^{i_2}},\frac{1}{x^{i_3}};\infty,1\right)+A\left(\frac{1}{x^{i_1+i_2}},\frac{1}{x^{i_3}};\infty,1\right) &+A\left(\frac{1}{x^{i_1}},\frac{1}{x^{i_2+i_3}};\infty,1\right)+A\left(\frac{1}{x^{i_1+i_2+i_3}};\infty,1\right).
\langle\sigma_{+}(0)\sigma_{-}(\tau)\rangle=\frac{1}{4}(e^{-\frac{\Gamma}{2}\tau}+\frac{1}{2}e^{-\frac{3\Gamma}{4}\tau}e^{-i\Omega_{R}\tau}+\frac{1}{2}e^{-\frac{3\Gamma}{4}\tau}e^{i\Omega_{R}\tau})e^{-i\omega\tau}
G_{M N}d x^{M}d x^{N}= H_{p}^{-{\frac{4 ( n - 1 )}{( p + n ) \Delta_{p}}}}\left[ - d t^{2}+ d{\bf x}\cdot d{\bf x}\right] + H_{p}^{\frac{4 ( p + 1 )}{( p + n ) \Delta_{p}}}\left[ d \tilde{y}^{2}+ d \tilde{s}_{n}^{2}\right] ,
\frac{1}{r}\int_{r-t}^{r+t}\widetilde{\Omega}_s[u_0](x) K_0(t,r;s;\mu,\nu^2)\, \mathrm{d}s = \int_{0}^{t}\frac{1}{r}\Big[ \Omega_{s+r}[u_0](x) -\Omega_{s-r}[u_0](x)\Big] K_0(t,r;s+r;\mu,\nu^2)\, \mathrm{d}s \xrightarrow [r\to 0]{}2 \int_{0}^{t}\frac{\partial}{\partial s}\, \Omega_{s}[u_0](x) K_0(t,0;s;\mu,\nu^2)\, \mathrm{d}s.
\Delta = \sum_{i = 1}^{n - 1}a_{2 i}z_{2}^{i}\zeta^{2 n - 2 i}+ a_{2 n}z_{2}^{n}+ z_{1}\sum_{i = 0}^{n - 2}b_{2 i + 1}z_{2}^{i}\zeta^{2 n - 2 i - 3},
V_d(r) \geq \omega_d \int_{a}^r \sinh(t)^{d-1}dt \geq \frac{\omega_d}{2 \cdot 2^{d-1}}\int_{a}^r e^{t(d-1)}dt = \frac{\omega_d}{2 (d-1) 2^{d-1}}[e^{r(d-1)}- e^{a (d-1)}] &\geq \frac{\omega_d}{4 (d-1) 2^{d-1}}e^{r (d-1)}.
Q=A_{1}\sqrt{\frac{2}{\rho}\cdot\frac{p_{1}-p_{2}}{(\frac{A_{1}}{A_{2}})^{2}-1}}=A_{2}\sqrt{\frac{2}{\rho}\cdot\frac{p_{1}-p_{2}}{1-(\frac{A_{2}}{A_{1}})^{2}}}
(a_{ij})=\left(\begin{array}{cc}\frac{1}{1-{\mu}^2m^2}0 0 \frac{m^2}{\left(1-{\mu}^2m^2\right)^2}\end{array}\right)b_i=\left(\begin{array}{c}0 -\frac{\mu m^2}{1-{\mu}^2m^2}\end{array}\right),i,j=1,2.
\Delta{\cal L}_{D}\ = \m^{\dagger}m \G \left({\frac{A_{D}}{\Lambda}},{\frac{A_{D}^{\dagger}}{\Lambda}}, \tau_{0}, \tau_{0}^{\dagger},{\frac{M M^{\dagger}}{\Lambda^{2}}},{\frac{\tilde{M}M^{\dagger}}{\Lambda^{2}}},{\frac{\tilde{M}\tilde{M}^{\dagger}}{\Lambda^{2}}}\right) \Bigg |_{D}
D[E,\phi] =[E,\sum_{i=1}^k e_i\cdot \partial_{e_i}\phi + e_y\cdot\partial_{e_y}\phi - \frac{k}{2}e_y\cdot \phi] &=[E,\sum_{i=1}^k y e_i\cdot \partial_i \phi + y e_y\cdot \partial_y \phi -\frac{k}{2}e_y\cdot \phi].
B ({\bf f})^{\left(\underset{{{\{m_1,m\}}}}{\bf t}{{, s_m}},{\bf s}\right)}&= [f_1, f_2]^{({{s_2}},s_1)}&= \displaystyle Ad_{e^{{{s_2}}f_2}e^{s_1f_1}}\left[f_1\,,\,f_2 \right] &= e^{{{s_2}}f_2}e^{s_1f_1}\left[f_1\,,\,f_2 \right]e^{-s_1f_1}e^{-{{s_2}}f_2}{{.}}
\mathrm{Li}_{2}(1-e_{\lambda}(-t)) =-\int_{0}^{1-e_{\lambda}(-t)}\frac{1}{x}\log_{\lambda}(1-x)dx &=\int_{0}^{t}\frac{-t}{e_{\lambda}(-t)-1}e_{\lambda}^{1-\lambda}(-t)dt &=\sum_{n=1}^{\infty}(-1)^{n-1}\beta_{n-1,\lambda}(1-\lambda)\frac{t^{n}}{n!}.
\dot{s}\varepsilon(z) \sum_{x\in \bar{\mathbb{F}}_q}a_x \varepsilon(x) \xi_1 = \sum_{x\in \bar{\mathbb{F}}_q}a_x( \varepsilon(x(z^2-z)^{-1}-z^{-1})+\varepsilon(xz^{-1})-\varepsilon(x(z-1)^{-1}) )\xi_1
\alpha_{i}= \sqrt{\frac{1}{2}\left( 1-\eta_{i}\frac{\gamma}{\sqrt{\gamma^{2}+1}}\right)}, \hspace{0.5cm}\beta_{i}= \eta_{i}\sqrt{\frac{1}{2}\left( 1+\eta_{i}\frac{\gamma}{\sqrt{\gamma^{2}+1}}\right)}.
\frac{n(n-1)(n-2)}{j}\left[y_j(2)-\frac{1}{j+1}\sum_{i=1}^{j+1}h_{j+1}(i)\right] =2(j-1)(n-j)-j^2-(1-2n)j-(n-2)b_n-2-u_n &=-3j^2+(1+4n)j-(n-2)b_n-2(n+1)-u_n &=-f_n(j)\ge0,
C_{l-1}^{\lambda+2}(t)-C_{l-3}^{\lambda+2}(t)=C_{l-1}^{\lambda+2}(1)-C_{l-3}^{\lambda+2}(1)-2(l-2+\lambda+2)\,\int_t^1C_{l-2}^{\lambda+2}(\tilde t)\,d\tilde t.
\partial_{\alpha}f^{p}\left( x_{0}\right) =\left( -1\right)^{p}{\displaystyle\sum\limits_{j=0}^{p-1}}\frac{\left( -1\right)^{j}F_{\alpha}\left( x_{j},x_{j+1}\right)}{F_{y}\left( x_{j},x_{j+1}\right)}{\displaystyle\prod\limits_{i>j}^{p-1}}\frac{F_{x}\left( x_{i},x_{i+1}\right)}{F_{y}\left( x_{i},x_{i+1}\right)}
g_{\dot{x}^{1}\dot{x}^{1}}=\frac{\cos^{2}t}{a}, g_{\dot{x}^{2}\dot{x}^{2}}=\frac{\sin^{2}t}{a}, g_{x^{1}\dot{x}^{2}}=g_{\dot{x}^{1}x^{2}}=0.
H_a = \left\{\left( \begin{array}{cccc}s t_1 t_2 t_3 0 s 0 t_1 0 0 s a t_2 0 0 0 s \end{array}\right) : s \in \mathbb{R}\setminus \{0 \}~,t_1,t_2,t_3 \in \mathbb{R}\right\}~, a \in \{-1,0,1 \}~.
\Big(\sum_{k=0}^n(g(S_k^{{(0)}}(\Delta))-\lambda)\Big)^+ \leq \Big(\sum_{k=1}^r(\omega_k^{{(0)}}+\delta)\Big)^++(\eta_r^{{(0)}}D'-\delta r)^+\leq \Big(\sum_{k=1}^r(\omega_k^{{(0)}}+\delta)\Big)^++D'\eta_r^{{(0)}}{\bf 1}(\eta_r^{{(0)}}\geq \delta' r),
\begin{cases}\,v \circ \gamma \!\!\!\!=\, v\, \gamma(\lambda) \,w \circ \gamma \!\!\!\!=\, \gamma(\lambda)^{-1}w \,X \circ \gamma \!\!\!\!=\, X \,+\, v\, \gamma'(\lambda)\gamma(\lambda)^{-1}w \,Y \circ \gamma \!\!\!\!=\,\, Y \ .\end{cases}
0 = (\boldsymbol{u}(t), \nabla \psi)_{\Omega}= (\boldsymbol{u}_0^c, \nabla \psi)_{\Omega}+ \sum_{k=0}^{\infty}[(\boldsymbol{u}_k^c, \nabla \psi)_{\Omega}\cos(k \omega t) + (\boldsymbol{u}_k^s, \nabla \psi)_{\Omega}\sin(k \omega t)]
\int_{B_{1/2}}|f_k(x)\Phi(x) |\,dx \le \|f_k\|_{L^{N}(B_{1/2})}\|\Phi \|_{L^{N'}(B_{1/2})}&\le \frac{C(N)}{k}\|\Phi \|_{H^{1}(B_{1/2})}\to 0.
\psi_{r_1,r_2}(x_1-y_1,x_2-y_2,x_3-y_3)=r_1^{-2}r_2^{-2}\psi^{(1)}\Big({x_1-y_1\over r_1}\Big)\psi^{(2)}\Big({x_2-y_2\over r_2},{x_3-y_3\over r_1r_2}\Big),
2^{3}6^{2}V^{2}= \left| \begin{array}{c c c}{{2 l_{1}^{2}}}&{{- l_{1 2}^{2}+ l_{1}^{2}+ l_{2}^{2}}}&{{- l_{1 3}^{2}+ l_{1}^{2}+ l_{3}^{2}}}\\{{- l_{1 2}^{2}+ l_{1}^{2}+ l_{2}^{2}}}&{{2 l_{2}^{2}}}&{{- l_{2 3}^{2}+ l_{2}^{2}+ l_{3}^{2}}}\\{{- l_{1 3}^{2}+ l_{1}^{2}+ l_{3}^{2}}}&{{- l_{2 3}^{2}+ l_{2}^{2}+ l_{3}^{2}}}&{{2 l_{3}^{2}}}\\ \end{array}\right|
\mathcal{A}^{c_1, c_2, \ldots, c_n}(t) =A^n_0+A^n_1 e^{-c_nt}+A^n_2 e^{-(c_n+c_{n-1})t}+\ldots+A^n_n e^{-(c_n+\ldots+c_1)t},
\sum_{p_{n+1}\le X}f(d_n)= \sum_{\substack{d_n\le y p_{n+1}\le X}}f(d_n)+\sum_{\substack{p_{n+1}\le X y<d_n\le \log X}}f(d_n)+\sum_{\substack{d_n>\log X p_{n+1}\le X}}f(d_n) = \frac{X}{\log^2X}(1+o(1))\int_{2}^yf(x)\,dx &+O\left(\frac{X\int_{y}^{\log X}f(t)\,dt}{\log^2X}+\frac{X\log \log X}{\log^2X}f(y)\right)+O\left(\frac{Xf(\log X)}{\log X}\right).
[D(1,s+1):D(m,p)]_{q=1}=[D(1,s):D(m,p-1)]_{q=1}+ &+(1-\delta_{2\widetilde{p_0},m}-\delta_{2\widetilde{p_0}.m+1}-\delta_{\widetilde{p_0},m})[D(1,s):D(m,p+1)]_{q=1}&+(1-\delta_{2p_0,m}-\delta_{2p_0,m-1})[D(1,s):D(m,p)]_{q=1}
\nu\iint_{Q_t}|\mathbf{b}_{xx}|^2 \omega_\delta^n dxds =\iint_{Q_t}\mathbf{b}_t\cdot \mathbf{b}_{xx}\omega_\delta^n dxdt+\iint_{Q_t}(u\mathbf{b})_x\cdot \mathbf{b}_{xx}\omega_\delta^n dxds \quad-\iint_{Q_t}\mathbf{w}_x\cdot \mathbf{b}_{xx}\omega_\delta^n dxds.
[ d e t ( \delta_{\mu \nu}+ L_{\mu \nu}) ]^{\frac{1}{2}}= \frac{[ 1 + \frac{1}{8}f_{i j}^{2}+ \frac{1}{1 6}\frac{1}{4 !}\epsilon^{i j k l}\epsilon^{m n p q}f_{m i}f_{n j}f_{p k}f_{q l}]^{2}}{[ - d e t ( \eta_{a b}+ \frac{1}{2}F_{a b}) ]^{\frac{1}{2}}}
\begin{aligned}A_{3}= \frac{1}{(q-1)^2q^4}(q^{m-1}-1)(q^5 - q^m - q^{3 + m}+ q^{4 + m})= \frac{1}{(q-1)^2q^4}(q^{m-1}-1)(q^5 + q^m(q^4-q^3-1))>0 \end{aligned}
AI :=\Big\{\Big({\begin{matrix}~a b~ ~ c d ~ \end{matrix}}\Big)\Big({\begin{matrix}~n_1~ ~ n_2 ~ \end{matrix}}\Big) \, modN = \Big({\begin{matrix}~ an_1 \oplus bn_2~ ~cn_1 \oplus dn_2 ~ \end{matrix}}\Big):(n_1,n_2)^t \in I \Big\}\subset \mathbb{Z}_N^2,
P_p(x_0)\ge h^{K_1}\left(\frac{\varepsilon}l\right)^{K_1}\prod_{i=1}^{K_1}\left(1+\kappa (i-1)\right), P_e(x_0)\ge h^{K_2}\left(\frac{\delta}l\right)^{K_2}\prod_{i=1}^{K_2}\left(1+\kappa (i-1)\right).
Z_{0}= \frac 1{( 2 \pi )^{N}}\int_{M}\mathrm{e}^{H ( x )}\sqrt{d e t \omega_{i j}}d^{2 N}x = \frac 1{( \pi )^{N}}\int_{{\cal M}}\exp ({H ( x ) + \frac 1 2 \omega_{i j}\theta^{i}\theta^{j})}d^{2 N}x d^{2 N}\theta ,
\mu_{\mathrm{I I}}( \eta ) = \frac{1}{n}\sqrt{\frac{\pi}{\epsilon \vert \eta_{\mathrm{i}}\vert}}\cos \biggl ( n \eta - n \eta_{1}+ \frac{2 \pi}{\epsilon}\ln \biggl \vert \frac{\eta_{1}}{\eta_{\mathrm{i}}}\biggr \vert \biggr ) .
\partial_{- 2}\theta^{\alpha 1}= \partial_{- 2}\lambda_{+}^{\alpha 1}\equiv \partial_{- 2}\lambda_{+}^{\alpha}= 0 , \ \partial_{+ 2}\theta^{\alpha 2}= \partial_{+ 2}\lambda_{-}^{\alpha 2}\equiv \partial_{+ 2}\lambda_{-}^{\alpha}= 0 .
\sin\left(\frac{\tilde{p}_{1}\cdot k}{2}\right)\sin\left(\frac{\tilde{p}_{2}\cdot k}{2}\right)\sin\left(\frac{\tilde{p}_{3}\cdot k}{2}\right) = - \frac{1}{4}\left(\sin \tilde{p}_{1}\cdot k + \sin \tilde{p}_{2}\cdot k + \sin \tilde{p}_{3}\cdot k\right)
\frac{\left\langle b\varphi (x)e^{a\cdot \varphi (0)/b}\right\rangle_b}{G(a/b)}=-2a\log r+\frac{\left\langle b\varphi (0)e^{a\cdot \varphi(0)/b}\right\rangle_b}{G(a/b)}+O(r^\sigma ).
b^i_t = \tilde{g}^{ik}\partial_x^{-1}\frac{\delta h}{\delta b^k}= \tilde{g}^{ik}\left( - \frac{\partial h}{\partial b^k_x}+ \partial_x \frac{\partial h}{\partial b^k_{xx}}\right) = w^i_j(\mathbf{b}_x) b^j_{xx}.
\parallel v\parallel_{H^{s+2}(\Omega_f)}+\parallel q\parallel_{H^{s+1}(\Omega_f)}\leqslant C\parallel v_t\parallel_{H^{s}(\Omega_f)}+C\parallel w_{\nu_{\mathcal{N}}}\parallel_{H^{s+\frac{1}{2}}(\Gamma_c)},s=0,1,2
L\left(t,\textbf{x},\dot{\textbf{x}},\Lambda\right)=L_0\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=1}^M\lambda_j\,L_j\left(t,\textbf{x},\dot{\textbf{x}}\right)-\displaystyle\sum_{j=M+1}^{N-1}\lambda^0_j L_j\left(t,\textbf{x}, \dot{\textbf{x}}\right)
\left[ i \gamma_{\mu}\partial_{\mu}+{\frac{\mu_{4}^{2}c}{\mu_{3}\hbar}}{\frac{( 1 + \gamma^{5})}{2}}+{\frac{\mu_{3}c}{\hbar}}{\frac{( 1 - \gamma^{5})}{2}}\right] \tilde{\psi}= 0 .
F^{\lambda_{1}\lambda_{2}, \lambda_{3}\lambda_{4}, \lambda_{5}\lambda_{6}}F_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}, \lambda_{3}^{\prime}\lambda_{4}^{\prime}, \lambda_{5}^{\prime}\lambda_{6}^{\prime}}= \left( \delta ( \lambda_{1}^{\prime}- \lambda_{1}) \delta ( \lambda_{2}^{\prime}- \lambda_{2}) , \right.
S [ B_{( p + 2 )}] = \int d^{d}x \ \sqrt{| g |}\ \left[{\textstyle \frac{( - 1 )^{( p + 2 )}}{2 \cdot ( p + 3 ) !}\frac{1}{e_{( p + 1 )}^{2}}}H_{( p + 3 )}^{2}+{\textstyle \frac{( - 1 )^{( p + 1 )}}{2 \cdot ( p + 2 ) !}\frac{1}{e_{( p )}^{2}}}B_{( p + 2 )}^{2}\right] \, ,
u(x,z) = \frac{e^{kx}}{2k}\left( 1 + \frac{A_1}k + \frac{A_2}{k^2}+O(\lvert k\rvert^{-3}) \right), v(x,z) = \frac{e^{kx}}{2}\left( 1 + \frac{B_1}k + \frac{B_2}{k^2}+ O(\lvert k\rvert^{-3}) \right),
\rho^\star(\theta^\sharp,X) =D^g\theta(J\theta^\sharp,JX)+\frac{1}{2}g(D^g_{JX}\theta,J\theta)+\frac{1}{2}g(D^g_X\theta,\theta)+g(d\theta,N_X), =\frac{1}{2}D^g\theta(J\theta^\sharp,JX)+\frac{1}{2}g(D^g_X\theta,\theta)+g(d\theta,N_X), =-\frac{1}{2}(d\theta)^{J,-}(\theta^\sharp,X)+g(d\theta,N_X).
e_H(A'_0 \cup A)-e_H(B'_0 \cup B) = e_H(A') +d_H(v,A') -(e_H(B')-d_H(v,B')) & = (|A'|-|B'|)D/2 +D =(|A'_0 \cup A|-|B'_0 \cup B|)D/2.
\sum_{x_s \in \mathcal{A}}\int_0^1 \Big|\prod_{i=1}^{s-1}U_{\mathcal{A}-x_s}(d_i \alpha) - \prod_{i=1}^{s-1}\delta U_{[1,N]-x_s}(d_i \alpha)\Big|\,d \alpha \gg \delta^{s}N^{s-2}- \delta N.
\bar{N}_{0 0}^{r s}= \delta^{r s}+{\frac{\sqrt{\alpha_{r}\alpha_{s}}}{\alpha_{3}}}- \mu \sqrt{\alpha_{r}\alpha_{s}}\, \epsilon^{r t}\epsilon^{s u}\alpha_{t}\alpha_{u}B^{T}\Gamma_{+}^{- 1}B \quad r , s , t , u = 1 , 2
{\cal R}( 0 ) = \rho \sin \eta \left[ \begin{array}{c c c c}{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}\\ \end{array}\right] = \rho \sin \eta \cdot P ,
\liminf_{n\to\infty}\frac{1}{b_c^{(n)}}\log P\left(\frac{n-A_n^*}{f_2(n)}\in O\right) \geq-\inf_{x\in O\cap (0,\ell_2^{-1})}H(\ell_2 x)\wedge\inf_{z\in O\cap (\ell_2^{-1},+\infty)}H(\ell_2z) &=-\inf_{y\in O\cap [0,\infty)}H(\ell_2 y)=-\inf_{y\in O}I_2(y),
\sigma(x_{[5]}) = \sum_{i \in [q], j \in [t]}\lambda_{ij}\Big(\sum_{\{a,b,c\}\in \binom{[4]}{3}}\tau_{i}(x_a, x_b, x_c) \tilde{\gamma}_i(x_{[4] \setminus \{a,b,c\}}, x_5)\Big) &+\sum_{i \in [s_3]}\Big(\sum_{\{a, b\}\in \binom{[4]}{2}}\tilde{\beta}''_{i}(x_a, x_b, x_5) \gamma''_i(x_{[4] \setminus \{a,b\}})\Big).
\left( \nabla^{2}-{\frac{[ 1 - K ( r ) ]}{r^{2}}}\sigma^{a}l^{a}-{\frac{[ 1 - K ( r ) ]^{2}}{2 r^{2}}}\right)_{B}^{A}\Phi^{B}( x ) = - ( E^{2}- m^{2}) \Phi^{A}( x )
\frac{1}{g_{a}^{2}( p^{2})}= \frac{k_{a}}{g_{a}^{2}( M_{X}^{2})}+ \frac{b_{a}}{1 6 \pi^{2}}\log \frac{M_{X}^{2}}{p^{2}}+ \frac{\Delta_{a}}{1 6 \pi^{2}}+ \frac{{\tilde{\Delta}}_{a}}{1 6 \pi^{2}},
F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}=[\begin{matrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{matrix}]
F(w,\mathfrak{p}, \mathfrak{p}', \mathfrak{p}'') =\lambda \mathfrak{p}-\inf_{\pi}\left\{\frac{1}{2}\sigma^2\left(\varepsilon (\mathfrak{p}')^2+\mathfrak{p}''\right)\pi^2+(\mu-r)\mathfrak{p}'\pi+\left(rw-c\right)\mathfrak{p}'\right\}&=\lambda \mathfrak{p}\geq 0.
\Gamma\left(\frac{T_+}{T_-}\right) = \alpha_-\frac{T_+}{T_-}+ 2 \alpha_-\left(\frac{2T_-}{T_+}+ \frac{T_{\rm i}}{T_+}\right)\frac{T_+}{T_-}+ 3\left(1 - \frac{T_+}{T_-}\right)> 0.
{\bf P}\bigl[\beta_{M,N}(a,b)=1\bigr] = \prod\limits_{n_1,\cdots ,n_M=0}^\infty \Bigl[\frac{b_0+\Omega}{\prod\limits_{j_1=1}^N b_0+b_{j_1}+\Omega}\frac{\prod\limits_{j_1<j_2}^N b_0+b_{j_1}+b_{j_2}+\Omega}{\prod\limits_{j_1<j_2<j_3}^N b_0+b_{j_1}+b_{j_2}+b_{j_3}+\Omega}\cdots\Bigr].
\sum_{0\le t<n,t\neq i-1}\!\!\!\!{\dim}_F(\{f_{i,j}(\beta^{r^t})\}_{j\in[l]}) \le (n-1)\frac{l}{r}+(r-1)\sum_{t=0}^{i-2}\frac{l}{r^{i-t}}+(r-1)\sum_{t=i}^{n-1}\frac{l}{r^{n-t+1}}&=l\Big(\frac{n-1}{r}+\frac{r^{i-1}-1}{r^i}+\frac{r^{n-i}-1}{r^{n-i+1}}\Big) &< \,l\frac{n+1}{n-k}.
\frac 1{r^{2}}\frac d{d r}\left( r^{2}\frac{d R_{q , l}}{d r}\right) - \frac{l ( l + 1 ) - q^{2}}{\alpha^{2}r^{2}}R_{q , l}- \frac{2 M}{\alpha^{2}}\left( \frac K r - E \right) R_{q , l}= 0 \ .
\overline{N}(a^{\ast})J_{1}=0 \overline{N}(a^{\ast})J_{2}=c_{k}^{-1}\pi_{e_{k}a}i \overline{N}(a^{\ast})J_{3}=c_{k}^{-1}\pi_{e_{k}a}j'\sigma_{2}\overline{N}(a^{\ast})J_{4}=0
\det D_{S_{\gamma}}^{2}=2^{2\zeta_{\Sigma_{S_{\gamma}}}(0)}\tau(X_{I}^{\iota},\Sigma_{S_{\gamma}};\gamma|_{X_{I}^{\iota}})^{-2}=2^{\frac{g-1}{3}}\tau(X_{I}^{\iota},\Sigma_{S_{\gamma}};\gamma|_{X_{I}^{\iota}})^{-2}.
\mathbb{P}\left[\textbf{e}(\omega) \in dx\right]=\frac{2x^2 \sqrt{(z-y)^3}}{\sqrt{2\pi (z-\omega)^3(\omega-y)^3}}e^{-\frac{x^2}{2(z-\omega)}-\frac{x^2}{2(\omega-y)}}dx
S_{\mathrm{c l}}= \mathrm{t r}\left( \hat{\bar{\sigma}}^{\dot{\alpha}}\bar{D}_{\dot{\alpha}}D_{\alpha}\hat{\sigma}^{\alpha}\right) + \mathrm{t r}\left({\hat{\bar{\sigma}}}^{\dot{\alpha}}\bar{D}_{\dot{\alpha}}\left( e^{- V}- 1 \right) D_{\alpha}{\hat{\sigma}}^{\alpha}\right) .
E_{\theta}\sum_{i=1}^{n}Var(\theta_{i}|X_i) \lesssim \tilde{q}_n\log\big(\frac{1}{\tau^{2a}}\big)+(n-\tilde{q}_n)\tau^{2a}\sqrt{\log\big(\frac{1}{\tau^{2a}}\big)}\mbox{as}\tau \rightarrow 0.
\mathbb E |E|^{K}= \mathbb E 1 \cdot |E|^n \cdot |E|^{1/a_1}\cdot |E|^{1/a_2}\cdots & \le \left(\mathbb E 1^r\right)^{1/r}\cdot \left(\mathbb E |E|^{2n}\right)^{1/2}\cdot \left(\mathbb E |E|^2\right)^{1/2a_1}\cdots & = \left(\mathbb E |E|^{2n}\right)^{1/2}\cdot \left(\mathbb E |E|^2\right)^{(K-n)/2}
(b_{14}^2+b_{23}^2)\cos^2(\phi-\psi) = 2(b_{14}+b_{23})^2c_{\phi}c_{\psi}s_{\phi}s_{\psi}+ \big{(}b_{13}^{(1)}\big{)}^2+\big{(}b_{24}^{(1)}\big{)}^2 \leq \frac{1}{2}(b_{14}+b_{23})^2 + \delta_1^2S^2(B).
\begin{array}{lll}\lim_{m\rightarrow\infty}\frac{h^0(Z,\mathcal O_{mA})}{m^d}= \lim_{m\rightarrow \infty}\frac{\chi(\mathcal O_{mA})}{m^d}&= \lim_{m\rightarrow \infty}\frac{\chi(\mathcal O_Z)-\chi(\mathcal O_Z(-mA))}{m^d}&= \lim_{m\rightarrow\infty}\frac{-\chi(\mathcal O_Z(-mA))}{m^d}&= \frac{-((-A)^d)}{d!},\end{array}
[\widetilde L]_s - [\widetilde L]_u = ([L]\circ \tau)_s - ([L]\circ \tau)_u \leq ([L]\circ \tau)_s + \tau_s - ([L]\circ \tau)_u - \tau_u = ([L]_{\tau_s}+ \tau_s) - ([L]_{\tau_u}+ \tau_u) = s-u.
f_n\left(\frac{n+4}{3}\right) =-n^2-3n+4+(n-2)b_n+2(n+1)+u_n &>-n^2-3n+4+(n-2)b_n+2(n+1)+(b_n-2)(n-2)\;\;\mbox{(by (\ref{ae6}))}&=(n-2)\left(2b_n-(n+5)\right)>0\;\;\mbox{(by (\ref{ae8})).}
\left( \mu \frac{\partial}{\partial \mu}+ \beta \left( g^{2}\right) \frac{\partial}{\partial g^{2}}- \gamma_{2}\left( g^{2}\right) \int d^{2}x J \frac{\delta}{\delta J}+ \eta \left( g^{2}, \zeta \right) \frac{\partial}{\partial \zeta}\right) E = 0
\|w\|_{L^\infty(\Omega)}\leq C\|u\|_{L^p(\Omega)}\leq C\left(\int_\Omega \gamma(v)u^2dx\right)^{1/2}\left(\int_\Omega \gamma^{-\frac{p}{2-p}}(v)dx\right)^{\frac{2-p}{2p}}.
E_{\pm}(q_{0},h^{2})=-\frac{h^{8}}{2^{5}c^{2}}+\frac{1}{\sqrt{2}}q_{0}h^{2}-\frac{c^{2}(3q_{0}^{2}+1)}{2h^{4}}-\frac{\sqrt{2}c^{4}q_{0}}{8h^{10}}(17q_{0}^{2}+19)+O(\frac{1}{h^{16}})
\lfloor\frac{-2mp^i}{q-1}\rfloor+\lfloor\frac{mdp^i}{q-1}\rfloor+\lfloor\frac{-m(d-1)p^i}{q-1}\rfloor-\lfloor\frac{-mp^i}{q-1}\rfloor+1 &=\sum_{h=1}^{d-2}\lfloor\langle\frac{hp^i}{d-1}\rangle-\frac{mp^i}{q-1}\rfloor+\sum_{\substack{h=1 h\neq\frac{d}{2}}}^{d-1}\lfloor\langle \frac{-hp^i}{d}\rangle+\frac{mp^i}{q-1}\rfloor.
\phi_{\tau}-d\phi_{xx}= \sigma(u_n)f(\tau,x,u_n)-\bar{f}(W) &\leq \sigma(u_n) f(\tau,x, u_n)-\sigma(W)f(\tau,x, W) &=b(\tau,x)\phi\;\; \mbox{for $(\tau,x)\in \Omega_n$.}
s y_1 \dots y_{i-1}= y_1 \dots y_{i-1}s + [s, y_1 \dots y_{i-1}] = y_1 \dots y_{i-1}s + \sum_{j=1}^{i-1}y_1 \dots y_{j-1}[s, y_j] y_{j+1}\dots y_{i-1}
H_{D}^{F}= \int{d^{3}\sigma \Big [ \lambda^{\mu}( \tau , \vec{\sigma}){\cal H}_{\mu}^{\ast}( \tau , \vec{\sigma}) - A_{\tau}( \tau , \vec{\sigma}) \Gamma ( \tau , \vec{\sigma}) + \mu_{\tau}( \tau , \vec{\sigma}) \pi^{\tau}( \tau , \vec{\sigma}) \Big ]}.
D \ge e^{t(1-q)^2}\left(1+ tq^2 \right) q(s-q) -\left(1- tq(1-q) + \frac{1}{2}t^2 q^2(1-q)^2 \right)^2 q(s-q) = \beta_0 + \beta_1 t + \beta_2 t^2+ \beta_3 t^3 + \beta_4 t^4 \, ,
{\mathcal D}_0^{-1}\simeq \widetilde{\phi}_1\phi^{\dagger}_1\lambda_1 P_L - \widetilde{\phi}_1 \widetilde{\phi}^{\dagger}_1\widehat{/\!\!\!\partial}P_R + \phi_1\widetilde{\phi}^{\dagger}_1 \lambda_1 P_R - \phi_1\phi^{\dagger}_1 \widehat{/\!\!\!\partial}P_L -\phi_0\phi^{\dagger}_0 \widehat{/\!\!\!\partial}P_L.
\mathrm{Mant}_{G,b,\mu}(e(J_b)LJ( \delta^{\frac{1}{2}}_{G,P_b}I^{M_b}_{M_S}(\rho)))=\mathrm{Mant}_{G,b, \mu}(\mathrm{Red}_b(I^G_{M_S}(\rho))).
\int d^{6}\xi \left( \sum_{a = 0}^{5}\left| ( \partial_{a}+ i A_{a}- i A_{a}^{\prime}) \chi \right|^{2}+ \biggr ( \frac{1}{4 g_{\mathrm{Y M}, p}^{2}}+ \frac{1}{4 g_{\mathrm{Y M}, p^{\prime}}^{2}}\biggl ) \sum_{I = 1}^{3}( \chi^{\dagger}\tau^{I}\chi )^{2}\right) \ ,
\mathrm{R e}~ a_{i}= \mathrm{R e}~ d_{i}, ~ ~ \mathrm{I m}~ a_{i}= - \mathrm{I m}~ d_{i}, ~ ~ \mathrm{R e}~ b_{i}= \mathrm{R e}~ c_{i}, ~ ~ \mathrm{I m}~ b_{3}= - \mathrm{I m}~ c_{i}( i = 2 , 3 )
{\frac{2 B}{a^{3 \gamma + 2 n}}}+{\frac{3 ( 3 \gamma + 4 n - 2 ) k c_{0}^{2}}{4 \pi G ( 3 \gamma + 2 n - 2 ) a^{2}}}\leq{\frac{( 3 \gamma + 4 n ) c_{0}^{2}\Lambda}{4 \pi G ( 3 \gamma + 2 n )}}.
\int_{-\infty}^{\infty}\exp\left(\pm i \omega(4\pi y_2^{1/2}\sqrt{1+u^{-2}}, \rho )\right) \exp\left( \pm i \omega(4\pi y_1^{1/2}\sqrt{1+u^{2}}, \rho)\right) & \times \tilde{h}(\mu) f^{\pm}_M\left(4\pi y_2^{1/2}\sqrt{1+u^{-2}}, \rho\right) f^{\pm}_M\left(4\pi y_1^{1/2}\sqrt{1+u^{2}}, \rho \right) (\mu) d\rho,
\limsup_{\eta\rightarrow0}\limsup_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}_{N}\exp\left\{aNA^{N}(\eta)\right\}\le0 \limsup_{\eta\rightarrow0}\limsup_{\epsilon\rightarrow0}\limsup_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}_{N}\exp\left\{a\sum_{i=1}^{N}\int_{0}^{\eta}\rho_{\epsilon,i}\left(x^{N}(t)\right)dt\right\}\le0
P=\left(\left(\left(I_n\otimes y_2^{j_2}\right)\left(I_n\otimes y_1^{j_1}\right)p(x,y)\left(I_n\otimes x_1^{k_1}\right)\left(I_n\otimes x_2^{k_2}\right)\right)_{j_1,k_1=-N}^N \right)_{j_2,k_2=-M}^M