harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
B_1: = \bigg(\int_0^{\infty}\bigg(\int_x^{\infty}[\underline{u}]^q(t) w(t)\,dt\bigg)^{r / p}[\underline{u}]^q(x) V_*^{- r / p}(x) w(x)\,dx \bigg)^{1 / r}, B_2: = \bigg(\int_0^{\infty}W^{r / p}(x) \bigg[\sup_{x \le \tau < \infty}\underline{u}(\tau)V_*^{-1 / p}(\tau)\bigg]^{r}w(x)\,dx \bigg)^{1 / r}.
\frac{\sigma^{2}}2 p^{( + )}= p^{( + )}\frac{\sigma^{2}}2 = p^{( 1 )}, \hspace{0 . 3 i n}\frac{\sigma^{2}}2 p^{( - )}= p^{( - )}\frac{\sigma^{2}}2 = p^{( \widetilde{1})},
l^{\prime}(\beta)=\sum_{j}(\sum_{i\in H_{j}}X_{i}-\sum_{l=0}^{m_{j}-1}\frac{\sum_{i:Y_{i}\ge t_{j}}\theta_{i}X_{i}-\frac{l}{m_{j}}\sum_{i\in H_{j}}\theta_{i}X_{i}}{\sum_{i:Y_{i}\ge t_{j}}\theta_{i}-\frac{l}{m_{j}}\sum_{i\in H_{j}}\theta_{i}}),
\Gamma = W - \int d^{4}x \, d^{4}\theta \, J V - \int d^{4}x \, d^{2}\theta \, \Big ( j \, \phi + \tilde{j}\, \tilde{\phi}\Big ) - \int d^{4}x \, d^{2}\bar{\theta}\, \Big ( j^{*}\phi^{*}+ \tilde{j}^{*}\tilde{\phi}^{*}\Big ) ,
\Big(\frac{1}{n}\sum_{i=1}^n\sum_{t=1}^T\|[g_{t}(x_{i,t})]_+\|\Big)^2\le\varepsilon_3T+\frac{n\varepsilon_4KT}{\gamma_0}+\frac{n\varepsilon_4(F\sigma+2G_1^2)}{\sigma\gamma_{0}(1-c)}T^{2-c}.
\lim \limits_{P_0 \rightarrow \infty}P_s(E_{12})= \frac{1}{2}\left\lbrace1- \sqrt{\frac{\bar{\gamma}|d_{\mathrm{min}}|^2}{\bar{\gamma}|d_{\mathrm{min}}|^2+2}}\sum \limits_{l=0}^{R}\binom{2l}{l}\left( \frac{1}{4+2\bar{\gamma}|d_{\mathrm{min}}|^2}\right)^l\right\rbrace
\delta_{\xi}\left< T_{z \overline{z}}\left( z , \overline{z}\right) \right> ={\frac{1}{9 6}}\left( 2 \partial_{z}^{2}\partial^{z}\xi^{z}\left( z , \overline{z}\right) + 2 \partial_{\overline{z}}^{2}\partial^{\overline{z}}\xi^{\overline{z}}\left( z , \overline{z}\right) \right) .
\phi ( x ) = \langle I N | \tilde{T}( e^{- i \int^{t}d^{4}x^{\prime}J ( x^{\prime}) \Phi ( x^{\prime})}) \Phi ( x ) T ( e^{i \int d^{4}x^{\prime \prime}J ( x^{\prime \prime}) \Phi ( x^{\prime \prime})}) | I N \rangle |
x\hat{K_{i}}+a(1-\hat{K_{i}})\stackrel{\tau}{\mapsto}(\sum_{j=0}^{p_{i}^{n_{i}}-1}x_{j}, \sum_{j=0}^{p_{i}^{n_{i}}-1}x_{j}\zeta_{p_{i}}^{j},...,\sum_{j=0}^{p_{i}^{n_{i}}-1}x_{j}\zeta_{p_{i}^{n_{i}}}^{j},a(1-\hat{K_{i}})),
\phi\left (\bigotimes\limits_{i=1}^{m-1}X_iE_{j_ij_i}X_i^*\otimes B\right )=W_{X}\left (\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes\varphi_{j_1,\ldots,j_{m-1},X}(B)\right )W_{X}^*
\sum_{i=1}^{n}E_{\theta_{i}}\big(T_{\tau}(X_{i})-\theta_{i}\big)^2= \sum_{i:\theta_i\neq 0}E_{\theta_{i}}\big(T_{\tau}(X_{i})-\theta_{i}\big)^2 + \sum_{i:\theta_i=0}E_{\theta_{i}}\big(T_{\tau}(X_{i})-\theta_{i}\big)^2
\left((1-\frac{1}{\gamma^2}\partial_x^2) v(t)\right) \circ \varphi(t) \cdot \varphi_x(t)^2 = (1-\frac{1}{\gamma^2}\partial_x^2) v_0 + \int_0^t \frac{3\gamma-3}{\gamma}\frac{\varphi_t(s) \varphi_{tx}(s)}{\varphi_x(s)}\;ds
[a+\epsilon/2,b] \subset\cup_{i=1}^{k}(a_{i},b_{i}+\epsilon/2^{i+1}) \textit{i,e.,}~b-a-\epsilon/2 \leq \sum_{i=1}^{k}(b_{i}-a_{i})+\epsilon\sum_{i=1}^{k}\frac{1}{2^{i+1}}<\sum_{i=1}^{k}(b_{i}-a_{i})+\epsilon/2 \textit{i,e.,}~~~~~~b-a< \sum_{i=1}^{\infty}(b_{i}-a_{i})+\epsilon.
\pi^{-1}([1 : 0]) = \{[1:0]\}\times g \cdot X_{i,j}^{k,l}\pi^{-1}([0: 1]) =\{[0:1]\}\times Y_{i+k-n-1,j+l-1-i-k+n}^{i+k-n-1}\cup \{[0:1]\}\times Y_{i+k-n,j+l-1-i-k+n}^{i+k-n}
{\mit \Gamma}_{\mathrm{c u s p}}( \theta ; \alpha_{s}) = - \frac{\alpha_{s}N_{c}}{\pi}\frac{1}{\sqrt{1 + x}}\left\{\sqrt{1 + x}- \left( 1 + \frac{x}{2}\right) \ln \frac{\sqrt{1 + x}+ 1}{\sqrt{1 + x}- 1}\right\}\, ,
\partial_{\pm 2}\alpha_{A B}= - \frac 1 4 \alpha_{A C}\Omega_{\pm 2}^{( i ) ( j )}\gamma_{C B}^{( i ) ( j )}, \p a r t i a l_{\pm 2}\beta_{\dot{A}\dot{B}}= - \frac 1 4 \beta_{\dot{A}\dot{C}}\Omega_{\pm 2}^{( i ) ( j )}\tilde{\gamma}_{\dot{C}\dot{B}}^{( i ) ( j )}.
(*)\begin{cases}c_{n-1}=t b_{k-1}c_{n-2}=tb_{k-2}+{t \choose 2}b_{k-1}^2 \ \vdots c_{n-k+1}=tb_{1}+\sum\limits_{i_1+2i_2+\cdots+(k-2)i_{k-2}=k-1}^{}d_{i_1, i_2, \ldots, i_{k-2}}\b_{k-1}^{i_1}b_{k-2}^{i_2}\ldots b_{2}^{i_{k-2}},\end{cases}
[ \hat{D}_{\alpha}, \hat{D}_{\beta}] f = - ( \partial_{\alpha}A_{\beta}^{\mu}- \partial_{\beta}A_{\alpha}^{\mu}) \partial_{\mu}f + ( A_{\alpha}^{\mu}\partial_{\mu}A_{\beta}^{\nu}- A_{\beta}^{\mu}\partial_{\mu}A_{\alpha}^{\nu}) \partial_{\nu}f .
T_{w(1,m)}(E_{N,k}(-s)) = \begin{cases}(-1)^{m-1}E_{2,1}(-s+m-1) , (-1)^{(m-1)N}(- E_{2,3}(-s+2m-2)) , (-1)^{m-1}(- E_{2,k+1}(-s+m-1)) , E_{2,2}(-s) + \delta_{s,0}(m-1)c .\end{cases}
\begin{aligned}(\rho_\alpha-\alpha (k+d-2)) x_{v}\leq(\rho_\alpha-\alpha d(v)) x_{v}= (1-\alpha)(k-1)x_{u_1}+(1-\alpha)\sum_{uv\in E(H)}x_u \leq(1-\alpha)(k-1)x_{u_1}+(1-\alpha)(d-1)x_v, \end{aligned}
\psi (\theta^a) = \sum_{i=0,1,..,3,5,..,d}\quad \sum_{\{a_1< a_2<...<a_i\}\in \{0,1,..,3,5,..,d\}}\alpha_{a_1, a_2,...,a_i}\theta^{a_1}\theta^{a_2}\cdots \theta^{a_i}.
\Phi(\tau,\theta,\overline{\theta})=Z(\tau)+\theta^a\chi_a(\tau)+\frac{i}{2}\overline{\theta}\theta\dot{Z}(\tau)+\theta\theta F(\tau)-\frac{i}{4}\theta\theta \overline{\theta}_a\dot{\chi}^a-\frac{1}{16}\overline{\theta}\overline{\theta}\theta\theta\ddot{Z}(\tau)
\Phi_{1}^{( 1 )}( u_{1}, g ) = 2 u^{2}\Phi_{0}^{( 1 )}( u_{1}, g ) \Phi_{1}^{( 1 )}( u_{1}, g ) + g u_{1}\partial_{u_{1}}^{2}\Phi_{1}^{( 1 )}( u_{1}, g ) + u_{1}^{2}\Phi_{0}^{( 2 )}( u_{1}, u_{1}, g ) \ .
\Big|\sum_{j=1}^N \frac{\lambda_j}{\pi}Re\Big\{\frac{D_tZ-\dot{z}_j}{c_0^j(\alpha-\omega_0^j)^2}\Big\}\Big|\leq \frac{\sum_{j=1}^N |\lambda_j|}{\pi}\max_{1\leq j\leq N}|c_0^j|^{-1}(\|D_tZ\|_{\infty}+|\dot{z}_j|)(\inf_{\alpha\in \mathbb{R}}|\alpha-\Phi(z_j)|)^{-2}
\mathcal{M}=\biggl\{\begin{pmatrix}3 3 4 2\end{pmatrix},\begin{pmatrix}3 3 2 2\end{pmatrix},\begin{pmatrix}0 3 4 2\end{pmatrix},\begin{pmatrix}0 3 2 2\end{pmatrix}\biggr\}\,,
\varpi_{[a,i]}\cdot \varpi_{[i+1,b]}= \frac{1}{(i-a+1)!}\cdot \frac{1}{(b-i)!}\cdot \varpi_{a}\varpi_{a+1}\cdots\varpi_{b}= \binom{b-a+1}{i-a+1}\varpi_{[a,b]}
\left(\mu_{\rm b}{\bf I}_{N_{\rm T}}+{\bf Q}_{\rm s}- \gamma_{\rm b}{\bf Q}_{\rm n}\right)\bar{\bf H}_{\rm b}{\boldsymbol\Psi}_{\rm b}\bar{\bf H}_{\rm b}^H = \mu_{\rm b}\left[{\bf 0}_{N_{\rm T}}\,\, \hat{\bf h}_{\rm b}\right]{\boldsymbol\Psi}_{\rm b}\bar{\bf H}_{\rm b}^H.
X_{\mu \nu}^{\bot}X_{\rho \sigma}^{\bot}= \frac{1}{2}\left( \, \pi_{\mu \nu}\pi_{\sigma \rho}- \pi_{\mu \sigma}\pi_{\nu \rho}- \pi_{\mu \rho}\pi_{\nu \sigma}\, \right) Q_{0}^{2}+ \frac{i}{4}\left( \, \pi_{\mu \sigma}\varepsilon_{\nu \rho \lambda}+ \pi_{\nu \rho}\varepsilon_{\mu \sigma \lambda}\, \right) e^{( 0 ) \lambda}Q_{0}.
\psi_\lambda(x,n,y)=\begin{cases}n \textnormal{when}(x,n,y)\in Y_\lambda -n \textnormal{when}(x,n,y)\in \mathcal{G}^{0}_A\setminus Y_\lambda -|n|-\kappa(x,n,y) \textnormal{when}(x,n,y)\in \mathcal{G}_A\setminus \mathcal{G}^0_A, \end{cases}.
\lim \limits_{r\rightarrow0}\sup \limits_{x\in{\mathbb{R}^{n}}}\varphi_{1}(x,r)^{-1}\, \left \Vert f\right \Vert_{L_{p}\left( B\left( x,r\right)\right)}=0\lim \limits_{r\rightarrow0}\sup \limits_{x\in{\mathbb{R}^{n}}}\varphi_{2}(x,r)^{-1}\, \left \Vert T_{\Omega}f\right \Vert_{L_{p}\left( B\left( x,r\right) \right)}=0.
\Bigl \{b^{r}( k , \sigma ) , b^{s \dagger}( p , \sigma^{\prime}) \Bigr \}= \Bigl \{d^{r}( k , \sigma ) , d^{s \dagger}( p , \sigma^{\prime}) \Bigr \}= 1 6 \pi^{3}k^{+}\delta_{r s}\delta_{\sigma \sigma^{\prime}}\delta^{3}( k - p ) \; .
\begin{cases}\displaystyle\frac{d}{dt}|H|_{D^{2,q}}\leq C(|\nabla u|_\infty+F+1)(1+|\rho|_{D^{2,q}}+|P|_{D^{2,q}}+|H|_{D^{2,q}})+F, begin{align*}8pt]\displaystyle\frac{d}{dt}|P|_{D^{2,q}}\leq C(|\nabla u|_\infty+F+1)(1+|\rho|_{D^{2,q}}+|P|_{D^{2,q}}+|H|_{D^{2,q}})+F.\end{cases}
\begin{aligned}\xi_{A \circ B}(s_n,f_n) = \frac{1}{\log(s_n)}\int_{f_n}^{s_nf_n}\rho_A(y) \frac{\rho_B(y \rho_A(y))}{y}\mathrm{d}y & \sim \lambda(A) \frac{1}{\log(s_n)}\int_{f_n}^{s_nf_n}\frac{\rho_B(\lambda(A) y)}{y}\mathrm{d}y.\end{aligned}
\Phi := \begin{pmatrix}(a_{1,1}, b_{1,1}) (a_{1,2}, b_{1,2}) \ldots (a_{1,K}, b_{1,K}) (a_{2,1}, b_{2,1}) (a_{2,2}, b_{2,2}) \ldots (a_{2,K}, b_{2,K}) \vdots \vdots \vdots \vdots (a_{L,1}, b_{L,1}) (a_{L,2}, b_{L,2}) \ldots (a_{L,K}, b_{L,K})\end{pmatrix}
(x\ast y) \ast \alpha^{2}(z)-\alpha^{2}(x)\ast (y\ast z)= \alpha^{2}((xy)\alpha(z)-\alpha(x)(yz)) = (-1)^{|x||y|}\alpha^{2}((yx)\alpha(z)-\alpha(y)(xz)) = (-1)^{|x||y|}((y\ast x) \ast \alpha^{2}(z)-\alpha^{2}(y)\ast (x\ast z)),
K_{\nu/2}(\sqrt{\nu}|\omega|) = \frac{2^{\frac{\nu-2}{2}}\Gamma\left(\frac{\nu+1}{2}\right)}{(\sqrt{\nu}|\omega|)^{\frac{\nu}{2}}\sqrt{\nu \pi}}\int_{\mathbb{R}}\left(1+\tfrac{n^2}{\nu}\right)^{-\tfrac{\nu+1}{2}}e^{i\omega n}~\mathrm{d}n.
d\psi_1'(z)-\frac{c}{2}\psi_1(z) =dy'(z)-cy(z)+\frac{c^2}{4d}v(z) begin{align*}0.2cm] =\left(\frac{sv(z)}{q(u(z))}+\frac{c^2}{4d}-s\right)v(z) begin{align*}0.2cm] \geq\frac{sv^2(z)}{q(u(z))}>0.
\frac{\partial}{\partial\overline{\tau}}\chi_{\gamma}(\overline{\tau})=\epsilon\left(f_{-}^{-1}\circ\gamma\right)\rho(\gamma)\nu f_{-}-\epsilon \left(f_{-}^{-1}\circ\gamma\right)(\nu\circ\gamma)\overline{\gamma'}\rho(\gamma)f_{-}\equiv 0.
[x^{+}\otimes a^{-},s\otimes c] =x^{+}\diamond s\otimes\frac{[a^{-},c]_{C}}{2}+[x^{+},s]\otimes\frac{(a^{-}\circ c)_{E}}{2},{}[x^{-}\otimes a^{+},s\otimes c] =x^{-}\diamond s\otimes\frac{[a^{+},c]_{E}}{2}+[x^{-},s]\otimes\frac{(a^{+}\circ c)_{C}}{2}.
a_{0}( x ) = \lambda +{\frac{2}{\sqrt{L_{1}L_{2}}}}\sum_{n_{1}, n_{2}}[{\frac{g ( \vec{p})}{\mu}}\sin p_{1}x_{1}\sin p_{2}x_{2}e^{- i p^{0}x^{0}}+ c . c ]_{p^{0}= | \vec{p}|}\, .
\langle \Phi^\ell_1 \Phi^{\ell'}_2 \rangle = \frac{\sqrt{(2\ell+1)(2\ell'+1)}}{4\pi^2}\int_{|t|<1}\frac{dtd\bar t}{(1-|t|^2)^2}\int^{2\pi}_{-2\pi}d\psi\ \Phi^\ell_1 \bar\Phi^{\ell'}_2 \, ,
r^{lk}(L):=\max_{\underline{a}\in L}r^{lk}(\underline{a}), \Theta^{lk}_{L}(\underline{a}):=\mbox{ad}^{r^{lk}(L)}T^{lk}(\underline{a})S^{lk}(\underline{a}), d^{lk}_{L}:=\mbox{rk}\,\Theta^{lk}_{L}(\underline{a}).
\left(z_{4}sz_{1}s\right)^{-1}z_{4}z_{1}=s^{-1}z_{1}^{-1}s^{-1}z_{1}\mbox{and}z_{4}\left(z_{4}sz_{1}s\right)^{-1}z_{1}=z_{4}\left(s^{-1}z_{1}^{-1}s^{-1}\right)z_{4}^{-1}z_{1}.
| \mathrm{i n}\rangle = | p \k , s \rangle = b^{*}( p ) \a^{*}( k , s ) | 0 \rangle \qquad \qquad | \mathrm{o u t}\rangle = | p^{\prime}\k^{\prime}, s^{\prime}\rangle = b^{*}( p^{\prime}) \a^{*}( k^{\prime}, s^{\prime}) | 0 \rangle
N_0(L,S,U) & = C_2^S C_0^UC_0^L +C_0^U(C_1^S)^2 + 2C_0^UC_1^UC_1^S + (C_1^U)^2 & - C_0^S(C_1^U)^2 + C_2^S(1-C_0^S)\sigma_v^2/A^2 + (C_1^S)^2\sigma_v^2/A^2
\frac{\varepsilon}{\lambda_k^{\beta_k}}\lambda_k(y^k_i - z^k_i) = O\left(\varepsilon\displaystyle\sum_{j = 1}^2\left(\frac{1}{\lambda_j^{\beta_j + \sigma}}+ |y^j - z^j|^{\beta_j + \sigma}\right)\right) + O(\frac{\varepsilon_{12}}{\lambda_k}), k = 1,2,i = 1,\cdot\cdot\cdot, N,
f(x,y_{\bar u + \theta(u - \bar u)}) - f(x,\bar y) = \frac{\partial f}{\partial y}(x,y_1)(y_{\bar u + \theta(u - \bar u)}- \bar y)\,\, y_1 = \bar y + \theta_1(y_{\bar u + \theta(u - \bar u)}- \bar y), &f(x,y_u) - f(x,\bar y) = \frac{\partial f}{\partial y}(x,y_2)(y_u - \bar y) y_2 = \bar y + \theta_2(y_u - \bar y).
\sum_{K}\prod_{1\le j,\ell\le N}\frac{\|H\|_{\infty}^{k_{j,\ell}}\prod\limits_{i=1}^{N}c_{i}!}{k_{j,\ell}!}=\sum_{K}\prod_{i=1}^{N}\binom{c_{i}}{k_{1,i},\dotsc,k_{N,i}}\,\|H\|_{\infty}^{\sum\limits_{\ell=1}^{N}k_{\ell,i}}=\prod_{i=1}^{N}\sum_{c_i\ge0}\left(N\|H\|_{\infty}\right)^{c_i}=\left(1-N\|H\|_{\infty}\right)^{-N},
x^nD_nh(x) =x^nD_n\big(x^c\ell_h(x)\big)=x^n\sum_{k=0}^n\binom{n}{k}D_k\big(x^c\big)D_{n-k}\ell_h(x) =\sum_{k=0}^n\binom{n}{k}c_kh(x)\frac{x^{n-k}D_{n-k}\ell_h(x)}{\ell_h(x)},
[\partial_t^2+ia\partial_{\alpha}, \mathfrak{H}]u= 2[z_{tt}, \mathfrak{H}]\frac{\bar{z}_{t\alpha}}{z_{\alpha}}+2[z_t, \mathfrak{H}] \frac{\bar{z}_{tt\alpha}}{z_{\alpha}}-\frac{1}{\pi i}\int \Big(\frac{z_t(\alpha,t)-z_t(\beta,t)}{z(\alpha,t)-z(\beta,t)}\Big)^2 \bar{z}_{t\beta}d\beta
y^2x^2 = 1-q^2(q+q^{-1})\nu_z + q^4\nu_z^2, z^2y^2 = 1-q^2(q+q^{-1})\nu_x + q^4\nu_x^2, x^2z^2 = 1-q^2(q+q^{-1})\nu_y + q^4\nu_y^2.
\begin{matrix}1&2&3\\ 2&3&1\\ 3&1&2\\ &(3)&\end{matrix}\begin{matrix}2&3&1\\ 1&2&3\\ 3&1&2\\ &(3)&\end{matrix}\begin{matrix}2&1&3\\ 3&2&1\\ 1&3&2\\ &(3)\end{matrix}\begin{matrix}3&2&1\\ 2&1&3\\ 1&3&2\\ &(2)&\end{matrix}\begin{matrix}1&3&2\\ 3&2&1\\ 2&1&3\\ &(1)&\end{matrix}.
\int_{\tilde{X}}fd\tilde{\boldsymbol{\beta}}_{\tilde{\nu}}(w)=\int_{G/\Gamma}\int_{X}f_{z}d\boldsymbol{\beta}_{\nu}\left(\tau(z)^{-1}.w\right)dm_{G/\Gamma}(z),\mbox{where}w\in G/P,f\in L^{\infty}\left(\tilde{X}\right),
\int_\gamma \frac{\Gamma(t + 1/4; \tau, \sigma)}{\Gamma(t - 1/4; \tau, \sigma)}\frac{\theta_0(t + 1/2; \tau)}{\theta_0(t - 1/4; \tau)}\frac{\theta_0(t + 1/2; \sigma)}{\theta_0(t - 1/4; \sigma)}dt = -(1 + i) \frac{\Gamma(1/4; \tau, \sigma)}{\Gamma(3/4; \tau, \sigma)}\frac{1}{(\tau; \tau)(\tau+1/2; 2\tau)}\frac{1}{(\sigma; \sigma)(\sigma+1/2; 2\sigma)},
\left( 1 + \eta \frac{\mu}{8}\right) F_{k+1}\leq \left( 1- \sum_{i=1}^{a-1}\epsilon_i \right) F_k + \sum_{i=1}^{a-1}\epsilon_i \, F_{k-iK}+ 4KQ \, \epsilon_{a-1}\sum_{j=k-aK}^k F_j,
\biggl \{\frac{( x + y ) ( 1 - x - y )}{M_{a}^{2}}\left( \frac{Z_{a}}{2}\right) K_{1}( Z_{a}) + \frac{( x + y - 1 ) ( x + y - 3 )}{M_{b}^{2}}\biggl [ \left( \frac{Z_{b}}{2}\right) K_{1}( Z_{b}) - \frac{1}{2}\biggr ] \biggr \}
A_{1,1}(k) = \int_0^t s \,\phi_{2}(k) e^{-isk^2}\int\overline{\mathcal{K}_{\#,R}'(x,k)}\partial_x \big[ u_{M_{1}(s)}\overline{u_{M_{2}}(s)}u_{M_{3}}(s)\big]\,dx\,ds.
g(v)-g(v')=(v-v')\cdot \nabla g(v')+|v-v'|^2\int_0^1 (1-\kappa)D^2g(v'+\kappa(v-v'))\cdot(\frac{v-v'}{|v-v'|},\frac{v-v'}{|v-v'|}) d\kappa,
R(0)\xrightarrow{(gr)^{\otimes m}}(B^{0})^{\otimes m}\hookrightarrow (B^{\otimes m})^{0}\intertext{which factors as}R(0)\xrightarrow{r^{\circ m}=0}R(0)\xrightarrow{g^{\otimes m}}(B^{0})^{\otimes m}\hookrightarrow (B^{\otimes m})^{0}.
\frac{1}{B_{k-1}}-\frac{1}{B_k}=\frac{B_k^2-B_{k-1}^2}{B_{k-1}B_k(B_{k-1}+B_k)}&\leq \frac{\sigma_k^2}{2B^3_{k-1}}\leq\left(\frac{8}{7}\right)^{3/2}\frac{\sigma_k^2}{2B_k^3}\leq\frac{4}{3}\frac{\sigma_k^2}{B^3_k}.
\Omega(\alpha,\beta)= \begin{cases}O(1) \mbox{if}\alpha=u=0, O\left(X^{4+2\varepsilon}|m^{\ast}\tilde{m}|^2|u|^{-1}s^{-2}\right) \mbox{if}0<|u|\le K, O\left(X^{-C}\right) \mbox{if}|u| > K,\end{cases}
Z_{E H}={\frac{1}{4}}{\frac{1}{( \eta \bar{\eta})^{4}}}\int d p_{x}d p_{y}q^{p_{x}\bar{p}_{x}+ p_{y}\bar{p}_{y}}\sum_{i}\left|{\frac{\theta_{i}}{\eta}}\right|^{4}+{\frac{1}{4}}\sum_{i , j}\left|{\frac{\theta_{i}}{\theta_{j}}}\right|^{4}.
\mathfrak{C}_2=\mathop{\sum\sum}_{\substack{d_2|q_2 d_2'|q_2'}}d_2d_2'\mathop{\mathop{\sideset{}{^\star}\sum}_{\substack{\alpha\bmod{q_2}n_1\alpha\equiv -m\bmod{d_2}}}\;\mathop{\sideset{}{^\star}\sum}_{\substack{\alpha'\bmod{q_2'}n_1\alpha'\equiv -m'\bmod{d_2'}}}}_{\bar{\alpha}q_2'-\bar{\alpha}'q_2\equiv n_2\bmod{q_2q_2'}}\; 1.
\frac{1}{T}\int_{t\in S_1(T)}\log|\zeta(\sigma+it)-a|dt = \int_{-\log_2T}^{\log_2T}\int_{\log|a|+\delta}^{\log_2 T}\Phi_1(u,v)\frac{\partial^2 h(u,v)}{\partial u\partial v}dudv- \frac{\textup{meas}(S_1(T))}{T}h(\log_2T,\log_2T) & + \frac{1}{T}\int_{t\in S_1(T)}\Big(h\big(\log_2T, \arg\zeta(\sigma+it)\big)+h\big(\log|\zeta(\sigma+it)|,\log_2T\big)\Big)dt,
P_{\lambda_n}:\ \ \ \varphi_{\mathcal{E},p}(\theta)\ \ \to\varphi_{\mathcal{E},p}\Big(\bigcup_{j=1}^{n}\theta_{\lambda_j}\Big) (a_s)_{s \in \theta}\to (a_s)_{s\in\bigcup_{j=1}^{n}\theta_{\lambda_j}}.
T_{G,v}QT_{F,v}^*(y) = \int_{B_{\mathbb{R}^2}}v(x)(Qv\pi_Fy)(x)d\lambda(x) & = \int_{B_1}\frac{1}{\lambda(B_1)}de_2d\lambda(x)+\int_{B_2}\frac{1}{\lambda(B_2)}ce_1d\lambda(x) & = de_2+ce_1 & = y.
|a-g(p)| \le\frac{1}{2}\Big(\max \big\{g(p)\colon p\in\cup_{i=1}^n\{x_i,y_i,u_i,v_i\}\big\}\qquad-\min \big\{g(p)\colon p\in\cup_{i=1}^n\{x_i,y_i,u_i,v_i\}\big\}\Big)\le 1
\frac{\pi^2}{2}\int_0^u \frac{1}{L(s)^2}\, du =\frac{\pi^2}{2}t^{1/3}\left(\frac{1+\beta}{u_t}\right)^{1/3}\left(\int_{\frac{\beta u_t}{1+\beta}}^{\frac{(u+\beta t)u_t}{t+\beta t}}\frac{1}{l(s)^2}\, ds\right).
\mu = \left( \begin{array}{c c c}{{\mu_{1 1}}}&{{}}&{{\mu_{1 3}}}\\{{}}&{{\mu_{2 2}}}&{{\mu_{2 3}}}\\{{\mu_{3 1}}}&{{\mu_{3 2}}}&{{}}\\ \end{array}\right) = \begin{array}{c c c}{{\ominus}}&{{}}&{{\ominus}}\\{{}}&{{\oplus}}&{{\oplus}}\\{{\ominus}}&{{\oplus}}&{{}}\\ \end{array}
d_{A B C}\, h^{A}h^{B}h^{C}= 3 \Big \{h^{1}\, \big ( h^{2}\big )^{2}- h^{1}\, \big ( h^{\mu}\big )^{2}- h^{2}\, \big ( h^{i}\big )^{2}+ \gamma_{\mu i j}\, h^{\mu}\, h^{i}\, h^{j}\Big \}\ .
\aligned \frak m^{\epsilon}_{1;\alpha_+,\alpha_-}([r_-]) &=\sum_{r_+ \in R_{\alpha_+}}[(ev_-,ev_+)^{-1}((r_-,r_+)) &\qquad\cap(\mathcal M(\alpha_-,\alpha_+)^{\boxplus\tau_0},\widehat{\mathcal U^+}(\alpha_-,\alpha_+),\widehat{\frak O}(\alpha_-,\alpha_+),\widehat{{\frak s^{+ \epsilon}}}(\alpha_-,\alpha_+))][r_+].\endaligned
[E_{\xi + \alpha_k}, E_{\xi^\prime}^*]_q = c_{k, \xi}^{-1}E_k \triangleright [E_\xi, E_{\xi^\prime}^*]_q + c_{k, \xi}^{-1}c_{k, \xi^\prime}^\prime q^{(\xi - \alpha_k, \alpha_k)}(E_\xi E_{\xi^\prime - \alpha_k}^* - q^{-(\xi, \xi^\prime + \alpha_k)}E_{\xi^\prime - \alpha_k}^* E_\xi).
\delta(H')\ge\delta(H)-\binom{n-1-\delta(G_{1/3})}2 \ge\delta(H)-\binom{n-1-g_{.8}(1/3)(n-1)}2 &\ge .8\binom{n-1}2-\binom{.3(n-1)}2\ge.709\frac{n^2}2,
\begin{array}{ll}\alpha_{_\nabla}= \mathrm{Re}(q_{1})(\alpha_{1}-\gamma_{1})+\mathrm{Im}(q_{1})(\alpha_{\bf{i}}+\gamma_{\bf{i}})+t\beta_{1}&+\mathrm{Re}(q_{2})(\hat\alpha_1-\hat\gamma_1)+\mathrm{Im}(q_{2})(\hat\alpha_{\bf i}+\hat\gamma_{\bf i})+\mathrm{Re}(q_{3})\hat\beta_1+\mathrm{Im}(q_{3})\hat\beta_{\bf i}.\end{array}
{\mathcal{A}}_{\mathcal{B}}(J)=\sum\limits_{l=1}^{n_b}\left[m_b^{(l)}\left({\bf{s}}_l^{\dag}{\bf{s}}_l+{\bf{p}}_l^{\dag}{\bf{p}}_l\right)+i\;{\bf{p}}_l^{\dag}\hat{J}^{\dag}{\bf{s}}_l+i\;{\bf{s}}_l^{\dag}\hat{J}{\bf p}_l\right]
J^{w}(\textbf{X}_{minRSSU}^{(n)}) \geq J^{w}(\textbf{Z}_{minRSSU}^{(n)}) = -\frac{1}{2}\prod_{i=1}^{n}[-2J^w(Z_{i:i})] &\geq -\frac{1}{2}[-2J^w(Z_{1:1})]^n = -\frac{1}{2}\left(\frac{\lambda^2 \Gamma(m+1)}{(2\lambda)^{m+1}}\right)^n.
\hat{f}(0)=\mu\{0\}-\frac{1}{\pi}\sum_{j=1}^n \Im(\frac{\hat{\mu}(j)}{j}) &=\lim_{n\to \infty}\frac{1}{n}\sum_{j=1}^n \hat{\mu}(j)-\frac{1}{\pi}\sum_{j=1}^n \Im(\frac{\hat{\mu}(j)}{j}).
S = - \frac{1}{g}\int_{D 6}( 1 + \frac{1}{4}) \left( F_{\mu \nu}F^{\mu \nu}+ 2 i \overline{{\theta}}\Gamma^{\mu}\partial_{\mu}\theta + 2 \delta_{i , j}\partial_{\mu}X^{i}\partial^{\mu}X^{j}\right)
h_0 =q_0^2\alpha^2-2q_0q_1\alpha -2p^1q_0-q_1^2 h_1 =-2q_0q_1\alpha^2+(6q_1^2+2q_0p^1)\alpha +2p^1q_1 h_2 =-p^1q_0\alpha^2+3(p^1q_1+p^0q_0)\alpha +2(p^1)^2-3p^0q_1.
e^{-\frac{\Delta}{c_1}}\int_{\sigma_{\mathrm{w}}^2}^{\sigma_{\mathrm{w}}^2+\sigma_{\mathrm{a}}^2}\left(\frac{1}{v}\right)^{n}e^{-\frac{z^{(0)}}{v}}e^{-\frac{v}{\zeta}}dv &\qquad\qquad=\int_{\sigma_{\mathrm{w}}^2}^{\sigma_{\mathrm{w}}^2+\sigma_{\mathrm{a}}^2}\left(\frac{1}{v}\right)^{n}e^{-\frac{(z^{(0)}+\Delta)}{v}}e^{-\frac{v}{\zeta}}dv.
\begin{cases}M^{x}_{1}(\tau^{*}_{1},\tau^{*}_{2}) = \sup\limits_{\tau_{1}\in \mathcal{T}}M^{x}_{1}(\tau_{1},\tau^{*}_{2}) M^{x}_{2}(\tau^{*}_{1},\tau^{*}_{2}) = \sup\limits_{\tau_{2}\in \mathcal{T}}M^{x}_{2}(\tau^{*}_{1},\tau_{2}).\end{cases}
t_{i_{1}i_{2}\cdot\cdot\cdot i_{m}}^{(m)}=\Omega_{j_{1}j_{2}\cdot\cdot\cdot j_{2m-2}i_{m}}^{(2m-1)}f_{i_{1}}^{j_{1}j_{2}}\cdot\cdot\cdot f_{i_{m-1}}^{j_{2m-2}j_{2m-3}}
\begin{aligned}0 = P_{-}\left( \frac{q}{\sigma_{-}}\right) + \frac{\phi}{\sigma_{-}}+ P_{-}\left( \frac{z^{n}\psi}{\sigma_{-}}\right) 0 = P_{+}\left(\frac{q}{z^{n}\sigma_{+}}\right) + P_{+}\left( \frac{\phi}{z^{n}\sigma_{+}}\right) + \frac{\psi}{\sigma_{+}}.\end{aligned}
\Xi_{v}\left( s_{1},..,s_{n};s_{f}\right) :=\log e^{\left\langle v,X\left(s_{1}\right) \right\rangle}\mathbb{P}_{q,p;s_{1}^{\ast}}^{}\left(\Omega_{1}\right) \mathbb{P}_{q,p}\left( \left\{s_{1}\right\}|\left({\textstyle\bigcap_{j=2}^{n}}\left\{s_{j}\right\}\right) \cap\left\{s_{f}\right\}\right) \ .
A_\infty = A_H + g_\infty^{-1}\bar\partial g_\infty - (g_\infty^{-1}\bar\partial g_\infty)^* &= A_H + 2 \Im \bar\partial\log\bigl((q\bar q)^{1/8}k^{1/2}\bigr) \begin{pmatrix}i 0 0 -i\end{pmatrix}
\left( \sum_{j_{1},\cdots,j_{m}=1}^{\infty}\left\vert T(e_{j_{1}},\cdots,e_{j_{m}})\right\vert^{\frac{p}{p-m}}\right)^{\frac{p-m}{p}}\leq\left( \sqrt{2}\right)^{m-1}\left\Vert T\right\Vert
\begin{array}{r c l}{{Q_{\Psi_{0}}( A \star \phi )}}&{{=}}&{{( Q_{\Psi_{0}}A ) \star \phi - A \star ( Q_{\Psi_{0}}\phi )}}\\{{}}&{{=}}&{{\cal{I}\star \phi}}\\{{}}&{{=}}&{{\phi ,}}\\ \end{array}
\overline{{J}}_{\dot{\alpha}}^{0}= -{\frac{i}{2}}\overline{{\sigma}}^{0 \dot{\beta}\beta}( \overline{{D}}_{\dot{\alpha}}R_{\beta \dot{\beta}}- 2 \varepsilon_{\dot{\alpha}\dot{\beta}}\overline{{D}}^{\dot{\gamma}}R_{\beta \dot{\gamma}}) \bigr |_{\theta = \overline{{\theta}}= 0}.
{}\, =\eta\bigl(d^*(\psi\otimes\operatorname{id}\otimes\varphi)[(\Delta(c^*)\otimes1)(\operatorname{id}\otimes\Delta)(\Delta(r^*y))\Delta_{23}(s)(1\otimes b\otimes1)]\bigr) &=\eta\bigl(d^*(\psi\otimes\operatorname{id}\otimes\varphi)((\Delta(c^*)\otimes1)(\operatorname{id}\otimes\Delta)[\Delta(r^*y)(1\otimes s)])b\bigr).
\sum_{n\geq 0}\frac{q^{n^2+2(i-1)n}(-q;q^2)_n}{(q^2;q^2)_n}= \frac{1}{(q^{2i-1};q^8)_\infty(q^4;q^8)_\infty (q^{9-2i};q^8)_\infty}.
V_{|m| -1/2}(0) =\alpha_m^{-1}; V_{|m| -1/2}(0)-V_{|m| -1/2}\big(\mathrm i(\lambda -1)\big) < 0; V_{|m| -1/2}\big(2\mathrm i(\lambda -1)\big)- V_{|m| -1/2}\big(\mathrm i(\lambda -1)\big) \geqslant 0; V_{|m| -1/2}\big(2\mathrm i(\lambda -1)\big) \geqslant 0.
\aligned\| (1 - \chi(\frac{x}{R})) u_{0}\|_{B_{1,2}^{1/2}}+ \| (1 - \chi(\frac{x}{R})) u_{1}\|_{B_{1,2}^{-1/2}}= R \| (1 - \chi(x)) u_{0}(Rx) \|_{B_{1,2}^{1/2}}+ R^{2}\| (1 - \chi(x)) u_{1}(Rx) \|_{B_{1,2}^{-1/2}}.\endaligned
S_{0,0}X_{0,4}-X_{0,4}S_{4,4}=(c_{0,4}X_{0,0}S_{0,4}-S_{0,4}X_{4,4})+(X_{0,2}S_{2,4}+X_{0,3}S_{3,4})-(S_{0,1}X_{1,4}+S_{0,2}X_{2,4}).
-\int_0^T(\theta, \phi_t)dt-\int_0^T(u\theta, \nabla\phi)dt+\nu\int_0^T(\Lambda^{\frac{1}{2}}\theta, \Lambda^{\frac{1}{2}}\phi)dt=(\theta_0, \phi(x,0))+\int_0^T(f, \phi)dt.
\mathcal{C}(p,j,s_1,s_2) = \frac{\sigma_{k}(p^j,s_1+1) \sigma_{\ell}(p^j,s_2+1)}{p^{j(s_1+s_2+1)}}- \frac{\sigma_{k}(p^{j+1},s_1+1) \sigma_{\ell}(p^{j+1},s_2+1)}{p^{(j+1)(s_1+s_2+1)+1}}.
l_{0}^{e}+l_{1}^{e}+l_{2}^{e}+l_{3}^{e}=2l_{0}-1,l_{0}^{e}+l_{1}^{e}-l_{2}^{e}-l_{3}^{e}=-2l_{1}-1, & l_{0}^{e}-l_{1}^{e}+l_{2}^{e}-l_{3}^{e}=-2l_{2}-1,l_{0}^{e}-l_{1}^{e}-l_{2}^{e}+l_{3}^{e}=-2l_{3}-1.
\alpha^\prime = \lim_{k\to\infty}2 \left(E \bigg[ \left\| \frac{\boldsymbol{w}_k}{\epsilon_{\rm r}+ |\boldsymbol{w}_{k-1}|}\right\|_{1}\bigg] \right. &- E \left. \bigg[ \left\|\frac{\boldsymbol{w}}{\epsilon_{\rm r}+ | \boldsymbol{w}_{k-1}|}\right\|_{1}\bigg] \right) .
T_k[V_{k+1}] (x_k, r_k) := \inf_{(u,r^{\prime})\in F_k(x_k, r_k)}\, \biggl\{c(x_k,u) \, \,+ & \ \sum_{x_{k+1}\in S}\, Q(x_{k+1}|x_k,u)\, V_{k+1}(x_{k+1}, r^{\prime}(x_{k+1})) \biggr\},
\chi_0 = x_0 +[-c_0,c_0]v,\chi_1 = f^{-1}\left([-R,R]\times\{L\}\right), \chi_{2,\pm}= f^{-1}\left(\{\pm R\}\times [-L',L']\right),\chi_2= \chi_{2,+}\cup \ \chi_{2,-}, \chi_{3,\pm}= q_\pm+[-1,1]v_+,\chi_3= \chi_{3,+}\cup \ \chi_{3,-},
\mathbb{E}|S_N-S_{N,1}|^2 =\sum^N_{n=1}\sum^{m_0}_{j=1}\mathbb{E}[\mathcal{P}_{n,n-j}^2]+\sum^N_{n=1}\sum^{m_0}_{i,j=1}\mathbb{E}[\mathcal{P}_{n,n-j}\mathcal{P}_{n-j+i,n-j}] &\leq \sum^N_{n=1}\mathbb{E}[K^2(X_n)]+\sum^N_{n=1}\sum^{m_0}_{i,j=1}\left(\mathbb{E}[\mathcal{P}^2_{n,n-j}]+\mathbb{E}[\mathcal{P}^2_{n-j+i,n-j}]\right) &\leq c_1 N.
\Gamma_{\infty}^{( 1 )}= \frac{i}{2}\mathrm{t r}\ln \Big ( D^{\Lambda}\prod_{i}( D_{M_{i}}^{\Lambda})^{a_{i}}\Big ) - i \ \mathrm{t r}\ln \Big ( G \prod_{i}( G_{m_{i}})^{b_{i}}\Big ) .
\begin{array}{lcr}ds^2={1\over2}K_{\rho\sigma}[d\phi_\rho d\phi_\sigma+a_\rho a_\sigma \sum\limits_{i=1}^3 d\Omega_{\rho,i}d\Omega_{\sigma,i}]; H=-2a_\rho d\Omega_\rho,\end{array}
W^{(1)}-W^{(3)}= b_0 y+( 120 b_0 + b_1 )y^5+( 5555 b_0 + 74 b_1 + b_2 )y^9 &+( 87440 b_0 + 1382 b_1 + 28 b_2 + b_3 )y^{13}&+(- 2666610 b_0 - 38670 b_1 - 675 b_2 - 18 b_3 )y^{17}+ \cdots.
\sum_{j=1}^{n+1}\| U^j \|_{C^{\ell+t_j, \alpha}_{\phi, \varphi_j}(\Omega)}\le C \left(\sum_{j=1}^{n+1}\| (L U)_j \|_{C^{\ell-s_j, \alpha}_{\phi, \phi^{t_j+s_j}\varphi_j}(\Omega)}+ \sum_{j=1}^{n+1}\| U^j \|_{L^2_{\phi^{-n}\varphi_j^2}(\Omega)}\right),
\sum_{x \in N(X,\varepsilon)}\vert \eta(x)\vert \leq D\left( \sum_{x \in N(X,\varepsilon)}\sum_{\lbrace y \colon \delta(y,x)=1 \rbrace}\vert \eta(x)-\eta(y)\vert \varrho \left(\vert (x,y)\vert\right)\right)
{\rm D}\bar{J}(A_r,B_r,C_r)[(A'_r,B'_r,C'_r)] = 2{\rm tr}( A_r'^T (-QP-Y^TX) ) + 2{\rm tr}(B_r'^T (QB_r+Y^TB) ) &+2{\rm tr}(C_r'^T(C_rP-CX) ).