harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\max_i\|x_i - x\|^p = \max_i \|y_i-y\|^p + \|z\|^p \ge \bigl(\max_i \|y_i-\frac{y}{\|y\|}\| - \| \frac{y}{\|y\|}- y\| \bigr)^p + \|z\|^p \ge (\alpha + \varepsilon - 1 + \|y\| )^p + \|z\|^p \ge (\alpha+\varepsilon-1)^p + \|y\|^p + \|z\|^p = (\alpha+\varepsilon-1)^p + 1.
D^{2}\ \rightarrow \ \partial_{t}^{2}- \left( \frac 1 r \partial_{r}r \partial_{r}+ \frac{1}{r^{2}}\partial_{\varphi}^{2}+ \partial_{z}^{2}\right) \ + \ \frac{2 i \beta \Theta ( r - R )}{r^{2}}\ \partial_{\varphi}\ + \ \frac{\beta^{2}\Theta^{2}( r - R )}{r^{2}}\ .
E_{0}= \Bigl ({\frac{4 R T \lambda^{2}}{\eta}}\Bigr )^{{\frac{1}{2}}}\sum_{n \neq 0 , 1}{\frac{1}{| n |^{{\frac{1}{2}}}}}= \Bigl ({\frac{4 R T \lambda^{2}}{\eta}}\Bigr )^{{\frac{1}{2}}}\Bigl [ 2 \, \zeta ( - 1 / 2 ) - 1 \Bigr ] .
\max_{\tilde{p}\!, p_1\!, p_2\!}\quad \alpha\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}}{\sigma^2_n}\!\Big)
\mathbb{E}(\sum_{k=1}^{n}\mathbb{E}_{k-1}X_{n,k})^{2}=n\cdot cov(X_{n,0},X_{n,2})+2\sum_{j=2}^{n}\sum_{k=1}^{j-1}\mathbb{E}(X_{n,0}X_{n,j-k+2}) \leq2n\max_{2\leq j\leq n}\sum_{k=2}^{j}\mathbb{E}(X_{n,0}X_{n,k}).
\begin{array}{c}ds^{2}=\frac{V(r)[d\chi -(2n\cos \theta -a\sin^{2}\theta )d\phi ]^{2}+\sin^{2}\theta \lbrack ad\chi -(r^{2}-n^{2}-a^{2})d\phi ]^{2}}{[r^{2}-(n+a\cos\theta )^{2}]}+[r^{2}-(n+a\cos \theta )^{2}](\frac{dr^{2}}{V(r)}+d\theta^{2})\end{array}
a^{-1}\psi=\tilde{a}(\mathrm{m})^{-1}\psi-\frac{\langle\mathbf{1}|\tilde{a}(\mathrm{m})^{-1}\psi\rangle}{\langle\mathbf{1}|\tilde{a}(\mathrm{m})^{-1}\mathbf{1}\rangle}\tilde{a}(\mathrm{m})^{-1}\mathbf{1},
\pi_{x_-}(\{y_+\})\geq 0 = \pi_{x_-}^{\uparrow}(\{y_+\}), \pi_{x_-}(\{z_+\})\geq 0 = \pi_{x_-}^{\uparrow}(\{z_+\}), \pi_{x_-}^{\uparrow}(\{y_-\}) = \frac{\nu(\{y_-\})}{\mu(\{x_-\})}\geq \pi_{x_-}(\{y_-\}), \pi_{x_-}^{\uparrow}(\{z_-\}) = \frac{\nu(\{z_-\})}{\mu(\{x_-\})}\geq \pi_{x_-}(\{z_-\}).
\textbf{Y}_{0,i}=\sqrt{\rho}\textbf{H}_{0,i}\textbf{F}_{0,i}c_{0,i}\textbf{X}_{0}+\sqrt{\rho}\sum\limits_{k=1}^{N_u-1}\textbf{H}_{0,i}\textbf{F}_{k,i}c_{k,i}\textbf{X}_{k}+\textbf{N}_{0,i},
\omega^{\omega^{\omega^{\epsilon_{0}+1}}}=\omega^{{\epsilon_{0}}^{\omega}}=\omega^{{\epsilon_{0}}^{1+\omega}}=\omega^{(\epsilon_{0}\cdot{\epsilon_{0}}^{\omega})}={(\omega^{\epsilon_{0}})}^{{\epsilon_{0}}^{\omega}}={\epsilon_{0}}^{{\epsilon_{0}}^{\omega}},
f=-\partial_{1}\xi=-\partial_{1}\left(\int_{0}^{1}\frac{1}{A(x)}dx\right)^{-1}\left(\int_{0}^{1}\frac{1}{A(x)}dx\right)\xi=-\partial_{1}\left(\int_{0}^{1}\frac{1}{A(x)}dx\right)^{-1}\partial_{1,c}u.
\alpha\Bigl[1+\frac{1}{N}\sum_{i\in I^{(i_{0})}}\bigl(M_{t+h}^{i}-M_{t-}^{i}\bigr)\Bigr] \leq\alpha\Bigl[1+\frac{1}{N}\sum_{i\in I^{(i_{0})}}M_{t+h}^{i}\Bigr] \leq\alpha+\beta^{1/2}\left(\frac{1}{N}\sum_{i=1}^{N}\bigl(M_{t+h}^{i}\bigr)^{2}\right)^{1/2}.
y_s^{(m)}=\sum_{k=0}^{\min\left(J-1,m-1\right)}\rho^k\mathbf{a}_{r,m-k}\mathbf{x}&+\sum_{k=0}^{\min\left(\lceil\frac{J}{2}\rceil-1,m-1\right)}\rho^k z_r^{(m-k)}+\frac{w^{(m)}}{f},
\tilde{\Sigma}_\pm B\varphi =\tilde{\Sigma}_\pm\gamma^{(2n+1)}(1-P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\gamma^{(2n+1)}SD_a\tilde{\Theta}_\pm(1-P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\gamma^{(2n+1)}S(1-P_{\rm F}-D_aP_{\rm F}D_a)D_a\tilde{\Theta}_\pm\varphi &=-\gamma^{(2n+1)}(1-P_{\rm F}-D_aP_{\rm F}D_a)\tilde{\Sigma}_\pm\varphi=-B\varphi,
z_{1,\delta}(x)= \partial_\nu\left[\delta^\frac{4-n}{2}\varphi_0\left(\frac{e_n-x}{\delta}\right)- U_{\delta,e_n}(x)\right],z_{2,\delta}(x)= \partial_\nu \left[ U_{\delta,-e_n}(x) -\delta^\frac{4-n}{2}\varphi_0\left(\frac{e_n+x}{\delta}\right)\right].
\left[ - \nabla_{\mathrm{\boldmath ~ \scriptstyle{~ \x i}~},{D}}^{2}- \lambda \,{\mathcal W}^{({D})}( \mathrm{\boldmath ~ \x i ~}) \right] \, \Phi ( \mathrm{\boldmath ~ \x i ~}) = \eta \, \Phi ( \mathrm{\boldmath ~ \x i ~}) \; .
\phi(\mathbf{t}_{ij}(a)) =\tilde{t}_{ij}(a), \phi(\mathbf{u}_{kl}(a)) =\tilde{u}_{kl}(a), \phi(\mathbf{v}_{kl}(a)) =\tilde{v}_{kl}(a), \phi(\mathbf{w}_{kl}(a)) =\tilde{w}_{kl}(a), \phi(\mathbf{f}_{i'k'}(a)) =\tilde{f}_{i'k'}(a), \phi(\mathbf{g}_{k'i'}(a)) =\tilde{g}_{k'i'}(a).
\mathcal{G}:=\bigg\{\bigg(\Big(f\widehat{w}^{-\frac1{r_{m+1}'}}\Big)^{r_m}, \Big(\Big(\prod_{i=1}^{m-1}\|f_i\|_{L^{p_i}(w_i)}\Big)f_m\widehat{w}^{-\frac1{r_m}}\Big)^{r_m}\bigg):(f,f_1,\dots,f_m)\in\mathcal{F}\bigg\},
P_1 \doteq \prod_{k=0}^{\infty}(t_k\theta_k)^{p^{-{k}}}= t^{\frac{p}{p-1}}(1-\xi^{\sigma_1}) \prod_{k=1}^\infty \left[ \xi^{\sigma_k}- \xi^{\sigma_{k+1}}\right]^{\frac{1}{p^k}}
\sum_{j=1}^m |a(t_{j})-a(t_{j-1})|^2 \lesssim \Big(\sum_{j=1}^m \|a(t)\|^2_{\textup{L}_t^2((t_{j-1},t_{j}),dt/t)}\Big)^{1/2}\Big(\sum_{j=1}^m \|ta'(t)\|^2_{\textup{L}_t^2((t_{j-1},t_{j}),dt/t)}\Big)^{1/2}& \leq \|a(t)\|_{\textup{L}^2_t((2^i, 2^{i+1}),dt/t)}\,\|ta'(t)\|_{\textup{L}^2_t((2^i, 2^{i+1}),dt/t)}
P_0 \circ P_1 \circ \ldots \circ P_{Q-1}=\left( (d\exp_{\gamma_0(0)})^{-1}\circ (d\exp_{\gamma_1(0)})^{-1}\circ \ldots \circ (d\exp_{\gamma_{Q-1}(0)})^{-1}\right)_*.
\langle \Psi_{1}(x_{1})\cdots\Psi_{n}(x_{n})\rangle=\displaystyle{\frac{F(\hat{X}_{1(i)}{\cdot\hat{X}_{1(j)}})}{\displaystyle{\prod_{l<m}(x_{lm}^{2})^{-\frac{1}{(n-1)(n-2)}(\eta_{1}+\cdots+\eta_{n})+\frac{1}{n-2}(\eta_{l}+\eta_{m})}}}}\,,
h\alpha_n = \left( \begin{array}{cc}ap^n bp^{-n}cp^n dp^{-n}\end{array}\right) = \left( \begin{array}{cc}d^{-1}p^n \ast 0 dp^{-n}\end{array}\right) \left( \begin{array}{cc}1 0 cd^{-1}p^{2n}1 \end{array}\right) =: g_1g_2,
a(b(x)) \in g(b(x))\ast g'(b(x))\subseteq [g(f(x)\ast f'(x))]\ast [g'(f(x)\ast f'(x))] \subseteq[g(f(x))\ast g(f'(x))]\ast [g'(f(x))\ast g'(f'(x))] =(g\circ f)(x)\ast(g\circ f')(x)\ast(g'\circ f)(x)\ast(g'\circ f')(x).
J_4' = -\int u^{(n)}\triangle_j (-\triangle)^{1-s}u^{(n+1)}\frac{\triangle_j u^{(n+1)}}{|\triangle_j u^{(n+1)}|}\leq -\int u^{(n)}(-\triangle)^{1-s}|\triangle_j u^{(n+1)}| \leq -\int (-\triangle)^{1-s}u^{(n)}|\triangle u^{(n+1)}| \leq 2^{-j\alpha}\|u^{n}\|_{B_{1,\infty}^{r+2-2s}}\|u^{(n+1)}\|_{B_{1,\infty}^{\alpha}}.
r\colon (A\times B)\times(A\times B)\to(A\times B)\times(A\times B), &r((a_1,b_1)(a_2,b_2))=( (a_2,\alpha_{a_1^{-1}}(b_2)),(a_1,\alpha_{a_1^{-1}}(b_2^{-1})b_1b_2)).
a_{i}=\int_{S_{\infty}\cap_{j=1,j\neq i}^{k}D_{\Sigma_{j}}}^{}\sqrt{-1}\partial\bar{\partial}f(s)=\int_{S_{\infty}\cap_{j=1,j\neq i}^{k}D_{\Sigma_{j}}}^{}f'(s)\omega_{\Sigma_{i}}=2\pi m,
\rho(I_{d,s})=\rho_{ic}(I_{d,s})=\rho_{ic}((I_{d,s})^\vee)=\frac{d(s-d+1)}{s}=\frac{d}{\widehat{\alpha}(I_{d,s})}=\frac{s-d+1}{\widehat{\alpha}((I_{d,s})^\vee)}.
e^{ik_{1}\cdot X}e^{ik_{2}\cdot X}\sim \left( \tau -\tau '\right)^{2\alpha 'G^{\mu \nu}k_{1\mu}\cdot k_{2\nu}}\times \left[ e^{-\frac{i}{2}\Theta^{\mu \nu}k_{1\mu}\cdot k_{2\nu}}\right] \times e^{i\left( k_{1}+k_{2}\right) \cdot X}+\cdots ,
m(\kappa)^\dagger =\sum_{j=-1}^{\beta-5}\kappa^jD_j+\mathcal O(\kappa^{\beta-4}), D_j =C_j+\sum_{\genfrac{}{}{0pt}{}{j_1\ge 0,j_2\ge 0,j_3\ge 0}{j_1+j_2+j_3=j+1}}C_{j_1}A_{j_2}C_{j_3},
\gamma = \left[ ( 1 + m^{2}k^{2}r^{4}) ( 1 + m^{2}k^{2}s^{4}) \right]^{\frac{1}{4}\left( 1 + \frac{i m}{k}\right)}\sqrt{( 1 - i m k r^{2}) ( 1 - i m k s^{2})},
\Psi = \exp\left\{\frac{\theta}{2}\Gamma_{\underline{r}\underline{\theta}}-A-{1\over 4}B +\int^r{\rm d}\rho{\rm e}^{-B(\rho)/2}\left({1\over 2a}-{\rm i}\Gamma_7{e\over 2}\Gamma_{\underline{r}\underline{\theta}}A_\theta(\rho)\right)\right\}\psi~,
\gamma(h_{1}k_{1}h_{2}k_{2}) = \gamma(h_{1}h_{2}k_{1}^{h_{2}}k_{2}) &= \iota(h_{1}h_{2})^{-1}&= \iota(h_{2})^{-1}\iota(h_{1})^{-1}&= \gamma( h_{2}k_{2}) \gamma(h_{1}k_{1}) .
\begin{alignedat}{3}\phi(e_1) = \alpha_1 e_1 + \alpha_2 e_2, \phi(e_2) = \beta_1 e_1 + \beta_2 e_2, \phi(e_3) = \gamma e_3 , \phi(e_4) = \delta_4 e_4 + \delta_5 e_5, \phi(e_5) = \epsilon_4 e_4 + \epsilon_5 e_5, \phi(e_6) = \zeta e_6 ,\end{alignedat}
\bigl|L_\mu\bigl(\hat\Delta_\mu(x),x\bigr)- L_\mu(x)/2\bigr| \le \sum_{i,j}K_{b_n}(x_{i,j}-x) \mathbf{1}_{Y_{i,j}=\hat\mu(x)}=\mathrm{O}_\mathrm{p}(1),
S = - \frac{L}{4 \pi \lambda^{2}}\int_{0}^{\infty}\frac{d t}{e^{2 \pi t}- 1}~ \left( ( B \lambda^{2}- i t ) \frac{( \lambda^{2}m^{2}+ v^{2})}{v}\ln \frac{\lambda m - i v}{\lambda m + i v}+ c . c . \right)
A_{n}= \left[ \begin{array}{c c c c c}{{u_{0}}}&{{q_{1}}}&{{0}}&{{. . .}}&{{0}}\\{{- 1}}&{{u_{1}}}&{{q_{2}}}&{{. . .}}&{{0}}\\{{0}}&{{- 1}}&{{u_{3}}}&{{. . .}}&{{0}}\\{{}}&{{.}}&{{.}}&{{.}}&{{}}\\{{0}}&{{. . .}}&{{0}}&{{- 1}}&{{u_{n}}}\\ \end{array}\right]
\Bigr\lvert\frac{1}{N}\sum_{n\leq N}\mu(n) f(T^nx)\Bigr\rvert= \Bigr\lvert\sum_{i=1}^l c_i\frac{1}{N}\sum_{n\leq N}\mu(n) g_i(T^nx)\Bigr\rvert \leq \left(\sum_{i=1}^l \lvert c_i\rvert\right) \epsilon.
\nu^m(T(t)\geq a) =\frac{t}{\sqrt{2\pi a}}\e^{mt}\int_1^\infty s^{-\frac{3}{2}}\exp\left\{-\tfrac{1}{2}(\tfrac{t^2}{as}+m^2as)\right\}ds & \leq \frac{t}{\sqrt{2\pi a}}\e^{mt}\int_1^\infty s^{-\frac{3}{2}}\exp\left\{-\tfrac{1}{2}\cdot 2\sqrt{\tfrac{t^2}{as}\cdot m^2as}\right\}ds & = \sqrt{\frac{2}{\pi a}}\t.
\begin{aligned}\mathrm{T}_e(X,Y,Z) = \sum_i s_it_iu_i + \frac 12 \sum_{\{i,j,k\}=\{1,2,3\}}(x_iy_jz_k) & + \frac 12 \sum_{i\not=j}s_i(y_j\overline{z_j})+ \frac 12 \sum_{i\not=j}t_i(x_j\overline{z_j})+ \frac 12 \sum_{i\not=j}u_i(x_j\overline{y_j}).\end{aligned}
K ( z , \bar{z}) = - \log \left[ i \bar{Z}^{I}\frac{\partial}{\partial Z^{I}}F ( Z ) - i Z^{I}\frac{\partial}{\partial \bar{Z}^{I}}\bar{F}( \bar{Z}) \right] = - \log \left[ i \langle \bar{v}( \bar{z}) , v ( z ) \rangle \right] \ .
T_H^{\rm BTZ}= (\rho_+^2- \rho_-^2)/2 \pi R^2 \rho_- = T_H, ~~{1 \over T_{L/R}^{\rm BTZ}}={1 \over T_H^{\rm BTZ}}\left ( 1 \pm{\rho_+ \over \rho_-}\right ) ={1 \over T_{L/R}}.
\nabla\Big(\mathbb{G}^{(\mathrm{k})}- \mathbb{G}^{(0)}\Big)(\mathrm{x},\mathrm{y}) = \mathrm{C}|\mathrm{x}-\mathrm{y}|\log{|\mathrm{x}-\mathrm{y}|}+ \mathrm{D}|\mathrm{x}-\mathrm{y}| + \mathcal{O}(|\mathrm{x}-\mathrm{y}|^2),
c_{s_{\alpha}}(\chi) = \dfrac{1 - q^{-1}\mathbf{z}^{n_{\alpha}\alpha^{\vee}}}{1 - \mathbf{z}^{n_{\alpha}\alpha^{\vee}}}\, ,c_{s_{\alpha}w}(\chi) = c_{s_{\alpha}}(\prescript{w}{}{\chi}) c_{w}(\chi).
T= \sum_{d_1,\ldots,d_k \le x}g_k(d_1,\ldots,d_k) \Big(\frac{x}{d_1\cdots d_k}P_{2k-1}\Big(\log \frac{x}{d_1\cdots d_k}\Big) + O\Big(\Big(\frac{x}{d_1\cdots d_k}\Big)^{\theta_{2k}+ \varepsilon}\Big),
\hat \Phi \equiv\left( \begin{array}{c}\hat \phi^+(x) \frac 1{\sqrt 2}(\hat H(x) + i\hat \chi(x)) \end{array}\right) \qquad \tilde{\hat\Phi}\equiv i\tau_2{\hat \Phi}^* = \left( \begin{array}{c}\frac 1{\sqrt 2}(\hat H(x) - i\hat \chi(x)) -\hat \phi^-(x) \end{array}\right) \/.
A_{1}^{\left( 0 \right)}={\frac{{\Phi}}{{2 \pi r^{2}}}}x^{2}, \; A_{2}^{\left( 0 \right)}= -{\frac{{\Phi}}{{2 \pi r^{2}}}}x^{1}, \; A_{0}^{\left( 0 \right)}= A_{3}^{\left( 0 \right)}= 0 , \; r^{2}= \left( x^{1}\right)^{2}+ ( x^{2})^{2}.
\min\limits_{\{\mathbf{x}_{ij}\}}\sum_{i=1}^N\bigg[\sum_{j=1}^M(E^l_{ij}x^l_{ij}+E^A_{ij}x^a_{ij}+E^C_{ij}x^c_{ij}) &\ \ +\rho_i\max\{T^{L}_{i}, T^{{A}_{(U)}}_{i}, T^{{C}_{(U)}}_{i}\}\bigg]\allowdisplaybreaks \eqref{eq_placement}eqref{xy_new_CAP}.\allowdisplaybreaks
\psi_{\pm}^{(0)}(r_{1},r_{2})=\frac{1}{\sqrt{2}}[\psi_{n_{1},l_{1},m_{1}}(r_{1})\psi_{n_{2},l_{2},m_{2}}(r_{2})\pm\psi_{n_{2},l_{2},m_{2}}(r_{1})\psi_{n_{1},l_{1},m_{1}}(r_{2})]
\psi^-_N=\frac{1}{[2r(r+r_3)]^{\frac{1}{2}}}\left( \begin{array}{c}-r_1+ir_2 r+r_3 \end{array}\right ) = \left( \begin{array}{c}-e^{-i\phi}\sin\frac{\theta}{2}\cos\frac{\theta}{2}\end{array}\right )
V(n,y_j)\ll U(j,n):=\frac{1}{\widetilde{y}_j}\bigg(\frac{\widetilde{y}_{j}}{\widetilde{y}_{0}}\bigg)^{-1/\ell^K}\sum_{r=0}^{+\infty}\bigg| \sum_{\substack{|\lambda|=r \lambda_1 < g_{j,n}(r)}}a(\lambda)\bigg|^2
d_{G,p}^{p}(u,v)=\mathbb{E}[\sup_{s\in\lbrack0,T]}|u_{s}-v_{s}|^{p}+(\int_{0}^{T}|\mathcal{D}_{x}(u_{s}-v_{s})|^{2}ds)^{\frac{p}{2}}+\int_{0}^{T}|{\mathcal{A}}_{G}u_{s}-{\mathcal{A}}_{G}v_{s}|^{p}ds].
\int_{\left\lbrace 1<u_n\leq k+1 \right\rbrace}|\nabla T_k(G_1(u_n))|^2 = \int_{\left\lbrace 1<u_n\leq k+1 \right\rbrace}|\nabla u_n|^2 & \geq \int_{\left\lbrace |\nabla u_n|>t, 1<u_n\leq k+1 \right\rbrace}|\nabla u_n|^2 &\geq t^2m(\{|\nabla u_n|> t,1<u_n\leq k+1\}).
\Pi_{\mu\nu}(k,m)=-\frac{g^2}{3\pi}\Lambda g_{\mu\nu}+\left(g_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}\right)\Pi^{(1)}(k^2,m)+i\epsilon_{\mu\nu\lambda}k^{\lambda}\Pi^{(2)}(k^2,m)
\begin{aligned}C_{N,\beta,d,D}= \int_{\mathbb{R}^{D \cdot N(d)}}\exp\left(-\sum_{i=1}^{D}\sum_{k=1}^{N(d)}\frac{(x^{(i)}_k)^2}{2(2\beta\lambda_k)^{-1}}\right) \prod_{i=1}^{D}\prod_{k=1}^{N(d)}dx^{(i)}_k &= \frac{1}{(2\beta)^{DN(d)/2}}\prod_{k=1}^{N(d)}\frac{1}{\lambda_k^{D/2}}. \end{aligned}
\displaystyle\frac{z-x}{\vert z-x\vert}\cdot\frac{p-x}{\vert p-x\vert}=\frac{z-z'}{\vert z-x\vert}\cdot\frac{p-x}{\vert p-x\vert}+\frac{z'-x}{\vert z-x\vert}\cdot\frac{p-x}{\vert p-x\vert}\le\frac{z'-x}{\vert z-x\vert}\cdot\frac{p-x}{\vert p-x\vert}.
\begin{array}{rlll}-m\left( \| w\|^2 \right)\Delta w =0 \; \; \; \mathbb R^{2}_{+}-m\left( \| w\|^2 \right)\left(\frac{\partial w}{\partial y}+V(x)w(x,0)\right) = f(w(x,0))\; \; \mathbb R\end{array}
(f_1 * f_2)(w,z) = \left| \{(u,v) \in \mathcal{O}_1 \times \mathcal{O}_2 : \sigma(u) + \sigma(v) = \sigma(w) z_1 z_2 [ \sigma(u), \sigma(v) ]^{1/2}= z \}\right| &= \left| \{(u,v) \in \mathcal{O}_1 \times \mathcal{O}_2 : u + v = w z_1 z_2 [ v, u ]^{1/2}= z \}\right|.
{\Delta_{1}u}^{\alpha ( i , j )}= \frac{u^{\alpha ( i + 1 , j )}- u^{\alpha ( i , j )}}{x_{1}^{( i + 1 )}- x_{1}^{( i )}}, \qquad{\Delta_{2}u}^{\alpha ( i , j )}= \frac{u^{\alpha ( i , j + 1 )}- u^{\alpha ( i , j )}}{x_{2}^{( j + 1 )}- x_{2}^{( j )}}.
\|e_C^{(s-1)k+q}\| \leq (1 - \frac{q}{k}) \|e_C^{(s-1)k}\| + \frac{q}{k}\|e_C^{sk}\| + \frac{\|u''(\xi)\|_{\infty}}{2}\tau_C^2 \leq \bar{C}(\tau_{C}^{2}+ h_1^2 + h_2^2 ), 1 \leq s \leq N, 1 \leq q \leq k-1.
\left( \sum_{n \leq \alpha}+ \sum_{n > \phi}\right) \psi_{0 , n}( x ) \psi_{0 , n}^{\dagger}( y ) = \delta ( x - y ) - \sum_{\alpha < n \leq \phi}\psi_{0 , n}( x ) \psi_{0 , n}^{\dagger}( y ) .
Z_{M,1}=\sup_{t\in S_{M}}\left[\frac{|\nu(t-u_M)|}{\lVert t-u_M\lVert_{2}^2+4x_{M}^2}\right]\leq C^\star\left[ C\frac{\varphi(2x_M)}{x_M^2\sqrt N}+ 2\sqrt{\frac{z_M+z}{x_M^2N}}+2C_{\mathcal F,\infty}^3\frac{z_M+z}{x_M^2N}\right].
-2 \Phi(N) \rho(N-1)^{2}\Delta A(N-1)^{2}+2 \Phi(N+1) \rho(N)^{2}\Delta A(N)^{2}&\ \ \ \ = \sum_{i=1}^{M}\lambda_{i}A(N)^{i}-\alpha_{1}b(N) +2 \omega_{1}b(N)A(N)-\beta (\rho^{2}(N-1) \Phi(N)+\rho^{2}(N)\Phi(N+1)),
\sigma_{R}(t)\ge\Biggl[ \sum_{i=2}^{n}\sum_{j=2}^{i}\Bigl(\max\lbrace\left|\frac{\partial f_{i}}{\partial x_{j}}(t,x_{1},\ldots,x_{i},u)\right|:|(x_{1},\ldots,x_{i},u)|\le 2\beta(t,R)+\bar{u}(t)+\xi\rbrace\Bigr)^{2}\Biggr]^{\frac{1}{2}},t\ge 0.
\begin{gathered}\int_0^x \rho^{\nu+1}({q^2\rho^2}/{x^2}; q^2)_{\gamma}J_\nu (u\rho; q^2) \,d_q \rho = x^{\nu-\gamma+1}u^{-\gamma-1}(1-q)(1-q^2)^\gamma \Gamma_{q^2}(\gamma + 1)J_{\gamma+\nu +1}(u x; q^2). \end{gathered}
\varepsilon^3 = \varepsilon (\alpha\eta\beta\mu\gamma + b_1 \alpha\eta\beta\mu\gamma \varepsilon) = \varepsilon \alpha\eta\beta\mu\gamma , \varepsilon^3 = (\alpha\eta\beta\mu\gamma + b_1 \alpha\eta\beta\mu\gamma \varepsilon) \varepsilon = \alpha\eta\beta\mu\gamma \varepsilon .
|B+B| = |A+A|^n & \geq (|A-A|+1)^n & > |A-A|^n + n|A-A|^{n-1}& \geq |A-A|^n + n(2k-1)^{n-1}& \geq |A-A|^n + k^n & = |B-B| + |B|.
\chi(\omega_{C_1}\otimes h_{1,*}\omega_X)\le h^0(C_1,\omega_{C_1}\otimes h_{1,*}\omega_X)\le h^0(C_1,h_{1,*}(\omega_X^2)) = h^0(X,\omega_X^2)=\chi(\mathcal{O}_X)+K_X^2\le 1+9=10
\begin{array}{rcl}S_{{\rm IIB}}= -T_{M_2}l\int d^2\xi\sqrt{|{\tilde g}_{ij}-\frac{1}{{\tilde k}^2}(\partial_i \varrho+{\tilde B}^{(1)}_i)(\partial_j \varrho+{\tilde B}^{(1)}_j)|}& & -\frac{T_{M_2}l}{2}\int d^2\xi\epsilon^{ij}\left[\tilde{B}^{(2)}{}_{ij}+2\partial_{i}\varrho\ \left({\tilde A}^{(1)}{}_{i}+m\pi\alpha^{\prime}b_{j}\right)\right]\, . \end{array}
S_{a,\epsilon}: y=h_\epsilon(x,z,\epsilon)=h_0(x,z)+\mathcal O(\epsilon),\,(x,z)\in \mathcal U^-, S_{r,\epsilon}: y=h_\epsilon(x,z,\epsilon)=h_0(x,z)+\mathcal O(\epsilon),\,(x,z)\in \mathcal U^+
\gamma\cdot\limsup\Vert x_{f(k)}-x^*\Vert^2 =\gamma\cdot\limsup\Vert x_{f(k)}-y_{\theta_{f(k)}}\Vert^2 \ge\limsup\Vert x_{f(k+1)}-y_{\theta_{f(k)}}\Vert^2 =\limsup\Vert x_{f(k+1)}-x^*\Vert^2 =\limsup\Vert x_{f(k)}-x^*\Vert^2.
\begin{aligned}\dot{G}(t)=-\int_{0}^{t}\left[E(t+\tau,t-\tau)-E(t-\tau,t+\tau)\right]d\tau \ \int_{B}\biggl[\rho\left[\dot{u}_{i}(2t)u_{i}^{0}+u_{i}(2t)v_{i}^{0}\right]+\frac{1}{\beta}\gamma\theta(2t)\theta^{0}+\beta k_{ij}\Theta_{,j}(2t)\theta_{,i}^{0}\biggr]dv,\end{aligned}
{d\over dt}(p\cdot\hat{H})= \sum_{\rho\in\Delta}{g_{|\rho|}^{2}\over |\rho|^{2}}(\rho\cdot\hat{H})\left[y_{|\rho|}(\rho\cdot q,w) x_{|\rho|}(-\rho\cdot q,w)-x_{|\rho|}(\rho\cdot q,w)y_{|\rho|}(-\rho\cdot q,w)\right].
G(U_n(t_n)) =(4-2d)\omega Q(U_n(t_n))+(3-d)\mathbf{c}\cdot \mathbf{P}(U_n(t_n)) \ =(4-2d)\omega Q(U_n(0))+(3-d)\mathbf{c}\cdot \mathbf{P}(U_n(0)) \ =(4-2d)\omega Q(\Phi_{\omega,\mathbf{c},n}) +(3-d)\mathbf{c}\cdot \mathbf{P}(\Phi_{\omega,\mathbf{c},n}) +O(\delta_n).
u_{m+ k - 1}(t, 1) = M_{k-1}\Big(u_{k+ 1}\big(t, x_{k+1}(- \tau_{m + k - 1}, 0, 0) \big), \dots, u_{k+ m-2}\big(t, x_{k+ m - 2}(-\tau_{m + k -1}, 0, 0) \big) \Big),
\sum_{n=0}^\infty \frac{q^{n^2+n}}{(q;q)_n}= \frac{1}{(q^2; q^5)_\infty (q^3 ; q^5)_\infty}=\frac{(q;q^5)_\infty (q^4;q^5)_\infty (q^5;q^5)_\infty}{(q;q^5)_\infty (q^2;q^5)_\infty (q^3;q^5)_\infty (q^4; q^5)_\infty (q^5;q^5)_\infty}= \Big( \frac{1}{(q;q)_\infty}\Big) \Big( (q;q^5)_\infty (q^4;q^5)_\infty (q^5;q^5)_\infty \Big).
F \left({\bf w}, u \right) = I_{2d}- u \widehat{M}_A ({\bf w}) \ \ \and \ \ \ \widehat{M}_A({\bf w})=\sum_{j=1}^{d}\Big( e^{i w_j}P_{2j-1}A + e^{-i w_j}P_{2j}A \Big) ,
g_{\theta}\equiv \bar{\zeta}\tilde{p}d_{\theta}+ \bar{d}_{\theta}\tilde{p}\zeta = - i ( \bar{\zeta}\tilde{p}p_{\theta}+ \bar{p}_{\theta}\tilde{p}\zeta ) - 4 k^{2}m^{2}( \theta \zeta + \bar{\zeta}\bar{\theta}) \approx 0 \, ,
c_{m}=\int_{\mathcal{D}}\underset{(m)}{\cdots}\int_{\mathcal{D}}\frac{1}{\Vert \mathbf{x}_{1}-\mathbf{x}_{2}\Vert^{\alpha}}\frac{1}{\Vert \mathbf{x}_{2}-\mathbf{x}_{3}\Vert^{\alpha}}\cdots \frac{1}{\Vert \mathbf{x}_{m}-\mathbf{x}_{1}\Vert^{\alpha}}d\mathbf{x}_{1}\dots d\mathbf{x}_{m}.
\overline{\left(\begin{pmatrix}\partial_0 0 0 \partial_0 \end{pmatrix}+ \begin{pmatrix}0 1 \Delta 0 \end{pmatrix}\right)}\begin{pmatrix}u w \end{pmatrix}= \begin{pmatrix}\int_0^\cdot \sigma(u(r))dW(r) 0 \end{pmatrix}.
\begin{array}{l}^{W_k=W\left[ \left( F(u_k,V_k(\sigma^{+}))+F(u_k,V_k(\sigma^{-}))-F(u_k,v_{k-1})\right) ,u_k\right]}_{\bar W_k=\bar W\left[ \left( \bar F(U_k(\sigma^{+}),v_k)+\bar F(U_k(\sigma^{-}),v_k)-\bar F(u_{k-1},v_k)\right) ,v_k\right] .}\end{array}
\beta_\epsilon=\frac{\epsilon+(1-\epsilon)\beta_\epsilon}{c_g^2 \ln \tfrac{2}{\beta_\epsilon}}\le \frac{\epsilon+(1-\epsilon)\tfrac{\epsilon}{\epsilon_g}}{c_g^2\ln \tfrac{2\epsilon_g}{\epsilon}}<\frac{(1+\epsilon_g^{-1})\epsilon}{c_g^2\ln \tfrac{2\epsilon_g}{\epsilon}}.
\|u\|_{W^{s,p,q}(\Omega)}=\left(\|u\|_{L^{p}(\Omega)}+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{q^{Q+sp}(y^{-1}\circ x)}dxdy\right)^{1/p},\,\,\,u\in W^{s,p,q}(\Omega).
\tilde{\Delta}\rho (k)=\frac{1}{\pi}\Theta_{(\Lambda_{K},\Lambda_{K}+\frac{1}{2}\delta (\Lambda_{K}))}(|k|)\ln \left[ \sqrt{\frac{\delta (\Lambda_{K})}{2(|k|-\Lambda_{K}}}+\sqrt{\frac{\delta (\Lambda_{K})}{2(|k|-\Lambda_{K})}-1}\right]
2 a c^3 + 2 b^3 d -b^2 c^2 - 3 a b cd= b^2c^2(\dfrac{2ac}{b^2}+\dfrac{2bd}{c^2}-1-\dfrac{3ad}{bc}) = b^2c^2\cdot\dfrac{2 (n+1) (-k^2+kn-k-1)}{(k+1) (k+2) (n-k+1)(n-k)}.
F(s) =\frac{2e^{\gamma}}{s}\left(1+\int_2^{s-1}\frac{\log(t-1)}{t}\mathrm{d}t+\int_2^{s-3}\frac{\log(t-1)}{t}\left(\int_{t+2}^{s-1}\frac{1}{u}\log\frac{u-1}{t+1}\mathrm{d}u\right)\mathrm{d}t\right)
\nabla\phi=\frac{\hat{e}_{1}}{h_{1}}\frac{\partial\phi}{\partial q^{1}}+\frac{\hat{e}_{2}}{h_{2}}\frac{\partial\phi}{\partial q^{2}}+\frac{\hat{e}_{3}}{h_{3}}\frac{\partial\phi}{\partial q^{3}}
\int_{0}^{2 \pi}d \theta \int_{0}^{\infty}d r r \Bigl ( \mathrm{e}^{-{\frac{\phi ({\bf r}_{i j})}{k T}}}- 1 \Bigr ) ={\frac{2}{N}}\Bigl \{b -{\frac{a}{( R T )^{2}}}\Bigr \}\ .
\left[ z_{1}, \, \bar{z}_{1}\right] = \left[ z_{2}, \, \bar{z}_{2}\right] = \left[ z_{3}, \, \bar{z}_{3}\right] = \left[ z_{4}, \, \bar{z}_{4}\right] = - \frac{\zeta}{2}\, , \quad \textrm{o t h e r s a r e z e r o}.
\sqrt{\frac{25}{4}+4\kappa^2\tilde{\omega}_{m,\pm}^2}= \frac{\left|8\left(m+\frac12\right)\pm\sqrt{25+4\left(29-4\left(m+\frac12\right)^2\right)\kappa^2+64(\kappa^2)^2}\right|}{2(1+4\kappa^2)}.
\varphi(\sigma^s(\omega,y)) = \gamma(1_G, \widehat{\phi}(\mathfrak{C}((\omega^1,y^1),1_{H_1},1_{H_2}))) & = \gamma(s, \widehat{\phi}(\mathfrak{C}((\omega^0,y^0),1_{H_1},1_{H_2}))) & = T^s(\gamma(1_G,\widehat{\phi}(\mathfrak{C}((\omega^0,y^0),1_{H_1},1_{H_2})))) & = T^s(\varphi(\omega,y)).
^0 \langle m \neq p \mid p \rangle = \{\sum_{n}\!^{\prime}[ 1 + Q ( H_{0}- E_{p})^{- 1}Q V Q ]_{m n}^{- 1}\frac{V_{n p}}{( E_{p}- E_{n}^{0})}\}^{0}\langle p \mid p \rangle
2\omega_{12}+\theta_{12}=\sum_i\frac{-B^1_{11,i}}{\mu}\omega_i=\sum_i\frac{B^1_{22,i}}{\mu}\omega_i=\sum_i\frac{B^2_{12,i}}{\mu}\omega_i, &B^1_{12,i}=0,~~B^2_{11,i}=B^2_{22,i}=0.
R_{0\cdot H,\beta}= c.H-r\beta.H & C_{0\cdot H,\beta}= \left(c.H-r\beta.H\right)\beta + \left(s-c.\beta + r\,\frac{\beta^2}{2}\right) H & S_{0\cdot H,\beta}= c.H\, \frac{\beta^2}{2}+ s \beta.H - (c.\beta) \cdot (\beta.H).
N_{E/F}(\mathfrak{A})=\mathcal{O}_F \Longleftrightarrow (\forall\,\mathfrak{p}\in\mathbb{P}_F)\,\sum_{\mathfrak{P}\mid\mathfrak{p}}\,f(\mathfrak{P}/\mathfrak{p})\cdot v_{\mathfrak{P}}(\mathfrak{A})=0.
\check{\pi}\big(\|\vartheta-\check{\theta}\|^2 \le\sigma^2 \chi^2_{|\hat{I}|,\alpha}|X\big)\ge\check{\pi}\big(\|\vartheta-\check{\theta}\|^2\le(1-|\hat{I}|/en)\sigma^2\chi^2_{|\hat{I}|,\alpha}|X\big)=1-\alpha.
Z_m(A|J) ={\rm exp}\Bigr\{\! - (i/\hbar) z_Q^{-1}\bigr( J_i A^i - \hbox{$\frac{1}{2}$}m^2 g_{ij}A^i A^j \bigr) \Bigr\}\,{\rm exp}\Bigr\{- m^2 g_{ij}A^i \frac{\delta}{\delta J_j}\Bigr\}Z_m(0|J).
\mathcal{F}_y := \left\{(x,r) \in q_\xi^{-1}(y) \cap I_{(\omega_1 , \ldots , \omega_M)}\times ( 0 , \infty ) : \frac{\mu \left(\Sigma_r ( \xi , x ) \cap R_{(\omega_1 , \ldots , \omega_M)}\right)}{\mu \left( \Sigma_r ( \xi , x ) \right)}< e^{-n}\right\}.
\left[ \Omega_{1}, \Omega_{1}\right]^{*}= \frac{1}{3}f_{a \left[ d \right.}^{e}f_{\left. b c \right]}^{a}\int d^{3}x \left( \epsilon^{0 i j k}A_{i}^{d}A_{j}^{b}A_{k}^{c}\eta_{e}\right) .
\begin{array}{l}{\delta_{b}x^{\mu}={b^{\mu}}_{\nu}n^{\nu},}\\{\delta_{b}A_{a}^{\mu}= -{b^{\mu}}_{\nu}\displaystyle \left( \varepsilon_{a b}\frac{g^{b c}}{\sqrt{- g}}{\Pi_{c}}^{\nu}- \xi_{a}n^{\nu}+ i ( \bar{\theta}\Gamma^{\nu}\partial_{a}\theta ) \right) .}\\ \end{array}
D_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}}+i\sigma^{\mu}_{\alpha{\dot\alpha}}{\overline\theta}^{\dot\alpha}\partial_{\mu}\, , \hspace{1cm}{\overline D}_{\dot\alpha}= -\frac{\partial}{\partial{\overline\theta}^{\dot\alpha}}-i\theta^{\alpha}\sigma^{\mu}_{\alpha{\dot\alpha}}\partial_{\mu}
\varphi(0)=0, \Lambda(0)=\Lambda_0,( \left(\frac{\partial \Lambda(t)}{\partial t_{\lambda}}\right)_{t=0},\left(\frac{\partial \varphi(t)}{\partial t_{\lambda}}\right)_{t=0})=(\pi_{\lambda},\eta_{\lambda}),\,\,\,\,\, \lambda=1,...,m,
\int \, d^{3}x d z \left[{\frac{1}{2}}\, ( \partial_{\mu}\phi )^{2}\, + \,{\frac{\phi^{2}}{4}}\mathrm{l n}( \phi^{2}) - \phi^{2}\, F_{\mu \nu}F^{\mu \nu}\, + \, . . . \right] ,
{\cal R}_{z\doteq z_n=\cdots=z_0}F={\rm Res}_{z=z_0}{\cal R}_{z_n=\cdots=z_0}\partial_{\bar{z}}^{-1}F-\sum_{i=0}^N{\cal R}_{z_n=\cdots=z_0}{\rm Res}_{z=z_i}\partial_{\bar{z}}^{-1}F.
\mathcal{P}_{\rm out}^D ={\rm Prob}\left\{\mathbb{P}_{e,D|p,h}>\mathbb{P}_{e,t}\right\}&={\rm Prob}\left\{\sum_{i=1}^2\sum_{j=1}^2 g_{ij}h_{ij}< \mathcal{A}_{th2}\right\},
\Re\Big\{\sum_{n=1}^{\infty}z_n^m \Big\}\leq \frac{1}{\alpha^m}- \frac{1}{(\alpha+1-\beta_1)^{2m}}+ \Re\Big\{\frac{1}{(\alpha+i\gamma')^{2m}}- \frac{1}{(\alpha +i\gamma' + 1-\beta_1)^{2m}}\Big\},
dV_{ht}=\sigma_{ht}(x)\theta^{0}\wedge \theta^{1}\wedge ...\wedge \theta^{n-1},~\ \ \sigma_{ht}(x)=\dfrac{1}{Vol(\mathbb{B})}\underset{E_{x}^{t_{0}}}{\int}\left\vert \det g(x,y)\right\vert d^{n}y.
J_{ini}(W_0)=\frac{1}{2}\int_{\mathbb{T}}\sum_{k=1}^3\frac{(h^W_k(u))^2}{\rho_{k-1}(u)}+\frac{(\sum_{k=1}^3h^W_k(u))^2}{1-\sum_{k=1}^3\rho_{k-1}(u)}du.
|I_3|\leq \frac\mu{8}\sum_{q\geq -1}\lambda_q^{2s+2\alpha}\|b_q\|_2^2+ \frac\nu{8}\sum_{q\geq -1}\lambda_q^{2s+2}\|u_q\|_2^2 &+C_{\nu,\mu}\left(\sum_{q\geq -1}\lambda_q^{2s}\|b_q\|_2^2\right)^{\gamma_1}+C_{\nu,\mu}\left(\sum_{q\geq -1}\lambda_q^{2s}\|u_q\|_2^2\right)^{\gamma_2},