harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|X_+(t)|^2+|X_0(t)|^2+|X_-(t)|^2 &=\sum_{k+\ell\leq s}\left[ |z^{(k,\ell)}(t)|^2+|\bar{z}^{(k,\ell)}(t)|^2+\sum_{i\in I_1}\left(|\xi_{i,1}(t)|^2+|\xi_{i,2}(t)|^2 \right)+\right.\left.\sum_{i\neq 0}|\xi_i(t)|^2 \right] &= \sum_{k+\ell\leq s}\| q^{(k,\ell)}(t) \|^2_G.
\begin{array}{ll}2{G_{f}}(\nabla_{X_i^I}\!\!\!\!\! Y_i^I,Z_k^K) \!=\!X_i^I({G_{f_1f_2}}(Y_i^I,Z_k^K))+Y_i^I({G_{f_1f_2}}(X_i^I,Z_k^K))\!-\!Z_k^K({G_{f_1f_2}}(X_i^I,Y_i^I)) &\!\!\!\!+{G_{f_1f_2}}([X_i^I,Y_i^I],Z_k^K)+{G_{f_1f_2}}([Z_k^K,X_i^I],Y_i^I)+{G_{f_1f_2}}([Z_k^K,Y_i^I], X_i^I).\end{array}
H_P = \int(\frac{8\pi^{2}}{g^{2}}(\pi^{a}_{i}+\frac{1}{8\pi}\epsilon_{ij}A^{j,a})^{2}+ \frac{g^{2}}{64\pi^{2}}G^{a}_{ij}G^{a}_{ij})d^{2}x
\partial_\mu^2 \Theta^i(\mu,\sigma) = \sigma^{-1}\partial_\sigma \Theta^i(\mu,\sigma), \partial_\mu\partial_\sigma \Theta^i(\mu,\sigma) = 0, \partial_\sigma^2 \Theta^i(\mu,\sigma) = \sigma^{-1}\partial_\sigma \Theta^i(\mu,\sigma).
{\rm Im\,}\varphi(x, y,s)\geq c\inf_{w'\in\Lambda}\Big({\rm Im\,}\psi(\infty,\hat x,(w',\alpha_{2n-1}(x)+s\beta_{2n-1}(x),1)) &+|d_{w'}\bigr(\psi(\infty, \hat x, (w',\alpha_{2n-1}(x)+s\beta_{2n-1}(x),1))-<\hat y, (w',\alpha_{2n-1}(x)+s\beta_{2n-1}(x),1)>)|^2\Big),
||H f_k||^2_{L^2(1,+\infty)}=\int^{+\infty}_1\Big|\int^{1/2+2^{-k}}_{1/2-2^{-k}}\frac{f_k(y)}{x-y}dy\Big|^2dx &\leq \int^{+\infty}_1\frac{1}{|x-\frac{3}{4}|^2}\Big(\int^{1/2+2^{-k}}_{1/2-2^{-k}}|f_k(y)|dy\Big)^2dx &\leq 2^{-k+1}\int^{+\infty}_1\frac{1}{|x-\frac{3}{4}|^2}dx\lesssim 2^{-k},
\tilde{\eta}=(\eta |\mathcal{I}_{i,n}^{(l)}|)^{p_l/(2 p_l \wedge 1)}, a=\mathbb{E}\big[\big(\int_{t_{i-1,n}^{(l)}}^{t_{i,n}^{(l)}}\|\tilde{\sigma}_s-\tilde{\sigma}_s(r)\|^2ds \big)^{p_l \vee \frac{1}{2}}\big| \mathcal{S}\big]
\Psi_{0}\left(T_{0}\right) =\frac{\left(\lambda T_{0}+1\right)^{p}-1}{\lambda^{p}}\Psi_{1}\left(T_{0}, T_{1}\right) =\frac{1}{\lambda^{p}}\left[\frac{\left(\lambda T_{1}+F\left(T_{0}\right)\right)^{p}}{\lambda T_{0}+1}-G\left(\frac{\left(\lambda T_{0}+1\right)^{p}-1}{\lambda^{p}}\right)\right]
\small\tilde{Z}^{n}(t) = E_n( T -t_m ) Y_m^{n}+ \int_{t_m}^t E_n( T -t_m ) F_n (Y_m^{n})\, \mbox{d}s + \int_{t_m}^t E_n( T -t_m ) B_n \, \mbox{d}W(s)
\psi\big(as_v^{\Lambda^i}b\big)=\psi(a)\psi\big(s_v^{\Lambda^i}\big)\psi(b)=\psi(a)\Big(\sum_{\mu\in v\Lambda^{e_i}}\Theta_{s_\mu^\Lambda,s_\mu^\Lambda}\Big)\psi(b) =\sum_{\mu\in v\Lambda^{e_i}}\Theta_{\psi(a)s_\mu^\Lambda,\psi(b^*)s_\mu^\Lambda}&=\sum_{\mu\in v\Lambda^{e_i}}\Theta_{\phi(a)s_\mu^\Lambda,\phi(b^*)s_\mu^\Lambda}.
P_{\mathcal{S}}(C_\mathcal{\mathcal{L}}+ \mathcal{L}^{\perp}) = C_\mathcal{\mathcal{L}}+P_{\mathcal{S}}( \mathcal{L}^{\perp}) \subseteq C_\mathcal{\mathcal{L}}+ \mathcal{L}^{\perp},
q^{2l}\prod_{{\stackrel{r=1}{r\not= s}}}^l \frac{q\lambda_r^2-q^{-1}\lambda_s^2}{q^{-1}\lambda_r^2-q\lambda_s^2}=\left [\frac{(1- \Delta^2 \mu \lambda_s^2 q)\left(1-\Delta^2\frac{\mu}{\lambda_s^2}q^{-1}\right)}{(1-\Delta^2 \mu \lambda_s^2 q^{-1})\left(1-\Delta^2\frac{\mu}{\lambda_s^2}q\right)}\right]^{N/4}\, .
T_{{\bar{m}}{\bar{n}}{\bar{s}}}\rightarrow T_{{\bar{m}}{\bar{n}}{\bar{s}}}+ \nabla_{\bar{m}}\nabla_{\bar{n}}\Lambda_{\bar{s}}\hspace{1 c m}H_{\alpha \nu}\rightarrow H_{\alpha \nu}+ W_{\alpha \nu s}G^{s{\bar{s}}}\Lambda_{\bar{s}}\ ,
G_{D}[ x_{2}, x_{1}] = \sum_{n_{1}}\sum_{n_{2}}. . . \sum_{n_{D - 1}}\, e^{i ( n_{1}\theta_{1}+ n_{2}\theta_{2}. . . + n_{D - 1}\theta_{D - 1})}\, G_{n_{1}n_{2}. . . n_{D - 1}}[ x_{2}, x_{1}] ,
\nabla \phi_\mathrm{pd}(x^{(k)},z^{(k)}) -\nabla \phi_\mathrm{pd}(x^{(k+1)},z^{(k+1)})\in \begin{bmatrix}A^T z^{(k+1)}+ \partial f(x^{(k+1)})+\nabla h(x^{(k+1)}) -A x^{(k+1)}+ \partial g^\ast(z^{(k+1)}) \end{bmatrix}.
\ln{\cal Z}_{B}= \frac{4 \pi V}{h^{3}}\left( \frac{\sqrt{\pi}}{2}( \frac{2 m}{\beta})^{3 / 2}z + \sqrt{\pi}( \frac{2 m}{\beta})^{3 / 2}\delta ( q ) z^{2}+ . . . \right) ,
\sum_{n=0}^\infty c(2n)q^n=\frac{2E_2(2\tau)-E_2(\tau)}{(q;q)_\infty}, &\sum_{n=0}^\infty c(2n+1)q^n=\frac{24}{(q;q)_\infty}\sum_{n=0}^\infty \sigma(2n+1)q^n.
\begin{pmatrix}Z_k Q_k \end{pmatrix}(t_2) = S_k(t_2,t_1) \begin{pmatrix}Z_k Q_k \end{pmatrix}(t_1) + \int_{t_1}^{t_2}S_k(t_2, t) \begin{pmatrix}c_k^{+}Z_{k+1}+ c_{k}^{-}Z_{k-1}d_k^{+}Z_{k+1}+ d_{k}^{-}Z_{k-1}\end{pmatrix}dt
\{\, x , \, y , \, z \}\, \in \,{\cal M}_{\Gamma}\, \left( \, t_{1}, . . . , t_{r}\, \right){}~ ~ \longrightarrow ~{\tilde{W}}_{\Gamma}\, \left( \, x , \, y , \, z , \, ; \, t_{1}, \, . . . . \, , t_{r}\, \right) ~ = ~ 0 ,
||F_{x,f}(X)-\alpha_{g_{0}}^{-1}(x) || = \Big| \Big| \int_{g \in U}\alpha_{g}^{-1}(x)f(g)1_{X}(g)dg - \int_{g \in U}\alpha_{g_{0}}^{-1}(x)f(g)dg ~\Big |\Big| =\Big|\Big| \int_{g \in U}(\alpha_{g}^{-1}(x)-\alpha_{g_{0}}^{-1}(x))f(g)dg ~\Big|\Big| \leq \int_{g \in U}|| \alpha_{g}^{-1}(x)-\alpha_{g_{0}}^{-1}(x)|| f(g) dg \leq \epsilon
\mathbb{P}\Big(d(w_{j-1}, w_{j}) \geq L \textup{for some}j \in I_{k, \alpha}\Big) \leq \sum_{j = 1}^\alpha \mathbb{P}\Big(d(w_{j-1}, w_{j}) \geq L \Big) & \leq \sum_{j = 1}^\alpha \mathbb{P}\Big(d(e, g_{1}) \geq L\Big) & \leq \alpha \cdot \frac{\delta}{2 \alpha}= \frac{\delta}{2}
( D s )_{\beta \gamma}= \delta_{\beta , \gamma}( D s )_{\beta}, \quad ( D s )_{\beta}= - i \sum_{\lambda \in \Delta_{s}, \ \beta \cdot \lambda = 1}\left[ g_{s_{1}}z_{d}^{( 1 / 2 )}( \lambda \cdot q , \xi ) + g_{s_{2}}z_{d}^{( 1 / 4 )}( \lambda \cdot q , \xi ) \right] ,
\mathcal{J}'(\overline{u};h) =\partial_{y}J(\overline{y},\overline{u}) S'(\overline{u};h) +\partial_{u}J(\overline{y},\overline{u})h &=\langle \overline{y}-g, S'(\overline{u};h) \rangle + \alpha \langle \overline{u}, h\rangle,
Y = \operatorname{reshape}\left(X, \begin{pmatrix}n_1 \frac{\bar n}{n_1 n_d}n_d\end{pmatrix}\right) \; \in \mathbf R^{n_1 \,\times \,\bar n/(n_1n_d)\,\times\, n_d}, \quad \bar n := \prod_{i=1}^d n_i, X = \operatorname{reshape}\left(Y, \begin{pmatrix}n_1 \dots n_d\end{pmatrix}\right).
Q^{3}| \Phi^{\prime}\rangle = \int d y^{-}d y_{\bot}\left\{f^{3 a b}A_{\bot}^{a}( y_{\bot}, y^{-}) \Pi_{\bot}^{b}( y_{\bot}, y^{-}) + \rho_{m}^{3}( y_{\bot}, y^{-}) \right\}| \Phi^{\prime}\rangle = 0 \, .
\begin{aligned}(Z^n)_{-k}= 0, \left(\sum_{i=0}^{m+n-2}\lambda_i Z^{m-i}\right)_{-k}= 0, \left(\sum_{i=0}^{m+n-2}\lambda_i Z^{m-i}\right)_{1-n}+ F_{1-n}= 0.\end{aligned}
\tilde{C}^{(q,r)}(b;h)= \dfrac{\Xi^{(q,r)}(b;h)-\dfrac{\rho^{(q)}_{b}(b;h)}{W^{(q)}(b)}Z^{(q)}(b;\Phi(q+r))}{\dfrac{Z^{(q)}(b;\Phi(q+r))}{W^{(q)}(b)}-\dfrac{r}{\Phi(q+r)}}.
\sum_{i \neq j}\int f_i \ast \hat \varphi_i \overline{f_j \ast \hat \varphi_j}d\sigma \lesssim R^{n+1}\sum_{i=1}^{N}\sum_{j=1}^{N}|z_i-z_j|^{-\rho}\| f_i\|_{2}\| f_j\|_{2}&\lesssim R^{n+1}N \max_{i,j}|z_i-z_j|^{-\rho}\sum_{i=1}^{N}\| f_i\|_{2}^2.
\textrm{P}_{\textrm{co},\textrm{n}}(\beta_t) = 1- 2 \pi \lambda_\textrm{R}\int_{0}^{\infty}\!\! e^{- \left[ \theta \left(\frac{\beta_t}{\rho}\right)^{\frac{2}{\alpha}}+\pi \lambda_\textrm{R}\right] r^2}\mathcal{L}_{W}\left( \frac{\beta_t}{\rho}r^\alpha \right) r \mathrm{d}{r},
X_{n}={{\cal D}\kern - 0 . 1 5 e m \raise 0 . 1 7 e x \llap{/}\kern 0 . 1 5 e m \relax}Y_{n}/ \lambda_{n}, \qquad Y_{n}={{\cal D}\kern - 0 . 1 5 e m \raise 0 . 1 7 e x \llap{/}\kern 0 . 1 5 e m \relax}^{\dagger}{X_{n}}/ \lambda_{n}.
\int_{X}\left(\int_{X\setminus B\left(0,2|x|\right)}\left(|y|^{\frac{\sigma-l-1}{2}}e^{-\langle \rho, y\rangle}e^{|\rho|\frac{|y|}{2}}e^{-\frac{\xi}{2}|y|}\right)u(y)dy\right)^{q}|x|^{-\beta q}dx \leq C\left(\int_{X}|x|^{\alpha p}|u(x)|^{p}dx\right)^{\frac{q}{p}}.
\hat{G}^{\xi = 0}[ a , b ; E ] = \frac{e^{- \sqrt{2 E}| a - b |}}{\sqrt{2 E}}+ \frac{\frac{\sqrt{2 E}}{v^{r e n}}+ 1}{\frac{\sqrt{2 E}}{v^{r e n}}- 1}\frac{e^{- \sqrt{2 E}( | a | + | b | )}}{\sqrt{2 E}}
\omega(t)+\frac{1}{2\pi i}\int_\Gamma \omega(\zeta)\,d\ln \left ( \frac{\zeta-t}{\overline{\zeta}-\overline{t}}\right )- \frac{1}{2\pi i}\int_\Gamma \overline{\omega(\zeta)}\, d\, \left (\frac{\zeta-t}{\overline{\zeta}-\overline{t}}\right ) =f(t),t=x+iy\in \Gamma,
C_{\mathrm{GN}}={}\tfrac{d-2s_c}2 (1-s_c)^{-\frac{2(1-s_c)}{d-2s_c}}(\tfrac{d}2)^{-\frac{d}{d-2s_c}}\mathfrak{I}(\omega)^{-\frac{2}{d-2s_c}}\omega^{\frac{2(1-s_c)}{d-2s_c}}={}\tfrac{d-2s_c}2 (1-s_c)^{-\frac{2(1-s_c)}{d-2s_c}}(\tfrac{d}2)^{-\frac{d}{d-2s_c}}S_\omega(\Phi)^{-\frac{2}{d-2s_c}}\omega^{\frac{2(1-s_c)}{d-2s_c}}.
F [ \phi , \psi ; z^{0}] = \int d z^{1}\left( \phi \left( \partial_{1}\psi -{\frac 1 g}\omega_{1}( z ) \right) - \frac{2 m}{g^{2}}e^{g \phi}e_{1}^{a}( z ) V_{a}( \psi ) \right) \
\kappa_M(z,\bar{z}'):=\frac{K_M(z,\bar{z}')}{\sqrt{K_M(z)K_M(z')}}=(\tilde{e}_{\bar{z}},\tilde{e}_{\bar{z}'})=\frac{(e_{\bar{z}},e_{\bar{z}'})}{\|e_{\bar{z}}\|\|e_{\bar{z}'}\|}.
16c_5(q,1,p_0) = \hat W_{r_0}(q-2) - 4 \hat W_{r_0}(q-1) + 6\hat W_{r_0}(q) - 4\hat W_{r_0}(q+1) + \hat W_{r_0}(q+2) &= \hat W_{r_0}(q-2)-2\hat W_{r_0}(q)+\hat W_{r_0}(q+2) = :q_1.
{\cal I}^I_{\rm 1-loop}=-\frac{iV^{(8)}B^4}{2\pi}\sum_w\frac{T_2}{w^I G_{IJ}w^J}\bigg[ \sum_{ij}e^{2\pi i (a_i+a_j)_I w^I}(q_i+q_j)^4-\sum_i e^{4\pi i a_{iI}w^I}(2 q_i)^4\bigg]
C_{\varepsilon}( \beta ) = \frac{\prod_{j \in I_{- \varepsilon}}( u_{j}+ 1 )^{[ \beta^{\vee}: \alpha_{j}^{\vee}]}}{\prod_{i \in I_{\varepsilon}}u_{i}^{\operatorname*{m a x}( [ \beta^{\vee}: \alpha_{i}^{\vee}] , 0 )}}
U G^{- 1}V , \, \, U = \left( \begin{array}{c c}{{P_{L}}}&{{i P_{R}}}\\{{i P_{R}}}&{{P_{L}}}\\ \end{array}\right) , \, \, V = \left( \begin{array}{c c}{{P_{R}}}&{{i P_{L}}}\\{{i P_{L}}}&{{P_{R}}}\\ \end{array}\right) .
V(r) = \frac{L(L+1)}{r^2}- \frac{Q}{r}f + \kappa B_1 \frac{r}{f}+ \kappa B_2 \frac{1}{f^2}+ \kappa B_3 \frac{r}{f^3}+ \kappa B_4 \frac{1}{f^4}, B_4>0,
dz_j \cdot 1 = 0, d\overline{z}_j \cdot 1 = \sqrt{2}d\overline{z}_j, dz_i \wedge dz_j \wedge d\overline{z}_k \cdot 1 = 0, d\overline{z}_i \wedge d\overline{z}_j \wedge dz_k \cdot 1 = \sqrt{2}(\delta_{ki}d\overline{z}_j - \delta_{kj}d\overline{z}_i).
\chi_{-\varepsilon ,\lambda}\left( \theta_{-\varepsilon}\right) =\delta\chi_{\varepsilon ,\lambda}\left( \theta_{-\varepsilon}+\right) ,\text{\ \ \}\chi_{-\varepsilon ,\lambda}^{\prime}\left( \theta_{-\varepsilon}\right) =\delta \chi_{\varepsilon ,\lambda}^{\prime}\left( \theta_{-\varepsilon}+\right) .
x_{1}= T^{*3(j-1)}x -\alpha_{0}e_{0}= \sum_{i=1}^{3}\alpha_{i}e_{i}, y_{1}= T^{*3(j-1-r)}y -\beta_{0}e_{0}= \sum_{i=1}^{2}\beta_{i}e_{i}, z_{1}= T^{*3(n-2j-2+r)}z -\gamma_{0}e_{0}= \sum_{i=1}^{2}\gamma_{i}e_{i}.
\Pi_{o d d}( p ) = - \frac{4 m g^{2}}{\beta}\sum_{n = - \infty}^{\infty}~ \int_{0}^{1}d x \int \frac{d^{2}l}{( 2 \pi )^{2}}[ \frac{1 - \frac{1}{2}\{e^{i l \tilde{p}}+ e^{- i l \tilde{p}}\}}{( M_{1}^{2}+ ( ( 2 n + 1 ) \frac{\pi}{\beta}+ i \mu )^{2}}~ ] ,
\dfrac{p_{1}}{p_0}=\dfrac{(\gamma+1-2\tilde{b})\beta_{i}-(\gamma-1)}{(\gamma+1)-(\gamma-1+2\tilde{b})\beta_{i}},~~~~\tan{\delta_i}=\dfrac{({\beta_i}-1)\tan{\phi_i}}{1+{\beta_i}{\tan}^2{\phi_i}},
\begin{array}{c}\{m+1\}=1+q\{m\}=\{m\}+q^m,\ \{m+n\}=\{m\}+q^m\{n\},\q^m\{-m\}=-\{m\}, \{m\}=0 \ \Leftrightarrow \q^{m}=1.\end{array}
\dfrac{1}{\sqrt{d_2}}\; \Phi \left.\left(\vec{\alpha}=\left\langle - \frac{1}{2}, - \frac{1}{2}\right\rangle, \vec{\beta}=\left\langle - \frac{1}{2}, - \frac{1}{2}\right\rangle\; \right| \, \mathbf{v}\right) \;,
\int_{G_1}u^2dsdt =\int_{\widehat{T(G_1)}}w^2dsd\tau d\lambda\le C_\epsilon \int_{\widehat{T(G_1)}}|\widetilde\nabla w|^2dsd\tau d\lambda &\le C_\epsilon\int_{G_1}(tu_t^2+u_s^2)dsdt.
{\cal G}({\bf x}) \approx{\beta m \over 4\pi^2 \vert{\bf x}\vert}\;K_1(m \vert{\bf x}\vert ) \approx{\beta m^{1/2}\over 2^{5/2}\pi^{3/2}\vert{\bf x}\vert^{3/2}}\;e^{-m \vert{\bf x}\vert}\;,
I_{n}(\delta_{0}) =\sum_{t=1}^{n}\sigma^2_t(\delta_{0})\begin{bmatrix}x_{t}x_{t}^\mathrm{T}x_{t}(m\pi)_{t-J_\phi}^\mathrm{T}0 (m\pi)_{t-J_\phi}x_{t}^\mathrm{T}A_{\phi\phi,t}A_{\phi\theta,t}0 A_{\theta\phi,t}A_{\theta\theta,t}\end{bmatrix}
D_uG_m(hx,z) \Dh(x) = D_uG_{n_0}(\ell x, G_{m-n_0}(hx,z)) \D\ell(x) & + D_sG_{n_0}(\ell x, G_{m-n_0}(hx,z)) \D_uG_{m-n_0}(hx,z) \Dh(x).
g(t_2, \xi) - g(t_1, \xi)=\sum_{k_1,k_2,k_3\in \mathbb{Z}}\sum_{(\iota_1, \iota_2,\iota_3)\in \mathcal{S}}J^{\iota_1, \iota_2,\iota_3}_{k_1,k_2, k_3}+ \int_{t_1}^{t_2}e^{i L(t,\xi) + i t \Lambda(\xi)}\widehat{R}(t,\xi) d t,
N_d(\mathrm{CH}(S))\cap\mathrm{An}(R) \subseteq \mathrm{An}(R)\cap\bigcup_{i=1}^N N_{C+d}(\mathrm{Cone}(x, B(y_i, \varepsilon)))\setminus B(x, R) \subseteq \mathrm{An}(R)\cap\bigcup_{i=1}^N \mathrm{Cone}(x, B(y_i, \varepsilon(1+AC'))),
\|\nabla\Delta^{-1}f\|_{L^2}^2= \|\nabla f_2\|_{L^2}^2=-\langle f,f_2\rangle=-\langle F,F_2/\psi_y\rangle=\langle \nabla F_1,\nabla(F_2/\psi_y)\rangle \leq \|\nabla F_1\|_{L^2}\|\nabla (F_2/\psi_y)\|_{L^2}.
Ps_{n}^{m}\left({z,\gamma^{2}}\right) =V_{n}^{m}\left( \gamma\right)\frac{\sin\left\{{\gamma z-{\frac{1}{2}}\pi n}\right\}}{\gamma z}\left\{{1+{O}\left({\frac{1}{z}}\right)}\right\}\quad\left({z\rightarrow\infty}\right) .
f(x_+) - f(x_-) = \frac{z-m}{z-y}\left(\varphi(x_+-y) -\varphi(x_--y) - \varphi(x_+-m)+\varphi(x_--m)\right) +\frac{m-y}{z-y}\left(\varphi(z-x_+) -\varphi(z-x_-) - \varphi(x_+-m)+\varphi(x_--m)\right).
0 =\sum_{u=1}^m h_u\theta_{ujk}(z_j)+B_{jk}(z_k)g_k(z_k)-g_j(z_j),\,\,\,\,\, 0 =\sum_{u=1}^mh_u\Lambda_{uj}'(z_j)-[\sum_{r,s=1}^n \Lambda_{M_{0j}}^{r,s}(z_j)\frac{\partial}{\partial z_j^r}\wedge \frac{\partial}{\partial z_j^s},\sum_{\alpha=1}^n g_j^{\alpha}(z_j)\frac{\partial}{\partial z_j^{\alpha}}],\,\,\,\,\,
\zeta^\wedge_{L, p}(s) = \int_{\substack{(A,\nu)\in \dot{H}v_p(A) \ge 0 v_p(A) + |v_p(\nu)| \geq 0}}\, \lvert \det A \rvert_p^{\, 5s-12}\, \lvert \nu \rvert_p^{\, -s+6}\theta_0 \big( (A,\nu)^\rho \big) \, \mathrm{d}\mu_p(A,\nu).
D_\zeta(\zeta_1,\zeta_2) \partial_\zeta D_z(\zeta,\zeta') F(z) =\partial_{\zeta_1}D_z(\zeta',\zeta_1) D_\zeta (z,\zeta_2) F(\zeta)+ \partial_{\zeta_2}D_z(\zeta',\zeta_2) D_\zeta (\zeta_1, z) F(\zeta),
\limsup_{\eta\rightarrow0}\limsup_{\epsilon\rightarrow0}\limsup_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}_{N}\exp\left\{a\left(\int_{0}^{\eta}\sum_{i=1}^{N}\rho_{\epsilon,i}\left(x^{N}(t)\right)dt-NA^{N}(\eta)\right)\right\}\le0
\tilde{Z}^{\Lambda ,\Lambda_0}(\varrho \, ;J) = \int d\mu_{\Lambda ,\Lambda_0}(\phi ) e^{- \frac{1}{\hbar}\bigl (\tilde{L}^{\Lambda_0,\Lambda_0}(\varrho \, ;\phi ) + \tilde{I}^{\Lambda_0, \Lambda_0}(\varrho ) \bigr ) +\frac{1}{\hbar}\langle \phi , J \rangle}.
\frac{1}{|hv^{\perp}\mp t v|}\left( hv^{\perp}\mp tv \right) =\frac{1}{|hv^{\perp}\mp t v|}\left( \frac{t}{r_0}(h v^{\perp}\mp r_0v)+ h \left(1- \frac{t}{r_0}\right) v^{\perp}\right), = \frac{1}{|hv^{\perp}\mp t v|}\left( \frac{t|h v^{\perp}\mp r_0v|}{r_0}n_{B^{\perp}}(\alpha^{\pm})+ h \left(1- \frac{t}{r_0}\right) v^{\perp}\right).
k =1 d_1^{-2^{1-1}}=d_1^{-1}\equiv d_4P_{32}(1)=d_4d_2d_1= & =d_4P_{32}(2)d_1\pmod{2}, k =2 d_1^{-2^{2-1}}=d_1^{-2}\equiv d_4P_{32}(2)=d_4d_2\pmod{2}.
\lim_{h\to 0}\mathbb{E}\left[\left(\frac{M(x+h)-M(x)}{h}- \frac{d}{dx}M(x) \right)^2 \right] & = \lim_{h \to 0}\mathbb{E}\left[ \left(\frac{d}{d x}M(\alpha(x,h)) -\frac{d}{d x}M(x)\right)^2 \right] & \leq \lim_{h\to 0}\mathbb{E}[K^2|h|^{2\delta}] = 0.
{\sum_{n \le N}\pi(n) R_{X}^{n}}- \log N= -0.26842 \cdots \approx B_{X}, \prod_{n \leq N}\left( 1- R_{X}^{n}\right)^{\pi(n)}= 0.18447 \cdots \approx \frac{e^{-\gamma}}{C_{X}}\cdot \frac{1}{N}= 0.18457 \cdots.
p \leq 1, n \geq (24 \cdot 2^{\binom{\ell}{2}}t^t q \ell! R^t)^2, n > (144 \ell!^2 \ell)^{2\ell}, n > \ell^{40\ell^2(\ell+1)}, n > (1000 q)^{8\ell(\ell+1)}R^{4k\ell(\ell+1) + 4t\ell}\binom{t}{k}^{4\ell}.
\begin{aligned}\dfrac{|b_{1,\delta}(x_a)|}{2}= \, \left|\int_0^1 h(x_a+\delta u(t)) u(t)\,dt\right| = \,\left|\int_0^1 (h(x_a+\delta u(t))-h(x_a))u(t)\,dt\right| \le \,\int_0^1 \left|h(x_a+\delta u(t))-h(x_a)\right|\,dt \le L_r \delta. \end{aligned}
\begin{aligned}\alpha(q-1)\theta\frac{n-p}{p}+\theta=\frac{n\theta}{p}, \alpha \theta \frac{n-p}{p}+1=\theta, &\alpha^{-\alpha\theta\frac{n-p}{p}}R^{-\alpha(q-1)\frac{n-p}{p}\theta}C^{\frac{n-p}{p}\theta}=1, \frac{1}{\alpha}=q-1.\end{aligned}
\int_{\{\Psi<-t_1\}}|\tilde F-(1-b_{t_0,B}(\Psi))f_{t_0}|^2 \leq \int_{\{\Psi<-t_1\}}|\tilde F-(1-b_{t_0,B}(\Psi))f_{t_0}|^2e^{-\Psi+v_{t_0,B}(\Psi)}\leq (e^{-t_1}-e^{-t_0-B})\int_{\{\Psi<-t_1\}}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|f_{t_0}|^2e^{-\Psi},
(\alpha, \beta){=}\begin{cases}{\pm}3 \mbox{iff}\ \alpha{=}\pm\beta \;\;\,2 \mbox{iff}\ \alpha{-}\beta{=}\gamma\ (\mbox{root}\ \gamma\perp h){-}2 \mbox{iff}\ \alpha{+}\beta{=}h{-}1 \mbox{iff}\ \alpha{+}\beta{=}h{+}\gamma\ (\mbox{root}\ \gamma\perp h)\end{cases}
S_4^{\prime}(\phi_s)=\biggl<S_4(\phi_s,\phi_f)\biggr>_{0f}+\frac{1}{2}\biggl(\biggl<S_4^2(\phi_s,\phi_f)\biggr>_{0f}-\biggl<S_4(\phi_s,\phi_f)\biggr>^2_{0f}\biggr)+\cdots .
\phi_{( 0 )}^{\prime \prime}+ 3 \alpha_{( 0 )}^{\prime}\phi_{( 0 )}^{\prime}+ \beta_{( 0 )}^{\prime}\phi_{( 0 )}^{\prime}- \gamma_{( 0 )}^{\prime}\phi_{( 0 )}^{\prime}= - \sqrt{2}b \left( b^{2}- \frac{2}{3}\right) \sigma^{2}e^{2 ( \gamma_{( 0 )}- \sqrt{2}b \phi_{( 0 )})}.
M_{2}(\mathcal{NC}(G))=\frac{1}{2}(2^{12k}-7\cdot2^{10k}-2^{9k}+21\cdot2^{8k}-26\cdot2^{6k}-2\cdot2^{5k}+15\cdot2^{4k}+3\cdot2^{3k}+6\cdot2^{2k}-8\cdot2^{k}).
\bigg\|\frac{T^{*2K}x}{x_{2K}w_{2K-1}\cdots w_{0}}-e_{0}\bigg\|^{2}\leq \frac{\mu^{2}\delta}{w_{0}^{2}w_{1}^{2}}\sum_{i=0}^{\infty}w_{2i+2K+1}^{2}~\bigg|\frac{x_{2i+2K+2}}{x_{2K}}\bigg|^{2}\leq \frac{\mu^{2}\delta}{w_{0}^{2}w_{1}^{2}}\sum_{i=0}^{\infty}w_{2i+2K+1}^{2}< C\epsilon,
\langle \varphi^{2}(x)\rangle^{(ab)}\approx - \frac{(b-a)^{1-D}}{(4\pi )^{D/2}\Gamma (D/2)}\int_{m_0}^{\infty}dy\, \frac{(y^2-m_0^2)^{D/2-1}}{c_b(y)e^{2y}/c_a(y)-1}F(y,r) ,\quad a,b\to \infty, b-a={\mathrm{const}}.
V\equiv\sum_{\mu\in{\mathcal R}}g_{|\rho|}(\rho\cdot\mu)(\rho^\vee\!\cdot\nu) \frac{\ell_\mu^j-\ell_{s_\rho(\mu)}^j}{\ell_\mu-\ell_{s_\rho(\mu)}}\, \ell^k_{s_\rho(\nu)}\check{s}_\rho=0.
h^1({N_{X'}}_{|_{Y_j}}(K_{Y_j}+D_j-3H))=h^0({N^*_{X'}}_{|_{Y_j}}(3H-D_j))=h^0(N^*_{Y_j}(3H-D_j))-2=h^1(N_{Y_j}(K_{Y_j}+D_j-3H))-2=0.
q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}\theta(\lambda)c_{s,1}\theta(\mu_{n_s^{-1}}(-1)\mu) =q_\lambda^{1/2}q_s^{-1}q_{\lambda\mu_{n_s^{-1}}(-1)}^{-1/2}E(\lambda\mu_{n_s^{-1}}(-1))c_{s,-1}E(\mu) & = dE(\lambda\mu_{n_s^{-1}}(-1))c_{s,-1}E(\mu)
\|v\|_{F^{s,0}(T)}^2 \lesssim \|v(0)\|_{H^{s,0}}^2 + \|v\|_{F^{s,0}(T)}^3 + \|v\|_{F^{0,0}(T)}\|v\|_{F^{s,0}(T)}\|u_2\|_{F^{2s,0}(T)}.
P_{A_\alpha}(M \wedge \mathrm{TAQ}(A_\alpha/R)) \simeq A_\alpha \wedge_{P_R X}P_{P_RX}(M \wedge \mathrm{TAQ}(P_RX/R)) \simeq A_\alpha \wedge_{P_RX}P_RX \otimes_R M \xrightarrow{\simeq}\left(A_\alpha \otimes_R M\right).
\begin{aligned}\int_{a\in \sigma( f^{-n}(V))}\mathrm{Vol}(\Gamma_n(a)) da \ge \int_{a\in \sigma( f^{-n}(V))}\mathrm{Card}\left(\Psi^j_n\cap \sigma^{-1}(a)\right) da = \mathrm{Vol}(\sigma(\Psi^j_n)) \ge e^{n(h_\nu-2\varepsilon)}e^{-n(2m_j\chi^+_j+\ldots+2m_q\chi^+_q)}e^{-8kn\varepsilon}.\end{aligned}
R_{z \bar{z}z \bar{z}}^{\prime}\equiv R_{z \bar{z}z}^{\prime}{}^{z}g_{z \bar{z}}^{\prime}= - 2 g_{z \bar{z}}g_{z \bar{z}}^{\prime}+ C_{z z z}g^{z \bar{z}}\bar{C}_{\bar{z}\bar{z}\bar{z}}\, .
{}^{({\cal C}^{\cal R}_H)}\Psi_{Y,X}\circ(\triangleleft u) \circ{}^{({\cal C}^{\cal R}_H)}\Psi_{X,Y}={}^{\cal C}\Psi_{Y,X}\circ ((\triangleleft u)\otimes(\triangleleft u)) \circ{}^{\cal C}\Psi_{X,Y}
(w_\nu(\omega_k) - w_\lambda(\omega_k))c_{\lambda,\mu}^{\nu,d}= \sum_\eta 2^{A(\lambda, \eta)}c_{\eta, \mu}^{\nu,d}- \sum_\xi 2^{A(\xi, \nu)}c_{\lambda, \mu}^{\xi,d}+ (c_{\lambda^*, \mu}^{\nu, d-1}- c_{\lambda, \mu}^{\nu^+, d-1}) + (c_{\lambda^{**}, \mu}^{\nu, d-1}- c_{\lambda, \mu}^{\nu^{++}, d-1}) \/,
\det\begin{pmatrix}\mu^{1}_{2,1}+O(u) \mu^{1}_{2,2}+O(u) O(\log u) \frac{\sqrt{u}}{2}[\mu^{2}_{2,1}+O(u)] \frac{\sqrt{u}}{2}[\mu^{2}_{2,2}+O(u)] -\frac{\mu^{1}_{2,2}+o(1)}{\sqrt{u}}\frac{\sqrt{u}}{2}[\mu^{3}_{2,1}+O(u)] \frac{\sqrt{u}}{2}[\mu^{3}_{2,2}+O(u)] -\frac{\mu^{2}_{2,2}+o(1)}{\sqrt{u}}\end{pmatrix}.
e^{-it\Delta}v_n^j(t) = e^{ix\xi_n^j}e^{-it\Delta}\Psi^j_{[h_n^j]}(t) = e^{ix\xi_n^j}\bigl (e^{-i(h_n^j)^2 t \Delta}\Psi^j ((h_n^j)^2t)\bigr)_{\{h_n^j\}}.
F_{i-\frac{1}{2}}^{*}=\frac{1}{2}\{[F(u_{i-\frac{1}{2}}^{R})+F(u_{i-\frac{1}{2}}^{L})]-a_{i-\frac{1}{2}}[u_{i-\frac{1}{2}}^{R}-u_{i-\frac{1}{2}}^{L}]\}.
\sum_{f = 0}^{d}\sum_{{\sigma_{f}\in P_{f}, \sigma_{d - f}\in P_{d - f}}\atop{\sigma_{f}\cup \sigma_{d - f}= \sigma_{d}}}( - 1 )^{l ( \sigma_{f})}{}_{l ( \sigma_{f}) - 1}C_{l ( \sigma_{d}) - 1}S ( \sigma_{f}) S ( \sigma_{d - f}) = ( - 1 )^{l ( \sigma_{d})}S ( \sigma_{d})
\Gamma_g^{\alpha \beta \gamma}\; (x,y,z) = \Gamma^{\alpha \beta \gamma}_r \;(x,y,z) -{i \over 4\pi^2}\; \varepsilon^{\beta i \gamma j}{\partial \over \partial x_\alpha}{\partial \over \partial y^i}G(x,y){{\partial}\over{\partial z^j}}\delta (y-z).
( z_{1},{\cal Z}_{1},{\cal F}_{1}) \times ( z_{2},{\cal Z}_{2},{\cal F}_{2}) = ( z_{1}z_{2}, \, z_{1}{\cal Z}_{2}+ z_{2}{\cal Z}_{1}, \, z_{1}{\cal F}_{2}+ z_{2}{\cal F}_{1}- 2{\cal Z}_{1}{\cal Z}_{2}) .
h ={\mathrm{e}}^{a_{0}L_{0}}\;{\mathrm{e}}^{a_{1}L_{1}}\;{\mathrm{e}}^{b_{- 1}J_{- 1}}\;{\mathrm{e}}^{b_{0}J_{0}}\;{\mathrm{e}}^{b_{1}J_{1}}\;{\mathrm{e}}^{b_{2}J_{2}}\; \tilde{h}\; ,
Q_{[\pi]}= \bigcup_{t \in T}D_{t \cdot \pi}= \bigcup_{s, t \in T}\Big( D_{t \cdot \pi}\cap s \cdot M \Big) = \bigcup_{s, t \in T}\Big( D_{s t \cdot \pi}\cap s \cdot M \Big) = \bigcup_{s, t \in T}s \cdot (D_{t \cdot \pi}\cap M) = \bigcup_{s, t \in T}\Big( C_{t \cdot \pi}\cap s \cdot M \Big) = \bigcup_{t \in T}C_{t \cdot \pi}.
\begin{array}{r c l c r c l c r c l c r c l}{\alpha_{6}}&{=}&{\frac{i}{6 !}}&{;}&{\alpha_{4}}&{=}&{\frac{1}{2 4 !}}&{;}&{\alpha_{0}}&{=}&{\frac{1}{4 8}}&{;}&{\alpha_{2}}&{=}&{\frac{i}{1 6}}\\ \end{array}
\sup_{\theta \in A \times W_n^{(\eta)}}\ln(n)^q\left| \frac{1}{n}\sum_{i=1}^n \left( f(X_{\frac{i-1}{n}}, \delta, \alpha) - f(X_{\frac{i-1}{n}}, \delta_0, \alpha_0) \right) h(z^n_i(\theta), \alpha) \right| \to 0,
\dfrac{b_{2n+1}}{b_n}= 2 \sqrt{p}b_{n}+ 2\left( a_1 - b_0\sqrt{p}\right)^{n+1}= -2 \sqrt{p}b_{n}+ 2\left( a_1 + b_0\sqrt{p}\right)^{n+1}= 2a_{n+1}.
\frac{\theta}{2^{2n+1}}\geq M\mu(y) =\sup_{0<\delta<4s-\|y\|_\infty}\frac{\mu(D_\delta(y))}{\kappa_n \delta^{2n+1}}&\ge\sup_{2r_i<\delta<4s-\|y\|_\infty}\frac{\mu(D_\delta(y))}{\kappa_n \delta^{2n+1}}&\geq\sup_{2r_i<\delta<4s-\|y\|_\infty}\frac{\mu(D_{r_i}(x_i))}{\kappa_n \delta^{2n+1}}=\frac{\mu(D_{r_i}(x_i))}{\kappa_n (2r_i)^{2n+1}}>\frac{\theta}{2^{2n+1}},
\mathcal{R}_m(P) = \begin{cases}1 P\geq \overline{P}_h H_2\left( \frac{1}{2}+ \frac{1}{2 \epsilon}\frac{1 - P\delta^{-2}(1-\epsilon^2)}{1 + P\delta^{-2}(1-\epsilon^2)}\right) \frac{\delta^2}{(1+\epsilon)^2}\leq P<\overline{P}_h 0 P<\frac{\delta^2}{(1+\epsilon)^2}.\end{cases}
E_{0}(\mathbf{z})B_{n,\mathbf{m}}^{(d)}(\mathbf{z}\mid{\boldsymbol{\omega}})= \sum_{i=1}^{r}B_{n,\mathbf{m}_{i}}^{(d)}(z\mid{\boldsymbol{\omega}}) \left(m_{i}+\frac{d}{2}(r-i)\right) h_{-,i}^{(d)}(\mathbf{m}).
\Lambda^{( 3 )}( g_{2}, g_{3})^{\alpha}\: + \: \Lambda^{( 3 )}( g_{1}, g_{2}g_{3})^{\alpha}\: = \: g_{3}^{*}\Lambda^{( 3 )}( g_{1}, g_{2})^{\alpha}\: + \: \Lambda^{( 3 )}( g_{1}g_{2}, g_{3})^{\alpha}\: + \: d \log \gamma_{\alpha}^{g_{1}, g_{2}, g_{3}}
c(n):=\frac{1}{\Gamma(\frac{1}{2})^{3}}\frac{\Gamma(n_{1}+\frac{1}{2})\Gamma(n_{2}+\frac{1}{2})\Gamma(n_{3}+\frac{1}{2})}{\Pi_{_{i=1}}^{3}\Gamma(n_{4}-n_{i}+1)\cdot\Pi_{1\leq j<k\leq3}\Gamma(n_{j}+n_{k}-n_{4}+1)}.
\mathcal{P}_{\mathfrak{b}_{a}}^{ext}\left( \mathrm{v},\upsilon\right)=\int_{\mathfrak{b}_{a}}(\sigma\cdot\nabla_{y}\mathrm{v}+z_{a}\cdot\upsilon+\mathcal{S}_{a}\cdot\nabla_{y}\upsilon)\: d\mu(y),
a_i^{[t]}d_i^{[t]}+2 b_i^{[t]}=0 \quad\forall \;i \in \mathcal{I},\; \forall t \in \mathcal{T}_1 a_i^{[t]}d_i^{[t]}- 2 b_i^{[t]}= 0 \quad\forall \;i \in \mathcal{I},\; \forall t \in \mathcal{T}_2.
\begin{aligned}\limsup_{N\rightarrow \infty}\left\Vert \left( f_N^{(s)}(0) - \int_{\mathcal{P}(\mathbb{R}^{2d})}h_N^{(s)}(0;h_0) d\pi (h_0)\right)\mathbf{1}_{Z_s \in \mathcal{G}_s \cap\hat{\mathcal{U}}_s^{\eta (\varepsilon)}}\mathbf{1}_{E_s (Z_s) \leq R^2}\right\Vert_{L^\infty_{Z_s}}& = 0\end{aligned}
\hat{A}\geq \frac{1-\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-\frac{2\beta}{\alpha}\vert L_{A\Omega_1}\vert -2\vert U_{A\Omega_1}\vert \geq \frac{1-\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-\frac{2\beta}{\alpha}\vert B_{A\Omega_1}\vert:= T.
S_2(...) = \frac{N^{3/4-i(t+\nu)}\,\eta_f(p_2)}{(p_1q)^{1/2}\,p_2^{1/4}}\,\sum_{\pm}\sum_{m \ll M_0}\frac{\lambda_f(m)}{m^{1/4}}e\left(-m \frac{\overline{(a+bq) p_2}}{q}\right) \mathcal{I}^{\pm}(m,u,q,p_1,p_2).
\phi(c(x)) = \frac{1}{\omega_C}1 \wedge c(x) \wedge c(x)^2 = \frac{1}{\omega_C}1 \wedge c(x) \wedge c\left(\frac{\bar x^2 \gamma}{t}\right) = \frac{1}{\omega_I}x \wedge \frac{\bar x^2 \gamma}{t}