harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
P(R_{n}(f^*_{\eta}) - R_{n}(f^*) \ge R(f^*_{\eta}) - R(f^*) + \eta) &=P\left(R_{n}(f^*_{\eta}) - R_{n}(f^*) - (R(f^*_{\eta}) - R(f^*)) \ge \eta\right) &\le \exp\left(-\frac{n}{4}\left(\frac{\eta^{2 - \beta}}{B}\land 3\eta\right)\right) &\le \exp\left(-\frac{n\eta^{2 - \beta}}{4B}\right).
s_{\alpha_{l_1}}\dots s_{\alpha_{l_{p}}}s_{\alpha_{i-1}}s_{\alpha_{l_{p+2}}}\dots s_{\alpha_{l_k}}(\beta_i) = s_{\alpha_{l_1}}\dots s_{\alpha_{l_{p}}}s_{\alpha_{i-1}}(\beta_i) = s_{\alpha_{l_1}}\dots s_{\alpha_{l_{p}}}(\beta_{i-1}) = \beta_{i-1}\ne \beta_n.
\sup_{P\in\mathcal{D}_{\mu}}\Big(\frac{1}{|P|}\int_P\sum_{k=\mu}^{\infty}{2^{skq}\big|\Pi_kT_{[a]}f(x) \big|^q}dx \Big)^{1/q}&\lesssim\sup_{0\leq k\leq \mu-1}{\Vert 2^{k(s+m)}\Pi_kf \Vert_{L^{\infty}}}+\sup_{R\in\mathcal{D}_{\mu}}\Big(\frac{1}{|R|}\int_R\sum_{k=\mu}^{\infty}{2^{(s+m)kq}\big|\Pi_kf(x) \big|^q}dx\Big)^{1/q}.
0 = tr(e_{[-1,n]}x^*x) = tr((e_{n+1}e_{n+2}..e_{2n+1})e_{[-1,n-1]}(e_{2n}..e_{n+1})x^*x) = tr(e_{[-1,n-1]}(e_{2n}..e_{n+1})(e_{n+1}..e_{2n+1})x^*x)~~~~~~~(\textrm{since}~ x^*x\in N) = tr(e_{[-1,n-1]}({\tau}^{n-1}e_{2n}e_{2n+1})x^*x).~~~~~~~~~~~(\textrm{by~equation}(3.4))
\mathcal{F}_{n}^{\left( j\right)}\left( x\right) \nabla^{2}\mathbb{K}_{n}^{(j)}\left( x\right) +\mathcal{G}_{n}^{\left( j\right)}\left( x\right)\nabla \mathbb{K}_{n}^{(j)}\left( x\right) +\mathcal{H}_{n}^{\left( j\right)}\left( x\right) \mathbb{K}_{n}^{(j)}\left( x\right) =0,
G_{[i]}:= \bigcup_{u_i=0}^{s_i-1}\bigcup_{q_i=0}^{p_i-1}G_{i-1}\beta^{u_i t_i}\alpha_i^{q_i}, W_{[i]}:= \bigcup_{u_1=0}^{s_1-1}\dots \bigcup_{u_{i-1}=0}^{s_{i-1}-1}\bigcup_{q_1=0}^{p_1-1}\dots \bigcup_{q_{i-1}=0}^{p_{i-1}-1}\Big(W_i \beta^{\sum_{j=1}^{i-1}u_j t_j}\prod_{j=1}^{i-1}\alpha_j^{q_j}\Big).
\sum_{\mu \in \Lambda}\gamma_{\mu}\langle \mu | : e^{B_{1}( z )}: e^{A_{1}( z )}| \chi \rangle V_{0}( \psi , z ) = V_{1}( \psi , w - z ) \sum_{\mu \in \Lambda}\gamma_{\mu}\langle \mu | : e^{B_{1}( z )}: e^{A_{1}( z )}| \chi \rangle \, ,
V(t, \overline{x}(t)+h)-V(t,\overline{x}(t)) +\langle \overline{p}(t),h\rangle &\leq \langle \overline{p}(t),h\rangle + \langle -\overline{p}(T),x_{h}(T)-\overline{x}(T)\rangle + c \mid x_{h}(T)-\overline{x}(T)\mid^2.
\Gamma_{c}= \left( \begin{array}{c c c}{\displaystyle{\gamma_{c b}^{a}}}&{\displaystyle{{\frac{1}{2}T_{g}^{a}{_c}}}}\\{\displaystyle{- \frac{{\dot{g}}_{a c}}{\sin^{2}\theta}}}&{\displaystyle{\mathrm{c o t}\theta \partial_{c}\theta}}\\ \end{array}\right)_{a , b , c , = 1 \cdots M}
(R^{k}f)_{p_{1}q_{1}\dots p_{m-k}q_{m-k}}^{i_{1}\dots i_{k}}:= \alpha (p_{1}q_{1}) \dots \alpha (p_{m-k}q_{m-k}) \frac{\partial^{m-k}f^{i_{1}\dots i_{k}}_{p_{1}\dots p_{m-k}}}{\partial x^{q_1}\dots \partial x^{q_{m-k}}}
{F_{{\Gamma_{k|X \ge Y}}}}\left({{z}}\right)= 1 - \frac{\lambda_k+\lambda_{\bar k}}{\lambda_{\bar k}}{e^{-{\lambda_k}{\varphi_k}}}+ \frac{\lambda_k}{\lambda_{\bar k}}e^{-\left(\lambda_k+\lambda_{\bar k}\right)\varphi_k},{F_{{\tilde\Gamma_{k|X < Y}}}}\left({{z}}\right)= 1-e^{-\left(\lambda_k+\lambda_{\bar k}\right)\tilde\varphi_k},
\varphi_{s,j,i}:\Omega_{b_{j}}\times\Omega\rightarrow\Omega_{b_{i}}\times\Omega &\varphi_{s,j,i}\left(\omega_{j},\omega\right)=\left(f_{s,j,i}\left(\omega_{j,}X_{s_{i}}\left(\omega\right)\right),\omega\right).
\nabla_x \,\omega(y_1,y_2) + \sum_{r=1}^2 \partial_{y_r}\left( \Psi(x,y_r) \omega(y_1,y_2)\right) dy_r = \omega(x,y_1)\omega(x,y_2), \nabla_x \,\nu_a(y)+ \partial_y\left( \Psi(x,y) \nu_a(y)\right) dy =\omega(x,y) \nu_a(x).
\sigma_{1}(t): T^{*}M|_{Q,P}\times \mathbb{R}\rightarrow T^{*}M\times \mathbb{R}&;({\sigma_{1}^{*}}(t)\Xi^{\alpha}:=\epsilon^{\alpha},{\sigma_{1}^{*}}(t)\Psi_{\alpha}:=\epsilon_{\alpha},{\sigma_{1}^{*}}(t)Q^{i},{\sigma_{1}^{*}}(t)P_{i},{\sigma_{1}^{*}}(t)u=t) &\mapsto (\Xi^{\alpha},\Psi_{\alpha},Q^{i},P_{i},u).
M=\begin{matrix}A\\C\\G\\T\end{matrix}[\begin{matrix}3&6&1&0&0&6&7&2&1\\ 2&2&1&0&0&2&1&1&2\\ 1&1&7&10&0&1&1&5&1\\ 4&1&1&0&10&1&1&2&6\end{matrix}].
\partial^\nu \Lambda_{\nu \mu}= \varepsilon_{\mu \nu}n^\nu \bar{c}^a \, \frac{\delta S}{\delta \phi^a}- A^a_\mu \, \frac{\delta S}{\delta c^a}+ \partial_\mu \bar{c}^a \, \frac{\delta S}{\delta b^a}+ \mbox{total deriv.}\quad .
\eta(\tau_n^q\tau_j, \tau_i)=\eta(\tau_n^{q-1}\tau_j, \eta(\tau_n,\tau_i))=\eta(\tau_n^{q-1}\tau_j,\tau_n^{q-1+D(i)}\tau_{i+1})=\eta(\tau_j,\tau_n^{D(i)}\tau_{i+1})
(\ln f)^{\ast}(v) =\sup_{y\in I}[vy-\ln f(y)]=\lim_{y\rightarrow-\infty}[vy-\ln f(y)]=(\lim_{y\rightarrow-\infty}y)\left( v-\lim_{y\rightarrow-\infty}\frac{\ln f(y)}{y}\right) & =(-\infty)\left( v-\lim_{y\rightarrow-\infty}\frac{f^{\prime}(y)}{f(y)}\right) =(-\infty)(v-1)=\infty\quad\forall v<1.
\smashoperator{\lim_{\nu\to \infty}}\,(x^{}_\nu+\sqrt{-1}\, y^{}_\nu;\xi^{}_\nu) = (x^{}_0;\xi^{}_0), \smashoperator{\lim_{\nu\to \infty}}\,|y^{}_\nu|\,|\eta^{}_\nu| = 0.
\left[ \begin{array}{l}{m}\\{n}\\ \end{array}\right]_{q}= \frac{( q ; q )_{m}}{( q ; q )_{n}( q ; q )_{m - n}}, ( p ; q )_{m}= \prod_{k = 0}^{m - 1}( 1 - p q^{k}) , ( p ; q )_{0}= 1 .
\psi(x_1) = (2\pi)^{-3/2}\int d^3p_1 (m/E_1)^{1/2}[b_{p_1r_1}u^{r_1}(p_1)e^{ix_1.p_1}+ d^*_{p_1r_1}v^{r_1}(p_1) e^{-ix_1.p_1}] ;
\theta_\varepsilon = \varepsilon + \frac{1}{2}\Bigl( \frac{\partial E_\sigma}{\partial \dot q^\nu}- \frac{d}{dt}\frac{\partial E_\sigma}{\partial \ddot q^\nu}\Bigr) \omega^\sigma \land \omega^\nu - \frac{\partial E_\sigma}{\partial \ddot q^\nu}\dot \omega^\sigma \land \omega^\nu.
\Re \dfrac{zF_{z}(z)-\overline{z}F_{\overline{z}}(z)}{F(z)}=\Re \dfrac{zf_{z}(z)-\overline{z}f_{\overline{z}}(z)}{f(z)}+\Re \dfrac{zR_{z}(z)-\overline{z}R_{\overline{z}}(z)}{R(z)}.
\big(u+vR_0(\lambda-\kappa^2)\;\!v\big)^{-1}=\big(J_0(\kappa)+S\big)^{-1}+\frac1{\kappa^2}\;\!\big(J_0(\kappa)+S)^{-1}SJ_1(\kappa)^{-1}S\big(J_0(\kappa)+S\big)^{-1}.
\left(\tfrac{\partial y(t)}{\partial t},\phi\right)+2\alpha \left(D\tfrac{\partial y(t)}{\partial t}, D\phi\right)+2\nu\left(Dy(t),D\phi\right)+b\left( \phi, y(t), \sigma\left(y(t)\right)\right)-b\left(y(t), \phi, \sigma\left(y(t)\right)\right) &=\left( u(t),\phi\right)\mbox{for all}\ \phi\in V.
{\mathfrak q}_{0,}^{Sp(p,q)\times Sp(1)}={\mathfrak l}_{0,}^{Sp(p,q)\times Sp(1)}+{\mathfrak u}_{0,}^{Sp(p,q)\times Sp(1)}\supset{\mathfrak q}_0^{Sp(p,q)\times Sp(1)}
\rho^{XQ^{\prime}Y}:=[I^{X}\otimes E^{Q}](\rho^{XQ})=\sum_{x=1}^{n}\sum_{y=1}^{m}p_{x}q_{y|x}|x\rangle\langle x|\otimes\rho_{y|x}\otimes|y\rangle\langle y|.
\biggl\{-4z^3(1-z) \frac{d^2}{dz^2}+ 2[(4a+3)z^3 - (4a-4b_1-2l-d+3)z^2 - 4(b_1-2b_2)z - 8b_2] \frac{d}{dz}+ [2a(2a+1) - A]z^2 + [- 4a^2 + 8ab_1 + 2a(2l+d-1) - 2b_1 - \epsilon]z + B_1 - 2b_1(4a-2b_1-2l-d-1) + 4b_2(4a-3) \biggr\}\phi(z) = 0
Z_{ft}(\tilde{K}) = \int D \tilde{\Psi}\, e^{-L + \tilde{\Psi}Q \tilde{K}}\,, Q =\begin{pmatrix}0 -1 0 1 0 0 0 0 1\end{pmatrix}, \tilde{K}=\begin{pmatrix}\tilde{k}\tilde{\bar{k}}\tilde{j}\end{pmatrix}\,.
\lim_{\substack{\nu \to \mu s \to k^+}}\mathbf{P}(\nu,s) \geq \lim_{\varepsilon \to 0}\liminf_{\nu \to \mu}\mathbf{P}(\overline{\nu}^\varepsilon,k) & \geq \sup_{n \geq 1}\frac{1}{n}\log\left(\frac{\int \varphi^k(A)d\mu_{nd'}^{0}(A)}{K\left(\int \varphi^k(A)d\mu_{n}^{0}(A)\right)^{d'-1}}\right)=\mathbf{P}(\mu^0,k)
KS(\textbf{x}\# [ev_t]) = \Lambda_{pt}\# j^H(\textbf{x}\# [ev_t]) = \Lambda_{pt}\# j^H(\textbf{x}) \# j^H([ev_t]) & = \Lambda_{pt}\# j^H(\textbf{x}) \# [ev_t^H] & = KS(\textbf{x}) \# [ev_t^H].
\left|\lim_{i\to\infty}F_{\Omega}(u_i)-|h'(0)|^{2-\beta}\lim_{i\to\infty}F_{B_1}(v_i)\right|=\left|\lim_{i\to\infty}A_1^i(\delta)-\chi(0)\lim_{i\to\infty}F_{B_1}(v_i)\right|.
\begin{aligned}2\psi(\mu-g)\omega^2 \leq \psi(\mu-g)\omega^2+c\frac{|\ln D-g|}{R^4}+\frac{\psi\omega^2}{2}+\frac{c}{(\tau-t_0+T)^2}&\,\,\,\,\,\,+\frac{c}{R^4}+c\frac{\alpha^2}{R^2}+cK^2+c(a+(n-1)K)^2+\frac{ca^2g^2}{(\mu-g)^2}.\end{aligned}
Q(y)=\frac{\kappa_0}{y^2}+\frac{g(y)}{y^4}+O\bigg(\frac{1}{y^6}\bigg),\;\;|y|\rightarrow+\infty, &Q'(y)=-\frac{2\kappa_0}{y^3}+O\bigg(\frac{1}{|y|^5}\bigg),\;\;|y|\rightarrow+\infty.
W(\lambda_i)=\frac{\eta}{\lambda_i}+\frac{\sqrt{a\lambda_i+\eta^2}}{\lambda_i}+\frac{a}{2N}\sum_{j}\frac{1}{\sqrt{a\lambda_j+\eta^2}}\, \frac{1}{\sqrt{a\lambda_j+\eta^2}+\sqrt{\vphantom{\lambda_j}a\lambda_i+\eta^2}}.
e^{z_1+z_2}\left[1+u_{11}z_1+u_{12}z_2\right]^{i}\left[1+u_{21}z_1+u_{22}z_2\right]^{k}=\sum_{m,n=0}^{\infty}\frac{S_{m,n}(i,k)}{m!n!}z_1^{m}z_{2}^{n}
G(z)=G(z_{1},...,z_{d})=E(z_{1}^{X_{1}}\cdot\cdot\cdot z_{d}^{X_{d}})=\sum_{x_{1},...,x_{d}=0}^{\infty}p(x_{1},...,x_{d})z_{1}^{x_{1}}\cdot\cdot\cdot z_{d}^{x_{d}},
\phi_m=N_m e^{{\nu \over 2}k \rho}\left[J_{\nu-1}\left({2m \over k}\right)Y_{\nu}\left({2m \over k}e^{{k \over 2}\rho}\right)-Y_{\nu-1}\left({2m \over k}\right)J_{\nu}\left({2m \over k}e^{{k \over 2}\rho}\right)\right]
\frac{1}{n}H(J,K|W_1,W_2,Y^n,\mathcal{C}_n) = \sum_{\mathfrak{C}_n}p(\mathfrak{C}_n)\frac{1}{n}H(J,K|W_1,W_2,Y^n,\mathcal{C}_n = \mathfrak{C}_n) \leq \sum_{\mathfrak{C}_n}p(\mathfrak{C}_n) (\frac{1}{n}+ \mathrm{Pr}(\mathcal{E})\frac{1}{n}\log(|\mathcal{J}||\mathcal{K}|)) \leq \epsilon_n,
\textbf{Z}_1\otimes\textbf{P}\textbf{C}=\textbf{g}_{1}-\varphi(x_0)\textbf{I}_{N_t\times1}, \textbf{Z}_2\otimes\textbf{P}\textbf{C}=\textbf{g}_{2}-\varphi(x_{N_h})\textbf{I}_{N_t\times1},
(x_1,x_2,x_3,x_4)^{\tau_1}= ( x_3, x_4 , x_1, x_2 ) (x_1,x_2,x_3,x_4)^{\tau_2}= ( x_4, x_3, x_2, x_1 ) (x_1,x_2,x_3,x_4)^{\tau_3}= ( x_2, x_1, x_4, x_3 ),
V \rho_{\gamma^1}^Y(-)=\Gamma^Z_{\gamma'}\widetilde{V}\rho_{\gamma^1}^Y(-)=\Gamma^Z_{\gamma'}\Gamma^Z_{\gamma'}\rho_{\gamma^2}^Y(-) \Gamma^Z_{\gamma'}\widetilde{V}=\rho_{\gamma^2}^Y(-)V,
\xi = \frac{\phi_1 -\phi_d}{\theta^*_1-\theta^*_d}+ \frac{(\theta_1-\theta_0)(\theta^*_d-\theta^*_0) - (\theta_d-\theta_0)(\theta^*_1-\theta^*_0)}{\theta^*_1-\theta^*_d}, \xi^* = \frac{\phi_d -\phi_1}{\theta_1-\theta_d}+ \frac{(\theta^*_1-\theta^*_0)(\theta_d-\theta_0) -(\theta^*_d-\theta^*_0)(\theta_1-\theta_0)}{\theta_1-\theta_d}.
(-1)^k \Big(n \frac{(3d^2+1)n^2-6d^2n+2d^2}{12d}-2\binom{n}{3}\Big)\sum_{0 \le s \le \frac{kd}{n}-1}\binom{d}{s}(-1)^{s}(k-\frac{n}{d}s) &-(-1)^k\binom{n}{2}\sum_{0 \le s \le \frac{kd}{n}-1}\binom{d}{s}(k-\frac{n}{d}s - 1)(-1)^{s}-(-1)^k\binom{n}{2}^2 \sum_{0 \le s \le \frac{kd}{n}-1}\binom{d}{s}(-1)^{s}.
\mathring E\{\bar W(\psi)\}= E\{W(\psi)\}+\lambda^{ar}\lambda^{st}\beta_a(2\lambda_{rs,t}+\lambda_{rst})-2\lambda^{ar}\mathring E(\mathring b_a \mathring l_r)-\lambda^{ab}(\beta_{ab}+\beta_a\beta_b)+O_p(n^{-3/2}).
c ( | x | ) = | x |^{4}\int e^{- i k x}\frac{{\cal F}( k^{2})}{k^{4}}d^{4}k = 4 \pi^{2}| x |^{3}\int_{0}^{\infty}\frac{J_{1}( k | x | )}{k^{2}}{\cal F}( k^{2}) d k ,
\mathbb{E}_{\lambda}^d\xi_t(O)\leq e^{-t}\sum_{n=0}^{\infty}\frac{t^n\lambda^nM^{2n}(d+1)^n}{n!E{\widetilde{\rho}}^2}{\widehat{E}}[(E{\widetilde{\rho}}^2)^{|X_n|}]
{\frac{1}{( 2 \pi \tau )^{d / 2}\sqrt{\mathrm{d e t}( \nu )}}}\mathrm{e x p}( - \tau [{\frac{1}{2}}({\frac{z_{j}}{\tau}}- \nu_{j}) \nu_{j k}^{- 1}({\frac{z_{k}}{\tau}}- \nu_{k}) + \nu_{o}] )
M_{\mathrm{NaCl}}=\underset{\epsilon \rightarrow 0+}{\lim}\left( \dfrac{1}{2}\int_{\epsilon}^{+\infty}\left\lbrace\vartheta_{3}(0\vert i v)-\vartheta_{4}(0\vert i v) \right\rbrace^{3}dv -\dfrac{1}{\sqrt{\epsilon}}\right)= \sum_{\vert k\vert +\vert m\vert +\vert p\vert \neq 0}\dfrac{(-1)^{k+m+p}}{\left( k^{2}+m^{2}+p^{2}\right)^{1/2}}.
\begin{aligned}M(x,y) = c_{0}c_{0}'\int_{0}^{\infty}(n+\{x\}+z)^{-1-2\beta}(n+\{y\}+z)^{-1+2\beta}dz + o\left( (n+\{x\})^{-\frac{1}{2}-\beta-\delta}(n+\{y\})^{-\frac{1}{2}+\beta-\delta}\right).\end{aligned}
P_{\epsilon}:= - W + \frac{1}{2}\epsilon \partial_{\xi}W - \frac{1}{8}\epsilon^2 \partial_{\xi}^2 W- \frac{1}{2}\epsilon^2 W^p + \frac{1}{48}\epsilon^3 \partial_{\xi}^3 W + \frac{1}{4}\epsilon^3 p W^{p-1}\partial_{\xi}W.
\Lambda_{ij}\left(\begin{array}{ccc}1 0 0 \phi_j \tau 0 \beta_j -2\phi_j^\star\tau_j 1\end{array}\right)=\left(\begin{array}{ccc}1 0 0 \phi_i \tau 0 \beta_i -2\phi_i^\star\tau_i 1\end{array}\right)\widetilde{\Lambda}_{ij}.
\phi^{4}= \left( \begin{array}{c c c c}{k^{1}t \otimes 1_{N_{1}\times N_{1}}}&{0}&{\dots}&{0}\\{0}&{k^{2}t \otimes 1_{N_{2}\times N_{2}}}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{\dots}&{\dots}&{k^{K}t \otimes 1_{N_{K}\times N_{K}}}\\ \end{array}\right)
\bar{X}^{n}_{1}(t) = \bar{\xi}^{n}+ \int_{0}^{t}b\left(s,\bar{X}^{n}_{1}(s),\mathfrak{m}(s),\tilde{u}^{n}_{1}(s)\right)ds &+ \int_{0}^{t}\sigma\left(s,\bar{X}^{n}_{1}(s),\mathfrak{m}(s)\right)dW^{n}_{1}(s),t\in [0,T].
\bigl \langle \hat P_r - \mathbb E \hat P_r, \mathbb E \tilde P_r - P_r \bigr\rangle= \bigl \langle \hat P_r - \mathbb E \hat P_r, P_r W_rP_r\bigr\rangle+\bigl \langle \hat P_r - \mathbb E \hat P_r, T_r\bigr\rangle,
\mathcal{S}=\Sigma +\displaystyle\int d^4x\left( b^a\partial A^a+h_\nu^a\partial_\mu B^{a\mu \nu}+\omega^a\partial \xi^a+h^{a\mu}\partial_\mu e^a+\omega^a\lambda^a+-\lambda^a\partial \overline{\xi}^a\right) \;,
\frac{z^{2^{m+1}}}{1-z}\leq \frac{2^{n-1}}{\xi^{n+1 \choose 2}}\left(1 - \frac{\xi^{n+1 \choose 2}}{2^{n-1}}\right)^{2^{m+1}}\leq \frac{2^{n-1}}{\xi^{n+1 \choose 2}}\exp\left(-\frac{\xi^{n+1 \choose 2}}{2^{n-1}}\right)^{2^{m+1}}= \frac{2^{n-1}}{\xi^{n+1 \choose 2}}\exp\left(-2^{m-n+2}\xi^{n+1 \choose 2}\right) \leq e^{-\sqrt{n}},
\begin{aligned}\int_{0}^{T}\int_{\min_{j}\{|x-x_{j}|\}\geq \frac{1}{500}}|\nabla u|^{2}\leq \int_{0}^{T}\int_{\min_{j}\{|x-x_{j}|\}\geq \frac{3}{5}}|\nabla u|^{2}+\int_{0}^{T}\sum_{j=1}^{m}\int_{\frac{1}{500}\leq |x-x_{j}|\leq \frac{3}{5}}|\nabla u|^{2}:= E_{1}+E_{2}.\end{aligned}
\phi(x,\lambda) =\theta^{^{\prime}}(1,\lambda)h(\lambda)\varphi(x,\lambda)+\frac{1}{2}(\theta(1,\lambda)-\varphi^{^{\prime}}(1,\lambda))(h(\lambda)\theta(x,\lambda)+g(\lambda)\varphi(x,\lambda))- & \ -\varphi(1,\lambda)g(\lambda)\theta(x,\lambda),
I_{2}( x_{-}, x_{+}| a ) \equiv \frac{1}{2}x_{-}D_{-}( x_{-}| \mathrm{L}) + ( x_{-}- a \mathrm{L}) D_{+}( x_{+}| \mathrm{L}) + \frac{1}{2 4 \mathrm{L}}x_{-}^{3}+ I_{2}( 0 , 0 | a ) ,
g = e^{xL_{-1}}e^{t(G_{-1/2}^{+}+G_{-1/2}^{-})}\left( \prod_{n\geq 3}e^{\xi^{\pm}_{n-1/2}G_{n-1/2}^{\pm}}e^{u_nL_n}e^{\phi_{n-1}U_{n-1}}\right)e^{\phi_1 U_1}e^{\xi^{+}G_{3/2}^{+}+\xi^{-}G_{3/2}^{-}}e^{u_2 L_2}.
g_{\mathrm{e f f}}^{2}= g_{f}^{2}u^{5}, \quad \tau_{D}= \left({\frac{1}{4 \pi}}\left[ g_{f}^{2}\left({\frac{u}{\cal R}}\right)^{5}\left({\frac{1 6}{\pi}}\right) \right]^{1 / 6}\right)^{- 2},
\begin{cases}\begin{aligned}\sum_{i,j,k,p}(\bar h^{p^{\ast}}_{ijk})^2=0, \ \ \bar \lambda=0, \ \ \bar\lambda_1\bar\lambda_2\neq 0, & S<\sup H^2=\bar H^2 \leq 3S-2 \ \ \ \S<\sup H^2<\dfrac43S,\end{aligned}\end{cases}
\tilde{q}_1(t,z,\zeta) = i\frac{\partial W_{13}}{\partial x}(t,x^{(1)}- x^{(3)})\cdot \xi^{(1)}+ \frac{1}{2}\Delta_xW_{13}(t,x^{(1)}- x^{(3)}) & - i A^{(1)}(t,x^{(1)})\cdot \frac{\partial W_{13}}{\partial x}(t,x^{(1)}- x^{(3)}).
N_6=-2c_6\int_{\mathbb R}b_k |D|\coth(\sqrt{\mu_2}|D|)(\zeta_k \nu_{n_k}+\varphi_k \nu_{n_k}) + \zeta_k \nu_{n_k}|D|\coth(\sqrt{\mu_2}|D|)(\varphi_k \nu_{n_k})dx,
\langle e^{\rho(u)},\partial_u (\widetilde{W}^{(u)})e^{\rho(u)}\rangle = \langle e^{\rho(u)},W (H^{(u)})^{-1}T(u)^{-1}e^{2\rho(u)}T(u)^{-1}(H^{(u)})^{-1}W e^{\rho(u)}\rangle &= \langle e^{\rho(u)}, \widetilde{W}^{(u)}e^{2\rho(u)}\widetilde{W}^{(u)}e^{\rho(u)}\rangle = \sum_{i\in V}(\widetilde{W}^{(u)}e^{\rho(u)})_i^2 e^{2\rho_i(u)},
\idotsint\limits_{t_1,\dots,t_k\ge 0, t_m=0}F_2(t_1,\dots,t_k)^2\prod_{i\ne m}dt_i\le k^2\Bigl(\int_0^\infty\frac{\psi(t/U_k)^2dt}{(1+T_kt)^2}\Bigr)^{k-2}\Bigl(\int_0^\infty\frac{\psi(t/2)^2dt}{(1+T_kt)^2}\Bigr)\ll k^2T_k^2 J_k(F).
\frac{\|F(\nabla v)\|_{L^{\infty}(\Omega)}}{\|v\|_{L^{\infty}(\Omega)}}=\min_{\varphi\in W_{0}^{1,\infty}(\Omega)\setminus\{0\}}\frac{\|F(\nabla \varphi)\|_{L^{\infty}(\Omega)}}{\|\varphi\|_{L^{\infty}(\Omega)}}=\Lambda_1(\infty,\Omega)= \frac{1}{\rho_{F}(\Omega)}.
\sum_{j\in S_0}(b_j^{(w)})^2 =\frac{\sum_{j\in S_0}(a_j^{(w)})^2}{\left(\sum_{j\in S_0}a_j^{(w)}\right)^2}=\frac{\sum_{j\in S_0}\|z_j\|^4}{\left(\sum_{j\in S_0}\|z_j\|^2\right)^2}&=\left(2+\frac{\left(\sum_{j\in S_0}\|z_j\|^2\right)^2-2\sum_{j\in S_0}\|z_j\|^4}{\sum_{j\in S_0}\|z_j\|^4}\right)^{-1}.
\frac{1}{1 6 \pi}\int_{B_{e}}\left( 2 \Lambda - F^{2}\right) \sqrt{g}d^{4}x = \beta_{R N}\Lambda \left[ \frac{r_{s}^{3}- r_{h}^{3}}{1 2}\right] \, + \, \frac{q^{2}}{4}\beta_{R N}\left( \frac{1}{r_{h}}- \frac{1}{r_{s}}\right)
(\beta^g\phi)_{g_2,g_1}=\sum_{h\in G}\beta^g_{g_2,h}\phi_{h,g_1}=\phi_{g^{-1}g_2,g_1}\qquad (\phi\beta^g)_{g_2,g_1}=\sum_{h\in G}\phi_{g_2,h}\beta^{g}_{h,g_1}=\phi_{g_2,gg_1}.
H^{i}(z^{i,(l)})=\begin{bmatrix}\begin{array}{c}r_{\textrm{dual}}^{i}(z^{i,(l)}) (r_{\textrm{primal,1}}^{i}(z^{i,(l)}),r_{\textrm{primal,2}}^{i}(z^{i,(l)})) S^{i,(l)}\Lambda^{i,(l)}\hat{e}\end{array}\end{bmatrix},
\mathrm{div}_w\tilde{E}^i=\left(\frac{\mathrm{d}\imath_{\tilde{E}^i}w}{w}\right)=\left(\frac{\mathrm{d}\imath_{\hat{E}^i}v}{w}\right)=\left(\frac{\mathrm{d}\imath_{\hat{E}^i}v}{v}\right)\left(\frac{v}{w}\right)=\mathrm{div}_v\hat{E}^i\left(\frac{v}{w}\right)\,.
2 \sum_{k=1}^d\sum_{i=1}^{k-1}\sum_{j=i+1}^{k-1}\frac{1}{((x_k-x_i)+\epsilon^2)((x_k-x_j)+\epsilon^2)}= 2\sum_{k=1}^d\sum_{i=k+1}^{d}\sum_{j=i+1}^{d}\frac{1}{((x_j-x_k)+\epsilon^2)((x_j-x_i)+\epsilon^2)}.
(C_{z}M_f C_{-{z}}g)(w) = e^{\alpha \langle w,{z}\rangle - \frac{\alpha}{2}|{z}|^2}(M_f C_{-{z}}g) (w - z) &= e^{\alpha \langle w, z \rangle - \frac{\alpha}{2}|z|^2}f(w - z) (C_{-z}g)(w - z) &= f(w-z)g(w)
f_0 = \frac{d^3 t^{12}+ 99 d^2 t^8 - 297 d t^4 - 27}{18 (t (d t^4 + 3))^2}= \pm \sqrt{d}+ \frac{(d t^4 \mp 6 \sqrt{d}t^2 - 3)^3}{18 (t (d t^4 + 3))^2}.
\nabla k^j = \nabla e^{jh}= e^{jh}\frac{e^{j\mathbf x}-1}{j\mathbf x}\nabla (jh) = e^{jh}\frac{e^{j\mathbf x}-1}{j\mathbf x}(j k^{-1}) \frac{\log \mathbf y}{\mathbf y-1}\nabla k &= k^{j-1}\frac{\mathbf y^j-1}{\mathbf y-1}(\nabla k).
\sum_{i=1}^m \int_t^{\infty}\int\limits_{\{| \nabla_i P_s f | > M \|\nabla_i f \|_1 \}}|\nabla_i P_s f|^2 d\mu \, ds \leq \frac{e^{-ct}}{M^{\frac{1}{q(t)}}}\| f \|_2^2.
\frac{d}{dt}(\sum_nP_n z^2_n) = -2 c_0 \sum_n P_n z_n - c_1 \sum_nP_nz^2_n + \sum_n \frac{\partial \beta_n}{\partial \mu}z_n^2 + \sum_B\frac{\partial \beta_B}{\partial \mu}z_B^2.
{\rm tr}_{l}(f)={\rm ev}_{a}\circ (1_{\overline{a}}\otimes f)\circ{\rm coev}_{\overline{a}}\textrm{and}{\rm tr}_{r}(f)={\rm ev}_{\overline{a}}\circ (f\otimes 1_{\overline{a}})\circ{\rm coev}_{a}
\frac{\beta(4-a\beta)}{4(1-a(N+1))}=\lim_{k\to+\infty}\frac{1}{2\pi}\int_{\mathbb{R}^2}e^{u_k}= \lim_{k\to+\infty}\frac{1}{2\pi}\int_{\mathbb{R}^2}\varepsilon_{1,k}\vert x \vert^{2N}e^{v_k}=4m;
{\cal{V}}= - \frac{T V^{2}}{8 \pi^{4}\lambda}\frac{m}{N ( N + m )}+ \frac{1}{2 \pi^{2}}( \frac{1}{N}( B^{0})^{2}+ \sum_{k}( B^{k})^{2}) + \frac{1}{\pi^{2}}\sum_{i}\sum_{j > i}| X^{( i j )}|^{2}) + \frac{1}{\pi^{2}}\sum_{i}| X^{i}|^{2}
[J^k_0,\Psi^{ab}(z)]|\Psi^{cd}\rangle~+~\Psi^{ab}(z)J^k_0|\Psi^{cd}\rangle=z^{-2\Delta_O+\Delta_\phi}~C^{ab,cd}_nf^{kn}_e|\phi^e\rangle~+~...
\sim \sum_{r=0}^t \frac{\big({_{\neq k}}\Lambda_{\neq d}\big)^r}{r!}\frac{\big({_{\neq d}}\Lambda_{\neq k}\big)^{t-r}}{(t - r)!}\;\; \sim \frac{2^t}{t!}\big({_{\neq k}}\Lambda_{\neq d}\big)^t &\sim \frac{2^t}{t!}\big( w^* (1 - \rho^{d+1}) (1 - \rho^{k+1}) \big)^t n^t .
\mathrm{E}_0\Pi\left(\|D^{\boldsymbol{r}}f-D^{\boldsymbol{r}}f_0\|_\infty>M(n/\log{n})^{-\frac{\alpha^{*}\{1-\sum_{l=1}^d(r_l/\alpha_l)\}}{2\alpha^{*}+d}}\middle|\boldsymbol{Y}\right)\leq\frac{(\log{n})^d}{n^\xi}.
C\left(\mu, \Psi \right) :=\inf_{L>\max\left\{1,\left[{\Psi}\left(({C_Y}\cdot K)^{-1}\right)\right]^{-1}\right\}}c_\mu 2^{n_\mu}\frac{L}{1-{C_Y}\cdot K\cdot{\Psi}^{-1}\left(\frac{1}{L}\right)},
|\dot{x}_i(t)-\mu_i(t)|\lesssim\frac{1}{t^{5/4-\delta_0}}, &\bigg|\dot{\mu}_i(t)+\frac{\alpha_i}{t^{\frac{3}{2}}}-\sum^n_{j=1}\frac{m_{ij}x_j(t)}{t^2}+\sum^n_{\substack{k,j=1, j\not=i}}\frac{b_{ijk}\alpha_k}{(\alpha_i-\alpha_j)^3t^2}\bigg|\lesssim \frac{1}{t^{9/4-\delta_0}},
0 = \frac{-1}{2\pi i}\int_{\mathcal{C}_{\{1-\varrho_{1}, \ldots, 1-\varrho_{N}\}}}dz \frac{z - \varrho_{k}}{1 - z - \varrho_{k}}\frac{1}{\sqrt{2z - 1}}\frac{1}{1 - z}e^{- \frac{\eta_{-}}{1 - z}v}, k>N.
\frac{\partial^n}{\partial x^n}V(y,x) = \delta_n^0 + O_n \Big( x^{-n/2}(y/x)^{\sqrt{x}}\Big) \frac{\partial^n}{\partial x^n}V(y,x) \ll_n x^{-n/2}(x/y)^{\sqrt{x}},
D^mk_u = A_u^{-1}\left( Dk_u(R)D^mr+D^mk_u(R)\left(DR \right)^{\otimes m}+ \mathcal{P}_m(k_u,R) \right) & - A_u^{-1}\left( Dg_u(K{}{})D^mK{}{}+D^mg_u(K{}{})\left(DK{}{}\right)^{\otimes m}+ \mathcal{P}_m(g_u,K{}{}) \right) &=f_{m,2}+ A_u^{-1}\left( Dk_u(R)D^mr + D^mk_u(R) (DR)^{\otimes m}- Dg_u(K{}{})D^mK{}{}\right),
h(\nu,\zeta) ={}2^{-1}(\alpha_1 - \alpha_2) \sin^2 2\nu \cos2\zeta-2^{-1}(\alpha_1 + 3\alpha_2) \sin^2 2\nu &+2\alpha_3 \cos \eta \cos 2\nu +2\alpha_3 \sin \eta \sin 2\nu \cos\zeta +3\alpha_1+\alpha_2+r,
d \Pi^{\underline{m}}= - i d \Theta^{1}\sigma^{\underline{m}}d \Theta^{1}- i d \Theta^{2}\tilde{\sigma}^{\underline{m}}d \Theta^{2}, d d \Theta^{1 \underline{\mu}}= 0 , d d \Theta_{\underline{\mu}}^{2}= 0 , \qquad ( I I A )
C_{1,k}= (A_{3k,k}+ \{a \}) \cup \left(\left\{3k a - 6 \left\lfloor \frac{3k}{3}\right\rfloor \right\}+ \left\{b \right\}\right), C_{2,k}= C_{1,k}+ \{a \}, C_{3,k}= (C_{2,k}+ \{a \}) \cup \left( \left\{3 k a - 6 \left\lfloor \frac{3k}{3}\right\rfloor \right\}\right) + \left\{2 c \right\}.
1\otimes q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}\theta(\lambda n_s^{-1}(\mu))T_{n_s^{-1}}= 1\otimes E(n_s^{-1}(\mu))E(\lambda n_s^{-1}) &= (w\widetilde{\chi})(E(n_s^{-1}(\mu)))\otimes E(\lambda n_s) & = (sw\widetilde{\chi})(E(\mu))\otimes E(\lambda n_s) = 0.
\gamma_{i j}= \Pi_{i}^{a}\, \Pi_{j}^{b}\, \eta_{\underline{{{a b}}}}\quad ; \quad G_{i j}= V_{i}^{\underline{{{a}}}}\, V_{j}^{\underline{{{b}}}}\, \eta_{\underline{{{a b}}}}\quad ; \quad \widehat{G}= \eta \, G \, \eta
Z^{(n+1)}_t = Z_0 + \int_0^t \int_{\mathbb{D}\times \mathbb{D}\times \Xi}\alpha(Z_s^{(n)}, v_s, \xi) d{\tilde{N}}_{\mu_{n}}& + \int_0^t \int_{\mathbb{D}\times \mathbb{D}\times \Xi}\hat{\alpha}(Z_s^{(n)}, v_s, \xi)\beta(|X_s^{(n)}- y_s|) d\mu_{n}Q(d\xi) ds X^{(n+1)}_t = X_0 + \int_0^t Z_s^{(n+1)}ds
D_{n+2}^1(n+2;2)D_n^2 -D_{n+1}^2D_{n+1}^1(n+1;2) =D_{n+1}^1D_n^3, D_{n+1}^0(1;2)D_{n+2}^0-D_{n+2}^0(1;2)D_{n+1}^0 =-D_{n+2}^{-1}(1;2)D_{n+1}^1, D_{n+2}^{-1}(n+2;2) D_{n+1}^2-D_{n+3}^{-1}(n+3;2)D_{n}^2 =D_{n+2}^{-1}(1;2)D_{n+1}^1.
\overline\tau_{\theta}:= \inf \{t \geq \theta,\,\, A_t > A_{\theta}\,\,{\rm or}\,\, C_{t^-}>C_{\theta^-}\}; \overline\sigma_{\theta}:= \inf \{t \geq \theta,\,\, A'_t > A'_{\theta}\,\,{\rm or}\,\, C'_{t^-}>C'_{\theta^-}\}.
\langle (\mathcal{E}_{k}^{\alpha}- \mathcal{E}_{k_n}^{\alpha})\varphi,\varphi\rangle_{L^2}= \langle (\mathcal{E}_{k}- \mathcal{E}_{k_n})\partial^{\alpha}\varphi,\partial^{\alpha}\varphi\rangle_{L^2}= \sum_{i=n+1}^{+\infty}\lambda_i \langle \phi_i,\partial^{\alpha}\varphi\rangle_{L^2}^2 \geq 0
{\cal P}_{x^{-}}e^{i\int_{-L}^{L}dy^{-}\big[A_{-}(x^{+},y^{-},{\vec x}_{\perp})-i{\vec \alpha}_{\perp}\big({\vec \partial}_{\perp}+i{\vec A}_{\perp}(x^{+},y^{-},{\vec x}_{\perp})\big)\big]}= \pm 1\,\,,
\delta(H[V_i]) \geq \alpha \binom{n-1}{2}- \kappa n^2 \geq \alpha \frac{n^2}{2}- \kappa n^2 - \alpha \frac{n}{2}& = \big(\alpha -\kappa/2\big)\frac{n^2}{2}-\alpha \frac{n}{2}\geq \big(\alpha -\kappa\big)\frac{n^2}{2}& \geq \big(\alpha -\kappa\big) \binom{|V_i|-1}{2}.
H_0(x,y,t)=\frac{e^{n^{2}t}}{2^{n-2}\pi^{n+1}}\int_{\rho}^{\frac{\pi}{2}}\frac{-d(\cos u)}{\sqrt{\cos^{2}\rho -\cos^{2}u}}\left( -\frac{1}{\sin u}\frac{d}{du}\right)^{n}\left[ \Theta_{n+1}(t,u)\right]
\psi_{k,j}^{\ast}(Q^{\prime}) \leq \Phi_{k,n}(\psi_{n,1}^{\ast}(Q))=(\psi_{n,1}^{\ast}(Q)-1)\frac{k+1}{k}\frac{n}{n+1}+1, 1\leq j\leq k-n, \psi_{k,k-n+j}^{\ast}(Q^{\prime}) \leq \Phi_{n,k}(\psi_{n,j}^{\ast}(Q))= (\psi_{n,j}^{\ast}(Q)-1)\frac{k+1}{k}\frac{n}{n+1}+1, 1\leq j\leq n.
\bigtriangleup\big(x_\mu^{\rm R}\big)=\big(x_\mu^{\rm R}-ia_\mu Z\big)\otimes1=Z^{-1}\otimes x_\mu^{\rm R}\bigtriangleup\big(p_\mu^{\rm R}\big)=p_\mu^{\rm R}\otimes1+Z\otimes p_\mu^{\rm R}
\mathcal{L}_1^{\pm}(n^2_1n_2,u,q,p_1) = d{\pi}^{3}\int_{0}^\infty V_1^{\pm}(z) z^{-1/3+i\nu}\, e\left(\frac{Nuz}{p_1qQ}\pm \frac{3(Nz n_1^2 n_2)^{1/3}}{p_1q r^{1/3}}\right)dz.
\lim_{L_1\to0}\tfrac{1}{L_1}(E(L_1,\ell_\nu,\ell_\mu)+E(L_1,\ell_\nu,\ell_{\mu'}))=1-\frac{\sinh\frac{\ell_\nu}{2}}{\cosh\frac{\ell_\nu}{2}+1}=2(e^{\frac{1}{2}\ell_\nu}+1)^{-1}.
\bigl(\tfrac1{q_1},\tfrac1{r_1}\bigr) = \bigl(\tfrac1{q_2},\tfrac1{r_2}\bigr) := \bigl(\tfrac{dp-2}{2(p+2)}, \tfrac{2d+4-dp}{2d(p+2)}\bigr),\quad\bigl( \tfrac1{\tilde{q}}, \tfrac1{\tilde{r}}\bigr):=\bigl(\tfrac{4-(d-1)p}{p(p+2)}, \tfrac{2dp-4}{dp(p+2)}\bigr).