harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\langle \mathbf{j},\nabla\overline{q}\right\rangle_{\mathbb{R}^{3}}+s^{2}\left\langle p_{j}^{\operatorname*{ext}}u,\overline{q}\right\rangle_{\mathbb{R}^{3}}=0\quad\forall q\in H^{1}\left( \mathbb{R}^{3}.\mathbb{A}_{j}^{\operatorname*[ext]}\right) .
K_{s}( x ) = \sqrt{\frac{\pi}{2 x}}e^{- x}\left( 1 + \sum_{k = 1}^{\infty}\frac{1}{( 2 x )^{k}}\frac{\Gamma \left( s + k + \frac{1}{2}\right)}{k ! \Gamma \left( s - k + \frac{1}{2}\right)}\right) \ .
P(\xi_1', \dots, \xi_n') = P(\eta_1(\xi_1^{+}-\xi_1^{-}) + \xi_1^{-}, \dots, \eta_n(\xi_n^{+}-\xi_n^{-}) + \xi_n^{-}) = \sum_{S\subset [n], |S| = d}\left (a_{S}\prod_{i\in S}(\xi_i^{+}-\xi_i^{-})\right )\prod_{i\in S}\eta_i + Q,
\begin{aligned}p_{(v_i,k)}=\sum_{i_1=i}^n s_{(e_{ii_1},m_i+k)}s_{(e_{ii_1},m_i+k)}^* &= \sum_{i_1=i}^n s_{(e_{ii_1},m_i+k)}p_{r((e_{ii_1}m_i+k))}s_{(e_{ii_1},m_i+k)}^* &= \sum_{i_1=i}^n s_{(e_{ii_1},m_i+k)}p_{(v_{i_1},m_i+k)}s_{(e_{ii_1},m_i+k)}^*. \end{aligned}
\frac{1}{2\pi i}\int_{{1}/{2}+iT}^{{1}/{2}+2iT}\chi(s)^{-1/2}\zeta(s)ds = \frac{1}{2\pi i}\left(\int_{{1}/{2}+iT}^{1+\delta +iT}+ \int_{1+\delta+iT}^{1+\delta +2iT}+ \int_{1+\delta+2iT}^{{1}/{2}+2iT}\right)\chi(s)^{-1/2}\zeta(s) ds = I_{1}+I_{2}+I_{3}.
|g^{\alpha+1/2}_{\alpha,r}(t)| =\bigg| C_1t^{\alpha+3/2}j_{\alpha+1}(rt) + C_2 t^{\alpha+3/2}j_{\alpha+2}(rt) +C_3 \int_0^t s^{\alpha+1/2}j_{\alpha+2}(rs)\, ds\bigg| &\lesssim \frac{1}{r^{\alpha+3/2}}+\frac{1}{tr^{\alpha+5/2}}+ \int_0^t s^{\alpha+1/2}|j_{\alpha+2}(rs)|\, ds
C_{G'}= \sup_{t\in[0,1]}\left\| G'(\mathcal I_{M_\ell}u_{h_{\ell-1}}+ t \mathcal I_{M_\ell}\triangle u_\ell)\right\| C_{G''}= \sup_{s,t\in[0,1]}\|G''(\xi(s,t))\|.
\left(\prod^{k}_{i=1}|\overline{W}_{i-1}T^{\varepsilon_i}_{\alpha_i}{\overline{W}\,}^{-1}_i|_\mathfrak{U}\right)^{-1}\left(\prod^{m}_{i=k+2}|\overline{W}_{i-1}T^{\varepsilon_i}_{\alpha_i}{\overline{W}\,}^{-1}_i|_\mathfrak{U}\right)^{-1}.
\Phi_{Q,s}= \sum_{\substack{(a,q)=1 q \sim Q}}\tau_{-a/q}\gamma^{(s)}(2^s N \,\cdot\, )= \bigg( \sum_{\substack{(a,q)=1 q \sim Q}}\delta_{a/q}\bigg) \ast \gamma^{(s)}(2^s N \,\cdot\, ).
g_{\tau \tau}=g_{\xi \xi}=\frac{\widetilde{a}^{2}(r)}{\left( \cosh \tau-\cos \xi \right)^{2}},\qquad g_{ss}=\frac{\widetilde{a}^{2}(r)\sin^{2}\xi}{\left( \cosh \tau -\cos \xi \right)^{2}},
g_n=Tg_{s_{k-1}+r},\quad\mbox{when}s_k+1\le n\le s_k+n_k. g_n=T^*g_{r+1-k}\quad\mbox{when}s_k+n_k+1\le n\le s_{k+1}-1. g_n=e_k\quad\mbox{when}n= s_{k+1}.
\frac{1}{m}\int f_m^\phi\,d\mu = \lim_{n \to \infty}\frac{1}{m}\int f_m^{\phi_n}d\mu_n & \geq \liminf_{n \to \infty}\inf_{k \geq 1}\frac{1}{k}\int f_k^{\phi_n}\,d\mu_n &=\liminf_{n \to \infty}\Lambda(\phi_n,\mu_n) &=\lim_{n \to \infty}\log\varrho\left(L_0^{\phi_n},L_1^{\phi_n}\right) &=\log\varrho\left(L_0^{\phi},L_1^{\phi}\right)
\lim_{N \to \infty}n_N \theta_N \cdot \frac{\theta_N}{\sigma_N}\leq \lim_{N \to \infty}\left(1+\frac{\sigma_N}{\theta_N^2}\right) \cdot \frac{\theta_N^2}{\sigma_N}=\lim_{N \to \infty}\frac{\theta_N^2}{\sigma_N}+1 \leq 2,
H = 1 + ( \sqrt{\alpha^{\prime}})^{{\frac{x - \tilde{d}}{\beta}}}( g_{s})^{{\frac{2 ( k - 1 )}{\beta}}}\left[ g_{f}^{2}\, \left({\frac{u}{\cal R}}\right)^{\tilde{d}}\, \left({\frac{d_{p}}{c_{p}}}\right) \right]^{- 1 / \beta}\, .
\left\langle L_Fe_{k},e_{j}\right\rangle = \begin{cases}\sum_{l=1}^n \alpha_l(k) \, a_{l,(\alpha(j)-\alpha(k))_l}\textrm{if}|\alpha(j) |\geq |\alpha(k)| 0 \textrm{if}|\alpha(j) |< |\alpha(k)|.\end{cases}
e^{\chi_{1}{\cal X}_{1}}= e^{a \chi_{1}}\left( \begin{array}{c c c}{{e^{- a \chi_{1}}}}&{{0}}&{{0}}\\{{0}}&{{\cos{\eta_{1}}}}&{{- \sqrt{\frac{n_{3}}{n_{2}}}\sin{\eta_{1}}}}\\{{0}}&{{\sqrt{\frac{n_{2}}{n_{3}}}\sin{\eta_{1}}}}&{{\cos{\eta_{1}}}}\\ \end{array}\right) ,
H_1 = \exp \left[i \theta_1 \left( \begin{array}{ccc}1 0 0 0 -1 0 0 0 0 \ \end{array}\right) \right] \quad ; \quad H_2 = \exp \left[i \theta_2 \left(\begin{array}{ccc}1 0 0 0 1 0 0 0 -2 \end{array}\right) \right]
\ln \frac{\mathcal{Z}_{q,m=0}(\hat g)}{\mathcal{Z}_{q,m=0}(g)}= \frac{\mathbf{c}_{\rm mat}}{96\pi}S^{{\rm cl},0}_{{\rm L}}(\hat g,g) +\frac{q^2(1-\textbf{h})}{4\pi}S^{\rm cl}_{\rm M}(\hat g,g).
N_d' =\#\{(x, y)\in \mathbb{F}_q^2: x^d+ax^{d-1}+b-y^2=0\}&=\sum_{\substack{y\in\mathbb{F}_q y\neq\pm\sqrt{b}}}N(x^d+ax^{d-1}+b-y^2=0)+2\cdot N(x^{d-1}(x+a)=0) &=\sum_{\substack{y\in\mathbb{F}_q y\neq\pm\sqrt{b}}}N(x^d+ax^{d-1}+b-y^2=0)+4.
T_{f g}\equiv \frac{1}{2}( Q_{0})_{f \beta_{0}}( Q_{1})_{\alpha_{1}\beta_{1}}( Q_{2})_{\alpha_{2}\beta_{2}}\cdots ( Q_{r})_{\alpha_{r}g}\epsilon^{\beta_{0}\alpha_{1}}\epsilon^{\beta_{1}\alpha_{2}}\cdots \epsilon^{\beta_{r - 1}\alpha_{r}}.
\lim_{N \rightarrow \infty}\max_{\delta N_k \leq l \leq (1-\delta)N_k}\max_{\mu, \nu}\left \vert \mathbb{E}^V \theta[\frac{N}{\pi}\int_{I}\tilde{G}_{\mu \nu}(z) \mathcal{X}(E)] dE-\mathbb{E}^V \theta[\frac{N}{\pi}\int_{I}\tilde{G}_{\mu \nu}(z)q(\operatorname{Tr}f_E(Q_1))]dE \right\vert =0.
\begin{array}{l}{{A = - \sqrt{\frac{2}{k}}( A^{+ 3}- A^{- 3}+ i W )}}\\{{\}}\\{{\bar{A}= \sqrt{\frac{2}{k}}( A^{+ 3}- A^{- 3}- i W )}}\\{{\}}\\{{Q = \frac{2}{\sqrt{k}}Q^{+}}}\\{{\}}\\{{\bar{Q}= \frac{2}{\sqrt{k}}Q^{-}}}\\ \end{array}
\tilde{y}_{S,m}(\mathbf{X})=y_{\emptyset,G}+{\displaystyle\sum_{{\textstyle{\emptyset\ne u\subseteq\{1,\cdots,N\}\atop 1 \le |u| \le S}}}}{\displaystyle \sum_{k=1}^{m}}{\displaystyle \sum_{{\textstyle{|\mathbf{j}_{|u|}|=k\atop j_{1},\cdots,j_{|u|}\neq0}}}}\tilde{C}_{u\mathbf{j}_{|u|}}\psi_{u\mathbf{j}_{|u|}}
P(a,b) =\frac{\gamma}{2}\left(h_1^2(1-a)+\frac{a}{1-a}+\log(1-a)+\xi(1,1)-\xi(a,b)\right) &+\frac{1-\gamma}{2}\left(h_2^2(1-b)+\frac{b}{1-b}+\log(1-b)+\xi(1,1)-\xi(a,b)\right).
d s_{D}^{2}= H^{- \frac{d - 2}{D - 2}}\left[ - f d t^{2}+ \sum_{i = 1}^{p}( d x^{i})^{2}+ H R_{T}^{2}\left( f^{- 1}A d R^{2}+ \frac{A}{K^{d - 2}}d v^{2}+ K R^{2}d \Omega_{d - 2}^{2}\right) \right]
S_{WZW}=\mathrm{Tr}\int d^2z\left[(g^{-1}\partial g)(g^{-1}\bar{\partial}g)-\int_0^1dt(\hat{g}^{-1}\partial_t\hat{g})[(\hat{g}^{-1}\partial\hat{g}),(\hat{g}^{-1}\bar{\partial}\hat{g})]\right],
\mathcal{S (}\Sigma \mathcal{) =}\int d^{4}x \frac{\delta \Sigma}{\delta \varphi^{i}}\frac{\delta \Sigma}{\delta \varphi^{* i}}+ \frac 1 2 C_{A B}^{C}\varepsilon^{A}\varepsilon^{B}\frac{\partial \Sigma}{\partial \varepsilon^{C}},
{\mathcal{E}}_t={\mathcal{E}}:= \Big\{\varphi: \mathcal{X}\longrightarrow \Gamma,\varphi \Big\}, &{\mathcal{E}}^{(t)}={\mathcal{E}}^t:= \Big\{\varphi: [\![0,t]\!] \times \mathcal{X}\longrightarrow \Gamma,\varphi \Big\}.
\int_{B_\frac{R+r}{2}}\left(\int_{\mathbb{R}^N\setminus B_R}\frac{|u_h(x)-u_h(y)|^{p-2}+|u(x)-u(y)|^{p-2}}{|x-y|^{N+s\,p}}\,dy\right)^\frac{p}{p-2}\,dx &\le C\,\left(\frac{1}{s\,(R-r)^{s\,p}}\right)^\frac{2}{p-2}\,\int_{B_\frac{R+r}{2}}\int_{\mathbb{R}^N\setminus B_R}\Big(|(\mathcal{V}_{p})_h|^2+|\mathcal{V}_{p}|^2\Big)\frac{1}{|x-y|^{N+s\,p}}\,dx\,dy,
\psi(x)=\frac{1}{2\pi i}\int_{c-iT}^{c+iT}\left(-\frac{\zeta'(s)}{\zeta(s)}\right) \frac{x^s}{s}\, ds+O^*\left(\sum_{n=1}^\infty\left(\frac{x}{n}\right)^c\Lambda(n)\min\left(0.501,\frac{1}{\pi T\left|\log\frac{x}{n}\right|}\right)\right).
\pi_{be_{k}}\beta(-c_{k}^{-1}i_{2}\sigma_{1}\overline{\pi}_{3}(n_{2})-c_{k}^{-1}\sigma_{3}\overline{\pi}_{4}(n_{3})) =\pi_{be_{k}}\beta(-c_{k}^{-1}n_{2}-c_{k}^{-1}n_{3}) &=-c_{k}^{-1}bn_{2}-c_{k}^{-1}bn_{3}
\tilde{\Pi}^{(1)}(p) = \frac{4\pi}{\kappa}\frac{C_2 (G)}{2p^2}\mu \int \frac{d^3 k}{(2\pi)^3}\frac{(k.p)^2 - k^2 p^2}{k^2 (k+p)^2 (k^2 + \mu^2)}
\begin{array}{l c r}{M^{+{\frac{n ( n + 1 )}{2}}}= ( \eta^{+})^{{\frac{n ( n + 1 )}{2}}}= N^{+{\frac{n ( n + 1 )}{2}}}}{S^{+ ( n + 1 )}= ( \eta^{+})^{n + 1}.}\end{array}
h_i(w, n) := \begin{cases}\min(w_i, \dotsc, w_{i+t-2}) -1 \sigma_t = 1 n- \max(w_i, \dotsc, w_{i+t-2}) \sigma_t = t w_{b}- w_{a}-1 1 < \sigma_t < t, \sigma_a = \sigma_{t}-1, \sigma_b = \sigma_{t}+1. \end{cases}
\Delta v+k_2^2v=0 \ \mbox{in}\ \mathbb{R}^2\setminus \overline{\Omega_2}, \Delta v+k_1^2v=0 \ \mbox{in}\{\Omega_2}, v^{-}=v^{+}, S_1\frac{\partial v^-}{\partial n}=S_2\frac{\partial v^+}{\partial n}\ \mbox{on}\partial \Omega \mbox{outgoing radiation condition with}k_2 \mbox{in}\mathbb{R}^2\setminus\overline{\Omega_2}.
e^{-m_i(x)s+f_{m_i(x)}(x)}=\mu_i(\overline{B}(x,\gamma_i))\leq\hat{\mu}(\overline{B}(x,\gamma_i))\leq C\mu_i(\overline{B}(x,\gamma_i))=Ce^{-m_i(x)s+f_{m_i(x)}(x)},~~\forall x\in K_i.
H_NS_N(i,j) =\frac{1}{J_N}(-J_{N-3}(J_2+J_3)+(-1)^{N-2}J_0J_2-J_{N-2}(J_1+J_2) ~+(-1)^{N-1}J_0J_1+2J_{N-1}) =\frac{1}{J_N}(-4J_{N-3}-2J_{N-2}+2J_{N-1}) =0.
\mbox{with}\;\;\overline{{\cal{Y}}^a}({\bf{r}})={\textstyle \frac{\partial_{j}}{\partial^{2}}\overline{{\cal A}_{j}^{\alpha}}({\bf r})}=r_{a}\,\frac{\overline{\cal{N}}}{gr}\;\;\mbox{and}\;\;{\cal{X}}^a({\bf{r}})={\textstyle\frac{\partial_i}{\partial^2}}A_i^\alpha({\bf{r}})=r_{a}\,\frac{\cal{N}}{gr}\,.
\sum_{\substack{r_1,\ldots,r_k d_i\mid r_i\forall i}}\frac{y_{r_1,\ldots,r_k}}{\prod_{i=1}^k\varphi(r_i)}=\frac{\lambda_{d_1,\ldots,d_k}}{\prod_{i=1}^k\mu(d_i)d_i}.
\begin{array}{rl}\displaystyle \varphi (\sigma ,\tau )= \phi (z)+\bar{\phi}(\bar{z})\: , \phi (z)= \frac{1}{2}\varphi_{0}-ip_{+}\log z+i\sum_{k\neq 0}\frac{1}{k}a_{k}z^{-k}\: , \bar{\phi}(\bar{z})= \frac{1}{2}\varphi_{0}-ip_{-}\log \bar{z}+i\sum_{k\neq 0}\frac{1}{k}\bar{a}_{k}\bar{z}^{-k}\: ,\end{array}
T_1=\left(\begin{array}{c|c|c|c|c|c}0 W_0 \hline-W_0' 0 W_1 \hline -W_1' 0 W_2 \hline -W_2' \ddots \ddots \hline \ddots 0 W_{l}\hline -W_{l}' 0\end{array}\right).
\tau^*(\mu_R,\mu_T) \le \tau_U(\mu_R,\mu_T) \le \tau_U(\mu_R,1/N_T) \le \tau_U(1,1/N_T)+\frac{\tau_U(1/4,1/N_T)-\tau_U(1,1/N_T)}{3/4}(1-\mu_R) = \frac{\tau_U(1/4,1/N_T)}{3/4}(1-\mu_R).
\sum\limits_{\begin{smallmatrix}{{l}_{1}}+\cdots +{{l}_{4}}={{p}^{r}}{{l}_{1}},\cdots ,{{l}_{4}}\in{\mathcal{P}_{p}}\end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}}}\equiv -\frac{4!}{5}{{p}^{r}}{{B}_{p-5}}\pmod{{p}^{r+1}}.
r^{\prime}_{\xi}(\omega) = \frac{-(1 - \omega)^{\alpha}+ (\xi - \omega) \alpha (1 - \omega)^{\alpha - 1}}{(1 - \omega)^{2\alpha}}& = - \frac{1}{(1 - \omega)^{\alpha}}\left( 1 - \alpha \frac{\xi - \omega}{1 - \omega}\right).
G(r)=\int_{x_1^2+\cdots +x_d^2 \leq r^2}\frac{1}{(2\pi)^{d/2}|\Sigma|^{1/2}}\exp\left( -\frac{1}{2}(\mathbf x-\mathbf\mu)^\top\Sigma^{-1}(\mathbf x-\mathbf\mu)\right)d\mathbf x.
\| (\widetilde{w}_{2n}-\widetilde{w}_{2n-2})x_i \| = \| w_{2n}v_{2n}\widetilde{w}_{2n-2}v_{2n}^*x_i- \widetilde{w}_{2n-2}v_{2n}v_{2n}^*x_i\| \leq \| w_{2n}[v_{2n}, \widetilde{w}_{2n-2}]v_{2n}^*x_i\| +\| (w_{2n}-1)\widetilde{w}_{2n-2}x_{i}\| & < \frac{1}{2^{2n-1}}
\delta_{\ell k}1(\ell\le n_1)+A_h(\ell,k) &=\frac 1{(2\pi i)^4}\int_{\Gamma_{d_1}}dz\int_{\Gamma_{d_3}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{n_1,\mu_1,\xi_1}(z)G_{\Delta n,\Delta\mu,\Delta\xi}(w)}{G_{k,\mu_1,\xi_1}(\zeta)G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)}\frac 1{(z-\zeta)(w-\omega)}\left(\frac 1{z-w}-h\right),
F_{1}=[\begin{matrix}1&2&3\\ 4&5&6\\ 7&8&9\end{matrix}]F_{2}=[\begin{matrix}10&11&12\\ 13&14&15\\ 16&17&18\end{matrix}]F_{3}=[\begin{matrix}19&20&21\\ 22&23&24\\ 25&26&27\end{matrix}]
\delta K = \delta \left( h^{a}{}_{b}\nabla_{a}n^{b}\right) = \delta h^{a}{}_{b}\nabla_{a}n^{b}+ h^{a}{}_{b}( \delta C )^{b}{}_{a c}n^{c}+ h^{a}{}_{b}\nabla_{a}\delta n^{b}\, ,
\begin{array}[t]{r}|k|k|^{2\rho}+\ell|\ell|^{2\rho}+m|m|^{2\rho}|\lesssim |k|\,|\ell|^{\rho}\,|m|^\rho+|\ell|\,|m|^{\rho}\,|k|^\rho+|m|\,|k|^{\rho}\,|\ell|^\rho\,, [5pt]\forall (k,\ell,m)\,;k+\ell+m=0\,,\end{array}
\frac{1}{2\lambda f^2}\int\;d^4X\,\frac{d^4q}{\left(2\pi\right)^4}\left(\varphi(X)\right)^2\tilde\pi_0(q)\left[\partial_q^2+ \frac{4i\lambda}{q^2-\Sigma^2(q)}\right]\tilde\pi_0(q),
\bar{\mu}_{\alpha}= \alpha f(\alpha) - c_{\bar{\alpha}}A_{\bar{\alpha}}\mbox{if $\alpha$ is not a border loop}, \!\!\!\!\!\!\!\!\!\!\! \bar{\mu}_{\alpha}= \alpha^2 - c_{\bar{\alpha}}A_{\bar{\alpha}}- b_i B_{\bar{\alpha}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mbox{if $\alpha$ is a border loop}.
\begin{aligned}{}\|v-L_\nu\|_{L^{\infty}(B(\sigma^\nu))}\leq \sigma^\nu\omega(\sigma^\nu)~~~~~~|b_\nu|\leq C, |a_{\nu+1}-a_\nu| \leq C\sigma^{\nu}\omega(\sigma^{\nu}),~~~~|b_{\nu+1}-b_\nu|\leq C\omega(\sigma^\nu).\end{aligned}
\lim_{n\rightarrow \infty}J_{m,n}^{\,(1,N)}(\mathbf{s})=\lim_{n\rightarrow\infty}\mathbf{E}\left[ \mathbf{s}^{\mathbf{Z}(m,n)}|\mathbf{Z}(n)\neq\mathbf{0},\mathbf{Z}(0)=\mathbf{e}_{1}\right] =s_{1}.
\mathcal{S}^* = \arg \min_{\mathcal{S}\in \mathbb{S}}\left\{\mathcal{C}(\mathcal{S})\right\}= \arg\min_{\mathcal{S}\in \mathbb{S}}\left\{\max_{u\in\mathcal{M}}\left\{\mathcal{C}_u(\mathcal{S}) \right\}\right\}\;.
\displaystyle\lim_{a\to0}\sup_{|Q|=a}\frac{1}{|Q|}\int_Q|b(x)-b_Q|\,dx=0, \displaystyle\lim_{a\to\infty}\sup_{|Q|=a}\frac{1}{|Q|}\int_Q|b(x)-b_Q|\,dx=0, \quad\quad{}\displaystyle\lim_{|y|\to\infty}\frac{1}{|Q|}\int_Q|b(x+y)-b_Q|\,dx=0, \mbox{for each}Q.
a_k(\mathcal{Q}):=\begin{bmatrix}\frac{1}{2\pi}\int_0^{2\pi}\mathcal{Q}(x_1,0) e^{-i k x_1}\dx_1 \frac{1}{2\pi}\int_0^{2\pi}\mathcal{Q}(x_1,1) e^{-i k x_1}\dx_1 \end{bmatrix}(k\in \mathbb{Z}).
\log\prod_p\big(1+r(p)^2\omega(p)\big) \leqslant\sum_p r(p)^2\omega(p)=A^2\sum_{A_0^2\leqslant p\leqslant N^{c/A_0^2}}\frac{\omega(p)}p &\leqslant A^2b_{\omega 2}\log\frac{c\log N}{2A_0^2\log A_0}+O_{\omega}\bigg(\frac{A^2}{\log A_0}\bigg).
\mathcal{E}^{( c l )}( C_{r - 1}^{\vee}) = \mathcal{E}^{( c l )}( B_{r}) = \frac{r m \cos ( \pi / 4 - \pi / 4 r )}{\pi b^{2}\cos ( \pi / 4 r )}; \quad w_{0}= w_{r}= 0
{\gamma_i}''(t)=0, \;\;\; a_1{\gamma_i}(0)-b_1{\gamma_i}'(0)=1,\;a_2{\gamma_i}(1)+b_2{\gamma_i}'(1)=0, &{\delta_i}''(t)=0, \;\;\;\; a_1{\delta_i}(0)-b_1{\delta_i}'(0)=0,\;\, a_2{\delta_i}(1)+b_2{\delta_i}'(1)=1,
M_{a_i}\lesssim \| a^{\prime}\| M_{U_i^*}+ \| a^{\prime \prime}\|[U_i^*]^2 & \lesssim \| a^{\prime}\| (M_{u_i^*}+ [w_i^*]_{2\alpha}+ M_{q^*_i}) + \| a^{\prime \prime}\|([u^*_i]_{\alpha}+ [w_i^*]_{\alpha}+ [q^*_i]_{\alpha})^2 & \lesssim M_{u_i^*}+ [w_i^*]_{2\alpha}+ \|u_i^*\|_{\alpha}+ [w_i^*]_{\alpha}+ N_0 + N_0^{int}.
\begin{aligned}\sum_{m=-\infty}^{\infty}\left(a;q\right)_{m}I_{\nu+m}^{(3)}(2z;q)\left(-qz\right)^{m}=\frac{(-aqz^{2},-1/(az^{2});q)_{\infty}z^{\nu}}{(q/a,-q^{\nu}/(az^{2});q)_{\infty}}.\end{aligned}
v_{n-1}\tfrac{\|z^{n-1}\|_{2}}{\|z^{n}\|_{2}}(- \mu_{2}+ n) + u_{n-1}\tfrac{\|z^{n-1}\|_{1}}{\|z^{n}\|_{1}}(-\mu_{1}+ n)= (\mu_{1}- \mu_{2})(\lambda + \mu_{2}+ \mu_{1}- 1) + (\lambda + 2n - 1), n \in \mathbb{Z}.
\delta S=-m\int\eta_{\mu\nu}\frac{d\delta x^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}d\tau=-m\int\eta_{\mu\nu}\frac{d\delta x^{\mu}}{d\tau}u^{\nu}d\tau=-m\int\eta_{\mu\nu}[\frac{d}{d\tau}(\delta x^{\mu}u^{\nu})-\delta x^{\mu}\frac{d}{d\tau}u^{\nu}]d\tau
E_k(x)=-6\ln \Gamma(x+1)+2\ln 2\pi +6 x(\ln x-1)+ \ln(\Phi_0(x)+\mathrm{MC}_k(x)), &G_k(x)=E_k(x)-E_k(x+1)=6\left(1-x \ln(1+\frac 1x)\right)+\ln\frac{\Phi_0(x)+\mathrm{MC}_k(x)}{\Phi_0(x+1)+\mathrm{MC}_k(x+1)}.
\nu_{W(g_0,g_1)}(z_0) = \nu_{\dfrac{( a^j_0\frac{g_1}{g_0}-a^j_1 )'}{a^j_0}}(z_0) =\nu_{\dfrac{(G_j/g_0)'}{a^j_0}}(z_0) = \nu_{G_j}(z_0) - 1.
\left|\left(f(y_1)-f_0(y''_1)(y_1)_n\right)-\left(f(y_2)-f_0(y''_1)(y_2)_n\right)\right| &=\left|(f(y_1)-P_{y''_1}(y_1))-(f(y_2)-P_{y''_1}(y_2))\right|\lesssim r(y_1)^{1+2\alpha-\epsilon}d_G(y_1,y_2)^{\epsilon}.
\hat{\mathbb{E}}_{\tau+}[\sum_{i=1}^nX_iI_{A_i}] =\hat{\mathbb{E}}_{\tau+}[\sum_{i=1}^n\sum_{j=1}^m\xi_j^iI_{B_j^i}I_{A_i}] &=\hat{\mathbb{E}}_{\tau+}[\sum_{i=1}^n\sum_{j=1}^m\xi_j^iI_{A_i\cap B_j^i}] & =\sum_{i=1}^n\sum_{j=1}^m\hat{\mathbb{E}}_{\tau+}[\xi_j^i]I_{A_i\cap B_j^i}.
(1 \otimes F_1) \otimes w \in (1 \otimes F_2) \otimes F_{\Delta}w - F_{\Delta}(1 \otimes F_2) \otimes w + Z & = (1 \otimes F_2) \otimes F_{\Delta}w - ((E_1 F_1 + E_2 F_2) \otimes F_1) \otimes w & + (1 \otimes F_1) \otimes H_{1_{\Delta}}w - \frac{1}{2}(1 \otimes F_1) \otimes w + Z.
E[v^\Phi] \leq F_L[\rho^\Phi] - \int_{\Omega}v^\Phi\,d\rho^\Phi & \leq E[v_\Phi] + \varepsilon & \leq F_L[\rho] - \int_{\Omega}v^\Phi\,d\rho +\varepsilon & = F_L[\rho] - \int_{\Omega}v\,d\rho + \int_\Omega (v^\Phi -v)\,d\rho + \varepsilon & \leq E[v] + \int_\Omega (v^\Phi -v)\,d\rho + 2\varepsilon.
{\widetilde T}_{\mu \nu}( p ) = T_{\mu \nu}( p ) -{\widehat M}_{P}^{D - 4}\biggl [ p^{2}H_{\mu \nu}- p_{\mu}p_{\nu}( \chi + g ) +{\frac{p^{2}}{2}}\eta_{\mu \nu}\chi \biggr ] ~ .
\left\vert \mathcal{R}^k \right\vert \lesssim \sum_{j=1}^{6}L_{7,j}= \mathcal{O}\left(k^{2}\, \left\lVert[P_0]\right\rVert^2 \, \sum_{j = 1}^\aleph \lvert Q_j \rvert^2 \left( \aleph \, k + d^{-3}\, \lVert N\rVert_{\mathcal{L}(\mathbb{L}^2(\Omega); \mathbb{L}^2(\Omega))}\right) \right).
\widehat{(S^r{f})}(\omega) = \Big(\frac{\sin(\frac{2\rho+1}{2\lambda}\omega)}{(2\rho+1)\sin(\frac{1}{2\lambda}\omega)}\Big)^r \hat{f}(\omega)=\Big(\frac{\sin(\frac{1}{2\lambda'}\omega)}{\frac{\lambda'}{\lambda}\,\sin(\frac{1}{2\lambda}\omega)}\Big)^r \hat{f}(\omega).
\begin{aligned}\mathcal{A}(p_0) =2^n\int_{\partial X}dV_{\hat{g}^{p_0}}=2^n\int_{\partial X}e^{-n(s_{p_0}-s_E)|_{\partial X}}dV_{\hat{g}^E}=\int_{\mathbb{S}^{n-1}\times\mathbb{S}^1(\lambda)}e^{nb_{\gamma_+}(t_0,w,\theta)}d\Theta_{n-1}d\Theta_\lambda. \end{aligned}
\frac{\varepsilon}{6d}\ge\exp\left(-\sum_{i=p+1}^n\lambda_i(1-\tau)\right) >\exp\left(\sum_{i=p+1}^n\log(1-\lambda_i(1-\tau))\right) &=\exp\left(\log\left(\prod_{i=p+1}^n(1-\lambda_i(1-\tau))\right)\right).
Y_{4}^{-2}=\frac{1}{r^{4}}[\frac{9}{4\pi}\cdot\frac{5}{32}]^{1/2}(7z^{2}-r^{2})(x-iy)^{2}=[\frac{9}{4\pi}\cdot\frac{5}{32}]^{1/2}(7cos^{2}\theta-1)(sin^{2}\theta e^{-2i\varphi})
\underset{\ell \uparrow \infty}{\lim}\int_0^T{\langle \xi_{\ell}(t), (\bar{A})^{-1}\eta_{\ell}(t)\rangle_Y \dot{\beta}(t)dt}= \int_0^T{\langle \xi(\cdot), \eta_*(t, \cdot)\rangle_{\textsl{H}^{-1}}\dot{\beta}(t)dt}.
\frac{1}{2}f'' (3-D) -\frac{1}{2}f \frac{s''}{s}+ f' \frac{s'}{s}\frac{(3-D)}{4}+ \frac{1}{4}f \left(\frac{s'}{s}\right)^2- f\left(\frac{f'}{f}\right)^2 \left(\frac{D^2-9D+18}{8}\right) = -\alpha^2 f \frac{(D-2)^2+(D-2)}{8}
\begin{array}{r c l}{}&{}&{\Delta ( k_{i}) = k_{i}\otimes k_{i},}\\{}&{}&{\Delta ( X_{i}^{\pm}) = X_{i}^{-}\otimes 1 + k_{i}^{- 1}\otimes X_{i}^{-},}\\{}&{}&{\Delta ( X_{i}^{+}) = X_{i}^{+}\otimes 1 + k_{i}^{- 1}\otimes X_{i}^{+}.}\\ \end{array}
x \triangleright \varphi_1 \otimes \ldots \otimes \varphi_g \otimes v_1 \otimes \ldots \otimes v_n = \Psi_{g,n}(x) \cdot \varphi_1 \otimes \ldots \otimes \varphi_g \otimes v_1 \otimes \ldots \otimes v_n
Z_2^{\rm{L}}=\ln\bigg(\frac{1}{\underbrace{\mathbb{E}\Big[\sum\nolimits_{i \in \Phi /{o}}{{P_t}{G_i}\beta{{\left|{{X_{i,o}}}\right|}^{-{\alpha_i}}}}\Big]}_{\overline{\Lambda}}+N_o}\bigg).
\Theta_{a}^{I}|_{l o o p s}= - \frac{i}{2}( \omega + \beta_{a}^{I}) \langle t^{I}+ \bar{t}^{\bar{I}}\rangle^{- 1};{}\omega = - \frac{C}{8 \pi^{2}}, \beta_{a}^{I}= \frac{b_{a}^{I}}{8 \pi^{2}}.
L_a \left({K_a}^{-1}{(1 - |t'|^2)^{-1/2}}\right) = 1, \;\K_a = \frac{\pi}{2}\int_{0}^{2 \pi}\left( \cos^2 \theta + \frac{\sin^2 \theta}{a^2}\right)^{-1/2}d \theta.
|\hat\psi^{(k)}(\xi) - \hat\psi^{(k)}(\xi-\lambda)| = |\lambda||\hat\psi^{(k+1)}(\xi-\xi^\ast)| \leq C |\lambda|(1+|\xi-\xi^\ast|)^{-r}\leq C \sigma\varepsilon(1+|\xi|)^{-r}.
M_{0, \varphi}^{+}(x) =2 i \left(\begin{array}{cc}\frac{-4 + 2 e^{i \varphi x}+ 2 e^{- \varphi}}{x + i}\frac{6 - 4 e^{i \varphi x}- 2 e^{- \varphi}}{x + i}& \frac{- 3 + e^{i \varphi x}+ 2 e^{- \varphi}}{x + i}\frac{4 - 2 e^{i \varphi x}- 2 e^{- \varphi}}{x + i}\end{array}\right),
R_{b c}^{\bar{b}^{\prime}c^{\prime}}( \beta_{2}) R_{a c^{\prime}}^{\bar{a}^{\prime}d}( \beta_{1}) S_{\bar{b}^{\prime}\bar{a}^{\prime}}^{\bar{b}\bar{a}}( \beta_{1}- \beta_{2}) = S_{a b}^{a^{\prime}b^{\prime}}( \beta_{1}- \beta_{2}) R_{a^{\prime}c}^{\bar{a}c^{\prime}}( \beta_{1}) R_{b^{\prime}c^{\prime}}^{\bar{b}d}( \beta_{2}) .
\begin{array}{l}\frac{\partial C_{i,j}}{\partial P_{i,j}}=\frac{\frac{\rho_{i}\gamma_{i,j}}{2}(1-p_{f}(\lambda_{j}))(1-p_{f}(\lambda_{i}))}{1+(\frac{\gamma_{i,j}}{q_{i,j}})P_{i,j}}-\eta (\sum_{l=1}^{L}\Phi_{i,j}^{s,(l)}){{\rm \; \; \; \; \; \; \; \; \; \;}-\kappa (\sum_{l=1}^{L}\Phi_{i,j}^{r,(l)})=0.}\end{array}
\begin{aligned}\nu_i(h)-\nu\left(\left(\partial_{b_i}Q_i\right)^{p^e}d_{j,i}Q_{i}^{j-p^{e}}\right)=\nu_i(h)-\left(p^e\nu_i(\partial_{b_i}Q_i)+\nu(d_{j,i})+(j-p^{e})\nu(Q_i)\right)= = p^{e}\nu(Q_i)-p^{e}\nu(\partial_{b_i}Q_i)=\frac{b_ip^{e}}{b_i}(\beta_i-\nu_i(\partial_{b_i}Q_i))=\frac{b}{b_i}(\beta_i-\nu_i(\partial_{b_i}Q_i)).\end{aligned}
\begin{aligned}\eta_U-\eta_S =(1-P_U^U)\cdot D_E''\cdot U - (1-P_S^S)\cdot D_E''\cdot S= =D_E''\cdot(U-S)-D_U''\cdot U+D_S''\cdot S= =D_E''\cdot(U-S)-(U-S)\cdot D_2''=D''(U-S).\end{aligned}
L({\bf\dot R},{\bf\ddot R},{\mbox{\boldmath$\rho$}}_b ,{\mbox{\boldmath$\dot\rho$}}_b) =\Gamma^{-1}F({\mbox{\boldmath$\alpha$}}_b,{\mbox{\boldmath$\beta$}}_b,{\mbox{\boldmath$\gamma$}})\,, b=\overline{1,N-1};
\begin{aligned}|x_{i}(t^\prime+1) - x_{i}^*|= |G_{i}(\hat{\mathbf{x}}(t)) - x^*_{i}|\leq ~\Vert G(\hat{\mathbf{x}}(t)) - \mathbf{x}^* \Vert_\infty \leq ~\gamma \max_{j}\big|x_{j}(\tau_{j}(t^\prime))- x^*_{j}\big|.\end{aligned}
d_p(u_0,u_1)^p=\lim_{k\rightarrow\infty}d_p(u_0^k,u_1)^p=\lim_{k\rightarrow\infty}\int_M|\dot{u}_1^k|^p\omega_{u_1}^n\wedge\eta=\int_M|\dot{u}_1|^p\omega_{u_1}^n\wedge\eta
F_1(N) = \sum_{n=1}^{N}n\gamma(n) = NF(N) - \sum_{n=1}^{N-1}F(n) & = N(1+O(N^{-\epsilon})) KN^u - \sum_{n=1}^{N-1}Kn^u(1+O(n^{-\epsilon})) & = \frac{Ku}{u+1}(1+O(N^{-\epsilon}))N^{u+1}.
I = \frac{1}{16\pi G}\int d^{10}x \sqrt{g}\left[e^{-2\phi}\Big( R + 4 \partial_\mu \phi \partial^\mu \phi -\frac{1}{12}H_3^2 \Big) - \sum_p \frac{1}{2 (p+2)!}F_{p+2}^2\right] + I_{WZ}
\delta(\sigma^{*}_{3}(t)I)=\int^{t_{2}}_{t_{1}}\left[-\dot{P}_{1}-Q^{1}\right]\delta Q^{1}dt+\left[\dot{Q}^{1}-P_{1}\right]\delta P_{1}dt+\left[P_{1}\delta Q^{1}\right]^{t_{2}}_{t_{1}}
\limsup\limits_{y\uparrow1}\limsup\limits_{x\rightarrow\infty}\mathcal{J}_1 \leqslant (1+\varepsilon) \max\limits_{1\leqslant k\leqslant K\atop k\notin\mathcal{K}}\biggl\{\limsup\limits_{y\uparrow1}\limsup\limits_{x\rightarrow\infty}\frac{\overline{F}_{\xi_{k}}(\mathit{xy})}{\overline{F}_{\xi_{k}}(x)}\biggr\}& = 1+\varepsilon,
Z_{ij}= U_{iR,jR}- U_{n+1-iL,jR}= \delta_{ij}- F(\lambda_{jR}|\lambda_{iR}) + F(\lambda_{jR}|\lambda_{n+1-iL}), Y_{ij}= U_{iR,jR}+ U_{n+1-iL,jR}= \delta_{ij}- F(\lambda_{jR}|\lambda_{iR}) - F(\lambda_{jR}|\lambda_{n+1-iL}).
(d F^{1,a})_{\gamma}|_{Y_{1}^{k,a}}:Y_{1}^{1,a}\cap V_{1}^{k,a}= Y_{1}^{k,a}\rightarrow D_{1}^{k,a}(V_{1}^{k,a}) = Z_{2}^{0,a}\cap V_{2}^{k-1,a}
\sum_{\substack{m = -p_1 + p_2 - p_3 p_j \in I_j p_1 \equiv a_1 \pmod{k^2}p_2 \equiv a_2 \pmod{k^2}}}1 \asymp \frac{X^2}{\log^3 X}\Big ( \frac{1}{k \varphi(k)^3}+ o(1) \Big )
\check{h}_{N}^{( 1 )}= \displaystyle \frac{1}{\check{q}_{N}^{( 0 )}}, \check{h}_{i}^{( 1 )}= \displaystyle \frac{\check{e}_{i}^{( 0 )}\check{h}_{i + 1}^{( 1 )}+ 1}{\check{q}_{i}^{( 0 )}}, i = 1, \dots , N - 1.
X_{1}= L \; \mathrm{c o s}\psi \ , \ \X_{2}= L \; \mathrm{s i n}\psi \; \mathrm{c o s}\theta \ , \ \X_{3}+ i X_{4}= L \; \mathrm{s i n}\psi \; \mathrm{s i n}\theta \; e^{i \phi}\ ,
\sum_{k=0}^{\frac{n-1}{2}}(-1)^k [3k+1]\frac{(q;q^2)_k^3}{(q;q)_k^3}\equiv [n]q^{\frac{(n-1)^2}{4}}(-1)^{\frac{n-1}{2}}\pmod{[n]\Phi_n(q)^2}.
J_n^{(p)}\ge \sum_{l_1=0}^{n-(p-1)k}\sum_{h_2=1}^{k}\sum_{h_3=h_2+1}^{h_2+k}...\sum_{h_p=h_{p-1}+1}^{h_{p-1}+k}1_{\tilde{G}_h}(\phi^{l_1}\omega).
x(t,y) = x_0(y) \exp \big( \int_0^t \{\pi(s,y) b_0(s,y)-\frac{1}{2}\pi^2(s,y)\sigma_0^2(s,y) &+\lambda\ln(1+\pi(s,y)\gamma_0(s,y)) -\lambda \pi(s,y) \gamma_0(s,y)\}ds &+\int_0^t \pi(s,y) \sigma_0(s,y)dB(s) &+ \int_0^t \ln(1+ \pi(s,y) \gamma_0(s,y)) d\tilde{N}(s) \big).
\begin{array}{c}{h = \left\{A^{-}, A^{+}\right\}_{q}= 2 \left\{\left( \frac{1}{\cos \frac{\omega x}{2}}\cdot \cosh \frac{i}{2}\frac{d}{d x}\right)^{2}- \cosh \frac{\omega}{4}\right\}=}\\{= \frac{\stackrel{\wedge}{P}}{2}+ V \left( x \right)}\\ \end{array}
\tau ([ev_{(X_1,...,X_n)}^{\sigma h\otimes I_k}(\partial_{X_{i_{m+1}}}Q(a))\#S^{(m+1,i_{m+1})}]cS^{(m+1,i_{m+1})}_{j_{m+1}}b) &=\tau(cS^{(m+1,i_{m+1})}_{j_{m+1}}bS^{(m,i_m)}_{j_m}a_{m-1}...a_1S^{(1,i_1)}_{j_1}a[ev_{(X_1,...,X_n)}^{\sigma h}(\partial_{nX_{i_{m+1}}}^{|n|+1}P))]\#(S^{(1,i_1)},...,S^{(m+1,i_{m+1})})),
{\cal L}= 2\sqrt{2}\epsilon^{\mu\nu\rho}[\eta^{4}A_{\rho}- 2i(\eta^{2}-\frac{1}{2}\vert\phi\vert^{2})\phi (D_{\rho}\phi)^{*}]F_{\mu\nu}+4(\eta^{2}- \vert\phi\vert^{2})^{2}\vert D_{\mu}\phi\vert^{2}-V
P(x,t) := \sum_{n=0}^\infty p_{n-1}(x) \frac{t^n}{n!}= \frac{e^{(1-x)t}\left(\arcsin(x^{1/2}e^{(1-x)t}) - \arcsin(x^{1/2}) \right)}{x^{1/2}(1-x e^{2(1-x)t})^{1/2}}.