harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_{u u u}={{\epsilon^{2}}}\frac{6 4 ( - 3 u + 4 m^{2})}{- 2 5 6 u^{2}( u - m^{2}) + 3 2 \Lambda^{3}m ( 9 u - 8 m^{2}) - 2 7 \Lambda^{6}}+{\cal O}( \epsilon^{3}) .
\mathbb{E}(X_p) = 1-\frac{1}{p}+ \sum_{i=0}^{\infty}\tau_k(p^i) \Big(\frac{1}{p^i}-\frac{1}{p^{i+1}}\Big) = \Big( 1-\frac{1}{p}\Big) \sum_{i=0}^{\infty}\frac{\tau_k(p^i)}{p^{i}}= \Big( 1-\frac{1}{p}\Big)^{-(k-1)}.
|S^-(\alpha)| \leq\sum_{r=1}^{\alpha k}q^{r(k-r+1)}q^{r(r+1)/2}=\sum_{r=1}^{\alpha k}q^{-\frac{r^2}{2}+(k+\frac32)r}\leq &\leq\alpha k q^{(-\frac{\alpha^2}{2}+\alpha)k^2+\frac32\alpha k},
P(\inf\{r \ge 0: \overline{Y}^{2,N}_{r}= \bigtriangleup \}> t \,|\, \overline{Y}^{2,N}_{0}= (y,0) )= \overline{\mathbb{P}}_{\overline{\xi^{*}}}^{h}(\inf\{r \ge 0: (\overline{X}_{r})^{\mbox{\tiny $J$}}_{i}= (\overline{X}_{r})^{\mbox{\tiny $J$}}_{j}\}> t)
\langle \varphi^{2}( x ) \rangle = \int d^{3}p \left[ | \psi_{0}( z_{0}) |^{2}| \chi_{0}|^{2}+ \int_{- \frac{H}{\Delta + 2}}^{\infty}d m \; | \psi_{m}( z_{0}) |^{2}| \chi_{m}|^{2}\right] .
\mathrm{\mathrm{}^* \!}F_{\mu \nu}( x ) = 4 \pi \{\mathrm{\small \frac{1}{2}}\epsilon_{\mu \nu \rho \sigma}[ \partial^{\sigma}{\tilde{\lambda}}^{\rho}( x ) - \partial^{\rho}{\tilde{\lambda}}^{\sigma}( x ) ] \},
b_j:=\left(\mathbb{E}_{\pi^*}\left[\sum\limits_{\ell=1}^{j-1}\textsf{N}_\ell(0,2L^{-2}_*)|^{4}\right]\right)^{1/2}=\left(\mathbb{E}_{\pi^*}\left[\textsf{N}_\ell(0,2(j-1)L^{-2}_*)|^{4}\right]\right)^{1/2}=2(j-1)L^{-2}_*
((BAB^{-1})^{-1})^t \Theta (BAB^{-1})^{-1}=(B^{-1})^t(A^{-1})^tB^t\Theta BA^{-1}B^{-1}= (B^{-1})^t(A^{-1})^t\Theta' A^{-1}B^{-1}= (B^{-1})^t\Theta' B^{-1}= \Theta.
\varprojlim_n D_{LT}(V/\pi_L^n V) = \varprojlim_n (\mathbf{A}\otimes_{o_L}V/\pi_L^n V)^{H_L}= (\varprojlim_n (\mathbf{A}\otimes_{o_L}V/\pi_L^n V))^{H_L}= (\mathbf{A}\otimes_{o_L}\varprojlim_n V/\pi_L^n V)^{H_L}= (\mathbf{A}\otimes_{o_L}V)^{H_L}= D_{LT}(V) \ .
\begin{array}{llll}\mathcal{P}_{n,d}= \{p: \exists q\in\mathbb{R}[x] \ \mbox{of degree}\ \hat{d}=\hat{d}(n,d)\ \mbox{s.t.}\ \ \ q \ \mbox{is sos,}\ \ \ pq \ \mbox{is sos}\}.\end{array}
I_{2}=\int_{\eta}^{\lambda(t-s)}\bigl(K_{0}(z)\bigr)^{k}\,dz<\int_{\eta}^{\lambda(t-s)}\bigl(K_{0}(\eta)e^{\eta-z}\bigr)^{k}\,dz\,\bigl[K_{0}(\eta)e^{\eta}\bigr]^{k}\bigl(\lambda(t-s)-\eta\bigr).
|\tilde\lambda_1 - (\chi_1 - \chi_2)| + \sum_{i=2}^n |\tilde\lambda_i - \chi_1|\leq C(|\textrm{err}_1| + |\textrm{err}_2|) \leq C[r^{3-\mu}(a^{-1}+ r^{\mu -1}) + r^2]\chi_1.
\Big( \sum_{i=1}^{N}\|f_i \|^2_{L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))}\Big)^{1/2}\le \Big( \sum_{i=1}^{N}\|f_i \|^r_{L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))}\Big)^{1/r}&\le \Big(\int_{\mathbb R^n}\Big(\sum_{i=1}^{N}\|f_i\|_{L_t^{r}(\mathbb R)}^{r}\Big)^{q/r}dx\Big)^{1/q}&=\|f\|_{L_{x}^{q}(\mathbb R^n;L_{t}^{r}(\mathbb R))}.
B_{n,\mathbf{m}}^{(d)}(\vert{\boldsymbol{\omega}}\vert\mathbf{1}-\mathbf{z}\mid{\boldsymbol{\omega}})= (-1)^{|\mathbf{m}|}B_{n,\mathbf{m}}^{(d)}(\mathbf{z}\mid{\boldsymbol{\omega}}).
\frac{n\log 2}{-\log L_{n}}= \frac{n \log 2}{-\left(n-n_{k}+s_{2k}\right)\log q}\leq -\left(\frac{\log 2}{\log q}\right)\frac{n_{k+1}}{s_{2k+1}}= -\left(\frac{\log 2}{\log q}\right)\frac{\sum_{i=1}^{k+1}a_{2j-1}}{\sum_{i=1}^{2k+1}a_{i}},
\real\frac{z \mathtt{h}_{a, \nu}'(z)}{\mathtt{h}_{a, \nu}(z)}\geq 1-\frac{(a\nu+1-a)(1-a^{a/2})}{2}- a^{a/2}\sum_{n=1}^\infty \frac{|z|}{\mathtt{j}_{\nu, n}^2-|z|}=\frac{|z| \mathtt{h}_{a, \nu}'(|z|)}{\mathtt{h}_{a, \nu}(|z|)}=\beta
\left\|A\left(A^{\intercal}A\right)^{\dagger}\right\| = \left\|U_A\Sigma_A V_A^{\intercal}\left(V_A\Sigma_A^2V_A^{\intercal}\right)^{\dagger}\right\| = \left\|U_A\Sigma_A^{\dagger}V_A^{\intercal}\right\| \leq \sigma_{\min}^{-1}(A).
\bullet \ \begin{tabular}{p{4.5in}}If $D$ is a chamber, then any nonempty proper subset ${\mathcal X}= \{\, \pm \psi_{i_1}\, , \,\dots \, , \, \pm \psi_{i_k}\, \}$ of $\Xi (D)$ is permissible. \end{tabular}\hskip 0.22in
\ell\frac{\partial}{\partial \ell}[\phi(\ell k_1,\ell^\frac{3}{2}k_2)]=\ell k_1\frac{\partial\phi}{\partial k_1}(\ell k_1,\ell^\frac{3}{2}k_2)+\frac{3}{2}\ell^\frac{3}{2}k_2\frac{\partial\phi}{\partial k_2}(\ell k_1,\ell^\frac{3}{2}k_2)=:\delta\phi_\ell(k)
p_{c}\left(U,T_{1},T_{2}\right)\approx \exp\left(-\bar{L}(T_1,T_2)\right)\left(\sum_{k=0}^{U-1}\frac{\bar{L}(T_1,T_2)^{k}}{k!}+U\sum_{k=U}^{\infty}\frac{\bar{L}(T_1,T_2)^{k}}{(k+1)!}\right)\;.
\sum_{{\bf n}\in \mathbb{Z}^k_{\geq 0}}\frac{q^{\frac12{\bf n}^\top A{\bf n}}{\bf x}^{{\bf n}}}{(q)_{n_1}\cdots (q)_{n_k}}=\sum_{{\bf m}\in \mathbb{Z}^{\ell}_{\geq 0}}\frac{q^{B ({\bf m})}{\bf x}^{U{\bf m}}}{(q)_{m_1}\cdots (q)_{m_\ell}},
\mathcal{Q}_{t}(x_{t-1}, \xi_{[t-1]}, D_{t-1})=\mathbb{E}_{\xi_t, D_t}\Big[ \mathfrak{Q}_t ( x_{t-1}, \xi_{[t-1]}, D_{t-1}, \xi_t , D_t ) | D_{t-1}, \xi_{[t-1]}\Big]
E[(I_{[S,T]}(\varphi_n) - I_{[S,T]}(\varphi_m) )^2] = E[(I_{[S,T]}(\varphi_n - \varphi_m) )^2] & = E \Bigg[ \int^T_S (\varphi_n - \varphi_m)^2 dt \Bigg].
\int_{\mathcal{X}\times \mathcal{Y}}\hat{K}_\tau \varphi(x, y) p_t(dx, dy) = \int_{\mathcal{Y}}\int_\mathcal{X}(\hat{K}^{\tau,y}\varphi(\cdot, y))(x) \nu_t^{\phi(y)}p_t(\mathcal{X}, dy) = 0 \hat{K}_\tau \hat{F}_\tau(x, y) = f_\tau(x, y) = K_\tau F_\tau (x, y).
L_{ij}=\left(\begin{array}{cccc}0 k_{1}-k_{3}-k_{1}+k_{2}k_{3}-k_{2}-k_{1}+k_{3}0 k_{1}-k_{3}k_{1}-k_{2}-k_{1}0 k_{2}-k_{3}+k_{2}k_{3}-k_{2}0\end{array}\right)
\int_\Omega G_\Omega(x,y)\,\delta(x)^{1-s}\chi_{(0,\eta)}(\delta(x))\;dx\ \leq\ \left\lbrace\begin{aligned}C\eta^{1+\sigma}\delta(y)^{s-\sigma}\hbox{if}\delta(y)\geq\eta & C\eta\delta(y)^s \hbox{if}\delta(y)<\eta.\end{aligned}\right.
m_\lambda = \begin{cases}(n+1)^2 \textit{if}\;\, k = 0, \; n_1 = n_2 = n, 2(n+1)^2 \textit{if}\;\, k \neq 0, \; n_1 = n_2 = n, 2(n_1+1)(n_2+1) \textit{if}\;\, k = 0, \; n_1 \neq n_2, 4(n_1+1)(n_2+ 1) \textit{if}\;\, k\neq 0, \; n_1 \neq n_2.\end{cases}
j^{\mu \nu}=\frac{L_p}{2\pi}\frac 1{\sqrt{g_x}}\epsilon^{\mu\nu \lambda \rho}\epsilon_{ab}\frac \partial{\partial \phi^c}\frac\partial{\partial \phi^a}(\ln \Vert \phi \Vert )\partial_\lambda \phi^c\partial_\rho \phi^b.
\frac{1}{\sqrt{N}}\sqrt{\frac{\pi}{\pi \! + \! g N}}\Phi_{0}^{( 1 )}\; = \; \frac{1}{2 \sqrt{\pi}}\left[ \frac{\theta}{N}- \operatorname{a r c s i n}\left( \frac{e^{2}}{2 m c}\sqrt{\frac{N}{\pi \! + \! g N}}\Phi_{0}^{( 1 )}\right) \right] \; .
Z_{0}^{\mathrm{Q E D}}= \int_{- \infty}^{\infty}\prod_{x_{\perp}}d \tilde{a}_{3}\left( x_{\perp}\right) \cos^{2}\left( \tilde{a}_{3}\left( x_{\perp}\right) \right) \exp \left\{- \frac{2 \ell}{g^{2}L}\sum_{y_{\perp}, \delta_{\perp}}( \tilde{a}_{3}( y_{\perp}+ \delta_{\perp}) - \tilde{a}_{3}( y_{\perp}) )^{2}\right\}
\Bigl ( \,{\frac{w^{\prime \prime}( s )}{2 \, w ( s )}}\, - \,{\frac{w^{\prime \prime}( t )}{2 \, w ( t )}}\, \Bigr ) \, w ( s + t ) ~ = ~ w ( s ) \, w^{\prime}( t ) \, - \, w^{\prime}( s ) \, w ( t ) ~ .
\mathrm{G r}_{2 , 1}= \left( \begin{array}{c c c c c}{\sqrt{Q}}&{1}&{0}&{\alpha_{1}}\\{1}&{\sqrt{Q}}&{1}&{z}\\{0}&{1}&{\sqrt{Q}}&{1}\\{\alpha_{1}^{- 1}}&{z}&{1}&{\sqrt{Q}}\\ \end{array}\right)
\tau_{j-1}(y) \in V_{j}\cap \overline{B}_{r-(2+36C_1(m) \delta +24 \delta)\sum \limits_{k=j}^\infty r_k}(x) \subset V_j \cap \overline{B}_{r-r_j(2+36C_1(m)\delta+24\delta)}(x) \subset V_j \cap \overline{B}_{r-2r_j(1+6\delta)}(x).
\operatorname{grad}\operatorname{div}\textbf{\textit{u}}= g^{ij}\Bigl( \frac{\partial^2 u^k}{\partial x_j \partial x_k}+ \Gamma^l_{kl}\frac{\partial u^k}{\partial x_j}+ \frac{\partial \Gamma^l_{kl}}{\partial x_j}u^k \Bigr) \frac{\partial}{\partial x_i}.
W_\alpha=-i\lambda_\alpha+\theta\cdot\sigma^\mu\cdot\bar{\theta}\,\partial_\mu\lambda_\alpha-\frac{i}{4}\theta^2 \bar{\theta}^2\partial^2 \lambda_\alpha+\theta_\alpha D-\frac{i}{2}\theta^2\sigma^\mu{}_{\alpha\dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\partial_\mu D
(x^1,\cdots,x^{2n},x^{2n+1}) \cdot(\tilde{x}^1,\cdots,\tilde{x}^{2n},\tilde{x}^{2n+1}) &=\left(x^1+\tilde{x}^1,\cdots,x^{2n}+\tilde{x}^{2n},x^{2n+1}+\tilde{x}^{2n+1}+\frac{1}{2}\sum_{i=1}^{n}(x^{i}\tilde{x}^{n+i}-\tilde{x}^{i}x^{n+i})\right).
1+\frac{1}{3}+\cdot\cdot\cdot+\frac{1}{2a-1}-\frac{1}{2}-\frac{1}{4}-\cdot\cdot\cdot-\frac{1}{2b}+\frac{1}{2a+1}+\cdot\cdot\cdot+\frac{1}{4a-1}-\frac{1}{2b+2}-\cdot\cdot\cdot
N(t,h) =N(t,h)-N_2(t,h) + N_2(t,h) &\le 2\epsilon \log N(t,h) +N(t',h) &\le \frac{N(t,h)}{N(t,h)-2\epsilon \log N(t,h)}N(t',h) &\le 2 N(t',h),
K_{m n}= \frac{( Z^{\prime})^{2}}{4 Z^{5 / 2}}\left\{- 4 p ( r ) I_{m n}+ w ( r ) W_{m n}+ [ p ( r ) - q ( r ) ] f^{- 1}( r ) \frac{y^{m}y^{n}}{r^{2}}\right\}
h''_{\ell}(w) = \frac{a_{\ell}}{k_{\ell}w (a_{\ell}- w)}- \frac{A_{\ell}B^2}{(1+Bw)^2}&= \frac{a_{\ell}g_{\ell}(w)}{k_{\ell}w (a_{\ell}- w)(1+Bw)^2},
F(kp_i)=\sum_{d|kp_i}\mu(d)N(d)=\sum_{d|k}\mu(d)(N(d)-N(dp_i))=\sum_{d|k}\frac{\mu(d)(p+1)}{2d}\left(1-\frac 1{p_i}\right)+\sum_{d|k}\mu(d)(\xi(d)-\xi(dp_i)).
\omega_{4}^{3}= - i \{\frac{1}{2}d \ln ( \frac{d \overline{{{g}}}_{2}}{d \overline{{{z}}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{{g}}}_{2}- \overline{{{g}}}_{2}d g_{2})}{( 1 - g_{2}\overline{{{g}}}_{2})}\},
\langle \hat{\Phi}^{*}( x ) \hat{\Phi}( x ) \rangle_{R e n .}= \frac 1{4 8 \pi^{2}r^{2}}\frac{1 - \alpha^{2}}{\alpha^{2}}- \frac 1{8 \pi^{2}\alpha r^{2}\ln ( \frac{2 r}R e^{-{\mathcal{C}}+ \alpha / \delta})}\ .
S = - \frac{1}{2}\int d^{2}x d y \, \Bigl ( i \bar{\psi}\gamma^{\mu}\frac{\partial \psi}{\partial x^{\mu}}- \frac{1}{2}\bar{\psi}\gamma_{5}\frac{\partial \psi}{\partial y}+ \frac{\partial \phi}{\partial x^{\mu}}\frac{\partial \phi}{\partial x_{\mu}}- \frac{1}{4}\frac{\partial \phi}{\partial y}\frac{\partial \phi}{\partial y}+ F^{2}\Bigr )
a_1 = e^{-\alpha_1^\sharp(x;\tau)/2},b_1 = e^{\alpha_1^\sharp(x;\tau)/2}e^{\Psi^\sharp_1(x, \rho; \tau)}, a_2 = e^{-\alpha_2^\sharp(x;\tau)/2},b_2 = e^{\alpha_2^\sharp(x;\tau)/2}e^{\Psi^\sharp_2(x, \rho; \tau)}.
\Pi_{\delta_\varepsilon (t),\xi}^\bot(u_0 - W_{\delta_\varepsilon(t),\xi}+\phi_{\delta_\varepsilon(t),\xi}-i^\ast (f_\varepsilon(u_0-W_{\delta_\varepsilon(t),\xi}+\phi_{\delta_\varepsilon(t),\xi})))=0.
\biggl|\frac{d}{dr}I_{\frac{d+1}{2},\frac{1}{2}}\biggl(1- \frac{r^2}{4}\biggr) \biggr|= \frac{(1-r^2/4)^\frac{d-1}{2}}{\mathrm{B}_{(d+1)/2,1/2}}\leq \frac{1}{\mathrm{B}_{(d+1)/2,1/2}}.
|\Omega(u)|=\int_{\Omega(u)}dV\leq \frac{1}{A(u)^{n}}\int_{\Omega(u)}|u|^{n}dV \leq \frac{1}{A(u)^{n}}\int_{M}|u|^{n}dV=2^{\frac{1}{n-1}}\tau^{-1}.
\chi_{m}^{\ell , s}( \tau , z ) = \sum_{r \in{\bf Z}_{k}}c_{m - s + 4 r}^{\ell}( \tau ) \: \theta_{2 m + ( k + 2 ) ( - s + 4 r ) , 2 k ( k + 2 )}\left( \tau , \frac{z}{k + 2}\right) .
\max \quad\sum_{m=1}^{2}\big[p_2^mr \min\{x_1^m,1-x_2^m\}- & w_dp_1^m\max\{\left(1-x_1^m\right)d_{1,2},\left(1-x_1^m-x_2^m\right)d_{1,0}+x_2^md_{1,2}\}\big] & s.t. x_1^1 + x_1^2 \leq 1
\frac{d\Pr\left[ X_{j}=I\right]}{dt}=\beta\sum_{k=1}^{N}a_{kj}\Pr\left[X_{k}=I\right] -\delta\Pr\left[ X_{j}=I\right] -\beta\sum_{k=1}^{N}a_{kj}\Pr\left[ X_{j}=I,X_{k}=I\right]
{\bf L}\Lambda^{\mathfrak{m}}\left(E_{R}(k)\right) \simeq{\bf L}\Lambda^{\mathfrak{m}}\left({\bf R}\Gamma_{\mathfrak{m}}(D)\right) \simeq{\bf L}\Lambda^{\mathfrak{m}}(D) \simeq D\otimes_{R}^{\bf L}\widehat{R}^{\mathfrak{m}}\simeq D\otimes_{R}\widehat{R}^{\mathfrak{m}}.
\langle V_{i_{1}}\dots V_{i_{n}}\rangle_{g}= \int{\cal D}\rho{\cal D}X e^{- S^{*}- g^{i}\int_{\Sigma}d^{2}\sigma V_{i}+ Q^{2}\partial \rho{\overline{\partial}}\rho - Q^{2}\rho \int_{\Sigma}d^{2}\sigma \rho R^{( 2 )}}V_{i_{1}}\dots V_{i_{n}}
\tilde A_{h,\tilde h}= e^{-m\hat h\tilde h s_{\tilde h}}D_{s_{\tilde h}}^j (\hat h\tilde h)^{m-j}P_{m-j}(\hat h x',e^{\hat h\tilde h s_{\tilde h}}x',y,(\hat h\tilde h)^{-1}D_{Y_{\tilde h}})\bigl(\delta(s_{\tilde h})\delta(Y_{\tilde h})\bigr)\cdot (\hat h\tilde h)^{-n}.
e^{- i \psi_{p + 1}}= \frac{a_{p}+ e^{- i \psi_{p}}}{\bar{a}_{p}+ e^{i \psi_{p}}}= e^{- i 2 \alpha_{p}}\frac{\mid a_{p}\mid + e^{- i \chi_{p}}}{\mid a_{p}\mid + e^{i \chi_{p}}}
a\frac{N^2}{|Q_0|}= (a^3\bar{\rho})^{1/3}N\bar{\rho}^{2/3}= (a^3\bar{\rho})^{1/3}\frac{N^{5/3}}{|Q_0|^{2/3}}\le (a^3\bar{\rho})^{1/3}\int_{Q_0}\rho_{\psi}^{5/3},
\Vert (\mathcal{A}_{D,\varepsilon}-\zeta I)^{-1}-(\mathcal{A}_D^0-\zeta I)^{-1}-\varepsilon K_D(\varepsilon ;\zeta )\Vert_{L_2\rightarrow H^1}\leqslant \widehat{C}_{13}\left(\varepsilon^{1/2}\vert \zeta \vert^{-1/4}+\varepsilon\right) , \zeta \in \gamma ,0<\varepsilon \leqslant\varepsilon_1,
\begin{cases}\Delta Q_{b}-Q_{b}+ib\Lambda Q_{b}+|Q_{b}|^{4/d}Q_{b}=0, P_{b}\equiv Q_{b}e^{i\frac{b|y|^{2}}{4}}>0 B_{R_{b}},Q_{b}(0)\in (Q(0)-\epsilon^{*}(\eta),Q(0)+\epsilon^{*}(\eta)), Q_{b}(R_{b})=0.\end{cases}
\partial_t \widetilde{\mathrm L}^2 \leqslant s^2 \frac{1}{2\pi}\oint_{S^1}\left(s^2-\partial_t \widetilde{\mathrm L}^2\right)\,\mathrm dt <\varepsilon \frac{1}{2\pi}\oint_{S^1}\partial_t\widetilde{\mathrm L}\,\mathrm dt = 0
[\begin{matrix}c_{2}&s_{1}s_{2}&c_{1}s_{2}\\s_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}&-c_{3}s_{1}-c_{1}c_{2}s_{3}\\ -c_{3}s_{2}&c_{3}c_{2}s_{1}+c_{1}s_{3}&c_{3}c_{2}c_{1}-s_{3}s_{1}\end{matrix}]
E_{1}\left(\tau,\epsilon,T\right)=\left|\int_{\mathcal{B}\left(T\right)}\widehat{S}_{\epsilon}\left(\omega\right)\int_{\mathbb{R}^{nd}}\sum_{m\in\mathbb{Z}^{nd}\setminus\left\{0\right\}}\vartheta_{T,y-Tm}\left(\omega\right)\widehat{\zeta}_{\tau}\left(y\right)dyd\omega\right|.
S=\int d^{6}x\left[{\frac{1}{12}}{\epsilon}^{0ijklm}F_{0ij}F_{klm}-{\frac 1 6}F_{klm}F_{klm}+{\frac 1 6}G^{{\mu}{\nu}{\sigma}}(F_{{\mu}{\nu}{\sigma}}-{\partial}_{[{\mu}}A_{{\nu}{\sigma}]})\right],
\| A(t)^{1/2}F(t) - (t^2 S)^{1/2}P \| \le C_5 t^2, |t| \le t^0; C_5 = \beta_5 \delta^{-1/2}\bigl(\|X_1\|^2 + c_*^{-1/2}\|X_1\|^3\bigr).
S = \frac 1{2 \kappa^{2}}\int d^{d}x \sqrt{- g}\ \left[ \R - \frac 1 2 \partial_{\mu}\Phi \partial^{\mu}\Phi - \frac 1 2 g_{I J}( \phi ) \partial_{\mu}\phi^{I}\partial^{\mu}\phi^{J}- V ( \phi ) \ \right] ,
\chi_{0}(q)= 1 + q^2 + q^3 + q^4 + q^5 + 2\,q^6 + 2\,q^7 + 3\,q^8 + 3\,q^9 + 4\,q^{10}+ 4\,q^{11}+ 6\,q^{12}+ O(q^{13})
S_P[{\bar x};{\hat g}] ={{1}\over{4\pi\alpha^{\prime}}}\sum_{n=-\infty}^{\infty}{{2n\pi}\over{{\rm sinh}(2n\pi/l)}}\left [ (|x_n^i|^2 + |x_n^f|^2){\rm Cosh}(2n\pi/l) -2{\rm Re}(x_n^i \cdot x_n^{f*}) \right ] \quad .
\omega(\Phi_t^1(x_1^\star, 0),0,t)= \int_0^t\frac{\rho_0(x_1^\star,0)}{\Phi_s^1(x_1^\star,0)}ds=\frac{\rho_0(x_1^\star,0)}{x_1^\star}\int_0^tD(s)ds=\varphi_{\delta}(x_1^\star)\int_0^tD(s)ds.
\begin{array}{l}{\cal X}^0\vert n,z\rangle_0=q^{n}z\vert n,z\rangle_0~,{\cal D}_0\vert n,z\rangle_0=\displaystyle\frac{q^{1-n}}{q-1}z^{-1}\vert n-1,z\rangle_0~, \mu_0\vert n,z\rangle_0=\vert n-1,z\rangle_0~,\end{array}
E_{\theta_0}[P(\theta_j^0\xi_j \leq B_{j,n}^{(\ell)}\big| N_j )] = E_{\theta_0}[P(\theta_j^0\xi_j \leq B_{j,n}^{(\ell)}\big| N_j )\mathbf{1}(N_j>0)]+ O(e^{-(nF_j)}) &=\Phi(-\alpha\mp \theta_j^0) \pm \frac{n\mu_{j,3}}{6\alpha^2}\phi(-\alpha \mp \theta_j^0) +O(\alpha^2/\sqrt{nF_j}).
\begin{array}{rcl}fe = \langle e,S(f)\rangle\langle 1,{\bar t}^{-1}\rangle 1\cdot \bar{t}^{-1}+\langle t,S(1)\rangle\langle 1,\bar{t}^{-1}\rangle ef +\langle t,S(1)\rangle\langle e,f\rangle t\cdot 1 = -\langle e,f\rangle{\bar t}^{-1}+ef+\langle e,f\rangle t~.\end{array}
\{(x_1,\dots,x_{\tilde{n}})\mid0\le x_i\le1, x_i\in I_{n,m}\,(^\forall i),\sum x_i\in\mathbb{Z}\}= \mathop{\cup}_{\sigma\in S_{\tilde{n}}}\{(x_1,\dots,x_{\tilde{n}})\mid0\le x_{\sigma(1)}\le \dots\le x_{\sigma({\tilde{n}})}\le1, x_i\in I_{n,m}\,(^\forall i),\sum x_i\in\mathbb{Z}\},
J^{(4)}(4;1,1,1,1)={\textstyle{1\over12}}\; \mbox{i}\pi^2 \;\frac{\Omega^{(4)}}{V^{(4)}}= \frac{2 \; \mbox{i}\pi^2}{m_1 m_2 m_3 m_4}\; \frac{\Omega^{(4)}}{\sqrt{D^{(4)}}}.
x_{2i-1,j}+ x_{2i-1,j+1}+ x_{2i,j}+ x_{2i,j+1}=x_{2i-1,j}+ x_{2i-1,j+1}+ (\tfrac{1}{2}S - x_{2i,n+1-j}) + (\tfrac{1}{2}S - x_{2i,n-j}) = a_j +S -a_j =S.
\sum_{i=0}^{[\frac{2n+1-\nu}{2}]}{{2n+1-2i}\choose{\nu}}A_i - q^{n-\nu+1/2}\sum_{i=0}^{[\frac{\nu}{2}]}{{2n+1-2i}\choose{2n+1-\nu}}A_i =0\qquad(\nu=0,1,\cdots, 2n+1).
V ( k , k \,^{\prime}; \tau ) = \frac{m}{\alpha}\, \left( e^{- \alpha ( k + k \,^{\prime}) \tau}\, \, \delta ( k - k \,^{\prime}- i ) + \frac{e^{\alpha ( k + k \,^{\prime}) \tau}}{4 \, k \, k \,^{\prime}}\, \, \delta ( k - k \,^{\prime}+ i ) \right) .
S= -\frac{1}{2\pi}\int d^2\sigma\left [ \partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu}-i\; \bar{\psi}^{\mu,j}\;\rho^{\alpha}\;\partial_{\alpha}\; \psi_{\mu,j}+ i\; \bar{\phi}^{\mu,k}\;\rho^{\alpha}\;\partial_{\alpha}\; \phi_{\mu,k}\right ]
\Pi_{2}\left(\displaystyle \sum_{b \in T_{k}, r \in L(k)}r^{-1}\overline{N}(^{\ast}b)\left(\pi_{br}(x)\right)\right)= -\displaystyle \sum_{b \in T_{k}, r \in L(k)}r^{-1}\gamma' \xi_{be_{k}}\pi_{br}(x)
\frac{dv}{v}=\frac{dM}{M}-\frac{vdv}{2c_{p}T}=\frac{dM}{M}-\frac{dv\frac{v^{2}}{c_{p}T}}{2v}=\frac{dM}{M}-\frac{dv\frac{M^{2}c^{2}}{c_{p}T}}{2v}
\left\vert v_{p}-v_{q}\right\vert \wedge u =P_{\left( x-te\right)^{+}}\left[ \left( v_{q}-v_{p}\right) \wedge u\right] +P_{\left( x-te\right)^{+}}^{d}\left[ \left( v_{q}-v_{p}\right) \wedge u\right] & \leq P_{\left( x-te\right)^{+}}u+{\textstyle\sum\limits_{k=p+1}^{q}}\left( t^{k}/k!\right) e.
S_{GF}^{(0)}= -\frac{1}{4}\int dx^4\sqrt{-\bar{g}}\Bigl[\bar{g}^{\mu \nu}\bar{g}^{\lambda \sigma}f_{\mu \lambda}^{(0)}f_{\nu \sigma}^{(0)}+\frac{I_2}{I_1}\bar{g}^{\mu \nu}a_{\mu}^{(0)}a_{\nu}^{(0)}\Bigr].
\begin{aligned}\|\nabla u-\nabla v\|_{L^{p(\cdot)}(\Omega)}\le 2\|d(\nabla u,SO(n))\|_{L^{p(\cdot)}(\Omega)}+2(C+1)\|M_\Omega(d(\nabla u,SO(n)))\|_{L^{p(\cdot)}(\Omega)}&\le C\|d(\nabla u,SO(n))\|_{L^{p(\cdot)}(\Omega)}\end{aligned}
Y_t^{b,n}=\xi_T^a -\int_t^T g^b(s,Y_s^{b,n}, Z_s^{b,n})+\lambda_s^n \frac{1-e^{\alpha(\xi^a_s-Y_s^{b,n})}}{\alpha}ds-\int_t^TZ_s^{b,n}\cdot dW_s,
h\sum_{i,j,k}\int \partial_i\partial_j \xi_k \partial G_{h/2}\ast u \, \partial_j\partial_k G_{h/2}\ast u \,dx= -\frac h2 \sum_{i,j,k}\int \partial_i\partial_j (\nabla \cdot \xi ) \partial_i G_{h/2}\ast u \, \partial_j G_{h/2}\ast u \,dx,
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}]\}.
S [ \psi , \phi , \eta ] = \int d^{D}x \left\{- \frac{1}{2}\sum_{j = 1}^{N}\bar{\psi}^{j}( \gamma \cdot \partial + * \phi * ) \psi^{j}+ \frac{N}{2 \lambda}\phi^{2}- \bar{\eta}\psi \right\}
\frac{d(V_i)}{d(V_3)}= \lim_{i\tau \nearrow 0}\frac{\sum_{j=1}^4 S_{j,i}\chi^{}_{V_j}(-\frac{1}{\tau})}{\sum_{k=1}^4 S_{k,3}\chi^{}_k (-\frac{1}{\tau})}= \lim_{q \rightarrow 0}\frac{\sum_{j=1}^4 S_{j,i}\chi^{}_{j}(q)}{\sum_{k=1}^4 S_{k,3}\chi^{}_k (q)}= \frac{S_{3,i}}{S_{3,3}}.
\begin{aligned}\eta\bigl(f\bigr)(s) = \Bigl( \operatorname{diag}\bigl[\overbrace{c(f(0)), \dots, c(f(0))}^a, f(t_1(s)), \dots, f(t_k(s)), \underbrace{c(f(1)), \dots, c(f(1))}_b \bigr] \Bigr), \end{aligned}
U^{\prime}(\lambda) = \frac{\lambda J_{0}(x) + J_{1}(x)}{1-{\lambda}^{2}}= J_{0}(g + \lambda) [ (g - \lambda) (g + \lambda)]^{-1}=J_{0}(g - \lambda)^{-1}= - (g + \lambda)^{-1}J_{0}
\left| \int_s^t L_N \pi_{r}^N(\phi_j)dr\right|^2 = \left( \int_s^t \left(\pi_{r}^N(A\phi_j) + R_3(\phi_j,N,r)\right) dr\right)^2 &\leq 2\left( \int_s^t \pi_{r}^N(A\phi_j) dr\right)^2 + 2\left(\int_s^t R_3(\phi_j,N,r,\sigma)dr\right)^2.
r_{k,\ell}(t,h) = \sum_{\substack{i_1,i_2,i_3 \ge 0 i_1+i_2+i_3 = k-1}}\frac{\alpha_{i_1,k}(\log t)^{i_3}}{i_2! i_3!}\sum_{\substack{j_1,j_2,j_3 \ge 0 j_1+j_2 +j_3 = \ell-1}}\frac{\alpha_{j_1,\ell}\mathcal{D}_{k,\ell}^{(i_2,j_2)}(0,0) (\log t)^{j_3}}{j_{2}! j_3!}.
I^{ij}=\int\limits_{\mathbb T^d}\! \int\limits_{\mathbb R^d}\! a (\xi\!-\!q) \mu (\xi, q ) \big( (\xi - q ) + ( \varkappa_1 (\xi) - \varkappa_1 (q)) \big)^i \big( (\xi - q ) + ( \varkappa_1 (\xi) - \varkappa_1 (q)) \big)^j v_0(\xi) dq d\xi.
\mathcal{C}_{+}^{\mathrm{local}}= \left\{z = z_{1}+ r e^{i\frac{2\pi}{3}}: 0 \leq r < \eta_{-}^{-1}\delta\right\}, \mathcal{C}_{+}^{1}= \left\{z = \Re{z_{2}}+ iy: \Im{z_{2}}< y < \Im{z_{3}}\right\}, \mathcal{C}_{+}^{2}= \left\{z \in \mathcal{C}_{\tau, 0}: 1 < \Re{z}< \Re{z_{2}}~~\Im z \geq 0\right\}.
- \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{4}F_{\mu \nu}^{\prime}F^{\mu \nu}= \frac{- 1}{8 g^{2}}({\cal F}_{\ast}^{\mu \nu}{\cal F}_{\mu \nu}- i{\cal F}_{\ast}^{\mu \nu}{\cal F}_{\mu \nu}i ) .
\beta_{i_0,j}=\frac{1}{q_\ell}\Big(\beta_{i_0-\ell,j+1}-\sum_{s=0}^{\ell-1}q_s\beta_{i_0-\ell+s,j}\Big) &=\frac{1}{q_\ell}\Big(\sum_{s=i_0-\ell}^{i_0+j\ell}q_{i_0-\ell,j+1,s}\gamma_s-\sum_{s=0}^{\ell-1}q_s\Big(\sum_{u=i_0-\ell+s}^{i_0+s+(j-1)\ell}q_{i_0-\ell+s,j,u}\gamma_u\Big)\Big)
[0,1)^{d+1}\times \mathbb{R}^{d+1}\times [0,1) \times \mathbb{R}\to[0,1) \times \mathbb{R}(x_1,\dots,x_{d+1},y_1,\dots,y_{d+1},x,t) \mapsto\left( Tx, t + \sum_{i=0}^d y_i \cdot 1_{[x_0 + \cdots + x_i,x_0 + \cdots + x_{i+1})}(x) \right)
\wedge_{i}\psi_B^{-1}(b_3^i) = \pm \nu(\mathcal{E}_3)^{-1}(\wedge_{i}\psi_B^{-1}\theta_3(a_3^i))\wedge (\wedge_{i}\psi_B^{-1}\tau_3^{-1}(c_3^i)) = \pm \nu(\mathcal{E}_3)^{-1}P \wedge Q.
m_{\mathrm ADM}(g) = \frac{1}{2(n-1)\omega_{n-1}}\lim_{r\to \infty}\int_{|x|=r}\sum_{i,j=1}^n \left( \frac{\partial g_{ij}}{\partial x_i}- \frac{\partial g_{ij}}{\partial x_j}\right) \frac{x_j}{|x|}\, d\sigma.
S_{\mathrm{{S u g r a}}}^{( D )}( \tilde{\hat{g}}, \tilde{\hat{B}}, \tilde{\hat{\phi}}) = S_{\mathrm{S u g r a}}^{( D )}( \hat{g}, \hat{B}, \hat{\phi}) + \int d^{D}x \A ( \hat{g}, \hat{B}, \hat{\phi}) \partial_{\underline{{{x}}}}B ( \hat{g}, \hat{B}, \hat{\phi}) \, ,
|F_1\cap F_2| =g_1(F_1\cap F_2) \mbox{and}k(F_1)+k(F_2) =\min\{k(F_1\cup F_2), |V(F_1\cup F_2)|-2\}+\min\{k(F_1\cap F_2), |V(F_1\cap F_2)|-2\}.
r(x) - r(y) = \sum_{j=1}^N \frac{x^\frac{1}{2^j}- y^\frac{1}{2^j}}{2^{j-1}}\le \left(x^\frac{1}{2}- y^\frac{1}{2}\right) \sum_{j=1}^N \frac{1}{2^{j-1}}&= \frac{\left(2 - \frac{1}{2^{N-1}}\right)(x-y)}{\left(x^\frac{1}{2}+ y^\frac{1}{2}\right)}< \frac{2^N - 1}{2^{N}\sqrt{y}}.
\widehat{u}\, :=\, \Big( \int^{T}_{0}\mathbb E \big[ \widetilde{X}_{t}^{2}\vert \mathcal F_{T}^{X}\big]{\mathrm d}t \Big)^{-1}\cdot \mathbb E \Big[ \int^{T}_{0}X_{t}\widetilde{X}_{t}{\mathrm d}t + \int^{T}_{0}\widetilde{X}_{t}{\mathrm d}X_{t}\, \Big \vert \, \mathcal F_{T}^{X}\Big] \,
\int_{1}^{\infty}d l F ( l ) = -{\frac{1}{\epsilon}}\left( \frac{d}{d \epsilon}\right)^{( d + 1 ) / 2}{\frac{\left[ \sqrt{\epsilon^{2}+ 1}- \epsilon \right]^{( d - 1 ) / 2}}{\sqrt{\epsilon^{2}+ 1}}}= - F ( 1 ) .
S = - \tau_{0}\int d \xi \left\{( 1 - v^{2})^{1 / 2}( 1 - \phi ) -{\frac{1}{2}}( 1 - v^{2})^{- 1 / 2}( h_{0 0}^{I I A}+ 2 h_{0 i}v^{i}+ h_{i j}v^{i}v^{j}) - C_{0}- C_{i}v^{i}\right\}
\left(x-\omega(a)\right) R_n(x)= -\left(a-\beta_{2n+1}\right) b_n(x)+\gamma_{2n}b_{n-1}(x)+P_{n+1}(x)+\alpha_{2n+1}P_{n}(x) & \left(x-\omega(a)\right) a_n(x) =b_{n+1}(x)+\alpha_{2n+2}b_n(x)-\left(a-\beta_{2n+2}\right)P_{n+1}(x)+\gamma_{2n+1}P_{n}(x)
H = \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \quad , \quad E = \left( \begin{array}{c c}{{0}}&{{1}}\\{{0}}&{{0}}\\ \end{array}\right) \quad , \quad F = \left( \begin{array}{c c}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{array}\right) \, .
\sum_{j=i}^{n-1}\binom{n(d+1)-i-2}{j-i}(-1)^{n-1+j}\,\, \left.{^{\mathbf{y}_{j}}\delta^{j}_{k_1}}\alpha(n;d,2d,\ldots,(n-1) d, k_1) \right|_{k_1=n d -1}&= \sum_{j=i}^{n-1}\binom{n(d+1)-i-2}{j-i}(-1)^{n-1}\,\, \left.{^{\mathbf{z}_{j}}\Delta^{j}_{k_1}}\alpha(n;k_1,2d,3d,\ldots,n d) \right|_{k_1=d+1}.
-q^{(n-1)^2-{n+1\choose 2}}=-q^{1-n}q^{\frac{n(n-3)}{2}}&\equiv -q^{1-n}\left(1-\frac{(n-3)(1-q^n)}{2}\right) &\equiv -q^{1-n}+\frac{(n-3)q(1-q^n)}{2}\pmod{\Phi_n(q)^2}.
[\beta_{q'}^{(\sigma)}, \hat{\beta}_{q'}^{(\sigma)}]=\sqrt{q'}\,e^{-\frac{\epsilon}{2}(\beta^{(\sigma)2}-\beta^{(\sigma)\dagger 2})}\equiv \sqrt{q'}{\cal S}_2(\epsilon),\quad q'\equiv e^{-\epsilon}=1/q\,{,}
[{\cal L}_{n,m},{\cal L}_{k,l}] = \sum_{s=0}^{{\rm Min}(m,k)}{(m+1)!(k+1)! \over (m-s)!(k-s)!(s+1)!}{\cal L}_{n+k-s,m+l-s}- (m \leftrightarrow l, n \leftrightarrow k) \,.
1 - u_{11}v_{11}= 2 - (u_{11}- w_1)v_{11}- w_1\bigl(v_{11}+w_1^*\bigr), 1 - v_{11}u_{11}= 2 - \bigl(v_{11}+ w_1^*\bigr) u_{11}+ w_1^*(u_{11}- w_1)
\int_{{\bf p}}\frac{p_{\mu}}{\left( p^{2}+ 2 p \cdot P - M^{2}\right)^{\alpha}}= \frac{- i \pi^{n / 2}P_{\mu}}{( 2 \pi )^{d}\Gamma ( \alpha ) \left( - P^{2}- M^{2}\right)^{\alpha - n / 2}}\Gamma \left( \alpha - \frac{n}{2}\right) ,
{\sf d}_{ij}=D_{ij},{\sf c}_{i}X =\{c: \exists a\in X, a\equiv_{i}c\},{\sf s}_{[i,j]}X =\{c: \exists a\in X, a\equiv_{ij}c\},{\sf s}^{i}_{j}X =\begin{cases}{\sf c}_{i}(X\cap D_{ij}), X, \end{cases}
\lim_{i\rightarrow \infty}\left\{\int_0^TL(\widehat{\gamma}_i(t),\dot{\widehat{\gamma}}_i(t))+F(\widehat{\gamma}_i(t),m^{\eta_i}(t))\,dt+G(\widehat{\gamma}_i(T),m^{\eta_i}(T))\right\}=\int_0^TL(\gamma(t),\dot{\gamma}(t))+F(\gamma(t),m^\eta(t))\,dt+G(\gamma(T),m^\eta(T)).