harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Omega_{+ 2}^{[ - 2 ] ( i )}= \tilde{\Omega}_{+ 2}^{[ - 1 0 ] ( i )}( \theta^{+})^{8}, \Omega_{- 2}^{[ + 2 ] ( i )}= \tilde{\Omega}_{- 2}^{[ + 1 0 ] ( i )}( \theta^{-})^{8},
\omega_{0}( x ) = 2 \int_{0}^{\infty}{\frac{x \rho ( \lambda ) d \lambda}{x^{2}- \lambda^{2}}}= \int_{0}^{\infty}\rho ( \lambda ) d \lambda \left({\frac{1}{x - \lambda}}+{\frac{1}{x + \lambda}}\right) = \int_{- \infty}^{\infty}{\frac{\rho ( \lambda ) d \lambda}{x - \lambda}},
H(x,F)=:\frac{R(F_1,t)}{R(F,t)}=\frac{(1-F(x))\left( \int_{x}^{uep(F)}\int_{u}^{uep(F)}1-F(t)dt du \right)}{\left( \int_{x}^{uep(F)}1-F(t)dt \right)^2}.
g = \left( \begin{array}{c c c c c}{- 1}&{0}&{0}&{\dots}&{0}\\{0}&{( g_{i j})}&{\dots}&{0}&{0}\\{0}&{\dots}&{1}&{\dots}&{0}\\{0}&{\dots}&{\dots}&{\ddots}&{0}\\{0}&{0}&{\dots}&{\dots}&{1}\\ \end{array}\right)
\int_{0}^{\infty}d r r^{\frac{n}{2}}( c^{2}+ r^{2})^{m}J_{\frac{n - 2}{2}}( r | p | ) = ( 2 / | p | )^{m + 1}\frac{c^{\frac{n}{2}+ m}}{\Gamma ( - m )}K_{\frac{n}{2}+ m}( c | p | ) ,
M^{n,\epsilon,u}_a(t_1,t_2,t_3;1)=\int_{t_1}^{t_2}(t_3-\zeta)^{-a}S_1(t_3-\zeta) \Psi^\epsilon_2 \big(\bar{X}(\zeta), Y_n^{\epsilon,u}(\zeta)\big)P_ndw_2(\zeta)
\Delta(e_{0})v_{n,m}=v_{n+1,m}, \Delta(e_{1})v_{n,m}=x[n]_{q}[\mu-n+1]_{q}v_{n-1,m}-\frac{xq^{-\mu +2n+m-1}[m]_{q}}{q-q^{-1}}v_{n,m-1}, \Delta(h_{0})v_{n,m}=2(n+m)v_{n,m},\Delta(h_{1})v_{n,m}=-2(n+m)v_{n,m}.
\begin{array}{ll}\!\! \!\! A_{ij}(x_0,u(x_0),Du(x_0))-A_{ij}(x_0,u_0(x_0),Du_0(x_0)) = \!\! \!\! D_uA_{ij}(x_0,\hat u(x_0), Du(x_0))(u-u_0)(x_0) + D_{p_k}A_{ij}(x_0,u_0(x_0),\hat p(x_0))D_k(u-u_0)(x_0),\end{array}
\overset{1}{R}(X,Y,AZ,AW) = \overset{g}{R}(X,Y,AZ,AW) - \overset{1}{D}(X,Y) F(AZ,AW) = \overset{g}{R}(X,Y,AZ,AW) - \overset{1}{D}(X,Y) F(Z,W),
\Gamma_{\lambda,\,\underline{\beta}}=\bigcup_{\ell\in\mathbb N}\Gamma_{\lambda,\,\underline{\beta}}(\ell) \subset \mathbb N\times \mathbb Z^N, \ \ Gamma_{\lambda,\,\underline{\beta}}(\ell) =\{(\ell,\nu(s/s_0^\ell)); s\in \mathrm{H}^0(G/P,\mathcal L_\lambda^{\otimes \ell})\}.
\frac{1}{\beta^{\Lambda}\sum_{i,j\in I_{0}, \beta_{ij}=\beta}1}\sum_{i,j\in I_{0}, \beta_{ij}=\beta}\prod_{x\in \Lambda}\hbox{Tr}_{\mathcal{H}_{x}}\left(h_{x, i}h_{x, j}^* b_{x}\right)
\sum_{n=0}^{\infty}\hat{\phi}^{(+)}_n(x)\hat{\phi}^{(+)}_n(y) =\delta_{xy}, \sum_{n=0}^{\infty}\hat{\phi}^{(-)}_n(x)\hat{\phi}^{(-)}_n(y) =\delta_{xy}, \sum_{n=0}^{\infty}\hat{\phi}^{(+)}_n(x)\hat{\phi}^{(-)}_n(y)=0.
\begin{array}{c}{\langle e_{a_{k}. . . a_{1}}\mid = ( \mid e_{a_{1}. . . a_{k}}\rangle )^{+}}\\{\langle e_{a_{k}. . . a_{1}}\mid e_{b_{1}. . . b_{k}}\rangle = \delta_{a_{1}b_{1}}. . . \delta_{a_{k}b_{k}}}\\ \end{array}
P^{(\lambda_{j})}(\lambda) = E_{\lambda_{j}}(\lambda) \Phi_{\mathrm{HG}}(r^{3/2}f_{\lambda_{j}}(\lambda);\beta_{j})(s_{j}s_{j+1})^{-\frac{\sigma_{3}}{4}}e^{\frac{2}{3}(r\lambda)^{3/2}\sigma_{3}},
\mathrm{\mathrm{H}}^{s,p}_0(\Omega)= \overline{\mathrm{C}_c^\infty(\Omega)}^{\lVert \cdot \rVert_{\mathrm{\mathrm{H}}^{s,p}(\mathbb{R}^n)}}\quad\mathrm{B}^{s}_{p,q,0}(\Omega)= \overline{\mathrm{C}_c^\infty(\Omega)}^{\lVert \cdot \rVert_{\mathrm{B}^{s}_{p,q}(\mathbb{R}^n)}}
\int_{2D}|Df|^2 \; =\; \int_{2D\setminus \bigcup_{k=m-1}^{\infty}Q_k}|Df|^2 + \int_{\bigcup_{k=m-1}^{\infty}Q_k}|Df|^2 &\leq\;{\mathrm{Area}(2D)}+ C\sum_{k=m-1}^{\infty}k^{2p}{\mathrm{Area}(Q_k)}=\;{\mathrm{Area}(2D)}+ C\sum_{k=m-1}^{\infty}k^{2p}2^{-2k}.
\tau_{h}({\bf w})^1_{s,t}:={\bf w}^1_{s,t}+{\bf h}^1_{s,t},\tau_{h}({\bf w})^2_{s,t}:={\bf w}^2_{s,t}+{\bf h}^2_{s,t}+ \int_s^t{\bf w}^1_{s,u}dh_u + \int_s^t{\bf h}^1_{s,u}dw_u.
n \alpha \hbar = \frac{1}{2 \pi}\oint_{C_{1}}\frac{p d y}{1 - y^{2}}= \frac{1}{2 \pi}\left( \oint_{\gamma_{0}}\frac{p d z}{1 - z^{2}}- \oint_{\gamma_{1}}\frac{p d y}{1 - y^{2}}- \oint_{\gamma_{2}}\frac{p d y}{1 - y^{2}}\right) .
\left\Vert \partial_{t}^{s}\widetilde{u}\right\Vert_{H^{2,1}\left(Q_{\varepsilon ,T}\right)},\left\Vert \partial_{t}^{s}\widetilde{m}\right\Vert_{H^{2,1}\left( Q_{\varepsilon ,T}\right)}\leq C\delta^{1-\rho},\forall \delta \in \left( 0,\delta_{0}\right) ,s=0,1,2.
\dot I_N(t) \le C \bigg( \varphi_0 + \frac{\|\nabla\varphi\|_\infty}{r^{2d}\varphi_0}+ \frac{\|\nabla\varphi\|_\infty^2}{r^{5d}\varphi_0^2}\bigg) I_N(t) + \bigg(\frac{1}{r^{2d}N}+ \frac{1}{r^{4d}\varphi_0 N^2}+ \frac{\varphi_0}{(r^{3d}N)^{1/2}}+ \frac{\|\nabla\varphi\|_\infty^2}{r^{5d}\varphi_0^2 N^{1/2}}\bigg)\,.
Z = < \exp \left( \int d \tau \left[ - \frac{1}{2 \pi}\xi \dot{\xi}+ \frac{\sqrt{\alpha^{\prime}}}{2 \pi}\xi \psi^{1}u - \frac{1}{8 \pi}u^{2}( X^{1})^{2}+ i A_{a}( X^{a}) \partial_{\tau}X^{a}+ \frac{i \alpha^{\prime}}{2}\theta_{a b}\psi^{a}\psi^{b}\right] \right) > \ ,
\bigl(\check R_{V_1,V_2}\otimes_{V_3}\bigr)\circ\bigl(_{V_2}\otimes \check R_{V_1,V_3}\bigr)\circ\bigl(\check R_{V_2,V_3}\otimes_{V_1}\bigr)=\bigl(_{V_1}\otimes \check R_{V_2,V_3}\bigr)\circ\bigl(\check R_{V_1,V_3}\otimes_{V_2}\bigr)\circ\bigl(_{V_3}\otimes \check R_{V_1,V_2}\bigr)\ .
\mathcal{L}(B_v) := \begin{cases}\{u\in B_v : \textnormal{dist}(u,v) = r \}\textnormal{if}B_v \textnormal{is a tree}, \{u\in B_v : \textnormal{dist}(u,v) \geq r \textnormal{and}\textnormal{dist}(u,C_v) \geq r-h \}\textnormal{if}C_v \textnormal{exists}, \end{cases}
J_f(s)_{ij}: = s_j \cdot \frac{\partial}{\partial s_j}\Big(s_i\cdot\frac{\partial}{\partial s_i}\big(\log f\big)\Big)(s) &=s_is_j\frac{\partial^2}{\partial s_i\partial s_j}\big(\log f\big)(s)+\delta_{ij}\cdot s_j\cdot\frac{\partial}{\partial s_i}\big(\log f\big)(s)
S_{\psi_\alpha}^{(r)}(s)= (\psi_\alpha-\xi)\left(\frac{2\pi n}{|t|}\right)\frac{\chi_f^{(r)}}{\chi_f}(1-s)+ &+\sum_{j=1}^l\psi_\alpha^{(j)}\left(\frac{2\pi n}{|t|}\right)\left(-\frac{2\pi n}{|t|}\right)^j\gamma_{j}^{(r)}\left(s,\frac{1}{|t|}\right),
E_\varphi(F_2) = E_{\varphi_m}(F_2) \leq \iint G(p,q) d \mu_2(p) d \mu_2(q) + \int \varphi_m d \mu_2 \leq \iint G(p,q) d \mu_1(p) d \mu_1(q) + \int \varphi_m d \mu_1 + \varepsilon & = E_{\varphi_m}(F_1) + \varepsilon = E_\varphi(F_1) + \varepsilon.
\frac{\partial}{\partial\xi}\frac{\delta^2\Gamma}{\delta\psi(y)\delta\bar\psi(x)}=+\frac{\delta^3\Gamma}{\delta\psi(y)\delta\bar K\delta \chi}\frac{\delta^2\Gamma}{\delta\bar\psi(x)\delta\psi}+\frac{\delta^2\Gamma}{\delta\psi(y)\delta\bar\psi}\frac{\delta^3\Gamma}{\delta\bar\psi(x)\delta K\delta\chi}
A=\frac{V_{o}}{V_{i}}=\frac{[\frac{M}{C}+(a\frac{M}{C})(\beta^{2}-1)+\frac{\beta^{2}+2\beta+3}{4(\beta^{2}+\beta+1)}]}{(\frac{N}{C}+\frac{3\beta^{2}+2\beta+1}{4(\beta^{2}+\beta+1)})}
\widehat{\Phi}(k) := \begin{cases}\widetilde{\Phi}(k) k \in E_1 \cup E_2 \cup E_3 \tau_{e_1}\Theta \, \widetilde{\Phi}(t_{e_1}c (k)) k \in E_4 \tau_{e_2}\, \widetilde{\Phi}(t^{-1}_{e_2}(k)) k \in E_5 \Theta \, \widetilde{\Phi}(c(k)) k \in E_6.\end{cases}
(u_{+}-v_{-}-\frac{1}{2}h)(u_{-}-v_{+}+ \frac{1}{2}h)H_{i}^{-}(u)H_{i+1}^{+}(v) &=(u_{+}-v_{-}+\frac{1}{2}h)(u_{-}-v_{+}-\frac{1}{2}h)H_{i+1}^{+}(v)H_{i}^{-}(u).
B(u_{n})=\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx\le C\left\|u_{n}\right\|_{\frac{12}{5}}^{4}\le C\left\|u_{n}\right\|_{2}^{3}\left\|\nabla u_{n}\right\|_{2}^{3}.
q_{2}\left(\mathcal F_2^c,{\bf p}\right)=\sum_{n\in \mathcal F_2^c}a_{n}\sum_{k=1}^{K_2^c}\Pr [K_{2,n,0}^c=k^c]\int_{0}^{\infty}q_{k,n,D_{2,\ell_0,0}}\left({\bf p},d\right)f_{D_{2,\ell_0,0}}(d){\rm d}d.
T_{q, k}((\Phi_u*\Phi_v)_{q}^{(g_1,g_2;h_1,h_2)})(y)=T_{q,\mathbf{s}_1}(\Phi_u)(\mathcal Z_{g_1}(y,\cdot)) T_{q,\mathbf{s}_2}(\Phi_v)(\mathcal Z_{g_2}(y,\cdot))
\zeta ( k ) = \exp \left( \frac{2 \pi i}{l ( \partial ( \rho^{2}( k ) ) )}\left( \sum_{\beta = 1}^{\nu - 1}l ( \rho^{1}( k_{\beta}) ) + z ( k_{\nu}) \right) \right) , \hspace{0 . 2 i n}\nu = 1 , . . . , q ( k ) ,
\hat{\gamma}_{ij}d\hat{x}^id\hat{x}^j=\left(1+\textstyle{\hat{k}\over 4}\delta_{mn}\hat{x}^m\hat{x}^n\right)\delta_{ij}d\hat{x}^id\hat{x}^j={{d\hat{r}^2}\over{1-\hat{k}\hat{r}^2}}+\hat{r}^2\left(d\hat{\theta}^2+\sin^2\hat{\theta}d\hat{\phi}^2\right).
\left| \dfrac{d^{n+m}}{dx^{n+m}}\Big(e^{t \, d/dx}\phi \Big)(x) \right| \leqslant \sum_{k=0}^{\infty}\dfrac{t^k}{k!}c(\phi, m,j) M^{k+n}= c \, M^n \, e^{t \, M},
\int_I|(f_n'\circ g)g^{(k+1)}-(f_0'\circ g)g^{(k+1)}| = \int_I (|f_n'-f_0'|\circ g)\cdot |g^{(k+1)}| &\leq \|f_n'-f_0'\|\int_I |g^{(k+1)}|\rightarrow 0.
\tau_{jl}=\pi+\mbox{i}\; \mbox{Arch}(-c_{jl})= \pi + \frac{\mbox{i}}{2}\ln\left(\frac{k_{jl}^2-m_j^2-m_l^2+\sqrt{\lambda(m_j^2,m_l^2,k_{jl}^2)}}{k_{jl}^2-m_j^2-m_l^2-\sqrt{\lambda(m_j^2,m_l^2,k_{jl}^2)}}\right) .
0=\sum_{l=1}^{n}C_{\varepsilon}^{l}\left\langle Z_{\varepsilon,\xi(y)},\left.\frac{\partial}{\partial y_{h}}\phi_{\varepsilon,\xi(y)}\right\vert_{y=0}+\left. \frac{\partial}{\partial y_{h}}W_{\varepsilon,\xi(y)}\right\vert_{y=0}\right\rangle_{\varepsilon}=-\frac{C}{\varepsilon}\sum_{l=1}^{n}C_{\varepsilon}^{l}\left( \delta_{hl}+o\left( 1\right)\right)
\psi_{CW,m}^r : \varprojlim_n o_{L_n}^\times \longrightarrow L_m u \longmapsto \frac{1}{r!\pi_L^{rm}}\left(\partial_{\mathrm{inv}}^{r-1}\Delta_{LT}g_{u,t_0}\right)_{|Z=t_{0,m}}\ .
\frac{(f_{1},\ldots,f_{p})}{(g_{1},\ldots,g_{p})}:=\left(\frac{f_{1}}{g_{p}},\frac{f_{2}\,g_{1}}{g_{p}},\frac{f_{3}\,g_{2}}{g_{p}},\ldots,\frac{f_{p}\,g_{p-1}}{g_{p}}\right).
- 2 G_{\mu \nu}{\frac{\delta \Gamma}{\delta G_{\mu \nu}}}+ \beta^{i}{\frac{\delta \Gamma}{\delta \phi^{i}}}= - 2 \sqrt{G}({\cal W}_{d}+ \nabla_{\mu}{\cal J}_{d}^{\mu}) - \beta^{i}{\frac{\delta S_{\mathrm{l o c}; 0}}{\delta \phi^{i}}}.
c_{3}\Pr_x[T_{V(H)}> \frac{t}{2}] \le \Pr_x[T_{R_{i-1}}<T_{V(H)}] \le \frac{|\mathrm{BMP_i}|}{|\mathrm{GMP_i}\cup \mathrm{BMP_i}|}= \frac{|\mathrm{BMP_i}|}{2^{2^{2n+i-2}}}\le 2^{-c_2 2^{2n+i}}.
\begin{gathered}D^{i}= \varepsilon^{i j}E_{j}+ \varepsilon^{i j k}w_{j}H_{k}, B^{i}= \mu^{i j}H_{j}- \varepsilon^{i j k}w_{j}E_{k}, \varepsilon^{i j}= - \frac{\sqrt{-g}}{g_{0 0}}g^{i j}, \mu^{i j}= - \frac{\sqrt{-g}}{g_{0 0}}g^{i j}, w_{i}= \frac{g_{i 0}}{g_{0 0}}. \end{gathered}
T_{s-1}=\frac{f^{(s)}(\tau)}{s!p^{s}}\left[E\{(\|X_{1}-X_{2}\|_{2}^{2}-p\tau)^{s}\}+E\{(\|Y_{1}-Y_{2}\|_{2}^{2}-p\tau)^{s}\}-2E\{(\|X_{1}-Y_{1}\|_{2}^{2}-p\tau)^{s}\}\right]
\Phi_i^{-1}(0) \subset \bigcup_{j=1}^k B_{10 m_i^{-1/2}}(x_j)\quad\Phi_i^{-1}(0)\cap B_{10 m_i^{-1/2}}(x_j)\neq\emptyset,\quad\forall j\in\{1,\ldots,k\}.
h((u_{kl}^t)^*u_{ij}^s) = \delta_{st}\delta_{jl}\frac{(\rho_s^{-1})_{ik}}{d_s}, h(u_{kl}^t (u_{ij}^s)^*) = \delta_{st}\delta_{ki}\frac{(\rho_s)_{jl}}{d_s}.
H = \sqrt{g_{t t}}\left[{\frac{P_{\phi}^{2}}{h ( r ) r^{2}}}+{\frac{P_{r}^{2}}{g_{r r}}}+{\frac{P_{\rho}^{2}}{g_{\rho \rho}}}+{\frac{( \rho P_{\phi}- N \rho^{p})^{2}}{g_{\phi \phi}}}\right]^{\frac{1}{2}},
\lambda_{2n}^\pm= 2n\pi -i \sum_{k=1}^{N+1}\frac{1}{(2i\lambda_{2n}^\pm)^k}\int_0^1\varphi_1r_k\,dt + \sqrt{e_2e_3+ h_{2n}^\pm}+ \frac{\ell^2_n}{n^{N+1}}.
\frac{s^{\ast}}{q}= \frac{L_{n,2}^{\ast}(q+s^{\ast})}{q}- \frac{L_{n,1}^{\ast}(q)}{q}\leq (\beta^{\ast}+\epsilon_{1})\frac{q+s^{\ast}}{q}- \alpha^{\ast}+\epsilon=\beta^{\ast}-\alpha^{\ast}+ \beta^{\ast}\frac{s^{\ast}}{q}+\epsilon_{2}
W_{2}( R ) ={\frac{1}{\sqrt{2}R}}B J_{1}B^{- 1}= \left( \begin{matrix}{0}&{0}&{0}&{0}\\{0}&{0}&{- i / R^{2}}&{i}\\{0}&{i}&{0}&{0}\\{0}&{i / R^{2}}&{0}&{0}\\ \end{matrix}\right) ,
\|u\|_{\widetilde{L}^p_{T}(\dot{B}_{2,1}^s)}\lesssim\|u\|^\theta_{\widetilde{L}^{p_1}_{T}(\dot{B}_{2,1}^{s_1})}\|u\|^{1-\theta}_{\widetilde{L}^{p_2}_{T}(\dot{B}_{2,1}^{s_2})}, \, \frac{1}{p}=\frac{\theta}{p_1}+\frac{1-\theta}{p_2}, \, s=\theta s_1+(1-\theta)s_2
\begin{array}{l c l}{{0}}&{{=}}&{{{{\mathcal L}_{n}}( u ) \star K [ a ] - K [ a ] \star{{\mathcal L}_{n}}( w )}}\\{{}}&{{=}}&{{( u_{3}- w_{3}+ 2 n \theta a_{2}^{\prime}) p^{n - 3}}}\\{{}}&{{+}}&{{( u_{4}- w_{4}+ ( n - 3 ) \theta ( w_{3}^{\prime}- u_{3}^{\prime}) + 2 n \theta a_{3}^{\prime}+ 4 n \theta^{2}a_{2}^{\prime \prime}) p^{n - 4}}}\\{{}}&{{}}&{{\vdots}}\\ \end{array}
\Xi^{\lambda}(K_{a}F_{a}\otimes1\otimes E_{a}K_{a}^{-1}\otimes K_{a})=\sum_{i,j,m,n,o}(2c_{j}^{i}-\delta_{j}^{i})\pi(K_{a}F_{a})_{m}^{j}c_{n}^{m}\pi(E_{a}K_{\lambda})_{o}^{n}c_{i}^{o}.
\int_{\mathbb{R}^{2}}\nabla u_{n}\nabla \varphi dx+\lambda_{n}\int_{\mathbb{R}^{2}}u_{n}\varphi dx+\left[ V_{1}^{\prime}(u_{n})-V_{2}^{\prime}(u_{n})\right] \varphi -\int_{\mathbb{R}^{2}}f(u_{n})\varphi dx=o(1)\Vert \varphi\Vert .
y_t=y_T+\int_t^Tg(x_s,y_s,z_s,\tilde{z}_s,\gamma_{(s,e)})ds-\int_t^Tz_sdW_s-\int_t^T\tilde{z}_sd\xi_s-\int_t^T\int_{\mathcal{E}}\gamma_{(s,e)}\tilde{N}(de,ds).
g_{s}l_{s}^{D - p - 1}~ k_{1}. k_{3}~ k_{2}. k_{4}~ \tilde{P}( \zeta ) \int d^{D - p - 1}p_{1}~ \int d^{D - p - 1}p_{2}~ \frac{q^{2}e^{i ({\vec{p}_{1}}\cdot{\vec{\epsilon}_{1}}+{\vec{p}_{2}}\cdot{\vec{\epsilon}_{2}})}}{( q^{2}+ p_{1}^{2}) ( q^{2}+ p_{2}^{2})}~ \delta^{p + 1}\left( \sum_{i}k_{i}\right) ~ .
\frac{d}{d\eta}\, \int_{\alpha_0}^\eta \frac{g(\zeta) \, \sin\zeta \, d\zeta}{\sqrt{\cos\zeta - \cos\eta}}= - \frac{q\, \sqrt{\pi}\, (d-2) \, \Gamma((d-1)/2)}{2^{(d-2)/2}\, \Gamma(d/2)}\, \frac{\sin\eta}{1-\cos\eta}\, \sqrt{\frac{1-\cos\alpha_0}{\cos\alpha_0-\cos\eta}}, \alpha_0\leq\eta\leq\pi.
nR_{n}^{\lambda}(x)=\frac{2x}{\sqrt{1+x^2}}(n+\lambda-1)R_{n-1}^{\lambda}(x)-(n+2\lambda-2)R_{n-2}^{\lambda}(x),\;\;\; n\geq 2; begin{align*}2pt] R_{0}^{\lambda}(x)=\frac{1}{(1+x^2)^{\frac{\lambda+1}{2}}},R_{1}^{\lambda}(x)= \frac{2\lambda x}{(1+x^2)^{1+\frac{\lambda}{2}}}.
x_{\rm c}[n]=\begin{cases}\kappa_{\rm t}, \mathrm{if}x[n]\geq \kappa_{\rm t}, x[n], \mathrm{if}\kappa_{\rm b}<x[n]< \kappa_{\rm t}, \kappa_{\rm b}, \mathrm{if}x[n]\leq \kappa_{\rm b}, \end{cases}
\, \ \ \mathrm{o r \ \a s}\ \ \, \ \{\cal{L}}_{B I}= \beta^{2}\, \Biggl [ \, \ \ 1 - \sqrt{1 + \frac{1}{2 \beta^{2}}\, ({\bf B}^{2}-{\bf E}^{2}) - \frac{1}{1 6 \beta^{4}}\, ({\bf E}\cdot{\bf B})^{2}}\, \ \ \, \Biggr ] \, .
C_1 = [\emptyset, 1, \emptyset ,\emptyset,\emptyset,2,\emptyset,\emptyset,10,\emptyset], C_2 =[8, \emptyset, \emptyset, 14, 16, \emptyset, \emptyset,\emptyset, \emptyset, \emptyset],C_3 =[\emptyset, \emptyset, 19, \emptyset, \emptyset, \emptyset, 25, \emptyset, \emptyset, 27].
s-\frac{\delta'}{1+\delta}\ =\s- ((1+\epsilon)(1+ub))^{\dagger}&=\s-\frac{\epsilon'}{1+\epsilon}- \frac{(ub)'}{1+ub}&=\ub' - \frac{(ub)'}{1+ub}&=\ \frac{u^2bb' - u'b}{1+ub}.
\langle\phi(x+f),\phi(y+g)\rangle_{\theta_2}=\langle x+\theta'(x)+f,y+\theta'(y)+g\rangle_{\theta_2}= \theta'(x)(y)+f(y)+(-1)^{|x||y|}\theta'(y)(x)+(-1)^{|x||y|}g(x) = 2\langle x, y\rangle_{\theta'}+\langle x+f,y+g\rangle_{\theta_1}.
(x+y)^{n}=(\begin{matrix}n\\ 0\end{matrix})x^{n}y^{0}+(\begin{matrix}n\\ 1\end{matrix})x^{n-1}y^{1}+(\begin{matrix}n\\ 2\end{matrix})x^{n-2}y^{2}+\cdot\cdot\cdot+(\begin{matrix}n\\n-1\end{matrix})x^{1}y^{n-1}+(\begin{matrix}n\\n\end{matrix})x^{0}y^{n},
\epsilon_{1eff}^{*}(\omega)=\epsilon_{2}^{*}\frac{(\frac{r_{2}}{r_{1}})^{3}+2\frac{\epsilon_{1}^{*}-\epsilon_{2}^{*}}{\epsilon_{1}^{*}+2\epsilon_{2}^{*}}}{(\frac{r_{2}}{r_{1}})^{3}-\frac{\epsilon_{1}^{*}-\epsilon_{2}^{*}}{\epsilon_{1}^{*}+2\epsilon_{2}^{*}}}
( R^{d}{}_{e}K^{e a}) f^{b}{}_{c d}- ( R^{d}{}_{e}K^{e b}) f^{a}{}_{c d}= K^{a a^{\prime}}K^{b b^{\prime}}K_{c c^{\prime}}( f^{c^{\prime}}{}_{a^{\prime}b^{\prime}})_{R}.
L(e_{ii})-L(e_{jj}) =[L(e_{ij}),e_{ji}]+[e_{ij},L(e_{ji})] &=\sum_{x<i}C_{xi}^{ii}e_{xi}+C_{ij}^{ij}e_{ii}-C_{ij}^{ij}e_{jj}-\sum_{y>j}C_{jy}^{jj}e_{jy}&+C_{ji}^{ji}e_{ii}+\sum_{y>i}C_{iy}^{ii}e_{iy}-\sum_{x<j}C_{xj}^{jj}e_{xj}-C_{ji}^{ji}e_{jj}.
\mathbf{T}(r_1)\cdots \mathbf{T}(r_k)=\begin{pmatrix}p_k(r_1,\dots,r_k) p_{k-1}(r_1,\dots,r_{k-1}) p_{k-1}(r_2,\dots,r_k) p_{k-2}(r_2,\dots,r_{k-1})\end{pmatrix}.
T = d i a g \{\tilde{q}^{( p - 1 )^{2}/ 4 - p / 8}, \tilde{q}^{( p - 2 )^{2}/ 4 - p / 8}, \tilde{q}^{( p - 2 )^{2}/ 4 - p / 8}, \ldots \tilde{q}^{2^{2}/ 4 - p / 8}, \tilde{q}^{2^{2}/ 4 - p / 8}, \tilde{q}^{1^{2}/ 4 - p / 8}\}
\begin{aligned}\frac{1}{2}\Big( \psi^{-0}_{n-1,p}\psi^{-0}_{n,p}\rho^{0+}_{n,p}-2\psi^{-0}_{n-1,p}\rho^{0+}_{n-1,p}\psi^{-0}_{n,p-1}+\rho^{0+}_{n-2,p}\psi^{-0}_{n-1,p-1}\psi^{-0}_{n,p-1}\Big) = - \varphi^{-+}_{n-1,p}\psi^{-0}_{n,p-1}& = - \psi^{-0}_{n-1,p}\varphi^{-+}_{n,p}\,.\end{aligned}
\frak M=\Bigl\{(\mu_0,\mu_1,\dots,\mu_{m-1}):\mu_i\in\Bbb Z,\ \mu_i\ge 0,\ \sum_{i=0}^{m-1}\mu_i=m,\ \sum_{i=0}^{m-1}i\mu_i\equiv 0\pmod m\Bigr\},
\beta_i\triangleq{m\choose i}\left[\frac{2}{n(n-1)}\right]^i\left[1-\frac{2}{n(n-1)}\right]^{m-i}\sum_{l=\lfloor i/2\rfloor+1}^i{i\choose l}[2p(1-p)]^l[(1-p)^2+p^2]^{i-l},
\Delta^\wedge = \partial^2_t + \frac 1{t+R}\partial_t + \frac 1{(t+R)^2}\partial_\theta^2 &= \partial^2_t + \frac 1R \sum_{m \ge 0}\left(\frac tR\right)^m \partial_t + \frac 1{R^2}\left(\sum_{m \ge 0}\left(\frac tR\right)^m\right)^2 \partial_\theta^2.
\delta \bar{\psi}_{\rho}= \bar{\epsilon}\left( \overleftarrow{\partial}_{\rho}- \frac{1}{2}\, \omega_{\rho m n}\, \sigma^{m n}+ \frac{1}{2}\, \alpha^{a}A_{\rho}^{a}\right) - \frac{1}{2 \sqrt{2}}\, e^{\phi}\, \bar{\epsilon}
\left\lbrace \begin{tabular}{l l}$u_t + u_{xxx}+ av_{xxx}=0$, in $(0,L)\times (0,T)$, $v_t +\frac{r}{c}v_x+\frac{ab}{c}u_{xxx}+ \frac{1}{c}v_{xxx}=0$, in $(0,L)\times (0,T)$, $u(0,t) =u(L,t) =u_{x}(L,t) = 0$, in $(0,T)$, $v(0,t) =v(L,t) =v_{x}(L,t) = 0$, in $(0,T)$, $u(x,0)=u^0(x), v(x,0) = v^0(x)$, in $(0,L)$.\end{tabular}\right.
-4\left\langle \sin^{2}t \cdot h , h \right\rangle_{\mathbb{S}^{1}}=\sum_{k=0}^{m}\sum_{s=0}^{1}\sum_{k_{1}=0}^{m}\sum_{s_{1}=0}^{1}c_{k,s}\overline{c_{k_{1},s_{1}}}\left\langle -4 \sin^{2}t\cdot Y_{2k,s},Y_{2k_{1},s_{1}}\right\rangle_{L^2(\mathbb{S}^{1})}.
M^{2}= - g^{4}C_{2}( R ) C ( R^{\prime}) \int_{q}\int_{k 5 5}{\frac{N ( k , k^{5},{\hat{k}}^{5}, q )}{( k^{2}) ( k^{2}- ( k^{5})^{2}) ( k^{2}- ({\hat{k}}^{5})^{2}) ( q^{2}- m_{1}^{2}) ( ( q - k )^{2}- m_{2}^{2})}}
{\rm lim}_{L\rightarrow +\infty}\frac{k_{\gamma}^{L}}{\sqrt{L}}=\frac{|\frac{d}{dt}(\omega(\dot{\gamma}(t)))|}{\left(\frac{\dot{\gamma}_1}{\gamma_1}\right)^2+\dot{\gamma}_3^2},~~if ~~\omega(\dot{\gamma}(t))= 0~~and~~\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0.
\sum_{k=0}^n{n\choose k}2^k F_{jk}(\pm \sqrt{5}F_j)^{n-k}B_{n-k}(x) = n F_j \Big ((\pm\sqrt{5}F_j x + L_{j})^{n-1}+ (\pm\sqrt{5}F_j (x-1) + L_{j})^{n-1}\Big ).
A^T \Sigma^{-1}+\Sigma^{-1}A+\sum_{k=1}^m N_k^T \Sigma^{-1}N_k + \sum_{i, j=1}^v H_i^T \Sigma^{-1}H_j k_{i j}\leq -\Sigma^{-1}BB^T \Sigma^{-1}, A^T \Sigma+\Sigma A+\sum_{k=1}^m N_k^T \Sigma N_k +\sum_{i, j=1}^v H_i^T \Sigma H_j k_{ij}\leq -C^T C,
\sum_{\rho \in \Delta_{+}}{g_{| \rho |}}| \rho \cdot \mu | = \sum_{\rho \in \Delta_{+}}{g_{| \rho |}}| g ( \rho ) \cdot \lambda | = ( \sum_{\rho \in \Delta_{+}}{g_{| \rho |}}\rho ) \cdot \lambda = 2 \varrho \cdot \lambda ,
Z =(\pi\times \pi)_*\left[(\widetilde{\varphi}\times \widetilde{\varphi})^{-1}(\Delta_{\mathbb{P}^2})-\Delta_{\widetilde{A}}-\sum_{(i,j)\in I^2}\mu_{i,j}\cdot E_i\times E_j \right] &=(\pi\times \pi)_*(\widetilde{\varphi}\times \widetilde{\varphi})^{*}(\Delta_{\mathbb{P}^2})-\Delta_{A}-\sum_{(i,j)\in I^2}\mu_{i,j}\cdot D_i\times D_j,
{\cal L}_{f}=i(\alpha' p^{+})\left\{\overline{\Theta}\Gamma_{+}\partial_{\tau}\Theta+\overline{\Theta}\Gamma_{11}\Gamma_{+}\partial_{\sigma}\Theta+(\alpha' p^{+})\overline{\Theta}\Gamma_{+}\tilde{\Omega}\Gamma_{-}\Gamma_{+}\Theta\right\}.
\langle \Psi_{1}( x_{1}) \cdots \Psi_{n}( x_{n}) \rangle = \displaystyle{\frac{F ( \hat{X}_{1 ( i )}{\cdot \hat{X}_{1 ( j )}})}{\displaystyle{\prod_{l < m}( x_{l m}^{2})^{- \frac{1}{( n - 1 ) ( n - 2 )}( \eta_{1}+ \cdots + \eta_{n}) + \frac{1}{n - 2}( \eta_{l}+ \eta_{m})}}}}\, ,
f =\sum_{|I|=p,|J|=q,k,\lambda}f_{IJk\lambda}dz_I\wedge d\overline{z}_J\otimes e_k\otimes b_\lambda\otimes \widehat{b}=\sum_k f_k \otimes e_k, f_k =\sum_{|I|=p,|J|=q,\lambda}f_{IJk\lambda}dz_I\wedge d\overline{z}_J\otimes b_\lambda\otimes \widehat{b}.
\frac{\partial g_2(\alpha)}{\partial \alpha}= n\frac{\Gamma(n+\sqrt{n}+2) \Gamma(x+\gamma\sqrt{n}+ 2)}{\Gamma(n+\sqrt{n}+3) \Gamma(x+\gamma\sqrt{n}+ 1)}\left(\psi(x+\gamma\sqrt{n}+2) - \psi(x+\gamma\sqrt{n}+1) \right).
F_{S}= \left( F_{S}^{( 0 )}, F_{S}^{( 1 )}\left( k_{1}\right) , F_{S}^{( 2 )}\left( k_{1}, k_{2}\right) , \ldots , F_{S}^{( n )}\left( k_{1}, k_{2}, \ldots , k_{n}\right) , \ldots \right) ,
R_1 = \frac{N\widetilde{w}(1) |\zeta(1+2it)|^2 \log N}{2\zeta(2)}\Big[ 1+ \frac{1}{i \eta \log N}+ O\Big(|\eta| \log N + \frac{(\log T)^{2/3+\varepsilon}}{\log N}+ |\eta| (\log T)^{2/3+\varepsilon}\Big) \Big].
\allowdisplaybreaks (\l,a)\, (\l',a') \,(\l,a)^{-1}=(\l\l', \l'a + \l a'+ aa') \, (\l^{-1},\overline{a}) =\big (\l',\l^{-1}\l'a + \l^{-1}\l a'+ \l^{-1}aa' + \l'a\overline{a}+ \l a'\overline{a}+ aa'\overline{a}+\l\l'\overline{a}\big ).
\mathbf{E}^G_x\left(\left(\eta_i-1/{\mu_x^G}\right)^2\right)=\frac{2\left(1-\frac{1}{\mu_y^GR_G(x,y)}\right)R_G(x,y)}{\mu_x^G}+\frac{1}{\mu_x^G\mu_y^G}-\frac{1}{(\mu_x^G)^2}\leq \frac{2R_G(x,y)}{\mu_x^G}.
\tau \circ \phi_{\tilde{\rho}}=d(\rho)^{-1}\tau \circ \tilde{\rho}\circ \phi_{\tilde{\rho}}=d(\rho )^{-1}\tau \circ \tilde{\rho}\circ \tilde{\rho}^{-1}\circ E_{\tilde{\rho}}= d(\rho )^{-1}\tau \circ E_{\tilde{\rho}}= d(\rho )^{-1}\tau .
d(x_n,x_{n+k}) \leq \sum_{i=n}^{n+k-1}d(x_i,x_{i+1}) \leq\sum_{i=n}^{n+k-1}d(G.x_i,G.x_{i+1}) + 2^{-i}\leq 2^{2-n}(1-2^{-k})\,,
\overline{q}(x,y)=(x_{\overline{q}},y_{\overline{q}})=(x-\frac{\psi_{\overline{q}-1}\psi_{\overline{q}+1}}{\psi_{\overline{q}}^{2}},\frac{\psi_{2\overline{q}}}{2\psi_{\overline{q}}^{4}})
d s^{2}= - e^{\frac{\Phi - \xi}{2}}d t^{2}+ e^{\frac{\Phi - \xi}{2}}d \vec{x}^{2}+ e^{\frac{\Phi + \xi}{2}- 5 \eta}d \rho^{2}+ e^{\frac{\Phi + \xi}{2}- \eta}d \Omega_{5}^{2}
\frac{R(m^2, b) R(q_1^{d_1}\cdots q_\kappa^{d_\kappa}, b)}{\varphi(b)}\ =\X \frac{R(m^2, b') R(m^2, q_1^{c_1}\cdots q_\kappa^{c_{\kappa}}) R(1, b')}{\varphi(b')},
[\begin{matrix}x\\y\end{matrix}]\rightarrow[\begin{matrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{matrix}][\begin{matrix}x\\y\end{matrix}]=[\begin{matrix}\frac{(x-y)}{\sqrt{2}}\\ \frac{(x+y)}{\sqrt{2}}\end{matrix}],
Z_{j,0}=\sum_{n\hbox{\small{odd}}}^\infty z_j^{(n)}, Z_{j,1}=\sum_{n\hbox{\small{even}}}^\infty z_j^{(n)}, & W_{j,0}=\sum_{n\hbox{\small{odd}}}^\infty w_j^{(n)}, W_{j,1}=\sum_{n\hbox{\small{even}}}^\infty w_j^{(n)},
J_{8\pi}^h(u) \ge \frac{1}{2}\int_{S^2}|\nabla u|^2 dv_{g_0}+ 2\int_{S^2}u \;dv_{g_0}-8\pi \log\left(\frac{1}{4\pi}\max_{S^2}h \int_{S^2}e^{u}dv_{g_0}\right)=
(R_q \widehat \chi)(p) = p^{-z^0}(R_q \widehat \chi)(I) \overset{\eqref{aux.MqtoRq}}{=\joinrel=}p^{-z^0}\frac{(Me^{-q})(z^0)}{\Gamma(z^0_1+\cdots+z^0_n)}= 0, p \in \mathbb R^n_+.
\int_{\Gamma^0}\frac{(N\cdot \nu^t_{\theta})}{\frac{\partial\sigma}{\partial s}}(x) f(x) d\mathcal{H}^{n-1}(x) =\lim_{\varepsilon \downarrow 0}\frac 1\varepsilon \int_{A^{\varepsilon}}f(x) d\mathcal{H}^n(x).
\frac{1}{t}\left[ \frac{1}{(a + t b)^{n+p}}- \frac{1}{a^{n+p}}\right] = -\frac{n+p}{\xi^{n+p+1}}b \leq -\frac{n+p}{a^{n+p+1}}b +\frac{C_{n,p,m,M}}{a^{n+p+1}}\cdot t
Y_iY_j = Y_j Y_i,T_\pi^n = Y_1\cdots Y_n\quad\hbox{and}Y_1Y_n^{-1}= T_0T_{n-1}\cdots T_1\cdots T_{n-1}. &\begin{array}{l}T_iY_i = Y_{i+1}T_i+(t^{\frac12}-t^{-\frac12})Y_i, T_i Y_{i+1}=Y_iT_i - (t^{\frac12}-t^{-\frac12})Y_i, \end{array}\qquad\hbox{and}T_i Y_j = Y_j T_i\quad\hbox{if $j\not\in\{i,i+1\}$.}
S_{BH}= \frac{2\pi A_9}{\kappa^2}= \frac{2\pi A_2}{\kappa^2_4}=\frac{32\pi^2m^2}{\kappa^2}l_1S_{23}S_{45}S_{67}\cosh\alpha \cosh\beta_1 \cosh \beta_2 \cosh \beta_3
U_1(x_1,x_2,t)= t^{p/2}\sum\limits_{n=1}^{N}t^{-n/2}\widetilde{S}_n(\eta_1,\eta_2) + V_1(\mu,\eta_1,\eta_2,t) + O(\sigma^{-\rho_4 N}), \quad\rho_4>0,
P_N^p\Big(\eta_t(x)=1,\eta_t(y)=0\Big|A_{x,y}(T)\Big) &=P_N^p\Big(\eta_t(x)=1\Big|A_{x,y}(T)\Big)P_N^p\Big(\eta_t(y)=0\Big|A_{x,y}(T)\Big) &\geq\frac{\Big[P_N^p\big(\eta_t(x)=1\big)-P_N^p\big(A^c_{x,y}(T)\big)\Big]\Big[P_N^p\big(\eta_t(y)=0\big)-P_N^p\big(A^c_{x,y}(T)\big)\Big]}{\Big[P^p_N\big(A_{x,y}(T)\big)\Big]^2}
|\mathcal{B}(s_1,s_2)| \le \sum_{g \mid h}\frac{1}{g^{\sigma_1+\sigma_2+1}}\prod_{(p,g)=1}\Big(1+O \Big( \frac{k \ell}{p^{\sigma_1+\sigma_2+2}}+ p^{-3+a_1}+ p^{-3+a_2}\Big) \Big) \prod_{p^{\alpha}\mid \mid g}\tau_{k}(p^j) \tau_{\ell}(p^j) \Big(1+O_{\ell}(p^{-1+\max(a_1,a_2)}) \Big).
[ (u^s + \tilde{u}_\epsilon), \partial^{\alpha}]\partial_x \tilde{w}_\epsilon = \sum\limits_{\beta \le \alpha,\, 1\le|\beta|}C^\beta_\alpha \,\,\partial^{\beta}( u^s + \tilde{u}_\epsilon) \partial^{\alpha - \beta}\partial_x \tilde{w}_\epsilon.
< \psi^{l}( x , y ) H ( x_{1}, y_{1}) \psi^{l}( x , y ) > ={\cal A}( l ) \sum_{n_{1}= 0}^{r}\sum_{n_{2}= n_{1}}^{r}\sum_{m_{1}= 0}^{s}\sum_{m_{2}= 0}^{s}\sum_{m_{3}= 0}^{s - m_{1}}c_{m_{1}, m_{2}, m_{3}}^{n_{1}n_{2}}a_{m_{1}, m_{2}, m_{3}}^{n_{1}n_{2}}( l ) .