harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta^{-\frac{n-4}{2}}\int_{{B}\backslash(\omega_+\cup \omega_-)}\varphi_0\Big(\frac{e_n-x}{\delta}\Big)U_{\delta,-e_n}^p\leq \left(\delta^{-\frac{n-4}{2}}\right)\left(C2^{n-3}\delta^{n-3}\right)\left(|{B}| 2^{n+2}\delta^{\frac{n+2}{2}}\alpha_n^p\right)=O(\delta^n),
\begin{array}{c}{{\{Q^{a \alpha}, \bar{Q}_{b \beta}\}= 2 \delta^{a}{}_{b}\gamma^{\mu}{}^{\alpha}{}_{\beta}P_{\mu}\, ,}}\\{{{}}}\\{{{}[ M_{\mu \nu}, Q^{a}] = i \textstyle{\frac{1}{2}}\gamma_{[ \mu}\gamma_{\nu ]}Q^{a}\, ,}}\\{{{}}}\\{{{}[{P}_{\mu}, Q^{a \alpha}] = 0 \, .}}\\ \end{array}
S = M^{6}\int d^{8}x \, R_{8}+ M^{6}\; L^{2}\int d^{8}x \, R_{8}^{2}+ \cdots + M^{6}r_{c}^{4}\int d^{4}x \, R_{4}+ \lambda_{2}\int d^{4}x \, R_{4}^{2}+ \cdots \, .
\mathcal{R}= -\int_B e^{-i\varphi}\big[i(A+\nabla\vartheta)\cdot(u_1\nabla u_2-u_2\nabla u_1) &+\Big((A+\nabla\vartheta)\cdot(A_1+A_2)-q+(A+\nabla\vartheta)\cdot\nabla\varphi\Big) u_1u_2\big] \, dx.
={\displaystyle\sum_{i,j}}\eta \left ({\phi}_{i},{\phi}_{j}, \xi_{1}, \ldots ,{\xi}_{q-2}\right )\left ({\phi}_{1}, \ldots ,\widehat{{\phi}_{i}}, \ldots ,\widehat{{\phi}_{j}}, \ldots ,{\phi}_{p+2}\right )=0
\frac{1}{\sqrt{2}}\begin{pmatrix}1 0 0 1 \end{pmatrix}, \frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{-1}0 0 -\sqrt{-1}\end{pmatrix}, \frac{1}{\sqrt{2}}\begin{pmatrix}0 1 -1 0 \end{pmatrix}, \frac{1}{\sqrt{2}}\begin{pmatrix}0 \sqrt{-1}\sqrt{-1}0 \end{pmatrix}
\ddot{x}=-\frac{ag}{1+a^2}\frac{\dot{x}}{\sqrt{\dot{x}^2+\dot{y}^2}},\qquad\ddot{y}=-\frac{ag}{1+a^2}\frac{\dot{y}}{\sqrt{\dot{x}^2+\dot{y}^2}},\ddot{z}=-\frac{a^2g}{1+a^2}.
\tau_l^d(\lambda,T,\alpha_0,\alpha_1,R_c,\sigma^2) =\frac{R_c^2}{R_c^d}\frac{V_2}{V_d}\tau^{2D}_l(\lambda^\prime,T,\alpha^\prime_0,\alpha^\prime_1,R_c,{\left(\sigma^\prime\right)}^{2}).
e_if_i^{(m)}= f_i^{(m)}e_i - f_i^{(m-1)}\frac{q^{-m+1}q^{2k_i}-q^{m-1}q^{2k_{i+1}}}{q^2-1}+ (1-q^{-2})\left(q^{m-1}f_{i}^{(m-1)}f_{\overline{i}}+ \frac{q^{2m-2}}{2}q^{k_i}F_{\overline{i}}f_i^{(m-2)}\right)e_{\overline{i}},
\sum\limits_{\ell=0}^{2}\frac{d}{dt}\langle\nabla^{\ell}u^{\epsilon},\epsilon\nabla^{\ell+1}\phi^{\epsilon}\rangle+\frac{3}{4}\|\nabla\phi^{\epsilon}\|_{H^2_{\frac{P'(\rho^{\epsilon})}{\rho^{\epsilon}}}}^2\lesssim D( u^{\epsilon},\eta^{\epsilon},\tau^{\epsilon})+\|\mathrm{div}u^{\epsilon}\|_{H^2}^2,
\mu_n C_{Sn}(i) = \alpha_n^{\rm R}(i) - \beta_n^{\rm R}(i)+\phi(i)-\lambda(i) \left( 1-\mu_m\right)C_{mD}(i) = \alpha_m^{\rm T}(i) - \beta_m^{\rm T}(i)+\phi(i)-\lambda(i) .
F_{\log,\Lambda}(x) =\lim_{s\to 0}\Gamma(\frac{s}{2})F_{s,\Lambda}(x)=2\left(\frac{d}{ds}F_{s,\Lambda}(x)\right)\bigg|_{s=0}= 2\zeta_\Lambda'(0;x) +\frac{2\pi^{d/2}}{d}|\Lambda |^{-1},
\int_{|y-x|=r}|y|^{-\kappa}dS(y) \leq \frac{\pi}{(|x|-r)^{\kappa-1}}\int_0^\pi \left(1 + \frac{4 r^2 \theta^2}{\pi^2 (|x|-r)^2}\right)^{-\kappa/2}\frac{2r}{\pi (|x|-r)}d\theta \leq \frac{\pi}{(|x|-r)^{\kappa-1}}\int_{0}^\infty (1+\tau^2)^{-\kappa/2}d \tau \leq \frac{C(\kappa)}{(|x|-r)^{\kappa -1}}.
g_{{\bf{k}},{\bf{q}}}({\bf{x}})= -e^{-i{\bf{q.x}}}(\frac{1}{2 \mbox{}N\mbox{}\epsilon_{{\bf{q}}}})\Lambda_{{\bf{k}}}(-{\bf{q}})\omega_{{\bf{k}}}({\bf{q}}) + i\mbox{}U_{{\bf{q}}}({\bf{x}})\Lambda_{{\bf{k}}}(-{\bf{q}})= -f^{*}_{{\bf{k}},{\bf{q}}}({\bf{x}})
\psi_{N, j}:= t_{C_{1,N}^{j}}t_{C_{2,N}^{j}}t_{C_{3,1}^{j}}t_{C_{3,2}^{j}}\cdots t_{C_{3,N-1}^{j}}t_{C_{4,1}^{j}}t_{C_{4,2}^{j}}\cdots t_{C_{4,N-1}^{j}}t_{C_{5,1}^{j}}t_{C_{5,2}^{j}}\cdots t_{C_{5,N-1}^{j}}.
T_{kk+1}f(T_{1k},T_{1k+1})=f(T_{1k},T_{1k+1})T_{kk+1}=T_{1k}^rT_{1k+1}^sT_{kk+1}= T_{kk+1}T_{1k}^rT_{1k+1}^rT_{1k+1}^s=T_{kk+1}f(T_{1k}T_{1k+1},T_{1k+1}).
\int_{\Omega^*}f^*(y) \dy = \int_{\Omega}f^*(Y(\cdot,v,Dv))\det DY(\cdot,v,Dv) > \int_{\Omega}tf(\cdot) + (1-t)f^*(Y(\cdot,u_0,Du_0))\det DY(\cdot,u_0,Du_0) = \int_{\Omega^*}f^*.
\int_{\partial \mathcal{O}_{(x,0), \rho}\cap [-1, 1]^2}|x_2|^\gamma d\sigma \le \Delta u(\overline{\mathcal{O}_{(x,0), \rho}}\cap [-1, 1]^2) = \int_{V}\nabla u \cdot \vec{\eta}d\sigma \le \int_{V}C |x_2|^\gamma d\sigma.
\begin{aligned}D_{k_{ii}}= 0 D_{k_{ij}}= D_{k_{ji}}\textstyle\sum_i D_{k_{ij}}= 0 D_{c_{ii}}= 0 \textstyle\sum_i D_{c_{ij}}= 0 D_{e_{ii}}= 0 D_{e_{ij}}= D_{e_{ji}}\end{aligned}
\bar \partial (H_{\infty}^{-1}\partial H_\infty)= - \begin{pmatrix}\partial_{\bar z}(r^{-1/2}\partial_z r^{1/2}) 0 0 \partial_{\bar z}(r^{1/2}\partial_z r^{-1/2}) \end{pmatrix}d z \wedge d \bar z = 0.
M_{11}=\left(\begin{array}{ccc}1 * 0 1 0 0 * * 0\end{array}\right),\,\,M_{12}=\left(\begin{array}{ccc}0 * 1 0 1 * * * 0\end{array}\right), M_{21}=\left(\begin{array}{ccc}0 0 0 0 0 * * * 0\end{array}\right),\,\,M_{22}=\left(\begin{array}{ccc}0 0 0 0 0 * * * 1\end{array}\right).
\displaystyle\sum_{l=1}^{p-q}z_{il}\overline{a_{ku}^{il}\left(Z\right)}+\displaystyle\sum_{l=1}^{p-q}\overline{z}_{jl}a_{uk}^{jl}\left(Z\right)=0, \mbox{for all $i,j,k,u=1,\dots,q$.}
{\cal Z}_{T}= \int \! \left[ d^{N}\vec{x}( t ) \sqrt{\det g ( \vec{x}( t ) )}\right] \exp \left[ -{\frac{N r}{2}}\int_{- T / 2}^{T / 2}\! d t \Bigl ( g_{i j}{\frac{d x^{i}}{d t}}{\frac{d x^{j}}{d t}}+{\vec{x}}^{2}\Bigr ) \right] ,
\int_{1}^\infty s^{1-\mu}|df(s)| =\sum_{k=0}^\infty \int_{2^k}^{2^{k+1}}s^{1-\mu}|df(s)|\lesssim \sum_{k=0}^\infty (2^k)^{1-\mu}\frac{1}{2^k}\int_{2^k/\lambda}^{\lambda 2^k}|f(t)|\, dt &\asymp \sum_{k=0}^\infty \int_{2^k/\lambda}^{\lambda 2^k}t^{-\mu}|f(t)|\, dt \lesssim \int_{1/\lambda}^\infty t^{-\mu}|f(t)|\, dt,
\begin{aligned}\delta^{f}_r(c_i,c_j)= F^f_r((c_{i-1}, c_i]\times [c_j, c_{j+1}))= F^f_r (c_{i-1}, c_{j+1}) + F^f_r (c_{i}, c_{j})- F^f_r (c_{i-1}, c_{j})- F^f_r (c_{i}, c_{j+1}).\end{aligned}
\Bigg\{\begin{bmatrix}1 0 0 1 \end{bmatrix}, \begin{bmatrix}0 1 -1 1 \end{bmatrix}, \begin{bmatrix}-1 1 -1 0 \end{bmatrix}, \begin{bmatrix}0 \sqrt{-1}\sqrt{-1}0 \end{bmatrix}, \begin{bmatrix}\sqrt{-1}0 \sqrt{-1}-\sqrt{-1}\end{bmatrix}, \begin{bmatrix}\sqrt{-1}-\sqrt{-1}0 -\sqrt{-1}\end{bmatrix}\Bigg\}.
\begin{array}[c]{rl}i) A:H\rightarrow2^{H}\ \alpha\text{-maximal monotoneoperator,}\medskip ii) f\in L^{1}\left( 0,T;H\right) ,\medskip iii) M\in C\left( \left[ 0,T\right] ;X\right) ,\;\;M\left(0\right) =0,\medskip iv) u_{0}\in H\end{array}
I^1_5=(\frac{\partial^{\alpha_1}\widetilde{E}\cdot\partial^{\alpha-\alpha_{1}}\nabla_{v}F}{\sqrt{\mu}},\frac{\partial^{\alpha}F}{\sqrt{\mu}}),I^2_5=(\frac{\partial^{\alpha_1}\bar{E}\cdot\partial^{\alpha-\alpha_{1}}\nabla_{v}F}{\sqrt{\mu}},\frac{\partial^{\alpha}F}{\sqrt{\mu}}).
\|u(t)-e^{-it{\mathcal K}_\lambda}\phi\|_{H^1_\lambda}=\Big\|\int_t^{\infty}e^{i(t-s)\mathcal{K}_{\lambda}}\mathcal{N}[u] ds\Big\|_{H_{\lambda}^1}&\lesssim\|\mathcal{N}[u]\|_{\mathcal{W}_{\lambda}'([t,\infty])}&\lesssim \|u\|^{2p-1}_{\mathcal{W}_{\lambda}([t,\infty])}\rightarrow 0
\left\Vert \omega_{s1}^{\nu}\right\Vert_{L^{2}}\left( t\right) +\left\Vert\omega_{n1}^{\nu}\right\Vert_{L^{2}}=e\left( t\right) & \leq e^{-\frac{1}{2}\nu t}e\left( 0\right) +Cd\nu\int_{0}^{t}e^{-\frac{1}{2}\nu\left( t-s\right)}\left\Vert \omega_{n2}^{\nu}\right\Vert_{L^{2}}\left( s\right) ds\leq Cd\nu.
e^{-\int_{\tau_j}^{t \wedge{\bar \rho}_j}G_j(v) \: dv}X^{[\tau_j, \xi_j]}_j(t \wedge{\bar \rho}_j) = \int_{\tau_j}^{t \wedge \xi_j \wedge{\bar \rho}_j}\mu X_{j-1}(u) e^{-\int_{\tau_j}^u G_j(v) \: dv}\: du + Z_j^{[\tau_j, \xi_j]}(t \wedge{\bar \rho}_j).
a^{(1)}\otimes a^{(2)}= \phi(\delta_{g}) =\sum_{hk = g}\delta_{h}\otimes \delta_{k}= \!\!\!\!\sum_{(h, k) \in \mathcal{M}_{g}}\delta_{h}\otimes \delta_{k}&= \!\!\sum_{\mathcal{O}: \mathrm{Inv}(\mathcal{O}) = g}\sum_{(h, k) \in \mathcal{O}}\delta_{h}\otimes \delta_{k},
S ={\sqrt{\Delta({\mathbb K})}}\sum_{d\vert\frac{m}{p}}\left(\mu(d) \lambda^{(n/d)^s}+\mu(pd) \lambda^{n/(dp)}\right) & ={\sqrt{\Delta({\mathbb K})}}\sum_{d\vert\frac{m}{p}}\pm \lambda^{(n/(pd))^s}\left(\lambda^{(n/pd)^s(p^s-1)}-1\right).
\bar q^t - q^{2 \tilde n}- \rho \bar q^t (1 - q^{2 \tilde n})= (1-\rho) \bar q^t - q^{2 \tilde n}+ \rho \bar q^t q^{2 \tilde n}\ge (1-\rho) \bar q^t - q^{2 \tilde n}\ge 0.
I_{2}= - n^{3}\int_{\sin^{- 1}( 1 / n )}^{\pi / 2}{\frac{\sin^{3}\theta}{\sqrt{n^{2}\sin^{2}\theta - 1}}}| T_{r g}|^{2}e^{- 2 \omega x \sqrt{n^{2}\sin^{2}\theta - 1}}d \theta ,
\delta \chi_a\;=\;-i\,\sqrt{\,\frac{1}{2}}\,\gamma^{\mu}\,(\partial_{\mu}\, \Phi)\,\varepsilon_a\;+\;A(\Gamma_{45})_{ab}\, \varepsilon^b\;+\;\frac{1}{2}\,\sqrt{\,\frac{1}{6}}\,\gamma^{\, \mu\nu}\,\left(\,H_{\,\mu \nu a b}\;-\;\sqrt{\,2}\,h_{\,\mu \nu a b}\,\right)\,\varepsilon^b
|\Psi\rangle=\sum_{i_{1},i_{2},\alpha_{1},\alpha_{2}}\Gamma_{\alpha_{1}}^{[1]i_{1}}\lambda_{\alpha_{1}}^{[1]}\Gamma_{\alpha_{1}\alpha_{2}}^{[2]i_{2}}\lambda_{\alpha_{2}}^{[2]}|i_{1}i_{2}\rangle|\Phi_{\alpha_{2}}^{[3..N]}\rangle
J_{0}(\alpha,\beta,Z):=-c_{1}(\xi|_{Z},\tau)+Q_{\tau}(Z)+\sum_{i}\sum_{k=1}^{m_{i}-1}\mu_{\tau}(\alpha_{i}^{k})-\sum_{j}\sum_{k=1}^{n_{j}-1}\mu_{\tau}(\beta_{j}^{k}).
C P T : \Psi ( \vec{x}, t ) \rightarrow - i L^{0}e^{i ( \alpha_{P}+ \alpha_{C}+ \alpha_{T})}\gamma^{0}\gamma_{5}\bar{\Psi}^{T}( - \vec{x}, - t ) ~ , L^{0}= \left( \begin{array}{c c}{{0}}&{{e^{i \eta}I_{4}}}\\{{e^{- i \eta}I_{4}}}&{{0}}\\ \end{array}\right)
0 = \frac{1}{m}\frac{\delta S_{\mathrm{e f f}}}{\delta z_{\rho}}= - \left[ \frac{d^{2}z^{\rho}}{d \tau^{2}}+ \Gamma_{\mu \sigma}^{\rho}\frac{d z^{\mu}}{d \tau}\frac{d z^{\sigma}}{d \tau}\right] + \frac{1}{m}\frac{\delta \Delta S_{m}}{\delta z_{\rho}},
O(2^{\omega(D)}xL_y^2 e^{-C{\sqrt{\log x}}}) =O( 2^{2.5 \log \log y}x(\log x)^4 e^{-C{\sqrt{\log x}}}) &= O\big( (\log x)^4(\log y)^2 x e^{-C{\sqrt{\log x}}}\big) &=o\left(\frac{\pi(x)}{D(\log y)^3}\right)=o\left(\frac{\pi_D(x)}{\log y}\right).
\sum_{i=1}^{s}\frac{1}{12}{(\frac{11}{12})}^{i-1}=\frac{1}{12}[{(\frac{11}{12})}^{0}+{(\frac{11}{12})}^{1}+...]=\frac{1}{12}[{(1-\frac{11}{12})}^{-1}]=1
\frac{\partial \widehat{H}(t,x)}{\partial y}=\frac{\partial H}{\partial y}(t,x,\hat{Y}(t,x,z),\hat{Y}(t,.,z)(x),\hat{u}(t,z),\hat{p}(t,x,z),\hat{q}(t,x,z),\hat{r}(t,x,z,.))
S_1 = \left(\begin{array}{ccccc}0 0 0 0 0 0 0 -\frac{1}{2}0 0 0 -\frac{1}{2}1 0 0 0 0 0 1 0 0 0 0 0 0 \end{array}\right) \;\;\;\; B = \left(\begin{array}{cccc}0 0 1 \end{array}\right).
+16.7\,a_1 \,a_2 \,a_3\,a_4 + 121\,a_3^2\,a_4^2 -12.0\,a_1 \,a_2^2\,a_3+15.0\,a_3 \,a_4^2\,a_1 -98.5\,a_3^2\,a_4 \,a_2).
E[g(W,\theta_{0},\tilde{\gamma})] =E[c^{\prime}p(Z)\{\bar{\gamma}(Z)-\gamma_{0}(Z)\}]=-E[E[\alpha_{0}(X)|Z]\{\bar{\gamma}(Z)-\gamma_{0}(Z)\}] & =E[\alpha_{0}(X)\{Y-\bar{\gamma}(Z)\}]=-c^{\prime}\Pi^{-1}E[a(X)\{Y-\bar{\gamma}(Z)\}]=0.Q.E.D.
|a(x_{h_k}+h_k^{\frac{1}{2}}x, \xi_{h_k}+h_k^{\frac{1}{2}}\xi;h_k)-a(x_{h_k}, \xi_{h_k};h_k)| &=|h_k^{\frac{1}{2}}\int_0^1 ((x\cdot \partial_x+\xi \cdot \partial_{\xi})a)(x_{h_k}+th_k^{\frac{1}{2}}x, \xi_{h_k}+th_k^{\frac{1}{2}}\xi;h_k)dt| &\lesssim h_k^{\frac{1}{2}}h_k^{\alpha_j -\varepsilon -\delta}
\int_{M}(A-b)u^{2}dV_{g}\leq\int_{M}Au^{2}dV_{g}-\mu\int_{M}\left[a|\nabla_{g}u|_{g}^{2}+|u|^{2}\right] dV_{g}+\int_{M}a\left\vert \nabla_{g}u\right\vert_{g}^{2}dV_{g}& \leq\frac{A-\mu}{A}\int_{M}\left[ a|\nabla_{g}u|_{g}^{2}+A|u|^{2}\right]dV_{g}=\frac{A-\mu}{A}\left\Vert u\right\Vert_{g,a,A}^{2}.
\Phi_{0 , 2}^{( 0 , 3 )}= a_{1 2}^{*}V^{( 0 , 3 )}\Phi_{1 , 2}^{( 0 , 3 )}= a_{2 2}^{*}a_{1 2}^{*^{2}}V^{( 0 , 3 )}
\int_0^1 e^{i\xi_n(1-2t)}p_{N+1}(t)\,dt = \frac{1}{2i\xi_n}p_{N+1}(0) 2i \sin\xi_n + \frac{1}{2i\xi_n}\int_0^1 e^{i\xi_n(1-2t)}p_{N+1}'(t)dt \, .
A_{i}\rightarrow{\frac{\exp ( - i \arg ( i c_{t}t_{c}+ d_{t}) )}{( i c_{t}t + d_{t})}}A_{i}, A_{i}\rightarrow{\frac{e^{- i \arg ( i c_{t}t_{c}+ d_{t})}}{( i c_{t}t + d_{t})}}A_{i},
D^2\Pi(p)(X_1^\top,X_2^\top) = D_{X_1^\top}(X_2^\top) - (D_{X_1^\top}X_2^\top)^\top = (D_{X_1^\top}X_2^\top)^\bot = A_p(X_1^\top,X_2^\top), D^2\Pi(p)(X_1^\top,X_2^\bot) = - (D_{X_1^\top}X_2^\bot)^\top.
\sum_{j=j_0-N}^{j_0+N}\sum_{k\in{E}}P_{R_{j,k}}=P_{I_{j_0+N}^1}\otimes P_{I_{j_0+N}^2}\otimes \cdots\otimes P_{I_{j_0+N}^d}&-P_{I_{j_0-N}^1}\otimes P_{I_{j_0-N}^2}\otimes\cdots\otimes P_{I_{j_0-N}^d}.
\hat{\mu}^\psi_{\Lambda,\epsilon}(d\phi, dt) := \frac{1}{Z^\psi_{\Lambda,\epsilon}(\alpha)}e^{-A^\psi(\phi, t)}\prod_{jk \in E(\Lambda)}\Big(f_\alpha(e^{t_{jk}})e^{t_{jk}}dt_{jk}\Big)\prod_{j \in \Lambda}d\phi_j,
\begin{array}{c}g_{+}(-\infty )=g_{0}(+\infty )=2\chi_{\infty}(S-2S^{+})+2\pi k_{W}^{+}g_{-}(+\infty )=g_{0}(-\infty )=-2\chi_{\infty}(S-2S^{-})+2\pi k_{W}^{-}\end{array}
\lefteqn{\psi_U \Big( \sum \limits_{{i},{j}}\,{f}_{{i}}{g}_{{j}}\otimes f'_{{i}}g'_{{j}}\Big) \: = \: \sum \limits_{{i},{j}}\,{f}_{{i}}{g}_{{j}}\otimes f'_{{i}}g'_{{j}}\otimes 1}.
\begin{aligned}\mathbf{E}_x(T_l; T_l>T_r) =\mathbf{E}_x(T_r+T_l\circ \theta_{T_r}; T_l>T_r) &=\mathbf{E}_r(T_r; T_l>T_r)+\mathbf{E}_rT_l \cdot \frac{x-l}{r-l}. \end{aligned}
\big|\partial_{\dot{\lambda}}\Psi[\lambda,\xi,\dot{\lambda},\dot{\xi},\phi][\dot{\bar{\lambda}}](x, t)\big|\lesssim \frac{t_0^{-\varepsilon}}{R^{a-2s}}\|\dot{\bar{\lambda}}(t)\|_{n-4s+1+\sigma}\left(\sum_{j=1}^k\frac{\mu_0^{-\frac{n-6s}{2}-1+\sigma}(t)}{1+|y_j|^{a-2s}}\right),
\theta^{2}\Delta E_{n}^{(1)}(even)=\frac{\theta^{2}V_{0}}{6}[1-4ma^{2}V_{0}(V_{0}-n\pi-\frac{\pi}{2})(\frac{1}{3}-\frac{1}{2(n+\frac{1}{2})^{2}\pi^{2}})] .
g_q({\boldsymbol\lambda},K)=\frac{1}{\sqrt{K\pi}}\int_{[0,K\pi]_{-\alpha}}\sum^{\left\lfloor q/2\right\rfloor}_{\ell=0}b_{q-2\ell}a_{2\ell}(h^{\otimes q-2\ell}_K(s,\cdot)\otimes h^{\prime\,\otimes 2\ell}(s,\cdot))({\boldsymbol\lambda})v_K(s)ds,
\frac{K_i-K_i^{-1}}{q_i-q_i^{-1}}\otimes K_i + K_i^{-1}\otimes \frac{K_i-K_i^{-1}}{q_i-q_i^{-1}}= \frac{K_i\otimes K_i - K_i^{-1}\otimes K_i^{-1}}{q_i-q_i^{-1}}.
U_{0}\left(I_{E_{0}}- LBTA\right) = I_{E_{0}}- LBTA + LBTU\left(I_{E}- ALBT\right)A &=I_{E_{0}}- LBTA + LBT\left(I_{E}- S\right)A &=I_{E_{0}}- LBTSA = I_{E}- S_{0}.
\partial_\beta \| \phi_{1, b, \beta}\|^{2}_{L^2_{\rho_\beta}}=\left(2\frac{1}{\beta}+\frac{d+2}{2\beta}- \left( \frac{2}{\beta}\left(\frac{d}{2}-\gamma+2 \right) - \frac{1}{\beta}\left(\frac{d}{2}-\gamma+1 \right) \right) \right) \|\phi_{1, b,\beta}\|_{\rho_\beta}^2+O(b^{1-\frac{\epsilon}{2}})
\left|K_\gamma^\perp\right|= \prod_{i=0}^N\left(\prod_{e\in N^{p_i}_0}\frac{\langle\eta,\alpha(e)\rangle}{\langle\eta,\alpha(\theta_{p_ip_{i+1}}(e))\rangle}\right)=\frac{\prod_{i=0}^N\prod_{e\in N^{p_i}_0}\langle\eta,\alpha(e)\rangle}{\prod_{i=0}^N\prod_{e\in N^{p_{i+1}}_0}\langle\eta,\alpha(\theta_{p_ip_{i+1}}(e))\rangle}.
\partial_{j}\partial_{k}=e^{-2t}(\omega_{j}\omega_{k}\partial_{t}^{2}+\omega_{j}\Omega_{k}\partial_{t}+\omega_{k}\Omega_{j}\partial_{t}+(\delta_{jk}-2\omega_{j}\omega_{k})\partial_{t}+\Omega_{j}\Omega_{k}-\omega_{j}\Omega_{k}).
B = - \int \partial_xu(\partial_x^3u)^2 \chi(x+\nu t) \; dx + 3 \int u(\partial_x^3u)^2 \chi'(x+\nu t) \; dx &- \int \partial_xu(\partial_x^2u)^2\chi''(x+\nu t) \; dx- \int u(\partial_x^2u)^2\chi'''(x+\nu t) \; dx.
\frac{S^\chi_{(\lambda, \lambda'),(\mu, \mu')}}{S^\chi_{(0, 0),(\mu, \mu')}}= \frac{e^{2\pi i \left((\lambda', \mu+\rho)+(\lambda, \mu'+\rho)\right)}\chi_\mu(\lambda)\chi'_{\mu'}(\lambda')}{\chi_\mu(0)\chi'_{\mu'}(0)}.
\begin{aligned}{d\xi_k \over dt}= 2\mu_k ,{d\delta_k \over dt}= 2\mu_k^2 +2\nu_k^2 ,{d\nu_k \over dt}= 8\nu_0^3 \sum_{n}^{}e^{-\Delta_{kn}}i(R_{kn}-R_{kn}^*),{d\mu_k \over dt}= -8\nu_0^3 \sum_{n}^{}e^{-\Delta_{kn}}(R_{kn}+R_{kn}^*).\end{aligned}
w\,\partial_w \big|_{w = z_l}\, f(w) = f(z_l) \sum_{j(\neq l)}^N \frac{z_l}{z_{lj}}+ \sum_{i (\neq l)}^N \frac{z_l}{z_{il}}\, \Biggl( \, \prod_{k(\neq i,l)}^N \! \frac{z_{lk}}{z_{ik}}\Biggr) \, f(z_i) \, .
\int_1^\infty (x-1)^{\mu-1}(x^2+\beta)^\nu dx = \frac{\Gamma(\mu)\Gamma(-\mu-2\nu)}{\Gamma(-2\nu)}{_2F_1}\left(-\frac{\mu}{2}-\nu,\frac{1-\mu}{2}-\nu;\frac{1}{2}-\nu;-\beta\right).
u \ast' v \overset{\hphantom{(0.00)}}{=}\mu(T(u)) v, u \diamond' v \overset{\hphantom{(0.00)}}{=}\rho_\mu(T(u))v = \mu(\partial_1(T(u))) \alpha_2(v) - \mu(\partial_2(T(u))) \alpha_1(v) \overset{\eqref{eq:oop2}}{=}\mu(T(\alpha_1(u))) \alpha_2(v) - \mu(T(\alpha_2(u))) \alpha_1(v).
\int^{2\pi}_{0}d\sigma X^{i}(\sigma)\hat{P}_{i}(\sigma) =x^{i}_{0}\hat{p}^{i}_{0}+\frac{g_{ij}}{2\sqrt{\alpha'}}\sum_{n=1}^{\infty}\left\{\left( \chi_{n}^{i}\alpha^{j}_{-n}+\bar{\chi}^{i}_{n}\tilde{\alpha}^{j}_{-n}\right) +\left(\bar{\chi}_{n}^{i}\alpha^{j}_{n}+\chi_{n}^{i}\tilde{\alpha}^{j}_{n}\right) \right\}~.
\{\psi^{1}(x),\psi^{2}(x),\psi^{3}(x)\}=(q^{1},q^{2},q^{3})\equiv(r,\theta,z)=\{\sqrt{x_{1}^{2}+x_{2}^{2}},tan^{-1}(x_{2}/x_{1}),x_{3}\}
\tau_{\alpha_{11}}\circ \tau_{\beta_{21}}=\tau_{\beta_{11}}\circ \tau_{\alpha_{21}}, \tau_{\alpha_{21}}\circ \tau_{\beta_{12}}=\tau_{\beta_{21}}\circ \tau_{\alpha_{12}},\tau_{\alpha_{12}}\circ \tau_{\beta_{21}}=\tau_{\beta_{12}}\circ \tau_{\alpha_{21}}.
\begin{array}{r c l}{\delta_{\omega}V^{a}}&{=}&{\omega^{b}\partial_{b}V^{a}- V^{b}\partial_{b}\omega^{a}+{\textstyle{\frac{1}{2}}}\partial_{b}\omega^{b}V^{a},}\\{\delta_{\omega}X}&{=}&{\omega^{a}\partial_{a}X .}\\ \end{array}
(r_1^{x_1},r_2^{x_2},\ldots,r_d^{x_d}):=(\underbrace{r_1,r_1,\ldots,r_1}_{x_1},\underbrace{r_2,r_2,\ldots,r_2}_{x_2},\ldots,\underbrace{r_d,r_d,\ldots,r_d}_{x_d}) \, .
lim_{t\rightarrow-3}\frac{t^{2}-9}{2t^{2}+7t+3}=lim_{t\rightarrow-3}\frac{(t-3)(t+3)}{(2t+1)(t+3)}=lim_{t\rightarrow-3}\frac{t-3}{2t+1}=\frac{-6}{-5}=\frac{6}{5}
u_{\mathbf{b}}^{\ast}=b_{k}^{\ast}b_{k-1}^{\ast}\dots b_{1}^{\ast}=u_{\mathbf{b}^{\ast}}\mathbf{b}^{\ast}:=\left( b_{k}^{\ast},b_{k-1}^{\ast},\dots,b_{1}^{\ast}\right) .
-\frac{\partial^2}{\partial t^2}\left[\log(\cosh(t)-1) \right] = - \frac{\partial}{\partial t}\frac{\sinh(t)}{\cosh(t)-1}& = - \frac{\cosh(t)\,(\cosh(t)-1) - \sinh(t)\sinh(t)}{(\cosh(t)-1)^2}& = \frac{\cosh(t)}{(\cosh(t)+1)^2}> 0.
\left.\begin{array}{ll}Q(u,v)= \displaystyle\dfrac{a_2}{m_1}\int_{\mathbb{R}^N}|\nabla u|^2dx+\dfrac{a_1}{2m_2}\int_{\mathbb{R}^N}|\nabla v|^2dx+\dfrac{N}{2}a_1a_2\int_{\mathbb{R}^N}v u^2dx,\end{array}\right.
\overline{f}\frac{df}{dx}w_{\alpha-1,\beta}+ A P_n^{(\alpha,\beta)}(x) \frac{\overline{f}}\mu w_{\alpha-1,\beta}+ P_{n-1}^{(\alpha,\beta)}(x)\overline{f}w_{\alpha-1,\beta}= \frac 1\mu |f|^2 w_{\alpha-1,\beta},
Z_{x|n}(r) =\min\left\{\frac{r^k}{\phi^k(T_{x|n})}:\; k=0,1,\ldots, d\right\}\mbox{and}Z_{x|n}(r) \leq \frac{r^t}{\phi^t(T_{x|n})}\mbox{for all}t\in [0,d].
\begin{aligned}\ \dS^1_i(t)= \mu_i^1(t)S_i^1(t)dt+\sum_{j=1}^{n_1}\sigma_{ij}^1(t)S_i^1(t)dW_j^1(t)\quad(i=1,\ldots,n_1), \ \dS^2_i(t)= \mu_i^2(t)S_i^2(t)dt+\sum_{j=1}^{n_2}\sigma_{ij}^2(t)S_i^2(t)dW_j^2(t)\quad(i=1,\ldots,n_2),\end{aligned}
( G_{O M}^{- 1})_{a b}={\frac{\left( ( 2 K^{2}- 1 ) - 2 K^{2}\sqrt{1 - K^{- 2}}\right)^{\frac{1}{6}}}{K}}\, \left[ \hat{\eta}_{\hat{a}\hat{b}}+ \,{\frac{1}{4}}( \hat{{\cal H}}^{2})_{\hat{a}\hat{b}}\right]
\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}- \nabla \cdot ( a \nabla u) = g(x,t),~ \mbox{in}~~ \Omega_T, u(x,0) = f_0(x), ~~~u_t(x,0) = f_1(x)~ \mbox{in}~~ \Omega, \partial_{n}u =-\partial_{t}u ~\mbox{on}~ \partial \Omega_T,
\frac{j_{0,k}}{j_{0,N+1}}= \frac{k-\frac{1}{4}}{N+\frac{3}{4}}+ e_{k,N}, |e_{k,N}|\leq \frac{1}{8(N+\frac{3}{4})(k-\frac{1}{4})\pi^2}, 1\leq k\leq N.
F_0 =-x_2^3x_3+x_1^4+bx_1^2x_3^2+c x_1x_3^3+d x_3^4,\; \textrm{when}\textrm{Char}(K)\ne 2, F_0 =-x_2^3x_3+x_1^4+ax_1^3 x_3+bx_1^2x_3^2+cx_1 x_3^3+d x_3^4,\; \textrm{when}\textrm{Char}(K)=2,
\int_{\Omega}|\nabla \varphi_{0\varepsilon}|^2 + \varepsilon \int_{\Omega}|\Delta \varphi_{0\varepsilon}|^2 = \int_{\Omega}\nabla \varphi_0 \cdot \nabla \varphi_{0\varepsilon}\leq \frac{1}{2}\int_{\Omega}| \nabla \varphi_0|^2 + \frac{1}{2}\int_{\Omega}| \nabla \varphi_{0\varepsilon}|^2.
C_n(0) =\sum_{y_n\in\{0,1\}}\log\Big(\frac{q_n(y_n|0,0)}{\nu^{\pi^*}_n(y_n|0)}\Big)q_n(y_n|0,0)=\log\Big(\frac{q_n(0|0,0)}{\nu^{\pi^*}_n(0|0)}\Big)q_n(0|0,0)+\log\Big(\frac{q_n(1|0,0)}{\nu^{\pi^*}_n(1|0)}\Big)q_n(1|0,0) &=\alpha_n\log\big(\frac{1-c_0(n)}{c_0(n)}\big)+\log\big(\frac{1}{1-c_0(n)}\big)-H(\alpha_n).
\psi_{( 2 ) i}^{\gamma}({\bf{r}}) = - f^{\alpha \beta b}\, f^{b \delta \gamma}\,{\cal{X}}^{\alpha}({\bf{r}}) \,{\cal{Q}}_{( 2 ) i}^{\beta}({\bf{r}}) \,{\cal{X}}^{\delta}({\bf{r}}) = - f^{\alpha \beta b}\, f^{b \delta \gamma}\,{\cal{X}}^{\alpha}({\bf{r}}) \, [ a_{i}^{\beta}({\bf{r}}) +{\textstyle \frac{2}{3}}x_{i}^{\beta}({\bf{r}}) ] \,{\cal{X}}^{\delta}({\bf{r}}) \; ,
{\tilde Q}_{(em)}=\int d^3x [i({\vec{\tilde W}}^{'}_{+}\cdot{\vec{\tilde \pi}}^{({\tilde W}_{+}){'}}-{\vec{\tilde W}}^{'}_{-}\cdot{\vec{\tilde \pi}}^{({\tilde W}_{-}){'}})+{\tilde j}^o_{(em)}](\vec x,x^o),
w^*_i(n,k) =\frac 12 \cdot (w^*_{i-1,n-1}+ w^*_{i,n-1}) |\widehat{W}_{i,n}| \leq\left(1+\frac{1}{\sqrt{n-1}}\right)\cdot \frac 12\cdot \left(|\widehat{W}_{i-1,n-1}| +|\widehat{W}_{i,n-1}|\right)\:.
\rho(C_0\times \dots \times C_{T-1}\times B) =\int_{B}\rho_1(C_0\times \dots \times C_{T-1}| m)\rho_2(dm) & =\int_{B}\int_{\mathcal{Z}}\otimes_{t=0}^{T-1}\delta_{\alpha_t(z,m^{(t+1)})}(C_t)\nu(dz)\rho_2(dm),
G_{m,k,r}^{(n)}({\boldsymbol{h}},\{p_{ij}\})=\sum_{I \subset [1,n-1] \atop |I|=r}{1 \over \prod\limits_{i \in I}p_{in}}\sum_{J \subset [1,n] \atop |I|+m=|J|+k}{n- | I \cup J | \choose n-m -|I|}{\tilde h}_{J},
\Phi^{( I )}= -{\frac{e^{2}}{8 \pi^{2}m}}\arctan \biggl \{\tan \biggl [{\frac{\Theta}{2}}+{\frac{\pi}{4}}\bigl ( 1 - \mathrm{s g n}( m ) \bigr ) \biggr ] \biggr \}\qquad \left( F = \frac{1}{2}\right) .
1- \overline{\Psi(z_l,x_{1i},x_{2i},x_{3i})}\Psi(z_k,x_{1j},x_{2j},x_{3j}) &=(1-\overline{z_l}z_k)\overline{\gamma(\lambda_i,z_l)}\gamma(\lambda_j,z_k)+(1-\overline{\lambda_i}\lambda_j)\langle U(\lambda_j)\eta(z_k,\lambda_j),U(\lambda_i)\eta(z_l,\lambda_i)\rangle_{\cal{H}}
e(G) \le e(H)+e(G[X\cup U])-\max\{0, |U|-1\}\le \begin{cases}3k-6 +e(G[X\cup U]) \, |U|=0 2k-3+e(G[X\cup U]) \, |U|=1 k+e(G[X\cup U]) \,|U|=2. \end{cases}\medskip
\sum_{b\in B}x_b & x_b \leq \sum_{i\in H}w_{i,b}b\in B x_b \leq \sum_{j\in C}z_{j,b}b\in B \sum_{b\in B}w_{i,b}= 1 i\in H & \sum_{b\in B}z_{j,b}= 1 j\in C & \sum_{i\in H}w_{i,b}\cdot h_i = \sum_{j\in C}z_{j,b}\cdot c_j b\in B & x_b, w_{i,b}, z_{j,b}\in\{0,1\}b\in B, i\in H, j\in C
2x^{\tilde \varepsilon x/ 2}h_s(i)^{(\tilde\varepsilon x)^2/64}=2 \big( x h_s(i)^{\tilde \varepsilon x / 32}\big)^{\tilde \varepsilon x /2}\underbrace{\le}_{\mbox{by \eqref{eq: Probhx0}}}2 n^{- \tilde \varepsilon x / 2}h_s(i)^{( \tilde \varepsilon x )^2 /128}.
\int_{\mathbb B^N_{+}}(1-\sum_{j=1}^{N}r_j)^{\alpha\cdot(\mathbf a+\mathbf 1)}r^{\mathbf c}dV = \int_{0}^{1}{r_N}^{c_N}\int_{W_{r_N}}(1-\sum_{j=1}^{N}r_j)^{\alpha\cdot(\mathbf a+\mathbf 1)}\prod_{j=1}^{N-1}r_j^{c_j}dr_1\dots dr_{N-1}dr_N,
\delta Z_{A\phi}=-\frac{g^2}{M_W(h)+M_{A\phi}(h)}\frac{1}{\pi}+\left(\frac{g^4 h^2}{M_W(h)^3}+ \frac{(\frac{\lambda}{6}+g^2)^2 h^2}{M_{A\phi}(h)^3}\right) \frac{1}{48 \pi}.
p\left( x|\theta\right) ={\displaystyle\prod\limits_{k=1}^{l}}p\left( x_{k}|\mu_{k}\sigma_{k}\right) p\left(x_{k}|\mu_{k}\sigma_{k}\right) \overset{}{=}\frac{1}{\sqrt{2\pi\sigma_{k}^{2}}}\exp\left[ -\frac{\left( x_{k}-\mu_{k}\right)^{2}}{2\sigma_{k}^{2}}\right]
\frac{(-a-p)_{p+k}(p-a)_{p+k}}{(1)_{p+k}^2}= \frac{\Gamma_p(-a+k)}{\Gamma_p(-a-p)}\cdot\frac{\Gamma_p(2p-a+k)}{\Gamma_p(p-a)}\cdot\frac{\Gamma_p(1)^2}{\Gamma_p(p+k+1)^2}\cdot\frac{(-p)\cdot p}{p^2}\equiv -\frac{(-a)_k^2}{(1)_{k}^2}\cdot\big(1+2p(H_{p-1-a+k}-H_k)\big)\pmod{p^2}.
\dfrac{|u(x) - u(x_0)|}{d_\beta (x,x_0)^\alpha}=\dfrac{|u(x) - u(x_0)|}{|z|^{\alpha \beta}}\ge \dfrac{\varepsilon |s|_h^{2\gamma}(x)}{|z|^{\alpha \beta}}\ge \dfrac{\varepsilon}{C}\dfrac{|z|^{2\gamma}}{|z|^{\alpha \beta}}.
[ \frac{e}{n + \frac{1}{2}}]^{n + \frac{1}{2}}\approx \frac{\sqrt{2 \pi}}{n !}\frac{e^{\frac{1}{2}}}{( 1 + \frac{1}{2 n})^{n + \frac{1}{2}}}\approx \frac{\sqrt{2 \pi}}{n !}
K_{\hat\tau}^n \ = \Y_{\hat\tau}^n - Y_0^n - g(\hat X) - \int_0^{\hat\tau}f_s(\hat X,\hat I_s)\,ds + \int_0^{\hat\tau}\int_\Lambda R_s^n(b)\,\hat\theta(ds\,db) &\ + \int_0^{\hat\tau}Z_s^n\,d\hat W_s + \int_0^{\hat\tau}\int_U L_s^n(z)\,(\hat\pi(ds\,dz)-\lambda_\pi(dz) ds).
\vert \cos < \overrightarrow{n_{1}}, \overrightarrow{n_{2}}> \vert = \frac{\vert \overrightarrow{n_{1}}- \overrightarrow{n_{2}}\vert}{\vert \overrightarrow{n_{1}}\vert \vert \overrightarrow{n_{2}}\vert}= \frac{1}{\sqrt{2 + 4 t^{2}}}
F\left( f\right) \left( k\right) \equiv \hat{f}\left( k\right) \doteq \int_{\mathbb{R}^{d}}f\left( x\right) \mathrm{e}^{-ik\cdot x}\mathrm{d}x=\frac{(-1)^{n}}{k_{j_{1}}^{2}\cdots k_{j_{n}}^{2}}F\left( \frac{\partial^{2n}f}{\partial x_{j_{1}}^{2}\cdots \partial x_{j_{n}}^{2}}\right) \left( k\right)\ ,
\delta{\cal L}= \mu \left( \psi_{R}^{( i ) \dagger}\psi_{R}^{( i )}+ \psi_{L}^{( i ) \dagger}\psi_{L}^{( i )}\right) + \mu_{5}\left( \psi_{R}^{( i ) \dagger}\psi_{R}^{( i )}- \psi_{L}^{( i ) \dagger}\psi_{L}^{( i )}\right)