harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta_{( 1 )}^{c}{\cal L}_{A P E G T}= \int d^{4}x{\frac{C_{2}\sigma^{2}g^{2}}{8 \pi^{2}}}* B_{\mu \nu}^{i}( x ) I^{\mu \nu , \alpha \beta}\left[ f_{0}( \alpha ) - f_{1}( \alpha ){\frac{\partial_{\lambda}\partial^{\lambda}}{M_{A}^{2}}}+ f_{2}( \alpha ){\frac{( \partial_{\lambda}\partial^{\lambda})^{2}}{M_{A}^{4}}}\right] * B_{\alpha \beta}^{i}( x ) .
\kappa_\lambda =\frac{n_\lambda}{r!}\sum_{g,h, \tau}(h) \delta^{a_{\tau(1)}}_{b_{\tau\rho(1)}}\cdots \delta^{a_{\tau(r)}}_{b_{\tau\rho(r)}}v_{a_1}\otimes \cdots \otimes v^{b_r},
|u_A-u_E| = \frac{\|u_A - u_E\|_{L^q(A \cap E)}}{\mu(A\cap E)^{1/q}}&\le \frac{\|u-u_A\|_{L^q(A)}+ \|u-u_E\|_{L^q(E)}}{\mu(A\cap E)^{1/q}}\le \frac{2Q}{\mu(A\cap E)^{1/q}}.
e_\lambda(x, y) ~=~4 \int_r^\infty \frac{\sinh s}{\sqrt{\cosh 2s -\cosh 2r}}\cdot \frac{-1}{2 \pi \sinh s}\partial_s \frac{-1}{2 \pi \sinh 2s}\partial_s \cdot \cdot \left( \frac{-1}{2 \pi \sinh s}\partial_s \right)^{2d-2}\frac{\sin (s \cdot \sqrt{\lambda - (2d+1)^2})}{\pi s}\,ds\,.
\sum_{n=1}^\infty c_{1,n}T_1^n\left(\{x_\tau^0\}_{\tau=1}^{r}\right)\equiv \sum_{n=1}^\infty c_{1,n}T_1^n\left(\{x_\tau^0 \tilde{f}_0^n\}_{\tau=1}^{r}\right) , \sum_{n=1}^\infty c_{2,n}T_2^n\left(\{x_\theta^0\}_{\theta=r+1}^{r+s}\right) \equiv \sum_{n=1}^\infty c_{2,n}T_2^n\left(\{x_\theta^0\tilde{f}_0^n\}_{\theta=r+1}^{r+s}\right).
Z_1(s) = \zeta(s) \zeta(3s) \zeta^2(5s), Z_2(s) = \zeta(2s) \zeta^2(3s) \zeta(4s) \zeta^2(5s), Z_3(s) = \zeta^2(3s) \zeta^2(4s) \zeta^4(5s), Z_4(s) = \zeta^2(4s) \zeta^4(5s) \zeta^2(6s) \zeta^6(7s),
\begin{dcases}\rho_t + V_0\rho_x + Q_0u_x =0 \ (0,+\infty)\times (0,L) , u_t - \frac{\nu}{Q_0}u_{xx}+ V_0 u_x +a\gamma Q_0^{\gamma-2}\rho_x =0 \ (0,+\infty)\times (0,L) .\end{dcases}
\frac{\partial^2 l}{\partial \theta_i^2}= -\frac{2}{\theta_i^2}- \frac{1}{\phi};\frac{\partial^2 l}{\partial \theta_i \partial \phi}= \frac{\theta_i}{\phi^2};\frac{\partial^2 l}{\partial \phi^2}= \frac{3}{2\phi^2}- \frac{\theta_i^2}{\phi^3};\frac{\partial^3 l}{\partial \theta_i^3}= \frac{4}{\theta_i^3}.
(-1)^* \cdot \biggl( -\sum_{l=0}^{\infty}\frac{\tilde{z}_i^l}{l!}\, s_{1,2,l+2}\biggr)^{g-e}\cdot \prod_{(j,l) p_{i,j,l}= 1}{}\frac{\tilde{z}_i^{l-1}}{(l-1)!},
\rho_2 ={1\over 2}\tau_1 \psi^2 + i \tau_2{\cal F}, \quad \rho_4 ={1 \over 24}i \tau_2\psi^4 +{1 \over 2}\tau_1{\cal F}\psi^2 +{1 \over 2}i \tau_2{\cal F}^2, \quad{\rm etc.},
\omega\cdot (\mathtt{l}(j_1)+\mathtt{l}(j_2)+\mathtt{l}(j_3))=\overline{\omega}\cdot (\mathtt{l}(j_1)+\mathtt{l}(j_2)+\mathtt{l}(j_3))+(\omega-\overline{\omega})\cdot (\mathtt{l}(j_1)+\mathtt{l}(j_2)+\mathtt{l}(j_3)) =j_1^3+j_2^3+j_3^3+O(\varepsilon^2)
\varepsilon(\tfrac{1}{2},\tau[a]\otimes \tau') = \prod_v\varepsilon(\tfrac{1}{2},\tau_v[a]\otimes \tau'_v) = + 1; \varepsilon(\tfrac{1}{2},\tau\otimes \tau'[a']) = \prod_v\varepsilon(\tfrac{1}{2},\tau_v\otimes \tau'_v[a']) = + 1.
\mathcal G (k) \leq C:= \mathcal G (0) + 2s\left( \sum_{j=0}^{\infty}\left( j + \alpha -1\right) \| g_j\|\right) \left( \sqrt{\frac{\mathcal G (0)}{\alpha -1}}+ \frac{2s}{\alpha -1}\sum_{j=0}^{\infty}\left( j + \alpha -1\right) \| g_j\| \right) .
\delta_k(sx_1\wedge...\wedge sx_p)=\sum_{i_1<\dots<i_k}\varepsilon\, h_k(sx_{i_1}\wedge...\wedge sx_{i_k})\wedge sx_1\wedge...\widehat{sx}_{i_1}...\widehat{sx}_{i_k}...\wedge sx_p.
I(t) \ge \int_{0}^\infty e^{-\frac{n-1}{t}(y^1-(\alpha- \beta)t)^2}\prod_{j=2}^n\left( \int_{y^j\ge0, |y^j-y^1|\le M/2}e^{-\frac{1}t (y^j-y^1)^2}dy^j\right) dy^1 &\ge e^{-\frac{M^2}{4t}}\int_{0}^\infty e^{-\frac{n-1}{t}(y^1-(\alpha- \beta)t)^2}h(y^1) dy^1,
g \: = \: dr^2 + r^2h + h_{0}+ r^{-2}h_{2}+ \cdots + r^{-2i}h_{2i}+ \cdots f \: = \: - \frac14 r^2 + f_{0}+ r^{-2}f_{2}+ \cdots + r^{-2i}f_{2i}+ \cdots,
X \smallsetminus{\downarrow}^{\mathbb{X}}p^{-1}[Y \smallsetminus (U \cap V)] = ( X \smallsetminus{\downarrow}^{\mathbb{X}}p^{-1}[Y \smallsetminus U]) \cap (X \smallsetminus{\downarrow}^{\mathbb{X}}p^{-1}[Y \smallsetminus V]),
\vec b=(\nabla y) (\bar{\mathcal{G}}_{2\times 2})^{-1}\left[\begin{matrix}\bar{\mathcal{G}}_{13}\bar{\mathcal{G}}_{23}\end{matrix}\right] + \frac{\sqrt{\det \bar{\mathcal{G}}}}{\sqrt{\det\bar{\mathcal{G}}_{2\times 2}}}\,\vec \nu.
\frac{P}{Q}= B_2 + \frac{B_1}{(1-\alpha x_1)}+ \frac{B_0}{(1-\alpha x_1)^2}= B_2 + \sum_{n=0}^\infty B_1 \alpha^n x_1^n + (n+1)B_0 \alpha^n x_1^n.
b^{N_1}\bigg( \prod_{i=1}^{m-1}(ad c^{N_i}b^{N_{i+1}}) \bigg) ad c^{N_m}\equiv_I b^{k_0}\bigg( \prod_{i=1}^{m-1}(ad (bc)^{k_i}) \bigg) ad c^{k_m}&\equiv_I b^{k_0}\bigg( \prod_{i=1}^{m-1}(ad b^{k_i}c^{k_i}) \bigg) ad c^{k_m}.
\sup_{(x)_m = \prod_{i=1}^k (x_i)^{\frac{1}{k}}_m}\Bigl[ \rho^{\frac{1}{k}}\Bigl( \sum_{i,j =1, i < j}^{k}\langle x_i, x_j \rangle \Bigr) \Bigr]= \rho^{\frac{1}{k}}\Bigl( \frac{k(k-1)}{2}|x|^2 \Bigr).
\varphi^{2d-1}g_e=\varrho^{-1}t^{n_e+1-p^{2d-1}}g_e \varphi^{2d-1}g_f=\varrho^{-1}t^{n_f+1-p^{2d-1}}g_f \gamma(x)g_e-x^{s_e}g_e\in t\cdot k_{\mathcal E}^+\cdot g_e \gamma(x)g_f-x^{s_f}g_f\in t\cdot k_{\mathcal E}^+\cdot g_f.
\vert\vert f_k\vert\vert_{L^2(m)}^2=k!\sum_{(i_1,\dots, i_k)\in (\{-N,\dots, N\}\setminus\{0\})^k}\big\vert\alpha_{i_1,\dots ,i_k}^{(k)}\big\vert^2\sigma_{i_1}^2\dots \sigma_{i_k}^2.
\mathbb{V}^{\kappa}_{\lambda}\hookrightarrow (\mathbb{M}^{\kappa}_{\lambda})^{\vee}\to \dotsb \to \bigoplus_{\ell(w)=i}(\mathbb{M}^{\kappa}_{w \cdot \lambda})^{\vee}\to \dotsb \to (\mathbb{M}^{\kappa}_{w_0 \cdot \lambda})^{\vee}\to 0.
{\mathcal L}_{e f f}\sim h \rho_{0}q \sum_{\alpha = 1}^{M}\epsilon_{k l}\dot{r}_{k}^{( \alpha )}R_{l}^{( \alpha )}- \frac{h^{2}\rho_{0}q^{2}}{2 \pi m}\sum_{\alpha = 1}^{M}\ln | \frac{r^{( \alpha )}}{\xi}| ,
l_1(\alpha\cdot \xi) =(d\alpha)|_{\wedge K}\cdot \xi+ (-1)^{a}\alpha \cdot l_1(\xi) l_2( \xi, \alpha\cdot \psi) =(-1)^{k}(\iota_{\xi}d\alpha)|_{\wedge K}\cdot \psi+ (-1)^{ka}\alpha \cdot l_2(\xi,\psi) l_3(\xi, \psi, \alpha\cdot \phi) =(-1)^{k+l+1}\iota_{\xi}\iota_{\psi}d\alpha\cdot \phi + (-1)^{(k+l+1)a}\alpha \cdot l_3(\xi, \psi,\phi).
\int_{\Omega_i}\varepsilon \nabla u^* \cdot \nabla \varphi_\epsilon\, dx-\int_e \varepsilon \nabla u^* \cdot \nu \bar{\varphi}\, ds+\eta_{e}\int_e u^* \bar{\varphi}\, ds =\int_{\Omega_i}f \varphi_\epsilon\, dx+\eta_{e}\int_e \hat{u}\bar{\varphi}\, ds.
\boldsymbol{E}[\sum_{j=1}^{T-2}\sum_{k=j+1}^{T-1}\sum_{l= k+1}^T X_j X_k X_l] \leq (\boldsymbol{E}[T])^{\frac{1}{4}}(\boldsymbol{E}[T^3])^\frac{1}{4}(\boldsymbol{E}[T^4])^\frac{1}{2}.
\beta^{-1}\partial_{q^i}(V\partial_{q^j}\beta+\beta\tilde F_j -\beta\partial_t\psi_j)(\tilde\gamma^{-1})^{ik}H_{k\ell}(\tilde\gamma^{-1})^{j\ell}-(2\hat{b}_+^i(\tilde\Sigma^{-1})_{ik}b_-^k+\nabla\cdot b_-) = 0
\int_E h dx = \int_{E \cap B_a}h dx + \int_{E \setminus B_a}h(|x|)dx &\ge \int_{E \cap B_a}h dx +\int_{E \setminus B_a}h(|T(x)|)dx &= \int_{E \cap B_a}h dx+\int_{B_a \setminus E}h(|x|)dx &=\int_{B_a}h dx.
S_\hbar=S_\hbar(t)= \left[\begin{array}{cc}S^{-,-}_\hbar S^{-,+}_\hbar & S^{+,-}_\hbar S^{+,+}_\hbar\end{array}\right]=\left[\begin{array}{cc}A_\hbar B_\hbar B_\hbar A_\hbar\end{array}\right],
(E_{i - 1}^{*}\triangleright \tilde{e}_{i - 1}, \tilde{e}_{i}) =(K_{i - 1}F_{i - 1}\triangleright \tilde{e}_{i - 1}, e_{i})= - q^{1/2}(K_{i - 1}\triangleright \tilde{e}_{i}, e_{i}) & = - q^{1/2}q^{-1}(\tilde{e}_{i}, \tilde{e}_{i}) = - q^{-1/2}.
\begin{aligned}\phi_{X+Y}(d\vec{1}) =(e^{id\vec{1}\cdot (X+Y)})=(e^{id\vec{1}\cdot X}e^{id\vec{1}\cdot Y}) &=(e^{id\vec{1}\cdot X})(e^{id\vec{1}\cdot Y})=\phi_{X}(d\vec{1})\phi_{Y}(d\vec{1}).\end{aligned}
q^{\varepsilon,\delta,h}_r =\frac{g^{\delta,h}(q^{\varepsilon,\delta,h})}{\varepsilon}q^{\varepsilon,\delta,h}_{rr}= \frac{( g^{\delta,h}(q^{\varepsilon,\delta,h}) )_r}{\varepsilon}=\frac{g^{\delta,h}_q ( q^{\varepsilon,\delta,h})}{\varepsilon}q^{\varepsilon,\delta,h}_r.
\begin{array}{ll}Z_{12}= \frac{1}{s}(\sum_{l=1}^{g(s)}e^{-4\pi i (l-1)/ns}+e^{4\pi i/n}\sum_{l=g(s)+1}^{s}e^{-4\pi i (l-1)/ns}) \times(\sum_{p=0}^{p_{max}}e^{2 \pi i p/n}\sum_{u(p)> j \geq l(p)}e^{-2\pi i(j-1)/M}),\end{array}
V ( z ) = [ 1 -{\frac{z}{Q_{p}}}{\frac{1 +{\frac{( Q_{p}b )^{\frac{n}{2}}}{2 ( x^{2}+ 4 Q_{p}z )^{\frac{n}{2}}}}}{( 1 +{\frac{( Q_{p}b )^{\frac{n}{2}}}{( x^{2}+ 4 Q_{p}z )^{\frac{n}{2}}}})^{2}}}] +{\frac{J_{\perp}^{2}}{2 m_{f}Q_{p}^{2}}}{\frac{1}{{( 1 +{\frac{( Q_{p}b )^{\frac{n}{2}}}{( x^{2}+ 4 Q_{p}z )^{\frac{n}{2}}}}})^{2}}},
\hat{\varDelta}= \frac{1}{2}\max(R_{n1}, R_{n2}), &\hat{\theta}_1 -\theta_1 = \frac{Q_{n2}- Q_{n1}}{v_2 - v_1}, \hat{\theta}_0 -\theta_0 = \frac{Q_{n1}v_2 - Q_{n2}v_1}{v_2 -v_1};
\begin{gathered}q:=y_{i_1}=y_{i_2}, i_1,i_2\in I_{F_2}\cup \bar J_{F_2}, y_k<y_i, i\in I_{F_2}\cup \bar J_{F_2}, k\in (I\cup \bar J)\setminus (I_{F_2}\cup \bar J_{F_2}), y_{\ell}=-\infty, \ell\in L.\end{gathered}
\mathfrak{D}(\xi,\xi') =\mathfrak{D}_{int}(\xi,\xi')=\sum_{k=1}^{n}|w'_j\circ w_i^{'-1}(z')-w_j\circ w_i^{-1}(z)|^2, \mathfrak{D}_0(\xi,\xi') =\mathfrak{D}_{int,0}(\xi,\xi')=\sum_{k=1}^{n}|w_j\circ w_i^{-1}(z')-w_j\circ w_i^{-1}(z)|^2,
[\begin{matrix}x^{\prime}\\y^{\prime}\end{matrix}]=[\begin{matrix}cos(-\pi/2)&sin(-\pi/2)\\ -sin(-\pi/2)&cos(-\pi/2)\end{matrix}][\begin{matrix}7\\ 7\end{matrix}]=[\begin{matrix}0&-1\\ 1&0\end{matrix}][\begin{matrix}7\\ 7\end{matrix}]=[\begin{matrix}-7\\ 7\end{matrix}].
=|\alpha|^{2}\frac{1}{Z}Tr\{e^{-\beta\hbar\omega a^{\dagger}a}\}+\frac{1}{Z}Tr\{a^{\dagger}ae^{-\beta\hbar\omega a^{\dagger}a}\}=|\alpha|^{2}+\frac{1}{Z}\sum_{n=0}^{\infty}ne^{-n\beta\hbar\omega},
T_{36}(x)=\sqrt{\frac{1}{2}-\frac{1}{2}\sqrt{1-4\left(\frac{768-x}{768}-e^{i\pi/3}\frac{x(1536-x)}{768D_1^{1/3}}+e^{-i\pi/3}\frac{D_1^{1/3}}{768}\right)}},
\int(u_{1}^{\prime}\int v_{1}dx)dx=\int(8\pi(\frac{M}{2\pi RT})^{3/2}e^{\frac{-Mx^{2}}{2\pi RT}})dx=(8\pi(\frac{M}{2\pi RT})^{3/2}e^{\frac{-Mx^{2}}{2\pi RT}})
\pi_{1}-\pi_{0}=\sum_{\omega\in S_{1}}\sum_{\alpha,\alpha^{\prime}}\Pi(A,\omega,\alpha,x)\Pi(A,\omega,\alpha^{\prime},x)-\sum_{\omega\in S_{0}}\sum_{\alpha,\alpha^{\prime}}\Pi(A,\omega,\alpha,x)\Pi(A,\omega,\alpha^{\prime},x).
P\sp{(\alpha,\alpha)}_k = \frac{(k+2\alpha+1)(k+2\alpha+2)}{(2k+2\alpha+1)(2k+2\alpha+2)}P\sp{(\alpha+1,\alpha+1)}_k - \frac{k+\alpha}{2(2k+2\alpha+1)}P\sp{(\alpha+1,\alpha+1)}_{k-2},
J^{w}(\textbf{X}_{RSS}^{(n)})=-\frac{1}{2}\prod_{i=1}^{n}\left(-2J^{w}(X_{(i:n)i})\right) =- \frac{Q_n}{2}\prod_{i=1}^{n}E\left(\Lambda_X^{w}(B_{2i-1:2n-1})\right),
F_T(x)=\frac{(1 - 2 x) (1 - 6 x + 12 x^2 - 10 x^3 + 2 x^4) -x^2 (1 - 2 x + 2 x^2)^2 C(x)}{1 - 9 x + 30 x^2 - 49 x^3 + 38 x^4 - 8 x^5 - 4 x^6}
\langle{\varphi}\,|\,V^{1/2}(\rho +h(\mathfrak{e}))^{-1}V^{1/2}{\varphi}\rangle =\int_{\Gamma^{\ast}}\frac{|\mathcal{F}^{\ast}\circ V^{1/2}({\varphi})(p)|^{2}}{\rho +\mathfrak{e}(p)}\mathrm{d}\mu^{\ast}(p).
\int_0^x t^\mu K_\nu(at)\,\mathrm{d}t =\frac{2^{\mu-1}}{a^\mu}\Gamma\bigg(\frac{\mu-\nu+1}{2}\bigg)\Gamma\bigg(\frac{\mu+\nu+1}{2}\bigg)G_{\mu,\nu}(ax), \int_x^\infty t^\mu K_\nu(at)\,\mathrm{d}t =\frac{2^{\mu-1}}{a^\mu}\Gamma\bigg(\frac{\mu-\nu+1}{2}\bigg)\Gamma\bigg(\frac{\mu+\nu+1}{2}\bigg)\tilde{G}_{\mu,\nu}(ax),
\langle c_kv^k,r^k\rangle =\langle x^k+\alpha_k(x^k-x^{k-1})-x^{k+1},y^k-x^{k+1}\rangle & =\langle x^k-x^{k+1},y^k-x^{k+1}\rangle+\alpha_k\langle x^k-x^{k-1},y^k-x^{k+1}\rangle & =\frac{1}{2}\|x^{k+1}-x^k\|^2+\frac{1}{2}\|r^k\|^2-\frac{1}{2}\|x^k-y^k\|^2+\alpha_k\langle x^k-x^{k-1},y^k-x^{k+1}\rangle.
\frac{\partial E}{\partial z^i_{\epsilon}}=2b^2\Big[z^3_{\tilde{\epsilon}}\sum\limits_{\tau}^{}(-1)^{{\tilde{\epsilon}}+\tau}z^i_{\tilde{\tau}}z^3_{\tau}+\delta_{i3}\sum\limits_{\tau ,k}^{}(-1)^{\epsilon +\tau}z^3_{\tau}z^k_{\tilde{\tau}}z^k_{\tilde{\epsilon}}\Big],
\mu(u,e^Te^S)\leq \mu(\frac{u}{2},e^T)\mu(\frac{u}{2},e^S)\leq\mu(\frac{u}{2},e^{T_+})\mu(\frac{u}{2},e^{S_+}) =e^{\mu(\frac{u}{2},T_+)+\mu(\frac{u}{2},S_+)}&\leq e^{\mu(\frac{u}{2},T)+\mu(\frac{u}{2},S)},
[e^{(n)}_{\{m\},i}\otimes^n 1_N][e^{(n)}_{\{m'\},j}\otimes^n 1_N] =\delta_{i,j}\delta(\{m\},\{m'\})\frac{n!}{f^{(n)}_{\{m\}}}e^{(n)}_{\{m\},i}\;,
f_{\mathbf{M^{\ast\ast}}_{-\xi_{\boldsymbol{s}k}}}(\boldsymbol{v})=c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast\ast}\overline{\mathcal{G}}_{n}\left\{\frac{1}{2}\boldsymbol{v}^{T}\boldsymbol{v}+\frac{1}{2}\xi_{\boldsymbol{s}k}^{2}\right\},~\boldsymbol{v}\in \mathbb{R}^{n-1},
\nabla_{d\pi L_g\xi}t= \pi_V d_{d\pi L_g\xi}\phi(t)=\pi_V d_{d\pi L_g\xi}\left([g],{T}g\pi_0(g^{-1}w)\right) = \pi_V\left([g],{T}g\xi \pi_0(g^{-1}w)-{T}g \pi_0(\xi g^{-1}w)\right) = \left[g, \pi_0\left(g^{-1}T^2g\xi \pi_0(g^{-1}w)\right)-\pi_0\left(g^{-1}T^2g\pi_0 (\xi g^{-1}w)\right)\right].
r_\varepsilon |\bar{w}'_\varepsilon(r_\varepsilon)|= O\left(\|\bar{w}'_\varepsilon\|_{L^\infty([0,r_\varepsilon])}\frac{{\mu_\varepsilon(r_\varepsilon/\mu_\varepsilon)^3}}{1+(r_\varepsilon/\mu_\varepsilon)^3}\right)+o\left(\frac{(r_\varepsilon/\mu_\varepsilon)^2}{1+(r_\varepsilon/\mu_\varepsilon)^2}\right)\,.
\begin{array}{r c l}{{\hat{J}_{\pm}^{\prime}D_{M K}^{J}}}&{{=}}&{{\sqrt{[ J \mp M ] [ J \pm M - 1 ]}D_{M \pm 1 , K}^{J}~ ,}}\\{{\hat{J}_{3}^{\prime}D_{M K}^{J}}}&{{=}}&{{K D_{M K}^{J}~ , ~ \hat{J}_{z}^{\prime}D_{M K}^{J}= M D_{M K}^{J}~ .}}\\ \end{array}
\begin{aligned}f_1 (\tau_1) \le{C \over u_0}\left [\int_{-10}^{u_0}\int_u^{2 T - u}\int_{S^2}|\Box \phi| r^3 |L(r \phi)| d \omega d v d u + \int_{-20}^0 \int_{\Sigma_t}|\Box \phi| |\partial_t \phi| d x d t + \Vert \phi_0 \Vert_{H^1}^2 + \Vert \phi_1 \Vert_{L^2}^2 \right ]. \end{aligned}
\rho[Z,P] = \min_{\eta(\cdot)}\int_0^1 \int_0^1 \Big[\eta(x) + \frac{1}{\alpha}\big(x - \eta(x)\big)_+\Big]\;dy\;dx &= \int_0^1 \min_{\eta}\Big[\eta + \frac{1}{\alpha}\big(x - \eta\big)_+\Big]\;dx = \int_0^1 x \;dx= \frac{1}{2},
\begin{array}{l}{\bf M}_{{\cal N}}(E) \equiv \left(\renewcommand{\arraystretch}{1.3}\begin{array}{cc}\begin{array}{cccc}2 E +{\cal N}- 1 0 2 0 2 g (1-{\cal N}) 2E +{\cal N}- 3 0 6 0 2 g ( 2 -{\cal N}) 2E +{\cal N}- 5 0 0 0 2 g ( 3 -{\cal N}) 2E +{\cal N}- 7\end{array}\cdots \vdots \ddots \end{array}\right) \end{array}
M_{1,Q}f \leq \overline M_{1,Q}f + N^{- \epsilon}A_{\tau}f , \lvert E\rvert^{-1}\langle \overline M_{1,Q}f, g \rangle \lesssim N^{1+ \epsilon /2}\langle f \rangle_{E}^{\frac1{\bar p + \epsilon}}\langle g \rangle_{E}^{\frac 1{\bar q + \epsilon}}.
\phi_1=\frac{\lambda}{m^2}{\bigg [}\left(\frac{1}{2\sqrt{3}}+\frac{3}{4\pi}\right)\frac{1}{\cosh^2(mx/2)}-\frac{\sqrt{3}}{4}(m\partial_m+2\lambda\partial_{\lambda}){\bigg]}\phi_{\rm kink}.
~~~\left(\frac{1}{j-1}+\left(\frac{1}{j-1}\right)^2+\cdots\right)\cdot\frac{j}{2}\cdot\binom{n}{k}+ \frac{j}{2}\cdot \frac{1}{(j-1)^2}\cdot \binom{n}{k}&<\left(\frac{j}{2(j-2)}+\frac{j}{2(j-1)^2}\right) \binom{n}{k}< \binom{n}{k}
E[W^k_{\ell\Delta,t}(j)|{\cal F}_{(\ell+1)\Delta}] =E\left[E\left[W^k_{\ell\Delta,t}(j)|X_{\ell,\Delta},{\cal F}_{(\ell+1)\Delta}\right]|{\cal F}_{(\ell+1)\Delta}\right] &=p^k_{j,+}(t-\ell\Delta) E\left[X_{\ell,\Delta}|{\cal F}_{(\ell+1)\Delta}\right] & =\Delta \nu p^k_{j,+}(t-\ell\Delta) Z_0(\ell\Delta),
G^{m, n}_{p, q}\left(z\Big\vert{A_{1}\ldots A_{p}\atop B_{1}\ldots B_{q}}\right) = \mathcal{M}^{-1}\left[\frac{\prod_{j=1}^{m}\Gamma(B_{j}+ s)\, \prod_{j=1}^{n}\Gamma(1 - A_{j}- s)}{\prod_{j=m+1}^{q}\Gamma(1 - B_{j}- s) \prod_{j=n+1}^{p}\Gamma(A_{j}+ s)}; z\right],
{{y_{M}=}}\sqrt{P_M \alpha_{M,k^*}}{\bf g}^H_{M,k^*}{\bf f}_{j^*}s_{M,k^*}+ \sqrt{P_F \alpha_{F,k^*}}{\bf g}^H_{F,k^*}{\bf w}_{i^*}s_{F,k^*}+n_M,
\mathbb{U}_{\vec{c},{\vec{c}}^{\prime}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2})=\frac{1}{4\pi}\displaystyle{\sum_{l=0}^{2j}}\frac{(c_{l}^{n})^{\prime}}{c_{l}^{n}}(2l+1)P_{l}(\mathbf{n_{1}}\cdot\mathbf{n}_{2})\ .
|p^+(-T^-)-p^-(-T^-)| \leqslant|p^+(-T^-)-p^+(-T^+)|+|\mu^+\nu^+-\mu^-\nu^-| &\leqslant c_4\Big((T^+-T^-)+|y^+-y^-|\Big) &\leqslant c_5|\xi^+(-T^-)-y^-|+c_4|y^+-\xi^+(-T^-)| &\leqslant c_5|\xi^+(-T^-)-y^-|+c_6(T^+-T^-) &\leqslant c_7|\xi^+(-T^-)-y^-|.
(U,\X,\Y,\Z): &=(\frac{U_1+(U_2)^{\eta^{\ast}}}{2}\frac{\tilde{X}+(\tilde{X})^{\eta^{\ast}}}{2},\ \frac{\tilde{Y}+(\tilde{Y})^{\eta^{\ast}}}{2},\ \frac{\tilde{Z}+(\tilde{Z})^{\eta^{\ast}}}{2}).
E_{2}^{pq}=R^p\lim_{\leftarrow\atop m}(H^q({\mathfrak X},{\mathcal F}_m))\Rightarrow H^{p+q}({\mathfrak X},\lim_{\leftarrow\atop m}{\mathcal F}_m)=H^{p+q}({\mathfrak X},{\mathcal F})
\left[ i \gamma^{\mu}\partial_{\mu}- m{\frac{1 \pm b_{1}\pm b_{2}}{a}}\right] \times \left[ i \gamma^{\nu}\partial_{\nu}+{\frac{a}{2 m}}\partial^{\nu}\partial_{\nu}+ m{\frac{b_{1}^{2}- 1}{2 a}}\right] \Psi_{1 , 2}= 0 .
\Pr \{\hat{f}_{\rm det}(X) = w_1 \}= \Pr \{X \in A_{D}(y_1) \}+ \gamma, & \Pr \{\hat{f}_{\rm det}(X) = w_i \}= \Pr \{X \in A_{D}(y_i) \}(i=2, \ldots, k^*).
(T+S)^\dagger -T^\dagger =(I+ST^\dagger)^{-1}T^\dagger-(I+ST^\dagger)(I+ST^\dagger)^{-1}T\dagger =[I-(I+ST^\dagger-)](I+ST^\dagger)^{-1}T^\dagger = (-ST^\dagger)(I+ST^\dagger)^{-1}T^\dagger .
\nabla P_{E_{a_{k_j}}}[i] = \frac{1}{M\sqrt{2\pi}\sigma_{n}\sqrt{{\boldsymbol w}^\emph{H}_{D_j}[i]{\boldsymbol w}_{R_j}[i]}}\sum^{M}_{j=1}exp\big(-\frac{c^2_j[i]}{2}\big)sgn(s_j)\Re[{\boldsymbol w}^\emph{H}_{R_j}[i]{\boldsymbol h}_{k_j}[i]s_j],
- \alpha^{-1}\big(\int_0^s U(u,x)f'(u)\mathrm{d}u+f(s)U(s,x)-f(0)U(0,x)\big) =- \int_0^s \int_0^1 f(u)A(x,y)U(u,y)\mathrm{d}y\mathrm{d}u+\beta \int_0^s f(u) \mathrm{d}\xi_2(u,x) .
-\frac{b-a}{b+a}\|h\|^2 =\|h\|^2- \frac{2b}{b+a}\|h\|^2 \leq \|h\|^2- \frac{2}{b+a}\langle S_{x,\tau}h,h \rangle =\left\langle \left(I_\mathcal{H}-\frac{2}{b+a}S_{x,\tau}\right)h, h\right\rangle &\leq \|h\|^2- \frac{2a}{b+a}\|h\|^2=\frac{b-a}{b+a}\|h\|^2.
(p_0,x_0,t_0) \circ_{\mathcal{L}}(p,x,t)= \Big( p\sqrt{|p_0|^2+1}+p_0 \sqrt{|p|^2+1}, x_0+x\sqrt{|p_0|^2+1}+p_0t, t_0+t\sqrt{|p_0|^2+1}+p_0 \cdot x \Big ).
\langle \alpha, \alpha \rangle = 2m(\alpha) \langle \alpha, \alpha \rangle^{2}+ 2m(2\alpha) \langle \alpha, 2\alpha \rangle^{2}+ 2n(\alpha) \langle \alpha, \alpha \rangle^{2}+ 2n(2\alpha) \langle \alpha, 2\alpha \rangle^{2}&= 2\langle \alpha, \alpha \rangle^{2}(m(\alpha)+4m(2\alpha ) + n(\alpha)+4n(2\alpha)).
\sigma_{\mathcal{Z}_s^{(i)}}^2=\frac{\sum_{j\in\mathcal{M}}\left(\beta_j^{(i)}\right)^2|\alpha_{ij}|^2\sigma_{Z_{s,j}^{(i)}}^2}{\sum_{j\in\mathcal{M}}\beta_j^{(i)}|\alpha_{ij}|^2},\, \sigma_{\mathcal{Z}_0^{(i)}}^2=\sum_{j\in\mathcal{M}}\left(\beta_j^{(i)}\right)^2\sigma_{Z_{0,j}^{(i)}}^2,
(T\phi)_x \Pi_M^{\sharp}((T\phi)_x^*\psi_1, \ldots, (T\phi)_x^*\psi_{n-1}) = (T\phi)_y \Pi_M^{\sharp}((T\phi)_y^*\psi_1, \ldots, (T\phi)_y^*\psi_{n-1})
(a , u) \ltimes_{R \oplus S}(b, v) =~ (R(a), S(u)) \ltimes_{R \oplus S}(b, v) ~+~ (a, u) \ltimes_{R \oplus S}(R(b), S(v)) =~ \big( R(a) \cdot b + a \cdot R(b),~ R(a) \cdot v + S(u) \cdot b + a \cdot S(v) + u \cdot R(b) \big) =~ \big( a \ast_R b,~ a ~{\cdot}_S~ v + u ~{\cdot}_S ~ b \big).
\frac{\partial}{\partial t}\int_{V}\rho EdV=\int_{S}u_{i}\sigma_{ij}dS_{j}+\int_{V}\rho u_{i}F_{i}dV-\int_{S}k\frac{\partial T}{\partial x_{j}}dS_{j}-\int_{S}\rho Eu_{j}dS_{j}+\int_{V}\rho HdV
\ll \frac{\exp(O(k))}{(\log\log X)^{2k}}\sum_{p_1,\dots,p_k,p'_1,\dots,p'_k \in I}\frac{\prod_{l=1}^r (p_{i_l}1_{p_{i_l}=p'_{j_l}})}{p_1 \dots p_k p'_1 \dots p'_k}1_{p_{i_{l_0}}| p_1 + \dots + p_{i_{l_0}-1}- p'_1 - \dots - p'_{j_{l_0}-1}}.
\begin{aligned}(1 - \phi_u d) \sum_k F^{ij}u_{ijk}v_k &= 0.5 w_t - (1 - \phi_u d) \sum_k v_k \left[ F^{ij}\chi_{ij,k}- (\psi)_k \right] & + (\phi_{uu}d) u_t \sum_k u_k v_k + u_t d \sum_k \phi_{ku}v_k + \phi_u u_t \sum_k d_k v_k.\end{aligned}
C^{\mathrm{dl},k}_{2}\left( N_{\mathrm{sct}}, N_{\mathrm{scr}}\right) = \left(\frac{T_{\mathrm{dl}}}{T}\right) \min\left\lbrace c^k_{\mathrm{b2s},1}\left( N_{\mathrm{scr}}\right), \right . \,\,\,\,\,\, \left. c^k_{\mathrm{s2u},2}\left( N_{\mathrm{sct}}\right) \right\rbrace.
\begin{pmatrix}a_{\beta,1}a_{\beta,2}u^e a_{\beta,3}u^e a_{\beta,4}\end{pmatrix}\begin{pmatrix}x^p_{\sigma^{-1}\circ \beta}y^p_{\sigma^{-1}\circ \beta}\end{pmatrix}=\lambda_\beta\begin{pmatrix}x_\beta y_\beta\end{pmatrix}
(WSW^*)^*(EW^*+\tilde{H})^*(EW^*+\tilde{H})(WSW^*) = (EW^*+S^*\tilde{H}WSW^*)^*(EW^*+S^*\tilde{H}WSW^*) &\stackrel{\mathcal{L}}{=}(EW^*+\tilde{H})^*(EW^*+\tilde{H})
\hat{G}(t,\xi) = \eta(|\xi|^2) \hat{\phi}(\xi)|\xi| \left( \hat{\psi}_\xi(t|\xi|)-\hat{\psi}_\xi(0)\right)+ \eta(|\xi|^2) \hat{\phi}(\xi)|\xi| \hat{\psi}_\xi(0)+ (1-\eta(|\xi|^2)) \hat{\phi}(\xi)|\xi| \hat{\psi}_\xi(t|\xi|) &= R^{(a)}(\xi, t|\xi|)+R^{(b)}(\xi) + R^{(c)}(\xi, t|\xi|).
\|t\circ P\circ Q\|_{\mathcal{Q}^{inj}}= \|I_H\circ t\circ P\circ Q\|_{\mathcal{Q}}= \|s\circ I_G\circ P\circ Q\|_{\mathcal{Q}}\leq \|s\| \cdot\|I_G\circ P\|_{\mathcal{Q}}\cdot \|Q\|^m &= \|I_H\circ t\| \cdot\|I_G\circ P\|_{\mathcal{Q}}\cdot\|Q\|^m\leq \|t\| \cdot\|P\|_{\mathcal{Q}^{inj}}\cdot\|Q\|^m.
\gamma_1 = \frac{- \int_\Omega \psi_1^2 \left\vert \nabla \left( \frac{u_{2,\lambda}}{\psi_1}\right)\right\vert^2 - \lambda (p-2) A(u_{2,\lambda}) + \lambda (2-q) B(u_{2,\lambda})}{\int_\Omega u_{2,\lambda}^2 + \int_{\partial \Omega}u_{2,\lambda}^2}.
\#\{A_{\mu m, l}\}= \sum_{i=0}^{m-1}\#\{A_{\mu m, l}^{(i)}\}\leq \sum_{i=0}^{m-1}\textrm{max}\{\mu - i, 0 \}= \sum_{i=0}^{\mu}\textrm{min}\{i, m\}= H_{[m]}(\mu m).
H_{2,1}^1 =\int_{\theta_{0}}^{\frac{\pi}{2}}\int_{\frac{1}{2}}^{\frac{3}{2}}\frac{dr d\theta}{|\sin(\theta+\varphi)|^{\tau}|re^{i\theta}-1|^{q}}& \leq \frac{1}{(1-\cos\theta_{0})^{\frac{q}{2}}}\int_{\theta_{0}}^{\frac{\pi}{2}}\frac{d\theta}{|\sin(\theta+\varphi)|^{\tau}}\leq \frac{C_{1}(\tau)}{(1-\cos\theta_{0})^{\frac{q}{2}}},
\langle e^{X^{0}( t_{1})}\cdots e^{X^{0}( t_{n})}\rangle = \prod_{i < j}| e^{i t_{i}}- e^{i t_{j}}|^{2}= 4^{\frac{n ( n - 1 )}{2}}\prod_{i < j}\sin^{2}({\frac{t_{i}- t_{j}}{2}}) .
p^{\ast}\big( \mathsf{1}^{\otimes i-1}\otimes v \otimes \mathsf{1}^{\otimes k+2-i}\big) = - v \otimes \mathsf{1}^{\otimes k+2}+ \mathsf{1}^{\otimes i}\otimes v \otimes \mathsf{1}^{\otimes k+2-i}
\partial_{t}\tilde{\textbf{u}}^{\varepsilon}+ \sum_{j=1}^{d}\textbf{P}J_{\varepsilon}A_{j}(J_{\varepsilon}(\tilde{\textbf{u}}^{\varepsilon}+\bar{\textbf{u}}))\partial_{x_{j}}J_{\varepsilon}\tilde{\textbf{u}}^{\varepsilon}=0,
\mu(|e^{z\langle l,\chi\rangle}-1|)\geq |\mu(|e^{z\langle l,\chi\rangle}-e^{iy\langle l,\chi\rangle}|)-\mu(|e^{iy\langle l,\chi\rangle}-1|)| = |\mu(|e^{x\langle l,\chi\rangle}-1|)-\mu(|e^{iy\langle l,\chi\rangle}-1|)|=\infty.
m_{\nu}=P^{-1}m_{0,\nu}P=\frac1d\mbox{\scriptsize $\left(\!\begin{array}{cccc}2a\!-\!d -ac \!b^2\!+\!bc\!-\!ac\! \!-2b\!-\!2c\! 4 d\!-\! 2c -2b\!-\!2c 4 4 2a\!+\!2b 2a\!-\!d 4 2a\!+\!2b \!b^2\!+\!ab\!-\!ac\! -ac d\!-\! 2c\end{array}\!\right)$}.
X''_1\mathcal{E}_{\pi|_{V}}((T_1, \ldots, T_n)|_{V}) = \mathcal{E}_{\pi|_{W''_1}}((T_1, \ldots, T_{\theta - 1}, T_\theta L_{\mathcal{E}_{\pi|_{V}}((T_1, \ldots, T_n)|_{V})}, T_{\theta + 1}, \ldots, T_n)|_{W''_1}).
\begin{array}{c}H_{l}s_{l}\left( r,\chi \right) =\left\{2\sinh{}^{2}\frac{i}{2}\frac{d}{dr}+\frac{l\left( l+1\right)}{2r\left( r+i\right)}e^{i\frac{d}{dr}}\right\}s_{l}\left( r,\chi \right) = =\left( \cosh \chi -1\right)s_{l}\left( r,\chi \right)\end{array}
f^{J_3 n_3 \sigma_3}_{J_1 n_1 \sigma_1;~J_2 n_2 \sigma_2}=(-1)^{2J_1(J_2-n_2-1)}{\tilde f}^{J_3 n_3 \sigma_3}_{J_1 n_1 \sigma_1;~J_2 n_2 \sigma_2}.
\begin{array}{rl}\alpha_0 \!\! := \frac{1}{2}(\sigma_{01}+ \sigma_{02}- \sigma_{12}) \ \alpha_1 \!\! := \frac{1}{2}(\sigma_{01}+ \sigma_{12}- \sigma_{02}) \ \alpha_2 \!\! := \frac{1}{2}(\sigma_{02}+ \sigma_{12}- \sigma_{01}) \;, \end{array}
\alpha \neq 0 q(s)=\frac{1}{G(s,K)+F(s,N)}G(s,K) =\frac{\frac{1}{x_0}.\underset{k=0}{\overset{K}{\sum}}\beta_{2.k+1}.\Gamma (s+2.k+1)}{-s.\Gamma (s)}F(s,N)=\frac{\underset{n=1}{\overset{n=N}{\sum}}\left(\frac{n}{x_0}\right)^{-s}}{x_0.s}
\bigg\| \frac{f}{|x|^2}\bigg\|_{2}^2 = -\frac{1}{\mathbf{R}_{3}}\mathrm{Re}\bigg\langle \frac{f}{|x|^2}, \Delta_r f \bigg\rangle - \frac{1}{\mathbf{R}_{3}}\bigg\|\partial_r \frac{f}{|x|}+ \mathbf{H}_{3}\frac{f}{|x|^2}\bigg\|_{2}^2
\hspace{- 2 0 m m}c ( n ) \equiv \Gamma ( - n ) e^{-{\frac{n ( n + 1 ) T}{2}}}\prod_{i = 1}^{n}\int d^{2}z_{i}\prod_{i < j}| z_{i}- z_{j}|^{2}\prod_{i , j}{\frac{1}{| 1 - z_{i}\bar{z}_{j}e^{- T}|^{{\frac{2}{Q^{2}}}+ 1}}}.
\langle \,p_{1}\,|\,\mathcal{O}(0)\,|\,p_{2},\dots ,p_{n}\,\rangle_{conn.}^{in}=\,\mathcal{O}_{n}(\theta_{1}+i\pi ,\theta_{2},\dots ,\theta_{n})=\mathcal{O}_{n}(\theta_{2},\dots ,\theta_{n},\theta_{1}-i\pi )
\frac{\log(2c_1h_0^{\alpha}\varepsilon^{-1})}{\alpha \log(s)}\leq L < \frac{\log(2c_1h_0^{\alpha}\varepsilon^{-1})}{\alpha \log(s)}+ 1 = \frac{\log(2c_1(h_0 s)^{\alpha}\varepsilon^{-1})}{\alpha \log(s)}.
u_{\ell,k}=\frac{h_{\ell,k}+h_{k,\ell}}{2},\,v_{\ell,k}=\frac{h_{\ell,k}-h_{k,\ell}}{2i},\,u_{\ell,k}^{(0)}=\frac{h_{\ell,k}^{(0)}+h_{k,\ell}^{(0)}}{2},\,v_{\ell,k}^{(0)}=\frac{h_{\ell,k}^{(0)}-h_{k,\ell}^{(0)}}{2i}.