harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e_{-1}\mapsto D_{p_1 - p_2,u_1}+ D_{u_2, v_1}, e_3 \mapsto -D_{v_1, v_2}= 4D_{p_1 - p_2,u_3}, e_0 \mapsto 2D_{u_3, v_3}+ 3D_{u_2,v_2}, e_4 \mapsto 3D_{v_1, u_3}, e_1 \mapsto D_{p_1 - p_2,v_1}+ D_{u_1,v_2}, e_5 \mapsto 5D_{v_2,u_3}, e_2 \mapsto 3D_{u_1, u_3}= 2D_{p_1 - p_2,v_2},
T_{j,m}(f,g)(x)= &\int f*\Phi_{aj+m}\left(x-\frac{t^a+\epsilon_P(t)}{2^{aj}}\right)g*\Phi_{bj+m}\left(x-\frac{t^b+\epsilon_Q(t)}{2^{bj}}\right)\rho(t)\,dt,
\left\| \Lambda^2 f_n(t)\right\|\leq \left\| \Lambda^2 f_{0}\right\|+ \rho ((\left\Vert \Lambda_{1}f_0\right\Vert)\int_0^t\left\Vert \Lambda^{2}f_n(s)\right\Vert ds \; \forall t \geq 0; \; n=2,3,...,
A=[\begin{matrix}1&2\\ 3&4\end{matrix}]=\frac{5}{2}[\begin{matrix}1&0\\ 0&1\end{matrix}]+\frac{i}{2}[\begin{matrix}0&i\\ -i&0\end{matrix}]+\frac{3}{2}[\begin{matrix}-1&0\\ 0&1\end{matrix}]+\frac{5}{2}[\begin{matrix}0&1\\ 1&0\end{matrix}]=\frac{5}{2}I+\frac{i}{2}\sigma_{y}+\frac{3}{2}\sigma_{z}+\frac{5}{2}\sigma_{x}
\langle \bar{P}^{n}\cdots \rangle ={\frac{\partial^{n}}{\partial \bar{t}^{n}}}\, \langle \cdots \rangle = \left({\frac{\partial t_{s}}{\partial \bar{t}}}\,{\frac{\partial}{\partial t_{s}}}\right)^{n}= \left( -{\frac{1}{\langle P P \rangle_{0}}}\,{\frac{\partial}{\partial t_{s}}}\right)^{n}\ \langle \cdots \rangle \ ,
\frac{\partial \mathcal{L}}{\partial \rho}=-\mu\;\mathrm{tr}\left(\frac{\mathbf{HSH}^{\rm H}}{\sigma^2\,\ln2}\left(\mathbf{I}_{N_R}+\left(1-\rho\right)\sigma^{-2}\mathbf{H}\mathbf{S}\mathbf{H}^{\rm H}\right)^{-1}\right) +\mathrm{tr}\left(\mathbf{HSH}^{\rm H}\right)=0,
\langle\nabla_{X_{1}}\nabla_{X_{2}}X_{2}-\nabla_{X_{2}}\nabla_{X_{1}}X_{2},X_{1}\rangle= \lambda_{1}\lambda_{2}+c\nu^{2}+\displaystyle\frac{\nu^{2}(b_{1}^{2}+b_{2}^{2})}{(\lambda_{2}-\lambda_{1})^{2}}.
\phi^+_i(u) = q^{h_i}\big(1 - \kappa_q \, e'_{\delta, \, \alpha_i}(- o_i u)\big), \phi^-_i(u^{-1}) = q^{-h_i}\big(1 + \kappa_q \, f'_{\delta, \, \alpha_i}(- o_i u^{-1})\big).
\left|f(x, t)\right|\lesssim \frac{t_0^{-\varepsilon}}{R^{a-2s}}\|\bar{\lambda}_1\|_{1+\sigma}\sum_{j=1}^k\frac{\mu_j^{-2s}\mu_0^{\frac{n-2s}{2}-1+\sigma}}{1+|y_j|^{a}}.
\prod_{\ell\in\lambda}([1]-\psi_{r,\ell}\psi_{r,\ell}^*)*[tk - (1-t)k_0] = \prod_{\ell\in\lambda}\left[t([1]-\psi_\ell^*\psi_\ell)*k + (1-t)([1]-\psi_{r,\ell}\psi_{r,\ell}^*)*k_0 - t(r^2-1) \psi_\ell\psi_\ell^* * k\right] \geq 0.
A\otimes B= \begin{pmatrix}A\, b_{11}A\, b_{12}\cdots A\, b_{1s}A\, b_{21}A\, b_{22}\cdots A\, b_{2s}\vdots \vdots \ddots \vdots A\, b_{r1}A\, b_{r2}\cdots A\, b_{rs}\end{pmatrix}.
Z(s)=\int{\cal D}\psi{\cal D}\bar{\psi}{\cal D}b_\mu ~F(\partial b)\delta(\varepsilon^{\mu \nu \rho}f_{\nu \rho}(b))~e^{-\int d^3x~\bar{\psi}(\partial \!\!\!/+i(s\!\!\!/+b\!\!\!/)+m)\psi}\;.
\dot{\rho}=-i[H,\rho]+2\kappa(a\rho a^{\dagger}-\frac{1}{2}(a^{\dagger}a\rho+\rho a^{\dagger}a))+2\gamma(\sigma\rho\sigma^{\dagger}-\frac{1}{2}(\sigma^{\dagger}\sigma\rho+\rho\sigma^{\dagger}\sigma))
h(x) \le \Bigl(g_1(x)+\frac{2}{6^2}(3^2 x-4)(2-2^2x)\Bigr)\Bigr|_{x=\frac{41}{90}}+ \frac1{10\cdot 6^2}- \eta &=-\frac{15967737560279378662084583483719591}{2955720136220085149075016830709099000}<0,
|\xi_{i}- \xi_{j}|^{2}=|\xi_{i}|^{2}+ |\xi_{j}|^2 - 2 \left\langle \xi_{i}, \xi_{j}\right\rangle, =2 \; . \; \frac{2p-1}{2p}- 2\; . \; \frac{2p-1}{2p}cos \left( \frac{\overline{d}_{ij}}{t}\right), =2 \; . \; \frac{2p-1}{2p}- 2 \; . \; \frac{2p-1}{2p}\;. \; \frac{2p-1-d_{ij}}{2p-1}, =4 \;.\; d_{ij}\;.\; \frac{1}{2p}.
V_iV_jf(\underline{z})=R_{q^{d-i}}M_{z_i}R_{q^{d-j}}M_{z_j}f(\underline{z})=q^{d-j}R_{q^{d-i}}z_iz_jf(q^{d-j}\underline{z})=q^{3d-2i-j}z_iz_jf(q^{2d-i-j}\underline{z})
ds^{2}= H_{1}^{3/4}H_{2}^{3/4}H_{3}^{1/4}\left[ H_1^{-1}H_2^{-1}H_3^{-1}(-dt^2 + ds^{2}) + H_1^{-1}d{\vec{x}}^{2}+ H_{2}^{-1}d{\vec{y}}^{2}\right] ~,
\begin{array}{l}{{\mathrm{H}_{2 j}^{\left( k \right)}= \varepsilon^{j}\left( 1 - \varepsilon \right)^{j}}}\\{{\mathrm{H}_{2 j + 1}^{\left( k \right)}= \varepsilon^{j}\left( 1 - \varepsilon \right)^{j + 1}}}\\ \end{array}\; \forall \; 0 \leq j \leq \frac{k - 1}2
[ K_m^{ij}, K_n^{lp}] = i (K_{m+n}^{ip}\, \delta^{jl}+K_{m+n}^{jl}\, \delta^{ip}-K_{m+n}^{il}\, \delta^{jp}- K_{m+n}^{jp}\,\delta^{il}) +2mk\delta_{pl}^{ij}\delta_{m+n,o}
\mathbf{x}_{\textrm{rls}}^{(t)}\triangleq\underset{\mathbf{x}}{\arg\min}\;\frac{1}{2}\mathbf{x}^{T}\mathbf{G}^{(t)}\mathbf{x}-(\mathbf{b}^{(t)})^{T}\mathbf{x}=\mathbf{G}^{(t)\dagger}\mathbf{b}^{(t)},
i \hbar{\frac{\partial \Phi}{\partial t}}( 1 , \cdots , N ) ={\frac{1}{2 m}}\sum_{p = 1}^{N}\{i \hbar \partial_{i}^{( p )}-{\frac{e}{c}}\sum_{r = 1 ( \ne p )}^{N}{\cal A}_{i}( p , r ) \}^{2}\Phi ( 1 , \cdots , N ) .
\omega_1(s,t) = -\frac{2}{3}t - \frac{2}{3}s + \frac{1}{3}\alpha + \frac{2}{3}\beta + \frac{\gamma}{3}, \omega_2(s,t) = \frac{1}{6}t - \frac{1}{3}s + \frac{1}{6}(\alpha - \beta) - \frac{\gamma}{3}, \omega_3(s,t) = -\frac{1}{2}t + \frac{1}{2}(\alpha - \beta), \omega_4(s,t) = s, \omega_5(s,t) = t.
\left\vert h(x)\right\vert =\left\vert \left( v+\varphi u\right) \left(x\right) \right\vert =\left\vert v(x)\right\vert \leq w(x)=-\int_{\mathbb{R}^{n}}\Gamma_{\overline{a}}(y^{-1}\circ x)f(y)dy\ & =-\int_{B_{R}\setminus B_{3}}\Gamma_{\overline{a}}(y^{-1}\circ x)f(y)dy.
(r s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\mu}s_{\nu^*})(s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\mu}s_{\nu^*})^* =(s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\mu}s_{\nu^*})^*(r s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\mu}s_{\nu^*}).
- \frac 1 2 \Omega^{+ 2 i}e^{- 2}- \frac 1 2 \Omega^{- 2 i}e^{+ 2}- 2 i \pi_{q}^{1 +}\gamma_{q \dot{q}}^{i}\pi_{\dot{q}}^{1 -}- 2 i \pi_{q}^{2 +}\gamma_{q \dot{q}}^{i}\pi_{\dot{q}}^{2 -}= 0
\theta_{0}= \frac{1}{2}i \sum_{k = 1}^{K}L_{k}\ln \left[ \frac{z - z_{k}}{z^{*}- z_{k}^{*}}\right] \ + \frac{1}{4}\sum_{k = 1}^{K}\sum_{n_{k}= 1}^{N_{k}}\left[ \theta_{k , n_{k}}( z ) + \theta_{k , n_{k}}^{*}( z^{*}) \right] .
\P_n(G, \sigma) = \prod_{v \in V(G)}\pi_{\sigma_v}\prod_{(u, v) \in E(G)}\frac{M_{\sigma_u,\sigma_v}}{n}\prod_{(u, v) \not \in E(G)}\left(1 - \frac{M_{\sigma_u,\sigma_v}}{n}\right).
F^0_j(\lambda) = \lambda^{\kappa_j}_1 \ldots \lambda^{\kappa_j}_{n-1}, \S_j(\lambda) = \lambda^{\kappa_j}_{n}, \R_j(\lambda) = \lambda^{\kappa_j \kappa_1^{-1}}_{1}F^1_j(\lambda) = \lambda^{\kappa_j \kappa_1^{-1}}_{2}\ldots \lambda^{\kappa_j \kappa_1^{-1}}_{n \vphantom{2}}\; .
=\frac{\partial}{\partial t}(\Psi(x,t)\Psi^{*}(x,t))-\frac{ih}{2m}\frac{\partial}{\partial x}(\Psi^{*}(x,t)\frac{\partial}{\partial x}\Psi(x,t)-\Psi(x,t)\frac{\partial}{\partial x}\Psi^{*}(x,t))
\left( (A^2)^k Ab \right)^t \! \left((A^2)^\ell b \right) = \left( A^{2k+1}b \right)^t \! \left( A^{2\ell}b \right) = (-1)^{k+\ell+1}\! \left( A^{k+\ell}b \right)^t \!A \left( A^{k+\ell}b \right) = 0
S=\int d^{6}x\left[{\frac 1 6}F_{{\mu}{\nu}{\sigma}}(A)F^{{\mu}{\nu}{\sigma}}(A)+\frac{1}{2({\partial}_{\lambda}a)({\partial}^{\lambda}a)}{\partial}^{\mu}a{\cal F}_{{\mu}{\nu}{\sigma}}(A){\cal F}^{{\nu}{\sigma}{\rho}}(A){\partial}_{\rho}a\right],
q \frac{dT}{dq}=\frac{\theta_{3}^{4}}{16}\frac{\theta_{2}^{4}\theta_{4}^{4}}{\theta_{3}^{8}}\left(1-2\frac{\theta_{2}^{4}}{\theta_{3}^{4}}\right)=\frac{\theta_{2}^{4}\theta_{4}^{4}}{16\theta_{3}^{4}}\left(1-2\frac{\theta_{2}^{4}}{\theta_{3}^{4}}\right).
\Phi_\iota^{-1}\circ\varphi_\iota^{-1}|_{\Phi_\iota\circ\varphi_\iota(U_\iota\cap U_\kappa)}=\Phi_\kappa^{-1}\circ\varphi_\kappa^{-1}|_{\Phi_\kappa\circ\varphi_\kappa(U_\kappa\cap U_\iota)},
A_z(n, \chi_0) = q^n\frac{n^{z-1}}{\Gamma(z)}F_d(1/q,z) + O_{A}(q^n n^{\Re z-2}(1+\log m)^{K_A}) &= \left(\prod_{p|d}\left(1+\frac{z}{q^{\deg p}}\right)^{-1}\right) F(1/q,z) q^n\frac{n^{z-1}}{\Gamma(z)}+ O_{A}(q^n n^{\Re z-2}(1+\log m)^{K_A}).
M_1^\top M_1 =\begin{pmatrix}r_1^\top r_1 r_1^\top r_2 \vdots r_1^\top r_m\end{pmatrix}(r_1^\top r_1,r_2^\top r_1,\ldots,r_m^\top r_1)=\begin{pmatrix}m J_m O \cdots O O m J_m \cdots O \vdots \vdots \ddots \vdots O O \cdots m J_m \end{pmatrix},
R(\kappa) =\sum_{j=0}^{\beta-2}\kappa^jG_j+\mathcal O(\kappa^{\beta-1}); G_j =G_{0,j}-\sum_{\genfrac{}{}{0pt}{}{j_1\ge 0,j_2\ge 0,j_3\ge 0}{j_1+j_2+j_3=j}}G_{0,j_1}vA_{j_2}v^*G_{0,j_3}.
|\hat v(q)-\hat v(q')|=|v(f^\ell z)-v(f^\ell z') \le |v|_{C^\eta}d(f^\ell z,f^\ell z')^\eta\le |v|_{C^\eta}C_1' d(Gz,Gz')^\eta \le |v|_{C^\eta}C_1' d_\theta(Gz,Gz')= |v|_{C^\eta}C_1' \theta^{-1}d_\theta(z,z').
\widehat{H}= \displaystyle \frac{2 \pi}{L}\left( \widehat{L_{0}}+ \widehat{\overline{{{L}}}_{0}}- \displaystyle \frac{c}{1 2}\widehat{I}+ \lambda \displaystyle \frac{L^{2 - h}}{\left( 2 \pi \right)^{1 - h}}\widehat{B}\right) \: ,
\inf\left\{\mathcal{E}_f(V)\ :\V\ge0,\ \int_\Omega V^{-p}\,dx\le 1\right\}=-\frac{1}{2}\sup_{u\in W^{1,2}_0(\Omega)\setminus\{0\}}\frac{\langle f,u\rangle^2}{\displaystyle\int_\Omega|\nabla u|^2\,dx+\left(\int_\Omega|u|^{2p/(p+1)}\right)^{(p+1)/p}}.
Nr^{2}(JO_{n}^{(3)})) =\frac{1}{49}\left(\begin{array}{c}\left(2^{n+1}-2\right)^{2}+\left(2^{n+2}+3\right)^{2}+\left(2^{n+3}-1\right)^{2}+\left(2^{n+4}-2\right)^{2}+\left(2^{n+5}+3\right)^{2}+\left(2^{n+6}-1\right)^{2}+\left(2^{n+7}-2\right)^{2}+\left(2^{n+8}+3\right)^{2}\end{array}\right) &=\frac{1}{49}\left(21845\cdot 2^{2n+2}+2^{n+10}+41\right).
H_Z(z;t) = \sum_{n=0}^{+\infty}p(n;\lambda t)\,H_Y(z;n) =\sum_{n=0}^{+\infty}p(n;\lambda t) \left[{\bf 1}_{\{z\geq 0\}}{\rm e}^{-\mu n}+\sum_{m=1}^{+\infty}p(m;\mu n)\,F^{(m)}_X(z) \right], z\in\mathbb{R}, \;\; t>0.
d+o(1) =\L(\omega_{k})=\left(\frac{1}{p}-\frac{1}{q}\right)\int_{\mathbb{G}}|\omega_{k}(x)|^{q}dx &\geq\left(\frac{1}{p}-\frac{1}{q}\right)\int_{\mathbb{G}}|\psi_{k}(x)|^{q}dx &=\left(\frac{1}{p}-\frac{1}{q}\right)\mu_{k}^{-q}\int_{\mathbb{G}}|\mu_{k}\psi_{k}(x)|^{q}dx+o(1) &=\mu_{k}^{-q}\L(\mu_{k}\psi_{k})+o(1).
M =\begin{pmatrix}m_0 m_{-1}m_{-2}\ldots \ldots m_{-k}m_1 m_0 m_{-1}m_{-2}\ldots m_{-(k-1)}\vdots \ddots \ddots \ddots \ddots \vdots m_l \ldots \ldots \ldots \ldots \cdot\end{pmatrix}
\int_0^{2/|\xi|^{\gamma}}e^{-3i\Phi(\xi,\eta)}\phi(|\xi|^{\gamma}\eta) d\eta = \int_0^{2/|\xi|^{\gamma}}e^{-3i\eta\xi^2}\phi(|\xi|^{\gamma}\eta) + e^{-3i\eta\xi^2}\left(e^{3i\xi\eta^2 - 3i\eta^3/4}-1\right)\phi(|\xi|^{\gamma}\eta) d\eta = O(|\xi|^{-2}) + O\left(\int_0^{|\xi|^{-\gamma}}|\eta|^2|\xi|d\eta\right) = O(|\xi|^{1-3\gamma})
\begin{aligned}|O*\beta(\mathcal{A}K)| = |O*b(\mathcal{A}K)|, |O*\gamma(\mathcal{A}K)| = |O*c(\mathcal{A}K)|, |(\mathcal{A}K)_{+++}|+|(\mathcal{A}K)_{-++}| = |(\mathcal{A}K)_{--+}|+|(\mathcal{A}K)_{+-+}|,\end{aligned}
\ln P(z,e^{2\pi i\frac{h}{k}-w}) =\frac{1}{k}\sum_{r=1}^{k}z^r\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(s) \Phi(z^k,s+1, \frac{r}{k}) D_{rh,k}(s)( k w)^{-s}ds :=\frac{1}{k}\sum_{r=1}^{k}z^r\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}J(s;z,r,h,k, w)ds.
\chi'_k(r) \leq O(\rho_{k}^{-\frac{1-\delta}{2}}) \chi'((\log \rho_k)^{-3/2}) = O(\rho_{k}^{-\frac{1-\delta}{2}}) (\log\rho_k)^{3}e^{-(\log \rho_k)^{3/2}}.
\left[\cos(\kappa_j x) + \frac{\xi}{\kappa_j}\sin(\kappa_j |x|)\right] e^{-i\kappa_j t}= \frac{\xi L}{2}\left\{\frac{1-i\kappa_j t}{Z_j^2}+ 2\sum_{n=1}^\infty \frac{\cos(k_n x)}{Z_j^2-(\pi n)^2}\left[ \cos(k_n t) - i \frac{\kappa_j}{k_n}\sin(k_n t)\right]\right\}
\Gamma_{k}[ \phi ] = - W_{k}[ J , \varphi_{0}^{k}] + \int J ( \varphi_{0}^{k}+ \phi_{k}^{\Lambda}) = -{\mathcal W}_{k}[ J_{k}^{\Lambda}, \varphi_{0}^{k}] + \int J_{k}^{\Lambda}\phi_{k}^{\Lambda},
\frac{d}{dt}\mathfrak{E}(\theta)=\int_{\mathbb{T}^n}\theta_{t}\log \theta dx=\int_{\mathbb{T}^n}((1-\delta)u+\delta\mathcal{R}\theta)\cdot \nabla\theta dx=-\delta\left\|\Lambda^{1/2}\theta\right\|_{L^2}^2.
E_x[e^{uZ_t}]= \big(1-\frac{\sigma^2}{2a}u(1-e^{-at})\big)^{-\frac{2a\theta}{\sigma^2}}\cdot \exp \Big (\frac{xue^{-at}}{1-\frac{\sigma^2}{2a}u(1-e^{-at})}\Big) .
R(\Lambda,Q) = -\left( 1 - e^{-\Lambda \frac{Q}{2}}\right) \log_2 \left( 1-e^{-\Lambda \frac{Q}{2}}\right) - \frac{e^{-\Lambda \frac{Q}{2}}}{\log(2)}\left[\log\left( \frac{1-e^{-\Lambda Q}}{2}\right) +\frac{\Lambda Q}{2}- \frac{\Lambda Q}{\left( 1-e^{-\Lambda Q}\right)}\right]
\mathcal{Z}_{m,n}^\gamma (z,\bar z) = (-1)^{m}(m\wedge n )!(\gamma+m+1)_n |z|^{|m-n|}e^{i[(n-m)\arg z]}\mathrm{P}^{(|m-n|,\gamma)}_{m\wedge n}(1-2 |z|^2).
X_2X_1 =q^{-2}X_1X_2, X_3X_1{}=X_1X_3-(q+q^{-1})X_2, X_3X_2=q^{-2}X_2X_3, X_4X_1{}=q^2X_1X_4-q^2X_3, X_4X_2=X_2X_4-\frac{q^2-1}{q+q^{-1}}X_3^2, ~~ X_4X_3=q^{-2}X_3X_4.
A^{S_1G_1S_2}(z)=\frac{1}{z^{4/3}}\begin{pmatrix}0 1 0 0 0 1 -t 0 0 \end{pmatrix}\hphantom{A^{S_1G_1S_2}(z)=}{}+\frac{1}{z}\begin{pmatrix}-p_1q_1-\theta^0_1-\frac{2}{3}0 0 0 p_2q_2-\frac{1}{3}0 0 0 p_1q_1-p_2q_2-\theta^0_2-1 \end{pmatrix}+\cdots.
\left[ \begin{array}{c}T_{n+2}(x) T_{n+1}(x) T_{n}(x)\end{array}\right]=\left[ \begin{array}{ccc}x^{2}x 1 1 0 0 0 1 0\end{array}\right]^{n}\left[ \begin{array}{c}T_{2}(x) T_{1}(x) T_{0}(x)\end{array}\right]
M_v (s,x_0) :={1 \over (1 + s)^2}\chi_{|\tau| \le 5}\Vert \partial \phi \Vert_{H^{11}(\Sigma_0)}+ \sup_{|\alpha| \le 10}\sup_{\psi_1}\int_0^s \int_{\Sigma_t}|\partial^\alpha (\Box \phi)| |(\partial_t + \partial_x) \psi| d x d t.
\mathcal{J}(y(v),v) \geq \frac{1}{2}\|\eta-y_d\|^2 + \frac{\lambda}{2}\|v-u_d\|^2 + \int_\Omega\left(\nabla y - \nabla \eta\right) \cdot \nu \nabla{p_\eta}\, d\boldsymbol{x}= \frac{1}{2}\|\eta-y_d\|^2 + \frac{\lambda}{2}\|v-u_d\|^2 + \int_\Omega\left(\nu \nabla y - \nu \nabla \eta\right) \cdot \nabla{p_\eta}\, d\boldsymbol{x}.
H(\Delta Y^{(n)}_{2}(\Delta_n)|M_2) =\sum_{i=1}^n h(Y^{(n)}_{2}(t_{n, i})-Y^{(n)}_{2}(t_{n, i-1})|Y_{2, t_{n, 0}}^{(n), t_{n, i-1}}, M_2) & \geq \sum_{i=1}^n h(\sqrt{N_2}B_{2}(t_{n, i})-\sqrt{N_2}B_{2}(t_{n, i-1})) &=\sum_{i=1}^n \log (2 \pi e N_2 \delta_n),
\partial_{\mu}^{z}\Delta V_{i}^{\mu}( z , u , v ) = - \frac{i g_{1}}{2}[ \delta^{4}( z - u ) - \delta^{4}( z - v ) ] ( Y_{R}^{i}L + Y_{L}^{i}R ) \Delta \Sigma^{i}( u - v ) \, ,
\|\tilde g\|_{W^{k, 1}(\Gamma(\delta) \cap A_j)}\le \begin{cases}C(\delta d_j^{N-1}) d_j^{2-m-k-N}(m+k+N>2), C(\delta d_j^{N-1}) |\log d_j| (N=2, m=k=0).\end{cases}
\begin{array}{cc}h_1(r)= 5 \textup{in}\ [2M,R_1], h_1(r)= 1 \textup{in}\ [2R_1,\infty), |h_1'(r)|\leq \frac{5}{\log 2}r^{-1}\textup{in}\ [R_1,2R_1].\end{array}
H_{-}= A^{\dagger}(\omega)A(\omega) = |x|^{\frac{i\omega}{2}}O^{\dagger}O\,|x|^{-\frac{i\omega}{2}}, H_{+}= A(\omega) A^{\dagger}(\omega) = |x|^{-\frac{i\omega}{2}}OO^{\dagger}\,|x|^{\frac{i\omega}{2}}.
\mathcal{R}_{n,\alpha}(\mathcal{F}_\beta, \Phi \circ \rho, \varepsilon) = \inf_{Q \in \mathcal{Q}_\alpha}\inf_{\tilde{f}}\sup_{P_{f_\varepsilon}\in \mathcal{P}_\varepsilon (\mathcal{F}_\beta)}\mathbb{E}_{P_{f_\varepsilon},Q}\Big\{\Phi \circ \rho (\tilde{f},f)\Big\},
-\hbar \sum_{r \geq 0}X_{i,r}^+ u^{-r-1}= A_i(u - \dfrac{1}{2}\hbar(i-1))^{-1}B_i(u - \dfrac{1}{2}\hbar(i-1)), -\hbar \sum_{r \geq 0}X_{i,r}^- u^{-r-1}= C_i(u - \dfrac{1}{2}\hbar(i-1)) A_i(u - \dfrac{1}{2}\hbar(i-1))^{-1}.
\frac{3}{4}\Lambda_{k+m+1}\le\frac{1}{4}{\textstyle\sum_{j=k}^{k+m}}\,\Xi_j+{bc}a^{-1}\big[bc(1-\theta)\big]^{\frac{1-\theta}{\theta}}{\textstyle\sum_{j=k}^{k+m}}\,\Xi_j^{\frac{1-\theta}{\theta}}.
A_{j+1}\leq c_0^j \left[ \omega(r) \right]^{(2-p)(1-1/\kappa) + (p-1)(2-1/\kappa) - p - 2 (1-1/\kappa)}A_j^{2-1/\kappa}= c_0^j \left[ \omega(r) \right]^{-(2-1/\kappa)}A_j^{2-1/\kappa}\,,
r_{n,p}={\pi^{-\frac{n}{2 p}}\left(\frac{2 \Gamma \left(\frac{1}{2}(n+p+2)\right)}{\Gamma \left(\frac{n+2}{2}\right) \Gamma \left(\frac{p+1}{2}\right)}\right)^{-\frac{n}{p}}}.
\frac{d}{dr}\log H_\infty(x_0,r) = \frac{2N_\infty(x_0,r)}{r}\le \frac{C}{r}\Longrightarrow H_\infty(x_0,r_2) \le H_\infty(x_0,r_1) \left(\frac{r_2}{r_1}\right)^{2C}
A=\mathfrak{G}(X,x_0)=\{\alpha\in G:\alpha x_0=x_0\}, \ \ F = \mathfrak{G}(Z,z_0)=\mathfrak{G}(Z^*,z^*_0) = \{\alpha\in G:\alpha z_0=z_0\}=\{\alpha\in G:\alpha z^*_0=z^*_0\}.
H^s(\Psi(Z(s(\lambda)))) = \rho(\Lambda)^{-(m,\ldots, m) + d(\lambda)}\sum_{\nu \in C_\lambda}H^s (\Psi(Z(s(\nu)))) &= \rho(\Lambda)^{-(m,\ldots, m) + d(\lambda)}\rho(\Lambda)^{(m,\ldots, m)}H^s(\Psi(Z(\lambda))) &= \rho(\Lambda)^{d(\lambda)}H^s(\Psi(Z(\lambda))).
\hat{H}_n(k_x, k_y) = \sin n k_x \sigma_x + \sin k_y \sigma_y + (u + \cos n k_x + \cos k_y)\sigma_z &=\left(\begin{array}{cc}u + \cos n k_x + \cos k_y \sin n k_x - i \sin k_y \sin n k_x + i \sin k_y - u - \cos n k_x - \cos k_y\end{array}\right),
\mathcal{L}(\varepsilon ; t;{\mathbf F}) := \int_0^t e^{-c_{*}(t-\widetilde{t})/2}(\varepsilon^2 +t-\widetilde{t})^{-1/2}\Vert \mathbf{F}(\cdot ,\widetilde{t})\Vert_{L_2(\mathcal{O})}\,d\widetilde{t},
X[i] :=X_1\wedge \cdots \wedge \widehat{X_i}\wedge \cdots \wedge X_{k+1}X[i,j] :=X_1\wedge \cdots \wedge \widehat{X_i}\wedge \cdots\wedge\widehat{X_j}\wedge \cdots \wedge X_{k+1}.
\|\tilde{V}^j_n\|^2_{L^2}= \lambda_j^2 \|V^j(\cdot- x^j_n)\|^2_{L^2}= M, \|\tilde{v}^l_n\|^2_{L^2}= (\lambda^l_n)^2 \|v^l_n\|^2_{L^2}= M.
\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}\overline{C}(m,n)z^{m}q^{n}=\dfrac{(-q;q^{2})_{\infty}(-zq^{2};q^{2})_{\infty}(-z^{-1}q^{2};q^{2})_{\infty}}{(q;q^{2})_{\infty}(zq^{2};q^{2})_{\infty}(z^{-1}q^{2};q^{2})_{\infty}}.
\mathcal{H}(h)(\boldsymbol{\lambda}_{1}, \boldsymbol{\lambda}_{2})=\int_{\mathbb{R}^{2d}}K\left(\sum_{i=1}^{4}\boldsymbol{\lambda}_{i},\mathcal{D}\right) h\left(\boldsymbol{\lambda}_{3}, \boldsymbol{\lambda}_{4}\right)G_{\alpha}(d\boldsymbol{\lambda}_{3}) G_{\alpha}(d\boldsymbol{\lambda}_{4}).
\mathfrak{E}_{q_1,q_2,q_2'}=\sum_{\alpha\bmod{q_1q_2q_2'}}K_{q_1}(\alpha\ell\bar{q}_2^3)K_{q_2}(\alpha\ell\bar{q}_1^3) K_{q_1}(-\alpha\ell\bar{q_2'}^3)K_{q_2'}(-\alpha\ell\bar{q}_1^3) \times \sum_{n\in \mathbb{Z}}W\left(\frac{\alpha+nq_1q_2q_2'}{N}\right).
\begin{array}{lll}\lambda_2(\lambda_2-\lambda_3)b_3+2\lambda_2(\lambda_2-\lambda_3)a_3+\lambda_2b_3a_2-\lambda_2b_3a_3-\lambda_2a_3^2+\lambda_2a_2a_3\ & \ \ =(1-\lambda_3)(\lambda_2-\lambda_3),\end{array}
\mathbf{d}_{1}=(0,\ldots,0,z-a_{0}^{(0)}) \mathbf{d}_{2}=(0,\ldots,0,-a_{0}^{(1)},z-a_{1}^{(0)}) \mathbf{d}_{3}=(0,\ldots,0,-a_{0}^{(2)},-a_{1}^{(1)},z-a_{2}^{(0)})
\lim_{N\to\infty}\left \|\sum\limits_{j=N}^{\infty}\frac{\lambda_j \chi_{Q_j}}{\omega(Q_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}=0,\ \lim_{N\to\infty}\left \|\sum\limits_{j=N}^{\infty}\frac{\mu_j \chi_{P_j}}{\omega(P_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}=0.
\textbf{y}_{r_{p,q}d,k}(2i)= \textbf{h}_{r_qd}^k \hat{b}^*_{r_pd,k}(2i-1)-\textbf{h}_{r_pd}^k \hat{b}^*_{r_qd,k}(2i)+\textbf{n}(2i).
\delta S = \int_{M}d^{2}X \mathrm{}\left( \frac{\partial{\cal L}}{\partial \Lambda_{A}}\mathrm{}\delta \Lambda_{A}+ \frac{\partial{\cal L}}{\partial \partial_{\alpha}\Lambda_{A}}\mathrm{}\delta \left( \partial_{\alpha}\Lambda_{A}\right) \right) \mathrm{.}
L_0(J)-L_0(I)=-|\omega_k|, &L_{1}^{2}(J)-L_{1}^{2}(I)\leq |K_{b0}||\omega_{k}|+\sum\limits_{i=2,3,5}|K_{bi}||\alpha_i|, &L_{i}^{2}(J)-L_{i}^{2}(I)=-|\alpha_i|,i=2,3,4,5. &Q(J)-Q(I)\leq (|K_{b0}||\omega_{k}|+\sum\limits_{i=2,3,5}|K_{bi}||\alpha_i|)L(I_0).
{\cal L}_{M}= - \frac{1}{4}G_{\mu \nu}^{a}G^{\mu \nu , a}-{\frac{1}{2}}\kappa_{1}^{2}D_{\mu}\phi^{\tilde{a}}D^{\mu}\phi^{\tilde{a}}-{\frac{1}{2}}\kappa_{2}^{4}D_{[ \mu}\phi^{\tilde{a}}D_{\nu ]}\phi^{\tilde{b}}D^{[ \mu}\phi^{\tilde{a}}D^{\nu ]}\phi^{\tilde{b}}- V ( \phi^{4}) \,
\Psi_{( 3 , 3 , 2 )}= \prod_{i < j}( z_{i}^{( 1 )}- z_{j}^{( 1 )})^{3}\prod_{i < j}( z_{i}^{( 2 )}- z_{j}^{( 2 )})^{3}\prod_{i , j}( z_{i}^{( 1 )}- z_{j}^{( 2 )})^{2}\Psi_{0}.
\mathcal{F}\big[R_0f\big](\xi)=\frac{1}{2\pi}\int_{\mathbb{R}}(\xi-\eta_1)\big[\,\mathrm{sgn}(\xi)-\,\mathrm{sgn}(\xi-\eta_1)\big]\widehat{f}(\xi-\eta_1)\widehat{h}(\eta_1)\, d\eta_1.
\mathbb{T}^{(i)}\left(\mathcal{A}_{j_1\dots j_n}\right)_{j_1,\dots, j_n=1}^{|\mathcal{E}|}=\left(\mathcal{A}_{j_1\dots j_n}\right)_{j_1,\dots,j_{i-1},j_{i+1},j_{i},j_{i+2},\dots,j_n=1}^{|\mathcal{E}|}
\bar{B_2}=\frac{b_{0,\sup}(a_{1,\inf}-\frac{k\chi_1}{d_3})+\frac{k\chi_2}{d_3}a_{0,\sup}}{(a_{1,\inf}-\frac{k\chi_1}{d_3})(b_{2,\inf}-\frac{l\chi_2}{d_3})-\frac{lk\chi_1\chi_2}{d_3^2}}.
\tilde{m}_{0}^{\ast}:=K^{\ast -1}m_{0}K^{\ast}=\left( \begin{array}{cc}0 -\kappa_{e}^{\frac{1}{2}}T\kappa_{o}^{-{\frac{1}{2}}}-\kappa_{o}^{-{\frac{1}{2}}}\bar{T}\kappa_{e}^{\frac{1}{2}}0\end{array}\right) =-\tilde{m}_{0}\,.
\mu^\ast(\pi\rtimes\sigma) = M^\ast(\pi)\rtimes\mu^\ast(\sigma) \intertext{and}\mu^\ast(\delta([\nu^a\rho,\nu^b\rho])\rtimes\sigma) =\!\!\sum_{i=a-1}^b \sum_{j=i}^b \sum_{\tau,\sigma'}\delta([\nu^{-i}\tilde\rho,\nu^{-a}\tilde\rho])\times\delta([\nu^{j+1}\rho,\nu^b\rho])\times\tau\otimes{}&\mkern140mu \otimes \delta([\nu^{i+1}\rho,\nu^j\rho])\rtimes\sigma'.
\widehat{R}_j^i (s) = \left| \int_0^s \left( h_j^i (z) - h^i(z) \right) g^i\left(\frac{1}{N}\sum_{n=1}^{N}\delta_{X^n (z)}\right) \, dz \right| &+ M \int_0^T \left| g^i_j\left( \frac{1}{N}\sum_{n=1}^{N}\delta_{X^n (z)}\right)- g^i\left( \frac{1}{N}\sum_{n=1}^{N}\delta_{X^n (z)}\right)\right| \, dz.
\omega_{+}^{- Y^{\prime}}( x , \theta^{+}, u ) = \tau_{+}^{- Y^{\prime}}( x , u ) + \theta_{+}^{+ A^{\prime}}l_{A^{\prime}}^{-- Y^{\prime}}( x , u ) + i ( \theta_{+}^{+})^{2}\mu_{-}^{- 3 Y^{\prime}}( x , u ) \; .
\frac{w}{z}=\frac{w\overline{z}}{|z|^{2}}=\frac{(u+vi)(x-iy)}{x^{2}+y^{2}}=\frac{ux+vy}{x^{2}+y^{2}}+\frac{vx-uy}{x^{2}+y^{2}}i.
Tr(p_o) tr(\xi\xi^*) = Tr(\xi\xi^*p_o)=Tr(\xi (p_o\xi)^*)=\sum_{k,l\geqslant 0}Tr(\xi(x^k_l)^*) & = \sum_{k,l\geqslant 0}Tr(\xi (y^k_l)^*p_o) p_oy^k_l=x^k_l & = Tr(p_o) \sum_{k,l\geqslant 0}tr(\xi (y^k_l)^*)=0.
[\Delta^d_\perp]^{-1}(x_\perp) = - \int \frac{d^dk_\perp}{(2\pi)^d}\frac{e^{i k_\perp \cdot x_\perp}}{k_\perp^2}= -\frac{1}{2^{2-d//2}}\frac{1}{(2 \pi)^{d/2}}\frac{1}{(x_\perp)^{d/2-1}}\Gamma(d/2 -1)
I_{n-2}(\lambda, 2\lambda, 2\lambda) = \int\limits_{0<x_2<\cdots<x_{n-1}<1}\prod_{i<j}^n |x_i-x_j|^{\lambda 2^{j-i}}\Big\vert_{x_1=0, x_n=1}dx, J_{n-2}(4\lambda, 2\lambda, 2\lambda) = \int\limits_{0<x_2<\cdots<x_{n-1}<1}\prod_{i<j}^n |x_i-x_j|^{\lambda 2^{n-(j-i)}}\Big\vert_{x_1=0, x_n=1}dx .
\vec{\sigma}\cdot\vec{p}\,\,\tilde\xi_{\lambda}(\vec{p})=2p\lambda\,\tilde\xi_{\lambda}(\vec{p}) \,, \quad \vec{\sigma}\cdot\vec{p}\,\,\tilde\eta_{\lambda}(\vec{p})=-2p\lambda\,\tilde\eta_{\lambda}(\vec{p})\,.
T^{-1}A T = \begin{pmatrix}A_1 0 A_6 0 A_2 A_3 A_4 A_5 0 0 A_7 0 0 0 A_8 A_9 \end{pmatrix}, T^{-1}B = \begin{pmatrix}B_1 B_2 0 0 \end{pmatrix}, C T = \begin{pmatrix}C_1 0 C_2 0 \end{pmatrix},
\sum_{n=0}^{|c|-1}e^{\pi i\frac{an^{2}+bn}{c}}=|\frac{c}{a}|^{\frac{1}{2}}e^{\pi i\frac{|ac|-b^{2}}{4ac}}\sum_{n=0}^{|a|-1}e^{-\pi i\frac{cn^{2}+bn}{a}}
\langle \overline{{{n_{m}, \ldots , n_{1}}}}| = \, ( - 1 )^{\sum_{a}\bar{n}_{a}( a - 1 )}\langle \bar{n}_{m}, \ldots , \bar{n}_{1}| = \langle \bar{0}| ( \hat{\eta}^{m})^{n_{m}}\cdots ( \hat{\eta}^{1})^{n_{1}}
\lbrack K]_{q\bar{q}}=\left( \begin{array}{ll}q\bar{q}+q^{-1}\bar{q}^{-1}-1 -(1+\bar{q}^{2}+\bar{q}^{-2}) q\bar{q}^{3}+q^{-1}\bar{q}^{-3}\end{array}\right)
\nabla_{X_2^v}Y_2^{v}=(\nabla{\hskip -0.2cm^{^{^{2}}}}_{X_2}Y_2)^v+f_{_{1}}^hB{\hskip -0.1cm_{_{f\!_{_{2}}}}}\!\!(X_2,Y_2)^hgrad(f_{_{1}}^h)
\bigg(\sum_{l=0}^{\infty}\|T_kf_j\|^t_{L^r(K)}\bigg)^{1/t}\leq Cq^{-k}\|\Omega\|_{H^1(\mathfrak{D}^*)}\bigg(\sum_{l=0}^{\infty}\|f_j\|^t_{L^r(K)}\bigg)^{1/t}.
\sum_{j=1}^{u_k-1}|I_{n_k}[a_1, \cdots , a_{n_k-1}, j]| \asymp \left(\sum_{j=1}^{u_k-1}|I_1(j)| \right) \cdot I_{n_k-1}[a_1 , \cdots , a_{n_k-1}],
(***) \leq \frac{A_{2}}{n^{(p+2)\alpha}}+ 16 \binom{p+1}{2}\left( 1+c_{\gamma}C\right)\gamma_{n}^{2}\mathbb{E}\left[ \left\| Z_{n}-m \right\|^{2p}\right] + 32\binom{p+1}{3}\left( 1+c_{\gamma}C\right)^{2p-4}\gamma_{n}^{3}\mathbb{E}\left[ \left\| Z_{n}-m \right\|^{2p}\right].
\sigma_{\mathrm{t o t}i}^{( 1 )}= \frac{( 2 \pi )^{4}\hbar^{2}}{v_{i}}\frac{1}{\pi}\mathrm{I m}R_{i i}^{( 1 )}\exp \left[ - \frac{1}{\hbar}\int_{- \infty}^{+ \infty}d t \Gamma^{( 1 )}( t ) \right] .
\begin{aligned}1 -3x_2 + (-3x_1)(-3x_2 + 1) + (-3x_1 + 1) (-x_2^3 + 3x_2^2) + (-x_2^3 + 3x_2^2) \geq &(-0.75x_1 + 0.5)\bigl((1-x_2)^3+0.75x_2 \bigr) + \bigl((1-x_1)^3+ 0.75x_1-0.5\bigr)0.5 & + (1-x_1)^3(-0.75x_2).\end{aligned}
\rho_1\sharp \rho_2(\xi)=\sum_{|\alpha|<N}\frac{(-i)^{|\alpha|}}{\alpha !}J\left[(s^\alpha \rho_1)\natural 1(\xi;\cdot,\cdot)\delta^\alpha \rho_2(\xi)\right]+\sum_{|\alpha|=N}(s^\alpha \rho_1)\sharp_0\widetilde{r}_{N\alpha}(\xi) .
\mathbb{R}_{12}(u_1,u_2|v_1,v_2)\,\mathrm{L}_1(u_1,u_2)\,\mathrm{L}_2(v_1,v_2)= &=\mathrm{L}_2(v_1,v_2)\,\mathrm{L}_1(u_1,u_2)\,\mathbb{R}_{12}(u_1,u_2|v_1,v_2),
Tr\left(\mathcal{M}\,\mathcal{F}_A\right) =\,Tr\left(\left(\frac{\gamma_i y^i}{|g(y,y)|^{1/2}}+\mathbb{I}_4\right)\cdot \left( \gamma_j y^j-\,A_j\, y^j\,\mathbb{I}_4\right) \right) & =\,4\,\left(\frac{\eta_{ij}y^iy^j}{|g(y,y)|^{1/2}}+ A_j\,y^j\right) & =\,4\,\left(\mp|g(y,y)|^{1/2}+\,A_j\,y^j\right),