harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal W}( q , p ) = - \frac{1}{2^{1 1}\pi^{4}}\left[ \frac{1}{\epsilon^{2}}+ \frac{1}{\epsilon}\left( 2 \log \frac{4 \pi \mu^{2}}{- ( p + q )^{2}}+ 8 - \frac{1 1}{2}\gamma \right) + \mathrm{f i n i t e ~ p a r t}\right] , \,
g^{(n)}(0) = \sum\limits_{m=0}^n \binom{n}{m}B^{(f)}_{n-m}(q)\frac{d^{m+r}}{dt^{m+r}}|_{t=0}\bigl[e^{-b_0 t}\prod\limits_{j=1}^{M-1}(1-e^{-b_j t})\bigr], & = (-1)^r \sum\limits_{p=0}^{M-1}(-1)^p\sum\limits_{k_1<\cdots<k_p=1}^{M-1}\bigl(b_0+\sum b_{k_j}\bigr)^r \,B^{(f)}_n\bigl(q+b_0+\sum b_{k_j}\bigr).
\tau_{H}= -m\left\{\cot\langle \alpha_{1}, H\rangle +\cot\langle 2\alpha_{1}, H\rangle+3 \frac{\cot \langle \alpha_{1}, H\rangle \cot \langle 2\alpha_{1}, H\rangle -1}{\cot \langle \alpha_{1}, H\rangle +\cot \langle 2\alpha_{1}, H\rangle}\right\}(2\alpha_{1}+\alpha_{2}).
{\cal H}_{\mu \nu}= c_{1}F_{\mu \nu}+ c_{2}\tilde{F}_{\mu \nu}+{\frac{c_{3}}{2}}\epsilon_{\mu \nu \alpha \beta}F_{\alpha \beta}+{\frac{c_{4}}{2}}\epsilon_{\mu \nu \alpha \beta}\tilde{F}_{\alpha \beta}\, ;
-[M^k,\,\mathcal{L}] f =\Big(M^k\Gamma(\sqrt\mu, f)-\Gamma(\sqrt\mu, M^kf)\Big)+\Big(M^k\Gamma(f, \sqrt\mu)-\Gamma(M^kf, \sqrt\mu)\Big) &=\sum_{1\leq j\leq 6}\Big(M^kL_j(\sqrt\mu, f)-L_j(\sqrt\mu, M^kf)\Big)+\sum_{1\leq j\leq 6}\Big(M^kL_j(f, \sqrt\mu)-L_j(M^kf, \sqrt\mu)\Big)
\mathcal{Z}_{m,n}^{\gamma}(z,\bar{z}) = (\gamma+1)_{m+n}\overline{z}^m z^n{_2F_1}\left( \begin{array}{c}-m , -n \gamma+1 \end{array}\bigg | 1-\frac 1{|z|^2}\right)=(\gamma+1)_{m+n}\overline{P_{m,n}^{\gamma}(z, \bar z)}
\alpha^3 =\smashoperator[r]{\sum_{k\,\equiv\, 0,-1}}(-1)^k(2k+1)X^{\frac{k^2+k}{2}}+\sum_{\mathclap{k\,\equiv\, 1,-2}}(-1)^k(2k+1)X^{\frac{k^2+k}{2}}+\sum_{\mathclap{k\,\equiv\, 2}}(-1)^k(\underbrace{2k+1}_{\mathclap{\equiv\,0}})X^{\frac{k^2+k}{2}}&\equiv\alpha_0+\alpha_1\pmod{5},
L_2^1=\frac{1}{4}(A_1Kx_1+B_1Kx_2+C_1K), &R_0^1=\frac{1}{4}(A_1x_1\sum_{l+k\leq 2}M_{lk}x_1^{l}x_2^k+B_1x_2\sum_{l+k\leq 2}M_{lk}x_1^{l}x_2^k+C_1\sum_{l+k\leq 2}M_{lk}x_1^{l}x_2^k &\quad-(A_3x_1+B_3x_2+C_3)), &A_0^1=-\frac{E}{4}, B_0^1=-\frac{J}{4}, D_0^1=-\frac{K}{4},
S(\bar{\gamma}_n)\cdot\bar{\gamma}_{n-1}/\gamma_n-\bar{\gamma}_n\cdot S(\bar{\gamma}_{n-1}/\gamma_n)=-\bar{\gamma}_n\cdot\bar{\gamma}_{n-1}/\gamma_n+\bar{\gamma}_n\cdot\bar{\gamma}_{n-1}/\gamma_n=0.
\frac{|z_{n+1}|}{|u_{n+1}|^N}=\frac{|z_n|}{|u_n|^N}\left|1-z_n^ru_n^m+z_n^ru_n^mO\left(u_n,z_nu_n^{-1}\right)\right| &=\frac{|z_n|}{|u_n|^N}\left|1-z_n^ru_n^m+z_n^ru_n^mO(u_n)\right|.
\Vert\tilde{y}- y \Vert^2_{L^2(U)}\le \sum\nolimits^4_{j=1}C\Vert y - (\bar{R}_j\,\cdot + \bar{c}_j)\Vert^2_{L^2(U)}\le C C_m^2 \lambda^2 (1+ \vartheta) (\gamma + \delta_4 + \epsilon \Vert W \Vert_*).
\begin{array}{l}{\{| H_{1 3}( H_{2 2}- \lambda_{j}) - H_{2 3}H_{1 2}|^{2}+}\\{| H_{2 3}( H_{1 1}- \lambda_{j}) - H_{1 3}H_{2 1}|^{2}+}\\{\left[ | H_{1 2}|^{2}- ( H_{1 1}- \lambda_{j}) ( H_{2 2}- \lambda_{j}) \right]^{2}\}^{- 1}\times}\\ \end{array}
-128(n-3)(n+2)(2n+1)(2n+3)^2(2n+5)(63n^3+390n^2+785n+506)Y_{n}& +16(n+1)(2n+3)(2n+5) (63n^6-933n^5-7645n^4-17421n^3-13730n^2-2538n &-252)Y_{n+1}+4(n+1)(n+2)(2n+1)(2n+5)(504n^5+2805n^4+5464n^3+4575n^2 &+1400n-60)Y_{n+2}-(n+1)(n+2)(n+3)^2(2n+1) (2n+3) (63n^3+201n^2+194n &+48) Y_{n+3}=0,
M^{\leq 1}= \mathrm{span}(u_1^1, v_1^1, w_1^1), M^{\leq 2}= M, M^1 \cong M^{\leq 1}, M^2 = M^{\leq 2}/M^{\leq 1}\cong \mathrm{span}(\overline{v_1^2 + v_2^1}).
\begin{cases}c_{p,q}^j=0, \forall p\in \{1,\cdots, j-1\}, q\in \{p,\cdots,j-1\}c_{p,q}^j=0, \forall p\in \{1,\cdots,j-1\}, q\in \{j+1,\cdots,n\}c_{p,q}^j=0, \forall p\in \{j+1,\cdots, n\}, q\in\{p,\cdots,n\}\end{cases}
{Rb_{n}(R)\over\sqrt{\alpha}r}M_{\lambda+\nu/2,\ |\nu|/2}(\alpha r^{2}/R^{2})\mathop{\longrightarrow}\limits_{R\rightarrow\infty}{(kr/2)^{|\nu|}\over\Gamma(1+|\nu|)}e^{i(n-|\nu|/2)\pi}{}_{0}F_{1}(1+|\nu|;-(kr/2)^{2})\ .
\mathbb{E}_{T}\sup_{g\in S(r)}\Big[\Big(\frac{1}{n_r}\sum_{i\in A_{r k}}\sum_{\ell=1}^{n/m}\big(f_0(T_{\ell}^{(i)})g(T^{(i)}_\ell)-\mathbb{E}_{T}f_0(T_{\ell}^{(i)})g(T^{(i)}_\ell)\big)\Big)^2- c_1 2^r/n_r \Big]_+ &\quad\leq c_2\frac{1}{n_r}e^{-c_3 2^{r}}+c_4\frac{2^{r}}{n_r^2}e^{-\sqrt{n_r}}\lesssim \frac{1}{n_r}e^{-(c_3 2^{r}\wedge \sqrt{n_r})}.
a^n_m = \frac{c^n_m(r) \hat y_n'(r) - d^n_m(r) \hat y_n(r)}{\hat j_n(r) \hat y_n'(r) - \hat j_n'(r) \hat y_n(r)}\mbox{and}b^n_m = \frac{c^n_m(r) \hat j_n'(r) - d^n_m(r) \hat j_n(r)}{\hat y_n(r) \hat j_n'(r) - \hat y_n'(r) \hat j_n(r)}.
(\sigma_{f(1)}-\sigma_{f(2)})^{2}+(\sigma_{f(2)}-\sigma_{f(3)})^{2}+\cdots+(\sigma_{f(n)}-\sigma_{f(n+1)})^{2}+(\sigma_{f(n+1)}-\sigma_1)^{2}=4\sum\limits_{i\in I}\mu_{f(i)}\sigma_{f(i)}.
{W}\,(\varphi_1, \varphi_2) =\sqrt{2}\left[ \frac{m^2}{\lambda}\,\varphi_1-\frac{\lambda}{6}\,\varphi_1^3 +\frac{\lambda}{2}\,\varphi_1\varphi_2^2+\frac{\mu}{\sqrt{2}}\,\varphi_2^2\right]\,,
T(r)\geqslant \sum_{i=s,t,l}(N(r,\min\{\nu_{(f^u,H_i),\leqslant k_i};1\leqslant u\leqslant 3\})-N^{(1)}_{(f,H_i),\leqslant k_i}(r)) &+ 2\sum_{i=1}^qN^{(1)}_{(f,H_i),\leqslant k_i}(r)+o(T(r)).
\sigma_\infty(N,w) =\lim_{\epsilon\to 0^+}(2\epsilon)^{-1}\int_{|N(x)-1|\le \epsilon}w(x)\, dx &=\lim_{\epsilon\to 0^+}(2\epsilon)^{-1}\int_{1-\epsilon}^{1+\epsilon}\left(\int_{N^{-1}(y)}w\, d\tau^{(y)}_\infty\right)\, dy=\int_{N^{-1}(1)}w\, d\tau^{(1)}_\infty,
T^X_{\alpha \beta}= \frac{\Omega^{\delta}}{2}( (\partial_{\alpha}- ie A_{\alpha}) X (\partial_{\beta}+ ie A_{\beta}) X^* - \frac{1}{2}g_{\alpha \beta}g^{\mu \nu}( \partial_{\mu}- ieA_{\mu}) X (\partial_{\nu}+ ieA_{\nu}) X^*),
(\widetilde{x}_1(t),\ldots,\widetilde{x}_n(t))^\intercal=T(x_1(t),\ldots,x_n(t))^\intercal, &(\widetilde{\mu}_1(t),\ldots,\widetilde{\mu}_n(t))^\intercal=T(\mu_1(t),\ldots,\mu_n(t))^\intercal, &(\widetilde{\alpha}_1,\ldots,\widetilde{\alpha}_n)^\intercal=T(\alpha_1,\ldots,\alpha_n)^\intercal.
\begin{pmatrix}\alpha_1^{2}+ \alpha_{2}\alpha_{3}\alpha_{1}\alpha_2 + \alpha_{2}\alpha_{4}0 \alpha_1\alpha_3+\alpha_3 \alpha_4 \alpha_{2}\alpha_{3}+\alpha_4^{2}0 \alpha_1\alpha_5+ \alpha_3 \alpha_6 + \Delta \alpha_5 \alpha_2\alpha_5+\alpha_4\alpha_6+\Delta \alpha_6 \Delta^{2}\end{pmatrix}=\begin{pmatrix}1 0 0 0 1 0 0 0 1\end{pmatrix}.
\sup_{0\leq t\leq T}|Law (X(t)) - Law (Y(t))| =: ||Flow(X) - Flow(Y)||_{\infty}&\leq 2 T^\ast\sqrt{d}e^{T^\ast C_1}\left[C_2 + C_3 (K_2 +T^\ast C_4) e^{T^\ast C_5}\right]||m-n||_{\infty},
S_{f}^{\star}= \int d^{4}x \biggl[\bar{\psi}(i{\ooalign{\hfil/\hfil\crcr$\nabla$}}- m_{f}) \psi - n\bar{\psi}\gamma_{\mu}\psi \partial^{\mu}\theta + \frac{1}{2}\frac{2\pi}{e}W^{\mu \nu}\omega^{F}_{\mu \nu}- \frac{1}{4}\frac{1}{|\phi|^{2}}(\bar{\psi}\gamma_{\mu}\psi )(\bar{\psi}\gamma^{\mu}\psi ) \biggr],
E_{in}^{(4)}(z) = \left( \mathbb{I}- \frac{A_n^{(3)}(z)-A_n^{(3)}(0)}{n^9 z}\right)E_{in}^{(3)}(z) E_{out}^{(4)}(z) = \left( \mathbb{I}- \frac{A_n^{(3)}(0)}{n^9 z}\right)^{-1}E_{out}^{(3)}(z)
F_{n}( z g_{1}, \cdots , z g_{n}) = \hspace{- 5 m m}\sum_{{\scriptstyle k_{1}+ 2 k_{2}+ \cdots + n k_{n}= n}\atop{\scriptstyle k_{1}\geq 0 , \cdots , k_{n}\geq 0}}\frac{n !}{k_{1}! \cdots k_{n}!}\left( \frac{z g_{1}}{1 !}\right)^{k_{1}}\left( \frac{z g_{2}}{2 !}\right)^{k_{2}}\cdots \left( \frac{z g_{n}}{n !}\right)^{k_{n}}.
\begin{array}[c]{ll}\beta^{0,\theta}=\displaystyle\frac{\big{[}J(u^{\theta}(\cdot))+\theta\big{]}^+}{J^{\theta}(u^{\theta}(\cdot))}, \beta^{j,\theta}=\displaystyle\frac{-\big{[}-X^{\theta}(t_j)\big{]}^+}{J^{\theta}(u^{\theta}(\cdot))},\j=1,2,\cdots,n. \end{array}
\Bigg|\int_{S_{k,0,T}}\int_{(\mathbb{R}^d)^{k+1}}\Biggl(\prod_{i=1}^kh(y_i) \Biggr)f(z)J_1 dz \prod_{j=1}^k dy_j \prod_{i=1}^k ds_{i}\Bigg| &\leq3 C_T \frac{1}{(k-1)!}\|h\|^k\|f\|T^{k}\frac{\log n}{n}.
A_{1}^{\left( 0 \right)}={\frac{{\Phi}}{{2 \pi r^{2}}}}x^{2}, A_{2}^{\left( 0 \right)}= -{\frac{{\Phi}}{{2 \pi r^{2}}}}x^{1}, A_{0}^{\left( 0 \right)}= A_{3}^{\left( 0 \right)}= 0 , r^{2}= \left( x^{1}\right)^{2}+ ( x^{2})^{2}.
(I+\tilde{T})^2 = \begin{pmatrix}I -I I I+2T\end{pmatrix}\begin{pmatrix}I -I I I+2T\end{pmatrix}&=\begin{pmatrix}0 -2(I+T) 2(I+T) 4T(I+T)\end{pmatrix}=2\begin{pmatrix}0 -I I 2T\end{pmatrix}\begin{pmatrix}I+T 0 0 I+T\end{pmatrix},
|\partial^\alpha \Box \phi| = \left |\sum_{\mu+\beta + \gamma = \alpha}(\partial^\mu f) m(d \partial^\beta \phi,d \partial^\gamma \phi) \right | \le C \Vert f \Vert_{C^k}\sum_{\mu+\beta + \gamma = \alpha}(|\partial \partial^\beta \phi| |\overline{\partial}\partial^\gamma \phi|)
(h + L_X\bar g)_{ij}= \tfrac{2}{n-2}c\delta_{ij}|x|^{2-n}+ O^{2,\alpha}(|x|^{\max\{-2q, 1-n \}}) v + X(\bar u) = -c |x|^{2-n}+ O^{2,\alpha}(|x|^{\max\{-2-q, 1-n \}})
\begin{aligned}M_w^B(Z, \lambda, N, \frac{\varepsilon}{4},G, d) \ge \inf_{\mathcal{G}_w(\mathcal{U})}\left\{\sum_{(w_{\mathbf{U}}, \mathbf{U}) \in \mathcal{G}_w(\mathcal{U})}e^{-\lambda \mathfrak{l}(\mathbf{U})}\right\}&\ge M_w(Z, \lambda, N, \varepsilon,G, d).\end{aligned}
\zeta(s)=2\left(\frac{\tau_2}{2\pi}\right )^{2s}\sum_{Z^2}\left [\left (n_1-g h_1+\frac{1}{2}-\tau_1\left ( n_2-g h_2+\frac{1}{2}\right )\right )^2+\tau_2^2 \left (n_2-g h_2+\frac{1}{2}\right )^2\right ]^{-s},
{\cal F}_2\left(a, \epsilon_1, \epsilon_2\right)=\frac{20 a^2 + 7 \epsilon_1^2 + 16 \epsilon_1 \epsilon_2 + 7\epsilon_2^2}{\left(4a^2 - \epsilon^2 \right)^2 \left(4a^2 - \left( 2\epsilon_1 + \epsilon_2\right)^2 \right) \left(4a^2 - \left(\epsilon_1 + 2\epsilon_2 \right)^2 \right)}.
a ={\partial_{\nu_1}}{\partial_{\bar{\nu_1}}}\log{\langle \nu \vert \nu \rangle}, b ={\partial_{\nu_1}}{\partial_{\bar{\nu_2}}}\log{\langle \nu \vert \nu \rangle}, c ={\partial_{\nu_2}}{\partial_{\bar{\nu_2}}}\log{\langle \nu \vert \nu \rangle}
\left(\begin{matrix}x=0 x=\infty \,\left( \frac12 \right) \begin{matrix}0 0 \theta^0 \end{matrix}\overbrace{\begin{matrix}1 0 t_2 0 \theta^\infty_1 0 0 0 \sqrt{t_1}\theta^\infty_2/2 0 0 0 -\sqrt{t_1}\theta^\infty_2/2 \end{matrix}}\end{matrix}\right) ,
s_1=\frac{\gamma_3-\gamma_1-(\beta_2-\beta_1)(\beta_3-\beta_1)}{\gamma_3}, s_2=\frac{\gamma_2}{\beta_3-\beta_1},s_3=\frac{\gamma_3}{\beta_3-\beta_1}, w=\frac{a_1}{4s_3}-\frac{\gamma_2}{\gamma_3},w_{\lambda}=\frac{\lambda}{(1+\lambda)^2}\, \frac{a_1}{s_3}-\frac{\gamma_2}{\gamma_3}.
T[ \pi(\psi) f] \sqrt{d\mu_\pi}= T \pi_{univ}(\psi) ( f \sqrt{d\mu_\pi}) = \pi_{univ}(\psi) T(f \sqrt{d\mu_\pi}) = \pi_{univ}(\psi)( g \sqrt{d\mu_\pi}) = [\pi(\psi)(g)] \sqrt{d\mu_\pi}= \pi(\psi) T \left( f \sqrt{d\mu_\pi}\right),
\widetilde{a}(e, v) = \widetilde{a}(u_h-z, v) - \int_{{\cal E}_h^0}[u-z]\cdot \{\nabla_h v\}ds - \int_{{\cal E}_h^i}\beta\cdot[u-z] [\nabla_h v]\, ds+ \int_{\Gamma_2}g(\lambda-\lambda_h)v\,ds.
-\frac{1}{8}\left[ \left( \nabla^\mu\nabla_\mu + \frac{1}{10}\nabla^\gamma \nabla_\gamma-72 \right) h_2^I + \nabla^\gamma \nabla_\gamma h^{\prime I}- 192\sqrt{2}\nabla^\gamma \nabla_\gamma b^I \right] Y^I =0,
{\delta^{\prime}}_{Q + S}x^{\alpha \beta}= 0 \, , \qquad{\delta^{\prime}}_{Q + S}\lambda^{4 \alpha}= ( x^{\alpha \beta}+ 2 i A_{4 4}\lambda^{4 \alpha}\lambda^{4 \beta}) \xi_{\beta}^{4}\, , \qquad{\delta^{\prime}}_{Q + S}( 1_{i}^{I}, 2_{i}^{I}, 3_{i}^{I}) = 0 \, .
P_{A_{r}, s}^{{\bf V}_{i}}( x ) = \prod_{1 \leq j_{1}< \cdots < j_{i}\leq r + 1}\Bigl ( x - \sin ( \bar{q}_{j_{1}}+ \cdots \bar{q}_{j_{i}}) \Bigr ) = P_{A_{r}, s}^{{\bf V}_{r + 1 - i}}( x ) .
\frac{c_1+\lambda_1}{w_1B}\left(\sum_{j \in \mathcal{S}}w_jd_j^{ne}\right)^2 =\sum_{j \in \mathcal{S}/\{1\}}w_jd_j^{ne}, &~~\vdots \frac{c_{|\mathcal{S}|}+\lambda_{|\mathcal{S}|}}{w_{|\mathcal{S}|}B}\left(\sum_{j \in \mathcal{S}}w_jd_j^{ne}\right)^2 =\sum_{j \in \mathcal{S}/\{|\mathcal{S}|\}}w_jd_j^{ne},
\beta x+ (\beta+\alpha(\tau-1))y=0, &(\beta+\alpha-c\alpha)x + ((\beta+\alpha)\tau-c\alpha)y=0, &-\beta^2+\alpha^2 - \tau \alpha^2 = 1, &-\beta^2+\alpha^2 - \tau \alpha^2 - \alpha\beta(\tau-c)=1, &-\tau \beta^2 - 2(\tau-c)\alpha\beta - (c^2-2c+\tau)\alpha^2=\tau', &\mu-x^2-2xy-\tau y^2 = \mu'
\widetilde{\omega_d}(f,t)=\begin{cases}\sup_{0\leq x\leq t\leq y\leq d(X),x\neq y}\frac{(t-x)\omega_d(f,y)+(y-t)\omega_d(f,x)}{y-x}0\leq t\leq d(X) \omega_d(f,d(X)) t > d(X)\end{cases}
J^{(1,1)}_2\leq 8\sum\limits_{n=2}^{\infty}p_n(T_*)\eta^2_*(n)+8\sum\limits_{n=2}^{\infty}\frac{p_n(T_*)}{\widetilde{n}}\left(\frac{C^{(2)}_{\textsf{KMT}}}{n^\vartheta}\right)^{1/2}\sum_{j=1}^{\widetilde{n}}a_j\,,
U_t =U_{t_k}+\int_{t_k}^{t}\bigl(AS_{\Delta t}U_{t_k}+S_{\Delta t}G'(X_k).U_{t_k}\bigr)dr+\int_{t_k}^{t}S_{\Delta t}\bigl(\sigma'(X_k).U_{t_k}\bigr)dW(r), U_{t_{k+1}}=S_{\Delta t}U_{t_k}+\Delta tS_{\Delta t}G'(X_k).U_{t_k}+S_{\Delta t}\bigl(\sigma'(X_k).U_{t_k}\bigr)\Delta W_k, U_{t_{\ell+1}}=h.
\bigl( \epsilon^{\rm hom}_{\rm stiff}\bigr)^{-1}\xi\cdot\xi = \inf_{\substack{v \in [L^2(Q)]^3_{\rm sol}, \langle v\rangle = 0}}\ \ \int_{Q_1}\epsilon_1^{-1}\left(v+\xi\right) \cdot \left(v+\xi\right),\ \ \ \xi\in{\mathbb R}^3.
[ x^{I}, p^{J}] = i \delta^{I J}, [ \alpha_{n}^{i}, \alpha_{m}^{j}] = \omega_{n}\delta^{i j}\delta_{n + m , 0}, [ \alpha_{n}^{i^{\prime}}, \alpha_{m}^{j^{\prime}}] = \omega_{n}^{\prime}\delta^{i^{\prime}j^{\prime}}\delta_{n + m , 0}.
norm(\pi^{\prime})=\prod_{u=\rho_1}^{k_1-1}\prod_{r=0}^{\rho_1-1}s_{u-r}\cdot \prod_{u=k_1}^{k_2-1}\prod_{r=0}^{\rho_2-1}s_{u-r}\cdot \prod_{u=k_2}^{k_3-1}\prod_{r=0}^{\rho_3-1}s_{u-r}\cdots \prod_{u=k_{m-1}}^{k_m-1}\prod_{r=0}^{\rho_m-1}s_{u-r},
E\psi=-\frac{\hbar^{2}(1-\lambda)}{2(L_{1}+L_{2})}\frac{d^{2}\psi}{dQ_{c}^{2}}-\frac{\hbar^{2}(1-\lambda)}{2\mu}\frac{d^{2}\psi}{dQ_{d}^{2}}+\frac{1}{2}\mu\omega^{2}Q_{d}^{2}\psi
-\big((z-B')^{-1}f_0\big)(x_1)(z-A)^{-1}e_1+(z-A)^{-1}x_2 =\big(-\big((z-B')^{-1}f_0\big)(x_1)+\alpha_1\big)(z-A)^{-1}e_1+(z-A)^{-1}\widetilde{x}_2
\mathcal{Z =}\int D A_{i}D \varphi^{+}D \varphi \exp \left[ - \int d^{3}x \left( \frac{1}{4}F_{i j}F_{i j}+ \left( \mathcal{D}_{i}{\varphi}\right)^{+}\left( \mathcal{D}_{i}{\varphi}\right) + V ( \varphi ) \; \right) \right] \; ,
w ( x^{3}, \upsilon ) = \frac{\zeta}{\upsilon}\tanh x^{3}, ~ ~ s ( x^{3}, \upsilon , \bar{\upsilon}) = - 2 \ln{\upsilon}+ \sum_{k = 1}^{\infty}2^{2 k}( \cosh{x^{3}})^{- 2 k}g_{2 k}( \upsilon , \bar{\upsilon}) ,
\begin{aligned}r_{ij}^nw_ndx^i\otimes dx^j:= \{2R_i{}^k{}_j{}^n\nabla_kw_n +R_{i}{}^k{}_j{}^mw_n(-\Gamma_{km}^n)+\nabla^k({R_{ikj}}^n)w_n +g^{kl}{R_{imj}}^nw_n (-\Gamma_{lk}^m) +R_i{}^m w_n (-\Gamma_{mj}^n)\}dx^i\otimes dx^j.\end{aligned}
\left|u(x) - b(x_0) w_{1/2}\left(\frac{\nu_{x_0}\cdot (x- x_0)}{(\nu_{x_0}\cdot A( x_0) \nu_{x_0})^{1/2}},\frac{x_{n+1}}{(a^{n+1,n+1}( x_0))^{1/2}}\right)\right| \leq C \left(c_\ast+\left\| \nabla'' g \right\|_{C^{0,\alpha}(B_{1/2}'')}\right) |x - x_{0}|^{\frac{1}{2}+\beta}.
\left[1 - e^{-p_1 (t_2 \wedge \cdots \wedge t_g)}\right]^{M_1}= \sum_{k_1=0}^{M_1}(-1)^{k_1}\binom{M_1}{k_1}e^{-k_1 p_1 (t_2 \wedge \cdots \wedge t_g)}.
h_{1,2,4}(x) =4(3x+1)(3x^3+27x^2+33x+1)\equiv 1\pmod{3}, h_{0,2,2}(x) =5 x^2+10 x+1\equiv 1\pmod{5}, h_{2,2,2}(x) =x^2+10 x+5\equiv x^2\pmod{5}
\Phi^{(\pm)}=\lim_{\ln r\rightarrow\pm\infty}\frac{1}{2\pi}\int\limits_0^{2\pi}d\theta V_\theta(r,\theta),\quad\Phi^{(\pm)}_K=\lim_{\ln r\rightarrow\pm\infty}\frac{1}{2\pi}s\beta\int\limits_0^{2\pi}d\theta \omega_\theta(r,\theta),
F^i_{\: \: j \beta \gamma}= \partial_\beta C^i_{\: \: j \gamma}- \partial_\gamma C^i_{\: \: j \beta}+ \sum_k C^i_{\: \: k \beta}C^k_{\: \: j \gamma}-\sum_k C^i_{\: \: k \gamma}C^k_{\: \: j \beta},
x_{i}^{\gamma}= a^{\frac{\gamma}{n - s}}\epsilon^{i \gamma}\left( F_{0}( \zeta ) + \frac{\gamma}{n - s}\sum_{\kappa = 1}^{n - s - 1}\epsilon^{- i n \kappa}\mu_{\kappa}l^{\kappa}F_{\kappa}( \zeta ) \right)
G(m,N,a,\varepsilon, s) \leq 6 \cdot C_1C_2^{N-2}\varepsilon ({m \over 3})^{1-ds\over a}\sum_{k=1}^{({m(1+\epsilon) \over 2})^{{1 \over a}}}k^{-ds}&\leq 6 \cdot 3^{ds-1 \over a}C_1C_2^{N-2}\varepsilon m^{1-ds\over a}\zeta(ds) &=C_1C_2^{N-1}\varepsilon m^{1-ds\over a},
(B_\Gamma \omega_n, \phi_{nj})=\int_{j/n}^{(j+d+1)/n}B_\Gamma\omega_n(\phi (s)) \overline{\phi_{nj}(\phi (s))}ds\approx\sum_{k=1}^{24}w_kB_\Gamma \omega_n(\phi (s_k)) \phi_{nj}(\phi(s_k)),
\eta_{( 0 , 0 ) ( 0 , 0 )}= 1 , \quad \eta_{( 1 , 0 ) ( 0 , 1 )}= - \eta_{( 0 , 1 ) ( 1 , 0 )}= i , \quad \eta_{( 1 , 1 ) ( 1 , 1 )}= - 1 .
{\cal L}= - \frac{1}{4}F^{\mu \nu}F_{\mu \nu}- \frac{\kappa}{2}\epsilon^{\mu \nu \rho}A_{\mu}\partial_{\nu}A_{\rho}+ \sum_{a}\bar{\psi}_{a}[ \gamma_{a}^{\mu}( i \partial_{\mu}+ q_{a}A_{\mu}) - m_{a}] \psi_{a},
\begin{array}{c}{< O_{k}O_{l}P >_{c}= ( \frac{1}{\rho^{2}}\frac{\partial \rho}{\partial t_{l}}+ \frac{1}{\rho^{2}}\frac{\partial \rho}{\partial \mu}\frac{\partial \mu}{\partial t_{l}}) < O_{k}P >}\\{- \frac{1}{\rho}( < O_{k}O_{l}P > + \frac{\partial \mu}{\partial t_{l}}< O_{k}P P > ) .}\\ \end{array}
g(\omega) := c_3\delta^{-1}LN^{-\beta}\frac{f_0^2(\omega/\delta)(\omega/\delta)}{B_N(f_0)\widetilde{f}_0(\omega)}\sin(\omega x_0/\delta)\Big[\widehat{\phi}\big(\tfrac{6x_0}{\pi}(\tfrac{\omega}{\delta}-N)\big)+ \widehat{\phi}\big(\tfrac{6x_0}{\pi}(\tfrac{\omega}{\delta}+N)\big)\Big].
{\hat{S}}= - T_{\mathrm{M 9}}\int d^{9}\xi \, \, |{\hat{k}}|^{3}\sqrt{| \mathrm{d e t}\left({\hat{\Pi}}+ |{\hat{k}}|^{- 1}l_{p}^{2}{\hat{\cal K}}^{( 2 )}\right) |}\, , \,
\begin{aligned}\Psi_h=\{v\in H^k(\mathcal{T}_h):v|_K\in P_k(K),K\in\mathcal{T}_h\}, &\Lambda_h=\{v\in L^2(\mathcal{T}_h):v|_K\in P_k(K),K\in\mathcal{T}_h\}, &\Theta_h=\{v\in [L^2(\mathcal{T}_h)]^d:v|_K\in RT_k(K),K\in\mathcal{T}_h\}, &\Sigma_h=\{v\in L^2(\partial\mathcal{T}_h):v|_e\in P_k(e),e\in\partial\mathcal{T}_h\},\end{aligned}
\sigma(E)=\frac{\pi e^{2}\hbar}{2mc\epsilon_{0}n_{r}E}\sum_{n}^{N_{fs}}\frac{1}{N_{p}}\sum_{l}^{N_{p}}\Delta E_{0,n}(R_{l})f_{0,n}(R_{l})g(E-\Delta E_{0,n}(R_{l}),\delta)
g_{R}^{2}( \mu ) ={\frac{g_{R}^{2}( \mu_{0})}{1 +{\frac{2 2}{3}}{\frac{C_{2}( G )}{( 4 \pi )^{2}}}g_{R}^{2}( \mu_{0}) \ln{\frac{\mu}{\mu_{0}}}}}={\frac{1}{{\frac{2 2}{3}}{\frac{C_{2}( G )}{( 4 \pi )^{2}}}\ln{\frac{\mu}{\Lambda_{\mathrm{Q C D}}}}}}
P=\stackrel{1}{P^{i}_j}\theta^{j}\otimes\frac{\partial}{\partial y^{i}}+\stackrel{2}{P^{j}_i}k_j\otimes\frac{\partial}{\partial p_{i}}+\stackrel{3}{P_{ij}}\theta^{j}\otimes\frac{\partial}{\partial p_{i}}+\stackrel{4}{P^{ij}}k_j\otimes\frac{\partial}{\partial y^{i}},
{\mathrm{I n d}}D = \frac{1}{8 \pi^{2}}\int [ R^{a b}\mathrm{\tiny \wedge}R_{a b}+ 2 ( T^{a}\mathrm{\tiny \wedge}T_{a}- R_{a b}\mathrm{\tiny \wedge}e^{a}\mathrm{\tiny \wedge}e^{b}) ] ,
\sigma v_i =\begin{cases}\lambda v_i, 1 \leq i \leq \ell, \lambda^{-1}v_i, \ell +1 \leq i \leq 2 \ell, \end{cases}\mathcal{N}v_i = \begin{cases}v_{i+1}, 1 \leq i \leq \ell -1, -v_{i+1}, \ell+1 \leq i \leq 2\ell -1, 0, i = \ell, 2\ell. \end{cases}
{\cal L}= \bar{\psi}_i(i{\not\!\partial}-e{\not\!\!A}-m)\psi_i - \chi^*_\mu \bar{\psi}^c_i\gamma^\mu\tau\psi_i - \chi_\mu \bar{\psi}_i \gamma^\mu\tau^T \psi^c_i + \frac{N}{g}\chi^*_\mu \chi^\mu ~.
({\cal D}^{2}- D_{1}\bar{D}_{1}+ \frac{1}{2}D_{1}^{2}\bar{P}_{1 1}- \frac{1}{2}P_{1 1}\bar{P}_{1 1}+ d \lambda^{2}) | \chi \rangle + \lambda ( \bar{D}_{2}- D_{1}\bar{P}_{1 2}) | \Phi \rangle = 0 \, .
\Omega_{b \nu}( a z , b z ) = \frac{\bar{I}_{\nu}^{( a )}( a z ) / \bar{I}_{\nu}^{( b )}( b z )}{\bar{K}_{\nu}^{( a )}( a z ) \bar{I}_{\nu}^{( b )}( b z ) - \bar{K}_{\nu}^{( b )}( b z ) \bar{I}_{\nu}^{( a )}( a z )}.
T =\int \frac{{\bf 1}_E(y)}{|y-x|^m}\; d\mu(y)=\int_{0}^\infty \mu(\{y\in E:\; |x-y|^{-m}\geq u\})\; du &=\int_{0}^\infty \mu(E\cap B(x, u^{-1/m})) \; du=m\int_{0}^\infty \mu(E\cap B(x, u)) u^{-m-1}\; du &=m\int_{r_1^{1-\delta}}^\infty \mu(E\cap B(x, u)) u^{-m-1}\; du,
\frac{\beta_{j^{\prime}; j}}{\beta_{j^{\prime}; j}^{( 1 0 )}}= b_{0}^{1 / 2}( \frac{2 ( s^{\prime}- s ) - | m | + | m^{\prime}| b_{0}}{( 2 s^{\prime}+ \frac{3}{2}) + ( | m^{\prime}| - | m | - 2 s - \frac{3}{2}) b_{0}}) \, ,
\lambda_i\geq \dfrac{2b}{1-e^{\frac{2b}{\sqrt{a^2+1}}C_0}}-a-b=-a+b+\dfrac{e^{\frac{2b}{\sqrt{a^2+1}}C_0}}{1-e^{\frac{2b}{\sqrt{a^2+1}}C_0}}2b=:-a+b+\delta.
\begin{array}{c}\int_{\Omega}(\nabla u_{n}\nabla \varphi_{1}+u_{n}\varphi_{1})\,\mathrm{d}x=\int_{\Omega}f(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))\varphi_{1}\,\mathrm{d}x\end{array}
\Omega_{i}\;=\; -\textstyle{\frac{1}{4\sqrt{3}}}\gamma^{+}\left[ \lambda_{j}\gamma^{j}\gamma_{i}+\lambda_{i}\right] \; ,\;\Omega^{-}\;=\; \textstyle{\frac{1}{4\sqrt{3}}}\lambda_{i}\gamma^{i}\left( \gamma^{+}\gamma^{-}+1 \right) \; .
\log \frac{m_t^{(n)}(x)}{\pi(x)}= e^{-\frac{\sigma^2}{2}t}\log \frac{m_0(x)}{\pi(x)}&- \int_0^t \frac{\sigma^2}{2}e^{-\frac{\sigma^2}{2}(t-s)}\left( \frac{2}{\sigma^2}\frac{\delta F}{\delta m}(m_s^{(n-1)},x) - \operatorname{KL}(m_s^{(n-1)}|\pi) \right) ds \,.
\int_{B^{\complement}(x,\rho) \cap B}\frac{dy}{|x-y|^{n+\beta}}= \int_{B \cap [B(x,2\rho)\setminus B(x,\rho)]}\frac{dy}{|x-y|^{n+\beta}}\geq \frac{|E|^{-\beta/n}}{(\omega_n \theta)^{1+\frac{\beta}{n}}2^{n+\beta}}.
C(f,\rho)t\ell \sum_{j=0}^{\ell-1}\Big(\sum_{i=0}^j m^i\Big)^2 \frac{(K^{j+1})^{\alpha-1}}{n^{2-\alpha}}\leq C(f,\rho) t\ell^2 m^2 n^{-(3-2\alpha) -\delta(\alpha-1)}.
X(n)_{s,t}=(1,\int_{s<s_{1}<t}dX(n)_{s_{1}},...,\int_{s<s_{1}<\cdot\cdot\cdot<s_{\lfloor p\rfloor}<t}dX(n)_{s_{1}}\otimes\cdot\cdot\cdot\otimes dX(n)_{s_{\lfloor p\rfloor}})
A_i^{\prime}= e^{-iga_{0}x_{0}\tau_{3}}\Omega_{{\rm D}}A_{i}^{(1)}\Omega_{{\rm D}}^{\dagger}e^{iga_{0}x_{0}\tau_{3}}+ e^{-iga_{0}x_{0}\tau_{3}}s_{i}e^{iga_{0}x_{0}\tau_{3}}+ \partial_{i}a_{0}x_{0}\tau_{3},
\left(\bigsqcup_{a \in \mathcal{A}}^{P_A}G_a\right) \wedge \left(\bigsqcup_{b \in \mathcal{B}}^{P_B}G_b\right) = \bigsqcup_{(a,b) \in \mathcal{A}\times \mathcal{B}}^{P_AP_B}G_a \wedge G_b.
U_{2k+1,2a+1}(x;q)-U_{2k+1,2a-1}(x;q) &=(1+xq)(xq)^{2a+1}\overline{U}_{2k+1,2k-2a}(xq;q)+(1+xq)(xq)^{2a-1}\overline{U}_{2k+1,2k-2a+2}(xq;q),
B_j =\sum_{a=0}^{l/s_i-1}\lambda_{j,a_j}e_a^{(l/s_i)}(e_{a(j,a_j\oplus 1)}^{(l/s_i)})^T j\in[n-1], B_n =\sum_{a=0}^{l/s_i-1}\Big(\prod_{q=s_ia_n}^{s_ia_n+s_i-1}\lambda_{n,q}\Big)e_a^{(l/s_i)}(e_{a(n,(s_ia_n\oplus s_i)/s_i)}^{(l/s_i)})^T.
g(\tau)=\frac{\vartheta_2^4(\tau)}{\vartheta_3^4(\tau)+\vartheta_4^4(\tau)},~~~~~f(\tau)=\frac{\vartheta_2^4(\tau) \vartheta_4^4(\tau)}{(\vartheta_2^4(\tau)-\vartheta_4^4(\tau))^2}~\,,
\tilde{\Phi}(\xi',\eta') = \xi_{N,M}{\xi'}^2 + \eta_{N,M}{\eta'}^2 - \frac{{\xi_{N,M}}^3 +{\eta_{N,M}}^3}{4}, \tilde{F}(\xi',\eta') = 2 \xi' \eta' + \frac{3}{2}\, \xi_{N,M}\, \eta_{N,M}.
\sum_{m,n\in\mathcal{M}'}r(m)r(n)\Phi\left(T\log{\frac{m}{n}}\right) \le \sum_{m \in \mathcal{M}'}r(m)^2 + \sum_{\substack{m,n \in \mathcal{M}' m\not=n}}r(m)r(n)\Phi\left(T\log{\frac{m}{n}}\right).
\Gamma [ A ] = - \frac{1}{4 \pi^{2}}\int_{0}^{\infty}\int d^{4}x \left\{g^{2}s^{2}{\cal F}_{2}( \frac{\Re C o s h ( g s X )}{\Im C o s h ( g s X )}) - 1 \right\}e x p ( - m^{2}s )
P_{R}=\frac{\partial L}{\partial \dot{R}}=-\frac{3\pi R\dot{R}}{2G},\hspace{0.2cm}P_{\phi}=\frac{\partial L}{\partial\dot{\phi}}=\frac{2\pi^{2}}{G_{\phi}}R^{3}\dot{\phi},\hspace{0.2cm}P_{a}=2\pi^{2}mR^{3}\dot{n}_{a}
\begin{cases}\varphi_n(0,\cdot) \rightarrow \varphi(0,\cdot), \varphi_n(L,\cdot) \rightarrow \varphi(L,\cdot) L^2(0,T), \psi_n(0,\cdot) \rightarrow \psi(0,\cdot), \psi_n(L,\cdot) \rightarrow \psi(L,\cdot) L^2(0,T).\end{cases}
{y^{[j]}}({t_2}) ={h^{[j1]}}({t_2})u_2^{[1]}+{h^{[j2]}}({t_2})\frac{{{h^{[22]}}({t_1})}}{{{h^{[22]}}({t_2})}}u_1^{[2]}+{h^{[j3]}}({t_2})\frac{{{h^{[23]}}({t_1})}}{{{h^{[23]}}({t_2})}}u_1^{[3]}.
A_{g_1,\dotsc,g_l}(K,\varphi)(x) =\frac{1}{|K|}\int_KT_{g_1}^t\varphi(x)\dotsm T_{g_l}^t\varphi(x)dt\forall x\in X &=\frac{1}{|K|}\int_K\varphi(T_{tg_1}x)\dotsm\varphi( T_{tg_l}x)dt.
\exists\ \{\nu_{i}\}_{i\in I(\bar{x})}\geq 0\ \forall_{k\in \mathbb{N}\backslash W}|\bar{x}_k-x_k|^{p-2}{(\bar{x}_k-x_k)}=- \sum_{i\in I(\bar{x})}\lambda_k^i\nu_i, \forall_{k\in W}|\bar{x}_k-x_k|^{p-2}{(\bar{x}_k-x_k)}=0.
L^{\prime}= \sum_{n > 0}\left( \frac 1 2 | \partial \phi |^{2}- \frac{M^{2}n^{2}}{2}| \phi_{n}|^{2}- 2 \lambda \phi_{n}\sum_{n > 0}| \phi_{n}|^{2}\right) + \frac{\lambda}{3 !}\sum_{n , n^{\prime}}{}^{\prime}\phi_{n}\phi_{n^{\prime}}\phi_{n + n^{\prime}}^{*},
\frac{\#\mathrm{H}^0(G,U_N)}{\#\mathrm{H}^{-1}(G,U_N)}=\frac{(\ker(\Delta):\mathrm{im}(\mathcal{N}))}{(\ker(\mathcal{N}):\mathrm{im}(\Delta))}=\frac{(U_K:N_{N/K}(U_N))}{(E_{N/K}:U_N^{\sigma - 1})},
V^{\alpha \dot{\alpha}}= \left( \begin{matrix}{{V^{1 \dot{1}}}}&{{V^{1 \dot{2}}}}\\{{V^{2 \dot{1}}}}&{{V^{2 \dot{2}}}}\\ \end{matrix}\right) ={\frac{1}{\sqrt 2}}\, \left( \begin{matrix}{{V^{1}+ V^{4}}}&{{V^{2}- V^{3}}}\\{{V^{2}+ V^{3}}}&{{- V^{1}+ V^{4}}}\\ \end{matrix}\right) \ ,
I_{\varepsilon}\left(t_{\varepsilon}(W_{\varepsilon,\xi})W_{\varepsilon,\xi}\right)=\left(\frac{1}{2}-\frac{1}{p}\right)\|W_{\varepsilon,\xi}\|_{\varepsilon}^{2}t_{\varepsilon}^{2}+\omega\left(\frac{1}{4}-\frac{1}{p}\right)t_{\varepsilon}^{4}G_{\varepsilon}(W_{\varepsilon,\xi})