harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\partial{}U(\mathbf{p},\Omega)}{\partial\Omega}= \sum_{k=1}^{K}\frac{A_d\sigma_{}^2{}p_k}{\ln2(P_c+P_{})\left[(\sigma_N^2\Omega+\sigma_{}^2)\|\mathbf{f}_k\|^2+A_d{}p_k\Omega\right](\sigma_N^2\Omega+\sigma_{}^2)}\triangleq{}\sum_{k=1}^{K}\frac{\mathsf{C}}{\mathsf{P}(\Omega)},
\alpha''(x_1, x_2, x_3, \dots, x_k) + \alpha''(x_2, x_1, x_3, \dots, x_k) = \sum_{i \in [s]}\gamma_i(x_1,x_{I_i}) \gamma'_i(x_2, x_{I'_i}) \delta_{i, 1}(x_{J_{i,1}}) \cdots \delta_{i, d_i}(x_{J_{i,d_i}})
\frac{\partial \widetilde{R}_{1,k}}{\partial\alpha}= - \Big(1-\frac{\tau}{T}+\frac{\alpha\tau}{T}\Big) \cdot \frac{1}{ln2}\cdot \frac{1}{\alpha+g}+ \frac{\tau}{T}\textmd{log}_2 \bigg(1+\frac{(1-\lambda) \beta_{1,k}^2}{\frac{(1-\lambda)\alpha}{\lambda \tau}c_1 + \frac{1}{M}\big(c_2 + \beta_{1,k}\sigma_n^2\big)}\bigg),
\Pi_{X_1,X_2}(e^{t}x,e^{t}y) = - \left( e^{t}y \right)^{2}\partial_{(e^{t}x)}\wedge \partial_{(e^{t}y)}=-e^{2t}y^{2}e^{-2t}\partial_{x}\wedge \partial_{y}= - y^2 \partial_x \wedge \partial_y &= \Pi_{X_1,X_2}(x,y),
I_0 = \frac{1}{2\pi}\int_{-\infty+ic}^{\infty+ic}\left(\frac{1}{s-1}+ \frac{1}{s}\right)\Bigl( \Phi_1(\phi;z)\overline{\Phi_1(\phi;\bar{z})}+ \Phi_1(\phi;-z)\overline{\Phi_1(\phi;-\bar{z})}\Bigr) \, dz,
\left[a_{ij}(p^+ ),a^\dagger_{lk}(q^+)\right]=\left\{b_{ij}(p^+ ),b^\dagger_{lk}(q^+)\right\}=\delta (p^+ -q^+)\left(\delta_{il}\delta_{jk}-\frac{1}{N}\delta_{ij}\delta_{kl}\right)
L_{ww}= w_1^2L_{11}+ 2w_1w_2L_{12}+ w_2^2L_{22}, =2 (w^1)^2{w^1}_{\substack{x_1}}+ 2w^1w^2({w^1}_{\substack{x_2}}+{w^2}_{\substack{x_1}}) + 2(w^2)^2{w^2}_{\substack{x_2}}.
G_{k,l+1}=\frac{k!}{(k+l+1)!}v^{k+l+1}g_0+\sum_{m=1}^l\frac{(-1)^m}{(m-1)!(l+1-m)!}\frac{v^{l+1-m}}{k+m}g_{k+m}&-\left\{\frac{k!}{(k+l+1)!}+\sum_{m=1}^l\frac{(-1)^m}{(m-1)!(l+1-m)!}\frac{1}{k+m}\right\}g_{k+l+1}.
T_{5}( k , k^{\prime}; a , b ) = \frac{T_{0}^{5}}{g}\sqrt{\left[ ( k - \chi_{B 0}k^{\prime})^{2}g^{2}+ k^{2}\right] \left[ ( a - \chi_{0}b )^{2}+ b^{2}g^{- 2}\right]}
[E_{i,k+1},E_{i+1,l-1}]_{q^{-2}}-(q_1+q_3)[E_{i,k},E_{i+1,l}]+q_1q_3[E_{i,k-1},E_{i+1,l+1}]_{q^{2}}=0\,, &[F_{i+1,l-1},F_{i,k+1}]_{q^{-2}}-(q_1+q_3)[F_{i+1,l},F_{i,k}]+q_1q_3[F_{i+1,l+1},F_{i,k-1}]_{q^{2}}=0\,.
{d \widetilde A^K_{i}(t)\over dt}< \bar p e^{\varepsilon_K\log K}\sum_{\ell=-\lfloor 1/2\delta_K\rfloor}^{1/\delta_K-1-\lfloor 1/2\delta_K\rfloor}h_{K}G(h_{K}\ell) \exp(\log K (\widetilde A^K_{i+\ell}(t)- \widetilde A^K_{i}(t)) - 2 \bar p.
\partial_0^{-1}v(t) = \int_0^t \int_0^s \cos((-\Delta)^{1/2}(s-r))\sigma(u(r))dW(r) d s = \int_0^t \int_r^t \cos((-\Delta)^{1/2}(s-r)) ds \sigma(u(r))dW(r) = \int_0^t (-\Delta)^{-1/2}\sin((-\Delta)^{1/2}(t-r)) \sigma(u(r))dW(r) = u(t).
\begin{cases}\gamma_1=K_{21}\beta_1+\alpha_1+O(1)\Delta'(\alpha_{5},\boldsymbol{\beta}^{*}), &\sigma'_{i}=K_{2i}\beta_1+\beta_i+\sigma_i+O(1)\Delta'(\alpha_{5},\boldsymbol{\beta}^{*}),i=2,3, &\gamma_4=\alpha_4+\beta_4, &\gamma_5=K_{25}\beta_1+\alpha_5+\beta_5+O(1)\Delta'(\alpha_{5},\boldsymbol{\beta}^{*}),\end{cases}
R+1>r_{i+1}=r_1- \Lambda_0 \sqrt{a_0}\cdot \sum_{k=1}^i \sqrt{t_k}&\geq R+3- \Lambda_0 \sqrt{a_0t_i}\cdot\sum_{k=0}^{\infty}(1+\mu)^{-k}&=R+3- \sqrt{t_i}\cdot \frac{\Lambda_0 \sqrt{a_0}(1+\mu)}{\mu}.
\big\vert D^{2n}g(x)(h_1,\dots, h_{2n})\big\vert \le\sum_{j=0}^{n}2^{n+j}\binom{2n}{2j}\frac{(2(n-j))!}{2^{n-j}(n-j)!}2^{n+j}\vert\vert x \vert\vert^{2j}\sum_{k=0}^{+\infty}\frac{C\tau^{k+n+j}}{k!}\vert\vert x \vert\vert^{2k}&\le C (2n)!\sum_{j=0}^{n}\frac{(4\tau)^{n+j}}{(n-j)!}\frac{\vert\vert x \vert\vert^{2j}}{(2j)!}e^{\tau \vert\vert x \vert\vert^2}.
V=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{r}r^{\prime2}sin\theta dr^{\prime}d\theta d\varphi=2\pi\int_{0}^{\pi}\int_{0}^{r}r^{\prime2}sin\theta dr^{\prime}d\theta=4\pi\int_{0}^{r}r^{\prime2}dr^{\prime}=\frac{4}{3}\pi r^{3}.
\mathcal{S}= \int{d^{4}x \sqrt{- \overline{{g}}}\left( \overline{{\mathcal{R}}}- \frac{3}{2}\left[ \overline{{\nabla}}\phi \right]^{2}- \frac{3}{2}m^{2}\left[ 1 - e^{- \phi}\right]^{2}\right)},
\mu_{2j-1}=\prod_{\ell=1}^m\bigg(\frac{L_\ell\Gamma(\frac{L_\ell}{2})\Gamma(\frac{L_\ell+1}{2})}{2\sqrt{\pi}}\bigg)^{1/2}\frac{\Gamma(j-\frac12)}{\Gamma(\frac{L_\ell}2+j-\frac12)}.
\hat{q}( t ) = \sqrt{\frac{\hbar}{2 m \omega}}\left[ a ( t ) + a^{\dagger}( t ) \right] = \sqrt{\frac{\hbar}{2 m \omega}}\left[ a ( t_{0}) \, e^{- i \omega ( t - t_{0})}+ a^{\dagger}( t_{0}) \, e^{i \omega ( t - t_{0})}\right] \ ,
(1) \ \g(t) = (e+t)^{a+b\sin(\log\log(e+t))}-e^a\b>0, a\geq 1+b\sqrt{2}&(2) \ \g(t) = t^\alpha(\log(a+t))^\beta \qquad\qquad\qquad\quad\, \ \alpha,\beta>0, a\geq 1,
P(\xi_{n+1}^{(N,1)}=y^{1},\cdot\cdot\cdot,\xi_{n+1}^{(N,N)}=y^{N}|\xi_{n}^{(N)})=\prod_{i=1}^{N}K_{n+1,\eta_{n}^{N}}(\xi_{n}^{(N,i)},y^{i}),
S^T=M_0^2\int\limits_0^{1/T}dt\int\limits_0^R dr\left[1+\frac{1}{2}{\bf u}\left(1-\frac{\alpha}{M_0^2}\triangle\right)(-\triangle){\bf u}\right]-\beta\int\limits_0^{1/T}dt\,\ddot{\bf u}\acute{\bf u}
S=\int dt\int_0^L dx\left\{\partial_0A^1\pi_1+\frac{1}{2}i\psi^\dagger\partial_0\psi-\frac{1}{2}i\partial_0\psi^\dagger\psi-{\cal H}-A^0\phi+\partial_1(A^0\pi_1)\right\}\ ,
T^*_{\pm}\kappa^*(-t)(i\sigma)\kappa(-t)T_{\pm}=e^{\mp t}\Big[ \pm \sin(\alpha)\Big(C^2(t)+(1-(2z)^2)S^2(t)\Big)-2\cos(\alpha)C(t)S(t) &+i\cos(\alpha)\Big(C^2(t)+(1-(2z)^2)S^2(t)\Big)\widetilde{I}\mp 2i \sin(\alpha)C(t)S(t)\widetilde{I}&+4zi\cos(\alpha)S^2(t)\widetilde{J}\mp 4z\sin(\alpha)S^2(t)\widetilde{K}\Big]{~.}&
f_{\beta,\gamma}= T_{u_j,v_j}(b_j,f_{j+1}), \begin{cases}u_j=\overline\beta\beta_j, v_j=\overline\gamma\gamma_j, \end{cases}\begin{pmatrix}\beta_j \gamma_j \end{pmatrix}= \Theta(\alpha_j)^\dag \begin{pmatrix}\beta \gamma \end{pmatrix}= \begin{pmatrix}\beta\alpha_j+\gamma\rho_j \beta\rho_j-\gamma\overline\alpha_j \end{pmatrix},
\Phi_{\mbox{div}}(\omega,D)={\Gamma\left(1-\frac D2\right) \over (4\pi)^{D/2}}\,\,{m^{D-4}\over \kappa}\int_{\Sigma}\left[2\left(m^2-\left(\frac 16 -\xi\right)R\right)-{\omega^2\over \kappa^2}{\cal P}\right],
N_{\delta}(G(h_i)) \leq 2m+\delta^{-1}\sum\limits_{j=0}^{m-1}R_{h_i}[j\delta,(j+1)\delta] &\leq 2(\delta^{-1}+1)+ \sum\limits_{j=0}^{m-1}h_{q}\delta^{\sigma-1}&\leq \delta^{\sigma-2}(4+2mh_{q}).
\begin{cases}u_{tt}- \Delta u + a(x) \, u_t = 0, \,, (x,t) \in \Omega \times(0, + \infty) begin{align*}0.8mm]u = 0 \,, \mbox{on}\partial \Omega \times (0,+ \infty), begin{align*}0.8mm]u(x,0) = u^0(x), \, u_t(x,0) = u^1(x) \,, \mbox{on}\Omega,\end{cases}
\frac{c_{T_3,T_4}}{c_{T_1,T_2}}=\frac{\omega_2(s_2^2 b/a)^{-1}}{\omega_2(b/a)}\,, \frac{c_{T_2,T_4}}{c_{T_1,T_3}}= \frac{\omega_2(s_2^2 a/b)^{-1}}{\omega_2(a/b)}\,,
{\cal V}\equiv\left( \begin{array}{c}U_\alpha^T begin{align*}1mm] \bar U^{\alpha\,T}\end{array}\right) \equiv\left( \begin{array}{cc}e_\alpha^A h_{A\alpha}begin{align*}1mm] g^{\alpha \bar\beta}\bar e^A_{\bar \beta}g^{\alpha \bar\beta}\bar h_{A\bar \beta}\end{array}\right) \ ,
\lim_{k\to\infty}\frac{i(\beta_-^j,\sigma_-^k)}{i(\beta_+^j,\sigma_-^k)}=\lim_{k\to\infty}\frac{i(\beta_-^j,\sigma_-^k)}{i(\beta_-^j,\sigma_+^k)}\frac{i(\beta_-^j,\sigma_+^k)}{i(\beta_+^j,\sigma_-^k)}&=\lim_{k\to\infty}\frac{i(\beta_-^j,\sigma_-^k)}{i(\beta_-^j,\sigma_+^k)}&>a &> \underset{t \to \infty}{\limsup}\frac{l_t(\beta_-^j)}{l_t(\beta_+^j)}.
~_{V^{*}}\langle\Delta(u|u|^{p-2}+au)+\phi(t)u,-\Delta(I-\frac{\Delta}{n})^{-1}u\rangle_{V}&=-\langle u|u|^{p-2}+au,nu-n\int_{0}^{\infty}{\rm{e}}^{-t}P_{\frac{t}{n}}u\d t\rangle_{L^{2}}+\phi(t)\|u\|_{n}^{2}&\leq -C_{1}\|u\|_{n}^{2}.
\min \ \sum_{i=1}^M \sum_{j=1}^M X_{ij}\pi^{(O)}_{ij}{\rm s.t.\}X_{ij}= X_{ji}&\min_{1 \leq i \neq j \leq M}\mathcal{K}_{ij}\geq K &X_{ij}\in \{0,1\},\ 1 \leq i,j\leq M.
\Phi(|z|)=\Phi_\gamma(|z|)= |z|^{\gamma}\enskip \mbox{for some $\gamma>-3$ and}& b(\cos \theta)\theta^{2+\nu}\ \rightarrow K\ \ \mbox{as}\ \ \theta\rightarrow 0+ \enskip \mbox{for some $0<\nu<2$ and $K>0$.}
\frac{d}{dt}\Bigg(\exp\Bigg(-\int_{t_0}^tA(\tau)d\tau\Bigg)\varphi(t)\Bigg) =\exp\Bigg(-\int_{t_0}^tA(\tau)d\tau\Bigg)\big(\frac{d}{dt}\varphi(t)-A(t)\varphi(t)\big) &\leq\exp\Bigg(-\int_{t_0}^tA(\tau)d\tau\Bigg)\psi(t),
j(E) = \frac{432t(t^2-9)(t^2+3)^3(t^3-9t+12)^3(t^3+9t^2+27t+3)^3(5t^3-9t^2-9t-3)^3}{(t^3-3t^2-9t+3)^9(t^3+3t^2-9t-3)^3}
F_0^2-F_1^2 = (1- \frac{1}{E^2+1})-(1-\frac{1}{E^2_f +1}) = \frac{1}{E^2_f +1}- \frac{1}{E^2 +1}= \frac{1}{E^2_f +1}(1 - (E_f^2+1)\frac{1}{E^2 +1})
c(p_1,p_2)=\max\bigg( \sum\limits_{k=1}^{p_1}(-1)^{k-1}c(p_1-k,p_2){a_2-cp_2+k-1\choose k}, \sum\limits_{j =1}^{p_2}(-1)^{j -1}c(p_1,p_2- j ){a_1-bp_1+ j -1\choose j}\bigg).
g_1 =(1,2,3)\,, g_2 =(1,4,7)(2,5,8)(3,6,9)\,, g_3 =(1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)\,.
\mathbb{E}_2=\frac{\sqrt{8}}{8}(e_1\otimes e_1\otimes e_1\otimes e_2+e_1\otimes e_1\otimes e_2\otimes e_1+e_1\otimes e_2\otimes e_1\otimes e_1+e_2\otimes e_1\otimes e_1\otimes e_1 -e_2\otimes e_2\otimes e_2\otimes e_1-e_2\otimes e_2\otimes e_1\otimes e_2-e_2\otimes e_1\otimes e_2\otimes e_2-e_1\otimes e_2\otimes e_2\otimes e_2)
[X,a_0](a_1, \dots , a_{k-1}, bc) = [X,a_0](a_1, \dots , a_{k-1}, b)c + (-1)^{p(b)p(c)}[X,a_0](a_1, \dots , a_{k-1}, c)b - [X,a_0](a_1, \dots , a_{k-1}, 1)bc
A\Vert K^{*}f\Vert^{2}= A\langle K K^{*}f,\f\rangle\leq\langle U_{\sigma}U_{\sigma}^{*}f,\f\rangle=\Vert U_{\sigma}^{*}f\Vert^{2}& = \sum_{j\in\sigma}\Vert\Lambda_{j}(CC')^{\frac{1}{2}}f\Vert^{2}+\sum_{j\in\sigma^{c}}\Vert\Omega_{j}(CC')^{\frac{1}{2}}f\Vert^{2}
\left(q;q\right)_{\infty}=\frac{1}{\sqrt{2\pi\log q^{-1}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q^{2}}\right)dx}{\left|\left(-q^{1/2}e^{ix};q\right)_{\infty}\right|^{2}}
\chi_{0,3}(x,y,z|\rho)=\sum_{i\geq0}\rho^{i}U_{i}(x)U_{i}(y)U_{i}(z)= ((1+\rho^{2})^{3}+16\rho^{3}xyz-4\rho^{2}(1+\rho^{2})(x^{2}+y^{2}+z^{2}))/w_{3}(x,y,z|\rho),
\bar{H}(y) = \frac{1}{2}\langle y, (Id + P(M_{ij}) NP^* y \rangle_Y + \langle P^* y , s \rangle - \frac{1}{N}\log \int_{Px=0}\exp \left( -H_{(M_{ij})}(x,y) \right) \mathcal{L}(dx).
H = N \left[ - \frac{1}{2 4}\epsilon_{i j k}( \hat{\theta}^{i}- \frac{1}{2}\theta^{i}) ( \hat{\theta}^{j}- \frac{1}{2}\theta^{j}) ( \hat{\theta}^{k}- \frac{1}{2}\theta^{k}) - \frac{1}{3 2}\epsilon_{i j k}\theta^{i}\theta^{j}( \hat{\theta}^{k}- \frac{1}{2}\theta^{k}) \right] \, .
e^{C\left( \gamma ,\varepsilon \right)}\left\Vert \tilde{u}\right\Vert_{L^{2}\left( O\left( r,\varepsilon \right) ;H\right)}\leq N_{\gamma,\varepsilon},O\left( r,\varepsilon \right) =O_{\frac{r}{8}}\times \left[ \frac{1-\varepsilon}{2},\frac{1+\varepsilon}{2}\right]
\underline u_{0n}\in C^2([\underline g_{0n}, \underline h_{0n}]),\; \underline u_{0n}(\underline g_{0n})=\underline u_{0n}(\underline h_{0n})=0,\; \underline u_{0n}(x)>0 \mbox{for}x\in (\underline g_{0n}, \underline h_{0n}),
\Gamma \lbrack A \rbrack = \mathrm{t r}\int_{0}^{\infty}{\frac{d T}{T}}\, \mathrm{e}^{- m^{2}T}\int{\cal D}x \,{\cal P}\mathrm{e x p}\left[ - \int_{0}^{T}\! \! \! d \tau \left({\frac{1}{4}}{\dot{x}}^{2}+ i g A_{\mu}\dot{x}^{\mu}\right) \right]
I_{\tilde{N}_p^t\tilde{S}_t}(\chi_{\boldsymbol{t}}^{(\pm)})/\tilde{N}_p^t\cong\begin{cases}S_{t_0}\times S_{t_1}\times\dots\times S_{t_{(p-1)/2}}t-t_0 A_{t_0}\times S_{t_1}\times\dots\times S_{t_{(p-1)/2}}t-t_0 \end{cases}
\Bigg(\sum_{v\in\mathbb{N}^n}\frac{c(v,q)}{\mathit{gl}(v,q)}X^v\Bigg) \!\circ \mathrm{Exp}\Bigg(\sum_{v\in\mathbb{N}^n\backslash\{0\}}\!\frac{s(v,q)}{1-q}X^v\Bigg) = 1,
\frac{1}{\lambda^t}M^t \vec v = \frac{1}{\lambda^t}(\lambda^t \vec v + \sum_{m=0}^{t-1}\lambda^{t-m-1}M^{m}\vec u) &= \vec v + \frac{1}{\lambda}\sum_{m=0}^{t-1}\frac{\lambda_u^{m}\cdot{m}^{d_u}}{\lambda^{m}}\frac{1}{\lambda_u^{m}\cdot{m}^{d_u}}M^{m}\vec u
{\hat{s}}^{2}={\frac{1}{4}}({\hat{a}}_{( 1 ) ( 1 )}-{\hat{a}}_{( 2 ) ( 2 )})^{2}+{\frac{1}{2}}({\hat{a}}_{( 1 ) ( 2 )}{\hat{a}}_{( 2 ) ( 1 )}+{\hat{a}}_{( 2 ) ( 1 )}{\hat{a}}_{( 1 ) ( 2 )}) ,
H(s,t; x_1, x_2, x_3) = A_B(s,\sqrt{-t}(x_1 + x_2)/2) \, A_B (s,\sqrt{-t}(x_1 - x_2)/2) &\times A_B(s,\sqrt{-t}x_3) \, G((x_1 + x_2)/2, (x_1 - x_2)/2, x_3)\;.
-\langle1,f'(\phi)\phi_{k}+ \mathcal{K}*\phi_{k}\rangle = \langle\Psi_{2}^{0},\mathcal{A}_{1}\phi_{k}\rangle &= \langle\Psi_{2}^{0},\mathcal{A}_{1}\phi_{k}+ \mathcal{A}_{2}\Phi_{1}^{0}\rangle.
m=2\pi R_{1}\times \tilde{R}_{7}\times{|a_{D}(u)| \over \alpha}\times{1\over gl_{s}^{4}}={R_1 \over R_{11}l_s^{2}}{2 \pi |a_{D}(u)|\over \alpha}={1 \over g'_s l_s^{'3}}{2 \pi \tilde{R}_7 |a(u)|\over \alpha},
\frac{1}{2y}\bigg(A+\frac{\sqrt{mn}}{m}B\bigg)=\frac{\alpha}{4t^\alpha}\mathrm{e}^{-\frac{\alpha(x^2+y^2)}{4t^\alpha}}\bigg(\frac yx\bigg)^{\frac{c+\sqrt{mn}-1}{2}}I_{\frac{c+\sqrt{mn}-1}{2}}\bigg(\frac{\alpha\sqrt{xy}}{2t^\alpha}\bigg),
a := \left( \int_0^{1}(1-p y)^{-2}y^{j+1}dy\right)\left(\int_0^1 (1-py)^{-1}y^{j-1}dy\right), b := \left(\int_0^1 (1-py)^{-2}y^{j}dy\right) \left(\int_0^{1}(1-p y)^{-1}y^{j}dy\right).
\frac{\partial^2}{\partial s \partial t}[(\exp{(sX+tY)}v,\exp{(sX+tY)}v)_{z_0}]|_{s=t=0}= \frac{1}{2}((XY+YX)v,v)_{z_0}+\frac{1}{2}(v,(XY+YX)v)_{z_0}+(Yv,Xv)_{z_0}+(Xv,Yv)_{z_0}= 2(Xv,Yv)_{z_0}+2(Yv,Xv)_{z_0}.
[\begin{matrix}K&B^{T}\\B&0\end{matrix}][\begin{matrix}U^{n+1}\\P^{n+1}\end{matrix}]=[\begin{matrix}F^{n}+\frac{1}{2\delta t}M(4U^{n}-U^{n-1})\\ 0\end{matrix}],\Sigma=[\begin{matrix}K&B^{T}\\B&0\end{matrix}].
{w}_{\beta}^{~ \alpha}= - 2 d \theta^{\alpha}D_{\beta}\phi - 2 d \theta_{\beta}D^{\alpha}\phi - 2 \Pi^{\alpha \dot{\gamma}}\partial_{\beta \dot{\gamma}}\phi + 2 \Pi_{\beta \dot{\gamma}}\partial^{\alpha \dot{\gamma}}\phi \; .
Var(\hat{\pi}^a_{\tilde{h}^*, T_n}(x)) = \frac{2}{T_n^2}\int_0^{T_n}\int_0^{t}k(t,s) 1_{s < t}\, \big(1_{|t -s| \le 3 \Delta_n}& + 1_{3 \Delta_n \le |t -s| \le D}+ 1_{D \le |t -s| \le T_n}\big) ds dt & = \sum_{j = 1}^3 \hat{I}_j.
H_{n , n + 1}= ( \cos \gamma ){\cal P}_{1}+ \frac{\sin 3 \gamma}{\sin 2 \gamma}{\cal P}_{2}+ \Bigl ( \frac{\sin 3 \gamma}{\sin 2 \gamma}+ ( \sin \gamma ) \frac{\cos 3 \gamma}{\sin 3 \gamma}\Bigr ){\cal P}_{3}.
lnL(\mu,\sigma^{2})=\sum_{i=1}^{n}lnf(x_{i}|\mu,\sigma^{2})=-\frac{n}{2}ln(2\pi)-\frac{n}{2}ln\sigma^{2}-\frac{1}{2\sigma^{2}}\sum_{i=1}^{n}(x_{i}-\mu)^{2}.
\phi'(z) = \frac{M_d(z) - zM_d'(z)}{M_d(z)^2},\phi''(z) = \frac{-zM_d(z)M_d''(z) - 2M_d(z)M_d'(z) + 2z(M_d'(z))^2}{M_d(z)^3}.
d(m,n) = \sum_{\underset{r \not \in D}{1 \leq r \leq m}}u_d(m-r,n+r) + f_d(m-r,n+r), f(m,n) =\sum_{\underset{r \not \in E}{1 \leq r \leq m}}d(m-r,n) + u(m-r,n), u(m,n) =\sum_{\underset{r \not \in C}{1 \leq r \leq m}}d_u(m-r,n-r) + f_u(m-r,n-r).
[ \tilde{\Sigma}( x ) , \tilde{\Sigma}( y ) ] = [{\partial}^{+}\tilde{\Sigma}( x ) ,{\partial}^{+}\tilde{\Sigma}( y ) ] = 0 , \; [ \tilde{\Sigma}( x ) ,{\partial}^{+}\tilde{\Sigma}( y ) ] = i{\delta}( x^{--}y^{-}) .
A_{k}(\theta)=M_{2k}(\theta)\prod_{j=0}^{k-1}\cos \pi(\theta+j \alpha)=\left(\begin{array}{cc}\tilde{Q}_{2k}(\theta) -\tilde{P}_{2k-1}(\theta+\alpha) \cos \pi \theta -\tilde{Q}_{2k-1}(\theta) -\tilde{P}_{2k-2}(\theta+\alpha) \cos \pi \theta \end{array}\right)
h(x) \le \Bigl(g_2(x)+\frac{2}{6^3}(2^3x-3)(11-3^3x)\Bigr)\Bigr|_{x=\frac{607}{1512}}+ \frac1{10\cdot 6^3}- \eta &=-\frac{115305379420414389410460002655935827}{212811849807846130733401211811055128000}<0,
g_{i\bar{j}}=\Big( \begin{array}{cc}\frac{1}{1-|\xi^{1}|^{2}}\frac{1}{1-\xi^{1}\bar{\xi}^{2}}\frac{1}{1-\xi^{2}\bar{\xi}^{1}}\frac{1}{1-|\xi^{2}|^{2}}\end{array}\Big).
A(T) = \begin{bmatrix}O D_{l(T)}O \cdots O O D_{l(T)}^T O D_{l(T)-1}\cdots O O O D_{l(T)-1}^T O \cdots O O \vdots \vdots \vdots \ddots \vdots \vdots O O O \cdots O D_2 O O O \cdots D_2^T O \end{bmatrix},
( \mathcal Q_0 + \ell R_b(\ell, \lambda(\ell) )^{-1}= \mathcal Q_0^{-1}+ \ell \mathcal Q_0^{-1}R_b(\ell, \lambda(\ell)) \mathcal Q_0^{-1}+ \mathcal O(\ell^2) begin{align*}4pt] = \mathcal Q_0^{-1}+ \ell \cdot b(0) \cdot I + \mathcal O(\ell^2 \cdot \ln \ell) .
\int_0^{1+3m/2-\sigma+\epsilon}\frac{n_{X_1}(r,\chi)}{r}dr &\quad\quad= \frac{1}{2\pi}\int_0^{2\pi}\log{|X_1( 1+3m/2 + (1+3m/2-\sigma+\epsilon)e^{i\theta}, \chi )|}d\theta &\quad\quad\quad- \log{|X_1(1+3m/2, \chi)|}.
Z^{++}_{n,p}=\frac{(n-p+a_{12}-a_{123}+j^{(3)}+1)(n-p+a_{12}-a_{123}-j^{(3)})}{(n-p+a_{12}-a_{123}+j^{(3)})(n-p+a_{12}-a_{123}-j^{(3)}-1)}\,.
\overline X_u \cong_{ht}\begin{cases}S^2 \vee \bigvee_{i\in I}S^{d}\textnormal{if}n < N<2n \left(\bigvee_{i=1}^{e-1}S^1\right)\vee\left( \bigvee_{i=1}^r S^{1}\right) \textnormal{if}N = n e' \textnormal{points}\textnormal{if}N = n-1, \end{cases}
{\cal Z}_{_0}= \int D \overline{{\xi}}D \xi \exp \left\{i \int d^{2}x \left[ - i \overline{{\xi}}( x ) \partial \! \! \! / \partial \! \! \! /^{\dagger}\partial \! \! \! / \xi ( x ) + m \overline{{\xi}}( x ) \partial \! \! \! / \partial \! \! \! / \xi ( x ) \right] \right\}.
\frac{\partial S}{\partial t}+H[f, \delta S/\delta f,t]-4CP^{-1/2}\sum_{a,b}\int dx\,\left(K^{ab}_x\frac{\delta^2 P^{1/2}}{\delta f^a_x \delta f^b_x}+\frac{\delta K^{ab}_x}{\delta f^a_x}\frac{\delta P^{1/2}}{\delta f^b_x}\right) =0 .
\sum_{\substack{s \geq i, t \geq j (s,t) \neq (i,j)}}( C_k )_{is}( X_{k+1})_{st}( D_k )_{tj}= \sum_{t > j}( C_k )_{ii}( X_{k+1})_{it}( D_k )_{tj}+ \sum_{\substack{s>i}}( C_k )_{is}\underbrace{\sum_{t>j}( X_{k+1})_{st}( D_k )_{tj}}_{:=(X_k^D)_{sj}},
\sum_{n=0}^{p-1-k}\binom kn(-\frac12)^n=\sum_{n=0}^{p-1-k}\binom{-k+n-1}n(\frac12)^n &\equiv\sum_{k=0}^{p-1-k}\binom{p-1-k+n}n(\frac12)^n=2^{p-1-k}\equiv\frac1{2^k}\pmod p
\mathcal M_{n}= e^{\beta_n \Delta_n}\frac{\lambda_n}{\beta_n}\frac{(x \star \nu) \; (\mathrm{\mathrm{sgn}}(x) \star \nu)}{|x| \star \nu}\big(\frac{e^{\beta_n \Delta_n}-1}{\beta_n \Delta_n}-1\big),
\forall\,i=1,2, \sum_{k=0}^{4}w^{k}_{s_i}\leq t_{s_i}, & \forall\,i=1,2, w^{1}_{r,}+w^{2}_{r,}+w^{i}_{r,}\leq t_r, & t_{s_1}+ t_{s_2}+ t_r \leq 1.
\frac{\partial}{\partial t}k(t) = \Delta_{L,\bar{g}_{\varepsilon(t),\delta(t)}}k(t) - Q_{\bar{g}_{\varepsilon(t),\delta(t)}}(k(t)) + \beta(t) \, k(t) - \Big ( \frac{\partial}{\partial t}\bar{g}_{\varepsilon(t),\delta(t)}+ 2 \,_{\bar{g}_{\varepsilon(t),\delta(t)}}\Big ) + \beta(t) \, \bar{g}_{\varepsilon(t),\delta(t)}.
\langle \phi^{-1}(-Ad_{g_1^{-1}}X), U_1\rangle_1 + \langle \phi^{-1}X, U_2\rangle_1 = -\langle Ad_{g_1^{-1}}X, U_1\rangle_0 + \langle X, U_2\rangle_0 =-\langle X, Ad_{g_1}U_1\rangle_0 + \langle X, U_2\rangle_0 = 0
G ( p )_{\mu \nu}= - i \left[ \int_{0}^{\infty}d s \frac{\rho_{1}( s )}{p^{2}- s^{2}+ i \epsilon}{\cal P}_{\mu \nu}( p ) - i M \epsilon_{\mu \nu \rho}\frac{p^{\rho}}{p^{2}}\int_{0}^{\infty}d s \frac{\rho_{2}( s )}{p^{2}- s^{2}+ i \epsilon}\right]
L_1 =f(1) = (a_0+a_4)+(a_1+a_5)+(a_2+a_6)+(a_3+a_7), L_2 =f(-1) = (a_0+a_4)-(a_1+a_5)+(a_2+a_6)-(a_3+a_7), H = |f(i)|^2= ((a_0+a_4)-(a_2+a_6))^2 + ((a_1+a_5)-(a_3+a_7))^2, K = |f(w_8)f(-w_8)|^2=((a_0-a_4)^2-(a_2-a_6)^2 +2(a_1-a_5)(a_3-a_7))^2 \;\;\;\;\;\;\;\;+ ((a_3-a_7)^2-(a_1-a_5)^2+2(a_0-a_4)(a_2-a_6))^2.
\left(\Pi_{\lambda_1,\lambda_2}g\right)^{\lambda_1,1-\lambda_2}_{i+\frac{1}{2},j-\frac{1}{2}}= g^{\lambda_1,1-\lambda_2}_{i+\frac{1}{2},j-\frac{1}{2}}, \mbox{if}\left(i,j\right) \in (\Bbbk_2^-,\Bbbk_1^+)
{\cal C}\left[ \begin{array}{c}{{\alpha}_{1}\ \cdots \ \alpha_{g}}\\{{\beta}_{1}\ \cdots \ \beta_{g}}\\ \end{array}\right] ={\cal C}\left[ \begin{array}{c}{{\alpha}_{1}}\\{{\beta}_{1}}\\ \end{array}\right]{\cal C}\left[ \begin{array}{c}{{\alpha}_{2}}\\{{\beta}_{2}}\\ \end{array}\right] \cdots{\cal C}\left[ \begin{array}{c}{{\alpha}_{g}}\\{{\beta}_{g}}\\ \end{array}\right]
\|\mathbf{W}^{\mathrm{RF}}- \mathbf{P}_\mathrm{p}\|_\mathrm{F}^2 = 2R - 2\mathrm{Tr}\left\lbrace \Re\left[ \mathbf{W}^{\mathrm{RF}}\mathbf{P}_\mathrm{p}^\mathrm{H}\right] \right\rbrace,
\tilde{u}( x ) = ( 2 \pi )^{- 3 / 2}\int \frac{d^{3}p}{\sqrt{2 p^{0}}}\Bigl ( - c_{1}( \vec{p}) e^{- i p x}+ c_{2}^{+}( \vec{p}) e^{i p x}\Bigl ) \quad ,
\sum_{i\in\mathcal{I}}\Phi_{i,T}(X) = I_2, ~~~ \sum_{i\in\mathcal{I}}\partial_{x_j}\Phi_{i,T}(X) = \mathbf{0}_{2\times2}, ~~~ \sum_{i\in\mathcal{I}}\partial_{x_jx_k}\Phi_{i,T}(X) = \mathbf{0}_{2\times2}, ~~j, k=1,2.
|\overline{P}(t)\rangle=\frac{1}{2q}(|P_{L}\rangle e^{-\frac{i}{\hbar}(m_{L}-\frac{i}{2}\gamma_{L})t}-|P_{H}\rangle e^{-\frac{i}{\hbar}(m_{H}-\frac{i}{2}\gamma_{H})t})=-\frac{p}{q}g_{-}(t)|P\rangle+g_{+}(t)|\overline{P}\rangle
g^{\ast}(u,v)\leq G(u,v):=\inf\bigg\{\sum_{n\geq1}\gamma_{n}(\ln\gamma_{n}-1)\mid\gamma_{n}\geq0,\u=\sum_{n\geq1}n\gamma_{n},\v=\sum_{n\geq1}(-1)^{n}\varsigma_{n}\gamma_{n}\bigg\}
\sum_{r = 1}^{\omega}\frac{e(A_r)}{|E(G_n)|}= 1 - \sum_{r = 1}^{\omega}\frac{\xi_r}{|E(G_n)|}= 1 - \frac{\omega \cdot O(n^{\max\{1-1/m,pA_1\}}\log^5 n)}{\Theta(n)}&= 1 - O \left( n^{\max \{-1/m,-1+pA_1\}/ 2}\log^{9/2}n \right).
{\cal A}_{\mu}( t ) = \int_{0}^{1}d \lambda f ( \lambda , t^{2}) \sum_{\sigma \sigma^{\prime}, \, \pm}\overline{{\psi}}_{\sigma}^{\pm}( ( \lambda - 1 ) t ) \gamma^{\mu}\psi_{\sigma^{\prime}}^{\pm}( \lambda t ) \quad .
\frac{1}{2}\frac{d}{dt}\|A u^N\|_{L^2}^2 + \nu \|A^{1+s/2}u^N\|_{L^2}^2 = -\langle A^2 u^N,u^N\cdot \nabla u^N\rangle -\langle A^2 u^N,\mathcal{U}^\alpha(u^N,u^N)\rangle.
\rho_{M, N}= \rho_{M, N}(\theta) = \begin{cases}\frac{N \sin\theta \sin\frac{\theta}{M + 1}}{\pi \sin(1-\frac{1}{M + 1})\theta}, \theta \in (0, \pi), 2^{\frac{1}{3}}\Big(\frac{N}{M+1}\Big)^{\frac{2}{3}}, \theta = 0. \end{cases}
[U_u-\tilde{u}_{\tilde{\sigma}}X_{\sigma u}](u_0-\Delta u)= [X_{0,0}\tilde{u}_0-(D_{\mu}X_{\rho})(D_{\mu}X_{\rho})\Delta \tilde{u}] -\tilde{u}_{\tilde{\sigma}\tilde{\rho}}(D_{\mu}X_{\rho})(D_{\mu}X_{\sigma}) + \tilde{u}_{\tilde{\sigma}}(X_{\sigma,0}-V\cdot X_{\sigma})+V\cdot U-U_0, \rho\ne \sigma,
0 = g a_{0}+ a_{0}^{3}+ 2 a_{0}\Sigma_{2}+ 2 \Sigma_{2}a_{0}+ \sum_{n > 0}{\frac{1}{n}}\left( a_{n}^{\dagger}a_{0}a_{n}+ a_{n}a_{0}a_{n}^{\dagger}- a_{0}\right) + 6 \Sigma_{3}\; .
|m_u^\gamma(A)-m_u^\gamma(B(y,2^{-i}))| \le m_{|u-m^\gamma_u(A)|}(B(y,2^{-i})) &\le C 2^{-is}\sum_{k=i-4}^{i-2}m_{g_k}^{\gamma/C}(B(y,2^{-i+3})) &\le C 2^{-is}\sum_{k=i-4}^{i-2}M_{\gamma/C}g_k(x).
\mathbb E_{P_\sigma}\left[\left(\psi\left(\frac{1}{\sqrt{\sigma(A)}}W^{(\sigma)}_A\right)\right)\left(\psi\left(\frac{1}{\sqrt{\sigma(B)}}W^{(\sigma)}_B\right)\right)\right]=[\psi]\left(\frac{\sigma(A\cap B)}{\sqrt{\sigma(A)\sigma(B)}}\right).
T_{1}= \frac{- N_{f}/ 2 + N_{c}}{2 \pi i}, \ \T_{0}= - \frac{1}{4 \pi i}\sum_{i = 1}^{N_{f}}m_{i}, \ \ \frac{\partial F}{\partial T_{1}}= 2 u - \frac{1}{2}\sum_{i = 1}^{N_{f}}m_{i}^{2}
\sideset{}{^\star}\sum_{a,b,c\bmod{q_i}}e_{q_i}(a+b+c)\sideset{}{^\star}\sum_{\chi_i\bmod{q_i}}\chi_i(rabc)\bar{\chi_i}(n\ell) = \sideset{}{^\star}\sum_{a,b,c\bmod{q_i}}e_{q_i}(a+b+c)\left\{\phi(q_i)1_{rabc\equiv n\ell\bmod{q_i}}-1\right\}= \phi(q_i)\sideset{}{^\star}\sum_{a,b\bmod{q_i}}e_{q_i}(a+b+n\ell\overline{rab})-c_{q_i}(1)^3.
\left(\begin{matrix}\gamma^{(1)}\gamma^{(2)}\end{matrix}\right)'=\Lambda_\beta\left(\gamma^{(1)},\gamma^{(2)}\right):=\left(\begin{matrix}-\lambda_1\sin\beta -\lambda_2\cos\beta \lambda_1\cos\beta -\lambda_2\sin\beta\end{matrix}\right) \left(\begin{matrix}\gamma^{(1)}\gamma^{(2)}\end{matrix}\right).
\begin{array}{l}T_i=L_{\bar b}L_{a}-L_{\bar a}L_{b}, T_{i+1}=R_{\bar b}R_{a}-R_{\bar a}R_{b}, T_{i+2}=R_{\bar ab-\bar ba}+L_{b}L_{\bar a}-L_{a}L_{\bar b},\end{array}
\int e^{3i\mu^2/4}e^{ia\ln|\eta^2-\mu^2/\eta|}d\mu = e^{i a\ln |\eta|^2}\int_{|\mu| \le |\eta|^{3/2}/2}e^{3i\mu^2/4}\left( 1 + \frac{\mu^2}{|\eta|^3}\phi \left( \frac{\mu^2}{|\eta|^3}\right) \right) d\mu & = e^{i a\ln |\eta|^2}|\eta|^{3/2}\int_{|z| \le 1/2}e^{3i|\eta|^3 z^2/4}\left( 1 + z^2 \phi ( z) \right) dz
\Gamma \left[ \Phi \right] =\frac{1}{2}\int_{p}\Phi_{p}\Phi_{-p}C^{-1}(p,\Lambda )+S\left[ \Phi \right] +\frac{1}{2}{tr}\ln \left.\left( C^{-1}+\frac{\delta^{2}S}{\delta \phi \delta \phi}\right) \right|_{\phi =\Phi}+\cdots
\tilde{S}_{3}(U)(x, t) = \sum_{n = 1}^{\infty}\left(\int_{0}^{t}e^{L_{n}(t - s)}\exp_{\tau}(D_{n}, t - \tau - s) U_{n}(s) \mathrm{d}s\right) \sin(\tfrac{\pi n}{l}x).
e=1-f=z^3\left(\begin{matrix}\alpha^{6}1 \alpha^{5}\alpha^{6}\end{matrix}\right) + z^2 \left(\begin{matrix}\alpha^{3}\alpha^{2}\alpha^{2}\alpha^{6}\end{matrix}\right) + z\left(\begin{matrix}\alpha^{4}\alpha^{4}1 0\end{matrix}\right) + \left(\begin{matrix}1 0 \alpha^{6}0\end{matrix}\right).
T_n^{(s)}(x) =2^s\sum_{0\le j\le (n-s)/2}n(n-1-j)^{\underline{s-1}}\binom{s+j-1}{s-1}T_{n-s-2j}(x) &-[\![n-s]\!]\;2^{s-1}n\,((n+s)/2-1)^{\underline{s-1}}\binom{(n+s)/2-1}{s-1}.
\lambda_{k}^{\epsilon, v_k}= \inf \Bigl \{\operatorname{Re}\bigl\{\lambda\bigr\}\, \bigl \vert \, \lambda \in \operatorname{Sp}\bigl\{\mathcal{L}_{k}^{\epsilon, v_k}\bigr\}\Bigr\}, \,\,\, k =1,2, \ldots, n.
|\langle n_1,\ldots,n_r;\bar{n}_1,\ldots,\bar{n}_r| \hbox{\boldmath $\eta$};M;C_r\rangle|^2={M!\over{n_1!\cdots n_r!\bar{n}_1!\cdots \bar{n}_r!}}|\eta_1|^{2n_1}\cdots|\eta_r|^{2n_r}|\eta_{-1}|^{2\bar{n}_1}\cdots|\eta_{-r}|^{2\bar{n}_r}.