harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{s=1}^{2^n-1}f_\mathcal{A}(s) =\sum_{j=0}^{n-1}\sum_{s=2^j}^{2^{j+1}-1}f_\mathcal{A}(s)\approx\sum_{j=0}^{n-1}c(\mathcal{A},1)3^j &=c(\mathcal{A},1)\left(\frac{3^n-1}{2}\right)\approx \frac{1}{2}c(\mathcal{A},1)3^n.
\mathcal{H}_{l(T)}= x - 1 , \mathcal{H}_j = x - (c_j + 1) - \dfrac{c_j}{\mathcal{H}_{j+1}}(\forall j \in \overline{2, l(T)-1}) , \mathcal{H}_1 = x - c_1 - \dfrac{c_1}{\mathcal{H}_2},
\gamma=\sum_{k=2}^{\infty}(-1)^{k}\frac{\lfloor log_{2}k\rfloor}{k}=\frac{1}{2}-\frac{1}{3}+2(\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{7})+3(\frac{1}{8}-\cdot\cdot\cdot-\frac{1}{15})+\cdot\cdot\cdot
K_{200}(k^2) = \int_{-\pi/2+3\epsilon/2}^{\pi/2-\epsilon/2}d \theta_2 \int_{-\pi/2+\epsilon/2}^{\theta_2-\epsilon}d \theta_1~{\cal F}(\theta_1)^{k^2-1}{\cal F}(\theta_2)^{k^2-1}\left| 3 \sin \frac{\theta_2 - \theta_1}{3}\right|^{-2 k^2}.
[ i \hbar \gamma^{\mu}\partial_{\mu}-{\frac{M c}{\phi ( x )}}{\frac{1 + \gamma^{5}}{2}}- M c \phi ( x ){\frac{1 - \gamma^{5}}{2}}+ i \hbar \gamma_{-}^{\mu}{\frac{1}{\phi ( x )}}\partial_{\mu}\phi ( x ) ] \psi^{D}( x ) = 0 \, .
\omega_{r}^{( 2 )}= 4 8 \rho^{3}d t_{0}\int_{S_{r}^{2}}{\frac{\rho d x_{0}^{i}+ d \rho \, ( x - x_{0})^{i}}{( t_{0}^{2}+ \rho^{2}+ ({\bf x}-{\bf x}_{0})^{2})^{4}}}\varepsilon_{i j k}d x^{j}d x^{k}.
{_{p}F_{q}}\left({a_{1}, \ldots, a_{p}\atop b_{1}, \ldots, b_{q}}; x\right) = \sum_{n=0}^{\infty}\frac{x^{n}}{n!}\frac{\prod_{j=1}^{p}(a_{j})_{n}}{\prod_{j=1}^{q}(b_{j})_{n}},
\begin{pmatrix}K_3 K_4\end{pmatrix}= \lim_{x_1\to0}\sqrt{2\pi x_1}\begin{pmatrix}t_3 D\end{pmatrix}=F\sqrt{\frac{2}{\pi a}}\frac{1}{H_{33}H_{44}-H_{34}^2}\begin{pmatrix}\delta_3H_{33}H_{44}-\gamma H_{34}^2 (\gamma-\delta_3)H_{33}H_{34}\end{pmatrix}.
u_{k}^{T}u_{k^{\prime}}^{*}=\sum_{n=0}^{N-1}(e^{\frac{i2\pi}{N}kn})(e^{\frac{i2\pi}{N}(-k^{\prime})n})=\sum_{n=0}^{N-1}e^{\frac{i2\pi}{N}(k-k^{\prime})n}=N\delta_{kk^{\prime}}
\tilde{U}_{n}=|\frac{iP_{n}^{\prime}}{\rho n\omega}[\frac{J_{0}(\Lambda_{n})-1}{J_{0}(\Lambda_{n})}]|\Rightarrow P_{n}^{\prime}=\tilde{U}_{n}|i\rho n\omega[\frac{J_{0}(\Lambda_{n})}{1-J_{0}(\Lambda_{n})}]|
Z_{1}\left( a , \epsilon_{1}, \epsilon_{2}\right) = \left( \epsilon_{1}\epsilon_{2}a_{2 1}\left( a_{1 2}+ \epsilon \right) \right)^{- 1}+ \left( \epsilon_{1}\epsilon_{2}a_{1 2}\left( a_{2 1}+ \epsilon \right) \right)^{- 1}.
_{}(_i (1_n^{*3}\otimes1_1)) = \tfrac{1}{16}( 1_n^{\ast 3}\otimes 1_1+3\cdot1_n^{\ast 2}i_n^{\ast}\otimes i_1-3\cdot1_n^{\ast}i_n^{\ast 2}\otimes 1_1-i_n^{\ast 3}\otimes i_1+ + j_n^{\ast 3}\otimes j_1+3\cdot j_n^{\ast 2}k_n^{\ast}\otimes k_1-3\cdot j_n^{\ast}k_n^{\ast 2}\otimes j_1-k_n^{\ast 3}\otimes k_1).
\left( \stackrel{[0,0]}{z}^{A},\stackrel{[0,0]}{z}^{A_{1}}\right) ,\;\left( \stackrel{[1,0]}{\cal P}_{1a_{0}},\stackrel{[0,1]}{\cal P}_{2a_{0}}\right),\;\left( \stackrel{[1,1]}{\lambda}_{a_{0}}\right) ,
-c_1(k-1+\lambda)\frac{\Gamma(n+1-\delta_{ij})}{\Gamma(n+1-\delta_{k-1, j})}\sum_{m=1}^nh(m) \rightarrow (-1)^{i-k+1}\frac{k-1+\lambda}{i-k+1}\prod_{d=k}^{i-1}\left(\frac{\lambda+d}{i-d}\right) &= (-1)^{i-k+1}\prod_{d=k-1}^{i-1}\left(\frac{\lambda+d}{i-d}\right).
\|O_1y_1+y_2\| \le \|O_1y_1+z_2\|+\|z_2-y_2\| &\le(1+\alpha)\|y_1+z_2\|+\|z_2-y_2\| &\le(1+\alpha)\|y_1+y_2\|+(2+\alpha)\|z_2-y_2\| &<(1+\alpha)\|y_1+y_2\|+(2+\alpha)\alpha\|y_2\| &\le(1+\alpha(1+(2+\alpha)A))\|y_1+y_2\|.
s_d - | Z_f| =\dim_k \left( \overline{S}/(x_0, \dots, x_{m-1},x_m^{d-r})\right)_d &=\dim ( \overline{S}/(x_0, \dots, x_{m-1}))_d - \dim ( \overline{S}/(x_0, \dots, x_{m-1}))_{r}+ \dim ( \overline{S}/(x_0, \dots, x_{m-1}))_0 &=\left( \binom{d+n-m}{d}-n+m-1 \right)- \binom{n+r-m}{r}+1.
{R}_{M}f =\sum\limits_{k=-\infty}^\infty\sum\limits_{\{j\in \Bbb Z:\, |k-j|>{M}\}}{D}_k{D}_j =\sum\limits_{\{\ell \in \Bbb Z:\,|\ell|> M\}}\sum\limits_{k=-\infty}^\infty D_kD_{k+\ell}f =\sum\limits_{k=-\infty}^\infty D_k(I-S_{k+M})f +\sum\limits_{k=-\infty}^\infty D_kS_{k-M-1}f.
\Delta(\delta_{\varphi}) = \Delta(1)(\delta_{\varphi}\otimes \delta_{\varphi})\Delta(1),S(\delta_{\varphi}) = \delta_{\varphi}^{-1},\varepsilon(\delta_{\varphi}a) = \varepsilon(a\delta_{\varphi}) = \varepsilon(a),\forall a\in A,
W ( x ) ={\cal W}( x , 0 ) = \langle \, 0 | \phi ( x ) \phi ( 0 ) | 0 \, \rangle = \Delta_{+}( x ) ={\displaystyle \int \! \! \frac{\mathrm{d}^{4}k}{( 2 \pi )^{4}}}\, \, \delta ( k^{2}- m^{2}) \, \theta ( k^{0}) \mathrm{e}^{- \mathrm{i}k x}\, .
\begin{array}{lll}v f_{v'u'}= v (q^{-1}u'v') = q^{-1}(q^2 u'v + (q-q^3) v'u) v' = q u' v v' + (1-q^2) v' u v' ~= q^2 u' v' v + (q^2-q^4) (v')^2 u, f_{vv'}u' = (qv'v) u' = qv' (q^2 u'v + (q-q^3) v'u) = q^2 u'v'v + (q^2-q^4) (v')^2 u , \end{array}
D_{\mathcal{L}}=\sum_{Y\in \mathcal{I}^G(X)}v_{\mathcal{L}}(\mu_Y) Y + \sum_{D\in \mathcal{D}_X}v_{\mathcal{L}}(\rho(D)) \overline{D}+\sum_{D\in \mathcal{D}\setminus \mathcal{D}_X}n_{\mathcal{L},D}\overline{D}.
G^{11}_{\hat\mu\hat\nu}=\left( \begin{array}{cc}e^{-\frac{2}{3}\phi}G_{\mu\nu}+e^{-\frac{2}{3}\phi}C_{\mu}C_{\nu}e^{\frac{1}{3}\phi}C_{\mu}e^{\frac{1}{3}\phi}C_{\nu}e^{\frac{4}{3}\phi}\end{array}\right).
\Psi_{k\ell}(x) =\frac{1}{8\pi \nu}\left(\frac{\delta_{k\ell}}{\|x\|}+ \frac{x_k x_\ell}{\|x\|^3}\right), \Theta (x)= \frac{x}{4\pi \|x\|^3}, x\in \mathbb{R}^3.
\left(\frac{1}{2}~ \mathbb{E}^Q_{n, 0}\left| \frac{d\mathbb{Q}_{n}}{d\mathbb{Q}_{n, 0}}- 1 \right| \right)^2 \leq \mathbb{E}^Q_{n, 0}\left( \frac{d\mathbb{Q}_{n}}{d\mathbb{Q}_{n, 0}}- 1 \right)^2 = \mathbb{E}^P_{n, 0}\left( \frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n, 0}}\right)^2 - 1,
X \Psi_{n_1 n_2 m}(\mu,\nu,\varphi;\delta_1,\delta_2) =- \alpha\frac{n_1 - n_2 + \frac{\delta_1 - \delta_2}{2}}{n + \frac{\delta_1+\delta_2}{2}}\,\Psi_{n_1 n_2 m}(\mu,\nu,\varphi;\delta_1,\delta_2),
i_{k+1}(B_{I,[m]}) ={\rm argmax}_{i \in I\backslash I_k(B_{I,[m]})}|\langle v_{k+1}(B_{I,[m]}) , B_{i,[m]}^\top\rangle| &={\rm argmax}_{i \in [n]\backslash I_k(B)}|\langle v_{k+1}(B) , B_{i,[m]}^\top\rangle|=i_{k+1}(B).
V_2(R_A, R_B) =V_2(L_1(A)+L_2(A) , L_1(B)+L_2(B))=V_2(L_1(A), L_2(B))+V_2(L_2(A), L_1(B)) &=\frac{1}{2}\left( |\pi_1(A)| | \pi_2(B)|+|\pi_2(A)| |\pi_1(B)| \right).
k =\begin{cases}\tilde{k}M_0 \setminus B(p_0, 2 t^{1/4}) (1 - \chi_{t}(r) )\phi_{t}^* \tilde{k}+ \chi_{t}(r) t^{2}k_{1}(1/2) t^{-1/4}< r < 2 t^{-1/4}t^{2}k_{1}r < (1/2) t^{-1/4}\end{cases},
\psi_{(y_1,y_2)}^{\prime\prime}(t) = (\kappa_1 + \kappa_2 + 1)t^{\kappa_1 + \kappa_2}\left[F_{(y_1, y_2)}(t) - \lambda(\kappa_1 + \kappa_2 + 2)\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}dz\right] + t^{\kappa_1 + \kappa_2 + 1}F_{(y_1, y_2)}^{\prime}(t).
\langle{\cal A}\rangle_0 \lim_{\tau^\prime \rightarrow \infty}\langle{\cal U}_{\tau^\prime}({\scriptstyle{\cal AB}}){\cal B}\rangle_0 \; - \; \langle{\cal A}\rangle_0\lim_{\tau^\prime \rightarrow \infty}\langle{\cal U}_{\tau^\prime}({\scriptstyle{\cal B}}){\cal B}\rangle_0 = 0 \; .
\begin{aligned}\int_0^{G(\tau)}U(y)\, dy = \Bigl(\log\frac{e^2\|G\|_\infty}{G(\tau)}\Bigr)^{-1}\, , G(\tau)U(G(\tau)) = \Bigl(\log\frac{e^2\|G\|_\infty}{G(\tau)}\Bigr)^{-2}\, , \ \mbox{and}U(y)+y U'(y) = 2 y^{-1}\Bigl(\log\frac{e^2\|G\|_\infty}{y}\Bigr)^{-3}> 0\mbox{for all}0<y<\|G\|_\infty\, .\end{aligned}
g_{\le \iota}= \sum_{\ell=-1}^\iota \sum_{m=1}^{M_{\ell}}\chi_{m}^{(\ell)}g_{m}^{(\ell)}+ \sum_{\ell=-1}^\iota \sum_{m=1}^{M_{\ell}}(1-\chi_{m}^{(\ell)}) g_{m}^{(\ell)}=: g_{\le \iota}^{(1)}+ g_{\le \iota}^{(2)}.
\lim_{t\to 0}\frac{\pi^{-}(a_t\cdot\psi_t - a\cdot\psi)}{t}= \pi^{-}\left(\lim_{t\to 0}\frac{a_t\cdot\psi_t - a_t\cdot\psi}{t}+ \lim_{t\to 0}\frac{a_t - a}{t}\cdot\psi\right) & = \pi^{-}( b\cdot\phi + a\cdot\psi).
\varphi( \bar{X}_\tau ) = \varphi( S_{t_0, T}X_{t_0}) + \int_{t_0}^\tau ( \mathcal{L}^S_{s, T}\varphi )( X_s, Y_s, Z_s ) \, ds+ \int_{t_0}^\tau \varphi'( S_{s, T}X_s ) \, S_{s, T}\, Z_s \, dW_s .
{\mathrm d}\overline{X}_{t, n}\, =\, b(t, \overline{X}_{t, n}, u \cdot \delta_{\overline{X}_{t,n+1}}+ (1 - u) \cdot \mathcal L_{\overline{X}_{t,n}}){\mathrm d}t +{\mathrm d}W_{t, n}\, ; t \ge 0 \, ,
\frac{\dim(\mathbb{Z}(\mathcal{T}_\xi)\rtimes\mathcal{S})}{n+1}= d^{2n}(-1)^{\lvert\Phi^+\rvert}\xi^{-\langle 2\rho,2\rho\rangle}\left(\sum_{w\in\mathfrak{S}_{n+1}}(-1)^{l(w)}\xi^{\langle 2\rho,w\bullet 0\rangle}\right)^{-2}.
\bigg\langle \tau_1, \tau_2, \rho \ \bigg\vert\ \begin{matrix}\tau_1 \rho \tau_2=\rho^2 \tau_2 \rho^{n}\tau_1=\rho^{n+1}\tau_2 \rho^n \tau_1=\tau_1 \rho \tau_2 \rho^{n-1}\end{matrix}\ \bigg\rangle
\left\lbrace\begin{array}{llr}\partial_t u + \partial_x u +\partial^3_x u - \partial^5_x u+u^p \partial_x u +a(x)u= 0, \mbox{in}\Q_{T}, u(t,0)=u(t,L)= \partial_x u(t,0)=\partial_x u (t,L)=\partial^2_xu (t,L) = 0, \mbox{on}\ [0,T], u(0,x) = u_{0}(x), \mbox{in}\ [0,L],\end{array}\right.
|\boldsymbol{v}|_{\boldsymbol{H}^{\emph{\textbf{curl}},\frac{1}{2}}}^2 := T\|\emph{\textbf{curl}}\, \boldsymbol{v}_0^c\|_{\Omega}^2 + \frac{T}{2}\sum_{k=1}^{\infty}\Big( k\omega \|\boldsymbol{v}_k\|_{\Omega}^2 + \|\emph{\textbf{curl}}\, \boldsymbol{v}_k\|_{\Omega}^2 \Big)
f(S_1, S_2, S_3) = \frac{\exp\left(-i(\mu_1 + S_1 \cdot \lambda - S_3 \cdot \lambda)v_1 -i (\mu_2 + S_2 \cdot \lambda - S_3 \cdot \lambda)v_2\right)}{(-1)^{|S_2|}(v_1-v_2)^{|S_1|+|S_2|-2}(2v_1+v_2)^{|S_1|+|S_3|-2}(v_1+2v_2)^{|S_2|+|S_3|-2}}
\overline{\epsilon}(p_{Y|X,S},p_S)=\max\limits_{p_{\hat{X}|S}\in\mathcal{P}}\sum\limits_{x\in\mathcal{X}}\min\limits_{s\in\mathcal{S}}p_{\hat{X}|S}(x|s), &\overline{q}(p_{Y|X,S},p_S)=\max\limits_{p_{\hat{X}|S}\in\mathcal{P}}\min\limits_{x\in\mathcal{X}_+,s\in\mathcal{S}}\frac{p_{\hat{X}|S}(x|s)}{\sum_{s'\in\mathcal{S}}p_{\hat{X}|S}(x|s')}.
T_{i}f\left( x\right) :=\frac{\partial}{\partial x_{i}}f\left(x\right) +\kappa\sum_{j=1,j\neq i}^{N}\frac{f\left( x\right) -f\left(x\left( i,j\right) \right)}{x_{i}-x_{j}}, \Delta_{\kappa}f\left( x\right) :=\sum_{i=1}^{N}T_{i}^{2}.
\begin{cases}v_{t}(t,x) +v_{x}(t,x)b_{t}+ \frac{1}{2}v_{xx}(t,x)\widetilde{\sigma}_{t}\ \ \ \ \ \ + E\big[f(t,x,v(t,\eta_{t}),v(t,x),v_{x}(t,x)\sigma_{t})\big]=0, \ \ \ (t,x)\in [0,T)\times \mathbb{R}; v(T,x)=g(x), \ \x\in \mathbb{R}, \end{cases}
{\displaystyle \left(\partial^{2}\theta^{k}_{\mathbf{A}},\mathbf{v}\right)+D(\widetilde{\theta}^{k}_{\mathbf{A}},\mathbf{v})= I_1^{k}(\mathbf{v})+ I_2^{k}(\mathbf{v})+ I_3^{k}(\mathbf{v})+ I_4^{k}(\mathbf{v}).}
f_{\min,\Delta(n,r)}- f_{\min,\Delta_n}\le ((1+s)^d - (1+ds))f_{\min,\Delta_n}\le (1+\epsilon){d \choose 2}s^2f_{\min,\Delta_n}=\frac{(1+\epsilon){d \choose 2}}{r^2(x^*_{\min})^2}f_{\min,\Delta_n}.
\frac{\phi(\lambda x)}{\phi(\lambda)}\geq \frac{\phi(\lambda b^{n-1})}{\phi(\lambda)}\geq a^{n-1}=b^{(n-1)\frac{\log{a}}{\log{b}}}\geq b^{-\frac{\log a}{\log b}}x^{\frac{\log a}{\log b}}\,.
\begin{aligned}-\frac{d}{dt}(F(\Gamma_t)-F(\Gamma))^\theta =\theta (F(\Gamma_t)-F(\Gamma))^{\theta-1}B_{\Gamma_t}(u,u) \ge \theta/c\, (F(\Gamma_t)-F(\Gamma))^{\theta-1}\|dF_{\Gamma_t}\|_{Y'_{\Gamma_t}}\|[u]_{\Gamma_t}\cdot\nu_t\|_{Y_{\Gamma_t}}\ge \theta/c\, \|[u]_{\Gamma_t}\cdot\nu_t\|_{Y_{\Gamma_t}}. \end{aligned}
\big\Vert (1 - u_j^* \Phi(u_j))(x - \Phi^\omega(x))\xi_{\varphi^\omega}\big\Vert_{L^2(M^\omega)}= \big\Vert (u_j - \Phi(u_j))(x - \Phi^\omega(x))\xi_{\varphi^\omega}\big\Vert_{L^2(M^\omega)}&\leq \big\Vert [u_j - \Phi(u_j),x]\xi_{\varphi^\omega}\big\Vert_{L^2(M^\omega)}&= \big\Vert [x,\Phi(u_j)]\xi_{\varphi^\omega}\big\Vert_{L^2(M^\omega)}.
g_1 = \gcd(d,b-a) \mbox{so $g_1 \le b-a < d$}g_2 = \gcd(d,c-b) \mbox{so $g_2 \le c-b < d$}g_3 = \gcd(d,c-a) \mbox{so $g_3 \le c-a < d$}g_4 = \gcd(c,b-a) \mbox{so $g_4 \le b-a < c$}g_5 = \gcd(c,d-b) \mbox{so $g_5 \le d-b < c$}g_6 = \gcd(c,d-a) \mbox{so $g_6 \le d-a \le c$}
\Delta\frac{x^\perp}{|x|^2}=0, x^\perp\cdot\nabla\frac{x^\perp}{|x|^2}=\frac{(x^\perp)^\perp}{|x|^2}, \mbox{div}\;\frac{x^\perp}{|x|^2}=0 \mbox{in $\mathbb R^2\setminus\{0\}$}
\cap_{n=1}^{\infty}A_n^c =\left\{\omega \in \Omega: X_n(\omega) \in B_n^c \forall n \in \mathbb{N}\right\}&= \left\{\omega \in \Omega : \phi^{-1}\left((X_n(\omega))_{n=1}^{\infty}\right) \in \phi^{-1}\left(\times_{n=1}^{\infty}B_n^c \right)\right\}&= X^{-1}(B)
r_{m+1}= (1-2^{-m-3})r_m\,, s_{m+1}= (1-2^{-m-3})s_m\,, \vartheta_{m+1}=\vartheta_m + \vartheta_0 2^{-m-2}, \mu_{m+1}=\mu_m - \mu_0 2^{-m-2}\,,K_m=4^m K_0 .
\sum_{n\leq x}\Phi_k(n)^{-z}n^{\beta z}=\frac{1}{2\pi i}\int_{a-i T}^{a+i T}F_{k, \beta}(s,z)\frac{x^s}{s}\,dx +O\left( \frac{x^a}{T}\sum_{n=1}^{\infty}\frac{\Phi_k(n)^{-b}}{n^{a- b\beta}|\log \frac{x}{n}|}\right).
f_{D4}\sim \frac{1}{\sqrt{|x_1x_2x_3|}}e^{-\frac{|x_1x_2x_3|}{yz}}\quad \mbox{or}\quad\frac{1}{\sqrt{|x_1x_2x_3|}}e^{-\frac{|x_1x_2x_3|}{y\bar z}}
{\hat\theta}(\lambda)e^{iT\lambda^{1/2}Re^{i\theta}}\int w(y,\zeta_0)\phi(x_1)\psi(x'')dx_1dx'' ={\hat\theta}(\lambda)e^{iT\lambda^{1/2}R^{1/2}e^{i\theta}}\Big(\psi(x'')dx''\Big)\int w(y,\zeta_0)\phi(x_1)dx_1.
S^C_{s,\tiny\mbox{div}}[g,\beta_H,\epsilon]={1 \over \pi}\int_{\Sigma}\left\{{1 \over 360 \epsilon^2}-{1 \over 1440}\left[2{\cal R}-{\cal Q}+30\left(\frac 16-\xi\right) R-30m^2\right]\ln{\epsilon^2 \over l^2}\right\}~~~,
h_{\pi^{(0)}(1)}^{(0)}+\dots+ h_{\pi^{(0)}(l')}^{(0)}+h_1^{(0)}&>c_1(\pi^{(0)})+\dots+c_{l'}(\pi^{(0)})+(d-v) &=c_1(\pi')+\dots+c_{l'+1}(\pi'),
\sum_{s=1}^{m}\, \mu_s \,^{c}_{0}{D}^{\alpha_s}_{t}u(x,t) - a(l(u)) \: \Delta u(x,t) =f(x,t) \mbox{in}\Omega\times(0,T], u(x,t) =0 \mbox{on}\partial\Omega\times(0,T], u(x,0) =u_0(x) \mbox{in}\Omega,
\begin{pmatrix}2d 2 -d (d + \frac{1}{2}(k+1)) 2d -1 0 0 -2 d + \frac{1}{2}(k+1) \end{pmatrix}\cdot\begin{pmatrix}H_{d,k}I_{d,k}T_{d,k+1}\end{pmatrix}=
\frac{1}{2\pi\mathrm{i}n}\oint_{|z|=1}\frac{f'_{n}(z)}{f_{n}(z)}\mathrm{d}z=\frac{1}{n}\sum_{j=1}^{n}\frac{-1}{2\pi}\int_{0}^{2\pi}\frac{\partial}{\partial x}\arg(E_{j}(x)-E)\mathrm{d}x=0.
\hat w - u \geq \hat w - \hat w(0,0) - U_1 \hat \xi - U_2 \epsilon - \frac{1}{2}U_3 \frac{\epsilon^2}{|\zeta_0|}- U_\beta ((L/d)^2 + \epsilon^2)^{(\beta+1)/2}\geq -2 \| w \|_{L^\infty}- U_2 \epsilon - U_\beta ((L/d)^2 + \epsilon^2)^{(\beta+1)/2}, \textrm{on}\{\hat \zeta = \epsilon \}.
\sum\limits_{k=1}^{(j-1)/2}(j+2-2k)(pq)^{k-1}=\frac{1}{pq-1}\left(\frac{2pq}{pq-1}\left(\frac{3}{2}(pq)^{\frac{j-1}{2}}-\frac{1}{2}(pq)^{\frac{j-3}{2}}-1\right)-j\right).
\begin{aligned}\frac{d}{dt}\Gamma_{\gamma_0, \omega}^+(t) = \dot\phi(t) \big(B_{+, \omega}(t)\big)^2 - \big(B_{-, \omega}(t)\big)^2 + 2\int_{L^+_{\gamma_0}(t)}f \partial_t f ~ d\theta =:{\dot \phi}(t) \Big[ \big(B_{+, \omega}(t)\big)^2 - \big(B_{-, \omega}(t)\big)^2 \Big] +{\mathcal I}_3, \end{aligned}
\mathbb P(i_{k+1}= i) = \begin{cases}\frac{1}{n}|I_{\mathrm{kink}}| = n \frac{1}{2n}|I_{\mathrm{kink}}| < n i \in I_{\mathrm{kink}}\frac{1}{2n}+ \frac{1}{2 (n - |I_{\mathrm{kink}}|)}|I_{\mathrm{kink}}| < n i \not \in I_{\mathrm{kink}}\end{cases}
u\left( x,k_{0}\right) \mid_{\Gamma_{b}=}g_{0}\left(x_{1},x_{2},x_{0}\right) ,\partial_{n}u\left( x,k_{0}\right) \mid_{\Gamma_{b}=}g_{1}\left( x_{1},x_{2},x_{0}\right) ,\forall x_{0}\in \left[ 0,1\right] .
\lambda_u^0:= \frac{1}{2}A\cdot \big( \frac{u(A+A',B+B')-u(A+A',B)-u(A+A',B')}{A+A'}-\frac{u(A,B+B')-u(A,B)-u(A,B')}{A}\big) &-((A,B)\leftrightarrow (A',B')).
u_n(S_n,x) \sim\sum_{k=1}^K \frac{\iota_k}{(\lambda_k^\infty)^{3/2}}W_{\boldsymbol{\ell}_k}\left(\frac{x-{\boldsymbol{\ell}_k}S_n-\mathbf{y}_k^\infty}{\lambda_k^\infty}\right) ,
\int_{I^k}\phi_{k,n}(y_1,\cdots,y_k)dy_1\cdots dy_k\geq \int_{I_1^k}\phi_{k,n}(y_1,\cdots,y_k)dy_1\cdots dy_k = S(I)^k(\ln n)^{-k}\int_{(-l_n,0)^k}\phi_{k,n}\big(\gamma_n(u_1),\cdots,\gamma_n(u_k)\big)\prod_{j=1}^k\beta_n(u_j)du_1\cdots du_k.
\begin{aligned}\left|\int_{[0,1)^d}f(M_{n}x)f(M_{n+k'}x)\,dx\right| \:\:\: \leq \:\:\: \left|\int_{[0,1)^d}(f-g_k)(M_{n}x)f(M_{n+k'}x)\,dx\right| \:\:\: +\left|\int_{[0,1)^d}g_k(M_{n}x)g_k(M_{n+k'}x)\,dx\right| \:\:\: +\left|\int_{[0,1)^d}g_k(M_{n}x)(f-g_k)(M_{n+k'}x)\,dx\right| \leq \:\:\: 2C||f||_2q^{-k}\end{aligned}
H^k(V,T_V \otimes \mathcal{O}(-Y_{\vert{V}})) = H^k(V,\Omega_V^{n-1}\otimes \mathcal{O}(-Y_{\vert{V}}- K_{\mathbb{P}\vert{V}})) & \subset H^k(V, \Omega_V^{n-1}\otimes \mathcal{O}(-mY_{\vert{V}}- mK_{\mathbb{P}\vert{V}}))
g = \left\| \begin{array}{c c c}{{g_{0}^{0}}}&{{{\bf a}^{T}}}&{{g_{4}^{0}}}\\{{{\bf b}}}&{{r}}&{{{\bf c}}}\\{{g_{0}^{4}}}&{{{\bf d}^{T}}}&{{g_{4}^{4}}}\\ \end{array}\right\| \
= \frac{i m^{2}}{\pi^{2}}\Bigl ( g^{\mu \nu}- \frac{k^{\mu}k^{\nu}}{k^{2}}\Bigr ) \frac{1}{k^{2}\sqrt{1 - 4 m^{2}/ k^{2}}}\log \frac{\sqrt{1 - 4 m^{2}/ k^{2}}+ 1}{\sqrt{1 - 4 m^{2}/ k^{2}}- 1}\quad , \quad k^{2}> 4 m^{2}, k^{0}> 0 .
[|D|^{p+1},x] = |D|^{p+1}x-x|D|^{p+1}= \left(\sum_{k=0}^{p+1}\binom{p+1}{k}\delta^{p+1-k}(x)|D|^k\right)-x|D|^{p+1}= \sum_{k=0}^p\binom{p+1}{k}\delta^{p+1-k}(x)|D|^k.
u_i = (p_1^{d_1}\ldots p_i^{d_i})^{e_i}(p_1^{e_i}\ldots p_i^{e_i})^{-d_i}v_i = (p_{i+1}^{e_{i+1}}\ldots p_h^{e_h})^{d_i}(p_{i+1}^{d_{i+1}}\ldots p_h^{d_h})^{-e_i},
(\widetilde{X}_1 \widetilde{X}_2 \widetilde{X}_c)^2 \widetilde{C^\prime}_4 \widetilde{C}_4 (\widetilde{Y}_{0,n})^2 \widetilde{Y}_{1,n}\widetilde{Y}_{2,n}\widetilde{C}\widetilde{Y^\prime}_{1,n}\widetilde{Y^\prime}_{2,n}\widetilde{C^\prime}= \partial^2.
\frac{dc_0}{d\tau}=\mu h_0\frac{b+c_0}{1+c_0}-\frac{\Gamma}{K_1}\frac{c_0}{K+c_0}+\Lambda\theta_0 \frac{d\theta_0}{d\tau}=0\implies \theta_0=\theta_{00}={\rm constant,}\,\,\,\frac{dh_0}{d\tau}=0\implies h_0=h_{00}={\rm constant}.
\mathcal{E}(k_n) = &\mathrm{tr}\Big\{\big(\delta - k_n\circ \overline{k_n})^{-1}\circ\Big(k_n\circ H[n-1] \circ\overline{k_n}+ \frac{1}{2}k_n\circ\overline{\Theta[n-1]}+ \frac{1}{2}\Theta[n-1]\circ \overline{k_n}\Big)\Big\}\le 0.
(-1)^{s_1+\cdots+s_k}\det \begin{pmatrix}\binom{t_1}{-s_1}\cdots \binom{t_k}{-s_k}\binom{t_1}{1-s_1}\cdots \binom{t_k}{1-s_k}\vdots \vdots \binom{t_1}{k-1-s_1}\cdots \binom{t_k}{k-1-s_k}\end{pmatrix},
\tilde{X}^{n}_{1}(t) = \xi^{n}_{1}+ \int_{0}^{t}b\left(s,\tilde{X}^{n}_{1}(s),\tilde{\mu}^{n}(s),\tilde{u}^{n}_{1}(s)\right)ds &+ \int_{0}^{t}\sigma\left(s,\tilde{X}^{n}_{1}(s),\tilde{\mu}^{n}(s)\right)dW^{n}_{1}(s),t\in [0,T].
4 ( 2 - a^{2}z )^{2}+ ( D - 1 ) a^{2}z ( 2 - a^{2}z )^{2}+ 4 k^{2}z ( 4 - a^{2}z ) = \xi z ( 4 - a^{2}z ) ( 2 - a^{2}z )^{2}
\frac{1}{n}\sum_{x}\sup_{s \in [0,T]}| \Phi_{n}(s, \tfrac{x}{n}) | \leq M_1 \quad\textrm{and}\quad\| \Phi_{n}\|_{\infty}:= \sup_{(s,u) \in [0,T] \times \mathbb{R}}| \Phi_n (s,u)| \leq M_2,
\sum_{i=1}^{\ell(\lambda)}\left((\lambda_i-i)^m-(-i)^m\right) =: \textsc{C}_\lambda(m)=\sum_{k=1}^{m-1}\frac{(-1)^{m-k}}{(m-k)!}\frac{(m-1)!}{(k-1)!}\Phi_k(\lambda)
\begin{array}{c}{\big [ \hat{\partial}_{\mu}, \hat{\phi}\big ] \ = \ \int \mathrm{d}^{D}x \ \hat{\partial}_{\mu}\phi ( x ) \hat{\Delta}( x ) ,}\\{\big [ \hat{\partial}_{\mu}, \hat{\Delta}( x ) \big ] \ =}\\{h a t{\partial}_{\mu}\hat{\Delta}( x ) .}\\ \end{array}
L_M(w\,|\,a) = -\frac{1}{M!}B_{M, M}(w\,|\,a)\,\log(w) + \sum\limits_{k=0}^M\frac{B_{M,k}(0\,|\,a) (-w)^{M-k}}{k!(M-k)!}\sum\limits_{l=1}^{M-k}\frac{1}{l}+ R_M(w\,|\,a), R_M(w\,|\,a) = O(w^{-1}), \,|w|\rightarrow\infty, \, |\arg(w)|<\pi.
DG(\lambda,b,\Omega,0,0)(h_{1},h_{2})=\left(\begin{array}{c}D_{f_{1}}G_{1}(\lambda,b,\Omega,0,0)h_{1}+D_{f_{2}}G_{1}(\lambda,b,\Omega,0,0)h_{2}D_{f_{1}}G_{2}(\lambda,b,\Omega,0,0)h_{1}+D_{f_{2}}G_{2}(\lambda,b,\Omega,0,0)h_{2}\end{array}\right).
\int_0^t\int_{D}p_D(t-s,x,y)u_s(y) F(\delta s,\delta y)= \sum_{n=0}^\infty \int_0^t\int_{D}I_n\big(p_D(t-s,x,y)h_n(., s,y)\big) F(\delta s,\delta y) = \sum_{n=0}^\infty I_{n+1}\Big(\widetilde{p_D(t-s,x,y)h_n(., s,y)}\Big),
\begin{array}{r l l}{{\delta ( D_{\mu}^{i j}[ A ] Q^{\mu , j}- a^{i})}}&{{=}}&{{\delta ( D_{\mu}^{i j}[ A ] Q^{\mu , j}+ D_{\mu}^{i j}[ A ] \delta Q_{\mu}^{j}- a^{i}- \delta a^{i})}}\\{{}}&{{=}}&{{\delta ( D_{\mu}^{i j}[ A ] Q^{\mu , j}- a^{i})}}\\ \end{array}
T = \left( \begin{array}{c c c c c}{{T_{( 1 )}}}&{{}}&{{}}&{{0}}&{{}}\\{{}}&{{T_{( 2 )}}}&{{}}&{{}}&{{}}\\{{0}}&{{}}&{{\cdots}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{T_{( \tilde{N}_{c})}}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{0}}\\ \end{array}\right) ,
\begin{aligned}\uppercase\expandafter{\romannumeral2}=\sum\limits_{j}\frac{\mu_{j}}{\omega(P_{j})^{1/p}}(M(T(b_{j}))(x)\chi_{2\sqrt{n}P_{j}^{*}}(x)+(M(\chi_{P_{j}})(x))^{\gamma}\chi_{(2\sqrt{n}P_{j}^{*})^{c}}(x))\end{aligned}
\{g ( z , \bar{z}) , J_{0}( z_{1}) \}= \frac{1}{2}\, ( t_{0}\, g ( z , \bar{z}) ) \, \delta ( z - z_{1}) , ~ \{g ( z , \bar{z}) , \bar{J}_{0}( \bar{z}_{1}) \}= \frac{1}{2}\, ( g ( z , \bar{z}) \, t_{0}) \, \delta ( \bar{z}- \bar{z}_{1}) ,
\alpha_n = \dfrac{\rho_{n}\rho_{n-1}(1-c_{n-1})\pm \sqrt{\rho_{n}^2\rho_{n-1}^2(1-c_{n-1})^2-4\omega^2d_n\rho_{n}\rho_{n-1}}}{2\rho_{n}}, n \geq 1.
\hat{k}_{\theta}^{*}( M ) \left( \begin{array}{c}{{\zeta_{1}( \theta )}}\\{{\zeta_{2}( \theta )}}\\ \end{array}\right) ~ = ~ \pm i \left( \begin{array}{c}{{\zeta_{1}( \theta )}}\\{{\zeta_{2}( \theta )}}\\ \end{array}\right) .
\tilde K_{a,b}(u;m+2) = (\tilde d_m+ ud/du) \tilde K_{a,b}(u;m) , \tilde K_{a,b+1}(u;m) = (b^{-1}d/du) \tilde K_{a,b}(u;m) , \tilde K_{a,b+1}(u;m) = (1+b^{-1}ud/du) \tilde K_{a+1,b}(u;m) .
\widetilde\Lambda_h^{e,\triangle}:=\span\{\tilde\mu_{h,i}^e:\,\alpha_{i}^e < \alpha_{\rm{stab}}\}, \widetilde\Lambda_h^{e,\Pi}:=\span\{\tilde\mu_{h,i}^e:\,\alpha_{i}^e \ge \alpha_{\rm{stab}}\}.
(q-1)\frac{q}{q-1}\frac{q^{h}}{n}\frac{(\log n)^{k-1}}{(k-1)!}\left(H\left( \frac{k-1}{\log n}\right)+O_A\left( \frac{k}{(\log n)^2}\right)\right) &=\frac{q^{h+1}}{n}\frac{(\log n)^{k-1}}{(k-1)!}\left(H\left( \frac{k-1}{\log n}\right)+O_A\left( \frac{k}{(\log n)^2}\right)\right).
\begin{array}{c}PR'\left(\sqrt{z\over w}\right)\left(q^{S_3}\otimes{E\over z}+{E\over w}\otimes q^{-S_3}\right)= =\left(q^{S_3}\otimes{E\over w}+{E\over z}\otimes q^{-S_3}\right)PR'\left(\sqrt{z\over w}\right)\end{array}
\index{TauberienTheorem}\liminf_{N \to \infty}{1 \over N}\sum_{m=0}^{N-1}a_m \leq \liminf_{\beta \uparrow 1}(1 - \beta) \sum_{m=0}^{\infty}\beta^m a_m &\leq \limsup_{\beta \uparrow 1}(1 - \beta) \sum_{m=0}^{\infty}\beta^m a_m \leq \limsup_{N \to \infty}{1 \over N}\sum_{m=0}^{N-1}a_m
I_{1}\ = \int \delta^{3}\!\left({\vec{r}_{1}-\vec r}\right)\,\delta \left(r_{1}-t\right)\frac{d^{3}r_{1}}{4\pi\,r_{1}}=\frac{1}{4\pi r}\delta\left(r-t\right)=\frac{1}{2\pi}\delta\left(\tau^2\right)
Z(X_1+X_2|Y^2) = \frac{Z(X|Y)^2}{2}\mathbb{E}_{y_1,y_2\sim p(y)}\sqrt{f(y_1)^2+f(y_2)^2-4}\ge \frac{Z(X|Y)^2}{2}\sqrt{(\mathbb{E}_{y_1\sim p(y)}f(y_1))^2+(\mathbb{E}_{y_2\sim p(y)}f(y_2))^2-4}\ge Z(X|Y)\sqrt{2-Z(X|Y)^2}.
\sum_{m=1}^{\left\lfloor \sqrt{n}\right\rfloor}(-1)^{m-1}p(n-m^2) = \sum_{m=1}^{\left\lfloor \sqrt{\frac{n}{2}}\right\rfloor}(-1)^{m-1}p(n-2m^2) = \frac{p(n) - p_{\mathcal{DO}}(n)}2,
\partial_{\mu}< T T R > = ( N_{c}^{2}- 1 ) \cdot \left[ \mathrm{\vspace{2 e x}\mathrm{1}\vspace{0 e x}}\right] + 3 \cdot ( N_{c}^{2}- 1 ) \cdot \left[ - \frac{1}{3}\right] = 0
\Theta =\begin{pmatrix}0 \theta_0 \theta_1 \theta_2 \cdots -\theta_{2}-\theta_{1}begin{align*}3pt] 0 \theta_0 \theta_1 \theta_2 \cdots -\theta_{2}& 0 \theta_0 \theta_1 \ddots \vdots 0 \theta_0 \ddots \theta_2 & 0 \ddots \theta_1 & \ddots \theta_0 begin{align*}2pt] 0\end{pmatrix}.
\langle \langle \Omega_{0}({\bf k}_{n}, t ) \Omega_{0}({\bf k}_{n^{\prime}}, t^{\prime}) \rangle \rangle_{T}= \delta_{n , - n^{\prime}}\langle \langle \Omega_{0}({\bf k}_{n}, t ) \Omega_{0}( -{\bf k}_{n}, t ) \rangle \rangle_{T}.
\textbf{R}_{C}(1,2)=\int_{0}^{\infty}\frac{x\cos(2\pi x)}{\{-1+\exp{(2\pi\sqrt{x})}\}}dx=\frac{1}{64}\left(\frac{1}{2}-\frac{3}{\pi}+\frac{5}{\pi^{2}}\right),
H = \left[ \begin{array}{c c}{1 \}&{\ 0}\\{0 \}&{\ - 1}\\ \end{array}\right] \ , E_{+}= \left[ \begin{array}{c c}{0 \}&{\ 1}\\{0 \}&{\ 0}\\ \end{array}\right] \ , E_{-}= \left[ \begin{array}{c c}{0 \}&{\ 0}\\{1 \}&{\ 0}\\ \end{array}\right] \ .
\sup_{0\le j\le N}\ \Vert P_{n}T^{\,j}P_{l}\,x\Vert \le \dfrac{1}{4}\,\beta_{l}\,\cdot \Bigl(\prod_{i=\Delta^{(k)}-N+1}^{\Delta^{(k)}-1}\!\!\!|w_{i}^{(k)}|\ \Bigr)\,\cdot\, \Vert P_{l}\,x\Vert .
V_{U}((r,t),(s,u)) = \frac{\sigma_{U}(r,s)}{(2-d(r,s))(3-d(r,s))}[ \overline{c(s,r)}t^{3-d(r,s)}+c(r,s) u^{3-d(r,s)}-C(r,s;t-u) \vert t-u \vert^{3-d(r,s)}]
\dot w(t) \leqslant \sum_{j=0}^N L_j w(t)+L\delta\sum_{j=1}^N L_j|\tau_j(t,u(t))-\tau_j(t,\tilde u(t))| \leqslant L w(t)+L\delta\sum_{j=1}^N L_jL_{\tau_j}|u(t)-\tilde u(t)| & \leqslant\Bigl(1+\delta\sum_{j=1}^N L_jL_{\tau_j}\Bigr)L w(t)= L_{\delta}w(t),
U_1(x_3): =\frac{1}{T}\int_{0}^{T}<u_1(t)>(x_3)dt &=\frac{1}{T}\int_{0}^{T}\int_{-\infty}^{t}K(x_3,t-\sigma)p_1(\sigma)d\sigma dt &\overset{\xi=t-\sigma}{=}\int_{0}^{\infty}K(x_3,\xi)P_1(\xi)d\xi &=\sum_{k=1}^{\infty}\frac{2((-1)^k-1)}{\Pi_1 k \pi}\left(\int_{0}^{\infty}e^{-\nu(\frac{k\pi}{h})^2 \xi}P_1(\xi)d\xi \right)\sin\frac{k\pi x_3}{h},