harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widehat c_{n,\rho,q}^{(k)}(z) =\rho^n\underbrace{\int_0^1\cdots\int_0^1}_k\left(\frac{-x_1\cdots x_k+z}{\rho}\right)_n d_q x_1\cdots d_q x_k &=\rho^n n!\underbrace{\int_0^1\cdots\int_0^1}_k\binom{(-x_1\cdots x_k+z)/\rho}{n}d_q x_1\cdots d_q x_k\,.
m_{ij}= c'_{\lambda_i}\, c'_{\lambda_j}+ \sum_{m>0}\beta^m\, c'_{\lambda_i+m}\,c'_{\lambda_j}+ \sum_{\ell>0}\sum_{m\geq 0}(-1)^\ell \left( \binom{\ell+m-1}{m}+ \binom{\ell+m}{m}\right) \beta^m \, c'_{\lambda_i+\ell+m}\, c'_{\lambda_j-\ell}
\begin{array}{c|c}\mbox{Physical spacetime:}z \mbox{Covering:}u=z^{1/q}\hline{\displaystyle{c \over 24\pi}\int_z (R+2)}{\displaystyle{c/q \over 24\pi}\int_u (R+2)}\Downarrow \Downarrow \{L_{n+r/q},\c \}\{Q_n,\ \{\displaystyle{c \over q}\}}\end{array}
\sum_{d \mid a}\mathcal{F}_d(x - 1) = \sum_{d \mid a}\prod_{i = 1}^r \binom{x - 1 + \delta_i}{\delta_i}= \prod_{i = 1}^r \sum_{\delta_i = 0}^{a_i}\binom{x - 1 + \delta_i}{\delta_i}.
B_{n}=\sum_{k=0}^{n}\left( -1\right)^{k}\frac{k!}{k+1}\genfrac{\{}{\}}{0pt}{}{n}{k},E_{n}\left( 0\right) =\sum_{k=0}^{n}\left( -1\right)^{k}\frac{k!}{2^{k}}\genfrac{\{}{\}}{0pt}{}{n}{k}.
T_{B}^{A}= \frac{\delta ( y )}{b_{0}}\mathrm{d i a g}( V_{\star}+ \rho_{\star}, V_{\star}- p_{\star}, V_{\star}- p_{\star}, V_{\star}- p_{\star}, 0 ) + \frac{\delta ( 1 - y )}{b_{0}}\mathrm{d i a g}( - V + \rho , - V - p , V - p , V - p , 0 ) ,
\Phi_{m,d+j-m}^{(j)}(P,\vartheta Q;A\times\vartheta B) = \sum_{F \in{\cal F}_{m}(P)}\sum_{G \in{\cal F}_{d+j-m}(Q)}f_j(n(P,\vartheta Q;F,\vartheta G)) [F, \vartheta G]\lambda_F(A)\lambda_G(B).
u \star v = \exp \Big ({\frac{\kappa}{2}}\omega^{i j}{\frac{\partial \phantom{x^{i}}}{\partial x^{i}}}{\frac{\partial \phantom{x^{j}}}{\partial{\tilde{x}}^{j}}}\Big ) u ({\bf x}) v ({\bf{\tilde{x}}}) \Big \vert_{{\bf x}={\bf{\tilde{x}}}}
x_t =\frac{x^{-\tilde{\theta}}}{\tilde{\theta}}+\int_0^tf(s,\frac{y_s^{-\tilde{\theta}}}{\tilde{\theta}})ds+\int_0^t\tilde{\sigma}y_s^{-(\tilde{\theta}+1)}dB_s^H &=\frac{x^{-\tilde{\theta}}}{\tilde{\theta}}+\int_0^tf(s,\frac{y_s^{-\tilde{\theta}}}{\tilde{\theta}})ds+\int_0^t\tilde{\sigma}\tilde{\theta}^{\theta}(x_s)^{\theta}dB_s^H,
(\psi_{0,n},\psi_{1,n})=r^k\sum_{j=1}^J\left(\frac{1}{\lambda^k_{jn}}U^j_L(\frac{r}{\lambda_{jn}},\frac{-t_{jn}}{\lambda_{jn}}),\frac{1}{\lambda_{jn}^{k+1}}\partial_tU^j_L(\frac{r}{\lambda_{jn}},\frac{-t_{jn}}{\lambda_{jn}})\right)+r^k(w^J_{0,n},w^J_{1,n})+(m,0).
\mathrm{J}_2=\int_{\vert x-x(t)\vert<R_0}\{p*(3uu_x^2+2u^3)\}u\Psi'+\int_{\vert x-x(t)\vert>R_0}\{p*(3uu_x^2+2u^3)\}u\Psi'=:\mathrm{J}_2^1+\mathrm{J}_2^2.
\frac{d}{du}\left(\begin{array}{c}\partial\beta/\partial v \partial t/\partial v \end{array}\right)=\left(\begin{array}{cc}a_{11}a_{12}a_{21}a_{22}\end{array}\right)\left(\begin{array}{c}\partial\beta/\partial v \partial t/\partial v\end{array}\right)\textrm{for}v=0.
\vert C_{a}\rangle = \left[{\frac{2^{d ( J_{a})}V ( J_{a})}{V ( R J_{a})}}\right]^{\frac{1}{2}}\sum_{{\widetilde p}\in 2 \Lambda ( J_{a})}\sum_{{\widetilde w}\in{\widetilde \Lambda}( R J_{a})}\sum_{\bf n}\zeta_{\bf n}V_{L}({\bf n}) V_{R}({\bf n}) \vert{\widetilde w},{\widetilde p}\rangle .
\widetilde{C}(z)_{11}=\sum_{k\geq 0}(-1)^k(z^{-6k-2}+z^{-6k-4}), \widetilde{C}(z)_{12}=\sum_{k\geq 0}(-1)^kz^{-6k-3}, \widetilde{C}(z)_{21}=\sum_{k\geq 0}(-1)^k(z^{-6k-2}+z^{-6k-4}), \widetilde{C}(z)_{22}=\sum_{k\geq 0}(-1)^k(z^{-6k-1}+z^{-6k-5}).
\Delta_{k-1}(d\overline{E})-d_\mathcal{D}\Delta_k(\overline{E}) = \frac{(-1)^{k+1}}{(k+1)!(2\pi i)^k}\int_{(\mathbb{P}^1)^k}C_{k+1}\Big(d_\mathcal{D}\Delta\big(f,h,\omega^X,\omega^Y,{\rm tr}_k\circ\lambda(\overline{E})\big),\log\mid z_1\mid^2,\cdots,\log\mid z_{k}\mid^2\Big).
\log d(K) = r_1\frac{\pi}{2}+ n(\gamma + \log 8\pi) - n \int_0^{\infty}\frac{1-F(x)}{2 \sinh \frac{x}{2}}\, dx - r_1 \int_0^{\infty}\frac{1-F(x)}{2 \cosh \frac{x}{2}}\, dx - 4 \int_0^{\infty}F(x) \cosh \frac{x}{2}\, dx + \sum_{\rho}\Phi(\rho) + 2 \sum_{\mathfrak P}\sum_{m=1}^{\infty}\frac{\log N\mathfrak P}{N\mathfrak P^{m/2}}F(m \log N\mathfrak P)
1+\varpi^{l-1}X^{\prime}+2^{-1}\varpi^{2l-2}X^{\prime 2}\pmod{\frak{p}^r}= 1+\varpi^{-1}X+2^{-1}\varpi^{2l-2}X^2+\varpi^lT\pmod{\frak{p}^r}= (1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^2)(1+\varpi^lT) \pmod{\frak{p}^r},
d_{r}^3\gamma(x,y,z,t) =\gamma(d(x,y),\alpha(z),t)+\gamma([x,z],\alpha(y),t)+\gamma([y,z],\alpha(x),t) + \gamma\left( x,y,[\alpha(z),t] \right)+\gamma\left( y,z,[\alpha(x),t] \right)+\gamma\left( x,z,[\alpha(y),t] \right)
u =c_1+c_2y-c_3x+c_4z+ \frac{1}{2c_6b_1(c_6 + x)}(c_5^2 c_6b_1y + c_5^2 c_7b_1x + 2c_5c_6c_7b_1z - 2c_5c_6b_1yz - 2c_6c_7y - c_6c_7b_1z^2+ c_6y^2 + c_6b_1yz^2 - c_7^2 x).
n(\rho_{-},\rho_{+},t)=tH''(\rho_{+})\frac{j^{tL(\frac{\cdot}{t})}(tH'(\rho_{+}))}{j^{tL(\frac{\cdot}{t})}(tH'(\rho_{-}))}K^{tL(\frac{\cdot}{t})}(tH'(\rho_{-}),tH'(\rho_{+}))
\sum_{m=n}^{\infty}\sum_{k=n}^{\min\{\ell,m\}}A_{m,k}=\sum_{k=n}^{\ell}\sum_{m=k}^{\ell}A_{m,k}+\sum_{k=n}^{\ell}\sum_{m=\ell+1}^{\infty}A_{m,k}=\sum_{k=n}^{\ell}\sum_{m=k}^{\infty}A_{m,k},
\mathrm{E}_\theta\hat{\pi}(I \in\mathcal{G}_1| X) =\sum_{I\in\mathcal{G}_1}\mathrm{E}_\theta\hat{\pi}(I|X)\le e^{-M}\sum_{I\in\mathcal{I}}(\tfrac{ne}{|I|})^{-c_4|I|}\le H'_0 e^{-M}.
{\bf A}^{-1}=({\bf T}+{\bf P}{\bf Q}^T)^{-1}={\bf T}^{-1}- [{\bf T}^{-1}{\bf P}({\bf 1}+{\bf Q}^T{\bf T}^{-1}{\bf P})^{-1}{\bf Q}^T{\bf T}^{-1}],
\lim_{z\rightarrow 0}\frac{\ln|\varphi(|z|)|}{\ln\|(z-T)^{-1}e_0\|_p}=\lim\limits_{z\rightarrow 0}\frac{k\ln|z|+\ln\big|\sum\limits_{j=0}^{n}c_j|z|^j\big|}{\ln\|(z-T)^{-1}e_0\|_p}&=\lim\limits_{z\rightarrow 0}\Big(\frac{k\ln|z|}{\ln\|(z-T)^{-1}e_0\|_p}+\frac{\ln\big|\sum\limits_{j=0}^{n}c_j|z|^j\big|}{\ln\|(z-T)^{-1}e_0\|_p}\Big).
\Delta^{\mu \nu , \lambda \sigma}( x - y ) = < 0 | \; T ({\bf a}^{\mu \nu}( x ) \; \;{\bf a}^{\lambda \sigma}( y ) \; ) | 0 > \; = \; \frac{1}{( 2 \pi )^{4}}\int d^{4}p \; e^{i \; p \cdot ( x - y )}\; \Delta^{\mu \nu , \lambda \sigma}( p ) ,
\mathcal{T}(q^{\pm 1/2})=\left( -1\right)^{\mathsf{N}}X_{q}M(1),\mathcal{T}(iq^{\pm 1/2})=\left( -1\right)^{\mathsf{N}}X\frac{(\zeta_{+}+1/\zeta_{+})}{(\zeta_{+}-1/\zeta_{+})}\frac{(\zeta_{-}+1/\zeta_{-})}{(\zeta_{-}-1/\zeta_{-})}_{q}M(i).
P({\cal M}{\rm YB}_3,s,t)= 1+3s t+s^2 t^3, P({\cal M}{\rm YB}_4,s,t)=1+6s t+s^2 \big(3t^2+4t^3\big)+s^3 t^6, P({\cal M}{\rm YB}_5,s,t)=1+10s t+s^2\big(15t^2+10t^3\big)+s^3\big(10t^4+5t^6\big)+s^4 t^{10}.
\min \sum_{i=1}^{n}{c_i \cdot x_i}\mbox{such that}0 \le x_i \le u_i\,, \ \mbox{$\forall \, i=1, \dots, n$}\mbox{and}a_i \le \sum_{j=1}^{i}{q^{i-j}\cdot x_j}\le b_i\,, \ \mbox{$\forall \, i=1, \dots, n$}
J^{\alpha}(Q_c)=\frac{(3\alpha-1)^{1-\frac1{2\alpha}}}{6\alpha}c^{\frac1{2\alpha}-1}\|Q_c\|_{L^2}=\frac{(3\alpha-1)^{1-\frac1{2\alpha}}}{6\alpha}\|Q\|_{L^2}\, .
\Psi_{M-1,2}^\alpha(t) =\int_t^{t_M}\bigg[\int_{t_{M-1}}^t e^{-\lambda_\alpha(t-\tau)}d\tau \bigg]^2ds =\frac{(1-e^{-2\lambda_\alpha(t-t_{M-1})})^2}{\lambda_\alpha}k &\le \frac{1-e^{-2\lambda_\alpha(t-t_{M-1})}}{\lambda_\alpha}k^2\le \frac{1-e^{2\lambda_\alpha k}}{\lambda_\alpha}k^2.
\Sigma=4\pi\left\vert f\left( k_{\mathrm{o}}\right) \right\vert^{2}\approx\frac{4\pi\rho^{2}k_{\mathrm{o}}^{4}L^{6}}{9}=\frac{16\pi\mu^{2}V^{2}L^{6}}{9\hbar^{4}}\approx4\pi a_{\mathrm{s}}^{2}.
\|v^{\Sigma}_{\Lambda}(t,\cdot)\|_{L^{\infty}}\le \|v^{\Sigma}(t,\cdot)\|_{L^{\infty}}+ \|v^{\Sigma}_{\Lambda^c}(t,\cdot)\|_{L^{\infty}}& = \|v(t,\cdot)\|_{W_h^{\rho, \infty}}+ \|v^{\Sigma}_{\Lambda^c}(t,\cdot)\|_{L^{\infty}}\, ,
\sum_{F_{g,h}\in \Gamma(F_{e,f})}\frac{1}{A_{g,h}}=\sum_{\alpha=0}^{\ell}\sum_{\{g,h\}\in Y_{e,f}(\alpha)}\frac{1}{A_{g,h}}\le \frac{2ckn^{2k-2}}{\delta n^{2k-2}}+ \sum_{\alpha = 1}^{\ell}\frac{2ck^{\ell+1}n^{2k-\alpha-1}}{\delta n^{2k - \alpha -1}}=\frac{2ck}{\delta}(1+ \ell k^{\ell}) < \frac{1}{\mathrm{e}}\enspace ,
\overline{R}_2=2\bar{P}_2-\sum\limits_{j=1}^{\infty}\Big(\bar{P}_2\bar{u}_j-3(j+2)\bar{u}_1\bar{u}_{j+1}+(j+3)\bar{u}_{j+2}-\bar{p}_{j+2}^{(-1)}\Big)\frac{\partial}{\partial \bar{u}_j}.
\sum\limits_{\begin{smallmatrix}{{l}_{1}}+\cdots +{{l}_{5}}={{p}^{r}}{{l}_{1}},\cdots ,{{l}_{5}}\in{\mathcal{P}_{p}}\end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}}}\equiv -\frac{5!}{6}{{p}^{r-1}}{{B}_{p-5}}\pmod{{{p}^{r}}}.
\begin{array}{l l}{[ h_{i}, h_{j}] = 0 ,}{[ e_{i}, f_{j}] = \delta_{i j}h_{i},}{\mathrm{~ [ ~}h_{i}, e_{j}\mathrm{~ ] ~}= A_{i j}e_{j},}{\mathrm{~ [ ~}h_{i}, f_{j}\mathrm{~ ] ~}= - A_{i j}f_{j},}\end{array}
\int_{A_j(0)\cap\mathcal{O}}u_{(x, 0), r}^2dV \ge\int_{r_j}^{R_j}\int_{\partial B_\rho^n(0,0)\cap\mathcal{O}}u_{(x, 0), r}^2 d\sigma d\rho =\int_{r_j}^{R_j}H(\rho,(0, 0), u_{(x, 0), r})d\rho \ge \int_{r_j}^{R_j}\frac{\rho^{n-1}}{r_1^{n-1}}H(r_1,(0,0), u_{(x, 0), r})d\rho \ge C(n)\frac{j^{n-1}}{r_1^{n-1}}H(r_1, (0, 0), u_{(x, 0), r}).
Ax^{k_1}+p(x)-(A(x^{*})^{k_1}+p(x^{*}))\left(\frac{x^{*}}{x}\right)^{-k_1}=\frac{1}{l_2}\left[\frac{1-\theta}{-\left(\frac{K}{A(x^{*})^{k_1}+p(x^{*})}+\theta\right)}-l_1\right].
\begin{array}{ccc}\{I_{k},I_{\ell}\}_{m}= 0 \{J_{\ell},I_{k}\}_{m}= (k+\ell+m-2)I_{k+\ell+m-2}\{J_{k},J_{\ell}\}_{m}= (\ell-k)J_{k+\ell+m-2}\end{array}
I_0+2I_2=-\frac{\Gamma\left(2-\frac{D}{2}\right)}{(4\pi)^{\frac{D}{2}}}\int_0^1 dx 2 (1-x)\sum_{n}\left[\left(1-\frac{4}{D}\right )\frac{1}{[k_0^2+m^2]^{2-\frac{D}{2}}}+\frac{4}{D}\frac{2-D/2}{\Gamma(3)}\frac{1}{[k_0^2+m^2]^{2-\frac{D}{2}}}\right].
f(x,k_n,t,\xi_1,\xi_2) \leq 0 \leq f(x,h_n,t,\xi_1,\xi_2), g(x,s,\hat{k}_n,\xi_1,\xi_2) \leq 0 \leq g(x,s,\hat{h}_n,\xi_1,\xi_2)
\begin{aligned}\prod_{j>r}(1-q^j)\prod_{k>s}(1-q^k)= \,\exp\!\left[-\frac{q^r+q^s}{1-q}+O\!\left(\!\rho^r+\rho^s+\frac{\rho^{2r}+\rho^{2s}}{1-\rho}\right)\right] = \,\exp\!\left(\!\!-\frac{q^r+q^s}{1-q}+O\bigl(n^{-1/2}(h^2+w^2)\bigr)\right).\end{aligned}
\prod_{j=1}^k \frac{\eta_j(0,t_j-t_{j-1}]^{\ell_j}}{\ell_j!}e^{-\eta_j(0,t_j-t_{j-1}]}= \prod_{j=1}^k \P \big( N_j(0,t_j-t_{j-1}]=\ell_j \mid \eta_j\big),
\int_1^T|\psi(s)|^2dt =\int_1^T\prod_{j=1}^k |\zeta(\sigma_j+ia_jt)|^2dt & \leq \prod_{j=1}^k \left(\int_1^T |\zeta(\sigma_j+ia_jt)|^{m(\sigma_j)}dt\right)^{\frac{2}{m(\sigma_j)}}\left(\int_1^T 1 dt\right)^{1-\sum_{j=1}^k \frac{2}{m(\sigma_j)}}begin{align*}1ex] \ll T^{1+\varepsilon}.
\Pr_{r_{E^{m+n}}p_{X^n}}\left\{\bigcup_{i=m+1}^{m+n}\left\{\sum_{\ell=1}^{i-m}X_{\ell}> \sum_{\ell=1}^i E_\ell \right\}\right\}\le \frac{e^{0.4}}{n\ln n},
[\delta_{1},\delta_{2}] \lambda_{i}= 2 i D_{\mu}\lambda_{i}\overline{\epsilon}_{1}^{j}\gamma^{\mu}\epsilon_{2 j}+ \frac{a}{2}\gamma^{\mu \nu}\lambda_{i}M_{kj}\overline{\epsilon}_{1}^{k}\gamma_{\mu \nu}\epsilon_{2 j}
\sum_{j = j_{\delta_1}+ 1}^{j_{\delta_2}}\frac{1}{t_j}\le \frac{c}{\Delta_n}\log(\frac{j_{\delta_2}}{j_{\delta_1}}) = \frac{c}{\Delta_n}\log(\frac{\Delta_n j_{\delta_2}}{\Delta_n j_{\delta_1}}) = \frac{c}{\Delta_n}\log(\frac{\delta_2}{\delta_1}).
a(2-q)t_{\max}^{2}\|w\|^{2}+ \epsilon (4-q)t_{\max}^{4}\|w\|^{4}-({2^*_\alpha}-q)t^{2^*_\alpha}_{\max}\int_{\Omega\times\{0\}}{|w(z,0)|^{2^*_\alpha}}dz, = 0.
\lim_{t\to+\infty}\xi'_{b_{i_k-1}}(t)=\lim_{t\to\infty}\frac{-u_t(\xi_{b_{i_k-1}}(t), t)}{u_r(\xi_{b_{i_k-1}}(t), t)}=\frac{c_kU_k'(r^0_{i_k-1})}{U_k'(r^0_{i_k-1})}=c_k.
\det [ - G_{a b}] = Z_{p}^{-{\frac{( p + 1 )}{2}}}\left( 1 - Z_{p}\sum_{m = p + 1}^{9}v_{m}^{2}\right) = Z_{p}^{-{\frac{( p + 1 )}{2}}}\left( 1 - Z_{p}v^{2}\right) \ .
I \, m ({\cal R}e s{\cal A}) = \operatorname*{l i m}_{p^{2}\rightarrow \mu^{2}}{\ - \frac 1{2 a_{1}}( \left| \tau_{\mu \nu}\right|^{2}- \frac 1 2 \left| \tau_{\mu}^{\, \, \, \mu}\right|^{2}) .}
I_{\mu}^{\infty}(v_j) \geq c_{\mu}^{\infty}&=\inf_{\gamma \in \Gamma^{\infty}}\max_{t\in [0,1]}I_{\mu}^{\infty}(\gamma(t)), &\geq \inf_{\gamma \in \Gamma^{\infty}}\max_{t\in [0,1]}I_{\mu}(\gamma(t)) &\geq \inf_{\gamma \in \Gamma}\max_{t\in [0,1]}I_{\mu}(\gamma(t)) &=c_{\mu}.
\int_T^{2T}\prod_{1\le i\le r}(\cos(t\log p_i))^{a_i}\prod_{r+1\le i\le s}(\cos(2t\log p_i))^{a_i}dt =Tg(n) + \sideset{}{'}\sum_{\ell_1,\ldots,\ell_s}\prod_{1\le i\le s}\frac{1}{2^{a_i}}{a_i \choose \ell_i}\int_T^{2T}(*) dt.
\log_q(|C|) \leq \sum_{i=j(t)}^t m_in_i=\hat{m}(N_t-N_{j(t)}+n_{j(t)}) &\leq\hat{m}(N_t-N_{j(t)}+\hat{m})\leq \hat{m}(N_t-\eta N_t+1+\hat{m}).
\sigma_n^2(2) =\frac{(n-2)^2(n-1)(8n+1)}{32400n^2(n+1)^2}=\frac2{90^2}- \frac{11}{6480 n}+ \frac{161}{32400n^2}+ O(n^{-3}), \sigma_n^2(3) =\frac{(n-2)(n-1)(16n^5-96n^4+359n^3-269n^2-963n-370)}{5832000 \cdot n^4(n+1)^3}&=\frac2{90^3}- \frac{1}{30375 n}+ \frac{1207}{5832000 n^2}+ O(n^{-3}).
r_{H}\approx \frac{1}{1 6}3^{3 / 8}2^{1 / 4}\left( 3 3 5 1 7 9 7 + ( 1 7 1 ) ( 2 6 5 9 ) \sqrt{5 7}\right)^{1 / 8}A^{- 1 / 8}r_{0},
\lefteqn{\sup_{T_j \leq i < T_{j+1}}X_i I\{T_j = s, T_{j+1}=t \}I\{D^+_{T_j}\}}& \leq \lambda^{max}(U_{T_j}+ M^{abs}_{{T_j},T_{j+1}^-}) I\{T_j = s, T_{j+1}=t \}I\{D^+_{T_j}\}.
- i{\cal M}_{f i}- ( i{\cal M}_{f i})^{*}= ( g )^{2}( 2 \pi ) \delta ( P^{2}- M^{2}) - 2 g^{2}I m \left({{\frac{\Pi ( P^{2})}{( P^{2}- M^{2}+ i \varepsilon )^{2}}}}\right)
\frac{\partial L}{\partial x^i}=\frac{d}{dt}\frac{\partial L}{\partial v^i}=\frac{\partial}{\partial t}\frac{\partial L}{\partial v^i}+v^j\frac{\partial}{\partial x^j}\frac{\partial L}{\partial v^i}+\frac{dv^j}{dt}\frac{\partial^2 L}{\partial v^j v^i}.
\Delta \phi_{min}= \Delta \phi_r, ~~\Delta \phi_{max}= \Delta \phi_{\theta},~~\mathrm{for}~\mathbf{z}_{\phi,r}= \mathbf{z}_I \Delta \phi_{min}= \Delta \phi_{\theta}, ~~\Delta \phi_{max}= \Delta \phi_r,~~\mathrm{for}~\mathbf{z}_{\phi,r}= -\mathbf{z}_I
\frac{1}{T}\int_T^{2T}2^{-k}\left(\sum_{p\leq Y}\left(a(p)p^{-it}+b(p)p^{it}\right)\right)^kdt &=\frac{1}{T}\int_T^{2T}2^{-k}\sum_{\ell=0}^k\begin{pmatrix}k \ell\end{pmatrix}\left(\ell !\sum_{\Omega(m)=\ell}a(m)g(m)m^{-it}\right)\left((k-\ell)!\sum_{\Omega(n)=k-\ell}b(n)g(n)n^{-it}\right)dt.
|F_N(p_0)|=\left|\sum_{j=1}^mQ_j(p_0)\right|\le\sum_{j=1}^m\left|Q_j(p_0)\right|=C_NN^\frac14\sum_{j=1}^m\left|\sin\phi_j\right|^N\le c_1N^\frac14\sum_{j=1}^m(\cos\alpha_j)^N,
\left[ a_{\alpha}, a_{\beta}^{\dagger}\right] = \delta_{\alpha \beta}\ , \left[ a_{\alpha}, a_{\beta}\right] = \left[ a_{\alpha}^{\dagger}, a_{\beta}^{\dagger}\right] = 0 \ , \ \left( \alpha , \beta = 1 , \cdots N + 1 \right) \ ,
e_{\alpha_1}f v_\lambda = \langle \alpha_1^\vee, \lambda \rangle e_{-(\sigma - \alpha_1)}v_\lambda, e_{\alpha_1}e_{-\sigma}v_\lambda = -3 e_{-(\sigma - \alpha_1)}v_\lambda, e_{\alpha_2}f v_\lambda = (\langle \alpha_2^\vee, \lambda \rangle + 1) e_{-(\sigma - \alpha_2)}v_\lambda, e_{\alpha_2}e_{-\sigma}v_\lambda = e_{-(\sigma - \alpha_2)}v_\lambda
\tan \mbox{\boldmath\Large $\left( \right.$}\! \! \! \delta_{0}^{(D_{0})}(k) - ( D/2 - 1 )\, \pi/2\mbox{\boldmath\Large $\left. \right)$}\! \!=\frac{\pi}{\ln \left( k^{2}/|E_{_{\rm (gs)}}| \right)}\; .
\frac{1}{2}\sum_{\ell=1}^n d_\ell-\sum_{d_\ell\geq\delta}d_\ell =\frac{1}{2}\left(\sum_{d_\ell<\delta}d_\ell-\sum_{d_\ell\geq\delta}d_\ell\right)\geq\frac{\delta d}{2}
\left(I_{N^{p}}-\mbox{Aux}_{p}A\right) Z+B\overline{Z}=V\left(z_{1},z_{2},\dots,z_{N}\right),\quad\mbox{where $A$, $B\in \mathcal{M}_{N^{p}\times N^{p}}\left(\mathbb{C}\right)$},
\langle D_x X^x_t, h \rangle = h + \int_0^t \nabla_x b(s,X^x_s) \langle D_x X^x_s, h \rangle ds + \int_0^t \nabla_x \sigma(s,X^x_s) \langle D_x X^x_s, h \rangle dW_s,
\underset{\bar{{\bf w}}\in \mathcal{W}}{\min}~ \underset{\bar{\bf x}\in \mathcal{X}}{\max}~ \phi^{(n)}(\bar{\bf w},\bar{\bf x}) = \underset{\bar{{\bf x}}\in \mathcal{X}}{\min}~ \underset{\bar{\bf w}\in \mathcal{W}}{\max}~ \phi^{(n)}(\bar{\bf w},\bar{\bf x})
2 \omega + \frac{m_{1}^{2}}{2 p_{1}}- \frac{m_{2}^{2}}{2 p_{2}}- \frac{m_{3}^{2}}{2 p_{3}}= \delta E_{D}^{( 4 )}+ \frac{\xi}{M_{D}}\left( p_{2}p_{3}+ ( p_{2}+ p_{3}) \omega + \omega^{2}\right)
m(\mu_{\varepsilon}(t)) - \mathcal{N}\partial_t \varphi_{\varepsilon}(t)= - \nu \Delta \varphi_{\varepsilon}(t) + \xi_{\varepsilon}(t) + \pi(\varphi_{\varepsilon}(t)) - \gamma \chi_{\varepsilon}(t) \ \ \textrm{in $H$.}
\begin{aligned}\| \mathbf{K_{e}}f(x) \|_{2,-\beta,n}= \| (1+x)^{\beta}\mathbf{K_{e}}f(nx) \|_{2}&\le \int_{0}^{\infty}(1+x)^{\beta}(n+nx)^{-\frac{1}{2}-\beta-\delta}dx \cdot n^{\frac{1}{2}+\beta-\delta}\cdot \| f \|_{2,-\beta,n}&= c'n^{-2\delta}\|f \|_{2,-\beta,n}.\end{aligned}
\Vert v_{i}(C^{*}\pi_{W_{i}}C^{\prime})^{\frac{1}{2}}f \Vert^{2}=\sum_{i \in I}v_{i}^{2}\langle \pi_{W_{i}}C^{\prime}f, \pi_{W_{i}}C f\rangle\leq B \Vert f \Vert^{2}.
\frac{\partial f_j}{\partial[\tilde{v}_i]_{j,\alpha}}=-\frac{1}{m|g|}\sum_{\substack{t=1 (t\neq j)}}^n[\tilde{w}_i]_{\alpha,t}\sum_{h=0}^{m-1}[\tilde{v}_{i-h-1}\tilde{w}_{i-h-1}]_{t,j}\frac{\lambda_t^{m-h-1}\lambda_j^h}{\lambda_t^m-\lambda_j^m}.
h_{\mu,g}\bigl(f^0,g^0\bigr)=\frac{|A_N(e^{i\lambda})(1-e^{i\lambda\mu})^{n}f^0(\lambda)- \lambda^{2n}C^{\mu,0}_{N}(e^{i\lambda})|}{|1-e^{i\lambda\mu}|^{n}p^0(\lambda)},