harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S ( d = 1 1 ) = - T \int d^{p + 1}\hat{x}| \hat{k}|^{\beta}| \hat{k}^{\prime}|^{\gamma}\sqrt{\mid \partial_{i}\hat{X}^{\hat{\mu}}\partial_{j}\hat{X}^{\hat{\nu}}\hat{\Pi}_{\hat{\mu}\hat{\nu}}\mid}\, .
P_{n}(\lambda x)= \sum_{k=0}^{\lfloor n/2 \rfloor}a_{\lambda, n, k}\dfrac{d^{k}}{dx^{k}}P_{n-k}(x)= \sum_{k=0}^{\lfloor n/2 \rfloor}b_{\lambda, n, k}P_{n-2k}(x),
c_{1b1}=(-135\nu^7+693\nu^6+417\nu^5+4821\nu^4)-64349\nu^3+134023\nu^2-48061\nu+12527 \ge -64349\nu^3+134023\nu^2-48061\nu+12527\ge 1000 (-67\nu^3+134\nu^2-67\nu+12) =\frac{1000}{27}\left[ 56+ 67(4-3\nu) (1-3\nu)^2\right]>0
[\begin{matrix}1&0&0&0\\ 0&\frac{1+e^{i\pi\alpha}}{2}&\frac{1-e^{i\pi\alpha}}{2}&0\\ 0&\frac{1-e^{i\pi\alpha}}{2}&\frac{1+e^{i\pi\alpha}}{2}&0\\ 0&0&0&1\end{matrix}]
(i) \,\,{\bf 1}\in \langle A_{\alpha_{1n}}{\bf 1}\mid n\geq2\rangle, (ii) \,\, S_{11}^L(\mu_\alpha)=\sum_{n=2}^\infty\left(1-2\sqrt{\alpha_{1n}(0)\alpha_{1n}(1)}\right)=:\sum_{n=2}^\infty\left(1-c_{1n}\right)= \infty, (iii) \,\, \mu_\alpha\perp \mu_{inv}.
\|P_{N_k}\xi\| \geq \left\|P_{N_k}\left(\sum_{j=1}^k c_j\xi_j\right) \right\|-\left \|P_{N_k}\left(\sum_{j=k+1}^\infty c_j\xi_j\right) \right\| &\geq c_k \|P_{N_k}\xi_k\|-\sum_{j=k+1}^\infty c_j \|P_{N_k}\xi_j\| &\geq c_k \|P_{N_k}\xi_k\|\left(1-\sum_{j=k+1}^\infty \frac{1}{2^j}\right)\geq \frac{c_k}{2}\|P_{N_k}\xi_k\|>0.
d \bar Y_t = \left[ \mathrm{Diag}(\bar Y_t) - \bar Y_t \bar Y_t^T \right] \Gamma^T d B_t + D^T \bar Y_t dt + \left[ \mathrm{Diag}(\bar Y_t) - \bar Y_t \bar Y_t^T \right]\left( a - \Sigma \bar Y_t \right)dt,
|F^{(pr)}(t)| =\left|\left(f(t)- \sum_{k=1}^{p-1}b_k\frac{t^{kr}}{(kr)!}\right)^{(pr)}\right|\le (pr)!\left(\frac{1+\varepsilon}{\varepsilon}\right)^{pr}\frac{C_2A_2^pM_p}{(pr)!}=C_3A_3^pM_p.
y\widetilde{Q}_n(y;x,t,s)=\widetilde{Q}_{n+1}(y;x,t,s)+\mathcal{\widetilde{A}}_{n}(x, t,s)\widetilde{Q}_n(y;x,t,s) &+\mathcal{\widetilde{B}}_{n}(x,t,s)\widetilde{Q}_{n-1}(y;x,t,s), n \geq0,
\Phi(r)=\frac{q}{4\pi\epsilon}\sum_{l=0}^{\infty}\frac{r_{<}^{l}}{r_{>}^{l+1}}(\frac{4\pi}{2l+1})\sum_{m=-l}^{l}Y_{lm}(\theta,\phi)Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})
R_n=\Big\{\vec{x}=(x_0,\ldots,x_n)\in L_n:x_{i+1}-x_i\in \{\pm e_j:1\leq j\leq d-\lfloor\frac{d}{\log d}\rfloor\}i &\lfloor\log d\rfloor \nmid(i+1); x_{i+1}-x_i\in \{e_j:d-\lfloor \frac{d}{\log d}\rfloor+1\leq j\leq d\}&i\lfloor\log d\rfloor \mid(i+1)\Big\},
U ( x ) = \overline{\psi}^{a}( x ) \psi^{a}( x ) \overline{\psi}^{b}( x ) \psi^{b}( x ) - \overline{\psi}^{a}( x ) \gamma_{s}\psi^{a}( x ) \overline{\psi}^{b}( x ) \gamma_{s}\psi^{b}( x ) .
c_k = \tau\sum_{n=0}^{M}\frac{a_{n}}{(k-n)!}&= \tau \left(\frac{a_0}{k!}+ \frac{a_1}{(k-1)!}+ \cdots + \frac{a_M}{(k-M)!}\right) &= \tau \bigg(\frac{q_1}{k!0!}+ \frac{2q_2-q_1}{1!(k-1)!}+ \cdots &\qquad+ \frac{(M+1)q_{M+1}-Mq_M}{M!(k-M)!}\bigg)
\begin{aligned}0\le r_- \le q \le r_+ \le p + q \ , \r_- \le \nu_3 \le r_+ \ , \nu_1 = p+2q-(r_++r_-) \ , \ \nu_2 = (r_++r_-)-\nu_3 \ , \J = \dfrac{1}{2}(r_+-r_-) \ ,\end{aligned}
f^{(j,{\zeta},{\tau_1},{\tau_2},c)}:=\begin{cases}\overline{\partial}\Big(\chi{}(\cdot{},{\zeta})\cdot{}c_j\Big) C_{\zeta}({\tau_2}) 0{\Omega}_{\zeta}({\tau_2})\setminus{}C_{\zeta}({\tau_2})\end{cases}
\int f\,d(\Phi_{k})_{*}(u_{d_{k}}\otimes u_{G_{k}}\otimes \mu_{\delta,\varepsilon})=\frac{1}{d_{k}}\sum_{j\in \mathcal{B}_{k}}\int_{W_{k,\delta,\varepsilon}}e^{2\pi i [\sigma_{k}(\widetilde{\alpha})\zeta](j)}\,d\mu_{\delta,\varepsilon}(\zeta).
{\rm det}(II^L)=-\frac{L}{4}-\langle e_1,\nabla_H(\frac{X_3u}{|\nabla_Hu|})\rangle-\overline{p}[X_1(\overline{p})+X_2(\overline{q})]+O(L^{-\frac{1}{2}})~~{\rm as}~~L\rightarrow +\infty.
\begin{matrix}lim_{x\rightarrow0}\frac{ln(1+x)}{x}&=&1\\lim_{x\rightarrow0}\frac{ln(1+ax)}{bx}&=&\frac{a}{b}\\lim_{x\rightarrow0}\frac{log_{c}(1+ax)}{bx}&=&\frac{a}{blnc}\end{matrix}
\left(\frac{1+Tt}{T}B_\frac{T^2 t}{1+Tt}:t\geq 0;\,\mathbb{P}_{0\to \mu}^{\delta,T}\right) &~~\stackrel{\mathcal{L}}{=}\left(\frac{1+Tt}{T}\left(T-\frac{T^2 t}{1+Tt}\right)X_\frac{\frac{T^2 t}{1+Tt}}{T\left(T-\frac{T^2 t}{1+Tt}\right)}:t\geq 0;\,\mathbb{P}_0^{\delta,\mu}\right) &~~=\left(X_t:t\geq 0;\,\mathbb{P}_0^{\delta,\mu}\right).
a_{n+1}\left(\lambda_{n+1}p_n-\hat{q}_{n-1}\right)+\left(\lambda_{n+1}p_{n-1}-\hat{q}_{n-2}\right) =0 a_{n+1}\left(\lambda_{n+1}q_n-a_1\hat{q}_{n-1}-\hat{p}_{n-1}\right)+\left(\lambda_{n+1}q_{n-1}-a_1\hat{q}_{n-2}-\hat{p}_{n-2}\right) =0
R^{\frac{1}{2}}( \theta )_{\left| \left. a_{k}\dots \frac{1}{2}, a_{1}, \frac{1}{2}\right| n_{k}\dots , m_{1}, n_{1}\right\rangle}^{\left| \left. b_{k}\dots \frac{1}{2}, b_{1}, \frac{1}{2}\right| n_{k}\dots , m_{1}, n_{1}\right\rangle}= R_{a_{1}b_{1}}^{\frac{1}{2}}( \theta ) \prod_{i = 1}^{k - 1}f_{b_{i}b_{i + 1}}^{a_{i}a_{i + 1}}( w_{m_{i}}, \nu_{n_{i + 1}}, \theta )
{\cal L}_{\mathrm{r e l}}^{( 2 )}= \frac{1}{2}\biggl ( \mu + \frac{g^{2}\lambda}{8 \pi r}\biggr ) \, \dot{{\bf r}}^{2}+ \frac{1}{2}\left( \frac{g^{2}\lambda}{8 \pi}\right)^{2}\biggl ( \mu + \frac{g^{2}\lambda}{8 \pi r}\biggr )^{- 1}( \dot{\psi}+{\bf w}({\bf r}) \cdot \dot{{\bf r}})^{2}.
J_1:=\big\{(k_1,\ldots,k_{n+1})\in\mathbb{Z}^{n+1}:\,\widetilde{k}_{n+1}\leq\max(0,\widetilde{k}_n)+10n\big\}, &J_2:=\big\{(k_1,\ldots,k_{n+1})\in\mathbb{Z}^{n+1}:\,k_1=\widetilde{k}_{n+1}\geq\max(0,\widetilde{k}_n)+10n\big\}.
= \frac{1}{2}+ \frac{1}{3}- \frac{1}{3}+ \frac{1}{4}+ \frac{1}{4}- \frac{1}{5}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{6}+ \frac{1}{7}- \frac{1}{7}+ \frac{1}{8}
\sum_{i=1}^{k+1}2a_i\langle u,e_i\rangle_\delta\int_{\partial_F X}dB_{(st_V(\delta),\theta)}(v)d(\mu(x_i))(\theta)+\int_{\partial_F X}DdB_{(st_V(\delta),\theta)}(D_\delta(st_V)(u),v)d\left(\sum_{i=1}^{k+1}a_i^2\mu(x_i)\right)(\theta)\equiv 0
H_N(\alpha,\beta;q) = \sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}\frac{(\dim \mathsf{V}^\lambda)^2}{d!}\omega_\alpha(\lambda) \Omega_{\frac{q}{N}}^{-1}(\lambda) \Omega_{\frac{1}{N}}^{-1}(\lambda) \omega_\beta(\lambda).
{\cal{R}}_{G B}^{2}={\frac{4 8 M^{2}( r^{2}- A^{2}{c o s}^{2}\theta ) [ ( r^{2}+ A^{2}{c o s}^{2}\theta )^{2}- 1 6 r^{2}A^{2}{c o s}^{2}\theta ]}{( r^{2}+ A^{2}{c o s}^{2}\theta )^{6}}}\,
|L_{g''}(T(\gamma^*))|_{g''}\leq C r^{-1}, |L_{g''}(\psi^* (T(\gamma^*))) |_{g''}= |L_{\psi^* g'}(\psi^*(T(\gamma^*))) |_{\psi^* g'}= |L_{g'}(T(\gamma^*)) |_{g'}\circ \psi \leq C r^{-1}.
\sum_v w(Z_v) \le \sum_v w(Z_v)\sum_{A\ni v}x_A = \sum_A x_A\sum_{v\in A}w(Z_v) &\le\sum_A x_A\sum_{v\in A}w(A) = \sum_A x_A\,|A|\,w(A) \le k\sum_A x_A\,w(A) =k\sigma^*.
[z^2-(\kappa^{(t)})^2]\tilde{\cal H}_{k-1}^{(s,t+1)}(z)=\tilde{\cal H}_{k+1}^{(s,t)}(z)+V_{k}^{(s,t)}\tilde{\cal H}_{k-1}^{(s,t)}(z),k=1,2,\dots,2m,
a(S) =C_S(N_1)^{-3/2}\,\det(S)^{\frac k2}\prod\limits_{p |{N_1}}\sum\limits_{i=0}^{b_p}p^{\frac i2}\Big(\frac{\alpha_p^{a_p-i+1}-\alpha_p^{-a_p+i-1}}{\alpha_p-\alpha_p^{-1}}- \delta_p p^{-\frac 12}\frac{\alpha_p^{a_p-i}-\alpha_p^{-a_p+i}}{\alpha_p-\alpha_p^{-1}}\Big).
l^{*}={\frac{r}{\gamma + 1}}+ \left({\frac{1}{\gamma ( \gamma + 1 )}}{\frac{A_{1}}{A_{0}}}-{\frac{1}{\gamma^{2}}}{\frac{B}{A_{0}}}\right) +{\frac{1}{\gamma ( \gamma + 1 )}}{\frac{B}{A_{0}}}\ln r + . . . .
\int_{-1}^{1}\int_{0}^{\sqrt{1-\rho^{2}cos^{2}\theta sin^{2}\phi}}\int_{0}^{\sqrt{1-\rho^{2}cos^{2}\theta sin^{2}\phi-1-\rho^{2}sin^{2}\theta sin^{2}\phi}}\rho^{2}sin\phi d\rho d\phi d\theta
\sum_{k=0}^\infty (f_X(x_0+k\lambda(J)) -f_A(x_0+(k-1)\lambda(J)) ) \geq \sum_{k = 0}^{k_0}(f_X(x_0+k\lambda(J))-f_A(x_0+(k-1)\lambda(J)) ) = f_X(x_0 + k_0 \lambda(J)) + \sum_{k=0}^{k_0-1}((f_X(x_0+k\lambda(J))-f_A(x_0+k\lambda(J)) \geq f_X(x_0 + k_0 \lambda(J)).
S_z^D=\int d^4 x \left(\int d^4 \theta K(z,z^{\dagger})\int d^2 \theta l z \left\{w-U(A_D)+\frac{\Lambda^4 zl}{w}f\left( \frac{A_D^2}{w},\frac{\Lambda^4}{w^2}\right) \right\}+h.c.\right)
y_{1}=-w\frac{\delta q_{/1}+q_{/2}(\kappa \,w^{2}+\delta '\,v)}{\delta^{2}}, y_{2}=\frac{\delta^{2}\,q-w^{2}\,v\,q_{/2}}{\delta w}, y_{3}=-\frac{v\,q+w^{2}\,q_{/2}}{w}.
y_m = y_{\frac{k}{2}-p}= \frac{1}{2 \tan(\frac{\pi p}{k})}= \frac{1}{2 \frac{\pi p}{k}(1 + O(\frac{p^2}{k^2}))}= \frac{k}{2 \pi p}\Big(1 + O(k^{-2}p^2)\Big).
Q(n,n-2p) - Q(n,n) = \frac{1}{6}(n-1)((n-1)^2 + 3(n-2p) - 1) - \frac{1}{6}(n-1)((n-1)^2 + 3n -1) &= \frac{1}{6}(n-1)((n-1)^2 +3n-6p-1 -((n-1)^2 +3n-1)) &= \frac{1}{6}(n-1)(3n -6p - 3n) &= (1-n)p,
(\Phi\circ\mathcal{J})(t) =\Phi(r(t),\sigma_{}(t),u_{}(t)) &=(r(t)\theta_{1}(\sigma_{}(t)),\dots,r(t)\theta_{n}(\sigma_{}(t)),\tau_{1}(u_{}(t)),\dots ,\tau_{n}(u_{}(t))),
g^{*}_{\lambda}(u)(x) = \Bigl[ \int_{0}^{\infty}\int_{\mathbb{R}^{n}}\bigl(\frac{t}{t+|y|}\bigr)^{\lambda n}t^{1-n}|\partial_{t,x}U(x-y,t)|^{2}dydt \Bigr]^{\frac12}
\vert s(x) - s_M^{\gamma}(x) \vert \leq \underbrace{\vphantom{\sum_{n=1}^{\infty}}c \mathcal{A}N P_1P_2}_{C}\underbrace{\sum_{n=M}^{\infty}\frac{(\sqrt{2}\varepsilon^2 L)^{n}}{\gamma^n \sqrt{n!}}}_{T_{M}}
K = \sum_{\alpha}\left( p_{\alpha}^{\bar{z}}\dot{z}_{\alpha}+ p_{\alpha}^{z}\dot{{\bar{z}}}_{\alpha}+ J_{\alpha}\cos \theta_{\alpha}\dot{\phi}_{\alpha}\right) + \int d^{2}z{\frac{\kappa}{2}}\left( \dot{A}_{z}^{a}A_{\bar{z}}^{a}- \dot{A}_{\bar{z}}^{a}A_{z}^{a}\right)
S^{(0)}_{\rm grav}=\frac{1}{2}\int d^4x[\partial_\sigma\gamma_{\lambda\rho}\partial^\sigma\gamma^{\lambda\rho}-\partial_\lambda\gamma^{\rho\kappa}\partial_\kappa\gamma^\lambda_\rho-\frac{1}{4}\partial_\rho\partial^\rho\gamma-\frac{1}{\alpha}\partial_\rho\gamma^\rho_\lambda\partial_\kappa\gamma^{\kappa\lambda}],
\mathbb{P}[X^n=\mathbf{x}_2, Y^n=\mathbf{y}_1|X^n\in\mathcal{A}, Y^n\in \mathcal{B}] =\mathbb{P}[X^n=\mathbf{x}_1, Y^n=\mathbf{y}_2|X^n\in\mathcal{A}, Y^n\in \mathcal{B}] &=\frac{p_{21}^{n/2}p_{12}^{n/2}}{2(p_{11}^{n/2}p_{22}^{n/2}+p_{12}^{n/2}p_{21}^{n/2})}.
I\ge (c_2-c_1)c_3+(a_2-a_1)a_3 \ge \frac23 [(a_2+c_2) - (a_1+c_1)] = \frac83(K_{13}-K_{12})=\frac83 [ K_{13}+ sK_{12}- (1+s)K_{12}] \ge \frac83 [K_s - (1+s)\epsilon_0+(1+s)\epsilon] \ge \frac83 [\frac{\sqrt{2}}2 - \frac{\sqrt{4+2\sqrt{2}}}4+(1+s)\epsilon] > \frac83 \epsilon.
\vec{a}^{\vec{x}}a_j^{-k}= s_{1,2}^{x_1}s_{1,3}^{x_2}\cdots s_{1,m+1}^{x_m}\cdots s_{1,j+1}^{x_j}\cdots s_{1,m+2}^{x_{2m+1}}\cdot s_{j-m+1,m+2}^{-k}&=s_{1,2}^{x_1}s_{1,3}^{x_2}\cdots s_{1,m+1}^{x_m}\cdots s_{j-m+1,j+1}^{x_j-k}\cdots s_{1,m+2}^{x_{2m+1}}.
\begin{aligned}\frac{\mathrm{d}}{\mathrm{d}t}q(0,t) =-\alpha \left(e^{\theta}-1\right)q(0,t), \frac{\mathrm{d}}{\mathrm{d}t}q(n,t) =-\alpha \left(e^{\theta}-1\right)q(n,t)+\alpha\sum_{j=1}^{n}\frac{\theta^{j}}{j!}q(n-j,t),\n\ge 1,\end{aligned}
\hat{\mathbf{\Sigma}}_{l}=\frac{w_{l}}{\mu}\left(\mathbf{\Omega}_{l}^{-1}-\left(\mathbf{\Omega}_{l}+\mathbf{H}_{l,l}\mathbf{\Sigma}_{l}\mathbf{H}_{l,l}^{\dagger}\right)^{-1}\right)l=1,\ldots,L;
I_1 = \sum_{\frac{\eta_1 n}{2N}\leq k \leq \frac{2\eta_2 n}{N}}\mathbb{P}(N_l = k)\mathbb{E}(T_l| N_l =k) = \sum_{\frac{\eta_1 n}{2N}\leq k \leq \frac{2\eta_2 n}{N}}B(k; n,p_l) \Delta(k,q_l)
A_{kj}^i:=\{\sigma_i<+\infty\}\cap \{z_i(u_k,\sigma_i)-z_i(u_{k-1},\sigma_i)= \varepsilon,\ldots,z_i(u_{j+1},\sigma_i)-z_i(u_j,\sigma_i)=\varepsilon, z_i(u_j,\sigma_i)-z_i(u_{j-1},\sigma_i)>\varepsilon\},2\leqslant j\leqslant k-1,3\leqslant k\leqslant n,1\leqslant i\leqslant n-1,
\mathrm{Po}(N)\simeq(\mathcal{I}_N^G/\mathcal{P}_{\mathbb{Q}})/(\mathcal{P}_N^G/\mathcal{P}_{\mathbb{Q}})\simeq\mathcal{I}_N^G/\mathcal{P}_N^G=\mathcal{I}_N^G/(\mathcal{P}_N\cap\mathcal{I}_N^G)\simeq(\mathcal{I}_N^G\cdot\mathcal{P}_N)/\mathcal{P}_N
\frac{1}{\epsilon}\psi(X-\epsilon\xi,T) = \frac{1}{\epsilon}\psi(X,T) - \epsilon\xi\frac{1}{\epsilon}\psi_{X}(X,T) + \frac{1}{2}\epsilon^{2}\xi^{2}\frac{1}{\epsilon}\psi(X-\epsilon\xi,T) + \epsilon^{2}\xi^{2}\frac{1}{\epsilon}R_{\psi}(X-\epsilon\xi,T) &= \theta - k\xi + \frac{1}{2}\epsilon\xi^{2}k_{X}+ \epsilon\xi^{2}R_{\psi}(X-\epsilon\xi,T),
U_{\vec{c},\vec{c}^{\prime}}^{j}=W_{\vec{c}^{\prime}}^{j}\circ(W_{\vec{c}}^{j})^{-1}:(Poly_{\mathbb{C}}(S^{2})_{\leq n},\star_{\vec{c}}^{n})\longrightarrow(Poly_{\mathbb{C}}(S^{2})_{\leq n},\star_{\vec{c}^{\prime}}^{n}) V_{\vec{c},\vec{c}^{\prime}}^{j}=(W_{\vec{c}}^{j})^{-1}\circ W_{\vec{c}^{\prime}}^{j}:M_{\mathbb{C}}(n+1)\rightarrow M_{\mathbb{C}}(n+1)
(f_0)_2(r)=c_0'(f_0)_1(r)\int\frac{e^{\frac{r^2}{4}}r^{1-m}}{((f_0)_1(r))^2}=ce^{\frac{r^2}{4}}r^{-(m+2\lambda)}\int e^{-\frac{r^2}{4}}r^{m+4\lambda+1}(1+O(\frac{1}{r^2}))=p(r)
\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}
\begin{aligned}P_k'(r_k,q_1') \le P_k'(r_k,N') = P_k(r_k,q) \le P_k(i,s_{k,q}) &\le P_k(i,s_{k,q}+2)-2 = P_k'(i,1)-2 \le P_k'(i,q_2')-2.\end{aligned}
\frac{z-w}{qz-q^{-1}w}\ell^-_{1j}(z)\ell^-_{j,j+1}(w)+\frac{(q-q^{-1})z}{qz-q^{-1}w}\ell^-_{jj}(z)\ell^-_{1,j+1}(w)= \frac{z-w}{qz-q^{-1}w}\ell^-_{j,j+1}(w)\ell^-_{1j}(z)+\frac{(q-q^{-1})w}{qz-q^{-1}w}\ell^-_{jj}(w)\ell^-_{1,j+1}(z).
m(l)= \sum_{j=0}^{\infty}m_j(l)\leq \left(\frac{Cr\sigma_{r,X}(rl)}{t}\right)^{rl}+\sum_{j= 1}^\infty\left(\frac{C2^{5j}r\sigma_{r,X}(rl)}{t}\right)^{rl}\exp(-rl2^{j-3}).
\begin{array}{l}\underset{i=1}{\overset{m}{\sum}}v_{i}=0\mbox{and}v_{i}\in N_{C_{i}}(x^{*})\mbox{for all}i\in\{1,\dots,m\}\quad\mbox{implies}v_{i}=0\mbox{for all}i\in\{1,\dots,m\}\end{array}
\textbf{C}=[\textbf{C}_{-1},\textbf{C}_0,\ldots,\textbf{C}_{N_h+1}]^T, &\textbf{C}_l=[c_{0,0,l},c_{0,1,l},\ldots,c_{0,2M,l},c_{1,0,l},\ldots,c_{1,2M,l},\ldots,c_{2^k-1,2M,l}], &\textbf{H}(x)=[B_{-1}(x),B_{0}(x),\ldots,B_{N_h+1}(x)],
\begin{array}[c]{ll}-d{p}^{10}(t)= \{p^{10}(t)-2\beta^0\bar{X}^{10}{(t)}\}dt,\t\in(t_{i-1},t_{i}), &p^{10}(t_{i})= -2\beta^0\bar{X}^{10}(t_{10})1_{i=10}(i)-\beta^i+p^{10}(t_{i}^{+}), i=1,2,\ldots,10,\end{array}
( \bar{\nabla}_{\mu}+ 2 \partial_{\mu}\bar{\phi}) ({\cal G}^{\mu \nu}- \bar{G}_{\alpha}^{~ ~ \nu}h^{\alpha \mu}- \bar{G}_{~ ~ \beta}^{\mu}h^{\beta \nu}) + \bar{G}^{\mu \nu}( \delta \Gamma_{\sigma \mu}^{\sigma}( h ) + 2 ( \partial_{\mu}\varphi ) ) = 0 ,
(\widehat{A}^{(1)}_{\varepsilon}G) (\eta) = -(|\eta|m + \varepsilon E^{-}(\eta)) G(\eta):= - E(\eta)G(\eta), begin{align*}.2cm] (\widehat{A}^{(2)}_{\varepsilon}G) (\eta) = \varepsilon\int_{\mathbb{R}^d}E^{+}(y,\eta) G (\eta\cup y) dy,
\vartheta_{1}( \tau , \nu ) = \sqrt{- 1}( y^{- 1 / 2}- y^{1 / 2}) q^{1 / 8}\prod_{n = 1}^{\infty}( 1 - q^{n}) ( 1 - q^{n}y ) ( 1 - q^{n}y^{- 1}) ,
\psi_{w}(u)=E[w_{t}e^{iux^{*}}]=\frac{E[w_{t}e^{iux_{1t}}]}{E[e^{iux_{1t}}]}exp\int_{0}^{u}i\frac{E[x_{2t}e^{ivx_{1t}}]}{E[e^{ivx_{1t}}]}dv
\sum_{\deg(D) = N}\mu_k(D) + \sum_{j=1}^{2g}\sum_{\ell=0}^{k-1}\frac{Z(\gamma_{j, \ell}^{-1})}{k\gamma_{j, \ell}^{1-k}Z'(\gamma_j^{-1})}\gamma_{j, \ell}^{N+1}- \frac{q^{N}}{\zeta(k)}\frac{q^{-g}h}{(1-q^{-1})}.
\Pi^\mu_\nu ={\rho\over12}\left[\begin{array}{ccc}-(\rho+2\delta\rho) 2a(\rho+P)(v +B )_{|j}& -2a^{-1}(\rho+P)v^{|i}\left\{2P+\rho+2(1+P/\rho)\delta\rho + 2\delta P\right\}\delta^i_j- (1+3P/\rho)\delta\pi^i_j\end{array}\right] \,.
bK_{G\left( x,b;c\right)}^{-}\left( x\right) =\left( \frac{x}{b}\right)^{x/b}\exp\left( -\frac{x}{b}\right) \left\{\frac{\Gamma\left(x/b+1\right)}{\gamma\left( x/b+1,c/b\right)}\right\}\left\{\frac{1}{\Gamma\left( x/b+1\right)}\right\}.
\frac{\hat y_1^h - x_1}{h^2}= u - x_2v_2' - x_3v_3' + \psi_1^h\,, \frac{\hat y_i^h - hx_i}{h^2}= \frac{v_i}{h}+ wx_i^\bot + \psi_i^h\,,i=2,3\,,
|\{xy \in G_a \mid xy \}| \sim tda^2p = \frac{a^2t(1-c)}{2c}; |\{xy \in G_a \mid xy \}| \sim t^2a^2q = \frac{a^2t(1-c)}{2c}; |\{xy \in G_a \mid xy \}| = 2t\binom{a}{2}\sim a^2t.
P = \sum_{n=0}^{\infty}\sum_{\mu_1,\cdots,\mu_n=1}^{N}\int{\frac{{d\theta_1}\cdots{d\theta_n}}{{n!}(2\pi)^n}| V_{\mu_1}(\theta_1)\cdots V_{\mu_n}(\theta_n) \rangle \langle V_{\mu_1}(\theta_1)\cdots V_{\mu_n}(\theta_n)|}
Z_{0,n}^{\epsilon}=\check{Z}_{0,n}^{\epsilon}+ \check{Z}_{0,1}^{\epsilon}\epsilon Z_{1,n}^{\epsilon}+\sum_{\chi =3}^{n-2}\check{Z}_{0,\chi}^{\epsilon}\epsilon^2 Z_{\chi,n}^{\epsilon}+ \check{Z}_{0,n-1}^{\epsilon}\epsilon Z_{n-1,n}^{\epsilon}.
\delta_{\epsilon}\lambda = \left( \not \! \partial \phi +{\textstyle \frac{1}{2 \cdot 3 !}}\Gamma_{1 1}\not \! \! H \right) \epsilon +{\textstyle \frac{5 i}{4}}m e^{\phi}\epsilon -{\textstyle \frac{3 i}{8}}e^{\phi}\not \! G^{( 2 )}\Gamma_{1 1}\epsilon \, ,
\mathcal{R}_{n,\alpha}(\theta(\mathcal{P}), \Phi \circ \rho, \varepsilon) = \inf_{Q \in \mathcal{Q}_\alpha}\inf_{\hat{\theta}}\sup_{P_\varepsilon \in \mathcal{P}_\varepsilon(\mathcal{P})}\mathbb{E}_{P_\varepsilon,Q}\left[\Phi \circ \rho\left\{\hat{\theta}, \, \theta(P)\right\}\right],
\rho_n= \sup_{t\in\mathbb{R}}|P(T_X\leq t)-P(T_Y\leq t)| \lesssim (\psi^2+\psi\beta)\phi(M_{x},M_y)+(\psi^3+\psi^2\beta+\psi\beta^2)\frac{(2M+1)^{2}}{\sqrt{n}}(\bar{m}_{x,3}^3+\bar{m}_{y,3}^3) &+\psi\varphi(M_{x},M_y)\sigma_j\sqrt{8\log(p/\gamma)}+\gamma+(e_{\beta}+\psi^{-1})\sqrt{1\vee\log (p\psi)}.
- \frac{\hat{y}_{2 3}}{\hat{y}_{1 2}\hat{y}_{3 4}}( \sum_{i = 0}^{4}c_{n}( s , t ) v^{n}= ( 1 + v ) ( s + v ( 1 + s - t ) + v^{2}) O ( \frac{1}{( \hat{y}_{1 2}\hat{y}_{3 4})^{2}})
\begin{aligned}(d/dt)|_{t=0}\langle G(x,\Psi(y,t))[D\Psi(y,t)]^{-1}(d/dt)\Psi(y,t)\rangle J\Psi(y,t), N(y)\rangle &\qquad= \langle \nabla_{y}G(x,y),X(y)\rangle X(y)+G(x,y)X(y)\mathrm{div}(X(y)) &\qquad= X(y)\mathrm{div}_{y}(G(x,y)X(y)).\end{aligned}
B.\big(r(h)+ \frac{i}{2n}(C.r(h))\big)= B.r(h)+\frac{i}{2n}(BC.r(h)) \equiv \frac{(-1)}{2n}hC.r(h)+ihr(h) \equiv ih\big(\frac{(-1)}{2ni}C.r(h)+r(h)\big) = ih\big(r(h)+\frac{i}{2n}C.r(h)\big)
\begin{array}{c}h^0x|_C=\sum\limits_{i=1}^sh(c_i)I_{A_i\cap C}=\sum\limits_{i=1}^s \sum\limits_{k=1}^\sigma h(c_i)I_{A_i\cap B_k\cap C}= \sum\limits_{i=1}^s \sum\limits_{k=1}^\sigma h(d_k)I_{A_i\cap B_k\cap C}=\sum\limits_{k=1}^\sigma h(d_k)I_{B_k\cap C}\ \P^*-\mbox{a.s.}\end{array}
\begin{aligned}\Delta\rho(x,t) = \varepsilon^{-2}\partial_r^2\eta(y,t,r)-\varepsilon^{-1}H(y,t)\partial_r\eta(y,t,r) -r|W(y,t)|^2\partial_r\eta(y,t,r)+\Delta_\Gamma\eta(y,t,r)+O(\varepsilon). \end{aligned}
T_n \coloneqq \sum_{i=1}^k \beta_i (\alpha_i^n - c_i^n) = \sum_{\nu=1}^p \bigg[ \sum_{j=k_\nu}^{k_{\nu+1}-1}\beta_j \alpha_j^n - \Big( \sum_{j=k_\nu}^{k_{\nu+1}-1}\beta_j \Big) c_{k_\nu}^n \bigg].
\mathbb{P}_{a,b,r,p}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( z\right)=\sum_{k=0}^{rp+\sigma}\sum_{i=0}^{rp+\sigma}\binom{rp+\sigma -i}{rp+\sigma -k}A_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,i\right)\left( z-1\right)^{rp+\sigma -k}.
\theta_{m,20}(q)= \sum_{p=-\infty}^{+\infty}~\sum_{\ell=0}^9q^{20(10p+\ell)^2 + (10p+\ell)m + \frac{m^2}{80}}= \sum_{\ell=0}^9 \theta_{10m + 400 \ell,2000}(q),
\kappa^2 = - \frac{1}{4}\lim_{R\rightarrow R_0}\Big[ g^{tt}g^{RR}(\partial_R g_{tt})^2 \Big]= \frac{(d-3)^2}{4 R_0^2 R_T^2 \cosh^2 \alpha \, A|_{R=R_0}}\ .
\left[{\frac{d^{4}}{d r^{4}}}+{\frac{2}{r}}{\frac{d^{3}}{d r^{3}}}- \left( M^{2}+{\frac{l ( l + 1 )}{r^{2}}}\right){\frac{d^{2}}{d r^{2}}}-{\frac{2}{r}}M^{2}{\frac{d}{d r}}\right] c_{l}( M r ) = 0 ,
t r_{n}' =\left(\frac{1}{R_{n}}+\frac{1}{R_{n}-1}\right)r_{n}^{2}+(2n+\alpha)\frac{R_{n}}{R_{n}-1}r_{n}+n(n+\alpha)\frac{R_{n}}{R_{n}-1}, t R_{n}' =t R_{n}^2+(2n+\alpha-t)R_{n}+2r_n.
{\cal N}_{Q}[ \tilde{\cal H}_{x}] ={\frac{1}{2}}\tilde{\Pi}_{x}^{2}+{\frac{1}{2}}( \nabla \tilde{\phi}_{x})^{2}-{\frac{1}{2}}I_{0}[ Q^{2}] +{\frac{1}{4}}Q^{2}I_{1}[ Q^{2}] +{\cal N}_{Q}[ V ( \tilde{\phi}_{x}) ]
\left( L_{n}^{1 / 2}( x ) \right)_{i k}K_{n}( y ) \left( K ( y - x + \frac{i}{2}) \right)_{k j}= \left( K ( y - x + \frac{i}{2}) \right)_{i k}K_{n}( y ) \left( L_{n}^{1 / 2}( x ) \right)_{k j},
D_\alpha u_t = D_\alpha[\rho(Q\ast u - u) ] - D_\alpha (\mathbf{u}\cdot \nabla u) & = D_\alpha[\rho(Q\ast u - u) ] - [ D_\alpha(\mathbf{u}\cdot\nabla u) - \mathbf{u}\cdot \nabla D_\alpha u] - \mathbf{u}\cdot \nabla D_\alpha u.
P =(x_4+b_{4,3}x_3+b_{4,1}x_1+\varepsilon_4)(x_2+b_{2,1}x_1+\varepsilon_2)(x_0+\varepsilon_0) Q =(x_3+a_{3,0}x_0+\gamma_3)(x_2+a_{2,0}x_0+\gamma_2)(x_1+a_{1,0}x_0+\gamma_0)
\frac{|\nabla u_n(\frac{x_n-x_n'}{t_n})|^2}{u_n^2(\frac{x_n-x_n'}{t_n})}\to \frac{|\nabla \bar u(\bar x) |^2}{\bar u^2(\bar x)}=\frac{1}{((\bar x\cdot e)^+)^2}\in (0,+\infty),
B.u= B\big(f(h)\big(r(h)+\frac{i}{2n}(C.r(h))\big) \big) \equiv \big(f^{\prime}(h)B+f^{\prime\prime}(h)C\big)\big(r(h)+\frac{i}{2n}(C.r(h))\big) \equiv ihf^{\prime}(h)(r(h)+\frac{i}{2n}C.r(h))-2nif^{\prime\prime}(h)(r(h)+\frac{i}{2n}C.r(h)) \equiv \big(ihf^{\prime}(h)-2nif^{\prime\prime}(h)\big)\big(r(h)+\frac{i}{2n}C.r(h)\big)
\boldsymbol{H}_R=\left[\begin{matrix}1 -c_1 \cdots \cdots -c_{n-1}0 \ddots \ddots \ddots \vdots 0 \ddots \ddots \ddots \vdots \vdots \ddots \ddots \ddots -c_1 0 \cdots 0 0 1 \end{matrix}\right] ,
( \Phi^{\prime}{}^{n}, \Phi_{m}^{\prime}{}^{*}) = \delta_{m}^{n}, \qquad ( \Phi^{\prime}{}^{n}, \Phi^{\prime}{}^{m}) = ( \Phi_{n}^{\prime}{}^{*}, \Phi_{m}^{\prime}{}^{*}) = 0 ,
\left|\sum_{k=1}^m \mu(k) \left(\left\lfloor \frac{n}{k}\right\rfloor - \left\lfloor \frac{n'}{k}\right\rfloor\right)\right| = \left|\sum_{t=1}^q \left( M(n/t) - M(n'/t) \right)\right| \leq \sum_{t=1}^q \left( \left\lfloor \frac{|d-d'|}{t}\right\rfloor +1 \right) \leq q + |d-d'| \sum_{t=1}^r \frac{1}{t}.
b_0(\lambda,\alpha) =\exp{(2\pi i x' (A^{-1}\tau_0'\delta^{-1})\alpha)}, b_t(\lambda,\alpha) =b(x',\alpha)=\exp{(2\pi i x' (A^{-1}\Im(\tau')t\delta^{-1})\alpha})=q^{\langle\alpha,\check{\phi}(\lambda)\rangle}, b'(\lambda,\alpha) =\exp{(-\pi i x' S\mathfrak{d}^{-1}\alpha)}.
\left(1-vux-\frac{xv}{1-v}\right)F(x,u;v) &=1-x+x(1-x)uK+xuF(x,u;0)-xu-vux+\frac{x}{1-v}F(x,u;1) &+\frac{ux^3}{(1-x)^2}\left(\frac{F(x,u;0)+F(x,u;1)-1}{(1-ux)(1-uvx)}-1\right) &+\frac{ux^3}{(1-x)^2(1-ux)(1-vux)}\left(\frac{1}{1-x}F\Big(x,u;\frac{1}{1-x}\Big)-\frac{x}{1-x}-F(x,u;1)\right).
f(\eta \cdot \lambda)= f \begin{pmatrix}r\lambda_1+ f \lambda_3 r\lambda_2+ f \lambda_4 a\lambda_1+ \sigma \lambda_3 a\lambda_2+ \sigma \lambda_4 \end{pmatrix}= \phi(r\lambda_2+ f \lambda_4) + a(\lambda_1)+ \sigma (\lambda_3) = \phi(r \lambda_2)+\phi(f \lambda_4)+ a(\lambda_1)+ \sigma (\lambda_3).
A:=\sum\limits_{i=1}^\infty A_i \in (\mathcal{X};\mathcal{Y})(E_1,\ldots,E_n;F){\rm ~~and~~}\|A\|_{(\mathcal{X};\mathcal{Y})}^q\le\sum\limits_{i=1}^\infty \|A_i\|_{(\mathcal{X};\mathcal{Y})}^q.
\left( \begin{matrix}{u}\\{v}\\{w}\\ \end{matrix}\right)_{t_{1}}= \left( \begin{matrix}{0}&{u}&{- u}\\{v}&{0}&{- v}\\{w}&{w}&{- 2 w}\\ \end{matrix}\right) \left( \begin{matrix}{u}\\{v}\\{w}\\ \end{matrix}\right)_{x}
I_1=\int_{\Pi}\varrho u_1\partial_t\varphi dxdt, I_2=\int_{\Pi}\varrho u_1u_i\partial_{x_i}\varphi dxdt, I_3=\int_{\Pi}p\partial_{x_1}\varphi dxdt, I_4=-\int_{\Pi}\mathbb S_{i1}\partial_{x_i}\varphi dxdt, I_5=\int_{\Pi}\varrho f_1\varphi dxdt.
h^{(q)}(0) = \frac{1}{\Gamma(-q)}\int_0^1 t^{-1-q}\left( h(t) - \sum_{k=0}^m \frac{d^k}{ds^k}h(s)\Big|_{s=0}\frac{t^k}{k!}\right) \, dt &+ \frac{1}{\Gamma(-q)}\int_1^\infty t^{-1-q}h(t) \, dt + \frac{1}{\Gamma(-q)}\sum_{k=0}^m \frac{1}{k!(k-q)}\frac{d^k}{dt^k}h(t)\Big|_{t=0};
e_{r(\alpha)}= \langle \delta_\alpha, \delta_\alpha \rangle = \langle h(\delta_\alpha), h(\delta_\alpha) \rangle = (z_1, \ldots, z_k)^* (z_1, \ldots, z_k) &= (\overline{z_1}, \ldots, \overline{z_k}) (z_1, \ldots, z_k) = ( |z_1|^2, \ldots, |z_k|^2).
\begin{aligned}M_\otimes(A)(((0,i),(0,j))) = A((0,(i,j)))\;, M_\otimes(A)(((1,i),(0,j))) = 0\;, M_\otimes(A)(((0,i),(1,j))) = 0\;, M_\otimes(A)(((1,i),(1,j))) = A((1,(i,j)))\;.\end{aligned}
\sum_{k_{1}+k_{2}+\cdot\cdot\cdot+k_{m}=n}(\begin{matrix}n\\k_{1},k_{2},...,k_{m}\end{matrix})x_{1}^{k_{1}}x_{2}^{k_{2}}\cdot\cdot\cdot x_{m}^{k_{m}}=(x_{1}+x_{2}+\cdot\cdot\cdot+x_{m})^{n}
\mathbb{E}\sum_{x \in V_{\sigma}(\mathbf{X}_0)}f(\mathbf{X}_0, x) &\qquad= \int_0^\infty \dots \int_0^\infty \mathbb{E}f( \mathbf{X}_0 \cup \{\sigma_1 z_1\}\oplus\tilde{\mathbf{X}}^{z_2,\dots,z_l}, \{\sigma_1 z_1\}\oplus \tilde{x}^{z_2,\dots,z_l})\,dz_1\dots\,dz_l,
\Delta\mathbf{K}_k = \widetilde{\boldsymbol{\Delta}}_k^H\widetilde{\mathbf{H}}_k + \widetilde{\mathbf{H}}_k^H\widetilde{\boldsymbol{\Delta}}_k + \widetilde{\boldsymbol{\Delta}}_k^H\widetilde{\boldsymbol{\Delta}}_k,
\delta \psi_{\mu}={\frac{1}{\sqrt{2 \pi G_{N}}}}\hat{\nabla}_{\mu}\epsilon ={\frac{1}{\sqrt{2 \pi G_{N}}}}[ \nabla_{\mu}-{\frac{\sqrt{G_{N}}}{4}}F_{\nu \rho}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}] \epsilon .
\mathcal{P}_k\varphi(\epsilon)=\left(\begin{array}{cc}\mathcal{P}^+_k\varphi(\epsilon) \mathcal{P}^-_k\varphi_k(\epsilon)\end{array}\right)=\left(\begin{array}{cc}\int_0^{+\infty}\psi_{k,\varepsilon}(r)\varphi(r)r^{n-1}dr \mathcal{C}\left(\int_0^{+\infty}\psi_{k,-\varepsilon}(r)\varphi(r)r^{n-1}dr\right)\end{array}\right)