harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Pr_{\rho\sim P_\epsilon}[\rho\notin D_k(H\rho^\intercal)] \leq 2e^{-\frac{\sqrt{N}}{3\epsilon}}+\frac{N}{k^*}\max_{j\in B}\left\{\Pr_{v\sim \mu_t}[|v H|=j] \cdot \frac{2^N}{\binom{N}{j}}-1\right\}&+\frac{2^{h(\epsilon)N+N^{\frac{4}{5}}}}{k^*}\max_{j\notin B}\left\{\Pr_{v\sim \mu_t}\big[|v H|=j\big]\cdot 2^{2\epsilon N\log|1-\frac{2j}{N}|}\right\}.
d(b_{10}) = U c_{10}' + \sigma_1 b_{12}' + \sigma_3 b_{33}d(b_{25}) = \sigma_{123}b_8 + \sigma_{123}b_{24}' + U c_{25}' d(a_{29}) = U \sigma_3 b_6 + U \sigma_{123}b_{17}' + \sigma_3 a_{18}+ U^2 a_{21}d(a_{34}) = U^2 a_2' + U \sigma_{123}b_{12}' + \sigma_3 a_{13}.
| f^n(X_s,Z^n_s) - f(X_s,Z_s) | \leq | f^n(X_s,Z^n_s) - f^n(X_s,Z^{n-1}_s) | + | f^n(X_s,Z^{n-1}_s) - f(X_s,Z_s) | \leq c \| Z^n_s - Z^{n-1}_s \| + | f(X_s,Z^{n-1}_s) - f(X_s,Z_s) |.
\begin{aligned}\sup\limits_{t \in [0,T_0]}\big(\|w_1(t,\cdot) - w_2(t,\cdot)\|_{C^{0}([0,1]}+\|z_1(t,\cdot) - z_2(t,\cdot)\|_{C^{0}([0,1])}\big) + \|h_1-h_2\|_{C^1([0,T_0])}+ \|g_1-g_2\|_{C^1([0,T_0])}&\leq K_1(\|w_{01}-w_{02}\|_{C^2([0,1])}+\|z_{01}-z_{02}\|_{C^2([0,1])}+|h_{01}-h_{02}|+|g_{01}-g_{02}|),\end{aligned}
\begin{cases}\partial_{t}a_{k}-\Delta a_{k}=-\mathrm{div}(u^{c,\gamma}\otimes a_{k-1}+a_{k-1}\otimes u^{c,\gamma})-\nabla\pi_{k-1}, \mathrm{div}a_{k}=0, \pi_{k-1}=(-\Delta)^{-1}\partial_{i}\partial_{j}(u^{c,\gamma}\otimes a_{k-1}+a_{k-1}\otimes u^{c,\gamma}), a_{k}(x,0)=w_{0}.\end{cases}
(x^-_{m_2}\cdots x^-_{m_n},x^-_{s_1}x^-_{s_2}\cdots x^-_{s_{n-1}}\Omega_\psi(-m_1-p)x_{s_{n}}^- ) &=\delta_{m_1+p,s_{n}}(x^-_{m_2}\cdots x^-_{m_n},x^-_{s_1}x^-_{s_2}\cdots x^-_{s_{n-1}})\in\mathbb Z[\![q]\!].
A :=\left\{\, | \{x\in D \, : \, \tilde{X}_{t_0}(x) \ge -1) \}|\ge (\log s)^{-\kappa}\, \right\}, B_n :=\left\{\, | \{x\in D \, : \, \tilde{X}_{t_0}(x) \ge n) \}|\ge n^{-1}(\log n)^{-2}\, \right\}, \bar B_n := B_n\setminus B_{n-1}, \n\ge n_0+1 \bar B_{n_0}=B_{n_0}\setminus A.
\left | \frac{ds}{dz}\right | = \frac{\sqrt 2}{1-u-v+u^2+uv+v^2}&= \frac{\sqrt 2 (\gamma^2 + \gamma\delta + \delta^2)}{\gamma^2 + \gamma\delta + \delta^2 + \alpha^2 + \alpha\beta + \beta^2 - \alpha\gamma-\beta\delta-\beta\gamma}
\mu u(t)= \gamma_1(t)(H_1(\alpha_1[u])+\chi_{(\tau_1,1]}(d_{11}I_1+e_{11}N_1)(u(\tau_1))) &+\delta_1(t)(L_1(\beta_1[v])+\chi_{[0,\tau_1]}(d_{12}I_1+e_{12}N_1)(u(\tau_1)))+ F_1(u,v)(t).
h(x_1, y_1, z_1)=h(x_1, y_1, z_1') = \begin{cases}\left(\frac1{ac}, \frac1{ac}, \frac1{a^2}- \frac{2a}{c}\lambda \right)z_1=z_1', \left(\frac1{ac}, \frac1{ac}, \frac1{a^2}- \frac{a}{c}\lambda \right) z_1\neq z_1'.\end{cases}
|\Gamma | (1-o(1))^{n}\left(m !\right)^{\ell - 1}p^{n-s}\geq (1-o(1))^{n}\frac{(n-s)!}{(m!)^{\ell}}(m!)^{\ell -1}p^{n}=(1-o(1))^{n}\frac{(n-s)!}{m!}p^{n}=(1-o(1))^n n!p^n
a_0 + a_1\cos(\frac{2k\pi}{N}) + \dots a_{\lfloor \frac{N}{2}\rfloor}\cos(\frac{2k\lfloor \frac{N}{2}\rfloor\pi}{N}) + b_1 \sin(\frac{2k\pi}{N}) + \dots b_{\lfloor \frac{N-1}{2}\rfloor}\sin(\frac{2k\lfloor \frac{N-1}{2}\rfloor\pi}{N})= 0.
{\cal M}_0^{(r)}\sim{\textstyle{1\over 2}}\, (n_1 + n_2 ) \hat Q_o (0)-{\textstyle{1\over 2}}\, \left[ 2\cdot 2^{r/2}\cdot k_1 k_2 + 2 \right] (m+\bar m) \hat Q_v(\zeta\tau) \,.
\hat{H}_{5}=4\mu_{B}^{2}\sum_{i>j}\{-\frac{8\pi}{3}(s_{i}\cdot s_{j})\delta(r_{ij})+\frac{1}{r_{ij}^{3}}[s_{i}\cdot s_{j}-\frac{3(s_{i}\cdot r_{ij})(s_{j}\cdot r_{ij})}{r_{ij}^{2}}]\}
\Delta_1n(x,y) = n(2x+y)+n(2x-y) &-\frac{4n(x)n(y)}{\left(4n(y)^{\frac{2}{9}}-n(x)^{\frac{2}{9}}\right)^9}\Big[256n(y)+2304n(x)^{\frac{2}{9}}n(y)^{\frac{7}{9}}+2016n(x)^{\frac{4}{9}}n(y)^{\frac{5}{9}}&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+336n(x)^{\frac{6}{9}}n(y)^{\frac{3}{9}}+n(x)^{\frac{8}{9}}n(y)^{\frac{1}{9}}\Big]
\begin{array}{c}{{\displaystyle \int \frac{d^{4}k_{1}}{( 2 \pi )^{4}}\int \frac{d^{4}k_{2}}{( 2 \pi )^{4}}D_{m}^{2}( k_{1}) D_{m}( k_{2}) D_{m}( k_{1}+ k_{2}) \overline{{{D ( k_{1}+ p )}}}K_{1}.}}\\ \end{array}
\mathbb{P}\left(\bigcup_{d\geq 2}\bigcup_{m\geq 1}\mathcal{F}_{m,d.\varepsilon}\right)=\mathbb{P}\left(\bigcup_{d\geq 2}\bigcup_{m\in\{1,\ldots,12\cdot 2^d-1\}}\mathcal{F}_{m,d.\varepsilon}\right).
T = \left( \begin{array}{c c c c c}{{1}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{1}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{1}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{- 1}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{- 1}}\\ \end{array}\right) ,
D^{\beta}v = \begin{cases}v{,}|\beta|=0 D_iv{,}|\beta|=1 D_jD_iv{,}|\beta|=2\end{cases},\, D^{\beta}H = \begin{cases}H{,}|\beta|=0 D_iH{,}|\beta|=1 D_jD_iH{,}|\beta|=2\end{cases},\, K^{\beta}= \begin{cases}K{,}|\beta|=0 K^i{,}|\beta|=1 K^{ji}{,}|\beta|=2\end{cases}
R= \begin{pmatrix}R_1 R_3 R_2 R_4 R_5 R_1 \end{pmatrix}=\begin{pmatrix}F_d P_1^T E_e E_{k-2l_1}P_5^T E_e E_{k-2l_1}P_2^T E_{n-2k-2l_2}F_d P_4^T E_e F_d P_3^T E_{n-2k-2l_2}F_d P_1^T E_e \end{pmatrix}.
2(q - \log_q N) & \ge 2\frac{(q-1)^2}{q}- \frac{3(q-1 - t(q-1- \frac{(q-1)^2}{q}))^2}{(2t(q-1 - \frac{(q-1)^2}{q})+q-1)\ln \frac{N}{\epsilon}}& = \frac{2(q-1)^2}{q}- \frac{3(q-1)(1-t/q)^2}{(1+2t/q)\ln \frac{N}{\epsilon}}.
\begin{aligned}\int_{\ell_n}^{M_n}\frac{x+1}{(\log(x+1))^{\beta-1}}dx= \sum_{\ell=0}^k\frac{(M_n+1)^2}{2B_\ell (\log(M_n+1))^{\beta+\ell-1}}\qquad+O\left(\frac{M_n^2}{(\log M_n)^{\beta+k}}\right).\end{aligned}
\beta_{n}^{R^{[1]}}= q+\alpha_{1}+ \left(p+\beta \right)\beta \; , n \geq 0 \alpha_{n}^{R^{[1]}}= \frac{n(n+3)}{(n+1)(n+2)}\gamma(p+2\beta) \; , & \gamma_{n}^{R^{[1]}}= \frac{n(n+5)}{(n+2)(n+3)}\gamma^{2}\; , \; n \geq 1.
\mathcal A_p \;(\delta p,\delta\b u):= \mathcal A_{p\mu\rho}(\delta p,0,0,\delta \b u)=\begin{pmatrix}\mathcal L_p \;(\delta p,\delta\b u) \mathcal B \;(\delta\b u)|_{\partial\Omega}\end{pmatrix}=\begin{pmatrix}\mathcal S 0\end{pmatrix}.
\begin{alignedat}{1}m_h(\dot u_h(t),v_h) + a_h(u_h(t),v_h) = m_h(f(t),v_h) \forall \,v_h\in V_h (0<t\le T) m_h(u_h(0),v_h) = m_h(u_{0},v_h) \qquad\ \ \, \forall \,v_h\in V_h . \end{alignedat}
-4 \int_{t}^{\infty}\frac{\lambda_{0,1}''(s)}{1+s-t}ds + 4 \alpha \log(\lambda_{0,0}(t)) \lambda_{0,0}''(t) - 4 \int_{t}^{\infty}\frac{\lambda_{0,0}''(s) ds}{(\lambda_{0,0}(t)^{1-\alpha}+s-t)(1+s-t)^{3}}&=E_{v_{3},ip,01}+E_{v_{3},ip,f}+E_{\lambda_{0,1}}
Q_k^{\Pi_2}(t) =\sum_{i=1}^k Q_i^{\Pi_2}(t)-\sum_{i=1}^{k-1}Q_i^{\Pi_2}(t) &\geq \sum_{i=1}^{k}Q_i^{\Pi_1}(t)-kn(N)-\sum_{i=1}^{k-1}Q_i^{\Pi_1}(t) &=Q_k^{\Pi_1}(t)-kn(N).
\begin{aligned}\|v^{k+1}-v^*\|^2_{H}-\|v^k-v^*\|^2_{H}= A^k-A^{k-1}+B^k-B^{k-1}+2(C^k-C^{k-1}) = A^k-A^{k-1}+2(C^k-C^{k-1})+\frac{1-\tau^k}{(\tau^k)^2}E^k-\frac{1-\tau^{k-1}}{(\tau^{k-1})^2}E^{k-1}&+\frac{1}{\tau^k}(D^k-D^{k-1})-\frac{1}{\tau^{k-1}}(D^{k-1}-D^{k-2})+D^{k-1}-D^{k-2}.\end{aligned}
\phi^{(2)}_c(\tau) =2\tan^{-1}\frac{1}{\alpha}\left(1 +\sqrt{1-\alpha^2}{\frac{e^{-\sqrt{1-\alpha^2}\omega_0(\tau-\tau_0)}+f^{(2)}(\alpha)}{e^{-\sqrt{1-\alpha^2}\omega_0(\tau-\tau_0)}-f^{(2)}(\alpha)}}\right)
\sigma_m=\min\left\{\left(\frac{c_6+c_8}{c_6\,c_8}+\frac{4}{c_2^2}\right)^{-\frac{2\,p\,(p+1)}{p-1}},\c_5\,4^{-\frac{p\,(p+1)}{p-1}},\ \left(\frac{2\,c_7\, c_2^\frac{p-1}{p+1}}{c_4}\right)^2 4^{-\frac{p\,(p+1)}{p-1}}\right\}.
\omega_2 ~=~ - \left[ 4(\mu - 1)(\mu - 2) \hat{\Psi}(\mu) ~+~ \frac{(16\mu^4 - 96\mu^3 + 188\mu^2 - 136\mu + 25)}{2(2\mu - 3)(\mu - 1)}\right] \eta_1^2 ~+~ \Pi_2
R ( \theta ) = \frac{\sin \left( \frac{2 \pi}{5}- \frac{3 i \theta}{5}\right) \sin \left( \frac{\pi}{5}- \frac{3 i \theta}{5}\right)}{\sin \left( \frac{2 \pi}{5}+ \frac{3 i \theta}{5}\right) \sin \left( \frac{\pi}{5}+ \frac{3 i \theta}{5}\right)}.
1 45 =3^2\cdot5 66 =2\cdot3\cdot11 120 =2^3\cdot3\cdot5 22 =2\cdot 11 54 =2\cdot3^3 99 =3^2\cdot11 144 =2^4\cdot3^2 32 =2^5 55 =5\cdot11 110 =2\cdot 5\cdot 11 176 =2^4\cdot11
\prod_{k = 1}^{\infty}\exp \left[ - 2 k \left( ( a_{k}^{B}a_{- k}^{B}+ b_{k}^{B}b_{- k}^{B}) \frac{1 + q^{2 k}}{1 - q^{2 k}}- ( a_{k}^{B}b_{- k}^{B}+ b_{k}^{B}a_{- k}^{B}) \frac{2 q^{k}}{1 - q^{2 k}}\right) \right] .
a^1_1=b_3a^{\overline{2}}_{\overline{2}}, a^1_2=\epsilon b_3a^{\overline{2}}_{\overline{1}}, a^2_2=b_3a^{\overline{1}}_{\overline{1}}, \epsilon a^2_1=b_3a^{\overline{1}}_{\overline{2}}.
\begin{aligned}\bigg\|\sum_{j=1}^3 L_j^2 f \bigg\|_{2}^2 + 2\mathbf{R}_3 \sum_{j=1}^3 \bigg\|\frac{L_j f}{|x|}\bigg\|_{2}^2 =\sum_{k = 1}^\infty \big(\mu_k( \mu_k-\tfrac{3}{2})\big) \bigg\| \frac{P_k f}{|x|^2}\bigg\|_{2}^2 \geq 0,\end{aligned}
K(s_\tau^\Lambda)=\sum_{(\mu,\nu)\in G\times H}c_{(\mu,\nu)}\Theta_{s_\mu^\Lambda , s_\nu^\Lambda}(s_\tau^\Lambda) =\sum_{(\mu,\nu)\in G\times H}c_{(\mu,\nu)}s_\mu^\Lambda{s_\nu^\Lambda}^*s_\tau^\Lambda &=\sum_{\substack{(\mu,\nu)\in G\times H (\alpha,\beta)\in \Lambda^{\min}(\nu,\tau)}}c_{(\mu,\nu)}s_{\mu\alpha}^\Lambda{s_\beta^\Lambda}^*=0.
\intop_{0}^{\infty}\frac{\sin(2\pi xyt)}{\sigma\left(y\right)e^{2\pi y}-1}\,dy=\frac{\sigma_{p}(2\pi xt)}{2}+\frac{1}{4}\,\frac{1}{1+\frac{1}{\pi p}}-\frac{1}{4\pi xt},
T_{w_{x}}^{-1}\sum_{a=1}^{r}\sum_{l=1}^{c_{a}}q^{l-1}(T_{c_{1}+\cdots +c_{a-1}+l-1}^{-1}\cdots T_{c_{1}+\cdots +c_{a-1}+1}^{-1}).f(w_{x}x-v_{c_{1}+\cdots +c_{a-1}+1}).
2(u_0U + u_0 V + v_0 U) + 2 u_0 V \sum_{j=1}^{\lfloor m/2 \rfloor}\frac{(-\chi')^{2j}}{(2j)!}= 0, 2(v_0V + v_0 U + u_0 V) + 2 v_0 U \sum_{j=1}^{\lfloor m/2 \rfloor}\frac{(-\chi')^{2j}}{(2j)!}= 0.
e_j^{(i+1)}= e_j^{(i)}+ (q^{2}-1)[x_{i}e_{j-1}^{(i)}- x_{i}^{2}e_{j-2}^{(i)}+ x_{i}^{3}e_{j-3}^{(i)}- \dots] e_j^{(i+1)}+ x_i e_{j-1}^{(i+1)}= e_j^{(i)}+ q^{2}x_i e_{j-1}^{(i)}.
\Delta ( m , \kappa^{2}) = 4 \left( m + \frac 1 2 \right)^{2}\left[ 2 5 + 4 \left( 2 9 - 4 \left( m + \frac 1 2 \right)^{2}\right) \kappa^{2}+ 6 4 ( \kappa^{2})^{2}\right]
\delta~(b)=~\frac{1}{4\pi}~\int_0^\infty \frac{dt}{t}e^{-b^2 t / 2\pi\alpha'}~[\prod_{i=1}^l\Theta_3(0,8i\pi^2\alpha' t /L_i^2)]~[\prod_{i=p+1}^c\Theta_3 (0,it L_i^2/2\pi \alpha')] \cdot (B \times J) ~~.
\{G_{Q}( \eta ) , G_{Q}( \rho ) \}^{*}= G_{Q}( [ \eta , \rho ] ) + \int_{\partial \Sigma}\omega \mathrm{\tiny \wedge}\omega \mathrm{\tiny \wedge}\dots \mathrm{\tiny \wedge}\omega \mathrm{\tiny \wedge}\eta_{a}d \rho^{a},
\Theta_{a}^{I}|_{l o o p s}= - \frac{i}{2}\, ( \omega + \beta_{a}^{I}) \, \langle t^{I}+ \bar{t}^{\bar{I}}\rangle^{- 1}~ ;{}~ ~ \omega = - \frac{C}{8 \pi^{2}}~ , \beta_{a}^{I}= \frac{b_{a}^{I}}{8 \pi^{2}}.
X(k_{1},k_{2},...,k_{r})=\sum_{n_{1}=0}^{N-1}\sum_{n_{2}=0}^{N-1}\sum_{n_{r}=0}^{N-1}x(n_{1},n_{2},n_{3})cas(\frac{2\pi}{N}(n_{1}k_{1}+n_{2}k_{2}+n_{3}k_{3}))
\Gamma_{\operatorname{ar}}(m) := \prod_{p^2\mid m}\left(-\frac{p^2-1}{p+2}\right)\prod_{\substack{p\mid m p^2\nmid m}}\left(1+\frac{p+p^{1/2}+1}{p^{3/2}+p^{1/2}+1}\right)^{-1};
- i \Sigma_{a b}( q , p ) = - ( i e )^{2}\epsilon_{a}\epsilon_{b}\int{\frac{d^{4}k}{( 2 \pi )^{4}}}{\frac{1}{i}}\left[ D^{( 0 )}\right]_{b a}^{\rho \sigma}( k ) \gamma_{\sigma}i S_{b a}^{( 0 )}({p}{\mkern - 9 . 0 m u /}- k ) \gamma_{\rho}\, .
(1-e^{i\theta})F_{N}(0,e^{-i\theta};e^{i\theta}) = (1-e^{i\theta}q^{N})+\sum_{n=1}^{N}\begin{bmatrix}N n\end{bmatrix}\frac{(q)_{n}q^{n^2}}{(1-2q\cos{\theta}+q^2) \cdots (1-2q^n\cos{\theta}+q^{2n})},
D_k(x) =\mathop{\textnormal{argmax}}_{\substack{\{z_1,...,z_k\}\subseteq \mathbb{F}_2^N{Hz_i}^\intercal=x \textnormal{for all}i}}\{P_\epsilon(z_1)+...+P_\epsilon(z_k)\}=\mathop{\textnormal{argmin}}_{\substack{\{z_1,...,z_k\}\subseteq \mathbb{F}_2^N Hz_i^\intercal=x \textnormal{for all}i}}\{|z_1|+...+|z_k|\}.
\frac{d}{dz}[z^{1-a_{p}}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|z)]=-z^{-a_{p}}G_{p,q}^{m,n}(\begin{matrix}a_{1},...,a_{p-1},a_{p}-1\\b_{q}\end{matrix}|z),n<p.
\widetilde A_0(x) = (-x_2, x_1)\ \int_0^1 \widetilde B_0( t\, x_1, t\, x_2)\, t\, dt = (-x_2, x_1)\ \int_0^1 \widetilde B_0( t\, |x|)\, t\, dt = \frac{(-x_2, x_1)}{|x|}\ \int_0^{|x|}\, \widetilde B_0(s)\, s\, ds.
n^2\int_{\mathbb{D}}|z|^{2n-2}(1-|z|^2)d|\mu|(z) \lesssim n^2\int_{\mathbb{D}}|z|^{2n-2}(1-|z|^2)dA(z) &=2n^2\int_0^1 r^{2n-1}(1-r^2)dr &=2n^2\left(\frac{1}{2n}-\frac{1}{2n+2}\right) &=\frac{4n^2}{4n^2+4n}.
\left\{p\leq x : \chi(p)=\zeta, \sin((\nu+1)\theta_p)>\epsilon\right\}\subset\left\{p\leq\frac{1}{\epsilon^2}: \chi(p)=\zeta\right\}\cup\{p\leq x : p\in\mathbb{P}_{>0}(\zeta,\nu)\}.
\Phi(r) = \Psi\left( \varphi \left( \frac{a + b\sqrt{-2}}c\right)\right)=\Psi\left(1 - \frac{2b}c, \frac ac, 1 - \frac ac\right) &=\frac{((c - 2b)c, a^2 + 2b^2 - 2bc, a^2 + 2b^2 + c^2 - 2(a+b)c)}{a^2 + 2b^2 + c^2 - (a + 2b)c}&=\frac{(|\beta|^2 - 2b, |\alpha|^2 - 2b, |\alpha|^2 + |\beta|^2 - 2(a+b))}{|\alpha|^2 + |\beta|^2 - (a + 2b)}.
[\partial_s\gamma(s,t),\partial_t\gamma(s,t)]=D_t\partial_s \gamma(s,t) =\nabla_{\partial_t\gamma(s,t)}\partial_s\gamma(s,t)-\nabla_{\partial_s\gamma(s,t)}\partial_t\gamma(s,t) &=D_t\partial_s \gamma(s,t)-D_s\partial_t \gamma(s,t)=0,
\tau \circ F\big((b_m\cdots b_2b_1)(c_1c_2\cdots c_n)\big) &=\delta_{i(1),k(1)}\,\tau\circ F\big((b_m\cdots b_2)E_{i(1)}(b_1c_1)(c_2\cdots c_n)\big) &=\delta_{i(1),k(1)}\delta_{m,n}\delta_{i(2),k(2)}\cdots\delta_{i(m),k(m)}&\cdot \tau\circ E_{i(m)}(b_m\,E_{i(m-1)}(b_{m-1}\cdots E_{i(2)}(b_2E_{i(1)}(b_1c_1)\,c_2)\cdots c_{m-1})\,c_m).
\int_{\mathbb{C}\mathbb{P}^{n-1}\times\mathbb{C}\mathbb{P}^{n-1}}|\langle \tau_\alpha, \tau_\beta\rangle|^{2m}\, d(\mu\times\mu)(\alpha,\beta)=\int_{\mathbb{C}\mathbb{P}^{n-1}}\int_{\mathbb{C}\mathbb{P}^{n-1}}|\langle \tau_\alpha, \tau_\beta\rangle|^{2m}\, d \mu(\alpha)\, d \mu(\beta)\geq \frac{1}{{d+m-1\choose m}}, \forall m \in \mathbb{N}.
\lim_{n\to\infty}F_{s_{\tilde S^*_{0,\lfloor nt_1\rfloor}},s_{\tilde S^*_{\lfloor nt_1\rfloor+1,\lfloor nt_2\rfloor}}}(u_1,u_2)=e^{-u_1^2t_1\slash2}e^{-u_2^2(t_2-t_1)\slash2}.
S_{eff}={1\over 16\pi}\,\int d^4 x\, \sqrt{-g}\,\, (- R +{1\over 2}{\partial_\mu \lambda \partial^\mu \bar{\lambda}\over ({\rm Im}\lambda)^2}- \sum_{n=1}^{N}{F^{(n)}_{\mu\nu}{}^\star \tilde{F}^{(n)}{}^{\mu\nu}}),
\widetilde{\Phi}_{c}^{2}\simeq \frac{4 ! 4 N \widetilde{h}^{2}\widetilde{m}^{2}}{( 4 \pi )^{2}}\left( \frac{N \widetilde{h}^{4}}{( 4 \pi )^{2}}- \widetilde{\lambda}\right)^{- 1}\simeq \frac{9 6 \xi \widetilde{m}^{2}}{\widetilde{h}^{2}}, \R_{c}\simeq \frac{2 \widetilde{m}^{2}}{\widetilde{\xi}}.
A(s) = 2^{s-1}\left( e^{i \frac{\pi}{2}(s-1)}+ e^{-i \frac{\pi}{2}(s-1)}\right) \frac{\Gamma \left(\frac{s+1}{2}\right) \Gamma \left(1 - \frac{s}{2}\right) \Gamma \left(s-\frac{1}{2}\right)}{{\pi}\Gamma(s+1)},
d_{\mu_\lambda\circ \pi_\lambda^{-1}}(\pi_\lambda(\omega)) = HD(\mu_\lambda\circ \pi_\lambda^{-1}) = \frac{h_\mu(\mathcal S)}{\chi_\mu}\ge \frac{h_\mu(\mathcal S)}{\log 2 + \mathop{\sum}\limits_{n \ge 1}\frac{\log (n+2)}{2^n}}
\mathcal{E}_{min}= \frac{1}{(4 \pi i)^2}\int \limits_{-i \infty}^{i \infty}\int \limits_{-i \infty}^{i \infty}A_{\nu}(m_1,m_2) \overline{A_{\nu}(n_1,n_2)}\frac{\left|F^{\#}\left(\nu_1, \nu_2\right)\right|^2}{\prod \limits_{k=1}^3 \left|\zeta \left(1+3\nu_k\right)\Gamma\left(\frac{1+3\nu_k}{2}\right)\right|^2}d\nu_1 d\nu_2
W_1(x) = \frac{x^3}{3}-x, W_2(x) = \frac{1}{4}(x^4-4 x^2+2), W_3(x) = \frac{1}{5}(x^5-5 x^3+5 x), W_4(x) =\frac{1}{6}(x^6-6 x^4+9 x^2 -2)
\widetilde U_{e x}^{i n s t}( l ) = - \frac{e^{2}}{2 ( 2 \pi )^{3}}\frac{( \overline{{u}}( p_{1}, \lambda_{1}) \gamma^{\mu}u ( p_{1}^{\prime}, \lambda_{1}^{\prime}) )}{\sqrt{2 p_{1}^{+}}\sqrt{2 p_{1}^{' +}}}\frac{( \overline{{v}}( p_{2}^{\prime}, \lambda_{2}^{\prime}) \gamma^{\nu}v ( p_{2}, \lambda_{2}) )}{\sqrt{2 p_{2}^{' +}}\sqrt{2 p_{2}^{+}}}\frac{\eta_{\mu}\eta_{\nu}}{q^{+ 2}}\, .
\begin{array}{rcl}I_{2}= |\vec{\Gamma}\cdot\vec{\Gamma}|^{2}=\sum_{n=1}^{N}|\Gamma^{(n)}|^{2}\, , & I_{4}= -\frac{1}{4}{\rm det}\left[\left(\begin{array}{c}\vec{\Gamma}\bar{\vec{\Gamma}}\end{array}\right)\left(\begin{array}{cc}\vec{\Gamma}\ \ bar{\vec{\Gamma}}\end{array}\right)\right]\, .\end{array}
\frac{\partial}{\partial x_\mu}(\left|x-a\right|+\left|x-b\right|+\left|x-c\right|)=\frac{(x-a)_\mu}{\left|x-a\right|}+\frac{(x-b)_\mu}{\left|x-b\right|}+\frac{(x-c)_\mu}{\left|x-c\right|}
\hat{\partial}^{+}\eta^{I}\bar{\alpha}^{I}| \Phi_{s}\rangle + \Bigl ( \hat{\partial}^{+}- \frac{2 \alpha^{+}}{2 s + d - 4}( \bar{\alpha}\hat{\partial}+ ( s - 1 ) \bar{\alpha}^{z}) \Bigr ) | \Lambda \rangle = 0 \, .
q^{\alpha^{2}/2}A_{q}\left(abq^{-\alpha-1}\right)=\int_{-\infty}^{\infty}\frac{\left(abq^{-1/2}e^{-ix};q\right)_{\infty}\exp\left(\frac{x^{2}}{\log q^{2}}+i\alpha x\right)}{\sqrt{\pi\log q^{-2}}}dx,
\tilde{\eta}= \frac{\| H^i(z^{i})+(H^{i})'(z^{i})\tilde{p}^{i}\|}{\| H^i(z^{i})\|}, \epsilon^i =\frac{(1-\beta)(1-\tilde{\eta}^{i})\|H^i(z^{i})\|}{\|\tilde{p}^{i}\|}, \eta_{\textrm{min}}^i =\textrm{max}\left\{\tilde{\eta}, 1-\frac{(1-\tilde{\eta})\delta^i}{\|\tilde{p}^{i}\|}\right\},
\int_0^1 f(x)^2 \, dx \leq n^2 X^2 + \frac{1}{n^2 \pi^2}\int_0^1 f'(x)^2 \, dx + 2|X|\sqrt{n}\sum_{k=0}^{n-1}\sqrt{\frac{1}{n^2 \pi^2}\int_{\frac{k+1/2}{n}}^{\frac{k+3/2}{n}}f'(x)^2 \, dx}
{\underbrace{\X_{\sigma^{\prime}i j}\}}_{T}= -{\frac{1}{E_{j}- E_{i}}}\left({\frac{d \,}{d \sigma}}f_{\sigma i j}\right) \int_{\sigma}^{\infty}d \sigma^{\prime}X_{\sigma^{\prime}i j}+{\frac{1}{E_{j}- E_{i}}}\left( 1 - f_{\sigma i j}\right) X_{\sigma i j}.
H_{p,p^{\prime}}(1,c)=\frac{4\pi}{\sqrt{c}}\,\sum_{n=1}^{\infty}\frac{p^{\prime2}+\lambda_{n}^{\prime2}}{p^{\prime}\left(p^{\prime}+\frac{1}{\pi}\right)+\lambda_{n}^{\prime2}}\cdot\frac{\lambda_{n}^{\prime-1}}{\sigma\left(\sqrt{c}\lambda_{n}^{\prime}\right)e^{2\pi\sqrt{c}\lambda_{n}^{\prime}}-1}
\vec{\phi}= ( \underbrace{\frac{W + \bar{W}}{| W |^{2}+ 1}}_{\phi_{1}}, \underbrace{i \frac{- W + \bar{W}}{| W |^{2}+ 1}}_{\phi_{2}}, \underbrace{\frac{| W |^{2}- 1}{| W |^{2}+ 1})}_{\phi_{3}},
F_1(y_1,y_2)= \bigg\{\frac{2}{\sqrt{{y_1}^2+{y_2}^2}}(y_1,y_2)+(v_1,v_2)\bigg |~ (v_1,v_2)\in[0,1]\times [0,1]\bigg\}, &F_2(x_1,x_2)= \bigg\{\frac{\sqrt 2}{\sqrt{{x_1}^2+{x_2}^2}}(x_1,x_2)+(v_1,v_2)\bigg |~ (v_1,v_2)\in[0,1]\times [0,1]\bigg\}.
\begin{aligned}\int_X \Big( D\log \left( R \hat{R}^T(X) \right) \left[ (\xi)_{\times}R \hat{R}^T(X) \right] p(X|R) &+ \log (R \hat{R}^T(X) ) D_2p \left( X\mid R \right) [(\xi)_\times R] \Big) dX = 0\end{aligned},
\frac{\partial \eta}{\partial t}=\frac{3}{2}\sqrt{\frac{g}{\ell}}\left(\frac{\sigma}{3}\frac{\partial^3 \eta}{\partial x^3}+\frac{2\alpha}{3}\frac{\partial \eta}{\partial x}+\frac12 \frac{\partial \eta^2}{\partial x}\right)
\begin{array}{c c c}{{A^{\sigma}( \mu )}}&{{\longrightarrow}}&{{A^{\sigma}( \mu + \nu )}}\\{{\downarrow}}&{{\,}}&{{\downarrow}}\\{{{\bf C}[ \sigma^{\vee}] ( \mu + \chi )}}&{{\longrightarrow}}&{{{\bf C}[ \sigma^{\vee}] ( \mu + \nu + \chi )}}\\ \end{array}
\nabla_{fX}\omega(Y) =(-1)^{(k-1)(l-1)}L_{fX}i_Y\omega-\omega(\nabla_{fX}Y) &=(-1)^{(k-1)(l-1)}df\wedge i_Xi_Y\omega+(-1)^{(k-1)(l-1)}fL_Xi_Y\omega-f\omega(\nabla_{X}Y) &=(-1)^{(k-1)(l-1)}df\wedge i_{Y\wedge X}\omega+f\nabla_X\omega(Y) &=f\nabla_X\omega(Y),
\Psi_{N}( \beta ) = ( 2 \pi )^{- N}\int_{0}^{2 \pi}\cdots \int_{0}^{2 \pi}\prod_{j < k}\left\vert \exp ( i \theta_{j}) - \exp ( i \theta_{k}) \right\vert^{\beta}d \theta_{1}\cdots d \theta_{N}
\dfrac{\displaystyle\prod_{p'=1}^{i-1}L_{-,p'}^{(s')}}{\displaystyle\prod_{p'=1}^{i}(\nu_{i,p}^{(s)}- \nu_{i,p'}^{(s')})}= \dfrac{\displaystyle\prod_{p'=1}^{i-1}\Big( \hbar'(p - p' -1 + \lambda_{i-1,p'}^{(s')}) - (a_s - a_{s'}) \Big)}{\displaystyle\prod_{p'=1}^{i}\Big( \hbar'(p - p' -1 + \lambda_{i,p'}^{(s')}) - (a_s - a_{s'}) \Big)}.
F\left(\begin{array}{c}p q u z \varepsilon \end{array}\right)=\left(\begin{array}{c}p- 2c_0\frac{u^2+z^2}{u^4+c_0^2}+2c_0-\varepsilon h_1(u,z,\varepsilon) q-\varepsilon h_2(u,z,\varepsilon) \hat{H}(p,q,u,z,\varepsilon) \end{array}\right),
\left\{\phi_{a}^{( 1 )}, \phi^{( 1 )}\right\}= \left\{\phi_{a}^{( 1 )}, T \right\}= \left. \left\{T , \phi_{3 , n}^{( 1 )}\right\}\right|_{\phi = T = 0}= \left. \left\{T , T \right\}\right|_{\phi = T = 0}= 0 \; , \; \; a = 1 , 2 \, .
\partial_\xi = \frac{1}{d}\partial_q, \partial_\zeta = \frac{c\sqrt{\rho}}{d}\partial_p \partial_{\xi}^2 = \frac{1}{d^2}\partial_q^2, \partial_\xi \partial_\zeta = \frac{c \sqrt{\rho}}{d^2}\partial_p \partial_q, \partial_\zeta^2 = \frac{c^2}{d^2}\rho \partial_p^2 + \frac{1}{2}\frac{c^2}{d^2}\rho_p \partial_p,
a = \left[\begin{array}{c}9 7 1 4 9 9 \end{array}\right],B = \left[\begin{array}{cccccc}8 0 1 3 9 9 0 0 9 5 2 6 1 9 4 1 1 8 3 5 1 0 8 0 9 2 1 8 2 1 9 6 8 0 1 8\end{array}\right],d = \left[\begin{array}{c}5 8 6 9 9 0 \end{array}\right].
B^{(1)}_{eff}(\vec{R}_{p_{1}})=\frac{2\pi\kappa\sigma_{1}}{q_{1}}\varepsilon_{ij}\partial_{i}v_{j}^{(2)}(\vec{R}_{p_{1}})\;\;,\;\; \partial_{i}=\frac{\partial}{\partial R^{i}_{p_{1}}}\;\;.
\det \left[ H^{(m,n)}_{\texttt{b}; j,k}( \boldsymbol{\xi}) \right]_{1\leq j,k\leq n}= \det \Bigl( \bigl[ 2(m+n) +u_{q_0}(\xi_j)+ u_{q_1}(\xi_j) \bigr]_{1\leq j\leq n}\Bigr) &= \prod_{1\leq j\leq n}\bigl(2(m+n) +u_{q_0}(\xi_j)+ u_{q_1}(\xi_j) \bigr) .
\mathbb{E}\left[ (\sum_{r=0}^{t-1}\hat{\gamma}^r_{p,r}\sigma_p^{r}\breve{Z}_p^{r}+ (\sigma_p^t)_{\perp}Z_p)^2 \right] = \mathbb{E}\left[ (\sigma_p^t\breve{Z}_p^t)^2 \right]=(\sigma_p^t)^2,\quad\forall p=1,...,P.
(N_{-}^{-1}\psi_1)_{x}-ik[N_{-}^{-1}\psi_1, \sigma_3]=N_{-}^{-1}(U-U_{-})\psi_1, (N_{-}^{-1}\psi_1)_{x}-4ik^3[N_{-}^{-1}\psi_1, \sigma_3]=N_{-}^{-1}(V-V_{-})\psi_1,
[y^k]T(y, z) =\frac1k [T^{k-1}]\Bigl(\frac{1-z}{1-z-zT}\Bigr)^{k}=\frac1k[T^{k-1}]\Bigl(\frac1{1-\frac{z}{1-z}T}\Bigr)^k &=\frac1k \binom{-k}{k-1}(-1)^{k-1}\Bigl(\frac{z}{1-z}\Bigr)^{k-1}=\frac{1}{k}\binom{2k-2}{k-1}\Bigl(\frac{z}{1-z}\Bigr)^{k-1}.
\frac{\psi_1{}_{(\alpha_1,\alpha_2,\beta_3+s_3)}\prod_{j=3}^{p-2}\psi_1{}_{(\beta_j+s_{j},\alpha_j,\beta_{j+1}+s_{j+1})}\psi_1{}_{(\alpha_{p-1},\beta_{p-1}+s_{p-1},\alpha_p)}}{\psi_1{}_{(\alpha_1,\alpha_2,\beta_3)}\prod_{j=3}^{p-2}\psi_1{}_{(\beta_j,\alpha_j,\beta_{j+1})}\psi_1{}_{(\alpha_{p-1},\beta_{p-1},\alpha_p)}}= \psi_{0,p}[\alpha,\beta,s]
\widetilde{E}_1(x,y,z) = \frac{d}{ds}|_{s=0}(s,0,0)(x,y,z)=\partial_x=e^{-\lambda_1 z}E_1, \widetilde{E}_2(x,y,z) = \partial_y= e^{-\lambda_2 z}E_2, \widetilde{E}_3(x,y,z) = \lambda_1 x \, \partial_x + \lambda_2 y \, \partial_y+ \partial_z= \lambda_1 x e^{-\lambda_1 z}E_1 + \lambda_2 y e^{-\lambda_2 z}E_2+ E_3.
\frac{d^2}{dt^2}\lambda(h,0) \leq \int_M \langle \frac{1}{2}\triangle_L h+^* h,h \rangle -\frac{1}{2}h_{ij}h_{ac}H_{iab}H_{jcb}dV_g &=\int_M \langle \frac{1}{2}\triangle h+^* h,h \rangle- \langle\mathring{R}h,h \rangle-R_{ij}h_{ik}h_{jk}dV_g.
\left| \sum_{t=0}^k P_t(\lambda) \right| \geq |\lambda^k + \lambda^{k-1}| - \sum_{i=1}^{\lfloor k/2 \rfloor}\left(\Theta(d^i) |\lambda|^{k-2i}+ \Theta(d^i) |\lambda|^{k-2i-1}\right) & \geq |\lambda^k + \lambda^{k-1}| - \sum_{i=1}^{\lfloor k/2 \rfloor}\left(\Theta(|\lambda|^{k-(i/2)}) + \Theta(|\lambda|^{k-1 - (i/2)})\right) \geq \Theta(|\lambda^k|),
f_{P}^{( n , i_{1}, i_{2}, \dots , i_{k})}\mathrm{t r}[ ( b^{\dagger})^{i_{1}}a^{\dagger}\dots a^{\dagger}] \dots \mathrm{t r}[ ( b^{\dagger})^{i_{k}}a^{\dagger}\dots a^{\dagger}] | 0 > ,
\delta_1(\tau)=\frac{1}{8}(\theta_2^4+\theta_3^4),~~\varepsilon_1(\tau)=\frac{1}{16}\theta_2^4\theta_3^4, &\delta_2(\tau)=-\frac{1}{8}(\theta_1^4+\theta_3^4),~~\varepsilon_2(\tau)=\frac{1}{16}\theta_1^4\theta_3^4.
\mathcal{H}W_{l}(s;yl^c) = \omega_{\pi_{0},l}(l^{-\frac{c}{2}})l^{\frac{c}{2}}\sum_{\mu_l\in{}_l\mathfrak{X}_{-\frac{c}{2}}'}\mu_l(y^{-1})\epsilon(\frac{1}{2},\mu_l^{-1})^2[\mathcal{M}W_l(s;\cdot)](\mu) = \omega_{\pi_{0},l}(l^{-\frac{c}{2}})l^{\frac{c}{2}}\mathcal{L}_{s,-\frac{c}{2}}(yb^2),
\underset{x^\prime \in \mathbb{R}^d}{\arg \max}\left[ \sum_{1\leq \ell \leq 2}\varphi_\ell(x^{\prime}, \lambda_{\mathrm{D}, \ell}^{\star})\right] = \underset{x^\prime \in \mathbb{R}^d}{\arg \max}\ \varphi_\ell(x^{\prime},\lambda_{\mathrm{D}, \ell}^{\star}).
\mu_{1}:= q^{k (m - k - l - n + 1)}a^{m - k - l - n}\left(\frac{c}{a b}\right)^{k}\frac{(a)_{k}(a q / c)_{k - m}}{(a q / b)_{k - l}}, \allowdisplaybreaks \mu_{2}:= q^{l (m - k - l - n + 1)}b^{m - k - l - n}\left(\frac{c}{a b}\right)^{l}\frac{(b)_{l}(b q / c)_{l - m}}{(b q / a)_{l - k}}.
0=\Delta_f\phi=\frac{\partial^2 \phi}{\partial t^2}-k_g\frac{\partial \phi}{\partial t}+\frac{\partial^2 \phi}{\partial x}-\frac{\partial f}{\partial t}\frac{\partial \phi}{\partial t}-\frac{\partial f}{\partial x}\frac{\partial \phi}{\partial x}.
\begin{cases}\dot{u}^\varepsilon (t,x) = \Delta u^\varepsilon (t,x)+\displaystyle{\frac{1}{\varepsilon}}f(u^\varepsilon (t,x) ),\ \ \t > 0,\x\in \mathbb{R}, u^\varepsilon (0,x) = u_0^\varepsilon (x),\ \ \x\in \mathbb{R},\end{cases}
\intop_\Omega\bigg|{1\over r}\psi_{1,r}\bigg|^2dx+{1\over 2}\intop_{-a}^a\psi_{1,r}^2\bigg|_{r=R}dz\le c(|\psi_{1,zz}|_{2,\Omega}^2+|\omega_1|_{2,\Omega}^2).
Y_j = \frac{X_{j,1}(\tau_{j+1})}{X_{j,1}(\tau_{j+1}) + X_{j,2}(\tau_{j+2})}= \frac{e^{-\int_{\tau_j}^{\tau_{j+1}}G_j(v) \: dv}X_{j,1}(\tau_{j+1})}{e^{-\int_{\tau_j}^{\tau_{j+1}}G_j(v) \: dv}X_{j,1}(\tau_{j+1}) + e^{-\int_{\tau_j}^{\tau_{j+1}}G_j(v) \: dv}X_{j,2}(\tau_{j+2})}.