harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{Pr}\left(\gamma_{\mathrm{R}_i\mathrm{D}}>z \right) = \mathrm{Pr}\! \left(\! |h_{\mathrm{SR}_i-\mathrm{SD}}|^2 > \frac{z \left(P_{\mathrm{PT}}|h_{\mathrm{PT-SD}}|^2 + N_0 \right)}{P_{\mathrm{ST}}}\right) &= \exp\left( - \frac{z \left(P_{\mathrm{PT}}y + N_0 \right)}{\Omega_{\mathrm{SR-SD}}P_{\mathrm{SR}}}\right).
\phi(q)=\sum_{n\ge0}\frac{q^{n^{2}}}{(-q^{2};q^{2})_{n}}=\frac{1}{\prod_{n>0}(1-q^{n})}\sum_{n\in Z}\frac{(-1)^{n}(1+q^{n})q^{n(3n+1)/2}}{1+q^{2n}}
\delta \left( x\right) =d\left( x,Tx\right) , A\left( x\right) =B\left( x,\frac{\lambda \delta \left( x\right)}{2}\right) \cap B\left( Tx,\frac{\lambda \delta \left( x\right)}{2}\right) , AA\left( x\right) =\bigcap_{y\in A\left( x\right)}B\left( y,\frac{\lambda^{2}\delta \left( x\right)}{2}\right) \cap A\left( x\right) .
F \! = \! (f,u_0) \in \mathcal{F}^{k,\mathbf{s}(s,\lambda,\lambda',\delta)}_{T, \varLambda^q,{\mathcal D}_{d\oplus d^*}}\times C^{2s+k+1,\lambda,\delta}_{\varLambda^q}\cap \mathcal{S}_{d^\ast}
\|u\|_{L_2(P)}=\|\phi_{P,*}u\|_{L_2(J\times B)}\leq C\| |\nabla_{g_C}(\phi_{P,*}u)|_{g_C}\|_{L_2(J\times B)}=C\| |\nabla_{g_P}u|_{g_P}\|_{L_2(P)}.
S_{\alpha}(x_{0})=c.l.s.\left\{A^{i}\Gamma_{1}\left( H \right),\thinspace i=0,\thinspace 1,\thinspace \mathellipsis \right\}=c.l.s.\left\{{(A-\alpha I)}^{i}\Gamma_{1}\left( H \right),\thinspace i=0,\thinspace 1,\thinspace \mathellipsis \right\}.
\begin{array}{c c c c}{{\mathrm{l o n g ~ r o o t s :}}}&{{\pm \, \beta_{k l}~ = ~ \pm ( 2 \epsilon_{j k l}e_{j}- e_{k}- e_{l})}}&{{}}&{{( 1 \leq k < l \leq 3 )}}\\{{\mathrm{s h o r t ~ r o o t s :}}}&{{\pm \, \alpha_{k l}~ = ~ \pm ( e_{k}- e_{l})}}&{{}}&{{( 1 \leq k < l \leq 3 )}}\\ \end{array}~ ,
{\cal S}_{m}^{( 4 )}= \int d^{4}x \sqrt{- g_{( 4 )}}\left[ ( \partial_{\mu}\varphi ) ( \partial_{\nu}\varphi ) g_{( 4 )}^{\mu \nu}+ m^{2}e^{- 2 \sigma ( z_{i})}\varphi^{2}+ \dots \right]
dX_t^{(j)}=\bigg(f^{(j)}(X_t^{(j)})+\nu \big(e^{\rho_t^{(j)}}X_t^{(j-1)}-2X_t^{(j)}+e^{\varrho_t^{(j)}}X_t^{(j+1)}\big)\bigg)dt &+\sum\limits_{i=1}^mc_i^{(j)}X_t^{(j)}\circ dW_t^{(i)},j=1,\dots ,N,
c^k_{\mathrm{s2u},1}\left( N_{\mathrm{sc}}\right) = \mathsf{E}\left\lbrace B \sum^{S_{\mathrm{s2u}}\left( k \right)}_{s=1}\log_2\left[ \frac{\gamma_{\mathrm{s2u},s}\left( k \right)}{\gamma_{\mathrm{s2u},0}\left( k \right)}\right] \right\rbrace.
T^{\mu\nu}=(\begin{matrix}T^{00}&T^{01}&T^{02}&T^{03}\\T^{10}&T^{11}&T^{12}&T^{13}\\T^{20}&T^{21}&T^{22}&T^{23}\\T^{30}&T^{31}&T^{32}&T^{33}\end{matrix}),
\hat K_{i j}(x, 0) = \sum_{(r, s) \not \in{\cal J}}c_{i j r s}(B) \hat K_{r s}(x, 0) + \sum_{(r, s) \not \in{\cal J}}\frac{\partial c_{i j r s}(B)}{\partial B_{pq}}K_{r s}(x, 0) \mbox{for}x \in (0, 1),
f(x_1, \ldots, 1, \ldots, x_n) = \operatorname{sgn}\left( \sum_{j =0, \; j \neq i}^n w_j x_j + w_i - \theta \right) f(x_1, \ldots, 0, \ldots, x_n) = \operatorname{sgn}\left( \sum_{j = 0, \; j \neq i}^n w_j x_j - \theta \right).
\begin{aligned}\Gamma_i^j (\psi_1\psi_2)= \sum_{k}(\Gamma_{i}^{k}(\psi_1) \circ \widetilde{\psi_2})\Gamma_{k}^{j}(\psi_2) , &\Gamma_i^j (\psi^{-1}) \circ \widetilde{\psi}=(\Gamma^{-1})_{i}^{j}(\psi).\end{aligned}
\frac{\partial{{L}_{D}}^{( i , j )}}{\partial u^{\alpha ( i , j )}}- \Delta_{1}( \frac{\partial{{L}_{D}}^{( i - 1 , j )}}{\partial ( \Delta_{1}u^{\alpha ( i - 1 , j )})}) - \Delta_{2}( \frac{\partial{{L}_{D}}^{( i , j - 1 )}}{\partial ( \Delta_{2}u^{\alpha ( i , j - 1 )})}) = 0 ,
f_{k,n}:=T_{\frac{k+1}{2^n}}(f) - T_{\frac{k}{2^n}}(f),\ \k=0,1,\dots, 2^n N-1, f_{k,n}:=T_{\frac{k}{2^n}}(f) - T_{\frac{k+1}{2^n}}(f),\ \k=-2^n N,\dots, -1.
\langle m \rangle ={\frac{\tilde{\psi}_{r}}{\tilde{\phi}}}={\frac{\left( e^{2 h}- 1 \right) \left( 3 + ( 2 + \sqrt{7}) e^{2 h}\right)}{\left( 1 + ( - 1 + \sqrt{7}) e^{2 h}+ ( 2 + \sqrt{7}) e^{4 h}\right)}}\ .
P(r,t)=A\frac{\lambda+1}{\lambda}\frac{(e^{t/\tau})^{b/2D}}{1-e^{-t/\tau}}(\frac{sinh(\frac{t}{2\tau})}{\lambda})^{\frac{b}{D}+1}(\frac{4\lambda^{2}}{(\lambda+1)^{2}e^{t/\tau}-4\lambda})^{\frac{b}{D}+\frac{1}{2}}.
\left\| f_*\left( \Phi_1(e^{i\theta}),\dots,\Phi_{n}(e^{i\theta}) \right) \right\| = \max_{\theta}\{|f_*(\lambda_1,\dots,\lambda_{n})|: (\lambda_1,\dots,\lambda_{n}) \in\sigma_T(\Phi(e^{i\theta}),\dots,\Phi_{n}(e^{i\theta})) \}.
\mathbb{E}\left[ \rho_2 ( \widetilde{Y}_2, y_2) \right] - 1 \leq \mathbb{E}\left[ \boldsymbol{d}_{\mathcal{S}_2}(\widetilde{S}_2, s_2)^{p_2}\right] = \boldsymbol{W}_{p_2}\left(\mu_2 , \delta_{s_2}\right)^{p_2}.
{\bf K}( \vec{\varrho}_{f}t_{f}, \vec{\varrho}_{i}t_{i}) \sim \int{\cal D}\left[ \vec{\varrho}\right]{\it e x p}\left[ \frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\left[ L_{\theta}+ L_{0}\right] d t \right] ~ ~ ,
\begin{array}[c]{ll}dW_{xx}(s,\bar{X}^{t,x;\bar{u}}(s)) &= \left\{W_{sxx}(s,\bar{X}^{t,x;\bar{u}}(s))+W_{xxx}(s,\bar{X}^{t,x;\bar{u}}(s))b(s)\right. &\left.+\frac{1}{2}W_{xxxx}(s,\bar{X}^{t,x;\bar{u}}(s))(\sigma(s))^{2}\right\}ds +W_{xxx}(s,\bar{X}^{t,x;\bar{u}}(s))\sigma(s)dB(s).\end{array}
\sigma^{1}= \left( \begin{array}{}{c ]{c c}0}&{1}\\{1}&{0}\\ \end{array}\right) , \quad \sigma^{2}= \left( \begin{array}{}{c ]{c c}0}&{- i}\\{i}&{0}\\ \end{array}\right) , \quad \sigma^{3}= \left( \begin{array}{}{c ]{c c}1}&{0}\\{0}&{- 1}\\ \end{array}\right)
I = \frac{k}{\pi}\int d t d \theta \left[ \partial_{-}\beta \partial_{\theta}\beta + \partial_{\theta}\alpha \partial_{-}\gamma \exp{( 2 \beta )}\right] + \frac{k}{\pi}\int d t d \theta T r \left( h^{- 1}K ( t ) \partial_{-}h \right) - \frac{k}{4}\int d t ( K^{3})^{2}
\langle x_{\sigma(UV)}, x_{\sigma(UW)}\rangle =\overline{f(UV)}f(UW)\langle x_V, x_W \rangle, \langle x_{\sigma(UV)}, \tau_{\sigma(UW)}\rangle =\overline{f(UV)}f(UW)\langle x_V, \tau_W \rangle, \langle \tau_{\sigma(UV)}, \tau_{\sigma(UW)}\rangle =\overline{f(UV)}f(UW)\langle \tau_V, \tau_W \rangle
\sum_{n=-\infty}^{+\infty}\int\frac{d^{d}t}{(2\pi)^{d}}log(4\pi^{2}n^{2}+t^{2}+\xi^{2})-\int\frac{d^{d}t}{(2\pi)^{d}}\sqrt{t^{2}+\xi^{2}}
d_{p,\mu}\|u_{n,R}^*\|^p_{X^{1,p}_\infty}\geq\int_0^\infty\exp_p\left(\mu|u_{n,R}^*|^{p'}\right)r^{\alpha_0}\mathrm dr &\quad+\sum_{j=1}^\infty\left(\dfrac{1}{\|u^*_{n,R}\|^{p'j}_{X^{1,p}_\infty}}-1\right)\dfrac{\mu^{p-1+j}}{\Gamma(p+j)}\|u_{n,R}^*\|^{p'(p-1+j)}_{L_{\alpha_0}^{p'(p-1+j)}}.
c_{2i+3}=(-1)^{i+2}\frac{2^{s}q-1}{2^{s-1}q+i+1}{2^{s-1}q+i+1\choose 2i+3}2^{2i+2}= (-1)^{i+2}\frac{(2^{s}q-1)(2^{s-1}q+i)!}{(2i+3)!(2^{s-1}q-i-2)!}\cdot 2^{2i+2}= -\frac{(2^{s-1}q+i)(2^{s-1}q-i-1)}{(2i+2)(2i+3)}\cdot 2^{2}\cdot c_{2i+1}= -\frac{(2^{s-1}q+i)(2^{s-1}q-i-1)}{(i+1)(2i+3)}\cdot 2\cdot c_{2i+1}.
\sum_{i=1}^ms_i^{r-1}\le \frac{n^r}{(\log n)^{1/(r-1)}}+\sum_{k=1}^\infty n^r\left(\frac{1}{k}-\frac{1}{k+1}\right)\left(\frac{\log (k+1)}{\log n}\right)^{1/(r-1)}&= \frac{n^r}{(\log n)^{1/(r-1)}}+ \sum_{k=1}^\infty\frac{n^r(\log (k+1))^{1/(r-1)}}{k(k+1)(\log n)^{1/(r-1)}}&= \frac{c'_1n^r}{(\log n)^{1/(r-1)}},
i \tilde{\Delta}( \rho , \sigma ) \equiv \sum_{{\bf p},{\bf p}^{\prime}}\frac{1}{\sqrt{2 E_{p}V}\sqrt{2 E_{p^{\prime}}V}}\langle a_{{\bf p}^{\prime}}^{\dagger}a_{\bf p}\rangle e^{- i ( P \cdot \rho - P^{\prime}\cdot \sigma )},
G_{{D}}^{( + )}({\bf r},{\bf r}^{\prime}; k ) ={\mathcal G}_{{D}}^{( + )}({\bf R}; k ) \stackrel{\scriptscriptstyle ( r \rightarrow \infty )}{\sim}- \frac{1}{4 \pi}\left( \frac{k}{2 \pi}\right)^{({D}- 3 ) / 2}\, e^{i \gamma_{{D}}}\, \frac{e^{i k r}}{r^{({D}- 1 ) / 2}}\, e^{- i{\bf k}^{\prime}\cdot{\bf r}^{\prime}}\; ,
f^{(N_{j-1}+\ell)}_{\theta^{-(N_{j-1}+\ell)}\omega}\left(X_{\bar s}\left (\theta^{-(N_{j-1}+\ell)}\omega\right)\right)= f_{\theta^{-\ell}\omega}^{(\ell)}\left(X_{\bar s^j}\left(\theta^{-\ell}\omega\right)\right)
=\frac{1}{2}\hat f^\mu \hat f_\mu -\epsilon^{\alpha\beta\gamma}\hat f_\alpha\partial_\beta (\hat A_\gamma +\hat a_{\gamma})+\frac{m}{2}\epsilon^{\alpha\beta\gamma}(\hat A_{\alpha}\partial_{\beta}\hat A_{\gamma}+2\hat a_{\alpha}\partial_{\beta}\hat A_{\gamma})+~O(\theta^{2}).
\sum_{k=1}^m 2 x_+^T(s)\Sigma^{-1}N_{k}x_+(s) u_k(s) \leq x_+^T(s) \Sigma^{-1}x_+(s) \left\|u(s)\right\|_{2}^2 +\sum_{k=1}^m x_+^T(s) N_k^T\Sigma^{-1}N_{k}x_+(s),
\sum_{l=0}^k c_e(k-l,l;s)={(a^2c^2/q,s^2;q^2)_k\over (q^2,q^3s^2/a^2c^2;q^2)_k}(q^3/a^2c^2)^k{}_4\phi_3\left[{a^2/q,c^2/q,q^{2k}s^2,q^{-2k}\atop-s,-qs,a^2c^2/q};q^2,q^2\right].
\bar{\xi}_\ell(\theta)=0, \bar{\xi}_\ell(\tilde{\theta})=0, \ell =1, \ldots, j-1, \bar{\xi}_{j}(\theta) = (1-w_{jj}) r^j \eta_j, \bar{\xi}_{j}(\tilde{\theta}) = (1-w_{jj}) r^{j+1/2}\eta_j.
\left\Vert f\right\Vert_{C_{1-\gamma ;\psi}\left[ a,b\right]}=\left\Vert\left( \psi \left( t\right) -\psi \left( a\right) \right)^{1-\gamma}f\left(t\right) \right\Vert_{C\left[ a,b\right]}=\underset{t\in \left[ a,b\right]}{\max}\left\vert \left( \psi \left( t\right) -\psi \left( a\right) \right)^{1-\gamma}f\left( t\right) \right\vert.
\begin{array}[c]{rl}\hat{X}(r):= X^{s,x^{\prime};\bar{u}}(r)-\bar{X}^{t,x;\bar{u}}(r),\ \hat{Y}(r):= Y^{s,x^{\prime};\bar{u}}(r)-\bar{Y}^{t,x;\bar{u}}(r), \hat{Z}(r):= Z^{s,x^{\prime};\bar{u}}(r)-\bar{Z}^{t,x;\bar{u}}(r),\ \hat{\Theta}(r):= \left( \hat{X}(r),\hat{Y}(r),\hat{Z}(r)\right) .\end{array}
\mathcal{L}_{\mathcal{I}^{({\rm CCU})}_{\rm d \rightarrow u}}(s)= \mathcal{L}_{\mathcal{I}^{({\rm CEU})}_{\rm d \rightarrow u}}(s)= \exp\left( - \frac{\pi^{2}}{2}\lambda \sqrt{s P_{\rm{d}}}\right),
{\cal T}|{\bf b}+\frac{1}{2}; n_1,n_2,\ldots,n_b\rangle=\sum_{i=0}^{2^{b-1}}(-)^{p_i+1}|{\bf{p_i+1/2}}; \tau_i(n_b,n_{b-1},\ldots, n_1),\frac{1}{2}\rangle,
\mathbb{E}\sup_{n}|\sum\limits_{i=1}^{n}{\mathbb{E}}(X_{i}|{\mathcal{F}}_{0})|\leq\sum_{k\geq1}\mathbb{E}\sup_{n}|\sum\limits_{i=1}^{n\wedge N_{k}}e_{-N_{k}+i}\mathbf{1}_{A_{k}}|+\sum_{k\geq1}N_{k}\varepsilon_{k}.
I_{1}=\int_{T_{\epsilon}^{2}}d^{2}z \partial_{\bar{z}}\overline{\rho}\partial_{z}\rho = \frac{1}{2i}\int_{T_{\epsilon}^{2}}\bar{\partial}\overline{\rho}\wedge\partial \rho = \frac{1}{2i}\int_{|z|=\epsilon}dz\overline{\rho}\partial_{z}\rho ,
\lambda_2=\frac{e^{q_3-q_1}}{1+e^{q_3-q_2}+e^{q_2-q_1}+e^{q_3-q_1}}=\frac{1}{e^{q_1-q_3}+e^{q_1-q_2}+e^{q_2-q_3}+1}<1/4,
u^1_{{\bf v}',\psi({\bf x})}\circ \Phi^{(1)}_{1,\psi({\bf x}),\delta}=u^2_{{\bf v},{\bf x}}\circ \tilde{\psi}_{\bf x}^{-1}\circ \Phi^{(1)}_{1,\psi({\bf x}),\delta}=u^2_{{\bf v},{\bf x}}\circ \Phi^{(2)}_{1,{\bf x},\delta}\circ \Psi_{\bf x}.
p^{\beta_2}(2,t)=\sum_{k=2}^{\infty}(-\lambda)^k\underset{\Omega^{k}_{2}}{\sum}\frac{t^{k_0\beta_0+k_1\beta_1+k_2\beta_2}}{\Gamma\left(k_0\beta_0+k_1\beta_1+k_2\beta_2+1\right)},
\underline{\Delta}G =t\left(2f_{ij}^{2}+2f_{i}f_{jji}+2Ric'_{ij}f_{i}f_{j}+2\alpha Ric'_{ij}f_{ij}-\alpha\partial_{t}(\underline{\Delta}f)\right) &\geq t\left[\left(2f_{ij}^{2}+\alpha Ric'_{ij}f_{ij}\right)+2f_{j}f_{jji}-2\rho_{1}\left\|\overline{\nabla}f\right\|^{2}-\alpha\partial_{t}(\underline{\Delta}f)\right],
-\left(\frac{4}{q^2}\right)^{1+\epsilon}\frac{1}{2\epsilon}\sum_{j=0}^{\infty}\frac{\Gamma(1+\epsilon+j)}{1+2\epsilon+2j}~\frac{1}{j!}\left(-\frac{4y}{q^2}\right)^j+\frac{1}{y^{\epsilon+1/2}}\frac{1}{\sqrt{q^2}}\frac{\sqrt{\pi}\Gamma(\epsilon+1/2)}{2\epsilon}.
\phi(y)=\sum_{\substack{z_1,,,z_t,z_{t+1}z_{i+1}\in I(z_i')}}G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'}),
h( [A, B]_m) = J_6( -J_6([J_6\tilde{A},J_6\tilde{B}]-[J_6\check{A},J_6\check{B}]) + i ( -J_6([J_6\tilde{A},J_6\check{B}]+[J_6\check{A},J_6\tilde{B}]) ) ) \,.
- i \left( \begin{matrix}{{a^{\alpha}}}&{{b}}\\ \end{matrix}\right) \left( \begin{matrix}{{\langle U_{\alpha}, \bar{U}_{\bar{\beta}}\rangle}}&{{\langle U_{\alpha}, V \rangle}}\\{{\langle \bar{V}, \bar{U}_{\bar{\beta}}\rangle}}&{{\langle \bar{V}, V \rangle}}\\ \end{matrix}\right) \left( \begin{matrix}{{\bar{a}^{\bar{\beta}}}}\\{{\bar{b}}}\\ \end{matrix}\right) = 0 \ .
\frac{u_{+}-v_{-}+\frac{1}{2}h}{u_{+}-v_{-}-\frac{1}{2}h}k_{i+1}^{+}(u+\frac{1}{2}hi)k_{i+1}^{-}(v+\frac{1}{2}hi+\frac{1}{2}h)^{-1}= \frac{u_{-}-v_{+}+\frac{1}{2}h}{u_{-}-v_{+}-\frac{1}{2}h}k_{i+1}^{-}(v+\frac{1}{2}hi+\frac{1}{2}h)^{-1}k_{i+1}^{+}(u+\frac{1}{2}hi).
f = P u + b \cdot D_v u + c u, &r_0 = r, r_n = r + (R - r) \sum_{k = 1}^n 2^{-k}, n \ge 1, &\zeta_n (t, v) = \xi \big(2^{2(n+1)}(R - r)^{-2}(- r^2_{n}- t)\big) \, \, \xi \big(2^{(n+1)}(R - r)^{-1}(|v| - r_{n})\big) & \times \xi \big(2^{3(n+1)}(R-r)^{-3}(|x| - r^{3}_{n})\big),
\begin{aligned}e^{-t}C_p\left|\dot{\xi}(t)\right|^{q}\leq e^{-t}\phi\left(\xi(t)\right)-e^{-t}D\phi\left(\xi(t)\right) \cdot \dot{\xi}(t) \Longrightarrow C_p\left|\dot{\xi}(t)\right|^{q}\leq \phi\left(\xi(t)\right)-D\phi\left(\xi(t)\right) \cdot \dot{\xi}(t). \end{aligned}
\phi_n(t;x)= (\Lambda^{\frac{1}{2}}\partial_x^nu)(X(t;x),t) = \int_{-\infty}^{+\infty}\frac{(\partial_x^nu)(X(t;x),t)-(\partial_x^nu)(y,t)}{|X(t;x)-y|^{\frac{3}{2}}}dy
E_{i}:=\{1,2,\ldots,N\}\backslash\{i\},E_{ij}:=E_{i}\backslash\{j\}, p_{i}(x) :=\prod\limits_{j\neq i}(x_{i}-x_{j}) =\sum\limits_{\ell=0}^{N-1}(-1)^{\ell}x_{i}^{N-1-\ell}\sum\limits_{J\subset E_{i},\#J=\ell}x^{\varepsilon_{J}}.
\beta=\frac{(1-\alpha\cos(\omega\tau_\alpha))^2+(\omega+\alpha\sin(\omega\tau_\alpha)^2}{1-\alpha\cos(\omega\tau_\alpha)}= (1-\alpha\cos(\omega\tau_\alpha))+\frac{(\omega+\alpha\sin(\omega\tau_\alpha)^2}{1-\alpha\cos(\omega\tau_\alpha)}
~~\left( (pd_{p}+qd'_{q}-d_{p}d'_{q})-2d'_{q}\cdot\frac{pq-e}{p-d'_{q}}\right)^{2}- \left( (pd_{p}+qd'_{q}-d_{p}d'_{q})^2-4d_{p}d'_{q}(pq-e) \right) = ~~ \frac{4d'^{2}_{q}(pq-e)}{(p-d'_{q})^{2}}\cdot (qd'_{q}-e) < 0,
\tilde{S}= \int d^{2}{x}\left\{g^{2}( 1 + \varphi_{1}^{2})^{2}( \partial \tilde{\varphi_{0}})^{2}+ 8 \theta g^{2}\partial_{a}\tilde{\varphi_{0}}\partial^{a}\varphi_{1}+ \frac{( 1 + 1 6 \theta^{2}g^{4})}{g^{2}( 1 + \varphi_{1}^{2})^{2}}( \partial \varphi_{1})^{2}\right\}
{\mathcal A}_{0}V_{\pm k}=\big(\pm i\sqrt{\mu_k}\big)V_{\pm k},\,\,\,{where}\,\, V_{\pm k}=\frac{v_k}{\sqrt{2}}\Big[\frac{1}{\sqrt{\mu_k}}, \pm i\Big],\hbox{for all}k\in \mathbb N^*.
X_1=\{(-1)\};\quad X_2=\{(-2,1),(0,-1)\}; X_3=\{(1,-2,0),(-2,1,0),(-1,-1,1),(0,0,-1)\}; X_4=\{(0,0,0,-1),(-1,-1,1,0),(-1,0,-1,1),(1,-1,-1,0), (-2,1,0,0),(0,-2,0,1),(0,1,-2,0)\}.
|P_{m+1}| \geq|P_{m}\cup\{1+p:p\in P_{n}\}|-|C\cap X_{\mu+m+1}| &\geq 1+|P_{m}|-|C\cap X_{\mu+m+1}| &\geq 2+l_{m}-(1+l_{m}-l_{m+1}) &=1+l_{m+1}
\delta \eta_{\alpha_{2 k + 1}}^{*}= Z_{\; \; \alpha_{2 k + 1}}^{\alpha_{2 k}}\Phi_{\alpha_{2 k}}^{*}+ A_{\alpha_{2 k + 1}}^{\; \; \alpha_{2 k + 2}}\Phi_{\alpha_{2 k + 2}}^{*}, \; k = 0 , \cdots , b ,
\mu_3([z^1_1,\ldots,z^1_{d_1}],[z^2_1,\ldots,z^2_{d_2}])=\mu_3(p_2 \circ p_1(z^1_1,\ldots,z^1_{d_1},z^2_1,\ldots,z^2_{d_2})) &=\iota_3^{\ast}(\Phi(z^1_1,\ldots,z^1_{d_1},z^2_1,\ldots,z^2_{d_2})) &=\iota_3^{\ast}(-\frac{1}{2}(2\lambda_1,\ldots,2\lambda_1,2\lambda_2,\ldots,2\lambda_2)+(\lambda_1,\ldots,\lambda_1,\lambda_2,\ldots,\lambda_2)) &=\iota_3^{\ast}(0)=0.
\beta = \frac{2}{a}\sqrt{\left( 1 - \frac{2(m-1)(1-a(N+1))}{N}\right)^2 + \frac{4(m-1)ma}{N}(1-a(N+1))}& + \frac{2}{a}\left( 1 - \frac{2(m-1)(1-a(N+1))}{N}\right)
\lim_{k \to \infty}\frac{p_i(x^k)}{k^4}= \lim_{k \to \infty}\frac{k^4(p_i(x^k)-p_i(\tilde x^k))}{k^8}+ \lim_{k \to \infty}\frac{p_i(\tilde x^k)}{k^4}= \lim_{k \to \infty}\frac{p_i(\tilde x^k)}{k^4}\ge 0.
C_{32}(p,u,v) \sim \frac 1\eta \sum_{J \le 0}2^{-J (J+1)/2 - J + J\tilde u}\sum_{L =0}^{-J}\xi_{L+1}(1/2)\, 2^{-L}\sum_{\ell\ge -J-L+1}\frac{(-1)^\ell}{\ell!}\frac 1{\ell + J+L - \tilde u}.
g_{0 0}= - H^{- \frac{1}{2}}, g ( r ) = H^{- \frac{1}{2}}, g_{S}( r ) = H^{\frac{1}{2}}r^{2}, g_{r r}( r ) = H^{\frac{1}{2}}, H = 1 + \frac{e^{\Phi_{0}}Q}{2 r^{4}}
F(a,p,k;b,t,s;x)= \sum_{n=0}^{\infty}\frac{\prod_{i=1}^{r}\:_{p_{i}}(a_{i})_{n,k_{i}}}{\prod_{j=1}^{q}\:_{t_{j}}(b_{j})_{n,s_{j}}}\frac{x^{n}}{n!}.
\big\Vert{T_{[a]}f_L}\big\Vert_{F_p^{{s},q}}=\Big\Vert \Big(\sum_{l=0}^{\infty}{2^{{s}lq}\Big| \sum_{n=M}^{L}{\phi_l\ast d_{{t_n}}}\Big|^q}\Big)^{{1}/{q}}\Big\Vert_{L^p}\gtrsim \Big\Vert \Big(\sum_{l=M}^{L-M}{2^{{s}{t_l}q}\Big| \sum_{n=M}^{L}{(\phi_{{t_l}}+\phi_{{t_l}+1})\ast d_{{t_n}}}\Big|^q}\Big)^{{1}/{q}}\Big\Vert_{L^p}.
0 = C_{i_p,\beta\;j_q,\gamma}^{k_s,\alpha}C_{l_t,\rho\;m_u,\sigma}^{i_p,\beta}\omega^{j_q,\gamma}\wedge\omega^{l_t,\rho}\wedge\omega^{m_u,\sigma}\quad(\alpha,\beta,\gamma,\rho,\sigma=1,\ldots,N)\;,
f^{( \pm )}( \nu ; k ; x ) = e^{\mp i k x}+ e^{\mp i k x}\sum_{n = 0}^{\infty}( \frac{1}{\pi})^{n + 1}\int_{1}^{\infty}d \alpha_{o}. . . \int_{1}^{\infty}d \alpha_{n}\frac{( \prod_{j = 0}^{n - 1}D ( \nu ; \alpha_{j}) e^{- 2 \alpha_{j}x})}{[ \prod_{j = 0}^{n - 1}( \alpha_{j}+ \alpha_{j + 1}) ] ( \alpha_{o}\pm i k )}
\begin{aligned}\sum_{n\ne0}\widehat{\Delta}(an)^{2}\leq\frac{1}{a}\sum_{|m|\leq a}\left(1-\frac{|m|}{a}\right)-1=\frac{1}{a}\left(2\lfloor a\rfloor+1-\frac{\lfloor a\rfloor(\lfloor a\rfloor+1)}{a}\right)-1 =\frac{1}{a}\left( \left(1+\lfloor a\rfloor-a\right)+\frac{\lfloor a\rfloor}{a}\left(a-\lfloor a\rfloor-1\right) \right) <\frac{1}{a},\end{aligned}
|u_{n,m}(x,t)| \leq C\left( 1+D+\left( \frac{|\mu_{n}|(\Omega_{T})+(|\sigma_{n}|\otimes \varphi_{2,m})(\Omega \times (-T,T))}{D^{N}}\right)^{m_{3}}\right) & ~~~~+C(\varphi_{1,n}\varphi_{2,m})\ast \mathbb{I}_{2}^{2D}[|\mu|+|\sigma |\otimes \delta_{\{t=0\}}](x,t).
Pr(\mathbb{W}|\alpha,\mathbb{Z})=\prod_{k=1}^{K}DirMult(\mathbb{W}_{k}|\mathbb{Z},\alpha)=\prod_{k=1}^{K}[\frac{\Gamma(\sum_{v}\alpha_{v})}{\Gamma(\sum_{v}n_{v}^{k}+\alpha_{v})}\prod_{v=1}^{V}\frac{\Gamma(n_{v}^{k}+\alpha_{v})}{\Gamma(\alpha_{v})}]
{s d e t}~ e_{M}^{~ A}={s d e t}^{- 1}~ e_{A}^{~ M}= d e t^{- 1}[ e_{a}^{~ m}- e_{a}^{~ \mu}( e_{\mu}^{~ \alpha})^{- 1}e_{\alpha}^{~ m}] ~ d e t ~ e_{\alpha}^{~ \mu},
\left| V_1 \cup \cdots \cup V_n \right| = |V_1| + \sum_{j=2}^{n}\left| V_j \setminus \left( V_1 \cup \cdots \cup V_{j-1}\right) \right| &= |V_1| + \sum_{j=2}^{n}\left( |V_j| - \left| (V_j \cap V_1) \cup \cdots \cup (V_j \cap V_{j-1}) \right| \right).
{\cal S}_{\mathrm{p r o b e}}= - M_{1}\int d^{2}\sigma \left[ \frac{R_{9}}{\sqrt{2 \alpha^{\prime}}}- \frac{R_{9}^{2}}{2 \alpha^{\prime}}\frac{Q_{1}}{r^{2}}+ \frac{2 V}{( 4 \pi^{2}\alpha^{\prime})^{2}}\frac{Q_{1}}{r^{2}}\right] .
V_{\mathrm{s t a t i c}}^{( 0 , 0 )}( R ) = - ( d - 2 ) \cdot{\frac{1}{R^{7 - p}}}V_{p}2^{2 - 2 p}\pi^{( 5 - 3 p ) / 2}{\alpha^{\prime}}^{3 - p}\Gamma ({\frac{7 - p}{2}}) \quad ,
D^{(2)}(E,\mathbf{g}) =\Bigl( \prod_{0\le i<j \le 2}(g_j-g_i) \Bigr)\bigl( D^{[0]}(E) D^{[1]}(E\Omega^3) q^{-g_1}-D^{[1]}(E) D^{[0]}(E\Omega^3) q^{-g_0}\bigr).
\lim_{n\to\infty}\frac{1}{n}\left( |X_n| - \frac{m_2-\lambda}{m-1+\lambda}\sum_{k=0}^{n-1}I_{\{\langle X_k\rangle=1\}}- \frac{m_1-\lambda}{m-1+\lambda}\sum_{k=0}^{n-1}I_{\{\langle X_k\rangle=2\}}\right) =0
4 \geq \frac{|N|}{|M|}\frac{|M|}{|M \cap N|}-\frac{8|N|}{|M|}= \frac{|N|}{|M \cap N|}- \frac{8|N|}{|M|}\geq \frac{|N|}{|M \cap N|}-\frac{8}{\gg(K^N)-1}=\frac{|N|}{|W \cap N|}-\frac{8}{\gg(K^N)-1}.
p(f;\Delta) = \frac{c_{p,n_1,n_2}}{|\Delta|^{n_1/2}}\frac{|F|^{(n_1-p-1)/2}}{|I+F|^{n/2}}\,_1F_0(\tfrac{n}{2};I-\Delta^{-1}, F(I+F)^{-1}) \prod_{j<j'}^p (f_j - f_{j'}),
\begin{aligned}\frac{1}{\tau^k}[f(x)-f(\breve{x}^{k})]-\frac{1}{\tau^{k-1}}[f(x)-f(\breve{x}^{k-1})]+(u-\widetilde{u}^k)^TF(\widetilde{u}^k) &+c^k(A(x-\widetilde{x}^k))^T(A\breve{x}^{k-1}-b)\geq(v-\widetilde{v}^k)^TQ^k(v^k-\widetilde{v}^k),~\forall u,\end{aligned}
f\left(t\right)=f\left(0\right)\sum_{k=0}^{\infty{}}\frac{{\left({E\left(i\hslash{}\right)}^{-\beta{}}t^{\beta{}}\right)}^k}{\Gamma\left(\beta{}k+1\right)}=f\left(0\right)E_{\beta{}}\left({\left(\frac{t}{i\hslash{}}\right)}^{\beta{}}E\right),
\tilde{E}^{-}_{\beta}= \begin{cases}\hat{E}^{-}_{\beta}\ \beta=[1],[2],[1,2] [2]_{v}!\hat{E}^{-}_{\beta}\ \beta=[1,2,2] [3]_{v}!\hat{E}^{-}_{\beta}\ \beta=[1,2,2,2],[1,2,1,2,2] \end{cases}.
\sum_{k=0}^{\infty}\frac{\left(\frac12\right)_k^2}{(1)_k^2}u_0^{2k}\sum_{k=0}^{\infty}\frac{\left(\frac12\right)_k^2}{(1)_k^2}u_0^{2k}(A+B k) = \frac{1}{\pi},
\sum_{|\lambda/\mu|=n}\frac{2^{|\lambda|-\ell(\lambda)-|\mu|+\ell(\mu)}f_{\lambda/\mu}H_\mu}{H_{\lambda}}\left(\sum_{\square\in\lambda}\binom{c_{\square}+1}{4}-\sum_{\square\in\mu}\binom{c_{\square}+1}{4}\right) =\ \frac23\binom{n}{3}+\frac23|\mu|\binom{n}{2}+\frac1{12}\left( q_2(\mu)+|\mu|^2-2|\mu|\right)n.
\begin{aligned}\nabla_k U_{ij}\equiv \, \nabla U (e_i, e_j, e_k) = \nabla_{kij}u + \nabla_k A^{ij}(x, t, \nabla u) \equiv \, \nabla_{kij}u + A_{x_k}^{ij}(x, t, \nabla u) + A^{ij}_{p_l}(x, t, \nabla u) \nabla_{kl}u, \end{aligned}
\eta_a^\lambda (\mathsf{P}) =\Xi^{\lambda}(K_a F_a \otimes 1 \otimes E_a K_a^{-1}\otimes K_a) -\Xi^{\lambda}(K_{a}F_{a}\otimes 1 \otimes 1 \otimes E_{a}) -\Xi^{\lambda}(K_{a}\otimes F_{a}\otimes E_{a}K_{a}^{-1}\otimes K_{a}) + \Xi^{\lambda}(K_{a}\otimes F_{a}\otimes 1 \otimes E_{a}).
\frac{t}{e^{t}-1}=\sum\limits_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!},\ |t|<2\pi, \frac{te^{xt}}{e^{t}-1}=\sum\limits_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!},\ |t|<2\pi,
\sum_{n = 1}^{\infty}\frac{1}{( n^{2}+ p^{2})^{k}}= \frac{\sqrt{\pi}\Gamma ( k - \frac{1}{2})}{2 \Gamma ( k ) p^{2 k - 1}}- \frac{1}{2 p^{2 k}}+ \left( \frac{\pi}{p}\right)^{k}\frac{e^{- 2 \pi p}}{\Gamma ( k )},
[\lambda(xg), \lambda(yt)]= \,\, [xh_g+g\beta_x, yh_t+t\beta_y] = \,\, x[h_g, y]h_t+y[x, h_t]h_g +t[x, \beta_y]h_g+tx[h_g, \beta_y]+x[h_g, t]\beta_y &+g[\beta_x, y]h_t+yg[\beta_x, h_t]+y[g, h_t]\beta_x+t[g, \beta_y]\beta_x+tg[\beta_x, \beta_y]+g[\beta_x, t]\beta_y.
\sum_{a \in [n] \setminus \{1\}}(\Pi_1 x_{a,1})\otimes x_{a,2}\otimes x_{a,3}=\sum_{a \in [n] \setminus \{1\}}(\alpha_{a}\Pi_1 x_{a,1})\otimes y_{\sigma(a),2}\otimes y_{\sigma(a),3}.
\liminf_{n\to\infty}sR-\frac{1}{n}sD_{1+s}(P_{W^{n}X^{n}Y^{n}}\|P_{W^{n}}\pi_{X^{n}Y^{n}}) \qquad+\frac{1}{n}sD_{1+s}(P_{X^{n}Y^{n}}\|\pi_{X^{n}Y^{n}}) =sR-s\left(\frac{(1-\epsilon)^{2}}{1+\epsilon'}I_{Q}(XY;W)+\frac{4\epsilon}{1-\epsilon'}H_{Q}(XY)\right).
\omega_{\pm}\equiv{\cal Q}_{\pm}( \mp \infty ) = \pm 2 \pi \displaystyle \frac{p - 1}{p + 1}( S - 2 S^{\pm}) + 2 \pi \displaystyle \frac{p}{p + 1}\left( \displaystyle \frac{\alpha}{\pi}- \delta + 2 k_{W}^{\pm}- 2 k_{\pm}\right) .
\hat{\mathbb{E}}_t[\varphi(X_{t+t_1}^{x},X_{t+t_2}^{x},\cdots,X_{t+t_m}^{x})] =\hat{\mathbb{E}}_t[\hat{\mathbb{E}}_{t+t_{m-1}}[\varphi(X_{t+t_1}^{x},X_{t+t_2}^{x},\cdots,X_{t+t_m}^{x})]] &=\hat{\mathbb{E}}_t[\overline{\varphi}_{m-1}(X_{t+t_1}^{x},X_{t+t_2}^{x},\cdots,X_{t+t_{m-1}}^{x})],
w_{3}= ({\bf n}_{2}\cdot{\bf n}_{1}) , ~ ~ w_{2}= ({\bf n}_{3}\cdot{\bf n}_{1}) , ~ ~ \bar{w}_{3}= ({\bf n}_{2}\cdot{\bf n}_{4}) , ~ ~ \bar{w}_{2}= ({\bf n}_{3}\cdot{\bf n}_{4}) , ~ ~ w_{1}= ({\bf n}_{2}\cdot{\bf n}_{3}) ,
E = ( \mathrm{a n a l y t i c \i n \}\Delta ) -{\frac{9 \pi^{3}}{3 2 \sqrt 2}}{\frac{1}{z_{c}^{5}}}\left({\frac{1 - 3 z_{c}^{2}}{5 - 6 z_{c}^{2}}}\right)^{2}{\frac{\Delta^{2}}{\log \Delta}}+ ( \mathrm{h i g h e r \o r d e r \t e r m s}) \ ,
{\frac{1}{\sqrt{2}}}( \hat{\lambda}_{1}\pm i \hat{\lambda}_{2}) = e^{\pm i ( H_{4}- \kappa + i \phi )}( z ) , \quad{\frac{1}{\sqrt{2}}}( \hat{\lambda}_{3}\pm i \hat{\lambda}_{4}) = e^{\pm i ( H_{5}- \kappa + i \phi )}( z ) .
{}_2F_1\bigg[\begin{matrix}\frac{1}{4}\frac{1}{4}&1\end{matrix}\bigg|\,4z(1-z)\bigg]_{p-1}\equiv\begin{cases}P_{\frac{p-1}{2}}(1-2z)\pmod{p^2}, p\equiv1\pmod{4}, P_{\frac{3p-1}{2}}(1-2z)\pmod{p^2}, p\equiv3\pmod{4}.\end{cases}
\nu_{\pi_k}= \frac{v^{S_{k+1}}- v^{S_k}}{b^{S_{k}}-b^{S_{k+1}}}& = \min \, \left\{\frac{v^{S_{k}\setminus \{j\}}- v^{S_k}}{b^{S_{k}}- b^{S_{k}\setminus \{j\}}}: j \in S_k\right\}& = \max \, \left\{\frac{v^{S_k}- v^{S_k \cup \{j\}}}{b^{S_k \cup \{j\}}- b^{S_k}}: j \in N^{\{0, 1\}}\setminus S_k\right\}.
\begin{aligned}v_{x_n}=\frac{1}{h_z}, v_{x_i}=-\frac{h_{x'}}{h_z}, v_t=-\frac{h_{t}}{h_z}v_{x_nx_n}=-\frac{h_{zz}}{h_z^3}, v_{x_ix_i}=-\frac{1}{h_z}\left(\frac{h_{x_i}^2}{h_z^2}h_{zz}-\frac{2h_{x_i}}{h_z}h_{x_iz}+h_{x_ix_i}\right) \forall i=1,\cdots,n-1.\end{aligned}
E_{m}= \frac{1}{i L}\frac{\partial}{\partial A_{m}}- \frac{e}{2 \pi i m}\left( j_{+}{}^{m}+ ( j_{-}{}^{m})^{\dagger}\right) , m \neq 0 , \quad E_{0}= \frac{1}{i L}\frac{\partial}{\partial c},
L(x^{k+1},\lambda)- L(x^{k+1},\lambda^{k+1}) =g^*(\lambda^{k+1})-g^*(\lambda)-(x^{k+1})^TA^T(\lambda-\lambda^{k+1}) \leq (\nabla g^*(\lambda^{k+1})+Ax^{k+1})^T(\lambda^{k+1}-\lambda)+\frac{L}{2}\|\lambda^k-\lambda^{k+1}\|_2^2 \ ,
a_0= \frac{1}{2}(\alpha_1-\alpha_2), a_1 =\beta_1, a_2=\frac{1}{2}(\alpha_1+\alpha_2), a_3=\beta_2, b_0= \frac{1}{2}(\alpha_3-\alpha_4), b_1 =\beta_3, b_2=\frac{1}{2}(\alpha_3+\alpha_4), b_3=\beta_4.
2S_2L \le \frac{1}{2}\frac{h^2(\tilde x)}{l(\tilde x)-h(\tilde x)}\left(\frac{3}{2}+4\frac{l(\tilde x)-h(\tilde x)}{h^2(\tilde x)}\right) &<\frac{1}{2}\frac{(3S_2L)^2}{L\left(\frac{1}{2}-2S_2\right)}\left(\frac{3}{2}+4\cdot\frac{1}{8}\right) &=2\frac{9S_2^2L}{1-4S_2},