harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta(l) = \sum_{\substack{0 \le l_1 \le \frac{l}{2}0 \le l_2 \le \frac{m-2l}{2}}}\Delta(l,l_1,l_2)< \sum_{\substack{0 \le l_1 \le \frac{l}{2}0 \le l_2 \le \frac{m-2l}{2}}}\Lambda(l_1,l_2,l,m)q^{\frac{mn}{4}+(\frac{3m}{4}-l)s}<cq^{\frac{mn}{4}+\frac{1}{2}(m^2-2lm+2l^2)+(\frac{3m}{4}-l)s}.
\frac12\|u(t)\|_2^2+\Gamma_2\int_0^t\|\Delta u\|_2^2\,ds+\beta\int_0^t\|u\|_4^4\,ds= \frac12\|u_0\|_2^2+\int_0^t\int_{\mathbb{R}^n}fu\,dxds &\qquad\strut +\Gamma_0\int_0^t\int_{\mathbb{R}^n}u\Delta u\,dxds-\int_0^t\int_{\mathbb{R}^n}uMu\,dxds+\int_0^t\int_{\mathbb{R}^n}uN(u)\,dxds
\begin{pmatrix}\alpha''_{i-1}0\end{pmatrix}= \begin{pmatrix}h_{i}0\end{pmatrix}\begin{pmatrix}\psi_i -\beta_{i-1}0 -\phi_{i-1}\end{pmatrix}+ \phi_{i}\begin{pmatrix}h_{i-1}0\end{pmatrix}.
\mathcal{J}_i(\tau_i,\tau_{-i})=\mathbb{E}\left\{e^{-\beta\tau_i}\Bigg[ \frac{x_i(\tau_i)}{l_1+l_2\left(\frac{1}{N}\sum\limits_{j=1}^Ne^{-\beta\tau_j}x_j(\tau_j)\right)}-K\Bigg]\right\}.
\begin{aligned}A =t^8-5t^5+7t^2 B =-t^9+3t^6-2t^3+6 C =t^{10}-4t^7+7t^4-3t D_f =t^{18}+10t^{15}-41t^{12}+80t^9-72t^6+20t^3+12 \Delta =3t^{18}+30t^{15}-123t^{12}+240t^9-216t^6+60t^3+36 \end{aligned}
\begin{array}{c}[(2,\chi_{1}^{1}) \oplus (-1,\chi_{2}^{1})]\otimes ((a+2i)-k',\chi_1^{2}~$or$~ \chi_1^{4}) =((a+2(i+1))-k',\chi_1^{2}~$or$~ \chi_1^{4}) \oplus ((a+2(i-1))-(k'-1),\chi_{2}^{k'-1}))\end{array}
\sum_{p=0}^{N-1}{\cal{R}}_p(\lambda_{N,m}){\cal{R}}_p(\lambda_{N,l})=f(\lambda_{N,m})f(\lambda_{N,l})\sum_{p=0}^{N-1}U_p\left(\frac{2+\lambda_{N,m}}{2}\right)U_p\left(\frac{2+\lambda_{N,m}}{2}\right).
b_{(i, \alpha, r_1, r_2), (i, \alpha, r_1, r_2)}= a_{(i, \alpha, r_1, r_2), (i, \alpha, r_1, r_2)}- \underset{d^{r''_1, r''_2}_{l, \gamma}> d^{r_1, r_2}_{i, \alpha}}{\sum}b_{(i, \alpha, r_1, r_2), (l, \gamma, r''_1, r''_2)}
B^{\top}(t)\bar{\bf w}^{(\epsilon)}(t) =B^{\top}(t){\bf w}^{(\epsilon)}(t) & \geq{\bf a}(t)-\mbox{\boldmath $\epsilon$}+\int_{t}^{T}K^{\top}(s,t){\bf w}^{(\epsilon)}(s)ds={\bf a}(t)-\mbox{\boldmath $\epsilon$}+\int_{t}^{T}K^{\top}(s,t)\bar{\bf w}^{(\epsilon)}(s)ds.
Y^{k,s,x}= g(X_T) + \int_{\cdot}^T f\left(r,X_r,Y^{k-1,s,x},\sqrt{\frac{d\langle M^{k-1,s,x}\rangle}{dV}}(r)\right)dV_r -(M^{k,s,x}_T - M^{k,s,x}_{\cdot}).
\left\{\begin{array}{c}I_c^{xx}I_c^{xy}I_c^{yy}\end{array}\right\}= \int\limits_\Omega \left\{\begin{array}{c}(x-x_c)^2 (x-x_c)(y-y_c) (y-y_c)^2 \end{array}\right\}~\mathrm{d}\Omega
\Delta_E^{M\times N}(v_ik_j) =(\lambda^{(0)}_i+\kappa^{(2)}_j)v_ik_j, \Delta_E^{M\times N}(\alpha_i\odot\beta_j) =(\lambda^{(1)}_i+\kappa^{(1)}_j)\alpha_i\odot\beta_j, \Delta_E^{M\times N}(w_ih_j) =(\kappa^{(0)}_i+\lambda^{(2)}_j)w_ih_j,
f(s,y) := \eta y\cdot \nabla w(s,y) + \eta N w(s,y) & - \frac{1}{s}\left[ |\nabla w(s,y)|^{q_*}- p \eta (N+1) w(s,y) + (p-2) \eta (N+1) y\cdot \nabla w(s,y) \right]
I(t) = - i \frac{\Gamma(\pm t - 1/8) \Gamma(\pm t + 3/8) \Gamma(\pm t + \sigma/2 - 1/8) \Gamma(\pm t + \sigma/2 + 3/8) \Gamma(\pm t + \tau/4) \Gamma(\pm t + \tau/4 + 1/2)}{\Gamma(\pm 2t)}.
\left. \frac{{\cal V}_{\mathrm{e f f}}}{N}\right|_{\mathrm{s p i r}}= \frac{m^{2}}{4 \pi}\ln \left( \frac{m^{2}}{m_{0}^{2}}\right) + \frac{m_{0}^{2}- m^{2}}{4 \pi}- \frac{2}{\beta}\int \frac{\mathrm{d}k}{2 \pi}\ln \left( 1 + \mathrm{e}^{- \beta \epsilon ( k )}\right) - \frac{\mu^{2}}{2 \pi}\ ,
{\mathbf Q}_{\rm CFIV}= 2\int_{0}^{\infty}dE E^{\frac{k}{2}}{\rm e}^{-A(\theta)}&=16 \pi \cos \frac{\pi}{k+2}\int_0^{\infty}dE E^{\frac{k}{2}}{\rm Ai}^{(k)}(E) \frac{d}{dE}{\rm Ai}^{(k)}(E).
D_{U^{\alpha}}=\frac{(1-|\eta|^2)^{\alpha\cdot\mathbf 1}}{\pi(1-w\bar \eta)^{2+\alpha\cdot\mathbf 1}}\Big(I+\sum_{j=1}^{n}\alpha_j(I+z_j\frac{\partial}{\partial z_j})\Big), &D_{V^{\gamma}}=\frac{e^{(\gamma\cdot\mathbf 1)(w\bar{\eta}-|\eta|^{2})}}{\pi}\Big(\sum_{j=1}^{n}\gamma_j(I +z_j\frac{\partial}{\partial z_j})\Big).
\frac{1}{T}\sum_{t=0}^{T-1}\|\nabla g(\theta_t)\|^2 \leq \delta^2 + \frac{2L(g(\theta_0)-g(\theta^T))}{T}&\leq \frac{\varepsilon^2}{16}+ \frac{2L\Delta_g}{T}\leq \frac{25}{144}\varepsilon^2.
(I_{+}+ I_{-}) \wedge \frac{d \chi_{\gamma_e}(\zeta)}{\chi_{\gamma_e}(\zeta)}= \frac{iR}{4}(L^{+}_1 + L^{-}_1 + L^{+}_2 + L^{-}_2) \wedge \frac{d \chi_{\gamma_e}(\zeta)}{\chi_{\gamma_e}(\zeta)}= \left[ 2\pi i A^{inst}+ \pi i V^{inst}\left( \frac{1}{\zeta}dc - \zeta d\overline{c}\right) \right] \wedge \xi_e
g_{nMN}=\sum_{k=0}^{\infty}C^{*k}_{\xi}C^{k}_{\eta}e_{kM}\otimes e_{k-n,N}(\frac{q^{-n}}{1-q^{2}})^{k}q^{k^{2}}\sqrt{\frac{[-n]_{q^{2}}!}{[-n+k]_{q^{2}}![k]_{q^{2}}!}}
(\lambda, m-d(\lambda))(\mu, m - d(\mu))^* = (\lambda, m-d(\lambda))\Big(\sum_{\alpha \in s(\lambda)\Lambda^{n-m}}(\alpha, m)(\alpha,m)^*\Big)(\mu, m - d(\mu))^* = \sum_{\alpha \in s(\lambda)\Lambda^{n-m}}(\lambda\alpha, m-d(\lambda))(\mu\alpha, m-d(\mu))^* \in B_n.
\begin{array}{c c c}{\partial_{\sigma}\varrho = 0 \hfill}&{\Longrightarrow}&{\varrho = \rho ( x , \tau ) , \hfill}\\{\partial_{\sigma}\theta = 1 \hfill}&{\Longrightarrow \hfill}&{\theta = \Theta ( x , \tau ) + \sigma , \hfill}\\ \end{array}
\mathfrak{u}=\sum_{l\in\mathbb{Z}}e^{ilt}\Bigg[\hat{u}^{+}_{0,l}\left( \begin{array}{c}\frac{e^{2|l|r}}{\sqrt{z}}-\mbox{sign}(l)\frac{e^{2|l|r}}{\sqrt{\bar{z}}}\end{array}\right)+\hat{u}^{-}_{0,l}\left( \begin{array}{c}\frac{e^{-2|l|r}}{\sqrt{z}}\mbox{sign}(l)\frac{e^{-2|l|r}}{\sqrt{\bar{z}}}\end{array}\right)\Bigg] +\mbox{higher order term}
P^\chi_{out}\!\leq\! \Big(\mathrm{Pr}\big\{|g_m||h^\chi_m|\!\leq\! \sqrt{\frac{\tau_\chi}{P_{BS}}}\big\}\Big)^M \cdot\mathrm{Pr}\big\{|h^\chi_d|\!\leq\! \sqrt{\frac{\tau_\chi}{P_{BS}}}\big\}.
v_p^x(b_0a_0(x))-v_p^x(b_ja_j(x)) = v_p^x((n+1)!a_0(x))-v_p^x\bigg(\frac{(n+1)!}{(j+1)!}a_j(x)\bigg) \leq v_p((n+1)!) - v_p\bigg(\frac{(n+1)!}{(j+1)!}\bigg) = v_p((j+1)!).
f ( x ; u , \gamma , \tau ) \equiv{\frac{s i n h \big ({\frac{\pi i}{\tau}}( 1 - \gamma ) x \big ) s i n h \big ({\frac{2 \pi i}{\tau}}\gamma u x \big ) s i n h \big ({\frac{\pi i ( N - 1 )}{\tau}}\gamma x \big )}{x s i n h \big ({\frac{\pi i}{\tau}}x \big ) s i n h \big ({\frac{\pi i N}{\tau}}\gamma x \big )}}\ .
\rho_{m}^{\mathrm{x}}-\left(\rho_{m}^{\mathrm{x}}\right)^{2}=0, \forall \mathrm{x}\in\{\mathrm{t}, \mathrm{r}\}, m \in \mathcal{M}, &0 \leq \rho_{m}^{\mathrm{t}}, \rho_{m}^{\mathrm{r}}\leq 1, m \in \mathcal{M}.
c_{jj}= \frac 89 \delta_j^2 - \frac 29 \delta_j (\omega_{j+1}^2 - \omega_j^2) - \frac 2{35}\big(\omega_{j+1}^3 - \omega_j^3 - 3\omega_{j+1}\omega_j(\omega_{j+1}- \omega_j)\big), 1\le j\le N, c_{jk}= \frac 89 \delta_j \delta_k - \frac 29 \delta_j (\omega_{k+1}^2 - \omega_k^2), 1\le j<k\le N.
I_7= \int \Big(\partial^\zeta_x(\nabla \phi^2 \cdot Q(v))-\nabla \phi^2 \cdot \partial^\zeta_x Q(v)\Big)\cdot \partial^\zeta_x u \leq |\partial^\zeta_x(\nabla \phi^2 Q(v))-\nabla \phi^2 \partial^\zeta_x Q(v)|_2|\partial^\zeta_x u|_2.
{\cal Z}_{T}= \int \! \left[ d^{N}\vec{x}( t ) \, \sqrt{\det g ( \vec{x}( t ) )}\right] \, \exp \left[ -{\frac{N r}{2}}\int_{- T / 2}^{T / 2}\! d t \Bigl ( g_{i j}\,{\frac{d x^{i}}{d t}}{\frac{d x^{j}}{d t}}+{\vec{x}}^{2}\Bigr ) \right] \, ,
\frac{1}{2}\tilde{j}(v_1(0) - \varphi) + \frac{1}{2}\tilde{j}(v_2(0) -\varphi) >\tilde{j}\Big( \frac{v_1(0) + v_2(0)}{2}- \varphi \Big) \ge \tilde{j}(v_1(0) - \varphi) + \Big(v_1'(0),\frac{v_2(0) - v_1(0)}{2}\Big) = \frac{1}{2}\tilde{j}(v_1(0) - \varphi) + \frac{1}{2}\tilde{j}(v_2(0) - \varphi),
\phi_{ij}(x)=\log\frac{\mu-\lambda+\lambda\mu x}{x},\phi_{k\ell}(x)=\log\frac{x}{\lambda-\mu+(\lambda+\alpha)(\mu+\alpha)x}, &\phi_{i\ell}(x)=\log\frac{1+\lambda x}{1+(\mu+\alpha)x}.
< \hat{{\cal{J}}}_{\mu_{1}\cdots \mu_{s}}^{( s )}( x ) ~ \hat{{\cal{J}}}_{\nu_{1}\cdots \nu_{s}}^{( s )}( 0 ) > ={\prod}_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}\left({\frac{c_{s}( t )}{x^{4}}}\right) .
\left\langle J_{(ab,\alpha)},J_{(cd,\beta)},P_{(e,\gamma)}\right\rangle =\alpha_{\delta}K_{\alpha\beta\gamma}^{\delta}\left\langle \tilde{J}_{ab},\tilde{J}_{cd},\tilde{P}_{e}\right\rangle , & =\frac{1}{8}\left( \alpha_{0}K_{\alpha\beta\gamma}^{0}+\alpha_{1}K_{\alpha\beta\gamma}^{1}\right) \varepsilon_{abcde}.
\mu_b (\alpha):=\sum_{h=1}^\infty r_{\alpha}(h)= \sum_{a=0}^\infty \frac{1}{b^{a\alpha}}\sum_{k=b^a}^{b^{a+1}-1}1=\sum_{a=0}^\infty \frac{(b-1)b^a}{b^{a\alpha}}=\frac{b^{\alpha}(b-1)}{b^\alpha - b}.
\begin{array}{c}w_{1,\varepsilon}=\frac{1}{8}{\varphi_{2,\varepsilon}^2 + \frac{1}{4}\sin^2\varphi_{1,\varepsilon}-\frac{c_0}{2}\sin\varphi_{1,\varepsilon}}+O(\varepsilon)e^{-\left(\sqrt{1-4c_0^2}+O(\varepsilon)\right)|x|}, w_{2,\varepsilon}=O(\varepsilon)e^{-\left(\sqrt{1-4c_0^2}+O(\varepsilon)\right)|x|}, \end{array}
K_{0}( p_{1 j}^{2}; m_{0}, \epsilon ) \equiv K_{0 0}( p_{1 j}^{2}; m_{0}^{2}, \epsilon ) , \Delta K ( p_{1 j}^{2}; m_{0}, M , \epsilon ) = \sum_{l = 1}^{\infty}\sum_{m = - l}^{l}K_{l m}( p_{1 j}^{2}; M_{l}^{2}, \epsilon ) ,
\det \mathcal{A}_{B_n}= \prod_{\substack{J \in \overline{2^{[\pm n]}}|J| \geq 2}}\Big(1- \prod_{\{i,j\}\in \binom{J}{2}}a_{H_{i,j}}^2\Big)^{2^{n-|J|+1}\,(|J|-2)!\,(n-|J|+1)!}& \prod_{\substack{I \in 2^{[n]}|I| \geq 1}}\Big(1- \prod_{i \in I}a_{H_i}^2\prod_{\{i,j\}\in \binom{I}{2}}a_{H_{i,j}}^2\, a_{H_{-i,j}}^2\Big)^{2^{n-1}\,(|I|-1)!\,(n-|I|)!},
E_2(\lambda_1,\lambda_2,\lambda_3)\wedge v_2= z_1^2 \lambda_1 +z_1z_2 \lambda_2+z_1z_3 \lambda_3, E_3(\lambda_1,\lambda_2,\lambda_3)\wedge v_3= z_1z_2 \lambda_1 +z_2^2 \lambda_2+z_2z_3 \lambda_3, (E_2(\lambda_1,\lambda_2,\lambda_3)+E_3(\lambda_1,\lambda_2,\lambda_3))\wedge (v_2+v_3)= z_1z_3 \lambda_1 +z_2 z_3 \lambda_2+z_3^2 \lambda_3,
\mathcal{N}_S :=\sum_{m \ge 2}((2m-3)N_{2m}-N'_{2m})+\sum_{m \ge 2}((2m-2)N_{2m+1})-1 =\sum_{\ell \ge 3}(\ell -3) N_{\ell}- \sum_{m \ge 2}N'_{2m}-1.
X+\sqrt{2}Y = \left( \frac{\alpha_1+\alpha_2}{2}+ \frac{\sqrt{2}}{2}(\beta_1+\beta_2)\right)^2+ \left( \frac{\alpha_1-\alpha_2}{2}+ \frac{\sqrt{2}}{2}(\beta_1-\beta_2)\right)^2 + \left( \frac{\alpha_3+\alpha_4}{2}+ \frac{\sqrt{2}}{2}(\beta_3+\beta_4)\right)^2+ \left( \frac{\alpha_3-\alpha_4}{2}+ \frac{\sqrt{2}}{2}(\beta_3-\beta_4)\right)^2.
(-1)^n \frac{\Gamma(b - a - n) \Gamma(a + n)}{n! \Gamma(c - a - n)}\, (-z)^{-a-n}= \frac{\Gamma(a) \Gamma(b - a)}{\Gamma(c - a)}\, (-z)^{-a}\ \frac{(1 + a - c)_n (a)_n}{(1 + a - b)_n n!}\, (-z)^{-n}
L \left( \frac{1}{2}, \chi_c \right) = 2\sum_{0 \neq \mathcal{A}\subset \mathcal{O}_K}\frac{\chi_c(\mathcal{A})}{N(\mathcal{A})^{1/2}}V \left(\frac{\pi N(\mathcal{A})}{N(c)^{1/2}}\right).
\sum_{i \in M_P}\sum_{j \in N_i}a_{ij}\tilde x_{ij}+ (b-s) \sum_{ij \in \bar P}\tilde x_{ij}\leq \sum_{i \in M_P}\sum_{j \in N_i}a_{ij}\tilde x_{ij}+ (b-s) |\bar P| & \leq s + (b-s) |\bar P| = b + (|\bar P| - 1) (b-s) & = b + (|M_P - M_0| - 1) (b-s).
\mu_i \sum_{t=1}^k \lambda_{1,t}\equiv x_1q_1+x_2q_2+x_3q_3+x_4q_4 &\equiv x_1q_1 + x_2(a\mu_i-q_1)+x_3(b\mu_i-q_1)+x_4(n\mu_i-(a+b)\mu_i+q_1) &\equiv q_1(x_1+x_4-x_2-x_3)+a\mu_i(x_2-x_4)+b\mu_i(x_3-x_4)+\mu_inx_4.
\beta ( r )_{N u t}^{2}= \frac{\lambda \, c \, \, 2^{n}\, r \, ( r - L ) \, L^{n}}{( n + 1 ) ( r + L )^{n}}\, \mathrm{{A p p e l l}}F_{1}[ n + 1 , 2 , - n , n + 2 , 1 - \frac{r}{L}, \frac{L - r}{2 L}] .
dV(t,X(t)) = LV(t,X(t))dt+\partial_xV(t,X(t))\cdot \sigma I(t)S(t) dW_t &+\int_U\left( V(X(t-)+J(u))- V(X(t-))\right)\check{N}(dt,du) & = LV(t,X(t))dt+\sigma S(t) dW_t+\int_U\left(\log(1+JS)\right)\check{N}(dt,du),
\left[ 2 i{\left( - 1 \right)}^{n - 1}{\epsilon}_{1}\cdot p_{1}P^{n - 1}\left( p_{1}^{2}, p^{2}\right) \right] \frac{{\left( - 1 \right)}^{n}i}{p^{2 n}-{\left( - m^{2}\right)}^{n}}\left[ 2 i{\left( - 1 \right)}^{n - 1}{\epsilon}_{2}\cdot p_{2}P^{n - 1}\left( p^{2}, p_{2}^{2}\right) \right] +
\mathbf{p}E_{mj}=E_{mj}+\sum_{i=j+1}^{m-1}\sum_{s=2}^{i-j+1}\sum_{I\in\mathcal{I}_{ji}(s)}E_{i_2i_1}\cdots E_{i_si_{s-1}}E_{m i_s}\frac{1}{\prod_{\ell\in I,\ell\neq j}(h_j-h_\ell)},
t_{\beta}t_{\gamma}\cdot t_b t_c t_a t_b \cdot t_{\alpha}t_{\beta}(a) = t_{\beta}t_{\gamma}\cdot t_b t_c t_a t_b (a) = t_{\beta}t_{\gamma}(c) = c, t_{\beta}t_{\gamma}\cdot t_b t_c t_a t_b \cdot t_{\alpha}t_{\beta}(\alpha) = t_{\beta}t_{\gamma}\cdot t_b t_c t_a t_b (\beta) = t_{\beta}t_{\gamma}(\beta) = \gamma,
(A_{+,n}- z I)^{-1}= \big[I + (A_- - z I)^{-1}\chi_n(A_-) B_+ \chi_n(A_-)\big]^{-1}(A_- - z I)^{-1}, & (A_+ - z I)^{-1}= \big[I + (A_- - z I)^{-1}B_+ \big]^{-1}(A_- - z I)^{-1},
\left.{\frac{z_{1}d z_{2}- z_{2}d z_{1}}{z_{0}^{*}}}\right|_{S}= \left.{\frac{z_{0}d z_{2}- z_{2}d z_{0}}{z_{1}^{*}}}\right|_{S}= \left.{\frac{z_{1}d z_{0}- z_{0}d z_{1}}{z_{2}^{*}}}\right|_{S}
\ll 2^{-k}\frac{(2k)!}{k!}[V(u,u')^k]+O\left(\frac{Y^{4k}}{T}\right) &=2^{-k}\frac{(2k)!}{k!}\left(\frac{1}{2}\sum_{p\leq Y}\frac{1}{p}(u^2+(u')^2+2uu'\cos(|h-h'|\log p))\right)^{k}+O\left(\frac{Y^{4k}}{T}\right).
Minimize_{x,y\in\mathbb{R}^{n}}\ \ \alpha_J^Ty+2b_J^Tx s.t.\ \ \alpha_k^Ty+2b_k^Tx+c_k \le 0\ \ (\forall\,k\in\{1,...,m\}) s.t.\ \x_i^2-y_i \le 0,\ \ \forall\,i
-\Delta(c+[a,c](a-bd^{-1}c)^{-1}) =-\Delta(c(a-bd^{-1}c)+[a,c])(a-bd^{-1}c)^{-1}&= -\Delta \,(- cbd^{-1}+a)c(a-bd^{-1}c)^{-1}&= -\Delta \,( -cb +ad)d^{-1}c(a-bd^{-1}c)^{-1}&=-d^{-1}c(a-bd^{-1}c)^{-1}&=-(ac^{-1}d-b)^{-1},
\left<F,G\right>_{H^2(D'_\beta)}:=\big<\widetilde{F},\widetilde{G}\big>_{L^2(\partial D'_\beta)}=\sum_{k=1}^4 \int_{E_k}\widetilde{F}(\zeta_1,\zeta_2)\overline{\widetilde{G}(\zeta_1,\zeta_2)}\d\zeta_1 d\zeta_2.
\left<\mathcal W_\Psi f,\mathcal W_\Psi h\right> &=\int_{\mathcal S^n}\!\int_{\mathcal S^n}\!\int_0^\infty \frac{\sum_{l,k}\bigl|a_l^k(\Psi_\rho)\bigr|^2}{N(n,l)}\,\frac{\lambda+l}{\lambda}\,C_l^\lambda(x\cdot y)\,\alpha(\rho)\,d\rho\,\overline{f(x)}\,d\sigma(x)\,g(y)\,d\sigma(y).
D(s, S_f \times \overline{S_g}) = W(s; f, \overline{g}) + \frac{1}{2\pi i}\int_{(\gamma)}W(s-z; f, \overline{g}) \zeta(z)\frac{\Gamma(z) \Gamma(s - z + k - 1)}{\Gamma(s + k - 1)}\; dz,
\gamma =\{\{i_{2sp+1},\overline{i_{2sp+1}}\},\ldots,\{i_{(2s+t)p},\overline{i_{(2s+t)p}}\}\}\delta = \{[i_{(2s+t)p+1},\overline{i_{(2s+t)p+1}}],\ldots,[i_{rp},\overline{i_{rp}}]\}.
H_{l , c}= - p_{u}= \frac{1}{2}( E - p_{y}) = \frac{p_{A}^{2}}{4 p_{v}}+ \frac{1}{4 \alpha^{\prime}p_{v}}( \hat{N}_{L}+ \hat{N}_{R}) + f ( \hat{J}_{L}+ \hat{J}_{R}) ~ .
\frac{\iota_n}{n^{t/2}}\sum\limits_{j=s+1}^{p}\frac{\vert u_{2,j-s}\vert^{t}}{\vert \widehat{\beta}_{2,j}\vert^{\varsigma}}= \iota_n n^{(\varsigma -t)/2}\sum\limits_{j=s+1}^{p}\frac{\vert u_{2,j-s}\vert^{t}}{\vert \sqrt{n}\widehat{\beta}_{2,j}\vert^{\varsigma}}= \iota_n n^{(\varsigma -t)/2}\Omega_P (\Vert \mathbf{u}_2 \Vert_{t}^{t}),
T_{N,2}=\sum^N_{n=1}\sum^{\infty}_{j=m_0+1}\Big( \mathbb{E}\big[K(X_n+a_j\widetilde{\varepsilon}_{n-j})|\varepsilon_{n-j}\big]-\mathbb{E}\big[K(X_n+a_j\widetilde{\varepsilon}_{n-j})\big]\Big).
\langle \mu^*(J a P_{\hat{1}}b J) \hat{1}, \hat{1}\rangle ={\rm Tr}(\mu^*( Ja J P_{\hat{1}}JbJ ) P_{\hat{1}}) ={\rm Tr}(\mu^* (P_{\hat{1}}) JbJ P_{\hat{1}}Ja J) &= \overline{{\rm Tr}( \mu( aP_{\hat{1}}b) P_{\hat{1}})}= \langle \hat{1}, \mu(a P_{\hat 1}b) \hat{1}\rangle.
T_{\mathrm{m i n}}< T < T_{\mathrm{m a x}}, \quad \mathrm{w h e r e}\quad T_{\mathrm{m i n}}^{2}= 1 / 2 \pi^{2}\alpha^{\prime}, \quad T_{\mathrm{m a x}}^{2}= 2 / \pi^{2}\alpha^{\prime}\quad .
\vcenter{\halign{$\hfil#$ $#=\Biggl\{z\in\Delta^n\hfil$ ${}\Biggm|\hfil#$ $\biggl\{{\displaystyle\frac{|x_j-z_j|^2}{1-|z_j|^2}}\biggm| |x_j|=1\biggr\}<R\Biggr\}$#\hfil\cr E_O(x,R){}\max\limits_j \thinspace;\cr\noalign{\vskip2\jot}F_O(x,R){}\min\limits_j \thinspace.\cr}}
\mathcal{F}^n_0\times \mathbb{Q}_+(y)^n\buildrel \tau^{B[0]}_{k_0,\varepsilon_0}\over \rightarrow\mathcal{F}^n_0\times \mathbb{Q}_+(y)^n\buildrel \tau^{B[1]}_{k_{1},\varepsilon_{1}}\over\rightarrow\cdots\buildrel \tau^{B[T-1]}_{k_{T-1},\varepsilon_{T-1}}\over \rightarrow\mathcal{F}^n_0\times \mathbb{Q}_+(y)^n,
\left\|\Lambda^m(1\otimes x)(D^\theta+i\lambda)^{-p-1}\right\|_1 \leq 2\sum_{k=1}^p\left\|\Lambda^{m-1}(1\otimes [D_k^\theta,x])(D^\theta+i\lambda)^{-p-1}\right\|_1 +\sum_{k=1}^p\|\Lambda^{m-1}(1\otimes [D_k^\theta,[D_k^\theta,x]])(D^\theta+i\lambda)^{-p-1}\Big\|_1.
g_{\varepsilon}(t,k):=\left(\tilde{a}_{\varepsilon}-a_{\varepsilon}\right)(t) \mathcal{H}_{\hbar,V}\tilde{u}_{\varepsilon}(t, k)+\left(\tilde{q}_{\varepsilon}- q_{\varepsilon}\right)(t) \tilde{u}_{\varepsilon}(t, k)+(f_{\varepsilon}-\tilde{f}_{\varepsilon})(t,k).
\tilde\rho^\eta_{e^{2\pi i t}}\circ \psi(s_\gamma^{a}i_\eta(f) s_\gamma^{*b}) = \rho^\eta_{e^{2\pi i t/N}}(s_\eta^{aN}i_\eta(f) s_\eta^{*bN}) = e^{2\pi i t(a-b)}s_\eta^{aN}i_\eta(f) s_\eta^{*bN}= \psi \circ \rho^\gamma_{e^{2\pi i t}}(s_\eta^{aN}i_\eta(f) s_\eta^{*bN}).
U(z,z') \Psi := \Psi + (-i) \int_{z'}^z d\zeta_1 \, H_1(\zeta_1) \Psi + (-i)^2 \int_{z'}^z d\zeta_1 \int_{z'}^{\zeta_1}d\zeta_2 \, H_1(\zeta_1) H_1(\zeta_2) \Psi + \cdots
\|h\|_{0,q}:=\sup_{(W,\varphi)\in\Omega_{L,q,1}}\left |\int_M h \, \varphi \right | & \|h\|^*_{1,q}:=\sup_{(W,\varphi)\in\Omega_{L,q+1,d}}\left |\int_M h \, \varphi\right | &\|h\|^-_{1,q}:=a\|h\|_{0,q}+\|h\|^*_{1,q},
G_0 =G_{0,0}-G_{0,0}v(I-m_1^\dagger M_2)M_0^\dagger (I- M_2m_1^\dagger)v^*G_{0,0}&-G_{0,0}v\bigl(Q+m_1^\dagger m_1m_1m_1^\dagger &\qquad-m_1^\dagger m_1-m_1 m_1^\dagger \bigr)v^*G_{0,0}&-G_{0,0}vm_1^\dagger v^*G_{0,2}-G_{0,2}vm_1^\dagger v^*G_{0,0}&\quad+G_{0,0}vm_1^\dagger M_4m_1^\dagger v^*G_{0,0}.
\breve{A}_{1}= \beta \int \sqrt{h}d^{D - 1}x \left( \frac 1 6 \breve{R}- V \right) = \beta \int \sqrt{h}d^{D - 1}x \left( \frac 1 6 \bar{R}-{\frac{1}{2 4}}F^{\alpha \beta}F_{\alpha \beta}- V \right) ,
\sum_{x,y \in X}\langle K(x,y)\xi(y) | \xi(x)\rangle & =\sum_{w,z \in X\setminus Y_1}\langle K(w,z)\xi(z) | \xi(w)\rangle & + \sum_{x, x' \in Y_1}\langle K(x,x')\xi(x') | \xi(x)\rangle & + \sum_{x \in Y_1, z \in X \setminus Y_1}2\mathfrak{Re}\langle K(x,z)\xi(z) | \xi(x)\rangle.
\Omega \equiv \sum_{A}r_{A}\, ( d \psi_{A}+ \cos \theta_{A}\, d \psi_{A}) = \frac{i}{4}\sum_{A}\left( \xi_{A}^{*}d \xi_{A}- \xi_{A}d \xi_{A}^{*}- \zeta_{A}^{*}d \zeta_{A}+ \zeta_{A}d \zeta_{A}^{*}\right) ,
\mathbf{II}\leq C_1 \Lambda\int_0^{r_0}\frac{r^n}{t^{n/2}}\cdot \exp\left(-\frac{r^2}{C_1t}\right)\frac{r}{t}dr =C_1\Lambda \int^{r_0t^{-1/2}}_0 r^{n+1}dr \leq C'(n,r_0,v_0,g_0,\delta ) t^{-1-\frac{n}2}
P_{\textrm{out,F}}^{\textrm{ISANC}}=P_{\textrm{out,F}}^{\textrm{CSANC}}= \textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{FDF=0}\right\rbrace \times \textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NDN=0}\right\rbrace &+\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NHF=0,FDF=0,NDF=1,NDN=1}\right\rbrace.
\int_{0}^{\frac{\pi}{2}}sin^{n-2}xcos^{2}xdx=[\frac{sin^{n-1}x}{n-1}cosx]_{0}^{\frac{\pi}{2}}+\frac{1}{n-1}\int_{0}^{\frac{\pi}{2}}sin^{n-1}xsinxdx=0+\frac{1}{n-1}W_{n}.
\|E^{k+\frac{1}{2}}\triangle^{\alpha}D^{\beta}u\| \lesssim \|\dot{D}D^k\triangle^{\alpha}D^{\beta}u\| &\lesssim\sum_{\iota+\nu=k+\alpha}\|\dot{D}\triangle^{\iota}D^{\nu}D^{\beta}u\| &\lesssim\sum_{\iota+\gamma=m}\|\dot{D}\triangle^{\iota}D^{\gamma}u\| &\lesssim \|\dot{D}\triangle^mu\|.
S^{( N )}\left[{\bf r}_{1}, \dots{\bf r}_{N - 1}\right] ({\bf r}^{\prime \prime},{\bf r}^{\prime}; t^{\prime \prime}, t^{\prime}) = \sum_{j = 0}^{N - 1}\left[ \frac{M \left({\bf r}_{j + 1}-{\bf r}_{j}\right)^{2}}{2 \epsilon}- \epsilon V ({\bf r}_{j}, t_{j}) \right] ,
\sum_{j=0}^\infty\left|\mathbb{P}(X>j)-\sum_{k=j+1}^\infty\sum_{i=0}^\infty\mathbb{P}(X=i)P_{i,k}\right|\leq \sum_{j=0}^\infty\sum_{i=0}^\infty\mathbb{P}(X=i)\left|\sum_{k=j+1}^\infty\left(Q_{i,k}-P_{i,k}\right)\right|.
S_1 = \alpha^4 b_1+\alpha^3 b_2 +\alpha^2 b_8 + \alpha b_{9}+ b_{10}, S_2 = \alpha^4 b_{11}+\alpha^3 b_{12}+\alpha^2 b_{13}+ \alpha b_{14}+ b_{15}, S_5 = \alpha^4 b_{21}+\alpha^3 b_{22}+\alpha^2 b_{23}+ \alpha b_{24}+ b_{25}, S_6 = \alpha^4 b_{26}+\alpha^3 b_{27}+\alpha^2 b_{28}+ \alpha b_{29}+ b_{30}, S_7 = \alpha^4 b_{31}+\alpha^3 b_{32}.
\int_0^1 F\bigl(d_{[\lambda,W]}(x)\bigr)\;\mathrm{d}x =\lambda F(0)+\int_{\lambda}^1 F\left((1-\lambda)d_{W}\left(\frac{x-\lambda}{1-\lambda}\right)\right)\;\mathrm{d}x &=\lambda F(0)+(1-\lambda)\int_0^1 H\bigl(d_W(x)\bigr)\;\mathrm{d}x\,,
S^2(f,g) (x) \geq \int_{0}^{1}|x-xy|^{-1/2}\left(-\log(|x-xy|)\right)^{-1}\frac{dy}{\sqrt{1-y^2}}&\geq \frac{\sqrt{x}}{\sqrt{2}}\int_{0}^{1}(x-xy)^{-1}\left(-\log(x-xy)\right)^{-1}dy &\geq \frac{1}{\sqrt{2x}}\int_{0}^{x}u^{-1}\left(-\log(u)\right)^{-1}du=+\infty,
y_k^c(x)=\cos (kx)+R_k(x), \quad\|R_k\|_{L^\infty}= O\Big(\frac1k\Big) k\to+\infty. \lambda_k^c =k^2+\widetilde a +\frac{1}{2}a_{c,2k}+O\Big(\frac1k\Big) k\to+\infty.
curl\vec{v}=(\frac{\partial v_{z}}{\partial y}-\frac{\partial v_{y}}{\partial z})\vec{e}_{x}+(\frac{\partial v_{x}}{\partial z}-\frac{\partial v_{z}}{\partial x})\vec{e}_{y}+(\frac{\partial v_{y}}{\partial x}-\frac{\partial v_{x}}{\partial y})\vec{e}_{z}=\nabla\times\vec{v}
\Big\langle \tau_{0}(\gamma_1) \dots \tau_{0}(\gamma_n) \Big\rangle_{g,\beta}^{X, \textup{red}}=\mathsf{N}_{g, \beta}\cdot \int_{t_{\ast}([X] \otimes \beta^{\otimes n})}\gamma_1 \otimes \ldots \otimes \gamma_n \,.
v_j(C_m(t)) = \frac{\Gamma(\frac{m}{2})\,t^j}{2\,\Gamma(\frac{j+1}{2})\Gamma(\frac{m-j+1}{2})\,(1+t^2)^{(m-2)/2}}, j=1,\ldots,m-1, v_m(C_m(t)) = \frac{\Gamma(\frac{m}{2})}{\sqrt{\pi}\,\Gamma(\frac{m-1}{2})}\int_0^t \frac{\tau^{m-2}}{(1+\tau^2)^{n/2}}\,d\tau .
(-1)^{j+r-1-l}\frac{1}{2}S(n,r)\binom{2n-j}{n}\frac{\binom{2(j+r-1-l)}{j+r-1-l}\binom{2(n-j+l+1)}{n-j+l+1}\binom{n-j}{j+r-l-1}}{2\binom{2n-j+l+1}{n}}
P(\{\sup_{0\leq t\leq T}|\mathbb{\hat{E}}_{t}[\xi^{n}]-M_{t}| >\varepsilon\}) =P(\{\sup_{0\leq t\leq T}|\mathbb{\hat{E}}_{t}[\xi^{n}]-\mathbb{\hat{E}}_{t}[M_{T}]|>\varepsilon\}) & \leq P(\{\sup_{0\leq t\leq T}\mathbb{\hat{E}}_{t}[|\xi^{n}-M_{T}|]>\varepsilon\}) & \leq\frac{1}{\varepsilon}\mathbb{\hat{E}}[|\xi^{n}-M_{T}|].
j^{\epsilon,s}(\{u_n\}_{n=0}^N)=\sum\limits_{n=0}^{N-1}j(u_n)\chi_n(x_3), J^{\epsilon,s}(\{u_n\}_{n=0}^N)=\sum\limits_{n=0}^{N-1}J(u_n)\chi_n(x_3),
\sum_{i}p_{i}^{\prime}=\sum_{i}m_{i}v_{i}^{\prime}=\sum_{i}m_{i}(v_{i}-V_{c})=\sum_{i}m_{i}v_{i}-\sum_{i}m_{i}\frac{\sum_{j}m_{j}v_{j}}{\sum_{j}m_{j}}=\sum_{i}m_{i}v_{i}-\sum_{j}m_{j}v_{j}=0.
L_\tau^{k+1}f|_{I_n}(t_{n,\mu}) = f(t_{n,\mu})\,, \operatorname{for}\;\; \mu = 0,\ldots, k\,, \partial_t L_\tau^{k+1}f|_{I_n}(t_{n-1}) = \partial_t f(t_{n-1})\,,
S = \int d t d^{2}x \left[ - h b_{0}{\tilde{J}}_{0}- h b_{k}{\tilde{J}}_{k}+ \frac{m}{2 \rho_{b}}{\vec{J}}_{b}^{2}- \frac{\hbar^{2}}{8 m \rho_{b}}\partial_{j}\rho_{b}\partial_{j}\rho_{b}- V ( \rho_{b}) \right] ,
\partial_{t}u^{\varepsilon}=\mathcal{J}_{\varepsilon}^{2}\partial_{x}^{5}u^{\varepsilon}-\mathcal{J}_{\varepsilon}(2(\mathcal{J}_{\varepsilon}u^{\varepsilon})(\mathcal{J}_{\varepsilon}u^{\varepsilon}_{x}))+\omega\mathcal{J}_{\varepsilon}^{2}u^{\varepsilon}_{x}.
M\left((1+x^2)(1+x(x^3+1))+t \:h(x) +y\left((1+x^3)(1+x^2)+ t\:h(x)\right)\right) = 2^4(5+6t), & M\left((1-x)+t \: h(x) +y\left((1+x^2)(1+x^3)+t\:h(x)\right)\right)=-2^6(1+3t),
B_2(w,z,x;t)=2 \widetilde{f}_2(w,z,x;t)=f_2(w,z,x;t)+f_2(z,w,x;t) & =\int_{0<\theta<r<t}G_{t-r}(x-z)G_{r-\theta}(z-w)drd\theta+\int_{0<r<\theta<t}G_{t-\theta}(x-w)G_{\theta-r}(w-z)drd\theta.
f ( p , k ) ={\frac{2 k^{3}( 4 \alpha^{3}p^{3}+ 4 \alpha^{2}p^{4}+ \alpha^{2}p^{2}k^{2}- 3 p^{4}k^{2}+ \alpha^{2}k^{4}+ 3 \alpha p k^{4}+ 3 p^{2}k^{4})}{( 3 \alpha ( \alpha p + p^{2})^{3})}}\Theta ( p - k ) ,
|\partial^\alpha Ta(x)-\partial^\alpha Ta(x+h)| &\quad=\left|\int_0^1h\cdot\nabla\partial^\alpha Ta(x+th)\,dt\right| &\quad\lesssim|h|\int_0^1\max_{i\in\{1,\ldots,n\}}|\partial^{\alpha+e_i}Ta(x+th)|\,dt &\quad\lesssim|h|\int_0^1|x+th|^{-M}\,dt\lesssim|h|^{N^{**}}|x|^{-M}.
\frac{1}{g(E_i,E_i)}Jd\varphi([E_i^\prime,E_i])=\frac{1}{g(E_i^\prime,E_i^\prime)}\nabla_{E_i^\prime}^\varphi d\varphi(E_i^\prime)+\frac{1}{g(E_i,E_i)}\nabla_{E_i}^\varphi d\varphi(E_i)
\bar{R}_k = \sum_{i \in \mathcal{I}}\pi_i R_k^i = \sum_{i \in \mathcal{I}}\pi_i \sum_{b \in \mathcal{B}}\alpha_{kb}\left(\frac{W \bar{r}_{kb}^i}{\sum_{j \in \mathcal{K}}\alpha_{jb}}\right).
B_\lambda(s)=i^{nk}\,\frac{\pi^{n(n+1)/2}}{\prod_{m=1}^{n}(m-1)!}2^{-n(2n+1)s+3n/2}\,\gamma_n\Big((2n+1)s-\frac12+k\Big)\prod_{j=1}^n\frac{\beta(\ell_j,s)}{\beta(k-j,s)}.
\tilde{a}^\mu (\vec{k})^K =\tilde{U}a^\mu (\vec{k})^K \tilde{U}^{-1}= a^\mu (\vec{k})^K + i [\tilde{Q}, a^\mu (\vec{k})^K] =a^\mu (\vec{k})^K + q^\mu (\vec{k})^* \, .
\frac{\Delta(A,C)}{\Delta(A',C')}=\prod_{i,j}\frac{\left(a+\left\lfloor\frac{\lfloor xN\rfloor+\lfloor zN\rfloor}{2}\right\rfloor\right)+\gamma_j-\alpha_i}{\left(a+\left\lfloor\frac{\lfloor xN\rfloor+\lfloor zN\rfloor}{2}\right\rfloor\right)+\gamma'_j-\alpha'_i}.
\frac{d}{ds}E_0(\mu(\cdot,s))=\left.\int_M\phi\partial_s\rho e^{-f}\, dV\right|_{t=t'}^{t''}-\int_{t'}^{t''}\int_M\left[\partial_t\phi+\frac12|\nabla\phi|^2 -\frac12 R^{H,f}\right] \partial_s\rho e^{-f}\, dVdt.