harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{\bf Q}_{e_j}=\kappa\tilde{\bf C}_{x_j}=\kappa\begin{bmatrix}{\rm diag}(\mathbf{C}_{x_j}) \hat{\mathbf{C}}_{x_j}\hat{\mathbf{C}}^{*}_{x_j}{\rm diag}(\mathbf{C}^{*}_{x_j})\end{bmatrix}.
\hat h^n_{n-j}(\theta) =\sum_{s=0}^{\infty}\int_{-1}^1 h^n_{n-j}(t)(-i\theta)^s \frac{t^s}{s!}dt &=(-i\theta)^{n+2}\sum_{r=0}^{\infty}(-i\theta)^{2r}\int_{-1}^1 h^n_{n-j}(t)\frac{t^{n+2+2r}}{(n+2+2r)!}dt
P_{r,N}(A(t)) = N^{-r/2}\sum_{j_1, j_2, \cdots, j_r =1}^N \big[G_{j_r}(t_r) \otimes B_{j_r}(t_r),[\ldots,[G_{j_1}(t_1) \otimes B_{j_1}(t_1), A(t)]\ldots ]\big].
s_1 = 2 s_2 = 3 s_3 = 7 s_4 = 43 s_5 = 1807 s_6 = 3263443 s_7 = 10650056950807 s_8 = 113423713055421844361000443 s_9 = 12864938683278671740537145998360961546653259485195807.
\sum_{\vert k\vert \leq n}s_H(k) = \frac{1}{4}\sum_{\vert k\vert \leq n}\big( \vert k+1\vert^{2H}+ \vert k-1\vert^{2H}- 2 \vert k \vert^{2H}\big) = \frac{1}{2}\big( \vert n+1\vert^{2H}- \vert n \vert^{2H}\big)
p_0+ap_1+a^2 p_2 = f(a), \cr p_1+(b-a)^{-1}(b^2-a^2) p_2 = (b-a)^{-1}(f(b)-f(a)), \cr \{(c-a)^{-1}(c^2-a^2)-(b-a)^{-1}(b^2-a^2)\}p_2 = (c-a)^{-1}(f(c)-f(a))-(b-a)^{-1}(f(b)-f(a)),
\begin{aligned}0 \geq \sum_p\frac{F^{kk}}{20K}(|u_{pk}|^2 + |u_{p\bar k}|^2) +\frac{1}{2}\psi'' F^{kk}|u_k|^2 + \psi' F^{kk}u_{k\bar k}&- 2\psi'^2\delta^{-1}F^{11}K - C\mathcal{F}+ C_\epsilon\psi'F^{kk}|u_k|^2 + \epsilon C\psi'\mathcal{F}. \end{aligned}
\sum_{j=2}^{m}- 2\beta_{j}\partial_{s_{k}}\log \Lambda_{j}= -2 \sum_{j=2}^{m}\beta_{j}\partial_{s_{k}}(\beta_{j}) \log \big(4\sqrt{r(x_{\smash{j}}-x_{1})}\big) -2\sum_{j=2}^{m}\beta_{j}\sum_{\substack{\ell = 2 \ell \neq j}}^{m}\partial_{s_{k}}(\beta_{\ell})\log(\widetilde{T}_{\ell,j}).
\lambda^2-\lambda\bigl\{(a_2 + a_3)\tilde{m}_1^2+ (a_1 + a_3)\tilde{m}_2^2+(a_1 + a_2)\tilde{m}_3^2)\bigr\}+\bigl(a_1a_2\tilde{m}_3^2+a_2a_3\tilde{m}_1^2+a_1a_3\tilde{m}_2^2\bigr) =0.
\frac{E |S_n(x_{n_i})|^k}{b_n^k \epsilon^k}= \frac{1}{2^k}\frac{E |S_n(x_{n_i}) - \tilde{S}_n(x_{n_i})+ S_n(x_{n_i}) + \tilde{S}_n(x_{n_i})|^k}{b_n^k \epsilon^k},
\int_{G_{N}}f(x)\sum_{i=1}^{N}M_{\epsilon,i}(x)dx =\sum_{i=1}^{N}\int_{G_{N}}\left[\nabla_{i}f(x)\right]V_{i}(x)dx+\epsilon\sum_{i\neq j}\int_{F_{ij}}\left(f_{ij}(x)+f_{ji}(x)\right)dS_{ij}(x) \le(N-1)\epsilon\sum_{i=1}^{N}\int_{G_{N}}\left|\nabla_{i}f(x)\right|dx+\epsilon\sum_{i\neq j}\int_{F_{ij}}\left(f_{ij}(x)+f_{ji}(x)\right)dS_{ij}(x)
\mathcal{G}(u)(r,\zeta):= 1+\frac{1}{4\pi}\int_{-1}^1\int_0^{R_0}K(r,\zeta,r',\zeta')u_{\sharp}(r',\zeta')^{\nu}r'^2dr'd\zeta' + &-\frac{1}{2}\int_{-1}^1\int_0^{R_0}u_{\sharp}(r',\zeta')^{\nu}r'dr'd\zeta'.
\begin{cases}\big[(e_1+ ie_2)\cdot \nabla_{t,s}\big((at+bs)-i(bt-as)\big)\big]=(a-ib)+i(b+ia) =0, begin{align*}2mm]\left((e_1+ie_2)\cdot\nabla_{t,s}\right)^2\big((at+bs)-i(bt-as)\big) =0.\end{cases}
\mathcal{L}_{I_{cc}}(\frac{n \eta_s \Gamma r_0^{\alpha_{s,c}}}{P_{c}G_0})=\exp\bigg(-2\pi\lambda_B\bigg (\sum_{j \in \{L,N\}}\sum_{i=1}^3 p_{G_i}\times &\bigg(\int_0^{\infty}\bigg(1-1/\bigg(1+\frac{n \eta_s \Gamma r_0^{\alpha_{s,c}}G_i}{G_0N_jt^{\alpha_{j,c}}}\bigg)^{N_j}\bigg)p_{j,c}(t)tdt\bigg)\bigg)\bigg)
\mathbb{P}_{\!x}\{X_{t_1}\in \mathrm{d}x_1,\dots,X_{t_n}\in \mathrm{d}x_n\}=\left(\prod_{k=1}^n p(t_k-t_{k-1};x_{k-1},x_k)\right)\mathrm{d}x_1\dots\mathrm{d}x_n.
X_{t \wedge \tau_m}= S_{t \wedge \tau_m}X_0 + \left(G_1 b^{n}(X_{\cdot \wedge \tau_m})\right)(t \wedge \tau_m) + \tfrac{\sin(\pi \theta)}{\pi}\big(G_\theta Y^{n, m, \theta}\big)(t \wedge \tau_m),t \in [0, T].
{\cal L}= \sum_{i , \bar{j}= 1}^{n}\left\{G_{i \bar{j}}\, \partial_{\mu}\Phi^{i}\, \partial_{\mu}\bar{\Phi}^{\bar{j}}+ G^{i \bar{j}}\frac{\partial{W}}{\partial \Phi^{i}}\, \frac{\partial \bar{W}}{\partial \bar{\Phi}^{\bar{j}}}\right\}+ \mathrm{f e r m i o n s}\, ,
\sup_{-\rho^{2}\leq t<0}\int_{B(\rho )}|u|^{2}dx+ \int_{Q (\rho)}|\nabla u|^{2}dxdt &\lesssim \left(\int_{Q(2\rho)}|u|^{10/3}dxdt\right)^{3/5}+ \int_{Q(2\rho)}|u|^{2}u \cdot \nabla \phi dxdt &+ \int_{Q(2\rho)}(\Pi-\overline{\Pi}_{2\rho}) u \cdot \nabla \phi dxdt
\left|\frac{1}{4\pi}\int\limits_{\partial\sum_{n}}\left(f_0(M)\right)'_{\bar{n}(M)}\, \frac{1}{\rho_{M_0}(M)}\, dS(M) \right| \leq C\alpha_{n}\cdot 2^{n}\cdot 2^{-n}=C\alpha_{n}
A(0,\theta) = \frac{\pi \alpha}{2}- \frac{\alpha}{2}\left[ \frac{\cos(8g \pi \theta)}{(2g) 2 \pi \theta}- \frac{\sin(8g \pi \theta)}{2g}\right] + O\Big(g^{-1}\theta \alpha^3 + g^{-2}\theta^{-2}\alpha \Big) .
\lim_{n\to\infty}F_n^{(1)}(z)=\frac{P_0^{(1)}(z)}{(z-\l_0)^{k_1}},\lim_{n\to\infty}F_n^{(2)}(z)=\frac{P_0^{(2)}(z)}{(z-\l_0)^{k_2}}
\|u\|_{L^{2^\ast}(B_r(x_0))}=\|\Psi_\lambda\ast f\|_{L^{2^\ast}(B_r(x_0))}\leq \|\Psi_\lambda\|_{L^{\frac{N}{N-2},w}(B_{r+r_0}(x_0-y_0))}\|f\|_{(2^\ast)'},
\sum_{j=1}^n \theta_{\alpha}\left(x - \frac{j}{n}\right) = n\sum_{k \in \mathbb{Z}}e^{-\pi \alpha k^2 n^2}e^{2 \pi i k n x}=n + 2 n e^{-\pi \alpha n^2}\cos\left( 2 \pi n x \right) + \mathcal{O}(n e^{-4 \pi \alpha n^2}).
A=\begin{bmatrix}a b\pi^n c d\end{bmatrix}= \underbrace{\begin{bmatrix}e f\pi^n g h \end{bmatrix}}_B \underbrace{\begin{bmatrix}p q\pi^n r s \end{bmatrix}}_C = \begin{bmatrix}ep + fr \pi^n (eq + fs) \pi^n gp + hr gq\pi^n + hs \end{bmatrix},
\frac{\mathrm{d}}{\mathrm{d}t}\left(\alpha\|\nabla^3u\|_{L^2}^2+\kappa\|\nabla^3\tau\|_{L^2}^2\right) +\kappa\mu\|\nabla^4\tau\|_{L^2}^2+\kappa\beta\|\nabla^3\tau\|_{L^2}^2 \leq B_1\varepsilon_0 \Big( \|\nabla^3 u \|_{L^2}^2 + \|\nabla^3 \tau \|_{L^2}^2\Big).
\frac{\|\alpha/\pi \|_2^2-1}{\|\alpha/\pi \|_2^2}-\pi(A) =2 \sum_x \pi(x) \left( \mathbb{P}_x[T_A \le t_{\mathrm{med}}] \right)^2-2 \frac{1}{\|\alpha/\pi\|_2^2}(1-e^{-\lambda_1 t_{\mathrm{med}}})^{2}.
\hat H^{(q)}_I(\hat x,\hat p)=\frac{f^2\hat x^2}{6}\bigl\{\hat x V'(\hat x) \partial^2_{\hat x}+ \bigl[\hat x V^{''}(\hat x) +3 V^{'}(\hat x)\bigr]\partial_{\hat x}+{\textstyle\frac{1}{3}}\hat x V^{'''}(\hat x) +{\textstyle\frac{3}{2}}V^{''}(\hat x)\bigr\}.
\sum_{r=1}^{\lfloor p^{l}/t\rfloor}(p^{l}-tr)^{2k}\equiv \frac{t^{2k}}{2k+1}\{\frac{2k+1}{t}p^{l}B_{2k}-B_{2k+1}(\frac{s}{t})\}\pmod{p^{3l-1}},
b_h(u,v_h) = \int_{Q}\left(\partial_{t}u v_h + \theta h \partial_{t}u \partial_{t}v_h + \nabla_x u \cdot \nabla_x v_h -\theta h \partial_{t}\nabla_x u \cdot \nabla_x v_h \right) \,dxdt + \theta h \int_{\Sigma_T}\nabla_x u \cdot \nabla_x v_h \,ds
\tilde{deg}T \equiv deg [(J^1_3)^{n_{13}}(J^1_2)^{n_{12}}(J^2_3)^{n_{23}}(J^3_2)^{n_{32}}(J_d)^{n_{d}}({\tilde J}_d)^{n_{\tilde d}}(J^3_1)^{n_{31}}(J^2_1)^{n_{21}}] \ =
\sum_{n=2}^{\infty}\Vert \alpha_{n}(x(u,0)) \Vert \vert z\vert^{n}\leq c^{\prime}\Vert \alpha_{0}(x(u,0)) \Vert \left( \frac{\vert z\vert}{\delta_{0}}\right)^{2}\left( 1-\frac{\vert z\vert}{\delta_{0}}\right)^{-2\lambda-2}.
\overline{\left [\begin{matrix}p 1 a_{21}a_{22}+p\end{matrix}\right ]}=\left [\begin{matrix}p 1 a_{21}a_{22}+p\end{matrix}\right ]+\left (v^{- a_{22}-1}\ \overline{[p+1]}- v^{a_{22}+1}[p +1]\right )\left [\begin{matrix}p+1 0 a_{21}-1 a_{22}+p+1\end{matrix}\right ].
+{\frac{i}{2}}\sum_{i = 2}^{h}\int_{{\cal C}_{i}}{\log}| L_{i}^{\prime}|^{2}{\frac{\overline{{L}}_{i}^{\prime \prime}}{\overline{{L}}_{i}^{\prime}}}d \bar{z}+ 4 \pi \sum_{i = 2}^{h}{\log}\left|{\frac{( 1 - \lambda_{i})^{2}}{\lambda_{i}( a_{i}- b_{i})^{2}}}\right| ,
\lambda_{k}= \lambda_{(s)}(\mathbf{x}^{k})=2(1-\beta) \varphi_{s}(\mathbf{x}^{k})/(\|\mathbf{d}^{k}_{s}\|^{2}L_{s}) \geq2(1-\beta) \varphi_{s}(\mathbf{x}^{k})/(\rho^{2}L),
\nabla^{2}\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{\beta}(t-s, y-x)H(s,x,\nabla f^{*}(t,x)) dx ds = \int_{0}^{t}\int_{\mathbb{R}^{d}}\nabla_{y}G_{\beta}(t-s, y-x)\nabla_{x}H(s,x,\nabla f^{*}(s,x))dx ds.
D_1^\alpha\big(\sum_k (f \ast \psi_k \cdot g \ast \psi_k)\ast \varphi_k(x)\big)=\sum_n \sum_k c_{n, k}2^{k \alpha}\left( f \ast\psi_{k, n}\cdot g \ast\psi_{k, n}\right) \ast \tilde \varphi_k(x) &=\sum_n \sum_k c_{n, k}\big( \big(D_1^\alpha f\big) \ast \tilde{\tilde{\psi}}_{k, n}\cdot g \ast\psi_{k, n}\big) \ast \tilde \varphi_k(x).
0 = \int_{T^d}(u(T,x)-\tilde u(T,x))(m(T,x)-\tilde m(T,x)) dx - \int_{T^d}(u(0,x)-\tilde u(0,x))(m(0,x)-\tilde m(0,x))\Bigr) dx & + \int_{t=0}^T E [ m(t,\cdot),\nabla u(t,\cdot), \tilde m(t,\cdot), \nabla \tilde u(t,\cdot)] dt .
c^k_{\mathrm{s2b},3}\left(N_{\mathrm{sct}}\right) = \mathsf{E}\left\lbrace B \sum_{i=1}^{N_{\mathrm{sct}}}\log_2\left[ \frac{P_{\mathrm{sc}}}{N_{\mathrm{sct}}}\left(B\sigma^2_{\mathrm{bs}}\|\mathbf{g}^\mathrm{T}_j \|^{2}\right)^{-1}+1 \right] \right\rbrace,
0=\frac{\partial{\cal L}}{\partial W_\mu^I}=2\partial_\nu\frac{\partial{\cal L}}{\partial{\cal F}^I_{\mu\nu}}= 2\partial_\nu\left( \frac{\partial{\cal L}}{\partial{\cal F}^{+I}_{\mu\nu}}+ \frac{\partial{\cal L}}{\partial{\cal F}^{-I}_{\mu\nu}}\right)\ ,
\mathbf x'^r = h_r^{-1}\left([h_r\Theta \mathbf f^r - h_r \mathbf d^r] \bmod h_r \Lambda_{s,m}+ \mathbf s^r\right) =[\Theta \mathbf f^r - \mathbf d^r] \bmod \Lambda_{s,m}+ h_r^{-1}\mathbf s^r.
\lim_{n\rightarrow\infty}\frac{k_j+k_j^{3/4}}{n}= R_j\frac{1-\frac{\prod_{k=1}^K\delta_k}{\delta_j}}{1-\prod_{k=1}^K\delta_k}, \lim_{n\rightarrow\infty}\frac{n_1}{n}= \max_{j\in\{1,\dots,K\}}\frac{R_j(1-\frac{\prod_{k=1}^K\delta_k}{\delta_j})}{(1-\delta_j)\frac{\prod_{k=1}^K\delta_k}{\delta_j}(1-\prod_{k=1}^K\delta_k)}.
u_R(x_1) \le c_R(x_1,\overline x_2, \dots, \overline x_N) - \frac{N-1}Nc_R(\overline x_1,\ldots, \overline x_N) & \le \frac{N(N-1)}{2}f(\frac{\alpha}2)- \frac{N-1}Nc_R(\overline x_1,\ldots, \overline x_N) =: M,
| \Psi_{( n , 0 )}\rangle = \int_{0}^{P^{+}}{\frac{d q_{1}\dots d q_{n}}{\sqrt{q_{1}. . . q_{n}}}}\delta ( q_{1}+ . . . + q_{n}- P^{+}) \hspace{1 m m}f^{( n , 0 )}( q_{1}. . . q_{n}) \mathrm{t r}[ a^{\dagger}( q_{1}) . . . a^{\dagger}( q_{n}) ] | 0 \rangle .
d^{nq+n_j}G_{r_n(x)}\left(\tilde{a}_j\circ r_n(x)\right) =G_{r_n(x)}\left(F_{r_n(x)}^{n_j+nq}\left(\tilde{a}_j\circ r_n(x)\right)\right) & = G_{r_n(x)}\left(\sigma_j\circ a^{n,j}(x)\right)+\log|u_{n,j}(x)|,
C_{\underline{{\alpha}}\underline{{\beta}}}= C^{\underline{{\alpha}}\underline{{\beta}}}= \left( \begin{array}{c c}{{0}}&{{\delta_{\beta}^{\alpha}\delta_{q}^{r}}}\\{{- \delta_{\alpha}^{\beta}\delta_{r}^{q}}}&{{0}}\\ \end{array}\right) \, .
T(2n,k)=\frac{(-1)^{n-k}\pi^{2n}}{4^n (2n)!}\sum_{p=k}^n \binom{2n}{2p}\binom{p}{k}E_{2n-2p}; &T^{\star}(2n,k)=\frac{(-1)^n\pi^{2n}}{4^n (2n)!}\sum_{q=k}^n \binom{2n}{2q}\binom{q}{k}E_{2q}.
C ( 1 , 1 ) = \sum_{n = 0}^{\infty}\frac{( - 1 / 2 )^{3 n}}{n + \frac 1 2}\left( \frac{1}{n + \frac 1 2}- 3 \log 2 - \sum_{m = 1}^{n}\frac{3}{m}\right)
\psi_{k,k+1}^{\ast}(Q^{\prime})=\frac{L_{k,k+1}^{\ast}(q^{\prime})}{q^{\prime}}\geq -\frac{(k-n+1)\Phi_{k,n}(\psi_{n,1}^{\ast}(Q))+\Phi_{k,n}(\psi_{n,2}^{\ast}(Q))}{n-1}-O(q^{\prime -1}),
\frac{\vec N_{k,t}}{V_t}\longrightarrow \frac{1}{1-\alpha}\frac{\Gamma(k)\Gamma\Big(1+\frac{1}{1-\alpha}\Big)}{\Gamma\Big(k+1+\frac{1}{1-\alpha}\Big)}=\frac{1}{1-\alpha}B\Big(k,1+\frac{1}{1-\alpha}\Big),
V_{\varphi}(t,x^{(t)},m^{(t)}) & =\inf_{w_t \in \widehat{\mathcal{R}}_t}\mathbb{E}_{\varphi}\left[\sum_{s=t}^{T-1}f(s,X_s,\mu_s,w_t(s,X^{(s)},\mu^{(s)}))+ F(X_T,\mu_T)| X^{(t)}=x^{(t)}, \mu^{(t)}=m^{(t)}\right],
\phi \left( x_{1}, \cdots , x_{2 l}\right) = \sum_{n_{1}, \cdots , n_{1}\geq 0}\varphi_{\left( n_{1}, \cdots , n_{l}\right)}P_{\left( n_{1}, \cdots , n_{l}\right)}\left( x_{1}, \cdots , x_{2 l}\right) ;
I_{11}=\{d+b_5+b_9+1,\ldots,d+a_1-b_6-b_8,d+a_1+1,\ldots,\overline{d+a_1+1}, & \qquad\overline{d+a_1-b_6-b_8},\ldots,\overline{d+b_5+b_9+1}\}.
\int_{u}^{u+T_2(\epsilon)}M(s-u,s)\,ds =\int_{0}^{T_2(\epsilon)}M(v,v+u)\,dv=\int_{0}^{T_2(\epsilon)}\int_{[v,v+u]}\mu(dw)\,dv &=\int_{[0,T_2(\epsilon)+u]}\left( w\wedge T_2(\epsilon) - (w-u)\vee 0\right) \mu(dw) &=\int_{[0,T_2(\epsilon)]}w\mu(dw) &\qquad+\int_{(T_2(\epsilon),T_2(\epsilon)+u]}\left(T_2(\epsilon) - (w-u)\vee 0\right) \mu(dw).
\Bigl [ \prod_{a}\Delta^{2}( \varphi_{a}) \Bigr ] \, \times \, \exp \Bigl \{-{\frac{1}{4}}{\frac{N}{L}}\Bigl ( \sum_{l = 1}^{L - 1}{\frac{1}{\sin^{2}{\frac{\pi l}{L}}}}+{\frac{1}{1 2}}\Bigr ) \sum_{a , \alpha}\varphi_{a , \alpha}^{2}+ O ( \varphi^{4}) \Bigr \}~ ~ .
\langle \Psi , \Phi \rangle = \int_{0}^{\infty}d H \Psi_{+}^{*}( H ) \Phi_{+}( H ) + \int_{0}^{\infty}d H \Psi_{-}^{*}( H ) \Phi_{-}( H ) + \sum_{n = 0}^{\infty}\Psi_{n}^{*}\Phi_{n}.
T^{(m+n)}_{cl}(z)=\sum_{i=1}^{p-1}{1-q_i^2\over 2h(z-z_i)^2}-{1\over 2\pi h}\sum_{i=1}^{p-3}\left({1\over z-z_i}+{z_i-1\over z}-{z_i\over z-1}\right){\partial S^{(0,m+n)}_{cl}\over\partial z_i}.
d\{f_1,f_2\}=-dL_{x_1}f_2 +(-1)^{(|x_1|-1)(|x_2|-1)}dL_{x_2}f_1 &= -(-1)^{|x_1|-1}L_{x_1}df_2 -(-1)^{(|x_1|-1)(|x_2|-1)+|x_2|}L_{x_2}df_1 &= +(-1)^{(|x_1|-2)(|x_2|-1)}L_{x_2}df_1 -(-1)^{|x_1|-1}L_{x_1}df_2 &= \{df_1,f_2\}- (-1)^{(|x_1|-1)(|x_2|-1)}\{df_2,f_1\}&= \{df_1,f_2\}- (-1)^{(|f_1|+n-1)(|f_2|+n-1)}\{df_2,f_1\}\;.
G_{k,1}=\int_0^vv'^kg_0(v')dv' &=\frac{1}{k+1}\left(v^{k+1}g_0-\int_0^vv'^{k+1}\frac{dg_0}{dv}(v')dv'\right) &=\frac{1}{k+1}\left(v^{k+1}g_0-g_{k+1}\right),
\begin{pmatrix}2 \xi_1^2 + \xi_2^2 - \partial_3^2 \xi_1 \xi_2 - \mathrm{i}\xi_1 \partial_3 begin{align*}2pt] \xi_1 \xi_2 \xi_1^2 + 2 \xi_2^2 - \partial_3^2 - \mathrm{i}\xi_2 \partial_3 begin{align*}2pt]- \mathrm{i}\xi_1 \partial_3 - \mathrm{i}\xi_2 \partial_3 \xi_1^2 + \xi_2^2 - 2 \partial_3^2 . \end{pmatrix}\hat u (\xi, x_3 ) = \omega \hat u (\xi, x_3 ) ,
\gamma_1 = \{e^{it}\colon \theta-2\delta \le t\le \theta - \delta \}&\gamma_2 = \{e^{it}\colon \theta-\delta \le t\le \theta - \delta/2 \}&\gamma_3 = \{e^{it}\colon \theta+\delta/2 \le t\le \theta + \delta \}&\gamma_4 = \{e^{it}\colon \theta+\delta \le t\le \theta + 2\delta\}
(\varepsilon'_\#\alpha_\#)(v) =\begin{cases}\alpha(v) \alpha(v)\in V_1(\varepsilon'); x_{e'}^{-1}\alpha(v)x_{e'}\alpha(v)\not\in V_1(\varepsilon').\end{cases}(\alpha_\#\varepsilon_\#)(v) =\begin{cases}\alpha(v) v\in V_1(\varepsilon); \alpha_\#(x_{e}^{-1}vx_{e}) v\not\in V_1(\varepsilon).\end{cases}
\int_{-\infty}^\infty u^h \,d\sigma_x(u) = \frac{1}{x}\sum_{n \leq x}\bigg( \frac{P_n(x) - D(x)}{\sqrt C(\log\log x)^{3/2}}\bigg)^h = \frac{M_h(x)}{x C^{h/2}( \log\log x )^{3h/2}}.
\beta' =\Bigl(\prod_{k=1}^\infty(1-X^{2k})(1+X^{2k-1})^2\Bigr)'=\beta\sum_{k=1}^\infty\Bigl(2\frac{(2k-1)X^{2k-2}}{1+X^{2k-1}}-\frac{2kX^{2k-1}}{1-X^{2k}}\Bigr) \gamma' =\Bigl(\prod_{k=1}^\infty(1-X^{2k})(1+X^{2k})^2\Bigr)'=\gamma\sum_{k=1}^\infty\Bigl(2\frac{2kX^{2k-1}}{1+X^{2k}}-\frac{2kX^{2k-1}}{1-X^{2k}}\Bigr)
(\{y_j=S_{x,\tau}^{-1}x_j+Ve_j-V\theta_\tau S_{x,\tau}^{-1}x_j\}_{j\in \mathbb{J}},\{\omega_j=S_{x,\tau}^{-1}\tau_j+Ue_j-U\theta_xS_{x,\tau}^{-1}\tau_j\}_{j\in \mathbb{J}})
\left\vert \widetilde{\varphi}_{0}^{-}\left( z(\rho)\right) \right\vert=c_{l}\left\vert \frac{1}{2}-i\rho\right\vert^{l+1}\left\vert J_{l}\left(\rho\right) \right\vert \leq\operatorname*{Const}\left( \left( 1+\left\vert \rho\right\vert\right)^{\frac{5}{2}l-\frac{1}{2}-2\alpha}\mathbf{L}(\rho)+1\right)
\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{FDF=0}\right\rbrace =1-e^{-\frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^{2}\left(2^{R}-1\right)}{\lambda_{\textrm{BF}}\left(P_{\textrm{F}}-P_{\textrm{N}}\left(2^{\textrm{R}}-1\right)\right)}}.
\aligned -L_{2}y(t) = -L_{2}y\left(t_{1}\right)+\int_{t_{1}}^{t}q(s)y^{\gamma}\left(\sigma(s)\right)s\geq \int_{t_{1}}^{t}q(s)y^{\gamma}\left(\sigma(s)\right)s &\geq y^{\gamma}\left(\sigma(t)\right)\int_{t_{1}}^{t}q(s)s,\endaligned
((x+z)^n+c,0)(z,x+z)^l=((x+z)^n+c,0)((x+z)^l,z(x+z)^{l-1}) = ((x+z)^{n+l},z(x+z)^{n+l-1})+ c((x+z)^l,z(x+z)^{l-1})) \in \textrm{Diag}(A)
\langle 0_{M}| S^{( + )}( \epsilon ) | 0_{M}\rangle = - 2 \sum_{\Omega , \vec{k}}\left[ \sinh^{2}\epsilon ( \Omega ) \ln \sinh^{2}\epsilon ( \Omega ) - \cosh^{2}\epsilon ( \Omega ) \ln \cosh^{2}\epsilon ( \Omega ) \right] \, ,
V_{t}-\frac{V_{1}}{\frac{2}{N^{2}}}=(1-\frac{2}{N^{2}})(V_{t-1}-\frac{V_{1}}{\frac{2}{N^{2}}})=(1-\frac{2}{N^{2}})^{t-1}(V_{1}-\frac{V_{1}}{\frac{2}{N^{2}}})
\Delta = \frac{2(2+\rho-\rho^2)-2e^{-\rho}(1+\rho)+\rho e^{\rho}(2+3\rho)}{2\lambda\left(1+\rho e^{\rho}\right)}\mathbb{E}(P_i) = \frac{1}{\lambda}+\frac{2-e^{-\rho}}{\mu}
\int_\Omega f_n \nabla \phi_n \cdot \bar u \dx = \int_\Omega \delta \nabla \bar u : \nabla \phi_n \otimes \nabla \phi_n \dx - \int_\Omega \delta (\nabla \cdot \bar u) \tfrac{1}{2}|\nabla \phi_n|^2 \dx \ .
{\cal L}_{4 F}^{P}= \frac{1}{4}\sum_{a = 1}^{2}\sum_{Q , Q^{\prime}}( - 1 )^{a + 1}g_{Q^{\prime}Q}^{\prime}{( \bar{Q^{\prime}}i \gamma_{5}Q^{\prime})}^{( a )}{\bar{Q}i \gamma_{5}Q )}^{( a )},
M(K_k) =\#(K_k)=2^{1-\binom k2}\binom Nk \pm \delta N^k, M(K_{k+1}) =\#(K_{k+1})=2^{1-\binom{k+1}2}\binom N{k+1}\pm \delta N^{k+1}, M(K_{k+2}-e) =\#(K_{k+2}-e)=2^{2-\binom{k+2}2}\binom N{k+2}\binom{k+2}2 \pm \delta N^{k+2},
\delta^{L}\Phi (X)=\frac{i}{2}\omega^{\alpha \beta}(X)\star \left(\mathcal{D}_{\alpha \beta}\Phi \right) (X)=\frac{i}{2}\left( \omega^{\alpha \beta}(X)\star \mathbf{Q}_{\alpha \beta}\right) \star \Phi(X)\equiv i\omega_{L}(X)\star \Phi (X).
\gamma=\frac{n+1}2 \left( \frac 1 2 -\frac 1 q \right) -\frac m{2 \mu_\ast (m+2)}, \frac 1{m+2}- \frac{2(n+1)}{\mu_\ast (m+2)(n-1)}< \gamma < \frac 1{m+2}.
n_i\cdot q_j + n_j\cdot q_i = a_{ii}^{-1/2}a_{jj}^{1/2}r_{ij}+ a_{jj}^{-1/2}a_{ii}^{1/2}r_{ji}- 2a_{ii}^{-1/2}a_{jj}^{-1/2}a_{ij}= a_{ii}^{-1/2}a_{jj}^{-1/2}\left[r_{ij}a_{jj}+ r_{ji}a_{ii}- 2a_{ij}\right] = 0.
\mathcal{S}_1=\frac{\psi_1 T_{\rm s}}{\sqrt{2\pi}}\,\mathcal{W}_2+\psi_{2}\,T_{\rm s}\,e^{-\frac{\psi_{2}\tau_{\rm d}}{N_{\rm a}}+\frac{\psi_1^2\tau_{\rm d}^2}{2N_{\rm a}^2}}\left[Q\left(\kappa_{\rm b}+\frac{\psi_{1}\tau_{\rm d}}{N_{\rm a}}\right)-Q\left(\kappa_{\rm t}+\frac{\psi_{1}\tau_{\rm d}}{N_{\rm a}}\right)\right].
\lVert h_{T_0^c}\rVert_{2,p}^p \leq \omega\lVert h_{T_0}\rVert_{2,p}^p+(1-\omega)\lVert h_{\tilde{T}\cap T_0^c}\rVert_{2,p}^p+(1-\omega)\lVert h_{\tilde{T}^c\cap T_0^c}\rVert_{2,p}^p &=\omega\lVert h_{T_0}\rVert_{2,p}^p+(1-\omega)\lVert h_S\rVert_{2,p}^p,
\Theta_{J}:=(x_{i+\lfloor d_1/2\rfloor},x_{i+\lfloor d_1/2\rfloor}+1)\times (y_{j+\lfloor d_2/2\rfloor}, y_{j+\lfloor d_2/2\rfloor+1}),{\bf x}^{r_1,r_2}_{J}:=(x_{i+\lfloor d_1/2\rfloor}+\frac{r_1 h_x}{d_1},y_{j+\lfloor d_2/2\rfloor}+\frac{r_2 h_y}{d_2}).
S(f)=\sigma^{2}\frac{2^{n}\pi^{n/2}\Gamma(\nu+\frac{n}{2})(2\nu)^{\nu}}{\Gamma(\nu)\rho^{2\nu}}(\frac{2\nu}{\rho^{2}}+4\pi^{2}f^{2})^{-(\nu+\frac{n}{2})}.
\begin{array}{l}\int_0^{\frac{1}{2}}Q(x)[\cos(2kx)+\int_0^xK(x,t)\cos(2kt)dt]dx \qquad+\int_{\frac{1}{2}}^bQ(x)[2A_2\cos(2kx)+2A_3\cos2k(x-\frac{1}{2}) \qquad\quad+2A_4\cos2k(x-1)+\int_0^xK(x,t)\cos(2kt)dt]dx=0.\end{array}
\lim_{x\to 0^+}\frac{G(x)}{\exp(e^{1/x})e^{-1/x}x^2}=\lim_{y\to \infty}\frac{\frac{1}{g(1/y)}y^{-2}}{\exp(e^y)y^{-2}+\exp(e^y)\left(-2e^{-y}y^{-3}- e^{-y}y^{-2}\right)}&=\lim_{y\to \infty}\frac{1}{1-e^{-y}(2y^{-1}+1)}=1.
G(z , \bar{z}) =\lim_{R\rightarrow\infty}\mid{R}\mid^{4\tilde\bigtriangleup_{4}}\langle\Phi_{H.W.}^{(\lambda_{1}, \mu_{1})}(0 , 0)\Phi_{H.W.}^{(\lambda_{2},\mu_{2})}(z , \bar{z}) \Phi_{T_{z}^{3}, Y^{3}}^{(\lambda_{3}, \mu_{3})}(1 ,1)\tilde\Phi_{L.W.}^{(\lambda_{4}, \mu_{4})}(R , R) \rangle =
\begin{aligned}\left|B_{\frac{i+j}{n}}(\omega)-B_{\frac{i+j-1}{n}}(\omega)\right| \leq\left|B_{t+(\frac{i+j}{n}-t)}(\omega)-B_{t}(\omega)\right|+\left|B_{t}(\omega)-B_{t+(\frac{i+j-1}{n}-t)}(\omega)\right| \leq\left[\left(\frac{i+j}{n}-t\right)+\left(\frac{i+j-1}{n}-t\right)\right]M \leq\frac{(2j+1)M}{n}.\end{aligned}
\lim_{\xi \to \xi_1^+}\phi_{1,\beta}(\xi)=1, \ \lim_{\xi \to \{\xi_\beta^1\}^-}\phi_{1,\beta}(\xi)=\beta,\hbox{and}\lim_{\xi \to \{\xi_\beta^2\}^+}\phi_{\beta,\gamma}(\xi)=\beta \ \lim_{\xi \to \{\xi_\gamma^1\}^-}\phi_{\beta,\gamma}(\xi)=\gamma.
\hat \varphi^{(4)}_n = \chi_{[0, \rho_0]}\Bigl( \sqrt{p_y^2 +k_n^2}\Bigr) \hat \varphi^{(4)}_n + \chi_{(\rho_0 , \infty)}\Bigl( \sqrt{p_y^2 +k_n^2}\Bigr) \hat \varphi^{(4)}_n = \chi_{[0, \rho_0]}\Bigl( \sqrt{p_y^2 +k_n^2}\Bigr) \hat \varphi^{(4)}_n + \mathcal{O}(1),
s_0^z(z_1)=-z_1,s_0^z(z_i)=z_i\;(i\geq 1), \quad\sum_{r=0}^\infty s_0^z(Q_r(x))u^r=\frac{1+z_1u}{1-z_1u}\prod_{i=1}^\infty\frac{1+x_iu}{1-x_iu}.
A_{\bar S S}(\tau:t_0) =\sum_{k=t_0}^{\tau-2}A_{\bar S}(\tau-2)\cdots A_{\bar S}(k+1)A_{\bar S S}(k)A_S(k:t_0)\cr{=}\sum_{k=t_0}^{\tau-2}\tilde A(\tau-1:k+1)A_{\bar S S}(k)A_S(k:t_0)\cr \stackrel{(a)}{\geq}e^{-M(\gamma)\lambda}\sum_{k=t_0}^{\tau-2}\tilde A(\tau-1:k+1) A_{\bar S S}(k),
\overline{Q}_{k,i}(x;q) =H_{k,i}(-1/q;xq;q)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{kn}q^{kn^2+kn-in}(1-x^iq^{(2n+1)i})(-xq^{n+1})_{\infty}(-q)_n}{(q)_n(xq^{n+1})_\infty},
p(x) =\left(\begin{array}{cccccc}1 x x^2 x^3 x^4 x^5\end{array}\right)(A_{(1,0,a,0,b,0,c,0,d)})^{-1}\mathbf{v_\gamma}(5,4) &=x\Big(x^4+\frac{ad-bc}{b^2-ac}x^2+\frac{c^2-bd}{b^2-ac}\Big).
\int_{S^{1}}|\widehat{g_{p}d\sigma}(Rx)|^{2}\, d\mu x = \int_{S^{1}}|\widehat{g_{p}d\sigma}(Rx)|^{2}\mu \ast P_{p}(x) \, d\sigma x & \leq \|\mu \ast P_{p}\|_{L^{\infty}}\int_{S^{1}}|\widehat{g_{p}d\sigma}(Rx)|^{2}\, d\sigma x
\tau_{0}(y)=\sum_{\sigma_{1}=0}^{1}..\sum_{\sigma_{4}=0}^{1}Y_{\sigma_{1}..\sigma_{4}}^{(0)}(t_{1}Z_{1})^{\sigma_{1}}(t_{2}Z_{2})^{\sigma_{2}}(t_{5}Z_{5})^{\sigma_{3}}(t_{8}Z_{8})^{\sigma_{4}}
\begin{bmatrix}0 0 0 0 0 0 0 1 -1 1 1 2 2 1 1 1 1 -1 2 2 4 3 2 2 2 1 -1 3 2 6 4 3 3 3 1 -1 4 4 10 7 5 5 6 1 -1 5 4 10 7 5 5 6 1 -1 6 5 13 9 7 7 8 1 -1 7 7 16 11 8 9 10 1 -1\end{bmatrix}.
\dot{h_1}/h_1=\frac{1}{2}h_2/h_1, \dot{h_2}/h_2=-\frac{1}{2}h_2/h_1-h_2/h_3 \dot{h_3}/h_3=\frac{1}{2}h_4/h_3+h_2/h_3, \dot{h_4}/h_4=-\frac{1}{2}h_4/h_3
\sum_{\substack{\mathbf{r},\mathbf{s}\in\mathcal{D}'_k r_m=s_m=1 r=s}}\frac{y^{(m)}_{\mathbf{r}}}{\phi_\omega(r)^2}\prod_{p|rs}S_p^{(m)}(\mathbf{r},\mathbf{s})\Bigl(y^{(m)}_{\mathbf{r}}+O\Bigl(\frac{T_k}{k}(Y_\mathbf{r}+Y_{\mathbf{s}})\frac{\phi(a_m W B)}{a_m W B}(\log{A})(\log\log{R})\Bigr)\Bigr).
\begin{array}{ll}\mbox{maximize}\sum_{i=1}^n H_i(\hat{x}_i) \mbox{subject to}x_t \in F_t\subseteq \mathbb{R}_+^n~\forall t\in[m] &\hat{c}_i^T \hat{x}_i\leq 1~\forall i\in[n]\end{array}
{W}_{\alpha}= \frac{1}{8}\; \bar{D}^{2}\, \left({e}^{- V}\! D_{\alpha}{e}^{V}\right) = i \left( \lambda_{\alpha}+ i \theta_{\alpha}D - \theta^{\beta}\, G_{\alpha \beta}- i \theta^{2}{\cal D}_{\alpha \dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}\right) \, .
V_{n}=\frac{S_n^{2-c}}{n}+\frac{S_{n+1}^{2-c}}{n+1}-\frac{S_n^{2-c}}{n}\sqrt{1-\frac{1}{n}}-\frac{S_{n+1}^{2-c}}{n+1}\sqrt{1+\frac{1}{n}},
e^{-\lambda_{1}t}h(x)e^{-w(x,t)}=\Pi_{x}\left[e_{a+g}(t)h(\xi_{t})e^{-l(\xi_{t})}\right]+\Pi_{x}\big[\int_{0}^{t}{\rm d}s\,e_{a+g}(s)q(\xi_{s})h(\xi_{s}) &\quad\cdot \pi^{h}(\xi_{s},e^{-\lambda_{1}(t-s)}e^{-w_{t-s}})\int_{[0,+\infty)}e^{-\theta\pi(\xi_{s},V_{t-s}f)}\eta(\xi_{s},{\rm d}\theta)\big].
-\sin\psi\csc\eta\cdot\left(\frac{F[p,q](\frac{i}{\mu_{0}})w_{1}[p,q](\frac{i}{\mu_{0}})w_{3}[p,q](\frac{i}{\mu_{0}})}{w_{2}[p,q](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{2}\gamma^{0}\partial_{\phi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
{\frac{\partial}{\partial x^{\alpha}}}\Gamma_{r}^{\alpha \beta \gamma}\ ( x , y , z ) = \{\frac{i}{4 \pi^{2}}}\varepsilon^{\beta i \gamma j}{\frac{\partial}{\partial y^{i}}}{\frac{\partial}{\partial z^{j}}}\delta ( x - y ) \delta ( x - z ) .
*\wedge\Bigl(\delta\Omega, \Omega\Bigr)=-\frac{1}{4\pi}\Bigl[{\bf F}_{\mu\nu}\nabla_\sigma{\bf F}^{\sigma\nu}+(*{\bf F})_{\mu\nu}\nabla_\sigma (*{\bf F})^{\sigma\nu}\Bigr]dx^\mu\otimes\varepsilon^1\wedge\varepsilon^2.
{{\psi}_{k}^{i n}}^{\prime}S^{2}= \frac{n_{s}\sqrt{k_{s}}}{\sqrt{\pi}}2^{- n_{s}}\Gamma \left( - n_{s}\right) a_{s}^{m}e^{l \phi_{s}/ 2}\left( \frac k{k_{s}}\right)^{n_{s}}= c o n s t .
\Pr\Big(\max_{k\in I_{2}}||S_{k}|-a_{k}|>11t\Big) = \sum_{k\in I_{2}}\Pr(\tau=k)\leq 5\sum_{k\in I_{2}}\Pr(\tau=k,||S_{n}|-a_{n}|>t) &\leq 5\Pr(||S_{n}|-a_{n}|>t) \leq 5p.
a(1+i(x-x')) =\frac{\Gamma(i(\bar x'-\bar x))}{\Gamma(1+i(x-x'))}& =\frac{\Gamma(i(\nu'-\nu)+(n-n')/2)}{\Gamma(1-i(\nu'-\nu)+(n-n')/2)}=(-1)^{n-n'}\frac{\Gamma(i(\nu'-\nu)-(n-n')/2)}{\Gamma(1-i(\nu'-\nu)-(n-n')/2)}\,.
\begin{cases}\|\sum_{l\le j\epsilon}T_a^{j, l}\|_{L^r\to L^s}\lesssim_{\epsilon}2^{jm - jn(\frac{1}{s}- \frac{1}{r})}\|T_a^{j, l}\|_{L^r\to L^s}\lesssim_{\epsilon}2^{-jn(\frac{1}{s}-\frac{1}{r})}2^{10n(m-n)(j+l)}, l>j\epsilon.\end{cases}
w(x,y)\equiv\mathcal{P}_y^su(x)=\frac{2^{1-s}}{\Gamma(s)}\sum_{k=1}^{\infty}(\lambda_k^{1/2}y)^s\mathcal{K}_s(\lambda_k^{1/2}y)\langle u,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}\psi_{k}(x),
\Delta E_{n,k}=\sin^2 \phi \cos^2\phi \, \left [ \sum_{n_j=1}\frac{4}{E_{n,k}^{(0)}-E_{n-1,k-1}^{(0)}}+ \sum_{n_j\geq 2}\frac{4(n_j-2)}{E_{n,k}^{(0)}-E_{n-1,k+1}^{(0)}}\right. &+\left . \sum_{m_j=1}\frac{4}{E_{n,k}^{(0)}-E_{n+1,k-1}^{(0)}}+\sum_{m_j\geq 2}\frac{4(m_j-2)}{E_{n,k}^{(0)}-E_{n+1,k+1}^{(0)}}\right ]\ .