harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{H}= -i \vec{\alpha}\cdot \hat{e}_j \frac{\partial}{\partial \xi^j}- \frac{i}{2 \Delta}\vec{\alpha}\cdot \vec{\kappa}- \frac{i}{\Delta}\vec{\alpha}\cdot \hat{e}_3 \Bigl( \frac{\partial}{\partial s}+ \frac{1}{2 \Delta}\frac{\partial ( \vec{\kappa}\cdot \vec{\xi})}{\partial s}\Bigr) + \beta m\ .
df(U_\tau)= \partial_af(U_\tau)dU_\tau^a+\tfrac{1}{2}\partial_a\partial_bf(U_\tau)dU_\tau^adU_\tau^b = \partial_af(U_\tau)D^a_\tau d\tau+\partial_af(U_\tau)H^a_i(U_\tau){dW}_\tau^i &+\left(\partial_af(U_\tau)\partial_bH_{i}^a(U_\tau)H_i^b(U_\tau)+\partial_a\partial_bf(U_\tau)H^a_i(U_\tau)H^b_i(U_\tau)\right)d\tau.
\displaystyle \int_{0}^{1}y_\delta(x_a,\tau)\,d\tau = \dfrac{a_{0,\delta}(x_a)}{2}\displaystyle \int_{0}^{1}\left ( y_\delta({x_a},\tau)-\bar{y}_a \right ) u(\tau)\,d\tau = \dfrac{b_{1,\delta}(x_a)}{2}
0=\lim_{r\to 0^+}r g'(r) =-\lim_{r\to0^+}\left\{\frac{n-2}{m}r^{-\frac{n-2}{m}}f(r^{-1})+ r^{-\frac{n-2}{m}-1}f'(r^{-1})\right\}&=-\frac{n-2}{m}D_A-\lim_{r \to\infty}r^{\frac{n-2}{m}+1}f'(r),
d s^{2}\approx - d \tilde{U}d \tilde{V}+ d X_{\mu}d X^{\mu}+ \left[ \frac{R^{\prime \prime}}{R}X_{i}^{2}+ \frac{( W^{2}R^{3})^{\prime \prime}}{W^{2}R^{3}}X_{4}^{2}\right] d \tilde{U}^{2},
\mathcal{M}(f)= \frac{1}{(2\pi)^{n/2}}\*\sum_{m=1}^n\* \frac{1}{m!}\*\sum_{\substack{(n_1, \ldots, n_m): n_1+\ldots +n_m=n+1, \n_1, \ldots, n_m\geq 1}}\* \frac{(n+1)!}{n_1!\cdots n_m!}\times \int_{L_{\pi}}\hat{f}(t_1, \ldots, t_n)\* \prod_{j=1}^m c_{n_j}(t_{M_{j-1}+1}, \ldots, t_{M_j-1}, -t_{M_{j-1}+1}-\ldots - t_{M_j-1}) \*d\lambda,
\ddot{z}_{n}=\frac{\ddot{\theta}_{n}}{\cos^{2}\theta_{n}}+\frac{2~\dot{\theta}_{n}^{2}~\sin \theta_{n}~}{\cos^{3}\theta_{n}}=\frac{\ddot{\theta}_{n}+2~\dot{\theta}_{n}^{2}~\tan \theta_{n}}{\cos^{2}\theta_{n}}~.
N_{1}=(\begin{matrix}0&1&1&1&0\\ 1&0&1&0&1\\ 1&1&0&1&1\\ 1&0&1&0&1\\ 0&1&1&1&0\end{matrix}),N_{2}=(\begin{matrix}1&0&1&0&1\\ 1&0&1&0&1\\ 0&0&0&0&0\\ 1&0&1&0&1\\ 1&0&1&0&1\end{matrix})
\nu^{\int}(y) := e^{-y_2}\int_{\mathbb{R}}\frac{1}{(4 \pi a_{tr}(y) y_2 )^{\frac{1}{2}}}\nu_{\partial}(s, 0) e^{\frac{-|y_1 - s|^2}{4 y_2 a_{tr}(y)}}\, \textrm{d}s.
{}_5 F_4 \left[ \left. \begin{matrix}a \frac{1}{2}a+1 c d e \frac{1}{2}a 1+a-c 1+a-d 1+a-e \end{matrix}\right| 1 \right] =\frac{\Gamma(1+a-c)\Gamma(1+a-d)\Gamma(1+a-e)\Gamma(1+a-c-d-e)}{\Gamma(1+a)\Gamma(1+a-d-e)\Gamma(1+a-c-e)\Gamma(1+a-c-d)}.
\mathcal{M}\left\lbrace{}^{\Psi}B_{p}^{(\alpha,\beta)}(x,y)\right\rbrace =\int_{0}^{\infty}p^{s-1}\int_{0}^{1}t^{x-1}(1-t)^{y-1}{}_{0}\Psi_{1}\left( \alpha,\beta;-\frac{p}{t(1-t)}\right) dtdp.
R_1 \le I(X_1;Y_1) + I(X;Y_1|W,X_1) * \stackrel{(a)}{\le}I(X_1;Y_1|W) + I(X;Y_1|W,X_1) * \stackrel{(b)}{=}I(X;Y_1|W), R_2 \le I(W;Y_1|X_1) * \stackrel{(c)}{\le}I(W;Y_1|X_1,X_2) * \stackrel{(d)}= I(W;Y_1|X), R_1 + R_2 \le I(X_1;Y_1) + I(X;Y_1|W,X_1) + I(W;Y_1|X_1) * = I(X,W;Y_1),
T_2-T'_2 \ll M^2\sum_{1\leq m_2\leq Q}m_2^{-1}\sum_{\stackrel{1\leq m_3\leq Q}{G(0,m_2,m_3)=0}}\sum_{d\leq Q}d^{-\frac{1}{2}+\varepsilon}&\ll M^2\sum_{1\leq m_2\leq Q}m_2^{-1}\sum_{d\leq Q}d^{-\frac{1}{2}+\varepsilon}&\ll M^2Q^{\frac{1}{2}+\varepsilon}.
\begin{array}{c}{{\nabla_{\rho}\nabla^{\rho}\Lambda^{\mu_{2}\ldots \mu_{s}}+ m^{2}\Lambda^{\mu_{2}\ldots \mu_{s}}= 0 ,}}\\{{\mathrm{w i t h}\; \; m^{2}= ( s - 1 ) ( s + d - 3 )}}\\ \end{array}
D^2 =\frac{1}{2}[D,D]=\frac{1}{2}[\mathcal{L}_K+\imath_L,\mathcal{L}_K+\imath_L] &=\frac{1}{2}\mathcal{L}_{[K,K]_{FN}}+\imath_{[K,L]_{FN}}+\mathcal{L}_{\imath_LK}+\frac{1}{2}\imath_{[L,L]_{RN}},
\mathfrak{Z}=\textrm{the ray class group of $K$ modulo $\mathfrak{C}p^\infty$,}Z^- =\textrm{the maximal quotient of $\mathfrak{Z}$ where complex conjugation acts as $-1$}, Z_p^- =\textrm{the maximal $p$-profinite quotient of $Z^-$}.
L = \exp\bigl(\mathcal{N}(0, 4\log 2)\bigr) \;(), X_2 = \beta^{-1}_{2,2}\bigl(a=(1,1), \,b_0=2, \,b_1=b_2=1/2\bigr), X_3 = \frac{2}{y^3}\,dy, \;y>1 \;(), Y = \frac{1}{y^2}\,e^{-1/y}\,dy,\;y>0 \;().
\{j_{0}^{a}( x ) , \psi_{\pm}^{b}( y ) \}= f^{a b c}\psi_{\pm}^{c}\delta ( x{-}y ) \, , \qquad \{j_{1}^{a}( x ) , \psi_{\pm}^{b}( y ) \}= \pm{\frac{1}{2}}f^{a b c}\psi_{\pm}^{c}\delta ( x{-}y ) \ .
D(Q) =\{f\in \ell^2 (X,m) : \lim_{t\downarrow 0}\frac{1}{t}\langle f-e^{-tL}f,f\rangle <\infty\}Q(f,g) =\lim_{t\downarrow 0}\frac{1}{t}\langle f-e^{-tL}f,g\rangle.
K_{2,3}(z,z'')= \Big(\int_{2r''<r'<\frac{r}2}+\int_{\frac{r}2\leq r'\leq2r}+\int_{r'>2r}\Big)G(z,z')Q(z',z'')\;d\mu(z') = K_{2,31}(z,z'')+K_{2,32}(z,z'')+K_{2,33}(z,z'').
u(n_{A})^{2}\propto(\frac{\partial n_{A}}{\partial R_{AB}})^{2}u(R_{AB})^{2}=n_{A}^{2}\frac{(R_{A}-R_{B})^{2}}{(R_{A}-R_{AB})^{2}(R_{AB}-R_{B})^{2}}u(R_{AB})^{2}
\parallel X\parallel_p =(E(\mid X\mid^p))^{\frac{1}{p}}\Rightarrow E\left(\left|\frac{XY}{\parallel X\parallel_p \parallel Y\parallel_q}\right|\right) \leq 1 \Rightarrow~~~~~~~~~~~~~~~~ E(|XY|) \leq \parallel X\parallel_p \parallel Y\parallel_q.
\frac{1}{g_{\Lambda}}- \frac{1}{4}\, \mathrm{L o g}\left( \frac{\Lambda^{2}}{M^{2}}\right) + \frac{1}{6}\frac{\Lambda^{2}}{M^{2}}e^{- 4 / g_{\Lambda}}+ O \left( \frac{\Lambda^{4}}{M^{4}}e^{- 8 / g_{\Lambda}}\right)
h ( \underline{{{\theta}}},{\underline{{{z}}}}) = \prod_{1 \le i < j \le n}F ( \theta_{i j}) \prod_{i = 1}^{n}\prod_{j = 1}^{m}\tilde{\phi}( \theta_{i}- z_{j}) \prod_{1 \le i < j \le m}\tau ( z_{i}- z_{j})
f'_n(x) = 2x \cdot \sum_{\nu=1}^{n}\alpha_{\nu}^{-1}\sum_{\kappa=1}^{\nu}(-1)^{\nu-\kappa}h_{\kappa-1}^2(x) &= 2x \cdot\sum_{\kappa=1}^{n}h_{\kappa-1}^2(x)\sum_{\nu=0}^{n-\kappa}(-1)^{\nu}\alpha_{\nu+\kappa}^{-1}.
\tilde \Psi^{+}=-i\frac 1r(2k+1)^{-1/2}\left(\begin{array}{c}(k+m+1/2)^{1/2}Y_k^{m-1/2}(\vartheta ,\varphi )B(r) (k-m+1/2)Y_k^{m+1/2}(\vartheta ,\varphi )B(r)\end{array}\right)
\alpha_0 = \alpha_0 (s_0,s_1,s_2), v_1 = v_1(s_0,m_1,\partial_\theta s_1,\partial_\varphi s_1 ) v_2 = v_2(s_0,s_1,m_1,m_2,\partial_\theta s_1,\partial_\varphi s_1,\partial_\theta s_2,\partial_\varphi s_2).
\tilde{\delta}(x) =t(\mathcal M)-{\lim_{n \to \infty}\delta(xz_n)}=t(\mathcal M)-{\lim_{n \to \infty}\delta_1(xz_n)}&=t(\mathcal M)-{\lim_{n \to \infty}\delta_1(x)z_n}=\delta_1(x)
a(s) =c_d\int_{I(s)}w\xi^{d-1}\,d\xi+O(e^{-d\theta_*s}) =c_d\int_{(1+t)^{\frac{1}{2}-\theta_*}}^\infty u_*(r,t)U_d(r)r^{d-1}\,dr+O(e^{-d\theta_*s}) =\frac{c_d}{c_*}\int_{(1+t)^{\frac{1}{2}-\theta_*}}^\infty u_*(r,t) \nu_d(r)r^{d-1}\,dr+o(1)
G=\left(\begin{array}{cccc}\mu \begin{array}{ccc}0 0 0 \end{array}e \begin{array}{c}0 0 0 \end{array}P_v \begin{array}{c}0 0 0 \end{array}P_h e \begin{array}{ccc}0 0 0 \end{array}\gamma 0 P_h 0 P_h\end{array}\right)
E\hat{V}_{d-2}(Z\cap A) = e^{-\gamma E V_d(K)}\sum\limits_{l=1}^{2^{n^d}-2}w_l^{(d-2)}\Big(\gamma c_2(B_l,W_l)E V_{d-2}(K) &\quad+\frac{\gamma^2}{2}c_3(B_l,W_l)\left( E V_{d-1}(K)\right)^2\Big)+O(a),
\gamma_t = \left( \,\kappa\frac{-y_{u}}{\sqrt{x_{w}^{2}+y_{w}^{2}}},\,\kappa\frac{x_{u}}{\sqrt{x_{w}^{2}+y_{w}^{2}}},\,y(t,u)x_{t}(t,u)\,\right) & + \left( \,0,0,\,-y(t,0)x_{t}(t,0)+\int_{0}^{u}\kappa\sqrt{x_{w}^{2}+y_{w}^{2}}dw\right) .
\sum_{x\in A_{1}}|x-y|^{-s}= 2^{-s}\sum_{j=1}^{2^{n_{1}+1}-1}\left(\sin \frac{\pi j}{2^{n_{1}+1}}\right)^{-s}-2^{-s}\sum_{j=1}^{2^{n_{1}}-1}\left(\sin \frac{\pi j}{2^{n_{1}}}\right)^{-s}= \frac{1}{2^{n_{1}+1}}\,\mathcal{L}_{s}(2^{n_{1}+1})-\frac{1}{2^{n_{1}}}\,\mathcal{L}_{s}(2^{n_{1}}).
\partial_{\tau}\hat{w}=\frac{\partial_{zz}^{2}\hat{w}}{1+\left(\partial_{z}\hat{w}\right)^{2}}+\left(n-1\right)\left(\frac{\partial_{z}\hat{w}}{z}-\frac{1}{\hat{w}}\right)+\frac{\frac{1}{2}+\sigma}{2\sigma\tau}\left(-z\,\partial_{z}\hat{w}+\hat{w}\right)
G_{0}^{( 2 )}( \vec{\rho}, v ) = - \frac{g}{L}\frac{1}{( 2 \pi )^{d - 1}}\sum_{n = 1}^{\infty}\cos ( \frac{n \pi v}{L}) \int d^{d - 1}p \frac{e^{i \vec{p}. \vec{\rho}}}{( \vec{p}^{2}+ ( \frac{n \pi}{L})^{2}+ m^{2})}.
d_A(g_t',g_{t+1}') = d_A(g_t,g_{t+1}) \leq d_A(g_t,\widehat{W}(t))+d_A(\widehat{W}(t),\widehat{W}(t+1))+d_A(\widehat{W}(t+1),g_{t+1}) \leq 2\eta + 1 + 2\eta = 4\eta+1,
{\rm rank}\left(\begin{array}{ccccc}\scriptstyle{d_{i,1}}\cdots \scriptstyle{N-1}\cdots \scriptstyle{d_{i,N-1}}& \scriptstyle{d_{N,1}d_{i,1}}\cdots \scriptstyle{2N}\cdots \scriptstyle{d_{N,N-1}d_{i,N-1}}& \scriptstyle{d_{N,1}}\cdots \scriptstyle{N-1}\cdots \scriptstyle{d_{N,N-1}}\end{array}\right)=2.
\bar{E}_{n}\Bigl |_{n \to \infty}\sim \frac{c \xi^{2}}{4 \pi a^{2}}\left[ \frac{1 0 - 3 \xi^{2}}{9 6 0 n^{2}}- \frac{2 8 2 2 4 - 7 3 4 4 \xi^{2}+ 7 2 0 \xi^{4}}{1 5 4 8 2 8 8 0 n^{4}}+ O \left( \frac{1}{n^{6}}\right) \right]{.}
\|I_{N-\gamma_i}(|u|^2)\|_{\cdot{H}^{s}_{\frac{2N}{\gamma_i}}}= \|(-\Delta)^{s/2}\left(I_{N-\gamma_i}(|u|^2)\right)\|_{L^{\frac{2N}{\gamma_i}}}= \|I_{N-\gamma_i}\left((-\Delta)^{s/2}(|u|^2)\right)\|_{L^{\frac{2N}{\gamma_i}}}\lesssim \| (-\Delta)^{s/2}(|u|^2)\|_{L^{\frac{2N}{2N-\gamma_i}}}= \| |u|^2\|_{\cdot{H}_{\frac{2N}{2N-\gamma_i}}^s}.
\psi_{\mathrm{r o t}}( x ) = e^{i \gamma_{5}\chi}\psi_{\mathrm{i n}}( x ) = \left( \begin{matrix}{{\nu}}&{{i \mu}}\\{{i \mu}}&{{\nu}}\\ \end{matrix}\right) \psi_{\mathrm{i n}}( x ) = \left( \begin{matrix}{{\nu + i \mu \lambda}}\\{{i \mu + \lambda \nu}}\\ \end{matrix}\right) e^{i k x},
\overline{Y}=\frac{K[X]+3\alpha K^{2}[X]^{2}+3\alpha{}^{3}K^{3}[X]^{3}+\alpha{}^{6}K^{4}[X]^{4}}{1+4K[X]+6\alpha K^{2}[X]^{2}+4\alpha{}^{3}K^{3}[X]^{3}+\alpha{}^{6}K^{4}[X]^{4}}
M^k = \left( \begin{array}{cccc}{\rm e}^{km}k{\rm e}^{(k-1)m}M_{12}\left( \frac{1-{\rm e}^{km}}{1-{\rm e}^{m}}\right) M_{13}\left(M^{k}\right)_{14}0{\rm e}^{km}0 \left( \frac{1-{\rm e}^{km}}{1-{\rm e}^{m}}\right)M_{24}0 0 1 0 0 0 0 1\end{array}\right)\, ,
I_{m n}^{\prime}= \frac{1}{A^{4}}\int \frac{d^{d}k}{{( 2 \pi )}^{d}}\frac{k^{2}k_{m}k_{n}}{( k^{2}+ a_{2}) ( k^{2}+ a_{3}) ({( k + p )}^{2}+ a_{2}) ({( k + p )}^{2}+ a_{3})}
I-A^*B = I - (I+X^*)^{-1}(X^*-I)(Y-I)(Y+I)^{-1}= I - (I+X^*)^{-1}[X^*Y-X^*-Y+I](Y+I)^{-1}= (I+X^*)^{-1}\bigl[(I+X^*)(Y+I) - X^*Y+X^*+Y-I\bigr](Y+I)^{-1}= (I+X^*)^{-1}[2X^*+2Y](Y+I)^{-1}.
\int_{B_R}|u^+|^{\alpha}\,dx = \alpha \int_0^\infty t^{\alpha-1}|\{x \in B_R \;|\; u^+(x) \geq t \}| \,dt \leq |B_R| s^\alpha + \alpha \int_s^\infty t^{\alpha-1}|\{x \in B_R \;|\; u^+(x) \geq t \}| \,dt
W_-(q^{\frac{1}{2}},y) = \frac{y^2 + y + 1}{y}\, q^\frac{1}{3}\left( 1 + \frac{2y^2 + 3y + 2}{(y+1)^2}q + \ldots \right) W_+(q^{\frac{1}{2}},y) = \frac{(y+1)^2 \, y}{y^2+y+1}q^{-\frac{2}{3}}\left(1 + \frac{y^4 + 4 y^3 + 3 y^2 + 4 y + 1}{(y+1)^2}q + \ldots \right) \,.
P_{\mu}(x;q,t) P_{\nu}(x;q,t) = \sum_{\lambda}f^{\lambda}_{\mu\nu}(q,t) P_{\lambda}(x;q,t),c_{\nu}(q,t) P_{\nu}(x;q,t) = \sum_{\lambda}K^{\lambda}_{\nu}(q,t) S_{\lambda}(x;t).
-{\bf I}\partial \sb \tau \sp 2 + g e \sb 0 F \partial \sb \tau = \exp \left( \frac{g e \sb 0}{2}F \tau \right) \left( -{\bf I}\partial \sb \tau \sp 2 + \frac{g \sp 2 e \sb 0 \sp 2}{4}F \sp 2 \right) \exp \left( - \frac{g e \sb 0}{2}F \tau \right) ,
t^{-\frac12}\tau^\vee_i = D_{i,i+1}+t^{-1}(D_{i+1,i}-1)+\frac{(1-t^{-1})Y^{-1}_{i+1}Y_i}{1-Y^{-1}_{i+1}Y_i}\qquad\hbox{and}t^{\frac12}\tau^\vee_i = t (D_{i,i+1}-1) + D_{i+1,i}+ \frac{(1-t)Y^{-1}_i Y_{i+1}}{1-Y^{-1}_iY_{i+1}}.
a\equiv a_D=4\left \{(\omega_n+i\mu_U)^2+{\stackrel{\rightharpoonup}{l}}^2+m_U^2+(\omega_n+i\mu_U)[\Omega_m+i(\mu_D-\mu_U)]+\stackrel{\rightharpoonup}{l}\cdot \stackrel{\rightharpoonup}{p}+m^2_U(m_D^2-m_U^2)/(m_U^2+m_D^2)\right \}.
g(t,\theta,x):=\frac{V_t (\theta \eta_t f) (x)}{\eta_t \phi(x)}\xrightarrow[t\to \infty]{}G(\theta):= \Big( \frac{1}{1+\theta^{-(\gamma_0 - 1)}}\Big)^{\frac{1}{\gamma_0 - 1}},x\in E, \theta \geq 0.
\sup_{\epsilon}\int_{X_j}|(1-v'_{\epsilon}(\Psi))fF^{1+\delta}|_{h_i}^2\le c_i(\sup_{X_j}|F|^{2+2\delta}) \int_{X_j}\mathbb{I}_{\{\Psi<-t_0\}}|f|_{\hat{h}}^2<+\infty,
\langle u^{\tau(a)}, v^{\tau(a)}\rangle = \langle u_1 + u_0 + u_{\frac{1}{4}}- u_{\frac{1}{32}}, v_1 + v_0 + v_{\frac{1}{4}}- v_{\frac{1}{32}}\rangle = \langle u_1, v_1 \rangle + \langle u_0, v_0 \rangle + \langle u_{\frac{1}{4}}, v_{\frac{1}{4}}\rangle + \langle u_{\frac{1}{32}}, v_{\frac{1}{32}}\rangle
h^{-1}(\varsigma)= \left(\begin{matrix}h^{z\bar{z}}h^{z\bar{w}}h^{w\bar{z}}h^{w\bar{w}}\end{matrix}\right) = \left(\begin{matrix}\frac{P}{\mu}+\frac{P^2|\eta|^2}{2k}-\frac{P^2\eta}{2k}-\frac{P^2\bar{\eta}}{2k}\frac{P^2}{2k}\end{matrix}\right).
A^+_m(c,d)= \frac{1}{2\pi i}\int_{|s|=r}\alpha^+_{m,c}(s) \sum_{n\geq 1}\left(e^s \zeta_c^d q\right)^n \, ds= \frac{1}{2\pi i}\zeta_c^d q\int_{|s|=r}\frac{\alpha^+_{m,c}(s)e^s}{1-e^s \zeta_c^d q}\, ds.
G(u(t)) \le G(v(t)) = G(|\Delta_{0}|) + \int_{0}^{t}G'(v(s)) v'(s) \d s = G(|\Delta_{0}|) + \frac{5\kappa_{R}}{2}\int_{0}^{t}\frac{\rho(u(s))}{\rho(v(s))}\d s \le G(|\Delta_{0}|) + \frac{5\kappa_{R}}{2}t,
\left. \begin{array}{l c}{\alpha_{n}^{\mu}| \Omega \rangle = 0}&{n > 0}\\{b_{n}| \Omega \rangle = 0}&{n \ge 0}\\{c_{n}| \Omega \rangle = 0}&{n > 0}\\{p^{\mu}| \Omega \rangle \propto \alpha_{0}^{\mu}| \Omega \rangle = 0 .}&{}\\ \end{array}\right.
P^{\mathrm{T}}_{i,out}= \sum\limits_{k=1}^{K}\upsilon^{\mathrm{T}}_{i,k}\textrm{Pr}\bigg(\Big(P_i = \frac{\varepsilon_k}{T}\Big) \cap \Big(\log_2(1+\gamma_i)<R\Big)\bigg) ~~~~~~= \sum\limits_{k=1}^{K}\pi_{i,k}\upsilon^{\mathrm{T}}_{i,k}F_{H^U_i}\Big(\frac{K(2^{R}-1)(\alpha P_H+N_0)T}{kC}\Big).
J_k(H)=J_k\Bigl(F+O\Bigl(\frac{T_k\log\log{R}}{\log{R}}F_2\Bigr)\Bigr) =J_k(F)+O\Bigl(\frac{T_k\log\log{R}}{\log{R}}J_k(F_2)\Bigr) &=J_k(F)\Bigl(1+O\Bigl(\frac{k^2T_k\log\log{R}}{\log{R}}\Bigr)\Bigr).
\bar{x}_n - x_n^0 = \frac{- \varepsilon_1 \bar{\xi}_1 + |\hat{x}| \bar{\xi}_2}{\sqrt{\varepsilon_1^2 + |\hat{x}|^2}}\leq |D \rho (0, \tilde{x})| |\bar{\xi}_1| + |\bar{\xi}_2| \leq C \epsilon_0 \delta^2 b_\alpha
|Z^t_i| = \big(d^t(a_i + \xi_i, a'_i + \xi_i) + |a_i - a'_i|\big)\,\big| d^t(a_i + \xi_i, a'_i + \xi_i)- |a_i - a'_i|\big| &\lesssim (\delta + |t|) (|a_i - a'_i| + \delta + |t|),
dY_1=\omega_{12}Y_2-\omega_1\left(A_1Y+N\right)+\mu\omega_2\xi_1+\mu\omega_1\xi_2, &dY_2=-\omega_{12}Y_1-\omega_2\left(A_1Y+N\right)+\mu\omega_1\xi_1-\mu\omega_2\xi_2, &d\left(A_1Y+N\right)=2A_1[\omega_1Y_1+\omega_2Y_2]
{\cal A}[\Sigma]=\int_{\Sigma}\omega=\int_{\Sigma}{\cal L}\left(z,\bar{z}, \varphi(z,\bar{z}),v=\partial_z \varphi,\bar{v}=\partial_{\bar{z}}\varphi\right)\,{\mbox{d}}z \wedge\mbox{d}\bar{z}\,.
|s\rangle\leftrightarrow[\begin{matrix}1\\ 0\\ \vdots\\ 0\\ 0\\ \end{matrix}],|s-1\rangle\leftrightarrow[\begin{matrix}0\\ 1\\ \vdots\\ 0\\ 0\\ \end{matrix}],...,|-(s-1)\rangle\leftrightarrow[\begin{matrix}0\\ 0\\ \vdots\\ 1\\ 0\\ \end{matrix}],|-s\rangle\leftrightarrow[\begin{matrix}0\\ 0\\ \vdots\\ 0\\ 1\\ \end{matrix}]
\left({\rm m}_i F_p \right) \left( \phi_0, \cdots, \hat{\phi}_{i-1},\phi_i, \cdots, \phi_p \right) = F_p \left( \phi_0, \cdots,\phi_{i-2}, \phi_i, \phi_i, \cdots, \phi_p \right) +O(\hbar)
\begin{aligned}\phi_{\mathrm{t},m}^\star = \sqrt{\alpha_{\mathrm{t},m}^\star}e^{\jmath\theta_{\mathrm{t},m}^\star}, \phi_{\mathrm{r},m}^\star = \sqrt{1 - \alpha_{\mathrm{t},m}^\star}e^{\jmath\theta_{\mathrm{r},m}^\star}, \forall m \in \mathcal{M}. \end{aligned}
\omega \to \exp ( \alpha E_{e_{1}- e_{2}}+ \beta E_{e_{1}+ e_{2}}+ \gamma E_{e_{1}}) \omega \exp ( - \tilde{\alpha}E_{e_{1}- e_{2}}^{+}- \tilde{\beta}E_{e_{1}+ e_{2}}^{+}- \tilde{\gamma}E_{e_{1}}^{+}) ,
s_{l+1}=s_{l}+\varphi_\delta(s_{l}) \leq s_{l-1}+ \varphi_\delta(s_{l-1}) (1+(1+R_\delta(\sigma))) &\leq s_{0}+ \varphi_\delta(s_0) \sum_{m=0}^{l}(1+R_\delta(\sigma))^m \leq \delta^2 \frac{(1+R_\delta(\sigma))^{l+1}-1}{R_\delta(\sigma)}.
f(\overline{x}_T) - f(x^*) =\tilde{O}\left( \sqrt{8M_2\sqrt{d}\Delta\mathcal{D}_\Psi}+ \sqrt{\frac{32M_2\mathcal{D}_{\Psi}da_{q}\Delta}{T^{\frac{\kappa}{(1+\kappa)}}}}+ \frac{2\sqrt{d}a_{q}M_2 \mathcal{D}_{\Psi}}{T^{\frac{\kappa}{1+\kappa}}}\right).
P_{R}= \frac{\partial L}{\partial \dot{R}}= - \frac{3 \pi R \dot{R}}{2 G}, \hspace{0 . 2 c m}P_{\phi}= \frac{\partial L}{\partial \dot{\phi}}= \frac{2 \pi^{2}}{G_{\phi}}R^{3}\dot{\phi}, \hspace{0 . 2 c m}P_{a}= 2 \pi^{2}m R^{3}\dot{n}_{a}
F_{x^Cy^0}y^C-F_{x^0}=u\left(\varphi_z-2\phi_{x^0}\right), F_{x^Cy^j}y^C-F_{x^j}=u\left(\left[-s(\varphi_s-\frac{2}{r}\phi_r)-z(\varphi_z-2\phi_{x^0})\right]u^j+\left[\varphi_s-\frac{2}{r}\phi_r\right]x^j\right).
A_{a_{1}}( \theta_{1}) A_{a_{2}}( \theta_{2}) = \sum_{n = 2}^{\infty}\, \sum_{\theta_{1}^{\prime}{<}\dots{<}\theta_{n}^{\prime}\atop{}}\! S_{~ a_{1}a_{2}}^{b_{1}\dots b_{n}}( \theta_{1}, \theta_{2}; \theta_{1}^{\prime}\dots \theta_{n}^{\prime}) A_{b_{1}}( \theta_{1}^{\prime}) \dots A_{b_{n}}( \theta_{n}^{\prime}) ~ ,
\varphi=\begin{bmatrix}v b \end{bmatrix},F=\begin{bmatrix}(d+B_{\infty})\otimes b-(w+u_{\infty})\otimes v (d+B_{\infty})\otimes v-(w+u_{\infty})\otimes b \end{bmatrix},{\mathcal{L}}=\begin{bmatrix}\nu 0 0 \mu \end{bmatrix}\Delta.
[Z_j]_{\alpha, \beta}= \sum_{i = 1}^{n(T, j+1)}[D_{j+1}^T]_{\alpha, i}\ [R_{j+1}^{-1}]_{i, i}\ [D_{j+1}]_{i, \beta}= \sum_{i = 1}^{n(T, j+1)}[D_{j+1}]_{i, \alpha}\ [D_{j+1}]_{i, \beta}\ [R_{j+1}^{-1}]_{i, i}\, .
\mathbf{x}=\sum_{\mu =1}^8 \tilde{\sigma}_{\mu}x^{\mu}=\left( \begin{array}{@{\,}cc@{\,}}z_2 z_1 -\bar{z}_1 \bar{z}_2 \end{array}\right)\,,\quad \mathbf{x}'=\sum_{\mu =1}^8 \tilde{\sigma}'_{\mu}x^{\mu}=\left( \begin{array}{@{\,}cc@{\,}}z_4 z_3 -\bar{z}_3 \bar{z}_4 \end{array}\right)\,,
\dot u^\pm_n\equiv \dot u^\pm_N:=\dot u_1^\pm-\partial_2\hat\varphi\dot u^\pm_2\,,\quad\dot H^\pm_n\equiv \dot H^\pm_N:=\dot H_1^\pm-\partial_2\hat\varphi\dot H^\pm_2\,,\quad\mbox{on}\,\,\,\Gamma_T\,.
\left( k^{2}+ m^{2}\right)^{- z}= \frac 1{\Gamma ( z )}\int_{0}^{\infty}d y \, y^{z - 1}\mathrm{e x p}\left[ - y \left( k^{2}+ m^{2}\right) \right] \, , \qquad \mathrm{R e}\, z > 0 \, ,
+\,x_{n+1}^2\,\Biggl\{(n+2-a)\;p_{a-2}(x)\;p_{b -2}(x)- \sum_{i=a+b-n-2}^{a}\,(a+ b-2i) \;p_{i-2}(x)\;p_{a+b - 2-i}(x)\Biggr\}\,.
\biggl\Vert \varepsilon \sum_{l=1}^d g^\varepsilon b_l\left([\Lambda^\varepsilon]S_\varepsilon b(\mathbf{D})D_l +[\widetilde{\Lambda}^\varepsilon ]S_\varepsilon D_l \right) (B^0-\zeta \overline{Q_0})^{-1}\biggr\Vert_{L_2\rightarrow L_2}\leqslant \mathfrak{C}_7\varepsilon c(\phi),
\varphi_{n,c}^{\alpha}(\cos(\frac{\theta}{\lambda}))=A_n\,\chi^{1/4}_{n,\alpha}(c)\frac{\sqrt{S(\cos(\frac{\theta}{\lambda}))}J_{0}\left(\chi^{1/2}_{n,\alpha}(c)S(\cos(\frac{\theta}{\lambda}))\right)}{(\sin(\frac{\theta}{\lambda}))^{\frac{1}{2}}r_n\left(\cos(\frac{\theta}{\lambda})\right)^{1/4}}+R_n(\cos(\frac{\theta}{\lambda}))
\mathcal{P}_{\textrm{out}}=\begin{cases}1-\sum\limits_{m=1}^M \mathcal{V}^{(M,m)}\xi^m J_{+}^{(m)}\textrm{if~}\mu_{\textrm{D}}^2\Gamma_0\leq \rho-1 1-\sum\limits_{m=1}^M\mathcal{V}^{(M,m)}\xi^m\big(I_+^{(m)}+I_-^{(m)}\big) \textrm{if~}\mu_{\textrm{D}}^2\Gamma_0>\rho-1 \end{cases}
\begin{aligned}E_sE_{s-1;s;s+1}+E_{s-1;s;s+1}E_s=-\sigma_s\sigma_{s+1}(e_se_{s-1;s;s+1}+e_{s-1;s;s+1}e_s)=0, &F_sF_{s-1;s;s+1}+F_{s-1;s;s+1}F_s=-\sigma_{s-1}\sigma_{s}(f_sf_{s-1;s;s+1}+f_{s-1;s;s+1}f_s)=0.\end{aligned}
\sup_{0<s<r<T}M_1(s,r) \le C^k \sup_{0<s<r<T}\int_{[0<\tau_1<\dots<\tau_k<r-s]}\prod_{j=1}^k (r-s-\tau_j)^{-\beta_0}(1+(\tau_j-\tau_{j-1})^{-1+\beta_0}) d\tau &<\infty.
I_{2}=\sum_{m_{1}\geq 1}\sum_{r=1}^{m_{1}}\sum_{m_{2}\geq 1}\int_{\Delta_{0,t}^{m_{1}+m_{2}}}\mathcal{H}_{m_{1}+m_{2}}^{X}(u)du_{m_{1}+m_{2}}...du_{1}
\partial_{y_i}G_2(y)=-\frac{y_{i}}{N\omega_{N}}\int_{\partial B(0,3)}\frac{\phi(z)}{|y-z|^{N}}dS(z)+ \frac{2-\|y\|^{2}}{2\omega_{N}}\int_{\partial B(0,3)}\frac{(y_i-z_i)\phi(z)}{|y-z|^{N+2}}dS(z), \forall y\in B(0,3).
\begin{aligned}\mathbf{h}_t(i)=-\sum_{\zeta\in \mathbb{F}_q}\left(s_i=\zeta\mid \mathbf{y}_t,\dots,\mathbf{y}_0\right) \log_2\left(s_i=\zeta\mid \mathbf{y}_t,\dots,\mathbf{y}_0\right). \end{aligned}
K^{\alpha}_{R,m}(z,w) = \frac{(\alpha+1)}{\pi R^{2}}\left\{\sum\limits_{n=0}^{m-1}(\alpha+2)_{n}\frac{(z\overline{w})^{n}}{n!R^{2n}}+ \frac{(z \overline{w})^{m}}{(m!)^{2}}{_3F_2}\left( \begin{array}{c}1 , 1 , \alpha+2 m+1 , m+1 \end{array}\bigg | \frac{z \overline{w}}{R^{2}}\right) \right\}.
\lim_{n\to\infty}\lim_{m\to\infty}u(t+\alpha_n+\beta_m,x;u^*(g),g) =\lim_{n\to\infty}u(\alpha_n,x;u^*(g^*\cdot t),g^*\cdot t) &=\lim_{n\to\infty}u(t,x;u^*(g^*\cdot\alpha_n),g^*\cdot \alpha_n) &=u^*(g^{**}\cdot t)(x)
\begin{aligned}y_{p,\lambda}(x) =Y_{\lambda}(x,p)-e^{x}\phi_{p,\lambda}(x) &=\sum_{n=2}^{\infty}\Big((1)_{p,\lambda}+(2)_{p,\lambda}+\cdots+(n-1)_{p,\lambda}\Big)\frac{x^{n}}{n!},(p\in\mathbb{N}).\end{aligned}
S_{\chi}= \int_{t^{\prime}}^{t^{\prime \prime}}d t \left( p_{i}\dot{q}^{i}+ \mu_{a}\dot{\lambda}^{a}+ i \zeta_{a}\dot{\eta}^{a}+ i \zeta^{\times}{\!}_{a}\dot{\eta}^{\times a}- H - i \{\Omega , \phi \}\right) .
u_{i_n}(s_n+t_n;s_n,u^0) =u_{i_n}(s_n+t_n;s_n+t_n-\tilde T,u(s_n+t_n-\tilde T;s_n,u^0)) &=u_0(s_n+t_n;s_n+t_n-\tilde T,u_{\cdot+i_n}(s_n+t_n-\tilde T;s_n,u^0)) &\ge u_0(s_n+t_n;s_n+t_n-\tilde T,\tilde u^n){\rm for}\,\, n\gg 1.
\Lambda_{\mu}( p , q ; \xi ) = \frac{- 1}{1 6 \pi^{2}}\gamma_{\alpha}\gamma_{\mu}\gamma_{\beta}\frac{\partial^{2}}{\partial p_{\alpha}\partial q_{\beta}}\int d^{3}x \, d^{3}y \, e^{i ( p \cdot x - q \cdot y )}e^{- e^{2}\xi \left| x - y \right| / 8 \pi}\frac{1}{| x |^{3}| y |^{3}}.
\stackrel{P}{\cal F}{}\! \!_{0 , p}^{( \pm )}\left( \begin{array}{c}{\! \! \alpha \! \!}\\{\! \! \beta \! \!}\\ \end{array}\right) = \sum_{s \in{\cal S}^{p - 3}}\stackrel{P}{K}{}\! \!_{0 , p}^{( \pm )}\left( \begin{array}{c}{\! \! \alpha \! \!}\\{\! \! \beta , s \! \!}\\ \end{array}\right) \stackrel{\alpha}{\omega}_{0 , p}\! \! \lbrack \beta + s \rbrack ,
F_m (\pi;x)=1+ \sum_{k=1}^{\infty}\frac{(-1)^k}{k!}\int_{x}^{\infty}\cdots \int_{x}^{\infty}\det[{K_{\mathrm{Airy}}(\pi; u_i, u_j)}]_{i,j = 1}^{k}du_1 \cdots du_k,
\left(E^{k}{}_{l}\right)^{i}{}_{j}=\delta^{k}{}_{j}\delta^{i}{}_{l}-g^{ki}g_{lj},\left(H^{A}{}_{B}\right)^{C}{}_{D}=\delta^{A}{}_{D}\delta^{C}{}_{B}-\eta^{AC}\eta_{BD}.
\delta\left({\tilde{\sigma}_{3}^{*}}(t)I\right)=\int^{t_{2}}_{t_{1}}\frac{\partial L_{T}}{\partial \Xi^{\alpha}}\delta\Xi^{\alpha}dt+\frac{\partial L_{T}}{\partial \zeta^{\alpha}}\delta\zeta^{\alpha}dt+\left[\Psi_{\alpha}\delta\Xi^{\alpha}\right]^{t_{2}}_{t_{1}}.
\begin{aligned}e^{-x(\alpha\cos\theta-\beta \sin\theta)}e^{-y(\alpha\sin \theta+\beta \cos \theta)}\left[1+\frac{x}{\alpha}\cos\theta+\frac{y}{\alpha}\sin\theta\right]^{i}\left[1-\frac{x}{\beta}\sin\theta+\frac{y}{\beta}\cos\theta\right]^{k}&=\sum_{m,n=0}^{\infty}C_{m,n}(i,k)\,\frac{x^{m}y^{n}}{\sqrt{m!n!}}.\end{aligned}
ds_{3Dc}^{2}=\frac{1}{1-\rho^{2}}\frac{d\mu_{x}^{2}}{\sigma_{x}^{2}}+\frac{2-\rho^{2}}{1-\rho^{2}}\frac{d\sigma_{x}^{2}}{\sigma_{x}^{2}}+\frac{2-\rho^{2}}{1-\rho^{2}}\frac{d\sigma_{y}^{2}}{\sigma_{y}^{2}}-\frac{2\rho^{2}}{1-\rho^{2}}\frac{d\sigma_{x}d\sigma_{y}}{\sigma_{x}\sigma_{y}}
P(F_{\ell}; J \times \mathbb{R}^{n-1}) \leq P (\widetilde{E}; J \times \mathbb{R}^{n-1}) \leq \liminf_{k \rightarrow \infty}P( E^{k}; J \times \mathbb{R}^{n-1}) &= \liminf_{k \rightarrow \infty}P( F_{\ell^{k}}; J \times \mathbb{R}^{n-1}) = \lim_{k \rightarrow \infty}P (F_{\ell^{k}}; J \times \mathbb{R}^{n-1}) &= P (F_{\ell}; J \times \mathbb{R}^{n-1}),
\sum_{k=0}^m\tbinom{m}{k}(-1)^k (n+k+1)a_{n+k}+ m \sum_{k=0}^{m-1}\tbinom{m-1}{k}(-1)^ka_{n+k+1}&= \sum_{k=0}^m\tbinom{m}{k}(-1)^k(n+k+1-k)a_{n+k}&= (n+1)\sum_{k=0}^m\tbinom{m}{k}(-1)^ka_{n+k}.
\sum_{n=0}^{p-1}R_{\mathbf{s}_{n}}(\tau) = \sum_{n=0}^{p-1}\sum_{i=0}^{p^m-1-\tau}\omega_{q}^{f_{n,i}-f_{n,i+\tau}}&=\sum_{i=0}^{p^m-1-\tau}\sum_{n=0}^{p-1}\omega_{q}^{f_{n,i}-f_{n,j}}(),
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\Phi(u_{\pm}(x)-u_{\pm}(y))(u_{\pm}(x)-u_{\pm}(y))K(x,y)dxdy =\int_\Omega |u_{\pm}|^{p^{\ast}_{s}}dx+\lambda \int_{\Omega}f(x,u)u_{\pm}dx
B_{L,k,1}=\sum_{L\le d < 2L}\sum_{\substack{p \mid k,\,\phi_e(q) e \mid d e >d/T}}\log \gcd(p^{\nu_p(k)},\phi_e(q)) , B_{L,k,2}=\sum_{L\le d < 2L}\sum_{\substack{p \mid k,\,\phi_e(q) e \mid d e \le d/T}}\log \gcd(p^{\nu_p(k)},\phi_e(q)).
\begin{array}{l}{\rm SINR}_{k,j}^m= 2y_{k,j}^m \sqrt{{p_{k,j}^m \left|G_{k,j}^m\right|^2}}-y_{k,j}^m \left({{\sum\limits_{j'\ne j}\sum\limits_{k'\ne{{k}}}{p_{k',j'}^{{m}}{{{\left|G_{k',j}^m\right|^2}}}}}+ N_{\rm RF}B_{\rm RF}}\right),\end{array}
T : \Upsilon_{l , m}= - i \sigma_{y}\Upsilon_{l , m}^{*}= \left( \begin{array}{r}{- \beta_{l , m}^{*}}\\{\alpha_{l , m}^{*}}\\ \end{array}\right) = ( - 1 )^{l - m}\left( \begin{array}{c}{\alpha_{l , - m}}\\{\beta_{l , - m}}\\ \end{array}\right) .
\kappa = \frac{k}{m_{f}}; L^{2}= \frac{m_{i}^{2}}{m_{f}^{2}}= \frac{\left[ T_{i}^{2}- T_{c}^{2}\right]}{\left[ T_{c}^{2}- T_{f}^{2}\right]}; \tau = m_{f}t ; \vec{x}= m_{f}\vec{r}