harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}d\tilde{Z}(t,z)=[-\pi(t,z)b_0(t,z)+\pi^2(t,z)\sigma^2_0(t,z)]\tilde{Z}(t,x,z)dt-\pi(t,z)\sigma_0(t,z)\tilde{Z}(t,z)dB(t) \tilde{Z}(0,z)=\int_D\frac{1}{\alpha(x)}dx.\end{cases}
2\frac{(q^2;q^2)_\infty^2}{(-q;q^2)_\infty^2}\sum_{n=-\infty}^\infty\frac{(-1)^nq^{n(3n+1)}}{1+q^{2n}}= \sum_{n=-\infty}^\infty\frac{(6n+1)q^{3n^2+2n}}{1-q^{6n+1}}-3\sum_{n=-\infty}^\infty \frac{(2n-1)q^{3n^2+2n-2}}{1-q^{6n-3}}.
\epsilon_{i}^{1}=\sup_{j\geq i}\sup_{1\leq\lambda\leq2}\left|\frac{L\left(j\lambda\right)}{L\left(j\right)}-1\right| \epsilon_{i}^{2}=\sup_{t\geq i}\left|\frac{g\left(t\right)}{L\left(t\right)t^{-\alpha}}\right|.
\lim_{M\to\infty}N_1^{1/3}c_2(\ell,k) =\frac{\alpha}{(2\pi i)^2}\int_{\Gamma_{d_3}}dw\int_{\Gamma_{-d_4}}d\omega\frac{e^{\frac 13w^3-\Delta\nu w^2-(\Delta\lambda+\alpha y)w}}{(w-\omega)e^{\frac 13\omega^3-\Delta\nu \omega^2-(\Delta\lambda+\alpha x)\omega}}&=\phi_2(x,y),
p(t, y|x) = e^{-t}\frac{1}{2\pi}\int_{-\pi}^{\pi}dk e^{-ik(y-x)}\sum_{n=0}^{\infty}\frac{t^n}{n!}(\cos k)^n &= e^{-t}\sum_{n=0}^{\infty}\frac{t^n}{n!}\frac{1}{2\pi}\int_{-\pi}^{\pi}dk \left( e^{-ik}\right)^{y-x}\left( \frac{e^{ik}+e^{-ik}}{2}\right)^n.
{\bf J}={\frac{1}{2}}\left( \begin{array}{c c}{p^{\dagger}}&{q^{\dagger}}\\ \end{array}\right){\bf \sigma}\left( \begin{array}{c}{p}\\{q}\\ \end{array}\right) +{\frac{1}{2}}\left( \begin{array}{c c}{r^{\dagger}}&{s^{\dagger}}\\ \end{array}\right){\bf \sigma}\left( \begin{array}{c}{r}\\{s}\\ \end{array}\right)
\frac{d}{dt}(P(t)y,w)_Y +(Ay, P(t)w)_Y + (P(t) y, Aw)_Y + (Ry, R\hat{y})_Y = begin{align*}1mm] \; \big( (B_0^* + B_1^* A^*) P y, [I + B_1^* R^* R B_1]^{-1}[ (B_0^* + B_1 A^*) P(t)w] \big)_U\,,
E(c_{\alpha\beta\gamma})= \left(\mu_\alpha+\mu_\beta+\mu_\gamma-\mu_1\right)c_{\alpha\beta\gamma}&\quad+p_{\alpha-1}c_{(\alpha-1)\beta\gamma}+p_{\beta-1}c_{\alpha(\beta-1)\gamma}+p_{\gamma-1}c_{\alpha\beta(\gamma-1)}.
(\mathcal{J}f)(\lambda) = \, t[\lambda]^{-1}(T_{w_{\lambda}}f)(\lambda_+)=t_j^{-1}t[s_j\lambda]^{-1}(T_{w_{s_j\lambda}}T_{j}f)((s_j\lambda )_+) \stackrel{}{=}\, t_j^{-1}\sum_{\mu\in P,\,\mu \preceq s_j\lambda}J_{s_j\lambda ,\mu}(T_{j}f) (\mu )= \sum_{\mu\in P,\, \mu \preceq \lambda}J_{\lambda ,\mu}f(\mu ),
\widehat{\mathrm{G}}= c_{0}\left\{\frac{1}{l}a \sin ( a l ) - a^{2}\cos ( a l ) \right\}+ c_{1}\left\{\frac{1}{l}a \cos ( a l ) + a^{2}\sin ( a l ) \right\}\, , \quad a = \pm \frac{( q - \bar{q})}{2}\, .
ab=ab|\begin{matrix}1&0\\ 0&1\end{matrix}|=a|\begin{matrix}1&0\\ 0&b\end{matrix}|=|\begin{matrix}a&0\\ 0&b\end{matrix}|=b|\begin{matrix}a&0\\ 0&1\end{matrix}|=ba|\begin{matrix}1&0\\ 0&1\end{matrix}|=ba,
K_t^U :=\exp\left(\sum_{0<s\le t}\log(1+F(X_{s-}^U,X_s^U))-\int_0^t \int_U F(X_s^U,y)J^X(X_s^U,y)dy\, ds-\int_0^t q_U(X_s^U)ds\right) &=\exp\left(\sum_{0<s\le t}\log(1+F(X_{s-}^U,X_s^U))\right)\, .
u_{k+1}^\delta = \begin{cases}u_k^\delta\int_\Sigma \frac{a_{[k]}(s,\cdot) \, y_{[k]}^\delta(\cdot)}{(A_{[k]}u_k^\delta)(s)}\, ds d(y_j^\delta, A_j u_k^\delta) > \tau \gamma \delta_{[k]}, u_k^\delta , \end{cases}
\mathcal{M}_{|\Delta_n|^2}(s) \approx \frac{\left({\alpha_n}/{b_{0,n}}\right)^{\alpha_n}}{\Gamma(\alpha_n)}& \times \sum_{j=0}^Jw_j\frac{{t_j}^{\alpha_n-1}}{2{t_j}s+1}\exp\left(-\frac{\tilde{A}_ns}{2{t_j}s+1}-\frac{\alpha_n{t_j}}{b_{0,n}}\right)
(\partial_{t}^+\phi_h^n,\phi_h^n)_{\mathcal{T}_h}+\epsilon\|\partial_{t}^+u_h^n\|^2_{\mathcal{T}_h}=\epsilon^{-1}(\partial_{t}^+f^n(u_h^n),\phi_h^n)_{\mathcal{T}_h}.
\begin{aligned}\dot{x}= \omega+\xi(y,z,\omega,\sigma,\mu)+f(x,y,z,\omega,\sigma,\mu), \dot{y}= \sigma+\eta(y,z,\omega,\sigma,\mu)+g(x,y,z,\omega,\sigma,\mu), \dot{z}= Q(\omega,\mu)z+\zeta(y,z,\omega,\sigma,\mu)+h(x,y,z,\omega,\sigma,\mu).\end{aligned}
\begin{array}{lll}\lambda_3(1-\lambda_3)b_1+2\lambda_3(1-\lambda_3)a_1+\lambda_2\lambda_3b_1a_1-\lambda_2\lambda_3b_1a_3+\lambda_2\lambda_3a_1^2-\lambda_2\lambda_3a_1a_3\ & \ \ =(\lambda_2-1)(1-\lambda_3),\end{array}
C(X)({\mathbb F}_q) =C(X)\cap{\mathbb A}^{n+1}({\mathbb F}_q)=Z_{\overline{\mathbb F}_q}\langle f_i,g_j\rangle\cap Z_{\overline{\mathbb F}_q}\langle x_k^q-x_k: 0\leq k\leq n\rangle &=Z_{\overline{\mathbb F}_q}(J),\qquad.
({\rm Mass})^2 ={1\over 4\rho_2^{(0)}}\left[{1\over 2}(\hat{\alpha}_L)^2 + N_L - 1\right]={1\over 4\rho_2^{(0)}}\left[{1\over 2}(\hat{\alpha}_R)^2 + N_R -\delta\right] ~.
\langle G_{\varphi}(x), x-x_p \rangle \geq \varphi(x^+)-\min \varphi +\frac{t}{2}\|G_{\varphi}(x)\|^2 \geq \varphi(x^+)-\min \varphi +\frac{t\epsilon}{2}(\varphi(x)-\varphi(x^+)) = \frac{t\epsilon}{2}(\varphi(x)-\min \varphi)+(1-\frac{t\epsilon}{2})(\varphi(x^+)-\min \varphi) \geq \frac{t\epsilon}{2}(\varphi(x)-\min \varphi),
R^{\sf SCF}_{\sf sym}(\mathbf{H}) = \min_{\ell = 1, \dots, L}\log\left( \frac{{\sf SNR}}{{\sf SNR}L_{\ell, \ell}^2}\right) &\!\!\!\!\!\!\!\!\!- \log\left( 2 \pi e G(\mathcal{L}') \right) - \log\left( \frac{\mu(\mathcal{L}, P_e)}{2 \pi e}\right),
\varphi_{3}(s,t_{0})= \left\lbrace \left[ \frac{\partial v(s,t_{0})}{\partial s}\right]^2 + \left[ \frac{\partial w(s,t_{0})}{\partial s}\right]^2 \right\rbrace^{-1/2}\frac{\partial v(s,t_{0})}{\partial s}.
{\cal J}_{i}^{0}\equiv - \frac \partial{\partial x}\psi^{i}- 2 \varepsilon^{i j k}\psi^{j}\pi_{k}= - \sqrt{1 -{\bf \varphi}^{2}}\varpi_{i}- \varepsilon^{i j k}\varphi^{j}\varpi_{k}\equiv J_{i}^{0}.
a^{(l, \sigma)}_-(y) \stackrel{\eqref{eq:thmaboutrelationbetweenboxdimensionandRokhlindimension-aminus}}{=}\left( \Lambda \left( \xi^{(l, \sigma)}(y) \right) - \left( \frac{1}{2}- 8\varepsilon \right) \right) \cdot 8L \, \leq 0 \; .
\sum_{n=1}^{N}L_n^{1/2}n^{5/4}\ll\sum_{m=0}^{\ell-1}p_{m+1}^{9/4}(p_{m+1}-p_m)^{3/2}&\leqslant\sum_{m=0}^{\ell-1}p_{m+1}^{11/4}(p_{m+1}-p_m) &\leqslant p_{\ell}^{11/4}\sum_{m=0}^{\ell-1}(p_{m+1}-p_m) &\leqslant p_{\ell}^{15/4}.
p \cdot P \biggl( \frac{|\sum_{k=1}^n x_{k1}x_{k2}- n \bar{x}_1 \bar{x}_2|}{(\sum_{k=1}^nx_{k1}^2)^{1/2}(\sum_{k=1}^n x_{k2}^2)^{1/2}}> C_0\sqrt{(\log p)/n}\biggr)\rightarrow0.
\sum_{m=0}^{N}(-y)^m\begin{vmatrix}e_{N+1-m}e_{N+2-m}e_{N+3-m}\cdots e_{N+\ell-m}e_0 e_1 e_2 \cdots e_{\ell-1}0 e_0 e_1 \cdots e_{\ell}\vdots \vdots \vdots \cdots \vdots 0 0 0 \cdots e_1\end{vmatrix}.
D_tZ(\alpha,t)=\sum_{j=1}^2\frac{\lambda_j i}{2\pi}\frac{1}{\overline{\alpha-z_j(t)}}=\frac{\lambda i}{2\pi}\frac{\overline{z_1(t)-z_2(t)}}{\overline{(\alpha-z_1(t))(\alpha-z_2(t))}}
\mathbf{y}_l = \mathbf{H}_l^{\mathrm{T}}\mathbf{x}^{\mathrm{RF}}_l + \mathbf{n}_l = \mathbf{H}_l^{\mathrm{T}}\mathbf{W}^{\mathrm{RF}}\mathbf{W}^{\mathrm{BB}}_l \mathbf{s}_l + \mathbf{n}_l = \mathbf{H}_l^{\mathrm{T}}\mathbf{W}_l \mathbf{s}_l + \mathbf{n}_l,
r_k(1,\tilde{\lambda}) =H(m_k,\delta_k+n_k)-H(u_0,v_0) &=H(0,\delta_k)+A(m_k,n_k)-H(u_0,v_0) &=H(0,\delta_k)-H(u_0,v_0)+A\big(u_0+m_k-u_0,m_k\lambda-\delta_k+v_0-u_0\lambda\big) &=H(0,\delta_k)-H(u_0,v_0)+A(u_0,v_0)-A(0,\delta_k)+ (m_k-u_0) A(1,\lambda) &=H(0,\delta_k)-H(u_0,v_0)+A(u_0,v_0)-A(0,\delta_k)+ (m_k-u_0)c(1,\tilde{\lambda})
ds^2_X(V,V) = \frac{\left(|Vs_0(x)|^2 + |Vs_1(x)|^2 \right) |s_0(x)|^2 - \left(s_0(x) \overline{Vs_0(x)}\right)\left(Vs_0(x) \overline{s_0(x)}\right)}{|s_0(x)|^4}= \frac{|Vs_1(x)|^2}{|s_0(x)|^2}.
|-2z+\lambda_iz^2| \leq 2c_1+\lambda_ic_1^2 &\leq 2\left(\sqrt{\frac{1}{2}+\frac{1}{\lambda_i^2}}-\frac{1}{\lambda_i}\right)+\lambda_i\left(\sqrt{\frac{1}{2}+\frac{1}{\lambda_i^2}}-\frac{1}{\lambda_i}\right)^2=\frac{\lambda_i}{2}.
(\begin{matrix}\hat{a}_{a}^{\dagger}\\ \hat{a}_{b}^{\dagger}\end{matrix})=(\begin{matrix}r_{ac}^{*}&t_{ad}^{*}\\t_{bc}^{*}&r_{bd}^{*}\end{matrix})(\begin{matrix}\hat{a}_{c}^{\dagger}\\ \hat{a}_{d}^{\dagger}\end{matrix}).
\Big\langle \sum \limits_{i=1}^{m}\alpha_i \tilde{f}^{|u_i\rangle}\Big| \sum \limits_{i=1}^{m}\beta_j \tilde{f}^{|v_j\rangle}\Big\rangle_{{}_{\mathfrak{J}, t}}= \sum \limits_{i,j=1}^{m}\overline{\alpha_i}\beta_j \langle u_i | v_j \rangle_t,
l(2\lambda+l)\,C_l^\lambda(t) =2\lambda l\left[C_l^{\lambda+1}(t)-t\,C_{l-1}^{\lambda+1}(t)\right] &=2\lambda\left[l\,C_l^{\lambda+1}(t)-(l-1)t\,C_{l-1}^{\lambda+1}(t)-t\,C_{l-1}^{\lambda+1}(t)\right],
\left(\begin{array}{cc}\mu_1 0 0 \mu_2 \end{array}\right)\left(\begin{array}{cc}0 d_1 d_2 0 \end{array}\right)\left(\begin{array}{cc}\bar{\mu_1}^{-1}0 0 \bar{\mu_2}^{-1}\end{array}\right) =\left(\begin{array}{cc}0 1 1 0 \end{array}\right).
\begin{array}{c}{D^{B}H^{A}= 0 , \qquad \qquad D^{B}D^{A}\Phi_{A}= [ H^{A}, \Phi_{B}] ,}{D^{A}K^{B}= 0 , \qquad \qquad D^{A}D^{B}\Phi_{B}= [ K^{B}, \Phi_{A}] .}\end{array}
\begin{aligned}\kappa_\ast =\frac{n+3}{n-1}, \kappa_2=\frac{(n+1)^2-6}{(n-1)^2-2}, \kappa_1 =\frac{(n+1)^2}{(n-1)^2 +4}n \ge 3, \kappa_0 =1+\frac3n, \kappa_3=\frac{n+1}{n-3}n \ge 4.\end{aligned}
\left\Vert \Gamma^*\sharp Z\right\Vert_{\ell}\leq\frac{C_1}{2}e^{-\frac{\delta_1}{t}},\ \ \left\Vert \Gamma^*\sharp Z^2\right\Vert_{\ell}\leq\frac{C_1}{2^2}e^{-\frac{\delta_1}{t}},\cdots
[\begin{matrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{6}\end{matrix}]=[\begin{matrix}\frac{1}{E_{1}}&-\frac{\nu_{21}}{E_{2}}&0\\ -\frac{\nu_{12}}{E_{1}}&\frac{1}{E_{2}}&0\\ 0&0&\frac{1}{G_{12}}\\ \end{matrix}][\begin{matrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{6}\end{matrix}]
J^{( N )}( N - 1 ; 1 , \ldots , 1 ) = \frac{1}{\Lambda^{( N )}}\; \left( \prod m_{i}^{2}\right) \; \sum_{i = 1}^{N}\frac{1}{m_{i}^{2}}\; F_{i}^{( N )}\; J_{i}^{( N )}( N - 1 ; 1 , \ldots , 1 ) .
\Lambda_0 =\left(\frac{a_0(u)}{u_1}+\frac{a_1(u)}{u_1^2}+\frac{a_2(u)}{u_1^3}+\left(a_1(u)+\frac{2a_2(u)}{u_1}\right) \xi_1 + a_2(u)u_1\xi_1^2 \right)\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1}.
\int_{r=\widetilde{R}_2}(|\widetilde{u}|^2 +|\widetilde{h}\nabla \widetilde{u}|^2)\,d\sigma =\frac{1}{2}\int_{B(0,\widetilde{R}-\eta,\widetilde{R}+\eta)}(|\widetilde{u}|^2 + |\widetilde{h}\nabla \widetilde{u}|^2)\,d x.
\begin{aligned}\big ( \nabla F_{0}(s,r) - \nabla F_{0}(S,R) \big ) \cdot (s - S, r - R)^{\top}& = \nabla F_{0}(s,r) \cdot (s - S, r - R)^{\top}+ \nabla F_{0}(S,R) \cdot (S - s, R - r)^{\top}& \geq c_{3}| \nabla F_0(s,r)| + c_{3}| \nabla F_0(S,R)| - 2 c_4 & \geq c_3 | \nabla F_0(s,r) - \nabla F_0(S,R)| - 2 c_4.\end{aligned}
\|h_{2}\|_{L^{r_{2}}(\mathbb{R}^{n})}= \frac{\|a\|_{L^{r_{2}}(\mathbb{R}^{n})}}{|T_{2}^{*}(h_{1},g)(x_{0})|}\lesssim \frac{\|a\|_{L^{\infty}}|B(x_{0},r)|^{1/r_{2}}}{CN^{2n}}\lesssim \frac{r^{-n/p}r^{n/r_{2}}}{CN^{2n}},
\begin{array}{lcl}\Gamma_{\alpha}=\{(x, \alpha(x))\}\xrightarrow{1\times \alpha^{-1}}\{(x,x) \}=\Delta_{C \times C}\Gamma_{\beta}=\{(x, \beta(x))\}\xrightarrow{1\times \alpha^{-1}}\{(x, \alpha^{-1}\beta (x))\}=\Gamma_{\alpha^{-1}\beta}.\end{array}
\tilde{D}_{T}^{[ r , 0 ]}( x_{M}^{2}) \rightarrow \frac{\chi_{r , 0}\Lambda^{\frac{1}{2}}}{\left( 2 \pi \sqrt{x_{M}^{2}}\right)^{\frac{3}{2}}}e^{- \gamma_{r , 0}\sqrt{x_{M}^{2}}}\cos \left( \omega_{r , 0}\sqrt{x_{M}^{2}}+ \varphi_{r , 0}\right) ,
0 =a (t_\infty^\flat)^{p-1}\int_{\mathbb{R}^N}|u^\flat|^{p}dx-\frac{b(t_\infty^\flat)^{2\cdot p^\flat-1}}{p^\flat}\int_{\mathbb{R}^N}(K\ast |u^\flat|^{p^\flat}) |u^\flat|^{p^\flat}dx &=\left(a (t_\infty^\flat)^{p-1}-\frac{b(t_\infty^\flat)^{2\cdot p^\flat-1}}{p^\flat}\right) \int_{\mathbb{R}^N}|u^\flat|^{p}dx
(\mathbf{\tau}^{\ast},\mathbf{\nu}^{\ast})^{T}=\left(\begin{array}[c]{cc}J_{\varphi}J_{\theta}\sin\varphi -J_{\theta}\sin\varphi J_{\varphi}\end{array}\right) (\frac{\partial}{\partial\varphi},\frac{1}{\sin\varphi}\frac{\partial}{\partial\theta})^{T},
\varphi(0) =\alpha_0\varphi(0)+\beta_0\varphi(1)+\gamma_0(3-\mathbb{E}S), \varphi(1) =\alpha_1\varphi(0)+\beta_1\varphi(1)+\gamma_1(3-\mathbb{E}S), \varphi(2) =\alpha_2\varphi(0)+\beta_2\varphi(1)+\gamma_2(3-\mathbb{E}S).
R(m, j) = \frac{m!}{(m-j)!}B_{m-j}\left(H_m^{(1)}, -H_m^{(2)},\ldots, (-1)^{m-j-1}(m-j-1)!H_m^{(m-j)}\right) &= \frac{1}{j!}B_{j}\left(H_m^{(-1)}, \ldots, (-1)^{j}(j-1)!H_m^{(-j)}\right),
\gamma(\kappa, m, n)=\frac{n+1}4 -\frac{n+1}{4 \mu_\ast (m+2)}\cdot \frac{\mu_\ast (m+2)(n-1)+12 \mu_\ast +2m}{2n \kappa -(n+1)}-\frac m{2 \mu_\ast (m+2)}.
{}s_{1}\sim\frac{c_{2}\beta(t)^{2}}{tc_{1}}(1-e^{\alpha(t)\pi i}) =-\frac{4i\beta(t)^{2}}{\alpha(t)}\sin(\frac{\alpha(t)\pi}{2}) \sim \frac{2i}{uy^{\frac{1}{2}}}\sin{\frac{\alpha(t)\pi}{2}}
b_1(y_1,y_2) := \frac{1-\frac{1}{2}\sin(2\pi y_1)\sin(2\pi y_2)}{1+\frac{1}{4}(\cos(2\pi y_1)-2\sin(2\pi y_1))\sin(2\pi y_2)}, b_2(y_1,y_2) := \frac{1+\frac{1}{2}\sin(2\pi y_1)\sin(2\pi y_2)}{1+\frac{1}{4}(\cos(2\pi y_1)-2\sin(2\pi y_1))\sin(2\pi y_2)}
\langle u_\phi(t \wedge \tau), v \rangle + \int_0^{t \wedge \tau}\langle \nu Au_\phi + B(u_\phi,u_\phi) - \phi, v \rangle dt = \langle u_0,v\rangle + \sum_k\int_0^{t \wedge \tau}\langle g_k(u_\phi),v \rangle dW_k
\nabla f(\mathbf{x}^{k+1}) + \mathbf{E}_{}^T\boldsymbol{\alpha}^k + \frac{\rho}{2}\mathbf{E}_{}^T\mathbf{E}_{}\mathbf{x}^k + \frac{\rho}{2}(2\mathbf{D}+ \frac{2}{\rho}\mathbf{P})(\mathbf{x}^{k+1}- \mathbf{x}^k) = \mathbf{0},
\begin{cases}R_{i1}(P,X)=\int_{0}^{\tilde{t}_i}(1-t)\frac{d^2u^s}{dt^2}(Y_i(t,P,X))dt, &R_{i2}(P,X)=\int_{\tilde{t}_i}^{1}(1-t)\frac{d^2u^{s'}}{dt^2}(Y_i(t,P,X))dt, &R_{i3}(P,X)=(1-\tilde{t}_i)\int_{0}^{\tilde{t}_i}\frac{d}{dt}\left( (M^s(\widetilde{Y}_i)-I)\nabla u^s(Y_i(t,X))\cdot (P-X) \right)dt.\end{cases}
D_{F}^{N}( x ) = \frac{- i}{( 2 \pi )^{4}}\int \frac{S_{\vec{k}}( N + 1 )^{2}e^{i \vec{k}. \vec{r}- i k_{0}\, h ( N_{\vec{k}}) \, t}\, \, d^{4}k}{k^{2}+ m^{2}}- ( N \rightarrow N - 1 ) \, ,
e^{ih}=\prod_{k=1}^{n-1}\Big(\prod_{i=1}^N(u_i^{(k)},v_i^{(k)})\Big) (w^{(k)}_1,x^{(k)}_1)(w^{(k)}_1,x^{(k)}_1)(y^{(k)},z^{(k)})\cdot e^{ih_n}.
\begin{matrix}J_z = \sum\limits_{\mu = -1}^{1}\mu a_{\mu}^{\dagger}a_{\mu},J_+ = (J_-)^{\dagger}= \sum\limits_{\mu = -1}^{\mu = 0}\sqrt{(\mu + 2)(1 - \mu)}a_{\mu + 1}^{\dagger}a_{\mu}.\end{matrix}
\sum_{0\leq m<n<\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}}\right)}{p^{m(2-\alpha-w)}p^{n(1-\beta)}}= \frac{\tau_B(p )}{p^{1-\beta}}- \frac{\tau_B(p )}{p^{3-\beta-w}}+ O\left( \frac{1}{p^{1+\varepsilon}}\right).
f_\varepsilon'(t) = (f'(t)-2\varepsilon^2tf(t))e^{-\varepsilon^2t^2}, f_{\varepsilon}''(t) = (f''(t)-4\varepsilon^2tf'(t)+(4t\varepsilon^4-2\varepsilon^2)f(t))e^{-\varepsilon^2t^2}
f_{11}f_{15}t = f_{29}\widetilde{c}= f_{31}\widehat{b}\mbox{where}f_{28}\widetilde{c}\in f_{15}\overline{A}^\ast \mbox{and}f_{32}\widehat{b}\in f_{26}(\widehat{C}\cap f_{25}^{-1}f_{15}\overline{A}^\ast).
\mathbb{E}^{\mu}\left[\int_0^{\infty}\omega(X_s)ds\right] =\int_0^{\infty}\mathbb{E}^{\mu}\left[\omega(X_s)\right]ds &=\int_0^{\infty}\sum_{x\in\Gamma}\mathbb{P}^{\mu}(X_s=x)\omega(x)ds &=\sum_{x\in\Gamma}\omega(x)\int_0^{\infty}\mathbb{P}^{\mu}(X_s=x)ds &\leq \left(\int_0^{\infty}\mathbb{P}^{\mu}(X_s=id)ds\right)\sum_{x\in\Gamma}\omega(x)<\infty,
\begin{array}{c}{{V ( z_{1 2}^{\prime}) = U^{- 1}( z_{1}; g ) V ( z_{1 2}) U ( z_{2}; g ) \, ,}}\\{{{}}}\\{{V ( z_{2 1}^{\prime}) = U^{- 1}( z_{2}; g ) V ( z_{2 1}) U ( z_{1}; g ) \, .}}\\ \end{array}
\theta \left( \pi - g L a \right) \langle 0 , 0 ; 0_{\psi}| H_{C}| 0 , 0 ; 0_{\psi}\rangle + \theta \left( g L a - \pi \right) \langle 1 , 1 ; 0_{\psi}| H_{C}| 1 , 1 ; 0_{\psi}\rangle
d_n(X^-(fs_{i,k+m})) = d_n(X^-(f)s_{i,k+m}) = d_n(X^-(f))s_{i,k+m}+ (-1)^{p(f)}X^-(f) d_n(s_{i,k+m}) = X^-(d_n(f))s_{i,k+m}+ (-1)^{p(f)}X^-(f)Y_{n-i+1}, X^-(d_n(fs_{i,k+m})) = X^-(d_n(f)s_{i,k+m}) + (-1)^{p(f)}X^-(f d_n(s_{i,k+m})) &= X^-(d_n(f))s_{i,k+m}+ (-1)^{p(f)}X^-(f)Y_{n-i+1},
\det [G_{\lambda_{i}-i+j}(x|\tau^{1-j}t)]_{1\leq i,j\leq n}=\frac{\det[(x_{j}|t)^{n+\lambda_{i}-i}]_{1\leq i,j\leq n}}{\det[(x_{j}|t)^{n-i}]_{1\leq i,j\leq n}}
u(t,x) = \big( U(t)g \big)(x) = \; \frac{1}{\Gamma(\alpha)}\left( \frac{t}{2}\right)^{2\alpha}\int\limits_0^\infty r^{-\alpha}\exp^{-\frac{t^2}{4r}}e^{-rf(x)}g(x) \, \frac{dr}{r}= \; \frac{2 g(x)}{\Gamma(\alpha)}\left( \frac{t f(x)^{\frac{1}{2}}}{2}\right)^{\alpha}K_{\alpha}\left( t f(x)^{\frac{1}{2}}\right)
L = \begin{pmatrix}1-p_{10}-p_{11}\left(\frac{p_{01}}{1-p_{10}-p_{11}}e^t + \frac{p_{00}}{1-p_{10}-p_{11}}\right)(p_{10}+p_{11}) \left(\frac{p_{11}}{p_{10}+p_{11}}(e^t-1)+1\right)(p_{10}+p_{11}) p_{10}+p_{11}\end{pmatrix}\, .
I_{c s}= \frac{1}{4 \pi}T r \int_{M}\left[ \frac{1}{a_{+}}A^{+}\wedge \left( d A^{+}+ \frac{2}{3}A^{+}\wedge A^{+}\right) + \frac{1}{a_{-}}A^{-}\wedge \left( d A^{-}+ \frac{2}{3}A^{-}\wedge A^{-}\right) \right]
D_{q}^{n}f(x)=\frac{(-1)^{n}q^{-(\begin{matrix}n\\ 2\end{matrix})}}{(1-q)^{n}x^{n}}\sum_{r=0}^{n}(-1)^{r}(\begin{matrix}n\\r\end{matrix})_{q}q^{(\begin{matrix}r\\ 2\end{matrix})}f(q^{n-r}x).
\mathrm{SINR}_{D_{j}^{t},D_j^{r}}\!=\!\frac{\mathrm{SNR}_{D_{j}^{t},D_j^{r}}}{\mathrm{SNR}_{A_i,D_{j}^{r}}\!+\!\underset{D_k\in \mathcal D_u^i\backslash \{D_j\}}{\sum}\mathrm{SNR}_{D_{k}^{\,t},D_{j}^{r}}\!+\!1}.
\frac12\frac{d}{dt}\sum_{q\geq -1}\left(\lambda_q^{2s}\|u_q\|_2^2+\lambda_q^{2s}\|b_q\|_2^2\right) \leq -\nu\sum_{q\geq-1}\lambda_q^{2s+2}\|u_q\|_2^2-\mu\sum_{q\geq-1}\lambda_q^{2s+2\alpha}\|b_q\|_2^2+ I_1+I_2+I_3+I_4+I_5,
\|u\|_{\varphi_1}=\|v\|_{\varphi_2}\quad\Rightarrow\quad\left\|\frac{u}{\|v\|_{\varphi_2}}\right\|_{\varphi_1}=1\quad\Rightarrow\quad\varrho_{\varphi_1}\left(\frac{u}{\|v\|_{\varphi_2}}\right)=1=\varrho_{\varphi_2}\left(\frac{v}{\|v\|_{\varphi_2}}\right).
\mathbb{P}_{\!x}\{X_{\tau_{ab}}\in \mathrm{d}z,\tau_{ab}<+\infty\}/\mathrm{d}z=\sum_{k=0}^{N-1}(-1)^k H_k^-(x)\,\delta_a^{(k)}(z)+\sum_{k=0}^{N-1}(-1)^k H_k^+(x)\,\delta_b^{(k)}(z).
\begin{aligned}\lVert u^{\tau_k}(t) - u^{\tau_j}(t)\rVert \leq 2 \lVert u_{n_k+1}- u_{n_k}\rVert + \lVert u_{m_j+1}- u_{m_j}\rVert + t_{n_k}\lVert y_{n_k}\rVert \quad + t_{m_j}\lVert y_{m_j}\rVert + \left \lVert \sum_{\ell=n_k+1}^{m_j-1}t_\ell (w_\ell - r_\ell)\right\rVert. \end{aligned}
b_{n,d}=b_{n-1,d+r-2}+ (d-1) \cdot p_{n-1,d-2}+ d_{n-1,d-2}+ \sum\limits_{i=0}^{n-1}\sum\limits_{j=0}^{d-2}\bigl(p_{n-i-1,d-j-2}\cdot p_{i,j}+2\cdot s_{n-i-1,d-j-2}\cdot b_{i,j}\bigr).
\tilde{a}_{\tilde{n}}^{\alpha}= \int_{0}^{\pi}~ \frac{d \sigma_{-}}{\pi}\exp \left[ 4 i \tilde{n}\frac{e_{\mu}X_{R}^{\mu}( \sigma_{-})}{e_{\mu}P_{R}^{\mu}}\right] \xi_{i}^{\alpha}\partial_{-}X_{R}^{i}( \sigma_{-}) \, ,
\Bigl(a\wedge\Bigl\{\frac{f(\theta)}{2\theta^2}e_2+ \frac{g(\theta)}{4\theta^3}b\Bigr\}\Bigr)_{12}=0 \quad \Bigl(a\wedge\Bigl\{\frac{f(\theta)}{2\theta^2}e_2+ \frac{g(\theta)}{4\theta^3}b\Bigr\}\Bigr)_{13}=\beta
L_\lambda(\mathbf{x},\mathbf{z},\mathbf{y}) = f(\mathbf{x}) + g(\mathbf{z}) + \mathbf{y}^T(\mathbf{Ax}+\mathbf{Bz}- \mathbf{c})+ \frac{\lambda}{2}||\mathbf{Ax}+\mathbf{Bz}- \mathbf{c}||_2^2.
L ~=~ \frac{1}{2}(\partial_\mu \phi^{ai})^2 ~+~ \frac{u}{4!}\left( \phi^{ai}\phi^{ai}\right)^2 ~+~ \frac{v}{4!}\left[ \left( \phi^{ai}\phi^{bi}\right)^2 ~-~ \phi^{ai}\phi^{ai}\phi^{bj}\phi^{bj}\right]
G_\pm(t)u[x] =(HJ_a-J_aH_0)e^{-itH_0}\tilde P_\pm u[x] &=(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i\Phi(x,y,\xi;t)}s_a(x,\xi) p_\pm(y,\xi) u[y] d\xi.
\overline{\rm R}_{i'i'}=R_{i'i'}&-R(\frac{\nabla f}{|\nabla f|},e_{i'},e_{i'},\frac{\nabla f}{|\nabla f|})-\frac{1}{|\nabla f|^2}\sum_{k'}(R_{i'i'}R_{k'k'}-R_{i'k'}R_{k'i'}).
c+o(1) = \left(\frac{1}{p}-\frac{1}{2\cdot p^\sharp}\right) [u_n]_{s,p}^p+\left(\frac{1}{p_s^*}-\frac{1}{2\cdot 2^\sharp}\right) \int_{\mathbb{R}^N}(K\ast |u_n|^{p^\sharp}) |u_n|^{p^\sharp}dx &\geq \left(\frac{1}{p}-\frac{1}{2\cdot p^\sharp}\right) [u_n]_{s,p}^p.
L= \bar{\psi}i \gamma^{\mu}\partial_{\mu}\psi +{\bar \psi}\gamma^{\mu}(B_{\mu}+ A_{\mu}+ \epsilon_{\mu \nu}\partial^{\nu}\theta ) \psi+\partial^{\mu}\eta B_{\mu}+\frac{1}{2 \pi}\partial^{\mu}\theta\partial_{\mu}\theta + \frac{1}{\pi}\epsilon^{\mu\nu}A_{\mu}\partial_{\nu}\theta \quad.
\xi\frac{\partial}{\partial \xi},\,\,\,\,\,\xi^2\frac{\partial}{\partial \xi},\,\,\,\,\,\wp^{(k)}(u-p)\xi^2\frac{\partial}{\partial \xi}\in H^0(S, \Theta_S),\,\,\,\,\,k=0,...,n-2,
R(\theta_1,\theta_3)=\sqrt{\pi}\frac{\Gamma(\theta_1)\,\Gamma(\frac{1}{2}-(\theta_1+\theta_3))\,\Gamma(\theta_3)}{\Gamma(\frac{1}{2}-\theta_1)\,\Gamma(\theta_1+\theta_3)\,\Gamma(\frac{1}{2}-\theta_3)}\,.
u_j^\omega =g_{0,j}\psi(D)u_j+\sum_{n\in \mathcal I}g_{n,j}\psi (D-n)u_j + g_{-n,j}\psi (D+n)u_j &=g_{0,j}\psi(D)u_j+2{\rm Re}\sum_{n\in \mathcal I}g_{n,j}\psi(D-n)u_j, j=0,1.
W_{2r+1,1}(Z)= (q-1)\frac{q^{2r+1}-q^{2r}-q^r}{2}Z^{q^m-q^{m-1}-q^{m-r-1}-1}&+(q-1)\frac{q^{2r}+q^r}{2}Z^{q^m-q^{m-1}-q^{m-r-1}}&+(q-1)(q^m-q^{2r+1}+q^{2r})Z^{q^m-q^{m-1}-1}&+(q^m-q^{2r+1}+q^{2r})Z^{q^m-q^{m-1}}&+(q-1)\frac{q^{2r+1}-q^{2r}+q^r}{2}Z^{q^m-q^{m-1}+q^{m-r-1}-1}&+(q-1)\frac{q^{2r}-q^r}{2}Z^{q^m-q^{m-1}+q^{m-r-1}}.
f_{2g-L}(t)=\sum_{l=1}^{L}\frac{(-1)^{l-1}}{l}\sum_{\sum_{s=1}^lg_s=g}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}\prod_{s=1}^lh_{2g_s-I_s}^{\mathcal{L}_{\mathbf{I}_s}}(t).
\mbox{l.o.t.}:= -2[\partial_1\hat u_N-\hat\lambda\partial_1\hat H_N]\dot q^+\varphi-2[\partial_1\hat q]\varphi\partial_t\varphi-2[\partial_1\hat q](\hat u^-_2-\hat\lambda\hat H^-_2)\varphi\partial_2\varphi &-2[\partial_1\hat q](\partial_1\hat u^-_N-\hat\lambda^-\partial_1\hat H^-_N)\varphi^2\,.
({{\alpha}^{2 a_1}{z_1},{\alpha}^{2 a_2}{z_2},\cdots,{\alpha}^{2 a_4}{z_{d-1}},{\alpha}^{2 a_d}{z_d},{\alpha}^{a_{d+1}}{z_{d+1}},{\alpha}^{a_{d+2}}{z_{d+2}},{\alpha}^{-a}{\psi},{\alpha}^{-(d+1)a}{\phi}})\,\,\,,
T_{N,1}-T_{N,2}=\frac{1}{2\pi}\sum^N_{n=1}\sum^{\infty}_{j=m_0+1}\int_{\mathbb{R}}\widehat{K}(u) \prod^{j-1}_{k=1}\phi_{\varepsilon}(a_ku)(1-\phi_{\varepsilon}(a_j u))(e^{\iota u a_j \varepsilon_{n-j}}-\phi_{\varepsilon}(a_j u)) \mathbb{E}e^{\iota u \widetilde{X}_{n, j}}\, du.
\left|\frac{zf_2'(z)}{f_2(z)}\right| =\left|1+\frac{\rho_2}{(1-\rho_2^2)}\left( \frac{u\rho_2^2+4\rho_2+u}{\rho_2^2+u\rho_2+1}+\frac{v\rho_2^2+2\rho_2+v}{v\rho_2+1}+\frac{q \rho_2^2+4\rho_2+q}{\rho_2^2+q \rho_2+1}\right)\right| =1+\sin(1)
\begin{array}{l}[H,X^\pm]=X^\pm~, begin{align*}1mm][X^+,X^-]=[2H]~, begin{align*}1mm]\Delta(X^\pm)=X^\pm\otimes q^{-H}+q^H\otimes X^\pm~, \Delta(H)=H\otimes 1+1\otimes H~, \epsilon(X^\pm)=0=\epsilon(H)~, S(X^\pm)=-q^{\mp 1}X^\pm~,~~~~S(H)=-H~.\end{array}
\theta_{L_{36}}(q) = 1 + (42840 + 4096 \alpha)q^4 +(1916928 - 98304 \alpha)q^5+ \cdots, \theta_{S(L_{36})}(q) = \alpha q + (960 - 60 \alpha) q^3 + (3799296 + 1734 \alpha)q^5+ \cdots,
\begin{cases}\displaystyle\dot{\hat{u}}=-\frac{\nu}{\alpha^2}\hat{u}+\frac{\nu}{\alpha^2}G^{-2}\hat{u}-G^{-1}B(G^{-1}\hat{u},G^{-1}\hat{u})+G^{-1}g, \hat{u}(0)=\hat{u}_0=Gu_0.\end{cases}
m=k_4+\frac{k_1}{2}-1,\ \ \ \a=\left(k_5-\frac{k_1}{2}+\frac{k_2}{2k_3}\right)k_3,\ \ \ \ \sigma=2k_4+k_1\ \ \mbox{and}\ \ \rho=2(k_4+k_5)+\frac{k_2}{k_3}-1, k_{3}\ne 0.
\Gamma(f,h) = \frac{1}{2}\Big[\mathcal{L}^N(fh) - f\mathcal{L}^Nh - h\mathcal{L}^Nf\Big] = \frac{1}{\beta}\, \nabla^Nf\cdot\nabla^Nh \,, \Gamma_2(f,h) = \frac{1}{2}\Big[\mathcal{L}^N\Gamma(f,h) - \Gamma(f,\mathcal{L}^Nh) - \Gamma(\mathcal{L}^Nf, h)\Big]\,.
\begin{array}{rcl}Q_7^{'''}(x)\!\! \!\!\!=\!\!\! \!\!210(-47520\pi^2+475200)x^4+120(1024\pi^5+23760\pi^3 [1.0 ex]\!\! \!\!\! \!\!\! \!\!-285120\pi)x^3+60(498960\pi^2\!-\!4989600)x^2+24(-6963\pi^5 [1.0 ex]\!\! \!\!\! \!\!\! \!\!-249480\pi^3+2993760\pi)x-14968800\pi^2+149688000.\end{array}
\sum_{n = 1}^{\infty}\big| D^i \sigma_n(x) \big| \leq \frac{\kappa}{1 + |x|^{\eta}}\sum_{n, k = 1}^{\infty}\big|D^j [\sigma_n, \sigma_k](x) \big| \leq \frac{\kappa^2}{1 + |x|^{2\eta}}
\sup_{1\leq k\leq N}\langle v^k, v^k\rangle_a+h\sum_{k=0}^{N-1}\left\|\frac{v^{k+1}-v^k}{h}\right\|_{L^2(\Omega)}^2 \leq \|g\|_{L^2(Q_{\mathcal T})}^2+\langle v_0, v_0\rangle_a.
\bigl(B\gamma(u_m^+)\bigr)(\theta) =\bigl(q*\gamma(u_m^+)\bigr)(\theta) =\frac{1}{4i}(q_me^{im\theta}-q_{-m}e^{-im\theta}) +\frac{1}{\pi}\sum_{k=-\infty}^\infty \frac{q_{2km}}{(2k)^2-1}e^{2ikm\theta}.
\phi_{k,n}(y_1,\cdots,y_k)\leq n^k(2\ln n)^{\frac{k}{2}}\mathbb{P}(\xi^{(n)}(I_{n,k})=0) \leq n^k(2\ln n)^{\frac{k}{2}}\prod_{j=1}^k\mathbb{P}(\xi^{(n)}([y_j,y_j+G_n(x_j)/S(I)])=0).
d V ~ = ~ \left[ \mathrm{d e t}( - g ) ]^{1 / 2}\right] \mathrm{T r}_{U ( N )}~ \Delta e_{a_{0}}^{0 \prime}\cdots \Delta e_{a_{d - 1}}^{( d - 1 ) \prime}d \xi^{a_{0}}\cdots d \xi^{a_{d - 1}}\quad ,
\varliminf_{i \to \infty}\frac{\log_2 \lvert K P_i(G) \cap Z : P_i(G) \cap Z \rvert}{\log_2 \lvert Z : P_i(G) \cap Z \rvert}\le \varliminf_{k \to \infty}\frac{\log_2 \lvert K \varrho_k \cap Z \varrho_k \rvert}{\log_2 \lvert Z \varrho_k \rvert}= \lim_{k \to \infty}\frac{(2m-1) 2^{k-n-1}+ 1}{2^{k-1}+1}= (2m-1)/2^n.
\sum_{2r_0+r_1=3d_1-d_2+\lambda}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}\frac{(d_2+2)!}{2^{r_2}(d_1+\lambda)!}{n_1 \choose d_1+1}{n_2 \choose d_2+2},