harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\varepsilon^{-2}\{-k_d\partial_r^2\eta_0(r)\}+\varepsilon^{-1}\{-V_\Gamma\partial_r\eta_0(r)+k_dH\partial_r\eta_0(r)-k_d\partial_r^2\eta_1(r)\}+\partial^\circ\eta_0(r)-V_\Gamma\partial_r\eta_1(r)+k_dr|W|^2\partial_r\eta_0(r) -k_d\Delta_\Gamma\eta_0(r)+k_dH\partial_r\eta_1(r)-k_d\partial_r^2\eta_2(r) = f+O(\varepsilon).
p(0,y) = - p({\rm L},y) =\delta p \,\,\,{\rm for}\,\,\, 0 \leq y \leq H \,,{\bf{u}}(x,0) ={\bf{u}}(x,H) = 0 \,\,\,{\rm for}\,\,\, 0 \leq x \leq{\rm L}\, .
\begin{aligned}\hat{F}(s,\pi,\psi) =\hat{F}(T_\psi(s),0,\psi), \hat{G}(s,\pi,\psi) =-\hat{H}(T_\psi(s),0,\psi), \hat{H}(s,\pi,\psi) =-\hat{G}(T_\psi(s),0,\psi)\end{aligned}
\begin{array}{c c l}{{\tilde{T}^{\, ( + )}}}&{{=}}&{{i^{\, M}\, T \, \left( P_{\, ( + )}g_{\, 2 M - 2}^{\ast}g_{\, 2 M - 1}^{\ast}\, - \, i P_{\, ( - )}g_{\, 2 M - 2}g_{\, 2 M - 1}\right)}}\\{{\tilde{T}^{\, ( - )}}}&{{=}}&{{i^{\, M}\, T \, \left( P_{\, ( + )}g_{\, 2 M - 2}g_{\, 2 M - 1}\, + \, i P_{\, ( - )}g_{\, 2 M - 2}^{\ast}g_{\, 2 M - 1}^{\ast}\right)}}\\ \end{array}
\dfrac{-\sum_{n\geq 1}\dfrac{R(n)^{t}}{n^{s}}\ln n}{\sum_{n\geq 1}\dfrac{R(n)^{t}}{n^{s}}}=\sum_{p}\dfrac{p^{s}\ln p(p^s-1)-(p^s-1+p^t)p^s\ln p}{(p^s-1)(p^s-1+p^t)}=\sum_{p}\dfrac{(p^s-1-p^s+1-p^t) p^s \ln p}{(p^s-1)(p^s-1+p^t)}
\int_{\mathbb R^2}(\Delta a)(x) (\nabla F)(x) \cdot \nabla^{\perp}a(x) dx >0, \int_{\mathbb R^2}(\Delta a)(x) x_1 \partial_1 F(x) dx =0, \int_{\mathbb R^2}(\Delta a)(x) x_2 \partial_2 F(x) dx=0, \int_{\mathbb R^2}(\Delta a)(x) (x_1 \partial_2 F(x) + x_2 \partial_1 F(x) )dx =0.
\int_{0}^{\infty}dk\j_{l_{1}}\left( kR\right) j_{l}\left( ka\right)j_{l^{\prime}}\left( ka\right) & =\frac{\pi^{3/2}}{8a}\delta_{l+l^{\prime},l_{1}}\left( \frac{a}{R}\right)^{l+l^{\prime}+1}\frac{\Gamma\left( \frac{1}{2}+l+l^{\prime}\right)}{\Gamma\left( \frac{3}{2}+l\right) \Gamma\left( \frac{3}{2}+l^{\prime}\right)},
\text E (x_j) = \sum_{m=1}^{M}a_m\mathcal P({x_j = a_m}|{\bf y}),{\text C}(x_j) = \sum_{m=1}^{M}[a_m - \text E(x_j)][a_m - \text E(x_j)]^*\mathcal P({x_j = a_m}|{\bf y}),{\underline{\text C}}(x_j) = \sum_{m=1}^{M}[a_m -\text E(x_j)]^2 \mathcal P({x_j = a_m}|{\bf y}).
\int\limits_{\mathbb{R}^{N}}\left\vert v\right\vert^{q}\frac{dx}{\left\vert x\right\vert^{\theta}}=d^{1+\frac{p-1}{p}q}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert D_{N,d,p}v\right\vert^{q}}{\left\vert x\right\vert^{N+\theta d-Nd}}dx
\big\|C\star \psi \star S\big\|_{L^1\cap L^\infty}\leq c\|E\|_{L^1\cap L^\infty}\Big[\frac{1}{m^4_t}{\bf 1}_{d=2}+ \Big(\frac{1}{m_t^{1/2}}+ \frac{1}{m_t^{5/2}}\Big){\bf 1}_{d=3}\Big] + c\Big(\frac{1}{m^4_t}+\frac{1}{m_t^{6}}\Big){\bf 1}_{d=2}+ c\Big(\frac{1}{m_t}+\frac{1}{m_t^{9/2}}\Big){\bf 1}_{d=3}.
\begin{aligned}\widehat{b}_i^{(1,2)}-\widehat{b}_i^{(N,1)}= Q_i^{K_1}, \widehat{b}_i^{(2,3)}-\widehat{b}_i^{(1,2)}= Q_i^{K_2}, \dots = \dots \widehat{b}_i^{(N-1,N)}-\widehat{b}_i^{(N-2,N-1)}= Q_i^{K_{N-1}}, \widehat{b}_i^{(N,1)}-\widehat{b}_i^{(N-1,N)}= Q_i^{K_N}, \end{aligned}
G_{s}^{\overbrace{\kappa \cdots \kappa}^{m}\overbrace{w \cdots w}^{n}\mu_{1}\cdots \mu_{s - n - m}}= \kappa^{m - 2}\left( - w \right)^{n}g_{s , ( s - n - m )}^{\mu_{1}\cdots \mu_{s - n - m}},
\begin{array}{c}{2 \Im m \displaystyle \int \displaystyle \frac{d x}{i}\varphi^{\prime}( x + i \epsilon ) \log ( 1 + f ( x + i \epsilon ) ) =}\\{= - 2 \Re e \displaystyle \int_{\Gamma}\displaystyle \frac{d u}{u}\log ( 1 + u ) - \displaystyle \frac{1}{2}\displaystyle \int d x G ( x ) \left( F^{2}( + \infty ) - F^{2}( - \infty ) \right)}\\ \end{array}
[\delta ( \epsilon_1),\delta ( \epsilon_2)] \zeta^A =\frac 12\partial_a\zeta^A \bar{\epsilon}_2 \gamma^a \epsilon_1 + \frac 14 \Gamma^A\bar{\epsilon}_2 \epsilon_1 -\frac 14 \gamma_a \Gamma^A \bar{\epsilon}_2\gamma^a \epsilon_1 \,.
-\int_{0}^{T}\int_{\Omega}\left\{\left|u-k\right|\,\frac{\partial\beta}{\partial t}+\displaystyle\sum_{j=1}^{d}F_{j}(u)\,\frac{\partial\beta}{\partial x_{j}}\right\}\,dx\,dt\leq \int_{\Omega}\left|u_{0}-k\right|\beta(0)\,dx
{H}^{2}(B(e,r) \cap \partial B_{\rho}\cap \Sigma) = \mathcal{H}^{2}(B(\rho e,R) \cap \partial B_{\rho}\cap \Sigma), &= \rho^{2}\mathcal{H}^{2}(B( e,\frac{R}{\rho}) \cap \mathbb{S}^{d-1}\cap \Sigma), &=\pi R^{2}, &=\pi (2 \rho^{2}+ \rho (r^{2}-1)-\rho^{3}).
v\left( \overline{\theta}_{h}\right) =\int_{\overline{\theta}_{h}}^{b}\phi^{-1}\left( \int_{\xi}^{b}h-\int_{y}^{b}h\right) dy\geq\int_{\overline{\theta}_{h}}^{b}\phi^{-1}\left( \int_{\overline{\theta}_{h}}^{y}h\right) dy.
E\prod_{i=\tau+1}^{n-1}[\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda\rho(S_i)\rho(S_{i+1})}]=E\prod_{i=\tau+1}^{n-1}[\frac{\lambda\rho(\widehat{S}_i)\rho(\widehat{S}_{i+1})}{1+\lambda\rho(\widehat{S}_i)\rho(\widehat{S}_{i+1})}],
\left(x\right)_{n}=\sum_{l=0}^{n}\dbinom{n}{l}C_{l}^{\left(r\right)}\left(x\right)D_{n-l}^{\left(r\right)}=\sum_{l=0}^{n}\dbinom{n}{l}D_{n-l}^{\left(r\right)}\left(x\right)C_{l}^{\left(r\right)}.
Q(x_1,\dots,x_d) =Q(y_1-\sum_{i=2}^d x_i,x_2,\dots,x_d) &=Q\circ S\circ r(x_2,\dots,x_d) &=Q\circ S\circ r(x_2',\dots,x_d') &=Q(x_1',\dots,x_d').
\left\{f_k(u)\left(\prod_{\scriptsize{j\ne k}\atop\scriptsize{1\le j\le \lambda}}q_j\right)(u-1)^2+(-1)^{\lambda}\left(\prod_{\scriptsize{j\ne k}\atop\scriptsize{1\le j\le \lambda}}p_j\right)u\right\}\cdot(u^a-1)(u^b-1)=\pm u^l (u-1)(u^c-1)
x\left(\begin{array}{c}P_0(x) P_1(x) P_2(x) \vdots\end{array}\right)=\left(\begin{array}{ccccc}a_0 \mathbb{I}_p b_1 a_1 \mathbb{I}_p &b_2 a_2 \mathbb{I}_p & \ddots \ddots \ddots\end{array}\right)\left(\begin{array}{c}P_0(x) P_1(x) P_2(x) \vdots\end{array}\right).
\{\rm Im}{\cal A}^{(oo\to oo)}_{\rm annulus}\ \ \stackrel{v\to1_{-}}{\sim}\ \ \ \Bigl\vert{\cal A}^{(oo\to oo)}_{\rm disk}(\pi/2) \Bigr\vert^2 \sim e^{- 2s \alpha^\prime\log 2}\ .
\begin{aligned}F(x^{(\ell)}) - \eta^k \ge \frac{2\varphi(\alpha)}{1 + 2\varphi(\alpha)}\big( F(x^{(\ell)}) - F(x^*)\big) \ge \frac{2\varphi(\alpha)}{1 + 2\varphi(\alpha)}\big( F(x^{(\ell)}) - F(x^{(\ell +1)})\big).\end{aligned}
\widetilde{\mathcal{R}}_{}\! =\! \bigcup_{P_{U|\widetilde{X}}}\!\Big\{\left(R_s,R_\ell,R_w\right)\!\colon\! &0\leq R_s\leq I(U;Y), &R_\ell\geq I(U;X) - I(U;Y), &R_w\geq I(U;\widetilde{X})- I(U;Y)\Big\},
\Gamma^{\infty}(\hat{x},y;a) = \frac{k^2_p}{\omega^2}\frac{^{i\frac{\pi}{4}}}{\sqrt{8\pi k_p}}^{-ik_p\hat{x}\cdot y}(\hat{x}\cdot a) \hat{x}+ \frac{k^2_s}{\omega^2}\frac{^{i\frac{\pi}{4}}}{\sqrt{8\pi k_s}}^{-ik_s\hat{x}\cdot y}(\hat{x}^{\bot}\cdot a) \hat{x}^{\bot},
\mathcal{I}_{\alpha}(\vec{f})(x) = \int_{{(\mathbb{R}^n)}^m}\frac{\prod_{i = 1}^{m}f_i(x - y_i)}{{|(y_1, \cdots, y_m)|}^{mn - \alpha}}\,d\vec{y},\quad \ \ 0< \alpha<mn,
\Gamma= \Gamma_0 + \Gamma_1=\beta \sum_{N=1}^K \frac{N_N v^2_N}{2R}-\frac{15 l^9_P}{16 R^3}\sum_{M<N, \N=1}^K \int_0^{\beta}dt \frac{N_MN_M v^4_{MN}}{(v^2_{MN}{t}^2 + r^2_{MN})^{\frac{7}{2}}}
T^{\nabla^{E}}(u, v, w) = -{\mathcal H}(X, Y, Z) - (R(X, Y), t)^{\mathcal G}- (R(Y ,Z), r)^{\mathcal G}- (R(Z, X), s)^{\mathcal G}+ ( [r,s]^{\mathcal G}, t)^{\mathcal G},
G_t^p(x,y,N) =\frac{\Big[1-\frac{P_N^p\big(A^c_{x,y}(T)\big)}{P_N^p\big(\eta_t(x)=1\big)}\Big]\Big[1-\frac{P_N^p\big(A^c_{x,y}(T)\big)}{P_N^p\big(\eta_t(x)=0\big)}\Big]}{P_N^p\big(A_{x,y}(T)\big)}&\geq\frac{\Big[1-\frac{P_N^p\big(A^c_{x,y}(T)\big)}{p}\Big]\Big[1-\frac{P_N^p\big(A^c_{x,y}(T)\big)}{1-f(T,p)}\Big]}{P_N^p\big(A_{x,y}(T)\big)}
\frac{x^{2}}{U^{2}\delta^{*}}\frac{1}{\rho}\frac{\partial P}{\partial y}=\frac{1}{2}\eta f^{(3)}+\frac{1}{2}f^{\prime\prime}-\frac{1}{4}ff^{\prime}+\frac{1}{4}\eta f^{\prime2}+\frac{1}{4}\eta ff^{\prime\prime},
\mathbf{p}_i^{[k+1]}= \underset{\mathbf{p}_i}{}\, f_i(\mathbf{p}_i,k+1) + \boldsymbol{\lambda}_i^{[k]T}\mathbf{p}_i + \frac{\rho}{2}\|\mathbf{p}_i - \mathbf{q}_i^{[k]}\|^2
\begin{array}{l}W_{q}=-\frac{1}{2}\int dx v(x)\int_{0}^{\infty}\frac{d\tau}{\tau}\tau^{-\frac{2}{1+\gamma}}\cr \Big(\exp(-\frac{\tau}{2}Q)\frac{u_{+}\tau}{\sinh u_{+}\tau}\frac{u_{-}\tau}{\sinh u_{-}\tau}\cosh (u_{+}\tau+u_{-}\tau)\cosh(u_{+}\tau-u_{-}\tau) -1 +\frac{\tau}{2}Q \Big)\end{array}
\begin{array}{cc}\mathring{K}_5[H_1,H_2]=0 \textup{in}\ [2M,R_1]\cup [2R_1,\infty), \mathring{K}_5[H_1,H_2]\geq -\frac{30C_0}{(\log 2)r}|\nabla_L H_1|^2-\frac{3C_0}{(\log 2)r}|\nabla_L H_2|^2 \textup{in}\ [R_1,2R_1].\end{array}
Y^T_t:= \mathcal S(T)Y_0+\int_0^t \mathcal S(T-s)\alpha_sds+\int_0^t \mathcal S(T-s)\sigma_sdW_s = \tilde Y_0+\int_0^t \tilde{\alpha}_sds+\int_0^t \tilde{\sigma}_sdW_s,t\in [0,T],
\begin{array}{r c l}{{\cal A}( r , \omega ; \omega_{\bot})}{=}{\displaystyle \frac{1}{\pi}\int_{- \infty}^{\infty}d \omega^{\prime}\frac{\mathrm{I m}~{\cal A}( r , \omega^{\prime}; \omega_{\bot})}{\omega^{\prime}- \omega - i \epsilon}.}\end{array}
V_2V_1^*f(z_1,z_2)=M_{z_2}M_{z_1}^*R_{\overline q}f(z_1,z_2)=M_{z_2}M_{z_1}^*f(\overline qz_1,\overline qz_2)=M_{z_1}^*z_2f(\overline qz_1,\overline qz_2) &V_1^*V_2f(z_1,z_2)=M_{z_1}^*R_{\overline q}M_{z_2}f(z_1,z_2)=\overline qM_{z_1}^*z_2f(\overline qz_1,\overline qz_2).
S_R = - \int d^2x\,\sqrt{g}\,\left\{\frac{1}{2}Zg^{\mu\nu}\partial_\mu \Phi \partial_\nu \Phi + CR +V +\frac{1}{\epsilon}\left( - \frac{V'}{C'}+ \frac{ZV}{2{C'}^2}\right) \right\}.
Rf(\omega, \omega \cdot u) =\frac{-2\pi^{-n/2}}{\Gamma(\frac{n}{2})}P_{(n+1)/2}(\Delta_S)\left\{\int\limits_{S^{n-1}}\int\limits_0^\pi Cf(u,\beta,\psi)\log{\frac{1}{|\omega\cdot \beta|}}\sin\psi d\psi d\beta \right\}+\frac{\Gamma(\frac{n+1}{2})}{\pi^{(n+1)/2}}\int\limits_{S^{n-1}}\int\limits_0^\pi Cf(u,\beta,\psi)\sin\psi d\psi d\beta,
\frac{1}{r}\Bigl(\frac{v}{|B|}\Bigr)^{\frac{1}{N}}=\Bigl(\frac{|B_g(y_0,r)|_g}{r^{N}|B|}\Bigr)^{\frac{1}{N}}= 1 - \frac{1}{6N(N+2)}S(y_0)r^2+o(r^2) &= 1 -\frac{1}{6N(N+2)}S(y_0)\Bigl(\frac{v}{|B|}\Bigr)^{\frac{2}{N}}+o\Bigl(\frac{v}{|B|}\Bigr)^{\frac{2}{N}}
+{\sum_{m = 1}^{\infty}}{\sum_{\alpha > 0}}\frac{\alpha^{2}}{2}\left[ E_{\alpha}^{( m )}\otimes E_{- \alpha}^{( - m )}+ E_{- \alpha}^{( m )}\otimes E_{\alpha}^{( - m )}\right] , \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
I_{(b, \tau)}=R_{k, \beta}\left(\frac{1}{(1-\delta)}\right)y^{\frac{1}{(1-\delta)}}+\frac{x}{2\pi i}\left( \int_{d-i \tau}^{d+i \tau}+\int_{d+i \tau}^{b+i \tau}+\int_{b-i \tau}^{d-i \tau}\right) R_{k, \beta}(z) \frac{(\alpha x^\delta)^{z}}{z(1-z+\delta z)}dz.
\begin{aligned}{\widetilde{\Delta}}r= \widetilde{\rm div}(\widetilde{\nabla}r)=\sum_{i=1}^n\widetilde{Hess}(r)(E_i,E_i)=\sum_{i=1}^n\langle\widetilde{\nabla}_{E_i}T,E_i\rangle_T = \sum_{i=1}^n\langle\nabla^T_{E_i}T,E_i\rangle_T=\sum_{i=1}^n Hess(r)(E_i,E_i),\end{aligned}
h_{1}= - f_{\; b c}^{a}\left( \frac{1}{2}A_{j}^{b}A_{k}^{c}B_{a}^{j k}+ \left( \partial^{l}B_{0 l}^{b}\right) \left( B_{a}^{0 i}A_{i}^{c}+ \eta_{a}^{\left( 2 \right) i}{\cal P}_{i}^{\left( 2 \right) c}+ \eta_{a}{\cal P}^{c}\right) \right) ,
E_{\mu}(z;q,t) =f_{\mu}(z;q,t)+\sum_{\substack{\nu \in \sigma(\mu) \nu \prec \mu}}c_{\mu,\nu}(q,t)f_{\nu}(z;q,t), f_{\mu}(z;q,t) =E_{\mu}(z;q,t)+\sum_{\substack{\nu \in \sigma(\mu) \nu \prec \mu}}d_{\mu,\nu}(q,t)E_{\nu}(z;q,t),
\Delta_2=\left({{e}^{q_2}}-{{e}^{q_1}}\right) \left({{e}^{q_3}}-{{e}^{q_2}}\right) \left({{e}^{q_2}}+{{e}^{q_1}}\right) \left({{e}^{q_3}}+{{e}^{q_2}}\right).
\omega_0(z) = J_{14}, \omega_1(z) = \frac{J_{24}b_1 q_{21}(0) - J_{13}b_1 q_{12}(1)}{b_1-b_2}, \omega_0(-z) = J_{32}, \omega_1(-z) = \frac{J_{31}b_2 q_{12}(0) - J_{42}b_2 q_{21}(1)}{b_2-b_1}.
M_4 C_5^{-1}C_4^{-1}=(C_1 C_2 + A_1 A_2) C_3 - M_4 C_5^{-1}A_5 B_5^{-1}B_4^{-1}A_4 C_4^{-1}=(C_1 C_2 + A_1 A_2) C_3 + (A_1 B_1^{-1}M_1 B_5^{-1}B_4^{-1}+ C_1 C_2 A_3)(-B_4 B_5 M_1^{-1}B_1 A_2 C_3) = C_1 C_2 C_3 -C_1 C_2 A_3 B_4 B_5 M_1^{-1}B_1 A_2 C_3.
\psi(y) = y^{-\frac{3}{4}}g\Bigl( \frac{m^2 y}{N}\Bigr) e \Bigl( \frac{2 \sqrt{y p}}{c}- \frac{T^2 c}{4 \pi^2 \sqrt{y p}}\Bigr) \widehat{k^*}\Bigl( \frac{M T c}{2 \pi^2 \sqrt{y p}}\Bigr).
ln(x)=\int_{1}^{x}\frac{1}{u}du=\int_{1}^{x}\int_{0}^{\infty}e^{-tu}dtdu=\int_{0}^{\infty}\int_{1}^{x}e^{-tu}dudt=\int_{0}^{\infty}\frac{e^{-t}-e^{-tx}}{t}dt
\begin{aligned}\nu_1(A) \leq \liminf_{n \rightarrow \infty}\frac{1}{h_2(r_n)}\int_A \int_X 1_{B(x,r_n)}(y) \nu_1(\mathrm{d}x) \nu_2(\mathrm{d}y) & = \liminf_{n \rightarrow \infty}\frac{1}{h_2(r_n)}\int_A \nu_1(B(y,r_n)) \nu_2(\mathrm{d}y) & = \liminf_{n \rightarrow \infty}\frac{h_1(r_n)}{h_2(r_n)}\nu_2(A).\end{aligned}
\begin{array}{c}{{\left\{Q_{\alpha}, Q_{\beta}\right\}= \left( S_{A}\right)_{\alpha \beta}}}\\{{\left( S_{A}\right)_{\alpha \beta}= \left( C \gamma^{m}\right)_{\alpha \beta}\, \, P_{m}+ \left( C \gamma^{m_{1}m_{2}}\right)_{\alpha \beta}\, \, \, Z_{m_{1}m_{2}}+ \left( C \gamma^{m_{1}\cdots m_{5}}\right)_{\alpha \beta}\, \, \, X_{m_{1}\cdots m_{5}}.}}\\ \end{array}
r =(d_p(u)+\varepsilon)^{-1/p'}\left\|(\lambda_1\delta_{(x_1,y_1)},\ldots,\lambda_n\delta_{(x_n,y_n)})\right\|_{p'}^{Lw}, s =(d_p(v)+\varepsilon)^{-1/p'}\left\|(\lambda'_1\delta_{(x'_1,y'_1)},\ldots,\lambda'_m\delta_{(x'_m,y'_m)})\right\|_{p'}^{Lw},
\forall\, g \in H, \forall\, \psi \in \mathcal{O}(H), \widetilde{g}\widetilde{h}= \widetilde{gh}, g\widetilde{h}= \widetilde{h}g, \widetilde{h}\psi = \psi\big(S^{-1}(h'')?\big)\widetilde{h'}.
\partial_{m^{2}}\psi_{n}^{( m )}( \xi ) = \frac{2 - n}{4 m^{2}}\psi_{n}^{( m )}( \xi ) + \frac{1}{2}i \xi ( m \sqrt{\xi})^{- n / 2}Z_{1 - n / 2}^{\prime}( i m \sqrt{\xi})
\|\pi_{N}^{\lambda}u-u\|_{H^1(\mathbb R)}\le \big\|\sqrt{1+x^2}\, (\pi_{N}^{\lambda}u-u)'\big\|_{L^2(\mathbb R)}+\|\pi_{N}^{\lambda}u-u\|_{L^2(\mathbb R)}& \le cN^{1-m}|u|_{{\mathbb B}^m_{\lambda}(\mathbb R)},
R_{\ell,\ell'}^{k,\kappa}= d_k - \sum_{k'\leq \kappa-1}R_{\ell,\ell'}^{k,k'}, R_{\ell,\ell'}^{\kappa,k}= d_k - \sum_{k'\leq \kappa-1}R_{\ell,\ell'}^{k',k}, R_{\ell,\ell'}^{\kappa,\kappa}= d_{\kappa}- \sum_{k\leq \kappa-1}d_k + \sum_{k,k'\leq \kappa-1}R_{\ell,\ell'}^{k,k'}.
K \wedge K = \frac{1}{8}\frac{\bar{\kappa}^{4}}{\lambda^{4}}( \hat{I}_{4})^{2}\epsilon ( x^{1 1})^{2}+ 3 6 d C^{\prime}\wedge d C^{\prime}+ \frac{6}{\sqrt{2}}\frac{\bar{\kappa}^{2}}{\lambda^{2}}\epsilon ( x^{1 1}) \hat{I}_{4}\wedge d C^{\prime}
(t-s)^{2\beta}\int_{s}^{t}e^{-\lambda (t-r)}(r-s)^{-\alpha}(t-r)^{\alpha-1}\mathrm{d}r\cr =(t-s)^{2\gamma}(t-s)^{2(\beta-\gamma)}\int_{0}^{1}e^{-\lambda (t-s)(1-v)}v^{-\alpha}(1-v)^{\alpha-1}\mathrm{d}v\leq(t-s)^{2\gamma}K(\lambda)
N(h)=\left[(h^{0})^{3}-\frac{3}{2}h^{0}\delta_{ij}h^{i}h^{j}+\frac{1}{\sqrt{2}}(h^{1})^{3}+\frac{3}{2\sqrt{2}}h^{1}[(h^{2})^{2}+\ldots+(h^{n})^{2}]\right].
\frac{f_1}{\eta^{2k_0}}= q^{\frac{r}{n}+\frac{e-1}{3}}\left(1+\sum_{x\geq 1}\frac{\prod_{i = 0}^{x-1}(3r+(2i+e-1)n)}{(3n)^xx!}(qg(q))^x\right)\times &\left(1+((e-1)n+3r)\sum_{m\geq 1}\left\{\frac{2^{8m}\prod_{i=2m+e}^{3m+e-2}(in+3r)}{m!n^m}\right\}(1+qg(q))^mq^{m}\right).
\lambda ={iq_1 A_q^{1/2}- q_2\over iq_2 A_q^{1/2}+ q_1}={q_1 q_2 (A_q -1)+ i (q_1^2 + q_2^2) A_q^{1/2}\over q_1^2 + q_2^2 A_q}\ .
\tau_{T_i,R_i}^C=\textstyle\min\left\lbrace\left(1-t_e-t_{d,T_i}\right)\,\log_2\left(1+\frac{\theta\eta_{_{T_i}}P_0 \rho_{_{T_i}}H_{T_i}^2\,t_e}{\sigma^2\left( 1-t_e-t_{d,T_i}\right)}\right)\right., &\qquad\qquad\qquad\;\;\textstyle\left.t_{d,T_i}\,\log_2\left(1+{P_0\,\phi_{_{R_i}}H_{R_i}}{\sigma^{-2}}\right)\right\rbrace.
\begin{aligned}B_n =\frac{2n\pi (-1)^n}{\sinh\left(\frac{n\pi}{2M}\right)}\int_{0}^{\infty}\frac{(1-x)\sin\left(M\log x\right)}{x\left[M^2\log^2(x)-\pi^2n^2\right]\left(e^{x}+1\right)}dx \end{aligned}
S_{5}= T_{5}\int_{W_{6}}\sqrt{- d e t G_{i j}}-{\frac{T_{5}}{4}}\int_{W_{6}}h_{3}\wedge * h_{3}- T_{5}\int_{W_{6}}C_{6}+{\frac{T_{5}}{2}}\int_{W_{6}}C_{3}\wedge d b_{2}.
d \Pi^{\underline{{m}}}= - i d \Theta^{1}\sigma^{\underline{{m}}}d \Theta^{1}- i d \Theta^{2}\tilde{\sigma}^{\underline{{m}}}d \Theta^{2}, d d \Theta^{1 \underline{{\mu}}}= 0 , d d \Theta_{\underline{{\mu}}}^{2}= 0 , \qquad ( I I A )
\|\theta_0\|_{L^1\left(\mathbb{T}^{n}\right)}\geq\|\theta^\epsilon_0\|_{L^1\left(\mathbb{T}^{n}\right)}=\int_{\mathbb{T}^{n}}\theta^\epsilon(x_t-\eta)d\eta\geq\int_{\mathcal{U}_2}\theta^\epsilon(x_t-\eta)d\eta\geq \frac{\theta^\epsilon(x_t)}{2}\left|\mathcal{U}_2\right|
\mathcal{R}_{f_1, f_2}(y, T) \ll T^{-10}+ \frac{1}{T}\Bigg( \frac{(q + T/Q)T}{q (\log T)^{1/9}}\Big) + T^{\varepsilon}\Big(\frac{(q + T/Q)T}{q T^{\gamma/4}}\Big) + \Big(q + \frac{T}{Q}\Big)T^{3\gamma}\Bigg) &\ll \frac{1}{(\log T)^{1/9}}+ \frac{T}{qQ}+ \frac{q}{T^{1 - 3\gamma - \varepsilon}}.
( \bar{A}\Gamma_{\hat{a}}B ) ( \bar{C}\Gamma^{\hat{a}}D ) = - \frac{1}{2}( \bar{A}\Gamma_{\hat{a}}e_{l}D ) ( \bar{B}\Gamma^{\hat{a}}e_{l}C ) + \frac{1}{2}( \bar{A}\Gamma_{\hat{a}}e_{l}C ) ( \bar{B}\Gamma^{\hat{a}}e_{l}D ) ,
\left( n_{1}-1\right) a_{1}+\left( n_{k}-1\right) a_{k}<\left(l-1\right) b_{1}+lb_{k}& =\left( l-1\right) \frac{n_{1}a_{1}}{l}+l\frac{n_{k}a_{k}}{l+1}& =n_{1}a_{1}-\frac{n_{1}a_{1}}{l}+n_{k}a_{k}-\frac{n_{k}a_{k}}{l+1}.
(\phi_*b_+)(t,x)=\left(0,\frac{1}{m}\gamma(t,q)(-p-\psi(t,q))\right), &(\phi_*b_-)(t,x) = \left(\frac{1}{m}(-p-\psi(t,q)),-\frac{1}{m}\delta^{ij}(-p_i-\psi_i(t,q))\nabla_q \psi_j(t,q)+\nabla_qV(t,q)-\tilde F(t,q,-p)\right).
\kappa_1(t,0)=\int_0^1\frac{r\,dr}{{{r}^{2}+2\,r\,\cos t+1}}+ \kappa_0(t,0) &\qquad\qquad=\frac{\arctan\left(\frac{\cos t}{\sin t}\right)}{\tan t}-\varphi(+\infty,t)=\frac{\arctan\left(\frac{1}{\tan t}\right)}{\tan t}-\frac{\pi}{2\tan t}.
\beta_1= \beta_2= \alpha_5=\beta_5=\gamma_5=0, \delta_5=-2\beta_3, \alpha_3 - \alpha_6 = \beta_3-\beta_6 = \gamma_3-\gamma_6 =0, \delta_3-\delta_6=-2\beta_4, \delta_5 = -2\beta_6.
\pi_Y^i\circ \Gamma\circ \pi^{2(d_X-p)+s+i}_X =({}^t\pi^{2(d_X-p)+s+i}_X\otimes\pi^i_Y)_*\Gamma & =(\pi^{2p-s-i}_X\otimes \pi^i_Y)_*\Gamma & = \left( (\sum_{j=0}^{2d_X}\pi_X^j)\otimes \pi_Y^i\right)_*\Gamma & = (\Delta_X\otimes \pi^i_Y)_*\Gamma & = \pi^i_Y\circ \Gamma,
\sum_{n=0}^{N}\frac{(1/c)_n q^n}{(q)_n}\frac{(q^{-N})_n}{\left(\frac{q^{-N}}{c}\right)_n}= \frac{(q^{-N})_N}{\left(\frac{q^{-N}}{c}\right)_N c^N}= \frac{(q)_N}{(cq)_{N}}.
K_{\delta,\xi}^{\bot}=\left\{\phi \in H^2(M)\slash \left\langle \phi , Z_{\delta,\xi}\right\rangle_{P_g}=0\and\ \left\langle \phi , Z_{\delta,\xi,\omega}\right\rangle_{P_g}=0,\ \forall \omega\in T_\xi M \right\}.
M_1= q^{\max\{n_2,n_4\}-n_3}(q^{n_3-n_1}-(q-1)^{n_3-n_1})\times &\sum_{\begin{subarray}{I}(u_{11},...,u_{1r_1})\in (\mathbb{F}_{q}^*)^{r_1}, a_{11}u_{11}+...+a_{{1r_1}}u_{1r_1}=b_1.\end{subarray}}N\big(\mathrm{x}^{E^{(1)}_i}=u_{1i}, 1\leq i\leq r_1\big). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3.12)
\Delta_{[ N + 2 k ] \times [ 2 k ]}\rightarrow \left( \! \begin{array}{c c}{1_{[ N ] \times [ N ]}}&{0_{[ 2 k ] \times [ N ]}}\\{0_{[ N ] \times [ 2 k ]}}&{\bar{\Lambda}_{[ 2 k ] \times [ 2 k ]}}\\ \end{array}\! \right) \Delta_{[ N + 2 k ] \times [ 2 k ]}\Lambda_{[ 2 k ] \times [ 2 k ]},
P ( t ) \equiv \sum_{L}b_{L}\, t^{L}= \frac{\prod_{j = 1}^{\kappa}( 1 - t^{2 j})}{\prod_{j = 1}^{N}( 1 - t^{2 j}) \prod_{j = 1}^{\kappa - N}( 1 - t^{2 j})}\, .
B=\limsup_{q\rightarrow \infty}\sup_{f\in L^{p}_{Q/p}(B(x_{0},r))\backslash\{0\}}\frac{\left\|\frac{f}{|x|^{\frac{\beta}{q}}}\right\|_{L^{q}(B(x_{0},r))}}{q^{1-1/p}\|f\|_{L^{p}_{Q/p}(B(x_{0},r))}}.
\left( \begin{array}{c}{{\hat{\alpha}( k )}}\\{{\hat{\beta}( k )}}\\ \end{array}\right) = \left( \begin{array}{c}{{{a}( k )}}\\{{i}}\\ \end{array}\right) \hat{\xi}( k ) + \left( \begin{array}{c}{{- i{b}( k )}}\\{{1}}\\ \end{array}\right) \hat{\eta}( k )
S(n,m):=\left\{(\{s_j\}_{j=1}^{m+1}, \{t_k\}_{k=1}^{n-m})\ \Bigg|\ \begin{array}{l}1=s_1<s_2<\cdots<s_{m+1}\le n, 1=t_1<t_2<\cdots<t_{n-m}\le n, \{s_j\}_{j=2}^{m+1}\cup \{t_k\}_{k=2}^{n-m}=\{2,3,\cdots,n\}, \{s_j\}_{j=2}^{m+1}\cap \{t_k\}_{k=2}^{n-m}=\emptyset.\end{array}\right\}.
S=-\frac{1}{2\kappa_5^2}\int_{{\cal M}_5}d^5x \sqrt{-g}\left[ R+\frac{1}{2V^2}\partial_\alpha V\,\partial^\alpha V+\frac{1}{3V^2}\alpha^2\right] -\sum_{i=1}^2 \frac{\sqrt{2}}{\kappa_5^2}\alpha_i\int_{{\cal M}_4^{(i)}}d^4 \xi_{(i)}\sqrt{-g}\, V^{-1}\,,
\sup_{\gamma \in \bar{\mathcal{P}}}\sup_{\pi \in \Pi\left(\mu_1, \mu_2, \gamma\right)}\int_{\mathcal{V}\times \mathcal{S}}\phi_\lambda \, d \pi = \sup_{\pi \in \mathcal{G}_{\lambda}}\int_{\mathcal{V}\times \mathcal{S}}\phi_\lambda \, d \pi.
N_d = \sum_{\substack{d_1+d_2=d d_1, d_2>0}}N_{d_1}N_{d_2}\left( \binom{3d-4}{3d_1-2}d_1^2d_2^2 - \binom{3d-4}{3d_1-1}d_1^3 d_2 \right)
\Big\| x - \sum_{m=1}^M \alpha_m x_{\sigma^m}\Big\|_{\widetilde{p}}< \varepsilon \mbox{and}\|x_{\sigma^m}\|_{h^{1_c}_{\widetilde{p}\widetilde{w}}(\sigma^m)}\le (1+\varepsilon) \lim_{\sigma,\mathcal{U}}\|x_\sigma\|_{h^{1_c}_{\widetilde{p}\widetilde{w}}(\sigma)}
\Psi_2(v_{\delta})=-c_1\,\partial_x (v_{\delta}^2)-\frac{c_2}{3}\partial_{xx}[(\partial_x^{-1}v_{\delta})^2]+\frac{c_2}{3}\pi_0[v_{\delta}^2]+c_3 \pi_0[(\partial_x^{-1}v_{\delta})^2]
m_{1}^{2}+ m_{2}^{2}= \gamma^{4}/ \eta^{2}, \quad m_{1}^{2}m_{2}^{2}= \gamma^{4}, \quad m_{1}^{2}- m_{2}^{2}= ( \gamma^{4}/ \eta^{2}) \sqrt{1 - 4 \eta^{4}/ \gamma^{4}}.
D_{\Delta_1 \Delta_2 \Delta_3 \Delta_4}(x_1,x_2,x_3,x_4) = \int_{z_0>0}\frac{d^5z}{z_0^5}\prod_{i=1}^4 \frac{z_0^{\Delta_i}}{(z_0^2+(\vec{z}-\vec{x_i})^2)^{\Delta_i}}.
\sum_{\alpha=1}^\infty \frac{|\varphi_\alpha(y)-\varphi_\alpha(z)|^2}{\lambda_\alpha}&\le \sum_{\alpha=1}^\infty \frac{8\wedge 2\lambda_\alpha |y-z|^2}{\lambda_\alpha}\le \int_0^\infty \frac{8}{\pi^2 u^2}\wedge 2|y-z|^2 du &\le \int_0^{\frac{2}{\pi |y-z|}}2|y-z|^2du+ \int_{\frac{2}{\pi |y-z|}}^\infty \frac{8}{\pi^2 u^2}du=\frac{8|y-z|}{\pi}.
g_{j}( \zeta ) = \sum_{n = 1}^{\infty}\overline{{g}}_{n}\overline{{\lambda}}_{n}^{2}\left[ \frac{d^{2}}{d y^{2}}\int_{\mathrm{a r c c o s h}( \large{y})}^{\infty}\frac{u d u}{\sqrt{\cosh u - y}}\frac{1}{\sinh^{j}( u / 2 )}\right]_{y = 1 + n^{2}\zeta^{2}/ 2},
\int \frac{x^2}{(x^2 q^{-1}(1-q);q^2)_\infty}\sin(x;q)d_qx = \frac{q}{(\frac{x^2}{q}(1-q);q^2)_\infty}\Big(x \sin( \frac{x}{q};q) - \cos (q^\frac{-1}{2}x;q)\Big),
\gamma_2=\lim_{x \rightarrow \infty}\left(\sum_{\substack{n \leq x n\in \mathcal{N}_2}}\frac{1}{n}-\kappa_2\log^{\alpha_2}x\right)=c_0\left(x_0\right)+\int_{x_0}^{\infty}\frac{\left(\alpha_2\kappa_2+o(1)\right)}{t(\log t)^{1-\alpha_2}}d t
\left\langle (-\mbox{div}(a(x)\nabla ( \phi_i \phi_j)), \phi_i \phi_j \right\rangle = \int_{\Omega}{a(x) ( \nabla (\phi_i \phi_j) \cdot \nabla (\phi_i \phi_j) ) dx}&\leq \left( \max_{x \in \Omega}{a(x)}\right) \| \nabla (\phi_i \phi_j)\|_{L^2}^2.
F_{\zeta\zeta}(\zeta(\alpha,t),t)=(\frac{\partial_{\alpha}}{\zeta_{\alpha}})^2F(\zeta(\alpha,t),t)=\frac{1}{\zeta_{\alpha}^2}\mathfrak{F}_{\alpha\alpha}-\frac{\zeta_{\alpha\alpha}}{\zeta_{\alpha}^3}\mathfrak{F}_{\alpha}.
\begin{array}{l}S_{10}:=(-5-s_{21}+s_{22}-5s_{23}+s_{31}-s_{32}-3s_{33})/2 begin{align*}2mm]\quad+(s_{11}-7s_{13}+s_{11}s_{13}+s_{11}s_{23}-s_{13}s_{21}+s_{13}s_{22})/3 begin{align*}2mm]\quad +(s_{11}^2-s_{12}^2+s_{13}^2-s_{11}s_{21}+s_{12}s_{22}+s_{13}s_{23})/6 begin{align*}2mm]\quad +(s_{11}^2-s_{12}^2)s_{13}/9+(s_{11}^3-s_{12}^3)/27, \end{array}
\sum_{j=1}^2 \frac{\lambda_j}{\pi}\frac{1}{(0-z_j)^2}\dot{z}_j(t)=\frac{\lambda i}{4\pi}\frac{1}{x}\sum_{j=1}^2 \frac{\lambda_j}{\pi}\frac{1}{(0-z_j)^2}=\frac{\lambda i}{4\pi x}\frac{\lambda}{\pi}\frac{4xyi}{(x^2+y^2)^2}=-\frac{\lambda^2 y}{\pi^2(x^2+y^2)^2}.
v_\varepsilon(p)-u(p) \leq \sup_\Omega (v_\varepsilon -u)=\limsup_{j\to+\infty}[v_\varepsilon(p_j) -u(p_j)] & = \limsup_{j\to+\infty}v_\varepsilon (p_j) -u(p) &\leq v_\varepsilon(p)-u(p) .
\left( V , 1 , 1 \right) = \left( \begin{array}{c c c}{1 + \frac{V^{2}}{2 a}}&{- \frac{V^{2}}{2 a}}&{- V^{T}}\\{\frac{V^{2}}{2 a}}&{1 - \frac{V^{2}}{2 a}}&{- V^{T}}\\{- V}&{V}&{{\bf 1}_{n - 2}}\\ \end{array}\right) \quad ,
{\Phi}^{(I)}=-\frac{s\,e^{2}F(1-F)}{2\pi |m|}\left[\frac{1}{6}(F - \frac{1}{2}) +\frac{1}{4\pi}\int_{1}^{\infty}\frac{dv}{v{\sqrt{v-1}}}\frac{C_{F}v^{F}-C_{F}^{-1}v^{1-F}}{C_{F}v^{F}+ 2 + C_{F}^{-1}v^{1-F}}\right]
\Bigl| \int u_{1,c_1}\overline{v}_{2,c_2}w_{3,\pm,c_3}dt dx \Bigr| \sim \Bigl| \int \bigl( \widetilde{u}_{1,c_1}* \widetilde{w}_{3,\pm,c_3}\bigr) \overline{\widetilde{v}}_{2,c_2}d\tau d k \Bigr| = \Bigl| \iint \widetilde{u}_{1,c_1}(\tau-\tau', k-k')\widetilde{w}_{3,\pm,c_3}(\tau',k') d\tau' dk' \overline{\widetilde{v}}_{2,c_2}(\tau,k) d\tau d k \Bigr|.
x_0^{-1}\delta\left(\frac{x_1-z}{x_0}\right) Y_{W_3}(v, x_1) I(w_1\otimes w_2) - x_0^{-1}\delta\left(\frac{-z+x_1}{x_0}\right) I(w_1\otimes Y_{W_2}(v, x_1)w_2) = z^{-1}\delta\left(\frac{x_1-x_0}{z}\right) I(Y_{W_1}(v,x_0)w_1\otimes w_2)
\begin{aligned}\frac{u}{(1+3a_+ u^3 t)^{1/3}}\leq \widetilde \Psi_t (u,\theta_n^0,G_n^0) \leq\frac{u}{(1+3a_- u^3 t)^{1/3}}, \frac{1}{(1+3a_+ u^3 t)^{1/(3\alpha)}}\leq \partial_u \widetilde \Psi_t (u,\theta_n^0,G_n^0) \leq\frac{1}{(1+3a_- u^3 t)^{\alpha/3}}.\end{aligned}