harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Big(\int_{0}^1{\int_{\Omega}{\Big\Vert \Big(\sum_{l=M}^{L-M}{\big| \sum_{n=1}^{L-l}{2^{-{s}{t_n}}{\Gamma}_{{t_l}}\ast h_{{n+l}}^{\omega,t}}\big|^q}\Big)^{{1}/{q}}\Big\Vert_{L^p}^p}d\lambda}dt\Big)^{{1}/{p}}&\gtrsim L^{-({s}+d-{d}/{q})/(2d)}\big(\log{L}\big)^{{1}/{2}}.
0 \leq \lim_{n\rightarrow \infty}\frac{1}{2\pi n\epsilon}\int_{0}^{2\pi}\bigg(\log|f_{n}(\mathrm{e}^{\mathrm{i}(x-\mathrm{i}\epsilon)})|-\log|f_{n}(\mathrm{e}^{\mathrm{i}x})|\bigg)\mathrm{d}x &= \frac{1}{\epsilon}(L(E,-\mathrm{i}\epsilon)-L(E,\mathrm{i}0))=-\omega^{-}(E,\mathrm{i}0).
4\gamma_{n}\left( \phi+l_{n-1}\right) \left( \phi+l_{n}\right) -\left(n-2\gamma_{n}\right)^{2}x^{2}=4\gamma_{n}\left( \phi+l_{n}+l_{n-1}-\frac{l_{n}l_{n-1}}{z^{2}}\right) \phi =\left[ 4\gamma_{n}\left( C_{n}+l_{n-1}\right) -\left( n-2\gamma_{n}\right)^{2}\right] \phi.
D^{(d)}_{n,i}:= \begin{cases}\left. (-1)^{i}\Delta_{k_1}^{i}\alpha(n;k_1,2d,3d,\ldots,nd)\right|_{k_1=d+1}, i=0,\ldots,n-1, \left. (-1)^{i}\,\,{^{\mathbf{z}_i}\Delta_{k_1}^{i}}\alpha(n;k_1,2d,3d,\ldots,nd)\right|_{k_1=d+1}, i=-n,\ldots,-1.\end{cases}
\begin{cases}\Delta \bar w_{1n}+e^{\bar w_{2n}}\left(1-\displaystyle r_n^2d_n^{-2}e^{\bar w_{1n}}\right)=0, \Delta \bar w_{2n}+e^{\bar w_{1n}}\left(1-\displaystyle r_n^2d_n^{-2}e^{\bar w_{2n}}\right)=0,\end{cases}B_{\frac d{2d_n}}(0)
\tilde{F}_{\varepsilon}\left( x,t\right) =\left[ \sqrt{\alpha_{\varepsilon}\beta_{\varepsilon}}\mu_{\varepsilon}\left( t\right) \right]^{\frac{n}{2}+2}V_{1}\left( \sqrt{\alpha_{\varepsilon}\beta_{\varepsilon}}\mu_{\varepsilon}\left( t\right) x,\beta_{\varepsilon}t\mu_{\varepsilon}\left( t\right) \right) ,
\left([\Re_A(T)]^{\sharp_A}\right)^2 + \left([\Im_A(T)]^{\sharp_A}\right)^2 =\frac{1}{2}\left[(T^{\sharp_A})^{\sharp_A}T^{\sharp_A}+ T^{\sharp_A}(T^{\sharp_A})^{\sharp_A}\right] &= \frac{1}{2}\left(TT^{\sharp_A}+ T^{\sharp_A}T\right)^{\sharp_A}.
4 \int_{S^2}\beta ( \epsilon_{pqi}\tilde X^q \tilde \nabla \tilde X^i) = \int_{S^2}\tilde \nabla^a \alpha_{ab}( \epsilon_{pqi}\tilde X^q \tilde \nabla \tilde X^i) = 0 5 \int_{S^2}D\beta ( \epsilon_{pqi}\tilde X^q \tilde \nabla \tilde X^i) = \int_{S^2}\tilde \nabla^a D \alpha_{ab}( \epsilon_{pqi}\tilde X^q \tilde \nabla \tilde X^i) = 0.
X ( \alpha_{1}, \alpha_{2}, \alpha_{1}^{\prime}, \alpha_{2}^{\prime}) = \exp \left[ i \alpha_{1}q^{1}{\frac{\partial}{\partial q^{1}}}+ i \alpha_{2}q^{2}{\frac{\partial}{\partial q^{2}}}+ i \alpha_{1}^{\prime}\overline{{{q}}}^{1}{\frac{\partial}{\partial \overline{{{q}}}^{1}}}+ i \alpha_{2}^{\prime}\overline{{{q}}}^{2}{\frac{\partial}{\partial \overline{{{q}}}^{2}}}\right]
\mathbf{K}_{i,n,l}(r\theta) = \frac{\varepsilon^{-d+1}}{\rho^\alpha}\int_{\mathbb{R}^d}\mathbf{1}_{2^{i-l-1}\rho<|r\theta-y|\leq 2^{i-l}\rho}\mathbf{K}_{n}(r\theta-y)\varphi_{\zeta\rho}(y)\frac{\langle\phi^{e,\varepsilon}(y/\rho),\eta^{\kappa}_{y_{\tau}}\rangle}{|y|^{d-\alpha}}&~~~~\times\left(\psi(2^{-i}\rho^{-1}y)-\psi(2^{-i+1}\rho^{-1}y)\right)dy.
\Phi^0_j(q_j, t) = -(1+o(1))\int_{t_0}^{t}\frac{\dot{\mu}_j(\tilde{s})}{\mu_j(\tilde{s})}\mu_j^{-(n-2s)}(\tilde{s})F\left(\frac{(t-\tilde{s})^{\frac{1}{2s}}}{\mu_j(\tilde{s})}\right)d\tilde{s}= c(1+o(1))
[\gamma^{2}-\frac{(1-\gamma^{2})^{2}c^{2}}{\gamma^{2}v^{2}}]x^{2}-[2\gamma^{2}v+2\frac{(1-\gamma^{2})c^{2}}{v}]tx+y^{2}+z^{2}=[c^{2}\gamma^{2}-v^{2}\gamma^{2}]t^{2}
\|v(a,\cdot)\|_{L^2}^2-\|v(b,\cdot)\|_{L^2}^2 = - 2\,\Re e\int_a^b{}_{\dot H^{-1}}\langle\partial_sv(s,\cdot),v(s,\cdot)\rangle_{\dot H^1}\,{\rm d}s & = 2\,\Re e \int_a^b \int_{{\mathbb{R}}^n}A(s,x)\nabla v(s,x)\cdot\overline{\nabla v(s,x)}\,{\rm d}x\,{\rm d}s & \ge 2\lambda\,\|\nabla v\|_{L^2(a,b;L^2)}^2.
-\int_0^L (\psi_x + \psi_{xxx})b dx -\int_0^L \psi_x \varphi_1\varphi_2 dx &+ \int_0^L \left(c_1 \varphi_1 -\sqrt{3}c_1\varphi_2-2 qa +2 q c\right)\psi dx=0,
\left[ \delta ( \omega - \omega^{\prime}) \delta ( p_{1}^{\prime}+ k_{1}^{\prime}- p_{1}- k_{1}) \right]^{2}={\frac{T L_{1}}{( 2 \pi )^{2}}}\delta ( \omega - \omega^{\prime}) \delta ( p_{1}^{\prime}+ k_{1}^{\prime}- p_{1}- k_{1})
q^{x}\cdot \Omega_{, \alpha r}\Omega_{, \beta s s}+ q^{- x}\cdot \Omega_{, \beta s}\Omega_{, \alpha r s}- ( q^{x}\cdot \Omega_{, \alpha s}\Omega_{, \beta r s}+ q^{- x}\cdot \Omega_{, \beta r}\Omega_{, \alpha s s}) ={\cal O}( s , \alpha , \beta ; q )
S_E={1\over{2\kappa^2_D}}\int d^Dx\sqrt{-g^E}\left[{\cal R}_{g^E}-{4\over{D-2}}(\partial\phi)^2-{1\over{2\cdot (p+2)!}}e^{2a_p\phi}F^2_{p+2}-{1\over{2\cdot D!}}e^{2a\phi}F^2_D\right],
\int_M \varphi P_g \varphi d\mu_g = \int_M\Big(|\nabla \Delta \varphi|_g^2-2T_2(\nabla \Delta \varphi,\nabla \varphi)-\frac{n-2}{2}\sigma_1(A)(\Delta_g \varphi)^2-T_4(\nabla \varphi,\nabla \varphi)+\frac{n-6}{2}Q_g \varphi^2\Big) d\mu_g.
B_1: = \bigg(\int_0^{\infty}\bigg(\int_0^x [\overline{u}]^q(t) w(t)\,dt\bigg)^{r / p}[\overline{u}]^q(x) V^{- r / p}(x) w(x)\,dx \bigg)^{1 / r}, B_2: = \bigg(\int_0^{\infty}W_*^{r / p}(x) \bigg[\sup_{0 < \tau \le x}\overline{u}(\tau)V^{- 1 / p}(\tau)\bigg]^{r}w(x)\,dx \bigg)^{1 / r}.
\ \varphi_k(0)=0,\ \varphi_k'\left(\lambda\right)=-\sum_{j=1}^{k-1}\frac{\lambda^{j-1}}{(j-1)!}e^{-\lambda}+\sum_{j=0}^{k-1}\frac{\lambda^j}{j!}e^{-\lambda}=\frac{\lambda^{k-1}}{(k-1)!}e^{-\lambda}
\begin{cases}-\Delta u = \lambda (-m_\infty u - a_\infty u^{p-1}) \mbox{in $D$}, \frac{\partial u}{\partial \mathbf{n}}= \lambda b_0 u^{q-1}\mbox{on $\Gamma_1$}, u=0 \mbox{on $\Gamma_0$}. \end{cases}
\widetilde{Y}=\big(\widetilde{y}_{ij}\big)_{i,j=1}^{r},\widetilde{y}_{ij}=\det X^{i,i+1,\ldots,i+s}_{j,r+1,\ldots,n}\,\big(\det X^{i+1,\ldots,i+s}_{r+1,\ldots,n}\big)^{-1}\quad(i,j=1,\ldots,r),
O_{F_1,F_2,\varepsilon}\left( B^{d\eta_d +\varepsilon}\right) = O_{F_1,F_2,\varepsilon}\left( N^{\eta_d(1+d\tau)+\varepsilon}\right)= O_{F_1,F_2,\varepsilon}\left(N^{\eta'_{d,F_1,F_2}+ \varepsilon}\right).
\partial_{\hat \xi}u(\hat \xi, 0) = U_1 + U_\beta \partial_{\hat \xi}H(\hat \xi, 0) = \partial_{\hat \xi}\hat w(0,0) + U_\beta (1+\beta) \hat \xi |\hat{\xi}|^{\beta-1}\cos{((1+\beta) \frac{\pi}{2})}.
\pi_{\varphi_1\bar{\otimes}\varphi_2}(a\otimes b) = \pi_{\varphi_1}(a)\otimes\pi_{\varphi_2}(b), a \in M_1, b \in M_2, \lambda^{\varphi_1\bar{\otimes}\varphi_2}(t) = \lambda^{\varphi_1}(t)\otimes\lambda^{\varphi_2}(t), t \in \mathbb{R}.
U_{\mp}(y, t) = \frac{1}{2 \pi i}\int_{(1000)}y^{-u}F(u) \frac{\gamma_{\mp}\Bigl(\dfrac{1}{2}+ u, t\Bigr)}{\gamma_-\Bigl(\dfrac{1}{2}, t\Bigr)}\frac{d u}{u^2}.
\rho (A^{-1})\leq \widehat{\lambda}_{k+1}=\max \left( \frac{A^{-1}\mathbf{x}_{k+1}}{\mathbf{x}_{k+1}}\right) =\overline{\lambda}_{k}-\min\left( \frac{\mathbf{x}_{k}+A^{-1}\mathbf{f}_{k}}{\mathbf{y}_{k+1}}\right) .
\left|r\frac{d}{dr}\left[{}_2\!F_{1}\left(\lambda,\lambda+1;2\lambda+1;\tilde{A}\right)\right]\right|\lesssim \frac{|1-z\bar{w}|^2}{|1-zw|^2}\frac{1-|zw|^2}{|1-z\bar{w}|^2}=\frac{1-|zw|^2}{|1-zw|^2},z=re^{i\theta}.
\chi_{V_{c,h}}(q)=\frac{q^{h-\frac{c}{24}}}{\prod_{n=1}^{\infty}(1-q^{n})}=\frac{q^{h-\frac{c-1}{24}}}{\eta(q)}=q^{h-\frac{c}{24}}(1+q+2q^{2}+3q^{3}+5q^{4}+\cdot\cdot\cdot),
\Phi_t: N\otimes_{\rm max}N^\circ \rightarrow\mathbb{C}\Phi_t (x\otimes y^\circ):=\frac{1}{(\xi_\tau|T_t\xi_\tau)_{L^2(N,\tau)}}(i_\tau (y^*) | T_t (i_\tau (x))_{L^2(N,\tau)}\, ,
<BR>i[M^{i0},M^{j0}]=3D\epsilon^{ij}(M-\Delta=),~~~\Delta=3D{{m^3}\over{4\pi=20}}<BR>\{(\int Q_1)^2-(\int Q_2)^2\}+{m\over{4\pi}}\{(\int P_1)^2-(\int=20P_2)^2\},<BR><BR>
\frac{\partial E}{\partial z^i_{\epsilon}}v^i= 2b^2f(x^1) \cos\left( x^2-\theta\right) \Big[ -\delta_{\epsilon 1}f(x^1)f'(x^1) \cos\left( x^2-\theta\right) +\delta_{\epsilon 2}\left\lbrace 1+f'^2\left( x^1\right) \right\rbrace \sin\left( x^2-\theta\right) \Big].
\prod_{1 < a < q_1/2 \atop (a, q_1) =1}{\xi}_a^{d_a}~=~ \prod_{1 < b < q_2/2 \atop (b, q_2) =1}{\xi}_b^{-e_b}\prod_{1 < c < q_3/2 \atop (c, q_3) =1}{\xi}_c^{-f_c}
\begin{array}{llllc}\underline{{\mbox{irrep of}}~SU(3)}\alpha=1 \phantom{6}{\bf Q}^{L}_{1}=0 \phantom{6}{\bf 1}\alpha=2{\bf Q}^{L}_{1}=\tilde{e}_{2},-\tilde{e}_{1},\tilde{e}_{1}- \tilde{e}_{2}{\bf 3}\alpha=3{\bf Q}^{L}_{1}=-\tilde{e}_{2},\tilde{e}_{1},\tilde{e}_{2}- \tilde{e}_{1}{\bf{\overline{3}}}\end{array}
\phi b^{m_0}xb^{n_k}\sim \phi b^{m_0}xb^{n_k+1}-\sum_{s\in S_b}\phi b^{m_0}xb^{n_k+1}s &\sim ... &\sim \phi b^{m_0}xb^{n_k+(m_k-n_k)}-\sum_{i=1}^{m_k-n_k}\sum_{s\in S_b}\phi b^{m_0}xb^{n_k+i}s &=\phi b^{m_0}xb^{m_k}-\sum_{i=n_k+1}^{m_k}\sum_{s\in S_b}\phi b^{m_0}xb^{i}s.
\partial_{r}v_{1}(t,r)+\partial_{r}v_{2}(t,r) = \int_{t}^{\infty}\frac{\lambda''(s) ds}{1+s-t}-\frac{b}{t^{2}\log^{b}(t)}+ E_{\partial_{r}v_{1}}(t,r)+E_{\partial_{r}v_{2}}(t,r)
\int_{\mathfrak{m}}\big|S(\alpha)\big|d\alpha\le \int_{\mathfrak{m}}\int_{[0,1]^3}\Big|\mathcal{F}(\alpha,\boldsymbol{\beta})\mathcal{H}_1(\alpha,\boldsymbol{\beta})\mathcal{H}_2(\alpha,\boldsymbol{\beta})\mathcal{H}_3(\alpha,\boldsymbol{\beta})\prod_{j=1}^{n-6}f_j(\alpha,\boldsymbol{\beta})\Big|d\boldsymbol{\beta}d\alpha.
\int_{\{r=r_0\}}|\hat{\mathcal{H}}^{i+1}_s|^2i_{\vec{n}}dVol(M)\leq \gamma_{_T}^4\varepsilon^4 P^{4i}\kappa_1^4 (\frac{\mathfrak{r}}{T^{i+1}})^{\frac{15}{16}}s^4\leq \varepsilon^2 P^{2i}\kappa_1^2 (\frac{\mathfrak{r}}{T^{i+1}})^{\frac{1}{2}}s^2
\begin{aligned}\|u\|_{L^{\infty}(\Omega_1)}\leq 1; \|f\|_{C^{-1,\alpha}(0)}\leq \delta; |g(x,t)|\leq \frac{\delta}{2}|(x,t)|^{1+\alpha},~~~~\forall~~(x,t)\in (\partial\Omega)_1; \|(\partial \Omega)_1\|_{C^{1,\alpha}(0)}\leq \frac{\delta}{2C_0}, \end{aligned}
\begin{aligned}{\textbf{I}_S}\left(\boldsymbol{\psi},{{\textbf{W}}}\right)^{-1}= \frac{{{\sigma_z^2}}}{{2{{\lvert\textbf{s}\rvert}^2}}}\left\{{{{\bf{I}}_{i{p_1}}}+{{\bf{I}}_{i{p_2}}}\left(\beta \right)}\right\},\end{aligned}
\lambda^{-2\gamma}\! \int_{\mathbb{R}^4}\!f_{1,m_1}\left(x+\widetilde{P}_1(t+s),y\right)\overline{f_{1,m_1}\left(x+\widetilde{P}_1(t),y\right)}&f_{2,m_2}\left(x,y+\widetilde{P}_2(t+s)\right)\overline{f_{2,m_2}\left(x,y+\widetilde{P}_2(t)\right)}\widetilde{\zeta}_\mathbf{m}(x,y,t,s)\,\mathrm{d}x\mathrm{d}y\mathrm{d}t\mathrm{d}s,
\left\langle \hat{M}_0^{n}\right\rangle(t) = \lambda_n t, \left\langle \hat{M}_i^{n}\right\rangle(t) = \frac{1}{n}\int_0^t(\hat{Q}^n_i(s)-\hat{Q}^n_{i+1}(s))ds, \left\langle \hat{M}_k^{n}\right\rangle(t) = \frac{1}{n}\int_0^t\hat{Q}^n_k(s)ds.
\mathbf{d}^{\overline{\mathcal{A}}}:\mathcal{F}(U,\mathfrak{g})\rightarrow\Omega^1(U,\mathfrak{g}),\quad\mathbf{d}^{\overline{\mathcal{A}}}\eta=\mathbf{d}f+\left[\,\overline{\mathcal{A}},f\right]
K(x_0,x_1,y_0,y_1,t_0,t_1) = x_{{0}}x_{{1}}y_{{0}}y_{{1}}- &t \left( d_{{-1,-1}}{x_{{1}}}^{2}{y_{{1}}}^{2}+d_{{-1,0}}{x_{{1}}}^{2}y_{{0}}y_{{1}}+d_{{0,-1}}x_{{0}}x_{{1}}{y_{{1}}}^{2}+d_{{0,0}}x_{{0}}x_{{1}}y_{{0}}y_{{1}}+d_{{0,1}}x_{{0}}x_{{1}}{y_{{0}}}^{2}+d_{{1,1}}{x_{{0}}}^{2}{y_{{0}}}^{2}\right).
|Z| \geq \left|\bigcup_{p\in[r+1]}N_{i}(w_p,Z) \right| \geq (r+1) \frac{n}{r}- \sum_{1 \leq q <p \leq r+1}|N_{i}(w_q,Z) \cap N_{i}(w_p,Z)| \geq |Z| \Big( \frac{r+1}{r}- \frac{\binom{r+1}{2}}{r^3}\Big) > |Z|.
\frac{f^{\prime}(z)f^{\prime}(w)}{(f(z)-f(w))^{2}}-\frac{1}{(z-w)^{2}}=\frac{\partial^{2}}{\partial z\partial w}log\frac{f(z)-f(w)}{z-w}=-\sum_{m,n\ge1}mnc_{mn}z^{m-1}w^{n-1}
|\xi -\eta|^p =\big( 1+ |\xi| + |\xi-\eta| \big)^{\frac{p(2-p)}{2}}\big( 1+ |\xi| + |\xi-\eta|\big)^{\frac{p(p-2)}{2}}|\xi -\eta|^p \leq \tau \, 3^{-p}\big( 1+ |\xi| + |\xi-\eta|\big)^p + C_p \tau^{\frac{p-2}{p}}\big(1+ |\xi| + |\xi-\eta|\big)^{p-2}|\xi-\eta|^2.
\begin{array}{c}G_{\rho \sigma \nu}^{abcd}=\beta_1(G_{\rho \sigma \nu}^{(1)}+G_{\rho\sigma \nu}^{(2)})-\beta_2(G_{\rho \sigma \nu}^{(1)}+G_{\rho \sigma \nu}^{(3)}) +\beta_3(G_{\rho \sigma \nu}^{(3)}-G_{\rho \sigma \nu}^{(2)})=0\end{array}
[ \beta_{n}^{1}* \alpha_{n}^{1}] = [ \beta_{n}^{0}* l_{n}* \alpha_{n}^{0}] = [ \beta_{n}^{0}* \alpha_{n}^{0}* l_{n}] = [ \beta_{n}^{0}* \alpha_{n}^{0}] [ l_{n}] = [ \beta_{n}^{1}* \alpha_{n}^{1}] [ l_{n}] .
W_{ijkl}W_{ipkq,t}W_{jplq,t}= \, W^{+}_{ijkl}W^{+}_{ipkq,t}W^{+}_{jplq,t}+W^{-}_{ijkl}W^{-}_{ipkq,t}W^{-}_{jplq,t}&+2W_{ijkl}W^{-}_{ipkq,t}W^{+}_{jplq,t}+W^{-}_{ijkl}W^{+}_{ipkq,t}W^{+}_{jplq,t}+W^{+}_{ijkl}W^{-}_{ipkq,t}W^{-}_{jplq,t}\,.
\mathbb{T'}_L =(\Delta')^{-1}\begin{pmatrix}d- \Delta' (a-bd^{-1}c)^{-1}([d,b]-[d,a]a^{-1}b)(d-ca^{-1}b)^{-1}-c+ \Delta' (ac^{-1}d-b)^{-1}([b,c]-[a,c]a^{-1}b)(d-ca^{-1}b)^{-1}\end{pmatrix}.
\tau^1= \inf\{t\ge 0:\Y\notin\Pi_{1}\}, \tilde \tau^2= \inf\{t\ge \tau^1 :\Y\notin \Pi_2\}, \tau^2= \inf\{t\ge 0:\Y\notin \Pi_2\}, \tilde\tau^3= \inf\{t\ge \tau^2:\Y\notin \Pi_3\}, \tau^3= \inf\{t\ge 0:\Y\notin \Pi_3\},
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-m+1)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m-1}(q_v^{2i}-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-m)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}^{-1}\leq q_v^{1/2}\cdot\frac{q_v^{(n_{v,\nu_v}-1)/2}+1}{q_v^{n+1}-1}.
MO_r(f)(z)=\left(\widehat{|f|^2_r}(z)-|\widehat{f_r}(z)|^2\right)^{1/2}&=\left( \frac{1}{2|D(z,r)|^2}\int_{D(z.r)}\int_{D(z,r)}|f(u)-f(v)|^2 dV(u) dV(v)\right)^{1/2}.
\tilde{g}^-_j\tilde{e}^{(-1)}_{ji}=\sum_{s\geq 1}\sum_{j_1<\ldots<j_s=j}(-1)^{s-1}\tilde{f}^{(0)}_{j_s j_{s-1}}\cdots \tilde{f}^{(0)}_{j_2 j_1}\cdot \left([z]T^-_{j_1i}(z)\right).
\begin{array}{lcl}0 = -\alpha \log_1(\pi_2) + \gamma \log_1(u(K)), k\lambda_1+\mu_1 a_1 = -\beta , 0 = \alpha \log_1(\pi_2) - \gamma \log_1(u(K)), k\lambda_2 + \mu_2 a_2 = -\beta.\end{array}
\begin{aligned}(({\rm int}N_1^r)\setminus M_1 \cup ({\rm int}N_2^r)\setminus M_2) = V_R(\Sigma\times\left[ 0,1\right], c_1,c_2) + \frac{1}{4}\int_{N_1^r}Hda + \frac{1}{4}\int_{N_2^r}Hda + 2r\pi\chi(\Sigma).\end{aligned}
\frac{P_{B}(t)-P_{\infty}(t)}{\rho_{L}}=R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}(\frac{dR}{dt})^{2}+\frac{4\nu_{L}}{R}\frac{dR}{dt}+\frac{2\sigma}{\rho_{L}R}
P_j(u_{K}\vert u_0) =\sum_{u_{1}}\ldots \sum_{u_{K-1}}P_j(u_{K}, u_{K-1}, \ldots, u_{1}\vert u_0) &=\sum_{u_{1}}\ldots \sum_{u_{K-1}}\prod_{i=1}^{K}P_j(u_{i}\vert u_{i-1}, \ldots, u_0) &\stackrel{}{=}\sum_{u_{1}}\ldots \sum_{u_{K-1}}\prod_{i=1}^{K}P_j(u_{i}\vert u_{i-1})\,.
\begin{aligned}\frac{d}{dz}\omega(Y_1,Y_2) =\omega(Y_1,A(\lambda,z)Y_2)+\omega(A(\lambda,z)Y_1,Y_2) & = \langle Y_1,JA(\lambda,z)Y_2\rangle+\langle A(\lambda,z)Y_1,JY_2\rangle & = \langle Y_1,\left[JA+A^TJ\right]Y_2\rangle. \end{aligned}
\mathcal{K}_{\nu,p}(x)=\intop_{0}^{\infty}\,\frac{y^{-\nu-\frac{1}{2}}(y+1)^{-\nu-\frac{1}{2}}}{\sigma\left(\frac{x}{2\pi}\,(2y+1)\,\right)e^{(2y+1)x}-1}dy,\,\,\,\,\,x>0,\,\,(\nu)<\frac{1}{2},
\begin{array}{lcl}p~ \partial_{\sigma}V ={\displaystyle 2 \sum_{j=2}^{D-1}\partial_{\tau}X^j \partial_{\sigma}X^j}p~ \partial_{\tau}V ={\displaystyle \frac{\alpha}{\tau^2}~(X^2-Y^2) + \sum_{j=2}^{D-1}\left\{(\partial_{\tau}X^j)^2 + (\partial_{\sigma}X^j)^2 \right\}}\end{array}
{\cal L}^{(0)}= -\frac{1}{\rho^{2}}\left[r\left(\partial_{\mu}\bar{w}\partial^{\mu}w +|m|^{2}|w|^{2}\right)+\frac{\theta}{2\pi i}\varepsilon^{\mu\nu}\partial_{\mu}\bar{w}\partial_{\nu}w\right]
k_1\phi_{1,s}\ln^{\frac12}\left(\frac{2}{\phi_{1,s}}\right) \leq w \leq k_2 \phi_{1,s}\ln^{\frac12}\left(\frac{2}{\phi_{1,s}}\right),\; \; q=1 &k_1 \phi_{1,s}^{\frac{2}{q+1}}\leq w \leq k_2 \phi_{1,s}^{\frac{2}{q+1}},\; \; q>1
E[M^{n,1}_{s_1}M^{n,2}_{s_1}M^{n,1}_{s_2}M^{n,2}_{s_2}]=E[M^{n,1}_{s_1}M^{n,2}_{s_1}M^{n,2}_{s_2}E[M^{n,1}_{s_2}| \mathcal{F}_{\eta_n(s_2)}]]=0.
R(\tau) = \frac{q^{1/5}}{1 + \frac{q}{q+\frac{q^{2}}{1+\frac{q^{3}}{1+\cdots}}}}, S(\tau) = q \prod_{n=1}^{\infty}\frac{(1 - q^{5n})^{6}}{(1-q^{n})^{6}},
\det(g)(x) =1-\frac{1}{3}R_{ij}x^ix^j - \frac{1}{6}R_{ij,k}x^ix^jx^k &- \left(\frac{1}{20}R_{ij,kl}+ \frac{1}{90}R_{aijb}R_{aklb}-\frac{1}{18}R_{ij}R_{kl}\right)x^ix^jx^kx^l + O(r^5),
t=\frac{2vsin\theta}{g}\cdot\frac{1}{2-\tilde{v}^{2}}(1+\frac{1}{\sqrt{2-\tilde{v}^{2}}\tilde{v}sin\theta}arcsin\frac{\sqrt{2-\tilde{v}^{2}}\tilde{v}sin\theta}{\sqrt{1-(2-\tilde{v}^{2})\tilde{v}^{2}cos^{2}\theta}})
d_M={\left(\begin{array}{cccc}w_{2,1}w_{2,2}\cdots w_{2,n-1}w_{3,1}w_{3,2}\cdots w_{3,n-1}\vdots \vdots \cdots \vdots w_{n,1}w_{n,2}\cdots w_{n,n-1}\end{array}\right),}
f(c_{1,i}) = \begin{cases}i, i=2,3,\ldots n-2 n-1, i=1 1, i=n-1 2n, i=n \end{cases}\intertext{and}f(c_{2,i}) = \begin{cases}i+n-1, i=2,3,\ldots n n, i=1 \end{cases}.
H_{N}(\sigma)=\beta\sum_{p=2}^{\infty}\frac{\beta_{p}}{N^{\frac{p-1}{2}}}\sum_{i_{1},\ldots,i_{p}=1}^{N}g_{i_{1},\ldots,i_{p}}\sigma_{i_{1}}\cdots\sigma_{i_{p}}+h\sum_{i=1}^{N}\sigma_{i},
R_{c}=\log_{2}\left ( \det\left [\left ( \rho_{f}\hat{\textbf{G}}_{cc}^T\textbf{P}_{c}\textbf{P}_{c}^{H}\hat{\textbf{G}}_{cc}^* \right )\textbf{R}_{c}^{-1}+\textbf{I}_{K_c}\right ]\right )
\widetilde{C}_{X^{\ast}}(\begin{pmatrix}y_1 0 0 1\end{pmatrix}, \begin{pmatrix}y_2 0 0 1\end{pmatrix}) =\widetilde{C}_{X^{\ast}}([y_1,g_1], [y_2,g_2]) &=\nu( y_2, g_1)\widetilde{c}_{X^{\ast}}(g_1^{y_2}, g_2) &=(y_1^{m}, y_2)_F=(y_1,y_2)_F.
Q_f = f_{\mu_1 \dots\mu_r}\Pi^{\mu_1}\psi^{\mu_2}\dots \psi^{\mu_r}+ \frac{i}{r+1}(-1)^{r+1}f_{[\mu_1 \dots\mu_r;\mu_{r+1}]}\psi^{\mu_1}\dots \psi^{\mu_{r+1}}.
\int_{0}^{\frac{\pi}{2}}sin^{2m}(x)dx=\int_{0}^{\frac{\pi}{2}}cos^{2m}(x)dx=\frac{1\times3\times5\times\cdot\cdot\cdot\times(2m-1)}{2\times4\times6\times\cdot\cdot\cdot\times2m}\cdot\frac{\pi}{2}form=1,2,3...
\tilde{g}_{k}( \eta ) = \frac{i}{2 \Omega_{k 0}}\int_{\eta_{0}}^{\eta}d \eta^{\prime}\left( e^{2 i \Omega_{k 0}( \eta - \eta^{\prime})}- 1 \right) \Delta \Omega^{2}( \eta^{\prime}) \left[ 1 + \tilde{g}_{k}( \eta^{\prime}) \right] \, .
\frac{\partial \omega^H}{\partial p_{\parallel}}= -2p_{\parallel}\left. \frac{\partial\mathcal{F}}{\partial (p_{\parallel}^2)}\right/ \frac{\partial \mathcal{F}}{\partial \omega},\frac{\partial \omega^H}{\partial p_z}= - b\sin p_zb\left/ \frac{\partial\mathcal{F}}{\partial \omega}\right..
\begin{aligned}\pi(\mathcal F_2^{(1)}) =\{F\in\pi(\mathcal F):[d-1]\subseteq F,F\cap\{d,d+1\}\neq\emptyset,d+2\notin F\}&\cup\{F\in\pi(\mathcal F):1\notin F,[2,d-1]\subseteq F,\{d,d+1\}\subseteq F,d+2\notin F\}\end{aligned}
|\dot{y}_R(t)|= \Big| 2 R \Im \int \nabla \varphi \left( \frac{x}{R}\right) \cdot \nabla u \bar{u}dx\Big| \le R \| \nabla \varphi \|_\infty \| \nabla u \|_2 \| u \|_2 \le CR t>0.
U(c,z) =\frac{1}{z}\,{}_{1}F_{2}\left[\begin{matrix}1 1-z,\ -\beta-z\end{matrix};c\right] ,V(c,z) =\frac{1}{z ( z+\beta )}\,{}_{1}F_{2}\left[\begin{matrix}1 1-z,\ 1-\beta-z\end{matrix};c\right] .
\hat w - u = \hat w - \hat w(0,0) - U_1 \hat x - U_2 |\zeta_0| - \frac{1}{2}U_3 |\zeta_0| - U_\beta H & \geq -2 \| w \|_{L^\infty}- (L/d) |U_1| + |U_2| \sigma + \frac{1}{2}|U_3| \sigma - U_\beta ((L/d)^2 + \sigma^2)^{(\beta+1)/2}\textrm{on}\{\hat \zeta = |\zeta_0| \}.
f = f^{(+)}- f^{(-)}&= f^{(+)}- x_{u_1}x_{u_3}\frac{f^{(-)}}{x_{u_1}x_{u_3}}&= f^{(+)}-x_cx_d\frac{f^{(-)}}{x_{u_1}x_{u_3}}- (x_{u_1}x_{u_3}-x_cx_d)\frac{f^{(-)}}{x_{u_1}x_{u_3}}
\sum_{w\in M_{K}}N_{w}\log(r_{w,\alpha}^{-1}) = \sum_{w\in M_{K}\setminus M'_{K,\alpha}}N_{w}\log(d(P_{1,w},P_{2,w})^{-1}) + \sum_{w\in M'_{K,\alpha}}N_{w}\log(\rho_{\alpha}^{-1}) & \le \sum_{w\in M_{K}}N_{w}\, \max(0,\log(d(P_{1,w},P_{2,w})^{-1})) + \sharp M_{\alpha}\, \log(\rho_{\alpha}^{-1}).
B_{q,1}= \int_{-\infty}^{\infty}\frac{\pi}{\pi^{2}+ 4d^{2}\omega^{2}}\left|\frac{\sin (\alpha \omega)}{\tanh(d\, \omega)}\right|\, \mathrm{d}\omega, B_{q,2}= \left|\int_{-\infty}^{\infty}\frac{2d}{\pi^{2}+ 4d^{2}\omega^{2}}\frac{\omega \cos(\alpha \omega)}{\tanh(d\, \omega)}\, \mathrm{d}\omega\right|.
\int d \beta \, \mathrm{e}^{- \sum_{k}\beta_{k}^{2}/ 2 \hat{x}- l \sum_{k}t_{k}\beta_{k}}{\frac{\prod_{i = 1}^{n}\prod_{k < l}2 \sinh{\frac{\beta_{k}- \beta_{l}}{2 p_{i}}}}{\prod_{k < l}\Bigl ( 2 \sinh{\frac{\beta_{k}- \beta_{l}}{2}}\Bigr )^{n - 2}}}
F_m(z,s) = f_m(z,s) + \frac{2|m|^{1/2-s}\sigma_{2s-1}(|m|)}{(2s-1)\Lambda(2s)}y^{1-s}+ 2\sqrt y \sum_{n\neq 0}\Phi(m,n;s) K_{s-\frac 12}(2\pi|n|y)e(nx),
\Phi_j(f_0,f_1,f_2,\ldots,f_{N-1}) =f_j+\exp_K (-t_1f_0)\ldots \exp_K(-(t_j-t_{j-1})f_{j-1})g_j\exp_K(t_j-t_{j-1})f_{j-1})\ldots\exp_K t_1f_0,
-q^{n/2}[\#X(F_{q^n})-(q^n+1)] =\sum_{d\mid u}\left( e_{d}\sum_{j\in I_d}w_d^{-jt}\right) &=\sum_{d\mid u}\left( e_{d}\sum_{j\in I_d}w_d^{-j}\right) &=-q^{m/2}[\#X(F_{q^m})-(q^m+1)].
F(w) = \frac1{2\pi\mathrm{i}}\int_{\Gamma}F(\zeta) \Big( \frac1{\zeta-(\zeta_0+\mathrm{i}\tau)}- \frac1{\zeta-(\zeta_0-\mathrm{i}\tau)}\Big)\mathrm{d}\zeta = \int_\Gamma F(\zeta)K_{\mathrm{i}\tau}(\zeta,\zeta_0) \,\mathrm{d}\zeta.
\iota\varepsilon_\#(v) =\begin{cases}\iota(v) v\in V_1(\varepsilon); \iota(x_e^{-1}vx_e) v\not\in V_1(\varepsilon).\end{cases}&=\begin{cases}v v\in V_1(\varepsilon); \overline{x_e}^{-1}v^{-1}\overline{x_e}v\not\in V_1(\varepsilon),\end{cases}
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-m+1)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m-1}(q_v^{2i}-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-m)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}^{-1}\leq \frac{q_v^{n_{v,\nu_v}/2}+1}{q_v^{n+1}-1}.
\Bigl( V(\psi\otimes\bar{w},z)\;\left( |\chi\rangle_{l}\otimes|\bar{u}\rangle_{l}\right) \Bigr)_{m}: =\lim_{\bar{z}\rightarrow 0}\bar{z}^{\bar{\Delta}_{j}+ \bar{\Delta}_{l}-\bar{\Delta}_{m}}\;\Bigl( \phi(\psi\otimes\bar{w};z,\bar{z})\;\left( |\chi\rangle_{l}\otimes|\bar{u}\rangle_{l}\right) \Bigr)_{m},
f_{X}(X;\Psi,\nu,\delta)=\frac{\Gamma_{p}(\frac{\nu+\delta+p-1}{2})}{\Gamma_{p}(\frac{\nu}{2})\Gamma_{k}(\frac{\delta+p-1}{2})|\Psi|^{\frac{\nu}{2}}}|X|^{\frac{\nu-p-1}{2}}|I_{p}+X\Psi^{-1}|^{-\frac{\nu+\delta+p-1}{2}}
\liminf_{\ell\to\infty}\frac{\log \mu(B_{r_\ell}(x))}{\log r_\ell}=\liminf_{\ell\to\infty}\frac{\sum_{i=1}^\ell \log t_i}{d\sum_{i=1}^{\ell+1}\log s_i - \log t_{\ell+1}}.
d^{-2n}\sum_{\substack{n_1 + \cdots + n_d = n n_i \ge 1, \; 1\le i\le d}}{2n \choose 2n_1, \ldots, 2n_d}\prod_{j=1}^d n_j^{-3/2}\le c_3\exp(-c_4n) + \sum_{\substack{n_1+\cdots + n_d=n n_i \in [\frac{n}{d},\, \frac{3n}{d}], \; 1\le i\le d}}\P (S_i = 2 n_i, \ 1\le i\le d) \prod_{j=1}^d n_j^{-3/2}\, .
S_{\rm C}= \frac{1}{e^2}\int d^4x\,\sqrt{g}\, \Bigr\{( \nabla^\mu T_{\mu\rho}^- - i T_{\mu\rho}^- V^\mu )( \nabla_\nu T^{\nu\rho}_+ - i V_\nu T^{\nu\rho}_+ ) \Bigr\},
\langle \chi,\alpha_0\rangle =\frac{1}{|N_G(P)|}\left( 3^k(q-1)(1+q-1)+\frac{q(q-1)}{2}(\chi(T)+\overline{\chi(T)})+ \frac{q^2(q-1)}{3}\cdot\varepsilon\cdot 3^k(1+\omega+\overline{\omega})\right)=0
E(B)={(k_bT)^2\over 2\pi v}\{{1\over 2}x{\rm ln}y+L(1-e^{-x}\}={(k_bT)^2\over 2\pi v}\{{B\over 2}x^2+{1\over 2}x{\rm ln}(1-e^{-x})+L(1-e^{-x})\}.
\varphi_\alpha(\lambda,\omega) \sim{1\over (4\pi)^{d/2}}\sum_{n=0}^\infty\left[a'_{n}(\omega)~{\lambda^{d/2-n+\alpha-1}\over \Gamma\left(\frac d2 -n +\alpha\right)}+b'_{n}(\omega)~{\lambda^{(d-1)/2-n+\alpha-1}\over \Gamma\left({d-1 \over 2}-n +\alpha\right)}\right],
a_{n,k}=(k-1)!(n-1)!\sum_{\ell_1=1}^{n-1}\frac1{\ell_1}\sum_{\ell_2=1}^{\ell_1-1}\frac1{\ell_2}\dotsm \sum_{\ell_{k-3}=1}^{\ell_{k-4}-1}\frac1{\ell_{k-3}}\sum_{\ell_{k-2}=1}^{\ell_{k-3}-1}\frac1{\ell_{k-2}}.
f=T_1+T_{s_0}+\sum_{n=1}^{\infty}q^{-2n}\left(T_{(s_1s_0)^n}+T_{s_0(s_1s_0)^n}-q\left(T_{(s_0s_1)^n}+T_{s_1(s_0s_1)^n}\right)\right)
H(z_1')-H(z_2') =\sum_{i=1}^N (H(\tilde\gamma(t_i))-H(\tilde\gamma(t_{i-1}))) &=\sum_{i=1}^N \left(\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,p_*\tilde\gamma(t_j))-\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,p_*\tilde\gamma(t_{j-1}))+2k_j\pi\right) &=\sum_{i=1}^N 2k_i\pi.
\mathrm{T r}e^{- \beta P \bar{\cal K}}= \left[ \frac{m L}{\sqrt{4 \pi \beta}}+ \frac{1}{\sqrt{\pi}}\sum_{n = 1}^{\infty}\frac{2^{n}\beta^{n - \frac{1}{2}}}{1 \cdot 3 \cdot 5 \cdot . . . \cdot ( 2 n - 1 )}\right] e^{- \beta}- 1
F_{\mu \nu}^{( 0 )}: = i ( k_{\mu}\zeta_{\nu}- k_{\nu}\zeta_{\mu}) e^{i k \cdot x}+ i ({k^{\prime}}_{\mu}{\zeta^{\prime}}_{\nu}-{k^{\prime}}_{\nu}{\zeta^{\prime}}_{\mu}) e^{i k^{\prime}\cdot x}
( e_{a}^{( 1 )}g_{b}^{( 1 )}- e_{b}^{( 1 )}g_{a}^{( 1 )}) + ( e_{a}^{( 2 )}g_{b}^{( 2 )}- e_{b}^{( 2 )}g_{a}^{( 2 )}) = 2 \pi \hbar n , \; \; n \in Z \; .
f(t):=\begin{cases}\cos \sqrt{k}t-\dfrac{\theta}{\sqrt{k}}\sin \sqrt{k}t, k> 0, 1-\theta t, k=0, \cosh \sqrt{-k}t-\dfrac{\theta}{\sqrt{-k}}\sinh \sqrt{-k}t, k<0,\end{cases}t\geq 0.
\mathcal{T}^\mathcal{F}_\star(X,Y)=\langle X\otimes_{\mathcal{X}_\star}Y,\mathrm{T}^\mathcal{F}\rangle_\star,~ \mathrm{R}^\mathcal{F}_\star(X,Y,Z)=\langle X\otimes_{\mathcal{X}_\star}Y\otimes_{\mathcal{X}_\star}Z, \mathrm{R}^\mathcal{F}\rangle_\star
E_{\delta}(gA_i',A_j'h)\gg\delta^{-3d}\|1_{gA_i'}*1_{A_j'h}\|_2^2 =\delta^{-3d}\|1_{A_i'}*1_{A_j'}\|_2^2 &\geqslant\delta^{O(\varepsilon)}N(A_i',\delta)^{\frac{3}{2}}N(A_j',\delta)^{\frac{3}{2}}&\geqslant\delta^{O(\varepsilon)}N(gA_i',\delta)^{\frac{3}{2}}N(A_j'h,\delta)^{\frac{3}{2}}