harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f_{Y}(y)=\frac{\delta^{r\sigma^{-1}}}{\chi(\sigma^{-1})}I^{m,n}_{p,q}\Bigg[\delta^{r\sigma^{-1}}y\left|\begin{matrix}(a_i-r\sigma^{-1}A_i+\sigma^{-1}A_i,r\sigma^{-1}A_i,\alpha_i)_{1,p}(b_j-r\sigma^{-1}B_j+\sigma^{-1}B_j,r\sigma^{-1}B_j,\beta_j)_{1,q}\end{matrix}\right.\Bigg],\ \y>0,
\delta \psi_{M}= D_{M}\varepsilon +{\frac{1}{6 4}}~ e^{3 \phi / 4}( \Gamma_{M}\,^{M_{1}M_{2}}- 1 4 \delta_{M}\,^{M_{1}}\Gamma^{M_{2}}) \Gamma^{1 1}\varepsilon F_{M_{1}M_{2}}
d_k\big( \overline{\sigma_t(\delta)},\overline{\gamma(t+\delta)}\big) = d_k \big( \widehat{\sigma_t(\delta)},\overline{\gamma(t+\delta)}\big) d_k\big( \overline{\gamma(t)},\overline{\gamma(t+\delta)}\big) = d_k \big( \overline{\gamma(t)},\overline{\gamma(t+\delta)}\big) ,
W_2(Q_{(\sigma)}) = c^{1/2}\alpha\frac{\|Q_{(\sigma)}\|_{L^2}^3\||\nabla|^\sigma Q_{(\sigma)}\|_{L^2}}{\|Q_{(\sigma)}\|_{L^4}^4}&= c^{1/2}(4\sigma-1)^{\frac{1-\sigma}{2\sigma}}\alpha \frac{\|Q_{(\sigma)}\|_{L^2}^{\frac{4\sigma-1}{\sigma}}\||\nabla|^\sigma Q_{(\sigma)}\|_{L^2}^{\frac{1}{\sigma}}}{\|Q_{(\sigma)}\|_{L^4}^4},
K_{p}( z ) \stackrel{( z \rightarrow 0 )}{\sim}\frac{1}{2}\left[ \Gamma ( p ) \left( \frac{z}{2}\right)^{- p}+ \Gamma ( - p ) \left( \frac{z}{2}\right)^{p}\right] \left[ 1 + O ( z^{2}) \right] .
\begin{array}{l}H_{2}=\dfrac{1}{(1+3h+2\alpha_{1}+2\alpha_{3})}\{-\dfrac{1}{2}q_{1}^2p_{1}-\dfrac{1}{2}q_{1}q_{2}p_{2}+2q_{2}p_{1}p_{2}+q_{1}p_{1}^2+\dfrac{1}{2}t_{2}q_{1}p_{1}+\dfrac{1}{2}q_{2}p_{1}-t_{2}p_{1}^2 begin{align*}10pt]\phantom{H_{1}=}-\dfrac{1}{2}\alpha_{1}q_{1}+(1+h+2\alpha_{1}-\alpha_{2}+2\alpha_{3})p_{1}+\eta t_{1}p_{2}\}.\end{array}
S = \frac{1}{1 6 \pi}\int d^{4}\! x ( - g^{\mathrm{I V}})^{1 / 2}\left[ R^{\mathrm{I V}}- 2 \Lambda - \frac{1}{2}\sum_{i = 1}^{N}( \nabla^{\mathrm{I V}}f_{i})^{2}\right] ,
\E^{\ast}[ S_{n,l}^{\ast}(x,y) ]^2 \leq \ \frac 1{(n-l+1)}C \sum_{i=1}^{n-l+1}S_{l,i}^2(x,y) & \ + \frac 1{d_l^{2}}C l^2 \left ( F(x,y) - \tilde F_{n,l}(x,y) \right )^2 \ + \frac 1{d_l^{2}}C J_m^2(x,y)/(m!)^2 l^2 \left ( \tilde \mu_{n,l}(H_m)\right )^2,
p_{{\bf a}}=\begin{cases}1{\bf a}=\emptyset -u{\bf a}=(0) -uv\cdot p_{{\bf a}'}{\bf a}=(0,z,{\bf a}') 0{\bf a}=(0,0,{\bf a}') (2u+v)\cdot p_{{\bf a}'}{\bf a}=(z,{\bf a}')\end{cases}
\Gamma_{1}= \left\{(\sin\eta) e^{i\delta},|\delta|\in(\tfrac{\pi}{2}-\eta ,\pi]\right\},\Gamma_{2}=\left\{1-te^{\pm i\eta},t\in(r,\cos\eta ]\right\}, \Gamma_{3}= \left\{1+re^{i\delta}, |\delta|\in[0,\pi-\eta )\right\},
V (D) = \begin{bmatrix}D e^{-\frac 12 DD^*}-e^{-\frac 12 D^*D}\frac{I - e^{-D^*D}}{D^*D}D^* \end{bmatrix},V (D)^{-1}= \begin{bmatrix}\frac{I - e^{-D^*D}}{D^*D}D^* - e^{-\frac 12 D^*D}e^{-\frac 12 DD^*}D \end{bmatrix};
G_i(t) = G(z_1^*,\ldots, z_{i-1}^*, t,z_{i+1}^*,\ldots z_n^*)=-|v_i|+\frac{r-\sum_{j\ne i}z_j^*(|v_j|-|v_i|)}{\sum_{j\ne i}|z_j^*|+t}.
\Theta_{\mathrm{D}}(\delta)=\left[\min_{\gamma \in \Sigma_{\mathrm{D}}(\delta)}\int_{\mathcal{S}_1 \times \mathcal{S}_2}g \, d \gamma, \max_{\gamma \in \Sigma_{\mathrm{D}}(\delta)}\int_{\mathcal{S}_1 \times \mathcal{S}_2}g \, d \gamma \right],
\frac{1}{4}\leq C_1 R^{-d/q}\|w_{\bar{L}}\|_{L^q(B_R)}\leq C_1 R^{-d/q}\|w_{\bar{L}}\|^{1/q}_{L^1(B_R)}\|w_{\bar{L}}\|^{1-1/q}_{L^\infty(B_R)}\stackrel{\eqref{eq:e6}}{\leq}C_1 R^{-d/q}2^{p-p/q}\|w_{\bar{L}}\|^{1/q}_{L^1(B_R)},
\sum_{t = 1}^{N}\sum_{n = 1}^{\infty}V_{m n}^{r t}V_{n 0}^{t s}= V_{m 0}^{r s}, \ \sum_{t = 1}^{N}\sum_{m = 1}^{\infty}V_{n m}^{r t}V_{m l}^{t s}= \delta_{n , l}\delta_{r , s}.
{_{3}}F_{2}\left(\begin{array}{ccc}\chi_4, \chi_4, \chi_4 \varepsilon, \varepsilon\end{array}\mid 1\right) =\frac{\chi_8(-1)}{q^2}[J(\chi_8,\chi_8)J(\chi_8,\chi_8^2)+\overline{J(\chi_8,\chi_8)}J(\chi_8,\chi_8^2)] &=\frac{1}{q^2}J(\chi_8,\chi_8^2)\times 2 Re(J(\chi_8,\chi_8))\times \chi_8(-1) &=\frac{1}{q^2}[-2u(-p)^t].
W^r_0(f)=\frac{(-1)^{\frac{n-1}{2}}}{(2\pi)^{n-1}}\int_{\mathbb{R}^{(n+1|n-1)}}\exp\left(-2i\pi\left(p_0f+p.\left(\frac{td_yf}{\Lambda}-\eta\right)+\sum_{j\neq r, k}\frac{\partial^2_{y^jy^k}f}{\Lambda^2}\Pi_jdy^k\right)\right)dp_0dp d\Pi.
\{Q_{a}\, , \, Q_{b}\}\, = \, 2 \, i \, P^{\mu}\, ( \gamma_{\mu})_{a b}\, + \, 2 \, i \, \epsilon^{i \, j}\, \int \, d^{2}\vec{x}\, ( \partial_{i}\, A ) \, ( \gamma_{j})_{a b}\, \left( m \, A \, + \, \frac{\lambda}{2}A^{3}\right) \, .
\log e\frac{1-p}{2}\left(\frac{h_1+h_2-2\rho\sqrt{h_1h_2}}{1-\rho^2 + \mathsf{P}(h_1+h_2-2\rho\sqrt{h_1h_2})}- \frac{h_1}{h_1\mathsf{P}+1}\right)\mathsf{P}= 0,
G_{a b , a^{\prime}b^{\prime}}G_{c d , c^{\prime}d^{\prime}}G^{a b , c d}F_{i}^{a^{\prime}b^{\prime}, c^{\prime}d^{\prime}, e^{\prime}}= \frac{( N - 2 ) ( N + 1 )}{2}\delta_{i}^{e^{\prime}},
\tilde{\nabla}_i\Phi =-\frac{n-4}{n-2}u^{-2\frac{n-3}{n-2}}\tilde{\nabla}_iu, \tilde{\nabla}_j\tilde{\nabla}_i\Phi =-\frac{n-4}{n-2}\left( u^{-2\frac{n-3}{n-2}}\tilde{\nabla}_j\tilde{\nabla}_iu - 2\frac{n-3}{n-2}u^{-\frac{3n-8}{n-2}}\tilde{\nabla}_ju\tilde{\nabla}_iu \right),
\limsup_{t\to 0}\, \frac{f(x+te_x) - f(x- at u_x)}{t}&\quad=\limsup_{t\to 0}\frac{f(x+t e_x)-f(x)}{t}-\lim_{t\to 0}\frac{f(x-{at}u_x)-f(x)}{t}= 1 + a^2.
S [ B , \tilde{B}, F ] = \int d^{5}x \left\{{\textstyle \frac{1}{2 \cdot 3 !}}H^{2}-{\textstyle \frac{1}{4}}m^{2}B^{2}+{\textstyle \frac{1}{4}}\epsilon \partial \tilde{B}( F + m B ) \right\}.
d \langle A_{i}\rangle_{t}= \frac{1}{4}\sum_{k=1}^{m}\left(U^{(i)}B_{t}\right)^{2}_{k}dt = \frac{1}{4}\vert U^{(i)}B_{t}\vert^{2}dt= \frac{1}{4}\vert B_{t}\vert^{2}dt.
S_{\mu}(c,z) =\sum\limits_{n=0}^{\infty}\frac{\mu_{n}(c)}{z^{n+1}}=\sum\limits_{n=-1}^{\infty}\frac{\mu_{n+1}(c)}{z^{n+2}}=\frac{1}{z}\sum\limits_{n=-1}^{\infty}\frac{\mu_{n+1}(c)}{z^{n+1}}=\frac{1}{z}\left[ \mu_{0}(c) +\sum\limits_{n=0}^{\infty}\frac{\mu_{n+1}(c)}{z^{n+1}}\right] .
\mathbf{T}^l_{\varepsilon_1,i,j}(\mu_l)(x)=\sup_{\rho\in (0,2R),e\in S^{d-1}}\frac{\varepsilon_1^{-d+1}}{\rho}\left|\left(\frac{1}{|.|^{d-1}}\Theta^{\varepsilon_1,e}_{l,\rho}(.)\right)\star \mathbf{K}_j^i\star\mu_l(x)\right|~~\forall~~x\in \mathbb{R}^d,
\partial_{x_i}g(s_n) = \frac{1}{\sqrt n}\Bigr(\partial_xf -\frac{Y_{n,i}}{2}\partial_zf\Bigr)(s_n), \partial_{y_i}g(s_n) = \frac{1}{\sqrt n}\Bigr(\partial_yf + \frac{X_{n,i}}{2}\partial_zf\Bigr)(s_n), \partial_{z_i}g(s_n) = \frac{\beta}{\sqrt n}(\partial_zf)(s_n).
Z_m = \sum_{\mu=0}^{L-1-m}\left\{\|\nabla \partial_t^\mu v\|_2^2 + \sum_{2 \leq |a| \leq m+1}\|\partial_t^\mu \nabla^a v\|^2_2 + \|\partial_t^{\mu+1}v\|^2_{H^m}\right\}&\leq C\left (Z_0 + \sum_{\mu=0}^{L-1-m}\sum_{2 \leq |a| \leq m+1}\|\partial_t^\mu \nabla^a v\|^2_2 + Z_{m-1}\right).
\lim_{j \to \infty}{\delta_t \int_{\Omega}{f_{m,j}(u^n(x)) \dx}}= \lim_{j \to \infty}{\int_{\Omega}{g_{m,j}(x) \dx}}= \int_{\Omega}{g_m(x) \dx}= \delta_t \int_{\Omega}{f_m(u^n(x)) \dx}.
V_2V_1^* =(R_{\overline q}\otimes U^*P+R_{\overline q}M_z\otimes U^*P^\perp)(R_{\overline q}\otimes U^*P^\perp + R_{\overline q}M_z^*\otimes U^*P) &=R_{\overline q}^2\otimes U^*PU^*P^\perp +R_{\overline q}^2M_z^*\otimes U^*PU^*P+R_{\overline q}M_zR_{\overline q}\otimes U^*P^\perp U^*P^\perp &\quad+R_{\overline q}M_zR_{\overline q}M_z^*\otimes U^*P^\perp U^*P
M_1e' = Mw_1e_0 \cong Mw_1e_0 \otimes_{e_{0}\Lambda e_{0}}e_{0}\Lambda e_{0}\cong Mw_1e_0\otimes_{S}e_{0}\Lambda e_{0}= M_1 \otimes_{S}e_{0}\Lambda e_{0}\cong M_1\otimes_{S}\Lambda e_{0}= Pe_0 =0.
\det(\lambda I-K) = \det\Big(\lambda I-D\Big)\Big(1+ \frac{l^{T}(\lambda I-D)^{-1}Dl}{\Vert l\Vert^{2}}\Big) = \Big(\Pi_{i=1}^{n}(\lambda- d_i)\Big)\cdot \sum_{i=1}^{n}\frac{\lambda l_{i}^{2}}{\Vert l \Vert^{2}(\lambda-d_i)}.
\|\alpha(t \wedge \tau_m(\lambda)) - \alpha (s \wedge \tau_m(\lambda))\| = \begin{cases}\|\alpha(\tau_m(\lambda)) - \alpha(s)\|, t \geq \tau_m(\lambda)\geq s, \|\alpha(t) - \alpha(\tau_m(\lambda))\|, t \leq \tau_m(\lambda) \leq s, \|\alpha(t) - \alpha(s)\|, t, s \leq \tau_m(\alpha), 0, t, s \geq \tau_m(\alpha).\end{cases}
H''_k(t_k)=\frac{A''_k(t_k)}{B_k(t_k)}-2\frac{A'_k(t_k)}{B^2_k(t_k)}B'_k(t_k)-\frac{A_k(t_k)}{B^2_k(t_k)}B''_k(t_k)+2\frac{A_k(t_k)(B'_k(t_k))^2}{B^3_k(t_k)}.
|h^{(n+1)}(x)| = | (h')^{(n)}(x) | = 2 | (h^2 gg')^{(n)}(x)| \leq 2h^2(x)(gg')^{(n)}(x) + 2\sum_{k=0}^{n-1}\binom{n}{k}\Big|(h^2)^{(n-k)}(x) \, (gg')^{(k)}(x)\Big|.
-\frac{d}{dt}\mathcal{T}_t(x^\mu_k)=\,\big\{\mathcal{T}_t(H_t), \mathcal{T}_t(x^\mu_k)\big\},-\frac{d}{dt}\mathcal{T}_t(y^\mu_k) =\,\big\{\mathcal{T}_t(H_t), \mathcal{T}_t(y^\mu_k)\big\}.
E_1 \biggl( \int_0^{T_{n\varepsilon}}\bigl|\xi^0_s\bigr| \,ds \bigg| T_{n\varepsilon}< T_0\biggr) = \sum_{k=1}^{n\varepsilon}\frac{1}{\widehat P_k(T_k^+>T_{n\varepsilon})}\frac{k}{q(k)},
B_n(Kf_0, \epsilon) = \Bigl\{f\in \mathcal{F}:{}-\int p_{Kf_0}\log\frac{p_{Kf}}{p_{Kf_0}}\, d\mu \leq n\epsilon^2, \int p_{Kf_0}\Bigl(\log\frac{p_{Kf}}{p_{Kf_0}}\Bigr)^2\, d\mu \leq n\epsilon^2, \Bigr\},
F(r) = f_1(r)+\frac{d+2}{d+3}\frac{r_{ps}r_s^{d+1}}{r^{d+2}}\chi(r)\Bigl( (\psi_N * a)(h(r))-Q_2(r)-a(h(r))\Bigr) &=g(r) + \frac{d+2}{d+3}\frac{r_{ps}r_s^{d+1}}{r^{d+2}}\Bigl[(1-\chi)(r)a(h(r)) + \chi(r)\Bigl((\psi_N*a)(h(r)) - Q_2(r)\Bigr)\Bigr].
g(z)=\left(\begin{matrix}a^*(z) -b(z) b^*(z) a(z)\end{matrix}\right)\left(\begin{matrix}e^{\chi(z)}0 0 e^{-\chi(z)}\end{matrix}\right)\left(\begin{matrix}d^{*}(z) -c^{*}(z) c(z) d(z)\end{matrix}\right),
\zeta_{T(E)}|_{S(F)}(t)=\prod\limits_{\substack{\alpha\in \Phi_{\mathrm{sym,ram}}/\Gamma_{E}, \alpha^{\mathrm{op}}\in \Phi_{\mathrm{sym,ram}}/\Gamma_F}}\omega_{E_\alpha/F_\alpha}(\iota_{\alpha}\alpha(t)).
d S^{2}= [ d u d v - \mu^{2}\sum_{i = 1}^{4}{z^{i}}^{2}d u^{2}+ \sum_{i = 1}^{4}d z^{i}d z^{i}] +{H_{5}^{\prime}}( d y^{2}+ y^{2}d{\Omega_{3}}^{2}) .
\begin{array}{rl}\bar{\cal{H}}\,m_{(2,1;4,2)}= \varepsilon_{(2,1;4,2)}\, m_{(2,1;4,2)}+2\beta \, m_{(3,1;3,2)}-4\beta \, m_{(3,2;2,2)}+4\beta \, m_{(2,1;3,3)}\cr +8\beta \, m_{(2,1;3,2,1)}+24\beta \, m_{(2,1;2,2,2)}+ 4\beta \, m_{(2,1;4,1,1)}\, . \end{array}
\begin{array}{c}{0 = \left[ - ( \Gamma^{0}+ \cos \alpha \Gamma^{\phi}) \Gamma ( e^{\psi}) + \sin \alpha \right] \psi =}\\{= \left[ \Gamma ( e^{\psi}) ( \Gamma^{0}+ \cos \alpha \Gamma ( e^{\phi}) ) + \sin \alpha \right] \psi =}\\{= \left[ \Gamma ( e^{\psi}) ( \Gamma ( e^{| |}) + \cos \alpha \Gamma ( e^{\phi}) ) + \sin \alpha \right] \psi}\\ \end{array}
W_2(u,z_1) = z\left(\lambda_-K_\mu(uz)+\pi i I_\mu(uz)\right)\left(\sum_{s=0}^{N-1}\frac{A_s(z)}{u^{2s}}+g_2(u,z_1)\right) \hphantom{W_2(u,z_1) =}{}+\frac{z}{u}\left(-\lambda_-K_{\mu+1}(uz)+\pi i I_{\mu+1}(uz)\right) \left(\sum_{s=0}^{N-1}\frac{B_s(z)}{u^{2s}}+zh_2(u,z_1)\right).
D(e_1)=dx\otimes(0,r_3,r_2)+du\otimes(0,r_2,r_1) D(e_2)=dx\otimes(-1,p_3,p_2)+du\otimes(0,p_2,p_1) D(e_3)=dx\otimes(0,q_3,q_2)+du\otimes(-1,q_2,q_1).
\begin{aligned}\xi*\ddot{u}_{i}=u_{i}-v_{i}^{0}t-u_{i}^{0}, l*\dot{\theta}=\theta-\theta^{0}l*\dot{\eta}=\eta-\eta^{0}, \zeta*(\theta+\beta\dot{\theta})=\theta-\theta^{0}e^{-t/\beta}, \chi*(\theta+\beta'\dot{\theta})=\theta-\theta^{0}e^{-t/\beta'},\end{aligned}
\dim_H E_\varphi(\Phi) \leq \liminf_{\ell\to \infty}{\sum_{j=1}^\ell \log 3\varepsilon e^{\beta^j/a}/a \over d \sum_{j=1}^{\ell+1}\log e^{\beta^j/a}-\log 3\varepsilon e^{\beta^{\ell+1}/a}/a}= \liminf_{\ell\to \infty}{\sum_{j=1}^\ell{\beta^j/a}\over d \sum_{j=1}^{\ell+1}{\beta^j/a}-{\beta^{\ell+1}/a}}={1 \over d\beta-\beta+1}.
G_{r}^{( f )}( \omega ) = \frac{1}{\sqrt{7}}\{( \tilde{\chi}\Gamma_{i}\chi \sigma_{i})_{r}+ ( \tilde{\chi}[ \Gamma_{j}, \Gamma_{k}] \chi \sigma_{j k})_{r}+ ( \sigma_{i}\sigma_{j}\sigma_{i j})_{r}+ ( \sigma_{i j}\sigma_{j k}\sigma_{k i})_{r}\},
\langle u , Z_s - Z_{\pi(s)}\rangle = b_{i,\ell_s}\langle u , \zeta(V_{s,i}) - \zeta(V_{\pi(s),i}) \rangle + b_{j,\ell_s}\langle u , \zeta(V_{s,j}) - \zeta(V_{\pi(s),j}) \rangle &= (b_{i,\ell_s}- b_{j,\ell_s})\langle u , \zeta(V_{s,i}) - \zeta(V_{s,j}) \rangle > 0.
\prod_{k=1}^{m}\left\|\frac{1}{\det\nabla \phi_k}\right\|_{L^{\infty}}\leq \prod_{k=1}^{m}\frac{1}{1-C\tau_0|M_k|^{1/2}}=\prod_{k=1}^{m}\left(1+\frac{C\tau_0|M_k|^{1/2}}{1-C\tau_0|M_k|^{1/2}}\right) \leq \prod_{k=1}^{m}\left(1+2C\tau_0|M_k|^{1/2}\right)\leq \exp\left(\sum_{k=1}^{m}2C\tau_0|M_k|^{1/2}\right)\leq 2.
S_{\ell}(\alpha) = \sum_{1\leq m^{\ell}\leq N}\Lambda(m) e(m^{\ell}\alpha), T_{\ell}(\alpha) = \sum_{m^{\ell}\le N}e(m^{\ell}\alpha) , f_{2}(\alpha) =\frac{1}{2}\sum_{1 \leq m\leq N}m^{-1/2}e(m\alpha), U(\alpha,H) = \sum_{1\leq m\leq H}e(m\alpha),
V_{e f f}( z ) = \frac{1 5}{4 \big ( | z | + \ell \big )^{2}}- \frac{3}{\ell}\delta ( z ) \bigg [ 1 - \frac{2 \ell \gamma}{9}\left( \partial_{z}\psi ( 0_{+}) + \frac{3}{2 \ell}\psi ( 0_{+}) \right) \bigg ] ,
DF(r(k),0)[h] = \sum_{m=1}^{\infty}K_{m}(r(k))\left(a_{m-1}- 2\left(\frac{1+r(k)}{1-r(k)}\right)a_{m}+ a_{m+1}\right) \sin(2mx)= \sum_{m=1}^{\infty}b_{m}\sin(2mx)
\left(s^{\prime}z_{1}s^{\prime}z_{2}^{i}\right)^{-1}z_{2}\left(s^{\prime}z_{1}s^{\prime}z_{2}^{i-i}\right) =\left[\left(s^{\prime}z_{1}s^{\prime}z_{2}^{i}\right)^{-1},z_{2}\right]\mbox{and}z_{2}\left(s^{\prime}z_{1}s^{\prime}z_{2}^{i}\right)^{-1}\left(s^{\prime}z_{1}s^{\prime}z_{2}^{i-i}\right) =e.
\max\left(\frac{h}{6},\frac{h}{2}\frac{b+1}{2b+1}\right) \le a < \min\left( \frac{h}{2}-\frac{h}{c+3}, \frac{\frac{h}{2}\left(\frac{1}{b}-1\right)}{\frac{1}{b}-2}\right) r(N) = \frac{1}{N^{3a-\frac{h}{2}}\log^{\frac{h}{2}}(N)}\rightarrow 0.
E\left(Y_{N_2}^{-t}\right) \leq E\left(X_1^{-t}+2^{-(N_2 -2)t}\right) = E\left(X_1^{-t}\right) + 2^{-(N_2 -2)t}\leq(1 + \dfrac{1}{2}\varepsilon)E\left(X_1^{-t}\right),
\| D_{j}\|^{p}_{\mathcal{H}_{p}}= \|P_\gamma\|_{L_p(\mathbb{T}^{N-\lambda})}^p = \int_{\mathbb{T}^{N - \lambda}}{\left|\int_{\mathbb{T}^{\lambda}}{P(u,v) u^{-\gamma}du}\right|^{p}dv}& \leq \int_{\mathbb{T}^{N-\lambda}}\int_{\mathbb{T}^{\lambda}}|P(u,v)|^p du dv = \|P\|_{L_p(\mathbb{T}^{N})}^p = \| D\|_{\mathcal{H}_{p}}^{p}\,.
E_{\mathrm{b o u n d}}= \frac{1}{2 e^{2}}\left( \frac{m_{W}m_{H}}{m_{Z}}\right)^{2}\left[ 2 g^{\prime}\Phi_{B}- \left( \frac{m_{W}m_{H}}{m_{Z}}\right)^{2}A \right] .
\int_{G_d}\Phi^{(j)}(K \cap gM,A\cap gB) \mu (dg) &=\int_{SO_d}\int_{{\mathbb R}^d}\Phi^{(j)}(K \cap (\vartheta M)^x,A\cap(\vartheta B)^x)\, \lambda(dx)\nu (d\vartheta ) = \Phi^{(j)}(K,A) \lambda_M(B) + \sum_{m=j+1}^{d}\int_{SO_d}\Phi_{m,d+j-m}^{(j)}(K,\vartheta M;A\times\vartheta B)\nu (d\vartheta ) .
\langle\mathfrak{A}(\psi),\zeta\rangle =\nu\int_0^1\frac{\partial \psi(x)}{\partial x}\frac{\partial \zeta(x)}{\partial x}\d x-\frac{\alpha}{\delta+1}\int_0^1\mathfrak{p}(\psi(x)+\varphi(x))\frac{\partial \zeta(x)}{\partial x}\d x &\quad-\beta\int_0^1\mathfrak{c}(\psi(x)+\varphi(x))\zeta(x)\d x,
\frac{\partial^2q}{\partial u\partial v}+\frac{1-r^2+2qr}{(1-q^2)r-q(1-r^2)}\frac{\partial q}{\partial u}\frac{\partial q}{\partial v}=0, \frac{\partial^2 r}{\partial v\partial u}+\frac{-(1-q^2)-2qr}{(1-q^2)r-q(1-r^2)}\frac{\partial r}{\partial u}\frac{\partial r}{\partial v}=0.
\mathrm{Hess}_{\mathbb{C}}(\psi)(\exp(x)) =\begin{pmatrix}\frac{1}{4}\mathrm{Hess}_{\mathbb{R}}(\psi)(x) 0 0 0 M_{\alpha_{(1)}}(x) 0 0 0 \ddots \vdots \vdots \vdots \ddots 0 0 0 M_{\alpha_{(\frac{n-r}{2})}}(x) \end{pmatrix},
\mathrm{M a g n e t i c ~ N S ~ 5 ~ n e a r - h o r i z o n}: \quad \hat{x}^{\mu}= \frac{\hat{\ell}x^{\mu}}{\sqrt{\alpha^{\prime}}}, \, h a t{u}= \frac{\hat{\ell}r}{( \alpha^{\prime})^{3 / 2}}, \ \theta , \, h a t{\ell}^{2}= g^{- 1}\alpha^{\prime}\ .
I_{W}^{\ast}= \frac{L}{2}\int_{\bf R}d t \int_{\Sigma}\eta_{a b}\{-^{( 2 )}A^{( + ) a}\wedge^{\mathrm{~}( 2 )}\dot{A}^{( + ) b}+^{( 2 )}A^{( - ) a}\wedge^{\mathrm{~}( 2 )}\dot{A}^{( - ) b}\}.
u\left( t\right) :=\left( \frac{\partial^{2}}{\partial\bar{\alpha}^{2}}H^{\mathrm{cl}}\right) \left( \alpha\left( t\right) \right)\in\mathbb{C}v\left( t\right) =\left( \frac{\partial^{2}}{\partial\alpha\partial\bar{\alpha}}H^{\mathrm{cl}}\right) \left(\alpha\left( t\right) \right) \in\mathbb{R}.
{\cal E}\equiv M = \int\limits_{-\infty}^{\infty}dx \left[ \frac{1}{2}{\dot{\phi}}^2 + \frac{1}{2}{{\phi}'}^2 + \frac{\lambda}{4}\left(\phi^2 - \frac{m^2}{\lambda}\right)^2 + \varepsilon \frac{m^3}{\sqrt \lambda}\phi \right]= \frac{2{\sqrt 2}m^3}{3 \lambda}
\frac{c}{x}\bigg(-\tau_t+\frac{1-\alpha}{t}\tau\bigg)+nx^k\eta_2-t^{1-\alpha}\xi_{1t}-2\eta_{1x}+\xi_{1xx}+\frac{c}{x^2}\xi_1+\frac{c}{x}\xi_{1x}-mx^k\phi_2=0,
\theta_{d - 1}= \frac{J}{l \sqrt{2}}\left[ \sqrt{1 + E}\, E l l i p t i c E \left( \frac{T}{l}, \frac{1}{E + 1}\right) - \frac{E}{\sqrt{1 + E}}\, E l l i p t i c F \left( \frac{T}{l}, \frac{1}{E + 1}\right) \right] \; ,
\Delta(v,v_*,w,w_*):= 2(v-w)\cdot (b(v-v_*)-b(w-w_*)) + \|\sigma(v-v_*)-\sigma(w-w_*)\|^2 = \Delta_1(v,v_*,w,w_*)+\Delta_2(v,v_*,w,w_*),
M^{2}\phi = \left[ M_{T}^{2}+ \frac{r}{r_{0}^{\prime}}p_{9}^{2}- \frac{1}{r_{0}^{\prime}r}\partial_{r}r^{2}\partial_{r}+ \frac{1}{r_{0}^{\prime}r}L ( L + 1 ) \right] \phi .
\psi_{\scriptscriptstyle{l}}(k,r)=\frac{e^{- i \pi W/2}\, e^{i \pi |l|}}{2 \sqrt 2}\,\left(H_{\scriptscriptstyle{W}}^{(-)}(kr) +e^{i \pi (W - |l|)}\, e^{2 i \delta_l(k)}\,H_{\scriptscriptstyle{W}}^{(+)}(kr)\right)\,,
\frac{\partial}{\partial\omega}L[f(\omega,t)]=\frac{\partial}{\partial\omega}\int_{0}^{\infty}f(\omega,t)e^{-st}dt=\int_{0}^{\infty}\frac{\partial}{\partial\omega}(f(\omega,t))e^{-st}dt=L[\frac{\partial}{\partial\omega}(f(\omega,t))]
\begin{matrix}y=[\begin{matrix}2\\ 1\\ -3\\ 2\\ 2\\ \end{matrix}],&X=[\begin{matrix}1&-1&0&0\\ 0&0&1&-1\\ 0&-1&0&1\\ 1&0&0&-1\\ 0&1&-1&0\\ \end{matrix}],&r=[\begin{matrix}r_{A}\\r_{B}\\r_{C}\\r_{D}\\ \end{matrix}],&e=[\begin{matrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \end{matrix}]\\ \end{matrix}
- \frac{4}{\Gamma ( \frac{1}{2}d ) \Gamma ( 1 - \frac{1}{2}d )}\frac{( v^{2})^{\frac{1}{2}d - 1}}{1 + v^{2}}{}_{2}F_{1}\left({\textstyle{1 , \frac{1}{2}d - \frac{1}{2}; \frac{1}{2}d ; \frac{v^{2}}{1 + v^{2}}}}\right){\cal I}_{0}\ ,
B^{\alpha+\beta}_R =-C_{\alpha,\beta}R^{-2(\alpha+\beta)}\int^R_0\left(\int^t_0B^\alpha_rdr\right)\frac{d}{dt}[(R^2-t^2)^{\beta-1}t^{2\alpha+1}]dt &=C_{\alpha,\beta}\int^1_0\varphi(t)M^\alpha_{Rt}dt,
\phi(t)= 1+\|v_2(t)\|_2^4+\|\partial_zv_2(t)\|_2^4+\|v_2(t)\|_2^2\|\nabla_Hv_2(t)\|_2^2 +\|\partial_zv_2(t)\|_2^2 \|\nabla_H\partial_zv_2(t)\|_2^2+\|T_2(t)\|_2^4+\|\partial_zT_2(t)\|_2^4 +\|T_2(t)\|_2^2\|\nabla_HT_2(t)\|_2^2+\|\partial_zT_2(t)\|_2^2 \|\nabla_H\partial_zT_2(t)\|_2^2,
\sum_{t=0}^{2}q^{4\binom{t}{2}+3t}j(-q^{4t+2};q^3)m\Big (-q^{7-4t},-1;q^{12}\Big ) &=\overline{J}_{1,3}\Big ( m(q^2,-1;q^3)+\Theta_2(q)\Big ) +\frac{1}{4}\overline{J}_{0,3}\Big (\mu(q^3)+\frac{J_{6,12}^2}{J_{3}^3}\Big).
B^{ij}=\partial_x\circ F^{ij}\circ \partial_x = \partial_x^{}(g^{ij}\partial_x^{}+ c^{ij}_ku^k_x + c^\alpha w^i_{\alpha k}u^k_x\partial_x^{-1}w^j_{\alpha h}u^h_x)\partial_x^{}
\mathbf{z}^w =\mathbf{e}^{w\log \mathbf{z}}=\mathbf{e}^{(u+iv)(\ln r+i\theta )}=\mathbf{e}^{(u\ln r-v\theta )+i(u\theta +v\ln r)}=(e^{u\ln r-v\theta}, u\theta +v\ln r)
d_{ESF}^+(v) =d_{ESPS_{h,i,j}}^+(v)+\sum_{(h',j')\in ([\ell^*]\times [f])\setminus \{(h,j)\}}d_{ESPS_{h',i,j'}}^+(v) \qquad\qquad+\sum_{(h',i',j')\in [\ell^*]\times ([K]\setminus \{i\})\times [f]}d_{ESPS_{h',i',j'}}^+(v) &=1+(\ell^*f-1)\cdot 0+\ell^*(K-1)f\cdot 1=1+\ell^*(K-1)f.
\sum_{k=0}^{n}\frac{1}{k!(n-k)!}\sum_{\pi\in S_{n}}(-1)^{|a_{\pi}|}\Phi^{n-k+1}(\Phi^{k}(a_{\pi(1)},...,a_{\pi(k)}),a_{\pi(k+1)},...,a_{\pi(n)})=0.
\sigma_{0}\left( A\left( \beta\right) \right) =\sigma\left( A\left(\beta\right) \right) \cap D_{0}\left( \beta\right) , \sigma_{1}\left( A\left( \beta\right) \right) =\sigma\left( A\left(\beta\right) \right) \cap\cup_{j=1}^{N_{R}}D_{j}\left( \beta\right).
V^{\prime}=T_{s_{i}}^{-1}\left(0\right)\cap T_{s_{j}^{\prime}}^{-1}\left(0\right)\cap T_{s_{k}^{\prime\prime}}^{-1}\left(0\right)=T_{(s_{i},s_{j}^{\prime},s_{k}^{\prime\prime})}^{-1}\left(0\right).
N_{(2,3)}(n) = 8\,\sigma(\frac{n}{2}) - 32\,\sigma(\frac{n}{8}) + 8\,\sigma(\frac{n}{3}) - 32\,\sigma(\frac{n}{12}) + 64\, W_{(1,3)}(n) + 1024\, W_{(1,3)}(\frac{n}{4}) - 256\, \biggl( W_{(3,8)}(n) + W_{(1,12)}(n) \biggr),
w^{(4)}_1 = s_2 s_4 s_5 s_3 s_4 s_2 = (s_2s_4s_5 s_4s_2)(s_2s_4s_3 s_4s_2) = s_{\alpha_2 + \alpha_4 + \alpha_5}s_{\alpha_2 + \alpha_4 + \alpha_3}.
{\check{c}}_{\gamma , \sigma \dot{\beta}}^{\enspace \enspace \enspace \, \sigma \dot{\sigma}}= \bar{c_{a}}^{\sigma \dot{\sigma}}E_{\gamma}{c_{\sigma \dot{\beta}}}^{a}+{c_{\sigma \dot{\beta}}}^{a}\bar{c_{b}}^{\sigma \dot{\sigma}}{c_{\gamma a}}^{b}-{c_{\gamma \dot{\beta}}}^{\dot{\sigma}}
Y(t)=\left(I+\sum_{k=1}^{\infty}Y_k(-t)^{-3k/2}\right)\begin{pmatrix}(-t)^{1/4}0 0 0 e^{\frac{2\sqrt2}{9}(-t)^{3/2}}0 0 0 e^{-\frac{2\sqrt2}{9}(-t)^{3/2}}\end{pmatrix},
R (b_k): \sum_l u_{i_1l}\ldots u_{i_kl}=\delta_{i_1i_2}\delta_{i_2i_3}\ldots\delta_{i_{k-1}i_k}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\Sigma ( z ) = \frac{\int_{- \infty}^{z}\sqrt{- g^{( 4 )}}d z^{\prime}\left({T^{t}}_{t}-{T^{z}}_{z}-{T^{r}}_{r}-{T^{\phi}}_{\phi}\right) \int d x d y}{\sqrt{g^{( 2 )}( z )}\int d x d y}
\begin{aligned}\int_{\Omega_\ell}(A(X_2)\nabla u_\ell).\nabla u_\ell \geq \int_{\Omega_\ell}(A_{22}\nabla_{X_2}u_\ell).\nabla_{X_2}u_\ell-\big(A_{11}^{-1}A_{12}\nabla_{X_2}u_\ell\big).A_{12}\nabla_{X_2}u_\ell &\geq \Lambda^1\int_{\Omega_\ell}u_\ell^2\,,\end{aligned}
\tau_1(x)=x\cdot\left[\begin{array}{ccc}1 0 0 0 1 1 0 0 1\end{array}\right]+e_1,\;\tau_2(x)=x\cdot\left[\begin{array}{ccc}1 0 1 0 1 0 0 0 1\end{array}\right]+e_2,\;\tau_3(x)=x\cdot\left[\begin{array}{ccc}1 0 0 0 1 0 0 0 1\end{array}\right]+e_3,
\Pr[\hat{M}= M] \leq \sum_k \binom{n}{k}\frac{s!}{(k+s)!}(1-y)^{n-k}(1-x)^{k(k+s)}e^{k \theta}\leq \sum_k \binom{n}{k}(1-y)^{n-k}e^{-k^2 x}\frac{e^{k \theta}}{k!}\leq \sum_k \binom{n}{k}(1-y)^{n-k}e^{-(2 \ell k - \ell^2)x}e^{k \theta}= e^{\ell^2 x}(1-y+e^{\theta-2\ell x})^n,
\mathcal{V}_r(\sigma)(z) =\sup_K\sup_{N_0<\ldots<N_K}\Big(\sum_{j=0}^{K-1}\|\sigma_{N_{j+1}}(z)-\sigma_{N_j}(z)\|_{}^r\Big)^{\frac1r}, \mathcal{V}_r(\gamma)(z) =\sup_K\sup_{N_0<\ldots<N_K}\Big(\sum_{j=0}^{K-1}d(\gamma_{N_j}(z),\gamma_{N_{j+1}}(z))^r\Big)^{\frac1r},
\left\| M_{n+1}\right\|^{2}= \left\| M_{n}\right\|^{2}+ 2 \left\langle M_{n}, \xi_{n+1}\right\rangle + \left\| \xi_{n+1}\right\|^{2}& \leq \left\| M_{n}\right\|^{2}+ 2 \left\langle M_{n}, \xi_{n+1}\right\rangle + M^{2}.
\| L_{\xi p_k}- L_{\xi_n p_k}\| = \sup_{a \in N, \| a \|_2 \leq 1}\| (\xi - \xi_n) p_k a \| = \sup_{a \in N, \| a \|_2 \leq 1}\varphi_n( p_k a^*a p_k ) \to 0.
\frac{1}{(1-t^2)^d}= \sum_C t^{\lambda_C}P^T_t(C), \frac{1}{(1-t^2)^{d+1}}= \sum_{\tilde{C}}t^{\tilde{\lambda}_C}P^{\tilde{T}}_t(\tilde{C}), \frac{1}{(1-t^2)^{d+1}}= \sum_{\hat{C}}t^{\hat{\lambda}_C}P^{\hat{T}}_t(\hat{C}).
f_\lambda(\lambda) =N\lambda^{N-1}+\sum_{k=1}^{N-1}(-1)^k(N-k)\lambda^{N-k-1}\sum_{1\le j_1<\cdots<j_k\le N}\underset{1\le l,m\le k}{{\rm det}}(H_{j_lj_m}), f_{\lambda \lambda}(\lambda) =N(N-1)\lambda^{N-2}+\sum_{k=1}^{N-2}(-1)^k(N-k)(N-k-1)\lambda^{N-k-2}\sum_{1\le j_1<\cdots<j_k\le N}\underset{1\le l,m\le k}{{\rm det}}(H_{j_lj_m}).
\frac{1}{\lambda_u}\sum_{j=0}^{t_0-1}\left(\frac{\lambda}{\lambda_u}\right)^j \left(\frac{{t-j-1}}{t}\right)^{d_u}\frac{1}{\lambda_u^{t-j-1}\cdot ({t-j-1})^{d_u}}M^{t-j-1}\vec u_{j+1}
\left(\partial_{s}-\partial_{yy}^{2}+\frac{1}{y}\left(2\left(n-1\right)-\frac{y^{2}}{2}\right)\partial_{y}-\frac{1}{y^{2}}\left(2\left(n-1\right)+\frac{y^{2}}{2}\right)\right)\left(\frac{k}{c_{2}}e^{-\lambda_{2}s}\varphi_{2}\left(y\right)\right)=0
P^{-1}{}_{\!\!\mu_1 \cdots \mu_k, \nu_1 \cdots \nu_k}(p) \, P_{\nu_1 \cdots \nu_k, \lambda_1 \cdots \lambda_k}(p) ={\cal E}^A{}_{\!\!\mu_1 \cdots \mu_k, \lambda_1 \cdots \lambda_k}\,
\left( \begin{array}{c c}{S^{i}\sigma_{i}}&{0}\\{0}&{S^{i}\sigma_{i}}\\ \end{array}\right) \equiv \frac{1}{2}h^{- 1}\left( \begin{array}{c c}{\sigma_{3}}&{0}\\{0}&{\sigma_{3}}\\ \end{array}\right) h ,
{G_{EG-PC}}={{\sum\limits_{l = 1}^N{{{\bigg{|}{{\bf{w}}_r^T{{\bf{C}}_l}{{\bf{w}}_t}}\bigg{|}^2}}}}}\big{/}\left({{\left({{\bf{w}}_t^H{{\bf{w}}_t}}\right)\left({{\bf{w}}_r^H{{\bf{w}}_r}}\right)}}\right).
\frac{z_0^1(B^1;k)}{\frac{\sin\{kB^1\}}{k}}=\frac{{\rm Res}\{z_0^1(B^1;k)/\frac{\sin\{kB^1\}}{k};\gamma_j\}}{(k-\gamma_j)}+{\mbox{higher order terms}},\, \mbox{for}k\mbox{near}\gamma_j.
\partial_{\mu}^{x}\, U ( x , x^{\prime}) = i g \left[ A_{\mu}( x ) - \! \! \int_{0}^{1}\! \! d s \, s \,{\frac{d \omega^{\nu}}{d s}}\U ( x , \omega ) \, F_{\nu \mu}( \omega ) \, U ( \omega , x ) \right] U ( x , x^{\prime})
|\nabla_g u|^2_g =g^{ij}\frac{\partial u^{\alpha}}{\partial x^i}\frac{\partial u^{\alpha}}{\partial x^j}=e^{\frac{|x|^2}{2(m-2)}}|\nabla_{g_0}u|^2_{g_0}; & dV_g =\sqrt{detg}dV_{g_0}=e^{-\frac{m}{4(m-2)}|x|^2}dV_{g_0}.
I_t^{-1}= I_{t-1}^{-1}- l_{gt}I_{t-1}^{-1}X_{t-m}^{t-1}(1+ l_{gt}(X_{t-m}^{t-1})^T I_{t-1}^{-1}X_{t-m}^{t-1})^{-1}(X_{t-m}^{t-1})^T I_{t-1}^{-1}.