harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle W(\ell_1)^{1/2}\ell_{1}|\Delta_n W(\ell_2)^{1/2}\ell_{2}\rangle_{n}=W(\ell_1)^{1/2}W(\ell_2)^{1/2}{}_n\langle \ell_{2}, \ell_{1}\rangle &=\delta_{\ell_1=\ell_2}W(\ell_1)^{1/2}W(\ell_2)^{1/2}W(\ell_1)^{1/2}W(\overline{\ell_{2}})^{\frac{1}{2}}&=\delta_{\ell_1=\ell_2}W(\ell_1).
\sum_{n=0}^{N}\frac{(q;q^2)_{n}(q^2;q^2)_N(zq^2;q^2)_{N-n}z^n q^{2n}}{(zq;q^2)_{n+1}(q^2;q^2)_{N-n}(zq^2;q^2)_{N+1}}&=\frac{1}{(1-zq^{2N+3})}\sum_{n=0}^{N}\left(\frac{(q;q)_{2n}z^{2n}q^{2n^2+3n}}{(zq;q)_{2n+1}}+\frac{(q;q)_{2n+1}z^{2n+1}q^{(2n+1)(n+2)}}{(zq;q)_{2n+2}}\right)\frac{(q^2;q^2)_N(zq^5;q^2)_N}{(q^2;q^2)_{N-n}(zq^5;q^2)_{N+n}}.
T_{n_v}^*T_{n_w}= E_{o_-\cdot v^{-1}}(n_v)E_{o_-}(n_w) & = E_{o_-\cdot v^{-1}}(n_vn_w)\in T_{n_{w_Gw_P}}+ \sum_{a < w_Gw_P}T_{n_a}C[Z_\kappa].
\begin{aligned}|B^b| = n - |A^*| - |C^*| - |\{x_i,\dots,x_k\}| - \left|\bigcup_{j=1}^{i-1}P_j\right| \geq n - n_i - 4(\epsilon \xi |R| + \epsilon)n - (\epsilon \xi |R| + \epsilon)n - (k-i+1) - \sum_{j=1}^{i-1}n_j &\geq \sum_{j=i+1}^{k}n_j - 6(\epsilon \xi |R| + \epsilon)n.\end{aligned}
\begin{aligned}0 = -cn^2+(h_1-2c)an^2-h_1\sum\limits_{k=-(n-1)}^{n-1}y_{n-k}\left(a(n^2-k^2)\right) + & = -cn^2+(h_1-2c)an^2-h_1\sum\limits_{k=-(n-1)}^{n-1}\sqrt{a(n^2-k^2)}f\!\left(\dfrac{n-k}{\sqrt{a(n^2-k^2)}}\right) & + & =\left(-c+(h_1-2c)a-h_1 R_n(a)\right)n^2 + ,\end{aligned}
\bigg( P(\xi)=\mathsf{D}_{z_{1}}\cdot P(\xi)=\dotsb=\mathsf{D}_{z_{1}}^{p}\cdot P(\xi)=0 \bigg) \Rightarrow \bigg( \Lambda_{z_{1}}( \mathsf{V}) (\xi) = \Lambda_{z_{1}}( \mathsf{U}) (\xi) \bigg).
{\cal L}_{\eta}| \Phi_{p h}\rangle = ( \eta \hat{\partial}+ \frac{1}{2}\hat{\partial}^{I}\eta^{J}M^{I J}+ \frac{1}{2}\eta^{I}M^{z I}) | \Phi_{p h}\rangle - \hat{\partial}^{+}\eta^{I}\alpha^{I}( \bar{\alpha}^{-}| \Phi \rangle ) \Bigr |_{\alpha^{+}= 0}^{\vphantom{5 p t}},
g(u) = \xi(1)-\xi(u)-\frac{q(1-u)}{[A_2(1-q)+\Delta][A_1q+A_2(1-q)+\Delta]}+\frac{1-u}{A_2[A_2(1-q)+\Delta]}-\frac1{A_2^2}\log\Big(1+\frac{A_2(1-u)}{\Delta}\Big).
\frac{1}{2}\frac{d}{dt}\sum_{\ell\le k\le N-1}\|\nabla^k f\|^2_{L^2_{x,p,\mathcal{I}}}+ \sum_{\ell\le k\le N-1}\|\nabla^k \{I-P\}f\|^2_{L^2_{x,p,\mathcal{I}}}\cr \le C\sum_{\ell\le k\le N-1}\sum_{|{\alpha_1}|\le k}\left\|\|\nabla^{|\alpha_1|}f\|_{L^2_{p,\mathcal{I}}}\|\nabla^{k-|\alpha_1|}f\|_{L^2_{p,\mathcal{I}}}\right\|_{L^2_x}^2.
\tilde{H}_{n , k}= n^{- k}\, \prod_{s = 0}^{k}( s \, n - 1 \, ) \sum_{m = 0}^{\lfloor \frac{k}{2}\rfloor}\frac{1}{m ! \, ( k - 2 \, m ) ! \, n^{m}}\prod_{l = 0}^{m}{\frac{1}{l \, n - 1}}\, v^{m n}\, u^{k - 2 m}
\mathbf E\big[M_{\tau+N,n}^2\big] =\sum_{S=1}^{N}S^{-|n|}\mathbf E\big[(Y_n(\tau+S)-Y_n(\tau+S-1))^2\big] &+\sum_{S=1}^{N}\sum_{S\neq R=1}^{N}S^{-|n|/2}R^{-|n|/2}\mathbf E\big[(Y_n(\tau+S)-Y_n(\tau+S-1))(Y_n(\tau+R)-Y_n(\tau+R-1))\big] &=\sum_{S=1}^{N}S^{-|n|}\mathbf E\big[(Y_n(\tau+S)-Y_n(\tau+S-1))^2\big]
\lim_{i \to \infty}\frac{r_{i}}{\tilde r_{i}}=+\infty \quad\mbox{or}\lim_{i \to \infty}\frac{r_{i}}{\tilde r_{i}}=0; 0< \liminf_{i \to \infty}\frac{r_{i}}{\tilde r_{i}}\leq \limsup_{i \to \infty}\frac{r_{i}}{\tilde r_{i}}<\infty .
{}\mathcal{T}_{\Gamma}(\overline{x}) =\!\big\{h\in\mathbb{X}\ |\g'(\overline{x})h\in\mathcal{T}_{K}(g(\overline{x}))\big\},\quad \mathcal{N}_{\Gamma}(\overline{x})=\mathcal{\widehat{N}}_{\Gamma}(\overline{x}) =\!\big\{\nabla g(\overline{x})\lambda\ |\ \lambda\in\mathcal{N}_{K}(g(\overline{x}))\big\}.
S_q (y, M)= \sum_{\alpha ,\beta\gamma, Z, KK}g^2_{\alpha}(M) g^2_{\beta}(M){(v^{\alpha}_e v^{\beta}_e+a^{\alpha}_e a^{\beta}_e)(v^{\alpha}_l v^{\beta}_l + a^{\alpha}_la^{\beta}_l) \over (s -m^2_{\alpha}+ i\Gamma{_\alpha}m_{\alpha})(s-m^2_{\beta}- i\Gamma_{\beta}m_{\beta})}\ .
\begin{array}{rcl}G_{\sigma}^{(n)}(\zeta'',\zeta_{2n-1}, \epsilon x^{-1}\zeta_{2n-1})^{\varepsilon''\, ss'}= s^{(1-\epsilon )/2}\delta_{s+s',0}G_{\epsilon\sigma}^{(n-1)}(\zeta'')^{\varepsilon''}~~~~ (s,s'=\pm ). \end{array}
{\cal L^{\prime}}_{h}^{M^{\prime}}( V_{h}, \phi ) = \frac{1}{1 6}\int d^{2}\theta \frac{1}{g_{h}^{2}}W^{a}( V_{h}) W^{a}( V_{h}) + \mathrm{h . c .}+ \int d^{4}\theta \sum_{i}\phi^{\dagger}e^{2 V_{h}^{i}}\phi
\tilde{D}_{d a , b c}^{l i , j k}\equiv{\frac{1}{N}}\sum_{\tilde{m}= 0}^{P - 1}\tilde{B}( l - j - Q \tilde{m}) \mathrm{e}^{i{\frac{2 \pi}{P}}( d - a ) \tilde{m}}\delta_{a b}\delta_{c d}\delta_{l i}\delta_{j k},
T^2 = \begin{bmatrix}0 0 0 0 0 -1 1 0 0 1 -1 0 0 0 0 0 \end{bmatrix}^2 = \begin{bmatrix}0 0 0 0 0 2 -2 0 0 -2 2 0 0 0 0 0 \end{bmatrix}= -2T
\mathcal{H}_{N}(X_{N}) = N^{2}\left(\mathcal{E}^{\neq}({\rm emp}_{N}) + \int_{M}V \, d{\rm emp}_{N}\right) = N^{2}\left(\mathcal{E}(\mu_{\theta}) + 2 \mathcal{G}(\mu_{\theta},{\rm emp}_{N}- \mu_{\theta}) + \mathcal{E}^{\neq}({\rm emp}_{N}- \mu_{\theta}) + \int_{M}V \, d{\rm emp}_{N}\right)
\langle \hat{\bf p}, j_{3}, [{\mathsf s}, j ] , n | S | \hat{\bf p}^{\prime}, j_{3}^{\prime}, [{\mathsf s}^{\prime}, j^{\prime}] , n^{\prime}\rangle = 2 \hat{E}( \hat{p}) \delta^{3}( \hat{p}- \hat{p}^{\prime}) \delta ({\mathsf s}-{\mathsf s}^{\prime}) \delta_{j_{3}j_{3}^{\prime}}\delta_{j j^{\prime}}\langle n \| S_{j}({\mathsf s}) \| n^{\prime}\rangle \, ,
= \frac{3}{2}\times \frac{1}{2}\times \frac{4}{3}\times \frac{2}{3}\times \cdots \times \frac{2 0 1 6}{2 0 1 5}\times \frac{2 0 1 4}{2 0 1 5}\times \frac{2 0 1 7}{2 0 1 6}\times \frac{2 0 1 5}{2 0 1 6}
\int_U{|\omega(y)| \over |x - y|^s}\, dy = \int_{U \setminus B_r(x)}{|\omega(y)| \over |x - y|^s}\, dy + \int_{U\cap B_r(x)}{|\omega(y)| \over |x - y|^s}\, dy & \leq{\|\omega\|_{L^1(U)}\over r^s}+ \left({\gamma_n \over n-qs}\right)^{1/q}r^{(n - qs)/q}\|\omega\|_{L^p(U)}.
= \frac{-\xi\cdot(\xi-2\eta)}{4|\xi|^2}\underbrace{(1+\cos(\angle(\xi-\eta, \eta)))\Big((\xi-\eta)\times\eta\Big)}_{}- \frac{\xi}{4 |\xi|}\cdot \underbrace{\Big(\frac{\xi-\eta}{|\xi-\eta|}+\frac{\eta}{|\eta|}\Big) \Big( (\xi - \eta) \times \eta \Big)}_{}.
\widetilde{U}_{s_n,t_n}(0, \Delta) = \int_{\mathbb{R}}U_{s_n,t_n}(x, \Delta) \,dx &= \int_{\mathbb{R}}\mathbf{1}_{(s_n,t_n)}(x) \,dx + O\left(\int_{\mathbb{R}}K_{s_n,t_n}(x, \Delta) \,dx\right) &= 2\pi \delta+ O\left(\frac{1}{\Delta}\right),
\lim_{s\rightarrow\infty}\left\Vert f\right\Vert_{L^{2}(K,\rho_{s})}=\left\Vert f\right\Vert_{L^{2}(K)}\lim_{s\rightarrow\infty}\left\Vert F\right\Vert_{L^{2}(K_{\mathbb{C}},\mu_{s,\tau})}=\left\Vert F\right\Vert_{L^{2}(K_{\mathbb{C}},\nu_{t})}.
I^{(6)}= \int d^6 x \sqrt{- g^{(6)}}(e^{2 \psi - 2 \phi}( R^{(6)}+3 (D \psi )^2 - 8 D_{\alpha}\psi D^{\alpha}\phi + 4 (D \phi )^2 ) - \frac{1}{12}H^{(6) 2}) ,
P_{x,r}^U(A_k^c | U_{k-1}^- = z)= P_{x,r}^U\Big( |U_k^- - U_{k-1}^-| > \Big(\frac{\epsilon^{1/3}x^{2/3}}{z}\Big) \cdot z \Big| U_{k-1}^- = z \Big) & \leq c_1 \exp\Big( -c_2 \frac{(\epsilon^{1/3}x^{2/3})^2}{2x}\Big) = c_1 \exp\Big(-\frac{c_2}{2}\epsilon^{2/3}x^{1/3}\Big).
\sum_{x\in X}\gamma_b(x)\phi_{a}(x) =\sum_{c\preceq b}\gamma_b(x^c)\phi_{a}(x^c) &=\sum_{c\preceq b}\gamma_b(x^c) 1_{a\preceq c}&=1_{a\preceq b}\sum_{a\preceq c\preceq b}\gamma_b(x^c).
\sum_{k = 2}^\infty \mathbb{E}_p\big(\Phi_A U_{k-2}(a_E,p)\big)t^k = \mathbb{E}_p \Big(\Phi_A \sum_{k=2}^\infty U_{k-2}(a_E,p)t^k\Big) = \mathbb{E}_p \Big( \frac{\Phi_A t^2}{pt^2 - a_E t + 1}\Big) = \frac{t^2}{(1+pt^2)}\cdot \mathbb{E}_p\left(\Phi_A\sum_{\ell=0}^\infty \left(\frac{a_Et}{1+pt^2}\right)^\ell \right).
\tilde{H}{}^{0f}_a=\frac{1}{2}\left[-\frac{d^2}{dx^2}+W_{-,\nu}^2-\nu-3-\frac{1}{2}(W'_{-,\nu}-\nu+1)\right]-\frac{1}{4}(W'_{-,\nu}-\nu+1)\cdot\sigma_i\otimes \sigma_i
f(y)=A e^{-y^2/4}M\left( \frac{a}{2}+\frac{1}{4},\frac{1}{2},\frac{y^2}{2}\right ) +Bye^{-y^2/4}M\left( \frac{a}{2}+\frac{3}{4},\frac{3}{2},\frac{y^2}{2}\right ),
\psi_{- 2 q}^{2 -}= - \frac{i}{2}\nabla_{-}\chi_{q}^{2}, \ \psi_{+ 2 q}^{2 -}= - \frac{i}{4}\Omega_{+}^{- 2 i}\gamma_{q \dot{q}}^{i}\chi_{\dot{q}}^{2},
\frac{\sup_{\|X_i\|_{\ell_\mathbb{R}^2}=1}\sum_{i,j=1}^la_{ij}\langle X_i,X_j\rangle}{\|P_{A_k}\|_{M_p(\mathbb{Z}^l)}}\geq\frac{2k(k-1)^2}{(2k(k-1)(2k-3))^{1-\frac{2}{p^\prime}}\big(\frac{2k(k-1)(2k-1)}{3}\big)^{\frac{2}{p^\prime}}}.
P A \left( I - P \right) x =\langle A^{\star}e_0 \otimes e_+, \left( I - P \right) x \rangle \, e_0 \otimes e_++\langle A^{\star}e_0 \otimes e_-, \left( I - P \right) x \rangle \, e_0 \otimes e_- .
\begin{array}{c}u=u_0+\frac{p^{+}}2\left[ (\sigma^{+}+f(\sigma^{+}))+(\sigma^{-}-g(\sigma^{-}))\right] v=v_0+\frac{p^{-}}2\left[ (\sigma^{+}-f(\sigma^{+}))+(\sigma^{-}+g(\sigma^{-}))\right]\end{array}
\Lambda ( x) - \hat{\Lambda}( x) = \hat{a}(n-1) \bigl\lbrack a(n)-a(n-1) - b(n-1) \bigr\rbrack + b(n) \bigl\lbrack \hat{a}(n) + \hat{b}(n) - \hat{b}(n-1) \bigr\rbrack. \!\!\!
\eta^k = \widetilde{F}^k(z^{k+1}) + \frac{\rho}{2}\big\| z^{k+1}-x^k\big\|^2 \le\widetilde{F}^k(z^{k+1}) + \frac{1}{2}\big[F(x^k) - \widetilde{F}^k(z^{k+1})\big] &= \frac{1}{2}\big[F(x^k) + \widetilde{F}^k(z^{k+1})\big].
Q_n(x) = \sum_{k\ge 0}q_{n,k}x^k = \sum_{k\ge 0}(q_{n-1,k}+ q_{n-2,k}+ q_{n-2,k-1})x^k = \sum_{k\ge 0}q_{n-1,k}x^k + \sum_{k\ge 0}q_{n-2,k}x^k + x\sum_{k\ge 0}q_{n-2,k-1}x^{k-1}= Q_{n-1}(x) + (1+x)Q_{n-2}(x).
\|u\|_{E^{\alpha,p}}^p =\int_0^T|_0D_t^\alpha u(t)|^pdt=\sum_{i=1}^n|\lambda_i|^p\int_{\mathbb{I}_i}|_0D_t^\alpha u_i(t)|^pdt &=\sum_{i=1}^n|\lambda_i|^p\int_0^T|_0D_t^\alpha u_i(t)|^pdt=\sum_{i=1}^n|\lambda_i|^p\|u_i\|_{E^{\alpha,p}}^p &=\sum_{i=1}^n|\lambda_i|^p,\ \ \forall u\in E_n.
\frac{\partial^{n-1}\alpha}{\partial u^{n-1}}(u,v)=\frac{d^{n-1}\alpha_i}{dv^{n-1}}(u)+\int_0^v\left\{\sum_{i=0}^{n-1}\binom{n-1}{i}\frac{\partial^{i+1}t}{\partial u^i\partial v}\frac{\partial^{n-1-i}\tilde{A}}{\partial u^{n-1-i}}\right\}(u,v')dv'.
\begin{aligned}h(v) + \frac{1}{2t}\lVert v-u \rVert^2 + \langle v-u,g\rangle \leq h(z)+\frac{1}{2t}\lVert z-u\rVert^2+\langle z-u,g\rangle -\frac{1}{2t}\lVert v-z\rVert^2. \end{aligned}
B^{( 1 )}= \left( \begin{array}{c c c c}{J^{( 1 )}}&{}&{}&{}\\{}&{J^{( 1 )}}&{}&{}\\{}&{}&{J^{( 1 )}}&{}\\{}&{}&{}&{J^{( 1 )}}\\ \end{array}\right) , \makebox [ 3 m m ]{}J^{( 1 )}= i \frac{\sqrt{e}}{\lambda}\left( \begin{array}{c c}{}&{- 1}\\{1}&{}\\ \end{array}\right) .
\bar{P}_{\mathrm{RX}}^{\mathrm{GA}}(\mathbf{u}_{*}) =\mu(\mathbf{u}_{*})+\sum_{i=1}^{N}\beta_{i}\int C(\mathbf{x}_{*},\mathbf{x}_{i})\, p(\mathbf{x}_{*})\,\mathrm{d}\mathbf{x}_{*}
\left\langle \Psi_{1}{}\left|\right.\Psi_{2}{}\right\rangle=\sum_{\alpha, \beta \in \Omega}\frac{1}{N(\alpha)N(\beta)}\int{\rm Tr}\left(f_{1}^{\alpha \beta +}f_{2}^{\alpha \beta}\right)P_{\lambda}dq^{1}\ldots dq^{n},
\sup_{\delta < |t| \leq \lambda}\left \lvert \frac{\phi(s + \mathrm{i}t)}{\phi(s)}\right \rvert = \sup_{\delta < |t| \leq \lambda}\left \lvert \prod_{j = 1}^{w_n}\frac{\tilde{\phi}_j(s + \mathrm{i}t)}{\tilde{\phi}_j(s)}\right \rvert \cdot 1 \leq c_2^{w_n}= o(n^{-1/2}),
H_{\phi}\Phi_{\nu}(\phi) = \epsilon_{\nu}^2 \Phi_{\nu}(\phi), H_{\phi}= - \partial_{\phi}^2 + \frac{\alpha^2 - 1/4}{\cos^2\phi}+ \frac{\beta^2 - 1/4}{\sin^2\phi}- 2 \partial_{\phi}^2 \log P^{(-\alpha-1,\beta-1)}_{m_1}(-\cos2\phi), \epsilon_{\nu}= \alpha + \beta + 1 + 2\nu, \nu=0, 1, 2, \ldots,
\chi(\theta)= \limsup_{m\to\infty}([A_*(\theta)^m]_{jj})^{\frac{1}{m}}\ge \limsup_{m\to\infty}([A_*(\theta)^{n_0 m}]_{jj})^{\frac{1}{n_0 m}}\ge c^{\frac{1}{n_0}}e^{\frac{\theta}{n_0}}.
= \; \Bigg |{\; \atop{\mathrm{d e t}\atop{\scriptstyle i , j = 1 , . . . , 2 n_{a}}}}\Bigg ( \frac{1}{w_{i}^{( a )}- z_{j}^{( a )}}\Bigg ) \Bigg | \; \times \; \Bigg |{\; \atop{\mathrm{d e t}\atop{\scriptstyle i , j = 1 , . . . , 2 n_{b}}}}\Bigg ( \frac{1}{w_{i}^{( b )}- z_{j}^{( b )}}\Bigg ) \Bigg |
F(t;x,y,1,u,z,v) =\frac{xzvt^2(1-yr)(ytuv(1-z)+z)}{(1-ytu)(1-tuv(y-yr+1))(tux+y^{-1}-tu)}\quad+\frac{tx(y-yzr+z)}{(y-yr+1)(tux+y^{-1}-tu)}F(t;x,y-yr+1,1,u,z,v) \quad-\frac{yu^2vt^2z(1-v)(yr-1)(ytuv(1-z)+z)}{(1-ytu)(1-tuv(y-yr+1))}F(t;x,y,1,u,1,v).
\prod_{a=1,2}\prod_{j=1}^W\exp\Big\{-M\big(r_{j,a}-\log r_{j,a}-1\big)\Big\}\cdot r_{j,a}^{\ell_{j,a}}\cdot \Big(1+\mathfrak{L}^{-\frac12}\Big)^{O(W^2)}&\leq \max_{b=1,2}\max_{k=1,\ldots,W}\prod_{a=1,2}\prod_{j=1}^W\exp\Big\{-M\big(r_{j,a}-\log r_{j,a}-1\big)\Big\}\cdot r_{j,a}^{\ell_{j,a}}\left(1+r_{k,b}^{-1}\right)^{O(W^2)}=e^{O(W^2)}.
g l ( 3 n ) \downarrow g l ( 3 ) + g l ( n ) \quad{\{}\overline{{{1^{K}}}}{\}}\cdot{\{}1^{L}{\}}\downarrow \sum_{\rho \vdash K , \sigma \vdash L}{\{}\overline{{{\rho^{\prime}}}}{\}}\cdot{\{}\sigma^{\prime}{\}}\times{\{}\overline{{{\rho}}}{\}}\cdot{\{}\sigma{\}},
\delta{\cal L}_{t o t a l}= \omega \left[ J^{\mu , a}( D_{\mu}c^{a}) - i \overline{{{\eta}}}^{a}( -{\frac{1}{2}}f^{a b c}c^{b}c^{c}) + i F^{a}\eta^{a}+ H^{\mu}{\frac{\partial{\cal L}_{t o t a l}}{\partial n^{\mu}}}\right]
{\displaystyle V_{\underline{{\alpha}}}^{( 1 )}= V_{\underline{{\alpha}}}- \frac{i{\cal P}_{\underline{{{\alpha \dot{\beta}}}}}\bar{V}^{\underline{{{\dot{\beta}}}}}}{( m - i e W / 2 )}; \ \, b a r V_{\underline{{{\dot{\alpha}}}}}^{( 1 )}= \bar{V}_{\underline{{{\dot{\alpha}}}}}- \frac{i{\cal P}_{\underline{{{\beta \dot{\alpha}}}}}V^{\underline{{\beta}}}}{( m + i e \bar{W}/ 2 )},}
\chi^{(2, 2k+3)}_{1,i}(q) =\chi^{(2, 2k+3)}_{1,2k+3-i}(q)=\displaystyle\sum_{n_1 \geq \cdots \geq n_{k}\geq 0}\frac{q^{n_1^2 +\cdots + n_{k}^2 +n_i +\cdots +n_{k}}}{(q)_{n_1 -n_2}\cdots (q)_{n_{k-1}-n_{k}}(q)_{n_{k}}},
X_{z_{\pm}}= - i Q_{\pm}^{- 1}\displaystyle \displaystyle \frac{\partial}{\partial \overline{{z}}_{\pm}}, X_{p_{\pm}}= i \displaystyle \displaystyle \frac{\partial}{\partial z_{\pm}}- i Q_{\pm}^{- 1}\displaystyle \frac{\partial p_{\pm}}{\partial z_{\pm}}\displaystyle \displaystyle \frac{\partial}{\partial \overline{{z}}_{\pm}},
{\cal M}_{ii \rightarrow jj}\sim g^2 \sum_{k}c_{iik}c_{jjk}\int_0^{\infty}dt e^{-t (p^2 + \lambda_k)}= g^2 \int_0^{\infty}dt e^{-t p^2}\sum_{k}c_{iik}c_{jjk}e^{-t \lambda_k}
\bigg\| \bigg(\sum_j \bigg(\sum_k (f^{jk})^s\bigg)^{\frac{t}{s}}\bigg)^{\frac{1}{t}}\bigg\|_{L^p(v)}\lesssim\prod_{i=1}^m\bigg\| \bigg(\sum_j \bigg(\sum_k (f_i^{jk})^{s_i}\bigg)^{\frac{t_i}{s_i}}\bigg)^{\frac{1}{t_i}}\bigg\|_{L^{q_i}(v_i)}.
(y(g),\gamma,\delta){\tau_{i_j}}= \sum_{E\subseteq\{i_1,\ldots,i_s\}}(\prod_{e\in E}y_e)(g^{\tau_E\tau_{i_j}},\gamma,\delta) &= y_j\sum_{E\subseteq\{i_1,\ldots,i_s\}}(\prod_{e\in E}y_ey_j)(g^{\tau_E\tau_{i_j}},\gamma,\delta) = y_j(y(g),\gamma,\delta).
S_{( L - P , - Q )}= T_{( L - P , - Q )}{\Omega}_{p}{\int}_{t_{-}}^{0}d t{\rho}^{p}\sqrt{1 +{\dot{\rho}}^{2}}- \frac{Q{\Phi}}{p + 1}{\Omega}_{p}{\int}_{t_{-}}^{0}d t{\rho}^{p + 1},
f_{d-k}= v_{d-3-k}^n+v_{d-3-k}z^{t-n}-(v_{d-3-k+1}-v_{d-3-k+2}+\cdots+(-1)^{d+d-3-k}(v_{d-3})+(-1)^{d+d-3-k+1}x)^n+v_{d-3-k}z^n-v_{d-3-k}x^n.
{U}^{\prime}\left({-{\tfrac{1}{2}}\gamma\alpha^{2},0}\right) =-\pi^{-1/2}2^{\left({\gamma\alpha^{2}+1}\right) /4}\Gamma\left({{\frac{1}{4}}\gamma\alpha^{2}+{\frac{3}{4}}}\right) \sin\left({{\frac{1}{4}}\gamma\alpha^{2}\pi+{\frac{3}{4}}\pi}\right) .
G_{1}^{( m , k )}= \Phi ( \vec{x}_{r}, \vec{x}_{s}) \left( \frac{1}{\mathcal{S}_{+}}- \frac{1}{\mathcal{S}_{-}}\right) = \Phi ( \vec{x}_{r}, \vec{x}_{s}) \left( \frac{\sinh \mathcal{U}}{\cosh \mathcal{U}- \cos \mathcal{T}}\right) .
Q_0(\mathbf{v,v'|u_0}) := \exp(\, w_1 \tilde{D}(\mathbf{u_0,v}) + w_2 [\tilde{D}(\mathbf{u_0,v'}) - \tilde{D}(\mathbf{u_0,v})] &- n \gamma_0(w_1,w_2) \, ) F(\mathbf{v'}) W^n(\mathbf{v|u_0})
\sum_{t}\operatorname{pr}(y|x,\mathbf{t})\operatorname{pr}\bigl(\mathbf{t}|x'\bigr)=\frac{\sum_{t}\operatorname{pr}(y|x,\mathbf{t})\operatorname{pr}(\mathbf{t})-\operatorname{pr}(x,y)}{\operatorname{pr}(x')}.
\mathcal{H}= \left( a_{i}^{\dagger}\, , \, b_{j}\right) \left( \begin{array}{l l}{\mathcal{E}_{i l}}{\mathcal{F}_{j l}^{\dagger}}{\mathcal{F}_{i m}}{- \mathcal{E}_{j m}^{T}}\end{array}\right) \left( \begin{array}{l}{a_{l}}{b_{m}^{\dagger}}\end{array}\right) .
\Delta_a(P_{a,j}f) = j(j+a-1)(P_{a,j}f), \Delta_a(P_{a,\nu,\varepsilon}f) = -\left(\tfrac{1-a}{2}+i\nu\right)\left(\tfrac{1-a}{2}-i\nu\right)(P_{a,\nu,\varepsilon}f).
\frac{d}{dt}\Phi_t(x_1^0,x_2)=(-\Phi_t^1(x_1^0,x_2)J(t)+x_2\omega(\Phi_t^1(x_1^0,{x_2}),{0},t),0),\Phi_{t=0}(x_1^0,x_2)=(x_1^0,x_2).
h^{j}_{s}= \eta^t{\begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}}_{s}+ \eta^t_s \begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}, \ \ \ \ \ \ \ \h^{j}_{t}= \eta^t{\begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}}_{t}+ \eta^t_t \begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix},
a{\mathbin \mathrm{\vphantom{X}\hspace{1 . 1 p t}\hspace{- 0 . 5 p t}\_\hspace{- 0 . 3 e m}\_\v r u l e w i d t h 0 . 5 p t \hspace{1 . 7 p t}}}b = g ( a , b ) ,
\mathbf{W}_b = \begin{bmatrix}\mathbf{w}_{b_1}\mathbf{0}\cdots \mathbf{0}\mathbf{0}\mathbf{w}_{b_2}\cdots \mathbf{0}\vdots \vdots \ddots \vdots \mathbf{0}\mathbf{0}\cdots \mathbf{w}_{b_K}\end{bmatrix}\in \mathbb{C}^{KQ \times K},
\hat{I}( m , q ) = 2 \pi \sigma r_{s}^{2}\sqrt{V ( r_{s})}\beta_{H}- \frac{q^{2}}{r_{s}r_{+}}( r_{s}- r_{H}) \beta_{H}- \frac{( r_{s}^{3}- r_{H}^{3}) \beta_{H}}{\l^{2}}\quad .
M_{\kappa,t}(z) = (1-z) \frac{1}{2i\pi}\int_{\gamma_{\kappa}}\frac{K_{2t}(y)[(K_{2t}(y))^2-1]}{[t(K_{2t}(y))^2 + (2-t)][wK_{2t}(y) - \kappa]}\frac{dw}{R(z,w)}
\Theta = \frac{\kappa}{2 \pi}\int_{D}d^{2}x ~ \epsilon^{i j}\left< \partial_{i}g^{- 1}\partial_{j}g g^{- 1}d g \right> + \frac{\kappa}{2 \pi}\oint_{\partial{D}}d \varphi \left< g^{- 1}\partial_{\varphi}g g^{- 1}d g \right> ,
\frac{2^{p/2}\int_0^1 | w'(t) |^p (1-t)^{n+\frac{p}{2}-\frac{3}{2}}\,dt}{\int_0^1 | w(t) |^p (1-t)^{n-\frac{3}{2}}\,dt}= \frac{2^{-p/2}\int_0^1 | z'(t) |^p (1-t)^{2n-2}\,dt}{\int_0^1 |z(t)|^p (1-t)^{2n-2}\,dt}
R_{D\!F}=\left( \begin{array}{cccc}x=0 0 a+c+1 2a+2c+g+2 x=1 0 b+c+1 2b+2c+g+2 x=\infty \ -2c\ \ -a-b-2c-g-1\ \ -2a-2b-2c-g-2\end{array}\right).
\tilde{g}_{p}= \,_{\textbf{\textit{f}}(p)}^{-1}\circ \mathcal{G}\circ_{p}:B (0_{p},r)\rightarrow B (0_{\mathcal{F}(p)},\varrho/2) \quad\tilde{g}_{p}^{-1}=_{p}^{-1}\circ \mathcal{G}^{-1}\circ_{\mathcal{F}(p)}:B (0_{\mathcal{F}(p)},r)\rightarrow B (0_{p},\varrho/2),
y(t) := \int (\partial_xu)^2 \chi_n(x+\nu t) \; dx + \int_0^t\int (\partial_x^3u)^2 \chi_n'(x+\nu\tau) \; dxd\tau &\leq \int (\partial_xu_0)^2(x)\chi_n(x) \; dx + \left|\int_0^t A+B \; d\tau\right| &\leq c_0 + c_1 \int_0^t \int (\partial_xu)^2 \chi_n(x+\nu\tau) \; dxd\tau &\leq c_0 + c_1 \int_0^t y(\tau) \; d\tau.
{\cal L}_{c . t .}^{( 1 )}={\frac{\sqrt{- g}}{( 4 \pi )^{2}\epsilon}}\lbrack{\frac{1}{1 2 0}}R^{2}+{\frac{7}{2 0}}R_{\mu \nu}R^{\mu \nu}-{\frac{5 3}{1 8 0}}\epsilon^{\alpha \beta \gamma \delta}\epsilon_{\mu \nu \rho \sigma}{R^{\mu \nu}}_{\alpha \beta}{R^{\rho \sigma}}_{\gamma \delta}\rbrack
\frac{N_\ell^+}{D_\ell^-}+\tilde{\lambda}_{\ell+1}^{1,+}-C\alpha^\ell\leq \frac{N_\ell^++N_\ell^-}{D_\ell^-}\leq \frac{N_\ell^++N_\ell^-}{D_\ell^++D_\ell^-}=\lambda_\ell^1\,.
\int_{{\cal L}_{d - 1}^{d}}f ( H ) d \nu_{d - 1}( H ) = c_{3}\int_{{\cal L}_{2}^{A}}\int_{{\cal L}_{d - 1}^{L}}f ( H ) [ H , A ]^{2}d \nu_{d - 1}^{L}( H ) d \nu_{2}^{A}( L ) .
x\oplus y=[\frac{x_{1}y_{1}}{\sum_{i=1}^{D}x_{i}y_{i}},\frac{x_{2}y_{2}}{\sum_{i=1}^{D}x_{i}y_{i}},...,\frac{x_{D}y_{D}}{\sum_{i=1}^{D}x_{i}y_{i}}]=C[x_{1}y_{1},...,x_{D}y_{D}]\forall x,y\in S^{D}
\begin{array}{r@{\,\,\,=\,\,\,}l@{\qquad}l}\displaystyle\int_0^\infty \psi_{\ell j}(r) \psi_{\ell j'}(r)\,dr \displaystyle\delta_{jj'}\mbox{for the bound states and}begin{align*}3mm]\displaystyle\int_0^\infty \psi_\ell^*(k,r) \psi_\ell(k',r)\,dr \displaystyle\frac{\pi}{2}\delta(k-k') \mbox{for the continuum states.}\end{array}
a_{(0,-\alpha_2)}=\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}f(r_1e^{i\theta_1},r_2e^{i\theta_2})r_2^{\alpha_2}e^{i\alpha_2\theta_2}\,d\theta_1\,d\theta_2,
X^{(0)}(t) = \eta (t) \, \mathbb{I}_{[-r,0]}(t) + \eta(0) \, \mathbb{I}_{(0,T]}(t), X^{(n+1)}(t) = \begin{cases}\eta (t) (-r \le t \le 0), \displaystyle \eta (0) + \int_0^t A_0 (s, X_s^{(n)}) \, \mbox{d}s + \int_0^t A (s, X_s^{(n)}) \, \mbox{d}W(s) (0 < t \le T) \end{cases}
\tilde{M}\equiv \omega^{-\frac{1}{2}}M \omega^{-\frac{1}{2}}= \big [ \tilde{M}_{jk}\big ] ,\ \ \ \tilde{\tilde{M}}\equiv C_1^{*}\tilde{M}C_1 = \bigl [ \tilde{\tilde{M}}_{jk}\bigl ],
3 c (q_1^2 q_2 - 3 q_2^2 + q_1 q_3) = 4 (q_1^3 - 3 q_1 q_2 + 3 q_3) (3 q_1^2 q_2 + 7 q_2^2 - q_1 q_3 - 10 q_4), &c q_3^2 = 0, &(4 q_2^2 + q_1 q_3 + 20 q_4) ( 2 q_1 q_2^2 + q_1^2 q_3 + 3 q_2 q_3 + 10 q_1 q_4) = 0.
\sum_{j,k}\theta_{jk}^\alpha d_{j+1}\leq\sum_j d_{j+1}K_j [\theta_j/K_j]^\alpha=\sum_j [d_{j+1}\theta_j ]^\alpha [d_{j+1}K_j]^{1-\alpha}&\leq \left(\sum_j \theta_{j}\,d_{j+1}\right)^\alpha \left(\sum_{j}d_{j+1}K_j\right)^{1-\alpha}&\leq C(w+W)^\alpha \left(\sum_{j}[w(2^{j+1}-1)]^{1-d}\right)^{1-\alpha}&\leq C' (w^\alpha+W^\alpha) w^{(1-d)(1-\alpha)}
U^+_1 = \Big( U \setminus \{u_1\}\Big) \cup \{q_i^{-1}u_1, q_{s}u_1, q_s^{-1}u_1, q_i u_1 \}W^+_1 = W \cup \{u_1, q_i^{-1}q_{s}^{-1}u_1, q_i q_s u_1 \},
\tau_{q,p}\left( x\right) :=\lim_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}_{q,p}\left\{0\longleftrightarrow\left\lfloor nx\right\rfloor\ ,\ \left\vert \mathbf{C}_{\{0,\left\lfloor nx\right\rfloor \}}\right\vert<\infty\right\}
Q_n(t)=Q_n^{(0)}(t)+(Q_n(0)-n)^++A_n(t)-\int_0^t\int_0^t \mathbf{1}_{\{s+x\le t\}}\,d\,\sum_{i=1}^{{{\hat A}_n}(s)}\mathbf{1}_{\{\eta_i\le x\}}\,,
\Delta q =\Delta \frac{d}{dt}\int^t_0 k_{\alpha}(t-s)p(s,x)ds &= \frac{d}{dt}\Delta \int^t_0 k_{\alpha}(t-s)p(s,x)ds &=\frac{d}{dt}\int^t_0 k_{\alpha}(t-s)\Delta p(s,x)ds=D^{1-\alpha}\partial^{\alpha}_tp=\frac{\partial p}{\partial t}.
\lim_{n\to \infty}\frac{1}{n}\sum^{n-1}_{j=0}\varphi \circ f_{\theta^{-j}\omega}^{(j)}(X_{s^{\prime \prime}}(\theta^{-j}\omega)) = \int \varphi \circ h(\omega) dm = \int \varphi d\left[ \left( h(\omega) \right)_* m\right].
- \Delta u_{a,R_{n}}+ \frac{5}{3}u_{a,R_{n}}^{7/3}- \phi_{a,R_{n}}u_{a,R_{n}}= 0, &- \Delta \phi_{a,R_{n}}+ a^{2}\phi_{a,R_{n}}= 4 \pi \left( m_{R_{n}}- u_{a,R_{n}}^{2}\right).
\frac{\langle \varphi^2 \rangle_0}{M_5^3}\sim \frac{C_0^2 \lambda_0 H^2}{M_5^3}\left( \frac{p}{H}\right)^{\frac{3 \Delta+8}{\Delta+2}}\sim \left. \frac{\langle \varphi_4^2 \rangle}{M_{p,eff}^2}\right \vert_{t=t_p}.
\left( \begin{array}{c}{{L}}\\{{M - g x g^{- 1}{\bf 1}_{B}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\tilde{g}}}&{{0}}\\{{0}}&{{\rho \cdot g}}\\ \end{array}\right) \left( \begin{array}{c}{{L}}\\{{M - x{\bf 1}_{B}}}\\ \end{array}\right) g^{- 1}\cdot \rho^{- 1}
J(y_1, y_2) = \frac{1}{p}\| (y_1, y_2)\|_{1,p}+ \frac{1}{q}\| (\nabla y_1, \nabla y_2)\|_{q,\eta}- \frac{1}{1 - \nu}\int_{\Omega}\left[a_1\vert y_1\vert^{1 - \nu}+ a_2\vert y_2\vert^{1 - \nu}\right]dz - \lambda\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}dz.
= \left(\dfrac{q^{d-r}(1 - q^{-1})}{(1 - q^{-2})^{r}}\right)\displaystyle\sum_{d' > d/N}\#\mathbb{F}_{q}[x]_{d'}^{2}q^{-Nd'}& = \left(\dfrac{q^{d-r}(1 - q^{-1})^{2}}{(1 - q^{-2})^{r}}\right)\displaystyle\sum_{d' > d/N}q^{(1-N)d'}& \leq \left(\dfrac{q^{d-r}(1 - q^{-1})^{2}}{(1 - q^{-2})^{r}}\right)\dfrac{q^{(1-N)d/N}}{1 - q^{1-N}},
{\cal L}= \frac{1}{2}\left\{\partial_m \phi_i'\partial^m \phi_i'+ \imath \bar{\psi}_i \gamma^{m}\partial_m \psi_i -{\cal W}_i{\cal W}_{ikl}\phi_k' \phi_l' -{\cal W}_{ik}{\cal W}_{il}\phi_k' \phi_l'-{\cal W}_{ij}\bar{\psi}_i \psi_j \right\},
\mu \big( T^{(r+1)r_n}A \cap A^c \big) \leq \sum_{i=1}^{r+1}\mu \big( T^{ir_n}A \cap T^{(i-1)r_n}A^c \big) < \big( r+1 \big) \big( 1-\beta \big) \mu (A) = \Big( \big( r+1 \big) - \big( r+1 \big) \beta \Big) \mu(A) = \Big( 1 - \big( ( r+1) \beta - r \big) \Big) \mu(A) < \Big( 1 - \mu(A) \Big) \mu(A) .
\int_{{\cal M}_{\beta}}R^{\mu \nu \lambda \rho}R_{\mu \nu \lambda \rho}= \alpha \int_{{\cal M}_{\beta_{H}}}R^{\mu \nu \lambda \rho}R_{\mu \nu \lambda \rho}+ 8 \pi ( 1 - \alpha ) \int_{\Sigma}R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}+ O ( ( 1 - \alpha )^{2}) ,
\phi ( r , \theta ) \stackrel{r \gg 1}{\longrightarrow}\mathrm{e}^{- i p r \cos \theta}+ \mathrm{e}^{i \pi / 4}{\cal A}_{\mathrm{A B}}( |{\bf p}| , \theta ) \frac{\mathrm{e}^{i p r}}{\sqrt{r}}
\Upsilon:=\exp\bigg\{\int_t^T\int_t^T \alpha_{H_0}|u-v|^{2H_0-2}\big[ q(B^1_u, B^1_v)+2 q(B^1_u, B^2_v) +q(B^2_u, B^2_v)\big] dudv \bigg\}\,.
\frac{\partial f_k}{\partial[\tilde{w}_i]_{\alpha,j}}=-\frac{[\tilde{v}_i]_{k,\alpha}}{m|g|}\sum_{h=0}^{m-1}[\tilde{v}_{i-h-1}\tilde{w}_{i-h-1}]_{j,k}\frac{\lambda_j^{m-h-1}\lambda_k^h}{\lambda_j^m-\lambda_k^m}
\sum_{\substack{z \ \equiv \l_1 \ (\mathrm{mod}\ \lceil 2N^{\epsilon}\rceil) s\ \equiv \l_2 \ (\mathrm{mod}\lceil N^{\epsilon}\rceil)}}p(x-z,t-s)^2 \le \sum_{z\in Z}\sum_{s=0}^t p(z,s)^2,
(dd^c \max(\sum_{k=1}^j \psi_k , a_j (j+1) \psi ))^n \geq 1_{\Omega \cap \{\psi < -\frac{1}{j+1}\}}(dd^c (\sum_{k=1}^j \psi_k ))^n & \geq \sum_{k=1}^j 1_{\Omega \cap \{-\frac{1}{k}\leq \psi < -\frac{1}{k+1}\}}(dd^c \psi_k )^n \geq \sum_{k=1}^j \min (f_k, j) (dd^c \psi_k)^n \QB(\Omega).