harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{E}[u(V-\mathbb{E}V)] =\mathbb{E}\Big[u(F(e)-\mathbb{E}F(e))\Big] =\mathbb{E}\Big[u(F(e)-\mathbb{E}F(\tilde e))\Big] & =\mathbb{E}\Big[u(\mathbb{E}[F(e)-F(\tilde e)\mid e])\Big] \leq\mathbb{E}\Big[\mathbb{E}\Big[u(F(e)-F(\tilde e))\mid e\Big]\Big] &=\mathbb{E}\Big[u(F(e)-F(\tilde e))\Big].
-\frac{e^{i\frac{\pi}{2}{\rm sign}\left(\frac{H}{P}\right)}}{\sqrt{8K|H|}}{\rm sign}(P)e^{-\frac{i\pi}{2K}\phi}\sum_{j=0}^{\infty}\frac{1}{j!}\left.\left(\frac{\pi}{2iK}\frac{P}{H}\right)^{j}\left[\partial_{\epsilon}^{(2j)}\frac{\prod_{i=1}^{3}2i\sin\frac{\epsilon}{p_{i}}}{2i\sin\epsilon}\right]\right|_{\epsilon=0}.
\lambda_i^2+\lambda_j^2+\lambda_k^2-2\lambda_i\lambda_j-2\lambda_i\lambda_k-2\lambda_k\lambda_j+3\lambda_i \lambda_j\lambda_k \qquad{}= (\lambda_i-\lambda_j)^2+(\lambda_k-\lambda_j)^2+\lambda_j\lambda_k(\lambda_i-1)+\lambda_j (\lambda_k-\lambda_j)+2\lambda_i\lambda_k(\lambda_j-1) \geq 0
\frac{1}{\eta}\tilde{G}_{\mu \nu}(z)=\sum_{\beta \neq \alpha^{\prime}}\frac{\zeta_{\beta}(\mu)\zeta_\beta(\nu)}{(E-\lambda_{\beta})^2+\eta^2}+ \frac{\zeta_{\alpha^{\prime}}(\mu)\zeta_{\alpha^{\prime}}(\nu)}{(E-\lambda_{\alpha^{\prime}})^2+\eta^2}.
|0_M\rangle = \prod_\sigma\prod_p \left( 1+\ldots + \frac{1}{n!}\,\tanh^n\epsilon(p)\, b_p^{(\sigma)\dagger}\bar{b}_{\tilde p}^{(-\sigma)\dagger}+\ldots \right)\, \left[\prod_p\,\cosh^{-2}\epsilon(p)\right] |0(\epsilon)\rangle\,{,}
L_{A}=\cfrac{1}{\cfrac{1}{L_{A_0}}+\cfrac{1}{1+ \cfrac{1}{\cfrac{1}{L_{A_1}-1}+\cfrac{1}{L_{A_2}-1}}}}=\cfrac{1}{\cfrac{1}{\frac{5}{4}}+\cfrac{1}{1+ \cfrac{1}{\cfrac{1}{\frac{5}{2}-1}+\cfrac{1}{\frac{7}{2}-1}}}}=\frac{155}{204}.
\frac{1}{\tau}=\sum_{k^{\prime}}S_{k^{\prime}k}^{Op}=\frac{2\pi}{\hbar}Z_{DP}^{2}\frac{\hbar\omega}{2V\rho c^{2}}(N_{q}+\frac{1}{2}\pm\frac{1}{2})\sum_{k^{\prime}}\delta_{k^{\prime},k\pm q}\delta[E(k^{\prime})-E(k)\pm\hbar\omega]
< W_{1}W_{2}W_{3}> = \frac{< [ W_{1}, [ W_{2}, W_{3}] ] >}{( p_{1}- 2 m_{1}) ( p_{3}+ 2 m_{1}+ 2 m_{2})}+ \frac{< [ W_{2}, [ W_{1}, W_{3}] ] >}{( p_{2}- 2 m_{2}) ( p_{3}+ 2 m_{1}+ 2 m_{2})}.
S^N(K) =\frac{1}{N-1}\sum_{k=1}^{N-1}K \left( \frac{k}{N-1}\right) &= \frac{K(1)-K(0)}{2(N-1)}+ \frac{1}{N-1}\left(\frac{K(0)+K(1)}{2}+\sum_{k=1}^{N-2}K \left( \frac{k}{N-1}\right) \right)\;.
C^k_{\rm diff}(G):=\left\{\begin{array}{l}c:G^{\times(k+1)}\to\mathbb{C}~{\rm smooth}, ~c(gg_0,\ldots,gg_k)=c(g_0,\ldots,g_k),\ \forall g,g_0,\ldots,g_k\in G\end{array}\right\}.
\frac{1}{\eta}(x^k-x^{k+1}) + A^T(\lambda^k-\lambda^{k+1}) \in \partial f(x^{k+1})- A^T\lambda^{k+1}\frac{1}{\eta}(\lambda^k-\lambda^{k+1})+A(x^k-x^{k+1}) \in \partial g^*(\lambda^{k+1})+ Ax^{k+1}
\sum_{i\neq j}|x_i-x_j|^{-s}+\frac{1}{2}\sum_{v\in C_L\setminus\{0\}}(|x_i-x_j+v|^{-s}-|v|^{-s}) = -(s/d)(d-s-2)D_L\sum_{i\neq j}|x_i-x_j|^2+o(D_L).
F_1 (x, y,\eta) ) = y_1-x_1, F_i (x,y,\eta) = y_i-x_i+ p_i (x,y,\eta) (i = 2,\ldots,n), F_{n+k}(x,y,\eta) = \eta_k+ q_k (x,y,\eta), (k = 1,\ldots,p),
\|\tilde{D}-\tilde{D}^{\prime}\| \leq (\kappa(ST)\kappa(S^{\prime}T^{\prime}))^{1/2}\|i\sigma M-i\sigma M^{\prime}\| &= (\kappa(ST)\kappa(S^{\prime}T^{\prime}))^{1/2}\|M-M^{\prime}\|
\widetilde{T}(f)(z) = -\iint_D \left( \frac{f(\zeta)}{\zeta - z}+ \frac{z\overline{f(\zeta)}}{1-z\overline{\zeta}}\right) d\xi \,d\eta &= -\iint_D \left( \frac{f(\zeta)}{\zeta - z}- \overline{\left(\frac{f(\zeta)}{\zeta-z}\right)}\right) d\xi \,d\eta ,
M_\mu(\alpha,\varepsilon,\delta,T,f) =\lim_{n\to\infty}M_\mu(n,\alpha,\varepsilon,\delta,T,f), \underline{M}_\mu(\alpha,\varepsilon,\delta,T,f) =\liminf_{n\to\infty}R_\mu(n,\alpha,\varepsilon,\delta,T,f), \overline{M}_\mu(\alpha,\varepsilon,\delta,T,f) =\limsup_{n\to\infty}R_\mu(n,\alpha,\varepsilon,\delta,T,f).
\left(1-c+(a^{p^m}-a)^p\left(1-\dfrac{1-c}{(1-c)^{p^m}}\right)\right)^p\beta^p + \left((a-a^{p^m})\left(1-\dfrac{(1-c)}{(1-c)^{p^m}}\right)\right)^{p^m}\left(\dfrac{-(1-c)}{(1-c)^{p^m}}\right)\beta=0.
K(L,r) =\int_{\frac{1-r^2}{1+r^2}}^{1}s^{2L-n-1}\int_0^{\alpha(s,r)}(2\cos\theta-2\cos\alpha(s,r))^{n-2}d\theta\; ds+\cdots =\int_{\frac{1-r^2}{1+r^2}}^{1}\int_0^{\alpha(s,r)}s^{2L-n-1}\sum_{j=0}^{n-2}\binom{n-2}{j}(-1)^j (2\cos\alpha(s,r))^j (2\cos\theta)^{n-2-j}d\theta\; ds+\cdots
\mathbb{E}\left[\exp(Y_{j_1})P_{(p,M)}\left[\forall j\leq j_1: Z_j\geq\frac{\exp(Y_j)}{j}\right] ,\forall i\leq j_1: U_i\geq\kappa, B(m,k)\right] = c_5 \mathbb{P}[B(m,k)]
-\mathcal{L}_v w = -\partial_x \left( a_{11}(v)\ \partial_x w + a_{13}(v)\ \partial_\eta w \right)-\partial_y \left( a_{22}(v)\ \partial_y w + a_{23}(v)\ \partial_\eta w \right) & -\partial_\eta \left( a_{31}(v)\ \partial_x w + a_{32}(v)\ \partial_y w+ a_{33}(v)\ \partial_\eta w \right) & - b_1(v)\partial_x w -b_2(v)\partial_y w -b_3(v)\partial_\eta w\ .
\|\Phi_{\#}^{\infty}\|^2_{\#}- \|\Phi_{\#}^y\|^2_{\#}= - 4 \alpha \int_{\mathbb{R}^3}\mathrm{d}k \; \left\| \frac{P u_{\alpha}}{(h_{\alpha}-e_{\alpha}+|k|)^{1/2}}\right\|_{L^2}^2 f_y(k) +O(\alpha^3 L^2 e^{-L}).
R_n(\bar z)^* H_{1,n}^{-1}\left\{-a u_{1,n}v_n^* - v_n u_{4,n}\right\}\Sigma_{4,n}= R_n(\bar z)^* (1-aT_n^*) \Sigma_{4,n}= (I-aT_n^*) R_n(\bar z)^* \Sigma_{4,n}.
g_r({\boldsymbol{X}}_{n_1,\ldots,n_{k-1}}):=s_{(1^{n_k}, r)}^{(q_1,\ldots,q_{k-1})}({\boldsymbol{X}}_{n_1,\ldots,n_{k-1}}) -q_{k-1}e_{r-n_{k-1}}^{(q_1,\ldots,q_{k-2})}({\boldsymbol{X}}_{n_1,\ldots,n_{k-2}}),
J_{\nu}^{(3)}(z;q):= \frac{(q^{v+1};q)_\infty}{(q;q)_\infty}\sum_{n=0}^{\infty}\dfrac{(-1)^n q^{\frac{n(n+1)}{2}}}{(q,q^{v+1};q)_n}(z)^{2n+\nu},z\in \mathbb C.
\| \bot\| = \bot & \| (\varphi \wedge \psi)(\overline{a})\| = \| \varphi(\overline{a})\| \wedge \| \psi(\overline{a})\| & \| (\varphi \vee \psi)(\overline{a})\| = \| \varphi(\overline{a})\| \vee \| \psi(\overline{a})\|
v_{3}(t,r) = -\frac{1}{2\pi}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\int_{0}^{2\pi}d\theta \frac{F_{0,1}(s,\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)})(r+\rho \cos(\theta))}{\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)}}
X(\zeta)=\Re\left( \imath\big(\zeta+\frac1{\zeta}\big) \,,\, \zeta-\frac1{\zeta}\,,\,\frac{\imath}2 \big(\zeta^2-\frac1{\zeta^2}\big) \,,\, \frac12 \big(\zeta^2+\frac1{\zeta^2}\big) \right)
{1 \over -(\gamma\cdot D)^2 +M^2}={1 \over -{\bf D}^2 +\Phi^2 +M^2}+{1 \over -{\bf D}^2 +\Phi^2 +M^2}\left({i\over 2}\gamma_\mu \gamma_\nu G_{\mu\nu}\right){1 \over -{\bf D}^2 +\Phi^2 +M^2}+ \cdots
\chi_{B_{3R}^{d_{\mathrm{CC}}}(0)}(F^{(\iota)}\psi)(\sqrt L) f = \chi_{B_{3R}^{d_{\mathrm{CC}}}(0)}\bigg(\sum_{\ell =-1}^\iota + \sum_{\ell = \iota + 1}^\infty \bigg) F_\ell^{(\iota)}(L,U) f & =: g_{\le \iota}+ g_{> \iota}.
\mathcal{M}(\varrho_{\varepsilon}^{(u_{\varepsilon}+E(f_{\varepsilon}),u_{\varepsilon}+E(f_{\varepsilon}))},\varepsilon \in (0,\bar{\varepsilon})) =\mathcal{M}(\varrho_{\varepsilon}^{(u_{\varepsilon},u_{\varepsilon})})+\mathcal{M}(\varrho_{\varepsilon}^{(E(f_{\varepsilon}),E(f_{\varepsilon}))})+\{0\}&=\{\delta_{u}^{S^{1}}+\delta_{f_{0}}\}.
\begin{aligned}I\!I_1 \le (\|\varrho\|_{L^3}\|{\rm div}u\|_{L^6}+\|u\|_{L^3}\|\nabla \varrho \|_{L^6})\|\nabla^2 \varrho\|_{L^2}&\lesssim (\|\varrho\|_{H^1}+\|u\|_{H^1})(\|\nabla^2 \varrho\|_{L^2}^2+\|\nabla^2 u\|_{L^2}^2) &\lesssim \delta (\|\nabla^2 \varrho\|_{L^2}^2+\|\nabla^2 u\|_{L^2}^2).\end{aligned}
2 \mathrm{Re}\langle \kappa\Phi_{\#}^y\,|\, H_y R_y^{\#}\rangle = (e_{\alpha}+ \alpha \|\lambda_y\|^2)\, 2 \mathrm{Re}\langle \kappa \Phi_{\#}^y\,|\, R^{\#}_y\rangle &+ 2 \alpha\mathrm{Re}\langle A^-_y \kappa \Phi_{\#}^y\,|\, A^-_y R^{\#}_y\rangle + 2 \alpha \mathrm{Re}\langle \kappa \Phi_{\#}^y\,|\,V_y R^{\#}_y \rangle,
\left.\frac{\partial F(t_{\mathtt{D}_{i}},\beta_{\mathtt{D}_{i}},\psi_{\mathtt{D}_{i}}^{(n)})}{\partial\beta_{\mathtt{D}_{i}}}\right|_{\psi_{\mathtt{D}_{i}}^{(n)}=t_{\mathtt{D}_{i}}/\beta_{\mathtt{D}_{i}}}=\frac{\partial f(t_{\mathtt{D}_{i}},\beta_{\mathtt{D}_{i}})}{\partial\beta_{\mathtt{D}_{i}}}.
\mathfrak{a}_{(n)}\circ r_n(x) =(f_{r_n(x)}^{qn},\ldots,f_{r_n(x)}^{qn})\left(\mathfrak{a}_{(0)}\circ r\circ L^{-n}_0(x)\right) & = (f_{r_n(x)}^{qn},\ldots,f_{r_n(x)}^{qn})\circ \phi_{r_n(x)}\left(L^{-n}_0(x)\right) & = \phi_{r_n(x)}\left(L^n_{r_n(x)}\circ L^{-n}_0(x)\right)
\mathfrak{CM}\bigl(\frac{q}{\tau}\,|\,\frac{1}{\tau},\tau\lambda_1,\tau\lambda_2\bigr) \,\Bigl(\frac{\Gamma(1-\tau)}{\pi\,\Gamma(\tau)}\Bigr)^{\frac{q}{\tau}}\Gamma(1-\frac{q}{\tau}) = \mathfrak{CM}(q\,|\,\tau,\lambda_1,\lambda_2) \Bigl(\frac{\Gamma(1-\frac{1}{\tau})}{\pi\,\Gamma(1/\tau)}\Bigr)^{q}\Gamma(1-q).
S_{\mathrm{G F}}= \int d^{4}x \{B_{\mu}D_{\nu}B^{\nu \mu}+ \varepsilon_{a b}B^{a}D_{\mu}C^{\mu b}- \frac{1}{2}B_{\mu}B^{\mu}- \frac{1}{2}\varepsilon_{a b}B^{a}B^{b}\},
\begin{cases}\|\sum_{l\le j\epsilon}T_a^{j, l}\|_{L^r\to L^s}\lesssim_{\epsilon}2^{jm}2^{-jn(\frac{1}{s}-\frac{1}{r})}2^{j(n(1-\rho)/2 + \varepsilon)(\frac{2}{s}-1)}\|T_a^{j, l}\|_{L^r\to L^s}\lesssim_{\epsilon}2^{-jn(\frac{1}{s}-\frac{1}{r})}2^{10n(m-n)(j+l)}, l>j\epsilon.\end{cases}
P_m(x) =p_{m0}\,f_0(x)+p_{m1}\,f_1(x)+\cdots+p_{mm}\,f_m(x), begin{align*}4pt]Q_n(x) =\,q_{n0}\,g_0(x)\,+\,q_{n1}\,g_1(x)\,+\,\cdots\,+\,q_{nn}\,g_n(x),
R_{1212}=\frac{\left(3{{e}^{3 q_3+2 q_2}}+2{{e}^{2 q_3+ 3 q_2}}-2{{e}^{q_3+4 q_2}}-2{{e}^{5 q_2}}\right){{e}^{2 q_1}}+\left(-{{e}^{2 q_3+2q_2}}+{{e}^{q_3+3q_2}}+{{e}^{4 q_2}}\right){{e}^{3 q_1}}}{(e^{q_1}+e^{q_2})\Delta_1},
2\,g((\nabla_{X}{f})Y,Z) = N^{\,(5)}(X,Y,Z), 0 = N^{\,(5)}(X,Y,Z) + N^{\,(5)}(Y,Z,X) + N^{\,(5)}(Z, X, Y) , 0 = N^{\,(5)}({f}X,Y,Z) + N^{\,(5)}({f}Y,Z,X) + N^{\,(5)}({f}Z, X, Y) .
\tau \biggl( \prod_{i = 1}^{N}D_{\zeta_i}(\theta_i)^{-1}\biggr) = \prod_{i = 1}^{N}D_{\zeta_i}(\theta_i)^{-1}\frac{\theta_i^{m(e^\mathsf{T}A)_i}}{(\theta_i; \zeta_i)_{e_i}^m}
\sum_{k=1}^{n}\frac{k}{(k+1)!}=\sum_{k=1}^{n}(\frac{k+1}{(k+1)!}-\frac{1}{(k+1)!})=\sum_{k=1}^{n}(\frac{1}{k!}-\frac{1}{(k+1)!})=1-\frac{1}{(n+1)!}
\left\|W Z^{\top}\right\| \leq 4 \sigma_{w,i}\sqrt{N_{i}(2 n_x+n_u) \log \frac{9|\widehat{\mathcal{C}}_1| T}{\delta}}\sum_{t=0}^{T-1}\left\|(\Sigma_{t}^{(i)})^{\frac{1}{2}}\right\| .
- \bigg(\sum_{k=1}^mR^h(\tau(\varphi),d\varphi(e_k))d\varphi(e_k)\bigg)^{\top}&+\sum_{j,k=1}^mh(\tau(\varphi),B_{\varphi}(e_j,e_k))\,B_{\varphi}(e_j,e_k) &-\sum_{k=1}^mR^h(\tau(\varphi),d\varphi(e_k))\,d\varphi(e_k)=0,
\sigma_n = \frac{r_n^{\gamma-2}}{M_n^{\gamma-1}}, g_n(y,s) = \frac{r_n^{\gamma}}{M_n^{\gamma}}f_n\left(\bar x_n + r_n y, \bar t_n + \frac{r_n^\gamma}{M_n^{\gamma-1}}s\right)
{w(T,{X}_T)}-w(s,{X}_s)-\int^T_s \nabla w(r,{X}_r)\d{W}_r &= -w(s,{X}_s)-\int^T_s \nabla w(r,{X}_r)\d{W}_r &= \int^T_s \nabla u(r,{X}_r)b(r,{X}_r)\d r &= \int^T_s{Z_r b(r,{X}_r)}\d r ,
\|\mathcal G\|^\frac{1}{p-1}_{q_0,B_2(0)}=R^{\frac{N(q_0-1)}{q_0(p-1)}}\|f\|^\frac{1}{p-1}_{q_0,B_{2R}(0)}\leq M R^{\frac{N(q_0-1)}{q_0(p-1)}},
A_{k+1}f(x_{k+1}) + D_h(x, z_{k+1}) = (A_{k+1}- \alpha_k) \Big(f(x_k) + A_k^{-1}D_h(x, z_k)\Big) + \alpha_k f_k(x) &= A_k \Big(f(x_k) + A_k^{-1}D_h(x, z_k) \Big)+ (A_{k+1}- A_k) f_k(x) & \leq A_k f(x_k) + D_h(x, z_k)+ (A_{k+1}- A_k) f(x).
32 \alpha (2 + \beta) (-\alpha^2 + 2 \alpha (1 + \beta) + (1 + \beta) (\beta + 4 s)) &+ (-\alpha^2 + 2 \alpha (5 + \beta) + (5 + \beta) (4 + \beta + 4 s)) (32 \alpha (1 + \beta) + (-\alpha^2 + 2 \alpha (1 + \beta) + (1 + \beta) (\beta + 4 s)) &\times (-\alpha^2 + 2 \alpha (3 + \beta) + (3 + \beta) (2 + \beta + 4 s)))=0
\int{\frac{[ d \delta \phi ]}{\mathrm{V o l}( \mathrm{W e y l})}}\int_{0}^{\infty}d l ~ J_{b}( l ;{\hat{g}}) e^{-{\frac{1 0 - d}{4 8 \pi}}S_{S L}[ \phi , \zeta , g ]}( \mathrm{d e t}^{\prime}\Delta )_{{\hat{g}}}^{- 5}\quad ,
T_{4,4}= \sum_{j\in\{\pm\}}B\int e^{i\Theta(\eta,\nu)}m_2(\eta,\nu)\theta_j(\nu/\eta)d\nu &+ B\int e^{i\Theta(\eta,\nu)}m_2(\eta,\nu)(1-\theta_+-\theta_-)(\nu/\eta)d\nu.
Z_{\scriptscriptstyle S D}= \frac{1}{N_{\scriptscriptstyle S D}}\int_{\Omega^{\scriptscriptstyle 1}( M )}\mathcal{D}f \hspace{4 p t}e^{- i \hspace{3 p t}\bigl \langle f , \hspace{4 p t}( \mathrm{1 \m k e r n - 5 m u I \!}\hspace{2 p t}+ \ast d_{\scriptscriptstyle 1}) f \bigr \rangle}
\left(I_{{q'}^{2}}+\mathcal{L}\left(W',Z'\right)\right)\cdot\overline{\left(I_{{q'}^{2}}+\mathcal{L}\left(W',Z'\right)\right)}^{t}=\overline{\left(I_{{q'}^{2}}+\mathcal{L}\left(W',Z'\right)\right)}^{t}\cdot \left(I_{{q'}^{2}}+\mathcal{L}\left(W',Z'\right)\right).
\frac{1}{n}=1-\frac{\Psi}{2}-\frac{\Psi^3-3\Psi^2}{2(2+\Psi-\Psi^2+2\sqrt{1+\Psi})}\cr =1-\int_{\mathbb{R}^3}\int_0^\infty f\sqrt{F_E^0}\phi(\mathcal{I}) \,d\mathcal{I}\,dp-\frac{\Psi_1}{2}-\frac{\Psi^3-3\Psi^2}{2(2+\Psi-\Psi^2+2\sqrt{1+\Psi})}.
[\sigma]_{C^{\alpha}(B^g(p,R))}= \sup \left\{\frac{|\sigma(x) - \mathbf{P}^g_{x'x}(\sigma(x'))|_{h(x)}}{d^g(x,x')^\alpha}: x, x' \in B^g(p,R), \;x\neq x',\;\mathbf{P}^g_{xx'}\;\right\}
\frac{1}{k+1}\big((a+\frac{d_i}{2})^{k+1}-(a-\frac{d_i}{2})^{k+1}\big)=\sum_{t=0}^{k}\frac{c_t}{t+1}\big((a-d_j)^{t+1}-(a+d_j)^{t+1}\big).
{\bf u}= ({\bf A}_1^\frac{1}{2})^{-}{\bf x}- ({\bf A}_1^\frac{1}{2})'({\bf A}_1+{\bf A}_2)^{-}({\bf x}+{\bf y}){\bf v}= ({\bf A}_2^\frac{1}{2})^{-}{\bf y}- ({\bf A}_2^\frac{1}{2})'({\bf A}_1+{\bf A}_2)^{-}({\bf x}+{\bf y}).
\begin{array}{rl}\delta\dfrac{dx_\alpha}{dt}-\dfrac{d}{dt}\delta\,x_\alpha=\displaystyle\sum_{k=1}^{s_2}E_k(L_\alpha )\delta y_k,\alpha=1,\ldots,s_1, &\delta\dfrac{dy_m}{dt}-\dfrac{d}{dt}\delta\,y_m=0,m=1,\ldots,s_2.\end{array}
\mathcal{L}_{A_f+X_f}(\mathbb{J}+J), = \mathcal{L}_{A_f}(\mathbb{J})+\mathcal{L}_{A_f}(J)+\mathcal{L}_{X_f}(\mathbb{J})+\mathcal{L}_{X_f}(J), = \mathcal{L}_{A_f}(J)+\mathcal{L}_{X_f}(J).
E_{mn}(x,y,z)=E_{0}\frac{w_{0}}{w}H_{m}(\frac{\sqrt{2}x}{w})H_{n}(\frac{\sqrt{2}y}{w})exp[-(x^{2}+y^{2})(\frac{1}{w^{2}}+\frac{jk}{2R})-jkz-j(m+n+1)\zeta]
h=h_{SU(N)}(g) =e^{iK_1\phi_1}\sin^{R_1}(\psi_1)\cos^{S_1}(\psi_1)\cdots e^{iK_{N-1}\phi_{N-1}}\sin^{R_{N-1}}(\psi_{N-1})\cos^{S_{N-1}}(\psi_{N-1}) &\quad\cdot h_{SU(N-1)}(g_{SU(N-1)}) e^{iL_{N-1}\omega_{N-1}}.
\sum_{\alpha\in \Phi^-(W_I)}T_{w\alpha}- \sum_{\alpha\in \Phi^-(W_J)}T_\alpha =\!\! \sum_{\alpha\in \Phi^-(W_I)\setminus \Phi^-(W_I\cap w^{-1}W_J w)}\!\!\!\!T_{w\alpha}- \!\!\sum_{\alpha\in \Phi^-(W_J)\setminus \Phi^-(W_J\cap w W_I w^{-1})}\!\!\!\!T_\alpha.
R_{ijkl}h_{il}h_{jk}=2\Big[(h_{11}h_{22}-h^2_{12})+(h_{11}h_{33}-h^2_{13})+(h_{22}h_{33}-h^2_{23})\Big], R_{jl}h_{lk}h_{jk}=2(h_{11}^2+h_{22}^2+h_{33}^2+2h_{12}^2+2h_{13}^2+2h_{23}^2).
\langle w, v \rangle = \langle w_1, v_1 \rangle + \langle w_2 , v_2 \rangle \langle (w_1, 0), v \rangle = \langle w_1, v_1 \rangle \ge 0 , \langle (0, w_2), v \rangle = \langle w_2 , v_2 \rangle \ge 0 .
(q x_1 - y_1) z_1 = (q-1)(q^d-1) \big( q^{d-1}(q+1)y_1^{-1}- \tau \big), (q x_1 - y_1) z_d = q^{d-1}(q-1)(q^d-1) \big( (q+1)x_1 - \tau \big).
\inf_{Q}\int_{\Theta}\chi^2(P_{\theta}\| Q) d w(\theta) \leq \int_{\Theta}\chi^2(P_{\theta}\| P_{\theta^*}) d w(\theta) &= \int_{U(\theta^*)}\chi^2(P_{\theta}\| P_{\theta^*}) d w(\theta) &\leq \sup_{\theta \in U(\theta^*)}\chi^2(P_{\theta}\| P_{\theta^*})\leq \exp(\rho^2 t^2_{\theta^*}) - 1
M ( a , b , z ) = 1 + \frac{a z}{b}+ \frac{a ( a + 1 ) z^{2}}{b ( b + 1 ) 2 !}+ . . . + \frac{a ( a + 1 ) . . . ( a + n - 1 ) z^{n}}{b ( b + 1 ) . . . ( b + n - 1 ) n !}+ . . . ,
P( X^{r, \mathbf{w}}_n =g ) = E (P( X^{r, \mathbf{w}}_n =g \, |\,X^{r, \mathbf{w}}_i, \, i<n )) = \sum_{k \in \mathcal{A}}\theta_k(\lambda P_{(k)}) (g)= (\lambda \hat P) (g) = \lambda (g) .
\frac{\partial G^{z}_{\mathfrak{uv}}}{\partial w_{aB}}=-G^{z}_{\mathfrak{u}a}G^{z}_{B\mathfrak{v}}, \frac{\partial G^{z}_{\mathfrak{uv}}}{\partial \overline{w_{aB}}}=-G^{z}_{\mathfrak{u}B}G^{z}_{a\mathfrak{v}},
\begin{aligned}\bar{x}^{k+1}=\bar{x}^k+\gamma_1^k \bar{\zeta}_w^k-\lambda^k \frac{(u\otimes{I_d})^T}{m}y^k \bar{y}^{k+1}=(1-\alpha^k)\bar{y}^k +\gamma_2^k \bar{\xi}_w^k+\bar{g}^{k+1}-(1-\alpha^k)\bar{g}^k\end{aligned}
J_p(\Gamma(y)-\Gamma(x))-J_p(1-\Gamma(x)) \ge C\,\frac{(\Gamma(y)-1)}{\Big((\Gamma(y)-\Gamma(x))^2+(1-\Gamma(x))^2\Big)^\frac{2-p}{2}}&\ge \frac{C}{2^\frac{2-p}{2}}\,\frac{(\Gamma(y)-1)}{(\Gamma(y)-\Gamma(x))^{2-p}}\ge C\,\Gamma(y)^{p-1}\, \left(1-\frac{1}{\Gamma(y)}\right).
\begin{array}{lll}\{\theta^0,\theta^0\}=-2\theta^1\theta^2, \{\theta^0,\theta^1\}=-2\theta^0\theta^1, \{\theta^0,\theta^2\}=2\theta^0\theta^2, \{\theta^1,\theta^1\}=0, \{\theta^1,\theta^2\}=0, \{\theta^2,\theta^2\}=0.\end{array}
H_D = \int{d^3\sigma \Big[ -A_\tau (\tau , \vec{\sigma})\Gamma^\ast (\tau , \vec{\sigma}) + \lambda^\mu (\tau , \vec{\sigma}){\cal H}^\ast_\mu (\tau , \vec{\sigma}) +\mu_\tau (\tau , \vec{\sigma}) \pi^\tau (\tau , \vec{\sigma}) \Big]}
\int_0^vv'\hat{I}_{n,0}(v')dv' =\frac{v^2}{2}\left(\hat{I}_{n,0}-\hat{I}_{n,2}\right), \int_0^vv'\hat{I}_{n,1}(v')dv' =v^2\left(\hat{I}_{n,1}-\hat{I}_{n,2}\right),
K(n_i, y; n_j, z) = - f_{n_j - n_i}(z - y) 1_{\{i < j\}}& + \sum_{k, l = 1}^d \int_{y}^{\infty}f_{n_m - n_i}(u-y)u^{k-1}du (A^{-1})_{lk}f_{n_j-d+l}(z - x_l)
\frac{n(n-1)(n-2)}{j}\left[y_j(3)-\frac{1}{j+1}\sum_{i=1}^{j+1}h_{j+1}(i)\right] =(j-1)(j-2)-j^2-(1-2n)j-(n-2)b_n-2-u_n &=(n-2)(2j-b_n)-u_n &<(n-2)(2j-b_n)-(b_n-2)(n-2)\;\mbox{(by (\ref{ae6}))}&=2(n-2)(j+1-b_n)\le0,
I_{n+\nu,n}(z) = \frac{n! (n+\nu)!}{\sqrt{\pi n}}\, \left( \frac{2}{z}\right)^{2n+\nu+1}\left( b(z) + \frac{b_\nu (z)}{n}+{\mathcal{O}}\left( \frac{1}{n^2}\right)\right)
I ( g ,{\bf A}) = I_{W Z N W}+ \frac{k}{2 \pi}\int d^{2}z t r ({\bf A}\overline{{{\partial}}}g g^{- 1}- \overline{{{{\bf A}}}}g^{- 1}\partial g +{\bf A}\overline{{{{\bf A}}}}- g^{- 1}{\bf A}g \overline{{{{\bf A}}}})
\beta_{\lambda}= \frac{1 2 N}{( t r m^{2})^{2}}( t r m_{\perp}^{4}( e \beta_{e}- 2 e^{2}\sum_{i}\frac{m_{i}\beta_{m_{i}}}{t r m^{2}}) + 2 e^{2}\sum_{i}m_{i}\beta_{m_{i}}( m_{i}^{2}- \frac{t r m^{2}}{N}) )
L = c_{0}s + c \sqrt{{\dot{\bf e}}_{3}^{2}- k_{2}^{2}}+{\bf p}({\dot{\bf x}}- s{\bf e}_{1}) + \sum_{i}{\bf p}_{i - 1}({\dot{\bf e}}_{i - 1}- k_{i - 1}{\bf e}_{i}+ k_{i - 2}{\bf e}_{i - 2}) - \sum_{i , j}d_{i j}({\bf e}_{i}{\bf e}_{j}- \delta_{i j}) ,
\partial_z \phi_1|_{z=-h_b}= - \mathcal{F}^{-1}\Big{(}\frac{\cosh(\frac{z}{h_b(X)}\sqrt{\mu}|\xi|)}{\cosh(\sqrt{\mu}|\xi|)}\hat{G}(\xi)\Big{)}(X)|_{z=-h_b(X)}= -G(X).
c_{2i+2}& =\frac{1}{2}\frac{\left( N\kappa-\kappa+m+1+3i\right)}{N\kappa+m+2+i}c_{2i}-\frac{1}{4}\frac{\left( 2i-1\right) i}{\left( N\kappa+m+2+i\right) \left( N\kappa+m+1+i\right)}c_{2i-2}.
|i,k_{1}...k_{n}\rangle=C_{0}|f,0\rangle+\sum_{m=1}^{\infty}\int d^{4}p_{1}...d^{4}p_{m}|f,p_{1}...p_{m}\rangle\langle f,p_{1}...p_{m}|i,k_{1}...k_{n}\rangle.
\mathbf{A}=\left[ \renewcommand{\arraystretch}{0.85}\begin{array}{cccc}\mathbf{C}_{0,0}\mathbf{C}_{0,1}\ldots \mathbf{C}_{0,t-1}\mathbf{C}_{1,0}\mathbf{C}_{1,1}\ldots \mathbf{C}_{1,t-1}\vdots \vdots \vdots \vdots \mathbf{C}_{t-1,0}\mathbf{C}_{t-1,1}\ldots \mathbf{C}_{t-1,t-1}\end{array}\right] ,
\overline{{{\mathrm{\Large ~ v ~}}}}_{D}( x ) \equiv{\frac{2 ( 4 \pi )^{D / 2}}{{\mathrm{t r}}{\bf 1}\, \Lambda^{D}}}{\mathrm{\Large ~ v ~}}_{D}( m ) ={\frac{( 4 \pi )^{D / 2}}{{\mathrm{t r}}{\bf 1}\, g^{2}\Lambda^{D - 2}}}\left( x -{\frac{\epsilon}{\Lambda^{2}}}\right) + \int_{1}^{\infty}\! \! d \tau \, \tau^{-{\frac{D}{2}}- 1}\left\{{\mathrm e}^{- \tau x}G_{D}( \tau{\cal F}) - 1 \right\}\ .
\begin{cases}\langle(U'(t), -U(t)), (\Phi(t), \Phi'(t))\rangle= \langle(U_1, -U_0), (\Phi_0, \Phi_1)\rangle & \displaystyle+\int_0^t \int_{\Gamma_1}(DH(\tau), \Phi(\tau) ) d\Gamma d\tau, \forall t\geqslant 0\end{cases}
H=\int d^{3}x\frac{N}{\sqrt{det(q)}}(\tilde{\pi}^{2}+\tilde{E}_{i}^{a}\tilde{E}^{bi}\partial_{a}\varphi\partial_{b}\varphi+det(q)V(\varphi))+N^{a}\tilde{\pi}\partial_{a}\varphi
G^{E}( 2 \pi n T , k ) = - \pi \, C_{\cal{O}}\left[ \psi \left({\frac{1 + n}{2}}-{\frac{i k}{4 \pi T}}\right) + \psi \left({\frac{1 + n}{2}}+{\frac{i k}{4 \pi T}}\right) \right] \, .
\frac{1}{2}m(\frac{dr}{d\tau})^{2}=[\frac{E^{2}}{2mc^{2}}-\frac{1}{2}mc^{2}]+\frac{GMm}{r}-\frac{L^{2}}{2\mu r^{2}}+\frac{G(M+m)L^{2}}{c^{2}\mu r^{3}},
g = \frac{1}{\sqrt{f^{1}- f^{2}}}\left( \begin{array}{c c}{\operatorname{e x p}( \alpha ) f^{1}}{\operatorname{e x p}( - \alpha ) f^{2}}{\operatorname{e x p}( \alpha )}{\operatorname{e x p}( - \alpha )}\end{array}\right) .
\sum_{n=0}^{\infty}\frac{(-1)^{n}}{3^{n}(2n+1)}=1-\frac{1}{3^{1}\cdot3}+\frac{1}{3^{2}\cdot5}-\frac{1}{3^{3}\cdot7}+\frac{1}{3^{4}\cdot9}-\cdot\cdot\cdot=\sqrt{3}arctan\frac{1}{\sqrt{3}}=\frac{\pi}{2\sqrt{3}}
\mathbf{P}_{N, \beta}\left( \overline{\mathbf{P}}_{N}(X_{N}) \in B( \overline{\mathbf{P}}, \delta) \right) = \mathbf{\Pi}_{N}\left( \overline{\mathbf{F}}_{N}(C) \in B( \overline{\mathbf{P}}, \delta) \Big| |C| = N \right).
B_{2i-1}^2B_{2i}v\ =\ ([2]B_{2i-1}B_{2i}B_{2i-1}-B_{2i}B_{2i-1}^2+B_{2i})v\cr =\ [2][\mu_i]B_{2i-1}B_{2i}v-([\mu_i]^2-1)B_{2i}v.
g_1({\bf u}) = \varphi_1(x) \left( \eta_1 \dfrac{u_1}{1 + u_1}+ \, \eta_2 \dfrac{u_6^{p_3}}{1 + u_6^{p_3}}\right), g_3({\bf u}) = \varphi_2(x) \left( \eta_3\dfrac{u_2}{1 + u_2}+ \, \eta_4 \dfrac{u_5^{p_1}}{(\lambda_{9}+ u_4)(1 + u_5^{p_1})}\right), g_4({\bf u}) =- \varphi_2(x)\, \eta_5 \dfrac{u_4^{p_2}}{1 + u_4^{p_2}}.
E_{D}(L_{i},a)= \frac{1}{2}\left(\prod_{i=1}^{D-2}L_i\right)\frac{1}{(2\pi)^{D-2}}\int_{0}^{\infty}dk_{1}\dots\int_{0}^{\infty}dk_{D-2}\sum_{n=1}^{\infty}\left[(k_{1})^{2}+\dots+(k_{D-2})^{2}+ \left(\frac{n\pi}{a}\right)^{2}\right]^{\frac{1}{2}}.
C^q_0(x)+\sum_{k=1}^{m^+}\sum_{j=1}^{m_k}C^q_{kj}(x)\left(\frac{\eta_k}{\eta_k+ s}\right)^j=\frac{1}{\psi_q^+(s)}\sum_{k=1}^{M}C^q_{k}\frac{e^{-\beta_{k,q}x}}{s+\beta_{k,q}}, \ \x\geq 0,
\log{h(z;\boldsymbol{\xi})}=\log{(f_0a_0)}+\log{(1+\sum_{r=1}^{\infty}\frac{f_r}{f_0}z^{-r})}+\log{(1+\sum_{r=1}^{\infty}\frac{a_r}{a_0}z^{r})}+\log{b(z)}&=\phi_0+\sum_{s}\log{|z_s|}+\sum_{r=1}^{\infty}\phi_r(\boldsymbol{\xi}) z^{-r}+\sum_{r=1}^{\infty}\alpha_r(\boldsymbol{\xi}) z^{r}+\sum_{r=1}^{\infty}\beta_r z^r
S_1(p,q,m_2;b,c,d) =\displaystyle\sum_{\alpha,\beta=1}^{p}\displaystyle\sum_{h=1}^{p-1}\left(\dfrac{h}{p}\right)e\left(\dfrac{b\alpha+c\beta+d\overline{q}(m_2\alpha^2\beta-h)}{p}\right) &=\left(\displaystyle\sum_{h=1}^{p-1}\left(\dfrac{h}{p}\right)e\left(\dfrac{-d\overline{q}h}{p}\right)\right)S_2(p,q,m_2;b,c,d),
\frac{1}{\lambda_{r}}\sigma_{0}- \frac{2}{\pi}\sigma_{0}+ \frac{1}{\pi r}\frac{\displaystyle \Gamma \left( \frac{3}{2}+ i m r \right) \Gamma \left( \frac{3}{2}- i m r \right)}{\displaystyle \Gamma \left( 1 + i m r \right) \Gamma \left( 1 - i m r \right)}= 0 ,
I_{k,s,e}(\lambda) = \left\{(A,B) \in I_{k,s,e}\mid \exists \, \Sigma \in X_{\lambda}, A \subset \Sigma \subset B \right\}, I_{k,s,e}^+(\lambda) = \left\{(A,B) \in I_{k,s,e}\mid \exists \, \Sigma^+ \in X_{\lambda}^+, A \subset \Sigma^+ \subset B \right\}.
h_{nm}^{LOS}(t)=\sqrt{P_p\exp(j\Phi_{p}).G_R(\theta_{p}).\sigma_{p}.\mathcal{P}_{p}.G_T(\phi_{p})}. \exp{(\frac{j2\pi}{\lambda}(d_s\sin(\theta_{p})+d_m\sin(\phi_{p}+\Upsilon_{p})))}. \exp{(\frac{-j2\pi V_m}{\lambda}\cos(\vartheta_v-\theta_{p}))}
\frac{5}{4^4 24}\big( 150 a^3N^{-1/4}-45\,a^2N^{-6/4}+21 a N^{-11/4}\big) &\ge \frac{5}{4^4 24}\big( a^3N^{-1/4}+a^2N^{-6/4}+17aN^{-11/4}\big).
\sum_{k = 0}^{\infty}{'}( b k )^{\nu}K_{\nu}( b k ) = \frac{\sqrt{\pi}}{b}2^{\nu}\Gamma ( \nu + 1 / 2 ) \sum_{n = 0}^{\infty}{'}\left[ \left( \frac{2 \pi n}{b}\right)^{2}+ 1 \right]^{- \nu - 1 / 2}
S_{0}= \sum_{l}{\frac{1}{2 \sigma_{l}^{2}}}\phi_{l}^{a}\phi_{- l}^{a}+ \sum_{l}{\frac{1}{2 \tau_{l}^{2}}}f_{l}^{a}f_{- l}^{a}+ \sum_{r}{\frac{1}{g_{r}}}\bar{\psi}_{r}^{a}\psi_{r}^{a}.
\|t^{1-\frac{1}{2r}}u^n\|_{L^{2r}_t L^{\infty}_x}+\|t^{\gamma_1}u^n\|_{L^{2r}_t L^{p_3}_x}+ \|t^{\gamma_2}u^n\|_{L^{\infty}_t L^{p_3}_x}+ &+\|t^{\beta_1}\nabla u^n\|_{L^{2r}_t L^{p_2}_x}+\|t^{\beta_2}\nabla u^n\|_{L^{\infty}_t L^{p_2}_x}\leq \|t^{\alpha_1}F^n\|_{L^{2r}_t L^{p_1}_x}.
(e_1,\dots,e_j) =(y_1,\dots,y_j)Y_1^{-1}-(e_{j+1},\dots,e_r)Y_2Y_1^{-1}\textup{, and hence}(e_1,\dots,e_j,e_{j+1},\dots,e_r) =(y_1,\dots,y_j,e_{j+1},\dots,e_r) \begin{pmatrix}Y_1^{-1}0 -Y_2Y_1^{-1}I\end{pmatrix}.
E[g(X_{1},...,X_{d})]=\int_{\mathbb{R}^{d}}g(x_{1},...x_{d})\cdot c(F_{1}(x_{1}),...,F_{d}(x_{d}))\cdot f_{1}(x_{1})\cdot\cdot\cdot f_{d}(x_{d})dx_{1}\cdot\cdot\cdot dx_{d}.
\eta = \frac{3}{64k_*v}\left[6v^{1/3}(v-2) - 8 + (v-2)\log \frac{(1-v^{1/3})^3}{1-v}+2 \sqrt{3}(2-v) \tan^{-1}\frac{2v^{1/3}+1}{\sqrt{3}}\right]
I_2\leq \frac{|d_2(t)|^2}{2\varepsilon_2}+\frac{\varepsilon_2}{2}\|u\|^2+M_1\|u\|^2+\frac{M_2}{2}(u^2(t,1)- u^2(t,0)) = \frac{|d_2(t)|^2}{2\varepsilon_2}+\bigg(\frac{\varepsilon_2}{2}+M_1\bigg)\|u\|^2+\frac{M_2}{2}(u^2(t,1)- u^2(t,0)).
J_1(\beta,N,T,\mathbf{R_1}) \leq CN\int \int_{\hat{\mathbf{R}}_1 \cap \{t-s>\delta\}}\frac{1}{2\pi (t-s)}e^{-\frac{\beta^{2/3}N^{2/3}(t-s)^2}{4(t-s)}}ds dt + CN\int \int_{\hat{\mathbf{R}}_1 \cap \{t-s \leq \delta\}}ds dt.