harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{t=1}^k LT(q_t)(ux_{i_t}+ p_{i_t}) = \sum_{t=1}^k LT(q_t)p_{i_t}= \sum_{t=1}^k c_tM(q_t)p_{i_t}& = \sum_{t=1}^{k-1}\left( c_1 + \dots + c_t \right)( M(q_t)p_{i_t}- M(q_{t+1}) p_{i_{t+1}}) & = \sum_{t=1}^{k-1}\left( c_1 + \dots + c_t \right)\frac{M(q_t)}{x_{i_{t+1}}}( x_{i_{t+1}}p_{i_t}- x_{i_t}p_{i_{t+1}}).
\phi_{(j,N)}f=\left( f-\mathbb{E}\left( f\;|\;A_{j}^{(N)}\right) \right)1_{A_{j}^{(N)}}, \phi_{(N)}f=\sum_{j=1}^{m_{N}}\phi_{(j,N)}f=f-\sum_{j=1}^{m_{N}}\mathbb{E}(f\;|\;A_{j}^{(N)})1_{A_{j}^{(N)}}.
\begin{matrix}p(x)&\propto&exp[\int^{x}\frac{dx}{x}(2D\mu_{\alpha}(\frac{6}{b}x)-1)],whereD=\frac{4ab}{3\sigma^{2}}\\ &=&N_{\alpha}(\frac{6}{b}x)^{D}x^{D-1}\end{matrix}
\begin{array}{r c l}{{C}}&{{=}}&{{C^{( 0 )}+ C^{( 1 )}+ C^{( 2 )}+ \ldots}}\\{{}}&{{}}&{{}}\\{{G}}&{{=}}&{{G^{( 0 )}+ G^{( 1 )}+ G^{( 2 )}+ \ldots}}\\{{}}&{{}}&{{}}\\{{\Lambda^{( \cdot )}}}&{{=}}&{{\Lambda^{( 0 )}+ \Lambda^{( 1 )}+ \ldots \, ,}}\\ \end{array}
q^{-k(d-k)}~U(k, i_0, \ldots, i_d, \ell_0, \ell_1, \ldots, \ell_k)=\left( \prod_{t=0}^k |E_{i_{\ell_t}}| \right) \left( q^{-\sum\limits_{t=1}^k (\ell_t-t)}\right).
I_{2,\max}(K^4_\omega) =(X_1^{}X_2^2X_3^2, X_1^2X_2^2X_3^{},X_1^{}X_3^2X_4^2,X_1^2X_3^{}X_4^2,X_1^2X_2^2X_4^2,X_2^2X_3^2X_4^2)S &=(X_1^{},X_2^2)S\bigcap(X_1^2,X_3^2)S\bigcap(X_1^{},X_4^2)S &\qquad\bigcap(X_2^2,X_3^{})S\bigcap(X_2^2,X_4^2)S\bigcap(X_3^{},X_4^2)S
(II)=\int_{\Delta}\frac{\alpha_n^{p+1}\delta^n}{(\delta^2+|x'|^2)^{n-\frac{1}{2}}}\int_0^{\frac{1-\sqrt{1-|x'|^2}}{\sqrt{\delta^2+|x'|^2}}}\frac{1}{(1+y_n^2)^n}\, dy_ndx'.
=\int_{\frac{2\pi n}{\delta_X}}^{\frac{2\pi n}{\delta}}\overline{\varphi_X\left(\frac{2\pi n}{t}\right)}\varphi_Y\left(\frac{2\pi n}{t}\right)\frac{(\log \frac{t}{2\pi})^\alpha}{t^\beta}dt\ll n^{1-\beta}(\log n)^\alpha.
\frac{1}{k!}\bigl(E_x \bigl[\bigl(I_{T}(h)\bigr)^k-\bigl(I_{T,n}(h)\bigr)^k \bigr]f(X_T) \bigr) &\!= \!\!\int_{S_{k,0,T}}\int_{(\mathbb{R}^d)^{k+1}}\Biggl(\prod_{i=1}^kh(y_i)\Biggr)f(z) (J_1\,{+}\,J_2\,{+}\,\cdots\,{+}\,J_{2k-1}\,{+}\,J_{2k})dz\prod_{j=1}^k dy_j \prod_{i=1}^k ds_{i}.
P\left(\mu^N\in O\right)=\mathbb{E}_{\hat{P}^{N,F,G,H}_{f_1, f_2, f_3}}\left(\frac{dP}{dP^N_{f_1, f_2, f_3}}\frac{1}{\mathcal{U}^N_{F,G,H}\left(T, \xi^N\right)}1_{\{\mu^N\in O\}}\right).
E_{0}\big[T_{\tau}(X_i)^{2}\big]=E_{0}\big[T_{\tau}(X_i)^{2}1\{|X_i|\leqslant \zeta_{\tau}\}\big]+E_{0}\big[T_{\tau}(X_i)^{2}1\{|X_i| >\zeta_{\tau}\}\big],
\begin{aligned}e^{-itH}(f-Qf) =e^{-itH}\sum_{j=1}^M\Omega_jf &\sim U_a(t)\lim_{\mbox{\scriptsize supp}\psi_a\downarrow\mathcal{T}_a}\lim_{t\to\infty}U_a(t)^*\varphi(H^a)(\psi_a(H^a)-P^a)P_a^{{\varepsilon}_a}(t)e^{-itH}f.\end{aligned}
\frac{1}{q}\Phi^-_k(q,0)[ \kappa^-(q+\alpha,0)^{-1}]_{i,k}= \frac{\pi_k}{\pi_i}\frac{1}{q}\hat\Phi^+_j(q,0)[ \hat\kappa^+(q+\alpha,0)^{-1}]_{i,k}.
d s^{2}= e^{\sigma}( d \rho^{2}+ \rho^{2}d \phi^{2}+ \sum_{i , j = 1}^{d - 2}( \gamma_{i j}( \theta ) + h_{i j}( \theta ) \rho^{2}) d \theta^{i}d \theta^{j}+ . . . ) \equiv e^{\sigma}d \tilde{s}^{2}
p(t) =t(t-\alpha_1)(t-\zeta\alpha_1)\cdots (t-\zeta^{k-1}\alpha_1) \cdots (t-\alpha_{2n/k})\cdots (t-\zeta^{k-1}\alpha_{2n/k}) &=t(t^k-\alpha_1^k)(t^k-\alpha_2^k)\cdots(t^k-\alpha_{2n/k}^k) &=\sum_{q=0}^{\frac{2n}{k}}(-1)^qe_q(\alpha_1^k,\dots, \alpha_{2n/k}^k)t^{2n-qk+1},
\delta^{s}S \simeq \int_{t_{1}}^{t_{2}}d t \epsilon \left[ U_{_{( \nu )}}^{s}\Omega_{_{( \nu )}}- y^{s}\Omega_{_{( 1 )}}- z_{_{( \nu )}}^{s}\left({\frac{\partial \Omega_{_{( \nu )}}}{\partial \pi}}\pi - \Omega_{_{( \nu )}}\right) \right] .
\frac{1}{r^k}\leq \varphi_0'(r) \leq \frac{e^{k\| s^{-2}\Phi_1(s)\|_{L^1(1, \infty)}}}{r^k}&\le \begin{cases}\frac{C}{r^{k}}\delta = 0 1 begin{align*}6pt]\frac{C (\log(h^{-1}))^{\kappa C}}{r^k}0 < \delta < 1\end{cases}, h \in (0, h_\delta], \, 1 \le r \le a.
g_1\mapsto g_1\left[\begin{array}{ccccc}0 0 0 1_t 0 0 1 0 0 0 0 0 1 0 0 -1_t 0 0 0 0 0 0 0 0 1\end{array}\right]xg_1=\left[\begin{array}{ccccc}0 0 0 -1_t 0 0 0 0 0 u 0 0 1 0 0 1_t 0 0 0 0 0 0 0 0 v\end{array}\right].
\mathcal N_{\mathbf T(u)}(w_s(u,0)) =\mathcal N_{\mathcal T_D(g(\bar x))}(\nabla g(\bar x) u; w_s(u,0)) &\subset \mathcal N_{\mathcal T_D(g(\bar x))}(\nabla g(\bar x) u) \cap [w_s(u,0)]^{\perp}=\mathcal N_D(g(\bar x);\nabla g(\bar x)u)\cap[w_s(u,0)]^\perp
\langle[\Phi^{\sqrt{\Delta_{\Phi}^{\Psi}}}](x),[\Phi^{\sqrt{\Delta_{\Phi}^{\Psi}}}](x)\rangle=[\varphi]_{\Delta_{1}}(\langle x,x\rangle)=[\psi](\langle x,x\rangle)=\langle[\Psi](x),[\Psi]\rangle(x),
x_{j_{k+1}}^{-}(q^{t_l +1}z)_{0}x_{j_k}^{-}(z)_\bullet \ldots x_{j_2}^{-}(z)_\bullet x_{j_1}^{-}(z)_\bullet \mathcal{Y}_i (z)\neq 0, &x_{j_{k+1}}^{-}(q^{t}z)_{0}x_{j_k}^{-}(z)_\bullet \ldots x_{j_2}^{-}(z)_\bullet x_{j_1}^{-}(z)_\bullet \mathcal{Y}_i (z)= 0 t\neq t_l +1.
u_\pm =\frac{1-\sqrt{1-\frac{4y((q\sqrt{x}\pm1)^2+\sqrt{x}(2q-1)(\mp2-\sqrt{x}))}{((1\mp\sqrt{x}q)(y+1)\pm2\sqrt{x}y)^2}}}{\frac{2y(1+\sqrt{x}(q-1))}{((1\mp\sqrt{x}q)(y+1)\pm2\sqrt{x}y)^2}}.
R_1 =\lfloor c_1a_2a_3 \rfloor,S_1=\lfloor c_1a_1a_3 \rfloor, T_1=\lfloor c_1a_1a_2 \rfloor , R_2 =\lfloor c_2a_2a_3 \rfloor,S_2=\lfloor c_2a_1a_3 \rfloor, T_2=\lfloor c_2a_1a_2 \rfloor , R_3 =\lfloor c_3a_2a_3 \rfloor,S_3=\lfloor c_3a_1a_3 \rfloor, T_3=\lfloor c_3a_1a_2 \rfloor .
E_q \left[ \left( \sum_{i \in I}(X_i - x_i^o) - (Y_i - y_i^o) \right)^2 \right] = \phi^2(i_0/n) \frac{\theta'(i_0/n)}{\theta(i_0/n)}\frac{(i_1-i_0)}{\beta n}+ O \left( \frac{(\log n)^2}{n}\right),
Z_n(\lambda,T) := \sum_{N \geq 0}\frac{T^N}{N!}\int_{\mathbb{T}_n^N}g_N^\lambda(x_1,\ldots,x_N) \mathrm{d}x_1 \ldots \mathrm{d}x_N = \sum_{k \geq 0}\sum_{0 \leq \ell \leq n}T^{nk}e^{-\lambda \ell}\mathrm{Vol}^{(n)}_{k,\ell}.
N_\theta^{2 (1 - 2 k)}\frac{t^{2 (1 - \theta)}}{t^2 + \lambda^{2 k}}\leq\left(\frac{\alpha^{1 - \theta}}{\alpha + \lambda}\right)^{2k}\quad\alpha =t^{1/k}\left( \frac{1 - \theta}{\theta}\right)^{1 - 1/(2k)}.
\Theta\left(f_{1},f_{2},T\right)=2^{1-n}d_{Q}T^{nd-2\left(n-1\right)}\int_{F\left(\left[0,\infty\right)^{2n}\right)}f_{2}\left(u_{2}-u_{4},\dots,u_{2n-2}-u_{2n}\right)\overline{\Psi}_{f_{1}}\left(u,T\right)du,
\sum_{k\in\mathbb{T}(v,j)}\int\nolimits_{\gamma(t_{j},\varepsilon)}a_{k}(t)\Psi_{k,t}(x)dt=\sum_{k\in\mathbb{T}(v,j)}\int\nolimits_{(t_{j}-\varepsilon,t_{j}+\varepsilon)}a_{k}(t)\Psi_{k,t}(x)dt,
\left(1+\frac{xv}{1-v}\right)B(x,v)+\frac{xv^2}{1-v}A(x,v) &\qquad\qquad\qquad=\frac{(1-v+xv)v^2}{1-v}A(x,1)-\frac{xv^2(1-2v+xv)}{(1-x)(1-v)}B(x,1)-\frac{x^2v^2(1-v+xv)}{1-x}, &-\left(1+x+\frac{xv}{1-v}\right)B(x,v)+\frac{(1-xv)(1-v+xv)}{xv(1-v)}A(x,v) &\qquad\qquad\qquad=\frac{(v-x)v^2}{1-v}A(x,1)+\frac{xv^2(x-v)}{(1-x)(1-v)}B(x,1)+\frac{xv^2(1-x+x^2)}{1-x}.
\frac{(2m-1)\#}{m\#}\le(\begin{matrix}2m-1\\m\end{matrix})=\frac{1}{2}((\begin{matrix}2m-1\\m-1\end{matrix})+(\begin{matrix}2m-1\\m\end{matrix}))<\frac{1}{2}(1+1)^{2m-1}=2^{2m-2}.
\frac{C_0 \epsilon_0}{(\kappa' )^{2/3}}\kappa_1^{1/12}+ C_0{(\kappa' \kappa_1 )^{1/3}}M^{1/3}= \frac{C_0 \epsilon_0}{(\kappa')^{2/3}}\kappa_1^{1/12}+ C_0 (\kappa'\kappa_1)^{1/3}M^{1/3}.
\begin{array}{ll}\delta Q^S(F^{\nu}_{k},F^{\nu}_{k})\ast^g \Gamma^v_{\nu}=\left( Q^S(F^{\nu}_{k},F^{\nu}_{k})-Q^S(F^{\nu}_{k-1},F^{\nu}_{k-1})\right) \ast^g \Gamma^v_{\nu}.\end{array}
\mu^*H =-\mu^*\lambda_2= -\lim_{T \to \infty}\frac{1}{T}\int_0^T \lambda_2(\hat X_s, 0) ds - \lim_{T \to \infty}\frac{1}{T}\int_0^T \left( \lambda_2(X_s, Y_s) - \lambda_2(\hat{X}_s, 0)\right) ds
\frac{d \phi_{1}}{d x}= \pm \frac{\partial W}{\partial \phi_{1}}= \pm \left( 2 \phi_{1}^{2}+ \sigma \phi_{2}^{2}- \frac{1}{2}\right) \hspace{1 c m}\frac{d \phi_{2}}{d x}= \pm \frac{\partial W}{\partial \phi_{2}}= \pm 2 \sigma \phi_{1}\phi_{2}\qquad .
m_{\varphi{\bar{\varphi}}}^{2}\sim G_{\varphi{\bar{\varphi}}}^{- 1}\, \left( G^{i{\bar{\jmath}}}R_{i{\bar{\jmath}}\varphi{\bar{\varphi}}}- G_{\varphi{\bar{\varphi}}}\right){\frac{m_{3 / 2}^{4}}{M_{p}^{2}}}{\cal J}\ ,
\begin{array}{l}{{3 ( 3 , 2 )_{( 1 , - 1 , 0 , 0 , 0^{3})}+ 3 ( \bar{3}, 1 )_{( - 1 , 0 , 1 , 0 , 0^{3})}+ 3 ( \bar{3}, 1 )_{( - 1 , 0 , 0 , 1 , 0^{3})}}}\\{{+ 3 ( 1 , 2 )_{( 0 , 1 , 0 , 0 , \underline{{{- 1 , 0 , 0}}})}+ 3 ( 1 , 1 )_{( 0 , 0 , - 1 , 0 , \underline{{{1 , 0 , 0}}})}+ 3 ( 1 , 1 )_{( 0 , 0 , 0 , - 1 , \underline{{{1 , 0 , 0}}})},}}\\ \end{array}
A(k_j,\;p_l) =(k^2)^{-i}\left[(k-p)^2\right]^{-j}, \;\; h(k_j\!\cdot\!n,\;p_l\!\cdot\!n) =(k\cdot n)^{-l},\;\; f(k_j\!\cdot\!n^*,\;p_l\!\cdot\!n^*) = (k\cdot n^*)^m,
M_{2}(\mathcal{NC}(G)) =\dfrac{1}{2}\Big{(}p^{4}q^{4}-7p^{3}q^{3}+41p^{2}q^{2}-51pq+3p^{4}q^{2}+13q^{2}-16p^{3}q^{2}+14pq^{2}+2p^{2}q^{3}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-16pq^{3}+8p^{2}q-9q+2pq^{4}+2p^{3}q+p^{4}q+18 \Big{)}
\mathcal I_p^{\sharp}(\mathbf h, \breve{\mathbf g}, \pmb{\phi}) = \frac{\zeta_p(2)^{2}(p^2+w_p)(p+w_p)}{p^3}\frac{p-w_p}{\zeta_p(2)(p^2+w_p)}= \zeta_p(2)\frac{(p+w_p)(p-w_p)}{p^3}= \frac{p^2(p^2-1)}{(p^2-1)p^3}= \frac{1}{p}.
\bigg(\sum_{\substack{r \mid q^{\ell}- 1 r \nmid q-1}}\frac{1}{r}\bigg) \frac{q^\ell}{\ell\Phi(W)}\sideset{}{'}\sum_{\substack{d_1, \ldots, d_k e_1, \ldots, e_k}}\frac{\lambda_{d_1 \ldots, d_k}\lambda_{e_1 \ldots, e_k}}{\prod_{i = 1}^k \Phi([d_i, e_i])},
\bar{P}^{\mathrm{PNC}}_i-\beta_i\bar{R}^{\mathrm{PNC}}_i =\gamma, i=1,2 \bar{P}^{\mathrm{PNC}}_3-\beta_{3,c}\bar{R}^{\mathrm{PNC}}_{3,c}-\beta_{3,p}\bar{R}^{\mathrm{PNC}}_{3,p}=\gamma, 2^{R_1^{\mathrm{PNC}}}\ln2 \left(\frac{1}{g_{1r}}+\frac{1}{g_{2r}}\right)-\beta_1 = 0, \frac{2^{R_2^{\mathrm{PNC}}}}{g_{2r}}\ln2 -\beta_2 = 0,
\begin{aligned}\int_{\Omega}\ast dw\wedge\eta'=(-1)^{n-1}\langle w,\delta\eta'\rangle_{L^{2}\Lambda^{0}(\Omega)}+(-1)^{n-1}\int_{\partial\Omega}\mathrm{tr}(w)\wedge\ast\boldsymbol{n}(\eta') =0, \end{aligned}
|\!|\varphi^{\sharp}F|\!|'_{K,k}\le \widetilde C_{\varphi,\varphi(K),k}|\!|F|\!|'_{\varphi(K),k}, \widetilde C_{\varphi,\varphi(K),k}= \sup_{\alpha,|\alpha|\le k}\left(\sum_{\beta\le \alpha}L^{k-|\alpha|}\:C_{\varphi,\varphi(K),|\alpha|}\:\operatorname{diam}(\varphi(K))^{|\alpha|-|\beta|}\right).
m_e(\lambda\pm ) =m_e^0(\lambda\pm )+l(\lambda\pm)\langle m_e^0(\lambda\pm ),u\rangle m_1(\lambda\pm ) &= \frac{m_e^0(\lambda\pm)+l(\lambda\pm)\big(\langle m_e^0(\lambda\pm),u\rangle m_1^0(\lambda\pm )-\langle m_1^0(\lambda\pm),u\rangle m_e^0(\lambda\pm )\big)}{1-l(\lambda\pm)\langle m_1^0(\lambda\pm ),u\rangle}.
\frac{1}{\lambda_u^t \cdot t^{d_u+1}}M^t \vec v = \frac{1}{\lambda_u^t \cdot t^{d_u+1}}\left(\lambda^t \vec v + \sum_{j=0}^{t-1}\lambda^{j}M^{t-j-1}\vec u_{}\right) &= \frac{1}{t^{d_u+1}}\vec v + \frac{1}{\lambda}\sum_{j=0}^{t-1}\frac{({t-j-1})^{d_u}}{t^{d_u+1}}\frac{1}{\lambda^{t-j-1}\cdot ({t-j-1})^{d_u}}M^{t-j-1}\vec u_{}
{\cal L}_{I}= - \frac{2}{9}\Omega^{\rho , \mu \nu \alpha \beta}\Omega_{\mu , \rho \nu \alpha \beta}+ \frac{2}{9}\Omega^{\mu \nu \alpha}\Omega_{\mu \nu \alpha}- \frac{1}{9}\Omega^{\rho , \mu \nu \alpha \beta}E_{\mu \nu \alpha \beta , \rho}- \frac{4}{9}\Omega^{\mu \nu \alpha}E_{\mu \nu \alpha}
e_m(z_1,z_2) = \left(\frac{(2m)!}{(m!)^2}\right)^{1/2}z_1^m z_2^m f_{j,m}(z_1,z_2) = \left(\frac{(2m+j)!}{m!(m+j)!}\right)^{1/2}z_1^{j+m}z_2^m \intertext{and}g_{j,m}(z_1,z_2) = \left(\frac{(2m+j)!}{m!(m+j)!}\right)^{1/2}z_1^m z_2^{j+m}.
\left| \mathcal{B}_{p h y s}\right\rangle = \int_{- \infty}^{\infty}d \rho \sum_{m \in Z}\ \sum_{r \in Z_{+}}e^{- i \rho \Theta}\left| B_{p^{+}= m + \alpha , p^{-}= - \frac{r}{m + \alpha}, p_{2}= \rho}\right\rangle .
1(\xi^{1}_{1},\xi^{1}_{2})=\sum_{k'\in\mathbb{Z}}\widehat{\varphi_{k'}}(\xi^{1}_{1})\widehat{\psi_{k'}}(\xi^{1}_{2}) +\sum_{k'\in\mathbb{Z}}\widehat{\psi_{k'}}(\xi^{1}_{1})\widehat{\widetilde{\psi_{k'}}}(\xi^{1}_{2}) +\sum_{k'\in\mathbb{Z}}\widehat{\psi_{k'}}(\xi^{1}_{1})\widehat{\varphi_{k'}}(\xi^{1}_{2})
\widetilde{T}\begin{pmatrix}x y \end{pmatrix}= \begin{pmatrix}z+a_2 z^2 \dots + a_i z^i +a_{i+1}(x)z^{i+1}+\dots \delta x + b_1xz + b_2 x z^2 +\dots +b_j x z^j + b_{j+1}(x) z^{j+1}+\dots \end{pmatrix},
\frac{\dot{a}_{0}^{2}}{a_{0}^{2}}+ \frac{\ddot{a}_{0}}{a_{0}}= - \frac{k_{( 5 )}^{4}}{3 6}\rho ( \rho + 3 p ) - k_{( 5 )}^{2}\frac{T_{5 5}}{3 b_{0}^{2}}
\nabla\times\hat{B}=\mu_{0}\hat{J}+i\mu_{0}\epsilon_{0}\omega\hat{E}\Rightarrow-\frac{B^{r}}{r}+\frac{dB^{(1)}}{dr}+\frac{B^{(1)}}{r}=\mu_{0}J+i\omega\mu_{0}\epsilon_{0}E.
\left(\frac{1}{g}\right)^{(p)}(x)=\frac{p!}{g^{p+1}(x)}\sum_{\left\{m_i\right\}}\frac{(-1)^{p-p_0}(p-p_0)!}{\prod_{j=1}^p(j!)^{m_j}m_j!}\prod_{j=0}^p\left[g^{(j)}(x)\right]^{m_j},
\delta_\epsilon A_\mu^a=\left( D_\mu \right)_{\;\;b}^a\epsilon^b,\;\delta_\epsilon B_{\mu \nu}=\partial_{\left[ \mu \right.}\epsilon_{\left. \nu \right]}+2g\epsilon_a\partial_{\left[ \mu \right.}A_{\left.\nu \right]}^a,
c_j^{kl}(\tilde{A}) =\bar{\gamma}\left(c_j^{kl}(A) -\bar{\gamma}\,\bar{a}_{kl}\sum_{s,t=1}^n c_{st}\int_Y r A e_j\cdot \nabla v^{st}\right) &=\bar{\gamma}\left(c_j^{kl}(A) -\bar{\gamma}\,\bar{a}_{kl}\sum_{s,t=1}^n c_{st}\,c_j^{st}(A) \right) &=0,
0\ge\bar{p}_{j_2}\ge\bar{p}_{i_0}\ge \bar{p}_{j_1}\ge 0\;\;\; \Rightarrow\;\;\; \bar{p}_{i_0}= 0\;\;\; \Rightarrow\;\;\; p_{i_0}=q_{i_0}=0 \;\;\; \Rightarrow\;\;\; s_{i_0}=0,
r(\partial_rV^{(\alpha,a)}+\partial_r H^{(\alpha,a)}\cdot\omega) &=\omega\cdot\Large[r\nabla V^{(\alpha,a)}+(r\partial_r H^{(\alpha,a)}\cdot\omega)\omega\Large] &=\omega\cdot\Large[r\nabla V^{(\alpha,a)}+r\nabla\cdot H^{(\alpha,a)}\omega-(\Omega H^{(\alpha,a)}\cdot\omega^\perp)\omega \Large],
\frac{1}{5}\times [ ( 0 - 5 )^{2}+ ( 1 - 5 )^{2}+ ( 5 - 5 )^{2}+ ( 9 - 5 )^{2}+ ( 1 0 - 5 )^{2}] = \frac{1}{5}\times [ 2 5 + 1 6 + 0 + 1 6 + 2 5 ] = \frac{2}{5}\times 8 2
\frac{1}{N}\begin{bmatrix}\sum_{i}\cos^2(\theta_t^i) -\sum_{i}\cos(\theta_t^i)\sin(\theta_t^i) -\sum_{i}\cos(\theta_t^i)\sin(\theta_t^i) \sum_{i}\sin^2(\theta_t^i) \end{bmatrix}\begin{bmatrix}\kappa_1 \kappa_2 \end{bmatrix}
\mathcal B'=\{e_{t_1}*e_{t_2}*\ldots *e_{t_{n-r}}*f_{k_1}*f_{k_2}*\ldots * f_{k_r};~r\in[0;n],~ 0\leqslant t_1\leqslant t_2\leqslant\ldots\leqslant t_{n-r},~0 < k_1\leqslant k_2\leqslant\ldots\leqslant k_{r}\}
I_{2}= - \frac{d}{d s}\left( s g ( s ) \int_{\Gamma}\frac{d z}{2 \pi i}z^{s}\left( \frac{\tanh ( \omega - z ) - \sigma}{\omega^{\prime}- 2 z}+ \frac{\sigma}{\omega^{\prime}- 2 z}\right) \right)_{s = 0}.
\boldsymbol{A}=\frac{1}{3}\left[\begin{matrix}(y-z)^2 (y-z)(z-x) (y-z)(x-y) (y-z)(z-x) (x-z)^2 (z-x)(x-y) (y-z)(x-y) (z-x)(x-y) (x-y)^2\end{matrix}\right ] , \boldsymbol{B}=\frac{1}{3}\big ( (x-y)p_z+(y-z)p_x+(z-x)p_y \big)\left[\begin{matrix}0 1 -1 -1 0 1 1 -1 0\end{matrix}\right ] .
\begin{array}{c}{\overline{{A}}_{\nu}^{1}\longrightarrow W_{\nu}^{-},}\\{\overline{{A}}_{\nu}^{2}\longrightarrow W_{\nu}^{+},}\\{\overline{{A}}_{\nu}^{3}\longrightarrow Z_{\nu},}\\{\overline{{A}}_{\nu}^{0}\longrightarrow A_{\nu}.}\\ \end{array}
B^h[W,W] = \frac{1}{2\pi}\int B^h(\xi,\eta)\hat W(\xi)\hat W(\eta) e^{i(\xi+\eta)\alpha}\,d\xi d\eta B^a[W,\bar W] = \frac{1}{2\pi}\int B^a(\xi,\eta)\hat W(\xi)\bar{\hat W}(\eta) e^{i(\xi-\eta)\alpha}\,d\xi d\eta.
0 = v_p(f^{n-1}_{\alpha, \beta}(\beta + \alpha B_{d,k}(1))) = v_p(f^{n-1}_{\alpha, \beta}(\beta) + h_{n-1}(\beta, \alpha B_{d,k}(1))) \geq \min \{v_p(f^{n-1}_{\alpha, \beta}(\beta)), v_p(h_{n-1}(\beta, \alpha B_{d,k}(1))) \}\geq \{\infty, v_p(\alpha) \}= v_p(\alpha)
\left\langle \pi,\int_X\imath_X(x)dP \right\rangle =\lim_{n\to \infty}\left\langle \varphi_n,\int_X\imath_X(x)dP \right\rangle=\lim_{n\to \infty}\int_X\langle \varphi_n,\imath_X(x)\rangle dP & =\int_X\lim_{n\to \infty}\langle \varphi_n,\imath_X(x) \rangle dP=\int_X\langle \pi,\imath_X(x) \rangle dP
G ( \omega , \mathrm{\boldmath k}, \rho \Lambda ; v , e^{2}) = \exp{\left\{\int^{\rho}\frac{d \rho^{\prime}}{\rho^{\prime}}\gamma \right\}}G ( \omega , \mathrm{\boldmath k}, \Lambda ; v_{e f f}( \rho ) , e_{e f f}^{2}( \rho ) )
{\mathbf b}= \left(\, \begin{array}{ccc}0 b_{12}0 0 0 b_{23}b_{31}0 0\end{array}\,\right)\ ,\qquad{\mathbf d}= \left(\, \begin{array}{ccc}0 d_{12}0 0 0 d_{23}d_{31}0 0\end{array}\,\right)
{}{}\sum_{i=1}^{M_R}\frac{l_i}{N_0 + P l_i}- \lambda \eta + \mu -\nu = 0 &\mu P = 0, \mu \geq 0, P \geq 0, &\nu (P-P_{max}) = 0 , \nu \geq 0, P-P_{max}\leq 0 &\lambda (E\left\{\eta P \right\}- Q) = 0, \lambda \geq 0, E\left\{\eta P \right\}\leq Q
I_{B}= \frac{1}{\widetilde C_{B}}( - 1 )^{\frac{S - \widetilde{S}}{2}}\: \sum_{m_{j}^{\prime}= 0}^{2 k_{j}+ 3}\delta_{\frac{M - \widetilde{M}}{2}+ \sum \frac{K^{\prime}( m_{j}^{\prime}+ 1 )}{2 k_{j}+ 4}}^{( K^{\prime})}\prod_{j = 1}^{r}\: n_{L_{j}\widetilde{L}_{j}}^{- m_{j}^{\prime}- 1}\: ,
{{\cal{G}}_{\rho}}^{{( 4 )}}= -{\textstyle \frac{3}{4}}({\textstyle \frac{\partial^{2}}{\partial t^{2}}}{A}( t ) )^{2}+{\textstyle \frac{3}{2}}({\textstyle \frac{\partial}{\partial t}}{A}( t ) ) ({\textstyle \frac{\partial^{3}}{\partial t^{3}}}{A}( t ) ) +{\textstyle \frac{9}{4}}({\textstyle \frac{\partial^{2}}{\partial t^{2}}}{A}( t ) ) ({\textstyle \frac{\partial}{\partial t}}{A}( t ) )^{2}
Ad(O)_{;k'}= \ad(\Psi_{k'}) Ad(O_{<k-C}) + \int_{k-C}^{k'}ad(\Psi_{k'}) ad(\Psi_l) Ad(O_{<k-C}) dl \ + \int_{k-C'}^{k'}\int_{l'}^{k'}ad(\Psi_{k'})ad(\Psi_l)ad(\Psi_{l'}) Ad(O_{<l'})dl dl' = \ \mathcal{L}+ \mathcal{Q}+ \mathcal{C}.
\| v \|_{H, \varepsilon}:= \min_{w \in H}{\{\frac{1}{2\varepsilon}\| w-v \|^2_H + \| w \|_H \}}= \int_0^{\| v \|_H}\min{\{s/ \varepsilon, 1 \}}\ds\textrm{for every $v \in H$}.
|R_m(x,h)| \leq \|w^{(m)}\|_{L^{\infty}([0,1])}\int_0^h \frac{(h-t)^{m-1}}{(m-1)!}\, \mathrm{d}t = \frac{h^m}{m!}\|w^{(m)}\|_{L^{\infty}([0,1])}.
\begin{aligned}6 \cdot 7 + 7^2 + O(7^3), 3 \cdot 7 + 3 \cdot 7^2 + O(7^3), 1 \cdot 7 + 4 \cdot 7^2 + O(7^3), 2 \cdot 7 + 3 \cdot 7^2 + O(7^3). \end{aligned}
\sum_{\ell=1}^n (2+p_\ell-p_{\ell-1})^2\leq 9\sum_{\ell=1}^n (p_\ell-p_{\ell-1})^2\leq p_n^{4/3+o(1)}=n^{4/3+o(1)}, 16(p_{n+1}-p_n)\sum_{\ell=1}^n (p_\ell-p_{\ell-1})^2\leq p_n^{5/8+o(1)}p_n^{4/3+o(1)}=p_n^{47/24+o(1)}.
L_1 = K_1 T(\phi_R)^{-1}\cdots T(\phi_0)^{-1}= T(\phi) T(\phi_R)^{-1}\cdots T(\phi_0)^{-1}-I, L_2 = K_2 T(\tilde{\phi}_R)^{-1}\cdots T(\tilde{\phi}_0)^{-1}=T(\tilde{\phi}) T(\tilde{\phi}_R)^{-1}\cdots T(\tilde{\phi}_0)^{-1}-I,
{\cal V}( z ) = - \left( \frac{2 \alpha^{2}- m^{2}}{| 1 - \alpha^{2}z^{2}|}- \frac{\alpha^{2}}{| 1 + \alpha z |}\delta ( 1 + \alpha z ) - \frac{\alpha^{2}}{| 1 - \alpha z |}\delta ( 1 - \alpha z ) \right)
[ L_{\Lambda_{\alpha}}, L_{\Lambda_{\alpha}}]=[H_{\Lambda_{\alpha}}\oplus V_{\Lambda_{\alpha}},H_{\Lambda_{\alpha}}\oplus V_{\Lambda_{\alpha}}] \subset [H_{\Lambda_{\alpha}}, V_{\Lambda_{\alpha}}] + [V_{\Lambda_{\alpha}}, H_{\Lambda_{\alpha}}] +\Sigma_{\beta,\delta \in \Lambda_{\alpha}}[L_{\beta},L_{\delta}].
\lim_{\Delta\to 0}\frac{\mathbb E_{t,u}(f(\sigma_{t+\Delta},t+\Delta))-f(u,t)}{\Delta}= \lim_{\Delta\to 0}\frac{\mathbb E_{t,u}(f(\sigma_{t+\Delta},t+\Delta))-\mathbb E_{t,u}(f(\sigma_{t+\Delta},t))}{\Delta}+\lim_{\Delta\to 0}\frac{\mathbb E_{t,u}(f(\sigma_{t+\Delta},t))-f(u,t)}{\Delta}= f_{t}(u,t)+h(u,t)
\begin{aligned}-\Delta w_t + \left[\frac{f_1(u_t)-f_1(u)}{u_t-u}-\lambda\right] w_t = \frac{f_2(u_t)}{u_t}u_t -\frac{f_2(u)}{u}u\geq \frac{f_2(u)}{u}w_t\geq 0, \end{aligned}
\begin{array}{l}{{f_{-}{\bf U}_{a + 1}{\bf X}_{a + 2 , a}^{' a + 1}+ c_{1 a}( u_{-}) f_{+}{\bf X}_{a + 2 , a}^{a + 1}{\bf U}_{a - 1}^{'}+ c_{2 a}( u_{+}) f_{-}{\bf X}_{a + 2 , a}^{a + 1}{\bf D}_{a + 1}^{'}}}\\{{\mathrm{~}= f_{+}{\bf X}_{a + 2 , a}^{' a + 1}{\bf U}_{a - 1}}}\\ \end{array}
S^{{\rm cl},0}_{{\rm L}}(\hat g,g):= \int_M\big(|d\omega|^2_{g}+ 2K_{g}\omega \big){\rm dv}_{g}, S^{{\rm cl}}_{{\rm AY}}(\hat g,g):= \int_M\big(\tfrac{1}{4}\phi\Delta_{g}\phi + \frac{\phi}{V_{g}}\big){\rm dv}_{g}.
\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)=\int_{\bigcup\gamma_{cd}{\cal F}}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)=\sum_{(c,d)=1}\int_{\gamma_{ab}{\cal F}}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)\;.
\hat{L}^{( \phi^{N})}: = a^{2}\dot{\varphi}_{0}^{2}\partial_{y}{\frac{1}{a^{2}\dot{\varphi}_{0}^{2}}}\partial_{y}a^{2}-{\frac{2 \kappa}{3}}a^{2}\dot{\varphi}_{0}^{2}.
(\Delta \gamma)_{12}= -i \hbar (\Gamma_i^{(2)}-\Gamma_i^{(1)}) d x^i +1\otimes \frac{1}{2}\omega_{ij}\left((\Gamma_{s}^{(2)})^i_{kl}- (\Gamma_{s}^{(1)})^i_{kl}\right) y^j y^k d x^l
\delta W = -{\alpha (1 + \alpha)^2 \over 1536}{\kappa^4 \over \lambda^6}\int_{M^{10}}d^{10}x \epsilon^{A_1 A_2 \cdots A_{10}}{\rm tr}\epsilon F_{A_1 A_2}{\rm tr}F_{A_3 A_4}F_{A_5 A_6}{\rm tr}F_{A_7 A_8}F_{A_9 A_{10}}.
w_-\in V,w_-= \begin{cases}w \mbox{in}N, 0 \mbox{in}\Omega\setminus N, \end{cases}\nabla w_-= \begin{cases}\nabla w \mbox{a.e.~in}N, 0 \mbox{a.e.~in}\Omega\setminus N. \end{cases}
S_{4}^{\prime}( \phi_{s}) = \biggl < S_{4}( \phi_{s}, \phi_{f}) \biggr >_{0 f}+ \frac{1}{2}\biggl ( \biggl < S_{4}^{2}( \phi_{s}, \phi_{f}) \biggr >_{0 f}- \biggl < S_{4}( \phi_{s}, \phi_{f}) \biggr >_{0 f}^{2}\biggr ) + \cdots .
\sum_{(t)^2\in (Y^n)^2}(t)^2=\sum_{\substack{(t_1)^2\in (Y^{p})^2,t_2\in Y^{q}p+q+1=n}}(t_1)^2\vee t_2 &+ \sum_{\substack{(t_1)^1\in (Y^{p})^1,(t_2)^1\in (Y^{q})^1 p+q+1=n}}(t_1)^1\vee (t_2)^1 +\sum_{\substack{t_1\in Y^p, (t_2)^2\in (Y^{q})^2 p+q+1=n}}t_1\vee (t_2)^2
R_{2k}^{(-1)}(x-1) = \gamma_n^{(0)}{}_3F_2 \left( \begin{matrix}-n, \delta+\frac{x}{4}, \delta-\frac{x}{4}-\frac{N-1}{2}, \eta+1+\frac{N}{2}\end{matrix}; 1 \right), R_{2k+1}^{(-1)}(x-1) = \gamma_n^{(1)}(x+2\eta-2\xi){}_3F_2 \left( \begin{matrix}-n, \delta+\frac{x}{4}, \delta-\frac{x}{4}-\frac{N-1}{2}, \eta+1+\frac{N+1}{2}\end{matrix}; 1 \right),
5 2 8_{B}= \frac 1 2 \left[ \left( 1 2 0 , 1 , 1 \right) + \left( 1 2 0 , 1 , 1 \right)^{\prime}\right] + \left( 1 0 , 3 , 1 \right) + \left( 1 2 6_{+}, 3 , 1 \right) .
\frac{1}{2}\frac{d}{dt}\Big[ \, a\frac{d}{dt}{\Vert u_n -u_m\Vert}^2 \, \Big] = \frac{d}{dt}\big[ \, a(u_n^{\prime}-u_m^{\prime}, u_n - u_m) \, \big] = \big({\big( a(u_n^{\prime}- u_m^{\prime})\big)}^{\prime}, u_n - u_m\big) + a{\Vert u_n^{\prime}- u_m^{\prime}\Vert}^2 \ge a{\Vert u_n^{\prime}- u_m^{\prime}\Vert}^2.
\left.\begin{array}{ll}\displaystyle Q(\xi^{\tau},-\eta^{\tau})=\tau^{N-6}\dfrac{a_2}{m_1}\int_{\mathbb{R}^N}|\nabla \xi|^2dx+\tau^{N-6}\dfrac{a_1}{2m_2}\int_{\mathbb{R}^N}|\nabla \eta|^2dx-\frac{N}{2}\tau^{N-6}a_1a_2\int_{\mathbb{R}^N}\eta\xi^2dx begin{align*}0.5cm] \tau^{N-6}\mathcal{Q}(\xi,-\eta).\end{array}\right.
\begin{aligned}\widehat{V}_{r_i,r_j}=M_{r_i}V_{r_i,r_j}M^{-1}_{r_j}, \widehat{f}_{r_i}=M_{r_i}f_{r_i}, \widehat{K}^\delta_{r_i}=M_{r_i}K^\delta_{r_i}M^{-1}_{r_i}, \widehat{g}_{r_i}=g_{r_i}M_{r_i}^{-1}, \widehat{K}_{r_i,r_j}=M_{r_i}K_{r_i,r_j}M^{-1}_{r_j}. \end{aligned}
w(x;\alpha, \beta,N) = \binom{\alpha+x}{x}\binom{N+\beta-x}{N-x}, x=0,1,\ldots,N, h_n (\alpha,\beta,N)= \frac{(-1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_n n!}{(2n+\alpha+\beta+1)(\alpha+1)_n(-N)_n N!}.
U_{n,q}\left( e_{m+1};z\right) =\frac{q^{m}z\left( 1-z\right)}{\left[n+m\right]_{q}}D_{q}U_{n,q}\left( e_{m};z\right) +\frac{q^{m}\left[n\right] z+\left[ m\right]_{q}}{\left[ n+m\right]_{q}}U_{n,q}\left(e_{m};z\right) .
Q_k(\hat{u},\hat{u})=\sum_{\substack{h\in \mathbb{Z}^2\setminus\{0\}|h|^2,|k-h|^2 \in \{\lambda, \mu_+,\mu_-\}}}i(\hat{u}(h,t)\cdot k)\hat{u}(k-h,t), |k|^2 \in\{\lambda, \mu_+,\mu_-\}
b_{2k}(m;y, z) \leq \binom{2k}{2\ell}b_{2\ell}(m_1^2;y,z)b_{2k-2\ell}(m_0;y,z) \leq \binom{2k}{2\ell}\binom{2\ell}{\ell}\left(b_{\ell}(m_1;y,z)\right)^2b_{2k-2\ell}(m_0;y,z) \leq \frac{(2k)!}{k!}\binom{k}{\ell}b_{\ell}(m_1;y,z)b_{2k-2\ell}(m_0;y,z),
2\ell^{\prime}\left( 0,a,\chi\right) =-\overline{E}_{0,\overline{\chi}}\left( 0\right) \log\left( 2k\right) +2\sum_{n=1}^{2k-1}\left(-1\right)^{n}\chi\left( n\right) \log\Gamma\left( \frac{n+a}{2k}\right).
N_V(0,5,5n+4)=N_V(1,5,5n+4)=\cdots=N_V(4,5,5n+4) =\dfrac{p(5n+4)}{5}, N_V(0,7,7n+5)=N_V(1,7,7n+5)=\cdots=N_V(6,7,7n+5) =\dfrac{p(7n+5)}{7}, N_V(0,11,11n+6)=N_V(1,11,11n+6)=\cdots=N_V(10,11,11n+6) =\dfrac{p(11n+6)}{11}.
\phi(yd^2y)(z) = 2\,{z}^{-4}+{z}^{-3}+\frac{1}{6}\,{z}^{-2}-{\frac{1}{90}}-{\frac{1}{240}}\,z+{\frac{1}{30240}}\,{z}^{2}+{\frac{1}{3024}}\,{z}^{3}+O\left({z}^{4}\right), \phi(dydy)(z) =3\,{z}^{-4}+{z}^{-3}+{\frac{1}{240}}-{\frac{1}{240}}z-{\frac{1}{1008}}{z}^{2}+{\frac{1}{3024}}{z}^{3}+O\!\left({z}^{4}\right),
d \big( \Psi(y_1,s),\Psi(y_2,t) \big) = d \big({\tilde{F}}_0(y_1),{\tilde{F}}_{t-s}(y_2) \big) &\le d \big({\tilde{F}}_0(y_1),{\tilde{F}}_0(y_2) \big) +d \big({\tilde{F}}_0(y_2),{\tilde{F}}_{t-s}(y_2) \big) &= d_Y(y_1,y_2) +|t-s| \le \sqrt{2 \Big( d_Y^2(y_1,y_2) +|t-s|^2 \Big)}.
(\kappa_1,\lambda_{1,1},\lambda_{1,2})=\max_{\kappa,\lambda_{1},\lambda_{2}}\mathrm{R}\left(f(x)-\frac{c}{\Phi_0(\nu;x)+\frac{\kappa_0}{x^2+\lambda_{0,1}x+\lambda_{0,2}+\frac{\kappa}{x^2+\lambda_{1}x+\lambda_{2}}}}\right).