harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|\bar x^*_j + \beta \bar d_j\|^2 - (x^*_{j0}+ \beta d_{j0})^2 = \|\bar x^*_j\|^2 + 2 \beta \bar d_j^T \bar x^*_j + \beta^2 \|\bar d_j\|^2 - \left((x^*_{j0})^2 + 2\beta d_{j0}x^*_{j0}+\beta^2 d_{j0}^2\right)=0,
\phi^*\Big( \frac{\phi(t)}{Lt}\Big)=\sup_{s\ge 0}\Big(s\frac{\phi(t)}{Lt}- \phi(s) \Big)\le \max \bigg\{\frac{\phi(t)}{L}, \sup_{s>t}\Big(s\frac{\phi(t)}{Lt}- \phi(s) \Big)\bigg\}.
\log\biggl(\frac{(d-1)^{d-1}}{d^{d/2}}\biggr) =\left(1+2x/n\right)\log\left(1+\frac{2x}{n}\right)-\left(1+x/n\right)\log\left(2+\frac{2x}{n}\right) =-\log2+\left(1-\log2\right)\frac{x}{n}+\frac{3}{2}\left(\frac{x}{n}\right)^{2}+O\left(\left(\frac{x}{n}\right)^{3}\right)
D^m r = A_c D^mk_c + Dg_c(K{}{})D^mK{}{}+D^mg_c(K{}{})\left(DK{}{}\right)^{\otimes m}+ \mathcal{P}_m(g_c,K{}{}) \quad- Dk_c(R)D^mr - D^mk_c(R)\left(DR \right)^{\otimes m}- \mathcal{P}_m(k_c,R) &=f_{m,1}+ Dg_c(K{}{})D^mK{}{}+ Dk_c(R)D^mr,
T r ( W [{\cal C}] ) = T r ( e^{- \oint_{\cal C}A_{\ 2}^{0}( x^{2}) T_{0}d x^{2}}) = T r ( e^{\oint_{\cal C}\frac{4 \pi}{k}J^{0}( x^{2}) T_{0}d x^{2}}) = 2 \cos ({\frac{2 \pi J_{0}^{0}}{k}}) .
M_{n+1}(\xi;\underline{\eta}): =\frac{1}{n+1}\big[\widetilde{M}_{n+1}(\xi;\underline{\eta})+\sum_{j=1}^n\widetilde{M}_{n+1}(\xi;\pi_j(\underline{\eta}))\big] &=\frac{-i(\xi-\eta_1-\ldots-\eta_{n+1})}{n+1}\int_{\mathbb{R}}e^{-i\xi\rho}\prod_{l=1}^{n+1}\frac{e^{i\eta_l\rho}-1}{\rho}\,d\rho,
\boldsymbol{\alpha}=\frac{\boldsymbol{\pi}D}{\boldsymbol{\pi}D \mathbf{1}}\boldsymbol{\pi}=\frac{\boldsymbol{\alpha}(-C)^{-1}}{\boldsymbol{\alpha}(-C)^{-1}\mathbf{1}}=\lambda^*\boldsymbol{\alpha}(-C)^{-1},
\begin{array}{lcl}\tilde{\mathcal L}_{-2}^{(-3,-1)}={v_{1}\star p}^{-3}+v_{0}\star p^{-2}+v_{-1}\star p^{-1}&= v_{-1}{p}^{-1}+(v_{0}+\theta v'_{-1})p^{-2}+(v_{1}+2\theta v'_{0}+\theta^{2}u''_{-1})p^{-3}\end{array}
{\frac{1}{\pi \Delta_{\alpha \beta}}}\left[ 1 -{\frac{1}{3}}( \pi \Delta_{\alpha \beta})^{2}\right] ={\frac{1 + ( 2 \pi \alpha^{\prime}B )^{2}q_{\alpha}q_{\beta}}{( 2 \pi \alpha^{\prime}B ) ( q_{\alpha}- q_{\beta})}}~ ,
P\left(\frac{M(t)}{t}>\sqrt{2\beta m}+\varepsilon\right) =P\left(\frac{M(t)}{t}>\sqrt{2\beta m}+\varepsilon\mid \mathcal{E}^c\right)P(\mathcal{E}^c) &+P\left(\frac{M(t)}{t}>\sqrt{2\beta m}+\varepsilon;\ \mathcal{E}\right).
E_F^0:= F^0 \cup F^1 \cup \Gamma_F, E_F^1:= \left\{(e,f)\in F^1\times F^1: s(f)\subseteq r(e) \right\}&\cup \left\{(e,[v])\in F^1\times F^0: [v]\subseteq r(e) \right\}&\cup \left\{(e,\omega)\in F^1\times \Gamma_F: \omega_i=1 \mathrm{~~whenever~~}e=e_i \right\},
C_{j_{1}, m_{1}, j_{2}, m_{2}}^{j_{3}, m_{3}}= \frac{N ( j_{3}, m_{3})}{N ( j_{1}, m_{1}) N ( j_{2}, m_{2})}\frac{j_{2}m_{1}- j_{1}m_{2}}{\sqrt{j_{3}( j_{3}+ 1 )}}
\delta \Omega_{\alpha}^{\Lambda}= 2 \gamma^{\mu}\nabla_{\mu}X^{\Lambda}\epsilon_{\alpha}+{\frac{1}{2}}\gamma^{\nu \lambda}{\cal F}_{\nu \lambda}^{+ \Lambda}\epsilon_{\alpha \beta}\epsilon^{\beta}+ 2 i \gamma^{\mu}A_{\mu}\epsilon_{\alpha}\ .
M_{\mu\nu}=-\frac{\hbar^{2}}{2m}\int_{z>z_{0}}({\psi_{\nu}^{T}}^{*}\frac{\partial^{2}}{\partial z^{2}}\psi_{\mu}^{S}-\psi_{\mu}^{S}\frac{\partial^{2}}{\partial z^{2}}{\psi_{\nu}^{T}}^{*})dxdydz,
\mathcal{S}= \left\lbrace \mathbf{A}\in \mathbb{C}^{N_\mathrm{T}\times N_{\mathrm{RF}}}| \varphi \in \{\frac{2b\pi}{2^{B}}\}_{b=1}^{2^{B}}, a_{mn}=e^{j\varphi}\right\rbrace
\begin{array}{l c l l}{{\mathrm{f l o o r}}}&{{\mathrm{S U ( 2 / 1 )}}}&{{\mathrm{f i e l d}}}&{{\mathrm{h e l i c i t y}}}\\{{\mid \mathrm{g n d}>}}&{{( 0 ~ + \frac{5}{2})}}&{{A_{\mu}}}&{{- 1}}\\{{\mid \mathrm{2 n d}>}}&{{( 1 ~ + \frac{5}{2})}}&{{2 \Psi_{\mu}}}&{{- \frac{3}{2}}}\\{{\mid \mathrm{3 r d}>}}&{{( 0 ~ + \frac{7}{2})}}&{{e_{\mu}^{a}}}&{{- 2 .}}\\ \end{array}
G^{\mu \nu \mu^{\prime}\nu^{\prime}}= g^{\mu \mu^{\prime}}g^{\nu \nu^{\prime}}+ g^{\mu \nu^{\prime}}g^{\nu \mu^{\prime}}-{\frac{2}{D}}g^{\mu \nu}g^{\mu^{\prime}\nu^{\prime}}+ C g^{\mu \nu}g^{\mu^{\prime}\nu^{\prime}}\: .
\frac{\binom{\alpha(\ell,m)}{k-\ell}\binom{\tau-\alpha(\ell,m)}{\ell}\binom{n}{m}}{\alpha(\ell,m)\big(\tau-\alpha(\ell,m)\big)\binom{k}{\ell}}= \frac{n!}{k!}\left(\frac{\binom{\alpha(\ell,m)}{k-\ell}}{\alpha(\ell,m)}\frac{(k-\ell)!}{(n-m)!}\right)\! \left(\frac{\binom{\tau-\alpha(\ell,m)}{\ell}}{\big(\tau-\alpha(\ell,m)\big)}\frac{\ell!}{m!}\right)\!.
\sigma(\sum B_i\mathrm{log}^i(x))A = A\sum B_i\mathrm{log}^i(x) + \delta(A) \sum q^i\mathrm{log}^i(x)\sigma(B_i)A = \sum \mathrm{log}^i(x)AB_i + \mathrm{log}(x)\partial(A), \intertext{whence the equality of $\mathrm{log}^i(x)$ terms for $i=1$:}q\mathrm{log}(x)\sigma(B_1)A = \mathrm{log}(x)(AB_1 +\partial(A)),
a_i = a_{i \bmod n}, a_ia_{i+1}= \frac{1}{64}(a_i+a_{i+1})+s, a_ia_{i+2}= \frac{1}{64}(a_i+a_{i+2}) + \begin{cases}s_2 i\equiv_2 0, s_2^f i\equiv_2 1. \end{cases}
\frac{d^{2}P}{d z^{2}}+ \left[ - \frac{f^{\prime \prime}}{f^{\prime}}+ f^{\prime}\cot f \right] \frac{d P}{d z}+ f^{2}\left[ l ( l + 1 ) - \frac{m^{2}}{\sin^{2}f}\right] P = 0 ~ .
A_0 = \big\{\big( T(e), T(e) \big) \}, \big( T(g^\Uparrow), T({g'}^\Uparrow) \big) \ \leadsto \ \big( T(g), T(g') \big) \textrm{for}g, g' \in G_{L(n+1)}, \g^\Uparrow \leftrightarrow g'^\Uparrow, \, g \leftrightarrow g', \n \geq 0.
A_{0}(\textbf{w}+\bar{\textbf{u}})\partial_{t}{\textbf{w}}+ A_{1}(\textbf{w}+\bar{\textbf{u}})\partial_{x}{\textbf{w}}= A_{0}(\textbf{w}+\bar{\textbf{u}})D(\textbf{u}, \bar{\textbf{u}}){\textbf{w}}.
\Phi ({\bf x}, \theta ) = \chi + \bar{\theta}\lambda + \frac 1 2 \bar{\theta}\theta F , D_{\alpha}= \frac{\partial}{\partial \bar{\theta}_{\alpha}}- ( \hat{\partial}\theta )_{\alpha}, \bar{D}_{\alpha}= \frac{\partial}{\partial \theta_{\alpha}}- ( \bar{\theta}\hat{\partial})_{\alpha}.
f(\alpha^*) = = \bigg[\bigg( \frac{p(1-x)}{x(1-p)}\bigg)^{x}\bigg( x \frac{1-p}{1-x}+ 1-p \bigg)\bigg]^{mt}& = \bigg[\bigg( x \frac{1-p}{1-x}+ 1-p \bigg) \bigg( \frac{p}{x}\bigg)^x \bigg( \frac{1-p}{1-x}\bigg)^x \bigg]^{mt}& = \bigg[\bigg( \frac{p}{x}\bigg)^x \bigg( \frac{1-p}{1-x}\bigg)^{1-x}\bigg]^{mt}
\limsup_{n\to\infty}\frac{1}{n+p\left(\frac{\varepsilon}{4}\right)}\log s \big( n+p\left(\frac{\varepsilon}{4}\right),\frac{\varepsilon}{4}, \overline{V}\big) \ge \limsup_{n\to\infty}\frac{1}{n+p\left(\frac{\varepsilon}{4}\right)}\log s(n,\varepsilon, \overline{U}) & = \limsup_{n\to\infty}\frac{1}{n}\log s(n,\varepsilon, \overline{U}).
\cfrac{\ell_{X_1}(\beta)}{\ell_{X_2}(\beta)}\leq K_4 \cdot \max \{1, cfrac{\ell_{X_1}(\alpha)}{\ell_{X_2}(\alpha)}, cfrac{\ell_{X_1}(\alpha')}{\ell_{X_2}(\alpha')}, cfrac{\ell_{X_2}(\gamma)}{\ell_{X_1}(\gamma)}\}.
U_{S}(v_{S},v_{S}^{\prime})=\int_{u_{B}=\underline{B}}^{\overline{B}}p(v_{S}^{\prime},u_{B})f_{B}(u_{B})du_{B}-v_{S}\int_{u_{B}=\underline{B}}^{\overline{B}}t(v_{S}^{\prime},u_{B})f_{B}(u_{B})du_{B}
U_{2}(t) = A_{2}V_{m+1}(t) + \int_t^{t_0}G_{2}(t, s) V_{m+1}(s) \, ds + \int_t^{t_0}H_{2}(t, s) U_{2}(s) \, ds \mbox{for}t_2 \le t < t_1,
\mathbb{W}_{\infty}:=\begin{cases}\left(\left(\mathbb{W}\oplus \mathbb{W}^{\ell,\left(\frac{-}{\ell}\right)}\right)\otimes_{\Lambda}\mathcal{R}\right)[f_\infty] \textrm{if $r=1$ and $\psi_\ell$ is the trivial character,}\left(\mathbb{W}\otimes_{\Lambda}\mathcal{R}\right)[f_\infty] \textrm{otherwise.}\end{cases}
\sup_{\delta\leq y\leq -2\delta +L}\|u_{\delta,\epsilon}v_{\delta,\epsilon}\|_{L^2(0,T)}\leq \sup_{\delta\leq y\leq -2\delta +L}\|v_{\delta,\epsilon}\|_{L^2(0,T)}\|u_{\delta,\epsilon}\|_{L^{\infty}(0,T)}&\leq \sup_{\delta\leq y\leq -2\delta +L}\|v_{\delta,\epsilon}\|_{L^2(0,T)}\|u_{\delta,\epsilon}\|_{H^1(0,T)}&\leq 1+\delta^{-1}.
J(p)=\sum_{n=0}^{p-1}\left( \frac{D(n)}{p}\right)= \left( \frac{a}{p}\right) \sum_{n=0}^{p-1}\left( \frac{(n+b/2a)^2-(b^2/4a^2-c/a)}{p}\right) = \left( \frac{a/4}{p}\right) \sum_{n=0}^{p-1}\left( \frac{(2an)^2-\Delta}{p}\right),
P(\chi_{P}^{2}(\{p_{i}\})>T)\sim C\int_{\sum_{i=1}^{m-1}y_{i}^{2}>T}\{\prod_{i=1}^{m-1}dy_{i}\}\prod_{i=1}^{m-1}exp[-\frac{1}{2}(\sum_{i=1}^{m-1}y_{i}^{2})]
\left\|\sum_{j\in\mathbb{S}}\widehat{A}_jU_jy\right\|^p=\left\|\sum_{j\in\mathbb{S}}\widehat{A}_j\widehat{L}_j\left(\sum_{k\in\mathbb{S}}L_kU_ky\right)\right\|^p=\sum_{j\in\mathbb{S}}\left\|\widehat{L}_j\left(\sum_{k\in\mathbb{S}}L_kU_ky\right)\right\|^p=\sum_{j\in\mathbb{S}}\|U_jy\|^p .
F_7(x,y) = 218954190636 x^6 - 547175054385 yx^5 + 500968274391 y^2x^4 - 206106707898 y^3x^3 + 37428532188 y^4x^2 - 2465893775 y^10x + 29807035y^6 G_7(x,y) = 218954190636 x^6 - 620159784597 yx^5 + 659031715782 y^2x^4 - 325814595210 y^3x^3 + 75233403960 y^4x^2 - 7055472502 y^5x + 173231474y^6
\intertext{Combining the remark before of this Theorem with Lemma \ref{thm:2.2}and Proposition \ref{prop}, we get}\sum_{n \le x}J_{k-1}^r(n) =x^k\sum_{d^r \le x}\frac{\mu(d)}{d^{rk}}+\Omega(x^{k-1}) &=\frac{x^k}{k\zeta(rk)}+\Omega(x^{k-1}).\intertext{This proved the lemma.}
Cl_{2m}(\frac{q\pi}{p})=\frac{1}{(2p)^{2m}(2m-1)!}\sum_{j=1}^{p}sin(\frac{qj\pi}{p})[\psi_{2m-1}(\frac{j}{2p})+(-1)^{q}\psi_{2m-1}(\frac{j+p}{2p})]
\frac{\cos (\eta - \frac{1}{2})x \cos (\eta' -\frac{1}{2})y}{2 \cos \frac{1}{2}x \cos \frac{1}{2}y}-\frac{\cos^2 (\eta - \frac{1}{2})x}{2 \cos^2 \frac{1}{2}x}= O(|\eta - \eta'|+ |x-y| ).
\begin{array}{l}-\beta^2\cdot A_{hk}=-(\alpha_h^2+\alpha_k^2)A_{hk}+2\alpha_h\alpha_kJ'A_{hk}J -\beta^2\cdot B_{hk}=-(\alpha_h^2+\alpha_k^2)B_{hk}+2\alpha_h\alpha_kJ'B_{hk}J.\end{array}
\sum_{j=1}^{\infty}\frac{A_{q}(qz_{j})}{z_{j}A_{q}'(z_{j})}T_{n}\left(\pm q^{-1/2}z_{j}^{-1/2}\right)T_{m}\left(\pm q^{-1/2}z_{j}^{-1/2}\right)=-2q^{n(n-1)/2}\delta_{mn},
\begin{aligned}d_X(F(x,y),F(u,v))\leq a\d_X(x,F(u,v))+ \b\d_X(u, F(x,y))+c\d_X(x,u); & \forall x\geq_{P_1}u,\y\leq_{P_2}v;\ 2b+c<1\end{aligned}
\zeta_i=\begin{cases}1 d_i=1, d_{i,S_1^*}+d_{i,S_3^*}+d_{i,V_1}i\in S_1^*, d_{i,V_1}i\in S_2^*, d_{i,S_1^*}+d_{i,V_1}i\in S_3^*.\end{cases}
\det \left( M'_{\Xi, \{x_1,x_3\}}\right)|_{\Pi^*}= l_1 l_3 x_1^{l_1}x_2^{l_2}x_3^{l_3-1}, \ \det \left( M'_{\Xi, \{x_2,x_3\}}\right)|_{\Pi^*}= l_2 l_3 x_1^{l_1+1}x_2^{l_2-1}x_3^{l_3-1}.
= \sum_{p\in a\cap\gamma}\delta_{y,w}\varepsilon(p;a,\gamma)\varepsilon(y;\gamma,\eta)\left|\gamma_x^p a_p\gamma_p^y\right|\left|\eta\right| &+\sum_{p\in a\cap\gamma}\delta_{x,z}\varepsilon(p;a,\gamma)\varepsilon(x;\gamma,\eta)\left|\eta\right|\left|\gamma_x^p a_p\gamma_p^y\right|,
\mathcal{Q}=\left\{\mathbf{q}_{\tilde{v}}\left| \mathbf{q}_{\tilde{v}}=\left[ f_{\tilde{v}}(-N_{\rm cp}), f_{\tilde{v}}(-N_{\rm cp}+1), \ldots, f_{\tilde{v}}(N-1) \right]^{\rm T}, \tilde{v}\in\mathcal{U}_{2V}\right. \right\}.
\int_{\partial\Omega}e^{-i\beta(\lambda z-\frac{\overline{z}}{\lambda})}[\frac{\partial u}{\partial w}+\beta(\lambda\frac{dz}{ds}+\frac{1}{\lambda}\frac{d\overline{z}}{ds})u]ds=0,\lambda\in\mathbb{C}\backslash\{0\}.
T(\mathcal{M}) \coloneqq (\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{M})^{\varphi = 1}\mathcal{M}(T) \coloneqq (\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}\otimes_{\mathbf{Z}_p}\mathcal{T})^{G_{\widetilde{L}_{\infty}}}.
\boldsymbol{D}^{-\sigma}u(x) := \dfrac{1}{\Gamma(\sigma)}\int_{-\infty}^{x}(x-s)^{\sigma -1}u(s) \,{\rm d}s, \boldsymbol{D}^{-\sigma *}u(x) := \dfrac{1}{\Gamma(\sigma)}\int_{x}^{\infty}(s-x)^{\sigma-1}u(s) \,{\rm d}s,
\mathcal{B}_v(\lambda) \otimes \mathcal{B}_w(\mu) = \bigsqcup_{\pi \in \mathcal{B}_w(\mu)^\lambda}\left( \bigcup_{a_1, \ldots, a_l \geq 0}\tilde{f}_{i_1}^{a_1}\cdots \tilde{f}_{i_l}^{a_l}( C(\pi, e) ) \setminus \{0 \}\right).
{\frac{\mathrm{C o t}( \pi z )}{i}}={\frac{2 e^{i \pi z}}{e^{i \pi z}- e^{- i \pi z}}}- 1 ={\frac{e^{i \pi z}+ e^{- i \pi z}}{2 i \quad \mathrm{S i n}( \pi z )}}\quad ,
\begin{array}{rl}SU(2N_{f})^3 2N_c d^{(3)}(2N_f), SU(2N_{f})^2 U(1)_R 2\frac{N_c'(N_c'-1)-N_c(N_c+1)}{N_f}d^{(2)}(2N_f), U(1)_R^3 N_c(2N_c+1)+N_c'(2N_c'-1) +4\frac{[N_c'(N_c'-1)-N_c(N_c+1)]^3}{N_f^2N_c^2}-4\frac{N_c'(N_c'-1)^3}{N_c^2}, U(1)_R -N_c(2N_c+3)+N_c'(2N_c'-1).\end{array}
E_{g_{1}}=V_{c}^{*}V_{ab}V_{ab}^{*}V_{c}= V_{c}^{*}V_{b}(V_{a}V_{a}^{*})V_{b}^{*}V_{c}\leq V_{c}^{*}V_{b}V_{b}^{*}V_{c}\leq E_{g_{2}}.
{{\mathbf{\Omega}}_{pF}}= \{\left.{\left({{{\sin}^2}{\theta_1}, \cdots ,{{\sin}^2}{\theta_k}}\right)}\right| \sum\limits_{i = 1}^k{{{\sin}^2}{\theta_i}}\leq{\delta^2},{{\sin}^2}{\theta_k}\leq \cdots \leq{{\sin}^2}{\theta_1}\leq 1 \}.
p_1 = c_d\int_{|y|< \frac{3}{2}r}\partial_{y_i}\bigg( \frac{1}{|x-y|^{d-2}}\bigg) \phi u^j \partial_{y_j}u^i dy; p_2 = c_d\int_{\frac{3}{2}r \leq |y|<\rho}\partial_{y_i}\bigg( \frac{1}{|x-y|^{d-2}}\bigg) \phi u^j \partial_{y_j}u^i dy.
M_h(x) = \sum_{n \leq x}\big( P_n(x) - D(x) \big)^h = \sum_{n \leq x}\big( \log 2 \cdot F_{\omega_0}(n) + S_1 + S_2 \big)^h &= \sum_{\substack{h_0,h_1,h_2\ge0 h_0+h_1+h_2=h}}\binom h{h_0,h_1,h_2}\sum_{n \leq x}\big( \log 2 \cdot F_{\omega_0}(n) \big)^{h_0}S_1^{h_1}S_2^{h_2},
L_{T}= \sigma^{*}_{2}(t)\left(\Theta_{1}\dot{\Theta}^{1}+P_{1}\dot{Q}^{1}-H_{T}\right) = P_{1}\dot{Q}^{1}-\frac{1}{2}(Q^{1})^{2}-\frac{1}{2}(P_{1})^{2}+\frac{1}{2}(\Theta^{1})^{2}+\frac{1}{2}(\Theta_{1})^{2}, &\therefore L_{T}=P_{1}\dot{Q}^{1}-\frac{1}{2}(Q^{1})^{2}-\frac{1}{2}(P_{1})^{2}+{constant},
\Omega_{1}={\frac{B_{1}D_{1}}{C_{1}B_{2}}}+ \theta_{1}{\frac{D_{2}}{C_{1}}}\ \mathrm{a n d}\ \tilde{\Omega}_{1}={\frac{B_{2}D_{1}}{C_{2}B_{1}}}+ \theta_{1}\Gamma{\frac{D_{2}}{C_{2}}}
\{\sigma(t),\sigma(u)\}=-2(\xi_3({\mathbf{x}}_2,{\mathbf{x}}_6,t)\xi_5({\mathbf{x}}_2,{\mathbf{x}}_6,u)-\xi_3({\mathbf{x}}_2,{\mathbf{x}}_6,u)\xi_5({\mathbf{x}}_2,{\mathbf{x}}_6,t)).
y_k^n e_k = \sum_{p=0}^{n-1}(-1)^{n-1+p}\sum_{\gamma}E_{\gamma}(x_1, \ldots, x_{k-1}) E_{n-p-\gamma}(x_k, \ldots, x_n) . y_k^p e_k &= \sum_{p=0}^{n-1}(-1)^{n-1+p}E_{n-p}(x_1, \ldots, x_n). y_k^p e_k
\varphi_{n}(x_{1}+x_2+\cdots+x_{l})= \sum_{\substack{k_{1}, k_2,\ldots, k_{l}\geq 0 k_{1}+k_2+\cdots +k_{l}=n}}\binom{n}{k_{1}, k_2,\ldots, k_{l}}\prod_{t=1}^{l}\varphi_{k_{t}}(x_{t})
\int_0^1 f(s)cos(N\pi s) ds = \sqrt{2}\sum_{p=1}^{\infty}f_p \int_0^1 sin(p\pi s)cos(N\pi s)ds &= \sqrt{2}\sum_{p \neq N}\dfrac{p}{p^2-N^2}\big[ 1+(-1)^{p+N+1}\big]f_p
\Psi(x)=\left(f(x)+{\cal F}^{\mu}_-(x)\widehat{b}{}^+_\mu+\epsilon_{\mu\nu\lambda}{\cal F}{}^\mu_+(x)\widehat{b}{}^{+\nu}\widehat{b}{}^{+\lambda}+\tilde{f}(x)\epsilon_{\mu\nu\lambda}\widehat{b}{}^{+\mu}\widehat{b}{}^{+\nu}\widehat{b}{}^{+\lambda}\right)|0\rangle,
\sum_k \lambda_k^{2\beta}\frac{1}{(1+\lambda_k\Delta t)^{2n}}{=}\sum_k \lambda_{k}^{-\frac12-2\kappa}\big|(-A)^{\frac14+\kappa+\beta}S_{\Delta t}^n e_k|_{L^2}^{2}\le C_\kappa t_{n}^{-\frac12-2\kappa-2\beta},
[g_1,g_2] =h_1(k_1h_2k_1^{-1})\left( (k_1k_2k_1^{-1})h_1^{-1}(k_1k_2^{-1})\right)\cdot\left( (k_1k_2k_1^{-1}k_2^{-1}) h_2^{-1}(k_2k_1k_2^{-1}k_1^{-1})\right)\cdot (k_1k_2k_1^{-1}k_2^{-1}) &=[h_1,h_2][h_2h_1h_2^{-1},k_1][k_1,h_2h_1][h_2h_1,k_1k_2k_1^{-1}][k_1k_2k_1^{-1},h_2][h_2,[k_1,k_2]][k_1,k_2].
\widehat{f}(y;s)=\prod_{i=1}^m y_i^{-\lambda_{m+1-i}+m+1-i-n \beta}\sum_{\alpha\in Q^+}\widehat{c}_\alpha(s) y^\alpha, &\widehat{\mathcal D}_y \widehat{f}(y;s)=\sum_{i=1}^m \langle \widehat{\alpha}s_i^{-1};\widehat{\alpha}\widehat{t}^{m-i}\rangle \widehat{f}(y;s).
\frac{R_{x}}{R_{s}}=\frac{R_{2}}{R_{1}}+\frac{R_{par}}{R_{s}}\cdot\frac{R_{1}^{\prime}}{R_{1}^{\prime}+R_{2}^{\prime}+R_{par}}\cdot(\frac{R_{2}}{R_{1}}-\frac{R_{2}^{\prime}}{R_{1}^{\prime}})
R_{\mu \nu}-{\frac{1}{2}}g_{\mu \nu}~^{( 4 )}R = - \ell_{P}^{2}\left( \frac{\delta \stackrel{\sim}{\cal L}_{m}^{\prime}}{\delta g^{\mu \nu}}-{\frac{1}{2}}g_{\mu \nu}\stackrel{\sim}{\cal L}_{m}^{\prime}\right) - \ell_{P}^{2}e^{- 1}\frac{\delta{\cal L}_{g}}{\delta g^{\mu \nu}}.
\sum_{u\sim U}T_2(u) =P^{-2}\sum_{p_1\neq p_2}\sum_{u\sim U}\left|\sum_{k,\ell,v}\left(\dfrac{m_1u(m_2k^2v-\ell q)}{p_1p_2}\right)\right| &\leq \sum_{u\sim U}\max_{\substack{p_i\nmid 2m_1m_2u \widehat{P}< p_1 < p_2\leq 2\widehat{P}}}\left|\sum_{k,\ell,v}\left(\dfrac{(m_2k^2v-\ell q)}{p_1p_2}\right)\right|.
\sum_{m = 0}^{\infty}\frac{q^{m ( m + \sigma )}}{( q )_{m}}= \frac{1}{( q )_{\infty}}\sum_{j = - \infty}^{\infty}\left\{q^{1 0 j^{2}+ ( 1 - 4 \sigma ) j}- q^{1 0 j^{2}+ ( 1 1 - 4 \sigma ) j + 3 - 2 \sigma}\right\}\qquad \sigma = 0 , 1 ,
W^0_{l+2}(p,p_1,\dots,p_{l+1})= K_p(q,\bar q)\sum_{k=0}^{l-1}W^0_{k+2}(q,\dots,p_{k+1})W^0_{l-k+1}(\bar q,p_{k+2},\dots,p_{l+1}) =\underset{t_1\in Y^p,t_2\in Y^q, p+q+1=l}{\sum}t_1\vee t_2
\begin{array}{rcl}\displaystyle \sum_{J<I}(-1)^{|J|}a_2({\overline{\sigma_J}}) = \displaystyle\sum_{J<I}(-1)^{|J|}a_2(\overline{{({\bf 1}_{\mathcal{G}\cup \mathcal{R}})_J}}) &= (-1)^{|I|}\{|x_{I}| a_2(U_{T_{I}^\epsilon}) + a_2(U_{\mathcal{G}'}) \}&= - (x_{I}+ x_{i_1i_2}x_{i_1i_3}+ x_{i_1i_3}x_{i_2i_3}). \end{array}
\begin{aligned}D_8 = 8 p^{9m}(|\nu_1\nu_3'p^{2m}|_p^{-1},p^{\ell+m})^{1/2}(|\nu_2\nu_2'p^{2m}|_p^{-1},p^{\ell+m})^{1/2}(|\nu_3\nu_1' p^{2m}|_p^{-1},p^{\ell+m})^{1/2}& \cdot (\ell+m+1)^3 (\varrho+m+1)(r+m+1)^2(\sigma+m+1)^2.\end{aligned}
{\rm ind}_{-\varGamma}(U) = (\dim \mathcal{T}_-(U,\varGamma)- \dim \mathcal{B}_-(U,\varGamma)) & - (\dim \mathcal{T}_+(U,\varGamma)- \dim \mathcal{B}_+(U,\varGamma)) & = -{\rm ind}_\varGamma (U).
H_e=1, \,\,\, \,\,\, H_rH_s=H_{rs}, \,\,\, \,\,\, r, s\in \mathbb{S}_4, H_j a_i = -a_{jij}H_j, \,\,\, \,\,\, i, j\in \mathcal{O}^4_2, a^2_{(12)}=0, a_{(12)}a_{(34)}+ a_{(34)}a_{(12)}= \Gamma(1 - H_{(12)}H_{(34)}), a_{(12)}a_{(23)}+ a_{(23)}a_{(13)}+ a_{(13)}a_{(12)}= \Lambda(1 - H_{(12)}H_{(23)}),
X_{2j-1}=\frac{\partial}{\partial x_{2j-1}}- \frac{1}{2}x_{2j}\frac{\partial}{\partial z}, \; \; X_{2j}= \frac{\partial}{\partial x_{2j}}+ \frac{1}{2}x_{2j-1}\frac{\partial}{\partial z}, \; \; Z= \frac{\partial}{\partial z},
\widehat{K''}_\mathrm{a-cpt-d}^\mathbb{R}=\sum_{k''=0}^{K_l}k''p_{K''|\widehat{K'}}(k'')=\sum_{k''=0}^{K_l}k''\binom{\widehat{K'}+k''}{k''}\xi^{k''}(1-\xi)^{\widehat{K'}},
\oint_{0}\frac{d \zeta}{\zeta^{n - 2}}\chi - \oint_{\infty}\frac{d \zeta}{\zeta^{n - 2}}\chi = \frac{1}{2 k_{1}}\oint_{\Gamma_{1}}\frac{d u}{\zeta^{n - 2}( u )}\log \frac{\rho ( u )}{\xi ( u )},
g_i(x;a,b) =\frac{f^{(i)}(A+x+2a\sqrt x)-f^{(i)}(A+x-2a\sqrt x)}{\sqrt{x}}&=\frac{1}{\sqrt{x}}f^{(i)}(A+x+2a\sqrt xs)\bigg|_{s=-1}^{s=1}&=\frac{1}{\sqrt{x}}\int_{-1}^1f^{(i+1)}(A+x+2a\sqrt xs)2a\sqrt x\,ds &=2a\int_{-1}^1f^{(i+1)}(A+x+2a\sqrt xs)\,ds,x>0.
S = \int d^{4}x \sqrt{g}\left({\frac{m^{8}v_{1}v_{2}v_{3}}{( 2 \pi )^{7}\lambda^{2}}}R +{\frac{1}{4}}{\frac{m_{s}^{2}v_{2}}{( 2 \pi )^{3}\lambda}}F^{2}+{\frac{1}{4}}{\frac{m_{s}^{2}v_{1}}{( 2 \pi )^{3}\lambda}}\tilde{F}^{2}+ \dots \right)
2g(y^*)-g(y^+)-g(y^-) =2G(\xi(-T^*))-g(y^+)-g(y^-)+2g(y^*)-2G(\xi(-T^*)) &\leqslant C|y^--y^+|^2+2g(y^*)-2G(\xi(-T^*)) &\leqslant C|h|^2+2g(y^*)-2G(\xi(-T^*)).
L = \! \int d^{2}x \left( i \, \psi^{\dagger}\dot{\psi}\! + \! i \, \chi^{\dagger}\dot{\chi}\! - \!{\frac{1}{2}}| \nabla \psi |^{2}\! - \!{\frac{1}{2}}| \nabla \chi |^{2}\! - \!{\frac{v}{4}}[ \psi^{\dagger}\! * \! \psi^{\dagger}\! + \! \chi^{\dagger}\! * \! \chi^{\dagger}] \! * \! \left[ \psi \! * \! \psi \! + \! \chi \! * \! \chi \right] \right) ,
\frac{1}{\mathrm{M}_p(a,b)}=(\max\{a,b\})^{-1}\sum_{k=0}^\infty \prod_{i=0}^{k-1}\frac{(\frac1p+i)^2}{\frac2p+i}\frac{1}{k!}\left[1-\left(\frac{\min\{a,b\}}{\max\{a,b\}}\right)^p\right]^k,
\int \frac{dw}{2 \pi}e^{-iw(t'-t)}\frac{1}{2 \bar{k}}\left( e^{i \bar{k}|x'-x|}+K_a(w) e^{-i \bar{k}(x'+x)}\right), ~~~~ \bar{k}=\sqrt{w^2-m_a^2}
\textbf{p}_{j,k}= [\underbrace{p_{j,k}(1),\cdots,p_{j,k}((1-\alpha) \tau)}_{\textrm{TM Pilots}}, \underbrace{p_{j,k}((1-\alpha) \tau+1),\cdots,p_{j,k}(\tau)}_{\textrm{TS Pilots}}]^T.
\begin{aligned}\||W|^{1/2}u\|_{L^2}^2(t)= \int |W|(t)|u|^2(t)\,dx \leq \int |W_1|(t)|u|^2(t)\,dx+\int |W_2|(t)|u|^2(t)\,dx \leq \|W_1\|_{L^{p/(p-2)}}(t)\|u\|^2_{L^{p,2}_{i,j}}(t)+\|W_2\|_{L^\infty}(t)\|u\|^2_{L^2}(t).\end{aligned}
|u_\xi(x)| = \Big|u_\xi(0)+|x|\int_0^1u_{\xi\xi}(tx)\,dt\Big| &\le |u_\xi(0)|+|x|\int_0^1\frac{C}{\, |tx|^\alpha \,}\,dt &\le |D u(0)|+\frac{C}{\, 1-\alpha \,}\, |x|^{1-\alpha}.
aQ=32( 11, \tfrac{1}{2}(- x_1 + x_3 + x_4 + 2 x_5 + x_6 + x_7-234), x_1 + x_3 + x_4 + x_6 + x_7 + x_8-234, &117 - x_3 - x_5 - x_8,\tfrac{1}{2}(x_1 + x_3 + 3 x_4 - x_6 - x_7), 2 x_1 - x_3 + x_5, &x_1 + x_3 - x_4 + x_6 - x_7, x_1 + x_3 - x_4 - x_6 + x_7,-2 (x_1 + x_3) + x_8, & 351 - 3 x_1 - x_4 - x_5 - x_6 - x_7 - x_8).
\begin{array}{l l}{z = \displaystyle \frac{1}{\sqrt{m^{*}}}\left( \sqrt{B}\big ( x_{1}+ i x_{2}\big ) + \frac{1}{\sqrt{B}}\big ( - i p_{1}+ p_{2}\big ) \right) , \hfill}\\{w = \displaystyle \frac{1}{\sqrt{B}}\big ( - i p_{1}- p_{2}\big ) , \hfill}\\ \end{array}
\mathbf{\hat{y}}_{k}=\mathbf{U}_{k}^{H}\left(\mathbf{y}_{k}-\mathbf{H}_{k}^{H}\mathbf{W}\mathbf{H}_{k}\mathbf{s}_{k}\right)=\mathbf{\mathbf{\hat{y}}}_{k,S}+\mathbf{\mathbf{\hat{y}}}_{k,\sigma},
\hat{\mathbb{E}}[|\hat{\mathbb{E}}_{\tau+}[X]-\hat{\mathbb{E}}_{(\tau\wedge T)+}[XI_{\{\tau\leq T\}}]|] =\hat{\mathbb{E}}[|\hat{\mathbb{E}}_{\tau+}[X]-\hat{\mathbb{E}}_{(\tau\wedge T)+}[XI_{\{\tau\leq T\}}]|I_{\{\tau> T\}}] &\leq C_Xc({\{\tau> T\}})\rightarrow 0,\ \ \ \ \T\rightarrow\infty.
\partial_2\varphi =\frac{\hat{H}^{+}_2\dot{H}^{+}_{N}+\hat{H}^{-}_2\dot{H}^{-}_{N}+(\hat{H}^{+}_2\partial_1\hat{H}^{+}_{N}-\hat{H}^{-}_2\partial_1\hat{H}^{-}_{N})\varphi}{(\hat{H}^{+}_2)^2+(\hat{H}^{-}_2)^2}&=\hat d_1\dot H^+_N+\hat d_2\dot H^-_N+\hat d_3\varphi\,,\quad\mbox{on}\,\,\,\Gamma_T\,,
\Phi_{tt}(0,c_0) = \int_\Omega \{f_{uu}(x,c_0) (1 + v_t (0,c_0))^2 + f_u(x,c_0) v_{tt}(0,c_0) \}& + \int_{\partial \Omega}b \{g''(c_0) (1 + v_t (0,c_0))^2 + g'(c_0) v_{tt}(0,c_0) \}.
A = \{z:z^q=-x^q\mbox{and}z^r=x^r\mbox{for}\{q,r\}\in\mathfrak{E}J',~q\in J'\}, B = \{z:z^q=x^q\mbox{and}z^r=x^r\mbox{for}\{q,r\}\in\mathfrak{E}J',~q\in J'\}.
\sum_{n = 0}^{\infty}\frac{n !}{( n + \ell ) !}( - 1 )^{n}L_{n}^{\ell}( x ) L_{n}^{\ell}( y ) \, t^{n}= \frac{( x y t )^{- \ell / 2}}{1 + t}e^{t ( x + y ) / ( 1 + t )}J_{\ell}( 2 \sqrt{x y t}/ ( 1 + t ) ) \, ,
NLP = \int_0^T \int \langle D \rangle^{N-1}u_0^3 \langle D \rangle^{N-1}\partial_z \partial_i \Delta^{-1}\left[ (u^j_0 \partial_j u^i_0) + ( u^j_{\neq}\partial_j u^i_{\neq})_0 \right] & = NLP_0 + NLP_{\neq}.
\alpha^{a}= \sqrt{\frac{m}{2}}\left[ \hat{A}^{a}+ \frac{i}{m}\hat{\pi}^{a}\right] , \{\alpha^{a}}^{\dagger}= \sqrt{\frac{m}{2}}\left[ \hat{A}^{a}- \frac{i}{m}\hat{\pi}^{a}\right] ,
X=\sqrt{{(\frac{R_{E}+y_{obs}}{y_{atm}})}^{2}cos^{2}z+\frac{2R_{E}}{y_{atm}^{2}}(y_{atm}-y_{obs})-{(\frac{y_{obs}}{y_{atm}})}^{2}+1}-\frac{R_{E}+y_{obs}}{y_{atm}}cosz.
\sum_{k=0}^{\infty}\frac{\binom{3k}{k}\binom{6k}{3k}}{x^k}(6H_{6k}-3H_{3k}-2H_{2k}+H_k)=\bigg(\log\frac{x}{x-432}\bigg)\sum_{k=0}^{\infty}\frac{\binom{3k}{k}\binom{6k}{3k}}{x^k},
\& W^c_{(k)}:= \frac{1}{\lambda^2N_{\Lambda}^2}\psi_{(k_1)}\Phi_{(k)}k_1, \ \k\in \Lambda_u \cup \Lambda_B, D^c_{(k)}:=\frac{1}{\lambda^2N_{\Lambda}^2}\psi_{(k_1)}\Phi_{(k)}k_2, \ \k\in \Lambda_B.
\Psi_{\alpha}^{-}=|i,k_{1}...k_{n}\rangle=C_{0}|f,0\rangle+\sum_{m=1}^{\infty}\int d^{4}p_{1}...d^{4}p_{m}C_{m}(p_{1}...p_{m})|f,p_{1}...p_{m}\rangle,
\bar F_{\lambda}(\zeta,\bar{\Lambda}) = ka_0- a_1 \sigma_1(\varepsilon,\zeta)^2 \bar\Lambda_1^2- a_1 \sigma_1(\varepsilon,\zeta) (\bar\Lambda')^T Q(\varepsilon,\zeta) \bar\Lambda' & +\sigma_1(\varepsilon,\zeta)^2\, \mathcal Poly_4(\varepsilon,\zeta,\bar \Lambda)+ \theta_\lambda(\zeta,\bar \Lambda) ,
\dim_F M_{(2a,b,c)}= \dfrac{2^nn!}{4^aa!2^{b+c}(b+c)!}\times \binom{b+c}{b}&= \dfrac{2^{2a}\times (2a)!}{4^aa!}\times \dfrac{n!}{(2a)!b!c!}&= \dim_F FS \times [C_2 \wr S_n: C_2 \wr S_{(2a, b, c)}].
-\int_0^s \Theta( u ,x) f'(u) du+ f(s){\Theta}(s,x)-f(0){\Theta}(0,x) = -\alpha \int_0^{s}\int_0^1 f(u)A(x,y)\Theta(u,y)dydu + \alpha \beta \int_0^s f(u) d\xi_1(s,x).
\mathbb{E}\left[ \Vert x_n-x\Vert_2^2\right] =\mathbb{E}\Big[\sum_{j=1}^{d}(\langle b_j,f\rangle_n-\langle b_j,f\rangle)^2 \Big] =\frac{1}{n}\sum_{j=1}^{d}\operatorname{Var}(b_j(X)f(X))\leq \frac{\varphi^2d}{n}\Vert f\Vert^2,
A_{3}( z , \bar{z}) = \frac{1}{\theta}\partial_{\bar{z}}\left( \frac{\chi_{3}}{\theta}\right) = - \frac{1}{2 \cosh ( l )}\, , \quad \bar{A}_{3}( z , \bar{z}) = \frac{1}{\theta}\partial_{z}\left( \frac{\chi_{3}}{\theta}\right) = - \frac{1}{2 \cosh ( l )}\, .
\begin{array}{r c l}{{X_{a}\to \tilde{X}_{a}}}&{{=}}&{{U ( t ) X_{a}U^{- 1}( t ) \, ,}}\\{{X_{0}\to \tilde{X}_{0}}}&{{=}}&{{U X_{0}U^{- 1}- \dot{U}U^{- 1}\, ,}}\\ \end{array}
\mu^{\prime}= \frac{1}{2}N x^{2}\phi^{2}+ e^{2 \gamma \phi}\left[ N ( K^{2}+ H^{2}) + \frac{1}{8 x^{2}}\left( \left( K^{2}+ H^{2}- 4 \right)^{2}+ 1 2 K^{2}H^{2}\right) \right] \ ,
\begin{aligned}\mathsf{M}^n= \{\mathsf{m}_1,\cdots,\mathsf{m}_n\}= \left\{g\bigl(\xi_0\bigr),g\bigl(\xi_1^{\mathsf{M}}\bigr),\cdots,g\bigl(\xi_{n-1}^{\mathsf{M}}\bigr)\right\}.\end{aligned}