harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Xi^{\prime \prime}+ 3 W^{\prime}\Xi^{\prime}+ 2 \sqrt{2}\frac{b}{\kappa}\left( \frac{\Delta}{8}+ \delta \right) \lambda_{0}^{2}e^{2 W - 2 \sqrt{2}b \kappa \Xi}+ 2 \frac{b}{\kappa}\lambda_{0}e^{W - \sqrt{2}b \kappa \Xi}\delta ( z - z_{0}) = - 4 \sqrt{2}\frac{b}{\kappa}\frac{\delta}{\Delta}\lambda_{0}^{2},
\hat{\Delta}^{(m,n)}_{\texttt{b};\lambda}:= \Biggl(\sum_{\mu\in \Lambda^{(m,n)}_{\texttt{b}}}\left| P_{\texttt{b};\mu}\bigl(\boldsymbol{\xi}^{(m,n)}_{\texttt{b};\lambda};q,q_0\bigr) \right|^2 \delta^{(m,n)}_{\texttt{b};\mu}(q) \Biggr)^{-1}.
\widetilde{S}^{m}_t=X_0 +\int_0^t b\big(r,\widetilde{S}^{m},\overline{\beta}^{m}_r, \widetilde{\gamma}^m_r\big) \mathrm{d}r+\int_0^t \sigma \big(r,\widetilde{S}^{m},\overline{\beta}^{m}_r, \widetilde{\gamma}^m_r\big)\mathrm{d}W_r,
\int_0^1 \left( A_s(N;x)-\mu(C_s)N \right)^2 \d\mu = \sum_{0\le i,j< N}\left( \int_0^1 I_s(T^i x) I_s(T^j x) \d\mu \right) - \mu(C_s)^2 N^2 &=\Sigma_1+\Sigma_2+\Sigma_3-\mu(C_s)^2N^2,
|\alpha_1 q u_1+\cdots+\alpha_d q u_d-p|^{d(\rho)/[K:\mathbb{Q}]}=|\rho(\alpha_1)\rho(q u_1)+\cdots+\rho(\alpha_d)\rho(q u_d)-\rho(p)|_\rho &=|\rho(\alpha_1)q\rho(u_1)+\cdots+\rho(\alpha_d)q\rho(u_d)-p|_\rho.
\Pi(E_{b})(z,w) = \frac{1}{\pi}\frac{\varphi_z(b) \overline{\varphi'_w(b)}- \varphi'_z(b) \overline{\varphi_w(b)}}{z-\bar{w}}&= \frac{1}{\pi}\int_0^{b}\varphi_{z}(t) \overline{\varphi_{w}(t)}dt.
\rho\Lambda^2 \left(\frac{1}{g}-1\right)= - \rho^3\ln\frac{(\Lambda l)^2}{2}+\gamma\rho^3 + \frac{\rho}{l^2}\ln\frac{(\rho l)^2}{4\pi}+ \frac{2\rho}{l^2}\ln\Gamma\left(\frac{(\rho l)^2}{2}\right) + O\left(\frac{1}{\Lambda}\right),
n_{1}=(a-1)(l+1)+br+l(l+1)(r-l-1)-\sum^{l}_{i=2}in_{i}\geq(a-1)(l+1)+br+l(l+1)(r-l-1)-\sum^{l}_{i=2}i(i-1)n_{i}=(a-1)(l+1)+br+l(l+1)(r-l-1)-b(b-1)-l(l+1)(a+2b-2) =-a(l^{2}-1)-b(b-1)-b(2l^{2}+2l-r)+(l+1)(rl-l^{2}+l-1) \geq 0\;,
\overline{J}_p(u,v)= \int_\Omega f(x,u,v, \nabla u) dx +\int_{J_u}K_p(x,0, u^+, u^-,\nu_u)d\mathcal{H}^{N-1}+\int_\Omega f_p^\infty(x,u,0,\tfrac{dDu}{d|D^c u|}) d|D^c u|,
| Y_{\scriptscriptstyle{W}}^{2}( k r ) - J_{\scriptscriptstyle{W}}^{2}( k r ) | \, \sin^{2}\Delta_{\scriptscriptstyle{l}}\leq \left( e^{a | | B | | / \sqrt \pi}+ 1 \right)^{2}\, \frac{( M a / 2 )^{2 W}}{\Gamma^{2}( W + 1 )}\, ( 1 + \mathrm{O}( ( M a )^{2}, ( M a )^{2 W}) ) \, ,
P_{c,\mathrm{N}}(T) \approx\sum_{n=1}^{N_\mathrm{N}}(-1)^{n+1}{N_\mathrm{N}\choose{n}}\times \int_{0}^{\infty}\mathrm{e}^{-\frac{n \eta_\mathrm{N}x^{\alpha_\mathrm{N}}T\sigma^2}{C_\mathrm{N}M_\mathrm{r}M_\mathrm{t}}-W_n(T,x)-Z_n(T,x)}\hat{f}_\mathrm{N}(x)\mathrm{d}x.
Z_{o n e - l o o p}= \int{\frac{d^{2}\tau}{( \mathrm{I m}\tau )^{{\frac{D}{2}}+ 1}| \eta |^{2 4}}}\sum_{\alpha , \beta}C \Big [ \begin{array}{c}{{\alpha}}\\{{\beta}}\\ \end{array}\Big ] \, \prod_{f \in F^{\prime}}\Theta^{1 / 2}\Big [ \begin{array}{c}{{\alpha ( f )}}\\{{\beta ( f )}}\\ \end{array}\Big ] ( \tau ) .
\omega_k =\begin{cases}I \mbox{if}k =1, \cr s_1s_2\cdots s_{k-1}\mbox{if}2 \leq k \leq n,\cr s_1s_2\cdots s_{n-1}s_{n}s_{n-1}\cdots s_{2n-k+1}\mbox{if}n < k \leq 2n.\cr\end{cases}
\sum_{\gamma\in\mathcal{S}_{2}(n+1,0)}w(\gamma)=\sum_{k=0}^{p}a_{-k-q}^{(k)}\Big(\sum_{\gamma\in\mathcal{S}_{2}(n,k)}w(\gamma)\Big)=\sum_{k=0}^{p}a_{-k-q}^{(k)}\langle\mathcal{B}_{q}^{n}e_{k},e_{0}\rangle=\langle\mathcal{B}_{q}^{n+1}e_{0},e_{0}\rangle
\le\mathbb{E}_{X^{n}}\{\frac{1}{M}\sum_{m}Tr\{(I-\Pi_{\rho_{X^{n}(m)},\delta})\rho_{X^{n}(m)}\}+\|\rho_{X^{n}(m)}-\Pi_{\rho,\delta}^{n}\rho_{X^{n}(m)}\Pi_{\rho,\delta}^{n}\|_{1}\}
\lim_{L\to \infty}{\mathbb E}\langle( m^p- \langle m^p \rangle)^2 \rangle=\lim_{L\to \infty}\frac{1}{|{\cal B}_p|^2}\sum_{X,Y \in{\cal B}_p}\mathbb E( \langle \sigma_X\sigma_{Y}\rangle-\langle \sigma_X\rangle \langle \sigma_{Y}\rangle) \leq \lim_{L\to\infty}\frac{1}{\beta\Delta_p \sqrt{|{\cal B}_p|}}=0,
\sum_{r=1}^{a-1}\left(\sum_{i=1}^{k-1}x_i\right)_r= m\left(\sum_{i=0}^{k-1}i\binom{k-1}{i}+\sum_{j=1}^{k-1}\left(\sum_{i=0}^{k-j-1}(i+2)\binom{k-j-1}{i}\right)\right) &\ \ +\left(\sum_{i=0}^{k-2}i\binom{k-2}{i}+\sum_{j=1}^{k-2}\left(\sum_{i=0}^{k-j-2}(i+2)\binom{k-j-2}{i}\right)\right) = m(2^{k-1}k-1)+2^{k-2}(k-1)-1.
y_R(t)= S(t-\tau)z_R(\tau)-\int_\tau^t S(t-s)(i\nu z(s) +\epsilon(\gamma z(s) -\mu \overline{z}(s)))\d s -\tfrac{1}{2}\int_\tau^tS(t-s)(y_R(s)F_\Phi)\d s-i\int_\tau^tS(t-s)(y_R(s)\d W(s)),
4 \omega(T_{\theta}) < \max \left\{u^{\theta}(0,t) - v_{\theta}(0,t) - (2D T_{\theta}+ \delta)t \, \mid \, t \in [T_{\theta},T] \right\}&= u^{\theta}(0,T_{\theta}) - v_{\theta}(0,T_{\theta}) - (2D T_{\theta}+ \delta)T_{\theta}&< 4 \omega(T_{\theta}),
\int_{\mathbb{T}_\alpha}K_1^{k}(x,y)dy =\int_{\mathbb{T}_\alpha}\frac{r(\lambda(x,y,E))}{\sum_{i=0}^{\alpha-1}f_iu_+^2(\alpha k+i,y,\lambda(x,y,E),E)}&=\int_{\mathbb{T}_\alpha}\frac{R_{-}^{2}(\alpha k+\alpha-1,x,\lambda(x,y,E),E)}{\sum_{i=0}^{\alpha-1}f_iu_-^2(\alpha k+i,x,\lambda(x,y,E),E)}r(\lambda(x,y,E))dy &=1,
\frac{i}{2}\int \frac{d^4k}{(2\pi)^4}\,\frac{e^2}{k^2 \Big(1 + (-1)^n k^{2n}/\Lambda^{2n}\Big) (k+p)^2}\to- \frac{1}{2}\int \frac{d^4k}{(2\pi)^4}\frac{e^2}{k^2 \Big(1 + k^{2n}/\Lambda^{2n}\Big) (k+p)^2}.
e_1=\begin{bmatrix}0 0 0 0 0 -1 0 1 0\end{bmatrix},e_2=\begin{bmatrix}0 0 -1 0 0 0 1 0 0\end{bmatrix},e_3=\begin{bmatrix}0 -1 0 1 0 0 0 0 0\end{bmatrix},
\frac{1}{\frac{1}{2}(n_{\alpha}+n_{\beta})+\mathrm{i}(q_\alpha-q_\beta)}= \int_0^\infty d\omega_\alpha \mathrm{e}^{-( \frac{1}{2}(n_{\alpha}+n_{\beta})+\mathrm{i}(q_\alpha-q_\beta) )\omega_\alpha}
\mathrm{P}_{m,1}=\mathrm{P}\left( z_m<\frac{\epsilon_1}{\rho}\right) =e^{-\lambda_c\pi \left(\frac{\rho}{\epsilon_1}\right)^{\frac{2}{\alpha}}}\sum^{m-1}_{k=0}\frac{(\lambda_c\pi)^{k}\left(\frac{\rho}{\epsilon_1}\right)^{\frac{2k}{\alpha}}}{k!}.
\left|\det \left(N_1^{1/3}V\right)\right|\le C^{L+1}(L+1)^{(L+1)/2}\prod\limits_{j=1}^re^{-c(-x_j)^{3/2}}\prod\limits_{j=1}^se^{-c(-x'_j)^{3/2}}\prod\limits_{j=1}^re^{-cy_j^{3/2}}\prod\limits_{j=1}^te^{-c{y'}_j^{3/2}}.
\delta_w F_{u_0}(\eta) (x) = \frac{d}{dr}F_{u_0}(\eta + rw) (x) \Big\vert_{r=0}= \int_{\mathbb{R}^2}DK_2\big( \eta(x) - \eta(y) \big) (w(x)-w(y)) \omega_0(y) dy
\gamma\alpha =R(h)\alpha=R(h)R(\varepsilon_N)\eta_{R(N)}\alpha=R(h)R(\varepsilon_N)RL(\alpha)\eta_P & =R(h\varepsilon_NL(\alpha))\eta_P=R(\varepsilon_ML(\beta))\eta_P=R(\varepsilon_M)RL(\beta)\eta_P=R(\varepsilon_M)\eta_{R(M)}\beta=\beta.
\|F_i(x_i, u) - F_i(x_i,z)\| = \sqrt{\sum_{l=1}^\mathcal{L}\|F_{il}(x_{il},u_l)-F_{il}(x_{il},z_l)\|^2}\leq \sqrt{2}\sqrt{\sum_{l=1}^\mathcal{L}(C_l^2+M_l^2{\rm cap}_{il}^2)}\,\|u - z\|.
\min_{\{\textbf{Q}_k\}}\frac{\frac{1}{\omega}\sum_{k=1}^K P_k(\textbf{Q}_k)+P_c+\kappa\phi\left(B\sum_{k=1}^K\theta_k(\textbf{Q}_k)\right)}{B\sum_{k=1}^K\theta_k(\textbf{Q}_k)}
H(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}}{(q;q)_{n}}=\frac{1}{(q^{2};q^{5})_{\infty}(q^{3};q^{5})_{\infty}}=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+\cdot\cdot\cdot
\Delta \zeta(x) =\tilde\alpha'' (|x|)+\frac{n-1}{|x|}\tilde\alpha'(|x|)= \frac{1}{n}W'(\tilde \alpha(|x|)+(n-1)\tilde\alpha''(\xi), &= \frac{1}{n}W'(\tilde\alpha(|x|)+\frac{n-1}{n}W'(\tilde \alpha(\xi))\leq W'(\zeta(x)), .
v \;=\;w_1\;\dotsb\;w_{\ell-1}\;x{=}v_\ell\;v_{\ell+1}\,\dotsb\;y{=}v_j\; \dotsb\;v_m w \;=\;w_1\;\dotsb\;w_{\ell-1}\;y{=}w_\ell\;w_{\ell+1}\;\dotsb\;x{=}w_k\;\dotsb\;w_m
2 - (K + 1)(1 - X_T)^2 = \frac{K+ 1}{2K + 1}(1 + X_T)^2 - \frac{2}{2K+ 1}\left[ (K + 1) X_T - K \right]^2 &\le \frac{K + 1}{2K + 1}(1 + X_T)^2 .
\bigg(R^{-d}\int_{B_{R}(y)}|R\nabla u|^2\bigg)^\frac{1}{2}\lesssim R^{-d}\int_{B_{2R}(y)}|u-\bar u|\stackrel{|y|=2R}{\le}R^{-d}\int_{B_{4R}(0)}|u-\bar u|.
\frac{1}{2}Ric^X(\partial_r)\cdot -\sum_{i=1}^n\varepsilon_iW_r(s_i)\cdot\widetilde{\nabla}^{h_r}_{s_i}-\frac{1}{2}\partial_r\cdot\sum_{i=1}^n\varepsilon_iW_r(s_i)\cdot W_r(s_i)\cdot = \, \frac{1}{2}J\partial_r\cdot -rh(P,\widetilde{\nabla}^h) + O(r^2).
b(u,u,u) = 0,~~~\,\, u \in \mathrm{V}, & b(u,w,w) = 0,~~~ u \in \mathrm{V}, w \in H^1, & \langle B(u,u), \mathrm{A}u\rangle_{\mathrm{H}}= 0, u \in \mathrm{D}(\mathrm{A}).
\begin{aligned}dI(V;X,\alpha) =\lim_{t\to 0}\frac{1}{t}\Big(\int_{V(t)}\alpha(\xi_{t}x)-\int_{V}\alpha(x)\Big) =\lim_{t\to 0}\frac{1}{t}\Big(\int_{V}\xi_{t}^{\ast}\alpha(\xi_{t}x)-\int_{V}\alpha(x)\Big) =\int_{V}\lim_{t\to 0}\frac{1}{t}\Big(\xi_{t}^{\ast}\alpha(\xi_{t}x)-\alpha(x)\Big) =\int_{V}\mathfrak{L}_{X}\alpha, \end{aligned}
\left\| F^\ast(z){c_1 \choose c_2}\right\|^2 + \left\| \Theta^\ast(z){c_1 \choose c_2}\right\|^2 = |c_1 + c_2 |^2 + | b_1(z)c_1|^2 + | b_2(z)c_2|^2 &\ge (|c_1| - |c_2 |)^2 + \delta^2 \min\{| c_1|^2 , | c_2|^2 \}&\ge \frac{\delta^2}{16}\, ( | c_1|^2 + | c_2|^2 )
\lbrack b+C,\overline{T}_{\Omega}]f =\left( b+C\right) \overline{T}_{\Omega}f-\overline{T}_{\Omega}(\left( b+C\right) f) & =b\overline{T}_{\Omega}f+C\overline{T}_{\Omega}f-\overline{T}_{\Omega}\left( bf\right) -C\overline{T}_{\Omega}f & =[b,\overline{T}_{\Omega}]f.
\int_{(-\epsilon)}\int_{(\epsilon/2)}\left|\zeta(1+w)\widetilde{W}(1-w)\mathcal{H}(z,-w)\frac{e^{\delta z}-e^{-\delta z}}{2\delta z}\right|\,|dz|\,|dw| & \ll \int_{(-\epsilon)}\int_{(\epsilon/2)}|w|^{-98}|z|^{\varepsilon-1}\min\left\{1, \frac{1}{\delta|z|}\right\}\,|dz|\,|dw| \ll \left( \frac{1}{\delta}\right)^{\varepsilon}.
\widetilde G_{n0}\le \sqrt{\dfrac{2}{\pi}}\sum_{l=1}^\infty \left|{\langle{u}_T,{\phi_{2ln}}\rangle}\right|\le \sqrt{\frac{2}{\pi}}\sum\limits_{l = 1}^\infty \frac{\|{u}_T\|_{H^\delta(\Omega)}}{2^{\delta}l^{\delta}n^{\delta}},
\chi_{\alpha \beta}^{\phi}( t , t^{\prime}; k ) = \int \frac{d^{3}k^{\prime}}{( 2 \pi )^{3}}g_{\alpha \beta}^{\phi}( t , t^{\prime}; k^{\prime}) g_{\alpha \beta}^{\phi}( t , t^{\prime}; k^{\prime}+ k ) \, ,
a_{0l}^{1}(\varphi(x)) = \frac{\lambda}{\mu^{l}}a_{0l}^{1}(x)\,\,\,(0 \leq l \leq p), \,\,\,\, \frac{\mu^{l}}{\lambda}a_{0l}^{1}(\varphi(x)) = a_{0l}^{1}(x)\,\,\,(l \geq p+1)
\underline{C}:=\underline{\varphi}^{-1}(\underline{A})=\Gamma_{H,H^*}^{-1}((\varphi^*)^{-1}(\underline{A}))=\Gamma_{H,H^*}^{-1}\bigcap_{s\in\underline{S}}(R^*)^{-s}(W^*)^c=\bigcap_{s\in\underline{S}}\Gamma_{H,H^*}^{-1}(R^*)^{-s}(W^*)^c.
2 5 \times ( 4 0 \times 9 ) = 2 5 \times 4 0 \times 9 = 9 0 0 0 0 ( 2 ) 6 6 \times 2 8 0 \times 5 = 6 6 \times ( \boxed{2 0}\times \boxed{5}) = \boxed{6 6 0 0}
v_1'=\frac{v_1 - c\tanh \theta_1}{1 - \tanh \theta_1 \frac{v_1}{c}}~, ~~~~~v_2'=\frac{v_2 - \tanh \theta_1 \dot\lambda_3}{1 - \tanh \theta_1 \frac{v_1}{c}}~, ~~~~~v_3'=\frac{v_3 + \tanh \theta_1 \dot\lambda_2}{1 - \tanh \theta_1 \frac{v_1}{c}}~.
\int^{\infty}_0\|(-A)^{-\alpha}T(t) x\|^2 dt =\int^{\infty}_0\sum_{n=1}^\infty\left|\left(-\frac{1}{n}+ in\right)^{-\alpha}e^{\left(-\frac{1}{n}+ in\right)t}x_n\right|^2 dt &=\frac{1}{2}\sum_{n=1}^\infty\frac{n^{1+2\alpha}}{(1+n^4)^{\alpha}}|x_n|^2.
(\overline{\rho}(aw_2,w_2a,w_1,w_2) w_1 \overline{\lambda}(aw_2,w_2a,w_1,w_2))^{\omega+1}\quad\quad=\overline{\rho}(aw_2,w_2a,w_1,w_2) w_1 \overline{\lambda}(aw_2,w_2a,w_1,w_2),
d\pi_\mu^\infty(X_j) = -\pi_\mu(w_0)d\pi_\mu(\overline{X}_j)\pi_\mu(w_0), d\pi_\mu^\infty(Y_j) = -\pi_\mu(w_0)d\pi_\mu(\overline{Y}_j)\pi_\mu(w_0), d\pi_\mu^\infty(T) = \pi_\mu(w_0)d\pi_\mu(\overline{T})\pi_\mu(w_0),
\sum_{k=1}^\infty|A_{n_k}f(x)-A_{n_{k-1}}f(x)| \leq \sum_{k=1}^\infty\sum_{j\in J(k)}|A_{m_j}f(x)-A_{m_{j-1}}f(x)|. &= \sum_{j=1}^\infty |A_{m_j}f(x)-A_{m_{j-1}}f(x)|.
\mathcal{M}_{|H^R|}(t) = \mathcal{L}\{f_{|H^R|}\}= \Big(\frac{e^{-(K_h+K_g)}}{(\beta^R)^2\beta_h\beta_g}\Big)^M\cdot \frac{2(K^R_d+1)}{\Omega^R_d}\cdot e^{-K^R_d}t^{-2M-2}+o(t^{-2M-2}).
\texttt{log}_2\prod_{m=1}^M\left(1+\frac{(\widetilde{\textbf{Q}}_k')_{m}(\mathbf{\Lambda}_k)_{m}}{\sigma^2}\right)=\texttt{log}_2\left|\textbf{I}_N+\frac{\beta\textbf{H}_k\textbf{Q}_k^{\star}\textbf{H}_k^H}{\sigma^2}\right|.
\big( S^{\wedge, (l)}\:V\: S^\vee_0 \big)(x,y) &=-2 \sum_{n=0}^\infty \frac{1}{n!}\int_{-\infty}^0 \alpha^{l}\:(\alpha - \alpha^2)^n \: (\Box^n V) \big|_{\alpha y + (1-\alpha) x}\:d\alpha \; S^{\bowtie, (n+l+1)}(x,y) \:.
H_{n}= \int d{\bf x}\int d k d k^{\prime}e^{i k x}e^{i k^{\prime}x}\phi ( k ) \phi ( k^{\prime}) ( k_{0}^{2}- k_{0}k_{0}^{\prime}) \sum_{s + t = n - 1}( i^{2})^{s + t + 1}( k^{2})^{s}( k^{2})^{t}- \int d{\bf x}{\cal L}_{n}
\mathcal{L}_1^{\mu}[\beta b] = -b \mathrm{F_3}+ O(\mu\beta^2), \mathcal{L}_2^{\mu}[\beta b] = -\frac{1}{2}b\mathrm{F}_4 + \frac{\beta^2}{6}b^3 \mathrm{F}_3 + O(\mu\beta^4),
R_2=\lambda^{1r}\lambda^{st}l_{rs}l_t+{\textstyle{1\over2}}\lambda^{1r}\tau^{st}l_{rs}l_t -{\textstyle{1\over2}}\lambda^{1r}\lambda^{su}\nu^{tv}\lambda_{rst}l_ul_v-{\textstyle{1\over 6}}\lambda^{1r}\tau^{su}\tau^{tv}\lambda_{rst}l_ul_v.
0 \geq \sum_{p=1}^n u^{p\bar{p}}G_{p\bar{p}}= \sum_{p=1}^n u^{p\bar{p}}\left ( \frac{\beta_{p\bar{p}}}{\beta}- \frac{|\beta_p|^2}{\beta^2}+ u u_{p \bar p}+ \vert u_p\vert^2 + B\right).
\mathbb{E}\left\lbrace \dot Z^{q}_{\mathbf{H}}(s,\mathbf{x}) \dot Z^{q}_{\mathbf{H}}(t,\mathbf{y}) \right\rbrace = H(2H-1)|t-s|^{2H-2}\displaystyle \prod_{i=1}^{d}(H_{i}(2H_{i}-1) \cdot |x_i-y_i|^{2H_i-2}).
\langle \omega_{n-1}\otimes\omega_{n-2}\otimes\dots\otimes\omega_{0},\tilde\omega_{n-1}\otimes\tilde\omega_{n-2}\otimes\dots\otimes\tilde\omega_{0}\rangle_{F^{\otimes n}}=\prod \langle \omega_{a},\tilde \omega_{a}\rangle_{F}.
I_1 = \{\,1, 2, 3, 4, 5\,\}, I_2 = \{\, 6\,\}, I_3 = \{\, 7, 8\,\}, J_1 = \{\,1 \,\}, J_2 = \{\, 2\,\}, J_3 = \{\, 3\,\}, J_4 = \{\, 4, 5, 6, 7, 8\,\},
\frac{1}{\sqrt{2}\sqrt{\pi}}\int_0^{\infty}\frac{ds}{s^{\frac{3}{2}}}\exp^{-sM_0}\int_0^s ds_1 M(s_1)_1\int_0^{s_1}ds_2 M(s_2)_1 \int_0^{s_2}ds_3 M(s_3)ds_3
w^{(1)}(\eta) = - 2 c_\beta S(\eta)^{-1}\cdot \boldsymbol{\Sigma}^{-1}\boldsymbol{\mu},f^{(1)}({\bf{x}}) ={\bf{x}},w^{(2)}(\eta) = c_\beta S(\eta)^{-1}\cdot\rm{vec}(\boldsymbol{\Sigma}^{-1}),f^{(2)}({\bf{x}}) =\rm{vec}({\bf{x}}{\bf{x}}^\top).
\frac{1}{(t+k+2)(t+k+3)(t+k+4)}+\frac{1}{(t-k)(t-k-1)(t-k-2)}= \frac{1}{(t-k-1)(t-k-2)(t+k+2)}+ \frac{1}{(t-k-2)(t+k+2)(t+k+3)},
\left(\begin{array}{c}x' t'\end{array}\right) =\left(\begin{array}{cc}\frac{1}{\sqrt{1-\beta^2}}\frac{\beta}{\sqrt{1-\beta^2}}\frac{\beta}{\sqrt{1-\beta^2}}\frac{1}{\sqrt{1-\beta^2}}\end{array}\right) \left(\begin{array}{c}x t\end{array}\right)
|u_{R}(\ell) - u_{R}(m)| \leq N_{\ell,m}^{1/2}\bigg( \sum_{i = 1}^{N_{\ell,m}}|D_{\rho_{i}}u_{R}(\ell_{i})|^{2}\bigg)^{1/2}\leq C|\ell - m|^{1/2}\| Du_{R}\|_{\ell^{2}_{\mathcal{N}}(\L)}\leq C C_{*}\lambda |\ell - m|^{1/2}= C_{1}\lambda |\ell - m|^{1/2}.
\delta_\Phi (\psi_1, \ldots,\psi_{n-q}) = & (\psi_{2}\phi_1-\phi_{1+q}\psi_1,\mbox{}\psi_{3}\phi_2-\phi_{2+q}\psi_2,\ldots,\psi_{n-q}\phi_{n-(q+1)}-\phi_{n-1}\psi_{n-(q+1)})
x(t) = c_1\,\sin \left (\sqrt{\lambda}\, \int_0^t \frac{1}{p_h(s,0)}\, ds \right ) + c_2 \, \cos \left (\sqrt{\lambda}\, \int_0^t \frac{1}{p_h(s,0)}\, ds \right ), y(t) = d_1\,\sin \left (\sqrt{\lambda}\, \int_0^t \frac{1}{p_h(s,0)}\, ds \right ) + d_2 \, \cos \left (\sqrt{\lambda}\, \int_0^t \frac{1}{p_h(s,0)}\, ds \right ),
\mathcal{E}(x',\xi',\lambda,z;h)=\sum_{j,k,l,m\in\mathbb{N}}(h^{2/3}T)^j(h^{2/3}\langle\lambda\rangle^{-1/2})^k(h^{1/3}\langle\lambda\rangle^{-1})^lh^m\mathcal{E}_{0,j,k,l,m}(x',\xi',\lambda,z)
\frac{1}{[ v \theta / \pi ]_{H_{\scriptscriptstyle 0}}^{\frac{1}{2}}}\frac{1}{[ \operatorname*{d e t}( \theta / \pi ) ]_{H^{\mu}\phi_{\scriptscriptstyle 0}}^{\frac{1}{2}}}\frac{1}{[ \operatorname*{d e t}( \theta / 4 \pi ) ]_{H_{\scriptscriptstyle 0}B^{\mu}}^{\frac{1}{2}}},
\frac{1}{\phi_{0}m^{2}}\frac{\partial^{2}\phi}{\partial t^{2}}- \frac{1}{\phi_{0}m^{2}}\frac{\partial^{2}\phi}{\partial x^{2}}+ \frac{\eta_{a}}{\phi_{0}m^{2}}\frac{\partial \phi}{\partial t}= - \frac{\phi}{\phi_{0}}+ \frac{\alpha \phi_{o}}{m^{2}}\frac{\phi^{2}}{\phi_{o}^{2}}- \frac{\lambda \phi_{0}^{2}}{m^{2}}\frac{\phi^{3}}{\phi_{0}^{3}}+ \frac{\xi_{a}}{\phi_{0}m^{2}}.
A, B = \left[ \begin{smallmatrix}* * 0 0 0 * 0 0 * * * * 0 * 0 * \end{smallmatrix}\right], \ \a= \left[ \begin{smallmatrix}* 0 * 0 \end{smallmatrix}\right], \ \b = \left[ \begin{smallmatrix}* * 0 0 \end{smallmatrix}\right].
R(h_1) =R(h_2)+R(h_1-h_2)=R(h_2)+(D-C)\left[\frac{\beta(t)}{2(\pi t)^2}-\frac{\cos(2\pi T t)}{2(\pi t)^2}\hat{\varphi}(t)\right] &=(C-D)\left[\frac{\cos(2\pi T t)}{2(\pi t)^2}\hat{\varphi}(t)\right]+B,
J^k_{1,1,2}=\int_{t_{k-1}}^{t_k}\left\{\int_{t_{k-1}}^u(p-2)|X^N(r)-X^N(s)|^{p-4}\left(X^N(r)-X^N(s),b(X^N(r))\right)dr\right\}&\times\left(X^N(t_{k-1})-X^N(s), \sigma(X^N(t_{k-1}))\frac{\Delta B_k}{\Delta}\right)du,
O_F=\left(\begin{array}{cc}-\frac{d^2}{dz^2}+\frac{\partial^2}{\partial\phi^2}V \frac{\partial^2}{\partial \chi\partial\phi}V \frac{\partial^2}{\partial \phi\partial\chi}V -\frac{d^2}{dz^2}+\frac{\partial^2}{\partial\chi^2}V\end{array}\right)_{\vert\phi =\phi(z), \chi =\chi(z)}
\big\vert\langle RT^{*n}x,y\rangle\big\vert =\bigg\vert\int_\mathbb{T}\lambda^n \langle x,E(\lambda)\rangle\overline{\langle y,E(\lambda)\rangle}\,d\mu(\lambda)\bigg\vert &=\Bigg\vert\sum_{q=0}^{+\infty}c_{q+n}\overline{c_q}\langle x,e_{q+n}\rangle\overline{\langle y,e_q\rangle}\Bigg\vert &\le M^2 t^n\vert\vert x\vert\vert\,\vert\vert y\vert\vert
\frac{\big\langle \psi_N, \sum_{j=1}^N -\Delta_{x_j}\psi_N\big\rangle}{\|\psi_N\|^2}\le \big\langle \phi_N, H_N^\mathrm{eff}\phi_N\big\rangle &\qquad-\frac{N(N-1)}{2}\big\langle\phi_N,\Big\{(H_{N-2}^\mathrm{eff}-4\pi\mathfrak{a}N)_{1,\dots,N-2}\otimes u_\ell(x_{N-1}-x_N) \Big\}\phi_N \big\rangle &+ C N^{-\varepsilon}.
S^{( 2 )}\simeq \frac K 2 \, 4 l n 2 \, \langle \Psi_{g}| Q | \Psi \rangle ( 2 \pi )^{2 6}\int d^{2 6}k \, d^{2 6}k^{\prime}\, T ( - k ) T ( k ) [ k^{2}- 1 ] .
\sum\limits_{k=0}^{+\infty}\sum\limits_{p=0}^{1}\varepsilon^{p\alpha +k -1}\, \left(w_{p\alpha +k -1}^{(i)}(x_i) + u_{p\alpha +k -1}^{(i)}\Big( x_i, \frac{\overline{x}_i}{\varepsilon}, t \Big) \right)
\alpha_{n-j}^{(n)}f \left( ( \mathbf{a}\cdot \mathcal{B})_{J^{*}}- ( \mathbf{a}\cdot \mathcal{B})_{J}\right) = \alpha_{j}^{(n)}f \left( ( \mathbf{a}\cdot \mathcal{B})_{J^{*}}- ( \mathbf{a}\cdot \mathcal{B})_{J}\right).
v^i(x_1^\ast,y_1^\ast) = \sum\limits_{x_2,\dots,x_{k-1}\in X}p(x_{k-1}|x_{k-2},\mu_{k-2}^\ast)\dots p(x_2|x_1^\ast,\mu_1^\ast) &\sum\limits_{\substack{x_k\in X y_k\in C_i}}p(x_k|x_{k-1},\mu_{k-1}^\ast) q(y_k|y_{k-1}^\ast,\nu(y_{k-1}^\ast))v^i(x_k,y_k).
\langle \tilde{\Theta}_\pm S\varphi, \tilde{\Theta}_\pm\tilde{\Theta}_\pm D_a\varphi\rangle =\pm \langle \tilde{\Theta}_\pm S\varphi,D_a\varphi\rangle &=\pm \langle S\tilde{\Theta}_\pm \varphi,D_a\varphi\rangle &=\pm \langle D_a\tilde{\Theta}_\pm \varphi,S\varphi\rangle=\langle \tilde{\Theta}_\pm D_a\varphi,S\varphi\rangle,
\widehat{Q}_0(x) = \widehat{\tau}_0, \widehat{Q}_k(x) = \widehat{\tau}_k \cdot x \cdot \widehat{Q}_{k-1}(x) - \widehat{a}_{k-1, k}\widehat{Q}_{k-1}(x)- \widehat{a}_{k-2, k}\widehat{Q}_{k-2}(x) - \cdots \widehat{a}_{0, k}\widehat{Q}_{0}(x)
[k;r_{k+1},\dots,r_s]:=\frac{1}{k!}\left|\begin{array}{ccc}x_1 \cdots x_s \cdot \cdots \cdot x_1 \cdots x_s y_1^{p^{r_{k+1}}}\cdots y_s^{p^{r_{k+1}}}\cdot \cdots \cdot y_1^{p^{r_{s}}}\cdots y_s^{p^{r_{s}}}\end{array}\right|.
[\begin{matrix}y_{1}\\y_{2}\\y_{3}\\y_{4}\\y_{5}\\y_{6}\\y_{7}\end{matrix}]=[\begin{matrix}1&0&0\\ 1&0&0\\ 1&0&0\\ 0&1&0\\ 0&1&0\\ 0&0&1\\ 0&0&1\end{matrix}][\begin{matrix}\mu_{1}\\ \mu_{2}\\ \mu_{3}\end{matrix}]+[\begin{matrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\\ \epsilon_{7}\end{matrix}]
[y_1, y_1]=y_4+\theta_1y_5, [y_1, y_2]=\frac{\alpha_3}{\alpha_1}y_4+\theta_2y_5, [y_2, y_2]=\frac{\beta_2\gamma_1}{\alpha_1\beta_6}y_5, [y_1, y_3]=\frac{\beta_4}{\beta_6}y_5, [y_3, y_1]=y_5, [y_2, y_3]=y_4+\theta_3y_5, [y_3, y_2]=-y_4.
{\cal L}_{0}= ( \partial^{\mu}+ i g_{0}A^{\mu}) \phi \, ( \partial_{\mu}- i g_{0}A_{\mu}){\bar{\phi}}- m_{0}^{2}\phi \,{\bar{\phi}}- \frac{\lambda_{0}}{4}( \phi \,{\bar{\phi}})^{2}- \frac{1}{4 \Omega_{D - 2}}\, F_{\mu \nu}F^{\mu \nu}.
R_{2}( x , a , b , d ) = \frac{1}{\left( 2 \pi \right)^{d - 2}b}\sum_{n^{\prime}= 1}^{\infty}\int d^{d - 2}p \frac{1}{\sqrt{\vec{p}^{\, 2}+ m^{2}+ ( \frac{n^{\prime}\pi}{b})^{2}}}\frac{\cosh ( ( a - 2 x ) \sqrt{\vec{p}^{\, 2}+ m^{2}+ ( \frac{n^{\prime}\pi}{b})^{2}})}{\sinh ( a \sqrt{\vec{p}^{\, 2}+ m^{2}+ ( \frac{n^{\prime}\pi}{b})^{2}})}.
{\bf B}(z) = R \circ J \circ T_{e_N}\circ S_2 \circ J \circ T_{-e_N}(z), \,\,R= \begin{pmatrix}1 & \ddots & 1 & -1\end{pmatrix},\,e_N = \begin{pmatrix}0 \vdots 0 1\end{pmatrix},
\alpha_{a}^{- 1}( \mu ) = \alpha_{a}^{- 1}( \Lambda ) +{\frac{{\widetilde b}_{a}}{4 \pi}}\int_{( \xi \Lambda )^{- 2}}^{( \xi \mu )^{- 2}}{\frac{d t}{t}}\sum_{{\bf m}}\exp ( - \pi t M_{\bf m}^{2}) ~ .
\Lambda^m(x)=-c^{-1}\int_0^x\int_0^{x_1}\left(f^5+\delta f^1_{x_2}\right)dx_2dx_1+c^{-1}\frac{x}{L}\int_0^L\int_0^{x_1}\left(f^5+\delta f^1_{x_2}\right)dx_2dx_1.
{\cal A}= \partial_{\phi}- \partial_{\phi}\log \chi_{m_1}(\phi), \chi_{m_1}(\phi) = (\cos\phi)^{-\alpha-\frac{1}{2}}(\sin\phi)^{\beta-\frac{1}{2}}P^{(-\alpha-1,\beta-1)}_{m_1}(-\cos2\phi),
A^{-1}\sharp_{\nu}B^{-1}\leqslant A^{-1}\sharp_{\nu}B^{-1}+ \sum_{k=0}^{\infty}r_{k}(A^{-1}\sharp_{\frac{m_k}{2^k}}B^{-1}-2 A^{-1}\sharp_{\frac{2m_k+1}{2^{k+1}}}B^{-1}+ A^{-1}\sharp_{\frac{m_k+1}{2^k}}B^{-1}) & \leqslant A^{-1}\nabla_{\nu}B^{-1}.
V_{K_{0}}(t)=V_{\overline{3_{1}}}(t)V_{4_{1}}(t) =(t+t^{3}-t^{4})(t^{-2}-t^{-1}+1-t+t^{2}) =t^{-1}-1+2t-3t^{2}+3t^{3}-2t^{4}+2t^{5}-t^{6}.
\int_{\mathfrak{M}}S_0(\alpha)S_1(\alpha)S_2(\alpha)S_3(\alpha)d \alpha &-\sum_{q\leq P}\frac{{\mu}^2(q)}{{\varphi}^4(q)}\sum_{\stackrel{a=1}{(a,q)=1}}^{q}\int_{-\frac{1}{qQ}}^{\frac{1}{qQ}}\sum_{m_j\in I_j}e(\lambda F(m_0,m_1,m_2,m_3))d\lambda &\ll BL^{-D}.
\psi_{> R}^{(l)}= \bigoplus_{i\in I^{(l)}}\psi_i \colon C^*_r( Y \rtimes \mathbb{R}) \to A_{> R}^{(l)}\varphi_{> R}^{(l)}= \sum_{i\in I^{(l)}}\varphi_i \colon A_{> R}^{(l)}\to C^*_r( Y \rtimes \mathbb{R}) \; .
I_1 \le \Bigg(\int_{I(x_0,Cr)}\bigg\{\sum_{k=k_0+1}^\infty{\mathfrak R}^{-k\beta}\big|D_k(a)(x)\big|\bigg\}^{p'}d\omega(x)\Bigg)^{1/p'}&\lesssim \Bigg(\int_{I(x_0,Cr)}\bigg\{\sum_{k=k_0+1}^\infty{\mathfrak R}^{-k\beta}\bigg\}^{p'}d\omega(x)\Bigg)^{1/p'}&\leq C r^\beta \omega(I)^{1/p'}.
U_{(a_{1,2},a_{2,2},\dots,a_{p-1,p-1)}}=\{\alpha \in Aut_{\Lambda}(R(2)\oplus R(\hat{1})) \mid k(\alpha)= \; \; (a_{1,2}+p\mathbb{Z},a_{2,2}+p\mathbb{Z},a_{3,3}+p\mathbb{Z},\dots ,a_{p-1,p-1}+p\mathbb{Z})\}.
e(G') =e(S)+e(\bar{S})+e(\bar{S},S) &=e(S)+\frac{1}{2}\sum_{x\in \bar{S}}\left(d_S(x)+d_{G'}(x)\right) &\leq \binom{s}{2}+\frac{1}{2}\left(s-1+\left\lceil\frac{k}{2}\right\rceil-1\right)(n-s).
\mu(\partial_{C\cup\bar Z}\Omega)= \mu(\partial_{C\cup\bar Z}\Omega\cap \bar Z)=\mu(\partial_{\bar Z}\Omega)\le \mu(\partial_{\bar Z}C) = \mu(\partial_{\bar Z}(C\cap \bar Z))=\mu(\partial_{\bar Z}(B_r(x)\cap Z))\le \mu(\partial B_r(x))=0.
E_{A}= \left( \begin{array}{c c}{E_{1}}&{E_{2}}\\ \end{array}\right){\cal E}_{A};{\cal E}_{A}= \left( \begin{array}{c}{\nu_{A}^{c}}\\{1}\\ \end{array}\right){\frac{1}{\sqrt{1 + \vert \nu_{A}\vert^{2}}}},
1 - \mathfrak{m}_{n}= \frac{1}{2}\frac{1 - |\tau_{n}\hat{\alpha}_{n-1}|^2}{1 + \mathcal{R}e(\tau_{n}\hat{\alpha}_{n-1})}= \frac{1}{2}\frac{|1 - \tau_{n+1}\hat{\alpha}_{n-1}|^2}{1 - \mathcal{R}e(\tau_{n+1}\hat{\alpha}_{n-1})}, n \geq 1,
S_{\lambda}^{-{\alpha_{\Re}}}:=\int_{W_a^+}e^{-{\mathrm{S}}(w^+)}(w^+)^{\alpha_{\Re}}\;\mathcal{D}_\lambda w^+=\int_{W_a^+}\sum_{m=0}^\infty a_m(\mathrm{V}(w^+))^m e^{-\mathrm{Q}(w^+)}(w^+)^{\alpha_{\Re}}\;\mathcal{D}_\lambda w^+
f(x)=f(a)+\frac{f^{\prime}(a)}{1!}(x-a)+\cdot\cdot\cdot+\frac{f^{(k)}(a)}{k!}(x-a)^{k}+\int_{a}^{x}\frac{f^{(k+1)}(t)}{k!}(x-t)^{k}dt.
\int_{B_r}|\nabla u|^2-|\nabla v|^2 = \int_{B_{r}\cap \Omega_0^+}|\nabla u|^2-|\nabla v|^2 &\qquad=\int_{D^+}|\nabla u|^2-|\nabla v|^2 + \int_{B_{\kappa r}\cap\Omega_0^+}|\nabla u|^2,
g_0(\frac{\partial\Psi}{\partial x_1},\frac{\partial\Psi}{\partial x_2}) = -Vg_0(I_1T,\frac{\partial\Psi}{\partial x_2}) = -V\omega_1(T,\frac{\partial\Psi}{\partial x_2}) =Vdx_1(\frac{\partial\Psi}{\partial x_2}) = 0.
\alpha(t)=\left(\int_{B_2}|h(V_t,\cdot)|^2\phi_{\rm rad}^2\, d\|V_t\|\right)^{\frac12},\,\,\mu(t)=\left(\int_{B_2}{\rm dist}\, (\cdot, J)^2\, d\|V_t\|\right)^{\frac12},
\| U^{*}(\Phi^{\prime}(f\otimes a) \oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\rho (p\otimes a)\oplus \cdots \oplus\bigoplus_{k=1}^m f(z_k)\rho (p\otimes a)}^m) U & - \Psi^{\prime}(f\otimes a)\oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\rho (p\otimes a) \oplus \cdots \oplus \bigoplus_{k=1}^m f(z_k)\rho (p\otimes a)}^m\| < \frac{\varepsilon}{7}
\dot{\phi}^{(1)}=\{\phi^{(1)},H_{T}\}\approx \frac{1}{2}(q^{3})^{2}&\therefore \phi^{(2)}:=q^{3}:\approx0, &\phi^{(3)}:=\dot{\phi}^{(2)}=\{\phi^{(2)},H_{T}\}\approx p_{1}&\therefore \phi^{(3)}:=p_{1}:\approx0.