harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\psi}_n^{N,h}(y) :=\frac{1}{\sqrt{\lambda_n^{N,h}}}h^2\sum\limits_{n=1}^{N}\sum\limits_{K\in \mathcal{T}_h}\sum\limits_{x_j\in I_{K}}\log\kappa(x_j,y_n)\phi_n^{N,h}(x_j)L_n(y).
\left. \begin{array}{l}{\delta_{a}\bigl ( [ S_{a}]^{- 1}[ S_{2}] \bigl ) = \delta_{a}\bigl ( ( D_{2}) ( \gamma_{2}) ( D_{1})^{- 1}\bigl ) = 0 ,}\\{\delta_{b}\bigl ( [ S_{b}] [ S_{1}] \bigl ) = \delta_{b}\bigl ( ( D_{1}) ( \gamma_{1}) \bigl ) = 0 ,}\\ \end{array}\right.
\Gamma_j\circ \Phi'|_{\oplus_{l=1}^{r}\mathfrak{h}(Y)(l)}\circ\eta'|_{\oplus_{l=1}^{r}\mathfrak{h}(Y)(l)}= \Phi|_{\oplus_{l=1}^{r}\mathfrak{h}(Y)(l)}\circ \eta|_{\oplus_{l=1}^{r}\mathfrak{h}(Y)(l)}.
Q_1 =\left[\begin{array}{cccc}2c_1\!-\!2\epsilon m_1 \epsilon \frac{m_1c_1}{k_1}0 0 * 2\epsilon \frac{m_1^2}{k_1}0 0 0 0 2c_2\!-\!2\epsilon m_2 \epsilon \frac{m_2c_2}{k_2}0 0 * 2\epsilon \frac{m_2^2}{k_2}\end{array}\right]
\begin{array}{rl}\| sA(T\!-\!\mu)^{-1}\| \le \|A(T\!-\!\mu)^{-1}\| \!\!\! \le \big\|\sqrt{a^2 \!+\! b^2 |T|^2}\, (T-\mu)^{-1}\big\| = \sup\limits_{t\in\sigma(T)}\!\dfrac{\sqrt{a^2 \!+\! b^2t^2}}{|t-\mu|}.\end{array}
M_{1}(\mathcal{NC}(G)) =19n(19n-1)^{2}-4(19n-1)\dfrac{(61n^{2}-19n)}{2}+4n(4n-1)^{2}+15n(3n-1)^{2}&= 6859n^{3}-2318n^{3}+64n^{3}+135n^{3}= 4740n^{3}
J_{\mu}:= \sum_{n,m\geq0}\sum_{\substack{\mathfrak{a}\in \mathfrak{g}^{\otimes n}\mathfrak{b}\in \mathfrak{g}^{\otimes m}}}\sum_{a,b\in \mathfrak{g}}(\mathfrak{a}\otimes (a\otimes b-\varepsilon(a,b)b\otimes a)\otimes \mathfrak{b}- \mu^{n+m}\alpha_{T}(\mathfrak{a})\otimes [a,b]_{\mathfrak{g}}\otimes \alpha_{T}(\mathfrak{b})),
x_{1,2j-1}+ x_{1,2j}+ x_{m,(n+1)-(2j-1)}+ x_{m,(n+1)-2j}&= x_{1,2j-1}+ x_{1,2j}+ \bigl( \tfrac{1}{2}S - x_{m,2j-1}\bigr) + \bigl( \tfrac{1}{2}S - x_{m,2j}\bigr) &= \bigl( n_1 m +1 \bigr) + S - \bigl( n_1 m +1 \bigr) =S.
\varrho=\varrho_m(1+\zeta e^{-2\psi})+\varrho_re^{-2\psi}+\varrho_{\psi},\ \ \ \ \ \ \ \ \ \ \ \ \wp=\textstyle{1\over 3}\varrho_re^{-2\psi}c^2+\varrho_{\psi}c^2.
\int_{B_{\rho}(x_0)}\nabla v_{i,n}\cdot \nabla \varphi = \int_{B_{\rho}(x_0)}g_{i,n}\varphi - M_n |v_{i,n}|^{p-1}v_{i,n}\sum_{j \neq i}a_{ij}|v_{j,n}|^{p+1}\varphi
const \times \prod_{i=1}^{N}\cos^{2N-2}\left(\frac{\theta_i}{2}\right)\prod_{1 \le i < j \le N}^{}\left(\tan\left(\frac{\theta_j}{2}\right)-\tan\left(\frac{\theta_i}{2}\right)\right)^2d\theta_1\cdots d\theta_N,
u_1^{\frac{N}{2}, N}(A_{0,k}) = (0,0) u_1^{\frac{N}{2}, N}(B_{0,k}) = \cos(k\pi) \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) u_1^{\frac{N}{2}, N}(C_{0,k}) = \cos(k\pi) \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right),
\begin{array}{r c l r c l}{{\hat{b}_{5}}}&{{=}}&{{v^{( 0 )}= 0 \, ,}}&{{\hat{b}_{i}}}&{{=}}&{{b_{i}\, , \hspace{1 c m}}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{\hat{a}^{( 2 )}{}_{i 5}}}&{{=}}&{{b_{i}+ v^{( 1 )}{}_{i}\, ,}}&{{\hat{a}^{( 2 )}{}_{i j}}}&{{=}}&{{a^{( 2 ) \prime}{}_{i j}\, ,}}\\ \end{array}
\left\lbrace \begin{aligned}E_{n+1}(\alpha_*,M)=F(E_n,\alpha_*,M,\kappa_n) F(E_n,\alpha_*,M,\kappa_n)\geq \xi_+, E_{n+1}(\alpha_*,M)=2\xi_+-F(E_n,\alpha_*,M,\kappa_n) F(E_n,\alpha_*,M,\kappa)< \xi_+,\end{aligned}\right.
\rho_h^{\mathrm{BEM}}(V) := \sum_{j = 1}^{N_h}\big\| \varrho_{h}^j(V) \big\|^2 + \frac{1}{h}\sum_{j = 1}^{N_h}\big\| P_h^{j-1}\varrho_h^j(V) \big\|^2,
G_{o}( K ; \vec{x}, \phi ; \vec{x^{\prime}}, \phi^{\prime}) = - \frac{1}{( 2 \pi )^{4}}\sum_{n = - \infty}^{n = + \infty}\int d^{3}p \frac{e^{i \vec{p}. ( \vec{x}- \vec{x^{\prime}})}e^{i n ( \phi - \phi^{\prime})}}{[ p^{2}+ n^{2}/ R^{2}- K^{2}- i \epsilon ]},
\mathbb{E}{\left( \mathbf{\theta}[\diamond_i, q](k) \cdot \mathbf{\theta}[\diamond_j, q](k) \right)}&=\mathbb{E}{\left( \sum_{*}{\alpha_{* \diamond_i}\mathbf{A}[*, q](k)}\sum_{*}{\alpha_{* \diamond_j}\mathbf{A}[*, q](k)}\right)}&= \sum_{*}{\alpha_{* \diamond_i}\alpha_{* \diamond_j}\cdot \sigma^2}
y_{11}= \phi^2\left[(\phi\Omega)_z^2+\phi\phi_{zz}(z(\phi\Omega)_z+s(\phi\Omega)_s)-(r^2-s^2)(\phi^2(\phi_{ss}\phi_{zz}-\phi_{sz}^2)-\Omega\delta_2)\right] y_{12}= y_{21}=\phi^3\left[\phi_{sz}(\phi\Omega)_z-\phi_{zz}(\phi\Omega)_s\right] y_{22}= -\phi^4(\phi_{ss}\phi_{zz}-\phi_{sz}^2).
\begin{aligned}\iint_{L^{k}(r)}|\nabla u|^2 dxds \leq C r^{2q-4}m(6r)m(r)^{\tau}\sum_i \iint_{Q(x_{i},t_{i};\,r)\cap \tilde{K}(r )}|\nabla u|^q dxds &=C r^{2q-4}m(r)^{\tau+1}\iint_{\widetilde{L}(r)}|\nabla u|^q dxds.\end{aligned}
\sum\limits_{i=1}^kI(U^m_{\mathcal{A}_i};\hat{S}^m_{a(i)}|U^m_{\cup_{j=1}^{i-1}\mathcal{A}_j})\leq\sum\limits_{i=1}^kI(U^m_{\mathcal{A}_i};Y^{\rho m}_{a(i)}|U^m_{\cup_{j=1}^{i-1}\mathcal{A}_j}).
\vartheta - h d t , \qquad \mathrm{w i t h}\qquad \vartheta = ( p_{i}- \frac{1}{2}{B_{c}}\, \varepsilon_{i j}\, x_{j}) d x_{i}+ \frac{1}{2}\theta \varepsilon_{i j}\, p_{i}d p_{j}= P_{i}d Q_{i},
Y_{t}^{n}=x_{0}+\int_{0}^{t}b\left( Y_{\pi_{n}\left( s\right)}^{n}\right)ds+\int_{0}^{t}G\left( Y_{\pi_{n}\left( s\right)}^{n}\right)dL_{s}^{0},t\in \lbrack 0,1).
\frac{1}{2}\frac{\mu_{0}^{2}}{m_{0}g^{2}N}\int \mathrm{d}^{3}{\bf r}\, \phi^{2}\left( r \right) = \left( \frac{m_{0}}{\vert E \vert}- \frac{\vert E \vert}{m_{0}}\right) \beta \, ,
\leq \int \prod_{a = 1}^{N - 1}\prod_{i = 1}^{n_{a}}d^{D}z_{i}^{( a )}\prod_{b < c}^{N - 1}\bigg | f^{( b , c )}\bigg | \prod_{d = 1}^{N - 1}\Big | \! \Big | \! \Big | f^{( d , N )}\Big | \! \Big | \! \Big |_{N}
( ( \delta_{\theta_{2 1}\gamma_{2 1}}) ( \delta_{\theta_{3 1}\gamma_{3 1}}\delta_{\theta_{3 2}\gamma_{3 2}}) \cdots ( \delta_{\theta_{k - 1 , 1}\gamma_{k - 1 , 1}}\cdots \delta_{\theta_{k - 1 , k - 2}\gamma_{k - 1 , k - 2}}) ) .
\sum_{t=0}^z(-1)^t\binom{x}{t}\binom{y-t+1}{z-t}= \sum_{t=0}^z(-1)^t\binom{x}{t}\binom{y-t}{z-t}+ \sum_{t=0}^{z-1}(-1)^t\binom{x}{t}\binom{y-t}{z-t-1}.
\sum_{k=1}^6\mathcal{N}_2^k =\sum_{X\in\{A,B\}}X_{++}(u,u) + X_{+-}(u,\bar{u}) + X_{-+}(\bar{u},u) + X_{--}(\bar{u},\bar{u}) + |\partial_x|^{1/2}O_{3,1/2}.
P(z, t)=\sum_{n=0}^{\infty}z^{n}p_{n}(t), Q(z, t)=\sum_{n=0}^{\infty}z^{n}q_{n}(t) \,, P^{*}(z, t)=\sum_{n=0}^{\infty}z^{n}p_{n}^{*}(t), \qquad Q^{*}(z, t)=\sum_{n=0}^{\infty}z^{n}q_{n}^{*}(t)\,,
\chi_i = \sum_{j=1}^{m}\left( \left( 1 - \psi_{i-j+1}\right)^2 \left( 1 - \phi_{i-j+1}\right)^2 +{\psi_{i-j+1}}^2 \left( 1 - \phi_{i-j+1}\right)^2 + \vphantom{U}\right. \qquad\qquad\left.\vphantom{U}\left( 1 - \psi_{i-j+1}\right)^2{\phi_{i-j+1}}^2 +{\psi_{i-j+1}}^2{\phi_{i-j+1}}^2 \right) .
\Phi_{\ell}(n+1) = \Phi_{\ell}(n) - \frac{p\ell}{n}\mathbb{P}(D_k(n)=\ell | D_k(n_0)=a) = \Phi_{\ell}(n) - \frac{p\ell}{n}\cdot \Big(\Phi_{\ell}(n)- \Phi_{\ell-1}(n)\Big)
\Gamma_{\Phi_{1}(x;h)}=C_{H}\sum_{\alpha=1}^{d}\int_{[0,1]^{2}}J_{1}J_{s}^{-1}V_{\alpha}(\Phi_{s})V_{\alpha}^{*}(\Phi_{t})(J_{t}^{-1})^{*}J_{1}^{*}|t-s|^{2H-2}dsdt,
Z(n+1,k)-Z(n,k)=\begin{cases}0, \mathrm{if}\ \eta(n)+\ell(n)\notin[k:p-1],\ell(n)\in[0:k-1], begin{align*}0.5em]1, \mathrm{if}\ \eta(n)+\ell(n)\in[k:p-1],\ell(n)\in [0:k-1], begin{align*}0.5em]0, \mathrm{if}\ \eta(n)+\ell(n)\notin [p:p+k-1],\ell(n)\in[k:p-1], begin{align*}0.5em]-1, \mathrm{if}\ \eta(n)+\ell(n)\in [p:p+k-1],\ell(n)\in[k:p-1].\end{cases}
S = \int d^2 x \hspace{1mm}\mbox{tr}\left(-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}D_\mu \phi D^\mu \phi +i \bar{\Psi}\gamma^\mu D_{\mu}\Psi -2ig\phi\bar{\Psi}\gamma_5\Psi \right),
{\cal C}^{\prime \prime}+ \biggl \{\frac{k^{2}}{1 + r^{2}}\biggl [ 1 + r^{2}\biggl ( - \frac{\eta}{\eta_{1}}\biggr )^{- 3 / 8}\biggr ] - \frac{2}{\eta^{2}}\biggr \}{\cal C}= 0 ,
{\overline{R}}_{e^{*}}=\mathbb{E}\left[{{\log}_2}\left({1 +{\gamma_{e^{*}}}}\right) \right] &=\frac{1}{{\ln 2}}\int_0^\infty{\frac{{\left({1 -{F_{{\gamma_{{e^*}}}}}\left( x \right)}\right)}}{{1 + x}}dx},
N(U_n)=\left|{\rm Re}(u_{3n},\nabla (u_{1n}\cdot \overline{u_{2n}})_{L^2}\right|\le \|\nabla \cdot u_{3n}\|_{L^2}\|u_{1n}\|_{L^4}\|u_{2n}\|_{L^4}\le C\|U_n\|_{L^4}^2.
[\begin{matrix}0&0\\ 0&0\end{matrix}],[\begin{matrix}1&0\\ 0&1\end{matrix}],[\begin{matrix}2&0\\ 0&2\end{matrix}],[\begin{matrix}1&1\\ 1&2\end{matrix}],[\begin{matrix}2&2\\ 2&1\end{matrix}],[\begin{matrix}0&1\\ 2&0\end{matrix}],[\begin{matrix}0&2\\ 1&0\end{matrix}],[\begin{matrix}1&2\\ 2&2\end{matrix}],[\begin{matrix}2&1\\ 1&1\end{matrix}]
\frac{|\beta_{n+1}|}{(n+1)!}\leq \frac{C(\beta)^{n}}{n}u^{n-1}[z^n]\sum\limits_{m=1}^n{n \choose m}(T_{\mathrm{Pen},1}(z))^m &= \frac{C(\beta)^{n}}{n}[z^n]\left(1+\frac{1}{u}T_{\mathrm{Pen},1}(uz)\right)^n.
E_{in}^{(3)}(z) = \left( \mathbb{I}- \frac{A_n^{(2)}(z) - A_n^{(2)}(0)}{n^6z}\right) E_{in}^{(2)}(z), z \in D(0,r_n) E_{out}^{(3)}(z) = \left( \mathbb{I}- \frac{A_n^{(2)}(0)}{n^6 z}\right)^{-1}E_{out}^{(2)}(z), z \in A(0; r_n, R),
\int_{0}^{T}{\bf a}^{\top}(t)\bar{\bf z}^{(\epsilon)}(t)dt =\int_{0}^{T}{\bf a}^{\top}(t){\bf z}^{(\epsilon)}(t)dt=\sum_{j=1}^{q}\int_{0}^{T}z_{j}^{(\epsilon)}(t)a_{j}(t)dt & =\lim_{r\rightarrow\infty}\sum_{j=1}^{q}\int_{0}^{T}z_{j}^{(k_{r})}(t)a_{j}(t)dt=\lim_{r\rightarrow\infty}\int_{0}^{T}{\bf a}^{\top}(t){\bf z}^{(k_{r})}(t)dt=M.
+ \left[ 2 i{\left( - 1 \right)}^{n - 1}{\epsilon}_{2}\cdot p_{1}P^{n - 1}\left( p_{1}^{2}, q^{2}\right) \right] \frac{{\left( - 1 \right)}^{n}i}{q^{2 n}-{\left( - m^{2}\right)}^{n}}\left[ 2 i{\left( - 1 \right)}^{n}{\epsilon}_{1}\cdot p_{2}P^{n - 1}\left( q^{2}, p_{2}^{2}\right) \right] +
\frac{\mathrm{d}}{\mathrm{d}s}\Gamma(1/s,x^s) =-\frac{1}{s^2}\Big[\log (x^s) \Gamma(1/s,x^s)+x^s T(3,1/s,x^s)\Big]-x\exp(-x^s)\log(x) &\triangleq\tilde\psi(s,x),
\pi_{\mathbf{0}}^{(n)}(\sigma ,\gamma )(t)\stackrel{d}{=}\hat{\pi}_{n}(N(n^{2}\gamma t)/n^{2})+\dfrac{1}{n}\Big [\dfrac{n^{2}\gamma t-R_{N(n^{2}\gamma t)}}{R_{N(n^{2}\gamma t)+1}-R_{N(n^{2}\gamma t)}}Z_{N(n^{2}\gamma t)+1}\Big ] .
\dim \big( GS_{\bf u,v}^{\bf n,r}(\Delta)\big)= (r_1+1)(r_2+1)J_{NT}+(r_2+1)(n_1-2r_1-2)E_{hor}&+(r_1+1)(n_2-2r_2-2)E_{ver}+(n_1-2r_1-2)(n_2-2r_2-2)N,
\eta_n(B)=\frac{1}{\mu(A'_n)}\mu(B\cap A'_n)\le\frac{1}{\mu(A'_n)}\mu\Big( \bigcup_{m\in \mathcal{I}_n\atop B\cap B_m\ne\varnothing}B_m\Big)\le \frac{\mu((1+\theta)B)}{\mu(A'_n)}<\frac{C^2(1+\theta)^s\mu(B)}{C^{-2}5^{-s}(1-1/2n)},
\MoveEqLeft \int e^{-3i \Phi(\xi,\eta)}f(\eta) g(\eta-\xi) \varphi_4(\eta/\xi) d\eta & = \int e^{-3i \xi \mu^2}f(\psi_1(\mu/\xi)\xi) g((\psi_1(\mu/\xi)-1)\xi) \varphi_4(\psi_1(\mu/\xi)) \psi'_1(\mu/\xi) d\mu & =: \int e^{-3i \xi \mu^2}h(\xi,\mu) d\mu.
F_{\gamma_{eq}}\left(x\right)= 1-\left[1-F_{\left|f\right|^2/\left(\rho^2+N_0\right)}\left(x\right)\right]\left[1-F_{\gamma^*}\left(x\right)\right] = 1-\left[1-\frac{\gamma\left(d,\frac{\lambda x}{\theta}\right)}{\Gamma\left(d\right)}\right]\left[1-\left(1-e^{-\frac{x}{\bar{\gamma}}}\right)^N\right]
\left<W_{R\times T}\right>=\frac{1}{N_c}\sum\limits_n^{}\left<\Psi_{ij}(0)\left|n\left>\right<n\right|\Psi_{ji}^\dagger(T)\right>=\frac{1}{N_c}\sum\limits_n^{}\left|\left<\Psi_{ij}(0)|n\right>\right|^2{\rm e}^{-E_nT},
H_{(X_+,D_+)}(y_1,y_2) = e^{\frac{\xi_+\log y_1+ h_+\log y_2}{2\pi i}}\sum_{d_1,d_2\geq 0}\frac{y_1^{d_1}y_2^{d_2}[\textbf{1}]_{(d_2,\cdots,d_2)}}{\Gamma(1+\frac{h_+}{2\pi i}+d_2)^{r^\prime+1}\Gamma(1+\frac{\xi_+-h_+}{2\pi i}+d_1-d_2)^{r^\prime+1}\Gamma(1+\frac{\xi_+}{2\pi i}+d_1)}.
dq_t= \tilde\gamma^{-1}(t)\left(-\partial_t\psi(t,q_t)-\nabla_qV(t,q_t))\right) dt+(\tilde\gamma^{-1}\sigma)(t)\circ dW_t, dq^\prime_t= \tilde\gamma^{-1}(t^*)\left(\partial_t\psi(t^*,q^\prime_t)-\nabla_{q}V(t^*,q^\prime_t)\right) dt+(\hat{\phi}_*(\tilde\gamma^{-1}\sigma))^i_\rho(t^*)\circ d\tilde W^\rho_t.
\frac{1}{\Delta^\sigma}\frac{\partial^2}{\partial x_\alpha\partial x_\beta}(\Delta^\sigma) =\sigma^2\cdot\tau(c_\alpha,x_2^{-1})\cdot\tau(c_\beta,x_2^{-1})+ \sigma\cdot\tau(c_\alpha,\frac{\partial x_2^{-1}}{\partial x_\beta}) &=\sigma^2\cdot\tau(c_\alpha,x_2^{-1})\cdot\tau(c_\beta,x_2^{-1})- \sigma\cdot\tau(c_\alpha,Q_{x_2^{-1}}c_\beta),
{||f||}^2\leq \displaystyle \sum_{m \in \mathfrak{D}}\displaystyle \sum_{p=0}^{q-1}|s_{p,m}| \displaystyle \sum_{k \in \mathfrak{D}}\sum_{j=0}^{q-1}|\overline{t_{j,k}}||\langle{T_{Am}\varphi_p, T_{Ak}e_{\alpha_j}}\rangle|.
(\rho_z|_{\mathbb{T}^2})(\omega_1,\omega_2) = \textrm{diag}(\omega_1^2, \omega_2^2, \omega_1^{-2}, \omega_2^{-2}, \omega_1\omega_2, \omega_1\omega_2^{-1}, \omega_1^{-1}\omega_2, \omega_1^{-1}\omega_2^{-1}, 1, 1),
\big\|{\rm\bf Def}(\mathbf{U})\big|\mathbb{L}_p(\Omega)\big\|:=\left[\sum_{j,k=1}^n \big\|\mathbf{D}_{jk}(\mathbf{U})\big|\mathbb{L}_p(\Omega)\big\|^p\right]^{1/p},\mathbf{U}\in\mathbb{W}^1_p(\Omega).
\begin{aligned}(\phi_0' (1), \phi_0 (1))_{\mathbb{R}^n}=(\phi_0' (1), P_{D_1}\phi_0 (1) + P_{N_1}\phi_0 (1) + P_{R_1}\phi_0 (1))_{\mathbb{R}^n}&= (P_{N_1}\phi_0' (1) + P_{R_1}\phi_0' (1), \phi_0 (1))_{\mathbb{R}^n}= 0,\end{aligned}
\tilde{P}:=-\lambda_{c}\tilde{Q}_{1}\frac{\partial V_{c}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]}\tilde{Q_{1}}^{\top}=I-\tilde{Q_{1}}^{-\top}\hat{A}_{1}^{\top}\left(\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\right)^{-1}\hat{A}_{1}\tilde{Q_{1}}^{-1}.
w_{n}^{\mathrm{s u s y}}\left( F_{n}\right) = \int \prod_{i = 1}^{n}d^{2}x_{i}\prod_{i = 1}^{n}d \xi_{i}\, \, w_{n}^{\mathrm{s u s y}}\left( x_{1}, \xi_{1}; \ldots ; x_{n}, \xi_{n}\right) F_{n}\left( x_{1}, \xi_{1}; \ldots ; x_{n}, \xi_{n}\right) \, \, .
X_{j}= \frac{\partial}{\partial \overline{x}_{j}}-\frac{1}{2}\sum_{s=1}^{n}\left( \sum_{i=1}^{m}U^{(s)}_{ji}\overline{x}_{i}\right)\frac{\partial}{\partial \widehat{x}_{s}}, j=1, \ldots, m, & Z_{i}= \frac{\partial}{\partial \widehat{x}_{i}}, i=1, \ldots, n.
K(Y_1,Y_2) = K(Y_1,Y_3) = - \frac{v' h'}{vh}+ \frac{v^2}{16h^4}&K(Y_2,Y_3) = - \frac{1}{4h^2}- \frac{3v^2}{16h^4}- \left( \frac{h'}{h}\right)^2 &K(Y_1,Y_4) = - \frac{v''}{v}\hskip 80pt K(Y_2,Y_4) = K(Y_3,Y_4) = - \frac{h''}{h}
\Phi_i=\left[\begin{array}{c|c}0 I_{n_{p_i}-1}\hline -\hat{p}_{i,n_{p_i}}\mathbf{\hat p}_i \end{array}\right],\; \Psi_i=\begin{bmatrix}1 0_{1\times (n_{p_i}-1)}\end{bmatrix}
I(X_{\pi(2)};Y_1|X_{\pi(1)},W_{\sigma(1)}) = I(X_{\pi(2)};Y_1|X_{\pi(1)}) +I(W_{\sigma(1)};Y_1|X_{\pi(1)},X_{\pi(2)}) - I(W_{\sigma(1)};Y_1|X_{\pi(1)}) \stackrel{(d)}{=}I(X_{\pi(2)};Y_1|X_{\pi(1)}),
M(\vec w,k-1)=\begin{pmatrix}c_{w_1}M_{11}\cdots c_{w_1+(k-1)p}M_{1k}c_{w_1-1}M_{21}\cdots c_{w_1+(k-1)p-1}M_{2k}\vdots \ddots \vdots c_{w_1-k+1}M_{k1}\cdots c_{w_1+(k-1)(p-1)}M_{kk}\end{pmatrix}.
B = \frac{2}{\pi}\int f^{\prime}\sin^{2}f \mathrm{d}r \frac{1}{4 \pi}\int \frac{( | R_{z}|^{2}- | R_{{\bar{z}}}|^{2}) ( 1 + | z |^{2})^{2}}{( 1 + | R |^{2})^{2}}\frac{2 i \mathrm{d}z \mathrm{d}{\bar{z}}}{( 1 + | z |^{2})^{2}}.
f_{n-7}= A_{2-n}A_{4-n}A_{6-n}\Big(\sum_{k=1}^{n-7}\psi_k\Big)+K_{6-n}\Big(\sum_{k=1}^{n-7}\psi_k\Big)+f=O(r^{n-6}) := b_{n-6}+O(r^{n-5}).
\mathfrak{S}_{2,4,2}= -\sum_{(i,j,k)\in [n]_1^3}\sum_{\pi \in T_n^{(i,j)}\cap T_n^{(1,k)}}\mathrm{sgn}(\pi) \binom{\pi(1)+1}{2}\mathcal{R}_{i+1}\mathcal{R}_{\pi(i)+1}\mathcal{R}_{\pi(1)-1}Q_{R}(\pi;1,i).
{\cal L}= \frac{1}{2}\{[\dot{X}(t)+g\xi(t)Y(t)]^{2}+ [\dot{Y}(t) -g\xi(t)X(t)]^{2}+ [\dot{Z}(t) - \xi(t)]^{2}\}- U(X(t)^{2}+Y(t)^{2})
\begin{alignedat}{1}Y_{0}=Y_{0}(A) =[1\,2\,3][4\,5\,6], Y_{1}=Y_{1}(A) =[1\,2\,4][3\,5\,6], Y_{2}=Y_{2}(A) =[1\,2\,5][3\,4\,6], Y_{3}=Y_{3}(A) =[1\,3\,4][2\,5\,6], Y_{4}=Y_{4}(A) =[1\,3\,5][2\,4\,6], Y_{5}=Y_{5}(A) =[1\,2\,3][1\,4\,5][2\,4\,6][3\,5\,6]-[1\,2\,4][1\,3\,5][2\,3\,6][4\,5\,6],\end{alignedat}
K_{\mu \rho}( x , x^{\prime}) = \frac{\partial^{2}}{\partial x_{\mu}\partial x_{\rho}^{\prime}}n^{b}( x ) n^{c}( x^{\prime}) \int_{0}^{+ \infty}d s \int D z \mathrm{e}^{- \frac 1 4 \int_{0}^{s}\dot{z}^{2}d \lambda}\left[ P{\,}\exp \left( i g \int_{0}^{s}d \lambda \dot{z}_{\alpha}B_{\alpha}\right) \right]^{b c},
\bar{\Phi}_A(\theta) = \bar{\phi}_A - \theta^a \phi^*_{A a}-\theta^2 \eta_A,\qquad\frac{\delta}{\delta \bar{\Phi}_A(\theta)}=\theta^2 \frac{\delta}{\delta \bar{\phi}_A}+\frac{\delta}{\delta \phi^*_{A a}}\theta_a -\frac{\delta}{\delta \eta_A}.
\hspace{. 7 4 5 i n}= - q ( \frac{z_{1}}{z_{2}})^{- \delta_{i 1}}r ( \frac{z_{1}}{z_{2}}) ( 1 \otimes q^{- 2 \overline{{{\rho}}}}) \overline{{{R}}}_{V^{*}V}( \frac{z_{1}}{z_{2}})
\sum_{\substack{1 \leq a, b_1, b_2, \ldots, b_s \leq n (a,n)=1}}(a-1,b_1,\ldots, b_s,n)\chi(a) \lambda_1(b_1)\cdots \lambda_s(b_s)= \varphi(n)\sigma_s\left((\frac{n}{d},w_1,\ldots, w_s)\right),
\begin{array}{rcl}{\hat c}^{(6)}{}_{ijklm5}= -5(2\pi\alpha^\prime) \partial_{[i}v^{(0)}c^{(4)}{}_{jklm]}-5{\textstyle\frac{m}{2}}(2\pi\alpha^\prime)^{3}v^{(0)}b_{[i}\partial_j b_k\partial_l b_{m]}& & +v^{(5)}{}_{ijklm}\, , \end{array}
\begin{aligned}\Bigg| \dfrac{\partial \epsilon_1(\eta,t)}{\partial \eta}\Bigg|_{\eta = \eta_a}\dot{\eta}_a\Bigg| \le \,\gamma|a_{0,\delta}(x_a)/2-\bar{y}_a| &+\gamma \left|\left.\dfrac{\partial\varepsilon_\delta(x,\bar{y}_a,t)-b_{1,\delta}(x)/2}{\partial x}\right|_{x = x_a}\right|\dfrac{|b_{1,\delta}(x_a)|}{2}.\end{aligned}
{j^{(b)}}^T_\mu \; := \; T_{\mu \nu}\overline{\psi}^{(b)}\gamma_\nu \psi^{(b)}\; \; \; \; , \; \; \; \; T_{\mu \nu}\; := \; \delta_{\mu \nu}- \frac{\partial_\mu \partial_\nu}{\triangle}\; ,
\frac{\operatorname*{diam}_i B_{\sigma'}}{\operatorname*{diam}_i B_{\sigma}}\leq \frac{\operatorname*{diam}_i B^{oct}_{\sigma'}+ C h}{\operatorname*{diam}_i B^{oct}_{\sigma}}= \frac{1}{2}+ C \frac{h}{\operatorname*{diam}_i B^{oct}_{\sigma}}.
\left\lbrace\begin{array}{ll}W''-\Delta W+ \overline A_1W=0 \hbox{in}(0, +\infty)\times \Omega, W= 0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu W+ \overline B_1W= C_1DH \hbox{on}(0, +\infty)\times\Gamma_1\end{array}\right.
p_x=\sum_{i=3}^np_{xi}=\frac{2}{n(n-1)(n-2)(n-3)}\sum_{i=3}^n(i-2)(n-i) &=\frac{2}{n(n-1)(n-2)(n-3)}\sum_{j=1}^{n-2}j(n-2-j) &=\frac{2}{n(n-1)(n-2)(n-3)}[(n-2)(n-1)(n-2)/2-(n-2)(2n-3)(n-1)/6] &=\frac{1}{3n}
\begin{cases}D_{0^+}^{\alpha,\beta}x_{1}(t) =f_{1}(t,x_{1},x_{2},..,x_{n}), D_{0^+}^{\alpha,\beta}x_{2}(t) =f_{2}(t,x_{1},x_{2},..,x_{n}), &\cdots D_{0^+}^{\alpha,\beta}x_{n}(t) =f_{n}(t,x_{1},x_{2},..,x_{n}), I_{0^+}^{1-\gamma}x_{i}(0^+) =x_0, \gamma=\alpha+\beta-\alpha\beta, i=1,2,..,n,\end{cases}
\mathbf{P}_{R_s}=\sum\limits_{t_k\in{\cal T}_s}{\bf p}^{(1)}_k \otimes{\bf p}^{(2)}_k \otimes{\bf p}^{(3)}_k, \mathbf{P}_{R_l}=\sum\limits_{t_k\in{\cal T}_l}{\bf p}^{(1)}_k \otimes{\bf p}^{(2)}_k \otimes{\bf p}^{(3)}_k.
F(\underline U(t,x))=K_2 \frac{\underline h(t)-|x|}{\underline h(t)}\Theta \Big([\nabla F({\bf 0})]^T+o(1){\bf I}_m\Big)\succeq K_2\frac{\underline h(t)-|x|}{\underline h(t)}\frac{3}{4}\lambda_1\Theta=\frac{3}{4}\lambda_1 \underline U(t,x),
\frac{d}{dt}\sum_{i=1}^{N}\|v_i\|_p^p + C\sum_{i=1}^{N}\|\nabla(v_i^{p/2})\|_2^2 \leq C\sum_{i=1}^{N}\left(\|v_i\|_{\mu+p-1}^{\mu+p-1}+ \|v_i\|_{p+{\delta}}^{p+{\delta}}+ \|v_i\|_p^p\right).
\frac{2\delta}{1+2\gamma\sigma+\sigma\beta\gamma^2}=\frac{\sqrt{(1-2\gamma\sigma+\gamma^2\sigma\beta)(1+2\gamma\sigma+\gamma^2\sigma\beta)}}{1+2\gamma\sigma+\gamma^2\sigma\beta}&=\sqrt{\frac{1-2\gamma\sigma+\gamma^2\sigma\beta}{1+2\gamma\sigma+\gamma^2\sigma\beta}}&=\sqrt{1-\frac{4\gamma\sigma}{1+2\gamma\sigma+\gamma^2\sigma\beta}}.
\Gamma_{t,t+\delta}(t+\delta) = \begin{pmatrix}(\cosh(L \delta) - 1) \theta(t) \theta(t)^{T}+ I \sinh(L \delta) \theta(t) \sinh(L \delta) \theta(t)^{T}\cosh (L \delta) \end{pmatrix}&= \begin{pmatrix}I L \theta(t) \delta L \theta(t)^{T}\delta 1 \end{pmatrix}+ o(\delta),
L^{1,1}_h(\omega) =e_i^*\wedge \theta_1 \otimes e^*_j\wedge \theta_2, L^{-1,-1}_h(\omega) =i_{e_i}\theta_1 \otimes i_{e_j}\theta_2, L^{1,-1}_h(\omega) =e_i^*\wedge \theta_1 \otimes i_{e_j}\theta_2, L^{-1,1}_h(\omega) =i_{e_i}\theta_1 \otimes e^*_j\wedge \theta_2.
\begin{aligned}{F}_{i}(t) =f_{0,i}+\int_0^tQ(s,{F}_{i}(s))ds + \int_0^t\left[ a\left( \left\| \Lambda{F}_{i}(s)\right\| +\int_0^s\Delta (\tau,{F}_{i}(\tau))d\tau \right) -a(\left\| \Lambda f_0\right\| )\right] \Lambda{F}_{i}(s)ds, t \geq 0.\end{aligned}
\sum_{\lambda\in \Lambda_{m,\mu}}\log| 4\pi e^\gamma \cdot Q(\lambda_z) | =- \sum_{\lambda\in \Lambda_{m,\mu}}\mathrm{CT}_{s=0}\left[\int_1^\infty e^{4\pi v Q(\lambda_z)}\, \frac{dv}{v^{s+1}}\right]
f(\mathbf{z})=\displaystyle -\sum_{j=1}^{n}(z_j\overline{z_{n+1}}-z_{n+1}\overline{z_{j}})^{2}+\sum_{1\leq j<k \leq n}(z_j \overline{z_{k}}-z_{k}\overline{z_{j}})^2
D_1 \leq n^2 \sum_{i=1}^{2n}\int_0^u \frac{J \d ( N_{1;i}+ N_{2;i})}{Y^4}\cdot \mathcal{O}(1) = \frac{1}{n}\int_0^u \frac{J}{(Y/n)^3}\d (\hat A_1 + \hat A_2) \cdot \mathcal{O}(1)
\int_{\Sigma ( \tilde{x}^{0})}d^{3}{\bf \tilde{x}}\hspace{0 . 2 5 e m}\Delta_{\pm}( x^{\prime \prime}- \tilde{x}) \stackrel{\leftrightarrow}{\partial_{0}}\Delta_{\mp}( \tilde{x}- x^{\prime}) = 0
\int_{1}^{\infty}\xi^{-2-1/\beta}C\xi^{-d/\alpha}t^{-\beta/\alpha - \beta d/\alpha}d\xi \le t^{-\beta/\alpha - \beta d/\alpha}C\int_{1}^{\infty}\xi^{-2-1/\beta - d/\alpha}d\xi \le t^{-\beta/\alpha - \beta d/\alpha}C \left(1 - \lim_{K \rightarrow \infty}\frac{1}{K}\right) = C t^{-\beta/\alpha - \beta d/\alpha}.
\left(D_{-}^{\alpha,\beta,\gamma}f\right)(x)=\left(-\frac{d}{dx}\right)^{[\operatorname{Re}(\alpha)]+1}\left(I_{-}^{-\alpha+[\operatorname{Re}(\alpha)]+1,-\beta-[\operatorname{Re}(\alpha)]-1,\alpha+\gamma}f\right)(x),
\Phi_{\xi}(0) = \sum_{v\in W_0}C(v\xi )=\sum_{v\in W_0}\prod_{\alpha\in R^+_0}\frac{1-t_{\alpha}e^{-i\langle v\xi ,\alpha\rangle}}{1-e^{-i\langle v\xi ,\alpha\rangle}}&\stackrel{\star}{=}\sum_{v\in W_{0}}t_v =\prod_{\alpha\in R_0^+}\frac{1-t_{\alpha}e_t (\alpha)}{1-e_t (\alpha)}> 0 ,
\nabla_{K_{n}}(z) =\nabla_{K_{n-1}}(z)+z\nabla_{J}(z) =(\nabla_{K_{n-2}}(z)+z\nabla_{J}(z))+z\nabla_{J}(z) \cdots =\nabla_{K_{0}}(z)+nz\nabla_{J}(z) =(-z^{4}+1)+nz(-4z^{3}+2z)=-(4n+1)z^{4}+2nz^{2}+1.
\int_{\mathcal{V}}\phi(v) \mathcal{L}\phi(v)\,{\rm d}\mu(v)=\int_{\mathcal{V}}|\phi(v)|^2\,{\rm d}\mu(v)- \iint_{\mathcal{V}\times\mathcal{V}}\phi(v) \phi(w)\,{\rm d}\mu(w)\,{\rm d}\mu(v).
\big\{\frac{S^{(i)}_{\tau^{N,i},N+\tau^{N,i}}}N< r_i\Big\}\supset \big(\cap_{n=1}^{2M}\big\{Z_n^{N,i}\in\{0,-1\}\big\}\big)\cap\big(\cup_{n=1}^M\{Z_{2n}^{N,i}=-1\}\big).
\begin{array}{c}{{\displaystyle{\sum_{n}\sim V \, \int \, \frac{d^{3}k}{( 2 \, \pi )^{3}}\, + \, S \, \int \, \xi \, \frac{d^{3}k}{( 2 \, \pi )^{3}\, k}+ \dots}}}\\ \end{array},
c_{\delta^{\prime}}:={\displaystyle \int \limits_{\delta^{\prime}}^{\infty}}\left( 1+\ln \frac{t}{r}\right) \sup_{x\in{\mathbb{R}^{n}}}\varphi_{1}\left(x,t\right) t^{-\frac{n}{p}+\frac{n}{s}-1}dt<\infty
\Big \|{\sigma}_\varepsilon^{(\varepsilon^{-2}s_{n + 1},\; 0)}(\cap_{j=1}^n \{\mathcal{M}_{s_j \varepsilon^{-2}}= \nu_j\}) - e^{(s_{n + 1}- s_n) Q}\mathcal{P}_{\nu_n}\dots e^{(s_2 - s_1) Q}\mathcal{P}_{\nu_1}e^{s_1 Q}\Big \|_{2,}\leq C n \varepsilon.
\mathbb{G}\left(w(x,t)\right) ={\bf\Phi}_{M_n,n}(\widetilde u_T )(x,t)-\widetilde \Phi_{M_n,n}( \widetilde g)(x,t)-\sum_{p=0}^{M_n}\Bigg[ \int_t^T e^{(s-t)p^{2\beta}}F_p(w)(s)ds \Bigg]\phi_p(x).
\varphi_{k,0}(x) = \cos kx - \varkappa_kx\sin kx + \delta_x(k+\varkappa_k) + \theta_k(x), \varphi_{k,1}(x) = \cos kx + \delta_x(k), \tilde \varphi_{k,0}(x) = \cos kx - \varkappa_kx\sin kx + \theta_k(x), \tilde \varphi_{k,1}(x) = \cos kx.
\det \left(\rho(F(X,Y)\right) = \det \begin{pmatrix}f(\lambda) g(\lambda^{2^{n-2}-1}) g(\lambda) f(\lambda^{2^{n-2}-1})\end{pmatrix}& =f(\lambda)f(\lambda^{2^{n-2}-1})-g(\lambda)g(\lambda^{2^{n-2}-1}):=k(\lambda).
I_n^{(1)}:= \int_{|\eta'| \leq R/2}d^3 \eta' \int d^3 p_\zeta \; \bigl| V_{13}(\alpha' \eta')\bigr| \,{\tilde t}^{\; 2}_n (p_\zeta) \,{\tilde J}^2 (\eta', p_\zeta) , I_n^{(2)}:= \int_{|\eta'|>R/2}d^3 \eta' \int d^3 p_\zeta \; \bigl| V_{13}(\alpha' \eta')\bigr| \,{\tilde t}^{\; 2}_n (p_\zeta) \,{\tilde J}^2 (\eta', p_\zeta) .
g_{j_{r}}|_{{\textstyle\bigcup_{i=1}^{\tilde{m}}}\Omega_{i}}=\tilde{b}_{1}g_{j_{1}}|_{{\textstyle\bigcup_{i=1}^{\tilde{m}}}\Omega_{i}}+\cdots+\tilde{b}_{r-1}g_{j_{r-1}}|_{{\textstyle\bigcup_{i=1}^{\tilde{m}}}\Omega_{i}}
\int\int\int\int p(x_j,y,f_{1:j-1},x_{1:j-1})\log p(x_j|y,f_{1:j-1},x_{1:j-1})dx_jdyd f_{1:j-1}dx_{1:j-1}&\qquad-\int\int p(x_j,x_{1:j-1})\log p(x_j|x_{1:j-1})dx_j dx_{1:j-1}&= \int\int\int p(x_j,y,x_{1:j-1})\log p(x_j|y,x_{1:j-1})dx_jdydx_{1:j-1}&\qquad-\int\int\int p(x_j,y,x_{1:j-1})\log p(x_j|x_{1:j-1})dx_j dy dx_{1:j-1}&=I(X_j;Y|X_{1:j-1}),
\left( \mathcal{A}^{\dag}-\mathcal{P}_{M}\mathcal{A}^{\dag}\mathcal{P}_{M}\right) \Omega_{n}=c_{\mathcal{A}}\left( n-l\right) \left[1-1_{n\leq M}\cdot1_{n-l\leq M}\right] \Omega_{n-l}& =c_{\mathcal{A}}\left( n-l\right) 1_{n>M\wedge\left( M+l\right)}\Omega_{n-l}n\in\mathbb{Z}.
\frac{\Delta_{jk}(mc^{2})^{2}L}{4\hbar cE}=\frac{GeVfm}{4\hbar c}\times\frac{\Delta_{jk}m^{2}}{eV^{2}}\frac{L}{km}\frac{GeV}{E}\approx1.27\times\frac{\Delta_{jk}m^{2}}{eV^{2}}\frac{L}{km}\frac{GeV}{E},
\mathbb{E}[\mathbb{Z}_n(\theta_0)\big|\mathcal{D}_1] = \sum_{i\in\mathcal{D}_2}\mathbb{E}[(\widehat{g}(X_i) - g_0(X_i))(\widehat{m}(X_i) - m_0(X_i))\big|\mathcal{D}_1].