harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\exp \left( i e \eta [ x_{\perp}; \pi ] \right) a_{-}( x_{\perp}) \exp \left( - i e \eta [ x_{\perp}; \pi ] \right) = a_{-}( x_{\perp}) - \frac{e}{2 L}{\cal G}_{( \perp )}[ x_{\perp}, x_{\perp}; 0 ] .
\tfrac{1}{\pi i}\partial\overline{\partial}H(X)=\tfrac{1}{2}\omega_{\mathrm{Hdg}}+\tfrac{1}{(g!)^2}\left(\int_{pr_{g+1}}\gamma^*\omega_0^{g+1}-\int_{pr_{g+1}}\gamma^* \left(\delta_{\Theta_{\alpha}}\right) \gamma^*\omega_0^{g}\right).
\Psi_{(P,Q)}\, u = -u_a(\cdot,\cdot)-m(P,Q,\cdot,\cdot)u(\cdot,\cdot), D(\Psi_{(P,Q)}) =\{u\in L^1\left((0,a_m), W^{1,1}(0,a_m)\right)\,|\, u(\cdot,0)= B(u)\},
D_{1}\eta_{2}^{\alpha_{1}}= - \pi^{\alpha_{1}}+ \frac 1 2 C_{\; \; \beta_{1}\gamma_{1}}^{\alpha_{1}}\eta_{1}^{\beta_{1}}\eta_{2}^{\gamma_{1}}+ \frac 1 2 C_{\; \; \beta_{1}\gamma_{3}}^{\alpha_{1}}\left( \eta_{2}^{\beta_{1}}\eta_{1}^{\gamma_{3}}+ \eta_{1}^{\beta_{1}}\eta_{2}^{\gamma_{3}}\right) ,
\frac{1}{2}\Delta_g |\nabla^2 u|_g^2 = |\nabla^3 u|_g^2+\langle\nabla \Delta \nabla_i u,\nabla \nabla^i u\rangle_g+u_{,ij}(R^l_{ijk}u_{,lk}+R_{j}^l u_{,il}+R_{ijk,k}^lu_{,l}+R_{ijk}^lu_{,lk}).
\left.\begin{array}{ll}\mbox{nilpotence}Q_B^2=0, \mbox{odd derivation}Q_B(A*B)=(Q_BA)*B+(-1)^AA*(Q_BB), \mbox{partial integrability}\int Q_B(\ldots)=0, \int A*B=(-1)^{AB}\int B*A, \end{array}\right.
\int_0^h (t+h-\tau)^{-\delta}d\tau = (1-\delta)^{-1}[(t+h)^{1-\delta}-t^{1-\delta}] & \le h t^{-\delta}\le h^{2\gamma}t^{-2\gamma}h^{1-2\gamma}t^{2\gamma-\delta}\le h^{2\gamma}t^{-2\gamma}T^{1-\delta},
\psi = [\widetilde{\sigma}^g(\widetilde{u}), e_- \cdot w + e_+ \cdot w], \alpha = [\sigma^g(u),\underbrace{e_+^{\flat}\wedge \widetilde{\alpha}_+ + e_- \wedge e_+ \wedge \widetilde{\alpha}_{\mp}+ \widetilde{\alpha}_0 + e_-^{\flat}\wedge \widetilde{\alpha}_-}_{=: \widetilde{\alpha}}]
\frac{d}{dt}\int_{\Sigma^t}\chi(x) d\mathcal{H}^k = -\int_{\Sigma^t}\chi(x) |\vec{H}_{\Sigma^t}(x)|^2 d\mathcal{H}^k + \int_{\Sigma^t}\nabla \chi(x) \cdot \vec{H}_{\Sigma^t}(x) d\mathcal{H}^k.
(\psi^* \lambda)(rx) [u] = \lambda(\psi(rx))[ d\psi(rx) u ] = \lambda(r\tilde{\psi}(x))[d\psi(x)u] = r \lambda(\tilde{\psi}(x))[d\tilde{\psi}(x) v + \sigma Y(\psi(x))] = r \lambda(\tilde{\psi}(x))[d\tilde{\psi}(x) v] = r (\tilde{\psi}^* \lambda)(x)[v] = r \lambda(x)[v] = r \lambda(x)[v+\rho Y(x)] = \lambda(rx)[u].
M[a_1,a_2]=\sum\limits_{k=0}^{|a_1|}\sum\limits_{\ell=0}^{|a_2|}{a_1\brack k}_{v^c}{a_2\brack \ell}_{v^b}v^{bck\ell}X^{(b\ell-a_1,ck-a_2)}.
\lim_{\lambda \to \infty}\left[ \mathrm{R e}\, \Gamma ( \varphi , \lambda ) \right]_{\lambda - d e p}\; = \; \varphi_{1}^{2}\lim_{\lambda \to \infty}\frac{\pi}{2}\, \sum_{n_{1}}\, \frac{\tau_{2}^{2}\lambda^{2}}{( \lambda^{2}+ \tau_{2}^{2}n_{1}^{2})^{\frac{3}{2}}}\,
\mathrm{pr}^{F}(C_{Y}D_{X}(C_{Z})) - C_{Y}\mathrm{pr}^{F}(D_{X}C_{Z}) &= \mathrm{pr}^{F}(C_{Y}\mathrm{pr}^{F}D_{X}(C_{Z})) +\mathrm{pr}^{F}(C_{Y}A^{F}_{X}(C_{Z})) - C_{Y}\mathrm{pr}^{F}(D_{X}(C_{Z})) &= \mathrm{pr}^{F}(C_{Y}A^{F}_{X}(C_{Z})),
\{X_t=y,\,N_t=\Gamma,\,\tau=t\}&=\bigcup_{o=y_0,\ldots,y_t=y \atop 1=n_0,\ldots,n_{t-1}=\Gamma(y_{t-1})}\begin{array}{c}\{X_1=y_1,\ldots, X_t=y_t\}\cap \{N_1(y_0)=n_0,\ldots N_t(y_{t-1})=n_{t-1}\}\cap\,\{N_t=\Gamma\}\cap\{f_t(U(y_0,n_0),\ldots,U(y_{t-1},n_{t-1}))=1\}.\end{array}
\nu(B(k)) = \nu(\beta(\beta-2^{n-k+1})(\beta-2\cdot 2^{n-k+1})\cdots (\beta-(j_k-1)2^{n-k+1})) = \nu(\beta) + \nu(\beta-2^{n-k+1}) + \nu(\beta-2\cdot 2^{n-k+1}) + \cdots + \nu(\beta-(j_k-1)2^{n-k+1})
\int_0^1\zeta_1(a,x)\zeta_1(b,x)\,dx=\int_0^1\zeta_1(a+b,x)\,dx+\sum_{n=1}^\infty\int_0^1\left\{\frac{\zeta_1(a,n+x)}{(n+x)^{b}}+\frac{\zeta_1(b,n+x)}{(n+x)^{a}}\right\}dx,
\begin{array}{cc}(BRu)_{j}(x,t)=-\displaystyle \sum_{k\neq j}\sum_{l=1}^{n}\int_{0}^{1}\int_{x_j}^{x}d_{j}(\xi,x,t)b_{jk}(\xi,\omega_j(\xi))c_{k}(x_k,\xi,\omega_j(\xi)) \displaystyle \times r_{kl}(\eta,\omega_k(x_k,\xi,\omega_j(\xi)))u_{l}(\eta,\omega_k(x_k,\xi,\omega_j(\xi)))\,d\xi d\eta, \;\;\; j\le n. \end{array}
\Delta^{( 2 )}p^{-}= \frac{g^{2}}{4 \pi}\int_{0}^{p^{+}}\frac{d k^{+}}{p^{+}- k^{+}}\frac{\left( \frac{m}{p^{+}}+ \frac{m}{k^{+}}\right)^{2}}{p^{--}\frac{m^{2}}{k^{+}}- \frac{\lambda^{2}}{p^{+}- k^{+}}}.
\textbf{c}_{q,j}^{(n)}=\frac{1}{n!(q-1)!(n-q)!}\sum_{i=0}^{2n-2-j}Q_{2n-i}(q)(-1)^i(n+1-q)^{2n-2-j-i}&=\frac{\binom{n}{q-1}}{(n!)^2}\sum_{i=j+2}^{2n}(-1)^iQ_i(q)(n+1-q)^{i-2-j}.
\int_{[\tilde{Q}^{\epsilon_{+}}_{n+k}([E^{ss}/G],\overline{\beta})]^{}}[\mathcal{D}'_{r}](\sum_{i=0}^{\infty}[\mathcal{D}'_i])^{s-1}=\sum_{\overline{\beta}'}\dfrac{\prod_{a=1}^r\mathfrak{r}_{n+k+a}}{r!}\int_{Q_{\overline{\beta}'}}p_*((\sum_{i=0}^{\infty}[\mathcal{D}'_i])^{s-1}),
\begin{array}{c}a_0 + b_0\Re\tau_d = - \frac{1}{2}|z_0|^2\Re v_2 - \frac{1}{2}\tilde{t}_0 \sqrt{d}\Im v_2 b_0\Im\tau_d = - \frac{1}{2}|z_0|^2\Im v_2 + \frac{1}{2}\tilde{t}_0 \sqrt{d}\Re v_2 \end{array}
\mathcal{E}_{1}(Y;\cdot) := |\nabla u|^{2}+ u^{10/3}+ \frac{1}{2}\phi(m-u^{2}), \mathcal{E}_{2}(Y;\cdot) := |\nabla u|^{2}+ u^{10/3}+ \frac{1}{8\pi}|\nabla \phi|^{2}.
{\textstyle \frac{1}{2}}\left( M_{\alpha}^{}\otimes M_{- \alpha}^{}\, - \, M_{- \alpha}^{}\otimes M_{\alpha}^{}\right) \, = ~ M_{\alpha}^{-}\otimes M_{\alpha}^{+}\, - \, M_{\alpha}^{+}\otimes M_{\alpha}^{-}~ = ~ M_{\alpha}^{-}\otimes M_{\alpha}^{}\, - \, M_{\alpha}^{}\otimes M_{\alpha}^{-}
\begin{array}{cc}\alpha_{1}(U,V) =g(\overset{\mu_{0}}{h}(U,V),\tilde{J}N), \alpha_{2}(U,V) =g(\overset{\mu_{0}}{h}(U,V),\tilde{J}E), \alpha_{3}(U,V) =g(\overset{\mu_{0}}{h}(U,V),N),\end{array}
\left|P_{X^{n}Y^{n}}-\pi_{X^{n}Y^{n}}\right| \geq1-4\exp\left(-n\frac{\frac{1}{n}\Omega^{(\alpha,\lambda)}(\{Q_{i}\}_{i=1}^{n})-\lambda\alpha R}{1+(1+\bar{\alpha})\lambda}\right).
\frac{e^{-4\pi \tau}}{1 - e^{-4\pi \tau}}\sim- \frac 14 \frac{q^2}{p^{(2)2}}\;,\;\;\frac{e^{-4\pi l^\prime}}{1 - e^{-4\pi l^\prime}}\sim - \frac 14 \frac{k^2}{p^{(1)2}}\;.
Q = S\left[\frac{1}{2\pi\sqrt{-1}}\oint_\Gamma D(z)dz \right]S^{-1}= S\left[ \begin{array}{cc}I_s 0 0 0 \end{array}\right] S^{-1}= S_{(:, 1:s)}(S^{-1})_{(1:s, :)}.
\exp(X(c_1)) \bmod \overline{\mathbb Q}^\times =\pi^{\frac{1}{4}}p_{F(\sqrt{-1})}(\mathrm{id},\mathrm{Inf}^{-1}(\tfrac{1}{4}\tau_1+\tfrac{1}{4}\tau_2-\tfrac{1}{4}\tau_3-\tfrac{1}{4}\tau_4)).
j'(t)= -\frac{1}{t^2\varepsilon^2}\int_{\Omega}\left(p(tu-q\ln\frac{1}{\varepsilon}\right)_+^{p-1}tu^2-\left(tu-q\ln\frac{1}{\varepsilon}\right)_+^{p}u)dx = \frac{1}{t^2\varepsilon^2}\int_{\Omega}\left(tu-q\ln\frac{1}{\varepsilon}\right)_+^{p-1}u[(1-p)(tu-q\ln\frac{1}{\varepsilon})-pq\ln\frac{1}{\varepsilon}]dx \leq 0.
u_{i}^{\u}= \frac{1}{\sqrt{1 + t \bar{t}}}\left( \begin{array}{c c}{{t}}&{{1}}\\{{- 1}}&{{\bar{t}}}\\ \end{array}\right) , \u_{u}^{\i}= \frac{1}{\sqrt{1 + t \bar{t}}}\left( \begin{array}{c c}{{\bar{t}}}&{{- 1}}\\{{1}}&{{t}}\\ \end{array}\right) ,
\sum_{k=m-l+1}^{m+1}\binom{2m-k+1}{m}\binom{k-1}{m-l}h^l(1+h)^{m-l}=\binom{2m+1}{l}h^l(1+h)^{m-l}&=\left(\binom{2m}{l}+\binom{2m}{l-1}\right)h^l(1+h)^{m-l},
M_{3}=-\frac{1}{8}+\frac{1}{2\sqrt{2}}-\frac{4\pi}{3}-\frac{\ln(2)}{4\pi}+\frac{\Gamma(1/8)\Gamma(3/8)}{\pi^{3/2}\sqrt{2}}& -2{\displaystyle\sum\limits_{i,j,k=-\infty}^{\infty}}{}^{\prime}\frac{(-1)^{i+j+k}}{\sqrt{i^{2}+j^{2}+k^{2}}\left( e^{8\pi\sqrt{i^{2}+j^{2}+k^{2}}}-1\right)}
A_{\ell,k}^p(u) = \big\{m_k^{\ell}(x_k^{i}) + \sqrt{\Delta}\sigma_k^{\ell p}(x_k^i) u + \sum_{p' \not=p}\sqrt{\Delta}\sigma_k^{\ell p'}(x_k^i) Z_{k+1}^{p'}\in (v^{\ell,j_{\ell}-}, v^{\ell,j_{\ell}+}) \big\}.
\sum_{M\geq1}g_r^{(d)}(M)z^M = z^d\sum_{s\geq0}z^{s}\sum_{n\geq0}\binom{n+s}{n}(z^r)^n = z^d\sum_{s\geq0}\frac{z^s}{(1-z^r)^{s+1}}&=\frac{z^d}{1-z^r}\cdot \frac{1}{1-\frac{z}{1-z^r}}=\frac{z^d}{1-z-z^r}.
\mathbb{E}\left[C_1^(a_)\right] = \int_0^\infty\int_0^{x_2}\frac{2}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}&\cdot[\log_2(1+\xi x_1) -\log_2\left(1 + (\sqrt{1+\xi x_2}-1)\frac{x_1}{x_2}\right)]dx_1dx_2.
{\mathbf V}_{j}^\ell=({{\mathbf V}_j^\ell}^+,{{\mathbf V}_j^\ell}^-)=\Big(\big({\varphi_{j1}^{\ell}}^{+},{\varphi_{j2}^{\ell}}^{+}\big),\, \big({\varphi_{j1}^{\ell}}^{-},{\varphi_{j2}^{\ell}}^{-}\big)\Big).
P'_{0}(1) = \lim_{z \rightarrow 1}\dfrac{\left[ \lambda(1-z) +\gamma \right]P_{0(z)}- \mu p_{11}}{\xi (1-z)}= \lim_{z \rightarrow 1}\dfrac{-\lambda P_{0}(z)+ \left[ \lambda(1-z) +\gamma \right]P^{'}_{0}(z)}{-\xi}= \dfrac{-\lambda P_{0}(1)+\gamma P^{'}_{0}(1)}{-\xi}.
\sum_{n=m+1}^{N}\mu\left( z_{n}\right) =\sum_{n=d\left( z_{0}\right)^{2}-RC+1}^{d\left( z_{0}\right)^{2}}\frac{1}{d\left( z_{0}\right)+n+d\left( z_{0}\right)^{2}}& \approx\log\frac{d\left( z_{0}\right) +2d\left( z_{0}\right)^{2}}{d\left( z_{0}\right) +2d\left( z_{0}\right)^{2}-RC+1}\approx\frac{RC}{d\left( z_{0}\right)^{2}},
h(\dot{\mathcal P}(s),\frac{\partial}{\partial \theta})=h(\dot{\gamma}(s)+W,\frac{\partial}{\partial \theta})=h(\dot{\gamma}(s),\frac{\partial}{\partial \theta})+h(W(s),\frac{\partial}{\partial \theta})=\Bigl(\frac{d\theta}{ds}+\mu\Bigr)m^2.
\omega_{2n+1|0}= \begin{bmatrix}\omega_{2n-1|0}\tilde{I}_{2n-1}\omega_{2n-1|0}\omega_{2n-1|0}- 2 \cdot1_{2n-1}\tilde{I}_{2n-1}\omega_{2n-1|0}- 2\cdot 1_{2n-1}\end{bmatrix}
\tilde{b}\bigg[\sum\limits_{\sigma\in \mathcal{S}_n}\alpha(\gamma_{\sigma})\sum\limits_{j=0}^{n-1}\alpha(\chi_{\sigma(1)}\chi_{\sigma(2)}\ldots, \chi_{\sigma(j)})\mathcal{U}_{\sigma(j+1)}\chi_{\sigma(j+2)},\ldots,\chi_{\sigma(n)}\bigg] f(\chi_1,\ldots,\chi_n) =0
(\partial_{1,j}+ \partial_{2,j})V_1 =V_1 \Lambda^j, (\partial_{1,j}+\partial_{2,j})\tilde V_2 =\tilde V_2 (\Lambda^\top)^j, (\partial_{1,j}+ \partial_{2,j})L_1 =0, (\partial_{1,j}+\partial_{2,j})L_2 =0.
P_1\theta = \Delta\theta - q{\sf U}_\infty^{q-1}\theta = - f({\sf U}_\infty) - \sum_{i=1}^N \tfrac{f^{(i)}({\sf U}_\infty)}{i!}(\theta-\theta_L)^i + \sum_{i=2}^N \tfrac{f_2^{(i)}({\sf U}_\infty)}{i!}(\theta-\theta_L)^i.
\langle y\rangle_{T^{*3}}= ~\{y, T^{*3}y ,\ldots, T^{*3(j-r)}y\}~,~~~\langle y\rangle_{T^{*3}}=j-r+1, \langle z\rangle_{T^{*3}}= ~\{z, T^{*3}z ,\ldots, T^{*3(n-2j-3+r)}z\}~,~~~\langle z\rangle_{T^{*3}}=n-2j-2+r,
J_{{\mu_0}{\mu_1}}^{ab}\left( x\right) =\int d\tau d\sigma \sum_iz_i^{ab}\,\delta^d\left( x_\mu-X^i_\mu\left( \tau ,\sigma \right) \right) \,\,\partial_\tau X_{[{\mu_0}}^i\left( \tau ,\sigma \right) \,\partial_\sigma X_{{\mu_1}]}^i\left( \tau,\sigma \right) \,\,.
\Delta_{n}(\lambda,b) := B_{n}^{2}(\lambda,b)-4bC_{n}(\lambda,b) & = \Big(b\big[\Omega_{n}(\lambda)+\Omega_{n}(\lambda b)\big]-(1+b^{2})\Lambda_{1}(\lambda,b)\Big)^{2}-4b^{2}\Lambda_{n}^{2}(\lambda,b).
\mathcal{B}_{a}(x) =\frac{1}{\Gamma\left(\frac{a}{\nu}\right)}\int_{0}^{\infty}t^{\frac{a}{\nu}-1}e^{-t}h_{t}(x)dt &=\frac{1}{\Gamma\left(\frac{a}{\nu}\right)}\int_{0}^{|x|^{\nu}}t^{\frac{a}{\nu}-1}e^{-t}h_{t}(x)dt+\frac{1}{\Gamma\left(\frac{a}{\nu}\right)}\int_{|x|^{\nu}}^{\infty}t^{\frac{a}{\nu}-1}e^{-t}h_{t}(x)dt &=:J_{1}+J_{2}.
\int e^{\sum_{m=4}^NJ_m^0(\psi)}d\mu_{C_{\le 0}}(\psi) =\int\int e^{\sum_{m=4}^NJ_m^0(\psi+\psi^1)}d\mu_{C_0}(\psi^1)d\mu_{D_{\le -1}}(\psi) &=\int e^{\log(\int e^{\sum_{m=4}^NJ_m^0(\psi+\psi^1)}d\mu_{C_0}(\psi^1))}d\mu_{D_{\le -1}}(\psi) &=\int e^{J^{-1}(\psi)}d\mu_{D_{\le -1}}(\psi).
\sum_{n=1}^{\infty}R^{E_n}_{M_D(\cdot, z)}(x_0)<\infty \textrm{if and only if}\sum_{n=1}^{\infty}2^{nd}\frac{\Phi(2^{-n})}{g(A_{2^{-n}}(z))^2}R^{E_n}_{g}(x_0)<\infty\, .
0 = \bar{\Phi}(\rho,\theta_i) - \bar{\Phi}(0,\theta_i) = \int_0^{\rho}\partial_{r_i}\bar{\Phi}(r_i,\theta_i)\,dr_i = \int_0^{\rho}\left(\partial_{r_i}\bar{S}_i \bar{\psi}^{(i)}+ r_i \partial_{r_i}^2\bar{S}_i \bar{\psi}^{(i)}+ r_i \partial_{r_i}\bar{S}_i \partial_{r_i}\bar{\psi}^{(i)}\right)\,dr_i.
\displaystyle\sum\limits_{n1,n2}C^{n_1 n_2}_{n_{12}j}\psi_{n_1}^{\mu_1}(\rho \cos \phi)\psi_{n_2}^{\mu_2}(\rho \sin \phi) = e^{-\rho^2/2}\displaystyle\sum\limits_{n=0}^{n_{12}+j}C^{n_1,n_2}_{n_{12}, j}H_{n}^{\mu_1}(\rho\cos\phi)H_{n_{12}+j-n}^{\mu_2}(\rho\sin\phi),
\ll k^2\sum_{a\leqslant N}\sum_{b\leqslant N}\sum_{c\leqslant N}\sum_{d\leqslant N}\left|a(a)\,a(b)\,a(c)\,a(d)\right|(abcd)^{-3/4}\cdot\frac{k\,M^{3/2}}{\sqrt{\vphantom ba}+\sqrt b+\sqrt{\vphantom bc}+\sqrt d}&\ll k^3\,M^{3/2}\left(\sum_{n\leqslant N}\left|a(n)\right|\,n^{-7/8}\right)^{\!4}\ll k^3\,M^{3/2}\,(N^{1/8})^4=k^3\,M^{3/2}\,N^{1/2}.
A-B=2m(t_2-t_1)-(mn+2c)(2t_1-t_0-t_2)+{2\over m}\Bigl({1\over t_2}-{1\over t_1}\Bigr)+{n\over m}\Bigl({1\over t_0}+{1\over t_2}-{2\over t_1}\Bigr).
K(L,1) =2^{n-2}\int_0^1\int_0^{\arccos s}\sum_{k=0}^\infty s^{2L+2Lk-n}\sum_{j=0}^{n-2}\ \binom{n-2}{j}(-1)^j s^j (\cos\theta)^{n-2-j}(s+\frac 1s-2\cos\theta)d\theta\; ds = 2^{n-2}\sum_{k=0}^\infty \sum_{j=0}^{n-2}\ \binom{n-2}{j}(-1)^j\int_0^{\pi/2}(\cos\theta)^{n-2-j}\int_0^{\cos\theta}s^{2L(1+k)-n+j}(s+\frac 1s-2\cos\theta)\; dsd\theta.
\textbf{err}^{(2)}= \mathcal{O}\Bigg(\delta^{7}\sqrt{\mathcal{K}^{(\mathrm{T_0})}_{\mathrm{r}}}\frac{1}{|\xi-\cdot|^{2-2\mathrm{r}}}\Big\Vert|\mathrm{E}|^{2}\Big\Vert_{\mathrm{L}^2(\Omega)}\Bigg).
1\geq\left\langle\lambda(\delta)x, \left(1+(1+\tilde{\Psi}_y(\delta))\frac{\tilde{c}(S,n)}{G(x)}\delta^{\frac{2}{n+1}}\right)y\right\rangle=\lambda(\delta) \left(1+(1+\tilde{\Psi}_y(\delta))\frac{\tilde{c}(S,n)}{G(x)}\delta^{\frac{2}{n+1}}\right)\quad.
\begin{bmatrix}a_1 a_2 \vdots a_n \end{bmatrix}= A_n^{-1}\left( \underset{:= B_{b,m}}{\underbrace{b_{m+1}- \sum_{s = \pm 1}\sum_{k=1}^{\lfloor \frac{\sqrt{24m+1}-s}{6}\rfloor}(-1)^{k+1}b_{m+1-k(3k+s)/2}}}\right)_{0 \leq m < n}.
\vec{D}=\left( \begin{array}{c}D_{1}D_{2}\vdots D_{r}\end{array}\right), \vec{F}=\left( \begin{array}{c}F_{1}F_{2}\vdots F_{s}\end{array}\right),\vec{e}=\left( \begin{array}{c}1 0 \vdots 0\end{array}\right),\vec{Z}=\left( \begin{array}{c}Z_{1}Z_{2}\vdots Z_{r}\end{array}\right).
H_N =N \left[\sum_{k=0}^{N-1}(-1)^{k+N-1}\small{\left(\begin{array}{c}N-1 k\end{array}\right)}\frac{1}{\left(N-k\right)^2}\right], S_N =N \left[\sum_{k=0}^{N-1}(-1)^{k+N-1}\small{\left(\begin{array}{c}N-1 k\end{array}\right)}\frac{1}{\left(N-k\right)^3}\right]
j_X^{(1)}\circ \phi_X(c) = \sum_i \Theta_{j_X(\phi_X(c)x_i),j_X(x_i)}= \phi_{X \oplus Y}(j_A(c)) \sum_i \Theta_{j_X(x_i),j_X(x_i)}= \phi_{X \oplus Y}(j_A(c)) P_X = \phi_{X \oplus Y}\circ j_A(c),
\Phi_T(M_{af_1+bf_2})= \Phi_T(aM_{f_1}+bM_{f_2})= a\Phi_T(M_{f_1})+b\Phi_T(M_{f_2})= = M_{ag_1}+aR_1+M_{bg_2}+bR_2 = M_{ag_1+bg_2}+aR_1+bR_2
S_N =\sum_{k=1}^N \frac{1}{k}H_k=\sum_{k=1}^N\frac{\log k}{k}+ \gamma H_N+ \sum_{k=1}^N\frac{\epsilon_k}{k}, &\stackrel{N\nearrow}{\approx}\frac{\log^2 N}{2}+\gamma_1+\gamma \log N+\gamma^2+ \sum_{k=1}^N\frac{\epsilon_k}{k},
\left\Vert p_{t}^{(\mathrm{ref})}\epsilon_{t}\right\Vert_{2}= \frac{1}{\Lambda}\left\Vert \phi_{t}^{(\mathrm{ref})}\right\Vert_{2}\left|\left\langle \phi_{t}^{(\mathrm{ref})},\epsilon_{t}\right\rangle \right| \leq \frac{1}{\Lambda^{1/2}}\left|\left\langle \phi_{t}^{(\mathrm{ref})},\epsilon_{t}\right\rangle \right|.
J^{( N )}\left( N ; 1 , \ldots , 1 \right) = 2 \mathrm{i}^{1 - 2 N}\pi^{N / 2}\frac{\Gamma \left({\textstyle{\frac{N}{2}}}\right)}{\prod m_{i}}\int_{0}^{\infty}\ldots \int_{0}^{\infty}\prod \mathrm{d}\alpha_{i}\delta \left( \alpha^{T}\| c \| \alpha - 1 \right) .
-A_{l}(z)=A_{k}(z)\frac{f(z+c_{k})}{f(z+c_{l})}+...+A_{l-1}(z)\frac{f(z+c_{l-1})}{f(z+c_{l})}+...+A_{1}(z)\frac{f(z+c_{1})}{f(z+c_{l})}+A_{0}(z)\frac{f(z)}{f(z+c_{l})}.
\begin{aligned}x_{i_m}^*(t_{q+1}) - x_{m+1}^*(t_{q+1}) ={}x_{i_m}^q(t_{q+1}) - x_{m+1}^q(t_{q+1}) begin{align*}2pt]\ge{}x_{i_m}^q(t_{q}+ \Delta \widetilde{t}_q ) - x_{m+1}^q(t_{q}+ \Delta \widetilde{t}_q ) = 0.\end{aligned}
\lim_{j\rightarrow\infty}\lim_{k\rightarrow \gamma_j}\frac{z_0^1(B^1;k)}{z_0^2(B^1;k)}=\lim_{j\rightarrow\infty,\,|\Gamma_j|\neq0}\frac{\lim_{k\rightarrow k_j}\frac{z_0^1(B^1;k)}{\sin\{k B^1\}/k}}{\lim_{k\rightarrow k_j}\frac{z_0^2(B^1;k)}{\sin\{k B^1\}/k}}.
{\mathcal M}= \left( \begin{array}{c}{\frac 1 2 ( M + M^{*})}\\{\frac 1{2 i}( M - M^{*})}\\ \end{array}\right) , \quad{\mathcal Z}= \left( \begin{array}{c}{\frac 1 2 ( Z + Z^{*})}\\{\frac 1{2 i}( Z - Z^{*})}\\ \end{array}\right) .
1540=(\begin{matrix}1540\\ 1\end{matrix})=(\begin{matrix}1540\\ 1539\end{matrix})=(\begin{matrix}56\\ 2\end{matrix})=(\begin{matrix}56\\ 54\end{matrix})=(\begin{matrix}22\\ 3\end{matrix})=(\begin{matrix}22\\ 19\end{matrix})
\begin{aligned}1 = \frac{u_{i,i}}{v_{i,i}}\cdot t_{i+1}\cdots t_{j-1}\cdot \frac{v_{j,j-1}}{u_{j,j-1}}= \frac{u_{i,i}}{v_{i,i}}\left(\frac{v_{i,i}}{u_{i,i}}\frac{u_{i,i+1}}{v_{i,i+1}}\right) \cdots \left(\frac{v_{i,j-2}}{u_{i,j-2}}\frac{u_{i,j-1}}{v_{i,j-1}}\right) \frac{v_{j,j-1}}{u_{j,j-1}}= \frac{u_{i,j-1}}{v_{i,j-1}}\frac{v_{j,j-1}}{u_{j,j-1}},\end{aligned}
N^*(1) =\int_{[x,1)}\![S^*(y)-S^*(x)]m^*(y)\,dy=\int_{[x,1)}\!\left[\frac{1}{M(x)}-\frac{1}{M(y)}\right]s(y)M^2(y)\,dy &=\int_{[x,1)}\!s(y) \frac{M^2(y)}{M(x)}\,dy-\int_{[x,1)}\!s(y) M(y)\,dy &\leq \frac{M(1)}{M(x)}\int_{[x,1)}\!s(y)M(y)\,dy-\int_{[x,1)}\!s(y) M(y)\,dy<\infty
\|c\|^2 \leq \sum_{u=1}^P \sum_{r=0}^{t-1}\|b_u^r\|^2 + 2\sum_{u=1}^P\sum_{r=0}^{t-1}\sum_{l=0}^{t-1}\hat{\gamma}^t_{u,r}\hat{\gamma}^t_{u,l}(b_u^r)^*b_u^l + 2\sum_{u=0}^P (\sigma_u^t)^2_{\perp}\|Z_u^{'t}\|^2 + \|w\|^2.
\mathcal{I}_1\ge \frac{c^2_2}{p\,2^{p+1}}\,\left\|\frac{|u|^\frac{2}{p+1}\, U^\frac{1}{p+1}}{\displaystyle\left(\int_\Omega u^2\, U\, dx\right)^\frac{1}{p+1}}-\frac{1}{U^\frac{p}{p+1}}\right\|^{p+1}_{L^{p+1}(\Omega)},
\gamma^{'}_{\xi}=\sum_{j=1}^n\Big(x_j^{'}(s)\mathring{e}_j+y_j^{'}(s)\mathring{e}_{n+j}\Big), \gamma^{'}_{T}=\sum_{j=1}^n \Big(z^{'}(s)+x_iy_i^{'}-y_ix_i^{'}\Big)T,
L_{1 / 4}= \frac{1}{2}h_{i j}( y ) \dot{y}^{i}\dot{y}^{j}+ \frac{1}{2}L_{\alpha \beta}( y ) ( \dot{\chi}^{\alpha}+ w_{i}^{\alpha}( y ) \dot{y}^{i}) ( \dot{\chi}^{\beta}+ w_{j}^{\beta}( y ) \dot{y}^{j}) - \frac{1}{2}L_{\alpha \beta}( y ) a^{\alpha}a^{\beta}.
A = C_{b,\gamma}^2 \, \frac{1}{b}\, \frac{1 - e^{-\frac{2 \gamma}{b}}}{1-e^{-2 \gamma a}}\, e^{-2 \gamma a}B = C_{b\gamma}^2 \, \frac{1}{b}\, \frac{1 - e^{-\frac{2 \gamma}{b}}}{1-e^{-2 \gamma a}}.
\Phi(x)=\frac{1}{2}+\varphi(x)(x+\frac{x^{3}}{3}+\frac{x^{5}}{3\cdot5}+\frac{x^{7}}{3\cdot5\cdot7}+\frac{x^{9}}{3\cdot5\cdot7\cdot9}+\cdot\cdot\cdot)
\left( \frac{d}{dt}\right)^{\ell}\vartheta_{3}(t)\simeq \frac{(-1)^{\ell}\ell !}{t^{1+\ell}}+\sum\limits_{j=\ell}^{+\infty}\frac{\left( -1\right)^{j}B_{2j+2}}{(j+1)(j-\ell)!}t^{j-\ell}
\mathcal{\tilde{F}}_{n}^{\left( j\right)}\left( x\right) \Delta \nabla \mathbb{K}_{n}^{(j)}\left( x\right) +\mathcal{\tilde{G}}_{n}^{\left(j\right)}\left( x\right) \Delta \mathbb{K}_{n}^{(j)}\left( x\right) +\mathcal{\tilde{H}}_{n}^{\left( j\right)}\left( x\right) \mathbb{K}_{n}^{(j)}\left( x\right) =0,
\bar{\nu}\cdot\nabla\rho_\eta^\varepsilon = -k_d^{-1}\Bigl(\overline{V_\Gamma\eta}\Bigr)+\varepsilon\Bigl(\bar{g}_1\bar{\zeta}_0-\bar{g}_0\bar{\zeta}_1\Bigr)+d\Bigl(\bar{\zeta}_1-\bar{\zeta}_0\Bigr) \quad\overline{Q_{\varepsilon,T}}.
\tau(\mathbf{S}_{\varphi}, \bar x)=\limsup_{x\stackrel{{\rm bd}(\mathbf{S}_{\varphi})}\longrightarrow \bar x}\mathbf{e}\big(\varphi'_\mathbf{H}(x,\cdot)^{-1}(-\infty, 1], \mathbf{T}^\mathbf{B}(\mathbf{S}_{\varphi}, x)\big).
\varepsilon_{\mu\nu\lambda\rho}\partial_\lambda\partial_\rho\theta_i^{\rm sing}(x)=2\pi\Sigma_{\mu\nu}^i(x)\equiv2\pi\int\limits_{\Sigma_i}^{}d\sigma_{\mu\nu}\left(x^{(i)}(\xi)\right)\delta\left(x-x^{(i)}(\xi)\right).
\|t\partial_tG(m,t)\|_{\mathbb{B}}= \|te^{-2t}(I_{m+1}(2t) - 2I_{m}(2t) + I_{m-1}(2t))\|_{\mathbb{B}}&\le C\frac{S_1^{1/2}+S_2^{1/2}}{\Gamma(m-1/2)},
\int x \cos(x;q)d_qx = - q^{-\frac{1}{2}}x \sin( q^{-\frac{1}{2}}x;q) - \cos(\frac{x}{q};q), & \int x \sin(x;q)d_qx = \frac{x}{q}\cos( q^{-\frac{1}{2}}x;q) - \sin(\frac{x}{q};q),
\gamma_6 = \alpha_3^2 = q^2 \left(s-6qs+15q^2s-20q^3s+15q^4s-6q^5s+q^5-2q^6+2q^6s\right)^2 \le q^2 \left(s+6qs+15q^2s+20q^3s+15q^4s+6q^5s+q^5+2q^6+2q^6s\right)^2 \le q^2s^2 \left(1+6q+15q^2+20q^3+15q^4+6q^5+q^4p+2q^5p+2q^6\right)^2 \le (68)^2 q^2s^2\le 2^{13}q^2 s^2\, .
[M^\bullet]= [Z^\bullet]*[U_2^\bullet]^{-1}=[H(M^\bullet)]* [ U_1^\bullet]*[U_2^\bullet]^{-1}= [H(M^\bullet)]*[ K_{\Im d_{U_1^\bullet}}]*[K_{\Im d_{U_2^\bullet}}]^{-1}= [H(M^\bullet)\oplus K_{\Im d}].
f(G_{i+1})-f(G_i) \geq \frac{1}{K-i}(f(G_i\oplus \bar{O}_{K-i})-f(G_i)) &\geq\frac{1}{K-i}\cdot\frac{K-i}{K}(f(O_K)-\eta f(G_i)) &= \frac{1}{K}(f(O_K)-\eta f(G_i)),
T_1 = -\int (\partial_{k}\sigma^{ij})(\partial_{i}\partial_{x_{k}}h) (\partial_j h)\varrho\,m_0 - \int (\partial_{k}\sigma^{ij})(\partial_{x_{k}}h)(\partial_{j}h) \partial_{i}(\varrho\,m_0)=: T_{11}+T_{12}.
I = ( I^{( 1 )}| I^{( 2 )}) = ( I_{1}^{( 1 )}, I_{2}^{( 1 )}, \ldots , I_{m_{1}}^{( 1 )}| I_{1}^{( 2 )}, I_{2}^{( 2 )}, \ldots , I_{m_{2}}^{( 2 )})
\overline F\big({\left\|{\sigma ( \cdot ,z, \cdot )}\right\|_{L^\infty \left({\mathbb{R}_+\times \mathbb{R}^2\times\mathbb{R}_+}\right)}}\big)= (1/A_0) \Big[ \left\|{\sigma(\cdot,z,\cdot)}\right\|_{L^\infty\left({\mathbb{R}_+\times \mathbb{R}^2 \times\mathbb{R}_+}\right)}\times
+{i\over 2}\sum_{i=2}^h\int_{{\cal C}_i}{\log}|L_i'|^2{{\overline L_i''}\over{\overline L_i'}}d\bar z+4\pi\sum_{i=2}^h{\log}\left|{(1-\lambda_i)^2 \over \lambda_i(a_i-b_i)^2}\right|,
\begin{matrix}{{\zeta^{+}= \eta_{z}^{+}\mathrm{e}^{\varphi}d z ,}}&{{\zeta^{-}= \eta_{z}^{-}\mathrm{e}^{\varphi}d z ,}}&{{\tilde{\zeta}_{+}= \eta_{+ \bar{z}}\mathrm{e}^{\varphi}d \bar{z},}}&{{\tilde{\zeta}_{-}= \eta_{- \bar{z}}\mathrm{e}^{\varphi}d \bar{z},}}\\ \end{matrix}
\exp(x):=\left\{0,\,\exp(x')\cdot[x-x']_{n},\,\exp(x'')[x-x'']_{2n+1}\right\}\mid\left\{\frac{\exp(x'')}{[x''-x]_{n}},\,\frac{\exp(x')}{[x'-x]_{2n+1}}\right\},
-2T_0^3T_2 + 37T_0^2T_1T_2 + 67T_0^2T_2^2 + 2T_0T_1^3 - 10T_0T_1^2T_2 + 114T_0T_1T_2^2 + 166T_0T_2^3 + 4T_1^4{}- 168T_1^3T_2 + 42T_1^2T_2^2 + 369T_1T_2^3 - 45T_2^4 = 0
\{{\cal P}_{0 , 0}^{a}, X^{\pm}\}= \{{\cal P}_{0 , 0}^{a}, A \}= \{\bar{{\cal P}}_{0 , 0}^{\bar{a}}, X^{\pm}\}= \{\bar{{\cal P}}_{0 , 0}^{\bar{a}}, A \}= 0 ,
\phi =X^\flat\wedge T^\flat-(JX)^\flat\wedge (JT)^\flat , &={e^u}\left(dx\wedge\left(\frac{1}{2}(\textbf{H}-1)d\theta+d\varphi\right) +dy\wedge \left(\frac{1}{2}(\textbf{H}+1)d\theta+d\varphi\right) \right),
( \nabla v_{\alpha q}^{\underline{\alpha}}) v_{\underline{\alpha}}^{\beta r}={\frac{1}{4}}\delta_{q}^{r}\hat{\Omega}^{a b}( \gamma_{a b})_{\alpha}^{\beta}+{\frac{1}{4}}\delta_{\alpha}^{\beta}\hat{\Omega}^{i j}( \gamma_{i j})_{q}^{r},
A_{a}= - i \kappa \ \xi \ \bar{\psi}^{1}\gamma_{a}\psi^{2}+ \kappa \ \theta \ \bar{\psi}^{1}\gamma_{5}\gamma_{a}\psi^{2}-{\frac{1}{2}}\kappa \ \varphi \ ( \bar{\psi}^{1}\gamma_{5}\gamma_{a}\psi^{1}- \bar{\psi}^{2}\gamma_{5}\gamma_{a}\psi^{2}) + . . . \ , \,
F_{\nu}^{ext}= \, - \frac{\alpha_f \hbar c}{2 \pi}\int d^3r \vec{R}_{\nu\mu}^{ext}\vec{J}^\mu = \int \hspace{0.5mm}{^*}\hspace{-0.8mm}\vec{F}_{\nu\mu}^{ext}d^3r \vec{j}^\mu .
\frac{1}{(1-u^2)^{r+1}}\begin{bmatrix}\frac{u^4}{1-u}((1-u)^r -1) - \frac{u^3}{1-u}((1-u^2)^r-1) -\frac{u^6}{1-u^3}((1-u)^r-1) +\frac{u^3}{1-u^3}((1-u^4)^r -1) \end{bmatrix}
K^{W}\; : = \; \mathrm{d i a g}\left( \frac{\pi}{\pi + g N}\frac{1}{- \triangle + \frac{e^{2}N}{\pi + g N}}\; , \; \frac{1}{- \triangle + 1}\; , \; . \; . \; . \; . \; , \; \frac{1}{- \triangle + 1}\right) \; .
\mathcal{M}_3 = \mathcal{M}^{(s-2)}_{0,\ldots, 0,\underset{i}{k_i+1},0,\ldots,0, \underset{j}{k_j}, k_{j+1},\ldots,k_{n}}, \mathcal{M}_4 = \mathcal{M}^{(s)}_{0,\ldots, 0,\underset{i}{k_i-1},0,\ldots,0,\underset{j}{k_j+1},k_{j+1},\ldots,k_{n}}.
v_{R,1}(s) \leqslant \frac1{(n-1)^2}\int_s^{+\infty}\frac{(p'+1/2)R^{1-1/p}-s_0^{1-1/p}/(p-1)}{t^2}dt &=\frac1{(n-1)^2}\frac{(p'+1/2)R^{1-1/p}-s_0^{1-1/p}/(p-1)}{s}
\frac{\Delta\rho}{\rho}+\frac{\nabla \Psi\cdot\nabla\rho}{\rho}+\Delta \Psi=\Delta\log\frac{\rho}{e^{-\Psi}}+\Big|\nabla\log\frac{\rho}{e^{-\Psi}}\Big|^{2}-\nabla \Psi\cdot\nabla\log\frac{\rho}{e^{-\Psi}},
(\lambda -3 ) v(\phi^2) +\phi^2 v'(\phi^2) =\frac{(N-1) v'(\phi^2)}{\left(2+2V'(\phi^2)\right)^{3/2}}+ \frac{v'(\phi^2) + 2\phi^2 v''(\phi^2)}{\left(2+ 2V'(\phi^2) + 4 \phi^2 V''(\phi^2)\right)^{3/2}}
\mathrm{e i t h e r}\; \; \tilde{{\cal S}}^{a b}= - \frac{i}{4}[ d x^{a}\; \wedge + \; e^{a}, \; d x^{b}\; \wedge + \; e^{b}] = - \frac{i}{4}[ \tilde{\gamma}^{a}, \tilde{\gamma}^{b}] , \quad \mathrm{o r}\quad \tilde{\tilde{\cal S}}{}^{a b}= \frac{i}{4}[ d x^{a}\; \wedge - \; e^{a}, \; d x^{b}\; \wedge - \; e^{b}] .
\begin{bmatrix}\begin{bmatrix}\mathbf{x}_{S}^{1}\mathbf{x}_{S^c}^{0}\end{bmatrix}\begin{bmatrix}\mathbf{x}_{S}^{0}\mathbf{x}_{S^c}^{1}\end{bmatrix}\end{bmatrix}\begin{bmatrix}\mathbf{B}^S \mathbf{N}^S\end{bmatrix}= \mathbf{e}_i.
S_{-}( \omega ) = \frac{( m + i k ) \left[ ( \omega + G \phi_{1}( 0 ) ) f^{\ast}( 0 ) +{f^{\prime}}^{\ast}( 0 ) \right]}{\omega \bigl [ ( \omega + G \phi_{1}( 0 ) ) f ( 0 ) +{f^{\prime}}( 0 ) \bigr ]}\, .