harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_{t}\phi=\frac{\partial}{\partial{t}}\left(\frac{\left\|\overline{\nabla}{f}\right\|^{2}}{(1-f)^{2}}\right)=\frac{\partial_{t}\left\|\overline{\nabla}{f}\right\|^{2}}{(1-f)^{2}}+\frac{2\left\|\overline{\nabla}{f}\right\|^{2}\partial_{t}f}{(1-f)^{3}}.
\rho = \sum_{a^{\prime}}\left[ \frac{i \omega_{a^{\prime}}}{\sqrt{\Im_{a^{\prime}}}}\lambda_{a^{\prime}}\pi_{a^{\prime}}+ \left( \omega_{a^{\prime}}^{2}G_{a^{\prime}}+ \frac{i}{2}b_{a^{\prime}}\right) \bar{\eta}_{a^{\prime}}\right] ,
\mathrm{var}[Z_i] \stackrel{(a)}{=}\mathbb{E}[X^2]\mathbb{E}[Y^{-2}]=\int_{\sqrt{\mu_i/\vartheta_i}}^{\infty}y^{-2}f_Y(y)\mathrm{d}y\stackrel{(b)}{=}2\int_{\sqrt{\mu_i/\vartheta_i}}^{\infty}\frac{1}{y}e^{\frac{\mu_i}{\vartheta_i}-y^2}\mathrm{d}y\stackrel{(c)}{=}e^{\frac{\mu_i}{\vartheta_i}}\mathrm{Ei}\Big(\frac{\mu_i}{\vartheta_i}\Big),
\sum_{i=1}^{2}\alpha_{i}^{J_{0}}\chi_{B(y_{i},2^{J_{0}}r)}= \alpha_{1}^{J_{0}}\chi_{B(y_{i},2^{J_{0}}r)}+ \alpha_{2}^{J_{0}}\chi_{B(y_{i}, 2^{J_{0}}r)}-\alpha^{J_{0}}\chi_{B(\frac{y_{1}+y_{2}}{2}, 2^{J_{0}+1})}+ \alpha^{J_{0}}\chi_{B(\frac{y_{1}+y_{2}}{2}, 2^{J_{0}+1})}&= f_{1}^{J_{0}+1}+ f_{2}^{J_{0}+1}&= \sum_{i=1}^{2}f_{i}^{J_{0}+1}.
\psi^{(11/2)}_{1A}(\tau,z) :=2i\frac{\theta_1(\tau,4z)}{\theta_1(\tau,z)\theta_1(\tau,2z)}\eta(\tau)^3 \psi^{(11/2)}_{2A}(\tau,z) :=-2i\frac{\theta_1(\tau,4z)}{\theta_2(\tau,z)\theta_2(\tau,2z)}\eta(\tau)^3
N_{t}= -\frac{\kappa_{t}}{\kappa}N+\frac{\kappa_{ss}}{\kappa}B-2\frac{\kappa_{s}\tau}{\kappa}N-\tau_{s}N+\kappa\tau T-\tau^{2}B. &= -\Big(\frac{\kappa_{t}}{\kappa}+2\frac{\kappa_{s}\tau}{\kappa}+\tau_{s}\Big)N+\tau\kappa T+\Big(\frac{\kappa_{ss}}{\kappa}-\tau^{2}\Big)B.
R_2 =\Big\{(x_1,x_2,...,x_{k}) \mid x_k= m;\x_{k-1}=0;\ \ 0\leq x_i\leq 2\ \ \ 1\leq i\leq k-2, &\ \ \ \ \ \ \ \ \ \ \ \x_i=2,\ \ \ \x_j=0\ \ \ \j< i\leq k-2\Big\}.
c_{1}(E_{1}^{(0)}+H^{\prime}-E)\langle\Phi_{2}^{(0)}|\Phi_{1}^{(0)}\rangle+c_{2}(E_{2}^{(0)}+H^{\prime}-E)\langle\Phi_{2}^{(0)}|\Phi_{2}^{(0)}\rangle=0
\left\{{\rm e}^{-\frac{A}{2}}\not\hspace{-0.1cm}{\partial}_x+\Gamma^r\left(\partial_r+{D_1\over 4}A'+{D_2\over 4}B'\right)+{\rm e}^{-{B\over 2}}\not\hspace{-0.1cm}{\Delta}_y\right\}\Psi=0,
\sum_{\substack{p\leq x p\in \mathcal{P}_u}}\log \left(1-\frac{1}{p}\right)^{-1}=\sum_{\substack{p\leq x p\in \mathcal{P}_u}}\sum_{k\geq 1}\frac{1}{k p^k}=\sum_{\substack{p\leq x p\in \mathcal{P}_u}}\frac{1}{p}+\sum_{\substack{p\leq x p\in \mathcal{P}_u}}\sum_{k\geq 2}\frac{1}{k p^k}.
\Pi_{3 B}~ = ~ - \, \frac{2 \pi^{4 \mu}}{( \mu - 1 )^{3}\Gamma^{4}( \mu ) \Delta}\left[ 1 - \frac{\Delta ( \mu - 1 )}{2}\left( 3 \Theta + \frac{1}{( \mu - 1 )^{2}}\right) \right]
\P(x^1_{1:n},x^2_{1:n},y_{1:n}) = \prod_{t=1}^n Q(y_t| x^1_t,x^2_t)q^1_t(x^1_t|x^1_{1:t-1},y_{1:t-1})\times & q^2_t(x^2_t|x^2_{1:t-1},y_{1:t-1}),
S_{{\mathrm{e f f}}}= \frac 1 2 \int d \tau \left[ \dot{\vartheta}^{2}- \frac{( \pi_{u}- \pi_{v})^{2}}{4 \sinh^{2}\frac{\vartheta}{2}}+ \frac{( \pi_{u}+ \pi_{v})^{2}}{4 \cosh^{2}\frac{\vartheta}{2}}\right] \, ,
|v_{i,n}(x) - \bar v_{i,n}(x)|= \frac{|u_{i,n}(x_n+r_n x)|}{L_n r_n^{\alpha}}|\eta(x_n)-\eta (x_n+r_n x)| \le \frac{l M r_n^{1-\alpha}}{L_n}|x|,
|M^i_{[\alpha, \psi_{\alpha}]}| \leq A^i_{[\alpha, \psi_{\alpha}][\alpha, \psi_{\alpha}]}\;\;\;\;\;{\rm and}\;\;\;\;\; M^i_{[\alpha, \psi_{\alpha}]}= A^i_{[\alpha, \psi_{\alpha}][\alpha, \psi_{\alpha}]}\;\;\;\;{\rm mod}\;\; 2 \;\;\;\; .
\psi (a \cdot u) = \phi(a) \cdot \psi(u) ~~~ ~~~ \psi (u \cdot a) = \psi(u) \cdot \phi(a), &\psi \circ H = H' \circ ( \phi \otimes \phi), &\phi \circ T = T' \circ \psi, ~ a \in A, u \in M.
i_Z^!\circ p_{13*}\circ (p_{12}\circ p_{23})^!(-) =q_{T*}\left(e(Y)^{-2}\cdot_2 (i_{Z^{(2)}})^!_{i_{Y^4}}\circ (p_{12}\circ p_{23})^!(-)\right) & = q_{T*}\left(e(Y)^{-2}\cdot_2 (p_T)^!_{p_{12}\times p_{23}}\circ i_{Z^2}^!(-)\right) & = q_{T*}\left(e(Y)^{-1}\cdot_2 p_T^*\circ i_{Z^2}^!(-)\right),
\Delta_0(r^\alpha\varphi) = r^{\alpha-2}A_\alpha\varphi, \Delta_0^2(r^\alpha\varphi) = \Delta_0(r^{\alpha-2}A_\alpha\varphi)=r^{\alpha-4}A_{\alpha-2}A_\alpha\varphi, \Delta_0^3(r^\alpha\varphi) = r^{\alpha-6}A_{\alpha-4}A_{\alpha-2}A_\alpha\varphi.
\gamma (\beta\sqrt{-\triangle}) =\frac{2 \pi^2}{\beta \sqrt{-\triangle}}+\frac{8 \pi}{\beta \sqrt{-\triangle}}\left( \sum_{k=1}^{\infty}{\mathrm arctan}\Big( \frac{\beta\sqrt{\triangle}}{4 \pi k}\Big) - \int_{0}^{\infty}\!{\mathrm d}\kappa\,{\mathrm arctan}\Big( \frac{\beta\sqrt{\triangle}}{4 \pi \kappa}\Big)\right).
\begin{aligned}(f_{\mathcal{S}})_\lambda(s_1, s_2) \leq \sup_{(y_1^\prime, y_2^\prime, x^\prime ) \in \mathcal{S}}\left\{f_1(y_1^\prime) + f_2(y_2^\prime) - \sum_{1 \leq \ell \leq 2}\lambda_\ell c_{Y_\ell}\left( y_\ell, y_\ell^\prime \right) \right\}&= (f_1 )_{\lambda_1}(y_1) + (f_2 )_{\lambda_2}(y_2).\end{aligned}
\det M(s^{(r)},{p}^{(r)})^2 =\det M(s^{(r)},{Q'}^{(r)})^2\det M({Q'}^{(r)},p^{(r)})^2 &=1\times \frac{1}{r^{2c_{r,n}}{{\prod_{\rho \in CP_{r,n}}\prod_{i \geq 1}\rho_{i}^2}}}
\partial_{\zeta_i'}A(\phi;\mu',\zeta') =-\partial_{\zeta_i'}T( N(\phi;\mu',\zeta') + E;\mu',\zeta')-T( \partial_{\zeta_i'}N(\phi;\mu',\zeta') ;\mu',\zeta') & - T( D_{\zeta_i'}E;\mu',\zeta') .
\widetilde{\Phi}^2_c\simeq \frac{4!4N\widetilde{h}^2\widetilde{m}^2}{(4\pi)^2}\left(\frac{N\widetilde{h}^4}{(4\pi)^2}- \widetilde{\lambda}\right)^{-1}\simeq \frac{96\xi\widetilde{m}^2}{\widetilde{h}^2}, \ \ \ \ \R_c \simeq\frac{2\widetilde{m}^2}{\widetilde{\xi}}.
\chi_1|_{I_K}= \prod_{\tau \in J}\omega_\tau^{a_\tau + 1}\prod_{\tau \not\in J}\omega_\tau^{b_\tau}, \chi_2|_{I_K}= \prod_{\tau \not\in J}\omega_{\tau}^{a_\tau + 1}\prod_{\tau \in J}\omega_{\tau}^{b_\tau}
\frac{d}{dt}M_{\chi_R}[u(t)]=\langle u(t), [(-\Delta)^{s_1}+(-\Delta)^{s_2},i\Gamma_{\chi_R}]u(t)\rangle+ \langle u(t), [-|u|^{p-2}u,i\Gamma_{\chi_R}]u(t)\rangle.
\widetilde{{S}}_{\lambda^3}\widetilde{{S}}_{\lambda^3}=(y_{4}-y_{2})(y_{4}-y_{3})\widetilde{{S}}_{\lambda^3}+(y_{4}-y_{3})\widetilde{{S}}_{\lambda^1}+\widetilde{{S}}_{\lambda^0}=(Y_{\lambda^5}-Y_{\lambda^3})(Y_{\lambda^2}-Y_{\lambda^1})\widetilde{{S}}_{\lambda^3}+(Y_{\lambda^2}-Y_{\lambda^1})\widetilde{{S}}_{\lambda^1}+\widetilde{{S}}_{\lambda^0}.
\int_0^t \gamma(E(e,s))ds \leq \int_0^{t^*(e)}\gamma(E(e,s))ds + \int_{t^*(e)}^t\gamma(E(e,s))ds \ , &\leq \int_0^{t^*(e)}\gamma(E(e,s))ds + Lk(|e|)|e|\int_{t^*(e)}^t\exp(-\lambda s) ds \ , &\leq \int_0^{t^*(e)}\gamma(E(e,s))ds + \frac{Lr}{\lambda}:= c(e) \ .
P=\sum_{\vec{k},\vec{l}}c_{\vec{k}\vec{l}}\partial_{x_{1}}^{k_{1}}\partial_{x_{2}}^{k_{2}}\cdot\cdot\cdot\partial_{x_{n}}^{k_{n}}x_{1}^{l_{1}}x_{2}^{l_{2}}\cdot\cdot\cdot x_{n}^{l_{n}}.
\frac{1-\int_{U_-}p(t,|x-y|)dx}{t}\ge \frac{1-\int_{B(y,R)}p(t,|x-y|)dx}{t}= \frac{1}{t}\left( 1 - \P^y (X_t \in B(y,R)) \right) = \frac{\P^0 (|X_t| \ge R)}{t}.
\psi_x\big((a\cdot\delta_u)*(a\cdot\delta_u)^*\big) =\psi_x\big(aa^*\cdot\delta_{uu^{-1}}\big) &=\|\xi(x)\|^{-2}\big(\pi(aa^*)\xi(x)\big|\xi(x)\big) &=\|\xi(x)\|^{-2}\big(\pi(a^*)\xi(x)\big|\pi(a^*)\xi(x)\big) &\geq 0
B(t,x; \eta) \ = \ \big\{(s,y) \in (0,T) \times H: \max \{|x-y|, |t-s|\}< \eta\big\}, \partial B(t,x; \eta) \ = \ \big\{(s,y) \in (0,T) \times H: \max \{|x-y|, |t-s|\}= \eta\big\}.
\begin{aligned}|\sqrt{x_ix_j}u_{x_ix_j}(P)| \leq C \eta^{\alpha/2}[\sqrt{x_ix_j}u_{x_ix_j}]_{C^{\alpha}_{WF}([0,T]\times\bar S_{n,m})}+2\eta^{-1}\|\sqrt{x_ix_j}u_{x_i}\|_{C([0,T]\times\bar S_{n,m})},\end{aligned}
\! \! \!{\cal E}_{n}^{( 1 )}\propto \! \! \! \int_{\infty}^{\infty}\Pi_{j = 1}^{n}d z_{j}( \chi_{n}^{( 0 )})^{*}( \sum_{i < j}^{n}\delta ( z_{i}- z_{j}) \partial_{z_{i}}) \chi_{n}^{( 0 )}\propto \! \! \! \int_{\infty}^{\infty}\Pi_{j = 1}^{n}d z_{j}( \chi_{n}^{( 0 )})^{*}\chi_{n}^{( 0 )}( \sum_{i \neq k}^{n}\mathrm{s i g n}( z_{i}- z_{k}) ) = 0 ,
\mathsf{E}_{\mathcal{R}}\left( J,\omega \right)^{2}=\mathsf{E}_{\mathcal{R}}^{1}\left( J,\omega \right)^{2}+...+\mathsf{E}_{\mathcal{R}}^{n}\left(J,\omega \right)^{2}=\mathsf{E}\left( J,\omega \right)^{2}.
\mathfrak{X}_{\mathfrak{p}}^{\psi}= \mathfrak{X}_{\mathfrak{p}}\otimes_{\mathbb{Z}_p[\Delta]}\O_{\psi},\mathfrak{X}_{\overline{\mathfrak{p}}}^{\psi}= \mathfrak{X}_{\overline{\mathfrak{p}}}\otimes_{\mathbb{Z}_p[\Delta]}\O_{\psi}.
G_2(x)-G_2(x+1) = \sum_{k=0}^{\infty}\frac{1}{(k+x+u)\left(k+x+\frac{u+v}{2}\right)\left(k+x+v\right)}&-\sum_{k=0}^{\infty}\frac{1}{(k+1+x+u)\left(k+1+x+\frac{u+v}{2}\right)\left(k+1+x+v\right)}= \frac{1}{(x+u)\left(x+\frac{u+v}{2}\right)\left(x+v\right)}.
D_{v}(e_{k}) = -\epsilon_{k}\sum_{j}\omega_{jk}(v) e_{j}= 2\sum_{j<p}\omega_{pj}(v) (\epsilon_{p}e_{p}^{*}\otimes e_{j}-\epsilon_{j}e_{j}^{*}\otimes e_{p}) (e_{k}) & = 2\sum_{j<p}\omega_{pj}(v) (e_{p}\wedge e_{j}) (e_{k}),
\begin{array}{l}a^q(x_1^{2q}+\ldots+x_{r-1}^{2q})+(b-b^q)(x_1^{q+1}+\ldots+x_{r-1}^{q+1})+m_1^qx_1^{q}+ +\ldots+m_{r-1}^qx_{r-1}^{q}+d^q-a(x_1^2+\ldots+x_{r-1}^2)-m_1x_1+\ldots+ -m_{r-1}x_{r-1}-d=0 .\end{array}
\frac{1}{n}\sum_{l=1}^n|p_l(i,j)-p(i,j)| \leq \frac{1}{2^k}\sum_{l=1}^{2^{k+1}-1}|p_l(i,j)-p(i,j)| \leq \frac{1+2+3+\cdots+(k+1)}{2^k}\frac{1}{6}=\frac{1}{2^k}\frac{(k+2)(k+1)}{2}\frac{1}{6}\rightarrow 0 \ (k\rightarrow \infty).
f_3:\Gamma_2 \oplus \Gamma_1 = \Gamma_2 \oplus Y \oplus \Lambda_1 \phantom{xxxxxxxxxxxxxxxxxxxx}&\cong \underbrace{f_2(\Gamma_2 \oplus Y)}_{\prec_\exists \, \Gamma_2}\oplus \underbrace{f_1(\Lambda_1)}_{= \,\Gamma_1}&\prec_\exists \Gamma_2 \oplus \Gamma_1
\lim_{x \rightarrow 1^-}(1-x) \sum_{n=0}^{\infty}a_n x^n = \lim_{n \rightarrow \infty}a_n &= \lim_{x \rightarrow 1^-}(1-x) G(x) &= \lim_{x \rightarrow 1^-}(1-x) \frac{g(x)}{\prod_{n=0}^{\infty}\left(1-xq^{Nn}\right)}&= \frac{g(1)}{\prod_{n=1}^{\infty}\left(1-q^{nN}\right)}.
G_{D}^{( n )}({\bf r^{\prime \prime}},{\bf r^{\prime}}; E ) = \sigma^{n}\, \left[ G_{D}^{( 0 )}({\bf 0},{\bf 0}; E ) \right]^{n - 1}\, G_{D}^{( 0 )}({\bf r}^{\prime \prime},{\bf 0}; E ) \, G_{D}^{( 0 )}({\bf 0},{\bf r}^{\prime}; E ) \; ,
\frac{\tilde{C}_{1 2 3}}{C_{1 2 3}^{( \mathrm{v a c )}}}= - \frac{J}{\sqrt{J_{1}J_{2}}}\frac{1}{n^{2}\pi^{2}}\sin^{2}( \frac{n \pi J_{1}}{J}) \cdot ( 1 - \frac{\l^{\prime}n^{2}}{2}) ,
\begin{cases}\begin{aligned}\bar h^{1^*}_{11k}+\bar h^{1^*}_{22k}=0, &\bar h^{2^*}_{11k}+\bar h^{2^*}_{22k}=0, &\bar \lambda_1\bar h^{1^{\ast}}_{11k}+4\bar \lambda \bar h^{2^{\ast}}_{11k}=0.\end{aligned}\end{cases}
\|u_i\|_2\leq\sqrt{\|u_i\|_0}\cdot \|u_i\|_\infty =\sqrt{k(t-1)-\ell}\cdot \|u_i\|_\infty & \leq\sqrt{k(t-1)}\cdot \|u_i\|_\infty & \leq\alpha\sqrt{k/(t-1)}.
\begin{array}{l}(n-2)fh\varphi,_{x_ix_j}-rf\varphi h,_{x_ix_j}-mh\varphi f,_{x_ix_j}-mh\varphi,_{x_i}f,_{x_j}-mh\varphi,_{x_j}f,_{x_i}-rf\varphi,_{x_i}h,_{x_j}-rf\varphi,_{x_j}h,_{x_i}=0, \ \ \ \forall i,j=1,\ldots,n, \i\neq j,\end{array}
b'(p) \geq (1-\frac{1}{k})^{-\frac{p}{2k}}[ 1 + \frac{2-k-k^2}{2k(k-1)}] = (1-\frac{1}{k})^{-\frac{p}{2k}}[ 1 - \frac{k+2}{2k}] \geq 0
\sum_{k = 1}^n \mathfrak a(t_k) \ge \mathfrak a( 1 - \sum_{k = 1}^n (-1)^k t_k), , \sum_{k = 1}^n \mathfrak a(t_k) \ge \mathfrak a(- \sum_{k = 1}^n (-1)^k t_k), .
D_{2,+}(x) = \begin{pmatrix}0 1 0 1 0 0 0 0 1 \end{pmatrix}D_{2,-}(x)\begin{pmatrix}0 1 0 1 0 0 0 0 1 \end{pmatrix}, x > 0, D_{2,+}(x) = \begin{pmatrix}1 0 0 0 0 1 0 1 0 \end{pmatrix}D_{2,-}(x)\begin{pmatrix}1 0 0 0 0 1 0 1 0 \end{pmatrix}, x < 0.
r(t,y)= \sum_{\substack{i,j=1 j\not=i}}^n\frac{A_{ij}(t,y-x_{i}(t))}{x^2_{ij}(t)}+\sum_{\substack{i,j=1 j\not=i}}^n\frac{B_{ij}(t,y-x_{i}(t))}{x_{ij}^3(t)}\varphi_{ij}(t,y),
B'_n(v) =C_{n-2}(v^{n-2}+v^{n-3})+2(B'_{n-1}(v)-C_{n-3}v^{n-3}) &+v(B_n(v)-b(n;n-3)v^{n-4}-b(n;n-2)v^{n-3}-b(n;n-1)v^{n-2}).
\frac{f(-1+h+a)}{2 r_x}= \begin{cases}- \frac{1}{r_x}+ h \left(1 + \frac{1}{r_x}\right) \mbox{if $a - 2 + h \ge 0$}, \frac{1}{r_x}\mbox{if $a - 2 + h \leq 0$}\end{cases}
\| X^k-X^i \|^p_{S^p_{ad}(\Omega; \mathcal{D}_t)}\leq \| X^2-X^1 \|^p_{S^p_{ad}(\Omega; \mathcal{D}_T)}\sum_{j=\min\{k,i\}}^{\infty}\frac{(LTC)^{j-1}}{(j-1)!}\to 0 \, ,\mbox{as}k , \, i \to \infty \, ,
\begin{array}{c}b^Ab_A=0\quad ,\quad b^A\sim ab^A\quad ,\quad h^{ABC}\sim ah^{ABC}\quad,\quad a\in R\backslash \left\{0\right\}h^{ABC}=-\frac 16\epsilon^{ABCDEF}h_{DEF}\qquad ,\qquad b_Ah^{ABC}=0 h^{ABC}h_{CDE}=\frac 14\delta^{[A}{}_{[D}b^{B]}b_{E]}\end{array}
|\langle\mu,\mu\rangle-2i\,\langle\mu,\xi+i\eta\rangle| =|\langle\mu,\mu\rangle+2\,\langle\mu,\eta\rangle-2i\langle\mu,\xi\rangle| &\geq \langle\mu,\mu\rangle-2\|\mu\|\,\|\eta\| &> \frac 12\langle\mu,\mu\rangle.
C_{+}={\frac{\ell^{4}}{4 ( r_{+}^{2}+ r_{( - )}^{2})^{2}}}\left[ \omega^{2}r_{+}^{2}- J \mu \omega + \left( - M +{\frac{r_{+}^{2}}{\ell^{2}}}\right) \mu^{2}\right] .
\begin{aligned}k^{\infty}_{j+1}(T) \leq k^{\infty}_{0}(T)+C \tilde{C}_{T}\Big[k^{u}_{j}(T) k^{\nabla y}_{j}(T)+ k^{\nabla y}_{j}(T)^2(k^{\infty}_{j}(T)+ \lvert \overline{b}\rvert ) \Big] &\leq k^{\infty}_{0}(T)+C \tilde{C}_{T}\Big[k^{q}_{j}(T)^2 (1+k^{\infty}_{j}(T)+ \lvert \overline{b}\rvert ) \Big].\end{aligned}
\begin{array}{rcl}\dot{x}_1 = - p_1 x_1 + p_2 x_2 - p_5 x_1 x_3 \dot{x}_2 = p_1 x_1 - p_2 x_2 - 2 p_3 x_2^2 + 2 p_4 x_3 + p_5 x_1 x_3 \dot{x}_3 = p_3 x_2^2 - p_4 x_3 \end{array}
t_2(2^mn+4) = t_2(2^{m-1}n+2)+t_2(2^{m-1}n+1) = t_2(2^{m-2}n+1) + t_2(2^{m-2}n)-2t_2(2^{m-2}n) = -2t_2(2^{m-3}n) -t_2(2^{m-3}n)-t_2(2^{m-3}n-1) = -3t_2(2^{m-3}n)-t_2(2^{m-3}n-1) = -3(t_2(2^{m-4}n+t_2(2^{m-4}n-1))+2t_2(2^{m-4}n-1) = -3t_2(2^{m-4}n)-t_2(2^{m-4}n-1) = \ldots = -3t_2(n)-t_2(n-1).
\delta \Psi_{\hat{a}}= \partial_{\hat{a}}\epsilon +{\frac{1}{8}}H^{- 1}( \partial_{\hat{a}}H ) \epsilon +{\frac{1}{8}}W \gamma^{\underline{{u x y}}}\gamma_{\underline{{\hat{a}}}}( i \sigma_{2}) \epsilon = 0 \,
\lambda^-(\mu)=\lim_{t\rightarrow\infty}\dfrac{1}{t}\log\parallel\psi^*_t|\mathcal{E}^s_x\parallel<0,~~~\lambda^+(\mu)=\lim_{t\rightarrow\infty}\dfrac{1}{t}\log m(\psi^*_t|\mathcal{F}^u_x)>0
W_a(f,g) =\frac{-W_a(f,k)W_a(g,h)+W_a(f,h)W_a(g,k)}{W_a(h,k)}=\frac{-\vec f_a(k)\vec g_a(h)-\vec f_a(h)\vec g_a(k)}{W_a(h,k)}.
G(z_1,z_2,z_3,z_4) = \sum_{h,k \leq T^\theta}\frac{a_h \overline{a_k}}{[h,k]}B_{z_1,z_2,z_3,z_4}\left(\frac{h}{(h,k)}\right)B_{z_3,z_4,z_1,z_2}\left(\frac{k}{(h,k)}\right),
\sqrt{2}V_{L}^{\vphantom{T}}L \partial_{i}V_{R}^{T}= - e^{- ( A + \eta / 2 )}E_{i}^{( L )}\hat{n}- e^{A - \eta / 2}{\frac{\hat{g}_{i j}}{\sqrt{\hat{g}}}}( B_{j}^{( L )}+ \epsilon_{j k l}E_{k}^{( L )}C_{l}) \hat{m}\ .
u_2 (t) = M(t) D(t) \mathcal{F}M(t) \mathcal{F}^{-1}v_2(t) &= M(t) D(t) v_2 (t) + O(t^{-3/4 + C_4 \varepsilon_1^2}) &= M(t) D(t) e^{-3 i \lambda_1 (\Phi_1-\theta_1)}w + O(t^{-3/4 + C_4 \varepsilon_1^2}).
\phi^1 :=du_{xx}-a_{1}dx-a_{2}dy-a_{3}dz, \phi^2 :=du_{xy}-a_{4}dx-a_{5}dy-a_{6}dz, \phi^3 :=du_{yy}-a_{7}dx-a_{8}dy-a_{9}dz, \phi^4 :=du_{xz}-a_{10}dx-a_{11}dy-a_{12}dz, \phi^5 :=du_{yz}-a_{13}dx-a_{14}dy-a_{15}dz.
a^2I(a,b) =\frac1a K_4'\left(\frac{b}{a}\right) &=\frac{2}{\sqrt{a^2+3b^2}}K_4\left(\sqrt{\frac{a^2-b^2}{a^2+3b^2}}\right) &=\frac{2}{\sqrt{a^2+3b^2}}I\left(1,\sqrt[4]{\frac{8(a^2+b^2)b^2}{(a^2+3b^2)^2}}\right) &=\frac{a^2+3b^2}{4}I\left(\frac{\sqrt{a^2+3b^2}}{2},\sqrt[4]{\frac{(a^2+b^2)b^2}{2}}\right).
\le C_2 \left\| \sum_{1 \le n \le k}\frac{4 w_n \mathbf{1}_{v_{n}}}{\mu v_{n}}\right\|_p= 4 C_2 \left\| \sum_{1 \le n \le k}\frac{w_n \mathbf{1}_{v_{n}}}{\mu v_{n}}\right\|_p.
S = \sum_{k}S_{k}={\frac{\cal A}{4 8 \pi}}\left( \sum_{k}c_{k}m_{k}^{2}\ln m_{k}^{2}- \ln \Lambda \sum_{k}c_{k}m_{k}^{2}+ \Lambda ( 1 - \ln \epsilon^{2}\Lambda ) \sum_{k}c_{k}\right) = S^{B H}~ .
\left( \begin{array}{c|c|c}\chi_1 \chi_2 \chi_3\end{array}\right) = \left( \begin{array}{ccc}1 1 2 0 7 7\end{array}\right) \left( \begin{array}{c|c|c}\chi_1' \chi_2' \chi_3'\end{array}\right) = \left( \begin{array}{ccc}1 2 3 0 7 7\end{array}\right)
P = -\frac{3}{8}g^2 + \rho T \left( 1 -\frac{2}{3}\frac{\rho T}{g^2}+ \frac{16}{9}\left( \frac{\rho T}{g^2}\right)^{2}-\frac{145}{27}\left( \frac{\rho T}{g^2}\right)^{3}+\frac{1472}{81}\left( \frac{\rho T}{g^2}\right)^{4}+ \ldots \right)
\Psi^l=\Psi^l\left(z_i,X,X_{z_{i}},X_{z_{i_1}^{j_1},z_{i_2}^{j_2}},\dots,X_{z_{i_1}^{j_1},z_{i_2}^{j_2},z_{i_3}^{j_3},\dots,z_{i_n}^{j_n}}\right)
||x||_p+||y||_p\leq 2^{\frac{1}{q}}\Big ( ||x||^p_p+||y||^p_p\Big )^{\frac{1}{p}}=2^{\frac{1}{q}}\Big ( \sum_{j=1}^{n}(|x_j|^p+|y_j|^p)\Big )^{\frac{1}{p}}.
\int_{\mathbb{R}}\xi_i(u_i) du_i = 1 , begin{align*}3pt] \int_{\mathbb{R}}u_{i}\xi_i(u_i) du_i = 0 , begin{align*}3pt] \int_{\mathbb{R}}u_i^{2}\xi_i(u_i) du_i = 1 .
(\nabla_{e_i}\Phi)(\phi e_i,\xi) =g(\xi\times e_i,\nabla_{e_i}(\xi\times\xi))+g(\nabla_{e_i}\xi,\xi\times(\xi\times e_i)) &=-g(\nabla_{e_i}\xi,e_i) &=g(\xi,\nabla_{e_i}e_i).
P_{T}= -\langle u_{0},f'(u_{0})u_{X}\rangle - \langle u_{0},\mathcal{K}*_{\theta}u_{0X}- \mathcal{K}_{1}*_{\theta}(k_{X}u_{0\theta}+ ku_{0X\theta}) + \frac{1}{2}kk_{X}\mathcal{K}_{2}u_{0\theta\theta}\rangle.
\left[ \begin{array}{ccc}1 \frac{1}{3}\frac{1}{5}\frac{1}{1+n}\frac{1}{3(3+n)}\frac{1}{5(5+n)}\frac{1}{(1+2n)(1+n)}\frac{1}{3(3+2n)(3+n)}\frac{1}{5(5+2n)(5+n)}\end{array}\right]\left[ \begin{array}{c}a_0 a_2 a_4 \end{array}\right]=\left[ \begin{array}{c}0 0 0\end{array}\right].
\bar{\cal R}^{\frac{1}{2}\, z \, \omega_{1}\, , \, \frac{1}{2}\, z \, \omega_{2}}\, = \, c \left( \begin{array}{c c c c}{{t_{1}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{s^{- 1}}}&{{w}}&{{0}}\\{{0}}&{{0}}&{{s t_{1}t_{2}^{\, - 1}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- t_{2}^{\, - 1}}}\\ \end{array}\right) \, ,
N_d = (3d-4)! \sum_{d_1+d_2=d}N_{d_1}N_{d_2}\left( -d_1^3 d_2 \frac{1}{(3d_1-1)!}\frac{1}{(3d_2-3)!}+ d_1^2 d_2^2 \frac{1}{(3d_1-2)!}\frac{1}{(3d_2-2)!}\right)
\delta K_{\mu \nu}= \frac{1}{2}e_{\mu}^{( 0 ) M}e_{\nu}^{( 0 ) N}({\cal L}_{\delta n}g_{M N}^{( 0 )}+{\cal L}_{\delta Z}{\cal L}_{n^{( 0 )}}g_{M N}^{( 0 )}- 2 n^{( 0 ) L}\delta \Gamma_{L M N}) .
t^\lambda\prod_{i=1}^{k}t_i^{-i+1}(t_i-1)^{i-1}\begin{vmatrix}1 t_1 \cdots t_1^{k-1}1 t_2 \cdots t_2^{k-1}\vdots \vdots \ddots \vdots 1 t_k \cdots t_k^{k-1}\end{vmatrix}.
\Vert S(t) \Delta g \Vert_{H_p^{\sigma}(\mathbb{R}^n)}= \Vert \Lambda_1 (t) (I-\Delta) g \Vert_{H_p^{\sigma}(\mathbb{R}^n)}\leq C t^{-\frac n 2(1-\frac 2p)}\Vert (I-\Delta)g \Vert_{L^{p'}(\mathbb{R}^n)}\leq C t^{-\frac n 2(1-\frac 2p)}\Vert g \Vert_{H^2_{p'}(\mathbb{R}^n)},
\int_{0}^{\infty}\frac{sinu}{u}du=lim_{s\rightarrow0}\int_{s}^{1}\frac{sinu}{u}du+lim_{t\rightarrow\infty}\int_{1}^{t}\frac{sinu}{u}du=lim_{s\rightarrow0}f(s)+lim_{t\rightarrow\infty}g(t)=F+G
\begin{aligned}\|Q_{g(t_j)}- \lambda_\infty f \|_{H^k(M )}\leqslant C \|Q_{g(t_j)}- \lambda_\infty f \|_{H^{2k}(M )}^{1/2}\|Q_{g(t_j)}- \lambda_\infty f \|_{L^2(M )}^{1/2}\end{aligned}
T_1 = \sum_{d \leq u}\sum_{m \leq X/d}\mu(d) (\log m) g(dm), T_2 = \sum_{m \leq uv}\sum_{r \leq X/m}a(m) g(mr), a(m) = \sum\limits_{d \leq u}\sum\limits_{\substack{n \leq v dn=m}}\mu(d) \Lambda(n) T_3 = \sum_{m > u}\sum_{v < n \leq X/m}b(m) \Lambda(n) g(mn), b(m) = \sum\limits_{\substack{d \leq u d | m}}\mu(d).
Z(s)=\underset{n=1}{\overset{n=N}{\sum}}n^{-s}-\frac{x_0^{1-s}}{1-s}-\frac{x_0^{-s}}{\Gamma (s)}.\underset{K\rightarrow \infty}{\lim}\underset{k=0}{\overset{K}{\sum}}\beta_{2.k+1}.\Gamma (s+2.k+1)
\begin{aligned}K_{0 2}(\psi) w= (\partial_u \nabla \mathcal{H})(T_{\delta})[w]+2 \varepsilon^{b-1}(\partial_u \nabla \mathcal{H})(T_{\delta})[U_1 w]+ &+\varepsilon^{2 (b-1)}U_1^T (\partial_u \nabla\mathcal{H})(T_{\delta}) [U_1 w]+2\,\varepsilon^{2 b-3}U_2 [w, \cdot]^T (\nabla\mathcal{H})(T_{\delta}).\end{aligned}
\frac{m - m_1}{m_1}\geq \frac{n-t-1}{n-t}\frac{(k-t)^2\rho(\rho + t)}{(k-t)^2(t+1+\rho)^2}= \frac{n-t-1}{n-t}\frac{\rho(t + \rho)}{(t + 1 + \rho)^2}.
I(a,b) = \int_{\o}\psi\left( a \left( x + \frac{y}{2a}\right)^2 + b \left( x + \frac{y}{2a}\right) \right) dx = \psi\left( \frac{y^2}{4a}+ \frac{by}{2a}\right) \int_{\o}\psi(ax^2 + xy+bx) \, dx = \psi\left( \frac{y^2}{4a}+ \frac{by}{2a}\right) I(a,b).
a \, \mathcal{H}\, b \Rightarrow aa^{-1}=bb^{-1}a^{-1}a=b^{-1}b, & \Rightarrow (aa^{-1})\theta= (bb^{-1})\theta (a^{-1}a)\theta = (b^{-1}b)\theta, & \Rightarrow (a\theta)(a\theta)^{-1}= (b\theta)(b\theta)^{-1}(a\theta)^{-1}(a\theta)=(b\theta)^{-1}(b\theta) & \Rightarrow a\theta \, \mathcal{H}\, b\theta,
\rho_{n}(x)=\frac{1}{d_{0}^{n-1}}\frac{\rho(x)}{(P_{n-1}(x)\frac{\varphi(x)}{2}-Q_{n-1}(x))^{2}+\pi^{2}\rho^{2}(x)P_{n-1}^{2}(x)}.
\bar{\Delta}_{\rm Proactive}=\frac{\mathbb{E}\sum_{t\in \mathcal{I}_j}\Delta{\left(t\right)}}{\mathbb{E}T^{\rm Proac}_j}=\frac{\mathbb{E}T^{\rm Proac}_j}{2\mathbb{E}\left(T^{\rm Proac}_j\right)^2}+\mathbb{E}\tau^{\rm Proac}_{V_j}-\tau_{\rm f}-\frac{1}{2}.
-\frac{e^{\pi i \alpha}}{2\pi i}\int_{-x_{k}}^{-x_{k-1}}(\Phi_{-}\sigma_{+}\Phi_{-}^{-1})(ru;r\vec{x},\vec{s})du = \frac{\partial_{s_{k}}\Phi_{1}(r\vec{x},\vec{s})}{r}.
A \ll \frac{X_4}{MH}\sum_{h=1}^H \sum_{\frac{X_3}{M_1}< n, n+h \leq \frac{X_4}{M}}\Lambda(n) \Lambda(n+h) \frac{2H \log^3 X}{h}&\ll \frac{X_4}{MH}\left( \frac{X_3}{M_1}\log \frac{X_3}{M_1}\right) \left( 2H \log^3 X \right) \log H \ll \frac{X^2 \log^5 X}{M^2}.
G \partial_1^2 F``=" \partial_1^2 (F G) - 2 \partial_1 F \partial_1G - F \partial_1^2 G ``=" - 2 (\partial_1 (F \partial_1 G ) - F\partial_1^2 G )+ \partial_1^2 (FG) - F \partial_1^2G `` = " \partial_1^2 (FG) - 2 \partial_1 ( F \partial_1 G ) + F \partial_1^2G.
\lim_{\sigma_{1 2}\rightarrow \infty}N_{S p ( 4 )_{2}}= \left( \begin{array}{c c c c}{0}&{0}&{0}&{0}\\{0}&{- 1 / 2}&{- 1}&{- 1 / 2}\\{0}&{- 1}&{- 1}&{- 1}\\{0}&{- 1 / 2}&{- 1}&{- 1 / 2}\\ \end{array}\right) .
F^{\prime}\left(g_{s,1,2}\right) =F^{\prime}\left(g_{s,2,3}\right)=F^{\prime}\left(g_{s,3,1}\right)=F\left(s\right) \mbox{and}F^{\prime}\left(g_{s,2,1}\right) =F^{\prime}\left(g_{s,3,2}\right)=F^{\prime}\left(g_{s,1,3}\right)=F\left(s\right)^{-1}
D^{\varphi}_f(x) :=D_f^{\phi}(f(z_1)\dots f(z_n)x) &=1+c(f)p_1\prod_{i=1}^nf(z_i)x+ c(f)p_2f\prod_{i=1}^nf(z_i)^2x^2 + \dots + c(f)f^{r_0-1}\prod_{i=1}^nf(z_i)^{r_0}x^{r_0}.
e^{\mu_1s}(\epsilon_3,\psi_1)_{L_y^2(B_{4{\sf R}_{\sf in}})}- (\epsilon_0,\psi_1)_{L_y^2(B_{4{\sf R}_{\sf in}})}= \int_0^s e^{\mu_1s_1}(\mathcal{R}_1+\mathcal{R}_2,\psi_1)_{L_y^2(B_{4{\sf R}_{\sf in}})}ds_1.
\delta g_{ij}= \frac{d}{dt}\big\langle \*{r}_i, \*{r}_j \big\rangle\bigg|_{t=0}= 2 \big \langle (\delta\*{r})_i, \*{r}_j \big\rangle = 2 \langle u_i \*{N}+ u\*{N}_i , \*{r}_j \rangle &= 2u_i \langle \*{N}, \*{r}_j \rangle + 2u \langle \*{N}_i, \*{r}_j \rangle = -2uh_{ij}.
\mathcal A_{s}(\mathrm M) g \, :=\, u \int_{\mathbb R^{2}}\widetilde{b}(s, y_{1}, y_{2}) g^{\prime}(y_{1}) \mathrm M_{s}({\mathrm d}y_{1}{\mathrm d}y_{2}) + (1-u) \int_{\mathbb R^{2}}\widetilde{b}(s, y_{1}, y_{2}) g^{\prime}(y_{1}) \mathrm m_{s}({\mathrm d}y_{1}) \mathrm m_{s}({\mathrm d}y_{2})
\| F \|_{\infty , \epsilon}\leq \sum_{| \alpha | \leq \sigma}\frac{1}{\alpha !}A_{\epsilon , | \alpha |}| F_{\alpha}( 0 ) | + \sum_{| \alpha | = \sigma + 1}\frac{1}{\alpha !}B_{\epsilon , \sigma + 1}\| F_{\alpha}\|_{\infty , \epsilon},
u_1 =e^{-\tau x_1}(e^{-i\tau r}|g|^{-1/4}c^{1/2}e^{i\lambda(x_1+ir)}b(\theta)+r'+r_0)=u_1'+e^{-\tau x_1}r_0, u_2 =e^{\tau x_1}(e^{-i\tau r}|g|^{-1/4}c^{1/2}e^{-i\lambda(x_1-ir)}+\overline{r''}),
\arraycolsep=1.4pt\def\arraystretch{1.3}\left. \begin{array}{cc}K_1=r_1^2-r_2^2, \K_2=r_1^2-r_3^2, H = 2r_1r_2r_3 \cos(\theta_1+\theta_2+\theta_3).\end{array}\right.
\lbrace S_{i}, S_{j}\rbrace = \varepsilon_{i j}^{\ \k}S_{k}, \ \, l b r a c e R_{i}, S_{j}\rbrace = \varepsilon_{i j}^{\ \k}R_{k}, \ \, l b r a c e R_{i}, R_{j}\rbrace = - d ( M ) \varepsilon_{i j}^{\ \k}S_{k}.
\operatorname{p}_f[\pi,\eta]\leq \sup_{\epsilon\in \mathbb{U}_X}\limsup_{i\in I}\frac{P_{f_{A_i}}(\epsilon_{F_i})}{\theta(KF_i)}\leq \sup_{\epsilon\in \mathbb{U}_X}\lim_{i\in I}\frac{P_{f_{A_i}}(\epsilon_{KF_i})}{\theta(KF_i)}= \sup_{\epsilon\in \mathbb{U}_X}\operatorname{p}_f[\pi,\epsilon].
\Delta^{p,+}_{a,b,\lambda,\mu}:= \psi_p(\lambda a + \mu b) - \lambda \psi_p(a) - \mu \psi_p(b), \Delta^{p,*}_a := \psi_p( a^*) - \psi_p(a)^*, \Delta^{p,\cdot}_{a,b}:= \psi_p(a b) - \psi_p(a) \psi_p(b).
[\omega_0^{(k,l)}: \omega_1^{(k,l)}] = [p \xi_0^k\xi_0^l +q\xi_1^k\xi_1^l - \xi_0^k\xi_1^l : -(p \xi_0^k\xi_0^l +q\xi_1^k\xi_1^l - \xi_0^l\xi_1^k)]
\int_r^{r_{\delta}}H'(t)/H(t)dt =\int_r^{r_{\delta}}(\frac{n+1-2s}{t}+O(1))dt+\int_r^{r_{\delta}}2N(t)/tdt &\le (n+1-2s)\ln(r_\delta/r)+C(r_\delta-r)+2(\gamma+\delta)\ln(r_\delta/r) &\le (n+1-2s+2\gamma+\delta)\ln(r_\delta/r)+C.
\widetilde{W}_{\nu}( \eta , \eta^{\prime}) = ( r r^{\prime})^{- \frac{d}{2}- \nu}{W}_{\nu}^{\infty}( [ \tau , \mathrm{e}] , [ \tau^{\prime}, \mathrm{e}^{\prime}] ) = \frac{1}{2 \pi^{\frac{d}{2}}}\frac{\Gamma ( \nu + \frac{d}2 )}{\Gamma ( \nu + 1 )}\frac{1}{{[ - ( \eta - \eta^{\prime})^{2}]}^{\frac{d}{2}+ \nu}}.