harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S^3_{r}(C_{p,q}(K)) = \begin{cases}S^3_{p/q}(K) \# L(q,p), ~r = pq, S^3_{r/q^2}(K), ~ m = npq \pm 1, ( S^3 \backslash N(K) ) \cup_{T}SFS_r, otherwise. \end{cases}
\begin{cases}\pi_0=\tfrac{1-pq_0}{pq_0}\pi_T. K=0 \pi_0=\tfrac{1-q_0}{q_0}\pi_T, \;\;\; \pi_i=\tfrac{(1-p)^i}{q_i}\pi_T, \;i=1,\ldots,K-1, \;\;\; \pi_K=\tfrac{(1-p)^K}{pq_K}\pi_T. K\geq 1,\end{cases}
[L^p_{\overline{\sigma}}(\Omega), D(A_p)]_{\theta}= [L^p(\Omega)\cap L^p_{\overline{\sigma}}(\Omega), H_{per,b.c.}^{2,p}(\Omega)\cap L^p_{\overline{\sigma}}(\Omega)]_{\theta}&= [L^p(\Omega), H_{per,b.c.}^{2,p}(\Omega)]_{\theta}\cap L^p_{\overline{\sigma}}(\Omega).
\lim_{m \to \infty}\frac{1}{m}\sum_{j = -m/2+1}^{m/2}\frac{e^{-2\pi i \delta \frac{j+\theta_1/2}{m}}}{1 - \exp \left\{- 2 \pi i \frac{j + \theta_1/2}{m}\right\}+ \frac{z}{m}}= \frac{\left( (-1)e^{-z}\right)^{\delta}}{1 - (-1)^{\theta_1}e^{-z}}.
\bar{D}(r_s) = \frac{\sum_{i=0}^{K-1-s}\binom{K}{s+1+i}(N-1)^iN}{\binom{K-2}{s-1}+\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^iN}&=\frac{\frac{\sum_{i=0}^{K-1-s}\binom{K}{s+1+i}(N-1)^i}{\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^i}}{\frac{\binom{K-2}{s-1}}{\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^iN}+1}&=\frac{\psi_1(N,K,s)}{\psi_2(N,K,s)+1}.
y^7 + (6t^4+6t^3+6t^2+12t+1)y^5 + (t^8 + 2t^7+3t^6+6t^5+t^4+5t+4)y^3 + (6t^{12}+5t^{11}+10t^{10}+7t^8+2t^7+3t^6+9t^5+3t^4+2t^3+6t^2+t+4)y + (11t^{14}+6t^{13}+12t^{12}+10t^{11}+5t^{10}+8t^9+6t^8+2t^7+2t^6+10t^5+7t^4+12t^3+3t^2+3t+9) = 0
\left ( \begin{array}{c}\tilde{w}_{1}(x) \vdots \tilde{w}_{N}(x) \end{array}\right ) = \sum_{k \in \mathbb{Z}}\tilde{B}_{k}\left ( \begin{array}{c}\tilde{f_{1}}(2x - k) \vdots \tilde{f}_{N}(2x - k) \end{array}\right ) .
\langle Q|\hat{\psi}_{sp}(x,t)|Q+1\rangle = \int\frac{d\omega\,dk}{(2\pi)^{2}}\frac{{\cal A}(k)}{k^{2}+m^{2}\sin^{2}\left(\frac{\lambda^{2}\bar{Q}}{4}\right)}\delta(\omega-M(Q+1)+M(Q)),
\begin{array}{cccccc}\phi^1_1 \phi^2_1 \ldots \phi^a_1 \ldots \phi^m_1 \vdots \vdots \vdots \vdots \phi^m_{N_m}\vdots \phi^2_{N_2}\vdots \phi^1_{N_1}\ \phi^a_{N_a}\end{array}
\sum_{j=-\infty}^{\infty}e^{-2\pi i \beta b j}\left(s+1-\sqrt{(s+1)^2 -1}\right)^{|\ell+jm|}&= \frac{\sinh\left((m-\ell)\cosh^{-1}(s+1)\right)+e^{2\pi i \beta}\sinh\left(\ell\cosh^{-1}(s+1)\right)}{\cosh\left(m\cosh^{-1}(s+1)\right)-\cos2\pi \beta}.
\langle\Psi_{0}|G_{2}|\Psi_{0}\rangle=\frac{1}{2}\sum_{ij}[\langle\psi_{i}\psi_{j}|\frac{e^{2}}{r_{ij}}|\psi_{i}\psi_{j}\rangle-\langle\psi_{i}\psi_{j}|\frac{e^{2}}{r_{ij}}|\psi_{j}\psi_{i}\rangle]
\begin{aligned}\frac{\tilde\mu-\mu}{2}\sum_{i=1}^N\frac{\|x_i-y_i\|^2}{a_i^2}\leq \frac{\tilde\mu}{2}\|x^*-v_0\|^2+\frac{\rho M^2N(N+3)}{4}+\frac{NrM^2}{2}+\sum^N_{i=1}\frac{\delta_i}{a_i}&+2\sum_{i=1}^N\frac{\varepsilon_i}{a_i^2}+\sqrt{2\tilde\mu}\sum_{i=1}^N \|x^*-v_{i}\|\cdot\sqrt{\frac{\delta_i}{a_i}}.\end{aligned}
\langle \cdot \,|\, \cdot\rangle_{\#}:=\langle \cdot\,|\, (h_{\alpha}- e_{\alpha}+ H_f)\,|\, \cdot \rangle, \langle \cdot\,|\, \cdot \rangle_{*}:= \langle \cdot \,|\, H_f \,|\,\cdot\rangle,
H_{y}=-L\frac{\partial T}{\partial x}^{TM}+\frac{1}{j\omega\mu}\frac{dL}{dz}\frac{\partial T}{\partial y}^{TE}=-L\frac{\partial T}{\partial x}^{TM}-\frac{k_{z}}{\omega\mu}L\frac{\partial T}{\partial y}^{TE}(31)
Q_{n}(T,x)=2\pi\lambda\sum_{k=1}^{4}b_k \int_{x}^{\infty}F\left(N_\mathrm{L},\frac{n\eta_\mathrm{L}\bar{a}_kTx^{\alpha_\mathrm{L}}}{N_\mathrm{L}t^{\alpha_\mathrm{L}}}\right)p(t)t\mathrm{d}t,
\left( \hat{L}_{\hbar}-L_{\hbar}^{n}\right)^{2}=\left( \sum_{\ell=0}^{nm-1}(-\hbar)^{\ell}C\partial^{2\ell}\right)^{2}\preceq_{\mathcal{S}}\delta\left( -\hbar\right)^{mn}\partial^{4mn}+C_{\delta}I
\begin{array}{l}|v(y'+Re_d,\omega) - \mu| = |v(Re_d,\tau_{y'}\omega) - \mu| \leq |v(Re_d,\tau_{y'}\omega) - \mu_N|+ C(\log R)^{1/2}R^{-\hat{\beta}/2}\leq C(\log R)^{1/2}R^{-\hat{\beta}/2}, \end{array}
\frac 1 N \int D ( A_{\mu}^{a}, \bar{C}^{a}, C^{a}, \bar{\psi}, \psi ) \{K^{a}( x ) + \partial_{x}^{\mu}({\cal D}_{\mu}^{a b}( x ) C^{b}( x ) ) \}e^{i S + E S T}= 0
\tilde{Z}_{\lambda ,\mu}(x|t)=\sum_{d=0}^{n}q^{d}\sum_{|\mathcal{T}|=\lambda /d/\mu}\prod_{1\leq i,j\leq n}\prod_{\ell =\lambda_{i+1}^{(j)}[d_{j}]+\rho_{i+1}+1}^{\lambda_{i}^{(j-1)}[d_{j-1}]+\rho_{i}-1}(x_{j}-t_{\ell}),
\begin{array}{ll}Z= \lim \limits_{\lambda \rightarrow 0}(\frac{\lambda}{2\pi})^n \displaystyle\int{\cal D}u{\cal D}\psi{\cal D}\chi \exp (- \frac{(s, s)}{2\lambda}- \frac{1}{\lambda}g_{ab}\chi^a \frac{\partial s^b}{\partial{u^i}}\psi^i+ \frac{1}{2\lambda}G_{ijab}\psi^i\psi^j\chi^a\chi^b) ~= \nu(D).\end{array}
\frac{\int_{\mathbb{D}}fe^{-\varphi}}{\int_{\mathbb{D}}e^{-\varphi}}=\frac{\int_{\mathbb{D}}fe^{-\psi}-\varepsilon\int_{\mathbb{D}}f\widetilde{\eta}}{\int_{\mathbb{D}}e^{-\psi}-\varepsilon\int_{\mathbb{D}}\widetilde{\eta}}=\frac{\int_{\mathbb{D}}fe^{-\psi}}{\int_{\mathbb{D}}e^{-\psi}}=f(0),\ \ \ \forall\,f\in{A^2(\mathbb{D})},
\begin{array}{l}W_A= \{(d_1,d_2) \,|\, a_i\leq d_i\}Z_A=\left\lbrace\begin{array}{c}(d_1,d_2)\,|\, \forall j=1,\dots, m \exists\, u,v \mbox{with}u+v=1 \mbox{such that}a_1\leq d_1\leq a_1+b_{2j}-1+v \quad\mbox{and}a_2\leq d_2\leq a_2+b_{1j}-1+u\end{array}\right\rbrace.\end{array}
z_j^+ := \mathbf{p}B_j^+(h_1-h_j)\cdots(h_{j-1}-h_j),(1\leq j\leq n), z_j^- := \mathbf{p}B_j^-(h_j-h_{j+1})\cdots(h_j-h_n),(1\leq j\leq n), z_{ij}^+ := \mathbf{p}\{B_i^+,B_j^+\}\pi_{ij}^+ z_{ij}^- := \mathbf{p}\{B_i^-,B_j^-\}\pi_{ij}^-,
|\mu_1(t)-\hat{\mu}_1(t)| =\frac{1}{2}\Big|\Big(e^{(a-2k)t}+e^{at})x_1+( e^{at}-e^{(a-2k)t})x_2\Big)-e^{at}x_1\Big| &=\frac{1}{2}|(e^{at}-e^{(a-2k)t})(x_2-x_1)| &= \frac{1}{2}e^{at}|x_2-x_1|(1-e^{-2kt}) & \leq |x_2-x_1|k e^{at}t & \leq k\,|x_2-x_1| \frac{1}{|a| e},
\lim_{j \to \infty}\int_B |\nabla u^\lambda_j|^{p-2}\nabla u^\lambda_j \cdot \nabla \varphi_j u^\lambda_j\, dx = c \int_S \varphi d\nu - \int_B \varphi d\mu - \lambda \int_B |u^\lambda_0|^p \varphi dx.
\gamma_{st}= (\lvert\gamma_{u}\rvert^{-1}\gamma_{u})_{t}&=-\lvert\gamma_{u}\rvert^{-3}\langle\gamma_{tu},\gamma_{u}\rangle\gamma_{u}+\lvert\gamma_{u}\rvert^{-1}\gamma_{tu}&=-\lvert\gamma_{u}\rvert^{-3}\langle\kappa B_{u},\gamma_{u}\rangle\gamma_{u}+\gamma_{ts}&=\kappa\langle B_{s},T\rangle\gamma_{s}+\gamma_{ts}&=\gamma_{ts},
\int_{\mathbb{R}^2}d^{\alpha}(x,0) |\partial^i_1 \partial_2^j \psi_T(x )|\,\textrm{d}x = (T^{\frac{1}{4}})^{\alpha - i - 2j}\int_{\mathbb{R}^2}d^{\alpha}(\hat{x},0) |\partial^i_1 \partial_2^j \psi_1(\hat{x})|\,\textrm{d}\hat{x}.
u^{0}\in H^{1}(\Omega), \varphi \in L^{2}\big((-\tau, 0), H^{2}(\Omega)\big), \gamma \in H^{3/4}\big((0, T), L^{2}(\partial \Omega)\big) \cap L^{2}\big((0, T), H^{3/2}(\partial \Omega)\big), f \in L^{2}\big((0, T), L^{2}(\Omega)\big)
c_{2}(\xi,\tau)=c_{2a}(\xi,\tau)+\frac{Pe}{4}(r^{2}-\frac{r^{4}}{2})\frac{\partial c_{1a}}{\partial\xi}+\frac{Pe^{2}}{32}(\frac{r^{2}}{6}+\frac{r^{4}}{2}-\frac{5r^{6}}{8}+\frac{r^{8}}{8})\frac{\partial^{2}c_{0}}{\partial\xi^{2}}
\forall (x,\theta)\in (T^*G \setminus \{0\})\cong (G\times \mathfrak{g}^{*}\setminus \{0\}),\,\sigma_{\textnormal{loc}, \mathcal{L}_{G}}(x,\theta)=(\theta,\theta)_{g}=:\|\theta\|_{g}^2.
(r + 1) d_2 - (r - 3)(g_2 - 1) = [(r + 1) d_2 - r g_2 - r(r + 1)] + r(r + 2) + 3(g_2 - 1) &\geq - r^2 + 8r - 3 + r(r + 2) - 3 &= 10r - 6.
\|\varphi\,\big|\,\mathbb{W}_p^m(\mathcal{C})\,\|:=\left[\|\varphi\,\big|\,\mathbb{L}_p(\Omega)\,\| +\sum\limits_{j=1}^n\|\mathcal{D}_j\varphi\,\big|\,\mathbb{L}_p(\mathcal{C})\|_p\right]^{1/p}.
\left( -i\varepsilon \delta \theta \int d^{4}z\{\partial^{0}A_{0}+\frac{1}{\theta^{2}}\partial_{i}A_{i}\}(z)\int d^{4}x\{(1-\theta )\partial^{0}A_{0}+(\frac{1}{\theta}-1)\partial_{i}A_{i}\}^{\alpha}(x)M_{0}^{'-1\alpha \beta}(x,z;\alpha_{0})\right)
cov\{\hat{m}_{4y}(\lambda,\lambda,0),\hat{m}_{2y}(\beta)\hat{m}_{2y}(\beta)\}= 2\big[\hat{m}_{2y}(\beta)\sum_{\tau = -K_N}^{K_N}[\hat{m}_{6y}(\lambda,\lambda,0,\tau ,\tau+\beta) - \hat{m}_{4y}(\lambda,\lambda,0)\hat{m}_{2y}(\beta)]\big].
g_{j}^{(net)}=\sum_{i\ne j}g_{i}=\frac{1}{m_{j}}\sum_{i\ne j}F_{i}=-G\sum_{i\ne j}m_{i}\frac{\hat{R}_{ij}}{|R_{i}-R_{j}|^{2}}=-\sum_{i\ne j}\nabla\Phi_{i}
q\equiv -\frac{1}{16\pi^2}\int d^4x\mbox{}trG_{\mu\nu}\tilde{G}_{\mu \nu}=\frac{1}{2}\int d\sigma_1 d\sigma_2 d\tau_1 d\tau_2\epsilon_{\mu \nu \alpha \beta}\dot{z}_{1\mu}z\prime_{1\nu}\dot{z}_{2\alpha}z\prime_{2\beta}\delta(z_1-z_2)
E = [ P_{1 1}^{2}+ [ 2 L_{1}L_{2}T^{( 2 )}+ V_{2}]^{2}]^{1 / 2}\approx P_{1 1}+ \frac{2 L_{1}L_{2}T^{( 2 )}V_{2}}{P_{1 1}}+{\frac{2 ( L_{1}L_{2}T^{( 2 )})^{2}}{P_{1 1}}}
f(x)=\xi_{1}f(a_{1})+\xi_{2}f(a_{2})+\xi_{3}f(a_{3})= \xi_{1}a_{1}+\xi_{1}\alpha_{3}a_{3}+\xi_{2}a_{2}+\xi_{2}\beta_{3}a_{3}+\xi_{3}a_{3}= \xi_{1}a_{1}+\xi_{2}a_{2}+(\xi_{1}\alpha_{3}+\xi_{2}\beta_{3}+\xi_{3})a_{3}.
s_1(a)s_1(b) = s_1(ab) + s_2(a+b) - s_2(a) - s_2(b) s_1(a)^2 = s_1(a^2) + 2s_2(a) s_1(a)s_1(ab)s_1(b) = s_1(a^2b) s_1(b) +s_1(a) s_1(ab^2) - s_1(a^2b^2)+ s_2(ab) + s_2(a)s_2(b).
{\cal A}_{\beta}= \frac{1}{8{\pi}^{2}}[ - 2 ( \beta / l^{2}) l^{- 2}e \cdot R \cdot e + ( \beta / l^{2})^{2}R^{a b}{\mathrm{\tiny ~ \wedge ~}}R_{a b}+ 2 ( \beta / l^{2})^{2}l^{- 2}T \cdot T ] + O [ ( \beta / l )^{- 2}) ] .
\langle \sigma_{I}| = \lim_{\widetilde{\xi}^{1}\rightarrow 0}\left(\frac{1}{\widetilde{\xi}^{1}}\right)^{2h_{\sigma_{I}}}\, \langle 0 | ~\sigma^{-}_{I}\left(-\frac{1}{\widetilde{\xi}^{1}}\right)~,
\left|\sum_{i=1}^n \int_{(i-1)\Delta_n}^{i\Delta_n}\frac{\rho}{\Delta_n^{(\mathfrak H+\epsilon)}}\int_{2-(i\Delta_n-s)}^2 x^{\mathfrak H+\epsilon}dx ds\right| \leq \rho \Delta_n^{1-(\mathfrak H+\epsilon)}2^{\mathfrak H+\epsilon}.
{\cal A}_{\mu}~ = ~ \left( \begin{array}{l}{B_{\mu A}}{V^{A}{}_{\mu}}\end{array}\right){}~ ~ ~ ~ ~ ~ ~ ~ ~ ~{\cal F}_{\mu \nu}~ = ~ \left( \begin{array}{l}{H_{\mu \nu A}}{V^{A}{}_{\mu \nu}}\end{array}\right)
E[R_s]\leq\frac{1}{2(L-1)}\sum_{\ell=2}^{L}\left(\log\left(\frac{1}{1+\frac{\sigma^2_{h_{\ell,r}}}{\sigma^2_{h_{\ell-1,r}}}}+\frac{P\sigma^2_{h_{\ell-1,r}}}{N_0}\right)+\log\left(\frac{1}{1+\frac{\sigma^2_{h_{\ell-1,r}}}{\sigma^2_{h_{\ell,r}}}}+\frac{P\sigma^2_{h_{\ell,r}}}{N_0}\right)\right),
C_{(n)}(i,j):=C(X_i,Y_j),\C^+_{>0,(n)}(i,j):=C_{>0}^+(X_i,Y_j), & C^+_{\le 0,(n)}(i,j):=C_{\le 0}^+(X_i,Y_j),\ (\forall i,j\in \{1,2,\cdots,n\}).
\sup_{\pi \in \mathcal{G}_{\lambda}}\int_{\mathcal{V}\times \mathcal{S}}\phi_\lambda \, d \pi =\sup_{\pi \in \Gamma\left( \Pi(\mu_1, \mu_2) , \phi_\lambda \right)}\int_{\mathcal{V}\times \mathcal{S}}\phi_{\lambda}\, d \pi
\langle \Delta_{\mathcal{X}},\nabla_{\mathcal{X}\mathcal{X}}^2L(\overline{\mathcal{X}}, \overline{\lambda},\overline{\mathcal{Y}})\Delta_{\mathcal{X}}\rangle -\Upsilon_{G(\overline{\mathcal{X}})}\big(\overline{\mathcal{Y}},G'(\overline{\mathcal{X}})\Delta_{\mathcal{X}}\big)=0.
y(t) = e^{At}y(0) + \int_0^t e^{A(t-s)}\big(B_0 g(s)+B_1 g_t(s)\big)\,ds= begin{align*}1mm] = e^{At}y(0) + \int_0^t e^{A(t-s)}B_0 \Big(g(s) + \frac{b}{c^2}g_t(s)\Big)\,ds\,.
\int dx \left(\,{1\over 2}\, \tilde{g}^{ij}{\delta^2 \Psi \over \delta \phi^i \delta \phi^j}\, +\,{1\over 2}\, g_{ij}\partial_x \phi^i \partial_x \phi^j\Psi \, \right)\, =\,i{\partial \over \partial t}\Psi\, .
|I_{3}| \leq\limsup_{t\rightarrow\infty}K\left(\frac{a(t)}{t}|\theta|\right)^{3}\left(\int_{0}^{t}\gamma_{0}(r)dr\right)\frac{t}{a(t)^{2}}&=\limsup_{t\rightarrow\infty}K\frac{a(t)}{t}|\theta|^{3}\frac{\int_{0}^{t}\gamma_{0}(r)dr}{t}&\leq\limsup_{t\rightarrow\infty}K\frac{a(t)}{t}|\theta|^{3}\overline{\gamma}_{0}&=0.
p((g - 1)(l_1 + l_2)) = \prod_{i = 0}^{2g}[(-g + i - 1)l_1 + (g - i - 1)l_2], & \alpha_{1,0}((g - 1)(l_1 + l_2)) = -2(2g - 1)(l_1 + l_2), & \alpha_{1,1}((g - 1)(l_1 + l_2)) = (g - 1)(g - 2)(l_1 + l_2)^2 - 4g(g - 1)l_1l_2.
B ( z ) = i \frac{8 m_{D}}{g_{m}^{2}}\frac{\mathrm{e}^{- m_{D}| z |}}{1 + \mathrm{e}^{- 2 m_{D}| z |}}, ~ ~ \psi ( z ) = 4{\,}\mathrm{s g n}{\,}z \cdot \operatorname{a r c t a n}\left( \mathrm{e}^{- m_{D}| z |}\right) .
\begin{aligned}-\frac{\omega(x,t)}{2}+\int_{0}^{t}\int_{\partial\Omega}\frac{\partial\varepsilon_{n, a}(x-\xi,b(t,\tau))}{\partial\nu(\xi)}a(\tau)\omega(\xi,\tau)dS_{\xi}d\tau &-\int_{0}^{t}\int_{\partial\Omega}\varepsilon_{n, a}(x-\xi,b(t,\tau))a(\tau)\frac{\partial \omega(\xi,\tau )}{\partial\nu(\xi)}dS_{\xi}d\tau=0,\end{aligned}
E_m=2(2m)!\left(\frac{2}{\pi}\right)^{2m+1}\left/\left(1+\frac{1}{3^{2m+1}}\right)\left(1-\frac{1}{5^{2m+1}}\right)\left(1+\frac{1}{7^{2m+1}}\right)\ldots,\right.
2g\big(\tilde{\nabla}_XY-\nabla_XY,LZ\big) ={\bf B}((\nabla_X\xi)_Y+(\nabla_Y\xi)_X,\xi_Z) \hphantom{2g\big(\tilde{\nabla}_XY-\nabla_XY,LZ\big) =}{}+{\bf B}((\nabla_X\xi)_Z-(\nabla_Z\xi)_X,\xi_Y)+{\bf B}((\nabla_Y\xi)_Z-(\nabla_Z\xi)_Y,\xi_X).
\begin{aligned}f_1 = \sigma_{k-1}(\lambda|1) \geq \delta (-\lambda_n) \sigma_{k-2}(\lambda|1n) &\geq \frac{\delta}{\mu_1}\lambda_1 \sigma_{k-2}(\lambda|1n) \geq \frac{\delta\mu_2}{\mu_1}\frac{k-1}{n-1}\sigma_{k-1}(\lambda|n) &\geq \left( \frac{\mu_2}{\mu_1}\right)^2 \frac{k - 1}{(n-1)(n-2+k) (n-k+1)}\sum_i f_i.\end{aligned}
\frac{1}{s}\int_0^T\iint\limits_{\mathbb{R}^N\times \mathbb{R}^N}\frac{H(x,y,(u+s\psi)(x,t)-(u+s\psi)(y,t))-H(x,y,u(x,t)-u(y,t))}{|x-y|^N}\,dx\,dy\,dt &=\int_0^T\iint\limits_{\mathbb{R}^N\times \mathbb{R}^N}\int_0^1 \frac{D_\xi H(x,y,(u+s\sigma\psi)(x,t)-(u+s\sigma\psi)(y,t))\cdot(\psi(x,t)-\psi(y,t))}{|x-y|^N}\,d\sigma\,dx\,dy\,dt
\begin{aligned}\chi(Y) = 6\frac{2Lt+3 L St^2 - S^2t^2}{(1+S t) (1+6 L t- 2 St )}c_t(TB) &=12 L t + 6 t^2 (2 c_1 L - 12 L^2 + 5 L S - S^2) + & \ \ + 6 t^3 (-12 c_1 L^2+5 c_1 L S-c_1 S^2+2 c_2 L+72 L^3-54 L^2 S+15 L S^2-S^3)+\cdots \end{aligned}
\hat x_k^*:=\frac{x_k^*}{\|x_k^*\|_{X^*}},~~\hat y_k^*=\frac{y_k^*}{\|x_k^*\|_{X^*}},~~\operatorname{and}~~\hat\varepsilon_k=\frac{\varepsilon_k}{\|x_k^*\|_{X^*}}
\begin{cases}\dot{\bar z}_i=\bar f_i(\bar z_i, e_i, v, w,\mu) \dot{\hat \eta}_i=F_i\hat\eta_i+F_iG_ie_i- G_i\bar g_i(\bar z_i,e_i, v, w,\mu) \dot{e}_i=\bar g_i(\bar z_i,e_i, v, w,\mu)-\Psi_i\hat \eta_i-\Psi_iG_ie_i+u_i^r \dot{k}_i=-\lambda k_{i}+\rho_i(e_{vi})e_{vi}^2, \dot{\theta}_i=|e_{vi}|.\end{cases}
H_{g - Q}= - \int d{\bf r}d{\bf x}\left[{\sf K}_{g}^{b}({\bf r}) \left( 4{\pi}| ({\bf r}-{\bf x}) | \right)^{- 1}{\cal K}_{0}^{b}({\bf x}) +{\cal K}_{0}^{b}({\bf x}) \left( 4{\pi}| ({\bf r}-{\bf x}) | \right)^{- 1}{\sf K}_{g}^{b}({\bf r}) \right] \, .
\varphi_{i}^{\frac{4}{n-2}}= u^{\frac{4}{n-2}}_{a_{i}}(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{2}+O(r^{n-2}(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{2}).
N_{dr}(m+1)-N_{dr}(m) =a^2+ad+\left(\sum_{i=1}^kx_i^{\prime}-\sum_{i=1}^kx_i\right)a & \geq\left(a+d+\sum_{i=1}^{k-1}x_i^{\prime}-\sum_{i=1}^{k-1}x_i\right)a & \geq(a+d-k)a\geq 0.
\Big\| \sum_{k=1}^L q_k\widetilde{\phi_k}\Big\|_{H^2}^2 \le\Big( \sum_{k=1}^L\| q_k\widetilde{\phi_k}\|_{H^2}\Big)^2 \le\Big( \sum_{k=1}^L \|q_k\|_\infty \|\phi_k\|_b\Big)^2 &\le C \Big( \sum_{k=1}^L \|q_k\|_{H^2}\|\phi_k\|_b\Big)^2 &\le C \Big( \sum_{k=1}^L\|q_k\|_{H^2}^2 \Big)\Big( \sum_{k=1}^L \|\phi_k\|_b^2 \Big).
\psi (x) = \widetilde\phi (x) - \phi (x) = t_1 \left({K^{-\frac{1}{{R^2}}}- 1}\right)\cos \left({\frac{x}{R}}\right)+ t_2 \left({K^{- \frac{4}{{R^2}}}- 1}\right)\cos \left({\frac{{2x}}{R}}\right) + \cdots,
A^1_n:=\int_{\widehat{I}\cap B_{2n+1}^c}\widehat{f}'(\widehat{x})^2d\widehat{x}+\sum_{k: \widehat{b}_k>2n+1\widehat{a}_k<-2n-1}\frac{\left(\widehat{f}(\widehat{a}_k)-\widehat{f}(\widehat{b}_k)\right)^2}{|\widehat{b}_k-\widehat{a}_k|},
Q(x,y) = \begin{vmatrix}-2 0 kx+my 0 -2 lx+ny kx+my lx+ny 2xy\end{vmatrix}&=8xy+2(kx+my)^2+2(lx+ny)^2 &=(2k^2+2l^2)x^2+(8+4km+4ln)xy+(2m^2+2n^2)y^2.
\mu_l = f\left( \sum_{k \in \mathcal{K}}\left\vert w_{lk}\right\vert^{2}, \tau_1 \right) = \frac{c_1}{\sum_{k \in \mathcal{K}}\left\vert w_{lk}\right\vert^{2}+ \tau_1}~ , \nu_{lk}= f\left(\left\vert w_{lk}\right\vert^{2}, \tau_2\right) = \frac{c_2}{\left\vert w_{lk}\right\vert^{2}+ \tau_2}
L(\phi) - L(\phi_{B}) \leq 2 \sum_{i=0}^{1}E \left[ \Big| p(\textbf{Z},i)P(Y=i \vert \textbf{Z}) - f_{i}(\textbf{Z}) \Big| \right] + 2 \sum_{i=0}^{1}E \left[ \Big| (1-q(\textbf{X},i))P(Y=i \vert \textbf{X}) - g_{i}(\textbf{X}) \Big| \right].
p_1b_1 = \prod_{l=1}^{N_1}p_{1,l}, b_1^{-1}p_2 b_2 = \prod_{l=1}^{N_2}p_{2,l}, \dots, b_{r-1}^{-1}p_r b_r = \prod_{l=1}^{N_r}p_{r,l}.
{\cal L}^{(2+1)scal}_{1/m^3}=-\frac{\alpha}{720 m^3}\left[6 F^{\nu \lambda}F_{\nu\lambda,\mu}^{~~~~\mu}+4F^{\nu \lambda,\mu}F_{\nu\lambda,\mu}+F^{\nu \lambda,}_{~~~\lambda}F_{\nu\mu,}^{~~~\mu}\right],
4!h_4 = p_{1111}+6p_{211}+3p_{22}+8p_{31}+6p_4, 4!S_4 =\Psi^{1111}+3\Psi^{112}+2\Psi^{121}+6\Psi^{13}+\Psi^{211}+3\Psi^{22}+2\Psi^{31}+6\Psi_4,
(\delta\times\delta)_!\Delta_!F(U_{I}\subseteq M, V_{J}\subseteq M) \overset{(\ref{lem:diagonal push formula 1})}{\simeq}\begin{cases}\Delta_!F(U_I\subseteq M, V_J\subseteq M) |I|=|J|=1 0 \end{cases}&\overset{(\ref{lem:diagonal push formula 2})}{\simeq}\begin{cases}F(U_I\subseteq M) U_I=V_J,\, |I|=|J|=1 0 \quad\end{cases}
\left({\boldsymbol\tau}(D^\star{\boldsymbol u}),B D^\star{\boldsymbol\varphi}\right)+{\cal R}e \left(B{\boldsymbol u}\cdot \nabla^\star{\boldsymbol u},{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right)\mbox{for all}\{\boldsymbol\varphi}\in{\boldsymbol V}_B^p.
P\{T_1 < T_2\}= p_2 M_2 \int_0^1 \left(1 - x^{p_1}\right)^{M_1}\left(1 - x^{p_2}\right)^{M_2 - 1}x^{p_2 - 1}dx &= -\int_0^1 \left(1 - x^{p_1}\right)^{M_1}\left[\left(1 - x^{p_2}\right)^{M_2}\right]' dx.
D\Phi_\epsilon(y,k) = \int_X (f_y,0) \phi_\epsilon(s_y-k) d{\mathcal H}^m +\frac{1}{\epsilon}\int_0^\epsilon dt \int_{X(y,k+t)}f \frac{(-s_{yy},1)}{|D_x s_y|}d{\mathcal H}^{m-1}.
\int\limits_{B_{3\cdot 2^{-n}\|M_1M_2\|}(M_2)}\frac{\omega(d(M))}{d^2(M)}\, dm_3(M)= &=\sum_{k=0}^{\infty}\int\limits_{\sigma_{nk}}\frac{\omega(d(M))}{d^2(M)}\, dm_3(M) = &=\sum_{k=k(n)}^{\infty}\int\limits_{\sigma_{kn}}\frac{\omega(d(M))}{d^2(M)}\, dm_3(M).
(X,Y)_{\lambda,{\bf k}}=\frac{1}{2}\int{\mbox{Tr}}_{\bf k}\bigg({\delta_r X \over \delta{\cal A}}{\delta_l Y \over \delta{\cal A}_\lambda^*}-{\delta_r X \over \delta{\cal A}_\lambda^*}{\delta_l Y \over \delta{\cal A}}\bigg),
\begin{cases}\rho_t+\rho_x +u_{x}=0 (0,T)\times (0,1), u_{t}-u_{xx}+u_x +\rho_x =0 (0,T)\times (0,1), \rho(t,0)=\rho(t,1) t\in (0,T) , u(t,0)=0, \ \u(t,1)=q(t) t\in (0,T) , \rho(0,x)=\rho_0(x),\ \u(0,x)=u_0(x) x\in(0,1). \end{cases}
\lambda_{112}\lambda_{qrr}[a_{(1\,r)},a_{(2\,q)}]_c=\lambda_{112}\lambda_{qrr}[a_{(1\,r)},[a_2,a_{(3\,q)}]_c]_c &=\lambda_{112}\lambda_{qrr}[[a_{(1\,r)},a_2]_c,a_{(3\,q)}]_c+\lambda_{112}\lambda_{qrr}\chi_2(g_{(1\,r)})\big(a_2[a_{(1\,r)},a_{(3\,q)}]_c &-\chi_{(3\,q)}(g_2)\chi_{(1\,r)}(g_2)[a_{(1\,r)},a_{(3\,q)}]_ca_2\big).
\delta_\eta q^a_i=\{q^a_i,\eta\phi\}=\eta\epsilon^{ab}q^b_i\ \ ,\ \ \delta_\eta p^a_i=\{p^a_i,\eta\phi\}=\eta\epsilon^{ab}p^b_i\ \ ,\ \ \delta_\eta\lambda=\dot{\eta}\ \ ,
\frac{1}{|x-x^{\prime}|}=\sqrt{\frac{\pi}{2RR^{\prime}(\chi^{2}-1)^{1/2}}}\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{\Gamma(m+1/2)}P_{-\frac{1}{2}}^{m}(\frac{\chi}{\sqrt{\chi^{2}-1}})e^{im(\varphi-\varphi^{\prime})}
\sigma^{;\alpha \beta}=g^{\alpha \beta}-\frac{1}{3}R^{\alpha ~~\beta~~}_{~~\gamma~~\delta}\sigma^{;\gamma}\sigma^{;\delta}+\frac{1}{12}R^{\alpha~~\beta~~}_{~~\gamma~~\delta;\rho}\sigma^{;\gamma}\sigma^{;\delta}\sigma^{;\rho}+O(\sigma^2)~.
\tilde{\Lambda}_0(x) :=\frac{1}{\varepsilon}\left(\cosh^{-1}\left(|v|\cosh(\lambda+\varepsilon \theta(x))-\frac{1}{\sqrt{c}}\tanh(\beta( \lambda+\varepsilon \theta(x)))\sinh(\lambda+\varepsilon \theta(x))\right) \right) &-\frac{1}{\varepsilon}\left(\cosh^{-1}\left(|v|\cosh(\lambda)-\frac{1}{\sqrt{c}}\tanh(\beta \lambda)\sinh(\lambda)\right) \right).
F= i \frac{dg_{1}\wedge d\overline{g}_{1}}{(1- g_{1}\overline{g}_{1})^{2}}- i \frac{dg_{2}\wedge d\overline{g}_{2}}{(1+ g_{2}\overline{g}_{2})^{2}}= - \frac{1}{2}(R^{1}_{2}+R^{3}_{4})~.
f(z) = g(z)^{2}= z+1 \mbox{and}E(z) = P^{(\infty)}(z)e^{-\frac{\pi i \alpha}{2}\theta(z)\sigma_{3}}N^{-1}f(z)^{\frac{\sigma_{3}}{4}}(ax)^{\frac{1}{4}\sigma_{3}}.
D^{\mu\nu\lambda\sigma}(x-y) = 16 \pi G \left(\eta^{\mu\lambda}\eta^{\nu\sigma}+ \eta^{\mu \sigma}\eta^{\nu \lambda}-{2 \over 9}\eta^{\mu \nu}\eta^{\lambda \sigma}\right) \int{d^{11}k \over (2 \pi)^{11}}{e^{i k \cdot (x-y)}\over - k^2}\,.
\Gamma_{1,fermions}\left[ B \right] = \frac{1}{2}\int\limits_0^\infty{\frac{{dT}}{T}}\int\limits_{x(0) = x(T)}{Dx(t)\exp ( - \frac{1}{4}}\int\limits_0^T{dt\dot x^2})Tr_L \Phi^{\left[{1/2}\right]}\left[{\dot x}\right]Tr_c P\exp (ig\int\limits_0^T{dt\dot x \cdot B)}
\begin{array}{c}\Delta u|_{t=\tau_1}=I_1(u(\tau_1)), \;\,\Delta u'|_{t=\tau_1}=N_1(u(\tau_1)),{\tau_1}\in (0,1), \Delta v|_{t=\tau_2}=I_2(v(\tau_2)), \;\,\Delta v'|_{t=\tau_2}=N_2(v(\tau_2)),{\tau_2}\in (0,1),\end{array}
\int_{0}^{\infty}\left(F_{\chi^2_{n}(\frac{\rho_2^2}{\sigma_j^2})}(y)-F_{\chi^2_{n}(\frac{\rho_1^2}{\sigma_j^2})}(y)\right) dy = \frac{\rho_1^2 - \rho_2^2}{\sigma_j^2}.
\Bigl |\frac{x}{|x|^2}-\bar x \Bigr|^2 = \frac{1}{|x|^2}-\frac{2x\cdot\bar{x}}{|x|^2}+2=\frac{|x|^2-2x\cdot\bar{x}+|\bar{x}|^2-1}{|x|^2}+1=\frac{|x-\bar x|^2-1}{|x|^{2}}+1=1.
\psi([\xi,\eta]) =\sum_j [\xi,\eta](a_j)\varphi(b_j) &=\sum_j (\xi\otimes \eta)((\delta r)(a_j))\varphi(b_j) &=\sum_{i,j}\xi([a_j,a_i])\eta(b_i)\varphi(b_j)+\sum_{i,j}\xi(a_i)\eta([a_j,b_i]))\varphi(b_j).
\hat{\rho}_{\tau}(k)=\frac{\sum_{t=k+1}^{T}\psi_{\tau_{1}}(y_{1t}-\hat{q}_{1,t}(\tau_{1}))\psi_{\tau_{2}}(y_{2,t-k}-\hat{q}_{2,t-k}(\tau_{2}))}{\sqrt{\sum_{t=k+1}^{T}\psi_{\tau_{1}}^{2}(y_{1t}-\hat{q}_{1,t}(\tau_{1}))}\sqrt{\sum_{t=k+1}^{T}\psi_{\tau_{2}}^{2}(y_{2,t-k}-\hat{q}_{2,t-k}(\tau_{2}))}},
h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}+\beta-\alpha^*-\beta^*}\mathcal{G}(2-\alpha-\beta+\alpha^*+\beta^*,\alpha,\beta-\alpha^*-\beta^*;A,B_{-\alpha^*-\beta^*},h,k) = \mathcal{K}(0,0,2-\alpha-\beta+\alpha^*+\beta^*;A,B_{-\alpha^*-\beta^*},\alpha,\beta-\alpha^*-\beta^*,h,k).
q_{k}^{\varphi}:=1-p_{k}^{\varphi},\qquad\left(p_{k}^{\varphi}\Psi\right)(x_{1},\ldots,x_{N}):=\frac{\varphi(x_{k})}{\Vert \varphi\Vert_2}\int d^{3}x_{k}\,\frac{\varphi^{*}(x_{k})}{\Vert \varphi\Vert_2}\Psi(x_{1},\ldots,x_{N}),\qquad1\leq k\leq N.
w_{k\overline l}^-{''}-\left[\frac{\overline l\left(\overline l+1\right)}{r^2}-\frac{\kappa}{r}\frac{V'}{2m+V}-\frac{1}{2}\frac{V''}{2m+V}+\frac{3}{4}\left(\frac{V'}{2m+V}\right)^2-V^2-2mV-k^2\right]w_{k\overline l}^-=0\,,
A = z \prod_{i=1}^h F_{k,i}(A) \prod_{i=h+1}^{k+1}F_{k,i}(A) \prod_{i=1}^{h-1}G_{k,i}(A)^{-1}\prod_{i=h}^{k}G_{k,i}(A)^{-1}&= z \prod_{i=1}^{k+1}F_{k,i}(A) \prod_{i=1}^{k}G_{k,i}(A)^{-1}.
\left\vert x\left( t\right) -x\left( s\right) -m\left( t\right) +m\left(s\right) \right\vert^{2}=2\int_{s}^{t}\left\langle m\left( r\right)-m\left( s\right) ,dk\left( r\right) \right\rangle & +2\int_{s}^{t}\left\langle x\left( s\right) -x\left( r\right) ,dk\left(r\right) \right\rangle .
\left|\log\left( \frac{\frac{zf_1'(z)}{f_1(z)}}{2-\frac{zf_1'(z)}{f_1(z)}}\right)\right| =\left|\log\left( \frac{1+\frac{e-1}{e+1}}{2-\left(1+\frac{e-1}{e+1}\right)}\right)\right|=\left|\log(e)\right|=1.
(\psi_0-\psi_1)(x_0,x_1,x_2) =x_0 f^{[2]}(x_0,x_1,x_2)-x_1f^{[2]}(x_0,x_1,x_2) &=f^{[1]}(x_0,x_2)- f^{[1]}(x_1,x_2) &=(\tilde\psi_0-\tilde\psi_1)(x_0,x_1,x_2).
\left( \begin{array}{c c}{{D^{2}e^{2 V}}}&{{\sqrt{2}\bar{\Phi}}}\\{{\sqrt{2}\Phi}}&{{\bar{D}^{2}e^{- 2 V}}}\\ \end{array}\right) \left( \begin{array}{c}{{Q}}\\{{\widetilde{Q}^{\dagger}}}\\ \end{array}\right) = 0 .
\mathcal{M}(u^1,\dots,u^r) = \sum_{\substack{i_1 \in \{1,\dots,m_1\}\vdots i_r \in \{1,\dots,m_r\}}}\sum_{\substack{|\alpha_1| = k_1 \vdots |\alpha^r| = k_r}}c_{i_1,\dots,i_r,\alpha^1,\dots,\alpha^r}\prod_{j=1}^r \partial^{\alpha^j}u^j_{i_j},
w_{g}w_{h}xw_{h}^{*}w_{g}^{*}=e_{g}w_{hg}xw_{hg}^{*}e_{g}~~(\textrm{by Lemma \ref{w-relations}}) =e_{g}\nu(j_{hg}(x)) = e_{g}\nu(\beta_{g}^{-1}(d)) =\nu(1_{\Omega g})\nu(\beta_{g}^{-1}(d)) =\nu(\beta_{g}^{-1}(d)1_{\Omega g})
\begin{aligned}|\underline{\vec z_{k+d-1}}\wedge\underline{\vec z_{k+d}}| = \bigg|\vec b_{k+d-1}\wedge\sum_{i=k+1}^{k+d-2}\nu_i\vec b_i\bigg|= = |\vec b_{k+d-1}|\bigg(\sum_{i=k+1}^{k+d-2}\nu_i^2|\vec b_i|^2\bigg)^{1/2}= \lambda_k''\cdot3|\underline{\vec z_{k+d-1}}|^{2+\beta}; \hskip2.1mm \end{aligned}
\tilde{T_{a}}\Theta_{k}\cong \Theta_{k}+ \frac{1}{M}C^{l}\left[ \begin{array}{c c c}{{D_{l}^{( 0 )}}}&{{0}}&{{0}}\\{{0}}&{{D_{l}^{\ast ( 0 )}}}&{{0}}\\{{0}}&{{0}}&{{\frac{\partial}{\partial s^{l}}}}\\ \end{array}\right] \Theta_{k},
{\cal O}_{\pm}^{a}( \{x \}) : = ( - 1 )^{\frac{N ( N - 1 )}{2}}\left[ \frac{1}{N !}\sum_{\pi}\mathrm{s i g n}( \pi ) \prod_{b = 1}^{N}\overline{{\psi}}_{\stackrel{{\scriptstyle 1}}{{\scriptstyle 2}}}^{( \pi ( b ) )}( x^{( b )}) \right]