harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tau^b = (d_1+1-\lambda)!{d_1+1 \choose \lambda}{d_2+1 \choose d_1+\lambda}{n_1 \choose d_1+1}{n_2 \choose d_2+1}&+ \sum_{2r_0+r_1=\lambda}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}\frac{(d_2+2)!}{2^{r_2}\lambda!}{n_1 \choose d_1+1}{n_2 \choose d_2+2},
\phi_{32}(z; \tau) =-\sum_{n_1, n_2 \in Z}\left(\frac{-4}{n_1(n_2-1)}\right) \zeta_{1}^{\frac{n_1}{4}}\zeta_{2}^{\frac{n_2}{4}}q^{n_{1}^2 + n_{2}^2}&= \theta \left(\frac{z_1}{2}; 8 \tau \right) \theta_{4}\left(\frac{z_2}{2}; 8 \tau \right).
\sup_{R_{y,x}}N(\epsilon\lVert (F\circ\phi_{y})/(w_{\theta}\circ\phi_{x})\rVert_{L_{2}(R_{y,x})}, \mathcal{F}/w_{\theta}, L_{2}(R_{y,x})) &=\sup_{Q_{y,x}}N(\epsilon\lVert (F\circ\phi_{y})/(w_{\theta}\circ\phi_{x})\rVert_{L_{2}(Q_{y,x})}, \mathcal{F}/w_{\theta}, L_{2}(Q_{y,x}))
\Psi_{[b]}(x) =\sum_{c_1' \delta(x) n^{-1/2}\ \leq \p \ \leq \c_2'\delta(x) n^{-1/2}}\left[\prod_{i=1}^n \left( \psi\left(\frac{x_i-b_i}{p}\right) + 1\right)\right].
T_wT_{w'}=T_{w'w}\quad\mbox{for}w,w'\in W^{(1)}\mbox{with}\ell(w)+\ell(w')=\ell(ww'), T^2_{s_i}=T_{s_i}\tau_{i}\quad\mbox{where}\tau_{i}=\sum_{t\in\overline{T}_i}T_{t}\quad\mbox{for}0\le i\le d.
\lim_{\varepsilon \to +0}\varepsilon \biggl( \frac{\zeta(s_1+s_2,a) - a^{-s_1-s_2}}{2}+\sum_{n=1}^\infty \frac{\Phi_0 (s_2 \,|\, n,a)}{(n+a)^{s_1}}\biggr) =0
p_{ij}= \bar\alpha_i + \bar\beta_j\quad\mbox{for some $\bar\alpha_1,\ldots,\bar\alpha_k,\bar\beta_1,\ldots,\bar\beta_k$ in the individual EM model, and}p_{ij}= 1 - \bar\gamma_i\bar\delta_j\quad\mbox{for some $\bar\gamma_1,\ldots,\bar\gamma_k,\bar\delta_1,\ldots,\bar\delta_k$ in the mass EM model.}
\begin{cases}dy(t) = Ay(t)dt + Fy_t dt+ f(t)dB(t),\hskip 15pt t\ge 0, y(0) = \phi_0,\hskip 5pt y_0=\phi_1,\hskip 5pt \Phi= (\phi_0, \phi_1)\in{\cal X}, \end{cases}
\sum_{j = 2}^{3}\phi_{j}^{a}f_{1 t}^{j}= - \sum_{j = 2}^{3}[ \phi_{j t}^{a}f_{1}^{j}+ \phi_{j 1}^{a}f_{t}^{j}+ \sum_{k = 2}^{3}( \phi_{j k}^{a}f_{t}^{k}) f_{1}^{j}] - \phi_{1 t}^{a}\quad \quad a = 1 , 2 , 3
\langle v, p \rangle = \langle \gamma(p), p \rangle = - \left[ \frac{\langle \eta(p), p_0 \rangle}{1 + \langle p, p_0 \rangle}\right] \langle p + p_0, p \rangle + \langle \eta(p), p \rangle = - \langle \eta(p), p_0 \rangle.
\mathcal G_{c,i_0+1}(\phi_{i_0}(n,k)U_{n,k}) = B_{i_0+1}(n,c) (k+c)\phi_i (n,k)U_{n,k}= 0, \mathcal G_{c+1,i_0+1}(\psi_{i_0}(n,k)U_{n,k}) = B_{i_0+1}(n,c+1)(k+c)\psi_i (n,k)U_{n,k}=0.
\lim_{n\rightarrow \infty}\frac{\dim_kL_{nm}}{n^{q(L)}}=\sum_{t=1}^{[k':k]}\lim_{n\rightarrow \infty}\frac{\#(S(L)_{nm}^{(t)})}{n^{q(L)}}=\sum_{t=1}^{[k':k]}\frac{{\rm vol}_{q(L)}(\Delta(S(L)^{(t)}))}{{\rm ind}(S(L)^{(t)})}
R_1(v)=(k-1)M_{12}(f_k)=(k-1)M_{21}(f_k^{-1})=m_k(k-1)(M_{22}(f_k^{-1})+M_{22}(h_k^{-1}))= = m_k(k-1)(M_{22}(f_k)+M_{22}(h_k)).
f_{[j,j+1]}= (b_j \oplus f_{j+1}) \, \Theta(\alpha_j)^\dag = \begin{pmatrix}\alpha_jb_j \rho_jb_j \rho_jf_{j+1}-\overline\alpha_jf_{j+1}\end{pmatrix}, \rho_j = \sqrt{1-|\alpha_j|^2},
L=\!\int d^2x\left( i\,\psi^\dagger \dot\psi\! +\! i\,\chi^\dagger \dot\chi\!-\!{1\over 2}| \nabla\psi|^2\!-\!{1\over 2}| \nabla\chi|^2 \!-\!{v\over 4}[\psi^\dagger\! *\! \psi^\dagger \!+\!\chi^\dagger\! *\! \chi^\dagger]\!*\! \left[\psi\! *\! \psi\!+\! \chi\! *\! \chi\right]\right),
\Psi_{\varsigma,K}=\frac{1}{\sqrt{\varepsilon}}\|\boldsymbol{\sigma}_K^{\varsigma}\|_{\boldsymbol{L}^{2}(K)}+\texttt{C}_{K}\|\boldsymbol{\mathsf{osc}}_{K}^{\varsigma}\|_{\boldsymbol{L}^{2}(K)}.
E_1(f_x,s)= \prod_{i=1}^g{(1-\beta_i^{-1}(f_x)p^{s+g-k})}, E(f_x,s) = E_1(f_x,s)\prod_{i=1}^g\frac{1}{(1-\beta_i(f_x)p^{k-s-g-1})}.
D_{ij}:=a_{i1}a_{j2}-q^{-1}a_{i2}a_{j1}D_{in}:=a_{i1}a_{n2}-q^{-1}a_{i2}a_{n1}&D_{55}:=a_{51}a_{52}D_{66}:=a_{61}a_{62}&D_{56}=a_{51}a_{62}-q^{-1}a_{52}a_{61}
\kappa := \tfrac{2 \nu}{1 + \nu}\quad\beta :=\tfrac{2 - \kappa}{2}\left(\tfrac{\tilde\beta}{1 - \kappa}\right)^{\frac{1 - \kappa}{2 - \kappa}}a^{\frac{1}{2 - \kappa}}\text{for some $0 < \tilde\beta \leq 1$ and $a > 0$,}
\exp(t)\ddot h(t) = \frac{d}{dt}\big(\exp(t)\dot h(t)\big)-\exp(t)\dot h(t) \exp(t)\dot u(t) = \frac{d}{dt}(\exp(t)u(t))-\exp(t)u(t) \exp(t)\dot h(t) = \frac{d}{dt}\big(\exp(t) h(t)\big)-\exp(t) h(t)
Dl_n(f)g = \langle Dl_n(f),g\rangle_R, Dl_\infty(f)g = \langle Dl_\infty(f),g\rangle_R, D^2l_n(f)gh = \langle D^2l_n(f)g, h\rangle_R, D^2l_\infty(f)gh = \langle D^2l_\infty(f)g, h\rangle_R.
\mathbb{E}\left[\mathbf{s}^{\left(\right)^H}\mathbf{s}^{\left(\right)}\right]= \mathbb{E}\left[s_c^*s_c\right]+\mathbb{E}\left[s_1^*s_1\right]\cdots+\mathbb{E}\left[s_M^*s_M\right],=M+1.
F^{a{\mu}_{1}{\cdots}{\mu}_{p+1}}=\frac{(-1)^p}{(p+1)!}{\Gamma}^{ab}{\epsilon}^{{\mu}_{1}{\cdots}{\mu}_{p+1}{\nu}_{1}{\cdots}{\nu}_{p+1}}G^{b}_{{\nu}_{1}{\cdots}{\nu}_{p+1}}.
\sum_{i}B_{ij}^{(u)}w_{i}^{(u)}=\sum_{i\in I_{\leq}}B_{ij}^{(u)}d_{i}^{(u)}+\sum_{i\in I_{>}}B_{ij}^{(u)}\rho_{u}& =\sum_{i\in I_{\leq}}B_{ij}^{(u)}d_{i}^{(u)}+\sum_{i\in I_{>}}B_{ij}^{(u)}d_{i}^{(u)}=\sum_{i}B_{ij}^{(u)}d_{i}^{(u)}& \geq a_{j}^{(u)}+\sum_{u=v+1}^{N}(t_{v}-t_{v-1})K_{ij}^{(v,u)}w_{i}^{(v)}\mbox{(by (\ref{copteq247}))}.
g^{(q)}(x,a) =-\tilde{r}^{(q)}(x)+\frac{\mathbb{W}^{(q)}(x-b)}{\mathbb{W}^{(q)}(a-b)}\tilde{r}^{(q)}(a) +\frac{\tilde{r}^{(q)}(b)+g^{(q)}(b,a)}{Z^{(q)}(b)}\left(r^{(q)}(x)-\frac{\mathbb{W}^{(q)}(x-b)}{\mathbb{W}^{(q)}(a-b)}r^{(q)}(a)\right).
\times e^{-{i}[ \tilde{\bf r}^{\prime \prime}\times \tilde{\bf r}+ \tilde{\bf r}^{\prime \prime \prime}\times \tilde{\bf r}^{\prime \prime \prime \prime}]}f ( \tilde{\bf r}^{\prime \prime \prime}) g ( \tilde{\bf r}^{\prime \prime \prime \prime}) h ( \tilde{\bf r}^{\prime \prime}) .
\partial_{x_jx_k}I_{h,T}\mathbf{u}(X)=\sum_{i\in\mathcal{I}^{s'}}\partial_{x_jx_k}\Phi_{i,T}(X)(\mathbf{E}_i^s(X) + \mathbf{F}_i^s(X)) + \sum_{i\in\mathcal{I}}\partial_{x_jx_k}\Phi_{i,T}(X)\mathbf{R}_i^s(X), ~s = \pm,
\biggl ( \int_{\mathrm{I m z = 0 \a x i s}}+ \int_{C_{R}}){\frac{{\bar{z}}^{c}( 1 -{\bar{z}})^{d}z^{1 + a}}{1 + a}}\_{2}F_{1}( 1 + a , - b , a + 2 ; z ) ,
F(x,t) =\int_0^t f(x,s)ds=\int_0^t \frac{f(x,s)}{\vert s\vert^{g^+-1}}\vert s\vert^{g^+-1}ds \geq \frac{f(x,t)}{\vert t\vert^{g^+-1}}\int_0^t \vert s\vert^{g^+-1}ds=\frac{1}{g^+}f(x,t)t \geq 0\ \ \ \ (\t\leq s<0\ \f(t)<0).
z(t) =\bar{z}(t)+\textbf{z}(v(t),w) x(t) =\bar{x}(t)+\textbf{x}(v(t),w) &=b(w)U_{d}^{-1}\xi(t)+\textbf{x}(v(t),w) &=b(w)U_{d}^{-1}(\bar{\xi}(t)+\hat{\xi}(t))+\textbf{x}(v(t),w) \eta(t) =\bar{\eta}(t)+\theta(v(t),w)+C\bar{x}(t) &=\bar{\eta}(t)+\theta(v(t),w)+Cb(w)U_{d}^{-1}(\bar{\xi}(t)+\hat{\xi}(t)).
ln\gamma_{s}=\frac{h-\nu}{\nu}ln(1+\frac{br_{a}}{55.5})-\frac{h}{\nu}ln(1-\frac{br_{a}}{55.5})+\frac{br_{a}(r_{a}+h-\nu)}{55.5(1+\frac{br_{a}}{55.5})}
{\cal G}^{MN}{\cal R}_{MN}=g^{\mu\nu}R_{\mu\nu}+4\sum_{i=1}^3 \nabla^2\ln r_i+2\sum_{i=1}^3 \partial^\mu \ln r_i\partial_\mu r_i+4\sum_{i,j=1}^3\partial^\mu\ln r_i\partial_\mu \ln r_j\,,
\lim_{y \rightarrow 1}\left\{\frac{\frac{d}{d y}\left[ R_{I}( y \sqrt{v - \epsilon}) \right]}{R_{I}( y \sqrt{v - \epsilon})}\right\}= \lim_{y \rightarrow 1}\left\{\frac{\frac{d}{d y}\left[ R_{E}( y \sqrt{\epsilon}) \right]}{R_{E}( y \sqrt{\epsilon})}\right\}.
\begin{aligned}M_j^2 = \left(~ |r_i |+ |c_i |\cos\left(\beta+\dfrac{2\pi}{3}j\right)\right)^2 &~~~~~+\left(|c_i |\sin\left(\beta+\dfrac{2\pi}{3}j\right)\right)^2 &= |r_i |^2+ |c_i |^2+2 |r_i | |c_i |\cos\left(\beta+\dfrac{2\pi}{3}j\right), \quad\forall j = 0, 1 ,2\end{aligned}
\mathcal{R}_m(t)=\sum^N_{j=1}\sum^k_{\ell=1}L_{\ell,j}\Bigr(\chi_j(x)e^{-m\varphi_j(z)-im\theta}\Bigr)P_{\ell,j}\Bigr(e^{-t\Box^+_{B_j,m}}(z,w)\Bigr)e^{m\varphi_j(w)+im\eta}\widetilde\chi_j(y),
\int_{0}^{2\pi}\frac{\cos(ky)}{(1+r^2)+(r^2-1)\cos(y)}dy = \frac{\pi}{r}\left(\frac{1-r}{1+r}\right)^{|k|}, \int_{0}^{2\pi}\frac{\sin(ky)}{(1+r^2)+(r^2-1)\cos(y)}dy = 0
\Delta^{a}\equiv \left( \nabla_{\lambda}- i g \nu^{a}\overline{{{A}}}_{\lambda}\right)^{2}= \nabla^{2}+ i g \nu^{a}H ( x_{1}\partial_{2}- x_{2}\partial_{1}) -{\frac{g^{2}( \nu^{a})^{2}}{4}}H^{2}( x_{1}^{2}+ x_{2}^{2}) .
c_{n m}= \prod_{j = 0}^{m - 1}\Big ( e^{i k}q^{2 n + 1 / 2}\frac{\lambda^{1 / 2}+ \tau \kappa \rho^{- 1}q^{- j - 1 / 2}}{\lambda^{1 / 2}+ \kappa \rho q^{j + 1 / 2}}\Big ) ,
v_{\Lambda \Omega}= \frac{1}{f_{\Lambda}}{\rm{sgn}}(\sigma_{R_{ij}^{(\ell)}\Lambda}^a) \, \frac{v_{\Lambda_{ij}\left(R_{ij}^{(\ell)}\Lambda \right)_{ij}}}{f_{\left(R_{ij}^{(\ell)}\Lambda\right)_{ij}}}f_{\Omega}\, \#(\Omega,\Lambda) \, .
\int_0^t \left\|u\right\|_{L^2_s}^2 \left\|u(s)\right\|_2^2 \exp\left(\int_s^t \left\|u(w)\right\|_2^2dw\right) ds\leq \left\|u\right\|_{L^2_t}^2 \left[-\exp\left(\int_s^t \left\|u(w)\right\|_2^2dw\right)\right]_{s=0}^t &=\left\|u\right\|_{L^2_t}^2\left(\exp\left(\int_0^t \left\|u(s)\right\|_2^2ds\right)-1\right).
(1-t_1a_1)(1-\rho a_1(t_1)a_2(t_2))(1-t_2a_2) = \frac{h_{a_1}(t_1)+h_{a_2}(t_2) - 1}{h_{a_1}(t_1)h_{a_2}(t_2)}\ (1 - t(a_1+a_2)).
\int_{0}^{T}\ln\frac{4\pi(r(t)-2)(1-\tau)}{r(t)^{2}}dt=\frac{T}{\frac{1}{3}-\frac{1}{q}}\int_{3}^{q}\frac{1}{r^{2}}\ln\frac{4\pi(r-2)(1-\tau)}{r^{2}}dr,
||\varphi(a)||_2^2 = \tau(( a\otimes p + \beta(a) \otimes p^{\perp})^*( a\otimes p + \beta(a) \otimes p^{\perp})) & = \tau(a^*a\otimes p + \beta(a^*a)\otimes p^{\perp}) & = \frac{1}{2}\tau(a^*a) + \frac{1}{2}\tau(a^*a) & = ||a||_2^2.
\textit{\textbf{I}}_{1}(t) = \hat{\textbf{h}}~\!\!^H_{1,k}\big(\textbf{h}_{1,k}-\hat{\textbf{h}}_{1,k}\big) x_{1,k}(t), \textit{\textbf{I}}^{\prime}_{1}(t) = \hat{\textbf{h}}~\!\!^H_{1,k}\big(\textbf{h}_{1,k}-\hat{\textbf{h}}_{1,k}\big) s_{1,k}(t).
1 + d\bigg(\frac{-(1 - 3 d D) + \sqrt{(1 - d D )^{2}+ 2 a d^{2}C}}{2 d}- 2 D \bigg) & = 1 + d\bigg(\frac{-(1 - d D) + \sqrt{(1 - d D )^{2}+ 2 a d^{2}C}}{2 d}\bigg) & > 1 + d\bigg(\frac{-(1 - d D) + | 1 - d D |}{2 d}\bigg) \geq 1.
i(\lambda,n)i(*)= (s_{\lambda},n)(s_{\delta},1)=\sum_{\mu\in\mathcal{A}_{\delta}}\#\Gamma_{\Delta}(\lambda,\mu)(s_{\mu},n+1) = \sum_{\mu\in\mathcal{A}_{\delta}}e\big((\lambda,n),(\mu,n+1)\big)i(\mu,n+1).
g_1(t) = 5 \sin 10 t, g_2(t) = 5 \cos 10 t, g_3(t)+ g_7(t) = -1 , g_4(t) = 1, g_5 = 1, g_6(t) + g_7(t) = 1, g_8(t) = g_9(t) = 0 .
\Gamma_{\mathrm{N , \{\n u_{j}\}}}= \frac{\hbar}{2}\left( - \frac{\lambda}{4}\right)^{\mathrm{N_{1}+ N_{2}}}\left[ \Phi \cdots \Phi \right]_{\star_{\mathrm{N_{1}}}}\left( - P \right) \widetilde{{\cal K}}_{\mathrm{N}- d / 2}\left( P \circ P + 1 / \Lambda^{2}\right) \left[ \Phi \cdots \Phi \right]_{\star_{\mathrm{N_{2}}}}( + P ) ~ ,
s_\lambda = \det\left[\begin{array}{c}h^{(j-1)}_{\lambda_i-i+1}\end{array}\right]_{1\leq i,j\leq\ell(\lambda)}\equiv\det\left[\begin{array}{c}h^{(0)}_{\lambda_i-i+j}\end{array}\right]_{1\leq i,j\leq\ell(\lambda)}\,.
\hat{H}_{\mathrm{I I E}}= \frac{1}{2 m}[ \hat{F}^{k}\!_{i}( \hat{f}= 0 ) , \hat{\Pi}_{k}- \hat{A}_{k}]_{+}\hat{g}^{i j}[ \hat{F}^{l}\!_{j}( \hat{f}= 0 ) , \hat{\Pi}_{l}- \hat{A}_{l}]_{+}+ V ( \hat{q}) +{\hbar}^{2}\hat{Q}\; ,
J_{\delta}\setminus \{(\imath_0, r_0+\frac{1}{2}), (\imath_0-1, r_0), (\imath_0, r_0-\frac{1}{2})\}=J_{\beta(\imath_0, r_0)(\delta)}\setminus \{(\imath_0-1, r_0+\frac{1}{2}), (\imath_0, r_0), (\imath_0-1, r_0-\frac{1}{2})\}.
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(288n+216)q^n \equiv f_6^3\cdot\dfrac{f_1^2}{f_2}&\equiv f_6^3\left(\dfrac{f_9^2}{f_{18}}-2q\dfrac{f_3f_{18}^2}{f_6f_9}\right)~(\textup{mod}~4),
\begin{array}{l c c c l}{{\hat{G}_{0}\equiv G_{0}\otimes \sigma_{3},}}&{{}}&{{}}&{{}}&{{\hat{G}_{- 1}\equiv G_{- 1}\otimes \sigma_{3},}}\\{{\hat{G}_{- \frac{1}{3}}\equiv G_{- \frac{1}{3}}\otimes \sigma_{3},}}&{{}}&{{}}&{{}}&{{\hat{G}_{- \frac{2}{3}}\equiv G_{- \frac{2}{3}}\otimes \sigma_{3}}}\\ \end{array}
f\star g \equiv \sum_{n=0}^{\infty}\frac{1}{n!}\left( \frac{i}{2}\right)^n \nabla_{A_1}\cdots \nabla_{A_n}f\ \Omega^{A_1B_1}\cdots\Omega^{A_nB_n}\ \nabla_{B_1}\cdots\nabla_{B_n}g &= f \exp \left(\frac{1}{2}i\Omega(\overleftarrow{\nabla},\overrightarrow{\nabla}) \right) g.
\int_0^1 (\Psi_d^{(\kappa)}\Omega^{(\kappa)})(y)\, T^{(\kappa)}\, (\Psi_d^{(\kappa)})^*(1-y)^{\alpha(\kappa)}y^{\beta(\kappa)}dy = \int_0^1 \Psi_d^{(\kappa)}(y)\,T^{(\kappa)}\, (\Psi_d^{(\kappa)}\Omega^{(\kappa)})^* (1-y)^{\alpha(\kappa)}y^{\beta(\kappa)}dy.
|Z_3| = \left| \frac{1}{\log x}\sum_{\rho}\frac{x^{\rho-s}-x^{2(\rho-s)}}{(s-\rho)^2}\right| \leq \frac{x^{\frac{1}{2}-\sigma}+x^{1-2\sigma}}{\log x}\sum_{\gamma}\frac{1}{(\sigma-\beta)^2+(t-\gamma)^2}.
a'' = \frac{2nQ^4}{a}- 2n \frac{(a')^2}{a}+ \left(f' +2n \frac{Q'}{Q}\right) a' b'' = 2\frac{n+1-Q^2}{b}- 2n \frac{(b')^2}{b}+ \left(f'- \frac{Q'}{Q}\right) b'.
\sum_{\nu \in{\cal R}}( L^{-})_{\mu \nu}\, e^{W}= \left( p \cdot \mu - i \omega q \cdot \mu + i \sum_{\rho \in \Delta_{+}}{\frac{\rho \cdot \mu}{\rho \cdot q}}\right) \, e^{W}= \mu \cdot \! \left( p + i{\frac{\partial W}{\partial q}}\right) e^{W}= 0 ,
\Xi_{1}( \kappa ) = \lim_{N \to \infty}\mu_{1 , N}^{1 1}( \kappa ) = \lim_{N \to \infty}\left[ \frac{\sinh \frac{\pi \kappa}{2}}{\sinh \frac{\pi N \kappa}{4}}\, e^{\frac{\pi N \kappa}{4}}- e^{\frac{\pi \kappa}{2}}\right] = - e^{- \frac{\pi | \kappa |}{2}}.
\Sigma(1,4)= 1234, 1235, 1236, 1237, 1245,1246, 1247, 1256, 1257,1345,1346,1347,1356,1357 \Sigma(2,4)= 1234, 1235, 1236, 1237, 1245,1246, 1247, 1256, 1257 \Sigma(3,4)= 1234, 1235, 1236, 1237 \Sigma(4,4)= 1234
h_{-L}^{\mathcal{L}}(t)=\sum_{l=2}^{l}\frac{(-1)^{l}}{l}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}\prod_{s=1}^lh_{-I_s}^{\mathcal{L}_{\mathbf{I}_s}}(t).
2\sqrt{2}e^{x^2/2}(x^2-3)\int_0^{x/\sqrt{2}}e^{-t^2}dt+2x \leq 2\sqrt{2}e^{x^2/2}(x^2-3)\frac{x}{\sqrt{2}}e^{-x^2/2}+2x &=2x(x^2-3)+2x\leq -4x+2x\leq 0,\forall x\in[0,1].
d\mu_{n}\left(y\right)=\frac{1}{2}\left(1+\lambda_{n}^{2}\right)e_{\lambda_{n}}\left(y\right)dy+\frac{1}{2}\left(\left(1-i\lambda_{n}\right)\delta_{0}+\left(1+i\lambda_{n}\right)e^{-i\lambda_{n}}\delta_{1}\right)
\Gamma^{(1,0)}=\left(\begin{array}{cccc}0 0 0 -i 0 0 -i 0 0 0 0 0 0 0 0 0 \end{array}\right), \Gamma^{(0,1)}=\left(\begin{array}{cccc}0 -i 0 0 0 0 0 0 0 0 0 0 0 0 i 0 \end{array}\right)
D^{\beta-1/2,\infty}_\infty = \left\{f \in B^{\beta-1/2,\infty}_\infty, \, f\geq 0, \, \mathrm{supp}(f) \subseteq [0,1], \, \int_{[0, 1]}f(x)\, \mathrm{d}x = 1 \right\},
\mathrm{m}_q = \frac{1}{2}c_q(n) \int_M \left(L_{(q)}+ (2q-1)! \left\langle \left[ T_{(2q-1)\alpha}, A_{\beta}\right] \cdot e_{\alpha}^{\top}, e_{\beta}^\top \right\rangle \right) \frac{1}{\sqrt{G}}dM,
\|F(t)-F(t_1)\| =\left\|\int\limits_{t_1}^{t}f(s)ds\right\| &\leq \left\|\int\limits_{t_1}^{t}f(s)ds-\sum\limits_{i=1}^{j}f(s_i)\Delta_i\right\|+\left\|\sum\limits_{i=1}^{j}f(s_i)\Delta_i\right\| &< \frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon
E^{+}( - h_{\lambda}, z_{2}) E^{-}( - h_{\mu}, z_{1}) = \left( 1 - \frac{z_{1}}{z_{2}}\right)^{\langle \lambda , \mu \rangle}E^{-}( - h_{\mu}, z_{1}) E^{+}( - h_{\lambda}, z_{2}) ,
\Psi_{i , g_{1}}^{[ L ] [ M ]}( x_{1}) \Psi_{j , g_{2}}^{[ M ] [ N ]}( x_{2}) \ = \ \sum_{k}( x_{1}- x_{2})^{h_{i}+ h_{j}- h_{k}}\Psi_{k , g_{1 2}}^{[ L ] [ N ]}( x_{2}) C_{( i , g_{1}) ( j , g_{2}) ( k , g_{1 2})}^{[ L ] [ M ] [ N ]}+ \dots
\phantom{{}={}}\frac1{n^2(n-3)}\Big\{(4 n \ell-n^2) \dfrac{\ell-Z_{\{3,4\}}(\ell)}{n-2}- 3\ell^2 - n^2 \dfrac{\ell-Z_{\{3,4\}}(\ell)}{n-2}Z_{\{3,4\}}(\ell) + n\ell Z_{\{3,4\}}(\ell) \Big\}&=\frac1{n^2(n-2)(n-3)}\Big\{F_{n1}+ F_{n2}\Big\},
\lambda=\bigoplus_{1\leq k\leq n}\bigoplus_{1\leq i_{1},\ldots,i_{k}\leq n}(a_{i_{1}i_{2}}a_{i_{2}i_{3}}\cdots a_{i_{k}i_{1}})^{1/k}=\mathop\mathrm{tr}A\oplus\cdots\oplus\mathop\mathrm{tr}\nolimits^{1/n}(A^{n}).
\mathcal{H}= \left\lbrace f:\mathbb{R}^{d}\rightarrow \mathbb{R}: \int_{\mathbb{R}^{d\cdot q}}\left( J(f) (\mathbf{y}_{1}, \ldots ,\mathbf{y}_{q}) \right)^{2}d\mathbf{y}_{1}\ldots d\mathbf{y}_{q}< \infty \right\rbrace
\delta \left( Z_{\;\;\beta_2}^{\alpha_1}\eta_{\alpha_1}^{*}+\frac 12C_{\beta_2}^{\alpha_0\beta_0}\Phi_{\alpha_0}^{*}\Phi_{\beta_0}^{*}\right) =D_{\;\;\beta_2}^{\alpha_2}\Phi_{\alpha_2}^{*},
I(\mathbf{V}^n_1;Y^n_1|\mathbf{V}^n_2) &=h(Y^n_1|\mathbf{V}^n_2)-h(Y^n_1|\mathbf{V}^n_1,\mathbf{V}^n_2) &\leq\frac{n}{2}\log(2\pi e(\alpha P+N_1))-h(Y^n_1|\mathbf{S}^n_1,\mathbf{S}^n_2) &=\frac{n}{2}\log(2\pi e(\alpha P+N_1))-h(Z^n_1) &=\frac{n}{2}\log\frac{\alpha P+N_1}{N_1}.
\|a_{1}\|_{L_{t}^{\infty}L_{x}^{p}}+\big\|\nabla|a_{1}|^{\frac{p}{2}}\big\|^{\frac{2}{p}}_{L_{t}^{2}L_{x}^{2}}\leq \|w_{0}\|_{L^{p}}\begin{cases}1+\frac{1}{\sqrt{2}}, p=2, 1+\sqrt[p]{\frac{p-2}{4p}}, p>2.\end{cases}
\dot{x}= \omega+\xi(y,z,\omega,\sigma,\mu)+f(x,y,z,\omega,\sigma,\mu), \dot{y}= \sigma+\eta(y,z,\omega,\sigma,\mu)+g(x,y,z,\omega,\sigma,\mu), \dot{z}= \widehat{Q}(\omega,\sigma,\mu)z+\widehat{\zeta}(y,z,\omega,\sigma,\mu)+h(x,y,z,\omega,\sigma,\mu),
|T_kS_kx - TSx| = |T_kS_kx - T_kSx + T_kSx -TSx| & \leq |T_kS_kx - T_kSx| + |T_kSx -TSx| & = |T_k(S_kx-Sx) | + |(T_k-T)Sx| & = |S_kx-Sx| + |(T_k-T)Sx|,
\int_{\Omega}\left(\begin{array}[c]{c}\left( \frac{\partial a^{ijkl}}{\partial u_{pq}}\left( M^{\ast}(x)\right)u_{ik}(x+he_{m})v_{pq}(x)+a^{ijkl}(D^{2}u(x))v_{ik}(x)\right) \times\left[ \zeta^{4}v_{jl}+4\zeta^{3}\zeta_{j}v_{l}+4\zeta^{3}\zeta_{l}v_{j}+4v(\zeta^{3}\zeta_{jl}+3\zeta^{2}\zeta_{j}\zeta_{l})\right] (x)\end{array}\right) dx=0.
\|Q_{B,L}^n(f) - Q_{D,T}^n(f)\| \leq C_2 d((B,L),(D,T))^\alpha v_\alpha (f) \sum_{i=0}^{\infty}k_\alpha(B^i,L^i) &\leq C d((B,L),(D,T))^\alpha v_\alpha(f).
k_{25r}=\Psi\left(k_r,k_{r/25}\right)=\sqrt{\frac{1}{2}-\frac{1}{2}\sqrt{1-4(k_{r}k'_{r})^2\cdot Y\left(\sqrt[3]{\frac{(p-1)^5}{11+6p+6p^2+p^3+p^4}}\right)^{12}}}
F_{\nu}^{e x t}= \, - \frac{\alpha_{f}\hbar c}{2 \pi}\int d^{3}r \vec{R}_{\nu \mu}^{e x t}\vec{J}^{\mu}= \int \hspace{0 . 5 m m}{^*}\hspace{- 0 . 8 m m}\vec{F}_{\nu \mu}^{e x t}d^{3}r \vec{j}^{\mu}.
Z_f(s, \chi, A_1) = q^{-(\omega+1)}\times \int_{B_1}\chi(ac(\pi^{k+r+l}x^p+\pi^ly^rz^l\mathbb{H}_r^k(y, t\pi z-y)|\pi^{k+r+l}x^p+\pi^ly^rz^l\mathbb{H}_r^k(y, t\pi z-y)|^s|dxdydz| = q^{-(\omega+1)-ls}\times \int_{B_1}\chi(ac(\pi^{k+r}x^p+y^rz^l\mathbb{H}_r^k(y, t\pi z-y)|\pi^{k+r}x^p+y^rz^l\mathbb{H}_r^k(y, t\pi z-y)|^s|dxdydz|,
\lim_{b \uparrow \infty}\frac{K_a^{(q,r)}(b)}{W^{(q+r)}(b)}= \frac 1{\Phi(q+r)}\Big( \tilde{Z}^{(q,r)}(-a )- \psi'(0+){Z^{(q)}(-a, \Phi(q+r))}\Big).
f'' = \frac{a''}{a}+ 2n \frac{b''}{b}a'' = 2n\left(\frac{a^3}{b^4}- \frac{a'b'}{b}\right) + a'f' b'' = \frac{2n+2}{b}- 2 \frac{a^2}{b^3}-\frac{a'b'}{a}- (2n-1)\frac{(b')^2}{b}+ b'f',
u^{\ell+1}= \Phi_{,1}^\tau({u^\ell}) = e^{\tau \mathcal{L}}\Big( u^\ell + \tau \mathcal{B}\left( F(u^\ell) \cdot \varphi_1\big(\tau \mathcal{A}\big) G(\overline{u^\ell})\right)\Big) \quad \varphi_1(z) = \frac{e^{z}-1}{z}
I'(u)[u_1] =I'(u)[v^+]=I'(v)[v^+]+\lambda\int_{\mathbb R^3}\phi_3(v^+)^2dx=0, I'(u)[u_2] =I'(u)[v^-]=I'(v)[v^-]+\lambda\int_{\mathbb R^3}\phi_3(v^-)^2dx=0, I'(u)[u_3] =0,
\limsup_{c\to 0}c^2\kappa(\|M\|_\infty,C_R,c) = 2 C_R\Vert M \Vert_\infty <\infty, \limsup_{C_R\to \infty}\tfrac{\kappa(\|M\|_\infty,C_R,c)}{C_R}= 2 \big( 1 + \tfrac{\Vert M \Vert_\infty}{c}\big)^2 + 1<\infty, \limsup_{\|M\|_\infty\to \infty}\tfrac{\kappa(\|M\|_\infty,C_R,c)}{\|M\|_\infty^2}= \tfrac{2C_R}{c^2}<\infty.
\iota(\Delta x_m)=\Delta x_m \sqrt{u_{yy}^{m,n}},\iota(\Delta y_{m,n})=(\Delta y_{m,n}- u_{m,n}\, \Delta x_m)\sqrt{u_{yy}^{m,n}}, \iota(\delta y_{m,n})= \delta y_{m,n}\sqrt{u_{yy}^{m,n}},
F^C_{\mbox{div}}(\beta,\mu)=-{1 \over 48\kappa \beta^2}\int_{\Sigma}\left[2c\mu^2+\ln{\mu^2 \over m^2}\left({1 \over 60}{4\pi^2 \over \kappa^2\beta^2}{\cal P}+\left(\frac 16-\xi\right) R-m^2\right)\right].
2a_{11}a_{21}\left( 1-\frac{a_{11}a_{21}}{a_{12}a_{22}}\right) \cos t=\frac{a_{11}^{2}a_{21}^{2}}{a_{12}^{2}a_{22}^{2}}(a_{12}^{2}+a_{22}^{2})-(a_{11}^{2}+a_{21}^{2})
\limsup_{M\to\infty}\; \sup_{n\in \mathbb{N}}\int_{\mathcal{X}}\mathbf{1}_{\{\|\phi\|_{\mathcal{X}}^{2}\geq M\}}\cdot \|\phi\|_{\mathcal{X}}^{2}\, [\Theta^{n}]_{\mathcal{X}}(d\phi) = 0.
\sum_{k=1}^{\infty}\frac{sinkt}{k^{2}}=-\frac{1}{2}\int_{0}^{t}ln(2-2cosx)dx\sum_{k=1}^{\infty}\frac{coskt}{k^{2}}=\frac{\pi^{2}}{6}-\frac{\pi t}{2}+\frac{t^{2}}{4}
|a(t_{j})-a(t_{j-1})|^2 \leq \big|a(t_{j})^2-a(t_{j-1})^2\big| \leq \Big| \int_{t_{j-1}}^{t_{j}}t(a(t)^2)' \frac{dt}{t}\Big| =\Big| \int_{t_{j-1}}^{t_{j}}2a(t)ta'(t) \frac{dt}{t}\Big|.
\frac{\partial}{\partial s}\tilde \psi =\log \frac{( \omega_{\varphi_{t, \delta}+\tilde\psi}^\delta)^n}{(\omega_{\varphi_{t, \delta}}^\delta)^n}+\mu\tilde \psi - h_{t,\delta}, & \tilde \psi_{0,\delta}=\tilde\psi_{\delta}(0,\cdot)= 0.
S_{k,n}(\beta) ={\frac{k+1}{n}}\sum_{a=0}^{n}{\binom{n}{a}}{\binom{n+k+a}{n-1}}\beta^{n-a}={\frac{k+1}{k+1+n}}{\binom{2n+k}{n}}+ \cdots+{\binom{n+k}{n}}\beta^{n}.
\Phi\,=\,(\,1-G^RK\,)^{-1}\,\Psi\,=\,(\,1+G^RK^T\,)\,\Psi\,=\,(\,1+G^0K^T-G^{\,\delta}K^T\,)\,\Psi\,=\,G^0K^T\,\Psi
\forall f,f'\in C^{\infty}(T^{*}G):\ \{\tilde{r}^{*}f,\tilde{r}^{*}f'\}_{T^{*}G^{\times2}}((\theta_{2},g_{2}),(\theta_{1},g_{1})) = \tilde{r}^{*}\left(\{f,f'\}_{T^{*}G}\right)((\theta_{2},g_{2}),(\theta_{1},g_{1})),
\delta^{k_{2}^{j}-1}(\varphi_{rest_{1}}^{j}) =\varphi_{rest_{1},[k_{2}^{j}-1]}^{j}(x,v_{1},v_{1,[1]},\ldots) \delta^{k_{2}^{j}}(\varphi_{rest_{1}}^{j}) =\varphi_{rest_{1},[k_{2}^{j}]}^{j}(x,v_{1},v_{1,[1]},\ldots,u_{rest_{1}})
\mathbb{P}_{x}[\tau<\zeta]=\mathbb{P}_{x}[\tau<\zeta,\,\lim\nolimits_{n\to\infty}X_{\tau_{n}}=X_{\tau}=\lim\nolimits_{n\to\infty}X_{\sigma_{n}}]=\mathbb{P}_{x}[\emptyset]=0.
Ric_{\widetilde{g}}(X_{i},X_{j}) = \frac{(n-2)\varphi_{,x_{i}x_{j}}}{\varphi}-\frac{m}{f}\left[f_{,x_ix_j}+\frac{\varphi_{,x_j}}{\varphi}f_{,x_i}+\frac{\varphi_{,x_i}}{\varphi}f_{,x_j}\right],\ \forall\i\neq j
\rho_{\frac{s}{2}}\left( x\right) =\sinh \frac{i}{2}\frac{d}{dx}\rho\left( x\right) \qquad \qquad\rho_{\frac{c}{2}}\left( x\right) =\cosh \frac{i}{2}\frac{d}{dx}\rho \left( x\right)
A(|x|) = \frac12\, \mathrm{C}''(|x|) - \left( |x| + \frac{1}{|x|}\right) \mathrm{C}'(|x|) + 2\, \mathrm{C}(|x|)\,, B(|x|) = \mathrm{C}''(|x|) - \left( 2\, |x| + \frac{1}{|x|}\right) \mathrm{C}'(|x|) + \frac14\, \frac{\left[\mathrm{C}'(|x|) \right]^2}{\mathrm{C}(|x|)}.
\tilde{\cal L}= \frac{1}{2}\, \partial_{\mu}\phi^{A}\partial^{\mu}\phi^{A}+ \frac{1}{2}\, \lambda \, \bigl ( \phi^{A}\phi^{A}- \mathrm{e}^{\eta^{1}}\bigr ) - \frac{1}{8}\, \mathrm{e}^{\eta^{1}}\partial_{\mu}\eta^{1}\partial^{\mu}\eta^{1}
C_{13}^4=\frac{\lambda_4}{\lambda_1\lambda_3},C_{23}^4=\frac{\lambda_4}{\lambda_2\lambda_3},C_{14}^3=\frac{\lambda_3}{\lambda_1\lambda_4}, C_{24}^3=\frac{\lambda_3}{\lambda_2\lambda_4}, C_{34}^2=\frac{\lambda_2}{\lambda_3\lambda_4},C_{34}^1=-\frac{\lambda_1}{\lambda_3\lambda_4}.
\int_{K_{\mathbb{C}}}f_{\pi,A}(g)\mu_{s,\tau}(g)~dg =\sum_{n=0}^{\infty}\frac{1}{2^{n}n!}(\Delta_{s,\tau})^{n}f_{\pi,A}(e), & =\mathrm{Tr}(\pi(x)e^{D_{\pi,s,\tau}/2}A)
\begin{aligned}\left|(h(x_a+\delta u(\tau))\pm h(x_a))u(\tau)-\dfrac{b_{1,\delta}(x_a)}{2}\right| \le \left|(h(x_a+\delta u(\tau))- h(x_a))u(\tau)\right|+\left|\dfrac{b_{1,\delta}(x_a)}{2}\right| +|h(x_a)||u(\tau)|.\end{aligned}
\langle x|y\rangle =e^{-\frac{|x|^{2}}{2}}\sum_{m=0}^{\infty}\frac{(\overline{x})^{m}}{\sqrt{m!}}e^{-\frac{|y|^{2}}{2}}\sum_{n=0}^{\infty}\frac{y^{n}}{\sqrt{n}!}\langle m|n\rangle &=e^{-\frac{|x|^{2}}{2}-\frac{|y|^{2}}{2}}\sum_{n=0}^{\infty}\frac{(\overline{x}y)^{n}}{n!}=e^{-\frac{1}{2}(|x|^{2}+|y|^{2})+\overline{x}y}.
\varphi^{\prime}(x)=\left(\begin{array}{c}\chi^{\prime}(x) \xi^{\prime}(x)\end{array}\right) ,\hspace{0.3in}\beta_\mu^{\prime}=-\left(\begin{array}{cc}0 \alpha_\mu^{\prime}\overline{\alpha}_\mu^{\prime}0\end{array}\right) .
K_t - \tilde K_t + \int_0^t\int_U \big(L_s(z) - \tilde L_s(z)\big)\,\lambda_\pi(dz)ds = \int_0^t \big(Z_s - \tilde Z_s\big)\,d\hat W_s, \int_0^t\int_U \big(L_s(z) - \tilde L_s(z)\big)\,\hat\pi(ds\,dz) = \int_0^t\int_\Lambda \big(R_s(b) - \tilde R_s(b)\big)\,\hat\theta(ds\,db).
\left\|f^t\right\| \geq\sup_{0\neq\varphi\in F^*}\frac{\left\|f^t(\varphi)\right\|_\infty}{\left\|\varphi\right\|}\geq\frac{\left\|\phi\circ f\right\|_\infty}{\left\|\phi\right\|}=\left\|\phi\circ f\right\|_\infty &\geq \left|\phi(f(x))\right|=\left\|f(x)\right\|>\left\|f\right\|_\infty-\varepsilon .
\|K_{V}^{-1}\|_{\mathcal{L}(L^2(\mathbb{R}^{2d}))}=\|\displaystyle\int_0^{+\infty}e^{-tK_{V}}dt\|_{\mathcal{L}(L^2(\mathbb{R}^{2d}))}\le\displaystyle\int_0^{+\infty}\|e^{-tK_{V}}\|_{\mathcal{L}(L^2(\mathbb{R}^{2d}))}dt &\le\frac{c}{\sqrt{B}}~.
a_{ii}^{-1/2}A^{1/2}e_i \cdot a_{jj}^{1/2}A^{-1/2}Re_j = a_{ii}^{-1/2}a_{jj}^{1/2}\left(e_i\cdot A^{1/2}A^{-1/2}Re_j\right) = a_{ii}^{-1/2}a_{jj}^{1/2}\left(e_i\cdot Re_j\right) = a_{ii}^{-1/2}a_{jj}^{1/2}r_{ij},
\langle \overline{x_1}\cdots \overline{x_{k-1}}|\mathcal{B}^\prime(z)| \overline{y_1}\cdots \overline{y_{k}}\rangle= (-1)^{k+1}(-1)^{j-1}(z^{\overline{y_j}}-z^{-\overline{y_j}}),