harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\partial_{z}\Phi =\tau+R_1(Tb-\kappa \tau)-R_2 T n+\mathcal O(R_1^2,R_2^2), \partial_{R_1}\Phi = n+\mathcal O(R_1,R_2), \partial_{R_2}\Phi =b+\mathcal O(R_1,R_2),\end{cases}
\Omega=-{\cal H}\,\mbox{d}z\wedge\mbox{d}\bar{z}+\bar{p}\,\mbox{d}z\wedge\mbox{d}\varphi+p\,\mbox{d}\varphi\wedge\mbox{d}\bar{z}\quad\mbox{with}\quad{\cal H}:=pv+\bar{p}\bar{v}-{\cal L}\,.
\delta x : = \overline{{\varepsilon}}\psi + \overline{{\psi}}\varepsilon , \qquad \delta \psi : = - ( \mathrm{i}\dot{x}+ \Phi ( x ) ) \varepsilon , \qquad \delta \overline{{\psi}}= ( \mathrm{i}\dot{x}- \Phi ( x ) ) \overline{{\varepsilon}},
IV=({\bf E}\cdot \nabla)(\omega\,k+\nabla \varphi) =E_1(\varphi_{x_1x_1},\varphi_{x_1x_2},\varphi_{x_1x_3})+E_2(\varphi_{x_2x_1},\varphi_{x_2x_2},\varphi_{x_2x_3})+E_3(\varphi_{x_3x_1},\varphi_{x_3x_2},\varphi_{x_3x_3}) &=({\bf E}\cdot \nabla \varphi_{x_1},{\bf E}\cdot \nabla \varphi_{x_2},{\bf E}\cdot \nabla \varphi_{x_3}).
\left ( 2 \sum_{i=1}^m n_i p_i Y_i - \frac{1}{2}E(Y)\right)^2 \leq \,\, \left(\frac{n-1}{2}\right) \sum_{i=1}^m n_iY_i^2 (2 p_i - q_i^2Y_i)^2 +\frac{1}{4}E(Y)^2 & +\sqrt{\frac{n-1}{2}}\, E(Y) \left (\sum_{i=1}^m n_iY_i^2 (2 p_i - q_i^2Y_i)^2 \right )^{1/2}.
\left[ H_{2}\left( \alpha\left( t\right) :a,a^{\dag}\right) ,a\right] =-v\left( t\right) a-u\left( t\right) \left( \alpha\left( t\right)\right) a^{\dag}\left[ H_{2}\left( \alpha\left( t\right) :a,a^{\dag}\right) ,a^{\dag}\right] =\bar{u}\left( t\right) a+v\left( t\right) a^{\dag}.
\Pi^{p 1 2}= d X^{p 1 2}- \frac{1}{2}\theta_{\alpha}( \Gamma^{p 1 2})^{\alpha \beta}d \theta_{\beta}\longrightarrow \Pi^{p}= d X^{p}- \frac{1}{2}\theta_{\alpha}( \gamma^{p})^{\alpha \beta}d \theta_{\beta},
\Big [ \begin{array}{c c}{{a_{1}}}&{{a_{2}}}\\{{b_{1}}}&{{b_{2}}}\\ \end{array}\Big ] \sim \Big [ \begin{array}{c}{{a_{1}}}\\{{b_{1}}}\\ \end{array}\Big ] \times \Big [ \begin{array}{c}{{a_{2}}}\\{{b_{2}}}\\ \end{array}\Big ] .
\begin{cases}\phi_\varepsilon\left(x\right)+\psi_\varepsilon \left(x\right)\geq-\varepsilon \forall x\in\mathbb{R}\phi_\varepsilon\left(x_{\varepsilon}\right)+\psi_\varepsilon \left(x_{\varepsilon}\right)\leq\varepsilon\end{cases}.
\|u\|_{L^\infty((0,T),H^1)}^2+\|u\|_{L^2((0,T),H^3)}^2+\|u\|_{L^4((0,T),L^4)}^4 &\quad\le C(1+Te^{\omega T})\left(\|u_0\|_{H^1}^2+\|f\|_{L^2((0,T),L^2)}^2\right).
I_{X_+}(y,z)=ze^{t_+/z}\sum_{d\in\mathbb K_{+}}y^{d}\left(\prod_{i=0}^{m}\frac{\prod_{a\leq 0,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}{\prod_{a \leq D_i\cdot d,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}\right)\textbf{1}_{[-d]},
\mathsf{Z}_{\ell}^k:=\sum_{i=\ell}^k\mathsf{B}_i,\widetilde x^k:=\frac{\sum_{i=0}^k\mathsf{B}_i x^i}{\mathsf{Z}_0^k},\widetilde x^k_\ell:=\frac{\sum_{i=\ell}^k\mathsf{B}_i x^i}{\mathsf{Z}_\ell^k},
B_{0i}(T=0)= - \frac{2 v^2}{\pi \kappa}\epsilon_{0ij}p^j\frac{\partial}{\partial m_{\sigma}^2}\left\{\frac{1}{m_{\sigma}^2 -M^2}\left[\left(M \frac{e^{- \tilde p M}}{(\tilde p )^2}+ \frac{e^{-\tilde p M}}{(\tilde p )^3}\right) - (M \rightarrow m_{\sigma})\right] \right\}
\alpha ( q_1 , q_0 ) = q_0 , \beta ( q_1 , q_0 ) = q_1 , m \bigl( ( q_2 , q_1 ) , ( q_1 , q_0 ) \bigr) = ( q_2 , q_0 ) , \epsilon (q) = ( q, q ) , i ( q_1 , q_0 ) = ( q_0, q_1 ) ,
f_{k,\ell}(p^{\alpha}) = \frac{\sum_{j=0}^{\alpha}\Big( \frac{\sigma_{k-1}(p^j,1)\sigma_{\ell-1}(p^j,1)}{p^{j}}- \frac{\sigma_{k-1}(p^{j+1},1)\sigma_{\ell-1}(p^{j+1},1)}{p^{j+2}}\Big)}{\Big( 1-\frac{1}{p}\Big)^{k-1}+ \Big( 1-\frac{1}{p}\Big)^{\ell-1}- \Big ( 1-\frac{1}{p}\Big )^{k+\ell-2}}.
\tilde{\pi}_1(\bar{a}) =\frac{1}{2}[\hat{v}(1),\hat{g}(\bar{a})]=-\frac{1}{2}[\hat{v}(1),[\hat{g}(1),\tilde{h}(a,1)]] &=-0-\frac{1}{2}[\hat{g}(1),[\hat{v}(1),\tilde{h}(a,1)]]=(-1)^{|a|}[\hat{g}(1),\hat{v}(a)].
\sigma_m(s) = \bigg(\sum_{j=1}^{m-1}R_j +(1-e^{-is})R_m, t_m(s) \bigg) \mbox{where}t_m(s) = 2\big(\frac{\pi}{2}-1\big)\big(\sum_{j=1}^{m-1}\|R_j\|^2\big)+2(s-\sin s) \|R_m\|^2 -2 \sum\limits_{j=2}^{m-1}( \langle \sum\limits_{k=1}^{j-1}R_k, R_j \rangle )-2 \sin s \langle \sum_{j=1}^{m-1}R_j , R_m \rangle.
x^2 wxa_4 = - x^2w\left(s-\frac{1}{6}x^3\right)wb_1 & = x^2cwcb_1 + x^2swsb_1 + \frac{1}{6}x^2wx^3wb_1 & = x^2wb_1 + \frac{1}{6}x^2w(q-6b_1) & = x^2wb_1 - x^2 wb_1 = 0
I_8= -\int (v\cdot \nabla (\partial^\zeta_x u))\cdot \partial^\zeta_x u \leq Cc_3 |\partial^\zeta_x u|^2_2, I_{9}= -\frac{\delta-1}{\delta}\int (\nabla (\phi^2+\eta^2) \cdot Q(\partial^\zeta_x u)) \cdot \partial^\zeta_x u\leq \frac{\alpha}{20}|\sqrt{\phi^2+\eta^2}\nabla (\partial^\zeta_x u) |^2_2+Cc^2_0|\partial^\zeta_x u|^2_2.
\mathcal{D}_{\mu,\kappa}\left(\boldsymbol{\lambda}\right) =\mathcal{D}_{0}\left(\boldsymbol{\lambda}\right)+\sum_{i\in \mathcal{I}\setminus 0}\mathcal{D}_{i,\mu}\left(\boldsymbol{\lambda}\right)-\frac{\kappa}{2}\left\|\boldsymbol{\lambda}\right\|^{2},
S_\mathfrak{a}(m,n; c) = \sum_{\left(\begin{smallmatrix}a \cdot c d \end{smallmatrix}\right) \in \Gamma_\infty \backslash \sigma_\alpha^{-1}\Gamma_0(N) / \Gamma_\infty}e\left( m \frac{d}{c}+ n \frac{a}{c}\right)
\hat{\eta}_{A B}= \mathrm{d i a g}( - c^{2}, \ 1 , \ 1 , \ 1 , \ 1 , \ 1 , \ 1 , \ 1 , \ 1 , \ 1 , \ 1 , \ 1 , \ 1 , \ 1 ) \hspace{2 . 4 c m}
[{\cal D}\mu ] ={\cal D}A_{\mu}{\cal D}\phi{\cal D}\theta{\cal D}\xi \prod_{\beta = 1}^{2}\delta \left( \Gamma_{\beta}[ A_{0}+ \xi , A_{1}, \theta ] \right) \det \mid \{\widetilde{\Omega}_{\alpha}, \Gamma_{\beta}\}\mid .
q(x)=\begin{pmatrix}\frac{1+\sqrt{1-xx^*}}{2}\frac x 2 \frac{x^*}{2}\frac{1-\sqrt{1-xx^*}}{2}\end{pmatrix}\in \begin{pmatrix}pAp pA(1-p) (1-p)Ap (1-p)A(1-p)\end{pmatrix}.
e_0(t, x) = \phi_0\left(\langle 2\phi_{0, x}+ \varphi_{0, x}, \varphi_{0, x}\rangle- \langle 2\phi_{0, t}+ \varphi_{0, t}, \varphi_{0, t}\rangle\right) \;\;\;\;\;\; - \varphi_0 \left(|\tilde \phi_{1, t}|^2- |\tilde \phi_{1, x}|^2 \right)+ \mathbf{1}_{\omega}\left(f_0^{p^{\perp}}- f_{0}^{\tilde \phi^{\perp}_1}\right).
(\mathcal{H}h)(\vec{x}, \vec{\nu})=\sum_{\begin{subarray}{c}(\vec{y}, \vec{\mu}) \in \mathcal{S}_{k_{1}, \ldots , k_{r}}(\vec{y}, \vec{\mu})\not=(\vec{x}, \vec{\nu}) \end{subarray}}q(\vec{x}, \vec{\nu}| \vec{y}, \vec{\mu})\left\{h(\vec{y}, \vec{\mu})-h(\vec{x}, \vec{\nu})\right\}(h \in F(\mathcal{S}_{k_{1}, \ldots , k_{r}})),
f ( r , z ,{\bar{z}}) * h ( r , z ,{\bar{z}}) = e^{{\frac{i}{2}}[ \Theta^{r z}( \partial_{r}^{1}\partial^{2}- \partial_{r}^{2}\partial^{1}) + \Theta^{r{\bar{z}}}( \partial_{r}^{1}{\bar{\partial}}^{2}- \partial_{r}^{2}{\bar{\partial}}^{1}) ]}f ( r_{1}, z_{1},{\bar{z}}_{1}) h ( r_{2}, z_{2},{\bar{z}}_{2}) |_{1 = 2}.
\int_{\Delta_{r - 2}}e^{u_{1}x_{1}+ \cdots + u_{r - 2}x_{r - 2}}\, \left( \int_{0}^{\alpha}e^{u_{r - 1}x_{r - 1}+ u_{r}( \alpha - x_{r})}d x_{r - 1}\right) \, d x_{1}\cdots d x_{r - 2},
\lim_{x\to0}\Omega_1(x)= \lim_{x\to0}\frac{\Gamma(1-ax+bx)}{\Gamma(1-ax)}\cdot\frac{(1-a-ax)(2-a-ax)\cdots(-ax)}{(1-a-ax+b+bx)\cdots(-ax+bx)}=\frac{a\cdot(1-a)_b}{a-b}.
f^{( \kappa )}( z ) = \frac{1}{\kappa}\Bigl ( 1 - \exp ( - \kappa \tan^{- 1}z ) \Bigr ) = z - \frac{\kappa}{\sqrt{2}}\frac{z^{2}}{\sqrt{2}}+ \frac{\kappa^{2}- 2}{2 \sqrt{3}}\frac{z^{3}}{\sqrt{3}}+ \cdots .
\pi_T(\bar{u}) =\sum_{sH\in G/H}P_{sH}V_{sH}S_{sH}\pi(1_{sH}\bar{u})S_{sH}^{-1}V_{sH}^*P_{sH}&=\sum_{sH\in G/H}P_{sH}V_{sH}\left[S_{sH}\pi(1_{sH}u)S_{sH}^{-1}\right]^*V_{sH}^*P_{sH}=\pi_T(u)^*.
\omega_{-}= \left( \begin{array}{c c}{{\Sigma_{-}^{( 1 )}}}&{{0}}\\{{I_{n}}}&{{\Sigma_{-}^{( 2 )}}}\\ \end{array}\right) , \qquad \omega_{+}= \left( \begin{array}{c c}{{0}}&{{\Gamma^{( 1 ) - 1}\Gamma^{( 2 )}}}\\{{0}}&{{0}}\\ \end{array}\right) ,
\mathcal{B}(s_1,s_2) = \sum_{g \mid h}\frac{1}{g^{s_1+s_2+1}}\sum_{e=1}^{\infty}\frac{\mu(e) \sigma_{k}(ge,s_1+1) \sigma_{\ell}(ge,s_2+1)}{e^{s_1+s_2+2}}.
\lim_{K\to\infty}\biggl( \limsup_{n\to\infty}E \biggl[\int_0^1\int_0^1 \,\mathrm{d}s \,\mathrm{d}t\,\Gamma_n(s,t) \mathbf{1}_{\{\Gamma_n(s,t)>K\}}\biggr] \biggr) =0.
\Pi(z)=\left({\begin{array}{*{20}c}{z^{-1/2}\Sigma^{-1/2}}0 0{I}\end{array}}\right) \Pi_o(z) \left({\begin{array}{*{20}c}{z^{-1/2}\Sigma^{-1/2}}0 0{I}\end{array}}\right).
\begin{array}{rcl}T^* ={\cal D}T{\cal D}^{-1}&= \left(\begin{array}{ccccc}t_{11}t_{12}-t_{13}t_{14}t_{15}t_{21}t_{22}-t_{23}t_{24}t_{25}-t_{31}-t_{32}t_{33}-t_{34}-t_{35}t_{41}t_{42}-t_{43}t_{44}t_{45}t_{51}t_{52}-t_{53}t_{54}t_{55}\end{array}\right)~.\end{array}
{\cal L}_{f}= \sqrt{2}i ( \alpha^{\prime}p^{+}) \left\{\theta_{+}^{\dagger}( \partial_{\tau}+ \partial_{\sigma}) \theta_{+}+ \theta_{-}^{\dagger}( \partial_{\tau}- \partial_{\sigma}) \theta_{-}+ 2 \theta_{+}^{\dagger}\tilde{\Omega}_{-}\theta_{-}+ 2 \theta_{-}^{\dagger}\tilde{\Omega}_{+}\theta_{+}\right\}
|g_N(z)| = \exp\Big( P_{f_N}(z)\Big) \leq \exp\Big(\frac{C_N}{1-|z|}\Big) \leq \exp \Big( \frac{C_N}{d}\log(1/G(1-|z|)) \Big) = \Big(G(1-|z|)\Big)^{-C_N/d}.
\omega_{p, k}^{\epsilon}: c(1,1)=up^{-k}. \omega_{2, k}^{\epsilon}: c(1,1)=u2^{-k-1}. E_k: c(\vec{e}_i,\vec{e}_i)=0, \c((1,1),(1,1))=c(\vec{e}_1,\vec{e}_2)=2^{-k}. E_k: c(\vec{e}_i,\vec{e}_i)=2^{-k}, \c((1,1),(1,1))=2c(\vec{e}_1,\vec{e}_1)+c(\vec{e}_1,\vec{e}_2)=2^{-k}.
L^{P}=mA^{P}\cdot A^{P}H=L-\partial V L =m(A\cdot AH+A\cdot\phi\overleftrightarrow{\partial}H-\frac{m_{H}^{2}}{2}\phi^{2}H) V^{\mu}=m(A^{\mu}\phi H+\frac{1}{2}\phi^{2}\overleftrightarrow{\partial^{\mu}}H),Q_{\mu}=m(A^{\mu}uH+u\phi\overleftrightarrow{\partial_{\mu}}H)
\frac{1}{2}\ge\sum_{i=b_n+1}^{n}\frac{1}{i-2}=\sum_{i=b_n-1}^{n-2}\frac{1}{i}>\int_{b_n-1}^{n-1}\frac{dx}{x}=\log\left(\frac{n-1}{b_n-1}\right).
\big\langle \mathfrak{Re}\langle x,\cdot\rangle,\mathfrak{Re}\langle y,\cdot \rangle\big\rangle_{L^2(m)}=\big\langle \mathfrak{Im}\langle x,\cdot\rangle,\mathfrak{Im}\langle y,\cdot \rangle\big\rangle_{L^2(m)}=\frac{1}{2}\mathfrak{Re}\langle Rx,y\rangle
\begin{aligned}\gamma_s^{(N)}(L) = \frac{\xi}{2N}\left.\frac{d}{dq}(q - q^{-1}) \big(V_{s}(L) + V_{2N-s}(L)\big)\right|_{q=\xi}&=\frac{N}{\pi\, \sqrt{-1}}\, (\xi - \xi^{-1}) \left.\frac{d}{d m}{V}_m(L)\right|_{\genfrac{}{}{0pt}{}{m=s}{q = \xi}}. \end{aligned}
f(x) = (x-a_1)^{e^1-1}\ldots (x-a_k)^{e^k-1}\Big((x-a_1)\ldots(x-a_k)h'(x) + e^1(\widehat{x-a_1})(x-a_2)\ldots(x-a_k)h(x) + e^2(x-a_1) (\widehat{x-a_2})\ldots(x-a_k)h(x) + e^k(x-a_1)\ldots (x-a_{k-1}) (\widehat{x-a_k}) \Big)
h(n) =\begin{cases}\frac{2n}{ab}-2 \mbox{if}n \equiv 0 \bmod{ab}, \frac{2(n-a)}{ab}-1 \mbox{if}n \equiv a \bmod{ab}, \frac{2(n-b)}{ab}-1 \mbox{if}n \equiv b \bmod{ab}, \frac{2(n-a-b)}{ab}\mbox{if}n \equiv a+b \bmod{ab}.\end{cases}
J_6: y=\frac{vy_L(2v-v_L)+v^2y_L}{2v_L(2v-v_L)}+\frac{(v-v_L)}{2v_L(2v-v_L)}\sqrt{\left( v y_L-4\frac{v_L^2}{y_L}\right)^2+4v_L^3\frac{(y_L^2-4v_L)}{y_L^2}}
(a)_L:=(a;q)_L := \prod_{n=0}^{L-1}(1-aq^n). \intertext{Some abbreviations of the notation we are going to use are}(a_1,a_2,\dots,a_k;q)_L := (a_1;q)_L(a_2;q)_L\dots (a_k;q)_L, (a;q)_\infty := \lim_{L\rightarrow\infty}(a;q)_L,|q|<1.
c_{v,p}=\frac{dE_{p}}{dT}|_{v}=\frac{9k_{B}}{m}(\frac{T}{T_{D}})^{3}n\int_{0}^{T_{D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx(x=\frac{\hbar\omega}{k_{B}T}),
\tau_{0}= \frac 1 2 \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) , \quad \tau_{1}= \frac 1 2 \left( \begin{array}{c c}{0}&{1}\\{- 1}&{0}\\ \end{array}\right) , \quad \tau_{2}= - \frac 1 2 \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) .
\widehat{\varphi_{k}}(\xi):=\hat{\varphi}(\frac{\xi}{2^{k}}), \,\,\,\,\,\, \widehat{\psi_{k}}(\xi):=\hat{\psi}(\frac{\xi}{2^{k}})=\widehat{\varphi_{k+2}}(\xi)-\widehat{\varphi_{k+1}}(\xi)
\mathsf{V}_{\rho,1}:= \frac{\log^2 e}{4}\left(\frac{p\cdot2h_{\rho}^2\mathsf{P}^2+ 4h_{\rho}\mathsf{P}}{(1 + h_{\rho}\mathsf{P})^2}+ \bar{p}\frac{2h_1^2\mathsf{P}^2}{(1+h_1\mathsf{P})^2}\right)
\left(-(1-\kappa r^2) \frac{d^2}{dr^2}- (d-1-d\kappa r^2)\frac{1}{r}\frac{d}{dr}+ \frac{l(l+d-2)}{r^2}- \frac{Q}{r}\sqrt{1-\kappa r^2}-{\cal E}\right) R(r) = 0,
w_{l}=z_{l}\overline{z}_{\tau\left(l\right)}+\lambda_{l}\left(z_{l}z_{\sigma\left(l\right)}+\overline{z}_{l}\overline{z}_{\sigma\left(l\right)}\right) +\displaystyle\sum_{k\geq 3}\varphi_{k}^{\left(l\right)}(z,\overline{z}),\quad\mbox{for all $l=1,\dots,N\leq N'$,}
A_{i}=g_{ij}A^{j}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Longrightarrow\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\partial_{;m}A_{i}=g_{ij}\partial_{;m}A^{j}
M_p( k_1 k_2 k_2 k_3 , k_3 k_4 k_1 k_4 ) & =2 \cos \big[ (t-s) (E_1 - E_3) \big] \big( \delta(p - k_3 ) - \delta (p - k_1) \big)\Phi ( k_1 k_2 k_3 k_2 )\Phi ( k_1 k_4 k_3 k_4 )
\bar{R}^\sigma_{~\lambda \mu \nu}(\bar{\Gamma}) =\bar{R}^a_{~b \mu \nu}(\bar{\Omega}) e_a^{~\sigma}e^b_{~\lambda}=( \partial_\mu \bar{\Omega}^a_{~b \nu}-\partial_\nu \bar{\Omega}^a_{~b \mu}+ \bar{\Omega}^a_{~\alpha \mu}\bar{\Omega}^\alpha_{~b \nu}- \bar{\Omega}^a_{~\alpha \nu}\bar{\Omega}^\alpha_{~b \mu})e_a^{~\sigma}e^b_{~\lambda}
\mathfrak{O}_{0,n-1}=\mathfrak{O}_{0,n}\sqcup\big(\mathfrak{O}_{0,n-1}\setminus\mathfrak{O}_{n-1,n}\big)\,,\qquad\neg\mathfrak{O}_{0,N}=\bigsqcup\limits_{k=1}^{N}\,\left(\mathfrak{O}_{0,k-1}\setminus\mathfrak{O}_{k-1,k}\right)\,.
\hat h^2(t)- \hat h^1(t) = \int_0^t P({t-r}) \left( ( \nabla \hat \gamma^2(r)- \nabla \hat \gamma^1(r) )^* \hat b(r) \right) \mathrm dr & + \int_0^t P({t-r}) \left ( \hat f(r, \hat \gamma^2(r), \nabla\hat \gamma^2(r) -\hat f( r, \hat \gamma^1(r), \nabla \hat \gamma^1(r) ) \right) \mathrm dr.
\frac{1-C_a}{1+C_b}=\frac{1- \big( (1+\alpha)\varkappa^2+(1+1/\alpha) \Lambda_1^2(\epsilon_3 + \widehat\Lambda_3 \varkappa^2)\big)}{1+(1+1/\alpha) \Lambda_1^2 \Lambda_3}=\theta_0(\varkappa,\alpha)<1.
\frac{\partial}{\partial{t}}{U}_{k}( \rho ) = \frac{1}{2}\int \frac{{d}^{d}{q}}{( 2 \pi )^{d}}\frac{\partial}{\partial{t}}{R}_{k}({q}) \left( \frac{{N}- 1}{{M}_{0}}+ \frac{1}{{M}_{1}}\right)
(1+yX_1)(1+yX_2) (1+y\tilde{X}_1)(1+y\tilde{X}_2)(1+y\tilde{X}_3)= \prod_{i=1}^5 (1+T_i y) - \frac{q}{1-q}y^{4}\tilde{X}_1 \tilde{X}_2 \tilde{X}_3 (X_1+X_2 + y X_1 X_2) \/.
{\left\|{{{\left[{\frac{{{\partial^{{N_t}+ 1}}}}{{\partial{t^{{N_t}+ 1}}}}\psi (y,t)}\right]}_{t = \zeta_{{t_f},{N_t},j}^{(\alpha ),t}}}}\right\|_{{L^\infty}[0,L]}}\le{B_{\max,j}}\in \mathbb{R}^+\, \forall j.
( t^{2}+ 1 ) \sigma_{t t}+{\frac{1}{\gamma - 2}}( t^{2}+ 1 ) \sigma_{t}^{2}+{\frac{2 \gamma}{\gamma - 2}}t \sigma_{t}+{\frac{2 \gamma}{\gamma - 2}}={\frac{2 k ( \gamma - 1 )}{\gamma - 2}},
\hat{\delta}\hat{A}_{\mu}( x ) = \delta ( A_{\mu}( x ) +{\frac{1}{2}}\theta^{\sigma \nu}A_{\nu}( x ) F_{\sigma \mu}( x ) +{\frac{1}{2}}\theta^{\sigma \nu}A_{\nu}( x ) \partial_{\sigma}A_{\mu}( x ) ) ,
K_j(x) = \frac 1{(2\pi)^d}\int_{\mathbb R^d}( (1+|\xi|^2)^{\frac s2}-1 ) \phi(\frac{\xi}{2^j})e^{i \xi \cdot x}d\xi & = \frac 1{(2\pi)^d}\cdot 2^{jd}\int_{\mathbb R^d}( (1+|2^j \tilde \xi|^2)^{\frac s2}-1) \phi(\tilde \xi) e^{i \tilde \xi \cdot 2^j x}d\tilde \xi.
\left( \begin{array}{c}{{\tilde{t}}_{1}}{{\tilde{t}}_{2}}\end{array}\right) = 2 \pi i \left( \begin{array}{c c}{1}{1 / 2}{0}{- 1 / 2}\end{array}\right) \left( \begin{array}{c}{t_{1}}{t_{2}}\end{array}\right) \, \, \, ,
\lambda_1(t) = \lambda_1(\tau_1) \lambda_2(t) = \lambda_2(\tau_1) + \lambda_0 b_M (\tau_1 - t) + \int_t^{\tau_1}\nu_2(s) ds \lambda_3(t) = e^{k_E(\tau_1 - t)}\bigl(\lambda_3(\tau_1) - a_E b_E \lambda_1(\tau_1) + \lambda_0 \frac{b_E}{k_E}\bigr) + a_E b_E \lambda_1(\tau_1) - \lambda_0 \frac{b_E}{k_E}.
CB(C_1\otimes_{(e)h}CB(B, A)\otimes_{(e)h}C_2, CB(B,C_1\otimes_{(e)h}A\otimes_{(e)h}C_2)) &=CB(B,CB(C_1\otimes_{(e)h}CB(B, A)\otimes_{(e)h}C_2, C_1\otimes_{(e)h}A\otimes_{(e)h}C_2))
\frac{2{\rm Li}_{(k_1, k_2,\ldots, k_r)}(1-(ab)^{-t})}{(a^{-t}+b^{t})^r}c^{rxt}=\sum_{n=0}^{\infty}{E}^{(k_1, k_2,\ldots, k_r)}_n(x;a,b,c)\frac{t^n}{n!}.
\|\L_{T_{g_{n-1}}^{\varepsilon}\circ \cdots \circ{T_{g_0}^{\varepsilon}}}h\|_{0,q}\leq A \theta^n\|h\|_{0,q}+ B\|h\|_{0,q+1}; &\|\L_{T_{g_{n-1}}^{\varepsilon}\circ \cdots \circ{T_{g_0}^{\varepsilon}}}h\|_{1,q}\leq A \theta^n\|h\|_{1,q}+ B\|h\|_{0,q+1}.
(k, g) \iota(h) = (k, g) (1, W(1, 1)^{-1}h) begin{align*}1mm] = (k,\, W(k, 1) V(1)(g) W(1, 1)^{-1}h) begin{align*}1mm] = (k, \,W(k, 1) W(1, 1)^{-1}g W(1, 1) W(1, 1)^{-1}h) begin{align*}1mm] = (k,\, W(k, 1) W(1, 1)^{-1}g h) = (k, g h),
\Sigma_{1}^{(t+1)}=\frac{\sum_{i=1}^{n}T_{1,i}^{(t)}(x_{i}-\mu_{1}^{(t+1)})(x_{i}-\mu_{1}^{(t+1)})^{\top}}{\sum_{i=1}^{n}T_{1,i}^{(t)}}
(f_1,f_2,\cdots,f_{L})=(\widetilde{f}_1,\widetilde{f}_2,\widetilde{f}_3,\widetilde{f}_1,\widetilde{f}_2,\widetilde{f}_3,\cdots,\widetilde{f}_1,\widetilde{f}_2,\widetilde{f}_3).
\lim_{\epsilon \to 0}\, \left\{\frac{d}{dr}G\left (r,r^\prime\right) \vert_{r=r^\prime+\epsilon}- \frac{d}{dr}G\left (r,r^\prime\right) \vert_{r=r^\prime-\epsilon}\right\}= \frac{1}{r^3 g(r)^2}
\bigg(\sum_{k\in\mathbb{Z}}\|f_k\|_{L_{t,x}^2(|V|^{-1})}^2\bigg)^{1/2}=\bigg(\iint\sum_{k\in\mathbb{Z}}|f_k|^2|V|^{-1}dxdt\bigg)^{1/2}&=\bigg\|\bigg(\sum_{k\in\mathbb{Z}}|f_k|^2\bigg)^{1/2}\bigg\|_{L_{t,x}^2(|V|^{-1})}&\leq C\|f\|_{L_{t,x}^2(|V|^{-1})}.
D^m[f_1 \circ f_2 \circ f_3](x) = D[f_1 \circ f_2](f_3(x)) D^m f_3(x) + D^m[f_1 \circ f_2 ](f_3(x)) \left( Df_3(x) \right)^{\otimes m}& + \mathcal{P}_m(f_1 \circ f_2, f_3)(x).
d(12d - 27)+3d(d-2) = s + 30 \sum \delta_p + \sum_I (4m_p+4l_p-15) + \sum_J(10m_p+c_p-15), d(12d - 27) = s + 24 \sum \delta_p + \sum_I (3m_p+3l_p-12) + \sum_J(7m_p+c_p-12).
\chi_{_K}= \left(\begin{array}{ccc}-(1-Z)-{4M_K^2a^2\cos^2\theta\over\rho^2(1-Z)}0 -{2M_Ka\cos\theta\over\rho(1-Z)}0 1 0 -{2M_Ka\cos\theta\over\rho (1-Z)}0 -{1\over 1-Z}\end{array}\right).
\big[\theta_i^{(a)},\theta_j^{(b)}\big]_{\pm}=\sum_{a=1}^{r}\lambda_{a}^2 \big[\theta_i^{(a)},\theta_j^{(a)}\big]_{\pm}+ \sum_{1 \le a < b \le r}\lambda_{a}\lambda_b\big(\big[\theta_i^{(a)},\theta_j^{(b)}\big]_{\pm}+\big[\theta_i^{(b)},\theta_j^{(a)}\big]_{\pm}\big),
\int_{0}^{1}g_{\mathrm{hyp},s}(z,\sigma_{q}w)du=\frac{4\pi v^{1-s}}{2s-1}\bigg(\delta_{p,q}y_{p}^{s}+\alpha_{p,q}(s)y_{p}^{1-s}+\sum_{n\not = 0}\alpha_{p,q}(n,s)W_{s}(n\sigma_{p}^{-1}z)\bigg).
Y ( T , U ) = \int_{\cal F}\frac{d^{2}\tau}{\Im \tau}\Gamma_{2 , 2}( T , U ) \left( - 6 \left[{\overline{{\Omega}}}_{2}^{\phantom{2}}- \frac{1}{8 \pi \Im \tau}\right] \frac{\overline{{\Omega}}}{\overline{{\eta}}^{2 4}}- \frac{\overline{{j}}}{8}+ 1 2 6 \right) \ ,
\sum_{\substack{S\subseteq B_1\cup\{v_1\}|S|=k+1}}p_{H_1,S}(x)-\sum_{\substack{S\subseteq B_1 |S|=k+1}}p_{H_1,S}(x)=\sum_{\substack{S\subseteq B_2\cup\{v_2\}|S|=k+1}}p_{H_2,S}(x)-\sum_{\substack{S\subseteq B_2 |S|=k+1}}p_{H_2,S}(x).
\left\|{A}\xi- \hat{A}\xi \right\|_2^2=\langle\bold{B}^{-1}\vec{\bold{a}}, \vec{\bold{a}}\rangle=\langle\overline{\bold{\Phi}}\bold{\Phi}^{'}\vec{\bold{a}}, \vec{\bold{a}}\rangle=\langle\bold{\Phi}^{'}\vec{\bold{a}}, \bold{\Phi}^{'}\vec{\bold{a}}\rangle=\langle\bold{A}\vec{\varphi}, \bold{A}\vec{\varphi}\rangle=\|\bold{A}\vec{\varphi}\|^2,
\begin{array}[c]{rl}H(t,x,p,q,u)= p^{\intercal}b(t,x,u)+\left\langle q,\sigma(t,x,u)\right\rangle +f(t,x,u), & (t,x,p,q,u)\in\lbrack0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times U,\end{array}
\|\theta(t)\|_\infty \lesssim \frac{\|\theta(0)\|_2}{(\nu T)^{\frac{1}{\alpha}}}e^{-\frac{\nu}2(2\pi\lambda_0)^\alpha (t-T)}+\lambda_0^{\frac2p -\alpha}\frac{\|f\|_p}{\nu}&\lesssim \frac{\|\theta(0)\|_2}{(\nu t)^{\frac{1}{\alpha}}}+\lambda_0^{\frac2p -\alpha}\frac{\|f\|_p}{\nu},
g_{L+1}(\mathbf{x})T_{j}p(\mathbf{x}) =\sum_{\ell=0}^{d-1}\sum_{\|\alpha\| \leq \ell}c_{\ell,\alpha}T^{\alpha}(g_{L+1+\ell})(\mathbf{x})=\sum_{\ell=0}^{d+1}\sum_{\|\alpha\| \leq \ell}c_{\ell,\alpha}'T^{\alpha}(g_{L+\ell})(\mathbf{x}).
dY(t) = (k_1(X(t)) -k_2(X(t))) dt + dW_1(t)+ (1 - \zeta(X(t))) dL_+^Y(t) + (\eta(X(t))-1) dL_-^Y(t) dU(t) = (k_1(X(t)) + k_2(X(t))) dt + dW_2(t) + \bar \zeta(X(t)) dL_+^Y(t) + \bar \eta(X(t)) dL_-^Y(t),
\frac{\theta^2}{k}\left|\frac{k}{\theta}-\frac{\bar{k}}{\bar{\theta}}\right|^2 \leq C_1 \frac{\theta^2}{k}\left(\frac{k^2}{\theta^2}+1\right) &= C_1 \left(k+\frac{\theta^2}{k}\right) &\leq C \hat{e}(\xi,\theta)
\frac{{r}^3}{c^3}P{r_{{\rm{INS}}}}\left({- 1 - \frac{{{\rho_t}}}{c}}\right) + \frac{{r}^2}{c^2}\left({- 2P{r_{{\rm{INS}}}}- 1}\right) + \frac{r}{c}\left({2P{r_{{\rm{INS}}}}+ P{r_{{\rm{INS}}}}\frac{{{\rho_t}}}{c}- 2}\right) + P{r_{{\rm{INS}}}}- 1 = 0
\bigl( (\nabla^S)^* \nabla^S -2\i \, \nabla^S_A + |A|^2 + \mathcal{R}-\i\, \div A +\i \, dA\cdot & + 1+ \i \, dV\cdot \nu\cdot +V^2 - A_0^2 + dA_0\cdot \bigr)\psi_n= (D_{A,V, A_0}+\i)\phi_n.
\int^{\infty}_{0}\int_{\mathbb{T}}\left[\theta^{\epsilon}_{t}+\left(\mathcal{H}\theta^{\epsilon}\right)\theta^{\epsilon}_{x}+\Lambda^{\gamma}\theta^{\epsilon}+\delta \theta^{\epsilon}\Lambda \theta^{\epsilon}\right]\psi dxdt= \int_{\mathbb{T}}\theta^{\epsilon}_{0}(x)\psi(x,0)dx
\| \tilde{g}_i - \tilde{g}_j \|_{L_\nu^1}= \int_{[0,\frac{1}{2}]}|2 g_i(2t)-2 g_j(2t)|\nu(dt) + \int_{[\frac{1}{2},1]}|2g_j(2t-1) - 2 g_i(2t-1)|\nu(dt) = \int_{[0,1]}|2g_i(t)-2g_j(t)|\nu (dt) =2 d(i,j)
a_1 \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2 \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+a_3 \frac{d b_2}{b_2}\wedge \frac{db_3}{b_3}= a_1' \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2' \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+ \frac{d z_2}{z_2}\wedge \frac{dz_3}{z_3}
s_i^2=1, (s_is_{i+1})^3=1,(s_is_j)^2=1 \mbox{~for~}|i-j|\geq 2, s_ia_i=a_{i+1}s_i,s_ia_j= a_js_i \mbox{~for}~j\neq i,i+1, a_j^2=z, a_ia_j=za_ja_i \mbox{~for~}i\neq j,
|k^3 - \alpha k_1^3 -\alpha k_2^3| = |3\alpha k(k_1 - c_1k)(k_1 - c_2k)| \gtrsim |k| \cdot \max\{|k_1 - c_1k|, |k_1 - c_2 k|\}\gtrsim |k|\cdot (c_1 - c_2)|k| \gtrsim |k|^2.
c\frac{d}{dt}\left( (1 - \frac{1}{c^{2}}|\frac{d{\bf x}_{1}}{dt}|^{2})^{- 1/2}\right)= \frac{1}{c}\sum_{i = 1}^{3}\frac{dx_{1}^{i}}{dt}\frac{d}{dt}\left( (1 - \frac{1}{c^{2}}|\frac{d{\bf x}_{1}}{dt}|^{2})^{- 1/2}\frac{dx_{1}^{i}}{dt}\right).
\begin{aligned}f(x) = -\tfrac{1}{N}\sum_{i=1}^N y_i \log \big(\sigma(A_i x)\big) + (1-y_i) \log \big(1-\sigma(A_i x)\big) + \beta \sum_{i=1}^N e_{\lambda,\sigma}l_0([x]_i) \nabla f(x) = \tfrac{1}{N}A^T \big(\sigma(A_i x) - y\big) + \beta \nabla e_{\lambda,\sigma}l_0(x) \end{aligned}
\mathfrak{B}(n;q_1,q_2,q_2')=\sum_{\alpha_2\bmod{q_2q_2'}}K_{q_2}(\alpha_2\ell\bar{q}_1^3)K_{q_2'}(-\alpha_2\ell\bar{q}_1^3)e_{q_2q_2'}(\alpha_2\bar{q}_1 n).
\rho(k,\omega)=\frac{1}{Z}\sum_{\alpha,\alpha^{\prime}}2\pi\delta(E_{\alpha}-E_{\alpha^{\prime}}-\omega)|\langle\alpha|\psi_{k}^{\dagger}|\alpha^{\prime}\rangle|^{2}(e^{-\beta E_{\alpha^{\prime}}}-\zeta e^{-\beta E_{\alpha}}),
\| A_\mu (t,f) \Psi \| \le \sum_{m,n =0}^\infty \frac{|t|^{m+n}}{m! n!}C^{m+n}C_N (N_\Psi + m+n+1)^{1/2}\cdots (N_\Psi +1)^{1/2}\| \Psi \| \times \| \hat{f}\|_{N,0}.
P_{\pi}g=[\begin{matrix}e_{\pi(1)}\\e_{\pi(2)}\\e_{\pi(3)}\\e_{\pi(4)}\\e_{\pi(5)}\end{matrix}][\begin{matrix}g_{1}\\g_{2}\\g_{3}\\g_{4}\\g_{5}\end{matrix}]=[\begin{matrix}g_{1}\\g_{4}\\g_{2}\\g_{5}\\g_{3}\end{matrix}].
{\cal V}= -{\frac{f_{\pi}^{2}}{2}}{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{i}}-{\frac{1}{4 e^{2}}}\biggl [{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{i}}\;{\cal R}_{\mathrm{j}}\cdot{\cal R}_{\mathrm{j}}-{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{j}}\;{\cal R}_{\mathrm{j}}\cdot{\cal R}_{\mathrm{i}}\biggr ] .
L_B=-{1\over 2}\left(\, \partial_\mu\phi \,\right)^2 +{m\over 4!}\epsilon^{\mu\nu\rho\sigma}\, \phi\,\partial_{\,[\,\mu}A_{\nu\rho\sigma\,]}-{1\over 2\cdot4!}F^2{}_{\mu\nu\rho\sigma}(A)
\mu_{-}= \left( \begin{array}{c c}{{I_{m}}}&{{0}}\\{{\mu_{- 2 1}}}&{{I_{m}}}\\ \end{array}\right) , \qquad \mu_{+}= \left( \begin{array}{c c}{{I_{m}}}&{{\mu_{+ 1 2}}}\\{{0}}&{{I_{m}}}\\ \end{array}\right) ,
S_{eff}= \int\!d^4x~[ -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}m^{2}( A_{\mu}+ \partial_\mu \rho )^2 - A^{\mu}\partial_{\mu}B + \frac{1}{2}\alpha (B)^2+ \dot{{\overline{\cal P}}}\frac{1}{\partial_{i}\partial^{i}}\dot{{\cal P}}-{\overline{\cal P}}{\cal P}]
f(G_K) \geq \frac{1}{K}f(O_K)+\left(1-\frac{\eta}{K}\right)f(G_{K-1}) &\quad\vdots &\geq \frac{1}{K}f(O_K)\sum\limits_{i=0}^{K-1}\left(1-\frac{\eta}{K}\right)^i &=\frac{1}{\eta}\left(1-\left(1-\frac{\eta}{K}\right)^K\right)f(O_K).
b(\xi-\eta,\eta) = \frac{(|\xi-\eta| + |\eta| - |\xi|) A(\xi-\eta,\eta) - 2 A(\eta,\xi-\eta)|\xi-\eta|^{1/2}|\eta|^{1/2}}{- (|\xi-\eta| + |\eta| - |\xi|)^{2}+ 4 |\xi-\eta||\eta|},
\lim_{\lambda\in\Lambda}\frac{P_{n+1,k}(z)}{P_{n,k}(z)}=\widetilde{F}_{k}^{(\rho)}(z)=\frac{S_{k}^{(\rho)}(z)}{S_{k}^{(\rho)}(\infty)}\,\frac{\phi_{k}(z)}{\phi_{k}'(\infty)},z\in\mathbb{C}\setminus[a_{k},b_{k}].
g_{-}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{\frac{x_{-}}{\sqrt{2}}}}&{{1}}\\ \end{array}\right) g_{0}= \left( \begin{array}{c c}{{e^{\frac{x}{2}}}}&{{0}}\\{{0}}&{{e^{- \frac{x}{2}}}}\\ \end{array}\right) g_{+}= \left( \begin{array}{c c}{{1}}&{{\frac{x_{+}}{\sqrt{2}}}}\\{{0}}&{{1}}\\ \end{array}\right) .