harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\begin{array}{ccl}x_{1}' = 0 x_{2}' = \left(\frac{1-\mu}{2-\mu}x_{1}+\frac{1-\mu}{3-\mu}x_{2}\right)\left(y_{1}+y_{2}\right) y_{1}' = \left(\frac{1-\mu}{2-\mu}x_{1}+\frac{1-\mu}{3-\mu}x_{2}\right)\left(y_{1}+y_{2}\right) y_{2}' = \left(\frac{\mu}{2-\mu}x_{1}+\frac{1+\mu}{3-\mu}x_{2}\right)\left(y_{1}+y_{2}\right).\end{array}\end{cases}
-4\left\langle \sin^{2}t\cdot h , h \right\rangle_{\mathbb{S}^{1}}=\Sigma_{0}+\Sigma_{1}\leq \left( \mu^{\ast}-2\right) \left( c_{0}^{t}\overline{c_{0}}+ c_{1}^{t}\overline{c_{1}}\right) .
u^0(x,t) \ge\frac{1}{2\pi}\int_{0}^{t}\frac{\rho d\rho}{\sqrt{t^2-\rho^2}}&\quad\times\int_{|\omega|=1}\left\{\frac{f(x+\rho\omega)}{1+|x+\rho\omega|}-|\nabla f(x+\rho\omega)|+g(x+\rho\omega)\right\}dS_{\omega}
\Phi\sim B \left[\frac{(\sqrt{2\mu k}u)^{\nu+1/2}}{2^{\nu}}\frac{(\sqrt{E_b})^{\nu+\frac{1}{2}}e^{i\frac{\nu\pi}{2}}}{\Gamma(1+\nu)}-\frac{(\sqrt{2\mu k}u)^{-\nu+1/2}}{2^{-\nu}}\frac{(\sqrt{E_b})^{-\nu+\frac{1}{2}}e^{i\frac{\nu\pi}{2}}}{\Gamma(1-\nu)}\right].
|I_{1}(x)| \leq\sum_{k=-\infty}^{-1}\int_{2^{k}R\leq |xy^{-1}|<2^{k+1}R}\frac{|xy^{-1}|^{\alpha-Q}|f(y)|}{(1+|xy^{-1}|)^{\gamma}}dy &\leq \sum_{k=-\infty}^{-1}\frac{(2^{k}R)^{\alpha-Q}}{(1+2^{k}R)^{\gamma}}\int_{2^{k}R\leq |xy^{-1}|<2^{k+1}R}|f(y)|dy &\leq CMf(x)\sum_{k=-\infty}^{-1}\frac{(2^{k}R)^{\alpha-Q+Q/p_{1}}(2^{k}R)^{Q/p_{1}^{'}}}{(1+2^{k}R)^{\gamma}}.
\begin{array}{r c l}{{\mathrm{t r}_{( N )}\left( J_{k}\Delta ( x ) \right)}}&{{=}}&{{\frac{1}{N}e^{2 \pi i k_{i}x^{i}/ L},}}\\{{\frac{1}{N}\sum_{x}\Delta ( x )_{\mu \nu}\Delta ( x )_{\lambda \rho}}}&{{=}}&{{\delta_{\mu \rho}\delta_{\nu \lambda},}}\\{{\frac{1}{N}\mathrm{t r}_{( N )}\left( \Delta ( x ) \Delta ( y ) \right)}}&{{=}}&{{N^{2}\delta_{x , y ( \mathrm{m o d}\L )}.}}\\ \end{array}
V^{u}\bigl(t, x\bigr) = [V_1^{u}\bigl(t, x\bigr), V_2^{u}\bigl(t, x\bigr), \ldots, V_n^{u}\bigl(t, x\bigr)]^T, \,\, (t, x) \in [0, T] \times \mathbb{R}^d \,\, \& \,\, u \in U.
B^{[i]}_{h,m-j}\coloneqq\frac{\tilde{b}^{[i]}_{h,1,m-j}D g^{[i]}_{h+1,m-j+1}}{\sum_{r=1}^{m-j}\tilde{b}^{[i]}_{r,1,m-j}D g^{[i]}_{r+1,m-j+1}}, h=1,\dots,m-j,
\widetilde D_\Omega^*S(\bar x|\bar y)(\bar y^*):=w^*\mbox{-}\limsup\limits_{\stackrel{\stackrel{(x,y)\stackrel{\operatorname{gph}S|_\Omega}{\longrightarrow}(\bar x,\bar y)}{y^*\rightharpoonup^* \bar y^*}}{\varepsilon\downarrow0}}\widehat D_{\Omega,\varepsilon}^* S(x|y)(y^*).
\langle \psi,\hat z\psi\rangle_{L^2(\mathbb{R}^d)}=z\,\Rightarrow \, \langle \psi,\mathcal{T}^*_\zeta\circ \hat z\circ \mathcal{T}_\zeta\psi\rangle_{L^2(\mathbb{R}^d)}=z-\zeta \ ,
P :=\mathbb{P}[G(m_1,n_1)\le v_1, G(m_2,n_2)\le v_2] &=\sum\limits_{u=-\infty}^{v_1}\sum\limits_{\substack{x\in W_{n_2}x_{n_1}=u}}\sum\limits_{\substack{y\in W_{n_2}y_{n_2}\le v_2}}\det\left(\Delta^{j-i}w_{m_1}(x_j)\right)_{1\le i,j\le n_2}\det \left(\Delta^{j-i}w_{m_2-m_1}(y_j-x_i)\right)_{1\le i,j\le n_2},
Y^{[ i ]}= \int_{w_{-}}^{w_{+}}d w e^{- 3 A ( w )}\int_{w_{-}}^{w}d w^{\prime}e^{3 A ( w^{\prime})}\int_{w_{-}}^{w^{\prime}}d w^{\prime \prime}e^{- A ( w^{\prime \prime})}F_{( T )}^{[ i ]}( w^{\prime \prime}) ,
r_{n+2}=3\sum_{j=0}^{n}(\begin{matrix}2n+2\\ 2j+2\end{matrix})r_{n-j}\sum_{k=0}^{j}(\begin{matrix}2j+1\\ 2k+1\end{matrix})r_{k}r_{j-k},r_{0}=1,r_{1}=0,
\langle p(v), \dot{p}(v)\rangle =\dot{r}(\chi(v)) \forall v\in V \mbox{and}\langle p(u), \dot{p}(v)\rangle+\langle p(v), \dot{p}(u)\rangle =\dot{r}(\chi(u))+\dot{r}(\chi(v)) \forall uv\in E(G),
\mathbf{R}^\mathrm{C}[\iota]=\sqrt{\rho_\mathrm{s}}\mathbf{G}_\mathrm{s}[\iota]\mathbf{S}^\mathrm{C}[\iota]+\mathbf{L}^\mathrm{C}[\iota]+\mathbf{N}^\mathrm{C}[\iota]
\frac{\dim \Phi_{p(q+1)j}}{\dim \Phi_{pqj}}\Bar{\delta}^{\Phi\Phi}_{ppq(q+1)j}+ \frac{\dim \Phi_{(p-1)qj}}{\dim \Phi_{pqj}}\Bar{\delta}^{\Phi\Phi}_{p(p-1)qqj}= 1,
\left|h_1^2+\frac{\partial_x\xi(a_N,b_N)}{\gamma_N}-\frac{a_N}{(1-a_N)^2}\right| \leq\frac{L\log^2 N}{N}, \left|h_2^2+\frac{\partial_y\xi(a_N,b_N)}{1-\gamma_N}-\frac{b_N'}{(1-b_N)^2}\right| \leq\frac{L\log^2 N}{N}.
\langle h_{1},\Phi_{E}(f)h_{2}\rangle=\overline{\langle\Phi_{E}(f)h_{2},h_{1}\rangle}=\int_{X}\overline{f}(x)\overline{\langle E(dx)h_{2},h_{1}\rangle}=\int_{X}\overline{f}(x)\langle h_{1},E(dx)h_{2}\rangle
A^{(2)}(y) = \frac{y^{\frac{1}{2}}-y^{-\frac{1}{2}}}{\phi_{-2,1}(q^2,y^2)^\frac{1}{2}\, \tilde\Delta(q^2)^\frac{1}{2}}\,, B^{(2)}(y) = \frac{\tilde\eta(q)^2}{\theta_3(q,y)}\,, C^{(2)}_{11}(y) = \frac{-\theta_3(q,y)}{\theta_2(q,y)}\,.
B = \left(\frac{\pi}{2}+ \gamma + \log{8\pi}\right) - \log \operatorname{rd}(K) - \mathcal{G}(F) + 2 \sum_{\mathfrak P}\sum_{m=1}^{\infty}\frac{\log N\mathfrak P}{N\mathfrak P^{m/2}}F(m \log N\mathfrak P) > 5.3721 - 4.8412 - 1.2642 + 0.7023 + 0.0331 = 0.0021.
\frac{\mathrm{d}}{\mathrm{d}t}\rho_t(\eta) + \sum_{\tilde{\eta}\in\mathcal{X}}\bigl(\psi_t(\tilde{\eta}) -\psi_t(\eta)\bigr)\hat{\rho}_t(\eta, \tilde{\eta})Q(\eta, \tilde{\eta}) = 0 \qquad\forall \eta\in\mathcal{X},
\langle g^{\mu \nu}({\bf x}_{F}) g^{\sigma \rho}({\bf y}_{F}) \rangle = - D^{\mu \nu ; \sigma \rho}({\bf x}_{F}-{\bf y}_{F}) = - C^{\mu \nu ; \sigma \rho}\vert{\bf y}_{F}-{\bf x}_{F}\vert^{- 4 \gamma}
\lambda^{\max}(u(r)v)=\lambda^{\max}(\sigma(r)u(-r)u^{-}(r^{-1})v) =-\lambda^{\min}(u(-r)u^{-}(r^{-1})v) &=-\lambda^{\min}(u^{-}(r^{-1})v)\geq -\lambda^{\max}(u^{-}(r^{-1})v)= -\lambda^{\max}(v).
\lambda_3 =\langle e_m+f_2, e_2+f_m, e_1, e_3\rangle, \,\,\, \lambda_4=\langle e_m+f_3, e_3+f_m, e_1, e_2\rangle, \textrm{and}\lambda_5 =\langle e_m+f_2+f_3, e_2+f_m, e_3+f_m, e_1\rangle.
\chi_\lambda(gx)=\begin{cases}\chi_{\lambda_0}^+(g x_0s_0)\overline{\chi}_{\lambda\backslash\{0\}}(x's')+\chi_{\lambda_0}^-(g x_0s_0)\overline{\chi}_{\lambda\backslash\{0\}}(x's') \text{if}x's'\in\tilde{A}_{t-t_0}, 0 \end{cases}
F_s(V) \ge (1-\frac{V}{V_0})F_s(0) + \frac{V}{V_0}F_s(V_0) ={{\frac{V}{V_0}F_s(V_0)=}}\beta_0 V, \textrm{and}\beta_0V_0 = F_s(V_0) \ge \frac{V_1-V_0}{V_1 - V}F_s(V) + \frac{V_0-V}{V_1 - V}F_s(V_1).
\Gamma_k^{(1)}\left({\bf C}_{x_{B_k}},{\bf C}_{x_{j}}\right)= \frac{1}{\alpha_k}\log\left(1+\frac{{\bf h}^{H}_{kB}{\bf C}_{x_{B_k}}{\bf h}_{iB}}{\sum_{m=1,m\neq k}^{K}{\bf h}^{H}_{kB}{\bf C}_{x_{B_m}}{\bf h}_{kB}+\sum_{j=1}^{J}{\bf h}^{H}_{kj}{\bf C}_{x_{j}}{\bf h}_{kj}+\sigma^{2}_w+\sigma_n^{2}}\right),
\partial_{x_n}u(\hat{x})_{\big|\{x_n=0\}}= \partial_{x_n}\big(c \circ F_\gamma\big)(\hat{x})_{\big|\{x_n=0\}}= \nabla c(x^\prime,\gamma(x^\prime))\cdot(-\mathbf{n}(x^\prime,\gamma(x^\prime))) = 0,
b_{\varepsilon',\hat{T}}((\hat{\ell})_{\sf p})+b_{\varepsilon'',\hat{T}}((\hat{\ell})_{\sf p})=b_{\varepsilon',\hat{T}}((\hat{\ell})_{\sf q})+b_{\varepsilon'',\hat{T}}((\hat{\ell})_{\sf q})
W(G_{2}(n,d,x,1)) =W(G_{2}(n,d,x,0))+1, W(G_{2}(n,d,x,2)) =W(G_{2}(n,d,x,0))+6, W(G_{2}(n,d,x,-1)) =W(G_{2}(n,d,x,0))+3.
\sum_{c \bmod{\varpi^{l-1}}}\widetilde{e}\left(\frac{c}{\varpi^{l-k-1}}\right) = N(\varpi^k) \sum_{c_2 \bmod{\varpi^{l-k-1}}}\widetilde{e}\left( \frac{c_2}{\varpi^{l-k-1}}\right)=0,
T_0 \phi_\xi = \sum_{v\in W_0}\frac{t_0-1}{1-e^{i\langle v\xi,{\alpha_0}\rangle}}C(v\xi)e^{iv\xi}+ \sum_{v\in W_0}\frac{1-t_0 e^{i\langle v\xi,-{\alpha_0}\rangle}}{1-e^{i\langle v\xi,-{\alpha_0}\rangle}}C(s'_0v\xi) e^{ic\langle v\xi,{\alpha_0}\rangle}e^{iv\xi}.
\begin{cases}\dot{S}(t) = \Lambda - \lambda_H S(t) - \mu S(t) begin{align*}0.2 cm]\dot{I}_H(t) = \lambda_H S(t) - (\rho_1 + \mu)I_H(t) + \alpha_1 A(t) begin{align*}0.2 cm]\dot{A}(t) = \rho_1 I_H(t) - \alpha_1 A(t) - (\mu + d_A)A \, ,\end{cases}
\langle \hat{t}( \frac{p_{1}}{\delta}, \ldots , \frac{q}{\delta}) , \stackrel{\vee}{\phi}\rangle = \delta^{3 - 3 l}\int d^{3 r - 3}\tilde{p}d^{3 v - 3}\tilde{q}\hat{t}_{1}^{[ \mu ]}( \frac{\tilde{p}}{\delta}) \hat{t}_{2}^{[ \nu ]}( \frac{\tilde{q}}{\delta}) \psi_{[ \mu \nu ]}^{0}( \tilde{p}, \tilde{q}) .
\partial_{k}\Gamma_{k}[ \phi ] = \lim_{\delta k \to 0}\frac{1}{2 \delta k}\left[ \ln \det \frac{\delta^{2}\Gamma_{k}}{\delta \phi_{k}^{\Lambda}\delta \phi_{k}^{\Lambda}}- \ln \det \frac{\delta^{2}\Gamma_{k}}{\delta \phi_{k - \delta k}^{\Lambda}\delta \phi_{k - \delta k}^{\Lambda}}\right] ,
D_{0}= D_{1}=\ \ldots \ =D_{n_{0}-1}= 0 \ , \D_{n_{0}}= (-1)^{\frac{n_{0}(n_{0}+1)}{2}}s_{n_{0}}^{n_{0}+1}= (-1)^{n_{0}(n_{0}+1)}t_{n_{0}}= t_{n_{0}}\ .
\frac{\langle \langle \psi^\prime|e^{-\hat H \tau_f}\hat O_1(\tau_1)\hat O_2(\tau_2)\ldots \hat O_n(\tau_n)e^{\hat H\tau_i}|\psi\rangle\rangle}{\langle \langle \psi^\prime|e^{-\hat H(\tau_f-\tau_i)}|\psi\rangle\rangle}\,.
\left| \{K_{\rho}: (n(\cdot, t)-\mu_{\pm}\pm \delta k)_{\pm}> 0 \}\right| &\leq \left[ \left(\frac{j}{j-1}\right)^2 (1-\nu) + \frac{C(c_0,\alpha) j}{\sigma^2 (j-1)^2}+ C(\delta) + d \sigma\right]|K_{\rho}|.
u_{2}=0, \v_{2}=-u_{3}v_{2}, \w_{2}=u_{3}-u_{3}w_{2}, \u_{3}=-w_{2}, &u_{1}-v_{2}(w_{2}-1)-\frac{1}{2}w_{2}(w_{2}-1) = v_{3}+w_{3}v_{2}-v_{3}w_{2}.
\int_{\Sigma}\left(\dfrac{n}{2}+|{\bf H}|^2+|\nabla \log v|^2\right)\phi^2e^{\frac{|x|^2}{4}}d\sigma =-\int_{\Sigma}\phi^2(\mathcal{L}\log v)e^{\frac{|x|^2}{4}}d\sigma &=\int_{\Sigma}\left<\nabla \phi^2, \nabla \log v\right>e^{\frac{|x|^2}{4}}d\sigma
F(u)=\left(\prod_{j=1}^m\frac{[v_j-v_{j-1}+\frac{1}{2}-\pi_j+u]}{[-\pi_j+u]}\right)\frac{[v_0-v_m-\frac{m}{2}+1-\pi_\mu+u]}{[1-\pi_\mu+u]}.
\| Af\|^2 = \langle f, \mathcal{T}\Pi Af \rangle - \| \mathcal{T}\Pi A f \|^2 \leq \|(1-\Pi)f \| \| \mathcal{T}\Pi Af \| - \| \mathcal{T}\Pi A f \|^2 \| Af\|^2 +\| \mathcal{T}\Pi A f \|^2 \lesssim \frac{\|(1-\Pi)f \|^2}{2\varepsilon}+ \frac{\varepsilon}{2}\|\mathcal{T}\Pi Af \|^2
l=-\frac{n}{2}log2\pi\sigma^{2}+\sum_{i=1}^{n}log[e^{-\frac{(x_{i}-\mu)^{2}}{2\sigma^{2}}}(1+e^{-\frac{(x_{i}+\mu)^{2}}{2\sigma^{2}}}e^{\frac{(x_{i}-\mu)^{2}}{2\sigma^{2}}})]
M_{p}^{2}= M_{5}^{3}\int_{- \infty}^{\infty}e^{2 A_{0}( y )}\, d y = \frac{4 M_{5}^{3}\sqrt{\pi}}{a ( k^{2}- 1 )}\frac{\Gamma \left( ( k^{2}+ 3 ) / 4 ) \right)}{\Gamma \left( ( k^{2}+ 1 ) / 4 \right)}\; ,
\Omega^{,\infty}_{I\to K}:= \bigg\{s = \{(n,s_n)\}_{IN\leq n \leq KN}: s_{LN}\in R(L,J_L^G)\ \forall I\leq L\leq K, s \subset \cup_{I\leq L<K}B(L,J_L^G) \bigg\},
-\beta F^{\mbox{{\tiny String}}}(\beta)=-\frac{N^2}{8}\int_0^{\infty}\frac{dt}{2}\,t^{-1}\left [\theta_3\left(0,\frac{2 \pi it}{\beta^2}\right)-\theta_2\left(0,\frac{2 \pi it}{\beta^2}\right)\right]f(t)
\frac{( k - 1 ) ! k !}{( 2 k + 1 ) !}{\}_{2}F_{1}( k , \k + 1 ; \ 2 k + 2 ; \u ) = \frac{1}{k ! ( k + 1 ) !}\frac{d^{k}}{d u^{k}}( 1 - u )^{k + 1}\frac{d^{k}}{d u^{k}}\frac{\log ( 1 - u )}{u}
\!\!\!\!\!\!\! \mathcal{O}_{\rm d}(R_d)= &\!\!\!\!\!\!\! 1-\int\limits_{0}^{R_{\rm M}}f_{R}(r_o)e^{\frac{N_o (1-2^{\frac{R_{d}}{B_{\rm d}^{\alpha{\rm D}}}})}{P_{\rm d}r_o^{-\eta}}}\mathcal{L}_{I_{\rm d}}\left(\frac{2^{\frac{R_{d}}{B_{\rm d}^{\alpha{\rm D}}}}-1}{P_{\rm d}r_o^{-\eta}|\mathcal{I}_{\rm d}(\alpha)|^2}\right)d r_o,
P^{'}_{c}(z) - \left[ ( \frac{\lambda}{\xi}- \frac{1}{z}\left( \frac{\mu j}{\xi}- c\right) \right]P_{c}(z) = - \frac{\gamma}{\xi (1-z)}P_{c-1}(z) + \frac{\mu}{\xi (1-z)}\sum_{n=1}^{c}nz^{n}p_{c,n}- \frac{c(\xi - \mu)}{\xi z}\sum_{n=0}^{c}z^{n}p_{c,n}+ \frac{1}{z}\sum_{n=1}^{c}n z^{n}p_{c,n}.
k z^{-k+1}r(z) + z^{-k+2}\dot{r}(z) - z^{-k+1}r(z) (\lambda (z) +1) - \eta (z) r(z)^{2}-\frac{\beta (z)}{2}z^{-2k+2}& +\eta (z) \gamma (z) z^{-2k+1}=0.
\widehat{h}= \displaystyle \frac{\widehat{H}}{{\cal M}}= \displaystyle \frac{2 \pi}{l}\left( \widehat{L_{0}}+ \widehat{\overline{{L}}_{0}}- \displaystyle \frac{c}{1 2}\widehat{I}+ \kappa \displaystyle \frac{l^{2 - h}}{\left( 2 \pi \right)^{1 - h}}\widehat{B}\right) \: .
\begin{aligned}G_{\epsilon}(t_1,\cdots, t_k) =\Big(\int^{1}_{0}K(t_1,r)\theta_{\epsilon}(r)dr,\cdots, \int^{1}_{0}K(t_k,r)\theta_{\epsilon}(r)dr\Big)^{\tau}&=\Big(G_{\epsilon}(t_1),\cdots,G_{\epsilon}(t_k)\Big)\end{aligned}
\hat{\cal L}_{0}= - \int d^{3}u d^{3}v \sum_{\tau = \pm}\hat{\phi}^{( \tau ) \dagger}({\bf x}; x_{0}) \hat{\tau}_{3}D_{\tau}( x ;{\bf u},{\bf v}) \hat{\phi}^{( \tau )}({\bf v}; x_{0}) ,
I_{N - n}\longrightarrow \frac{1}{n !}\biggl \{\sum_{i = 1}^{N}q_{i}, \dots \biggl \{\sum_{i = 1}^{N}q_{i}, \prod_{k = 1}^{N}p_{k}\biggr \}\dots \biggr \}= I_{N - n}( g = 0 ) ,
\left( \begin{array}{c}U^{+}_{k,l}U^{-}_{k,l}\end{array}\right)=\left( \begin{array}{c}u^{+}_{k,l}(2l)^{-k+\frac{1}{2}}I_{k-\frac{1}{2}}(2lr)-u^{-}_{k,l}(2l)^{k+\frac{1}{2}}I_{-k+\frac{1}{2}}(2lr) -u_{k,l}^{+}(2l)^{-k+\frac{1}{2}}I_{k+\frac{1}{2}}(2lr)+u_{k,l}^{-}(2l)^{k+\frac{1}{2}}I_{-k-\frac{1}{2}}(2lr)\end{array}\right)
[v\otimes b,v'\otimes b']=[\left[\begin{array}{cc}0 v\otimes b v^{t}\otimes b 0\end{array}\right],\left[\begin{array}{cc}0 v'\otimes b' v'^{t}\otimes b' 0\end{array}\right]]=\left[\begin{array}{cc}v(v')^{t}\otimes bb'-v'v^{t}\otimes b'b 0 0 (v)^{t}v'\otimes[b,b']_{A^{-}}\end{array}\right].
{E}^{0,i}\big[e^{- \alpha\overline{m}_{\mathbf{e}_q}- \beta \overline\xi_{\mathbf{e}_q}}, J_{\overline{m}_{\mathbf{e}_q}}= j\big]= \Phi^+_j(q,0)[\kappa^+(q+\alpha,\beta)^{-1}]_{i,j}.
I(u) = \frac{1}{\gamma \alpha}(\log |x|)^{1-\alpha}\left[1 - \exp \left\{- \gamma (\log |x|)^\alpha \left[ 1- \left( 1 + \frac{\log u}{\log |x|}\right)^\alpha \right] \right\}\right] & \leq \frac{1}{\gamma \alpha}(\log |x|)^{1-\alpha}\left[1 - \exp \left\{2 \gamma \alpha (\log |x|)^{\alpha-1}\log u \right\}\right] & \leq - 2 \log u,
V_{2}( \mathrm{x ) = \frac{1}{3 2}\mathrm{x^{6}- \frac{\o m e g a}{1 6 \mathrm{g}}\mathrm{x^{4}+ ( \frac{\o m e g a^{2}}{3 2 \mathrm{g^{2}}}+ \frac{3 j}{4}+ \frac{3}{8}) \mathrm{x^{2}- \frac{\o m e g a}{4 \mathrm{g}}( 5 j + \frac{1}{2})}}}}
\frac{d}{dt}\bigl(\mathcal{E}(\overline{D}_s)-\mathcal{E}(\overline{D}_0)\bigr)_{|_{t=s^+}}=\frac{d}{dt}\Bigl(\int_{\Delta_D}\check{g}_t\Bigr)_{|_{t=s^+}}\forall s\in [0,1[.
f_Y \left( y \right)= \left({\frac{m_\mathrm{I}}{\bar \gamma_\mathrm{I}}}\right)^{m_\mathrm{I}L}\frac{y^{m_\mathrm{I}L-1}}{\Gamma \left({m_\mathrm{I}L}\right)}\, \exp\left(-\frac{m_\mathrm{I}}{\bar{\gamma}_\mathrm{I}}y\right),
\det [D^2v - A(\cdot,v,Dv)] = \det \big[D^2u_m - A(\cdot,u_m,Du_m) \quad+k(I - A_{ij,u}(|x|^2-r^2)-A_{ij,p_k}\cdot x)\big], B(\cdot,u,Du) = B(\cdot,u_m,Du_m)+\frac{k}{2}B_u(|x|^2-r^2)+kB_{p_k}\cdot x.
\sum_{1\le i\le n}\partial_{\theta_i}\Delta_C =\sum_{1\le r\le k}(\partial_{\theta_{i_r}}+\partial_{\theta_{j_r}})\Delta_C + \sum_{i\notin \bigcup_{r=1}^k C_r}\partial_{\theta_i}\Delta_C =\sum_{1\le r\le k}(\partial_{\theta_{i_r}}+\partial_{\theta_{j_r}})\Delta_C =0\,.
E = - 2 \pi^{4}L^{2}( N \alpha^{\prime})^{5 / 2}\lim_{r \to \infty}\left( \nu \left( R^{3}e^{- 2 \Phi}\right)^{\prime}- \sqrt{\nu_{B P S}}\left( R_{B P S}^{3}e^{- 2 \Phi_{B P S}}\right)^{\prime}\right) .
\int_{- \infty}^{\infty}\left( \frac{1}{2}\phi^{2}+ U ( \phi ) \right) d y \rightarrow \int_{- \infty}^{\infty}\left( \frac{- 3 \xi^{2}}{8 \kappa^{2}}+ \frac{3 \xi}{2 \kappa^{2}}\delta ( y ) \right) d y \; .
\sup_{x,y}\Big|(G_t*f_j) (x) - (G_t*f_j) (y)\Big| = \sup_{x,y}\left| \sum_{k\ne 0}e^{-4\pi^2 k^2 t}a_k^{(j)}\left[e^{i2\pi k x}-e^{i2\pi k y}\right]\right| &\le 4 e^{-4\pi^2 t}\left(\sum_{k \ne 0}\left[a_k^{(j)}\right]^2\right)^{1/2}
\Delta_{0}^{( \widetilde{1})}= \frac 1{2 m^{2}}i p^{( \widetilde{1})}\left( i p^{( \widetilde{1})}- \varepsilon m \right) \left( 1 - \sigma_{p}^{( \widetilde{1}) 2}\right) = \Psi_{0}^{( \widetilde{1})}\cdot \overline{{{\Psi}}}_{0}^{( \widetilde{1})}
\begin{aligned}\varphi_t(x_q, t_q)+F(w_q(x_q, t_q), \nabla\varphi(x_q, t_q), \nabla^2\varphi(x_q, t_q)-C_q\varepsilon I, W_\star[x_0, t_0]) &\geq \lambda \frac{u(y_q, t_q)^{q-1}}{\varphi(x_q, t_q)^{q-1}}D_{1, q}+ (1-\lambda) \frac{u(z_q, t_q)^{q-1}}{\varphi(x_q, t_q)^{q-1}}D_{2, q},\end{aligned}
(T_sT_tT_s\cdots T_s)_m=(T_sT_tT_s\cdots T_t)_{m+1}=(T_tT_sT_t\cdots T_t)_mT_t=q_t(T_tT_sT_t\cdots)_{m-1}+(1-q_t)(T_tT_sT_t\cdots)_m
\begin{aligned}j(\rho) =\begin{pmatrix}-\nabla\rho^{A}+\frac{1}{N^\gamma}\Big(\rho^{A}(1-\rho^{A})(E_{A}-E_{C})-\rho^{A}\rho^{B}(E_{B}-E_{C})\Big) -\nabla\rho^{B}+\frac{1}{N^\gamma}\Big(\rho^{B}(1-\rho^{B})(E_{B}-E_{C})-\rho^{A}\rho^{B}(E_{A}-E_{C})\Big)\end{pmatrix}.\end{aligned}
\hat H^{(q)}_I(\hat x,\hat p)=\frac{f^2}{6}\bigl\{\hat x^3 V'(\hat x) \partial^2_{\hat x}+ \bigl[\hat x^3 V^{''}(\hat x) +3 \hat x^2 V^{'}(\hat x)\bigr]\partial_{\hat x}+{\textstyle\frac{1}{3}}\hat x^3 V^{'''}(\hat x)+{\textstyle\frac{3}{2}}\hat x^2 V^{''}(\hat x)\bigr\}.
\mathbb E^{(t)}(Z_t(Z_t-1)s^{Z_t-2}, T\in dt_1\mid Z_{t-t_2}=y)/dt_1 &=y\psi_{t_2}^{'}(s) \psi_{t_2}(s)^{y-1}\Phi^{''}(\psi_{t_1}(s))\psi_{t_1}^{'}(s) &=y\psi_{t_2}^{'}(s) \psi_{t_2}(s)^{y-1}\psi_{t_1}^{'}(s) \sum_{n\geq 2}n(n-1) \mu (n)\psi_{t_1}(s)^{n-2}.
\underline{\Lambda}^1_{\lambda}- \underline{\Lambda}^1_{\lambda '}= \underline{\Lambda}^1_{\lambda}- p +p -\underline{\Lambda}^1_{\lambda '}\subseteq (\underline{\Lambda}^1_{\lambda}- \underline{\Lambda}^1_{\lambda}) + (\underline{\Lambda}^1_{\lambda '}- \underline{\Lambda}^1_{\lambda '}) \subseteq 2(\underline{\Lambda}_0 - \underline{\Lambda}_0)
I_{2}=-\int\nabla P\cdot(\partial_{t}\mathbf{u}+ \mathbf{u}\cdot\nabla\mathbf{u})dx = \int P\partial_{j}u_{i}\partial_{i}u_{j}dx \leq C\|P\|_{\rm BMO}\|\partial_{j}u_{i}\partial_{i}u_{j}\|_{\mathcal{H}^1},
(1-\alpha)(x) = x\cdot 0 = f_k(x,0\cdot x) = x + \sum_{j=1}^k (-1)^j\alpha^j(x-(1-\alpha)^{-1}\alpha(x)), x = f_{k-1}(x,0\cdot x)\cdot 0 = (1-\alpha)(x+\sum_{j=1}^{k-1}(-1)^j\alpha^j(x-(1-\alpha)^{-1}\alpha(x))).
\Gamma^{a}_{bc}(z)=\widetilde{\lambda}^a_d(z)\left(\frac{\partial \lambda^d_b(z)}{\partial z^c}+\frac{1}{2}C^{d}_{pq}\lambda^p_b(z)\lambda^q_c(z)\right)=\frac{1}{2}\widetilde{\lambda}^a_d(z)\left(\frac{\partial\lambda^d_b(z)}{\partial z^c}+\frac{\partial\lambda^d_c(z)}{\partial z^b}\right),
\min_{x\in \mathcal{X}}\left\{\sum_{(i,j,r,s) \in [n]^4}\overline{c}_{i,j}d_{r,s}x_{i,r}x_{j,s}+ \max_{\mathcal{S}\subseteq [n]^2:|\mathcal{S}|\le \Gamma}\left\{\sum_{(i,j)\in \mathcal{S}}\sum_{r,s\in [n]}\Delta \overline{c}_{i,j}d_{r,s}x_{i,r}x_{j,s}\right\}\right\}.
F_p(\gamma) = \frac{1}{2}\sum_{i,j=1}^n \Omega_{ij}^\sigma(x) \int_{\gamma}(z_i - x_i) dz_j + \frac{1}{3}\sum_{i,j,k=1}^n \left( (\partial_k \Omega_{ij}^\sigma)(x) + [ \omega_k^\sigma(x), \Omega_{ij}^\sigma(x) ]\right) \int_\gamma (z_i - x_i) (z_k- x_k) dz_j ,
\frac{\partial h}{\partial \dot{x}^{2}}=\frac{(1-b^2)^{\frac{3}{2}}}{2}x^{1}+\lambda\left( \frac{\dot{x}^{2}}{\sqrt{(\dot{x}^{1})^{2}+(\dot{x}^{2})^{2}}}+b\sin (t+c)\right).
\lim_{\theta \downarrow 0}\frac \partial{\partial \theta}Z^{(q)\prime}(x, \theta ) = \lim_{\theta \downarrow 0}\frac \partial{\partial x}\frac \partial{\partial \theta}Z^{(q)}(x, \theta )= Z^{(q)}(x) - \psi'(0+) W^{(q)}(x) = \frac \partial{\partial x}\Big(\lim_{\theta \downarrow 0}\frac \partial{\partial \theta}Z^{(q)}(x, \theta ) \Big).
F_{n}\left({\vec z}\right) = \sum_{{\vec c}}D_{{\vec c},n}\theta\left[\begin{array}{c}{\vec 0}\frac{{\vec c}}{\ell}\end{array}\right]\left({\vec z}-\frac{\tau}{\omega\kappa}\left(n+\frac{\beta}{2\pi}\right){\hat e}+\frac{\alpha}{2\pi\omega\kappa}{\hat e},\Omega\right)
[\bar \epsilon Q \ , \bar Q \epsilon' ]^* = \bar \epsilon (\sl +\Gamma^{11}Z){\sl \over 2{\bf P}^2}(\sl +\Gamma^{11}Z) \epsilon'=\bar \epsilon \Gamma^{\hat m}{\bf P}_{\hat m}\epsilon' = \bar \epsilon(\sl +\Gamma^{11}Z) \epsilon' \ .
W_{\mbox{eff}}=\int d^4x \left[ -\frac{1}{4}F^2_{\mu\nu}-\frac{1}{2}m^2{\left(A_\mu-{\partial_\mu\phi^2}\right)}^2 +i\bar\psi\gamma^\mu D^+_\mu \psi\right] +M_1 +S_{\mbox{gf}}
(I-\tilde T^2)\begin{pmatrix}f g\end{pmatrix}= \begin{pmatrix}-2I -2T 2T 4T^2-2I\end{pmatrix}\begin{pmatrix}f g\end{pmatrix}= \begin{pmatrix}-2f - 2Tg 2Tf + 4T^2g -2g\end{pmatrix}=\begin{pmatrix}0 0\end{pmatrix},
e_C+e_{C'}-e_Ce_{C'}=(f_1+f_0)+(f_2+f_0)-(f_1+f_0)(f_2+f_0) &\textstyle=f_1+f_2+2f_0 -f_0=f_1+f_2+f_0=\sum_{\epsilon\in E_C\cup E_{C'}}\epsilon.=e_{C+C'}.
\nu =-|Z|^{-6}\frac{1}{2}(|Z_0|^2+|Z_1|^2)(|Z_2|^2+|Z_3|^2)^2,\mu_1=|Z|^{-2}(|Z_3|^2-|Z_2|^2) f \mu_2+i \mu_3 = 2 i |Z|^{-2}Z_2 \overline{Z_3}f,f=\frac{1}{4}|Z|^{-2}(-2(|Z_0|^2+|Z_1|^2)+(|Z_2|^2+|Z_3|^2)).
\sum_{n=0}^{\infty}\frac{q^{\binom{n+1}{2}}}{\left(q;q\right)_{n}}\left(-x\right)^{n}A_{q}\left(-q^{n}\right)=\frac{\left(-q;q\right)_{\infty}}{\left(-qx;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}\left(-1\right)^{n}\prod_{k=0}^{2n-1}\left(x-q^{k}\right)}{\left(q^{2},-q,-q^{2};q^{2}\right)_{k}},
\overline{\beta_{ik}}= \lim_{K\to \infty}\frac{1}{NK}\sum_{n=1}^{N}{\sum_{k=1,k\neq i}^{K}{\beta_{ni}\beta_{nk}}}&= \lim_{K\to \infty}\frac{1}{K}\sum_{k=1,k\neq i}^{K}{\beta_{1i}\beta_{1k}}&= \beta_{1i}\overline{\beta}
\hat{q}_{\beta}(n,t)=\sum_{k=1}^{n}\sum_{\Lambda_{n}^{k}}k!\prod_{j=1}^{n-k+1}\frac{((1-p)^{j}/j)^{x_{j}}}{x_{j}!}\left(\frac{-\lambda t^{\beta}}{\ln p}\right)^{k}E_{\beta,k\beta+1}^{k+1}(-\lambda t^{\beta}),
\alpha_n(t) = \alpha_n\left(\frac{1}{2}+ j \left(\omega + j \sigma\right),\rho(t)\right) = n \sqrt{2\pi}\exp\left(-\frac{1}{2}\left[1+\gamma_{\alpha}\right]t\right) \exp\left(\frac{j}{2}\omega_{\alpha}t \right).
u_2(t) = M(t) D(t) e^{-3 i \lambda_1 |W_1|^2 \log t}\begin{pmatrix}1 0 \end{pmatrix}P Q(t) P^{-1}\begin{pmatrix}W_2 \overline{W_2}\end{pmatrix}&+ O(t^{-3/4 + C_1 \varepsilon^2_1 + 2\mu C_2 \varepsilon^2_1 + C_4 \varepsilon_1^2}).
F^{ij}v'_{ij}= F^{ij}[a^l u_{lij}+{a^l}_{,ij}u_l +2{a^l}_{,i}u_{lj}+ b_{ij}] \leq F^{ij}[a^l u_{lij}+2{a^l}_{,i}u_{lj}] + C_{8}\sum\limits_{i = 1}^n{{F^{ii}}}.
e^{\prime}(\rho,x)=e^{i\rho x}\left( \frac{z-1}{2(z+1)}-\frac{1}{2}\int_{x}^{\infty}q(t)dt+(z+1)\sum_{n=0}^{\infty}\left( -1\right)^{n}z^{n}d_{n}(x)\right)
{}_3\phi_2\begin{bmatrix}q^{-N}, a, b ;q,q c, q^{1-N}/t\end{bmatrix}= \frac{(at)_n}{(t)_n}{}_3\phi_2\begin{bmatrix}q^{-N}, c/b, a ;q, btq^{N}c, at\end{bmatrix}
\lim_{\epsilon \rightarrow 0}\left(\int_{\epsilon}^{t/2}dy \frac{e^{-y}}{y\log^{a}(\frac{t}{i y})}\right) = \int_{0}^{t/2}dy \frac{e^{-y}\sin(a\tan^{-1}(\frac{\pi}{2(\log(t)-\log(y))}))}{y((\log(t)-\log(y))^{2}+\frac{\pi^{2}}{4})^{a/2}}
\frac{1}{2 s L}\int_{- s L}^{s L}d x^{-}\varphi_{i}^{n}( x , s L ) \varphi_{j}^{m}( x , s L ) = \frac{1}{2 L}\int_{- L}^{L}d x^{-}\varphi_{i}^{n}( x , L ) \varphi_{j}^{m}( x , L ) \quad ,
(\mu'_{*})^{1/4}-\mu_{*}^{1/4}= \big(\nabla \mu^{1/4}\big) (v_*)\cdot (v'_*-v_*) & + \int_0^1(1-\tau) \big( \nabla^2 \mu^{1/4}\big)(v_*+ \tau(v'_*-v_*) ) (v'_*-v_*)^2 d\tau\,.
{\rm X}^{l m 0}\partial_i{\rm D}^{l m}- \partial_m ({\rm X}^{m l 0}{\rm D}_{l i}) -\partial_l({\rm X}^{m l 0}{\rm D}_{m i}) - q{~}\partial_i \partial_l \partial_m{\rm X}^{l m 0}= 0.
1=\lim_{z\rightarrow-\infty}\underline{u}(z)\leq\liminf_{z\rightarrow-\infty}u(z)\leq\limsup_{z\rightarrow-\infty}u(z)\leq\lim_{z\rightarrow-\infty}\overline{u}(z)=1, begin{align*}0.2cm] 0=\lim_{z\rightarrow-\infty}\underline{v}(z)\leq\liminf_{z\rightarrow-\infty}v(z)\leq\limsup_{z\rightarrow-\infty}v(z)\leq\lim_{z\rightarrow-\infty}\overline{v}(z)=0,
x\mapsto x y^{11},y\mapsto y^3\Longrightarrow\mathcal{I}_{1,2,\beta_1,\beta_2}=12\int_{\mathcal{D}}\check{f}_{t,s}(x,y)^s x^{\beta_1}y^{{4(5\beta_1+3\beta_2+24s)}}\frac{dx}{x}\frac{dy}{y},
\Psi_1=\sum_{N=0}^{\infty}\sum_{l=0}^{N}\sum_{n=-l}^{l}\left\{_1a_{N l n}\phi^{(+)}_{N l n}(t,r,\theta,\phi) +_1\! a_{N ln}^{\dagger}\phi^{\ast(+)}_{N l n}(t,r,\theta,\phi)\right\}
\begin{array}{lcl}F_{-1\mu}+ F_{n \mu}= (X_{-1\lambda}+ X_{n\lambda}){X_\mu}^\lambda +(X_{n, -1}X_{-1, \mu}-X_{-1, n}X_{n, \mu}) -\frac{1}{2}n(X_{-1, \mu}+ X_{n \mu})\cr \cr = P_\lambda({X_\mu}^\lambda + (D+1 - \frac{n}{2}) \delta^\lambda_\mu)\end{array}
d_xT(x_0,y) = -\frac{1}{e^{-{\psi(x_0,y)}}}\left( d \phi_{x_0}\int_{-\infty}^y e^{-{\psi(x_0,v)}}dv - \int_{-\infty}^y e^{-{\psi(x,v)}}d_x{\psi}(x_0,v) dv \right) &= \Gamma(x_0,y)
\sum_{j=0}^{i_0}\Phi(x,\kappa^j \rho)\lesssim \sum_{j=0}^{i_0}\Psi(y_0, 5\kappa^j \rho)+\|Du\|_{L^\infty(\Omega_{5\rho}(y_0))}\int_0^{5\rho}\frac{\varrho^\sharp_0(t)}{t}\,dt.
\int_{[0,1]}\!\pi(x) R_t(x) f(x)\,dx = \int_{[0,1]}\!\Pi(dx)\,R_t(x) f(x) &= \int_{[0,1]}\!\Pi_t(dx)\,f(x) &= \int_{[0,1]}\!\Pi_t^*(dx)\int_{[0,x]}\!\pi^{(x)}(y)f(y)\,dy &= \int_{[0,1]}\!\int_{[y,1]}\!\Pi^*_t(dx)\,\pi^{(x)}(y)f(y)\,dy.
u_{4,s}(t,x) = \frac{1}{2\pi}\int_{B_{s-t}(x)}\frac{\left(\int_{|y|}^{\infty}v_{4,c}(s,q) dq\right)}{\sqrt{(s-t)^{2}-|y-x|^{2}}}dy=\frac{1}{2\pi}\int_{B_{s-t}(0)}\frac{\left(\int_{|z+x|}^{\infty}v_{4,c}(s,q) dq\right)}{\sqrt{(s-t)^{2}-|z|^{2}}}dz
A * B = \left. e^{\frac{\theta}{2}( \partial_{y}\partial_{\bar{y}^{\prime}}- \partial_{y^{\prime}}\partial_{\bar{y}})}A ( x : y , \bar{y}) B ( x : y^{\prime}, \bar{y}^{\prime}) \right|_{y = y^{\prime}, \bar{y}= \bar{y}^{\prime}}