harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
i_{\ast}[\mathcal{E}_{d, v}] =\left(1-q_1^{-1}\right)^{d-1}\left(1-q_2^{-1}\right)^{d-1}\cdot & h_{\ast}\left(\mathcal{O}\left(\chi\right)\prod_{j\geq i+2}(1-q^{-1}\mathcal{O}(\beta_j-\beta_i))\prod_{j>i}(1-q_1^{-1}\mathcal{O}(\beta_i-\beta_j))(1-q_2^{-1}\mathcal{O}(\beta_i-\beta_j)) \right),
\left( \frac{\partial_{x}w_{n}}{\rho_{n}}\right)(x,t) \le \max_{\mathbb{T}}\frac{\partial_{x}w_{n}^{0}}{\rho_{n}^{0}}=: \mathcal{M}_{n}^{0}, ~\forall ~(x,t) \in \mathbb{T}\times [0,T^{*}).
P^{-}={\frac{\mu^{2}q_{0}^{2}}{2}}+ \mu^{2}\Sigma_{2}+{\frac{\lambda q_{0}^{4}}{4 ! d}}+{\frac{\lambda q_{0}^{2}\Sigma_{2}}{2 ! d}}+{\frac{\lambda q_{0}\Sigma_{3}}{d}}+{\frac{\lambda \Sigma_{4}}{d}}\; .
\mathcal{V}_s^\eta = \left\{(Z_s,Z_s^\prime) \in \overline{\mathcal{D}_s}\times\overline{\mathcal{D}_s}\left| \begin{aligned}\inf_{1 \leq i \neq j \leq s}|v_i-v_j^\prime| > \eta & \textnormal{and}& \inf_{1\leq i \leq s \; : \; v_i \neq v_i^\prime}|v_i-v_i^\prime| > \eta\end{aligned}\right.\right\}
\begin{aligned}\dot{x}= (b_{20}x+1) (4 b_{11}x^2-b_{11}y^2-2 b_{20}x y-2 y)/2:=P_3(x,y), \dot{y}= x+b_{20}x^2+b_{11}x y-b_{20}y^2- (b_{11}b_{20}/2 ) y^3-b_{20}^2 x y^2+2 y b_{11}b_{20}x^2:=Q_3(x,y).\end{aligned}
b_j = \ \binom{j+n-1}{n-1}- \ell \sum^r_{i=1}\binom{j + d_i - d+ n-1}{n-1}+ (r-1) \ell \binom{j}{d}& \ + \sum_{k=2}^\ell (-1)^k \binom{\ell}{k}\binom{j - k (d - d_1)+ n-1}{n-1}begin{align*}1ex] \ + \binom{\ell}{2}\binom{j + 2 d_2 - 2 d + n-1}{n-1}+ \ell (\ell -1) \binom{j + d_1 + d_2 - 2 d+ n-1}{n-1}= \a_j.
=\frac 1{m!}\epsilon^{\mu_1\cdot \cdot \cdot \mu_k\mu_{k+1}\cdot \cdot \cdot \mu_n}\epsilon_{A_1\cdot \cdot \cdot A_m}J(\frac\phi v)\frac{\partial v^{A_1}}{\partial x^{\mu_{k+1}}}\cdot \cdot \cdot \frac{\partial v^{A_1}}{\partial x^{\mu_n}},
-\partial_{z}^{\mu}[\delta^{(4)}\left( x-z\right) \delta^{(4)}(y-z)\gamma_{\mu}]=i\delta^{(4)}(y-z)\overline{S}^{-1}(x,y)-i\delta^{(4)}\left( x-z\right) \overline{S}^{-1}(x,y)
\begin{pmatrix}r k_s k_s \end{pmatrix}= \Lambda = \Theta^{}(\Lambda) = \begin{pmatrix*}[l]A_c k_c +g_c \circ K{}{}-k_c \circ (A_c +r) A_u^{-1}k_u \circ (A_c + r) - A_u^{-1}g_u \circ K{}{}A_s k_s\circ (A_c + r)^{-1}+ g_s \circ K{}{}\circ (A_c +r)^{-1}\end{pmatrix*}.
\|u - u_h\|_{L^2(\Omega)}\lesssim h \min_{j=1,\ldots,J}\{h^{\min\{k,s-1\}}, h^{-\beta_j + s_j-1}\}& \lesssim \min_{j=1,\ldots,J}\{h^{\min\{k+1,s\}}, h^{\alpha_j + s_j-1-\varepsilon}\}.
\dot{Q}_{B}= \frac{q^{2}\; \dot{c}\; ( p^{2}- m^{2})}{1 + q^{2}}( 1 - q^{2}) \quad \mathrm{a n d}\quad \dot{Q}_{r}= \frac{d}{d \tau}( \lambda \; e ) \; \frac{p^{2}- m^{2}}{1 + q^{2}}( 1 - q^{2}) .
L_{x_1x_2-x_1^2-x_2^2}^{(2)}(y) = \left(\begin{array}{rcc}y_{11}-y_{20}-y_{02}y_{21}-y_{30}-y_{12}y_{12}-y_{21}-y_{03}y_{21}-y_{30}-y_{12}y_{31}-y_{40}-y_{22}y_{22}-y_{31}-y_{13}y_{12}-y_{21}-y_{03}y_{22}-y_{31}-y_{13}y_{13}-y_{22}-y_{04}\end{array}\right).
x_2^{(j+1)}=\frac{(x_1^{(j+1)})^2 +x_3}{x_2^{(j)}}=x_1^{(j+1)}\frac{x_1^{(j+1)}}{x_2^{(j)}}+\frac{x_3}{x_2^{(j)}}>x_1^{(j+1)}.
g(r) = \sum_{e,f}a_{1,e,f}\otimes a_{2,e,f}\otimes \cdots \otimes x_i y_i b_{e,f}\otimes \cdots \otimes a_{N,e,f}&= \sum_{e,f}a_{1,e,f}\otimes a_{2,e,f}\otimes \cdots \otimes b_{e+1,f+1}\otimes \cdots \otimes a_{N,e,f}= 0.
\frac{1}{Z}= ( \det A )^{1 / 2}= \frac{\det A}{( \det A )^{1 / 2}}= \int \prod_{i}d \phi_{i}d \bar{\theta}_{i}d \theta_{i}\; e^{\bar{\theta}_{i}A_{i j}\theta_{j}+ \phi_{i}A_{i j}\phi_{j}}
q(\tilde{r}_K)\otimes q(\tilde{r}_K) =\psi^g(r_K)\otimes\psi^g(r_K)=\psi^g(\mu^*(r_K))=\mu^*\psi^g(r_K) =\mu^*(q(\tilde{r}_K))=q(\mu^*(\tilde{r}_K))=q(\tilde{r}_K\otimes 1 + 1\otimes \tilde{r}_K + \tilde{r}_K\otimes\tilde{r}_K)
\frac{p_{n}\#}{(p_{n}-1)\#}=e^{-\sum_{k=1}^{n}ln(1-\frac{1}{p_{k}})}=\sum_{m=0}^{\infty}\frac{1}{m!}(\sum_{l=1}^{\infty}\sum_{k=1}^{n}\frac{1}{lp_{k}^{l}})^{m}
K_{c m}\left( \, x - x_{0}\ ; s \, \right) = \left( \,{\frac{\pi M_{0}}{i s}}\, \right)^{D / 2}\exp \left[ \, i \,{\frac{M_{0}}{2 s}}\, \left( \, x - x_{0}\, \right)^{2}\, \right] \ .
p_S(\textbf{y}_k| \boldsymbol{\psi}, \textbf{W}_k) ={\frac{1}{\pi^{3}\sigma_z^{6}}e^{- \frac{{\left\| \textbf{y}_k-\lvert \textbf{s}\rvert \beta \textbf{W}_k^\textbf{a}(\textbf{x}) \right\|}_2^2}{\sigma_z^2}}}.
\frac{1}{(2\pi i)^{\frac{n}{2}}}\int_{\delta^{0}}res\left(\frac{\omega_\beta}{f}\right) =\left(\sum_{j=0}^{d-2}n_{j,0}\zeta_d^{j(\beta_n+1)}\right)\frac{B\left(\frac{1}{2},\dots,\frac{1}{2},\frac{\beta_n+1}{d}\right)B\left(A_{\beta'},A_{\beta'}\right)}{(2\pi i)^{\frac{n}{2}}}F\left(a,b,c;\frac{1}{\lambda}\right),
L_{\mathrm{e f f}}={\frac{f^{2}}{2 g_{Y M}^{2}}}\left[ 1 + \sum_{L , m = 1}^{\infty}c_{L m}\left({\frac{g_{Y M}^{2}f^{2}}{x^{7 - p}}}\right)^{L}\left({\frac{f^{2}}{x^{4}}}\right)^{m - L}\right]
\begin{pmatrix}1 -1 1 1\end{pmatrix}\underset{\mbox{Wronskian}}{\underbrace{\begin{pmatrix}h\left(0\right) h\left(1\right) h'\left(0\right) h'\left(1\right)\end{pmatrix}}}=\begin{pmatrix}0 h\left(1\right)-h'\left(1\right) h\left(0\right)-h'\left(0\right) 0\end{pmatrix}
\Vert \varphi^k\Vert^2 = |\varphi (0)|^{2 k}+ \int_{\mathbb{D}}\omega (z) \, k^2 \, |\varphi (z)|^{2 (k - 1)}|\varphi ' (z)|^2 \, dA(z) & \leq \rho^{2k}(1+ k^2 \rho^{- 2}) \, \Vert \varphi \Vert^2.
-\int_{\Omega}(x\cdot\nabla u)\Delta udx=\int_{\Omega}(x\cdot\nabla u)(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy)|u|^{2_{\mu}^{\ast}-1}dx+\lambda \int_{\Omega}(x\cdot\nabla u)udx.
\alpha=(\frac{e^{2}}{4\pi\epsilon_{0}d})/(\frac{hc}{\lambda})=\frac{e^{2}}{4\pi\epsilon_{0}d}\times\frac{2\pi d}{hc}=\frac{e^{2}}{4\pi\epsilon_{0}d}\times\frac{d}{\hbar c}=\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}.
0 = 4 ( 1 + q ) \left( \frac{d \phi}{d A}\right)^{2}- 2 \frac{P^{\prime}( A )}{P ( A )}\frac{d \phi}{d A}- \frac{f_{0}^{2}}{2 P^{2}( A )}+ \frac{2 Q + \mu e^{- 4 ( 1 + q ) \phi}/ Q}{4 P ( A )}
\varphi_{\gamma,n,b}\left(x\right)=\dfrac{1}{k_{\gamma,n}}\dfrac{\sqrt{V_{\gamma-1}\left(np^{-\gamma}+bp^{-\gamma}\right)}}{V_{\gamma-1}\left(np^{-\gamma}\right)}\left(f_{\gamma,n,0}\left(x\right)+\dfrac{k_{\gamma,n}V_{\gamma-1}\left(np^{-\gamma}\right)}{V_{\gamma-1}\left(np^{-\gamma}+bp^{-\gamma}\right)}f_{\gamma,n,b}\left(x\right)\right),
P(x,y)-\sqrt{-3}Q(x,y) =(x-\sqrt{-3}y)^3(L_1(x,y)+\sqrt{-3}L_2(x,y)) &=(x^3-9xy^2)L_1(x,y)-9(y^3-x^2y)L_2(x,y) &+((x^3-9xy^2)L_2(x,y)+3(y^3-x^2y)L_1(x,y))\sqrt{-3}
\begin{aligned}X_1(t,x,v)-X_2(t,x,v)=\int_{0}^{t}\int_{0}^{s}E_1(\tau,X_1(\tau,x,v))-E_2(\tau,X_2(\tau,x,v)) & +V_1(\tau,x,v) \wedge B(\tau,X_1(\tau,x,v))-V_2(\tau,x,v) \wedge B(\tau,X_2(\tau,x,v))d\tau ds\end{aligned}
c_0^{q-2}\varphi (c_0) = \int_\Omega m + c_0^{p-2}\int_\Omega a + c_0^{q-2}\int_{\partial \Omega}b=0, & c_0^{q-1}\varphi' (c_0)= (2-q)\int_\Omega m + (p-q)c_0^{p-2}\int_\Omega a = 0.
\biggl \{\lambda_{\epsilon}^{u}\in \Sigma^{\epsilon}\, \Bigl \vert \, \exists \,\omega_i >0, \,\, i=1,2, \ldots, n, \,\, \min_{\lambda_{\epsilon}^{u}\in \Sigma^{\epsilon}}\bigl\langle \omega,\, \lambda_{\epsilon}^{u}\bigr\rangle = \bigl\langle \omega,\, \lambda_{\epsilon}^{u^{\ast}}\bigr\rangle \biggr\}
{\cal L}=\partial_{+}\sigma\partial_{-}\sigma+\partial_{+}\pi\partial_{-}\pi-\frac{1}{2}(\partial_{\bot}\sigma)^2-\frac{1}{2}(\partial_{\bot}\pi)^2-\frac{1}{2}\mu^2(\sigma^2+\pi^2)-\frac{\lambda}{4}(\sigma^2+\pi^2)^2,
\nonumber \left( C_{\gamma_{1}\gamma_{2}\gamma}^{N_{1}N_{2}N}w_{\gamma_{1}}w_{\gamma_{2}}w_{\gamma}\right)^{2}= \int d g \, D_{0 0}^{( N_{1})}D_{0 0}^{( N_{2})}\overline{{{D_{0 0}^{( N )}}}}.
A ( k_{j}, \; p_{l}) = ( k^{2})^{- i}\left[ ( k - p )^{2}\right]^{- j}, \; \; h ( k_{j}\! \cdot \! n , \; p_{l}\! \cdot \! n ) = ( k \cdot n )^{- l}, \; \; f ( k_{j}\! \cdot \! n^{*}, \; p_{l}\! \cdot \! n^{*}) = ( k \cdot n^{*})^{m},
\begin{aligned}\frac{\phi_{k+1}(x) - \phi_{k+1}(y)}{x-y}= \alpha_k\phi_k(x) + (\alpha_kx+\beta_k)\alpha_{k-1}b_k[\phi_k](y)\phi_{k-1}(x) &\!\!\!+ \sum_{j=0}^{k-2}\alpha_j\left[(\alpha_kx+\beta_k)b_{j+1}[\phi_k](y) + \sum_{s=j+2}^k\gamma_{k,s}b_{j+1}[\phi_{s-1}](y)\right]\phi_j(x).\end{aligned}
p^{\beta_n}(n,t)=p^{\beta_n}(n,0)-\lambda(I_t^{\beta_n}p^{\beta_n}(n,t)-I_t^{\beta_{n-1}}p^{\beta_{n-1}}(n-1,t)),\ \ 0<\beta_n\leq 1,\ \lambda>0,\n\geq 0,
\bigtriangleup\big(x_\mu^{\rm L}\big)=x_\mu^{\rm L}\otimes Z=1\otimes\big(x_\mu^{\rm L}+ia_\mu Z\big)\bigtriangleup\big(p_\mu^{\rm L}\big)=p_\mu^{\rm L}\otimes Z^{-1}+1\otimes p_\mu^{\rm L}
\begin{array}[c]{c}\frac{1}{2}\Delta_{b}\varphi_{0}^{2}=\left\vert \nabla_{b}\varphi_{0}\right\vert^{2}+\varphi_{0}\Delta_{b}\varphi_{0}=\left\vert \nabla_{b}\varphi_{0}\right\vert^{2}-\left\langle \nabla_{b}\varphi,\nabla_{b}\varphi_{0}^{2}\right\rangle -\frac{a\lambda_{1}}{f+a}\varphi_{0}^{2}.\end{array}
\phi(F(C)) = \ \phi[\lambda(F(C))] \geq \ \phi(-v) = \ \phi\big[D\big(\tau\lambda(F(A))+(1-\tau)\lambda(F(B))\big)\big] \geq \ \tau \phi[\lambda(F(A))] +(1-\tau) \phi[\lambda(F(B))] = \ \tau \phi[F(A)] +(1-\tau) \phi[F(B)].
\frac{\det^{\prime}( - \partial_{\tau}^{2}+ V^{\prime \prime})}{\det ( - \partial_{\tau}^{2}+ \omega_{+}^{2})}= \frac{e^{( \omega_{-}- \omega_{+}) ( T / 2 - \tau_{I})}}{2 \omega_{-}F_{+}F_{-}}\int_{- \infty}^{\infty}d \tau F^{2}( \tau ) .
H_{\rm{new}}\propto\sum_{m=\frac{1}{2}}^{\Lambda}b^{\dag}_m b_m\sum_{l=1}^{\Lambda +\frac{1}{2}}\sum_{n=\frac{1}{2}}^{\Lambda}R^{lmn}(\zeta) C_1^{lmn}(\zeta) \delta^n_{m+l}\;.
\begin{array}{r l}J_1= \int_{\partial B_0}\vert x-y\vert^{1-n}\vert\mathcal{L}_\xi V_\alpha(x)\vert\,d\sigma, J_2= \int_{\partial B_0}\vert x-y\vert^{1-n}\vert\mathcal{L}_\xi Z_\alpha(x)\vert\,d\sigma.\end{array}
\vec{D}=\left( \begin{array}{c}D_{1}D_{2}\vdots D_{r}\end{array}\right), \vec{F}=\left( \begin{array}{c}F_{1}F_{2}\vdots F_{s}\end{array}\right),\vec{c}=\left( \begin{array}{c}c_{1}c_{2}\vdots c_{r}\end{array}\right),\vec{Z}=\left( \begin{array}{c}Z_{1}Z_{2}\vdots Z_{r}\end{array}\right).
\exp(x_1X_1+y_1Y_1+x_2X_2+y_2Y_2+zZ)\cdot\exp(x_1'X_1+y_1'Y_1+x_2'X_2+y_2'Y_2+z'Z)= &\exp((x_1+x_1')X_1+(y_1+y_1')Y_1+(x_2+x_2')X_2+(y_2+y_2')Y_2 &+\left(z+z'+\tfrac{1}{2}(x_1y_1'-y_1x_1'+x_2y_2'-y_2x_2')\right)Z)
\begin{array}{lcl}H^{(t)}_{+}:= \inf\{s \ge 0 \colon\; X^{(\hat{Z},t)}_s -\hat{Z}= \ell_L \}, H^{(t)}_{-}:= \inf\{s \ge 0 \colon\; X^{(\hat{Z},t)}_s -\hat{Z}= - \ell_L +1 \}\end{array}
\begin{array}{c c l}{{( 0 ~ 0 ~ - \frac{7}{2})}}&{{\rightarrow}}&{{( 0 ~ - \frac{7}{2}) ( + \frac{7}{2}) \oplus ( 0 ~ - \frac{5}{2}) ( + \frac{3}{2}) ,}}\\{{( 0 ~ 0 ~ + \frac{3}{2})}}&{{\rightarrow}}&{{( 0 ~ + \frac{5}{2}) ( - \frac{7}{2}) \oplus ( 0 ~ + \frac{3}{2}) ( - \frac{3}{2}) .}}\\ \end{array}
c_*(k):=\inf_{\xi\in Q_\ell^*}d_k^+(\xi)=\inf_{\xi\in Q_\ell^*}\inf_{\substack{V\,\textrm{subspace of}\,\Lambda^+_\xi H_\xi^{1/2}\dim V=k}}\sup_{u_\xi\in V\setminus \{0\}}\frac{\left(|D_{\xi}|u_\xi,u_\xi\right)}{\|u_\xi\|_{L^2_\xi}^2}.
\int_0^{\infty}s^{z-1}e^{-s^2x^2}\,ds = x^{-z}\int_0^\infty t^{z-1}e^{-t^2}\,dt = x^{-z}\int_0^\infty u^{\frac{z-1}{2}}e^{-u}\frac{u^{-\frac{1}{2}}}{2}\,du = \frac{x^{-z}}{2}\Gamma\left(\frac{z}{2}\right).
f_\rho - [n/n]_{f_\rho}= \frac{R_n}{Q_n}= \frac1{w_\Delta}\frac{\Psi_n^*}{\Psi_n}\frac{1+\upsilon_{n1}+\upsilon_{n2}(\gamma_n^*/\gamma_n)\big(\Psi_{n-1}^*/\Psi_n^*\big)}{1+\upsilon_{n1}+\upsilon_{n2}(\gamma_n^*/\gamma_n)\big(\Psi_{n-1}/\Psi_n\big)}.
j(u)(z)) =\mathbb{E}[\int_0^T(\int_D f(t, x,y(t,x,z),u(t,x,z),z)\mathbb{E}[\delta_Z(z)|\mathcal{F}_t]dx)dt +\int_Dg(x,y(T,x,z),z)\mathbb{E}[\delta_Z(z)|\mathcal{F}_T]]dx.
\begin{aligned}\varepsilon^2(\partial_tW_\varepsilon-\Delta_HW_\varepsilon -\varepsilon^{\alpha-2}\partial_z^2W_\varepsilon+U_\varepsilon\cdot\nabla W_\varepsilon+U_\varepsilon\cdot\nabla w+u\cdot\nabla W_\varepsilon) &+\partial_zP_\varepsilon=-\varepsilon^2(\partial_tw-\Delta_Hw-\varepsilon^{\alpha-2}\partial_z^2w+u\cdot\nabla w),\end{aligned}
h_{n}= - \frac{1}{2}\frac{1}{N^{2}}\frac{\partial^{2}}{\partial \lambda_{n}^{2}}+ \frac{1}{2}\{g^{2}\lambda_{n}^{6}- 2 g \lambda_{n}^{4}+ ( 1 + 2 g ) \lambda_{n}^{2}- 1 \}
\mathrm{e}^{- S_{\mathrm{e f f}}[ A_{0}]}\equiv \sum_{B_{M}}\int [ d \vec{a}] < \vec{a}\vert \mathrm{e}^{- H / T}\mathrm{e}^{i \frac{\kappa}{T}\int A_{0}( B_{M}+ B / 2 )}\vert \vec{a}+ \vec{\nabla}A_{0}/ T >
\begin{aligned}f(x) \,\rightarrow\,\min g_i(x) \,\leq\,0 \qquad i\in\mathcal M h_j(x) \,=\,0 \qquad j\in\mathcal P G_l(x)\,\leq 0\,\lor\,H_l(x) \,\leq\,0 \qquad l\in\mathcal Q. \end{aligned}
\partial_y^k\Phi_b=C_p\tilde{b}b_c\partial_y^kQ_b+O\big{(}|\tilde{b}|^2\partial_y^kQ_b+b_c^{2}\mathbf{1}_{[-2,-1]}(b_cy)+e^{-\frac{1}{10b_c}}\mathbf{1}_{[1,2]}(b_cy)\big{)},
\widehat{Q}_B\widehat{\Phi}(0)=\oint\frac{dz}{2\pi i}j_B(z)\cdot t\xi ce^{-\phi}(0)\otimes\sigma_3\sigma_1=-it\left(\frac{1}{2}\xi c\partial ce^{-\phi}+\eta e^{\phi}\right)(0)\otimes\sigma_2,
l_{s}^{- 2}d s^{2}={\frac{U^{2}}{\lambda^{1 / 2}}}\eta_{i j}d x^{i}d x^{j}+{\frac{\lambda^{1 / 2}}{U^{2}}}d U^{2}+ \lambda^{1 / 2}d \Omega_{5}^{2}.
Q(M,P_i,P_j,\tau)= \left\{e^{\frac{1}{24}E_2(\tau)A}\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}\left[\bigotimes_{n=1}^{\infty}S_{q^n}(\widetilde{T_CM})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^m}(\widetilde{L_C})\right]\right. &\left.\cdot\varphi(\tau)^{16}{\rm ch}(\mathcal{V}_i){\rm ch}(\mathcal{V}_j)\right\}^{(14)}.
L_{bulk}f(\overrightarrow{\eta})=\sum_i \eta_i (m+\eta_{i+1}) \left( f(\overrightarrow{\eta}^{i,i+1}) -f(\overrightarrow{\eta}) \right)+\sum_i \eta_{i+1}(m+\eta_{i}) \left( f(\overrightarrow{\eta}^{i+1,i}) -f(\overrightarrow{\eta}) \right).
\Phi_1(u,v) = \frac{1}{T}\{t\in S_1(T) : \log|\zeta(\sigma+it)|\leq u \arg\zeta(\sigma+it)\leq v\}\tilde{\Phi}_1(u,v) = \mathbb{P}\big(F_1 \log |\zeta(\sigma, X)|\leq u \arg\zeta(\sigma, X)\leq v\big).
{\frac{3}{b^{2}}}{\frac{a^{\prime}}{a}}\left({\frac{a^{\prime}}{a}}+{\frac{n^{\prime}}{n}}\right) -{\frac{3}{c^{2}n^{2}}}\left[{\frac{\dot{a}}{a}}\left({\frac{\dot{a}}{a}}-{\frac{\dot{n}}{n}}\right) +{\frac{\ddot{a}}{a}}\right] -{\frac{3 k}{a^{2}}}= -{\frac{8 \pi G_{5}}{c^{4}}}\Lambda ,
e_{\Sigma_3}(B_r) = \, \left(\frac{(q^2-1)^r}2+\frac16 (q-1)^{2r}+ \frac13 (q^2+q+1)^r - (q-1)^r \right) T + \left(- \frac{(q^2-1)^r}2+\frac16 (q-1)^{2r}+ \frac13 (q^2+q+1)^r\right) S +\left( \frac13 (q-1)^{2r}- \frac13 (q^2+q+1)^r -(q-1)^r+1\right) V ,
u \mbox{is a global weak solution of \eqref{eq1}in}L^\infty(L^p) ; begin{align*}4pt] \exists !\,u_0\in L^p({\mathbb{R}}^n) \mbox{such that}u(t,\cdot)=\Gamma(t,0)u_0\mbox{in}L^p({\mathbb{R}}^n)\mbox{for all}t>0.
\begin{aligned}L_{11}(q) = \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6n^2 + 2n - 4j^2 -j}+ q^{6n^2 + 10n + 4 - 4j^2 - j}& + \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{6n^2 + 16n + 10 - 4j^2 - 3j}+ q^{6n^2 + 8n + 2 - 4j^2 - 3j}\end{aligned}
\psi_+^r(x)=\sum_{\sigma=\pm 1/2}\int_{k^+>0}{dk^+ d^2k_\perp \over 16\pi^3 k^+}\Bigl[b^r(k,\sigma) u_+(k,\sigma) e^{-ik\cdot x}+d^{r\dagger}(k,\sigma) v_+(k,\sigma)e^{ik\cdot x}\Bigr] \;,
q(t) =\ \Phi_{22}(t)\Phi_{22}^{-1}(T) \int_0^T \Phi_{22}(T-s)(I-\Gamma)^TQ\int_0^se^{A(s-\tau)}B\nu_i (\tau) d\tau d s &-\int_0^t \Phi_{22}(t-s)(I-\Gamma)^TQ\int_0^se^{A(s-\tau)}B\nu_i (\tau) d\tau d s=: q_1(t)-q_2(t) .
V_{r}=\left\{f:(-1,1)\rightarrow\mathbb{C}\right. f,f^{\prime},\ldots,f^{(r-1)}\in AC_{\mathrm{loc}}(-1,1); & \left. f^{(j)}\in L^{2}((-1,1);(1-x)^{j-2}(1+x)^{\beta+j}),\text{}j=0,1,\ldots,r\right\},
Z_0(k_s;l_{1s},l_{2s})=\frac{(-1)^{2l_{1s}+2l_{2s}-1}(1-k_s)^2}{4^{2l_{1s}}k_s^{3/2}[1+(-1)^{2l_{2s}}\sqrt k_sk_s^{l_{1s}}]^{2-2l_{1s}}}.
\int \! \delta \xi \, d s T r \left[{\bar{\eta}}_{\mu}( g^{\mu \nu}\delta_{\rho}\delta^{\rho}- \delta^{\mu}\delta^{\nu}) \eta_{\nu}\right] = - \int \! \delta \xi \, d s T r \left[ ( \delta_{\nu}{\bar{\eta}}_{\mu}- \delta_{\mu}{\bar{\eta}}_{\nu}) ( \delta^{\nu}\eta^{\mu}- \delta^{\mu}\eta^{\nu}) \right] ,
\cos \angle \mathbf u_i\mathbf z'\mathbf u_j=\frac{\|\mathbf u_i-\mathbf z'\|^2+\|\mathbf u_j-\mathbf z'\|^2-\|\mathbf u_i-\mathbf u_j\|^2}{2\|\mathbf u_i-\mathbf z'\|\|\mathbf u_j-\mathbf z'\|}=\frac{2(r')^2-1}{2(r')^2}\leq 0
\int_0^{+\infty}(\mathcal{S}_L(\tau ) L f, \mathcal{S}_L(\tau) f )_{L^2_v}d\tau = -\int_0^{+\infty}\frac d{d\tau}\Vert \mathcal{S}_L(\tau) f \Vert_{L^2}^2 d\tau = \Vert f \Vert_{L^2}^2-\lim_{\tau \to \infty}\Vert \mathcal{S}_L(\tau) f \Vert_{L^2}^2 = \Vert f \Vert_{L^2}^2,
\begin{array}{lcl}a_1\mu_1 = -\alpha \log_1(\pi_2) + \gamma \log_1(u(K)), k\lambda_1+\mu_1 a_1 = -\beta , a_2\mu_2 = \alpha \log_1(\pi_2) - \gamma \log_1(u(K)), k\lambda_2 + \mu_2 a_2 = -\beta.\end{array}
I({K'}_B^C;W_1Y_2^{n_1}S^n\Theta_2U_C|1_B^C) \stackrel{}{=}I({K'}_B^C;Y_2^{n_1}S^{n_1}|1_B^C) &\leq I({K'}_B^C1_B^C;Y_2^{n_1}S^{n_1}) &\leq I({K}_B;Y_2^{n_1}S^{n_1}).
\Gamma_\rho(t) = \Bigg\{\begin{bmatrix}x w(t,x) \cos \varphi w(t,x) \sin \varphi\end{bmatrix}\, \Bigg| \, x \in [\tilde{x_0}(t),\tilde{x_5}(t)], \varphi \in [0,2\pi] \Bigg\}
\partial_t \mathsf{G}_{t_1-\tau,x-z}Q^T \mathsf{G}^T_{\tau-t_2, z-y}=\frac{Q^T\mathsf{G}^T_{t_1 - t_2,x-y}- \mathsf{G}_{t_1 - t_2,x-y}Q^T}{2}\,.
e^{\pm,1}_k=\int_0^1(1-\tau)L{''}_{(U^{a,\pm}+V^{\pm}_k+\tau\delta V^{\pm}_k;\Psi^{a,\pm}+\Phi^{\pm}_k+\tau\delta \Phi^{\pm}_k)}((\delta V^{\pm}_k,\delta \Phi^{\pm}_k),(\delta V^{\pm}_k,\delta \Phi^{\pm}_k))d\tau
\frac{1}{\lambda_4}= \lambda_1\lambda_2\lambda_3 = \sqrt{\frac{s_{+}(a)s_{+}(c)}{s_{+}(b)}}\sqrt{\frac{s_{+}(a)s_{+}(b)}{s_{+}(c)}}\sqrt{\frac{s_{+}(b)s_{+}(c)}{s_{+}(a)}}= \sqrt{s_+(a)s_+(b)s_+(c)}.
\sum_{j=1}^N \int_{I_j}v_x^2 dx \leq \frac{k(k+1)^2(k+2)}{h^2}\sum_{j=1}^N \int_{I_j}v^2 dx, & \sum_{j=1}^{N-1}[v]^2_{j+1/2}\leq \frac{4(k+1)^2}{h}\sum_{j=1}^N \int_{I_j}v^2 dx, & \sum_{j=1}^{N-1}\{v_x\}^2_{j+1/2}\leq \frac{k^3(k+1)^2(k+2)}{h^2}\sum_{j=1}^N \int_{I_j}v^2 dx.
v_{ea}^2(f_{avg}^*,g) =\lim_{\beta\uparrow 1}(1-\beta) v_\beta^2(f_\beta^*,g) &\leq \lim_{\beta\uparrow 1}(1-\beta)v_\beta^2(f_\beta^*,g_\beta^*) &=v_{ea}^2(f^*_{avg},g^*_{avg}).
\sum\limits_n \big|\langle d_n,Ae_n \rangle\big| \leq \sum\limits_n \int_M \big|\langle d_n,f(z)\rangle\langle g(z), e_n\rangle\big| \, dz &=\int_M \sum\limits_n\big|\langle d_n,f(z)\rangle\langle g(z), e_n\rangle\big| \, dz.
\sum_{i=k}^K\lambda^{k-1}_i\sigma_i =\sum_{i=1}^m\lambda^{k-1}_i\sigma_i &=\left(\sum_{i=1}^m\lambda^{k-1}_iP^i-Q^K,Q^0-Q^K\right) &=(Q^{k-1}-Q^K,Q^0-Q^K) &=(Q^{k-1}-Q^K,Q^{k-1}-Q^K) &>0,
\textrm{minimize}\quad \sum_{i=1}^T b_i \textrm{s.t.}\quad \alpha \sum_{i\in V_{t-1}\setminus U}b_i+ \sum_{i\in U\cap V_{t-1}}b_i + (T-|U\cap V_{t-1}|)b_{t}\ge 1, &b_t\ge 0,
\hat\Phi\left(\begin{pmatrix}a b\end{pmatrix},s\right) := \begin{pmatrix}\langle z_1^*, \Phi(a u_\ell + b w_\ell, p(s))\rangle \langle z_2^*, \Phi(a u_\ell + b w_\ell, p(s))\rangle\end{pmatrix}.
K_{\theta}( x_{1}, x_{2}, x_{3}) = \frac{1}{\pi^{2}\theta^{2}}\, \exp \left[ - \frac{2 i}{\theta}\epsilon_{\alpha \beta}\Bigl ( x_{1}^{\alpha}x_{2}^{\beta}+ x_{2}^{\alpha}x_{3}^{\beta}+ x_{3}^{\alpha}x_{1}^{\beta}\Bigr ) \right]
\bar\pi_{n,g,h}= \sum_{j=0}^{g}\sum_{\substack{k_{2}',\dots,k_{h}',k_{2}'',\ldots, k_{n-h}'' \sum_{i}ik_{i}' = j, \ \sum_{i}ik_{i}'' = g-j}}\!\!\!\!({e_{2}'}^{k_{2}'}\cdots{e_{h}'}^{k_{h}'}\otimes{e_{2}''}^{k_{2}''}\cdots{e_{n-h}''}^{k_{n-h}''})U_{k_{2}'\ldots,k_{h}'}U_{k_{2}'',\ldots,k_{n-h}''}
\frac{\partial\vec{V}}{\partial x^{i}}=\frac{\partial}{\partial x^{i}}(v^{j}\frac{\partial\vec{\Psi}}{\partial x^{j}})=\frac{\partial v^{j}}{\partial x^{i}}\frac{\partial\vec{\Psi}}{\partial x^{j}}+v^{j}\frac{\partial^{2}\vec{\Psi}}{\partial x^{i}\partial x^{j}}.
A = \langle (1,0,0,0)\rangle \ \ \B = \langle (0,1,0,0)\rangle C = \langle (0,0,1,0)\rangle \ \ \D = \langle (0,0,0,1)\rangle W = \langle (0,1,1,1)\rangle \ \ \X = \langle (1,0,1,1)\rangle Y = \langle (1,1,0,1)\rangle \ \ \Z = \langle (1,1,1,0)\rangle.
| \varphi |_\alpha := \sup_{x,\tilde{x}\in (0,1)}\frac{| \varphi (x) - \varphi ( \tilde{x}) |}{|x - \tilde{x}|^\alpha}\mbox{and}| \varphi |_{\mathrm{pw}, \alpha}:= \max_{i \in \{0 , \ldots , \ell -1\}}\sup_{x,\tilde{x}\in I_i^\circ}\frac{| \varphi (x) - \varphi ( \tilde{x}) |}{|x - \tilde{x}|^\alpha}.
h(t,\mathbf{\Phi},\mathbf{\Theta})=\sum\limits_{n=1}^N \sum\limits_{k=1}^{M_n}a_{nk}\delta(t-\tau_{nk})\delta(\mathbf{\Phi}-\mathbf{\Phi}_{nk})\delta(\mathbf{\Theta}-\mathbf{\Theta}_{nk}).
\alpha_k = \alpha_{k,K^{*}}\leq \frac{1}{{\binom{k-1}{l-1}}}\sum_{\{L_i \subseteq K^{*},\; |L_i|=l\}}\alpha_{l,L_i}\leq \frac{1}{{\binom{k-1}{l-1}}}\sum_{i=1}^{\binom{k}{l}}\alpha_{l,L_i}.
K_R(z|c,d)= \left(\begin{array}{ccc}1 0 0 0 1 0 0 0 1\end{array}\right)+\frac{1-z^2}{(cz-1)(dz-1)}\left(\begin{array}{ccc}cd 0 1 0 0 0 -cd 0 -1 \end{array}\right).
\frac{h}{N}\Bigg(|F_0^l|+\sum_{n=2}^{\infty}|T_0^{l,(n)}|\Bigg)\le M^{(a_1+a_2+a_4)l}\alpha^{-3}, M^{-(a_1+a_2+a_4)l+ra_3l}\sum_{m=2}^N c_0^{\frac{m}{2}}M^{\frac{a_1l}{2}m}\alpha^m \Bigg(\|F_m^l\|_{l,r}+\sum_{n=2}^{\infty}\|T_m^{l,(n)}\|_{l,r}\Bigg)\le 1, &\ (\forall r\in \{0,1\}).
\int\limits_0^\infty{{x^{- \frac{s}{2}- 1}}\left[{\bar R\left({- x}\right) - \bar R\left( x \right)}\right]}dx = 2{\left({- 1}\right)^{\frac{{10 - s}}{4}}}\frac{{\Gamma \left({1 - \frac{1}{2}s}\right)}}{{\zeta \left( s \right)}}\sin \frac{{\pi s}}{4}
\begin{cases}\Delta_{X,z}^{\mu}\phi_0 = 0 \ \ \mathrm{in}\ \ \mathcal{S}_b, \phi_0|_{z=0}= \psi, \ \ \big[\partial_z\phi_0 - \mu\beta \nabla_X b \cdot \nabla_X \phi_0 \big]\big|_{z=-1+\beta b}= \mu \beta \nabla_X \cdot \mathcal{L}_{1}^{\mu}[\beta b]\nabla_X\psi, \end{cases}
({\cal B}_V)^{\it{approx}}_{cl}= \sum_{{\vec n},{\vec m}}\sum_{s,s'}\left[([a^{s}_{\vec n}]^*_{cl}\bar{u}^s_{\vec n}+[b^{s\dagger}_{-{\vec n}}]^*_{cl}\bar{v}^s_{-{\vec n}})([a^{s'}_{\vec m}]_{cl}\gamma^0 u^{s'}_{\vec m}+[b^{s'\dagger}_{-{\vec m}}]_{cl}\gamma^0 v^{s'}_{-{\vec m}})\right]F({\vec n}-{\vec m},X).
\prod_{i=1}^{n}\left( \det (A_i ) \right)^\frac{1}{n}\leq \frac{1}{n^n}\ \left(\sum_{i=1}^n \left(\det(A_i)\right)^\frac{1}{n}\right)^n \leq \frac{1}{n^n}\ \left( \det \left( \sum_{i=1}^n A_i \right) \right).
|c_{n}(x)| \leq Cx^n\int_{0\leq \xi_1\leq \xi_2\leq \dots \leq \xi_{n+1}=x}\left [\prod_{j=1}^n\rho(\xi_j)\right]d\xi_1d\xi_2\dots d\xi_n & \leq C \frac{x^n}{n!}\left(\int_{[0,x]}\rho(\xi)d\xi\right)^n\stackrel{\emph{def}}{=}C \frac{x^n}{n!}(M(x))^n,
(\mathcal{Z}_1,\mathcal{Z}_2)\in{\rm gph}\,D\mathcal{N}_{K}\big(\mathcal{X}|\mathcal{Y}\big) \ \Longrightarrow\ \Pi'_{K}\big(\mathcal{X}+\mathcal{Y};\mathcal{Z}_1+\mathcal{Z}_2\big)=\mathcal{Z}_1.
F_T(x) =1+xF_T(x)+xC(x)\big(F_T(x)-1\big)+x^2\big(F_T(x)-1\big) &+\frac{x}{1-x}\Big(xC(x)\big(F_T(x)-1\big)-x\big(F_T(x)-1\big)\Big) &+\frac{x^2}{(1-x)^2}\Big(x^2C(x)\big(F_T(x)-1\big)+x^2\Big)+x\Big(-xF_T(x)-xC(x)\big(F_T(x)-1\big)+F_T(x)-1\Big)\,.
\lim_{n \to \infty}\frac{1}{\sigma^2(s)}n^{2h(s)-3}E|\zeta_{n}(s,t)|^2= \frac{\frac{1}{2}(c(s)+\overline{c(s)})}{(3-2h(s))(1-h(s))}t^{3-2h(s)}\leq \max\{t,t^2\}\left( \frac{1}{(1-h(s))^2}+\frac{1}{(1-h(s))(2h(s)-1)}\right).
(A) = \langle w, B_\lambda(0) v_0 + v_1 \rangle \leq \left\| \frac{w}{d_0(\cdot)}\right\|_2 \| d_0(\cdot) \{B_\lambda(0) v_0 + v_1 \}\|_2 \leq C_\lambda \| \nabla w \|_2 \| d_0(\cdot) \{B_\lambda(0) v_0 + v_1 \}\|_2 \leq \frac{1}{4}E(w(t)) + C_\lambda \| d_0(\cdot) \{B_\lambda(0) v_0 + v_1 \}\|_2^2.
\mathcal{S}:=\sum\limits_{p=0}^{n-1}\gamma_{p}^{(\nu ,n)}\left(2\sum\limits_{m=0}^{+\infty}\left( m+\nu +\frac{n}{2}\right)^{2p+1}e^{-\left( m+\frac{n}{2}+\nu \right)^{2}t}\right) .
q_0(h)-q_4(h)=\sum\limits_{r=0}^{\min(n_1,\Delta n)}\sum\limits_{s=0}^{n_1}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s}}\sum\limits_{\substack{\mathbf{d}\in [n_1+2,n_2]_<^r \mathbf{d'}\in [n_1+2,n_2]_<^{\Delta n}}}\det M_h=0
s_i(\sum_{\gamma\in Q^+}\dim (J^-_{z,-\gamma}\cap T_i(U_z^-))e(-\gamma)) = \sum_{\gamma\in Q^+}\dim (J^-_{z}\cap T_i(U_{z,-\gamma}^-))e(-\gamma) = \sum_{\gamma\in Q^+}\dim (J^-_{z,-\gamma}\cap T_i^{-1}(U_z^-))e(-\gamma),
\sum_{t=-\infty}^{\infty}q^{(t+a(\mu\pi))(\frac{1}{2}-s)}c_{t,l}(\mu) = \zeta_F(1) q^{-(l-a(\mu))(\frac{3}{2}-s)-\frac{a(\mu)}{2}}\epsilon(\frac{1}{2},\mu)^{-1}.
\big( f_1(\tau,\theta)\cdot y_{\gamma_{m1}}\big) \widehat{\ast}\big( f_2(\tau,\theta)\cdot y_{\gamma_{m2}}\big) = f_1(\tau,\theta-\langle \gamma_{m2}, - \rangle \tau/2)f_2(\tau,\theta + \langle \gamma_{m1}, - \rangle \tau/2) \cdot y_{\gamma_{m1}+\gamma_{m2}},
M= 4\pi \int_0^\infty dr~r^2\{\frac{F_\pi^2}{8}[ F'^2 +2\frac{sin^2F}{r^2}] +\frac{1}{2e^2}~\frac{sin^2F}{r^2}~[\frac{sin^2F}{r^2}+ 2F'^2]\}
Z_{\gamma_1,\gamma_2}^N(t) =A_N(\gamma_1,\gamma_2) \int_{\mathbb{R}^2}' \left[ \int_0^t (s,x_1)_N^{\gamma_1}(s,x_2)_N^{\gamma_2}B(dx_1)B(dx_2)ds\right] &=A_N(\gamma_1,\gamma_2) \int_0^t\left[ \int_{\mathbb{R}^2}' (s,x_1)_N^{\gamma_1}(s,x_2)_N^{\gamma_2}B(dx_1)B(dx_2)\right] ds,,
G_j^{\Psi}(X, \varphi' + \zeta ; \, u_j) \leq \prod_{k=1}^M \sup_{t\in [0,1]}g_{j+ \frac{k-1}{M}}(X_{k/M}, \xi_k + 1_{k=1}t u_j ) G_{j+1}(\bar{X}, \varphi')