harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{Q ( x , T , T^{*}) ={\left({\frac{B}{A}}e^{- ( \beta - \alpha ) x}\right)}^{2}\left( ( 1 + 2 \beta x )^{2}- \widetilde{E}- 1 \right) + ( 1 + 2 \alpha x )^{2}- \widetilde{E}- 1}
\frac{\partial \psi}{\partial t}= \left( \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) \, \frac{\partial}{\partial x}+ \left( \begin{matrix}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{matrix}\right) \right) \psi \ .
\lambda^{i}={\frac{2 ( D - 2 )}{\kappa^{D - 1}}}{\frac{1}{W}}{\frac{\partial W}{\partial \phi}}e^{-{\frac{1}{2}}A}\tilde{\eta}_{-}^{i}\ , \qquad \psi_{r}^{i}= -{\frac{2 ( D - 2 )}{\kappa^{D - 1}}}\left({\frac{1}{W}}{\frac{\partial W}{\partial \phi}}\right)^{2}e^{-{\frac{1}{2}}A}\tilde{\eta}_{-}^{i}\ , \qquad \psi_{\mu}^{i}= 0 \ .
G'(r_n) \ll r_n^{n-2}I_{n-1}(f)\left (r_0\sqrt{\frac{2\pi e}{n}}nr_n-\sqrt{\pi}\sqrt{\ln n}\right ) & = \sqrt{\pi}r_n^{n-2}I_{n-1}(f)\left ( r_0\sqrt{2e}-\sqrt{\ln n}\right ) & <0
j_{a}^{\mu}( x ) ={\frac{1}{( 2 \pi )^{4}}}\int d^{4}k e^{- i k \cdot x}j_{a}^{\mu}( k ) ={\frac{1}{( 2 \pi )^{4}}}\int d^{4}k e^{- i k \cdot x}\sum_{n = 1}^{\infty}j_{n , a}^{\mu}( k ) ,
S^{\mathrm{e f f}}= - i \ln \int \prod_{X^{M}}d B_{\bar{N}}d \sigma d \sigma^{\dagger}\delta \left( \sqrt{- g}\nabla_{\bar{N}}B^{\bar{N}}\right) \prod_{X_{/ \! /}}\delta ( \widetilde C^{M}) \exp \left[ i \int \sqrt{- g}{\cal L}_{2}d^{6}x \right] .
g_1(t,p) = \begin{cases}-\beta_1 t+\alpha \mathbb{E}\left(\left[\sum\limits_{j=1}^{N_p}X_j\!-\!T_1\!-\!t\right]^+\right) +\beta_2 \mathbb{E}\left(\left[\sum\limits_{j=1}^{N_p}X_j-T_1\right]^+\right), t<0, \beta_2 t+(\alpha+\beta_2)\mathbb{E}\left(\left[\sum\limits_{j=1}^{N_p}X_j-T_1-t\right]^+\right), t>0. \end{cases}
\begin{aligned}\epsilon^{-1}[J_1(u_1^\epsilon(\cdot),u_2(\cdot))- J_1(u_1(\cdot),u_2(\cdot))]=\epsilon^{-1}\mathbb{E}\bigg[\int_0^T\Big(Z^{u_1^\epsilon}(t)l_1^{u_1^\epsilon}(t)-Z(t)l_1(t)\Big)dt + \Big(Z^{u_1^\epsilon}(T)\Phi_1^{u_1^\epsilon}(x(T))-Z(T)\Phi_1(x(T))\Big)+\Big(\gamma_1^{u_1^\epsilon}(y(0))-\gamma_1(y(0))\Big)\bigg]\geq 0.\end{aligned}
\Lambda^m(1\otimes x) = \sum_{k=1}^p\Big(1\otimes \frac{2D_k^\theta}{(1-\Delta^\theta)^{\frac12}}\Big)\cdot \Lambda^{m-1}(1\otimes [D_k^\theta,x]) \quad-\sum_{k=1}^p\Big(1\otimes\frac1{(1-\Delta^\theta)^{\frac12}}\Big)\cdot \Lambda^{m-1}(1\otimes [D_k^\theta,[D_k^\theta,x]]).
24{\rm Cl}_2(\theta_{5,3})-12{\rm Cl}_2(2\theta_{5,3})-8{\rm Cl}_2(3\theta_{5,3})+6{\rm Cl}_2(4\theta_{5,3})=Z_{15}=15\sum_{k=1}^7\left(\frac{k}{15}\right){\rm Cl}_2\left(\frac{2\pi k}{15}\right)
\Sigma_3' =\sum_{\substack{q_2\sim Q_2 (q_2,10)=1}}\sum_{\substack{a_1< d_1d_2q_1q_2 (a_1,d_1d_2q_1q_2)=1}}\sup_{|\gamma|\le E/Y}F_{V}\Bigl(\frac{a_1}{d_1d_2q_1q_2}+D'\gamma\Bigr)^2, \Sigma_5' =\sum_{\substack{q_2\sim Q_2 (q_2,10)=1}}\sum_{\substack{a_2< d_1q_1q_2 (a_2,d_1q_1q_2)=1}}\sup_{|\gamma|\le E/Y}F_V\Bigl(\frac{a_2}{d_1q_1q_2}+D' V\gamma\Bigr)^2.
Y_{m,n}\left( 1, 1- \frac{\lambda}{m}, \frac{T}{m}\right) = \frac{1}{2}\sum_{\theta \in \{0,1\}^2}\prod_{w^n = (-1)^{\theta_2}}\prod_{j=0}^{m-1}\left| 1- \frac{\lambda}{m}+ \exp \left\{2 \pi i \frac{j + \theta_1 /2}{m}\right\}+ \frac{Tw}{m}\right|.
\cup_{l=0}^{r-1}(X+l(v_i-v_e)) \supseteq \cup_{l=0}^{r-1}(X_j+l(v_i-v_e)) &= \cup_{l=0}^{r-1}(X_0+j(v_i-v_e)+l(v_i-v_e)) &= \cup_{l=0}^{r-1}(X_0+(j+l)(v_i-v_e)) & = \cup_{t=0}^{r-1}(X_0+t(v_i-v_e)) &=\mathbb{F}_r^m.
H((V,W),x,t) = \left(\begin{array}{c}(1-v_0^2) V - (\frac{A(x)^2}{2}+\frac{B(x)^2}{2}+ A(x) B(x) \cos(\eta t))(V+v_0) -v_0 V^2 - \frac{V^3}{3}- W \varepsilon(V - \gamma W)\end{array}\right),
Q^{N}_{\omega}(B\times R\times D) \doteq \frac{1}{N}\sum_{i=1}^{N}\delta_{X^{N}_{i}(\cdot,\omega)}(B)\cdot \delta_{\rho^{N,i}_{\omega}}(R)\cdot \delta_{W^{N}_{i}(\cdot,\omega)}(D),\; \omega\in \Omega_{N},
\begin{array}{c c c c}R_1: \left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right) \to \left(x^{2,3},x^{4,5},-x^{6,7},-x^{8,9}\right) R_2: \left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right) \to \left(-x^{2,3},-x^{4,5},x^{6,7},x^{8,9}\right) R_3: \left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right) \to \left(-x^{2,3},x^{4,5},-x^{6,7},x^{8,9}\right) \end{array}
\begin{array}{l}Y_{2}R_{12}K_{1}R_{21}= R_{12}K_{1}R^{-1}_{12}Y_{2}+ q^2R_{12}{\cal P}\quad, \tilde{Y}_{2}R_{21}^{-1}K_{1}R_{21}= R_{12}K_{1}R_{21}\tilde{Y}_{2}+ q^{-2}R_{12}{\cal P}\quad.\end{array}
\int_{\textstyle\prod_{t=1}^m [b_{t,1}, b_{t,2}]}\ \frac{du_1du_2\ldots du_m}{\displaystyle \sum_{t=1}^m\left(u_t-c_t\right)^{2s}\left(u_t-d_t \right)^{2s}}&=\int_{\textstyle\prod_{t=1}^m\ [b_{t,1}-\eta_t,b_{t,2}-\eta_t]}\ \frac{dv_1dv_2\ldots dv_m}{\displaystyle \sum_{t=1}^m\left(v_t+\gamma_t\right)^{2s}\left(v_t-\gamma_t \right)^{2s}}
A_{\mu}^{( c l )}= \left( \begin{array}{c c c}{X_{\mu}^{( 1 )}}&{}&{}\\{}&{X_{\mu}^{( 2 )}}&{}\\{}&{}&{X_{\mu}^{( 3 )}}\\ \end{array}\right) = \sum_{a = 0}^{2}X_{\mu}^{a}\otimes U^{a}.
\begin{cases}\begin{aligned}\bar \lambda_1\lim_{m\rightarrow\infty}\langle X,e_1\rangle(p_m)+\bar \lambda \lim_{m\rightarrow\infty}\langle X,e_2 \rangle(p_m)=0, &\bar \lambda\lim_{m\rightarrow\infty}\langle X,e_1\rangle(p_m)+\bar \lambda_2 \lim_{m\rightarrow\infty}\langle X,e_2 \rangle(p_m)=0.\end{aligned}\end{cases}
\frac{\lambda_{2i+1}}{a}> 1-\frac{1}{4}\cdot \frac{\left \lfloor 4(MN-1)N\sin^2 \frac{\pi}{2(2N-1)}\right \rfloor+1}{4(MN-1)N\sin^2 \frac{\pi}{2(2N-1)}}& > 1-\frac{1}{4}\cdot\frac{4}{3}=\frac{2}{3}.
M_\lambda (0) = (\sqrt\lambda A + q I - (1/\sqrt\lambda) AQ)^{-1}+ P_{\lambda,1}\Psi_\lambda (q) M_\lambda (0) = (I - P_{\lambda,1}\Psi_\lambda (q))^{-1}(\sqrt\lambda A + q I - (1/\sqrt\lambda) A Q)^{-1}= (\frac{1}{\sqrt\lambda}(q A^{-1}-\Theta U(q)) + O(1/\lambda))^{-1}(\frac{1}{\sqrt\lambda}A^{-1}+O(1/\lambda))
x_{i_1}(1+x_{i_2})(1+x_{i_3}) = x_{i_1}+ x_{i_1}x_{i_2}+ x_{i_1}x_{i_3}+ x_{i_1}x_{i_2}x_{i_3}x_{i_2}(1+x_{i_1}) (1+x_{i_3}) = x_{i_2}+ x_{i_1}x_{i_2}+ x_{i_2}x_{i_3}+ x_{i_1}x_{i_2}x_{i_3}x_{i_1}x_{i_3}(1+x_{i_2}) = x_{i_1}x_{i_3}+ x_{i_1}x_{i_2}x_{i_3}x_{i_2}x_{i_3}(1+x_{i_3}) = x_{i_2}x_{i_3}+ x_{i_1}x_{i_2}x_{i_3}
\frac{1+\sum_{i=0}^hk_1(i)}{r+k_1(0)}a^{\widehat{P}^{Cont,+r}}+ \frac{1+\sum_{i=0}^hk_2(i)}{s+k_2(0)}b^{\widehat{P}^{Cont,+s}}&=1+\sum_{j=1}^h\left(\frac{k_1(j)}{k_1(0)+k_1(j)}a^{\widehat{P}^{Cont}_j}+ \frac{k_2(j)}{k_2(0)+k_2(j)}b^{\widehat{P}^{Cont}_j}\right).
z\frac{\partial}{\partial z}\frac{(q)_{N}}{(zq)_{N}(z^{-1}q)_{N}}=z\sum_{n=1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(-1)^{n+1}(q)_nq^{n(n+1)/2}}{(q)_{n+N}}\left(\frac{1-q^n}{(1-zq^n)^2}-\frac{1-q^n}{(z-q^n)^2}\right).
\tilde{E}:= \sup_{\tilde{x}\in B_{\frac{1}{4}},\, 0<\tilde{r}<\frac{1}{4}, \frac12<\tilde{t}\leq 1}\frac{\tilde{\mu}_{\tilde{t}}(B_{\tilde{r}}(\tilde{x}))}{\tilde{r}^{n-1}}\leq 2^{n-1}l^{1-n}E_1.
subject\,to\begin{array}{ccc}a_{1}\cos\omega_{1}T_{0}+b_{1}\sin\omega_{1}T_{0}= a_{2}\cos\omega_{2}T_{0}+b_{2}\sin\omega_{2}T_{0}, a_{1}= a_{2}\cos\omega_{2}T+b_{2}\sin\omega_{2}T.\end{array}
\widetilde{\mu}\left(\pi_{\mathrm{N}}^* ([X/u])\right) = \widetilde{\mu}\left([g_{u,1}]\oplus \pi_{\mathrm{N}}^* ([X/1])\right) & = \widetilde{\mu}\left([g_{u,1}]\right)+ \widetilde{\mu}\left([\pi]\right) & = \mu\left([X/u]\right) - \mu\left([X/1]\right) + \mu\left([X/1]\right) & = \mu\left([X/u]\right).
C_{\mu \alpha}=\overline{\Psi}(x)\Gamma_\mu \overline{\Gamma}_\alpha \Psi (x)\hspace{0.3in}\Theta_{\mu [\alpha \beta]}=\overline{\Psi}(x)\Gamma_\mu \overline{\Gamma}_{[\alpha}\overline{\Gamma}_{\beta ]}\Psi (x)
L_{4}^{F}= \frac{\kappa^{2}}{3 6}B_{\mu \nu \alpha \beta}^{F}B_{\mu_{1}\nu_{1}\alpha_{1}\beta_{1}}^{F}G^{\mu \mu_{1}; \nu_{1}\nu}K^{\alpha \alpha_{1}; \beta_{1}\beta}- \frac{\kappa^{2}}{1 2}C_{\mu \nu \rho ; \alpha \beta}^{F}C_{\alpha_{1}\beta_{1}}^{F \mu \nu \rho}K^{\alpha \alpha_{1}; \beta_{1}\beta}.
\nu_x \frac{q\nu_y\nu_z - q^{-1}\nu_z\nu_y}{q-q^{-1}}= \nu_x - \nu_x x^2 = \nu_x - q^{-2}\nu_y - q^2\nu_z + q^{-1}(1-x^2) +q\nu_y\nu_z = \nu_x - q^{-2}\nu_y - q^2\nu_z + q^{-1}\frac{q\nu_y\nu_z - q^{-1}\nu_z\nu_y}{q-q^{-1}}+ q\nu_y\nu_z = \nu_x - q^{-2}\nu_y - q^2\nu_z + \frac{q^2\nu_y\nu_z - q^{-2}\nu_z\nu_y}{q-q^{-1}}.
\gamma(0,h) =\{\left\vert t\right\vert =h,\operatorname{Im}t\geq0\},\text{}\gamma(\pi,h)=\{\left\vert t-\pi\right\vert =h,\operatorname{Im}t\geq0\}, \gamma(t_{j},\varepsilon) =\{\left\vert t-t_{j}\right\vert =\varepsilon,\operatorname{Im}t\geq0\}
\begin{array}{l}{{[{H}_{\pi_{\alpha}}] : ={\dot{u}}^{\alpha}( x ) - \frac{\partial{H}}{\partial \pi_{\alpha}},}}\\{{[{H}_{u^{\alpha}}] : ={\dot{\pi}}_{\alpha}( x ) + \frac{\partial{H}}{\partial u^{\alpha}}- \nabla_{a}( \frac{\partial{H}}{\partial ( \nabla_{a}u^{\alpha})}) .}}\\ \end{array}
\frac{\partial \mathcal{S}}{\partial T}=\frac{2l\pi^2\Xi^2r_{+}^5(\Xi^2+1)(r_{+}^4+q^2l^4)}{3[4r_{+}^8(\Xi^2-1)+\Xi^2(q^2l^4-r_{+}^4)^2+4q^2r_{+}^4l^4]}.
|\frac{\partial^{k_{1}+\cdot\cdot\cdot+k_{n}}f(\zeta_{1},\zeta_{2},...,\zeta_{n})}{{\partial z_{1}}^{k_{1}}\cdot\cdot\cdot\partial{z_{n}}^{k_{n}}}|\le\frac{Mk_{1}\cdot\cdot\cdot k_{n}!}{{r_{1}}^{k_{1}}\cdot\cdot\cdot{r_{n}}^{k_{n}}}
g(a_1,a_2,\dots,a_k) =\left(\max_{0\le j\le a_1-1}m_j\right)-a_1\,, n(a_1,a_2,\dots,a_k) =\frac{1}{a_1}\sum_{j=0}^{a_1-1}m_j-\frac{a_1-1}{2}\,, s(a_1,a_2,\dots,a_k) =\frac{1}{2 a_1}\sum_{j=0}^{a_1-1}(m_j)^2-\frac{1}{2}\sum_{j=0}^{a_1-1}m_j+\frac{a_1^2-1}{12}\,,
E = \frac{m_{p}( \dot{r}^{2}+ r^{2}\dot{\phi}^{2})}{\sqrt{1 - H_{0}( \dot{r}^{2}+ r^{2}\dot{\phi}^{2})}}+ m_{p}H_{0}^{- 1}[ \sqrt{1 - H_{0}( \dot{r}^{2}+ r^{2}\dot{\phi}^{2})}- 1 ] .
\dot{\eta}^{\bar{u}}(t)= - \left( \frac{\partial f}{\partial x}(x^{(\bar{u},x_{0})}(t),\bar{u}(t))\right)^T\eta^{\bar{u}}(t) + \sum_{i=1}^p \mu^{\bar{u}}_i(t)\frac{\partial g_i}{\partial x}^T(x^{(\bar{u},x_{0})}(t),\bar{u}(t))
\max_{a\in \mathcal{A}(s)}\left[ g(s, a) + \tau\sum_{\mathbf{H}'}\mathrm{Pr}(\mathbf{H}'|\mathbf{H})h^{(n)}(s')\right]= g(s, A_1^*, A_2^*) + \tau\sum_{\mathbf{H}'}\mathrm{Pr}(\mathbf{H}'|\mathbf{H})h^{(n)}(B_1', B_2', \mathbf{H}').
0\leq \mathbb{P}(\xi^{(n)}(I_{n,k})=\xi^{(n)}(J_{n,k,j})=0)=\mathbb{P}(\xi^{(n)}(I_{n,k}\cup J_{n,k,j})=0) \leq \mathbb{P}(\xi^{(n)}(J_{n,k,j}\cup \left(\cup_{i\neq j}I(z_i, F_n(x_0))\right))=0) \leq \mathbb{P}(\xi^{(n)}(J_{n,k,j})=0)\prod_{i\neq j}\mathbb{P}(\xi^{(n)}(I(z_i, F_n(x_0)))=0) = D_n((z_{j+1}-z_j)/2)(D_n(F_n(x_0)/2))^{k-1}.
\lim_{R \rightarrow \infty}R \| (1 - \chi(x)) P_{\leq 0}(u_{0}(Rx)) \|_{B_{1,2}^{1/2}}+ R^{2}\| (1 - \chi(x)) P_{\leq 0}(u_{1}(Rx)) \|_{B_{1,2}^{-1/2}}= 0.
G_{(\sigma)}(z,z')= \sum_\Gamma\frac{\exp\pi i[\Omega_\Gamma(\{\sigma_s\})+\sum_s2l_{1s}\sigma_s(J_{(o)s}(z)-J_{(o)s}(z'))]}{[z-g_\Gamma(z')][c_\Gamma z'+d_\Gamma]^3}
H(r) := \frac{1}{\Gamma(r)}\, E\left[S_N^{(r)}\right] = E\left[\frac{\Gamma(r + S_N)}{\Gamma(r) \, \Gamma(S_N)}\right]= \sum_{k = N}^{\infty}\frac{\Gamma(k + r)}{\Gamma(r) \, (k-1)!}\, P\{S_N = k\}.
L_{1}(w,b,\xi,\alpha,\beta)=\frac{1}{2}w^{T}w+c\sum_{i=1}^{N}\xi_{i}-\sum_{i=1}^{N}\alpha_{i}\{y_{i}[w^{T}\phi(x_{i})+b]-1+\xi_{i}\}-\sum_{i=1}^{N}\beta_{i}\xi_{i},
\frac{\sin^2 ((1-\phi)R^{u_\eta})}{\sin^2 R^{u_\eta}}= 1- 2\eta d \tan \frac{d}{2}+ O(\eta^2) \frac{\sin^2 ((1-\psi)R^{u_{1-\eta}})}{\sin^2 R^{u_{1-\eta}}}= 1- 2\eta d \tan \frac{d}{2}+ O(\eta^2).
f_{q}(1) =q_{0}+q_{1}i+q_{2}j+q_{3}k, f_{q}(i) =-q_{1}+q_{0}i-q_{3}j+q_{2}k, f_{q}(j) =q_{2}-q_{3}i+q_{0}j-q_{1}k, f_{q}(k) =q_{3}+q_{2}i+q_{1}j+q_{0}k.
\partial_t\tilde{\chi}~=~\alpha H^{-\alpha-1}\Delta\tilde{\chi}+\alpha H^{-\alpha-1}|A|^2\tilde{\chi}+(1-\alpha)H^{-\alpha}e^{-\frac t{2^{\alpha}}}\cosh u-\frac 1{2^{\alpha}}\tilde{\chi}.
V_2V_1^*(z^n\otimes \xi) =\overline q^{2n}z^n\otimes U^*PU^*P^\perp\xi+\overline q^{2n-2}z^{n-1}\otimes U^*PU^*P\xi &\quad+\overline q^{2n+1}z^{n+1}U^*P^\perp U^*P^\perp\xi+\overline q^{2n-1}z^n\otimes U^*P^\perp U^*P\xi
\partial_t v_i - \partial_\alpha \big\langle \nu, \frac{\partial G}{\partial\Xi^A}( \lambda_\Xi ) \, \frac{\partial\Phi^A}{\partial F_{i\alpha}}(\lambda_F) \big\rangle = 0 \intertext{and for $A=1,\dots , 19$}\partial_t \Phi^A (F) - \partial_\alpha \big ( \frac{\partial\Phi^A}{\partial F_{i\alpha}}(F) v_i \big ) = 0
2\lambda_n\beta_n\langle\nabla\Psi_1( x_n),u- x_{n+1}\rangle=2\lambda_n\beta_n\langle\nabla\Psi_1( x_n),u- x_n\rangle+2\lambda_n\beta_n\langle\nabla\Psi_1( x_n),x_n- x_{n+1}\rangle.
\begin{aligned}\sum_{m=-\infty}^{\infty}(q^{-\ell};q)_{m}2^{m}q^{\ell(|m|-m)/2}S_{\ell}\left(-q^{-|m|-\ell};q\right) =\left(-\frac{1}{2}\right)^{\ell}q^{\binom{\ell+1}{2}}\frac{\left(q^{\ell}/2;q\right)_{\infty}}{\left(q,q/2;q\right)_{\ell}}.\end{aligned}
\frac{1}{2}\sum_{r=3}^{R}d_{\sqrt{r^{2}-1}}(r^{2}-1) \leq \frac{1}{2}\sum_{r=3}^{R}d_{r}(r^{2}-1) <\frac{1}{2}\sum_{r=3}^{R}d_{R}(r^{2}-1).
A_{1}=\mathbb{E}^{Q}\exp\left\{-2\int_{0}^{t}\hat{b}(s,\,Z_{s}^{x,y})dW_{s}-\int_{0}^{t}\hat{b}^{2}(s,\,Z_{s}^{x,y})ds\right\}A_{2}=\mathbb{E}^{Q}\exp\left\{-2\int_{0}^{t}\hat{b}(s,\,Z_{s}^{x,y})\frac{y-Z_{s}^{x,y}}{t-s}ds\right\}
H_{3, \le 1}=\int_{\mathbb{T}}\left\{c_1(v_x^3+3\,v_x^2\,z_x)+c_2(v_x^2\,v+2\,v_x\,v\,z_x+v_x^2\,z)+c_3(v^3+3\,v^2\,z)\right\}\,dx, &H_{3, \geq 2}=\int_{\mathbb{T}}\{c_1(z_x^3+3\,z_x^2\,v_x)+c_2(z_x^2\,z+z_x^2\,v+2\,z_x\,z\,v_x)+c_3(z^3+3\,v^2\,z)\}\,dx, & H_{4, 0}=\int_{\mathbb{T}}\{c_4\,v_x^4+c_5\,v_x^3\,v+c_6\,v_x^2\,v^2+c_7\,v^4\}\,dx.
\int_0^T \int_{\Omega}\big(E^2(|\dot{u}|)|\dot{u}|^2 + |\dot{u}|^2 \big) \mathrm{d}x \mathrm{d}t = \int_{Q_1}\big(E^2(|\dot{u}|)|\dot{u}|^2 + |\dot{u}|^2\big) \mathrm{d}x \mathrm{d}t &+\int_{Q_2}\big(E^2(|\dot{u}|)|\dot{u}|^2 + |\dot{u}|^2\big) \mathrm{d}x \mathrm{d}t.
E(a^{(i)}) \geq \frac12\sum_{j=1}^k (a^{(i)}_j)^2- 2\log(x^{(i)}_{j_1}) -2 \log(x^{(i)}_{j_3}+ x^{(i)}_{j_3+1})-\ldots & -2\log(x^{(i)}_{1}+ \ldots+ x^{(i)}_{k-1})
\int\left(\frac{1}{2}|v|^2+\hat{e}-\frac{1}{2}|\bar{v}|^2-\bar{\hat{e}}\right)(x,0)\varphi(0)\:dx + \int_0^T\int \left(\frac{1}{2}|v|^2+\hat{e}- \frac{1}{2}|\bar{v}|^2-\bar{\hat{e}}\right)\varphi^{\prime}(t)\:dx\:dt =-\int_0^T\int (r-\bar{r})\varphi(t)\:dx\:dt.
\uppercase\expandafter{\romannumeral2}\leq \exp\left({n\tau(\int_{\Omega}\log X dP - \frac{\delta}{2})}\right)E\left(\exp(-\tau\sum_{k=1}^n\log X_k)\right) = \exp\left({n\tau(\int_{\Omega}\log X dP - \frac{\delta}{2})}\right)E\left(X_1^{-\tau}X_2^{-\tau}\cdots X_n^{-\tau}\right).
\widetilde{c}_{jl}\equiv \cos\widetilde{\tau}_{jl}= \frac{(n_j \cdot n_l)}{\sqrt{n_j^2 n_l^2}}= \frac{(-1)^{j+l}\; D_{jl}^{(N-1)}}{\sqrt{D_{jj}^{(N-1)}\; D_{ll}^{(N-1)}}}.
\| f(M)-f(M')\|_F^2 = \sum_{i,k=1}^d ( f( \mu_i) - f( \mu'_k))^2 \langle p_i, q_k\rangle^2 & \leq \frac{1}{L^2}\sum_{i,k=1}^d ( \mu_i - \mu'_k)^2 \langle p_i, q_k\rangle^2 = \frac{1}{L^2}\| M-M'\|_F^2.
|\mu|\leq c \sqrt{\frac{n+1}{m+1}}\frac{1}{\Lambda_{m,n}|\eta|}\frac{1}{2\sqrt{n+1}(\sqrt{n+1}+\sqrt{m+1})\sqrt{\eta^2+2m+2}}= c \frac{1}{|\eta|(\sqrt{n+1}+\sqrt{m+1})^2},
\nu^{\pm}_c(Y_1,Y_2,Y_3,Y_4)=\nu^{\pm}_c(Y_1,Y_2,Y_3)+\nu^{\pm}_c(Y_1,Y_3,Y_4)=\nu^{\pm}_c(Y_1,Y_2,Y_4)+\nu^{\pm}_c(Y_2,Y_3,Y_4),
{}_2F_1\bigg[\begin{matrix}\alpha \beta &\gamma\end{matrix}\bigg|\,1\bigg]_{c}-{}_2F_1\bigg[\begin{matrix}-2 \beta &\gamma\end{matrix}\bigg|\,1\bigg]_{c}=\Psi(sp)-\Psi(0)\equiv\Psi'(0)\cdot sp\equiv\psi'(0)\cdot sp\pmod{p^2}.
\sqrt{\alpha}\left(\sum_{m,n=1}^{\infty}(-1)^{m}\,K_{0}\left(\left(2n-1\right)m\,\alpha\right)-\frac{\log(2)}{4}\right)=\sqrt{\beta}\left(\sum_{m,n=1}^{\infty}(-1)^{m}K_{0}\left(\left(2n-1\right)m\,\beta\right)-\frac{\log(2)}{4}\right).
\begin{array}{ccc}K^{-1}_{ab}= p \left(\begin{array}{cc}\sqrt{g}{1 \over \sqrt{g}}{1 \over \sqrt{g}}\sqrt{g}\end{array}\right) + (1-p)\left(\begin{array}{cc}{1 \over \sqrt{g}}\sqrt{g}\sqrt{g}{1 \over \sqrt{g}}\end{array}\right)\end{array}
{\rm exp}\left( \tbinom{n_1,\ldots,n_r}{z_1,\ldots,z_r}\right) =\sum_{m=1}^r\sum_{(i_1,i_2,\ldots,i_m)}\frac{1}{i_1!i_2!\cdots i_m!}\tbinom{n_{i_1}',n_{i_2}',\ldots,n_{i_m}'}{z_{i_1}',z_{i_2}',\ldots,z_{i_m}'},
\frac{\partial}{\partial R_{{\lambda}{\tau}}}\int d^Dp\,\,\, e^{-\frac{1}{2}p\cdot R \cdot p}= -\frac{1}{2}(2\pi)^{D/2}\left[\frac{\partial}{\partial R_{{\lambda}{\tau}}}(\det R)\right] (\det R)^{-3/2}
d s_{5}^{2}= P \left( \frac{d r^{2}}{\Delta}+ d \theta^{2}\right) - \Delta s i n^{2}\theta d \tau^{2}+ \left( \frac{\Delta - a^{2}s i n^{2}\theta}{\Sigma}\right) \left( d u^{2}+ d v^{2}\right) .
\frac{|x|}{|z|}\phi'(|x|)=\lambda\left(\frac{|x|}{|z|}\right)^{1-q}=1-\left(\frac{R}{|z|}\right)^{1-q}+\lambda\left(\frac{R}{|z|}\right)^{1-q}=(q-1)\log_{q}\frac{R}{|z|}+\lambda\left(\frac{R}{|z|}\right)^{1-q}
o_I\big((u\ast_{g_1g_2}v)\ast_{g_1g_2, g_2}w_1-u\ast_{g_1g_2, g_2}(v\ast_{g_1g_2, g_2}w_1\big))\mid_{M^2(0)}= o_{M^3}(u\ast_{g_1g_2}v)o_I(w_1)\mid_{M^2(0)}-o_{M^3}(u)o_I(v\ast_{g_1g_2, g_2}w_1)\mid_{M^2(0)}= o_{M^3}(u)o_{M^3}(v)o_I(w_1)\mid_{M^2(0)}-o_{M^3}(u)o_{M^3}(v)o_I(w_1)\mid_{M^2(0)}= 0
\hat{r}(w)=\frac{\hat{g}(w)}{i\bar{k}}+\frac{-i\bar{w}\hat{g}(w)}{(i\bar{k})^2}+...+\frac{(-i\bar{w})^{n-1}\hat{g}(w)}{(i\bar{k})^{n}}+\frac{1}{(i\bar{k})^{n}}\frac{(-i\bar{w})^n\hat{g}(w)}{i\bar{k}+i\bar{w}}.
\begin{aligned}U_N(\vec u) \sim U_N(\vec u^s) + \frac{1}{2N}\sum_{k=1}^N \lambda^s_k a_k^2 \vec \phi^{(k)}\cdot \vec \phi^{(k)}\textrm{as}\ \vec u \to \vec u^s & = E_N^s + \frac{1}{2}\sum_{k=1}^N \lambda_k^s a_k^2 \end{aligned}
\alpha ( y ){\frac{\partial^{2}L}{\partial y^{2}}}+ \alpha^{\prime}( y ){\frac{\partial L}{\partial y}}+ \beta ( y ) L = \alpha ( x ){\frac{\partial^{2}L}{\partial x^{2}}}+ \alpha^{\prime}( x ){\frac{\partial L}{\partial x}}+ \beta ( x ) L .
\frac{v(\theta,t)}{e^{2\theta t}}=He^{-2\theta t}\int_0^ts^{2H-1}e^{\theta s}ds+H\int_0^ts^{2H-1}e^{-\theta s}ds\to\frac{H\Gamma(2H)}{\theta^{2H}},\quad t\to\infty.
\int_{S^1}d(0,x)^2~H_t(x)~d\sigma(x) = \int_{(-\pi, \pi]}d(0,\theta)^2~(\sum_{i \in \mathbb{Z}}G_t(\theta + 2\pi i) )d\sigma(\theta) & < \int_{(-\infty, \infty]}d(0,x)^2~G_t(\theta + 2\pi i) d\lambda(x) & = t.
\begin{cases}\partial_t \omega^j + (u^j \cdot \nabla ) \omega^j = (\omega^j \cdot \nabla )u^j, 0<t\le 1, u^j=-\Delta^{-1}\nabla \times \omega^j, \omega^j \Bigr|_{t=0}= f^j \in C_c^{\infty}(\mathbb R^3). \end{cases}
\left\|\bar{\textbf{\textit{T}}^*}\right\|_p = ( \sum_{n|(\exists p_m | v_{n,m}^t \in q_t)}((1-\epsilon_{t,n})T^*_n+\epsilon_{t,n}T_n)^p +\sum_{n|( \nexists p_m | v_{n,m}^t \in q_t)}(T_n)^p)^{1/p}
\tilde{\theta}_h(x)(hD- 3 d\varphi(x))(v^0)^3 = 3(v^0)^2 \tilde{\theta}_h(x)(hD-d\varphi(x))v^0 & = 3(v^0)^2 Op_h^w(\tilde{\theta}_h(x)(\xi-d\varphi(x)))v^0 - \frac{3}{2i}h \tilde{\theta}'_h(x)(v^0)^3\, ,
-\int_0^T\int_\Omega \varphi_i^L(u^{(\delta)})\psi g_m'dxdt = 2m\int_{T-1/m}^{T-1/(2m)}\int_\Omega \varphi_i^L(u^{(\delta)})\psi dxdt \to\int_\Omega \varphi_i^L(u^{(\delta)}(\cdot,T))\psi(\cdot,T)dx\quad\mbox{as}m\to\infty.
S^{(0)}_2={1\over2}x^i\Lambda_{ij}x^j+y^*_{\alpha a}D^\alpha_{2\beta}c^{\beta a}+\bar{y}_\alpha D^\alpha_{4\beta}B^\beta+\varepsilon^{ab}c^*_{\alpha a|b}D^{-1\alpha}_{2\gamma}D^\gamma_{4\beta}B^\beta.
\left\langle \phi_{2}({\bf x}) , t_{2}\left| \right. \phi_{1}({\bf x}) , t_{1}\right\rangle = \left\langle \phi_{2}({\bf x}) \left| \hat{T}\exp \left[ - \frac{i}{\hbar}\int_{t_{1}}^{t_{2}}\hat{H}d t \right] \right| \phi_{1}({\bf x}) \right\rangle ,
\delta_{\kappa_1}p_1^\mu =2\bar{\kappa}_1\left( \begin{array}{c}\lambda_2^2\gamma^\mu \,\gamma \cdot p_1\ -\gamma \cdot \lambda_2\,\,\gamma \cdot p_1\,\lambda_2^\mu +\lambda_2\cdot p_1\,\,\gamma \cdot \lambda_2\,\,\gamma^\mu \,\end{array}\right) \dot{\theta}_1.
T_\psi^{F,U_1,U_2}(X,Y) =\sum_{i,j,k=1}^{n}\psi(z_i,z_k^{(1)},z_j^{(2)})P_i X P_k^{(1)}YP_j^{(2)}& = \sum_{j,k=1}^{n}\Bigl(\sum_{i=1}^n \psi(z_i,z_k^{(1)},z_j^{(2)})P_i\Bigr) X P_k^{(1)}YP_j^{(2)}& = \sum_{j,k=1}^{n}\psi(F, z_k^{(1)},z_j^{(2)})X P_k^{(1)}YP_j^{(2)},
C_\diamond=\tfrac{1}{\sqrt{2}}C_{1}\diamond C_2 \colon \mathcal{H}\to \mathcal{H}\oplus \mathcal{H}, \, C_\boxplus=\tfrac{1}{\sqrt{2}}C_1\boxplus C_2 \colon \mathcal{H}\oplus \mathcal{H}\to \mathcal{H}.
\alpha_{-s}[\rho_2(\xi)] =\sum_{|\alpha|<N}\frac{s^\alpha}{\alpha !}\partial_s^\alpha[\alpha_{-s}[\rho_2(\xi)]]_{s=0}+\sum_{|\alpha|=N}s^\alpha r_{N\alpha}(s,\xi) & = \sum_{|\alpha|<N}\frac{(-is)^\alpha}{\alpha !}\delta^\alpha \rho_2(\xi)+\sum_{|\alpha|=N}s^\alpha\alpha_{-s}[\widetilde{r}_{N\alpha}(-s,\xi)],
b^{k+1}_{i+1,1}= \sum_{l=1}^a (-1)^{l-1}e_l b^{k}_{i+1,l}=\sum_{l=1}^a (-1)^{l-1}e_l \left(h_{k+i+1-l}- \sum_{r=1}^i h_{i+1-r}b^k_{r,l}\right) & =\sum_{l=1}^a (-1)^{l-1}e_l (h_{k+i+1-l}) - \sum_{r=1}^i h_{i+1-r}\left(\sum_{l=1}^a (-1)^{l-1}e_l b^k_{r,l}\right) & =h_{(k+1)+(i+1)-1}- \sum_{r=1}^i h_{i+1-r}b^{k+1}_{r,1}.
\sum_{j>0}\int_{\mathcal L_{2j}}(cd,d)_p\chi_{\psi}(d)\chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh = -p(1-p^{-1})^2 \sum_{j>0}p^{-2j}= -p^{-1}\frac{(1-p^{-1})^2}{(1-p^{-2})}= \frac{1-p}{p(p+1)}.
|g_2 (\tau,r)| \lesssim_\delta 1 \cdot r^{-3}= r^{-3}; |g_3 (\tau,r)| \lesssim_\delta r^{-2}\cdot r^{-1}= r^{-3}; |g_4 (\tau,r)| \lesssim_\delta r^{-1}\cdot r^{-1}= r^{-2}.
\sum_{\substack{r=1 (r,n)=1}}^{\lfloor n/3\rfloor}\frac{1}{r}\equiv -\frac{3}{2}q_3(n)+\frac{3}{4}nq_3^2(n)+\frac{1}{3}J_{3}(n)n^{\phi(n)-1}\phi_{J_{3}}^{(2-\phi(n))}(n)\frac{B_{\phi(n)-1}(\frac{1}{3})}{\phi(n)-1}\pmod{n^2};
\underbrace{0 = d_{0,1}= \cdots d_{0,t_0}}_{=\tau_0}< \underbrace{d_{1,1}= \cdots =d_{1, t_1}}_{=\tau_1}< \cdots <\underbrace{d_{s-1,1}= \cdots =d_{s-1,t_{s-1}}}_{=\tau_{s-1}}<1 = \tau_{s}= \sigma_s.
\lim_{x\rightarrow 0^{\pm}}T(0,x) = A^{\pm}_{a},\lim_{x\rightarrow 0^{\pm}}\lim_{t\rightarrow 0}(N+iB)(t,x)e^{-ia^{2}log \frac{\sqrt{t}}{x}-ix^{2}/4t}=B^{\pm}_{a}.
-{\frac{3}{2}}\cos{4 \tilde{\theta}}+ 1 6 \tilde{k}^{4}\tilde{\gamma}^{4}- 1 2 \tilde{k}^{2}\tilde{\gamma}^{2}+ 1 9 2 \tilde{k}^{4}\tilde{\gamma}t - 6 9 1 2 \tilde{k}^{6}t^{2}+{\frac{3}{2}}.
\pi^n(a) \cdot f(\zeta \otimes \zeta') = \iota_n(\pi^n(a))(\zeta \otimes \zeta') = \pi^{n+1}(a)(\zeta \otimes \zeta') = a(n - k) \zeta \otimes \zeta' = (\pi^n(a) \zeta) \otimes \zeta' = f( \pi^n(a) \cdot (\zeta \otimes \zeta')).
\bigg\|{f(C^p)\over f(C)^p}\bigg\|_\infty=\bigg\|{g((B^{1/2}A^{-1}B^{1/2})^p)\over g(B^{1/2}A^{-1}B^{1/2})^p}\bigg\|_\infty=\bigg\|{g((A^{-1/2}BA^{-1/2})^p)\over g(A^{-1/2}BA^{-1/2})^p}\bigg\|_\infty
N \left({{\star_{i=1}^{m-1}}G},\alpha^{m}\right) + N \left({\star_{i=1}^{m-2}G},\alpha^{m-1}\right) \leq N\left({{\star_{i=1}^m}G},\alpha^{m+1}\right) \leq N \left({{\star_{i=1}^{m-1}}G}, \alpha^{m}\right) + N \left({{\star_{i=1}^{m-2}}G},\alpha^{m-1}\right) + n
\sum_{1\leq |n|\leq Z}\frac{\chi_d(n) (1-e(\alpha n))}{n}= \sum_{\substack{1\leq |n|\leq Z P^{+}(n)\leq y}}\frac{\chi_d(n) (1-e(\alpha n))}{n}+\sum_{\substack{1\leq |n|\leq Z P^{+}(n)> y}}\frac{\chi_d(n) (1-e(\alpha n))}{n},
\mathbf{C}_k=\begin{bmatrix}\mathbf{h}^T(1,k) \mathbf{h}^T(2,k) \cdots \mathbf{h}^T(R,k) \mathbf{h}^T(2,k) \mathbf{h}^T(3,k) \cdots \mathbf{h}^T(R+1,k) \vdots \vdots \ddots \vdots \mathbf{h}^T(S,k) \mathbf{h}^T(S+1,k) \cdots \mathbf{h}^T(Q,k) \end{bmatrix}
\boldsymbol{P}_{M}\left( \boldsymbol{L}_{JK}u(x)\right) =\left(A_{JK}(u;K+p^{N}\mathbb{Z}_{p})\right) \Omega\left(p^{N}\left\vert x-J\right\vert_{p}\right) -A_{JK}\boldsymbol{P}_{M}u(x)\Omega\left( p^{N}\left\vert x-J\right\vert_{p}\right) .
\Phi_{q}(1,t) =\int_{\Omega}a\left( x\right) \left[ (t\phi_{1})\left\{\log(t\phi_{1})+\frac{w_{q}(1,t)}{t\phi_{1}}\right\}-w_{q}(1,t)\right]\phi_{1}& =t\int_{\Omega}a\left( x\right) \phi_{1}^{2}\log(t\phi_{1}) & =t\left\{(\log t)\int_{\Omega}a\left( x\right) \phi_{1}^{2}+\int_{\Omega}a\left( x\right) \phi_{1}^{2}\log\phi_{1}\right\}.
\frac{|(U(F))(x,y)-(U(F))(x',y)|}{d_2(x,x')}& = \frac{|h(y)F(\varphi(x,y),\tau(y))-h(y)F(\varphi(x',y),\tau(y))|}{d_2(x,x')}& = \frac{|F(\varphi(x,y),\tau(y))-F(\varphi(x',y),\tau(y))|}{d_2(\varphi(x,y),\varphi(x',y))},x,x'\in X_2, y\in Y_2
y_{+}\geq \frac{1}{3\beta}(2\beta +1-\alpha ) \geq \frac{1}{3\beta}(\beta +2+(\beta_{+}-1)-\alpha ) begin{align*}3mm] =\frac{1}{3\beta}\big\{\beta +2 + \frac{\alpha^{1/2}}{1-3\alpha}[\big((3(3-\alpha -\alpha^{2})\big)^{1/2}+5\alpha^{1/2}-(1-3\alpha )\alpha^{1/2}]\big\}begin{align*}3mm] \geq \frac{\beta+2}{3\beta},
\int_{t}^{\infty}\left| \frac{\widetilde{B(t-\tau)}}{(t-\tau)}\right| d\tau\leq \left( \sum_{m=0}^{n_{\tilde{r}}-1}\frac{1}{p^{m+1}}M(\lambda_{\tilde{r}},m+1) \frac{m!}{\alpha^{m+1}}\right),
q(\bar{y}^S)=q(\bar{y}^R)+q^{\prime}(\bar{y}^R)\left( \sum_{p=1}^4 \Delta \bar{y}^{(p)}(E) \right)+ \frac{1}{2}q^{\prime \prime}(\bar{y}^R)\left( \sum_{p=1}^3 \Delta \bar{y}^{(p)}(E) \right)^2+\frac{1}{6}q^{(3)}(\bar{y}^R)\left( \sum_{p=1}^2 \Delta \bar{y}^{(p)}(E) \right)^3 +\frac{1}{24}q^{(4)}(\bar{y}^R)\left( \Delta \bar{y}^{(1)}(E) \right)^4+o(N^{-2}).
i \frac{\partial}{\partial t}\langle q^{\prime}, t \mid q , 0 \rangle = - \frac{1}{2}\frac{\partial^{2}}{\partial q^{2}}\langle q^{\prime}, t \mid q , 0 \rangle + V ( q^{\prime}) \langle q^{\prime}, t \mid q , 0 \rangle ,