harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta W=\int_{t_{0}}^{t_{1}}(\sum_{i=1}^{n}F\cdot\frac{\partial h}{\partial q_{i}}\epsilon\dot{q}_{i})dt=\sum_{i=1}^{n}(\int_{t_{0}}^{t_{1}}F\cdot\frac{\partial h}{\partial q_{i}}\epsilon\dot{q}_{i}dt).
\forall z\in\mathbb R,\ \ \{\mathbb E}\left[\exp\left(iz\int_{\mathbb R^2}H(x,y)\, dZ_{x,y}\right) \right]= \Phi_{(c_0+c_1)\int_{\mathbb R^2}|H(x,y)|^\beta_+\, dxdy,(c_0-c_1)\int_{\mathbb R^2}|H(x,y)|^\beta_-\, dxdy,\beta}(z)).
\begin{aligned}\int_{-\theta^{-\alpha_{0}}R_j^2}^{t}\int_{B_{R_j}}\,u^{\,2b}\,\chi_{_{\left[u\leq l_j\right]}}f_3^2\eta_j^2\,dxd\tau\leq C\Lambda^{2\left(b-1\right)}\theta_0^{\,2}\left(\int_{-\theta^{-\alpha_{0}}R_j^2}^{t}\left|A_{l_j,R_j}^-(\tau)\right|^{\frac{\widehat{r}}{\widehat{q}}}\,d\tau\right)^{\frac{2}{\widehat{r}}\left(1+\kappa\right)}\end{aligned}
{D_m}{D_k}\Psi -{D_k}{D_m}\Psi = ({\delta_m}{\delta_k}\Psi -{\delta_r}\Psi G_{mk}^r) - ({\delta_k}{\delta_m}\Psi -{\delta_r}\Psi G_{km}^r) & ={\delta_m}{\delta_k}\Psi -{\delta_k}{\delta_m}\Psi = ({\delta_m}{\delta_k}-{\delta_k}{\delta_m})\Psi = \left[{{\delta_m},{\delta_k}}\right]\Psi.
q_{\infty}\left(\mathcal F_1^{c'},\mathcal F_2^{c'},\mathbf T^{'}\right)-q_{\infty}^{*}=\left(a_{1}-a_{n_2}\right)\left(f_{2,K_2^c,\infty}(T_{n_2}^*)-f_{1,K_1^c+\min\{K_1^b+\mathcal F_1^{b*}\},\infty}\right).
\mu_{\rm I}(\eta )=\frac{1}{\sqrt{2n}}\biggl\vert \frac{\eta}{\eta_{\rm i}}\biggr \vert^{1/2}\cos \biggl(\frac{2\pi}{\epsilon}\ln \biggl\vert \frac{\eta}{\eta_{\rm i}}\biggr \vert \biggr).
\mathcal{K}(a, b, m, n) = \sum_{l = 0}^{m + n - \frac{1}{2}}\binom{m + n - \frac{1}{2}}{l}\frac{(-1)^{l}}{2}\int \frac{u^{l - \frac{1}{2}}}{(1 - u + a)^{m}(1 - u + b)^{n}}{\rm d}u.
x_q^k =\frac{k}{q}+ \frac{\alpha(k/q)}{q^2}+ \varepsilon O(q^{-4}), \varphi_q^k =\frac{\mu(x_q^k)}{q}\left (1+ \frac{\beta_3(k/q)}{q^2}+ \varepsilon O(q^{-4}) \right ),
\sum_{0\leq m<n<\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}}\right)}{p^{m(\frac{1}{2}+s_1 -z)}p^{n(\frac{1}{2}+s_2+z)}}= \frac{\tau_B(p )}{p^{\frac{1}{2}+s_2+z}}+O\bigg( \frac{1}{p^{1+\varepsilon}}\bigg).
-\Box^{\ast}\left(\tau \left(u|\nabla\log u|^2+u\SS-2\Delta u\right)-u\log u-\frac{nu}2\log\tau\right) &=2\tau u|\alpha-\nabla\nabla\log u-\frac g{2\tau}|^2+\tau u\mathcal D_\alpha(-\nabla\log u).
\hat{A}= \hat{M}-\hat{N}\geq \hat{M}-\tilde{N}= 2(1-\theta)\Omega_3+\Omega_2-\vert\Omega_2-D_{A\Omega_1}\vert+\frac{1-(2\theta-1)\vert 1-\alpha\vert}{\alpha}D_{A\Omega_1}&-\frac{\alpha+\beta+(2\theta-1)\vert \alpha-\beta \vert}{\alpha}\vert L_{A\Omega_1}\vert-2\theta\vert U_{A\Omega_1}\vert.
g_{2}((\nabla \pi_{\ast})(Z,\omega W),\pi_{\ast}(X))-g_{2}((\nabla\pi_{\ast})(Z,\omega \phi W),\pi_{\ast}(X)) =g_{1}(\mathcal{T}_{Z}\omega W,\mathcal{B}X)+g_{1}(W,\varphi Z)\eta (X)
\sum\limits_{n=1}^{\infty}e^{-\frac{c}{2}n}\left\|r_2^n \right\| \leq \sum\limits_{q=1}^{2}\|\beta_q\|_{\mathrm{L_1}(0,\infty)}\sum\limits_{n=1}^{\infty}e^{-\frac{c}{2}n}\left( 2\int_{0}^{k}\left\|\mathbf{B}_qu_{t}(\zeta)\right\| d\zeta + k \int_{k}^{t_n}\left\|\mathbf{B}_qu_{tt}(\zeta)\right\| d\zeta \right).
|R_4| \leq \biggl\{\log \Bigl(\frac{n-1}{e^{\Psi(k)}}\Bigr) \! + \! \int_{\mathcal{X}_n}\! \! \! f(x) \biggl( |\log f(x)| + \frac{a(f(x))}{f(x)^{\frac{2}{d}}V_d^{\frac{2}{d}}}\biggr) \, dx \biggr\}\mathbb{P}\Bigl(\mathrm{B}_1 \geq \frac{a_n}{n-1}\Bigr) &=o(n^{-(3-\epsilon)}),
I_{2}=\frac{1}{2}\delta^{2}\int_{\mathbb{R}_{+}^{n}}\tilde{a}(\delta y)\left(U(y)\chi(\delta y)+\delta v_{q}(y)\chi(\delta y)\right)^{2}|g(\delta y)|^{1/2}dy =\frac{1}{2}\delta^{2}a(q)\int_{\mathbb{R}_{+}^{n}}U(y)^{2}dy+o(\delta^{2}) =\delta^{2}a(q)\frac{n-2}{(n-1)(n-4)}\omega_{n-1}I_{n-1}^{n}+o(\delta^{2})
{}I_{1}= \frac{1}{N - 3}\,{d^{3 - N}}{Y^{- 1}}\left( \cos^{2}\frac{\phi_{x}}{2}\right)^{\frac{3 - N}{2}}{}_{2}F_{1}\left( \frac{N - 3}{2}, \frac{5 - N}{2}; \frac{N - 1}{2}; \sin^{2}\frac{\phi_{x}}{2}\right) .
\frac{1}{r}E_0\left[g(x+\underline{X}'_{T'(r)}+\overline{X}_{T(r)})\right] = E_0\left[\int_0^{\infty}g(x+\overline{X}_t+\underline{X}'_{T'(r)})e^{-rt}\,dt\right] &=E_0\left[\int_0^{\infty}E_0\left[\left.g(x+\overline{X}_t+\underline{X}'_{T'(r)})\,\right\vert\,\mathcal{F}_t\right]e^{-rt}\,dt\right] .
\tilde{I}_{\mu_1 \ldots \mu_n}\sim \int_{\frac{- \pi}{a}}^{\frac{\pi}{a}}\frac{d^4l}{(2 \pi)^4}\sum_{r=- \infty}^{+ \infty}\sum_{j=0}^{n-1}\frac{A_j^{\mu_1 \ldots \mu_n}(l,Q,M_r)}{P^2(l+Q_j)+M^2r^2}, \quad Q_j=p_1+\ldots + p_j.
Z_{( r , s )}( q ) = \sum_{( r^{\prime}, s^{\prime}) , ( r^{\prime \prime}, s^{\prime \prime})}N_{r^{\prime}r^{\prime \prime}}^{( A_{g - 2})}{}^{r}N_{s^{\prime}s^{\prime \prime}}^{( A_{g - 1})}{}^{s}\chi_{r^{\prime}, s^{\prime}}( q ){\chi_{r^{\prime \prime}, s^{\prime \prime}}( q )}^{*}
\Xi(\omega):= \{(Q,\hat P)\in\mathfrak{P}(\Omega)\times\mathfrak{P}(\Omega):\E_Q|\Delta S_t(\omega,\cdot)|<\infty,\E_Q[y\Delta S_t(\omega,\cdot)]\leq 0,\ \forall y\in\mathcal{H}_t(\omega), &P(\omega)\ll Q\ll \hat P\in\mathcal{P}_t(\omega)\}.
E=\left\{\frac{2^{e_{k}}-3^{k+1}-3^{k}2^{2b_{0}+1}-\sum\nolimits_{j=1}^{k}3^{k-j}2^{2b_{0}+\sum\nolimits_{i=1}^{j}2b_{i}3^{i}+\sum\nolimits_{i=1}^{j}\upsilon_{i}+1}}{3^{k+2}}\right\},
- \pi \cdot K\tilde{H}(x) = \int_0^{x_0}\frac{\tilde{\theta}(y)}{y-x}\, dy + \int_{x_0}^{1/2}\frac{g(y)}{y-x}\, dy - \log \left( \frac12 - x \right) \approx \int_0^{x_0}\frac{\tilde{\theta}(y)}{y-x}\, dy + 1.
A(h_{+})+C(h_{+})=4\left( s-v\right) \times \left( \dfrac{\left( s-v\right) (t^{2}+s^{2})}{s^{2}}h_{+}^{2}+\dfrac{2w\left( vt-ws\right)}{s}h_{+}-\allowbreak w\left( vt-ws\right) \right)
\prod_{\substack{l,m,n \geq 1 l,m \leq n ; \, \gcd(l,m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{l m^2}{n^4}}= \exp\left\{\sum_{n=1}^{\infty}\left( \frac{n}{2}+ \frac{n^2}{2}\right) \left( \frac{n}{6}+ \frac{n^2}{2}+ \frac{n^3}{3}\right) \frac{z^n}{n^4}\right\}
{1 \over 2}\sum_{i=1,2}\left({S_i\over\Delta^3}\right)^2\left[\log \left({S_i \over\Delta^3}\right)-{3\over2}\right]-\left({S_1\over\Delta^3}+{S_2\over\Delta^3}\right)^2 \log \left({\Lambda \over \Delta}\right)+\sum_{n\geq3}\sum_{i=0}^n c_{n,i}\left({S_1\over\Delta^3}\right)^{n-i}\left({S_2\over\Delta^3}\right)^i,
\omega^i = \phi^j \, ({\rm 1}+ \frac{1}{2!}A^2 + \frac{1}{4!}A^4 +\cdots )_j^{\,\,\, i}- d\xi^{\beta}\, (\frac{1}{2!}A + \frac{1}{4!}A^3 +\cdots)_{\beta}^{\,\,\,\,i}
u(x_{0}, t_{0})-p\cdot x_{0}=\inf_{\partial_{p}G_{0}}u-a-p\cdot x_{0}\leq u(y,s)-p\cdot y+p\cdot (y-x_{0})-a &\leq u(y,s)-p\cdot y+a-a= u(y,s)-p\cdot y,
\Delta'(\mathbf{F}_{u})= \sum_{p=0}^n\sum_{u\leq v_{1}\times v_{2}}\mathbf{M}_{v_{1}}\otimes \mathbf{M}_{v_{2}}=\sum_{p=0}^n\sum_{u_{(1)}^p\leq v_{1}}\mathbf{M}_{v_{1}}\otimes \sum_{u_{(2)}^p\leq v_{2}}\mathbf{M}_{v_{2}}= \sum_{p=0}^n\mathbf{F}_{u_{(1)}^p}\otimes \mathbf{F}_{u_{(2)}^p}=\Delta(\mathbf{F}_{u}),
\nabla (\tilde{f}\Theta) (H, e, i) = \frac{1}{|e|}\sum_{j \in e}\tilde{f}(H, e, j) \Theta(H, e, j) = \frac{1}{|e|}\sum_{j \in e}f(H, j) \Theta(H, e, j) \stackrel{(a)}{\geq}\frac{1}{|e|}\min_{j \in e}f(H, j) = \tilde{f}_(H, e, i),
\bar{\alpha}: = \left( \begin{matrix}{\bar{d}_{1}}\\{\bar{d}_{2}}\\ \end{matrix}\right) , \quad \bar{\beta}: = \left( \begin{matrix}{\bar{d}_{1}}\\{- \bar{d}_{2}}\\ \end{matrix}\right) , \quad \bar{D}_{0}: = 2 \left( \begin{matrix}{\bar{d}_{1}}&{0}\\{0}&{\bar{d}_{2}}\\ \end{matrix}\right) .
R_{\lambda}= u_{\lambda}^{-\frac{n}{n-4}}\Big\{- \frac{4(n-1)}{(n-4)}\Delta_g u_{\lambda}- \frac{8(n-1)}{(n-4)^2}\frac{|\nabla_g u_{\lambda}|^2}{u_{\lambda}}+ R_g u_{\lambda}\Big\}.
z =W_Y(z)_--W_Y(z)_++2G_Y^X(z)_+\ , U'(z) =W_{(1)}(z)_-+G_X^Y(z)_+\ , U'(z) =W_{(1)}(z)_-+G_{X_1}^{X_2}(z)_++z\ , U'_+\left(z+t_2/t_3\right) =W_{(2)}(z)_-+W_{(2)}(z)_++W_{(2)}(-z)\ .
\min{\{d(p_{i-1},p_i) , d(p_i,p_{i+1})\}}= \min\{d(x_0,s_ix_0), d(x_0,s_{i+1}x_0) \}&\ge 2(s_i^{-1}x_0 \cdot s_{i+1}x_0)_{x_0}+18\delta +1 &= 2(p_{i-1}\cdot p_{i+1})_{p_i}+18\delta +1
\sum_{k = c \log n}^n \left| E \left[ \frac{X_k - x_k^o}{x_k^o}- \frac{Y_k - y_k^o}{y_k^o}\right] \right| = \sum_{k = c \log n}^n \phi(k/n)^{-1}|E (X_k - x_k^o) - (Y_k - y_k^o) | (1+o(1)) \le c'' \sum_{k = c \log n}^n (\log k)^5 k^{-3/2}= o(1),
\begin{aligned}\max~ R(\mathbf{F}_R, \mathbf{F}_B, \mathbf{W}_R, \mathbf{W}_B) s.t.~ ||\mathbf{F}_R\mathbf{F}_B||_F^2 = N_s, \mathbf{F}_R \in \mathcal{F}_R,~\mathbf{W}_R \in \mathcal{W}_R,\end{aligned}
\lambda ([x]_b \cdot b^n) = \left(\frac{(n \log b)^3}{\log (n \log b)}\right)^{1/5}, \implies [x]_b \cdot b^n \cdot \exp (-c \lambda ([x]_b \cdot b^n)) = \frac{b^n}{\frac{1}{[x]_b}\exp (c \left(\frac{(n \log b)^3}{\log (n \log b)}\right)^{1/5})}.
( 1 , 2 ) = \left( \begin{matrix}{0}&{1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{1}\\ \end{matrix}\right) , ( 2 , 3 ) = \left( \begin{matrix}{0}&{1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{1}\\ \end{matrix}\right) .
\max \Biggl\{\sum_{i \in N}\sum_{j \in V}w_{ij}z_{ij}: \exists ~ (x, y) \geq 0 \eqref{eq:obm_lp_pol1}\eqref{eq:obm_lp_pol2}z_{ij}=\sum_{t\in[n]}\sum_{S\subseteq V\backslash j}x_{i,j}^{t,S}\Biggr\},
(q\kappa)^{k - 1}|\xi_{A_1}- \xi_{B_1}| \leq |\sum_{j = 0}^{k}(-1)^{j}(e_{j}(B) - e_{j}(A))\xi_{A_1}^{k - j}| \leq \max_{j}|e_{j}(B) - e_{j}(A)|.
\sum_{i=0}^{\mu-1}a_{i,j}y_{i,j}= \sum_{i=0}^{\mu-1}a_i(x(E_j)) y^{(A)}_i(E_j) = \sum_{i=0}^{\mu-1}(a_iy^{(A)}_i)(E_j) = a(E_j) \ .
C(L,n)=\frac{L^2}{4\sqrt{\pi}(n-1)!}\sum_{k=1}^\infty \frac{\Gamma(Lk-\frac n2) \Gamma(Lk-\frac{n-1}2)}{(\Gamma(Lk+1))^2}\bigl(Lk+\dfrac{n(n-1)}2\bigr)\ .
\mathbf{A}=\mbox{Prod}\left(\mbox{Prod}\left(\mathbf{U},\mathbf{D}_{0},\mbox{\ensuremath{\mathbf{D}}}_{0}^{\top}\right),\mbox{Prod}\left(\mathbf{V},\mathbf{D}_{1},\mbox{\ensuremath{\mathbf{D}}}_{1}^{\top}\right)^{\top^{2}},\mbox{Prod}\left(\mathbf{W},\mathbf{D}_{2},\mbox{\ensuremath{\mathbf{D}}}_{2}^{\top}\right)^{\top}\right),
\Gamma(x,y) =-\frac{1}{e^{-{\psi}}}\left( d \phi(x) \int_{-\infty}^y e^{-{\psi(x,v)}}dv - \int_{-\infty}^y e^{-{\psi(x,v)}}d_x{\psi}(x,v) dv \right) &= \frac{1}{e^{-\psi}}\left( d \phi(x) \int_{y}^\infty e^{-{\psi(x,v)}}dv - \int_{y}^\infty e^{-{\psi(x,v)}}d_x{\psi}(x,v) dv \right) .
S_{m e a s}^{( 2 )}= - \frac{\Lambda^{2}}{4 \pi^{2}}\int \frac{d p d q}{( 2 \pi )^{4}}( 2 \pi )^{4}\delta^{4}( p + q ) M_{v}( p ) A^{\mu}( p ) A_{\mu}( q ) ,
\int_{{B}\backslash \omega_+}|\zeta_\delta(x)|U_{\delta,e_n}^p \leq C\alpha_n^p\delta^\frac{6-n}{2}(\log \delta)^\tau\int_{{B}\backslash \omega_+}(4\delta)^{\frac{n+2}{2}}=O(\delta^4(\log \delta)^\tau).
v_p([z^n]F(G,q;z)) \geq\min_{n_0+n_1p+\dots=n}\sum_{b_i\leq 0}\left(b_in_i-v_p\left(n_i!\right)\right) &\geq \min_{n_0+n_1p+\dots=n}\sum_{b_i\leq 0}\left(b_lp^{i-l}n_i-v_p\left((p^{i-l}n_i)!\right)\right) &=\min_{n_0+n_1p+\dots=n}\left(b_l\sum_{b_i\leq 0}p^{i-l}n_i-\sum_{b_i\leq 0}\left(v_p\left((p^{i-l}n_i)!\right)\right)\right).
A= Id+aE_{-2,0}=\small{\begin{array}{rrrr|c|rrrc}\dots 1 0 0 0 0 0 0 \dots \dots 0 1 0 a 0 0 0 \dots \dots 0 0 1 0 0 0 0 \dots \hline \dots 0 0 0 1 0 0 0 \dots \hline\dots 0 0 0 0 1 0 0 \dots \dots 0 0 0 0 0 1 0 \dots \dots 0 0 0 0 0 0 1 \dots \end{array}}
h(X_1+Z_2+X_2)-h(X_1) &=h(\sqrt{t}X_1+\sqrt{t}Z_2+\sqrt{t}X_2)-h(\sqrt{t}X_1) &=2m_2(-\frac1{2}\int p''\ln p)t+m_3(-\frac1{6}\int p'''\ln p)t^{3/2}&\quad+O(t^2);
\Gamma_{3}( s ) = \frac{1}{s - m^{2}}\left[ ( s - m^{2}) \frac{1}{2}\log \frac{C}{B}+ 3 m^{2}\log \frac{m^{2}}{D}- A + 1 6 \pi^{2}\alpha_{f}( s ) ( s - m^{2}) + \gamma_{f}( s ) \right] \; ,
\ln\left[{\cal L}^{{\rm I},{\cal R}}(\nu, x) \right] \sim\ln\left[ 1+\sum_{k \geq 1}{v_k(t(x))\over \nu^k}+{\alpha}{t(x) \over \nu}\left( 1+\sum_{k \geq 1}{u_k(t(x))\over \nu^k}\right) \right]= \sum_{n \geq 1}{{\cal U}_n^{{\rm I},{\cal R}}(t(x)) \over \nu^n}
\langle (x_\alpha - x) \otimes y_\alpha, (x_\alpha - x) \otimes y_\alpha \rangle \ = \ \langle x_\alpha - x , x_\alpha - x \rangle\otimes \langle y_\alpha ,y_\alpha \rangle \ = \ (\langle x_\alpha - x , x_\alpha - x \rangle\otimes 1) \cdot (1 \otimes \langle y_\alpha ,y_\alpha \rangle ).
K_{+,p'}^{(s')}= \hbar'( p - p' + \kappa_{i+1, p'}^{(s')}) - (a_s - a_{s'}), L_{+,p'}^{(s')}= \hbar'( p - p' + \lambda_{i+1, p'}^{(s')}) - (a_s - a_{s'}), K_{-,p'}^{(s')}= \hbar'( p - p' - 1 + \kappa_{i-1, p'}^{(s')}) - (a_s - a_{s'}), L_{-,p'}^{(s')}= \hbar'( p - p' - 1 + \lambda_{i-1, p'}^{(s')}) - (a_s - a_{s'})
\tilde S^i(t,q)= \beta^{-1}(t,q) \partial_{q^j}(\tilde\gamma^{-1})^{il}(t,q) (\tilde\gamma^{-1})^{jk}(t,q)H_{lk}(t,q) &-\frac{1}{2}\sum_{\xi}(\tilde\gamma^{-1})^{il}(t,q)\partial_{q^k}\sigma_{l\xi}(t,q) (\tilde\gamma^{-1}(t,q)\sigma(t,q))^k_\xi.
H^\prime= (t_{e_1}t_{\psi_2(A_2)}\cdots t_{\psi_2(A_{2g-1})})^{2g}t_{\psi_1(A_{2g})}\cdots t_{\psi_1(A_2)}t_{a_2}t_{a_2}t_{\psi_1(A_2)}\cdots t_{\psi_1(A_{2g})}&\cdot (t_{e_2}t_{\psi_3(A_2)}\cdots t_{\psi_3(A_{2g-1})})^{2g}t_{\psi_1(A_{2g})}\cdots t_{\psi_1(A_2)}t_{a_2}t_{a_2}t_{\psi_1(A_2)}\cdots t_{\psi_1(A_{2g})}\cdot t_{a_{g+1}}^{-1}.
|\partial y^{\alpha}\mathfrak Z(y,z)| + |\partial^{\alpha}_z \mathfrak Z (y,z)| \le C(S,\delta)\bigg(|y|S_{\tau}(y)\bigg)^{-1-|\alpha|_2}&|| \le C(S,\delta)\bigg(|y|S_{\tau}(y)\bigg)^{-1-|\alpha|/2}\|u_ku_{\ell}|_{B_{S_1}}\|_1 \le \|\nabla u_k \|_2 \|u_{\ell}\|_2 &x\in B^c_S, S > S_1, \alpha=1.
\|y^\prime_v\|_1=\tau(y^\prime_v)=\frac{2^{v+1}}{A_j}\int_{2^v}^{2^{v+1}}\left\|\left( \frac{\partial T_{s,j}(x_{v+j})}{\partial s}\right)^*\right\|_2^2ds+2\sum_{k=0}^{A_j-1}\|T_{{\gamma_k}, j}(x_{v+j})^*\|_2^2.
p_{-k}(a) =\sum_{\alpha_{1}+\alpha_{2}=a}p_{-2}(\alpha_{1}) p_{-(k-2)}(\alpha_{2}), p_{-k}(b) =\sum_{\beta_{1}+\beta_{2}=b}p_{-2}(\beta_{1})p_{-(k-2)}(\beta_{2}), p_{-k}(a+b) =\sum_{\gamma_{1}+\gamma_{2}=a+b}p_{-2}(\gamma_{1})p_{-(k-2)}(\gamma_{2}),
<\Phi(Z_1){\bar \Phi}(Z_2)>= \mbox{ln}(Z_{12}) -{\theta_{12}\bar\theta_{12}\over 2 Z_{12}}\;, \;\;<{\cal V}(Z_1){\bar{\cal V}}(Z_2)>= \mbox{ln}(Z_{12}) -{\theta_{12}\bar\theta_{12}\over 2 Z_{12}}\;.
\int_{B}|\nabla v|^{p-2}\Big(h_{ij}(v)\frac{\partial v^i}{\partial x_\alpha}\frac{\partial \Phi^j}{\partial x_\beta}+\frac{1}{2}\big(\frac{\partial h_{ij}}{\partial x_k}\circ v\big)\frac{\partial v^i}{\partial x_\alpha}\frac{\partial v^j}{\partial x_\beta}\Phi^k \Big)\delta^{\alpha\beta}dx=0
^{c}_{0}{D}^{\alpha}_{t}u(x,t)-a(l(u)) \: \Delta u(x,t) = f(x,t) \mbox{in}\Omega\times(0,T], u(x,t) = 0 \mbox{on}\partial\Omega\times(0,T], u(x,0) =u_0(x) \mbox{in}\Omega,
E(b,q,\beta)=\begin{cases}\frac{1}{q}\big(I(b)+E_1(b,q,\beta)\big), \ \textrm{if}\ -q/2<b\le q/2 \ \textrm{and}\b\not=0, \frac{1}{q}E_1(b,q,\beta), \ \textrm{if}\b=0, \frac{1}{q}I(b), \ \textrm{otherwise}.\end{cases}
D_{\mathcal{G}}(\Phi (u,T(u)),\Phi (w,T(w))) =D_{\mathcal{G}}((u,F(u)+T(u)),(w,F(w)+T(w))) = \lvert u-w \rvert + H_d(F(u)+T(u), F(w)+T(w)) \leq \lvert u-w \rvert + H_d(F(u),F(w))+ H_d(T(u),T(w)) \leq \lvert u-w \rvert + l \lvert u-w \rvert +H_d(T(u),T(w)) \leq (1+l) \left \{\lvert u-w \rvert + H_d(T(u),T(w))\right\}.
\Delta^e(x_i) =x_i\otimes 1+1\otimes x_i \Delta^e(y_i) =y_i\otimes 1+1\otimes y_i \varepsilon^e(x_i) =0 \varepsilon^e(y_i) =0 S^e(x_i) =-x_i S^e(y_i) =-y_i,
{\frac{\delta{\cal E}( h )}{h^{2}}}= \kappa_{0}\ln{\frac{h}{m}}+ \kappa_{1}\ln \ln{\frac{h}{m}}+ \kappa_{2}+{\cal O}{\mathrm{\Large ~ (}}\ln \ln{\frac{h}{m}}{\mathrm{\Large ~ /}}\ln{\frac{h}{m}}{\mathrm{\Large ~ )}}\, ,
z_{n-1}\cdot \frac{\sigma \cdot v_{n-1}(j_1)}{\sigma_{n-1}}+z_n \cdot \frac{\sigma \cdot v_n(j_1)}{\sigma_n}=\tilde{x}_{}(j_1) - c_{i_1}- \hat{u}, & z_{n-1}\cdot \frac{\sigma \cdot v_{n-1}(j_2)}{\sigma_{n-1}}+z_n \cdot \frac{\sigma \cdot v_n(j_2)}{\sigma_n}=\tilde{x}_{}(j_2) - c_{i_2}- \hat{w},
K_{a}= \sum_{b = 1 , b \neq a}^{N}\frac{1}{\sigma_{b}^{2}- \sigma_{a}^{2}}l_{a b}^{2}+ p_{a}^{2}+ ( 2 - \sigma_{a}^{2}) q_{a}^{2}- q_{a}^{2}\sum_{b = 1}^{N}q_{b}^{2}
(\kappa_{a,\omega,n}^{-1})^*\circ \gamma^\perp_{p,(-\delta,\delta)}(p_{a,\omega,n})=(0,c_{a,\omega,n},1),(\kappa_{a,\omega,n}^{-1})^*\circ \hat{\gamma}^\perp_{p,(-\delta,\delta)}(p_{a,\omega,n})=(c_{a,\omega,n},0,1).
\begin{array}{ll}(|\xi|^2+\tau^2|v|^2)_t + 2k_2 \xi\nabla\cdot v + 2k_2 v\cdot\nabla\xi + 2 |v|^2 begin{align*}6pt]= -2\gamma k_1 \xi^2\nabla\cdot v - 2k_1 \xi v\cdot\nabla \xi - 2 k_1 \tau^2 v\cdot\nabla v\cdot v+ \frac{2}{k_1}(\frac{1}{\bar{\varrho}}- \frac{1}{\varrho})\nabla\xi\cdot v.\end{array}
\int_X\Phi_1(|f|)d\mu =\displaystyle\sum^{\infty}_{n=1}\int_X\Phi_1(|x_n|)\chi_{E_n}&=\sum^{n_0}_{n=1}\Phi_1(|x_n|)\mu(E_n)+\sum\limits_{n\geq n_0}\Phi_1(| x_n|)\frac{\mu(F_n)}{\Phi_1(|x_n|)}<\infty.
K |_{x_{h}}= 9 ( N^{'}|_{x_{h}})^{2}( \frac{\sigma^{'}}{\sigma}|_{x_{h}})^{2}+ 6 N^{'}|_{x_{h}}N^{'^{\prime}}|_{x_{h}}\frac{\sigma^{'}}{\sigma}|_{x_{h}}+ ( N^{'^{\prime}}|_{x_{h}})^{2}+ 6 \frac{( N^{'}|_{x_{h}})^{2}}{x_{h}^{2}}+ \frac{1 2}{x_{h}^{4}}\ .
1-\int_{\Omega}\varphi(g_0)d\mu= \int_{\Omega}\varphi(1-g_0)d\mu=\int_{\Omega}\mathrm{sup}\varphi(1-z^{(n)})d\mu \leq \sum_{n=1}^\infty\int_{\Omega}\varphi(1-z^{(n)})d\mu=\sum_{n=1}^\infty(1-\int_{\Omega}\varphi(z^{(n)})d\mu) \leq 2^{-1}\int_\Omega\varphi(\mathcal C_{q_0}e_0)d\mu.
\alpha_B\pi^\flat(r)\widetilde{D}(\bar{r}) =\alpha_B\pi^\flat(r)\cdot(u-1)=\alpha_C\varphi^\flat(r)\cdot(u-1)=\alpha_C\varphi^\flat(r)u-\alpha_C\varphi^\flat(r) &=\alpha_C\psi^\flat(r)-\alpha_C\varphi^\flat(r)=\psi^\sharp\alpha_F(r)-\varphi^\sharp\alpha_F(r)=D\big(\alpha_F(r)\big)
f_{1,3,1}(q^2,q^2;q^2)+ q^3f_{1,3,1}(q^6,q^6;q^2) &=\frac{q^{-2}J_{4,8}J_{16,32}j(q^2;q^{16})j(q^{20};q^{32})}{j(-q^{6};q^{16})j(-q^{-2};q^{16})}+ \frac{q^{5}J_{4,8}J_{16,32}j(q^{10};q^{16})j(q^{36};q^{32})}{j(-q^{14};q^{16})j(-q^{6};q^{16})}.
E[x^{n}]=\frac{\lambda m^{n+1}}{\kappa+1/\kappa}(\sum_{i=0}^{n}\frac{n!}{(n-i)!}\frac{1}{(m\lambda\kappa)^{i+1}}-\sum_{i=0}^{n}\frac{n!}{(n-i)!}\frac{1}{(-m\lambda/\kappa)^{i+1}})
\hat{y}_- := \max_{k>0}\left(-{{\textstyle{\frac{1}{k}}}}\lfloor yk \rfloor +{{{\hat{y}^{\mkern1.7mu0}_{\mkern-.3mu-}\mkern-2mu(k)}}}\right),\;\;\;\;\hat{y}_+ := \min_{k>0}\left(-{{\textstyle{\frac{1}{k}}}}\lceil yk \rceil +{{{\hat{y}^{\mkern1.7mu0}_{\mkern-.3mu+}\mkern-2mu(k)}}}\right),
S_{1 2}= \frac{3 ( \mathrm{\boldmath \sigma}^{( 1 )}\cdot{\bf r}) ( \mathrm{\boldmath \sigma}^{( 2 )}\cdot{\bf r})}{r^{2}}- \mathrm{\boldmath \sigma}^{( 1 )}\cdot \mathrm{\boldmath \sigma}^{( 2 )}
\nu_y = \frac{\left(\nabla \psi(x), -1 \right)}{\langle x, \nabla \psi(x) \rangle - \psi(x)}= I^+ \left \{\, \left(\nabla \psi(x), \langle x, \nabla \psi(x) \rangle - \psi(x) \right) \, \right\}.
0 = f^* (K_X + \Delta) \cdot E_1= (K_Y + E_1) \cdot E_1 + \widetilde{\Delta}\cdot E_1 - a_1 E_1^2 + (1 - a_2) E_1 \cdot E_2 + (1 - a_3) E_1 \cdot E_3 + \sum_{i \not = 1, 2, 3}(1- a_i) E_1 \cdot E_i.
\int \frac{(-q^{n+3}x^2;q^2)_\infty}{(-x^2;q^2)_\infty}\widetilde{h}_n(x;q) d_qx= & \frac{(-q^{n+1}x^2;q^2)_\infty}{[n+1]_q(-x^2;q^2)_\infty}\left( q^n x \widetilde{h}_n(x;q)- q^{1-n}(1-q^n)\widetilde{h}_{n-1}(x;q)\right),
D(g_1v_2,g_1v_5,g_1v_7)D(g_1v_4,g_1v_6,g_1v_8)(D(g_1v_1,g_1v_3,g_1X)g_1Y+D(g_1v_1,g_1v_3,g_1Y)g_1X) & = c(g_1)^3D(v_2,v_5,v_7)D(v_4,v_6,v_8) g_1(D(v_1,v_3,X)Y+D(v_1,v_3,Y)X).
(P_k)_i=\begin{bmatrix}0 0 \eta_r U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\ldots \overbrace{0 \ldots 0}^{(k-1)}\eta_r \mu^{-\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r) \overbrace{0 \ldots 0}^{(k-3)}\end{bmatrix}^{T},
\phi_{\omega , k}( x ) ={\frac{1}{4 \pi^{3}\kappa^{1 / 2}}}\left( \sinh ( \pi \omega / \kappa ) \right)^{1 / 2}~ K_{i \omega / \kappa}( \rho ( m^{2}+ k_{j}^{2})^{1 / 2}) e^{- i \omega t}e^{- i k_{j}z^{j}},
\partial_{t}u - \mathrm{div}\,\big(\mathbf{L}\nabla u) = 0 (0, \infty) \times G \partial_{t}\mathbf{L}+ \mathbf{L}= \mathbf{F}(\nabla_{\sigma}u) (0, \infty) \times G, u(0, \cdot) = u^{0}, \mathbf{L}(0, \cdot) = \mathbf{L}^{0}G
\frac{d \sigma ( g \rightarrow \gamma )}{d \Omega^{\prime}}= \, \frac{k^{2}H_{0}^{2}b^{2}p^{2}}{4 ( 2 \pi )^{4}( p^{2}- \frac{m^{2}\pi^{2}}{a^{2}})}\left( 1 + 3 \frac{\omega}{p}\right) \sin^{2}\left( \frac{a p}{2}\right) 2 \sin^{2}\left[ \frac{c ( p + 2 k )}{2}\right]
S_{eff}= \int\left( -\frac{1}{4}\sum_n(\partial_{\mu}b^n_{\rho\sigma}\partial^{\mu}b^{n,\rho\sigma}+{m_n^{2}}b^n_{\rho\sigma}b^{n,\rho\sigma})-\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi\right) dx.
\sum_{k=0}^{\infty}\left( \left( 2k+1\right) \beta+r\mid\alpha\right)_{n}\frac{\left( -1\right)^{k}\left( 2\pi\right)^{2k}}{\left(2k+1\right) !}=\beta\sum_{j=1}^{\left\lfloor n/2\right\rfloor}S\left(n,2j+1;\alpha,\beta,r\right) \left( -1\right)^{j}\left( 2\pi\beta\right)^{2j}.
{\cal J}_i^\mu =\frac{-1}{1+4{\Phi}^2}\biggl( \left(\delta^{ij}+4\Phi^i\Phi^j\right) \varepsilon^{\mu \nu}\partial_\nu\Phi^j+2\varepsilon^{ijk}\Phi^j\partial^\mu \Phi^k\biggr) .
\frac{\mu\alpha}{\pi}{\rm Im}\left\{\frac{(e^{-i\pi}t)^{\alpha-1}}{\lambda+e^{-i\pi\alpha}\,t^\alpha}\right\}= \frac{\mu\alpha}{\pi}\frac{\lambda\,t^{\alpha-1}\sin\pi\alpha}{\lambda^2+2\lambda\,t^\alpha\cos\pi\alpha+t^{2\alpha}}
\begin{array}{rl}y(x,\lambda)\tilde{y}(x,\lambda)\!\!\!\! =A_1(\lambda)+A_2\cos(2kx)+A_3\cos2k(x-\frac{1}{2}) &\quad+A_4\cos2k(x-1)+\frac{1}{2}\int_0^xK(x,t)\cos(2kt)dt\end{array}
\begin{array}{ll}{[}J_i, J_j] = i{\epsilon_{ij}}^k J_k (i,j,k=1,2,3){[}J_i, S_j ] = i{\epsilon_{ij}}^k S_k{[}S_i, S_j] = -i{\epsilon_{ij}}^k J_k (2 \vec{J}^2 - C_2 - 4){[}C_2, J_i ] = [C_2 , S_i] =0 \mbox{with}\vec{J}^2 = J^2_1 + J^2_2 + J_3^2 \end{array}
\Pi ( \tau ) = \frac{2 n_{c}\, \rho_{c}^{2}\, \zeta}{m^{* 2}\, \tau^{4}\sqrt{\pi}}\int_{0}^{\infty}d \alpha \int_{0}^{\infty}d \beta \, \mathrm{e}^{- \zeta^{2}{t^{2}}}\left( \zeta^{2}t^{3}- \frac{3}{2}t \right) \; \cosh \alpha \; \cosh \beta \; ,
\frac{( 3 \gamma - 2 ) \kappa_{4 )}^{2}\rho_{0}}{6}a^{- 3 \gamma}+ \frac{( 3 \gamma - 1 ) \kappa_{( 4 )}^{2}\rho_{0}^{2}}a^{- 6 \gamma}+ \frac{2}{3}\sigma_{0}a^{- 6}- \frac{\Lambda_{( 4 )}}{3}a + \frac{2{\cal U}_{0}}{\lambda \kappa_{( 4 )}^{2}}a^{- 4}< 0 .
\forall \, p\leq r < q, \;\;\;\; \sharp\{p\leq s \leq r\,|\, D_{\lambda}(s) = \times \}> \sharp\{p\leq s \leq r\,|\, D_{\lambda}(s) = \circ \}\sharp\{p\leq s \leq q\,|\, D_{\lambda}(s) = \times \}= \sharp\{p\leq s \leq q\,|\, D_{\lambda}(s) = \circ \}.
{\zeta_{P \bar{\cal K}}}( s ) = \zeta_{\bar{\cal V}}( s ) + \frac{\Gamma ( s + \frac{1}{2})}{\sqrt{\pi}\, \Gamma ( s )}\left[ \frac{2}{3^{s + \frac{1}{2}}}{}_{2}F_{1}[{\textstyle \frac{1}{2}},{s + \textstyle \frac{1}{2}},{\textstyle \frac{3}{2}}, -{\textstyle \frac{1}{3}}] - \frac{1}{4^{s}}\frac{1}{s}\right]
\gamma= \kappa_0 \kappa_1 \kappa_2 \kappa_3 \Delta_2^2 = \sigma_2 \sigma_1^2 \sigma_2 \sigma_3 \sigma_4w_1 \sigma_2 \sigma_3 \sigma_2^{-1}w_2 \sigma_2 \sigma_4^{-1}\sigma_3^{-1}\sigma_1^2.
\sum_{t \in S_{2}}\mu_1(t^{-1}mt) = -\sum_{\substack{q \in F^{\times}}}\psi_0(q^{-1}(-{a_{21}}^{-1}a_{23}(a_{11}-1)+a_{13})) &= -\sum_{\substack{q \in F^{\times}}}\psi_{0}(q^{-1}(-\delta + a_{13})).
A_n ~=~\begin{pmatrix}b_{1}\sqrt{a_{1}}\sqrt{a_{1}}{b_{2}}\sqrt{a_{2}}\ddots \ddots \ddots & \sqrt{a_{n-2}}{b_{n-1}}\sqrt{a_{n-1}}& \sqrt{a_{n-1}}{b_{n}}\end{pmatrix}~.
\gamma^{\mu}\Psi_{,\mu}= -i\gamma^{\mu}\frac{e^{\gamma^{5}\alpha}}{\rho}\{Im(\overline{\Psi}_{,\mu}\Psi)- j_{\mu}-\gamma^{5}[Im(\overline{\Psi}_{,\mu}\gamma^{5}\Psi) -g_{\mu}] \}\Psi
T=-\frac{1}{2}(\partial X)^2 -\frac{1}{2}(\partial \phi)^2 -\frac{1}{\sqrt{2k}}\partial^2\phi-\frac{1}{2}(\partial H')^2 +\left\{Q_{U(1)},~ \frac{1}{\sqrt{2(k+2)}}\eta(-\partial u+i\partial v')\right\}.
\omega_{\mathrm{h}}^\mathrm{DL}= 2\pi\frac{d_\mathrm{h}}{\lambda^\mathrm{DL}}\cos{\phi}\sin{\theta}= \frac{\lambda^\mathrm{UL}}{\lambda^\mathrm{DL}}\omega_{\mathrm{h}}^\mathrm{UL}= \frac{F_\mathrm{c}^\mathrm{DL}}{F_\mathrm{c}^\mathrm{UL}}\omega_{\mathrm{h}}^\mathrm{UL},
0=\left( \frac{1}{2}\sigma^{2}\left( L_{Y_{I}}C_{x}\right)_{,x}+C^{x}L_{Y_{I}}C_{x}\right) -\frac{T_{I,t}}{T_{I}}\left( \frac{1}{2}\int L_{Y_{I}}C_{x}+C^{x}Y_{I}+2\psi_{I}+\frac{1}{2}\int Y_{I}\mbox{\rm d}x\right) +2f_{I,t}.
\Delta_{he_i}\Delta_{he_j}f(x) = f(x+he_i+he_j) - f(x+he_i) - f(x+he_j) + f(x) &= \frac{f(x+he_i+he_j)-2f(x+he_i)+f(x-he_i+he_j)}{2}&+ \frac{f(x+he_i+he_j)-2f(x+he_j)+f(x-he_j+he_i)}{2}&- \frac{f(x-he_i+he_j) - 2f(x) + f(x-he_j+he_i)}{2}&= \frac{\Delta_{he_i}^2f(x+he_i) + \Delta_{he_j}^2f(x+he_j)- \Delta_{h(e_i+e_j)}^2 f(x)}{2}
e_n=p_n= 1/4, f_n=q_n=-\kappa, g_n=\kappa^2+\omega^2/4, r_n=\kappa^2, t_n=-1/16, & s_n=0, u_n=\kappa/4, w_n=u_n\omega^2, v_n =-(\kappa^2/4+\omega^2/8), z_n=-\omega^2g_n/4, n \geq 1.
\langle a,b;\langle c;d\rangle\rangle= \langle a,b,c;d\rangle+\langle a,\langle b;c\rangle;d\rangle+\langle \langle a,b;c\rangle;d\rangle &+\langle a,c,b;d\rangle+\langle \langle a;c\rangle,b;d\rangle+\langle c,a,b;d\rangle.
L_{\alpha}(n^{3/2}z) = \begin{pmatrix}n^{-\frac{1}{2}}0 0 0 1 0 0 0 n^{\frac{1}{2}}\end{pmatrix}L_{\alpha}(z), L_{\alpha}(n^{3/2}f(z)) = \begin{pmatrix}n^{\frac{1}{2}}0 0 0 1 0 0 0 n^{-\frac{1}{2}}\end{pmatrix}L_{\alpha}(n f(z))