harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c}\langle e_{a_{k}...a_{1}}\mid = (\mid e_{a_{1}...a_{k}}\rangle)^{+}\langle e_{a_{k}...a_{1}}\mid e_{b_{1}...b_{k}}\rangle = \delta_{a_{1}b_{1}}...\delta_{a_{k}b_{k}}\end{array}
\sum_{Y < p \leq X}\frac{1-\left|\cos\left(\pi \left\Vert \frac{(t-t_1) \log p}{2\pi}\right\Vert\right)\right|}{p}\geq \left(1- \int_0^{1}|\cos (\pi t)| dt \right) \log \frac{\log X}{\log Y}+ O(1) &= \left(1-\frac{2}{\pi}\right) \log \frac{\log X}{\log Y}+ O(1)
A(t,k,r) =\binom{r-1}{t}\sum_{j=0}^t(-1)^j\binom{t}{j}\binom{t+j}{j}\Big(\frac{k+1}{r}\Big)^j &+\binom{r-1}{t}\sum_{j=1}^t(-1)^{j}\binom{t}{j}\binom{t+j}{j}\left(\frac{k...(k-j+1)}{(r-1)...(r-j)}-\Big(\frac{k+1}{r}\Big)^j\right)=:\binom{r-1}{t}(S_3+S_4).
c_{1}^{2}\left[ f_{2}^{2}f_{2}^{\prime \prime}-f_{2}\left( f_{2}^{\prime}\right)^{2}\right] f_{1}^{3}+\left[ f_{2}^{\prime \prime}+2ac_{1}f_{2}^{\prime}+a^{2}c_{1}^{2}f_{2}\right] f_{1}=0,
\langle T_{i}^{k}\rangle^{( b )}= \frac{\zeta \delta_{i}^{k}}{\pi \xi^{2}}\int_{0}^{\infty}d \omega \frac{A - B \omega / a}{A + B \omega / a}e^{- 2 \omega \ln ( a / \xi )}\tilde{F}_{0}^{( i )},
a=\xi\bar{\xi}'\cos\theta\cos\theta'+\eta\bar{\eta}'\sin\theta\sin\theta', &b=-\xi\bar{\xi}'\cos\theta\sin\theta'+\eta\bar{\eta}'\sin\theta\cos\theta', &c=-\xi\bar{\xi}'\sin\theta\cos\theta'+\eta\bar{\eta}'\cos\theta\sin\theta', &d=\xi\bar{\xi}'\sin\theta\sin\theta'+\eta\bar{\eta}'\cos\theta\cos\theta'.
\frac{\partial^2\bar{\psi}}{\partial r^2}(1,-e_N) =\left.\frac{d^2}{d\theta^2}\psi(\cos\theta,0,\ldots,0,\sin\theta)\right|_{\theta=0}=\left.\frac{d^2}{d\theta^2}\phi(\cos\theta,0,\ldots,0,\sin\theta)\right|_{\theta=0}=\frac{\partial^2\bar{\phi}}{\partial r^2}(1,-e_N)
W_{r,\le M}:=\; Q_{r,\le M}\big( Q_{r,\le M}(h_\mathrm{MF}-\mu_++2\lambda K_{11})Q_{r,\le M}\big)^{-1}Q_{r,\le M}W_{\ell,\le M}:=\; Q_{\ell,\le M}\big( Q_{\ell,\le M}(h_\mathrm{MF}-\mu_++2\lambda K_{22})Q_{\ell,\le M}\big)^{-1}Q_{\ell,\le M}.
(z-\mu^{(t)}){\cal H}_{k-1}^{(s,t+1)}(z)={\cal H}_{k}^{(s,t)}(z)+Q_k^{(s,t)}{\cal H}_{k-1}^{(s,t)}(z),s,t=0,1,\dots, &{\cal H}_{k}^{(s,t)}(z)={\cal H}_{k}^{(s,t+1)}(z)+E_{k}^{(s,t)}{\cal H}_{k-1}^{(s,t+1)}(z),s,t=0,1,\dots,
{\cal L}^{(p)}_{\rm WZ}= \epsilon^{i_1\ldots i_{p+1}}\sum_{k=0}^{(p-1)/2}\,\,a_{p,k}C_{i_1\ldots i_{p-2k}}(F^k)_{i_{p-2k+1}\ldots i_p}\partial_{i_{p+1}}T\,,
\Sigma_{\bar{X}^j \bar{Y}^j}= \mbox{cov}(Y_0^j + P^{'}(P P^{'})^{-1}\bar{V}^j + \bar{U}^j , Y_0^k + P^{'}(P P^{'})^{-1}\bar{V}^k ) &= \Sigma_{\bar{V}^j}P^{'}(P P^{'})^{-1}(P P^{'})^{-1}P^{'}.
z_{2}(x,y)=E_{1}(x,y)\int\frac{E_{2}(x,y)}{E_{1}(x,y)}F_{2}(\varphi_{2}(x,y))|_{y=\psi_{1}(x,\overline{y})}dx|_{\overline{y}=\varphi_{1}(x,y)};
\tilde D_\Theta = \{x\in V^1\cap V_\Theta: T^\#_\Theta x \in (j_\Theta^*\circ c)V_\Theta \}= \{x\in V^1\cap V_\Theta: T^\#_\Theta x\in (i^*_\Theta\circ j^*\circ c)V\}.
\frac{\hat{f}_2(k/n;Z,\kappa)}{f_2(k/n;X;\kappa)}= \frac{\hat{f}_2(k/n;X,\kappa)}{f_2(k/n;X;\kappa)}+ \frac{\hat{f}_2(k/n;u,\kappa)}{f_2(k/n;X;\kappa)}+\frac{\hat{f}_{1,1}(k/n;X,u,\kappa)}{f_2(k/n;X;\kappa)}.
\begin{array}{lll}\left.\partial\,^kR_n\right|_{\xi=1}=0 \quad k\in\mathcal{N}=\{n-p, \dots, n-1\}\left.\partial\,^kR_m\right|_{\xi=1}=0 k\in\mathcal{M}=\{m-p, \dots, m-1\}\end{array}
\begin{gathered}A = u\Big( \triangle\alpha - (\lambda+k-2)(\lambda+n-k)\alpha -2d^\ast\beta \Big) -r\dot{u}\left( 2\lambda +n-1\right) \alpha - r^2 \ddot{u}\,\alpha, B = u\Big( \triangle\beta - (\lambda+n-k-2)(\lambda+k)\beta -2d\alpha \Big) -r\dot{u}\left( 2\lambda +n-1\right) \beta - r^2 \ddot{u}\,\beta.\end{gathered}
\begin{array}{ll}E\cdot r_6 = E \cdot (vv'-q^{-\frac{1}{2}}v'v) = (K \cdot v)(E \cdot v') + (E \cdot v)v' - q^{-\frac{1}{2}}((K \cdot v')(E \cdot v) + (E \cdot v')v) & = q^{-1}vu' + uv' - q^{-\frac{1}{2}}(q^{-1}v'u + u'v) &= r_5 + q^{-1}r_4.\end{array}
\delta:=\frac{1}{1-\lambda_h\kappa_{m}^2h_{\mathcal{T}}^{2{m}}}-1=\frac{\lambda_h\kappa_{m}^2h_{\mathcal{T}}^{2{m}}}{1-\lambda_h\kappa_{m}^2h_{\mathcal{T}}^{2{m}}}=\lambda_h\kappa_{m}^2h_{\mathcal{T}}^{2{m}}(1+\delta)\in P_0(\mathcal{T}).
\begin{cases}-dY_{t}=f(t,\eta_{t},y_{t},z_{t},y_{t+\delta(t)},z_{t+\zeta(t)}) dt - Z_{t}dB_{t}^{H}, \ \ \t\in [0,T]; Y_{t}=g(\eta_{t}), \ \Z_{t}=h(\eta_{t}), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t\in[T,T+K]. \end{cases}
y_{t}= x_0-\sum_{t(l)<t/\Gamma}\nabla f(x_{t(l)})\Gamma+\sum_{t(l)<t/\Gamma}\epsilon(l)\Gamma= y_0-\sum_{\Gamma t(l)<t}\nabla f(y_{\Gamma t(l)})\Gamma+\sum_{t(l)<t/\Gamma}\epsilon(l)\Gamma.
K_j = K_j^2 = \left(\prod_{i: v_i \in I_j}P_i\right)^2 = \left(\prod_{i: v_i \in I_j}P_i \prod_{i: v_i \notin I_j}I\right)^2= \prod_{i=1}^{n}\hat{P}_i \prod_{i=0}^{n-1}\hat{P}_{n-i}.
|P_q^cY|_\alpha = |\zeta_q(Y)|\,|Y_q'|_\alpha\le C|Y|_\alpha\,|Y_q'|_\alpha\le C|Y|_\beta\, |Y_q'|_\alpha, |P_q^cY|_0 = |\zeta_q(Y)|\,|Y_q'|_0\le C|Y|_\alpha\,|Y_q'|_0\le C|Y|_\beta \,|Y_q'|_0.
\int_0^T\int_{\Omega}\nabla\Phi_{N}\cdot \nabla\Psi\,dxdt+\varepsilon\int_0^T\int_\Omega\nabla^{2m+1}\Phi_{N}\cdot\nabla^{2m+1}\Psi\,dxdt &=\int_0^T\int_\Omega\big(n_{N}-c(x) \big)\Psi\,dxdt,
\sum_{A}M_{A}\int_{x}^{\infty}d \theta \frac{d \epsilon_{A}^{>}}{d \theta}L_{A}^{>}( \theta ) = \sum_{A}M_{A}\int_{\epsilon_{A}^{>}( x )}^{\epsilon_{A}^{>}( \infty )}d \epsilon_{A}^{>}\ln ( 1 + e^{\epsilon_{A}^{>}( \theta )}) ,
5: \tilde{T}_k^\prime ={\cal R}[T_k^\prime] O(N_v^3) = O(L^9) \ , & 6: \tilde{T}_{k+1}={\cal O}[\tilde{T}_k^\prime] \leq O(N_d N_s) = O(L^4) \ , & 1: T_{k+1}={\cal R}^{-1}[\tilde{T}_{k+1}] O(N_v^3) = O(L^9) \ .
\hspace{0.5in}+\lambda\frac{1}{3^3}\frac{1}{2^{12}}\frac{1}{\pi^4}\big{[}\Omega_5\big{]}^{\frac{1}{2}}cot{\theta_4^0}\sum_{N=1}^{\infty}C_N^{\frac{3}{2}}(cos{\theta_4^0})C_N^{\frac{3}{2}}(cos{\theta_4})R\frac{1}{\epsilon}\int_Cd^4\eta D_c(\eta_1 - \eta)D_c(\eta - \eta_2)
\left|A\,\eta(x)-B\,\eta(y)\right|^p=\left|(A-B)\,\frac{\eta(x)+\eta(y)}{2}+(A+B)\,\frac{\eta(x)-\eta(y)}{2}\right|^p \le 2^{p-2}\,|A-B|^p\, (\eta(x)^p+\eta(y)^p) &+2^{p-2}\, (|A|^p+|B|^p)\, |\eta(x)-\eta(y)|^p.
\left\lbrace\begin{aligned}dQ^\epsilon(t) =f\bigl(Q^\epsilon(t)-P^\epsilon(t)\bigr)dt+\sigma\bigl(Q^\epsilon(t)-P^\epsilon(t)\bigr)d\beta(t), dP^\epsilon(t) =-\frac{P^\epsilon(t)}{\epsilon^2}dt+f\bigl(Q^\epsilon(t)-P^\epsilon(t)\bigr)dt+\sigma\bigl(Q^\epsilon(t)-P^\epsilon(t)\bigr)d\beta(t).\end{aligned}\right.
\frac1{\eta_i}:=\frac1{\delta_i}\Big(\sum_{j=1}^{m-1}\frac1{\delta_j}\Big)^{-1}=\frac1{\delta_i}\Big(\sum_{j=1}^{m+1}\frac1{\delta_j}-\frac1{\delta_m}-\frac1{\delta_{m+1}}\Big)^{-1}=\frac1{\delta_i}\Big(\frac{1-r}{r}-\frac1{\varrho}\Big)^{-1}
u_{1}= x_1x_2x_3^{7},\u_{2}= x_1^{3}x_2^{3}x_3^{3},\u_{3}= x_1x_2^{3}x_3^{5}, u_{4}= x_1x_2x_3^{2}x_4^{2}x_5^{3},\u_{5}= x_1x_2x_3^{2}x_4^{5},\u_{6}= x_1x_2^{2}x_3^{3}x_4^{3}.
h_1= x^4 + x y^3 + y^4 - x^2 y z - x y^2 z, h_2= x^4 - x^2 y^2 - y^4 + x^2 y z + y z^3, h_3= x^2y + xy^2 + x^2z - xyz + y^2z - xz^2 - yz^2 - z^3.
\|{\partial}(ab)-a{\partial}b\|_\tau^2=\|({\partial}a)b\|_\tau^2=\tau(\langle({\partial}a)b, ({\partial}a)b\rangle_{\mathcal A})=\tau(b^*\langle{\partial}a,{\partial}a\rangle_{\mathcal A}b) \phantom{\|{\partial}(ab)-a{\partial}b\|_\tau^2}\leq \|\langle{\partial}a,{\partial}a\rangle_{\mathcal A}\|_\infty \tau(b^*b),
f_0 =1\quad\mbox{and}f_k=zf_{k-1}+zg_{k-1}+zh_{k-1},k\geq 1, g_k =z\sum\limits_{\ell\geq k+1}f_\ell+z\sum\limits_{\ell\geq k+1}g_\ell +z\sum\limits_{\ell\geq k+1}h_\ell,k\geq 0, h_k =zf_k+zg_k+zh_k,k\geq 0.
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{align*}4pt] u \mbox{is a global weak solution of \eqref{eq1}such that}\tilde u\in L^p({\mathbb{R}}^n),
A^{ij}W_{i;j}=\dfrac{A^{ij}f_{i;j}}{2W}+\dfrac{A^{ij}A^{kl}}{W}\hat{u}_{l;j}\hat{u}_{k;i}+\dfrac{A^{ij}f_{i}f_{j}}{4W^{3}}+\dfrac{1}{\eta W}A^{ij}\eta_{i}f_{j}-A^{ij}N^{k}\hat{u}_{k;ij}.
||\widehat{T}(z_{0})\psi||_{\phi,S_{0}}=\int_{\mathbb{R}^{2n}}|W(\widehat{T}(z_{0})\psi,\phi)(z)|dz & =\int_{\mathbb{R}^{2n}}|W(\widehat{T}(z_{0})\psi,\widehat{T}(-z_{0})\phi)(z-z_{0})|dz & =||\psi||_{\widehat{T}(-z_{0})\phi,S_{0}}
v_{a}^{+}= e^{i \alpha}\sqrt{\frac{M}{m}}\left( \begin{matrix}{{e^{- w_{2}/ 2}u^{+}}}\\{{- e^{w_{2}/ 2}u^{+}}}\\ \end{matrix}\right) \hspace{0 . 5 c m}v_{b}^{-}= e^{i \delta}\sqrt{\frac{M}{m}}\left( \begin{matrix}{{- e^{w_{2}/ 2}u^{-}}}\\{{e^{- w_{2}/ 2}u^{-}}}\\ \end{matrix}\right) ,
F ( \phi^{i}, \phi_{i}^{*}, c_{a}^{*}, c_{a_{1}}^{*}) = \sum_{m , n = 0}\left( c_{a_{1}}^{*}\right)^{m}\left( y_{a_{1}}\right)^{n}F_{m , n}( \phi , \phi^{*}, c^{*}) \ .
(-1)^{p+1}d^{(k, i)\, +}_{p+1}= \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}}\, (-1)^{p-1}(-1)^{2}d^{(k, i)\, +}_{p-1}- \sqrt{\frac{(i+pk)!}{(i+pk+k)!}}(-1)^{p}d^{(k, i)\, +}_{p}
\frac{d^{2}}{dt^{2}}\bigg|_{t=0}V_k(g_t) =\frac{d^{2}}{dt^{2}}\bigg|_{t=0}\bigg[K-\frac{K}{2}|J|^{2}\bigg] =-K\frac{d}{dt}\bigg|_{t=0}\int_{\mathbb T}\langle \nabla\varphi, \nabla(\sigma\cdot J)\rangle g_{t}~ d\theta .
\begin{aligned}\mathcal{R}_kq_0(t) =\mathcal{K}_0(\mathcal{R}_{k-1}q_0(t)), \mathcal{R}_k\tilde{q}_0(t) =\mathcal{K}_0(\mathcal{R}_{k-1}\tilde{q}_0(t)), \mathcal{R}_kp(t) =\mathcal{K}_1(\mathcal{R}_{k-1}p(t)), \mathcal{R}_k\tilde{p}(t) =\mathcal{K}_1(\mathcal{R}_{k-1}\tilde{p}(t))\end{aligned}
\left\{\sum\limits_{\ell=m+1}^{\infty}a_{\ell}\cdot\left(\frac{t}{t-1}\right)^\ell+\sum\limits_{\ell=m+1}^{\infty}b_{\ell}\cdot \left(\frac{t}{t-\lambda}\right)^\ell | a_{\ell},b_{\ell}\in k\right\}
||D^{\alpha}_{\ast}(\mathcal{A}_3{\mathbf V})||^2_{L^2(\Omega_t)}\lesssim||\mathcal{A}_3{\mathbf V}||^2_{s,\ast,t}&\leq C(K)\Big(||{\mathbf V}||^2_{s,\ast,t}+||\dot{{\mathbf U}}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s+2,\ast,t}\Big).
Q_{+}= \int d x \left\{\bar{\psi}_{-}^{\bar{j}}( x ) \left( \frac{\partial}{\partial \bar{\Phi}^{\bar{j}}}( x ) + \frac{1}{2}G_{i \bar{j}}\frac{\partial \Phi^{i}( x )}{\partial x}\right) + \psi_{+}^{j}( x ) \Omega_{j}( \Phi ( x ) ) - \psi_{+}^{l}\bar{\psi}_{-}^{\bar{i}}\bar{\psi}_{+}^{\bar{j}}\frac{\partial G_{l \bar{j}}}{\partial \bar{\Phi}^{\bar{i}}}\right\}\, .
\begin{array}{l}\dot{S}_1=\displaystyle{\frac{1}{3}(1+3S_1)(3S_2-3S_1+2)}, \dot{S}_2=\displaystyle{\frac{1}{9}\left(27 S_2^2-27 S_1S_2-9S_2+3 S_1-2\right)}. \end{array}
\Gamma_{4,tree}^{\hat{\varepsilon}^0a_0...\hat{\varepsilon}^3a_3}\propto\left[ (\hat{\varepsilon}^0\cdot \hat{\varepsilon}^1) (\hat{\varepsilon}^2\cdot \hat{\varepsilon}^3) f^{a_0a_3e}f^{a_1a_2e}+ perm.\right]\neq 0.
|Tb_l|= |\int_{K_l}(\frac{x_j-y_j}{|x-y|^{n+1-\beta}}-\frac{x_j-\bar y_j}{|x-\bar y|^{n+1-\beta}}) b_l dy | \le C\delta_l\int_{K_l}\frac{1}{|x-y|^{n+1-\beta}}|b_l(y)| dy = C\delta_l\int_{R^n}\frac{|b_l(y)|\mathrm{I}_{K_l}(y)}{|x-y|^{n+1-\beta}}dy,
r_{c}:=\sqrt{\frac{6}{5}}[\frac{\sigma_{b}\sigma_{t}}{3\sigma_{b}\sigma_{t}+\sigma_{c}(\sigma_{b}-\sigma_{t})}];r_{t}:=\sqrt{\frac{6}{5}}[\frac{\sigma_{b}\sigma_{t}}{\sigma_{c}(2\sigma_{b}+\sigma_{t})}]
(I_i + f(u) E_i)(I_i + f(-u) E_i)=(I_j + f(u) E_j)(I_j + f(-u) E_j) =I_i + (f(u)+f(-u)) E_i + f(u)f(-u)E_i^2 = I_j + (f(u)+f(-u)) E_j + f(u)f(-u)E_j^2 =I_i + (f(u)+f(-u)) E_i + f(u)f(-u) \delta E_i =I_j + (f(u)+f(-u)) E_j + f(u)f(-u) \delta E_j
I(\overline{C}^{N};\overline{E}^{N}|\overline{S}^{N}=s^{N})=\frac{1}{2}\sum_{i=1}^{N}log_{2}(\frac{|s_{i}^{2}C_{U}+\sigma_{n}^{2}I|}{|\sigma_{n}^{2}I|}),
\sigma_{r e n}={\frac{M^{4}}{6 4 \pi^{2}}}[ \Psi ({\frac{3}{2}}+ \nu ) + \Psi ({\frac{3}{2}}- \nu ) - 2 \ln M \tau -{\frac{2}{3 ( M \tau )^{2}}}+{\frac{1}{1 5 ( M \tau )^{4}}}]
X\triangleright\mathsf{M}_{m}^{n}=\pi(S(X_{(1)}))\mathsf{M}_{m}^{n}\pi(X_{(2)}),\quad\mathsf{N}_{m}^{n}\triangleleft X=\pi(X_{(1)})\mathsf{N}_{m}^{n}\pi(S(X_{(2)})).
V_1 V_2^* = (I_{H^2}\otimes P^\perp U+ M_z\otimes PU)(I_{H^2}\otimes P U + M_z^*\otimes P^\perp U) & = I_{H^2}\otimes{P^\perp}U P U + M_z \otimes P U P U+ M_z^*\otimes{P^\perp}U{P^\perp}U + M_z M_z^* \otimes P U{P^\perp}U .
\begin{array}{c c}{{\mathrm{T r}[ A , \phi ]^{2}}}&{{= v^{2}\mathrm{T r}[ X ( \theta )^{- 1}A X ( \theta ) , \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{- 2}}\\ \end{array}\right) ]^{2}}}\\{{}}&{{= 1 8 v^{2}( ( A^{1}\sin \theta - A^{2}\cos \theta )^{2}+ ( A^{3})^{2}) ,}}\\ \end{array}
\log(e^U e^V) = \sum_{k > 0}\frac{(-1)^{k-1}}{k}\underset{\substack{r_i+s_i > 0, 1 \leq i \leq k}}{\sum}a(r_1,s_1,...,r_k,s_k) (ad U)^{r_1}(ad V)^{s_1}\cdots (ad U)^{r_k}(ad V)^{s_k-1}V.
n_{i}=|C_{m}\bigcap(C_{m}+1)|+|C_{m}\bigcap(C_{m+1}+1)|+|C_{m+1}\bigcap(C_{m}+1)|+|C_{m+1}\bigcap(C_{m+1}+1)| =(m,m)_{7}+(m,m+1)_{7}+(m+1,m)_{7}+(m+1,m+1)_{7}, =(m,m)_{7}+2(m,m+1)_{7}+(m+1,m+1)_{7}.
\sum_{i=1}^n \widehat{R}_{T,i}(\widehat{R}_{Y,i}- \theta_0\widehat{R}_{T,i}) = \sum_{i=1}^n R_{T,i}(R_{Y,i}- \theta_0R_{T,i}) + \sum_{i=1}^n (\widehat{R}_{T,i}- R_{T,i})(R_{Y,i}- \theta_0R_{T,i}) &\quad- \sum_{i=1}^n \widehat{R}_{T,i}(\widehat{R}_{Y,i}- R_{Y,i}- \theta_0(\widehat{R}_{T,i}- R_{T,i})).
\vec{a}= [{\frac{1}{6}}\sin{\phi_{1}}\sin{3 u}- ( \pm_{2}){\frac{1}{2}}\cos{\phi_{1}}\cos{( 2 u - \theta_{2})}+{\frac{1}{2}}\sin{\phi_{1}}\sin{( u - 2 \theta_{2})}] \, \hat{x}
\begin{array}{c}{{p_{( 1 )}\cdot p_{( 2 )}= \tilde{p}_{( 1 )}\cdot \tilde{p}_{( 2 )}= 2 t / \alpha^{\prime}, \qquad p_{( 1 )}\cdot D \cdot \tilde{p}_{( 2 )}= \tilde{p}_{( 1 )}\cdot D \cdot p_{( 2 )}}}\\{{\mathrm{a n d}\qquad p_{( 1 )}\cdot D \cdot \tilde{p}_{( 1 )}= p_{( 2 )}\cdot D \cdot \tilde{p}_{( 2 )}= 2 s / \alpha^{\prime},}}\\ \end{array}
= \sum_{\lambda\subset (n+1)^m}(-1)^{m(n+1)-|\lambda |}\Bigl( P_{\lambda}(x_1,\ldots ,x_m;q,t,\mathbf{t}) & \times \sum_{\substack{\mu\subset n^m, \, 0\leq r\leq m \mu\sim_{r}\lambda}}(-1)^{r+|\lambda|-|\mu|}C^{\lambda,m}_{\mu , r}(q,t,\mathbf{t}) P_{m^n-\mu^\prime}(z_1,\ldots ,z_n;t,q,\mathbf{t}) \langle z;t_0\rangle_{t,m-r}\Bigr) ,
\mathcal{L}_{K, (is_1, is_2+js_2)}(\alpha_1, \alpha_2\alpha_3)=\sum_{\substack{i_1>i_2>0 i_2\neq N}}\frac{\alpha_1^{q^{i_1}}\alpha_2^{q^{i_2}}}{(\theta^{q^{i_1}}-t)^{is_1}(\theta^{q^{i_2}}-t)^{is_2+js_2}}+\sum_{i_1>N=i_2>0}\frac{\alpha_1^{q^{i_1}}\alpha_2^{q^{i_2}}}{(\theta^{q^{i_1}}-t)^{is_1}(\theta^{q^{i_2}}-t)^{is_2+js_2}}.
(2d-1) \sum_{1\leq i<j\leq N}\lambda_i\lambda_j\|A(\beta_i-\beta_j)\|_2^2 =\sum_{i=1}^{N}\lambda_i\Bigl\|A(\sum_{j=1}^{N}\lambda_j\beta_j-d\beta_i)\Bigr\|_2^2 - \sum_{i=1}^{N}\lambda_i(1-d)^2\|A\beta_i\|_2^2,
\Phi ( \vec{k}, t ) = \int d^{3}k{\hat{\Phi}}( \vec{k}, t ) e^{i{\vec{k}}\cdot{\vec{x}}}, \quad \mathrm{w i t h}\quad{\overline{{\hat{\Phi}}}}( \vec{k}, t ) ={\hat{\Phi}}( - \vec{k}, t ) , \nonumber
B_j^+\Omega_\lambda=\sum_{i=1}^j\sum_{s=1}^{j-i+1}\sum_{I\in\mathcal{I}_{ij}(s)}(-1)^{\sum_{\alpha=j}^n(\alpha+1)(\lambda_\alpha-\lambda_{\alpha+1})}\frac{\lambda_i-\lambda_j+1}{\lambda_i+1}d_I^+(\mu)\Omega_{\lambda+\epsilon_i},
S\left[(m+\frac{\delta}{2})iv_\mu+q_\mu \right] = \frac{1+\gamma_3}{2}\, \frac{1}{(\mbox{$ \, -\frac{1}{2}\,$}\delta+iq_3+\frac{\left|{\bf q}\right|^2}{2m})-\frac{1}{4\pi}\ln \frac{\epsilon}{4}}\, + \, O\left(\frac{\ln m}{m}\right).
{\bar{\psi}}\gamma^{\mu}\stackrel{\leftrightarrow}{\nabla}_{\mu}\psi ={\bar{\psi}_{R}}\gamma^{\mu}\stackrel{\leftrightarrow}{\nabla}_{\mu}\psi_{R}+{\bar{\psi}_{L}}\gamma^{\mu}\stackrel{\leftrightarrow}{\nabla}_{\mu}\psi_{L}
\dot z(t) =(A+BK)z(t)+e^{Ar_1}BK[z(t-r_1-\tau_4(t))-z(t-r_1)],t\in[0,t_{-1}^*)\cup[t_k,t_k^*), \dot z(t) =(A+BK)z(t)+e^{Ar_1}BK[z(t-r_1-\mu(t))-z(t-r_1)]+e^{Ar_1}Be_3(t),\,t\in[t_{-1}^*,t_0)\cup[t_k^*,t_{k+1})
\int_{\Delta_\tau \cap \{|x_{N+1}|=L\}}e^{-\frac{|x|^2}{4}}\, |\langle \omega,\nu_{}\rangle| \leq C L^{-1}\int_{\Delta_\tau \cap \{|x_{n+1}|=L\}}e^{-\frac{|x|^2}{4}}\, |x_1^2+\hdots+x_n^2-2(n-1)| \leq C L^{-1}\int_{\Sigma \cap \{|x_{n+1}|=L\}}e^{-\frac{|x|^2}{4}}\, u^2.
\begin{cases}A_1^{-1}g\in W^{2,p}(\Omega)\mbox{for all}\g\in L^p(\Omega) \mbox{in case}\ \sigma > 0 \ \mbox{or}\p \leq 2^*, A_1^{-1}g\in W^{2,\rho}(\Omega)\mbox{for all}\g\in L^\rho(\Omega) \ \mbox{and}\ \rho \in [2^*,p] \mbox{in case}\ \sigma = 0 \ \mbox{and}\p > 2^*,\end{cases}
\Delta_{\kappa}(\vec{x},x_{0}) \sim{-i\kappa \cosh \left({\pi \kappa x^{0}\over 2}\right) \Gamma \left( 1 + i \kappa x^{0}\right) \over\left(2\pi\right)^{2}\sqrt{1 +{m^{2}\over \kappa^{2}}}\,|\vec{x}| \left( \kappa | \vec{x}|\right)^{1+ i \kappa x^{0}}}
(\frac{\partial}{\partial y_{j}}(\frac{\partial\Phi}{\partial y_{k}})_{\{y_{i\ne k}\}})_{\{y_{i\ne j}\}}=(\frac{\partial}{\partial y_{k}}(\frac{\partial\Phi}{\partial y_{j}})_{\{y_{i\ne j}\}})_{\{y_{i\ne k}\}}
W^s((0,x);F) = \{0\}\times W^s(x;P), W^u((0,x);F) = \bigcup_{-\infty<s<1}\{r\}\times \Phi^{\tilde P}_{t_+(r)}\big (W^u(x;P)\big ), W^s((1,y);F) = \bigcup_{0<s<+\infty}\{r\}\times \Phi^{P}_{t_-(r)}\big (W^s(y;\tilde P)\big ), W^u((1,y);F) = \{1\}\times W^u(y;\tilde P),
W_{-1}= w_{-1}\otimes w_0 - q^2 w_0 \otimes w_{-1}, W_1 = w_0 \otimes w_1 - q^2 w_1 \otimes w_0, W_0 = w_{-1}\otimes w_1 - w_1 \otimes w_{-1}- q^{-1}(q - q^{-1}) w_0 \otimes w_0.
v_{j+1}(t)=e^{it\Delta}(\chi_{j+1}u_{0})+i\int_{0}^{t}e^{i(t-\tau)\Delta}(\Delta \chi_{j+1}v_{j}+2\nabla \chi_{j+1}\nabla v_{j}-v_{j+1}|v_{j}|^{2})(\tau)d\tau,
\int_{-\infty}^{\infty}\exp\left(-c_1\left(a-\psi\right)^2-c_2\left(\psi-b\right)^2\right)d\psi \quad\quad\quad\quad\quad\quad= \left(\frac{\pi}{c_1+c_2}\right)^{\frac{1}{2}}\exp\left(-\frac{c_1 c_2}{c_1 +c_2}\left(a-b\right)^2\right)
\left(\tilde{\Phi}_{0}(\xi_{2},\!\xi_{3},\!t)\right)_{\alpha}=\left(\begin{array}{c}0 \left(\!\tilde{\Phi}_{0}(\xi_{2},\!\xi_{3},\!t)\right)_{2}0 \left(\tilde{\Phi}_{0}(\xi_{2},\!\xi_{3},\!t)\right)_{4}\end{array}\!\right)\hspace{0.1cm}.
S^{k(v)}I_{g(v), k(v)}(\alpha_1, \dots, \alpha_{k(v)}) = \int_U a^{I_v^1, \dots, I_v^{k(v)}}\frac{\partial \alpha_1}{\partial x^{I_v^1}}\dots \frac{\partial \alpha_{k(v)}}{\partial x^{I_v^{k(v)}}}.
\nabla^{A}\partial_{A}\Phi^{a}= -{\frac{( \partial^{A}\Phi^{b}) ( \partial_{A}\Phi^{b})}{\eta^{2}}}\Phi^{a}+{\frac{\sqrt{- h}}{\sqrt{- g}}}{\frac{\partial \sigma_{i}( \Phi^{a})}{\partial \Phi^{a}}}\delta ( y_{i}) \, ,
\begin{array}{c}S'=-\frac{i\beta}{4}\left\{\frac{1}{\epsilon},{\cal O}'\right\}=-\frac{i}{16}\left\{\frac{1}{\epsilon},\left\{\frac{1}{\sqrt{1+X^2}},[X,{\cal F}]\right\}\right\}+\frac{i}{32}\left\{\frac{1}{\epsilon},\left\{\frac{X}{(1+X^2)(1+\sqrt{1+X^2})},[X^2,{\cal F}]\right\}\right\}. \end{array}
\sum_{\substack{z_j^k<n, m<z_{j+1}^k mn=\square}}\frac{b_{k}(n;z_j,z_{j+1})b_{k}(m;z_j,z_{j+1})}{mn}&\leq k! \sum_{0\leq s\leq k/2}\binom{k}{s, s, k-2s}\left(\sum_{d}\frac{b_{k-2s}(d;z_j,z_{j+1})}{d^2}\right) \left(\sum_{\ell}\frac{b_{s}(\ell;z_j,z_{j+1})}{\ell^2}\right)^2 &\leq 3^k k!\left(\sum_{z_j<p<z_{j+1}}\frac{1}{p^2}\right)^{k},
J_p(s)=\sqrt{\pi}e^{-2\pi ip\tilde{t}_1}\sum_{n_2}\int_{0}^{\infty}\frac{dt}{t}\frac{t^s}{\sqrt{t}}e^{-\frac{\tilde{n}_2^2}{|\tau|^2}t- \frac{\pi^2}{t}p^2}.
M_{\Omega_1}= \frac{1}{\alpha}(D_{A\Omega_1}- \beta L_{A\Omega_1}), N_{\Omega_1}= \frac{1}{\alpha}[(1-\alpha)D_{A\Omega_1}+(\alpha-\beta) L_{A\Omega_1}+\alpha U_{A\Omega_1}]; &M_{\Omega_2}=D_{A\Omega_1}-U_{A\Omega_1}, N_{\Omega_2}= L_{A\Omega_1}.
1 + a_n^{p^j}+ \cdots + a_n^{(p-1)p^j}=\frac{a_n^{p^{j+1}}-1}{a_n^{p^j}-1}&\equiv p +\pi^{(p-1)\gamma_j}\delta_j^{(p-1)}({\rm mod}\ \pi^{e+1}).
\overline{H}\left(\bigsqcup_{a \in \mathcal{A}}^{P_A}G_a \right) = \lim_{n \rightarrow \infty}\frac{1}{n}H_\chi\left(\bigwedge_{a \in \mathcal{A}}G_a^{\wedge n T_{\overline{a}^n}(a)}\right).
U^{\mathrm{bf}}_{\mathrm{AP}}(\vec x;C) =e^{-\mathrm{i}{\theta\over 2}\tau^3}U^{\mathrm{bf}}_{\mathrm{AP}}(D(e^{\mathrm{i}{\theta\over 2}\tau^3})\cdot\vec x;e^{\mathrm{i}{\theta\over 2}\tau^3}C e^{-\mathrm{i}{\theta\over 2}\tau^3})e^{\mathrm{i}{\theta\over 2}\tau^3}
\overline{[A_{a,r}]}=\sum_{i=0}^r v^{-ia -{i+1 \choose 2}}\cdot \frac{\prod_{k=1}^i (1-v^{2(a+k)})}{\prod_{k=1}^{\lfloor \frac{i}2\rfloor}(1-v^{4k})}[A_{a+i, r-i}] = \sum_{i=0}^r \frac{\prod_{k=1}^i (v^{-a-k}-v^{a+k})}{\prod_{k=1}^{\lfloor \frac{i}2\rfloor}(1-v^{4k})}[A_{a+i, r-i}].
\varepsilon.\frac{1}{\varepsilon}\ell n (_q<\zeta_\ell|\zeta_{\ell-1}>_q)\cong\frac{\varepsilon}{2}(\frac{\Delta\bar{\zeta}_\ell}{\varepsilon}\_q<\zeta_\ell|T^-_i|\zeta_\ell>_q-\frac{\Delta \zeta_\ell}{\varepsilon}\_q<\zeta_\ell|T^+_i|\zeta_\ell>_q)\,
A_{\Gamma_{\tau}}=\frac{1}{\sqrt{1+\left|\partial_{z}\hat{w}\right|^{2}}}\left(\begin{array}{ccc}\frac{\partial_{zz}^{2}\hat{w}}{1+\left|\partial_{z}\hat{w}\right|^{2}}\frac{\partial_{z}\hat{w}}{z}\,I_{n-1}\frac{-1}{\hat{w}}\,I_{n-1}\end{array}\right)
\mathcal{E}_{m}(f) =\sum_{k=0}^{4^{m}-1}c(I_{k}^{(m)})\left(f\left(\frac{k}{4^{m}}\right)-f\left(\frac{k+1}{4^m}\right)\right)^{2}, \mathcal{E}(f) = \lim_{m\to\infty}\mathcal{E}_m.
\underline{L}+ L' ={r (x - a) - x r' \over r r'}\left [{x \over 2 r}(L - \underline{L}) -{y \over r}e_{x y}\right ] +{r y - r' y \over r r'}\left [{y \over 2 r}(L - \underline{L}) +{x \over r}e_{x y}\right ].
W_{\wedge}= \left\{\begin{array}{c}{L_{- 1}\quad L_{0}\quad L_{1}}\\{J_{- 2}\quad J_{- 1}\quad J_{0}\quad J_{1}\quad J_{2}}\\{\Lambda_{- 3}\quad \Lambda_{- 2}\quad \Lambda_{- 1}\quad \Lambda_{0}\quad \Lambda_{1}\quad \Lambda_{2}\quad \Lambda_{3}}\\{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots}\\ \end{array}\right\}
W = (n-2-(n-4))\rho_G - \bigoplus_{i=1}^L \lambda(i) - \bigoplus_{i=1}^{\frac{1}{2}(m_1p-n)}\lambda(i) = 2\rho_G - \Big(\bigoplus_{i=1}^L \lambda(i) + \bigoplus_{i=1}^{\frac{1}{2}(m_1p-n)}\lambda(i) \Big)
\psi_{l}\left(\frac{a\overline{b}}{l^{\frac{n_l}{2}}}m\right)e\left(\frac{a\overline{b}}{l^{\frac{n_l}{2}}}m\right) = \psi\left(\frac{a\overline{b}}{l^{\frac{n_l}{2}}}m\right) = 1.
\sum_{N\ge 1}(-1)^{N+1}y^N[z^N]\frac{(1-z)^{-x}}{1-tz}\bigl((1-z)t\bigr)^N = \frac{1}{2\pi i}\oint_{|z|=2/3}\frac{(1-z)^{-x}}{z(1-tz)}\frac{\tfrac{y(1-z)t}{z}}{1+\tfrac{y(1-z)t}{z}}\,dz = \frac{1}{2\pi i}\oint_{|z|=2/3}\frac{(1-z)^{-x}}{z(1-tz)}\cdot\frac{y(1-z)t}{z+y(1-z)t}\,dz.
\frac{\partial^{4}\varphi(\omega)}{\partial\omega^{4}}={(\frac{\lambda}{2\pi c})}^{4}(24\lambda\frac{\partial\varphi(\lambda)}{\partial\lambda}+36\lambda^{2}\frac{\partial^{2}\varphi(\lambda)}{\partial\lambda^{2}}+12\lambda^{3}\frac{\partial^{3}\varphi(\lambda)}{\partial\lambda^{3}}+\lambda^{4}\frac{\partial^{4}\varphi(\lambda)}{\partial\lambda^{4}})
A = \frac{\Gamma \left( 1 - n + m \alpha_{-}^{2}\right)}{\Gamma \left( - n + ( 1 + m ) \alpha_{-}^{2}\right) \Gamma \left( 1 - \alpha_{-}^{2}\right)}C_{n m , 1 2}^{n , m + 1}A_{n , m + 1}^{\alpha}
E(\max_{2^{a}\leq k\leq2^{b}}|s_{k}E^{k}X|^{2})\leq\sum\nolimits_{j=a}^{b-1}E(\max_{2^{i}\leq k\leq2^{i+1}}|s_{k}E^{k}X|^{2}) \ll\sum\nolimits_{j=a}^{b-1}\left( s_{2^{j+1}}^{\ast}\right)^{2}E|E^{2^{j}}X|^{2}\ll\sum\nolimits_{j=2^{a-1}}^{2^{b}}j^{-1}\left( s_{4j}^{\ast}\right)^{2}E|E^{j}X|^{2}.
\sigma_{X^*}(m) = \lim_{t\to \infty}\frac{\log \left| \kappa_{X^*}^{(m)}(t) \right|}{\log t}= \lim_{t\to \infty}\frac{\log \left| \kappa_X^{(m)}m I_{m-1}(t) \right|}{\log t}= \lim_{t\to \infty}\frac{\log \left| I_{m-1}(t) \right|}{\log t}.
\langle x''|(E-\hat{H}_{\pm})^{-1}|x'\rangle\simeq\frac{1}{{\rm i}\hbar\,\sqrt{|\dot{x}'\dot{x}''|}}\sum_{x_{\rm qc}}^{{\rm fixed}\,E}\exp\left\{\frac{{\rm i}}{\hbar}\,W[x_{\rm qc}]\mp{\rm i}\varphi [x_{\rm qc}]-{\rm i}(n_{R}+n_{L})\frac{\pi}{2}\right\}
E_{0}(z) - \frac{Q_{n}(z)}{R_{n}(z)}= \frac{\overline{\gamma}_{n}z^{n}}{\overline{r}_{n,n}}+ O\big(z^{n+1}\big) \mbox{and}E_{\infty}(z) - \frac{Q_{n}(z)}{R_{n}(z)}= \frac{\gamma_{n}}{r_{n,n}z^{n+1}}+ O\big((1/z)^{n+2}\big),
\left \lbrace \begin{array}{ll}\mathrm{div}\, T(\omega, Du )= 0 \mathrm{in}\{\mathbb{S}}[\Omega_Q]^-\,, u(x+qe_j) =u(x) +Be_j \textrm{$\forall x \in \overline{\mathbb{S}[\Omega_Q]^{-}}, \forall j\in \{1,\dots,n\}$}, T(\omega,Du(x))\nu_{\Omega_Q}(x)+ b(x)u(x)=g(x) \textrm{$\forall x \in \partial \Omega_Q$}\,, \end{array}\right.
=\frac{n+1}{4\pi i\hbar}\int_{S^{2}}{W_{\vec{c}}^{j}(P)}\star_{\vec{c}}^{n}[W_{\vec{c}}^{j}(H),W_{\vec{c}}^{j}(A)]_{\star_{\vec{c}}^{n}}\dS & =\frac{n+1}{4\pi i\hbar}\int_{S^{2}}\widetilde{W_{\vec{c}}^{j}(P)}[W_{\vec{c}}^{j}(H),W_{\vec{c}}^{j}(A)]_{\star_{\vec{c}}^{n}}\dS
x^{-1}(i +x) \sum_{j=1}^{\infty}\lambda_{l,j}(l+x)^{-j}\ = \ \ \lambda_{l,1}x^{-1}+ \sum_{j=1}^{\infty}\left( \lambda_{l,j+1}+ (i - l) \lambda_{l,j}\right) x^{-1}(l+x)^{-j}
\lim_n W_1(\upsilon_x^n, \delta_x) = \lim_n \sup_{\|g\|_{Lip_0(X)}\leq 1}\left\{\upsilon_x^n(g) - g(x)\right\}&= \lim_n \sup_{\|g\|_{Lip_0(X)}\leq 1}\langle g, \pi_n\circ L(\delta_x) - \delta_x\rangle &= \lim_n \|\pi_n( L(\delta_x)) - \delta_x \|_{\mathcal{F}(X)}=0.