harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
-\sum_{j=1}^N \frac{\lambda_j}{\pi}Re\Big\{\frac{D_tZ-\dot{z}_j}{(\alpha-z_j)^2}\Big\}= -\frac{\lambda}{\pi}Re\Big\{\frac{D_tZ(\alpha,t)}{(\alpha-z_1(t))^2}\Big\}= -\frac{\lambda}{\pi}Re\Big\{\frac{1}{(\alpha-z_1(t))^2}\frac{\lambda i}{2\pi\overline{\alpha-z_1(t)}}\Big\}= \frac{\lambda^2}{2\pi^2}\frac{y}{|\alpha-z_1(t)|^4}.
h_0 \Big(\left(\chi_{-j_k}\right)^{m_k}\dots \left(\chi_{-j_2}\right)^{m_2}\left(\chi_{-j_1}\right)^{m_1}|0\rangle \Big)=\Big( \sum_{j_i\in 2\mathbb{Z}+1/2}m_i - \sum_{j_i\in 2\mathbb{Z}-1/2}m_i\Big)\Big(\left(\chi_{-j_k}\right)^{m_k}\dots \left(\chi_{-j_2}\right)^{m_2}\left(\chi_{-j_1}\right)^{m_1}|0\rangle \Big).
\gamma_F(\frac{1}{2}\psi \circ q_5) = \gamma_F(\frac{1}{2}\psi)^2 \cdot \gamma_F(-J_1, \frac{1}{2}\psi) \cdot (r' \cdot J_2, J_1)_F = \gamma_F(J_1, \frac{1}{2}\psi)^{-1}\cdot (r' \cdot J_2, J_1)_F.
\mathcal S_{\boldsymbol\nu,\boldsymbol\mu}=\prod_{j=1}^3\mu_{j,\boldsymbol\cdot}!\,\mu_{\boldsymbol\cdot,j}!\times \bigl[\xi_j^{\mu_{\boldsymbol\cdot,j}}\bigr] f_{k_1}(\xi_j)^{\nu_j}\,\bigl[\eta_j^{\mu_{j,\boldsymbol\cdot}}\bigr]f_{k_2}(\eta_j)^{\nu_j}.
\int_{0}^{2\pi}\left(H_{n}\left(\rho\cos\theta\right)\right)^{2}\frac{d\theta}{2\pi}=\frac{\left(n!\right)^2}{\rho^{2n}}\sum_{j=0}^{n}\frac{\rho^{2j}}{j!j!}\left(L_{n-j}^{\left(2j-n\right)}\left(\rho^2\right)\right)^2.
\|\{\nabla \varphi_h \cdot \nu\}\|_{L_2(e)}\leq\Big\|\nabla \varphi_h \cdot \nu \Big|_{\Omega_i}\Big\|_{L_2(e)}+ \Big\|\nabla \varphi_h \cdot \nu \Big|_{\Omega_l}\Big\|_{L_2(e)}.
\frac{1}{N_{F}}\sum_{j=1}^{N_{F}}\tilde{X}^{j,n}= \sum_{j=1}^{N_{F}}\sum_{i=1}^{N_{F}}D_{i,j}X^{i,n}= \sum_{i=1}^{N_{F}}w^{i,n}X^{i,n}.
\sum_{i=1}^{m_1}T(n_i) < \left(\sum_{i=1}^{m_1}k_i\right) \cdot \frac{3}{4}\cdot \frac{1}{X_0}+ 2\cdot 3 \cdot \frac{1}{X_0}&=\left(\sum_{i=1}^{m_1}k_i + 8\right) \cdot \frac{3}{4}\cdot \frac{1}{X_0}.
\psi_{n,2}^{\ast}(Q) \leq \max_{i=1,2}\frac{L_{n,P_{i}}^{\ast}(q)}{q}\leq \frac{n+1}{n}\frac{1}{1+\widehat{w}_{n}(\xi)}- \frac{1}{n}+\epsilon_{1}= \frac{n-\widehat{w}_{n}(\xi)}{n(1+\widehat{w}_{n}(\xi))}+\epsilon_{1}.
\left\{\zeta_i^{(k)}, \zeta_{q}^{(p)}\right\}_t = O(e^{-\delta t}). \left\{\zeta_i^{(k)}, \varphi_{q}^{(p)}\right\}_t = \frac{1}{4}(\varepsilon(k-p)-1)(C-R) + O(e^{-\delta t}). \left\{\varphi_i^{(k)}, \varphi_{q}^{(p)}\right\}_t = O(e^{-\delta t}).
C_{\rho_1, \rho_2}(j)=\begin{cases}(2^j|\rho_1 -\rho_2|)^{-10n}|\rho_1-\rho_2|\geq 4\sqrt{n}, 2^{-nj/2}|\rho_1 - \rho_2| < 4\sqrt{n},\end{cases}
h_t(t)= -\frac{1}{108}\big(3t_3+4t_1^3+6t_1t_2\big)\big(5t_1^6-48t_1^4t_2+12t_3t_1^3+36t_1^2t_2^2+72t_3t_1t_2-36t_3^2-48t_2^3\big).
|\hat\psi^{(k)}(\xi-\lambda) -\hat\psi^{(k)}((1+\nu)(\xi-\lambda))| & =|\nu(\xi-\lambda)| |\hat\psi^{(k+1)}(\xi-\lambda+\xi^\ast)| & \leq C |\nu|(1+|\xi|)(1+|\xi-\lambda+\xi^\ast|)^{-r}& \leq C |\nu|(1+|\xi|)\big(1+(1-|\nu|)(|\xi|-|\lambda|)\big)^{-r}& \leq C \sigma\varepsilon(1+|\xi|)^{-r+1},
\hat{A}\geq \frac{1-(2\theta-1)\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-\frac{2\theta\beta-2(\theta-1)\alpha}{\alpha}\vert L_{A\Omega_1}\vert -2\theta\vert U_{A\Omega_1}\vert \geq \frac{1-(2\theta-1)\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-\frac{2\theta\beta}{\alpha}\vert B_{A\Omega_1}\vert\:= T.
f_\alpha(x|\mu=1) \equiv f_\alpha(x) = \frac{1}{\pi}{\rm Im}\int_0^\infty e^{-xt}\,e^{-(te^{-i\pi})^\alpha}\,dt = \frac{1}{\pi}\int_0^\infty e^{-xt}\,e^{-t^\alpha \cos\pi\alpha}\sin(t^\alpha \sin\pi\alpha)\, dt
|I_n^1| \le C\mathsf E\sum_{j=0}^{n-1}\frac{\sigma_{j+1}^2}{s_n^2}\left(|\widehat t_j-t_j|^{\alpha'/2}+|X_{j+1}-X_j|^{\alpha'}\right) \le C\mathsf E\sum_{j=0}^{n-1}\frac{\sigma_{j+1}^2}{s_n^2}\left(\left(\frac{\sigma_{j+1}}{s_n}\right)^{\alpha'}+ \left(\frac{\lambda_j|\xi_{j+1}|}{s_n}\right)^{\alpha'}\right).
G_{\lambda,q}^{''}\left(-\frac{B\gamma(q_{0})}{2\varphi(q_{0})},q_{0}\right)=\left(\begin{array}{cc}2\varphi(q_{0}) 0 0 -\frac{B^{2}\gamma(q_{0})}{2\varphi(q_{0})}\gamma''_{q}(q_{0})+\frac{B^{2}\gamma^{2}(q_{0})}{\varphi^{2}(q_{0})}\varphi''_{q}(q_{0})\end{array}\right)
\begin{aligned}(e \#\tilde p)(x,\xi)=1 &+\frac{1}{4}\int_0^1 (1-t) e^{\frac{iht}{2}\sigma(D_x,D_\xi; D_y,D_\eta)}(ih \sigma(D_x,D_\xi; D_y,D_\eta))^2(e(x,\xi)\tilde p(y,\eta))|_{\substack{y=x \eta=\xi}}dt.\end{aligned}
2^{-(E-\frac{n}{2})k_-}2^{-(F+\frac{n}{2}-\frac{n}{a})k_+}2^{-(D-\frac{n}{a})l}=2^{-(E-\frac{n}{2})k_-}2^{-[F-(J-\frac{n}{2})]k_+}2^{-(D-J)l},
( \bar{M}^{- 1})_{\gamma}^{\beta}= ( q^{{\frac{1}{2}}}- \bar{q}^{{\frac{3}{2}}}) \bar{a}_{+}^{\beta}\bar{a}_{-}^{\sigma}{\cal E}_{\sigma \gamma}+ \bar{q}^{\bar{N}+{\frac{3}{2}}}\delta_{\gamma}^{\beta}
\pi^{\top}z < w_0 +\sum_{i=1}^{n+1}\varepsilon_i w^i_0 \leq \inf\{w^{\top}x \, | \x\in C\}+ \sum_{i=1}^{n+1}\inf\{(\varepsilon_i w^i)^{\top}x \, | \x\in C\}&\leq \inf\{(w^{\top}+ \sum_{i=1}^{n+1}\varepsilon_i w^i)^{\top}x \, | \x\in C\}\leq \pi^{\top}x \ \ \forall x\in C,
\begin{array}{l}<\chi,\chi'>=\displaystyle \prod_{r=1}^{N}\delta_{n_{r}n'_{r}}\cdot\displaystyle \prod_{r=1}^{M}\delta_{m_{r}m'_{r}}\cdot\displaystyle \prod_{r=1}^{Q}\delta_{q_{r}q'_{r}}\cdot\displaystyle \prod_{r=1}^{T}\delta_{t_{r}t'_{r}}\cdot\displaystyle \prod_{r=1}^{4}\delta_{\nu_{r}\nu'_{r}}.\end{array}
H_{RWA}=-\hbar\Delta_{P}|1\rangle\langle1|+\hbar\Delta_{S}|3\rangle\langle3|+\frac{\hbar\Omega_{P}}{2}(|1\rangle\langle2|+h.c.)+\frac{\hbar\Omega_{S}}{2}(|3\rangle\langle2|+h.c.)
0 =\ln\rho_i+\tilde{\xi}_i\sigma_i^2+\ln(1+\tilde{\xi}_iI_{\ell{i}})+\sum_{k\ne{i},k\ne\ell}\ln(1+\tilde{\xi}_iI_{ki}) &=\ln\rho_i+\tilde{\xi}_i'\sigma_i^2+\ln(1+\tilde{\xi}_i'I_{\ell{i}}')+\sum_{k\ne{i},k\ne\ell}\ln(1+\tilde{\xi}_i'I_{ki}) &<\ln\rho_i+\tilde{\xi}_i\sigma_i^2+\ln(1+\tilde{\xi}_i'I_{\ell{i}}')+\sum_{k\ne{i},k\ne\ell}\ln(1+\tilde{\xi}_iI_{ki}),
w(x^{3}, \upsilon)= \frac{\zeta}{\upsilon}\tanh x^{3},~~~~~s(x^{3},\upsilon,\bar{\upsilon})=-2 \ln{\upsilon}+ \sum_{k=1}^{\infty}2^{2k}(\cosh{x^{3}})^{-2k}g_{2k}(\upsilon,\bar{\upsilon}),
\begin{cases}\alpha_5\gamma_3=\alpha_1\gamma_4 \alpha_7\gamma_3=\alpha_3\gamma_4 \beta_5\gamma_3=\beta_1\gamma_4 \beta_3\gamma_3=\alpha_1\gamma_5 \beta_7\gamma_3=\alpha_3\gamma_5 \gamma_1\gamma_3=\beta_1\gamma_5 \beta_3\gamma_4=\alpha_5\gamma_5 \beta_7\gamma_4=\alpha_7\gamma_5 \gamma_1\gamma_4=\beta_5\gamma_5\end{cases}
\begin{array}{l}{1^{0}\quad E_{j}( s ){\neq}E_{k}( s ) \quad \forall s \in [ 0 , 1 ] , \quad j{\neq}k ,}\\{2^{0}\quad \forall j \P_{j}( s )}\\{m b o x{i s d o u b l e c o n t i n u o u s l y}}\\{\qquad \qquad \quad \mathrm{d i f f e r e n t i a b l e f u n c t i o n o f}\s \in [ 0 , 1 ] ,}\\ \end{array}
{\cal H}_{C_{n}}^{u n t w i s t e d}={\frac{1}{2}}p^{2}+{\frac{g_{s}^{2}}{2}}\sum_{\alpha \in \Delta_{s}}\wp ( \alpha \cdot q ) +{\frac{g_{l}^{2}}{4}}\sum_{\Xi \in \Delta_{l}}\wp ( \Xi \cdot q ) ,
\sum_{i=1}^n\tilde{f}(a_i)\times \tilde{f}(b_i) \in \tilde{R}_2^0 \textrm{whenever}\sum_{i=1}^na_i\times b_i \in \tilde{R}^0_1\textrm{for}a_i,b_i \in \tilde{R}_1.
\begin{aligned}q_{t}(x,y) =\int_0^t\int_X \widetilde{p}_s(x,z)V(z)p_{t-s}(z,y)d\mu(z)ds &=\int_0^{t/2}\int_X\ldots+\int_{t/2}^t\int_X\ldots:= I_1+I_2.\end{aligned}
S_{4g,}= q^{2g+1}\sqrt{q}\sum_{\substack{f \in \mathcal{M}_{\leq 4g}d(f)}}\frac{d_4(f)}{|f|^{\frac{3}{2}}}\sum_{\substack{C \in \mathcal{M}_{\leq y}C | f^{\infty}}}\frac{1}{|C|^2}\bigg[ \sum_{V \in \mathcal{M}_{d(f)-2g-2+2d(C)}}G(V, \chi_f) - \frac{1}{q}\sum_{V \in \mathcal{M}_{d(f)-2g+2d(C)}}G(V, \chi_f) \bigg].
\mathrm{sign}(h_{n,k},\Delta_{k})=\begin{cases}\varepsilon_{n,k-1}\mbox{for}\,\,k\,\,\mbox{even}, \varepsilon_{n,k-1}\mbox{for}\,\,k\,\,\mbox{odd},\,\,\ell(n)<k, (-1)\varepsilon_{n,k-1}\mbox{for}\,\,k\,\,\mbox{odd},\,\,k\leq \ell(n).\end{cases}
( a )^{\prime}\ \x \rightarrow e^{- \chi \alpha / ( 2 a )}x , \ \Omega \rightarrow e^{- \chi \alpha / ( 2 a )}\Omega , \e^{2 \rho}\rightarrow e^{\chi \alpha / 2}e^{2 \rho}, \A \rightarrow A , \f \rightarrow f + \alpha ,
\Delta n \sum_{l = 1}^{\infty}( 2 l + 1 ) A_{l}= \left. \Delta n \lim_{\varepsilon \to 0}\left( \overline{{\cal D}}_{r r}- 2 \overline{{\cal D}}_{r \rho}+ \overline{{\cal D}}_{\rho \rho}\right) \right|_{r = 1 +{\varepsilon}/{2}\atop \rho = 1 -{\varepsilon}/{2}}+ 1{,}
|F_n| =\sum_{l=0}^n \bigg( \frac{1}{l+1}\prod_{k=1}^d (l^k+1)^{d-k+1}\bigg) \ge \sum_{l=1}^n l^{\big( \sum\limits_{k=1}^d k(d-k+1)\big) -1}&=\sum_{l=1}^n l^D \ge\sum_{l=[\frac{n+1}{2}]}^n l^D \ge (\frac{n-1}{2})^{D+1}.
\begin{aligned}= i\lambda - \rho_L = d\Lambda -\rho. begin{align*}1ex] = \Lambda - 2\rho({\mathfrak u}\cap{\mathfrak k}) = i\lambda - \rho({\mathfrak u}\cap{\mathfrak p}) + \rho({\mathfrak u}\cap{\mathfrak k}). \end{aligned}
\left( \begin{matrix}{{Q_{E}}^{U}}\\{{Q_{M}}^{U}}\\ \end{matrix}\right) ={\frac{e}{4 \pi}}\int d S_{k}\sqrt{- g}U ( x ) \left( \begin{matrix}{F^{k 0}}\\{\tilde{F}^{k 0}}\\ \end{matrix}\right) U^{- 1}( x )
\begin{aligned}L_0^{\parallel}=\p_0^+p_0^- +\sum_{n\neq0}\ (\ :\alpha_n^+\alpha_{-n}^-:+\n:c_{-n}b_n:)+\frac{2-q}{24}\;, L_0^{\perp}=\frac{1}{2}\sum_{n\in\mathbb{Z}}\sum_{i=2}^{q-1}\ :a_n^ia_{-n}^i:\;,\end{aligned}
||e^f-1||_2^2 =||f||^2_2+\sum\limits_{k=3}^{\infty}\dfrac{2^k-2}{k!}\int{f^k}dx &\leq||f||^2_2+\sum\limits_{k=3}^{\infty}\dfrac{2^k-2}{k!}\bigg(\pi\bigg[\dfrac{k}{2}-1\bigg]\bigg)^{\frac{k-2}{2}}||f||^k_{1,2},
f_R(a)=\int_{D_R}|(i\nabla+A_0)Z_a^R|^2\,dx-\int_{D_R}|(i\nabla+A_{\mathbf e})\tilde \varphi_a|^2\,dx+o(1),\quad|a|\to 0^+, f_R(a)=O(1) ,\quad|a|\to 0^+,
A^*\Psi=\left(\begin{array}{c}\partial_x^3 v-a_0\partial_x^5 v+a_1\partial_x v+ (-1)^k \displaystyle \int_0^\infty g(s)\partial_{x}^{2k}\zeta^t(x,s)ds -v+\dfrac{g'(s)}{g(s)}\zeta^t+\partial_s\zeta^t\end{array}\right)
\sigma_{3}^{(1)}: T^{*}M|_{\Theta}\times T^{*}M|_{Q,P}\rightarrow T^{*}M ; (\sigma_{3}^{*}\Xi^{a}:=\epsilon^{a},\sigma_{3}^{*}\Psi_{a}:=\epsilon_{a},\sigma_{3}^{*}\Theta^{\alpha},\sigma_{3}^{*}\Theta_{\alpha},\sigma_{3}^{*}Q^{i},\sigma_{3}^{*}P_{i})\mapsto(\Xi^{a},\Psi_{a},\Theta^{\alpha},\Theta_{\alpha},Q^{i},P_{i}).
= \sum\limits_{(M_b, \mu_b) \in \mathcal{I}^{G, \mu}_{M_b, b'}}(\mathrm{Ind}^G_{P_b}\circ \otimes^k_{i=1}(\mathrm{Mant}_{M_{b_i}, b'_i, \mu_{b i}}\circ \mathrm{Red}_{b'_i}) \circ (\delta_{P_b}\otimes \mathrm{Jac}^G_{P^{op}_b}))\otimes [1][|\cdot|^{\langle \rho_G, \mu_b \rangle - \langle \rho_G, \mu \rangle}].
\begin{aligned}2\int_0^T \int_{\omega}\xi^2 e^{2s\varphi}a (v_x)^2dxdt \le- 2 \int_0^T \int_{\omega}s \xi^2 \varphi_t e^{2s\varphi}v^2dxdt & -2\int_0^T \int_{\omega}\left(\xi^2e^{2s\varphi}a\right)_xvv_x\:dxdt + J. \end{aligned}
R=\frac{p_{\psi}^{2}}{2I_{3}}+\frac{p_{\psi}^{2}cos^{2}\theta}{2I_{1}sin^{2}\theta}+\frac{p_{\phi}^{2}}{2I_{1}sin^{2}\theta}-\frac{p_{\psi}p_{\phi}cos\theta}{I_{1}sin^{2}\theta}-\frac{I_{1}\dot{\theta}^{2}}{2}+Mglcos\theta.
\Pr \left[K_{2,n,0}^c=k^c|\right] = \sum_{\mathcal X\in g\left(\mathcal N_{i,-n},k^c-1\right)}\prod_{m\in \mathcal X}(1-\Pr[Y_{m,n,i}=0])\prod_{m\in \mathcal N_{i,-n}\setminus\mathcal X}\Pr[Y_{m,n,i}=0],k^c=1,\cdots, K_2^c.
\varprojlim_K C^\bullet(G_{L}, o_L[G_{L}/G_K] \otimes_{o_L}V) = C^\bullet_{}(G_{L}, \varprojlim_K (o_L[G_{L}/G_K] \otimes_{o_L}V)) = C^\bullet_{}(G_{L}, o_L[[\Gamma_L]] \otimes_{o_L}V) \ .
\begin{aligned}V = \sum_{i=1}^n x_i \left(\alpha_i(z) +\sum_{k=n_0+1}^n\left(\sum_{j=1}^n\gamma_{ikj}(z) \partial_{x_j}b_k + \sum_{l=1}^m\nu_{ikl}(z) \partial_{y_l}b_k\right)\ln x_k\right)\partial_{x_i}&\quad+\sum_{l,l'=1}^m\sum_{k=n_0+1}^n \beta_{ll'k}(z)\partial_{y_{l'}}b_k \ln x_k\partial_{y_l},\end{aligned}
\Psi(f)(\sigma(a)\circ \sigma(b)) =f(B_1(a_1))f(S(B_2(a_2)))f(\sigma(b)) &=f(B_1(a_1)S(B_2(a_2)))f(\sigma(b)) &=\Psi(f)(\sigma(a))\Psi(f)(\sigma(b)).
\int_{0}^{\infty}x^\mu J_\nu(a x)dx = 2^\mu a^{-\mu-1}\frac{\Gamma\left(\frac{1}{2}+\frac{1}{2}\nu +\frac{1}{2}\mu\right )}{\Gamma\left(\frac{1}{2}+\frac{1}{2}\nu- \frac{1}{2}\mu\right )}\quad - \Re \nu -1 <\Re \mu <\frac{1}{2}, \quad a>0
\sqrt{-G}\,G^{MN}R_{MN}=\sqrt{-g}\,[g^{\mu\nu}R_{\mu\nu}-\frac{D-3}{4(D-2)}g^{\mu\nu}\partial_\mu\ln\Phi\cdot\partial_\nu\ln\Phi-\frac{\Phi}{4}f_{\mu\nu}f^{\mu\nu}],
T_{4}=-\frac{2\pi i}{c}(cz+d)B_{0}(B)P_{2}(0,A_{d}), T_{5}=-\frac{2\pi i}{c\left( cz+d\right)}B_{0}(B)P_{2}\left(0,A\right) , T_{6}=2\pi i\sum\limits_{n=1}^{ck}f(-dn)P_{1}\left( \frac{n}{ck}\right)P_{1}\left( \frac{dn}{c},B_{-a}\right) +2\pi if(0)P_{1}\left(0,B_{-a}\right) .
x_2=\frac{a_1+b_2}{2}= c_1-c_2+c_3, x_5=\frac{a_2+b_3}{2}= -c_1+c_2+c_3-d_1+d_2, x_6=\frac{a_3+b_3}{2}= \frac{-c_1+c_2+c_3+c_4-d_1+2d_2-d_3}{2}, y_1=\frac{-a_1+b_1}{2}= c_2-c_3+d_1, y_3=\frac{-a_2+b_1}{2}= c_1-c_3+d_1, y_4=\frac{-a_2+b_2}{2}= c_1-c_2+d_1, y_6=\frac{-a_3+b_3}{2}=\frac{-c_1+c_2+c_3-c_4-d_1+2d_2+d_3}{2}.
\beta = \sin^{-1}\left( \frac{-\ddot{s}_i}{\sqrt{\dot{s}_i^2+\ddot{s}_i^2}}\right)= \sin^{-1}\left( \frac{-(p_{s_i}\sigma+q_{s_i})}{\sqrt{\dot{s}_i^2+(p_{s_i}\sigma+q_{s_i})^2}}\right).
\widehat{A^{s,\alpha}}(\xi) = i^{-|\alpha|}\partial_{\xi}^{\alpha}( i^{|\gamma|}\xi^{\gamma}) & = i^{|\gamma|-|\alpha|}\frac{\gamma !}{(\gamma-\alpha)!}\xi^{\gamma-\alpha}= \frac{\gamma !}{(\gamma-\alpha)!}i^{|\gamma-\alpha|}\xi^{\gamma-\alpha}.
c_{ih}= \binom{n}{h}^{-1}\sum_{r=\max(0,h+i-n-k)}^{\min(h-k,i-l-k)}(-1)^{i-l-k-r}\binom{i+\alpha+l-k}{r}\binom{i+\beta-l+k}{i-l-k-r}\binom{n-i}{h-k-r}, begin{align*}1ex] d_{hi}= g_{i,l,k}^{(\alpha,\beta)}\sum_{r=0}^{i-l-k}(-1)^{i-l-k-r}\binom{i+\alpha+l-k}{r}\binom{i+\beta-l+k}{i-l-k-r}\binom{n+i+\alpha+\beta}{h+\beta+k+r}^{-1},
\Delta(F_j x)=\sum_{\substack{a+b=n j\le a-1}}F_jx_{t_1,\dots,t_a}\otimes x_{t_{a+1},\dots,t_n}+\sum_{\substack{{a+b=n}j\le a+1}}x_{t_1,\dots,t_a}\otimes F_jx_{t_{a+1},\dots,t_n}
Z[j_{1},j_{2}] = Tr\left[\rho\bar{T}{e^{i\int d^4x(H_{s}+j_{2}\phi^*_{s}+j_{2}^*\phi_{s})}}T{e^{-i\int d^4x(H_{s}-j_{1}\phi^{\ast}_{s}-j_{1}^{\ast}\phi_{s})}}\right]
g = Z_{2}= Z_{0}^{- 2}\Biggl ( \frac{a_{0}+ Z_{0}}{\bar{a}_{0}+ Z_{0}^{- 1}}\Biggr ) \Biggl ( \frac{\mu_{+}+ Z_{0}}{\bar{\mu}_{+}+ Z_{0}^{- 1}}\Biggr ) \Biggl ( \frac{\mu_{-}+ Z_{0}}{\bar{\mu}_{-}+ Z_{0}^{- 1}}\Biggr )
\dot{\varrho}+ 3{\frac{\dot{a}_{0}}{a_{0}}}\left( \varrho +{\frac{\wp}{c^{2}}}\right) = -{\frac{\dot{G}_{4}}{G_{4}}}\varrho +{\frac{3 k c \dot{c}}{4 \pi G_{4}a_{0}^{2}}}-{\frac{3{\cal C}c \dot{c}}{4 \pi G_{4}a_{0}^{4}}}.
\hat{\cal{L}}_{SD}^S=\frac{1}{2}(\hat f^\mu +\hat h^\mu -\partial^{\mu}\hat \phi )( \hat f_\mu +\hat h_\mu -\partial_{\mu}\hat\phi )-\frac{1}{2m}\epsilon^{\alpha\beta\gamma}(\hat f_{\alpha}\partial_{\beta}\hat f_{\gamma}+2\hat h_{\alpha}\partial_{\beta}\hat f_{\gamma}).
\lim_{N\to\infty}\int \|z\|\, d\mu_N(z) = \lim_{N\to\infty}\int_{\|z\|\leq r}\|z\|\, d\mu_N(z) +\lim_{N\to\infty}\int_{\|z\|> r}\|z\|\, d\mu_N(z) & = \int_{\|z\|\leq r}\|z\|\, dP(z) + \lim_{N\to\infty}\int_{\|z\|> r}\|z\|\, d\mu_N(z).
\mbox{\mathversion{bold}$v$}_{0}\times \mbox{\mathversion{bold}$v$}_{0ss}=q^{0}_{1s}\mbox{\mathversion{bold}$v$}_{0}\times \mbox{\mathversion{bold}$e$}^{0}+ q^{0}_{2s}\mbox{\mathversion{bold}$v$}_{0}\times \mbox{\mathversion{bold}$w$}^{0}begin{align*}3mm] =-q^{0}_{1s}\mbox{\mathversion{bold}$w$}^{0}+q^{0}_{2s}\mbox{\mathversion{bold}$e$}^{0}.
M_s(\mu)+e^{-\beta}d M_{cd^{-1}s}(\mu)+\cdots +e^{-\beta (k+1)}d^{k+1}M_{c^{k+1}d^{-(k+1)}s}(\mu) &=M_s(\nu)+e^{-\beta}dM_{cd^{-1}s}(\nu)+\cdots +e^{-\beta (k+1)}d^{k+1}M_{c^{k+1}d^{-(k+1)}s}(\nu).
D_{a \alpha}\mathrm{v}^{\beta}{}_{\gamma}( \theta ) = \textstyle{\frac{2}{3}}\delta_{\alpha}{}^{\beta}D_{a \delta}\mathrm{v}^{\delta}{}_{\gamma}( \theta ) - \textstyle{\frac{1}{3}}\delta^{\beta}{}_{\gamma}D_{a \delta}\mathrm{v}^{\delta}{}_{\alpha}( \theta ) \, ,
S = \langle M \rangle k \left[ \frac{5}{2}- \ln \left( \frac{2 \langle M \rangle \beta^{3 / 2}}{\lambda \sqrt{\pi}}\right) + \frac{\alpha ( q ) \langle M \rangle}{\lambda \sqrt{\pi}}\beta^{3 / 2}+ . . . \right] .
\varphi_{\infty}(f;q) = \varphi_0(f;q) + \sum_{n=1}^{\infty}(\varphi_n(f;q) - \varphi_{n-1}(f;q)) &= 1 - \sum_{n=1}^{\infty}q^nf(n) \varphi_{n-1}(f;q) = \begin{array}{c}1- \sum_{(6)}\end{array}.
\pi_{TM,X_{C},Y_{V},c}({\bf x},{\bf y})=\left( \begin{array}{ccc|ccc}0 0 0 0 0 0 0 0 0 0 x^3 x^1 0 0 0 0 0 0 \hline 0 0 0 0 0 0 0 -x^3 0 0 0 0 0 -x^1 0 0 0 0 \end{array}\right).
I_1=\|V\|_{\infty}^2\int_{\mathbb{R}^d}\left(\int_{B_1(x)}\frac{|\psi(y)|}{|x-y|^{d-\alpha}}dy\right)^2dx, I_2=\int_{\mathbb{R}^d}|V(x)|\left(\int_{\mathbb{R}^d\setminus B_1(x)}\frac{|V(y)|^{1/2}|\psi(y)|}{|x-y|^{d-\alpha}}dy\right)^2dx.
f_{|\Delta_n|^2}(x) ={2^{{(1-\alpha_n)}/{2}}}{\Gamma(\alpha_n)}\left(\frac{\alpha_n x}{b_{0,n}}\right)^{{(\alpha_n-1)}/{2}}& \times K_{\alpha_n-1}\left(\sqrt{\frac{2\alpha_n x}{b_{0,n}}}\right).
\sum\limits_{j=1}^{\infty}\int\limits_{R_{j}}|f(z)|\,dz=\sum\limits_{j=1}^{\infty}\int\limits_{\mathbb{R}^{n+1}}\chi_{R_{j}}(z)|f(z)|\,dz= =\int\limits_{\mathbb{R}^{n+1}}\sum\limits_{j=1}^{\infty}\chi_{R_{j}}(z)|f(z)|\,dz \le N \|f|L_{1}(\mathbb{R}^{n+1})\|.
_{B_1, B_2^0, B_2^1}\f(B_1, B_2^0, B_2^1) & \frac{1}{2}\left( B_1 + \frac{B_1}{M}B_2^1 +(1 - \frac{B_1}{M}) B_2^0 \right) \leq B_, \ \ 0 \leq B_1 \leq M, \ 0 \leq B_2^1 \leq B_1, \ 0 \leq B_2^0 \leq M- B_1.
\lim\limits_{r\rightarrow\infty}\left|\sum\limits_{j=l+1}^{n}m_{j}f(\|Q_{1r}-Q_{jr}\|)+\sum\limits_{j=l+1}^{n}m_{j}\langle Q_{jr}-Q_{2r},\frac{Q_{1r}-Q_{2r}}{\|Q_{1r}-Q_{2r}\|}\rangle B_{jr}\right|<B
\int_{0}^t \int_{0}^{L}\partial_t h(x,s)q_s(x,s){\rm d}x{\rm d}s = -6\int_0^t \int_{0}^{L}\dfrac{\partial_t h(x,s)}{|h(x,s)|^2}{\rm d}x{\rm d}s = \left[ \int_{0}^{L}\dfrac{6}{h(x,s)}{\rm d}x\right]_{s=0}^{s=t}\,.
\left\|r_{1,2}(t;z)\right\|_{L^{2,1}}=\left\|r_{1,2}(z)\right\|_{L^{2,1}}, &\left\|z^{-2}r_{1,2}(t;z)\right\|_{L^{2,1}}=\left\|z^{-2}r_{1,2}(z)\right\|_{L^{2,1}}.
h(Y_A^n)-h(Y_B^n) \leq I(A^n;B^n) + I(A^n;Y_A^n|B^n) + I(B^n;Y_A^n|A^n) - I(B^n;Y_B^n) - I(A^n;Y_B^n|B^n) &\stackrel{(a)}{=}I(A^n;h_1 A^n + Z_A^n) + I(B^n;h_2 B^n+Z_A^n|A^n) - I(B^n;Y_B^n) - I(A^n; h_1 A^n +Z_B^n) &\stackrel{(b)}{=}I(B^n;h_2 B^n+Z_A^n) - I(B^n;h_1 A^n + h_3 B^n+Z_B^n),
\sum_{j=2}^{m}B_{\vec\tau, \vec s}^{(j,1)}(r)=\frac{\beta_{m-1}}{2\pi is_m}\partial_{\beta_{m-1}}\log\frac{\Gamma(1+\beta_{m-1})}{\Gamma(1-\beta_{m-1})}-\frac{\beta_m}{2\pi is_m}\partial_{\beta_{m}}\log\frac{\Gamma(1+\beta_{m})}{\Gamma(1-\beta_{m})},
\mbox{det}\left( 1 \!\! 1 \, \pm \, \hat{F}\right) = 1 \, - \, \frac{1}{2}Tr(\hat{F}^{2}) \,+ \, \frac{1}{8}\left[Tr(\hat{F}^{2})\right]^{2}- \frac{1}{4}\, Tr(\hat{F}^{4}) \, + \mbox{det}\, (\hat{F})
\Upsilon_{\varepsilon}:=\lambda\|\rho\phi_{\varepsilon}\|_{L^{4,\varepsilon}}^{4}+m^{2}\|\rho^{2}\psi_{\varepsilon}\|_{L^{2,\varepsilon}}^{2}+\|\rho^{2}\nabla_{\varepsilon}\psi_{\varepsilon}\|_{L^{2,\varepsilon}}^{2}
[ \ell_{\mu}^{2}, \, q \cdot \! \nu ] = - i \left[ 2 ( \mu \cdot \nu ) \, \ell_{\mu}+ \sum_{\rho \in \Delta_{+}}g_{| \rho |}( \rho \cdot \mu ) ( \rho^{\vee}\! \cdot \nu ) \frac{\ell_{\mu}^{2}- \ell_{s_{\rho}( \mu )}^{2}}{\ell_{\mu}- \ell_{s_{\rho}( \mu )}}\check{s}_{\rho}\right] .
\lim_{\lambda\to\infty}2^{n\lambda}\int_{\mathbb{R}^{n}}|R^{\alpha,\beta}(2^{\lambda}\mathbf{x}+\mathbf{z}, 2^{\lambda}\mathbf{y}+\mathbf{z})|E_{\alpha}(|\mathbf{x}-\mathbf{y}|)d\mathbf{y}&\leq \lim_{\lambda\to\infty}C_{2k+1}\int_{\mathbb{R}^{n}}e^{-M((2k+1)(\mathbf{x}-\mathbf{y}))}E_{\alpha}(2^{-\lambda}|\mathbf{x}-\mathbf{y}|)d\mathbf{y}=0,
X_k =\{(0,x)\mid x \in X_{k-1}\}\cup N_1(L_{1k2},-1), Y_k =\{(0,x)\mid x \in Y_{k-1}\}\cup N_1(L_{1k2},-1), Z_k =\{(0,x)\mid x \in Z_{k-1}\}\cup N_1(L_{1k2},-1).
\big| \big({\cal F}(v) -{\cal F}(\hat v) \big)_i(\tau, \xi)| \le{\cal N}\left( \int_t^\tau \big( |V(s, x_i(s, \tau, \xi))| + |V(s, 0)| \big)\, ds + \sum_{j=k+1}^{k+m}| V_{j}(t, 0)| \right) begin{align*}6pt] \le 2 \sqrt{n}{\cal N}\Big( L_1^{-1}e^{L_1 \tau + L_2}\| V \| + \sum_{j=k+1}^{k+m}| V_{j}(t, 0)| \Big).
\big(I+(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}A^{\dagger}\big)^{-1}(A+UV^{\ast}) =A+UV^{\ast}, (A+UV^{\ast})\big(I+A^{\dagger}UF_{S_{A}}U^{\ast}(A^{\dagger})^{\ast}\big)^{-1}=A+UV^{\ast}.
\Big(\frac{q^{n_1}-1}{q-1}+1\Big)\Big(\frac{q^{n_2}-1}{q-1}+1\Big) \leq q^{n_1}\frac{q^{n_2}-1}{q-1}+\frac{q^{n_1}-1}{q-1}+1 =\frac{q^{n_1+n_2}-1}{q-1}+1,
M^*_{m}= \sum^{m}_{k=N_6+1}M_k\mathbf{P}(T=k) = \sum^{m}_{k=N_6+1}M_k(\mathbf{P}(T>k-1)-\mathbf{P}(T>k)) &=M_{N_6+1}\mathbf{P}(T>N_6)-M_m\mathbf{P}(T>m)+\sum^{m-1}_{k=N_6+1}(M_{k+1}-M_k)\mathbf{P}(T>k).
V_{i j}^{a}\ = \ \frac{1}{2 \sqrt 2}\, \Sigma^{2}\, f_{x i j}^{a}\, K^{x}~ , \qquad T_{i j}^{a}\ = \ \frac 1 2 \, \Sigma^{- 1}\, L_{i}^{I}{}^{k}\, f_{x k j}^{a}\, K_{I}^{x},
xv' = f' - v = \frac{2}{v^{-2}+ 1}- v = \frac{-v (v - 1)^2}{1 + v^2}\;, \frac{v'}{v}= -\frac{v^2 - 2v + 1}{x(1 + v^2)}= -\frac{1}{x}+ \frac{2v}{x(1 + v^2)}\;.
\int_{0}^{\infty}dEe^{-Et}\rho_{\Omega}(E)=\int_{M}dx\int_{M}dy[\langle\Omega|V(t,x)V(0,y)|\Omega\rangle-\langle\Omega|V(0,x)|\Omega\rangle\langle\Omega|V(0,y)|\Omega\rangle]
\operatorname{p}_f^{(OW)}[\pi,\eta]=\limsup_{i\in I}\frac{P_{f_{A_i}}[\eta_{F_i}]}{\theta(A_i)}\leq \limsup_{i\in I}\frac{P_{f_{A_i}}(\mathcal{U}_{F_i})}{\theta(A_i)}\leq \operatorname{p}_f^{(OW)}(\pi).
\Phi_{W}=\int\frac{\rho(x^{\prime},t)\rho(x^{\prime\prime},t)(x-x^{\prime})_{i}}{|x-x^{\prime}|^{3}}(\frac{(x^{\prime}-x^{\prime\prime})^{i}}{|x-x^{\prime}|}-\frac{(x-x^{\prime\prime})^{i}}{|x^{\prime}-x^{\prime\prime}|})d^{3}x^{\prime}d^{3}x^{\prime\prime}
N_{\mathrm{c}3}=N_{3}-N_{\mathrm{d}3}-\binom{3}{2}N_{\mathrm{d}1}\left(N_{2}-N_{\mathrm{d}2}\right)+\left[\binom{3}{2}\binom{2}{1}N_{\mathrm{d}1}N_{\mathrm{d}1}-\binom{3}{1}N_{\mathrm{d}2}\right]\left(N_{1}-N_{\mathrm{d}1}\right)=3552.
|I_4|=|\int_{{\mathbb R}^3}\int_{{\mathbb R}^3}(E+v\times B)\cdot v\frac{(\partial^{\alpha}F)^2}{2\mu}\,dv\,dx|\leq C\|E\|_{L^{\infty}}\|\langle v\rangle^{\frac{1}{2}}\frac{\partial^{\alpha}F}{\sqrt{\mu}}\|^2,
G_1(s) =(n+p)\Big(p(P_q^{n,p-q}-P_{q-1}^{n+1,p-q})-s(P_{q-1}^{n,p-q+1}-pP_{q-2}^{n+1,p-q+1})\Big)-(n+1)\big(P_q^{n-1,p-q}-pP_{q-1}^{n,p-q}\big), G_2(s) =(n+p)P_q^{n,p-q}+P_q^{n-1,p-q}-pP_{q-1}^{n,p-q}.
d_{2}\equiv \frac{\pi}{2 m_{0}}\cdot \frac{m_{0}( \sinh{m_{0}x_{0}}- \sinh{m_{0}y_{0}}) + ( x_{0}- y_{0}) \cosh{\frac{\mathrm{L}m_{0}}{4}}}{\sinh{\frac{\mathrm{L}m_{0}}{2}}}.
1-\mathcal{S}_{j}\left(U,T_{1},T_{2},\beta\right)=\int_{0}^{\infty}\sum_{n=M_{j}}^{\infty}\mathcal{T}_{j,Y_{j}}\left(n,y,U,T_{1},T_{2},\beta\right)f_{Y_{j}}(y){\rm d}y.
j_{\chi}^{r}(t_{\mathrm{cyc}}^{s}(\gamma_{\chi \omega^{s}}))= j_{\chi}^{r}(u^{w_{\chi}\cdot s}\gamma_{\chi})= u^{w_{\chi}\cdot s}(u^{r}\gamma_{K})^{w_{\chi}}= (u^{r+s}\gamma_{K})^{w_{\chi}}= j_{\chi \omega^{s}}^{r+s}(\gamma_{\chi \omega^{s}})
\sum_{k=1}^{n}|a_{k}|^{\lambda_{a}}|b_{k}|^{\lambda_{b}}\cdot\cdot\cdot|z_{k}|^{\lambda_{z}}\le(\sum_{k=1}^{n}|a_{k}|)^{\lambda_{a}}(\sum_{k=1}^{n}|b_{k}|)^{\lambda_{b}}\cdot\cdot\cdot(\sum_{k=1}^{n}|z_{k}|)^{\lambda_{z}}.
\sum\limits_{1\leq n\leq N}f_k(n)f_k(n+1)=N\sum_{\substack{w_1\leq N,w_2\leq N+1 (w_1,w_2)=1}}\frac{g_k(w_1)g_k(w_2)}{(w_1w_2)^2}+O_{\varepsilon}(N^{\varepsilon}).
\frac{\mathrm{card}\, \mathcal{T}^i_n}{\mathrm{card}\, \Upsilon_n}\rightarrow 1 \mbox{\,\,\, and \,\,\,}\frac{\mathrm{card}\, \mathcal{S}^i_n}{\mathrm{card}\, \Upsilon_n}\rightarrow 1, \mbox{\,\,\, as \,\,\,}n\rightarrow \infty, \i=1, 2.
\frac{1}{\mu}\left( 1 - \frac{N_{f}}{N_{c}}\right) m^{\frac{2 N_{c}- N_{f}}{N_{c}- N_{f}}}+ m_{f}m^{\frac{N_{c}}{N_{c}- N_{f}}}- \Lambda^{\frac{3 N_{c}- N_{f}}{N_{c}- N_{f}}}= 0 .
\frac{1}{1- \mu +\mu \sqrt{1-z}}= \frac{1 -\mu - \mu \sqrt{1-z}}{1 -2\mu +\mu^2z}&= \frac{1}{2\mu -1}[1- \mu - \mu\sqrt{1-z}] \sum_{k=0}^{\infty}\biggl( \frac{\mu^2z}{2\mu -1}\biggr)^k.
\Psi_0(z) + \Psi_1(z) + \Psi_2(z) = 1 + \mathcal{O}(z^{-3/2}), \Psi_0(z) \Psi_1(z) + \Psi_0(z) \Psi_2(z) + \Psi_1(z) \Psi_2(z) = \frac{1}{z}+ \mathcal{O}(z^{-2}) \Psi_0(z) \Psi_1(z) \Psi_2(z) = \frac{1}{4z^2}+ \mathcal O(z^{-3})
|V^{\beta,n}(x',i) - v^\beta(x,i)| \ \leq \ \frac{L_2}{\gamma}|x - x'| + |V^{\beta,n}(x,i) - v^\beta(x,i)| \ \underset{\substack{n\rightarrow\infty x'\rightarrow x}}{\longrightarrow}\ 0
{\not \! \! D}^{2}= ( D_{\mu})^{2}- \frac{i}{4}[ \gamma^{\mu}, \gamma^{\nu}] F_{\mu \nu}= ( D_{\mu})^{2}+ \left( \begin{array}{c c}{{( E + B ) \sigma^{3}}}&{{0}}\\{{0}}&{{( - E + B ) \sigma^{3}}}\\ \end{array}\right)
u(t)\leq-\beta^{2^{m}-1}C_{3}^{2^{m}}(T-t)^{2^{m}-1}\prod_{k=1}^{m}(\frac{1}{2^{k}-1})^{2^{m-k}}\leq-C\left[\beta C_{3}(T-t)\prod_{k=1}^{m}(2^{-\frac{k}{2^{k}}})\right]^{2^{m}}
\lim_{L \to \infty}( \mathbb{E}\langle f R_{1, n+1}^p \rangle - \frac{1}{n}\mathbb{E}\langle f \rangle \mathbb{E}\langle R_{1,2}^p \rangle - \frac{1}{n}\sum_{a=2}^n \mathbb{E}\langle f R_{1,a}^p\rangle )= 0,
\mathfrak{A}(n;q_1,q_2,q_2')=q_1\sideset{}{^\star}\sum_{b\bmod{q_1}}\;\sideset{}{^\star}\sum_{\xi,d\bmod{q_1}}e_{q_1}(-\ell \bar{b}(\xi+\ell q_2)\bar{q}_2^2q_2'\bar{n}\overline{\xi}+b+(\xi+\ell q_2)\bar{q}_2'^2\bar{n}\bar{d}+d)+q_1.