harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
w_3 = -n(n-1)p_5 - nq_4, w_2 = - [2(n-1)p_5 + q_4] \sum_i z_i - n(n-1)p_4 - nq_3, w_1 = - [2(n-1)p_5 + q_4] \sum_i z_i^2 - 2p_5 \sum_{i<j}z_iz_j - [2(n-1)p_4 + q_3] \sum_i z_i{}- n(n-1)p_3 - nq_2,
\widehat{\mu}_{\min}(\overline F_{\! n_1+\cdots+n_\ell})\geqslant\sum_{i=1}^\ell\Big(\widehat{\mu}_{\min}(\overline F_{\!n_i})-\frac 32\nu(\Omega_\infty)\ln(E_{n_i})-\int_{\Omega}b_{n_i}(\omega)\,\nu(\mathrm{d}\omega)\Big).
G_p(r) :=b^{\frac{2}{p}}\cdot \nabla (\mu -\Delta )^{-\frac{1}{2}-\frac{1}{r}}, Q_p(q) \upharpoonright{\mathcal E}:=(\mu -\Delta )^{-\frac{1}{2}+\frac{1}{q}}|b|^{1-\frac{2}{p}}, T_p \upharpoonright{\mathcal E}:=b^{\frac{2}{p}}\cdot \nabla(\mu - \Delta)^{-1}|b|^{1-\frac{2}{p}}.
\chi^2g^{-1}*g^{-1}*\nabla^{g_0(t)}h*\nabla^{g_0(t)}h = \chi^2(1-\chi^2)g^{-1}*g^{-1}*\nabla^{g_0(t)}h * \nabla^{g_0(t)}h + g^{-1}*g^{-1}*\nabla^{g_0(t)}(\chi^2h)*\nabla^{g_0(t)}(\chi^2h) + g^{-1}*g^{-1}*\chi^2*\nabla^{g_0(t)}\chi*\nabla^{g_0(t)}\chi*h*h + g^{-1}*g^{-1}*\chi^3*\nabla^{g_0(t)}\chi*\nabla^{g_0(t)}h*h.
{\tilde c}_{\alpha\beta}^{\enspace \enspace \gamma}=A_{\alpha}^{\delta}A_{\beta}^{\sigma}(A^{-1})^{\gamma}_{\mu}c_{\delta \sigma}^{\enspace \enspace \mu}+(A^{-1})^{\gamma}_{\delta}(A_{\alpha}^{\sigma}E_{\sigma}A_{\beta}^{\delta}+ A^{\sigma}_{\beta}E_{\sigma}A_{\alpha}^{\delta})
|h_{t}(z) - h(z_0)| \leq |h_{t}(z) - h(z)| + |h(z) - h(z_0)| & = |h_{\tau_0}\circ\varphi_{t,\tau_0}(z) - h_{\tau_0}\circ\varphi_{0,\tau_0}(z)| + |h(z) - h(z_0)| & \leq C_3 |\varphi_{t,\tau_0}(z) - \varphi_{0,\tau_0}(z)| + |h(z) - h(z_0)|
\Psi ( \sigma_{1}, \sigma_{2}) = \sum_{s , t \in{\bf Z}}~ e^{i s \sigma_{1}/ ( n - m \theta )}e^{- i t \sigma_{2}/ n}\sum_{i = s b}^{n + s b}~ \sum_{j = m b t}^{n + m b t}c_{s , t , i , j}~ V^{i}U^{j}.
L_f^*(\chi,s) =L_f^*(\underline{\pi},s)|_{E(\pi_j) = \chi(c_j^*) \textrm{for all}j}, \ \C_f^*(\chi,s) =C_f^*(\underline{\pi},s)|_{E(\pi_j) = \chi(c_j^*) \textrm{for all}j}.
\{(L_1, \dots, L_m), (L'_{2m-k}, \dots, L'_m): L_m = L_m' & L_{i-1}\subset L_i, \dim L_i/L_{i-1}= 1, zL_i \subset L_{i-1}, L_i \in Gr^{\lambda^{(i)}(T)}i = 1, \dots ,m & L_{i-1}' \subset L'_i, \dim L'_i/L'_{i-1}=1, zL'_i \subset L'_{i-1}, L'_i \in Gr^{\lambda^{(i)}(T')}i = 2m-k+1, \dots, m \}
L_m(x) =\sum_{s=0}^{m-2}\binom{m-2}{s}x^mF_T(x)C(x)\big(C(x)-1\big)^{s}&+\sum_{s=1}^{m-1}\binom{m-1}{s}x^m\big(F_T(x)-1\big)C^2(x)\big(C(x)-1\big)^{s-1}&=x^mC^{m-1}(x)F_T(x)+\frac{x^mC(x)^{m+1}\big(F_T(x)-1\big)}{C(x)-1}-x^m\big(F_T(x)-1\big)C(x)^2\,,
\begin{aligned}P_\rho :\ X_c \to \mathcal{L}(X_c , X_c) x \mapsto A_c + \rho(x)\end{aligned}\begin{aligned}Q_\rho : \ X_c \mapsto \mathcal{L}(X_c,X_c) x \mapsto (P_\rho (T(x)))^{-1}\end{aligned}
L_{12}=-\Delta^{-1}\left(bc-bdb^{-1}a+\Delta\right)(c-db^{-1}a)^{-1}&=-\Delta^{-1}b-(c-db^{-1}a)^{-1}&=-\Delta^{-1}\left(b+\Delta(c-db^{-1}a)^{-1}\right),
E_{-\theta_{j}v_{j}}\phi_{\hbar}^{(4v_{j})}(\xi)=\phi_{\hbar}^{(4v_{j})}(\xi+\theta_{j,\xi}v_{j})=\hbar^{4}\phi^{(4v_{j})}(\hbar\xi+\hbar\theta_{j,\xi}v_{j})=\hbar^{4}E_{-\hbar\theta_{j}v_{j}}\phi^{(4 v_{j})}(\hbar\xi).
S = \int d^{d - 1}x \sqrt{| g |}\ \left[ R +{\textstyle \frac{1}{2}}\left( \partial \varphi \right)^{2}-{\textstyle \frac{1}{4}}e^{- a \varphi}F_{( 2 )}^{2}+{\textstyle \frac{1}{2}}\left({\cal D}\phi \right)^{2}-{\textstyle \frac{1}{2}}m^{2}e^{a \varphi}\right] ,
\setlength{\abovedisplayskip}{4 pt}\setlength{\belowdisplayskip}{4 pt}{\boldsymbol{W}_{t + 1,j}^{\alpha,out}}={\boldsymbol{W}_{t,j}^{\alpha,out}}+{\lambda^\alpha}\left({\boldsymbol{e}_{t,j}-\boldsymbol{y}_{t,j}\left(\boldsymbol{x}_{t,j}\right)}\right)\left[{{\boldsymbol{v}_{t,j}^\alpha};{\boldsymbol{x}_{t,j}}}\right]^{\mathrm{T}},
\frac{1}{2}\sum_{r , s = 1}^{3}p^{( r )}\cdot p^{( s )}\left( V_{3}^{\left[ r s \right]}\right)_{0 0}= \frac{1}{2}\sum_{r , s = 1}^{3}p^{( r )}\cdot p^{( s )}\mathcal{C}
h_n^+(x):= \begin{cases}0, a_k\leq x\leq a_k+\delta', h(a_k+\delta)\cdot \displaystyle{\frac{\ss_k(x)-\ss_k(a_k+\delta')}{\ss_n(a_k+\delta)-\ss_n(a_k+\delta')}}, a_k+\delta'<x<a_k+\delta, h(x), a_k+\delta\leq x<b_k .\end{cases}
\sum_{\substack{a\bmod{q}a\bar{a}\equiv 1\bmod{q}}}\chi(a)e(n\bar{a}/q) L_f\bigl(s+u,\tfrac{a}{q}\bigr)=\omega q^{1-2s - 2u}\frac{\gamma(1-s-u)}{\gamma(s+u)}\sum_{\substack{a\bmod{q}a\bar{a}\equiv 1\bmod{q}}}e(n\bar{a}/q)L_f\bigl(1-s-u,-\tfrac{\bar{a}}{q}\bigr),
(g_1, \,\ldots, \,g_n, \,g_{n+1}) \cdot (h_1Q, \; \ldots, \; h_{n-1}Q, \; h_n) = (g_1h_1Q, \; \ldots, \; g_{n-1}h_{n-1}Q, \; g_nh_ng_{n+1}^{-1}),
\tilde{\Phi}_{\lambda m}^{\mu V^{( l )}}( z ) = g_{\lambda}^{\mu V^{( l )}}( z ) \int_{0}^{s_{1}\infty}d_{p}t_{1}J^{S}( t_{1}) \cdots \int_{0}^{s_{l - m}\infty}d_{p}t_{l - m}J^{S}( t_{l - m}) \phi_{m}^{( l )}( z ) ,
F(n,k) =[3n+2k+1]\frac{(q;q^2)_{n}(q^{2k+1};q^2)_{n}^2 q^{-{n+1\choose 2}-(2n+1)k}}{(q;q)_{n}^2 (q^2;q^2)_{n}}, begin{align*}5pt]G(n,k) =-\frac{(1+q^{n+2k-1})(q;q^2)_{n}(q^{2k+1};q^2)_{n-1}^2 q^{-{n\choose 2}-(2n-1)k}}{(1-q)(q;q)_{n-1}^2 (q^2;q^2)_{n-1}},
E_{q}={\frac{1}{2 \pi g^{1 / 2}( L )}}\partial_{\bar{\beta}}\Gamma |_{\bar{\beta}= \beta_{H}}={\frac{1}{9 6 \pi}}\int_{x_{+}}^{L}{\frac{1}{g}}({\frac{4}{\beta_{H}^{2}}}- g^{2}( x ) ) d x -{\frac{1}{1 6 \pi g^{1 / 2}( L )}}g^{\prime}( L ) .
f_y(z,y)-Tz^{10}= y^6z^4-4y^5z^3-4y^5z^5+12y^4z^4+6y^4z^2+\frac{41}{4}y^4z^6-4y^3z-\frac{25}{2}y^3z^7 -12y^3z^3-\frac{99}{4}y^3z^5+4y^2z^2+\frac{75}{4}y^2z^4+\frac{59}{4}y^2z^6+y^2-\frac{17}{4}yz^3 +\frac{75}{4}yz^7+4yz^5+\frac{25}{4}yz^9-\frac{25}{2}z^8-\frac{25}{4}z^4-\frac{71}{4}z^6-Tz^{10}
B(N) = \sum_{1\leq n \leq N}2^{\omega(n)}\cdot \lambda_\mathcal{G}(n)=\sum_{1\leq n \leq N}\sum_{d|n}\mu^2_{\mathcal{G}}(d) \cdot \lambda_\mathcal{G}(n) &\stackrel{(**)}{=}\sum_{1\leq n \leq N}\sum_{d|n}\mu^2_{\mathcal{G}}(d) \cdot \lambda_\mathcal{G}\left(\frac{n}{d}\right) =\sum_{1\leq n \leq N}\left(\mu^2_{\mathcal{G}}* \lambda_\mathcal{G}\right)(n)
\begin{array}{l}\Lambda_{i}=\widehat{x}_{i}\otimes e^{\frac{\widehat{p}_{0}}{\kappa}}-1\otimes\widehat{x}_{i}, \Lambda_{0}=\widehat{x}_{0}\otimes1-1\otimes\widehat{x}_{0}+\frac{1}{\kappa}\widehat{x}_{i}\otimes e^{\frac{\widehat{p}_{0}}{\kappa}}\widehat{p}_{i}.\end{array}
\#\{\tau_{s_k}\in{\bf{S}}_{s_k}:f_{\tau_{s_k}}\not\equiv 0\}\prod_{i=1}^{k-1}\max_{\tau_{s_{i+1}}\in{\bf{S}}_{s_{i+1}}}\#\{\tau_{s_i}\in{\bf{S}}_{s_i}:\tau_{s_i}\subset\tau_{s_{i+1}},f_{\tau_{s_i}}\not\equiv 0\}.
\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\cdot\cdot\cdot\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{1}{1-x_{1}x_{2}\cdot\cdot\cdot x_{n-1}x_{n}}dx_{1}dx_{2}\cdot\cdot\cdot dx_{n-1}dx_{n}
\sum_{m=0}^{\delta^2}p^\#_m(o, y) \le \int_0^{10 \delta^2}p_t(o, y) \mathrm{d}t + \delta^2 \P( N\le 2\delta^2) & \le O( 1/ f(n))+ \delta^2 \exp ( - c \delta^2),
\tilde{h}_k^{(1)}\big(2 \ln (k)\big) = \gamma_{2\ln(k)}k^{\frac{1}{2\ln(k)}}q_p &= \Bigg[\frac{\Gamma\big(\ln(k)+\frac{1}{2}\big) k}{-\big(2\ln(k)\big)^{1-\ln(k)}\sqrt{\pi}\Gamma\big(\ln(k)\big) \ln(p)}\Bigg]^{\frac{1}{2\ln(k)}}&\sim \sqrt{2e\ln(k)}.
q_t(s_j,y) := P\Big( \big( S_{s_j}^1\big)_+ \ge \frac{\psi(t) s_j}{(1+\delta) t}- y \Big) \,, p_t(r,y) := P \Big( S_{r}^1 \ge \frac{\psi(t) (\delta t + r)}{(1+\delta)t}+ y \Big) \,
{\|y(x)-z(x)\|}_{C_{1-\gamma}[a,x_1]}={\|I_{a^+}^{\alpha}f(x,y(x))-I_{a^+}^{\alpha}f(x,z(x))\|}_{C_{1-\gamma}[a,x_1]}\leq A \frac{(x_1-a)^{\alpha}\Gamma(\gamma)}{\Gamma(\gamma+\alpha)}{\|y(x)-z(x)\|}_{C_{1-\gamma}[a,x_1]}
\begin{array}{r c l}{\Phi}&{=}&{\displaystyle \frac{1}{N}\sum_{n = 0}^{\infty}\left\langle \mathrm{T r}( u U + v V )^{n}\right\rangle}\\{}&{=}&{\displaystyle \frac{1}{N}\sum_{w ( u , v )}w ( u , v ) \left\langle \mathrm{T r}w ( U , V ) \right\rangle \ .}\\ \end{array}
I^{G}(\bar x,\bar y,\bar z) :=\{l\in\mathcal Q\,|\,G_l(\bar x)=\bar y_l\,\land\,H_l(\bar x)\neq\bar z_l\}, I^{H}(\bar x,\bar y,\bar z) :=\{l\in\mathcal Q\,|\,G_l(\bar x)\neq\bar y_l\,\land\,H_l(\bar x)=\bar z_l\}, I^{GH}(\bar x,\bar y,\bar z) :=\{l\in\mathcal Q\,|\,G_l(\bar x)=\bar y_l\,\land\,H_l(\bar x)=\bar z_l\}.
{\hat \pi}^i_a(0)\simeq m{\Delta{\hat x}^i_a\over\varepsilon}-{e\over 2}\sum_{b=1}^N\epsilon^{ij}[{\hat B}({\hat{\bf x}}_b(\varepsilon) )\delta_{ab}+{e\over\theta}\delta({\hat{\bf x}}_a (\varepsilon)-{\hat{\bf x}}_b(\varepsilon))]\Delta{\hat x}^j_b\,,
\tilde B_{ii}(k_1:k_0) \geq \eta_i 0\leq k_0\leq k_1\leq \tilde k_L,\cr \tilde B_{jj}(k_1:k_0) \geq \tilde \eta_j 0\leq k_0\leq k_1\leq \sigma,\cr \sum_{k=0}^{\sigma-1}B_{ji}(k) \geq \tilde \delta,
6 \frac{t_{1}^{4}}{4 !}+ t_{4}\frac{t_{0}^{3}}{3 !}+ 4 \frac{t_{0}^{2}}{2 !}\frac{t_{2}^{2}}{2 !}+ 6 t_{0}t_{2}\frac{t_{1}^{2}}{2 !}+ 3 t_{1}t_{3}\frac{t_{0}^{2}}{2 !}
\begin{aligned}y_jy_i ={\lambda}_{ji}y_iy_j 1\leq i<j\leq n y_jx_i ={\lambda}_{ij}x_iy_j 1\leq i<j\leq n x_jy_i ={q}_i{\lambda}_{ij}y_ix_j 1\leq i<j\leq n x_jx_i ={q}_{i}^{-1}{\lambda}_{ij}^{-1}x_ix_j 1\leq i<j\leq n x_iy_i-{q}_iy_ix_i =({q}_i-1)\left(1+\sum_{k=1}^{i-1}y_kx_k\right) \qquad 1\leq i\leq n.\end{aligned}
\mathcal{L}(w) =\begin{cases}-a +\chi\tilde{v}(x,t),t\ge 0\cr-a+\chi \tilde v(x,0), t<0\end{cases}&\leq -a + \frac{\chi a}{b-\chi}+\chi \varepsilon=\chi \varepsilon- \frac{a(b-2\chi)}{b-\chi}.
\sum_{n_{1}, n_{2}}\! \! \!^{\prime}f ( n_{1}, n_{2}) = \sum_{n_{1}= 1}^{\infty}\ \sum_{n_{2}= - \infty}^{\infty}\f ( n_{1}, n_{2}) + \sum_{n_{2}= 0}^{\infty}\f ( 0 , n_{2})
\begin{aligned}\|f_{j}\|_{C^{\alpha}(S_h)}\leq k^2\left(\|q\|_{C^{\alpha}(\overline S_h)}\sup_{S_h}\vert v_{j}\vert+\|v_{j}\|_{C^{\alpha}( \overline S_h)}\sup_{S_h}\vert q-1\vert\right). \end{aligned}
L=\frac{1}{2}(\partial_{\mu}\phi_{r})(\partial^{\mu}\phi_{r})-\frac{1}{2}m_{r}^{2}\phi_{r}^{2}-\frac{\lambda_{r}}{4!}\phi_{r}^{4}+\frac{1}{2}\delta_{Z}(\partial_{\mu}\phi_{r})(\partial^{\mu}\phi_{r})-\frac{1}{2}\delta_{m}\phi_{r}^{2}-\frac{\delta_{\lambda}}{4!}\phi_{r}^{4},
\aligned\sum_{k_1+k_2=k+1}\sum_{i=1}^{k-k_2+1}(-1)^*{\frak m}_{k_1}(x_1,\ldots,{\frak m}_{k_2}(x_i,\ldots,x_{i+k_2-1}),\ldots,x_{k}) = 0.\endaligned
(v_1^{p^e+1}f^{p^e}(a_1),v_2^{p^e+1}f^{p^e}(a_2),\dots,v_n^{p^e+1}f^{p^e}(a_n),f_{k-1}^{p^e})=(u_1g(a_1),u_2g(a_2),\dots,u_ng(a_n),-g_{n-k}),
x_1 =\cos\vartheta_1, x_2 =\sin\vartheta_1\cos\vartheta_2, x_3 =\sin\vartheta_1\sin\vartheta_2\cos\vartheta_3, \dots x_{n-1}=\sin\vartheta_1\dots\sin\vartheta_{n-2}\cos\vartheta_{n-1}, x_n =\sin\vartheta_1\dots\sin\vartheta_{n-2}\sin\vartheta_{n-1}\cos\varphi, x_{n+1}=\sin\vartheta_1\dots\sin\vartheta_{n-2}\sin\vartheta_{n-1}\sin\varphi,
[\delta(a), \delta(b)](c) =\delta(a)\delta(b)(c)-(-1)^{|a||b|}\delta(b)\delta(a)(c) &=\{a, \{b, c\}\}-(-1)^{|a||b|}\{b, \{a, c\}\}, \delta(\{a, b\})(c) =\{\{a, b\}, c\}, (\delta d)(a)(b) =\delta(d(a))(b)=\{d(a), b\}
c({\lambda,0}) = \inf\{I_{\lambda,0}(\mathbf{u}):\ \mathbf{u}\neq 0,\ \mathbf{u}\in \mathcal{N}_{\lambda,0}\}&\leq \inf\{I_{\lambda,0}(u_1,u_2,0,\ldots,0):\ (u_1,u_2)\neq (0,0),\ (u_1,u_2,0,\ldots, 0)\in \mathcal{N}_{\lambda,0}\}.
\begin{array}{l c r}{{P ( W_{2}) = ( y^{3}+ x^{3}+ z^{3}+ \mu x y z ) + w ( y^{2}+ x z ) + w^{2}( y + x^{{\frac{1}{2}}}z^{{\frac{1}{2}}}) + w^{3}.}}\\ \end{array}
R_{1}=R(x, qx, x, qx), R_{2}=R(x, qx, q^{2}x, x), R_{3}=R(x, q^{2}x, x, q^{2}x),R_{4}=R(x, qx, qx, q^{2}x), R_{5}=R(qx, q^{2}x, q^{3}x, x),\R_{6}=R(qx, q^{2}x, q^{2}x, x).
(\vartheta_{X}-\vartheta_{X'})(y,\nu,\ell) = \sum_{i=1}^r \langle \nu,u_i \rangle (\vartheta_X-\vartheta_{X'})(y,\rho_i,0) + \sum_{j=1}^{s-r}\langle (\nu,\ell),w_j \rangle (\vartheta_X-\vartheta_{X'})(y,\kappa(p_j)p_j,\kappa(p_j)).
0 =\rho_{z_{k}}+\sum_{l=1}^{n}\rho_{a_{l}}(A_{l})_{z_{k}}+\rho_{b}B_{z_{k}}0 =\rho_{z_{i}z_{k}}+\sum_{l=1}^{n}\rho_{z_{i}a_{l}}(A_{l})_{z_{k}}+\rho_{z_{i}b}B_{z_{k}}, 1 \leq i \leq n
A_l^k=\left(\frac{1}{\Gamma\left(\frac{n+1}{2}\right)}\prod_{\tau=1}^{n-1}\frac{2^{n-\tau+2k_\tau-2}\,(k_{\tau-1}-k_\tau)!\,(n-\tau+2k_{\tau-1})\,\Gamma^2(\frac{n-\tau}{2}+k_\tau)}{\sqrt\pi\,\Gamma(n-\tau+k_{\tau-1}+k_\tau)}\right)^{1/2},
p^{n_c}(2g_n-2) \geq p^n (2g_0-2) + \sum_{i=n_u+1}^n \varphi(p^i) \left( 1 + p^{i-n_u-1}\right) = p^n (2g_0-2) +p^n-p^{n_u}+ p^{n_u}\frac{p^{2(n-n_u)}-1}{p+1}.
\left( \phi_{W}\star \psi_{W}\right) \left( \hat{x}\right) \equiv \phi \left( \hat{x}\right) \psi \left( \hat{x}\right) = \frac{1}{\left( 2 \pi \right)^{n}}\int d^{n}k d^{n}q \, e^{- i ( k + q ) \hat{x}}e^{i k \hat{x}}\tilde{\phi}\left( q + k \right) \tilde{\psi}\left( - k \right) .
I_{\Psi}= - \int_{{\cal M}_{b}}d^{5}x \sqrt{- g}\left[ \frac{1}{2}g^{M N}\partial_{M}\Psi \partial_{N}\Psi + U ( \Psi ) \right] - \sum_{\sigma = \pm}\int_{\Sigma_{\sigma}}d^{4}y_{\sigma}\sqrt{- q_{\sigma}}V_{\sigma}( \Psi_{\sigma})
Color_1=\{1, 2, 3, 43, 44, 45, 46, 49, 51\}, &Color_2=\{4, 5, 6, 8, 11, 14, 18, 20, 21, 23, 28, 33, 36, 48\}, &Color_3=\{7, 13, 15, 17, 25, 26, 29, 31, 34, 37, 38, 40, 42, 47\}, &Color_4=\{9, 10, 12, 16, 19, 22, 24, 27, 30, 32, 35, 39, 41, 50\}.
\begin{array}{l l}{{K_{1}^{u}= 0}}&{{K_{1}^{v}= - \sin u}}\\{{K_{2}^{u}= 0}}&{{K_{2}^{v}= \cos u}}\\{{K_{3}^{u}= \sin v}}&{{K_{3}^{v}= 0}}\\{{K_{4}^{u}= - \cos v}}&{{K_{4}^{v}= 0 .}}\\ \end{array}
-\frac{\sin^3(\nu\pi)}{\pi^3}\frac{\mathfrak Q_{\nu}({\alpha})}{[P_\nu(1-2\alpha)]^{3}}={}\frac{2}{3}+O\left( \left[\frac{P_\nu(2\alpha-1)}{P_{\nu}(1-2\alpha)}\right]^2 \right), \alpha\to{}1-0^{+}.
\frac{dT(x,A)}{dx}=\frac{-v'(x)+\frac{v(x)v'(x)-F_{1}}{\sqrt{D(x)}}}{F_{1}}=v'(x)\frac{v(x)-\sqrt{D(x)}}{F_{1}\sqrt{D(x)}}-\frac{1}{\sqrt{D(x)}}.
h'(\tau) =-(4-d)(\sqrt{\omega}-\tau)^{3-d}\mu \left(1,\frac{\mathbf{c}}{\sqrt{\omega}}\right), h''(\tau) =(3-d)(4-d)(\sqrt{\omega}-\tau)^{2-d}\mu \left(1,\frac{\mathbf{c}}{\sqrt{\omega}}\right),
\frac{1}{\rho}\left( A_{iJKj}u_{j,K}- \beta_{Ji}F^3 -(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L}- B_{iJKj}F^1_{j})_{,K}\right)_{,J}= F^2_i, \frac{1}{a}\left( -\beta_{Ki}F^1_{i,K}+ (m_{IJ}F^3_{,J}+ M_{jLKI}u_{j,LK}+K_{IJ}\tau_{,J})_{,I}\right) = F^4.
{P_2}\left( n \right) = \frac{1}{{n - 3}}{{\bar B}_1}\left({\frac{1}{{{\delta^2}}},n - 1,n - 2}\right) \,\,\, + \frac{1}{{n - 1}}{{\bar B}_1}\left({\frac{1}{{{\delta^2}}},n - 3,n}\right) \,\,\, - \frac{2}{{n - 2}}{{\bar B}_1}\left({\frac{1}{{{\delta^2}}},n - 2,n - 1}\right)
\begin{array}{l}{\cal D}_{+k}{\cal X}^{+k}-{\cal X}^{+k}{\cal D}_{+k}=\mu_{+k}~,~~~~~~\mu_{+k}{\cal X}^{+k}=q^2{\cal X}^{+k}\mu_{+k}~, \mu_{+k}{\cal D}_{+k}=q^{-2}{\cal D}_{+k}\mu_{+k}~,~~~~~~\mu_{+k}\equiv1+(q^2-1){\cal X}^{+k}{\cal D}_{+k}~,\end{array}
\begin{aligned}\left|\sum_{j=1}^k\left\{\big[-(-\Delta)^{\frac{s}{2}}\eta_{j, R},-(-\Delta)^{\frac{s}{2}}\tilde{\phi}_j\big]+ \tilde{\phi}_j\big(-(-\Delta)^{s}-\partial_t\big)\eta_{j, R}\right\}\right| &\quad\quad\lesssim \frac{1}{R^{a-2s}}\|\phi\|_{n-2s+\sigma,a}\sum_{j=1}^k\frac{\mu_j^{-2s}\mu_0^{\frac{n-2s}{2}+\sigma}(t)}{1+|y_j|^{a}}.\end{aligned}
\partial_{z}A_{2}- A_{3}\partial_{z}\varphi_{0}+ \bar{A}_{1}A_{3}+ \bar{A}_{2}A_{2}= \partial_{\bar{z}}\bar{A}_{2}- \bar{A}_{3}\partial_{\bar{z}}\varphi_{0}+ A_{1}\bar{A}_{3}+ A_{2}\bar{A}_{2}\, .
\frac{\partial}{\partial\bar{\Lambda}_1}\left[ - 2 a_1 \bar\Lambda_1 + \frac{\partial}{\partial \bar \Lambda_1}\mathcal Poly_4 (0,\zeta_0,\bar \Lambda)\right](\bar{\Lambda}^0) =-2a_1\,+12a_2\lambda_0\Bigl(\sum_{i=1}^kP_{i1}(0,\zeta^0)^4\Bigr)(\bar{\Lambda}_1^0)^2 & =4a_1,
EB^2(t,s,x ,\omega, v)= \big( \sum_{i=1,2,3}- (\hat{v}+\omega)_2( \omega \times \mathbf{e}_3)\cdot \mathbf{e}_i (B\cdot(\omega \times \mathbf{e}_i))- (\hat{v}+\omega)_3 B_2, \sum_{i=1,2,3}(\hat{v}+\omega)_1( \omega \times \mathbf{e}_3)\cdot \mathbf{e}_i (B\cdot(\omega \times \mathbf{e}_i))
\nabla_{e_1}e_1 = m_{12}(-e_2+ e_3), \enskip \nabla_{e_1}e_2 = \frac{1}{2}e_6, \enskip \nabla_{e_1}e_3 = -e_5 2 \nabla_{e_1}e_6 = -e_2 + e_3 - e_4 = \nabla_{e_2}e_3 = \nabla_{e_3}e_3, \nabla_{e_2}e_2 = e_2 - e_3, \enskip \nabla_{e_2}e_6 = \frac{1}{2m_{12}}e_5.
\begin{aligned}\sum_{j\le j_0}\psi_j2^{jr}\Vert\dot{\Delta}_jR(f,g)\Vert_{L^2}\le C \Big( \sup_{j'\in\mathbb{Z}}2^{j'(r_3+\frac{N}{2}-\beta )}\Vert \dot \Delta_{j'}g \Vert_{L^2}\Big) \times\sum_{j\le j_0}\sum_{j'>j-4}2^{(j-j')(r+\frac{N}{2}-\beta )}2^{j'(r-r_3)}\Vert \dot\Delta_{j'}f \Vert_{L^2}\psi_j.\end{aligned}
A_{j}(v_1,...,v_{k-2j})=\int_0^\infty 2\beta^{-1}\sum_{\alpha=1}^{k-2(j-1)-1}\sum_{\delta>\alpha}A_{j-1}^{\alpha\delta}(e^{-t\tilde\gamma}v_1,...,e^{-t\tilde\gamma}v_{k-2j})dt,
y(1-y)\frac{\partial^{2}F_{3}(x,y)}{\partial y^{2}}+x\frac{\partial^{2}F_{3}(x,y)}{\partial x\partial y}+[c-(a_{2}+b_{2}+1)y]\frac{\partial F_{3}(x,y)}{\partial y}-a_{2}b_{2}F_{3}(x,y)=0
\begin{cases}\dot{\theta}=\partial_y H_{\varepsilon}(\theta, y, z), \dot{y}=-\partial_{\theta}H_{\varepsilon}(\theta, y, z), \dot{z}=\partial_x \nabla_z H_{\varepsilon}(\theta, y, z),\end{cases}H_{\varepsilon}:=\varepsilon^{-2 b}\,\mathcal{H}\circ A_{\varepsilon},
J_1(g) =[b,R_j](f_1)(g), J_2(g,\tilde g) =\big(b(g)-b_B\big)\big(R_j(f_2)(g)-R_j(f_2)(\tilde g)\big), J_3(g',g) =R_j\big((b-b_B)f_2\big)(g')-R_j\big((b-b_B)f_2\big)(g), J_4(g,\tilde g) =\big(b(g)-b_B\big)R_j(f_2)(\tilde g).
p_1(A_1) = (A_1-s_1)(A_1-t_1)(A_1-u_1) &= UW(Z - s_1)W^{-1}U^{-1}UW(Z - t_1)W^{-1}U^{-1}UW(Z - u_1)W^{-1}U^{-1}&= UW(Z - s_1)(Z - t_1)(Z - u_1)W^{-1}U^{-1}= UWO_3W^{-1}U^{-1}= O_3
\mathrm{d}\mu_{\mathrm{o n e - f l}}= \frac{1}{2^{1 1}\pi^{4}v^{2}}\, M_{\mathrm{P V}}^{5}\left( \frac{8 \pi^{2}}{g^{2}}\right)^{2}\! \exp \left( - \frac{8 \pi^{2}}{g^{2}}- 4 \pi^{2}| v |^{2}\rho_{\mathrm{i n v}}^{2}\right) \frac{\mathrm{d}\rho^{2}}{\rho^{2}}\, \mathrm{d}^{4}x_{0}\, \mathrm{d}^{2}\theta_{0}\, \mathrm{d}^{2}\bar{\beta}_{\mathrm{i n v}}\, \mathrm{d}^{2}\bar{\theta}_{0}\, .
\ell(\hat y_t,y_t) \leq \left(\frac{1}{\eta_{t+1}}-\frac{1}{\eta_{t}}\right) \log{D_{t}}-\frac{1}{\eta_{t+1}}\log \! \left( \sum_{d=1}^{D_t}\hat p_{d,t}^{\frac{\eta_{t+1}}{\eta_t}}e^{-\eta_{t+1}\ell(f_{d,t},y_t)}\right) + \frac{\eta_t}{8}
S_F(x-x')= \int \frac{d\omega}{2\pi}\frac{dk}{2\pi}\; \frac{i e^{-i\omega (t -t')}}{\omega{}^2 - k{}^2 - m^2 + i\epsilon}\left( \begin{array}{cc}m -i(\omega +k) i(\omega -k) m \end{array}\right) e^{ik(x -x')}.
D_{\! x}^{\sigma}\sum_{k = 0}^{\infty}a_{k}x^{k}= \! \! \sum_{k \, = \left\lceil \sigma \right\rceil}^{\infty}\! a_{k}\, \frac{\Gamma ( 1 \! + \! k )}{\Gamma ( 1 \! + \! k \! - \! \sigma )}\, x^{k - \sigma}\quad ( \sigma > 0 )
f^{(1)}(\boldsymbol\alpha) \triangleq -\log p^{(1)}(\mathbf R;\boldsymbol\alpha) -L\log\pi = \log|\boldsymbol\Sigma^{(1)}_{\boldsymbol\alpha}|+\left(\boldsymbol\Sigma_{\boldsymbol\alpha}^{(1)-1}\widehat{\mathbf \Sigma}_{\mathbf R}\right).
\displaystyle{\raisebox{1.1ex}{\scriptsize{$\diamond$}}}\mbox{\hspace{-0.33ex}}\tilde{S}^{(\epsilon_{j + 1})}(Q^{(j + 1)}) \simeq \frac{1}{2}\sum_{\rho = \pm}\frac{{\cal N}^{(\rho)}(Q^{(j + 1)})}{{\cal D}_\rho}\, ,
\tau ( x_{1}x_{2}. . . | y_{1}y_{2}. . . ) = \langle 0 | T [{\hat{\psi}}( x_{1}){\hat{\psi}}( x_{2}) . . .{\hat{\bar{\psi}}}( y_{1}){\hat{\bar{\psi}}}( y_{2}) . . . ] | \Phi \rangle
X_{0}^{A{A^{\prime}}}: = i{\sqrt 2}( \sigma^{A}\beta^{A^{\prime}}- \varsigma^{A}\alpha^{A^{\prime}}) = - i{\sqrt 2}({\bar{\sigma}}^{A^{\prime}}{\bar{\beta}}^{A}-{\bar{\varsigma}}^{A^{\prime}}{\bar{\alpha}}^{A}) ,
\lim_{u\to\infty}\frac{g_{\kappa,u}(g^{-1}(u)x)}{q(u)}=\lim_{u\to\infty}\frac{\sum_{i=0}^\kappa(g(M^i g^{-1}(u)x)-u)}{q(u)}=\sum_{i=0}^\kappa((M^{-1/\alpha})^i x^{-1/\alpha}-1) &=\frac{1-M^{-(\kappa+1)/\alpha}}{1-M^{-1/\alpha}}x^{-1/\alpha}-(\kappa+1)
I_{1}= \int_{T_{\epsilon}^{2}}d^{2}z \partial_{\bar{z}}\overline{{{\rho}}}\partial_{z}\rho = \frac{1}{2 i}\int_{T_{\epsilon}^{2}}\bar{\partial}\overline{{{\rho}}}\wedge \partial \rho = \frac{1}{2 i}\int_{| z | = \epsilon}d z \overline{{{\rho}}}\partial_{z}\rho ,
S_{\mathrm{\tiny{M e m b r a n e}}}= \int d^{3}\sigma \sqrt{h}[ \frac{1}{2}h^{\alpha \beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}g_{M N}- \frac{1}{2}- i \bar{\theta}\Gamma^{\alpha}\nabla_{\alpha}\theta
\|g\|_{X(w)'}=\sup_{\|f\|_{X(w)}=1}\int_\Omega\!|f|w|g|w^{-1}\,\mathrm{d}\mu=\sup_{\|h\|_{X}=1}\int_\Omega\!|h||g|w^{-1}\,\mathrm{d}\mu &=\|gw^{-1}\|_{X'}=\|g\|_{X'(w^{-1})}.
\int_{\mathbb{R}^{2N}}\frac{J_p(U(x)-U(y))\, \big(\psi_{\varepsilon}(U(x))-\psi_\varepsilon(U(y)\big)}{|x-y|^{N+s\,p}}\,dx\,dy=\mathcal{S}_{p,s}\, \int_{\mathbb{R}^N}U^{p^*-1}\,\psi_\varepsilon(U)\,dx.
|H| =n- \sum\limits_{i=1}^{2l+1}(|S_i|+|T_i|)-|V(C_{2l+1})| &\leq n-(2l+1)\left( \frac{n}{2(2l+1)}-2+\frac{n}{2(2l+1)}-1\right)-(2l+1) &\leq 2(2l+1).
= \; \mathrm{T r}\Big [ \frac{- \not{\! \partial}+ m}{- \triangle + m^{2}}\not{\! \! B}\frac{- \not{\! \partial}- m}{- \triangle + m^{2}}\not{\! \! B}\frac{- \not{\! \partial}+ m}{- \triangle + m^{2}}\not{\! \! B}\frac{- \not{\! \partial}- m}{- \triangle + m^{2}}\not{\! \! B}\Big ] \; .
z^\mu \frac{\rho}{z}z^{-\mu}= e^{(\mu \log z)}\frac{\rho}{z}=\sum_k \frac{1}{k!}((\mu \log z))^k \frac{\rho}{z}=\sum_k \frac{1}{k!}(\log z)^k \frac{1}{z}[\underbrace{\mu,[\mu,[\cdots,[\mu}_{},\rho]\cdots]] =\sum_k \frac{1}{k!}(\log z)^k \frac{1}{z}\rho =\rho
(e(t)+y(t))' \leq M_{1}G_{1}-ak_{1}y &= M_{1}G_{1}+M_{1}G_{1}-M_{1}G_{1}-ak_{1}y &\leq 2M_{1}G_{1}-\frac{M_{1}}{\bar{x}}e(t)-ak_{1}y &\leq 2M_{1}G_{1}-\bar{\mu}(e(t)+y(t))\textnormal{where}\bar{\mu}=\min\left\lbrace \frac{M_{1}}{\bar{x}},ak_{1}\right\rbrace ,
{I_5}= -\int_\Omega (v_z\cdot\nabla_H\bar v)\cdot v_z - \int_\Omega (v_z\cdot\nabla_H\tilde v)\cdot v_z &= -\int_\Omega (v_z\cdot\nabla_H\bar v)\cdot v_z + \int_\Omega \mathrm{div}_H\,v_z\, \tilde v\cdot v_z + \int_\Omega (v_z\cdot\nabla_Hv_z)\cdot\tilde v &=:{I_{51}+ I_{52}+ I_{53}},
g_{i j}\! = \! \bar{g}_{i j}- 2 s^{A}\bar{\kappa}_{i j A}+ s^{A}\! s^{B}( \bar{g}^{m n}\bar{\kappa}_{i m A}\bar{\kappa}_{j n B}+ g^{M N}\! A_{i M A}A_{j N B}\! )
x_{d}^{( 1 / 2 )}( u , \xi ) = \biggl [{\frac{\sigma ( 2 \xi - u )}{\sigma ( 2 \xi ) \sigma ( u )}}+{\frac{\sigma ( 2 \xi - u - \omega_{1})}{\sigma ( 2 \xi ) \sigma ( u + \omega_{1})}}\exp [ 2 \eta_{1}\xi ] \biggr ] .
g^{(1)}(t) =Q^1 \left(\exp(\pi(t)) \vee Q^1(\lambda(t)\vee \exp(\pi(t)))\right) & =Q^1\left(Q\left(\lambda(t)\vee \exp(\pi(t))\right) + \lambda(t)\vee \exp(\pi(t)) \vee Q^1(\exp(\pi(t)))\right) & = Q^1\left(\lambda(t)\vee \exp(\pi(t)) \vee g^{(0)}(t)\right).
\hat u(x, k)= -k^2\int_{\mathbb{R}^3}\left[1-c^{-2}(y)\right]\cdot \hat{u}(y, k)\cdot\Phi(x-y)\dy & -\frac{ik}{2\pi}\int_{\mathbb{R}^3}c^{-2}(y)\cdot f(y)\cdot \Phi(x-y)\dy,x\in\mathbb{R}^3.
\left( \begin{array}{l l}{a}{b}{c}{d}\end{array}\right) = \left( \begin{array}{l l}{\operatorname{c o s}\alpha / 2}{\operatorname{s i n}\alpha / 2}{- \operatorname{s i n}\alpha / 2}{\operatorname{c o s}\alpha / 2}\end{array}\right) .
(-1)^{n}\sum_{r=0}^{m}\frac{L_{m+n-r}^{(n)}(x)\pi_{r}(y)}{(n+m-r)!r!}=\sum_{r=0}^{m}(-1)^{r}(\begin{matrix}-n-1\\r\end{matrix})\frac{\pi_{n-r}(x+y)}{(m-r)!},
\left(\frac{I_T(snr)}{snr}\right)' =\frac{1}{snr}\left(I'_T(snr)-\frac{I_T(snr)}{snr}\right) &=\frac{1}{2 snr}\left(\int_0^T E[(X(s)-E[X(s)|Y_0^T])^2] ds-\int_0^T E[(X(s)-E[X(s)|Y_0^s])^2] ds \right) \leq 0,
(F_{\mu\nu})^{(+)}_{+1}= + (F_{\mu\nu})^{(-)}_{-1}\, , \,(F_{\mu\nu})^{(+)}_{-1}= + (F_{\mu\nu})^{(-)}_{+1}\, , \,(F_{\mu\nu})^{(+)}_{0}= - (F_{\mu\nu})^{(-)}_{0}\quad.
\frac{\zeta(1/5)^{2}\zeta(7/10)\zeta(9/10)}{\zeta(1/10)\zeta(3/10)\zeta(4/5)^{2}}=\frac{(5-\sqrt{5})(\sqrt{10}+\sqrt{5+\sqrt{5}})}{10\cdot2^{\frac{3}{10}}}
b_{0}.b_{1}b_{2}b_{3}b_{4}...=b_{0}+b_{1}(\frac{1}{10})+b_{2}(\frac{1}{10})^{2}+b_{3}(\frac{1}{10})^{3}+b_{4}(\frac{1}{10})^{4}+\cdot\cdot\cdot.
\left|\frac{u_{k_1+j}(x)}{u_{k_1}(x)}-\frac{u_{k_1+i}(x_0)}{u_{k_1}(x_0)}\right|\leq C r^{\alpha}\forall x\in \overline{B_{r}(x_0)\cap \Omega_2},\j=2,\ldots, \tilde l.
z(1-z)\mathsf{F}_{zz}\left(\tfrac{1-\sqrt{\delta}}{2},\tfrac{1-\sqrt{\delta}}{2};1 ;z\right)+\left[1-\left(2-\sqrt{\delta}\right)z\right] \mathsf{F}_z\left(\tfrac{1-\sqrt{\delta}}{2},\tfrac{1-\sqrt{\delta}}{2};1 ;z\right)-\left(\tfrac{1-\sqrt{\delta}}{2}\right)^2 \mathsf{F}\left(\tfrac{1-\sqrt{\delta}}{2},\tfrac{1-\sqrt{\delta}}{2};1 ;z\right)=0.
{J}_i(u_i, u_{-i}, f)=\ \mathbb{E}\left[\int_0^T \left(|x_i- (\Gamma x^{(N)}+\eta)|_Q^2 + u_i^T R u_i -\frac{1}{\gamma}|f(t)|^2\right) dt+x_i^T(T)Hx_i(T)\right],
\delta_{f}\left( \left\{g , h \right\}_{{\footnotesize P B}}\right) \equiv \left\{f , \left\{g , h \right\}_{{\footnotesize P B}}\right\}_{{\footnotesize P B}}= \left\{\left( \delta_{f}g \right) , h \right\}_{{\footnotesize P B}}+ \left\{g , \left( \delta_{f}h \right) \right\}_{{\footnotesize P B}}.
V \approx \frac{m}{4 \pi}g^{2}\{( 1 + \frac{m^{2}}{8M^{2}}) \phi+ \frac{m^{2}}{4M^{2}}\mbox{\boldmath $\sigma$}^{(1)}\cdot \mbox{\boldmath $\sigma$}^{(2)}\phi \}
u_{\alpha\beta}:= \varphi_{\alpha}\circ\varphi_{\beta}^{-1}-\mathrm{id}_{E|_{U_{\alpha\beta}\otimes\mathbb{C}[\epsilon]}}, v_{\alpha}:= (\varphi_{\alpha}\otimes\mathrm{id})\circ\nabla^v\circ\varphi_{\alpha}^{-1}-\nabla|_{U_{\alpha}}\otimes\mathrm{id}_{\mathbb{C}[\epsilon]}.