harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \frac{t}{(1+\lambda t)^{\frac{1}{\lambda}}-1}\right)^r (1+\lambda t)^{\frac{x}{\lambda}}= \sum_{n=0}^\infty \beta_{n,\lambda}^{(r)}(x) \frac{t^n}{n!},(\textnormal{see}\,\, [1,2]).
\Phi_a(x,\xi,z) =\varphi_a(x-z,\xi)-\varphi_a(x,\xi)+z\cdot\nabla_x\varphi_a(x,\xi) &=z\cdot \left(\int_0^1 \theta_1 \int_0^1 \nabla_x^2\varphi_a(x-\theta_1\theta_2 z,\xi) d\theta_2 d\theta_1\right)z .
\begin{array}{r l l l l l l}{\omega_{2}}&{=}&{\omega_{3}}&{=}&{\omega_{5}}&{=}&{1 ,}\\{\mathrm{a n d}}&{}&{\omega_{1}^{2}}&{=}&{\omega_{4}}&{=}&{- ( q - q^{- 1})^{2},}\\ \end{array}
A_i \geq e^{-a|1-y_i|^2}+ e^{-ay_i^2}\geq \frac{1}{2}e^{-\frac{a}{2}(|1-y_i|^2+y_i^2)}&=\frac{1}{2}e^{a(y_i-y_i^2-\frac{1}{2})}\geq \frac{1}{2}e^{-\frac{a}{2}}.
g(\nabla_{\partial_n}\partial_i, \partial_j) = \frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^n}+ \frac{\partial g_{jn}}{\partial x^i}- \frac{\partial g_{in}}{\partial x^j}\right) = \frac{1}{2}\frac{\partial g_{ij}}{\partial x^n}
|B(\tilde g_i)(X,z)| =\frac{1}{|z|}\Big| \tilde h_i(z+t_0(X)) \tilde Q(z+t_0(X),D_t \tilde h(z+t_0(X))) & - \tilde h_i(t_0(X)) D_t\tilde Q(t_0(X),D_t \tilde h(t_0(X))) z\Big| &\leq A_1 + A_2 + A_3
\hat{\mathbf{y}}_{\scriptscriptstyle{A}}= \hat{\mathbf{H}}_3 \,\mathbf{s}^p_{\scriptscriptstyle{A}}+ \hat{\mathbf{H}}_4 \,\mathbf{s}^p_{\scriptscriptstyle{B}}+ \hat{\mathbf{n}}_{\scriptscriptstyle{A}},
\dim_{\rm H}Y \geq \frac{h(f|_{W},Y)}{h(f|_{W})}\frac{(\chi - \epsilon)}{(\chi + \epsilon)}\dim_{\rm H}W = \frac{h(f^N|_{R},Y)}{h(f^N|_{R})}\frac{(\chi - \epsilon)}{(\chi + \epsilon)}\dim_{\rm H}W.
\langle v^1_{\tau,h}(t_n) , v^1_{\tau,h}(t_n)\rangle + \langle \nabla v^0_{\tau,h}(t_n), \nabla v^0_{\tau,h}(t_n)\rangle = \langle u_{1,h}, u_{1,h}\rangle + \langle \nabla u_{0,h}, \nabla u_{0,h}\rangle
{d+1 \choose |I|}\cdot \|A\| = \sum_{\sigma \in A}\frac{|\{F \in X(d)\,|\,\sigma \in F\}|}{|X(d)|}= \frac{|A| \cdot k_I^{[d]}}{|X \cap \prod_{i \in I}V_i|\cdot k_I^{[d]}}= \frac{|A|}{|X \cap \prod_{i \in I}V_i|}
{\cal H}_{S T}= \sqrt{\left( \Pi + \frac{\sqrt{1 + b^{2}}}{b}B \right)^{2}+ \left( \frac{\frac{\sqrt{1 + b^{2}}}{b}B}{R}\Pi - R \right)^{2}}\geq \Pi + \frac{\sqrt{1 + b^{2}}}{b}B .
\left\langle \left[ w\right]_{\operatorname*{D};j}\left( s\right) ,\gamma_{\mathbf{n};j}\left( s\right) \overline{\mathbf{m}}\right\rangle_{\mathbb{R}^{3}}=\left\langle \psi,\gamma_{\mathbf{n};j}\left( s\right) \overline{\mathbf{m}}\right\rangle_{\Gamma_{j}}\quad\forall\mathbf{m}\in\mathbf{H}\left(\mathbb{R}^{3},\operatorname*{div}\right) .
\Gamma \mapsto \big\{(s, \xi_s) \in \dot{T}^{*}\Sigma_0 \,|\, \exists (u, v; \xi_u, \xi_v) \in \Gamma, (s, u, v; \xi_s, \xi_u, \xi_v,) \in \mathrm{WF}(K) \big\}
H^{\frac{1}{8}}\lambda = e^{- i \omega t}r^{- \frac{5}{2}}\left\{F ( r ) \cdot \Psi_{- l}+ i G ( r ) \cdot ( \frac{\Gamma^{0}\Gamma^{i}x_{i}}{r}) \Psi_{- l}^{\pm}\right\}
h_{1}^{2}v^{2}\rightarrow h_{2}^{2}v^{2}+ ( 2 h_{2}h_{3}+ h_{3}^{2}\phi^{2}) ( v^{j}\phi^{j})^{2}\equiv g_{1}v^{2}+ g_{2}( v^{i}\phi^{i})^{2}~ ,
W_{F}=F\cdot|\psi^{-}\rangle\langle\psi^{-}|+\frac{1-F}{3}|\phi^{+}\rangle\langle\phi^{+}|+\frac{1-F}{3}|\psi^{+}\rangle\langle\psi^{+}|+\frac{1-F}{3}|\phi^{-}\rangle\langle\phi^{-}|
\mathrm{SINR}_{a,n_s,s_i}= \frac{|y_{a,n_s,s_i}- n_{a,n_s,s_i}|^2}{\sum_{n_s^{'}=1,n_s^{'}\neq n_s}^{|\mathbf{s}|}|y_{a^{'},n_s^{'},s_i}-n_{a^{'},n_s^{'},s_i}|^2+{n_{a,n_s,s_i}^2}},
(g,k_1) = (n^3-1,-n^7+3n^4-3n), (k_1,k_2) = (-n^7+3n^4-3n,-n^{18}+o(n^{18})), (k_2,k_3) = (-n^{18}+o(n^{18}), n^{47}+o(n^{47}), ...
\mathfrak{L}_p(\delta,\eta,\phi): =\frac{\partial}{\partial s}(D^{(s)}\psi_s)\Big|_{s=0}&=T(\psi)+D\big(\frac{\partial}{\partial s}\Big(\psi(t,z-s\eta+O(s^2))\Big)\Big|_{s=0}+O_{L^2_1}(1)+\phi\big).
-\frac{\delta\Gamma}{\delta V^{(2)}(\phi)}= C_{(n)}\sum_{l\,=\,0}^{\infty}g^{2(l+1)}\int d^n p \left[ \frac{1}{2p\cdot\partial}\,\partial^{\mu}V^{(2)}(\phi)\,\frac{\partial}{\partial p^{\mu}}\right]^l\,f_0(p)\,,
\sum_{i=1}^2 |w_{t,k_ti}|^2 p_{t,i}\le \frac{B_{t,k_t}+\alpha_t T_f(E_{k_t}-\tilde{p}_{t})}{(1-\alpha_t)T_f}+ E_{k_t}, & \sum_{i=1}^2 |w_{t,\bar{k}_ti}|^2 p_{t,i}\le \frac{B_{t,\bar{k}_t}+\alpha_t T_fE_{\bar{k}_t}}{(1-\alpha_t)T_f}+ E_{\bar{k}_t}.
\int_{0}^{1}\cdot\cdot\cdot\int_{0}^{1}(\sum_{k=1}^{2n}\frac{ln(t_{k})}{\prod_{i\ne k}(t_{k}-t_{i})})dt_{1}\cdot\cdot\cdot dt_{2n}=\frac{1}{2}(-1)^{n+1}(2\pi)^{2n}\beta_{2n}
\theta_k = \sum_{I \in \mathcal{I}_k}c^k_I\biggl(\sum_{e \in E}\Bigl(\prod_{i \in I}a_{ie_i}\Bigr) w_e \biggr), k = 1, \ldots, \kappa, & w \geq 0,\ \sum_{e \in E}w_e = 1,\ \lambda_{ij}= \sum_{e \in E:e_i = j}w_e, i =1, \ldots, d ,\; j =0, \ldots, n, f = A(\lambda),\ (\lambda_i, \delta_i) \in(\ref{eq:SOS2-log}), i = 1, \ldots, d.
P^{0,1}{\rm ad}\nu_i\Delta_0^{-1}\bar\partial^*\mu_1P^{1,0}\mu_2 \bar\partial\Delta_0^{-1}\star{\rm ad}\nu_j \star P^{0,1}\nu_i =i P^{0,1}{\rm ad}\nu_i\Delta_0^{-1}\bar\partial^*\mu_1P^{1,0}\mu_2 \bar\partial\Delta_0^{-1}{\rm ad}\nu_i \star\nu_j
E_{n_{x}, n_{y}}^{0}= m \omega^{2}\hbar^{2}\left( \frac{1}{m^{2}\omega^{2}\hbar^{2}}+ \frac{\theta^{2}}{4 \hbar^{4}}\right)^{\frac{1}{2}}\left( n_{x}+ n_{y}+ 1 \right) - \frac{m \omega^{2}\theta}{2}\left( n_{x}- n_{y}\right) .
\sum_{\ell =1}^{5}x_{\ell}^{\frac{D}{w_{\ell}}}+a_0\prod_{\ell =1}^{5}\left( x_\ell\right)+ \sum_{i =7}^{r+3}a_i\prod_{\ell =1}^{5}x_\ell^{n^\ell_i}=0,
|f_{\delta}(\tfrac{1}{\delta}, \delta )|^{-(\tfrac{1}{\delta^2}+ \delta^2)}\sim 1, |f_{\delta}(\tfrac{1}{\delta}, \delta )|^{-(\tfrac{1}{\delta^2}+ \delta^2)}= 0 \int_{C_1}|f_{\delta}|^2e^{-|\cdot|^2}\omega_o \le C/\sqrt{\delta},
S^{r e g}[{\mathcal A}] = \frac{\kappa}{2}\langle A_{\mu}| u ( \frac{\partial^{2}}{M^{2}}) R_{\mu \nu}| A_{\nu}\rangle + \frac{1}{4}\langle F_{\mu \nu}| v ( \frac{\partial^{2}}{M^{2}}) | F_{\mu \nu}\rangle ,
{\mathbf{\Omega}}= \left\{{{{\sin}^2}{\theta_1}+{{\sin}^2}{\theta_2}\leq{\delta^2},{{\sin}^2}{\theta_2}\leq{{\sin}^2}{\theta_1},0 \leq{{\sin}^2}{\theta_1},{{\sin}^2}{\theta_2}\leq 1}\right\}.
\int_{X \times X}\frac{|u(x)-u(y)|^p}{\rho(x,y) d(x,y)^{ps}}\, d \mu(x) \, d \mu(y) =ps \int_X \int_X \int_{d(x,y)}^{\infty}\frac{|u(x) - u(y)|^p}{\rho(x,y) t^{ps+1}}\,dt \, d \mu(x) \, d \mu(y) = ps \int_0^{\infty}\frac 1{t^{ps+1}}\iint_{\{d(x,y) \leq t\}}\frac{|u(x)-u(y)|^p}{\rho(x,y)}\,d\mu(x) \, d \mu(y).
\hat{\phi}(p) = 2 \pi \int_{-\infty}^\infty dt \int_0^\infty r^2\:dr \int_{-1}^1 d\cos \vartheta\;\phi(t,r)\: e^{i p^0 t - i |\vec{p}\,| r \cos \vartheta}&= \frac{2 \pi}{|\vec{p}\,|}\:\int_{-\infty}^\infty d\omega \int_0^\infty r\:dr\:\phi(t,r)\: e^{i p^0 t}\,\Big( e^{i |\vec{p}\,| r}- e^{-i |\vec{p}\,| r}\Big) \:,
G_{p,f}(u)-G_{p,f}(v_0) \ge d\left[\int_\Omega\langle\nabla v_0,\nabla \varphi\rangle\,dx+\left(\int_\Omega|v_0|^{2p/(p-1)}\,dx\right)^{-1/p}\int_\Omega|v_0|^{2/(p-1)}\,v_0\,\varphi\,dx-\langle f,\varphi\rangle\right] &+\frac{d^2}{2}\int_\Omega|\nabla\varphi|^2\,dx,
\sum_{d \geq 0}\left[ \frac{B_{d+1}(s/r)}{(d+1)!}\kappa_d -\sum_{i=1}^k \frac{B_{d+1}(\Theta^{\gamma_i})}{(d+1)!}\psi_i^d +\frac{1}{2}\sum_{\Gamma_{cut}}\frac{rB_{d+1}(\Theta^{\gamma^-})}{(d+1)!}\bar{\rho}_{\Gamma_{cut}*}\left( \sum_{\substack{i+j=d-1 i,j\geq0}}(-\psi_+)^i\psi_i^j\right)\right]
Q(a) = \bigcup_{b\in B}\Gamma(f)(a,b) \wedge P(b) & = \bigcup_{b\in B}\sigma(f(a),b)\wedge \rho(a,a) \wedge P(b) & = \bigcup_{b\in B}\sigma(f(a),b)\wedge \rho(a,a) \wedge P(f(a)) &= P(f(a)) \wedge \bigcup_{b\in B}\Gamma(f)(a,b) &= P(f(a)) \wedge \rho(a,a)
\renewcommand{\arraystretch}{1 . 2}\begin{array}{r c l}{{\chi_{0}^{\alpha}\omega}}&{{=}}&{{- \theta^{\alpha}\, ,}}\\{{\chi_{0}^{\alpha}\theta^{\beta}}}&{{=}}&{{d^{\alpha \beta}\Omega \, ,}}\\{{L_{0}\omega}}&{{=}}&{{\Omega \, .}}\\ \end{array}
< \dot{x}( \tau_{1}) \cdot \dot{x}( \tau_{2}) > = [ < \dot{x}( \tau_{1}) \cdot \dot{x}( \tau_{2}) >_{o}+ \dot{x}^{c l}( \tau_{1}) \cdot \dot{x}^{c l}( \tau_{2}) < 1 >_{o}] e^{- S^{c l}},
10^{\frac{W}{10}}= \int_0^{\frac{\lambda}{\eta_4\bar{\gamma}P}}\left(\lambda-{\eta_4}P x\right)f_{T}(x)\,\mathrm{d}x & = \lambda F_{T}\left(\frac{\lambda}{\eta_4\bar{\gamma}P}\right)-{\eta_4}P \int_0^{\frac{\lambda}{\eta_4\bar{\gamma}P}}xf_{T}(x)\,\mathrm{d}x,
M_{1,2}= \frac{P_3(1)}{L}\sum_{w \mid q}\varphi(w) \mu(q/w)\mathop{\sum \sum}_{\substack{b \leq y_2 b \equiv \pm n (w) (b,q)=1}}\frac{\Lambda(b) P_2[b]}{(bn)^{\frac{1}{2}}}V_1 \left( \frac{n}{q^{\frac{1}{2}}}\right).
g_{\nu_{l}}( \lambda a , \lambda r ) \equiv J_{\nu_{l}}( \lambda r ) \bar{Y}_{\nu_{l}}^{( a )}( \lambda a ) - \bar{J}_{\nu_{l}}^{( a )}( \lambda a ) Y_{\nu_{l}}( \lambda r ) , \quad \lambda = \sqrt{\omega^{2}- m^{2}},
\frac{1}{2 \pi i}\left( \int_{2-i t_j}^{2+i t_j}+ \int_{2+i t_j}^{\frac{1}{4}-\sigma+i t_j}+ \int_{\frac{1}{4}-\sigma+i t_j}^{\frac{1}{4}-\sigma-i t_j}+ \int_{\frac{1}{4}-\sigma-i t_j}^{2-i t_j}\right) \frac{L_f'}{L_f}(s+w) \Gamma(w) Y^w dw &= \frac{L_f'}{L_f}(s) + \sum_{-t_j < \gamma < t_j}\Gamma(\rho-s) Y^{\rho-s}.
\mathbb P_{\gamma^{-1}|I| \bar c_{\pm}(i,a)}(N_{i,j-1}^{\pm}=n)=e^{-\gamma^{-1}|I|\bar c_{\pm}(i,a)\Delta t}\frac{(\gamma^{-1}|I|\bar c_{\pm}(i,a)\Delta t)^n}{n!}.
\int\,\textrm{Re}\left(\chi_{F_j}\,T_{P}^{*}(1)\right)(\cdot) \geq \frac{1}{500}\,\int\chi_{F_j}(\cdot)\,\left|\,\int \psi_{P}(\cdot-y)\,\chi_{E(P)}(y)\,dy\,\right| &=\frac{1}{500}\,\int\,|\chi_{F_j}\,T_{P}^{*}(1)(\cdot)|\:.
\alpha\displaystyle Re\int_{R^n}(1+4\pi^2|\zeta|^2)^{\frac{\alpha-2}{2}}\hat{u}(\zeta)\bar{\hat{v}}(\zeta)d\zeta=[n(\displaystyle\frac{1}{q_1+1}+\frac{1}{q_2+1})-(n-\alpha)]\int_{R^n}u^{q_1+1}(x)dx.
\bigg|\frac{\partial}{\partial y}P_y(H^{s}u)(t,x)\bigg| = \bigg|\frac{\partial}{\partial y}P_y\bigg(\frac{1}{c_s}\int_0^\infty(P_\tau u(t,x)-u(t,x))\,\frac{d\tau}{\tau^{1+2s}}\bigg)\bigg| \leq C_s\bigg\{\int_0^y+\int_y^\infty\bigg\}\bigg|\frac{\partial}{\partial y}P_y(P_\tau u(t,x)- u(t,x))\bigg|\,\frac{d\tau}{\tau^{1+2s}}.
\frac{2(3+t+2u)(1+2t)u}{3(1+t)(\displaystyle\sum_{k=0}^{n-1}u^k)^2}& = \frac{2(2+t)(1+2t)u}{3(1+t)(\displaystyle\sum_{k=0}^{n-1}u^k)^2}+\frac{3(1+t)(1+u)u}{3(1+t)(\displaystyle\sum_{k=0}^{n-1}u^k)^2}+p_{1,3}
\vec{X}_{11}= \Bigg( 0, 1, - \frac{\lambda}{m-n}\bigg(\frac{1}{1+ B^{-1}}\bigg)\Bigg), \vec{X}_{12}= \Bigg( \frac{-\frac{1-m+n}{m-n}+1}{ec}\;,\;1\;,\; -\frac{\lambda}{m-n}\frac{\big(-\frac{1-m+n}{m-n}+1\big)\lambda + e}{e(1+C^{-1})}\Bigg).
D^{\rm grav}_{\mu\nu\lambda\rho}=(\eta_{\mu\lambda}\eta_{\nu\rho}+\eta_{\mu\rho}\eta_{\nu\lambda}-\eta_{\mu\nu}\eta_{\lambda\rho})\frac{-i}{(2\pi)^4}\int d^4k\frac{{\cal E}^2(k^2)}{k^2-i\epsilon}\exp[ik\cdot(x-y)],
\Gamma(\pi)\ge\frac{1}{N}\sum_{\ell=1}^N\left(\frac{\rho_\ell T_{\ell,\pi}^{av}}{2}+\frac{c_\ell}{T_{\ell,\pi}^{av}}\right)\implies\Gamma(\tilde{\pi}_{OF}^\star)\ge\frac{1}{N}\sum_{\ell=1}^N\left(\frac{\rho_\ell T_{\ell,\tilde{\pi}_{OF}^\star}^{av}}{2}+\frac{c_\ell}{T_{\ell,\tilde{\pi}_{OF}^\star}^{av}}\right),
\mathcal Q(\ell, \omega) = \frac1\ell \mathcal Q_0 - \frac{4 |\Gamma| \cdot |\partial_2 \psi_{\varkappa, 2}(0)|^2}{\sqrt{\Lambda -\omega}\cdot \sqrt{2 \zeta_1 ''(\varkappa)}}\; T_\ell^* \bigr( P_+ + P_- \bigr) T_\ell + R(\ell,\omega) .
f =\sum_{(i_1, i_2, i_3, i_4)\in{\{1,2,\ldots,k\}}^4}a_{i_1, i_2, i_3, i_4}x_{i_1}y_{i_2}z_{i_3}w_{i_4}, a_{i_1, i_2, i_3, i_4}\in K.
\eta_{r s}A_{-}^{r}B^{s}= 1 + 2 i h f^{- 1 / 2}\theta_{( 1 6 )}^{T}\dot{\theta}_{( 1 6 )}+ i f^{- 1 / 2}v_{i}\partial_{j}h \theta_{( 1 6 )}^{T}\gamma^{i j}\theta_{( 1 6 )}+ \cdots ,
v_l(x,i,s,\nu)=\begin{cases}f(x,z_l,\nu_{i,s}) \hbox{if}z_l\geq i e^{-\lambda(i-z_l)}f(x,z_l,\nu_{i,s}) \;\;\; +\int_0^{i-z_l}\lambda e^{-\lambda t}(f(x,i-t,\nu_{i,s})+v(i-t,i-t,s,\nu_{i,s}+\delta_{i-t})dt \hbox{if}z_l< i, \end{cases}
\begin{array}{c}{\rm Coeff}((\omega\phi)(t^{n}),t^{kn})={\rm Coeff}((\omega\phi)(t),t^{k}) nt^{n-1}{\rm Coeff}((\omega\phi)(t^n),t^{kn})={\rm Coeff}(\omega(\phi(t^n)),t^{n(k+1)-1}).\end{array}
{\cal L}= - \partial_{\mu}h^{\mu \nu}\partial_{\alpha}h_{\nu}^{\alpha}+ \frac{1}{2}\partial_{\alpha}h^{\mu \nu}\partial^{\alpha}h_{\mu \nu}+ \partial_{\mu}h^{\mu \nu}\partial_{\nu}h_{\alpha}^{\alpha}- \frac{1}{2}\partial_{\alpha}h_{\mu}^{\mu}\partial^{\alpha}h_{\nu}^{\nu}- \frac{M^{2}}{2}\left( h_{\mu \nu}h^{\mu \nu}- h_{\mu}^{\mu}h_{\nu}^{\nu}\right) .
{W^{d}_\nu}(Z,Z') = \frac{1}{2(2\pi)^{\frac{d}{2}}}\Gamma\left(\frac{d-1}{2}+i\nu\right) \Gamma\left(\frac{d-1}{2}-i \nu\right) \,\,((Z\cdot Z')^2-1)^{-\frac{d-2}{4}}\,P^{-\frac{d-2}{2}}_{-\frac{1}{2}+ i\nu}\left({Z\cdot Z'}\right),
\int_\tau^t{{1}}_{\{\zeta_0(s) >{\tilde{\zeta}}_0(s)\}}(\zeta_0(s) -{\tilde{\zeta}}_0(s)) (\eta_0'(s) -{\tilde{\eta}}_0'(s)) \, ds \le \int_\tau^t{{1}}_{\{\zeta_0(s) >{\tilde{\zeta}}_0(s)\}}(\zeta_0(s) -{\tilde{\zeta}}_0(s)) \eta_0'(s) \, ds & \le \int_\tau^t{{1}}_{\{\zeta_0(s) > 0\}}\zeta_0(s) \, \eta_0(ds) & = 0,
y(1-y)\partial_{y}^2 R(y) + \frac{1}{2}(2-5y)\partial_{y}R(y) + \left(\frac{\omega^2 r_{c}^2}{4y}- \frac{\nu(\nu+1)}{4(1-y)}- \frac{m^2 r_{c}^2}{4}\right)R(y) = 0.
(-\Delta)^{1/2}f(x) = C \int_{\mathbb R^n}\frac{f(x+y) - 2f(x) + f(x-y)}{|y|^{n+1}}d y &= C \int_{|y|>2t}\frac{f(x+y) - 2f(x) + f(x-y)}{|y|^{n+1}}d y &+ C \int_{|y|<2t}\frac{f(x+y) - 2f(x) + f(x-y)}{|y|^{n+1}}d y.
\big(\tau_{k + 1, \ell}^{(\alpha,\beta)}\big)^{2}=\tau_{k + 1, \ell}^{(\alpha + 1,\beta)}\tau_{k + 1, \ell}^{(\alpha - 1,\beta)}+\tau_{k + 2, \ell + 1}^{(\alpha - 1,\beta)}\tau_{k, \ell - 1}^{(\alpha + 1,\beta)}-\tau_{k + 2, \ell}^{(\alpha - 1,\beta)}\tau_{k, \ell}^{(\alpha + 1,\beta)}.
\int_{B_{1}}\phi\psi^{1+q}w^{2}dx + \frac{c_{n,\beta}}{4C\Lambda}g_{\alpha}*\int_{\rho'B_{1}}\int_{\rho'B_{1}}\frac{(w(s,x)-w(s,y))^{2}}{|x-y|^{n+2\beta}}\phi dxdy\leq g_{\alpha}*F.
[b(x, y) + \pi(x, y)\xi_n(y)] - [b(x, z) + \pi(x, z)\xi_n(z)] = [b(x, y) - b(x, z)] + \pi(x, y)\xi_n(y) - \pi(x, z)\xi_n(z) = -\pi(x, y)b(y, z) + \pi(x, y)\xi_n(y) - \pi(x, z)\xi_n(z) = \pi(x, y)[-b(y, z) + \xi_n(y) - \pi(y, z)\xi_n(z)] = \pi(z, y)[0] = 0.
\hat \gamma^n(\{x\}\times \{y\}\times [0,t]) = \int_0^t \exp(-s) \eta^n(x, y \mid s) ds \hat \theta^n(\{x\}\times \{y\}\times [0,t]) = \int_0^t \exp(-s) \eta^n_{(1)}(x \mid s)G(M^n(s))(x, y) ds.
h(s) = \sum_{m=1}^{\infty}\frac{1}{m}\phi(ms) & = \sum_{m \leq s^{-1/2}}\frac{\phi(ms)}{m}+ \sum_{s^{-1/2}< m \leq s^{-1}}\frac{\phi(ms)}{m}+ \sum_{m > s^{-1}}\frac{\phi(ms)}{m}& =: \psi_1(s)+\psi_2(s)+\psi_3(s).
K_n(x, y) = \frac{\mu}{2\pi i}\left( \frac{y}{x}\right)^{\nu}\oint_{\Sigma'}\frac{ds}{s - x}\int^{\infty}_y \frac{dt}{t - y}k_n(s, t) \left( \frac{s}{t}\right)^{\nu + \mu}\left( \frac{t - y}{s - x}\right)^{\mu},
E_{1,b}(Y, \Theta_1,\Theta_2) \ll \widehat{P}^n \widehat{Y}^{5/2-n/2+\varepsilon}\widehat{\Theta}_1\widehat{\Theta}_2\widehat{Z}^{1-n/2-\mu}\left(\widehat{V}^n+\widehat{Y}^{n/3}\right) = \widehat{Y}^{5/2+n/2+\varepsilon}\widehat{\Theta}_1\widehat{\Theta}_2\widehat{Z}^{1+n/2-\mu}+\widehat{P}^{n-5}\widehat{Y}^{5/2-n/6+\varepsilon}\widehat{Z}^{3-n/2-\mu}.
{\rm d}s^2=U_{IJ}(x)\,{\rm d}\vec{x}^I\cdot{\rm d}\vec{x}^J+(U^{-1}(x))^{IJ}\Big({\rm d}\varphi_I+\vec{W}_{IK}(x)\cdot{\rm d}\vec{x}^K\Big)\Big ({\rm d}\varphi_J+\vec{W}_{JL}(x)\cdot{\rm d}\vec{x}^L\Big )\ .
z_i = \frac{q^{d-i}(q^i-1)(q^{2d-i+1}-1)}{(q^{d-i}+1)(q^{d-i+1}+1)}(1 \leq i \leq d-1), z_d = \frac{(q^d-1)(q^{d+1}-1)}{q+1}.
z_{1,\delta}= O\left(\frac{\delta^\frac{4-n}{2}}{(1+|\frac{x'}{\delta}|)^{n-3}}\right) \partial B_1(0)\cap B_1(e_n),z_{1,\delta}= O\left(\delta^\frac{n-2}{2}\right) \partial B_1(0) \cap B_1^c(e_n).
\begin{aligned}-\iint h_s \partial_x u^\epsilon h^\epsilon \leq \Vert h_{s}\Vert_{L^\infty}\Vert u^\epsilon_x \Vert_{L^2}\Vert h^\epsilon\Vert_{L^2}&\leq L \Vert h_{s}\Vert_{L^\infty}\Vert u^\epsilon_x \Vert_{L^2}\Vert h^\epsilon_x\Vert_{L^2}= L \Vert h_{s}\Vert_{L^\infty}\Vert v^\epsilon_y \Vert_{L^2}\Vert g^\epsilon_y\Vert_{L^2},\end{aligned}
\int_{a_{k}}^{b_{k}}\psi_{n,k}(\tau)\,\tau^{s}\,d\mu_{k,j}(\tau)=0,\left\lceil\frac{\ell-j}{p+1}\right\rceil\leq s\leq \left\lfloor\frac{n+p\ell-1-j(p+1)}{p(p+1)}\right\rfloor,k\leq j\leq p-1.
\Delta_j f(x^j_k) = \frac{1}{\mu(Q^j_k)}\sum_{\{i:d(x^j_i,x^j_k)<C\delta^j\}}\left(f(x^j_i) - f(x^j_k)\right) \left(\mu(Q^j_k)+\mu(Q^j_i)\right) H^j_{ki}&= \sum_{\{i:d(x^j_i,x^j_k)<C\delta^j\}}\left(f(x^j_i) - f(x^j_k)\right) \left(1+\frac{\mu(Q^j_i)}{\mu(Q^j_k)}\right) H^j_{ki}.
\mathbb{E}_{\underline{\varepsilon}|\underline{X}}\exp\left\{\frac{1}{\gamma}\sum_{i=1}^n\varepsilon_ig(X_i)\right\}= \prod_{i=1}^n\mathbb{E}_{\varepsilon_i|X_i}\exp\left\{\frac{1}{\gamma}\varepsilon_ig(X_i)\right\}.
\binom{a+j}{j}\sum_{l=0}^{a}\binom{n-j+l}{l}\binom{n-j}{a+j-l}\binom{n-a-2j+l}{a-l}\sum_{k=a+j-l}^{n-(a+j-l)}\binom{n-2a-2j+2l}{k-j-a+l}G(n,k,a)
a_d (\ell, n, \lambda) = \binom{d+n-1}{n-1}- \ell \left [ \binom{k + n-1}{n-1}+ \binom{d - k + n-1}{n-1}\right ] & + \sum_{i=2}^\ell (-1)^i \binom{\ell}{i}\binom{d - i k + n-1}{n-1}+ \binom{\ell}{2}\binom{2 k - d +n -1}{n-1}+ \ell^2.
\Delta^{I}n_{3}=\Delta^{I}\Big(-\frac{\sqrt{\omega}}{\sqrt{\Phi}}\Big)=\lambda_{31}\Big(-\frac{au}{\sqrt{\Phi}}\Big)+\lambda_{32}\Big(-\frac{bv}{\sqrt{\Phi}}\Big)+\lambda_{33}\Big(\frac{\sqrt{\omega}}{\sqrt{\Phi}}\Big).
\begin{aligned}\partial_t \rho_1 = \partial_x \left( \partial_x \rho_1+ \rho_1 \partial_x \phi \right) , \partial_t \rho_2 = \partial_x \left( \partial_x \rho_2-2 \rho_2 \partial_x \phi \right) , -\partial_x^2 \phi = 12(x-0.5)^2 +\rho_1-2\rho_2 .\end{aligned}
\sum_{k=0}^\infty \frac{(1+aq^{4k+1})\,(a^2q^2;q^4)_k (bq^2;q^4)_k (cq^2;q^4)_k}{(1+aq)\,(a^2q^4/b;q^4)_k (a^2q^4/c;q^4)_k (q^4;q^4)_k}\biggl(\frac{aq}{bc}\biggr)^k =\frac{(a^2q^6;q^4)_\infty (aq^3/b;q^4)_\infty (aq^3/c;q^4)_\infty (a^2q^2/bc;q^4)_\infty}{(a^2q^4/b;q^4)_\infty (a^2q^4/c;q^4)_\infty (aq^5;q^4)_\infty (aq/bc;q^4)_\infty}.
E_{C}= - \frac{\hbar c}{2 \pi^{2}}\sum_{n = - \infty}^{\infty}\int_{0}^{\infty}d k_{z}\int_{k_{z}}^{\infty}\sqrt{y^{2}- k_{z}^{2}}\, d_{y}\, \ln \left[ 1 -{{\xi}^{2}}{{( y{{\partial}_{y}}({I_{n}}( a y ){{{K}}_{n}}( a y ) ) )}^{2}}\right] \, ,
S=\int[\frac{1}{4}F^{2}+\frac{1}{2}(D\theta)^{2}]=\int[\frac{1}{4}F^{2}+\frac{1}{2}(\partial\theta-HeA)^{2}]=\int[\frac{1}{4}F^{2}+\frac{1}{2}(\partial\theta-mA)^{2}]
\frac{d}{dx}D_n(x|a_1,\dots,a_r) =n!\sum_{l=0}^{n-1}\frac{(-1)^{n-l-1}}{l!(n-l)}D_l(x|a_1,\dots,a_r)\,, \frac{d}{dx}\widehat D_n(x|a_1,\dots,a_r) =n!\sum_{l=0}^{n-1}\frac{(-1)^{n-l-1}}{l!(n-l)}\widehat D_l(x|a_1,\dots,a_r)\,.
\hat{\Phi}^*_{v,w}(e^{v,w}_x) =m_{v\to w}^{-1}\star \left(\frac{\partial \bar{b}_{v,w}}{\partial \eta_x}\right) &=m_{w\to v}\star \left(m_{w\to v}^{-1}\star\left(m_{v\to w}^{-1}\star \left(\frac{\partial \bar{b}_{v,w}}{\partial \eta_x}\right)\right)\right) \Phi_{v,w}(e^{v,w}_x) =m_{v\to w}\star \left(\frac{\partial \log b_{v,w}}{\partial \theta_x}\right)
\mathcal{B}_\Sigma \;=\int d^Dx\,\left( \frac{\delta \Sigma}{\delta \varphi^i}\frac \delta{\delta \varphi^{i*}}\;+\;\frac{\delta \Sigma}{\delta\varphi^{i*}}\frac \delta{\delta \varphi^i}\right) \;\;,
\nabla_{X\wedge Y}Z =\widetilde{\nabla}_{X\wedge Y}Z+F^{k+l,m}(X\wedge Y,Z) &=X\wedge \widetilde{\nabla}_Y Z+(-1)^{kl}Y\wedge\widetilde{\nabla}_X Z+F^{k+l,m}(X\wedge Y,Z),
\alpha = \beta ( \mu_{U}- \mu_{D}) / 2 , \; \; \cosh \theta_{C}= \frac{1}{\sqrt{2}}\left[ \frac{S_{U D}( p )}{\sqrt{S_{U D}^{2}( p ) - R_{U D}^{2}( p )}}+ 1 \right]^{1 / 2}, \; \; \sinh \theta_{C}= \frac{1}{\sqrt{2}}\left[ \frac{S_{U D}}{\sqrt{S_{U D}^{2}( p ) - R_{U D}^{2}( p )}}- 1 \right]^{1 / 2}
\mathrm{( f )}+ \mathrm{( g )}= - \frac{d_{\Gamma}g^{2}}{2 x_{i j}^{4}}\sum_{k}\frac{x_{j k}^{\mu}x_{i k}^{\nu}+ x_{j k}^{\nu}x_{i k}^{\mu}- \delta^{\mu \nu}x_{j k}\cdot x_{i k}}{x_{j k}^{2}\, x_{i k}^{2}},
E_{n,\kappa}\;=\;mc^2\Big(1+\frac{(Z\alpha_\mathrm{f})^2}{\big(n+\sqrt{\kappa^2-(Z\alpha_\mathrm{f})^2}\,\big)^{\!2}}\Big)^{\!-\frac{1}{2}},n\in\mathbb{N}_0,\,\kappa\in\mathbb{Z}\!\setminus\!\{0\}\,.
\lambda((P\oplus_m P_t)[S]) =\lambda_{\widehat{\mathcal{E}}_{t}[S]}(x_{s_1},\dots, x_{s_k}) &=\lambda_{\widehat{\mathcal{E}}_t}(x'_1,\dots, x'_{m+m_t-1}) &=\lambda_{\mathcal{E}}(x'_1, \dots, x'_{m-1},y)+\lambda_{\mathcal{E}_{t}}\left(x'_{m},\dots, x'_{m+m_t-1}\right).
x_{2k}=\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}|t|^{-2\beta}e^{-ik t}dt=\frac{c_{k,-2\beta}}{2\pi}, x_{2k+1}=\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}|t|^{-2\beta}e^{ik t}dt=x_{2k},
\frac{d}{dz}[z^{-b_{q}}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|z)]=z^{-1-b_{q}}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{1},...,b_{q-1},b_{q}+1\end{matrix}|z),m<q,
\frac{d^m}{dz^m}\ln\left(\frac{1+z}{1-z}\right) = (m-1)! \left[\frac{(-1)^{m-1}}{(1+z)^m}+ \frac{1}{(1-z)^m}\right] =\frac{(m-1)!}{(1-z^2)^m}\sum_{s=0}^{m}\left[{{m}\choose{s}}\left((-1)^{m-1+s}+ 1\right)z^s\right],
r_2<\lim_{x\to b^-}[a_k(x)f^{(k)}(x)]^{(k-2)}=\lim_{x\to b^-}\left(\sum_{i=0}^{k-2}\binom{k-2}{i}a_k^{(k-2-i)}(x)f^{(k+i)}(x)\right).
\frac{1}{\lambda_{r}}- \frac{1}{( 2 \pi )^{2}}\left[ 3 \left( C_{u v}- \frac{2}{3}\right) - \left( C_{u v}- \ln \frac{m^{2}}{\sigma_{0}^{2}}\right) \frac{m^{2}}{\sigma_{0}^{2}}- \left( C_{u v}- \frac{1 3}{6}- \ln \frac{m^{2}}{\sigma_{0}^{2}}\right) \frac{1}{r^{2}\sigma_{0}^{2}}\right] = 0 \, .
d \psi(a(t),\hat{u}(t))[\partial_t(a(t),\hat{u}(t))] = \partial_t (\psi(a(t), \hat{u}(t))) = \mathcal{L}^c_{\psi(a(t), \hat{u}(t))}\dot{h}(t)h^{-1}(t).
\varphi ( x ) = \epsilon^{1 / 2}\Big \lbrack \left( \frac{C_{1}}{2 C_{2}}\right)^{1 / 2}\tilde{\varphi}_{0}+ \epsilon ( - \frac{D_{0}}{D_{1}}\tilde{\varphi}_{0}+ \varphi_{1 p}) \Big \rbrack \; ,
\limsup_{n\to\infty}\|T_xf^n(v)\|<1 \forall v\in E^s(x)\textrm{and}\|v\|=1\intertext{and}\liminf_{n\to\infty}\|T_xf^n(v)\|>1 \forall v\in E^{cu}(x)\textrm{and}\|v\|=1
\left[q^d\right]\sum_{j=0}^{N}(-1)^j\binom{N}{j}\left[t^{\ell+j}\right]\frac{1}{(1-yt)(1-t)^N-qt^N}+(-1)^{N+1}\left[q^{d-1}\right]\left[t^\ell\right]\frac{1}{(1-yt)(1-t)^N-qt^N}&=(-1)^N\left[q^d\right]\left[t^{\ell+N}\right]\frac{(1-t)^N}{(1-yt)(1-t)^N-qt^N}+(-1)^{N+1}\left[q^{d-1}\right]\left[t^\ell\right]\frac{1}{(1-yt)(1-t)^N-qt^N}.
\liminf_{t\to\infty}\frac 1t \int_0^t I(\bar t)d\bar t\geq\liminf_{t\to\infty}\frac 1t \int_0^t \frac{c_{\min}}{a_{\max}N+c_{\max}}(N-S(\bar t))d\bar t \geq \frac{c_{\min}}{(a_{\max}N+c_{\max})a_{\max}}\lambda >0.
\int_{B_{\frac{R+r}{2}}}\int_{\mathbb{R}^N\setminus B_R}\frac{|(\mathcal{V}_p)_h|^{2}+|\mathcal{V}_p|^2}{|x-y|^{N+s\,p}}\,dx\,dy\le \left(\frac{R+r}{R-r}\right)^N\,\left(\frac{R+h_0}{R-r}\right)^{s\,p}\,\frac{C}{s\,(R-s)^{s\,p}}\langle u\rangle^p_{\mathcal{X}^{p}_s(B_{\frac{R+r}{2}+h_0},B_{R+h_0})},
[X,\mathrm{Ad}(g^{-1})\cdot E_{\theta}] =\mathrm{Ad}(g^{-1})\cdot[\mathrm{Ad}(g)\cdot X,E_{\theta}] = \mathrm{Ad}(g^{-1})\cdot[w^{-1}\cdot X, E_{\theta}] = \theta(w^{-1}\cdot X)\, \mathrm{Ad}(g^{-1})\cdot E_{\theta}= (w\cdot\theta)(X)\, \mathrm{Ad}(g^{-1})\cdot E_{\theta}\,.
\begin{aligned}q_0 = \min\left\{p_0^{\prime\prime}, \max\left\{c_0(\nu q_0 + q_1), p_0^{\prime}\right\}\right\}q_1 = \min\left\{p_1^{\prime\prime}, \max\left\{c_1(q_0 + \nu q_1), p_1^{\prime}\right\}\right\}\end{aligned}
\int_0^u q(v) \: dv = \int_{\tau_{k^*+1}/a_N}^{\tau_{j^*(u)}/a_N}q(v) \: dv + \int_{\tau_{j^*(u)}/a_N}^u q(v) \: dv &\leq \frac{(1 + e/k_N)(j^*(u) - (k^*+1))}{k_N}+ \frac{e}{k_N}
K_{q}( \theta ) = \frac{( 1 ) ( 2 - B / 2 ) ( 1 + B / 2 )}{( 1 - E ( \sigma_{0}, \sigma_{1}, \beta ) ) ( 1 + E ( \sigma_{0}, \sigma_{1}, \beta ) ) ( 1 - F ( \sigma_{0}, \sigma_{1}, \beta ) ) ( 1 + F ( \sigma_{0}, \sigma_{1}, \beta ) )}
A_n(q) =\frac{1}{(q-1)q^{2n+1}}\left((q-2)q^{2n+1}+q^{3n}(q^n-1)+q^{2n+1}(q^n-1)+q^{2n+2}+(q-1) q^{3n+1}\right) &=2+q^n+\sum_{i=0}^{n-1}q^{i+n-1}+\sum_{i=0}^{n-1}q^i.
\partial_{t}A_{i}(x,t) + \frac{1}{2(t-T)}A_{i}(x,t) + \sum_{k=1}^{n}\frac{(x-X)^{k}}{2(t-T)}\partial_{k}A_{i}(x,t) =0 \intertext{and}\partial_{t}u(x,t) + \sum_{k=1}^{n}\frac{(x-X)^{k}}{2(t-T)}\partial_{k}u(x,t) =0
2g(w) M_n \big( g(w)\big) + g(w) f(w)^2 M_n' \big( g(w)\big) = \sum_{m=-\infty}^\infty I_{n+1,n}(w + 2\pi im), 2g(w) f(w) M_n \big( g(w)\big) + \Big\{6g(w)^2 f(w) + g(w) f(w)^3\Big\}\, M_n' \big( g(w)\big) + g(w)^2 f(w)^3 M_n'' \big( g(w)\big) = \sum_{m=-\infty}^\infty I_{n+2,n}(w + 2\pi im)\, .
{\tilde{c}}_{\alpha \beta}^{\enspace \enspace \gamma}= A_{\alpha}^{\delta}A_{\beta}^{\sigma}( A^{- 1})_{\mu}^{\gamma}c_{\delta \sigma}^{\enspace \enspace \mu}+ ( A^{- 1})_{\delta}^{\gamma}( A_{\alpha}^{\sigma}E_{\sigma}A_{\beta}^{\delta}+ A_{\beta}^{\sigma}E_{\sigma}A_{\alpha}^{\delta})
\begin{cases}a_{11}a_{22}-a_{12}a_{21}-1=0\,, a_{13}a_{24}-a_{14}a_{23}-1=0\,, a_{14}-a_{12}+a_{11}=0\,, a_{24}-a_{22}+a_{21}=0\,, a_{21}''a_{22}-a_{22}''a_{21}+a_{11}''a_{12}-a_{12}''a_{11}=0\,,\end{cases}
g_p(x^{(p)}, \xi^{(p+1)})=\big\{(x_p-\phi_p(x^{(p+1)}, \xi^{(p+1)})^2+\psi_p(x^{(p+1)}, \xi^{(p+1)})\big\}c_p(x^{(p)}, \xi^{(p+1)})