harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{k=0}^{m}\frac{(q;q^2)_k^2 (q;q^2)_{2k}}{(q^2;q^2)_{2k}(q^6;q^6)_k^2}[8k+1]q^{2k^2}\equiv0\equiv q^{-(n-1)/2}[n]\biggl(\frac{-3}{n}\biggr) \pmod{[n]}
\frac{1}{\alpha}\frac{|t|}{\eta^{1+\frac{2}{\alpha}}}\left|\frac{d p_N}{d u}\right|_{u= \frac{t}{\sqrt[\alpha]{\eta}}}\leq \frac{1}{\pi \alpha^2}\frac{|t|}{b^{1+\frac{2}{\alpha}}}\Gamma\left(\frac{2}{\alpha}\right),
\Omega(\vartheta(f_{\underline s})) =\int_{N'}\Omega_M(\vartheta(f_{\underline s})(w_M^{-1}n'))\chi(n')^{-1}\; dn' &=\int_{N'}\Omega_M (f_{\underline s}( \theta_N( w_M^{-1}n')))\chi(n')^{-1}\; dn'
\tilde{\theta}_i \left( \frac{\tilde{\theta}_i}{n\phi}- \frac{h_{ii}}{n}(\tilde{\theta}_i - \hat{\theta}_i) \right)- \tilde{\theta}_i \sum_{j \neq i}^{}\frac{h_{ij}}{n}(\tilde{\theta}_j - \hat{\theta}_j) - \frac{2}{n}=0
\mathfrak{h}_{p}=\left\{~u=\begin{pmatrix}0 \beta -\overline{\beta}\gamma\end{pmatrix}~\Bigr|~x\beta\overline{y}-y\overline{\beta}\overline{x}+w\beta\overline{z}-z\overline{\beta}\overline{w}+y{\gamma}\overline{y}+z{\gamma}\overline{z}=0~\right\}.
\mathrm{B}_{p}\equiv \exp \{\frac{1}{p}t_{p}( \overline{{\rho}}_{\mathrm{t o t , -}}( p ) \overline{{\rho}}_{\mathrm{t o t , +}}( - p ) - \overline{{\rho}}_{\mathrm{t o t , +}}( p ) \overline{{\rho}}_{\mathrm{t o t , -}}( - p ) ) \}.
\begin{aligned}\left({{\rm{P2}}}\right)\ \ \mathop{\max}\limits_{\left\{{{{\bf{b}}_l}}\right\}_{l = 1}^L}\ \{\Big|{\sum\nolimits_{l = 1}^L{{\bf{h}}_l^H{{\bf{Q}}_l}{{\bf{b}}_l}}}\Big|^2}{\rm{s.t.}}\ \ \sum\nolimits_{l = 1}^L{{{\left\|{{{\bf{Q}}_l}{{\bf{b}}_l}}\right\|}^2}}\le P. \end{aligned}
L_i \longmapsto \left[\begin{array}{cc}I 0 0 K_i\end{array}\right] \left[\begin{array}{cc}0 K_i K_i^{-1}0\end{array}\right]\left[\begin{array}{cc}I 0 0 K_i^{-1}\end{array}\right]= \left[\begin{array}{cc}0 I I 0\end{array}\right],
\frac{f_{z_0}(b_j)}{f_{z_0}(a_j)}=e^{G_{\Omega}(p(b_j),z_0)+i\widetilde{G_{\Omega}}(p(b_j),z_0)-G_{\Omega}(p(a_j),z_0)-i\widetilde{G_{\Omega}}(p(a_j),z_0)}&=e^{G_{\Omega}(p(b_j),z_0)-G_{\Omega}(p(a_j),z_0)+i\int_{p_*\gamma_j}\widetilde{d}G_{\Omega}(z,z_0)}.
\varphi(ab) =\varphi(\frac{a\circ b}{2}+\frac{[a,b]}{2})=\varphi(\frac{a\circ b}{2})+\varphi(\frac{[a,b]}{2})=(\frac{a\circ b}{4})^{+}+(\frac{a\circ b}{4})^{-}+(\frac{[a,b]}{4})^{+}+(\frac{[a,b]}{4})^{-}=\frac{1}{4}(a^{+}a^{+}+a^{+}a^{-}+a^{-}a^{+}+a^{-}a^{-})=(\frac{a^{+}+a^{-}}{2})(\frac{a^{+}+a^{-}}{2})=\varphi(a)\varphi(b),
\begin{aligned}\partial_t \Omega^i(\Gamma) =3\rho^{2/3}\partial_{\rho}(\Omega^i_{,1}(\Gamma)v), \;\;\;i=1,5,6,7 \, , \partial_t \Omega^i(\Gamma) =\rho^{-1/3}(\Omega^i_{,2}(\Gamma)+\Omega^i_{,3}(\Gamma))v \, ,\;\;\;i=2,3,4 \, ,\end{aligned}
\left( \begin{array}{c}{\phi_{A}}\\{\varphi_{A}}\\ \end{array}\right) = \left( \begin{array}{c c}{\frac{3}{4}}&{- \frac{\sqrt{7}}{4}}\\{- \frac{\sqrt{7}}{4}}&{- \frac{3}{4}}\\ \end{array}\right) \left( \begin{array}{c}{\phi_{B}}\\{\varphi_{B}}\\ \end{array}\right) .
\begin{cases}- k_{\varepsilon}\Delta \tilde{\Psi}_{\varepsilon}+ |\nabla \tilde{\Psi}_{\varepsilon}|^2 - \tilde{\lambda}_0 \underline{m}= 0 \Omega \tilde{\Psi}_{\varepsilon}= -k_{\varepsilon}\log(w)(\frac{\mathbf{x}-\mathbf{x}_{\varepsilon}}{k_{\varepsilon}}) \partial\Omega\end{cases}
\begin{multlined}[t][12.5cm]\frac{d_{\nu_{G}}2^{\frac{d_{G}}{2}}}{(\sqrt{2}\pi)^{d_{T}-1}}\left(\frac{k\lambda_{\nu_{T}}}{\pi}\right)^{d_{M}-\frac{d_{G}}{2}-\frac{(d_{T}-1)}{2}}\cdot\chi_{\nu_{G}}(g_{j}^{-1}) e^{-ik\vartheta_{j}\nu_{T}}\cdot \cdot\frac{e^{i\sqrt{k}(\theta_{2}-\theta_{1})\lambda_{\nu}}\cdot e^{H}}{\|\Phi_{T}(m)\|\mathcal{D}(m)}.\end{multlined}
= \sum_{p=1}^{r_i}\left(E_{\Lambda^n}\left[\frac{h(\textbf{A}_{p,i,n}\textbf{X}^n|\Omega^n,\Lambda^n=\lambda^n,W_{i+1},\ldots,W_K)}{r_ir_{i+1}}\right.\right. &\ \ \left.\left.-\frac{h(\textbf{C}_{i,n}\textbf{X}^n|\Omega^n,\Lambda^n=\lambda_n,W_{i+1},\ldots,W_K)}{r_ir_{i+1}}\right]\right) =0
\upsilon_{k+1}-\upsilon_k\leq2\Big[\Big(\frac{2}{n}\frac{1}{k}\sum_{i=1}^k\upsilon_i\Big)^2-\Big(1+\frac{4}{n}\Big)\frac{1}{k}\sum_{j=1}^k\Big(\upsilon_j-\frac{1}{k}\sum_{i=1}^k\upsilon_i\Big)^2\Big]^\frac{1}{2}.
\eta^2\left(v_i+\omega+\eta\sum_{i=1}^{n-1}l_i\right)\left(v_i-\omega+\eta\sum_{i=1}^{m-1}k_i\right) = \prod_{j\neq i}^{N-r}\frac{v_i-v_j-\eta}{v_i-v_j+\eta}, r<N.
\begin{array}{l}S_{+}\rightarrow d^2 S_{+}+2cd S_0 -c^2 S_{-}\,, S_{0}\rightarrow bdS_{+}+(1+2bc)S_0 -acS_{-}\,, S_{-}\rightarrow -b^2 S_{+}-2abS_0 +a^2 S_{-}\,.\end{array}
\P(X \geq \delta_i) = \P(Y \geq \delta_i') = \P(Y \geq 2^{-j}) = \sum_{i \in [j]}\P\left\{Y = 2^{-i}\right\}= \sum_{i \in [j]}r 2^i = 2r\left( 2^{j}- 1 \right) \leq 2r2^j.
\int_{-1}^{1}\frac{x(1-x^2)^{k-i}-t(1-t^2)^{k-i}}{x-t}dt\cr =x\int_{-1}^{1}\frac{(1-x^2)^{k-i}-(1-t^2)^{k-i}}{x-t}dt +\int_{-1}^{1}(1-t^2)^{k-i}.
u_t \,+\: \mbox{div}\;\!\mbox{\boldmath $f$}(x,t,u)\,+\: \mbox{div}\,\mbox{\boldmath $g$}(t,u)\:=\;\mu(t) \;\!\;\! \mbox{div}\,(\;\! |\;\!\nabla u\,|^{\:\!p-2}\,\nabla u \;\!),
\mathrm{trace}(-i\hbar\partial_{x^{k}}\lor\frac{1}{2\epsilon}(x^{\bot})^{k}\mathrm{OP}_{\hbar}^{T}((2\pi\hbar)^{2}\nu))=\underset{\mathbb{R}^{2}}{\int}\frac{1}{\epsilon}p\cdot q^{\bot}\nu(dqdp).
\mathcal{T}\left(\bar p + \kappa^{2}\left(p-\bar p\right)\right) \approx \mathcal{T}\left(\bar p\right) + \kappa^{2}\mathcal{T}'\left(\bar p\right) \left(p-\bar p\right) = \bar \tau + \kappa^{2}\mathcal{T}'\left(\bar p\right) \left(p-\bar p\right)\,,
\partial_{t}\mu_{S}= \cal A_{D,N}^{-1}{\rm RHS}= 2 ({\rm I}_{N_{B}}- \cal R )^{-1}\tilde{\cal A}_{-1}{\rm RHS}= 2 \sum_{k=0}^{+\infty}\cal R^k \tilde{\cal A}_{-1}{\rm RHS}.
S [ \Psi^{*}, \Psi ] \ = \{\frac{1}{2 \pi \rho}}\int_{{\bf R}\times S^{1}}d t d \varphi [{\frac{i}{2}}\rho ( \Psi^{*}{\dot{\Psi}}-{\dot{\Psi}}^{*}\Psi ) + i \Psi^{*}\partial_{\varphi}\Phi ]
T_{\lambda_{0,r}+1}\cdots T_{d-1}T_dT_{d-1}\cdots T_{\lambda_{0,r}+k}&= q_0^{-1}T_{\lambda_{0,r}+1}\cdots T_{d-1}(T_{d-1}\cdots T_1 X_1T_0^{-1}T_1^{-1}\cdots T_{d-1}^{-1}) T_{d-1}\cdots T_{\lambda_{0,r}+k}&= q_0^{-1}T_{\lambda_{0,r}+1}\cdots T_{d-1}T_{d-1}\cdots T_1 X_1T_0^{-1}T_1^{-1}\cdots T_{\lambda_{0,r}+k-1}^{-1}.
\int \frac{\mathrm{d}^{3}Q}{|{\bf Q}|}N ( |{\bf Q}| ) \left\{\frac{\mathrm{d}}{\mathrm{d}Q_{0}}\left( \frac{1}{Q_{0}}\mathrm{l n}( k \cdot Q ) \right) \right\}_{Q_{0}= |{\bf Q}|}= \pi \mathrm{l n}( T ) \mathrm{l n}( k^{2}) + \cdots ,
f(t,x)=\rho(t,x) \exp\left(\frac{1}{2}\int_{x-t}^{x+t}\frac{f_1(s)ds}{c_1+f_0(s)}+\frac{1}{2}\int_{0}^t \left[ \int_{x+s-t}^{x+t-s}G(s,y)dy\right]ds \right),
\omega_{n,j,k}^{-1}=-\sqrt{d_{n}}(x_k^{(n)}-i\omega)[u_{n}^{\mathcal{R}}(x_k^{(n)})]'u_{n-1}^{\mathcal{L}}(x_j^{(n)}) \neq 0, j,k = 1 \ldots n.
\begin{aligned}\widehat{\widetilde{P_j}F_k}(\tau) = \frac{1}{(2\pi)^{n+1}}{\phi}\Big(\frac{|\tau|}{2^{mj}}\Big) \lim_{R\rightarrow\infty}\int_{\mathbb{R}^n}e^{ix\cdot\xi}\bigg( \int_{\mathbb{R}}e^{it\tau}e^{it\Phi(\xi)}\psi\Big(\frac{t}{R}\Big) dt \bigg) \phi\Big(\frac{|\xi|}{2^k}\Big) \hat{f}(\xi) d\xi . \end{aligned}
\Pr[X_1=1|\mathcal{C}]=\Pr[X_2=1|\mathcal{C}]=2p(1-p)=2p-1, &\Pr[X_1=X_2|\mathcal{C}]=\Pr[X_1=1|\mathcal{C}]\Pr[X_2=1|\mathcal{C}]+\Pr[X_1=0|\mathcal{C}]\Pr[X_2=0|\mathcal{C}]=(2p-1)^2+(2-2p)^2=9-12p.
{\frac{m}{\sqrt{2}}}\Xi = \lim_{| z | \rightarrow \infty}\left\lbrace \left( \begin{matrix}{( \tau + 1 - \epsilon_{z}) \sqrt{2 z}}\\{\sqrt{2 z}}\\ \end{matrix}\right) - \left( \begin{matrix}{a_{D}( z , \tau , m )}\\{a ( z , \tau , m )}\\ \end{matrix}\right) \right\rbrace \cdotp
(Gu)(t)= T(t)u_{0}+\chi\int_{0}^{t}T(t-s)\nabla\cdot ( u(s) \nabla (\Delta-I)^{-1}u(s))ds & +(1+a)\int_{0}^{t}T(t-s)u(s)ds-b\int_{0}^{t}T(t-s)u^{2}(s)ds,
\int_{0}^{\infty}d t ~ t^{\frac{4 - p}{2}}~ e^{-{\frac{t b^{2}}{2 \pi \alpha^{\prime}}}}= \Gamma ({\frac{6 - p}{2}}) ~ ({\frac{2 \pi \alpha^{\prime}}{b^{2}}})^{\frac{6 - p}{2}}
\bar D_{3,a,4}= \frac{|X_1-X_2|}{|z|}\big| D_\xi Q(X_\tau,z+t_0(X_\tau),D_t \tilde h(X_\tau,t_0(X_\tau))) &\qquad\cdot \big [D^2_{X,t}\tilde h(X_\tau,z+t_0(X_\tau)) -D^2_{X,t}\tilde h(X_\tau,t_0(X_\tau)) \big]\big|
L_{k,2,3}=\int_{J_{2}}\left|\sum_{l=1}^{k-1}\int_{J_{3}}\frac{\left|\dot{\bar{h}}_{k}(u)\right||I_{l}|dv}{((1-v)|I_{l}|+|I_{l+1}|+\cdots+|I_{k-1}|+u|I_{k}|)^{H+\frac{1}{2}}}\right|^{2}|I_{k}|du \leq C_{H,l_{0}}\left|\sum_{l=1}^{k-1}\int_{J_{3}}\frac{|I_{l}|dv}{((1-v)|I_{l}|+|I_{l+1}|+\cdots+|I_{k}|)^{H+\frac{1}{2}}}\right|^{2}\cdot|I_{k}|,
\mathbb{S}_q(\ell_n) (x_{\beta^\vee}) = (1-x_{\beta + n \delta})^{-1}x_{\beta^\vee}, \mathbb{S}_q(\ell_n) (x_{\delta^\vee}) = \prod_{k=0}^{n-1}(1-(q^{\frac{1}{2}})^{1-n+2k}x_{\beta + n \delta})^{-1}x_{\delta^\vee}, \mathbb{S}_q(\ell_{\infty})(x_{\beta^\vee}) = x_{\beta^\vee}.
\sup_{\lambda > 0}\lambda \int_{\{(b-a)/2 < x < b/3 \,:\, (b-a)/b > \lambda\}}x^{-1}\, dx\lesssim\int_a^b x^{-1}\, dx=\log \frac{b}a \simeq \frac{b-a}a \simeq \frac{b-a}b,
R_2(n_1,n_2; x_1,x_2; \beta;N) = \frac{r_{n_1}(\beta_1-\beta_0-1, \beta_2-\beta_1-1, -x_2-1,\beta_1+x_2; x_1)}{(-N)_{n+m}(-N+\beta_0)_{n+m}(\beta_2-\beta_1)_{n}(\beta_3-\beta_1)_{m}}\qquad{}\times r_{n_2}(2n_1+\beta_2-\beta_0-1, \beta_3-\beta_2-1, n_1-N-1, N+n_1+\beta_2; x_2-n_1).
||-\frac{\sin(2(v_{1}+v_{2}+v_{3}+v_{4}))}{r^{3}}\left(\cos(2Q_{\frac{1}{\lambda(t)}}+2v_{5})-\cos(2Q_{\frac{1}{\lambda(t)}})\right)\vert_{r = R\lambda(t)}||_{L^{2}(R dR)}&\leq \frac{C}{t^{11/2}\log^{3b-3+\frac{5N}{2}}(t)}
\frac{dI_{\epsilon}}{dR}(R) = -\frac{4\pi}{R_{0}}\left(\frac{R^{2}-R^{2}_{0}}{RR_{0}}\right) \left(\frac{R^{2}+R^{2}_{0}}{R^{2}}\right)\int_{\epsilon}^{\infty}\frac{d\tau_{2}}{\tau_{2}^{1/2}}e^{-\pi \tau_{2}\left(\frac{R^{2}- R^{2}_{0}}{RR_{0}}\right)^{2}}
Q^{\infty}(\zeta) = \frac{1}{2}\begin{pmatrix}\gamma(\zeta)+\gamma(\zeta)^{-1}\frac{1}{i}\left(\gamma(\zeta)-\gamma(\zeta)^{-1}\right) -\frac{1}{i}\left(\gamma(\zeta)-\gamma(\zeta)^{-1}\right) \gamma(\zeta)+\gamma(\zeta)^{-1}\end{pmatrix},
\sum_{\ell=1}^\infty \frac{(\ell+1)^{1-\delta}}{\ell^2}\leq \sum_{\ell=1}^\infty \frac{\ell^{1-\delta}+1}{\ell^2}= \frac{\pi^2}{6}+ \sum_{\ell=1}^\infty\frac{1}{\ell^{1+\delta}}< \infty,
\Pi^{\varepsilon}_{\vec{\mathbb{P}}}(f_{1},f_{2})(x):=\sum_{\vec{P}\in\vec{\mathbb{P}}}\frac{C^{\varepsilon}_{Q_{\vec{P}}}}{|I_{\vec{P}}|^{\frac{1}{2}}}\langle f_{1},\Phi^{1}_{\vec{P}_{1}}\rangle\langle f_{2},\Phi^{2}_{\vec{P}_{2}}\rangle\Phi^{3}_{\vec{P}_{3}}(x),
\left( \begin{array}{c}\dot{\Phi}\Phi^{\prime}\end{array}\right) \;\; \rightarrow \;\; \left( \begin{array}{cc}cosh\,\theta sinh\,\theta sinh\,\theta cosh\,\theta\end{array}\right)\;\; \left( \begin{array}{c}\tilde{\dot{\Phi}}\tilde{\Phi}^{\prime}\end{array}\right)\;\;.
\begin{cases}\begin{aligned}\Delta^m u= e^u, &u^{(2i+1)}(0)=0, \forall\; 0\leq i\leq m-1, &\Delta^iu(0)= \Delta^iu_0(0), \forall\; 0\leq i\leq m-2, & \Delta^{m-1}u(0) = \Delta^{m-1}u_0(0) - \alpha, \alpha > 0.\end{aligned}\end{cases}
{}_{}\textrm{F}_{p}(a,b;c ;z)= \frac{1}{B(b, c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-zt)^{a}exp\left[-\frac{p}{t(1-t)}\right] dt, (p>0 ;\p=0 \and \ |arg(1-z)|<\pi ;\Re(c)>Re(b)>0)
n!{\|\tilde{h}_n(.,t,x)\|}_{\mathcal{H}^{\otimes n}}^2 \geq \frac{1}{n!}C_5e^{-2\mu_1 t}\int_{[0,t]^{2n}}\prod_{i=1}^n\gamma(t_i-s_i)\prod_{i=1}^n \Big(t+t_{i+1}-t_{i}\Big)^{-\beta/\alpha}d\textbf{t}d\textbf{s}.
H^{(0)}= \sum_{r = \pm1}\sum_{\vec{p}, p^{02}=\vec{p}^2}|p^0|\; ( b^{(0)+}_{\vec{p}, r}b^{(0)}_{\vec{p}, r}- d^{(0)+}{}_{\vec{p}, r}d^{(0)}_{\vec{p}, r}).
\frac{J_a^{(q,r)}(b,\theta )}{W^{(q+r)}(b)}= \frac{Z^{(q,r)}_a(b,\theta )}{W^{(q+r)}(b)}- \frac{r Z^{(q)}(-a, \theta) \overline{W}^{(q+r)}(b)}{W^{(q+r)}(b)}\xrightarrow{b \uparrow \infty}\tilde{Z}^{(q,r)}(-a,\theta ) - \frac{r Z^{(q)}(-a, \theta)}{\Phi(q+r)}.
\mathrm{t r}\left[ Q P^{, \rho}\right] = 0 . \qquad \mathrm{t r}\left[ Q P^{, \rho}P^{, \sigma}\right] = \frac{1}{k^{2}}P^{\rho \sigma}. \qquad \mathrm{t r}\left[ Q P^{, \rho \sigma}\right] = \frac{2}{k^{2}}P^{\rho \sigma}
\begin{array}{l l}{n \left( \frac{1 - k}{1 + k}\right) = 2 G \left( U - T - J^{2}- \nu S^{2}\right)}\\{B^{2}= 1 - 8 G \left( U + \frac{( J^{2}+ \nu S^{2})}{2}\right)}\\{m^{2}= 4 G ( J^{2}+ \nu S^{2}) .}\\ \end{array}
\tilde{g}_{k,\ell}(q_\ell) = q_\ell^{-n}\ \prod_{i=1}^{n+1}\,(1+c_iq_\ell)\ \prod_{i=1}^{k-1}\,(1-b_iq_\ell)\, \prod_{i=k+1}^n (1+a_iq_\ell) \,.
\Gamma_{s}( u^{- 1})^{T}\mathcal{C}\gamma^{a}= \Gamma_{s}( u^{- 1})^{T}\tilde{\mathcal{C}}\Gamma_{s}( P^{a}) = \tilde{\mathcal{C}}\Gamma_{s}( u ) \Gamma_{s}( P^{a}) .
Eq1 :\rho_{0}^{3}v_{0}^{2}+2\omega\frac{\rho_{0}v_{0}}{K_{0}}=4\mathfrak{S}_{0}Eq2 :\frac{\rho_{1}}{\rho_{0}v_{0}}-\omega\lbrack J_{3}\frac{1}{\rho_{0}^{2}v_{0}}+J_{4}\rho_{0}v_{0}]\ =J_{2}\ Eq3 :\rho_{0}(\rho_{1}^{2}-2h)+\frac{\omega^{2}}{\rho_{0}}-\frac{\omega}{2K_{0}}\rho_{0}v_{0}=J_{1}
q^{lm'(n-m')+(l-1)(n-m')^2-(l-1)n^2}\prod_{i=2}^{m'}q^{n(l_1-l_i)}\cdot q^{-m'(n-m')-nm'}&=q^{-n(l_1 + \ldots + l_{m'})}q^{m'nl}\cdot q^{lm'(n-m')+(l-1)(n-m')^2-(l-1)n^2-m'(n-m')-nm'}&=q^{-n(l_1 + \ldots + l_{m'})}q^{lm'(2n-m') +(l-1)(-2nm'+{m'}^2) -2nm' +{m'}^2}&=q^{-n(l_1 + \ldots + l_{m'})},
0 =[C_ju,((\rho\otimes\mu_l)\circ P)\otimes\mu_k] &=[u,-C_j((\rho\otimes\mu_l)\circ P)\otimes\mu_k] &=[u,-(l_j-k_j+k_{j+1})((\rho\otimes\mu_l)\circ P)\otimes\mu_k] &=[d_{k,l},-(l_j-k_j+k_{j+1})\rho] &=[-(l_j-k_j+k_{j+1})d_{k,l},\rho],
m(\{fg, a\}) = m(f\{g, a\}+(-1)^{|f||g|}g\{f, a\}) = m(f)[h(g)m(a)-(-1)^{|g||a|}m(a)h(g)]+(-1)^{|f||g|}m(g)[h(f)m(a)-(-1)^{|f||a|}m(a)h(f)] = [m(f)h(g)+(-1)^{|f||g|}m(g)h(f)]m(a)-(-1)^{|f||a|+|g||a|}m(a)[m(f)h(g)+(-1)^{|f||g|}m(g)h(f)] = h(fg)m(a)-(-1)^{|fg||a|}m(a)h(fg).
\Gamma_1'' = \Gamma_1\setminus \{b,(b_{c_1},b_{c_2}),\cdots \cdots (b_{c_{k 1}},b_{c_k})\}\Gamma_2'' = \Gamma_2+\{(c,b),(b_{c_1},b_{c_2}),\cdots \cdots (b_{c_{k 1}},b_{c_k})\}
f_1 \textstyle = \sum_{r - 2 > i(p-1) + 2 > 2}\Upsilon_i^{(1)}x^{i(p-1)+2}y^{r-i(p-1)-2}, f_2 \textstyle = \sum_{r - 2 > i(p-1) + 2 > 2}\Upsilon_i^{(2)}x^{i(p-1)+2}y^{r-i(p-1)-2},
e^{-\mathcal{M}_{\lambda}\left[\, \mathrm{Tr}\,\widetilde{\mathrm{F}};\alpha\right]}=\mathcal{M}_{\lambda}\left[\mathrm{E}^{-\mathrm{Tr}\,\widetilde{\mathrm{F}}};\alpha\right]\,,\;\;\;\;\;\alpha\in\mathbb{S}_\lambda\;.
\psi(x;y,z,t)\coloneqq\phantom{.}\Bigl((t=0)\land (x=z)\Bigr) \lor\Bigl((t=1)\land (\phi(x; y)\lor x=z)\Bigr) \phantom{.}\lor\phantom{.}\Bigl((t=2)\land (\neg \phi(x;y)\land x\ne z)\Bigr) \lor \Bigl((t=3)\land (x\ne z)\Bigl)
\left. \left( \overline{{{g}}}^{a 1}\partial_{a}X_{\mu}\mathcal{P}^{2}+ \frac{1}{3}\Pi^{( U ) a 1}\mathcal{P}^{\nu}\partial_{a}X^{\lambda}A_{\mu \nu \lambda}\right) \right|_{\sigma_{1}= 0 , \pi}\approx 0
\hat{S}=\left( \begin{array}{cccc}1 d_{X_1}d_{X_2}d_{X_3}d_{X_1}1 d_{X_1}d_{X_3}d_{X_1}d_{X_2}d_{X_2}d_{X_1}d_{X_3}-1 -1 d_{X_3}d_{X_1}d_{X_2}-1 -1 \end{array}\right).
\times c ( l_{0}, l_{1}; l_{0}, l_{1}+ 1 ) d ( l_{0}, l_{1}+ 1 ; l_{0}, l_{1}) c ( l_{0}, l_{1}+ 1 ; l_{0}+ 1 , l_{1}+ 1 ) d ( l_{0}+ 1 , l_{1}+ 1 ; l_{0}, l_{1}+ 1 ) ,
J [{\hat{\xi}}] ={\frac{1}{8 \pi G}}\int_{r = r_{+}}\! \! d^{n - 2}x \Bigl \{n^{a}\nabla_{a}{\hat{\xi}}^{t}\sqrt{\sigma}+{\hat{\xi}}^{a}\pi_{a}{}^{r}+ n_{a}{\hat{\xi}}^{a}K \sqrt{\sigma}\Bigr \}
P_{n}(\epsilon, q)\leq\sum\limits_{c = r}^{m}P\left\{\omega: \sup_{k\geq n}{k\choose r}^{-\frac{1}{(q + 1)}}\left|\left| C_{m, k, c}(\omega)\right|\right|_{{\cal{B}}\otimes{\cal{B}}}\geq\epsilon_{c}\right\},
\Psi = \sum_{i \; \bullet}c_{\bullet}\mathrm{\Large ~ e^{i{\bf ~ k ~ \cdot ~ r}_{i}}~}a_{i}^{+}\left| O \right\rangle + \sum_{i \; \circ}c_{\circ}\mathrm{\Large ~ e^{i{\bf ~ k ~ \cdot ~ r}_{i}}~}a_{i}^{+}\left| O \right\rangle
{\cal L}_{\rm YM}={\cal L}_{\rm YM}^{(i)}+{\cal L}_{\rm YM}^{(a)},\quad{\cal L}_{\rm YM}^{(i)}=-\frac14\left({\cal F}_{\mu\nu}^i\right)^2 ,\quad{\cal L}_{\rm YM}^{(a)}=-\frac14\left({\cal F}_{\mu\nu}^a\right)^2 .
\begin{aligned}A_J^*(A_Iw_\alpha-y) = A_J^*(A_Iw_\alpha-A_Iu_I^\dagger-Av) = A_J^*(A_I(A_I^*A_I)^{-1}A_I^*Av-Av-\alpha A_I(A_I^*A_I)^{-1}s^\dagger), \end{aligned}
\exp i \int d^{4}x \,{\cal L}_{c o r r}= \left( \mathrm{d e t}\,{\hat{D}}\right)^{- 1}= \exp - \, \mathrm{T r}( \ln{\hat{D}}) = \exp - \int d^{4}x \, \langle t ,{\bf x}\vert \, \ln{\hat{D}}\, \vert t ,{\bf x}\rangle ,
\|\sum_{\ell=2^n}^k \frac{\varepsilon_\ell(\omega)\ell^{it}}{\ell \sqrt{\log (\ell+1)}}\|_{\S^2}\le \sum_{\ell=2^n}^k \frac{1}{\ell \sqrt{\log (\ell+1)}}\le \frac{2}{\sqrt n}\underset{n\to +\infty}\longrightarrow 0\, .
\psi_m^{(1)}(n) = \sum_{\nu=1}^n \psi_m(n)\,, = \sum_{\nu=1}^n \left[ B_{1,\nu-1}+ m(m-1) B_{m, \nu-2}\right]\,, &= \sum_{\nu=1}^n B_{1,\nu-1}+ m(m-1) \sum_{\nu=1}^n B_{m, \nu-2}\,, &= B_{2,n-1}+ m(m-1) B_{m+1, n-2}\,, &= B_{2,n-1}+ \frac{m(m-1)}{m+1}(n-1) B_{m, n-1}\,.
\left\lbrace\begin{array}{rrr}\alpha_1^{'}x_1^2+(\alpha_2^{'}+\alpha_3^{'})x_1x_2+\alpha_4^{'}x_2^2-\lambda(x_1,x_2)x_1 = 0 \beta_1^{'}x_1^2+(\beta_2^{'}+\beta_3^{'})x_1x_2+\beta_4^{'}x_2^2-\lambda(x_1,x_2)x_2 = 0 \end{array}\right.
\tilde{f}_K (\tau_1, \xi_1 , \eta_1) = \tilde{f}( \tau_1 , \xi_1, K^{\frac{1}{2}}\eta_1), \tilde{g}_K (\tau_2, \xi_2, \eta_2) = \tilde{g}( \tau_2, \xi_2, K^{\frac{1}{2}}\eta_2), \tilde{h}_K (\tau, \xi, \eta) = \tilde{h}( \tau, \xi, K^{\frac{1}{2}}\eta).
\psi \mid U_l^* (\tau,z) = l^{-1}\underset{(D,r)}\sum \left( \underset{\substack{r' \bmod{2ml}r' \equiv r \bmod{2m}}}\sum c_\psi(l^2D,lr') \right) e\left( \frac{r^2-D}{4m}\tau + rz \right) .
X: x_{0}^{3}x_{1}+ x_{0}x_{1}^{3}+ x_{0}x_{2}^{3}+ x_{0}^{2}x_{1}x_{3}+ x_{1}x_{2}^{2}x_{3}+ x_{0}^{2}x_{3}^{2}+ x_{1}^{2}x_{3}^{2}+ x_{0}x_{3}^{3}= 0.
[ a_{m}^{\dagger}, \; \tilde{\psi}_{\dot{\alpha}}] = [ a_{m}^{\dagger}, \; \tilde{\psi}_{\dot{\alpha}}^{\dagger}] = [ \tilde{a}_{m}^{\dagger}, \; \psi_{\dot{\alpha}}] = [ \tilde{a}_{m}^{\dagger}, \; \psi_{\dot{\alpha}}^{\dagger}] = 0
< \bar{0}| \eta_{\hat{\alpha}\hat{\beta}}: \bar{T}^{\hat{\alpha}\hat{\beta}}: |_{P_{0}}| \bar{0}> = - < \bar{0}| : \bar{T}^{\hat{1}\hat{1}}: |_{P_{0}}| \bar{0}> = - \frac{1}{\sinh^{2}( \chi )}< \bar{0}| : T^{\hat{0}\hat{0}}: |_{P_{0}}| \bar{0}>
I_{\alpha}=-\frac1{1+t\epsilon}\lambda_{\alpha}^2\prod_{i=1}^d\frac{\alpha_i!}{k_i^{\alpha_i}}+\frac{1+u}{1-t\epsilon}\lambda_{\alpha}^2\prod_{i=1}^d\frac{\alpha_i!}{(k_i+u)^{\alpha_i}}.
D4(1-(\alpha^{-1}\beta-1)U,\widehat{U},\overline{U},-1+(\alpha^{-1}\beta-1)\widehat{\overline{U}};0,0,0)=0 ~\Leftrightarrow~ \cfrac{\widehat{U}}{\overline{U}}=\cfrac{(\beta-\alpha)U-\alpha}{(\beta-\alpha)\widehat{\overline{U}}-\alpha}\,,
\int_{S^{n-1}}(L_K z_1) z_2 dV_K = \frac{1}{n}\int_{S^{n-1}}z_1 h_K \SS(z_2 h_K,h_K,\ldots,h_K) d\theta & = V(z_1 h_K, z_2 h_K, h_K,\ldots,h_K) .
X^2_+(t)= \sum_{i\in I_1}|\xi_{i,1}(t)|^2+|\xi_{i,2}(t)|^2+\sum_{i\in I_2}|\xi_{i,3}(t)|^2+|\xi_{i,4}(t)|^2 + \sum |\xi^+_i(t)|^2+\sum_{i:\gamma^-_i>\gamma}|\xi^-_i(t)|^2, X^2_-(t)= \sum_{i:\gamma^-_i\leq \gamma}|\xi^-_i(t)|^2.
\Big | \sum_{k=1}^K \prod_{j=1}^{n-1}e_{j,k}e_{n}\Big |_{{Y(}X_{n+1})}\le \Big | \sum_{k=1}^K \prod_{j=1}^{n-1}e_{j,k}\Big |_{Y(Y(X_1, \dots, X_{n-1}))}|e_{n}|_{X_{n}}
\frac{1}{16}(f^1_{uuu}+f^1_{uvv}+f^2_{uuv}+f^2_{vvv})+\frac{1}{16\omega}(f^1_{uv}(f^1_{uu}+f^1_{vv})-f^2_{uv}(f^2_{uu}+f^2_{vv}) - f^1_{uu}f^2_{uu}+f^1_{vv}f^2_{vv})=-\frac{1}{8}<0.
\int d \beta \mathrm{e}^{- \sum_{k}\beta_{k}^{2}/ 2 \hat{x}- l \sum_{k}t_{k}\beta_{k}}{\frac{\prod_{i = 1}^{n}\prod_{k < l}2 \sinh{\frac{\beta_{k}- \beta_{l}}{2 p_{i}}}}{\prod_{k < l}\Bigl ( 2 \sinh{\frac{\beta_{k}- \beta_{l}}{2}}\Bigr )^{n - 2}}}
l^2 = \left(1 +{c_p \,N \over (r M_P)^{7-p}(R_S M_P)^2 (V_T M_P^p)}\right) R_S^2 = R_S^2 +{c_p \,N \over (r M_P)^{7-p}M_P^2 (V_T M_P^p)}
V_D = V_E = V_0 \int d^2 z \left| \Phi_{X_1}(z )\right|^2 \left|\Phi_{X_2}(z)\right|^2 ={V_0 \over \sqrt{2 \pi}}{1 \over L_y}e^{-(X_1-X_2)^2 / 2}.
G_1^{(2,0)}= u_{(3,0)}; G_1^{(1,1)}= 2u_{(2,1)}; G_1^{(0,2)}= u_{(1,2)}\, , G_2^{(2,0)}= u_{(2,1)}; G_2^{(1,1)}= 2u_{(1,2)}; G_2^{(0,2)}= u_{(0,3)}\, .
\frac{f'}{f}= - \frac{a_{k-1}}k - \left( \frac{k-1}2 \right) \frac{H''}{H'}+ H' \hbox{or}\frac{f'}{f}= - \frac{a_{k-1}}k - \left( \frac{k-1}2 \right) \frac{H''}{H'}- m \frac{H'}{H},
U_1^\star NU_1 = U_1^\star\begin{bmatrix}A_1 0 C_1 B_1\end{bmatrix}V_2^{-1}V_2\begin{bmatrix}0 A_2 B_2 C_2\end{bmatrix}V_3 V_3^{-1}\begin{bmatrix}B_3 C_3 0 A_3\end{bmatrix}U_1,
z^{(1)}_{n}(b) := b \ln(n/k) \bigg(\frac{[nx]-2}{n}-x\bigg)^{p}, z^{(2)}_{n}(b):= b \ln(n/k) \bigg(\frac{[n(x-\epsilon)]+1}{n}-x\bigg)^{p}, z^{(3)}_{n}(b) := b \ln(n/k) \bigg(\frac{[nx]-1}{n}-x\bigg)^{p}, z^{(4)}_{n}(b):= b \ln(n/k) \bigg(\frac{[n(x-\epsilon)]+2}{n}-x\bigg)^{p}.
A_{x}^{( n )}={\frac{e^{{\phi_{0}}}\, r_{0}\, \left({P^{( n )}}\,{\Delta}+{Q^{( n )}}\, ( \Sigma + r_{\pm}( u , v ) ) \right)}{{\sqrt{{M^{2}}-{| \Upsilon |^{2}}}}\, \left({{r_{\pm}( u , v )}^{2}}-{{{\Delta}}^{2}}-{\Sigma^{2}}\right)}},
\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}
p^{\nu_3}_{k-1}(3,t)=(-1)^{k-1}\underset{\Lambda^{k-1}_{3}}{\sum}\frac{\lambda_1^{k_1+1}\lambda_2^{k_2}\lambda_3^{k_3-1}t^{k_1\nu_1+k_2\nu_2+k_3\nu_3}}{\Gamma\left(k_1\nu_1+k_2\nu_2+k_3\nu_3+1\right)}.
\bar{T}_1=\mathrm{P}_{\theta}(\bar{A}^c_M) = \mathrm{P}\Big(\sum_{i=1}^{\bar{m}}\xi_{[i]}^2>\tfrac{(1+B)}{\beta}\bar{m}\log(\tfrac{en}{\bar{m}}) +\tfrac{M}{6}\Big) \le \exp\big\{-M\beta/6\big\}.
v_j^*R_j^*(\bar z)H_{1,j}^{-1}R_j(b)v_jQ_{3,j}^*(b) = P_{3,j}^*(\bar z), \widehat H_{1,j}= P_{1,j}(b)Q_{3,j}^*(b), I-(z-a)u_{1,j}^* R_j^*(\bar z)H_{1,j}^{-1}R_j(a)v_j=Q_{4,j}^{*}(z)Q_{4,j}^{*-1}(a), I-(z-b)u_{1,j}^* R_j^* (\bar z)H_{1,j}^{-1}R_j(b)v_j=Q_{3,j}^{*}(z)Q_{3,j}^{*-1}(b).
\widehat{\mathbf{v}}_{s,m}= \left[\sum\limits_{\ell=1}^L a_{s,m,\ell}\mathbf{t}_{m,\ell}\right] \bmod \Lambda_{C,m}\stackrel{(a)}{=}\left[\sum\limits_{\ell=1}^L a_{s,m,\ell}\left({\mathbf{y}}_\ell+\mathbf{q}_\ell \right) \right] \bmod \Lambda_{C,m}\stackrel{(b)}{=}\mathbf{v}_{s,m}
\|\hat{\Sigma}_t^{n}-\bar{\Sigma}_t^n\|_{L_{}(L^2([0,1]))}^2 = \int_0^1\int_0^1 \left(\hat{q}_t^n(x,y)-\bar{q}_t^n(x,y)\right)^2 dxdy = \sum_{j_1,j_2=0}^n \Delta_n^2 \left(\langle \left(SARCV_t^n- \int_0^t\Sigma_s ds\right) \delta_{x_{j_1}},\delta_{x_{j_2}}\right)^2 \leq 4 \|SARCV_t^n-\int_0^t\Sigma_s\|_{}^2,
\log(d^2/y) \geq \log( \frac{(m \log m)^{2/3}}{(m \log m)^{1/3}}) \approx \log m \log(m/d) = \log(\frac{m}{(m \log m)^{1/3}}) \approx \log m \log((m/d)^2/y) \geq \log( \frac{m^2}{(m \log m)^{2/3}\times (m \log m)^{1/3}}) \approx \log m
[X,Y]^{\bullet}_{Der(\mathcal{F})}(fg)=[X^i\partial_i,Y^j\partial_j]^{\bullet}_{Der(\mathcal{F})}(fg)=\{(X^i\partial_i)\bullet(Y^j\partial_j)-(Y^j\partial_j)\bullet (X^i\partial_i)\}(fg) &=\{X^iY^j\partial_i\circ\partial_j+X^i\rho(\partial_i)(Y^j)\partial_j-Y^j\rho(\partial_j)(X^i)\partial_i-X^iY^j\partial_j\circ\partial_i\}(fg).
F_{1}= - \frac{1}{2 4}\log \Bigl \{\biggl ( N_{1}+ 2 \varphi \frac{1}{y_{+}^{3 / 2}y_{-}^{1 / 2}}\biggr ) \biggl ( K_{1}+ 2 \varphi \frac{1}{y_{+}^{1 / 2}y_{-}^{3 / 2}}\biggr ) ( y_{+}- y_{-})^{4}\Bigr \}.
u \circ v = \sum_{s = 0}^{\infty}{\frac{\kappa^{2 s}}{2^{2 s}( 2 s + 1 ) !}}\omega^{i_{1}j_{1}}\ldots \omega^{i_{2 s}j_{2 s}}{\frac{\partial^{2 s}u}{\partial x^{i_{1}}\ldots \partial x^{i_{2 s}}}}{\frac{\partial^{2 s}v}{\partial x^{j_{1}}\ldots \partial x^{j_{2 s}}}}\, .
\begin{array}{lll}[J,K_1]=K_2, [J,K_2]=-K_1, [K_1, K_2]=-J, begin{align*}2pt] [J,P_0]=0 , [J,P_1]=P_2 , [J, P_2]=-P_1 , begin{align*}2pt][K_1,P_0]=P_1 , [K_1,P_1]=P_0 , [K_1, P_2]=0, begin{align*}2pt][K_2,P_0]=P_2 , \quad[K_2,P_1]= 0 , \quad[K_2, P_2]=P_0, begin{align*}2pt][P_0,P_1]=0, \quad[P_0, P_2]=0 , [P_1,P_2]=0 . \end{array}
S = - \frac{1}{2}\int \! d \xi^{+}d \xi^{-}[{({E_{0}}^{- 1}+ \Pi )^{- 1}}_{i j}( \partial_{+}g g^{- 1})^{i}( \partial_{-}g g^{- 1})^{j}+ \partial_{+}t \partial_{-}t ] ,
\!\!\!\!\!\!\!\!\!\!\! y_{v}(t)= A \sqrt{P_{v_o}r_o^{-\eta}h_o}s(t,B_{v}^{\alpha{\rm D}},b_{v}) + \sum_{j \in \tilde{\Psi}_{\rm d}}\mathfrak{I}^{({\rm d}\rightarrow v)}_{j}(t) + \sum_{j \in \tilde{\Psi}_{\rm u}}\mathfrak{I}^{({\rm u}\rightarrow v)}_{j}(t)+ \mathfrak{I}_{{\rm s}_{v}}(t) + n(t).
[o_1+o_2, a\mapsto \delta_1(a) +\delta_2(a)] =\ i(o_1+o_2) + \sum_{b\in \Sigma}\big(j(b)\cdot (\delta_1(a) + \delta_2(a))\big) =\ i(o_1)+i(o_2) + \sum_{b\in \Sigma}\big(j(b)\cdot \delta_1(a)\big) + \sum_{b\in \Sigma}\big(j(b)\cdot \delta_2(a))\big) =\ [o_1, \delta_1] + [o_2, \delta_2]
\left|\left\langle \mathbf{P}_{\varepsilon}^{\beta}v_{n}^{\varepsilon}\left(s\right),\phi_{j}\right\rangle -\left\langle \mathbf{P}_{\varepsilon}^{\beta}w_{n}^{\varepsilon}\left(s\right),\phi_{j}\right\rangle\right| \le CC_{1}\log\left(\gamma\left(T,\beta\right)\right)\sum_{k=1}^{n}\left|V_{k}^{\varepsilon}-W_{k}^{\varepsilon}\right|.