harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c c c c c}{{p_{j}= - p_{i},}}&{{}}&{{\alpha_{n}^{( j )}= - \alpha_{- n}^{( i )}= - \alpha_{n}^{( i ) \dagger},}}&{{}}&{{\bar{\alpha}_{n}^{( j )}= - \bar{\alpha}_{- n}^{( i )}= - \bar{\alpha}_{n}^{( i ) \dagger}.}}\\ \end{array}
2(\widetilde{\nabla}_{[\mu}T_{\nu ]\lambda}{}^{\rho}+T_{\lambda \lbrack\mu}{}^{\tau}T_{\nu ]\tau}{}^{\rho})=R_{\mu \nu \lambda}^{{\footnotesize (T)}}{}^{\rho}+\check{Z}_{\mu \nu \lambda}{}^{\rho},
(tv_it^{-1})\cdot(\vec{x},s) =(tv_i)\cdot(A^{-1}\vec{x},s-1)=t\cdot(A^{-1}\vec{x}+e_i,s-1)=(A\cdot A^{-1}\vec{x}+Ae_i,s-1+1) &= (\vec{x}+Ae_i,s)=(Av_i)\cdot (\vec{x},s).
Z^p_m(x,\zeta)=\left[ C^{n/2}_m(\frac{x\cdot\overline{\zeta}}{|x||\overline{\zeta}|})-C^{n/2}_{m-2p}(\frac{x\cdot\overline{\zeta}}{|x||\overline{\zeta}|})\right]|x|^m|\overline{\zeta}|^m,
M_{p_{e}, q_{m}}={\frac{2 \pi}{g_{\mathrm{Y M}}^{2}}}{\frac{q_{m}u}{\operatorname{c o s}\xi}}= u \sqrt{\left( \frac{2 \pi q_{m}}{g_{Y M}^{2}}\right)^{2}+ p_{e}^{2}}\, .
H(u,v)=\frac{a^{2}+b^{2}+\frac{4u^{2}}{a^{2}}+\frac{4v^{2}}{b^{2}}}{a^{2}b^{2}\sqrt{(1+\frac{4u^{2}}{a^{4}}+\frac{4v^{2}}{b^{4}})^{3}}}
S_+(z)=S_-(z)\begin{pmatrix}0 -1 1 0\end{pmatrix}, z\in (-\infty,z_0), &S_+(z)=S_-(z)\begin{pmatrix}1 0 -e^{\pi i\alpha}1\end{pmatrix}, z\in \Sigma'_1, &S_+(z)=S_-(z)\begin{pmatrix}1 0 -e^{-\pi i\alpha}1\end{pmatrix}, z\in \Sigma'_3, &S_+(z)=S_-(z)\begin{pmatrix}e^{-\pi i\alpha}-1 0 e^{\pi i\alpha}\end{pmatrix}, z\in (z_0,0).
a\left(\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}\right) +\frac{ib}{2}\left(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho}\right)\epsilon^{\rho\sigma\alpha\beta}=\frac{1}{2}\epsilon^{\alpha\beta\rho\sigma}\left(\partial_{\rho}B_{\sigma}-\partial_{\sigma}B_{\rho}\right).
\left. \begin{array}{l l l l r}{a = \sqrt{2 1 5 3 3}+ 1 1 9 \,}{\, \, \,}{b}{=}{- 1 9 4}{c = \sqrt{2 1 5 3 3}- 1 1 9 \,}{\, \, \,}{d}{=}{1 9 4}\end{array}\right.
\frac{dQ_{l}}{dt}=F_{a}C_{art}+F_{g}(\frac{Q_{g}}{P_{g}V_{g}})+F_{pn}(\frac{Q_{pn}}{P_{pn}V_{pn}})-(F_{a}+F_{g}+F_{pn})(\frac{Q_{l}}{P_{l}V_{l}})
f(w; k \lambda) = C_k \cdot d^{k w}\cdot \dfrac{\prod_{i=0}^{k r-1}\varGamma\left(w+\frac{i}{k r}\right)}{\prod_{i=1}^{r}\prod_{j=0}^{k-1}\varGamma\left(w+\frac{v_i + j}{k}\right)},
{(\mathbb{S}_{+}^n \cap \mathcal{S})}^{*}\cap \mathcal{S}= \mathbb{S}_{+}^n \cap \mathcal{S},{(\mathcal{L}\cap \mathcal{S})}^{\perp}\cap \mathcal{S}= \mathcal{L}^{\perp}\cap \mathcal{S}.
\delta{\cal L}_{\mathrm{a n o m a l y}}\ = \ \frac{\theta_{i}}{3 2 \pi^{2}}\ \sum_{a}\, b_{a}^{\prime i}\, F^{a}\widetilde{F}^{a}\ - \ \frac{\theta_{i}}{7 6 8 \pi^{2}}\b_{g r a v}^{\prime i}\, R \widetilde{R}\ ,
\frac{dy}{dx}=\frac{1}{\sqrt{1-(\frac{1}{x})^{2}}}\cdot(-\frac{1}{x^{2}})=-\frac{1}{x^{2}\sqrt{1-\frac{1}{x^{2}}}}=-\frac{1}{x^{2}\frac{\sqrt{x^{2}-1}}{\sqrt{x^{2}}}}=-\frac{1}{\sqrt{x^{2}}\sqrt{x^{2}-1}}=-\frac{1}{|x|\sqrt{x^{2}-1}}
k(y,\theta, (t_k-t_j)_{j,k=1,\dots,n},w)=\sum_{j,k=1}^n\theta_j \theta_k\int_{\mathcal{X}\times \mathbb{R}^d}g(x,v)g(x,t_k-t_j+v) h(y,-w-v)p(dx) dv.
||\lambda(t) \sqrt{\omega}\mathcal{F}(\sqrt{\cdot}(N(u_{1})-N(u_{2}))(t,\cdot \lambda(t)))(\omega \lambda(t)^{2})||_{L^{2}(\rho(\omega \lambda(t)^{2})d\omega)}&\leq C \frac{||y_{1}-y_{2}||_{Z}}{t^{4}\log^{\epsilon -b}(t)}
\mathbb{P}_{\Lambda,\rho,\beta,H}^{\vec{h}}(\omega_e=1|\omega_{G_t}=\omega^{\rho}_{G_t})\leq \mathbb{P}_{\Lambda,\rho,\beta,H}^{|\vec{h}|}(\omega_e=1|\omega_{G_t}=\omega^{\rho}_{G_t}).
{_iA_s}(F\cdot G)={_iA_s}\left(\frac{1}{|s|}\sum_{a=0}^{|s|-1}{_aA_s}(F)\cdot L_s^a\cdot G\right) = \frac{1}{|s|}\sum_{a=0}^{|s|-1}{_iA_s}\left({_aA_s}(F)\cdot L_s^a\cdot G\right) = \frac{1}{|s|}\sum_{a=0}^{|s|-1}{_aA_s}(F)\cdot{_iA_s}(L_s^a\cdot G) = \frac{1}{|s|}\sum_{a=0}^{|s|-1}{_aA_s}(F)\cdot{_{i-a}A_s}(G)
\varpi\bigl(z^{\left(t+1\right)}\bigr)=\sum_{i=1}^{n}x_{i}^{\:\left(t+1\right)}+\sum_{j=1}^{\nu}y_{j}^{\:\left(t+1\right)}=\bigl(\sum_{i=1}^{n}x_{i}^{\:\left(t\right)}\bigr)\bigl(\sum_{j=1}^{\nu}y_{j}^{\:\left(t\right)}\bigr)
\displaystyle\int_{B_\rho}\dfrac{|\tau_h(\tau_hv(x))|^{p}}{|h|^{(1+\gamma) p}}dx \leq c |h|^p \displaystyle\int_{B_R}\dfrac{|\tau_hDv(x)|^{p}}{|h|^{(1+\gamma) p}}dx = c \displaystyle\int_{B_R}\dfrac{|\tau_hDv(x)|^{p}}{|h|^{\gamma p}}dx \le c [Dv]_{B^{\gamma}_{p,\infty}(B_R)}^p,
B(q^2)=b_0(q^2)g^3+g^5{B_1(q^2,Q^2)\over \epsilon}+g^5B_2(q^2,Q^2)+g^7\left[{B_3(q^2,Q^2) \over \epsilon^2}+{B_4(q^2,Q^2)\over \epsilon}+B_5(q^2,Q^2)\right]+...
\textrm{P}_\textrm{so}^{l}(\beta_e) = 2 \pi \lambda_e \!\! \int_{0}^{\infty}\!\! e^{- \left( \theta \left( \frac{\beta_e}{\rho}\right )^{\frac{2}{\alpha}}+ \pi \lambda_e\right )r^2}\mathcal{L}_{W}\!\! \left(\frac{\beta_e}{\rho}r^\alpha \! \right) r \mathrm{d}{r},
\widetilde{S}_{V}( \widetilde{\Phi}) \equiv S_{W}( \Phi_{0}+ \widetilde{\Phi}) - S_{W}( \Phi_{0}) = - \frac{1}{g_{o}^{2}}\left[ \frac{1}{2}\langle \widetilde{\Phi}, Q \widetilde{\Phi}\rangle + \frac{1}{3}\langle \widetilde{\Phi}, \widetilde{\Phi}* \widetilde{\Phi}\rangle \right] ,
\|uv\|_{B^s_{p,r}}\lesssim \|u\|_{B^s_{p,r}}\|v\|_{B^s_{p,r}},\,\|f(\rho) \|_{B^s_{p,r}}\leq C(\|\rho\|_{L^\infty})\|\varrho\|_{B^s_{p,r}}\hbox{with}f(1)=0,\,s>\frac{d}{p}\hbox{or}s\geq \frac{d}{p}, r=1,
(M_{pk}^{p-1}-M_p^{p-1})\int_0^T\phi_p({_0D_t^\alpha}u(t))({_0D_t^\alpha}u_k(t)-{_0D_t^\alpha}u(t))dt &=(M_{pk}^{p-1}-M_p^{p-1})\langle I_1'(u),u_k-u\rangle &\rightarrow0\ \ \mbox{as}\k\rightarrow\infty,
R(\lambda+i0;x,y) =\tfrac{i}{2\sqrt{\lambda}}\bigl[e^{i|x-y|\sqrt{\lambda}}-\tfrac{q}{q-i\sqrt{\lambda}}e^{i(|x|+|y|)\sqrt{\lambda}}\bigr], R(\lambda-i0;x,y) = \tfrac{i}{2\sqrt{\lambda}}\bigl[e^{-i|x-y|\sqrt{\lambda}}-\tfrac{q}{q+i\sqrt{\lambda}}e^{-i(|x|+|y|)\sqrt{\lambda}}\bigr]
\int_{|z|>\frac{1}{\delta}}U^{\frac{4(n-1)}{n(n-2)}}|v_{q}|^{\frac{2(n-1)}{n}}dz \le C\int_{|z|>\frac{1}{\delta}}\frac{1}{|z|^{\frac{4(n-1)}{n}}}\frac{1}{|z|^{\frac{2(n-1)(n-3)}{n}}}dz \le C\int_{\frac{1}{\delta}}^{\infty}r^{-\frac{3n^{2}-12n+10}{n}}=O(\delta^{\frac{3n^{2}-11n+10}{n}})=o(\delta^{\frac{2(n-1)}{n}})
\int_{t_{M-1}}^t \phi^2_\alpha(t-s) ds=\begin{cases}\frac{1-e^{-2\lambda_\alpha (t-t_{M-1})}}{2\lambda_\alpha}, ; \frac{2\sqrt{\lambda_\alpha}(t-t_{M-1})-\sin(2\sqrt{\lambda_\alpha}(t-t_{M-1})}{2\lambda_\alpha^{3/2}}, ,\end{cases}
dm_{i,j}(x,y)=\prod_{k=1}^rd\mu_{q_{t_k}(0)-q_{s_k}(x_{i,j})}(x_{s_k},y_{t_k})\times\prod_{s:\, \forall k\,s\neq s_k}d\mu(x_s)\times\prod_{t:\,\forall k\,t\neq t_k}d\mu(y_t)
M(p)=exp(\int_{0}^{1}\int_{0}^{1}\cdot\cdot\cdot\int_{0}^{1}log(|p(e^{2\pi i\theta_{1}},e^{2\pi i\theta_{2}},...,e^{2\pi i\theta_{n}})|)d\theta_{1}d\theta_{2}\cdot\cdot\cdot d\theta_{n}).
\sum_{i=1}^n T_i\left ( (T_i^*)^{r-1}-(T_{i-1}^*)^{r-1}\right ) = \sum_{i=1}^n T_i (T_i^*)^{r-1}- T_{i-1}(T_{i-1}^*)^{r-1}- \sum_{i=1}^n Z_i (T_{i-1}^*)^{r-1}= T_n (T_n^*)^{r-1}- \sum_{i=1}^n Z_i (T_{i-1}^*)^{r-1}\, .
\frac{1}{\phi_{(\bar X_n,s_n^2)}(t_1, t2)}\left. \frac{\partial \phi_{(\bar X_n,s_n^2)}(t_1, t_2)}{\partial t_2}\right|_{t_2=0}=\frac{1}{\phi_{s_n^2}(t2)}\left. \frac{\partial \phi_{s_n^2}(t_2)}{\partial t_2}\right|_{t_2=0}.
\left| Q^{( 1 )}\vert E , p^{3}\rangle \right|^{2}= \langle E , p^{3}\vert \left( Q^{( 1 )}\right)^{2}\vert E , p^{3}\rangle = \langle E , p^{3}\vert H^{\prime}- P^{3}\vert E , p^{3}\rangle = E - p^{3}= 0 .
\mu(P)=\rho(s(\mu))(P)=\int_{S^1}s(\mu)(\kappa(\theta))P(-\sin\theta,\cos\theta)d\theta=\sum_{\frac{2-k}{2}\leq n\leq\frac{k-2}{2}}a_n(\mu)\int_{S^1}e^{2ni\theta}P(-\sin\theta,\cos\theta)d\theta.
H_{j}\; = \; \frac{i}{2}[ \: \eta \tau_{j}^{y}\tau_{j + 1}^{x}\: - \: \eta^{- 1}\tau_{j}^{x}\tau_{j + 1}^{y}\: + \: q \tau_{j}^{x}\tau_{j}^{y}\: + \: q^{- 1}\tau_{j + 1}^{x}\tau_{j + 1}^{y}] .
[ \hat{G}_{1}^{\prime}, \hat{G}_{2}^{\prime}] = 2 i{\bf 1}, \hspace{0 . 3 c m}[ \hat{G}_{3}^{\prime}, \hat{G}_{4}^{\prime}] = - 2 i{\bf 1}, \hspace{0 . 3 c m}[ \hat{N}_{1}^{\prime}, \hat{N}_{2}^{\prime}] = 2 i{\bf 1},
\gamma_{A}( g_{\mathrm{R}}) = - \frac{g_{\mathrm{R}}^{2}}{( 4 \pi )^{2}}\frac 1 2 \left[ \frac{1 7}6 - \frac{\alpha_{\mathrm{R}}}2 - \hat{\beta}_{\mathrm{R}}\right] ,
\ddot{x}= - \Phi ( x ) \Phi^{\prime}( x ) - \Phi^{\prime \prime}( x ) \overline{{\psi}}\psi , \qquad \dot{\psi}= - \mathrm{i}\Phi^{\prime}( x ) \psi , \qquad \dot{\overline{{\psi}}}= \mathrm{i}\Phi^{\prime}( x ) \overline{{\psi}}.
\left ( \begin{array}{c}\tilde{\varphi}_{0}\vdots \tilde{\varphi}_{p}\end{array}\right ) = D^{p}\mathcal{F}^{-1}\Big [ \prod_{j = 1}^{\infty}2^{-p}\tilde{\mathcal{A}}( e^{- i 2^{-j}\xi}) v \Big ] .
\gamma^i_{a\dot b}\tilde\gamma^j_{\dot b c}+ \gamma^j_{a\dot b}\tilde\gamma^i_{\dot b c}= 2\delta^{ij}\delta_{ac}\;, \qquad \tilde\gamma^i_{\dot ab}\gamma^j_{b\dot c}+ \tilde\gamma^j_{\dot ab}\gamma^i_{b\dot c}= 2\delta^{ij}\delta_{\dot a\dot c}\;.
{\tilde{\beta}}_{\mu \nu}^{G}= R_{\mu \nu}+ 2 \gamma \nabla_{\mu}\partial_{\nu}\Phi , \qquad{\tilde{\beta}}^{\Phi}= - R + 4 \gamma^{2}( \nabla_{\mu}\Phi ) ( \nabla_{\nu}\Phi ) G^{\mu \nu}- 4 \gamma \nabla^{2}\Phi + Q^{2}
\sum_{a=1}^{n-2}\sum_{m=1}^{n-a-1}\sum_{t=a+m}^{n-1}\binom{n-t+a-1}{a-1}=\sum_{a=1}^{n-2}\sum_{m=1}^{n-a-1}\left(\binom{n-m}{a}-1\right)=\sum_{a=1}^{n-2}\left(\binom{n}{a+1}-(n-a)\right) &=2^n-\binom{n+1}{2}-1
\begin{array}[l]{l}\max_{t\in \lbrack0,T]}\left \vert y^{}(t)\right \vert=O(\varepsilon^{}), \max_{t\in \lbrack0,T]}\left \vert X^{\varepsilon}(t)-\bar{X}(t)-y^{}(t)\right \vert =o(\varepsilon), \end{array}
\liminf_{n \to \infty}\frac{\log R_{n}(x)}{\varphi(n)}=\liminf_{i \to \infty}\frac{\log \ell_i}{\varphi(n_i)}= \liminf_{i \to \infty}\left( \frac{\log \ell_i}{\log n_i}\cdot \frac{\log n_i}{\varphi(n_i)}\right) &= \lim_{i \to \infty}\frac{\log \ell_i}{\log n_i}\cdot \liminf_{i \to \infty}\frac{\log n_i}{\varphi(n_i)}= A \cdot \frac 1\gamma = \alpha.
\hat{Q}'_k(\mu) = F(z^{k+1}) - \widetilde{F}^k(z^{k+1}) + \langle g^{k+1}- s^{k+1}, \hat{x}(\mu)-z^{k+1}\rangle = F(z^{k+1}) - \widetilde{F}^k(z^{k+1}) -\frac{\mu}{\rho}\big\|s^{k+1}- g^{k+1}\big\|^2 .
\limsup_{\epsilon\rightarrow0}\widetilde{\mathbb{Q}}_{\infty}^{\tilde{b},\tilde{\Gamma}}\left[\tilde{\rho}:\left|\mathfrak{I}_{\tilde{g},\tilde{\rho}}(T)-\mathfrak{I}_{\tilde{g},\tilde{\rho}}(0)-\int_{0}^{T}\mathfrak{K}_{\tilde{g},\tilde{\rho},\epsilon}(t)dt\right|>\delta\right]=0
\mathcal{E}_{n\cdot3}= \Bigl\{\max_{i=1, 2, 3}\bigl|V_{n,i}^2/n-1\bigr| \leq\varepsilon_{n1}n^{(\beta-1)/2},\max_{i=1, 2, 3}|\Delta_{n,i}| \leq\varepsilon_{n2}\Bigr\},
s(2^k+(-1)^kf,3f)= \frac{2^k+(-1)^kf}{36f}+\frac{f}{4(2^k+(-1)^kf)}-\frac{(-1)^k}{4}+\frac{1}{36(2^k+(-1)^kf)f}&+(-1)^ks(3\cdot 2^k,2^k+(-1)^kf)
(x_{\alpha\dot\alpha})=\left(\begin{array}{cc}z_2 z_1 -\bar{z}_1 \bar{z}_2 \end{array}\right), \qquad(\bar{x}^{\dot\alpha\alpha})=\left(\begin{array}{cc}\bar{z}_2 -z_1 \bar{z}_1 z_2 \end{array}\right).
\frac{1}{r_2^2}\int_{\{2a_0^f(\Psi;h)\Psi\le2\log r_2\}}|f|^2_h =\lim_{p\rightarrow2a_0^f(\Psi;h)+0}\frac{1}{r_2^2}\int_{\{p\Psi<2\log r_2\}}|f|^2_h &\ge G(0;c\equiv1,\Psi,h,I_+(h,2a_{z_0}^f(\Psi;h)\Psi)_{z_0},f).
\Delta S_0-8\exp(-2 T_0)S_0=4\exp(-2T_0)\left(T_0^2-T_0 \right)\,, &\Delta S_1-8\exp(-2 T_0)S_1=4\exp(-2T_0) \left(S_0+2S_0^2-4T_0 S_0+2S_0T_0^2-T_0^3+\frac{T_0^4}{2}\right)\,, &\Delta S_2-8\exp(-2 T_0)S_2=4\exp(-2T_0) T_0\,,
\bar{\Delta}_{\rm Reactive}=\frac{\mathbb{E}G^{\rm Reac}_j}{\mathbb{E}T^{\rm Reac}_j}=\frac{\mathbb{E}T^{\rm Proac}_j}{2\mathbb{E}\left(T^{\rm Reac}_j\right)^2}+\mathbb{E}\tau^{\rm Reac}_{V_j}-\tau_{\rm f}-\frac{1}{2}.
\pi_*\{L_j L_{\bar k}\textbf u_j,\textbf u_k\}= \partial_{t^j}\pi_*\{L_{\bar k}\textbf u_j,\textbf u_k\}-\pi_*\{L_{\bar k}\textbf u_j,L_{\bar j}\textbf u_k\}\equiv 0,
D^k\phi(Z)=\sum_{q\geq 2\vee k}\frac{|b_{q}|}{(q-k)!}H_{q-k}(Z)D^kL\phi(Z)=-\sum_{q\geq 2\vee k}q\frac{|b_{q}|}{(q-k)!}H_{q-k}(Z)
\lim_{\lambda \to \infty}\frac{P_{\lambda p(p+1)+\rho+1,k}(z)}{P_{\lambda p(p+1)+\rho,k}(z)}=\widetilde{F}_{k}^{(\rho)}(z),z\in\mathbb{C}\setminus[a_{k},b_{k}], k=0,\ldots,p-1,
Y_n=\begin{bmatrix}y x_{-n+1}x_{-n+2}\dots x_{-1}x_{n-1}y x_{-n+1}\dots x_{-2}x_{n-2}x_{n-1}y \dots x_{-3}\vdots \vdots \vdots \ddots \vdots x_1 x_2 x_3 \dots y\end{bmatrix}
I'[u]v =\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}dxdy &\quad\quad+ \int_{\mathbb{R}^N}V(x)|u|^{p-2}u v dx-\frac{1}{p_{r;s}^{\uparrow *}}\int_{\mathbb{R}^N}(K\ast g(u))g'(u)v dx -\varepsilon_W\int_{\mathbb{R}^N}W(x)f'(u)v dx.
\left(P^{\mu_1\nu_1\,\mu_2\nu_2}\right)^{-1}=d_A{\cal T}_{A\;{\mu_1\nu_1\,\mu_2\nu_2}}+d_B{\cal T}_{A\;{\mu_1\nu_1\,\mu_2\nu_2}}+d_C{\cal T}_{A\;{\mu_1\nu_1\,\mu_2\nu_2}}+\sum_{i=4}^{14}d_i T_{i\;{\mu_1\nu_1\,\mu_2\nu_2}}
\Psi( y, \pi)= \begin{pmatrix}\phi \left( \begin{pmatrix}y_1 \vdots y_r \end{pmatrix}, \frac{y_{r+1}- \pi_{1}y_r - \ldots - \pi_{r}y_1}{\sigma}\right) \begin{pmatrix}y_1 \vdots y_r \end{pmatrix}\chi \left( \left( \frac{y_{r+1}- \pi_{1}y_r - \ldots - \pi_{r}y_1}{\sigma}\right)^2 \right) \end{pmatrix}.
\mathcal{E}(\hat{g}) =\lim_{\rho\rightarrow\infty}\int_{S_{\rho}}\left(\partial_{caa}\hat{g}_{ii}(z) - \partial_{cij}\hat{g}_{ij}(z) \right)\frac{z^c}{\rho}d\omega_{\rho}=\omega_{n-1}8\gamma_n\alpha (n-1)(n-2).
L:=\frac{1}{2}\mu(\frac{\partial w}{\partial t})^{2}-\frac{1}{2}EI(\frac{\partial^{2}w}{\partial x^{2}})^{2}+q(x)w(x,t)=\frac{\mu}{2}\dot{w}^{2}-\frac{EI}{2}w_{xx}^{2}+qw\equiv L(x,t,w,\dot{w},w_{xx})
\Gamma_{WZ}[\widetilde{\phi}_1 \widetilde{\phi}_2]\ =\ \Gamma_{WZ}[\widetilde{\phi}_1] + \Gamma_{WZ}[\widetilde{\phi}_2] -\frac{i}{4\pi}\int_\Sigma Tr(\phi^{-1}_1d\phi_1 \,\phi_2d\phi^{-1}_2)\ .
\psi_{\rho_i}(p^{\dag}_i\mathcal{M}\otimes \rho_i^{+}N) =\psi_{f p_i}(p^{\dag}_i\mathcal{M}\otimes (fp_i)^{+}N) &\simeq \psi_{f}(p_{i+}(p^{\dag}_i\mathcal{M}\otimes p_i^+ f^{+}N)) &\simeq \psi_{f}(p_{i+}p^{\dag}_i\mathcal{M}\otimes f^{+}N) &\simeq 0
\partial_{a_2}C_2 = -\tfrac{1}{6}, \tfrac{6}{5}d(a_2,\gamma) \partial_{a_2}C_1= 2 \epsilon \big(C_2 + a_2\big), \tfrac{6}{5}d(a_2,\gamma) \partial_{a_2}C_0 = \epsilon C_1.
\Phi_{c}( p , J ) = \frac{1}{i}( 2 \pi )^{\, 4}\delta_{J ( - p )}\, W_{c}^{\, \alpha , \, \alpha_{0}}( J ) = \tilde{C}_{\alpha}^{\, \alpha_{0}}( p ) \, \Bigl \{\, ( 2 \pi )^{\, 4}\, \delta_{\Phi ( - p )}\, i \, L^{\, \alpha , \, \alpha_{0}}( \Phi ) \, \vert_{\, \Phi = i \tilde{C}_{\alpha}^{\, \alpha_{0}}\! J}\, + i \, J ( p ) \, \Bigr \}\ .
\tilde{\xi}(x\otimes y- \varepsilon(x,y) y\otimes x)= \tilde{\xi}(x\odot y- \varepsilon(x,y) y\odot x) = \tilde{\xi}(x)\cdot_{A}\tilde{\xi}(y) -\varepsilon(x,y) \tilde{\xi}(y)\cdot_{A}\tilde{\xi}(x) = \xi(x)\cdot_{A}\xi(y) -\varepsilon(x,y)\xi(y) \cdot_{A}\xi(x) =[\xi(x),\xi(y)]_{A}= \xi([x,y]_{A})=\tilde{\xi}([x,y]_{A}).
\big|E[\textit{S}^{*}(t)\textit{I}_{1}(t)]\big|^2 = \Big|E\big[(\phi_{1,k}-1) \sigma_{\hat{h}_{1,k}}^4\big] (M^2 + M) (1-\lambda) \Big|^2 & = \frac{(1-\lambda)^2 \beta_{1,k}^4 \big|E[\textbf{p}_{1,k}\textbf{s}_{1,k}^H]\big|^2}{\lambda^2}\cdot\frac{M^4}{\tau^2}+ \mathcal{O}\Big(\frac{M^4}{\tau^4}\Big) + \mathcal{O}\Big(\frac{M^2}{\tau^2}\Big).
\frac{1}{2}\frac{d}{dt}\big(\|v\|_{L^{2}}^{2}+\|w\|_{L^{2}}^{2}\big)+\|\nabla v\|_{L^{2}}^{2}+\|\nabla w\|_{L^{2}}^{2}+\int_{\mathbb{R}^{3}}(v+w)(v-w)^{2}dx=0.
( \bar{M}_{m})_{j \, k}= i g{\frac{( 1 - \delta_{j \, k})}{\sin^{2}[ ( j - k ) \pi / ( r + 1 ) ]}}- i g \delta_{j \, k}\sum_{l \ne j}{\frac{\cos [ ( j - l ) \pi / ( r + 1 ) ]}{\sin^{2}[ ( j - l ) \pi / ( r + 1 ) ]}}.
\omega_{(s_i,r_j)}=\frac{1}{T_{g,(s_i,r_j)}^{glm}}=\sqrt{\frac{A_{s_i,r_j}E_{r_j}+B_{s_i,r_j}E_{s_i}}{E_{s_i}^2 E_{r_j}p_{th}^{s_i}}}.
\{y_{i,i+1}y_{i+1,i+2}y_{i,i+1}-y_{i+1,i+2}y_{i,i+1}y_{i+1,i+2},\, i=1,\ldots,n-2 \}, \big\{T_{i,i+1}^2-\big(q-q^{-1}\big) T_{i,i+1}-1 \big\},
I_{\mu_{1}\cdots \mu_{n}}( x_{1}, \ldots , x_{N}) \equiv \int \frac{\mathrm{d}^{4}k}{( 2 \pi )^{4}}\frac{k_{\mu_{1}}\cdots k_{\mu_{n}}}{\prod_{i = 1}^{N}( k^{2}+ x_{i}) ( q^{2}+ x_{i})}.
F_{m}(x) \equiv \prod_{n=0}^{\infty}\left(1+x^{2^{n}}\right)^{m}\equiv \left(\prod_{n=0}^{\infty}\left(1+x^{2^{n}}\right)\right)^{m}\equiv \frac{1}{(1-x)^{m}}\equiv \sum_{n=0}^{\infty}{n+m-1\choose m-1}x^{n}\pmod{2}.
\lim_{k\to +\infty}\int_{\mathbb{R}^{N}}\big[\mathrm{H}_{u}(x, \zeta_{\lambda_{k}}\mathfrak{e}_1, \zeta_{\lambda_{k}}\mathfrak{e}_2 )\mathfrak{e}_1+\mathrm{H}_{v}(x, \zeta_{\lambda_{k}}\mathfrak{e}_1, \zeta_{\lambda_{k}}\mathfrak{e}_2 )\mathfrak{e}_2\big]\,dx=
P_{\mu}^{\varepsilon}\: P_{\mu^{\prime}}^{\varepsilon}\; = \; \frac{1}{4}\left( X \: t_{\mu}^{\varepsilon}\: t_{\mu^{\prime}}^{\varepsilon}\: X^{*}+ X \: t_{\mu}^{\varepsilon}\: X \: t_{\mu^{\prime}}^{\varepsilon}+ t_{\mu}^{\varepsilon}\: X^{*}\: t_{\mu^{\prime}}^{\varepsilon}\: X^{*}\right) .
\langle0|0\rangle_{S}=exp[\frac{i}{2}\int\frac{dp^{4}}{(2\pi)^{4}}S^{\mu_{1}\cdot\cdot\cdot\mu_{l}}(-p)\frac{\Pi_{\mu_{1}\cdot\cdot\cdot\mu_{l}\nu_{1}\cdot\cdot\cdot\nu_{l}}(p)}{p_{\sigma}p^{\sigma}-m^{2}+i\epsilon}S^{\nu_{1}\cdot\cdot\cdot\nu_{l}}(p)]
\frac{\partial^{2}f_{\omega}(x_{i})}{\partial^{2}x_{i}}=2\alpha(1+\delta) + \omega_{i}rh^{\beta}\frac{\partial^{2}\eta(x_{i}h^{-1})}{\partial^{2}x_{i}}\geq 2\alpha -\frac{\alpha}{\sqrt{d}}\geq \alpha.
{\mathcal{F}}_{\mathcal{A}}(J ,J'_{\sigma,k}) = \frac{{\mathcal{A}}_{j'_{\sigma(k)}}^{J\setminus J'_{\sigma,k}}}{x_{1}-x_{j'_{\sigma(k)}}-1}-\frac{\tilde{\mathcal{D}}_{j'_{\sigma(k)}}^{J\setminus J'_{\sigma,k}}}{x_{1}+x_{j'_{\sigma(k)}}}\,,
\sum_{m=0}^{2r}D[f_{2l+ml}]-\sum_{m=0}^{2r}D[f_{ml}]= 4A \sum_{m=0}^{2r}\frac{d}{d \tau}\left\{K_1 \sqrt{\dot x^2_{+ml}(0)}+K_2 \sqrt{x^2_{l+ml}(l)}\right\}.
w_* = \begin{pmatrix}\cos \theta_* \sin \theta_* \cos \varphi_* \sin \theta_* \sin \varphi_*\end{pmatrix},n_* = \begin{pmatrix}-\sin \theta_* \cos \theta_* \cos \varphi_* \cos \theta_* \sin \varphi_*\end{pmatrix}, p_* = \begin{pmatrix}0 - \sin \varphi_* \cos \varphi_*\end{pmatrix},
T^{(n)}{}_{\mu\nu}= G^{(n)}{}_{\mu}{}^{\rho_{1}\cdots\rho_{n-1}}G^{(n)}{}_{\nu\rho_{1}\cdots\rho_{n-1}}-{\textstyle\frac{1}{2n}}g_{\mu\nu}G^{(n)\, 2}\, ,
a_{ii}^{-1/2}a_{jj}^{-1/2}A^{1/2}e_i \cdot A^{1/2}e_j = a_{ii}^{-1/2}a_{jj}^{-1/2}\left(e_i\cdot A^{1/2}A^{1/2}e_j\right) = a_{ii}^{-1/2}a_{jj}^{-1/2}\left(e_i\cdot Ae_j\right) = a_{ii}^{-1/2}a_{jj}^{-1/2}a_{ij}.
d_{2^{n-1-k}-(2l+1)}^{2^k}\equiv d_{2l+1}^{2^k}\pmod{2}\longleftrightarrow &\longleftrightarrow q(k,2l+1)^{2^k}=d_{2l+1}^{-2^k}d_{2^{n-1-k}-(2l+1)}^{2^k}\in E d_{2^{n-1-k}-(2l+1)}^{2^{k-1}}\not\equiv d_{2l+1}^{2^{k-1}}\pmod{2}
\hat{\varphi}_{s+t}(x,\omega) = \tilde{\phi}_{0,s+t}(x,\omega) = \tilde{\phi}_{s,s+t}( \tilde{\phi}_{0,s}(x,\omega), \omega) = \tilde{\phi}_{0,t}( \tilde{\phi}_{0,s}(x,\omega), \theta_s\omega) = \hat{\varphi}_{t}(\hat{\varphi}_{s}(x,\omega),\theta_s\omega)
\frac{\|z_k\|_{\pi,n}}{\|z_k\|_{ah,2n}}>\frac{i_1^{2n}\sum_{j=2}^{k+1}{i_j^{2n}}}{i_1^{4n}k+i_{k+1}^{4n}}> \frac{i_1^{-2n}i_2^{2n}}{1+k^{-1}i_1^{-4n}i_{k+1}^{4n}}\to \infty ,
\langle H_y\rangle_{R_y}\geq |\kappa|^2 \|\Phi_{\#}^y\|_{\#}^2 + \|R^{\#}_y\|^2_* + \alpha \Big(C_3 - C\sum_{j=2}^6 \varepsilon_j \Big)\|R^{\#}_y\|^2 +(1- C\alpha) \|P R^{\#}_y\|^2+ O(\alpha^4 \log(\alpha^{-1})).
R_{(1,11)}(n) = 12\sigma(n) - 36\sigma(\frac{n}{3}) + 12\sigma(\frac{n}{11}) -36\sigma(\frac{n}{33}) + 144\, W_{(1,11)}(n) + 1296\, W_{(1,11)}(\frac{n}{3}) - 432\, \biggl( W_{(3,11)}(n) + W_{(1,33)}(n) \biggr).
f_*:=\inf f([\mathfrak{s},\infty)\times [\delta_1,u^{**}])>0, \, \, \partial_uf(s,u)>\frac{{\rm d}f_+^{\infty}(0)}{{\rm d}u}-\frac{2\gamma}{3},\, \, \forall (s,u)\in [\mathfrak{s},\infty)\times [0,\delta_1].
\liminf_{n \to \infty}\frac{\sum_{k=1}^n s_n a_n}{\sum_{k=1}^n s_n}\leq \liminf_{n \to \infty}\frac{\sum_{k=1}^n w_n a_n}{\sum_{k=1}^n w_n}&\leq \limsup_{n \to \infty}\frac{\sum_{k=1}^n w_n a_n}{\sum_{k=1}^n w_n}\leq \limsup_{n \to \infty}\frac{\sum_{k=1}^n s_n a_n}{\sum_{k=1}^n s_n}.
[ b_{{\bf{0}}}, b_{{\bf{0}}}^{\dagger}] = b_{{\bf{0}}}b_{{\bf{0}}}^{\dagger}- b_{{\bf{0}}}^{\dagger}b_{{\bf{0}}}= e x p ( - i \mathrm{~}X_{0}) \mathrm{~}N_{0}\mathrm{~}e x p ( i \mathrm{~}X_{0}) - N_{0}= 1
\sigma_{\ell - \ell_P}(E_o^P(w)) = (-1)^{\ell(w) - \ell_P(w)}\sigma(E^P_o(s))\sigma(E^P_{o\cdot s}(s^{-1}w)) & = (-1)^{\ell(w) - \ell_P(w)}\sigma(E^P_o(s)E^P_{o\cdot s}(s^{-1}w)) & = (-1)^{\ell(w) - \ell_P(w)}\sigma(E^P_o(w)).
\begin{array}{ll}\|w\|_{\mathcal{A}_{\ell}^{k}}+ \frac{1}{\sqrt{\beta + C_{\eta}}}\big\|w|_{y=0}\big\|_{A^{k}}begin{align*}12pt]\leq C(T,\lambda_p)\Big(\|w_0\|_{\mathcal{A}_{\ell}^{k}(\mathbb{R}_{+}^2,t=0)}+ \frac{1}{\max\{\sqrt{\beta -C_{\eta}},\sqrt{\delta}\}}\big\|w_0|_{y=0}\big\|_{A^k(\mathbb{R},t=0)}\Big),k\leq p,\end{array}
{\bf A}={\sum}_{\nu_1 =1}^{r_1}\ldots{\sum}^{r_d}_{{\nu_d}=1}\beta_{\nu_1, \ldots ,\nu_d}\,{\bf a}^{(1)}_{\nu_1}\otimes \ldots \otimes{\bf a}^{(d)}_{\nu_d},\ell=1,\ldots,d ,
{\widetilde{\gamma}_o}= \frac{{{P_S}G_\mathrm{M}^{S}G_\mathrm{M}L\left(r\right)}}{{\sum\nolimits_{i \in \Phi /o}\left(P_S G_i^{S}+P_A G_i^{A}\right) L\left(\left|X_i\right|\right) +{\sigma_o^2}}}.
\frac{1}{L}\sum_{l=1}^{L}I_{q,s^n}(\gamma_l) &\geq \frac{1}{L}\sum_{l=1}^{L}I_{{q'},s^n}(\gamma_l) - f(\delta) &\geq (1-\epsilon) \mu(I_{{q'},s^n})- f(\delta) &\geq (1-\epsilon) \mu(I_{q,s^n}) -(2-\epsilon) f(\delta)
\begin{aligned}\omega_6 = w(t_1w+t_2y)dx + w(t_0w+t_3x)dy-(t_0yw+t_1xw+(t_2+t_3)xy)dw, \omega_7 = w(t_1w+t_2z)dy + w(t_0w-t_2y+t_3z)dz-(t_0zw+t_1yw+t_3z^2)dw.\end{aligned}
D(-\lambda)=W(\hat c_0,c_0)+\sum_{n=1}^N\lambda^n \sum_{1\leq i_1<i_2<\dots i_n \leq N}m_{i_1}m_{i_2}\dots m_{i_n}(x_{i_n}-x_{i_{n-1}})(x_{i_{n-1}}-x_{i_{n-2}})\dots (x_{i_2}-x_{i_1}) c_0(x_{i_1})\hat c_0(x_{i_n}),
\int_{B_\rho(\zeta_i)}U_i^4\,U_j\,U_m =\frac{2}{5}\,a_3\,\mu_i\,\mu_j\,\mu_m\,G_\lambda(\zeta_i,\zeta_j)\,G_\lambda(\zeta_i,\zeta_m)+\mathcal{R}_{i,j,m}^5+\mathcal{R}_{i,j,m}^6.
\ln \frac{f(x)}{f(x+1)}+\ln\frac{\Phi_0(x)+\mathrm{MC}_2(x)}{\Phi_0(x+1)+\mathrm{MC}_2(x+1)}= \frac{\frac{9}{512}- \frac{\kappa_1}{8}}{x^5}+\frac{5 (-27 + 176 \kappa_1 + 64 \kappa_1 \lambda_1)}{2048 x^6}+O\left(\frac{1}{x^7}\right).
\left( \begin{matrix}{{X^{\Lambda}}}\\{{-{\frac{i}{2}}F_{\Lambda}}}\\ \end{matrix}\right) \Longrightarrow \left( \begin{matrix}{{X^{0}}}\\{{- i X^{1}}}\\ \end{matrix}\right) , ~ ~ \left( \begin{matrix}{{X^{1}}}\\{{- i X^{0}}}\\ \end{matrix}\right) .
\left[ \left({\frac{H_{1}^{\prime}}{H_{1}}}+{\frac{H_{2}^{\prime}}{H_{2}}}\right) +{\frac{2}{r}}\right] ( h_{1}^{\prime}+ h_{2}^{\prime}) = - 4 \left({\frac{H_{1}^{\prime}}{H_{1}}}-{\frac{H_{2}^{\prime}}{H_{2}}}\right) \varphi^{\prime},
\int_{{\cal M}_{g , N}}\! \! [ d m ] [ d z ] \langle \prod_{i = 1}^{N}\: \oint_{C_{z_{i}}}\! \! G^{-}\bar{G}^{-}\phi_{i}( z_{i}) \prod_{a ,{\bar{a}}= 1}^{3 g - 3}\: G^{-}( \chi_{a}) \bar{G}^{-}( \bar{\chi}_{\bar{a}}) \rangle_{\Sigma_{g , N}}\label 2
\inf \{S_\omega ({\bf u}) \ |\{\bf u}\neq0, \, V({\bf u}) = 0\}=\inf \{S_\omega ({\bf u}) \ |\{\bf u}\neq0, \, V({\bf u}) = 0,\, H({\bf u}) = \tfrac{d}{2(1-s_c)}\omega M({\bf u})\}.
\frac{\partial^su(x_i,y_j,t)}{\partial x^s}\cong\sum\limits_{m=0}^{M_x}{a_{im}^{(s)}u(x_m,y_j,t)}, &\frac{\partial^su(x_i,y_j,t)}{\partial y^s}\cong\sum\limits_{m=0}^{M_y}{b_{jm}^{(s)}u(x_i,y_m,t)},
\int_{\Omega}\frac{b(x)u\Delta u}{M(x,|\nabla u|_{2}^{2})}dx=-\int_{\Omega}\frac{|\nabla u|^{2}}{c+|\nabla u|_{2}^{2}}dx+\frac{1}{2}\int_{\Omega}u^{2}\left\{-div\left[\frac{\nabla c}{(c+|\nabla u|_{2}^{2})^{2}}\right]\right\}dx.
(217t^6+920t^5+972t^4+1436t^3+205t^2-172t+6)(-t+1)^2 > (217t^6+920t^5+972t^4+1436t^3+205t^2-172t+6)(-t+1)^2\biggr\rvert_{t=0.6263716633 - 10^{-10}}> 81.3892822256
\phi ( x ) \star \chi ( x ) ={\frac{1}{( 2 \pi )^{4}}}\int d^{4}p \d^{4}p^{\prime}\widetilde{\phi}_{\kappa}( p ) \widetilde{\chi}_{\kappa}( p^{\prime}) e^{i \Delta^{( 2 )}( p , p^{\prime}){x}}
\Delta z\approx(\begin{matrix}\frac{\partial z}{\partial x_{1}}&\frac{\partial z}{\partial x_{2}}&\frac{\partial z}{\partial x_{3}}&\cdot\cdot\cdot&\frac{\partial z}{\partial x_{p}}\end{matrix})(\begin{matrix}\Delta x_{1}\\ \Delta x_{2}\\ \Delta x_{3}\\ \vdots\\ \Delta x_{p}\end{matrix})Eq(8)
\left\|r_1^{\frac{1}{2}}\right\| \leq \int_0^{k}\left\|u_{tt}(\zeta)\right\| d\zeta, \left\|r_1^{n-\frac{1}{2}}\right\|\leq \frac{k}{2}\int_{t_{n-1}}^{t_{n}}\left\|u_{ttt}(\zeta)\right\| d\zeta, n\geq 2.
d\Sigma =\sqrt{-\det (h_{ij})}d\tau^0\wedge d\tau^1 = \sqrt{-\det \left( (g_{\mu\nu}\circ \iota) \frac{\partial x^\mu}{\partial \tau^i}\frac{\partial x^\nu}{\partial \tau^j}\right)}d\tau^0\wedge d\tau^1,
f(x)={\small \gamma \lambda \beta e}^{(x/\alpha )^{\beta}}(\frac{{\small x}}{{\small \alpha}})^{\beta -1}e^{-\lambda \alpha (e^{(x/\alpha )^{\beta}}-1)}\left[ 1-e^{-\lambda \alpha (e^{(x/\alpha )^{\beta}}-1)}\right]^{\gamma -1},x\geq 0.
q(c(z'))= q(c(z_1...z_{n-2})c(z_{n-1}'z_n'z_{n+1}') = q(c(z_1...z_{n-2})c_ac_bc_b) = q(c(z_1...z_{n-2}))ab^2 = q(c(z_1...z_{n-2}))ba = q(c(z))