harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
1+\frac{(1-r^2)^2}{|1-r^2w_1|^{2}}-\frac{1-r^2}{1-r^2w_1}w_1 -\frac{1-r^2}{1-r^2 \bar w_1}\bar w_1 =\left|1-\frac{(1-r^2) w_1}{1-r^2w_1}\right|^2 =\frac{|1-w_1|^2}{|1-r^2w_1|^{2}},
g(w\otimes(v\otimes v-q(v))) =\overline{w}g(v\otimes v-q(v))+g(w)(\overline{v\otimes v-q(v)}) &=\overline{w}g(v\otimes v-q(v))+0=\overline{w}(g(v\otimes v)-g(q(v))) &=\overline{w}g(v\otimes v)=\overline{w}(\overline{v}f(v)+f(v)\overline{v})=0,
Q_1(V,W) = \int_U \sum_{i,k,l}\left[ \frac12 \Gamma^i_{\: \: il}\partial_k (v^k w^l)+ \frac12 \Gamma^i_{\: \: ik}\partial_l (v^k w^l)- \Gamma^i_{\: \: kl}\partial_i (v^k w^l) \right] dx^1 \ldots dx^n.
y_{l}=\sum\limits_{k=(l-1)2\beta+1}^{(l-1)2\beta+\beta}(-x_{k}^2)+\sum\limits_{k=(l-1)2\beta+1}^{(l-1)2\beta+\beta}[-x_{k}(\xi_{k}+\xi_{k+\beta})] & +\sum\limits_{k=(l-1)2\beta+1}^{(l-1)2\beta+\beta}\xi_{k}\xi_{k+\beta}.
\frac{1}{2\pi i}\int_{-T_2}^{+T_2}\frac{e^{iEt}e^{-\alpha t^2}}{t-z}dt =\frac{1}{2\sqrt{\pi \alpha}}\int_{-\infty}^{+\infty}dw\; \exp[-(w-E)^2/(4\alpha)]F_{z,T_2}(w) =\frac{1}{2\sqrt{\pi \alpha}}\int_0^{+\infty}dw\; e^{iwz}\exp[-(w-E)^2/(4\alpha)] +R_1+R_2
\langle \mathcal{A}\phi, \phi \rangle = \int_0^{\infty}\left( \phi^{\prime}(y) \right)^2 \rho dy - (d-1) \int_0^{\infty}\frac{\phi(y)^2}{y^2}\rho dy \geq - \left(\alpha^2 - \left(d - 2\right) \alpha + (d-1) \right) \int_0^{\infty}\frac{\phi(y)^2}{y^2}\rho dy - \frac{\alpha}{2}\int_0^{\infty}\phi (y)^2 \rho dy.
e^{-F}= \int \prod_k D\mu[M_k]e^{-N^2 S[M]}= \int \prod_k D\mu[M_k] e^{-N^2 (S[M] - S_0[M])}e^{-N^2 S_0[M]}= e^{-F_0}\Big< e^{-N^2 (S - S_0)}\Big>_{S_0}\; .
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2 \frac{2^{\alpha}}{\sqrt{\pi}}(p-\frac{1}{2})! \times\int_{0}^{1}\xi^{\frac{\alpha}{2}}\sum_{s=0}^p\frac{p!}{s!(p-s)!}(-1)^s\frac{\xi^{s-\frac{1}{2}}}{(s-\frac{1}{2})!}\frac{(1-\xi)^{p-s-\frac{1}{2}}}{(p-s-\frac{1}{2})!}{\rm d}\xi
S_{N,M,1}+S_{N,M,2}=\sum_{k=1}^{M}\left(\frac{2^{n_{k}}}{2^{n_{1}}+\cdots+2^{n_{M}}}\right)^2\left(\mathcal{R}_{1}(2^{n_{k}})+2^{-n_{k}+1}\,(2\,\mathcal{R}_{1}(2^{n_{k}+1})-\mathcal{R}_{1}(2^{n_{k}})) \sum_{j=k+1}^{M}2^{n_{j}}\right)+O(\epsilon).
\mu(\epsilon):=\max_{\left\Vert q\right\Vert \le\epsilon}\left[\left(\intop_{0}^{1}\left\Vert \xi^{\prime}(sq)-\xi^{\prime}(0)\right\Vert \mathrm{d}s\right)^{2}+\left\Vert \nu(q)-\nu(0)\right\Vert^{2}\right]^{1/2}\to0,\quad\epsilon\to0,
\left\vert \sum\limits_{j=0}^{2}\varepsilon^{\frac{j}{2}}\left\vert \lambda\right\vert^{1-\frac{j}{2}}\xi^{j}\right\vert \leq C\left( 1+\varepsilon\xi^{2}\left\vert \lambda \right\vert^{-1}\right) .
\mu(A_m) \leq \sum_{\ell=1}^\infty\mu(C_{m, \ell}) \leq \sum_{\ell=1}^\infty 4^{\ell}\mu(B_{m, \ell}) \leq C_5 \sum_{\ell=1}^\infty 4^{\ell}e^{-c_1m \ell}\leq C_5 \sum_{\ell=1}^\infty e^{(2-c_1m) \ell}.
\int_{0}^{1}u^{\epsilon}_x\bar{v}_x+\int_{0}^{1}u^{\epsilon}_x\bar{v}+\int_{0}^{1}\rho^{\epsilon}_x\bar{v}+\lambda\int_{0}^{1}u^{\epsilon}\bar{v}=\int_{0}^{1}g\bar{v}.
W_{\pi}(g_{-2,1,v}) = q^{1-\frac{a(\chi)}{2}}\zeta_F(1)^{-1}\epsilon(\frac{1}{2},\chi^{-1}) \int_{-v^{-1}+\varpi^{1-a(\chi)}\mathcal{O}^{\times}}\chi(y_1)\chi^{-1}(y_1v+\varpi^{a(\chi)-1})\psi(y_1\varpi^{-1})d^{\times}y_1.
\varepsilon_{0}= \frac{E_{0}}{A L_{z}}= \frac{e B}{( 2 \pi )^{2}}\int_{- \infty}^{\infty}d p_{z}\left( \sqrt{m^{2}+ p_{z}^{2}}- 2 \sum_{n = 0}^{\infty}\sqrt{m^{2}+ p_{z}^{2}+ 2 e B n}\right) .
r(a, c(bc)) = r((ac)b,c)r(ac,b)r(a,c), r((ac)b,c) s(ac,b) = s(a,c(bc)) s(c,bc) r(b,c), s(c,a(cb)) r(a,cb) = r((ca)c,b)r(a,c)s(c,a) s((ca)c,b) = s(c,a(cb))s(a,cb)s(c,b)
\frac{(\omega_0+\sqrt{-1}\partial \bar{\partial}\phi_{\epsilon})^n}{\omega_0^n}=\frac{\exp(\dot{\phi_{\epsilon}}-F_0-\gamma \phi_{\epsilon}-\theta_X-X(\phi_{\epsilon}))}{\prod_{i=1}^d(\epsilon^2+|s_i|_{H_i}^2)^{(1-\beta)\tau_i}},
\langle E_{e f f}( x_{0}, x_{1}) \rangle \; = \; \frac{1}{6}\left[ \frac{\partial^{2}}{\partial x_{0}^{2}}\frac{\partial^{2}}{\partial y_{1}^{2}}- \frac{\partial^{2}}{\partial x_{0}\partial x_{1}}\frac{\partial^{2}}{\partial y_{0}\partial y_{1}}\right] \Delta_{\varphi}( x_{0}, x_{1}; y_{0}, y_{1}) |_{y_{\mu}\to x_{\mu}}\; ,
\Phi_g^X = \sum_{k_0, k_{r+1}, \ldots, k_m}\sum_{\beta \in H_2(X)}\langle\gamma_0^{k_0}, \gamma_{r+1}^{k_{r+1}}, \ldots, \gamma_m^{k_m}\rangle_{g,\beta}^X \frac{y_0^{k_0}}{k_0!}\frac{y_{r+1}^{k_{r+1}}}{k_{r+1}!}\cdots \frac{y_m^{k_m}}{k_m!}q^\beta\prod_{i=1}^r e^{y_i \int_\beta \gamma_i}
I_0=\{0,\dots,q\},\a_0=0, &I_1=\{q+1,\dots,(q+1)+q\},\a_1=q+1, &\dots &I_{l-1}=\{(l-1)(q+1),\dots,(l-1)(q+1)+q\},\a_{l-1}=(l-1)(q+1), &I_l=\{l(q+1),\dots,l(q+1)+(q-1)\},\a_l=l(q+1), &\dots &I_{p-1}=\{l(q+1)+(p-1-l)q,\dots,l(q+1)+(p-1-l)q+(q-1)\},\a_{p-1}=l(q+1)+(p-1-l)q.
\begin{pmatrix}K_3 K_4\end{pmatrix}=\lim_{x_1\to 0}\sqrt{2\pi x_1}\mathbf{t}^{(+)}(x_1) = G\sqrt{\frac{2}{\pi a}}\frac{1}{H_{33}H_{44}-H_{34}^2}\begin{pmatrix}(\gamma-\delta_4)H_{34}H_{44}\delta_4 H_{33}H_{44}-\gamma H_{34}^2\end{pmatrix}.
N_j\, =\sum_{kp+l=s_j}\binom nk\binom{-n}l(-1)^{k+l}\cr =\sum_{kp+l=s_j}(-1)^k\binom nk\binom{n+l-1}l\cr =\sum_k(-1)^k\binom nk\binom{n+s_j-kp-1}{n-1}.
h_1(i\alpha P(z_2)-P(z_2),z_2) = a_{m_0,n_0}(i\alpha-1)^{m_0}(P(z_2))^{m_0}\big(z_2^{n_0}+o(|z_2|^{n_0}\big), h_2(i\alpha P(z_2)-P(z_2),z_2) = b_{k_0,l_0}(i\alpha-1)^{k_0}(P(z_2))^{k_0}\big(z_2^{l_0}+o(|z_2|^{l_0}\big),
H_{(F_1,D_2+D_3+D_4)}(y_1,y_2)=e^{\frac{P\log y_1+ H\log y_2}{2\pi i}}\sum_{d_1,d_2\geq 0}y_1^{d_1}y_2^{d_2}\frac{1}{\Gamma(1+\frac{H}{2\pi i}+d_2)}[\textbf{1}]_{d_2-d_1,d_1,d_1}.
\left|PV\int_{B_{\delta(x)/2}(x)}\frac{\phi(x)-\phi(y)}{|x-y|^{N+2s}}\;dy\right|\ \leq\ \|\phi\|_{C^{2s+\alpha}(B_{\delta(x)/2}(x))}\int_{B_{\delta(x)/2}(x)}\frac{dy}{|x-y|^{N-\alpha}}\ & =\ \|\phi\|_{C^{2s+\alpha}(B_{\delta(x)/2}(x))}\frac{\omega_{N-1}}{\alpha}\left(\frac{\delta(x)}{2}\right)^\alpha
[ ( 3 , -{\frac{1}{4}}) , ( 2 , -{\frac{1}{2}}) ]_{( - 1 / 3 , 1 / 4 , ( 1 / 6 , - 5 / 6 )}~ ~ ; ~ ~ [ ( \bar{3},{\frac{1}{4}}) , ( 2 ,{\frac{1}{2}}) ]_{( 1 / 3 , - 1 / 4 , ( - 1 / 6 , 5 / 6 )}\ \ ,
\langle 0 \mid \left[{\alpha}_{m - n}{\alpha}_{n},{\alpha}_{- m - s}{\alpha}_{s}\right] \mid 0 \rangle = n ( m - n ) \left({\varepsilon}_{m - n}-{\varepsilon}_{- n}\right) \left({\delta}_{n - m - s}+{\delta}_{n + s}\right)
e^{it\sqrt{\mathcal{L}_V}}u_0 =\int_0^\infty(s\rho)^{-\frac{n-2}2}J_{\nu_0}(s\rho)e^{it\rho}(\mathcal{H}_{\nu_0}u_0)(\rho)\rho^{n-1}d\rho &=\int_0^\infty(s\rho)^{-\frac{n-2}2}J_{\nu_0}(s\rho)e^{it\rho}\chi(\rho)\rho^{n-1}d\rho.
-v_1z^{3n^2+n}= v_1z^{3n^2-n}(z^t-z^nx^n)-z^{3n^2}f_4+(v_1z^{3n})^n-\left(\left(\sum_{j=2}^{d-3}(-1)^{j}v_iz^{3n}\right)+(-1)^{d}(xz^{3n})\right)^n,
\int d^4 y\frac{\delta}{\delta{\cal O}(y)}\int d^4 x\,\underbrace{{\cal O}(x)\star{\cal O}(x)\star\ldots \star{\cal O}(x)}_{n \rm \; factors}= n \int d^4 x\,\underbrace{{\cal O}(x)\star{\cal O}(x)\star\ldots \star{\cal O}(x)}_{n-1 \rm \; factors}.
| u^a_0 (p, t) -\phi^a (p) | = \left| \int_M H(p, q, t) \big( \phi^a (q) - \phi^a (p) \big) dV_q \right| \leq \tilde{C}_\beta t^\beta ( || \nabla_H \phi^a ||_{C^0}+ || \phi^a ||_{C^0})
- \; \frac{1}{e}\; \frac{\lambda^2}{(x_1^2 + \lambda^2)^{\frac{3}{2}}}\;\left[ \mbox{arctan}\left( \frac{x_2}{\sqrt{x_1^2 + \lambda^2}}\right)\; - \; \pi\Big(\frac{1}{2}+ m\Big) \right] \; .
f(\mathcal{G}_{\epsilon}(x)) \leq f(x) + \langle \nabla f(x), \mathcal{G}_{\epsilon}(x)- x\rangle + \frac{1}{2\epsilon}\| \mathcal{G}_{\epsilon}(x) - x\|^2 &\overset{\eqref{eq:one}}{=}f(x) - \frac{\epsilon}{2}\|\nabla f(x)\|^2.
{\cal L}_{cs}=\varepsilon^{\mu \nu \kappa}\Gamma_{\kappa \lambda}^{\,\,\,\,\,\,\,\,\rho}(\partial_\mu \Gamma_{\rho \nu}^{\,\,\,\,\,\,\lambda}+\frac 23\Gamma_{\mu \sigma}^{\,\,\,\,\,\,\,\lambda}\Gamma_{\nu \rho}^{\,\,\,\,\,\,\sigma})
\phi(x_j)= \mu_j x'_j, \phi(y_j)= \nu_j y'_j, \mbox{if}\; \; \tau_j =1, &\phi(x_j)= \mu_j y'_j, \phi(y_j)= \nu_j x'_j, \mbox{if}\; \; \tau_j=-1.
\lim_{\lambda\rightarrow\infty}a_{\lambda p(p+1)+\rho}=a^{(\rho)}=(z-\widetilde{F}_{0}^{(\rho)}(z)) \prod_{i=\rho-p}^{\rho-1}\widetilde{F}_{0}^{(i)}(z),z\in\mathbb{C}\setminus[a_{0},b_{0}].
\mathcal{I}_3:=-\int_{\mathbb{R}^N\setminus B_R}\int_{B_R}\frac{\Big(J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big)}{|h|^{\gamma+t}}\,\delta_h u(y)\,\eta(y)^p\,d\mu(x,y),
\mathrm M_{t,n}\, :=\, \frac{\,1\,}{\,n\,}\sum_{i=1}^{n}\delta_{(X_{t,i}^{(u)}, X_{t,i+1}^{(u)})}\,, \mathrm m_{t, n}\, :=\, \frac{1}{\, n \,}\sum_{i=1}^{n}\delta_{X_{t,i}^{(u)}}\,, 0 \le t \le T\, ,
\sum_{(x_i-x_j) \parallel \; (1,0)}\big|\varphi_1(x_i) - \varphi_1(x_j)\big|^2 = \sum_{x_i, x_j}\big|\varphi_1(x_i) - \varphi_1(x_j)\big|^2 + \sum_{x_i, x_j}\big|\varphi_1(x_i) - \varphi_1(x_j)\big|^2 &= -2*8\begin{pmatrix}1 0 \end{pmatrix}\xi_2 \begin{pmatrix}1 0 \end{pmatrix},
\frac{d}{dx}A^*(x)=\frac{\partial_B H_A\;\partial_xH_B-\partial_xH_A\;\partial_BH_B}{\partial_AH_A\;\partial_BH_B-\partial_BH_A\;\partial_AH_B},\qquad\frac{d}{dx}B^*(x)=\frac{\partial_xH_A\;\partial_AH_B-\partial_AH_A\;\partial_xH_B}{\partial_AH_A\;\partial_BH_B-\partial_BH_A\;\partial_AH_B}.
\phi(g_1,\ldots,g_4) = \sum_{N_1\ldots N_4}\sum_\Lambda \phi_{\cal A}^{N_1\ldots N_4\,\Lambda}\sqrt{{\rm dim}_{N_1}\ldots{\rm dim}_{N_4}}\left({C}^{N_1\ldots N_4\,\Lambda}_{\beta_1\ldots\beta_4}D^{(N_1)}_{\beta_1\gamma_1}(g_1) \ldots D^{(N_4)}_{\beta_4\gamma_4}(g_4)\w_{\gamma_1}\ldots w_{\gamma_4}\ \right).
\widetilde{P}_{\pm}= \int_{0}^{x \pm}d y^{\pm}\, \widetilde{T}_{\pm \pm}( y^{\pm}) \qquad \qquad \widetilde{\Delta}_{\pm}= \int_{0}^{x \pm}d y^{\pm}\, y^{\pm}\, \widetilde{T}_{\pm \pm}( y^{\pm}) \ \ .
\big| B(x_1,x_3,x_1,x_3)+B(x_2,x_3,x_2,x_3)-2B(x_1,x_3,x_2,x_3) \big| = \big| A(x_1,x_3) \left( A(x_1,x_1)+A(x_2,x_2)-2A(x_1,x_2) \right)-2\left( A(x_1,x_3)-A(x_2,x_3) \right)^2 \big| \le C |x_1-x_2|^2,
\begin{aligned}\, \|\partial_t^j D^l_x D_v^{m+2}u\|_{L_{\infty}(Q_{1/2})}\leq N \delta^{-\theta}\|D_v^2 u - (D_v^2 u)_{Q_1}\|_{L_2 (Q_1)}+ N \delta^{-\theta}\sum_{k = 0}^{\infty}2^{-k}(|(-\Delta_x)^{1/3}u - ((-\Delta_x)^{1/3}u)_{Q_{1, 2^k}}|^2)_{Q_{1, 2^k}}^{1/2},\end{aligned}
\limsup_{n \to \infty}\|T_g(s_n)\|_{\infty}\geq \lim_{n \to \infty}\left(\tfrac{1}{2}\cos\left(\tfrac{\pi}{4}\right)e^{-C_1(\gamma,\beta)}\left(c - \tfrac{1}{n}\right) - \frac{e^{C_1(\gamma,\beta)}C_2(\beta)}{(1-r_{\beta}^2)^{\alpha}}r_{\beta}^{k_n}\right) &= \tfrac{1}{2}\cos\left(\tfrac{\pi}{4}\right)e^{-C_1(\gamma,\beta)}c > 0,
\limsup_{\delta \to 0}\limsup_{n \to \infty}\sup_{\tau \in \mathcal{T}^\delta}{{E}}\left[ \sum_{k=0}^\infty (k+2) \left|{\bar{B}}^n_k(\tau+\delta) -{\bar{B}}^n_k(\tau) \right| + \left|{\bar{\eta}}^n(\tau+\delta) -{\bar{\eta}}^n(\tau) \right| \right] = 0.
\begin{aligned}u\left(\xi(0)\right) = \int_0^{t}e^{-s}\left(C_p\left|\dot{\xi}(s)\right|^q + f\left(\xi(s)\right) \right) ds + e^{-t}u(\xi(t)) =\int_0^{t}e^{-s}\left(C_p\left|\dot{\xi}(s)\right|^q + f\left(\xi(s)\right) \right) ds +e^{-t}u(y)\end{aligned}
[\theta_,\theta_j] = [x_i+u_{ij},x_j+ u_{ji}]+ [x_i+x_j,u_{ij}] + \left [u_{ij},\sum_{k=1}^{n}x_k\right] \hphantom{[\theta_,\theta_j] =}{}+\sum_{k=1 \atop k \not= i,j}^{n}[u_{ij}+u_{ik},u_{jk}]+[u_{ik}, u_{ji}] +[x_i,u_{jk}]+[u_{ik},x_j]+ [x_k,u_{ij}].
f\left( re^{i\varphi}\right) =f\left( r\cos\varphi,r\sin\varphi\right) :={\displaystyle\sum_{k=0}^{\infty}}{\displaystyle\sum_{\ell=1}^{a_{k}}}f_{\left( k,\ell\right)}\left( r\right) Y_{\left( k,\ell\right)}\left( \varphi\right) .
{\cal L}_{\mathrm{e f f}}= \frac{N \tanh ( \beta v_{T}/ 2 )}{1 6 \pi v_{T}}v_{T}^{2}( \partial^{\mu}\theta - i \varphi^{*}\partial^{\mu}\varphi + 2 e A^{\mu})^{2}\, \, \, \, .
\|P^\prime(\cdot)\|_1=\|(1-B_s)^{-1}R^\prime(\cdot)\|_1 \leq \frac{1}{1-\|B_s\|_1}\|R^\prime(\cdot)\|_1 \leq 2C\rho^{\varepsilon_0}\langle s\rangle^{-\varepsilon_1}.
\sum_{n\geq1}F_2(\zeta_n)= \sum_{n\geq 1}\frac{F(\zeta_n,\log y+\zeta_n)}{R(\zeta_n)}+\sum_{n\geq1}\frac{F_1(\log y+\zeta_n)-F(\zeta_n,\log y+\zeta_n)}{R(\zeta_n)}.
\sum_{R\in G}^{6}\Gamma(R)_{11}^{*}\Gamma(R)_{22}=1^{2}+(1)(-1)+(-\frac{1}{2})(\frac{1}{2})+(-\frac{1}{2})(\frac{1}{2})+(-\frac{1}{2})^{2}+(-\frac{1}{2})^{2}=0.
\nu(n) := \mathbb{1}_{]-\infty, 0[}(n)\left(\displaystyle \sum_{k = 0}^{+\infty}\mathcal{U}(k)\,\rho(n - k)\right)+\mathbb{1}_{[0, +\infty[}(n)\left(\displaystyle \sum_{k = 0}^{+\infty}\mathcal{U}'(-k)\,\rho(n + k)\right).
[u_{ij},u_{ik}+u_{jk}]+[u_{ik},u_{jk}]+[z_i,u_{jk}]+[u_{ik},z_j]+[z_k,u_{jk}] \qquad{}=q_{ij}q_{jk}[\partial_{(ij)},\partial_{(jk)}]+q_{ik}[\partial_{(ik)},z_j]=0.
H^1(0,1) \ni (u,v)\mapsto \int_0^1 u'\overline{v'}\dx-\left(\begin{pmatrix}1 -1 -1 1 \end{pmatrix}\begin{pmatrix}u(1) u(0)\end{pmatrix}\mid \begin{pmatrix}v(1) v(0)\end{pmatrix}\right)\in \mathbb C\ ,
\bar{\Phi}_{\rm{dn}}|^{\tau_{i}}_{\tau_{i-1}}= \frac{\int_{\tau_{i-1}}^{\tau_{i}}\sum_{\beta=1}^{\kappa}\Phi_{\rm{st},\beta}(t) d t + \Omega_{\rm{m}}(\tau_{i})}{\tau_{i}-\tau_{i-1}},~i=1,\cdots,N_\tau.
\mathbb{E}[\mathcal{D}^{ij}(0) \mathcal{D}^{il}(-J,-1)] = \frac 12 \mathbb{E}[\mathcal{D}^{ij}(0) \mathcal{D}^{ij}(-J,-1)] = - \frac{p(\chi+k-2)}{2k(k-1)}.
\beta_k = \left( u^2 \right) \beta_k = \left( \sum_{i=1}^n \lambda_i \beta_i \beta_i^* \right) \left( \sum_{j=1}^n \lambda_j \beta_j \beta_j^* \right) \beta_k = \left( \sum_{i=1}^n \lambda_i^2 \beta_i \beta_i^* \right) \beta_k = \lambda_k^2 \beta_k.
\begin{aligned}C(X)\ddot{X}+Q(X)\ddot{A}= - Q'(X)\dot{X}\dot{A}-C'(X)\dot{X}^2+F(X)+K(X)A 2I(X)\ddot{X}+2C(X)\ddot{A}= - I'(X)\dot{X}^2-U'(X)+2F'(X)A &+K'(X)A^2+Q'(X)\dot{A}^2\end{aligned}
\begin{array}{ll}g_{\mu\nu}= \eta_{\mu\nu}+ h_{\mu\nu}, \Lambda(\phi) = \Lambda(\phi_0) + \Lambda'(\phi_0)\phi_{(1)}, T_{\mu \nu}= T_{(0)\mu\nu}+ T_{(1)\mu \nu}, \phi = \phi_0 + \phi_{(1)},\end{array}
\left (\begin{array}{c}M_1 M_2 M_3 M_4\end{array}\right )=\frac{1}{\sqrt{2}}\left (\begin{array}{crrr}1 1 -1 -1 1 -1 1 -1 1 1 1 1 1 -1 -1 1\end{array}\right )\left (\begin{array}{c}L_1 L_2 L_3 L_4\end{array}\right )
\begin{cases}\partial_t \psi - \Delta \psi + \div (\psi \otimes \psi) + \nabla \pi + \phi( \partial_t \eta + \Delta \eta) - 2 \div(\phi \otimes \nabla \eta) + (\psi \cdot \nabla \eta) \phi =0 \div \psi = - \nabla \eta \cdot \phi \, . \end{cases}
\bar{I}_{mod}[\mathbf{Q}] = \int_{\bar{\Omega}}D^3 \phi\left(\frac{|\bar{\nabla}\mathbf{Q}|}{D}\right) + \frac{D^3}{L}f_B\left(\mathbf{Q}\right) \bar{dV}= D \int_{\bar{\Omega}}\bar{\psi}\left(|\bar{\nabla}\mathbf{Q}|^2 \right) + \frac{1}{\bar{L}}\bar{f_B}\left(\mathbf{Q}\right) \bar{dV}
j^6_{\lambda}(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6) =b_\lambda(\xi_1) b_\lambda(\xi_2) b_\lambda(\xi_3)b_\lambda(\xi_4) b_\lambda(\xi_5) b_\lambda(\xi_6)+\tilde j^6_\lambda(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6),
X_j^{[r_2, t^*]}(t^* \wedge \tau) = \int_{r_2}^{t^* \wedge \tau}\mu X_{j-1}(u) e^{\int_u^{t^* \wedge \tau}G_j^*(v) \: dv}\: du + e^{\int_{r_2}^{t^* \wedge \tau}G_j(v) \: dv}Z_j^{[r_2, t^*]}(t^* \wedge \tau).
\psi ( x ) = Z^{a + \frac{1}{2}}\, \, \, r^{- ( l + 1 )}\, \, \, ( y - 1 )^{\frac{( \lambda - s )}{2}}\, \, \, ( y + 1 )^{\frac{- ( \lambda + s )}{2}}\, \, \, P_{n}^{( \lambda - s - \frac{1}{2}) , ( - \lambda - s - \frac{1}{2})}( y ) \, \, \, P_{k}( x )
\mathbf{V}_{N,\epsilon}^{\tilde{\Gamma}}(t,\,x) =\frac{1}{N^{2}}\sum_{\substack{1\le c_{1},\,c_{2}\le m i\in I_{c_{1}}^{N},\,j\in I_{c_{2}}^{N}}}\gamma_{c_{1},c_{2}}^{2}(t,\,x_{i})\left[\frac{1}{2\epsilon}\chi_{\epsilon}(x_{j}-x_{i})-\left(\delta^{+}(x_{j}-x_{i})+\delta^{+}(x_{i}-x_{j})\right)\right].
\mathbb{F}_{\geqslant \boldsymbol{u}}:= \left\{\x \in{\mathbb{F}}\ | \ \vartheta(x) \geqslant_{\boldsymbol{a}}\boldsymbol{u}\ \right\}\cup \left\{0 \right\}\mathbb{F}_{> \boldsymbol{u}}:= \left\{\x \in{\mathbb{F}}\ | \ \vartheta(x) >_{\boldsymbol{a}}\boldsymbol{u}\ \right\}\cup \left\{0 \right\}
\psi_{CP}= \psi_C \circ \psi_P : \left(\begin{array}{l}f_\pm^{q}f_\pm^{-q}\end{array}\right) (\theta, \phi)\mapsto \left(\begin{array}{l}(-1)^{(1 \pm 1) q}(f^{q}_\mp)^* (-1)^{(1 \mp 1) q}(f^{-q}_\mp)^* \end{array}\right) (\pi - \theta, \pi + \phi).
\mathbb E |N_{\infty}-N^n_{\infty}|^p \eqsim_{p}\mathbb E [N-N^n]_{\infty}^{\frac p2}=\mathbb E \bigl([N]^c_{\infty}+ [N]^q_{\infty}+ [N-N^n]^a_{\infty}\bigr)^{\frac{p}{2}}\geq \mathbb E \bigl([N]^c_{\infty}+ [N]^q_{\infty}\bigr)^{\frac{p}{2}}.
0 \in \begin{bmatrix}H\mathbf{x}+\mathbf{g}-A^T\mathbf{y}+ \partial_\mathbf{x}I_D(\mathbf{x},\mathbf{y}) +\rho (\mathbf{x}-\mathbf{x}_k) A\mathbf{x}-\mathbf{b}+\partial_\mathbf{y}I_D(\mathbf{x},\mathbf{y}) + \delta (\mathbf{y}-\mathbf{y}_k)\end{bmatrix}
{\cal F}(s)=\frac{\sqrt{\pi}}{\Gamma(s)}\sum_{\mu_k}\int^\infty_0 dt ~t^{s-3/2}\sum_{n\in Z\!\!\!Z}\exp \left [-\frac{\beta^2}{4\pi^2}(m^2+\mu^2_k) t -\frac{\pi^2 n^2}{t}-2\pi i \left(a+\frac{1}{2}\right) n \right].
\nabla\Delta_{\tilde{g}}\Phi =-c_n\nabla R_g + O_2(r^{-2\delta-3}), \tilde{g}(\nabla\Delta_{\tilde{g}}\Phi,\tilde{\nu}) =-c_n\tilde{g}(\nabla R_g,\nu_e) - c_n\underbrace{\tilde{g}(\nabla R_g,\tilde{\nu}-\nu_e)}_{O_1(r^{-2\delta-3})}+ O_2(r^{-2\delta-3}),
\frac{1}{n}\sum_{k\in N}e_{k,j+n-2}^n +\frac{1}{n}e^{n-1}_{i,j}+\frac{1}{n}e_{i,j+1}^{n-2}&+\left(1-\frac{1}{n}\right)\left[\frac{1}{n^2}e_{i,j}^{n-3}+\frac{1}{n}e_{i,j+1}^{n-3}\right]+\frac{1}{n}\sum_{\tau=1}^{n-4}e_{i,j+\tau+1}^{n-\tau-3}
S_1=\sqrt{\pi}\Gamma(\nu+1/2)\sum_{k=0}^\infty\frac{(\nu+1/2)_k}{k!}\bigg(\frac{\beta}{\alpha}\bigg)^{2k}=\frac{\sqrt{\pi}\Gamma(\nu+1/2)}{(1-\beta^2/\alpha^2)^{\nu+1/2}},
W_{\perp}(t)=\tilde{Z}(t) W_{\perp}(t_0) Z(t)^{-1}: (I-P(t))H \overset{Z(t)^{-1}}{\rightarrow}(I-P(t_0))H\overset{W_{\perp}(t_0)}{\rightarrow}(I-\tilde{P}(t_0))H\overset{\tilde{Z}(t)}{\rightarrow}(I-\tilde{P}(t))H,
\left\Vert f\right\Vert_{C_{\gamma ;\psi}\left[ a,b\right]}=\left\Vert\left( \psi \left( t\right) -\psi \left( a\right) \right)^{\gamma}f\left(t\right) \right\Vert_{C\left[ a,b\right]}=\underset{t\in \left[ a,b\right]}{\max}\left\vert \left( \psi \left( t\right) -\psi \left( a\right) \right)^{\gamma}f\left( t\right) \right\vert
\left| \frac{\partial x_k}{\partial x_0}\right| = \left| \lim \limits_{h \to 0}\frac{x_k(x_0 + h) - x_k(x_0)}{h}\right| &< \lim \limits_{h \to 0}\frac{c_n \cdot (3/5)^{k/2}|x_0 + h - x_0|}{|h|}&= c_n \cdot (3/5)^{k/2}.
a_{n}= \frac{( a_{n}+ 1 ) \cdot n}{n + 1}= a_{1}\cdot \frac{a_{2}}{a_{1}}\cdot m \cdot \frac{a_{n}}{a_{n - 1}.}= 1 \times \frac{2}{1}\times \frac{3}{2}\times m \frac{n}{n - 1}
E_{C}^{( 2 ) \; r e n}( R ) = \frac{1}{2 \pi}\int_{0}^{\infty}d \omega \ln \left( 1 - e^{- 2 R \sqrt{\omega^{2}+ \omega_{0}^{2}}}\right) = - \frac{\omega_{0}}{2 \pi}\sum_{n = 1}^{\infty}n^{- 1}K_{1}( 2 n \omega_{0}R ) ,
\rho^{Z,f\oplus 0}_t(S_c S_d) =\exp{(2\pi\sqrt{-1}t f)}S_c S_d =\rho^{A,f}_t(S_{a}), \rho^{Z,f\oplus 0}_t(S_d S_c) = S_d \exp{(2\pi\sqrt{-1}t f)}S_c = S_d \exp{(2\pi\sqrt{-1}t f)}S_d^* S_{b}.
\begin{aligned}\left| \int_{\lambda(A)f_n}^{\lambda(A)s_nf_n}\frac{\rho_B(u)}{u}\mathrm{d}u - \int_{f_n}^{s_nf_n}\frac{\rho_B(u)}{u}\mathrm{d}u \right| \leq \int_{\lambda(A)s_nf_n}^{s_nf_n}\frac{1}{u}\mathrm{d}u + \int_{\lambda(A)f_n}^{f_n}\frac{1}{u}\mathrm{d}u & = 2 \log\left( \frac{1}{\lambda(A)}\right),\end{aligned}
\begin{array}{lll}\mbox{\rm maximize}b^Ty \mbox{\rm subject to}U^T ( C - \sum^{m}_{i=1}y_i A_i ) U \in \mathbb{S}^{d}_{+}U^T ( C - \sum^{m}_{i=1}y_i A_i ) V = 0 & V^T ( C - \sum^{m}_{i=1}y_i A_i ) V = 0,\end{array}
{}_3\phi_2\!\left[\begin{matrix}(1-r)^j,b,c d,e\end{matrix};1-r,1-r\right] &=\frac{((1-r)/e;1-r)_j}{((1-r)bc/de;1-r)_j}\;{}_3\phi_2\!\left[\begin{matrix}(1-r)^j,d/b,d/c d,de/bc\end{matrix};1-r,1-r\right].
Q_{\mathrm{I}}= \frac{1}{\sqrt{2}}( e^{i \pi / 4}Q_{2}+ e^{- i \pi / 4}\overline{{Q}}_{\dot{2}}) , \qquad Q_{\mathrm{I I}}= \frac{1}{\sqrt{2}}( e^{- i \pi / 4}Q_{1}+ e^{i \pi / 4}\overline{{Q}}_{\dot{1}}) ,
N:= \Big\{0, 1, ..., i \Big\}, E:=\Big\{(1, 0), ... (i, 0) \Big\},L:N \to \{1, ..., d\}, h_0:= \{0\}\cup \Big\{j\in \{1, ..., i\}: a_j = 0\Big\}, H:= \Big\{\big\{j\in\{1, ..., i\}: a_j=k\big\}: k=1, ..., m[a]\Big\}.
\mathcal{A}\left(\begin{array}[c]{c}g_{+}g_{-}\end{array}\right) =\left(\begin{array}[c]{c}-\left( v\partial_{x}-\beta_{x}\partial_{v}\right) g_{+}+\partial_{x}\phi\partial_{v}\mu_{+}-\left( v\partial_{x}+\beta_{x}\partial_{v}\right) g_{-}-\partial_{x}\phi\partial_{v}\mu_{-}\end{array}\right) ,
\bar{x}_1 = \lambda_1 \hat{x}_1 + \lambda_2 \hat{x}_2 + \lambda_3 \hat{x}_4, \ \bar{x}_2 = \lambda_2 \hat{x}_3 + \lambda_3 \hat{x}_5 + \lambda_4 \hat{x}_7, \ \bar{x}_3 = \lambda_3 \hat{x}_6 + \lambda_4 \hat{x}_8 + \lambda_5 \hat{x}_9.
\sum_{k=1}^{n}(D_{k}+ D_{k+1})c'_k (H'_{k+1}-H'_{k}) &=\sum_{k=1}^{n}2 D_{k+1}c'_k (H'_{k+1}-H'_{k}) +(D_{k}-D_{k+1})c'_k (H'_{k+1}-H'_k).
JQ_{n}^{(3)}=\sum_{s=0}^{3}J_{n+s}^{(3)}e_{s}=J_{n}^{(3)}+\sum_{s=1}^{3}J_{n+s}^{(3)}e_{s},\ (J_{n}^{(3)}\textbf{1}=J_{n}^{(3)})
\mathcal{K}(z) = v_1 \mathcal{F}_1^{-1}g_z (|q_1|) \mathcal{F}_1\mathcal{F}_2^{-1}(p_2^2 +q_2^2 + z^2)^{-1}\chi_{(r_\varepsilon , R]}(|q_2|)\mathcal{F}_2 v_2
\left\{\Sigma_{\mu},\Sigma_{\nu}\right\}=\Sigma_{\lambda}C^{\lambda}\!_{\mu\nu},\left\{\widehat{\Sigma}_{\mu},\widehat{\Sigma}_{\nu}\right\}=-\widehat{\Sigma}_{\lambda}C^{\lambda}\!_{\mu\nu},\left\{\Sigma_{\mu},\widehat{\Sigma}_{\nu}\right\}=0,
F^{k}\!_{i}F^{l}\!_{j}\left(g^{ij}\left\{\begin{array}{c}a ak\end{array}\right\}-g^{jb}\left\{\begin{array}{c}i kb\end{array}\right\}\right)\partial_{l}+F^{l}\!_{i}g^{ij}(\partial_{k}F^{k}\!_{j})\partial_{l}=0\;,
J_{2}( m ( | z | + 1 / k ) ) \sim{\frac{m^{2}( | z | + 1 / k )^{2}}{8}}, \ \Y_{2}( m ( | z | + 1 / k ) ) \sim -{\frac{4}{\pi m^{2}( | z | + 1 / k )^{2}}}-{\frac{1}{\pi}}
\partial_\beta D (n||p_{\beta,\mu}) = \partial_\beta \ln \mathcal{Z}(\beta,\mu) + \sum_i n_i \varepsilon_i - \mu \sum_i n_i &= -\sum_j \frac{\varepsilon_j - \mu}{\exp\{\beta(\varepsilon_j - \mu)\}-1}+ \sum_i n_i \varepsilon_i - \mu \sum_i n_i.
\mathfrak{B}^r(\xi,t)=-\frac{\mathfrak{B_0}}{\sqrt{\tau}}=\begin{pmatrix}0 -i\eta\beta^r(\xi)e^{-t\varphi(\xi,0)}\tau^{-i\nu-\frac{1}{2}}i\eta\gamma^r(\xi)e^{t\varphi(\xi,0)}\tau^{i\nu-\frac{1}{2}}0 \end{pmatrix}
I_{21}= -2r^2\int_{0}^{2\pi}\frac{\cos(x)h(x)\sin(x)}{((1+r^2)+(r^2-1)\cos(x+y))}dy I_{22}= 2r^2\int_{0}^{2\pi}\frac{\cos(x)h(y)\sin(y)}{((1+r^2)+(r^2-1)\cos(x+y))}dy I_{23}= -2r^2\int_{0}^{2\pi}\frac{\cos(y)h(x)\sin(x)}{((1+r^2)+(r^2-1)\cos(x+y))}dy I_{24}= 2r^2\int_{0}^{2\pi}\frac{\cos(y)h(y)\sin(y)}{((1+r^2)+(r^2-1)\cos(x+y))}dy
s_{n-2}s_{n-1}(x) = \sum_{1\leq i\leq n}s_{n-2}(d_i^{-(n-i)}(x))+ \sum_{\substack{1<k\leq n 1\leq i_1<\cdots<i_k\leq n}}(-1)^{k+1}d_{i_1}^{-(n-i_1)}\cdots d_{i_k}^{-(n-i_k)}(x) .
Q_{3}= - \sum_{n = 0}^{\infty}\mathrm{{s g n}}( n + z ) b_{n}^{\dagger}b_{n}+ \sum_{n = 1}^{\infty}{\mathrm{s g n}}( n - z ) d_{n}^{\dagger}d_{n}+ \; ( z -{\frac{1}{2}}) \ .
\bar R_{Sk}^e(i-1) = \frac{i-2}{i-1}\bar R_{Sk}^e(i-2)+\frac{r_k^{\rm R}(i-1)}{i-1}C_{Sk}(i-1) , \bar R_{kD}^e(i-1) = \frac{i-2}{i-1}\bar R_{kD}^e(i-2)+\frac{r_k^{\rm T}(i-1)}{i-1}C_{kD}(i-1) ,
\epsilon=F_\epsilon(\beta_\epsilon)+\epsilon=(c_g^2\ln\tfrac{2}{\beta_\epsilon}-1+\epsilon)\beta_\epsilon \ge (c_g^2\ln\tfrac{2}{\beta_g}-1)\beta_\epsilon=\epsilon_g(\tfrac{1}{\beta_g}-1) \beta_\epsilon \ge \epsilon_g \beta_\epsilon,
c_{m , d}= 4^{1 - d / 2}\frac{\Gamma ( h_{+}) \Gamma ( h_{-})}{\Gamma ( \frac{d}{2}) \Gamma ( \frac{d}{2}- 1 )}\times \frac{\Gamma ( \frac{d}{2})}{2 ( d - 2 ) \pi^{d / 2}}= \frac{\Gamma ( h_{+}) \Gamma ( h_{-})}{( 4 \pi )^{d / 2}\Gamma ( \frac{d}{2})}.
\left|\int_{\mathbb{R}^2_+}\langle y \rangle^{2k+ 2\ell +2\alpha_2}[\partial^\alpha, (u^s+{u}^{n-1})]\partial_x w^{n}\partial^\alpha w^n dx dy\right| &\le C(1+ \|{w}^{n-1}\|_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}) \|w^n\|^2_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}.
\rho(t) = \begin{cases}r \cdot \ln(t) + \ln(|c_{n_1}|) + \sum_{i=1}^s \ln(|a_i|) \; t \le T_0; (r+s) \cdot \ln(t) + \ln(|c_{n_1}|) \; t \ge T_{\infty}. \end{cases}
\sum_{l=0}^{n}Q_{T,l}(x) =\sum_{l=0}^{n}T_{l}(x)+\textbf{i}\sum_{l=0}^{n}T_{l+1}(x)+\textbf{j}\sum_{l=0}^{n}T_{l+2}(x)+\textbf{k}\sum_{l=0}^{n}T_{l+3}(x) &=(T_{0}(x)+T_{1}(x)+\cdots+T_{n}(x))+\textbf{i}(T_{1}(x)+T_{2}(x)+\cdots+T_{n+1}(x)) &\ \ +\textbf{j}(T_{2}(x)+T_{3}(x)+\cdots+T_{n+2}(x))+\textbf{k}(T_{3}(x)+T_{4}(x)+\cdots+T_{n+3}(x)).
\Phi_B(T_{\delta})^2=\varepsilon^2 v_{\delta}^2+\varepsilon v_{\delta}\tilde{q}+\varepsilon^3 v_{\delta}\Psi_2(v_{\delta})+\tilde{\mathtt{Q}}_2(T_{\delta}), \Phi_B(T_{\delta})^3=\varepsilon^3 v_{\delta}^3+\tilde{\mathtt{Q}}_3(T_{\delta}),
J_3[H_1,H_2]:= \hat{J}_3[H_1,H_2]-\frac{1}{2}(\epsilon_1g)\nabla_a(2|H_1|^2+|H_2|^2)+\frac{1}{2}\nabla_a(\epsilon_1 g)(2|H_1|^2+|H_2|^2), \hat{J}_3[H_1,H_2]:= \tilde{J}_3[H_1,H_2]\ \ \textup{in}\ \ [3M,\infty), \hat{J}_3[H_1,H_2]:= \bar{J}_3[H_1,H_2]\ \ \textup{in}\ \ [2M,3M].