harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\iint_D f(a_1, a_2, b_1, b_2) \cdot \phi_{n-2}(-a)\phi_{n-2}(-b) \, dab = \iint_D f(a_1,a_2,b_1,b_2) \cdot \left(1 + \frac{2\alpha_1}{n}+ \frac{\alpha_2(a+b)}{n}- \frac{a^2+b^2}{2n}\right) \, dab + o(n^{-1}) \; .
\,\, (k^{(0)})^{d+1}\int_0^\infty G_{\alpha_0,\dots,\alpha_n}(k^{(0)}r,\dots,k^{(n)}r) (r^d dr) = \,\, \frac{\Gamma(d+1)}{\Gamma(\alpha_0+\dots+\alpha_n)}F^{(n)}_D(d+1;\alpha_1,\dots,\alpha_n;\alpha_0+\cdots+\alpha_n;\mathbf z_1,\dots,\mathbf z_n),
\widehat{y}_1^{\,[K,L]}(x) = (q-1)\alpha_1\sum_{n=K}^{L}c_n, c_n = q^n\xi\frac{\big(q^{\lambda+n+1}\xi / x, q^{n+1}\xi\alpha_1, q^n \xi\alpha_2 ; q\big)_{\infty}}{\big(q^{n+1}\xi / x, q^n \xi\beta_1, q^n \xi\beta_2 ; q\big)_{\infty}}.
\left| \frac{(A^+)^{n-1}(c_k^+)^a((c_k - 1)^+)^{n-a}}{(A^-)^{n-1}(c_k^-)^n}\right| = \left| A^{n-1}c_{k}^a(c_{k}- 1)^{n-a}\right| = |c_{k+1}-1| > 2 > 1,
qq_1^2\tilde\theta^{-1}\bigl(\tilde H_{0,1}\bigr) \equiv q_1\bigl([F_{1,1},F_{0,-1}]_{q^{2}}+q_1[F_{0,0},F_{1,0}]_{q^{2}}\bigr) &=qq_1^2\bigl(\theta^{-1}(H_{0,1})+q_1\theta^{-1}(H_{1,1})\bigr)\,.
\sqrt 2 \biggl ( B_{\underline{i}}^{( 1 )}\alpha_{I J}+ i B_{\underline{i}}^{( 2 )}\beta_{I J}\biggr ) \epsilon^{J}= e^{- \phi + 3 U}\bigl ( \partial_{\underline{i}}\overline{V}\bigr ) \gamma^{0}\epsilon_{I}.
\left\| h(t) \right\|^2_{L^2(\varrho\,m_1)}+ \left( t \left\| \nabla_\xi h(t) \right\|^2_{L^2(\varrho\,m_0)}+t^3\left\|\nabla_{x}h(t) \right\|^2_{L^2(\varrho\,m_0)}\right) \lesssim\mathcal{F}(t,h(t))\leq\mathcal{F}(0,h(0))=\left\| h(0) \right\|^2_{L^2(\varrho\,m_1)}.
\{\frac{(1+\theta_{1}i)^{2}}{1+{\theta_{1}}^{2}},\frac{(1-\theta_{1}i)^{2}}{1+{\theta_{1}}^{2}},\frac{(1+\theta_{2}i)^{2}}{1+{\theta_{2}}^{2}},\frac{(1-\theta_{2}i)^{2}}{1+{\theta_{2}}^{2}}\}.
Z[J,K]=e^{\frac{1}{\hbar}W[J,K]}=\int\!D\Phi\,e^{\frac{1}{\hbar}\left[-S[\Phi] +\frac{1}{2}\int\!\!J(x)\Phi^{2}(x)\,d^{n}x+\frac{1}{24}\int\!\!K(x)\Phi^{4}(x)\,d^{n}x\right]}.
\int_0^T \, dt \int_\Omega \Bigl\langle v (x,t), \frac{\partial \phi}{\partial t}(x,t) \Bigr\rangle \, dx\, - \, \int_0^T \, dt \int_\Omega \langle D v (x,t), D \phi (x,t) \rangle \, dx\; = \; 0 &\mathrm{for}\;^\forall \phi \, \in \, \dot{C}^\infty ( \Omega \times (0,T)) \; \mathrm{with}\; \div \phi \, = \, 0.
\Bigl[\prod_a \Delta^2(\varphi_a)\Bigr] \,\times\, \exp\Bigl\{-{1\over 4}{N\over L}\Bigl(\sum_{l=1}^{L-1}{1\over \sin^2{\pi l\over L}}+{1\over 12}\Bigr)\sum_{a,\alpha}\varphi_{a,\alpha}^2 + O(\varphi^4)\Bigr\}~~.
M_{c h a r g e d}={\frac{1}{1 6 \pi G_{D}^{\prime}}}\Omega_{D - 2}( D - 3 ) ( \sqrt{Q^{2}+{\frac{1}{4}}r_{0}^{2 ( D - 3 )}}+{\frac{D - 1}{2 ( D - 3 )}}r_{0}^{D - 3})
u(\tau_2) - u(\tau_1) \geq v(\tau_2)/4 - v( \tau_1 ) / 4= 2^{-2}(v(2 \tau_1) - v( \tau_1)) \geq 2^{-2}\cdot 2^{k-1}\tau_1 & \geq 2^{k -3}\cdot 2^{-3k -3}& = 2^{-2k - 6}.
V_{C K M}\sim \left( \begin{array}{c c c}{{1}}&{{- \kappa_{g}\Delta s}}&{{- \tau_{g}\Delta s}}\\{{\kappa_{g}\Delta s}}&{{1}}&{{\kappa_{n}\Delta s}}\\{{\tau_{g}\Delta s}}&{{- \kappa_{n}\Delta s}}&{{1}}\\ \end{array}\right) ,
\frac{\partial}{\partial t}\int_{-\infty}^{0}f(s,x-z,t,y)j(t,y)dy =\phi'(t)\left [ fj\right]_{-\infty}^{0}+\frac{1}{2}\left[j\partial_y f-f\partial_y j \right]_{-\infty}^{0}
f_{\textbf{Q}}:=f\left(\textbf{Q}\right)=\sum_{j=1}^sb_j\textbf{Q}^{\lambda_j}\mbox{and}f_{\tilde{\textbf{Q}}}:=f\left(\tilde{\textbf{Q}}\right)=\sum_{j=1}^sb_j\tilde{\textbf{Q}}^{\lambda_j}.
\exp Y (\mathbf{t}) = \begin{pmatrix}\mathrm{diag}(\cos t_1, \dotsc, \cos t_p) 0 - \mathrm{diag}(\sin t_1, \dotsc, \sin t_p) 0 I_{q - p}0 \mathrm{diag}(\sin t_1, \dotsc, \sin t_p) 0 \mathrm{diag}(\cos t_1, \dotsc, \cos t_p)\end{pmatrix},
W_{n}^{4}=\sum_{i=1}^{N}\left[2x_{i}^{n+2}\partial_{i}^{3}+3(n+2)x_{i}^{n+1}\partial_{i}^{2}+ (n+1)(n+2)x_{i}^{n}\partial_{i}-(n+2)x_{i}^{n+1}(1-\theta_{i}\Pi_{i})\partial_{i}^{2}\right]
\theta[\vec{X}]_{\hat{i}\hat{j}}=\frac{\omega}{\sqrt{2}}\frac{C^{\prime}(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u)}{C(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u)}diag(0,1,1)
{\cal O}_{k}= \delta_{k , 0}+ \sum_{j = 0}^{\infty}\frac{( L_{- 1})^{j}L_{k + j}}{( k + j + 1 ) !}, \hspace{1 e m}\bar{{\cal O}_{k}}= \delta_{k , 0}+ \sum_{j = 0}^{\infty}\frac{( \bar{L}_{- 1})^{j}\bar{L}_{k + j}}{( k + j + 1 ) !}.
\begin{array}{r l}{{}}&{{T r W ( w , z_{2}) W ( z_{1}, w ) W ( w , z_{1}) W ( z_{2}, w )}}\\{{=}}&{{T r R W ( z_{1}, w ) W ( w , z_{2}) R^{- 1}R W ( z_{2}, w ) W ( w , z_{1}) R^{- 1}}}\\{{=}}&{{T r W ( z_{1}, z_{2}) W ( z_{2}, z_{1})}}\\{{=}}&{{T r W ( z_{1}, z_{1})}}\\ \end{array}
\Biggl.\Biggl.+\frac{(i^2+i-3)\zeta_{i+1}^{(i_2)}\zeta_{i}^{(i_1)}-(i^2+3i-1)\zeta_{i}^{(i_2)}\zeta_{i+1}^{(i_1)}}{\sqrt{(2i+1)(2i+3)}(2i-1)(2i+5)}\Biggr)\Biggr] - \frac{1}{24}{\bf 1}_{\{i_1=i_2\}}{\Delta^3},
\begin{cases}\underline u_t-\Delta \underline u=\delta^*(q_{i_k}-\underline u), |x|\in [c_k^- t, c_k^+t], \; t>T_1, \underline u=q_{i_k}-\sigma, |x|\in \{c_k^-t, c_k^+t\}, \; t>T_1, \underline u=q_{i_k}-\sigma, |x|\in [c_k^-t, c_k^+t], \; t=T_1.\end{cases}
A(x,y) = \left(-\sin\left(x-\frac{y}{2}\right)(1+R(x)) + \frac{R(x)-R(x-y)}{2\sin\left(\frac{y}{2}\right)}\cos(x-y)\right)^{2}& + \left(\cos\left(x-\frac{y}{2}\right)(r+R(x)) + \frac{R(x)-R(x-y)}{2\sin\left(\frac{y}{2}\right)}\sin(x-y)\right)^{2}.
\Theta_{\tilde{\rho},\rho}(F):= \sum_{j=1}^m \int_{\mathbb{R}^4}F(y,x') F(x,x')F(y,y')F(x,y') begin{align*}-1ex] \tilde{\rho}_{2^{k_j}}(x'-y') ({\rho}_{2^{k_{j-1}}}-{\rho}_{2^{k_j}})(x-y) \,dx dy dx' dy'.
\omega_{V_\infty}( \begin{pmatrix}1 -1 \end{pmatrix}u(b)) \varphi = P(-x(b+\sqrt{-1})^{-1}) \psi(-\frac{1}{2}\langle x,x(b+\sqrt{-1})^{-1}\rangle) \cdot{\rm det}(\frac{b+\sqrt{-1}}{\sqrt{-1}})^2.
e^{-\frac{i\eta}{2}}(\begin{matrix}cos^{2}\theta+e^{i\eta}sin^{2}\theta&(1-e^{i\eta})e^{-i\phi}cos\theta sin\theta\\ (1-e^{i\eta})e^{i\phi}cos\theta sin\theta&sin^{2}\theta+e^{i\eta}cos^{2}\theta\end{matrix})
S^{\ast}_{k}f(x) \leq C \int_{d(x,y)<\rho(x')}{1\over{d(x,y)^{n}}}V(y') \chi_{B_{0}}(y)f(y) dy \, + & + C \int_{d(x,y)\ge\rho(x')}\Big({{\rho(x')}\over{d(x,y)}}\Big)^{k}{1\over{d(x,y)^{n}}}V(y') \chi_{B_{0}}(y)f(y) dy = \mathbf{A}(x)+\mathbf{B}(x).
< \overline{{T}}( k_{1}) T ( k_{2}) \overline{{A}}_{i}( k_{3}) >_{t r u n c}= - i ( 2 \pi )^{p + 1}\delta \left( \sum k \right) \: i \overline{{e}}\left( \widehat{\overline{{k}}}_{1}- \hat{k}_{2}\right)_{i}
v(m,\mathbf{j}') - v(m,\mathbf{j}) = (n-i)p - (n-i+1)r + s(j_i) - s(j_i - r) + s(j_{i+1}) - s(j_{i+1}+ p) \geq (n-i)p - (n-i+1)r + 1 -s(p) > \frac{n-i-1}{2}p - \frac{n-i+1}{2}- \lceil \log_2(p) \rceil \geq 0
\begin{bmatrix}0 1\end{bmatrix}_{1,0}\begin{bmatrix}1 0\end{bmatrix}_{1,0}=\begin{bmatrix}1 0\end{bmatrix}_{1,0}\begin{bmatrix}1 1\end{bmatrix}_{1,0}\begin{bmatrix}0 1\end{bmatrix}_{1,0}.
\begin{gathered}K v_{1}= q^{2}v_{1}, K v_{0}= v_{0}, K v_{-1}= q^{-2}v_{-1}, E v_{1}= 0, E v_{0}= [2]^{1/2}v_{1}, E v_{-1}= [2]^{1/2}v_{0}, F v_{1}= [2]^{1/2}v_{0}, F v_{0}= [2]^{1/2}v_{-1}, F v_{-1}= 0.\end{gathered}
P_{\nu_{e}\rightarrow \nu_{\mu}}\approx 2 s i n^{2}\theta_{\nu}c o s^{2}\theta_{\nu}\left[ 1 - c o s \left( \frac{m_{1}^{2}- m_{2}^{2}}{2 p}\right) \frac{c^{3}}{\hbar}t \right] \quad ,
\mathcal{N}(z_-,z_+,[0,1],\rho,H^{(k)},J) =\Bigg\{(\tau,v)\in [0,1]\times C^{\infty}(\mathbb{R}\times S^1\rightarrow M) \ \Bigg| \ \begin{matrix}v \textrm{\satisfies \}(\textrm{B}) \lim_{s\rightarrow \pm\infty}v(s,t)=x_{\pm}(t) [u_-\sharp v,x_-]=[u_+,x_+]\end{matrix}\Bigg\}
{V_{avg}^{\epsilon}}^{\prime}=\int_{-\epsilon}^{\epsilon}dxV(x)|\psi^{\prime}|^{2}=\frac{\epsilon^{2}|c|^{2}}{1+\frac{4}{3}|c|^{2}\epsilon^{3}}\int_{-\epsilon}^{\epsilon}dxV(x)\simeq2\epsilon^{3}|c|^{2}V(0)+\cdot\cdot\cdot,
|\phi \rangle =\exp \sum\limits_a\left(C_1\left(\left|\vec{p}\right| \right)A_{\vec{p},1}^{a+}A_{\vec{p},1}^{a+}+C_2\left(\left|\vec{p}\right| \right)A_{\vec{p},2}^{a+}A_{\vec{p},2}^{a+}+C_3\left(\left|\vec{p}\right| \right)\left(B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}+i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \right) \mid 0\rangle
\Phi^{(1)}(\tau,1+\rho e^{i\beta \pi}) \sim \frac{\pi}{\sqrt{\tau}}\rho^{-\frac{1}{2}}\left[1+\sqrt{2}C_{\beta}\left(\frac{\tau}{\rho}\right)^{\frac{1}{4}}\right]^{-2n}=:g^{(1)}(\tau,\rho),
G_k(ar) = \max\Big\{ar-\frac{1}{k},0 \Big\}&= \max\Big\{ar-a\frac{1}{k}+ \frac{a-1}{k},0 \Big\}&\leq \max\Big\{ar-a\frac{1}{k},0 \Big\}+ \frac{a-1}{k}&= aG_k(r) + \frac{a-1}{k}.
G(z\,|\,\tau) \triangleq\exp\left(a(\tau)\frac{z}{\tau}+b(\tau)\frac{z^2}{2\tau^2}\right)\frac{z}{\tau}\prod_{m, \,n=0}^\infty{}' \left(1+\frac{z}{\Omega}\right)\exp\left(-\frac{z}{\Omega}+\frac{z^2}{2\Omega^2}\right),
F(\theta,r'):=\Psi_2^{-1}\circ \Psi_1^{-1}\circ \Psi_0^{-1}\circ F_{q,N,\mu}\circ \Psi_0\circ\Psi_1\circ\Psi_2(\theta,r')=\bigg(\gamma+\theta+r',r'+\tilde\varepsilon(\theta,r')\bigg),
\widetilde{\mathcal{L}}(U^{(k+1)}) = \widetilde{\mathcal{R}}(U^{(k)}), k=0,\dots, N_T, \widehat{\mathcal{L}}(U^{(k+1)}) = \widehat{\mathcal{R}}(U^{(k)}), k=0,\dots, N_T,
\pi^+(P)+\pi^-(P)\geq \pi(P_0)-\pi(2^{29})+\pi^+(2^{29})+\pi^-(2^{29}) \geq \frac{P}{\log P}\left(1+\frac{1}{\log P}+\frac{2}{\log^2 P}+\frac{7.32}{\log^3 P}\right)-\pi(2^{29})+3285696>406550053.02
\Delta_{d}(r^d F(r)) = \sum_{h=0}^d c_{h,d}\big(r^d F\big(\frac{r}{2^h}\big) - 2^{d}\frac{r^d}{2^d}F\big(\frac{r}{2^{h+1}}\big) \big) &= \sum_{h=0}^d c_{h,d}r^d \big( F\big(\frac{r}{2^h}\big) - F\big(\frac{r}{2^{h+1}}\big) \big) &= \sum_{h=0}^{d}c_{h,d}r^d \Delta_{0}F\big(\frac{r}{2^h}\big),
\begin{array}{c}{{\psi_{A}( x ) = \psi_{A}^{\left( - \right)}( x ) + \psi_{A}^{\left( + \right)}( x ) , \quad \psi_{B}( x ) = \psi_{B}^{\left( - \right)}( x ) + \psi_{B}^{\left( + \right)}( x )}}\\{{\psi ( x ) = \psi_{A}^{\left( - \right)}( x ) + \psi_{B}^{\left( + \right)}( x )}}\\ \end{array}
(\widetilde f_1\otimes\widetilde f'_1)(\omega_{Z,Z'})= \sqrt 2\,\rho_{Z^{(1)}}\otimes\left[\rho_{\binom{4,2}{3,1}}+\rho_{\binom{4,1}{3,2}}+\rho_{\binom{3,2}{4,1}}+\rho_{\binom{3,1}{4,2}}\right]= 2\left[\rho^{(2)}_{Z^{(1)}}\otimes\sum_{M''=\emptyset,\binom{4}{3}}\rho^{(2)}_{\Lambda_{M''}}\right].
\ell_{\boldsymbol{i}}:= \ell_{i_1}\cdot \ldots \cdot \ell_{i_n}:= \ell^{F^\emptyset}_{i_1}\cdot \ell^{F^{i_1}}_{i_2}\cdot \ell^{F^{i_1i_2}}_{i_3}\cdot \ldots \cdot \ell^{F^{i_1\dots i_{n-1}}}_{i_n}
\begin{cases}c(x)\frac{\partial}{\partial y}\Phi_a(y;x)-d_A\frac{\partial^2}{\partial y^2}\Phi_a(y;x) = M_{11}\Phi_a+M_{21}\Phi_b,\Phi_a>0 begin{align*}5pt] c(x)\frac{\partial}{\partial y}\Phi_b(y;x)-d_B\frac{\partial^2}{\partial y^2}\Phi_b(y;x) = M_{12}\Phi_a+M_{22}\Phi_b,\Phi_b<0. \end{cases}
K(x,x') = (1 + o(1)) \frac{\zeta_W\left(1 + \frac{2 + ix + ix'}{r \log q}\right)}{\zeta_W\left(1 + \frac{1 + ix}{r \log q}\right)\zeta_W\left(1 + \frac{1 + ix'}{r \log q}\right)},
s_{r(F)}^\Sigma-s_\mu^\Sigma{s_\mu^\Sigma}^*=(s_{r(F)}^\Sigma-s_\mu^\Sigma{s_\mu^\Sigma}^*)\Bigg(\prod_{\lambda\in \mathrm{Ext}_{\Sigma\setminus \Sigma H_I}(\mu;F)}(s_{r(F)}^\Sigma-s_{\mu\lambda}^\Sigma{s_{\mu\lambda}^\Sigma}^*)\Bigg).
\left( \bar{a}_{\alpha}^{n^{\prime}}\bar{b}^{m^{\prime}\beta}-\frac{1}{C}\delta_{\alpha}^{\,\beta}\bar{a}^{n^{\prime}}\cdot\bar{b}^{m^{\prime}}\right) |0>,\quad\bar{a}_{\alpha_{1}}^{n_{1}^{\prime}}\cdots\bar{a}_{\alpha_{k}}^{n_{k}^{\prime}}|0>,\quad etc.
b ( \vec{\lambda}, m , l , w ) = \sum_{\vec{t}, \sum t_{i}= 0}\det_{\frac{w}{2}\times \frac{w}{2}}\left| \Biggl ({m \atop ( m / 2 ) + \frac{l + w}{2}t_{i}+ \lambda_{j}- \lambda_{i}}\Biggr ) \right| ,
0 =\gamma_0 \gamma_2 u_2^2, 0 =\dfrac{u_1}{\delta'}+ \gamma_0 u_2^2 + \gamma_1 \gamma_2 u_2^2, 0 =u_1^2 + \dfrac{u_2}{\delta'}+ \gamma_1 u_2^2 + \gamma_2^2 u_2^2+\dfrac{\alpha_2 \gamma_0 u_1 + \alpha_1 \gamma_0 u_2 + \alpha_2 \gamma_0 \gamma_2 u_2}{\delta'^2}.
\begin{cases}-\Delta \psi_1'(c_0) = \gamma_1'(c_0) \psi_1(c_0) \mbox{in $\Omega$}, \partial_{\mathbf{n}}\psi_1'(c_0) = \gamma_1'(c_0) \psi_1(c_0) \mbox{on $\partial \Omega$}. \end{cases}
A(\theta):= \sum\limits_{\ell\le|\nu|\le \hat \ell}a_\nu\left(e^{-i\theta_1}-1\right)^{\nu_1}\left(e^{i\theta_1}-1\right)^{\nu_2}\dots\left(e^{-i\theta_d}-1\right)^{\nu_{2d-1}}\left(e^{i\theta_d}-1\right)^{\nu_{2d}},\theta\in R_\pi,
=\frac{1}{\Gamma(\mu)}\sum_{m\le y}e^{-m/N}\sum_{n=1}^{m-1}\chi(n)\Lambda(n)(m-n)^{\mu-1}+ O\left(2^\mu\sum_{m\le y}\sum_{\substack{n=1 n\neq m}}^\infty\Lambda(n)\frac{e^{-n/N}}{|m-n|}+(\log N)\sum_{m\le y}\Lambda(m)e^{-m/N}\right).
V_{(\gamma_t,t]}= \sum_{\ell \in (\gamma_t,t]}G(0,S_\ell) \le \frac{3+o(1)}{2\pi}\sum_{\ell \in (\gamma_t,t]}||S_\ell^1|+1|^{-1}\le \frac{3+o(1)}{2\pi}\frac{(1+\delta) t}{\psi(t)}(\log_3 t)\le \frac{(1+2\delta) t}{2 h_3(t)}\,.
\partial v=N\partial_{z,\alpha}N^{-1}v =\gamma_{\alpha}\partial_z(\gamma_{-\alpha}v)=\gamma_{\alpha}(\gamma_{-\alpha}'v+\gamma_{-\alpha}\partial_zv) &=\gamma_{\alpha}(-\alpha\gamma_{-\alpha}v+\gamma_{-\alpha}\partial_zv) &=(\partial_z-\alpha) v.
\psi_{k}^{\prime \prime}+ 2 \frac{z^{\prime}}{z}\psi_{k}^{\prime}+ k^{2}\psi_{k}= 0 , \pi_{k}^{\prime \prime}- 2 \frac{z^{\prime}}{z}\pi_{k}^{\prime}+ k^{2}\pi_{k}= 0 .
\mathrm{Pf}(A) = \frac{1}{2^m m!}\sum_{\sigma\in S_{2m}}\mathrm{sgn}(\sigma)\prod_{i=1}^{m}a_{\sigma(2i-1),\sigma(2i)}, \mathrm{Hf}(A)= \frac{1}{2^m m!}\sum_{\sigma\in S_{2m}}\prod_{i=1}^{m}a_{\sigma(2i-1),\sigma(2i)},
\mathfrak{S}=\sum_{q\in\mathcal{Q}}\frac{1}{q^{\frac{3}{2}}}\sum_{n=1}^\infty \frac{\lambda_{\tilde\pi}(n,1)}{\sqrt{n}}V\left(\frac{n}{Xq^{\frac{3}{2}}}\right)\sideset{}{^\star}\sum_{\chi\bmod{q}}g(\chi)^3(1+\chi(-1))\bar\chi(n\ell).
\sqrt{\left\{X^{\mu}\ , X^{\nu}\right\}_{\mathrm{P B}}\left\{X_{\mu}\ , X_{\nu}\right\}_{\mathrm{P B}}}\longleftarrow \quad ( ? ) \quad \longrightarrow \sqrt{\mathrm{T r}\mathbf{F}_{\mu \nu}\mathbf{F}^{\mu \nu}}
L^F_4=\frac{\kappa^2}{36}B^F_{\mu\nu\alpha\beta}B^{F}_{\mu_1\nu_1\alpha_1\beta_1}G^{\mu\mu_1;\nu_1\nu}K^{\alpha\alpha_1;\beta_1\beta}-\frac{\kappa^2}{12}C^F_{\mu\nu\rho;\alpha\beta}C^{F\mu\nu\rho}_{~~~~~~\alpha_1\beta_1}K^{\alpha\alpha_1;\beta_1\beta}.
\displaystyle\lim_{n \rightarrow \infty}\frac{\beta_n}{\gamma^n}= \beta_0 + (p-1)\displaystyle\lim_{n\rightarrow \infty}\frac{\gamma^{n-1}-\gamma}{\gamma^n(\gamma-1)}= \beta_0 + (p-1)\frac{\gamma}{\gamma-1}= \frac{p_{s}^*(p_{s}^* - \theta')}{\theta'(p_{s}^* - \theta'p)}.
\eta_t^{(i)}= \frac{D}{G\sqrt{t}}, \ \eta_t^w = \frac{2 D^2}{\sqrt{(2 D^2 G^2 + 2 \ln m)t}}, \textrm{and}\eta_t^q = \frac{2 \ln m}{\sqrt{(2 D^2 G^2 + 2 \ln m)t}}
\langle \rho_{\Sigma}^{\otimes k}(m_1,...,m_k),\bigotimes_{l=1}^{m_1}h_{1,l}\otimes...\otimes \bigotimes_{l=1}^{m_k}h_{k,l}\rangle_{\mathcal H^m}= \prod_{j=1}^{k}\sum_{p\in\mathcal P(m_l)}\prod_{(x,y)\in p}\langle \Sigma_s h_{x,j},h_{y,j}\rangle.
\left[\left({\dot{R}\over R}\right)^2+\xi_{+}^2 R^{-{2\over 3}k^2}\right]^{1\over2}-\left[\left({\dot{R}\over R}\right)^2+\xi_{-}^2 R^{-{2\over 3}k^2}\right]^{1\over2}=-{\kappa_{5}^2\over3}[\lambda(\phi)+\rho].
\frac{\partial{}\varphi(\mathbf{p},\xi)}{\partial{}p_k}=\frac{\partial{}U(\mathbf{p})}{\partial{}p_k}+\xi\left(\frac{1}{p_k}-\frac{1}{P_{}-p_k}-\frac{T_k}{\Gamma_{}-T_k{}p_k}\right) &+\xi\sum_{m=1}^M\frac{-\Omega{}|g_{mk}|^2}{P_{}^{}-\Omega\left(\sum_{k=1}^K|g_{mk}|^2p_k+\sigma_N^2\right)},
\lim_{n\to\infty}\sup_{x\in\mathbb{R}}P\left(\left|\sup_{t\in[a_n,T-a_n]}|Z_n(t)|-x\right|\leq M(v_n+7\varepsilon)\right)=\lim_{n\to\infty}P(w_n(Z_n;n^{-1})>M\varepsilon)=0.
\left\vert f(\zeta)-\frac{p_{m}^{\prime}}{q_{m}^{\prime}}\right\vert=\vert f(\zeta)-f(p_{m}/q_{m})\vert\leq U\vert\epsilon_{m}\vert\leq q_{m}^{-m}\leq q_{m}^{\prime -m/\psi(m)}.
\bar{D}_{1}V Z_{2}= Z_{2}V \bar{D}_{1}+ 2 P_{-}\quad , \quad P_{-}= \frac{1}{2}\left( \begin{array}{c r r c}{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{- 1}}&{{0}}\\{{0}}&{{- 1}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right)
J_2= Q_3-h'fvu_1^2+\frac{2fG_{uu}}{A}vu_1^3w_1+\frac{h'^2}{2}\frac{Au_1^2}{w_1^2}+ \frac{h'^2}{1-G_u}\frac{Au_1^3}{v^2w_1}+\frac{c^{11}}{2}g'^2\frac{u_1^2}{w_1^2}A+\frac{c^{11}g'^2}{1-G_u}\frac{Au_1}{w_1}+O(u_1^2).
a_{0,1}(\ell,k)=\frac 1{(2\pi i)^4}\int_{\Gamma_{D_1}}dz\int_{\Gamma_{D_2}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{n_1,\mu_1,\xi_1}(z)G_{\Delta n,\Delta\mu,\Delta\xi}(w)}{G_{k,\mu_1,\xi_1}(\zeta)G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)(z-w)(z-\zeta)(w-\omega)},
|\mathbb{E}[f^{*}(x,\mathbf{h},\mathbf{z})\cdot x]|^2=|\mathbf{w}^{T}\mathbf{R}_{\mathbf{r}x}^*|^2=\mathbf{w}^{H}\mathbf{R}_{\mathbf{r}x}\mathbf{R}_{\mathbf{r}x}^{H}\mathbf{w},
q_{1}( t ) = \frac{1}{\sqrt{2 m_{1}\omega_{1}}}( e^{- i \omega_{1}t}( c A_{1}( 0 ) + s A_{2}( 0 ) ) + e^{i \omega_{1}t}( c A_{1}( 0 ) + s A_{2}( 0 ) )^{\dag}) ,
\int_{-\infty}^{+\infty}\tilde{J}_{\nu}(x)e^{-iqx}dx=\frac{\Gamma(\frac{\nu+1+n-iq}{2})}{\Gamma(\frac{\nu+1-n+iq}{2})}2^{n-iq}e^{iqln(k_{0}r_{0})}.
\begin{array}{c c c c}{{\bar{\varepsilon}_{i}\gamma^{A}\varepsilon^{\prime i}= 0}}&{{\bar{\rho}_{i}\tilde{\gamma}^{A}\rho^{\prime i}= 0}}&{{\bar{\varepsilon}_{i}\gamma^{[ A}\tilde{\gamma}^{\mu ]}\rho^{i}= 0}}&{{A = 4 , 5}}\\ \end{array}
M:= \sup_{\alpha_1}v(i,j,\hat{\mu}_{\alpha,\varepsilon}) \left[e^{\alpha_1\left(\left(\mu_{\alpha,\varepsilon}(j) - \hat{\mu}_{\alpha,\varepsilon}(j)\right)^- - \left(\mu_{\alpha,\varepsilon}(i) - \hat{\mu}_{\alpha,\varepsilon}(i)\right)^-\right) + c_{\alpha,\varepsilon}(i,j)}- 1\right] < \infty.
\left(D_{0+}^{0,\alpha',\beta,\beta',\gamma}f\right)(x)=\left(D_{0+}^{\gamma,\alpha'-\gamma,\beta'-\gamma}f\right)(x),\ \ \ \ \left(D_{-}^{0,\alpha',\beta,\beta',\gamma}f\right)(x)=\left(D_{-}^{\gamma,\alpha'-\gamma,\beta'-\gamma}f\right)(x).
\|(z-T)^{-1}\|=\big\|\sum\limits_{n=0}^{\infty}\frac{T^n}{z^{n+1}}\big\| \leq\sum\limits_{n=0}^{\infty}\frac{\|T^n\|}{|z|^{n+1}}=\sum\limits_{n=0}^{\infty}\frac{\beta_n}{|z|^{n+1}}&=\|(z-T)^{-1}e_{0}\|_1\leq \|(z-T)^{-1}\|.
\frac{n}{2}\log2\pi e N_2=h(Z_2^{(n)}) &=h(Y_2^{(n)}\mid X^{(n)})\overset{(e)}{=}h(Y_2^{(n)}\mid M_2,M_3,X^{(n)}) &\leq h(Y_2^{(n)}\mid M_2,M_3)\leq h(Y_2^{(n)}) \leq \frac{n}{2}\log2\pi e(P+N_2),
S_{\mathrm{{Q H}}}= \int_{\cal M}d^{3}x ~ \left( \psi^{\dagger}\, i \left( \partial_{0}- i A_{0}- i e a_{0}\right) \psi + \frac{1}{2 m_{e}}\psi^{\dagger}\left( \nabla - i A - i e a \right)^{2}\psi \right) + \frac{1}{m}S_{\mathrm{C S}}^{[ U ( 1 ) ]}[ A ]
U := \bigoplus_n U_n, (U(a,\lambda)\Psi)^{\pmb{b_k}}_n(\pmb{\theta}) := \exp \left(i \sum_{l=1}^n p_{b_{k_l}}(\theta_l) \cdot a\right) \Psi^{\pmb{b_k}}_n(\theta_1 - \l, \cdots, \theta_n - \l),
\begin{array}{ll}\eta = [\mu_i,\sigma_{ij}]^\top_{i,j\in\{1,\ldots,d\}, i\le j},Z(\eta) = S(\eta)^{\frac{1}{1-\beta}}M_{\eta,\beta},h({\bf{x}})\equiv 1, w(\eta)=\big[w^{(1)}(\eta),w^{(2)}(\eta)\big]^\top,f({\bf{x}}) = \big[f^{(1)}({\bf{x}}),f^{(2)}({\bf{x}})\big]^\top,\end{array}
\overline{\mathcal{F}}(\gamma_i \mid \mu_i) = \int_X \overline F(\sigma_i)d\mu_i + F'_{\infty}\gamma_i^{\perp}(X),{\mathcal{F}}(\gamma \mid \nu_X) = \int_{X\times X}F(\sigma)d\nu_X + F'_{\infty}\gamma^{\perp}(X\times X).
\|\psi\|_{L^1(T\wedge \omega_\phi)}:=\int_M -\psi T\wedge \omega_\phi \wedge (\omega^T)^{n-p-1}\wedge \eta =\|\psi\|_{L^1(T)}+\int_M -\psi dd^c_B(\phi T)\wedge (\omega^T)^{n-p-1}\wedge \eta
\mathrm{P_T}(\theta) = -\prod_{j = 1}^M \mu^{-1}_j \sum_{k = 1}^M \Hat{\mathrm{D}}_k \, \mathcal{L}_{\mathrm{I}}\left(\frac{\mu_k^{-1}\theta \mathrm{I}}{\mathrm{G}_{\mathrm{S,TX}}\,\mathrm{G}_{\mathrm{RX}}}\right),
\begin{tabular}{|c|c|c|c|}\hline$\nu$ $b$ $\rho(\nu,b)$ $T$ \hline $(4,4,1,1)$ $(1,3,2,2,2)$ $(4,3,1)$ $T(2,4)=T(4,1)=5$ \hline $(4,3,1)$ $(1,3,2,2)$ $(4,2)$ $T(2,3)=T(3,1)=4$ \hline $(4,2)$ $(1,3,2)$ $(4)$ $T(2,1)=T(2,2)=3$ \hline $(4)$ $(1,3)$ $(1)$ $T(1,2)=T(1,3)=T(1,4)=2$ \hline $(1)$ $(1)$ $T(1,1)=1$ \hline\end{tabular}
x_{n+1}=x_n-\frac{\big( e_n+C_2e_n^2+C_3e_n^3+O(e_n^4)\big)}{\big(N+2C_2Ne_n+\big(NC_2^2+NC_3-\frac{C_3}{4N}\big)e_n^2+O(e_n^3)\big)}, =x_n- \big( e_n+\big(-C_2^2+\frac{C_3}{4N^2}\big)e_n^3+O(e_n^4)\big).
[(U_1\circ d_0)f](\mu)=\sum_{i=1}^j (d_0f)(\mu;\delta_j)dx^{i}=\sum_{i=1}^j (d_0f)(\mu,\mu+\delta_i)dx^{i}= \sum_{i=1}^j (f(\mu+\delta_i)-f(\mu))dx^{i}= \sum_{i=1}^j \mathcal{D}_if(\mu)dx^{i}
\phi=\frac{(\Delta G_{W}^{TS\rightarrow D}-\Delta G_{M}^{TS\rightarrow D})}{(\Delta G_{W}^{N\rightarrow D}-\Delta G_{M}^{N\rightarrow D})}=\frac{\Delta\Delta G^{TS\rightarrow D}}{\Delta\Delta G^{N\rightarrow D}}
\eta_{M N}: = \eta_{A B}\overline{{{L}}}_{M}^{A}\overline{{{L}}}_{N}^{B}= g_{a b}\overline{{{L}}}_{M}^{a}\overline{{{L}}}_{N}^{b}+ g_{i j}\overline{{{L}}}_{M}^{i}\overline{{{L}}}_{N}^{j};
Ric(U_i, U_i) = \lambda_f - (m-n-1)^2 \left\{\lambda^2 \|\nabla (f + \log \lambda)\|^2 - \frac{\lambda^4}{4}\|grad_{\mathcal{H}}\frac{1}{\lambda^2}\|^2 + \lambda^2 g(\nabla f, \nabla_{\mathcal{H}}\frac{1}{\lambda^2})\right\},
\frac{\mathrm{d}^2}{\mathrm{d}t^2}\iota(\varphi_t)= \int ( \ddot{\varphi}- |\nabla \dot{\varphi}|_{\varphi_t}^2) (_{\varphi_t}\omega -n) \omega_{\varphi_t}^n + \int \dot{\varphi}_{,\alpha}\dot{\varphi}_{,\bar{\beta}}\omega_{\bar{\alpha}\beta}\omega_{\varphi_t}^n >0.
{1\over C_A}=\frac{2^{\frac{r}{2}}\prod(k_j+2)}{K\kappa_{\alpha}^A\kappa_{\widetilde{\alpha}}^A},\quad{1\over C_B}=\frac{2^{\frac{r}{2}}}{\kappa_{\alpha}^B\kappa_{\widetilde{\alpha}}^B},
C^{\mathrm{dl},k}_{3}\left( N_{\mathrm{sct}}, N_{\mathrm{scr}}\right) = \left(\frac{T_{\mathrm{dl}}}{T}\right) \min\left\lbrace c^k_{\mathrm{b2s},3}\left(N_{\mathrm{scr}}\right), c^k_{\mathrm{s2u},1}\left( N_{\mathrm{sct}}\right) \right\rbrace.
4\pi T_{\mu}{}^{\nu}=\left(\begin{array}[c]{cc}\left( \mathbf{E}\cdot\mathbf{D}+\mathbf{H}\cdot\mathbf{B}\right) /2 \left( \mathbf{E\times H}\right)_{q}\bigskip -\left( \mathbf{D\times B}\right)_{p}E_{p}D_{q}+H_{p}B_{q}-\delta_{pq}\left( \mathbf{E}\cdot\mathbf{D}+\mathbf{H}\cdot\mathbf{B}\right) /2\end{array}\right) .
\begin{cases}\begin{array}{ccl}x_{1}= 0 x_{2}= \frac{1-\mu}{3-\mu}x_{2}\left(y_{1}+y_{2}\right) y_{1}= \frac{1-\mu}{3-\mu}x_{2}\left(y_{1}+y_{2}\right) y_{2}= \frac{1+\mu}{3-\mu}x_{2}\left(y_{1}+y_{2}\right),\end{array}\end{cases}
(Q_{m}-\hat Q_{m}) - \eta \, k^2 \, a^{5-h}\, \sum_{j=1 \atop j \neq m}^\aleph{\bf{P}}_{0}\cdot \Upsilon_k(z_{m}, z_j) \cdot (Q_j-\hat{Q}_j)= i \, k \, a^{5-h}\,{\bf{P}}_{0}\cdot ( H^{Inc}(z_{m})-\hat H^{Inc}(z_{m}))+Error,
\Gamma_{1}=\{(\wedge^2_{13},\wedge^2_{24}),(\vee^2_{13},\vee^2_{24})), ((')^2_2,(')^2_1),(0,0), (1,1)\}; \Gamma_{2}=\{(\wedge^2_{13},\vee^2_{24}),(\vee^2_{13},\wedge^2_{24}), (x^2_2,x^2_1), (0,1),(1,0)\}.
\Gamma_{m}( D ; \alpha_{1}, . . . , \alpha_{m}) = \frac{1}{2}\Gamma ( \frac{1}{2}( \sum_{i = 1}^{m}\alpha_{i}- D ) ) \, G_{m}( \sum_{i = 1}^{m}\alpha_{i}; \alpha_{1}, . . . , \alpha_{m}) .
DF=\begin{bmatrix}a_1 f_{1x_1 x_1}+\cdots +a_m f_{m x_1 x_1}a_1 f_{1x_1 x_2}+\cdots +a_m f_{m x_1 x_2}f_{1x_1}\cdots f_{mx_1}a_1 f_{1x_2 x_1}+\cdots +a_m f_{m x_2 x_1}a_1 f_{1x_2 x_2}+\cdots +a_m f_{m x_2 x_2}f_{1x_2}\cdots f_{mx_2}\end{bmatrix}\,.
\frac{1}{2}\partial_t |\nabla H-h^S_{\nu\nu}HV|^2 = (\nabla_i H - h^S_{\nu \nu}H V_i) \cdot \Big( H h_{ij}\nabla_j H + \nabla_i(\Delta H +|A|^2 H) & - (\Delta H+|A|^2 H) h^S_{\nu \nu}V_i - (\partial_t h^S_{\nu \nu}) H V_i + O(H^2 +|\nabla H|) \Big).
{\cal L}_E=\bar \psi_a (i \partial \!\!\! / +m-\frac{e}{\sqrt{N}}A \!\!\! /)\psi_a-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}-i \frac{\kappa}{8\pi}\epsilon^{\mu \nu \rho}A_{\mu}\partial_{\nu}A_{\rho}.
\sigma^{(n)\vee}=\begin{pmatrix}1 1 1 \cdots 1 0 0 0 \cdots 0 0 -1 0 \cdots 0 0 1 0 \cdots 0 0 0 -1 \cdots 0 0 0 1 \cdots 0 \vdots \vdots \vdots \ddots \vdots \vdots \vdots \vdots \ddots \vdots 0 0 0 \dots -1 0 0 0 \cdots 1 0 0 0 \cdots 0 1 1 1 \cdots 1\end{pmatrix}
t_o\left( g,a,n\right) =\ln \left\{\frac{x_o^2\left( g,a,n\right)}{f\left[x_o\left( g,a,n\right) ,a,n\right]}\right\}=\quad ;\quad w_o\left(g,a,n\right) =\ln \left[ x_o\left( g,a,n\right) \right] =
d_{n,d}=d_{n-1,d+r-2}-\sum\limits_{i=1}^{r-2}q^{(i)}_{n-1,d+r-2-i}+\dfrac{d(d-1)}{2}s_{n-1,d-2}+2\sum\limits_{i=0}^{n-1}\sum\limits_{j=0}^{d-2}s_{n-i-1,d-j-2}\cdot d_{i,j}.
S_{1}= \frac{\sqrt{( y^{2}+ x_{0}^{2})^{3}+ ( y^{2}+ x_{0}^{2})^{2}I_{2}+ ( y^{2}+ x_{0}^{2}) ( I_{2}^{2}- I_{4}) / 2 + ( I_{3})^{2}}}{y^{2}+ x_{0}^{2}},
\mbox{Prod}\left(\mathbf{A}^{(1)},\cdots,\mathbf{A}^{(m)}\right)=\sum_{0\le k<n}\mbox{Prod}_{\boldsymbol{\Delta}^{(k)}}\left(\mathbf{A}^{(1)},\cdots,\mathbf{A}^{(m)}\right).
\mathbb{E}[Y_i] \ge 2^{-m}\binom{l}{\frac{l}{2}}(1-\sigma^{\frac{l}{2}})(1+\sigma^{\frac{l}{2}})^{m-1}\ge \frac{2^{l-m}}{l+1}(1-\sigma^{\frac{l}{2}})(1+\sigma^{\frac{l}{2}})^{m-1}\ge \frac{2^{l-m}}{l+1}(1-\sigma^{\frac{l}{2}}).
R_1=\,|m|-\delta+\delta{{2-g\over 4}+{x^2\over 4\delta}\over 1+|m|}\,{_1F_1({2-g\over 4}+1+{x^2 \over 4\delta},2+|m|;\delta)\over_1F_1\left({2-g\over 4}+{x^2\over 4\delta},1+|m|;\delta\right)}\,.