harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tan{g\beta \tilde{A}_{0}\over 2}= i\,{1-e^{ig\beta \tilde{A}_{0}}\over 1+ e^{ig\beta \tilde{A}_{0}}}= i\left[1 - 2 e^{ig\beta\tilde{A}_{0}}+ 2 (e^{ig\beta \tilde{A}_{0}})^{2}+ \cdots \right]
g_{1}(\mathcal{A}_{X}\omega W,\mathcal{B}Y)+\eta(Y)g_{1}(X,\omega W) =g_{2}((\nabla \pi_*)(X,Y),\pi_*\omega\phi W)-g_{2}((\nabla \pi_*)(X,\mathcal{C}Y),\pi_*\omega W)
Z(G_{4,1}\vee C^{(1)}\vee G_{4,1})= \begin{cases}Z(C(3,3,2))=23 (C^{(1)}=C(\dot1,\dot1,2)) Z(C(3,1,4))=19 (C^{(1)}=C(\dot1,1,\dot2)) Z(C(3,1,1,3))=25 (C^{(1)}=C(\dot1,1,1,\dot1)) Z(C(6,2))=13 (C^{(1)}=C(\ddot2,2)). \end{cases}
\frac{1}{n^{2}}\sum_{k=1}^{[nt]}\mathrm{var}(X_{n,k}^{2})+\frac{2}{n^{2}}\sum_{k=2}^{[nt]}\sum_{u=1}^{k-1}\mathrm{cov}(X_{n,k}^{2},X_{n,u}^{2})\leq\frac{2[nt]}{n^{2}}(\max_{0\leq k\leq n}\sum_{u=0}^{k}\mathrm{cov}(X_{n,0}^{2},X_{n,u}^{2})) \leq\frac{2t}{n}(\mathrm{var}(X_{n,0}^{2})+\sum_{u=0}^{n}\mathrm{cov}(X_{n,0}^{2},X_{n,u}^{2})).
\begin{aligned}\int_X -u dd^c v \wedge \Omega^{n-1}=\int_X -v dd^c u \wedge \Omega^{n-1}-2(n-1) \int_X v du \wedge d^c \Omega \wedge \Omega^{n-2}&+ (n-1)(n-2)\int_X -v u d\Omega \wedge d^c \Omega\wedge \Omega^{n-3}.\end{aligned}
\frac{d^2}{d\theta^2}= (\dot{x_\theta})_i(\dot{x_\theta})_j D_{x_ix_j}-g_z^2q_mq_nE^{n,j}E^{m,a}D_{x_j}(E_{ab})E^{b,i}D_{x_i}&+ g_zg_{j,z}E^{n,j}E^{m,i}q_mq_nD_{x_i}.
\frac{2 a \, e^{2 \gamma a}\left(e^{2 n \gamma a}- n \, e^{n \gamma a}- 1\right)}{\left(e^{a \gamma n}-1\right)^3}= 0\Longleftrightarrow e^{2 n \gamma a}- n \, e^{n \gamma a}- 1 = 0.
\int_M \langle \nabla f(0), \nabla \eta_0 \rangle \: d\mu_0 \: = \:\int_M \langle \nabla f(1), \nabla \eta_1 \rangle \: d\mu_1 =\int_M \langle \nabla \eta_1, \nabla \eta_1 \rangle \: d\mu_1.
J(x;k)=\begin{cases}\begin{pmatrix}|r(k)|^2 \overline{r(k)}\mathrm{e}^{-2ik^2 x}r(k) e^{2ik^2 x}0\end{pmatrix},k \in \mathbb{R}, \begin{pmatrix}-|r(k)|^2 -\overline{r(k)}e^{-2ik^2x}r(k) e^{2ik^2x}0\end{pmatrix}, k \in i\mathbb{R}. \end{cases}
F_{11}F_{22}-F_{12}^2 = \bigg(\frac{\phi^{(3)}(Y)}{\phi''(Y)^2}\bigg)^2\big(Y_{11}Y_{22}-Y_{12}^2\big) &\phantom{xx}{}+ \frac{\phi^{(3)}(Y)}{\phi''(Y)^2}\bigg(\frac{1}{\phi''}\bigg)''(Y)\big(2Y_1Y_2Y_{12}- Y_1^2Y_{22}- Y_2^2Y_{11}\big) \ge 0.
\sum_{k=0}^{\infty}z^k = \lim_{p \to \infty}\int_0^p e^{-t}\sum_{k=0}^{\infty}\frac{(zt)^k}{k!}\; dt=\lim_{p \to \infty}\int_0^p e^{-t(1-z)}\; dt & =\;\; \lim_{p \to \infty}\left[ \frac{e^{-p(1-z)}}{z-1}+ \frac{1}{1-z}\right] .
c_{-1}\,:\, k\to A\otimes A ~~,~~~ 1\mapsto \frac{1}{2}\left( \begin{pmatrix}1 0\end{pmatrix}\otimes \begin{pmatrix}1 0\end{pmatrix}+ \begin{pmatrix}0 1\end{pmatrix}\otimes\begin{pmatrix}0 1\end{pmatrix}\right) \ .
B := \{t \in \chi A : |z_1^* (f_k (t)) - z_1^* (f_k (t_1)) | < 1, \, k=1,\ldots, n\}, &D := \{z^* \in \Gamma_{z_0^*}: |z^* (f_k (t_1)) - z_1^* (f_k (t_1)) | < 1, \, k=1,\ldots, n\}.
((F)(E_{1}^{n}F_{2}^{m}) \otimes F_2) \otimes w_s = nm(E_{1}^{n-1}F_{2}^{m-1}H_2 \otimes F_2) \otimes w_s & - nm(m-1)(E_{1}^{n-1}F_{2}^{m-1}\otimes F_2) \otimes w_s &+ n(E_{1}^{n-1}F_{2}^{m}E_2 \otimes F_2) \otimes w_s &- m(E_{1}^{n}F_{2}^{m-1}F_1 \otimes F_2) \otimes w_s.
\alpha = \tfrac{1}{2}\big[(-C_{1}^{*})\sigma_1 + (-C_{2}^{*})\sigma_2 + (-C_{3}^{*})\sigma_3\big] \beta = \tfrac{1}{2}\big[(-M_{1}^{*})\sigma_1 + (-M_{2}^{*})\sigma_2 + (-M_{3}^{*})\sigma_3\big],
\log(|z-w|)d\log(|w|)d\log(|z|)d\log(|z-1|)d\log(|w-1|) = \log(|z-w|)d\arg(|w|)d\arg(|z|)d\arg(|z-1|)d\arg(|w-1|) \log(|w|)d\log(|z-w|)d\log(|z|)d\log(|z-1|)d\log(|w-1|) = \log(|w|)d\arg(|z-w|)d\arg(|z|)d\arg(|z-1|)d\arg(|w-1|).
\widetilde{\mathbf{v}}\left( k\right) =\mathbf{v}\left( k\right)+\int\frac{W\left\{\mathbf{v}\left( k,\cdot\right) ,\psi\left(\cdot,\mathrm{i}s\right) \right\}}{k^{2}+s^{2}}\mathrm{d}\mu\left(s\right) ,\k\notin\Sigma,
-\sum_{i,j=1}^n \big ( R^{(s)}(x) \big )^{-1}_{ji}D_z B_{ij}\big ( x, u^{(\tau)}(x), Du^{(\tau)}(x) \big ) \leq n \frac{\delta}{1+\delta^2}\beta_1, \ \forall x \in \overline{\Omega}.
\delta_x^3Y-\bigl(3r_1+\tfrac{\delta_xq}{q}\bigr)\delta_x^2Y+\bigl(2r_1^2-2\delta_xr_1+r_0+2r_1\tfrac{\delta_xq}{q}\bigr)\delta_xY +\bigl(\delta_xr_0-r_1r_0-r_0\tfrac{\delta_xq}{q}\bigr)Y=0.
[e_1, e_1]=\alpha_2e_5, [e_1, e_2]=\alpha_3e_4+ \alpha_4e_5, [e_2, e_1]=\alpha_5e_4+\alpha_6e_5, [e_2, e_2]=\beta_2e_5, [e_3, e_3]=\gamma_6e_5.
\lim_{r \rightarrow \infty}\partial_{r}\overline{v_{M}}(r) = \int_{0}^{\infty}\overline{y}(\xi) \lim_{r\rightarrow \infty}\partial_{r}\left(\frac{\phi(r,\xi)}{\sqrt{r}}\right) \chi_{\leq 1}\left(\frac{\xi}{M}\right)\rho(\xi) d\xi=0
h\left(x_1,x_2\right)j =\int_{\widehat{G^2}}{{\mathcal{F}}_sf\left(\omega_i,\omega_j\right)\omega_j{(x}_2)\omega_i(x_1)}{d^2}_{{\mu}_{\widehat{G^2}}}\left(\omega_i,\omega_j\right)j &=\int_{\widehat{G^2}}{{\mathcal{F}}_sf\left(\omega_i,\omega_j\right)\omega_j{(x}_2)j\overline{\omega_i(x_1)}}{d^2}_{{\mu}_{\widehat{G^2}}}\left(\omega_i,\omega_j\right).
N^2\big|\langle\psi_N, q_1p_2V_\beta(x_1-x_2)\widehat{m}q_1 q_2\psi_N\rangle\big|\le \,N^2\big|\langle\psi_N, q_1p_2U_{0,\beta}(x_1-x_2)\widehat{m}q_1 q_2\psi_N\rangle\big| &+N^2\big|\langle\psi_N, q_1p_2(\Delta_1h_{0,\beta}(x_1-x_2))\widehat{m}q_1 q_2\psi_N\rangle\big|
S_{\rm massive\D0-brane}=T\int d\tau\,\Big(e^{-3\hat{\phi}/4}\sqrt{-\dot{X}^{\hat{\mu}}\dot{X}^{\hat{\nu}}\hat{g}_{\hat{\mu}\hat{\nu}}}+\dot{X}^{\hat{\mu}}\hat{A}_{\hat{\mu}}+m\,\hat{V}_t\Big)\,,
\int_0^T |u'|^2 dx \le q\biggl(\int_{\{|u|> M\}}u^2 dx + \int_{\{|u|\le M\}}u^2\, dx\biggr) &\le q\biggl(\int_{\{|u|> M\}}\frac{F(u)}{\theta\alpha}dx + M^2|\{|u(x)|\le M\}|\biggr) &\le q\biggl(\int_{0}^T \frac{F(u)}{\theta\alpha}dx + M^2T\biggr) &\le q\biggl(\frac{q}{2\theta\alpha}\int_0^T |u'|^2 dx + M^2T\biggr),
[x_k^{\ell_k}] G_{k,1}^{-\ell_1}= [x_k^{\ell_k}] (F_{k,2}- x_k)^{-\ell_1}= [x_k^{\ell_k}] F_{k,2}^{-\ell_1}\Big( 1 - \frac{x_k}{F_{k,2}}\Big)^{-\ell_1}&= F_{k,2}^{-\ell_1}\binom{-\ell_1}{\ell_k}\Big( - \frac{1}{F_{k,2}}\Big)^{\ell_k}= \binom{\ell_1+\ell_k-1}{\ell_k}F_{k,2}^{-\ell_1-\ell_k}.
h(x+\rho_{\beta ,\varepsilon,k}^{-}(x))-h(x) \geq h(x+\rho)-h(x)\geq\sum_{i=1}^{n}\frac{h^{(i)}(x)}{i!}\rho^{i}&\geq \Im(\beta) + M_{\varepsilon,k}(x),\x\in[t,+\infty).
\mathbf{v}_{m}^{\left\langle j+1\right\rangle}\negmedspace\gets\negmedspace\left[\mathbf{v}_{m}-Q_{\Lambda_{d,m}}\negmedspace\left(\negthinspace\sum_{l\in\mathcal{S}^{\left\langle j\right\rangle}}a_{ml}\negmedspace\sum_{\mu=\pi_{c}\left(l\right)}^{j}\negmedspace\hat{\mathbf{t}}_{l,\mu}\negthinspace\right)\negthinspace\right]\negthickspace\bmod\Lambda_{s}
\sum_{k=0}^{n-1}\frac{(a^2q,q^4,q/a^2;q^3)_kq^{3k}}{(aq^3,q^3/a,q^3;q^3)_k}\equiv\frac{(1-a^2q)(1-q/a^2)(q^3;q^3)_{n-1-(n+1)/3}}{(aq^3,q^3/a;q^3)_{(n+1)/3}}q^{(n^2+5n-2)/3}.
w_1 = s_1 s_2 (s_3 s_2 s_4 (s_3 s_2 s_1 s_2 s_3) s_4 s_2 s_3) s_2 s_1 w_2 = s_4 s_3 s_2 s_3 s_4 s_3 s_2 s_3 s_2
\sigma_{1}^{2}\left( A\right) =\frac{pq+rs+\sqrt{(pq+rs)^{2}-4(q-s)prs}}{2}, \sigma_{2}^{2}\left( A\right) =\frac{pq+rs-\sqrt{(pq+rs)^{2}-4(q-s)prs}}{2}, \sigma_{3}\left( A\right) =\cdots=\sigma_{q}\left( A\right) =0.
\int_{\tau_{k^* + 1}}^{\tau_{j+1}}G_{k^*}(v) \: dv = \sum_{m=1}^{j-k^*}\bigg( \int_{\tau_{k^* + m}}^{\tau_{k^* + m + 1}}G_{k^* + m}(v) \: dv - sm(\tau_{k^*+m+1}- \tau_{k^*+m})\bigg) &\geq \bigg( \sum_{m=1}^{j-k^*}\int_{\tau_{k^* + m}}^{\tau_{k^* + m + 1}}G_{k^* + m}(v) \: dv \bigg) - s (j - k^*) (\tau_{j+1}- \tau_{k^*+1}).
(i_{\mathcal{N}}u)v_{\partial\Omega}=(i_{\mathcal{N}}u)v_{\Sigma}=u(\mathcal{N})v_{\Sigma}=\boldsymbol{n}(u)(\mathcal{N})v_{\Sigma}=(\partial\Sigma_{v})^{\flat}(\mathcal{N})v_{\Sigma}=\partial\Sigma,
\Delta m^{2}(\beta)-\Delta m^{2}(\infty)=\frac{\mu^{D-2}\lambda}{2(2\pi)^{D/2}}\sum^{\infty}_{n=1}\biggl(\frac{m}{\mu^{2}\beta n}\biggr)^{\frac{D}{2}-1}K_{\frac{D}{2}-1}(mn\beta)
\sum_{u\in S'}d_G(u,x_1)+\sum_{u\in S'}d_G(u,x_2) \leq 2\sum_{u\in S'}d_G(S,u)+4(n-7)=2\sigma(S)+4(n-7) &=\frac{1}{2}\bigg((n-7)^2+2(n-7)\bigg)+4(n-7).
A_4 = A_2^2\left( \frac{1}{2}+ \frac{2A_1^2 c_3}{\nu^2}+ \frac{A_3}{8}+ \left( c_2 \nu^3 \log^3T_0 + \frac{A_1^2/6000}{\nu^2 \log^2 T_0}\right) \frac{1}{T_0^{1-2\nu}}+ \frac{c_1^2 e^{-2\alpha}/6000}{\nu^2 T_0^{5\nu}\log^2 T_0}\right) + \frac{215}{16T_0},
2\left(n(i+q)-m\left(j+\frac q2\right)\right)d^1_{0,q}(m+n,i+j) =\left(n(i+q)-m\left(j+\frac{3q}2\right)\right)d^1_{0,q}(n,j) &\quad(m,i)\ne(0,-2q), 2nd^1_{0,q}(n,j-2q) =nd^1_{0,q}(n,j), d^1_{0,q}(m,i)+d^1_{0,q}(n,j) =0(m+n,i+j)\ne(0,-2q), d^1_{0,q}(-n,-j-2q)+d^1_{0,q}(n,j) =2d^0_{0,q}(0,-2q),
\int \frac{d^{3}q}{( 2 \pi )^{3}}\; d^{2}\theta_{1}\, d^{2}\theta_{2}\; \Phi ( - p , \theta_{1}) \, \left[ - \frac{e^{2}}{4}G_{3}( p , P , q ) D^{\alpha}( q ) D^{\beta}( q ) \Delta ( q ; 1 2 ) \right] \Delta_{\alpha \beta}( P ; 1 2 ) \, \Phi^{*}( p , \theta_{2}) .
U=\frac{1}{2}\int_{0}^{a}\int_{-b/2}^{b/2}D\{(\frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\partial y^{2}})^{2}+2(1-\nu)[(\frac{\partial^{2}w}{\partial x\partial y})^{2}-\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2}}]\}dxdy
\sum_{k=N}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}= - \frac{1}{2\pi i}\int_{c-i\infty \atop{{\rm Max}[0,2N-3]<c={\rm Re}\,s<2N-1}}^{c+i\infty}\; \frac{z^{-s}\, \zeta(-s)}{\sin(s\pi)}\; ds & + \;\; \frac{1}{2\pi i}\int_{C}\frac{1}{(2\pi z)^{2s-1}}\; \frac{\Gamma(2s-1) \zeta(2s)}{\sin(s \pi)}\; ds \;\;,
\left(\frac{\Lambda(f \otimes \mathrm{Ad}(g), k)}{\langle g,g\rangle^2 c^+(f)}\right)^{\sigma}= \frac{\Lambda(f^{\sigma}\otimes \mathrm{Ad}(g^{\sigma}), k)}{\langle g^{\sigma},g^{\sigma}\rangle^2 c^+(f^{\sigma})}.
W(G_{n,r,s}) =W(C_{2r})+W(K_{n-2r})+\sum_{u \in C_{2r}, v \in K_{n-2r}}d(u,v) &=r^3+\binom{n-2r}{2}+(n-2r)(r^2+1) &=\binom{n}{2}+(r-1)^2n -r(r-1)^2
L_{J^{r+1}\Xi}h\rho= \big(J^{r+2}\Xi_H + J^{r+2}\Xi_V\big)\rfloor (d_V h\rho + d_H h\rho) + (d_V+d_H)\big(\big(J^{r+1}\Xi_H + J^{r+1}\Xi_V\big)\rfloor h\rho\big) \hphantom{L_{J^{r+1}\Xi}h\rho}{}= d_{H}\big( J^{r+1}\Xi_{H}\rfloor h\rho\big)+J^{r+2}\Xi_{H}\rfloor d_{H}h\rho+ J^{r+2}\Xi_{V}\rfloor d_{V}h\rho.
A(x_1 \mid p_{[2..\ell]}) +8 \gamma_2 + O(\log n) \stackrel{(a)}{\leq}C(x_1 \mid p_{[2..\ell]}) + 16 \gamma_2 + O(\log n) & \stackrel{(b)}{\leq}n_1 + 16 \gamma_2 + O(\ell-2) \log n &\stackrel{(c)}{<}n_1 + \eta_1 +1 = \hat{n}_1.
\begin{aligned}(E'\circ E, v):0\rightarrow E^{0}\overset{v}\rightarrow E^{1}\overset{v}\rightarrow \cdots \overset{v}\rightarrow E^{l-1}\overset{u\circ v}\rightarrow E^{l+1}\overset{u}\rightarrow\cdots \overset{u}\rightarrow E^{l+m'}\rightarrow 0.\end{aligned}
{\cal I}_{n}( \kappa ) ={\frac{n}{( n + 1 ) \kappa}}+ \int_{0}^{1 / \kappa}\Bigl ({\frac{\displaystyle 1}{\displaystyle \sqrt{1 - e^{- x}}}}- 1 \Bigr ) ( 1 - \kappa x )^{1 / n}d x .
q_{1}(\xi-\eta, \eta) =\sum_{i=1,2,3}q_{1}^i(\xi-\eta, \eta),q_{1}^1(\xi-\eta, \eta)=\big((\xi-\eta)\eta |\eta|^{-1/2}+ |\eta|^{3/2}/2\big)\theta(\eta, \xi-\eta)
I_{E}^{\mathrm{b u l k}}= - \frac{1}{1 6 \pi G}\int_{\cal{M}}d^{D}x \sqrt{g}\Big ( R - \frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi - \frac{1}{2 ( p + 2 ) !}e^{a \phi}F_{p + 2}^{2}\Big )
J^{\mathrm{2 - l o o p}}( \phi_{B}) = \frac{\lambda_{0}\phi_{B}^{3}}{1 + \frac{9 \lambda_{0}}{1 6 \pi^{2}}\ln \frac{\Lambda^{2}}{3 \lambda_{0}\phi_{B}^{2}}}+ \frac{\lambda_{0}^{3}\phi_{B}^{3}}{2 5 6 \pi^{4}}\left[ 2 7 \ln \frac{\Lambda^{2}}{3 \lambda_{0}\phi_{B}^{2}}+ 5 4 + 4 3 2 \Omega ( 1 ) \right]
\max_{\substack{\tfrac{1}{\log X}\leq \sigma' \leq 2 |t'-t_1| \leq T_0}}\frac{|F_0(1+\sigma' + it')|^{1/2}}{|1+\sigma' + it'|}\ll \max_{|t'-t_1| \leq T_0}|F_0(1+\tfrac{1}{\log X}+ it')|^{1/2}+ 1 \ll \frac{(\log X)^{1/2}}{\exp(M(f; X)/2)}+ 1.
u_{\underline{{m}}}^{\underline{{a}}}\equiv ( 1 / 2 ( u_{\underline{{m}}}^{+}+ u_{\underline{{m}}}^{-}) , u_{\underline{{m}}}^{i}, 1 / 2 ( u_{\underline{{m}}}^{+}- u_{\underline{{m}}}^{-}) ) .
\rho^N(\bar{x}, t) \leq\int_{A_1}f_0^N(x,v)\d\bar{v}+C\int_{A_1^c}e^{-\lambda |v|^2}\d\bar{v}& \leq CN^{3\alpha}+ C\int_{A_1^c}e^{-\lambda |v|^2}\d\bar{v}.
\begin{bmatrix}-P^{-1}0 \lambda^t I_n I_n\end{bmatrix}\begin{bmatrix}-P \lambda^t P \lambda^t P Q(\lambda)\end{bmatrix}\begin{bmatrix}I_n -\lambda^t I_n 0 I_n\end{bmatrix}=\begin{bmatrix}I_n 0 0 P(\lambda)\end{bmatrix},
\int_{\mathbb{R}^N}\phi(t_R|\nabla w_R|)(t_R|\nabla w_R|)^2 dx+\int_{\mathbb{R}^N}\phi(t_R| w_R|)(t_R|w_R|)^2 dx \leq\int_{\mathbb{R}^N}m \Phi(t_R |\nabla w_R|) dx+\int_{\mathbb{R}^N}m\Phi(t_R|w_R|) dx &=m\int_{\mathbb{R}^N}\Phi(t_R|\nabla w_R|)+\Phi(t_R|w_R|) dx &\leq m\xi_1(t_R) \int_{\mathbb{R}^N}\Phi(|\nabla w_R|)+\Phi(|w_R|) dx,
-\frac{\partial^{2}lnL(\alpha,\beta|X)}{N\partial\alpha^{2}}=var[ln(X)]=\psi_{1}(\alpha)-\psi_{1}(\alpha+\beta)=I_{\alpha,\alpha}=E[-\frac{\partial^{2}lnL(\alpha,\beta|X)}{N\partial\alpha^{2}}]=lnvar_{GX}
\frac{1}{n}\sum_{k=0}^{n-1}\ln( \vert \varphi_{a,b}'(y_k)\vert ) = \frac{1}{n}\sum_{k=0}^{n-1}\ln\left( \left\vert \varphi_{0,1}'\left(\frac{y_k-a}{b}\right)\right\vert \right) = \frac{1}{n}\sum_{k=0}^{n-1}\ln\left( \left\vert \varphi_{0,1}'\left( \varphi_{0,1}^{(k)}\left(\frac{y_0-a}{b}\right) \right)\right\vert \right) \to \ln(2)
ds^2 = e^{-2k\mid y \mid}\left[ -\left( 1 - \frac{U_T^4}{k^4}e^{4 k \mid y \mid}\right) dt^2 +\sum^3_{i=1}dx^i dx^i \right] + \frac{dy^2}{1 - \frac{U_T^4}{k^4}e^{4 k \mid y \mid}}
A_{1}= - \frac{\sqrt{N ( \rho_{+}+ \rho_{-})}}{2 \sqrt 2}E_{1}\left( 1 - \Psi ( \alpha , \beta ) \right) \; \; A_{2}= \frac{\pi E \Lambda}{2 \sqrt 2}\sqrt{\rho_{+}- \rho_{-}}E_{1}
f_{0}(x_{n}^{1}) = (c_{n+1}^{11}I_{n+1}^{-1}(x_{n}^{1}) + \alpha_{n+1}^{11}y_{f_{0}}^{1}(I_{n+1}^{-1}(x_{n}^{1}))+ \beta_{n+1}^{11}z_{f_{0}}^{1}(I_{n+1}^{-1}(x_{n}^{1}))+ d_{n+1}^{11}, &\gamma_{n+1}^{11}z_{f_{0}}^{1}(I_{n+1}^{-1}(x_{n}^{1}))+ e_{n+1}^{11}I_{n+1}^{-1}(x_{n}^{1})+ f_{n+1}^{11}) & =(y_{n}^{1}, z_{n}^{1}) .
\begin{cases}\displaystyle \Delta v_1 +e^{u_2}(1-e^{u_1})=\frac{4(\epsilon N_1+\beta_1)}{(1+|x|^2)^2}, \displaystyle \Delta v_2 +e^{u_1}(1-e^{u_2})=\frac{4(\epsilon N_2+\beta_2)}{(1+|x|^2)^2},\end{cases}\quad\mathbb R^2.
\triangle \alpha_1 = \frac{\partial \tau_1}{\partial \xi}+ \frac{\partial \tau_2}{\partial \eta},\triangle \alpha_2 = \frac{\partial \tau_2}{\partial \xi}- \frac{\partial \tau_1}{\partial\eta}, \triangle \equiv \frac{\partial^2}{\partial \xi^2}+ \frac{\partial^2}{\partial\eta^2}.
\phi^{\prime}( \vartheta , \nu ) = \displaystyle \frac{1}{p + 1}\displaystyle \frac{2 \sin \displaystyle \frac{2 \nu \pi}{p + 1}}{\cosh \displaystyle \frac{2 \vartheta}{p + 1}- \cos \displaystyle \frac{2 \nu \pi}{p + 1}}.
\Phi = \left( \begin{array}{c c}{{- H C}}&{{0}}\\{{0}}&{{- H C}}\\ \end{array}\right) , \, \, \, \, \, \,{\cal K}= \left( \begin{array}{c c}{{K^{1 1}}}&{{K^{1 2}}}\\{{K^{2 1}}}&{{K^{2 2}}}\\ \end{array}\right) , \, \, \, \, \, \, \chi^{\mu}= \left( \begin{array}{c}{{K^{1 3}\psi^{3 \mu \dagger}}}\\{{K^{2 3}\psi^{3 \mu \dagger}}}\\ \end{array}\right) .
- i \bar{\kappa}_{( \alpha )}\gamma^{\mu}\kappa_{( \beta )}\partial_{\mu}= - i \tilde{\mathcal{C}}_{\alpha \gamma}\Gamma_{s}( T^{I})^{\gamma}{}_{\beta}\Gamma_{\mathrm{A d j}}( u^{- 1})^{a}{}_{I}e_{a}= - i \tilde{\mathcal{C}}_{\alpha \gamma}\Gamma_{s}( T^{I})^{\gamma}{}_{\beta}k_{( I )}\, ,
S_{B}= \frac{2 \pi}{e^{2}}\int d^{3}x ( K^{- 1})_{a b}\left( \frac{1}{4}F_{\mu \nu}^{a}F_{\mu \nu}^{b}+ \frac{1}{2}\partial_{\mu}\phi^{i a}\partial_{\mu}\phi^{i b}\right)
w_{d}(\boldsymbol{\zeta})= \limsup_{H\to\infty}\max_{\Vert L\Vert_{\infty}\leq H}-\frac{\log L(\boldsymbol{\zeta})}{\log H}, \widehat{w}_{d}(\boldsymbol{\zeta})= \liminf_{H\to\infty}\max_{\Vert L\Vert_{\infty}\leq H}-\frac{\log L(\boldsymbol{\zeta})}{\log H},
P_{N\beta}(\theta_1,\ldots,\theta_N) \, d\theta_1\cdots d\theta_N=C_{N\beta}\prod_{1\leq \ell<j\leq N}\left\vert \exp(i\theta_\ell)-\exp(i\theta_j)\right\vert^\beta \, d\theta_1\cdots d\theta_N
\sum_{i=1}^{m}\mathbb{I}\left\{\sum_{j \in T}w_{ij}\ge\frac{\epsilon(a-b)|T|}{n}\right\}\le m \exp \left( - \frac{\epsilon^2 (a-b)^2}{C\beta^2 a k^2}\right) + \frac{|T \Delta T^*|}{4\beta k}
Z=\int Ds_i D\phi DA_i\exp\left\{-\frac{1}{2}[ A_iG^{-1}A_i+ A_i^{\phi,s}G^{-1}A_i^{\phi,s}]- i\kappa [(s_i-\partial_i\phi)\epsilon_{ij}A_j+\epsilon_{ij}\partial_i s_j]\right\}
\gamma_{J}^{I}s^{L}= \delta_{J}^{L}s^{I}+ \sum_{L_{1}L_{2}= L}\delta_{J}^{L_{1}}s^{I L_{2}}+ \sum_{L_{1}L_{2}= L}\delta_{J}^{L_{2}}s^{L_{1}I}+ \sum_{L_{1}L_{2}L_{3}= L}\delta_{J}^{L_{2}}s^{L_{1}I L_{3}}.
z = \chi(t/\tilde{T})\left(W_t^\gamma T[u_0, \cdots, u_0]-\frac{f}{\bigtriangleup+i\gamma}\right) &\qquad-i\chi(t/\tilde{T})\int_0^tW_{t-s}^\gamma\left(\mathfrak{F}(v) + \mathcal{HL}_B[z, v, \cdots, v]\right)\,ds.
\tilde{\phi}= - \operatorname{l o g}\bigl ( \frac{\tilde{\rho}^{2}+ \frac{\tilde{Q}^{2}}{G M}\tilde{\rho}+ A^{2}\operatorname{c o s}^{2}\theta}{\tilde{\rho}^{2}+ A^{2}\operatorname{c o s}^{2}\theta}\bigr )
c_{1}=\frac{\Gamma(1/2)}{(2\pi)^{1/2}}\left[\int_{0}^{\infty}s^{-1/2}g_{1}(s)\mathrm{d}s\right]^{-1}&=\frac{1}{\sqrt{2}}\left[\int_{0}^{\infty}s^{-1/2}g_{1}(s)\mathrm{d}s\right]^{-1}.
\int_{[X_{0,n,0}]^{\rm virt}}c_1(\mathcal{L}_1)^{i_1}{\rm ev}_1^*(\phi_{\alpha_1}) \cdots c_1(\mathcal{L}_n)^{i_n}{\rm ev}_n^*(\phi_{\alpha_n}) = \int_{\overline{\mathcal{M}}_{0,n}}\psi_1^{i_1}\cdots \psi_n^{i_n}\int_X \phi_{\alpha_1}\cdots\phi_{\alpha_n}.
\langle r^{2}\rangle=\frac{\int r^{2}\cdot e^{-\frac{Yaln(r/a)}{4\pi k_{B}T}}d^{2}r}{\int e^{-\frac{Yaln(r/a)}{4\pi k_{B}T}}d^{2}r}\sim\frac{2-\frac{Y\cdot a}{4\pi k_{B}T}}{4-\frac{Y\cdot a}{4\pi k_{B}T}}.
U^{\prime}_{[4]\times[N]}= - \left( \! \begin{array}{cccc}U^{\prime}_{11}U^{\prime}_{12}\cdots U^{\prime}_{1N}U^{\prime}_{21}U^{\prime}_{22}\cdots U^{\prime}_{2N}U^{\prime}_{31}U^{\prime}_{32}\cdots U^{\prime}_{3N}U^{\prime}_{41}U^{\prime}_{42}\cdots U^{\prime}_{4N}\end{array}\! \right),
{\bf X_{L}}: ={\frac{\partial}{\partial t}}+ \dot{q}^{s}{\frac{\partial}{\partial q^{s}}}+ a^{s}( q , \dot{q}){\frac{\partial}{\partial \dot{q}^{s}}}+ \eta^{\mu}{\bf \Gamma}_{\mu}= :{\bf X}_{0}+ \eta^{\mu}{\bf \Gamma}_{\mu},
\phi^{\zeta\theta}(e_{vw}) := (\phi^\zeta)^\theta(e_{vw}) = \tau^\zeta(v,e)\theta(v) + \phi^\zeta(e_{vw}) + \tau^\zeta(w,e)\theta(w) &= \zeta(v)\tau(v,e)\theta(v) + \phi(e_{vw}) + \zeta(w)\tau(w,e)\theta(w) &= \tau(v,e)\theta^\zeta(v) + \phi(e_{vw}) + \tau(w,e)\theta^\zeta(w) &= \phi^{\theta^\zeta}(e_{vw}) = \phi^{\theta^\zeta \zeta}(e_{vw})
\small g_3 =2\eta_1+1\eta_2+8\eta_3 g_2 =g_3\eta_1+2\eta_2+1\eta_3 g_1 =g_2\eta_1+g_3\eta_2+2\eta_3 g_0 =g_1\eta_1+g_2\eta_2+g_3\eta_3
\sum_{n=0}^{\infty}\sqrt{\frac{2}{\pi}}x^{n+\frac{1}{2}}e^{x}K_{n-\frac{1}{2}}(x)\frac{t^{n}}{n!}=1+x\sum_{n=1}^{\infty}\theta_{n-1}(x)\frac{t^{n}}{n!}=e^{x(1-\sqrt{1-2t})}.
S_{n+1}f(\tfrac{1}{2})=\sum_{i=0}^{n}f\big(T^{i}(\tfrac{1}{2})\big)=\sum_{i=1}^{n}f\big(T^{i}(\tfrac{1}{2})\big)=\sum_{i=1}^{n}f\big(T^{i}\circ \mathcal I(\tfrac{1}{2})\big).
m_{n,d}(u) = (1-u^2)^{d-\frac{3}{2}}\frac{c_{d-1}}{(d-1)!}\sum_{k=0}^\infty \frac{(d-1)_k}{k!}\frac{C_{n+2k}^{d-1}(u)}{C_{n+2k}^{d-1}(1)},-1\leq u\leq1,
\int_{{\cal M}_{\alpha}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}=\alpha\int_{{\cal M}_{\alpha=1}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}+8\pi(1-\alpha)\int_{\Sigma}R_{\mu\nu\lambda\rho}n^{\mu}_in^{\lambda}_in^{\nu}_j n^{\rho}_j+O((1-\alpha)^2)~~~,
\chi_{1 / 2}^{1 , 2}| \Omega{\rangle}_{L}\otimes{\Psi}_{1 / 2}^{a}{\Psi}_{1 / 2}^{i *};{\Psi}_{1 / 2}^{a *}{\Psi}_{1 / 2}^{i}| \Omega{\rangle}_{R}\mathrm{a n d e x c h a n g e}\ \Psi \longrightarrow \Phi ,
g_{i}({\bf{q}}) = [ \sum_{{\bf{k}}}\frac{{\bar{n}}_{{\bf{k}}-{\bf{q}}/ 2}-{\bar{n}}_{{\bf{k}}+{\bf{q}}/ 2}}{({\tilde{\omega}}_{i}({\bf{q}}) - \frac{{\bf{k . q}}}{m})^{2}}]^{- \frac{1}{2}}
A_{n}\left( x;z\right) =\frac{\phi\left( x;z\right)}{2\gamma_{n}\left(z\right) C_{n}\left( x;z\right)},B_{n}\left( x;z\right)=\frac{n-2\gamma_{n}\left( z\right)}{2\gamma_{n}\left( z\right)C_{n}\left( x;z\right)}x,
K_{1}'(x) = \lambda P_n(0)\int_{z=x}^{\infty}\int_{s=0}^{z-x}V(z-s)f_{B}(s) \lambda e^{-\lambda(z-x)}ds dz - P_n(0)\int_{z=x}^{\infty}f_{B}(z-x) \lambda e^{-\lambda(z-x)}dz.
X_1 =\begin{pmatrix}1 1 1 \epsilon^{\frac{1}{3}}\epsilon^{\frac{1}{3}}e^{i2\pi\frac{1}{3}}\epsilon^{\frac{1}{3}}e^{i2\pi\frac{2}{3}}\epsilon^{\frac{2}{3}}\epsilon^{\frac{1}{3}}e^{i2\pi\frac{2}{3}}\epsilon^{\frac{1}{3}}e^{i2\pi\frac{4}{3}}0 0 0 \vdots \vdots \vdots 0 0 0\end{pmatrix}
W_{n_*{\scriptscriptstyle (K)}}^{{\scriptscriptstyle 0}}= \sum_{p=n_*{\scriptscriptstyle (K)}}^\infty \frac{\pi_p}{\lambda_{n_*{\scriptscriptstyle (K)}-1}\pi_{n_{*}\scriptscriptstyle{(K)}-1}}= I_{1}+ I_{2}+I_{3}.
\lim_{n \uparrow \infty}V_q^n(x):=\lim_{n \uparrow \infty}q\mathbb E_x\left[\int_{0}^{\infty}e^{-qt}e^{-p\int_{0}^{t}\rm{\bf{1}}_{\{X^n_s \leq b\}}ds}\rm{\bf{1}}_{\{X^n_{t}>y\}}dt\right]=V_q(x).
\lim_{M\rightarrow\infty}\sup_{n\in\mathbb{N}}\mathbb{P}\Big(\sup_{t\leq T}\Vert Y_n(t)\Vert_H >\sqrt{M}\Big) \leq \lim_{M\rightarrow\infty}\sup_{n\in\mathbb{N}}\mathbb{P}(\tau_{n}^M <T) =0 ,
\left\{\begin{array}{c}\partial_\mu(LA_{\nu}^{a})-\partial_\nu(LA_{\mu}^{a}) +\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c}L^{2}\left(\begin{array}{cc}0 1 -1 0\end{array}\right)\end{array}\right\}\left(\begin{array}{c}e^{a}f^{a}\end{array}\right)=0,
\Bigl( \frac{1}{16 \pi G}\Bigr)^2 S_1[h] = \Bigl( \frac{1}{G\mu^2}\Bigr)^2\int d^3x \sqrt{h(x)}\mu^3 \biggl\{c_1 +c_2 \frac{R(x)}{\mu^2}\biggr\}~~~ ,
Op(a)(v_1, \dotsc, v_n) = \frac{1}{(2\pi)^n}\int_{\mathbb{R}^n}e^{ix(\xi_1 + \dots + \xi_n)}a(x, \xi_1, \dotsc, \xi_n)\prod_{l=1}^n \hat{v}_j(\xi_l)\, d\xi_1 \dots d\xi_n \, .
\begin{cases}{}^C\left(t^{1-2H}\frac{\partial}{\partial t}\right)^{\alpha}u_{\alpha}(x,t)= H^{\alpha}\frac{\partial^2}{\partial x^2}u_{\alpha}(x,t),\alpha \in (0,1), \: x\in\mathbb{R}, \: t>0, u_{\alpha}(x,0)=\delta(x),\end{cases}
\frac{d}{dt}M_{\varphi_{R}}[\psi(t)] \leq8\|\Delta\psi(t)\|_{2}^{2}-8\mu\gamma_{q}\|\psi(t)\|_{q}^{q}-8\|\psi(t)\|_{4^*}^{4^*}&+(C\xi+R^{-16})\|\Delta\psi(t)\|^{2}_{2}+O(R^{-4}+\xi^{-\frac{q-2}{10-q}}R^{-\frac{4(q-2)(N-1)}{10-q}})(N = 5),
a a^{-1}= a^{-1}a = 1,y^{2}= x^{2}+ x^{3},b^{2}= a^{3}, ba = ab,ya = ay,bx = xb,yx = xy,by = -yb+2pb^{2}, a^2 x = - xa^{2}- axa - a^2 +\left( 1 + 3q \right) a^{3}, ax^2= -ax-xa-x^2a-xax+(2+3q)qa^3.
\big(\rho(a)-\pi^*(\psi(a))\big).f= \omega(a_f,a)-\pi(\psi(a),df) = \omega(a_f,a)+\psi(a)(\pi^*(df)) = \omega(a_f,a)-\omega(a,\lambda(\pi^*(df))) = \omega(a_f+\lambda(\pi^*(df)),a) = 0.
\lim_{h\rightarrow0}\frac{1}{\|h\|}_{\!H^1}\!\bigg|V(u+h)-V(u)-\int_{[0,1]}u'h' \bigg|= \lim_{h\rightarrow0}\frac{1}{2\|h\|}_{\!H^1}\!\bigg|\int_{[0,1]}(h')^2\bigg|=0\,.
A_{mn}(x):=(-1)^{m+n}\left( \sum_{k=1}^{N}\alpha_{k}^{+}e^{-2\tau_{k}x}\frac{\left( \frac{1}{2}-\tau_{k}\right)^{m+n}}{\left( \frac{1}{2}+\tau_{k}\right)^{m+n+2}}+\frac{1}{2\pi}\int_{-\infty}^{\infty}s^{+}\left(\rho\right) e^{2i\rho x}\frac{\left( \frac{1}{2}+i\rho\right)^{m+n}}{\left( \frac{1}{2}-i\rho\right)^{m+n+2}}d\rho\right) ,
\mathbf{log}\, \big( \mathbf{e}^{\mathbf{log}\, (a,\pi )}\ominus_\mathrm{exp}\mathbf{e}^{\mathbf{log}\, (a,-\pi )}\big) =\mathbf{log}\, \mathbf{e}^{\ln a+i\pi -\ln a+i\pi}=\mathbf{log}\,(e^0,2\pi )=\ln 1+2\pi i=2\pi i.
\lambda(r_-,p_+)=\int_{|z|=1}\Bigl( \ln \frac{r_-(z)}{z}\,\, d\ln \frac{q_+(\gamma_r(z))}{r^+(\gamma_r(z))}-\ln \frac{p_+(\gamma_p(z))}{\gamma_p(z)}\,\, d\ln \frac{q_-(z)}{p^-(z)}\Bigr).
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=(2^{(\dim M_2^{K_{\ell}}-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^{K_{\ell}}|})(2^{(\dim M_2^L-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^L|})^{-1}\leq 2^{(n_{2,\nu_2}-1)/2}+1
0 > (\alpha_1 - \alpha_2 ) \int_{r_1}^{\infty}w f_1 f_2 = \underset{r\to \infty}{\lim}(PW)(r) - (PW)(r_1) & = \underset{r\to \infty}{\lim}[P(r) (f_1(r) f_2'(r) - f_1'(r) f_2(r) )] - P(r_1) (f_1(r_1) f_2'(r_1) - f_1'(r_1) f_2(r_1) ).
\overline{\partial}(\Phi_1^{\mu+t \tilde\mu}\circ (\Phi_1^\mu)^{-1})=\left(\frac{t\tilde\mu}{1-\vert\mu\vert^2}\frac{\partial \Phi_1^{\mu+t\tilde\mu}}{\overline{\partial \Phi_1^\mu}}\right)\circ (\Phi_1^\mu)^{-1},
\left( \begin{array}{c}{{a_{D}( u )}}\\{{a ( u )}}\\ \end{array}\right) \rightarrow M_{i}^{H i g g s}\left( \begin{array}{c}{{a_{D}( u )}}\\{{a ( u )}}\\ \end{array}\right) = \left( \begin{array}{c}{{a_{D}( u e^{2 \pi i})}}\\{{a ( u e^{2 \pi i}) = - a}}\\ \end{array}\right) ,
\lim_{T\to\infty}\frac{1}{T}\mathbb{E}_x \Biggl(\int_0^T \sum_{k=1}^K \sum_{j=1}^{J_k}\sum_{a=0}^{1}C_{k}(j,a) X_{j,k}^{\pi,a}(t)\,\mathrm{d}t \Biggr),
Z_{\mathrm{O}}= \int{\cal D}q_{a}^{1}{\cal D}p_{a}^{n_{a}}\exp i \int d t [ p_{a}^{n_{a}}q_{a}^{1 ( n_{a})}- \hat{H}_{\mathrm{O}}( q^{1}, q^{1 ( s )}, p^{n}) ] ,
x^2 \nu_y = q^4 \nu_y x^2, x^2\nu_z = q^{-4}\nu_z x^2, y^2 \nu_z = q^4 \nu_z y^2, y^2\nu_x = q^{-4}\nu_x y^2, z^2 \nu_x = q^4 \nu_x z^2, z^2\nu_y = q^{-4}\nu_y z^2.