harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|u_{n,k,m}|\leq v_{n,k,m}\leq c_{1}\left( 1+D+\left( \frac{|\sigma|(\Omega )+|\mu |(\Omega_{T})}{D^{N}}\right)^{m_{3}}+\mathbb{I}_{2}^{2D}\left[ T_{n}(|\sigma |)\otimes \delta_{\{t=0\}}\right] \right) & +c_{1}\varphi_{m}\ast \mathbb{I}_{2}^{2D}\left[ \mu_{1,n}+\mu_{2,n}\right] .
\varphi^{\prime}( x ) = \left( \begin{array}{c}{{\chi^{\prime}( x )}}\\{{\xi^{\prime}( x )}}\\ \end{array}\right) , \hspace{0 . 3 i n}\beta_{\mu}^{\prime}= - \left( \begin{array}{c c}{{0}}&{{\alpha_{\mu}^{\prime}}}\\{{\overline{{{\alpha}}}_{\mu}^{\prime}}}&{{0}}\\ \end{array}\right) .
\phi(\alpha, \tau) = \widetilde \alpha \widetilde \tau{\widetilde \alpha}^{-1}\cdot{\widetilde \tau}^{-1}= \alpha(x_{i_1}\cdots x_{i_k}) \cdot (x_{i_1}\cdots x_{i_k})^{-1}=x_{\alpha(i_1)}\cdots x_{\alpha(i_k)}\cdot (x_{i_1}\cdots x_{i_k})^{-1}= 1.
\lim_{v\to 0^+}\int_{x-\delta}^{x+\delta}\frac{(x-t)\varphi(t)dt}{(x-t)^2+v^2}=\lim_{v\to 0^+}\int_{x-\delta}^{x+\delta}\frac{\vert (x-t)\varphi(t)\vert dt}{(x-t)^2+v^2}=+\infty
\pm(1-(1+\Phi_1(\delta))c_1\delta^{\frac{p}{n-1+p}})e_{i}\in \partial(B_p^n)_{\delta},1\leq i \leq n, \left(1-(1+\Phi_2(\delta))c_2\delta^{\frac{2}{n+1}}\right)\frac{1}{\sqrt[p]{n}}\sum_{i=1}^n \pm e_i\in \partial (B_p^n)_{\delta}
z=Hr=(\begin{matrix}1&0&1&0&1&0&1\\ 0&1&1&0&0&1&1\\ 0&0&0&1&1&1&1\\ \end{matrix})(\begin{matrix}0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\end{matrix})=(\begin{matrix}2\\ 4\\ 2\end{matrix})=(\begin{matrix}0\\ 0\\ 0\end{matrix})
\begin{array}{r l l}{{{\cal E}( A , B ) = \int_{M_{3}}< ( H^{A}\otimes I_{E}- I_{E}\otimes K^{B})}}&{{\wedge}}&{{( D^{A}\Phi_{A}\otimes I_{E}- I_{E}\otimes D^{B}\Phi_{B}) >}}\\{{- \int_{M_{3}}< ( D^{A}\Phi_{A}\otimes I_{E})}}&{{\wedge}}&{{( H^{A}\otimes I_{E}) >}}\\ \end{array}
q_3 = -3\cdot t_3, q_7 = -1575\cdot t_7, q_{11}= -9823275\cdot \left(t_{11}- t_3^2t_5 \right), q_5 = 45\cdot t_5, q_9 =\textstyle 99255\cdot \left(t_9 - \frac{1}{3}t_3 \right), q_{13}= 1404728325\cdot \left(t_{13}- t_3^2t_7 - t_3t_5^2 \right).
\Omega(s) = \int_{\Gamma_{0}(4k) \backslash H}\overline{f(z)}\theta(z) M(s,z) \frac{dx dy}{y^{2}}&=\frac{1}{2\sqrt{k}}\int_{0}^{\infty}\int_{\Gamma_{0}(4k) \backslash H}\overline{f(z)}\theta(z)\sum_{h_1,h_2 \in \mathbb{Z}}\exp\big( -\frac{4\pi t^2}{y}\big| \frac{h_1}{4\sqrt{k}+\sqrt{k}zh_2\big|^2\big) t^{s+1}}dt/t.
\begin{aligned}T^0(X,Y)+T^0(I_1X,I_1Y)+T^0(I_2X,I_2Y)+T^0(I_3X,I_3Y)=0, U(X,Y)=U(I_1X,I_1Y)=U(I_2X,I_2Y)=U(I_3X,I_3Y). \end{aligned}
F_q(\theta_k)=\frac{1}{2\pi\sigma_\eta^2}\frac{\operatorname{e}^{-\frac{\theta_k^2}{\sigma_\eta^2}}}{\operatorname{Q}\big(\frac{\theta_k}{\sigma_\eta}\big)\operatorname{Q}\big(-\frac{\theta_k}{\sigma_\eta}\big)},
\mathrm{P}_\mathrm{T}(\theta) = \mathbb{P}\left[\displaystyle\sum_{i=1}^M \mathrm{P}_i \, h_i \, d_{\mathrm{S},i}^{-\alpha_\mathrm{L}}> \theta \frac{\mathrm{W}+ \mathrm{I}}{\mathrm{G}_{\mathrm{S,TX}}\,\mathrm{G}_{\mathrm{RX}}}\right].
\phi_{lm}\left( \mathbf{D}\left( \alpha,\gamma,\beta\right) \mathbf{r}\right) =\sum_{m_{1}=-l}^{l}\left[ D^{\left( l\right)}\left(\alpha,\gamma,\beta\right) \right]_{mm_{1}}^{\ast}\phi_{lm_{1}}\left(\mathbf{r}\right) .
\epsilon_{i}^{a}( \theta ) + \Omega_{k}\ast L_{i}^{b}( \theta - \sigma_{b a}) = \sum_{j = 1}^{k - 1}I_{i j}^{s u ( k )}\Omega_{k}\ast ( \epsilon_{j}^{a}+ L_{j}^{a}) ( \theta ) , \quad a \neq b .
K ( ( x_{1}, y_{1}) , ( x_{2}, y_{2}) , t ) ={\frac{1}{4 \pi t}}( \exp ( -{\frac{( x_{1}- x_{2})^{2}+ ( y_{1}- y_{2})^{2}}{4 t}}) - \exp ( -{\frac{( x_{1}- x_{2})^{2}+ ( y_{1}+ y_{2})^{2}}{4 t}}) )
{{{\bf{\widehat h}}}_{kl}}\!=\! \sqrt{{\tau_p}{\rho_{\rm{p}}}}{{\bf{R}}_{kl}}{\bf{\Psi}}_{{t_k}l}^{- 1}{\bf y}_{{t_k}l}^{\rm{p}}\sim{\cal N}_{\mathbb C}\left({{\bf 0},{\tau_{\rm{p}}}{\rho_{\rm{p}}}{{\bf{R}}_{kl}}{\bf{\Psi}}_{{t_k}l}^{- 1}{{\bf{R}}_{kl}}}\right),
\exp^{\diamond}\big[ \int_0^{T_0}\int_{\mathbb{R}}(e^{ix\psi(s,\zeta)}-1)\tilde{N}(ds,d\zeta)] &= \exp\big[\int_0^{T_0}\int_{\mathbb{R}}ix\psi(s,\zeta)\tilde{N}(ds,d\zeta)- \int_0^{T_0}\int_{\mathbb{R}}(e^{ix\psi(s,\zeta)}-1-ix\psi(s,\zeta))\nu(d\zeta) \big]
\begin{aligned}\widetilde{\Pi}(\alpha)+\kappa(\alpha-1)^{1-\kappa}p(\alpha-1)\int_{\beta}^{\alpha-1}\frac{t^{\kappa-1}}{\sigma(t-1)}\,dt =2, \widetilde{\Xi}(\alpha)-\kappa(\alpha-1)^{1-\kappa}q(\alpha-1)\int_{\beta}^{\alpha-1}\frac{t^{\kappa-1}}{\sigma(t-1)}\,dt =0.\end{aligned}
A_0 = 0\ , \ \A_r =\frac{H_1}{2gr}\tau_\varphi\ , \ \A_\theta = \frac{\left(1-H_2\right)}{g}\tau_\varphi, \ \A_\varphi = -\frac{\sin\theta}{g}\left(H_3 \tau^{(2)}_r+\left(1-H_4\right)\tau^{(2)}_\theta\right)\ \ ,
n_{a\beta \gamma}^{b}=g \big{(}X_b, R(X_{\beta}, X_{\gamma})X_a \big{)}+ \omega_{a}^{\rho}(X_{\beta}) \omega^{b}_{\rho}(X_{\gamma})- \omega_{a}^{\rho}(X_{\gamma}) \omega^{b}_{\rho}(X_{\beta}) &=R_{\beta \gamma a}^b+ \omega_{a}^{\rho}(X_{\beta}) \omega^{b}_{\rho}(X_{\gamma})- \omega_{a}^{\rho}(X_{\gamma}) \omega^{b}_{\rho}(X_{\beta}).
I(M; Y^{(n)}(\Delta_n)) =\sum_{i=1}^{n}h(Y^{(n)}(t_{n, i})-Y^{(n)}(t_{n, i-1})|Y^{(n),t_{n, i-2}}_{t_{n, 0}})-\sum_{i=1}^{n}h(B(t_{n, i})-B(t_{n, i-1})) & \leq \sum_{i=1}^{n}h(Y^{(n)}(t_{n, i})-Y^{(n)}(t_{n, i-1}))-\sum_{i=1}^{n}h(B(t_{n, i})-B(t_{n, i-1})).
\int_K\frac{\partial^2 R_Ku}{\partial x_2^2}\frac{\partial^2 v_h}{\partial x_2^2}\;\mathrm{d}x_1\mathrm{d}x_2=\frac{1}{2}\int_K(a_6+a_8)\frac{\partial^3v_h}{\partial x_2^3}\;\mathrm{d}x_1\mathrm{d}x_2.
\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(i+j+1)(2j+3)}\equiv\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(i+j+1)(2j-1)}-4p^2+8p(-1)^{\frac{p-1}{2}}\pmod{p^3},
K_{i\omega}(\mu\rho)\simeq{i\pi \over 2\sinh \pi \omega}\left[{1 \over \Gamma(i\omega+1)}\left({\mu\rho \over 2}\right)^{i\omega}-{1 \over \Gamma(-i\omega+1)}\left({\mu\rho \over 2}\right)^{-i\omega}\right]~~~.
\begin{aligned}0 =\bold{pr}X\left(\frac{u'}{u}-u_1+u_{-1}\right)\Big|_{\frac{u'}{u}-u_1+u_{-1}=0}&=-\frac{\phi}{u}(u_1-u_{-1})+\frac{\phi_t}{u}+\phi_u(u_1-u_{-1})-(\xi_t+\xi_uu(u_1-u_{-1}))(u_1-u_{-1})-S\phi+S_{-1}\phi.\end{aligned}
(\widetilde{f_1}\circ \widetilde{f_2})(x)=\widetilde{f_1}(\widetilde{f_2}(x))=\widetilde{f_1}(f_2(x))=\widetilde{f_1}(F_2(x))=F_2(\widetilde{f_1}(x))=F_2(F_1(x)).
\partial_k\bigl(\alpha(\beta_1,\ldots,\beta_n)\bigr)=\sum_{i=1}^n(\partial_k\beta_i)a_i\beta_1^{a_1}\ldots\beta_i^{a_i-1}\ldots\beta_n^{a_n}=\sum_{i=1}^n(\partial_k\beta_i)(\partial_i\alpha)(\beta_1,\ldots,\beta_n).
\mathcal{N}(z_-,z_+,F_s,J'_s)\longrightarrow \mathcal{N}(\widetilde{f}(z_-),\widetilde{f}(z_+),G_s,J''_s) v(s,t)\longrightarrow u(s,t)=\phi_L^t(v(s,t))
\sum_{k=1}^\infty (-1)^k\frac{\rho_k}{k}Y^k =\sum_{l=1}^\infty\frac{(-1)^l}{l}\sum_{a_1+\ldots+a_n=l}\frac{l!}{a_1!\ldots a_n!}\sigma_1^{a_1}\ldots\sigma_n^{a_n}Y^{a_1+2a_2+\ldots+na_n}&=\sum_{k=1}^\infty\sum_{(1^{a_1},\ldots,k^{a_k})\in P(k)}(-1)^{a_1+\ldots+a_k}\frac{(a_1+\ldots+a_k-1)!}{a_1!\ldots a_k!}\sigma_1^{a_1}\ldots\sigma_k^{a_k}Y^k.
1-V_{\O_l}(z)\leq 1- V(z)\leq \exp{\left(-\frac{1}{2}\sum_{\substack{i=0 pi+j\geq k(l,h)}}^qv_{pi+j}^h(0)\right)}\leq \exp{\left(-\frac{1}{2p}\sum_{k\geq k(l,h)}^{pq}v_k^h(0)\right)}
W_{C D B}{}^{A}\equiv \varepsilon^{i j}f_{j C}^{X}f_{i D}^{Y}{\cal R}_{X Y B}{}^{A}= \frac 1 2 \varepsilon^{i j}f_{j C}^{X}f_{i D}^{Y}f_{k B}^{Z}f_{W}^{A k}R_{X Y Z}{}^{W}\, .
M^{-1}= \left(\begin{array}{cc}\mathbf{A}^{-1}+ \mathbf{A}^{-1}\mathbf{B}\mathcal{H}^{-1}\mathbf{C}\mathbf{A}^{-1}- \mathbf{A}^{-1}\mathbf{B}\mathcal{H}^{-1}- \mathcal{H}^{-1}\mathbf{C}\mathbf{A}^{-1}\mathcal{H}^{-1}\end{array}\right).
2\cdot[\begin{matrix}1&8&-3\\ 4&-2&5\end{matrix}]=[\begin{matrix}2\cdot1&2\cdot8&2\cdot-3\\ 2\cdot4&2\cdot-2&2\cdot5\end{matrix}]=[\begin{matrix}2&16&-6\\ 8&-4&10\end{matrix}]
t^\ell_+(\mu,\nu;a,x) = \frac{N(\mu)}{N(\nu)}\frac{\Gamma(\nu-2\ell)\Gamma(\mu-\nu)}{\Gamma(\mu-2\ell)}\, e^{a(\mu-\ell)}\, x^{\nu-\mu}_+\,, \qquad 2\ell < \Re(\nu) < \Re(\mu) \,.
\widehat{F}-\big(1-q^{\lambda}\big)I_3 = \begin{pmatrix}B_0-\big(1-q^{\lambda}\big) B_1 B_2 B_0 B_1-\big(1-q^{\lambda}\big) B_2 B_0 B_1 B_2-\big(1-q^{\lambda}\big) \end{pmatrix}
\widetilde{n^i}\,{\partial\over\partial n^i}T^j_{\k}=\Big({c_{_{\rm S}}^2\over c^2}-{c^2\over c_{_{\rm S}}^2}\Big)n^j\mu_k +\Big({c_{_{\rm S}}^2\over c^2}-1\Big)(\rho c^2 +P)g^j_{\k}\ .
\zeta(s,t)=\zeta(s)\zeta(t)+\frac{1}{2}[(\begin{matrix}s+t\\s\end{matrix})-1]\zeta(s+t)-\sum_{r=1}^{N-1}[(\begin{matrix}2r\\s-1\end{matrix})+(\begin{matrix}2r\\t-1\end{matrix})]\zeta(2r+1)\zeta(s+t-1-2r)
\big[\varphi(\mathbb{A}/ f \mathbb{A})\big]_{\mathbb{A}}=\det_{\mathbb{F}_q(z_1,\dots,z_n)[X]}(X-\varphi_{\theta}\mid \mathbb{A}/f\mathbb{A}\otimes_{\mathbb{F}_q(z_1,\dots,z_n)}\mathbb{F}_q(z_1,\dots,z_n)[X])_{|X=\theta}.
\beta_{1}=\{k(m+d)\mid 0\leq k\leq q+1\}, \beta_{2}=\{k(m+d)+(q+1)m+(q+e-1)d \mid 0\leq k\leq q+1\}, \beta_{3}=\{(q+1)m+(q+i)d\mid 2\leq i\leq e-2\}.
\begin{aligned}\Box \phi^I = (Q_2)^I_{J K}(d \phi^J,d \phi^K) + (Q_3)^I_{J K}(d^2 \phi^J,d \phi^K) + R^I_{J K}(d^2 \phi^J,d \phi^K), \phi^I (0,x) = \phi_0^I (x), \partial_t \phi^I (0,x) = \phi_1^I (x). \end{aligned}
z_{1}\mathcal{S}_{1,k}(z_{1},z_{2})=-s(z_{1})\mathcal{S}_{1,k}(z_{1},z_{2})-\frac{N-k}{N}s(z_{2})-\frac{N-k}{N}s(z_{2})\mathcal{S}_{1,k}(z_{1},z_{2})+O_{L^1}((d_{N}^{3}N^{-1})^{1/2})
\sum_{\mathcal{A}}\binom{n_{\mathcal{A}}}{s_{r-1}}\le (c+o(1))\cdot n^{(a_1+\cdots+a_{r-1})-\frac{a_1\cdots a_{r-1}}{s_1 \cdots s_{r-2}}}\cdot \binom{n}{s_{r-1}}.
\widetilde{\mathfrak{S}}_{t}f(x) =\Pi_{x}^{h}\left[f(\xi^{h}_{t})\right]-\Pi_{x}^{h}\left[\int_{0}^{t}q(\xi^{h}_{s})\widetilde{\mathfrak{S}}_{t-s}f(\xi^{h}_{s}){\rm d}s\right]+\Pi_{x}^{h}\left[\int_{0}^{t}q(\xi^{h}_{s})\pi^{h}(\xi^{h}_{s},\widetilde{\mathfrak{S}}_{t-s}f){\rm d}s\right].
\begin{cases}f(X) = 3 X^2 + 4 A; \phi(X) = X^4 - 2 A X^2 - 8 B X + A^2; g(X) = 3 X^3 - 5 AX - 27 B; \psi(X) = X^3 + AX + B; \Delta^\prime = 4 A^3 + 27 B^2.\end{cases}
\Omega^1_{\Gamma_N}\coloneqq\{x\in \Omega \colon \textrm{there exists}y\in \Omega \textrm{such that}b(y'')=y_{d-1}\textrm{and}t\in [0,1], & \textrm{satisfying}x=(y'', y_{d-1}- t(2L)^{-1}(y_d-a(y')),y_d)\}.
U\{a_1,...,a_l\}= \{a_i : 1\le i\le l,\{a_k\}_{k=1}^l \not\supset{\cal N}(a_i) \}\cup \{x\in{\mathbb{Z}^{d}}: x \not\in\{a_k\}_{k=1}^l y \in\{a_k\}_{k=1}^l (x,y)=1 \}.
\tau^*(\mu_R,\mu_T)\ge\tau_{L1}\triangleq\max\limits_{\substack{l=1,\cdots,\min\{N_T,N_R\}s_1=0,1,\ldots,l s_2=0,1,\ldots,N_R-l}}\frac{1}{l}\Big\{(s_1+s_2)-(N_T-l)s_2\mu_T &\left.-\left(\frac{2s_2+s_1+1}{2}\cdot s_1+s_2^2\right)\mu_R\right\},
\begin{aligned}g\{k_d^{-1}\partial^\circ\eta_0+k_d^{-2}V_\Gamma^2\eta_0-\Delta_\Gamma\eta_0-k_d^{-1}V_\Gamma H\eta_0-k_d^{-1}f\}= \nabla_\Gamma g\cdot\nabla_\Gamma\eta_0-k_d^{-1}(\partial^\circ g)\eta_0+k_d^{-2}gV_\Gamma^2\eta_0, \end{aligned}
\langle X_{\sigma(i)},D^* P P_iZ \rangle =\langle P\widetilde V_{\sigma(i)}, P_iZ \rangle =\langle \widetilde V_{\sigma(i)}, P_iZ \rangle-\kappa(P_iZ )_{\sigma(i)}=\langle \widetilde V_{\sigma(i)}, Z \rangle-\langle ({\rm I}-P_i)\widetilde V_{\sigma(i)}, Z \rangle-\kappa(P_iZ )_{\sigma(i)}.
U\left[ v\left( t\right) \right] =\frac{1}{2}\left( v\left( t\right),v\left( t\right) \right) ,W_{\mathrm{dis}}\left[ v(t)\right]=\beta\left( v\left( t\right) ,Bv\left( t\right) \right) ,\text{}W\left[ v\left( t\right) \right] =\operatorname{Re}\left( v\left(t\right) ,f\left( t\right) \right) .
\hat{\Omega}_{ij}{}^k{}_l=\hat{R}_{ij}{}^k{}_l+\varepsilon \hat{g}_{jl}\delta^k{}_i-\varepsilon \hat{g}_{il}\delta^k{}_j +\hat{K}_{jl}\hat{K}^{k}{}_i -\hat{K}_{il}\hat{K}^k{}_j-2\, \hat{K}_{ij}\hat{K}^k{}_{l},
{\cal A}_{t r i a n g l e , 2}^{( 1 )}= -{\frac{\pi}{m}}\Omega^{2}\left[{\frac{1}{\epsilon}}+ \ln{\frac{4 \pi \mu^{2}}{p^{2}}}- 2 \ln | 2 \sin{\frac{\theta}{2}}| - \gamma +{\cal O}( \epsilon ) \right] \ ,
d_i[g_1,\ldots,g_n] = \begin{cases}[g_2,\ldots,g_n] i = 0, [g_1,\ldots, g_{i-1},g_i g_{i+1}, g_{i+2},\ldots,g_n] 0<i<n, [g_1,\ldots,g_{n-1}] i = n.\end{cases}
\frac{d}{d\lambda}S_{\omega,\mathbf{c}}(\Phi^{\lambda})|_{\lambda=1}=\frac{d}{d\lambda}\left.\left(L(\Phi^{\lambda})+N(\Phi^{\lambda})+\omega Q(\Phi^{\lambda})+\mathbf{c}\cdot \mathbf{P}(\Phi^{\lambda})\right)\right|_{\lambda =1}&=2L(\Phi)+\left(\frac{d}{2}+1\right)N(\Phi)+\mathbf{c}\cdot \mathbf{P}(\Phi)
V_{g}^{( 0 )}( p , h ) ~ = ~ : \! h_{\mu \nu}( \partial X^{\mu}+ i p \cdot \psi \psi^{\mu}) ({\bar{\partial}}X^{\nu}+ i p \cdot{\widetilde \psi}{\widetilde \psi}^{\nu}) e^{i p \cdot \! X}\! : ~ ,
{F_Z}\left( z \right) = \frac{{{m_f}^{{m_f}- 1}{z^{{m_f}}}}}{{B\left({{m_f},{m_s}}\right)\left({{m_s}- 1}\right){^{{m_{f}}}}{{\bar z}^{{m_f}}}}}&\times{}_2{F_1}\left({{m_f},{m_f}+{m_s},{m_f}+ 1; - \frac{{{m_f}z}}{{\left({{m_s}- 1}\right)\bar z}}}\right).
\frac{\partial^{2}S_{n}}{\partial t_{1}^{+ 2}}= - \frac{S_{n}}{1 - S_{n}^{2}}( \frac{\partial S_{n}}{\partial t_{1}^{+}})^{2}- \frac{1}{t_{1}^{+}}\frac{\partial S_{n}}{\partial t_{1}^{+}}+ \frac{( n + 1 )^{2}}{t_{1}^{+ 2}}\frac{S_{n}}{1 - S_{n}^{2}}- 4 S_{n}( 1 - S_{n}^{2}) .
0 \leq \mathbf E[\limsup_{t\to\infty}X^{(p^\ast)}_t] \leq \inf_{p > p^\ast}\mathbf E[\limsup_{t\to\infty}X^{(p)}_t] = \inf_{p > p^\ast}\mathbf E[X^{(p)}_\infty] = \inf_{p > p^\ast}\Big(1 - \tfrac 1p\log\Big(\tfrac 1p\Big)\Big) = 0.
\Big\| \sup_{\epsilon > 0}\Big| \mathop{\mathop{\sum_{Q \in \mathcal{D}}}_{Q^a = F}}_{\ell(Q) > \epsilon}\epsilon_Q D_Q f \Big|\, \Big\|_{L^p(\mu)}^p \lesssim \mu(F), |f| \le 1,\, p \in (0, \infty).
\delta_{0 2}= \int_{r_{\mathrm{m i n}}}^{\infty}d r \left[ \sqrt{D_{2}E^{2}- p_{1 1}^{2}-{\frac{J^{2}}{r^{2}}}}- \sqrt{E^{2}- p_{1 1}^{2}-{\frac{J^{2}}{r^{2}}}}\right]
\mathcal A_{\omega_s}([u(s, \cdot), w_- \# u((-\infty, s] \times S^1)]) = - \int_{D^2}w_s^* \omega_s + \int_0^1 H(u(s,t),t)dt & = - \int_{D^2}w_s^*(\omega_0 + \kappa(s) \alpha) + \int_0^1 H(u(s,t), t)dt & = - \int_{D^2}w_s^* \omega_0 - \kappa(s) \int_{D^2}w_s^* \alpha + \int_0^1 H(u(s,t), t)dt.
\begin{array}{llllll}\phi(0) \multicolumn{2}{l}{\phi(0,\dots,\binom r2,\frac{r(3-r)}{2})}\multicolumn{2}{l}{\phi(0,\dots,\binom r2,r,\frac{r(3-r)}{2})}\dots \multicolumn{2}{l}{\phi(0,\binom r2,r,\dots,r,\frac{r(3-r)}{2})}\phi(0,\frac{r(3-r)}{2},\dots,\frac{r(3-r)}{2},r(2-r)). \end{array}
\phi_{\infty}^{f}= \sqrt{\frac{1}{2 \pi}}{\frac{e^{- i u}}{u^{3 / 2}}}\left\{\alpha e^{i ( \nu + 1 / 2 ) \pi / 2}+ \beta e^{- i ( - \nu + 1 / 2 ) \pi / 2}\right\}
B_{\gamma}\left( a\right) =\int_{\Theta_{d}}\hat{f}\left( -\gamma^{-1}\theta \right) \mathrm{e}^{-i\theta \cdot a}\mathrm{d}\left( \mathrm{F}_{\ast}\mathrm{P}\right) \left( \theta \right) +R_{f,a,\gamma}\ ,
\left|\phi(t)-\phi(s)\right| \le8\int_{0}^{\left|t-s\right|}\log\left\{1+\frac{4\left(M-T^{2}\right)}{u^{2}}\right\}dp(u) \le2\int_{0}^{\delta}u^{-\frac{3}{4}}\log\left(M+\frac{4M}{u^{2}}\right)du =8\delta^{\frac{1}{4}}\log M+2\int_{0}^{\delta}u^{-\frac{3}{4}}\log\left(1+\frac{4}{u^{2}}\right)du.
\begin{matrix}X_{k}&=&\sum_{m=0}^{N/2-1}x_{2m}e^{-\frac{2\pi i}{N}(2m)k}+\sum_{m=0}^{N/2-1}x_{2m+1}e^{-\frac{2\pi i}{N}(2m+1)k}\end{matrix}
\sum_{j_1=1}^m\sum_{j_2\cdots j_\ell=0}\prod_{r=1}^\ell\delta_{j_r}(A)=\sum_{j_1=1}^n \delta_{j_1}(A) \sum_{j_2\cdots j_\ell=0}\prod_{r=2}^\ell\delta_{j_r}(A)=(1-\delta(A))\sum_{j_2\cdots j_\ell=0}\prod_{r=2}^\ell\delta_{j_r}(A)
\frac{H_{2}}{\beta^2}=\sum_a T^{aa}_{2}-\sum_{a, b}T^{ab}_{0}T^{ba}_{1}+s\sum_a T^{aa}_{1}+\frac{1}{3}\sum_{a,b,c}T^{ab}_{0}T^{bc}_{0}T^{ca}_{0}-\frac{2s}{3}\sum_{a, b}T^{ab}_{0}T^{ba}_{0}+\frac{1}{6}\sum_{a,b}T^{aa}_{0}T^{bb}_{0}+\frac{2s^2-1}{6}\sum_a T^{aa}_{0}
\phi(-q^{\frac{j-i}{2}}x_{j-1}x_{j-1}\cdots x_{i})=\displaystyle \sum_{m_{i,j}\geq 0}(-1)^{m_{i,j}}\frac{q^{\frac{m_{i,j}(m_{i,j}-1)}{2}}(-q^{\frac{j-i}{2}}x_{j-1}x_{j-2}\cdots x_{i})^{m_{i,j}}}{(q)_{m_{i,j}}}.
\Big(\mathcal{K}_{\alpha_\mathrm{p}}-\mathcal{K}_\Big)\Big[\Phi^{\textbf{e}}(\xi,t;z,\cdot)\Big](\mathrm{y},\tau) = \int_{\partial\Omega}\frac{(\mathrm{y}-\mathrm{v})\cdot\nu_{\mathrm{v}}}{2\pi|\mathrm{y}-\mathrm{v}|^2}\Big[\varphi(\mathrm{v},\mathrm{y},\mathrm{t},\tau)-\Phi^{\textbf{e}}(\xi,t;z,\tau)\Big]d\sigma_\mathrm{v}
m_K(\varphi \otimes \psi) = m_K \circ (i_{H_{\mathrm{reg}}}\otimes i_{H_{\mathrm{reg}}})\!\left( \varphi \otimes 1 \otimes \psi \otimes 1 \right) = d_{H_{\mathrm{reg}}, H_{\mathrm{reg}}}(\varphi \otimes 1 \otimes \psi \otimes 1) = \psi(S(b_i)?b_j) \varphi(?a_ja_i) &= \varphi(a_j ? a_i) \psi(S(b_i)b_j ?)
j \neq k \implies \lim_{n \rightarrow \infty}\frac{\lambda_{j,n}}{\lambda_{k,n}}+ \frac{\lambda_{k,n}}{\lambda_{j,n}}+ \frac{|t_{j,n}- t_{k,n}|}{\lambda_{j,n}}+ \frac{|x_{j,n}- x_{k,n}|}{\lambda_{j,n}}= \infty.
\iint_{Q_{\rho}}u_t \varphi(x,t) \,dx\,dt = \left. -\int_{K_{\rho}\times \{t\}}F \zeta^{q}dx \right|_{t_0}^{t_1}+ q \iint_{Q_{\rho}}F \zeta^{q-1}\zeta_{t}\,dx\,dt.
M_{N}= \left( \begin{array}{rrrrrrrrrrrrrrrrrrr}1 0 1 0 -1 0 -1 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 -1 0 -1 0 1 0 1 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 \end{array}\right),
\mathcal{E}_f(V)-\mathcal{E}_f(U_0)=\big[J_{p,f}(u)-J_{p,f}(u_0)\big]+\frac{1}{2}\left[\int_\Omega Vu^2\,dx-\left(\int_\Omega|u|^{2p/(p+1)}\,dx\right)^{(p+1)/p}\right],
\left\|\frac{x+y}{2}\right\|^4 \leq 2\left\|\frac{x}{2}\right\|^4+2\left\|\frac{x}{2}\right\|^4+12\left\|\frac{x}{2}\right\|^2 \left\|\frac{y}{2}\right\|^2-\left\|\frac{x-y}{2}\right\|^4\leq \frac{1}{8}+\frac{1}{8}+\frac{3}{4}-\frac{\epsilon^4}{16}=1-\frac{\epsilon^4}{16}\leq (1-\delta)^4.
\bar{E}_{n}= \frac{c}{4 \pi a^{2}}\int_{0}^{\infty}y \, d y \, \left\{\ln \left[ 1 - \xi^{2}\sigma_{n}^{2}( y ) \right] + \frac{\xi^{2}}{4}\frac{y^{4}}{( n^{2}+ y^{2})^{3}}\right\}, \quad n = 1 , 2 , \ldots{.}
\mathcal{B}_{t}(f,g)=\sum_{\varsigma,\mu,\nu\in\{-1,1\}^{n\times 3}}K_{\varsigma_{1}}\otimes\ldots\otimes K_{\varsigma_{n}}(Z_{t,\varsigma,\mu}fZ_{t,\varsigma,\nu}g).
\min_{\mathbf{X}_1}~{\rm tr}\{\mathbf{X}_1\}{\rm s.t.}~{\rm tr}\{\mathbf{B}_1\mathbf{X}_1\}\geq \beta_1\sigma^2 &{\rm tr}\{\mathbf{B}_2\mathbf{X}_1\}\geq (2^{Q_s}-1)\sigma^2 &{\rm tr}\{\mathbf{B}_3\mathbf{X}_1\}\leq\sigma^2\left(\frac{1+\beta_1}{2^{Q_p}}-1\right) &{\rm rank}\{\mathbf{X}_1\}=1 &\mathbf{X}_1\succeq\mathbf{0},
R_1=\hbox{diag}(\lambda_1, \overline \lambda_1, ~ \lambda_1\lambda_2 \lambda_3, \overline \lambda_1 \overline \lambda_2 \overline \lambda_3, ~\ldots, ~ \lambda_1 \lambda_2 \cdots \lambda_{2k+1}, \overline \lambda_1 \overline \lambda_2\cdots \overline \lambda_{2k+1}, \ldots)
P_ixP_i =(u_i^*e_Nu_i)x(u_i^*e_Nu_i) &=u_i^*e_N(u_i x u_i^*)e_Nu_i &=u_i^*E_N(u_i x u_i^*)e_Nu_i &=u_i^*u_iE_N(x)u_i^*e_Nu_i &=0.
F^{( 1 )}= - i \int_{x_{0}}^{x}d x_{j}A_{j}^{( 1 )}- \frac{i}{2}\int_{x_{0}}^{x}d x_{j}\, \epsilon^{m l}k_{m}\, \partial_{l}A_{j}^{( 0 )}+ \frac{i}{2}\int_{x_{0}}^{x}d x_{j}\, \epsilon^{m l}A_{m}^{( 0 )}\, \partial_{l}A_{j}^{( 0 )}.
\textrm{Tr}(L(\mu))\mathcal{Q}_\mu=\mu^N\mathcal{Q}_{\frac{\mu}{\sqrt{\alpha}}}+\frac{\Delta}{\mu^N}\mathcal{Q}_{\mu\sqrt{\alpha}}, \alpha\doteq \frac{1}{1+\eta}=\frac{1}{1+\i \hbar}
\frac{d}{dt}p_k^f(t)\, = \, -f(\lambda_k) p_k^f(t)+ \sum_{r=1}^{k-1}p_r^f(t) \int_0^\infty \prod_{j=r}^{k-1}\lambda_j \sum_{m=r}^{k}\frac{e^{-\lambda_m s}}{\prod_{l=r , l \neq m}^{k}(\lambda_l-\lambda_m )}\nu (ds),
F_{\theta_1, \theta_2, \theta_3}= c_{\theta_1, \theta_2, \theta_3}\: I ,c_{\theta_1, \theta_2, \theta_3}= \frac{1}{4}\left(\cos(\theta_1-\frac{\pi}{3}) + \cos(\theta_2-\frac{\pi}{3}) + \cos(\theta_3-\frac{\pi}{3}) + \cos(\theta_4-\frac{\pi}{3})\right),
Q_{jk}= \dfrac{\left|\left\langle L_{F}e_k, e_j \right\rangle \right|^2}{4\left|\Re\left(\left\langle L_{F}e_j, e_j \right\rangle\right)\right|\left|\Re\left(\left\langle L_{F}e_k, e_k \right\rangle\right)\right| b_{jk}b_{kj}}
\int_{(0,+\infty)}h_n(t)\,d\|E_{\rho,\sigma}(t)\xi_\sigma\|^2 &\quad=\inf_{x(\cdot)}\int_{[1/n,n]}\bigl\{\sigma((1-x(s))^*(1-x(s)))+s^{-1}\rho(x(s)x(s)^*)\bigr\}(1+s)\,d\nu_n(s),
P_e =\Pr\left((M_{12},M_1,M_2)\neq(\hat{M}_{12},\hat{M}_1,\hat{M}_2)\right) &=\Pr\left((M_{12},M_1,M_2)\neq(\hat{M}_{12},\hat{M}_1,\hat{M}_2)|(M_{12},M_1,M_2)=(1,1,1)\right).
\begin{aligned}2\frac{|\nabla\psi|^2}{\psi}h^3 \omega \leq2D\cdot|\nabla\psi|^2\psi^{-3/2}\cdot\psi^{1/2}\omega &\leq\frac 35\psi\omega^2+c D^2\frac{|\nabla\psi|^4}{\psi^3}&\leq\frac 35\psi\omega^2+c\frac{D^2}{R^4}.\end{aligned}
\langle P(x),Q(y)\rangle=\sum_{j_1,...,j_{N-1}=0}^{\infty}\sum_{\substack{I\subset[N] N\notin I}}(-1)^{|I|}\tilde{\phi}\left(\vec{d}(x,y)+2(j_1,...,j_{N-1},0)+2\chi^I\right).
(a+b\sqrt d)(a-b\sqrt d) = \prod_{p \in S_\pm}(p)^{l_pn_p+r_p}\prod_{q \notin S_\pm}(q)^{r_q}= \prod_{p \in S_\pm}(\gamma_p)^{n_p}(\widetilde{\gamma_p})^{n_p}(p)^{r_p}\prod_{q \notin S_\pm}(q)^{r_q},
\varphi((u^{\beta}_{s,t})^*u^{\alpha}_{i,j})=\frac{\delta_{\alpha,\beta}\delta_{i,s}\delta_{j,t}(Q_{\alpha})_{i,i}^{-1}}{\mathrm{tr}(Q_{\alpha})}\mathrm{~and~}\varphi(u^{\beta}_{s,t}(u^{\alpha}_{i,j})^*)=\frac{\delta_{\alpha,\beta}\delta_{i,s}\delta_{j,t}(Q_{\alpha})_{j,j}}{\mathrm{tr}(Q_{\alpha})}
z^{2}H_{\nu}^{( 1 , 2 )}( z ) = \pm{\frac{1}{\sin{\nu \pi}}}\left[ -{\frac{2^{\nu}z^{2 - \nu}}{\Gamma ( 1 - \nu )}}\left( 1 + O ( z^{2}) \right) + e^{\mp \nu \pi i}{\frac{z^{\nu + 2}}{2^{\nu}\Gamma ( \nu + 1 )}}\left( 1 + O ( z^{2}) \right) \right] \, ,
|(g\tilde A_n\Delta\tilde A_n)\cap K| \leqslant\sum_{i=1}^{p_n}|(gA_n\Delta A_n)g_{n,i}^{-1}\cap K| \leqslant p_n\tau |(gA_n\Delta A_n)\cap L| \leqslant \tau |C|\frac{|(gA_n\Delta A_n)\cap L|}{|A_n\cap B|}.
l A_{1}+ n B_{1}= l U_{l}^{n}S t r \left( \ldots \phi^{i_{2 k - 1}^{\prime}}\ldots \underbrace{\partial_{a_{l}}\phi^{j_{l}}\phi^{i_{2 k}^{\prime}}}\ldots \right) + n U_{l}^{n}S t r \left( \ldots \phi^{i_{2 k - 1}^{\prime}}\ldots \underbrace{\phi^{i_{n}}\phi^{i_{2 k}^{\prime}}}\ldots \partial_{a_{l}}\phi^{j_{l}}\ldots \right)
I_W(a,b; p) = \int\limits_0^{\infty}\!dx x \exp(-p^2x^2) J_\nu(ax) J_\nu (bx) &= \frac{1}{2p^2}\exp[-(a^2+b^2)/4p^2] I_0 \!\left( \frac{ab}{2p^2}\right) \;,
\mathrm{Num}= -\left[L_X(s+1,t)L_V(s+1,t+1)+L_X(s,t)L_V(s+2,t+1)\right]L_X(s+1,t+1)L_V(s,t) &+L_X(s,t)L_V(s+1,t+1)\left[L_X(s+2,t+1)L_V(s,t)+L_X(s+1,t+1)L_V(s+1,t)\right] = L_X(s,t)L_V(s,t)\left[L_X(s+2,t+1)L_V(s+1,t+1)-L_X(s+1,t+1)L_V(s+2,t+1)\right] &+L_X(s,t)L_X(s+1,t+1)L_V(s+1,t)L_v(s+1,t+1) &-L_X(s+1,t)L_X(s+1,t+1)L_V(s,t)L_V(s+1,t+1)=I_1+I_2-I_3.
g(\pi') = \sum_{i=1}^{k-t-|R|+2}\bigl( h_i(\pi'_1 \dotsm \pi'_{k-|R|}, n-|R|) - h'_i(\pi'_1 \dotsm \pi'_{k-|R|}) \bigr) + \sum_{i=k-t-|R|+3}^{n-|R|}g_i(\pi')
\bar{\partial}P_{s}^{j , m}= \lambda [ \phi_{s - 1}^{l , m + 1 / 2}\otimes \overline{{\phi}}^{1 / 2 , - 1 / 2}- \phi_{s - 1}^{l , m - 1 / 2}\otimes \overline{{\phi}}^{1 / 2 , 1 / 2}]
\lambda ( R ) = -{\frac{1}{2 A_{2}^{2}}}\left( 1 + g A_{2}^{2}- \sum_{p \ge 1}p A_{p}\left( 1 -{\frac{p - 2}{p + 2}}g A_{2}^{2}\right) ( c_{p}- c_{2}+ \bar{\tau}\cdot \bar{g}_{p}) \right) \ .
BM=\begin{bmatrix}B_1S_1+B_2S_3 B_1S_2+B_2S_4 B_3S_1+B_4S_3 B_3S_2+B_4S_4\end{bmatrix}=\begin{bmatrix}B_1S_1+B_2S_3 0 0 B_3S_2+B_4S_4\end{bmatrix}
v_{A}=\frac{k_{A}e_{0}a}{1+\frac{a}{K_{m}^{A}}+\frac{a^{\prime}}{K_{m}^{A^{\prime}}}},v_{A^{\prime}}=\frac{k_{A^{\prime}}e_{0}a^{\prime}}{1+\frac{a}{K_{m}^{A}}+\frac{a^{\prime}}{K_{m}^{A^{\prime}}}}
\Lambda(\xi_2,\omega) = (1+(1+|\omega|)^{-\alpha}|\omega|)^{-1/(1-\alpha)}(1+|\omega|) & = (1+|\omega|)^{\alpha/(1-\alpha)}((1+|\omega|)^\alpha+|\omega|)^{-1/(1-\alpha)}(1+|\omega|) & \leq(1+|\omega|)^{-(1-\alpha)/(1-\alpha)+1}& = 1.
F(r)=\frac{Gm\rho}{8r^{3}}\int_{R-r}^{R+r}\frac{(r^{2}+s^{2}-R^{2})\sqrt{2(r^{2}R^{2}+r^{2}s^{2}+R^{2}s^{2})-s^{4}-r^{4}-R^{4}}}{s^{2}}ds
\tilde{e}'''_k:= \mathcal{B}'(S_{\theta_k}{\mathbf V}_k|_{x_1=0},S_{\theta_k}\psi_k)((\delta{\mathbf V}_k)|_{x_1=0},\delta\psi_k) &-\mathcal{B}'({\mathbf V}_{k+\frac{1}{2}}|_{x_1=0},\psi_{k+\frac{1}{2}})((\delta{\mathbf V}_k)|_{x_1=0},\delta\psi_k).
\nabla_x \left ( X^{-1}(x,s) - X^{-1}(x,t) \right )= \left ( \nabla_a X \circ X^{-1}\right ) (x, t) \left ( \nabla_a \left (X - \mathrm{Id}\right ) \right ) \left ( X^{-1}(x,t), t-s \right )
3 \mu(f,f,M) =\int f D_{\mu,M}(f) \;du= -\sum_{k=3}^\infty \lambda_k\int f f_k \frac{D_{\mu,M}(L)}{h_L}\;du = -\sum_{k=3}^\infty \lambda_k\int f_k^2 \frac{D_{\mu,M}(L)}{h_L}\;du \leq 0.
\begin{aligned}\dot{\beta}_{T,\eta}= \frac{K}{4\sqrt{3}}\bigg(-\beta_{T,\eta}^{2}+\frac{\sqrt{3}}{2}\beta_{T,\eta}- \frac{4\sqrt{3}M}{K}-\eta \bigg), t > T, & \beta_{T,\eta}(T)=R(T),\end{aligned}
\widehat{\boldsymbol{S}}_e=\left(\sum_{k=1}^N \boldsymbol{A}_{\Gamma^{(k)}_e}\boldsymbol{S}^{(k)}_e\boldsymbol{A}^T_{\Gamma^{(k)}_e}\right) \widehat{\boldsymbol{g}}_e=\sum_{k=1}^N \boldsymbol{A}_{\Gamma^{(k)}_e}\boldsymbol{g}^{(k)}_e.