harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C^{(pq)}:= \mathbb{E}_0\left[ \bar{U}^{(pl)}_h\bar{U}^{(ql)}_h \middle|\ \mathcal{F}_{n, l-1}\right] \; =\; \mathbb{E}_0\left[ \bar{U}^{(pl)}_h\bar{U}^{(ql)}_h \middle| \mathbf{X}^{(p)}, \mathbf{X}^{(q)}\right],
\left[\begin{array}{c}\iota(g_1z_2,g_2z_2) 1\end{array}\right]=R^{-1}\left[\begin{array}{cc}U(g_1z_1) 0 0 \mathfrak{J}\overline{U(g_2z_2)}\end{array}\right]B(g_1z_1,g_2z_2)^{-1}S
\frac{\partial(r,\theta,h)}{\partial(x,y,z)}=(\begin{matrix}\frac{x}{\sqrt{x^{2}+y^{2}}}&\frac{y}{\sqrt{x^{2}+y^{2}}}&0\\ \frac{-y}{x^{2}+y^{2}}&\frac{x}{x^{2}+y^{2}}&0\\ 0&0&1\end{matrix})
{\cal S}\equiv \sum_{\{n_{\bf k}\}}\left( \prod_{\bf k}N_{j_{\bf k}j_{\bf k}^{\prime}}^{m_{\bf k}n_{\bf k}^{\prime}}N_{i_{\bf k}i_{\bf k}^{\prime}}^{n_{\bf k}n_{\bf k}^{\prime}}i_{\bf k}! i_{\bf k}^{\prime}! j_{\bf k}! j_{\bf k}^{\prime}! \right) \langle \{n_{\bf k}\}\mid \, \rho \, | \{m_{\bf k}\}\rangle \, .
\lambda_{1,p}(\beta) \le \frac{\int_0^1 \frac{| w' |^p F_\beta^p}{|\beta'|_h^{p-1}}\,dt}{\int_0^1 | w |^p |\beta'|_h \,dt}\le \frac{\int_Z \frac{| w' |^p F_\beta^p}{|\beta'|_h^{p-1}}\,dt}{\int_Z | w |^p |\beta'|_h \,dt}= \frac{\int_0^1 \frac{| w' |^p F_\alpha^p}{|\alpha'|_h^{p-1}}\,dt}{\int_0^1 | w |^p |\alpha'|_h \,dt}
\begin{array}{c}\gamma_0 = \{\emptyset, \left[ 0, 1 \right] \}\cup \{\left[ 0, p) \mid p \in (0,1 \right] \}\cup \left. \{(q,1 \right] \mid q \in \left[ 0,1) \}\cup \{\left[ 0, r) \cup (s, 1 \right] \mid r, s \in (0, 1),\,\,r \le s \}\,. \right.\end{array}
\delta_{p_1p_2}\delta(k)=\langle{\varphi_{p_1},T_{Ak}\varphi_{p_2}}\rangle =\displaystyle \frac{1}{N^2}\langle{\widehat{\varphi_{p_1}}, \widehat{T_{Ak}\varphi_{p_2}}}\rangle = \displaystyle \frac{1}{N^2}\sum_{n \in{\mathbb{Z}}^2_N}\widehat{\varphi_{p_1}}(n)\overline{\widehat{\varphi_{p_2}}(n)}e^{2\pi i \langle{n, Ak}\rangle/N},
\widetilde \zeta_{j,N}=\sum_{\substack{k=1 k\,}}^{\min\{2j+1,4N-2j-1\}}\gamma^k \begin{pmatrix}j \frac{k-1}{2}\end{pmatrix}\begin{pmatrix}2N-j-1 \frac{k-1}{2}\end{pmatrix}=\sum_{\substack{\ell=0}}^{\min\{j,2N-j-1\}}\gamma^{2\ell +1}\begin{pmatrix}j \ell\end{pmatrix}\begin{pmatrix}2N-j-1 \ell\end{pmatrix}\,.
x_{2,1}= ( 0, 0, 1, 3, 0, 2, 3, 3, 1, 3, 0, 0, 1, 3, 0, 0), x_{2,2}= ( 0, 0, 1, 3, 0, 0, 3, 3, 1, 3, 2, 0, 1, 3, 0, 0), x_{2,3}= ( 0, 0, 1, 1, 0, 0, 3, 3, 1, 3, 0, 2, 3, 3, 0, 0).
\int_{\mathbb{T}^{N}}\{( x_{i}\mathcal{D}_{i}f(x))^{\ast}L(x)^{\ast}HL(x) g(x) -f(x)^{\ast}L(x)^{\ast}HL(x) x_{i}\mathcal{D}_{i}g(x) \}\mathrm{d}m(x) =0
\int_0^1 \overline{\xi(x)}\xi^\prime(x)dx = \frac{1}{2}\int_0^1\frac{d}{dx}|\xi(x)|^2dx + i \int_0^1 \Im (\overline{\xi(x)}\xi^\prime(x)) dx = i \int_0^1 \Im (\overline{\xi(x)}\xi^\prime(x)) dx,
\sum_{y_{t}}\Big(\log\Big(\frac{q_t(y_t|y^{t-1}_{t-M},x_t)}{\nu^{\pi}_{t}(y_t|y^{t-1}_{t-J})}\Big)+C_{t+1}(y^t_{t+1-J})\Big)q_t(y_t|y^{t-1}_{t-M},x_t)-s\gamma_t(x_t,y^{t-1}_{t-N})=1-\lambda_t(y^{t-1}_{t-J}),~\forall{x_t}\in{\cal X}_t.
C=\frac{\mu}{\lambda'^3 \sigma}\left(\frac{\omega}{6}+\frac{1}{2}\sqrt{\frac{\omega^2}{9}+\frac{8\lambda'\omega^{5/4}\theta^{1/4}}{3\sqrt{a_1'a_2'}H^{1/2}}+\frac{4}{3}\left( \frac{1}{a_1'}+\frac{1}{a_2'}\right)\frac{\lambda'\omega}{H}}\right)^2 , &C'=\sqrt{\frac{C\sigma \omega \theta}{\lambda'^3 \mu}}.
[y_{l},y_{(1\,l)}]_c =-\chi_{l-1l}(g_{(1\,l-2)})\Big((1-\xi)y_ly_{l-1\,l}-\lambda_{l-1ll}\xi^2\Big)y_{(1\,l-2)}&+\chi_{(1\,l-2)}(g_{l})y_{(1\,l-2)}\Big((1-\xi)y_ly_{l-1\,l}-\lambda_{l-1ll}\xi^2\Big) &=(1-\xi)y_ly_{(1\,l)}&+\lambda_{l-1ll}\xi^2\chi_{l-1l}(g_{(1\,l-2)})(1-\chi_{(1\,l-2)}(g_l)\chi_l(g_{(1\,l-2)}))y_{(1\,l-2)},
e^{adX}(D_1D_2) = \sum_0^\infty\frac{1}{n!}(ad X)^n(D_1D_2) = \sum_0^\infty\frac{1}{n!}\sum_{0\leq i\cdot j,~i+j = n}\frac{n!}{i!j!}(adX)^i(D_1)(adX)^j(D_2) & = e^{adX}(D_1D_2) = e^{adX}(D_1)e^{adX}(D_2)
\mathbf{T}=(T_{p-q})_{p,q=0}^{n-1}, AT_{j}=BT_{j-n}j=1,2,\cdots,n-1, \mathbf{U}=(U_{p-q})_{p,q=0}^{n-1}, AU_{j}=BU_{j-n}j=1,2,\cdots,n-1.
\mathbf{e}_{M_\mathrm{h}}\left[\omega \right] = \frac{1}{\sqrt{M_\mathrm{h}}}\left[1 \,\, e^{j\omega}\,\, \cdots \,\, e^{j\left(M_\mathrm{h}-1\right)\omega}\right]^\mathrm{T}, \,\, \omega \in \left[0,2\pi\right).
(z_{2}z_{1})^{t}u'\otimes(b'\alpha_{2})\alpha_{1}=-(z_{1}\circ z_{2})^{t}u'\otimes b'\frac{[\alpha_{1},\alpha_{2}]}{2}-[z_{1},z_{2}]^{t}u'\otimes b'\frac{\alpha_{1}\circ\alpha_{2}}{2}.
H^{0}\left( X, K_X\right) = H^{0}\left( \mathbb{P}^1 \times C, K_{\mathbb{P}^1 \times C}\right) \oplus \bigoplus_{\chi \ne \chi_{000}}{H^{0}\left( \mathbb{P}^1 \times C, K_{\mathbb{P}^1 \times C}+L_{\chi}\right)},
\frac{\left \|\overline{\nabla}\varphi\right \|^{2}}{\varphi}=\frac{\left \|\psi'\right \|^{2}\left \|\overline{\nabla}d\right \|^{2}}{\rho^{2}\varphi}\leq \frac{c_{2}}{\rho^{2}},\quad\frac{\partial \varphi}{\partial t}\leq c_{2}\rho_{1},
\big((A+B)^2-C^2\big)\big((A-B)^2-C^2\big) = \big((A+C)^2-B^2\big)\big((A-C)^2-B^2\big)= \big((B+C)^2-A^2\big)\big((B+C)^2-A^2\big) = A^4+B^4+C^4-2A^2B^2-2A^2C^2-2B^2C^2 \mbox{for any}\ \A,B,C\in \mathbb{R},
\iint_Q \rho_3^2 |y_{xx}|^2 dxdt \leq C \left(\|y_0\|_{H_0^1(I)}^2 + \iint_Q \rho^2 |G|^2 dxdt + \iint_{\mathcal{O}\times (0,T)}\rho_1^2 |f|^2 dxdt \right. &+ \left. \iint_Q \rho_0^2|y|^2 dxdt + \sum_{i=1}^2 \iint_Q \rho_0^2 |p^i|^2 dxdt \right)
\mathbb E_x^i u_n(T \wedge \tau_R,\omega_{T \wedge \tau_R}) = u_n(0,x)+\mathbb E_x^i \int_0^{T \wedge \tau_R}f(t,\omega_t)dt & + \mathbb E_x^i \int_0^{T \wedge \tau_R}\big[(b-b_n)\cdot \nabla u_n\big](t,\omega_t)dt
\begin{array}{c c c c c c c}{{\cal L}_{\mathbf{I}}K^{x}}&{=}&{\epsilon^{x y z}K^{y}W_{\mathbf{I}}^{z}}&{;}&{{\cal L}_{\mathbf{I}}\omega^{x}}&{=}&{\nabla W_{\mathbf{I}}^{x}}\\ \end{array}
\Lambda_{\mathcal{R}}(n)=\begin{cases}\prod_{i=1}^{\ell}\log{p_i}, n=p_1\cdots p_\ell(\frac{\log{p}_1}{\log{X}},\dots,\frac{\log{p_\ell}}{\log{X}})\in\mathcal{R}, 0, \end{cases}
P_1 = \frac{1}{6}\frac{1}{n}\sum_{i=1}^n \psi_i \psi_i^T\Omega^{-1}\bar{\psi}\Pi^T\Omega^{-1}\psi_i - \frac{1}{2}\frac{1}{n}\sum_{i=1}^n \psi_i \psi_i^T\Omega^{-1}\bar{\psi}\bar{\psi^T}\Omega^{-1}\psi_i \psi_i^T\Omega^{-1}\bar{\psi}\,.
\int_0^T \langle V'(\vec p)\widehat{\vec p}_1({u}_2^\prime(t)+Q{u}_2(t)),w(t)\big\rangle_X\d t = -\int_0^T \langle u_2, V'(\vec p)\widehat{\vec p}_1(w^\prime(t)+Qw(t))\big\rangle_X\d t &= \int_0^T \langle V'(\vec p)\widehat{\vec p}_2({u}^\prime(t)+Q{u}(t)), z(t)\big\rangle_X\d t.
{\frac{m}{\sqrt{2}}}\, \Xi = \lim_{| z | \rightarrow \infty}\left\lbrace \left( \begin{matrix}{{( \tau + 1 - \epsilon_{z}) \sqrt{2 z}}}\\{{\sqrt{2 z}}}\\ \end{matrix}\right) - \left( \begin{matrix}{{a_{D}( z , \tau , m )}}\\{{a ( z , \tau , m )}}\\ \end{matrix}\right) \right\rbrace \cdotp
U_{l_0}(x,0) =U_{l_0}(x_1+c\tau+\sigma^*,x'+\rho,\tau) &=\cdots=U_{l_0}(x_1+cn\tau+n\sigma^*,x'+n\rho,n\tau)\overset{n\rightarrow\infty}{\longrightarrow}p^-_{l_0}
dL=\sum_{i=1}^{n}(\frac{\partial L}{\partial q_{i}}dq_{i}+\frac{\partial L}{\partial\dot{q}_{i}}d\dot{q}_{i})+\frac{\partial L}{\partial t}dt=\sum_{i=1}^{n}(\dot{p}_{i}dq_{i}+p_{i}d\dot{q}_{i})+\frac{\partial L}{\partial t}dt,
\left[ L(n) ,L (m) \right] =\left( n-m\right) L (n+m)+\delta_{n,-m}\frac{n^{3}-n}{12}C, \left[ L (n) ,W (m )\right] =\left( n-m\right) W (n+m )+\delta_{n,-m}\frac{n^{3}-n}{12}C_{W}, \left[ W (n),W( m ) \right] = 0 , C, C_{W}\quad\mbox{are in the center of}\W(2,2).
D_I=\left\{\left(\begin{array}{cc}e^{-t_1}\mathrm{I}_2 0 0 e^{-t_2}\mathrm{I}_3 \end{array}\right), t_1>t_2\right\}, \textrm{where}\mathrm{I}_k \textrm{is the $k\times k$ identity matrix.}
\left\|\frac{f}{|x|^{k+\alpha}}\right\|_{L^{2}(\mathbb{G})}\leq\left[\prod_{j=0}^{k-1}\left|\frac{Q-2}{2}-(\alpha+j)\right|\right]^{-1}\left\|\frac{1}{|x|^{\alpha}}\mathcal{R}^{k}f\right\|_{L^{2}(\mathbb{G})},
I_2\leq \frac2{\delta^2}\|u^2\|_{L^\infty(L^{p'_1})}\|\sigma^1-\sigma^2\|_{L^2(L^{2p_1})}^2 + C_{d,p_1}\bigg(\sum_{i=1}^2\|u^i\|_{L^\infty(L^{p'_1})}\bigg) \big\| \nabla \sigma^1\big\|_{L^2(L^{2p_1})}^2.
D_{1 2}^{( l )}\stackrel{1}{\widehat{F}^{( l )}}\stackrel{2}{\widehat{F}^{( 1 )}}D_{1 2}^{( l ) \; - 1}= \left( \begin{array}{c c}{{\widehat{F}^{( l + 1 )}}}&{{0}}\\{{0}}&{{- \alpha \beta \widehat{F}^{( l - 1 )}}}\\ \end{array}\right) ,
Exp^+=\Big\{(\sum_l u_le^{ilt}, \sum_l -\mbox{sign}(l)u_le^{ilt})|\{u_l\}_{l\in\mathbb{Z}}\in \ell^2\Big\}\mbox{and}&Exp^-=\Big\{(\sum_l u_le^{ilt}, \sum_l \mbox{sign}(l)u_le^{ilt})|\{u_l\}_{l\in\mathbb{Z}}\in \ell^2\Big\}.
\frac{1}{\prod_{n = 1}^{\infty}n^{N_{n}}N_{n}!}\prod_{n = 1}^{\infty}\left( \frac{N_{n}!}{\prod_{s_{n}= 1}^{\infty}\left( s_{n}^{M_{n , s_{n}}}M_{n , s_{n}}! \right)}\right)
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m)/2}\cdot|M_2^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot|M_2^L|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}^{-1}\leq 2^{3/2}\cdot(2^{n_{2,\nu_2}}-1).
a_{ij}= \int_0^M\int_0^M(M-\max\{\eta,\eta'\})\phi_i(\eta)\phi_j(\eta')\,d\eta\,d\eta', b_{ij}= \int_0^M\int_0^M(M-\max\{\eta,\eta'\})\phi_{N+i}(\eta)\phi_j(\eta')\,d\eta\,d\eta', c_{ij}= \int_0^M\int_0^M(M-\max\{\eta,\eta'\})\phi_{N+i}(\eta)\phi_{N+j}(\eta')\,d\eta\,d\eta'.
\int_{B_{x_\alpha}(r_\alpha)}\left\{u_\alpha X_\alpha(\nabla u_\alpha) +\frac{n-2}{2n}(_g X_\alpha) u_\alpha^2\right\}dv_g ~~= \mu_\alpha^{2}\left(\int_{\mathbb{R}^n}\left(u_0 \langle \nabla u_0,x\rangle_{\xi}+\frac{n-2}{2}u_0^2 \right) dx+o(1) \right).
\bar\partial_B^*\bar\partial_B\phi= -\sum_a\nabla_{V_a}\nabla_{\bar V_a}\phi +\nabla_{H^{0,1}}\phi +\sum_{a,b}\bar\omega^a\wedge \bar V_b\,\lrcorner\, \nabla_{V_b}\nabla_{\bar V_a}\phi &-\sum_b \bar\omega^a\wedge H^{0,1}\,\lrcorner\, \nabla_{\bar V_a}\phi.
\begin{cases}\displaystyle\frac{dy(t)}{dt}=Ay(t) + \displaystyle\int^0_{-r}\beta(\theta)A y(t+\theta)d\theta + f \dot B(t),\,\,\,\,t\ge 0, y(0)=\phi_0,\,\,\,\,y_0=\phi_1,\,\,\,\Phi=(\phi_0, \phi_1)\in{\cal X},\end{cases}
\alpha^{2}- n^{2}= 2 n \int_{0}^{\infty}{\frac{1}{1 + \left({\frac{r}{r_{0}}}\right)^{2 n}}}{\frac{d a}{d r}}d r + 2 \int_{0}^{\infty}\left( 1 - h^{2}\right) r d r \, .
\textsf{P}\{\sum_{i\le N}(\max_{j\le n}\tilde{g}_{ij}^{2})\Vert B_{i}\Vert_{2}^{2}>t \}\le e^{-\lambda t}\prod_{i=1}^{N}\textsf{E}\exp(\lambda (\max_{j\le n}\tilde{g}_{ij}^{2})\Vert B_{i}\Vert_{2}^{2}).
P^{(\alpha,\beta)}_{d,k}(x)=\frac{1}{C_{\alpha}}\frac{(|\alpha|+1)_k}{k!}\sum^{k}_{n=0}A^{(d)}_{\alpha}(n)\frac{(-k)_n(k+|\alpha|+\beta+1)_n}{(|\alpha|+1)_n}\frac{(\frac{1-x}{2})^n}{n!}.
\Theta^{i j}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{- B^{- 1}\sinh \gamma}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- B^{- 1}\cosh \gamma}}\\{{B^{- 1}\sinh \gamma}}&{{0}}&{{B^{- 1}\cosh \gamma}}&{{0}}\\ \end{array}\right) .
\begin{aligned}\nabla(\lambda\nabla\cdot\b u)+2\nabla\cdot(\mu\;\varepsilon(\b u))-\rho\b u_{tt}=\b F\qquad\Omega \b u|_{\partial\Omega}= 0 \b u|_{\Omega\times\{t=0\}}= \b g \partial_t \b u|_{\Omega\times\{t=0\}}=\b h.\end{aligned}
\begin{cases}\Delta \bar{w}-\partial_t\bar{w}=0{\rm{in}}\ \Q_{R/4}\cr\bar{w}=1{\rm{on}}\ \ \partial_p(\overline{Q_{R/4}}\setminus\{x_n=0\}\cr\bar{w}=\frac{5}{8}{\rm{on}}\ \Q_{R/4}'. \cr\end{cases}
{\mathrm d}X_{t, i}^{(u)}\, =\, b\big( t, X_{t, i}^{(u)}, \widehat{F}_{t, i}^{(u)}\big){\mathrm d}t +{\mathrm d}W_{t, i}\, ; t \ge 0 \, , i \, =\, 1, \ldots , n-1 \, ,
Pdf^*({E}_{i}) =((\beta_i-\alpha_i)\bar{p}, -q\alpha_i\bar p )_{(\bar p,q\bar p)}, Jdf^*({E}_i) =\frac{1}{\sqrt{3}}\left((2\beta_i-\alpha_i)\bar p, q(\alpha_i+\beta_i)\bar p \right)_{(\bar p,q\bar p)}
2\ell(J)^{d}\geq \mu^{I}(J)=\mu_{m}^{n(I)}(J)=\mu_{m}^{n(I)+1}(J)\frac{\ell(I)^{d}}{\mu_{m}^{n(I)+1}(I)}\geq 2^{d-n-1}\ell(J)^{d}.
U_{2^{p_0}+\cdots+2^{p_s}}=2^{p_0-p_1}U_{2+2^{p_2-p_1}+\cdots+2^{p_s-p_1}}+4(1-{2^{p_0-p_1}})U_{2^{p_1}+\cdots+2^{p_s}}+1-2^{p_0-p_1}.
\partial_i V ={1\over 4}\bar Z^{CD}\bar Z^{AB}P_{ABCD,i}+{1\over 4}Z_{AB}Z_{CD}P^{ABCD}{}_{,i}+{1\over 2}\bar Z_I P^I_{AB,i}\bar Z^{AB}+{1\over 2}Z^I P_{I,i}^{AB}Z_{AB}=0 \ .
\langle f_{\underline{j}},\gamma_{-}(f_{a})e_{\underline{i}}\rangle_{k}=\sum_{r=1}^{k}\delta_{a,i_{r}}\delta_{\underline{j},\underline{i}\backslash i_{r}}\langle f_{\underline{i}\backslash i_{r}}\wedge f_{a},e_{\underline{i}}\rangle_{k}.
\sum_{j=1}^{n-1}|a_jp_j| \leq \sum_{j=1}^{n-1}|a_jU_j| + 2C\sum_{j=1}^{n-1}|a_j| &\leq N\sum_{j=1}^{n-1}|a_j| + 2C\sum_{j=1}^{n-1}|a_j| &\leq (N+2C)(n-1)\frac{C}{N^3}&\leq (N+2C)\frac{C}{N^2}
\hat{\eta}=\bigl(\mathbf{h}_{\mathrm{u}}^{\mathrm{T}}\,\mathbf{h}_{\mathrm{u}}\bigr)^{-1}\,\mathbf{h}_{\mathrm{u}}^{\mathrm{T}}\,\bigl(\mathbf{y}-\mathbf{1}^{\mathrm{T}}L_{0}\bigr),
(\dot{\mathrm{H}}^{s_0,p}_0(\Omega),\dot{\mathrm{H}}^{s_1,p}_0(\Omega))_{\frac{s-s_0}{s_1-s_0},q}= \dot{\mathrm{B}}^{s}_{p,q,0}(\Omega)=\dot{\mathrm{B}}^{s}_{p,q}(\Omega),
\left|\frac{(\alpha)_{m}}{(\gamma)_{m}}\right|=\left|\frac{(\alpha)_{j}}{(\gamma)_{j}}\cdot\frac{(\alpha+j)_{m-j}}{(\gamma+j)_{m-j}}\right|\leq C(\alpha,\gamma)\cdot \left|\frac{(\alpha+j)_{m-j}}{(\gamma+j)_{m-j}}\right|< C(\alpha,\gamma)
V^{(k)}_{\{z_{i}\}}=\begin{bmatrix}1 1 \dots 1 z_{1}z_{2}\dots z_{k}z_{1}^{2}z_{2}^{2}\dots z_{k}^{2}\vdots \vdots \ddots \vdots z_{1}^{k-1}z_{2}^{k-1}\dots z_{k}^{k-1}\end{bmatrix}.
\rho_{\alpha_{1}, \alpha_{2}}(\mu) = \frac{1}{lp}\int_{0}^{\infty}y^{-\frac{\mu+1}{lp}-1}h_{\alpha_{1}, \alpha_{2}}(y^{-\frac{1}{lp}}, t) dy & = \frac{1}{M}\sum_{r=0}^{\infty}\frac{(-1)^r}{r!}\, \frac{\Gamma(\frac{m r - \mu}{M})}{\Gamma(-\mu)}\, t^{(1-\frac{m}{M})r + \frac{\mu}{M}}, t >0.
\int \, \Big( \mathfrak{J}_L \, \big( \, U^{L}_{g_0}\, f \big)_x , g_x \, \Big) \, d\mu_{\mathfrak{G}/H}(x)= \int \, (\mathfrak{J}_L \, f_x , k_x ) \, d\mu_{\mathfrak{G}/H}(x),
f'(x) =\frac{1}{xp(x)}\int_0^{x}(h(t)-\Gamma_{r,\lambda}h)p(t)\,\mathrm{d}t &=-\frac{1}{xp(x)}\int_x^{\infty}(h(t)-\Gamma_{r,\lambda}h)p(t)\,\mathrm{d}t,
\delta X^{\hat{m}^{\prime}}= - \Lambda^{\hat{m}^{\prime}}{}_{\hat{n}^{\prime}}X^{\hat{n}^{\prime}}\, ; \qquad \Lambda_{\hat{m}^{\prime}\hat{n}^{\prime}}= - \Lambda_{\hat{n}^{\prime}\hat{m}^{\prime}}\, .
\begin{array}{rcl}G^{\pm\pm}= -4 x^{\pm}x^{\pm}\left(\phi'e^{(\phi - 2 \psi)}\right)'e^{-\phi}\,, G^{\pm y}= 0\,, G^{\pm\mp}= 4 \left(- \zeta\phi'e^{\phi}\right)'e^{-(\phi + 2 \psi)}\,, G^{yy}= 4 \left(- \zeta\psi'\right)'e^{-2\psi}\,.\end{array}
HP(\mathbb{C}[x,y]/(\partial P_{ab}/\partial x,\partial P_{ab}/\partial y)dx\wedge dy) = \frac{(1-t^{d-s})(1-t^{d-r})}{(1-t^{s})(1-t^{r})}t^st^r = \frac{(t^{s}-t^{d})(t^{r}-t^{d})}{(1-t^{s})(1-t^{r})}.
\frac{\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (-s^{(1)}_{k})_{k=0}^{m-1}\right)}{ab\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (1,0,\ldots,0) \right)}\stackrel{a\to 0+}{\longrightarrow}\infty.
S_{\mathrm{N B I}}= - \frac{\alpha}{4}\mathrm{T r}\left( Y^{- 1}[ A_{\mu}, A_{\nu}]^{2}\right) + V ( Y ) - \frac{\alpha}{2}\mathrm{T r}( \bar{\psi}\Gamma^{\mu}[ A_{\mu}, \psi ] ) ,
x_{3}p_{3}+ x_{4}p_{4}= 8 f + 4 a ~ ~ , ~ x_{3}q_{3}+ x_{4}q_{4}= 1 2 g ~ ~ , ~ x_{3}r_{3}+ x_{4}r_{4}= 6 k ~ ~ , ~ x_{3}s_{3}+ x_{4}s_{4}= 4 m
T_1 =\frac{d-c}{c}\sum_{n=1}^N \frac{\left(\frac{dq}{c}\right)_{n-1}(c q)_{N-n}(cq)^n}{(q)_n (c q)_N (q)_{N-n}}T_2 = \frac{d-c}{c}\sum_{n=1}^N \frac{\left(\frac{dq}{c}\right)_{n-1}(c q)_{N-n}(cq)^n}{(q)_n (c q)_N (q)_{N-n}}\left(-\sum_{k=1}^{n-1}\frac{q^kd/c}{1- q^kd/c}+ \sum_{k=1}^n \frac{q^k}{1-q^k}\right).
\sigma^{(i)}y = \sum_{s = 0}^i (-1)^s C_i^s (\sigma y^{(s)})^{(i - s)}, (\sigma^{(i)}y^{(k)})^{(k)}= \sum_{s = 0}^i (-1)^s C_i^s (\sigma y^{(s + k)})^{(i - s + k)}, ( (\sigma^{(i)}y^{(k)})^{(k)}, z ) = \sum_{s = 0}^i (-1)^{i + k}C_i^s ( \sigma y^{(s + k)}, z^{(i - s + k)}), z \in \mathfrak D.
\frac{\|p_{n+1}\|^2}{2}=\frac{\|p_n\|^2}{2}+ \alpha_*\frac{\beta^{(1)}(\kappa_n)}{\|p_n\|}+\alpha_*^2\left(\frac{\delta \beta^{(2)}(\kappa)}{\|p_n\|^4}+\frac{\delta \beta^{(4)}(\kappa)}{\|p_n\|^4}\right).
4K_X =\widehat{\pi}^*(4K_Y+2D_1+2D_2+2D_3+3D_{g,\i}+3D_{g,-\i}+3D_{g_1g, \i}+3D_{g_1g, -\i}) =\widehat{\pi}^*(-K_Y+\Phi+\Phi_1)+\widehat{\pi}^*(M_1+2M_2+2N_2+2M_3+2N_3)
\begin{cases}\Delta \eta_1(x) +\lambda\eta_1(x)+\lambda_1\eta(x)=\lambda\eta(x_0)cap(\omega)\lvert x-x_0\rvert^{-1},\;\;\;x\in\Omega begin{align*}0.3cm]\eta_1(x)=\eta(x_0)cap(\omega)\lvert x-x_0\rvert^{-1},\;\;\;x\in\partial\Omega\end{cases}
\int_{0}^{t}\int_{0}^{\pi}f(v(s,x)) \phi(x) \, \widetilde{W}(\textrm{d}s, \textrm{d}x) = M_t(\phi) = \langle \overline{v}_t, \phi \rangle - \int_{0}^{t}\langle v(s, \cdot), \phi'' \rangle \, \textrm{d}s, t \leq T,
\Tilde{\Phi}_L(\beta)\Tilde{\sigma}_j^{-1}(\xi) = \Tilde{\sigma}_i^{-1}h_i(\xi) = \Tilde{\sigma}_i^{-1}((\alpha,h_i)_{M,n}+\xi) = (\alpha,h_i)_{M,n}+ \Tilde{\sigma}_i^{-1}(\xi)
D_t v^j \coloneqq \alpha_j \delta_t v^j + \beta_j \delta_t v^{j-1}, \alpha_j = \frac{2\tau_j+\tau_{j-1}}{\tau_j+\tau_{j-1}}, \beta_j = - \frac{\tau_j}{\tau_j+\tau_{j-1}}\,, j=2,\dots,M.
\langle \mathbf{v}_1, \mathbf{z}\rangle \langle \mathbf{v}_2, \mathbf{z}\rangle = h^2 - \frac{(a + b + c)^2}3 &= \frac23\left((a - b)^2 + (b - c)^2 + (c - a)^2\right) &\ge \frac23.
k(\zeta)=\mathbf a_+(\zeta)\begin{pmatrix}1 -\bar{\zeta}\zeta 1\end{pmatrix}=\begin{pmatrix}1 0 \zeta 1\end{pmatrix}\begin{pmatrix}\mathbf a_+(\zeta) 0 0 \mathbf a_+(\zeta)^{-1}\end{pmatrix}\begin{pmatrix}1 -\bar{\zeta}0 1\end{pmatrix}\in SU(2).
\hat{R}(z) = \frac{\frac{-10 + 7 \sqrt{2}}{6(\sqrt{2}-1)}z^3 + \frac{3-2\sqrt{2}}{\sqrt{2}-1}z + 1}{\left(1- \frac{2-\sqrt{2}}{2}z \right)^2}| \hat{R}(\infty) | = \infty
h^{0}(\mathcal{E}\otimes\mathcal{E^{\vee}})-h^{1}(\mathcal{E}\otimes\mathcal{E^{\vee}})+h^{2}(\mathcal{E}\otimes\mathcal{E^{\vee}})-h^{3}(\mathcal{E}\otimes\mathcal{E^{\vee}})=-14
\delta_{\bf a}{\cal L}_{0}= \mu \dot{\bf x}\cdot \delta_{\bf a}\dot{\bf x}+ \frac{\mu}{R^{2}}{\bf x}\cdot \delta_{\bf a}{\bf x}= \frac{\mu}{R}{\bf a}\cdot \frac{d}{d t}\left[{\bf x}\sinh \left( \frac{t}{R}\right) \right] .
R^{-}_h[u_h](x,\theta) = \sup_{j=1,\ldots,n}\frac{u_h(x)-u_h(x-e_j)}{|e_j| \cos(\theta - \theta'_j)}R^{+}_h[u_h](x,\theta) = \inf_{j=1,\ldots,n}\frac{u_h(x+e_j)-u_h(x)}{|e_j| \cos(\theta - \theta'_j)}.
\psi_\lambda(g)=\sum^r_{i=0}\tilde{c}_i(t)\big(\zeta\bar{\zeta}'\big)^{\Lambda_1-i}\big(\xi\bar{\xi}'\eta\bar{\eta}'\big)^{\Lambda_2}\big(\gamma\bar{\gamma}'\big)^{\Lambda_3}\sum^i_{l=0}{i\choose l}{r-i \choose i-l}a^l(bc)^{i-l}d^{r-2i+l}.
\beta = ( U_{0}^{\dag}U_{0})^{- 1 / 2}= \left( \begin{array}{c c}{[ ( Z_{1}+ Z_{2}) f^{- 1}]^{- 1 / 2}}&{0}\\{0}&{[ ( Z_{1}+ Z_{2}+ \zeta ) f^{- 1}]^{- 1 / 2}}\\ \end{array}\right) .
\partial_t u^\varepsilon + \nu \eta_\varepsilon*\eta_\varepsilon* A^s u^\varepsilon +\mathcal{P}^\alpha \eta_\varepsilon*[(\eta_\varepsilon*u^\varepsilon)\cdot \nabla (\eta_\varepsilon*u^\varepsilon)+\mathcal{U}^\alpha(\eta_\varepsilon*u^\varepsilon,\eta_\varepsilon*u^\varepsilon)]=0
\mathcal{P}(x_k = a_m|{\bf y}_{C}) = \frac{\mathcal{P}(x_k = a_m |{\bf y}_{MS})\mathcal{P}(x_k = a_m|{\bf y}_{MR})}{\sum_{m}\mathcal{P}(x_k = a_m|{\bf y}_{MS})\mathcal{P}(x_k = a_m|{\bf y}_{MR})}
f(x)=\sum_{E=\{a_1,\ldots,a_r\}\in \mathcal{E}_2}\prod_{j=1}^{m_2}\frac{x_{i_j}^{E(j)}}{E(j)!}=\frac{\lambda_{\mathcal{E}_2}(x_{i_1}, \dots, x_{i_m})}{r!},
\lambda_{(0)}^{\alpha a A}{\lambda_{(0)\alpha a}}^{B}= \frac{3\cdot 2^6}{g^{2}}\frac{\rho_0^{4}}{\left( \rho_0^{2}+(x-x_{0})^{2}\right)^{4}}\: \zeta^{\alpha A}{\zeta_{\alpha}}^{B}\, .
\sum_{r=1}^{n+m+1}(-1)^{r+1}\sum_{1\leq k_1<\ldots<k_r\leq n+m+1}\left(\prod_{i=1}^{r}\tau_t(c_{k_i}) \right)\tau_t(c_1\cdots \hat{c}_{k_1}\cdots \hat{c}_{k_r}\cdots c_{n+m+1})+o(t),
\angle^+_{\gamma(0)}(\gamma,\eta) = \limsup_{t,s\to0^+}\angle^k_{\gamma(0)}(\gamma(t),\eta(s)) = \lim_{\varepsilon\to 0}\sup \big\{\angle^k_{\gamma(0)}(\gamma(t),\eta(s)) : 0<s,t\leq \varepsilon \big\}.
S = \frac 1 2 (^{+}\tau -^{-}\tau ) \int_{M}\varepsilon^{\mu \nu \alpha \beta}\,{\cal R}_{\mu \nu}^{a b}\,{\cal R}_{\alpha \beta}^{c d}\epsilon_{a b c d}- \frac i 2 (^{+}\tau +^{-}\tau ) \int_{M}\varepsilon^{\mu \nu \alpha \beta}\,{\cal R}_{\mu \nu}^{a b}\,{^{*}{\cal R}}_{\alpha \beta}^{c d}\, \epsilon_{a b c d},
Q(x) = \frac{x}{1-e^{-x}}= 1 + \frac{x}{2}+ \sum_{l \geq 1}\frac{(-1)^{l-1}B_{2l}}{(2l)!}x^{2l}= 1 + \frac{x}{2}+ \frac{x^2}{12}- \frac{x^4}{720}+ \ldots
\dfrac{M_{2}(\mathcal{C}(D_{2m}))}{|e(\mathcal{C}(D_{2m}))|}= \dfrac{(m-2)(m-3)^{3}+m}{(m-2)(m-3)+m}> \dfrac{(m-2)(m-3)^{2}+m}{2m-2}= \dfrac{M_{1}(\mathcal{C}(D_{2m}))}{|v( \mathcal{C}(D_{2m}))|}.
\frac{1}{b_{n}}\mathrm{cov}(K(\frac{x_{j}-X_{0}}{b_{n}}),K(\frac{x_{p}-X_{k}}{b_{n}})) =\frac{1}{b_{n}^{3}}\int{\displaystyle\int}K^{\prime}(\frac{x_{j}-u}{b_{n}})K^{\prime}(\frac{x_{p}-v}{b_{n}})H_{k}(u,v)dudv\leq||K^{\prime}||_{\infty}^{2}\frac{1}{b_{n}^{3}}\eta_{k}.
\left(1 - x \right) A_m L(x,y) = L \left( x \gamma , \frac ym \right) A_m - x L \left( x \gamma , \frac{yq}m \right) E \left(\frac{yq}m \right) A_m E(y)^{-1}&U\left( x \gamma, \frac{yq}m \right) A_m \left(1 - \frac qx \right) = A_m U(x,y) - \frac qx E \left( \frac{yq}m \right)^{-1}A_m E(yq) U(x,yq)
\delta S^{CTP}\left[ \phi^+,\phi^-\right] =-i\ln \int D\Phi^+D\Phi^-\;e^{i\left\{S_H\left( \Phi^+\right) -S_H\left( \Phi^+\right) +S_{lH}\left(\phi^+,\Phi^+\right) -S_{lH}\left( \phi^-,\Phi^-\right) \right\}}
\int_{\Omega_\ell}\big(A \nabla (\rho_{\ell^{'}}u_\ell)\big).\nabla (\rho_{\ell^{'}}u_\ell) - \int_{\Omega_\ell}u_\ell^2 (A\nabla \rho_{\ell^{'}}).\nabla \rho_{\ell^{'}}= \lambda_\ell^1\int_{\Omega_\ell}\rho_{\ell^{'}}^2u_\ell^2\,.
z\in\mathrm{Fix}(S_{N})=\mathrm{Fix}\left(\mathcal{T}_{\left\{C_{m,r}\left(N\right)\left(j\right)\right\}_{j=1}^{r}}\right)=\cap_{j=1}^{r}C_{f\left(\left(r-1\right)N+j-1\right)}\subset C_{f\left(\left(r-1\right)N+k\right)}=C_{p}.
c_k =\begin{cases}\tau\sum_{n=0}^{k}\frac{a_{n}}{(k-n)!}- (\tau-1)b_k 0 \le k \le M-1 \tau\sum_{n=0}^{M}\frac{a_{n}}{(k-n)!}\quad\,\,\,\,\, k>M-1\end{cases}
\frac{[ a^{\prime}]}{a}= - \frac{\kappa^{2}}{3}( \lambda_{-}( \phi ) + \rho ) b \mathrm{~ , ~}\frac{[ n^{\prime}]}{n}= \frac{\kappa^{2}}{3}( - \lambda_{-}( \phi ) + 3 p + 2 \rho ) b
\sum_{n=-\infty}^{\infty}g_{\mathbb{H},s}(\sigma_{p}^{-1}w,\gamma_{\infty}^{n}\sigma_{p}^{-1}z)=\frac{4\pi}{2s-1}\Im(\sigma_{p}^{-1}w)^{s}\Im(\sigma_{p}^{-1}z)^{1-s}+\sum_{n\neq0}\frac{1}{|n|}W_{s}(n\sigma_{p}^{-1}z)\overline{V_{\overline{s}}(n\sigma_{p}^{-1}w)}.
C^{\mathrm{ul},k}_{2}\left( N_{\mathrm{sct}}, N_{\mathrm{scr}}\right) = \left(\frac{T_{\mathrm{ul}}}{T}\right) \min\left\lbrace c^k_{\mathrm{u2s},1}\left( N_{\mathrm{scr}}\right), c^k_{\mathrm{s2b},2}\left(N_{\mathrm{sct}}\right) \right\rbrace.
\sum_{\substack{\lambda=1 \{\lambda:\, \varepsilon_j(\lambda,N,\Sigma)=\varepsilon_j(\lambda,N,\Sigma^{\ast})=h\}}}^{b^n}1=b^{j-1}(b^{n-j}-(\nu_j(N,\Sigma^{\ast})-\nu_j(N,\Sigma))).
\pi_i=g\bar\Gamma\psi_i'\sqrt{2V_{eff}-\rho_f-p_f}e^{3T}, \ \ \ \pi_f=g\bar\Gamma({n_f^2\theta_f'\over \rho_f+p_f})\sqrt{2V_{eff}-\rho_f-p_f)}e^{3T},
\Phi_{\left( k\right)}\left( \tau, \theta^{+}, \theta^{-}\right)=q_{0\left( k\right)}\left( \tau \right) +\theta^{+}\lambda_{-}W^{-k}+\theta^{-}\lambda_{+}W^k+\theta^{+}\theta^{-}\left( -W+\nu\frac{W^{\prime}}{2W}\right) .
f^{[2]}(1,1,z_2) =\frac{f^{[1]}(1,1)-f^{[1]}(1,z_2)}{1-z_2}=\frac{\frac{d}{dz}f(z)_{\vert z=1}-\frac{f(1)-f(z_2)}{1-z_2}}{1-z_2}=\frac{-g(z_2)}{1-z_2}=g^{[1]}(1,z_2).
\left[\begin{array}{c}p_{i0}^v p_{i1}^v\end{array}\right]\in\left\{\left[\begin{array}{c}0 1\end{array}\right],\left[\begin{array}{c}1 0\end{array}\right]\right\}~~i=1,\dots,|\mathcal{V}|, &p_{jk}^e=0.7,~\forall{e_{jk}}\in\mathcal{U}_e.
I_1 = \frac{1}{16\pi^2}\int \frac{d^4k}{(2\pi)^4}\,\frac{d^4l}{(2\pi)^4}\,\frac{1}{\Big(1+k^{2n}/\Lambda^{2n}\Big)\,\Big(1+l^{2n}/\Lambda^{2n}\Big)\,k^4 l^4}.
\frac{dy_i}{dz}= \sum^m_{j=1}(D\varphi )^{-1}_{ij}f_j(\varphi (y),z)+ \sum^m_{j=1}(D\varphi )^{-1}_{ij}g_j(\varphi (y),z) =: F_i(y,z) + G_i(y,z),
S(x): ={\sum_p}^* \left |a_{L_1}(p)\Delta_1(\log p)+\cdots+a_{L_m}(p)\Delta_m(\log p)\right|^2 &=\sum_{j=1}^m{\sum_p}^* |a_{L_j}(p)|^2|\Delta_j(\log p)|^2 &+\sum_{1\leq k\ne l\leq m}{\sum_p}^* a_{L_k}(p)\overline{a_{L_l}(p)}\Delta_k(\log p)\overline{\Delta_l(\log p)}.
\frac{L_{i}}{n_{i}}\sum_{l=1}^{n_{i}}t_{x_{l}}=-\frac{L_{t}}{n_{t}}\sum_{j=1}^{n_{t}}\overline{t}_{x_{j}}-\frac{S}{n_{\Omega}}\sum_{k=1}^{n_{\Omega}}b_{x_{k}}
h_{\dot{x}^1\dot{x}^1}= \frac{\lambda (\dot{x}^2)^2}{((\dot{x}^1)^2+(\dot{x}^2)^2)^{\frac{3}{2}}}, ~ h_{\dot{x}^1\dot{x}^2}=-\frac{\dot{x}^1\dot{x}^2}{((\dot{x}^1)^2+(\dot{x}^2)^2)^{\frac{3}{2}}}, ~h_{\dot{x}^2\dot{x}^2}= \frac{\lambda (\dot{x}^1)^2}{((\dot{x}^1)^2+(\dot{x}^2)^2)^{\frac{3}{2}}}.
\begin{array}{ll}\partial_y w^1 = \frac{1}{\sqrt{\pi}}\Big(\int\limits_{-\frac{y}{2\sqrt{t}}}^{+\infty}e^{-\xi^2}\partial_y w_0^1(2\sqrt{t}\xi + y) \,\mathrm{d}\xi+ \int\limits_{\frac{y}{2\sqrt{t}}}^{+\infty}e^{-\xi^2}\partial_y w_0^1(2\sqrt{t}\xi - y) \,\mathrm{d}\xi\Big).\end{array}