harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Bigl[ \varepsilon ({\bf r}, \omega ) \, \omega^2 \delta_{j m}-{\rm rot}_{j l}{\rm rot}_{l m}\Bigr] D_{m k}(\varepsilon,{\bf r},{\bf r}^\prime, \omega) = - \omega^2 \delta ({\bf r}-{\bf r}^\prime) \delta_{j k}
\frac{m_{i}}{m_{j}}=\frac{\sin \left[ (\eta_{k}-\eta_{j})\pi \vartheta_{h}+(\hat{\eta}_{k}-\hat{\eta}_{j})\pi \vartheta_{H}\right]}{\sin \left[(\eta_{i}-\eta_{k})\pi \vartheta_{h}+(\hat{\eta}_{i}-\hat{\eta}_{k})\pi\vartheta_{H}\right]}\;.
\begin{aligned}\lim_{n\rightarrow \infty}\int_I \left(u_n(x)-u(x)\right)^2m(dx) &=\lim_{n\rightarrow\infty}\int_W \left(\varphi\left(\ss_n(x)\right)-\varphi\left(\ss_\infty(x)\right)\right)^2m(dx) &= \int_W\lim_{n\rightarrow\infty}\left(\varphi\left(\ss_n(x)\right)-\varphi\left(\ss_\infty(x)\right)\right)^2m(dx) &=0. \end{aligned}
S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-\eta}[2h_{|\nu}^{\mu\nu}h_{\mu\lambda}^{|\lambda}-2h_{|\nu}^{\mu\nu}h_{\lambda|\mu}^{\lambda}+h_{\nu|\mu}^{\nu}h_{\lambda}^{\lambda|\mu}-h^{\mu\nu|\lambda}h_{\mu\nu|\lambda}]+S_{m}
49m_{\chi\chi}^1 =37\Longleftrightarrow 49m_{\chi\chi}^u=6\Longleftrightarrow 49m_{\chi\chi}^v=6, 49m_{\chi\chi}^1 =39\Longleftrightarrow 49m_{\chi\chi}^u=5\Longleftrightarrow 49m_{\chi\chi}^v=5, 49m_{\chi\chi}^1 =43\Longrightarrow 49m_{\chi\chi}^u=3, 49m_{\chi\chi}^1 =36\Longleftrightarrow\{49m_{\chi\chi}^u,49m_{\chi\chi}^v\}=\{3,10\}.
\log(p_{n+1})=\lim_{s\to\infty}\left(\frac{Q_n(as)}{\zeta(as)}\right)\left(1-\frac{Q_n(as)}{\zeta(as)}\right)^{-1}\left(\frac{Q_n^{'}(as)}{Q_n(as)}-\frac{\zeta^{'}(as)}{\zeta(as)}\right)
c_3=\left[\frac{n(n-1+p')}{p'-1}\cdot\frac{|B_{p'}^n|_n}{|B_{p'}^{n-1}|_{n-1}}\right]^{\frac{p'}{n-1+p'}}\cdot\frac{p'-1}{p'}, c_4=n^{\frac{2}{n+1}}G_{B_{p'}^n}((n^{-1/p'},\dots,n^{-1/p'}))\quad.
\begin{array}{cl}G^{(i)}_N (z, z'| z'_1 , \cdots , z'_N , z_N , \cdots , z_1 )^{j'_1 , \cdots , j'_N , j_N , \cdots , j_1}={}_{i}\langle B|A^{(i)}_{SW}(z, z')\phi_{j'_1}(z'-z'_1 )\cdots \phi_{j'_N}(z'-z'_N )\phi^{j_N}(z_N -z') \cdots \phi^{j_1}(z_1 -z') A^{(i)}_{SW}(z, z')|B\rangle_i . \end{array}
\cdot \> \left[ \frac{\Gamma(n_1+|m|+\delta_1+1)\Gamma(n_2+|m|+\delta_2+1)\Gamma(n+l+\delta_1+\delta_2+1)}{(n_1)!(n_2)!(n-l-1)!\Gamma(l+\delta_2+1)\Gamma(l+|m|+\delta_1+\delta_2+1)}\right]^{1/2}
\Psi \bmod e_{h+1}=\Psi' \bmod e_{h+1}= \pm |\det G|^2\sigma_0\wedge \bar{\sigma}_0\wedge \cdots \wedge \sigma_h\wedge \bar{\sigma}_h &= \pm |\det G|^2Ae_0\wedge f_0\wedge \cdots \wedge e_h\wedge f_h.
\partial_{J}^{I}u_{i}^{K}= \delta_{J}^{K}u_{i}^{I}-{\frac{1}{N}}\delta_{J}^{I}u_{i}^{K}, \qquad \partial_{J}^{I}u_{K}^{i}= - \delta_{K}^{I}u_{J}^{i}+{\frac{1}{N}}\delta_{J}^{I}u_{K}^{i}.
\setlength\arraycolsep{1pt}A=\begin{psmallmatrix}-a_0 0 0 0 0 w w_0 P -a-w^2 P 0 0 0 w w_0 P^{a}-1 -w^2 P^{a}-a-w^2P 0 0 w w_0 P^{\sigma}-w^2 P^{\sigma}0 -a-w^2P 0 w_0 (P+wP^w) -w^2 P^{w}0 0 -a-w^2P\end{psmallmatrix},
\begin{aligned}\log M^{- r}_{{\rm sym}^m}= -r\sum_{p\leqslant (m+1)r}\log D(p^{-1},{\rm sym}^m[g(\theta_{m,p}^-)]) & + \sum_{p>\sqrt{(m+1)r}}h_m^{-}((m+1)r/p) + O_m(r^{1/2})= : S_1 + S_2 + O_m(r^{1/2}).\end{aligned}
\Vert [\widetilde{\Lambda}^\varepsilon](S_\varepsilon -I)(B^0-\zeta\overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}&\leqslant 2 C_\Omega^{(p)}\Vert \widetilde{\Lambda}\Vert_{L_p(\Omega)}\Vert (B^0-\zeta\overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow H^1(\mathbb{R}^d)}.
\Gamma = \left\lbrace 1 - \sup_{\lbrace \cup_{j=1}^s U_j : U_j \in \mathfrak{U}_{i_j}\rbrace}\min_{H \in \mathcal{H}}\det(H^\intercal H) : \lbrace \mathfrak{U}_{i_1},\ldots,\mathfrak{U}_{i_s}\rbrace \in \mathfrak{P},~s=1,\ldots,r\right\rbrace.
\mathbb{X}: = \overline{\mathbb{Q}}[\mathcal{K}_1]\times\ldots\times\overline{\mathbb{Q}}[\mathcal{K}_\aleph]= \left\{[p_1,\ldots,p_\aleph],\:p_i\in\overline{\mathbb{Q}}[\mathcal{K}_i],\:i=1,\ldots,\aleph\right\},
\delta \alpha_1+\gamma \left( -\frac 12C_{\;\;bc}^aB_a^{\mu \nu}A_\mu^bA_\nu^c\right) =\partial_\mu \left( -\frac12C_{\;\;bc}^a\varepsilon^{\mu \nu \lambda \rho}A_\nu^bA_\lambda^c\eta_{a\rho}\right) .
M^{\prime}={\cal U}^{T}\left( \begin{matrix}{U_{5}}&{0}\\{0}&{U_{2 1}}\\ \end{matrix}\right){\cal U}M{\cal U}^{T}\left( \begin{matrix}{U_{5}^{T}}&{0}\\{0}&{U_{2 1}^{T}}\\ \end{matrix}\right){\cal U},
\begin{array}{l l l}{D_{u}V}&{=}&{U}\\{D_{u}U}&{=}&{e^{K}C_{u u u}G_{u{\bar{u}}}^{- 1}{\bar{U}}}\\{d_{u}{\bar{U}}}&{=}&{G_{u{\bar{u}}}{\bar{V}}}\\{d_{u}{\bar{V}}}&{=}&{0}\\ \end{array}
{\cal W}(M_{\alpha})=\int_{M_{\alpha}}H_{\alpha}^2dA_{\alpha}+\int_{\partial M_{\alpha}}\kappa^{\alpha}ds=\bar{\cal W}(\bar{M}_{\alpha})=\int_{\bar{M}_{\alpha}}\left(\bar{H}_{\alpha}^2+\bar{R}_{\alpha}\right) d\bar{A}_{\alpha}+\int_{\partial\bar{M}_{\alpha}}\bar{\kappa}^{\alpha}d\bar{s},
F_{n-1}-{(n-1)}^{p}\,\,\,\,\, \ = \ \sum_{i=1}^{n-1}\left(i^{p}\ - \ 2\sum_{j=0}^{p/2}\binom{p}{2j+1}i^{p-2j-1}\right) \cdot F_{n-1-i}, F_n - n^{p}\,\,\,\,\, \ = \ \sum_{i=1}^{n}\left(i^{p}\ - \ 2\sum_{j=0}^{p/2}\binom{p}{2j+1}i^{p-2j-1}\right) \cdot F_{n-i}.
\begin{matrix}Minimize&f^{T}x&+&g^{T}y&+&h_{k}^{T}z_{k}&&\\subjectto&Tx&+&Uy&&&=&r\\ &&&V_{k}y&+&W_{k}z_{k}&=&s_{k}\\ &x&,&y&,&z_{k}&\ge&0\end{matrix}
I_{}= \frac{\mathcal{M}^\oplus_{|\cdot|}(\beta_1,\beta_2,\beta_3;\boldsymbol{\eta},\boldsymbol{\zeta},\boldsymbol{\tau},\boldsymbol{\rho})^2}{\|(\boldsymbol{y}-\boldsymbol{\eta},\boldsymbol{p}-\boldsymbol{\zeta})\|_{\boldsymbol{H}^{\textbf{curl},\frac{1}{2}}}}
\Delta{K}^{s,\infty}= \big(\mu^s_1(\beta-1)-\mu^s_0(\alpha-1)\big)+H(\alpha)-H(\beta)+s+\log\Big(\frac{1+2^{\mu^s_1+\Delta{K}^{s,\infty}}}{1+2^{\mu^s_0+\Delta{K}^{s,\infty}}}\Big).
- \frac{i}{2}\int_{x_{0}}^{x}d x_{j}\, \, \epsilon^{m l}k_{m}\partial_{l}A_{j}^{( 0 )}= - \frac{i}{2}m \int_{x_{0}}^{x}\, d x_{j}\, ({\bf v}\times \nabla A_{j}^{( 0 )})_{3}.
0 \le \int_0^\infty\frac{d\sigma(t)}{t(1+t^2)}=\int_0^\infty\frac{d\widetilde\sigma(t)}{1+t^2}\overset{\eqref{eq:phi_form_2_cond}}{{}\le{}}- a - \sum_{i=1}^{\varkappa}\frac{\sigma_i}{\lambda_i}<\infty
\hat{y}_{i}\in[b_{0}+b_{1}x_{i}\pm W\sqrt{(\frac{1}{n-2}\sum_{j=1}^{n}e_{j}^{2})\cdot(\frac{1}{n}+\frac{(x_{i}-\overline{x})^{2}}{\sum_{j=1}^{n}(x_{j}-\overline{x})^{2}})}],
v_1(s):=\frac{\cos(2\zeta(s))-n^2}{n^2-1}, ~v_2(s):=\frac{2i\sin\zeta(s)\mathcal S(\cos(\zeta(s)),n)}{n^2-1}, &F(s):= e^{-i k_{+}\vert y \vert\cos(\zeta(s)+\theta_{\hat y})}.
\mathbb{E}\left[\frac{\left( \sum^K_{i=1}a_i \right)}{p_s}\mathbf{S}\right] = \left( \sum^K_{i=1}a_i \right) \frac{p_s}{p_s}\mathbf{\Sigma}_s= \sigma^2\mathbf{I}_m \sum^K_{i=1}a_i ,
\frac{\dim(\mathbb{Z}(\mathcal{T}_\xi)\rtimes\mathcal{S})}{n+1}\dim^{+}(\overline{\mathbf{1}}) = d^{2n}(-1)^{\lvert\Phi^+\rvert+n}\left(\sum_{w\in\mathfrak{S}_{n+1}}(-1)^{l(w)}\xi^{\langle 2\rho,w\bullet 0\rangle}\right)^{-2},
\mathbb{P}(N_\varepsilon^{x}> M) \leq \mathbb{P}(r(X^{x}_M)\geq \varepsilon)=\mathbb{P}(V(X^{x}_M)\geq \sqrt{\varepsilon}) \leq \frac{1}{\sqrt{\varepsilon}}\mathbb{E}\left\{V(X^{x}_M)\right\}=\frac{1}{\sqrt{\varepsilon}}P^MV(x) \leq \left(1-\frac{\beta^2(1-\delta)}{4d}\right)^M \frac{V(x)}{\sqrt{\varepsilon}}, x\in D.
q ( t^{\prime})^{- ( 1 + r )}+ 2 r \frac{t^{\prime \prime \prime}t^{\prime}- ( t^{\prime \prime})^{2}}{( t^{\prime})^{2}}+ \frac{( 1 + r )^{2}+ 4 r^{2}}{4}\left( \frac{t^{\prime \prime}}{t^{\prime}}\right)^{2}- 1 = 0 \, .
\mathcal{D}_\beta = \partial_\beta + \frac{1}{4}\partial_{\beta}x^+ \left[ \omega_{\rho \sigma +}\Gamma^{\rho \sigma}- \frac{1}{2 \cdot 5!}F_{\lambda \nu \rho \sigma \kappa}\Gamma^{\lambda \nu \rho \sigma \kappa}i\sigma^2 \Gamma_+ \right]
\begin{array}{l}{\delta \theta^{\alpha}= \epsilon^{\alpha}, \qquad \delta \bar{\theta}_{\alpha}= \bar{\epsilon}_{\alpha},}\\{\delta x^{\bar{\mu}}= - i \bar{\epsilon}_{\alpha}\tilde{\Gamma}^{{\bar{\mu}}\alpha \beta}\bar{\theta}_{\beta}+ i \epsilon^{\alpha}\Gamma_{\alpha \beta}^{\bar{\mu}}\theta^{\beta}, \qquad \delta x^{1 1}= 0 ,}\\ \end{array}
{\mathcal{M}}_\alpha= \frac{1}{2\alpha}\left( \alpha \mathcal I + \mathcal H \right)\left( \alpha \mathcal I + \mathcal S \right), ~~{\mathcal N}_\alpha=\frac{1}{2\alpha}\left( \alpha \mathcal I - \mathcal H \right)\left( \alpha \mathcal I -\mathcal S \right).
m_{1,5}=0, m_{2,5}=0, 1-m_{1,3}m_{6,8}=0, m_{6,8}+m_{1,3}m_{7,8}=0, 2m_{2,3}m_{6,8}+m_{7,8}=0, m_{4,5}-m_{7,8}=0, 1-m_{2,3}m_{7,8}=0, 1-m_{4,5}m_{6,8}=0, 2m_{6,8}+m_{4,5}m_{7,8}=0.
e_{(k)}H_{2k}^{-1}H_{2k+1}^{-1}H_{2k-1}H_{2k}e_{(k)}= e_{(k)}H_{2k}^{-1}H_{2k+1}H_{2k-1}^{-1}H_{2k}e_{(k)}+(q-q^{-1})e_{(k)}(H_{2k}-H_{2k}^{-1})e_{(k)}-(q-q^{-1})^{2}e_{(k)}^{2}= e_{(k)}H_{2k}^{-1}H_{2k+1}H_{2k-1}^{-1}H_{2k}e_{(k)}.
U:=\{t\in A:u(t-\varepsilon\alpha(\xi)),u(t-\varepsilon\beta(\xi))\notin\lbrack\tfrac{1}{2},\tfrac{3}{2}]\}~, & V:=\{t\in A:u(t-\varepsilon\alpha(\xi)),u(t-\varepsilon\beta(\xi))\notin\lbrack-\tfrac{3}{2},-\tfrac{1}{2}]\}~.
\sqrt{n}\int\left\Vert \hat{\Delta}_{\ell1}(w)\right\Vert F_{0}(dw) \leq\sqrt{n}\left\Vert \hat{\alpha}_{1\ell}-\alpha_{10}\right\Vert \left\Vert\hat{\gamma}_{1\ell}-\gamma_{10}\right\Vert & =O_{p}(\sqrt{n}\{n^{-d_{1}[(2\xi_{1}-1)/(2\xi_{1}+1)]2\xi_{2}/[2\xi_{2}+1]}[\ln(n)]^{2}+n^{-\xi_{3}/(2\xi_{3}+1)}\ln(n)+n^{-d_{1}}\}n^{-d_{1}}) & =o_{p}^{{}}(1).
u_{j}:=\left. \left( \mathsf{S}_{j}\left( s\right) u_{\operatorname*{N};j}^{\operatorname*{mult}}-\mathsf{D}_{j}\left( s\right)u_{\operatorname*{D};j}^{\operatorname*{mult}}\right) \right\vert_{\Omega_{j}^{-}},\quad1\leq j\leq n_{\Omega}.
\nabla_U(\sigma) = \nabla_U(\sigma^\beta (e_\beta|_U)) = d\sigma^\beta \otimes (e_\beta|_U) + \sigma^\beta \nabla_U(e_\beta|_U) & = d\sigma^\beta \otimes (e_\beta|_U) + \sigma^\beta (\nabla_M(e_\beta)|_U) & = d\sigma^\beta \otimes (e_\beta|_U) + \sigma^\beta (\omega^\alpha_\beta \otimes e_\alpha)|_U.
A_1^\dagger A_1 \vert n_1,n_2\rangle = n_1\vert n_1,n_2\rangle; A_2^\dagger A_2 \vert n_1,n_2 \rangle = n_2 \vert n_1,n_2\rangle; \vert n_1,n_2 \rangle = \frac{(A_1^\dagger)^{n_1}(A_2^\dagger)^{n_2}}{\sqrt{n_1!n_2 !}}\psi_{00},
w_iw_jw_{ij}=A^3\alpha^3(\alpha-1)d^{3\alpha-4}+4A^2B\alpha^2(\alpha-1)d^{2\alpha-3}x'\cdot\nabla_{x'}d &\qquad+2A^2B\alpha^2d^{2\alpha-2}|\nabla_{x'}d|^2+4AB^2\alpha(\alpha-1)d^{\alpha-2}(x'\cdot\nabla_{x'}d)^2 &\qquad+4AB^2\alpha d^{\alpha-1}[2x'\cdot\nabla_{x'}d+x_ax_bd_{ab}]+8B^3|x'|^2.
\hat{Q}= \int d x \colon \left[ \gamma^{\mu}\gamma^{0}\hat{\vec{\psi}}( x^{\mu}) \partial_{\mu}\hat{\vec{\psi}}( x^{\mu}) + i \gamma^{0}\hat{\vec{\psi}}( x^{\mu}) \vec{\nabla}\hat{W}( x^{\mu}) \right] \colon \qquad .
p_{x,y}(h)=4f(x,y)h^{2}+ 4\allowbreak{\large ((}\left( t-1\right) x+1{\large )(}\left( s-t\right) x-s{\large )}-\left( s-1\right) y{\large (}s+\left( s-t+2\right) x{\large ))}h+{\large (}\left( s-t\right) x-s{\large )}^{2}+\allowbreak 4s\left(s-1\right) xy
S(k):=\left(\begin{array}{cc}T(k) R_{+}(k) R_{-}(k) T(k)\end{array}\right),\ \S^{-1}(k):=\left(\begin{array}{cc}\overline{T(k)}\overline{R_{-}(k)}\overline{R_{+}(k)}\overline{T(k)}\end{array}\right).
\sum_{k \ge 1}\frac{1}{e^{l(\gamma)\cdot (k+\Re(s))-\varepsilon\cdot k}}= \frac{1}{e^{l(\gamma) \cdot \Re(s)}}\sum_{k \ge 1}\frac{1}{e^{k \cdot (l(\gamma)-\epsilon)}}\le \frac{1}{e^{l(\gamma) \cdot \Re(s)}}\cdot \frac{1}{e^{l(\gamma)-\epsilon}-1}.
\left[ T_{(A,0)},T_{(B,0)}\right] =\left( K_{00}{}^{k}-K_{00}{}^{k+1}\right) C_{AB}{}^{C}T_{(C,k)}, & =\left( K_{00}{}^{0}-K_{00}{}^{1}\right) C_{AB}{}^{C}T_{(C,0)}, & =C_{AB}{}^{C}T_{(C,0)},
2\frac{\partial \phi_1}{\partial x_4}\frac{\partial \phi_2}{\partial x_4}\cdot \cdot \cdot \frac{\partial \phi_{n-2}}{\partial x_{n}}=\pm \frac{\partial^2 \phi_1}{\partial x_4 \partial x_4}\frac{\partial \phi_2}{\partial x_5}\cdot\cdot\cdot\frac{\partial \phi_{n-3}}{\partial x_n}\phi_{n-2},
v_\mu(y_k) =\sum_{j=1}^Na_j\left (\hat{\delta}_{jk}(-(2\pi)^{-1}\log |y_k-y_j| -\mathcal{G}_{\mu}(y_k-y_j))+\frac{\delta_{jk}}{2\pi}(\log(\mu/2i)+\gamma)\right ) &=\sum_{j=1}^N[\Gamma_{\alpha,Y}(\mu)]_{kj}a_j,
\sqrt{x_ix_j}u_{x_ix_j}, \sqrt{x_i}u_{x_iy_l}, u_{y_ly_k}\in C^{0,\alpha}_{WF}([0,T]\times \bar U),\forall\, i,j\in I,\forall\, l,k=1,\ldots,m, \sqrt{x_i}u_{x_i x_j}, u_{x_jx_k}\in C^{0,\alpha}_{WF}([0,T]\times \bar U),\quad\forall\, i \in I, \quad\forall\, j,k \in I^c.
S_2= \varphi_t^{-1}(S_2) =\varphi_t^{-1}\big( \varphi_t(I_\Delta+(x_v^2:v \in V))+(\Theta,w)\big)_2 &=\big(I_\Delta+(x_v^2:v\in V)+(\varphi_t^{-1}(\theta_1),\theta_2,\theta_3,\varphi_t^{-1}(w))\big)_2.
\begin{aligned}-(\omega^0)'\otimes(\omega^0)'+(\omega^1)'\otimes(\omega^1)'+(\omega^2)'\otimes(\omega^2)' =0, -(\omega^0)''\otimes(\omega^0)''+(\omega^1)''\otimes(\omega^1)''+(\omega^2)''\otimes(\omega^2)'' =0 \end{aligned}
{\cal L}^{( 1 )}= \pi_{0}\dot{A}^{0}+ \pi_{i}\dot{A}^{i}+ \pi_{\theta}\dot{\theta}+ B \dot{N}+ \bar{\cal P}\dot{\cal C}+ \bar{\cal C}\dot{\cal P}+ \tilde{\Omega}_{2}\dot{\rho}-{\cal H}^{( 1 )}
\sum\frac{S_n(s,t)}{(s+t)^{n+1}}=\sum\frac{L_n(s,t)-L_n(t,s)}{(s-t)(s+t)^{n+1}}&=\frac1{s-t}\left[\frac1{t(s+t-1)}-\frac1{s(s+t-1)}\right]=\frac1{st(s+t-1)}.
GB = [\: 42c_{4}+7c_{5}+14c_{11}+3c_{12}, & 42c_{3}+28c_{5}-114c_{6}+198c_{7}+228c_{8}+117c_{9}-48c_{10}+4c_{11}+6c_{12}, & 56c_{2}-6c_{6}+6c_{7}+12c_{8}+c_{9}, & 168c_{1}-112c_{5}-182c_{6}+126c_{7}+84c_{8}-35c_{9}+24c_{10}-128c_{11}-12c_{12}\: ]
D_{n}(q)=(-1)^{n+1}\,\frac{nq^{n^{2}}}{(q;q)_{n}}-\sum_{k=1}^{n-1}(-1)^{k}\,\frac{q^{k^{2}}}{(q;q)_{k}}\,D_{n-k}(q),\n=1,2,3,\ldots.
H_N(\alpha,\beta;q) = \sum_{\lambda \vdash d}\frac{(\dim \mathsf{V}^\lambda)^2}{d!}\omega_\alpha(\lambda) \Omega_{\frac{q}{N}}^{-1}(\lambda) \Omega_{\frac{1}{N}}^{-1}(\lambda) \omega_\beta(\lambda) = \left\langle \omega_\alpha \Omega_{\frac{q}{N}}^{-1}\Omega_{\frac{1}{N}}^{-1}\omega_\beta \right\rangle.
\hat{\lambda}_{n,j}= \frac{\hat{A}_{n}(z_{n,j})}{R_{n}^{\prime}(z_{n,j})}= \frac{(1+c_1^2)(z_{n,j}- 1)}{4 d_1 z_{n,j}}\frac{Q_{n}(z_{n,j})}{R_{n}^{\prime}(z_{n,j})}, j =1, 2, \ldots n.
\begin{aligned}\tfrac{\sigma_0 L}{h_0 B}= 1 - \tfrac{1-\kappa h_0^2}{2\left[2-(1+\kappa)h_0\right]}= \tfrac{\kappa h_0^2 - 2(1+\kappa)h_0 + 3}{2\left[2-(1+\kappa)h_0\right]}\,\,, \tfrac{\alpha_0}{B}= \tfrac{1-\kappa h_0}{2-(1+\kappa)h_0}\end{aligned}
\bar{\nu}_{\lambda}(c) := \min\left\{\int_{S^2}A^* \omega_{\lambda}\, \bigg| \, \mbox{$A$ is an exponent of $c$}\right\}= \min\{\lambda g_0 + g_1\,| \, \mbox{$(g_0, g_1)$ is an exponent of $\vec{x}$}\}.
C^{FB,A.1}_{X^n\rightarrow{Y^n}}(\kappa)=\inf_{s\geq{0}}\sum_{y_{-1}\in\{0,1\}}\Big(K^s_0(y_{-1})\mu(y_{-1})+(n+1)\kappa\Big),~\mu(y_{-1})~\mbox{is fixed}.
\frac{d}{d\tau}\,=\,\sqrt{\dot x^2}\,\frac{d}{ds\,}{;}\qquad\frac{d^2 x_\mu}{ds^2\,}\,=\,\frac{\dot x^2\ddot x_\mu \,-\,(\dot x \ddot x)\,\dot x_\mu}{(\dot x^2)^2}\,{;}\qquad k_1\,\frac{\partial k_1}{\partial \ddot x_\mu}\,=\,-\,\frac{1}{\dot x^2}\,\frac{d^2 x_\mu}{ds^2\,}{;}
Q(r)=Q(\bar r)+\left[{dQ\over d\mu^2}\right]_{\bar r}\left(\mu^2(r)-\mu^2(\bar r)\right)+\left[{dQ\over d(\Gamma^{-2})}\right]_{\bar r}\left(\Gamma^{-2}(r)-\Gamma^{-2}(\bar r)\right)+...
\left(\bigcup_{i=0}^{2k}V_u(C_i)\right) \cup \left(\bigcup_{i=0}^{2k}V_v(C_i)\right) = \{u_m \mid m \in \mathbb{Z}_n\}\cup \{v_m \mid m \in \mathbb{Z}_n\}= V_u(G) \cup V_v(G)
d^-(u,v) - d^+(u,v) = \begin{cases}0 uv, vu \in E(X) uv, vu \notin E(X); d(u)-d(v) uv\in E(X) vu \notin E(X); d(v)-d(u) vu\in E(X) uv \notin E(X). \end{cases}
0<\sum_{i\in\mathcal{S},~j\notin\mathcal{S}}u_{ij}=\sum_{i\notin\mathcal{S},~j\in\mathcal{S}}u_{ij}=\sharp\mathcal{S}-\sum_{i\in\mathcal{S},~j\in\mathcal{S}}u_{ij}.
a^c_t=\begin{bmatrix}P Q Q P\end{bmatrix}, P=\begin{bmatrix}\cosh t 0 0 0 1 0 0 0 \cosh t\end{bmatrix}\otimes I,Q=\begin{bmatrix}0 0 \sinh t 0 0 0 \sinh t 0 0\end{bmatrix}\otimes I.
\frac{dz}{d\lambda}=\mathcal{B}R(\zeta)\frac{\prod_{j=0}^{M}\prod_{k=1}^{n_j}\left[\omega(\zeta,a_{k}^{(j)})\right]^{\beta_k^{(j)}}}{\prod_{j=0}^{M}\omega(\zeta,\gamma_{1}^{(j)}) \omega(\zeta,\gamma_{2}^{(j)})},
\sum_{l'}\bar{\Lambda}_\gamma(H^{-1})_{ll'}(1-e^{-2\beta_{l'}}\Delta_{l'}) = \bar{\Lambda}_\gamma(1-e^{-2\pi\gamma}\textit{\textbf{I}})^{-1}\left( \begin{array}{c}-e^{-2\pi\gamma}\textit{\textbf{I}}- e^{-2\beta_0}\Delta_0 1 + e^{-2\beta_0}\Delta_0 \end{array}\right).
\tilde{S}_{L}= \frac{1}{4 \pi \alpha^{\prime}}\int_{\Sigma}[ \tilde{g}^{\mu \nu}( \partial_{t}\tilde{x}_{\mu}\partial_{t}\tilde{x}_{\nu}- \partial_{\sigma}\tilde{x}_{\mu}\partial_{\sigma}\tilde{x}_{\nu}) + 4 \pi i \alpha^{\prime}\tilde{b}^{\mu \nu}\partial_{t}\tilde{x}_{\mu}\partial_{\sigma}\tilde{x}_{\nu}] ,
\int_k^{\infty}h\left(t,\left|x_0\right|+|x|\right)e^{\alpha x}dx =\frac{1}{\sqrt{2 \pi}\sqrt{t}}e^{k \alpha -\frac{\left(\left|x_0\right|+|k|\right){}^2}{2t}}&+\alpha e^{\frac{t \alpha^2}{2}-\alpha \left|x_0\right|}\left(1-\Phi \left(\frac{|x_0|+|k|-t \alpha}{\sqrt{t}}\right)\right) &=F_2(\alpha, t, x_0, 1),
\begin{aligned}S(B_2(k),B_3(k)) =y\cdot B_2(k)-x^{k-1}\cdot B_3(k) &=y\cdot \left ( \frac{y^{k+1}-1}{y-1}+x\cdot\frac{x^{k}-1}{x-1}\right )-x^{k-1}\cdot (xy-1) &=\frac{xy^{k+2}-y^{k+2}-x^k+x^ky^2-yx^{k-1}-xy^2+y+x^{k-1}}{(x-1)(y-1)}&=B_1(k)+\frac{x^{k-1}-1}{x-1}\cdot B_3(k).\end{aligned}
\mu^{\prime}= \alpha^{2}\left( e^{2 \gamma \psi}N ( K^{\prime})^{2}+ \frac{1}{2}N x^{2}( H^{\prime})^{2}+ \frac{1}{2 x^{2}}( K^{2}- 1 )^{2}e^{2 \gamma \psi}+ K^{2}H^{2}\right. \, \nonumber
P[\hat{K}_U \neq K_U \hat{K}_W \neq K_W] = P[J=0]P[ \hat{K}_U \neq K_U \hat{K}_W \neq K_W | J = 0] + P[J=1]P[ \hat{K}_U \neq K_U \hat{K}_W \neq K_W | J = 1]
-\sum_{i,j=1}^{m}\int_{\Omega}\partial_i H_j(u) |\nabla_{XY}u_i| |\nabla_{XY}u_j| \zeta_i \zeta_j \le \sum_{i=1}^{m}\int_{\Omega}\langle |\nabla_{XY}u_i| \zeta_i, |\nabla_{XY}u_i| \zeta_i\rangle_{XY}.
I_{\mathrm{m a t t e r}}= \frac{m^{2}}{4}\int d^{2}x \left\langle q \Biggl | \epsilon^{\mu \nu}[ D_{\mu}\Phi , D_{\nu}\Phi ] + \frac{m^{2}}{2}e ( q ) \Bigl [ \Phi , [ \Phi , q ] \Bigr ] \right\rangle \, ,
\Sigma(\lambda^{-1}A)=\Big[\frac{a_{1}}{\lambda},\frac{b_{1}}{\lambda}\Big] \cup \Big[\frac{a_{2}}{\lambda},\frac{b_{2}}{\lambda}\Big] \cup \cdots \cup \Big[\frac{a_{\ell-1}}{\lambda},\frac{b_{\ell-1}}{\lambda}\Big]\cup \Big[\frac{a_{\ell}}{\lambda},\frac{b_{\ell}}{\lambda}\Big]
\begin{array}{rcll}P(\eta_n=j, \zeta_n=-) = r_j, j=0,1,\ldots,u-1; P(\eta_n=u, \zeta_n=-) = r_-; P(\eta_n=u, \zeta_n=+) = r_0+r_1+\ldots+r_u-r_-; P(\eta_n=j, \zeta_n=+) = r_j, j=u+1,u+2,\ldots.\end{array}
R_{q,\ell}\ge \frac{\log\left( f_{3,2}\prod_{j=4}^q \left(j!{j^2-j+1\choose j}\right)\cdot \left(\prod_{i=3}^{\ell}(q!)^{q^{i-1}-2q^{i-2}+q^{i-3}}\right) \right)}{\log(q^\ell!)},
\left(\frac{d}{d\tau}\frac{\partial L}{\partial{\dot x}_\mu}-\frac{d^2}{d\tau^2}\frac{\partial L}{\partial{\ddot x}_\mu}+\frac{d^3}{d\tau^3}\frac{\partial L}{\partial{\stackrel{\ldots}{x}}_\mu}\right){\dot x}_\mu=0.
(i_{Y,g}\times X)^{*}(Y \times f)^{*}(F) = ((Y \times f)(i_{Y,g}\times X))^{*}(F) &=(i_{Y,g}\times f)^{*}(F) &=((i_{Y,g}\times Z)(Y \times f))^{*}(F) &=(e \times f)^{*}(i_{Y,g}\times Z)^{*}(F).
v_t-d_2\Delta v+\chi_2 \nabla v \cdot \nabla w & \leq v\left\{b_0-(b_2-\frac{\chi_2}{d_3}k)v+|\Omega|(b_3)_-\overline{u}-\left(|\Omega|(b_4)_++l\frac{\chi_2}{d_3}\right)\underline{v}+|\Omega|(b_4)_-\overline{v}\right\}&+v\left\{\left(|\Omega|((b_3)_-+|b_4|)+l\frac{\chi_2}{d_3}+k\frac{\chi_{1}}{d_{3}}\right)\epsilon\right\},
G^0_{\mu\nu}= \frac{\epsilon_{\mu\nu\lambda}p_\lambda+\delta_{\mu\nu}M+p_\mu p_\nu/M}{p^2+M^2}\;,~~D^0_{\mu\nu}(p)= \frac{\epsilon_{\mu\nu\lambda}p_\lambda}{p^2}\;, ~~{\rm and}~~\Gamma^0_{\mu\nu\lambda}=g\epsilon_{\mu\nu\lambda}
\displaystyle{E^{TE}(a) =\left. \frac{\hbar\, c}{a}\, \sum_{\nu=3/2}^{\nu_{0}}\nu \, \left[\zeta_\nu^{TE}(s,0^+) - \zeta_\nu^{TE}(s,x)\right]\right|_{s\rightarrow -1}}.
G_{\beta}( x , x^{\prime}) = G_{\infty}( x , x^{\prime}) + \frac{1}{8 \pi^{2}r r^{\prime}}\sum_{n \neq 0}\sum_{l = 0}^{\infty}( 2 l + 1 ) Q_{\nu_{l}- 1 / 2}( u_{\beta}) P_{l}( \cos \gamma ) \ ,
\tilde{\Pi}^{( 1 )}( p ) = \frac{4 \pi}{\kappa}\frac{C_{2}( G )}{2 p^{2}}\mu \int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{( k . p )^{2}- k^{2}p^{2}}{k^{2}( k + p )^{2}( k^{2}+ \mu^{2})}
\widetilde{\varphi}_{\mu}( x ) = \frac 1{3 !}\epsilon_{\mu}^{\nu \rho \sigma}\varphi_{\nu \rho \sigma}( x ) , \hspace{0 . 5 i n}\widetilde{\varphi}( x ) = \frac 1{4 !}\epsilon^{\mu \nu \rho \sigma}\varphi_{\mu \nu \rho \sigma}( x )
\widetilde c_{\pm}^{(a^{\pm1}b^k,fa)}(p,s)\overset{(\ref{eq:ctildeDef})}{=}c_{\pm}^{(b^{\pm1}a^k,\sigma(f)b)}(s,p^{-1})\overset{(\ref{eq:cpcmRightMultiplication})}{=}c_{\pm}^{(b^{\pm1}a^k,\sigma(f))}(s,p^{-1}s^{-1})\overset{(\ref{eq:ctildeDef})}{=}\widetilde c_{\pm}^{(a^{\pm1}b^k,f)}(ps,s).
\theta^+ = (\frac{1}{n}+\epsilon,\frac{1}{n}-\frac{\epsilon}{2n-1},\ldots,\frac{1}{n}-\frac{\epsilon}{2n-1};-1) \theta^- = (\frac{1}{n}-\epsilon,\frac{1}{n}+\frac{\epsilon}{2n-1},\ldots,\frac{1}{n}+\frac{\epsilon}{2n-1};-1)
\tilde{\cal H}= \omega \, q \cdot \!{\frac{\partial}{\partial q}}-{\frac{1}{2}}\sum_{j = 1}^{r}{\frac{\partial^{2}}{\partial q_{j}^{2}}}- \sum_{\rho \in \Delta_{+}}{\frac{g_{| \rho |}}{\rho \cdot q}}\rho \cdot \!{\frac{\partial}{\partial q}}.
q^x \cdot \Omega_{,\alpha r}\Omega_{,\beta ss}+ q^{-x}\cdot \Omega_{,\beta s}\Omega_{,\alpha rs}- (q^x \cdot \Omega_{,\alpha s}\Omega_{,\beta rs}+ q^{-x}\cdot \Omega_{,\beta r}\Omega_{,\alpha ss}) ={\cal O}(s,\alpha , \beta ;q)
G_m(D;\alpha_1,...,\alpha_m) = [\prod_{i=1}^m \Gamma(\alpha_i)]^{-1}(\prod_{i=1}^m \int_0^\infty dt_i \, t_i^{\alpha_i-1}) \, T^{-\frac{D}{2}}e^{-\frac{1}{T}\sum_{i<j}t_it_j x_{ij}^2}
\sum_{x \in N(X,\varepsilon)}\vert \eta(x)\vert \leq C_2M(3 \varepsilon)\left( \sum_{x \in \Gamma}\sum_{\lbrace y \colon \delta(y,x)=1 \rbrace}\vert \eta(x)-\eta(y)\vert \varrho \left(\vert (x,y)\vert\right)\right)
\sigma^k((z_1,\ldots,z_n)) =\tau_\mathbf{-a}^{k-k_0}(\sigma^{k_0}((z_1,\ldots,z_n))) &=-\tilde\tau_\mathbf{-a}^{k-k_0}(-\sigma^{k_0}((z_1,\ldots,z_n)))
( F^{( 1 )})^{2}= \frac{\kappa^{2}}{3 6}H_{\nu \rho}^{b c}\left\{\frac{\partial^{2}W^{( 0 )}}{\partial p_{\nu}\partial p_{\rho}}, \sigma_{b c}\right\}H_{\kappa \epsilon}^{g h}\left\{\frac{\partial^{2}W^{( 0 )}}{\partial p_{\kappa}\partial p_{\epsilon}}, \sigma_{g h}\right\}
B_n(f,x)=\sum_{k=0}^n f(\dfrac{k}{n}b_n)\left(\begin{array}{c}n k \end{array}\right) \left(\dfrac{x}{b_n}\right)^k \left(1-\dfrac{x}{b_n}\right)^{n-k}, ~~~~~ 0\leq x\leq b_n,
M_s(x,\theta) =k_{N,s}(1-|x|^2)^s\lim_{z\to\theta,z\in B}\int_0^1 \frac{t^{s-1}}{((1-|x|^2)_+(1-|z|^2)_+ t +|x-z|^2)^{\frac{N}{2}}}\dt &=k_{N,s}\frac{(1-|x|^2)^s}{|x-\theta|^{N}}\int_0^1 t^{s-1}\dt=\frac{k_{N,s}}{s}\frac{(1-|x|^2)^s}{|x-\theta|^{N}}
\langle (\pi_{2})^{(n)}(U) (I_{n}\otimes m_{2}) (\pi_{2})^{(n)}(U^{*}) x^{(2)}, x^{(2)}\rangle \leq c^{2}\sum\limits_{k=1}^{n}\langle (e_{11}\otimes m_{2})y_{k}, y_{k}\rangle+ \epsilon = c^{2}\sum\limits_{k=1}^{n}\langle f_{k}(I_{n}\otimes m_{2})f_{k}^{*}x^{(2)}, x^{(2)}\rangle +\epsilon.
\zeta^D_{\rm ren}(\alpha,R)=\frac{m^{{\scriptstyle \frac{D}{2}}-1}}{2^{D-2}\pi^{D-1}R^{{\scriptstyle \frac{D}{2}}-1}}\sum_{n=1}^\infty\frac{\cos(2n\pi\alpha)K_{{\scriptstyle \frac{D}{2}}-1}(2n\pi Rm)}{n^{{\scriptstyle \frac{D}{2}}-1}}.
\sum_{k=n/2+1}^{n-1}\int_{(k-1)/n}^{k/n}(Q(u) -q_{n,k})^2 du\le \frac{\pi}{6 n}\sum_{k=n/2+1}^{n-1}\frac{1}{(n-k)^2}\le \frac{\pi}{6 n}\sum_{m=1}^{\infty}\frac{1}{m^2}=\frac{\pi^3}{36 n}.
\int_0^t \frac{2\rho (e^{2\rho t}-1)}{2\rho \alpha e^{2\rho s}(e^{2\rho t}-1) + (e^{2\rho(t-s)}-1)(e^{-2\rho(t-s)}-1)}ds & \phantom{AAAAAAAA}=\frac{\sqrt{1-e^{-2\rho t}}}{2\sqrt{2\rho \alpha}}\log \left( \frac{(e^{\rho t}a_t-1)(e^{\rho t}a_t+1+a_t^2-e^{-2\rho t})}{(e^{\rho t}a_t+1)(-e^{\rho t}a_t+1+a_t^2-e^{-2\rho t})}\right)
|\lambda^*( F_0, \dots, F_k; K )| \leq H^{1/p'}\prod_{j=0}^k ( \| F_j \|_s^s )^{\tfrac{1}{k+1}\big( \tfrac{k}{p}+ \tfrac{1}{p'}\big)}&= H^{1/p'}\prod_{j=0}^k \| F_j \|_s,
\Delta_f({\textbf{p}}^i) = \frac{{f({\textbf{p}}^i) - \mathop{\min}\limits_{j = 1, \cdots ,W}f({\textbf{p}}^j)}}{\mathop{\max}\limits_{j = 1, \cdots ,W}f({\textbf{p}}^j)-\mathop{\min}\limits_{j = 1, \cdots ,W}f({\textbf{p}}^j)},
\left\|d_G(\cdot,y_0)^{-(1+4\alpha)}E^{y_0,ij}_1\right\|_{L^\infty(\mathcal{B}_{1}^+(y_0))}+\left[d_G(\cdot,y_0)^{-(1+4 \alpha -\epsilon)}E^{y_0,ij}_1\right]_{\dot{C}^{0,\epsilon}_\ast(\mathcal{B}_1^+(y_0))}\leq C.
B= ( -t_{1}+ c+ \alpha z) C_{1}+ ( -\frac{t_{2}}{2}+ \sum_{k\geq 0}c_{k}z^{k})C_{2}-\frac{z}{4}D + z(-\frac{t_{2}}{2}+\sum_{k\geq 0}c_{k}z^{k}) E,
Q_L^B(\hat\delta_0) = \begin{bmatrix}S_{\phi}(\hat \delta_{0}) S_{\theta}(\hat{\delta}_{0})\end{bmatrix}[\hat{G}_n-\hat{F}_n \hat{E}_n^{-1}\hat{F}_n^\mathrm{T}]^{-1}\begin{bmatrix}S_{\phi}(\hat{\delta}_{0}) S_{\theta}(\hat \delta_{0})\end{bmatrix}.
r\left( y,K\right) =\sup_{x\in A}d\left( y,x\right) , r\left( K\right) =\inf_{y\in M}\sup_{x\in K}d\left( y,x\right) , C\left( K\right) =\left\{y\in M:\sup_{x\in K}d\left( y,x\right) =r\left(K\right) \right\}