harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
| \alpha k^3 - k_1^3 - (n-k_1)^3 - \alpha (k-n)^3 | = \left| \alpha k^3 - (d_1k + \delta)^3 - [(1-d_1 + \mu)k - \delta]^3 + \alpha \mu^3 k^3 \right| =\left| k^3[\alpha - 1 + 3d_1 - 3d_1^2 + \mathcal{O}(\epsilon)] + \mathcal{O}(\epsilon)k^2 +\mathcal{O}(\epsilon^2)k + \mathcal{O}(\epsilon^3) \right| \gtrsim |k^3|.
\| [w_n ,a] \| = \| w_naw_n^* -a \| & = \| w_naw_n^* -z_na w_n^* + z_na w_n^*-z_naz_n^* + z_naz_n^* - z_nz_n^*a + z_nz_n^*a -a \| & \leq \| w_na -z_na \| + \| z_n \| \| w_na^*-z_na^* \| + \| z_n\| \|az_n^* - z_n^*a \| + \| z_nz_n^*a -a \| & \to 0
\frac{\Omega^{2}}{\pi G\rho}=\frac{4abc}{3(a^{2}-b^{2})}(a^{2}R_{J}(a^{2},b^{2},c^{2},a^{2})-b^{2}R_{J}(a^{2},b^{2},c^{2},b^{2}))
\nu^H =\sum_{j=1}^n \frac{r^2x_j-y_j(t-t_0)}{rR^2}X_j+\frac{r^2y_j+x_j(t-t_0)}{rR^2}Y_j=\frac{r^2\xi^H+(t-t_0)J\xi^H}{rR^2}\mbox{and}\eta =-J\nu^H=\frac{(t-t_0)\xi^H-r^2J\xi^H}{rR^2}.
\lim_{b\rightarrow-1}(1-b)F(b,0;b) = \sum_{j\geq 0}q^{{(3j^2+j)}/{2}}(1-q^{2j+1}) =1 + \sum_{n\geq 1}\frac{(-1)^n q^{2n-1}}{(-q;q^2)_n}= \sum_{n=0}^\infty \frac{(-1)^n q^{(n^2+n)/2}}{(-q;q)_n}.
\sum_{u,v}p_{U,V}(u,v)\boldsymbol{1}(\{u=v\}) = 1- \alpha \mbox{(cf.\ \eqref{defEpsilon})}\quad\mbox{and}\sum_{u,v}s_{U,V}(u,v)\boldsymbol{1}(\{u=v\}) = \frac{1}{|\mathcal{W}|};
\Sigma_{m}( \zeta ) = \lim_{N \rightarrow \infty}\sum_{r = 1}^{N}( \Delta_{m}( \zeta ) )^{r}= \lim_{N \rightarrow \infty}{\frac{1 - \Delta_{m}( \zeta )^{N}}{1 - \Delta_{m}( \zeta )}}\; .
\mathcal{F}_{\mu\nu}= \left( \begin{array}{cccc}0 \varepsilon_1 \varepsilon_2 \varepsilon_3 -\varepsilon_1 0 -{b}_3{b}_2 -\varepsilon_2{b}_3 0 -{b}_1 -\varepsilon_3 -{b}_2{b}_1 0 \end{array}\right)
A^{(p)}(t,z):= \sum_{w \in Asc(p)}t^{|w|}z^{|w|_0}= 1+ \sum_{n \geq 0}\binom{p+n-1}{n}\frac{zt}{(1-zt)^{n+1}}\prod_{i=1}^n (1-(1-t)^i).
\sigma^{1}= \left( \begin{array}{l l}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \quad \sigma^{2}= \left( \begin{array}{l l}{0}&{i}\\{- i}&{0}\\ \end{array}\right) , \quad \sigma^{3}= \left( \begin{array}{l l}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) .
F^{\mu}_{\lambda}(k+1)-F^{\mu}_{\lambda}(k) =\sum_{j}F^{\mu\cup j}_{\lambda}(k) &=\sum_{j}\frac{k!}{(k-\ell(\lambda)+\ell(\mu)+1)!(\alpha_{j}-\beta_{j}-1)!\prod_{i\neq j}(\alpha_{i}-\beta_{i})!}&=\frac{(k+1)!}{(k+1-\ell(\lambda)+\ell(\mu))!\prod_{i}(\alpha_{i}-\beta_{i})!}-\frac{k!}{(k-\ell(\lambda)+\ell(\mu))!\prod_{i}(\alpha_{i}-\beta_{i})!}.
p_{n+2}(x) = \sum_{m=0}^{n}\frac{i^m}{4\pi m!}\int_{\Omega}(c^{-2}-1)(y)p_{n-m}(y)|x-y|^{m-1}dy & \ \ \ \ - \frac{i^{n+2}}{8\pi^2(n+1)}\int_{\Omega}c^{-2}(y)(f_2-f_1)(y)|x-y|^n dy.
\pi_{\omega^{'}}(y^sz^s+c_1xy^{s-1}z^{s-1}w+\cdots +c_sx^sw^s)(e_0\otimes e_0\otimes \cdots \otimes e_0)=\pi_{\omega^{'}}(c_sx^sw^s)(e_0\otimes e_0\otimes \cdots \otimes e_0)\neq 0
\sum_{i=1, i \ne k}^{n-1}\frac{a_k}{a_i a_{n+k-i}}x_i' x_{n+k-i}',=-\sum_{i=1, i \ne k}^{n-1}\frac{a_k}{a_i a_{i-k}}(\sum_{p=1}^{n-1}a_i \zeta^{pi}x_{p}x_{n+k-i}) (\sum_{r=1}^{n-1}a_{n+k-i}\zeta^{(n+k-i)r}x_r)
\varphi(r)=A\frac{e^{-\sqrt{\kappa^{2}}r}}{r}+A^{\prime}\frac{e^{\sqrt{\kappa^{2}}r}}{2r\sqrt{\kappa^{2}}}=A\frac{e^{-\kappa r}}{r}+A^{\prime\prime}\frac{e^{\kappa r}}{r}=A\frac{e^{-\kappa r}}{r},
u(t,x)=\pi^{-\frac{1}{2}}t^{-\frac{1}{3}}(t^{-\frac{1}{3}}|x|)^{-\frac{1}{4}}\Re\left(e^{i\phi}\hat u_0(t^{-\frac{1}{2}}|x|^{\frac{1}{2}})\right)+O(t^{-\frac{1}{3}}(t^{-\frac{1}{3}}|x|)^{-\frac{7}{4}}),
C_3(y_0) =\sup_{t\ge y_0}\frac{\exp(C_1(y_0)R(t))-1}{R(t)}=\frac{\exp(C_1(y_0)R(y_0))-1}{R(y_0)}, C_4(y_0) =\sup_{t\ge y_0}\frac{1-\exp(-C_2(y_0)R(t))}{R(t)}=C_2(y_0).
\lambda = 6 \frac{\kappa^{2}}{\tilde{\kappa}^{4}}, \Lambda = \frac{4 \pi}{\tilde{M}_{p}^{3}}\left[ \tilde{\Lambda}+ \left( \frac{4 \pi}{3 \tilde{M}_{p}^{3}}\right) \lambda^{2}\right]
K_4 =-\sum\limits_{|\alpha|=0}^{3}\left\langle B({\bf{v}})\partial_y^2\partial_\tau^\alpha{\bf{v}}, \partial_\tau^\alpha{\bf{v}}\right\rangle-\sum\limits_{|\alpha|=0}^{3}\sum\limits_{|\beta|=1}^{|\alpha|}\left\langle\partial_\tau^\beta B({\bf{v}})\partial_\tau^{\alpha-\beta}\partial_y^2{\bf{v}}, \partial_\tau^\alpha{\bf{v}}\right\rangle &=:K_4^1+K_4^2.
\sum_{j=1}^m B_j \otimes \overline{B_j}=\sum_{j=1}^m \left(\sum_{i, k=1}^n u_{ij}u_{kj}\right) A_i \otimes \overline{A_k}=\sum_{i, k=1}^n \left(\sum_{j=1}^m u_{ij}u_{kj}\right) A_i \otimes \overline{A_k}&= \sum_{i, k =1}^n \delta_{i,k}A_i \otimes \overline{A_k}=\sum_{i=1}^n A_i \otimes \overline{A_i}.
L(\tilde{J_{k}})\leq L(J_{k})+M e^{-lkh}h\|Z_{0}\|_{\infty}\big(L(J_k)+1\big), &Q(\tilde{J_{k}})\leq Q(J_{k})+M e^{-lkh}h\|Z_{0}\|_{\infty}\big(L(J_k)+1\big)^2.
< O_{n}O_{m}>_{q}= \frac{1}{2^{n + m}}\sum_{k = 0}^{n}\sum_{l = 0}^{m}\left( \begin{array}{c}{{n}}\\{{k}}\\ \end{array}\right) \left( \begin{array}{c}{{m}}\\{{l}}\\ \end{array}\right) < W_{q}^{n - k , k}W_{- q}^{m - l , l}> .
T_{a,j,*}^\nu (h)=\chi_j(r)r^{-\frac{d-2}{2}}\int_0^\infty \frac{e^{it\rho^a}(r\rho/2)^\nu}{\Gamma(\nu+1/2)\pi^{1/2}}\int_{-1}^1 \sum_{n=0}^\infty \frac{(i\theta r\rho)^n}{n!}(1-\theta^2)^{\nu-1/2}d\theta &\times\rho^{\frac{-d+1+a}{2}}\chi_{\leq -j-5}(\rho)h(\rho)d\rho.
{{\cal G}^{- 1}}_{a b}^{c d}\eta_{c d}^{\mathrm{L L T}}= \left( k^{2}- \Lambda_{l}+ \frac{1}{2 r^{2}}( N - 1 ) ( N - 4 ) - \frac{N + d}{N + d - 2}\Lambda \right) \eta_{a b}^{\mathrm{L L T}},
S(u_i,u_j) =S(b'_i+c_i w_i,b'_j+c_jw_j) &=S(b'_i,b'_j)+c_iS(w_i, b'_j)+c_jS(w_j,b'_i)+c_ic_jS(w_i,w_j) &=S(b'_i,b'_j)+c_i\int_{\partial \Sigma}\phi (D_\eta w_i) b'_j+c_j\int_{\partial \Sigma}\phi (D_\eta w_j) b'_i &=S(b'_i,b'_j)+(c_i+c_j) \int_{\partial \Sigma}\phi b'_i b'_j &=0.
e^{(n-\nu-2)t}\int_X(S(t)^2+(\nu+3)S(t)+n)\omega(t)^n &=e^{(n-\nu-2)t}\int_XS(t)^2\omega(t)^n+(\nu+3)L(t)+ne^{(n-\nu-2)}[\omega(t)]^n &\le e^{(n-\nu-2)t}\int_XS(t)^2\omega(t)^n+C_2e^{-2t},
\zeta (t) = \frac{\gamma_{3}}{6 \gamma_{2}^{3/2}}+ \frac{\gamma_{4}\gamma_{2}- 3 \gamma_{3}^2}{24 \gamma_{2}^3}t + \frac{\gamma_{5}\gamma_{2}^2 - 10 \gamma_{4}\gamma_{3}\gamma_{2}+ 15 \gamma_{3}^3}{120 \gamma_{2}^{9/2}}t^2 + \ldots
\Delta f(u(r)\#\delta_h) = \sum_{l \in G}\Phi(ju(r)l^{-1}.(iu(l.r)^{-1}))(h)\Phi(l.iu(r)iu(l.r)^{-1})(l) u(r)\#\delta_{l^{-1}}\otimes u(l.r)\delta_{lh}\# u(r) &= (f\otimes f)\Delta(u(r)\#\delta_h)
\begin{array}{rcl}\left\Vert P_K\left(x^\nu - \frac{\nabla f(x^\nu) + \nabla g(x^\nu) \xi^\nu}{1 + \|\xi^\nu\|}\right) - x^\nu\right\Vert = 0, \enspace \max_i \left|g_i(x^\nu) \frac{\xi_i^\nu}{1 + \|\xi^\nu\|}\right| = 0, \enspace \max_i \{g_i(x^\nu)_+\}= 0, begin{align*}5pt]\end{array}
\max_{m\in\mathcal{M}_n}\mathbb{P}_{P^{(\mathcal{C}^\star_n)}}\big(\tilde{M}\neq m |M=m\big) \leq\epsilon \max_{m\in\mathcal{M}_n}\mathsf{D}\Big(P^{(\mathcal{C}^\star_n)}_{\mathbf{Z}|M=m}\Big|\Big|P^{(\mathcal{C}^\star_n)}_\mathbf{Z}\Big) \leq \epsilon.
\rho_{\phi}[\iota](x) = \rho_{J}[\bar{\iota}] (1, x) &=\sqrt{{\rm vol}_{g}(\iota_{*}e_{1}, \cdots, \iota_{*}e_{n}, -\xi, \phi \iota_{*}e_{1}, \cdots, \phi \iota_{*}e_{n})}
f'' =\frac{-2\lambda}{b}\bigg(\frac{\Pi}{\pi}g'+g\frac{\pi^2-\Pi\pi'}{\pi^2}-g\frac{\Pi b'}{\pi b}\bigg) &=\frac{-2\lambda}{b}\bigg(\frac{\Pi}{\pi}g'+g+\frac{\Pi}{\pi}g \frac{s'}{s}\bigg)\in C(0,1),
G_d:=-2[\bar{q}, \mathcal{H}]\frac{\partial_{\alpha}\bar{\mathfrak{F}}}{\zeta_{\alpha}}-2[\bar{\mathfrak{F}},\mathcal{H}]\frac{\partial_{\alpha}\bar{q}}{\zeta_{\alpha}}-2[\bar{q},\mathcal{H}]\frac{\partial_{\alpha}\bar{q}}{\zeta_{\alpha}}-4D_t q:=G_{d1}+G_{d2}+G_{d3}+G_{d4}.
{\mathrm{\boldmath ~ \a l p h a ~}}_{1}^{*}\cdot{\bf h}- \frac{{\mathrm{\boldmath ~ \a l p h a ~}}_{1}^{*}\cdot{\mathrm{\boldmath ~ \a l p h a ~}}_{2}^{*}}{4 \pi r_{1 2}}\sqrt{1 -{\bf v}_{2}^{2}}\sqrt{g_{2}^{2}+ q_{2}^{2}}.
\delta U_{-1}= [B_0^-,U_{-1}], \;\;\; \delta U_0 = \left([B_1^-,U_{-1}] + [B_0^-,U_0]\right)_{\oplus}, \delta U_1 = \left([B_2^-,U_{-1}]+[B_1^-,U_0]+[B_0^-,U_1]\right)_{\oplus}.
\begin{array}{lllll}r_1:=auv+bvu+cw^2, r_2:=avw+bwv+cu^2, r_3:=awu+buw+cv^2, r_4:=bu'v'+av'u'+cw'^2, r_5:=bv'w'+aw'v'+cu'^2, r_6:=bw'u'+au'w'+cv'^2, r_7:=uu'+u'u, r_8:=vu'+u'v, r_9:=wu'+u'w, r_{10}:=uv'+v'u, r_{11}:=vv'+v'v, r_{12}:=wv'+v'w, r_{13}:=uw'+w'u, r_{14}:=vw'+w'v, r_{15}:=ww'+w'w.\end{array}
e^{-w\left( 0,x\right)}=\Pi_{0,x}\left[ e^{-f\left( \tau ,\xi_{\tau}\right)}+k\int_{0}^{\tau}ds\left[ \varphi \left( s,\xi_{s};e^{-w\left(\tau -s,\xi_{s}\right)}\right) -e^{-w\left( \tau -s,\xi_{s}\right)}\right] \right] .
\psi^{\lambda}_{(0,0)}= \frac{1}{\kappa}\bigg[ \cos(2\xi(2\hat{\lambda}_1+3\hat{\lambda}_2)) + \cos(2\xi(\hat{\lambda}_1-\hat{\lambda}_2)) - \cos(2\xi(\hat{\lambda}_1+3\hat{\lambda}_2)) - \cos(2\xi(2\hat{\lambda}_1+\hat{\lambda}_2)) \bigg] & = - \frac{1}{8 \kappa \pi^2}J_{x,y}\left( \hat{\lambda}_2/2\kappa, (\hat{\lambda}_1+\hat{\lambda}_2)/2\kappa \right),
V \times\mathbb{R}^{n}=((V\setminus Q)\times Q)\cup((V\setminus Q)\times\hat{Q})\cup(Q\times Q)\cup(Q\times\hat{Q}) &\subset(\hat{S}{\times}Q)\cup((V{\setminus}Q){\times}S)\cup(\hat{S}{\times}\hat{S})\cup(Q{\times}Q)\cup (Q{\times}(S{\setminus}Q))\cup (Q{\times}\hat{S})~.
\begin{aligned}\left(x \frac{d}{dx}\right)^4 \log \left( \theta_3(x) \right) = x \, \psi(x) + 7 x^2 \, \psi'(x) + 6 x^3 \, \psi''(x) + x^4 \psi'''(x) & < \left(6.35 \, x - 275.8 \, x^2 + 1533 \, x^3 - 1558 \, x^4 \right) e^{- 2 \pi x}& + \left(-6 \, x + 140.7 \, x^2 - 330.6 \, x^3 + 193 \, x^4 \right) e^{-\pi x}.\end{aligned}
\begin{aligned}\Vert q_{1}-q_{2}\Vert^{2}_{H^{-1}(\Omega)}\leq \frac{C}{h^\frac{2n}{n+2}}e^\frac{18R}{h}\Vert \mathcal{N}_{q_{1}}-\mathcal{N}_{q_{2}}\Vert^{2}+ C h^\frac{4}{n+2}& \leq C e^\frac{20R}{h}\Vert \mathcal{N}_{q_{1}}-\mathcal{N}_{q_{2}}\Vert^{2}+ C h^\frac{4}{n+2}.\end{aligned}
\begin{aligned}\nabla |\mathring{A}|^2=2|\mathring{A}|^2H^{-1}\nabla H,~\nabla|A|^2=2|A|^2H^{-1}\nabla H, |\mathring{A}|H^{-1}\nabla H=\sqrt{2}\nabla\mathring{A}(\nu_1,\nu_1)=-\sqrt{2}\nabla\mathring{A}(\nu_2,\nu_2), 2|\nabla\mathring{A}(\nu_1,\nu_2)|^2=(\gamma_t+\frac{1}{2})|\nabla H|^2-\frac{2}{H}\mathring{h}_{ij}\nabla^iH\nabla^jH. \end{aligned}
\left\langle \tilde{\eta}^{\varepsilon},\varphi \right\rangle =\varepsilon^{\left( d/2\right) -1}\sum_{b\in \left( D^{\varepsilon}\right)^{\ast}}\nabla \varphi \left( b\right) \tilde{\eta}^{\varepsilon}\left( b\right) \varphi \in \mathcal{C}_{0}^{\infty}\left( D\right)
E\left(\beta \right)=E_{0}+\widetilde{E}\left(\beta \right)=\frac{1}{2r}\sum_{\ell =1}^{\infty}\ell^{2}\left(\ell^{2}+\mu_{eff}^{2}\right)^{1/2}+\frac{1}{r}\sum_{\ell =1}^{\infty}\frac{\ell^{2}\left(\ell^{2}+\mu_{eff}^{2}\right)^{1/2}}{e^{\beta \left(\ell^{2}+\mu_{eff}^{2}\right)^{1/2}/r}-1},
+ T_{1}({\bf{k}},{\bf{q}}) \sum_{{\bf{q}}_{1}}a_{{\bf{k}}+{\bf{q}}/ 2 -{\bf{q}}_{1}/ 2}^{\dagger}({\bf{q}}_{1}) a_{{\bf{k}}-{\bf{q}}_{1}/ 2}({\bf{q}}_{1}-{\bf{q}})
\hat{\mathfrak{k}}= \{(u(t), \sigma(u(-t))) \mid u \in L(\mathfrak{g}, \sigma)\}, \hat{\mathfrak{m}}= \{(u(t), - \sigma u(-t)) + \alpha c + \beta d \mid u \in L(\mathfrak{g}, \sigma), \ \alpha, \beta \in \mathbb{R}\}.
\rho_j^0=\frac{1}{\Delta x}\displaystyle\int_{x_{j-1/2}}^{x_{j+1/2}}\overline{\rho}(x)\,\d x \rho_j^n\simeq\frac{1}{\Delta x}\displaystyle\int_{x_{j-1/2}}^{x_{j+1/2}}\rho(t^n,x)\,\d x n>0.
\varepsilon^{i}\rightarrow{\varepsilon^{i}}= \varepsilon^{i}+ \delta \varepsilon^{i}={{W_{5}( p , q , r )}^{i}}_{j}\varepsilon^{j}= \varepsilon^{i}+ ( p \varepsilon^{1}+ q \varepsilon^{2}+ r \varepsilon^{3}) \frac{k^{i}}{\omega}
\int_{\Sigma}K_{g_{\phi_{\infty}}}d\mathrm{vol}_{g_{\phi_{\infty}}}=2\pi\chi(\Sigma)+2\pi(3-1)=2\pi\chi(\Sigma)+4\pi \int_{S^2}K_{g_{\vec{\chi}}}d\mathrm{vol}_{g_{\vec{\chi}}}=-8\pi \int_{S^2}K_{g_{\vec{\Psi}}}d\mathrm{vol}_{g_{\vec{\Psi}}}=4\pi,
M'_{|x|^2}(t) = 8s \|u(t)\|^2_{\dot{H}^s}- \frac{4d\alpha}{\alpha+2}\|u(t)\|^{\alpha+2}_{L^{\alpha+2}}= 4d\alpha E(u(t)) - 2(d\alpha-4s) \|u(t)\|^2_{\dot{H}^s}.
\sum_{i=1}^2[\mathfrak{a}_i^+(t)]^2\leq c_0\,\sup_{\tau\geq t}\bigg(\frac{1}{[\mathfrak{q}_1(\tau)-\mathfrak{q}_2(\tau)]^4}+\sum_{i=1}^2[\mathfrak{a}_i^-(\tau)]^2\bigg), &\|\epsilon(t)\|_{H^1}^2\leq C_0\,\sup_{\tau\geq t}\bigg(\frac{1}{[\mathfrak{q}_1(\tau)-\mathfrak{q}_2(\tau)]^4}+\sum_{i=1}^2[\mathfrak{a}_i^-(\tau)]^2\bigg).
M = \left( \kappa g \left[ \frac{1 6}{9}\langle \phi \rangle - \frac{1 5 6 8}{2 4 3}\langle \psi^{2}\rangle - \frac{1 7 6}{2 4 3}\langle \psi^{3}\rangle \right] \right) \approx ( 0 . 6 7 2 8 9 )
\hat{n}^{M}= \left[ \frac{a^{\prime}\, A}{( 1 - k a^{2})^{1 / 2}}, 0 , \dots , 0 , \frac{1}{A}[ 1 - k \, a^{2}+ ( a^{\prime}\, \, A )^{2}]^{1 / 2}\right] .
y_t = y_0 + \int_0^t\tilde{\vartheta}_s\,\mathrm{d}{s}+ 2c_H^{B,R}\int_0^t\tilde{\varphi}_s\delta B_s^{\frac{H}{2}+\frac{1}{2}}+ \int_0^t\tilde{\psi}_s\delta R^H_s
D_{c,h}\left( D_s^k |c'| \right) = D_s^k \left( \langle D_s h, v \rangle |c'|\right)- \sum_{j=0}^{k-1}D_s^j \big( \langle D_s h, v\rangle D_s^{k-j}|c'| \big) &= D_s^k \left( \langle D_s h, v \rangle |c'|\right) -\sum_{i=0}^{k-1}\binom{k}{i+1}D_s^i \langle D_s h, v\rangle D_s^{k-i}|c'|\,.
\delta \eta_{\alpha_{2 k + 1}}^{*}= \Phi_{\alpha_{2 k}}^{*}Z_{\; \; \alpha_{2 k + 1}}^{\alpha_{2 k}}+ \Phi_{\alpha_{2 k + 2}}^{*}A_{\alpha_{2 k + 1}}^{\; \; \alpha_{2 k + 2}}, \; k = 0 , \cdots , b ,
s\left( -c,d;A_{c},B_{-b}\right) -s\left( d,c;B_{b},A_{c}\right) & =P_{1}\left( 0,B_{-b}\right) P_{1}\left( 0,A_{-c}\right) -\frac{d}{c}B_{0}(A)P_{2}\left( 0,B_{b}\right) -\frac{c}{d}B_{0}\left( B\right)P_{2}\left( 0,A_{c}\right) -\frac{1}{2dc}B_{0}(B)B_{2}(A).
\frac{\partial \mathcal U}{\partial \beta}=\frac{w_c}{\beta + \frac{\sum_{i=1}^{n}b_i}{\sum_{i=1}^{n}a_i}}+\frac{w_d}{\beta+\frac{\sum_{j=1}^{m}(v_j-s_j)}{\sum_{j=1}^{m}(u_j+s_j)}}.
{\bar{{\cal G}}}_{\mathrm{( A )}}^{a}\left[ \Delta ( x ) \cdot \Gamma \right] = 0 ,{\cal G}_{\mathrm{( A )}}^{a}\left[ \Delta ( x ) \cdot \Gamma \right] = 0 , \mathrm{A}= \mathrm{I}, \mathrm{I I},
-\frac{1}{r}\int_{t+6r}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{(s-t)}\lambda''(s)\left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(-1-\rho^{2}+r^{2})^{2}+4r^{2}}}+F_{3}(r,\rho,\lambda(s))\right) &=-2r \int_{6r}^{\infty}dw \lambda''(t+w) w\left(\frac{1}{2(1+w^{2})}-\frac{1}{2(\lambda(t+w)^{2-2\alpha}+w^{2})}\right)+E_{4}
H_{2,1}^1 =\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\int_\gamma^{\frac{3}{2}}\frac{dr d\theta}{|\gamma+r\sin\theta|^\tau r^{q-1}|re^{i(\theta-\varphi)}-1|^q}, H_{2,1}^2 =\int_{\frac{\pi}{2}}^\frac{3\pi}{2}\int_\gamma^{\frac{3}{2}}\frac{dr d\theta}{|\gamma+r\sin\theta|^\tau r^{q-1}|re^{i(\theta-\varphi)}-1|^q}.
c(2k)=\left(\frac{-\lambda}{1-p^{-2}}\right)^{k-1}\frac{c(2)p^{k(k-1)}(p^2-1)^{k-2}}{(p^4-1)^2(p^6-1)^2\ldots(p^{2k-2}-1)^2 (p^{2k}-1)}.
\limsup_{b \rightarrow 0}\lim_{n \rightarrow \infty}\|n^{-1/2}N_n(\lfloor n^{1/2}a \rfloor)-n^{-1/2}N_n(\lfloor n^{1/2}(a+b)\rfloor)\|_{L^2}^2= &\limsup_{b \rightarrow 0}\lim_{n \rightarrow \infty}O(|n^{-1/2}\lfloor n^{1/2}a \rfloor-n^{-1/2}\lfloor n^{1/2}(a+b)\rfloor|)=o(1).
W_{11}=[\begin{matrix}\lambda&0&0&0\\ 0&\lambda&0&0\\ 0&0&\lambda&0\\ 0&0&0&\lambda\\ \end{matrix}]=\lambda I_{4},W_{22}=[\begin{matrix}\lambda&0\\ 0&\lambda&\\ \end{matrix}]=\lambda I_{2},W_{33}=[\begin{matrix}\lambda&0\\ 0&\lambda&\\ \end{matrix}]=\lambda I_{2},W_{44}=[\lambda]=\lambda I_{1}
a(x,\xi) \theta_h(x)\chi(h^{\beta}\xi) \gamma(\frac{x+p'(\xi)}{\sqrt{h}}) = a(x, d\varphi(x)) \theta_h(x) + a(x, d\varphi(x))\theta_h(x) \big[\chi(h^{\beta}\xi)\gamma(\frac{x+p'(\xi)}{\sqrt{h}})-1\big] & + b(x,\xi) \chi(h^{\beta}\xi)\gamma(\frac{x+p'(\xi)}{\sqrt{h}})(x+p'(\xi))\, .
z^{\alpha}K_{\alpha}( z ) = 2^{\alpha - 1}\Gamma ( \alpha ) \left[ 1 + \sum_{j = 1}^{\infty}\frac 1{j ! ( 1 - \alpha )_{j}}\left( \frac{z}2 \right)^{2 j}- \frac{\Gamma ( 1 - \alpha )}{\Gamma ( 1 + \alpha )}\left( \frac{z}2 \right)^{2 \alpha}\sum_{j = 0}^{\infty}\frac 1{j ! ( 1 + \alpha )_{j}}\left( \frac{z}2 \right)^{2 j}\right] ,
\left(A_0 + 2 \sum_{i=1}^x A_i\right) (p^x-2) + \sum_{i=1}^x A_i p^{i-1}= \left(2 \sum_{i=1}^x B_i\right) (p^x-2) + \sum_{i=1}^x B_i p^{i-1}.
\partial_s\Phi_{\beta,u,\gamma}^\delta =-\frac{\xi''(s)}{2}\bigl(\partial_{xx}\Phi_{\beta,u,\gamma}^\delta+\gamma(s) \bigl(\Phi_{\beta,u,\gamma}^\delta\bigr)^2\bigr),\,\,(s,x)\in [0,u)\times \mathbb{R},
\gamma_{p^k}(a) = -\delta(\gamma_{p^{k-1}}(a)) + \frac{d^{m^{k-1}}- (d-p)^{m^{k-1}}}{p}\cdot \frac{\phi(\gamma_{p^{k-1}}(a))}{d^{m^{k-1}}}= \frac{1}{p}\cdot \left( \gamma_{p^{k-1}}(a)^p - (d-p)^{m^{k-1}}\cdot \frac{\phi(\gamma_{p^{k-1}}(a))}{d^{m^{k-1}}}\right),
\|z\|_{L^{2\kappa}(Q_0)}\le\epsilon^k\|z\|_{L^{2\kappa}(Q_k)}+C\epsilon^{-(1-\theta)/\theta}\sum_{n=0}^{k-1}\epsilon^nD_n^{1/2\theta}\|z\|_{L^1(Q_{n+1})}\le\epsilon^k\|z\|_{L^{2\kappa}(Q_k)}+C\epsilon^{-(1-\theta)/\theta}\sum_{n=0}^{k-1}\epsilon^n\left[2^{2n}(\rho^{-2}+t^{-1})\right]^{1/2\theta}\|z\|_{L^1(Q_{n+1})}
\omega_{4}^{3}= - i \{\frac{1}{2}d \operatorname{l n}( \frac{d \overline{{g}}_{2}}{d \overline{{z}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{g}}_{2}- \overline{{g}}_{2}d g_{2})}{( 1 - g_{2}\overline{{g}}_{2})}\},
H''_{R,q}(t) = \frac{T''(t)}{(1-q)T(t)}- \frac{1}{1-q}\left( \frac{T'(t)}{T(t)}\right)^2 = \frac{1}{T(t)}H''_{T,q}(f(t)) - \frac{1}{1-q}\left( \frac{T'(t)}{T(t)}\right)^2.
S^{I}_{p,q}(N,z^{\beta}) \equiv (-1)^N z_{-M}^{\beta(N-1)}\, p^{-1}\int_{0}^{\infty}\frac{t^{N+q/p-1}\, e^{-t^{1/p}}}{t+z_{-M}^{-\beta}}\; dt - \frac{2 \pi i}{p}\; z_{-M}^{-\beta q/p}& \;\; \times \;\; \sum_{j=1}^M e^{-i(2j-1) q \pi/p}\, \exp \!\left( -z_{-M}^{-\beta/p}e^{-i(2j-1) \pi/p}\right) \;\;.
K ( t ,{\bf x}, s \, \vert \, t ,{\bf x}, 0 ) = \int{\frac{d \omega}{2 \pi}}\int{\frac{d p_{y}}{2 \pi}}\int{\frac{d p_{z}}{2 \pi}}\; \langle x \vert \, \exp - i ( - \omega^{2}- d_{x}^{2}+ ( p_{y}- q A )^{2}+{p_{z}}^{2}) s \, \vert x \rangle .
\label[Ieq]{eq:abstracttheoremsplitting}\big\langle f^{k,N}_t - f_t^{\otimes k}, \varphi_k \big\rangle = \big\langle f^{k,N}_t - F^{k,N}_t , \varphi_k \big\rangle + \big\langle F^{k,N}_t - \overline{F}{}^{k,N}_t , \varphi_k \big\rangle + \big\langle \overline{F}{}^{k,N}_t - f_t^{\otimes k}, \varphi_k \big\rangle.
{F_{{C_s}}}\left( x \right) =\Pr\left(\min\left({C_s^{ap}},{C_s^{sk}}\right)<x\right) &=1-\Pr\left(\min\left({C_s^{ap}},{C_s^{sk}}\right)>x\right) &=1-\Pr\left(C_s^{ap}>x\right)\Pr\left({C_s^{sk}}>x\right).
\mathrm{\mathrm{}~^{*}\! ~}F_{\mu \nu}( x ) \stackrel{?}{=}\partial_{\nu}\tilde{A}_{\mu}( x ) - \partial_{\mu}\tilde{A}_{\nu}( x ) + i \tilde{g}[ \tilde{A}_{\mu}( x ) , \tilde{A}_{\nu}( x ) ] ,
(-\frac{1}{2}ax^{2}+Jx)=-\frac{1}{2}a(x^{2}-\frac{2Jx}{a}+\frac{J^{2}}{a^{2}}-\frac{J^{2}}{a^{2}})=-\frac{1}{2}a(x-\frac{J}{a})^{2}+\frac{J^{2}}{2a}
{r_{H}}_{+}-{r_{H}}_{-}\simeq \frac{( p + 1 ) ( p + 2 )}{4 [ | \Lambda | - 2 \pi G_{D}( 1 -{f_{H}}^{2})^{2}]}\left[ \frac{4 \hat{R}}{p ( p + 1 )}\right]^{\frac{p + 1}{p}}> 0 .
\sum_{i+j=m}v^{-ai +\frac12 i(i+1)}\begin{bmatrix}a + i i\end{bmatrix}\frac{\prod_{k=1}^j (v^{-a-i-k}-v^{a+i+k})}{\prod_{k=1}^{\lfloor \frac{j}2\rfloor}(1 -v^{4k})}=v^{-am -\frac12 m(m+1)}\begin{bmatrix}a + m m\end{bmatrix}.
\chi_{jk}\; \equiv\;\chi_{jk}(x_1,\dots,x_j;y_1,\dots,y_k) u_0^{\otimes(N_1-j)}\; \equiv\;u_0(x_{j+1})\cdots u_0(x_{N_1}) v_0^{\otimes(N_2-k)}\; \equiv\;v_0(y_{k+1})\dots v_0(y_{N_2}) \,.
\frac{\binom{n-r}{i-k}\binom{r-\frac{1}{2}}{r-k}\binom{r-\frac{1}{2}}{k}}{\binom{n}{i}}=\frac{\binom{n-r}{i-k}\binom{2r}{r}\binom{2r}{2k}}{2^{2r}\binom{n}{i}}.
\xi_p \left \{\begin{matrix}0 1 a_{21}a_{22}\end{matrix}\right \}=\left \{\begin{matrix}p 1 a_{21}a_{22}+p\end{matrix}\right \}+ \left (v^{-a_{22}-3}\ \overline{[p]}+ v^{a_{22}+3}[p] \right ) \left \{\begin{matrix}p+1 0 a_{21}-1 a_{22}+ p+1\end{matrix}\right \}.
||\dot{{\mathbf U}}||^2_{s,\ast,T}=||{\mathbf V}+J_0{\mathbf V}||^2_{s,\ast,T}&\leq C(K)\Big(||{\mathbf V}||^2_{s,\ast,T}+||\dot{{\mathbf U}}||^2_{W^{1,\infty}_{\ast}(\Omega_T)}||\hat{W}||^2_{s,\ast,T}\Big) &\leq C(K)||{\mathbf V}||^2_{s,\ast,T}+TC(K)||\dot{{\mathbf U}}||^2_{W^{1,\infty}_{\ast}(\Omega_T)}||\hat{W}||^2_{s+1,\ast,T},
I-\cal{F}_Q(Y)^*\cal{F}_P(X)= \,Q^*_{21}(I-Y^*Q^*_{22})^{-1}(I-Y^*X)(I-P_{22}X)^{-1}P_{21}&\quad+\begin{bmatrix}I Q^*_{21}(I-Y^*Q^*_{22})^{-1}Y^* \end{bmatrix}(I-Q^*P)\begin{bmatrix}I X(I-P_{22}X)^{-1}P_{21}\end{bmatrix}.
X_1 = \bigg\{\psi(t),\t\in [0,1]\bigg|\ \psi(0) = \psi_0,\ \|\psi(t)\|_2 \leq 2C_1(1 + t)^{-\frac{d}{4}}\|\psi_0\|_1\ \{\rm and}\ &\sqrt{Q(\psi(t))}\leq 2C_1C_Q(1 + t)^{-\frac{d}{4}-\frac{\eta}{2}}\|\psi_0\|_1,\ 0 \leq t \leq 1 \bigg\}.
\nabla \widetilde{h}_\lambda(y_{i+1})^T(y_{i+1}-y_i)= 0, ~i=0, 1, \cdots; & \nabla \widetilde{h}_\lambda(y_{i+1})^T\widetilde{A}\widetilde{A}^T\nabla \widetilde{h}_\lambda(y_i)= 0, ~i=0, 1, \cdots.
\chi(E_{C, A}\otimes{E_{C, A}}^*) = \frac{{c_1(E_{C, A}\otimes{E_{C, A}}^*)}^2}{2}- c_2(E_{C, A}\otimes{E_{C, A}}^*) + (E_{C, A}\otimes{E_{C, A}}^*) \chi(\mathcal{O}_X).
\alpha = \left( \begin{array}{c c c}{{\omega_{2 m}}}&{{0}}&{{0}}\\{{0}}&{{\omega_{2 m}^{- 1}}}&{{0}}\\{{0}}&{{0}}&{{1}}\\ \end{array}\right) , ~ \beta = \left( \begin{array}{c c c}{{0}}&{{i}}&{{0}}\\{{i}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}\\ \end{array}\right)
\Xi_\Omega(E) =\,\lim_{\|t\|\to d_{\mathbb{K}min}}\,\frac{1}{t}\,\left(t\,\Omega'\star \,d\Phi_\mathcal{E}(0,E)-\,t\,\Omega'\star \,d\Phi_\mathcal{E}(0,E))\right) & =\,\Omega'\star\,d\Phi_\mathcal{E}(0,E)-\,\Omega'\star\,d\Phi^c_{\mathcal{E}}(0,E).
B_{\mathfrak a}(f\wedge g)(x_1,\cdots,x_{p+q})=\sum_{\sigma\in Sh(p,q)}B_{\mathfrak a}\left( f (x_{\sigma(1)},\cdots,x_{\sigma(p)}),g (x_{\sigma(p+1)},\cdots,x_{\sigma(p+q)})\right)
\mu_k =\alpha_k^2(1+\zeta_k^{-1}) \lambda_k =\alpha_k^2(1+\zeta_k)(1+\delta^2\zeta_k^{-1}) \rho_k^2 =(1+\zeta_k)(1-2m\alpha_k+\tilde M\alpha_k^2 (1+\delta^2\zeta_k^{-1}))
G^n_{k,\varepsilon}=\frac{i\sqrt{2}|\Gamma(\gamma+1+i\nu)|}{\Gamma(2\gamma+1)}e^{\pi\nu/2}(2\varepsilon r)^{\gamma-\frac{(n-1)}2}{\rm Im}\left\{e^{i(\varepsilon r+\xi)}{_1F_1}(\gamma-i\nu,2\gamma+1,-2i\varepsilon r)\right\}
\langle[A_1]_N({\bf x})\rangle =\frac{i}{\sqrt{2k\omega}}\left\{\frac{1}{2}(1 -\frac{m}{\omega})J_{N+1}(kr)e^{i(N+1)\varphi}+ \frac{1}{2}(1 +\frac{m}{\omega})J_{N-1}(kr)e^{i(N-1)\varphi}\right\}
\nu(B(e,r)) = \int_{0}^{r-1}\mathcal{H}^{2}(\partial B_{\rho}) d\rho + \int_{r-1}^{r+1}\mathcal{H}^{2}(B(e,r) \cap \partial B_{\rho}\cap \Sigma) d\rho , = 4\pi \int_{0}^{r-1}\rho^{2}d\rho +\int_{r-1}^{r+1}\pi (2 \rho^{2}+ \rho (r^{2}-1)-\rho^{3}) d\rho, &= \frac{4}{3}\pi r^{3}.
L^{\pm}(u)^{-1}=\begin{pmatrix}k_{1}^{\pm}(u)^{-1}+e_{1}^{\pm}(u)k_{2}^{\pm}(u)^{-1}f_{1}^{\pm}(u) -e_{1}^{\pm}(u)k_{2}^{\pm}(u)^{-1}-k_{2}^{\pm}(u)^{-1}f_{1}^{\pm}(u) k_{2}^{\pm}(u)^{-1}\end{pmatrix}
U_j(\underline{p},\underline{q}) \equiv \frac{gm_j}{\sqrt{16\pi^3}}\frac{1}{p^+\sqrt{q^+}}\,,\;\;V_{2s}(\underline{p},\underline{q}) \equiv \frac{g}{\sqrt{8\pi^3}}\frac{\vec{\epsilon}_{2s}^{\,*}\cdot\vec{p}_\perp}{p^+\sqrt{q^+}}\,,
S_{\omega}(v)=\frac{1}{2}\|\nabla v\|_{L^2}^2+\frac{\omega}{2}\|v\|_{L^2}^2-\frac{a}{p+1}\|v\|_{L^{p+1}}^{p+1}-\frac{b}{q+1}\|v\|_{L^{q+1}}^{q+1}, &K_{\omega}(v)=\|\nabla v\|_{L^2}^2+\omega \|v\|_{L^2}^2-a\|v\|_{L^{p+1}}^{p+1}-b\|v\|_{L^{q+1}}^{q+1}.
\sigma_{V}( p^{2}) = \frac{1}{m^{2}}\frac{1 + \frac{\Sigma_{A}}{m}}{\epsilon \left( 1 + \frac{\Sigma_{A}}{m}\right)^{2}+ 2 \left( 1 + \frac{\Sigma_{A}+ \Sigma_{B}}{2 m}\right) \frac{\Sigma_{B}- \Sigma_{A}}{m}},
\begin{array}{c c c c c}{E G}&{\leftarrow}&{E G \times M}&{\rightarrow}&{M}\\{\downarrow}&{}&{\downarrow}&{}&{\downarrow}\\{B G}&{\leftarrow}&{E G \times_{G}M}&{\rightarrow}&{M / G .}\\{}&{\pi}&{}&{\sigma}&{}\\ \end{array}
x_{k}=e^{-\frac{2k\pi i}{N-1}}-\frac{t}{N-1}\sum_{n=0}^{\infty}\frac{(te^{\frac{2k\pi i}{N-1}})^{n}}{\Gamma(n+2)}\cdot\frac{\Gamma(\frac{Nn}{N-1}+1)}{\Gamma(\frac{n}{N-1}+1)}
f_L(t) = \left( \frac{t_i - t}{t_i - t_{i-1}}\right) f (t_{i-1}) + \left( \frac{t - t_{i-1}}{t_i - t_{i-1}}\right) f (t_{i}) t \in [t_{i-1}, t_i) i = 1,\dots,N.
\Lambda^{AB}_N (\mu, \bar\mu) =\Lambda^{AB}_{N - 1}(\mu, \bar\mu)+ \frac{1}{2 \sqrt{2}}\left( \mu^A_{N - 2}\bar\mu^{B, N - 2}- \mu^B_{N - 2}\bar\mu^{A, N - 2}\right) .
F_0''(r) =\left\{(1-N)r^{-N}[U(r)]^{-2}-2r^{1-N}[U(r)]^{-3}U'(r)\right\}\int_0^r \tau^{N-1}U(\tau)^2\,d\tau+1 \to\frac{1-N}{N}+1=\frac{1}{N}\quad\mbox{as}r\to 0.
L(\gamma,Y)=-Y|\eta|+H^{1/q}\bigg(\left(\frac{\gamma}{\gamma+1}\right)^{p-2}(1+Y)^{p-1}\left(Y-\frac{1}{p-1}\right) &+\frac{(2-p)\gamma^{p-1}+\gamma^{p-2}}{p-1}\frac{Z}{|\zeta|^p}\bigg)^{1/p},
R_{H}=\prod_{i=1}^{N_f}\left[\frac{\sqrt{|m_i|^2+({\tilde m}_i+\frac{1}{2}M_W)^2}+{\tilde m}_i+\frac{1}{2}M_W}{\sqrt{|m_i|^2+({\tilde m}_i-\frac{1}{2}M_W)^2}+{\tilde m}_i-\frac{1}{2}M_W}\right]^{-k/2}\ .
\varphi_1|_{W_x}(\sum z_i\mathbf e_i')=\sum_{i=1}^n\Big(\lambda_{n+2}\frac{(\lambda_{n+3}-\lambda_i)^2x_i^2}{(\lambda_{n+3}-\lambda_{n+2})^2x_{n+2}^2}+\lambda_{n+3}\frac{(\lambda_{n+2}-\lambda_i)^2x_i^2}{(\lambda_{n+3}-\lambda_{n+2})^2x_{n+3}^2}+\lambda_i\Big)z_i^2