harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_{\mu}A_{\mu}= \frac{k q^{\prime}u_{\mu}^{\prime}}{c^{2}}\partial_{\mu}\left( \frac 1 r \right) = - \frac{k q^{\prime}u_{\mu}^{\prime}}{c^{2}}\left( \frac{R_{\mu}}{r^{3}}\right) = 0
K^{(2)}_{s_1,s_2;w}= \begin{pmatrix}P_{s_1}0 0 P_{s_2}\end{pmatrix}\,\begin{pmatrix}K_{\rm Ai}- \mathrm{e}^{-|w| H}(1- K_{\rm Ai}) \mathrm{e}^{|w|H}K_{\rm Ai}K_{\rm Ai}\end{pmatrix}\,.
\phi_{\chi_{j}\otimes \chi_{\bar{j}}}^{\alpha}\star \phi_{\chi_{k}\otimes \bar{\chi}_{k}}^{\beta}\ = \ \sum C \left( \chi_{j}, \chi_{\bar{j}}, \alpha \bigm | \chi_{k}, \chi_{\bar{k}}, \beta \bigm | \chi_{l}, \chi_{\bar{l}}, \gamma \right) \phi_{\chi_{l}\otimes \chi_{\bar{l}}}^{\gamma}\ .
S_{\rm BFV}=\int d\tau \left( p\dot{x}+ P_n\dot{\psi}^n+ \pi_a\dot{\lambda}_a + \overline{\theta}_a \dot{\theta}_a -T_a\lambda_a -\overline{B}_aB_a-\frac{i}{2}\overline{B}_2c_1\lambda_1\right) \,.
\tau_{t}\leq \tau_{t_{0}}\prod_{i={t_{0}}}^{t-1}\Big(1-\frac{2}{i}\Big)\leq \tau_{t_{0}}\prod_{i={t_{0}}}^{t-1}\Big(1-\frac{1}{i}\Big)\le \frac{(t_{0}-1)\tau_{t_{0}}}{t}\le \frac{2(t_{0}-1)\delta_{t_{0}}}{t}\enspace.
\binom{p}{r}\binom{q}{s}\frac{s! r! [p+q-r-s-1]!}{(p+q)!}\left[ N^{\prime}_{R,p-r,q-s-1}(u) + N^{\prime}_{R,p+1-r,q-s-2}(u)\right] & \qquad\leq3 K_{1}^{p-r+2}K_{2}^{q-s+2}\frac{M_{p,q}}{p+q+3}\left[K_{2}^{-1}+ K_{1}K_{2}^{-2}\right] ;
\varphi_\varepsilon^*(u) =\left(1-\frac{\lambda \mathbb{E}X^*}{c}\right)\left(1+\sum_{n=1}^{\infty}\left(\frac{\lambda \mathbb{E}X^*}{c}\right)^nF_I^{*n}(u)\right) &=\varphi_\varepsilon^*(0)\left(1+\sum_{n=1}^{\infty}\left(\psi_\varepsilon^*(0)\right)^nF_I^{*n}(u)\right),\,u\geqslant 0,
\sqrt{\frac{2 5}{4}+ 4 \kappa^{2}\tilde{\omega}_{m , \pm}^{2}}= \frac{\left| 8 \left( m + \frac 1 2 \right) \pm \sqrt{2 5 + 4 \left( 2 9 - 4 \left( m + \frac 1 2 \right)^{2}\right) \kappa^{2}+ 6 4 ( \kappa^{2})^{2}}\right|}{2 ( 1 + 4 \kappa^{2})}.
|k_h(x)-k_h(x')| =|\int_0^1 (\partial_x F(tD^\sigma u(x+he)+(1-t)D^\sigma u(x),x+the)- &\partial_x F(tD^\sigma u(x'+he)+(1-t)D^\sigma u(x'),x'+the)dt| &\le \|F\|_{C^2}(|D^\sigma u(x+he)-D^\sigma u(x'+he)|+|D^\sigma u(x)-D^\sigma u(x')|+|x-x'|) &\le C\|F\|_{C^2}([u]_{C^{\sigma+\alpha_1}(R_1)}|x-x'|^{\alpha_1}+|x-x'|).
T_{F}^{- 1}= \frac{i}{2 \sqrt{2}}\sum_{k}e^{- i H_{k}}\biggl [ ( 1 - i ) e^{i H_{k^{\prime}}}e^{i H_{\bar{k}^{\prime}}}+ ( 1 + i ) e^{i H_{k^{\prime}}}e^{- i H_{\bar{k}^{\prime}}}+ ( 1 + i ) e^{- i H_{k^{\prime}}}e^{i H_{\bar{k}^{\prime}}}+ ( 1 - i ) e^{- i H_{k^{\prime}}}e^{- i H_{\bar{k}^{\prime}}}\biggr ]
n(\Lambda,i,h) =\#\{0\leq k\leq f-1: \alpha\overline{p}^k\bar i=\bar h\}&=\#\{0\leq k\leq f-1: a{p}^ki\equiv h\pmod{e}\}&=\#\{0\leq k\leq f-1: a{p}^ki'\equiv h'\pmod{e'}\}\enspace.
V_{N}(s)-\frac{T-s}{T-t}V_{N}(t) =\frac{1}{\sigma\sqrt{N}}\sum_{i=1}^N\left\{\sum_{r=s+1}^T\left[\left(Y_{i,r}-\mu_i\right)-\frac{1}{T-t}\sum_{v=t+1}^T \left(Y_{i,v}-\mu_i\right)\right]\right\}&=\frac{1}{\sigma\sqrt{N}}\sum_{i=1}^N\sum_{r=s+1}^T\left(Y_{i,r}-\widetilde{Y}_{i,t}\right).
\begin{array}{c}\displaystyle\int_{\Omega}p\left|\nabla u\right|^{2}\geq p_{0}\int_{\Omega}\left|\nabla u_{0}\right|^{2}-2\pi p_{0}\left(\sum_{i=1}^{m}d_{i}^{2}\right)I\left(\dfrac{R}{R_{0}}\right)+ \displaystyle-2\pi \left(1-a^{2}\right)p_{0}\sum_{i\neq j}\left|d_{i}\right|\left|d_{j}\right|\log \frac{R}{|a_{i}-a_{j}|}-C,\end{array}
C_{n,g}\left( H\right) =\left( \begin{array}{ccccc}H_{k,1}H_{k,2}H_{k,3}... H_{k,n}H_{k,n-g+1}H_{k,n-g+2}H_{k,n-g+3}... H_{k,n-g}H_{k,n-2g+1}H_{k,n-2g+2}H_{k,n-2g+3}... H_{k,n-2g}\vdots \vdots \vdots \ddots \vdots H_{k,g+1}H_{k,g+2}H_{k,g+3}... H_{k,g}\end{array}\right) ,
( A^{L})_{i j}~ = ~ \mathrm{n u m b e r ~ o f ~ p a t h s ~ f r o m ~ j ~ t o ~ i ~ o f ~ l e n g t h ~ L ~}~ = ~ \mathrm{n u m b e r ~ o f ~ L ~ - p a t h s ~ f r o m ~ j ~ t o ~ i ~}\, .
\mathcal{E}_f(V)-\mathcal{E}_f(U_0) \ge \frac{1}{2}\,\left[\int_\Omega V\,u^2\,dx-\left(\int_\Omega|u|^{2p/(p+1)}\,dx\right)^{(p+1)/p}\right] &\ge\frac{1}{2}\, \int_\Omega V\, u^2\, dx\,\left[1-\left(\int_\Omega V^{-p}\, dx\right)^{1/p}\right]=\frac{1}{2}\, \left(\int_\Omega V\, u^2\, dx\right)\, |1-\lambda|,
R = \left( \begin{array}{c c c c c}{{N - 1}}&{{- 1}}&{{.}}&{{.}}&{{- 1}}\\{{- 1}}&{{N - 1}}&{{- 1}}&{{}}&{{.}}\\{{.}}&{{- 1}}&{{.}}&{{.}}&{{.}}\\{{.}}&{{}}&{{.}}&{{.}}&{{- 1}}\\{{- 1}}&{{.}}&{{.}}&{{- 1}}&{{N - 1}}\\ \end{array}\right) \; .
G= \begin{pmatrix}I_{k_1}A B_1 + u B_2 C_1 + u C_2 D_1 + u D_2 E_1 0 u I_{k_2}0 (1+u) C_3 u D_3 + (1+u) D_4 E_2 0 0 (1+ u) I_{k_3}(1+ u) C_4 (1+ u) D_5 E_3 0 0 0 (1+ u^2) I_{k_4}0 E_4 0 0 0 0 (u + u^2) I_{k_5}E_5\end{pmatrix},
\omega_{-}^{\psi}= \left( \begin{array}{c c}{{\Sigma_{-}^{( 1 )}- \chi}}&{{( \Sigma_{-}^{( 1 )}- \chi ) \, \chi - \chi \, \Sigma_{-}^{( 2 )}+ \partial_{-}\chi}}\\{{I_{n}}}&{{\Sigma_{-}^{( 2 )}+ \chi}}\\ \end{array}\right) .
\begin{multlined}[t][12.5cm]\widetilde{\Pi}_{\nu_{G},k\nu_{T}}(x,x)^{(j)}\sim \frac{d_{\nu_{G}}}{(2\pi)^{d_{T}}}k\int_{G}\int_{(-\pi,\pi)^{d_{T}}}\int_{0}^{+\infty}\chi_{\nu_{G}}(g^{-1})e^{ik\Psi(x,t,g,\vartheta)}\cdot s\left(\tilde{\mu}_{g^{-1}}^{G}\circ\tilde{\mu}_{-\vartheta}^{T}(x),x,kt\right)\gamma_{j}(g,\vartheta)d t d\vartheta dg\end{multlined}
\lim_{q\rightarrow\infty}d_{\textrm{max},1}(f) = \lim_{q\rightarrow\infty}\max_{b\in\{0,1\}^q}d_{\textrm{max},1}(f,b) & \leq \lim_{q\rightarrow\infty}\frac{2{q \choose q/2}}{2^q}& = \lim_{q\rightarrow\infty}\frac{2\sqrt{\frac{2}{\pi q}}2^q}{2^q}& = \lim_{q\rightarrow\infty}2\sqrt{\frac{2}{\pi q}}& = 0,
t_{a,b,c}x_{i}t_{a,b,c}^{-1}=\begin{bmatrix}1 0 0 a^{-1}bt^{i}1 0 a^{-1}cb_{i}b^{-1}ct^{ip}1 \end{bmatrix}=\begin{bmatrix}1 0 0 t^{i+j}1 0 a^{-1}cb_{i}t^{(i+j)p}1 \end{bmatrix}.
\begin{array}{r l}{{\theta_{6}+ \theta_{1 8}- \theta_{- 1 8}- \theta_{- 6}}}&{{= - i ~ q^{-{\frac{1}{1 2}}}{\frac{\eta^{3}}{\eta ( \tau / 2 ) \eta ( 2 \tau )}}}}\\{{\theta_{- 2}+ \theta_{1 0}- \theta_{2 2}- \theta_{- 1 4}}}&{{= q^{-{\frac{1}{1 2}}}{\frac{\eta^{3}}{\eta ( \tau / 2 ) \eta ( 2 \tau )}}.}}\\ \end{array}
\theta_{\rm ev}(\tau,z)=\sum_{n\in{\bf Z}}q^{\frac{1}{4}(2 n)^2}\zeta^{2 n}\,,\qquad\theta_{\rm od}(\tau,z)=\sum_{n\in{\bf Z}}q^{\frac{1}{4}(2 n+1)^2}\zeta^{2 n+1}
nA_{k,k-1}=- \mathbf{y}_{k-1}^2 \sqrt{1+\frac{a}{k-1}},nA_{k,k}= \mathbf{y}_{k-1}^2 + \mathbf{x}_{k}^2,n A_{k,k+1}=-\frac{\mathbf{x}_{k}^2}{\sqrt{1+\frac{a}{k}}}.
\Big| T_\Sigma(i\tau/2) - \sum_{l = 0}^k (i\tau)^{-(2j-1+l)}T_{\Sigma}^l \Big| \lesssim \sum_{\substack{k+1 \leq |\alpha|+|\beta| \leq 2j-1+k \max\{\alpha_l, \beta_l\}\leq [\frac{k}2]+1}}|\tau|^{-|\alpha| - |\beta|}\prod_k \| \partial^{\alpha_k}u_k \|_{l^{2j}_{\tau}DU^2}\| \partial^{\beta_k}v_k \|_{l^{2j}_{\tau}DU^2}.
\psi ( \varphi_{1}, \varphi_{2}) = \sum_{n_{1}, n_{2}= - \infty}^{+ \infty}\Theta_{n_{1}}( \varphi_{1}) \Theta_{n_{2}}( \varphi_{2}) \psi_{n_{1}, n_{2}}( \varphi_{1}, \varphi_{2}) \ ,
\xi_{n}= \sum_{\overline{A}\in \overline{\mathfrak{B}}_n ,\; \varepsilon (\overline{A})\ge 2}\left( -1\right)^{\varepsilon (\overline{A})}\left| \overline{A}\right| \prod_{i=0}^{n-2}\left[ \left( n-i\right) ! \right]^{\psi_i (\overline{A})}
\chi_{t o p}\; : = \left( \frac{g^{2}}{1 6 \pi^{2}}\right)^{2}\; \; \int d^{4}x \; \langle T \; q ( x ) \; q ( 0 ) \rangle \; \; , \; \; q ( x ) \; : = \; \mathrm{T r}\Big ( F ( x ) \tilde{F}( x ) \Big ) \; ,
E_{o_-}(\lambda n_w) = q_{\lambda n_w}^{1/2}q_{\lambda n_{ws}}^{-1/2}q_{n_s}^{-1/2}E_{o_-}(\lambda n_{ws})E_{o_-\cdot ws}(n_s), E_{o_-\cdot v}(\lambda n_w) = q_{\lambda n_w}^{1/2}q_{\lambda n_{ws}}^{-1/2}q_{n_s}^{-1/2}E_{o_-\cdot v}(\lambda n_{ws})E_{o_-\cdot vws}(n_s).
\bigl\langle\Sigma_{xy}-\widehat{\Sigma}_{xy},\widehat{U}_{1}^{\ast}\widehat{V}_{1}^{\ast^{\prime}}-\widehat{U}_{1}\widehat{V}_{1}^{\prime}\bigr\rangle=\tfrac{1}{2}\langle\Sigma-\widehat{\Sigma},W \rangle.
|I_0|\le{8\over\pi}\int_0^1{s^{-1/2}\over(\pi/2)^{p-1/2}}ds+{8\over\pi}\int_1^{+\infty}{s^{-1/2}\over(2s\cosh \b)^{p-1/2}}ds ={2^{p+7/2}\over\pi^{p+1/2}}+{1\over\pi(p-1)2^{p-7/2}(\cosh \b)^{p-1/2}},
\alpha =\pi+4\arctan\left( 4\frac{\Gamma(3/4)^{2}}{\Gamma(1/4)^{2}}\right) \ =\pi+4\arctan\left( \frac{1}{2}\frac{3^{2}-1}{3^{2}}\frac{5^{2}}{5^{2}-1}\frac{7^{2}-1}{7^{2}}\cdot\cdots\right) =4.8560553209...
T_{(-k)}(t)= g^{-1}_k\bigl(X(t)\bigr) = \Bigl(g_{k, h}^{-1}\bigl(X(t)\bigr) \Bigr)_{h\not =k}= \Bigl(g_{\cdot, h}^{-1}\bigl(t, X(t)\bigr) \Bigr)_{h\not=k}= \Bigl[v_h-v_k\Bigr]_{h\not = k}^{-1}\bigl(X(t)-v_kt\bigr).
\tilde{\eta}(X) :=\eta(\tilde{\xi})^{-1}\eta(X), \tilde{\Phi}(X) := \Phi(X - \tilde{\eta}(X)\tilde{\xi}), \tilde{g}_L(X,Y) := \eta(\tilde{\xi})^{-1}g_L(X-\tilde{\eta}(X)\tilde{\xi},Y-\tilde{\eta}(Y)\tilde{\xi}) + \tilde{\eta}(X)\tilde{\eta}(Y).
\delta_{\xi}g_{\mu \nu}={\cal L}_{\xi}g_{\mu \nu}= \xi_{\mu ; \nu}+ \xi_{\nu ; \mu}\; ; \; \delta_{\xi}h_{\mu \nu}={\cal L}_{\xi}h_{\mu \nu}= \xi^{\lambda}\partial_{\lambda}h_{\mu \nu}+ \partial_{\mu}\xi^{\lambda}h_{\lambda \nu}+ \partial_{\nu}\xi^{\lambda}h_{\mu \lambda}.
f(a,b;\mathbf{t})=\int_0^{t_1}\ldots\int_0^{t_m}|x_1-x_m|^a |x_2-x_1|^b |x_3-x_2|^a |x_4-x_3|^b \ldots &\times |x_{m-1}-x_{m-2}|^a |x_{m}-x_{m-1}|^b d\mathbf{x},
\|h-h_0\|_{L^1(D(z_0,\rho))}= \|h-h_{(r)}+h_1\|_{L^1(D(z_0,\rho))}\leq \|h-h_{(r)}\|_{L^1(D(z_0,\rho))}+\|h_1\|_{L^1(D(z_0,\rho))}\leq \Big(1+\frac{2\rho}{r}\Big)^2\Big(2A_0+\frac{C\rho}{r}A_0\Big) \leq C\frac{\rho^3}{r^3}A_0,
{}\,=\,(\operatorname{id}\otimes\operatorname{id}\otimes\varphi)\bigl((1\otimes c\otimes1)(\operatorname{id}\otimes\Delta)((y\otimes1)(\Delta a))\bigr), &{}\,=\,(\operatorname{id}\otimes\operatorname{id}\otimes\varphi)\bigl((y\otimes c\otimes1)(E\otimes1)\Delta_{13}(a)\bigr).
M := \left( \begin{array}{cccc}-\nu_1 \sigma^z \mu_1 S(\gamma_1) \mu_1 S(\gamma_1)^T -\nu_2 \sigma^z \ddots \ddots \ddots \mu_{n-1}S(\gamma_{n-1}) \mu_{n-1}S(\gamma_{n-1})^T -\nu_{n}\sigma^z \end{array}\right),
C = - \frac{4 i}{g}\frac{1}{I_{2}}[ B , F ] + \Omega \Bigl ( \frac{1}{2}a^{1}\lambda_{1}+ \frac{1}{2}a^{2}\lambda_{2}+ \frac{1}{2}a^{3}\lambda_{3}+ \frac{1}{2}a^{8}\lambda_{8}\Bigr ) \Omega^{\dagger}.
\int_{\mathbb{R}^{N}}\big[ U_k(x)+V_k(x)\big]\,dx = \int_{\mathbb{R}^{N}}\Phi_{R}(x)\big[ U_k(x)+V_k(x)\big]\,dx +\int_{\mathbb{R}^{N}}(1-\Phi_{R}(x))\big[ U_k(x)+V_k(x)\big]\,dx
\sum_{n = 0}^{\infty}\epsilon ( n ) = \int_{0}^{\infty}{d n}\, \epsilon ( n ) +{\frac{1}{2}}\, \epsilon ( 0 ) -{\frac{1}{1 2}}\, \epsilon^{\prime}( 0 ) +{\frac{1}{7 2 0}}\, \epsilon^{\prime \prime \prime}( 0 ) -{\frac{1}{3 0 2 4 0}}\, \epsilon^{( v )}( 0 ) + \cdots ,
P(x_0,x_1,x_2,x_{12})=0, P^{(1)}(x_0,x_2,x_3,x_{23})=0, P^{(2)}(x_0,x_3,x_1,x_{31})=0, P^{(3)}(x_3,x_{31},x_{23},x_{123})=0, P^{(4)}(x_1,x_{12},x_{31},x_{123})=0, P^{(5)}(x_2,x_{23},x_{12},x_{123})=0,
\Delta \Gamma_{V}^{( 2 )}= \mathrm{R e}\int d^{2}\theta \frac{d^{4}p}{( 2 \pi )^{4}}W_{a}( p ) C^{a b}W_{b}( - p ) \Big ( \tilde{f}_{1}+ \tilde{f}_{2}+ \tilde{f}_{3}\Big ) ,
\overline{\zeta}_q(-m)= \sum_{j=0}^{2m+1}(-1)^{j+1}\frac{(n+1-q)^j}{m}\binom{m+1}{j}\xi_q(-2m+j-1) &+(-1)^m m!\sum_{j=2m+2}^{2n+2m}(n+1-q)^j \frac{\Gamma(-m+j-1)(-2m-2+j)}{\Gamma(j+1)}\gamma_{n,q}(-2m+j-1),
\frac{q(\lambda(Hom))}{q(\lambda(Ext))}=\frac{[Ext^1_{\hat{\mathbb{Z}}}(P,\mathbb{Z})][\ker(\lambda(Ext^2))]}{[Ext^2_{\hat{\mathbb{Z}}}(P,\mathbb{Z})]}=[\mathrm{cok}(\hat{\lambda}(H^0(K_v)))].
\chi(gn_2^-) = (\tfrac{c'+d}{d})e^{\tfrac{-\pi i}{12}[-\tfrac{c+2d}{d}+\tfrac{b}{d}+12s(c+2d,d)]}&=(\tfrac{c'}{d})e^{\tfrac{-\pi i}{12}[-\tfrac{c}{d}+\tfrac{b}{d}+12s(c,d)]}e^{\tfrac{\pi i}{6}}&=\chi(g)\chi(n_2^-).
\bigg(\sum_{j=1}^N |p_{1,j}-q_{1,j}||w_j|\bigg)^2 \le \frac{C\|\nabla\varphi\|_\infty^2}{r^{4d}\varphi_0^2}\bigg( |x_1-y_1|^2 + \frac 1N\sum_{k=1}^N |x_k-y_k|^2\bigg)\frac 1N \sum_{j=1}^N|w_j|^2 \,.
(x,u)=\lim_{k\to\infty}\Big( t_k x_k + \sum_{j=0}^\infty \big(\alpha^j_k z^j - \beta^j_k z^j\big), \frac{1}{2}t_k v_k + \sum_{j=0}^\infty\big(\alpha^j_k f^j_k - \beta^j_k g^j_k\big) \Big)
\mu^{i*}_{\nu}((a,b)):=\overline{\mu}_{}^{i+}((a,b))+\overline{\mu}_{}^{i-}((a,b))+\mu^{IC}\left(\{(t,x) \ :\t>0,\x_{i,\nu}(t;0,a)<x<x_{i,\nu}(t;0,b)\}\right) \ .
\begin{aligned}\min_{\{\delta_{m,n}^{(k)}\}}~ \left|\left|\mathbf{Q}^{(k)}- \left(\{\delta_{m,n}^{(k)}\}\circ \frac{j}{\sqrt{N_t}}\{e^{j\phi_{m,n}^{(k)}}\}\right)\mathbf{F}_B^{(k)}\right|\right|_F^2 s.t.~ |\delta_{m,n}^{(k)}| \leq \bar{\delta}^{(k)}, \forall m,n,\end{aligned}
\bar{F}(x) = \rho_{1}f^{*}(\gamma)\bar{B}^{r}(x) + \rho_{2}\bar{V}^{r}(x) + \rho_{1}f^{*}(\gamma) \int_{0}^{x}\bar{F}_{\gamma}(x-u)dB^{r}(u), x \geq 0.
\frac{8 \pi^{2}}{g_{eff}^{2}(\mu)}=\frac{8 \pi^{2}}{g_{0}^{2}}+ 2 \ln \left( \frac{\Lambda}{\mu}\right) -2 \ln Z(\mu) =\frac{8 \pi^{2}}{g_{W}^{2}(\mu)}- 2 \ln Z(\mu)
\begin{aligned}\|\nabla_iu\|_{L^{p,2}_{i,j}}=\|\mathcal{R}_{i,j}\nabla_iu\|_{L^{p,2}_{d_{i,j}}}=\|(\nabla_{d_{i,j}}+\nabla_{D_{i,j}})\mathcal{R}_{i,j}u\|_{L^{p,2}_{d_{i,j}}}.\end{aligned}
\mod{\frac{I_1(\mu)}{G(\mu)}}=\frac{1}{\mod{\sin(\mu^{\frac{1}{2}})}}\mod{\frac{O(\mu^{-\frac{1}{2}})e^{-\mu+1+O(\mu^{-\frac{1}{2}})}+O(\mu^{-1})e^{O(\mu^{-\frac{1}{2}})}}{e^{-\mu+1}-1}}\leq\frac{C}{\mod{\mu}^{\frac{1}{2}}}\frac{1}{\mod{\sin(\mu^{\frac{1}{2}})}}\frac{\mod{e^{-\mu+1}}+1}{\mod{e^{-\mu+1}-1}},
\begin{array}{lll}u_{\overline{\nu_0}}=(1,0,0,0,0,1,0), u_{\overline{\nu_4}}=(0,0,0,0,1,0,1), u_{\overline{\nu_1}}=(0,1,0,0,0,1,0), u_{\overline{\nu_5}}=(-1,-1,-1,-1,-1,0,1), u_{\overline{\nu_2}}=(0,0,1,0,0,1,0), u_{\tau_1}=(0,0,0,0,0,1,0), u_{\overline{\nu_3}}=(0,0,0,1,0,0,1), u_{\tau_2}=(0,0,0,0,0,0,1).\end{array}
\quad\frac{1}{\lambda(t)^{2\sigma_c}}\int\chi\bigg{(}\frac{x-x(T)}{R(t)}\bigg{)}|u(t)|^2 &=\int\chi\Bigg{[}\frac{1}{A}\bigg{(}y+\frac{x(t)-x(T)}{\lambda(t)}\bigg{)}\bigg{]}|Q_{b}+\varepsilon|^2dy.
S\left(\underset{k}{\underbrace{a,\ldots,a}},N,1\right) =a_{N}\left(\prod_{l=1}^{k}\mathbf{S}_{a}\right)_{N,1}=a_{N}\left(\mathbf{S}_{a}\left(\prod_{l=1}^{k-1}\mathbf{S}_{a}\right)\right)_{N,1}=\frac{1}{a_{N}}\sum_{m=1}^{N}a_{N}\cdot a_{m}G\left(m+k-1,m\right) =G\left(N+k,N\right).
W_{\psi_{1},\varphi_{1}}f(z) = \psi_{1,*}(z'_{[j]})e^{-\left\langle z_{[j]}, (V^*b^0)_{[j]}\right\rangle}f\big(z_{[j]}+(V^*b^0)_{[j]}, Gz'_{[j]}+ (b^1)'_{[j]}\big) & = (T_{(V^*b^0)_{[j]}}W_{\psi_{1,*},\varphi_{1,*}})f(z),
r_1(t) =\int_0^t e^{(t-s)A}\frac{-\Delta tM_{\ell(s)}}{1+\Delta tM_{\ell(s)}}F({X}_{\ell(s)})ds r_2(t) =\int_0^t e^{(t-s)A}\bigl(F(\tilde{Y}(t_{\ell(s)})+\tilde{Z}_{x_0}(t_{\ell(s)}))-F(\tilde{Y}(s)+\tilde{Z}_{x_0}(t_\ell(s))\bigr)ds.
\{i_5=n\}= \bigcap_{k=1}^{n-1}\bigg\{1-a_{i_2}+\sum_{i=k, i \ne i_3}^\infty (1-b_i) >1\bigg\}\cap \bigg\{1-a_{i_2}+\sum_{i=n,i\ne i_3}^\infty (1-b_i) \leq 1\bigg\}.
\ell_\rho(\gamma\vert_{[0,t]})=\int_0^t \rho(y+s)\, ds =\begin{cases}\frac{1}{\beta-1}\left[\frac{1}{y^{\beta-1}}-\frac{1}{(y+t)^{\beta-1}}\right] y\ge 1, 1-y+\frac{1}{\beta-1}\left[1-\frac{1}{(t+y)^{\beta-1}}\right] y<1\le y+t, t y+t<1. \end{cases}
\alpha_{( \cdot ) , ( \cdot )}^{- 1 , 1}= A_{( \cdot ) , ( \cdot )}^{- 1 , 1}+ a A_{( \cdot ) , ( \cdot )}^{0 , 0}+ b B_{( \cdot ) , ( \cdot )}^{0 , 0}+ c B_{( \cdot ) , ( \cdot )}^{- 1 , 1}.
X=\boldsymbol{\mathrm{Proj}}_C(\bigoplus_{m\ge0}f_*L^{\otimes m})=\boldsymbol{\mathrm{Proj}}_C(\bigoplus_{m\ge0}f_*\mathcal{O}_X(mD))\cong \boldsymbol{\mathrm{Proj}}_C(\bigoplus_{m\ge0}f'_*L'^{\otimes m})=X'.
F_C^{(n)}(a,b;c_1,\dots, c_n;z_1, \dots, z_n) = \sum_{m_1, \dots, m_n\in \bold Z_{\geq 0}}\dfrac{(a,m_1+\cdots +m_n)(b,m_1+\cdots +m_n)z_1^{m_1}\cdots z_1^{m_1}}{(c_1,m_1)\cdots (c_n,m_n)m_1!\cdots m_n!},
L_{1}= 2 f_{G}^{d - 1}\sum_{i}\epsilon^{\mu_{1}\mu_{2}\cdots \mu_{d}}\epsilon_{a_{1}a_{2}\cdots a_{d + 1}}{\cal O}_{i , i + 1}^{a_{1}n}( x ) D_{\mu_{1}}{\cal O}_{i , i + 1 ; n}^{a_{2}}( x ) e_{\mu_{2}}^{a_{3}}( x , i ) e_{\mu_{3}}^{a_{4}}( x , i ) \cdots e_{\mu_{d}}^{a_{d + 1}}( x , i ) \, ;
\texttt{sum}=v'_1(1+\eta_N)+v'_2(1+\eta_{N-1})+\cdots +v'_n(1+\eta_1) &=v_1(1+\eta_{N+1})+v_2(1+\eta_{N})+\cdots +v_N(1+\eta_2) &=(a_1+\alpha_1)(1+\eta_{N+1})+(a_2+\alpha_2)(1+\eta_{N})+\cdots +(a_N+\alpha_N)(1+\eta_2)
\Pi_{\mu \nu}\equiv -{\frac{1}{4}}T_{\mu \lambda}T_{\nu}^{\lambda}+{\frac{1}{1 2}}T T_{\mu \nu}+{\frac{1}{8}}g_{\mu \nu}\left( T_{\lambda \kappa}T^{\lambda \kappa}-{\frac{1}{3}}T^{2}\right) .
\left|\underset{i = 1}{\stackrel{K}{\prod}}h(\sigma - \lambda_i, \frac{\xi}{\sqrt{K}})\right| \leq C \lambda^{K-2}\left(\frac{1}{1 + |\xi|/\sqrt{K}}\right)^2 &\leq C \lambda^{K-2}\frac{K}{K + \xi^2}&\leq C \lambda^{K-2}\frac{K}{1 + \xi^2}
\|T_Kf\|_{L^p(W)}\leq \sum_{j=1}^N \|(T_Kf_j) e_j\|_{L^p(W)}=\sum_{j=1}^N \|T_Kf_j\|_{L^p((w_{jj}^{(\frac{2}{p})})^\frac{p}{2})}\leq C\sum_{j=1}^N \|f_j\|_{L^p((w_{jj}^{(\frac{2}{p})})^\frac{p}{2})}\leq C'\|f\|_{L^p(W)},
\left[\begin{array}{cccccc}1 0 0 0 0 -2 0 1 0 0 0 -2 0 0 1 0 0 -15 0 0 0 1 0 -20 0 0 0 0 1 -20 \end{array}\right] \left[\begin{array}{c}1/15 1/15 0 2/3 2/3 8/15\end{array}\right]=\left[\begin{array}{c}-1 -1 -8 -10 -10\end{array}\right]\in 2\Delta_{(1,2,2,15,20,20)}\, .
\lambda_{1,p}(\gamma) \le \frac{\int_c^d \frac{| v' |^p F_\gamma^p}{|\gamma'|_h^{p-1}}\,dt}{\int_c^d | v |^p |\gamma'|_h \,dt}= \frac{\int_0^1 \frac{| w' |^p F_\beta^p}{|\beta'|_h^{p-1}}\,dt}{\int_0^1 | w |^p |\beta'|_h \,dt}
\langle m_i, m_j\rangle =\begin{cases}[2], \; j \equiv i \bmod d; z, \; i\equiv 0\equiv j-1 \bmod d; z^{-1}, \; i-1\equiv 0\equiv j \bmod d; 1, \; i\equiv j\pm 1\bmod d,\;; 0, .\end{cases}
x\circledS_{\nu}y:=x^{\ast}\left( \left( x^{\ast}\right)^{-1}y^{\ast}yx^{-1}\right)^{\nu}x=x^{\ast}\left\vert yx^{-1}\right\vert^{2\nu}x=\left\vert \left\vert yx^{-1}\right\vert^{\nu}x\right\vert^{2}.
\tan [ 2 ( t - t_{0}) ] = \frac{4 e - ( j^{2}- 1 + 2 e ) ( 1 - x^{2})}{\sqrt{- 2 e}( x^{2}- 1 ) \sqrt{2 e + ( 1 - x^{2}) ( 1 - j^{2}/ x^{2})}}.
\int\prod_i dY_i\delta(\sum_{i=1}^6w_iY_i - t)e^{-\sum_i\mu_iY_i}\ \stackrel{Y_i\rightarrow Y_i+ln\mu_i}{\longrightarrow}\int\prod_i dY_i\delta(\sum_{i=1}^6w_iY_i-t^\prime)e^{-\sum_iY_i},
q_{i,i}-\sum_{j=1,j\neq i}^{N}| q_{i,j}| = \sum_{j=1}^{N}q_{i,j}& = \sum_{j=1}^{N}\left( w_{i,j}+w_{i+1,j+1}-w_{i,j+1}-w_{i+1,j}\right) = w_{i,1}-w_{i+1,1}+w_{i+1,N}-w_{i,N}&\geq 0,
\mu^{l}= f_{b c}^{a}f_{d e}^{c}\left( \epsilon^{0 i j l}A_{j}^{e}\eta_{i}^{\left( 2 \right) d}-{\cal P}^{\left( 2 \right) l e}\eta^{d}\right) \left( B_{a}^{0 k}A_{k}^{b}+ \eta_{a}^{\left( 2 \right) k}{\cal P}_{k}^{\left( 2 \right) b}+ \eta_{a}{\cal P}^{b}\right) .
\begin{cases}u''-\Delta u =0 \hbox{in}(0, +\infty)\times \Omega, v''-\Delta v=0 \hbox{in}(0, +\infty)\times \Omega, u=v=0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu u +\alpha u=0 \hbox{on}(0, +\infty)\times \Gamma_1, \partial_\nu v +\beta v=0 \hbox{on}(0, +\infty)\times \Gamma_1\end{cases}
q_0(y): = \frac{pU(y)^{p-1}c_2b_j^{2s}}{(n-4s)c_{n,s}^{n-4s}c_1}+ \frac{b_j^{2s}}{(n-4s)c_{n,s}^{n-4s}}\left(Z_{n+1}(y) + \frac{n-2s}{2}\alpha_{n, s}\frac{1}{\left(1+|y|^2\right)^{\frac{n-2s}{2}}}\right)
{\bf{Cov}}\left[V_{\varepsilon}(u), \,V_{\varepsilon}(v)\right] = \begin{cases}-2 \, \log|u-v|, \, \varepsilon\ll|u-v|\leq 1, 2\left(\kappa-\log\varepsilon\right),\, u=v,\end{cases}& + O(\varepsilon),
\bigl\langle\Xi(X_1-X'_1),Y\bigr\rangle =\bigl\langle\Xi(X_1-X'_1),[Y_1,b]\bigr\rangle =-\bigl\langle\Xi(Y_1),[b,X_1-X'_1]\bigr\rangle-\bigl\langle\Xi(b),[X_1-X'_1,Y_1]\bigr\rangle =0
\frac{d^2 f}{d x^2}- \frac{(5 x^3+1)}{2 x (1-x^3)}\frac{d f}{d x}- \frac{l(l+6)}{4 x^2 (1-x^3)}f +\frac{\lambda^2}{(1-x^3)^2}\left( 1 + \frac{\rho_0^6}{(\omega R)^6 x^3}\right) f =0\,,
r(G(p_I)-k_1p_I-k_0)-\mu p_I(G'(p_I)-k_1)-\frac{1}{2}\sigma^2p_I^2G''(p_I) &\quad>rH(p_I)-\mu p_IH'(p_I)-\frac{1}{2}\sigma^2p_I^2H_{-}''(p_I)=0,
\frac{1}{\pi}\int_{0}^{\infty}\,d\eta_1\int_{-\infty}^{\infty}\,d\tau_1 \frac{1}{x-\tau_1-i\eta_1}\overline{\partial}\Psi_{f_n}(\tau_1,\eta_1) = f_n(x)+iH[f_n](x) .
\bar{N}_{n m}^{r s}= \frac{1}{n m}\oint_{z^{r}}\frac{d z}{2 \pi i}\oint_{z^{s}}\frac{d w}{2 \pi i}\frac{1}{( z - w )^{2}}( - 1 )^{n ( r - 1 ) + m ( s - 1 )}( f ( z ) )^{( - 1 )^{r}n}( f ( w ) )^{( - 1 )^{s}m}
H^{(1)}= - \partial_{\theta}^2 - \cot\theta \partial_{\theta}+ \frac{k^2}{\sin^2 \theta}\left(- \partial_{\phi}^2 + \frac{(\alpha+3/2)(\alpha+1/2)}{\cos^2 \phi}+ \frac{(\beta-1/2)(\beta-3/2)}{\sin^2 \phi}\right)
\widetilde{\underline{D}}\;\widetilde{\underline{x}}^{2\beta}M_{k}=2\beta\;\widetilde{\underline{x}}^{2\beta-1}M_{k},\widetilde{\underline{D}}\,\widetilde{\underline{x}}^{2\beta+1}M_{k}=2(\beta+k+\gamma_2)\,\widetilde{\underline{x}}^{2\beta}M_{k},
\omega(u^*xu) =\sum_{i=1}^n \langle u^* xub_i,b_i\rangle_{L^2(M,\phi)}=\sum_{i=1}^n \langle x ub_i,ub_i\rangle_{L^2(M,\phi)}=\sum_{i=1}^n \langle x P_Kub_i,P_Kub_i\rangle_{L^2(M,\phi)}&=\sum_{i=1}^n \langle P_Kx P_Kub_i,ub_i\rangle_{K}=\sum_{i=1}^n \langle P_Kx P_Kb_i,b_i\rangle_{K}=\sum_{i=1}^n \langle xb_i,b_i\rangle_{L^2(M,\phi)}=\omega(x).
\begin{aligned}p =\langle K(x,y), (1-\lambda_\delta(x,y))\chi_0(y)[\phi(y)-\phi(x)]\psi(x) \rangle + \langle K(x,y), \lambda_\delta(x,y)\chi_0(y)[\phi(y)-\phi(x)]\psi(x) \rangle =: p_{1,\delta}+p_{2,\delta}. \end{aligned}
\begin{aligned}\|u\|_{L^{\infty}(0,T;D_{A_{q}}^{1-\frac{1}{p},p})}+\|\partial_{t}u\|_{L^{p}(0,T;L^{q})}+\|u\|_{L^{p}(0,T;W^{2,q})}+\|p\|_{L^{p}(0,T;W^{1,q})}\leq C\left(\|u_{0}\|_{D_{A_{q}}^{1-\frac{1}{p},p}}+\|u\cdot\nabla u\|_{L^{p}(0,T;L^{q})}+\|\theta\|_{L^{p}(0,T;L^{q})}\right).\end{aligned}
\leq2\sqrt{1+\mathcal{L}^{2}}\int_{0}^{2\rho}\int_{S^{n-1}}\frac{1}{r^{(n-2)+\eta}}r^{n-1}drd\sigma(\omega) &=\frac{2}{2-\eta}(2\rho)^{2-\eta}\sqrt{1+\mathcal{L}^{2}}\sigma(S^{n-1}),
z{\cal H}_{k-1}^{(s+1,t)}(z)={\cal H}_{k}^{(s+1,t)}(z)+(q_{k}^{(s,t)}+e_{k}^{(s,t)}){\cal H}_{k-1}^{(s+1,t)}(z)+q_{k}^{(s,t)}e_{k-1}^{(s,t)}{\cal H}_{k-2}^{(s+1,t)}(z).
\sqrt{a_-(t)}\Vert u(t;x,f_1) - u(t; x,f_2)\Vert \le \C\int_0^{\infty}s\sqrt{a_-(s)}\Vert f_1-f_2\Vert \, ds = C{\Vert f_1 - f_2 \Vert}_Y, \ \ \forall t\ge 0.
\begin{aligned}\int_{-\infty}\phi_1^{-2}(x)dx=\infty,\ \ \int^{+\infty}\phi_1^{-2}(x)dx<\infty, &\int_{-\infty}\phi_2^{-2}(x)dx<\infty,\ \ \int^{+\infty}\phi_2^{-2}(x)dx=\infty, &\int_{-\infty}^{+\infty}\phi_3^{-2}(x)dx<\infty.\end{aligned}
\pi_{5}(a(2j,k)) =\pi_{5}\left(\sum_{i=1}^{\infty}a(2j-1,i)m(6i,k+i)\right) \geq\min_{i\geq1}\left(\pi_{5}(a(2j-1,i))+\pi_{5}(m(6i,k+i))\right) \geq\min_{i\geq1}\left(j+\left\lfloor\dfrac{5i-5}{2}\right\rfloor+\left\lfloor\dfrac{5k-i-1}{2}\right\rfloor\right).
Z_N = \frac{\prod_{j,k = 1}^{N}a(\lambda_j,\nu_k)}{v_N(\{\lambda\})}\det \left[\lambda_j^{k - 1}- (\lambda_j + 1)^{k - 1}\prod_{l = 1}^N\frac{b(\lambda_j,\nu_l)}{a(\lambda_j,\nu_l)}\right]_{j,k=1,\ldots,N}.
{\cal L}={\cal R}\sqrt{{\cal - G}}= - ( k^{i j}- h g^{i j}) \sqrt{g}\frac{d g_{i j}}{d s}- ( N{\cal H}_{0}+ N^{i}{\cal H}_{i}) - 2 \frac{d}{d s}( h \sqrt{g}) - \nabla_{i}\varphi^{i}.
|H[b]-h|=\lim_{n\to\infty}\left|H[b]-H[b_n] \right|\leq \left(\sup_{n}||b_n||^{5/3}_{L^{2}(\mathbb{R}^{3})}+||b||^{5/3}_{L^{2}(\mathbb{R}^{3})}\right)\varepsilon^{1/2}.
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{m^1}{n^2}}= \exp\left\{\sum_{n=1}^{\infty}\left( \frac{n}{2}+ \frac{n^2}{2}\right) \frac{z^n}{n^2}\right\}
b \varphi (a^0 ,a^1 ,\ldots ,a^{q+1}) := \sum_{j=0}^q (-1)^j (a^0 ,\ldots ,a^j a^{j+1}, \ldots , a^{q+1}) &+ (-1)^{q+1}(a^{q+1}a^0 ,a^1 ,\ldots ,a^{q}) , B \varphi (a_0 , \ldots , a_{q-1}) := \sum_{j=0}^{n-1}(-1)^{(n-1)j}( 1 , a_j , \ldots , a_{q-1}, a_0 , \ldots , a_{j-1}) .
e^{- F}= \int \prod_{k}D \mu [ M_{k}] e^{- N^{2}S [ M ]}= \int \prod_{k}D \mu [ M_{k}] e^{- N^{2}( S [ M ] - S_{0}[ M ] )}e^{- N^{2}S_{0}[ M ]}= e^{- F_{0}}\Big < e^{- N^{2}( S - S_{0})}\Big >_{S_{0}}\; .
p(x_j|y,f_{1:j-1},x_{1:j-1})=\frac{p(x_j,y|f_{1:j-1},x_{1:j-1})}{p(y|f_{1:j-1},x_{1:j-1})}=\frac{p(x_j,y|x_{1:j-1})}{p(y|x_{1:j-1})}=p(x_j|y,x_{1:j-1}).
\begin{cases}M_{C_j(i)}^\mu (t) :=\sum_i \int_{s}^{s+t}\frac{1}{|V(C_j(i),\phi)|}\int_{B_{3r_j/2}(x)}\tilde{g}^{\mu}_i (\tau,y,u;\omega) \phi_{B_{r_j}(x)}^2(y)\;dy dw^{i}_{\tau}, H_{C_j(i)}^\mu (t) := \frac{1}{|V(C_j(i),\phi)|}\int_{B_{3r_j/2}(x)}h^\mu (t +s,y;\omega) \phi_{B_{r_j}(x)}^2(y) \;dy,\end{cases}
(\widehat{X}_{t_{1}}\cdots \widehat{X}_{t_{r}}G(f))(x) =\left(\widehat{X}_{l-r+1}\cdots \widehat{X}_{l}\,(\widehat{T}_{l-r}\cdots \widehat{T}_{\sigma_{x}(t_{1})})\cdots (\widehat{T}_{l-1}\cdots \widehat{T}_{\sigma_{x}(t_{r})})\widehat{T}_{w_{x}}(f)\right)(w_{x}x),
\left|f\left(\sum_{j=1}^m x_j\right)\right| \leqslant \left\|\left(g_j(x_j)\right)_{j\in B}\right\|_{\ell_{p_{\zeta}}}+ \theta\Big\|\Big(f_q(\sum_{j\in E_q}x_j)\Big)_{q\in A_2}\Big\|_{\ell_{p_{\zeta}}}.
| A ( r ) |^{2}= \frac{\Gamma ( \frac{D}{2})}{\pi^{\frac{D}{2}+ 1}2^{D + 1 + 2 l}}\frac{| \Gamma ( D / 2 + l + i r ) |^{2}| \Gamma ( i r ) |^{2}}{| \Gamma ( 2 i r ) |^{2}}\: .