harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta_{\perp}\kappa_{1}= \kappa_{3}\kappa_{2}\Psi_{3}+ 2 \kappa_{2}\Psi_{2}{}^{\prime}+ \kappa_{2}^{\prime}\Psi_{2}+ \Psi_{1}{}^{\prime \prime}- \left( \kappa_{1}^{2}+ \kappa_{2}{}^{2}\right) \Psi_{1}.
\tilde{L}=-\frac{1}{8\pi}\left( \frac{4\pi}{g^{2}\bar{\tau}\tau}\tilde{F}_{\mu \nu}\tilde{F}^{\mu \nu}-\frac{i}{\bar{\tau}\tau}\frac{\theta}{2\pi}\frac{1}{2}\epsilon_{\mu \nu \rho \sigma}\tilde{F}^{\mu \nu}\tilde{F}^{\rho \sigma}\right).
E(t)=\begin{bmatrix}C(t) \Lambda^{-1/2}S(t) -\Lambda^{1/2}S(t) C(t) \end{bmatrix},E_h(t)=\begin{bmatrix}C_h(t) \Lambda_h^{-1/2}S_h(t) -\Lambda_h^{1/2}S_h(t) C_h(t) \end{bmatrix},
\forall\bigl(\alpha,\alpha'\bigr)\in\mathbb{R}_+^N\times\mathbb{R}_+^N, \Biggl(\bigoplus_{i=1}^N \bigl(\alpha_i+\alpha'_i\bigr) S_{\theta_i}\Biggr)=\Biggl(\bigoplus_{i=1}^N \alpha_iS_{\theta_i}\Biggr)\oplus\Biggl(\bigoplus_{i=1}^N\alpha'_i S_{\theta_i}\Biggr).
\delta_L \psi(x) = -{i \over 2}\epsilon_{ab}\sigma^{ab}\psi, \quad\delta_L{e^{a}}_{\mu}(x) = \epsilon{^a}_b e{^b}_{\mu}+{\kappa^{4}\over 4}\varepsilon^{abcd}\bar{\psi}{^j}\gamma_5 \gamma_d \psi{^j}(\partial_{\mu}\epsilon_{bc}).
\delta \left( \begin{matrix}{{g_{i j}}}\\{{\pi^{i j}}}\\ \end{matrix}\right) = \{\left( \begin{matrix}{{g_{i j}}}\\{{\pi^{i j}}}\\ \end{matrix}\right) , T_{\mu}\}F^{\mu}, \: \: \delta N^{\mu}= \dot{F}^{\mu}- U_{\beta \gamma}^{\mu}N^{\beta}F^{\gamma}.
\sum_a \big( |v_s^a|^2 + | \nabla v^a_s |^2 \big) (p, t) \leq e^{C_9'' ( t - \frac{T_3}{2})}\sup_{q \in M}\sum_a \bigg( |v_s^a (q , \frac{T_3}{2}) |^2 + | \nabla v^a_s (q , \frac{T_3}{2}) |^2 \bigg) = O(s),
Q_{\rho}^{\lambda}=\begin{cases}B_{\rho}(x_0)\times\left(t_0-\theta_{\lambda}^{-1}\rho^2,t_0\right), \qquad \theta_{\lambda}\geq 1 begin{align*}4mm] \theta_{\lambda}^{1/2}B_{\rho}(x_0)\times\left(t_0-\rho^2,t_0\right), 0<\theta_{\lambda}< 1, begin{align*}1mm]\end{cases}
\begin{array}{l}\displaystyle\int_{0}^{T}\int_{\mathbb{T}}s|w|^2e^{2s\varphi}dxdt+\int_{\mathbb{T}}s[|w|^2e^{2s\varphi}]_{t=0}dx+\int_{\mathbb{T}}s[|w|^2e^{2s\varphi}]_{t=T}dx \leq C_1\displaystyle\left(\int_{0}^{T}\int_{\mathbb{T}}|w_t+\frac{a}{b}w_x|^2e^{2s\varphi}dxdt+\int_{0}^{T}\int_{\omega}s|w|^2e^{2s\varphi}dxdt\right).\end{array}
\begin{array}{l c r}{\partial_{\sigma}^{2}X^{9}+ \sum_{\nu = 1}^{8}{\frac{1}{2}}( \{X^{\nu + ( - )^{\nu + 1}}, \{X^{\nu + ( - )^{\nu + 1}}, X^{9}\}\}+ \{X^{\nu}, \{X^{\nu}, X^{9}\}\}) = 0 ,}\\ \end{array}
I_-^\mu \big\{(1+x)^{1-\mu}\hat h_0(x)\big\}= \sum_{l=0}^{N-1}\frac{\Gamma(l+2-\mu)}{(l+1)!}\tilde \varrho_{l0}\,(1+x) P_l^{(0,1)}(x), & I_-^\mu \big\{(1+x)^{-\mu}(\hat h_0(x)-1)\big\}= \sum_{l=0}^{N-2}\frac{\Gamma(l+2-\mu)}{(l+1)!}\hat \varrho_{l+1,0}\,(1+x)P_l^{(0,1)}(x).
\check{u}_\alpha(x) = d_\alpha^{\frac{n-2}{2}}u_\alpha\left(\exp_{x_{1,\alpha}}(d_\alpha x) \right), &\check{h}_\alpha(x) = h_\alpha\left(\exp_{x_{1,\alpha}}(d_\alpha x) \right), &\check{g}_\alpha(x) = \left(\exp^\star_{x_{1,\alpha}}g \right) (d_\alpha x),
A_i\coloneqq A_{i,1}\bigtimes_{j=1}^{n-1}(t_im_{2(n+j-1)}m_{2(n+j)-1},t_im_{2(n+j)-1}m_{2(n+j)})A_{i,-j}(t_im_{-2(j-1)}m_{-2j+1},t_im_{-2j+1}m_{-2j})A_{i,j+1}.
\tilde{E}_{o u t}= E_{o u t}+ \tilde{E}_{c l a s s}^{o u t}= \frac{1}{2 a \mu}[ - 0 . 0 0 8 5 + \ln a ( \frac{- 1}{1 2 8}+ \frac{1}{8 \pi}) ] + \frac{h^{o u t}}{a}.
V_t^i := e^{\int_0^t\Lambda\big (s,\tilde \xi^{i,N}_{s},u^{S^N(\tilde{\xi})}_{s}(\tilde \xi^{i,N}_{s})\big )ds}\textrm{and}\tilde V_t^i := e^{\int_0^t\Lambda\big (r(s),\tilde \xi^{i,N}_{r(s)},\tilde v_{r(s)}(\tilde \xi^{i,N}_{r(s)})\big )ds}\ ,
f ( \vec{x}, \vec{p}, t ) = \int_{\Delta V}{\frac{d^{3}x^{\prime}\, d^{3}p^{\prime}}{( 2 \pi )^{3}}}\, W ( \vec{x}- \vec{x}^{\prime}, \vec{p}- \vec{p}^{\prime}, t ) \cdot g ( \vec{x}^{\prime}, \vec{p}^{\prime}) ,
\mathcal{H}W_{l}(s;yl^c) = l^{\frac{c}{2}}\sum_{\substack{\mu_l\in{}_l\mathfrak{X}\setminus\{1\}, c=-2a(\mu)}}\mu_l(y^{-1})\epsilon(\frac{1}{2},\mu_l^{-1}\pi_{0,l})[\mathcal{M}W_l(s;\cdot)](\mu_l) &+ l^{\frac{c}{2}}B_{\pi_{0,l}}(l^{-c})[\mathcal{M}W_l(s;\cdot)](1).
\Lambda_p = \lim_{j \to \infty}\lambda_{1,p}(\gamma_j)\le \liminf_{j \to \infty}\frac{\int_0^1 \frac{| v' |^p F_j^p}{L_h(\gamma_j)^{p-1}}\,dt}{\int_0^1 | v |^p L_h(\gamma_j) \,dt}\le \liminf_{j \to \infty}\frac{\int_0^b \frac{| w' |^p F_j^p}{L_h(\gamma_j)^{p-1}}\,dt}{\int_0^b | w |^p L_h(\gamma_j) \,dt}
\begin{aligned}\left |\frac{\partial^2}{\partial a^2}\zeta'_B(0;a,1,1)-2\left(\frac 1{12}-\zeta'_R(-1)\right)\frac 1{a^3}-\sum_{k=2}^{N-1}\left(1-\frac 1 k\right)B_{2k}\zeta_R(2k-1)a^{2k-3}\right| \leq \left(1-\frac 1 N\right)|B_{2N}\zeta_R(2N-1)|a^{2N-3};\end{aligned}
Z = \sum_{N = 1}^{\infty}e^{- \beta \frac{N \hbar}{2 R}}\int \left( \prod_{k , \lambda}d \Pi_{\lambda}^{k}\right) \left( \prod_{l , \mu}d \theta_{\mu}^{l}\right) < \Pi_{\lambda}^{k}\theta_{\mu}^{a}| e^{- \beta H \left( \Pi_{\lambda}^{k}, Y_{\lambda}^{k}, \theta_{\mu}^{a}\right)}| \Pi_{\lambda}^{k}\theta_{\mu}^{a}>
p^{\vert V \Gamma_0 \vert}( T_{w_1}^{(1)}P_{w_1}^\perp ) \ldots ( T_{w_r}^{(1)}P_{w_r}^\perp ) P_{V\Gamma_0}( T_{w_{s+1}}^{(1)}P_{w_{s+1}}) \ldots ( T_{w_{d}}^{(1)}P_{w_{d}}),
I=\begin{pmatrix}1 0 0 1 \end{pmatrix}, \;\xi=\begin{pmatrix}-1 0 0 -1 \end{pmatrix}, \; \eta=\begin{pmatrix}i 0 0 -i \end{pmatrix}, \; \zeta=\begin{pmatrix}0 i i 0 \end{pmatrix},
{W}_{\alpha}= \frac{1}{8}\bar{D}^{2}\left({e}^{- V}\! D_{\alpha}{e}^{V}\right) = i \left( \lambda_{\alpha}+ i \theta_{\alpha}D - \theta^{\beta}G_{\alpha \beta}- i \theta^{2}{\cal D}_{\alpha \dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}\right) .
\psi_{n,v,\varphi}:(-1,\infty) \rightarrow \mathsf{H}^{1}(\mathbb{R}^n\setminus \{0\};\mathbb{C}^{n-1}); \psi_{n,v,\varphi}(\lambda):=\frac{K_n\varphi}{1+\lambda -v}.
\Pi_{\mu \nu}^{a b \ ( 1 )}( p , M_{i}) = \lim_{M_{i}\rightarrow \infty}\frac{g_{0}^{2}c_{v}}{1 6 \pi^{2}}\left\{\frac{5}{3}\sum_{i = 1}^{N}c_{i}\log \left( \frac{M_{i}^{2}}{\kappa^{2}}\right) + \Pi_{f i n}^{( 1 )}\right\}( p^{2}g_{\mu \nu}- p_{\mu}p_{\nu}) \delta^{a b}
\operatorname{fix}(y,M^G) \le \sum_{2l_{i,-i}+l_{-1}+l_1=m}p_{l_{i,-i}}(E_{i,-i})p_{l_{-1}}(E_{-1}) p_{l_1}(E_1) &<\sum_{2l_{i,-i}+l_{-1}+l_1=m}cq^{l_{i,-i}(\frac{kb}{2}-3l_{i,-i})+l_{-1}(\frac{kb}{4}-\frac{3}{2}l_{-1}+\frac{1}{2})+l_{1}(\frac{kb}{4}+s-\frac{3}{2}l_{1}+\frac{1}{2})}&< q^{\frac{1}{8}m(2n-3m)+c'm}
\chi_{\dot{q}\dot{p}}^{1}= \chi^{1}\delta_{\dot{q}\dot{p}}+ \chi^{1 i j}\tilde{\gamma}_{\dot{q}\dot{p}}^{i j}+ \chi_{\dot{q}\dot{p}}^{1 i j k l}, \ \chi_{q p}^{2}= \chi^{2}\delta_{q p}+ \chi^{2 i j}\gamma_{q p}^{i j}+ \chi^{2 i j k l}\gamma_{q p}^{i j k l}.
J_4 \to 0, &J_6 \to 0, &-J_5 \to J_k = \int_{c-i\infty}^{c+i\infty}\frac{\Gamma(L-s)\Gamma(a - L +s)}{2\pi i \Gamma(b-L+s)}x^{s-L}ds, &I_{M,N}\to I_1 = \int_{c-i\infty}^{c+i\infty}\frac{\Gamma(-s)\Gamma(a + s)}{2\pi i \Gamma(b+s)}x^s ds &= \frac{\Gamma(a)}{\Gamma(b)}F[a,b;-x],
= \exp\left\{\sum_{n=1}^{\infty}\left( \frac{-y (y^2 + 4 y + 1) (y^n - 1) + 3 n (y^2 -1) y^{n+1}- 3 n^2 ((y-1)^2 y^{n+1}) + n^3 (y-1)^3 y^{n+1}}{(1-y)^4}\right) \frac{z^n}{n^4}\right\}
({\gamma^{\mu}})_{c b}={( \gamma^{\mu})^{a}}_{b}\epsilon_{a c}= \left( \begin{array}{c c c}{}&{\vline}&{}\\{0}&{\vline}&{{\sigma^{\mu{\dot{L}}}}_{M}}\\{}&{\vline}&{}\\ \hline{}&{\vline}&{}\\{{\sigma^{\mu \dot{M}}}_{L}}&{\vline}&{0}\\{}&{\vline}&{}\\ \end{array}\right) ,
(\alpha-3)u_{N-1}'(s)-(2s+1)u_{N-1}''(s)+2K_{N}(s) &=(\alpha-3)u_{N-1}'(s)+(s-1)u_{N-1}''(s)-3su_{N-1}''(s)+2K_{N}(s) &=-5u_{N-1}'(s)-3su_{N-1}''(s)-\xi_{N}(s)\ ,
\begin{aligned}\liminf_{\alpha\to 0}\left(-\frac{|\tilde{x}-\tilde{y}|^4}{\varepsilon^4}+2\alpha \max_{x\in K}|x|^2-\alpha(|\tilde{x}|^2+|\tilde{y}|^2)\right) \leq \liminf_{\alpha\to 0}\left(-\frac{|\tilde{x}-\tilde{y}|^4}{\varepsilon^4}+2\alpha \max_{x\in K}|x|^2\right)<0,\end{aligned}
K^{\prime}\, = \, K_{1 2}^{\prime}\, ( G_{3}^{0})^{- 1}\, + \, K_{2 3}^{\prime}\, ( G_{1}^{0})^{- 1}\, + \, K_{3 1}^{\prime}\, ( G_{2}^{0})^{- 1}\, + \, K_{1 2 3}^{\prime}\, ,
\Delta(D_{im}) =\sum_{\substack{k<5 k<l}}a_{ik}a_{ml}\otimes D_{kl}-q^{-1}\sum_{\substack{k<5 l<k}}a_{ik}a_{ml}\otimes D_{lk}& + (a_{i5}a_{m6}+q^{-1}a_{i6}a_{m5}) \otimes D_{56}& + (1+q^{-2})\sum_{5\leq k \leq 6}a_{ik}a_{mk}\otimes D_{kk}+q^{-1}\sum_{\substack{k\geq5 l<5}}a_{ik}a_{ml}\otimes D_{lk}\,.
J_{i,k}^{(s)}=\frac{1}{[2\Gamma(1/s)]^{k+1}}\sum_{j=0}^k\binom{k}{j}\,[\Gamma(1/s)]^{k-j}\sum_{m=0}^\infty c_{m,j}\,\Gamma\left(m+\frac{i+j+1}{s}\right).
{\cal P}_{n}^{\pm}(\beta )=N_{\pm}^{2}(\beta )\left[|T_{n}(\beta )|^{2}+|T_{n}(-\beta )|^{2}{\pm}\left(T_{n}^{*}(\beta )T_{n}(-\beta)+T_{n}(\beta )T_{n}^{*}(-\beta )\right)\right].
b_{14}^{(k)}\pm b_{23}^{(k)}= -\big{(}b_{14}^{(k-1)}\pm b_{23}^{(k-1)}\big{)}(\cos\phi_{k}\sin\psi_{k}\pm\sin\phi_{k}\cos\psi_{k}) &= -\big{(}b_{14}^{(k-1)}\pm b_{23}^{(k-1)}\big{)}\sin(\psi_{k}\pm\phi_{k}), k\geq 1.
C(x_k \mid x_V, p_{[k+1 ..\ell]}) \stackrel{(a)}{\geq}C(x_{V \cup \{k\}}, p_{[k+1..\ell]}) - C(x_{V}, p_{[k+1..\ell]}) - c \log n \stackrel{(b)}{\geq}A(x_{V \cup \{k\}}, p_{[k+1..\ell]}) - A(x_{V}, p_{[k+1..\ell]}) - 8\gamma_{k+1}- c \log n \stackrel{(c)}{\geq}n_k.
t = E{\int}{d r \, \Bigl ( 1 +{\frac{4 M}{r}}-{\frac{4 q^{2}}{r^{2}}}\Bigr ) \left({E^{2}\Bigl ( 1 +{\frac{4 M}{r}}-{\frac{4 q^{2}}{r^{2}}}\Bigr ) -{\frac{L^{2}}{r^{2}}}- m^{2}\sqrt{1 +{\frac{4 M}{r}}-{\frac{4 q^{2}}{r^{2}}}}}\right)^{- 1 / 2}}\ .
\left\langle e_{k},\mu_{\Upsilon}\left( \mathcal{X}\right)e_{q}\right\rangle_{\mathbb{R}^{d}}:=\frac{1}{4}\left( \mu_{e_{k}+e_{q}}\left( \mathcal{X}\right) -\mu_{e_{k}-e_{q}}\left( \mathcal{X}\right) \right) \ .
\mathcal{D}_{-}(\lambda )=\frac{(\lambda^{2}/q-q/\lambda^{2})}{(\lambda^{2}-1/\lambda^{2})}\mathcal{A}_{-}(\lambda^{-1})+\frac{(q-1/q)}{(\lambda^{2}-1/\lambda^{2})}\mathcal{A}_{-}(\lambda ),
\varphi\circ\bar\partial_E\circ\varphi^{-1}= d\bar z\left[\frac{\partial}{\partial\bar z}+\begin{pmatrix}-\bar Z\log u_1 -\bar Q/u_1u_2 0 0 \bar Z\log u_2 0 0 0 0\end{pmatrix}\right]
\begin{array}{r c l}\hat{\phi}_{(I)}(t,\vec{x}\,) = e^{i(t-t_0)\hat{H}_0}\,\hat{\phi}(t_0,\vec{x}\,)\,e^{-i(t-t_0)\hat{H}_0}\ , \hat{\pi}_{(I)}(t,\vec{x}\,) = e^{i(t-t_0)\hat{H}_0}\,\hat{\pi}(t_0,\vec{x}\,)\,e^{-i(t-t_0)\hat{H}_0}\ ,\end{array}
\sum_{k=\overline{k}}^{R}e^{-2\beta k}I_k \geq\sum_{k=\overline{k}}^{R}e^{-2\beta k}\widetilde{c}\ss e^{-\alpha (R-k)}Z_k= \widetilde{c}\ss e^{-2\beta R}\sum_{k=\overline{k}}^R Z_k > \widetilde{c}\ss e^{-2\beta R}\frac{\widetilde{\eta}}{2}(R-\overline{k}+1).
\frac{8 \pi^{2 \mu}a^{2}( \alpha - 1 ) a ( \beta ) Q_{\mu \nu}}{\Delta ( \alpha - 1 )^{2}\mu \Gamma ( \mu - 1 )}\left[ 1 + \Delta \left( B ( \beta ) - B ( \alpha - 1 ) - \frac{1}{( \alpha - 1 )}\right) \right]
c_{r , s}^{( l )}= q^{\frac{- \hat{c}}{2 4}}\sum_{m = 0}^{N / 2}C_{2 m}^{l - 1}\left\{\sum_{j \in{\bf Z}, m \equiv m_{r , s}( j )}q^{\alpha_{r , s}( j )}- \sum_{j \in{\bf Z}, m \equiv m_{r , - s}( j )}q^{\alpha_{r , - s}( j )}\right\}
y^2 + \sum_{i=1}^{d-1}(y^2)^{(1,2,3)}+ (y^2)^{(1,3,2)}= (1, 1, 0, \dots, 0) + (0, d-1, d-1, 0, \dots, 0) + (1,0,1,0, \dots, 0) = (2,0,\dots,0) =: 2e^1 \in (\mathbb{Z}_d)^\alpha \cap M
0=\frac{\partial \phi}{\partial t}\frac{\partial^2\phi}{\partial x\partial t}+\frac{\partial \phi}{\partial x}\frac{\partial^2\phi}{\partial x^2}=\frac{\partial \phi}{\partial t}\left(\frac{\partial^2\phi}{\partial x\partial t}+p_1\frac{\partial \phi}{\partial x}\right),
X(t) = \begin{cases}\eta(t) (-r \le t \le 0), \displaystyle \eta(0) + \int_0^t A_0 (s, X_s) \, \mbox{d}s + \int_0^t A (s, X_s) \, \mbox{d}W(s) (0 \le t \le T), \end{cases}
\frac{\partial \breve{C}_{i}}{\partial \lambda_{i}}=\frac{-(\mu_{i}+\tilde{R}_{i})}{\sqrt{4\pi M}\sigma_{n}^{2}}e^{-\frac{x_{i}^{2}}{2}}+\frac{\delta_{i}}{\sqrt{4\pi M\sigma_{n}^{2}(\sigma_{n}^{2}+2E(|s_{i}h_{i}^{(pr)}|^{2})}}e^{-\frac{y_{i}^{2}}{2}},
\sum_{k=n_0}^{n-1}\mu(I_k(c_j)\setminus I_{k+2}(c_j))\log{\frac{1}{|I_{k+2}(c_j)|}}\;\leq\; C_1\sum_{k=n_0}^{n-1}\frac{(k+2)}{q_{k+1}}\,.
{\sum_{k}}\frac{(-1)^{k}(L-k)\}{R(l_{1},l_{2},l_{3};k)}=(-1)^{L}{\sum_{k}}\frac{(-1)^{L-k}(L-k)\}{R(l_{1},l_{2},l_{3},L-k)}=(-1)^{L}\Sigma_{1}& =L{\sum_{k}}\frac{(-1)^{k}\}{R(l_{1},l_{2},l_{3};k)}-{\sum_{k}}\frac{(-1)^{k}\k}{R(l_{1},l_{2},l_{3};k)}=L\Sigma_{0}-\Sigma_{1}
\lambda_{1}= ( \frac{\theta_{1 1 0 0}\theta_{1 0 0 0}}{\theta_{0 1 0 0}\theta_{0 0 0 0}})^{2}, \; \lambda_{2}= ( \frac{\theta_{1 0 0 1}\theta_{1 1 0 0}}{\theta_{0 0 0 1}\theta_{0 1 0 0}})^{2}, \; \lambda_{3}= ( \frac{\theta_{1 0 0 1}\theta_{1 0 0 0}}{\theta_{0 0 0 1}\theta_{0 0 0 0}})^{2},
- i \bar{\kappa}_{( \alpha )}\gamma^{\mu}\kappa_{( \beta )}\partial_{\mu}= - i \tilde{\mathcal{C}}_{\alpha \gamma}\Gamma_{s}( T^{I})^{\gamma}{}_{\beta}\Gamma_{\mathrm{A d j}}( u^{- 1})^{a}{}_{I}e_{a}= - i \tilde{\mathcal{C}}_{\alpha \gamma}\Gamma_{s}( T^{I})^{\gamma}{}_{\beta}k_{( I )},
\Phi(R)=\oint_{\partial B_{R}(r_{0})}E_{0}\cdot dS=\oint_{\partial B_{R}(r_{0})}E_{B_{R}}\cdot dS+\oint_{\partial B_{R}(r_{0})}E_{C}\cdot dS=\oint_{\partial B_{R}(r_{0})}E_{B_{R}}\cdot dS
\begin{aligned}\mathbb{A}: =(24 c_1^2-12 c_4) D_S^5\{\mathrm{I}-2 D_{-2}U D_S^2 \}+(\frac{14}{3}c_2^2-4 c_6) D_S^3+(4 c_6-\frac{16}{3}c_2^2) \{D_S^3 U+D_S U D_S^2\}&+12 (c_2 c_3-c_7) D_S+(24 c_7-16 c_2 c_3) D_S U-6 c_3^2 D_S^{-1}.\end{aligned}
f^{\left(p\right)}_k+\sum_{l=1}^{p}\binom{p}{l}\frac{\left[\mathbf{D}\right]_{k,k}\underline{m}^{\left(l\right)}f^{\left(p-l\right)}_k}{1+\left[\mathbf{D}\right]_{k,k}\underline{m}\left(0\right)}=0
U' = U \cup \{q_i^{-1}\hat{z}_J, q_{s}\hat{z}_J, q_s^{-1}\hat{z}_1, q_i \hat{z}_1 \}W' = W \cup \{\hat{z}_J, q_i^{-1}q_{s}^{-1}\hat{z}_J, \hat{z}_1, q_i q_s \hat{z}_1 \},
\dfrac{M_{2}(\mathcal{NC}(G))}{|e(\mathcal{NC}(G))|}- \dfrac{M_{1}(\mathcal{NC}(G))}{|v( \mathcal{NC}(G) )|}~~~~~=\dfrac{q^{11}(q-5)+q^{8}(14q-35)+q^{5}(15q-12)+q^{4}(5q^{6}-4)+q^{3}(18q^{4}-5)+16q^{2}-10q+2}{(q^{5}(q^{2}-2q-2)+q^{2}(5q^{2}-4)+q^{3}+1)(q^{3}-q-1)}~~~~~:=\dfrac{f(q)}{g(q)}.
\left\vert \left( \nu_{\mathsf{k}}-a_{\mathsf{m}^{\ast},\mathsf{m}^{\ast}}\right) \right\vert \leqslant \sum_{\mathsf{n=1},\mathsf{n\neq m}^{\ast}}^{+\infty}r_{\mathsf{n,m}^{\ast}}=\left\vert a_{\mathsf{m}^{\ast},\mathsf{m}^{\ast}}\right\vert
\frac{\partial}{\partial s}= (x_1\dot{w}\ldots ,x_n \dot{w}) = \dot{w}\cdot (x_1, \ldots ,x_n) = (\dot{r}e^{i\phi_E}+r\,i\dot{\phi}_E e^{i\phi_E})\cdot (x_1, \ldots ,x_n) = (\dot{r}+r\,i\dot{\phi}_E )\cdot e^{i\phi_E}\cdot (x_1, \ldots ,x_n).
\left( \frac{2}{\lambda e^{t}+1}\right)^{\alpha}e^{xt}=\sum_{n=0}^{\infty}\mathcal{E}_{n}^{\left( \alpha\right)}\left( x;\lambda\right)\frac{t^{n}}{n!},\left( \left\vert t\right\vert <\left\vert\log\left( -\lambda\right) \right\vert \right) ,
\Lambda_{\pm}(1)=\Lambda (1) \pm \frac{(4\kappa)^{3/2}(1-k)^{\kappa-\frac{1}{2}}}{(2\pi)^{1/2}2^{\kappa}}\bigg[1+(1-k)\{\kappa(\frac{1}{2}-\ln 2)+\frac{1}{4}\}+O\bigg(\frac{1}{\kappa}\bigg)\bigg]
F( T)=-\frac{\xi_0^2x_0^4t}{8\pi a}\sum_{l=1}^\infty\left[ \frac 1{\nu (x_0^2+l^2)^2}+\sum_{n=1}^\infty\frac{2\nu^5}{(x_0^2+l^2+n^2t^2)^2(\nu^2+n^2t^2)^3}\right],
x_nx_1 = z, ~~~ x_{n-1}x_2 = -z, ~~~ \cdots ~~~ x_ix_{n-i+1}= (-1)^{n-i+2}z, ~~~\cdots ~~~x_2x_{n-1}= (-1)^nz, x_nx_2 = z, ~~~ x_{n-1}x_3 = -z, ~~~ \cdots ~~~ x_ix_{n-i+2}= (-1)^{n-i+2}z, ~~~\cdots ~~~x_2x_{n}= (-1)^nz, x_1x_n = (-1)^{n+1}z;
\int_{0}^{(x-2k t)/(2\sqrt t)}e^{iu^2}du=\left(\int_{0}^{(x-2k t)/(2\sqrt t)}e^{iu^2}du\right)\Bigl(\mu_-((x-2k t)/(2\sqrt t))+\mu_0(\cdot)+\mu_+(\cdot)\Bigr)\,.
\sum_{j = 0}^k{( - 1)^j G_{2j}^2}= \frac{5}{6}G_0^2 + \frac{1}{6}G_1^2 - \frac{1}{6}G_2^2 + \frac{{( - 1)^k}}{6}G_{2k + 1}^2 &+ \frac{{( - 1)^k}}{2}G_{2k + 2}^2 - \frac{{( - 1)^k}}{6}G_{2k + 3}^2\,,
L_{L1}={K \, v^2 \over 2\, R}~\left[1+\sum_{L = 1}^\infty \sum_{q=0}^{L-1}\, d_{LL}(0,q)~K^{q}~N^{-q}\left({N\,v^2\over R^2 M^9_p V_T~b^{7-p}}\right)^{L}~(1+S_{L1})\right]
G_{2k}(\tau) = 2\zeta(2k) + \frac{2(-1)^k(2\pi)^{2k}}{(2k-1)!}\sum_{m=1}^\infty\sum_{d=1}^\infty d^{2k-1}q^{md}&= 2\zeta(2k) + \frac{2(-1)^k(2\pi)^{2k}}{(2k-1)!}\sum_{d=1}^\infty \Big(\sum_{k \mid d}k^{2k-1}\Big) q^d
\widetilde{L}^{-1}M\widetilde{U}^{-1}=\left[ \begin{array}{cc}C^{T}C -C^{T}C+C^{T}\widetilde{Q}C^{T}C-\widetilde{Q}^{T}C -C^{T}C+C^{T}\widetilde{Q}+\widetilde{Q}^{T}C\end{array}\right].
|C^{\varepsilon}_{Q_{\vec{P}},\vec{n}_{1},\vec{n}_{2},\vec{n}_{3}}|\lesssim\prod_{j=1}^{3}\frac{1}{(1+|\vec{n}_{j}|)^{M}}\cdot C^{\varepsilon}_{|I_{\widetilde{P'}}|}
\Phi(e(b)) = \Phi((\hdots, G^2(b), G(b),\stackrel{\stackrel{0}{\downarrow}}{b}, f(b), F^2(b) , \hdots))= (\hdots, G^2(b), G(b), b, \stackrel{\stackrel{0}{\downarrow}}{f( b)}, F^2(b), \hdots ) = e( f(b))
J_{a}\propto \left| \begin{array}{c c c c c}{{m_{e}}}&{{m_{d}}}&{{m_{c}}}&{{m_{b}}}&{{m_{a}}}\\{{\epsilon_{e}- \epsilon_{a}}}&{{v_{d e}}}&{{v_{c e}}}&{{0}}&{{0}}\\{{0}}&{{\epsilon_{d}- \epsilon_{a}}}&{{0}}&{{0}}&{{v_{a d}}}\\{{0}}&{{0}}&{{\epsilon_{c}- \epsilon_{a}}}&{{v_{b c}}}&{{v_{a c}}}\\{{0}}&{{0}}&{{0}}&{{\epsilon_{c}- \epsilon_{a}}}&{{v_{a b}}}\\ \end{array}\right| \, .
\left\|\left(\mathbb{I}- \frac{1}{N}\mathbf{1}\mathbf{1}^T \right)\zeta^{k+1}(\ell)\right\|\le \left\|\mathbb{I}- \frac{1}{N}\mathbf{1}\mathbf{1}^T \right\|\,\|\zeta^{k+1}(\ell)\|=\|\zeta^{k+1}(\ell)\|.
\mathcal{L}= \beta^2 \sqrt{1+\frac{m^2}{\beta^2}A_\mu A^\mu}-\frac{1}{4}B_{\mu \nu}B^{\mu \nu}- \frac{\chi\theta}{2}\varepsilon^{\mu \nu \rho \lambda}\partial_\mu B_{\nu \rho}A_\lambda\; .
\mathrm{Sub}_\sigma\stackrel{I_{x_0}}{\to}I_{x_0}(\mathrm{Sub}_\sigma)=f_{\sigma,x_0}(\mathcal O(C)\otimes \mathrm{Sh}(\mathrm H_C^{\mathrm{dR}}))\stackrel{(\underline{f}_{\sigma,x_0})^{-1}}{\to}\mathcal O(C)\otimes \mathrm{Sh}(\mathrm H_C^{\mathrm{dR}}),
e^{i \Gamma_{C}[ \vec{\varphi}]}= \langle T_{C}\mathrm{e x p}\left( i \int_{C}\left[{\cal L}^{\prime}( \vec{\phi}( x ) ) - \frac{\delta \Gamma_{C}[ \vec{\varphi}]}{\delta \vec{\varphi}( x )}\cdot \{\vec{\phi}( x ) - \vec{\varphi}( x ) \}\right] \right) \rangle .
\hat\ell := L_\ell^- (p - \ell^) = 2 L_\ell^- p = 2 (\ell + \lceil (p-\ell)/2 \rceil) - p = 2 \ell + 2 \lceil (p-\ell)/2 \rceil - p = \begin{cases}\ell p-\ell , \ell+1 \end{cases}
\frac{\overline{{( \Delta N )^{2}}}}{\overline{{N}}^{2}}\sim \beta \sim \frac{M_{s}}{M}\ , \qquad \frac{\overline{{( \Delta{\cal R})^{2}}}}{\overline{{\cal R}}^{2}}\sim \frac{\beta}{\delta}\sim \frac{( R^{2}/ \ell_{s}^{2})}{( M / M_{s})}.
\tilde{F}^{[ n ]}( z + \phi ) \rightarrow e^{( n - 1 ) f ( z + \phi ) - n f ( z )}( \tilde{F}^{[ n ]}( z + \phi ) - \delta_{n , 1}h^{\bar{0}}( f ( z + \phi ) + \bar{f}( \bar{z}) )
S_{k}=\left\{\begin{array}[c]{c}\mathbf{x}_{k}:\mathbf{x}_{k-1}\in S_{k-1},\mathbf{x}_{k}=A\mathbf{x}_{k-1}+Bv_{k},y_{k}=C\mathbf{x}_{k-1}+D_{1}v_{k}, \left\vert v_{k}\right\vert \leq1,\left\vert y_{k}-z_{k}\right\vert\leq1.\end{array}\right\}
\Delta(t,x):= (1-\epsilon) \delta\big[\Phi'(x-\underline h(t))+\Phi'(-x-\underline h(t))\big] &+(1-\epsilon)\big[F(\Phi(x-\underline h(t)))+ F(\Phi(-x-\underline h(t)))\big]- F(\underline U(t,x)_+).
N^{( l )}( L ) \; : = \; \sum_{\{\alpha_{a}, \beta_{a}\}}\prod_{b = 1}^{l}\left[ \psi_{\alpha_{b}}^{( b )}( - L , 0 ) \; U \Big ({\cal C}( L ) \Big ) \; \overline{{{\psi}}}_{\beta_{b}}^{( b )}( + L , 0 ) \right] \; ,
T^{i j}= \kappa \, \eta^{i k}\eta^{j l}h_{\mu \nu}( \xi ) \, \partial_{k}\xi^{\mu}\partial_{l}\xi^{\nu}- \frac{\kappa}{2}\, \eta^{i j}\eta^{k l}h_{\mu \nu}( \xi ) \, \partial_{k}\xi^{\mu}\partial_{l}\xi^{\nu}- \kappa \, \eta^{\mu \nu}V ( \xi ) .
F^{kl}\phi_{;kl}- \dot{\phi}\leq \, \phi^{ij}\{-F^{kl}h_{rk}h^r_l W_{ij}- F^{kl}g_{kl}W_{ij}+ 2F h_i^k W_{kj}& \, +2F \epsilon g_{ij}- F^{kl,rs}W_{kl;i}W_{rs;j}\}+ F^{kl}\phi^{ij,rs}W_{ij;k}W_{rs;l}+ O (\phi).
E_{n}=\frac{4\pi}{\ln\left( \frac{1}{1-cos(m_{2})}\right)}\left(n+\frac{\theta}{2\pi}\right) \left( 1+\frac{l_{p}^{2}\mu_{0}^{2}\cos(m_{2})}{(1-\cos(m_{2}))\ln\left(\frac{1}{1-cos(m_{2})}\right)}+\ldots\right),
\dfrac{5a_0}{2}\|\partial_x^2 u\|^2 = -\partial_t\left(\dfrac{1}{2}\int_0^Lxu^2 dx\right)-\dfrac{3}{2}\|\partial_x u\|^2+\dfrac{1}{3}\int_0^Lu^3 dx+\dfrac{a_1}{2}\|u\|^2 & -(-1)^k\int_0^Lxu\int_0^\infty g(s)\partial_x^{2k}\eta^t(x,s)ds dx.
||\widehat{S}\psi||_{\phi,S_{0}}=\int_{\mathbb{R}^{2n}}|W(\widehat{S}\psi,\phi)(z)|dz & =\int_{\mathbb{R}^{2n}}|W(\psi,\widehat{S}^{-1}\phi)(S^{-1}z)|dz & =||\psi||_{\widehat{S}^{-1}\phi,S_{0}}
cov\{\hat{m}_{2y}(0),\hat{m}_{2y}(-\beta)\}= \frac{1}{N}\sum_{\tau = -K_N}^{K_N}[\hat{m}_{4y}(0,\tau,\tau-\beta) - \hat{m}_{2y}(0)\hat{m}_{2y}(-\beta)],
D(A) = \biggl\{x = \sum_{m=1}^{\infty}a_m e_m: \sum_{m=1}^{\infty}\gamma_m a_m e_m \biggr\}A \biggl( \sum_{m=1}^{\infty}a_m e_m \biggr) = \sum_{m=1}^{\infty}\gamma_m a_m e_m
\frac{\partial \ln(R_{\rm FA})}{\partial \lambda}=\frac{-1}{\lambda(1-\lambda)}+ \frac{w_{c}( \frac{w_{c}}{r}-1 )(1-\lambda)^{\frac{w_{c}}{r}-2}}{1-( 1-\lambda )^{\frac{w_{c}}{r}-1}},
\; Q_1(u_3 z_1+u_2z_1^2+z_1^4) = \; Q_0Q_1(x_1y_1z_1) = \; x_1^4y_1^2z_1+x_1^4y_1z_1^2+x_1^2y_1^4z_1+x_1^2y_1z_1^4+ x_1y_1^4z_1^2+x_1y_1^2z_1^4\not=0.
\theta(z,t)\theta(x,y)-\theta(y,t)\theta(x,z)-\theta(x,[y,z,t])+D(y,z)\theta(x,t)=0, \theta(z,t)D(x,y)-D(x,y)\theta(z,t)+\theta([x,y,z],t)+\theta(z,[x,y,t])=0,
\rho_{0,0,n}=n\pi+\frac{q_1(0)+(-1)^{n+1}q_1(1)}{n\pi}\sin n\pi a+\frac{\kappa_{0,0,n}}{n}, \{\kappa_{0,0,n}\}\in l_2,n\in\mathbb{Z}\!\setminus\!\{0\};
{\rm Corr}_{\rho,\Phi}^{f,g}(A,B) := \Phi\left(f(\rho) g(\rho) A^*B \right)-\Phi\left(f(\rho) A^* g(\rho) B \right){\rm I}_{\rho,\Phi}^{f,g}(A) :={\rm Corr}_{\rho,\Phi}^{f,g}(A,A)\,,
1 \le \sum_{e\in E}\rho(e)\eta(e) = \sum_{e\in E}\left(\sigma(e)^{1/p}\rho(e)\right) \left(\sigma(e)^{-1/p}\eta(e)\right) \le \left( \sum_{e\in E}\sigma(e)\rho(e)^p \right)^{1/p}\left( \sum_{e\in E}\hat{\sigma}(e)\eta(e)^{q}\right)^{1/q},
\int_{N_1^+}\phi(n) \psi(g_t nx) \;d\mu_x^u(n)= e^{-\d t}\sum_{i\in I(\L)}\int_{N_1^+}\rho_i(nn_i) \phi(g_{-t}nn_i g_t ) \psi(nx_i ) \;d\mu_{x_i}^u(n).
C_0^1=\delta^{11}C_{12}y^2=({w^1}_{\substack{x_2}}-{w^2}_{\substack{x_1}}) y^2 &C_0^2=\delta^{22}C_{21}y^1=({w^2}_{\substack{x_1}}-{w^1}_{\substack{x_2}}) y^1,
{\cal A}={\frac{1}{2}}\int_{0}^{\infty}{\frac{d \tau_{2}}{\tau_{2}^{1 4 - d / 2}}}{\frac{1}{\eta^{2 4 - d}( i \tau_{2}/ 2 )}}~ ~ \gamma_{1}( i \tau_{2}/ 2 ) \ .
\left\|\vec{t}\right\|_{\dot f_{\infty,q}^s(\mathbb{A})}=\|u\|_{\dot f_{\infty,q}^s}\sim\|u\|_{\dot f_{p,q}^{s,\frac{1}{p}}}=\left\|\vec{t}\right\|_{\dot f_{p,q}^{s,\frac{1}{p}}(\mathbb{A})}
\prod_{i = 1}^{+ \infty}( J_{- i}^{+})^{p_{i , +}}\prod_{i = 1}^{+ \infty}( J_{- i}^{0})^{p_{i , 0}}\prod_{i = 0}^{+ \infty}( J_{- i}^{-})^{p_{i , -}}| 1 \rangle
<{S_{\mbox{\scriptsize B}}}>={g\over{2\sqrt{2\alpha'}}}{\int_{-\infty}^{+\infty}}d\tau\int{d^{26}}k\int{\mathcal{D}_{\tilde{g}}}(\bar{X},Y)\exp\left\{-{S_0}+ik\cdot\left[{X_0}(\tau)+\bar{X}(\tau)\right]\right\}{L_{\mbox{\scriptsize B}}}(k,\tau),
\mathbf{W}=(\mathbf{C}_s+\mathbf{C}_w)^{\frac{1}{2}}\mathbf{C}_w^{-1}\mathbf{H}\mathbf{a}\mathbf{a}^H\mathbf{H}^H\mathbf{C}_w^{-1}(\mathbf{C}_s+\mathbf{C}_w)^{\frac{1}{2}}\;.
\begin{cases}\binom{\overline{s}+s}{s-(D-k_1+1)}c_{D-k_1+1}+\binom{\overline{s}+s}{s-(D-k_1)}c_{D-k_1}+\cdots+\binom{\overline{s}+s}{s}c_{0}=0 \vdots \binom{\overline{s}+D-k_2}{k_1-1-k_2}c_{D-k_1+1}+\binom{\overline{s}+D-k_2}{k_1-k_2}c_{D-k_1}+\cdots+\binom{\overline{s}+D-k_2}{D-k_2}c_{0}=0.\end{cases}
2 \int_{t}^{t+\frac{r}{6}}\frac{\lambda'''(s)}{r}(s-t) ds = \frac{2}{r}\left(\frac{r}{6}\lambda''(t+\frac{r}{6})-\lambda'(t+\frac{r}{6})+\lambda'(t)\right) &=\frac{2\lambda'(t)}{r}+ E_{\partial_{t}v_{1},1}(t,r)
\partial_t \tilde{u}_i - d_i \Delta \tilde{u}_i \ge f_i\big(t,x,\tilde{u}_i, [{\bf\tilde{u}}]_{a_i}, [{\bf 0}]_{b_i}\big), d_i \partial_{\nu}\tilde{u}_i \ge g_i\big(x,\tilde{u}_i, [{\bf \tilde{u}}]_{c_i}, [{\bf 0}]_{d_i}\big), i=1,\ldots,6.
\|\tilde{\ell_{1}}- \bar{\ell_{1}}\|\leq \frac{2\varepsilon}{\lambda_{1}^2}(\lambda_{1}+ \|x_{1}\|), \ \ \ \bar{a}_{1}-\tilde{a_{1}}=\frac{\varepsilon}{\lambda_{1}^2}\ \ \tilde{c_{1}}-\bar{c}_{1}\leq \frac{\varepsilon}{\lambda_{1}^2}\|x_{1}\|^2
F(\eta) = \frac{\Gamma((d-2)/2)}{2\, \pi^{(d+2)/2}}\, \frac{1}{\sin\eta}\,\csc^{d-3}\bigg(\frac{\eta}{2}\bigg) \, \frac{d}{d\eta}\int_\alpha^{\eta}\frac{g(\zeta) \sin \zeta \, d\zeta}{\sqrt{\cos\zeta - \cos \eta}}, \alpha \leq \eta \leq \pi,
\int_{2R_{1,i}}|T_{\mu, \varepsilon}(\varphi_{1,i}, 1_{4R_{1,i}}g_2)|\,d\mu \lesssim \mu(R_{1,i})^{1-1/r}\cdot \|\varphi_{1,i}\|_{L^{\infty}(\mu)}\mu(R_{1,i})^{1/p}\cdot \lambda^{1/2}\mu(R_{1,i})^{1/q}&= \lambda^{1/2}\|\varphi_{1,i}\|_{L^{\infty}(\mu)}\mu(R_{1,i}) \lesssim \lambda^{1/2}|\nu|(Q_{1,i}).
\mathbf{J}=(\mathbf{G}_1\diamond\mathbf{G}_2\diamond\mathbf{G}_3\diamond\mathbf{G}_4\diamond\mathbf{G}_5)^H(\mathbf{G}_1\diamond\mathbf{G}_2\diamond\mathbf{G}_3\diamond\mathbf{G}_4\diamond\mathbf{G}_5)