harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bar\partial(\Phi_1^{\mu})^{-1}= -\mu\circ(\Phi_1^{\mu})^{-1}\bar\partial\overline{(\Phi_1^{\mu})^{-1}}. \bar\partial\overline{(\Phi_1^{\mu})^{-1}}=\left(\frac{1}{1-\vert\mu\vert^2}\frac{1}{\overline{\partial \Phi_1^\mu}}\right)\circ (\Phi_1^\mu)^{-1}.
{\tilde x_u}\left( t \right) = \sqrt{{{\rm{2}}\mathord{\left/{\vphantom{{\rm{2}}T}}\right. \kern-\nulldelimiterspace}T}}{\mathop{\Re}\nolimits}\left({{x_u}\left( t \right){e^{j\left({2\pi f_u^{{\rm{RF}}}t + \varphi_u^{{\rm{RF}}}}\right)}}}\right)
e^{\sum_{i\in[n]}c_i x_i}(d+\mathrm{im}(\tilde\omega))e^{-\sum_{i\in[n]}c_i x_i}=d+\sum_{i\in[n]}x_i\cdot dc_i-\sum_{i<j\in[n],\alpha\geq 0}\Big({\theta(p_{ij}+\mathrm{ad}x_i|\tau)\over\theta(\mathrm{ad}x_i|\tau)\theta(p_{ij}|\tau)}-{1\over \mathrm{ad}x_i}\Big)(t_{ij}) =d+A_{\mathrm{KZB}}.
< t r \, G_{\cal A}( x , x ) >_{\cal A}= \int{\mathcal{D}Y}\, e^{-{\lambda_{2}}\oint{d}\tilde{s}}\, W = \int d L \, e^{- \lambda_{2}L}\int{\mathcal{D}Y}\, \delta ( L - \oint{d}\tilde{s}) \, W
\mathbb{P}[X_v=1,d_{Z'}(v)<r] = \binom{t_1}{r-1}p^{r-1}(1-p)^{t_1-r+1}&=(1+O(t_1^{-1})) \frac{t_1^{r-1}}{(r-1)!}p^{r-1}(1-p)^{t_1-r+1}.
\begin{array}{l c r}{\{X^{\mu}, X^{\nu}\}= \sum_{\stackrel{m_{1}, n_{1}}{m_{2}, n_{2}}}X_{m_{1}n_{1}}^{\mu}( \sigma ) X_{m_{2}n_{2}}^{\nu}( \sigma ) \{L_{m_{1}n_{1}}, L_{m_{2}n_{2}}\},}\\ \end{array}
\begin{gathered}X_m = f_m = f(x_m),Y_{m,n}= e_{m,n}= y_{m,n}\, f_{x}^m + g_m, U_{m,n}= u_{m,n}+ \frac{e_{x}^{m,n}}{f_{x}^m}= u_{m,n}+ \frac{\Delta e_{m,n}}{\Delta f_m}- \frac{\Delta y_{m,n}}{\Delta x_m}=u_{m,n}+ \frac{y_{m+1,n}}{\Delta f_{m}}\Delta\bigg( \frac{\Delta f_m}{\Delta x_m}\bigg) + \frac{\Delta g_m}{\Delta f_m},\end{gathered}
d_{W}\theta^{\alpha}+{\frac{1}{2}}C_{\beta \gamma}^{\alpha}\theta^{\beta}\wedge \theta^{\gamma}= u^{\alpha}\ , \{}d_{W}u^{\alpha}= - C_{\beta \gamma}^{\alpha}\theta^{\beta}\wedge u^{\gamma}\ ; \{}i_{W \, X_{\alpha}}\theta^{\beta}= \delta_{\alpha}^{\beta}\ , \{}i_{W \, X_{\alpha}}u^{\beta}= 0 \ ;
\mathbf{H}_{\mathcal{S}}= \mathbf{\Gamma}_{\mathcal{S}}\left(\mathbf{\Phi}_{\mathcal{S}}\hat{\mathbf{H}}_{\mathcal{S}}+\mathbf{\Omega}_{\mathcal{S}}\tilde{\mathbf{H}}_{\mathcal{S}}\right),
I_1(d,a,c)={1 \over 2(a^2+c^2-d^2)}\ln \left({c^2 \over (d+a)^2}\right)+{a \over (c^2-d^2)^{1/2}(a^2+c^2-d^2)}\arctan \left({(c^2-d^2)^{1/2}\over d}\right)
\lim_{n\rightarrow 0}\frac{1-P}{n^3(1+P)}= \lim_{n\rightarrow 0}\frac{1}{1+P}\lim_{n\rightarrow 0}\frac{1-P}{n^3}\lim_{n\rightarrow 0}=\frac 12\lim_{n\rightarrow 0}\frac{1}{3n^2}\frac{\partial}{\partial n}(1-P) = -\frac{1}{2}\psi''\left(x+\frac a3\right)=\sum_{k=0}^{\infty}\frac{1}{\left(k+x+\frac a3\right)^3}.
G=\left[\begin{array}{rrrrrrrrrrrrr}1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1\end{array}\right]
I = T_{( 1 )}\int d^{2}\zeta \Bigl [ \sqrt{1 - ( \partial_{0}Y^{m})^{2}- ( 2 \pi \alpha^{\prime}F_{0 1})^{2}}+{\widehat C}_{0 1}+ 2 \pi \alpha^{\prime}F_{0 1}{\widehat C}\Bigr ] \ .
\begin{array}{c}(\dot{x},\dot{y})=i(a\alpha\zeta^{\alpha-1},b\beta\zeta^{\beta-1})= (A\zeta^{\alpha-1},B\zeta^{\beta-1}) (i\dot{y}^*,-i\dot{x}^*)=(iB^*(\zeta^*)^{\beta-1},-iA^*(\zeta^*)^{\alpha-1})\end{array}
(R\pi_{HT\ast}\mathbb F_\ell)_{\mathfrak m,\bar{y}}= \varinjlim_i \left(R\lambda_{U_i/K_{p,i}\ast}\left( (R(\pi_{HT}/G(\mathbb Q_p))_\ast \mathbb F_\ell)_{\mathfrak m}\right)|_{U_{i,\bar{\eta}}/K_{p,i}}\right)|_{\bar{x}_i}\ .
{\bar \epsilon}Q =-{\rm Tr}\left\{\left({\bar \epsilon}\Gamma^I \Theta\right) \left(D_\tau X_I\right)+\left(2\Sigma^{0I}D_\tau X_I -R\Sigma^{IJ}[X_I,X_J]\right)\epsilon_{\alpha}\Theta_\alpha^{\dag}\right\},
(\sigma,v) = \sum_{k\geq K_0}a_k e^{\lambda^p_k (T-t)}(\xi_{\lambda^p_k}, \eta_{\lambda^p_k}) + \sum_{|k|\geq K_0}b_k e^{\lambda^h_k(T-t)}(\xi_{\lambda^h_k}, \eta_{\lambda^h_k}) ,
i\Theta_{E,h_E,x_0}=i\sum_{j,k,\lambda,\mu}c^E_{jk\lambda\mu}dz_j\wedge d\overline{z}_k\otimes e^*_\lambda\otimes e_\mu,\,\, i\Theta_{F,h_F,x_0}=i\sum_{j,k,\lambda,\mu}c^F_{jk\lambda\mu}dz_j\wedge d\overline{z}_k\otimes f^*_\lambda\otimes f_\mu.
(-1)^k s_k /k=\sum_{i_1+2i_2+\cdots+ki_k=k}(-1)^{i_1+\cdots+i_k}\frac{(i_1+\cdots+i_k-1)!}{i_1!\cdots i_k!}\sigma_1^{i_1}\cdots \sigma_k^{i_k}
[(h_1, \ldots, h_{m+1}), (k_1, \ldots, k_{n+1}) ]_c := &= \big( [h_1, k_1],~ [h_1, k_2] + [h_2, k_1],~ \ldots, \underbrace{[h_1, k_i] + [h_2, k_{i-1}] + \cdots + [h_i, k_1]}_{i},~ \ldots, [h_{m+1}, k_{n+1}] \big),
\mathcal{B}_{t}(f,g)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot(\xi+\eta)}\chi_{D_{\nu}}e^{t^{\frac{1}{\alpha}}(|\xi+\eta|_{1}-|\xi|_{1}-|\eta|_{1})}\chi_{D_{\mu}}\hat{f}(\xi)\chi_{D_{\varsigma}}\hat{g}(\eta)d\xi d\eta.
G^2 =\frac{|u(x)-u(\eta)|^{2s}}{|x-\eta|^{1+2(1+s)\beta-\alpha}}|\lambda u(x)+(1-\lambda)u(\eta)|^{1-s}&\leq C(s)\frac{|u(x)|^{1+s}+|u(\eta)|^{1+s}}{|x-\eta|^{1+2(1+s)\beta-\alpha}}= C(s)\frac{|u(x)|^{1+s}+|u(\eta)|^{1+s}}{|x-\eta|^{1- 2(1+s) \delta}},
{S_{e f f}}^{\prime}= \int{d^{4}x [ - \frac{1}{4}{F^{\prime}}^{2}- \frac{1}{2}{F^{\prime}}^{{\mu}{\nu}}S_{{\mu}{\nu}}^{\lambda}A_{\lambda}- 2 S_{\lambda}^{{\mu}{\nu}}A^{\lambda}S_{{\mu}{\nu}}^{\beta}A_{\beta}+ \frac{1}{2}{m^{2}}_{\gamma}A^{2}]}
\mathcal{K}_p:=\big\{K \neq K(p) \;\mid\;\, \zeta_p \in K \textrm{in case}\mathrm{char}(K)\neq p & \textrm{and}\sqrt{-1}\in K \textrm{in case}p=2 \textrm{and}\mathrm{char}(K)=0 \big\}.
\mathbf{E}[\|\mathcal{X}^{t+1}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}|\mathcal{X}^{t}]=\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}-\mathbf{E}_{i\sim \mathbf{p}^{t}}[f_{i}(\mathcal{X}^{t})],
\begin{array}{@{}l@{}}{\displaystyle \mathcal{E}(t) = \int_{\Omega}\left(\frac{1}{2}\big\vert \left[i\nabla + q\mathbf{A}\right]\Psi\big\vert^{2}+ V|\Psi|^{2}+ \frac{1}{2}\big|\frac{\partial \mathbf{A}}{\partial t}\big|^{2}+ \frac{1}{2}|\nabla \times \mathbf{A}|^{2}+ \frac{1}{2}|\nabla \phi|^{2}\right)\;\mathrm{d}\mathbf{x}.}\end{array}
[ \hat{S}_{+}, \hat{S}_{-}] = \displaystyle \frac{\partial G ( S_{3})}{\partial S_{3}}\vert_{S_{3}\to \hat{S}_{3}}\stackrel{d e f .}{=}[ [ \hat{S_{3}}] ] ~ , ~ [ \hat{S}_{3}, \hat{S}_{\pm}] = \pm \hat{S}_{\pm}~ ,
\begin{array}{cl}\null a_i X(T,t_i) [f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i},\overline{u}(t_i))] = \int_{t_i + b_i}^{t_i + b_i + a_i}X(T,t_i)[f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i},\overline{u}(t_i))]d \xi.\end{array}
\begin{cases}\nabla u_\alpha(x_\alpha)=0 ~~ \alpha, &d_g(x_\alpha,x)^{\frac{n-2}{2}}u_\alpha(x) \leq C~~ x\in B_{x_\alpha}(7\rho_\alpha) \alpha, &\lim_{\alpha\to +\infty}\rho_\alpha^{\frac{n-2}{2}}\underset{B_{x_\alpha}(6\rho_\alpha)}{\sup}u_\alpha=+\infty.\end{cases}
\sum_{n=0}^{m-1}F\left(n,\frac{m-1}{2}\right)=\sum_{n=0}^{m-1}[3n+m]\frac{(q;q^2)_{n}(q^{m};q^2)_{n}^2 q^{-{n+1\choose 2}-\frac{(2n+1)(m-1)}{2}}}{(q;q)_{n}^2 (q^2;q^2)_{n}}.
\lim_{\delta\rightarrow 0}\limsup_{N\rightarrow+\infty}\sup_{\sigma\in\mathcal{T}, t\leq \delta}\widetilde{P}_{f_1, f_2, f_3}^{N, \tilde{F}, \tilde{G}, \tilde{H}}\left(\left|\eta^N_{t+\sigma}(\vec{f})-\eta^N_{\sigma}(\vec{f})\right|>\epsilon\right)=0
{\textstyle\sum_{k=\widehat{k}}^{\nu}}\,\|x^{k+1}\!-\!x^k\|<(1/4){\textstyle\sum_{k=\widehat{k}}^\nu}\,\Xi_k+b\gamma^{-1}\varphi(\Phi(x^{\ell(\widehat{k})})\!-\!\Phi^*).
\epsilon\|\Delta_h u_h^n\|^2_{\mathcal{T}_h}=(\epsilon^{-1}f^n(u^n_h),\Delta_h u^n_h)_{\mathcal{T}_h}-(\phi_h^n,\Delta_h u^n_h)_{\mathcal{T}_h}&\le \epsilon^{-1}(\|u_h\|_{0,6}^3+\|u_h^n\|_{\mathcal T_h})\|\Delta_h u_h^n\|_{\mathcal{T}_h}+\|\phi_h^n\|_{\mathcal T_h}\|\Delta_h u_h^n\|_{\mathcal{T}_h}.
d_{\Sigma}=\max_{(d^{[{\sf d}]}_{11},\cdots,d_{KN}^{[{\sf d}]},d^{[{\sf u}]}_{11},\cdots,d_{KN}^{[{\sf u}]})\in\mathcal{D}}\left\{\sum_{k=1}^K\sum_{i=1}^{N}\left(d_{ki}^{[{\sf d}]}+d_{ki}^{[{\sf u}]}\right)\right\},
\dfrac{1}{M(\alpha,\beta)}\left\lVert \left( A_\pi^\alpha v +\pi_s A_\pi^\alpha v \right)\right\rVert = \dfrac{2 \left\lVert A_\pi^\alpha v \right\rVert}{M(\alpha,\beta)}\left\lVert \dfrac{A_\pi^\alpha v +\pi_s A_\pi^\alpha v}{2\left\lVert A_\pi^\alpha v\right\rVert}\right\rVert &\le \dfrac{2}{M(\alpha,\beta)}\left(1-\delta_E(\kappa)\right).
D_{i,j}=0, 1\leq j<i\leq k, D_{i,j}=D_{1,j-i+1}, 1\leq i\leq j\leq k, C_{i,n-2k}=0, 1\leq i\leq k-1, D_{i,j}E_2 =-D_{i,k+j}-D_{i+1,k+j}, 1\leq i\leq k-1, \ \ 1\leq j\leq k.
D_{z,A}:= \prod_{i=1}^mD_{z;A^{(i)}},\xi_{B}:= \prod_{i=1}^m\prod_{j=1}^q \xi_{i;\beta_j^{(i)}},\Delta_{AB}^{(\ell+1)}:= \prod_{j=1}^q \Delta_{\alpha_j^{(i)}\beta_j^{(i)}}^{(\ell+1)}.
\zeta\left(s;\not\!\!D^{\dag}\not\!\!D\right)={\mu}^4 \Omega\frac{ab}{4{\pi}^2}\left[ 2\sum_{n_a=1}^{\infty}\left(2n_aa+c\right)^{-s}+ 2\sum_{n_b=1}^{\infty}\left(2n_bb+c\right)^{-s}+\right.
T^{n,k}(x) =\int_{x}^{a}(1+T^{n,k}(y))f_{n,k}(y;x)dy+\int_{a}^{+\infty}f_{n,k}(y;x)dy &=\int_{x}^{a}T^{n,k}(y)f_{n,k}(y;x)dy+1
\left(\begin{matrix}x=0 x=\infty \, \left( \frac12 \right) \overbrace{\begin{matrix}0 0 0 0 1 \theta^0 \end{matrix}}\overbrace{\begin{matrix}0 \sqrt{-t_2}\theta^\infty_1/2 0 -\sqrt{-t_2}\theta^\infty_1/2 -t_1 0 \theta^\infty_2 \end{matrix}}\end{matrix}\right) ,
W(-i \{f\,,\,g\})\phi=\frac{1}{2}E_lE_k(w)w'\phi-\frac{1}{2}wE_kE_l(w')\phi &-E_k(w')wE_l(\phi)+E_l(w)w'E_k(\phi)-ww'[E_k,E_l](\phi)
F(2^{-i}) = \sum_{j=0}^i c_{i+1}(j) = \sum_{j=0}^{i/2}c_{i+1}(2 j) =\sum_{j=0}^{i/2}\frac{2^{(1-(i+1)) 2j}F(2^{2 j-(i+1)})}{2^{(1-2 j)(2 j)/2-1}(2 j)!}= \sum_{j=0}^{i/2}\frac{4^{j (j-i)}F\left(2^{2 j-i-1}\right)}{2^{j-1}(2 j)!}
\frac{1+g_{p+1}}{1-g_{p+1}}=-\Biggl(\frac{1+a_p}{2}\Bigl(\frac{1+g_p}{1-g_p}\Bigr)+\frac{1-a_p}{2}\Bigl(\frac{1-g_p}{1+g_p}\Bigr)\Biggr)
\left\{d_{+}\in L_{+}^{2}(\mathbb{S};\mathbb{C}):\max_{c_{+}\in L_{+}^{2}(\mathbb{S};\mathbb{C})}\mathfrak{f}_{\mathfrak{m}}\left( c_{-},c_{+}\right) =\mathfrak{f}_{\mathfrak{m}}\left( c_{-},d_{+}\right) \right\}
e^{-\frac{1}{n}lna_{n}}=1+\frac{1}{n}+\frac{1}{n}\sum_{i=1}^{K-1}\frac{1}{\prod_{k=1}^{i}ln_{(k)}(n)}+\frac{\rho_{n}}{n\prod_{k=1}^{K}ln_{(k)}(n)}.
\int_0^{\infty}\sqrt{\sum_{k=0}^\infty \min\{\alpha_{1,{\bf X}}(k), H(t)\}}\dt = \int_0^\infty \int_0^1{\bf 1}_{v \leq G_\alpha(H(t))}\dv \dt &= \int_0^1 \int_0^\infty{\bf 1}_{t \leq Q ( G^{-1}_\alpha(v))}\dt \dv &= \int_0^1 Q \circ G_{\alpha}^{-1}(v) \dv \, .
u\left( t,\varepsilon \right) =M\left( t,\varepsilon \right) f_{1}\left(\varepsilon \right) +N\left( t,\varepsilon \right) f_{2}\left( \varepsilon\right) +r_{\left[ 0,T\right]}F^{-1}\Phi \left( \xi ,\varepsilon \right) F\bar{f}\left( \xi \right) ,
{\rm I}_1 \leq c h^{(2-s-r)p}\int_{0}^{t_n}(t_n-t)^{(\frac{r\alpha}{2}+\gamma-1)p}\d t \leq c t_n^{(\frac{r\alpha}{2}+\gamma-1)p+1}h^{(2-s-r)p}.
\begin{array}{l l l}{V}&{= \frac{1}{2}\int_{M}d^{2}x \Omega}&{\left[ \frac{1}{2}F_{i j}F^{i j}+ D_{i}\phi \overline{{D^{i}\phi}}\right.}\\{}&{}&{\left. + \frac{1}{4}( | \phi |^{2}- 1 )^{2}\right]}\\ \end{array}
R_O =\sum_j \chi_j^2 (\mathring R_q - \mathring R_{q,j}) +\left(\sum_j \chi_j^2 \mathring R_{q,j}+ \sum_{j,k}\mathring{\mathcal Q}_{j,k}\right) + R_{O,{\rm high}}&=: R_{O,{\rm approx}}+ R_{O,{\rm low}}+ R_{O,{\rm high}}\,,
\sum_{\substack{a_{ij}\geq 1 \prod_{i=0}^{\ell-1}\prod_{j=0}^{\ell-1}a_{ij}\leq x}}\prod_{i=0}^{\ell-1}\prod_{j=0}^{\ell-1}|\lambda_{\pi}(a_{ij})|^2 \chi^j (a_{ij}) = x P_{\ell-1}(\log x) +O_{\pi,\chi}\Big(x^{\frac{4 \ell^2 -1}{4 \ell^2 +1}}\log^{\ell-1}x \Big),
f^i_n|_{B_R(\bar{x}_n)}\to f^i|_{B_R(\bar{x})}, g^i_n|_{B_R(\bar{x}_n)}\ \to g^i|_{B_R(\bar{x})}, f^{ij}_n |_{B_R(\bar{x})}\to f^{ij}|_{B_R(\bar{x})}\; \forall R>0 f^i(\bar{x})=g^i(\bar{x})=f^{ij}(\bar{x})=0
\frac{1}{q(x)}\int_{-\infty}^{x}\frac{2}{a(y)}q(y) dy \leq \ \frac{1}{q(x)}\int_{-\infty}^{x}\frac{2 b(y)}{a(y)}\frac{1}{b(x)}q(y) dy = \ \frac{1}{q(x)}\frac{1}{b(x)}\int_{-\infty}^{x}r(y) e^{\int_{0}^{y}r(u) du}dy = \ \frac{1}{q(x)}\frac{1}{b(x)}\big(q(x) - q(-\infty)\big) \leq \ \frac{1}{b(x)}.
{}_{I}\langle \ell , s | \tilde{q}^{H^{( c )}}| \ell^{\prime}, s^{\prime}\rangle_{I}= \delta_{\ell , \ell^{\prime}}\delta_{s , s^{\prime}}( - 1 )^{s}\chi_{\ell}^{( N - 2 )}( \tilde{q}) \chi_{s}^{S O ( 8 )}( \tilde{q}) ,
\mathcal{Q}=Z_3 \mathcal{Q}^{\rm R}, \quad \mathcal{Q}^{\rm R}:={1 \over 2}a_\mu^{\rm R}(x) a^\mu{}^{\rm R}(x) + \alpha^{\rm R}i \bar{C}^{\rm R}(x) C^{\rm R}(x) .
g_{i\bar{j}}\, \bar{\phi}^{\bar{j}}{''}+ \bar{\partial}_{\bar{k}}g_{i\bar{j}}\, \bar{\phi}^{\bar{j}}{'}\bar{\phi}^{\bar{k}}{'}+ \frac{1}{2}\left(A' - B' + 2C' \right)g_{i\bar{j}}\, \bar{\phi}^{\bar{j}}{'}= e^{B}\left( e^{-2C}\partial_i V_{\mathrm{BH}}+ \partial_i V \right) \;,
(A-5I)^{2}x_{3}=(\begin{matrix}0&0&2&-8\\ 0&0&0&4\\ 0&0&0&-3\\ 0&0&0&1\end{matrix})(\begin{matrix}x_{31}\\x_{32}\\x_{33}\\x_{34}\end{matrix})=(\begin{matrix}2x_{33}-8x_{34}\\ 4x_{34}\\ -3x_{34}\\x_{34}\end{matrix})\ne(\begin{matrix}0\\ 0\\ 0\\ 0\end{matrix}).
\underbrace{(t+t^{2}+t^{3}+\dots)(t+t^{2}+t^{3}+\dots)\dots (t+t^{2}+t^{3}+\dots)}_{k+1}=\frac{t^{k+1}}{(1-t)^{k+1}}=\sum_{u\ge k}{u \choose k}t^{u+1}.
\delta_{\lambda}(m,\tilde{m}) = \inf_{\pi,\sigma\in Perm(\mathbb N)}\sum_{i\geq 1}|m_{\pi(i)}^{\lambda}- \tilde m_{\sigma(i)}^{\lambda}| = \sum_{k\geq 1}|m_k^{\lambda}- \tilde{m}_k^{\lambda}|,
\omega = \frac{1}{\gamma^{2}}\, \int_{\tau}^{\tau + 2 \pi}\, \, \frac{1}{\psi^{2}( z )}\, \delta \psi^{\prime}( z ) \wedge \, \delta \psi ( z ) \, d z + \frac{1}{\gamma^{2}}\, \delta \lambda \wedge \, \frac{\delta \psi ( \tau )}{\psi ( \tau )}.
W_{\alpha \beta}( x - y ) = \left( \begin{array}{c c c c}{{0}}&{{0}}&{{\delta ( x - y )}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\delta ( x - y )}}\\{{- \delta ( x - y )}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- \delta ( x - y )}}&{{0}}&{{0}}\\ \end{array}\right) .
u(x,t) \,=\, \frac{\nu}{L}\,\tilde u\Bigl(\frac{x}{L}, \frac{\nu t}{L^2}\Bigr)~, \omega(x,t) \,=\, \frac{\nu}{L^2}\,\tilde \omega\Bigl(\frac{x}{L}, \frac{\nu t}{L^2}\Bigr)~, p(x,t) \,=\, \frac{\rho\nu^2}{L^2}\,\tilde p\Bigl(\frac{x}{L}, \frac{\nu t}{L^2}\Bigr)~.
\operatorname{lp}\int_{\rho>\epsilon}(-i\partial\overline{\partial}\log\rho)^{k}\wedge\alpha^{n+1-k}=\operatorname{lp}\epsilon^{-k}\int_{\rho=\epsilon}\vartheta\wedge({\rm d}\vartheta)^{k-1}\wedge \alpha^{n+1-k}=0.
\eqref{equ:Z3r1r203}= \sum_{j=r_1+1}^{r_1+r_2+2}\frac{(-1)^{r_1}}{r_2+1}\binom{r_2+1}{j-r_1-1}\frac{B_j}{j}\zeta(s+j-r_1-r_2-2) &+\sum_{k=1}^{r_2}\sum_{j=r_1+1+k}^{r_1+r_2+2}(-1)^{r_1+k}\binom{r_2}{k-1}\binom{r_2+1-k}{j-k-r_1-1}\frac{B_kB_{j-k}}{k(j-k)}\zeta(s+j-r_1-r_2-2).
{\cal{L}}_{DJ}^S=\frac{1}{2}(f^\mu -\partial^{\mu}\phi ) (f_\mu-\partial_{\mu}\phi )-\epsilon^{\alpha\beta\gamma}f_\alpha\partial_\beta A_\gamma +\frac{m}{2}\epsilon^{\alpha\beta\gamma}A_{\alpha}\partial_{\beta}A_{\gamma}.
D_{\zeta}^{2 s + \tau , 2 t + \tau^{\prime}}( x , x_{1}, \ldots , x_{n}) = ( - i )^{2 s + \tau + 1}\sigma_{2 s + \tau}^{+}\sum_{p = 0}^{t}x^{2 s - 2 p + \tau + 1 - \zeta}\hat{\sigma}_{2 p + \zeta}^{-}.
\frac{\partial u}{\partial t}(l,a,t)+\frac{\partial u}{\partial a}(l,a,t)=-\mu(E[u],l,a)u(l,a,t), & u(l,0,t)=\int_0^\infty\int_{\hat{l}}^\infty b(l,\hat{l})\beta(E[u],\hat{l},a)u(\hat{l},a,t)\, da\,d\hat{l}, & u(l,a,0)=u_0(l,a).
\int_{\mathcal{X}}\Psi(\nu, \mu) \left(\mathrm{d}x\right) = \int_{\mathcal{X}}\Psi(\nu, \mu) \left(x\right) \mathrm{d}x = \frac{1}{Z(\nu,\mu)}\int_{\mathcal{X}}\exp \left(- \frac{2}{\sigma^2}\frac{\delta F}{\delta \nu}(\nu, \mu, x) - U^{\pi}\left(x\right)\right) \mathrm{d}x = 1,
\int \prod_{x \in \Sigma_{g}}\! \! d c^{\alpha}( x ) c ( x_{i}) \left( 1 + S_{\mathrm{q}}( c ) \right) = \int \prod_{x \in \Sigma_{g}\setminus x_{i}}\! \! d c^{\alpha}( x ) e^{- S_{\mathrm{q}}},
\frac{1}{4}\gamma_{a b}\psi_{A}\overline{{\psi}}_{B}\gamma^{a b}\psi_{C}\epsilon^{A C}- \frac{1}{2}\gamma_{a}\psi_{A}\overline{{\psi}}_{B}\gamma^{a}\psi_{C}\epsilon^{A C}+ 3 \psi_{C}\overline{{\psi}}_{B}\psi_{A}\epsilon^{B C}= 0
\sum_{j=1}^{2^n}\begin{bmatrix}P_{n|s_0}^{-1}\end{bmatrix}_{j,k}\exp_2\left(-\sum_{i=1}^M \begin{bmatrix}P_{n|s_0}^{-1}\end{bmatrix}_{j,i}H(Y^n|X^n = x_i^n)\right).
{\frac{I^{2}}{c^{2}}}\left( 1 -{\frac{c^{2}B}{I^{3 / 2}}}\varrho \right) \ddot{\Phi}+{\frac{3 I}{c^{2}}}{\frac{\dot{a}_{0}}{a_{0}}}\dot{\Phi}\left( 1 +{\frac{B}{\sqrt{I}}}\wp \right) + V^{\prime}( \Phi ) = 0 .
v_{5,1,2}(t,r) = \frac{-1}{2\pi}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}&\int_{0}^{2\pi}d\theta \left(N_{2}(f_{v_{5}}^{\lambda_{1}})-N_{2}(f_{v_{5}}^{\lambda_{2}})\right)(s,\sqrt{r^{2}+\rho^{2}+2 \rho r \cos(\theta)}) \frac{(r+\rho \cos(\theta))}{\sqrt{r^{2}+2 r \rho \cos(\theta) + \rho^{2}}}
\begin{aligned}\sum_{i\in I_y}|u(y)-m^\gamma_u(2B_i)| \le\sum_{i\in I_y}\Big(|u(y)-m^\gamma_u(B(y,R))|+|m^\gamma_u(B(y,R))-m^\gamma_u(2B_i)|\Big) &\le C2^{-ls}\sum_{j=l-4}^l M_{\gamma/C}\,g_j(y) &\le C2^{-ks}\sup_{j\ge k}2^{(k-j)s}M_{\gamma/C}\,g_j(y) &\le Cd(x,y)^s\tilde g_k(y).\end{aligned}
c_i(f)\Phi(0,0)+c_{i-1}(d_{i,i-1}(f))\Phi(0,0) &-c_i(f) \mathcal{F}_\wedge(\Phi)(0,0)-c_{i-1}(d_{i,i-1}(f))\mathcal{F}_\wedge(\Phi)(0,0) &=c_i(f)+c_{i-1}(d_{i,i-1}(f))
<{\cal O}({\bf x}){\cal O}({\bf y})>\;\propto\;\sum_{\gamma_{1},\gamma_{2}\in\Gamma}\,\frac{|\gamma'_{1}({\bf x})|^{\Delta}|\gamma'_{2}({\bf y})|^{\Delta}}{|\gamma_{1}{\bf x}-\gamma_{2}{\bf y}|^{2\Delta}}\;=\;\sum_{\gamma_{1},\gamma_{2}\in\Gamma}\,\frac{|(\gamma_{1}^{-1}\gamma_{2})'({\bf y})|^{\Delta}}{|{\bf x}-\gamma_{1}^{-1}\gamma_{2}{\bf y}|^{2\Delta}}
D:=(\begin{matrix}d_{1,1}&d_{1,2}&\cdot\cdot\cdot&d_{1,M}\\d_{2,1}&d_{2,2}&\cdot\cdot\cdot&d_{2,M}\\ \vdots&\vdots&&\vdots\\d_{M,1}&d_{M,2}&\cdot\cdot\cdot&d_{M,M}\end{matrix}).
L_g(u) \leq \sum_{P^+(n) \leq u}\frac{g(n)}{n}= \prod_{p \leq u}\left(1+\frac{g(p)}{p}+ \sum_{k \geq 2}\frac{g(p^k)}{p^k}\right) \leq P_g(u)\prod_{p \leq u}\left(1+\sum_{k \geq 2}\frac{g(p^k)}{p^k}\right) \leq e^SP_g(u).
{\frac{g_{\tiny \mathrm{U V}}^{( a )}}{g_{\tiny \mathrm{I R}}^{( a )}}}={\frac{\sin{\frac{\pi a}{k + l + 2}}}{\sin{\frac{\pi}{k + l + 2}}}}{\frac{\sin{\frac{\pi}{k + 2}}}{\sin{\frac{\pi a}{k + 2}}}}={\frac{g_{( 1 , a )}^{[ l , k ]}}{g_{( a , 1 )}^{[ l , k ]}}},
S_{\mathrm{l i n e a r}}= m_{0}( h_{0 i , j}( 0 ) + C_{i , j}( 0 ) ){\frac{i}{8}}\theta \gamma^{i j}\theta + m_{0}( B_{i j , k}( 0 ) + C_{0 i j , k}^{( 3 )}( 0 ) ){\frac{i}{1 6}}\theta \gamma^{i j k}\theta +{\cal O}( \theta^{4})
T_\psi^{U_0,U_1,U_2}(X,Y)=\sum_{i,j,k=1}^{n}\psi(z_i^{(0)},z_k^{(1)},z_j^{(2)})P_i^{(0)}X P_k^{(1)}YP_j^{(2)}, X,Y\in B(\ell^2_n).
S_1=\{z:|\zeta-z|\leq 2|x-y|\}, & S_2=\Big\{z:2|x-y|\leq |\zeta-z|\leq\frac{4}{|x-y|}\Big\}& S_3=\Big\{z:|\zeta-z|\geq \frac{4}{|x-y|}\Big\}.
F(\lambda)=\left(\sum_{s \in S_\mu \setminus \lambda}|S_\mu(s-1) \setminus \lambda|\right)-\left(\sum_{t \in T_\mu \setminus \lambda}|T_\mu(t+1) \setminus \lambda|\right)+\frac{|\lambda|}{2}\left(|S_\mu \setminus \lambda| - |T_\mu \setminus \lambda|\right).
\int_{-\infty}^{\infty}\Phi_{z^{\prime}}^{\ast \;\mu}(\eta )\Phi_{z}^{\mu}(\eta )\;d\eta =\delta (z-z^{\prime})\,,\;\int_{-\infty}^{\infty}\Phi_{z}^{\ast \;\mu}(\eta^{\prime})\Phi_{z}^{\mu}(\eta)\;dz=\delta (\eta -\eta^{\prime})\,.
|v_t|^2_0 + \int_0^t |\partial_x v_r|^2_0 dr \lesssim (v_0^2,m_0) - 2 \int_0^t ( u_r^{(1)}\partial_x v_r + v_r \partial_x u^{(2)}_r ,v_r m_r) dr = (u_0^2,m_0) + \int_0^t ( (v_r)^2, \partial_x(u_r^{(1)}m_r)) dt + 2 \int_0^t ( (v_r)^2 \partial_x u^{(2)}_r , m_r) dr .
\tilde{e}(t)=e(t_{k})-e(t) \tilde{\eta}(t)=\eta(t_{k})-\eta(t) \tilde{\xi}(t)=\hat{\xi}(t_{k})-\hat{\xi}(t),~\forall t\in[t_{k},t_{k+1}),~k\in\mathbb{S}.
\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) .
(x_1,x_2,x_3) = \sigma^{(g_1,g_2,g_3)}(x_1,x_2,x_3) = (\sigma^{(g_1,1_{G_2},1_{G_3})}(x_1), \sigma^{(1_{G_1},g_2,1_{G_3})}(x_2) \sigma^{(1_{G_1},1_{G_2},g_3)}(x_3)).
\sum_{k_{1}=0}^{\infty}|\sum_{k_{2}=0}^{k_{1}}\cdot\cdot\cdot\sum_{k_{n}=0}^{k_{n-1}}a_{1,k_{n}}a_{2,k_{n-1}-k_{n}}\cdot\cdot\cdot a_{n,k_{1}-k_{2}}|
<\overrightarrow{\mu}>=\frac{e}{m}\biggl\{1+\frac{\alpha}{2\pi}-\frac{m^{2}e^{2}|\widetilde{q}|^{2}}{48\pi^{2}}\biggl[\ln\frac{m^{2}|\widetilde{q}|^{2}}{4}-1+2\gamma_{E}\biggr]\biggr\}\overrightarrow{s}+\frac{ie^{2}(\widetilde{q\hspace{-0.2cm}/})}{4\pi^{2}|\widetilde{q}|^{2}}\overrightarrow{\theta}
\{F, G \}_\pm = \pm C_{I J}^K \frac{\partial F}{\partial \mu_I}\frac{\partial G}{\partial \mu_J}\mu_K \pm \rho^i_I \biggl( \frac{\partial F}{\partial \mu_I}\frac{\partial G}{\partial q^i}- \frac{\partial F}{\partial q^i}\frac{\partial G}{\partial \mu_I}\biggr) ,
\underset{s=-\ell}{\mathrm{Res}}\zeta_{U}\left( s,z\right) =\frac{1}{\log a}\left( z^{\ell}+ \sum_{j=1}^{\ell}\frac{(-\ell)_j}{j!}(a-1)^{-j}z^{\ell-j}\right) &= \frac{1}{\log a}\sum_{j=0}^{\ell}\binom{\ell}{j}\left(\frac{1}{1-a}\right)^j z^{\ell-j}= \frac{1}{\log a}\left(z+\frac{1}{1-a}\right)^\ell.
H_{0_{B}}= H_{0}+ \int d^{3}x \left( \eta_{1 a}^{i}\mathcal{P}_{2 i}^{a}- \eta_{1 a}\mathcal{P}_{2}^{a}- \eta_{2 a}\partial^{i}\mathcal{P}_{2 i}^{a}+ \eta_{2 a}^{i}\partial_{i}\mathcal{P}_{2}^{a}\right) \equiv \int d^{3}x h_{0_{B}},
D_{nt}=\sum_{r,s\in \mathbb{Z}}b_{n,r,s}^{2\eta}b_{n,r,s}^{t-2\eta}\leq \bigg(\sum_{r,s\in \mathbb{Z}}b_{n,r,s}^{2}\bigg)^{\eta}\bigg(\sum_{r,s\in \mathbb{Z}}b_{n,r,s}^{(t-2\eta)/(1-\eta)}\bigg)^{1-\eta}.
{\widetilde{\cal K}}_{a}={\frac{( \sqrt{t})^{- 2 d}}{\eta^{2 d}({\mathrm{e}}^{- 2 \pi t})}}\left( ( 2 \sqrt{t})^{d ( J_{a})}V ( J_{a}) \right) \sum_{{\widetilde p}\in 2 \Lambda ( J_{a})}\exp ( - \pi t{\widetilde p}^{2}) \left({\frac{( \sqrt{t})^{d ( R J_{a})}}{V ( R J_{a})}}\right) \sum_{{\widetilde w}\in{\widetilde \Lambda}( R J_{a})}\exp ( - \pi t{\widetilde w}^{2}) ~ .
S [ q ] = \lim_{T \rightarrow \infty}\left\{\int_{- T}^{T}d t \frac 1 2 m \dot{q}^{2}- \frac 1 4 m \omega_{0}^{2}\ell_{0}^{2}T \left({\frac{1}{T}}\int_{- T}^{T}d t{\frac{q^{2}}{\ell_{0}^{2}}}\right)^{2}\right\},
\sum_{i=1,2}\|\varrho(b_i(t,y,z)-b_i(t,y,0))\partial_i\tilde{\zeta}_\mathbf{n}^\Psi\|{_{\mathcal{H}^{1,\i}}^2}\leq\Lambda(\frac{1}{c_0},Y_m(t)),~m\geq6.
\hat E_1(v) =-y_2=:E_1(\bar v), \hat E_2(v) =y_1-zx =y_1-(z_0-(y_1+y_2+y_3))(x_0+\xi_0-(\xi+y_1+y_2))=:E_2(\bar v) \hat E_3(v) =y_2-y_1r =y_2-y_1(r_0-(y_2+y_3))=:E_3(\bar v) \hat E_4(v) =y_3-zr =y_3-(z_0-(y_1+y_2+y_3))(r_0-(y_2+y_3))=:E_4(\bar v).
S_n(f_m,t_l)\to f_m(t_l)\qquad\,\,n \to+\infty,\,\,n\in\bigcup_{j=1}^{+\infty}\mathcal B_{m_1,j}, &1\leq l\leq m_1,\,\,1\leq m\leq m_1-1.
\langle 2 , m_{1}; 1 , m_{2}| L , M \rangle \: \{T_{2 , m_{1}}^{[ 0 ]}, P_{1 , m_{2}| 0}^{[ 1 ]}\}\qquad ( \langle 2 , m_{1}; 1 , m_{2}| L , M \rangle \: \{S_{2 , m_{1}}^{[ 0 ]}, P_{1 , m_{2}| 0}^{[ 1 ]}\})
\Delta_{a}(x) = \frac{x^{\frac14 +\frac{a}{2}}}{\pi\sqrt{2}}\sum_{n\leq N}\frac{\sigma_{a}(n)}{n^{\frac{3}{4}+\frac{a}{2}}}\cos\left(4\pi\sqrt{nx}-\frac{\pi}{4}\right)+ O\left(x^{\frac12 + \varepsilon}N^{-\frac12}\right),
4(\ell-k) \sum_{t=0}^{2 \ell}\beta_t^{(k)}p_t = 4(\ell - k)\varphi_0(p_k)= 2\varphi_0[h,p_k] =2(\ell-k) \alpha p_k+[h, \varphi_0(p_k)] =2(\ell-k) \alpha p_k+2(\ell-t) \sum_{t=0}^{2 \ell}\beta_t^{(k)}p_t,
m_{\pi_{0}}^{2}m_{\eta}^{2}m_{\eta^{\prime}}^{2}\ \propto \M ( m_{u}m_{d}+ m_{u}m_{s}+ m_{d}m_{s}) - m_{u}( m_{d}- m_{s})^{2}- m_{d}( m_{u}- m_{s})^{2}- m_{s}( m_{u}- m_{d})^{2}
\kappa = \displaystyle \frac{2 \Gamma ( h / 2 )}{\pi \Gamma ( 1 - h / 2 )}\left( \displaystyle \frac{\sqrt{\pi}}{2 \Gamma \left( \displaystyle \frac{1}{2 - h}\right) \Gamma \left( \displaystyle \frac{h}{4 - 2 h}\right)}\right)^{2 - h}.
{\overline{Q}}_a = \Big(\frac{{\mbox{\ae}}R^{-1/2}P}{\sqrt{6}}+ i \frac{\sqrt{k}\sqrt{6R}}{{\mbox{\ae}}}\Big ){\overline{\lambda}_a}- 3i{\mbox{\ae}}(6 R)^{-3/2}({\overline{\lambda}}{\overline{\lambda}})\lambda_a,
[D^{-1}+D^{-1}C(A-BD^{-1}C)^{-1}BD^{-1}](D-CA^{-1}B) = I-D^{-1}CA^{-1}B+D^{-1}C(A-BD^{-1}C)^{-1}B -D^{-1}C(A-BD^{-1}C)^{-1}[(BD^{-1}C-A)+A]A^{-1}B = I.
J_t(t;\mathbf{u},\mathbf{v}) =(\rho t^{-\beta}e^{-\delta t}\nabla \mathbf{u}(0),\nabla\mathbf{v})+\bigg(\rho\int_0^t(t-s)^{-\beta}e^{-\delta(t-s)}\nabla \mathbf{u}_t(s)ds,\nabla\mathbf{v}\bigg) &=\rho t^{-\beta}e^{-\delta t}a(\mathbf{u}(0),\mathbf{v})+J(t;\mathbf{u}_t,\mathbf{v})
\begin{array}{lcccl}\hat G_{0}\equiv G_{0}\otimes \sigma_3, \hat G_{-1}\equiv G_{-1}\otimes \sigma_3, \hat G_{-\frac{1}{3}}\equiv G_{-\frac{1}{3}}\otimes \sigma_3, \hat G_{-\frac{2}{3}}\equiv G_{-\frac{2}{3}}\otimes \sigma_3\end{array}
- \frac{t^2}{4\alpha \Gamma(-\alpha)}\int\limits_0^{\infty}r^{-\alpha-2}\exp^{-\frac{t^2}{4r}}T(r) x \, dr \; + \; \frac{1}{\Gamma(-\alpha)}\int\limits_0^{\infty}r^{-\alpha-1}\exp^{-\frac{t^2}{4r}}x \, dr = \, \frac{t^2}{4\alpha \Gamma(-\alpha)}\int\limits_0^{\infty}r^{-\alpha-2}\exp^{-\frac{t^2}{4r}}\big( x - T(r) x \big) dr.