harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle \tilde{\Theta}_\pm \tilde{\Theta}_\pm D_a\varphi,\tilde{\Theta}_\pm B\varphi\rangle =\mp \langle D_a\varphi,\tilde{\Theta}_\pm B\varphi\rangle =\pm \langle D_a\varphi,B\tilde{\Theta}_\pm \varphi\rangle =\mp \langle B\varphi,D_a\tilde{\Theta}_\pm \varphi\rangle =-\langle B\varphi,\tilde{\Theta}_\pm D_a\varphi\rangle,
\nabla^{2}p-\frac{1}{c_{0}^{2}}\frac{\partial^{2}p}{\partial t^{2}}+\frac{\delta}{c_{0}^{4}}\frac{\partial^{3}p}{\partial t^{3}}=-\frac{\beta}{\rho_{0}c_{0}^{4}}\frac{\partial^{2}p^{2}}{\partial t^{2}}
G^{- 1}= ( G^{A B}) = \left( \begin{array}{c c}{{\widetilde{{g}^{i j}}}}&{{g^{i l}\partial_{l}\theta_{\gamma}g^{\gamma \beta}}}\\{{g^{j l}\partial_{l}\theta_{\gamma}g^{\gamma \alpha}}}&{{- g^{\alpha \beta}}}\\ \end{array}\right)
\mathcal{S}_{p}=\inf_{u\in D^{1,p}(\mathbb{R}^N)\setminus\{0\}}\frac{\|\nabla u\|^p_{L^p(\mathbb{R}^N)}}{\displaystyle\left(\int_{\mathbb{R}^N}|u|^\frac{N\,p}{N-p}\, dx\right)^\frac{N-p}{N}},
\partial_{\rho}(f)(x_1,\cdots,x_{p+1}) =\sum_{1\leq i < j \leq p+1}(-1)^{i+j}f([x_i,x_j],x_1,\cdots,\widehat{x_i},\cdots,\widehat{x_j},\cdots,x_{p+1}) +\sum_{i=1}^{p+1}(-1)^{i+1}\rho(x_i)f(x_1,\cdots,\widehat{x_i},\cdots,x_{p+1}),
\left[ G_{\varepsilon}^{\prime}\left( W_{\varepsilon,\xi_{0}}+\phi_{\varepsilon,\xi_{0}}\right) -G_{\varepsilon}^{\prime}\left( W_{\varepsilon,\xi_{0}}\right) \right] \left[ \left( \frac{\partial}{\partial y_{h}}W_{\varepsilon,\xi(y)}\right)_{|_{y=0}}\right] =o(1),
\begin{array}{l l l}{{\Delta = + 1 \quad}}&{{: \qquad}}&{{E_{n}^{+}= E_{n + 1}^{-}> 0 \quad , \qquad E_{0}^{-}= 0 \quad ,}}\\{{\Delta = - 1 \quad}}&{{: \qquad}}&{{E_{n}^{-}= E_{n + 1}^{+}> 0 \quad , \qquad E_{0}^{+}= 0 \quad ,}}\\{{\Delta = 0 \quad}}&{{: \qquad}}&{{E_{n}^{-}= E_{n}^{+}> 0 \quad ,}}\\ \end{array}
[t^r] \sum_{\lambda\in \mathcal{P}}\prod_{i\geq 1}\left[\lambda_i-\lambda_{i+2}+1 \atop \lambda_{i+1}-\lambda_{i+2}\right]_{t}q^{{\lambda_1\choose 2}+|\lambda|}= [t^r] \prod_{i\geq 1}\frac{1}{(1-q^{2i-1})(1-tq^{2i})}.
Im\{\zeta(\alpha,t)-z_j(t)\}= Im\Big\{\zeta(\alpha,0)-z_j(0)-\int_0^t \dot{z}_j(\tau)d\tau\Big\}+Im\int_0^t D_{\tau}\zeta(\alpha,\tau)d\tau -\int_0^t b(\alpha,\tau)Im\{\partial_{\alpha}\zeta(\alpha,\tau)\}d\tau.
C(\gamma) \le \frac{N(N-1)}{2}f\left(f^{-1}\left(\frac{N^2(N-1)}{2}f(\beta)\right)\right) =\frac{N(N-1)}{2}\cdot\frac{N^2(N-1)}{2}f(\beta)=\frac{N^3(N-1)^2}{4}f(\beta),
(a_{n-k}a_{k+2})(a_{n-k-1}a_{k+1}) =a_{n-k}(a_{k+2}a_{n-k-1}a_{k+1}) & =a_{n-k}(a_{k+1}a_{n-k-1}a_{k+2}) &=(a_{n-k}a_{k+1})(a_{n-k-1}a_{k+2})=\lambda_{k+1}\lambda_{k+2}
I_{\psi_{R(\varepsilon)}}- A(n,p,\ell,\psi_{R(\varepsilon)}) \le \varepsilon I_0 &\le 2^{\frac{1}{p-1}}\| \psi^{\ell}\|_{L^1(\mathbb{R}^n)}R(\varepsilon)^n &= 2^{\frac{1}{p-1}}\|\psi_{R(\varepsilon)}^{\ell}\|_{L^1(\mathbb R^n)}.
\sum_{\substack{a,b \in [n] a \neq b}}\sum_{\substack{c,d \in [n] c \neq d}}\{a,b\}\cdot \{c,d\}=\frac{1}{3}\sum_{\substack{a,b,c,d \in [n] a,b,c,d \,\mathrm{distinct}}}\{a,b\}\cdot \{c,d\}+ \{a,c\}\cdot \{b,d\}+ \{a,d\}\cdot \{b,c\}.
{}4\cos^2 \Bigl(\frac{\pi}{m+2}\Bigr) \leq \frac{n^2}{r}< 4\cos^2\Bigl(\frac{\pi}{m+3}\Bigr) , m \in{\mathbb N}\,,{}Q \in J_m \equiv \left\{2 \cos \Bigl(\frac{\pi}{k+2}\Bigr)\,, k=1,\ldots,m \right\}.
{\mathcal L}_+=h^\mu\partial_\mu+\lambda^\alpha_{\mathbf i}D_\alpha^{\mathbf i}\ ,\qquad\qquad{\mathcal L}_-={\bar h}^\mu\partial_\mu+{\bar\lambda}^{\mathbf i}_{\dot{\alpha}}{\bar D}_{\mathbf i}^{\dot{\alpha}}\ .
\max_{{\hat z_n^{\rm R}}\ge 0,{\hat z_n^{\rm R}}\ge0}\ln \left(\frac{1}{2}+\frac{1}{2}{\rm{erf}}\big(\sqrt{\hat z_n^{\rm R}}-\sqrt{P/2}\big)\right) + \ln\left(\frac{1}{2}+\frac{1}{2}{\rm{erf}}\big(\sqrt{\hat z_n^{\rm I}}-\sqrt{P/2}\big)\right) \mathrm{s.t.}~{\hat z_n^{\rm R}}+{\hat z_n^{\rm I}}= B_n.
\mu^{N,i}_{\boldsymbol{\alpha}^*_t}+\frac{1}{N-1}\underset{j\neq i}{\sum}h(\mu^{N,j}_{\boldsymbol{\alpha}^*_t})=\Delta_{*,t}^{N,i}:=-R^{-1}B\lambda^{N,i}_{\boldsymbol{y}_t^{*}},
d_1= \textrm{Im}\int_{-\frac 12(\omega_1+\omega_2)}^{\frac 12(\omega_1-\omega_2)}\psi(t) \textrm{d}t= -\int_{-\frac 12(\omega_1+ \textrm{R}\omega_2)}^{\frac 12(\omega_1-\textrm{Re}\omega_2)}\frac{\partial u}{\partial x_2}\left(x_1-\frac{i \textrm{Im}\; \omega_2}2 \right) \;\textrm{d}x_1.
P_r(t) = \bigl[(1-x^2)\bigl(1+t^3+(t+t^2)(1+x^2+x^{-2})\bigr)^{r}\bigr]_0 = \displaystyle\sum_{k=0}^rR_k\binom{r}{k}(1+t^3)^{r-k}(t+t^2)^k,
\mathfrak{F}_{\lambda}(\varphi)=\frac{1}{p}\,[\varphi]^p_{W^{s,p}(\mathbb{R}^N)}-\frac{\lambda}{p}\,\int_\Omega \frac{|\varphi|^p}{d_\Omega^{s\,p}}\,dx-\int_B \varphi\,dx,\mbox{for every}\varphi\in \mathcal{X}^{s,p}_0(\Omega; d_{\Omega}).
\begin{aligned}\sum_{j \in \mathbb{Z}}\overline{W}_{j -1}( \overline{W}_j - \overrightarrow{W}^\ell_j ) \varphi_j = \sum_{j \in \mathbb{Z}}\overline{W}_{j -1}\sum_{i = 0}^{\ell -2}( W_{j + i}- W_{j + i +1}) \psi_{i + 1}\varphi_j & = \sum_{k \in \mathbb{Z}}F_k ( W_k - W_{k + 1}) \end{aligned}
( - 1 )^{F}= \psi_{1 1}( - 1 )^{\sum_{m = 1}^{\infty}\psi_{- m}\cdot \psi_{m}},{G}= - \gamma_{0}\beta_{0}- \sum_{m = 1}^{\infty}\left[ \gamma_{- m}\beta_{m}+ \beta_{- m}\gamma_{m}\right] .
dU_{jk}(t)=\sum_{\ell=1}^N S^{-1}_{j \ell}(t) dS_{\ell k}(t) = \dfrac{(S^{-1}(t)dM(t)S(t))_{jk}}{\Lambda_k(t)-\Lambda_j(t)},1 \leq j \not= k \leq N, \, t \geq 0.
w_{\alpha}(z):=\sum_{j=1}^{n}g^{\bar j \alpha}(p) \left(K(z, p)^{-1}\left. \frac{\partial}{\partial \overline{t_j}}\right|_{t=p}K(z, t) - \left. \frac{\partial}{\partial \overline{t_j}}\right|_{t=p}\log K(t, t)\right),
J_{1}^{k}| \Psi^{a b}\rangle = J_{1}^{k}J_{- 1}^{( a}| \phi^{b )}\rangle = - f_{d}^{k ( a}f_{e}^{b ) d}| \phi^{e}\rangle +{\frac{k}{2}}g^{k ( a}| \phi^{b )}\rangle .
\begin{aligned}\lVert \zeta_{\eta,n}+ \zeta_{C,n}\rVert = \tfrac{1}{t_n}\lVert \bar{z}_n - z_n + t_n g_n^\xi \rVert \leq 2 L_{\eta}(z_n) + 3\lVert g_n^\xi\rVert,\end{aligned}
L_F=\bar\psi i\gamma^\mu (\partial_\mu-ieA_\mu)\psi -m\bar\psi\psi+{g\over 2}\mid\bar\psi\gamma^\mu\psi\mid^2 -{{pe^2}\over 4}\mid A_{\mu\nu}\mid^2+{{qe^2}\over 2}\epsilon_{\mu\nu\lambda}A^\mu A^{\nu\lambda}+J_\mu A^\mu.
\left[ \psi ( x ) , \psi ( y ) \right]_{\pm}= \frac{1}{( 2 \pi )^{\frac{3}{2}}}\int \left( e^{- i p ( x - y )}\pm e^{i p ( x - y )}\right) \frac{d^{3}p}{2 \omega}= i \Delta^{\left( + \right)}( x - y ) \pm i \Delta^{\left( - \right)}( x - y )
\mathrm{Ad}(\exp H) X_{\alpha, j}= \cos \langle \alpha, H \rangle X_{\alpha, j}+ \sin \langle \alpha, H \rangle Y_{\alpha, j}, \mathrm{Ad}(\exp H) Y_{\alpha, j}= -\sin \langle \alpha, H \rangle X_{\alpha, j}+ \cos \langle \alpha, H \rangle Y_{\alpha, j}.
\displaystyle{T(H \gg H_0)=T_1\frac{24m^4}{\omega^3}\bigg[\frac{\omega_1}{\omega_2 \sqrt{4m^2-\omega_2^2}}\mbox{arctan}\left(\frac{\omega_2}{\sqrt{4m^2-\omega_2^2}}\right)+ \frac{\omega_2}{\omega_1 \sqrt{4m^2-\omega_1^2}}\mbox{arctan}\left(\frac{\omega_1}{\sqrt{4m^2-\omega_1^2}}\right)-\frac{\omega}{4m^2}\bigg]}
\textstyle \sum_{j = 1}^\mu F_{\omega_j}{\omega_j \choose u-1}+ \sum_{l=\alpha-m}^\alpha \sum_{r - m > i(p-1)+\alpha > m}C_l{r-\alpha+l \choose i(p-1) +l}{i \choose u-1}\textstyle = \delta_{u=\alpha+1}(1 + (-1)^\alpha F_{\omega_1}+ \delta_{L = 2\alpha}F_{\omega_\mu}).
\hat{S}_{\lambda \lambda^{\prime}}= \left[ \nabla_{\lambda}\; , \; \nabla_{\lambda^{\prime}}\right] \hat{1}+ \frac{1}{2}\left( \nabla_{\lambda}\hat{E}_{\lambda^{\prime}}- \nabla_{\lambda^{\prime}}\hat{E}_{\lambda}\right) + \frac{1}{4}\left( \hat{E}_{\lambda}\hat{E}_{\lambda^{\prime}}- \hat{E}_{\lambda^{\prime}}\hat{E}_{\lambda}\right)
L_{1}\big(\eta^{(1)}\big) =\left( \prod\limits_{j=1}^{N}x_{j}^{(1)}\right)^{-\gamma\kappa}\rho\big(z_{1}^{-\kappa},z_{1}^{\kappa}\big) \left\{\alpha_{0}( x (\widetilde{u},0)) +\sum_{n=1}^{\infty}\alpha_{n}(x(\widetilde{u},0)) z_{1}^{n}\right\}.
\big|\Xi_m^{-1}(z)\big| =\big|\overline{\Xi_m(\overline{z})}\big| =|\Re z|^{-\Im z}\Big(1 +O\big(|z|^{-1}\big)\Big)\ \textrm{holds for}z\in\mathfrak S^1\ \textrm{as}|z|\to \infty.
a_{f}(t)=-\frac{i}{\hbar}\int_{0}^{t}\langle f|H^{\prime}(t^{\prime})|i\rangle e^{i\omega_{fi}t^{\prime}}dt^{\prime}=-\langle f|F|i\rangle\frac{e^{i(\omega_{fi}-\omega)t}-1}{\hbar(\omega_{fi}-\omega)}
=\int_{\textstyle\prod_{t=1}^m\ [-(\eta_t-b_{t,1}),b_{t,2}-\eta_t]}\ \frac{dv_1dv_2\ldots dv_m}{\displaystyle \sum_{t=1}^m\left(v_t^2-\gamma_t^2\right)^{2s}}&\le\int_{\textstyle\prod_{t=1}^m\ [-(b_{t,2}-b_{t,1}),b_{t,2}-b_{t,1}]}\ \frac{dv_1dv_2\ldots dv_m}{\displaystyle \sum_{t=1}^m\left(v_t^2-\gamma_t^2\right)^{2s}},
Z(n+1,k)-Z(n,k)\leq Z(n+1,k-1)-Z(n,k-1)+\begin{cases}0, \mathrm{if}\ \ell(n)\in[0:k-2], begin{align*}0.5em]1, \mathrm{if}\ \ell(n)=k-1, begin{align*}0.5em]0, \mathrm{if}\ \ell(n)\in[k:p].\end{cases}
e^{{\cal I}( i \alpha_{b}+ j \beta_{b}+ k \gamma_{b}) + i \alpha_{r}+ j \beta_{r}+ k \gamma_{r}}({\cal I}c t + i x + j y + k z ) e^{{\cal I}( i \alpha_{b}+ j \beta_{b}+ k \gamma_{b}) - i \alpha_{r}- j \beta_{r}- k \gamma_{r}}\; \; .
\frac{q(q^2;q^2)_\infty^3}{(-q;q^2)_\infty^2}\omega(q) =-3\sum_{\substack{n=-\infty n\neq 0}}^\infty \frac{nq^{3n^2+2n}}{1+q^{6n}}-\sum_{n=-\infty}^\infty \frac{(3n+2)q^{3n^2+8n+4}}{1+q^{6n+4}}&=-3\sum_{n=-\infty}^\infty \frac{nq^{3n^2+2n}}{1+q^{6n}}+\sum_{n=-\infty}^\infty \frac{(3n+1)q^{3n^2+4n+1}}{1+q^{6n+2}}.
\mathbb{E}[Y\tilde{X}]= \mathbb{E}[\mathbb{E}[Y\tilde{X}|X]] = \mathbb{E}[\rho X\tilde{X}] = \rho\mathbb{E}[\mathbb{E}[ X\tilde{X}|\tilde{X}]] = \rho\mathbb{E}[\tilde{X}^2].
\chi_n = \omega_n - \sum_{k=0}^n \frac{1}{\sqrt{2k+1}}= \omega_n - \left(1-\frac{1}{\sqrt{2}}\right)\sum_{k=1}^{n}\frac{1}{\sqrt{k}}- \sum_{k=n+1}^{2n+1}\frac{1}{\sqrt{k}};
A_{p}(\chi)=\lim_{\zeta\to\chi}(\chi/\zeta) (1+\zeta/\chi)^{2}R_{p}(\chi)X(\zeta), &B_{p}(\chi)=\lim_{\zeta\to\chi}D_{\zeta}(\chi/\zeta)^2 (1+\zeta/\chi)^{4}R_{p}(\chi)X(\zeta).
\varphi = g_1*\Phi_1(\varphi)+\ldots+g_N*\Phi(\varphi) &= g_i*(\rho\cdot \Psi_1(\varphi))+\ldots+g_N*(\rho\cdot \Psi_N(\varphi)) &= f_1*\Psi_1(\varphi)+\ldots+f_N*\Psi(\varphi)
f''(h_0,\ldots, h_{n-1})(\nu) &=\sum_{i_0,\ldots, i_{n-1}}\langle f'(g_{i_0},\ldots, g_{i_{n-1}}), \nu|_{[i_0,\ldots, i_{n-1}]}\circ L\rangle &=\sum_{i_0,\ldots, i_{n-1}}\sum_{t_0,\ldots, t_{n-1}}\langle f(h_{t_0}, \ldots, h_{t_{n-1}}),(\nu|_{[i_0,\ldots, i_{n-1}]}\circ L)|_{[t_0,\ldots, t_{n-1}]}\pi \rangle.
\bigl \{T_{a}, T_{b}^{( 1 )}\bigr \}_{( \phi , \pi )}+ \bigl \{T_{a}^{( 1 )}, T_{b}\bigr \}_{( \phi , \pi )}+ \bigl \{T_{a}^{( 1 )}, T_{b}^{( 2 )}\bigr \}_{( \eta )}+ \bigl \{T_{a}^{( 2 )}, T_{b}^{( 1 )}\bigr \}_{( \eta )}= 0 .
H_{0}^{\prime}=\frac{1}{2}\int d^{3}x\left(B_{a}^{ij}F_{ij}^{a}-A_{i}^{a}A_{a}^{i}+\left( \partial^{i}B_{0i}^{a}\right) \left( \partial_{j}B_{a}^{0j}\right) \right) \equiv\int d^{3}x\,h_{0}^{\prime},
\begin{array}{l}(H^{n})_{W}= H^{n}+ \frac{1}{24}(2n-1)n(n-1)H^{n-2}+\frac{1}{5760}(20n^{2}-48n+7)\prod_{k=0}^{3}(n-k) H^{n-4}+\frac{1}{2903040}(280n^{3}-1596n^{2}+1874n-93)\prod_{k=0}^{5}(n-k)H^{n-6}+... \end{array}
tr_N(BA) =\sum_{u\in N}\langle u|BAu\rangle =\sum_{u\in N_e}\langle u|Bu\rangle s(u)=\sum_{u\in N_e}\langle Au|Bu\rangle =\sum_{u\in N}\langle Au|Bu\rangle= &=\sum_{u\in N}\langle u|ABu\rangle=tr_N(AB)\:,
\Omega_{i_{1}\dots i_{2 m - 1}}^{( 2 m - 1 )}= f_{[ i_{1}i_{2}}^{l_{i}}\dots f_{i_{2 m - 3}i_{2 m - 2}}^{l_{m - 1}}t_{i_{2 m - 1}] l_{i}\dots l_{m - 1}}\ .
+\frac{2}{9}Y(z)^6\cdot{}_2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{Y(z)^{12}}{27}\right)=\frac{\sqrt{3}\Gamma\left(\frac{1}{3}\right)^3}{2\pi \sqrt[3]{2}}.
\frac{\prod_{i=1}^{[\frac{k}{2}]+[\frac{n-k}{2}]}(1-t^{4i})}{\prod_{i=1}^{[\frac{k}{2}]}(1-t^{4i})\prod_{i=1}^{[\frac{n-k}{2}]}(1-t^{4i})}=\sum_{d\geq 0}p\Big(\big[\frac{k}{2}\big], \big[\frac{n-k}{2}\big]; d\Big)t^{4d}
C_k = \max_{p_X(x)}\left[ \mathbb{I}(X,X+N_{eq})-\mathbb{I}(X;Z)\right] & \geq \max_{p_X(x)}\mathbb{I}(X,X+N_{eq}) - \max_{p_X(x)}\mathbb{I}(X;Z) &= C_{BE}-C_{E}\triangleq C_{k,1}^{LB},
R=I+(\theta-\frac{\theta^{3}}{3!}+\frac{\theta^{5}}{5!}-\cdot\cdot\cdot)K+(\frac{\theta^{2}}{2!}-\frac{\theta^{4}}{4!}+\frac{\theta^{6}}{6!}-\cdot\cdot\cdot)K^{2},
|\psi (t)-\psi (u)| =1-\psi(u) =1-\dfrac{b_{2n}-u}{b_{2n}-a_{2n}}=\dfrac{u-a_{2n}}{b_{2n}-a_{2n}}& < \dfrac{u-t}{b_{2n}-a_{2n}}\leq \dfrac{t^{1/k}}{b_{2n}-a_{2n}}= \dfrac{t^{1/k}}{(a_{2n})^{2/k}}& < \dfrac{(a_{2n})^{1/k}}{(a_{2n})^{2/k}}=\dfrac{1}{(a_{2n})^{1/k}}<\dfrac{1}{2^{2n}}<\dfrac{1}{2^{2n-1}}.
\left| e^{\sqrt{2}\widehat{h}(x_0)}- e^{\sqrt{2}h(f(x_0))}\right| = f(x_0) \left| e^{\sqrt{2}\widehat{h}(x_0) - \sqrt{2}h(f(x_0))}- 1 \right| \lesssim f(x_0)^\frac{2\beta}{2\beta+1}(\log n/n)^{\frac{\beta}{2\beta+1}}.
d \mu_{\Lambda}= Z^{- 1}e^{- \int_{\Lambda}d^{2}x : \lambda \phi ( x )^{4}+ \sigma \phi ( x )^{2}- \frac{m^{2}}{2}( \phi ( x ) - \xi )^{2}:_{m_{0}}+ \delta V}d \mu_{m , \xi},
- \int_0^1 r^{d-1}(1-r^2)^{\mu + 1/2}\partial_r^2 f (r\xi) g(r\xi) \, dr & =\int_0^1 r^{d-1}\partial_r f (r\xi) \partial_r g(r\xi) (1-r^2)^{\mu + 1/2}\, dr & + \int_0^1 r^{d-1}\Big( \frac{d-1}{r}(1-r^2) - (2\mu + 1)r \Big)\partial_r f (r\xi) g(r\xi) (1-r^2)^{\mu - 1/2}\, dr.
I ( 0 , 0 , N - k ) \, = \, 2 \pi \int_{0}^{\infty}\frac{r}{( 1 + r^{2})^{N - k + 2}}\, d r \, = \, \frac{- \pi}{N - k + 1}\ \bigg [ \frac{1}{( 1 + r^{2})^{N - k + 1}}\bigg ]_{0}^{\infty}= \, \frac{\pi}{N - k + 1}
\begin{aligned}T_1(x,y):= \frac{\partial P_1}{\partial x}+ \frac{\partial Q_1}{\partial y}, D_1(x,y):= \frac{\partial P_1}{\partial x}\frac{\partial Q_1}{\partial y}-\frac{\partial P_1}{\partial y}\frac{\partial Q_1}{\partial x}\end{aligned}
s+\tau \leq \sigma \textmd{(regularity requirement from the energy conservation)}, &(\beta-\frac{1}{2})+=\tau \textmd{(Sobolev embedding in 1D)}, & \beta+s=\frac{d}{2}-d(1-\beta) \textmd{(radial Sobolev embedding)}, & (2+p)\beta+1=2+r \textmd{(the relation between $\beta$ and $r$)}.
\begin{array}{cc}^{\rho}D^{\beta(1-\alpha)}_{0_+}\Big(f(t,u(t),^\rho D^{\alpha,\beta}u(t))-p(t)u(t)\Big) =\delta_{\rho}\^{\rho}I^{1-\beta(1-\alpha)}_{0_+}\Big(f(t,u(t),^\rho D^{\alpha,\beta}u(t))-p(t)u(t)\Big)\in C_{1-\gamma,\rho}[0,T]. \end{array}
\overline{P}=\varepsilon^{\mu ,\mu}+\varepsilon^{\widetilde{\mu}, \widetilde{\mu}}\hspace{0.5in}P=\varepsilon^{0,0}+\frac12\varepsilon^{[\mu \nu ],[\mu \nu ]}+\varepsilon^{\widetilde{0},\widetilde{0}}
\ell_{2m+\tau}(y) := y^{(2m + \tau)}+ \sum_{k = 0}^{m-1}(-1)^{i_{2k}+ k}(\sigma_{2k}^{(i_{2k})}(x) y^{(k)})^{(k)}+ \sum_{k = 0}^{m + \tau - 2}(-1)^{i_{2k+1}+ k + 1}\bigl[(\sigma_{2k+1}^{(i_{2k+1})}(x) y^{(k)})^{(k+1)}+ (\sigma_{2k+1}^{(i_{2k+1})}(x) y^{(k+1)})^{(k)}\bigr], \: x \in \mathbb R_+,
X \cdot \nabla_X \varphi = a^2 e_1\cdot\nabla_{e_1}\varphi + b^2 e_2 \cdot\nabla_{e_2}\varphi + ab (e_1 \cdot \nabla_{e_2}\varphi + e_2 \cdot \nabla_{e_2}e_1) &=e_1 \cdot \nabla_{e_1}\varphi = e_2 \cdot \nabla_{e_2}\varphi.
\begin{array}{ccc}\alpha_1\to \alpha_1+\omega_1 \alpha_2, \ \ \ \beta_1\to c~ \beta_1+\omega_2\beta_2 \alpha_2\to c~ \alpha_2\phantom{\omega_2c \alpha_2}, \ \ \ \ \ \ \ \beta_2\to \beta_2\phantom{nnc+\omega_2 (\beta_2+1)}. \end{array}
\Delta A = \frac{3 2 \pi L}{\sqrt{4 - \beta^{2}}}\mathrm{a r c t a n h}\sqrt{\frac{2 - \beta}{2 + \beta}}\to 8 \pi L \log \frac{4}{\beta}\, \, \, \mathrm{a t}\, \, \, \beta \to 0 .
T_{\alpha}, \quad G_{a}= \frac{1}{1 2}\sigma_{a}^{\alpha \dot{\beta}}( R_{\, \, \dot{\beta}\alpha \gamma}^{\gamma}+ R_{\, \, \alpha \dot{\beta}\dot{\gamma}}^{\dot{\gamma}}) , \quad W_{\alpha \beta \gamma}= \sigma_{( \alpha \dot{\delta}}^{a}\sigma_{\beta}^{b \dot{\delta}}T_{a b \gamma )}
s ( \psi ) = \left( \frac{2}{k + 2}\right)^{\frac{1}{4}}\sum_{j = 0}^{k / 2}e^{\frac{- j^{2}}{k}}( - 1 )^{j}\, c o t^{1 / 2}\left( \frac{( 2 j + 1 ) \pi}{2 ( k + 2 )}\right) \sqrt{2 j + 1}\ \, \frac{\sin ( 2 j + 1 ) \psi}{\sin \psi}.
p^{\nu_3}(3,t)=\frac{\lambda_1}{\lambda_3}\sum_{k=2}^{\infty}(-1)^k\underset{\Lambda^{k}_{3}}{\sum}\frac{\lambda_1^{k_1}\lambda_2^{k_2}\lambda_3^{k_3}t^{k_1\nu_1+k_2\nu_2+k_3\nu_3}}{\Gamma\left(k_1\nu_1+k_2\nu_1+k_3\nu_3+1\right)},
{}_4\phi_3\!\left[\begin{matrix}(1-r)^j,1-a,b,c d,e,(1-r)^{j+1}(1-a)bc/de\end{matrix};1-r,1-r\right] &=\frac{((1-r)/e,(1-r)(1-a)bc/de;1-r)_j}{((1-r)(1-a)/e,(1-r)bc/de;1-r)_j}&\quad\;\times{}_4\phi_3\!\left[\begin{matrix}(1-r)^j,1-a,d/b,d/c d,de/bc,(1-r)^{j+1}(1-a)/e\end{matrix};1-r,1-r\right].
F_{2}^{\prime}=\gamma F_{2}+i\beta\gamma G_{3}=\frac{F_{2}-\dfrac{i}{c}\left( \mathbf{v\times G}\right)_{2}}{\sqrt{1-v^{2}/c^{2}}}, F_{3}^{\prime}=\gamma F_{3}-i\beta\gamma G_{2}=\frac{F_{3}-\dfrac{i}{c}\left( \mathbf{v\times G}\right)_{3}}{\sqrt{1-v^{2}/c^{2}}}.
\chi_{\sigma u}=\zeta[c]\int\!\frac{\rm d^3k}{(2\pi)^3}\exp(ik_\mu u_\mu)\exp(-ik_\mu\epsilon_{\mu\nu\lambda}\partial_\nu L^3_\lambda)\exp(i\int\!{\rm d^3x}\,j^c_\mu L^3_\mu)~.
\langle\frac{\delta S}{\delta Q_{\mu}^i(x)}\rangle=Z^{-1}\int [dQ][dC][d\overline{C}]\left(\frac{\delta (S_0+S_{fix}+S_{ghost})}{\delta Q_{\mu}^i(x)}+J^{\mu ,i}(x)\right)\exp (iS)=0.
\frac{d^2\Theta(s)}{ds^2}+\frac{2s+1}{s(1+s)}\frac{d\Theta(s)}{ds}+\left[\frac{(-\ell(\ell+1))s^2+(-\ell(\ell+1)-\gamma/2)s+(-[m^2+\beta+\gamma]/4)}{s^2(1+s)^2}\right]\Theta(s)=0.
\pi_{1+\sigma}= -q_2 \pi_{2+\sigma}= d_{1,0}- a_{1,1}q_2 \pi_{2+2\sigma}= 4d_{1,1}+2q_4 - 2q_2q_3 - q_2^3 + (6b_1^3 -18 b_1b_2 + 6b_3)q_1.
\widetilde{\mathcal{E}}(w)=\big(K_{\overline X}\otimes\overline{K}_{\overline X}\big)^{-w/(n+2)}, \mathcal{E}(w)=\big(K_M\otimes\overline{K}_M\big)^{-w/(n+2)}\cong\widetilde{\mathcal{E}}(w)|_M
\mathfrak{w}_{n,m}= \frac{1}{\tau}\int\limits^{t_{n}}_{t_{n-1}}\int\limits_{t_{m-1}}^{t_{m}}\rho_{\alpha}(t-s)dsdt =\frac{\left[(t_{n}-t_{m-1})^{\alpha+1}-(t_{n}-t_{m})^{\alpha+1}\right] -\left[(t_{n-1}-t_{m-1})^{\alpha+1}-(t_{n-1}-t_{m})^{\alpha+1}\right]}{\tau\Gamma(2+\alpha)}
\hat h \circ \hat g = \sum_{i=0}^m \sum_{j=i}^{n+m}h_i \star \delta_0 \varphi^{j-i,m-i}_0 \x^{m+n-j}= \sum_{i=0}^m \sum_{s=0}^{n+m-i}h_i \star \delta_0 \varphi^{s,m-i}_0 \x^{m+n-i-s}.
(1,0)^{\perp}=-\cos\theta_1JE_1=-\sqrt{2}J\nabla(\sin\theta_1), (i,0)^{\perp}=-\sin\theta_1JE_1=\sqrt{2}J\nabla(\cos\theta_1), (0,1)^{\perp}=-\cos\theta_2JE_2=-\sqrt{2}J\nabla(\sin\theta_2), (0,i)^{\perp}=-\sin\theta_2JE_2=\sqrt{2}J\nabla(\cos\theta_2),
\begin{array}{c}f_{1}(u)=\int \sqrt{1-\frac{\lambda^{2}}{c^{2}}\sin^{2}\left( \frac{u}{c}\right)}\cos \alpha (u)du, f_{2}(u)=\int \sqrt{1-\frac{\lambda^{2}}{c^{2}}\sin^{2}\left( \frac{u}{c}\right)}\sin \alpha (u)du.\end{array}
0\rightarrow U'\stackrel{\begin{pmatrix}C' D'\end{pmatrix}}{\longrightarrow}V\oplus V\stackrel{(A,B)}{\longrightarrow}W ,V \stackrel{\begin{pmatrix}A B\end{pmatrix}}{\longrightarrow}W\oplus W\stackrel{(C'',D'')}{\longrightarrow}U''\rightarrow0.
|B_r|\|S_{B_r}f\|_{L^2_{avg}}^4 = |B_r|^{-1}\left(\int\sum_{\tau}|f_\tau|^2 W_{B_r}\right)^2 \lesssim \sum_{j=1}^{1000}|B_r|^{-1}\left(\int\sum_{\tau}|f_{j,\tau}|^2 W_{B_r}\right)^2 =\sum_{j=1}^{1000}|B_r|^{-1}\left(\int|f_{j}|^2 W_{B_r}\right)^2 .
\tau_{\nu,\nu}^1 = (1- q^{-1}) \dfrac{\mathbf{z}^{-n \lceil B(\alpha,\nu)/n \rceil \alpha}}{1 - q^{-1}\mathbf{z}^{-n\alpha}}= \dfrac{1-v}{1-v \mathbf{z}^{-n\alpha}}\begin{cases}\mathbf{z}^{-n\alpha}1 \end{cases}
E[F^{(M)}|\mathcal{F}_{t}]=P_{t}F^{(M)}=\sum_{n=0}^{\infty}\frac{1}{2^{n}}\int_{t}^{T}\ldots\int_{s_{n-1}}^{T}(D_{s_{n}}^{2}\ldots D_{s_{1}}^{2}F^{(M)})(\omega^{t})\,\mathrm{d}s_{n}\ldots \,\mathrm{d}s_{1}.
\delta ( \tilde{g}( A_{\nu}^{\omega}) ) \equiv \frac{\lim_{\alpha \rightarrow 0}e^{- \frac{1}{2 \alpha}\int g ( A_{\nu}^{\omega}) d x}}{\lim_{\alpha \rightarrow 0}\int{\cal D}\Lambda e^{- \frac{1}{2 \alpha}\int d x \Lambda^{2}}}
I_g=\frac{1}{2\pi \kappa}\int d^2 x \epsilon^{\mu\nu}\Bigl(\eta_a (\partial_\mu e^a_\nu + \omega_\mu \epsilon^a_{~b}e^b_\nu ) +\eta_2 \partial_\mu\omega_\nu+\eta_3 (\partial_\mu a_\nu+\frac{1}{2}\epsilon_{ab}e^a_\mu e^b_\nu)\Bigr)\,.
\phi_{m_{1}, n_{1}}\times \phi_{m_{2}, n_{2}}= \sum_{m_{3}= | m_{1}- m_{2}| + 1}^{m_{1}+ m_{2}- 1}\sum_{n_{3}= | n_{1}- n_{2}| + 1}^{n_{1}+ n_{2}- 1}\phi_{m_{3}, n_{3}},
\tilde{\mathfrak{L}}[\tilde{u}](x) = \varphi_{0}(\psi_{0}(\tilde{u})) + \alpha^{2}\left[ A(\tilde{u})\left( \frac{d\tilde{u}}{dx}\right)^{2}+ B(\tilde{u})\frac{d^{2}\tilde{u}}{dx^{2}}\right],
\begin{array}{@{}l@{}}{\displaystyle \partial U^{k}= (U^{k}-U^{k-1})/\Delta t, \quad\partial^{2}U^{k}=(\partial U^{k}-\partial U^{k-1})/\Delta t,}begin{align*}2mm]{\displaystyle \overline{U}^{k}= (U^{k}+U^{k-1})/2, \widetilde{U}^{k}=(U^{k}+U^{k-2})/2,}begin{align*}2mm]\end{array}
\widehat{\mathbf{c}}_{n}=\begin{cases}\mathbf{1}, 1\leq n\leq p, (-a_{n-p-1},1,\ldots,1), n\geq p+1,\end{cases}\widehat{\mathbf{d}}_{n}(z) =(0,\ldots,0,z), n\geq 1.
\Lambda(r \otimes x\cdot f) = \ \sum r_{1}\sigma\bigl(S(\pi(r_{2}))(x\cdot f)\bigr) = \sum (x\cdot r_{1})\sigma\bigl(S(\pi(r_{2})\cdot x^{-1})(x\cdot f)\bigr)= \ \sum(x\cdot r_{1})\sigma \bigl(x\cdot\bigl(S(\pi(r_{2})) f\bigr)\bigr)=x\cdot \sum r_{1}\sigma \bigl(S(\pi(r_{2})) f\bigr)= \x\cdot\Lambda(r \otimes f)
Q_{M}^{\max}( N , \{\nu_{M}\}, r ) = Q_{M - 1 / 2}^{\max}( N - 2 \mu , \{\nu_{M}^{\prime}\}, r - 1 ) , \quad \mu = \sum_{M}\nu_{M}, \quad \nu_{M}^{\prime}= \nu_{M + 1 / 2}.
0=[b,L_{1,j_{0}}]=[\sum_{i\in I}b_{i}L_{\alpha,i},L_{1,j_{0}}]=\sum_{i\in I}b_{i}[L_{\alpha,i},L_{1,j_{0}}]=\sum_{i\in I}b_{i}k_{\alpha}L_{\alpha+1,1+j_{0}}.
S_{3,A,V}(x, t; B, \Psi_m(T)) = i\bigg[\int_0^{\Psi_m(T)(t)}A(x+B(s))\cdot dB(s) +\frac{1}{2}\int_0^{\Psi_m(T)(t)}(\nabla\cdot A)( x+B(s)) ds\bigg] &+ \int_0^t V\Big(x+B\big(\Psi_m(T)(s)\big)\Big) ds &=: i\Big[ S_{1,A}^m(x,t; B,T)+S_{2,A}^m(x,t; B,T)\Big]+S_{3,V}^m(x,t;B,T).
\sum_{j\ge2}2^{nj/{q'}}\int_{2t_k}^\infty \Bigl(\frac{t_k}{t}\Bigr)^{M}e^{-\frac{c}{2}(2^jtk/t)}\frac{{\rm d}t}{t}=\sum_{j\ge2}2^{nj/{q'}}\int_0^{2^{j-1}}2^{-jM}s^{M}e^{-\frac{c}{2}s}\,\frac{{\rm d}s}{s}&\le \Bigl(\int_0^\infty s^{M-1}e^{-\frac{c}{2}s}\,{\rm d}s\Bigr)\Bigl(\sum_{j\ge2}2^{nj/{q'}}2^{-jM}\Bigr)<\infty.
\underset{z_n = 0}{}\frac{I_n(z_1,\dots,z_n)}{z_n}= \gamma^{-1}\cdot I_{n-1}(z_1,\dots,z_{n-1}) &\underset{z_1 = \infty}{}\frac{I_n(z_1,\dots,z_n)}{z_1}= - q^r \cdot I_{n-1}(z_2,\dots,z_n)
\frac{\delta \phi}{\delta J^{(0)}}=-D^{-1}=\setlength{\unitlength}{1mm}\begin{picture}(13,10)(-2,0)\thicklines\put(4,1){\circle{8}}\put(-0.25,1){\circle*{1}}\put(8.25,1){\circle*{1}}\end{picture}.
E_{P}\left[ T_{P,a,\mathcal{G}}|M_{k},_{\mathcal{G}}\left(M_{k}\right) \right] =\frac{I_{a}(A)}{\pi_{a}(\mathbf{O}_{min};P)}E_{P}\left[ Y|M_{k},_{\mathcal{G}}\left(M_{k}\right) \right].
f = \max{(f_1, f_2)}= \max{(p_1 - q_1, p_2 - q_2)}&= \max{\left(p_1 + q_2 - (q_1 + q_2), p_2 + q_1 - (q_1 + q_2)\right)}& = \max{(p_1 + q_2, p_2 + q_1)}- (q_1 + q_2)
\frac12\frac{d}{dt}\|D^m B\|_{L^2}^2+\int_{\mathbb S^1}[D^m, B]J_x D^mB\, dx+\int_{\mathbb S^1}BD^m J_xD^mB\, dx &-\int_{\mathbb S^1}[D^m, J]B_xD^mB\, dx-\int_{\mathbb S^1}JD^m B_xD^mB\, dx+\mu \|\Lambda^{\frac{\alpha}{2}}D^mB\|_{L^2}^2=0.
\tau^{(r)}_j := \sum_{l=r}^{j-1}\sum_{p=0}^{2^{\Delta^{(l)}}-1}\frac{2^{l-\Delta^{(l)}}}{2^l +p2^{l-\Delta^{(l)}}}= \sum_{l=r}^{j-1}\sum_{p=0}^{2^{\Delta^{(l)}}-1}\frac{1}{2^{\Delta^{(l)}}+ p}=: (j-r) \log 2 + \lambda_{j}^{(r)},
\sigma^{-1}:\mathbb{R}^{n}\rightarrow H_{R}^{1(n)};\sigma^{-1}(u)=(\tau(u),x(u))=(R\frac{R^{2}+|u|^{2}}{R^{2}-|u|^{2}},\frac{2R^{2}u}{R^{2}-|u|^{2}}).
Z^{4}=t^{2}\partial_{t}+H+\frac{t}{2c}K^{1}-\left( \frac{1}{2}\int \frac{dx}{\sigma \left( x\right)}^{2}+\frac{1}{2c}\int \frac{dx}{\sigma \left(x\right)}-ct\int \frac{dx}{\sigma \left( x\right)}+\frac{c^{2}}{2}t^{2}\right) F\partial_{F}
( t^{- 1})^{\prime}= t^{\mathrm{i j}}{\frac{\partial \pi^{\alpha}}{\partial p^{\mathrm{i}}}}{\frac{\partial \pi^{\beta}}{\partial p^{\mathrm{j}}}}{\frac{\partial}{\partial \pi^{\alpha}}}\otimes{\frac{\partial}{\partial \pi^{\beta}}}\in T_{\pi ( \mathrm{p})}( \Sigma ) \otimes T_{\pi ( \mathrm{p})}( \Sigma )
{\cal F}= -\frac16 C_{ABC}t_A t_B t_C - \frac{\chi \zeta(3)}{4 \pi^3}+ \frac{1}{8 \pi^3}\sum\_{d_1, \ldots, d_h}n_{d_1, \ldots, d_h}Li_3(\exp( i \sum_A d_A t_A )),
(i+j+1)^{N+1}\mathfrak{d}_1^{N+2}\dot{\phi}(i+j)= (i+j+1)^{N}\mathfrak{d}_1^{N+1}\dot{\psi}(i+j) &+(N+1)(i+j+1)^{N}\mathfrak{d}_1^{N+1}\dot{\phi}(i+j+1),
\big|\xi(x(t)) - z(t)\big|^2 \le 3\bigg|\int_0^t \varphi(x(s))\,ds\bigg|^2 + 3L_b^2\bigg(\int_0^t \big|\xi(x(s)) - z(s)\big|\,ds\bigg)^2 + \frac{6}{\beta}\big|M(t)\big|^2 \,.
\sum_{i=0}^{\dim_{\mathbb{R}}X}(-1)^i Tr (\psi_i \phi^i ) = \sum_{b \in \mathbb{B}}(-1)^{\deg(b)}\langle\psi_* \phi^*(b) , b\rangle &= \sum_{b \in \mathbb{B}}(-1)^{\deg(b)}\langle\psi_* (b ), \phi^*(b)\rangle
\langle m_1,l_1,k-m_1,l_2 | 0,l_{12}\rangle = (-1)^{m_1}\sqrt{\binom{k}{m_1}\frac{(2l_2)_k}{(2l_1)_{m_1}(2l_2)_{k-m_1}}}\langle 0,l_1,k,l_2 | 0,l_{12}\rangle.