harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\breve{B}_1=\frac{2}{A^2(1+(1-\alpha)\rho_1)}\breve{B}_2=\frac{(3-\alpha+(1+\alpha_0)\rho_0)\rho_1+3+(1+\alpha_0)\rho_0}{A^2(1+(2-\alpha)\rho_1)+(1-\alpha)\rho_1^2)}\breve{B}_3=\frac{(1+(1-\alpha)\rho_1)^2(1+(1+\alpha_0)\rho_0)}{4(1+\rho_0)(1+(2-\alpha)\rho_1+(1-\alpha)\rho_1^2)}.
\Gamma_{1 , f e r m i o n s}\left[ B \right] = \frac{1}{2}\int_{0}^{\infty}{\frac{{d T}}{T}}\int_{x ( 0 ) = x ( T )}{D x ( t ) \exp ( - \frac{1}{4}}\int_{0}^{T}{d t \dot{x}^{2}}) T r_{L}\Phi^{\left[{1 / 2}\right]}\left[{\dot{x}}\right] T r_{c}P \exp ( i g \int_{0}^{T}{d t \dot{x}\cdot B )}
\psi\left( \xi\right) =\frac{m^{2}-1}{\left({z^{2}-1}\right) \left({z^{2}-\sigma^{2}}\right)}+\frac{\left({1-\sigma^{2}}\right) \left({6z^{4}-\left({3+\sigma^{2}}\right) z^{2}-2\sigma^{2}}\right)}{4\left({z^{2}-1}\right) \left({z^{2}-\sigma^{2}}\right)^{3}}.
\begin{array}{c}a^{\dagger}a = |\varphi (N)|, begin{align*}4mm]a a^{\dagger}= |\varphi (N+1)|, begin{align*}4mm]a a^{\dagger}- a^{\dagger}a = |\varphi (N+1)| - |\varphi (N)| =G_{1}(N), begin{align*}4mm]c(N) = e^{i \arg \varphi (N)}= \frac{\varphi (N)}{|\varphi (N)|}.\end{array}
\begin{aligned}\Vert D^{\tilde{\alpha}-(0,1)}(\partial_x u)\partial_yD^{\alpha-\tilde{\alpha}}u\Vert_{L^2_{l+k}}= \Vert D^{\tilde{\alpha}-(0,1)}(\partial_x u)\partial_yD^{\alpha-\tilde{\alpha}}u\Vert_{L^2_{l+1+(k-1)}}\leq C\Vert \partial_y u\Vert_{H^{m-1}_{l+1}}\Vert \partial_x u\Vert_{H^{m-1}_0}\lesssim E_u^2.\end{aligned}
2W^{+}_{ijkl}W^{+}_{ipqt,l}W^{+}_{jpqt,k}= 8 \Big(\lambda |d\lambda|^{2}+\mu|d\mu|^{2}+\nu|d\nu|^{2}&\quad\quad-\lambda(\mu-\nu)^{2}|a|^{2}-\mu(\nu-\lambda)^{2}|b|^{2}-\nu(\lambda-\mu)^{2}|c|^{2}\Big)\,.
\frac{1}{2\pi}\int_0^{2\pi}\left|\frac{e^{x+re^{it}}}{1+e^{x+re^{it}}}\right| r^{-n}\, \mathrm{d}t \leq \frac{1}{2\pi}\int_0^{2\pi}\frac{1}{\sin(r)}r^{-n}\, \mathrm{d}t = \frac{1}{\sin(r) r^n}.
X_{a b}={\frac{- e^{4}}{1 6 m^{2}( p k ) ( p k^{\prime})}}\mathrm{T r}\left[ \gamma^{\beta}( \gamma^{\mu}( p + k )_{\mu}+ m ) \gamma^{\alpha}( \gamma^{\nu}p_{\nu}+ m ) \gamma_{\beta}( \gamma^{\rho}( p - k^{\prime})_{\rho}+ m ) \gamma_{\alpha}( \gamma^{\lambda}{p^{\prime}}_{\lambda}+ m ) \right]
\begin{aligned}\widetilde X = \left\{w : \overline I \mapsto V \mid \partial_t w_{| I_n}(t_{n,\mu}) \in H\,,\; \mu = 0,\ldots,k\,, \; n=1,\ldots ,N \right\}\,, \widetilde Y = \left\{w : \overline I \mapsto H \mid \partial_t w_{| I_n}(t_{n,\mu}) \in H\,,\; \mu = 0,\ldots,k\,, \; n=1,\ldots ,N \right\}\end{aligned}
\begin{array}{l}\displaystyle{\vphantom{\Big(}\{{L_0}_\lambda w(a)\}_{z,\rho}=\sum_{j\in J^f_0}{\{{w(q^{j})}_{\lambda+\partial}w(a)\}_{z,\rho}}_\to w(q_{j})}\displaystyle{\vphantom{\Big(}=\sum_{j\in J^f_0}w(q_{j})w([q^{j},a])+\sum_{j\in J^f_0}(q^{j}|a)(\lambda+\partial)w(q_{j})+z\sum_{j\in J^f_0}(s|[q^{j},a])w(q_{j})\,.}\end{array}
\begin{array}{c}{{A_{~ \mu}^{\lambda}\eta_{\lambda \rho}A_{~ \nu}^{\rho}\propto \eta_{\mu \nu}}}\\{{{}}}\\{{B_{\alpha}^{\mu}B_{\mu \beta}= \bar{B}_{\dot{\alpha}}^{\mu}\bar{B}_{\mu \dot{\beta}}= A_{~ \nu}^{\mu}B_{\mu \alpha}= A_{~ \nu}^{\mu}\bar{B}_{\mu \dot{\alpha}}= B_{\alpha}^{\mu}\bar{B}_{\mu \dot{\alpha}}= 0}}\\ \end{array}
G_r(r,z,\bar r,\bar z) \, =\, -\frac{1}{\pi}\frac{z-\bar z}{r^{3/2}\,\bar r^{1/2}}\,F'(\xi^2)~, \xi^2 \,=\, \frac{(r-\bar r)^2 + (z-\bar z)^2}{r\bar r}~, begin{align*}1mm] G_z(r,z,\bar r,\bar z) \, =\, \frac{1}{\pi}\frac{r-\bar r}{r^{3/2}\,\bar r^{1/2}}\,F'(\xi^2) + \frac{1}{4\pi}\frac{\bar r^{1/2}}{r^{3/2}}\Bigl(F(\xi^2) - 2\xi^2 F'(\xi^2)\Bigr)~.
\frac{1}{\sqrt{2\pi}}\frac{\int_{c_1+\sqrt{n_1}G_m^{-1}(AQL)}\left[ 1 - \Phi( c_2 - z + (\sqrt{n_1}+ \sqrt{n_2}) G_m^{-1}(AQL) ) \right] e^{-z^2/2}\, dz}{1- \Phi( c_1 + \sqrt{n_1}G_m^{-1}(AQL) )}= 1-\alpha_2
R_{i t i t}= \frac{- a a_{t t}}{( 1 + \frac{K}{4}\vec{x}\vec{x})^{2}}, R_{i j i j}= \frac{a^{2}( K + a_{t}^{2})}{( 1 + \frac{K}{4}\vec{x}\vec{x})^{4}}; \; i \neq j
\Delta_2= \ \frac{q^2}{4\pi^2|n|}\int_{3X^2}^{4X^2}\big(e(x\beta)w'(x)\big)'\Big(xB_2\big(\frac{4\pi\sqrt{x|n|}}{q}\big)\Big)dx \ll \ \frac{q^2}{4\pi^2|n|}\int_{3X^2}^{4X^2}\big(|\beta|+x^{-1}\big)q^{1/2}x^{-1/4}|n|^{-1/4}dx \ll \ |\beta|q^{5/2}X^{3/2}|n|^{-5/4}+q^{5/2}X^{-1/2}|n|^{-5/4}.
\Big| \frac{D_tZ(\alpha,t)-D_t(\beta,t)}{\alpha-\beta}\Big|^2=\Big|\sum_{j=1}^N\frac{\lambda_j}{2\pi}\frac{1}{(\alpha-z_j)(\beta-z_j)}\Big|^2 = \sum_{j=1}^N\sum_{k=1}^N\frac{\lambda_j\lambda_k}{(2\pi)^2}\frac{1}{(\alpha-z_j)(\beta-z_j)\overline{(\alpha-z_k)(\beta-z_k)}}
\tilde{R}_{1 1}^{\prime}= \left( \frac{\tilde{R}}{R}\right)^{1 / 2}\frac{l_{p}^{3}}{R_{8}R_{9}}\ \ , \ \ \tilde{l}_{p}= \left( \frac{\tilde{R}}{R}\right)^{1 / 3}\frac{l_{p}^{2}}{( R R_{8}R_{9})^{1 / 3}}\ \ , \ \ \tilde{R}_{i}^{\prime}= \left( \frac{\tilde{R}}{R}\right)^{1 / 2}R_{i}.
D(n\lvert\{1,2\}) =D(n-2\lvert\{1,2\})+D(n-2\lvert\{1\})=D(n-2\lvert\{1,2\})+1 &=D(n-4\lvert\{1,2\})+D(n-2\lvert\{1\})+1=D(n-4\lvert\{1,2\})+2 &=D(n-6\lvert\{1,2\})+3=\cdots=D(n-2k\lvert\{1,2\})+k=0+k=\lfloor\frac{n-1}2\rfloor.
\begin{array}{c}\left \{\beta(j+u)= \beta(j)+\beta(u) \right \}_{j=1}^{k-u}\mbox{and}\ \ \beta(0)=0 \ \ \mbox{and}\ \ \delta_{n}^k=\beta(v)+\beta(u) \mbox{and}\ \ \left \{\beta(j+v) = \beta(j) + \beta(v) \right \}_{j=1}^{k-v}\end{array}
d_0=1+2=3,\d_{2^{n-1}}=1-2=-1,\d_{2^{n-2}}=d_{3\cdot2^{n-2}}=1+0=1, d_{2^{n-3}}=d_{7\cdot2^{n-3}}=1+\sqrt{2},\d_{3\cdot2^{n-3}}=d_{5\cdot2^{n-3}}=1-\sqrt{2}.
p\widetilde u(p,x,l)+b\widetilde u_x(p,x,l)+\widetilde u_{xxx}(p,x,l)-\lambda_l \widetilde u_x(p,x,l)=0, \widetilde u(p,0,l)=\widetilde \nu_0(p,l)\equiv \int_{\mathbb R_+}\!\! \int_0^Le^{-pt}\psi_l(y)\nu_0(t,y)\,dydt, \widetilde u_x(p,0,l)=\widetilde \nu_1(p,l),
\int_{0}^{\infty}\hat{P}_{m}(x^{2})\hat{P}_{n}(x^{2})\rho(x)\,\mbox{d}x+\sum_{k=0}^{N(\theta)}\mu(\{\lambda_{k}\})\,\hat{P}_{m}(\lambda_{k}^{\,2})\hat{P}_{n}(\lambda_{k}^{\,2})=\delta_{m,n}
d\boldsymbol{r}^{\prime}\boldsymbol{I}_{12}^{-}=(d\boldsymbol{x}+d\boldsymbol{y})\boldsymbol{I}_{12}^{-}=2d\boldsymbol{x}\boldsymbol{I}_{12}^{+}\boldsymbol{I}_{12}^{-}=0
i \partial_t c_k = \sum_{\substack{\ell,m,n \geq 0 k + \ell = m + n}}\frac{1}{2\pi}\frac{(k + \ell)!}{2^{k+\ell}\sqrt{k! \ell ! m ! n!}}\overline{c_\ell}c_m c_n, k\geq 0.
\phi(x)\cdot \alpha(h_1*h_2) =\phi(x)\cdot \int_D \alpha(h)\> d(\delta_{h_1}*\delta_{h_1})(h)= \int_D T_h\phi(x)\> d(\delta_{h_1}*\delta_{h_1})(h) &= T_{h_1}\circ T_{h_1}\phi(x) =\phi(x)\cdot \alpha(h_1)\alpha(h_2)
\mathcal{R}_\ell(v_\ell,q_\ell) = \mathcal{F}_\ell(v_\ell,q_\ell) -{\mathcal B}_\ell \big( (u_\ell^{(\nu_1)},p_\ell^{(\nu_1)}),(v_\ell,q_\ell) \big),\forall\, (v_\ell,q_\ell) \in V_\ell \times Q_\ell,
j_{n , a}^{\mu}( k ) = - i g f^{a b c}\int d \Omega{\frac{1}{v \cdot k + i 0^{+}}}\sum_{k^{\prime}+ k^{\prime \prime}= k}( v \cdot A ( k^{\prime}) )_{b}j_{n - 1 , c}^{\mu , \Omega}( k^{\prime \prime}) \ \n > 2 ,
N_{\mathrm{c}\,m+1}=N_{m+1}-N_{\mathrm{d}\,m+1}-\sum_{s=1}^{m-1}\left[\sum_{n=s}^{m}\binom{m+1}{m-n+1}N_{\mathrm{d}\,m-n+1}\mathcal{C}_{s}^{n}\right]\left(N_{s}-N_{\mathrm{d}s}\right) -\binom{m+1}{1}N_{\mathrm{d}1}\mathcal{C}_{m}^{m}\left(N_{m}-N_{\mathrm{d}m}\right).
L<\sigma^2\left[\frac{t}{w(t)}\left(1-\frac{r(t)}{t^2}\right)-\frac{\tau}{w(\tau)}\left(1-\frac{r(\tau)}{\tau^2}\right)\right]+\frac{\tau(t-\tau)}{tw(t)}\frac{1}{N}\sum_{i=1}^N\delta_i^2,
\eta^{( s_{1}, s_{2}, s_{3}, s_{4})}= \left[ 1 + \sum_{k = 1}^{5}u_{k}^{( s_{1}, s_{2}, s_{3}, s_{4})}\Gamma^{0}\left( \Gamma^{2 k}x^{2 k - 1}- \Gamma^{2 k - 1}x^{2 k}\right) \right] \eta_{0}^{( s_{1}, s_{2}, s_{3}, s_{4})},
\displaystyle\frac{1}{\lambda -x}=\displaystyle\frac{1}{\epsilon}=\displaystyle\frac{(-1)^{\delta}e^{iZ_{N}(x+\epsilon )}iZ'_{_{N}}(x+\epsilon )}{1+(-1)^{\delta}e^{iZ_{N}(x+\epsilon )}}+...={\cal R}_{\delta}(x+\epsilon )+...
\chi_{m,j}(\lambda) := -\log P[\mathrm{e}^{-\lambda \eta_{m,j}(1)}] &\ = \int_{0}^{\infty}(1 - \mathrm{e}^{-\lambda u}) n_{m,j}(T_0 \in du) &\ = \int_{0}^{\infty}P^m_x[ 1 - \mathrm{e}^{-\lambda T_0}]j(dx).
S = \left. \frac{\pi \chi R}{\left( 2 U^{\prime}e^{2 U}\right) \left( 1 - e^{2 U}R^{2}\right)}\left[ ( U^{\prime}R + 2 R^{\prime}) e^{2 U}-{\frac{2 R}{r}}e^{U}\right] \right|_{r = r_{h}}.
\tilde{\Psi}({\bf x}^{\boldsymbol{\alpha}}) ={\boldsymbol{\mu}}^{\boldsymbol{\alpha}}= \bigodot_{1 \leq i \leq m}{\mu_i}^{\alpha_i}= \bigodot_{1 \leq i \leq m}{\tilde{\Psi}\left( \phi ([L(\mu_i)]) \right)}^{\alpha_i}= \bigodot_{1 \leq i \leq m}{\tilde{\Psi}(x_i)}^{\alpha_i}
L_{\mu_1\oplus\nu_1}\bar\partial =\textnormal{ad}{\nu_1}-\mu\partial, L_{\mu_1\oplus\nu_1}\bar\partial=0, L_{\mu_1\oplus\nu_1}\bar\partial^* =0,L_{\mu_1\oplus\nu_1}\bar\partial^*=-\star\textnormal{ad}\nu_1\star -\partial^* \bar\mu_1.
\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\frac{1}{\left(\lambda_{n}^{2}+x^{2}\right)^{s}}=\frac{x^{-2s}}{\Gamma(s)}\,\frac{1}{2\pi i}\,\intop_{\mu-i\infty}^{\mu+i\infty}\Gamma(z)\,\Gamma(s-z)\,\zeta_{p}(2z)\,x^{2z}dz
\| E^j \|^2 - \| E^{j-1}\|^2 + \|E^j- E^{j-1}\|^2 = 2 h \big\langle \Delta f^j,E^j\big\rangle + 2 \big\langle \Delta g^{j-1}\Delta W^j + \varrho^j, E^j-E^{j-1}\big\rangle + 2 \langle \varrho^j,E^{j-1}\rangle.
\begin{aligned}A(x_0, x_1, x_2)y_0^2 + B(x_0, x_1, x_2)y_0y_1 + C(x_0,x_1,x_2)y_0y_2 \, + & D(x_0, x_1, x_2)y_1^2 + E(x_0, x_1, x_2)y_1y_2 + F(x_0,x_1,x_2)y_2^2, \end{aligned}
H_{\Delta_k}( s,t ;m) = \,\, \frac{1}{6m}2 (m+2) \Gamma \left(m/2+2\right) F_1\left(m/2+2;1,1;4;1-s t,1-s\right) \,\,-\frac{1}{6m}\Gamma \left(m/2+3\right) 2 F_1\left(m/2+3;1,1;5;1-st,1-s\right) \,\,-\frac{1}{6m}\Gamma \left(m/2+3\right) s F_1\left(m/2+3;1,2;5;1-s t,1-s\right).
\langle P(x),Q(y)\rangle =\sum_{\substack{k_1,...,k_N=0 m_1,...,m_N=0}}^{\infty}\left(\prod_{i=1}^N\left\langle\delta_{\omega_{x_i}(k_i)},\delta_{\omega_{y_i}(m_i)}\right\rangle\right)\left\langle A^*Be_{(k_1,...,k_N)}, e_{(m_1,...,m_N)}\right\rangle.
(\overline{f_1},\cdots,\overline{f_k}) = (\overline{\pi^{b_1}e_{r-k+1}},\cdots,\overline{\pi^{b_k}e_r}) \textrm{and}(f_{k+1},\cdots,f_{d}) = (\pi^{b_{k+1}}e_{r+1},\cdots, \pi^{b_d}e_{r-k+d});
\begin{array}{lllllllll}J^1 = \langle x_1+x_2 \rangle, J^{2}= \langle x_1 \rangle, J^{3}= \langle h_1+2h_2+x_1 \rangle, J^{4,\alpha}= \langle h_1+ \alpha h_2 \rangle,\end{array}
\prod_{i=1}^m\sqrt{k_i!}H_{k_i}(x_i) =\frac{\partial^{k_1+\ldots +k_m}}{\partial t_1^{k_1}\ldots\partial t_m^{k_m}}\exp\left( \sum_{i=1}^m t_ix_i-\frac{1}{2}\sum_{i=1}^m t_i^2 \right)\mid_{t_1 = \ldots = t_m= 0}.
{\mathcal L}_{\theta'}(z') =4z_1'z_2'\frac{{\mathcal L}_{\theta}(z){\mathcal L}_{-\theta}(z)}{4z_1^2z_2^2}&=(z_0^2-z_2^2){\mathcal L}_{\theta}(z){\mathcal L}_{-\theta}(z).
{\cal F}= \det \left[ \sum_{r = 0}^{\infty}c_{r , j}\left( t_{i}^{r + \frac{1}{2}}- t_{i}^{- ( r + \frac{1}{2})}\right) + \sum_{r = - \infty}^{- 1}c_{r , j}\left( t_{i}^{r + \frac{1}{2}}- t_{i}^{- ( r + \frac{1}{2})}\right) \right] .
\lim_{\epsilon\to 0^+}\int_{\{(x_1,y_1,z_1)\in\mathbb{R}^3 : |\det| > \epsilon\}}\limits \frac{f(x)g(y)h(z)}{\det\begin{pmatrix}1 1 1 x y z \end{pmatrix}}\delta(x_1+y_1+z_1)\, dx_1dy_1dz_1,
\mathbb{P}\left\{\left\|\sum_{k=1}^{\widehat{m}}\mathcal{R}_{a_k}\right\|\geq t\right\}\leq2n_1n_2\exp\left(\frac{-t^2/2}{\frac{12\mu c_s{\widehat{m}}r}{n^2}+\frac{5\mu c_s r}{n}t/3}\right).
{\cal Z}= \int D h_{\mu \nu}\exp \left\{- \int d^{4}x \left[ \frac{1}{1 2 \eta^{2}}H_{\mu \nu \lambda}^{2}+ g_{m}^{2}h_{\mu \nu}^{2}- 2 \zeta \cos \left( \frac{\left| h_{\mu \nu}\right|}{\Lambda^{2}}\right) \right] \right\}.
\sum_{n=1}^{\infty}\frac{(q)_{n-1}a^n}{(1-q^n)(a)_n}=\sum_{n=1}^{\infty}\frac{aq^{n-1}}{(1-aq^{n-1})^2}&=\sum_{m=1}^{\infty}m\left(\frac{a}{q}\right)^m\sum_{n=1}^{\infty}q^{mn}&=\sum_{m=1}^{\infty}\frac{ma^m}{1-q^m}.
v_{t}(t,x) =v_{xx}(t,x)+v(t,x)g(t,x, u(t,x;u_0,g)) & =v_{xx}(t,x)+v(t,x)g(t, x,v(t,x))+v(t,x)g(t, x,u(t,x;u_0,g))-v(t,x)g(t,x,v(t,x)) &> v_{xx}(t,x)+ v(t,x)g(t, x,v(t,x))\for \all \t>0,0\le x<l,
\|D[U^{n}]\|_{L^p(\Omega;L^q(0,T;\mathcal{L}_2^0))}^q =\|\sum_{j=0}^{n-1}\chi_{[t_{j}, t_{j+1})}(s) B_{n-j}P_h\|_{L^p(\Omega;L^q(0,T;\mathcal{L}_2^0))}^q = \|\sum_{j=0}^{n-1}\chi_{[t_{j}, t_{j+1})}(s) B_{n-j}P_h\|_{L^q(0,T;\mathcal{L}_2^0)}^q=\tau \sum_{j=0}^{n-1}\| B_{n-j}P_h\|_{\mathcal{L}_2^0}^q< \infty.
u_{x}(t,x;s) =\phi'_{\min}(x-y_{s})e^{-\int_{s}^{t}(1-f_{u}(\tau,u(\tau,x;s)))d\tau}+\int_{s}^{t}b(r,x;s)e^{-\int_{r}^{t}(1-f_{u}(\tau,u(\tau,x;s)))d\tau}dr,
\sum_{n=0}^{\infty}\textup{PDO}_\textup{t}(8n+7)q^n =8\dfrac{f_2^{14}f_3f_{6}^4f_{8}^2}{f_1^{14}f_4^3f_{12}^2}+16\dfrac{f_2^9f_3^2f_4^5f_{6}}{f_1^{13}f_{8}^2}+32q\dfrac{f_2^{8}f_3^3f_4f_{8}^2f_{12}^2}{f_1^{12}f_{6}^2},
A=\begin{pmatrix}\cos(\theta_0t) -\sin(\theta_0t) \cos(\theta_0t) \sin(\theta_0t) 0 0 \sin(\theta_0t) \cos(\theta_0t) -\sin(\theta_0t) \cos(\theta_0t) 0 0 0 0 0 0 \cos(2\theta_0t) \sin(2\theta_0t)\end{pmatrix}\ ,
E\max_{1\leq k\leq2^{r}}\left\vert s_{k}E^{k}X)\right\vert^{p}\ll\sum\nolimits_{i=0}^{r-1}\left( s_{2^{i+1}}^{\ast}\right)^{p}E|E^{2^{i}}X|^{p}\ll\sum\nolimits_{i=1}^{2^{r-1}}i^{-1}\left( s_{4i}^{\ast}\right)^{p}E|E^{i}X|^{p}\ll\sum\nolimits_{i=1}^{2^{r-1}}i^{-1}\left( s_{4i}^{\ast}\right)^{p}E|E^{i}X|^{p}.
\tilde w(x)\,dx =\frac 1 2\, w(y;-1,-q^{m+1},a^2,b^2;q^2)\,dy, x^2\tilde w(x)\,dx =\frac 1 8\, w(y;-q^2,-q^{m+1},a^2,b^2;q^2)\,dy,
\Vert b(\mathbf{D})(B^0(\vartheta) +\lambda_0 \overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}&\leqslant \alpha_1^{1/2}\Vert \mathbf{D}(B^0(\vartheta) +\lambda_0\overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}.
\tau_{\sqrt{t};t}f= \tau_{-\sqrt{t};t}f=f, (\tau_{1;t}f)(z_1,\dots,z_n)=f(z_1,\dots,z_n,1), (\tau_{-1;t}f)(z_1,\dots,z_n)=f(z_1,\dots,z_n,-1).
\inf_{\omega \in I_3}| h_{\xi}(\omega)| = | h_{\xi}(\omega_{\xi}^{\ast}+ \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}) | = \ldots = \frac{\frac{1-\alpha}{2}\xi^{\alpha}}{\xi - 1 - \frac{1}{2}\xi^{\alpha}}\sim |\xi |^{\alpha - 1}.
\left( \mathbf{curl}\,\sigma\left(\mathbf y_\rho\right)\times \mathbf r_\rho,\mathbf r_\rho\right)=b\left(r_\rho,r_\rho,\sigma\left(y_\rho\right)\right)-b\left(r_\rho,r_\rho,\sigma\left(y_\rho\right)\right)=0.
S = \int d^{2}x d^{4}\theta K ( \Phi , \overline{{{\Phi}}}) - \int d^{2}x d \theta^{+}d \theta^{-}W ( \Phi ) - \int d^{2}x d \overline{{{\theta}}}^{+}d \overline{{{\theta}}}^{-}\overline{{{W}}}( \overline{{{\Phi}}}) \quad ,
g(\gamma):=b_2-\frac{\gamma(b_1)_-}{\gamma+1}-\frac{(a_2)_-}{\gamma+1}-\frac{\chi_2l(\gamma-1)}{d_3\gamma}-\frac{\chi_2k(\gamma-1)}{d_3(\gamma+1)}-\frac{\chi_1l(\gamma-1)}{d_3\gamma(\gamma+1)}.
h_{1}= 0 , h_{3}= h_{\overline{{{3}}}}= \frac{2}{9}, h_{8}= \frac{1}{2}, h_{6}= h_{\overline{{{6}}}}= \frac{5}{9}, h_{1 0}= h_{\overline{{{1 0}}}}= 1 , h_{1 5}= h_{\overline{{{1 5}}}}= \frac{8}{9}\ \ .
\log \bigg( \frac{k_N}{sj X_j(t^*)}\cdot e^{sjt^*}\bigg( \frac{s}{\mu}\bigg)^{-j + 2k_N/3}\bigg) = -\log N + \frac{2k_N}{3}\log \bigg( \frac{s}{\mu}\bigg) + j \log j - j + o(k_N). &= - \frac{1}{3}\log N + j \log j - j + o(k_N),
\begin{aligned}\int_0^T\int_{\alpha}^1 \Big(s \Theta (z_x)^2 + s^3 \Theta^3 \left( \frac{x-x_0}{a}\right)^2z^2\Big) e^{2s\varphi}dxdt & \le k \int_0^T\int_{\alpha}^1 s \Theta (z_x)^2e^{2s\Phi}dxdt +k\int_0^T\int_{\alpha}^1 s^3\Theta^3 z^2e^{2s\Phi}dxdt & \le C \int_0^T \int_{\omega}\frac{v^2}{a}dxdt,\end{aligned}
(\ldots, x_1, x_0)(w_n) \Phi = (\ldots, x_1, x_0) \left( (p_n) \Psi \circ (u_n) \Psi \circ \delta \circ (s_n)\Psi \right) = (\ldots, x_1, x_0p_n) \left((u_n) \Psi \circ \delta \circ (s_n)\Psi \right).
\|L_1\| \lesssim \frac{E(\lambda)^\varrho}{|\Im\lambda|^{m + 2}}\varepsilon_0\Big( \sum_{k=E+1}^{E(\lambda)}k^{-\varrho}\|\partial_\nu\phi^{(1)}_k\|^2_{L^2(\partial B_R)}&\quad+ \sum_{k=E+1}^{E(\lambda)}k^{-\varrho}\|\partial_\nu\phi^{(1)}_k\|_{L^2(\partial B_R)}\|\partial_\nu(\Delta\phi^{(1)}_k)\|_{L^2(\partial B_R)}\Big).
M\geq u(x_v) =C_1\int_\Omega \ln \frac{1}{|x_v-y|}v^p(y)\mathrm{d}y-\int_\Omega h(x_v, y)v^p(y)\mathrm{d}y &\geq C_1\int_\Omega \ln \frac{1}{|x_v-y|}v^p(y)\mathrm{d}y-\gamma_2\int_\Omega v^p(y)\mathrm{d}y &\geq C_1\int_\Omega \ln \frac{1}{|x_v-y|}v^p(y)\mathrm{d}y-\gamma_{1}\gamma_2,
(X+Y)^n =(X+Y)(X+Y)^{n-1}=(X+Y)\sum\limits_{k\ge0}{n-1\brack k}_ww^{k(n-1-k)}X^kY^{n-1-k}=\sum\limits_{k\ge0}\left({n-1\brack k-1}_ww^{(k-1)(n-k)}+{n\brack k}_ww^{k(n-1-k)+2k}\right)X^kY^{n-k}=\sum\limits_{k\ge0}\left({n-1\brack k-1}_ww^{-n+k}+{n\brack k}_ww^k\right)w^{k(n-k)}X^kY^{n-k}=\sum\limits_{k\ge0}{n\brack k}_ww^{k(n-k)}X^kY^{n-k}.
\tilde s^{(2)}_{q1}(\tilde x, t)=-\frac{i}{12\mu}\bigl \{\bigl [ \partial_{\tilde x}\psi^{*}(\tilde x,t) \bigr ]\tilde x\partial_{\tilde x}^2\psi(\tilde x,t)-\bigl [ \partial_{\tilde x}^2\psi^{*}(\tilde x,t) \bigr ]\tilde x\partial_{\tilde x}\psi(\tilde x,t) \bigr \}.
\widehat{F}_2 = \frac{1}{2}x_1^2 \widehat{F}_3 = -\frac{1}{3}x_1^3 + x_1x_2 \widehat{F}_4 = \frac{1}{4}x_1^4 + x_1x_3 - x_1^2x_2 + \frac{1}{2}x_2^2
dq_t^m= \frac{1}{m}u_t^m dt, d(u^m_t)_i= \left(-\frac{1}{m}\tilde \gamma_{ik}(t,q_t^m) (u_t^m)^k+ F_i(t,x^m_t)\right)dt+\sigma_{i\rho}(t,q_t^m)dW^\rho_t,
Z^{( 1 )}= \frac{1}{2 !}\int_{1 \cdots 6}V_{1 2 3}V_{4 5 6}( - 2 ) \frac{\delta^{3}}{\delta D_{3 6}^{- 1}\delta S_{1 2}^{- 1}\delta S_{4 5}^{- 1}}Z^{( 0 )}.
\begin{array}{lll}A = \sqrt{\tau}E, b = \displaystyle{\frac{w+\sqrt{\tau}c}{1+\sqrt{\tau}}}, \gamma = \displaystyle{\frac{\beta(1+\sqrt{\tau})^2+\frac{1}{2}\sqrt{\tau}\langle c-w,E^{-1}(c-w)\rangle}{(1+\sqrt{\tau})^2(\tau+1)}}.\end{array}
p_{-2}(a)p_{-2}(b) =\sum_{\alpha_{1}+\alpha_{2}=a}\sum_{\beta_{1}+\beta_{2}=b}p(\alpha_{1})p(\alpha_{2})p(\beta_{1})p(\beta_{2}) &> \sum_{\gamma_{1}+\gamma_{2}=a+b}p(\gamma_{1})p(\gamma_{2}) &=p_{-2}(a+b),
\eta^{(1)}=x( u_{1}-z_{1},u_{1}+z_{1}), u_{1}=\frac{1}{2}e^{\mathrm{i}\theta}( 1+\zeta ) , z_{1}=\frac{1}{2}e^{\mathrm{i}\theta}(\zeta-1) , \eta^{(2)}=x ( u_{2}-z_{2},u_{2}+z_{2}), u_{2}=\frac{1}{2}e^{\mathrm{i}\theta}\big( 1+\zeta^{-1}\big), z_{2}=\frac{1}{2}e^{\mathrm{i}\theta}\big( 1-\zeta^{-1}\big) =\zeta^{-1}z_{1}.
V''(t)(V')^\frac{1}{n-k-1}\leq a(V')^{1-\delta_1}+cb t^\frac{k-1}{k}V^{\delta_2+\delta_3}(V')^{1-\delta_2}+ \tfrac{cb}{{1+\delta_3}}t^\frac{n-k-2}{n-k-1}(V^{1+\delta_3})' &+ c|H| t^\frac{k-1}{k}V^{\frac{1}{k}}(V')^{1-\delta_2}+ \alpha c|H| t^\frac{n-k-2}{n-k-1}(V^{1/\alpha})'
\big[\big[x_i^+,x_j^+\big],\big\{h_i,x_i^{-}\big\}\big]= \big\{\big[\big[x_i^+,x_j^+\big],h_i\big],x_i^-\big\}+ \big\{h_i,\big[\big[x_i^+,x_j^+\big],x_i^-\big]\big\}\hphantom{\big[\big[x_i^+,x_j^+\big],\big\{h_i,x_i^{-}\big\}\big]}{}= -\big\{\big[x_i^+,x_j^+\big],x_i^-\big\}- \big\{h_i,x_j^{+}\big\}= -\big\{T_i(x_j^+),x_i^-\big\}- \big\{h_i,x_j^{+}\big\}.
Z_1=d^3z_1+3cd^2z_2+3c^2dz_3+c^3z_4 Z_2=b^2dz_1+d(ad+2bc)z_2+c(bc+2ad)z_3+ac^2z_4 Z_3=bd^2z_1+b(bc+2ad)z_2+a(ad+2bc)z_3+a^2cz_4 Z_4=b^3z_1+3ab^2z_2+3a^2bz_3+a^3z_4 Z_5=\lambda z_1+\mu z_2+\nu z_3+\xi z_4+(ad-bc)z_5.
\frac{\partial X_{}^\sigma}{\partial p\>\!_i^\mu}\, \epsilon_{\nu \sigma \rho_1 \ldots \rho_{n-2}}^{}~ =~\frac{\partial X_{}^\sigma}{\partial p\>\!_i^\nu}\, \epsilon_{\mu \sigma \rho_1 \ldots \rho_{n-2}}^{}~,
\|\tilde\omega\|_{\nu_{pn}}^2 = \frac{1}{(p(1-p))^d}\sum_{t=0}^d \binom{d}{t}\frac{k^{\underline{t}}(n-k)^{\underline{d-t}}}{n^{\underline{d}}}(1-p)^{2t}p^{2(d-t)}= \frac{1}{(p(1-p))^d}\sum_{t=0}^d \binom{d}{t}p^t (1-p)^{d-t}(1-p)^{2t}p^{2(d-t)}\left(1 \pm O\left(\frac{d^2}{p(1-p)n}\right)\right) = 1 \pm O\left(\frac{d^2}{p(1-p)n}\right).
\ ||x^*||_1\,\varphi(t^*,x^*) - \mathbb{E}\left[L(\rho)X(\rho)\varphi \big(t^*-\rho,\mathfrak{X}(\rho)\big)\right] =\, \mathbb{E}\left[\int_0^{\rho}L(s)X(s)\, g\big(t^*-s,s,\mathfrak{X}\big)\, \mathrm{d}s \right] >\, 0\,.
F \begin{pmatrix}x y \theta \lambda\end{pmatrix}=\begin{pmatrix}x - a(\theta,\lambda) x^N + f_N(x,y,\theta,\lambda) + f_{\geq N+1}(x,y,\theta,\lambda) y + x^{N-1}B(\theta,\lambda) y+ g_{N}(x,y,\theta,\lambda)+ g_{\geq N+1}(x,y,\theta,\lambda) \theta +\omega + h_P(x,y,\theta,\lambda )+ h_{\geq P+1}(x,y,\theta,\lambda)\end{pmatrix}
\mathcal{M}_{f_{\lambda}}(s) =L\sum_{j=1}^{m}\sum_{i=1}^{m}\mathcal{D}\left(i,j\right)\Gamma\left(s+j-1\right)\Biggl( \theta_{n-m+i}^{n-m+s+j-2}& -\sum_{l=1}^{n-m}\sum_{k=1}^{n-m}\left[\mathbf{\Psi}^{-1}\right]_{k,l}\theta_{l}^{n-m+s+j-2}\theta_{n-m+i}^{k-1}\Biggr)
\sum_{k=1}^{m}|Z_{k}|^{2}\lambda_{k}+ \sum_{l=1}^{s}V_{l}W_{l}= 0, \sum_{k=1}^{m}|Z_{k}|^{2}+ \sum_{l=1}^{s}\bigl(|V_{l}|^{2}+ |W_{l}|^{2}\bigr) = 1,
\pi(h^{-1})\varepsilon(g) = \pi(h^{-1})\pi(g)\pi(g^{-1}) = \pi(h^{-1}g)\pi(g^{-1}) = \pi(h^{-1}g)\pi(g^{-1}h)\pi(h^{-1}g)\pi(g^{-1}) = \pi(h^{-1}g)\pi(g^{-1}h)\pi(h^{-1}gg^{-1}) = \varepsilon(h^{-1}g) \pi(h^{-1}r(g)) = \varepsilon(h^{-1}g) \pi(h^{-1}r(h)) = \varepsilon(h^{-1}g)\pi(h^{-1}),
\int_{- \varepsilon_{\alpha}}^{Q - \sum_{\beta = 1}^{\alpha - 1}p^{\beta}+ \varepsilon_{\alpha}}d p^{\alpha}\to \int_{\tilde{\varepsilon}_{\alpha}}^{2 \lambda - \sum_{\beta = 1}^{\alpha - 1}\tilde{p}^{\beta}- \tilde{\varepsilon}_{\alpha}}d \tilde{p}^{\alpha}\ .
G_{MN}dx^Mdx^N=H^{-{{4(n-1)}\over{(p+n)\Delta_p}}}_p\left[-dt^2+d{\bf x}\cdot d{\bf x}\right]+H^{{4(p+1)}\over{(p+n)\Delta_p}}_p\left[d\tilde{y}^2+d\tilde{s}^2_n\right],
\theta(n_s^{-1}(\lambda))T_{n_s^{-1}}- T_{n_s^{-1}}\theta(\lambda) = \sum_{k = 0}^{\langle \nu(\lambda),\alpha\rangle - 1}\theta(n_s^{-1}(\lambda)\mu_{n_s}(k)n_s^{-2})c_{n_s,k}& = \sum_{k = 1}^{\langle \nu(\lambda),\alpha\rangle}c_{n_s,k}\theta(\mu_{n_s}(-k)n_s^{-2}\lambda).
\frac{d}{dt}\int u^2\phi_N\,dx +\epsilon\int (\partial_x^2u)^2\partial_x\phi_N\,dx &\le \int u^2\,\Big(\partial_t\phi_N+\frac32\partial_x^5\phi_N+\frac{25}{4(5-\epsilon)}\, \frac{(\partial_x^3\phi_N)^2}{\partial_x\phi_N}\Big)\,dx &\;\;\;+2\int u F\phi_N\,dx.
\partial_{0,1}^{P}(\eta_{0,1}) =0, \partial_{1,0}^{\gamma}(\eta_{0,1})+\partial_{0,1}^{P}(\eta_{1,0}) =0, \partial_{1,0}^{\gamma}(\eta_{1,0})+\partial_{2,-1}^{\sigma}(\eta_{0,1}) =0.
W_{\bar{\mu}}:=\{f \in \mathbb{F}_{q^d}[\lambda_0,\ldots,\lambda_{\alpha-1}]: \mathrm{mdeg}(f) \leq (\mu_0,\ldots,\mu_{\alpha-1})\}(\bar{\mu}= (\mu_0,\ldots,\mu_{\alpha-1})).
\frac{\partial \chi^{\varepsilon}}{\partial t}+\displaystyle\sum_{j=1}^{d}f_{j}^{\prime}(c)\,\frac{\partial\chi^{\varepsilon}}{\partial x_{j}}= \frac{\partial m^{\varepsilon}}{\partial c}\,\,\mbox{in}\,\,\mathcal{D}^{\prime}\left(\Omega\times\mathbb{R}\times (0,T)\right).
\sum_{\substack{\log x<|x-n|\leq x \Lambda(n)\neq 0}}\frac{1}{|x-n|}=\int_{\log x}^{x}\frac{dp(u)}{u}=\left[\frac{p(u)}{u}\right]_{\log x}^{x}+\int_{\log x}^{x}\frac{p(u)}{u^2}\,du.
T_{12}=\frac{N_{1}N_{2}}{\det\left( 2\left( M_{1}+M_{2}\right) \sigma\right)^{d/2}}\exp\left( \frac{1}{4}\left( \lambda_{1}+\lambda_{2}\right)^{T}\left( M_{1}+M_{2}\right)^{-1}\left( \lambda_{1}+\lambda_{2}\right)\right)
P^{j}_=\sum_{k=1}^{K}\sum_{r=1}^{L}\sum_{f=1}^{L}g_{kf}^{\dagger}(h_{jf}^{\mbox{\tiny{p}}})^{\dagger}h_{jr}^{\mbox{\tiny{p}}}g_{kr}{\beta_k^{j}}^{(r,f)}.
\mathrm{R e}\, \chi_{T}^{\mathrm{D}}( \omega ) = - \frac{\hbar A}{4 \pi^{2}}\left( \frac{k_{B}T}{\hbar}\right)^{5}\int_{0}^{\Omega}d v \sqrt{v ( \Omega - v )}\int_{0}^{\infty}d u \, \frac{u^{2}}{e^{\frac{u}{2}}- 1}\left[ 1 +{\cal O}( \Omega^{2}) \right]
\left\Vert E_1\right\Vert_{H^1(\mathcal G)}\leq \sum_{j=1}^{\infty}\left\vert \lambda_j^{(1)}\right\vert \left\Vert a_j^{(1)}-\Pi_l(G_j^{(1)}, \tilde G_j^{(1)})\right\Vert_{H^1(\mathcal G)}\leq \varepsilon \sum_{j=1}^{\infty}\left\vert \lambda_j^{(1)}\right\vert & \leq \varepsilon C_0\left\Vert F \right\Vert_{H^1(\mathcal G)}.
S_{4*}= \int d^4 x \sqrt{-g}\left[ R +\frac{1}{4}{\rm Tr}\left[ \nabla P \nabla P^{-1}\right] -\frac{1}{2}\left( \nabla x \right)^2 -\frac{1}{2}e^{\sqrt{3}x}\nabla\tau^T P \nabla\tau \right] .
\Lambda^{\prime}( \lambda_{1}) = ( \lambda_{1}- \lambda_{2}) ( \lambda_{1}- \lambda_{3}) ; \Lambda^{\prime}( \lambda_{2}) = ( \lambda_{2}- \lambda_{1}) ( \lambda_{2}- \lambda_{3}) ; \Lambda^{\prime}( \lambda_{3}) = ( \lambda_{3}- \lambda_{1}) ( \lambda_{3}- \lambda_{2})
\langle \psi^{\dagger}P_{+}\psi \rangle_{\cal{R}}= \langle \psi^{\dagger}P_{+}\psi \rangle_{{\cal{R}}= 0}\cdot \big ( \frac{{\cal{R}}}{2 m_{\gamma}^{2}}\big )^{\frac{\pi}{2 \pi + g_{2}^{2}}}\exp \Big [ - \frac{\pi}{4 e^{2}}{\cal{R}}- \frac{\pi m_{\gamma}^{2}}{4 e^{2}}\gamma \Big ] \hspace{. 5 c m}\mathrm{f o r}\; \frac{{\cal{R}}}{m_{\gamma}}\rightarrow \infty .
\int_{\mathcal S^n}Y_l(y)\,f(x\cdot y)\,d\sigma(y) &=Y_l(x)\cdot\frac{(4\pi)^\lambda\,l!\,\Gamma(\lambda)}{(2\lambda+l-1)!}\int_{-1}^1 f(t)\,C_l^\lambda(t)\left(1-t^2\right)^{\lambda-1/2}dt.
\frac{(4\pi)^{k-1}}{\Gamma(k)}(n(n+\ell))^{\frac{k-1}{2}}\int_0^{\infty}y^k g(Yy) e^{-2\pi(2n+\ell)y}\frac{dy}{y^2}&=\frac{1}{k-1}\cdot g\left( \frac{Y(k-1)}{4\pi(n+\frac{\ell}{2})}\right)+O\left(\frac{1}{k^{1/2-\varepsilon}(n+\frac{\ell}{2})}\right),