harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Bigl| \int h (\phi_{c_1}(\xi_1, \eta_1) + \phi_{c_2}(\xi_2, \eta_2)) f (\phi_{c_1}(\xi_1, \eta_1) ) g (\phi_{c_2}(\xi_2, \eta_2)) d \xi_1d \eta_1 d\xi_2 d\eta_2 \Bigr| & \lesssim A^{\frac{1}{2}}N_1^{-2}\| f \circ \phi_{c_1}\|_{L_{\xi, \eta}^2}\|g \circ \phi_{c_2}\|_{L_{\xi, \eta}^2}\|h \|_{L_{\xi, \eta, \tau}^2},
\mathrm{dA}^i\mathrm{dN}^j_k=\delta^i_k\mathrm{dA}^j,\;\; \mathrm{dN}^i_l\mathrm{dN}^j_k=\delta^i_k\mathrm{dN}^j_l,\;\; \mathrm{dN}^i_l\mathrm{dA}_k^\star=\delta^i_k\mathrm{dA}_l^\star,\;\;\mathrm{dA}^i\mathrm{dA}_k^\star= \delta^i_k\mathrm{d}t.
D_{m,\lambda}^{(k)}(\lambda x)=m! \sum_{n=0}^m \sum_{i_1+i_2+ \cdots+i_n=m}\frac{D_n^{(k)}(x)}{n!}\left(^{\lambda}_{i_1}\right) \left(^{\lambda}_{i_2}\right) \cdots \left(^{\lambda}_{i_n}\right).
{\frac{c_{A}}{2 4}}\sum_{r}\left[ H_{1}, A_{i}^{c}( r ) \right] \left[ H_{1}, A_{i}^{c}( - r ) \right] \sum_{k}{\frac{1}{\omega_{k}^{3}}}= -{\frac{c_{A}}{2 4}}{\frac{1}{2 \pi^{2}}}l o g \left({\frac{\Lambda}{\mu}}\right) \int d^{3}x \, E^{2}
\int_{1}^{\infty}(u-1)^{\alpha -2}u^{-\alpha}\frac{g_{\infty}(x/u)}{g_{\infty}(x)}\mathrm{d}u=\frac{k\overset{M}{\ast}g_{\infty}(x)}{g_{\infty}(x)}\rightarrow \lbrack \gamma (\alpha -1)]^{-1},x\rightarrow \infty ,
F_{\lambda_0}( (x)) = - J \dot{( (x))}- \nabla H( (x)) + 2 \pi \lambda_0 (x) & = (-J \dot x ) + 2\pi - (\nabla H(x)) + 2\pi \lambda_0 (x) & = ( - J \dot x - \nabla H(x) + 2\pi (\lambda_0+1) x ) &= (F_{\lambda_0+1}(x)),
\Vert\nabla \tilde{y}- \nabla y \Vert^4_{L^4(U)}\le C\sum\nolimits^4_{j=1}\Big( \Vert \nabla y - \bar{R}_j \Vert^4_{L^4(U)}+ \frac{\vartheta (1+ \vartheta)}{\lambda^2}\Vert y - (\bar{R}_j\,\cdot + \bar{c}_j)\Vert^2_{L^2(U)}\Big) & \le C C_m^2 \delta_4 + C C_m^2 \vartheta (1+ \vartheta)^2 (\gamma + \delta_4 +\epsilon \Vert W \Vert_*).
D_{a,b}(\pi_{\lambda}\boxtimes\pi_{\hat{\mu}}) = D_{a,b}(D_a(x_1^{\lambda_1+a-1}\dots x_a^{\lambda_a})D_b(x_{a+1}^{\hat{\mu}_1+b-1}\dots x_{a+b}^{\hat{\mu}_b})) &=D_{a+b}(x_1^{\lambda_1+a-1}\dots x_a^{\lambda_a}x_{a+1}^{\hat{\mu}_1+b-1}\dots x_{a+b}^{\hat{\mu}_b}).
G^{(4)}_{\mathcal R}(x,y)= -\frac{ie^{i k_{+}\vert x \vert}}{2\pi}\int_{\mathcal{I}_{s_b,s_b+\infty}}\left[f(s_b)\sqrt{s-s_b}+ \frac{f(s_b)-f(0)}{s_b}(s-s_b)^{\frac 32}\right ]e^{-\vert x \vert s^2}ds &-\frac{ie^{i k_{+}\vert x \vert}}{2\pi}G^{(4)}_{\mathcal R,1}(x,y),
Q_{f}=f_{\mu_{1}\dots \mu_{r}}\Pi^{\mu_{1}}\psi^{\mu_{2}}\dots \psi^{\mu_{r}}+\frac{i}{r+1}(-1)^{r+1}f_{[\mu_{1}\dots \mu_{r};\mu_{r+1}]}\psi^{\mu_{1}}\dots \psi^{\mu_{r+1}}.
{\|I_{a^{+}}^{\alpha}f\|}_{C_{1-\gamma}[a,b]}={\|(x-a)^{1-\gamma}I_{a^{+}}^{\alpha}f\|}_{C[a,b]}\leq{\|f\|}_{C_{1-\gamma}[a,b]}{\|I_{a^{+}}^{\alpha}(x-a)^{\gamma-1}\|}_{C_{1-\gamma}[a,b]},
J(\boldsymbol{x}(t),\boldsymbol{u}(t),t)= 0.5\Big[ \boldsymbol{x}^T(t_f) F(t_f) \boldsymbol{x}(t_f)\Big] + 0.5 \Big[ \int_{t_0}^{t_f}\boldsymbol{x}^T(t)Q(t){\boldsymbol{x}}(t)+\boldsymbol{u}^T(t)R(t)\boldsymbol{u}(t) dt \Big]
\hat{I}_{=2,<\epsilon}(k):=\{\,\,\alpha_{k'}\,\,|\,\, k' \leq k, \,\, J(\alpha_{k'+1},\alpha_{k'})=2,\,\,\,\alpha_{k'+1},\,\,\alpha_{k'}\,\,\mathrm{satisfy\,\,2,3,4,5,6\,\, in\,\,Lemma\,\, \ref{mainlemma}}\}
I(p) = \frac{ck^2}{q^2(1 - q^k)}, &\frac{dI}{dp}= ck^2\left(\frac{2 - (2 + k)q^k}{q^3(1 - q^k)^2}\right), &\mathrm{E}\left[\frac{d^3\ell}{dp^3}\right] = \frac{kc}{q^3}\left(\frac{2k^2q^{2k}}{(1 - q^k)^2}+ \frac{3k(k - 1)q^k}{(1 - q^k)}+ k(k-3)\right).
\frac{d^k}{dx^k}h(f(x)) = \sum_{(m_1,...,m_k) \in \mathcal{M}_k}\frac{k!}{m_1!...m_k!}h^{(m_1+...+m_k)}(f(x)) \prod_{j=1}^k \left( \frac{f^{(j)}(x)}{j!}\right)^{m_j},
\Phi ( \tau , \theta , \overline{{{\theta}}}) = Z ( \tau ) + \theta^{a}\chi_{a}( \tau ) + \frac{i}{2}\overline{{{\theta}}}\theta \dot{Z}( \tau ) + \theta \theta F ( \tau ) - \frac{i}{4}\theta \theta \overline{{{\theta}}}_{a}\dot{\chi}^{a}- \frac{1}{1 6}\overline{{{\theta}}}\overline{{{\theta}}}\theta \theta \ddot{Z}( \tau )
\int_{\Lambda}\prod_{a=1}^N \left[ \prod_{i=1}^{n_a}d^2x_{i}^{(a)}d^2y_{i}^{(a)}\right]\left\langle \prod_{b=1}^N \prod_{j=1}^{n_b}\left[ : e^{+i 2\sqrt{\pi}\varphi^{(b)}(x_{j}^{(b)})}:_{Q^{0,S}}:: e^{-i 2\sqrt{\pi}\varphi^{(b)}(y_{b}^{(b)})}:_{Q^{0,S}}\right] \right\rangle_{Q^{0,S}}
W[{\Phi}]=\frac{1}{2}\int_{- \infty}^{\infty}dx^{+}\int_0^L dx_1^{-}\int_0^L dx_2^{-}{\Phi}^{\star,++}(x_1^{-},x^{+}|x_2^{-},x^{+})\cdot (p_{(1)}^{-}+ p_{(2)}^{-}+
J_{i,k,l}^{(\alpha,\beta)}(x) = \sum_{h=k}^{n-l}c_{ih}B_h^n(x) (i = k+l,k+l+1,\ldots,n), begin{align*}1ex] B_h^n(x) = \sum_{i=k+l}^{n}d_{hi}J_{i,k,l}^{(\alpha,\beta)}(x) (h = k,k+1,\ldots,n-l).
\frac{g'(t)}{g(t)}=(\ln g(t))' &=\left(\sum_{j=1}^r\ln(e^{a_j t}-1)-r\ln t-\biggl(\sum_{j=1}^r a_j\biggr)t\right)' &=\sum_{j=1}^r\frac{a_j e^{a_j t}}{e^{a_j t}-1}-\frac{r}{t}-\sum_{j=1}^r a_j\,.
\phi_{\alpha_{1}}( t ) = \sqrt{\frac{\pi}{4 H_{0}}}e^{- \frac{3}{2}H_{0}t}H_{\nu}^{( 1 )}( z ) , \phi_{\alpha_{2}}( t ) = \sqrt{\frac{\pi}{4 H_{0}}}e^{- \frac{3}{2}H_{0}t}H_{\nu}^{( 2 )}( z ) ,
\| \partial_1^j \partial_2^l (f^m_t - f^m_T) \| = \left\| \int^T_t \partial_1^j \partial_2^l \mathcal{A}f^m_s \, \textrm{d}s \right\| \lesssim \int^{T}_t e^{-\frac{s}{2}}(s^{\frac{1}{4}})^{-j - 2l +\alpha -4}\, \textrm{d}s & \lesssim (T^{\frac{1}{4}})^{-j - 2l + \alpha}+ (t^{\frac{1}{4}})^{-j - 2l + \alpha}
C = \sum_{m\in\mathcal{M}}\sum_{i\in\mathcal{N}}p_i^ms_m\Bigg[\sum_{j\in\mathcal{N}_i}rF_{j,i}^m &+ w_{s}d_{i,0}\max\Big\{0,1-\sum_{j\in\mathcal{N}_i^+}x_{j}^{m}\Big\}\Bigg],
Q = \left( \begin{array}{c c}{{0}}&{{q}}\\{{s q^{T}}}&{{0}}\\ \end{array}\right) \qquad \bar{Q}= \left( \begin{array}{c c}{{0}}&{{\bar{q}}}\\{{s \bar{q}^{T}}}&{{0}}\\ \end{array}\right) \, .
{E}_2(t, \tau) = -i \int_0^{\tau}e^{i (\tilde{\tau}-\tau) A(t)^{1/2}}F(t) \bigl(A(t)^{1/2}F(t) - (t^2 S)^{1/2}P \bigr)e^{-i \tilde{\tau}(t^2S)^{1/2}P}P \, d \tilde{\tau}.
\left|\left(\nabla\nabla \Delta^{(k)}Z^{(l)}\odot g, X^{(l)}\right) - 2\left(\Delta^{(k+1)}Z^{(l)}, Z^{(l)}\right)\right| \quad\leq C\|\nabla^{(k+1)}Z^{(l)}\|\left(\|h\| + \sum_{p=0}^k\|\nabla^{(p)}A\| + \sum_{p=0}^{k}\sum_{q=1}^{l}\|\nabla^{(p)}X^{(q)}\|\right)
\mathcal{L}^{(t)}\left[ x^{-n+1}Q_{n}^2(x; t) \right] = a_{n,n-1}(t) \sigma_{n,n}(t) + \sigma_{n,n}(t) \sum_{k=1}^{n+1}[\alpha_{k+1}(t) + \beta_{k}(t)],
R(x,x)=\sum\limits_{n=1}^{\infty}\lambda_n|\phi_n(x)|^2\quad\mbox{and}\|\log\kappa-\log\kappa_M\|^2_{L^2(\Omega,C({D}))}\leq \sum\limits_{n>M}\lambda_n\|{\phi_n}\|_{C(D)}^2.
\frac{{1 - p}}{{1 - \tilde p}}= \frac{1}{{1 -{\psi_0}+{\psi_1}\frac{p}{{1 - p}}}}\le \frac{1}{{1 -{\psi_0}+{\psi_1}\frac{q}{{1 - q}}}}= \frac{{1 - q}}{{1 - \tilde q}},
\int \! \frac{\mathrm{d^{3}}k}{( 2 \pi )^{3}}\frac{1}{8 ( \omega_{k}^{0})^{3}}= \frac{1}{2}\int \frac{\mathrm{d}^{4}k}{( 2 \pi )^{4}}\frac{- i}{( k^{2}- m^{2}+ i o )^{2}}.
F^{( n )}= d C^{( n - 1 )}= - ( - 1 )^{\frac{n ( n + 1 )}{2}}d \hat{\star}C^{( 1 1 - n )}= ( - 1 )^{\frac{n ( n - 1 )}{2}}\hat{\star}d C^{( 9 - n )}= ( - 1 )^{\frac{n ( n - 1 )}{2}}\hat{\star}F^{( 1 0 - n )},
m_{b}n_{b}+m_{a}n_{a}=m(n_{a}+n_{b})\Rightarrow m_{b}\frac{n_{b}}{n_{a}}+m_{a}=m(1+\frac{n_{b}}{n_{a}})\Rightarrow\frac{n_{b}}{n_{a}}=\frac{m-m_{a}}{m_{b}-m}
\vartheta^{-1}(\omega_0\omega_1\cdots \omega_{n-1}) = 2^{-\frac{n-1}2}\cdot(-1)^{\frac{(n-1)(n-2)}{2}}\xi_1\prod_{j=1}^{n-1}(\xi_j-\xi_{j+1})\otimes (\varsigma_1\varsigma_2\cdots\varsigma_{n-1}),
1_{N_B}(x_{\eta}^{-1}x g_{A}x^{-1}x_{\zeta})\beta_{i,j}(x_{\eta}^{-1}x g_{A}x^{-1}x_{\zeta})\phi(x) =1_N(g_A)1_{N_{B}}(x_{\eta}^{-1}x_{\zeta})\beta_{ij}(x_{\eta}^{-1}x_{\zeta})\phi(x).
\widetilde{Z}:=\left(s_0- \frac{S_{0}}{10},s_0+ \frac{S_{0}}{10}\right) \times\Omega,\widetilde{J}_1:=\left(s_0- \frac{S_{0}}{10},s_0+ \frac{S_{0}}{10}\right) \times \Gamma_1,
p_{j\beta}(t+1)=\sum_{\alpha=1}^{L}\sum_{i=1}^{N}T_{j\beta}^{i\alpha}p_{i\alpha}(t)=\sum_{\alpha=1}^{L}\sum_{i=1}^{N}(T^{t})_{j\beta}^{i\alpha}p_{i\alpha}(0)
\delta_h(t_n,\zeta)= (1-\theta)\int_{t_n}^{t_n+\zeta h}(f(t,x(t),x(qt))-f(t_n,x(t_n),x(qt_n))){\rm d}t &+\theta\int_{t_n}^{t_n+\zeta h}(f(t,x(t),x(qt))-f(t_{n+1},x(t_{n+1}),x(qt_{n+1}))){\rm d}t &+\int_{t_n}^{t_n+\zeta h}(g(t,x(t),x(qt))-g(t_n,x(t_n),x(qt_n))){\rm d}B(t),
I_{j}\simeq \Bigg(\sum_{m= j}^\infty (\delta\alpha)^m \sum_{\beta_{ns}= m}(\beta!)^{-1}\prod_{l=1}^n \Big( \frac{2}{\alpha s}\Big)^{\frac{\beta_l}{2}+\frac{1}{s}}\Big(\Gamma\Big( \frac{s\beta_l}{2}+1\Big)\Big)^{\frac{1}{s}}\Bigg)^p
I_n^{2}(\Pi;q) = \sum_{k=1}^{\lfloor{\frac{n}{2}}\rfloor}\sum_{0 < a_1 < \cdots < a_k < n}q^{k(k+1)}q^{k(n-2k-1) - (a_1 + a_2 + \cdots + a_k) + \frac{k(k+1)}{2}}I_{n-2k-1}(\Pi;q) &= \sum_{k=1}^{\lfloor{\frac{n}{2}}\rfloor}\sum_{0 < a_1 < \cdots < a_k < n}q^{k(n-2k-1) + \frac{3k(k+1)}{2}- (a_1 + a_2 + \cdots + a_k)}I_{n-2k-1}(\Pi;q).
||g_k-g||_{X^{*}}= \sup_{||\phi||_X\leq 1}\Big|\int_{\Omega}{f_n}\Big\{{\big(|v_k|+\frac{1}{n}\big)^{-\delta}}-{\big(|v|+\frac{1}{n}\big)^{-\delta}}\Big\}\phi\,dx\Big| &\leq n ||\phi||_{L^{p_s}(\Omega)}\Big|\Big|{\big(|v_k|+\frac{1}{n}\big)^{-\delta}}-{\big(|v|+\frac{1}{n}\big)^{-\delta}}\Big|\Big|_{L^{p_s^{'}}(\Omega)}.
\mathrm{fpr}_\Omega(x)< f(n,q):=\begin{cases}2/q^{2(m-2)}+1/q^{2m-1}+1/q^{2(m-1)}+1/q^2 \textrm{if}n=2m \textrm{and}m\geq 3, 1/q^{2m+1}+1/q^{2m+1}+1/q^{2m}+1/q^2 \textrm{if}n=2m+1 \textrm{and}m\geq 3.\end{cases}
z^{1/3}f_{1}(z) + z^{2/3}f_{2}(z) = -\varphi_{1}(z)-2\varphi_{2}(z) \omega^2 z^{1/3}f_{1}(z) + \omega z^{2/3}f_{2}(z) = \varphi_{2}(z)-\varphi_{1}(z).
\varphi ( \theta ) = - \frac{2 \cosh \theta \sin \frac{\pi}{\lambda}}{\sinh^{2}\theta + \sin^{2}\frac{\pi}{\lambda}}\, \sim \, - 4 \sin \frac{\pi}{\lambda}\, \mathrm{e}^{- \left| \theta \right|}+ O \left( \mathrm{e}^{- 2 \left| \theta \right|}\right) \, ,
\gamma^0=\left\{\begin{array}{cc}\sigma_3 0 0 -\sigma_3 \end{array}\right\}\ \ \ \ \ \gamma^1=\left\{\begin{array}{cc}i\sigma_2 0 0 -i\sigma_2 \end{array}\right\}\gamma^2=\left\{\begin{array}{cc}-i\sigma_1 0 0 i\sigma_1 \end{array}\right\}\ \ \ \ \ \gamma^3=\left\{\begin{array}{cc}0 1 -1 0 \end{array}\right\}.
\begin{aligned}p_{ij}= p_{ji}= v^{\ell(y)}+ v^{\ell(y)-2}+ \cdots + v^{\ell(y)-2(n-j)}+ + v^{\ell(y)-2i}+ v^{\ell(y)-2i-2}+ \cdots + v^{\ell(y)-2(n-j+i)}.\end{aligned}
|(\psi \circ \Psi) (x - x_i)| = |(\psi \circ \Theta \circ \pi)(x - x_i)| & = |(\psi \circ \Theta)(e_{M_1}^\perp(x - x_i))| &\leq \|e_{M_1}^\perp\|_{\psi \circ \Theta}\, \|x - x_i\|_{\psi \circ \Theta}\to 0 i \to \infty.
V(r,u) = - 2 (8 \pi^2 \alpha')^{-1/2}\int_0^{\infty}dl e^{-r^2 l/2\pi\alpha'}{l}^{1/2}{{{\rm tanh}(u) e^{-u^2 l/\pi}}\over{{\rm Sin}(ul)}}[ 22 + 2{\rm Cos}(2ul) ] \quad .
\mu_n(1) =\frac{(b_n+4)\mu(\tau_{n+1}=1)}{h_{n+1}(1)}+ \frac{b_n\mu(\tau_{n+1}=2)}{h_{n+1}(2)}, &=\frac{(b_n+4)\mu(\tau_{n+1}=1)}{(b_{n}+4)h_n(1)+(b_n+3)h_n(2)}+ \frac{b_n\mu(\tau_{n+1}=2)}{b_{n}h_n(1)+(b_n+1)h_n(2)}.
G_{I_{k;q}}:=\sum_{i\subset I_{k;q}}H_i+t\sum_{I_{1;i}\subset I_{k;q}}V^{(k, q-1)}_{I_{1;i}}+\dots+t\sum_{I_{k-1;i}\subset I_{k;q}}V^{(k, q-1)}_{I_{k-1;i}}\,;
\times D^{- 2}({\bf y}-{\bf z}) \cos \left( \phi_{\bf x}- \phi_{\bf y}+ \phi_{\bf z}\right) + D^{2}({\bf x}-{\bf y}) D^{- 2}({\bf x}-{\bf z}) D^{- 2}({\bf y}-{\bf z}) \cos \left( \phi_{\bf x}+ \phi_{\bf y}- \phi_{\bf z}\right) \Bigr ] .
\varepsilon^2 >\left\|\widetilde f - P_m^{(-\alpha-1,\beta-1)}(x) p(x) \right\|_{L^2((-1,1);W_{\alpha,\beta})}=\left\|f - \left(P_m^{(-\alpha-1,\beta-1)}(x)\right)^2 p(x) \right\|_{\mathcal H}.
\psi_{R_{1}f_{1}, R_{2}f_{2}}^{\alpha_{1}^{1}\alpha_{1}^{2}, \alpha_{2}^{1}\alpha_{2}^{2}}( x_{1}, x_{2}) = - \psi_{R_{2}f_{2}, R_{1}f_{1}}^{\alpha_{2}^{1}\alpha_{2}^{2}, \alpha_{1}^{1}\alpha_{1}^{2}}( x_{2}, x_{1}) .
0 = \phi(xy+yx-t^2)_1 &= 2 \left[ a_0 (b_1 x + b_2 y + b_3 t) + b_0 (a_1 x + a_2 y + a_3 t) - c_0c_3t\right] &= 2 \left[ (a_0b_1 + b_0a_1)x + (a_0b_2+b_0a_2)y + (a_0b_3+b_0a_3 - c_0c_3)t\right].
\langle \hat{{\cal O}}_{n}( \vec{z}) \hat{{\cal O}}_{n}( \vec{z}^{\prime}) \rangle \, = \, \frac{B_{{\hat{{\cal O}}_{n}}, \hat{{\cal O}}_{n}}}{( \vec{z}- \vec{z}^{\prime})^{2 \hat{\Delta}_{n}}}\,
\frac{\Gamma(k+1)}{\Gamma(kp/2+1)^{1/p}\Gamma(kp'/2+1)^{1/p'}}= \frac{k}{(kp/2)^{1/p}(kp'/2)^{1/p'}}\cdot\frac{\Gamma(k)}{\Gamma(kp/2)^{1/p}\Gamma(kp'/2)^{1/p'}}&= C_p\cdot \frac{\Gamma(k)}{\Gamma(kp/2)^{1/p}\Gamma(kp'/2)^{1/p'}}.
V ( d ) = - G_{\mathrm{N}}{\frac{1}{d}}- \sum_{n = - \infty , n \not = 1}^{+ \infty}G_{\mathrm{N}}{\frac{1}{d}}e^{+ i{\frac{| n |}{L}}d}\sim{\frac{1}{d^{1 + 1}}}\quad a s \quad d \rightarrow 0
\begin{cases}b_2=a_1^2+4a_2; b_4=2a_4+a_1a_3; b_6=a_3^2+4a_6; b_8=a_1^2a_6+4a_2a_6-a_1a_3a_4+a_2a_3^2-a_4^2.\end{cases}
\sup_{\substack{\alpha\neq \beta 0\leq t<T_0^*}}\Big |\frac{z(\alpha,t)-z(\beta,t)}{\alpha-\beta}\Big|+\sup_{\substack{\alpha\neq \beta 0\leq t<T_0^*}}\Big |\frac{\alpha-\beta}{z(\alpha,t)-z(\beta,t)}\Big|=\infty.
0\ge \int_{\mathbb{R}^N}\langle \nabla w_{T},\nabla (u_0-w_{T})_+\rangle\, dx+c_2^{2-q}\, \int_{\mathbb{R}^N}w_T^{q-1}\, (u_0-w_{T})_+\, dx &-\int_{\mathbb{R}^N}f\, (u_0-w_{T})_+\, dx=\int_{\mathbb{R}^N}(h_T-f)\, (u_0-w_{T})_+\, dx.
\sum_{i=1}^{q}\int_0^T \alpha (t) \theta_{i}(t) \frac{d}{dt}f_{i,\lambda}(u_{\lambda}(t)) dt = - \sum_{i=1}^{q}\int_0^T \frac{d}{dt}(\alpha \theta_{i})(t) f_{i,\lambda}(u_{\lambda}(t)) dt.
vw_{\varrho>>1}=\sqrt{2n+2}\left(\frac{2^{\frac{j+1}{2}}(1-2^{-(j+1)}+r_{j})}{2-2^{-(j+1)}}-\frac{2^{\frac{j}{2}}(1-2^{-j}+r_{j})}{2-2^{-j}}\right),
\frac{\partial}{\partial i}\log q(i,j) =\left[\frac{\Gamma'(i+\lambda+1)}{\Gamma(i+\lambda+1)}-\frac{\Gamma'(i+1)}{\Gamma(i+1)}\right]-\left[\frac{\Gamma'(i+j+1/c_1+\lambda+\mu+2)}{\Gamma(i+j+1/c_1+\lambda+\mu+2)}-\frac{\Gamma'(i+j+1)}{\Gamma(i+j+1)}\right] &= [f'(i+\lambda+1)-f'(i+1)]-[f'(i+j+1/c_1+\lambda+\mu+2)-f'(i+j+1)],
\alpha_{\eta}:=\lim\limits_{t\to 0^{+}}\frac{\log \left (\sup\limits_{u>0}\frac{\eta(t u)}{\eta(u)}\right )}{\log(t)}, \beta_{\eta}:=\lim\limits_{t\to +\infty}\frac{\log \left (\sup\limits_{u>0}\frac{\eta(t u)}{\eta(u)}\right )}{\log(t)}.
\int_{I_2}|\hat{\psi}(r_{\xi}(\omega)) |^2 \beta(\omega)\, d\omega \leq \int_{I_2}C^2 (1 + | r_{\xi}(\omega) |)^{-2r}\, d\omega & \leq \int_{I_2}C^2 (1 + | r_{\xi}(\omega_{\xi}^{\ast}) |)^{-2r}\, d\omega & = \frac{\alpha}{1 - \alpha}C^2 \xi^{\alpha}(1 + | r_{\xi}(\omega_{\xi}^{\ast}) |)^{-2r}.
\begin{array}{l}\displaystyle H_{1m}\cdot G_{1m}=(I_2-F_m)=(I_2-F_m)^2=H_{1m}\cdot (G_{1m}\cdot H_{1m})\cdot G_{1m}, H_{2m}\cdot G_{2m}=F_m=F_m^2=H_{2m}\cdot (G_{2m}\cdot H_{2m})\cdot G_{2m}, (H_{1m}\cdot G_{1m})(H_{2m}\cdot G_{2m})=(H_{2m}\cdot G_{2m})(H_{1m}\cdot G_{1m})=0.\end{array}
\xi_{(g,x)}*\xi_{(h,y)}=\sum_{f_1,f_2\in \Gamma}\delta_{f_1g,f_2}(f_1\otimes f_2h\otimes x^{f_1}(y\gamma)^{f_2})= \sum_{f\in \Gamma}f\otimes fgh\otimes x^{f}(y\gamma)^{fg}&=\xi_{(gh,x(y\gamma)^g)}.
\Gamma^{\phi^{-}}( p ) = - i / ( p^{2}+ i \varepsilon ) \left[ K_{Q}( p ) + H_{Q}( p ) - i S_{Q}( p ) - \frac{R_{Q}^{2}( p )}{K_{Q}( p ) + H_{Q}( p ) + i S_{Q}( p )}\right] ,
(F E_{1}^{n}F_{2}^{m}\otimes F_2) \cdot w_{p_1 - p_2}= mn(2p_2 + p_1 - m) (E_{1}^{n-1}F_{2}^{m-1}\otimes F_2) \cdot w_{p_1 - p_2}- (m+n + 1) (E_{1}^{n}F_{2}^{m}\otimes F_1) \cdot w_{p_1 - p_2}.
h_{1-L}^{\mathcal{L}^{(1)(L+1)}}(t)-h_{0}^{\mathcal{L}_{2,L+1}}(t)\prod_{\alpha=1,\alpha\neq2}^Lh_{-1}^{\mathcal{K}_\alpha}(t)=h_{1-L}^{\mathcal{L}^{(12)(L+1)}}(t)-h_{0}^{\mathcal{K}_2\otimes\mathcal{K}_{L+1}}(t) \prod_{\alpha=1,\alpha\neq2}^Lh_{-1}^{\mathcal{K}_\alpha}(t),
\lim\inf_{T\to\infty}{R_1^{\pi}(T)\over \log(T)}\ge \lim\inf_{T\to\infty}{R^{\pi}(\lfloor Tr_1\rfloor)\over \log(T)}& = \lim\inf_{T\to\infty}{R^{\pi}(\lfloor Tr_1\rfloor)\over \log(\lfloor Tr_1\rfloor)}\ge \beta.
\begin{array}{l c l}{{g^{a b}\left< T_{a b}\right>}}&{{=}}&{{\left( \eta^{a b}+ h^{a b}\right) \left< T_{a b}\right> = 2 \eta^{z \overline{{z}}}\left< T_{z \overline{{z}}}\right> + \cal O \left( h^{2}\right)}}\\{{\& =}}&{{4 \left< T_{z \overline{{z}}}\right> + \cal O \left( h^{2}\right) \; ,}}\\ \end{array}
[Y^\star,[Y^\star,Y]]=-2(1-r)(Y^\star)^2 +(n(1-r)-X)Y^\star + Y, [Y,[Y^\star,Y]]=-2(r-1)\{Y,Y^\star\}+ Y(X-n(1-r) )-2(k-n)(r-1)( X+k(r-1)) -Y^\star\left( X^2+2(1-r)(n-2k-1)X+(r-1)^2(2n+(n-2k)^2)\right).
b_{\mathbf t_l}=\begin{cases}T_{a_k,l}b_{\mathbf t_{l-1}}\hbox{if $ \lambda=\mu\cup p$ with $p=(k,\mu_k+1)$,}T^{-1}_{l,2f}\sum_{j=b_{k-1}+1}^{b_k}q^{b_k-j}T^{-1}_{j,2f-1}b_{\mathbf t_{l-1}}, \hbox{if $\lambda=\mu\setminus p$ with $p=(k,\mu_k)$.}\end{cases}
\Vert B_\varepsilon (\vartheta )^{1/2}{\mathcal J}_2( \varepsilon;\vartheta; \widehat{\zeta}) \Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \mathfrak{C}_5\varepsilon c(\phi )^2, & 0 < \vartheta \leqslant 1,0< \varepsilon \leqslant \vartheta^{-1},\widehat{\zeta}=e^{i\phi},0<\phi <2\pi,
[ \hat{D}_{\alpha}, \hat{D}_{\beta}] f = - ( \partial_{\alpha}A_{\beta}^{\mu}- \partial_{\beta}A_{\alpha}^{\mu}) \partial_{\mu}f + ( A_{\alpha}^{\mu}\partial_{\mu}A_{\beta}^{\nu}- A_{\beta}^{\mu}\partial_{\mu}A_{\alpha}^{\nu}) \partial_{\nu}f \, .
S = - \frac{1}{2 \kappa_{P}^{2}}\int d^{4}x \sqrt{- g}\left[ \frac{1}{2}R + \frac{1}{4}( \nabla \varphi )^{2}+ \frac{3}{4}( \nabla \beta )^{2}+ \frac{q_{5}}{2}e^{( \beta - \varphi )}( \nabla z )^{2}\right] ,
\lambda_{s,t}\cdot\sum_{a=\lfloor\beta n\rfloor}^{\lfloor\frac{r-1}{r}n\rfloor-1}\Delta(a)=\lambda_{s,t}\cdot\sum_{a=\lfloor\beta n\rfloor}^{\lfloor\frac{r-1}{r}n\rfloor-1}M(a)\left[H\left(\frac{n-a}{a}\right)+o(1)\right],
\real \left( \frac{zh_{{\nu}}^{\prime}(z)}{h_{{\nu}}(z)}\right) =1-\real\left( \sum_{n\geq1}\frac{z}{\gamma_{{\nu},n}^{4}-z}\right) \geq1-\sum_{n\geq1}\frac{\left\vert z\right\vert}{\gamma_{{\nu},n}^{4}-\left\vert z\right\vert}=\frac{\left\vert z\right\vert h_{{\nu}}^{\prime}(\left\vert z\right\vert )}{h_{{\nu}}(\left\vert z\right\vert )},
( \tilde{\pi}_{g})_{a b}\equiv ( \pi_{g})_{a b}+ \frac 1{2 ( \hat{p}+ \Pi_{1}) p_{1}}( p_{\psi}\Gamma^{\mu}\Gamma^{\nu}\theta ) ( \hat{p}^{\mu}+ \Pi_{1}^{\mu}){T^{\nu}}_{a b},
Z_{t}\, :=\, \exp \Big ( - \int^{t}_{0}b(s, X_{s}, F_{s}){\mathrm d}B_{s}- \frac{1}{\, 2\,}\int^{t}_{0}\lvert b(s, X_{s}, F_{s})\rvert^{2}{\mathrm d}s \Big) \, ; t \ge 0 \, ,
\bar{N}^{rs}_{nm}= \frac{1}{nm}\oint_{z^r}\frac{dz}{2 \pi i}\oint_{z^s}\frac{dw}{2 \pi i}\frac{1}{(z-w)^2}(-1)^{n (r-1) + m (s-1)}(f (z))^{(-1)^r n}(f (w))^{(-1)^s m}
Y_{1}= \left( \begin{array}{c c c c}{x_{1}}&{0}&{0}&{\cdots}\\{0}&{x_{1}}&{0}&{\cdots}\\{\vdots}&{0}&{x_{1}}&{\cdots}\\{\vdots}&{\vdots}&{0}&{\ddots}\\{0}&{0}&{0}&{0}\\ \end{array}\right) .
\left( \begin{array}{c}{{\tilde{t}}_{1}}\\{{\tilde{t}}_{2}}\\ \end{array}\right) = 2 \pi i \left( \begin{array}{c c}{1}&{1 / 2}\\{0}&{- 1 / 2}\\ \end{array}\right) \left( \begin{array}{c}{t_{1}}\\{t_{2}}\\ \end{array}\right) ,
Q_{j_0}- \eta \, k^2 \, a^{5-h}\, \sum_{j=1 \atop j \neq j_{0}}^\aleph{\bf{P}}_0 \cdot \Upsilon_k(z_{j_0}, z_j) \cdot Q_j \, = i \, k \, a^{5-h}\,{\bf{P}}_0 \cdot H^{Inc}(z_{j_0})+ Error,
\begin{array}{l}c_i=|\boldsymbol{v}|_i+\sqrt{gh_i}, c_{\rm max}=\max\limits_i \{c_i\}, c_{(i,j)}=\max\{c_i,c_j\}, \phi_{\rm min}=\min\limits_i \left\{\displaystyle\frac{\sigma_i}{\sum\limits_{j\in{\cal N}(i)}l_{(i,j)}}\right\}. \end{array}
r_1(u, \bar v) = \sum_{j=1}^n R(e_j, \bar e_j, u, \bar v), r_2(u, \bar v) = \sum_{j=1}^n R(e_j, u, \bar v, \bar e_j), r_3(u, \bar v) = \sum_{j=1}^n R(u, \bar v, e_j, \bar e_j), r_4(u, \bar v) = \sum_{j=1}^n R(u, e_j, \bar e_j, \bar v),
\langle M_{1,2}\psi, M_{2,1}\psi\rangle_{L^2(\Omega_T)}=-\frac{1}{2}s^2 \lambda^2 \int_{\Omega_T}\sigma \xi^2 \partial_t (\psi^2) \,\d x\,\d t = s^2 \lambda^2 \int_{\Omega_T}\sigma \partial_t \xi \xi \psi^2 \,\d x\,\d t
\frac{P_j}{P_i}= \left[\frac{(\sup(\mathcal{L}_j))^2+(\inf(\mathcal{L}_j))^2}{(\sup(\mathcal{L}_i))^2+(\inf(\mathcal{L}_i))^2}\right]^{\frac{\alpha}{2}}&= \left(\frac{\eta_j}{\eta_i}\right)^{-\frac{\alpha}{2}}\left[\frac{(\sup(\mathcal{L}_j))^4-(\inf(\mathcal{L}_j))^4}{(\sup(\mathcal{L}_i))^4-(\inf(\mathcal{L}_i))^4}\right]^{\frac{\alpha}{2}}.
f_{(\theta, r)}(x) = \exp\Bigl ( \langle \theta, z\rangle - \sum_{j}\gamma_j(\theta_j, r) (\alpha_j u_j\wedge r^{-1}) - \sum_{j}\xi_j(\theta, r) (\mu_j^{(r)}v_j\wedge r^{-1}) \Bigr ).
L = \mathrm{T r}\left\{- \frac{1}{4}G^{\mu \nu}G_{\mu \nu}+ \frac{1}{2}D_{\mu}S^{\dagger}D^{\mu}S \right\}+ \frac{1}{2}D_{\mu}\phi^{\dagger}D^{\mu}\phi - V ( S , \phi ) \, .
\left( L^{{M}}(\Omega),|\cdot |_{{M}}\right)^* = \left( L^{\tilde{M}}(\Omega),\|\cdot \|_{\tilde{M}}\right)\ \mbox{and}\ \left( L^{\tilde{M}}(\Omega),|\cdot |_{\tilde{M}}\right)^* =\left( L^{{M}}(\Omega),\|\cdot \|_{{M}}\right)
\int_{B(0,RC_{2}^{1/2}t^{1/2})^{c}}\frac{C_{1}}{t^{n/2}}\exp\left(-\frac{1}{C_{2}}\left(\frac{\vert\xi\vert^{2}}{t}\right)\right)d\xi=C\int_{B(0,R)^{c}}\exp\left(-\vert\xi\vert^{2}\right)d\xi,
T^{0 0}={\frac{2 c o s h^{2}( \gamma )}{g^{2}}}\left[ - C^{2}s e c h^{2}( C r ) \Big ( 1 - C r \; t a n h ( C r ) \Big )^{2}+{\frac{( C^{2}r^{2}s e c h^{2}( C r ) + 1 )^{2}}{2 r^{2}}}\right]
\tilde{f}_{\beta}(x,y)=(x^{mn_1}-y^{mn_1})^{n_2}+x^{ma}y^{n_1b}+\sum_{\nu\in N_{\beta}}t_{\nu}(x^{mn_1}-y^{mn_1})^{\ell_{\nu}}x^{ma_{\nu}}y^{n_1b_{\nu}}\tilde{\tilde{f}}_{\beta}(x,y)=(1-y^{mn_1})^{n_2}+x^{q}y^{n_1b}+\sum_{\nu\in N_{\beta}}t_{\nu}(1-y^{mn_1})^{\ell_{\nu}}x^{\frac{\nu+q(n_2-\ell_{\nu})}{n_2}}y^{n_1b_{\nu}}
\begin{aligned}\lim_j\|T^{-n_{k_j}}XJ_{\theta,1}^{-1}p\|^2 =\lim_j(\|T^{-n_{k_j}}XJ_{\theta,1}^{-1}p\|^2-\|pXJ_{\theta,1}^{-1}\chi^{-n_{k_j}}\|^2) + L(p) &=\|R^{-1}X_1 P_{L^2(\mu_{})}p\|^2+F(p) + L(p).\end{aligned}
\sum_{i=1}^{n-2}\int_s^t \int_s^{v_n}\cdots \int_s^{v_{n-i+1}}\prod_{k=n-i+1}^{n}\Phi(v_k-v_{k-1}) h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots dv_n &=\sum_{i=1}^{n-2}\int_s^t \int_s^{u}\Phi_i(u-r) h_{r}^{s,(n-i-1)}dr du,
\Biggl| \bigcup_{i=1}^{R}V_i \setminus \{0\}\Biggr| \leq \sum_{i=1}^{R}|V_i \setminus \{0\}| \leq \left( \frac{q^{k+1}-1}{q-1}- 1 \right) (q - 1) = q^{k+1}- q < q^{k+1}-1,
{a}_{\bot}^{\ell}( x_{\bot}) = \frac{1}{L}\int d y^{-}d y_{\bot}d ( x_{\bot}- y_{\bot}) \nabla_{\bot}\left( \nabla_{\bot}\cdot A_{\bot}^{3}({y}_{\bot}, y_{-}) \right) \frac{\tau^{3}}{2}.
\left[ \sqrt{\frac{2 5}{4}+ 4 \kappa^{2}\tilde{\omega}_{m}^{2}}- \left( m + \frac 1 2 \right) \right]^{2}\ge \left[ \sqrt{\frac{2 5}{4}}- \left( m + \frac 1 2 \right) \right]^{2}= [ 2 - m ]^{2}\ge 1
\overset{J}{V}(j)_2 \overset{I}{U}(i)_1 = (\overset{I}{a'_k})_1 \overset{I}{U}(i)_1 \overset{I}{S(a''_k)}_1 (\overset{J}{b'_k})_2 \overset{J}{V}(j)_2 \overset{J}{S(b''_k)}_2 = (\overset{I}{a_l})_1 \overset{IJ}{R}_{12}\overset{I}{U}(i)_1 \overset{IJ}{R}{^{-1}_{12}}\overset{J}{V}(j)_2 \overset{IJ}{R}_{12}\overset{J}{S(b_l)}_2
\tilde{S}_{Q 1}^{k i n}( B_{i}) = \frac{2}{N}\mathrm{T r}\, \sum_{j = - L / 2 + 1}^{L / 2}( 1 - \cos k_{j}) B_{j}^{2}= \tilde{S}_{Q 1 f}^{k i n}( B_{i})
\begin{aligned}\int_\Omega |\nabla T_{k}(u_p)|\, dx + \int_{\partial\Omega}(\lambda|T_{k}(u_p)|-g T_{k}(u_p)) d\mathcal H^{N-1}\le \int_\Omega f T_{k}(u_p)\, dx + \frac{p-1}{p}|\Omega| + \frac{p-1}{p}\int_{\partial\Omega}\lambda d\mathcal H^{N-1}.\end{aligned}
\oint_{C}dz \omega(z)=-(-z_{\infty})^{-(1-\vartheta_{3})}4\sin \pi \vartheta_{2}\sin \pi\vartheta_{1}e^{\pi i \vartheta_{1}}\frac{\Gamma(\vartheta_{1})\Gamma(\vartheta_{2})}{\Gamma(1-\vartheta_{3})},
\Gamma := \{\varphi(c_1,\ldots,c_m,d^1_{\varphi,\bar{c}},\ldots,d^n_{\varphi,\bar{c}}) : \varphi(\bar{x},\bar{y}) \in \Phi,~ |\bar{x}|=m,~ |\bar{y}|=n,~ \bar{c}\in C^m\}.
+\;\frac 1{\left( q-1\right) !}\left[ \,\left( -\delta_M\delta_N+\delta_N\delta_M\mp C_{MN}^{\;K}\,\cdot \delta_K\right) A_{\nu_2\ldots \nu_q}\right] \Lambda^{\nu_2\ldots \nu_q}\cdot \delta \left(\sigma -\sigma^{\prime}\right) \quad \cdots \quad ,
\begin{array}{l}\psi_{a}(z) \psi^{b}(w) = \frac{\delta_{a}^{b}}{(z-w)}+ \cdots \ \psi_{a}(z) \psi_{b}(w) = \psi^{a}(z) \psi^{b}(w) = 0 + \cdots\end{array}
B_1=\left\{\left.\frac{9+4\ell}{40}\right|\underset{\ell\neq 4}{0\leq\ell\leq 8}\right\}\cup\left\{\left.\frac{10+4\ell}{40}\right|\underset{\ell\neq 5}{1\leq\ell\leq 9}\right\}\cup\left\{\left.\frac{11+4\ell}{40}\right|\underset{\ell\neq 1,6}{0\leq\ell\leq 9}\right\},
\frac{\partial \Phi_{0 1}}{\partial y}\, - \, \varphi ( y ) \, \frac 1 i \, \frac{\partial \Phi_{0 1}}{\partial q}\, = \, \varphi ( y ) \bigl [ p \, - \, 2 i \bigl ( \delta ( q ) - \delta ( 1 - q ) \bigr ) \bigr ] \, \Phi_{0 1},
\begin{aligned}Q_{k,k+1}=\int_{\tau_k}^{\tau_{k}+1}\int_{\tau_k+1}^{\tau_{k}+2}R(s-u,\omega(s)-\omega(u))dsdu &=\int_0^1\int_0^1 R(s+1-u,\omega(\tau_k+1+s)-\omega(\tau_k+u)) dsdu &=\int_0^1\int_0^1 R(s+1-u,x_k(1)+x_{k+1}(s)-x_k(u)) dsdu.\end{aligned}
T(z)=\frac{1}{8\pi}\partial_{z}\varphi \partial_{z}\varphi =\sum^{\infty}_{k=-\infty}L_{k}z^{-k-2}\quad ,\quad \bar{T}(\bar{z})=\frac{1}{8\pi}\partial_{\bar{z}}\varphi \partial_{\bar{z}}\varphi =\sum^{\infty}_{k=-\infty}\bar{L}_{k}\bar{z}^{-k-2}