harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal J}_{\mu_{1}\cdots \mu_{s}}^{( s )}\rightarrow{\cal J}_{\mu_{1}\cdots \mu_{s}}^{( s )}+ \sum_{s^{\prime}= 0}^{s - 2}\h_{s , s^{\prime}}( g ) \ \Delta_{\mu_{1}\cdots \mu_{s}}^{( s , s^{\prime})},
L_{1}^{\ast\ast}=\frac{c_{1}\left[\xi_{p}^{3}\overline{G}_{(1)}\left(\frac{1}{2}\xi_{p}^{2}\right)-\xi_{q}^{3}\overline{G}_{(1)}\left(\frac{1}{2}\xi_{q}^{2}\right)\right]}{F_{Y}(\xi_{p},\xi_{q})},
H_d(x) =x^d H_0(x)+x^{d+2}\left(\frac{1}{(1-x)^d}-1\right)L &+\left(\frac{1}{(1-x)^d}-1\right)\sum_{t\ge 2}x^{t+1+d}\left(1+(L-1)\frac{1-1/(1-x)^t}{1-1/(1-x)}\right)\, .
\tilde{\Xi}_n^{c}(z_1,z_2, s)=\sum_{c> 0}\sum_{\substack{a,d, ad \equiv 1 \pmod c}}\frac{(\overline{\mu_{\gamma}(z_1,-z_2)-1/c})^n~ \overline{\mu_{\gamma}(z_1,-\bar{z_2})-1/c})^n}{|\mu_{\gamma}(z_1,-z_2)-1/c|^{2s}~ |\mu_{\gamma}(z_1,-\bar{z_2})-1/c|^{2s}}
\frac{(|\Phi'_+(x)|+|\Phi'_-(x)|)\,dx}{2\pi}= \frac{(\sqrt{r^2+1}+1)\,dx}{\pi\sqrt{r^2-x^2}(x^2+(\sqrt{r^2+1}+1-\sqrt{r^2-x^2})^2)}+ \frac{(\sqrt{r^2+1}+1)\,dx}{\pi\sqrt{r^2-x^2}(x^2+(\sqrt{r^2+1}+1+\sqrt{r^2-x^2})^2)},\x\in(-r,r).
S_{A}= - \frac{1}{4}\int d^{6}x \sqrt{^6 g}g^{A B}g^{M N}F_{A M}F_{B N}\simeq - \frac{1}{4}\epsilon^{2}\int_{1}^{a}d \phi \int d^{4}x \sqrt{- \eta}\eta^{\mu \nu}\eta^{\alpha \beta}F_{\mu \alpha}F_{\nu \beta}.
C(\alpha)=\sum_{\substack{0\le j_1,i_1,\dots,j_{q-1},i_{q-1}\le n-1\cr s(j_1,i_1)\oplus\cdots\oplus s((j_{q-1},i_{q-1})=\alpha}}a_{i_1}^{q^{j_1}}\cdots a_{i_{q-1}}^{q^{j_{q-1}}}.
t_8^{r_1\cdots r_8}M_{r_1r_2}N_{r_3r_4}\cdots N_{r_7r_8}= -6\, M_{t_1t_2}N_{t_2t_1}N_{mn}N_{nm}+ 24\,M_{t_1t_2}N_{t_2m}N_{mn}N_{nt_1}\, .
dq^\prime_t= \tilde\gamma^{-1}(t^*)\left(-\nabla_qV(t^*,q^\prime_t)+\tilde F(t^*,q_t)\right) dt &+\tilde S(t^*,q^\prime_t)dt+\hat{\phi}_*(\tilde\gamma^{-1}\sigma)(t^*,q^\prime_t) \circ d\tilde W_t.
\pi_{\ast}0^!_{\widetilde X}(\pi^*Z) \overset{\eqref{eq:saito:20}}{=}\pi_\ast e_\ast({\rm pr}_\pi^!Z\cdot_{{\rm id}_{T^\ast X\times_X\widetilde X}}d\pi^!(\widetilde X)) \overset{\eqref{eq:saito:50}}{=}\pi_\ast ({\rm pr}_\pi^!Z\cdot_{{\rm id}_{T^\ast X\times_X\widetilde X}}\widetilde X)+\pi_\ast ({\rm pr}_\pi^!Z\cdot_{{\rm id}_{T^\ast X\times_X\widetilde X}}K)
\overline{\lambda_{\uparrow L -}}\: \Lambda +\lambda_{\uparrow L -}\: \overline{\Lambda}\;=\; 2\:I_\uparrow\: \hat{Y}_L\:I_\downarrow\:\hat{Y}_R\: I_\uparrow \:\left( T^{(0)}_{[1]}\:T^{(0)}_{[1]}\:\overline{T^{(-1)}_{[0]}\: T^{(0)}_{[0]}}\:+\:T^{(-1)}_{[0]}\: T^{(0)}_{[0]}\:\overline{T^{(0)}_{[1]}\:T^{(0)}_{[1]}}\right) \;.
B_{0,1}^h=b_1b_2\cdots b_h, \ \ \B_{0,2}^h=b_1b_2\cdots b_h \cdot c_ha_{h+1}, \ \ 1\leq h\leq g; &B_{2k-1}^h=a_k \cdot b_kb_{k+1}\cdots b_{h+1-k}\cdot c_{h+1-k}a_{h+1-k}, \ \ 1\leq k\leq r, \ 1\leq h \leq g; &B_{2k}^h=a_k \cdot b_{k+1}b_{k+2}\cdots b_{h-k}\cdot c_{h-k}a_{h+1-k}, \ \ 1\leq k\leq r, \ 1\leq h \leq g.
\mathcal{D}(p_{_1}||p_{_0}) + \rho_0 \left[\int_{-\infty}^{\infty}p_{_1}(y) dy -1\right] &+ \rho_1 \left[\int_{-\infty}^{\infty}y^2 p_{_1}(y) dy - P_y\right] = \int_{-\infty}^{\infty}\mathcal{L}(y, p_{_1}(y)) dy - \tau,
[v_1, v_2]=4\kappa v_2, [v_1, v_3]=-4\kappa v_3, [v_1, v_4]=2\kappa v_4, &[v_1, v_5]=-2\kappa v_5, [v_2, v_3]=-8\kappa v_1-\frac{2}{\kappa}p v_6, [v_2, v_5]=-4\kappa v_4, &[v_3, v_4]=4\kappa v_5, [v_4, v_5]=2 \kappa v_6,
\ddot{\Phi}- \frac{2}{a}\, \dot{\Phi}_{, r}+ \frac{1}{a^{2}}\, \Phi_{, r r}+ H \dot{\Phi}= 8 \pi G \, \dot{\varphi}\left( \delta \dot{\varphi}- \frac{1}{a}\, \delta \varphi_{, r}\right) ,
\begin{array}{l c l}{{C ( \Lambda ) E_{- \alpha}( \mu ) \omega}}&{{=}}&{{\frac{2}{\Lambda - \mu}\left( E_{- \alpha}( \Lambda ) h_{\alpha}( \mu ) \right.}}\\{{}}&{{}}&{{\left. - E_{- \alpha}( \mu ) h_{\alpha}( \Lambda ) \right) \omega + E_{- \alpha}( \Lambda ) C ( \Lambda ) \omega}}\\ \end{array}
{\cal L}= - \frac{1}{4}F_{\mu \nu}^{2}+ \frac{1}{2}( \partial_{\mu}\varphi ) ( \partial^{\mu}\varphi ) - \frac{e}{{2 \sqrt \pi}}\varepsilon^{\mu \nu}F_{\mu \nu}\varphi + + m \sum \left({\cos \left({2 \pi \varphi + \theta}\right) - 1}\right) - A_{0}J^{0},
{\rm d}\equiv{\rm d}z^{{\cal A}}\frac{\partial~}{\partial z^{{\cal A}}}=e^{{\cal A}}D_{{\cal A}}=e^{A}\partial_{A}+{\rm d}\theta^{i\alpha}D_{i\alpha}+{\rm d}\tilde{\theta}^{\tilde{\imath}}_{\alpha}{\tilde{D}}^{\alpha}_{\tilde{\imath}}
\begin{array}{r c l}\delta_\epsilon A^a = \{A^a,Q_\epsilon\}=gf^{abc}\epsilon^b A^c\ \ \ , \delta_\epsilon \pi^a = \{\pi^a,Q_\epsilon\}=gf^{abc}\epsilon^b \pi^c\ \ \ , \delta_\epsilon \phi^a = -\dot{\epsilon}^a+gf^{abc}\epsilon^b\phi^c\ \ \ ,\end{array}
\begin{gathered}\cot(\phi)>0,\ \ \cot(\theta)<0,\ \ \theta\in\mathcal{A}\cup\mathcal{D}\ 0< \cot(\theta)<\cot(\phi), \ \ \theta\in\mathcal{B}\cup\mathcal{E}\cot(\phi)<\cot(\theta), \ \ \theta\in\mathcal{C}\cup\mathcal{F}\end{gathered}
\displaystyle w_{jj}= \Big(\frac{2h_ih_{ij}}{(\beta-h)^2}\Big)_j + \Big(\frac{2h^2_i h_j}{(\beta-h)^3}\Big)_j \displaystyle = \frac{2h^2_{ij}}{(\beta-h)^2}+ \frac{2h_ih_{ijj}}{(\beta-h)^2}+\frac{8h_ih_jh_{ij}}{(\beta-h)^3}+\frac{2h^2_i h_{jj}}{(\beta-h)^3}+ \frac{6h^2_i h^2_j}{(\beta-h)^4}.
\sum_{\alpha(1),\ldots,\alpha(k)=1}^n \delta_p(\alpha,\beta) u_{\alpha(1)i(1)}^{r_1}\cdots u_{\alpha(k)i(k)}^{r_k}=\sum_{\gamma(1),\ldots,\gamma(l)=1}^n \delta_p(i,\gamma) u_{\beta(1)\gamma(1)}^{s_1}\cdots u_{\beta(l)\gamma(l)}^{s_l}.
P_s[(\frac{v_t}{\gamma})^\frac1p](x) P_s[(\frac{v_t}{\gamma})^\frac1p \Gamma ( \log\, \frac{v_t}{\gamma})](x)-| P_s[(\frac{v_t}{\gamma})^\frac1p \nabla \log\, \frac{v_t}{\gamma}] |^2(x) = \frac{1}{A_x}\int_{\mathbb{R}^n}|\vec{\varphi}_x(y)|^2 \, d\mu(y)
\psi_N(t, y, w) = \int e^{\displaystyle i(y\cdot x + w\cdot v)}\, M_t^{(N)}(dx, dv) = \frac1N\sum_{j =1}^{N}e^{\displaystyle i\,(\, y\cdot x_j^{(N)}(t) + w\cdot u^{(N)}_j(t)\,)}.
\|z_{m}\|_{H}^2 + h\sum_{n=0}^{m-1}\|B_{1}^{1/2}z_{n+1}\|_{H}^2 + \|v_{m}\|_{V}^2 + h^2\sum_{n=0}^{m-1}\|z_{n+1}\|_{V}^2 \leq C_{5}
g_0(s)=\int_{0}^{\infty}\!\!\!\! \int_{0}^{\infty}\!\!\!\! \int_{0}^{\infty}\!\!\!\! \int_{0}^{\infty}e^{-\sum_{i=0}^3 \tau_i}\overset{\scriptstyle 3}{\underset{\scriptstyle i=1}\Pi}\tau_i^{\frac{a_i}4-1}\left(\int_{c-\i\infty}^{c+\i\infty}e^{-xt}dt\right)\ \frac{d\tau_0}{\tau_0}d\tau_1d\tau_2d\tau_3.
\left|\bar A(y_1)(z_1,z_1)-\bar A(y_2)(z_2,z_2)\right|\le c_1(1+|z_1|^2+|z_2|^2)\left(|y_1-y_2|+|z_1-z_2|\right), &\left|\bar f(y_1,z_1)-\bar f(y_2,z_2)\right|\le c_1(1+|z_1|+|z_2|)\left(|y_1-y_2|+|z_1-z_2|\right).
\| \widetilde{c}_{j l}\| = \left( \begin{array}{c c c c}{{1}}&{{\widetilde{c}^{\prime}}}&{{\widetilde{c}^{\prime}}}&{{\widetilde{c}^{\prime}}}\\{{\widetilde{c}^{\prime}}}&{{1}}&{{1 / 2}}&{{1 / 2}}\\{{\widetilde{c}^{\prime}}}&{{1 / 2}}&{{1}}&{{1 / 2}}\\{{\widetilde{c}^{\prime}}}&{{1 / 2}}&{{1 / 2}}&{{1}}\\ \end{array}\right) , \quad \widetilde{c}^{\prime}= - \sqrt{\frac{1 + 3 c}{2 ( 1 + 2 c )}}.
{\cal O}= \exp \left( - \sum_{n = 1}^{\infty}\alpha_{- n}A_{n}\right) : \exp \left[ \sum_{n , m = 1}^{\infty}\alpha_{- n}\left( C_{n m}- \delta_{n m}\right) \alpha_{m}\right] : \exp \left( - \sum_{n = 1}^{\infty}B_{n}\alpha_{n}\right) \mathrm{e}^{- \phi}
\sum_{N=0}^{\infty}\frac{B_{N}}{N!}|\mathbf{u}|^{N}=\sum_{N=0}^{\infty}B_{N}\sum_{|\mathbf{m}|=N}\Psi_{\mathbf{m}}^{(d)}(\mathbf{u}) =\sum_{\mathbf{m}\in \mathcal{P}}B_{|\mathbf{m}|}\Psi_{\mathbf{m}}^{(d)}(\mathbf{u}).
\mathcal{E}_{s,m\Lambda}^{\rm cp}(m^d N) =m^{-s}\mathcal{E}_{s,\Lambda}^{\rm cp}(m^d N), \zeta_{m\Lambda}(s)=m^{-s}\zeta_\Lambda(s), \mathcal{E}_{\log,m\Lambda}^{\rm cp}(m^d N) =\mathcal{E}_{\log,\Lambda}^{\rm cp}(m^d N), \zeta_{m\Lambda}'(0)=\log m +\zeta_\Lambda'(0),
\begin{pmatrix}{\dot{f}}_1^{\alpha}\vdots{\dot{f}}_N^{\alpha}\end{pmatrix}=\begin{pmatrix}\lambda_{11}^{\alpha}\cdots \lambda_{1N}^{\alpha}\vdots \ddots \vdots \lambda_{N1}^{\alpha}\cdots \lambda_{NN}^{\alpha}\end{pmatrix}\begin{pmatrix}f_1^{\alpha}\vdots f_N^{\alpha}\end{pmatrix},
D_{3,a}=\frac{1}{|z|}\big[ Q(X_1,z+t_0(X_1), D_t \tilde h(X_1,z+t_0(X_1) ) )- Q(X_1,z+t_0(X_1), D_t \tilde h(X_1,t_0(X_1) ) ) &- \big( Q(X_2,z+t_0(X_2), D_t \tilde h(X_2,z+t_0(X_2) ) )- Q(X_2,z+t_0(X_2), D_t \tilde h(X_2,t_0(X_2) ) ) \big) \big]
|x-y|= |x-2^{|j|+2N}e_1+2^{|j|+2N}e_1-2^{|k|+2N}e_1+2^{|k|+2N}e_1-y| \ge |2^{|j|+2N}e_1-2^{|k|+2N}e_1|-|y-2^{|k|+2N}e_1|-2^{-j}\ge C2^{2N}.
U_{2n,N-x}= U_{2n,N+x+1}= \frac{(-1)^n}{\sqrt{2}}\tilde Q_n(x;\alpha,\beta,N), U_{2n+1,N-x}= -U_{2n+1,N+x+1}= -\frac{(-1)^n}{\sqrt{2}}\tilde Q_n(x;\alpha+1,\beta,N)
~s_1^*=\mathcal{F}\Big(r_3^*-G_3^*\left(Z^*\right),l_3^*,l_3^{**}\Big),~s_2^*=\mathcal{F}\Big(r_4^*-G_4^*\left(Z^*\right),l_4^*,l_4^{**}\Big),
\textsf{E}\Vert BA\Vert^{p}=\textsf{E}\Vert BA\Vert^{p}\mathbb{I}_{(M\le C_{1}(\varepsilon))}+\sum_{k=1}^{\infty}\textsf{E}\Vert BA\Vert^{p}\mathbb{I}_{(2^{k-1}C_{1}(\varepsilon)<M\le 2^{k}C_{1}(\varepsilon))}.
\lambda\int_U u^{\alpha-1}\xi^2\frac{\partial u}{\partial t}dx =\int_U \nabla \cdot (K (|\nabla u|)\nabla u )u^{\alpha+\delta-1}\xi^2dx &=-(\alpha-\lambda)\int_U K (|\nabla u|)|\nabla u|^2u^{\alpha+\delta-2}\xi^2dx-2\int_U K (|\nabla u|)\nabla u\cdot \nabla\xi u^{\alpha+\delta-1}\xi dx.
{\frac{1}{2 \pi}}\int_{0}^{2 \pi}d \varphi \, \left[ R ( r , \varphi ) \, K_{1}( R ( r , \varphi ) ) \right]^{2}= \sum_{n = - \infty}^{\infty}\left[ r ( I_{n}\, K_{n})^{\prime}( r ) \right]^{2}\equiv F ( r ) ,
h_{-1-L}^{\mathcal{L}_+}(t) =h_{-1-L}^{\mathcal{L}}(t) h_{1-L}^{\mathcal{L}_+}(t)-h_{1-L}^{\mathcal{L}}(t) =h_{-L}^{\mathcal{L}_{10L+1}}(t) h_{3-L}^{\mathcal{L}_+}(t)-h_{3-L}^{\mathcal{L}}(t) =h_{2-L}^{\mathcal{L}_{10L+1}}(t) &\cdots
\mathfrak{C}(R):= \{\rho \in C([0,+\infty)\times[-1,1]) : \rho(0,\zeta)=\rho(0,0), &\;\rho (r,\zeta)=\rho (r,-\zeta)\;\;\mbox{for}\;\; \forall \zeta\in[-1,1], \;\; \;\;\rho(r,\zeta)=0\;\; \;\; r\geq R \}
L\left(q(t),\dot{q}(t),{_a^C D^{\frac{1}{2}}_t}q(t)\right)=\frac{1}{2}m\left(\dot{q}(t)\right)^2-U(q(t))+\frac{\gamma}{2}\left({_a^C D^{\frac{1}{2}}_t}q(t)\right)^2,
\Psi_1 \left( \boldsymbol{d}_{\mathcal{S}_1}(s_1, s_1^\prime)^p, \rho_2(y_1, y_1^\prime)^{p}\right) = \Psi_1 \circ \psi_1 \left( \boldsymbol{d}_{\mathcal{Y}_1}(y_1, y_1^\prime)^{p}, \boldsymbol{d}_{\mathcal{Y}_2}(y_2, y_2^\prime )^p , \boldsymbol{d}_{\mathcal{X}}(x, x^\prime)^p \right),
R_n^*({\bar z}) H_{4,n}^{-1}\left\{-b u_{4,n}v_n^* + v_n u_{2,n}^* \right\}\Sigma_{2,n}= R_n^*({\bar z}) (I-bT_n^*) \Sigma_{2,n}= (I-bT_n^*)R_n^*({\bar z}) \Sigma_{2,n}.
\mathcal{G}(10c+1) \oplus \mathcal{G}(1) = 6 = \mathcal{G}(8c+1) \oplus \mathcal{G}(2c+1) \mathcal{G}(10c+1) \oplus \mathcal{G}(4c+1) = 7 = \mathcal{G}(8c+1) \oplus \mathcal{G}(6c+1)
\frac{\left( x_{k}\right)_{j}^{2}}{\left( x_{k}\right)_{m}^{2}}=\frac{\det\left( A_{G\backslash\left\{j\right\}}-\lambda_{k}I\right)}{\det\left( A_{G\backslash\left\{m\right\}}-\lambda_{k}I\right)}=\frac{c_{A_{G\backslash\left\{j\right\}}}\left( \lambda_{k}\right)}{c_{A_{G\backslash\left\{m\right\}}}\left( \lambda_{k}\right)}
Q_k(z,a) = P_k ((z-a_1)^{-1}, ... , (z-a_N)^{-1}) , & Q_{k,j}(z,a) =(z-a_j)^{-1}\frac{\partial P_k}{\partial \xi_j}((z-a_1)^{-1}, ... , (z-a_N)^{-1}) Q(z,a) = \prod_{s=1}^N (z-a_{s})^{N!}\, ,
\theta_{(i)}\begin{cases}\leq \tilde{\theta}_{(i)}\leq \eta_{(i)},1\leq i\leq k_1; = \tilde{\theta}_{(i)},k_1<i<k_2; \geq \tilde{\theta}_{(i)}\geq \eta_{(i)},k_2\leq i \leq n.\end{cases}
\sqrt[\leftroot{-4}\uproot{5}5]{5abK_5}=\begin{cases}2^d 5^e r' s \sqrt[\leftroot{-2}\uproot{3}5]{K_5}, \textnormal{for FLT2 Case I}, 2^d 5^e r_0 s' \sqrt[\leftroot{-2}\uproot{3}5]{K_5}, \textnormal{for FLT2 Case II}.\end{cases}
\|u(t_n)-u^n\|+\gamma^{\frac12}\Big{(}\Delta t\sum_{k=1}^n\|\sigma(t_{k-\theta})-\sigma^{k-\theta}\|^2\Big{)}^{\frac12}+\Big{(}\Delta t\sum_{k=1}^n\|\nabla u(t_{k-\theta})-u^{k-\theta}\|^2\Big{)}^{\frac12}\leq C\Delta t^2.
\int_0^{1/2}(-u'(x))\Phi(x)\,dx\ge \Phi\left(\frac{1}{2}\right)\int_0^{1/2}(-u'(x))x\,dx=\Phi\left(\frac{1}{2}\right)\left(-\Big[u(x)x\Big]_0^{1/2}+\int_0^{1/2}u(x)\,dx\right).
\left| \frac{{\partial \varphi}_{m+j}}{{\partial x}_{i}}\right|\le \frac{l_{j}^{y}}{l_{1}^{y}}\frac{\left( M_{1}^{y}M_{j}^{y}-\mu_{1}^{y}\mu_{j}^{y}\right)}{{(\mu_{j}^{y})}^{2}}\frac{l_{1}^{x}}{\mu_{1}^{x}\sum\limits_{i=1}^m \frac{l_{i}^{x}}{M_{i}^{x}}},\thinspace j=1,\thinspace \mathellipsis ,\thinspace n;\thinspace i=1,\mathellipsis ,m,
\lambda_{G/H}\circ q (f) \xi (yH) = \int_G f(g) \xi(g^{-1}yH) \, dg = \int_{G/H}\int_H f(xh) \xi(h^{-1}x^{-1}y H) \, dh dx = \int_{G/H}\int_H f(xh) \xi(x^{-1}y H) \, dh dx
\sum_{j,g=0}^C\tbinom Cj\tbinom Cg\tbinom{2^t(B+1)-1+C-j-g}{2^tB+C+1-k_0}\tbinom{2^t(B+1)-(C-j-g)}{2^tB+C+1-k_0}\tbinom{2^t(B+1)-C-2}{C-j-2g}.
Z_{N}= -{\frac{\beta^{N + 1}}{\pi \omega x}}\int_{0}^{\infty}{\frac{d s}{s}}\Im \widetilde Z \oint_{C_{0}}{\frac{d w}{2 \pi i}}{\frac{1}{w^{N + 1}( w - \beta )}}F_{0}( w , s ) ,
Q_k^{n-1}(u,v,t) = (1-u^2)^{k/2}(1-v^2)^{k/2}P_k^{n-1}(\frac{t-uv}{\sqrt{(1-u^2)(1-v^2)}}), \lambda_{i,j}= \frac{n+2k}{n}(h^{n+2k}_i h^{n+2k}_j)^{1/2}
C_n^{(4)}=c^{\rm (in)}_{n,4}+ 2 \cdot c^{\rm (out)}_{n,2}+\sum_{k=1}^n \dfrac{n}{k}\cdot \mathfrak{d}^{\rm (in)}_{n-k,k}+\sum_{k=1}^{n-3}\sum_{l=1}^{n-k-2}\dfrac{1}{2}\cdot \dfrac{n}{k \cdot l}\cdot \tilde{s}^{(2)}_{n-k-l,k,l},
\varrho_{\lambda,{\cal O}}(u) = \int_{\cal O}\left( \big| \nabla u \big|^{p(x)}+ \big( \lambda V(x) + Z(x) \big) | u |^{p(x)}\right) \ge M \int_{\cal O}|u|^{p(x)}= M \varrho_{p(x), \cal{O}}(u),
\sup_{\delta\leq y\leq -2\delta +L}\|u_{\delta,\epsilon}|u_{\delta,\epsilon}|^2\|_{L^2(0,T)}\leq \sup_{\delta\leq y\leq -2\delta +L}\|u\|^2_{L^{\infty}(0,T)}\|u_{\delta,\epsilon}\|_{L^{2}(0,T)}&\leq \sup_{\delta\leq y\leq -2\delta +L}\|u_{\delta,\epsilon}\|_{H^1(0,T)}^2\|u_{\delta,\epsilon}\|_{L^{2}(0,T)}&\leq 1+\delta^{-2}
\lim_{s\rightarrow0}\Gamma(s)\left( \frac{1}{z^{s}}G(-1/z,s;B_{-b},A_{-c},0,0)-G(z,s;A_{-c},B_{b})\right) & =2\pi iP_{1}\left( 0,B_{-b}\right) P_{1}\left( 0,A_{-c}\right)-\frac{2\pi i}{z}B_{0}(A)P_{2}\left( 0,B_{b}\right) -2\pi izB_{0}\left(B\right) P_{2}\left( 0,A_{c}\right) .
n^\frac{1}{4}= z_1^\frac{1}{2}- \left(z_1^\frac{1}{2}- n^\frac{1}{4}\right) > b(z_3 - 1) - \frac{z_3 + 1}{2}- 1> b(z_3 - 1) - (z_3 - 1) + 3 = \ell(z_3 - 1),
\|a - T_{\mu}^{n}x\|^{2}= \langle a - T_{\mu}^{n}x , x_{2}^{*}\rangle= \mu_{t}\langle a - T_{t}^{n}x , x_{2}^{*}\rangle \leq \sup_{t}\|a - T_{t}^{n}x\|\|a - T_{\mu}^{n}x\| \leq \|a-x\|\|a - T_{\mu}^{n}x\|,
D\theta^{a}=\tau^{a}, D\tau^{a}=\beta_{b}^{a}\theta^{b}, \mathbf{F}_{b}^{a}=\digamma_{b}^{a}-\beta_{b}^{a}-D\beta_{b}^{a}\mathbf{m,}\digamma_{b}^{a}=d\alpha_{b}^{a}+\alpha_{c}^{a}\alpha_{b}^{c}.
\frac{d}{ds}\langle \gamma_k'(s),\nabla d\rangle = A_k(\gamma'_k(s),\gamma'_k(s))\langle\nabla d,\mathbf n_k\rangle+\nabla^2d(\gamma'_k(s),\gamma'_k(s))\rangle = A_k(\gamma'_k(s),\gamma'_k(s))\langle\nabla d,\mathbf n_k\rangle+\nabla^2d((\gamma'_k(s))^T,(\gamma'_k(s))^T),
\nabla_{x_1}x_2 = (\lambda / 2) x_n , \nabla_{x_2}x_1 = - x_2 - (\lambda / 2) x_n , \nabla_{x_1}x_n = - (\lambda / 2) x_2 , \nabla_{x_n}x_1 = - (\lambda / 2) x_2 , \nabla_{x_2}x_2 = x_1 , \nabla_{x_2}x_n = (\lambda / 2) x_1 , \nabla_{x_n}x_2 = (\lambda / 2) x_1 .
\delta \mathcal{P}_{a}^{( 1 )}= - \pi_{a}^{0}, \; \delta \mathcal{P}_{a}= \partial_{i}B_{a}^{0 i}, \; \delta P_{a}^{( 1 ) i}= p_{a}^{i}, \; \delta P_{a}^{i}= - \partial^{i}\varphi_{a},
\frac{1}{2}\int \int \mathrm{d}w \mathrm{d}x \left( \left( \partial_{\mu}\tilde{y}\right) \left( \partial_{\mu}\tilde{y}\right) + \left( \partial_{\mu}\tilde{z}\right) \left( \partial_{\mu}\tilde{z}\right) \right) = \frac{1}{2}\int \mathrm{d}t \left( \tilde{y}\left( n . \partial \tilde{y}\right) + \tilde{z}\left( n . \partial \tilde{z}\right) \right)
J_{i j}\star \varphi = 0 , \quad \varphi \varphi^{\dagger}\varphi = \left| \lambda \right| \varphi , \quad 0 = \left( 1 + V^{\prime}\left( - \left| \lambda \right|^{2}\right) \right)^{2}- 1 - \left| \lambda \right|^{2}.
r_a =\begin{dcases}a-\frac{1}{\mathit{e}}\ \frac{1}{\mathit{e}}< a \leqslant \frac{\mathit{e}+\mathit{e}^{-1}}{2}\mathit{e}-a \ \frac{\mathit{e}+\mathit{e}^{-1}}{2}\leqslant a <\mathit{e}.\end{dcases}
\sup\limits_{y\in(0,\infty)}\int\limits_{(0,\infty)}\left( \frac{z}{y}\right)^{\delta}z^{- \frac{1+2s}{2}}|\tilde{k}(z,y)|d z = \sup\limits_{y}y^{-\delta}y^{\frac{1-2s}{2}}\int\limits_{y}^{\infty}z^{-1 + \frac{2s-1}{2}+\delta}dz = const,
0=\lim_{r\to \infty}\frac{\sqrt{\sum_{i=p+1}^{m}\left(e^r_i\tilde{y}^r_{ii}-\mu_r\right)^2}}{\mu_r}=\lim_{r\to \infty}\sqrt{\sum_{i=p+1}^{m}\left(\frac{e^r_i}{\mu_r}\tilde{y}^r_{ii}-1\right)^2},
c_1(\rho) n^{2\alpha -2}\sum_{x,y}s(y-x)^2 \big(f\big(\tfrac{y}{n}\big) - f\big(\tfrac{x}{n}\big) \big)^4 + c_2(\rho) n^{2\alpha-2}\sum_{x}\Big(\sum_{y}s(y-x)\big(f\big(\tfrac{y}{n}\big)-f\big(\tfrac{x}{n}\big)\big)^2\Big)^2.
c_0 := 1 + \sum_{i=0}^{k-1}e(a_i)^2 \in I\quad\mbox{and}c_j := \sum_{i=0}^{j-1}\alpha e(a_i) e(a_{k-j+i}) + \sum_{i=j}^{k-1}e(a_i)e(a_{i-j}) \in I\quad\mbox{for all}j\in\{1,\ldots,\lfloor k/2\rfloor\}
\textbf{W}^{c}_{z}=\left( \begin{array}{llll}w^{(c)}_{11}w^{(c)}_{12}\cdots w^{(c)}_{1,M_z-1}w^{(c)}_{21}w^{(c)}_{22}\cdots w^{(c)}_{2,M_z-1}\vdots \vdots \ddots \vdots w^{(c)}_{M_z-1,1}w^{(c)}_{M_z-1,2}\cdots w^{(c)}_{M_z-1,M_z-1}\end{array}\right), \c=1,2,
[\Phi \wedge \Phi^{*_{H}}] = \begin{pmatrix}\alpha^2 - |z|^2\beta^2 + 2i \Im b^2 z 2 \alpha b -2 \beta \bar b \bar z -2 \alpha b z + 2 |z|^2 \beta \bar b |z|^2\beta^2 - \alpha^2 - 2i \Im b^2 z\end{pmatrix}dz \wedge d\bar z.
\delta_{j_1j_2}G_{i_1i_3\alpha}^{j_1j_2\eta}\delta^{i_1\alpha}\tilde\gamma_{\eta k}+2\delta^{i_1i_2}G_{i_1i_2i_3}^{j_1j_2j_3}\gamma_{j_2j_3}\delta_{j_1k}= -\int_0^\infty\frac{d}{dy}\left[\sum_{\alpha,\eta}(e^{-y\tilde\gamma})_{\alpha k}(e^{-y\tilde \gamma})_{\alpha\eta}(e^{-y\tilde\gamma})_{i_3\eta}\right]dy=\delta_{i_3k},
\begin{bmatrix}a[m],\begin{bmatrix}b[n],c[k]\end{bmatrix}\end{bmatrix}+\begin{bmatrix}c[k],\begin{bmatrix}a[m],b[n]\end{bmatrix}\end{bmatrix}+\begin{bmatrix}b[n],\begin{bmatrix}c[k],a[m]\end{bmatrix}\end{bmatrix}=(\sum_{i=1}^3\Theta_i) [k+m+n]=0,
\begin{array}[c]{rl}\mathcal{H}(t,x,p,q,P,v,u) = H(t,x,p,q,v)+\dfrac{1}{2}\sum\limits_{i=1}^{d}\left( \sigma^{i}(t,x,v)-\sigma^{i}(t,x,u)\right)^{\intercal}P\left( \sigma^{i}(t,x,v)-\sigma^{i}(t,x,u)\right) & -\dfrac{1}{2}\sum\limits_{i=1}^{d}\left( \sigma^{i}(t,x,u)\right)^{\intercal}P\left( \sigma^{i}(t,x,u)\right) .\end{array}
\Delta (C_i,C_i) = \frac{q-1}{2}\{0\}+(i,i)C_0+(i+1,i+1)C_1 & =\frac{q-1}{2}\{0\}+\frac{q-3}{4}(C_0+C_1) & =\frac{q-1}{2}\{0\}+\frac{q-3}{4}(\mathbb{F}_q-\{0\})) \ (i=1,2)
D_{\mu \nu}( k ) = \frac{- i}{k^{2}- \theta_{i n}^{2}}\left( P_{\mu \nu}- i \theta_{i n}\epsilon_{\mu \nu \rho}\frac{k^{\rho}}{k^{2}}\right) - i \lambda \frac{k_{\mu}k_{\nu}}{k^{4}}f ( k^{2}) ,
V_1(\tau ) = e^{( \tau - \tau_1) \Delta}(V_1(\tau_1)) + \int_{\tau_1}^{\tau}e^{(\tau- \tau') \Delta}\left( - 2 ( U \nabla \psi_1 ) + U \Delta \psi_1 + \psi_1\left( U^p - \hat U^p \right) \right)(\tau') d \tau'.
\omega_{n,1}\left({\nu ,p}\right) =2\int_{1}^{p}{\left\vert{\frac{\tilde{{F}}_{n}\left( q\right) dq}{q^{2}\left({1-q^{2}}\right)}}\right\vert}+\sum\limits_{s=1}^{n-1}{\frac{1}{\nu^{s}}\int_{1}^{p}{\left\vert{\frac{\tilde{{G}}_{n,s}\left( q\right) dq}{q^{2}\left({1-q^{2}}\right)}}\right\vert ,}}
\begin{aligned}\int_{B_t(0)}\frac{8}{(1+ |z|^2)^2}\Big( \frac{|z|^2}{(1+ |z|^2)^2}- \frac{(1- |z|^2 )^2}{2(1+ |z|^2)^2}\Big)\mathrm{d}z=\frac{4\pi t^2(t^2-1)}{(t^2+1)^3}.\end{aligned}
\Big\|\left(\prod_{l=1}^k \frac{\lambda_l^{1/2}(1+D^2)}{(1+\lambda_l+D^2)^2}\right)\Lambda^{m+k}(a) \left(\prod_{l=1}^k \frac{1}{1+\lambda_l+D^2}\right)(D+i\lambda)^{-p-1}\Big\|_1 \leq \|\Lambda^{m+k}(a)(D+i\lambda)^{-p-1}\|_1\prod_{l=1}^k \frac{1}{\lambda_l^{1/2}(1+\lambda_l)}.
H_n=\sum_{m=1,p^k\nmid m}^n \frac 1m +\frac 1{p^k}H_{n_1}=\frac b{p^{k-1}a}+\frac{u_{n_1}}{p^kv_{n_1}}=\frac{pbv_{n_1}+au_{n_1}}{p^kav_{n_1}},
\Delta^{(n+m+1)}(F_n^{\heartsuit}O)\subset \sum_{\substack{(i_1,\ldots,i_{n+m+1})\in\mathbb Z_{\geq 0}^{n+m+1}| i_1+\ldots+i_{n+m+1}=n}}F_{i_1}^{\heartsuit}O\otimes\cdots\otimes F_{i_{n+m+1}}^{\heartsuit}O.
\Phi^1(x): = \int_{A \backslash A_{H}^{\sigma}\times A_{H}^{\sigma}}\Phi(z_1^{-1}z_2x) \frac{dz_1dz_2}{du}f^1(x): = \int_{\widetilde{A}\backslash A_{G}^{\sigma}\times A_{G}^{\theta}}f(\widetilde{z}^{-1}\widetilde{z}_{\theta}x) \frac{d\widetilde{z}d\widetilde{z}_{\theta}}{dt}.
: \! \prod_{i = 1}^{\ell}V_{2 , 1}( q^{- \frac{k}{2}}t^{\frac{\ell + 1}{2}- i}z ) \prod_{j = 1}^{k}V_{1 , 2}( t^{- \frac{\ell}{2}}q^{\frac{k + 1}{2}- j}z ) \! : ,
0 =(C^n)_{-m-j}&=\sum_{i_1+\dots +i_n=-m-j}C_{i_1}C_{i_2}\dots C_{i_n}&= nC_{-m-n-j+1}+ \sum_{{i_1+\dots +i_n=-m-j}\atop{i_k\ne -m-n-j+1\ \forall k}}C_{i_1}C_{i_2}\dots C_{i_n},
\left[{\mathcal J}_{\ell}^{0}\left( t , \underline{{x}}\right) ,{\mathcal J}_{\ell}^{0}\left( t , \underline{{y}}\right) \right] = \frac{i \sigma_{H}}{2 \pi}\delta^{\prime}\left( \underline{{x}}- \underline{{y}}\right) ,
P =\sum_{d\geq 0}B_d t^d=\sum_{d\geq 0}{FB}_d t^d+\sum_{d\geq 0}{TB}_{d-1}t^d+\sum_{d\geq 0}{TB}_d t^d &=FP+\overline{TP}+\underline{TP}=FP+t\,\underline{TP}+\underline{TP}&=FP+(1+t)\,\underline{TP}
\begin{aligned}u^\varepsilon (y_3) \leq \int_0^{s_3 + s_4}L \big( \xi^\varepsilon (s, y_1, -\gamma ), -\gamma (\xi^\varepsilon (s, x, -\gamma )) \big) e^{-\lambda s}\, ds + u^\varepsilon (y_4) e^{-\lambda (s_3 + s_4)}\leq C(1 + \lambda ) \cfrac{4 \sqrt{h}}{c_0 \mu}+ u^\varepsilon (y_4).\end{aligned}
\widetilde{\theta}_2=\widetilde{\theta}_2(N, \mathcal{T}, \{x_j\}_{j=1}^{N}):=\theta+\frac{1-\theta}{l_1+l_2\left(\frac{1}{N}\sum\limits_{j=1}^N\int_0^{\tau_j}e^{-\beta t}f(x_j(t))dt\right)}.
\lim_{s\rightarrow \infty}\left( V(s)^\gamma -V(r_k / 16)^\gamma \right) V(s) \mathcal{P}_1(s) \leq c_8 \lim_{s\rightarrow\infty}\frac{V(s)^\gamma - V(r_k / 16)^\gamma}{V(s)}= 0,
f_0(\tau) =\sum_{j=1}^4\bigl(\alpha_j\log(\tau-\alpha_j)-\beta_j\log(\tau-\beta_j)\bigr) -\sum_{j=1}^3(\alpha_j-\beta_j)\log(\alpha_j-\beta_j)+(\beta_4-\alpha_4)\log(\beta_4-\alpha_4).
S(K)=\inf_{f\in H_0^p(Kdm_\infty)}\int_{\mathbb{T}^\infty}|1-f|^pK dm_{\infty}=\|\pi(1)\|_{\mathcal{Q}}^p=\sup_{g\in \mathbf{B}\cap \mathcal{Q}^{*}}\left|\int_{\mathbb{T}^\infty}gK dm_\infty\right|^p,
\phi_{-}(p^cy)(z)\phi(p^dy)(z) = \frac{(-1)^{c}c!B_0}{z^{c+1}}\left( \frac{(-1)^{d}d! B_0}{z^{d+1}}+ \sum_{n\geq 0}\frac{B_{n+d+1}}{n+d+1}\frac{1}{n!}z^n \right) = +(-1)^c\frac{B_0B_{c+d+2}}{(c+1)(c+d+2)}+ O(z)
\bigl(~_r\mathcal{F}_s(\alpha)\bigr)^{q^d}=\sum_{n=N}^{\infty}\biggl(\prod_{m=1}^{n+d-1}(\theta^{q^m}-t)^{c(m-n)q^{n+d-m}}\biggr)\alpha^{q^{n+d}}+\sum_{n=N-d+1}^{N-1}\biggl(\prod_{m=1}^{n+d-1}(\theta^{q^m}-t)^{c(m-n)q^{n+d-m}}\biggr)\alpha^{q^{n+d}}&+\sum_{n=0}^{N-d}\biggl(\prod_{m=1}^{n+d-1}(\theta^{q^m}-t)^{c(m-n)q^{n+d-m}}\biggr)\alpha^{q^{n+d}}.
\begin{aligned}\|d(\nabla v,SO(n))\|_{L^{p(\cdot)}(\Omega)}\le \|d(\nabla u,SO(n))\|_{L^{p(\cdot)}(\Omega)}+\|\nabla u-\nabla v\|_{L^{p(\cdot)}(\Omega)}&\le C\|d(\nabla u,SO(n))\|_{L^{p(\cdot)}(\Omega)}\end{aligned}
\Gamma ( T , S , \Phi ) \stackrel{\mathcal{G C P}[ G^{a}\mathcal{M}^{c}]}{\longmapsto}\Gamma^{\mathrm{G C P}}\equiv \Gamma ( T , S , G^{a *}( \mathcal{M}^{c *}( \Phi^{\mathrm{C P}}) ) ) .
J \epsilon_{a b 3}\epsilon_{i j}( - )^{\gamma - 1}\left[ S_{\gamma \alpha}^{( 2 )}( P ) \right]_{\rho \mu}\left( \gamma_{5}C \right)_{\rho \sigma}\left[ S_{\gamma \beta}^{( 2 ) \sigma \nu}( - P ) \right]_{\sigma \nu}.
\begin{array}{l}c_{2a}(u_{+})c_{2a}(u_{-}){\bf D}_{a+1}{\bf D}_{a+1}^{'}+c_{2a}(u_{-}){\bf X}^{a+1}_{a+2,a}{\bf X}^{'a+1}_{a,a+2}+c_{3a}(u^{'},u){\bf D}_{a+1}{\bf U}_{a-1}^{'}\mbox{}=c_{2a}(u_{+})c_{2a}(u_{-}){\bf D}_{a+1}^{'}{\bf D}_{a+1}+c_{2a}(u_{-}){\bf X}^{'a+1}_{a,a+2}{\bf X}^{a+1}_{a+2,a}+c_{3a}(u^{'},u){\bf U}_{a-1}^{'}{\bf D}_{a+1}\end{array}
v_{0}= u - w_{1}a^{- \gamma}- ( \mathrm{\frac{1}{1 2}}u^{\prime \prime}+ u^{2}) a^{- 2 \gamma}+ 2 ( \mathrm{\frac{1}{6}}w_{1}^{\prime \prime}+ u w_{1}) a^{- 3 \gamma}+ \mathrm{O}( a^{- 4 \gamma}) .
F_{n,h}(x,y)=\begin{cases}\frac{\frac{1}{nh}\sum_{i=0}^{n-1}\mathbb I(x\leq X_i \leq x+h, X_{i+1}\leq y)}{\frac{1}{nh}\sum_{i=0}^n \mathbb I(x\leq X_i\leq x+h)}\mbox{if}\ \ \mathbb I(x\leq X_i\leq x+h)\neq 0 \mbox{for some}0\leq i \leq n-1 0 \mbox{otherwise,}\end{cases}
K(x,y) = \frac{1}{\mu(B(y,\gamma d(x,y)))}\int_{B(y,\gamma d(x,y))}K(x,y)d\mu(z) &\leq \frac{1}{\mu(B(y,\gamma d(x,y)))}\int_{B(y,\gamma d(x,y))}\sup_{y\in B(y,\gamma d(x,y))}K(x,\eta)d\mu(z) &\leq \frac{H}{\mu(B(y,\gamma d(x,y)))}\int_{B(y,\gamma d(x,y))}\inf_{y\in B(y,\gamma d(x,y))}K(x,\eta)d\mu(z) &\leq H \widetilde{K}(x,y).
g\left(\omega\right) = \frac{1}{2}\sum_{m=1}^{M_\mathrm{h}}r_{mmk_l}+ \sum_{p=1}^{M_\mathrm{h}-1}\sum_{m=1}^{M_\mathrm{h}-p}r_{m\left(m+p\right)k_l}e^{-jp\omega},
\omega = w_{1}x^{1}d x^{2}\ldots d x^{N + 2}- w_{2}x^{2}d x^{1}\ldots d x^{N + 2}+ \ldots + ( - )^{N + 1}w_{N + 2}x^{N + 2}d x^{1}\ldots d x^{N + 1}~ .