harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_t a - \frac{1}{2}\nabla_x \cdot(\vert v \vert^2 v\mu^{1/2}, g_2)=0, &\partial_t b + \nabla_x (a+5c) + \nabla_x \cdot (v\otimes v \mu^{1/2}, g_2) =0, &\partial_t c + \frac{1}{3}\nabla_x \cdot b +\frac{1}{6}\nabla_x \cdot (\vert v \vert^2 v \mu^{1/2}, g_2) =0.
\frac{\partial \Phi_{n}^{*}( z^{- 1})}{\partial \bar{t}_{1}}= - \frac{S_{n}^{*}}{S_{n - 1}^{*}}\frac{h_{n}}{h_{n - 1}}( \Phi_{n}^{*}( z^{- 1}) - z^{- 1}\Phi_{n - 1}^{*}) .
||(g^{+}_3,g^{-}_3) ||_{H^{s+1}(\Gamma_T)}\leq C||(\mathcal{G}^{+},\mathcal{G}^{-}) ||_{H^{s+1}(\Gamma_T)}\leq C\Big(||\mathbf{f}||_{s+2,\ast,T}+||(g^{+}_1, g^{-}_1) ||_{H^{s+2}(\Gamma_T)}\Big).
v_{2l-1}:=\left(1,\cos \frac{2 \pi l}{n},\cos \frac{4 \pi l}{n},\ldots,\cos \frac{2 \pi l (n-1)}{n}\right), v_{2l}:=\left(0,\sin \frac{2\pi l}{n},\sin \frac{4\pi l}{n},\ldots,\sin \frac{2 \pi l(n-1)}{n}\right).
g_{1}(\mathcal{A}_{X}\omega W,\mathcal{B}Y)+\eta(Y)g_{1}(X,\omega W) =g_{2}((\nabla \pi_*)(X,Y),\pi_*\omega\phi W) &-g_{2}((\nabla \pi_*)(X,\mathcal{C}Y),\pi_*\omega W),
\hat v_0 (p + a) - \hat v_0 (p-a)= \frac 1{\pi}\int e^{-i(p \cdot x)}\left[e^{-i(a \cdot x)}- e^{i(a \cdot x)}\right] v_0 (x) d^2 x = [\min (1,|a|)] \hat f_a ( p) ,
\limsup_{n\rightarrow\infty}\frac{1}{|F_n|}\sum_{g\in F_n}\rho(gx,gy) \ge\limsup_{n\rightarrow\infty}\frac{1}{|\widehat{F_n}|}\sum_{g\in \widehat{F_n}}\rho(gx,gy)=\int \rho(x,y) d \lambda \ge \tau\lambda(W),
T(v_{i_{1}}\otimes v_{k_{1}}\otimes\cdot\cdot\cdot\otimes v_{k_{N}})=\sum_{i_{1}^{\prime},l_{1},...l_{N}}T_{i_{1}k_{1}...k_{N}}^{i_{1}^{\prime}l_{1}...l_{N}}v_{i_{1}^{\prime}}\otimes v_{l_{1}}\otimes\cdot\cdot\cdot\otimes v_{l_{N}}
\lim_{n \to \infty}n^{d(r,s)-3}E(S_{\lfloor nt \rfloor}(r)\overline{S_{\lfloor nt \rfloor}(s)})=\frac{c(r,s)+\overline{c(s,r)}}{(3-d(r,s))(2-d(r,s))}\sigma(r,s)t^{3-d(r,s)}.
\int_{\Sigma_r}\tau \Delta \ln (-\sigma^{ab}l_{ab}) d\Sigma_r = \int_{S^2}(a^i \tilde{X}^i) r \Delta [\frac{2}{r}-\frac{r^3}{45}|\alpha|^2 ] r^2 dS^2 +O(r^6) = -r^5 \int_{S^2}(a^i \tilde{X}^i) \tilde \Delta \frac{1}{90}|\alpha|^2 dS^2+O(r^6) = \frac{1}{45}r^5 \int_{S^2}(a^i \tilde{X}^i) |\alpha|^2 dS^2+O(r^6).
S_{e}D_{s}C^{T}\overline{v}= S_{e}D_{s}\left( \sum_{n=0}^{\infty}c_{nm}\overline{v_{n}}\right)_{m}&= \sum_{m = 0}^{\infty}s^{m}\left( \sum_{n=0}^{\infty}c_{nm}\overline{v_{n}}\right) e_{m}.
\left \{\widetilde{\phi}_n (b_1 \ldots b_d \M_{k,l}) \ \vert \b_i \in \mathcal{B}_{{\rm TL}_{\mu_i}(q)}\mbox{for all}i = 1, \ldots d, 1 \leq k, l \leq m_{\mu}, \mu \in{\rm Comp}_d(n) \right \}.
\Lambda_1=\left(\frac{p-q}{2(\alpha+\beta-q)}\right)^{\frac{p}{\alpha+\beta-p}}\left(\frac{\alpha+\beta-q}{\alpha+\beta-p}|\Omega|^{\frac{\alpha+\beta-q}{\alpha+\beta}}\right)^{-\frac{p}{p-q}}S^{\frac{\alpha+\beta}{\alpha+\beta-p}+\frac{q}{p-q}},
\begin{cases}-\frac{d^{2}\psi_{k}}{d\theta^{2}}+a(\theta )\psi_{k}=\mu_{k}\psi_{k}\psi_{k}(0)=\psi_{k}(2\pi ) \psi_{k}^{\prime}(0)=\psi_{k}^{\prime}(2\pi ),\end{cases}
v_M(\pi_L^{i_j}\cdot p^h\pi_M^{p^{j-h}\tilde{\phi}_{M/L}^k(c)}) \ge mi_{j-h}+p^{j-h}\tilde{\phi}_{M/L}^k(c) &=\tilde{\phi}_{L/K}^{j-h,m}(\tilde{\phi}_{M/L}^k(c)) &\ge\lambda_{M/K}^{j-h+k}(c).
Q^{n.4}_\tau=n^{2\alpha+1}\int^{\tau}_0 \gamma_s(s-\eta_n(s))^{2\alpha}(\sigma' \sigma)^2(X_{\eta_n(s)}) \, \left[\left(W_s-W_{\eta_n(s)}\right)^2 -(s-\eta_n(s))\right]\,ds.
\lambda(t)\left( \int_0^{t}a e^{b \tau}P(\tau) d \tau - T_{\delta}e^{bt}\right) = 0, \forall t \beta(t)\left( \int_{0}^{t}P(\tau) d\tau - \sum_{i=0}^{h(t)}E_i \right) = 0, \forall t
\|u\|_{L^\infty(\Omega)}\,\le\, C \|\omega_\theta\|_{L^p(\Omega)}^\sigma \,\|\omega_\theta\|_{L^q(\Omega)}^{1-\sigma}~, \hbox{where}\sigma \,=\, \frac{p}{2}\,\frac{q-2}{q-p}\,\in\, (0,1)~.
-\frac{\partial LD_{\alpha}I(p\|q)}{\partial q_{j}}=-\frac{1}{\alpha}\left[-\frac{1}{X}\left(\sum_{i}p^{\frac{1}{\alpha}}_{i}q^{1-\alpha}_{i}\right)^{\frac{1}{\alpha}-1}p^{\alpha}_{j}q^{-\alpha}_{j}+\frac{1}{Y}\left(\sum_{i}q_{i}\right)^{-\frac{1}{\alpha}}\right]
\mathrm{d i a g}\Big ( \frac{e^{2}N}{\pi \! + \! g N}+ \frac{\pi}{\pi \! + \! g N}4 m c \tilde{\lambda}\; , \; 4 m c \tilde{\lambda}\; , \; . \; . \; . \; . \; . \; , \; 4 m c \tilde{\lambda}\Big ) \; .
\tau^1=\left(\begin{array}{c c}0 1 1 0 \end{array}\right)\ \ ,\ \ \tau^2=\left(\begin{array}{c c}0 -i i 0 \end{array}\right)\ \ ,\ \ \tau^3=\left(\begin{array}{c c}1 0 0 -1 \end{array}\right)\ \ ,
\widetilde{m}(x,y )=\prod_{i=1}^{d}\widetilde{m}_i (x,y_i ),\ \widetilde{w}(x,y )=\sum_{i=1}^{d}\widetilde{w}_i (x,y_i ),\ \widetilde{H}(x,\Lambda )=\sum_{i=1}^{d}\widetilde{H}_i (x,\Lambda_{i}),
\left( \underset{i=1}{\overset{b}{\sum}}Z_i^2 \leq \frac{25 kN_\alpha}{C_\alpha}b \right) \leq \exp \left( - b \frac{\frac{25k N_\alpha}{C_\alpha}- 1 - \log \left( \frac{25k N_\alpha}{C_\alpha}\right)}{2}\right) \leq \alpha/8,
\frac{\left(q^{2},qz,q/z;q^{2}\right)_{\infty}}{(q;q)_{\infty}}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}(-1)^{n}}{(q;q)_{n}}\frac{A_{q}\left(q^{-n}z\right)}{z^{n}}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}(-1)^{n}}{(q;q)_{n}}z^{n}A_{q}\left(\frac{1}{q^{n}z}\right).
r_{1}:= -b_{1}^{3}+ 4b_{1}d_{2}- 3b_{3},r_{2}:= -b_{1}^{4}+ 5b_{1}^{2}d_{2}- 2b_{1}b_{3}- b_{2}^{2}+ 3b_{2}d_{2}- 6d_{2}^{2},
\begin{array}{l}\tau_1 \equiv \pi_0 + \eta^1 \approx 0 \tau_2\equiv E' + \phi' + \pi + A_1 +\eta^2 \approx 0 \tau_3 \equiv E + \eta^3 \approx 0 \tau_4 \equiv -\pi - \phi' - 2A_1 + A_0 + \eta^4 \approx 0 \end{array}
\begin{array}{l l}{{1 )}}&{{h = 3 / 8 , \quad a = 3 / 8 0 ,}}\\{{2 )}}&{{h = 7 / 8 , \quad a = 7 / 1 6 ,}}\\{{3 )}}&{{h = 3 / 8 + x , \quad a = 3 / 8 0 .}}\\ \end{array}
A_1(\iota_{y*}1)(\bar{w}) =\delta_{\bar{y},\bar{w}}\frac{e(T_yT^*\mathcal{B})}{\prod\limits_{\alpha\in R^+_P}(-y\alpha)}&=\delta_{\bar{y},\bar{w}}e(T_{\bar{y}}T^*\mathcal{P})\prod\limits_{\alpha\in R^+_P}(y\alpha-\hbar).
\hat{a}_{j_{0},...,j_{D-1}}=\sum_{i_{0}=0}^{n_{0}-1}\cdot\cdot\cdot\sum_{i_{D-1}}^{n_{D-1}-1}a_{i_{0},...,i_{D-1}}\prod_{d=0}^{D-1}\omega_{n_{d}}^{i_{d}j_{d}}
C_K^{(2)}= (4\alpha_1 M_1^2{c}_*^{-2}+\widetilde{M}_1^2 \check{c}_*^{-2})^{1/2}, &C_K^{(3)}=(\alpha_1 M_2^2\check{c}_*^{-2}+4\lambda_0^{-2}\Vert f\Vert^4_{L_\infty}\widetilde{M}_2^2)^{1/2}.
I_{n,i,22}^{21}(\theta) = - \frac{\ln \left(X_{\frac{i-1}{n}}\right) X_{\frac{i-1}{n}}}{\delta \alpha^2 X_{\frac{i-1}{n}}^{1/ \alpha}}k'_\alpha (z^n_i(\theta)) - \frac{X_{\frac{i-1}{n}}}{\delta X_{\frac{i-1}{n}}^{1/ \alpha}}f'_\alpha (z^n_i(\theta))
{\left|{\hat h_{\min}^{{\rm{rough}}}}\right|^{\rm{2}}}={\sum\limits_{{{\bf{r}}_n}\in{I_1}}^{}{\min \left({{{\left|{r_{n,1}^{}}\right|}^{\rm{2}}},{{\left|{r_{n,2}^{}}\right|}^{\rm{2}}}}\right)}}/{\left|{{I_1}}\right|}-{N_0}
C^{(0)}_{\mu\mu_1\ldots\mu_n}(x,y_1,\ldots,y_n) =\int(2\pi)^4\d(k+\sum p_i)C^{(0)}_{\mu\mu_1\ldots\mu_n}(p_1,\ldots,p_n) e^{ik\cdot x}e^{i\sum p_i\cdot y_i}\,,
\begin{array}{l l}{{\alpha_{1}={\frac{1}{2}}( e_{1}- e_{2}- e_{3}- e_{4}- e_{5}- e_{6}+ \sqrt 2 e_{7}) ,}}&{{\alpha_{2}= e_{1}+ e_{2},}}\\{{\alpha_{3}= - e_{1}+ e_{2},}}&{{\alpha_{4}= - e_{2}+ e_{3},}}\\{{\alpha_{5}= - e_{3}+ e_{4},}}&{{\alpha_{6}= - e_{4}+ e_{5},}}\\{{\alpha_{7}= - e_{5}+ e_{6},}}&{{\alpha_{0}= - \sqrt{2}e_{7}.}}\\ \end{array}
r(x,y,t)=-2\frac{\partial p}{\partial \nu_x}(x,y,t) -2\sum_{j\geq 1}\int_0^t\int_\Sigma r^j(x, z,s)\frac{\partial p}{\partial \nu_x}(z,y,t-s)dA(z)ds (x,t)\in \Sigma \times (0,+\infty ), \; y\in \overline{\Omega}.
\frac{1}{2\pi\mathrm{i}n}\oint_{|z|=1}\frac{f_{n}'(z)}{f_{n}(z)}\mathrm{d}z \geq\frac{1}{2\pi n\epsilon}\int_{0}^{2\pi}\bigg(\log|f_{n}(\mathrm{e}^{\mathrm{i}x})|-\log|f_{n}(\mathrm{e}^{\mathrm{i}(x-\mathrm{i}\epsilon)})|\bigg)\mathrm{d}x.
S=R\zeta-\frac{1}{2}g\zeta^{2}+\frac{1}{2}\zeta u_{b}^{2}-\frac{1}{6}\zeta^{3}u_{b}u_{b}^{\prime\prime}-\frac{1}{6}\zeta^{3}(u_{b}^{\prime})^{2}+\cdot\cdot\cdot.
(\mathcal{J}^{\ast}V)(x) =\int_{\lbrace z>1\rbrace}|V(y+z)-V(y)|\nu(\mathrm{d}z) \leqslant\int_{\lbrace z>1\rbrace}z^{\kappa}\nu(\mathrm{d}z)+c_{2}\nu(\lbrace z>1\rbrace)<\infty.
\left( \begin{matrix}{{m^{i}}}\\{{n^{i}}}\\{{p^{i}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{m}}&{{j}}&{{k}}&{{m^{4}}}&{{m^{5}}}&{{\cdots}}\\{{0}}&{{n}}&{{l}}&{{n^{4}}}&{{n^{5}}}&{{\cdots}}\\{{0}}&{{0}}&{{p}}&{{p^{4}}}&{{p^{5}}}&{{\cdots}}\\ \end{matrix}\right)
\|\omega\|_X^2 =\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx-\frac{\lambda}{2_\alpha^*-2}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx &\leq\lambda C(\|\omega\|_X^2+\|\omega\|_X^4),
\Delta_{P_{\alpha}}\left( \mathbf{O}\right) =b_{a}\left( \mathbf{O};P_{\alpha}\right) -E_{P_{\alpha}}\left[ b_{a}\left( \mathbf{O};P_{\alpha}\right) |O_{1}\right] -E_{P_{\alpha}}\left[ b_{a}\left( \mathbf{O};P_{\alpha}\right) |O_{2}\right] +E_{P_{\alpha}}\left[ b_{a}\left( \mathbf{O};P_{\alpha}\right) \right] =\alpha O_{1}O_{2}
\sum_{n=0}^{m-1}F\left(n,\frac{m-1}{2}\right)=F\left(m-1,\frac{m-1}{2}\right) =[2m-1]\frac{(q;q^2)_{m-1}(q;q^2)_{(m-1)/2}}{(q;q)_{m-1}(q^2;q^2)_{(m-1)/2}}begin{align*}5pt] ={2m-1\brack m-1}{m-1\brack \frac{m-1}{2}}_{q^2}\frac{[m]}{(-q;q)_{m-1}^2}.
l_2=\frac{1}{q_2}\sum_{j=3}^{n}l_j q_j \Big(\frac{\psi'(\beta_2 q_1)\psi'(\beta_1 q_j)+\psi'(\beta_1 q_1)\psi'(\beta_2 q_j)}{\psi'(\beta_1 q_1)\psi'(\beta_2 q_2)-\psi'(\beta_2 q_1)\psi'(\beta_1 q_2)}\Big)
D'(1/x) = g^2(x)C'(x) \pmod{x^{n-1}-1}&=\left(\sum_{r=0}^{n-2}\beta^{\pi^{-r}}x^r\right)^2\cdot \left(\sum_{w=0}^{m-1}C'_{2u+hw}x^{2u+hw}\right) \pmod{x^{n-1}-1}&=x^{2u}\cdot\left(\sum_{r=0}^{n-2}\beta^{2\pi^{-r}}x^{2r}\right)\cdot\left(\sum_{w=0}^{m-1}C_l^{2^w}x^{hw}\right) \pmod{x^{n-1}-1}
\omega_1 =-\frac{\sqrt{3}}{2}a_1a_2 + b_{12}c_{12}+ b_{13}c_{13}- b_{23}c_{23}; \omega_2 =\frac{\sqrt{3}}{2}a_2b_{12}- a_1c_{12}+\frac{1}{2}a_2c_{12}- b_{13}b_{23}+ c_{13}c_{23}; \omega_3 =\frac{\sqrt{3}}{2}a_2c_{12}+ a_1b_{12}-\frac{1}{2}a_2b_{12}+ b_{13}c_{23}+ b_{23}c_{13}.
I = \sum_{j=1}^{N}\int_{E_{\tau j}}\frac{\big\lvert\Psi'\big(u+\mathrm{i}a(u)+\mathrm{i}\tau\big) \big(1+\mathrm{i}a'(u)\big)\big\rvert}{\big\lvert\Psi\big(u+\mathrm{i}a(u)+\mathrm{i}\tau\big) - \alpha\big\rvert^q}\,\mathrm{d}u = \sum_{j=1}^{N}I_{\tau j}.
|A| =\sum_{Q\in\mathcal Q_B}|A\cap Q| + \sum_{Q\in\mathcal G_L}|A\cap Q| &\leq\sum_{Q\in\mathcal Q_B}\delta_1|Q| + \sum_{Q\in\mathcal G_L}\delta_2|Q| &\leq \delta_1 |B|+\delta_2 .
\beta = -\frac{Q}{2}+\sqrt{\frac{Q^{2}}{4}-2+2h},\,\,\,\,\,d = 1 - \frac{\frac{Q}{2}-\sqrt{\frac{Q^{2}}{4}-2+2h}}{\frac{Q}{2}- \sqrt{\frac{Q^{2}}{4}-2}},
dX_{k}^{\epsilon,v_k}(t) = f_k\bigl(X_{k}^{\epsilon, v_k}(t), v_{k}^{\epsilon}\bigl(X_{k}^{\epsilon, v_k}(t) \bigr)\bigr) dt + \sqrt{\epsilon}\sigma_k\bigl(X_{k}^{\epsilon, v_k}(t)\bigr) d W(t),
\Delta_n(C_n(t)) = \sup_{s,u \in C_n(t)}\|X_s - X_u\|_{2^n}\le L(r) \sup_{s,u \in C_n(t)}\left(2^{n/2}\|s-u\|_2 + 2^{n/r}\|s-u\|_\infty \right) & \le L(r) \left( 2^{n/2}\Delta_{d_2}(A_{n-1}(t)) + 2^{n/r}\Delta_{d_\infty}(B_{n-1}(t)) \right) & \le L(r) \left( 2^{(n-1)/2}\Delta_{d_2}(A_{n-1}(t)) + 2^{(n-1)/r}\Delta_{d_\infty}(B_{n-1}(t)) \right),
E_{k,\chi_{-2}}^{\infty}(\tau) =1-\frac{2k}{B_{k,8}}\sum_{n=1}^{\infty}\sigma_{k-1,\chi_{-2}}^{\infty}(n)q^{n}\intertext{and}E_{k,\chi_{-2}}^{0}(\tau) =\delta_{k,1}-\frac{2k}{B_{k,8}}\sum_{n=1}^{\infty}\sigma_{k-1,\chi_{-2}}^{0}(n)q^{n},
I_{N,M}^{}\left( f\right) :=\frac{2\pi R^{2}}{M\cdot N^{2}}{\displaystyle\sum_{j=1}^{N}}{\displaystyle\sum_{s=1}^{M}}\left( j-\frac{1}{2}\right) f\left( r_{j}\cos\frac{2\pi\left( s-\frac{1}{2}\right)}{M},r_{j}\sin\frac{2\pi\left( s-\frac{1}{2}\right)}{M}\right) .
\int_{-\infty}^\infty{dp\over 2\pi}e^{-\alpha_1 p^2}\, e^{-\alpha_2(p+k)^2}={1\over \sqrt{4\pi(\alpha_1+\alpha_2)}}exp\left[-k^2\left({\alpha_1\alpha_2\over \alpha_1+\alpha_2}\right) \right]
\langle (x_p\otimes g)\otimes (y_q\otimes h), (u_p\otimes g)\otimes (v_q\otimes h)\rangle = \langle (y_q\otimes h), \langle(x_p\otimes g), (u_p\otimes g)\rangle (v_q\otimes h)\rangle = \langle (y_q\otimes h), \beta_g^{-1}(\langle x_p,u_p\rangle) (v_q\otimes h)\rangle = \beta_h^{-1}(\langle y_q, \beta_g^{-1}(\langle x_p,u_p\rangle) v_q\rangle).
\|U_h^{n}\|^2+\Delta t\sum_{k=2}^n\|\nabla U_h^{k-\theta}\|^2+\gamma\Delta t\sum_{k=2}^n\|\Sigma_h^{k-\theta}\|^2 \leq C(\|U_h^{0}\|^2+\Delta t\sum_{k=1}^n\|g^{k}\|^2)+\Delta t\sum_{k=1}^n\|U_h^{n}\|^2+\Delta t\sum_{k=1}^n\|u_H^{n}\|^2.
a^{2341}=a^{2431},a_{1}^{2341}=a_{1}^{2431},a_{2}^{2341}=a_{2}^{2431},a_{3}^{2341}=a_{3}^{2431},a_{4}^{2341}=a_{4}^{2431}, a_{12}^{2341}=a_{12}^{2431},a_{13}^{2341}=a_{13}^{2431},a_{14}^{2341}=a_{14}^{2431},a_{23}^{2341}=a_{23}^{2431},a_{24}^{2341}=a_{24}^{2431}, a_{123}^{2341}=a_{123}^{2431},a_{124}^{2341}=a_{124}^{2431}
\tilde{{\cal H}}_{1}= ( c - b - a ) \, D^{- 1}b \, D^{- 1}c + ( b + 1 ) \, D^{- 1}a \, D^{- 1}b + c D^{- 1}a D^{- 1}c + \frac{1}{2}( D^{- 1}a )^{2}
\begin{cases}u_{tt}- u_{xx}+\frac{\mu}{1+t}u_t+\frac{\nu^2}{(1+t)^2}u=f(t,x), x\in \mathbb{R}, \t>0, u(0,x)=0, x\in \mathbb{R}, u_t(0,x)=0, x\in \mathbb{R},\end{cases}
|\Phi(\xi_1', \eta_1', \xi, \eta) | = |\xi_1' \xi(\xi_1' + \xi) + \eta_1' \eta (\eta_1' + \eta)|\leq 2^5 A^{-\frac{3}{2}}d^{-1}N_1^3, |F (\xi_1', \eta_1', \xi,\eta)| = |\xi_1' \eta + \xi \eta_1' + 2 (\xi_1' \eta_1' + \xi \eta)|\leq 2^5 A^{-1}d^{-1}N_1^2,
lnp_{\theta}(x)=ln\mathbb{E}_{z\sim q_{\phi}(\cdot|x)}[\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)}]\ge\mathbb{E}_{z\sim q_{\phi}(\cdot|x)}[ln\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)}]
|Q(\eta_\delta W) - Q(W)| \leq \int_{\delta^2\leq |z|\leq \delta}|\eta_\delta(d W)^N|^2 + |(Wd\eta_\delta)^N|^2 + 2|\eta_\delta(d W)^N|^2|(W d\eta_\delta)^N|^2 +O(\delta^2) = O(\delta^2)+ O\Big (\frac{1}{\log \delta^{-1}}\Big )=O\Big (\frac{1}{\log \delta^{-1}}\Big ).
e^{-( s- s_{0})\lambda_{n}}y^{2\lambda_{n}}=e^{- s\lambda_{l}}y^{2\lambda_{l}}e^{s_{0}\lambda_{l}}\left(ye^{-( s- s_{0})/2}\right)^{-2(\lambda_{l}-\lambda_{n})}\le e^{- s\lambda_{l}}y^{2\lambda_{l}}e^{s_{0}\lambda_{l}},
({\rm B}){\mathfrak G}_{\sigma_{\lambda}}(X,Y) ={\operatorname{DET}\Big| \prod\limits_{a=1}^{\lambda_{j}+n-j}(x_i +y_a + \beta x_i y_a) (1+\beta x_i)^{j-1}\Big|_{1 \le i,j \le n}\over \prod\limits_{1 \le i < j \le n}(x_i-x_j)}.
\lim_{n\to\infty}\int_{R^6}|\nabla_{x,t}U_j^L|^2(x,\frac{t_{jn}}{\lambda_{jn}})\,dx\lim_{n\to\infty}\int_{R^6}(\partial_tU_j^L)^2(x,\frac{t_{jn}}{\lambda_{jn}})\,dx=0.
S_{E}\, = \, M_{0}^{2}\, \int_{0}^{T}\! \! d t \int_{0}^{R}\! \! d r \left[ 1 \, + \, \frac{1}{2}{\bf u}\, \partial^{2}{\bf u}\right] \, + \, \sum_{a = 1}^{2}\, \frac{m_{a}}{2}\, \int_{t_{1}}^{t_{2}}\! \! d t \,{\dot{\bf u}}^{2}( t , r_{a}) \,{,}
\binom{n}{k}/\binom{n}{(k+r+1)/2}=\frac{(n-k+1)\cdots (n-(k+r+1)/2)}{(k+r+3)/2 \cdots k}\ge \frac{n-(k+r+1)/2}{(k+r+3)/2}&=\frac{3k+r-1}{k+r+3}\ge \frac{3k+(k-3)-1}{k+(k-3)+3}=\frac{2k-2}{k}>\frac{k-1}{k}+\frac{k-1}{2k-1}.
\mathcal{U}_{2k+1}=\mathrm{Cone}_{2k+1},\mathcal{U}_{2k+3}=\cup_{i=1}^{10}\mathcal{B}_i, \mathcal{U}_{2k+5}=\{S|S=S' \cup S''0\leq |S'| \leq 5S'\subset\mathcal{S}_7', S''\subset{[4k]}\}, \mathcal{U}_j = \mathcal{P}_n(j), j\geq 2k+7j,
G = \langle \sigma_{0}\rangle \times \left(\langle \sigma_{1}, \sigma_{2}\rangle \ltimes \left(\langle \sigma_{3}\rangle \times \langle \sigma_{4}\rangle \times \langle \sigma_{5}\rangle \right) \right) \cong \mathbb{Z}/ 2 \mathbb{Z}\times \left(S_{3}\ltimes \left(\mathbb{Z}/ 2 \mathbb{Z}\right)^{3}\right),
V_i \cdot{\cal N}_{\overline{\alpha V}}\equiv{\vec U}_i \cdot{\vec P}_{\overline{\alpha V}}- \sum_{r}V^r_i N^r_{\overline{\alpha V}}+ \sum_{\ell: ~right}T^\ell_i J^\ell_{\overline{\alpha V}}- \sum_{\ell: ~left}T^\ell_i J^\ell_{\overline{\alpha V}}~.
\Theta^0_{s}(\psi)=s\chi_0\left( \begin{array}{c}-i\frac{a^+(\eta_0)_t}{\sqrt{z}}i\frac{a^-(\bar{\eta}_0)_t}{\sqrt{\bar{z}}}\end{array}\right)+s\sigma(\chi_0)\mathfrak{c}^b_0-s\sigma(\chi_0)\mathfrak{h}^b_0-s\sigma(\chi_0)\mathfrak{k}^b_0.
\delta_{\mathsf{P,Ach}}(\mu,r) \leq \max\left\{\frac{(1-\mu M)K}{Mr}~,~ \frac{(1-\mu M)K}{\min\{M,K\}}+ \mu(M+K-1)\right\}= \frac{(1-\mu M)K}{\min\{M,K\}}+ \mu(M+K-1),
\pi_X(t) := \frac{\varphi (t)}{A_1 \big( X(t-r) \big) \, \tilde{X}(t)}, \ \ \pi_B (t) := L(t) - \pi_X(t) \, \tilde{X}(t), \ \V(t) := \pi_B(t) \, B(t) + \pi_X(t) \, X(t).
\sum_{\substack{Q\leq q \leq 2Q, q\,}}\sum_{\substack{1 \leq n \leq w, n \in{\mathfrak U}_q \,{\rm mod}\, q.}}b_n = \sum_{\substack{Q\leq q \leq 2Q, q\,}}\sum_{\chi\,{\rm mod}\, q}c_q(\chi) \sum_{1 \leq n \leq w}\chi( n) b_n .
-A(X)= \tfrac{1}{(g-1)!g!}\int_{X^{g-1}}\log\|\Lambda\|(P_1+\dots+P_{g-1})\Phi_{\Theta}^*\nu^{g-1}(P_1,\dots,P_{g-1}) = \tfrac{1}{(g!)^2}\int_{X^{g}}\log\|\Lambda\|(P_1+\dots+P_{g-1})\Phi^*\nu^{g}(P_1,\dots,P_g),
\ell(\lambda w) = \sum_{\alpha\in\Sigma^+,w^{-1}(\alpha) > 0}\lvert\langle\nu(\lambda),\alpha\rangle\rvert + \sum_{\alpha\in\Sigma^+,w^{-1}(\alpha) < 0}\lvert\langle\nu(\lambda),\alpha\rangle - 1\rvert.
f(z)=\sum_{n =0}^{\infty}\langle f, \varphi_{X,n}(z)\big\rangle\varphi_{X,n}(z) + \frac{1}{4\pi}\sum_{p\in \mathcal{P}_{X}}\int_{-\infty}^{\infty}\big\langle f,\mathcal{E}_{X,\mathrm{par},p}(z,1\slash 2+ir)\big\rangle \mathcal{E}_{X,\mathrm{par},p}(z,1\slash 2+ir)dr,
{\cal D}\bar{\cal M}\equiv{\cal D}\bar{B}_{\mu \nu}{\cal D}\bar{B}_{\mu 5}{\cal D}\bar{B}_{5 5}{\cal D}\bar{\zeta}_{\mu}{\cal D}\bar{\zeta}_{4}{\cal D}\bar{\zeta}_{\mu , \theta}{\cal D}\bar{\zeta}_{4 , \theta}{\cal D}\bar{\eta}{\cal D}\bar{\eta}_{, \theta},
\displaystyle{0= \lim_{\varrho \to 0^+}\frac{\int_{Q(x_0,\varrho)}|v|^p dx}{|Du|(Q(x_0,\varrho))}= \lim_{\varrho \to 0^+}\frac{\int_{Q(x_0,\varrho)}|v|dx}{|Du|(Q(x_0,\varrho))}.}
\left|u_0(x,t)\right|\le Ct^{-\frac{N+2A}{2}}U(|x|)\le \nu t^{-\frac{N+2A}{2}}, \left|u_{1,i}(x,t)\right|\le \nu t^{-\frac{N+2A}{2}}, |R_2(x,t)|\le \nu t^{-\frac{N+2A}{2}},
t^{1/2}T_i \sum_\mu f_\mu = \sum_{\mu:\ \mu_i < \mu_{i+1}}\left( tf_{s_i\mu}+(t-1)f_\mu \right) + \sum_{\mu:\ \mu_i = \mu_{i+1}}t f_\mu + \sum_{\mu:\ \mu_i > \mu_{i+1}}f_{s_i\mu}&= \sum_{\mu:\ \mu_i < \mu_{i+1}}tf_{s_i\mu}+ \sum_{\mu:\ \mu_i \le \mu_{i+1}}tf_\mu = t \sum_\mu f_\mu.
(\mathcal{W}_X^{-1})_{i}^j = (h^{-1}_X)^{jk}g_{ki}^X = (h^{-1}_Y)^{kj}\left(g_{ki}^Y+\frac{\langle Y,\partial_iY\rangle\langle Y,\partial_kY\rangle}{(1-|Y|^2)}\right)\sqrt{\frac{1-\langle N,Y\rangle^2}{1-|Y|^2}}.
H^{m}(s)= \begin{cases}\frac{1}{p+2}s^{p+2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \s \leq m, \frac{1}{2}m^p s^2 + \left(\frac{1}{p+2}- \frac{1}{2}\right) m^{p+2}\ \ \ \ \s\geq m.\end{cases}
\frac{d}{dt}||f_t||_{\chi, u_t}=\frac{\int_M \chi'(\frac{f_t}{||f_t||_{\chi,\phi_t}})\nabla_{\dot{u}_t}f_t d\mu_{u_t}}{\int_M\chi'(\frac{f_t}{||f_t||_{\chi, u_t}}) \frac{f_t}{||f_t||_{\chi, u_t}}d\mu_{u_t}}
(I_{0^+}^{\alpha}f)(x) = \frac{1}{\Gamma(\alpha)}\int_0^xf(t)(x-t)^{\alpha-1}\d t\ ,\x\ge 0, (I_{1^-}^{\alpha}f)(x) = \frac{1}{\Gamma(\alpha)}\int_x^1f(t)(t-x)^{\alpha-1}\d t\ ,\x\le 1,
\sum_{k=0}^N\frac{(q^{-N},b_1q^{n_1},\cdots,b_mq^{n_m};q)_k}{(q,b_1,\cdots,b_m;q)_k}q^k=(-1)^N\frac{(q;q)_Nb_1^{n_1}\cdots b_m^{n_m}}{(b_1;q)_{n_1}\cdots(b_m;q)_{n_m}}q^{\binom{n_1}{2}+\cdots+\binom{n_m}{2}}.
\det ({\cal M}_{N}^{R}) = [ \alpha ( h + 1 / 4 8 - c / 2 4 ) ]^{P_{1 / 2}( N )}\prod_{r , s}[ ( h - h_{r , s}^{R}) ( h - h_{s , r}^{R}) ]^{P_{1 / 2}( N -{r s / 4})}{}~ ,
\sum_{i=1}^{n}v_{i}^{q+1}a_{i}^{qj+l}&=\frac{q+1}{2}\sum_{i=1}^{\frac{q-1}{a}}\omega^{ai(q+1)}(\omega^{a(2i+1)(qj+l)}+\omega^{a(2i+2)(qj+l)}) &=\frac{q+1}{2}\omega^{at(q+1)}\sum_{i=1}^{\frac{q-1}{a}}\omega^{ai(q+1)(t+1)}+\frac{q+1}{2}\omega^{at\frac{q+1}{2}}\sum_{i=1}^{\frac{q-1}{a}}\omega^{ai(q+1)(t+1)}&=0.
J_2' = [D_{s}v(x,G_n(hx,0)) -D_{s}v(x,G_n(hx,z)) ]\D_uG_n(hx,0) \Dh(x), J_2'' = D_{s}v(x,G_n(hx,z)) \ [D_uG_n(hx,0) - D_uG_n(hx, z)] \Dh(x).
\Big( k \sum_{n = 1}^N \big\| E^n \big\|_{L^2(\Omega;V)}^2 \Big)^{\frac{1}{2}}\le \Big( k \sum_{n = 1}^N \big\| \Theta^n \big\|_{L^2(\Omega;V)}^2 \Big)^{\frac{1}{2}}+ \Big( k \sum_{n = 1}^N \big\| \Xi^n \big\|_V^2 \Big)^{\frac{1}{2}}
\sigma_{3}: T^{*}M|_{Q,P}\rightarrow T^{*}M &;(\sigma_{3}^{*}\Xi^{a}:=\epsilon^{a},\sigma_{3}^{*}\Psi_{a}:=\epsilon_{a},\sigma_{3}^{*}\Theta^{\alpha}:=\epsilon^{\alpha},\sigma_{3}^{*}\Theta_{\alpha}:=\epsilon_{\alpha},\sigma_{3}^{*}Q^{i},\sigma_{3}^{*}P_{i}) &\mapsto (\Xi^{a},\Psi_{a},\Theta^{\alpha},\Theta_{\alpha},Q^{i},P_{i})
S_{\mathrm{N B I}}= - \frac{\alpha}{4}\mathrm{T r}\left( Y^{- 1}[ A_{\mu}, A_{\nu}]^{2}\right) + V ( Y ) - \frac{\alpha}{2}\mathrm{T r}~ ( \bar{\psi}\Gamma^{\mu}[ A_{\mu}, \psi ] ) ,
B_{\psi}(\psi)=\sum_{n=0}^{\frac{N}{2}-1}w_{n}^{*}e^{j(n-\frac{N-1}{2})\psi}+\sum_{n=\frac{N}{2}}^{N-1}w_{n}^{*}e^{j(n-\frac{N-1}{2})\psi}
\|x_{n+1}-y\|^2 =\big\|Q_C\big(x_n+\eta_n(U_{r_n}x_n-x_n)\big)-y\big\|^2 &\leq \|x_n+\eta_n(U_{r_n}x_n-x_n)-y\|^2 &\leq \|x_n-y\|^2 -\eta_n(2-\eta_n)\big(r_n+\|x_n-Tx_n\|\big)^2 &\leq \|x_n-y\|^2 - \eta_n(2-\eta_n)r_n^2.
\begin{aligned}\int_{\mathbb{R}^{n}}\varepsilon_{n, b}(x,t)dx=\frac{1}{\big(4\pi a_{1}(t)\big)^{\frac{n}{2}}}\int_{\mathbb{R}^{n}}e^{-\frac{|x|^{2}}{4a_{1}(t)}}dx &=\prod_{i=1}^{n}\frac{1}{\pi^{\frac{n}{2}}}\int_{-\infty}^{\infty}e^{-\xi_{i}^{2}}d\xi_{i}=1, t>0.\end{aligned}
\vec{g}_{1}= \frac{\vec{f}_{1}+\vec{f}_{2}}{\|\vec{f}_{1}+\vec{f}_{2}\|},\ \vec{g}_{2}= \frac{\vec{f}_{1}+\vec{f}_{3}}{\|\vec{f}_{1}+\vec{f}_{3}\|},\ \vec{g}_{3}= \frac{\vec{f}_{1}+\vec{f}_{4}}{\|\vec{f}_{1}+\vec{f}_{4}\|}.
\begin{pmatrix}a b c d e f 1 1 1\end{pmatrix}\begin{pmatrix}a b c 1 e f 0 1 1\end{pmatrix}\begin{pmatrix}a b c 1 1 f 0 0 1\end{pmatrix}\begin{pmatrix}1 b c 0 e f 0 1 1\end{pmatrix}\begin{pmatrix}1 b c 0 1 f 0 0 1\end{pmatrix}.
\int_{x_k}^{x_{k+1}}\frac{L_B s(x)}{2\sqrt{x}e^{I(x)}}\circ dW(x)= \int_{x_k}^{x_{k+1}}\frac{L_B s(x)}{2\sqrt{x}e^{I(x)}}d\overleftarrow{W}(x) - \int_{x_k}^{x_{k+1}}\frac{L_B s(x)}{2\sqrt{\beta}x e^{I(x)}}dx.
\left\Vert D^{\alpha}u\right\Vert_{L^{2}\left( \omega_{\delta}\right)}\leq C^{k}\sum_{i=0}^{k}\binom{k}{i}\left( \frac{k}{\delta}\right)^{\left( k-i\right) dm}\left\Vert Q^{i}u\right\Vert_{L^{2}\left( \omega\right)}
\bigg\| \int_0^t S_{t-s}\alpha(\xi(s;x)) ds \bigg\| \leq \int_0^t \| S_{t-s}\alpha(\xi(s;x)) \| ds \leq M \int_0^t e^{\beta(t-s)}\| \alpha(\xi(s;x)) \| ds \leq MB \int_0^t e^{\beta(t-s)}ds = MB \varphi_{\beta}(t)
\sum_{\lambda}\epsilon^{\mu}( \lambda ) \epsilon^{* \nu}( \lambda ) = - g_{\perp}^{\mu \nu}+{\frac{1}{q^{+}}}\bigl ( \eta^{\mu}q_{\perp}^{\nu}+ \eta^{\nu}q_{\perp}^{\mu}\bigr ) +{\frac{{\bf q}_{\perp}^{2}}{( q^{+})^{2}}}\eta^{\mu}\eta^{\nu},
\begin{aligned}\epsilon_4 = 8C_4 \epsilon_5 = 80\kappa C_4 \epsilon_6 = 528\kappa^2C_4 + 12C_6 - 48\theta_{2,2,2}\epsilon_7 = 2912\kappa^3C_4 + 168\kappa C_6 - 672\kappa \theta_{2,2,2}-56 \theta_{3,3,1}\end{aligned}
\begin{aligned}\sum_{k=j}^m\binom{m}{k}\binom{k}{j}(-1)^{k-j}= \sum_{k=j}^m(-1)^{k-j}\frac{m!}{k!(m-k)!}\frac{k!}{(k-j)!j!}= \frac{m(m-1)\cdots (m-j+1)}{j!}\sum_{k=j}^m\frac{(m-j)!(-1)^{k-j}}{(m-k)!(k-j)!}= \frac{m(m-1)\cdots (m-j+1)}{j!}(1-1)^{m-j}=0,{\quad}j<m.\end{aligned}
\sum_{|\alpha|=l}\iint_{\mathbb{R}^{n+1}\setminus (\mathbb{R}^{n}\times \frac{1}{2}B^{1})}\gamma(x,t)|D^\alpha f(x,t)|\,dx\,dt \le &\qquad\le C\sum_{m\in\mathbb Z^n}\gamma_{0,m}\|\varphi\mid L_1(\widetilde CQ_{0,m})\|\le C\sum_{m\in\mathbb Z^n}\gamma_{0,m}\|\varphi\mid L_1(Q^n_{0,m})\|,
\alpha_{1}:= \frac{x_{1}^{q}+x_{2}^{q}}{2}-x_{0}, \alpha_{l}:= \frac{x_{l+1}^{q}- x_{l-1}^{q}}{2}, \enspace l = 2,...,Q-1, \alpha_{Q}:= x_{1}- \frac{x_{Q-2}^{q}+x_{Q-1}^{q}}{2},
\lim\limits_{n\to \infty}\int_{{\mathbb{R}}^3}\frac{F(x,u_{n})}{\|u_{n}\|_{E}^{4}}\,dx=\lim\limits_{n\to \infty}\Big(\int_{\Omega}+\int_{\mathbb{R}^{3}\setminus \Omega}\Big)\frac{F(x,u_{n})}{\|u_{n}\|_{E}^{4}}\,dx=+\infty.
\textbf{y}_{r_{p,q}d,k}(2i-1)= \textbf{h}_{r_pd}^k \hat{b}_{r_pd,k}(2i-1)+\textbf{h}_{r_qd}^k \hat{b}_{r_qd,k}(2i)+\textbf{n}(2i-1),
\left( \begin{array}{c c c c c}{{}}&{{4 ( 2 \Delta^{2}+ \Delta ) + 4 ( 4 \Delta + 1 ) \theta}}&{{}}&{{6 ( \Delta + \theta )}}&{{}}\\{{}}&{{6 ( \Delta + \theta )}}&{{}}&{{4 ( \Delta + \theta ) + \frac{c}{2}}}&{{}}\\ \end{array}\right) \left( \begin{array}{c}{{b^{( 1 , 1 )}}}\\{{b^{( 2 )}}}\\ \end{array}\right) = 0 ,
y_{1t}^-(1) = x_{\beta_{1t}}^-(1)=[\,x_t^-(0),\, y_{1(t-1)}^-(1)\,]_{\langle t, t+1\rangle\cdots \langle t, n\rangle \langle t, n-1\rangle\cdots \langle t, 1\rangle}, y_{1t}^+(-1) = x_{\beta_{1t}}^+(-1)=[\,y_{1(t-1)}^+(-1),\, x_{t}^+(0)\,]_{\langle t+1, t\rangle^{-1}\cdots \langle n, t\rangle^{-1}\langle n-1, t\rangle^{-1}\cdots\langle 1, t\rangle^{-1}}.
J_{s,t}= I + \sum_{k=1}^n\int_s^t \nabla A_{k}(Z_r)\cdot J_{s,r}\mathrm{d}W_{k,r}+\int_s^t \nabla B(Z_r) \cdot J_{s,r}\mathrm{d}r, &= I + \sum_{k=1}^n \int_s^t \nabla A_k (Z_r) \cdot J_{s,r}\circ \mathrm{d}W_{k,r}+ \int_s^t \nabla A_0 (Z_r) \cdot J_{s,r}\mathrm{d}r.