harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
u_1^{f'}(x)\big|_{D_{T_2}}=u_1^f(x)\big|_{D_{T_2}},\ \ \ \frac{\partial u_1^{f'}(x)}{\partial x_0}\Big|_{D_{T_2}}=\frac{\partial u_1^{f}(x)}{\partial x_0}\Big|_{D_{T_2}}.
\mathcal{S}^{b}_{x}\int_{D}G_{D}(x,z)\mathcal{S}^{b}_{z}G_{D}(z,y)dz=\int_{D}\mathcal{S}^{b}_{x}G_{D}(x,z)\mathcal{S}^{b}_{z}G_{D}(z,y)dz,\forall x,y\in D,\x\not=y.
\begin{cases}dp(t,x,z) = -[\frac{1}{2}\frac{\partial^2}{\partial x^2}p(t,x,z)+\pi(t,z)\{a_0(t,z)p(t,x,z) + b_0(t,z)q(t,x,z)\}]dt+ q(t,x,z) dB(t); t\in[0, T] p(T,x,z) = \frac{\partial U}{\partial y}(x,Y(T,x,z),z)\mathbb{E}[\delta_Z(z)|\mathcal{F}_T] p(t,x,z) =0 ; \forall (t,x,z)\in[0,T]\times \partial D\times \mathbb{R}.\end{cases}
\int_{\Omega}\varphi^{\prime}\overline{w}\; \d x + \int_{\Omega}\nabla \varphi \cdot \overline{\nabla w}\; \d x = - 2 \int_{\Omega}(u \cdot \nabla) d \cdot \overline{(d w)}\; \d x + 2 \int_{\Omega}\lvert \nabla d \rvert^2 \varphi \overline{w}\; \d x.
\langle \iota_{p,g}(T^*)x\otimes g, y\otimes g\rangle = \langle (T^*x)\otimes g, y\otimes g\rangle = \beta_{g^{-1}}(\langle T^* x,y\rangle)= \beta_{g^{-1}}(\langle x, Ty\rangle) = \langle x\otimes g, (Ty)\otimes g\rangle = \langle x\otimes g, \iota_{p,g}(T)y\otimes g\rangle.
V_{1}(z_{1})V_{2}(z_{2})=\sum_{i}C_{12i}(z_{1}-z_{2})^{\Delta_{i}-\Delta_{1}-\Delta_{2}}(\overline{z}_{1}-\overline{z}_{2})^{\overline{\Delta}_{i}-\overline{\Delta}_{1}-\overline{\Delta}_{2}}(V_{i}(z_{2})+\cdot\cdot\cdot).
\left\|\int_0^t E_h(t-s)P_h[b(\widehat{u}(s))-b(\widehat{u}_h(s))]ds \right\| & \leq \int_0^t \|E_h(t-s)P_h[b(u(s))-b(\widehat{u}_h(s))]\|ds & \leq \int_0^t \|\widehat{u}(s)-\widehat{u}_h(s)\|ds=\int_0^t \|e_h(s)\|ds.
h_\ast \mathcal F_{\chi_H}=h_\ast h^\ast\mathcal F\simeq \mathcal F\otimes h_\ast\Lambda\simeq \mathcal F\otimes\bigoplus_{\psi}\mathcal F_{\psi}\simeq\bigoplus_{\psi}\left(\mathcal F\otimes\mathcal F_{\psi}\right)
( f_{\lambda}, f_{\lambda^{\prime}}) = \Omega_{N - 1}\int_{0}^{\infty}f_{\lambda}^{*}f_{\lambda^{\prime}}( \sinh \sigma )^{N - 1}d \sigma = \Omega_{N - 1}\int_{1}^{\infty}v_{\lambda}^{*}v_{\lambda^{\prime}}d x \: , \nonumber
\mathcal{R}(x)= \sum_{\rho}\frac{1}{(n_\rho - 1)!}\lim_{s\rightarrow \frac{k-\rho}{2}}\frac{{\rm d}^{n_\rho -1}}{{\rm d}s^{n_\rho -1}}\Bigg\{\left(s- \frac{k-\rho}{2}\right)^{n_\rho}\frac{\Gamma(s)}{\zeta(k-2s)x^s}\Bigg\}.
V_i^{L}=\lim\limits_{\delta p_i \to 0}\frac{E_\kappa(\vec{p}+\delta\vec{p}) - E_\kappa(\vec{p})}{\delta^L p_i}=e^{-\frac{E(\vec{p})}{\kappa c^2}}\frac{\partial E_\kappa (\vec{p})}{\partial p_i}\ = \e^{-\frac{E(\vec{p})}{\kappa c^2}}V_i \, .
{\ \chi_{n}^{\mu \upsilon}= \left( \eta_{n_{1}}^{\mu}\sin n \tau - \eta_{n_{2}}^{\mu}\cos n \tau \right) \left( \eta_{n_{1}}^{v}\sin n \tau - \eta_{n_{2}}^{v}\cos n \tau \right) n^{2}\cos^{2}n \sigma -}
1 - \frac{\sigma_{k}(p,s_1+1) \sigma_{\ell}(p,s_2+1)}{p^{s_1+s_2+2}}= 1- \frac{k \ell}{p^{s_1+s_2+2}}+ O_{k,\ell}(p^{-3-3 \sigma_1}+p^{-3-3 \sigma_2}) = 1- \frac{k \ell}{p^{s_1+s_2+2}}+ O_{k,\ell}( p^{-3+3\max(a_1,a_2)}) \sigma_1 \ge -a_1, \sigma_2 \ge -a_2.
A_{t+2,k}= \begin{pmatrix}2 4 \cdots 2^t \frac{2^{t+1}- 1}{3}a_{n-k+1}1 2 \cdots 2^{t-1}2^t a_{n-k+2}0 1 2 \cdots 2^{t-1}a_{n-k+3}\vdots \ddots \ddots \ddots \vdots \vdots \vdots \ddots 1 2 a_{n-k+t+2}0 \cdots \cdots 0 1 a_{n-k+t+3}\end{pmatrix}
\frac{2\epsilon_f+r}{h(\gamma \bar\Delta)}\left[ \sum_{k=0}^{T-1}1 \right] - \left[ \sum_{k=0}^{T-1}(1 - \Theta_k)(1 - \Lambda_k) + \Theta_k \Lambda_k' \right] < \left( \frac{2\epsilon_f+r}{h(\gamma \bar\Delta)}- \frac{1}{2}\right) T + \frac{1}{2}\left|\log_\gamma\frac{\bar\Delta}{\delta_0}\right| + \frac{1}{2}.
\mathcal S(t) \triangleq \bigg\{(\mathbf{x},\mathbf{y})| \mathbf{x},\mathbf{y}\in \{0,1\}^{P-1}, x_p+y_p\leq 1, p\in\mathcal{P}\backslash \{P\}, \sum_{p\in\mathcal{P}\backslash \{P\}}(x_p+2y_p)=t \bigg\}, h(\mathbf{z},t)\triangleq \sum_{(\mathbf{x},\mathbf{y})\in \mathcal S(t)}\prod_{p=1}^{P-1}2^{x_p}z_p^{x_p+2y_p},\ \mathbf{z}\in \mathbb{R}^{P-1}_{++},
\bigtriangleup(\hat{x}_\mu)=\hat{x}_\mu\otimes1+{\hat{\mathfrak{I}}}=Z^{-1}\otimes\hat{x}_\mu-a_\mu p^{\rm L}_\alpha\otimes\hat{x}_\alpha+{\hat{\mathfrak{I}}}=O_{\mu\alpha}\otimes\hat{x}_\alpha+{\hat{\mathfrak{I}}}, \bigtriangleup\big(p^{\rm L}_\mu\big)=p^{\rm L}_\mu\otimes Z^{-1}+1\otimes p^{\rm L}_\mu+{\hat{\mathfrak{I}}}.
E^u_\lambda(t_0) =\{u(t_0)\in E:\, Ju'(t)+S_\lambda(t)u(t)=0,\,\, \lim_{t\rightarrow-\infty}u(t)=0\}\subset E, E^s_\lambda(t_0) =\{u(t_0)\in E:\, Ju'(t)+S_\lambda(t)u(t)=0,\,\, \lim_{t\rightarrow+\infty}u(t)=0\}\subset E
{\cal J}\Pi_{{\cal S}_+^p}(\overline{M})H = P \left[\begin{array}{cc}P_\alpha^T HP_{\alpha}\Theta_{\alpha\gamma}\circ P_\alpha^T HP_{\gamma}[7pt] P_\gamma^T HP_{\alpha}\circ{\Theta}_{\gamma \alpha}0\end{array}\right] P^T\, \forall \, H\in{\cal S}^p\, .
P\{X_{a,b}\le x\}=\int_{0}^{x}abt^{a-1}(1-t^{a})^{b-1}dt=\int_{0}^{x^{a}}b(1-t)^{b-1}dt=P\{Y_{1,b}\le x^{a}\}=P\{Y_{1,b}^{1/a}\le x\}.
x_1^{(j+2)}=\frac{(x_2^{(j+1)})^2 +x_3}{x_1^{(j+1)}}=x_2^{(j+1)}\frac{x_2^{(j+1)}}{x_1^{(j+1)}}+\frac{x_3}{x_1^{(j+1)}}>x_2^{(j+1)}.
b(\theta)=b(\theta,\tau_0)=\frac{\sum_{i\in I_*^c}\theta_i^2}{\sigma^2+\sigma^2|I_*|\log(en/|I_*|)}=\frac{\sum_{i=i_*+1}^{n}\theta_{[i]}^2}{\sigma^2+\sigma^2i_*\log(en/i_*)}=\frac{B(I_*,\theta)}{\sigma^2+V(I_*, \theta)}.
\frac{y}{x + 2}= \sqrt{2 - \lambda}\frac{\prod_{i = 1}^{N}( v - a_{i}) + \prod_{i = 1}^{N}( v + a_{i})}{\prod_{i = 1}^{N}( v - a_{i}) - \prod_{i = 1}^{N}( v + a_{i})}.
\|\partial_\lambda^k\mathcal{P}_\lambda^\#(z)\|_{\mathcal{L}(\mathcal{B}_{\lambda,r}^\#,\mathcal{H}_{\lambda,r}^\#)}\leqslant \; C_k\langle\lambda\rangle^{-k}\|\partial_\lambda^k\mathcal{E}_\lambda^\#(z)\|_{\mathcal{L}(\mathcal{H}_{\lambda,r}^\#,\mathcal{B}_{\lambda,r}^\#)}\leqslant \; C_k\langle\lambda\rangle^{-k}.
\begin{cases}T_x^2=T_x,\T_y^2=1, \forall x\in X,\ \forall y\in Y, T_zT_{z'}=T_{z'}T_z, \forall z,z'\in Z, T_xT_y=T_x, \textrm{if $x>y$ in $Z$ (by Lemma~\ref{lem:01=0})}.\end{cases}
\tilde T_6(z/2) = \frac{i}{(2\pi)^{\frac12}}\int_{\xi_1+\xi_2+\xi_3=0}\frac{1}{z^2(z+\xi_1)(z+\xi_2)}\left(\frac{1}{z}+\frac{1}{z-\xi_3}\right) \hat u(\xi_1) \hat u(\xi_2) \hat u(\xi_3) d\xi_1d\xi_2,
\begin{aligned}\mathbf{C}^{n,k}(x)= \left[ \left(\log (P(x)) \right)^2-\log(P(x))\right] &+\frac{1}{n}\left(-\log (P(x))\left[k-1 \right]+\frac{1}{2}\left(\log (P(x))\right)^2-\frac{1}{2}\left(\log (P(x))\right)^3\right)+{\rm o}\left(\frac{1}{n}\right).\end{aligned}
\begin{cases}2c_1+c_2 =2(w_1+w_3\kappa)-2w_3\theta\kappa\geq c, c_3 =w_2+w_4\kappa\geq c, \det(M) =c_3(c_3+c_4)P_\theta-\frac{\left(c_3P_\theta+(c_3+c_4)-(2c_1+c_2)\right)^2}{4}\geq c\end{cases}
\left[\Phi(x+v),\Phi(y+w) \right]_{\theta'}&=d'\left[x-h(x)+v,y-h(y)+w \right]_{\theta'}&=[x,y] +\rho(x)w+\rho(y)v-\rho(x)h(y)-\rho(y)h(x)+\theta'(x,y) &=[x,y] +\rho(x)w+\rho(y)v-\rho(x)h(y)-\rho(y)h(x)+\theta(x,y)+d^1(h)(x,y) &=[x,y] +\rho(x)w+\rho(y)v+\theta(x,y)-h\left([x,y] \right) &=\Phi\left( [x+v,y+w]_{\theta}\right) .
\Lambda(k,\ell):=i\,\left(-{\rm sgn}\,k+{\rm sgn}\,\ell\right)=\begin{cases}-2i\,,\qquad\mbox{if}\,\,k>0\,\,\mbox{and}\,\,\ell<0\,, \mp i\,,\qquad\mbox{if}\,\,\pm k>0\,\,\mbox{and}\,\,\ell=0\,, 0\,,\qquad\mbox{if}\,\,k\ell>0\,,\end{cases}
|\varphi'|^2 = \left|\tfrac\mu{2\zeta}s^{\mu-1}\widetilde\varphi - \tfrac1{2p}s^{\mu-1-\frac1p}\partial_1\widetilde\varphi\right|^2 + \left|\tfrac12 s^{\mu-1}\partial_\zeta\widetilde\varphi \right|^2,
\displaystyle \int_{0}^{T}\Phi_{t}\circ dW_{t}=\displaystyle \mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{i=0}^{n-1}\Phi(\frac{B_{t_{i}}+B_{t_{i+1}}}{2},t_{i}) (B_{t_{i+1}}-B_{t_{i}})
[z,t]\circ [z', t'] =[z_{1},\cdots,z_{n}, t] \circ [z_{1}',\cdots,z_{n}', t'] &: =\Big[z_{1}+z_{1}',\cdots, z_{n}+z_{n}',t+t'+2 \mbox{Im}\big(\sum_{j=1}^{n}z_{j}\bar{z}'_{j}\big) \Big].
\dim(V_\lambda) \; \; \;=\; \prod_{1\leq i<j\leq m}\Bigl( \frac{\lambda_i - \lambda_j}{j-i}+1\Bigr) \; \; \cdot \prod_{1\leq i<j\leq m}\Bigl( \frac{\lambda_i + \lambda_j}{2m+1-i-j}+ 1 \Bigr).
K_{2}^{( 1 )}= \frac{1}{3 2 \pi^{2}}\int_{0}^{\infty}\frac{d s}{s^{2 - \epsilon}}\exp ( i s A ) = \frac{1 - \xi}{1 6 \pi^{2}}A^{1 - \epsilon}( \frac{2}{\epsilon}- \gamma + 1 )
h_{\i}^{\underline{{a}}}\, \dot{\xi}_{\alpha}^{\underline{{a}}}- \varepsilon_{i j}\, B_{\alpha}^{\underline{{a}}}\, \xi_{\alpha}^{\underline{{a}}}\, \dot{x}^{j}-{\cal E}_{j ; \alpha}^{\underline{{a}}}\, \xi_{\alpha}^{\underline{{a}}}= 0 .
m(r,\Phi^\alpha_{ij})\leqslant\sum_{u=1}^3m(r,F_u^{ij})+2\sum_{u=1}^{3}m\biggl(\dfrac{\mathcal{D}^{\alpha}(F_u^{ji})}{F_v^{ji}}\biggl)+O(1)\leqslant\sum_{u=1}^3m(r,F_u^{ij})+o(T_f(r)).
T\colon X \;\longrightarrow\;\; \Biggr( \bigoplus_{\omega\leqslant \alpha<\lambda}(P_{\alpha+1}-P_\alpha)(X) \Biggr)_{\!\!\ell_p},Tx = \bigl(P_{\alpha+1}x - P_{\alpha}x\bigr)_{\omega\leqslant \alpha<\lambda}(x\in X)
\partial_{+}\partial_{-}( e^{- 2 \phi}) = - \lambda^{2}+ \frac{\mu^{4}}{1 6 \beta}{}~ \mathrm{s e c h}^{4}\left[ \frac{\mu}{2 \sqrt 2}( \alpha ( x^{+}- x_{0}^{+}) - \frac{1}{\alpha}( x^{-}- x_{0}^{-}) ) \right] ,
< O_{k_{0}}O_{k_{1}}. . . O_{k_{p - 1}}P >_{0}= - \mid \ln \mu \mid \left( \begin{array}{c}{2 \Sigma}\\{\Sigma}\\ \end{array}\right) \frac{1}{\pi \sqrt{2 | t_{2}|}}\frac{1}{( - 4 t_{2})^{\Sigma}}\frac{\partial^{p - 1}}{\partial t_{0}^{p - 1}}t_{0}^{\Sigma},
\kappa= \sup_{\substack{\theta \in \mathbb{R}^d \mathbb E ( \langle \theta, X \rangle^2 ) > 0}}\frac{\mathbb E \bigl( \langle \theta , X \rangle^4 \bigr)}{\mathbb E \bigl( \langle \theta, X \rangle^2 \bigr)^2}.
\int_{\{-t_1\le\Psi<-t_2\}}|\tilde{F}|^2_ha(-\Psi)=\frac{G(T_1;c)}{\int_{T_1}^{+\infty}c(t)e^{-t}dt}\int_{t_2}^{t_1}a(t)e^{-t}dt
I_1 \leq \frac2\delta \|u^2\|_{L^\infty( L^q)}\|b^1-b^2\|_{L^1(L^p)}+ C'_{d,p}\bigg(\sum_{i=1}^2\|u^i\|_{L^\infty(L^q)}\bigg) \big\| \nabla b^1\big\|_{L^1(L^p)}.
{\hat \nabla}_a{\hat \nabla}_a{\hat A}= -\sum_{J=0}^N \sum_{m=-J}^J A_{Jm}[{\hat S}_a,[{\hat S}_a,{\hat Y}_m^J]]= -\sum_{J=0}^N \sum_{m=-J}^J J(J+1)A_{Jm}{\hat Y}_m^J.
\tau_{n+1,\omega}^{I}(\chi) \coloneqq \sup\big\lbrace\tau:\ \tau^{I}_{n,\delta_1,\omega}(\chi)\leq \tau \leq b \ \ \ \W_{\mathbf{X}(\omega),\gamma,\delta_1}^{{\gamma-\delta_1}}(\tau_{n,\omega}^{I}(\chi),\tau)\leq \chi\big\rbrace.
D_{\alpha}= \partial_{\alpha}- \frac{1}{2}i \partial_{\alpha \dot{\beta}}\widetilde{\vartheta}^{\dot{\beta}}, \qquad \widetilde{D}_{\dot{\beta}}= - \widetilde{\partial}_{\dot{\beta}}+ \frac{1}{2}\vartheta^{\alpha}i \partial_{\alpha \dot{\beta}}
\int_{\Omega}\omega(x)^{p(x)}|Xu(x)|^{p(x)-1}|Xv(x)|\,dx =\int_{\Omega}\omega(x)^{p(x)}|Xu(x)|^{p(x)-1}|Xv(x)|\omega(x)^{-1}\omega(x)\,dx &\leq c\||Xu|^{p(\cdot)-1}\omega(x)^{p(\cdot)}\|_{L^{p'(\cdot)}_{\omega^{-1}}(\Omega)}\|Xv(x)\|_{L^{p(\cdot)}_{\omega}(\Omega)},
\hat{w}(f) =\left[\sum_{k,l=1}^n\hat{( a_{i,k}u_{k,l}\overline{a_{l,j}})}(f) \right]_{i,j=1}^m &=\left[\sum_{k,l=1}^na_{i,k}\hat{u}_{k,l}(f) \overline{a_{l,j}}\right]_{i,j=1}^m=A \hat{u}(f) A^*=( \theta_A \circ \hat{u}) (f).
\mathrm{E}\bigg[\Big\|\int_t^{t+h}\Psi \Big( \int_t^s \Phi \,\mathrm{d}W_r^K\Big)\, \mathrm{d}W_s^K -\sum_{i,j \in \mathcal{J}_K}\bar{I}_{(i,j)}^{Q,(D)}(h) \; \Psi \big(\Phi \tilde{e}_i, \tilde{e}_j\big) \Big\|_H^2\bigg] \leq 2C \sum_{i=1}^L \mathrm{E}\Big[\big( \tilde{A}_{(i)}^Q(h) -\tilde{A}_{(i)}^{Q,(D)}(h)\big)^2\Big].
\sum\limits_{n=1}^{\infty}|\ln\lambda_{m (n+1)-1}-\ln\lambda_{m n}| \leq \sum\limits_{n=1}^{\infty}|\ln\lambda_{m (n+1)-1}-\ln\mu^{(2)}_{m (n+1)-1}| +\sum\limits_{n=1}^{\infty}|\ln\mu^{(2)}_{m (n+1)-1}-\ln\lambda_{mn}|.
|\int_{\mathcal{Y}^{N'+1}}g(y_{[n,n+N']})P^{\mu}(dy_{[n,n+N']}|Y_{[0,n-1]}) & -\int_{\mathcal{Y}^{N'+1}}g(y_{[n,n+N']})P^{\nu}(dy_{[n,n+N']}|Y_{[0,n-1]})|+\frac{2}{3}\epsilon &\leq \|g\|_{\infty}\frac{\epsilon}{3 \|g\|_{\infty}}+\frac{2}{3}\epsilon=\epsilon
\begin{cases}\tau\left(1-\gamma\left(\frac{\lambda}{c_{1}}\right)\right)=-\lambda s\,\frac{\mu}{c_{1}^{2}}\gamma\left(\frac{\lambda}{c_{1}}\right)-t \lambda\tau\left(1-\gamma\left(\frac{\lambda}{c_{1}}\right)\right)=s\left(\left\Vert x\right\Vert +\mu-\mu\gamma\left(\frac{\lambda}{c_{1}}\right)\right)-\lambda t\end{cases}
\phi(p_v) = \phi\Big(\sum_{e \in vE^1}s_e s^*_e\Big)= \sum_{e \in vE^1}e^{-\beta}\phi(p_{s(e)}) &=\sum_{e \in vE^1}e^{-\beta}\psi(p_{s(e)})=\psi(p_v).
\mathbb{E}\big(|\widetilde{\nu}_{n+1}(A)-\widetilde{\nu}_{n}(A)|^p\big)\le 2N^{p-1}\sum_{j=1}^N\sum_{S\in \mathcal{S}_n^j(A)}\mathbb{E}\Big(\Big| \int_{S\cap A}U_S(x)V_S(x)\, \nu(dx)\Big|^p \Big),
A^{i_{1}\ldots i_{n}}A^{j_{1}\ldots j_{n}}\bar{B}_{k_{1}\ldots l_{m}}\bar{B}_{l_{1}\ldots l_{m}}D_{i_{1}j_{1}, \ldots , i_{n}j_{n}}^{k_{1}l_{1}, \ldots , k_{m}l_{m}}
\sup_{t\in[0,T]}M_z(t) \geq \frac{\lambda^{2\theta_0}}{N\nu^z}-\frac{N^{z-1}}{\lambda^{2\alpha_0-2\theta_0(1+\alpha_0)}\nu^{(z-1)(1+{\alpha_0})}}&=\frac{\lambda^{2\theta_0(1+\alpha_0)}}{\nu^{(z-1)(1+\alpha_0)}}\left[\frac{\nu^{\alpha_0z-1-\alpha_0}}{N\lambda^{2\alpha_0\theta_0}}-\frac{N^{z-1}}{\lambda^{2\alpha_0}}\right].
\varphi_{n}(x;q^{-1})=\frac{1}{2}q^{-(n-1)^{2}/4}\left(\mathcal{E}_{q}(x)\mathcal{E}_{q}(-q^{n/2}x)-(-1)^{n}\mathcal{E}_{q}(-x)\mathcal{E}_{q}(q^{n/2}x)\right).
\sum_{i=1}^N \sum_{i' = 1}^{\tilde N_i}(\mu + \eta)(\tilde I_i^{i'}) \leq \sum_{i=1}^N \sum_{i' = 1}^{\tilde N_i}\mu(\tilde I_i^{i'}) + \sum_{i=1}^N \eta(I_i) \leq \sum_{i=1}^N (1-K)\mu(I_i) + \epsilon_1 = (1-K)\mu(I) + \epsilon_1.
\|f\|_{C^1}: =\|f\|_{L^{\infty}}+\|\nabla f\|_{L^{\infty}}&=\|f\|_{\mathfrak{E}}+\Big\|\sqrt{\Big(\frac{\partial f}{\partial r}\Big)^2+\frac{1-\zeta^2}{r^2}\Big(\frac{\partial f}{\partial \zeta}\Big)^2}\Big\|_{\mathfrak{E}}.
\frac{1}{|Q|}\int_Q{\sum_{k=3}^{\infty}{2^{km}\mathfrak{M}_{\sigma,2^k}\Pi^*_kf(x)}}dx \lesssim \frac{1}{|Q|}\sum_{P\in\mathcal{A}_Q}{\int_{3P}{\sum_{k=3}^{\infty}{2^{km}\mathfrak{M}_{\sigma,2^k}\Pi_k^*f(x)}}dx}&\lesssim \sup_{R\in\mathcal{D}_2}{\frac{1}{|R|}\int_R{\sum_{k=3}^{\infty}{2^{km}\mathfrak{M}_{\sigma,2^k}\Pi_k^*f(x)}}dx}.
S+\gamma[\frac{DS}{Dt}-\Delta V\cdot S-S\cdot(\Delta V)^{T}]=\mu(h,d)[B+\gamma(\frac{DB}{Dt}-\Delta V\cdot B-B\cdot(\Delta V)^{T})]-gA+C_{1}(gA-\frac{C_{2}I}{\mu(h,d)^{2}})
\gamma ( a ; b_{1}, \ldots , b_{l}) : = ( ( \underbrace{a^{1}, \ldots , a^{1}}_{m_{1}{\ \mathrm{t i m e s}}}) + b_{1}, ( \underbrace{a^{2}, \ldots , a^{2}}_{m_{2}{\ \mathrm{t i m e s}}}) + b_{2}, \ldots , ( \underbrace{a_{l}, \ldots , a_{l}}_{m_{l}{\ \mathrm{t i m e s}}}) + b_{l}) .
\inf_{t\in T}\bigl(X(t) - h_{1}(t)\bigr) \le X(s) - h_{1}(s) = X(s) -h_{2}(s) + h_{2}(s) - h_{1}(s) &\le X(s) - h_{2}(s) + \|h_{2}-h_{1}\|_{\infty}s\in T.
\left|\frac{\left(x_{N+1}^{(n),m}-x_{N-1}^{(n),m}\right)\prod_{i=1}^{N}(x^{(n),m}_{i+1}-x^{(n),m}_i)\Delta_N(\xi^{(n),m})f\left(\xi^{(n)}\right)}{\Delta_{N+1}(x^{(n),m})}\right|
\left\arrowvert\int_0^1 d\rho\;\rho^{N-2}\int_0^{D\rho^2}\frac{dh}{\left(\rho^2+h^2\right)^{s+N/2}}-\frac{D}{1-2s}\right\arrowvert\ = =\ \left\arrowvert\int_0^1 d\rho\;\left[\rho^{N-2}\int_0^{D\rho^2}\frac{dh}{\left(\rho^2+h^2\right)^{s+N/2}}-D\rho^{-2s}\right]\right\arrowvert\leq \widetilde\beta,
{\cal L}_G =-\frac{1}{4}(B_{\mu\nu}-Q_{\mu\nu})^2+\frac{1}{2}(A_\mu+\partial_\mu\theta)^2+\frac{1}{2m}\epsilon_{\mu\nu\rho\sigma}(B^{\mu\nu}-Q^{\mu\nu})\partial^\rho(A^\sigma+\partial^\sigma\theta),
L_{0}| \kappa , s \rangle ( w ) = \left\{e^{2 w}\left[ \frac{s}{2}+ \frac{i \kappa}{4}\right] + e^{- 2 w}\left[ \frac{s}{2}- \frac{i \kappa}{4}\right] \right\}| \kappa , s \rangle ( w ) .
S_N(x)={1\over\sqrt{1-\beta}}\sum_{l=0}^N\beta^l\sum_{n=0}^lc_n{\Gamma(n/2+l-1/2)\over\Gamma(3n/2-1/2)\Gamma(l-n+1)}\left[{g\over x(x^2-1)}\right]^n.
\Vert F\Vert_{L^{2}(K_{\mathbb{C}},\mu_{s,\tau})}^{2}=\left(e^{\Delta_{s,\tau}/2}|F|^{2}\right) (e) & =\left( e^{\Delta_{s,\tau}/2}e^{\tau\partial^{2}/2}e^{\bar{\tau}\bar{\partial}^{2}/2}(f_{\mathbb{C}}\bar{f}_{\mathbb{C}})\right) (e).
K(T;0,0) = \sum_{n=-\infty}^{\infty}\frac{1}{Z}\int_{\varphi^{(n)}(0)=0}^{\varphi^{(n)}(T)=0}{\cal D}\varphi^{(n)}(t) \exp \left\{-\int_{0}^{T}L(\varphi^{(n)},\dot{\varphi}^{(n)}) dt \right\}
\langle V(x)\rangle=N^{-1}\lim_{T\rightarrow\infty}\int dA_0\langle 0|e^{iTH}e^{{\frac{2\pi i}{g}}\int_C dy^i\epsilon_{ij}E_i(y)}e^{i\int A_0[\partial_iE_i-J_0]}e^{iTH}|0\rangle\,.
p_{1\zeta}= \sqrt{-\frac{\ddot{\omega}^H_{11*}}{\ddot{\omega}^0_{33*}}(p_{\xi}^2 + p_{\eta}^2) +2\frac{\delta \omega}{c\,\ddot{\omega}^0_{33*}}}, p_{2\zeta}=\sqrt{-\frac{\ddot{\omega}^E_{11*}}{\ddot{\omega}^0_{33*}}(p_{\xi}^2 + p_{\eta}^2) + 2\frac{\delta\omega}{c\,\ddot{\omega}^0_{33*}}},
\|Dv(t)\|_{p ;B_\rho}\le C\left(\|D^2 v(t)\|_{L^q(B_\rho)}^a[v(t)]_{\alpha;B_\rho}^{1-a}+ [v(t)]_{\alpha;B_\rho}\right),a=1-\frac{q}{N+2}<\frac{1}{\gamma}.
\| \Theta_{k+1}\|_{C^{1,\alpha}_\nu}\leq Q\sum_{m=2}^{k+1}{C_m\left(\sum_{I\in \mathcal{I}_{m,k}}{\|\varphi_1\|_{C^{1,\alpha}_{-1}}^{i_1}\|\varphi_2\|_{C^{1,\alpha}_{\nu}}^{i_2}\dots \|\varphi_{k}\|_{C^{1,\alpha}_{\nu}}^{i_{k}}}\right)}
\left( \prod\nolimits_{s=1}^{n}E_{s}\left( f^{(\alpha )}\right)_{p\left(\cdot \right) ,\omega}\right)^{1/n}n^{r}\precapprox K_{m}\left( f^{(\alpha +r)},n^{-1},p\left( \cdot \right) ,\omega \right)
t(v_1)-t (v_2) \leq \int^{v_1}_{v_2}(\int_{\tilde N_t\cap \{|z|\leq \frac12\}}H^2)^\frac12 (Area (\tilde N_t\cap \{|z|\leq \frac12\}))^{-\frac32}dv &=\int^{v_1}_{v_2}(\int_{K_t}H^2 )^\frac12 \cdot (Area(K_t))^{-\frac32}dv.
\left[ \hat{G}\left( \mathbf{k}\right) \right]_{lm,l^{\prime}m^{\prime}}=\hat{\omega}_{lm}^{\ast}\left( \mathbf{k}\right) \mathbf{\hat{G}}\left(\mathbf{k}\right) \hat{\omega}_{l^{\prime}m^{\prime}}\left( \mathbf{k}\right).
\limsup_{n \rightarrow \infty}\frac{\sum_{\gamma \in Q(n)}\exp \left( \ell_\psi(\gamma) \right)}{\sum_{\gamma \in P(n)}\exp \left( \ell_\psi(\gamma) \right)}= \limsup_{n \rightarrow \infty}\frac{\sum_{\gamma \in Q_*(n)}\exp \left( \ell_\psi(\gamma) \right)}{\sum_{\gamma \in P_*(n)}\exp \left( \ell_\psi(\gamma) \right)},
Z^*W_{\pi^0}(K)Z= e^{i 90 m}b_1 z_1^*z_1 + e^{i 90 m}b_2 z_2^*z_2+\frac12 e^{i 90 m}b_3 [z_1^*(z_1+z_2)+ z_2^*(z_1+z_2)]
\begin{cases}\eta^\zeta(\rho,\rho u)=\rho\int_{-1}^1 \zeta(u+\rho^{\vartheta}s)[1-s^2]_+^\Lambda ds, q^\zeta(\rho,\rho u)=\rho\int_{-1}^1 (u+\vartheta\rho^\vartheta s)\zeta(u+\rho^{\vartheta}s)[1-s^2]_+^\Lambda ds. \end{cases}
\{(k,l) \mid k,l\ne j,\alpha_k\ne\alpha_l\}= \{(k,l) \mid k,l\not\in C_J,\alpha_k\ne\alpha_l\}&\cup\{(k,l) \mid k\not\in C_J,l\in C_J,l\ne j\}&\cup\{(k,l) \mid k\in C_J,k\ne j,l\not\in C_J\},