harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{B_R}(1-c^{-2}(y))\cdot \Delta^{-1}(c^{-2}f)(y)\cdot v(y)\dy = \int_{B_R}(1-\widetilde{c}^{-2}(y))\cdot \Delta^{-1}(\widetilde{c}^{-2}\widetilde{f})(y)\cdot v(y)\dy
\begin{bmatrix}\dot p \dot v \end{bmatrix}= \begin{bmatrix}0 1 0 -b/m \end{bmatrix}\begin{bmatrix}\dot x \dot v \end{bmatrix}+ \begin{bmatrix}0 1/m \end{bmatrix}u + \begin{bmatrix}I_2 O_{2,3}\end{bmatrix}w ,
f_1(u, v)= \pi^{k+r}u^p+v^l\sum_{i=0}^k(-1)^i\dbinom{k+r}{k-i}\dbinom{i+r-1}{i}(\pi t)^{k-i}v^{k-i}= \pi^{k+r}u^p+(-1)^k\dbinom{k+r-1}{k}v^l+\sum_{i=0}^{k-1}(-1)^i\dbinom{k+r}{k-i}\dbinom{i+r-1}{i}(\pi t)^{k-i}v^{k+l-i}.
V'_1(u_n)[u_n - \bar{u}] = 4B_1(u_n^2, u_n(u_n - \bar{u})) = 4B_1(u_n^2, (u_n - \bar{u})^2) + 4B_1(u_n^2, \bar{u}(u_n - \bar{u})) &=4B_1(u_n^2, (u_n - \bar{u})^2)+o(1).
2(1-x^2)^3 \Bigl(\frac{\mu(x)}{\eta(x)^{3/2}}+\frac{2\sqrt{c}x}{(1-x^2)^{3/2}}\Bigr)\eta(x)^3\lambda''(x)=(1-x^2)^{3}\mu(x)^2 -4c x^2 \eta(x)^3=\nu(x), \quad\mbox{say}.
x_{i,j}+ x_{i,j+1}= S- a_j, x_{i,n+1-j}+ x_{i,n-j}= a_j, x_{m+1-i,j}+ x_{m+1-i,j+1}= S-a_j, \mbox{and}x_{m+1-i,n+1-j}+ x_{m+1-i,n-j}= a_j.
k_g =\frac{1}{2}\sum_{d|N_g}\mathrm{mult}_d(0,0) =\frac{1}{2}\sum_{d|N_g}\frac{1}{d}\sum_{k|d}\mu(d/k) c_{g^k}(0) & = \frac{1}{2}\sum_{d|N_g}\frac{1}{d}\sum_{k|d}\mu(d/k) \mathrm{tr}(g^k) - 2\sum_{d|N_g}\frac{1}{d}\sum_{k|d}\mu(d/k) &= -2 + \frac{1}{2}\sum_{k}b_k,
g\big(c(t,x)\big) \left( \frac{\partial_{xx}w(t,x)}{1 + |\partial_x w(t,x)|^2}- \frac{1}{w(t,x)}\right)\begin{cases}\leq -g\big(c_0(x)\big) \frac{1}{w_0(x)}+ \varepsilon, \geq -g\big(c_0(x)\big) \frac{1}{w_0(x)}- \varepsilon\end{cases}
r(X,Y) =\left(\rho^\nabla\right)^{J,+}(X,Y)+\sum_{i,k=1}^{2n}g\left(JX,N(e_i,e_k) \right)g\left(Y,N(e_i,e_k) \right) &+\sum_{i,k=1}^{2n}g\left(N(X,e_k),e_i \right)g\left(N(Y,e_k),Je_i \right),
h(x, y)=J_p\Big((M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))\Big)-J_p\Big((M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))+(\varepsilon+\varepsilon_0)\,\widetilde\Gamma(y)\Big),
p\left(\frac{k}{q_1}, \frac{l}{q_2}\right) = c^{q_1, q_2}_{k, l}+ \sum_{i=0}^{n_1}\sum_{j = 0}^{n_2}a_{i,j}\delta_{k,l}^{q_1, q_2}(i,j) & \leq c^{q_1, q_2}_{k, l}+ \sum_{i=0}^{n_1}\sum_{j = 0}^{n_2}|a_{i,j}| \delta_{k,l}^{q_1, q_2}(i,j) & \leq c^{q_1, q_2}_{k, l}+ \gamma_1 \frac{(q_1-1)}{q_1^2}+ \gamma_2 \frac{(q_2-1)}{q_2^2},
\begin{aligned}\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}\frac{\mathcal{F}_{(\eta,\zeta)}(z,q)}{\|(z,q)\|_{1,\frac{1}{2}}}\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}\frac{\mathcal{F}_{(\eta,\zeta)}(z,q)}{|(z,q)|_{1,\frac{1}{2}}}. \end{aligned}
d_h(w) = \ |\{(j,i) \mid 2\leq j<i\leq n-1, \w(j)>w(i) \}| \qquad+ |\{(1,i) \mid 1<i<n, \a>w(i) \}| \qquad+ |\{(j,n) \mid 1<j<n, \w(j)>b \}| = \ |\{(j,i) \mid 2\leq j<i\leq n-1, \w(j)>w(i) \}| +(a-1) + (n-b).
{\bf \Phi}= \left[ \begin{array}{c}{\, \, \varphi^{+}( x ) \,}{\, \, \varphi^{-}( x ) \,}\end{array}\right] , \, \, \,{\bf \Phi}^{T}= \Big [ \, \varphi^{+}( x ) \, \, \, \varphi^{-}( x ) \, \Big ] .
\mathcal{Q}(\lambda) = \left\{(x_1,\ldots,x_4) \in \mathbb{R}_{\geq 0}^4 \mid \begin{array}{c}x_1\leq m_1, x_1+x_2+x_3 \leq m_1 + m_2, x_3\leq m_2, x_1+x_2+x_4\leq m_1+m_2.\end{array}\right\},
\textrm{\ensuremath{\ensuremath{\underset{\mathbf{r}}{\textrm{max}}}}}\sum_{l=1}^{M}U_{l}(\mathrm{SINR}_{l}^{LR}(\mathbf{r}_{l}),\mathrm{SINR}_{l}^{RL}(\mathbf{r}_{l})) \textrm{s.t. \textbf{C1}and \textbf{C2}}
\psi_1((b^{\mathbf{n}})^{dh}) = (b^{\mathbf{n}})^d|_0, a^{x_1}, \dots, a^{x_{p-1}})^{h}&= (((b^{\mathbf{n}})^{d}|_0)^{h|_0}, (a^{x_1})^{h|_1}, \dots, (a^{x_{p-1}})^{h|_{p-1}}).
D_1:=\frac{\widetilde{C}_1^p\widetilde{C}_3(1-\mathrm{e}^{-1/2})\varepsilon^p}{(p+1)2^{p+1}},\ \Q_1:=\frac{\widetilde{C}_0^q\widetilde{C}_2\,\varepsilon^q}{2}
f_{j}(x;n,N,h)=\frac{\cos\left(2A_{N}\left(\frac{n}{x}\right)b_{j,N}+\frac{\pi (2h-1)j}{2N}\right)-e^{-2A_{N}\left(\frac{n}{x}\right)a_{j,N}}\cos\left(\frac{\pi (2h-1)j}{2N}\right)}{\cosh\left(2A_{N}\left(\frac{n}{x}\right)a_{j,N}\right)-\cos\left(2A_{N}\left(\frac{n}{x}\right)b_{j,N}\right)}.
X_{r s}= \left( \Gamma_{\mu}C \right)_{r s}X^{\mu}+ \left( \Gamma_{[ \mu \nu ]}C \right)_{r s}\, X^{[ \mu \nu ]}+ \left( \Gamma_{[ \mu_{5}- \mu_{s}]}C \right)_{r s}\, X^{[ \mu_{1}\ldots \mu_{5}]}\, ,
\left<W(C)\right>=\left<\exp\left[\frac{ig}{4}\int\limits_{\Sigma}^{}d\sigma_{\mu\nu}\left(f_{\mu\nu}+h_{\mu\nu}\right)\right]\right>_{a_\mu, j_\mu}=\left<W(C)\right>_{a_\mu}\left<\exp\left(\frac{ig}{4}\int\limits_{\Sigma}^{}d\sigma_{\mu\nu}h_{\mu\nu}\right)\right>_{j_\mu}.
\left\langle \frac{\partial \Phi}{\partial \theta_i}, \frac{\partial \Phi}{\partial \theta_j}\right \rangle = \lambda_i\delta_{ij}, \left\langle \frac{\partial^2 \Phi}{\partial \theta_i \partial \theta_j}, \Phi\right \rangle = - \lambda_i \delta_{ij},
\left(\begin{array}{cc}I-\widehat{\lambda}_{k}A -A\mathbf{x}_k -\mathbf{x}_k^{T}0\end{array}\right)\left(\begin{array}{c}\Delta\mathbf{y}_{k}\delta_{k}\end{array}\right) =\left[\begin{array}{c}\widehat{\lambda}_{k}A\mathbf{x}_{k}-\mathbf{x}_{k}0\end{array}\right].
W_{1,1}=\{U_0\}, \Gamma_{U_0}= \{l_0\}, U_0=(Y_{1,1}^{\mathrm{unf}}, X_{1,1}, \ldots, X_{1,B}), p(x_{1,1}, \ldots, x_{1,B}| y_{1,1}^{\mathrm{unf}})=\prod_{b\in [1:B]}p_{X_1}(x_{1,b})
\epsilon_{1}^{\mu}= ( 0 , \, \epsilon^{i j}\frac{{p}_{j}}{| \vec{p}|}) \qquad \qquad \epsilon_{2}^{\mu}= ( \frac{| \vec{p}|}{M}, \, \frac{w_{p}}{M}\frac{{p}^{i}}{| \vec{p}|}) .
Ker({\mathcal{L}_\mu^y}) =Span(\partial_\mu Q_\mu-\frac{1}{2\mu}\Lambda Q_\mu),{\mathcal{L}_\mu^y}(\partial_i Q_\mu) =-\mu \partial_i Q_\mu,{\mathcal{L}_\mu^y}(\Lambda Q_\mu) =-2\mu\Lambda Q_\mu.
\ell'_2(\ell_2'(x,y),z)_{|_m}+\circlearrowleft(x,y,z)=0\, \Longrightarrow\,\ell'_2([x,y]_\mathfrak{g},z)_{|_m}+\ell'_2(\eta(x,y),z)_{|_m}+\circlearrowleft(x,y,z)=0, &\Longrightarrow \,\nu(z)\left(\eta(x,y)_{|_m}\right)-\eta([x,y]_\mathfrak{g},z)_{|_m}+\circlearrowleft(x,y,z)=0.
\|A\|_1 = \sum_{\substack{1 \leq i \leq n 1 \leq j \leq k}}\|a^{ij}\|_1 = \sum_{\substack{1 \leq i \leq n 1 \leq j \leq k}}\sum_{\gamma \in \Gamma}|a^{ij}_\gamma|.
\aligned\frac{4(n-1)}{n-2}\Delta \psi +R\psi= -\frac{n-1}{n}\tau^{2a}t^{2N}t_0^2\psi^{N-1}&+\bigg[|\sigma+LW|^2+\bigg(2\max\big\{\|\varphi_0\|_{L^\infty},\,2\big\}-\|\varphi\|_{L^\infty}\bigg)_+\bigg(\|\sigma+LW_0\|^2_{L^\infty}+k_0^2\bigg)\bigg]\psi^{-N-1}, -\frac{1}{2}L^*LW= \frac{n-1}{n}t^Nt_0\psi^Nd\tau^{a}.\endaligned
\exp\left(-Z^{*}HZ\right) =\sum_{K}\prod_{1\le m,n\le N}\frac{\left(-h_{m,n}\right)^{k_{m,n}}\left(\overline{z_{m}}\right)^{k_{m,n}}\left(z_{n}\right)^{k_{m,n}}}{k_{m,n}!}=\sum_{K}\prod_{j=1}^{N}\left(\overline{z_{j}}\right)^{r_{j}}z_{j}^{c_{j}}\left(\prod_{1\le m,n\le N}\frac{\left(-h_{m,n}\right)^{k_{m,n}}}{k_{m,n}!}\right).
\langle{\cal O}\rangle_{A}= \langle{\cal O}\rangle_{L}+ \int_{0}^{1}d \kappa \int D \phi \biggl ( \tilde{\delta}_{1}[ \phi ] + \kappa \tilde{\delta}_{2}[ \phi ] \biggr ) \Theta^{\prime}[ \phi ]{\frac{\delta{\cal O}}{\delta \phi}}e^{i S_{\mathrm{e f f}}^{M}}.
T_{\textnormal{inner}}= \left\lceil \left(\tfrac{2^{p+2}(5p+1)}{p!}\tfrac{LD^{p-1}}{\mu_{\textnormal{Minty}}}\right)^{\frac{2}{p}}+ \left(\tfrac{2^{p+5}}{p!}\tfrac{LD^{p-1}}{\mu_{\textnormal{Minty}}}\right)^{\frac{2}{p+1}}\right\rceil,
\psi(u,t)\,\stackrel{\mbox{\tiny def}}{=}\,\frac{\|\psi(\cdot,0)\|_{L^2}}{\sqrt{c_f^{1,2,0}}}\times\frac{1}{x(t)}\,f(\frac{|u|}{x(t)})\,e^{{\bf i}\frac{\dot{x}(t)|u|^2}{4x(t)}},
\Phi_0(\lambda): =\sigma_1^{-1}\sqrt{\lambda}\sin(\sigma_1\sqrt{\lambda})[\cos(\sigma_2\sqrt{\lambda})+a] &=(1+a)\lambda\prod_{n=1}^\infty\left(1-\frac{\lambda}{\alpha_n}\right)\prod_{n=0}^\infty\left(1-\frac{\lambda}{\beta_n}\right).
\gamma_{c,k}=\frac{a_c^2\lvert \mathbf{\hat{h}}_k^{\textrm{T}}\mathbf{p}_c\rvert^2}{\sum\limits_{i=1}^K a_i^2\lvert \mathbf{h}_k^{\textrm{T}}\mathbf{p}_i\rvert^2+\sigma_w^2}.
r=\frac{n(\sum_{i=1}^{n}(X_{i}Y_{i}-(\frac{\sum_{i=1}^{n}X_{i}}{n})(\frac{\sum_{i=1}^{n}Y_{i}}{n})))}{n[\sqrt{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}]\times\sqrt{\sum_{i=1}^{n}(Y_{i}-\overline{Y})^{2}}]}
\xi^{H_{1},H_{2}}_{R}(x,t)=\sum_{i=1}^{M}\sum_{k=1}^{N}\frac{1}{\tau}\int_{I_{i}}\int_{D}\phi_{k}(y)\xi^{H_{1},H_{2}}(y,r)dydr\phi_{k}(x)\chi_{I_{i}}(t),
p(x,y,z+\phi_1) = y\left( a_{010}- \frac{a_{000}a_{011}}{a_{001}}\right) - x \left( \frac{a_{000}a_{101}}{a_{001}}+ y\frac{a_{000}a_{111}}{a_{001}}\right) + z\left( a_{001}+ a_{011}y + a_{101}x + a_{111}xy\right).
\frac{(z)_{a-k}}{(1-a-z)_{a-k}}=\frac{\Gamma(z+a-k)\Gamma(1-a-z)}{\Gamma(z)\Gamma(1-z-k)}=(-1)^a\cdot\frac{\Gamma(1-a-z)\Gamma(z+k)}{\Gamma(1-a-z+k)\Gamma(z)}=\frac{(-1)^a(z)_{k}}{(1-a-z)_{k}},
\sum_{r,s}e^{-i (r-s) \Omega_G t}\frac{\partial}{\partial t}\Bigl(b_{(n)}^{(r, s)}\Bigr) \hat{a}^{\dagger r}_0 \hat{a}^s_0= i \sum_{r,s}e^{-i (r-s) \Omega_G t}b_{(n-1)}^{(r, s)}[\hat{a}^{\dagger r}_0 \hat{a}^s_0, \hat{H}'].
\displaystyle \widetilde{\cal H}_{\scriptscriptstyle K}= \frac{1}{( k + 1 ) !}\Biggl ( \lambda_{j}K_{j_{1}j_{2}\cdots j_{k + 1}}^{j}+ i \bar{c}_{j}( \partial_{d}K_{j_{1}j_{2}\cdots j_{k + 1}}^{j}c^{d}) \Biggr ) c^{j_{1}}\cdots c^{j_{k + 1}}
\begin{array}{ccc}\nabla_{X_{1}}X_{2}= \displaystyle\frac{c\nu b_{2}}{(\lambda_{2}-\lambda_{1})}X_{1}, \nabla_{X_{2}}X_{1}= \displaystyle\frac{-c\nu b_{1}}{(\lambda_{2}-\lambda_{1})}X_{2}. \end{array}
k\geq n\log_2\begin{pmatrix}{\frac nt}\end{pmatrix}\begin{pmatrix}1-{\frac{\log_2\log_2\begin{pmatrix}{\frac nt}\end{pmatrix}}{\log_2\begin{pmatrix}{\frac nt}\end{pmatrix}}}\end{pmatrix}^{-1}
\mathbb{P}_{f_0,T}(\hat{j}=j) \lesssim \sum_{l=j}^{j_{\max}}\big( (\log_2 n)n_{l}^{1-\gamma}+e^{-(c/2)\tau^2l}\big) &\lesssim (\log_2 n)^2n_{j_{\max}}^{1-\gamma}+ e^{-(c/2)\tau^2j}\lesssim n^{-2}+e^{-(c/2)\tau^2j},
P^{( N_{f}= 1 , \nu = 1 )}( \lambda , m ) = \frac{\lambda}{2}e^{- \lambda^{2}/ 4}\frac{1}{m I_{1}( m )}\left[ t I_{2}( \lambda ) I_{3}( t ) - \lambda I_{2}( t ) I_{3}( \lambda ) \right]_{t = \sqrt{\lambda^{2}+ m^{2}}}\: .
\|\mathcal{A}_{1}(y,p^1,p^2,f)(0)\|_{H_0^1(I)}=\int_I |(y_{xt}(0)-(a(y_x(0,t,0))y_x(0))_{xx}-f_x(0){1}_{\mathcal{O}}+ \frac{1}{\mu}p_x(0){1}_{\omega}|^2 dx &\leq C (\|(y,p^1,p^2,f)\|_Y^2 + \|(y,p^1,p^2,f)\|_Y^4) < +\infty.
\mathfrak{J}:= \Big\{\big\{\mathrm{Tr}\,h_{\gamma}, \^2E_S[f] \big\}, \^2E_S[g]\Big\}+\Big\{\big\{^2E_S[f],^2E_S[g] \big\}, \mathrm{Tr}\, h_{\gamma}\Big\}+ \Big\{\big\{^2E_S[g],\mathrm{Tr}\, h_{\gamma}\big\},^2E_S[f] \Big\}\ .
\int \frac{d}{da_0}\frac{d}{da_0'}G (a_0 t, x) G(a_0' s, x-y) d x_{|a_0 = a_0'}= \frac{d}{da_0}\frac{d}{da_0'}G (a_0 t + a_0' s, y)_{|a_0 = a_0'}&= t s \; \partial_t^2 G(a_0 (t+s),y).
\left(\frac{d^n}{dz^n}M_zf\right)(z) =\frac{d^n}{dz^n}(zf(z))=\binom{n}{0}zf^{(n)}(z)+\binom{n}{1}f^{(n-1)}(z) &=(M_zf^{(n)})(z)+(n T_{z}f^{(n)})(z) &=\left(T\frac{d^n}{dz^n}f\right)(z)\,.
H_0^-(x) =\left(\frac{b-x}{b-a}\right)^{\!\!N}\sum_{\ell=0}^{N-1}\binom{\ell+N-1}{\ell}\!\left(\frac{x-a}{b-a}\right)^{\!\!\ell}&=\frac{(b-x)^N}{(b-a)^{2N-1}}\sum_{\ell=0}^{N-1}\binom{\ell+N-1}{\ell}(x-a)^\ell(b-a)^{N-1-\ell}.
\widetilde{\mathsf{V}}\cdot \big( \mathsf{D}_{t}^{q}\cdot P \big) \big\vert_{U_{x,p}}= H_{0}^{q}\,P + H_{1}^{q}\, \big(\mathsf{D}_{t}^{1}\cdot P\big) + \dotsb + H_{q}^{q}\, \big(\mathsf{D}_{t}^{q}\cdot P\big){\scriptstyle(q=0,1,\dotsc,p)}.
\mathrm{Soc}\big(H^3_{*}(\mathbb{P}^r,\mathcal{I}_{\mathbb{F}}(-d+r-3))\big) = \Bbbk(d-r), h^2(\mathbb{P}^r, \mathcal{I}_Y(j)) = 0 \mbox{for all}j \geq d-r, \mbox{and}h^2(\mathbb{P}^r, \mathcal{I}_X(j)) = 0 \mbox{for all}j \geq d-r+1
\psi\delta_0|_{S_1}= -2, \psi\delta_{2,1}|_{S_1}= 1, \delta_0^2|_{S_1}= 8, \delta_0\delta_{1,1}|_{S_1}= -2, \delta_0\delta_{2,1}|_{S_1}= -2, \delta_{1,1}\delta_{2,1}|_{S_1}= 1.
u(x)=m_1 \log \frac{1}{|x|}+\frac{1}{2\pi}\int_{|y|<\varepsilon}\left( \log\frac{1}{|x-y|}\right) (-\Delta u(y))\,dy+h_1 (x) v(x)=m_2 \log \frac{1}{|x|}+\frac{1}{2\pi}\int_{|y|<\varepsilon}\left( \log\frac{1}{|x-y|}\right) (-\Delta v(y))\,dy+h_2 (x) \quad 0<|x|<\varepsilon
|^2\mathbb{X}_N^{(f)}| = \sum_{\pi(N): N=\{N_1^{c_1};N_2^{c_2};\ldots N_f^{c_f}\}}\prod_{j=1}^f\binom{|^2\mathbb{X}_{N_j}^{(1)}|+c_j-1}{c_j};f\ge 2.
\L_3'=\L_1'\L_2^p=\L_1'\frac{\L_1^{p+p^2}}{(1+p)^p}=\left(\frac{\L_1^{1+p+p^2}}{(1+p)^p(1+p+p^2)}\right)'\quad\Rightarrow\L_3=\frac{\L_1^{1+p+p^2}}{(1+p)^p(1+p+p^2)}.
[ T_{\alpha}, T_{\beta}^{\dagger}] = \bar{U}_{\alpha \beta}^{\gamma}T_{\gamma}+ T_{\gamma}^{\dagger}\bar{U}_{\beta \alpha}^{\dagger \gamma}+ \bar{U}_{\alpha \delta}^{\gamma}\bar{U}_{\beta \gamma}^{\dagger \delta}( - 1 )^{\varepsilon_{\gamma}\varepsilon_{\delta}}.
|J_a ={\epsilon_a}, \; J_b ={\epsilon_b}, \; D_a = \epsilon_a \delta, \; D_b = -\epsilon_b \delta \rangle \equiv{1 \over 4}|a_{\epsilon_a}b_{\epsilon_b}\rangle +{\delta \over 4 \sqrt{3}}|a_{\epsilon_a}b_{\epsilon_b}9 \rangle
X^i(\tau,\sigma)=x^i+\frac{p^i}{2\pi}\,\tau +\frac{i}{2\sqrt{\pi}}\sum_{n\ne 0}\left(\frac{1}{n}\alpha_n^i\,e^{-in(\tau-\sigma)}+\frac{1}{n}\tilde{\alpha}_n^i\,e^{-in(\tau+\sigma)}\right),
F_{N, M}\Xi\left(\eta_{1}, \cdots, \eta_{M}\right)\lesssim e^{CM^{2}}e^{-\frac{\beta NM}{2}\int V(\eta)d\mu_{eq}(\eta)}e^{-\frac{\beta}{2}N\sum_{j=1}^{M}\mathcal{J}^{V;\boldsymbol{B}}(\eta_{j})}+O(e^{-c N^2}).
U^{n}(t)\geq I_2(t)=\frac{1+\sqrt{1-4C\max(t^{\frac{1}{2}-\frac{1}{q}},t^{\frac{1}{p}})U_0}}{2\max(t^{\frac{1}{2}-\frac{1}{q}},t^{\frac{1}{p}})}\geq\frac{1}{2\max(t^{\frac{1}{2}-\frac{1}{q}},t^{\frac{1}{p}})}.
S_{c l}^{I}= \frac{a^{n}}{n !}\int d^{d + 1}x \int d^{d}\vec{y}_{1}. . . d^{d}\vec{y}_{n}[ \lambda_{n}^{\prime}\Psi_{1}+ \lambda_{n}( \Psi_{2}+ n \frac{a^{\prime}}{a}+ \ln \frac{( x^{d})^{n}}{\prod_{i = 1}^{n}[ ( x^{d})^{2}+ | \vec{x}- \vec{y_{i}}|^{2}]}) ]
M_{i}= \left( \begin{array}{c c}{{a_{r}}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c c}{{a_{r - 1}}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c c}{{a_{1}}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right)
P(z) = \left(\mathbb{I}- \frac{A_n^{(3)}(0)}{n^9 z}\right)^{-1}\left(\mathbb{I}- \frac{A_n^{(2)}(0)}{n^6 z}\right)^{-1}\left(\mathbb{I}- \frac{A_n^{(1)}(0)}{n^3 z}\right)^{-1}N(z),z \in \partial D(0,R).
h_{d , s}( 1 \G e V ) = F_{u}\left( \frac{\alpha_{1}( M_{Z})}{\alpha_{1}( M_{S})}\right)^{\frac{1 2}{1 6 8}}\left( \frac{\alpha_{1}( M_{S})}{\alpha_{1}( M_{S t r})}\right)^{\frac{6}{1 9 8}}h_{d , s}( M_{S t r})
\langle A^* \xi, v \rangle = \big\langle \xi,T \big( \tilde{\alpha}(t_0 + s,\langle \zeta,h_0 - \phi(t_0) + v \rangle) - \tilde{\alpha}(t_0 + s,\langle \zeta,h_0 - \phi(t_0) \rangle) \big) \big\rangle - \langle \xi,\alpha(h_0 + v) - \alpha(h_0) \rangle.
Y_{m+1}= Y_m + a(Y_m) h + \sum_{j=1}^K b^j(Y_m) \Delta \beta^j_m + \sum_{i,j =1}^K \Big(\frac{\partial b^{l,i}}{\partial x_k}(Y_m)\Big)_{1\leq l,k\leq d}b^{j}(Y_m) \int_{t_m}^{t_{m+1}}\int_{t_m}^s \,\mathrm{d}\beta_r^i \,\mathrm{d}\beta_s^j
[X_{j},[X^{j},X^{i}]] +{i\over 2}\{b^{+j},b^{i}{}_{j}- b^{+-}\delta^{i}{}_{j}\}=0,\qquad [X_{i},[X^{i},X^{-}]] -{i\over 4}\{b^{ij},b_{ij}\}=0.
{\cal S}( \Sigma ) \, \, = \, \, \displaystyle{\int}d^{3}x \, \displaystyle{\sum_{\Phi = A_{\mu}^{a}, \, c^{a}, \, \Psi_{_A}, \, \varphi_{i}}^{}}{\displaystyle{\frac{\delta \Sigma}{\delta \Phi^{*}}}}{\displaystyle{\frac{\delta \Sigma}{\delta \Phi}}}\, + b \, \Sigma = 0 \, , \quad \mathrm{w i t h ~}\b = \displaystyle{\int}d^{3}x \, b^{a}{{\displaystyle{\frac{\delta}{\delta \bar{c}^{a}}}}}\, .
S_{d 3}= - \int d^{4}x \,{\frac{1}{4 g_{\mathrm{Y M}}^{2}}}\, \mathrm{T r}\Bigl ( F_{a b}F^{a b}+{\frac{2}{( 2 \pi \alpha^{\prime})^{2}}}D_{a}X^{\mu}D^{a}X_{\mu}+{\frac{1}{( 2 \pi \alpha^{\prime})^{4}}}[ X^{\mu}, X^{\nu}]^{2}\Bigr ) + \cdots ,
\alpha_\gamma = \begin{cases}\frac{1}{4}+\frac{1+\sqrt{8\gamma^2+1}}{16\gamma^2}, \mbox{if}~\gamma \ge \frac{1+\sqrt{2}}{2}, \frac{2\gamma+1}{8\gamma^2}+\frac{\sqrt{8\gamma+3}}{16\gamma^2}, \mbox{if}~\frac{1}{2}<\gamma< \frac{1+\sqrt{2}}{2}.\end{cases}
\mathbb E_x(A^\mu_{\tau_D})^2=2\mathbb E_x\int_0^{\tau_D}R^D\mu(X_t)\,dA^\mu_t\le 2\|R^D\mu\|_\infty \mathbb E_xA^\mu_{\tau_D}\le 2\|R^D\mu\|^2_\infty,
m(L)\geq\pi^{-4}\cdot\frac{q^\frac{3m+l-k}{2}}{2^3\cdot 3\cdot 5}\cdot (1-2^{-2})^2\cdot (1-q^{-2})^2\cdot \zeta(2)^2=\frac{q^\frac{3m+l-k}{2}}{2^9\cdot 3\cdot 5}\cdot (1-q^{-2})^2
\|V^{-1}\bigl(|Vw|^2 Vw\bigr)\|_{\dot H^1}\lesssim \| |Vw|^2 Vw\|_{H^1}+ t^{\frac12}\bigl|(Vw)|_{x=0}\bigr|^3 & \lesssim \|Vw\|_{L^\infty}^2 \|Vw\|_{H^1}+ t^{-\frac14}\|w\|_{H^1}^3 & \lesssim \bigl[\|w\|_{L^\infty}+ t^{-\frac14}\|w\|_{H^1}\bigr]^2 \|w\|_{H^1}+t^{-\frac14}\|w\|_{H^1}^3,
C_{3}^{-1}\mathrm{e}^{-n^{1+\mu}}\bigl |a_pn^{1+\mu}\bigr |^{-{(l_p-1)}/{a_p}}\leq\tau_{E}(s_n) \leq C_{3}\mathrm{e}^{-n^{1+\mu}}\bigl |a_pn^{1+\mu}\bigr |^{{(l_p-1)}/{a_p}}.
\mathbb E \|(F \star \bar{\mu})_{\infty}\|_{L^q(S)}^p = \mathbb E \Bigl\|\sum_{n\geq 0}(F_n \star \bar{\mu})_{\infty}\Bigr\|_{L^q(S)}^p = \mathbb E \Bigl\|\sum_{n\geq 0}d_n\Bigr\|_{L^q(S)}^p\eqsim_{p, q}\|(d_n)\|_{s_{p,q}}^p \eqsim_p (\|(d_n)\|_{S^p_q}+ \|(d_n)\|_{D^p_{q,q}}+ \|(d_n)\|_{D^p_{p,q}})^p.
(\varphi E(\lambda))(T_{n_w}) = \varphi(E(\lambda)T_{n_w}) & = \varphi(E(\lambda)T_{n_s}T_{n_{sw}}) & = \varphi((T_{n_s}- c_s)E(n_s^{-1}(\lambda))T_{n_{sw}}) & = (\varphi(T_{n_s}- c_s)E(n_s^{-1}(\lambda)))(T_{n_{sw}}).
-\frac{I_{a'}I_{a'}I_{b'}I_{b'}}{\Im_{a'}\Im_{b'}}\beta \frac{1}{2}G_{\omega_{a'}}(0) + \frac{I_{a'}I_{b'}I_{a'}I_{b'}}{\Im_{a'}\Im_{b'}}\frac{1}{8}g(\dot{\omega}_{a'},\dot{\omega}_{b'}) -\frac{I_{a'}I_{b'}I_{b'}I_{a'}}{\Im_{a'}\Im_{b'}}\left( \frac{1}{8}g(\dot{\omega}_{a'},\dot{\omega}_{b'}) - \frac{1}{2}\beta G_{\omega_a}(0)\right) ,
\widetilde{\mathbf{M}}=\Big(M_{T_1,T_2}^{1,2,5-\alpha}(\omega),M_{T_1,T_2}^{3,N,2-\alpha}(\omega),\dots,M_{T_1,T_2}^{\frac{N-1}{2},\frac{N+1}{2}+3,2-\alpha}(\omega),M_{T_1,T_2}^{\frac{N+1}{2},\frac{N+1}{2}+2,5-\alpha}(\omega)\Big)^{\top}
\tilde{\Psi}_{y}^{*}= - \tilde{\Psi}_{x x}^{*}- 2 u \tilde{\Psi}^{*}+ ({\theta}_{y}-{\theta}_{x}^{2}+{\theta}_{x x}) \tilde{\Psi}^{*}+ 2{\theta}_{x}\tilde{\Psi}_{x}^{*},
\Lambda(k)= \ \lambda_{j+l}(k),\k\in \biggl[k_j + \frac{l}{2}, k_j+ \frac{l+1}{2}\biggr], \l = 0, 1, \dots, n-1, begin{align*}0.2cm]\Lambda(k) = \ \Lambda(2k_j-k),\k\in \biggl[k_j-\frac{n}{2}, k_j\biggr].
\wedge^1F_1\otimes S(-s_1) \xrightarrow{\begin{bmatrix}T_0 T_1 T_2\end{bmatrix}}\wedge^1F_0\otimes S(-s_1)\xrightarrow{\begin{bmatrix}0 T_2 -T_1 -T_2 0 T_0 T_1 -T_0 0 \end{bmatrix}}F_0^*\xrightarrow{\begin{bmatrix}T_0 T_1 T_2\end{bmatrix}}F_1^*.
\mbox{Re}(e^{i\theta}T) =\frac{e^{i\theta}T+e^{-i\theta}T^{\ast}}{2}=\frac{1}{2}\{(\cos \theta +i\sin \theta )T+(\cos \theta -i\sin \theta)T^{\ast}\}& =\left( \cos \theta \right) \frac{T+T^{\ast}}{2}-\left( \sin \theta\right) \frac{T-T^{\ast}}{2i}=\left( \cos \theta \right) H-\left( \sin\theta \right) K.
\mathrm{P}_{2,i}^o= \sum^{i}_{m=1}\frac{\gamma\left(M-m+1,\max\{\xi_m, \frac{\epsilon_{2,m}}{\rho \beta_m^2}\}\right)}{(M-m)!}\times \prod^{m-1}_{n=1}\left[1-\frac{\gamma\left(M-n+1,\max\left\{\xi_n, \frac{\epsilon_{2,n}}{\rho \beta_n^2}\right\}\right)}{(M-n)!}\right].
y = C_{+}e^{- \gamma_{+}( t - \tau )}+ C_{-}e^{- \gamma_{-}( t - \tau )}\, , \ \gamma_{\pm}= \frac{1 \pm K}{2}\, , \K = \sqrt{1 - 4 \varepsilon}\, , \C_{\pm}= \mp \frac{\tilde{w}+ \gamma_{\mp}\tilde{u}}{K}\, .
4\int_0^T\|\partial_x u\|^2dt+5a_0\int_0^T\|\partial_x^2 u\|^2dt &=\int_0^L xu_0^2dx-\int_0^Lxu^2(x,T)dx-(-1)^k\int_0^T\int_0^L2 x u\int_0^\infty g(s)\partial_x^{2k}\eta^t(x,s)dsdxdt +a_1\int_0^T\|u\|^2 dt & \leq L\|u_0\|^2+a_1\int_0^T\|u\|^2 dt-(-1)^k\int_0^T\int_0^L2 x u\int_0^\infty g(s)\partial_x^{2k}\eta^t(x,s)dsdxdt .
T_{+ p - \dot{p}}^{- \dot{q}}= \frac{\chi}{2}\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{- \dot{p}}^{- 2 i}, \T_{+ q + p}^{- \dot{q}}= \frac{\chi}{2}\left( \tilde{\gamma}_{\dot{q}q}^{i}\Omega_{+ p}^{- 2 i}+ q \leftrightarrow p \right) .
\det \left(M'_{\Xi, \{x_{i_1}\}}\right)|_{\Pi^*}= (m_1-l_1) x_{i_1}^{m_1+l_1-1}x_{i_2}^{m_2+l_2}, \det \left(M'_{\Xi, \{x_{i_2}\}}\right)|_{\Pi^*}= (m_2-l_2) x_{i_1}^{m_1+l_1}x_{i_2}^{m_2+l_2-1}.
T^{n}([\begin{matrix}x_{1}^{n-1}\\x_{2}^{n-1}\\ \vdots\\x_{n-1}^{n-1}\\ 0\\ \end{matrix}]+(y_{n}-\epsilon_{x}^{n-1})\vec{b}^{n})=[\begin{matrix}y_{1}\\y_{2}\\ \vdots\\y_{n-1}\\y_{n}\end{matrix}].
\phantom{o}b_1 + m_2 a_2 +b_2 =0 =0 \ \ \ & \phantom{o}a_1 + m_1 b_1 + (2+4m_2) a_2 + 4 b_2 =0 \ \ \ & \phantom{o}a_1+ m_1 b_1+a_2 = 0 \ \ \ .
\overline{\nabla}_{Z_{i}}\widehat{W}=\sum_{k=1}^{n}\epsilon_{k}\overline{g}( \overline{\nabla}_{Z_{i}}\widehat{W},Z_{k})Z_{k},\;\;\;\overline{\nabla}_{\widehat{W}}Z_{j}=\overline{g}(\overline{\nabla}_{\widehat{W}}Z_{j}, \widehat{W})\widehat{W},
u(x)=\lim_{k\uparrow+\infty}u_k(x)=u_0(x)+\lim_{k\uparrow+\infty}\int_\Omega f(y,u_{k-1}(y))\,G_\Omega(x,y)\; dy= =u_0(x)+\int_\Omega f(y,u(y))\,G_\Omega(x,y)\; dy.
\{(r-s)z^{12}z_1^{r-s}-2(z_1^{r-s}z_2+\cdots + z_1z_2^{r-s})\}= z_{12}\{(r-s)z_1^{r-s}+ \sum_{\ell=1}^{r-s-1}2(r-s-\ell)z_1^{r-s-\ell}z_2^\ell\}\, .
\hat{T}_{M N}^{\varphi}={\frac{1}{4 \epsilon}}( \frac{1}{2}\hat{g}_{M N}\partial_{P}\varphi \partial^{P}\varphi - \partial_{M}\varphi \partial_{N}\varphi ) + \frac{V}{\epsilon}e^{- \varphi}\hat{g}_{M N}\: .
\begin{array}{llll}x\partial =\theta, \partial x =\theta+1, x^2\partial^2 =\theta(\theta-1), \partial^2x^2 =(\theta+1)(\theta+2), x^3\partial^3 =\theta(\theta-1)(\theta-2),\quad \partial^3x^3 =(\theta+1)(\theta+2)(\theta+3),\end{array}
K_{\Delta_k}(u;m) = [\sum_{l=1}^2\mathrm{I}^{(l)}(T;\tilde j_m)](u), \,\,\, H_{\Delta_k}(u,v;m) = [\sum_{l=1}^4\mathrm{II}^{(l)}(T;\tilde j_m)] (u,v),
\int_{0}^{\infty}x\sin(\mu x){}_{2}F_{1}\Big(a,b;\frac{3}{2};-c^2 x^{2}\Big)dx=2^{-a-b+1}\pi c^{-a-b}\mu^{a+b-2}\frac{K_{a-b}({\mu}/{c})}{\Gamma(a)\Gamma(b)}.
\phi^{s_1, s_2}_{2[r_1-1]+r_2-1}(x) = x^{2(r_1-1)+r_2-1}- \underset{l=1}{\overset{r_1-1}{\sum}}\underset{l'=1}{\overset{r_2-1+l}{\sum}}B^{s_1, s_2}_{2(r_1-1)+r_2-1, 2(r_1-1-l)+ r_2-1+l'}\phi^{s_1, s_2}_{2[r_1-1-l]+r_2-1+l'}(x)
\Phi^{\infty}= i \frac{\eta}{\sqrt{2}}\vec{p}\cdot \vec{\sigma}, \qquad \vec{p}= \left( \begin{array}{c}{\sin \tau \cos N \phi}\\{\sin^{2}\tau \sin N \phi + \cos^{2}\tau}\\{\sin \tau \cos \tau ( \sin N \phi - 1 )}\\ \end{array}\right)
\|e^{it_2 \mathcal{K}_{\lambda}}u(t_2)-e^{it_2 \mathcal{K}_{\lambda}}u(t_1)\|_{H_{\lambda}^1}= \Big\|\int_{t_1}^{t_2}e^{is\mathcal{K}_{\lambda}}\mathcal{N}[u]\Big\|_{H_{\lambda}^1}\lesssim \|\mathcal{N}[u]\|_{\mathcal{W}_{\lambda}{'}([t_1 ,t_2])}\lesssim \|u\|^{2p-1}_{\mathcal{W}_{\lambda}([t_1,t_2])}\rightarrow0
\mathcal{X}_d^{-\top}\mathcal{P}_d^{}\mathcal{X}_d^{-1}= \mathcal{J}\mathcal{J}= \begin{bmatrix}I 0 0 -I \end{bmatrix}\mathcal{X}_d^\top \mathcal{X}_d^{}= \mathcal{D}_d,
\sum_{j=1}^{\frac{n-1}2}\binom nj =\frac 12\sum_{j=1}^{n-1}\binom nj = \frac 12(2^n-2)=2^{n-1}-1 \sum_{j=1}^{h-1}\binom{2h}j+ \frac 12 \binom{2h}h = \frac 12 2^{2h}-1=2^{n-1}-1
\pi(\{\theta_{li}\}_{i=1,l=1}^{N,K},\sigma^{2},\{\alpha_{l}\}_{l=1}^{K},\{\beta_{lb}\}_{l=1,b=1}^{K,P},\{\omega_{l}\}_{l=1}^{K}|\{y_{ij}\}_{i=1,j=1}^{N,M_{i}})
-tc\rho_n^2 + \sum_{i>k_n}\log (1-2t\lambda_i)^{-1/2}= -tc\rho_n^2 - \frac{1}{2}\sum_{i>k_n}\log (1-2t\lambda_i) \leq -tc\rho_n^2 + \frac{1}{2}\sum_{i>k_n}\frac{2t\lambda_i}{1-2t\lambda_i}.
\langle{T_{\k}^{j}}\rangle - \langle T_{_{\ominus}k}^{j}\rangle \sim{\frac{\pi M^{2}}{2}}\Phi^{2}\Big ( ( 2 c^{- 2}- c_{_\mathrm{S}}^{- 2}) u^{j}u_{k}- \gamma_{_k}^{j}\Big ) \ln \Big \{{\frac{w_{\!_{\odot}}}{w}}\Big \}\ .
\sum_{s_1\geq\dots\geq s_{r-1}\geq0}\frac{q^{s_1^2+\dots+s_{r-1}^2+s_{i}+\dots+s_{r-1}}}{(q)_{s_1-s_2}\dots(q)_{s_{r-2}-s_{r-1}}(q^2;q^2)_{s_{r-1}}}=\frac{(q^{2r},q^{i},q^{2r-i};q^{2r})_\infty}{(q)_\infty},
\rho_{134}(S_2)^{2(k+1)}= \rho_{134}(S_2)^{2k}\rho_{134}(S_2)^{2}& = (S_2(S_2S_1)^kS_2^*+S_1(S_1S_2)^kS_1^*)(S_2(S_2S_1)S_2^*+S_1(S_1S_2)S_1^*) & = S_2(S_2S_1)^{k+1}S_2^*+S_1(S_1S_2)^{k+1}S_1^*
i \frac{d}{d t}\langle \psi^{( 0 )}| \hat{x}^{m}\hat{p}^{n}| \psi^{( 0 )}\rangle = \langle \psi^{( 0 )}| \left[ \hat{x}^{m}\hat{p}^{n}, \frac{1}{2 \mu}\hat{p}^{2}+ V ( \hat{x}) \right] | \psi^{( 0 )}\rangle = 0 .
\langle \tilde{J}_\mu^{(\alpha)}(X), \s_{\mu}(X)\rangle = \ell!\alpha^\ell ((\ell+1)\alpha + (n-1))(\alpha + (n-2))\cdots(\alpha + 1) \alpha = \ell\cdot \ell!(\alpha + (n-\ell-2))\cdots(\alpha + 1)\alpha^{\ell+2}+ \ell!(\alpha+(n-\ell-1))\cdots(\alpha + 1)\alpha^{\ell+1},
\langle RT^{*n}e_k,e_\ell \rangle =\int_{\mathbb{T}}\lambda^{n}\langle e_k,E(\lambda) \rangle\overline{\langle e_\ell , E(\lambda) \rangle}\, d\mu(\lambda)=\Big\langle e_k,\int_{\mathbb{T}}\lambda^{n}\overline{\langle e_\ell , E(\lambda)\rangle}E(\lambda)\,d\mu(\lambda) \Big\rangle,
\mathrm{rk}(L'_{2,0}) =n-1,\ \mathrm{rk}(L'_{2,1})=2, \mathrm{rk}(K_{\ell,v,0}) =n-1,\ \mathrm{rk}(K_{\ell,v,1})=1(v= 2,p_{k'+1},\cdots,p_k), \mathrm{rk}(L'_{v,0}) =n,\ \mathrm{rk}(L'_{v,1})=1(v\neq 2,p_{k'},\cdots,p_k).