harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{S_{n_1^p}}\left( f \right) = \frac{1}{{2T}}{S_{w_1^i}}\left({f - f'}\right) + \frac{1}{{2T}}{S_{w_1^i}}\left({f + f'}\right) + \frac{1}{{2T}}{S_{w_1^q}}\left({f - f'}\right) + \frac{1}{{2T}}{S_{w_1^q}}\left({f + f'}\right)
\delta(t_{k+1}) \leq\delta(t_{0})+0.5\sum^{k}_{l=0}|\beta^{\top}e(t_{l})|\mathsf{T}_{\max}&\leq\delta(t_{0})+0.5\beta_{\max}\sum^{k}_{l=0}\|e(t_{l})\|\mathsf{T}_{\max}.
\pi_* \widetilde{Q}(H) =-2\sum_{k=0}^{\infty}\alpha_k (-2L)^k +3\sum_{k=0}^{\infty}\alpha_k (-3L)^k &= -2\frac{Q(H)-\alpha_1 H-\alpha_0}{H^2}\Big|_{H=-2L}+3\frac{Q(H)-\alpha_1 H-\alpha_0}{H^2}\Big|_{H=-3L}&= -2\frac{Q(H)}{H^2}\Big|_{H=-2L}+3\frac{Q(H)}{H^2}\Big|_{H=-3L}+\frac{Q(0)}{6L^2}.
\begin{cases}\partial_t u(x,t)+\partial_x^3 u(x,t)-a_0\partial_x^5 u(x,t)+u(x,t)\partial_x u(x,t)+a_1 \partial_x u(x,t) +(-1)^k \displaystyle \int_0^\infty f(s)\partial_{x}^{2k}u(x,t-s)ds=0, \ (x,t)\in I\times (0,\infty), u(0,t)=u(L,t)=\partial_x u(0,t)=\partial_x u(L,t)=\partial_x^2 u(L,t)=0, t>0, u(x,-t)=u_0(x,t), x\in I,\t{\geq}0.\end{cases}
\int_{B_\delta}\tau_i \tau_j |\tau|^{-(k + 2s)}d\tau = \delta_{ij}\int_{B_\delta}\tau_1^2 |\tau|^{-(k + 2s)}d\tau = \delta_{ij}k^{-1}\int_{B_\delta}|\tau|^{2 - k - 2s}d\tau = \delta_{ij}k^{-1}|\mathbb S^{k - 1}| \frac{\delta^{2 - 2s}}{2 - 2s}.
P^*_{u_i,\nu}(A_i^{c}\triangle \hat\Omega_i)\le \sum\limits_{j=1}^s P^*_{u_i,\nu}((A_{ij}\times C_{ij})\triangle \hat\Omega_{ij}^c)<\frac{\varepsilon}{m}, \ \ \mbox{where}\ \A_i^{c}=\bigcup\limits_{j=1}^{r}\Omega_{ij}^c.
M_{1}(\mathcal{C}(A(n, p)))=p^{n}(p^{2n}-1)(p^{2n}-p^{n}-1)^{2}, M_{2}(\mathcal{C}(A(n, p)))=\dfrac{p^{n}(p^{2n}-1)(p^{2n}-p^{n}-1)^{3}}{2},
\left(1-Q_h^*Q_h\right)g(\xi) =g(\xi)-\sum_{\mu\in\{0,\pm1\}^n}\hat{\varphi}(h\xi)\overline{\hat{\varphi}(h\xi+\mu)}g(\xi+h^{-1}\mu) &=(1-|\hat\varphi(h\xi)|^2)g(\xi)-\sum_{0\neq\mu\in\{0,\pm1\}^n}\hat{\varphi}(h\xi)\overline{\hat{\varphi}(h\xi+\mu)}g(\xi+h^{-1}\mu).
\langle \Omega | T \left( \sin ( g L a_{3}\left( x_{\perp}\right) / 2 ) \sin ( g L a_{3}\left( y_{\perp}\right) / 2 ) \right) | \Omega \rangle \quad \sim \quad \langle \Omega | T \left[ u \left( x_{\perp}\right) u \left( y_{\perp}\right) \right] | \Omega \rangle \ . \nonumber \,
|A_{i,j}|_2^2= \int_{[y_i, y_i+a_i]}dx\int_{[y_j, y_j+a_j]}|K^{GUE(n)}(x,y)|^2dy = \int_{[y_i, y_i+a_i]}dx\int_{[y_j, y_j+a_j]}O((\ln n)^2)dy=a_ia_jO((\ln n)^2) \leq (G_n(C_0)/S(I))^2O((\ln n)^2)=O\left(\frac{\ln n}{n^2}\right)O((\ln n)^2)=O\left(\frac{(\ln n)^3}{n^2}\right).
J_2= \int_1^{m/2}\int_1^{n/2}\int_{1}^{\infty}\frac{dy_1dy_2dr}{(y_1^{\alpha_1}+y_2^{\alpha_2}+(m-r)^{\alpha_3})((y_1+k)^{\alpha_1}+(n-y_2)^{\alpha_2}+r^{\alpha_3})}\leq C\int_1^{m/2}\int_1^{n/2}\int_{1}^{\infty}\frac{dy_1dy_2dr}{(y_2^{\alpha_2}+m^{\alpha_3})(y_1^{\alpha_1}+r^{\alpha_3})}\leq C \left(m^{\alpha_3}\right)^{1/\alpha_1+1/\alpha_2+1/\alpha_3-2}.
J^{\pm}(u)= \begin{pmatrix}1 0 e_{1}^{\pm}(u) \ddots \vdots \ddots * \ldots e_{n-2}^{\pm}(u) 1\end{pmatrix}\begin{pmatrix}k_{1}^{\pm}(u) 0 &\ddots & \ddots 0 k_{n-1}^{\pm}(u)\end{pmatrix}&\begin{pmatrix}1 f_{1}^{\pm}(u) \ldots * &\ddots & \ddots f_{n-2}^{\pm}(u) 0 1\end{pmatrix}
\left|\mathbf{E}\Delta_{\nu_m,n}^*(Z_{\nu_m}^*)\right| \le \mathbf{E}\left|\Delta_{\nu_m,n}^*(Z_{\nu_m}^*)\right|\le \mathbf{E}\delta_{\nu_m,n}^*(Z_{\nu_m}^*) &=\rho_nB_n{\mathbf{P}}(Z_{\nu_m}^*>0)+2{\mathbf{E}}Z_{\nu_m}^*\frac{B_m^2}{B_n^2}+{\mathbf{E}}\big[Z_{\nu_m}^*;Z_{\nu_m}^*>3B_m\big].
\begin{aligned}T :=\#\lbrace q\in\mathbb{P}\mid v_q(R)=1\rbrace, R=q_1\cdots q_T, s_{2,4}:=\#\lbrace\ell\in\mathbb{P}\mid v_\ell(R)=1,\ \ell\equiv\pm 1\,(\mathrm{mod}\,5)\rbrace s_4 :=\#\lbrace\ell\in\mathbb{P}\mid v_\ell(R)=1,\ \ell\equiv +1\,(\mathrm{mod}\,5)\rbrace.\end{aligned}
\frac{\partial}{\partial t}G-\dot{\Phi}^{ij}\nabla_i\nabla_jG= (\dot{G}^{ij}\ddot{\Phi}^{kl,mn}-\dot{\Phi}^{ij}\ddot{G}^{kl,mn})\nabla_ih_{kl}\nabla_jh_{mn}&+(1+\alpha)\Phi\dot{G}^{ij}h^k_ih_{kj}+\kappa(1-\alpha)\Phi\dot{G}^{ij}g_{ij}.
\mathbf{H}(t, \cdot) = \exp\big(-t/\tau\big) \mathbf{H}^{0}+\int_{0}^{t}\exp\big(-(t - s)/\tau\big) \mathbf{F}\big(\nabla \mathbf{u}(s, \cdot)\big) \mathrm{d}\mathbf{s}.
(\mathbb{I}_{|\mathcal{E}|^n}-\boldsymbol{c}_i)\mathbb{Q}^{-1}\mathcal{A}^{(PQ^{-1},Q)}=(\mathbb{I}_{|\mathcal{E}|^n}-\boldsymbol{c}_i)\mathbb{Q}^{-1}\mathcal{A}^{(PQ^{-1},QT_i)}
\int_0^t\int_\Omega \rho \mathbf{w}\varphi_tdx ds -\int_\Omega \rho_0\mathbf{w}_0\varphi|_{t=0}ds -\int_0^t\int_\Omega\rho u\mathbf{w}\varphi_xdx ds =-\int_0^t\int_\Omega \mu \mathbf{w}_x\varphi_xdxds
\varepsilon_{p}^{D}( L ) = \frac{L^{p - D}}{2^{D + 1}}\sum_{q = 0}^{p - 1}( - 1 )^{q + 1}C_{p}^{q}\left( \sqrt{\pi}\right)^{q - D}\Gamma \left( \frac{D - q}{2}\right) A \left( 1 , \cdots , 1 ; \frac{D - q}{2}\right)
\int \frac{\delta \hat{\Gamma}_{c l}}{\delta \hat{\varphi}_{0}}\; , \; \int \left\{A \frac{\delta}{\delta A}+ c \frac{\delta}{\delta c}\right\}\hat{\Gamma}_{c l}\; , \; m_{H}\partial_{m_{H}}\hat{\Gamma}_{c l}\; , \; e \partial_{e}\hat{\Gamma}_{c l}\; ,
C_1:=\{1,9,2,4\},C_2:=\{1,9,3, 10\},C_3:=\{1,9,5, 7\},C_4:=\{1,9,6,8\}, &C_5:=\{2,4,3, 10\}, C_6:=\{2,4,5, 7\},C_7:=\{2,4,6, 8\},C_8:=\{3, 10,5, 7\}, &C_9:=\{5, 7,6,8\},C_{10}:=\{3, 10,6, 8\}.
\left( \prod_{s=1}^{l}\prod_{p=1}^{i}(u - \hbar'(p - 1) + a_s) \right) B_i(u) \xi_{\Lambda}= \sum_{(s,p) \in R_{\Lambda,i}}(-\gamma_{i,p}^{(s)}) \left( \prod_{(s',p') \neq (s,p)}\dfrac{u - \nu_{i,p'}^{(s')}}{\nu_{i,p}^{(s)}- \nu_{i,p'}^{(s')}}\right) \xi_{\Lambda + \delta_{i,p}^{(s)}},
K_n(x,y)=\frac{1}{(2\pi i)^2}\int_{-\frac{1}{2}+i\mathbb R}dv\oint_{\gamma}du \prod_{j=0}^r \frac{\Gamma(v+\nu_j+1)}{\Gamma(u+\nu_j+1)}\, \frac{\Gamma(u-n+1)}{\Gamma(v-n+1)}\, \frac{x^uy^{-v-1}}{v - u},
\left(1+\frac{\tau}{2}\eta^n+\frac{\tau^2}{2}\vartheta^n\right)\tilde{\phi}^{n+1}_{jk}- \frac{\tau^2}{4}\left(\Delta_h\tilde{\phi}^{n+1}\right)_{jk}= \mathcal{H}^n_{jk},j,k=0,1,\ldots,M-1,
\sup \left\{\varepsilon \alpha^{-1}f_{\ell_0,\ell_0+1}: \Gamma_{\ell_0}\cap C_{\varepsilon \alpha^{-1/2}}(\Pi(y)) \right\}\leq c' \varepsilon^{4-2\nu'}\alpha^{-2}\leq c' \alpha^{\tfrac{2-\nu'}{1-\nu}- 2}= c' \alpha^{\tfrac{2\nu-\nu'}{1-\nu}}.
\lim_{\theta\in\Theta}\mathcal{A}(F_{\theta},\centerdot)=\mathcal{A}(\centerdot)\textrm{in}C(E,Y)\intertext{with the property: for each $\varphi\in E$,}\lim_{\theta\in\Theta}A(F_{\theta},\varphi)(x)=\mathcal{A}(\varphi)(x)\ \forall x\in X\textrm{and}\mathcal{A}(\varphi)=\mathcal{A}(T_g\varphi)\ \forall g\in G.
H_{n , n + 1}= ( \cos \gamma ) \,{\cal P}_{1}+ \frac{\sin 3 \gamma}{\sin 2 \gamma}\,{\cal P}_{2}+ \Bigl ( \frac{\sin 3 \gamma}{\sin 2 \gamma}+ ( \sin \gamma ) \frac{\cos 3 \gamma}{\sin 3 \gamma}\Bigr ) \,{\cal P}_{3}\, .
M = 3 D \int d^{2}k ( a^{+}( k ){\frac{1}{i}}{\frac{\partial}{\partial \theta}}a ( k ) + b^{+}( k ){\frac{1}{i}}{\frac{\partial}{\partial \theta}}b ( k ) ) +{\frac{m}{\mid = m \mid}}\int d^{2}k ( a^{+}( k ) a ( k ) - b^{+}( k ) b ( k ) ) ,
dc^m(t) = \Big( -mc^m(t) + m \langle v^m(t), \psi_x(\cdot- \varphi^m(t)) \rangle c^m(t) &{}- m \langle R^m(t, v^m(t)), \psi(\cdot- \varphi^m(t)) \rangle \Big) dt -\epsilon m \langle \psi(\cdot- \varphi^m(t)), dW_t \rangle, c^m(0) = -\epsilon m \langle \eta, \psi\rangle.
\Gamma^{( P V )}( A ) = \frac{1}{2}\left. \frac{d}{d ~ s}( \zeta ( s ) - \zeta^{P}( s ) ) \right|_{s = 0}= i \frac{\pi}{2}\eta ( 0 ) - i \sum_{k = 0}^{\infty}( - 1 )^{k}\frac{m^{( 2 k + 1 )}}{2 k + 1}\eta ( 2 k + 1 ) ,
f(A)X\bar{f}(B) -f(B)X\bar{f}(A) & =\int_{0}^{2\pi}\int_{0}^{2\pi}\Big[\left( e^{i\alpha}-A\right)^{-1}(e^{i\alpha}+A)X(e^{-i\beta}+B^{\ast})\left( e^{-i\beta}-B^{\ast}\right)^{-1}& \,\,\,\,-\left( e^{i\beta}-B\right)^{-1}(e^{i\beta}+B)X(e^{-i\alpha}+A^{\ast})\left( e^{-i\alpha}-A^{\ast}\right)^{-1}\Big]d\mu (\alpha)d\mu (\beta ).
dX(t) = \alpha(X(t), R(t), u(t,Z))dt + \beta(X(t), R(t), u(t,Z))dv(t) &+\int_{\mathbb{R}}\gamma(X(t), R(t), u(t,Z), \zeta) \tilde{N}(dt, d\zeta); t\in[0, T ], &X(0) F(\cdot), \mathbb{E}[\phi(X(0))] =\int_{\mathbb{R}}\phi(x)F(x)dx; \phi\in C_0(\mathbb{R}).
k_{j}=f(t_{n}+c_{j}h_{n},y_{n}+u_{j}+h_{n}\sum_{i=1}^{s-1}a_{j,i}k_{i})-f(t_{n},y_{n})-f_{x}(t_{n},y_{n})u_{j}-f_{t}(t_{n},y_{n})c_{j}h_{n},
Z[T,(e_i,\vec x_i)]= \frac{\int [dA_0(\vec x)]{\rm ~e}^{-S_{\rm eff}[A_0]}~\prod_j{\rm ~e}^{ie_jA_0(\vec x_j)/T}}{\int [dA_0(\vec x)]{\rm ~e}^{-S_{\rm eff}[A_0]}}
\mathrm{SINR}_{l}^{RL}(\mathbf{r}_{l})=\frac{\mathrm{SNR}_{l}^{RL}}{1+\sum_{k:\textrm{\ensuremath{kl\in\mathcal{E}}}}[(1-r_{kl})\mathrm{INR}_{kl}^{RL}+r_{kl}\mathrm{INR}_{kl}^{LL}]},
\overline{M}(w;x_1,..,x_n)=M^{(2)}(w;x_1,..,x_n) K_n(1/w) M^{(1)}(w;x_1,..,x_n), &T(w;x_1,..,x_n)=_0 ( \overline{M}(w;x_1,..,x_n) \widetilde{K}_0 (w) ).
q_{j}^{(i)}=\frac{1}{\pi}\int_{0}^{\infty}d\omega \,\sinh \pi\omega \,f^{(i)}[\tilde{D}_{i\omega}(k\xi ,k\xi_{j})]-\int_{0}^{\infty}d\omega \frac{I_{\omega}(k\xi_{1})I_{-\omega}(k \xi_{2})}{D_{\omega}(k\xi_{1},k\xi_{2})}\frac{F^{(i)}[D_{\omega}(k\xi ,k\xi_{j})]}{I_{\omega}(k\xi_{j})I_{-\omega}(k \xi_{j})},
|P_{d}| \prec \frac{1}{N^{d-m_0}}(1-c_0)^{K}\Big(\frac{C_0}{N}\Big)^{m_0-2}(N\Psi)^{m_2}\sum_{\mathcal{J}^{(\mathrm{off})}_d}\prod_{i=1}^{m_1}|G_{x_i y_i}| \prec (1-c_0)^{K}N^2 \leq N^{-8}.
\Pi^{p 1 2}= d X^{p 1 2}- \frac{1}{2}\theta_{\alpha}( \Gamma^{p 1 2})^{\alpha \beta}d \theta_{\beta}\longrightarrow \Pi^{p}= d X^{p}- \frac{1}{2}\theta_{\alpha}( \gamma^{p})^{\alpha \beta}d \theta_{\beta}\, ,
F_{m,n}(t_{1},t_{2}):=\frac{1}{(m+1)(n+1)}\frac{(1-\cos (m+1)t_{1})(1-\cos(n+1)t_{2})}{\left( 4\sin \frac{t_{1}}{2}\sin \frac{t_{2}}{2}\right)^{2}},
\vert z, \bar z \rangle = \exp\{z_1a^\dagger_1 + z_2a^\dagger_2 + z_3a^\dagger_3 \cdots +\bar z_1 b^\dagger_1 + \bar z_2 b^\dagger_2 + \bar z_3 b^\dagger_3 + \cdots \}\vert 0 \rangle,
J_{5}=J_{3}+4\mathfrak{S}_{0}J_{4}=\frac{2u_{1}}{w_{0}}-\frac{2K_{1}}{K_{0}^{2}}J_{6}=-\frac{2J_{4}}{K_{0}}=\frac{K_{1}}{K_{0}^{2}w_{0}}a =4\mathfrak{S}_{0}J_{2}\ ,\b=J_{5}-\frac{2J_{2}}{K_{0}},c=J_{6}d=4\mathfrak{S}_{0}e=-\frac{2}{K_{0}}
P_{A_s}x = \left\{z\in \mathbb{R}^n \mid \exists\; \mathcal{I}\in \mathcal{I}_x\; \mbox{such that}\; z(k)=\begin{cases}x(k) \mbox{if}k \in \mathcal{I}, 0 \mbox{else}\end{cases}\right\}.
\mathbf{p}\{B_i^-,B_j^-\}=\sum_{k=i}^{n}\sum_{s=1}^{k-i+1}\sum_{I\in \mathcal{I}_{ik}(s)}\sum_{l=j}^{n}\sum_{t=1}^{l-j+1}\sum_{J\in \mathcal{I}_{jl}(t)}E^{e_I+e_J}\{B_k^-,B_l^-\}H_{IJ}^-.
\langle T_{zz}(z,\bar{z})T_{zz}(0,0)\rangle = \frac{\pi}{3\cdot 16}\int_0^\infty d\mu^2 \, \tilde{c}(\mu^2)\int \frac{d^2 p}{(2\pi)^2}\frac{e^{\frac{i}{2}(p\bar{z}+\bar{p}z)}}{p\bar{p}+\mu^2}\, \bar{p}^4,
{\bf \Omega}^\prime (x_1, \ldots , x_N)=\prod_{i=1}^{N/4}f(x_{4i-1}-x_{4i})\bar f(x_{4i-3}-x_{4i-2})\, \, \delta \left (\sum_{i=1}^{N/4}(x_{4i-3}-x_{4i-1})\right ) \, .
J_k:= -\frac{1}{k\pi}(\sqrt{\mod{k\pi}}-i\sqrt{\mod{k\pi}}+O(1))e^{-\alpha_{1,k}-i(2k\pi + \alpha_{2,k})+O(\mod{k}^{-1})}e^{-\frac{1}{2}-\frac{1}{\sqrt{|k|}}+i\sqrt{\mod{k\pi}}+O(\mod{k}^{-\frac{1}{2}})},
C_{K 3}^{f u l l}\equiv e_{A}\, \cap \, e_{B}= \left( \begin{matrix}{{\begin{matrix}{{\sigma_{1}}}&{{0}}&{{0}}\\{{0}}&{{\sigma_{1}}}&{{0}}\\{{0}}&{{0}}&{{\sigma_{1}}}\\ \end{matrix}}}&{{{\bf 0}}}\\{{{\bf 0}}}&{{\begin{matrix}{{- C ( E_{8})}}&{{{\bf 0}}}\\{{{\bf 0}}}&{{- C ( E_{8})}}\\ \end{matrix}}}\\ \end{matrix}\right) \ ,
GB1(y_{i};a_{i},b_{i},p_{i},\overline{p}-p_{i}+q)=\frac{|a_{i}|y_{i}^{a_{i}p_{i}-1}(1-(\frac{y_{i}}{b_{i}})^{a_{i}})^{\overline{p}-p_{i}+q-1}}{b_{i}^{a_{i}p_{i}}B(p_{i},\overline{p}-p_{i}+q)}
\Xi_{1}(\kappa)=\lim_{N\to\infty}\mu_{1,N}^{11}(\kappa)=\lim_{N\to\infty}\left[\frac{\sinh\frac{\pi\kappa}{2}}{\sinh\frac{\pi N\kappa}{4}}\,e^{\frac{\pi N\kappa}{4}}-e^{\frac{\pi\kappa}{2}}\right]=-e^{-\frac{\pi|\kappa|}{2}}.
v = \pm{\frac{\partial}{\partial x}}\operatorname{l n}\bigg [{\frac{W_{\tilde{C}\tilde{C}}( \tilde{\theta}_{1 , 2}\rightarrow \tilde{\theta}_{1 , 2}+ \pi / 2 )}{W_{\tilde{C}\tilde{C}}}}\bigg ] ~ .
t^2=\frac{\sum_{i=1}^k \|w_i\|_{\lambda_i}^2}{\sum_{i=1}^d \mu_i |w_i|_4^4+\sum_{j\neq i}b|w_i w_j|^2_2}\leq \frac{\sum_{i=1}^k \|u_i\|_{\lambda_i}^2}{\sum_{i=1}^d \mu_i |u_i|_4^4+\sum_{j\neq i}b|u_i u_j|^2_2}=1
\varphi(n,\phi) = \frac{\phi}{\ln 2}\int^{\infty}_{0}\frac{1}{1+x\phi}\left(1-\sum^{n-1}_{p=0}{n-1 \choose p}\gamma_n\right. \left. \times (-1)^p \frac{\left[\frac{\gamma(N-M+1,x)}{(N-M)!}\right]^{K-n+p+1}}{K-n+p+1}\right) dx.
\frac{1}{T}\int_{0}^{T}|\zeta(s+i \tau, \alpha)| \,d \tau= \frac{1}{T}\int_{t}^{T+t}|\zeta(\sigma+i \tau, \alpha)| \,d \tau \leq \frac{1}{T}\int_{-(T+|t|)}^{T+|t|}|\zeta(\sigma+i \tau, \alpha)| \,d \tau
2\pi f(x)=\int\limits_{[0,2\pi)}\sum\limits_{k\in\mathbb{S}}a_{k}(t)\Psi_{k,t}(x)dt+\sum_{k\in\mathbb{Z}\backslash\mathbb{S}}\int\limits_{[0,2\pi)}a_{k}(t)\Psi_{k,t}(x)dt,
R_e(r)P[\eta(r)<S_{th}]=\underset{t>0}\int \underset{i \in \phi}\prod\frac{1}{1+\{e^t-1,T\}r^\alpha d_i^{-\alpha}}-\underset{i \in \phi}\prod\frac{1}{1+\{e^t-1,T,\hat{S}_{th}\}r^\alpha d_i^{-\alpha}}.
{\sum\limits_{k=1}^{q}\int\limits_\gamma\left|\langle \nabla g_{k}^{H,\beta}(x),dx\rangle \right | ~+~ \sum\limits_{k=1}^{q}\int\limits_{\widetilde{\gamma}}\left|\langle \nabla g_{k}^{H,\beta}(y),dy\rangle \right | \over \parallel L_n(\sigma) - L_n(\sigma') \parallel_1 + \parallel L_n(\tau) - L_n(\tau') \parallel_1}~<1
\sum_{k=0}^n \binom{p+k}{p}\binom{q+n-k}{q}\binom{k}{c}\binom{n-k}{d}H_{p+k}H_{q+n-k}={}\binom{n+p+q+1}{n-c-d}\binom{p+c}{c}\binom{q+d}{d}[H_{p+q+c+d+1}^{(2)}-H_{n+p+q+1}^{(2)}+(H_{n+p+q+1}-H_{p+q+c+d+1}+H_{p+c})(H_{n+p+q+1}-H_{p+q+c+d+1}+H_{q+d})]
W_i(M) = \frac{G_i}{c^{i+n(m-i)}}, \mathrm{if}\,\, s(m-i)<2i, &W_i(M) = \frac{G_i}{c^{i+n(m-i)}}\log(M), \,\, \mathrm{if}\,\, s(m-i)=2i, &W_i(M) = \frac{G_i}{c^{i+n(m-i)}}M^{s(m-i)-2i}\mathrm{if}\,\, s(m-i)> 2i,
\psi_i^n(t) = \psi_{0 i}+ \int_0^t\biggl(\psi_{i-1}^n(s)\sum_{j=1}^{i-1}jV_{i-1,j}\psi_j^n(s) - \psi_i^n(s)\sum_{j=1}^{i}jV_{i,j}\psi_j^n(s) -\psi_i^n(s)\sum_{j=i}^{n-1}V_{i,j}\psi_j^n(s)\biggr)ds.
I_{p_1,q_1}\otimes u_{h_1}\oplus\cdots\oplus I_{p_m,q_m}\otimes u_{h_m}\oplus\begin{bmatrix}0 I_{k_{m+1}}I_{k_{m+1}}0\end{bmatrix}\otimes 1\oplus\cdots\oplus\begin{bmatrix}0 I_{k_l}I_{k_l}0\end{bmatrix}\otimes 1
\begin{cases}{}^C\left(t^{1-2H}\frac{\partial}{\partial t}\right)^{2\alpha}u_{2\alpha}(x,t)= H^{2\alpha}\frac{\partial^2}{\partial x^2}u_{2\alpha}(x,t),\alpha \in \left(0,\frac{1}{2}\right], u_{2\alpha}\left(x,0\right)=\delta(x),\end{cases}
\inf_{u}J(u)=P_{10}^{\ast}\left[ \left( x_{0}+\frac{a_{1}}{\gamma -{\mathcal{A}}_{0}}\right)^{2}-\frac{a_{1}^{2}{\mathcal{A}}_{0}}{2\gamma(\gamma -{\mathcal{A}}_{0})^{2}}\right] >0.
f_3 = \chi_{>3R}[ \chi_{>2R}, (\partial_k T_{g_{ext}^{kl}}\partial_l + T_{b_{ext}}\cdot\nabla)] (1-\chi_{> 3R}) \tilde w + [ \chi_{>2R}, (\partial_k T_{g_{ext}^{kl}}\partial_l + T_{b_{ext}}\cdot\nabla)] \chi_{> 3R}\tilde w .
\{T^{(0)}_{\alpha}(a),T^{(0)}_{\beta}(a) \}=T^{(0)}_{\gamma}(a)U^{\gamma}_{\alpha\beta}(a), \quad\{H_0(a),T^{(0)}_{\alpha}(a) \}=T^{(0)}_{\gamma}(a)V^{\gamma}_{\alpha}(a)
u\in L^{p,\infty}\cap L^{q,p}\hookrightarrow L^{\sigma,\tau}\ \Rightarrow\f\in L^{\frac{\sigma}{2},\frac{\tau}{2}}\sigma(\theta,p)=\frac{pN}{N+2\theta-2},\tau(\theta,p)=\frac{p}{1-\theta}.
|u_{x_ix_j}| =\left|\frac{1}{{\rm Vol}(B(x,r))}\int_{\partial B(x,r)}u_{z_i}(z)(\nu(z), e_j)dz\right| \le \frac{C(d)}{{\rm Vol}(B(x,r)){\rm dist}(x,\partial K)}\int_{\partial B(x,r)}u(z)|(\nu(z), e_j)|dz\le C(d)^2\frac{u(x)}{{\rm dist}(x,\partial K)^2}
-\frac{\partial L_{d}G\left(p\|q\right)}{\partial q_{j}}=\left(1-\alpha\right)\left\lbrace 1-\left[\left(\frac{a}{a-b}\right)\left(\frac{T_{j}}{p_{j}}\right)^{a-1}-\left(\frac{b}{a-b}\right)\left(\frac{T_{j}}{p_{j}}\right)^{b-1}\right]\right\rbrace
-\Box_x S^{(l)}(x,y) = -\frac{1}{2}\:\frac{\partial}{\partial x^k}\left( (y-x)^k \:S^{(l-1)}(x,y) \right) = 2 \:S^{(l-1)}(x,y) + \frac{1}{4}\:(y-x)^2 \:S^{(l-2)}(x,y) \:.
F_1(a^i,u^i,B^i,\nabla\Pi^i)= (a^2-S_ma^2)(\Delta\delta u-\nabla\delta\Pi)-\delta u\cdot\nabla u^1+\delta a(\Delta u^1-\nabla\Pi^1) &+(1+a^2)(\delta B\cdot\nabla B^2+ B^1\cdot\nabla \delta B)+\delta a( B^1\cdot\nabla B^1),
\int_0^{\pi}|\cos\left(x_j t \right)|^{2m}dt =\frac{1}{x_j}\int_0^{x_j\pi}|\cos s|^{2m}ds=\int_0^{\pi}|\cos s|^{2m}ds &=2\int_0^{\pi/2}|\cos s|^{2m}ds=\frac{(2m-1)!!}{(2m)!!}\pi
\psi_0(x)=\log(1+\alpha x),\psi_i(x)=\log \frac{x}{\lambda},\phi_{i0}(x)=\log\frac{1+\lambda x}{1+\alpha x}, \intertext{and}\psi_k(x)=\log(1+(\lambda+\alpha)x),\phi_{k0}(x)=\log\frac{x}{\lambda+\alpha(\lambda+\alpha)x}.
\frac{\partial E[S^*(\boldsymbol{\mu};0)]}{\partial \mu_k}= \lim_{\theta\to 0}\frac{E[f(\mathbf{W}^*(\boldsymbol{\mu}+\theta\cdot \mathbf{e}^k; 0), \mathbf{J}(0),\boldsymbol{\mu}+\theta\cdot \mathbf{e}^k)]-E[f(\mathbf{W}^*(\boldsymbol{\mu}; 0), \mathbf{J}(0), \boldsymbol{\mu})]}{\theta}.
r_{q+1,q+1}=r_{1,q+1}=1, r_{2,q+1}=2, r_{3,q+1}=3, r_{0,q+1}=0 r_{q+1,1}=r_{2,1}=\frac{q^2-q}{2}, r_{3,1}=r_{0,1}=\frac{q^2-cq}{2}, r_{1,1}=\frac{q^2+cq}{2}.
\mathbb P\begin{bmatrix}\begin{Vmatrix}{\frac 1{2k}}\sum_{s\in{\hat S}}{\tilde H}_{t,M}(xs)\end{Vmatrix}_{L^2}\leq{\frac 12}||{\tilde H}_{t,M}||_{L^2}\end{bmatrix}\geq 1-{\frac{2^{\frac{n+4}2}}{n!\eta}}\exp\begin{pmatrix}{\frac{-k}{16\ln 2}}\end{pmatrix}
\left[B(0)+{\frac d N}B_1 \right]^{2N}=~ I_{2N}+{\frac{2d}N}\widehat P + \sum_{j=0}^{2N-1}\zeta_{j,N}B(0)^{2j}B_2(0) + \sum_{j=1}^{2N-1}\eta_{j,N}B(0)^{2j}\,,
|\langle S_{f}^{\nu,\alpha}(g),h\rangle_{A_{\alpha}^{2}}| =|\langle h_{f}^{\nu}(g),h\rangle_{A_{\nu}^{2}}|\le\| h_{f}^{\nu}(g)\|_{A_{\nu}^{q}}\|h\|_{A_{\nu}^{q'}}&\lesssim \|h_{f}^{\nu}\|_{A_{\omega}^{p}\to A_{\nu}^{q}}\|g\|_{A_{\omega}^{p}}\|h\|_{A_{\widetilde{\nu}}^{q'}},g,h\in H^\infty.
\beta_{21}\left(\left|\eta\right|\right) =\beta_{20}\left(-\left|\eta\right|\right),\quad\beta_{22}\left(\left|\eta\right|\right)=\beta_{22}\left(-\left|\eta\right|\right), \beta_{33}\left(\left|\eta\right|\right) =\beta_{33}\left(-\left|\eta\right|\right),\quad\beta_{44}\left(\left|\eta\right|\right)=\beta_{44}\left(-\left|\eta\right|\right).
\times \prod_{a=1}^N \Big[ c^{(a)}\Big]^{n_a + m_a}\; \;\prod_{b=1}^N \left[\left( \frac{2e^{-\gamma}}{M^{(1)}}\right)^{\frac{\pi}{\pi+gN}\frac{1}{N}}\prod_{I=2}^N\Big( \frac{2e^{-\gamma}}{M^{(I)}}\Big)^{(U_{Ib})^2}\right]^{n_b + m_b}\; .
2|E(G)| = \sum_{f\in{F(G)}}\ell(f) = \sum_{f \in X}\ell(f) + \sum_{f \in Y}\ell(f) + \sum_{f \in F(G)\setminus{X\cup Y}}\ell(f) \geq 3|X| + 4|Y| + 5(|F(G)| - |X| - |Y|) = 5|F(G)| - 2|X| - |Y|.
\begin{array}{c}\mathring{\mathcal S_y}\subset \mathcal I_{r-1}\times \mathring{H^0(\mathcal O_{\mathbf P^n}(h_r))}\mathring{\mathcal S_y}=\{(c, g_r): c\in I_{r-1}, g_r(c(t^r_1))=\cdots=g_r (c(t^r_{h_rd+1}))=0\}\end{array}
||J(u)||_{X^*}= \sup_{||\phi||_X\leq 1}\big|<J(u),\phi>\big| &\leq\sup_{||\phi||_{X}\leq 1}\big|\int_{\Omega}\mathcal{A}(x,\nabla u)\cdot\nabla\phi\,dx\big| &\leq\sup_{||\phi||_X\leq 1}\big|\int_{\Omega}\big(w^\frac{1}{p'}|\nabla u|^{p-1}\big)\big(w^\frac{1}{p}|\nabla\phi|\big)\,dx\big| &\leq\;||u||_{X}^{p-1}.
\begin{array}{ll}\sum\limits_{\alpha,\beta \in \mathbb{F}_{p^m}}S(\alpha,\beta)^3=8p^{3m}+2p^{\frac{3m}{2}}(4( n_{1,1,0}-n_{-1,-1,0})+(\sqrt{q^*}+1)^3(n_{1,1,1}-n_{-1,-1,1})+ &(\sqrt{q^*}-1)^3(n_{1,-1,1}-n_{-1,1,1})+(p^d+1)^3(n_{1,1,2}-n_{-1,-1,2})+(p^d-1)^3(n_{1,-1,2}-n_{-1,1,2})) , \end{array}
\| X -\tau I\|_{\infty}-\left\| \frac{|X-\tau I| +|X^*-\overline{\tau}I|}{2}\right\|_{\infty}=\left\|\begin{pmatrix}|x| 0 0 |y| \end{pmatrix}\right\|_{\infty}-\frac{1}{2}\left\|\begin{pmatrix}|x| +|y| 0 0 |x|+|y| \end{pmatrix}\right\|_{\infty}&= \left| |x|-|y| \right| &=\varepsilon(X)