harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\varphi_h^{\mathcal{T}+ \mathcal{B}(y^*)}\left( \begin{bmatrix}q_0 p_0 \end{bmatrix}\right)= \begin{bmatrix}q_0 + \frac{h}{m}\phi_1 \big(h\Omega(q^*)\big) p_0 \exp(h \Omega(q^*)) p_0 \end{bmatrix},
\limsup_{N\to \infty}d^{N,k}\left( (8\pi^2)^{-1}N^2\log k+uN^2 \right)<1, &\liminf_{N\to \infty}d^{N,k}\left( (8\pi^2)^{-1}N^2\log k+uN^2 \right) >0.
\int_{\partial\Omega}K(x,y)\,\varphi_\gamma(y)\,d\sigma(y) = \int_{\partial\Omega}K(x,y)\,(\varphi_\gamma(y)-P_\gamma(y,z))\,d\sigma(y) &\qquad+\int_{\partial\Omega}K(x,y)\,\varphi_\gamma(z)\,d\sigma(y)
F^{{\mu}{\nu}{\sigma}}= - G^{{\mu}{\nu}{\sigma}}+{\lambda}^{{\rho}[{\mu}}{{\cal G}_{\rho}}^{{\nu}{\sigma}]}+{\frac{1}{3 !}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\lambda}_{{\theta}[{\rho}}{{\cal G}^{\theta}}_{{\eta}{\delta}]}.
\small\begin{aligned}P(A|A) = \frac{\beta+\alpha(1- \alpha -\beta)^t}{\alpha+\beta},~ P(B|A) = \frac{\alpha-\alpha(1- \alpha -\beta)^t}{\alpha+\beta}, P(A|B) = \frac{\beta- \beta(1- \alpha -\beta)^t}{\alpha+\beta},~ P(B|B) = \frac{\alpha+\beta(1- \alpha -\beta)^t}{\alpha+\beta}. \end{aligned}
K_n := \left(\frac14+C\Delta t\Bigg[ |\boldsymbol{u}(t_n)|_{W^{1,\infty}}+ h^{2z}|\boldsymbol{u}(t_n)|_{W^{1,\infty}}^2 + \frac{h^2}{\gamma}| \boldsymbol{u}(t_n)|^2_{W^{1,\infty}}\right. &\qquad\qquad\qquad\left. + \frac{h^{2z-2}}{\gamma}\|\boldsymbol{u}(t_n)\|_{\infty}^2 + h^{2z-2}\|\widetilde{\boldsymbol{u}}_h^n\|_{\infty}^2 \Bigg]\right)<1
\phi(g_1,\ldots,g_4) = \sum_{N_1\ldots N_4}\sum_\Lambda \left(\phi^{N_1\ldots N_4}_{\alpha_1\ldots\alpha_4}\D^{(N_1)}_{\alpha_1\beta_1}(g_1) \ldots D^{(N_4)}_{\alpha_4\beta_4}(g_4)\C^{N_1\ldots N_4\,\Lambda}_{\beta_1\ldots\beta_4}\right) \ \left({C}^{N_1\ldots N_4\,\Lambda}_{\gamma_1\ldots\gamma_4}w_{\gamma_1}\ldots w_{\gamma_4}\right).
g_M=\sum_{k\le M}\ln \int \prod_{a=0}^u dp_X(y^a)\exp\Big(\tau_{\pi_k}\sum_{a<b}^{0,u}y^a y^b\Big)=\sum_{k\le M}\ln \int \prod_{a=0}^u dp_X(y^a)\exp\Big(\tau_{k}\sum_{a<b}^{0,u}y^a y^b\Big)
\begin{aligned}\dot x(t) =Ax(t)+B_1w(t)+\chi(t)B_2K[C_2x(t-\tau(t))+D_2v(t-\tau(t))] &+(1-\chi(t))B_2K[C_2x(t-\bar{\eta}(t))+D_2v(t-\bar{\eta}(t))+e(t)], z(t) =C_1x(t)+\chi(t)D_1K[C_2x(t-\tau(t))+D_2v(t-\tau(t))] &+(1-\chi(t))D_1K[C_2x(t-\bar{\eta}(t))+D_2v(t-\bar{\eta}(t))+e(t)],\end{aligned}
\sum_{n=0}^{\infty}\frac{(\alpha)_n(\beta)_n}{(\gamma)_n(q)_n}z^n=\frac{(\alpha z)_{\infty}}{(z)_{\infty}}\sum_{n=0}^{\infty}\frac{(\alpha)_n(\gamma/\beta)_n(-\beta z)^nq^{n(n-1)/2}}{(\gamma)_n(\alpha z)_n(q)_n}
\begin{cases}\Psi(x_{1})=\psi(x_{2})=\lambda x_{2}\Psi(s_{1})=\psi(s_{2})=\mu s_{2}+x_{2}^{n+e_{2}}a(x_{2}) \Psi(y_{1})=\psi(y_{2})=\frac{\mu^{d}}{\lambda^{n}}y_{2}+W \Psi(z_{1})=\psi(z_{2})=\frac{\mu^{dm}}{\lambda^{nm+e_{2}}}z_{2}+\frac{(\frac{\mu^{d}}{\lambda^{n}}y_{2}+W)^{m}-\frac{\mu^{dm}}{\lambda^{nm}}y_{2}^{m}+x_{2}^{n+e_{2}}a(x_{2})}{\lambda^{e_{2}}x_{2}^{e_{2}}}\end{cases}
C_{i_1,\ldots,i_k}(1) = \sum_{r=1}^k C_{i_r}(1)C_{i_1,\ldots,\widehat{i_r},\ldots,i_k}+ \sum_{1\leq r<s \leq k}\widetilde{C}_{i_r,i_s}(1)C_{i_1,\ldots,\widehat{i_r},\ldots,\widehat{i_s},\ldots,i_k}.
\Bigl( D^s(f_j \tilde g_j ) \Bigr) (x) = \frac 1{(2\pi)^{2d}}\int 2^{js}\cdot |2^{-j}(\xi+\eta)|^s \chi(2^{-j}\xi) \chi(2^{-j}\eta) \widehat{f_j}(\xi) \widehat{\tilde g_j}(\eta) e^{i(\xi+\eta) \cdot x}d\xi d\eta,
\frac{E_{s}(\alpha_{N})}{N^{1+s}}= \sum_{k=1}^{t-1}\left(\frac{2^{n_{k}+1}}{N}\right)^{1+s}\left(\sum_{j=k+1}^{t}2^{n_{j}-n_{k}}\right)\frac{\mathcal{L}_{s}(2^{n_{k}+1})}{(2^{n_{k}+1})^{1+s}}+ \sum_{k=1}^{t}\left(\frac{2^{n_{k}}}{N}\right)^{1+s}\left(1-\sum_{j=k+1}^{t}2^{n_{j}-n_{k}+1}\right)\frac{\mathcal{L}_{s}(2^{n_{k}})}{(2^{n_{k}})^{1+s}},
\begin{cases}c_{\alpha + n +1}- c_{n+1}= b_{\alpha + n}- b_{\alpha + n +1}d_n = c_{\alpha + 1}+ (b_{\alpha + 1}- b_\alpha) = d b_{\alpha + n}= b_\alpha + (d_1 - c_{\alpha+1}) + \sum_{j=2}^n (c_j - c_{\alpha + j}) b = b_\alpha + d - c_{\alpha + 1}+ \sum_{j=2}^\infty (c_j - c_{\alpha+j})\end{cases}\ ,
\vartheta((\mathrm{W},\mathrm{X})+z) =\sum_{\alpha\in X_\xi}\bigg(\sum_\beta a_{\alpha\beta}b(z,\beta)\mathrm{W}^{\beta}\bigg)b(z,\alpha)\mathrm{X}^\alpha &=\sum_{\alpha\in X_\xi}\sigma_\alpha(\vartheta)(\mathrm{W}+z)b(z,\alpha)\mathrm{X}^\alpha.
\langle f_{j}|O|\psi\rangle=\sum_{i=1}^{n}c_{i}\langle f_{j}|O|e_{i}\rangle=\sum_{i=1}^{n}\langle f_{j}|e_{i}\rangle\langle f_{i}|h\rangle=\langle f_{j}|h\rangle,\forall j
b^{( n )}( x^{2}) = \int_{- \infty}^{x^{2}}d \lambda \, \int_{- \infty}^{\lambda}d \lambda^{\prime}\, H^{\prime \prime}( \lambda^{\prime}) \, b^{( n - 1 )}( \lambda^{\prime}) \quad \mathrm{f o r ~ n ~ = ~ 1 , ~ 2 , ~ 3 , ~ \dots ~ .}
\Delta S_{G}= - f_{g}^{2}\int d^{3}x a \sum_{n = 1}^{N}\frac{1}{4}\epsilon^{i j k}\epsilon_{A B C D}E_{i}^{A}( x , n ) R_{k 0}^{C D}E_{j}^{B}( x , n + 1 ) + \cdots .
(2 \cos(\tfrac{\pi}{3}+ x))^{-\ell}\leq (2 \cos(\tfrac{\pi}{3}+ \tfrac{\pi}{3\ell}))^{-\ell}= \Big[(2 \cos(\tfrac{\pi}{3}+ \tfrac{\pi}{2(3\ell/2)}))^{-3\ell/2}\Big]^{2/3}\leq (0.0658\dots)^{2/3}= 0.16\dots.
g^{S_0 \oplus S_1}_{(1, i)}= 1 + \beta \sum_{j \in S_1}p_{ij}g^{S_0 \oplus S_1}_{(1, j)}+ \beta \sum_{j \in N \setminus S_1}p_{ij}g^{S_0 \oplus S_1}_{(1, j)}= 1 + \beta \sum_{j \in S_1}p_{ij}g^{S_0 \oplus S_1}_{(1, j)},
\begin{aligned}\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}\bar{q}_{\beta}(0,t) =-\lambda \bar{q}_{\beta}(0,t), \frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}\bar{q}_{\beta}(n,t) =-\lambda \bar{q}_{\beta}(n,t)+\frac{\lambda}{(1-\rho)^{-r}-1}\sum_{j=1}^{n}\binom{r+j-1}{j}\rho^{j}\bar{q}_{\beta}(n-j,t),\n\ge 1,\end{aligned}
+ \frac{1}{2 N^{3}}\left[ \frac{2 \sin^{3}[ \widehat{\theta}_{t}] + \sin [ 2 \widehat{\theta}_{t}] ( 2 \cos [ \widehat{\theta}_{t}] - ( \cos [ \theta_{i}] + \cos [ \theta_{j}] ) )}{( \cos [ \widehat{\theta}_{t}] - \cos [ \theta_{i}] )^{2}( \cos [ \widehat{\theta}_{t}] - \cos [ \theta_{j}] )^{2}}\right]
I(a,c,t,s)= \sum_{n_{1}=0}^{\infty}\ldots \sum_{n_{s}=0}^{\infty}\frac{a^{n_{1}+\ldots + n_{s}}}{n_{1}! \ldots n_{s}!}\exp \left( \nu R^{c,p}(t,s,\mathbf{n}^{(s)}) - a(c-1) \right).
\sum_{r=0}^{m-1}\chi(r) C_{m,r}(n) =\frac{1}{m}\sum_{r=0}^{m-1}\chi(r)\sum_{j=1}^{m-1}\csc^{2n}\left(\frac{j\pi}{m}\right) e^{\frac{2\pi i r}{m}j}&= \frac{\tau(\chi)}{m}\sum_{j=0}^{m-1}\overline{\chi(j)}\csc^{2n}\left(\frac{j\pi}{m}\right) &= \frac{\tau(\chi)}{m}\overline{L_{X_m}(n,\chi)}.
M_1= \frac{Z_2 Z_3}{(Z_2-Z_1)(Z_3-Z_1)}, M_2= \frac{Z_1 Z_3}{(Z_1-Z_2)(Z_3-Z_2)}, M_3= \frac{Z_1 Z_2}{(Z_1-Z_3)(Z_2-Z_3)}.
\int_{\gamma_1 \cdots \gamma_r}\omega_1 \cdots \omega_k = \sum_{\substack{g \in \Delta(k,r) g = (n_1,\ldots,n_r)}}\Big(\int_{\gamma_1}\omega_1 \cdots \omega_{n_1}\Big) \Big(\int_{\gamma_2}\omega_{n_1+1}\cdots \omega_{n_1+n_2}\Big) \cdots \Big(\int_{\gamma_r}\omega_{n_1 + \cdots + n_{r-1}+ 1}\cdots \omega_k\Big).
\binom{m-z}{u_1,u_2,\ldots,u_z}\binom{n-m}{m-z}=\frac{1}{m!}\frac{\prod_{j=0}^{z+\rho-1}(m-j)\prod_{j=0}^{m-z-1}(n-m-j)}{\prod_{j=1}^z u_j!}.
c_1 := \max_{h\in(0,1]}\,\,\,\, 2 h - h^2 \frac{-\kappa}{1-\kappa}= 2 - \frac{-\kappa}{1-\kappa}c_2 := \max_{h\in[1,\bar{h}(\kappa)]}2 h - h^3 \frac{-\kappa}{2-(1+\kappa)h}\geq 2 - \frac{\kappa}{2-(1+\kappa)}= c_1.
|\begin{matrix}0&5&0\\ 8x_{1}&-2x_{3}cos(x_{2}x_{3})&-2x_{2}cos(x_{2}x_{3})\\ 0&x_{3}&x_{2}\end{matrix}|=-8x_{1}|\begin{matrix}5&0\\x_{3}&x_{2}\end{matrix}|=-40x_{1}x_{2}.
\begin{bmatrix}u^t \beta\end{bmatrix}\begin{bmatrix}Df(x,\lambda_1,\lambda_2) \bar{p}\bar{q}^t 0\end{bmatrix}=\begin{bmatrix}p^t 0\end{bmatrix}, \begin{bmatrix}Df(x,\lambda_1,\lambda_2) \bar{p}\bar{q}^t 0\end{bmatrix}\begin{bmatrix}v \beta\end{bmatrix}=\begin{bmatrix}q 0\end{bmatrix},
\frac{\mathcal{H}_{d-1}(W'(\omega)\cap O)}{|W'(\omega)\cap O|^\frac{d-1}{d}}\geq \frac{\sum_{l=1}^N \mathcal{H}_{d-1}(B(x_{j_l},\rho')\cap O)}{\sum_{l=1}^N |B(x_{j_l},\rho')\cap O|^\frac{d-1}{d}}\geq C_.
R_{\gamma A}=\tfrac{2}{1+2\gamma\sigma+\sigma\beta\gamma^2}\begin{bmatrix}\tfrac{1}{2}(1-\beta\sigma\gamma^2) -\gamma\sqrt{\sigma(\beta-\sigma)}\gamma\sqrt{\sigma(\beta-\sigma)}\tfrac{1}{2}(1-\beta\sigma\gamma^2)\end{bmatrix},
\sum^{N}_{n=1}\beta_n\sqrt{\sum^{K}_{i=1}\|\mathbf{w}^{}_{i,n}\|^2+\|\mathbf{w}^{}_{i,n}\|^2}+ \sum_{i=1}^{K}\|\mathbf{w}^{}_i\|^2 + \lambda\sum_{i=1}^{K}\|\mathbf{w}^{}_i\|^2.
S_{0}^{( D )}( k ) = \frac{1 - \lambda \, e^{i \pi \nu}\, \Xi ( - 2 \nu ) \, \left( E / \mu^{2}\right)^{\nu}}{1 - \lambda \, e^{- i \pi \nu}\, \Xi ( - 2 \nu ) \, \left( E / \mu^{2}\right)^{\nu}}\; ,
H_q =e^{i\beta}(I_q+\zeta_{6-q}(J_q-I_q))^{\otimes d}&=e^{i\beta}\left((1+(q-1)\zeta_{6-q})\left(\frac{1}{q}J_q\right)+(1-\zeta_{6-q})\left(I_q-\frac{1}{q}J_q\right)\right)^{\otimes d}.
\aligned\lim_{R \rightarrow \infty}R \| (1 - \chi(x)) P_{\geq 0}(u_{0}(Rx)) \|_{B_{1,2}^{1/2}}+ R^{2}\| (1 - \chi(x)) P_{\geq 0}(u_{1}(Rx)) \|_{B_{1,2}^{-1/2}}= \lim_{R \rightarrow \infty}R \sum_{j \geq 0}2^{j/2}\| (1 - \chi(x)) P_{j}(u_{0}(Rx)) \|_{L^{2}}+ R^{2}\sum_{j \geq 0}2^{-j/2}\| (1 - \chi(x)) P_{j}(u_{1}(Rx)) \|_{L^{2}}= 0.\endaligned
\partial_{\mu}\Psi_{a}= U_{e x t}\left( x^{\mu}\right) \widetilde{U}_{i n t}\left( x^{\mu}, x^{m}\right) \left( \partial_{\mu}+ i m v_{\mu \alpha}\sigma^{\alpha}+ i m v_{\mu c}\sigma^{c}\right) \psi_{a}^{r}\psi_{r}^{i n t}
\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}=\frac{a}{2(\Delta x)^{2}}((u_{i+1}^{n+1}-2u_{i}^{n+1}+u_{i-1}^{n+1})+(u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}))
\not \! p S ( p ) = 1 - i e^{2}\int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{1}{k^{2}( e^{2}/ \pi - k^{2}- i \epsilon )}\not \! k \gamma^{5}\left[ \gamma^{5}S ( p ) + S ( p + k ) \gamma^{5}\right] \; ,
\left\langle Z,x^{\star}\right\rangle_{E,E^{\star}}=\left\langle\int_{0}^{t}\Phi (s)dw(s),x^{\star}\right\rangle_{E,E^{\star}}=\int_{0}^{t}\left\langle dw(s),\Phi^{\star}(s)x^{\star}\right\rangle_{H}.
\mathcal{L}^{-1}\left(\frac{1}{x}\iint\limits_{\sqrt{2x}T_N}e^{-\frac{1}{2}\left( y^{2}+z^{2}\right)}\, dy \, dz\right)(\rho^{2}) = \int\limits_{0}^{2\pi}I_{T_N}(\rho\cos\theta,\rho\sin\theta) \, d\theta
\sum_{n=0}^{\infty}\frac{q^{\binom{n+1}{2}}\left(-x\right)^{n}}{\left(q;q\right)_{n}}A_{q}\left(-q^{n}\right)=\frac{\left(q^{3/2};q\right)_{\infty}}{\left(q^{3/2}x;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{\left(n^{2}+3n\right)/4}\prod_{k=0}^{n-1}\left(x-q^{k}\right)}{\left(q^{1/2},q^{3/4},-q^{3/4};q^{1/2}\right)_{n}}.
M = \frac{i}{\sqrt{{\mathrm{\boldmath \beta}}^{2}}}\left( \begin{matrix}{E_{\mathrm{\boldmath \beta}}}&{- E_{-{\mathrm{\boldmath \mu}}}}\\{E_{\mathrm{\boldmath \mu}}}&{E_{-{\mathrm{\boldmath \beta}}}}\\ \end{matrix}\right) .
\Delta t \sum_{n=1}^{m}(\partial_{t}^+f^n(u_h^n),\phi_h^n)_{\mathcal{T}_h}\le C \Delta t\sum_{n=1}^{m}\|\partial_{t}^+u_h^n\|_{\mathcal T_h}\|\phi_h^n\|_{L^6(\Omega)}&\le \epsilon \Delta t\sum_{n=1}^{m}\|\partial_{t}^+u_h^n\|_{\mathcal T_h}^2 + C\Delta t \sum_{n=1}^m \|\phi_h^n\|_{L^6(\Omega)}^2.
u_{j,\alpha}^{(\infty)}(n)=\frac{P_j(-\hbar \tilde{\partial}_t)\tau(n)}{\tau(n+1)^{\frac{1}{2}-\alpha}\tau(n)^{\frac{1}{2}+\alpha}}~,~~u_{j,\alpha}^{(0)}(n)=\frac{P_j(-\hbar \tilde{\partial}_{\bar{t}})\tau(n+1)}{\tau(n+1)^{\frac{1}{2}-\alpha}\tau(n)^{\frac{1}{2}+\alpha}}~.
(g \circ f)(\tau(a,b)) = (g\circ f)\left(-a,a+b+\frac{1}{4k}\right)= g\left(-a,-ka+2k\left(a+b+\frac{1}{4k}\right)\right) &=g\left(-a,ka+2kb + \frac{1}{2}\right)=(g \circ \tau_1)\left(a,ka+2kb\right)= g(a,ka+2kb)=(g \circ f)(a,b).
\log\epsilon\int_{\rho=\epsilon}(n+1)f_1G_2\vartheta\wedge({\rm d}\vartheta)^n+2\epsilon^{-n}\log\epsilon \operatorname{Re}\int_{\rho=\epsilon}if_1\overline{\partial}f_2\wedge({\rm d}\vartheta)^n+O(\epsilon\log\epsilon).
F =(\I(U_1;Y_1),\I(X_1,U_2;Y_2|U_1)-I(U_2;Y_3),\I(X_2;Y_3) \ ), H =(\ 0,\I(U_1;Y_1)+ I(X_1,U_2;Y_2|U_1)-I(U_2;Y_3),\I(X_2;Y_3) \ ).
\tilde{\Theta}^L_{\lambda, T}: = \tilde{\Theta}^{n; L}_{\lambda, T}= \Theta^{L_n}_{\lambda_n}\ \prod_{p = 1}^{n}\ \tilde{\Phi}^{[\frac{k}{2}| \gamma_{p}|^2]}(\tau, - \beta_{p}, \beta_{p}+ \gamma^\vee_{p}+ \sum_{i = 1}^{p -1}(\gamma^\vee_{p}| \gamma_i) \beta_i),
C_{3,j}:={}_{r^{j-1}}(C_{3,1}), &b_j := r^{j-1}(b_1), d_j := r^{j-1}(d_1), &V_{i,j}:={}_{r^{j-1}}(V_i) W_{i,j}:={}_{r^{j-1}}(W_i)
\int_0^T\|\nabla \beta_n(t)\|^2_{L^2}\,{\rm d}t \le \|\xi_0\|^2_{L^2}+ \int_0^T \left(\psi_n(t)+C_2 \|\zeta_n(t)\|^2_{L^4}\|\beta_n(t)\|^2_{L^2}\right)\,{\rm d}t &\le \|\xi_0\|^2_{L^2}+ C_2\left(\sup_{0 \le t \le T}\|\beta_n(t)\|^2_{L^2}\right)\int_0^T \|\zeta_n(t)\|^2_{L^4}\,{\rm d}t + \int_0^T \psi_n(t)\,{\rm d}t.
\Psi_{- n_{1}-{\frac{\beta}{2}}}^{\pm}\Psi_{- n_{2}-{\frac{\beta}{2}}}^{\pm}- ( - 1 )^{\beta}\Psi_{- n_{2}-{\frac{\beta}{2}}}^{\pm}\Psi_{- n_{1}-{\frac{\beta}{2}}}^{\pm}= 0
\begin{aligned}\limsup_{N\rightarrow\infty}\left\Vert\left( f_N^{(s)}(0,Z_s)-\left(f_N^{(2)}(0) \otimes f_0^{\otimes (s-2)}\right)(Z_s)\right)\mathbf{1}_{Z_s \in \mathcal{G}_s \cap \hat{\mathcal{U}}_s^{\eta(\varepsilon)}}\mathbf{1}_{E_s (Z_s) \leq R^2}\right\Vert_{L^\infty_{Z_s}}& =0\end{aligned}
t_h :=\frac{2h}{3}\cos\frac{\theta+4\pi}{3}+\frac{h}{3}\in\left( 0,\frac{2h}{3}\right) , T_h :=\frac{2h}{3}\cos\frac{\theta}{3}+\frac{h}{3}\in\left( \frac{2h}{3},h\right).
D_{\mu \nu}^{( 0 )}( p ) = \frac{1}{p^{2}+ \kappa^{2}}\left[ ( \delta_{\mu \nu}- \frac{p_{\mu}p_{\nu}}{p^{2}}) - \kappa \epsilon_{\mu \nu \lambda}\frac{p_{\lambda}}{p^{2}}\right] + \xi \, \frac{p_{\mu}p_{\nu}}{( p^{2})^{2}}
(Q_{n-2}) \int_0^T |u_{n-1}|^2 \,dt \le C\Big(\sum_{i=1}^{n-2}e_{1-i}(U_{n-i})(0) & + \sum_{p=1}^{n-2}|A^{-p/2}w_{1}^{n-1-p}(0)|^2 +\int_0^T |w_{1}^{n-2}|^2 \,dt\Big) \,.
n^{-2}g u^{p\bar p}u^{q\bar q}u_{p\bar p \bar j}u_{q\bar q j}- n^{-1}g u^{p\bar p}u^{q\bar q}|u_{p\bar q j}|^2 + n^{-1}g u^{p\bar p}u_{p \bar p j \bar j}=2 Re( g_{vj}v_{\bar j}) + g_v v_{j\bar j}+g_{j\bar j},
\lim_{n\to\infty}G_n^{(m)}(q)=(q)_{\infty}\sum_{0\leq k_m\leq\dots\leq k_2\leq k_1}\frac{q^{-2k_m}q^{\sum_{j=1}^m(k_i^2+2k_i)}}{(q)_{k_m}^2(q)_{k_1-k_2}\dots(q)_{k_{m-1}-k_m}}
G_k(s) = \sum_{\substack{L(\rho,\pi)=0 |s-\rho|\leq 200\eta}}\frac{1}{(s-\rho)^{k+1}}+\sum_{\substack{L(\rho',\pi\otimes\chi)=0 |s-\rho'|\leq 200\eta}}\frac{1}{(s+1-\beta_{\chi}-\rho')^{k+1}}+O\Big(\frac{m^2\eta\log(q QT)}{(200\eta)^k}\Big).
S_{0}( \phi ) = \frac{1}{2}\int \frac{d^{d}k}{( 2 \pi )^{d}}\left[ \sum_{\mu}\frac{4}{a^{d - 2}}\sin^{2}\left( \frac{a k_{\mu}}{2}\right) + m^{2}\right] \phi ( - k ) \phi ( k ) \ \ .
\iota(\{2\}^{k_{d}},3,\dots\{2\}^{k_{1}},3) =(-1)^{k_{1}+\cdots+k_{d}}x_{2k_{1}+3}\cdots x_{2k_{d}+3}, \theta(\{2\}^{k_{d}},3,\dots\{2\}^{k_{1}},3,\{2\}^{k_{0}}) =(-1)^{k_{0}+\cdots+k_{d}}x_{2k_{0}+2}x_{2k_{1}+3}\cdots x_{2k_{d}+3}.
\int_0^\theta \frac{S(\zeta)\,\tan^{d-3}(\zeta/2) \,d\zeta}{\sqrt{\cos \zeta - \cos\theta}}= \frac{\Gamma((d-2)/2)}{2\, \pi^{(d-2)/2}}\, \sin^{d-3}\bigg(\frac{\theta}{2}\bigg) \, (F_Q - Q(\theta)), 0 \leq \theta \leq \alpha.
\dot{\lambda}^{\bar{u}}= -\frac{\partial f}{\partial x}^T\lambda^{\bar{u}}- \mu^{\bar{u}}\frac{\partial g}{\partial x}= \left(\begin{array}{cc}0 2 -1 2\end{array}\right)\lambda^{\bar{u}}- \mu^{\bar{u}}\left(\begin{array}{c}0 1\end{array}\right)
\qquad - \;{\frac{\kappa}{8 \pi}}T r \Bigl ( [ u^{- 1}\delta u ] ( 0 ) \; \delta{\cal M}{\cal M}^{- 1}\; + \; [ \delta v v^{- 1}] ( 0 ) \;{\cal M}^{- 1}\delta{\cal M}\Bigr ) \quad ,
n'_{{\mathfrak{a}_{i+1}}}(s_i \cdot \mathfrak{b})=\begin{cases}n_{\mathfrak{a}_i}(\mathfrak{b}) + t^{-1}n_{\mathfrak{a}_i}(s_i \cdot \mathfrak{b}), \ell(s_i \cdot \mathfrak{b}) > \ell(\mathfrak{b}); n_{\mathfrak{a}_i}(\mathfrak{b}) + t n_{\mathfrak{a}_i}(s_i \cdot \mathfrak{b}), \ell(s_i \cdot \mathfrak{b}) < \ell(\mathfrak{b}); \end{cases}
D_k ({\vec x},{\vec x'}) = \sum_{n=-\infty}^\infty\int_0^\infty{q dq\over 2\pi\alpha}{e^{in(\theta-\theta')/\alpha}\over q^2+ k_4^2}J_{\left|n/ \alpha \right|}(q r) J_{\left|n/ \alpha \right|}(q r'),
D ( x , x^{\prime}) = \sum_{n e v e n}\int{\frac{d^{4}k}{( 2 \pi )^{4}}}\int_{0}^{\infty}{\frac{q d q}{\pi}}{\frac{e^{i k_{\mu}( x^{\mu}- x^{\mu})}e^{i n ( \theta - \theta^{\prime})}}{k_{4}^{2}+ q^{2}}}J_{n}( q r ) J_{n}( q r^{\prime}) .
D\left(\dfrac{S_n}{\sqrt{n/(1-2\alpha)}}\right) \leq \begin{cases}\dfrac{C_{\alpha}\log n}{\sqrt{n}}(-1<\alpha\leq 0), begin{align*}4mm]\dfrac{C_{\alpha}\log n}{n^{(1-2\alpha)/2}}(0\leq \alpha<1/2), begin{align*}4mm]\end{cases}D\left(\dfrac{S_n}{\sqrt{n\log n}}\right) \leq \dfrac{C_{1/2}\log \log n}{\sqrt{\log n}}(\alpha=1/2).
\hat{\tilde{\mathrm{P}}}_{\mathrm{R}}\equiv \hat{\mathrm{P}}_{\mathrm{R}}+ \frac{1}{2}\sum_{p > 0}\{[ \hat{v}_{\mathrm{P , +}}, \hat{\mathrm{G}}_{+}]_{+}+ [ \hat{v}_{\mathrm{P , -}}, \hat{\mathrm{G}}_{-}]_{+}\}
\begin{cases}-\Delta \psi_1(c_0) = \gamma_1(c_0) \psi_1(c_0) \mbox{in $\Omega$}, \partial_{\mathbf{n}}\psi_1 = \gamma_1(c_0) \psi_1(c_0) \mbox{on $\partial \Omega$}. \end{cases}
\mathrm{Ker}\, d_{2^{r+1}-1}\cap \left( D_1/(v_te_t ) \{x_3^2\}\oplus D_r\langle x_3^2 y_r, x_3 z_r \rangle \right) = D_1/(v_t e_t)\{x_3^2 \}\oplus D_r \langle x_{3}^2 y_{r+1}, x_3 z_{r+1}, \rangle.
\begin{aligned}\lim\limits_{\stackrel{x \to x_{0}}{x \in K}}\big\langle \nabla_{x}(S\varphi)(x,t),\nu(x_{0})\big\rangle=\frac{1}{2}\varphi(x_{0},t) &+\int_{0}^{t}\int_{\partial\Omega}\frac{\partial\varepsilon_{n, a}(x_{0}-\xi,b(t,\tau))}{\partial\nu(x_{0})}\varphi(\xi,\tau)a(\tau)dS_{\xi}d\tau,\end{aligned}
\frac{d^{n}}{dz^{n}}\left\lbrace{}^{\Psi}F_{p}^{(\alpha,\beta)}(a,b;c;z)\right\rbrace =\frac{(a)_{n}(b)_{n}}{(c)_{n}}\left[{}^{\Psi}F_{p}^{(\alpha,\beta)}(a+n,b+n;c+n;z)\right].
\overset{\widetilde{I}\widetilde{J}}{R}_{12}\overset{I}{B}_1\overset{\widetilde{I}\widetilde{J}}{(R')}_{12}\overset{J}{A}_2 = \overset{J}{A}_2\overset{\widetilde{I}\widetilde{J}}{R}_{12}\overset{I}{B}_1\overset{\widetilde{I}\widetilde{J}}{(R^{-1})}_{12},
|1-\alpha+\alpha\tfrac{1-\gamma \sigma}{1+\gamma \sigma}| \leq|1-\alpha|+|\alpha\tfrac{1-\gamma \sigma}{1+\gamma \sigma}| &=\alpha-1+\alpha\tfrac{\gamma \sigma-1}{1+\gamma \sigma}&\leq \alpha-1+\alpha\tfrac{\gamma \beta-1}{1+\gamma \beta}=r_\beta
h_K^-=\frac{w_K}{\Pi_{d_0}(p,H)}\left (\frac{p}{4\pi^2}\right )^{m/4}\prod_{\chi\in X_K^-}L(1,\chi')\leq w_K\left (\frac{pM_{d_0}(p,H)}{4\pi^2D_{d_0}(p,H)}\right )^{m/4}.
\mu_{n_s}(k) = n_s(\lambda_s)^k\lambda_s^{-k}, \mu_{n_s}(-k) = \lambda_s^kn_s^{-1}(\lambda_s)^{-k}, c_{n_s,k}= \lambda_s^k\cdot c_{n_s}, c_{n_s,-k}= n_s(\lambda_s)^k\cdot c_{n_s}.
\|u(t+h)-u(t)\|^2_{\mathbb H^\mu}\le 3 \|[S(t+h)-S(t)]\xi\|^2_{\mathbb H^\mu}& + 3\int_0^t \|D_h f(u)(t-\tau)\|^2_{\mathbb H^{\mu-1}}d\tau & + 3\int_t^{t+h}\|f(u(t+h-\tau),\mathcal Hu(t+h-\tau))\|_{\mathbb H^{\mu-1}}^2d\tau & = F_1(t) + F_2(t) + F_3(t),
\frac{\tau^{-1}}{\Gamma(1-\alpha)}\mathcal{D}_t^{(\alpha)}( \mu_1 \delta_t u_i^{n-\frac{1}{2}}+\mu_2 u_i^{n-\frac{1}{2}})+\psi(u_i^{n-\frac{1}{2}},u_i^{n-\frac{1}{2}})-\frac{h^2}{2}\psi(w_i^{n-\frac{1}{2}},u_i^{n-\frac{1}{2}}) -\lambda w_i^{n-\frac{1}{2}}=0,1\leq i \leq M-1,1\leq n\leq N,
\frac{1}{i}\frac{\delta^{2}S}{\delta \chi ( x ) \delta \chi ( y )}= - \frac{i}{2}{\cal G}( x , y ){\cal G}( y , x ) + \phi_{c}( x ){\cal G}( x , y ) \phi_{c}( y ) \equiv \frac{\Sigma ( x , y )}{2}
m_{j}(E) = \int_{E_j^-}^{E_j^+}\frac{f_j(E,x)}{\sqrt{(E_j^+-x)(x-E_j^-)}}dx &= \sum_{k = 1}^\infty a_k^j(E)\int_{E_j^-}^{E_j^+}\frac{x^k}{\sqrt{(E_j^+ -x)(E_j^- - x)}}dx &= \sum_{\vec{m}\in [1,2(n+1)]^n}c_{\vec{m}}E^{\vec{m}},
\overline{\zeta}_q'(0)= 6 \xi'_q(-1)-3(n+1-q)\xi_q'(0)-2\eta_q'(-1)+(n+1-q)\eta'_q(0)+2(n+1-q)\xi_q(0) &+\sum_{j=2}^{2n}(n+1-q)^j w_j\gamma_{n,q}(j-1).
f(t,t_1,t_2,t_3)=\frac{(t_1-t)(t_3-t_2)}{(t_2-t_1)(t-t_3)}\left(1+(t_2-t)(t_3-t_1)\left(\frac16(z'(0))^{-1}z'''(0)-\frac14((z(0)')^{-1}z(0)'')^2\right)\right)
\begin{array}{l}{\rho ( \theta ) \rho ( - \theta ) ( \operatorname{s i n}^{2}\frac{n \gamma}{1 6}+ \operatorname{s i n h}^{2}\frac{\theta}{2}) \operatorname{c o s h}^{2}\frac{\theta}{2}= 1 \vspace{3 m m}}{\rho ( \theta ) = \rho ( i \pi - \theta )}\end{array}\; .
\displaystyle\frac{c\lambda(\nu^{2}-b_{1}^{2})}{\gamma-\lambda}-\displaystyle\frac{\nu^{2}b_{1}^{2}}{(\gamma-\lambda)^{2}}+\displaystyle\frac{\nu^{2}b_{2}^{2}}{(\gamma-\mu)(\gamma-\lambda)}-\lambda\gamma-c(1-b_{1}^{2})=0,
\zeta(s,\mathbb{X}_\alpha)= \sum_{n=0}^{\infty}\frac{\mathbb{X}_\alpha(n)}{(n+\alpha)^s}\quad\zeta(s,\mathbb{Y}_\alpha)= \sum_{n=0}^{\infty}\frac{\mathbb{Y}_\alpha(n)}{(n+\alpha)^s}
\mathfrak{M}\left[{}^{M\!L\!\!}{B}_{p}(x,y) : s\right]= \int_{0}^{1}t^{x-1}(1-t)^{y-1}\left[\int_{0}^{\infty}p^{s-1}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right)dp\right]dt.
\Lambda^{+}(X) = \sum_{i\in\mathcal{I}^-}(A_i-\overline{X})^T \otimes \Phi^+_i(X) \overline{M}^{+}+ \sum_{i\in\mathcal{I}^+}(A_i-\overline{X})^T\otimes \Phi^+_i(X) - (X-\overline{X})^T\otimes I_2 ,
g ( \underline{{{x}}}) = \bigl ( \prod_{s = 1}^{l}\frac{\partial}{\partial y_{j_{s}}^{0}}\bigr )^{K}\sum_{| \underline{{{m}}}| \leq K - 1 , m_{j_{1}}^{0}= . . . = m_{j_{l}}^{0}= 0}( \frac{\partial}{\partial \underline{{{y}}}})^{\underline{{{m}}}}\nu_{\underline{{{m}}}}( \underline{{{y}}}) ,
F_{c_{(\mu, r)}}(x, y)=\begin{cases}0, (x, y)\notin c_{(\lambda, r)}, e^{-(\lambda-\mu)^2}, (x, y)\in c_{(\lambda, r)}(0<\lambda\neq \mu), 1, (x, y)\in c_{(\lambda, r)}(0<\lambda= \mu), \end{cases}
\theta_1 =\dfrac{3 ( 2 a c^3 + 2 b^3 d -b^2 c^2 - 3 a b c d)}{6 ac^3 +6 b^3 d-4 b^2 c^2}, ~ \theta_2= \dfrac{c^2 (3ac-b^2)^2}{6 ac^3 + 6 b^3 d-4 b^2 c^2}, ~ t=\dfrac{b (3bd-c^2)}{c (3ac-b^2)}.
\stackrel{(a)}{\geq}\sqrt{\frac{a_{n}D_2(\tau_C)}{v^*(D_1(\tau_C))^2}}\left(\sqrt{\frac{b_{n,1}}{s_1+\frac{\eta_n^*}{v^*}}}-\sqrt{\frac{b_{n,{2}}}{s_{2}}}\right) =T_{{\rm SVC},n,1}^*,
\begin{aligned}f(a)=\sum_{j=2}^{k_1}\left\{p_{j}a+\xi_{j}\right\}+\sum_{j=k_{1}+1}^{k}\left\{p_{j}a\right\}=\sum_{j=2}^{k_1}\left(p_{j}a+\xi_{j}\right)+\sum_{j\in\mathcal{K}_{0}^{+}}p_{j}a+\sum_{j\in\mathcal{K}_{0}^{-}}\left(1+p_{j}a\right) &=k_{0}^{-}+\sum_{j=2}^{k}p_{j}a+\sum_{j=2}^{k_1}\xi_{j},\end{aligned}
T(n_i)+T(n_{i+1})+T(n_{i+2}) <\left( \frac{85}{27}+ \frac{320}{243}\right)\cdot \frac{1}{X_0}= \frac{1085}{243}\cdot \frac{1}{X_0}&<6 \cdot \frac{3}{4}\cdot \frac{1}{X_0}= (k_i+k_{i+1}+k_{i+2}) \cdot \frac{3}{4}\cdot \frac{1}{X_0}.
\begin{cases}u_t - A_1u =h(t,x) \chi_{\omega}(x), \ (t,x) \in (0,T) \times (0,x_0), u(t,0)=(au_x)(t,x_0)=0, \t \in (0,T), u(0,x)=u_0(x)_{|_{[0,x_0)}},\end{cases}
f\bullet g \ (\mathbf{n})=\int_{S^{2}\times S^{2}}f(\mathbf{n}_{1})g(\mathbf{n}_{2}) \mathcal{L}(\mathbf{n}_{1}, \mathbf{n}_{2}, \mathbf{n})d\mathbf{n}_{1}d\mathbf{n}_{2}\ .
2 \int_X u du \wedge d^c \Omega \wedge \Omega^{n-2}=\int_X d(-u)^2 \wedge d^c \Omega \wedge \Omega^{n-2}&=\int_X - (-u)^2 dd^c \Omega \wedge \Omega^{n-2}- (n-2) \int_X (-u)^2 d\Omega \wedge d^c \Omega \wedge \Omega^{n-3}&\geq - B_\Omega \|u\|_\infty^2 \int_X \left[\omega^2 \wedge \Omega^{n-2}+ \omega^3 \wedge \Omega^{n-3}\right].
\int_0^1\rho^{-2s}(r) \int_1^\infty \, | G_{m,2}^+(\lambda, r,r')|^2\, \rho^{-2s}(r') \, r'\, dr' \, r \, dr = \frac{o(\lambda^{2\mu(\alpha)})}{1+m^2}\ , \lambda\to 0+.
X_{IV}=\sum_{t=2}^{a+1}\sum_{a+1 \ge p(t)>\cdots >p(2)>p(1)=1}\sum_{\ell(t)\in J_{p(t)}}\cdots \sum_{\ell(1) \in J_{p(1)}}K_{t}(w; z_{\ell(1)}, \ldots , z_{\ell(t)}){}\times \prod_{1\le s \le t}^{\curvearrowleft}Z_{\ell(s)}^{J_{p(s+1)-1}\cup \cdots \cup J_{p(s)}}(\vec{z})u(a+2, (a+1)^{k_{a+1}}, \ldots , 1^{k_{1}}),
\limsup_{t\to \infty}\|u(\cdot,t)\| \le \sup_{t\in\mathbb R^+}\int_0^t\omega(t-s,\lambda_1)\|p\|_\infty ds & = \|p\|_\infty\lambda_1^{-1}\sup_{t\in\mathbb R^+}(1-\omega(t,\lambda_1))=\|p\|_\infty\lambda_1^{-1}.
(U,V)_r = \frac{(m-r)! \, (n-r)!}{m! \, n!}\, \sum\limits_{i=0}^r \, (-1)^i \binom{r}{i}\, \frac{\partial^r U}{\partial x_1^{r-i}\, \partial x_2^i}\, \frac{\partial^r V}{\partial x_1^i \, \partial x_2^{r-i}}\, .
\overline{p}(n)=\frac{1}{2\pi}\underset{2 \nmid k}{\sum_{k=1}^{\infty}}\sqrt{k}\underset{(h,k)=1}{\sum_{h=0}^{k-1}}\dfrac{\omega(h,k)^2}{\omega(2h,k)}e^{-\frac{2\pi i n h}{k}}\dfrac{d}{dn}\biggl(\dfrac{\sinh \frac{\pi \sqrt{n}}{k}}{\sqrt{n}}\biggr),
|f(z)| \leqq |z| + \sum_{\nu=k+1}a_{\nu}|z|^{\nu}& \leqq |z| + |z|^{k+1}\; \sum_{\nu=k+1}a_{\nu}& \leqq |z| + |z|^{k+1}\;\dfrac{(1- \gamma) \Gamma(k+1+\mu)\Gamma(\tau+1)}{(1+\delta k) \Gamma(k+1+\tau) \Gamma(\mu+1)}&= |z| + |z|^{k+1}\;\dfrac{(1- \gamma) (\mu+1)_{k}}{(1+\delta k) \,(\tau+1)_{k}}
B_1 = A_F \frac{Z^{\frac{2}{d}}}{V^2}+ O_F \left( \left(\frac{Z}{V^d}\right)^{\frac{1}{d-1}}+ \frac{Z^{\frac{2}{d}}}{V^2 (\log Z)^6}+ \frac{Z^{\frac{1}{d}}}{V}(\log Z)^{6(d-1)}\right)
\frac{S_1}{S_2}\geq \sum_{k=1}^\infty a_k q_k & = \prod_{p \leq X}\left(1 - q_p p^{-1}\right)^{-1}& = \left( \prod_{p \leq X}(1 - p^{-1})^{-1}\right) \left( \prod_{p \leq X}\frac{p-1}{p-q_p}\right).
\alpha_{2}^{( 1 )}- \alpha_{2}^{( 2 )}\tau \rho + i \alpha_{1}^{( 2 )}\rho - i \alpha_{1}^{( 1 )}= 0 , \ \ \ \beta_{2}^{( 1 )}- \beta_{2}^{( 2 )}\tau \rho + i \beta_{1}^{( 2 )}\rho - i \beta_{1}^{( 1 )}= 0 .