harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\|e^{\pm,1}_k\|_{s,X}\le C\delta^2\theta_k^{L_1(s)}\Delta_k, s_0-1\le s\le s_1-1\cr\|\overline{e}_{\pm,k}^{(1)}\|_{s,X}\le C\delta^2\theta_k^{s+s_0-2\alpha-2}\Delta_k, s_0\le s\le s_1-1\cr\|\tilde{e}_{k}^{(1)}\|_{H^s(\omega_X)}\le C\delta^2\theta_k^{s+s_0-2\alpha-\frac 52}\Delta_k, s_0\le s\le s_1-\frac 12\end{cases}
\delta v(t,x)=\int_0^t\int_{{\mathbb R}^n}\sum_{ij}\left(a^{v^2}_{ij}-a^{v^2}_{ij}\right)(s,y)\frac{\partial^2 v^1}{\partial x_i\partial x_j}(s,y)p^{v^2}(t,x,s,y)dsdy,
E(1) = (1-\frac1\delta p_1) E'(2) = E(1)\left( 1 - \frac{\delta}{\delta-1}h_1 \right) E(1) E(2) = \frac12 E'(2)\left( 1+s_1-\frac1{\delta-2}p_2 \right) E'(2) E'(3) = E(2)\left( 1 - 2\frac{\delta-2}{\delta-3}h_2 \right) E(2) E(3) = \frac13 E'(3)\left( 1+2s_2-\frac1{\delta-4}p_3 \right) E'(3)
I_1 =\int_{\lambda_2}^{\lambda^*}Q^{\frac{1}{2}}(\eta,x)d\eta=\frac{1}{x}\int_{\sqrt{4a b+i}}^{T}\sqrt{t^2-(4a b+i)^2}\; (1+o(1))d t &=\frac{T_{*}^2}{2x}-\frac{4a b+i}{4x}-\frac{4a b+i}{2x}\ln(2T_*)+\frac{4a b+i}{4x}\ln(4a b+i)+o\left (\frac 1 x\right ).
Z ={V_{p + 1}}{{\left({\frac{\mathrm{T}}{2 \pi}}\right)}^{( p + 1 ) / 2}}\int d \tau{\frac{\mathrm{d e t}( \psi , B )}{\sqrt{\mathrm{d e t}\hat{( B , B )}}}}{\frac{\sqrt{\mathrm{D e t}^{\prime}{\hat{P}^{\dagger}}\hat{P}}}{\mathrm{V o l}( \mathrm{C K V})}}{{\left({\frac{{\mathrm{D e t}_{N}}^{\prime}\hat{\Delta}}{\int{d^{2}}\xi \sqrt{\hat{g}}}}\right)}^{- ( p + 1 ) / 2}}
\tan\theta = \frac{\tilde{b}}{\alpha'}= \frac{(2\pi)^2\tilde{b}^2}{\alpha'\tilde{V}_2}\frac{N_{p-2}}{N_p}\ \ \ \ \Longrightarrow \ \ \frac{N_{p-2}}{N_p}= \frac{\tilde{V}_2}{(2\pi)^2\tilde{b}}.
\Omega(a_0,a_1;b_0,b_1) = \left\langle{{\rm Det}(1 - a_1 U) \,{\rm Det}(1 - b_1 \bar U) \over{\rm Det}(1 - a_0 U) \,{\rm Det}(1 - b_0 \bar U)}\right\rangle \;.
M^{\prime}={\cal U}^{T}\left( \begin{array}{l l}{U_{5}}{0}{0}{U_{2 1}}\end{array}\right){\cal U}M{\cal U}^{T}\left( \begin{array}{l l}{U_{5}^{T}}{0}{0}{U_{2 1}^{T}}\end{array}\right){\cal U},
Y_{t\wedge \tau_m}= \sup_n \inf_k Y^{n, k}_{\tau_m}+\int_{t\wedge \tau_m}^{\tau_m}\big( f{(s, Y_s, Z_s )}dA_s + g_s d\langle N\rangle_s \big) -\int_{t\wedge \tau_m}^{\tau_m}\big(Z_s dM_s + dN_s\big).
{\sf M}\left\{\left(J^{*}[\psi^{(3)}]_{T,t}-\sum\limits_{j_1, j_2, j_3=0}^{p}C_{j_3 j_2 j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}\right)^2\right\}\le \frac{C}{p}
\operatorname{Tr}\left( \chi_I(H+V) \chi_{J^c}(H) \right) = \sum_j \left\langle \phi_j, \chi_{J^c}(H) \phi_j \right\rangle = \sum_j \left\langle \phi_j, \left( V \frac{\chi_{J^c}(H)}{(H - E_j)^2}V \right) \phi_j \right\rangle \leq \frac{\lVert V \rVert^2}{\operatorname{dist}(I, J^c)^2}\operatorname{Tr}\left( \chi_I(H+V) \right) .
\Omega_{0}( z ) = \Omega_{0}^{A}( z ) \, \Theta_{A}= I^{A B}X_{B}( z ) \, \Theta_{A}, \quad \mathrm{w h e r e ~}X_{A}= \langle \Theta_{A}, \Omega_{0}\rangle = \int_{\widetilde{\Theta}_{A}}\Omega_{0},
H(Y_2^n|W_1Y_2^{n_1}S^n\Theta_2U_C) = H(U_B^C|W_1Y_2^{n_1}S^n\Theta_2U_C) &=H({K'}_B^C|W_1Y_2^{n_1}S^n\Theta_2U_C) &\geq H({K'}_B^C|1_B^CW_1Y_2^{n_1}S^n\Theta_2U_C) &= H({K'}_B^C|1_B^C) - I({K'}_B^C;W_1Y_2^{n_1}S^n\Theta_2U_C|1_B^C)
\Delta H^\epsilon=d(d\widetilde d_\epsilon\times \widetilde d_\epsilon)=d\widetilde{d}_\epsilon\times d\widetilde{d}_\epsilon=\sum_{i,j=1}^3\frac{\partial \widetilde{d_\epsilon}}{\partial x_i}\times\frac{\partial \widetilde{d_\epsilon}}{\partial x_j}dx_i\wedge dx_j \{\rm{in}}\ \mathbb R^3.
P_{n+2}(x) =\Big(x-\omega(a)+\left(a-\beta \right)\left(a+p+\beta \right) - \alpha_{2}-\alpha_{1}\Big) P_{n+1}(x) - \Big(\alpha_{2}\alpha_{1}+\gamma \left(p+2\beta \right)\Big)P_{n}(x) - \gamma^2 P_{n-1}(x)\, .\;
M=M_{(\underbrace{m_1,\dots,m_1}_{r_1},\dots,\underbrace{m_l,\dots,m_l}_{r_l})}, pi=\underbrace{\tau_1\otimes\dots\otimes\tau_1}_{r_1}\otimes\dots\otimes\underbrace{\tau_l\otimes\dots\otimes\tau_l}_{r_l}
N(h(s),s):= g(\tilde{y}_1(s)+h(s),s)- g(\tilde{y}_1(s),s) - \frac{\partial g^{+\infty}}{\partial y_1}(P_1^{+\infty})h(s)= = h(s)\int_0^1 \Big[\frac{\partial g}{\partial y_1}(\tilde{y}_1(s)+\sigma h(s),s) -\frac{\partial g^{+\infty}}{\partial y_1}(P_1^{+\infty}) \Big] d\sigma
\hspace{-0.95in}\lambda =4\; \mbox{for}\; \left| \vec{p}\right| <\bar{m}/2 , \quad \lambda =3\ \; \mbox{for}\; \bar{m}/2<\left| \vec{p}\right|<\bar{m}, \quad \lambda =3 \; \mbox{or}\; 4 \; \mbox{for}\; \left| \vec{p}\right|>\bar{m}.
h^{\ast}(u,v) =\sup_{(x,y)\in\operatorname*{dom}h}[xu+yv-h(x,y)]=\sup_{y\in\mathbb{R}_{-}^{\ast}}\left( yv+\sup_{x\in\mathbb{R}}\left[xu-e^{x}[f(y)]^{3}\right] \right) & =\sup_{y\in\mathbb{R}_{-}^{\ast}}\left( yv+[f(y)]^{3}\exp^{\ast}\left(\frac{u}{[f(y)]^{3}}\right) \right) .
2 a_1 a_2 + 4 a_3 + a_4=0, 12 a_2 + 3 a_5 - a_1 (6 a_3 + 2 a_4)=0, 8 a_3 + 3 a_4 + 4 a_6 + a_1 (6 a_2+2 a_5), 4 a_2 + a_5 - 2 a_1 a_6=0.
S_3 = (1 + o(1))\int_{|\mathbf{x}| \leq \sqrt{r}}\int_{|\mathbf{x'}| \leq \sqrt{r}}\left(\prod_{j = 1}^k K(x_j, x_j')\right) \tilde{F}(\mathbf{x})\tilde{F}(\mathbf{x'})\;d\mathbf{x}\;d\mathbf{x'}.
s \Gamma^{\alpha}{}_{\beta \gamma}( u ) = \Gamma^{\alpha}{}_{\beta \gamma}( u ) - \left( \begin{array}{c c}{0}&{L^{i}{}_{j c}}\\{C^{a}{}_{b i}}&{0}\\ \end{array}\right) = \left( \begin{array}{c c}{L^{i}{}_{j k}}&{0}\\{0}&{C^{a}{}_{b c}}\\ \end{array}\right)
p_k^{\lambda}(n)=\frac{\sum_{i=1}^{\lfloor\frac{n-k}{2}\rfloor}{k+2i\choose k-1}p_{k+2i}^{\lambda}(n)+\sum_{i=1}^{\lfloor\frac{n-\ell(\lambda)}{2}\rfloor}\sum_{\mu\rhd_{2i+1}\lambda}\kappa_{\mu,\lambda}p_k^{\mu}(n)}{n+1-k-\ell(\lambda)}.
\frac{1}{\Gamma(k_j)}\biggl(\log\frac{1-x_{j+1}}{1-x_j}\biggr)^{k_j-1}=\frac{1}{(k_j-1)!}\biggl(\int_{x_{j+1}}^{x_j}\frac{dx}{1-x}\biggr)^{k_j-1}&=\int_{x_j>x_{j2}>\cdots>x_{jk_j}>x_{j+1}}\frac{dx_{j2}}{1-x_{j2}}\cdots\frac{dx_{jk_j}}{1-x_{jk_j}},
Y_{l,\pm\frac{1}{2},j,m}=\frac{1}{\sqrt{2(j\mp\frac{1}{2})+1}}(\begin{matrix}\pm\sqrt{j\mp\frac{1}{2}\pm m+\frac{1}{2}}Y_{l}^{m-\frac{1}{2}}\\ \sqrt{j\mp\frac{1}{2}\mp m+\frac{1}{2}}Y_{l}^{m+\frac{1}{2}}\end{matrix}).
\sum_{{{n\geq1},{n\neq{k}}}}\frac{1}{h_{\nu,n}^4-{h_{\nu,k}^4}}=-\frac{1}{2h_{\nu,k}^2}\sum_{{{n\geq1}}}\frac{1}{h_{\nu,n}^2+{h_{\nu,k}^2}}+\frac{\nu+3}{4h_{\nu,k}^4}-\frac{h_{\nu,k}^{\nu-4}}{\sqrt{\pi}2^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)\mathbf{H}_{\nu}'(h_{\nu,k})}.
X_{i,0}^\prime = t_{i-1}\left(\frac{1}{\varphi_{1}}+\sum_{\ell=2}^{i-1}\frac{t_{\ell-1}^2}{\varphi_{\ell}\varphi_{\ell-1}}\right)=\frac{t_{i-1}}{\varphi_{i-1}}.
c^2 \int_0^1 |\partial_x m|(\partial_x p)^2 \d x - 4c^2 \int_0^1 \frac{m^2 |\partial_x m| (\partial_x p)^2}{1+m^2}\d x = c^2 \int_0^1 |\partial_x m| (\partial_x p)^2 \left( 1 - \frac{4m^2}{1+m^2}\right) \leq c^2 \int_0^1 |\partial_x m| (\partial_x p)^2 \d x.
B_1: = \bigg(\int_0^{\infty}\bigg(\int_x^{\infty}[V_1^*]^q(t) w(t)\,dt\bigg)^{r / p}[V_1^*]^q(x) V_*^{- r / p}(x) w(x)\,dx \bigg)^{1 / r}, B_2: = \bigg(\int_0^{\infty}W_*^{r / p}(x) \bigg[\sup_{x \le \tau < \infty}[V_1^*](\tau)V^{-1 / p}(\tau)\bigg]^{r}w(x)\,dx \bigg)^{1 / r}.
\{{\cal P}_\alpha,{\cal J}_\beta\}=-4iP^2(J\gamma)_{\alpha\beta}+6i(PJ)(P\gamma)_{\alpha\beta}-4\Gamma_{\alpha\beta}(PJ)+\frac{i}{4}\left((1-\nu^2)P^2-16(PJ)^2\right)\epsilon_{\alpha\beta},
\begin{aligned}0 \geq - C_2 \beta \sum_i F^{ii}+ (A + \frac{M}{2}) \gamma_1 \sum_i F^{ii}- \frac{\beta}{2n}F^{11}U_{11}+ \gamma_2 \sum_{U_{ii}< 0}F^{ii}U_{ii}&\geq \frac{\gamma_1 (A + M)}{2}\sum_i F^{ii}- \frac{\beta}{2n}F^{11}U_{11}+ \gamma_2 \sum_{U_{ii}< 0}F^{ii}U_{ii},\end{aligned}
\sum_{m=0}^{\infty}\sum_{\ell_1=0}^{\min\{n_1,m\}}\sum_{\ell_2=0}^{\min\{n_2,m\}}A_{m,\ell_1,\ell_2}=\sum_{m=0}^{n_1}\sum_{\ell_1=0}^m\sum_{\ell_2=0}^mA_{m,\ell_1,\ell_2}+\sum_{m=n_1+1}^{n_2}\sum_{\ell_1=0}^{n_1}\sum_{\ell_2=0}^mA_{m,\ell_1,\ell_2}+\sum_{m=n_2+1}^{\infty}\sum_{\ell_1=0}^{n_1}\sum_{\ell_2=0}^{n_2}A_{m,\ell_1,\ell_2}.
B(z)=i\frac{8m_D}{g_m^2}\frac{{\rm e}^{-m_D|z|}}{1+{\rm e}^{-2m_D|z|}},~~\psi(z)=4{\,}{\rm sgn}{\,}z\cdot\arctan\left({\rm e}^{-m_D|z|}\right).
\mu_{\underline{m}}\left(\left|\begin{array}{cc}X Y x y\end{array}\right|^{\underline{k}-2}\right)=x^{\frac{\underline k-2}{2}-\underline m}y^{\frac{\underline k-2}{2}+\underline m},\left|\begin{array}{cc}X Y x y\end{array}\right|^{\underline{k}-2}:=\prod_\sigma \left|\begin{array}{cc}X_\sigma Y_\sigma x_\sigma y_\sigma\end{array}\right|^{k_\sigma-2}.
\left(\sum_{k=1}^nw(x_k)x_k\hat{B}^{(k)}(x)\right)\hat{Y}=\left(\sum_{k=1}^nw(x_k)x_k{\partial\over \partial x_k}\right)\hat{Y}= \operatorname{diag}\big[w(\hat{y}_1),w(\hat{y}_2),\dots,w(\hat{y}_n)\big] \hat{Y},
\Lambda_1^{n-\frac{1}{2}}-\sum\limits_{n=1}^{N}\Lambda_2^{n-\frac{1}{2}}-\sum\limits_{n=1}^{N}\Lambda_3^{n-\frac{1}{2}}-\sum\limits_{n=1}^{N}\Lambda_4^{n-\frac{1}{2}}=\sum\limits_{n=1}^{N}\Lambda_5^{n-\frac{1}{2}}.
\Pi_{\mu \nu}( k , m ) = - \frac{g^{2}}{3 \pi}\Lambda g_{\mu \nu}+ \left( g_{\mu \nu}- \frac{k_{\mu}k_{\nu}}{k^{2}}\right) \Pi^{( 1 )}( k^{2}, m ) + i \epsilon_{\mu \nu \lambda}k^{\lambda}\Pi^{( 2 )}( k^{2}, m )
E_0(\vec x;\vec\beta_0)=E(\vec y;\vec\beta_0)\prod_{j=2}^m\left[\left(\frac{2(x_1-x_j)}{x_1-2x_j}\right)^{\beta_j^2}e^{-2i\beta_j |x_1|\, |x_1-x_j|^{1/2}}\right]\ \Big(1+\Big( \frac{\log r}{r^{3/2}}\Big)\Big).
\begin{aligned}\frac{1}{\tau^k}[f(x)-f(\breve{x}^{k})]-\frac{1}{\tau^{k-1}}[f(x)-f(\breve{x}^{k-1})]+(u-\widetilde{u}^k)^TF(\widetilde{u}^k) &+c_1^k(A(x-\widetilde{x}^k))^T(A\breve{x}^{k-1}-b)\geq(v-\widetilde{v}^k)^TQ^k(v^k-\widetilde{v}^k), ~\forall u,\end{aligned}
\mathbb{P}(|\mathcal{C}_{\max}(\mathbb{G})|>k)\leq \mathbb{P}(Z_{> k}\geq k)\leq \frac{\mathbb{E}[Z_{>k}]}{k}=k^{-1}n\mathbb{P}(|\mathcal{C}(V_n)|>k) &\leq k^{-1}nc_0\mathbb{P}(r+S_t>0\forall t\leq k).
\sum_{i=0}^{n_j-1}\psi_A(z_i,k_j+i) \geq\left( \sum_{i=0}^{n_j-1}\sum_{\ell \in C_j}\psi_\ell(z_i,k_j+i) \right)- 2N^2(N-1) - \frac{N^2n_j}{2^j}
\begin{array}{l}D_{\mu}^{ij}[A](x)\langle T[j^{\mu ,j}(x)\frac{\delta j^{\beta,l}}{\delta A_{\lambda}^k}(y)]\rangle +ig\delta^4(x-y)(f^{bil}\langle\frac{\delta j^{\beta,k}}{\delta A_{\lambda}^b}(x)\rangle+f^{kia}\langle\frac{\delta j^{\beta,a}}{\delta A_{\lambda}^l}(x)\rangle)=0.\end{array}
B_{t}(m,\ell^{2n+1}d)-\ell^{\lambda-1}\left(\frac{(-1)^{\lambda}m}{\ell}\right)B_{t}(m,\ell^{2n-1}d) = B_{t}(m,\ell^{2n+2}(d/\ell))-\ell^{\lambda-1}\left(\frac{(-1)^{\lambda}m}{\ell}\right)B_{t}(m,\ell^{2n}(d/\ell)) = \ell^{(k-2)n}\, (B_{t}(\ell^{2n}m,\ell d)-B_{t}(\ell^{2n-2}m,d/\ell))\equiv 0\pmod{\ell^{(k-2)n}}.
I = \int d x^{+}d x^{-}( 4 \Omega \partial_{+}\partial_{-}\rho + \frac{\mu}{2}e^{2 \rho}\Omega^{1 - \lambda - \gamma / 4}+ \Omega^{\delta}\partial_{+}f \partial_{-}f - e^{- 2 \rho}\Omega^{( \gamma - 2 \epsilon ) / 4 + 1}F_{- +}^{2})
(\partial_{zz}u^z)(t,0) = C_1 \int \frac{r^2 -2z^2}{(r^2+z^2)^{\frac 52}}\frac 1r\partial_r ( r \omega^{\theta}) \cdot r dr dz & = C_1 \int \frac{3 r (r^2 - 4 z^2)}{(r^2 + z^2)^{\frac 72}}\omega^{\theta}(t,r,z) r dr dz.
{\bf{Cov}}\left[V_{\varepsilon}(\psi), \,V_{\varepsilon}(\xi)\right] = \begin{cases}-2 \, \log|e^{i\psi}-e^{i\xi}|, \, |\xi-\psi|\gg \varepsilon, 2\left(\kappa-\log\varepsilon\right), \psi=\xi,\end{cases}& + O(\varepsilon),
\tilde{\cal H}= \frac{1}{2}( \vec{E}^a \vec{E}^a + \vec{B}^a \vec{B}^a)+ \frac{1}{2}p^{\dagger}p + \frac{2}{(g\chi)^2}G^a_{\rm rad}G^a_{\rm rad}+ \frac{1}{2}(\vec{\nabla}\chi)^2+ \frac{g^2}{8}\chi^2 \vec{A}^a \vec{A}^a + V(\chi^2/2) \ .
\mathbb{E}[\|D(k)(v^k(\ell) - [y^{k}]_\ell\mathbf{1}) \|]=\mathbb{E}\left[\mathbb{E}[\|D(k)(v^k(\ell) - [y^{k}]_\ell\mathbf{1}) \|\mid\mathcal{F}_k]\right]\le \sqrt{\lambda}\,\mathbb{E}[\|v^k(\ell) - [y^{k}]_\ell\mathbf{1}\|].
0= \frac{1}{2}\langle \tau_{H}, \alpha_{2}\rangle\{(2m_{1}+m_{2}+2m_{3})(1-(\cot \langle \alpha_{2}, H \rangle)^{2}) &+(m_{2}+2m_{3})(1-(\tan \langle \alpha_{2}, H \rangle )^{2}) +4(m_{2}+2m_{3})\}(\alpha_{1}+2\alpha_{2}).
\int_{-\infty}^\infty \Big ({\rho}_{(1), \infty}^{(\rm cJ)}(x;\beta,p,q) |_{\beta = 2 \atop q=0}- 1 \Big ) e^{i \tau x}\, dx \mathop{\sim}\limits_{\tau \to 0}\pi \sum_{n=1}^\infty{(-1)^n c_{2n}\over (2n-1)!}| \tau |^{2n-1}.
[f\,w^{n+1\,}[v^{2}+\delta^{2}(\lambda^{2}+1)]+k\,v^{2}+\delta \,\acute{}\,v+\delta^{2}(k-\lambda)]\,du^{2}& +2\, \delta \,(f\, \lambda \,w^{n+1}-1)\,du\,dv+f\,w^{n+1\,}dv^{2}=0.
C=\begin{pmatrix}a_{1,5}a_{1,6}a_{1,7}a_{1,8}a_{2,5}a_{2,6}a_{2,7}a_{2,8}a_{3,5}a_{3,6}a_{3,7}a_{3,8}a_{4,5}a_{4,6}a_{4,7}a_{4,8}\end{pmatrix}.
\langle \mathrm{div}\, V^{6}, \phi\rangle_{L^{2}(\Omega)}\leftarrow \langle \mathrm{div}\, V^{6}_{n}, \phi\rangle_{(L^{2}(\Omega))^{2}}= \langle V^{6}_{n}, \nabla \phi\rangle_{L^{2}(\Omega)}\to -\langle V^{6}, \nabla \phi\rangle_{(L^{2}(\Omega))^{2}}.
\begin{aligned}PD^1_a: H^{BM}_r(\tilde M_a)\to H^{n-r}(\tilde M, \tilde M^a) PD^b_1: H^{BM}_r(\tilde M^b)\to H^{n-r}(\tilde M, \tilde M_b) PD : H^{BM}_r(\tilde M)\to H^{n-r}(\tilde M) PD^b_2: H^{BM}_r(\tilde M, \tilde M^b)\to H^{n-r}(\tilde M_b) PD^2_a: H^{BM}_r(\tilde M, \tilde M_a)\to H^{n-r}(\tilde M^a) \end{aligned}
{{E}}\sum_{k=0}^{K-1}\int_{(\tau,\tau+\delta] \times [0,1]}(K+3)\gamma(M) \ell(\varphi^n_k(s,y)) \,ds\,dy + K(K+3)M\delta + 5{\bar{U}}_K & \le (K+3)\gamma(M) M_0 + K(K+3)M\delta + 5{\bar{U}}_K.
\begin{cases}-\Delta \eta-k^2\eta=g\,\,\,\, \Omega\backslash K begin{align*}0.3cm] \frac{\partial \eta}{\partial n}=0\;\,\,\,\,\partial K begin{align*}0.3cm] \eta=0\,\,\,\,\partial\Omega,\end{cases}
L = (U_1 + U_2 - U_3 - U_4, U_3 + U_4 - U_5 - U_6), Q = (U_5-U_0, U_6 - U_2, U_1U_2 - U_3 U_4, U_3U_4 - U_5 U_6), N = Q + (U_1 - U_5).
f_{j}\eta := ( x_{1},\,\ldots,\,x_{p},\,r_{j}x_{p+1},\,\dots,\, r_{j}x_{q},\,r_{j}x_{q+1},\,x_{q+1},\,\ldots,\,x_{s}; & a_{0},\,\ldots,\,a_{p}=t_{0},\,\ldots,\,a_{q},\,t_{1},\, a_{q+1},\,\ldots,\,a_{s});
|c_{i,j}^{(k-2s,l-2r-2)}| \leq \dfrac{1}{2j}\left(|c_{i,j-1}^{(k-2s,l-2r-1)}| + |c_{i,j+1}^{(k-2s,l-2r-1)}|\right) \leq \dfrac{1}{2j}\frac{4V_{k,l}}{\pi^2}\Gamma_{0,0}[s](i)\left(\Gamma_{1,-2}[r](j)+\Gamma_{1,0}[r](j)\right)= \dfrac{4V_{k,l}}{\pi^2}\Gamma_{0,0}[s](i)\Gamma_{0,-2}[r](j).
\begin{aligned}\vert K_e^{w_{G_n}}(g)\Delta^{\frac{1}{2}-\delta}(g) \rvert = \vert K_e^{w_{G_n}}(c)\Delta^{\frac{1}{2}-\delta}(c) \rvert &\leq 2^{n^2-1}\cdot p^{3(n+4)(n-1)m}\cdot (\ell+(n-1)m+1)^{(n^2-1)}\cdot ((n-1) \ell+n)^{n^3}& \qquad\times p^{-(\frac{1}{4n^2-18n+22}-2\delta)(a_1+a_2+a_3+\cdots+a_{n-1})}.\end{aligned}
|(PXu, T_{2\delta}u)| = |(Xu, P^*T_{2\delta}u)|\leq \frac12(\|Xu\|^2+\|P^*T_{2\delta}u\|^2) & \leq \frac12 \|Xu\|^2+\|T_{2\delta}P^*u\|^2+\|[P^*,T_{2\delta}]u\|^2.
\begin{array}{ccccccc}L < \sum_{r=1}^{\infty}\ln (1-y^r) < -\sum_{r=1}^{\infty}y^r-(1/2)\sum_{r=1}^{\infty}y^{2r}= -y/(1-y)-y^2/2(1-y^2) = -y(2+3y)/2(1-y^2) =: L_0~.\end{array}
x =[x_1: \cdots :x_n :1] &=[\mathrm{tanh}(t_1)\mathrm{cos}(t_{n}){\displaystyle \prod_{k=2}^{n-1}}{\mathrm{sin}(t_{k})}: \cdots :\mathrm{tanh}(t_1)\mathrm{cos}(t_2):1],
\begin{aligned}\rho(L_{-1})\rho(L_{k+1}) = h(z)^k (L-\lambda-k-1)(L+(k+1)\lambda) P(L-\lambda-k,k) \rho(L_{0})\rho(L_{k}) = h(z)^k (L-k) (L+k\lambda ) P(L-\lambda-k,k) \rho(L_{1})\rho(L_{k-1}) = h(z)^k (L+\lambda-k+1) (L+(k-1)\lambda) P(L-\lambda-k,k) \end{aligned}
\sum_{\substack{a_1, a_2, \ldots, a_k =1 (a_1\cdots a_k,n)=1 (a_1+\cdots+a_k,n)=1}}^n (a_1+\cdots+a_k-1,n) = \varphi_k(n)\sigma_0(n)\;\;\;\;\;\;\;\;
t^i\ell_{i}(z_j)\cdot (Y_1Y_2\dots Y_{j-1}Y_j Y_{j+1}\dots Y_nW) =t^i\ell_{i}(z_j)\cdot (Y_jY_1\dots Y_{j-1}Y_{j+1}\dots Y_nW) &=t^i\cdot\tau^i(Y_j)Y_1\dots Y_{j-1}Y_{j+1}\dots Y_nW &=\tau^i(Y_jW)Y_1\dots Y_{j-1}Y_{j+1}\dots Y_n.
B^r(m) :=\sum_{x=1}^m \prod_{k=1}^x \frac{1-b(k)}{1+b(k)},B^l(m):=\sum_{x=1}^m \prod_{k=1}^x \frac{1-b(-k)}{1+b(-k)}, I^r(b) :=\sum_{x=1}^{\infty}\frac{1}{1+B^r(x)},I^l(b):=\sum_{x=1}^{\infty}\frac{1}{1+B^l(x)},
\Phi_{|D(v)}\simeq \{(t; [x_1, \dots, x_h:0 : \dots :0];[0: \dots :0: t y_{n-p+r+1}: \dots : t y_{n-p+r+h-1}: y_{h}: \dots :y_{n-p+r-h-1}: 0 \dots:0] \in \mathbb{C}\times \mathbb{P}^{n-1}\times \mathbb{P}^{n-1}\left| t \sum_{a=r+1}^{h-1}x_a y_{n-p+a}+ x_h y_h=0 \right\}.
Ae^{i\omega t+i\beta sin(\Omega t)}=Ae^{i\omega t}(J_{0}(\beta)+\sum_{k=1}^{\infty}J_{k}(\beta)e^{ik\Omega t}+\sum_{k=1}^{\infty}(-1)^{k}J_{k}(\beta)e^{-ik\Omega t}),
I_u(x_0,r_1) - I_u(x_0,r_0) =\int_{r_0}^{r_1}\frac{2\,t}{H_u^2(x_0,t)}\Big(H_u(x_0,t) \; E_u(x_0,t) - D_u^2(x_0,t) \Big)\,\d t
+ \sqrt{| g_{E}|}\textstyle{\frac{1}{2}}e^{\varphi}a^{2}\left\{\left[ \mathrm{T r}\left( Q \mathcal{M}\right) \right]^{2}- 2 \mathrm{T r}\left( Q \mathcal{M}Q \mathcal{M}\right) \right\},
U_{0}= \left( \begin{array}{c c}{{Z_{1}+ Z_{2}}}&{{0}}\\{{0}}&{{Z_{1}+ Z_{2}+ \zeta}}\\{{- \sqrt{\zeta + \rho^{2}}\bar{z}_{2}}}&{{\rho z_{1}}}\\{{- \sqrt{\zeta + \rho^{2}}\bar{z}_{1}}}&{{- \rho z_{2}}}\\ \end{array}\right)
\begin{array}{cl}\Delta_{(n+1 l-1)}^{(0)}(x_1,\cdots,x_m|z_1,\cdots,z_{n+1}|z_{n+2},\cdots,z_N) = \displaystyle\sum_{k=1}^{n+1}\frac{\prod_{i=n+2}^{N}(z_i-z_k q^{-2})}{\prod_{\scriptstyle{j=1}\atop \scriptstyle{j\ne k}}^{n+1}(z_k-z_j)q^{-1}}\Delta^{(n l)}(x_1,\cdots,x_m|z_1,\mathop{\hat{\cdots}}^{k},z_{n+1}|z_k, z_{n+2},\cdots,z_N), \end{array}
({\cal E}^{\underline{D-1}}_{\tau})^2+\sum_{i=1}^{i=D-2}({\cal E}^{i}_{\tau})^2=({\cal E}^{\underline0}_{\tau})^2\equiv (\partial_\tau X^{\underline 0}-i\partial_\tau\bar\Theta\Gamma^{\underline 0}\Theta)^2,
K_n(x)= (-1)^n n! \varphi^{n+1}\frac{\det\left[ \begin{cases}x^{j-1}(k=1) \partial_\lambda^{j+k-3}\varphi (k\geq 2)\end{cases}\right]_{j,k=1,\dots,n+1}}{\det\left[\partial_\lambda^{j+k-2}\varphi\right]_{j,k=1,\dots,n+1}}.
-\Delta_x \nabla_\eta W_{\geq m,1}(\eta,x)=-4\pi \sum_{k\in\mathbb{Z}^3\setminus\{0\}}ik e^{-i\eta\cdot k}\delta_k(x)+4\pi\sum_{\substack{|k|_\infty<m k\in\mathbb{Z}^3}}ixe^{i(2\pi k-\eta)\cdot x},
q[\frac{1}{16}(\sqrt{22}-3\sqrt{2})(\frac{1}{3}\sqrt[3]{6\sqrt{3}+2\sqrt{11}}-\frac{1}{3}\sqrt[3]{6\sqrt{3}-2\sqrt{11}}+\frac{1}{3}\sqrt{11}+1)^{4}]=exp(-\frac{1}{11}\sqrt{11}\pi)
S_1=\int d\tau \left( \begin{array}{c}p_1\cdot \dot{x}_1-p_1\cdot \lambda_2\,A_1 +\bar{\theta}_1\gamma^{\mu \nu}\dot{\theta}_1\,p_{1\mu}\lambda_{2\nu}- \frac 12e_1p_1^2 +\bar{\theta}_1\gamma^{11,\nu}\dot{\theta}_1\,m_1\lambda_{2\nu}-\frac 12e_1m_1^2\end{array}\right) .
\frac{d}{dt}[n(t,q(t,x))q_{x}(t,x)]=(n_t+n_xq_t)q_x+n(t,q)q_{tx}&=(n_t+(uv-u_xv_x)n_x+(uv-u_xv_x)_xn)(t,q(t,x))q_x(t,x) &=-\frac{1}{2}n(uv_x-vu_x)(t,q(t,x)).
\mathcal{H}^{(\beta-1+\frac{i\alpha}{2},\beta-1-\frac{i\alpha}{2})}_{n}(x)=\dfrac{(2\omega)^n(\beta+\frac{i\alpha}{2})_n}{(n+2\beta-1)_n}{}_2 F_1\left(-n,n+2\beta-1; \beta+\frac{i\alpha}{2};\dfrac{\omega-x}{2\omega}\right).
u_{i}^{n+1}=u_{i}^{n}-\frac{\Delta t}{2\Delta x}A[u_{i+1}^{n}-u_{i-1}^{n}]+\frac{\Delta t^{2}}{2\Delta x^{2}}A^{2}[u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}].
\sum_{b=1}^N \Big( a_\mu^{(b)},j_\mu^{(b)}\Big) =\sum_{I=1}^N \sum_{b=1}^N\Big( U_{I b}A_\mu^{(I)}, j_\mu^{(b)}\Big) :=\sum_{I=1}^N \Big( A_\mu^{(I)}, J_\mu^{(I)}\Big) \; ,
\Phi(z)=\begin{pmatrix}1 0 v(s) 1\end{pmatrix}\left(I+\begin{pmatrix}m_1+m_4 m_2 m_3 m_1-m_4\end{pmatrix}\right)e^{\frac{1}{4}i \pi \sigma_3}z^{-\frac{1}{4}\sigma_3}Ne^{sz^{1/2}\sigma_3},
x=\frac{1}{\pm\sqrt{1+t(\frac{1-X_{2}}{1+X_{2}})}},X_{2}=\frac{1}{\pm\sqrt{1+t_{2}(\frac{1-X_{4}}{1+X_{4}})}},X_{4}=\frac{1}{\pm\sqrt{1+t_{4}(\frac{1-X_{8}}{1+X_{8}})}}.
v_p(f^2_{\alpha, \beta}(1)) \geq \min \{v_p(\alpha) + v_p(d-r) + d v_p(\beta), v_p(\alpha) + (d-r) v_p(\beta), v_p(\beta) \}= \min \{v_p(d-r) + dv_p(\beta), (d-r)v_p(\beta)\}
f(x, y) = (4 w_1^2 x^2 + w_2^2)(1+xy^2) \sin(w_1 x^2) \sin(w_2 y) - 2 w_1 (1 + 2xy^2) \cos(w_1 x^2) \sin(w_2 y) - 2 w_2 xy \sin(w_1 x^2) \cos(w_2 y).
\left\lbrace\begin{array}{lll}\displaystyle \sum_{r=1}^p (u''_re_r- \Delta u_re_r + u_rAe_r)=0 \hbox{in}(T,+\infty) \times \Omega, \displaystyle \sum_{r=1}^p (\partial_\nu u_re_r + u_rBe_r) =0 \hbox{on}(T,+\infty) \times \Gamma.\end{array}\right.
\begin{pmatrix}a_1 b_1 c_1 d_1\end{pmatrix}\begin{pmatrix}\pi^{-1}-\pi \end{pmatrix}\begin{pmatrix}a_2 b_2 c_2 d_2\end{pmatrix}\begin{pmatrix}0 1\end{pmatrix}=\begin{pmatrix}\pi^{-1}a_1d_1-\pi b_1b_2 \pi^{-1}c_1d_2-\pi b_2d_1\end{pmatrix}
h(\lambda;\epsilon,\alpha) = \lambda\sum_{k=0}^\infty F_\alpha\left(\left(\frac{\epsilon^\alpha}{k}\right)^{1/(1-\alpha)}\right)\frac{e^{-\lambda}\lambda^k}{k!}= \lambda e^{-\lambda}+\lambda e^{-\lambda}\sum_{k=1}^\infty F_\alpha\left(\left(\frac{\epsilon^\alpha}{k}\right)^{1/(1-\alpha)}\right)\frac{\lambda^k}{k!}
\alpha_1(N)=c\frac{W[\psi_2,\psi_3]}{W[\psi_1,\psi_2,\psi_3]},\alpha_2(N)=-c\frac{W[\psi_1,\psi_3]}{W[\psi_1,\psi_2,\psi_3]},\alpha_3(N)=c\frac{W[\psi_1,\psi_2]}{W[\psi_1,\psi_2,\psi_3]}
\psi_t(u,x) \leq C_2^{1/4}|f_t(u,x)|+ \tilde{C}(1+3^p |f_t(u,x)|^p) |f_t(u,x)| & \leq \big(C_2^{1/4}+\tilde{C}\big) (|u|+|u|^{1+p}) \big(|f(t-x)|+3^p|f(t-x)|^{1+p}\big) & = (|u|+|u|^{1+p}) \, g(t-x).
\vec{\phi}_{\mathrm{T K 2 L}}( x ) = \frac{( - 1 )^{\alpha}}{4}( 1 - ( - 1 )^{\gamma}\tanh ( x + a ) ) \vec{e}_{1}+ \frac{( - 1 )^{\beta}}{4}( 1 + ( - 1 )^{\gamma}\tanh ( x + a ) ) \vec{e}_{2}
0=\frac{\partial}{\partial y}(y\beta(y))+\frac{1}{4}\frac{\partial^{2}}{\partial y^{2}}\beta(y)+\left(\frac{2}{\pi}\right)^{1/2}\left(\frac{64}{3}y^{6}-68y^{4}+46y^{2}-\frac{15}{4}\right)e^{-2y^{2}}
\ ||x^*||_1\,\varphi(t^*,x^*) - \mathbb{E}\left[L(\rho)X(\rho)\varphi \big(t^*-\rho,\mathfrak{X}(\rho)\big)\right] =\, \mathbb{E}\left[\int_0^{\rho}L(s)X(s)\, g\big(t^*-s,s,\mathfrak{X}\big)\, \mathrm{d}s \right] <\, 0\,,
\frac{d^{2}}{dx^{2}}e^{\frac{1}{x}}=\frac{d}{dx}(-\frac{1}{x^{2}}e^{\frac{1}{x}})=-\frac{-2}{x^{3}}\cdot e^{\frac{1}{x}}-\frac{1}{x^{2}}\cdot\frac{-1}{x^{2}}\cdot e^{\frac{1}{x}}=(\frac{2}{x^{3}}+\frac{1}{x^{4}})\cdot e^{\frac{1}{x}}
\begin{cases}|1-ta_{11}z_1|^{m_{11}}\cdot \ldots \cdot |1-a_{1j_{1}}z_{j1}|^{m_{1j_{1}}}\cdot \ldots \cdot |1-ta_{1n}z_n|^{m_{1n}}=r_1, \ldots |1-ta_{n1}z_1|^{m_{n1}}\cdot \ldots \cdot|1-a_{nj_{n}}z_{jn}|^{m_{nj_{n}}}\cdot \ldots \cdot |1-ta_{nn}z_n|^{m_{nn}}=r_n,\end{cases}
I_d(\xi) =\sum_{x\in\xi}d(x,\xi\setminus x)e^{C\beta_{s}}=D(\xi)e^{C\beta_{s}}, \intertext{and, analogously, taking into account that $\phi\geq0$,}I_b(\xi) =\sum_{x\in\xi}b(x,\xi\setminus x)e^{C\beta_{s-1}}\leq\sum_{x\in\xi}\frac{z(x)}{m(x)}d(x,\xi\setminus x)e^{C\beta_{s-1}}
A_{nh}:=[z^n](T_{h-1}-T_{h-2})(z) =[z^n]((T-T_{h-2})-(T-T_{h-1})) =[z^n]\frac{1-u^2}{u}\Bigl(\frac{u^{h}}{1-u^h}-\frac{u^{h+1}}{1-u^{h+1}}\Bigr) =[z^n]\frac{1-u^2}{u}\frac{u^h-u^{h+1}}{(1-u^h)(1-u^{h+1})}=[z^n]\frac{(1+u)(1-u)^2}{u}\frac{u^h}{(1-u^h)(1-u^{h+1})}
\int_0^1 U(s,x)^2 \mathrm{d}x-\frac{1}{d}\sum_{i=1}^d U(\frac{\lfloor sT \rfloor}{T},\frac{i}{d})^2 =o(1) \frac{1}{d}\sum_{i=1}^d \big( U^{d,T}(\frac{\lfloor sT \rfloor}{T},\frac{i}{d})^2-U(\frac{\lfloor sT \rfloor}{T},\frac{i}{d})^2 \big) =o(1).
\begin{aligned}\int_{\Omega}\frac{\delta\mathcal{F}}{\delta v}\wedge d(\partial\phi)= \big\langle\ast\frac{\delta\mathcal{F}}{\delta v},d(\partial\phi)\big\rangle_{L^{2}\Lambda^{1}(\Omega)}=(-1)^{n-1}\int_{\partial\Omega}\mathrm{tr}(\frac{\delta\mathcal{F}}{\delta v})\wedge\partial\phi_{\partial}. \end{aligned}
g_b(z^2) =\frac{1}{1+3z}\,g_b\biggl(z\,\frac{1-z}{1+3z}\biggr), g_c(z^3) =\frac{1}{1+2z+4z^2}\,g_c\biggl(z\,\frac{1-z+z^2}{1+2z+4z^2}\biggr) \intertext{and}g_5(z^5) =\frac{1}{1+3z+4z^2+2z^3+z^4}\,g_5\biggl(z\,\frac{1-2z+4z^2-3z^3+z^4}{1+3z+4z^2+2z^3+z^4}\biggr).
a^{4123}=a^{4132},a_{1}^{4123}=a_{1}^{4132},a_{2}^{4123}=a_{2}^{4132},a_{3}^{4123}=a_{3}^{4132},a_{4}^{4123}=a_{4}^{4132}, a_{12}^{4123}=a_{12}^{4132},a_{13}^{4123}=a_{13}^{4132},a_{23}^{4123}=a_{23}^{4132},a_{24}^{4123}=a_{24}^{4132},a_{34}^{4123}=a_{34}^{4132}, a_{123}^{4123}=a_{123}^{4132},a_{234}^{4123}=a_{234}^{4132}
{J}_2(t,\tau) := M A(t)^{-1/2}\sin(\tau A(t)^{1/2}) M^* \widehat{P}- M_0 (t^2 M_0 \widehat{S}M_0)^{-1/2}\sin (\tau (t^2 M_0 \widehat{S}M_0)^{1/2}) M_0 \widehat{P}.
\phi_{i j}( x^{-}, 0 ) = \frac{1}{\sqrt{2 \pi}}\int_{0}^{\infty}\frac{d k^{+}}{\sqrt{2 k^{+}}}\left( a_{i j}( k^{+}) \mathrm{e}^{- i k^{+}x^{-}}+ a_{j i}^{\dag}( k^{+}) \mathrm{e}^{i k^{+}x^{-}}\right) ,
\begin{aligned}d\tilde F_h(w) =dF_{\Gamma_h}d\Psi_h(w) &=\kappa\int_{\Gamma}\big(\Delta_{g(h)}H(h) + H(h)((H(h)^2/2-2K(h)) &\quad\quad\quad+ C_0(2K(h)-H(h)\,C_0/2))\big)(\nu_h\circ\phi_h)\cdot\nu\,w\,\sqrt{g(h)}\,dA.\end{aligned}
q(P)=\sum_{i=1}^{k}\sum_{j=1}^{k}\frac{|V_{i}||V_{j}|}{n^{2}}d(V_{i},V_{j})^{2}\le\sum_{i=1}^{k}\sum_{j=1}^{k}\frac{|V_{i}||V_{j}|}{n^{2}}=1
\partial^{\varsigma}(a+\gamma)^{-1}=(a+\gamma)^{-1}\sum_{k=1}^{|{\varsigma}|}\sum_{\substack{{\varsigma}={\varpi}^1+\dots +{\varpi}^k |{\varpi}^j|\geq 1}}C_{{\varpi}^1,\dots,{\varpi}^k}\prod_{j=1}^k((a+\gamma)^{-1}\partial_x^{{\varpi}_{j,1}}\partial_\xi^{{\varpi}_{j,2}}\partial_\lambda^{{\varpi}_{j,3}}a).