harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}K(t,s)= \sqrt{\frac{2H}{(1-2H)\beta(1-2H,H+\frac{1}{2})}}\cdot\left[\left(\frac{t}{s}\right)^{H-\frac{1}{2}}(t-s)^{H-\frac{1}{2}}-\left(H-\frac{1}{2}\right)s^{\frac{1}{2}-H}\int_{s}^{t}u^{H-\frac{3}{2}}(u-s)^{H-\frac{1}{2}}du\right],\end{aligned}
\{b_{2i-1}, b_{2i}\}\subset G_{\Gamma( \mathcal{B}_{\sum_{i=1}^{2n}\varepsilon_{i}2^{i-1}})}\ \mathrm{and}\a_{2i-1}, a_{2i}\not\in G_{\Gamma( \mathcal{B}_{\sum_{i=1}^{2n}\varepsilon_{i}2^{i-1}})}
\|w\| = \|AA^{-1}w\| \le \|A^\frac{1}{2}\| \cdot \|A^\frac{1}{2}A^{-1}w\|= \|A^{-1}w\|_A , \|w_c\| = \|A_c^{-\frac{1}{2}}A_c^\frac{1}{2}w_c\| \le \|A_c^{-\frac{1}{2}}\| \cdot \|A_c^\frac{1}{2}w_c\|= \kappa_c^\frac{1}{2}\|w_c\|_{A_c},
\omega^{1 2}= - i \{\frac{1}{2}d \ln ( \frac{d \overline{{{g}}}_{1}}{d \overline{{{z}}}_{1}}\frac{d z_{1}}{d g_{1}}) + \frac{( g_{1}d \overline{{{g}}}_{1}- \overline{{{g}}}_{1}d g_{1})}{( 1 - g_{1}\overline{{{g}}}_{1})}\},
\sum_{l \geq 0}\sum_{\mid m \mid \leq l}\int_{1}^{\infty}d \lambda u_{\lambda , l , m}( r , \theta , \phi ) \overline{{{u}}}_{\lambda , l , m}( r^{\prime}, \theta^{\prime}, \phi^{\prime}) = \Omega ( k ) \delta^{3}( k - k^{\prime}) .
{{\bf{X}}^{[i]}}(n) = \sum\limits_{j = 1}^N{\sum\limits_{a = 1}^A{s_a^{[ji]}{\bf{v}}_a^{[ji]}(n)}}= \sum\limits_{j = 1}^N{{\bf{V}}_j^{[i]}(n){{\bf{s}}^{[ji]}}},
\left| E_j - E_k \right| \leq \left| E_j - \xi^{(x_j)}\right| + \left| \xi^{(x_j)}- \xi^{(e_{i_j})}\right| + \left| \xi^{(x_k)}- \xi^{(e_{i_k})}\right| + \left| \xi^{(x_k)}- E_k \right| \leq 6 \sqrt K \tau_n = \epsilon_n \, .
H_n^{(1)}(z) = \sqrt{\frac{2}{\pi z}}^{i(z-\frac{n\pi}{2}-\frac{\pi}{4})}\left( 1+\mathcal{O}(\frac{1}{z})\right), \z \to \infty, H_n^{(1)\prime}(z) = \sqrt{\frac{2}{\pi z}}^{i(z-\frac{n\pi}{2}+\frac{\pi}{4})}\left( 1+\mathcal{O}(\frac{1}{z})\right), \z \to \infty,
\int_{\{z\in D:\psi(z)=r\}}|f|^2e^{-\varphi}\left(\frac{\partial \psi}{\partial v_z}\right)^{-1}|dz|\ge \int_{\{z\in D:\psi(z)=r\}}|F_0|^2e^{-\varphi}\left(\frac{\partial \psi}{\partial v_z}\right)^{-1}|dz|.
U_1 \subset B_\delta (f^{k^+}(\sigma (f^{n_0}(y_0))))\subset V_\Lambda, f^{k^+}(\bar x_0)\in U_1 \cap W^s(f^{k^+}(\sigma(f^{n_0}(y_0)))), f^{-k^+}(U_1)\subset V_0.
\det(g) = (\alpha_1 \delta_1 - \beta_1 \gamma_1) + (\alpha_2 \delta_2 - \beta_2 \gamma_2) \epsilon + (\alpha_1 \delta_2 + \alpha_2 \delta_1 - \beta_1 \gamma_2 - \beta_2 \gamma_1) \sqrt{\epsilon}&= (\alpha_1 \delta_2 + \alpha_2 \delta_1 - \beta_1 \gamma_2 - \beta_2 \gamma_1) \sqrt{\epsilon}.
\omega^{( S ) - 1}( x ) \{\tilde{A}_{i}^{( S )}( x ) - S ( x ) \tilde{A}_{i}^{( N )}( x ) S^{- 1}( x ) \}\omega^{( S )}( x ) = 2 \sqrt{6}\{{\cal T}_{i}^{( N )}( x ) -{\cal T}_{i}^{( S )}( x ) \},
{\bf P}_0 = \sum_{\nu=1}^{N}{z_\nu}\,{\cal W}_\nu (\widetilde{\bf P}_R)= \sum_{\nu=1}^{N}{z_\nu}\,{\cal W}_\nu (\widetilde{\mathbf{P}}_{R_s}+ \widetilde{\mathbf{P}}_{R_l}) =:{\bf P}_s +{\bf P}_l.
\begin{array}{r}\dfrac{U^{n+1}-U^{n}}{\Delta t}-\dfrac{U_{xx}^{n+1}+U_{xx}{}^{n}}{2}+k_{1}\dfrac{(UU_{x})^{n+1}+(UU_{x}^{n})^{n}}{2}+k_{2}\dfrac{(UV)_{x}^{n+1}+(UV)_{x}^{n}}{2}=0 \dfrac{V^{n+1}-V^{n}}{\Delta t}-\dfrac{V_{xx}^{n+1}+V_{xx}{}^{n}}{2}+k_{1}\dfrac{(VV_{x})^{n+1}+(VV_{x}^{n})^{n}}{2}+k_{3}\dfrac{(UV)_{x}^{n+1}+(UV)_{x}^{n}}{2}=0\end{array}
L_{\hbar}=L\left( \left\{\hbar^{l}b_{l,\hbar}(\sqrt{\hbar}\left(\cdot\right) )\right\}_{l=0}^{m}\right) =\sum_{l=0}^{m}(-\hbar)^{l}\partial^{l}b_{l,\hbar}(\sqrt{\hbar}\left( \cdot\right) )\partial^{l}\mathcal{S}.
s= (e_{h,1}e_{\alpha,2}-e_{h,2}e_{\alpha,1})e_{\alpha_2}; t= (e_{g,1}e_{\alpha,2}-e_{g,2}e_{\alpha,1})e_{\alpha_2}; r= (e_{g,2}e_{h,1}-e_{g,1}e_{h,2})e_{\alpha_2}
\int\limits^{\infty}_{0}\!dt\,Im\,T_{1}(t)=-\frac{m^2}{4}\int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t^{2}}\left(1-\frac{4m^{2}}{t}\right)^{\hspace{-3pt}\frac{1}{2}}=-\frac{1}{24}\, \, .
E_{F}= \int_{x_{1}}^{x_{2}}\left[{\frac{1}{2}}( \nabla \phi )^{2}+{\frac{1}{2}}( \nabla \psi_{1})^{2}+{\frac{1}{2}}( \nabla \psi_{2})^{2}+ V ( \phi , \psi_{1}, \psi_{2}) \right] \, d x \ .
\mathcal{I}(\mathbf{F}_R, \mathbf{F}_B) = \log_2 \left( \left| \mathbf{I}_{N_s}+ \frac{\gamma}{N_s}\mathbf{H}\mathbf{F}_R\mathbf{F}_B \mathbf{F}_B^H\mathbf{F}_R^H \mathbf{H}^H \right| \right).
D_0(x) = 1 D_1(x) = 2x D_2(x) = 2x+x^2 D_3(x) = 2x+2x^2 D_4(x) = x+4x^2+x^3 D_5(x) = 5x^2+4x^3+x^4 D_6(x) = 3x^2+9x^3+3x^4+x^5 D_7(x) = x^2+11x^3+10x^4+3x^5+x^6
Q(t_1,t_2,t_3)= \left(\widetilde{P}_1'\widetilde{P}_2'\right)(t_2)\left(\widetilde{P}_1'\widetilde{P}_2'\right)(t_3)\left(\widetilde{P}_1'(t_2)\widetilde{P}_2'(t_3)-\widetilde{P}_1'(t_3)\widetilde{P}_2'(t_2)\right)\cdot \left(\widetilde{P}_2''\widetilde{P}_1'(t_1)-\widetilde{P}_1''\widetilde{P}_2'(t_1)\right)
E\bigg[\int_0^{\infty}e^{-k\beta t}V(dt)\bigg] =E\bigg[\int_0^{\infty}\int_t^{\infty}k\beta e^{-k\beta s}ds\,V(dt)\bigg] &=E\bigg[\int_0^{\infty}k\beta e^{-k\beta s}\int_0^s V(dt)\,ds\bigg] &=E\bigg[\int_0^{\infty}k\beta e^{-k\beta s}V(s)\,ds\bigg] &=\int_0^{\infty}k\beta e^{-k\beta s}EV(s)\,ds.
\aligned\frac{1}{2}\mathcal{L}S= \sum_{i,j,k,p}(h_{ijk}^{p^{\ast}})^{2}+S(1-\frac{3}{2}S)+2H^{2}S-\frac{1}{2}H^{4}-\sum_{j,k,p,q}H^{p^{\ast}}h_{jk}^{p^{\ast}}H^{q^{\ast}}h_{jk}^{q^{\ast}}.\endaligned
P_{n}(y)=\frac{1}{H_{n-1}({\bf c})}\det\begin{pmatrix}c_{0}c_{1}\cdots c_{n}c_{1}c_{2}\cdots c_{n+1}\vdots \vdots \ddots \vdots c_{n-1}c_{n}\cdots c_{2n-1}1 y \cdots y^{n}\end{pmatrix},
a_{13}^{3142}=a_{13}^{3124}=a_{13}^{3412}=a_{13}^{3421}=a_{13}^{3214}=a_{13}^{3241}= a_{13}^{4312}=a_{13}^{4321}=a_{13}^{4231}=a_{13}^{2314}=a_{13}^{2341}=a_{13}^{2431}
\beta_{10}^{2}=\frac{4}{35}\frac{(r-5L)(5r^7+25r^6 L+97 r^5 L^2 +485r^4 L^3 +2495 r^3 L^4 + 12475 r^2 L^5 +62235 L^6 +7L^7)}{(r^2-L^2)^4}.
\int_{B_r}\int_{B_{r}}\frac{|v(x)-v(y)|^{2}}{|x-y|^{n+2\alpha}}+f_\varepsilon\left(\int_{\Omega^c\cap B_r}\chi_{\{v>0\}}\right) &=\int_{B_r}\int_{B_{r}}\frac{|v(x)-v(y)|^{2}}{|x-y|^{n+2\alpha}}+g_\sigma(v-\varphi)+\lim_{\delta\rightarrow0}f_\varepsilon\left(\int_{\Omega^c\cap B_r}h_\delta(v)\right).
\sum_{k=1}^{n}\binom{k+p}{p}H_k = \sum_{k=1}^{n}\sum_{j=1}^{k}\binom{k+p}{p}j^{-1}=\sum_{j=1}^{n}\sum_{k=j}^{n}\binom{k+p}{p}j^{-1}& = \sum_{j=1}^{n}j^{-1}\sum_{k=1}^{n}\binom{k+p}{p}- \sum_{j=1}^{n}j^{-1}\sum_{k=1}^{j-1}\binom{k+p}{p}& = \binom{n+p+1}{p+1}H_n - \sum_{j=1}^{n}\binom{j+p}{p+1}j^{-1},
S_{P}^{d + 1}= \int_{M_{d + 1}}\Bigl [{\frac{1}{2}}B_{d - 1}\wedge F +{\frac{1}{8}}B_{d - 1}\wedge{}^{\star}B_{d - 1}+{\frac{\mu^{2}}{2}}A \wedge{}^{\star}A \Bigl ] ,
G_j(x,y) = y_j^2 -{1\over 4}a_j x_j^2 - x_j y_j y_0 +{1\over 4}x_j^2 x_0-{1\over 4}\sum_{k=1\atop k\neq j}^{2m}{\left(x_j y_k - x_k y_j\right)^2\over a_j - a_k}\> .
K(h^{\frac{1}{2^n}},2)^{r_n}H_{\nu}(a,b) \leqslant a\nabla b -\sum_{k=0}^{n-1}r_{k}\big[H_{\frac{m_k}{2^k}}(a,b)-2H_{\frac{2m_k+1}{2^{k+1}}}(a,b)+H_{\frac{m_k+1}{2^k}}(a,b)\big] \leqslant K(h^{\frac{1}{2^n}},2)^{R_n}H_{\nu}(a,b),
\lim_{N{\rightarrow}\infty}\sum_{n = 1}^{N}\phi_{n}( x )^{\dag}\gamma_{5}\phi_{n}( x ) = \lim_{\Lambda \rightarrow \infty}\sum_{n = 1}^{\infty}\phi_{n}( x )^{\dag}\gamma_{5}f ( \lambda_{n}^{2}/ \Lambda^{2}) \phi_{n}( x )
\sup_{\delta < |t| \leq \frac{\pi}{h}}\left \lvert \frac{\phi(s + \mathrm{i}t)}{\phi(s)}\right \rvert = \sup_{\delta < |t| \leq \frac{\pi}{h}}\left \lvert \prod_{i = 1}^n \frac{\phi_i(s + \mathrm{i}t)}{\phi_i(s)}\right \rvert \leq c_1^n = o(n^{-1/2}).
\left\Vert\frac{1}{nS}\sum_{m=0}^{n-1}\int_0^S B(s+mS) K_1(I-P^d_{M_{k-1}})e^{-isH_d}E_{H}(B){\hat P}^{k-2}_{{\hat M}_d}ds e^{-imSH}\right\Vert.
\gamma_{\scriptscriptstyle{l}}= \Phi / 2 \pi - l + \sqrt 2 \pi \, b_{\scriptscriptstyle{l - 1}}\, a^{l}/ \psi_{\scriptscriptstyle{l}}^{<}( k , a ) + k^{2}[ ( \Phi / 2 \pi - l - \gamma_{\scriptscriptstyle{l}}) \, \chi_{\scriptscriptstyle{l}}( a ) - ( r \partial_{r}\chi_{\scriptscriptstyle{l}})_{a}] + \mathrm{O}( k^{4}) \, .
D \cdot c_{j,i}= D_{j,i}D_{j-1,i}^{-1}c_{j,i}= \chi_{j,i,i}(D) c_{j,i}; D \cdot a_{j,k,l,m}= D_{j,k}D_{j-1,l}^{-1}a_{j,k,l,m}= \chi_{j,k,l}(D) a_{j,k,l,m},
\frac{1}{2}+\frac{\xi^{-k}x_2/x_1}{1-\xi^{-k}x_2/x_1}=\frac{1}{2}\frac{1+\xi^{-k}x_2/x_1}{1-\xi^{-k}x_2/x_1},\ \ \ frac{1}{2}+\frac{\xi^{k}x_1/x_2}{1-\xi^{k}x_1/x_2}=-\frac{1}{2}\frac{1+\xi^{-k}x_2/x_1}{1-\xi^{-k}x_2/x_1}.
\begin{aligned}\widetilde V'_i=\widetilde V_{i+k}-x_i \widetilde V_{i+1}, \widetilde V'_{i+1}=\widetilde V_{i+k}+ y_{i}\widetilde V_{i+1}, \widetilde V'_{i+k-1}= \widetilde V_{i+2k-1}-x_{i+k-1}\widetilde V_{i+k}, \widetilde V'_{i+k}= \widetilde V_{i+2k-1}+ y_{i+k-1}\widetilde V_{i+k}.\end{aligned}
A_{3}=\begin{pmatrix}a_{6,6}\cdots a_{6,n}\vdots \cdots \vdots a_{n,6}\cdots a_{n,n}\end{pmatrix}\ \textrm{and}\A_{3}'=\begin{pmatrix}a_{6,6}' \cdots a_{6,n}' \vdots \cdots \vdots a_{n,6}' \cdots a_{n,n}'\end{pmatrix},
k_{( i ) \mu \nu}= -{\frac{\mu}{4}}( f_{~ \mu \lambda}^{Y}f_{~ ~ \nu}^{( i ) \lambda}+ f_{~ \nu \lambda}^{Y}f_{~ ~ \mu}^{( i ) \lambda}) +{\frac{1}{2 \mu}}( k_{( 5 ) \mu}k_{( i ) \nu}+ k_{( 5 ) \nu}k_{( i ) \mu}) .
(4f^2gg_z-2ff_zg^2)h-4f^2g^2h^2-2f^2g^2h_z-g^2ff_{zz}+ f^2gg_{zz}-2f^2g_z^2 +f_z^2g^2+f_zgfg_z+4V_2f^3g^2=0.
\sum_{y=1}^{\left \lfloor{R}\right \rfloor}(R^{2}- y^{2})^{1/2}\ge \frac{\pi R^{2}}{4}-\frac{R}{2}+(R^2-\lfloor R\rfloor^2)^{1/2}-\frac12(R^2-(\lfloor R\rfloor-1)^2)^{1/2}-\frac23 (R-\lfloor R \rfloor)^{3/2}(2R)^{1/2}.
A ( Z ) ={\frac{1}{3 2 0 \pi^{4}}}\left[{\frac{1}{( Z - 1 )^{4}}}+{\frac{1}{( Z + 1 )^{4}}}\right] ={\frac{1}{1 0}}\left[ ( \Delta_{1}^{\prime})^{2}+ ( \Delta_{2}^{\prime})^{2}\right] ,
XY+YZ+ZX & >2(1+\frac{16}{5}\delta)(1-\frac{16}{5}\eta)+(1-\frac{16}{5}\eta)^{2}& =3+\frac{32}{5}\delta+(\frac{16}{5})^{2}\eta^{2}-\frac{2\times16^{2}}{5^{2}}\delta\eta-\frac{64}{5}\eta & >3+(\frac{16}{5})^{2}\eta^{2}-\frac{32^{2}}{5^{3}}\delta^{2}+\frac{32}{5^{2}}\delta & >3(0<\delta<\frac{5}{32}).
\lim_{t_0\to -\infty}\overline{u}(t-t_0,0)=\frac{a-\chi\underline{u}_{0}}{b-\chi}\lim_{t_0\to -\infty}\underline{u}(t-t_0,0)=\frac{a-\chi\overline{u}_{0}}{b-\chi}.
\displaystyle\lambda_1=\inf\left\{\frac{\int_{\Sigma}|\nabla \varphi|^2e^{\frac{|x|^2}{4}}d\sigma}{\int_{\Sigma}\varphi^2e^{\frac{|x|^2}{4}}d\sigma}; \varphi\in C_0^{\infty}(\Sigma), \int_{\Sigma}\varphi^2e^{\frac{|x|^2}{4}}d\sigma\neq 0\right\}.
\quad \tilde{g}_{i,j}(x)\tilde{g}_{j,i}(1/x)=1, &\tilde{g}_{i,j}(x_2/x_1)=\frac{G^+_{i,j}(x_1,x_2)}{F^+_{i,j}(x_1,x_2)}=\frac{F^-_{i,j}(x_1,x_2)}{G^{-}_{i,j}(x_1,x_2)}i,j\in I.
\overline{a^{-1}((-1, 1)) \cap b^{-1}(0)}\backslash a^{-1}((-1, 1)) \cap b^{-1}(0) &=\overline{(a, b)^{-1}((-1, 1) \times \{0 \})}\backslash(a, b)^{-1}((-1, 1) \times \{0 \}) &\subset(a, b)^{-1}( \overline{(-1, 1) \times \{0 \}}) \backslash(a, b)^{-1}((-1, 1) \times \{0 \}) = \pi(W).
\left\{L(f)+\Gamma(f) \right\}\sqrt{F_E^0}=\frac{U_\mu p^\mu}{c p^0}\left\{\biggl(1-p^\mu q_\mu \frac{1+\frac{\mathcal{I}}{mc^2}}{bmc^2}\biggl)F_E-F\right\}\equiv \frac{1}{p^0}Q.
f(0)={}0=f'(0), f''(0)={}2g(J_1'(0),J_2'(0))=2g(W_1'(0),W_2'(0)), f'''(0)={}\partial_r[g\left(R^g(\partial_r,J_1(r))\partial_r,J_2(r)\right)+2g(J_1'(r),J_2'(r))+g\left(J_1(r),R^g(\partial_r,J_2(r))\partial_r\right)]\bigr|_{r=0}=0
{\cal L}={\frac{m}{2}}\, \epsilon^{\mu \nu \lambda}A_{\mu}^{a}( \partial_{\nu}A_{\lambda}^{a}+{\frac{g}{3}}f^{a b c}A_{\nu}^{b}A_{\lambda}^{c}) - F^{a}( \partial_{\mu}A^{\mu a}) + \partial^{\mu}\overline{{{c}}}^{a}( D_{\mu}c^{a})
\zeta_{\rho}(s)=-\frac{2\, R^{2s}}{\Gamma(s+1)\,\Gamma(-s)}\sum_{l=1}^{\infty}\nu^{1-2s}\int_0^{\infty}dk\,k^{-2s}\frac{d}{dk}\ln\Delta_{\rho,l}(\nu k), \quad \nu=l+\frac{1}{2}, \;\;\rho=\pm1
\mathbb{P}_{n_{l}}\left[ -M \le \log\left(Y_{n_{l}}\right) \le M \cap -M \le \left\{\sum_{i=1}^{k}\frac{2\mu_{i}W_{n_{l},i}- \mu_{i}^2}{2 \sigma_{i}^2}\right\}\le M \right] \ge 1- \frac{\delta}{50}.
e'-\sigma(e') =e-e\sigma(x)\sigma(e)-\sigma(e)+ex\sigma(e)=e-\sigma(e)+e(x-\sigma(x))\sigma(e) =e-\sigma(e)+\lambda^{-1}eu\sigma(e)=e-\sigma(e)+e\sigma(e)+1+\lambda^{-1}u-2e &=1-e-\sigma(e)+e\sigma(e)+\lambda^{-1}u=(1-e)(1-\sigma(e))+\lambda^{-1}u=\lambda^{-1}u.
e^{\textstyle - J \alpha_{-}\Phi ( \sigma , \tau )}= c_{J}a ( \varpi ) \sum_{m = - J}^{J}\, ( - 1 )^{J - m}\,{\frac{\lambda_{m}^{( J )}( \varpi )}{\lfloor \varpi + 2 m \rfloor}}\psi_{m}^{( J )}( \sigma ) \,{\overline{{\psi}}_{m}^{( J )}}( \sigma ) \bigr / a ( \varpi ) ,
\sum_{j , j^{\prime}; \mu , \mu^{\prime}}z_{j}^{( \mu )}\{J_{( \mu )}^{j , - j}, J_{( \mu^{\prime})}^{j^{\prime}, - j^{\prime}}\}^{*}( y ) t_{j^{\prime}, - j^{\prime}}^{( \mu^{\prime})}= [ R ( \vec{z}; y ) , y ]
\displaystyle \sum_{{\bf k}=(k_1,...,k_{n-1}) \in \mathbb{N}^{n-1}_{\geq 0}}\frac{{\bf x^{k}}}{\prod_{i=1}^{n-1}(q)_{k_i}}=\sum_{\eta}\frac{q^{{\rm codim}(\eta)}{\bf x^{\overline{dim}(\eta)}}}{\displaystyle \prod_{1\leq i\leq j\leq n-1}(q)_{m_{[i,j]}(\eta)}},
\begin{aligned}h(y) =\mathbf{E}_y(T_2+T_1\circ \theta_{T_2}) &=\mathbf{E}_yT_2+\frac{z-y}{z-x}h(x)+\frac{y-x}{z-x}h(z) &>\frac{z-y}{z-x}h(x)+\frac{y-x}{z-x}h(z). \end{aligned}
F_{\gamma_{SCHCN}}(\gamma) \approx \mathcal{L}^{-1}\left[\frac{N!}{N_c!{N_c}^{N-N_c}}\frac{\lambda_0{\lambda_{eq}}^N}{s^{N+2}}\right] &=\frac{1}{(N+1)N_c!{N_c}^{N-N_c}}\lambda_{0}{\lambda_{eq}}^N\gamma^{N+1}.
\mu_c \|\Pi_h u - u_h\|_h^2 \leq a_h(\Pi_h u - u_h,\Pi_h u - u_h) & = a_h(\Pi_h u - u,\Pi_h u - u_h) & \leq \mu_b \|\Pi_h u - u\|_{h,*}\| \Pi_h u - u_h\|_{h}.
\begin{array}{ll}\int\limits_0^t \int\limits_{\mathbb{R}^3}\partial_t\varphi_1\xi + \gamma k_1 v\cdot(\xi\nabla\varphi_1 +\varphi_1\nabla\xi)+\gamma k_1 \bar{p}v\cdot\nabla\varphi_1- k_1 \varphi_1 v\cdot\nabla \xi \,\mathrm{d}x\mathrm{d}s begin{align*}6pt]= -\int\limits_{\mathbb{R}^3}\xi_0(x,\tau) \varphi_1(\cdot,0)\,\mathrm{d}x.\end{array}
p_{0 \mu}= p_{\mu}- \frac{1}{2}\tilde{S}^{h_{1}h_{2}}\omega_{h_{1}h_{2}\mu}- \frac{1}{2}\tilde{S}^{k_{1}k_{2}}\omega_{k_{1}k_{2}\mu}- \frac{1}{2}\tilde{S}^{h_{1}k_{1}}\omega_{h_{1}k_{1}\mu},
\begin{aligned}\Bigl(B+A\tau-\frac12\Bigr)\gamma_{0,K}u_K = A(\partial_{R,K}u_K), \Bigl(B+A\tau+\frac12\Bigr)\gamma_{0,\Omega}u_\Omega = A(\partial_{R,\Omega}u_\Omega). \end{aligned}
q_1=\frac{q}{(d,|m^{\ast}|)}=\frac{k_3^3hd(m^{\ast})^2}{(d,|m^{\ast}|)}\mbox{and}L:=\max\left\{\left(\frac{|m^{\ast}\tilde{m}|}{k_2hd}\right)^{1/2}X^{3-\varepsilon}, \frac{|u|^{1/2}X^{4-\varepsilon}}{k_2k_3hd}\right\},
S_{P V}( \Phi , \Phi^{*}, \underline{{\Phi}}_{i}) = \frac{1}{2}\sum_{i = 1}^{N}{\underline{{\Phi}}}_{i}^{A}S_{A B}{\underline{{\Phi}}}_{i}^{B}- \frac{1}{2}M_{i}^{2}{\underline{{\Phi}}}_{i}^{A}T_{A B}{\underline{{\Phi}}}_{i}^{B},
T_{L}( t_{2}, t_{1})_{C}^{A}={\cal P}\exp [ \int_{t_{2}}^{t_{1}}d t \, O m e g a \circ H ]_{C}^{A}, \qquad T_{R}( t_{2}, t_{1})_{C}^{B}={\cal P}\exp [ \int_{t_{2}}^{t_{1}}d t \H \circ \Omega ]_{C}^{B}
(\int_{-\infty}^{\infty}e^{-x^{2}}dx)^{2}=\int_{-\infty}^{\infty}e^{-x^{2}}dx\int_{-\infty}^{\infty}e^{-y^{2}}dy=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^{2}+y^{2})}dxdy.
\mathbf{Err}_{i}^{\alpha}= \sum_{|\gamma|\leq |\beta|< |\alpha|, \beta +\gamma =\alpha}{\alpha \choose \beta}\mathcal{N}^{(\beta,\gamma)}_{i}+ \textup{commutator terms if $|\alpha_3|+|\alpha_4|=1$},{\alpha \choose \beta}:= \prod_{j=1}^4{\alpha_j \choose \beta_j}.
d = - \frac{\left\{\phi_{3}^{G}, \Phi_{1}\right\}}{\left\{\phi_{2}^{G}, \Phi_{1}\right\}}\, , \; \; v = - \frac{\left\{\phi_{3}^{G}, \Phi_{6}\right\}}{\left\{\Phi_{7}, \Phi_{6}\right\}}\, , \; \; u = - \frac{\left\{\phi_{3}^{G}+ v \phi_{1}^{\left( 1 \right)}, \phi_{4}^{G}\right\}}{\left\{\phi_{2}^{( 1 )}, \phi_{4}^{G}\right\}}\; .
b_{2,1}= \frac{1 - h_0 + h_{2,1}}{2}, b_{2,2}= \frac{1 - h_0 + 2 h_{1,1}- h_{2,2}}{2}, b_{2,3}= \frac{1 + h_0 - 2 h_{1,2}+ h_{2,3}}{2}, b_{2,4}= \frac{1 + h_0 - h_{2,4}}{2}.
d^2_{Q}(\theta,\gamma) =(d^2\theta,d_{r}^3\circ d_{r}^2\sigma -\frac{1}{2}B_{\mathfrak a}\left( d^1\tau\wedge (d^1\tau\circ \alpha)\right) + \frac{1}{2}B\left(d^1\tau\wedge (d^1\tau\circ \alpha) \right)) &=(0,0) .
\begin{array}{rcl}\hat{P}_6^{''}\!(x)\!\! \!=\! \!\!1920x^4\!-\!3840\pi x^3\!+\!\left(2880\pi^2\!-\!16128\right)x^2\! [1.0 ex]\!\! \! \! \!\! +\!\left(\!-960\pi^3\!+\!16128\pi \right)x+120\pi^4-4032\pi^2.\end{array}
l_1\left(BP\right) =\max\limits_{x\in P}\langle-h_d,x\rangle-\sum_{i=1}^{d-1}\min\limits_{x\in P}\langle h_i,x\rangle-\min\limits_{x\in P}\langle(-h_1-\cdots-h_d),x\rangle=l_1(AP).
\varprojlim_{\frac{1}{q}\operatorname{Tr}(L_{n+1}/L_n)}M_n \otimes_{L_n}L_{n+1}\cong \varprojlim_n M_{n_1}\otimes_{A}D(\Gamma_{n_1},A)/Z_{n}&=M_{n_1}\otimes_A \varprojlim_n D(\Gamma_{n_1},A)/Z_n = M_{n_1}\otimes_A D(\Gamma_{n_1},A)/\mathfrak{L}_{n_1}.
w'(EP) = \frac{-(n-0+(m+1)-0-1)}{2}+ w'(P) = -\frac{n+m}{2}+ v_{n,m}= -\frac{n+m}{2}- \sum_{k=D_{n,m}}^{D_{n,m}+m-1}k = -\sum_{k=D_{n,m}}^{D_{n,m}+m}k = -\sum_{k=D_{n,m+1}}^{D_{n,m+1}+(m+1)-1}k = v_{n,m+1}.
B^k A^k-(B^k A^k)^2 S =B^k A^k-(B^k A^k)^2 B^k T^2 A^K=B^k A^k-B^k (A^k B^k A^k) B^k T^2 A^k &=B^k A^k-B^k A^{k+2}T^2=B^k (I-A^2 T^2) A^k=B^k (I-A T) A^k &=B^k Q A^k=B^k Q^k A^k=B^k(QA)^k.
E_1(x,y)+E_2(x,y) \leq\displaystyle\sum_{\rho=0}^r\Gamma_{\rho,L}(x)\cdot\omega_{\rho}(F;\Lambda_{\rho,L}(x),0) &+\|L\|\cdot\displaystyle\sum_{\sigma=0}^s\Gamma_{\sigma,M}(y)\cdot\omega_{\sigma}(F;0,\Lambda_{\sigma,M}(y)).
\dim_{\kappa(b)}\pi_{i}(P\otimes_{A}B\otimes_{B}\kappa(b)) =\dim_{\kappa(b)}\pi_{i}(P\otimes_{A}\kappa(a)\otimes_{\kappa(a)}\kappa(b)) &=\dim_{\kappa(b)}\pi_{i}(P\otimes_{A}\kappa(a))\otimes_{\kappa(a)}\kappa(b) &=\dim_{\kappa(a)}\pi_{i}(P\otimes_{A}\kappa(a)),
\frac{A_{\sigma_n}(i)}{B_{\sigma_n}}\leq p_{\sigma_n}(i) < \frac{A_{\sigma_n}(i)+1}{B_{\sigma_n}}\;\; A_{\sigma_n}(1) = B_{\sigma_n}- \sum_{i\in\mathcal{I}_n^{\sigma}\setminus\{1\}}A_{\sigma_n}(i),
\sum_{x \in [1, p^{\alpha-1}]}\min\left(f(k), \sum_{i=1}^{r_k}\psi_{y_i}(x) \right) = f(k) \frac{p^{\alpha-1}-1}{2}+ \sum_{x=\frac{p^{\alpha-1}+1}{2}}^{p^{\alpha-1}-1}(-r_k x + p^{\alpha-1}r_k + b -a) + b = f(k)\frac{p^{\alpha-1}-1}{2}+ \frac{p^{\alpha-1}-1}{2}(b-a+p^{\alpha-1}r_k) &- \frac{r_k}{2}\frac{3p^{2\alpha-2}-4p^{\alpha-1}+1}{4}+ b.
\begin{array}{l}(x-1)^{\epsilon_4}h_2(x) =\begin{cases}(x-1)^{n-r+k_2}p_2(x)-(x-1)^{n-r+k_1-r_1+k_4}p_1(x)p_4(x) \mbox{if~}n-r+k_1>r_1, (x-1)^{r_1-k_1+k_2}p_2(x)-(x-1)^{k_4}p_1(x)p_4(x) \mbox{if~}n-r+k_1\le r_1,\end{cases}\end{array}
|\lambda_A\psi_A +\,\lambda_B\psi_B|^2 =\,|\lambda_A\,\psi_A|^2+\,|\lambda_B\psi_B|^2 +2 \Re (\lambda_A\lambda^*_B\psi_A\,\psi^*_B) & =\,|\lambda_A|^2\,|\psi_A|^2+\,|\lambda_B|^2\,\psi_B|^2 +2 \, \Re (\lambda_A\lambda^*_B\langle\psi_A|\psi^*_B\rangle).
\left(\frac{\gamma_i-Nx}{\alpha_i-x}\right)' =\frac{(-N)(\alpha_i-x)-(\gamma_i-Nx)(-1)}{(\alpha_i-x)^2}&=\frac{(\gamma_i-Nx)-N(\alpha_i-x)}{(\alpha_i-x)^2}&\le\frac{(N\alpha_i-Nx)-N(\alpha_i-x)}{(\alpha_i-x)^2}=0.
\begin{array}{l c l}{{{\cal C}_{c o n}}}&{{\equiv}}&{{\{8 6 4 \psi^{6}+ \phi = \pm 1 \},}}\\{{{\cal C}_{1}}}&{{\equiv}}&{{\{\phi = \pm 1 \},}}\\{{{\cal C}_{\infty}}}&{{\equiv}}&{{\{\phi , \psi = \infty \},}}\\{{{\cal C}_{0}}}&{{\equiv}}&{{\{\psi = 0 \}.}}\\ \end{array}
\begin{aligned}\omega^{(g)}_{2,0}=\sum_{r_1,r_2\ge0}D_{-x_1}^{r_1}D_{-x_2}^{r_2}[w_1^{r_1}w_2^{r_2}]e^{(w_1^3+w_2^3)\hbar^2/24}\tfrac{dz_1dz_2}{(z_i-z_j)^2-\frac{\hbar^2}{4}(w_i+w_j)^2}&=\sum_{i+k=g}\frac{1}{24^{i}i!4^k}(D_{-x_1}^{3}+D_{-x_1}^3)^{i}(D_{-x_1}+D_{-x_2})^{2k}\frac{dz_1dz_2}{(z_1-z_2)^{2(k+1)}}.\end{aligned}
\lambda(t)^{2(1-\sigma_c)}\widetilde{E}(t) \lesssim\lambda(t)^{2(1-\sigma_c)}\widetilde{E}(0)+b_c^{8}\lesssim\lambda(0)^{2(1-\sigma_c)}\widetilde{E}(0)+b_c^{8}&\lesssim \int|\varepsilon_y(0)+(Q_{b(0)})_y|^2\theta(\tilde{y})+b_c^{8}&\lesssim \int|(Q_{b(0)})_y|^2\theta(\tilde{y})dy+\|\varepsilon_y(0)\|_{L^2}^2+b_c^8 &\lesssim b_c^8,
S(\zeta)=\frac{\omega_\zeta(\zeta,\alpha)\omega(\zeta,\bar\alpha^{-1})-\omega_\zeta(\zeta,\bar\alpha^{-1})\omega(\zeta, \alpha)}{\prod_{j=1}^{M}\omega(\zeta,\gamma_{1}^{(j)}) \omega(\zeta,\gamma_{2}^{(j)})},
[\begin{matrix}2&3&1\\ 0&8&-2\end{matrix}]\circ[\begin{matrix}3&1&4\\ 7&9&5\end{matrix}]=[\begin{matrix}2\times3&3\times1&1\times4\\ 0\times7&8\times9&-2\times5\end{matrix}]=[\begin{matrix}6&3&4\\ 0&72&-10\end{matrix}]
\beta |f'(z^m)| +(1-\beta)|f(z^m)| + \sum_{n=1}^{\infty}|a_n|\phi_{n}(r) \leq \beta |f'_{0}(z^m)| +(1-\beta)|f_{0}(z^m)|+\phi_{1}(r) +\sum_{n=2}^{\infty}\left|\frac{f^{(n)}_{0}(0)}{n!}\right| \phi_{n}(r) \leq -f_{0}(-1) \leq d(0, \partial{\Omega}),
\sum_{n\ge 0}g\left(3^{2\alpha-1}n+\frac{3^{2\alpha}-1}{4}\right)q^n =\frac{E(q^{6})^5}{E(q^3)^4}\sum_{j\ge 1}d_{2\alpha-1}(j)X^j, \sum_{n\ge 0}g\left(3^{2\alpha}n+\frac{3^{2\alpha}-1}{4}\right) q^n =\frac{E(q^{2})^5}{E(q)^4}\sum_{j\ge 1}d_{2\alpha}(j)X^j.
G_r^{(2)}(x)=\sum^{\infty}_{n_0,n_1,\ldots,n_{\nu-2}=0}x^{\frac{r-1}{2}+\sum_{i=0}^{\nu-2}n_i(\nu-1-i)}\times\prod_{i=1}^{\nu-2}\left[\begin{array}{c}n_i+\tilde{n}_i+\tilde{V}_{i,r}n_i \end{array}\right] .
\begin{array}{c}{H_{l}s_{l}\left( r , \chi \right) = \left\{2 \sinh{}^{2}\frac{i}{2}\frac{d}{d r}+ \frac{l \left( l + 1 \right)}{2 r \left( r + i \right)}e^{i \frac{d}{d r}}\right\}s_{l}\left( r , \chi \right) =}\\{= \left( \cosh \chi - 1 \right) s_{l}\left( r , \chi \right)}\\ \end{array}
{\cal D}^{\alpha_{i}\widehat{e}}\widetilde{x}_{i}^{\mu}\circ \gamma [ \widetilde{\varphi}_{i}]^{- 1}= \mathrm{e}^{\frac{d}{1 6}( \widetilde{\varphi}_{i}( 0 ) + \widetilde{\varphi}_{i}( 1 ) )}{\cal D}^{\widehat{e}}\widetilde{x}_{i}^{\mu}, ( i \rightarrow f ) ,
[\begin{matrix}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{xy}\end{matrix}]=\frac{1}{1-\nu_{xy}\nu_{yx}}[\begin{matrix}E_{x}&\nu_{yx}E_{x}&0\\ \nu_{xy}E_{y}&E_{y}&0\\ 0&0&G_{xy}(1-\nu_{xy}\nu_{yx})\end{matrix}][\begin{matrix}\epsilon_{xx}\\ \epsilon_{yy}\\ 2\epsilon_{xy}\end{matrix}].
\alpha\left(v\alpha_{1}^{a_{1}}\alpha_{2}^{a_{2}}\alpha_{3}^{a_{3}}\right)\alpha^{-1}=\left(\alpha\cdot v\right)\alpha_{1}^{a_{1}b_{4}-a_{3}b_{3}+r_{3}a_{2}b_{1}^{-1}b_{4}+\frac{1}{2}a_{2}b_{4}\left(b_{1}^{-1}-1\right)}\alpha_{2}^{a_{2}b_{1}^{-1}b_{4}}\alpha_{3}^{a_{3}b_{1}},
D_{\pi,s,\tau}=\sum_{j=1}^{\dim\mathfrak{k}}\left[ \left( s-\frac{t}{2}\right) \pi_{\ast}(X_{j})^{2}+\frac{t}{2}\pi_{\ast}(Y_{j})^{2}-u\,\pi_{\ast}(X_{j})\pi_{\ast}(Y_{j})\right]
|I| \lesssim \frac{1}{\tau^{2j-1+l}}\|v^{-}\|_{l^{\frac{2j}{j^-+1}}_\tau DU^2}\|v^{+}\|_{l^{\frac{2j}{j^++1}}_\tau DU^2}\prod_{i = 2+j^-}^{2j-j^+ -1}\|v_i\|_{l^{2j}_\tau DU^2}
B_{n,\mathbf{m}}^{(d)}(\mathbf{z}+\omega_{j}\mathbf{1}\mid{\boldsymbol{\omega}})-B_{n,\mathbf{m}}^{(d)}(\mathbf{z}\mid{\boldsymbol{\omega}})= \sum_{i=1}^{r}B_{n-1,\mathbf{m}_{i}}^{(d)}(z\mid \widehat{{\boldsymbol{\omega}}}(j)) \left(m_{i}+\frac{d}{2}(r-i)\right) h_{-,i}^{(d)}(\mathbf{m}).
1 <p<p+\mu<N\theta\leq\frac{N\mu}{N-p}\leq s<N, 1 \leq q<r<\frac{Np}{N-p};a=\frac{\left[ \left( N-\theta\right)r-\left( N-s\right) q\right] p}{\left[ \left( N-\theta\right) p-\left(N-\mu-p\right) q\right] r}.
W_\mathcal{A}(f,g)(z) =e^{\pi i (Cz)\cdot z}|\det(I_{d\times d}-A_{11})|^{-1}e^{2\pi i((I_{d\times d}-A_{11}^T)^{-1}\xi)\cdot x}&\qquad\times V_{\tilde{g}}f(A_{11}^{-1}x,(I_{d\times d}-A_{11}^T)^{-1}\xi),
T=[\begin{matrix}1/2&0\\ 0&1/7\\ \end{matrix}]\{[\begin{matrix}0&0\\ -5&0\\ \end{matrix}]+[\begin{matrix}0&-1\\ 0&0\\ \end{matrix}]\}=[\begin{matrix}0&-1/2\\ -5/7&0\\ \end{matrix}].
\Big| \frac{1}{2m \ell n}\sum_{x}\Phi_{n}(s,\tfrac{x}{n}) \sum_{j=1}^{m-1}\int_{\Omega_2^j(x)}\sum_{z = x-\ell}^{x-1}[ \eta(z) - \eta(z - j \ell) ] \overrightarrow{\eta}^{\varepsilon n}(x+1) [f(\eta) - f( \eta_{1,x,j,z}) ] d\nu_{b}\Big|,
S t r \left( D_{a_{1}}\phi^{j_{1}}\ldots D_{a_{l}}\phi^{j_{l}}\phi^{i_{1}}\ldots \phi^{i_{n}}\underbrace{\phi^{i_{2 ( k + r )}^{\prime}}\phi^{i_{2 ( k + r ) - 1}^{\prime}}}\dots \underbrace{\phi^{i_{2}^{\prime}}\phi^{i_{1}^{\prime}}}\ldots F_{a_{p}a_{p - 1}]}\right) ,
f(v_{i,k}) =\begin{cases}6(k+1)+1-i, k=1,2 6k+1+i, 1 \leq i \leq 5, k=3 6k+1, i=6, k=3 6k+1-i, k>3, k-3\equiv_{5}1,2,3,4 6k+1+i, 1 \leq i \leq5, k>3, k-3\equiv_{5}0 6k+1, i=6, k>3, k-3\equiv_{5}0\end{cases}.
\frac{d\lambda}{dt}=-\lambda\big[2c(\lambda^2+\mu^2+\lambda\mu)+a\big]+ b\Big(\frac{1}{3}\lambda^2-\frac{2}{3}\mu^2-\frac{2}{3}\lambda\mu\Big), \frac{d\mu}{dt}=-\mu\big[2c(\lambda^2+\mu^2+\lambda\mu)+a\big]+ b\Big(\frac{1}{3}\mu^2-\frac{2}{3}\lambda^2-\frac{2}{3}\lambda\mu\Big).
\dot W^{1,\epsilon}_t :=\begin{cases}\mu_\eta t \in (4k\eta, (4k+2)\eta), -\mu_\eta t \in ((4k+2)\eta, (4k+4)\eta),\end{cases}\dot W^{2,\epsilon}_t :=\begin{cases}\mu_\eta t \in (2k\eta, (2k+1)\eta), -\mu_\eta t \in ((2k+1)\eta, (2k+2)\eta),\end{cases}
{\sf c}_it ={\sf c}_i(-{\sf c}_{m_1}\ldots{\sf c}_{m_n}(-d)) ={\sf c}_i-{\sf c}_{m_1}\ldots{\sf c}_{m_n}(-d) &={\sf c}_i\cdot -{\sf c}_i{\sf c}_{m_1}\ldots{\sf c}_{m_n}( -d) &= -{\sf c}_i{\sf c}_{m_1}\ldots{\sf c}_{m_n}( -d) &=-{\sf c}_{m_1}\ldots{\sf c}_{m_n}( -d) &=t