harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{x_u(z)}= \frac{1}{N}\sum\limits_{n=1}^N{\frac{{{q}{{\left|{{\lambda_u}}\right|}^2}}}{{\frac{1}{N}\sum\nolimits_{i=1}^U{\frac{{{q}}}{{1 +{x_i(z)}}}{{\left|{{\lambda_i}}\right|}^2}}+ \sum\nolimits_{k=1}^K{{p_{k,n}}{g_{k,n}}}+{\sigma^2}}}},
- T_{2}({\bf{k}},{\bf{q}}) \sum_{{\bf{q}}_{1}}a_{{\bf{k}}-{\bf{q}}/ 2 +{\bf{q}}_{1}/ 2}^{\dagger}({\bf{q}}_{1}) a_{{\bf{k}}+{\bf{q}}_{1}/ 2}({\bf{q}}_{1}-{\bf{q}})
f_j(\lambda_k)=\frac{\lambda_k^{-n_B+n+j-1}}{2^{\,-n_B+n+j}\sqrt{\pi}}&\times G^{3,0}_{0,3}\left(\begin{array}{c}\dfrac{n_B-n-j+1}{2},\, \dfrac{n_B-n-j+2}{2},\, 0 \end{array}\Bigg|\, \frac{\lambda_k^2}{4}\right).
{{\cal H}}_{\mathrm{c o l l}}= \frac{1}{2}{\cal I}^{\alpha \beta}I_{\alpha}I_{\beta}-{\cal I}^{\alpha \beta}I_{\alpha}\overline{{J_{\beta}^{( 2 )}}}+ \frac{1}{2}{\cal I}^{\alpha \beta}\overline{{J_{\alpha}^{( 2 )}J_{\beta}^{( 2 )}}}.
S_{G B}^{v o l u m e}= - 2 \left( ( e^{2 U})^{\prime}|_{r = r_{h}}\right)^{- 1}\left[ \int_{r_{h}}^{r_{0}}d r{\frac{\partial}{\partial r}}( e^{2 U})^{\prime}\left( 1 - ( e^{U}R^{\prime})^{2}\right) \right] \ .
p(t,g)= \mathcal{F}^{-1}_{\xi}[{\hat{p}(0,e^{-t}(\xi-\mu)+\mu)e^{-H_t(\xi-\mu)}}] =e^{i\mu g}\mathcal{F}^{-1}_{\xi}[{\hat{p}(0,e^{-t}\xi+\mu)e^{-H_t(\xi)}}] =e^{i\mu g}\mathcal{F}^{-1}_{\xi}[\hat{p}(0,e^{-t}\xi+\mu)]*(\frac{1}{\sqrt{2\pi}}\mathcal{F}^{-1}_{\xi}[e^{-H_t(\xi)}]).
\left|\ddot x_ke_2^\perp+\ddot y_ke_3^\perp\right|^2 = \left|\left( \ddot x_k(1-\dot x_k^2)-\ddot y_k \dot x_k\dot y_k \right) e_2 + \left( \ddot y_k(1-\dot y_k^2)-\ddot x_k\dot x_k\dot y_k \right) e_3 \right|^2+o(1) =|\ddot x_k\dot y_k - \ddot y_k\dot x_k|^2+o(1)=|\kappa|^2+o(1),
\left\Vert g\left( x,\xi\right) \right\Vert_{L^{2}(d\xi)}:=\left(\int_{\mathbb{R}}\left\vert g\left( x,\xi\right) \right\vert^{2}d\xi\right)^{1/2}\left\Vert g\left( x,\xi\right) \right\Vert_{L^{2}(dx\otimes d\xi)}:=\left( \int_{\mathbb{R}^{2}}\left\vert g\left( x,\xi\right) \right\vert^{2}dxd\xi\right)^{1/2}.
\frac{d N ( \omega )}{d \sigma}= \frac{1}{2 \pi}\frac{e^{2}}{\hbar c^{3}}( 2 a_{0}\omega_{0})^{3}\langle x^{2}\rangle \int_{- \infty}^{+ \infty}d s e^{- i s \omega / a_{0}\omega_{0}}\mathrm{c s c h}^{4}( s - i \epsilon ) ,
g^{-1}((-jg^{(h)}(c')(-a)^{-\frac{\ell_1p^h}{i}})_{h=0}^\infty) =-jg^{-1}(g(c') g([(-a)^{-\frac{\ell_1}{i}}])) &=-jc'[(-a)^{-\frac{\ell_1}{i}}]\equiv -jc(-a)^{-\frac{\ell_1}{i}}\bmod p.
\mathbb{P}[\gamma passestotheleftz_{0}]=\frac{1}{2}+\frac{\Gamma(\frac{4}{\kappa})}{\sqrt{\pi}\Gamma(\frac{8-\kappa}{2\kappa})}\frac{x_{0}}{y_{0}}_{2}F_{1}(\frac{1}{2},\frac{4}{\kappa},\frac{3}{2},-(\frac{x_{0}}{y_{0}})^{2})
\Vert \chi-\varphi\Vert^2 =\sum_{g\in G}^{}\vert \chi(g)-\varphi(g)\vert^2 w(g) =\sum_{n=1}^{+\infty}\vert \chi(h_n)-\varphi(h_n)\vert^2 w(h_n) &\leqslant \sum_{n=1}^{N_\varepsilon}\vert \chi(h_n)-\varphi(h_n)\vert^2 w(h_n)+\varepsilon
\left|\dot{\xi}(s)\right|=\left(\frac{1}{C_pq}\right)^\frac{1}{q-1}\left|Du(\xi(s))\right|^\frac{1}{q-1}=\left(\frac{1}{q^{-1}p^{-\frac{q}{p}}q}\right)^\frac{1}{q-1}\left|Du(\xi(s))\right|^\frac{1}{q-1}=p\left|Du(\xi(s))\right|^\frac{1}{q-1}.
\int_0^T |w_{n+q+1}|^2\,dt &\le C(T) \int_0^T \big( \sum_{l=0}^{q}|| \mathcal{\mathbf{B}}_{n+l}^{\ast}U_{n+l}||_{G_{n+l}}^2 + ||\mathcal{\mathbf{B}}_{n+q+1}^{\ast}W_{n+q+1}||_{G_{n+q+1}}^2\big)\, dt \,.
(1, A_\ell') = \begin{pmatrix}(1, T'_\ell) 0 -(1, [\ell]_*\langle \ell^{-1}\rangle) 0 0 (1, T'_\ell) 0 -(1, [\ell]_*\langle \ell^{-1}\rangle) \ell^{1 + j}0 0 0 0 \ell^{1 +j}0 0 \end{pmatrix}
\begin{aligned}\max_{\boldsymbol{\Theta},\mathbf{w}_m}\biggl|\sum_{n=1}^N g_n c_n h_n B_r(\theta_I) + h_d B_d(\theta_d)\biggr|^2 \textrm{s.t.}\|\mathbf{w}_m\|^2\leqslant 1,\:m=1,2, \phi_n\in [0,2\pi), \: \forall n=1,2,\ldots,N.\end{aligned}
Y_{l k}( \theta , \Phi ) = \sqrt{\frac{2 l + 1}{\pi}}\left( \sin \frac{\theta}{2}\right)^{| k |}P_{\: l - | \frac{k}{2}|}^{( | k | , 0 )}\! \Big ( \cos \theta \Big ) \mathrm{e}^{\mathrm{i}k \Phi}.
\int_{\mathbb{R}^{N}}\frac{u_k(x)|^{\alpha(x)}|v_k|^{\beta(x)}}{q(x)}\,dx +o(1) = \Phi(u_k,v_k)\geqslant d_5\min \{\|(u_k, v_k)\|^{\overline{p}}, \|(u_k, v_k)\|^{p^{-}}\}>0.
\sum_{u\in U}d_s(u)\, \le\, \sum_{u\in U}\tau^{r-s}d(u)+ 2^{r\choose2}\tau^{r-s-1}\sum_{\ell=0}^{r-s-1}2^{-{s+1+\ell\choose2}+\ell}\tau^{-\ell}\sum_{u\in U}d^{(2+\ell)}(u) &=\,\tau^{r-s}\mu(U)nd+2^{{r\choose2}-1}\tau^{r-s}\sum_{j=2}^{r-s+1}2^{-{s+j-2\choose2}-s+1}\tau^{1-j}\sum_{u\in U}d^{(j)}(u)\,.
f_n^{(\omega)}:X^k \to \mathbb{R}, f_n^{(\omega)}(\mu) = P_n^{(\omega)}g_{\mu}= \frac{1}{n}\sum_{i=1}^n \bigwedge_{j=1}^k d(\xi_i^{(\omega)},\mu_j) f_\infty:X^k \to \mathbb{R}, f_\infty (\mu) = P g_{\mu}= \int_X \bigwedge_{j=1}^k d(x,\mu_j) P( x)
\star_{\partial M}i^*_{\partial M}\left(\star d\widehat{\phi}\right)=(-1)^n\cdot \phi,\star_{\partial M}i^*_{\partial M}\left(d\left(\star d\widehat{\phi}\right)\right)=(-1)^{n-1}\cdot \star_{\partial M}d^{\star_{\partial M}}\phi.
\sum_{k=1}^n\sin^2(kx) \overset{\eqref{eq-2.2-2}}{=}\frac{n}{2}-\frac{1}{2}\sum_{k=1}^n\cos(2kx) begin{align*}5pt] \overset{\eqref{eq-2.10}}{=}\frac{n}{2}-\frac{1}{2}\left(\frac{\sin((2n+1)x)}{2\sin(x)}-\frac{1}{2}\right) begin{align*}5pt] =\frac{n}{2}-\frac{\sin((2n+1)x)}{4\sin(x)}+\frac{1}{4},
( B )_{j , j - 1}= ( B )_{j - 1 , j}= \frac{j ( ( 2 j_{0}+ 1 )^{2}- j^{2})}{4 ( j_{0}+ 1 ) \sqrt{4 j^{2}- 1}}\; \; , \; \; j = 1 , 2 , \ldots , 2 j_{0}.
A = (2a+2)(2a+3) + 2b_1(4a-2l-d+7-4z_1) + 16b_2z_1(1-z_1){}+ 24b_3z_1^2(1-z_1), B_1 = 2b_1(4a+2b_1+7) - 4b_2(4a-2l-d+9-4z_1) - 24b_3z_1(1-z_1), B_2 = - 4b_1^2 + 4b_2(4a+4b_1+9) - 6b_3(4a-2l-d+11-4z_1), B_3 = -16b_2(b_1-b_2) + 6b_3(4a+4b_1+11),
\left(q/a;q^{2}\right)_{\infty}\sum_{n=0}^{\infty}\frac{\left(aq;q^{2}\right)_{n}z^{n}}{\left(q;q\right)_{n}}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}A_{q}\left(q^{-2n}az\right)}{\left(q^{2};q^{2}\right)_{n}\left(-a\right)^{n}},
(a+b+c)^j =\sum\limits_{k\leq m\leq j}\binom{j}{j-m,m-k,k}a^{j-m}b^{m-k}c^k =\sum\limits_{k=0}^j\sum\limits_{m=k}^{j}\binom{j}{j-m,m-k,k}a^{j-m}b^{m-k}c^k
( w_{1j}, \ldots, w_{j-1,j})' = (I-P_{j-1})^{-1}P_{[1:j-1], j}, (w_{jj}, \ldots, w_{J,j})' = P_{[j:J],j}+ P_{[j:J], [1:j-1]}(I-P_{j-1})^{-1}P_{[1:j-1], j},
\|\partial_{z}r_{1,2}(t;z) \|_{L^2}= \big\| \partial_{z}r_{1,2}( z)e^{-2i\eta^2 t}- 2i\alpha t (1 -\frac{\beta^2}{4}z^{-2}) r_{1,2}(t;z) e^{-2i\eta^2 t}\big\|_{L^2}&\leq 2\alpha T \left\| r_{1,2}( z)\right\|_{L^2}+\left\|\partial_{z}r_{1,2}( z)\right\|_{L^2}+ \frac{1}{2}\alpha\beta^2 T \| z^{-2}r_{1,2}( z) \|_{L^2}.
{\cal Z}_{_0}= \int D\overline{\xi}D\xi \exp \left\{i \int d^2 x \left[ -i \overline{\xi}(x) \partial \!\!\!/ \partial \!\!\!/^\dagger \partial \!\!\!/ \xi(x) + m \overline{\xi}(x) \partial \!\!\!/ \partial \!\!\!/ \xi(x) \right] \right\}.
dj^n_i(t) = 2 \sqrt{j^n_i(t)(1 - j^n_i(t))}dB^n_i(t) + 2\Big((p - n + 1) + (p + q - 2n + 2)j^n_i(t)\Big) dt + d\Phi^n_i(t) - d\Psi^n_i(t)
\bigl(w_i,e^{i\theta_\infty}(\phi_i+\phi_e)- \nabla\cdot(\sigma_{i\infty}\nabla\phi_i) \bigr)_{L^2(\mathbb{R}^d)}&-\bigl(w_e,e^{i\theta_\infty}(\phi_i+\phi_e)-\nabla\cdot(\sigma_{e\infty}\nabla\phi_e)\bigr)_{L^2(\mathbb{R}^d)}=0.
S_j^{j-\pi(\ell)}(\ell)= \dfrac{P_j^{j-\pi(\ell)}(\ell)}{\omega \cdot \ell +\Omega_j -\Omega_{j-\pi(\ell)}}\mbox{if}\; 0<|\ell|\le K, S_j^{j-\pi(\ell)}(\ell)= 0 \mbox{otherwise}.
L(f, g, s) = \frac{N^{r + r' - 2s - 2}\pi^{2s + 1 - r'}(-i)^{r - r'}2^{2s + r - r'}}{\Gamma(s)\Gamma(s - r' + 1)}\big\langle f^*, g E^{(r - r')}_{1/N}(\tau, s - r + 1)\big\rangle_N
I\ni\lambda_\alpha(1+p^{t})\mu''-\mu''=\sum_{j=0}^\infty\mu''_j\left(((X_\alpha+1)^{1+p^{t}}-1)^j-X_\alpha^j\right)= =\sum_{j=1}^\infty\mu''_j\left((X_\alpha+X_\alpha^{p^{t}}+X_\alpha^{p^{t}+1})^j-X_\alpha^j\right)\ .
\int_{\mathbb{R}^d}\,\epsilon^d\, \bigg|\sum_{i=1}^{N}\,\hat{F_i}(\epsilon\lambda)\,e^{-2\pi i x_i\cdot\lambda}\bigg|^2\,d\mu(\lambda)\leq \left(G(\delta,\epsilon)+I(\delta,\epsilon)\right)^2 &\le\left(\sqrt{\mathcal{D}^{+}_N(\check{\mu})+\rho'}+\sqrt{\rho'}\right)^2\,\|f\|_2^2\leq (\mathcal{D}^{+}_N(\check{\mu})+\rho)\,\|f\|_2^2
s(N\cap K:_{M}a) =s_{1}s_{2}((N:_{M}a)\cap (K:_{M}a)) & \subseteq s_{1}(N:_{M}a)\cap s_{2}(K:_{M}a) &\subseteq (N+(0:_{M}a)\cap (K+(0:_{M}a)) &=(N\cap K)+(0:_{M}a).
\eta_{1}( \beta \sqrt{- \triangle}) = \frac{2 \sqrt{\pi}}{\beta}\int_{0}^{\infty}\frac{{\mathrm{d}}s}{s^{3 / 2}}\,{\mathrm{e}}^{- s m^{2}}\, \frac{f ( - s \triangle ) - 1}{s \triangle}, \ \beta \rightarrow 0 ,
e_1 L^2(M) = V_0 \oplus V_1 \oplus V_2 \quad V_0 := \bigoplus_{n=0}^\infty T_0 H^n \;\; , & V_1 := L^2(A e_1) \oplus \bigoplus_{n=0}^\infty (e_1 H \ominus L)H^n \;\; , V_2 := L \oplus \bigoplus_{n=0}^\infty (LH \ominus T_0)H^n \;\; .
\pi_{(1,1,1)}(x_1,x_2,x_3) =1, \pi_{(2,1)}(x_1,x_2,x_3) =x_1-x_2, \pi_{(3)}(x_1,x_2,x_3) =(x_1-x_2)(x_1-x_3)(x_2-x_3).
\big(\zeta_t(x),\theta_t(x)\big)\rightarrow (a,b) &\begin{cases}1 a=b=0, \delta a=\zeta_t(x)b=0, \gamma a=\zeta_t(x)+\theta_t(x)b=0, \lambda y\sim x, a=\zeta_t(x)b=\theta_t(x)+\zeta_t(y), 0 .\end{cases}
|F_g(f)| \leq \mathbb E \Bigl(\sum_{k=0}^{n}\mathbb E_{k-1}(\|f_k\|\|g_k\|)\Bigr) \leq \mathbb E\Bigl(\sum_{k=0}^{n}(\mathbb E_{k-1}\|f_k\|^q)^{1/q}(\mathbb E_{k-1}\|g_k\|^{q'})^{1/{q'}}\Bigr) \leq \|f\|_{H^{s_q^n}_p(X)}\|g\|_{H^{s_{q'}^n}_{p'}(X^*)}.
\begin{array}{lcr}x = \alpha_1 \displaystyle \frac{\alpha_3 \pm ||\alpha ||}{(\alpha_1^2 + \alpha_2{}^2 )}\ , y = \displaystyle -\alpha_2 \frac{\alpha_3 \pm ||\alpha ||}{(\alpha_1^2 + \alpha_2^2)}\ .\end{array}
W(x)-W(x_a)= \big(h'^2(x)+(m-1)\cos^2 h(x)\big)^{\frac{p}{2}-1}\big((p-1)h'^2(x)-(m-1)\cos^2 h(x)\big) &+(m-1)^{\tfrac{p}{2}}\cos^p h(x_a)\geq 0.
{res_J^{M}}{\Theta_J}([x_K^{M}]) ={res_J^{M}}({\Theta_J}[x_K^{M}]) &={res_J^{M}}(W_{M}/W_{K}) &=\sum_{d\in{{W_M}\bigcap{X_{KJ}}}}W_{J}/W_{J\cap{d^{-1}(K)}}&=\sum_{d\in{{W_M}\bigcap{X_{JK}}}}{W_{J}/W_{J\cap{d(K)}}}.
\frac{1}{(2\pi)^2}\int_{BZ}\frac{\L^\alpha}{\omega-\omega^\alpha}d\alpha =\frac{1}{(2\pi)^2}\int_0^{1}\frac{2\pi r'}{\omega-\omega_* + \frac{1}{2}c_\delta r'^2}dr' +B_2(\omega) &= -\frac{1}{2\pi c_\delta}\ln (\omega-\omega_*) + B_3(\omega).
\Omega_{n}\stackrel{\rm def}{=}\alpha_{n}+2\,P_{n}+2\,\phi_{n}+\frac{4\,||K^{2}||\,R_{n}(E_{n},E_{n})}{\sigma^{2}}+L(n,{\bf R)}\rightarrow 0,{\bf}\mbox{{\it as}}n\rightarrow \infty{\bf ,}
\liminf_{n\to\infty}\frac{g_+(b^n(x_0+v))}{n g_+(b^nx_0)}= \liminf_{n\to\infty}\frac{\nu\big((b^n(x_0+v),b^n(x_0+v)+1]\big)}{n\nu\big((b^n x_0,b^nx_0+1]\big)}\ge c_0,
H_{n,m}= U(N_{A}-N_{B})^2+\mu(N_{A}-N_{B})+t\left(A^{\dagger}B+A B^\dagger\right) &=U(N_{A}-N_{B})^2+\mu(N_{A}-N_{B}) +\sum_{i=1}^n\sum_{j=1}^m t_{i,j}(a_{i}b_{j}^\dagger+a_{i}^\dagger b_{j}),
\gamma(3) =-{\displaystyle\int\limits_{0}^{1}}\frac{(x-1)^{3}(3x^{8}-11x^{7}+13x^{6}+x^{5}-18x^{4}+14x^{3}+3x^{2}-3x-3)}{x^{10}-5x^{9}+10x^{8}-10x^{7}+x^{6}+11x^{5}-11x^{4}+2x^{3}+x^{2}-1}dx \ =0.8408530104...
\begin{aligned}\int_0^T\int_{0}^{\tilde\lambda_1}\left(s\Theta a(v_x)^2 + s^3 \Theta^3\frac{(x-x_0)^2}{a}v^2\right)e^{2s\varphi}dxdt\le C \int_0^T \int_{\omega}v^2dxdt,\end{aligned}
u = \frac{s_+}{\sqrt{2}}- \frac{\sqrt{2}k^2}{2}\,\frac{2b^2 + c^2\,s_+}{b^2\,(-b^2 + 4c^2\,s_+)}\,r^{-2}+ O(r^{-4}), v = -\frac{s_+}{\sqrt{6}}- \frac{\sqrt{6}k^2}{2}\frac{-b^2 + c^2\,s_+}{b^2\,(-b^2 + 4c^2\,s_+)}\,r^{-2}+ O(r^{-4}) .
\bar{G}\int \Pi_{j=1}^{m}\phi_1\left( x^j\right) \int_{A^c}\phi_m(w) \, Q(x,w) dP_0^m(w) dP_0^m (x) &\qquad\geq \int_{A^c}\phi_m(w) \int \phi_m(x) \, Q(x,w) dP_0^m (x) dP_0^m(w).
\mathrm{coef}\left( \left( T_{1}+T_{2}\right) h_{2m+1}^{-},\psi_{2m,1}\right) & =-\mathrm{coef}\left( h_{2m+1}^{-},\psi_{2m+1,0}\right) +\left(2N\kappa+2m+4\right) \mathrm{coef}\left( h_{2m+1}^{-},\psi_{2m+1,2}\right) & =-1+\left( N\kappa+m+2\right) \frac{g_{1}^{o}\left( N\kappa-\kappa+m\right)}{N\kappa+m+2}=N\kappa-\kappa+m.
T^{(2)}_{EN}(\hat{r}\rightarrow\infty)\approx[c_1\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}+c_2\frac{\Gamma(2-c)\Gamma(b-a)}{\Gamma(b-c+1)\Gamma(1-a)}]e^{-(1+2i\lambda)\log\hat{r}}.
\displaystyle-\frac{1}{2\lambda u}GF_x+\frac{w}{u}GF+FH=R(\lambda), \displaystyle\frac{1}{2\lambda v}GH_x+\frac{w}{v}GH+FH=R(\lambda), \displaystyle\frac{1}{\lambda u}FG_x+\frac{v}{u}F^2+G^2=R(\lambda).
-{1\over 2}\Bigl[ (\eta^{\mu_1\mu_2}\eta^{\nu_1\nu_2}- \eta^{\mu_1\nu_2}\eta^{\mu_2\nu_1}) (\eta^{\mu_3\mu_4}\eta^{\nu_3\nu_4}- \eta^{\mu_3\nu_4}\eta^{\mu_4\nu_3}) + (2\leftrightarrow 3), (2\leftrightarrow 4) \Bigr]
a^*_{1,1}=a^*_{0,1}=a^*_{0,2}=0, &a^*_{3,0}+3a^*_{3,1}+3a^*_{3,2}+a^*_{3,3}+6a^*_{2,1}+6a^*_{2,2}+2a^*_{2,3}+3a^*_{1,2}+a^*_{1,3}=\mu_R.
\sum_{\alpha\in I_1}D_{c_\alpha,\widehat{c}_\alpha}=\sum_{\alpha\in I}D_{c_\alpha,\widehat{c}_\alpha}-\sum_{\alpha\in I_2}D_{c_\alpha,\widehat{c}_\alpha}-\sum_{\alpha\in I_0}D_{c_\alpha,\widehat{c}_\alpha}.
\big\langle x\big\vert\, \exp -i\left[\left(-d_{x}^2-4q p_u x\right)s\right]\, \big \vert x\big\rangle= \left({1 \over{(4\pi is)^{1/2}}}\right)\;\exp -4i\left(q p_u xs +{1 \over 3}q^2 p_u^2 s^3\right).
{\hat{\lambda}}={\frac{\nu}{\xi a^{1 - \nu}( 1 - ( a / \ell )^{2 \nu})}}-{\frac{\pi \nu}{( 1 - ( a / \ell )^{2 \nu})}}-{\frac{C \nu^{2}a^{2 \nu}}{( 1 - ( a / \ell )^{2 \nu})^{2}}}
p_1 = [1,2] = x_1x_7-x_2x_6, p_2 = [1,3] = x_1x_8-x_3x_6, p_3 = [1,4] + [2,3] = x_1x_9-x_4x_6 + x_2x_8-x_3x_7, p_4 = [1,5] + [2,4] = x_1x_{10}-x_5x_6 + x_2x_9-x_4x_7, p_5 = [2,5] + [3,4] = x_2x_{10}-x_5x_7 + x_3x_9-x_4x_8, p_6 = [3,5] = x_3x_{10}-x_5x_8, p_7 = [4,5] = x_4x_{10}-x_5x_9.
A_m=v^2(y_m(v)+1)-\int_0^vv'^2\frac{dy_m}{dv}(v')dv',B_m=-\frac{1}{v}\int_0^vv'^3\frac{dy_m}{dv}(v')dv'+\int_0^vv'^2\frac{dy_m}{dv}(v')dv'.
s_\alpha\big|_{[x_{j-1},x_j]}(x) =y_{j-1}+\dfrac{y_{j}-y_{j-1}}{h_j}(x-x_{j-1})+O(\alpha^2) t_\alpha\big|_{[x_{j-1},x_j]}(x) =y_{j-1}+\dfrac{y_{j}-y_{j-1}}{h_j}(x-x_{j-1})+O(\alpha^2) .
S_{jk}=\frac{g^2(\mu_j,\lambda_k)}{f(\mu_j,\lambda_k)}e^{\phi^{\kappa}(\mu_j)+\phi^{\nu}(\lambda_k)}+\frac{g^2(\lambda_k,\mu_j)}{f(\lambda_k,\mu_j)}e^{\phi^{\kappa}(\lambda_k)+\phi^{\nu}(\mu_j)}
\begin{array}{ll}(x_c)'(y;\Delta y) ={\cal A}_c(y,\widehat{W}_1,\widehat{W}_2 )^{-1}\left(-{\rm D}F(x_c(y))^*\widehat W_1(\Delta Y/c)\right. begin{align*}6mm] \left. -{\cal J}h(x_c(y))^T (\Delta\mu)+{\rm D}g(x_c(y))^*\widehat{W}_2(\Delta\lambda)\right).\end{array}
\zeta(\Omega^{s_1-1}\omega_{p_1}\Omega^{s_2-1}\omega_{p_2}\ldots\Omega^{s_k-1}\omega_{p_k}):=\zeta\left(\begin{array}{rlcl}s_1, s_2 \ldots s_k \lambda^{p_1}, \lambda^{p_2-p_1}\ldots \lambda^{p_k-p_{k-1}}\end{array}\right)
\times\,\frac{\omega}{2i\pi\hbar\sin\omega\Delta t}\,e^{-i\frac{\pi}{2}|\ell|}\,e^{\frac{i\omega}{2\hbar}\frac{\cos\omega\Delta t}{\sin\omega\Delta t}(r^2_f+r^2_i)}\,J_{|\ell|}\left(\frac{\omega r_f\,r_i}{\hbar\sin\omega\Delta t}\right)\ \ \ .
\Big|\int_{B_r}\frac{\, 2u(x)-u(x+z)-u(x-z) \,}{|z|^{N+2s}}\,dz\Big| \le N \omega_N\sup_{\eta\in B_r(x)}|D^2 u(\eta)|\,\int_0^r \frac{dr}{\, r^{2s - 1}\,}&= N \omega_N\,\frac{\, r^{2 - 2s}\,}{\, 2 - 2s \,}\,\sup_{\eta\in B_r(x)}|D^2 u(\eta)|.
L = \frac{1}{2}\left( \partial_{\mu}\varphi \right)^{2}+ \frac{1}{2}m^{2}\varphi^{2}+ \frac{\lambda}{4 !}\varphi^{4}+ \frac{1}{2}\xi R \varphi^{2}+ \Lambda_{0}+ \kappa R + \widetilde{a}_{1}R^{2}.
\begin{cases}x_{1}^{\left(2t+2\right)}=\;\delta_{1}x_{2}^{\left(2t+1\right)}y^{\left(2t+1\right)}\medskip x_{2}^{\left(2t+1\right)}=\;\gamma_{2}x_{1}^{\left(2t\right)}y^{\left(2t\right)}\medskip y^{\left(2t+2\right)}=\;\delta x_{2}^{\left(2t+1\right)}y^{\left(2t+1\right)}y^{\left(2t+1\right)}=\;\gamma x_{1}^{\left(2t\right)}y^{\left(2t\right)}.\end{cases}
*_{i=1}^n \theta_i(a_1\cdots a_{j-1}a_j \cdots a_{m+1}) = \theta_{i_1}(a_1)\cdots \theta_{i_j}(a_{j-1}a_j)\cdots \theta_{i_{m+1}}(a_{m+1}) = \theta_{i_1}(a_1)\cdots \theta_{i_{j-1}}(a_{j-1})\theta_{i_j}(a_j)\cdots \theta_{i_{m+1}}(a_{m+1}).
\qquad \qquad \qquad \qquad \qquad \frac{d v}{d x}= - \frac{c \varepsilon^{2}H Z^{4}n}{m \nu_{0}^{2}v^{3}}\left({0 . 9 6 + \log \frac{h \nu_{0}v^{4}}{8 \pi c \varepsilon^{2}H Z^{4}}}\right) . \qquad \qquad \qquad \qquad ( 1 7^{\prime})
\overline{d}(X) = \mathrm{min}\{r \equiv 2\mu(X) \;\mathrm{mod}\; 2 \mid \exists \, x \in \tilde{H}^r_{\mathbb{Z}/2}(X), U^lx \neq 0 \; \mathrm{for\; all}\; l \geq 0 \}, \underline{d}(X) = \mathrm{min}\{r \equiv 2\mu(X)+1 \;\mathrm{mod}\; 2 \mid \exists \, x \in \tilde{H}^r_{\mathbb{Z}/2}(X), U^lx \neq 0 \; \mathrm{for\; all}\; l \geq 0 \}-1,
\frac{d}{dt}F_1(t) = \sum_{i=1}^m \int_{\Omega}(1+{\rm log}c_i)\partial_t c_idx & =- \sum_{i=1}^m \int_{\Omega}\nabla_x ({\rm log}c_i)\cdot (\nabla_x c_i+q_ic_i \nabla_x \psi)dx & = - \sum_{i=1}^m \int_{\Omega}\left( c_i^{-1}|\nabla_x c_i|^2 +q_i\nabla_x c_i \cdot \nabla_x \psi \right)dx
\Phi(\xi, \eta_1, \eta_2) = \frac{3\xi}{(1+\xi^2)^{5/2}}\zeta_1 \zeta_2 + O\bigg(\zeta_1^3 + \zeta_2^3\bigg)=\frac{3\xi}{(1+\xi^2)^{5/2}}\zeta_1 \zeta_2 + O\left([ \varrho_1(t)]^3 \right).
\gamma \, = \, ( 9 6 \pi )^{- 1}[ D - 2 6 - \frac{3}{2}\alpha^{\prime}({\cal R}- \frac{1}{1 2}H_{i j k}H^{i j k}+ 4 \partial_{i}C \partial^{i}C ) \, + \, 0 ( \alpha^{2}{\cal R}^{2}) ] .
L = A + \bar C_{\sigma_1}^{j_1,i_1}y^{\sigma_1}_{j_1i_1}+ \bar C_{\sigma_1 \sigma_2}^{j_1 j_2,i_1 i_2}y^{\sigma_1}_{j_1i_1}y^{\sigma_2}_{j_2i_2}+\cdots + \bar C_{\sigma_1 \dots \sigma_n}^{j_1 \dots j_n,i_1 \dots i_n}y^{\sigma_1}_{j_1 i_1}\cdots y^{\sigma_n}_{j_n i_n}
0=\frac{\delta \bar J_i}{2}={\mathbb E}\int_0^T\left\{(\tilde x_i-\Gamma \tilde m_i)^TQ[ x_i-(\Gamma m_i +\eta)]+ \tilde u_i^T R \hat u_i -\gamma \tilde p_i^T p_i\right\}dt +{\mathbb E}\tilde x_i^T(T) H x_i(T).
N-\Delta\varphi_{R}(r) =N-\varphi_{R}''(r)-\varphi_{R}'(r)\frac{N-1}{r}&=1-\varphi_{R}''(r)+N-1-\varphi_{R}'(r)\frac{N-1}{r}&=1-\varphi_{R}''(r)+(N-1)(1-\frac{\varphi_{R}'(r)}{r})\geq0
\lvert a_j(x)-a(x)\rvert \leqslant a(k\cdot 2^{-j})+2M\cdot 2^{-j}+M(x-k\cdot 2^{-j}) \phantom{\leqslant{}}- \big(a(k\cdot 2^{-j})-M(x-k\cdot 2^{-j})\big) = 2M\cdot 2^{-j}+2M(x-k\cdot 2^{-j}) \leqslant 4M\cdot 2^{-j}.
\Delta \xi_{1\epsilon}+|x|^{2\beta_2-4}e^{\xi_{2\epsilon}}(1-|x|^{2\beta_1}e^{\xi_{1\epsilon}})=0, \Delta \xi_{2\epsilon}+|x|^{2\beta_1-4}e^{\xi_{1\epsilon}}(1-|x|^{2\beta_2}e^{\xi_{2\epsilon}})=0,\quad\quad |x|\leq R_0.
A_{k}-9A_{k-2}&=2\times3^{k+1}[\frac{k^{2}-4k-3}{(k-1)k(k+1)(k+2)}]+8k[\frac{k^{2}+4k+1}{(k-1)k(k+1)(k+2)}]+[\frac{2k^{2}+40k+18}{(k-1)k(k+1)(k+2)}] &>0.
|G^{k+1}_{m_n}(x)| = \left| -x\int_{1}^{x}(-1)^kG^k_{m_n}(y)dm_n(y)+\int_{0}^{x}(-1)^kyG^k_{m_n}(y)dm_n(y) \right| &\leq \sup_{z \in [0,x_0]}|G^k_{m_n}(z)|\left(x_0|\tilde{m}_n(x_0)| + \int_{0}^{x_0}ydm_n(y)\right).
r^{\epsilon,\hat{v}}(t) = \bigl(r_{1}^{\epsilon,\hat{v}}(t), r_{2}^{\epsilon,\hat{v}}(t), \ldots, r_{n}^{\epsilon,\hat{v}}(t)\bigr) = \biggl(\int_0^t \chi_1\bigl(\zeta^{\epsilon}(s)\bigr) ds, \int_0^t \chi_2\bigl(\zeta^{\epsilon}(s)\bigr) ds, \ldots, \int_0^t \chi_n\bigl(\zeta^{\epsilon}(s)\bigr) ds \biggr),
G_{1}^{(2)}(\lambda_{1},\vec{r}_{1},z_{1},\vec{r}_{2},z_{2})=\frac{\lambda_{1}}{2}\int{}d^{d-1}r\int_{0}^{L}dz\,\,G_{0}^{(2)}(\vec{r}_{1}-\vec{r},z_{1},z)G_{0}^{(2)}(\vec{0},z)G_{0}^{(2)}(\vec{r}-\vec{r}_{2},z,z_{2}).
\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}q^{n}}{\left(q;q\right)_{n}}=\frac{\left(aq;q\right)_{\infty}}{\left(q;q\right)_{\infty}}=\frac{1}{\sqrt{2\pi\log q^{-1}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q^{2}}\right)dx}{\left(q/a,-q^{1/2}/(ae^{ix}),-q^{1/2}(ae^{ix});q\right)_{\infty}},
2\tau v_{ix}(k)(v_{i\bar t}(k))_x = 2\tau v_{ix}(k)(v_{ix}(k))_{\bar t}begin{align*}3mm] = v_{ix}^2(k)+ v_{ix}^2(k)-2v_{ix}(k)v_{ix}(k-1)+v_{ix}^2(k-1)-v_{ix}^2(k-1) begin{align*}3mm] = v_{ix}^2(k)+\Big(v_{ix}(k)-v_{ix}(k-1)\Big)^2-v_{ix}^2(k-1) begin{align*}3mm] = v_{ix}^2(k)-v_{ix}^2(k-1) +\tau^2 v_{ix\bar t}^2(k).
\overline Y=\left (Q_{a\cup_{b_U}}Q_{b_S}Q_{c\cup_{b_L}}\right) \left[ \begin{array}{ccc}I_{|a\cup_{b_U}|}0 0 0{\rm Diag}\,(w_{b_S}) 0 0 0 I_{|c\cup_{b_L}|}\end{array}\right ] \left ( \begin{array}{c}Q_{a\cup_{b_U}}^T begin{align*}4pt] Q^T_{b_S}begin{align*}4pt] Q_{c\cup_{b_L}}^T \end{array}\right)
T_1^{(1)}= \left[\begin{array}{ccccccc}\ddots \ddots 0 I 0 D_{T_1^*}T_1 -T_1^* D_{T_1}0 I 0 \ddots \ddots\end{array}\right] \ \ \textrm{and}\ \T_j^{(1)}= I_{\ell_2(\mathbb Z)}\otimes T_j, \ 2\leq j\leq n.
\mathbf{J}_{\mathcal{E},\mathcal{P}}(\mathbf{r})=\sum_{j,l,m}\frac{a_{l,m}^{(j)}}{(\mathcal{B}_{l,m}^{(j)},\mathcal{D}_{l,m}^{(j)})}\mathcal{D}_{l,m}^{(j)}(\mathbf{r}),
\mathbf{T^1_{k}}=\int_{B_{1/2}}\nabla (U_0 -V) \cdot \nabla ((2-\eta_r)(U_k-U_0))\,dx = \int_{B_{1/2}}(2-\eta_r) \nabla (U_0-V)\cdot \nabla (U_k-U_0)\,dx &-\int_{B_{1/2}}\big ( (U_k -U_0) \nabla (U_0-V) \cdot \nabla \eta_r \big )\,dx.
S ={\frac{\pi \bar{r}_{+}^{2}}{G}}+{\frac{1}{1 2}}( 1 - ({\frac{\bar{r}_{-}}{\bar{r}_{+}}})^{2}) \ln{\frac{L}{( \bar{r}_{+}- \bar{r}_{-})}}+{\frac{1}{6}}\ln{\frac{\bar{r}_{+}}{z_{0}}}.
h^i(x^i) =-\sum_{p\in \textrm{supp}(x^{i*})}x^{i*}_p\ln \frac{x^i_{p}}{x^{i*}_p}\geq - \ln \big(\sum_{p\in \textrm{supp}(x^{i*})}x^{i*}_p\frac{x^i_{p}}{x^{i*}_p}\big) \geq - \ln \big(\sum_{p\in \textrm{supp}(x)}x^i_{p}\big) =0,
\sigma^{ij}(\hat{\rho}) := \frac{1}{2}\mathrm{Tr}\left[\left\{\hat{R}^i-r^i(\hat{\rho}),\;\hat{R}^j-r^j(\hat{\rho})\right\}\hat{\rho}\right]\;,i,\,j=1,\,\ldots,\,2n\;,
|v_{5}(t,r)| \leq \begin{cases}\frac{C r}{t^{7/2}\log^{3b-3+\frac{5N}{2}}(t)}, r \leq \frac{t}{2}\frac{C \log^{4}(t)}{\sqrt{r}t^{3/2}}, r > \frac{t}{2}\end{cases}
\sum_{1\leq i_1,\ldots,i_{l-a_1-a_2}\leq p}(\mu_{i_1,\ldots,i_{l-a_1-a_2}}^{(1)}-\mu_{i_1,\ldots,i_{l-a_1-a_2}}^{(2)})^2=\|\mathcal{T}_{1,l}^{(a_1,a_2)}-\mathcal{T}_{2,l}^{(a_1,a_2)}\|_{\rm F}^2,
\!\!\!\!\!\!\mathbb{E}\left[{P_u}^{\frac{2}{\eta}}\right]=\frac{\rho^{\frac{2}{\eta}}\gamma\left(2, \pi \lambda R_{\rm M}^{2}\right)}{\pi\lambda\left( 1-\exp\left(- \pi \lambda R_{\rm M}^{2}\right)\right)}.
\rho (\theta ) = -{1\over \pi}\sum_{l=1}^L V_l\int_{\Gamma_l}{d \theta' \over{\cal Z}_l (\theta')}\log \left | 2 \sin \left({\theta' - \theta \over 2}\right)\right |+{1\over \pi}\sum_{l=1}^L \int_{\Gamma_l}{d\theta'\over{\cal Z}_l (\theta')}\log \left | 2 \sin \left({\theta' - \theta \over 2}\right)\right | \rho (\theta ') .
c(\gamma) =\int_0^{+\infty}\frac{g(\tau)}{\tau^{1+2s}}\,d\tau &=\int_0^{\frac{1}{4\sqrt{N}}}\frac{g(\tau)}{\tau^{1+2s}}\,d\tau+\int_{\frac{1}{4\sqrt{N}}}^{\frac{1}{2\sqrt{N}}}\frac{g(\tau)}{\tau^{1+2s}}\,d\tau+\int_{\frac{1}{2\sqrt{N}}}^{+\infty}\frac{g(\tau)}{\tau^{1+2s}}\,d\tau &=:I_1+I_2+I_3.
h_2 \left( I + \begin{pmatrix}0 b c 0 \end{pmatrix}p \right) h_2^{-1}= I + \frac{\begin{pmatrix}c \alpha \beta + b \alpha \gamma c \beta^2 - b \alpha^2 -c \alpha^2 + b \gamma^2 -c \alpha \beta - b \alpha \gamma \end{pmatrix}p}{-\alpha^2 - \beta \gamma}.
\mathcal{R}_{i,j}\nabla_i =(\nabla_{d_{i,j}}+\nabla_{D_{i,j}})\mathcal{R}_{i,j},\,\,\,\,\mathcal{R}_{i,j}\nabla_j =(\nabla_{D_{i,j}}-\nabla_{d_{i,j}})\mathcal{R}_{i,j}
\xi_k = E \big[ \left\{\boldsymbol{v}^T_{k}\boldsymbol{x}_k \boldsymbol{x}^T_{k}\boldsymbol{v}_k \right\}\big] = \left\{\boldsymbol{R}E \big[ \boldsymbol{v}_{k}\boldsymbol{v}^T_k \big]\right\}.
X_{1}=~\{x, T^{*3}x ,\ldots, T^{*3(3j-n+2)}x, y, T^{*3}y ,\ldots, T^{*3(3j-n+2)}y\}X_{2}=~\{T^{*3(3j-n+3)}x ,\ldots,T^{*3j}x, T^{*3(3j-n+3)}y ,\ldots, T^{*3j}y, z,\ldots, T^{*3(n-2j-3)}z\}.
Y_{1}=0.137;(\rho_{0},\rho_{1},v_{0})=(279.8\cdot\omega^{2},0.049\cdot\omega^{-1},0.00049\cdot\omega^{-3}) Y_{2}=0.390;(\rho_{0},\rho_{1},v_{0})=(0.510\cdot\omega^{2},-0.198\cdot\omega^{-1},0.776\cdot\omega^{-3})
\langle \psi^{\dagger}({\bf{0}}, 0 ) \psi ({\bf{x}}, t ) \rangle = e x p \mathrm{~}[ \mathrm{~}( \frac{1}{4 N_{0}}) \sum_{{\bf{q}}}f_{{\bf{q}}}({\bf{x}}, t ) \mathrm{~}] \langle \psi^{\dagger}({\bf{0}}, 0 ) \psi ({\bf{x}}, t ) \rangle_{f r e e}
\begin{cases}\|a\|_{L_{T}^{\frac{4p}{3}}L_{x}^{2p}}\leq C_{1}\|w_{0}\|_{L^{p}}, \|N(w_{1},w_{2})\|_{L_{T}^{\frac{4p}{3}}L_{x}^{2p}}\leq C_{2}T^{\frac{p-3}{2p}}\|w_{1}\|_{L_{T}^{\frac{4p}{3}}L_{x}^{2p}}\|w_{2}\|_{L_{T}^{\frac{4p}{3}}L_{x}^{2p}}.\end{cases}
-4 m^{d+1}\, \, \frac{q(\vec{x})}{|\vec{v}(\vec{x})|}\sum_{m=1, 3, 5, . .}^{d/2}{\frac{d}{2}\choose m}(\partial_{i}\hat{n}^{a}\,\partial_{i}\hat{n}^{a})^{(\frac{d}{2}-m)}|\vec{v}(\vec{x})|^{m}
\left\lbrace\begin{array}{rlll}\dot{y}(s) = \nabla_p H(y(s),p(s)) \mbox{for all}s\in [t,T], -\dot{p}(s) \in \partial_x^- H(y(s),p(s)) \mbox{a.e. in}[t,T],\end{array}\right.
\begin{aligned}\|\partial_t^{j}D^l_x D_v^{m}u\|_{L_{\infty}(Q_{r})}&\le N \delta^{-\theta}(\|D_v u\|_{L_2 (Q_{R})}+ \|D_x u\|_{L_2 (Q_{R})}), \, \, j \in \{0, 1\}, l + m \ge 1.\end{aligned}
\sum_{f\in \mathcal{B}_{2k}(p)}\omega_f \lambda_f(r) \lambda_f(s)= \delta_{rs}+2 \pi i^{2k}\sum_{p|c}\frac{\mathcal{S}(r, s, c )}{c}\,J_{2k-1}\left(\frac{4\pi \sqrt{rs}}{c}\right),
\frac{ds^2}{\alpha'\sqrt{4\pi g_sN}}=(1+\delta_1)\frac{d\rho^2}{\rho^2-\frac{b^2}{\rho^2}}+(1+\delta_2)(\rho^2-\frac{b^4}{\rho^2})dt^2+\rho^2dx_i^2+d\Omega_5^2,