harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sigma_{\mathrm{t o t}i}^{( 1 )}= \frac{( 2 \pi )^{4}\hbar^{2}}{v_{i}}\frac{1}{\pi}\mathrm{I m}R_{i i}^{( 1 )}\exp \left[ - \frac{1}{\hbar}\int_{- \infty}^{\infty}d t \Gamma^{( 1 )}( t ) \right] ,
{\frac{\partial^{2}G}{\partial q_{i}\partial q_{j}}}={\frac{\partial^{2}G}{\partial p_{i}\partial p_{j}}}\, , \quad{\frac{\partial^{2}G}{\partial q_{i}\partial p_{j}}}= - \,{\frac{\partial^{2}G}{\partial q_{j}\partial p_{i}}}.
\times \left[ \begin{array}{r r r}{p_{1 1}^{j}}&{p_{1 2}^{j}}&{p_{1 3}^{j}}\\{p_{2 1}^{j}}&{p_{2 2}^{j}}&{p_{2 3}^{j}}\\{p_{3 1}^{j}}&{p_{3 2}^{j}}&{p_{3 3}^{j}}\\ \end{array}\right] , \verb + + \scriptscriptstyle{( j = 1 , 2 , 3 )},
I^{(0)}(x,x')=\epsilon(x_0-x'_0)\delta({\bf x-x'})\frac{e^{i\frac{3\pi}{4}}e^{ig\Lambda}}{4\sqrt{\pi}}\int_{e^{i\pi}\infty}^{\infty}t^{-\frac{3}{2}}e^{-i(x_0-x'_0)^2 t}dt.
(1) \langle X, \nabla(\frac{F}{\varphi}) \rangle =\frac{\beta-1}{\beta}\sum_{i}\lambda_{i}^{-2}(\nabla_{i}F)^{2}, &(2) \lambda_{1}^{-2}h_{11j}= (\lambda_{1}^{-1}-\frac{\beta-1}{\beta}\lambda_{j}^{-1})\nabla_{j}\log F,~1\leq j\leq n,\qquad\qquad\qquad\qquad &(3) \nabla_{m}F =0,~2\leq m\leq \mu.
\hat{G}^{(10)}_{xx}\to \hat{G}^{(10)\,\prime}_{xx}={1\over{\hat{G}^{(10)}_{xx}}}, \ \ \ \ \hat{G}^{(10)}_{\mu\nu}\to \hat{G}^{(10)}_{\mu\nu}\ \ (\mu,\nu \neq x),\ \ \ \e^{2\hat{\Phi}}\to e^{2\hat{\Phi}^{\prime}}={{e^{2\hat{\Phi}}}\over{\hat{G}^{(10)\,\prime}_{xx}}}.
R := (f \times id_{\mathbb{F}_1 \times \mathbb{F}_2}) \circ l \circ (R_1 \times R_2) \circ l^{-1}\circ (f \times id_{\mathbb{F}_1 \times \mathbb{F}_2})^{-1}&: U \lhd \mathbb{F}_1 \times \mathbb{F}_2 \rightarrow U \lhd \mathbb{F}_1 \times \mathbb{F}_2
\int_{B}F(D\psi_n)\, dx =\int_{B}F\left(\int \varphi_{1/n}(x-y)\, Du(y)\, dy\right)\, dx &\le \int_{B}\left(\int \varphi_{1/n}(x-y) F(Du(y))\, dy\right) dx &\le \int_{B}\varphi_{1/n}*F(Du)\, dx.
J_s = \int_{\{\vec{w}\in \mathbb{R}^n\,:\, |\vec{w}|\le 1 \}}|\vec{w}|^s e^{-f(\vec{w})}\, d\vec{w}+\int_{\{\vec{w}\in \mathbb{R}^n\,:\, |\vec{w}| > 1 \}}|\vec{w}|^s e^{-f(\vec{w})}\, d\vec{w}= J_{s_1}+ J_{s_2}.
\sigma_r(HD_2) =\Theta(HD_2)^{r-2}\,\sigma(HD_2)\gtrsim A^{-2}\,\Theta(HD_2)^{r-2}\,\sigma(\{R\}) & \gtrsim A^{-2}\,B^{2(r-2)}\,\Theta(R)^{r-2}\,\sigma(\{R\}) \geq \frac1{B^{6-2r}}\,\sigma_r(\{R\}).
\sum_{i=0}^{n-1}|R_{5,i}|\lesssim \sum_{i=0}^{n-1}(t_{i+1}-t_i)^{2H_0}\leq \max_{0\le i\le n-1}(t_{i+1}-t_i)^{2H_0-1}\sum_{i=0}^{n-1}(t_{i+1}-t_i) \to 0,\,\n\to \infty.
k_{p}= d\frac{(bd-a_\sigma)^{2}e^{2h\sigma}+ 2\sigma e^{h\sigma}\left(2bd+h(b^2d^2-a_\sigma^2)\right) - (bd+a_\sigma)^{2}}{\left(bd+a_\sigma+e^{h\sigma}(bd-a_\sigma)\right)^{2}}\;, k_{i}= 2d\sigma^2e^{h\sigma}\frac{2bd+h(b^2d^2-a_\sigma^2)}{\left(bd+a_\sigma+e^{h\sigma}(bd-a_\sigma)\right)^{2}}\;.
\sum_{k=1}^\infty \alpha_k\left(\sum_{s=1}^{k}\beta^{k-s}\alpha_{s-1}\right) = \sum_{k=1}^\infty \left(\sum_{s=1}^{k}\beta^{k-s}\alpha_k\alpha_{s-1}\right) \leq \sum_{k=1}^\infty \left(\sum_{s=1}^{k}\beta^{k-s}\alpha_{s-1}^2\right).
y_n(x) = R_n(\lambda(x);\alpha,\beta,\gamma,\delta), \Lambda(x) =\lambda(x) = x(x+\gamma+\delta+1), A(n) = \frac{(n+\alpha+1)(n+\alpha+\beta+1)(n+\gamma+1)(n+\beta+\delta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}, C(n) = \frac{n(n+\alpha+\beta-\gamma)(n+\alpha-\delta)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}.
\mathcal{A}_{-1}u = \sum_{n = 1}^{\infty}\lambda_{n}\langle u, \phi_{n}\rangle_{X_{-1}}\phi_{n}, \tilde{\mathcal{A}}_{-1}u = \sum_{n = 1}^{\infty}(\alpha \lambda_{n}+ \beta) \langle u, \phi_{n}\rangle_{X_{-1}}\phi_{n}u \in X.
-\frac{\partial L_{d}D_{\beta}(p\|q)}{\partial q_{j}}=-\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]q^{\beta-1}_{j}+\left[\frac{a-1}{a-b}B^{a-2}-\frac{b-1}{a-b}B^{b-2}\right]p_{j}q^{\beta-2}_{j}
G(\zeta)=G(\zeta,1) =\lim_{z\to 1}\frac{2n^2z^{\frac{n-1}{2}}(1-z)^2-n(n-1)z^{\frac{n-1}{2}}(1-z^n)(1-z)^2-2(1-z^n)^2}{2(1-z^n)^2 (1-z)^2}begin{align*}5pt] =\frac{n^2-1}{24},
Y_s=\left(0,\frac{s-S^x_{N_s-1}}{X_{N_s}}\cdot (S_{N_s}^y-S^y_{N_s-1})+S^y_{N_s-1},\right. \left. \frac{s-S^x_{N_s-1}}{X_{N_s}}\cdot (S_{N_s}^z-S^z_{N_s-1})+S^z_{N_s-1}\right).
\mathbb{E}[|X_t(x)|^2] = |\exp(t\tilde A )\exp(\frac{t}{2}\alpha ) \exp(-\frac{t^2}{2}\beta) \exp(\frac{p(t)}{2}\Gamma )x|^2 &= |\sum_{j=1}^d e^{-\frac{a_{j}}{2}t - \frac{b_{j}}{2}t^2 - \frac{\gamma_{j}}{2}(t^3-p_\Gamma t)}\langle x, v_{j}\rangle \exp(t\tilde A ) v_{j}|^2.
||a_j^i|^2(X) \lambda_i (X) - |a_j^i|^2(X_1) \lambda_i(X_1)| \leq M ||a_j^i|^2(X) - |a_j^i|^2(X_1)| + \|\vec{x}_j\|_{z_0}^2|\lambda_i (X) - \lambda_i(X_1)| \leq M||\vec{x}_j||_{z_0}^2 \ \epsilon^2 + ||\vec{x}_j||_{z_0}^2\epsilon= ||\vec{x}_j||_{z_0}^2 (M\epsilon^2 + \epsilon).
\sum_{\gamma\in\mathcal{C}_{[n,r,s]}}w(\gamma)=a_{-s}^{(r+s)}\sum_{k\in\mathbb{Z}}\sum_{\ell\in\mathbb{Z}}A_{[k-1,r-1]}^{(1)}B_{[\ell-k-1,s-1]}^{(1)}W_{[n-\ell-1,0]}
u_{1}=-t^{-1}y-c_{2}\ln t+[-t^{-1}(x+y)-c_{2}\ln t+c_{2}/2]^{1/2}, begin{align*}4pt] u_{2}=-t^{-1}x-c_{2}\ln t-[-t^{-1}(x+y)-c_{2}\ln t+c_{2}/2]^{1/2}, begin{align*}4pt] p=-\frac{1}{2}t^{-2}(x+y)^{2}+c_{2}t^{-1}(1-\ln t)(x+y).
f^{(2)}= \epsilon p_\Gamma + \sum_{k=m+1}^{t}\sum_{i=1}^n M_{2k}x^{2ke_i}+ \sum_{k=m+1}^{t}\left(\sum_{|\alpha|=2k-1}f_\alpha x^\alpha + \sum_{|\alpha|=2k}f_\alpha x^\alpha\right) = g^{(1)}+ g^{(2)}+ g^{(3)},
E |\mu^N(t)- m(t)| + \frac CN \sum_{i=1}^N E|X^N_i(t) - Y^N_i(t)| \leq \frac{C}{\sqrt{N}}+2K_1 \int_0^t e^{2K(t-s)}\frac{C}{\sqrt{N}}ds \leq \frac{C}{\sqrt{N}}.
0 = \frac{\delta_{L}}{\delta C^{\alpha a}}\mathbf{S}_{m}^{a}( \Gamma_{m}) = - \mathbf{\tilde{Q}}_{m}^{a}\frac{\delta_{L}}{\delta C^{\alpha a}}\Gamma_{m}, \qquad 0 = \frac{\delta_{L}}{\delta C^{\alpha a}}\mathbf{D}_{A}( \Gamma_{m}) = \mathbf{\tilde{Q}}_{A}\frac{\delta_{L}}{\delta C^{\alpha a}}\Gamma_{m}.
l_1(t)=(t-(T-\delta_1))((T+\delta_1)-t), &\rho_1(x,t)=\frac{e^{\lambda\psi(x)}}{l_1(t)}, &\theta_1(x,t)=\frac{e^{\lambda\psi(x)}-e^{2\lambda\|\psi(x)\|_{C(\overline{\Omega})}}}{l_1(t)},
\lambda[T_{k+1}, T] = \sum_{\ell=1}^{q}\gamma_{k+1}^{(\ell)}[T_{k+1}^{\ell-1},T_{k+1}^{\ell}]+ \gamma_{k+1}^{(q+1)}[T_{k+1}^{q},T] \leq 2^{k+1}q+2^{k+1}
V_{\alpha}({\bf x})=\sum_{j\in Q_{\alpha}}-\frac{Z}{|x_j|}+\sum_{j,k\in Q_{\alpha}, j<k}\frac{1}{|x_j-x_k|}+\sum_{j\in Q_{\alpha}, k\not\in Q_{\alpha}}\frac{1}{|x_j-x_k|}\,.
[\Pi_{F^{(j)}}, \Pi_{G^{(k)}}]_(g_1, \cdots, g_{j+k-1}) = \cdots = \{\{\{\{F^{(j)}, G^{(k)}\}^{[1]}, g_1\}^{[1]}, \cdots \}^{[1]}, g_{j+k-1}\}^{[1]}= \Pi_{\{F^{(j)}, G^{(k)}\}^{[1]}}(g_1, \cdots, g_{j+k-1})
{\cal C}_{0}= i \, \int d{\bf{r}}\,{\textstyle{\cal X}^{\alpha}}({\bf r}) \, j_{0}^{\alpha}({\bf r}) \; , \; \; \mathrm{a n d}\;{\bar{\cal C}}= i \, \int d{\bf{r}}\, \overline{{{{\cal Y}^{\alpha}}}}({\bf r}) \, j_{0}^{\alpha}({\bf r}) \; .
|\beta| =\sum_{i=1}^n \beta_i=\sum_{i\in A^c}\frac{\alpha_i}{2}=\frac{|\alpha|}{2}- \sum_{i\in A}\frac{\alpha_i}{2}= \frac{m}{2}- l,\intertext{and}|\gamma| =\sum_{i=1}^n \gamma_i=\sum_{i\in A}\frac{\alpha_i-1}{2}= \frac{2l-|A|}{2}=l-k.
2 \pi^{2}\int_{0_{( V_{4})}}^{l}\partial_{r}[ \overline{{\Psi}}\Psi ]_{r}r^{3}d r = \left< [ \overline{{\Psi}}\Psi ] \right>_{l}6 \pi^{2}\int_{0}^{l}r^{2}d r = 1
\sum_{i=1}^{n}f\big(T^{i}\circ \mathcal I(\tfrac{1}{2})\big)=\sum_{i=-n}^{-1}f\big(\mathcal I\circ T^i(\tfrac{1}{2})\big)=-\sum_{i=-n}^{-1}f(T^i(\tfrac{1}{2}))=S_{-n}f(\tfrac{1}{2}),
d^2h(y)(w|z) =\lim_{\tau\downarrow 0}\frac{2[h_k(y+\tau w+\frac{1}{2}\tau^2z )-h_k(y)-\tau dh(y)(w)]}{\tau^2}=\langle w,\nabla^2h_k(y)w\rangle+\langle\nabla h_k(y),z\rangle =d^2h(y)(w)+\langle\nabla h_k(y),z\rangle
d_{P}^{2}(h_{(N)}^{-},h_{(N)}^{+})=\sum_{j=1}^{m_{N}}d_{P}^{2}(h_{(j,N)}^{-},h_{(j,N)}^{+})\leqslant d_{P}^{2}(g_{-},g_{+})+3m_{N}\varepsilon^{2}\leqslant4m_{N}\varepsilon^{2}.
E_{0}\left( \ell , \mu , e B \right) = - 2 \times \frac{1}{2}\times \left( \frac{e B L^{2}}{2 \pi}\right) \sum_{n = 0}^{\infty}\, \sum_{\alpha \in \left\{- 1 , 1 \right\}}\, I_{n \alpha},
e = \frac{a_0+ 1}{c_0}g \iff e = \frac{u_+}{v_+}g \iff (e,g)=\pm(u_+,v_+), e = \frac{a_0- 1}{c_0}g \iff e = \frac{u_-}{v_-}g \iff (e,g)=\pm(u_-,v_-).
\partial^m_a \Gamma_i = \frac{\Gamma^n}{\lambda_i}\{\delta_{nm}\psi_a^i - \phi^n_b \phi^m_c \psi_b^i \psi_c^i \frac{\psi_a^i}{\lambda_i^2}+ \phi^n_b \phi^m_c \sum_{j \neq i}\psi_b^j \frac{(\psi_c^i \psi_a^j + \psi_a^i \psi_c^j)}{(\lambda_i^2 - \lambda_j^2)}\}\vspace{-12pt}
V_N^tF_Nf= a^{1/2}N^{-1/4}(\sum_{n=0}^{N/2-1}f(naN^{-1/2})u(n-\sqrt{N}bt/2\pi)+ & \sum_{N/2}^{N-1}f((n-N)aN^{-1/2})u(n-\sqrt{N}bt/2\pi)) = a^{1/2}N^{-1/4}(\sum_{n=0}^{N/2-k-1}f((n+k)aN^{-1/2})u(n)+ & \sum_{n=N/2-k}^{N-k-1}f((n+k-N)aN^{-1/2})u(n)+ & \sum_{n=N-k}^{N-1}f((n+k-N)aN^{-1/2})u(n)).
X=\begin{pmatrix}X_1 X_2 X_1 \end{pmatrix}, \Q=\begin{pmatrix}Q_1 Q_2 Q_1 \end{pmatrix}, \P=\begin{pmatrix}P_1 0 \end{pmatrix}, \R=\begin{pmatrix}0 R_1 \end{pmatrix}
\left|\frac{zF_1'(z)}{F_1(z)}\right|=\left|1-\frac{\rho_1}{(1-\rho_1^2)}\left( \frac{u\rho_1^2+4\rho_1+u}{\rho_1^2+u\rho_1+1}+\frac{v\rho_1^2+4\rho_1+v}{\rho_1^2+v\rho_1+1}+\frac{q \rho_1^2+4\rho_1+q}{\rho_1^2+q \rho_1+1}\right)\right|=\frac{1}{3},
\begin{aligned}\sum\limits_{i = 1}^{N}{\left({{n_{i + 1}}-{n_i}}\right)}, &\sum\limits_{i = 1}^{N}{\left({{n_{i + 1}}-{n_i}}\right)\left({2{\tau_{\rm c}}+ 2\mathcal{T}+{n_{i + 1}}+{n_i}}\right)}.\end{aligned}
\left( \begin{array}{c}{\sigma^{i}[ \omega_{0}^{+ 0 i}\tilde{\epsilon}_{2}- i F^{+ 0 i}\tilde{\epsilon}^{1}]}\\{\sigma^{i}[ - \omega_{0}^{+ 0 i}\tilde{\epsilon}^{1}- i F^{+ 0 i}\tilde{\epsilon}_{2}]}\\ \end{array}\right) = 0 ,
\sum_{j|k}\mu(k/j) C_n(\omega_n^j) = \sum_{j|k}\mu(k/j) \binom{2j}{j}\geq \binom{2k}{k}- \sum_{\substack{j|k j < k}}\binom{2j}{j}\geq \frac{4^k}{\sqrt{\pi(k+1/2)}}- (2\sqrt{k}-1) \frac{4^{k/2}}{\sqrt{\pi(k/2)}}\geq 0.
r_b^k = r_b^{k-1}(1-\sin(\alpha_{k-1})) = \cdots = r_b^0 \prod_{j=0}^{k-1}(1-\sin(\alpha_j))= r_b^0 \nu_k, r_c^k = r_c^{k-1}(1-\sin(\alpha_{k-1})) = \cdots = r_c^0 \prod_{j=0}^{k-1}(1-\sin(\alpha_j))= r_c^0 \nu_k.
\sum_{k=N}^\infty\frac{(\alpha t)^k}{k!}k^\beta\exp\left((\ln k)^{\mu_1}\cdots(\ln_mk)^{\mu_m}\right)\asymp (\alpha t)^\beta\exp\left((\ln(\alpha t))^{\mu_1}\cdots(\ln_m(\alpha t))^{\mu_m}\right)\quad\mbox{as $t\to\infty$},
\left\| \nabla^{k}_{x}\left(\partial_{t}K_{3}(t)g- \mathcal{F}^{-1}\left[e^{-\frac{\nu t |\xi|^{2 \sigma}}{2}}\cos (t |\xi|)\right] \ast g\right)\right\|_{2}& \le C (1+t)^{-\frac{n}{4 \sigma}-1+\frac{1}{2 \sigma}-\frac{k}{2 \sigma}}\| g \|_{1}+C e^{-ct}\| \nabla^{k}_{x}g\|_{2},
\zeta_{c y l}^{D}( s ) = \frac{a^{s - 1}}{2 \sqrt{\pi}\Gamma \left( \frac{\displaystyle s}{\displaystyle 2}\right) \Gamma \left( \frac{\displaystyle 3 - s}{\displaystyle 2}\right)}\sum_{n = - \infty}^{\infty}\int_{0}^{\infty}d y \, y^{1 - s}\frac{d}{d y}\ln \left[ 2 y I_{n}( y ) K_{n}( y ) \right] .
\| Q_{<k-Cm}P_k \mathcal N(\phi_{k_1},\overline{Q_{> k-Cm}\phi_{k_2}})\|_{N[I]}\lesssim \ 2^{Cm}\| \chi_I^k \mathcal N(\phi_{k_1},\overline{Q_{> k-Cm}\phi_{k_2}})\|_{L^1 L^2}\lesssim \ 2^{Cm}\| \chi_I^k \phi_{k_1}\|_{L^\frac94 L^\infty}\| \Box \phi_{k_2}\|_{L^{\frac95}L^2}
\sum_{k=N}^\infty\frac{(\alpha t)^k}{k!}k^\beta(\ln k)^{\mu_1}\cdots(\ln_mk)^{\mu_m}\asymp(\alpha t)^\beta(\ln (\alpha t))^{\mu_1}\cdots(\ln_m(\alpha t))^{\mu_m}\quad\mbox{as $t\to\infty$},
A_{x}^{( n )}={\frac{e^{{\phi_{0}}}r_{0}\left({P^{( n )}}{\Delta}+{Q^{( n )}}( \Sigma + r_{\pm}( u , v ) ) \right)}{{\sqrt{{M^{2}}-{| \Upsilon |^{2}}}}\left({{r_{\pm}( u , v )}^{2}}-{{{\Delta}}^{2}}-{\Sigma^{2}}\right)}},
\left\vert \bar{\psi}(\gamma)\right\vert =|E[\bar{\alpha}(X)\{\gamma(X)-\gamma_{0}(X)\}]-E[\alpha_{0}(X)f_{0}(0|X)\{\gamma(X)-\gamma_{0}(X)\}] & +E[\{\partial f(\delta(X)|X)/\partial u\}\{\gamma(X)-\gamma_{0}(X)\}^{2}]| & \leq C\left\Vert \gamma-\gamma_{0}\right\Vert^{2}.Q.E.D.
\{\{x;y_1\cdots y_p\};z_1\cdots z_q\}= \sum \{x;z_1\cdots z_{i_1}\{y_1;z_{i_1+1}\cdots z_{j_1}\}z_{j_1+1}\cdots z_{i_p}\{y_p;z_{i_p+1}\cdots z_{j_p}\}z_{j_p+1}\cdots z_q\},
y=\varphi-\varphi_0\left(\!z=\alpha_0(\sigma,\bar{\sigma})+\!\!\sum^{\infty}_{n=1}\frac{{\hat{y}}^n}{n!}\alpha_n(\sigma,\bar{\sigma}),\,\bar{z}=\bar{\alpha}_0(\sigma,\bar{\sigma})+\!\!\sum^{\infty}_{m=1}\frac{{\hat{y}}^m}{m!}\bar{\alpha}_m(\sigma,\bar{\sigma})\right)\,.
\langle D_s P^{\operatorname{Arc^0}}(k), n \rangle = \langle D_s (an+bv), n \rangle= \langle (D_s a+b \kappa) n+ (D_sb-a\kappa)v, n \rangle &= D_s a+b \kappa \langle D_s^2 P^{\operatorname{Arc^0}}(k), v \rangle = \langle D_s^2 (an+bv), v \rangle=\langle D_s ((D_s a+b \kappa) n+ (D_sb-a\kappa)v), v \rangle &=-\kappa (D_s a+b \kappa) + D^2_s b- D_s(a\kappa)
p_{B}(S(f,P_1)-S(f,P_2)) =p_{B}\left(\sum\limits_{i=1}^{N_m}(f(s_i)-f(s_{i}^{\prime}))\right)(t_i-t_{i-1}) &=p_{B}\left(\sum\limits_{i=1}^{2^{m-1}}(h_{m}(s_i)x_{m}^{*}d(B_{m}^{(i)}))\right) &=p_{B}(x_{m}^{*})\sum\limits_{i=1}^{2^{m-1}}d(B_{m}^{(i)})) &=p_{B}(x_{m}^{*})2^{m-1}\left[2^{-(m-1)}\left(1-\sum\limits_{j=1}^{m}\dfrac{1}{3^{j}}\right)\right]>\frac{1}{2}.
B= \left[\begin{array}{c}0 Q\end{array}\right] M= \left[\begin{array}{cc}0 I -\bar{\Lambda}-\bar{\Lambda}\end{array}\right] Z= \left[\begin{array}{cc}Q(e_j-e_k) (Q (e_j-e_k))^{\top}0 0 0\end{array}\right]\,.
P_{3,j}^*(\bar z) = v_j^*R_j^*(\bar z)H_{1,j}^{-1}R_j(b)v_jQ_{3,j}^*(b) = v_{j-1}^*R_{j-1}^*(\bar z)H_{1,j-1}^{-1}R_{j-1}(b)v_{j-1}Q_{3,j}^*(b)+P_{1,j}^*(\bar z)\widehat H_{1,j}^{-1}P_{1,j}(b)Q_{3,j}^*(b).
a_{1}= 6 C_{2}| z_{t r}| \mathrm{,}a_{2}= \frac{2 7}{| z_{t r}|^{2}}+ 1 5 C_{2}\mathrm{,}a_{3}= \frac{8 1}{| z_{t r}|^{3}}+ \frac{2 0 C_{2}}{| z_{t r}|}.
\widetilde{\cal F}_{i j k}= \partial_{t_{i}}a_{I}\partial_{t_{j}}a_{J}\partial_{t_{k}}a_{K}{\cal F}_{I J K}, \hspace{1 0 m m}{\cal F}_{I J K}= \partial_{a_{I}}\partial_{a_{J}}\partial_{a_{K}}{\cal F}( a ) .
(u^2+4tu+4t+6+6)([2+2t+u]c_2-[2+t]b_2-uc_3) & =(u^2+2tu+2t+6u+4)([3+t+u]b_2-[1+2t]c_2-ub_3) & +2u(u+1)c_2+u^2b_3+u^2c_3
\sum\limits_{i=1}^{l}m_{i}B_{\sigma ir}Q_{ir}-B_{2\sigma r}=\frac{1}{2}\sigma\sum\limits_{i=1}^{l}\sum\limits_{j=1,\textrm{}j\neq i}^{l}\frac{m_{i}m_{j}(Q_{ir}+Q_{jr})}{(\sigma-\sigma(Q_{ir}\odot Q_{jr})^{2})^{\frac{1}{2}}(1+Q_{ir}\odot Q_{jr})},
\frac{1}{(2\pi )^{n}}\int_{^{(n)}_{\texttt{b}}}f(X_1,\ldots X_n) \frac{\sqrt{\rho_{\texttt{b}}(X_1,\ldots,X_n)}}{O_{\texttt{b}}(X_1,\ldots,X_n;q,q_0)}X_1\cdots X_n &= \sum_{\lambda\in\Lambda^{(m,n)}_{\texttt{b}}}f \bigl( \boldsymbol{X}^{(m,n)}_{\texttt{b};\lambda}\bigr) \hat{\Delta}^{(m,n)}_{\texttt{b};\lambda},
\phi(P_n) =\sum_{k=0}^n\sum_{x\in \Sigma_k}\phi(T_xPT_x^*)=\sum_{k=0}^n\sum_{x\in \Sigma_k}e^{-\beta k}\phi(PT_x^*T_x) &=\phi(P)\sum_{k=0}^ne^{-\beta k}d^k=(1-e^{-\beta}d)\sum_{k=0}^ne^{-\beta k}d^k
\psi = \frac{1}{N}\mathrm{tr}\left( \mathbf{I}_N + \frac{K}{N}\frac{1}{\phi}\mathbf{I}_N \right)^{-2}= \frac{1}{\left( 1 + \frac{\beta}{\phi}\right)^2}, \phi = \frac{1}{N}\mathrm{tr}\left[ \left( 1 + \frac{\beta}{\phi}\right) \mathbf{I}_N \right]^{-1}= \frac{1}{1+\frac{\beta}{\phi}}.
(\alpha, 1_T) \cdot (s_1, t_1) (s_2, t_2) \ldots (s_m, t_m) =(\alpha\cdot s_1, t_1) \cdot (s_2, t_2) \ldots (s_m, t_m) &= (\alpha\cdot s_1 s_2^{t_1}, t_1 t_2) \cdot (s_3, t_3) \ldots (s_m, t_m) &= (\alpha\cdot s_1 s_2^{t_1}s_3^{t_1 t_2}\ldots s_m^{t_1 t_2\ldots t_m}, t_1 t_2 \ldots t_m).
s_{2 r}^{(r)}\! =\! s_{r}d_{0}^{(r)}\! +\! s_{r+1}d_{1}^{(r)}\! +\! \ldots\! +\!s_{2r-2}d_{r-2}^{(r)}\! +\! s_{2r-1}d_{r-1}^{(r)}\, , \s_{2 r + m}^{(r)}\!= \!\sum\limits_{k=0}^{r-1}\s_{r + m + k}^{(r)}d_{k}^{(r)}\, , \m \geq 1 \, .
H^{1}(dP_9,{\cal O}_{dP_9}(6 \sigma|_{dP_9}- 5F))|_{b_{-8}}\stackrel{-m_1 g_2}{\rightarrow}H^{1}(dP_9,{\cal O}_{dP_9}(9 \sigma|_{dP_9}- F))|_{c_{-3}}.
\overset{0}{R}(X,Y)Z = \overset{g}{R}(X,Y)Z - \frac{1}{2}\overset{1}{D}(X,Y) A Z - \frac{1}{4}(\overset{2}{D}(X,Z) + \overset{3}{D}(X,Z)) A Y + \frac{1}{4}(\overset{2}{D}(Y,Z) + \overset{3}{D}(Y,Z)) A X + \frac{1}{4}\pi (Z) (\pi (Y) X - \pi (X) Y ),
q_{2}^{(i)}-q_{1}^{(i)}=\frac{1}{\pi}\int_{0}^{\infty}d\omega\,\sinh \pi \omega \,s^{(i)}-\int_{0}^{\infty}d\omega\frac{I_{\omega}(k\xi_{1})I_{-\omega}(k \xi_{2})}{D_{\omega}(k\xi_{1},k\xi_{2})}S^{(i)},
f(x) := (\theta-1) \big( a_2 x^{\frac{\gamma-\theta}{1-\theta}}- a_1 x - a_{-1}x^{\frac{\theta+1}{\theta-1}}+ a_0 x^{\frac{\theta}{\theta-1}}+ \frac{1}{2}b^2 \theta x^{-1}\big)
N_{m}^{( s )}( \bar{\beta}z + \bar{\alpha})^{- 2 s}\, \left( \frac{\alpha z + \beta}{\bar{\beta}z + \bar{\alpha}}\right)^{m}= \sum_{n = 0}^{\infty}\, T_{m n}^{( s )}N_{n}^{( s )}\, z^{n}.
z_{n+1}=z_{n}+\phi(t_{n},z_{n};h_{n})+h_{n}\sum_{j=1}^{s}b_{j}k_{j},withk_{i}=q(t_{n},z_{n};t_{n}+c_{i}h_{n},h_{n}\sum_{j=1}^{i-1}a_{ij}k_{j}),
q^{-k+1}+ q^{-2\lambda_{r+1}+k+1}(q_0q_1^{-1}-1) + \sum_{m=k+1}^{\lambda_{r+1}}q^{-2\lambda_{r+1}-k+2m}(q^{-1}-q) +\sum_{m=1}^{k-1}q^{-2\lambda_{r+1}-k+2+2m}(q^{-1}-q) + \sum_{m=1}^{k-1}q^{-2\lambda_{r+1}-k+2+2m}(q^{-1}-q)(q_0q_1^{-1}-1).
\begin{array}{c c c c c c c}{{\nabla_{i}\Phi}}&{{=}}&{{( \partial_{i}+{\frac{1}{2}}p \partial_{i}{\cal K}) \Phi}}&{{;}}&{{\nabla_{i^{*}}\Phi}}&{{=}}&{{( \partial_{i^{*}}-{\frac{1}{2}}p \partial_{i^{*}}{\cal K}) \Phi}}\\ \end{array}
\begin{gathered}\Big( \bigcap\limits_{j}V_j\big((a^1_{1}, \ldots, a^{k_1}_1), \ldots, (a^1_j,\ldots, a^{k_j}_j), \ldots, (a^1_r, \ldots, a^{k_r}_r)\big) \Big) \leq \sum\limits_{i,j}e^i_j -r\end{gathered}
\big\| P_s \big[f^\frac1p \big] \big\|_{L^q(\gamma)}=\big(\frac{(2\pi)^{\frac12(c_1+c_2)-1}}{\sqrt{1-e^{-2s}}}\big)^n\int_{\mathbb{R}^{2n}}e^{-\pi \langle x,\mathcal{Q}x\rangle}\prod_{j=1,2}f_j(x_j)^{c_j}\, dx
\partial_t \mathcal{S}= -\int_{L_t}(\theta_t- \hat{\theta}) (e^{-i\hat{\theta}}\Omega)= - \int_{L_t}(\theta_t- \hat{\theta}) (e^{i(\theta-\hat{\theta})})dvol_{L_t}= - \int_{L_t}(\theta_t-\hat{\theta}) \sin(\theta_t-\hat{\theta}) dvol_{L_t}.
(1+\frac{1}{x^{2}})Q(x)=\int_{x}^{\infty}(1+\frac{1}{x^{2}})\phi(u)du>\int_{x}^{\infty}(1+\frac{1}{u^{2}})\phi(u)du=-\frac{\phi(u)}{u}|_{x}^{\infty}=\frac{\phi(x)}{x}.
R = - \frac{1}{2 Q^{2}}\left[ \frac{3 5}{1 6}+ Q^{2}\frac{\Lambda_{7}}{M_{7}^{5}}+ \frac{7}{8}\sqrt{( \frac{5}{2})^{2}- 8 Q^{2}\frac{\Lambda_{7}}{M_{7}^{5}}}\right] .
\dot{x}h+\dot{x}g\,=\, 2, 2\dot{h}+\dot{v}h+{v}'g\,=\, 2,{{e}^{\nu}}{h}'-{{e}^{\lambda}}\dot{g}\,=\, 0, 2{g}'+{\lambda}'g+\dot{\lambda}h\,=\, 2.
\sum_{k=l}^{l+n}\delta_{k,k+1}^2+4(e_{l+n+1}^2-e_l^2)\leq 2^{6}\varepsilon^{-1}\Lambda_6^{2}(\mu_{l}^2-\mu_{l+n+1}^2)+\frac{\varepsilon}{4}\sum_{k=l}^{l+n}(\mu_{k+1}^2+\mu_{k}^2)+\frac{1}{2}\varepsilon C_{rl}^{-2}\sum_{k=l}^{l+n}e_{k+1}^2.
\frac{d}{d\Lambda}e^{-\frac{1}{\hbar}\left( L^{\Lambda ,\Lambda_0}(\varphi ) + I^{\Lambda ,\Lambda_0}\right)}= \frac{\hbar}{2}\langle \frac{\delta}{\delta \varphi}\,,\dot C^{\Lambda ,\Lambda_0}\frac{\delta}{\delta \varphi}\rangle \, e^{-\frac{1}{\hbar}\left( L^{\Lambda ,\Lambda_0}(\varphi ) + I^{\Lambda ,\Lambda_0}\right)}.
\frac{\partial\overline{u_{i}}}{\partial t}+\frac{\partial\overline{u_{i}}\overline{u_{j}}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{i}}+\nu\frac{\partial^{2}\overline{u_{i}}}{\partial x_{j}\partial x_{j}}-(\overline{\frac{\partial u_{i}u_{j}}{\partial x_{j}}}-\frac{\partial\overline{u_{i}}\overline{u_{j}}}{\partial x_{j}}).
\begin{aligned}j^A_{x,x+1}= \xi^{A}_{x}-\xi^{A}_{x+1}+\frac{E_{A}-E_{B}}{2N^\gamma}(\xi^{A}_{x}\xi^{B}_{x+1}+\xi^{B}_{x}\xi^{A}_{x+1}) &+\frac{E_{A}-E_{C}}{2N^\gamma}(\xi^{A}_{x+1}+\xi^{A}_{x}-2\xi^{A}_{x}\xi^{A}_{x+1}-\xi^{A}_{x}\xi^{B}_{x+1}-\xi^{B}_{x}\xi^{A}_{x+1})\,,\end{aligned}
w(x,y)=\frac{a^{2}}{2\pi^{2}D}\sum_{m=1}^{\infty}\frac{E_{m}}{m^{2}sinh\alpha_{m}}sin\frac{m\pi x}{a}(\alpha_{m}coth\alpha_{m}sinh\frac{m\pi y}{a}-\frac{m\pi y}{a}cosh\frac{m\pi y}{a}).
\left.\begin{array}{ll}\lambda_n p_n = \hat{q}_{n-1}\lambda_{n+1}p_{n+1}= \hat{q}_n \lambda_n q_n = a_1\hat{q}_{n-1}+\hat{p}_{n-1}\lambda_{n+1}q_{n+1}= a_1\hat{q}_n+\hat{p}_n\end{array}\right.
\begin{array}{l}P(|u_{0}(x)|) = \frac{g \left( |u_{0}(x)| \right) |u_{0}(x)|^{1_{2}^{*}}}{1_{2}^{*}}- \frac{2 \gamma}{1_{2}^{*}}\int_{0}^{|u_{0}(x)|}\frac{s^{1_{2}^{*}+ 1}\log^{\gamma -1}(2+ s^{2})}{2+ s^{2}}\; ds \cdot\end{array}
\left\{Q_{\alpha}^{a}, Q_{\beta}^{b}\right\}= \delta^{a b}\gamma_{\alpha \beta}^{\mu}\, \, P_{\mu}+ \sum_{p = 0 , 1 , \cdots}\gamma_{\alpha \beta}^{\mu_{1}\cdots \mu_{p}}\, \, \, Z_{\mu_{1}\cdots \mu_{p}}^{a b}.
((X_1 \otimes X_2).M).(Y_2 \otimes Y_1) \to (X_1.((X_2.M).Y_2).Y_1 l_{X_1\boxtimes Y_1, X_2\boxtimes Y_2,M}=b_{X_1,X_2.M,Y_2}n_{X_1.(X_2.M),Y_2,Y_1}m_{X_1,X_2,M}
\begin{array}{r c l}{{\displaystyle{\sum_{n = 1}^{N}}( k^{( n )})^{2}}}&{{=}}&{{0 \, ,}}\\{{}}&{{}}&{{}}\\{{\displaystyle{\sum_{n = 1}^{N}}| k^{( n )}|^{2}}}&{{=}}&{{\frac{1}{2}\, .}}\\ \end{array}
\|\mathcal{T}^{2}(U^{2})-\mathcal{T}^{2}(U^{1})\|_{s-1}^{2}=\|\mathcal{T}(V^{2})-\mathcal{T}(V^{1})\|_{s-1}^{2}&\leq C\left(T_{0}\sup_{0\leq t\leq T_{0}}\|\mathcal{T}(U^{2})-\mathcal{T}(U^{1})\|_{s-1}^{2}+\int_{0}^{T_{0}}\|(\hat{v}^{2}-\hat{v}^{1})(\tau)\|_{s}^{2}d\tau\right).
\eqref{eq:gwinvariants}=\sum_{\Gamma\in G_{n,d}}\frac{1}{|\mathrm{Aut}(\Gamma)|}\int_{F_\Gamma}\frac{i_\Gamma^*\left(\mathrm{ev}_1^*(\varphi_1)\psi_1^{a_1}\cdots\mathrm{ev}_n^*(\varphi_n)\psi^{a_n}e(R^0\pi_*f^*\O_{\P^4}(5))\right)}{e(N_{F_\Gamma})}.
Equ> p^s+1-(\tau+1)p^t- \frac{(p-\tau+1)p^{s}-p^{t+1}}{p^{t-1}+1}= \frac{p^{s-t-1}-(p-\tau)p^s-(\tau+1)p^{2t-1}+p^{t+1}-(\tau+1)p^t}{p^{t-1}+1}+1 > 0,
(D_{1}^{*})^{2,a}|_{G ({\rm Im}(D_{1}^{1,a}))}: G ({\rm Im}(D_{1}^{1,a})) \rightarrow (D_{1}^{*})^{2,a}G ({\rm Im}(D_{1}^{1,a})) = Y_{1}^{1,a}
aD'_1 exp(-i\frac{{\Gamma}_2}{|{\Gamma}_2|}{\pi}/4){\psi}_{j}^{\pm}=[1 + exp ({\mp}\frac{\sin{\theta}_j}{|{\sin \theta}_j|}2i{\theta}_j)](\frac{{\Gamma}_2}{|{\Gamma}_2|}){\psi}^{\pm}_j.
\mathcal{D}\left( T_{\alpha}\right) := \left\{x \in D \, \big| \lim\limits_{t \rightarrow 0+}- t^{1-2\alpha}U'(t)x \right\}, T_{\alpha}x : = \lim\limits_{t \rightarrow 0+}- t^{1-2\alpha}U'(t)x.
\widetilde{\Phi}_i:=\sum_{k\in\mathcal{I}_{e_i}}m_{e_i,k}^{*}\otimes\Phi_i(m_{e_i,k})\in \bigl(\mathbf{M}^{\pi_{\mathcal{S}(v_i)}}\otimes S_i\bigr)^{K_{v_i}}.
A_{( b )}= \frac{g^{4}}{\alpha^{4}}\int d^{2 6}k ~ \frac{1}{( P_{1}+ P_{4})^{2}- \frac{1}{\alpha^{\prime}}}\, \frac{1}{k^{2}- \frac{1}{\alpha^{\prime}}}\, \frac{1}{( P_{1}+ P_{4}- k )^{2}- \frac{1}{\alpha^{\prime}}}\, \frac{1}{( P_{1}+ P_{4})^{2}- \frac{1}{\alpha^{\prime}}},
\begin{array}{r c l}{{}}&{{}}&{{{\cal V}= \{f \in{\cal L}( V , W ) \vert f_{| W}~ \mathrm{i s ~ a ~ s c a l a r ~ o p e r a t o r}\}~ ,}}\\{{}}&{{}}&{{{\cal W}= \{f \in{\cal L}( V , W ) \vert f_{| W}= 0 \}~ .}}\\ \end{array}
T_{j}=\{\beta \bmod p^{t+1}| \beta \in T , \left\lfloor \frac{\beta}{p^{t+1}}\right\rfloor =j \}, T_{j,i}=\{\beta \in T | \left\lfloor \frac{\beta}{p^{t+1}}\right\rfloor =j, \beta \equiv i \bmod{p}\}.
\frac{qx+2-x}{(1-x)^2}\sum_{j=0}^\infty f_j \left(q, 2-\frac{1}{x}, 2-\frac{1}{x}\right) = \frac{2}{1-x}- \frac{1}{qx+2-2x}+ \frac{(qx+1-x)^2}{(1-2x)^2 (qx+2-2x)}\sum_{j=0}^\infty f_j \left(q, 3-\frac{1}{x}, 3-\frac{1}{x}\right).
\int_DU'(y(T,x,z))dx= \tilde{p}(0,z) \exp \big( - \int_0^T \{\Phi(s,z) +\frac{a_0(s,z)}{b_0(s,z)}\}dB(s) &+\frac{1}{2}\int_0^T\{\Phi^2(s,z) -\frac{a_0^2(s,z)}{b_0^2(s,z)}\}ds \big)=:\tilde{p}(0,z)\Gamma(T,z) .
M_{1}(\mathcal{NC}(G)) =(2mn-n)(2mn-n-1)^{2}-4(2mn-n-1)\dfrac{(mn-n)(mn-n-1)+mn(n-1)}{2}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+n(m-1)(mn-n-1)^{2}+mn(n-1)^{2}&= 5m^{3}n^{3}-9m^{2}n^{3}+4mn^{3}&= n^{3}(5m^{3}-9m^{2}+4m)
{\cal U}_{\sigma}({\bf p}) ={\frac{m}{\sqrt{2}}}\left( \begin{array}{l}{D^{J}\left( \alpha ({\bf p}) \right) \xi_{\sigma}}{D^{J}\left( \alpha^{- 1 \, \dagger}({\bf p}) \right) \xi_{\sigma}}\end{array}\right) \quad ,
\kappa_i =\lim_{k\to\infty}n_k^{-1}\log\Vert M^{n_k}_\omega v\Vert &\ge\lim_{k\to\infty}n_k^{-1}\log\left(\Vert M^{n_k}_\omega v_{n_k}\Vert -\Vert M^{n_k}_\omega (v_{n_k}-v)\Vert\right) &\ge \lim_{k\to\infty}n_k^{-1}\log\Vert M^{n_k}_\omega\Vert+\lim_{k\to\infty}n_k^{-1}\log\left(1-\Vert v_{n_k}-v\Vert\right) &\ge \kappa_\mu,
\begin{cases}-\Delta_y \chi -|p|^2 \chi_{zz}-2 p\cdot \nabla_{y}\chi_z +\bar c(p)\chi_z(y,z) -g(y, \chi(y,z)) =0 (y,z)\in [0,1]^{n+1}\chi(y,z+1)=\chi(y,z)+1 \chi(\cdot, z) \end{cases}
D_{14}= \frac{(at_3)^{1-s_1}(at_3+1)^{-s_3}}{s_1+s_3-1}+ \frac{s_3}{s_1+s_3-1}\int_{at_3}^\infty \frac{1}{u^{s_1}(u+1)^{s_3+1}}du.
\left\langle \partial\mathbb{A}\left( I+\mathbb{A}\right)^{-1}a,\left(I+\mathbb{A}\right)^{-1}a\right\rangle =-\left\langle a,\left(I+\mathbb{A}\right)^{-1}a\right\rangle \left\langle a,\left( I+\mathbb{A}\right)^{-1}a\right\rangle & =-\left\langle a,\left( I+\mathbb{A}\right)^{-1}a\right\rangle^{2}.
Q_0 (\sigma) = 0,Q_2 (\sigma) = 0, Q_0(\sigma) = -u \sigma_u + 4g_2 \sigma_{g_2}+ 6g_3 \sigma_{g_3}+ \sigma , Q_2(\sigma) = -\tfrac{1}{2}\sigma_{u,u}- \tfrac{1}{24}g_2 u^2 \sigma + 6 g_3 \sigma_{g_2}+ \tfrac{1}{3}g_2^2 \sigma_{g_3}