harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\substack{\{u,v\}\in\binom{U}{2}x_{uv}\geq \l}}x_{uv}\ge \frac{a_i\mu}{2}\overset{\eqref{eq:mui}}{\geq}\frac{1}{2}\cdot\frac{a_i}{4^{k^2}}(\rho n)^{k}p_{\rm I}^i\,.
E_{\ell^\infty \Gamma}( T_\xi^* T_\eta )(t)= E_{\ell^\infty \Gamma}(T_{\eta \otimes \overline{\xi}})(t)= \langle T_{\eta \otimes \overline{\xi}}\delta_t, \delta_t \rangle= \langle u_t^* (\eta \otimes \overline{\xi}) u_t, \eta \otimes \overline{\xi}\rangle.
\begin{array}{r c l}{I_{1}( \rho^{2}, 0 )}&{=}&{\displaystyle ( 4 \pi )^{- 2}( \pi \mu^{2}\rho^{2})^{2 - n / 2}\Gamma ( n / 2 - 2 )}\\{}&{=}&{\displaystyle ( 4 \pi )^{- 2}\left[ \Gamma ( n / 2 - 2 ) - \ln ( \pi \mu^{2}\rho^{2}) + O ( n / 2 - 2 ) \right] ,}\\ \end{array}
\varepsilon\partial_j [\Lambda^\varepsilon ]b(\mathbf{D})(S_\varepsilon -I)(B^0-\zeta\overline{Q_0})^{-1}=[(\partial_j\Lambda )^\varepsilon ](S_\varepsilon -I)b(\mathbf{D})(B^0-\zeta\overline{Q_0})^{-1}&+\varepsilon [\Lambda^\varepsilon ](S_\varepsilon -I)b(\mathbf{D})\partial_j (B^0-\zeta\overline{Q_0})^{-1}.
\sum_{\mathbf{r}=(r_1, \ldots, r_{t-1}) \in \mathcal{S}_{\mathsf{valid}}}|\gamma_{\mathbf{r}}| \le \sum_{\mathbf{r}=(r_1, \ldots, r_{t-1}) \in \mathcal{S}_{\mathsf{valid}}}2^{t(q_1 +\ldots + q_t) - \sum_{i=1}^{t-1}r_{i}}\le 2^{t(q_1 + \ldots + q_t +1)}.
\bar{Q}_{i}= \int d^{3}x \: \biggl \{\varepsilon_{i j k}\left( F_{0 j}- \tilde{F}_{0 j}\right) \chi_{k}+ \psi_{0}D_{i}\lambda + \psi_{i}D_{0}\lambda - \left( F_{0 i}- \tilde{F}_{0 i}\right) \eta + \varepsilon_{i j k}\psi_{j}D_{k}\lambda +{\frac{1}{2}}\left[ \phi , \lambda \right] \chi_{i}\biggl \}.
\sum_{(i,j_1,j_2)\in{\cal I}}K_{b_n}(x_{i,j_1}-x)K_{b_n}(x_{i,j_2}-x)= \mathrm{O}\bigl[ \min(h,M_n)nb_n k_n\iota_n\bigr], M_n=\max_{1\le i\le n}m_i
c_{t,x}({s,y}) =\frac{1}{\prod_{(l,w) \in \mathcal{S}(t,x)}P(Z_l^{w}= w -l e_d )}& \qquad\times \sum_{\sigma \in{\rm Sym}(\mathcal{S}(t,x))}{\rm sgn}(\sigma) P(Z_{\sigma_1(t,x)}^{\sigma_2(t,x)}= y +s e_d ) & \times \prod_{(h,z) \in \mathcal{S}(t,x) \backslash \{(t,x)\}}P(Z_{\sigma_1(h,z)}^{\sigma_2(h,z)}= z -h e_d),
\ln\left\lbrace\prod_{j\neq k}\frac{|\sin\pi(\theta-\theta_j)|}{|\sin\pi(\theta_k-\theta_j)|}\right\rbrace=\sum_{j\neq k}\ln|\sin\pi(\theta-\theta_j)|-\sum_{j\neq k}\ln|\sin\pi(\theta_k-\theta_j)|< (2q_n-1)(\frac{\ln{q_{n+1}/|\ell|}}{2q_n-1}+\epsilon).
\delta_{\epsilon}A_{( \sigma )}^{\alpha \beta}= \varepsilon^{\alpha \beta \gamma \delta}\partial_{\gamma}\epsilon_{\delta ( \sigma )}\equiv \stackrel{( 0 )}{R}_{( \sigma )}^{\prime \alpha \beta \delta ( \rho )}\epsilon_{\delta ( \rho )}, \delta_{\epsilon}B_{\alpha}^{( \sigma )}= 0 ,
\begin{array}{ll}g_{_L}(\widehat{\mathbf{f}}_\mu, \widehat{\mathbf{f}}_\nu) = g_{\mu\nu}g_{_L}(\widehat{\mathbf{f}}_\mu, \mathbf{e}_k) = 0 g_{_L}(\mathbf{e}_k,\mathbf{e}_l)=g_{0 \,kl}\, , \end{array}
{\cal L}_s=\frac{\alpha^2}2\partial_\mu \varphi^\alpha \partial^\mu\varphi^\alpha -\frac{\beta^2}4\partial_\mu \varphi^\alpha \partial_v\varphi^\beta \left\{\partial^\mu \varphi^\beta \partial^v\varphi^\alpha -\partial^\mu \varphi^\alpha \partial^\nu \varphi^\beta \right\}+\frac{{}_1}{{}^2}\mu \left( \varphi^\alpha \varphi^\alpha -1\right)
\Psi_{\mu_+}\Big(A_{\ell_i}\,\Big|\,gF_n\setminus\big[R^g_n\cup\bigcup_{j=i}^m A_{\ell_j}\big]\Big) =\Psi_{\mu^{(\ell_i-1)}K_{A_{\ell_i}}}\Big(A_{\ell_i}\,\Big|\,gF_n\setminus\big[R^g_n\cup\bigcup_{j=i}^m A_{\ell_j}\big]\Big) \;.
\mathbb{E}_{\pi_e}H^2(\omega_{e^c},U_e(\omega_e))\left(T_z(\omega_{e^c},U_e(\omega_e))-T_z(\omega_{e^c},U_e(\omega_e)-2^{-j})\right)^2\mathbf{1}_{\{U_e(\omega_e)\ge 2^{-j}\}}= \int_0^1 H^2(\omega_{e^c},x)\left(T_z(\omega_{e^c},x)-T_z(\omega_{e^c},x-2^{-j})\right)^2\mathbf{1}_{\{x\ge 2^{-j}\}}\,\mathrm{d}x.
\begin{pmatrix}g_1 g_2 \end{pmatrix}= \begin{pmatrix}f_1(2) f_1(4) f_2(2) f_2(4) \end{pmatrix}^{-1}\begin{pmatrix}f_1 f_2 \end{pmatrix}= \begin{pmatrix}\phantom{-}3 -1 -2 \phantom{-}1 \end{pmatrix}\begin{pmatrix}f_1 f_2 \end{pmatrix}.
J^{w_1}(\textbf{X}_{minRSSU}^{(n)}) =-\frac{(n!)^2}{2(n-1)!!}\prod_{i=1}^{n}E(\Lambda_X^{w_1}(B_{1,2i-1})) &\le -\frac{(n!)^2}{2(n-1)!!}\prod_{i=1}^{n}E(\Lambda_Y^{w_2}(B_{1,2i-1})) &=J^{w_2}(\textbf{Y}_{minRSSU}^{(n)}).
k \cdot \varphi_{k*}\left( \Delta a \right) = \varphi_{k*}\circ r_* \left( p_{k*}[S^1]\times a \right) &= r_{*}\circ (1\times \varphi_k)_* \left( [S^1]\times a \right) &= r_* ([S^1]\times k^i a) &= k^i \cdot \Delta (a),
\times \int_{\Omega}I_{A B}^{K}D_{R}E_{B}^{J}( h ) \biggl ( D_{S}E_{A}^{P}E_{C}^{L}( f ) D_{J - R - S + M + L + P + N + K}E_{D}^{M}( g ) - ( F \leftrightarrow G ) \biggr ) ,
\sum_{n\in\mathbb{Z}, m\ge0}(-1)^nq^{n^2+m(m+1)/2}=\sum_{n\ge0}s_{+}t(n)q^n=\sum_{n\ge0}(-1)^nq^{n(n+1)}\left(\sum_{0\le j\le n}q^{-j(j+1)/2}+\sum_{0\le j\le n-1}q^{-j(j+1)/2}\right).
\tilde{R}(x,t) = \sum_{n}m \omega_{n}^{2}\partial_{x}Q_{n}(x,t) \Big\{(q_{n0}- Q_{n0})\cos[\omega_{n}(t-t_{0})] + \frac{\dot{q}_{n0}}{\omega_{n}}\sin [\omega_{n}(t-t_{0})] \Big\},
B_{q_1,q_2}(x_1^i x_2^j; x_1,x_2) - x_1^ix_2^j = \sum_{k=0}^{q_1}\sum_{l=0}^{q_2}\delta^{q_1, q_2}_{k,l}(i,j) b_{k,l}^{(q_1,q_2)}(x_1,x_2).
D^{\mu_{1}\cdot \cdot \cdot \mu_{k}}( \partial n ) = \frac 1{m !}\epsilon^{\mu_{1}\cdot \cdot \cdot \mu_{k}\mu_{k + 1}\cdot \cdot \cdot \mu_{n}}\epsilon_{a_{1}\cdot \cdot \cdot a_{m}}\partial_{\mu_{k + 1}}n^{a_{1}}\cdot \cdot \cdot \partial_{\mu_{n}}n^{a_{m}}
|\mathcal{P}(Q; w^s)| = \int_{\{u^s = w^s \}\cap Q}- u^s_t \det D^2 u^s \,dx, &\geq \int_{\{u = w \}\cap \left\{u_{t}\leq -s\right\}\cap Q}- u^s_t \det D^2 u^s \,dx, &\geq \int_{\{u=w\}\cap Q}-u_{t}\det D^{2}u-Cs|Q| &=|\mathcal{P}(Q; w)|-Cs|Q|.
\begin{aligned}Q^N(\mathcal{M}^{-1}(\{s\}))< \Big(\sum_{\gamma=1}^{r_s}\binom{r_s}{\gamma}(2^{l-1})^\gamma e^{-c_0\gamma N}\Big)Q^N(s)= &\Big((1+2^{l-1}e^{-c_0N})^{r_s}-1\Big)Q^N(s),\s\in A_{}.\end{aligned}
T_{\alpha \beta}= G_{\mu \nu}( \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}-{\scriptstyle \frac{1}{2}}g_{\alpha \beta}g^{\alpha^{\prime}\beta^{\prime}}\partial_{\alpha^{\prime}}X^{\mu}\partial_{\beta^{\prime}}X^{\nu}) = 0 ,
-q^{-n/2}[\#X(F_{q^n})-(q^n+1)] =\sum_{d\mid u,\sqrt p \in \mathbb Q(w_d)}\left( e_{d,1}\sum_{j\in I_d^+}w_d^{-jt}+ e_{d,2}\sum_{j\in I_d^-}w_d^{-jt}\right) +\sum_{d\mid u,\sqrt p \not \in \mathbb Q(w_d)}\left( e_{d}\sum_{j\in I_d}w_d^{-jt}\right).
k_{1}\cdot(PA)^{t}+k_{2}\cdot(PB)^{t}+k_{3}\cdot(PC)^{t}\ge2\sqrt{k_{1}k_{2}k_{3}}(\frac{(PD)^{t}}{\sqrt{k_{1}}}+\frac{(PE)^{t}}{\sqrt{k_{2}}}+\frac{(PF)^{t}}{\sqrt{k_{3}}}).
A_{\mu}= \Bigl ( \bigl ({\frac{Q_{e}r}{r^{2}+ A^{2}\cos^{2}\theta}}\bigr ) , ~ 0 , ~ 0 ,{}~ - \bigl ({\frac{Q_{e}r A \sin^{2}\theta}{r^{2}+ A^{2}\cos^{2}\theta}}\bigr ) \Bigr )
\sum_{n=0}^{\infty}Y_{n}^{\left( k\right)}\left( \lambda \right) \frac{t^{n}}{n!}=\frac{2^{k}}{\left( \lambda -1\right)^{k}}\sum_{n=0}^{\infty}\left( -1\right)^{n}\left( \begin{array}{c}k+n-1 n\end{array}\right) \left( \frac{\lambda^{2}}{\lambda -1}\right)^{n}t^{n}.
\int_{0}^{1}\overline{\xi(x)}\xi^{\prime}(x)dx+ \int_{0}^{1}\overline{\xi(x)}\eta^{\prime}(x)dx =\lambda\int_{0}^{1}|\xi(x)|^2 dx \int_{0}^{1}\overline{\eta(x)}\eta^{\prime\prime}(x)dx+\int_{0}^{1}\overline{\eta(x)}\eta^{\prime}(x)dx+\int_{0}^{1}\overline{\eta(x)}\xi^{\prime}(x)dx =\lambda\int_{0}^{1}|\eta(x)|^2dx.
{\cal D}_{g_{\mu \nu}( w )}\nabla_{\sigma ( x )}^{g_{\kappa \lambda}( z )}\equiv \delta ( z - w ) \left[ t_{\kappa \lambda \ \ \ , \sigma \tau}^{\ \ \ \mu \nu}\nabla^{\tau ( x )}\right] \delta ( w - x ) +{\cal O}[ l^{- 1}] \; ,
\frac{d^+}{dt}\biggr\rvert_{t=0}f(t) =F'(\mu(A))\cdot \frac{d^+}{dt}\biggr\rvert_{t=0}\mu((1-t) \times_{p,\bar{\alpha}}A + t \times_{p,\bar{\alpha}}B) + F(\mu(A))-F(\mu(B)).
AA1=[\begin{matrix}(1+2\lambda+\beta)&-(\lambda-\alpha)&0&0\\ -(\lambda+\alpha)&(1+2\lambda+\beta)&-(\lambda-\alpha)&0\\ 0&-(\lambda+\alpha)&(1+2\lambda+\beta)&-(\lambda-\alpha)\\ 0&0&-2\lambda&(1+2\lambda+\beta)\end{matrix}],
\det (P_{R,m+1}(t)-I_{(m+1)\times (m+1)})= &=\det\left(\begin{matrix}p_{R,m+1,1}(t)-1 \begin{matrix}p_{R,m+1}(t) 0 \cdots 0 \end{matrix}\begin{matrix}p_{R,m+1}(t) 0 \vdots 0 \end{matrix}\boxed{P_{R,m}(t)-I_{m\times m}}\end{matrix}\right) &=(L-1)\det(P_{R,m}(t)-I_{m\times m})>0, \, \, \forall t\geq t_0.
\dot Q_t^{11}= A' Q_t^{11}+ Q_t^{11}A + Q_t^{11}B B' Q_t^{11}+ C' C\,, \dot Q_t^{12}= (A + B B' Q_t^{11})' Q_t^{12}\,, \Q_t^{21}= (Q_t^{12})'\,, \dot Q_t^{22}= (Q_t^{12})' B B' Q_t^{12}\,,
\left\{(t_1,\ldots, t_{i-1}, t_{i+1},\ldots, t_{n-1}, s, k) ~ \begin{tabular}{|l}$h\bar{h}=h_i$ $g_s\in c'(h, Y), g_k\in c'(\bar{h}, Y)$ $[t_1,\ldots, t_{i-1}, s, k, t_{i+1}, \ldots, t_{n-1}]\neq \emptyset$\end{tabular}\right\}
\frac{d N(t)}{dt}= \int_0^{x_{}}\int_0^x \frac{u}{x}\beta(u,x-u)n(t,u)n(t,x-u) du dx - \int_0^{x_{}}\int_0^{x_{}}\beta(x,v)n(t,x)n(t,v) dv dx &+ \int_{0}^{x_{}}\int_{x}^{x_{}}b(x,v) S(v) n(t,v) dv dx- \int_{0}^{x_{}}\int_0^x \frac{u}{x}b(u,x) S(x) n(t,x) du dx.
\big[ \mathfrak{A}_0, \mathfrak{A}_1 \big] \big( (\alpha_0, \beta_0), (\alpha, \beta) \big) = - \partial_{(\alpha, \beta)}\mathfrak{A}_0 \big( (\alpha_0, \beta_0), (\alpha, \beta) \big) \, \mathfrak{A}_1 \big( \alpha_0, \beta_0 \big) & = \begin{pmatrix}0 - \tilde{A}_1 ( \alpha_0) \, \exp (\alpha)\end{pmatrix}.
\sum_{k=0}^K T_{k+1}(x) \leq \frac{1}{\alpha}D_w(x,x_0)- \frac{1}{\alpha}\sum_{k=0}^K D_w(x_{k+1},x_k)+L\cdot \ell(\tau+1)\cdot \sum_{k=0}^K \sum_{i=1}^{\tau+1}i D_w(x_{k-\tau+i},x_{k-\tau+i-1}).
Z_{Y}( G , \lambda A , N ) = \int_{\Sigma ({\cal M})}d \nu \; \mathrm{e}^{- \frac{n \lambda A}{2}}\frac{( - 1 )^{i}N^{2 - 2 g}}{| S_{\nu}|}( - 1 )^{\sum_{j}x_{j}},
(T) =\int_{\mathbb{T}^n}\sum_{\xi\in\mathbb{Z}}\sigma_{e^{-tT^*T}-e^{-TT^*}}(x, e_{\xi})dx. &=\int_{\mathbb{T}^n}\sum_{\xi\in\mathbb{Z}}\sigma_{[T,S]}(x, e_{\xi})dx,
\mathrm{U}_{\mathrm{e}}(\xi,\mathrm{t}) =-\frac{\gamma_{\mathrm{p}}}{\gamma_{\mathrm{m}}}\frac{1}{\alpha_\mathrm{m}}\int_{0}^t\int_{\partial\Omega}\Phi(\xi,t;\mathrm{z},\tau)\gamma^{\textbf{int}}_{1}\mathrm{U}_{\mathrm{i}}(\mathrm{y},\tau)d\sigma_\mathrm{y}d\tau + \textbf{err}^{(1)}+ \textbf{err}^{(2)}.
\widehat{\mathcal{R}^1}[u, \cdots, u]_k := \sum_{\substack{\ell=1 odd}}^{p}\sum_{\substack{k=k_1-k_2+\cdots-k_{p-1}+k_p k_\ell = k}}\prod_{\substack{i=1 odd}}^{p}\widehat{u}_{k_i}\prod_{\substack{i=2 even}}^p\overline{\widehat{u}_{k_i}}= \frac{p+1}{4\pi}\widehat{u}_k\int_\mathbb{T}|u|^{p-1}\,dx.
0=-\sum_{i,j,p}\bar h_{ijl}^{p^{*}}\bar h_{ijk}^{p^{*}}=\sum_{i,j,p}\bar h_{ij}^{p^{*}}\bar h_{ijkl}^{p^{*}}=\bar h_{11}^{1^{*}}\bar h_{11kl}^{1^{*}}+3\bar h_{12}^{2^{*}}\bar h_{12kl}^{2^{*}}=\bar\lambda_1\bar h_{11kl}^{1^{*}}+3\bar \lambda_2\bar h_{22kl}^{1^{*}}.
J(u) =\displaystyle \int_{\Omega}\frac{B(x)}{p(x)}|\nabla u(x)|^{p(x)}dx+ \displaystyle\int_{\Omega}\frac{C(x)}{p(x)-1}|u(x)|^{p(x)-1}dx &+\displaystyle\int_{\Omega}\frac{A(x)}{p(x)}|u(x)|^{p(x)}dx+\displaystyle\int_{\Omega}\frac{D(x)}{p(x)+1}|u(x)|^{p(x)+1}dx
H_\lambda D\Bigl(\frac{\sum_{\square\in\lambda}c_{\square}^{2r}}{H_\lambda}\Bigr)= \sum_{\lambda^{i+}}\frac{H_\lambda}{H_{\lambda^{i+}}}\Bigl(\sum_{\square\in\lambda^{i+}}c_{\square}^{2r}-\sum_{\square\in \lambda}c_{\square}^{2r}\Bigr)=\sum_{\lambda^{i+}}\frac{H_\lambda}{H_{\lambda^{i+}}}x_i^{2r}.
\langle \phi, \chi(H_{\kappa,\eta}) e^{-itH_{\kappa,\eta}}\phi \rangle = \sum_{j=1}^n e^{-itz_j(\eta)}\langle \phi, (H_{\kappa,\eta}- z)^{-1}_c \phi \rangle|_{z=z_j(\eta)}+ \int_{\gamma}e^{-itz}\langle \phi, (H_{\kappa,\eta}- z)^{-1}_c - (H_{\kappa,\eta}- z)^{-1}\phi \rangle dz &+ \mathcal{E}_1
\left| \frac{1-e^{-z\tau}}{z\tau}\right| = \left| \frac{e^{\xi}-1}{\xi}\right| = \frac{|e^{-\rho\cos\theta}\cos(\rho\sin\theta)-1-\mathrm{i}e^{-\rho\cos\theta}\sin(\rho\sin\theta)|}{\rho}& \ge \frac{(e^{-2\rho\cos\theta}+1-2e^{-\rho\cos\theta})^{1/2}}{\rho}=\frac{e^{-\rho\cos\theta}-1}{\rho}\ge -\cos\theta >0
u_{1,j}= c_j u_{1, j_e}+ (1-c_j) u_{1, j_e+1}\mbox{and}v_{1,j}= c_j v_{1, j_e}+ (1-c_j) v_{1, j_e+1}\quad 1 \le j\le \tilde J, & u_{I,j}= c_j u_{I, j_e}+ (1-c_j) u_{I, j_e+1}\mbox{and}v_{I,j}= c_j v_{I, j_e}+ (1-c_j) v_{I, j_e+1}\tilde J+2\le j\le J,
|\beta_{n+1}| \leq C(\beta)^n u^{n-1}\sum\limits_{l=1}^n |C(n+1,l)| |\mathcal{T}_{\textrm{R},l}[n+1]| \frac{|\beta_{n+1}|}{(n+1)!}\leq [z^n]\frac{C(\beta)^n}{n+1}\sum\limits_{l=1}^{\infty}q_{\mathrm{R},l}|C(n+1,l)| \frac{\left(\frac{1}{u}T_{\mathrm{R},1}(uz)\right)^l}{l!}.
[ ( h r^{3}\partial_{r})^{2}+ \omega^{2}r^{6}{\mathcal H}- \frac{8 h r^{4}R^{4}}{( r^{2}+ R^{2})^{2}}( 1 + \frac{r_{0}^{2}}{R^{2}}) ] \tilde{\nu}= 0 ,
B_1 = x_2 y_1 + x_1 y_2 + a_{22}x_2 y_2 + a_{32}x_3 y_2 + a_{23}x_2 y_3 + a_{33}x_3 y_3 B_2 = -x_3 y_1 + a_{12}x_2 y_2 + a_{13}x_2 y_3, B_3 = -x_1 y_3 + a_{21}x_2 y_2 + a_{31}x_3 y_2.
\psi_{1t}=-\psi_{2ss}+\bigg\{-\frac{1}{2}\big((\psi_{1})^{2}+(\psi_{2})^{2}\big)+\frac{1}{2}\big((\psi_{1})^{2}+(\psi_{2})^{2}\big)\big\rvert_{s=0}\bigg\}\psi_{2}, begin{align*}3mm]\psi_{2t}= \psi_{1ss}- \bigg\{-\frac{1}{2}\big((\psi_{1})^{2}+(\psi_{2})^{2}\big)+\frac{1}{2}\big((\psi_{1})^{2}+(\psi_{2})^{2}\big) \big\rvert_{s=0}\bigg\}\psi_{1}.
\frac{\ell(f_1,\chi)\cdot \ell(f_2,\chi^{-1})}{\langle f_1,f_2\rangle}=\frac{\Lambda_F(2)\cdot\Lambda(1/2,\Pi,\chi)}{2\Lambda(1,\Pi,{\rm ad})}\cdot\prod_v\beta_v(f_{1,v},f_{2,v}),
{2\over \sqrt{v_{n-2}^2(1-u_{n-2})^2+4w_{n-2}v_{n-2}(1-u_{n-2})}}\log|{a_{n-1}^{-}\over a_{n-1}^{+}}|={2\over (1-u_{n-2})v_{n-2}}X\log{1+X\over 1-X}
c_{0}= \frac{H}{P}\sum_{j = 1}^{3}\frac{n_{j}}{p_{j}}+ l , \; P = p_{1}p_{2}p_{3}, \; H = P \sum_{j = 1}^{3}\frac{q_{i}}{p_{i}}= p_{1}p_{2}q_{3}+ p_{1}q_{2}p_{3}+ q_{1}p_{2}p_{3}.
\lambda \big((g, q_1 \ast q_2) \big) = g \cdot \partial'(q_1 \ast q_2) &= g \cdot \big(\partial'( q_1) \ast \big(\phi(e_{q_1}) \cdot \partial'(q_2) \big) \big) &= g \cdot \partial'(q_1) \ast \big( (g \phi(e_{q_1})) \cdot \partial'(q_2) \big) &=\lambda\big((g,q_1)\big) \ast \lambda \big( \big( g\phi(e_{q_1}) \big),q_2 \big).
\sup\left\{\int_{0}^{k}G[(M_{\mathcal{T}}\phi)^{\ast}(t)]h(t)dt:\phi\geq0\int_{X}\phi d\mu=f\phi^{\ast}\leq R\right\}= =\int_{0}^{k}G\left( \frac{1}{t}\int_{0}^{t}R(u)\chi_{\lbrack0,\sigma(f))}(u)du\right) h(t)dt.
\tilde{a}_{(n-1)+1-i}=\binom{n-1+1}{i}\left(\dfrac{a_{n+1-(i+1)}}{\binom{n+1}{i+1}}+\dfrac{a_{n+1-i}}{\binom{n+1}{i}}\right)=0 \left(0\leq i\leq n-1\right)
\begin{aligned}\frac{\partial v_{\infty}}{\partial t}(x, t) =d_{2}\left[\left(J_{2}* v_{\infty}\right)(x, t)-v_{\infty}(x, t)\right]+r_2v_{\infty}(x, t)\left(-1-v_{\infty}(x, t)\right) &\leq d_{2}\left[\left(J_{2}* v_{\infty}\right)(x, t)-v_{\infty}(x, t)\right]-r_2v_{\infty}(x, t),\end{aligned}
\big\| \big(\tfrac{\partial}{\partial y}\psi\big)( x, y ) \, v_1 \big\|_V= \left\| u_{0,1}( t, x ) \, F'( y ) \, v_1 \right\|_V\leq \tfrac{c_{-\vartheta}}{( T - t )^{\vartheta}}\, \| F'( y ) \|_{L( H, H_{-\vartheta})},
\lim_{n \to \infty}C_{p_n}(\varphi(z_n)) = \lim_{n \to \infty}\left( \sqrt[p_n]{\Lambda_1(p_n)}\sqrt[p_n]{\alpha(p_n)}\vert \varphi(z_n) + \sigma_n \vert^{\frac{\alpha(p_n)}{p_n-1}}v_{p_n}(x_0)^{\frac{\beta(p_n)}{p_n-1}}\right) &= \Lambda_{1,\infty}\vert\varphi(z_0)\vert^{\theta}v_{\infty}(x_0)^{1-\theta}
\left\Vert M\right\Vert =\max_{\zeta\in\sigma\left( M\right)}\left\vert\zeta\right\vert ,\left\Vert \left( \zeta I-M\right)^{-1}\right\Vert^{-1}=\operatorname{dist}\left( \zeta,\sigma\left(M\right) \right) ,
\sum\limits_{k=1}^{n-2}(\sum\limits_{l=1}^{(k,n-1-k)}q^{-\frac{3}{2}+l}) hX_{n-1-k}-\sum\limits_{k=1}^{n-3}(\sum\limits_{l=1}^{(k,n-2-k)}q^{-\frac{3}{2}+l}) hX_{n-1-k}-q^{-\frac{1}{2}}hX_1 = \sum\limits_{k=\frac{n}{2}}^{n-3}q^{-\frac{5}{2}+n-k}hX_{n-1-k}=\sum\limits_{k=\frac{n}{2}+2}^{n-1}q^{-\frac{1}{2}+n-k}hX_{n+1-k}.
\sum_{\rho\in R_\epsilon}(\rho) =\frac{1}{2\pi}\int_{0}^{T}\left[\log \left| -\frac{1}{2}-\epsilon+it\right|-\log \left| -\frac{1}{2}+\epsilon+it\right|\right]\dt+ \frac{1}{2\pi}\int_{\frac{1}{2}-\epsilon}^{\frac{1}{2}+\epsilon}\left[\arg \left(\sigma-1+iT\right)-\arg (\sigma-1)\right]\d\sigma
\|(q-\delta(n+1)-\epsilon) T_{f}(r,s) \le \sum_{j=1}^{q}\frac{1}{\deg Q_j}N_{f}^{[m]}(r,s;Q_{j}) &+c\left(m_0(\mathfrak{L};r, s)+{\rm Ric}_{p}(r,s)+\kappa\log^+ \mathcal{Y}\left(r^{2}\right)+\kappa\log^+r\right),
(\mathbf{P}(\tilde\sigma_2,\tilde\sigma_6)+X)_{|B'=0}=(P_{j5})_{|B'=0}\cdot{(\lambda_i)_{B'=0}\over(\lambda_3)_{B'=0}}-(P_{i5})_{|B'=0}\cdot{(\lambda_j)_{B'=0}\over(\lambda_3)_{B'=0}}
\|b\|_{\mathcal{BMO_B}}:=\sup_{B\in\mathcal{B}}\|b-b_B\|_{{\rm exp}\,L, B}=\sup_{B\in\mathcal{B}}\inf \left\{\lambda>0:\,\frac{1}{\mu(B)}\int_B e^{\frac{|b(x)-b_B|}{\lambda}}\,d\mu\leq 2 \right\}.
\left< T ( z ) \Phi_{i_{1}}( z_{1},{\bar{z}}_{1}) \ldots \right> = \sum_{k}\sum_{j_{k}}\left[ \frac{\Delta_{i_{k}}^{j_{k}}}{( z - z_{k})^{2}}+ \frac{\delta_{i_{k}}^{j_{k}}}{( z - z_{k})}\frac{\partial}{\partial{z_{k}}}\right] \left< \Phi_{j_{1}}( z_{1},{\bar{z}}_{1}) \ldots \right> \: .
\frac{d x^{\mu}}{d s}= \frac{d t}{d s}\frac{d x^{\mu}}{d t}, \frac{d^{2}x^{\mu}}{d s^{2}}= \frac{d t}{d s}\frac{d}{d t}\left( \frac{d t}{d s}\frac{d x^{\mu}}{d t}\right) ,
\begin{cases}u_{1,\infty}u_{2,\infty}= u_{3,\infty}u_{4,\infty}, u_{1,\infty}+ u_{3,\infty}= |\Omega|^{-1}M_{13}, u_{1,\infty}+ u_{4,\infty}= |\Omega|^{-1}M_{14}, u_{2,\infty}+ u_{3,\infty}= |\Omega|^{-1}M_{23},\end{cases}
2 \pi \alpha^{\prime}E = 1 - \frac{1}{2}\frac{\alpha^{\prime}}{\alpha_{e f f}^{\prime}}, \ \ \g = \frac{\alpha^{\prime}}{\alpha_{e f f}^{\prime}}, \ \ \g_{s}= G_{s}\sqrt{\frac{\alpha_{e f f}^{\prime}}{\alpha^{\prime}}},
\mathbb{P}_{\boldsymbol{f}}\left(\bigcup_{\theta \in B^c}\left\lbrace \bar{v}_{k}(\theta) \geq \frac{1}{2}\mathbb{E}_{\boldsymbol{f}}\left[\bar{v}_{k}(\theta) \right]\right\rbrace \right) \leq \sum\limits_{\theta \in B^c}\exp \left( \frac{\left( \mathbb{E}_{\boldsymbol{f}}\left[\bar{v}_{k}(\theta)\right]\right)^2}{8k\log^2\frac{1}{\alpha}}\right)
T_{q}\mathcal{C}' = \left\{\left.\left(\begin{array}{c}\xi \alpha \eta \end{array}\right)~~\right\vert~~ \left(\begin{array}{c}D\xi + (L_{u}\alpha)^{1,0}- (X_{\eta})^{1,0}\ast\overline{\partial}\eta^{1,0}- d\mu_{c}(\xi)\end{array}\right) = 0 \right\}
I_l=\int_0^{\pi}\dfrac{\cos{2nl\theta}\,d\theta}{(a_{2n}+\cos{2n\theta})^{2s}}=(-1)^l(2\rho^{2n})^{2s}\dfrac{\pi\sum_{m=0}^{2s-1}{2s+l-1 \choose m}{4s-m-2 \choose 2s-1}\left(\rho^{4n}-1\right)^m}{\rho^{2nl}\left(\rho^{4n}-1\right)^{4s-1}}.
\lim_{k \rightarrow 0}\tilde{I}_{D Z M}( k^{2}) \Big |_{C_{1}^{0}}= - 2 . 1 2 5 8 5 + 1 . 3 3 3 3 \log ( k ) - 0 . 2 3 8 6 6 3 k^{({\frac{4}{3}})}+ 0 . 0 4 6 3 0 4 k^{({\frac{8}{3}})}.
(*)\leq \sup_{x\neq y}|x-y|^{-\mu}r^{[l]-q-1}\sum_{|k|_1=[l]-q-1}\frac{1}{k!}\left|\int_{\mathbb{R}^m}\tilde{u}(z)z^k (D^kg(x-\theta rz)-D^kg(y-\theta rz))dz\right|.
\vec r_{GV}={1\over 2}{1\over p^2+2}{1\over 2p^2+1}\left( \begin{array}{c}{p^2\over 4}\sin{4u}+(p^2+1)^2\sin{2u}-{2\sqrt{2}\over 3}p\cos{3u}-2\sqrt{2}p(p^2+2)\cos{u}-{p^2\over 4}\cos{4u}-((p^2+1)^2-2)\cos{2u}\end{array}\right)
\begin{aligned}\d I ={}J\,\omega_1{}+ I_2\,\omega_2{}+ I_3\,\omega_3\,,{}\d J ={}-(K_3+KI)\,\omega_1{}+ J_2\,\omega_2{}+ J_3\,\omega_3\,,{}\d K ={}K_1\,\omega_1{}+ K_2\,\omega_2{}+ K_3\,\omega_3\,.{}\end{aligned}
| \Xi_{\beta_{1}}\rangle \star | \Xi_{\beta_{2}}\rangle = \exp \biggl ( \chi^{T}( 1 + \Phi{\cal K})^{- 1}\beta +{\frac{1}{2}}\beta^{T}{\cal K}( 1 + \Phi{\cal K})^{- 1}\beta \biggr ) | \Xi_{f}\rangle ,
|y^*(u_i - y_i)| \le |y^*(Tx_i-y_i)| + \frac{\delta}{2}= |y^*(T(x_i - y_i))| + \frac{\delta}{2}= |\varphi(y^*)(x_i - y_i)| + \frac{\delta}{2}< \frac{\delta}{2}+ \frac{\delta}{2}= \delta
f'(r) =-\frac r2+(n-1)\cdot\frac 1{\sqrt n}\tan\frac r{\sqrt n}&=-\frac{\sqrt n}2\cdot\frac r{\sqrt n}+\left(\sqrt n-\frac 1{\sqrt n}\right)\tan\frac r{\sqrt n}&\geq\frac{\sqrt n}2\left(\tan\frac r{\sqrt n}-\frac r{\sqrt n}\right)\geq 0,
-i\frac{y^2}{\sqrt{v}}e^{-\pi i \bar{\tau}b^2/2}e^{2\pi i (c\bar{\tau}+w)(bx-c|z|^2)}(c\bar{\tau}+w)e^{-\frac{\pi y^2}{v}(c\bar{\tau}+w)^2}&=-i\frac{y^2}{\sqrt{v}}(c\bar{\tau}+w) e^{-2\pi i \bar{\tau}(b/2-cx)^2}e^{2\pi i (bxw-cx^2w)}e^{-\frac{\pi y^2}{v}|c\tau+w|^2}.
s_0^{\star 2}=\frac{8P\tilde{k}_4A_0^2A_1^2+\tilde{k}_2A_0^2-4P\tilde{k}_4 A_1^4-\tilde{k}_2A_1^2}{8\tilde{k}_4A_0^2A_1^2-2\tilde{k}_4A_0^4-2\tilde{k}_4A_1^4}, s_1^{\star 2}=2P-s_0^{\star 2}.
2 \times \left\{A , B , C , D \right\}= \left\{A , \left\{B , C , D \right\}\right\}+ \left\{B , \left\{C , D , A \right\}\right\}+ \left\{C , \left\{D , A , B \right\}\right\}+ \left\{D , \left\{A , B , C \right\}\right\}\; ,
\begin{aligned}E_{\varphi}[\mu_\varepsilon] = E_{\varphi}[\mu]+ \varepsilon \Re D_{V,h}(\mu) + o(\varepsilon), E_{\varphi}[\mu^{\varepsilon}] =E_{\varphi}[\mu^{\varepsilon}_{-\varepsilon}]+\varepsilon \Re D_{V,h}\left(\mu^{\varepsilon}\right) + o(\varepsilon).\end{aligned}
\frac{\vec{k}^{2}}{m_{f}}\phi_{\sigma}( \vec{k}) + \sum_{\sigma^{\prime}}\frac{m_{f}}{( 2 \pi )^{3}}\int \frac{d^{3}k^{\prime}}{\sqrt{E_{k}E_{k^{\prime}}}}v_{O B E}( \sigma , \sigma^{\prime}, \vec{k}, \vec{k}^{\prime}) \phi_{\sigma^{\prime}}( \vec{k}^{\prime}) = \frac{M_{f u l l}^{2}- 4 m_{f}^{2}}{4 m_{f}}\phi_{\sigma}( \vec{k}) \; ,
\displaystyle\sqrt{x}\frac{M_{\alpha}(x)}{E_{\alpha}(x)}=\begin{cases}\sqrt{2x}J_{\alpha}(x) \mbox{if}0\leq x\leq X_{\alpha}\sqrt{x}\left(J^2_{\alpha}(x)+Y^2_{\alpha}(x)\right)^{1/2}\mbox{if}x\geq X_{\alpha}\end{cases}
\varphi_{k,\lambda}^{n-1}(z) = (n-1)! 2^{n-1}\frac{J_{n-1}(\sqrt{(2k+n)|\lambda|}\,|z|)}{(\sqrt{(2k+n)|\lambda|}\,|z|)^{n-1}}+ m_k( \sqrt{|\lambda|}|z|)
\|f\|\leq\|M_f-\Phi_T^{-1}(R)\|=\|\Phi_T^{-1}\circ\Phi_T(M_f-\Phi_T^{-1}(R)) \|=\|\Phi_T^{-1}(M_g+R-R) \|= =\|\Phi_T^{-1}(M_g)\|\leq \|\Phi_T^{-1}\|\|M_g\|=\|\Phi_T^{-1}\|\|g\|.
\begin{array}{ll}\frac{1}{2}\{g(\nabla_U \xi, V) + g(\nabla_V \xi, U)\}+ Ric^\nu (U, V) - (m-n)g(U, V)\|H\|^2 + \sum\limits_{j=1}^{n}g(\nabla_{X_j}H, X_j) g(U, V) + \mu g(U,V) = 0.\end{array}
{\cal F}= \left( \begin{array}{c c c c c}{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{F_{1}}}&{{0}}&{{0}}\\{{0}}&{{- F_{1}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{F_{2}}}\\{{0}}&{{0}}&{{0}}&{{- F_{2}}}&{{0}}\\ \end{array}\right) .
\Delta S_M^{\rm div}(\lambda=0) = \frac{2}{(4-d) 64 \pi^2}\int d^4x \sqrt{-g}\left[M_{\mu\nu\rho\sigma}M^{\mu\nu\rho\sigma}+ 2 M_{\mu\nu\rho\sigma}\left( P^{\rho\sigma\mu\nu}+ \frac{1}{6}R \delta^{\rho(\mu}\delta^{\sigma\nu)}\right) \right] .
\mathbb{Z}^n_{\sigma}= \{\lambda \in \mathbb{Z}^n: \sigma(\lambda, \mu) = 1 \ \forall \mu \in \mathbb{Z}^n\}, \mathbb{Z}^n_u = \{\lambda \in \mathbb{Z}^n: u(\lambda, \mu) = 0 \ \forall \mu \in \mathbb{Z}^n\}.
\mathrm{\Gamma_{\m u}^{1}( p^{\prime}, p ) = \frac{\a l p h a_{s}}{2 4 \p i}\g a m m a_{\m u}\Gamma ( 2 - \o m e g a - \s i g m a ) , ~ \o m e g a \to ( 3 / 2 )^{+}, \s i g m a \to ( 1 / 2 )^{+}~ .}
\begin{array}{c}{{0 = \left[ - ( \Gamma^{0}+ \cos \alpha \Gamma^{\phi}) \Gamma ( e^{\psi}) + \sin \alpha \right] \psi =}}\\{{= \left[ \Gamma ( e^{\psi}) ( \Gamma^{0}+ \cos \alpha \Gamma ( e^{\phi}) ) + \sin \alpha \right] \psi =}}\\{{= \left[ \Gamma ( e^{\psi}) ( \Gamma ( e^{| |}) + \cos \alpha \Gamma ( e^{\phi}) ) + \sin \alpha \right] \psi}}\\ \end{array}
{\cal M}_w={\cal D}_{\bf r}^{R_w}(w),\hbox{for any}\,\,w\in J_{NT},{\cal M}_R=\Big\{\xi_{ij}^R\Big\}_{i=r_1+1,r_2+1}^{n_1-r_1-2,n_2-r_2-2},\hbox{for any}\,\,R\in \Delta,{\cal M}=\bigcup_{w\in J_{NT}}{\cal M}_w\cup\bigcup_{e\in C}{\cal M}_e\cup \bigcup_{R\in \Delta}{\cal M}_R,
\begin{array}{ll}(\overline \nabla_XJ_\alpha )Y=\frac{1}{4n}\left[g(X,Y)p_\alpha +\overline \theta_\alpha (Y)X+g(X,J_\alpha Y)J_\alpha p_\alpha \right. \qquad\quad\left.+\overline \theta_\alpha (J_\alpha Y)J_\alpha X\right] .\end{array}
2\|\rho_m-\pi\|_V =\sum_{i=0}^{N/2}|\rho_m(0,i)-\pi_i| &\geq \sum_K |\rho_m(0,i)-\pi_i| &\geq \sum_K \pi_i-\sum_K\rho_m(0,i) &\geq 1-\frac{1}{k^2}-\frac{B}{(\sqrt{N}\lambda_1^m-k)^2}
A^{-1}=-\frac{1}{c_{0}}M_{3}=\frac{1}{40}[\begin{matrix}6&26&-14\\ -8&-8&12\\ 6&-14&6\end{matrix}]=[\begin{matrix}0.15&0.65&-0.35\\ -0.20&-0.20&0.30\\ 0.15&-0.35&0.15\end{matrix}]
Z=\int d\left[\psi,\bar{\psi}\right] \prod_{\mu=0}^{2}d\left[A_{\mu}\right]\prod_{c_{0}=1}^{N-1}d\left[A_{3}^{c_{0}}\right]\Delta_{\rm FP}\left[A\right]e^{i(S \left[A,\psi,\bar{\psi}\right]+S_{\rm gf}\left[A \right])}\ .
\mathfrak{T}_{3}:= \Bigg\{(l,m,s): l\in \mathbb{N}_{0}, m\in \bigg\{-l-\frac{1}{2},\ldots,l+ \frac{1}{2}\bigg\},s=\pm\frac{1}{2},\Omega_{l,m,s}\neq 0 \Bigg\}.
\left[ \frac \delta{\delta \Phi \left( y \right)}, w \left( x \right) \right] = - d_{\mathrm{W}}\left( \Phi \right) \delta \left( x - y \right) \frac \delta{\delta \Phi \left( x \right)}, \Phi = \left( b^{a}, h^{a}\right) ,