harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2}K_{ijk}+J_{j,ik}=\mathbb{E}[\frac{1}{2}\frac{\partial^{3}lnf_{\theta_{0}}(X_{t})}{\partial\theta_{i}\partial\theta_{j}\partial\theta_{k}}+\frac{\partial lnf_{\theta_{0}}(X_{t})}{\partial\theta_{j}}\frac{\partial^{2}lnf_{\theta_{0}}(X_{t})}{\partial\theta_{i}\partial\theta_{k}}].
\hat{G}_{lm,l^{\prime}m^{\prime}}\left( k\mathbf{\hat{e}}_{z}\right)=\int_{0}^{\infty}dR\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi R^{2}\sin\theta\exp\left( -ikR\cos\theta\right) G_{lm,l^{\prime}m^{\prime}}\left(\mathbf{R}\left( R,\theta,\phi\right) \right) .
\int_{SO(n)}\mathcal W_\Psi^L f(\rho,\hat x\tilde g)\,d\nu(\tilde g) =\frac{1}{\Sigma_n}\int_{SO(n)}\int_{\mathcal S^n}\Psi_\rho^L\left((\hat x\tilde g)^{-1}y\right)\,f(y)\,d\sigma(y)\,d\nu(\tilde g) &=\frac{1}{\Sigma_n}\int_{\mathcal S^n}\int_{SO(n)}\Psi_\rho^L(\tilde g^{-1}\hat x^{-1}y)\,d\nu(\tilde g)f(y)\,d\sigma(y).
\partial_t \rho=-\div_q\left[\frac{c\, p}{\sqrt{m^2c^2+|p|^2}}\rho\right]+\div_p\left[\left(\nabla_q V+\gamma\frac{c\,p}{\sqrt{m^2c^2+|p|^2}}\right)\rho \right]+\gamma\theta\Delta_p \rho.
\widehat{Y}(qx) = \bigg(F_{\infty}+\frac{F_1}{1-\alpha x}\bigg) \widehat{Y}(x) = \begin{pmatrix}q^{\mu}-q^{\mu}(1-\beta /\alpha ) begin{align*}1mm] \displaystyle \frac{(1-q^{\mu})\alpha x}{1-\alpha x}\displaystyle \frac{(-\alpha +(\alpha - \beta)q^{\mu})x+q^{\lambda}}{1-\alpha x}\end{pmatrix}\widehat{Y}(x).
H_{free}=\hbar\omega_{s}\sum_{j=1}^{2}\hat{a}_{sj}^{\dagger}\hat{a}_{sj}+\hbar\omega_{p}\sum_{j=1}^{2}\hat{a}_{pj}^{\dagger}\hat{a}_{pj}+\hbar\omega_{s}\hat{a}_{c}^{\dagger}\hat{a}_{c}
\alpha_{j}( A ) \equiv \sum_{m_{1}, \ldots , m_{j}\neq 0}\epsilon^{m_{1}+ \cdots + m_{j}}\oint \frac{d t_{1}}{2 \pi i}\cdots \oint \frac{d t_{j}}{2 \pi i}\mathrm{e}^{- N ( \widetilde{\Phi}_{m_{1}}( t_{1}) + \cdots + \widetilde{\Phi}_{m_{j}}( t_{j}) )}M_{1 2}M_{2 3}\cdots M_{j 1}
\Gamma_{\ell,m,n}=\left\{A=(a_{ij})\in \Gamma:\substack{a_{21}=a_{31}=a_{32}=a_{41}=a_{42}=a_{43}=0, a_{12}\in P^{\ell-m}, a_{13}\in P^{\ell-n}(t),a_{14}\in P^\ell(t),a_{23}\in P^{m-n},a_{24}\in P^m(t) ,a_{34}\in P^n(t), a_{11},a_{22},a_{33},a_{44}\in \mathbb{F}_q}\right\}/\{\lambda I:\lambda\in \mathbb{F}_q^\times\},
\int_\Omega\Phi(f_n(t))\, \mathrm{d}x- \int_\Omega\Phi(f_n(t_0))\, \mathrm{d}x=\int_{t_0}^t\Big\langle \partial_t f_n(\tau),\Phi'(f_n(\tau))\Big\rangle_{(H^1)',H^1}\, \mathrm{d}\tau\,.
\omega y^{\prime}+ 4 \left( e^{- \bar{\omega}\bar{z}}w_{-}+ e^{\bar{\omega}\bar{z}}\bar{w}\right) y^{2}+ 8 E y + 4 \left( e^{\bar{\omega}\bar{z}}w_{+}+ e^{- \bar{\omega}\bar{z}}w \right) = 0 .
A =\begin{bmatrix}a_{1,1}a_{1,2}a_{1,3}\varepsilon \varepsilon a_{2,1}\varepsilon \varepsilon \varepsilon a_{2,5}\varepsilon \varepsilon \varepsilon a_{3,4}\varepsilon \varepsilon a_{4,2}\varepsilon \varepsilon \varepsilon a_{5,1}\varepsilon \varepsilon a_{5,4}\varepsilon\end{bmatrix},
({{\alpha}^{2 a_{1}}{z_{1}},{\alpha}^{2 a_{2}}{z_{2}}, \cdots ,{\alpha}^{2 a_{4}}{z_{d - 1}},{\alpha}^{2 a_{d}}{z_{d}},{\alpha}^{a_{d + 1}}{z_{d + 1}},{\alpha}^{a_{d + 2}}{z_{d + 2}},{\alpha}^{- a}{\psi},{\alpha}^{- ( d + 1 ) a}{\phi}}) \, \, \, ,
\frac{X_j(t)}{X_{k^*}(t)}\leq \alpha (1 - \delta) e^{s(j - k^*)(t - \tau_{j+1})}\bigg( \frac{C_6 \mu}{s}\bigg)^{j - k^*}e^{s(j - k^*)(\tau_{j+1}- \tau_{k^*+1})}&= \alpha (1 - \delta) \bigg( \frac{C_6 \mu}{s}\bigg)^{j-k^*}e^{s (j - k^*)(t - \tau_{k^*+1})}.
\|P_m-f\|_{S_n^p}=|P_m(0)-f(0)|+|P'_m(0)-f'(0)|+\cdots &\phantom{=}+|P^{(n-1)}_m(0)-f^{(n-1)}(0)|+\|P^{(n)}_m-f^{(n)}\|_{H^p}&=\|p_m-f^{(n)}\|_{H^p}\,.
\sum_{i=1}^{n}\big(\deg_{\widetilde{X}}(v_{i})-2\big)\leq N\cdot\big(\deg_{\widetilde{Y}}(a) -2\big)\cdot (3-2)\leq N\cdot\big(\deg_{\widetilde{Y}}(a) -2\big)\cdot\big(\deg_{\widetilde{Z}}(b) -2\big).
\left|\mu(\rho r^2u_x)|_{x=0}R^2\frac{dR}{dt}\right|\leq C \left((E_0(0)+E_1(0))^\frac{1}{2}S(t)\left|\frac{dR}{dt}\right|+\int_0^tS(t-\tau)\left|\frac{d\mathfrak{R}}{dt}\right|(\tau)d\tau\left|\frac{dR}{dt}\right|\right),
\displaystyle\frac{{\cal J}_-}{{\cal J}_+}= 2 \displaystyle\frac{J_{l - \xi}(ka)J_{l + \Phi}(ka) - J_l(ka) J_{l + \Phi - \xi}(ka)}{J_{l - \xi}(ka) H^+_{l + \Phi}(ka) - J_l (ka) H^+_{l + \Phi - \xi}(ka)}-1.
Y_{l}=\X^{l}\oplus X^{-l}& =\left\{\cos\left( \alpha lx\right) \omega_{1}\left( y\right)+\sin\left( \alpha lx\right) \omega_{2}\left( y\right) ,\ \omega_{1},\omega_{2}\in L_{\frac{1}{K_{2}\left( y\right)}}^{2}\left(y_{1},y_{2}\right) \right\},
\tilde{\Psi}^{+}= - i \frac 1 r ( 2 k + 1 )^{- 1 / 2}\left( \begin{array}{c}{( k + m + 1 / 2 )^{1 / 2}Y_{k}^{m - 1 / 2}( \vartheta , \varphi ) B ( r )}\\{( k - m + 1 / 2 ) Y_{k}^{m + 1 / 2}( \vartheta , \varphi ) B ( r )}\\ \end{array}\right)
a_{i;n}(\rho)=\sum_{j_1=0}^{n}\sum_{j_2=0}^{n-j_1}...\sum_{j_{{N_\rho}-1}=0}^{n-\sum_{r=1}^{{N_\rho}-2}j_{r}}c_{i;j_1,j_2,...,j_{{N_\rho}-1},n-\sum_{r=1}^{{N_\rho}-1}j_{r}}\biggl(\prod_{d=1}^{{N_\rho}-1}\rho_{d}^{j_d}\biggr) \rho_{N_\rho}^{n-\sum_{r=1}^{{N_\rho}-1}j_{r}}
{\cal F}_{\mathrm{v a c}}( p ) \equiv \frac{d}{d{\alpha}_{- p}}{\cal A}_{\mathrm{v a c}; +}- \frac{d}{d{\alpha}_{p}}{\cal A}_{\mathrm{v a c}; -}= ( 1 - \mathrm{N}^{2}) \frac{e_{+}^{2}\mathrm{L}^{2}}{4{\pi}^{2}}\frac{1}{p}.
\mathcal{E}_{k,1}= \{\mathsf{supp}(x^n)\subseteq S_k \}\mathcal{E}_{k,2}= \{T_{j,k}\not\subseteq \cup_{i \in \mathsf{supp}(x^n)}T_{i,k}\mbox{~~for all~~}j \in S_k \setminus \mathsf{supp}(x^n) \}.
\Pi^{ab\ (1)}_{\mu\nu}(p,M_i)=\lim_{M_i \rightarrow\infty}\frac{g^2_0 c_v}{16 \pi^2}\left\{\frac{5}{3}\sum_{i=1}^N c_i\log\left(\frac{M_i^2}{\kappa^2}\right) + \Pi^{(1)}_{fin}\right\}(p^2g_{\mu\nu}- p_\mu p_\nu) \delta^{ab}
q ( r ) = \frac{1}{( 2 \pi )^{N + 1}a^{3}}\sum_{{\bf \alpha}}\frac{\lambda_{n , l}^{3}T_{n}( \lambda_{n , l})}{\sqrt{\lambda_{n , l}^{2}+ k^{2}a^{2}+ m^{2}a^{2}}}f_{n}^{( q )}[ J_{n}( \lambda_{n , l}r / a ) ] , \quad q = \varepsilon , p_{i},
\phi^{a}( w ) ~ \phi^{b}( z ) ={\frac{[ I ]}{( w - z )^{2 \Delta_{\phi}}}}~ + ~ A \left\{{\frac{f^{a b c}}{( w - z )^{\Delta_{\phi}}}}\phi^{c}( z ) ~ + ~{\frac{1}{( w - z )^{\Delta_{\phi}- 1}}}J_{- 1}^{a}\phi^{b}( z ){}~ + ~ . . . \right\}~ + ~ . . .
\{F , G \}= \frac{\partial F}{\partial x_{l}}\frac{\partial G}{\partial p_{l}}- \frac{\partial F}{\partial p_{l}}\frac{\partial G}{\partial x_{l}}+ i ( - 1 )^{F}\frac{\partial F}{\partial \psi_{l}}\frac{\partial G}{\partial \psi_{l}}\ .
{\cal L}_{F F}= \bar{\Psi}[ - i \frac{2}{3}L_{\mu}( \partial_{\mu}+ i a_{\mu}+ i C_{\mu}) + M ] \Psi - \frac{i}{8 \pi ( \alpha - 1 )}\epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda},
\begin{cases}(\imath_{k-1}, \imath_k, \imath_{k+1})=(\imath, \jmath, \imath), (\imath'_{k-1}, \imath'_k, \imath'_{k+1})=(\jmath, \imath, \jmath), \imath_s=\imath'_s\ \s\ \s\neq k, k\pm 1,\end{cases}
S_{ij}(i\pi -\theta )=S_{ij}(h\theta_{h}+H\theta_{H}-\theta)=\prod\limits_{x=1}^{h}\prod\limits_{y=1}^{H}\left\{x+h,y+H\right\}_{\theta}^{-\mu_{ij}(x+h,y+H)}\,\,.
F\left(g_{s,j,k}\right)\circ F\left(g_{e,i,j}\right)=F^{\prime}\left(s\right)^{\left(j,k,i\right)}=F^{\prime}\left(s\right)^{\left(i,j,k\right)}=F\left(g_{e,j,k}\right)\circ F\left(g_{s,i,j}\right).
\cdots\longrightarrow [\bigvee_{i=1}^{k}S \mathbb{S}_{i}^4, Cat]\stackrel{(S(g))^{*}}{\longrightarrow}[\mathbb{S}^{8}, Cat]\stackrel{f_{X}^{*}}{\longrightarrow}[X, Cat]\stackrel{\iota_{Cat}^{*}}{\longrightarrow}[\bigvee_{i=1}^{k}\mathbb{S}_{i}^4, Cat]\stackrel{g^{*}}{\longrightarrow}[\mathbb{S}^{7}, Cat],
f(P_x) \triangleq \mathcal{D}(p_{_1}||p_{_0}) - \mathcal{D}(p_{_0}||p_{_1}) &= \frac{P_y^2 - \sigma_w^4}{\sigma_w^2 P_y}- \log \frac{P_y^2}{\sigma_w^4}&=\frac{P_x + \sigma_w^2}{\sigma_w^2}- \frac{\sigma_w^2}{P_x + \sigma_w^2}- \frac{1}{2}\log(P_x + \sigma_w^2) + \log \sigma_w.
\int \frac{{\rm d}\Omega}{4\,\pi}\;\;{\cal L}_{\mu_1\mu_2\mu_3}^{a_1 a_2 a_3}\left(k_1,k_2,k_3,\hat Q\right)=-i\;f^{a_1\,a_2\,a_3}\;{\cal A}_{\mu_1\mu_2\mu_3}\left(k_1,k_2,k_3\right).
K(x,y)=\frac{1}{(2\pi\i)^2}\oint\oint \frac{(1-\sqrt{\zeta}z^{-1})^N}{(1-\sqrt{\zeta}z)^{M-1}}\frac{(1-\sqrt{\zeta}w)^{M-1}}{(1-\sqrt{\zeta}w^{-1})^{N}}\frac{dzdw}{(z-w)z^{x+1}w^{-y}}
\chi R_0(z_0) \chi = \sum_{j=1}^{\frac{d-1}{2}}\tilde{B}_j(r, -z_0 ) +2 i\Big( \frac{z_0}{2 \pi}\Big)^{d-1}\mathcal{E}^*_{\chi}(- \bar{z_0}) \mathcal{E}_{\chi}( -z_0)
\sum_{k=0}^n \binom{p+k}{p}\binom{q+n-k}{q}\binom{k}{c}\binom{n-k}{d}H_{p+k}={}\binom{n+p+q+1}{n-c-d}\binom{p+c}{c}\binom{q+d}{d}(H_{n+p+q+1}-H_{p+q+c+d+1}+H_{p+c})
(\alpha \otimes \Delta_c) \circ \Delta_c(l) =\alpha (l_1) \otimes T(l_{12}) \otimes T(\alpha (l_2))-\alpha (l_1) \otimes T(\alpha (l_2)) \otimes T(l_{12}) &-T(\alpha (l_2)) \otimes \alpha (l_1) \otimes T(l_{12})+T(\alpha (l_2)) \otimes T(l_{12}) \otimes \alpha (l_1).
{{\mathbf{\Omega}}_{p2}}= \{\left. \left({{\sin}^2}{\theta_1}, \cdots ,{{\sin}^2}{\theta_k}\right) \right|\sin{\theta_1}\leq \delta , 0 \leq{{\sin}^2}{\theta_k}\leq \cdots \leq{{\sin}^2}{\theta_1}\}.
L\ =\ \{0, 1, 2, 5, 8, 10, 11, 12, 14, 15, 16, 18, 23, 25, 26, 28, 29\}, R\ =\ \{0, 1, 3, 4, 8, 10, 11, 13, 14, 15, 17, 19, 20, 22, 23, 24, 28\}.
\bar{J}(\bar{\gamma}) =\sqrt{2}\int_{\bar{\gamma}}\bar{T}dt=\sqrt{2}\int_{\bar{\gamma}}(\bar{U}+h)dt=\int_{\bar{\gamma}}\sqrt{\bar{U}+h}d\bar{s}=\int d\bar{s}_{h,\omega}d\bar{s}_{h,\omega}^{2}=\bar{T}d\bar{s}^{2}=(\bar{U}+h)d\bar{s}^{2}=(U+h-\frac{\omega^{2}}{2\rho^{2}})d\bar{s}^{2}
\frac{1}{i \hbar}[ W_{1},W_{2}] =\{W_{1},W_{2}\}_{M}=\frac{2}{\hbar}\sin \frac{\hbar}{2}(\frac{\partial}{\partial p_{2}}\frac{\partial}{\partial \lambda_{1}}- \frac{\partial}{\partial p_{1}}\frac{\partial}{\partial \lambda_{2}}) W_{1}W_{2}
\mathcal{G}^{(n)}_{L-d}(x_i ; \bar{X}_i^{1,n}) = x_i^d \sum_{l=0}^{n-1}x_i^l \; \psi_{l,d}\sum_{\stackrel{1 \leq j_1 < \dots < j_{n-1-l}\leq n}{j_{\alpha}\neq i}}\prod_{\alpha=1}^{n-1-l}x_{j_{\alpha}}
f(x) = c_{1}\epsilon (x^{0}) \delta ((x^{0})^{2}- |{\bf x}|^{2}) + c_{2}\left( ((x^{0}+ i0)^{2}- |{\bf x}|^{2})^{- 1}+((x^{0}- i0)^{2}- |{\bf x}|^{2})^{- 1}\right) + c_{3}
\begin{array}{l}{{^{W_{k}= W \left[ \left( F ( u_{k}, V_{k}( \sigma^{+}) ) + F ( u_{k}, V_{k}( \sigma^{-}) ) - F ( u_{k}, v_{k - 1}) \right) , u_{k}\right]}}}\\{{_{\bar{W}_{k}= \bar{W}\left[ \left( \bar{F}( U_{k}( \sigma^{+}) , v_{k}) + \bar{F}( U_{k}( \sigma^{-}) , v_{k}) - \bar{F}( u_{k - 1}, v_{k}) \right) , v_{k}\right] .}}}\\ \end{array}
\phi(\emptyset) =x+y, \phi(p_1)=\phi(p_3) =2x-y, \phi(p_2)=\phi(p_4) =2y-x, \phi(p_1p_2)=\phi(p_2p_4)=\phi(p_3p_4) =x-2y, \phi(p_2p_3)=\phi(p_1p_3)=\phi(p_1p_4) =y-2x.
\Omega:= \mbox{\rm d}p^\mu \wedge \mbox{\rm d}x^1 \wedge \cdots \underset{{\mu^{\textrm{th}}{\textrm{place}}}}{\wedge \underbrace{\mbox{\rm d}\varphi}\wedge}\cdots \wedge \mbox{\rm d}x^n - \mbox{\rm d}H \wedge \mbox{\rm d}x^1 \wedge \cdots \wedge \mbox{\rm d}x^n\, .
{\cal E}={\frac{\pi \mu^{2}}{\lambda}}\int_{0}^{\infty}d r r \left[ h^{2}\! + ( \varphi^{\prime})^{2}+{\frac{f^{2}\varphi^{2}}{r^{2}}}+{\frac{\lambda}{2}}\left( \varphi^{2}- 1 \right)^{2}\right] .
\Vert B_{s,\tau}f\Vert_{L^{2}(\mathbb{C}^{d},\mu_{s,\tau})}^{2}=\left(e^{\frac{1}{2}\Delta_{s,\tau}+\frac{\tau}{2}\partial^{2}+\frac{\bar{\tau}}{2}\bar{\partial}^{2}}|f_{\mathbb{C}}|^{2}\right) (0).
|I_{2}(x)| \leq C\|f\|_{L^{p, \phi}(\mathbb{G})}\sum_{k=0}^{\infty}(2^{k}R)^{Q-\alpha+\beta}\frac{\left(\int_{2^{k}R\leq|xy^{-1}|<2^{k+1}R}K_{\alpha,\gamma}^{p_{2}}(xy^{-1})dy\right)^{1/p_{2}}}{\omega(2^{k}R)(2^{k}R)^{Q/p_{2}}}.
\Pi_{\epsilon}^{( \pm 1 )}= \frac 1{2 m_{2}}\sigma_{k}\left( \sigma_{k}\pm 1 \right) \left( m_{2}- B \right) \Lambda_{\epsilon}= \Psi_{\epsilon}^{( \pm )}\cdot \overline{{\Psi}}_{\epsilon}^{( \pm )}.
C\sum_{n=1}^{\infty}2^n\int_1^{2^{n}}\left[\left(1+ny+2^{n\alpha_3/\alpha_1}\right)^{1-2\alpha_1}-\left(2^n+ny+2^{n\alpha_3/\alpha_1}\right)^{1-2\alpha_1}\right]dy.
\frac{1}{u_0(x)^{p-1}}- (p-1) \int_0^t f(v(x,s))e^{(1-p)as}\,ds &\geq 2\varepsilon^{p-1}|x|^\alpha + \frac{1}{u_0(0)^{p-1}}-(p-1) \int_0^t f(v(0,s))e^{(1-p)as}\,ds &+(p-1) \int_0^t \big(f(v(0,s))-f(v(x,s))\big)e^{(1-p)as}\,ds.
\frac{\lambda_{n}}{\lambda_{n,2}}=\frac{\lambda_{n,2}}{\lambda_{n,3}}=\cdots=\frac{\lambda_{n,n+1}}{\lambda_{n,n+2}}=\frac{\widehat{\lambda}_{n}}{\widehat{\lambda}_{n,2}}=\cdots=\frac{\widehat{\lambda}_{n,n}}{\widehat{\lambda}_{n,n+1}},
Pr(Y_{i}=K)=1-\sum_{j=1}^{K-1}Pr(Y_{i}=j)=1-\sum_{j=1}^{K-1}Pr(Y_{i}=K)e^{\beta_{j}\cdot X_{i}}\Rightarrow Pr(Y_{i}=K)=\frac{1}{1+\sum_{j=1}^{K-1}e^{\beta_{j}\cdot X_{i}}}
\left((-1)^k \frac{\partial^k}{\partial z^k_2}\log p \right) ((-1)^k \frac{\partial^k}{\partial z^k_2}\log p) (t, x_2) = (-1)^k \frac{\partial^k}{\partial z^k_2}\left(\sum_{i=1}^{n}\log(z-\rho_i(t))\right)\Big|_{z=x_2}=- \sum_{i=1}^{n}\frac{(k-1)!}{(x_2-\rho_i(t))^k}.
\frac{d}{dt}\int_\Omega F_r^2(\overline U(x,t))dx=2\int_\Omega\big((\overline U_+^2(x,t)+r)^{\frac{1}{2}}-r\big) \overline U_+(x,t)(\overline U_+^2(x,t)+r)^{-\frac{1}{2}}\bar U_t(x,t)dx.
\hat{\Upsilon}_{a}= \frac{1}{4}\left( \hat{\omega}_{a}^{\ \underline{{{b c}}}}- 4 \hat{e}_{a}^{\ [ \underline{{{b}}}}\hat{k}^{\underline{{{c}}}]}\right) \hat{\Gamma}_{\underline{{{b c}}}}+ 2 \hat{k}_{a}\ ,
\gamma_{m}^{\mu \nu}= \left( \begin{array}{c c}{1}&{0}\\{0}&{1}\\ \end{array}\right) , \left( \begin{array}{c c}{0}&{- 1}\\{- 1}&{0}\\ \end{array}\right) , \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) .
\frac{\displaystyle\left(\int_{\Omega_n}|\phi|^q\,dx\right)^\frac{p}{q}}{\displaystyle\int_{\Omega_n}|\nabla \phi|^p\,dx}\simeq \left(\frac{1}{n}\right)^{(N-1)\,\frac{p-q}{q}}=|\Omega_n|^\frac{p-q}{q}.
= \; 8 m^{2}\mathrm{T r}\Big [ \frac{1}{- \triangle + m^{2}}\not{\! \! B}\frac{1}{- \triangle + m^{2}}\not{\! \! B}\frac{1}{- \triangle + m^{2}}\not{\! \partial}\not{\! \! B}\frac{1}{- \triangle + m^{2}}\not{\! \partial}\not{\! \! B}\Big ] \; ,
\sum_{j=1}^{m_0}\Xi_j \wedge \left( \sum_{t=0}^{p-1}\omega^t_{\mathbb{B}^n}\wedge (F_j^*\omega_{\mathbb{B}^{N_j}})^{p-1-t}\right) + ( \lambda - m_0) \omega^p_{\mathbb{B}^n}= \sum_{j=m_0+1}^m F_j^*(\omega_{\mathbb{B}^{N_j}}^p).
N^n(\alpha,\varpi) = \sum_{j=1}^{\lfloor n/k_n\rfloor}\sum_{i=(j-1)k_n+1}^{(j-1)k_n+m_n}1\Bigl(\bigl|\Delta_i^n X\bigr|\leq\alpha\sqrt{\widehat{V}{}^n_{j}}n^{-\varpi}\Bigr),
\rho_{m_{0}\rightarrow m_{1}+ \cdots + m_{N}}^{D}= \left( \prod_{i = 1}^{N}\int \frac{d^{n}\! \vec{\mu_{i}}}{( 2 \pi )^{n}}\right) \int d^{n}\vec{k}\, \, \mathrm{e}^{i \vec{k}. \sum_{i}\vec{\mu_{i}}}\, \rho_{m_{0}\rightarrow m_{1 \mu_{1}}+ \cdots + m_{N \mu_{N}}}^{D - n}
\lim_{|N|\to \infty}G^{\epsilon}(N+iy)=\lim_{|N|\to \infty}G^{\epsilon}_y(N)=\lim_{|N|\to \infty}\int_{\mathbb{R}}\widehat{G^{\epsilon}_c}(t)e^{-2\pi(y-c)t}e^{i2\pi Nt}\d t =0,
\Phi_{k}(G_k)=\langle x^{2^k},[y,x, \overset{2^k-1}{\ldots}, x], [y,x, \overset{2^k}{\ldots}, x], [y,x, \overset{2^k-3}{\ldots}, x,y]\rangle \Phi_{k+1}(G_k) \Phi_k(G_k)/\Phi_{k+1}(G_k) \cong C_2^{\,4}.
q = \frac 1{3 2 \pi^{2}}\int \, \epsilon_{\alpha \beta \gamma \delta}\, \mathrm{t r \,}\hat{F}^{\alpha \beta}\, \hat{F}^{\gamma \delta}\, d^{4}q = \frac 1{3 2 \pi^{2}}\int \, \epsilon_{a b c d}\, \mathrm{t r \,}\hat{F}^{a b}\, \hat{F}^{c d}\, \sqrt{g}d^{4}q .
L(s,\pi,\tau)=\prod_{i=1}^k\prod_{j=1}^l L(s,\delta_i,\delta_j') \ \ \ \ \ \ \epsilon(s,\pi,\tau,\psi)=\prod_{i=1}^k\prod_{j=1}^l \epsilon(s,\delta_i,\delta_j',\psi).
[ \textstyle{\frac{1}{2}}s_{b}^{a}t_{1}^{b}{}_{a}, \textstyle{\frac{1}{2}}s_{d}^{c}t_{2}^{d}{}_{c}] = \textstyle{\frac{1}{2}}s_{b}^{a}[ t_{1}, t_{2}]_{a}^{b}.
\Gamma_{n l o c}= \frac{1}{8 \pi}\int_{0}^{\infty}d \kappa [ \frac{1}{4 \lambda^{2}}C_{Q}( \kappa ) C_{Q}( \kappa ) \beta^{( 1 )}( \kappa ) - 2 C_{Q}( \kappa ) C ( \kappa ) \beta^{( 3 )}( \kappa )
\frac{d}{dt}T(\Omega_t) =\int_{\S^{n-1}}h_t|\overline{\nabla}U(hx+\nabla h,t)|^{2}K^{-1}dx &=\int_{\S^{n-1}}-f{\psi}(h)dx+\eta(t)\int_{\S^{n-1}}h|\overline{\nabla}U(hx+\nabla h,t)|^{2}K^{-1}dx &=0.
\sup_{\mathbf{\hat{x}}\in \Psi(k_0),\,\mathbf{k}\in \Pi(k_0)}\ \ \sum_{t=0}^\infty \tilde{\beta}_t\,\tilde{U}(\hat{x}_t) begin{align*}5pt] \hat{x}_t + k_{t+1}\leq f(k_t), t=0,1,\ldots, begin{align*}5pt] k_0 > 0\ ,
\xi^4 = \cos |\pi| \ , \ \xi^\alpha = \sin |\pi| \frac{\pi^\alpha}{|\pi|}\ ,\D_i\xi^\alpha= \partial_i \xi^\alpha +\varepsilon^{\alpha\beta\gamma}A_i^\beta \xi^\gamma \ , \ \D_i\xi^4= \partial_i \xi^4 \ ,
h(Y)=\frac{1}{2}\log (2 \pi e )+A^2-I &I= \frac{2}{\sqrt{2 \pi}A}\exp(-\frac{A^2}{2}) & \times \int_{0}^{\infty}\exp(-\frac{y^2}{2 A^2}) \cosh(y) \log(\cosh(y)) dy
\beta_{k+2}= ws_{i_k}s_{i_{k+1}}\alpha_{i_k}= ws_{i_k}(\alpha_{i_k}+\alpha_{i_{k+1}}) = w\alpha_{i_{k+1}}, \beta_{k+1}= ws_{i_k}\alpha_{i_{k+1}}= w(\alpha_{i_k}+ \alpha_{i_{k+1}}) = ws_{i_{k+1}}\alpha_{i_k}, \beta_k = w\alpha_{i_k}= ws_{i_{k+1}}(\alpha_{i_k}+ \alpha_{i_{k+1}}) = ws_{i_{k+1}}s_{i_k}\alpha_{i_{k+1}}.
\langle U, U^{\ast}\rangle_{\mathcal{H}}= \int_{\Omega}\big(\rho v_{i}v_{i}^* +a \theta \theta^* + A_{iKLj}u_{i,K}u^*_{j,L}+ C_{iIJKLj}u_{i,JI}u^*_{j,LK}+ K_{IJ}\tau_{,J}\tau^*_{,I}\big) \mathrm{d}x &+ \int_{\Omega}\big(M_{iJKL}u_{i,KJ}\tau^*_{,L}+ M_{jLKI}u^*_{j,KL}\tau_{,I}\big) \mathrm{d}x,
G_{T}W_{\mathrm{e f f}; a}^{\ast}\frac{\pi_{\lambda}}{\pi_{a}}\left( 1 - \alpha \right) = G_{S}W_{\mathrm{e f f}; \lambda}^{\ast}\left( 1 - \alpha \right) + \beta X_{B}+ M^{2}\big | \frac{\pi_{a}}{\Lambda}\big |^{2},
\mathbf{{x}}=\sum_{l=1}^{K}\mathbf{{P}}_{{l}}\mathbf{{s}}_{{l}}=\sum_{l=1}^{K}\sum_{\mathcal{A}'\subseteq\mathcal{K},|\mathcal{A}'|=l}\mathbf{{P}}_{\mathcal{A}'}{\mathbf{s}}_{\mathcal{A}'}.
n^{-[3-2d(s)]}\sum_{j=-\infty}^{nt}\Bigl[\sum_{k=nu+1}^{nt}v_{k-j}(s)\Bigr]^2\le\Bigl[\frac2{[1-d(s)]^2}+\frac1{2d(s)-1}\Bigr]\cdot|t-u|^{3-2d(s)}
G_{i+1}G_i E_{i+1}= E_i G_{i+1}G_i=E_i E_{i+1}&G_{i+1}E_i G_{i+1}=G_i^{-1}E_{i+1}G_i^{-1}&G_{i+1}E_i E_{i+1}=G_i^{-1}E_{i+1}&G_i E_i = E_i G_i = l^{-1}E_i &E_i G_{i+1}E_i = l E_i
S[\phi]\!\!=\!\!\int_0^\beta\!\!\! d\tau\int\!\!\! d^3 x\!\left(\partial_{\nu}\phi_{a}\partial^{\nu}\phi_{a}\!\!+\!m^2_{o}\phi^2\!\!+\!\frac{\lambda_{o}}{4N}\phi^4\!\!-\!\mu^2\phi^2\!\!+\!2i\mu(\phi_{2}\partial_{\tau}\phi_{1}\!\!-\!\phi_{1}\partial_{\tau}\phi_{2})\right)
\begin{gathered}(-\zeta;\zeta)_{d}=(1+\zeta^d)\prod_{j=1}^{d-1}(1+\zeta^j)=2,(-\zeta^2;\zeta^2)_{d}=(-\zeta^4;\zeta^4)_{d}=2, (-\zeta;\zeta^2)_{d}=(1+\zeta)(1+\zeta^3)\dotsb(1+\zeta^d)(1+\zeta^{d+2})\dotsb(1+\zeta^{2d-1})=(-\zeta;\zeta)_{d}=2.\end{gathered}
y^k_{xx}=f_{11}^k + f_{1i}^k y^i_x + f_{ij}^ky^i_x y^j_x + f^1_{ij}\, y^i_x y^j_x y^k_x\,,k=2,\dots,N\,,\,\,\, i,j\geq 2\,,f^k_{ij}=f^k_{ji}\,.
J =(V_t-\sigma\beta)\beta e^{-\beta t}+f(V)-f(V+\sigma\beta e^{-\beta t}) =(V_t-\sigma\beta)\beta e^{-\beta t}-f'(V+\theta )\sigma\beta e^{-\beta t}= \beta e^{-\beta t}\Big\{\big[-f'(V+\theta)-\beta\big]\sigma+V_t\Big\},
\alpha_{i}^{a}= \sqrt{\frac{\omega}{2 \hbar g}}\left[ \hat{q}_{i}^{a}+ i \frac{g}{\omega}\hat{p}_{i}^{a}\right] ,{\alpha_{i}^{a}}^{\dagger}= \sqrt{\frac{\omega}{2 \hbar g}}\left[ \hat{q}_{i}^{a}- i \frac{g}{\omega}\hat{p}_{i}^{a}\right] ,
f_{j}=\left( \begin{array}{c}f_{j}^1 f_{j}^2 \vdots f_{j}^r \end{array}\right)=\sum_{|\alpha|=n+1}\left( \begin{array}{c}f_{j, \alpha}^1 f_{j, \alpha}^2 \vdots f_{j, \alpha}^r \end{array}\right)\cdot e_j^\alpha
\lim_{n \to \infty}({\mathrm{M}}_{t,n}^{(k)}, 0 \le t \le T) \, =\, \delta_{({\mathrm{M}}_{t}^{(k)}, \,\, 0 \, \le \, t \, \le \, T)}\mathcal M (\Omega_{k}) \, .
(d_{u}\oplus d_{v})(x) = \lambda_{u}(2e_{\emptyset}- 2e_{u}) + 0 + \lambda_{\emptyset}(2e_{u}- 2e_{\emptyset}) + \mu_{\emptyset}(f_{v}- nf_{\emptyset}) & = (2\lambda_{u}- 2\lambda_{\emptyset})e_{\emptyset}+ (2\lambda_{\emptyset}- 2\lambda_{u})e_{u}+ \mu_{\emptyset}(f_{v}- nf_{\emptyset}),
dX_t = B(t,X_t,Y_t,Z_t,[X_t,Y_t,Z_t])dt + \sigma dW_t X_0 =\xi \in L^2(\Omega,\mathcal{F}_0,\mathbb{P};\mathbb{R}), dY_t = -F(t,X_t,Y_t,Z_t,[X_t,Y_t,Z_t])dt + Z_t dW_t Y_T = G(X_T,[X_T]).
{\Phi}(r,z,\omega)=\vert\nabla_x{c}(z,{X}^{M_{{\mathcal{P}}_n}}_{r-})-\nabla_x{c}(z,{X}^{M_{{\mathcal{P}}_m}}_{r-})\vert\vert D^Z{X}^{M_{{\mathcal{P}}_m}}_{r-}\vert_{l_2},
\frac{n^2\alpha_n^2}{8}-\frac{n^2G_n^2(x)}{32}=\frac{n^2(1+z/\ln n)^2G_n(x)^2}{32}-\frac{n^2G_n^2(x)}{32}= \frac{(z/\ln n)(2+z/\ln n)n^2G_n(x)^2}{32}= (z/\ln n)(2+z/\ln n)(\ln n+o(\ln n))\to2z
\int_{[0,1)^2}(f_n-A) \zeta \, dx = -\int_{[0,1)^2}(g_n \partial_1 \zeta + h_n \partial_2 \zeta) \, dx & \stackrel{n\to\infty}{\to}-\int_{[0,1)^2}(g \partial_1 \zeta + h \partial_2 \zeta) \, dx = \int_{[0,1)^2}(f-A) \zeta \, dx.
\overline{\chi}_{(\lambda_1,\lambda_2)}^\pm(s_1s_2)=\begin{cases}\chi^+_{\lambda_1}(s_1)\chi^+_{\lambda_2}(s_2) s_1\in\tilde{A}_{n_1}s_2\in\tilde{A}_{n_2}, \pm i\chi^+_{\lambda_1}(s_1)\chi^+_{\lambda_2}(s_2) \end{cases}
\frac{\left(n_{k+1}-1\right)\log 2}{-\log L_{n_{k+1}-1}}= \frac{\left(n_{k+1}-1\right)\log 2}{-\left(s_{2k+1}-1\right)\log q}=-\left(\frac{\log 2}{\log q}\right)\frac{\sum_{j=1}^{k+1}a_{2j-1}-1}{\sum_{i=1}^{2k+1}a_{i}-1},
p_{j}\left(\frac{e^{tA}x-x}{t}-Ax\right)\leqslant \frac{1}{t}\sum_{n=2}^{\infty}\frac{\big(t \, p^{X}_{j}(A)\big)^{n}}{n!}p_{j}(x)= \left( \frac{e^{t p^{X}_{j}(A)}-1}{t}-p^{X}_{j}(A) \right) p_{j}(x),
\begin{aligned}\mu_c^{\pm}(Y_1,Y_2,\dots,Y_m)= \mu_c^{\pm}(Y_1C_1,Y_2C_2,\dots,Y_mC_m), \nu_c^{\pm}(Y_1,Y_2,\dots,Y_m)= \nu_c^{\pm}(Y_1C_1,Y_2C_2,\dots,Y_mC_m).\end{aligned}
\lim_{n\to \infty}\frac 1 n \log \lVert A^n(x)v\rVert = \lim_{n\to \infty}\frac 1 n \log \lVert A^n(x)P(x) v\rVert &\le \lim_{n\to \infty}\frac 1 n \log \lVert A^n(x)P(x)\rVert &=\Lambda(\mu)<\lambda_i,
\mu_l^{[12]}=\mu_{l-1}^{[12]}+\mu_{l-1}^{[13]}+\delta_{l,2}, \mu_l^{[13]}=\mu_{l-1}^{[12]}+\mu_{l-1}^{[23]}+\delta_{l,3}, \mu_l^{[23]}=\mu_{l-1}^{[12]},
D^2 \sim \sum_{i, j}\mathcal{E}_i \mathcal{E}_j^* \otimes (\gamma_{-}(w_i) \gamma_{-}(w_j)^* + q^{-(\xi_i, \xi_j)}\gamma_{-}(w_j)^* \gamma_{-}(w_i)) & + \sum_{i, j, k, l}c_{i, j}^{k, l}\mathcal{E}_k \mathcal{E}_l^* \otimes \gamma_{-}(w_j)^* \gamma_{-}(w_i).
\sum_{k , l \in{\bf Z}}\exp \left[ - \frac{\pi R^{2}}{\alpha^{\prime}\tau_{2}}| ( 2 l + \delta ) + ( 2 k + \epsilon ) \tau |^{2}\right] ={\cal Z}_{( \epsilon , 0 )}+ ( - 1 )^{\delta}{\cal Z}_{( \epsilon , 1 )},
p_k^{(\rm id)}= \left(\frac{\mu_2^*}{\ln2\left(\nu_2^*-[\boldsymbol{\Lambda}]_{k,k}^2\right)}-\frac{\sigma^2}{[\boldsymbol{\Lambda}]_{k,k}^2}\right)^+\quad\forall\,k=1,2,\ldots, r.
\begin{cases}d_1(x_{p,k})=py_{p,k}, d_1(y_{p,k})=d_1(x_1)=0,\textrm{for $p$ a prime and}k\geq 0, d_1(\alpha\beta)=d_1(\alpha)\beta+(-1)^{\operatorname{deg}(\alpha)}\alpha d_1(\beta),\end{cases}
=H_{a}(\frac{2}{T}\cdot\frac{e^{j\omega_{d}T/2}(e^{j\omega_{d}T/2}-e^{-j\omega_{d}T/2})}{e^{j\omega_{d}T/2}(e^{j\omega_{d}T/2}+e^{-j\omega_{d}T/2})})
\frac 1{2\Gamma \left(-\frac 12\right)}\sum\limits_{j=0}^{d}\frac{2a_j}{j-d-1}+\frac 1{2\Gamma \left( -\frac 12\right)}r_{d+1}\left( -\frac 12\right) +\left. \frac{\mu^{s+1}}{\Gamma \left( \frac s2\right)}\frac{a_{d+1}}{s+1}\right\rfloor_{s=-1}=
\sup_{x\in\Omega}\int_{\Omega\,\cap\,B_{H_0}(x,1)}F(|z(y,t)|)\,dy \le MI(0)+C\int_0^t s^{\sigma-1}\sup_{x\in\Omega}\|z(s)\|_{L^{1+\delta}(\Omega\,\cap\,B_{H_0}(x,1))}^{1+\delta}\,ds +C\int_0^t s^{-\sigma}I(s)\,ds \le M+C\int_0^t s^{\sigma-1-\frac{\delta N}{2}}I(s)^{1+\delta}\,ds+C\int_0^t s^{-\sigma}I(s)\,ds
\widetilde{\mathbf{H}}_k^{(R)}= \widetilde{\mathbf{U}}_k^{(R)}\widetilde{\boldsymbol{\Sigma}}_k^{(R)}\left[\widetilde{\mathbf{V}}_k^{(R),1},\widetilde{\mathbf{V}}_k^{(R),0}\right]^H,
\begin{array}{l}\displaystyle c_0(a,b)=\frac{(1+1)2^{-1}\Gamma\left(\frac{1}2\right)}{\sqrt\pi(1-a-b)}=\frac1{1-a-b},\displaystyle c_1(a,b)=\frac{(a+b)2^{0\vphantom{A^A}}\Gamma(1)}{\sqrt\pi(-a-b)}=-\frac1{\sqrt\pi}, \displaystyle c_2(a,b)=-\frac{a^2+a+b^2+b}{2(1+a+b)}\,.\end{array}
P_{n}\Big(U_{n}(\theta_{n},\hat{c}_{n,\rho}(\theta_{n}))\neq \emptyset\Big) \ge P_{n}\Big(U_{n}(\theta_{n},c^{I}_{n,\rho}(\theta_{n}))\neq \emptyset\Big) &=\Pr(\mathfrak{W}(c_{\pi^*})\ne\emptyset)+\Big[ P_n\Big(U_{n}(\theta_{n},c^{I}_{n,\rho}(\theta_{n}))\ne\emptyset\Big)-\Pr\Big(\mathfrak{W}(c_{\pi^*})\ne\emptyset\Big)\Big].
\begin{array}{lll}a_{k}(t,x) := \Phi(t,x,u_{n_{k}}(t,x)) b_{k}(t,x) := g(t,x,u_{n_{k}}(t,x)) a(t,x) := \Phi(t,x,{u}(t,x)) b(t,x) := g(t,x,{u}(t,x)) \ . \end{array}
\cos d({P_{\eta'}}, S_{\eta}) \ge \cos ((1-\eta)x- (1-\eta')y) + \alpha_{1-\eta}\beta_{1-\eta'}(\cos d_{PS}- \cos (x-y)) &= \cos \left((1-\eta) (x-y) + (\eta' - \eta)x + (\eta'-\eta)(y-x)\right) + \alpha_{1-\eta}^2 (\cos d_{PS}- \cos (x-y)) &+ \alpha_{1-\eta}(\beta_{1-\eta'}-\alpha_{1-\eta})(\cos d_{PS}- \cos (x-y)).
\langle f,g\rangle = \frac{1}{2}\int_{G^2}((g(x_1,x_2)\overline{f(x_1,x_2)}+f(x_1,x_2)\overline{g(x_1,x_2)})d^2_{{\mu}_{G^2}}\left(x_1,x_2\right) &=Sc\int_{G^2}f\left(x_1,x_2\right)\overline{g\left(x_1,x_2\right)}{d^2}_{{\mu}_{G^2}}\left(x_1,x_2\right).