harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Tilde{\Phi}_2(\beta_0,\eta)=\bigg[\tfrac{2}{L_{2m}}\bigg[\frac{\tilde{N_2}}{L_{2m}}\bigg(\frac{\eta^{3-2\mu-\nu}-\beta_0^{3-2\mu-\nu}}{(2-\mu)(3-2\mu-\nu)}-\frac{\beta_0^{2-\mu}\eta^{1-\mu-\nu}-\beta_0^{3-2\mu-\nu}}{(2-\mu)(1-\mu-\nu)}\bigg)
{y}_{C_u}(t)= \sum_{k=1}^{K}({\bf h}_{u,k}^T{\bf F}_k{\bf s}_ke^{\mathrm{j}2\pi{f_k}t}\!+\!\sum_{m=1}^{M}{\bf h}_{u,m,k}^T{\bf\Phi}_m{\bf G}_{m,k}{\bf F}_k{\bf s}_ke^{\mathrm{j}2\pi{f_k}t}+n_{C_u}(t)),
\begin{array}[c]{l}\mathbb{E}\left[ \int_{0}^{T}\left\vert \delta\sigma_{x}(t)\xi^{2,\epsilon}(t)\right\vert^{2}I_{E_{\epsilon}}(t)dt\right] \leq\mathbb{E}\left[\sup\limits_{t\in\lbrack0,T]}\left\vert \xi^{2,\epsilon}(t)\right\vert^{2}\int_{E_{\epsilon}}\left\vert \delta\sigma_{x}(t)\right\vert^{2}dt\right] =o(\epsilon^{2}).\end{array}
\frac{hmn}{h+1}+\frac{m}{h+1}-m+h - tmh = \frac{1}{h+1}\left(hmn+m-m(h+1)+h(h+1)-(h+1)tmh \right) &= \frac{1}{h+1}\left(rhm+m-m(h+1)+h(h+1) \right) &=\frac{h}{h+1}\left(m(r-1)+h+1\right) >0
\phi ( g ) = \sum_{\Lambda}\ \phi_{\alpha \beta}^{\Lambda}\D_{\alpha \beta}^{( \Lambda )}( g ) = \int_{S O ( 3 )}d h \ \phi ( g h ) = \sum_{\Lambda}\ \int_{S O ( 3 )}d h \ \phi_{\alpha \beta}^{\Lambda}\D_{\alpha \gamma}^{( \Lambda )}( g ) \D_{\gamma \beta}^{( \Lambda )}( h ) .
\left( m + \frac{1}{2 r_{1}}+ \frac{1}{2 r_{2}}\right) d r^{2}+ \left( \frac{1}{r_{2}}- \frac{1}{r_{1}}\right) d r d R + \left( 2 M - \frac{2}{R}+ \frac{1}{2 r_{1}}+ \frac{1}{2 r_{2}}\right) d R^{2}
V_{Z}={\frac{2 ( n + 1 ) r^{3}+ 3 r_{o}r^{2}+ 9 r r_{o}^{2}/ 2 n + 9 r_{o}^{3}/ 4 n^{2}}{r^{4}( r + 3 r_{o}/ 2 n )^{2}}}( r - r_{o}) .
\bar{\alpha}(\Omega) = \inf\left\lbrace \dfrac{\mathrm{Vol}\left(\Omega\Delta B_\rho(x)\right)}{\mathrm{Vol}(B_{\rho})}:x\in\mathbb{R}^{n+1},\mathrm{Vol}(\Omega)=\mathrm{Vol}(B_{\rho})\right\rbrace
h^*\mathcal{G}_{\varphi}(M',\varphi_*D') \cong\mathcal{G}_{\Phi}(h^*M',h^*\varphi_*D')=\mathcal{G}_{\Phi}((h^*\varphi_*M)_{>\phi(s)},h^*\varphi_*D') &=\mathcal{G}_{\Phi}((\Phi_*H^*M)_{>\phi(s)},\Phi_*H^*D')=\Phi_*((H^*M)_{>s},H^*D'),
e_{1}= \frac{1}{3}( \theta_{2}^{4}( 0 , \tau ) + \theta_{3}^{4}( 0 , \tau ) ) , e_{2}= - \frac{1}{3}( \theta_{1}^{4}( 0 , \tau ) + \theta_{3}^{4}( 0 , \tau ) ) , e_{3}= \frac{1}{3}( \theta_{1}^{4}( 0 , \tau ) - \theta_{2}^{4}( 0 , \tau ) ) .
\mathtt{A}(s) := \langle A^n(s,u_n)-A^n(s,v_n), \psi_k' (w_n(s))\rangle, \mathtt{B}(s) := \sum_{i=1}^{n}\big(\psi''_k(w_n(s))(g(u_n(s))-g(v_n(s)))\phi_i,(g(u_n(s))-g(v_n(s))\phi_i)\big),
\int_{D_{2R}}g(z)\omega_{D\setminus D_{2R}}^x(dz) =\int_{|y|<R}\left(\int_{D_{2R}}M_D(z,y)\omega_{D\setminus D_{2R}}^x(dz)\right)\mu(dy) &=\int_{|y|<R}M_D(x,y)\mu(dy)=g(x).
\times\Big(\Theta [V_{\rho\sigma j}+\sqrt{2\kappa}(a_\sigma D_\rho A_j-a_\rho D_\sigma A_j+A_j f_{\rho\sigma})]+4(\overline{\Theta}_\rho +\sqrt{2\kappa}a_\rho\Theta) D_\sigma A_j\Big),
S = \int d^{4}x \sqrt{| g_{4}|}\left[ e^{- \Phi}\left( R + \left( \nabla \Phi \right)^{2}- 6 \left( \nabla \beta \right)^{2}- \frac{1}{2}e^{2 \Phi}\left( \nabla \sigma \right)^{2}\right) - \frac{1}{2}Q^{2}e^{- 6 \beta}\right] .
{\mathbb T}_{TM}^h(\omega)=S_{\epsilon^{-1}}^h+iA_{\epsilon^{-1}}^{1,h}-iA_{\epsilon^{-1}}^{2,h}+|{\boldsymbol k}|^2 M_{\epsilon^{-1}}^h-\left(\frac{\omega}{c}\right)^2 M^h.
L(\phi,\Lambda)=\sum_{m=1}^{\infty}{1\over(2m)!}\int{dp_1 ... dp_{2m}\over(2\pi)^{8m-4}}L_{2m}(p_1,...,p_{2m};\Lambda)\delta^4(p_1+ ... +p_{2m})\phi(p_1) ... \phi(p_{2m})
1-G_v\big((\alpha^*_nx+\beta^*_n)^{\frac{1}{p}}\big)= n^{-1}e^{-x}\left\{1-\frac{(1-v^{-1})^3(\log\log n)^{2}}{2\log n}\right. &\left.+\frac{(1-v^{-1})^{2}(1-\log 2\Gamma({1}/{v}) +x)\log\log n}{\log n}+o\big(\frac{\log\log n}{\log n}\big)\right\}.
\prod_{i = 1}^{m_{a}^{\prime}}\psi_{2}^{( a )}({y^{\prime}}_{i}^{( a )}) \prod_{i = 1}^{n_{a}}\overline{{{\psi}}}_{1}^{( a )}( x_{i}^{( a )}\! + \hat{\tau}) \prod_{i = 1}^{m_{a}}\psi_{2}^{( a )}( y_{i}^{( a )}\! + \hat{\tau}) \prod_{i = 1}^{n_{a}^{\prime}}\overline{{{\psi}}}_{1}^{( a )}({x^{\prime}}_{i}^{( a )}) \Big \rangle_{0}\; .
\Delta^{m-1}u(r_0)\leq \Delta^{m-1}u(r)+\frac{V(b)}{\omega_{2m-1}}\int_{r_0}^r s^{1-2m}ds \leq \Delta^{m-1}u(r)+\frac{V(b)}{\omega_{2m-1}}\int_{r_0}^\infty s^{1-2m}ds & =\Delta^{m-1}u(r)+ \frac{V(b)r_0^{2-2m}}{(2m-2)\omega_{2m-1}}.
X(t)=\widetilde{X}+e^{t \widetilde{A}^T}[e^{t \widetilde{A}}\widetilde{Z}e^{t \widetilde{A}^T}+ (X_0 - \widetilde{X})^{-1}-\widetilde{Z}]^{-1}e^{t \widetilde{A}^T}
q^{n-2}\operatorname{DET}\big| S_{2n-2-i-j}\big(q; q^{i-1}\beta\big) \big|_{1 \le i,j \le n-2}\qquad{}= q^{(n-2)(n-3)(4n-7)/6}\prod_{a=1}^{n-3}\big(1+q^{a-1}\beta\big)^{n-a-2}\left\{{(1+ \beta)^{n-1}-1 \over \beta}\right\}.
v(z) = \mathcal{T}(z)v_0 + iC_1\int_0^z e^{-C_2 z^\prime}\mathcal{T}(z-z^\prime)\big[|v(z^\prime)|^2v(z^\prime)\big]dz^\prime + i\frac{C_2^2}{4}\int_0^z\mathcal{T}(z-z^\prime)t^2v(z^\prime),
\|\boldsymbol{u}_k-\boldsymbol{u}_k'\|=\sum_{i=0}^{m-1}\|{u}_{k+i}-{u}_{k+i}'\| \leq \sum_{i=0}^{m-1}\|{u}_{m-1}-{u}_{m-1}'\|q^{k+i-m+1}\leq C_{m,q}q^k \|\boldsymbol{u}_0-\boldsymbol{u}_0'\|, 1\leq k\leq n.
\int_{0}^{B_{max}}(1-F(y)) \mathrm{d}y = \sum_{i=1}^{k}{p_i A_i}\le \sum_{i=1}^{k}{A_i \sum_{j=i}^{k}{p_j}}\le k\max_{1\leq i\leq k}{A_i \sum_{j=i}^{k}{p_j}}= k{\sup_{0\leq x \leq B_{max}}{x (1-F(x))}}.
\int_0^\infty s^{it}s^{\frac12}w_\alpha(s) \frac{ds}{s}= \int_0^\infty (is)^{it}(is)^{\frac12}w_\alpha(is) \frac{ds}{s}= i^{-\alpha+\frac12 + it}\int_0^\infty s^{-\alpha+\frac12 + it}(e^{-s}- \sum_{j=0}^{m-1}\frac{(-s)^j}{j!}) \frac{ds}{s}.
(I):=\big | \big( h^{(M)}(f(x)) - h^{(M)}(f(y)) \big) \prod_{j=1}^k \big( f^{(j)}(x) \big)^{m_j}\big|, &(II):=\big| h^{(M)}(f(y)) \big( \prod_{j=1}^k \big( f^{(j)}(x) \big)^{m_j}- \prod_{j=1}^k \big( f^{(j)}(y) \big)^{m_j}\big) \big| .
\frac{\partial y}{\partial\varphi}\cdot\frac{\partial x}{\partial\lambda}-\frac{\partial y}{\partial\lambda}\cdot\frac{\partial x}{\partial\varphi}=R\cdot R\cdot cos\varphi-0\cdot(-R\cdot\lambda\cdot sin\varphi)=R^{2}\cdot cos\varphi=s\cdot cos\varphi
\mathrm{P}\left( |h_n|^2 < \xi_1\right) + \mathrm{P}\left( |g_{n,1}|^2<\xi_1, |h_n|^2 >\xi_1\right) = \mathrm{P}\left(\min\{|g_{n,1}|^2, |h_n|^2\}<\xi_1\right).
\sum_{r\le \overline r_1}{k-j_0+L+M \choose r}\left( \frac qp \right)^{j_0r}= e^{\overline r_0}\Phi\left( \frac{\overline r_1 - \overline r_0}{\sqrt{\overline r_0}}\right)\left( 1 + O\left( \frac{\log\log n}{\log n}(L+M) \right)\right),
\deg m_{\mu_i ,\nu_i}\left( \frac{\gamma}{\sqrt{i}}\left( z^i_1+\ldots+z^i_r\right),-\frac{\gamma}{\sqrt{i}}\left( z_1^{-i}+ \ldots +z^{-i}_r \right)\right)= i \left( \mu_i-\nu_i \right).
\varphi (g_3 g^{n}_1 g_2) = \varphi \big( (g^{j}_1 g_3) (g^{i}_1 g_2) \big) \mbox{or}\varphi (g_3 g^{n}_1 g_2) = \varphi \big( (g^{i}_1 g_2) (g^{j}_1 g_3) \big) \,.
x_{+}(t) = x_{\alpha_{2}}(t^5) x_{\alpha_{5}}(t) x_{\alpha_{6}}(2t) x_{\alpha_{7}}(3t) x_{0000110}(4t^2) x_{0000011}(2t^2) x_{0000111}(3t^3), x_{-}(t) = x_{-\alpha_{2}}(t^5) x_{-\alpha_{5}}(3t) x_{-\alpha_{6}}(2t) x_{-\alpha_{7}}(t) x_{-0000110}(3t^2)x_{-0000011}(t^2) x_{-0000111}(3t^3)
\sum_{r=u}^{s}(\begin{matrix}s\\r\\ \end{matrix})(F^{*})^{r}(1-F^{*})^{s-r}=\sum_{t=0}^{l}(\begin{matrix}s\\t\\ \end{matrix})(F^{*})^{t}(1-F^{*})^{s-t}.
-\partial_1 u \lceil x_1, (\cdot)_{T}\rceil f + \big(\partial_1 u \lceil x_1, (\cdot)_{t}\rceil f\big)_{T-t}&\quad=-\sum_{\tau=T/2^k, \, k=1, \dots, n}\big(\partial_1 u \lceil x_1, (\cdot)_{\tau}\rceil f_{\tau}+\lceil \partial_1 u, (\cdot)_{\tau}\rceil \lceil x_1, (\cdot)_{\tau}\rceil f \big)_{T-2\tau}.
\int_0^\infty\int_{|y|\leq\rho}\left|K(O(at)x-y, t)-\frac{e^{-1/4t}}{8\pi t}\mathbb I\right|\,dy\,dt &=\int_0^\infty\int_{|z|\leq\rho}\left|K(x-z,t)-\frac{e^{-1/4t}}{8\pi t}\mathbb I\right|\,dz\,dt.
\Vert \nabla \bar{y}_i- \hat{R}\Vert^p_{L^p(\tilde{Q})}\le C\Vert \nabla y - \hat{R}\Vert^p_{L^p(\tilde{Q}\cap \hat{W})}+ C\sum\nolimits_{\Gamma_t \in{\cal F}(Q^k_i(q))}|X_t|_\infty^2 |A_t|^p & \le CC_m\delta_p(Q^k_i(q)\cap \hat{W}) + CC_m (\epsilon s^{-1})^{\frac{p}{2}-1}\epsilon |\partial \hat{U}_i \cap Q^k_i(q)|_{\cal H}
= [ ( 1 + \frac{1}{2}) ( 1 + \frac{1}{3}) ( 1 + \frac{1}{4}) \cdots ( 1 + \frac{1}{2 0 1 3}) ] \times [ ( 1 - \frac{1}{2}) ( 1 - \frac{1}{3}) ( 1 - \frac{1}{4}) \cdots ( 1 - \frac{1}{2 0 1 3}) ]
\left|\frac{1}{2\pi}\int_{\frac{1}{\sqrt{K}}|\xi| > \delta}{\underset{i = 1}{\stackrel{K}{\prod}}h(\sigma - \lambda_i, \frac{\xi}{\sqrt{K}}) d\xi}\right| \leq C K \lambda^{K-2}\int_{\mathbb{R}}{\frac{1}{1 + \xi^2}d\xi}&\leq \frac{C}{K}
\mathrm{a n d \s o \o n , \w h e r e}\qquad J [ 1 ] = \left( \frac{\pi}A \right)^{2 \omega}\frac{e^{- A H}}{D_{\|}D_{\perp}^{\omega - 1}}, \qquad \qquad \qquad \qquad \qquad \qquad
d s^{2}= X ( \frac{d \eta^{2}}{1 - \eta^{2}}- \frac{d \mu^{2}}{1 - \mu^{2}}) - ( 1 - \mu^{2})^{1 / 2}( 1 - \eta^{2})^{1 / 2}[ \chi d y^{2}+ \frac{1}{\chi}( d x - \lambda d y )^{2}] .
x=\mbox{sn}(k,w)\;\mbox{or}\;\mbox{sn}\, w,\;\;\;\sqrt{1- \mbox{sn}^2\, w}= \pm \mbox{cn}\, w,\;\;\;\sqrt{1-k^2 \mbox{sn}^2\, w}=\pm \mbox{dn}\,w,
\delta I_{CS}=\frac{1}{8\pi^2}\int_{T^2}d^2x\,\epsilon^{ij}{\rm Tr}[G_m\partial_{i}G_m^{-1}({\cal U}^{-1}_s\partial_j{\cal U}_{s}-{\cal U}_{s}\partial_j{\cal U}_s^{-1})].
e^{c \phi}e^{f(q^{a})}- e^{-2 c \phi}e^{-2 f(q^{a})}= e^{\phi}[\,c+ c_{1}+ Y_{0}X_{0}f(q^{a}) \,] - e^{-2 \phi}[\, c + c_{1}- Y_{1}X_{0}f(q^{a}) \,] .
M_n g=\frac{\widehat{g}(0)}{2}e_0+\sum_{j=1}^\infty \widehat{g}(j) M_n e_j =\frac{\widehat{g}(0)}{2}e_0+\sum_{j=1}^\infty \widehat{g}(j) e_{nj}\in L_s.
\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}g^{j,t}\|^2_{L^2(B^g(x^t_j,2 \sqrt{t}), \mu_2)}\leq C' \left( \|g_0^{j,t}\|^2_{L^2(C^{j,t}_0, \mu_2)}+\sum_{k\geq 1}e^{-c 2^k}\|g_k^{j,t}\|^2_{L^2(C^{j,t}_k, \mu_2)}\right).
(\tilde{f}_i (\pi))(t) := \begin{cases}\pi(t) (0 \leq t \leq t_0), s_i(\pi(t) - \pi(t_0)) + \pi(t_0) (t_0 \leq t \leq t_1), \pi(t) - \alpha_i (t_1 \leq t \leq 1). \end{cases}
V(\textbf{x})=\dfrac{4\pi}{\vert T\vert}\sum_{\textbf{k}\in \lbrace0,1\rbrace^{n}}(-1)^{\vert\textbf{k}\vert}\int_{0}^{+\infty}\Theta (\mathbf{x}-a\textbf{k}\vert v) dv.
{}~~~~~\times\Bigg\{\sum_{\sigma\in Z_N}\frac{w(\omega u'_2u_3,\sigma+j_2+j_3)w(qu_2'',\sigma)s(\sigma,j_1)}{w(q^{-1}u'_2,\sigma+j_2)w(u_2''u_3,\sigma+i_3)}\Bigg\}_0.
\begin{aligned}w(h) :=\sum_{(v,E)\in V(R)\times E(R)}h(v,E) &= \sum_{e\in \mathcal{F'}}\sum_{v\in V(R)}\sum_{v':g(v')=v}\sum_{E\in E(R)}w(e, E,v') &\ge 2(\alpha+\varepsilon' ) 4^{j}t\times \frac{2}{4^{j}}\ge 4(\alpha+\varepsilon' )t,\end{aligned}
F_{x_\perp \bar x_\perp}^{21}=- \frac{1}{2\pi |x_\perp|}e^{2ix\cdot\omega}\partial_{\bar x_\perp}(\bar x_\perp I)\sim e^{2ix\cdot \omega}e^{-2|\omega||x_\perp|}{(\omega_1+i\omega_2)\pi^{{1\over2}}\over{2^{{3\over2}}|\omega|^{{1\over2}}|x_\perp|^{{3\over2}}}}
x\cdot I(G-N[v_0]) = x \cdot (2x+1)^{2k+4}\cdot(3x^2+4x+1) &= x(3x^2+4x+1) \cdot \bigg[\sum_{i=0}^{2k+4}\binom{2k+4}{i}(2x)^{i}] &= x(3x^2+4x+1) \cdot [(2x)^{2k+4}+\dots] &= 3\cdot2^{2 k + 5}x^{2 k + 8}+\dots
\begin{aligned}(\tilde{\Delta}_E\tilde{h},\tilde{h})_{L^2(\tilde{g})}=2\int_{I}f^{n-2}\varphi^2dr\left\|\nabla \omega\right\|_{L^2(g)}^2 +(2n+6)\int_{I}(f')^2f^{n-2}\varphi^2dr\left\|\omega\right\|_{L^2(g)}^2 & +2\int_{I}(\varphi')^2f^{n}dr\left\|\omega\right\|_{L^2(g)}^2 -4\int_{I}f''f^{n-1}\varphi^2dr\left\|\omega\right\|_{L^2(g)}^2.\end{aligned}
f(\mathbf{x}_{0}+\varepsilon \mathbf{x})=O(\varepsilon^{\alpha}L(\varepsilon)) \ \ \ \mbox{as}\varepsilon\to 0^{+}\ \ \mbox{in}\mathcal{K}_{M,r}'(\mathbb{R}^{n}) \ (\mbox{resp.}\mathcal{S}'_{r}(\mathbb{R}^{n})),
L ( \zeta (J_0(w_{k+1}) - J_0(w_k))) = - \zeta' (J_0(w_{k+1}) - J_0(w_k)) \textnormal{in}\mathcal{M}\times (0,T), & (\zeta (J_0(w_{k+1}) - J_0(w_k)))(\cdot,0) = 0 \textnormal{on}\mathcal{M}
C(\Delta(r+2))-d-1+\left\lfloor \frac{17s-4}{6}\right\rfloor = \frac{2^{r+2}-1}{3}- (2^{r+2}-1)+\frac{17 \cdot 2^{r-1}-4}{3}=\frac{2^{r-1}-2}{3}\geq 0
[T_n, T_m] = -\sum_{l=1}^{\infty}r_l( T_{n-l}T_{m+l}- T_{m-l}T_{n+l}) - \frac{(1-q)(1-t^{-1})}{1-qt^{-1}}( q^nt^{-n}- q^{-n}t^{n}) \delta_{m+n,0}.
V_{p^{+},p^{-}}^{r,s}\left( z\right) =e^{iX^{-}p^{+}}\,\,e^{-\frac12X_2\left( 1-2is\right)}\,F_{r,s}\left( e^{-X_2}p^{-}p^{+}\right)e^{iX^{+}p^{-}}
H_{r,s}=\frac{h_{B}^{r+1}h_{A}^{s+1}}{(r+s+2)r! (s+1)!}+\sum_{\substack{0\leq u\leq r 0\leq v \leq s}}\frac{h_{u,v}h_{B}^{r-u}h_{A}^{s-v}}{(r-u+s-v+1)(r-u)!(s-v)!}.
SINR_{ijk}= \frac{{\bf u}^{T}_{ijk}{\bf T}_{ijk}{\bf u}_{ijk}}{{\bf u}_{ijk}^{T}{\bf F}_{ijk}{\bf u}_{ijk}},\ \forall\{i,j\}\in\mathcal{L},\ \forall k\in\mathcal{K},
| z , \gamma , n_{1}\rangle = e^{- \frac{1}{2}| z |^{2}}T^{\dagger}e^{- \frac{1}{4}\ln \gamma ( a_{2}^{2}- ( a_{2}^{\dagger})^{2})}e^{z a_{2}^{\dagger}}| 0 , n_{1}\rangle
x\phi_1 x^{\ast}=y\phi_2 y^{\ast}=\left[\begin{array}{ccc}0 0 e_t 0 \zeta I_r 0 e_t^{\ast}0 0\end{array}\right]\left[\begin{array}{ccc}0 0 e_t 0 0_r 0 e_t^{\ast}0 0\end{array}\right],0\leq t\leq m.
\big\| g_i \big\|_{k, L^2(Y, L^p \otimes F)}= \Big( p^{\frac{n - m + k}{2}}\cdot \sqrt{k! \cdot (2 \pi)^k}+O(p^{\frac{n - m + k - 2}{2}})\Big)\cdot \big\| B_{i, p}^{\perp}f \big\|_{L^2(X, L^p \otimes F)}.
\sqrt{a_{12}^{2}+a_{22}^{2}+2a_{12}a_{22}\frac{a_{12}^{2}+a_{22}^{2}-(a_{11}^{2}+a_{21}^{2})}{4a_{11}a_{21}}}=2^{-\frac{1}{2}}\sqrt{a_{12}^{2}+a_{22}^{2}+a_{11}^{2}+a_{21}^{2}}.
R_1(v)=m_1 \sum_{j=1}^k M_{12}(e_j^{-1}) - C= m_1 \sum_{j=1}^k M_{21}(e_j)-C =m_1R_2(v)-C= =m_1(k-1)(M_{22}(e_1)+M_{22}(f_1))-C.
\liminf_{n\to \infty}\frac{J_n^{(p)}}{n}\ge P (T^{i}_{Z_0}-T^{i-1}_{Z_0}\le ki=1,..,p-1, T^p_{Z_0}=\infty, \{Z_m\}_{m\in{\mathbb Z}}\not\supset{\cal N}( Z_{0}) Z_0\notin\{Z_m\}_{m=-\infty}^{-1}.),
\begin{array}{c}{P R^{\prime}\left( \sqrt{\frac{z}{w}}\right) \left( q^{S_{3}}\otimes{\frac{E}{z}}+{\frac{E}{w}}\otimes q^{- S_{3}}\right) =}{= \left( q^{S_{3}}\otimes{\frac{E}{w}}+{\frac{E}{z}}\otimes q^{- S_{3}}\right) P R^{\prime}\left( \sqrt{\frac{z}{w}}\right)}\end{array}
\partial_\tau v_g^n(y,\tau) - \frac{1}{2}\partial_{yy}v_g^n(y,\tau) + (y + a_n'(y)) \partial_y v_g^n(y,\tau) = \left | \partial_y w_g^n (y,\tau) \right |^2, \: \forall (y,\tau) \in \mathbb{R}\times [0,\infty),
L(\phi_i,v^i_a)=\frac{1}{2}\left(\sum_{a,b=1}^4 \eta_{ab}v^1_av^1_b-m^2\phi_1^2\right)+\frac{1}{2}\left(\sum_{a,b=1}^4 \eta_{ab}v^2_av^2_b-m^2\phi_2^2\right)-\frac{1}{4}g(\phi_1^2+\phi_2^2)^2.
I(r,t,\psi(\cdot))=\int_0^t\frac{\rho d\rho}{\sqrt{t^2-\rho^2}}\int_{|r-\rho|}^{r+\rho}\frac{\lambda^m\psi(\lambda) T_{m-1}\left(\Theta(\lambda,r,\rho)\right)d\lambda}{\sqrt{\lambda^2-(r-\rho)^2}\sqrt{(r+\rho)^2-\lambda^2}}
\Lambda (f) = \lim \limits_{n \to \infty}\frac{1}{n}\log \mathbb{E}[e^{n \nu_n(f)}] = \begin{cases}-\int \limits_0^1 \log \big(1 - \frac{G(x)}{2(1-x)}\big) \; dx K < 2 \infty K > 2 \end{cases}~,
\left(\prod_{n \in \mathbb{N}}X_n\right) / \omega = \frac{\lbrace (x_n)_{n\in\mathbb{N}}\in \prod_{n \in \mathbb{N}}X_n \ \vert \ \lim_\omega (d_n(o_n,x_n))<\infty \rbrace}{\sim_\omega}
\omega \widetilde{T}_{-}( q , p ) \rightarrow \left( \mathrm{e}^{- i \pi \widetilde{T}_{1}}\ast \widetilde{T}_{-}\right) ( q , p ) = \omega \widetilde{T}_{+}( q , p ) = \sum_{i = 1}^{N}\frac{p_{i}^{2}}{2}.
(d\tilde{\star}\tilde{\omega})_{i_1\dots i_{n-k+1}}(x^\prime,x^n)=\begin{cases}(d\star \omega)_{i_1\dots i_{n-k+1}}(x^\prime,|x^n|) i_l=n, \operatorname{sgn}(x^n)(d\star \omega)_{i_1\dots i_{n-k+1}}(x^\prime,|x^n|) i_l\neq n\end{cases},
\bigg| \frac{\tau_{j+1}^* - \tau_j^*}{a_N}- \frac{1}{q_j}\bigg| = \bigg| \frac{1}{k_N q(\tau_j^*/a_N)}- \frac{1}{q_j}\bigg| = \frac{1}{k_N}\bigg| \frac{1}{q(\tau_j^*/a_N)}- \frac{k_N}{q_j}\bigg|.
V_{n}(R)=\int_{0}^{R}\frac{2\pi^{n/2}}{\Gamma(\frac{n}{2})}r^{n-1}dr=\frac{2\pi^{n/2}}{n\Gamma(\frac{n}{2})}R^{n}=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}R^{n}.
(A_{0}(z),\ldots,A_{p-1}(z))=\frac{(1,\ldots,1)}{(0,\ldots,0,z-a_{0}^{(0)})+(A_{0}^{(1)}(z),\ldots,A_{p-2}^{(1)}(z),-\sum_{j=1}^{p}a_{0}^{(j)}A_{j-1}^{(1)}(z))}.
\chi_{1}( x , \tau ) = \sqrt{\frac{1}{\pi \alpha}}\sum_{n = 1}^{\infty}\left( \psi_{n}^{( 1 )}\cos{\frac{n \tau}{\alpha}}+ \psi_{n}^{( 2 )}\sin{\frac{n \tau}{\alpha}}\right) \left({\frac{x}{\epsilon}}\right)^{\frac{n}{\alpha}},
\partial_tf_1(J_t,L_t)|_{t=0^+}=\partial_t\phi(J_t,L_t)|_{t=0^+}&=\partial_t\psi(J_t,L_t)|_{t=0}&=\partial_1\psi(K,L)\partial_t J_t|_{t=0}+\partial_2\psi(K,L)\partial_t L_t|_{t=0}&=\partial_2\psi(K,L)\partial_t L_t|_{t=0}
\varepsilon(\sigma)\bigg(\sum_{F\subset E\atop\mu(F)=M}\prod_{(i,j)\in E-F}x_{\sigma(i)}\bigg)Q_\sigma+\varepsilon(\tau\sigma)\bigg(\sum_{F\subset E\atop\mu(F)=M}\prod_{(i,j)\in E-F}x_{\tau\sigma(i)}\bigg)Q_{\tau\sigma}
\begin{aligned}Q_1^{(n+1)}\circ \partial =\partial\circ Q_1^{(n)}\circ f^{(n)}, Q_2^{(n+1)}\circ \partial =\partial\circ\frac1{f^{(n)}}\circ Q_2^{(n)}, Q_1^{(n+1)}\circ Q_2^{(n+1)}\circ\partial =\partial\circ Q_1^{(n)}\circ Q_2^{(n)}. \end{aligned}
ds^2=-\left(1-\frac{r_0^2}{r^2}\right)\left(1+\frac{r_1^2}{r^2}\right)^{-1}\left(1+\frac{r_2^2}{r^2}\right)^{-1}\beta^2 dt^2+\frac{\Lambda^{-1}}{r^2}\left(1-\frac{r_0^2}{r^2}\right)^{-1}dr^2
P \lbrace \mathcal{T}(t) = x+v_0(t-s) \ |\ \mathcal{T}(s) =x\rbrace = \frac{e^{-\lambda (t-s)}}{2}\Bigl( 1+ \frac{|v_1s-x|}{(a_1-a_2)s}\, P\lbrace N(s)\ \ |\ \mathcal{T}(s) = x\rbrace \Bigr).
R_{\rm FA}= \frac{\sum\limits_{\mathbf{a}\in \mathbb{B}^{N\times 1}\atop ||\mathbf{a}||_{0}=\lambda N}\vert \hat{\mathcal{U}}_{ac}^{\mathbf{a}}({\mathbf{I}}) - \mathcal{U}_{ac}^{\mathbf{a}}\vert}{\sum\limits_{\mathbf{a}\in \mathbb{B}^{N\times 1}\atop||\mathbf{a}||_{0}=\lambda N}|\mathcal{U}^{\mathbf{a}}_{{ac}}|}\leq{\tau.}
\frac{d}{dt}\left\| \theta \right\|^{2}_{H^{2}}+\nu\left\|\Lambda^{\frac{\gamma}{2}}\theta\right\|^{2}_{H^{2}}\leq C\left(\left\|\theta_{x}\right\|_{L^{\infty}}+ \left\|\mathcal{H}\theta_{x}\right\|_{L^{\infty}}\right) \|\theta\|^{2}_{H^{2}}.
\frac{\partial}{\partial t}\hat{\rho}_{n}=-(\frac{i}{\hbar}\hat{H}_{A}^{\times}+n\gamma)\hat{\rho}_{n}-\frac{i}{\hbar}\hat{V}^{\times}\hat{\rho}_{n+1}+\frac{in}{\hbar}\hat{\Theta}\hat{\rho}_{n-1}
\cos \left ( \frac{2\pi (N-1)}{2N-1}+\psi\right )\geq 0 \rightarrow \left ( \frac{1}{2}+\frac{1}{2N-1}\right )\pi \leq \psi \leq \left ( \frac{3}{2}+\frac{1}{2N-1}\right ).
\begin{cases}d\bar{x}_{t,n}= x_{0,n-1}\bar{x}_{t,1}dt + x_{t,1}(x_{0,n-1}- x_{t,n-1})dt + a_n x_{t,n}+ x_{t,n-2}x_{t,n-1}dt d\bar{x}_{t,1}= x_{0,2}\bar{x}_{t,n}dt + x_{t,n}(x_{0,2}- x_{t,2})dt + x_{t,n-1}x_{t,n}dt - \sigma_1 dW_t^{(1)}.\end{cases}
\frac{1}{2^{g+1}}\sum_{c_2, \ldots, c_n}\frac{\psi_1^{g+1 - \sum_j c_j}}{(g+1- \sum_j c_j)! \prod_{j=2}^n c_j!}\cdot \frac{(2g+2-2\sum_j c_j)!}{\prod_j (b_j - 2c_j)!}\prod_{j=2}^n \psi_j^{c_j},
E(z,g) = t^{z+1}+ t^{1-z}\frac{\xi(z)}{\xi(z+1)}\qquad+\frac{4t}{\xi(z+1)}\sum_{m=1}^\infty\eta_{\frac{z}{2}}(m)K_{\frac{z}{2}}(2\pi m t^2)\cos 2\pi m u, \eta_{\frac{z}{2}}(m) =\sum_{ab = m}\left(\frac{a}{b}\right)^{\frac{z}{2}}.
\mathcal{A}= \{y_{a_{j_0}, a_{j_1}, a_{j_2},\dots,a_{j_{t-1}}}\; \forall\; \textbf{a}= (a_{j_0}a_{j_1}a_{j_2}\dots a_{j_{t-1}}) \in \mathbb{Z}_q^t\ : 0 \leq j_0<j_1<j_2< \dots <j_{t-1}<m \}.
\begin{array}{lll}l_1(i,j) = \{[\lambda,\omega\zeta_i\lambda,\mu,\omega\zeta_j\mu]\}, l_2(i,j) = \{[\lambda,\mu,\omega\zeta_i\lambda,\omega\zeta_j\mu]\}, l_3(i,j) = \{[\lambda,\mu,\omega\zeta_j\mu,\omega\zeta_i\lambda]\}.\end{array}
\sigma_{x}=\frac{1}{\sqrt{2}}(\begin{matrix}1&0\\ 0&-1\end{matrix}),\sigma_{y}=\frac{i}{\sqrt{2}}(\begin{matrix}1&0\\ 0&1\end{matrix}),\sigma_{z}=\frac{1}{\sqrt{2}}(\begin{matrix}0&1\\ 1&0\end{matrix}).
\begin{cases}\partial_t\mathbf{v}+(\mathbf{v}\cdot \nabla)\mathbf{v}=-\nabla P,\ \ D\times (0,T), \nabla\cdot \mathbf{v}=0,\ \ D\times (0,T), \mathbf{v}\cdot \mathbf{n}=v_n,\ \ \partial D\times (0,T), \mathbf{v}(x, 0)=\mathbf{v}_0(x), \ \ D,\end{cases}
\bigoplus_{\begin{subarray}{c}\boldsymbol{b}=(b_1,\ldots,b_r)\in\mathbb N^r |\boldsymbol{b}|=b_1+\cdots+b_r=\delta b_1\mu_1+\cdots+b_r\mu_r=t\end{subarray}}S^{b_1}(V_1/V_0)\otimes\cdots\otimes S^{b_r}(V_r/V_{r-1}).
\ln\left|\frac{Y^{+\pm}_+}{Y^{+\pm}_-}\right|\sim\ln\left|\frac{Y^{-\mp}_+}{Y^{-\mp}_-}\right|\sim\pm\ln 2\pm\ln\left|\frac{kq}{\Omega\sqrt{k^2-q^2}}\right|+{\textstyle \frac{3}{2}}\ln\left|\frac{k+q}{k-q}\right|.
D_{++}^{(CPV)ab}(x)=D_{++}^{(CPV)}(x)\delta^{ab}= -{i\delta^{ab}\over (2\pi)^2}\int d^2k\,e^{ikx}{\partial\over \partial k_-}CPV\left({1\over k_-}\right)=-{i\delta^{ab}\over 2}|x^-|\delta(x^+)\ ,
H_{\beta,h}^{\Lambda,\sigma}( \omega_{\Lambda}) := -\beta \left[ \sum_{\{x,y\}\subset \Lambda \atop x\sim y}\omega_x\omega_y + \sum_{x\in \Lambda, y\notin \Lambda \atop x\sim y}\omega_x \sigma_y \right]+ h \sum_{x \in \Lambda}\omega_x.
g^\varepsilon b({\mathbf D}) \left(\varepsilon \Lambda^\varepsilon b({\mathbf D}) e^{-\mathcal{A}_D^0t}\right) &=g^\varepsilon (b({\mathbf D}) \Lambda)^\varepsilon b({\mathbf D}) e^{-\mathcal{A}_D^0t}+ \varepsilon \sum_{i,j=1}^d g^\varepsilon b_i \Lambda^\varepsilon b_j D_i D_j e^{-\mathcal{A}_D^0t}.
\begin{cases}{}^C\left(t^{1-2H}\frac{\partial}{\partial t}\right)^{\alpha}u_{\alpha}(x,t)= H^{\alpha}\frac{\partial^2}{\partial x^2}u_{\alpha}(x,t),\alpha \in (0,1), \: x\in \mathbb{R}\t > 0, u_{\alpha}(x,0)=\delta(x),\end{cases}
C_F(\rho^a, \rho^b) = \Bigg\{\rho\in H^{1}([0,1];\mathcal P(G)), m\in L^2([0,1];\mathcal S^{N\times N}) \Bigg| (\rho(0),\rho(1))=(\rho^a,\rho^b), d\rho_i(t)+\underset{j\in N(i)}{\sum}m_{ij}dt + \underset{j\in N(i)}{\sum}(\Sigma_j-\Sigma_i)\theta_{ij}(\rho) dW^{\delta}(t)=0.\Bigg\}
\gamma_i = \frac{\rho_i\vert \textbf{h}_i^H \textbf{w}_i\vert^2}{\sum_{j=1,j\neq i}^{N_{ue}}\rho_j\vert \textbf{h}_i^H \textbf{w}_j\vert^2+\rho_0\vert \textbf{h}_i^H \textbf{w}_0\vert^2+\sigma_n^2}.
\mathbf{q}_p^{\mathrm{H}}(\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{H}_{\mathrm{e}})^{-1}\mathbf{q}_p =\sum_{i=1}^{P}\lambda_i^{-1}|\mathbf{q}_p^{\mathrm{H}}\mathbf{u}_i|^2 &=\lambda_p^{-1}\cos^2\theta_{p,p}+\sum_{i\neq p}\lambda_i^{-1}\cos^2\theta_{p,i},
\zeta(\,\cdot\,) = \eta(\,\cdot\, \cap K_n) + \eta(K_n^c) \delta_{\Delta_n}(\,\cdot\,), \theta(x,a) = \begin{cases}\phi(x,a), x\in K_n \int_{K_n^c}\phi(y,a) \nu_n(dy) , x = \Delta_n,\end{cases}
R_y(x) = \sum_i \left(\sum_j c_{ij}y^{q^j}\right) x^{q^i}= \sum_i r_i(y)x^{q^i}; L_x(y) = \sum_j \left(\sum_i c_{ij}x^{q^i}\right) x^{q^j}= \sum_j l_j(x)y^{q^j}.
\int_\Omega \tfrac{1}{2}\bar \rho|\bar u|^2+\varepsilon \tfrac{1}{2}\bar n|\bar v|^2+h_1(\bar \rho)+h_2(\bar n) - \delta\tfrac{1}{2}|\nabla \bar \phi|^2 \dx \Big|_{\tau=0}^{\tau=t}= \int_0^t \int_\Omega \varepsilon \bar v \cdot \bar e \dxd\tau + \int_0^t \int_\Omega \delta (\partial_\tau \bar \phi) \Delta \bar \phi \dxd\tau \ ,
M^\pm(\rho) = A_1^\pm[\rho]\rho_0[\rho] + A_0^\pm[\rho]\rho_1[\rho], &N^\pm(\rho) = \rho_1[\rho] A_0^\pm[\rho], &P^\pm(\rho) = \lambda_{0}A_1^\pm[\rho]\rho_0'[\rho] + \lambda_1 A_0^\pm[\rho]\rho_1'[\rho],
\sum_{i=-N}^N \big\| \|A_t^\phi (F,G)(x,y)\|_{\textup{V}_t^2([2^i,2^{i+1}],\mathbb{C})}\big\|^2_{\textup{L}^2_{(x,y)}(\mathbb{R}^2)}& \lesssim \prod_{\rho\in\{\phi,\psi\}}\Big( \sum_{i=-N}^N \int_{\mathbb{R}^2}\int_{1}^2 \big( A_{2^it}^{\rho}(F,G)(x,y) \big)^2 \frac{dt}{t}dx dy \Big)^{1/2}.
\Phi''(\vec p)[\widehat{\vec p}_1,\widehat{\vec p}_2]=F''(V(\vec p))[V'(\vec p)\widehat{\vec p}_1,V'(\vec p)\widehat{\vec p}_2]+ F'(V(\vec p))V''(\vec p)[\widehat{\vec p}_1,\widehat{\vec p}_2]
[ \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}\, .
A_{4}= \frac{g^{2}\sqrt{2}}{2 \pi}\int d^{2}z \ \langle \mathrm{l e f t p a r t}\rangle \ \langle \bar{G}_{1}\bar{G}_{2}\bar{G}_{3}\bar{G}_{4}\rangle \ \langle \prod_{i}e^{i / 2 \K_{i}X}e^{i / 2 \K_{i}\bar{X}}\rangle
\left(\sum_{j=1}^\infty \left(\sum_{n=1}^\infty |x_{n}^*(x_{j})|^{p}\right)^{{q}/{p}}\right)^{1/{q}}= \left(\sum_{j \in J}\|x_j\|^{q}\cdot \left( \sum_{n=1}^\infty \left|x_n^*\left(\frac{x_j}{\|x_j\|}\right)\right|^{p}\right)^{q/p}\right)^{1/q}& \leq \| (x_{j})_{j=1}^\infty\|_{q}\| (x_{n}^*)_{n=1}^\infty\|_{p,w},
\ell_{j}\left( s\right) \left( \mathsf{N}_{j}\left( s\right)\circ\mathsf{L}_{j}\left( s\right) v,w\right) =\left\langle \mathsf{L}_{j}\left( s\right) v,\overline{w}\right\rangle_{\mathbb{R}^{3}}=\ell_{j}\left( s\right) \left( v,w\right) \qquad\forall w\in H^{1}\left(\mathbb{R}^{3}\right) .
F_{\vec{\sigma}, a, \vec{z}}(\vec{t}) := \frac{f_{\vec{\sigma}, a, \vec{z}}(\vec{t})}{f_{\vec{\sigma}, a, \vec{z}}(\vec{0})}, f_{\vec{\sigma}, a, \vec{z}}(\vec{t}) := \Gamma (\sigma_1 +{\rm{i}}t_1) \Gamma (\sigma_2 +{\rm{i}}t_2) \Phi_2 (\sigma_1 +{\rm{i}}t_1, \sigma_2 +{\rm{i}}t_2, a, z_1,z_2).
{I_{D}}^{\mathrm r}\equiv - V T{\frac{{\mathrm{t r}}{\bf 1}}{2 ( 4 \pi )^{D / 2}}}\int_{1 / \Lambda^{2}}^{\infty}\! d \tau \tau^{-{\frac{D}{2}}- 1}\left\{{\mathrm e}^{- \tau ( m^{2}+ \epsilon )}G_{D}( \tau F ) - 1 \right\}\ .