harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{n,l}^{\ast}(x) = \S_{n,l}^{\ast}(x_{i_0(x)}(0),x_{i_1(x)}(1)) \ + S_{n,l}^{\ast}(x_{i_1(x)}(1),x_{i_2(x)}(2)) & \ + \cdots \ + S_{n,l}^{\ast}(x_{i_{K-1}(x)}(K-1),x_{i_K(x)}(K)) \ + S_{n,l}^{\ast}(x_{i_K(x)}(K),x),
\underset{n \rightarrow \infty}{\lim}\frac{\sqrt{{\rm{Var}}[\sum_{i = 1}^m \omega_i \bar{f}_{n,k}(t_i)]}}{\sqrt{\omega_1^2 + \cdots + \omega_m^2 + 2 \sum_{i < j}\omega_i \omega_j e^{-|t_i - t_j|}}}= 1.
G_{k}^{\pm,+}\left( \omega_{\pm}\right) =H_{k}^{\pm,+;\varepsilon}\left( \omega_{\pm}\right) -H_{k}^{\pm,+;0}\left( \omega_{\pm}\right), G_{k}^{\pm,-}\left( \omega_{\pm}\right) =H_{k}^{\pm,-;\varepsilon}\left( \omega_{\pm}\right) -H_{k}^{\pm,-;0}\left( \omega_{\pm}\right),
\mathbb{P}\left(\begin{array}{c}\mbox{There is $ \psi \in \ell^2(B_n) $ with $ H_n \psi = \lambda \psi $ and}\mbox{$\; | \lambda - E | \leq 2^{- n- 1}W $ such that $\; P_2( \psi) \leq \varepsilon^4 $}\end{array}\right) \leq C \, W \varepsilon
f^{\vec{\alpha}\beta\gamma}_{(\eta)}=f^{\alpha[1]\beta b[1]}\,\,f^{b[1]\alpha[2]b[2]}\,f^{b[2]\alpha[3]b[3]}\,\cdots\,\,f^{b[\eta-2]\alpha[\eta-1]b[\eta-1]}f^{b[\eta-1]\alpha[\eta]\gamma}\;,
\zeta_{\widetilde{L}_p}^\mathrm{iso}(s) = \prod_{\mathfrak{p}\mid p}\frac{1 + q_\mathfrak{p}^{\, 8d+2-4s}}{(1 - q_\mathfrak{p}^{\, 4 d +4-3s}) (1 - q_\mathfrak{p}^{\, 8d +3-4s}) (1 - q_\mathfrak{p}^{\, 12d-5s})}.
\nu_{d,p}(\mathbb J^c(U\Lambda U')) =\nu_{d,p}(\{A\in\mathcal V_{d,p}: U\Lambda^{1/2}U'A (A'U\Lambda U'A)^{-1/2}\in \mathbb G^c\}) &=\nu_{d,p}(\{C\in\mathcal V_{d,p}: U\Lambda^{1/2}A (A'\Lambda A)^{-1/2}V \in \mathbb G^c\}) &=\int_{\mathcal V_{d,p}}\mathbf{1}_{\mathbb G^c}(U\Lambda^{1/2}A (A'\Lambda A)^{-1/2}V)\;\nu_{d,p}(dA).
\bar{\gamma}_{ki}^{p}=\bar{\gamma}_{i}^{p}=\frac{\bar{\upsilon}_{i}^{p}}{{\displaystyle \sum_{j=1}^{L}\alpha_{j}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}\right]+\bar{\upsilon}_{i}^{p}\sum_{j\neq i}^{L}\kappa_{j}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}^{2}\right]}}
\mathcal B_r(n,a,b) := \left \{\lambda \vdash n-rab \ \Bigg | \ \begin{array}{l}\lambda\neq (ra, r(a-2)), r(a+b+1)\not \in \lambda, \lambda_1-\lambda_2 \leq 2r(a+b+1) \lambda_1 = 3r(a+b+1) \end{array}\!\!\right\}.
\pi_i =\frac{2(i+2)}{3(i+1)!}, A =\sum\limits_{i=1}^\infty\frac{1}{b_i\pi_i}=\frac{3}{2}\sum\limits_{i=1}^\infty\frac{(i+1)!}{(i+3)(i+2)}=\infty, S =\sum\limits_{i=1}^\infty\frac{1}{b_i\pi_i}\sum\limits_{j=i+1}^{\infty}\pi_{j}=\sum\limits_{i=1}^\infty\frac{(i+1)!}{(i+3)(i+2)}\sum\limits_{j=i+1}^{\infty}\frac{(j+2)}{(j+1)!}<\infty.
\Phi^{( I )}= -{\frac{e^{2}}{8 \pi^{2}m}}\operatorname{a r c t a n}\biggl \{\tan \biggl [{\frac{\Theta}{2}}+{\frac{\pi}{4}}\bigl ( 1 - \mathrm{s g n}( m ) \bigr ) \biggr ] \biggr \}.
0 \leq 1-\det ( -A)/\det ( -B) &\leq\left|( -B)^{-1/2}(B -A)( -B)^{-1/2}\right|_2^2 &\leq\left|( -B)^{-1/2}\right|_{\infty}^4|B -A|_2^2=\left|( -B)^{-1}\right|_{\infty}^2|B -A|_2^2,
X_{1,n+1}= a-X_{1,n}^2 + bX_{2,n}+ \eta_1 X_{2,n+1}= X_{1,n}+\eta_2 Y_{1,n+1}= a - (c X_{1,n}Y_{1,n}+ (1-c) Y_{1,n}^2)+bY_{2,n}Y_{2,n+1}= Y_{1,n}\eta_1 \sim Unif(-\alpha,\alpha) \eta_2 \sim Unif(-\alpha,\alpha),
\mathcal{B}_0\partial_t{\mathbf V}_\sigma+\mathcal{B}_1\partial_1{\mathbf V}_\sigma+\mathcal{B}_2\partial_2{\mathbf V}_\sigma+(\mathcal{B}_3+\partial_1\mathcal B_1){\mathbf V}_\sigma=\tilde{\mathcal F}_{\sigma}\,,
\Pi_j^{p,q}(z_j,s):=\Lambda_{N_j}^{p,q}(z_j,s)-\Lambda_{M_{0j}}^{p,q}(z_j)\ll A_1(s)=\frac{b_1}{16c_1}\sum_{v=1}^\infty\frac{c_1^v(s_1+\cdots +s_l)^v}{v^2}
a_0^{\kappa,J+\frac12}=-a^{\kappa,J+\frac12}, a_1^{\kappa,J+\frac12}=-\frac{(3J-2)\,a^{\kappa,J+\frac12}}{3}, a_\iota^{\kappa,J+\frac12}=\frac{2(J-\iota)\,a_{\iota-1}^{\kappa,J+\frac12}-a^{\kappa,J+\frac12}\binom{J}{\iota}}{2\iota+1},\qquad\iota=2,3,\dots,\kappa-1,
\widetilde{H}_{b}^{r,s}(M)\cong\left\{\varphi\in\mathcal{H}_{b}^{r,s}(M)\,|\,\varphi\wedge d\eta\in\overline{\partial}_b\left(F_b^{r+1,s}(M)\right)\right\}\cong\left\{\varphi\in\mathcal{H}^{r,s}_{b}(M)\,|\,\varphi\wedge d\eta=0\right\}.
F_B=\bigg [ 2 \epsilon^1_{ij}\epsilon^2_{il}p^j k^l -\epsilon^1_{ij}\epsilon^2_{ij}\vec k\cdot \vec p\bigg ]{\Gamma (1-2k_0^2) \Gamma (\vec k\cdot \vec p +k_0^2)\over\Gamma (\vec k\cdot \vec p-k_0^2+1)}\ ,
\dim_H E_\varphi(\Phi) = \liminf_{\ell\to\infty}\frac{\sum_{j=1}^\ell (1/c-1)\gamma^j}{d\sum_{j=1}^{\ell+1}1/c \gamma^j - (1/c-1)\gamma^{\ell+1}}= \frac{1-c}{d\gamma - (1-c)(\gamma-1)}.
\theta ( - \bar{R}_{1}, t ) , \kern 0 . 2 5 i n \theta ( \bar{R}_{2}, t ) , \kern 0 . 2 5 i n \phi ( - \bar{R}_{1}, t ) , \kern 0 . 2 5 i n \phi ( \bar{R}_{2}, t ) , \kern 0 . 2 5 i n r_{1}( t ) , \kern 0 . 2 5 i n r_{2}( t ) .
\mathrm{I}= \Gamma a_0\left(\prod_{k=1}^{m-2}b_k(a_k)\right)|D|^{n_1}, \mathrm{II}= |D|^{-n_1}\delta^2(a_{m-1})|D|^{n_1}, \mathrm{III}= |D|^{-n_1}\left(\prod_{k=m}^{p-1}b_{k+1}(a_k)\right)|D|^{n_1+n_2}.
p^{-1}R^{-p}= p^{-1}\left( \frac{\sqrt{\frac{\kappa}{CA}+\left( \frac{c_d'|\nabla f(a)|}{2CA}\right)^2}+ \frac{c_d'|\nabla f(a)|}{2CA}}{\frac\kappa{CA}}\right)^p & = \frac{c_{d,p}}{\kappa^p}\left( \sqrt{\tfrac{CA}{c_d'^2}\kappa +\left( \tfrac12 |\nabla f(a)| \right)^2}+ \tfrac12|\nabla f(a)| \right)^p.
V_{\mu_1\mu_2\mu_3}^{a_1 a_2 a_3}=-i g f^{a_1 a_2 a_3}\left[\eta_{\mu_1 \mu_2}(k_1-k_2)_{\mu_3}+ \eta_{\mu_2 \mu_3}(k_2-k_3)_{\mu_1}+ \eta_{\mu_3 \mu_1}(k_3-k_1)_{\mu_2}\right]
||\Delta U(\cdot,0)x||_{L^{\infty}(0,\tau_1;L^2)\cap L^{q}(0,\tau_1;L^{p})}\lesssim ||x||_{H^2}, ||\mu x||_{L^2}\lesssim ||x||_{L^2}\lesssim ||x||_{H^2}, ||g(x)||_{L^2}=||x||^{\alpha}_{L^{2\alpha}}\lesssim ||x||^{\alpha}_{H^2},
\mathcal{B}:=\!\left\{(p+1,p-1),(1,-1),(p+2,p),(2,0),(1,1),(p+1,p+1)\right\}, varepsilon_2(\boldsymbol{B}):=\varepsilon\left( \frac{B_2-3B_1}{2p},\frac{B_1}{p}\right).
\begin{aligned}U(\theta,M) =\int_{0}^{\infty}\left[\frac{1}{x^{1/2+\theta}}-\frac{1}{x^{1/2-\theta}}\right]\frac{\sin\left(M\log x\right)}{\left(e^{x}+1\right)}dx &=\sum_{n=1}^{\infty}A_n \sin(2n\pi\theta)\sinh(2n\pi M)\end{aligned}
\frac{d\gamma}{dt}=\frac{d}{dt}\frac{1}{\sqrt{1-v^{2}/c^{2}}}=\frac{2}{c^{2}}\frac{dx}{dt}\frac{d^{2}x}{dt^{2}}(1-\frac{1}{c^{2}}(\frac{dx}{dt})^{2})^{-3/2}
(X \xrightarrow{1_X \otimes \omega}X \otimes \overline{X}\otimes X \xrightarrow{\overline{\omega}^{\,*}\otimes 1_X}X) = 1_X, (\overline{X}\xrightarrow{\omega \otimes 1_{\overline{X}}}\overline{X}\otimes X \otimes \overline{X}\xrightarrow{1_{\overline{X}}\otimes \overline{\omega}^{\, *}}\overline{X}) = 1_{\overline{X}}.
\tilde \phi\big(Q^{\alpha_{n-1}}v_{0}^{(1)}\otimes Q^{\alpha_{n-2}}v_{0}^{(1)}\otimes\cdots\otimes Q^{\alpha_{0}}v_{0}^{(1)}\big)=Q_{n-1}^{\alpha_{n-1}}Q_{n-2}^{\alpha_{n-2}}\cdots Q_{0}^{\alpha_{0}}v_{0}^{(n)},
\Omega_{s}= - \mathrm{d}x^{a}\wedge \mathrm{d}p_{a}+ \Omega_{\mathrm{m}}\qquad \Omega_{\mathrm{m}}= \frac{s}2 \frac{\epsilon^{a b c}p_{a}\mathrm{d}p_{b}\wedge \mathrm{d}p_{c}}{( - p^{2})^{3 / 2}}\qquad ( p^{2}< 0 ) ,
u = \arg \min_{v \in{H}_0^1(\Omega)}F[v], F[v] = \int_{\Omega}k_x \left( \frac{\partial v}{\partial x}\right)^2 + \int_{\Omega}k_y \left( \frac{\partial v}{\partial y}\right)^2 - 2 \int_{\Omega}v f.
\int_{\Gamma_\infty}|\nabla_{\Gamma_\infty}w_\infty(x)|^2 \, d\mu_{\Gamma_\infty}\leq \liminf_{n\to +\infty}\int_{\Gamma_\infty}|\nabla_{\Gamma_\infty}w_n(x)|^2 \, d\mu_{\Gamma_\infty}, \lim_{n\to \infty}\int_{\Gamma_\infty}w_n(x)f(\tau_n(x)) \, d\mu_{\Gamma_\infty}=\int_{\Gamma_\infty}w_\infty(x)f(x) \, d\mu_{\Gamma_\infty}.
N_>(B,C) = |\{y \in B \mid y > x,\y \equiv_n x-1 \}| - |\{y \in B \mid y>x,\y \equiv_n x \}|, N_<(B,C) = |\{y \notin B \mid y < x-1,\y \equiv_n x-1 \}| - |\{y \notin B \mid y<x-1,\y \equiv_n x \}|.
\tau_{\beta, \lambda}(\lambda' \cdot v) = \tau_{\beta, \lambda}\omega_\gamma \tau_{\alpha, \lambda'}\omega_\gamma^{-1}v & = \omega_\gamma \tau_{\alpha, \lambda'}\omega_\gamma^{-1}\tau_{\beta, \lambda}v & = \lambda' \cdot (\tau_{\beta, \lambda}v)
{\mathrm Re}\partial_{p^2}\Gamma^T_{+-}(p^2)\big|_{p^2 = \kappa_W^2}= 1 \quad{\mathrm Re}\partial_{p^2}\Gamma^T_{ZZ}(p^2)\big|_{p^2 = \kappa_Z^2}= 1 \quad{\mathrm Re}\partial_{p^2}\Gamma^T_{AA}(p^2)\big|_{p^2 = \kappa_A^2}= 1
f(b) := \begin{cases}\frac{\int\limits_{0}^{\pi}\sin^{n-2}(t)dt}{\int\limits_{0}^{\pi}\frac{\sin^{n-2}(t)}{[\phi(b\cos (t))]^n}dt}, \mbox{if}dV=dV_{BH}, \frac{\int\limits_{0}^{\pi}\sin^{n-2}(t)T(b\cos (t))dt}{\int\limits_{0}^{\pi}\sin^{n-2}(t)dt}, \mbox{if}dV=dV_{HT}, \end{cases}
T^{ab}_{0}= \oint\frac{d\xi}{\xi}\phi_a^-(\xi)V_a^{-1}(\xi)V_b(\xi), T^{ab}_{1}=-\frac{1}{\beta}\oint\frac{d\xi}{\xi}\phi_a^-(\xi)V_a^{-1}(\xi)\phi_b^+(\xi)V_b(\xi) -\sum_c \oint\frac{d\xi d\eta}{\xi\eta}\frac{\frac{\eta}{\xi}}{1-\frac{\eta}{\xi}}\phi_a^-(\eta)V_a^{-1}(\eta)\phi_c^-(\xi)V_c(\eta)V_c^{-1}(\xi)V_b(\xi).
\frac{\partial^p}{\partial w^p}f(z,w)= p! \left( \prod_{b=1}^k \frac{1}{1 - wz^{n_b}}\right) h_p \left( \frac{z^{n_1}}{1 - wz^{n_1}}, \cdots, \frac{z^{n_k}}{1 - wz^{n_k}}\right),
C_{\textrm{N}}^{\textrm{NOMA}}=\begin{cases}\frac{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^2 \left( 2^R-1\right)}{P_{\textrm{N}}}, \textrm{if}k > 2^R, \frac{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^2 \left( 2^R-1\right)}{P_{\textrm{F}}-P_{\textrm{N}}\left( 2^R-1\right)}, \textrm{if}k \le 2^R,\end{cases}
\begin{cases}A(x,t)= I + \delta r^{\alpha}M(x,t), Du(x,t)= e_n +\delta r^{\alpha}w(x, t), g(x,t)= g(0,0) + \delta r^{\alpha}w_1(x, t), |u(x,t)| \leq C \delta r,\end{cases}
\hat{d}_n(\rho) = \frac1Nz_n^* \left( \frac{1-\rho}{1-(1-\rho)c_N}\frac1n\sum_{i=1}^{n-1}\frac{z_iz_i^*}{\hat{d}_i(\rho)}+ \rho I_N \right)^{-1}z_n &\leq \frac1Nz_n^* \left( \frac{1-\rho}{1-(1-\rho)c_N}\frac1n\sum_{i=1}^{n-1}\frac{z_iz_i^*}{\hat{d}_n(\rho)}+ \rho I_N \right)^{-1}z_n.
V_{e f f}= - \frac{\left( \rho_{1}+ \lambda_{1}\left( \frac{z_{v}}{z_{h}}\right)^{2 - \nu}\right)^{2}}{2 ( k z_{v})^{4}( \rho_{2}+ \lambda_{2}^{* ( 0 )})}+ \frac{\alpha^{2}}{( k z_{v})^{4}}P \left( ( z_{v}/ z_{h})^{- \epsilon}, \left( \alpha \ln ( z_{v}/ z_{h}) \right) \right)
H(\sigma) =\sum_{j=1}^3\frac{1}{m(\sigma_j)}\le\frac{1-\sigma_1}{3}+\frac{1-\sigma_2}{3}+\frac 14 &=\frac 14+\frac{a_1+a_2}{6a_3}\le\frac 14+\frac 16 \cdot \frac 32=\frac 12.
\int_{0}^{\infty}\! d t \, I m \, T_{1}( t ) = - \frac{m^{2}}{4}\int_{4 m^{2}}^{\infty}\! \! \frac{d t}{t^{2}}\left( 1 - \frac{4 m^{2}}{t}\right)^{\frac{1}{2}}= - \frac{1}{2 4}\enspace .
\alpha(\sigma_r)x-\alpha(\mu_r)x \leq\left(\Big|\frac{1}{r}\int^r_0s\frac{d}{ds}\alpha(\sigma_s)xds\Big|^{2}\right)^{\frac12}&\leq\left(\frac{1}{r^2}\int^r_0sds\int^r_0s\big|\frac{d}{ds}\alpha(\sigma_s)x\big|^2ds\right)^{\frac12}&=\frac{1}{\sqrt{2}}\left(\int^r_0s\big|\frac{d}{ds}\alpha(\sigma_s)x\big|^2ds\right)^{\frac12}\leq \frac{1}{\sqrt{2}}g_1(x).
{Z}^h(x) = \frac{1}{h^2}\Big(\nabla u^h(x', hx_3) - R^h(x')\big(Q_0(x') + hP_0(x', x_3)\big)\Big) \rightharpoonup Z \qquad\mbox{weakly in}L^2(\Omega,\mathbb{R}^{3\times 3}).
2(2n+1)f_0^2 f^n_{2n+2}= -(2n+3)f_1 f_4 f^n_{2n-3}\phantom{=}\ + \big(2(2n+1) f_0 f_4 - (n+2)f_1 f_3\big) f^n_{2n-2}\phantom{=}\ + \big((2n+1) f_0 f_3 - f_1 f_2\big) f^n_{2n-1}\phantom{=}\ + n f_1^2 f^n_{2n}.
\alpha_{t,C}(x) =\int_0^{tC^2}Cp_{tC^2-s}(0,0)\rho_{\theta_0^{Cx}}(s)ds, &= \int_0^{tC^2}\left(\frac{C}{\sqrt{tC^2-s}}\varphi(\sqrt{tC^2-s}) - C\Phi(-\sqrt{tC^2-s}) \right)\rho_{\theta_0^{Cx}}(s)ds, \x\neq 0,
f:B_{Q}\to\mathbb{M}_2(\mathbb{C}) &a+b\textbf{i}+c\textbf{j}+d\textbf{k}\mapsto M=\begin{pmatrix}a+b\textbf{i}c+d\textbf{i}-c+d\textbf{i}a-b\textbf{i}\end{pmatrix}{~.}
D_1 := \left[0,\frac{1}{2}\right]\times\left(\left[0,\frac{1}{2}\right]^{n-1}\backslash \left[0,k\right]^{n-1}\right)\times \left[-\frac{1}{2},\frac{1}{2}\right]^m, D_2 := \left[0, 1\right]\times\left([0,1]^n\backslash\left[0,k/2\right]^{n-1}\right)\times \left[-1,1\right]^m,
B_1: = \bigg(\int_0^{\infty}\bigg(\int_0^x [\Phi_1]^q(t) w(t)\,dt\bigg)^{r / p}[\Phi_1]^q(x) \Phi[v;p]^{- r / p}(x) w(x)\,dx \bigg)^{1 / r}, B_2: = \bigg(\int_0^{\infty}W_*^{r / p}(x) \bigg[\sup_{0 < \tau \le x}\Phi_1(\tau)\Phi[v;p]^{-1 / p}(\tau)\bigg]^{r}w(x)\,dx \bigg)^{1 / r}.
Z_{\scriptscriptstyle MCS}(J) = e^{i \alpha - \frac{i\pi}{4}\eta (0, \hskip 3pt \ast d_{\scriptscriptstyle 1})}\tau_{\scriptscriptstyle RS}(M)^{\scriptscriptstyle 1/2}e^{i \langle J, \hskip 3pt \frac{1}{\ast d_{\scriptscriptstyle 1}}\hskip4pt J \rangle}Z_{\scriptscriptstyle SD}(\hat{J})
\begin{aligned}0 \geq \frac{1}{2}\sum_i F^{ii}U_{ii}^2 + A F^{ij}(\underline{u}-u)_{ij}- A(\underline{u}- u)_t &- C_1 K \Big(1 + |\phi_{uuu}| K^{1/2}\Big) \sum_i F^{ii}- C_1 K.\end{aligned}
\max_{\tau_i,\tau_{-i}\in \mathcal{S}}\mathcal{J}_i(\tau_i,\tau_{-i})=\max_{\tau_i,\tau_{-i}\in \mathcal{S}}\mathbb{E}\left\{{C}_{\theta}(\tau_i,\tau_{-i})-e^{-\beta\tau_i}K\right\}
\begin{aligned}\mu^2\overline{\textbf{det}}(v \diamond v)=\overline{\textbf{det}}(x)\overline{\textbf{det}}(s), \mu \textbf{Tr}(v \diamond v)=\textbf{Tr}(x)\textbf{Tr}(s), &\mu \lambda_1(v \diamond v)= \lambda_1(x) \lambda_1(s).\end{aligned}
{\tilde{Z}}_{m}^{T}= \int \prod_{\mu ; <{i j}>}d P_{i j}^{\mu}e^{\{- \sum_{\mu ; <{i j}>}( P_{i j}^{\mu})^{2}- l n ( \sum_{\mu}( P^{\mu})^{2}) \}}\cdot{c o n s t r a i n t s},
(H^1(\mathbb{R}^d), B^{1+s}_{2,q}(\mathbb{R}^d))_{t,2}= H^{1+ts}(\mathbb{R}^d),\,\,\,\,\, (H^{-1}(\mathbb{R}^d), B^{-1+s}_{2,q}(\mathbb{R}^d))_{t,2}= H^{1+ts}(\mathbb{R}^d), (B^{-s_1}_{2,q}(\mathbb{R}^d), B^{-s_2}_{2,q}(\mathbb{R}^d))_{t,2}= H^{-(1-t)s_1 - t s_2}(\mathbb{R}^d).
S = \int d^{5}x \sqrt{- g}\left({\frac{R}{2}}- \Lambda_{b}+{\cal L}^{( M )}\right) + \sum_{j = 1 , 2 \ \mathrm{b r a n e s}}\int d^{4}x \sqrt{- g^{( j )}}\left({\cal L}_{j}^{( M )}- \Lambda_{j}\right) ~ ,
-\sum_{a_k\leq 0}\int_0^\infty g(x)x^{w-1}dx -\sum_{k=1}^{r_1+r_2}\frac{1}{2\pi}\int_{-\infty}^\infty \mathrm{Re}\Big[\frac{\Gamma'_k}{\Gamma_k}(\frac{1}{2}+it+w_k)\Big]\hat{g}(\frac{1}{2}+it)dt,
\tilde{g}^{k\overline{l}}\tilde{g}^{i\overline{j}}\overline{e}_{l}e_{i}(f)e_{k}\overline{e}_{j}(f) \geq ~~ \frac{1}{n}\left(\tilde{g}^{i\overline{j}}e_{i}\overline{e}_{j}(f)\right)^{2}\geq \left(1-\frac{\epsilon}{5}\right)\frac{(Lf)^{2}}{n}-\frac{C}{\epsilon}|\partial f|_{\tilde{g}}^{2}.
\mathrm{var}\big[\widehat{K}_{\!\mathrm{a-cpt-f}}^\mathbb{R}\big]\!=\! \upsilon^2\bigg(\!\Big(1\!-\!\frac{\pi}{4}\Big)\Big(1\!+\!\frac{1}{N_1\bar{\gamma}'}\Big)\!K\!+\!\frac{1}{2N_1\bar{\gamma}'}\!+\!\frac{1}{2N_2\bar{\gamma}_c}\bigg)=\frac{4-\pi}{\pi}K+\frac{1+\frac{N_1\bar{\gamma}'}{N_2\bar{\gamma}_c}}{\pi(1+N_1\bar{\gamma}')}
r_i^m = \bigg(\frac{pq_i}{\mathsf{d}(r_i)}\bigg)^m = \frac{\prod_{j=1}^k \mathsf{d}(r_j)^m}{\mathsf{d}(r_i)^m}\q_i\bigg(\frac p{\mathsf{d}(r_1) \dots \mathsf{d}(r_k)}\bigg)^m \in P.
\begin{aligned}\sup_{0< t< T}e^{-\frac{\omega t}{2}}t^{\frac{3}{2}(\frac{1}{p}-\frac{1}{q})}\int_{0}^{t}(t-s)^{-\frac{1}{2}-\frac{3}{2q}}s^{-3(\frac{1}{p}-\frac{1}{q})}\ \d s =\left(\sup_{0< t< T}e^{-\frac{\omega t}{2}}t^{\frac{1}{2}-\frac{3}{2p}}\right) B \big(1-3\big(\tfrac{1}{p}-\tfrac{1}{q}\big),\tfrac{1}{2}-\tfrac{3}{2q}\big), \end{aligned}
\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\frac{1}{\left(\lambda_{n}^{2}+x^{2}\right)^{s}}=\frac{\sqrt{\pi}x^{1-2s}}{2\Gamma(s)}\Gamma\left(s-\frac{1}{2}\right)-\frac{1}{2}\,\frac{x^{-2s}}{1+\frac{1}{\pi p}}+2\sin(\pi s)x^{1-2s}\,\intop_{1}^{\infty}\frac{(t^{2}-1)^{-s}}{\sigma(xt)e^{2\pi xt}-1}dt,
{Q_2}(M,P_i,P_j,\tau)= \left\{e^{\frac{1}{24}E_2(\tau)A_2}\widehat{A}(TM,\nabla^{TM}){\rm ch}\left[\bigotimes_{n=1}^{\infty}S_{q^n}(\widetilde{T_CM})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^{m-\frac{1}{2}}}(\widetilde{T_CM})\right]\right. &\left.\cdot\varphi(\tau)^{16}{\rm ch}(\mathcal{V}_i){\rm ch}(\mathcal{V}_j)\right\}^{(12)},
W_v(r) = 1 + \sum_{w \in c_v(r)}W_v(w) = 1 + \sum_{w \in c_r(r)- \{v\}}W_r(w) = n+1 - W_r(v) & < n+1 - \lfloor \frac{n+1}{2}\rfloor = \lceil \frac{n+1}{2}\rceil
\delta \, M_{\beta , \gamma}^2 = - \frac{2 \, G_N}{\pi \, \ell_s^2}\int \frac{d^d \, k \d \, \omega}{(2\pi)^{d+1}\, [\mbox{\boldmath$k$}^2 -\omega^2 - i \varepsilon]}\int d \tau_{12}\, d \sigma_1 \, d \sigma_2\, e^{-i \, \ell_s^2 M \omega \, \tau_{12}}\, \langle W (k,1,2) \rangle_{\beta , \gamma}\, ,
\mbox{\boldmath $\gamma$}_t(s) =\frac{d{\mathbf Z}_{t}}{d \ln\kappa}\,{\mathbf Z}_{t}^{-1}= \left( \begin{array}{cc}0 -g_B^2\, K_{Mm}(s) begin{align*}2mm]-g_B^2\, K_{Mm}(s) 0\end{array}\right) \:.
\int^{\infty}_{\tau_m}\|CT(t)x\|^2 dt =\sum_{n=m}^{\infty}\int^{\tau_{n+1}}_{\tau_n}\|CT(t)x\|^2 dt &\leq (2e)^2\sum_{n=m}^{\infty}\int^{\tau_{n+1}}_{\tau_n}\|\mu_n t e^{-\mu_n t}CT(t)x\|^2 dt &\leq (2e)^2 M_3\|x\|^2 \sum_{n=m}^{\infty}\frac{1}{|\log (2\mu_n)|^{2\beta}}.
\Delta_{\varphi_2;g_{i_2}}^{n_{2,i_2}+1}\cdots \Delta_{\varphi_r;g_{i_r}}^{n_{r,i_r}+1}h_i(x)= \Delta_{\varphi_1;g_{i}}^{n_{1,i}+1}\Delta_{\varphi_2;g_{i_2}}^{n_{2,i_2}+1}\cdots \Delta_{\varphi_r;g_{i_r}}^{n_{r,i_r}+1}f(x)=0
\prod_{j=r+1}^n \prod_{\ell=0}^{\lambda_r-\lambda_j-1}\frac{1 - t q^\ell t^{j-r}}{1- q^\ell t^{j-r}}=\prod_{c=m+1}^{\lambda_r}\prod_{j=\lambda_c'+1}^{n}\frac{1-tq^{\lambda_r-c}t^{j-r}}{1-q^{\lambda_r-c}t^{j-r}}=\prod_{c=m+1}^{\lambda_r}\frac{1-tq^{\lambda_r-c}t^{n-r}}{1-q^{\lambda_r-c}t^{\lambda'_c+1-r}}
d s^{2}= \left( 1 +{\frac{K}{r^{6}}}\right)^{- 2 / 3}\eta_{\mu \nu}d x^{\mu}d x^{\nu}+ \left( 1 +{\frac{K}{r^{6}}}\right)^{1 / 3}\delta_{m n}d y^{m}d y^{n},
R_{\mu}= \mu^{\frac 13}, L_\mu = \mu^{\frac 23}R_{\mu}= \mu^{\frac 27}(\ln \mu)^{-\frac{1}{7}}, L_\mu = \mu^{\frac37}(\ln\mu)^{\frac{2}{7}}, R_{\mu}= \mu^{\frac{2}{2n+1}}, L_\mu = \mu^{\frac{3}{2n+1}},
\sum_{k=0}^{\infty}\frac{\left(\frac12\right)_k^2}{(1)_k^2}(a+bk) x^{2k}=\sum_{k=0}^{\infty}\frac{\left(\frac12\right)_k^2}{(1)_k^2}\left( a(1+t)+b\frac{t(1+t)}{1-t}+ 2b\frac{(1+t)^2}{1-t}k \right) t^{2k}
L_{\pm1}=\{-1/2-\sigma\pm iv \mid v\in(\sqrt{|t|},\infty)\}, &L_{\pm2}=\{3/2-\sigma\pm iv \mid v\in(\sqrt{|t|},\infty)\}, &C_{1}=\{-1/2-\sigma+\sqrt{|t|}e^{-i\pi\theta}\mid \theta\in(1/2,3/2)\}, C_{2}=\{3/2-\sigma+\sqrt{|t|}e^{i\pi\theta}\mid \theta\in(-1/2,1/2)\}.
Z=1+(B_{0}-\frac{A_{0}}{RT}-\frac{C_{0}}{RT^{3}})\rho+(b-\frac{a}{RT})\rho^{2}+(\frac{\alpha a}{RT})\rho^{5}+\frac{c\rho^{2}}{RT^{3}}(1+\gamma\rho^{2})exp(-\gamma\rho^{2})
\frac{\partial^kF}{\partial v^k}(a,v)=\int_a^v\frac{\partial^kf}{\partial u^k}(v,v')dv'+\sum_{l=0}^{k-1}\binom{k}{l+1}\frac{\partial^{k-1}f}{\partial u^{k-1-l}\partial v^l}(v,v).
(q_{k+1}+\dots+q_n)^2 = (2m-S^+_k)^2 \leq (n-k)(q^2_{k+1}+\dots+q^2_n) &= (n-k)\Big(2m+\displaystyle\sum_{v_i\in V(G)}d^2_{v_i}-(q^2_1+\dots+q^2_k)\Big) &\leq (n-k)\Big(2m+\displaystyle\sum_{v_i\in V(G)}d^2_{v_i}-\frac{{S^+_k}^2}{k}\Big).
\left( \int d^{2}z d^{2}w{\frac{1}{z - w}}\psi^{\dagger}( z ) U^{q}( z ) \psi^{\dagger}( w ) U^{q}( w ) \mathrm{e}^{-{\frac{1}{4}}( | z |^{2}+ | w |^{2})}\right)^{N / 2}| 0 \rangle .
R_{\varphi}(x) =\left(\begin{array}{cc}e^{\frac{i \varphi x}{2}}0 & 0 e^{-\frac{i \varphi x}{2}}\end{array}\right),{F(x) =\left(\begin{array}{cc}p(x) q(x) & q(x) p(x)\end{array}\right).}
\frac{\partial \Omega}{\partial z^{\alpha}}=\frac{1}{n!}\frac{\partial h(x)}{\partial z^{\alpha}}\epsilon_{{\mu}{\nu}\cdots{\rho}}dx^{\mu}dx^{\nu}\cdots dx^{\rho}+\frac{1}{(n-1)!}h(x)\epsilon_{{\mu}{\nu}\cdots{\rho}}\frac{\partial dx^{\mu}}{\partial z^{\alpha}}dx^{\nu}\cdots dx^{\rho}.
\Lambda_\nu(\alpha,\beta)={}[P_\nu(1-2\alpha)]^2[P_{\nu}(2\beta-1)]^2\mathbb Z+P_\nu(1-2\alpha)P_\nu(2\alpha-1)P_{\nu}(1-2\beta)P_{\nu}(2\beta-1)\mathbb Z{}+[P_\nu(2\alpha-1)]^2[P_{\nu}(1-2\beta)]^2\mathbb Z,
\mathrm{exp}\left(\int_{P_{0}}^{P}\omega^{(3)}_{\hat{\nu}_{n+1}(x,t_{m}),\hat{\mu}_{n+1}(x,t_{m})}(P)\right) \mathop{=}\limits_{\zeta\rightarrow 0}\mathrm{exp}(w_{0}^{\infty\pm})(1+(M(x,t_{m})\pm\gamma_{n})\zeta+O(\zeta^{2})),~as~P\rightarrow P_{\infty\pm}.
\frac{\delta F}{\delta \psi}=D\frac{\partial \epsilon}{\partial \psi}- \left(D\frac{\partial \epsilon}{\partial \nabla\psi}\right) &= 2D \widehat{H}_e\psi+\frac{i\hbar}{M}D\boldsymbol{A}\cdot \nabla \psi +\frac{i\hbar}{M}\operatorname{div}\!\big(D(i\hbar\nabla+\boldsymbol{A})\psi\big),
T^{\mu}\_{\nu}( \eta ) = \left( \begin{array}{c c}{{H ( \eta ) c^{- 1}( \eta )}}&{{0}}\\{{0}}&{{T_{\beta}^{\beta}- c^{- 1}( \eta ) H ( \eta ) \}}\\ \end{array}\right) + c^{- 1}\left( \begin{array}{c c}{{\zeta}}&{{\alpha^{\prime}}}\\{{- \alpha^{\prime}}}&{{- \zeta \}}\\ \end{array}\right)
\mathbb{E}(\langle :X_T\otimes \ldots \otimes X_T:,\Phi \rangle -\rho_{\Phi}^T)^2=\mathbb{E}\langle :X_T\otimes \ldots \otimes X_T:,\Phi \rangle^2-\mathbb{E}(\rho_{\Phi}^T)^2.
\begin{aligned}\left\|\left(\frac{1}{N}\sum_{n=0}^{N-1}U_2^nXU_1^{-n}\right)U_1-U_2\left(\frac{1}{N}\sum_{n=0}^{N-1}U_2^nXU_1^{-n}\right)\right\| = =\frac{1}{N}\,\|XU_1-U_2^NXU_1^{-N+1}\| \leq \frac{2\|X\|}{N}\to 0,\end{aligned}
i_0^* \mathcal{W}=d\,i_0^* \Lambda=\sum_{1\le k<j\le \nu}A_{k\,j}(\varphi)\,d\varphi_k\wedge d \varphi_j, A_{k\,j}(\varphi):=\partial_{\varphi_k}a_j(\varphi)-\partial_{\varphi_j}a_k(\varphi).
(W(f) \phi )(y)=i\Bigl( \sum_{k=1}^n{\frac{d}{dt}}\bigl(E_k\,,\, v(y\exp(tE_k/2)\bigr)\bigr \vert_{t=0}\phi (T) &+{\frac{d}{dt}}\,\phi (y\exp (tv(y))\bigr \vert_{t=0}\Bigr).
\left( \begin{array}{cc}u i\alpha i\beta \bar{u}\end{array}\right) =\frac{1}{\sqrt{r_+^2 + r_-^2}}\left( \begin{array}{cc}e^{i (r_+ \phi+r_- t)}\sqrt{r^2 + r_-^2}i e^{r_- \phi-r_+ t}\sqrt{r_+^2 - r^2}i e^{- r_- \phi+ r_+ t}\sqrt{r_+^2 - r^2}e^{- i ( r_+ \phi + r_- t)}\sqrt{r^2 + r_-^2}\end{array}\right) .
\delta \eta_{(p)\mu_1\ldots \mu_{p-2k-1}}^{*}=-\left( \left(p-2k\right) \partial^\mu A_{(p)\mu \mu_1\ldots \mu_{p-2k-1}}^{*}+\partial_{\left[ \mu_1\right.}A_{(p)\left. \mu_2\ldots \mu_{p-2k-1}\right]}^{*}\right) .
P_{x\geq u}s^{u}e =\sum\limits_{n=1}^{\infty}P_{x\geq u}P_{x=ne}s^{u}e\leq\sum\limits_{n=0}^{\infty}P_{ne\geq u}P_{x=ne}s^{n}e & =P_{ne\geq u}\sum\limits_{n=0}^{\infty}P_{x=ne}s^{n}e=P_{ne\geq u}x^{s}.
\sigma^{; \alpha \beta}= g^{\alpha \beta}- \frac{1}{3}R_{~ ~ \gamma ~ ~ \delta}^{\alpha ~ ~ \beta ~ ~}\sigma^{; \gamma}\sigma^{; \delta}+ \frac{1}{1 2}R_{~ ~ \gamma ~ ~ \delta ; \rho}^{\alpha ~ ~ \beta ~ ~}\sigma^{; \gamma}\sigma^{; \delta}\sigma^{; \rho}+ O ( \sigma^{2}) ~ .
\eta ( x_{1}, x_{2}) = \int{\frac{d^{d}p_{1}}{( 2 \pi )^{d / 2}}}\int{\frac{d^{d}p_{2}}{( 2 \pi )^{d / 2}}}e^{i ( p_{1}x_{1}- p_{2}x_{2})}\eta ( p_{1}, p_{2})
\begin{aligned}F_{(1)}\left(z,G_{(1)}^Y(z)\right) =0\ ,F_{(1)}\left(G_{(1)}^Y(z)-z,G_{(1)}^Y(z)\right)=0\ , &F_{(2)}\left(z,G_{(2)}^Y(z)\right) =0\ ,F_{(2)}\left(G_{(0)}^Y(z)-z,G_{(0)}^Y(z)\right)=0\ .\end{aligned}
\|u_0\|_{X_h(0)}= h^{-\frac12}\|u_0\|_{L^2}^2 + h^{\frac{15}2}\|\partial_x^4u_0\|_{L^2}^2 + h^{-\frac92}\|y^2\partial_xu_0\|_{L^2}^2 + h^{-\frac12}\|(x\partial_x + y\partial_y)u_0\|_{L^2}^2.
\int_{\mathfrak{m}}\big|S(\alpha)\big|d\alpha \le \mathcal{I}_1^{1/2}\mathcal{I}_2^{1/2}\sup_{\substack{\alpha\in\mathfrak{m}\boldsymbol{\beta}\in [0,1]^3}}\Big|\prod_{k\not=i,j}f_k(\alpha,\boldsymbol{\beta})\Big|.
MPSK_{BER}=\frac{2}{\max(\log_2 M,2)}\sum_{k=1}^{\min(2,[M/4])}\left[\frac{1}{2}(1-\mu_k)\right]^{U}&.\sum_{\ell=1}^{U-1}\begin{pmatrix}U-1+\ell \ell\end{pmatrix}\left[\frac{1}{2}(1+\mu_k)\right]^\ell
\begin{aligned}\varphi(t;z) = \varphi(0;z) + \dot{\varphi}(0;z) t + \int_{0}^t (t-s) \ddot{\varphi}(s;z) \,ds &=z+X(z) t + \int_{0}^t (t-s)DX(\varphi(s;z)) X(\varphi(s;z))\, ds .\end{aligned}
\mathbf{E}\left[\mathbb{Y}_\alpha(n_1)^{m_1}\cdots \mathbb{Y}_\alpha(n_k)^{m_k}\right] = \int_{\Omega_2}\prod_{j=1}^{k}\mathbb{Y}_\alpha(n_j)^{m_j}\,d\mathbf{P}_2 &= \prod_{j=1}^{k}\int_{S_1}\omega_{n_j}^{m_j}\,d\mathbf{m}(\omega_{n_j})
t^{2}=\frac{n_{x}n_{y}}{(n_{x}+n_{y})(1-r^{2})}[(\frac{d_{1}}{s_{1}})^{2}+(\frac{d_{2}}{s_{2}})^{2}-2\rho(\frac{d_{1}}{s_{1}})(\frac{d_{2}}{s_{2}})]
\delta_{\hat{\chi}}\hat{L}_{\hat{\mu}_{1}\ldots \hat{\mu}_{r}}= m \hat{\lambda}_{\hat{\mu}_{1}}\hat{k}^{\hat{\nu}}\hat{L}_{\hat{\nu}\hat{\mu}_{2}\ldots \hat{\mu}_{r}}+ \ldots + m \hat{\lambda}_{\hat{\mu}_{r}}\hat{k}^{\hat{\nu}}\hat{L}_{\hat{\mu}_{1}\ldots \hat{\mu}_{r - 1}\hat{\nu}}.
[ V_{\alpha}, V_{\beta}] = \epsilon_{\alpha \beta}^{\! \gamma}V_{\gamma}, \qquad [ V_{\alpha}, V_{m}^{a}] = V_{m}^{b}( \sigma_{\alpha})_{b}^{~ a}, \qquad \{V_{m}^{a}, V_{m}^{b}\}= - m^{2}( \sigma^{\alpha})^{a b}V_{\alpha}
\|f\|_{L^p_k(\mathbb{R}^{n-1})}:=\|f\|_{L^p(\mathbb{R}^{n-1})}+\sum_{|\alpha|\le k}\|\partial^\alpha f\|_{L^p(\mathbb{R}^{n-1})},\forall\,f\in L^p_k(\mathbb{R}^{n-1}).
4 \sqrt{3}\pi^{3}\int_{0}^{\infty}\left( \sum_{n, s=1}^{\infty}\chi_{-3}(n) s^{2}e^{-2\pi u ns}\right)\left( \sum_{k, r=1}^{\infty}\frac{\chi_{-3}(kr)}{k}\left(e^{-\frac{2\pi kr}{9u}}- e^{-\frac{2\pi kr}{3u}}\right) \right) udu.
\gamma_2 = \frac{1}{|G|}\left( \sum\limits_{g\in G_{n-1}}\frac{-\mu_1(g)}{\big( \mu_1(g) - 1\big)^2}+ \sum\limits_{g\in G_{n-2}}\frac{1}{\big( \mu_1(g) - 1\big)\big( \mu_2(g) - 1\big)}\right),