harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d s^{2}= 2 d X^{+}d X^{-}- \mu^{2}\sum_{i = 1}^{8}( X^{i})^{2}( d X^{+})^{2}+ \sum_{i = 1}^{8}( d X^{i})^{2}+ ( d X^{9}+ \mu X^{8}d X^{+})^{2}.
\frac{[\mathrm{Td\,}(\nabla^{T^{1,0}U_j},T^{1,0}U_j)\wedge\mathrm{ch\,}(\nabla^{L^m},L^m)]_{2n}(z_j)}{dv_{U_j}(z_j)}=\frac{[\mathrm{Td_b\,}(\nabla^{T^{1,0}X},T^{1,0}X)\wedge e^{-m\frac{d\omega_0}{2\pi}}\wedge\omega_0]_{2n+1}(x_0)}{dv_X(x_0)},
2\Im(z_{+}) =2\sqrt{1-\left(\frac{\alpha(1+\lambda^{2})-(1-\lambda^{2})}{2\lambda\alpha}\right)^{2}}=\frac{\sqrt{(1-\lambda^{2})(\alpha(1+\lambda)-(1-\lambda))((1+\lambda)-\alpha(1-\lambda))}}{\lambda\alpha}=\frac{1-\lambda^{2}}{\lambda\alpha}\sqrt{\left(\alpha-\alpha_{0}\right)\left(\alpha_{0}^{-1}-\alpha\right)}.
\widetilde u (t,y) = Q^{-1}(t)u(t,X(t,y)), \widetilde \pi(t,y) = \pi(t,X(t,y)) \widetilde \ell(t) = Q^{-1}(t) \dot a(t), \widetilde \omega(t) = Q^{-1}(t) \omega(t),
\mathcal{P}(Q'; u):=\bigcup_{(x_{0},t_{0})\in Q'}\mathcal{P}((x_{0}, t_{0}); u) &= \left\{(p,h): \exists\,(x_{0},t_{0})\in Q',\,s.t. \min_{(x,s)\in Q,\,s\leq t_{0}}u(x,s)-p\cdot x=u(x_{0}, t_{0})-p\cdot x_{0}=h\right\}.
e_2(b) = \begin{psmallmatrix}0 0 0 & 1 \end{psmallmatrix}\otimes\begin{psmallmatrix}0 0 1 & 1 \end{psmallmatrix}\otimes\begin{psmallmatrix}0 1 1 & 1 \end{psmallmatrix}\otimes\begin{psmallmatrix}0 1 1 & 1 \end{psmallmatrix}\,.
\begin{aligned}\Bigl|\Bigl(\mu_\tau - \sum_j \delta_{x_j}/l\Bigr)(f \circ t_i^{m,n})\Bigr| < \varepsilon/3, \Bigl|\Bigl(\mu_{\tau'}- \sum_j \delta_{x_j'}/l\Bigr)(f \circ t_i^{m,n})\Bigr| < \varepsilon/3\end{aligned}
x(s;t,x(t;t_0,x,\lambda \xi ), \xi(t;t_0,x,\lambda \xi)) =x(s;t_0, x,\lambda \xi ), \xi (s;t,x(t;t_0,x,\lambda \xi ), \xi(t;t_0,x,\lambda \xi)) =\xi (s;t_0, x,\lambda \xi )
V_{9}=-\varepsilon\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})(\mu^{2})\sum_{i=1}^{d}\sum_{j=1}^{d}\left[\Theta_{p_{i}p_{j}}(t,x,\mu^{2},Dw^{2})\left(\frac{\partial^{2}(w^{1}-w^{2})}{\partial x_{i}\partial x_{j}}\right)\right]\dx,
K_{d i v}( t ) = \frac{2 V \pi^{\frac{m - 1}{2}}}{( 2 \pi )^{m - 1}\Gamma \left( \frac{m - 1}{2}\right)}\int_{0}^{\infty}d k k^{m - 2}e^{- ( M^{2}- S^{2}+ k^{2}) t}.
J=\int \limits_0^\infty \int \limits_o^\infty \frac{d\rho}{\rho}\frac{d\delta}{\delta}e^{-1/\rho -\rho /\delta}G (w_1,w_3,\rho ,\delta ) \cdot e^{-\rho\cdot w_1-\rho k\cdot w_3-\rho \delta k^2\cdot w_1w_3}
I_{s,N}\left(\alpha\right)\sim\sum_{n\in\mathbb{Z}}c_{n}e\left(n\alpha\right)\qquad\mathrm{where}c_{n}\coloneqq\begin{cases}\sin\left(2\pi ns/N\right)/\left(\pi n\right) \mathrm{if\,}n\neq0, 2s/N \mathrm{if\,}n=0,\end{cases}
\frac{d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}+3R^{4}=(\frac{d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}+R^{2})^{2}.
\psi\circ\pi\circ\Phi_{i}(q,p)=\psi(q)+ip^{k}\frac{\partial\psi}{\partial x^{k}}(q)-\frac{1}{2}p^{j}p^{k}\frac{\partial^{2}\psi}{\partial x^{j}\partial x^{k}}(q)+O(\left\vert p\right\vert^{3}),
\omega_{i}^{j}[u^{-}]\ :\R_{i}[u^{-}](\omega_{i}^{j}[u^{-}]) \in Z_{i}^{j}, j=1,\dots,J_{i},\omega_{i}^{0}[u^{-}]:=-\infty,\ \omega_{i}^{J_{i}+1}[u^{-}]:=+\infty
\bar{\mskip - 3 m u \bar{B}}( r ) = C_{2}+ D_{2}| \ln r | + g | \ln r | \biggl [ \int_{r}^{1}r^{\prime}| V ( r^{\prime}) | \bar{\mskip - 3 m u \bar{B}}( r^{\prime}) d r^{\prime}\biggr ] ,
Q^s_{\Sigma}(\phi, \phi) = Q_{\Sigma}(\phi, \phi) + s\int_{\Sigma}|A_{\Sigma}|^2\phi^2 = \int_{\Sigma}|\nabla^{\Sigma}\phi|^2 -{\big(}(1-s)|A_{\Sigma}|^2 + Ric(\nu, \nu){\big)}\phi^2
\widetilde{\mathcal K}(s,y) =\frac{i\zeta_1(0)}{|y|}\int_0^\infty\int_0^{2\pi}e^{i(|y|r+sr)}\zeta_2(\varphi)\zeta_3(r)r\,d\varphi dr &- \frac{i}{|y|}\int_0^\infty\int_0^{2\pi}\int_0^\pi e^{i(|y|r\cos\omega+sr)}\zeta_1'(\omega)d\omega \zeta_2(\varphi)\zeta_3(r)r\,d\varphi dr.
H_{nm}^{\{{\bf R}\}}(0,0)\approx\frac{(-1)^{\theta_{nm}}}{\sqrt{2\pi}}\Gamma\left(\frac{n+m+1}2\right)R_{11}^{n/2}R_{22}^{m/2}\left(\frac{|R_{12}|}{\sqrt{R_{11}R_{22}}}+1\right)^{\frac{n+m+1}2},
\int d^{d}x\left\{f+u^{\dagger}\Gamma (\tau \gamma_{0}\gamma^{i}\partial_{i})u+u^{\dagger}\Xi (\tau \gamma_{0}\gamma^{i}\partial_{i}){v}+{v}^{\dagger}\Pi (\tau \gamma_{0}\gamma^{i}\partial_{i})u+{v}^{\dagger}\Upsilon (\tau \gamma_{0}\gamma^{i}\partial_{i}){v}\right\}.
\begin{cases}\partial_t A-\varepsilon d_A\Delta A = AH_A\big(x,A,F_B(1)-\mathcal B\big), 0 < x < 1, \; t \geq 0, begin{align*}5pt] \partial_t \mathcal B-\varepsilon d_B\Delta \mathcal B = -(F_B(1)-\mathcal B)H_B\big(x,A,F_B(1)-\mathcal B\big),\end{cases}
\begin{gathered}\mathcal{X}_\epsilon ' = \int_0^t \frac{d}{d\epsilon}\mathcal{V}_\epsilon ds,\,\, \pi_\epsilon = \int_0^t \frac{d}{d\epsilon}\mathcal{T}_\epsilon ds + \sigma_{\epsilon, 0}', \mathcal{V}_\epsilon = \mathcal{V}(X_\epsilon, \tau_\epsilon), \, \mathcal{T}_\epsilon = \mathcal{T}(X_\epsilon, \tau_\epsilon). \end{gathered}
\frac{d^{i+1}\tilde{u}}{dv^{i+1}}=\left(\frac{d\tilde{a}_{i-1}}{dv}+\tilde{a}\tilde{a}_{i-1}\right)\tilde{u}+\frac{d\tilde{b}_{i-1}}{dv}+\tilde{a}_{i-1}\tilde{b},
\alpha^{(2n+1)}(z) := Q_{2,n}^*(\bar z)Q_{2,n}^{*-1}(a), \beta^{(2n+1)}(z) := -Q_{1,n+1}^*(\bar z)P_{1,n+1}^{*-1}(a), \gamma^{(2n+1)}(z) := -(b-z)(z-a)P_{2,n}^*(\bar z)Q_{2,n}^{*-1}(a), \delta^{(2n+1)}(z) := P_{1,n+1}^*(\bar z)P_{1,n+1}^{*-1}(a)
\tilde{\Lambda}^{2 \ell_{c}}={\frac{\Lambda_{1}^{2 n_{c}}\, \Lambda_{2}^{2 n_{c}}}{\det ( A_{m_{c}}^{2}) \det ( v^{2}I_{m_{c}}+ | A_{m_{c}}|^{2}) \, v^{2 \ell_{c}}}}.
e^{-1}{\cal L}= -{1\over 2}R -{1\over 4}G_{IJ}F_{\mu\nu}{}^IF^{\mu\nu J}-{1\over 2}g_{ij}\partial_{\mu}\phi^i \partial^\mu \phi^j+{e^{-1}\over 48}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^IF_{\rho\sigma}^JA_\lambda^K
\tilde{\Gamma}_{\alpha\beta}^{\mu}=\Gamma_{\alpha\beta}^{\mu}+\frac{1}{2}g^{\mu\nu}k^{2}\phi^{2}(A_{\alpha}F_{\beta\nu}+A_{\beta}F_{\alpha\nu}-A_{\alpha}A_{\beta}\partial_{\nu}ln\phi^{2}).
\begin{aligned}\alpha_{0, 1}=2\,c_2 \overline{v}_{xxx}-6\,c_3 \overline{v}_x, &\alpha_{0, 2}=\partial_x L_{\overline{\omega}}[\beta_2]-3 \partial_x [(\beta_1)_x \,(\beta_1)_{xxx}]-3 \partial_x [(\beta_1)_{xx}^2]+a_{0, 1}(\beta_1)_{xx}+(a_{0, 2})_x.\end{aligned}
\sum_{n = 1}^{\infty}\frac{1}{\left( 2 n \atop n \right)}\frac{1}{n^{c}}\prod_{a , b , i , j}\left[ \sum_{m = 1}^{n - 1}\frac{1}{m^{a}}\right]^{i}\left[ \sum_{k = 1}^{2 n - 1}\frac{1}{k^{b}}\right]^{j}
\mathcal{E}= O\!\left(\frac{1}{n}\right) O\!\left\Vert \!\left( \begin{matrix}\Psi_{12}\Psi_{22}\big(n^2f(z)\big)\!\! -\Psi_{12}^2\big(n^2f(z)\big) \Psi_{22}^2\big(n^2f(z)\big)\!\! -\Psi_{12}\Psi_{22}\big(n^2f(z)\big)\end{matrix}\right)\! \left(\frac{F^2}{w_+}(z) + \frac{F^2}{w_-}(z) - 2 \right)\! \right\Vert_{L^1(1, 1+1/n)},
\int \prod_{j = 1}^{g}d \nu_{j}d \bar{\nu}_{j}d \hat{\nu}_{j}d \hat{\bar{\nu}}_{j}\, \, \mathrm{e}^{\int_{\Sigma_{g}}{\cal L}_{5}}= \det \left({\frac{1}{\Omega - \bar{\Omega}}}\bar{\partial}\bar{\Omega}{\frac{1}{\Omega - \bar{\Omega}}}\partial \Omega \right) ,
u_{1}:=(s_{l-r}\cdots s_{\sigma_{x}(t_{1})})(s_{l-r+1}\cdots s_{\sigma_{x}(t_{2})})\cdots (s_{l-1}\cdots s_{\sigma_{x}(t_{r})}), u_{2}:=(s_{m+r-1}\cdots s_{\sigma_{y}(t_{r})-1})(s_{m+r-2}\cdots s_{\sigma_{y}(t_{r-1})-1})\cdots (s_{m}\cdots s_{\sigma_{y}(t_{1})-1}).
\det\left({A}_t(x,x')_{ij}\right)_{i,j=1}^{N+1}=\mathcal{P}^{(N+1)}_{s}(t)(x,x'), \det\left({D}_t(y,y')_{ij}\right)_{i,j=1}^{N}=\hat{\mathcal{P}}^{(N)}_{s}(t)(y,y').
{\cal H}_{T}= \frac{{\bf \pi}^{2}+{\bf B}^{2}}{2}+ ({\bf \theta}\cdot{\bf B}) \frac{{\bf B}^{2}-{\bf \pi}^{2}}{2}+ \left({\bf \theta}\cdot{\bf \pi}\right) \left({\bf \pi}\cdot{\bf B}\right)
\frac{N_n(|\eta|)}{n!}=\frac{m(m+n)^{n-1}}{n!}<\frac{m(m+n)^{n-1}}{n^ne^{-n}\sqrt{2\pi n}}= \frac{m}{m+n}\frac{1}{\sqrt{2\pi n}}e^n(1+\frac{m}{n})^n<e^{m+n}.
( \pi ({\vec{x}}) )^{*}\pi ({\vec{x}}) = \sum_{\vec{n}, \vec{m}}\frac{1}{X^{3}}{\sqrt{\kappa_{\vec{n}}\kappa_{\vec{m}}}}( q_{\vec{n}}^{*}q_{\vec{m}}) e x p \left( 2 \pi i ({\vec{n}}-{\vec{m}}) \cdot{\vec{x}}/ X \right) .
\nu(\gamma_1(t_1)-\gamma_1(t_2)) =\nu\int_{t_2}^{t_1}\gamma(s)ds\geq \nu(2\pi)^2 \int_{t_2}^{t_1}(\eta-ks)^2ds &\geq \nu(2\pi)^2k^2(t_1-t_2)^3/12\geq (a+1)\nu^{1/3}(t_1-t_2)-C,
w_{1}\left( x\right) :=(1-ztx_{1})^{-1-\kappa}\left( 1-ztx_{2}\right)^{-\kappa}(1-z^{-1}tx_{2})^{-1-\kappa}\left( 1-z^{-1}tx_{1}\right)^{-\kappa}, w_{2}\left( x\right) :=(1-ztx_{2})^{-1-\kappa}\left( 1-ztx_{1}\right)^{-\kappa}(1-z^{-1}tx_{1})^{-1-\kappa}\left( 1-z^{-1}tx_{2}\right)^{-\kappa},
(\tilde \gamma^{-1})^i_j=(\tilde\gamma^{-1})^{ik}\delta_{kj}=\left( \begin{array}{ccc}\gamma/(\gamma^2+B_0^2) -B_0/(\gamma^2+B_0^2) 0 B_0/(\gamma^2+B_0^2) \gamma/(\gamma^2+B_0^2) 0 0 0 \gamma^{-1}\end{array}\right)
L_{1}=I-F_{1}^{\ast}V\widehat{F}V^{\ast},L_{2}=-(I+F_{3}^{\ast}V)\widehat{F}V^{\ast},L_{3}=F_{1}^{\ast}V\widehat{F},L_{4}=(I+F_{3}^{\ast}V)\widehat{F}, R_{1}=I-U\widehat{E}U^{\ast}E_{1}^{\ast},R_{2}=-\widehat{E}U^{\ast}E_{1}^{\ast},R_{3}=U\widehat{E}(I-U^{\ast}E_{2}^{\ast}),R_{4}=\widehat{E}(I-U^{\ast}E_{2}^{\ast}).
\tau{}_{m}^{(1)}=\alpha{}_{2m}\alpha{}_{2m+1}=\frac{(2m)^{2}(2m+1)^{2}(x^{2}-(2m))(x^{2}-(2m+1)^{2})}{16(4m-1)(4m+1)^{2}(4m+3)}\sigma_{m}^{(1)}=\alpha{}_{2m+1}+\alpha{}_{2m+2}=\frac{(2m+1)^{2}(x^{2}-(2m+1)^{2})}{4(4m+3)(4m+1)}+\frac{(2m+2)^{2}(x^{2}-(2m+2)^{2})}{4(4m+5)(4m+3)}.
\left(\textsc{Q},\tilde{L}\left(t,\textbf{x},\textbf{y},\dot{\textbf{x}},\dot{\textbf{y}}\right),\{\dot{x}_\alpha-\Phi_\alpha\left(t,\textbf{x},\textbf{y},\dot{\textbf{y}}\right)=0,\alpha=1,\ldots,s_1\}\right),
\| N(u(|\cdot|)) - N(v(|\cdot|) ) \|_{H^{m_d}}& \lesssim \big ( \|u(|\cdot|) \|_{H^{m_d}}+\|v(|\cdot| ) \|_{H^{m_d}}\big ) \| u(|\cdot|) - v(|\cdot|) \|_{H^{m_d}}
\begin{gathered}\left\{(\widehat\rho_j)_{X'}:X'\to X_{x^{-1}(\delta(j))}\right\}_{j=1}^n\ , \left\{(\widehat\rho_j)_{X''}:X''\to X_{x^{-1}(\delta(j))}\right\}_{j=1}^n\ ,\end{gathered}
\sigma_{{a b}}^{{[ - 2 ]}}\equiv ( \sigma^{{0}}- \sigma^{{9}})_{{a b}}= \mathrm{~{\it ~ d i a g}}( 0 , 2 \delta_{{\dot{A}}{\dot{B}}}) = ( \tilde{\sigma}^{{0}}+ \tilde{\sigma}^{{9}})^{{a b}}= \tilde{\sigma}^{{[ + 2 ] a b}},
g^{i j}= \frac{e^{2}L N}{4 \pi T}\left( \begin{matrix}{N - 1}&{- 1}&{\ldots}&{- 1}\\{- 1}&{N - 1}&{\ldots}&{- 1}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{- 1}&{- 1}&{\ldots}&{N - 1}\\ \end{matrix}\right) .
\begin{aligned}\delta_{2r}(c) =I_1\times\cdots\times I_{i_r-1}\times[m_{i_r}+1]\times I_{i_r+1}\times\cdots\times I_{k}, \delta_{2r-1}(c) =I_1\times\cdots\times I_{i_r-1}\times[m_{i_r}]\times I_{i_r+1}\times\cdots\times I_{k}.\end{aligned}
Q^{(n)}(w|\zeta )^{\varepsilon}=\prod_{a\in A}\zeta_a \left( \prod_{j=1}^{a-1}(xz_j -w_a )\prod_{j=a+1}^{2n}(xw_a -z_j ) \right) \left/ \prod_{a,b\in A\atop a<b}(x^{-1}w_b -xw_a ). \right.
H_{\ell'}^?, S_{p_{i'},q_{i'},r_{i'}}, \Phi, S_{p_k,q_k,r_k}, z = \Psi_1, \Psi_2; & \vdash \Psi_1 \to S_{p_{k'},q_{k'},r_{k'}}; & \vdash \Psi_2 \to z.
\begin{gathered}R_T(x,U) - n = R_T(T^n x, U) = R_T(x, T^{-n}U), \frac{R_T(x,U)}{n}= R_{T^n}(x,U), R_{T^{n_1}\times \cdots \times T^{n_d}}\big((x_1, \ldots, x_d), U_1 \times \cdots \times U_d \big) = \bigcap_{i=1}^d R_{T^{n_i}}(x_i,U_i).\end{gathered}
R = \begin{bmatrix}-\beta(0) \beta(0) 0 0 \delta(1) -\beta(1)-\delta(1) \beta(1) 0 \vdots 0 \delta(2) -\beta(2)-\delta(2) \beta(2) \vdots 0 0 \delta(3) -\beta(3)-\delta(3) \cdots \cdots \ddots\end{bmatrix}.
\|x_j\|_2^2=\Big(\Big\|\sum_{\substack{k=1 k\ne j}}^t P_{jk}x_k\Big\|_2\Big)^2 \le \Big(\sum_{\substack{k=1 k\ne j}}^t \|P_{jk}\|_2 \|x_k\|_2\Big)^2 \le (t-1) \sum_{\substack{k=1 k\ne j}}^t \|P_{jk}\|_2^2 \|x_k\|_2^2.
\frac{3q-7}{6(q-1)}\|\nabla u(t)\|^2_{L^2}+\frac{p-q}{(p+1)(q-1)}\|u(t)\|^{p+1}_{L^{p+1}}+\frac{\omega}{2}\|u(t)\|^2_{L^2}&=I_\omega(u(t)) = S_\omega(u(t)) -\frac{2}{3(q-1)}G(u(t)) \leq S_\omega(u(t)) <m_\omega, \forall t\in I_{\max}.
\left|u_{k}\right| \rightarrow{2^{\nu - 3/2}\over \sqrt{2 k}}{\Gamma\left(\nu\right) \over \Gamma\left(3/2\right)}\left|C_{+}\left(k\right) + C_{-}\left(k\right)\right| \left({y \over 1 - \epsilon}\right)^{1/2 - \nu}.
{\cal I}_{a, J_t}^{-1}Y_t = \left({\cal I}_{a, J_t}^{-1}P^{J_{\tau_k}}_{t-\tau_k}{\cal I}_{a, J_{\tau_{k-1}}}\right) \left({\cal I}_{a, J_{\tau_{k-1}}}^{-1}S_k{\cal I}_{a, J_{\tau_{k-2}}}\right) \cdots \left({\cal I}_{a, J_{\tau_{1}}}^{-1}S_1{\cal I}_{a, J_{\tau_{0}}}\right) \left({\cal I}_{a, J_{0}}^{-1}P^{J_0}_{\tau_1}Y_0\right),
\Gamma_f^*\Gamma_f(b_mw_n) ={\bf P}_+\left(\overline{fb}\,b \Gamma_{f}(b_mw_n)\right)={\bf P}_+\left(\bar b{\bf P}_+(\bar f\Gamma_{f}(bb_mw_n))\right) &={\bf P}_+\left(\bar b\Gamma_f^*\Gamma_f(v_m)\right)=s_n^2(\Gamma_f){\bf P}_+\left(\bar bv_n\right)=s_n^2(\Gamma_f)b_mw_n.
< \rho_{-}> = \frac{k_{-}}{\sqrt{K}\alpha^{\prime}}, \; \; < P_{-}> = \frac{1}{2 \sqrt{K}\alpha^{\prime}}[ \frac{k_{-}^{2}- 2}{k_{-}}+ \frac{2}{k_{-}}\; \frac{E ( k_{-})}{K ( k_{-})}] ,
\varphi_{2n+1}(x;q^{-1})=q^{-n^{2}}\left(\mathcal{C}h_{q}(x)\mathcal{C}h_{q}(q^{n+1/2}x)-q^{1/2}\mathcal{S}h_{q}(x)\mathcal{S}h_{q}(q^{n+1/2}x)\right)
v_{xc,\alpha\beta}^{LDA}(r)=\frac{\delta E^{LDA}}{\delta\rho_{\alpha\beta}(r)}=\frac{1}{2}\delta_{\alpha\beta}\frac{\delta E^{LDA}[2\rho_{\alpha}]}{\delta\rho_{\alpha}}=-\delta_{\alpha\beta}(\frac{3}{\pi})^{1/3}2^{1/3}\rho_{\alpha}^{1/3}
\sum_{0 \leq a<b \leq 3}\frac{\kappa^2}{4}\int_{I^2 \times I^2}ds dt d\bar{s}d\bar{t}\e^{-\kappa^2|\sigma(s,t)-\sigma(\bar{s},\bar{t})|^2/8}|J_{ab}^\sigma|(s,t)|J_{ab}^\sigma|(\bar{s},\bar{t}) = 2\pi \sum_{0 \leq a<b \leq 3}\int_{I^2}\rho_\sigma^{ab}(s,t)|J_{ab}^\sigma|(s,t)\ds dt + O(\tilde{C}e^{-\kappa \bar{C}}),
\mathbb{E}\left \{\gamma^2 \right \}=\frac{\overline{\gamma}^2 \left( m(1+\mu)(1+2 \mathbb{E}\left \{\kappa_\theta \right \})+\mu \mathbb{E}\left \{\kappa_\theta^2 \right \}(1+m) \right)}{m \mu (1+K)^2}
\chi_{\ell}^{( \infty )}( r ) = H_{\nu}^{( 1 )}( k r ) \sim \sqrt{\frac{2}{\pi k}}e^{- i \frac{\pi}{2}( \ell - \frac{\Phi}{2 \pi}) - i \frac{\pi}{4}}\frac{e^{i k r}}{\sqrt{r}},
\liminf_{N \to \infty}\; \mathbb{E}\left[ \frac{1}{N}\displaystyle\sum\limits_{t = 1}^N \frac{1}{2}\log\left( 1 + g^\prime(t)\right) \right] & \ge \displaystyle\sum\limits_{j = 0}^{\infty}p(1-p)^j \frac{1}{2}\log(1 + p(1-p)^j E)
\vec{v}_k := \Big( a_1 = 2g - \frac{n+2+k}{2},\underbrace{1, \dots, 1}_{k}, \underbrace{\frac 12, \dots \frac 12}_{n-1-k}, \,a_{n+1}= \frac 3 2, \,a_{n+2}= -\frac 1 2 \Big).
\sum_{w\in D_n/D_{n-1}}\!\!\! w\cdot \frac{\prod_{1\le i\le 2n-3}\vartheta(z_n-y_i) \prod_{1\le i\le n}\vartheta(Y+z_i) \prod_{1\le i<n}\vartheta(Y-z_i)}{\prod_{1\le i<n}\vartheta(z_n+z_i)\vartheta(z_n-z_i)}\hphantom{\sum_{w\in D_n/D_{n-1}}\!\!\!}{}= \vartheta(2Y) \prod_{1\le i\le 2n-3}\vartheta(Y-y_i),
I_{g}= \frac{1}{2 \pi \kappa}\int d^{2}x \epsilon^{\mu \nu}\Bigl ( \eta_{a}( \partial_{\mu}e_{\nu}^{a}+ \omega_{\mu}\epsilon_{~ b}^{a}e_{\nu}^{b}) + \eta_{2}\partial_{\mu}\omega_{\nu}+ \eta_{3}( \partial_{\mu}a_{\nu}+ \frac{1}{2}\epsilon_{a b}e_{\mu}^{a}e_{\nu}^{b}) \Bigr ) \, .
g_{\mathrm{I^{\prime}}}\propto g_{\mathrm{h}}^{- 1 / 2}R_{\mathrm{9 h}}^{3 / 2}, \quad R_{\mathrm{9 I^{\prime}}}\propto g_{\mathrm{h}}^{1 / 2}R_{\mathrm{9 h}}^{1 / 2}\quad ,
a^{3}+2a^{2}(R_{F}-\sqrt{3\Gamma}r_{ZP})+aR_{F}(R_{F}-4r_{ZP}\sqrt{3\Gamma})=r_{ZP}\sqrt{3\Gamma})R_{F}^{2}[\frac{2S^{2}\Phi^{\prime2}+r_{ZP}^{2}(\Gamma-1)}{S^{2}\Phi^{\prime2}-0.5r_{ZP}^{2}}].
2 [ \dot{x}^{2}\acute{x}^{A A^{\prime}}- ( \dot{x}\acute{x}) \dot{x}^{A A^{\prime}}] = v \bar{\pi}^{A}\bar{\pi}_{B}\dot{x}^{B A^{\prime}}+ \bar{v}\pi^{A^{\prime}}\pi_{B^{\prime}}\dot{x}^{A B^{\prime}}.
\begin{array}{l}{x^{-}x^{0}= q x^{0}x^{-}~ , ~ ~ ~ ~ ~ x^{0}x^{+}= q x^{+}x^{0}~ ,}{x^{+}x^{-}- x^{-}x^{+}= ( q^{1 / 2}- q^{- 1 / 2}) x^{0}x^{0}~ .}\end{array}
H_k^{m-n}=8\lambda_n\lambda_mv\int\limits_0^{+\infty}\int\limits_0^{\min(1,\frac{R_{mk}}{R_n^{lb}})}[\sqrt{\frac{1-z^2}{(\frac{R_{mk}}{R_n^{lb}})^2-z^2}}+\sqrt{\frac{(\frac{R_{mk}}{R_n^{lb}})^2-z^2}{1-z^2}}]dz(R_n^{lb})^2\exp\{-\pi\sum\limits_{i=1}^N(\lambda_i(R_i^{lb})^2)\}dR_{mk}
\rho_{K^*}(u')u'v\leq h(K^*,v)=\rho_{K^*}(u)uv &\rho_{L^*}(u')u'v\leq h(L^*,v)=\rho_{L^*}(u)uv &\rho_{K^*}(u')u'v'=h(K^*,v')\geq \rho_{K^*}(u)uv' &\rho_{L^*}(u')u'v'=h(L^*,v')\geq \rho_{L^*}(u)uv'.
\mathrm{div}^{\overline{\nabla}}(T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W})=\mathrm{div}^{\nabla'}(T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W})-\overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W},T_{r}\overline{\nabla}_{\widehat{W}}\widehat{W}),
\|\Lambda_{k,g}[\sigma,\rho]\| \lesssim |\lambda \sigma^{n-3}\rho^{\gamma}|^{-\frac 1 4}\sigma^{\frac 3 8}\sigma^{\frac 3 8}(\rho\sigma)^{\frac 1 2}&= |\lambda \sigma^{n-8}\rho^{\gamma -2}|^{-\frac 1 4}.
0=\mathcal U(\Phi):=\partial_{s}\Phi+\left(\frac{1}{2}+\omega_{l}\right)\xi\partial_{\xi}\Phi-e^{2\omega_{l s}}\left(\partial_{\xi\xi}\Phi+\frac{d-1}{\xi}\partial_{\xi}\Phi-\frac{d-1}{2\xi^{2}}\sin(2\Psi)\right).
\Phi^{R_1\cdot R_2,\beta,D}_{p_1,\ldots,p_n}(\{\tau_i\}_{i=1}^{|Q|},Q)=\Phi^{(R_1,R_2),\beta_1+\beta_2,D_1\cup_{p}D_2}_{p_1,\ldots,p_n}(\{\tau_i\}_{i=1}^{|Q|},Q').
\mathrm{rank}_{\Lambda(G)}(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)) = \mathrm{rank}_{\Omega(G)}(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty) / p \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)) = 0
{\cal J}_{i}^{\mu}= \frac{- 1}{1 + 4{\Phi}^{2}}\biggl ( \left( \delta^{i j}+ 4 \Phi^{i}\Phi^{j}\right) \varepsilon^{\mu \nu}\partial_{\nu}\Phi^{j}+ 2 \varepsilon^{i j k}\Phi^{j}\partial^{\mu}\Phi^{k}\biggr ) .
G_{\perp \perp}= \frac{1}{2}\nu ( \nu - 1 ) \! \left( \frac{{a^{\prime}}^{2}}{a^{2}}- \frac{\dot{a}^{2}}{n^{2}a^{2}}- \frac{k}{a^{2}}\right) + \nu \! \left( \frac{n^{\prime}a^{\prime}}{n a}+ \frac{\dot{n}\dot{a}}{n^{3}a}- \frac{\ddot{a}}{n^{2}a}\right) .
\int_0^t p(u,x)du\geq \int_0^{1/\psi^*(|x|^{-1})}p(u,x)du\geq c^* \int_0^{1/\psi^*(|x|^{-1})}\frac{u \psi^*(|x|^{-1})}{|x|^d}du= \frac{c^*}{2|x|^d \psi^*(|x|^{-1})}\,.
\|e_h^{u^{m}}\|^2_{\mathcal{T}_h}+2 \Delta t\sum_{n=1}^{m}\|e_h^{\phi^{n}}\|^2_{\mathcal{T}_h}+ (\Delta t)^2\sum_{n=1}^{m}\|\partial_{t}^+ e_h^{u^{n}}\|^2_{\mathcal{T}_h}\le C(\epsilon)\big((\Delta t)^2+ h^{2(k+2)}\big).
G'(x) > \frac{1}{k(\log(x/k) - 1 - B(x/k))}\, \left( \sum_{j=1}^{m_2}\frac{j \cdot b_j}{\log^{j+1}x}- \sum_{i=1}^{m_1}\frac{i \cdot a_i}{\log^{i+1}x}\right)
\sum_{n\le x}\sum_{p\le Y}\nu_p(n)\log p = \sum_{n\le x}\sum_{p\le Y}\log p \sum_{\substack{j\ge1 p^j\mid n}}1 = \sum_{p\le Y}\log p \sum_{j\ge1}\sum_{\substack{n\le x p^j\mid n}}1 &\le \sum_{p\le Y}\log p \sum_{j\ge1}\frac x{p^j}= x \sum_{p\le Y}\frac{\log p}{p-1}\ll x\log Y
n_{P}=q^{2i+2}N(n,i)+\left(\mu_{n-i-3}(q^{2})q^{2i+4}\right)^{2}N(n-1,i+1) =\left[q^{2i+2}q^{2i+2}(q^{2}-1)\mu_{n-i-3}(q^{2})+\left(\mu_{n-i-3}(q^{2})q^{2i+4}\right)^{2}\right]N(n-1,i+1) =q^{4i+4}\mu_{n-i-3}(q^{2})\left[q^{2}-1+q^{4}\mu_{n-i-3}(q^{2})\right]N(n-1,i+1)\;.
B_{M L}= i ( - \pi )^{\omega}p^{-}( p^{2})^{\omega - 2}\sum_{n = 0}^{\infty}\frac{\Gamma ( 3 - \omega + n ) \Gamma ( \omega - 1 + n )}{\Gamma ( \omega + n ) \Gamma ( n + 2 )}( 1 - \eta^{n + 1})
\frac 1s-\frac 1{p}=\frac 1{q}-\frac 1{p}=\sum_{i=1}^m \Big(\frac1{q_i}-\frac1{p_i}\Big)=\frac1{q_m}-\frac1{p_m}=\frac 1{s_m}-\frac 1{p_m}
\alpha^t=h_1^{-t}\alpha^t=(h_1^{-1}\alpha)^t=(h_2h_1^{-1}h_2^{-1})^t=h_2h_1^{-t}h_2^{-1}=1\intertext{and}\alpha^p=\alpha^ph_2^p=(\alpha h_2)^p=h_1h_2^ph_1^{-1}=1,
= \; \frac{1}{\pi^{2}}\int d^{2}p \frac{1}{p^{2}+ m_{d}^{2}}\; \frac{1}{p^{2}}\; \sin^{2}( p_{1}L ) \; \bigg [ \frac{p_{2}}{p_{1}}e^{i p_{2}L}- \frac{p_{1}}{p_{2}}\Big ( 1 - e^{i p_{2}L}\Big ) \bigg ] \; = \; 0 \; .
D_{{\vec c},p-1}= D_{{\vec c}, 0}e^{-i2\pi\left(q\mid c \mid +N_{\Phi}c_{\gamma}\right)/\ell}e^{\frac{i\pi\tau}{\omega\kappa^2}\left(p-1\right)^{2}}e^{i\tau\beta N_{A}\left(p-1\right)/{\omega\kappa^2}}e^{-i\alpha N_{A}\left(p-1\right)/{\omega\kappa^2}}
(\nabla f) (\nabla \hat G(X)) = -p\,k^{p-1}(-\nabla \hat G (X))\, \, (\nabla k (-\nabla \hat G(X)) ) = \frac{-X}{h(X)}\, p \, k^{p-1}(-\nabla \hat G (x) ) = X p (-\beta)^{p-1}\, h^{[( \beta - 1 )(p - 1) - 1 ]}(X) \, k^{p-1}(\nabla h (X)) = X \, p\,\, (-\beta)^{p-1}h(X)^{-n}.
\|h\|^*_{2,q}:= \sup_{\underset{i=1,\dots, d}{(W,\varphi^i)\in\Omega_{L,q+2,d}}}\sum_{i=1}^{d}\int_M \partial_{x_i}h \, \varphi^i; \|h\|_{2,q}:= b\|h\|_{1,q}+\|h\|^*_{2,q}.
\frac{r\sqrt{2}}{R}\le\frac{1}{2}(sin\frac{A}{2}cos\frac{B}{2}+sin\frac{B}{2}cos\frac{C}{2}+sin\frac{C}{2}cos\frac{D}{2}+sin\frac{D}{2}cos\frac{A}{2})\le1
\int\limits_0^T{dt}\omega_{\mu \nu}[\dot x]\to \int\limits_0^T{dt\omega_{\mu \nu}}[\dot x^{cl}]+\frac{T}{2}\int\limits_0^T{dt[\ddot x_\mu}(t)\dot x_\nu(t) - \ddot x_\nu(t)\dot x_\mu(t)],
T \sum_{n = - \infty}^{\infty}\frac{1}{( ( ( 2 n + 1 ) \pi T )^{2}+ m^{2})^{p}}\sim \frac{m^{1 - 2 p}}{2 \sqrt{\pi}}\frac{\Gamma ( p - \frac{1}{2})}{\Gamma ( p )}- \frac{( 2 m T )^{1 - p}}{m \Gamma ( p )}e^{- \frac{m}{T}}+ \cdots
||\N^{(1)}_{(f,H_i),\leqslant k_i}(r)\geqslant \sum_{\underset{v\ne i,t}{v=1}}^{2n+2}N^{(1)}_{(f,H_v),\leqslant k_v}(r)-N^{(1)}_{(f,H_t),\leqslant k_t}(r)-2(N(r,S_i)+N(r,S_t))+o(T(r)).
{\Gamma^{0}}_{a}= \left\{\left( \begin{matrix}{\sigma_{3}}&{0}\\{0}&{\sigma_{3}}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{\sigma_{2}}&{0}\\{0}&{- \sigma_{2}}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{0}&{\sigma_{2}}\\{\sigma_{2}}&{0}\\ \end{matrix}\right) \right\}.
q_{a,b,c}=\frac{1}{12}(a-b)\Big(17(a^{3}+b^{3})+197ab(a+b)+12(a^{2}+b^{2}-16ab)c-222(a+b)c^{2}-168c^{3}\quad-362(a^{2}+b^{2})-1238ab+798(a+b)c+1674c^{2}+1075(a+b)-3774c+590\Big),
\mathbb{E}|X_N^{\mathrm{meso}}\left( (1-\phi_N)f \right)|^2 = 2 \int_0^{N\|f\|_{L^\infty}}x\mathbb{P}\left( \left|\sum_i (1-\phi_N)f(d_N\lambda_i)\right| \geq x \right)\,dx, &\leq N^2 \|f\|^2_{L^\infty(\mathbb{R})}\mathbb{P}\left( \left| \sum_i (1-\phi_N)f(d_N\lambda_i)\right| > 0 \right), &\leq N^2 e^{-N^c}\|f\|^2_{L^\infty(\mathbb{R})}\to 0;
\eta_{M, M-1}(q|a, b) = \exp\Bigl( \int\limits_0^\infty \frac{dt}{t}\Bigl[\frac{\bigl(\mathcal{S}_{M-1}e^{-xt}\bigr)(q\,|\,b)-\bigl(\mathcal{S}_{M-1}e^{-xt}\bigr)(0\,|\,b)}{\prod\limits_{i=1}^M (1-e^{-a_i t})}+ qe^{-t}\frac{\prod\limits_{j=1}^{M-1}b_j}{\prod\limits_{i=1}^M a_i}\Bigr]\Bigl),
f(w) = C^\frac1\alpha{\rm exp}\big[{-\frac{\alpha'}{\alpha}\frac{1}{\sqrt{e^{2s}-1}\sqrt{1-e^{-2s}}}\langle w, c \rangle + \frac{\alpha'}{\alpha}\frac{e^{-s}}{\sqrt{e^{2s}-1}\sqrt{1-e^{-2s}}}|c|^2}\big],
[D_t^2-iA\partial_{\alpha},\mathcal{H}]\partial_{\alpha}^k\tilde{\theta}=2[D_t\zeta,\mathcal{H}]\frac{\partial_{\alpha}\partial_{\alpha}^k\tilde{\theta}}{\zeta_{\alpha}}-\frac{1}{\pi i}\int \Big(\frac{\zeta(\alpha,t)-\zeta(\beta,t)}{\zeta(\alpha,t)-\zeta(\beta,t)}\Big)^2 \partial_{\beta}\partial_{\beta}^k \tilde{\theta}d\beta
\int_{x}^{\infty}\frac{t^{s-1}ln^{m}t}{e^{t}}dt=\frac{\partial^{m}}{\partial s^{m}}\int_{x}^{\infty}\frac{t^{s-1}}{e^{t}}dt=\frac{\partial^{m}}{\partial s^{m}}\Gamma(s,x)
\Big( \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)}\Big)^{1/2}\lesssim_{\lambda}C_2^{1/4}C_3^{1/4},
\|u\|_p^\gamma=C_5\, \|u\|_p^\gamma\left(\int_{|h|>2}\frac{dh}{|h|^{N+ps}}\right)^{\frac{\gamma}{p}}=\frac{C_5}{2^{\gamma/p}}\left(\int_{|h|>2}\frac{\|\delta_hu\|_p^p}{|h|^{ps}}\frac{dh}{|h|^{N}}\right)^{\frac{\gamma}{p}}\leq \frac{C_6}{2^{\gamma/p}}[u]_{s,p}^\gamma.
\langle f_{1}, T_{g}f_{2}\rangle = \langle f_{1}, P ( f_{2}g ) \rangle = \langle P f_{1}, f_{2}g \rangle = \langle f_{1}, f_{2}g \rangle = \langle f_{1}g^{*}, f_{2}\rangle &= \langle f_{1}g^{*}, P f_{2}\rangle = \langle P ( f_{1}g^{*}), f_{2}\rangle = \langle T_{g^{*}}f_{1}, f_{2}\rangle. \blacksquare
T_n(m) =\frac{(-1)^n}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^{2m}(t)\cos(2n t)dt &=\frac{(-1)^n}{\pi}\int_{0}^{\pi}\cos^{2m}(u)\cos(2n u)\cos(n\pi)du &=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\cos^{2m}(u)\cos(2n u)du &=\frac{1}{2^{2m}}\frac{\left(B(m+n+1,m-n+1)\right)^{-1}}{2m+1}=\frac{1}{2^{2m}}C_{2m}^{m-n},
\frac{1}{2}\left|z_1^\frac{1}{2}- n^\frac{1}{4}\right| = \frac{|z_1 - n^\frac{1}{2}|}{2|z_1^\frac{1}{2}+ n^\frac{1}{4}|}< \frac{\frac{z_2 - 1}{2}+ 0.652}{2|z_1^\frac{1}{2}+ n^\frac{1}{4}|}\le \frac{\frac{z_2 - 1}{2}+ 0.652}{2|\ell(z_2)^\frac{1}{2}+ \ell(\ell(z_2))^\frac{1}{4}|}.
\frac{1}{\varphi(M)}\sum_{\substack{q\leq\Omega (q,Mp_0)=1}}\frac{1}{\varphi(q)}\sum_{\xi\pmod{M}}\:\sum_{\psi\pmod{q}}\sum_{\substack{\lvert\gamma_{\psi^*\xi}\rvert\leq X^{1/2}\beta_{\psi^*\xi}>{1/2}}}\frac{X^{\beta_{\psi^*\xi}}}{\lvert\rho_{\psi^*\xi}\rvert}.
\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}S_{n}\left(x;q\right)z^{n}}{\left(cq;q\right)_{n}}=\frac{\left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}}\left(a;q\right)_{n}\left(-xz\right)^{n}}{\left(cq,az;q\right)_{n}}S_{n}\left(\frac{c}{xq^{n}};q\right).