harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal D}^{K}=\{{\cal D},\;\;\}_{\cal A}- t(\phi^a-tI_ag^{ab})\{I_a,\;\;\}_{\cal A}+\frac{t(1-t)}{2}f^c_{ab}c^ac^b\{I_a,\;\;\}_{\cal A}--tc^a \{Q_a,\;\;\}_{\cal A},
\sigma^K(x) = -\frac{1}{\pi i}\oint_{\Gamma_N}{\Bigg(\tilde \varphi(x, \mu)\varphi^K(x, \mu)-\frac{1}{2}\Bigg)\hat M(\mu)}d\mu - 2 \sum_{k=N+1}^{K}\sum_{j=0}^1 (-1)^j\alpha_{kj}\Bigg(\tilde \varphi_{k,j}(x)\varphi^K_{k,j}(x) - \frac{1}{2}\Bigg),
\begin{aligned}x = 2P + Q [ m_1 n_2 n_3 - m_2 n_3 n_1 - m_3 n_1 n_2 ] y = 2P + Q [ m_2 n_3 n_1 - m_3 n_1 n_2 - m_1 n_2 n_3 ] z = 2P + Q [ m_3 n_1 n_2 - m_1 n_2 n_3 - m_2 n_3 n_1 ] w = [m_1 n_2 n_3]^2 + [m_2 n_3 n_1]^2 + [m_3 n_1 n_2]^2 \end{aligned}
v(1,X_n)-v(0,0)=\sum_{j=0}^{n-1}\left(v(t_{j+1},X_{j+1})-v(t_j,X_{j+1})+v(t_j,X_{j+1})-v(t_j,X_j)\right) = \sum_{j=0}^{n-1}\left(v_t(\widehat t_j,X_{j+1})(t_{j+1}-t_j) +v_x(t_j,X_j)(X_{j+1}-X_j)+\frac{1}{2}v_{xx}(t_j,\widehat X_j)(X_{j+1}-X_j)^2\right),
{e^{v\over2}s_\alpha(u)\Big[e^{w\over2}s_\beta(z)(u+z)+e^{\beta z\over2}s(w)(m-w)\Big]+e^{\alpha u\over2}s(v)\Big[e^{w\over2}s_\beta(z)(p-v)-e^{\beta z\over2}s(w)n\Big]\over s(v-\alpha u) s(w-\beta z)}=0
I:= \big(1-\frac{\sigma^2}{2a}u(1-e^{-at})\big)^{-\frac{2a\theta}{\sigma^2}}\cdot \exp \Big (\frac{xue^{-at}}{1-\frac{\sigma^2}{2a}u(1-e^{-at})}\Big) II:= \exp \Big ( \int_0^t \int_0^{\infty}\Big(e^{\xi\psi(s,u)}-1\Big)\nu(d\xi)ds\Big)
{}\times \Bigg \{\sum_{\sigma \in Z_{N}}\frac{w ( x_{3}, x_{1 3}, x_{1}| d , h + \sigma ) w ( x_{4}, x_{2 4}, x_{2}| a , g + \sigma )}{w ( x_{8}, x_{5 8}, x_{5}| e , c + \sigma ) w ( x_{7}/ \omega , x_{6 7}, x_{6}| f , b + \sigma )}\Bigg \}_{0}
\sum_{\substack{(x,y)\in L_i^{s}\times L_{i+1}^{s}x<y}}w_{x,y}= \frac{|L_i|\left|L_{i+1}^{\leq s}\right|-\left|L_i^{\leq s-1}\right||L_{i+1}|}{|L_i||L_{i+1}|}.
p_{k}=\frac{a}{2}\left(\frac{\lambda_{k-1}^2}{\lambda_k}\right)+\frac{1}{2}\left(\frac{b}{\lambda_k}-c\right)-p_{k-1},~M_{1,k}=p_{k-1}-\frac{\alpha_k+b c\lambda_{k-1}}{2\lambda_k}-\frac{a\lambda_{k-1}^2}{2\lambda_k}.
\left[{\begin{array}{*{20}{c}}{{{\left|{{h_{11}}}\right|}^2}}{{{\left|{{h_{12}}}\right|}^2}}{{{\left|{{h_{21}}}\right|}^2}}{{{\left|{{h_{22}}}\right|}^2}}\end{array}}\right] = \left[{\begin{array}{*{20}{c}}{0.4475}{1.5760}{1.5406}{2.6081}\end{array}}\right].
\int_{M}a(x)\nabla_{g}W_{\delta_{\varepsilon}(t),\eta}\nabla\Phi_{\delta_{\varepsilon}(t),\eta}d\mu_{g}= -\int_{M}\nabla_{g}a(x)\nabla_{g}W_{\delta_{\varepsilon}(t),\eta}\Phi_{\delta_{\varepsilon}(t),\eta}d\mu_{g}-\int_{M}a(x)\Delta_{g}W_{\delta_{\varepsilon}(t),\eta}\Phi_{\delta_{\varepsilon}(t),\eta}d\mu_{g}
\frac{k_1^4+k_2^4}{(k_1^2+k_2^2+z^2)^4}= \frac{1}{(k_1^2+k_2^2+z^2)^2}- \frac{2 z^2}{(k_1^2+k_2^2+z^2)^3}+ \frac{z^4}{(k_1^2+k_2^2+z^2)^4}- \frac{2 k_1^2k_2^2}{(k_1^2+k_2^2+z^2)^4}, \frac{(k_1^2+k_2^2)^2}{(k_1^2+k_2^2+z^2)^4}= \frac{1}{(k_1^2+k_2^2+z^2)^2}-\frac{2z^2}{(k_1^2+k_2^2+z^2)^3}+\frac{z^4}{(k_1^2+k_2^2+z^2)^4}.
\lim_{\varepsilon\rightarrow0^{+}}\frac{1}{\varepsilon^{N}}\int_{Q_{\nu}^{+}( x_{0},\varepsilon)}\vert u(x) -u^{+}(x_{0})\vert dx=0,\qquad\lim_{\varepsilon\rightarrow0^{+}}\frac{1}{\varepsilon^{N}}\int_{Q_{\nu}^{-}(x_{0},\varepsilon)}\vert u(x) -u^{-}(x_{0})\vert dx=0,
A_{\alpha \beta}(\widetilde\phi^2) = - K^3\sum\limits_{l = 1}^\infty{\frac{{\left({- \ln K}\right)^l}}{{l!}}}\left\{{\frac{1}{2}g_{\alpha \beta}\sum\limits_{m =1}^{2l - 1}{\widetilde\phi^2_m}\phi_{2l - m}+ \delta_{\alpha,0}\delta_{\beta,0}\sum\limits_{m = 1}^l{\widetilde\phi^2_{2m - 1}\phi_{2l - 2m + 1}}}\right\}.
\rho^{\frac{\delta-1}{2}}-\overline{\rho}^{\frac{\delta-1}{2}}\in C([0,T_*]; H^3),\ \u\in C([0,T_*]; H^{s'}) \cap L^\infty([0,T_*]; H^3), &\rho^{\frac{\delta-1}{2}}\nabla^4 u \in L^2([0,T_*]; L^2),\ \u_t \in C([0,T]; H^1)\cap L^2([0,T] ; D^2),
x_1\!=\!\Big(1\!-\!\sqrt{1-c_{2n}^2}\Big)/2\!=\!c_{2n}^2\Big(2\big(1\!+\!\sqrt{1\!-\!c_{2n}^2}\big)\Big)^{-1},\,\,x_2\!=\!\Big(1\!+\!\sqrt{1\!-\!c_{2n}^2}\Big)/2.
(a)h=-\sum_{\alpha\in X}c_{\alpha}d_{h-\alpha}\varepsilon(h, \alpha)\alpha, &(b) \sum_{\gamma\perp h}\varepsilon(h, \gamma)\left\{\sum_{\alpha\in X}c_{\alpha}d_{h+\gamma-\alpha}\varepsilon(\alpha, h{+}\gamma)\right\}e^{\gamma}=0,
L_{geod}= p_\mu \dot{x}^\mu = \frac{\partial L_{geod}}{\partial \dot{x}^\mu}\dot{x}^\mu = p_i \dot{x}^i + p_t \dot{t}+ p_\sigma \dot{\sigma}= p_i \dot{x}^i + p_\sigma \dot{\sigma}+ \frac{p_t p_\sigma}{mc}.
\varepsilon_{k+1}= - i \left( R(\mathcal{V}_{k+1}^A) - R(\mathcal{V}_k^A) \right) + \varepsilon_k + e^{-(k+1) e_0 t}\left( \mathcal{L}-(k+1) e_0 \right) \mathcal{Z}_{k+1}^A & = -i \left( R(\mathcal{V}_{k+1}^A ) - R(\mathcal{V}_k^A) \right) + O\left( e^{-(k+2) e_0 t}\right).
\psi_{\mp}^{\mu}| p , \pm \rangle_{R R}^{0}= \psi_{\pm}^{i}| p , \pm \rangle_{R R}^{0}= 0 , \qquad \psi_{\pm}^{\alpha}\equiv{\frac{1}{\sqrt{2}}}\left( \psi_{0}^{\alpha}\pm i \tilde{\psi}_{0}^{\alpha}\right) .
U= \begin{pmatrix}0 1 0 \ldots 0 0 0 1 0 \ldots 0 \vdots \vdots \ddots \ddots \ldots 0 \vdots \vdots . 0 \ddots 0 . . . . 0 1 0 0 . . \ldots 0 \end{pmatrix}
k_v\bullet_v\bigl(\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t}\bigr):=\Bigl(\bigotimes_{e\in\mathcal{S}(v|s)}\textup{Ad}(k_v)X_e\Bigr)\otimes\Bigl(\bigotimes_{e^\prime\in\mathcal{S}(v|t)}\textup{Ad}(k_v)Y_{e^\prime}\Bigr).
(\psi_m\circ\varphi_{m-1}\circ\psi_{m-1}\circ\hdots\varphi_{n+1}\circ\psi_{n+1}) (\varphi_n(x)) =\psi_m(\varphi_n(x)) =\psi_m(\varphi_m(\varphi_{m,n}(x))) =\varphi_{m,n}(x).
\mathbb{E}(\mathbb{E}_{k-1}X_{n,k}\mathbb{E}_{j-1}X_{n,j}) =\mathbb{E}(X_{n,j}\mathbb{E}_{k-1}X_{n,k})=\mathbb{E}(X_{n,j-k+1}\mathbb{E}_{0}X_{n,1}) & =\mathbb{E}(\mathbb{E}_{0}X_{n,1}\mathbb{E}_{0}X_{n,j-k+1})=\mathbb{E}(X_{n,0}X_{n,j-k+2})
d e t \hat{K}={\frac{4 \coth^{2}({\frac{\omega}{2}}) \sinh \Bigl ( ( \nu - q_{+}){\frac{\omega}{2}}\Bigr ) \sinh ( q_{-}{\frac{\omega}{2}}) \sinh ({\frac{\omega}{2}})}{\sinh ({\frac{\nu \omega}{2}})}}\, .
\lambda_1Ap_2^{\lambda_1}+\frac{\exp((\mu-r)\delta)}{r-\mu}&>\lambda_1Ap_2^{\lambda_1}+\frac{\lambda_2-1}{\lambda_2-\lambda_1}\cdot\frac{\exp((\mu-r)\delta)}{r-\mu}&=\frac{1}{\lambda_2-\lambda_1}E'(p_2)>0.
\ell\left(\dfrac{V[[x_1, x_2]]}{I_n+(p^sx_1)}\right)= \ell\left(\dfrac{V[[x_1, x_2]]}{I_n+(p^s)}\right)+\ell\left(\dfrac{V[[x_1, x_2]]}{I_n+(x_1)}\right) \leq s \cdot \ell\left(\dfrac{V[[x_1, x_2]]}{I_n+(p)}\right)+\ell\left(\dfrac{V[[x_1, x_2]]}{I_n+(x_1)}\right).
(\alpha \gamma - \epsilon \beta \delta) + (-\alpha \delta + \beta \gamma ) \sqrt{\epsilon}= (\gamma - \delta \sqrt{\epsilon})(\alpha + \beta \sqrt{\epsilon}) &= (\gamma + \delta \sqrt{\epsilon})(\alpha - \beta \sqrt{\epsilon}) &= (\alpha \gamma - \epsilon \beta \delta) + (\alpha \delta - \beta \gamma) \sqrt{\epsilon}.
\Tilde{k}:= vec(\Tilde{K}) =[\Tilde{K}_{\bullet 1}^T \ldots \Tilde{K}_{\bullet n}^T]^T, \delta := \Big[\frac{a_{\bullet 1}}{\hat{c}_{11}}^T \ldots \frac{a_{\bullet n}}{\hat{c}_{nn}}^T \Big]^T.
\big[U^{(m)}{\tilde{\circ}}_{5}U^{(m)}\big]=\tilde{k}_p([\omega])\big(\big[H^{(2)}\big]-f_p([\omega])\big) \phantom{\big[U^{(m)}{\tilde{\circ}}_{5}U^{(m)}\big]=}{}+\tilde{l}_p([\omega]) \prod\limits_{i=1}^{p}([\omega]- h_{i,1})\prod\limits_{i=1}^{p-1}\big([\omega]- h_{mp+p+i,1}\big)\prod\limits_{i=1}^{m^2p}\bigg([\omega]-h_{p+\tfrac{i}{m},1}\bigg),
\bar{\Phi}= 2 \sqrt 2 r , \bar{F}_{t r}= Q e^{- 2 \sqrt 2 r}, \bar{T}= 0 ,{}\bar{G}_{\mu \nu}= \left( \begin{array}{c c}{{- f}}&{{0}}\\{{0}}&{{f^{- 1}}}\\ \end{array}\right) ,
{\rm Per}_{\mathbb{T}_1}(\tilde{A})={\rm Per}_{\mathbb{T}_1}(A),\,\,\,|\tilde{A}|=|A|,\,\,\,\lambda(\tilde{A})=\lambda(A)\,\,\,\,\,\,{\rm P}_E(\tilde{B})={\rm P}_E(A),
\geq \mathbb{E}(\liminf_{t \to \infty}\frac{1}{t}\sum_{j=0}^{t-1}\Pi_{C_\ell}(X_{C_1}(m),\ldots,X_{C_{\ell-1}}(m),\tilde{X}_{C_{\ell}}(m),X_{C_{\ell+1}}(m), \ldots,X_{C_{r}}(m))),
\sigma(\beta\alpha)_2(i) = (\beta\alpha)_2\sigma^{-1}(i) = \beta_4(\alpha_2(\tau^{-1}(i'))) + \beta_4^c(\beta_3(\tau^{-1}(i'))) &= \beta_4(\tau(\alpha)_2(i')) + \beta_4^c(\sigma(\beta)_3(i')) = (\sigma(\beta)\tau(\alpha))_2(i).
L_2^{\beta\gamma;\ (\lambda, \mu)}(z)=\lambda :\left(\partial_z\beta (z)\right)\gamma (z): +(\lambda +1):\beta (z)\left(\partial_z\gamma (z)\right): +\frac{\mu}{z}:\beta (z)\gamma (z): +\frac{(2\lambda +1)\mu -\mu^2}{2z^2},
K(h^{\frac{1}{2^{t-1}}},2)^{r_{t-1}}a\sharp_{\nu}b = a\nabla_{\nu}b - \sum_{k=0}^{t-2}r_{k}\big[\big(a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}\big)^{\frac{1}{2}}-\big(a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big)^{\frac{1}{2}}\big]^{2}= K(h^{\frac{1}{2^{t-1}}},2)^{R_{t-1}}a\sharp_{\nu}b,
(I^{[C]})^{[C]}= \{x\in A~ \mid ~\exists~ z\in I^{[C]}[x]\leq [z]\}&= \{x\in A~ \mid ~\exists~ z\in A y\in I [x]\leq [z]\leq [y]\}&= \{x\in A~ \mid ~\exists~ y\in I [x]\leq [y]\}= I^{[C]}
\int_0^1-\eqref{111.1a}\times\frac{1}{u_{\alpha\beta}}\bigg(\ln\frac{u_{\alpha\beta}^2}{\underbar{n}}\bigg)_-^{2k-1}dx,k=1,2,3,\cdots,\quad\ (\cdot)_-:=\min\{0,\cdot\}.
E_{\vec{z}}^{\vec{\mu}}(\vec{x}, \vec{\nu})=\lim_{\begin{subarray}{c}M \to \infty M' \to -\infty \end{subarray}}\prod_{i=1}^{k}\frac{z_{i}^{M'-1}}{(1+z_{i})^{M}}\langle \prod_{1\le i \le k}^{\curvearrowright}C_{\mu_{i}}^{[M', M]}(z_{i})\prod_{1\le i \le k}\beta_{\nu_{i}, x_{i}}^{*}\rangle_{[M', M]}.
Z^{( 0 )}\; = \; e^{- \frac{1}{2}\big ( \alpha^{( 1 )}, C_{m_{d}}\alpha^{( 1 )}\big )}\; \prod_{I = 2}^{N}e^{- \frac{1}{2}\big ( \alpha^{( I )}, C_{0}\alpha^{( I )}\big )}\; ,
\frac{P(Z_0(\Delta \ell_1)=n)}{P(W^k_{\ell_1\Delta,t}(j)=1,Z_0(\Delta \ell_1)=n)}= \frac1{P(W^k_{\ell_1\Delta,t}(j)=1|Z_0(\Delta \ell_1)=n)}&= \frac1{n\nu\Delta p^k_{j,+}(t-\Delta \ell_1)},
|I|=\Big|\frac{\lambda i(\dot{z}_1)^2}{2\pi}2x(t)\Big\{\frac{1}{(\zeta(\alpha,t)-z_1(t))^3(\zeta(\alpha,t)-z_2(t))}+\frac{1}{(\zeta(\alpha,t)-z_1(t))^2(\zeta(\alpha,t)-z_2(t)^2)}&+\frac{1}{(\zeta(\alpha,t)-z_1(t))(\zeta(\alpha,t)-z_2(t))^3}\Big\}\Big|,
\Phi_{1}\left(\left(\left\Vert x\right\Vert +\mu\right)\left|\left\Vert x\right\Vert +\mu\right|\left(1-\frac{\lambda^{2}}{c_{2}^{2}}\right)\right)=\Phi_{1}\left(\mu\left|\mu\right|\left(1-\frac{\lambda^{2}}{c_{1}^{2}}\right)\right).
{\frac{d^{2}y}{d \tilde{\lambda}^{2}}}-{\frac{1}{2}}{\frac{{\cal W}^{\prime}}{\cal W}}\left({\frac{d y}{d \tilde{\lambda}}}\right)^{2}-{\frac{1}{2}}\left[{\cal W}- \left({\frac{d y}{d \tilde{\lambda}}}\right)^{2}\right]{\cal W}^{- 1}\partial_{y}({\cal W}g_{\mu \nu}){\frac{d x^{\mu}}{d \tilde{\lambda}}}{\frac{d x^{\nu}}{d \tilde{\lambda}}}= 0 .
\Phi_1 (t) = \int^t_1 \tau^{-1}|v_1(\tau)|^2 \d\tau &= |\alpha|^2 \log t + \int^t_1 \tau^{-1}(|e^{3i\lambda_1 \Phi_1}v_1(\tau)|^2 - |\alpha|^2) \d\tau.
T_n \leq C_K \frac{\ln(n)^qw_n}{n}\sum_{i=1}^n \ \left( 1 + \frac{1}{X_{\frac{i-1}{n}}^q}\right) \sup_{(a,b) \in A, \; (\delta, \alpha) \in W_n^{(\eta)}}\left| h(z^n_i(\theta),\alpha) \right|{\bf 1}_{\Omega_K},
\lambda^{'}=(\overbrace{\alpha_{1},\cdots,\alpha_{1}}^{M_{1}},\overbrace{\alpha_{2},\cdots,\alpha_{2}}^{M_{2}},\cdots,\overbrace{\alpha_{s},\cdots,\alpha_{s}}^{M_{s}};\overbrace{\alpha_{1},\cdots,\alpha_{1}}^{N_{1}},\cdots,\overbrace{\alpha_{s},\cdots,\alpha_{s}}^{N_{s}})
V=\frac{\partial}{\partial\sigma^{2}}log[L(\sigma^{2},X)]=\frac{\partial}{\partial\sigma^{2}}[-log(\sqrt{2\pi\sigma^{2}})-\frac{(X-\mu)^{2}}{2\sigma^{2}}]=-\frac{1}{2\sigma^{2}}+\frac{(X-\mu)^{2}}{2(\sigma^{2})^{2}}
\langle \sigma_{I}| = \lim_{\widetilde{\xi}^{1}\rightarrow 0}\left( \frac{1}{\widetilde{\xi}^{1}}\right)^{2 h_{\sigma_{I}}}\, \langle 0 | ~ \sigma_{I}^{-}\left( - \frac{1}{\widetilde{\xi}^{1}}\right) ~ ,
\int_{\mathbb{R}^n}e^{-f(\vec{w})}\,d\vec{w}= \int_{\{\vec{w}\, : \, f(\vec{w}) \le 1 \}}e^{-f(\vec{w})}\,d\vec{w}\, + \, \sum_{j=1}^\infty \int_{\{\vec{w}\,:\, j \le f(\vec{w}) \le j+1\}}e^{-f(\vec{w})}\, d\vec{w}.
K(L,r) =\sqrt{\frac{\pi}2}\frac{(n-2)! 2^{n-2}}{\Gamma(n+1/2)}r^{2n-1}\Gamma(n+1/2) \sum_{k=0}^\infty\frac 1{(2L+2kL)^{n+1/2}}+\cdots =\frac{\sqrt{\pi}}8 (n-2)!\zeta(n+1/2) r^{2n-1}\frac 1{L^{n+1/2}}+\cdots
\begin{array}{l l}{{\begin{array}{l l}{{\frac{d}{d r}\sigma ( r ) = 0 ,}}&{{r = 0 ,}}\\{{\sigma ( r ) = 0 ,}}&{{r \rightarrow \infty ,}}\\ \end{array}}}&{{\begin{array}{l l}{{A ( r ) \neq 0 ,}}&{{r \rightarrow \infty ,}}\\{{A ( r ) = 1 ,}}&{{r = 0 .}}\\ \end{array}}}\\ \end{array}
\div \big(x_2^a\nabla u(x)\big) = &\sum_{k=0}^{m-1}\Big((m-2k)(m-2k-1)\, \alpha_k+ 2(k+1)(2k+1+a)\,\alpha_{k+1}\Big)\,x_1^{m-2k-2}x_2^{2k+a}=0
(\left\|\overline{\nabla}f\right\|^{2}-\partial_{t}f)^{2}=\frac{1}{\alpha^{2}}\left(\frac{G}{t_{0}}\right)^{2}+\frac{2(\alpha-1)}{\alpha^{2}}\left\|\overline{\nabla}f\right\|^{2}\left(\frac{G}{t_{0}}\right)+\frac{(\alpha-1)^{2}}{\alpha}\left\|\overline{\nabla}f\right\|^{4}.
\int_0^s|f(\xi)R(\xi)|^{p/2}d\xi=\int_0^s|f(\xi)|^{p/2}\cdot|R(\xi)|^{p/2}d\xi\le\left(\int_0^s|f(\xi)|^pd\xi\right)^{1/2}\left(\int_0^s|R(\xi)|^pd\xi\right)^{1/2},
\begin{array}{l c r}{R e g i o n I \qquad}&{\psi^{b}= s \left( \begin{array}{c}{E + m}\\{- \kappa}\\ \end{array}\right) e^{\kappa x}}&{\qquad x \leq - \xi}\\{R e g i o n I I I \qquad}&{\psi^{b}= s^{\prime}\left( \begin{array}{c}{E + m}\\{\kappa}\\ \end{array}\right) e^{- \kappa x}}&{\qquad x \geq \xi}\\ \end{array}
\chi_{V_v,\sigma_v}(a) = \varepsilon(\tfrac{1}{2},\tau_v[a]\otimes \tau'_v)\cdot \xi_{\tau_v[a]}(-1)^{N-1}\cdot \xi_{\tau_v'}(-1)^{N[a]}; \chi_{V'_v,\sigma'_v}(a') = \varepsilon(\tfrac{1}{2},\tau_v\otimes \tau_v'[a'])\cdot \xi_{\tau_v}(-1)^{N[a']}\cdot \xi_{\tau_v'[a']}(-1)^{N}.
e^{-\Phi}(-\det G_{\|})^{\frac{1}{2}}=g^{-1}Z^{-1}\Bigl(1-|v|^2 e^{\psi}Z\Bigr)^{\frac{1}{2}}\approx g^{-1}Z^{-1}-\frac{1}{2}g^{-1}e^{\psi}|v|^2\ .
0=4a \left(\frac{d\bar{\phi}}{dA}\right)^2-\frac{2}{P(A)}\frac{dP(A)}{dA}\frac{d\bar{\phi}}{dA}-\frac{\bar{f}_0^2}{2P^2(A)}+\frac{2 \bar{Q}+\mu e^{-4a \bar{\phi}}/\bar{Q}}{4P(A)},
\Big(\sum_{p\le x}\frac{\log p}{p^{1/2+it}}\Big(1-\frac{\log p}{\log x}\Big)\Big)^k = \sum_{\substack{\mu_1, \mu_2, \ldots, \mu_R \sum_r \mu_r = k}}\binom{k}{\mu_1\ \mu_2\ \cdots\ \mu_R}\prod_{r=1}^R\Big(\frac{\log p_r}{p_r^{1/2+it}}\Big(1-\frac{\log p_r}{\log x}\Big)\Big)^{\!\mu_r}&= \sum_n c(n) \, n^{-it}.
Z = \int{\cal D}h_{\mu\nu}{\cal D}\theta_{\mu}{\cal D}\varphi{\cal D}b_{\mu}{\cal D}b{\cal D}c_{\mu}{\cal D}\bar{c}_{\mu}{\cal D}c{\cal D}\bar{c}\exp i\int d^{4}x\, L_{{\rm T}, a=1}' ,
f\in \mathrm{ker}(I-T) \Leftrightarrow d_Bd_A^\ast f=f \Leftrightarrow d_B(d_A^\ast f-d_B^\ast f)=d_A(d_A^\ast f-d_B^\ast f)=0 & \Leftrightarrow d_A^\ast f-d_B^\ast f\in \mathcal{L}\cap (\ker d_A \cap \ker d_B)=\mathcal{L}\cap\mathcal{L}^\bot = \{0\}\Leftrightarrow d_A^\ast f=d_B^\ast f.
\underset{t\to+0}{\mbox{{\rm ess lim}}}\int_{{\bf R}^N}\int_{{\bf R}^N}G(x-y,t)\eta(x)\,dx\,d\mu(y) =\int_{{\bf R}^N}\eta(y)\,d\mu(y) +\lim_{t\to +0}\int_{{\bf R}^N}(\eta(y,t)-\eta(y))\,d\mu(y) =\int_{{\bf R}^N}\eta(y)\,d\mu(y).
V_{n m}^{>}( B ) \propto \frac{- 1}{l^{3}}\int_{0}^{1}d \eta \int_{0}^{1}d u \, u^{2 ( n - 2 )}\left[ F ( u ; 0 ) \right]^{n - 2 m}[ \widetilde{G}( u ) ]^{m}.
\begin{aligned}p_{[1^011^1]}\bigg(11^{2r}m+\frac{11-11^{2r+1}}{24}\bigg) \equiv 0\pmod{11^{2r}}, p\bigg(11^{2r+1}m+\frac{1-11^{2r+2}}{24}\bigg) \equiv 0\pmod{11^{2r+1}}.\end{aligned}
i(T_{00}) =0.6554010386...=\frac{1}{6}(3.9324062319...) & =\frac{\pi}{18\sqrt{3}}{\displaystyle\int\limits_{0}^{\pi}}\frac{\xi(x)\eta(x)}{\sqrt{b(x)\left( a(x)-c(x)\right)}}\sin\left(\frac{x}{2}\right) dx
M^1=\left[\begin{array}{cc}I_n -\frac{c}{2x_{n+1}}0 1\end{array}\right]\left[\begin{array}{cc}M_x-\frac{cc^*}{2x_{n+1}}0 0 2x_{n+1}\end{array}\right]\left[\begin{array}{cc}I_n 0 -\frac{c^*}{2x_{n+1}}1\end{array}\right].
\overline{\alpha}_{f}(x)= \limsup_{n\to \infty}h_{H_{X}}(f^{n}(x))^{1/n}=\limsup_{n\to \infty}h_{H_{X}}(\pi(g^{n}(y)))^{1/n}&\leq \limsup_{n\to \infty}h_{H_{Y}}(g^{n}(y))^{1/n}= \overline{\alpha}_{g}(y).
\underline U_t(t,x)=K_1K_2\frac{(1-\beta)\ln (t+\theta)+1}{(t+\theta)^{\beta}}\Theta+\frac{K_2\beta |x|}{(t+\theta)^{1+\beta}}\Theta,\ \ \ \underline h(t)-(t+\theta)^\beta<|x|\leq \underline h(t), &\underline U_t(t,x)={\bf 0}, \ |x|< \underline h(t)-(t+\theta)^\beta.
\delta^{k}_T(f)(x_{0},\dots,x_{k}) = & \sum_{0\leq s < t\leq k}(-1)^{t+|x_{t}|(|x_{s+1}|+\dots+|x_{t-1}|)}f\Big(\alpha(x_{0}),\dots,\alpha(x_{s-1}),[x_{s},x_{t}],\alpha(x_{s+1}),\dots,\widehat{x_{t}},\dots,\alpha(x_{k})\Big).
\left|\frac{\partial_1 u(y)}{|y-ste_1-x|^{N+2\sigma}}1_A(y)\right|\leq\frac{|\partial_1 u(y)|}{1+|y|^{N+2\sigma}}\frac{1+|y|^{N+2\sigma}}{|y-ste_1-x|^{N+2\sigma}}1_A(y)\leq K\frac{|\partial_1 u(y)|}{1+|y|^{N+2\sigma}}=:f(y),
\left|K_{\gamma}^\perp\right| =\left|K^\perp_{p_0p_1}\right|\cdot\left|K^\perp_{p_1p_2}\right|\cdots\left|K^\perp_{p_Np_0}\right| & = \frac{t_{p_0}}{t_{p_1}}\cdot\frac{t_{p_1}}{t_{p_2}}\cdots\frac{t_{p_N}}{t_{p_0}}& = 1,
\|\phi\|_{s,1}:= \frac{1}{r}\sum_{l=1}^r \|\phi^{(1)}\|_1 \ldots \|\phi^{(l-1)}\|_1 \|\phi^{(l)}\|_s \|\phi^{(l+1)}\|_1 \ldots \|\phi^{(r)}\|_1.
\chi(L_1) = \frac 16 L_1^3 + \frac 14 L_1^2.\,c_1(X_1)+ \frac1{12}L_1.\,\big(c_1(X_1)^2 + c_2(X_1)\big) + \frac 1{24}c_1(X_1).\,c_2(X_1).
N^{2}\log N =\sum_{k=1}^{p-1}\left(\sum_{j=k+1}^{p}2^{n_{j}-n_{k}}\right) \left(2^{2(n_{k}+1)}\log\left(2^{n_{k}+1}\right)+2^{2(n_{k}+1)}\log\left(\frac{N}{2^{n_{k}+1}}\right)\right) +\sum_{k=1}^{p}\left(1-\sum_{j=k+1}^{p}2^{n_{j}-n_{k}+1}\right)\left(2^{2n_{k}}\log \left(2^{2n_{k}}\right)+2^{2n_{k}}\log\left(\frac{N}{2^{2n_{k}}}\right)\right).
\mathbb{P}\left\{\mathcal{M}_i^{1, j}\right\}\leq \mathbb{P}\left\{\sum_{t=0}^{T-1}c^{(i)}_t V^{(i)}_t \leq \bar{t}\right\}+\mathbb{P}\left\{\sum_{t=0}^{T-1}d^{(i)}_t V^{(i)}_t > \bar{t}\right\},
\frac{d^2}{dt^2}(\|xu(t)\|^2_{L^2}+ 2 \|xv(t)\|^2_{L^2}) = 8 \left(\|\nabla u(t)\|^2_{L^2}+ \frac{1}{2}\|\nabla v(t)\|^2_{L^2}\right) - 20 \emph{Re}\int \overline{v}(t) u^2(t) dx.
n-10+\frac{1}{16n(n+2)(n-3)}\Big\{(n-2)(n-8)(n-10)(5n^2-2n-120) & +8(n-3)[(n-10)(-n^3+8n^2+28n-176)+8(-n^3+6n^2+30n-116)]\Big\}= n-10+\frac{1}{16n(n+2)(n-3)}\Big[-3n^5+2n^4+228n^3-264n^2-1760n-768\Big] = \frac{-3n^5+18n^4+52n^3-200n^2-800n-768}{16n(n+2)(n-3)}<0.
\mathbf{f}_{N_{\mathrm{F}}^{\mathrm{h}}}^n = \left[1 \,\, e^{-j2\pi n/N_\mathrm{F}^{\mathrm{h}}}\,\, \cdots \,\, e^{-j2\pi n \left(N_\mathrm{F}^{\mathrm{h}}- 1 \right)/N_\mathrm{F}^{\mathrm{h}}}\right]^\mathrm{T}.
b^{\dagger}_{{\bf{k+q/2}}}b_{{\bf{k-q/2}}}=N_{0}\delta_{{\bf{k}}, 0}\delta_{{\bf{q}}, 0}+[\delta_{{\bf{k+q/2}}, 0}(\sqrt{N_{0}})d_{{\bf{k}}}(-{\bf{q}}) + \delta_{{\bf{k-q/2}}, 0}d^{\dagger}_{{\bf{k}}}({\bf{q}})(\sqrt{N_{0}})]
c_1 =16(1-\nu)^2 c_{1a}+3(1-\mu)c_{1b},\quad c_{1a}=-27\nu^6+144\nu^5-102\nu^4+1620\nu^3-9883\nu^2+12484\nu+1732 c_{1b}=81\mu\nu^8-108\mu\nu^7+135\nu^8-1260\mu\nu^6-828\nu^7-12276\mu\nu^5+276\nu^6+84774\mu\nu^4 -4404\nu^5-157140\mu\nu^3+69170\nu^4+152628\mu\nu^2-198372\nu^3-156108\mu\nu+182084\nu^2 +27969\mu-60588\nu+73967
\begin{array}{lll}\delta e^{a}=\frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{a}\psi \hspace{1cm}\delta \omega^{ab}=-\frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{ab}\psi \delta \psi =D\epsilon \hspace{1cm}\delta A^{abcde}=\frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{abcde}\psi .\end{array}
\gamma ( z ) \sim 4 i \pi g_{s}^{B}\alpha^{\prime}\, \sum_{\epsilon = \pm 1}\left( \sum_{p = 1}^{N_{1}}\log \frac{( z + \epsilon z_{1 p})}{\Lambda}- \sum_{q = 1}^{N_{2}}\log \frac{( z + \epsilon z_{2 q})}{\Lambda}\right)
A_{1}=\left(\begin{array}{cc}0 2 0 0\end{array}\right),A_{2}=\left(\begin{array}{cc}0 0 2 0\end{array}\right),B_{1}=\left(\begin{array}{cc}2 0 0 0\end{array}\right),B_{2}=\left(\begin{array}{cc}0 0 0 2\end{array}\right).
u_{12}u_{23}-u_{13}u_{22}= u_{12}(u_{23}-u_{13}u_{21})-u_{13}(u_{22}-u_{12}u_{21}) & = u_{12}((1+a_{11})u_{23}-u_{13}u_{21})-u_{13}((1+a_{11})u_{22}-u_{12}u_{21}) &-a_{11}(u_{12}u_{23}-u_{13}u_{22}),
r_3\leqslant \sum\limits_{k^{\prime}=1}^{Q}\omega^{\prime}_{k^{\prime}}\log_2 (2\pi e\sigma_{k^{\prime}}^{\prime}/\omega^{\prime}_{k^{\prime}})-\sum\limits_{k=1}^{Q}\omega_k\log_2 (2\pi e\sigma_k).
\xi(\lambda ) =1-\lambda \left(\frac{1}{m+1}+\sum_{j=2}^\infty\frac{1}{j(m+j)}\prod_{\nu=1}^{j-1}\left(1-\frac{\lambda}{\nu(m+\nu)}\right)\right) &=\prod_{\nu=1}^\infty\left(1-\frac{\lambda}{\nu(m+\nu)}\right).
\begin{array}{rcl}f(x)\!\! \!\!>\!\! \!\!Q_{22}(x)\!=\!8x^2\!-\!1\!+\!8x^2 \underline T_{14}^{\cos ,0}(2x)\!+\! \underline T_{22}^{\cos ,0}(4x)\!+\!P_9(x) \overline T_{13}^{\sin ,0}(2x),\end{array}
det'\Bigl(\ast d_{\scriptscriptstyle 1}+ (\ast d_{\scriptscriptstyle 1})^{\scriptscriptstyle 2}\Bigr)=(-1)^{\psi}det'(\ast d_{\scriptscriptstyle 1})det'(\mathrm{1\mkern-5muI}+\ast d_{\scriptscriptstyle 1})
S_{{\underline{{{0}}}}{\underline{{{0}}}}}\, = \, - \frac{2 e^{4{\cal U}}}{( 8 \pi )^{2}r^{4}}\mathrm{I m}{\cal N}_{\Lambda \Sigma}( p^{\Lambda}p^{\Sigma}+ \ell ( r )^{\Lambda}\ell ( r )^{\Sigma})
A_\Lambda\|f\|^2\leq\sum_{i\in \mathbb{Z}}\|\Lambda_if\|^2=\sum_{i\in \mathbb{Z}}\|\Lambda_0 T^if\|^2 =\sum_{i\in \mathbb{Z}}\|\Lambda_0T^iT^{-n}T^nf\|^2 &=\sum_{i\in \mathbb{Z}}\|\Lambda_0T^{i-n}T^nf\|^2 &=\sum_{i\in \mathbb{Z}}\|\Lambda_iT^nf\|^2 &\leq B_\Lambda\|T^nf\|^2\leq B_\Lambda\|T\|^{2n}\|f\|^2,
\begin{aligned}\Big|\langle\boldsymbol{\nu},\eta_t\rangle -\bar\eta_t+ \langle\boldsymbol{\nu},q_{\alpha,x_\alpha}\rangle-\bar q_{\alpha,x_\alpha}\Big| \le c_3 \langle \boldsymbol{\nu}, \eta (\lambda | \bar U;x,t ) \rangle = c_3 \mathcal{H}( \boldsymbol{\nu},U,\bar{U};x,t)\;,\end{aligned}
N(\Phi)= \Pi^{\bot}\left\{i^{*}\left(f(W_{\delta,q}+\delta V_{\delta,q}+\Phi)-f(W_{\delta,q}+\delta V_{\delta,q})-f'(W_{\delta,q}+\delta V_{\delta,q})[\Phi]\right)\right\}R= \Pi^{\bot}\left\{i^{*}\left(f(W_{\delta,q}+\delta V_{\delta,q})\right)-W_{\delta,q}-\delta V_{\delta,q}\right\}
{\displaystyle \sum_{n=0}^{\infty}\left(\frac{x^{n}}{n!}\right)^{\gamma}}\leq\left(\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\right)^{2-\gamma}\left(\sum_{n=0}^{\infty}\left(\frac{x^{n}}{n!}\right)^{2}\right)^{\gamma-1}\leq(\mathrm{e}^{x})^{2-\gamma}(\mathrm{e}^{2x})^{\gamma-1}=\mathrm{e}^{\gamma x}
|u\rangle_{\tau}= N(u) \cdot \frac{\sqrt{u}}{\sqrt{1- u\tau}}\cdot exp\Bigl( \sum_{m=1}^{\infty}\left( \frac{m-u}{m+u}\right) a_{m}^{\dagger}\tilde a_{m}^{\dagger}e^{2m\tau}\Bigr)exp\Bigl(- \frac{u\hat q^{2}}{4(1-u\tau)}\Bigr) |0\rangle \, .
R(x_i, x_j) x_j = - x_i (\mbox{for $i=3, \ldots, n-1$ and $j \neq 2, i$}) , R(x_n, x_1) x_1 = \lambda x_2 - (1 - (1/4) \lambda^2) x_n, R(x_n, x_2) x_2 = (1/4) \lambda^2 x_n , R(x_n, x_i) x_i = (\lambda /2) x_2 - x_n (\mbox{for $i=3, \ldots, n-1$}) .
(T^{\ast}T)^2 =M_{\bar{u}E(|u|^{2})\chi_{S}(E(|w|^{2}))^{2}}EM_{u}, T^{\ast}T =M_{\bar{u}E(|w|^{2})}EM_{u}, T^{\ast^3}T^3 =M_{|E(uw)|^{4}E(|w|^2)}M_{\bar{u}}EM_{u}.
f_{j,\flat}=\sum_{m\in\mathbb{Z}^2}\widetilde{\eta}_mf_{j,m,\flat}, f_{j,\sharp}=\sum_{m\in\mathbb{Z}^2}\!\widetilde{\eta}_mf_{j,m,\sharp}, f_{j,}=\sum_{m\in\mathbb{Z}^2}\widetilde{\eta}_m\left[\eta_m\left(\psi^{(j)}\ast_jf_j\right)-\psi^{(j)}\ast_j\left(\eta_mf_j\right)\right].
H= \begin{pmatrix}\frac{i}{2}0 0 -\frac{i}{2}\end{pmatrix},E = \begin{pmatrix}0 \frac 1{2\sqrt{2}}-\frac1{2\sqrt{2}}0\end{pmatrix},V= \begin{pmatrix}0 \frac i{2\sqrt{2}}\frac i{2\sqrt{2}}0\end{pmatrix},
det'\Bigl(\ast d_{\scriptscriptstyle 1}+ (\ast d_{\scriptscriptstyle 1})^{\scriptscriptstyle 2}\Bigr)=e^{i \pi \psi}det'(\ast d_{\scriptscriptstyle 1})det'(\mathrm{1\mkern-5muI}+\ast d_{\scriptscriptstyle 1})
v_{1}^{1}=\varphi_{1,[2]}^{1}=2\bar{u}^{2}-x^{3}v_{1,[1]}^{1}=\varphi_{1,[3]}^{1}=2\bar{u}_{[1]}^{2}-2\bar{u}^{2}+x^{3}\,, \:\:\vdots
F_{i}\mapsto x_{i i}\left(\sum_{j=1}^{i}x_{j i}\partial_{x_{ji}}-\sum_{j=1}^{i-1}x_{j i-1}\partial_{x_{j i-1}}\right) -\sum_{j=i+1}^{n}x_{ij}\partial_{x_{i+1,j}}+\sum_{j=1}^{i-1}x_{j i}\partial_{x_{j i-1}}+\mu_{i}x_{i i}.
|x|\,|xe^\alpha-y| \ge h(n-1)c_4 n^{-g+1}\frac{|\alpha|^n}{4^n\,n!}\ge c_4^2|\alpha| n^{-2g}\binom{2n-2}{n-1}4^{-n}\ge c_5 n^{-2g-1}\ge c_5 (\log |x|)^{-2g-1},
\kappa=\frac{|\mathbb{E}_{x,\mathbf{z},\mathbf{h}}[f^*(x,\mathbf{h},\mathbf{z})\cdot x]|^2}{\mathcal{E}_\mathrm{s}\mathbb{E}_{x,\mathbf{z},\mathbf{h}}[|f(x,\mathbf{h},\mathbf{z})|^2]},
a_D={\sqrt 2\over \pi}\int_1^{{\cal G}(a)+B}{dx \sqrt{x-{\cal G}(a)-B}\over \sqrt{x^2-1}},\qquad a={\sqrt 2\over \pi}\int_{-1}^1{dx \sqrt{x-{\cal G}(a)-B}\over\sqrt{x^2-1}}.
\Delta_{\gamma_{1}}\Delta_{\gamma_{2}}~f\left( z\right) =\gamma_{1}\gamma_{2}~f\left( q^{2}z\right) -\left( \gamma_{1}+\gamma_{2}\right)~f\left( qz\right) +f\left( z\right) =\Delta_{\gamma_{2}}\Delta_{\gamma_{1}}~f\left( z\right) .
\mathbf{y}(v) =\chi_{\{v\geq 0\}}\int_0^{v}e^{\frac{y^2}{2}}[\frac12e^{-y^2}+N(y)-1]dy+\chi_{\{v< 0\}}\int_0^{v}e^{\frac{y^2}{2}}[-\frac12e^{-y^2}+N(y)]dy; \mathbf{z}(v) =\chi_{\{v\geq 0\}}e^{\frac{v^2}{2}}[\frac12e^{-v^2}+N(v)-1]+\chi_{\{v< 0\}}e^{\frac{v^2}{2}}[-\frac12e^{-v^2}+N(v)].
K_{y}=[\begin{matrix}0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\end{matrix}],K_{z}=[\begin{matrix}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{matrix}]