harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Psi_B=\,\sum^{N_B}_{l=1}\Big(n_{1l}(\varphi^{-1}_l(x),z_l)\,\exp(\imath \,\vartheta_{1l}(\varphi^{-1}_l(x),z_l))\break +n_{2l}(\varphi^{-1}_{l}(x),z_l)\,\exp(\imath \,\vartheta_{2l}(\varphi^{-1}_l(x),z_l))\Big)|\varphi^{-1}_{l}(x),z_l\rangle.
\begin{array}{r c l c c}{{\ell ( r )}}&{{=}}&{{r^{3}\, \frac{2 \, \sqrt{a^{2}+ 1}}{a^{2}- 1}\, \frac{d}{d r}\, \left[ H ( r ) \right]^{\frac{1 - a^{2}}{1 + a^{2}}}}}&{{;}}&{{\mathrm{e l e c t r i c ~ 0 ~ - b r a n e}}}\\{{p}}&{{=}}&{{- \frac{4}{\sqrt{a^{2}+ 1}}\, k}}&{{;}}&{{\mathrm{m a g n e t i c ~ s o l i t o n i c ~ 0 ~ - b r a n e}}}\\ \end{array}
( \alpha_{o p t}^{2})^{e v}= \sqrt{\frac{2 \mid \beta \mid^{2}t a n h \mid \beta \mid^{2}+ 2 \mid \beta \mid^{2}c o s 2 \theta_{1}+ 1}{2 \mid \beta \mid^{2}t a n h \mid \beta \mid^{2}- 2 \mid \beta \mid^{2}c o s 2 \theta_{1}+ 1}}( \alpha_{o p t}^{2})^{o},
\begin{array}{r c l c r c l c r c l c r c l}{{\alpha_{6}}}&{{=}}&{{\frac{i}{6 !}}}&{{;}}&{{\alpha_{4}}}&{{=}}&{{\frac{1}{2 \, 4 !}}}&{{;}}&{{\alpha_{0}}}&{{=}}&{{\frac{1}{4 8}}}&{{;}}&{{\alpha_{2}}}&{{=}}&{{\frac{i}{1 6}}}\\ \end{array}
\hat{A}= \hat{M}-\hat{N}\geq \hat{M}-\tilde{N}= \Omega_2-\vert\Omega_2-D_{A\Omega_1}\vert+\frac{1-\vert 1-\alpha\vert}{\alpha}D_{A\Omega_1}&-\frac{\alpha+\beta+\vert \alpha-\beta \vert}{\alpha}\vert L_{A\Omega_1}\vert-2\vert U_{A\Omega_1}\vert.
\sigma(x,y,z) = -(1+\gamma)(x^2+y^2+z^2+x^2y^2+y^2z^2+z^2x^2) &+ \gamma[(x+y)^2+(y+z)^2+(z+x)^2+(xy+yz)^2+(yz+zx)^2+(zx+xy)^2]\le0.
\hat{\mathbb{E}}_{(\tau\wedge T)+}[\varphi(X_{\tau\wedge T+t_1}^x,\cdots,X_{\tau\wedge T+t_m}^x)]=\hat{\mathbb{E}}[\varphi(X_{t_1}^y,\cdots,X_{t_m}^y)]_{y=X^x_{\tau\wedge T}}.
(\lambda C_\lambda)^{p-1}\cdot (\lambda (p-1) + n-p) = c_1 C_\mu^{k_1}\cdot (\lambda C_\lambda)^\alpha, (\mu C_\mu)^{p-1}\cdot (\mu (p-1) + n-p) = c_2 C_\mu^{k_2}\cdot (\lambda C_\lambda)^{k_3}.
\tau_{j}\left( r\right) =\left(\begin{array}[c]{cc}e^{\frac{2\pi ji}{n}}0 0 e^{-\frac{2\pi ji}{n}}\end{array}\right) \tau_{j}\left( s\right) =\left(\begin{array}[c]{cc}0 1 1 0\end{array}\right) .
X_1\cdot X_2=X_1+X_2+\frac{1}{2}[X_1,X_2]+\sum_{p\geq 3}\sum_{(i_1,\dots,i_p)\in \{1,2\}^p}c_{(i_1,\dots,i_p)}[X_{i_1},\dots,X_{i_p}].
\sum_{\{\rho_1,\rho_2:|\rho_1-\rho_2|\ge 4\sqrt{n}\}}C_{\rho_1,\rho_2}(j)\|F_{\rho_1}\|_{L_{y,s}^2(|(y,s)|^{\alpha p})}\|G_{\rho_2}\|_{L_{x,t}^2(|(x,t)|^{\alpha p})}&=\sum_{\{\rho_1,\rho_2:|\rho_1-\rho_2|\ge 4\sqrt{n},|\rho_1|\ge 1 \}}+ \sum_{\{\rho_1,\rho_2:|\rho_1-\rho_2|\ge 4\sqrt{n},|\rho_1|<1 \}}.
M_{n,n}= n \int_{\frac{n}{n+1}}^1 \left[Y_{n,n}-\ln n+\left(\ln n-\ln\frac{t}{1-t}\right)\right]^2 6t(1-t)\mathrm{d}t &=B_n^2\left(\frac{3}{n}+{\mathcal O}\left(\frac{1}{n^2}\right)\right)-B_n\left(\frac{3}{n}+{\mathcal O}\left(\frac{1}{n^2}\right)\right) +\frac{3}{2n}+{\mathcal O}\left(\frac{1}{n^2}\right)\ ,
T_5 \geq -Cs^2 \lambda^4 \iint_Q \xi\left|A \nabla \eta \cdot \nabla \eta\right|^2 |u|^2dx d t -C\lambda^2\iint_{Q}\xi \left| A\nabla u\cdot \nabla \eta \right|^2 dx dt &-C s^2 \lambda^3 \int_0^T \int_{\omega}\xi|u|^2dx d t- C\lambda \iint_Q \xi A \nabla u \cdot \nabla u dx d t.
\Gamma^{1 2}\pm \Gamma^{3 4}= 2 i \sigma_{3}^{\pm}, \, \, \, \, \Gamma^{2 4}\pm \Gamma^{3 1}= 2 i \sigma_{1}^{\pm}, \, \, \, \, \Gamma^{1 4}\pm \Gamma^{2 3}= - 2 i \sigma_{2}^{\pm},
cov[f_{d}(x),f_{d^{\prime}}(x^{\prime})]=\sum_{q=1}^{Q}v_{d,d^{\prime}}b_{q}k_{q}(x,x^{\prime})=v_{d,d^{\prime}}\sum_{q=1}^{Q}b_{q}k_{q}(x,x^{\prime})=v_{d,d^{\prime}}k(x,x^{\prime})
S ( h , A ) = S_{0}+ \frac{i k}{2 \pi}\int_{D \times R}d x d t d \tau \mathrm{T r}\partial_{\tau}[ A_{+}* \partial_{x}h * h^{- 1}] + \frac{i k}{4 \pi}\int_{D \times R}d x d t d \tau \mathrm{T r}\partial_{\tau}( A_{+}* A_{x}) ,
{\cal Z}^{(N_{f})}(\{\mu\}) =\det\left(\begin{array}{ll}A(\{\mu_f\}) A(\{-\mu_f\}) A(\{-\mu_f\}) A(\{\mu_f\})\end{array}\right)\Delta(\{\mu\})^{-1}, A_{ij}= (\mu_i)^{j-1}e^{\mu_i}.
\Lambda(f \otimes \mathrm{Ad}(g), k) = \mathcal C \cdot \mathcal I \cdot\frac{\langle f,f\rangle}{\langle h, h \rangle}|\langle (\mathrm{id}\otimes U_M) F_{\chi |\mathcal H \times \mathcal H}, g \times g\rangle|^2,
H^{(12+4)}_{1A,4}(\tau) :=H^{(12)}_{1A,4}(\tau)+H^{(12)}_{1A,8}(\tau), H^{(12+4)}_{3A,4}(\tau) :=H^{(6)}_{3A,2}(\tau/2)+H^{(6)}_{3A,4}(\tau/2),
q_{i+1}-q_i=\frac{\sum\limits_{j\ne i+1}\sum\limits_{k=i+1}^{j-1}m_jm_k v_j-\sum\limits_{j\ne i}\sum\limits_{k=i}^{j-1}m_jm_k v_j}{M}= \frac{\sum\limits_{j\ne i}m_im_j v_i-\sum\limits_{j\ne i}m_jm_i v_j}{M}.
\lim_{\varepsilon \rightarrow 0}\varepsilon^{\frac{n}{2}}\int_{\mathbb{R}^n}T(Pw_{\varepsilon\alpha})\overline{Q(x)}w_{\varepsilon \beta}dx=(\pi/\beta)^{n/2}\int_{\mathbb{T}^n}(A P)(x)\overline{Q(x)}dx.
\mathrm{U}_{\mathrm{e}}(\xi,t) =\frac{\gamma_{\mathrm{p}}}{\gamma_{\mathrm{m}}}\frac{1}{\alpha_\mathrm{m}}\Bigg[-\int_{0}^t\int_{\partial\Omega}\Phi^{\textbf{e}}(\xi,t;z,\tau)\gamma^{\textbf{int}}_{1}\mathrm{U}_{\mathrm{i}}(\mathrm{y},\tau)d\sigma_\mathrm{y}d\tau + \textbf{err}^{(1)}+ \textbf{err}^{(2)}\Bigg].
j_{3}(\alpha,m,L)=\langle \Omega|\bar{\chi}_{\alpha}\left(x\right)\gamma_{3}\chi_{\alpha}\left(x\right)|\Omega\rangle = \frac{1}{L}\sum_{k_{3}}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{2 k_{3}}{E_{k}}
| s_j^{\top}\,(\gamma_j\,s_j+J_{m_j}^\top r_{m_j}) | \; \leq \;\frac{4\,\|J_{m_j}\|^2\,\|J_{m_j}^\top r_{m_j}\|^2 + 2\theta_{in}\,\|J_{m_j}^\top r_{m_j}\|^2}{\gamma_j^2}\; = \; \frac{4 \|J_{m_j}\|^2+ 2 \theta_{in}}{\mu_j^2}.
T_1 =r_1s_1s_2s_3s_4s_2s_1,T_2=s_0s_2s_1s_3s_4s_2s_1s_3s_4s_2, T_3 =r_3s_3s_2s_1s_4s_2s_3,T_4=r_4s_4s_2s_1s_3s_2s_4
\| e^{(t-s) \Delta}F_1 (s) \|_{L^1}\le \| F_1 (s) \|_{L^1}= \int_{\{|x|<\sqrt s \}}s^{-\frac{\alpha +1}{\alpha}}\Bigl| f \Bigl( \frac{x}{\sqrt s}\Bigr) \Bigr|^{\alpha +1}dx = \int_{\{|x|< 1 \}}|f(x)|^{\alpha +1}dx
2\cdot3\cdot11\cdot23\cdot31\cdot47059\cdot2217342227\cdot1729101023519\cdot58254480569119734123\cdot8491659218261819498490029296021 &=4200017949707747062038711509670656632404195753751630609228764416142557211582098432545190323474818541298976556403,
\Omega(\xi,\eta,\zeta) = \Delta(\xi,\eta,\zeta)\Delta(\xi,-\eta,-\zeta)\Delta(\xi,-\eta,\zeta)\Delta(\xi,\eta,-\zeta) &= J(\xi)^2 + J(\eta)^2 + J(\zeta)^2 - 2J(\xi)J(\eta) - 2J(\eta)J(\zeta) - 2J(\zeta)J(\xi),
u=-\frac{\mu}{r}+\sum_{n=2}^{N_{z}}\frac{J_{n}P_{n}^{0}(sin\theta)}{r^{n+1}}+\sum_{n=2}^{N_{t}}\sum_{m=1}^{n}\frac{P_{n}^{m}(sin\theta)(C_{n}^{m}cosm\varphi+S_{n}^{m}sinm\varphi)}{r^{n+1}}
|b_{14}^{(k)}\pm b_{23}^{(k)}| = |\sin(\psi_{k}\pm\phi_{k})|\cdots |\sin(\psi_{1}\pm\phi_{1})|\cdot|b_{14}^{(0)}\pm b_{23}^{(0)}| \leq |b_{14}^{(0)}\pm b_{23}^{(0)}|,
e_{n}H_{n - 1 +{\frac{D}{2}}}^{( 1 ) \prime}( | \omega | r^{\prime}) + f_{n}H_{n - 1 +{\frac{D}{2}}}^{( 2 ) \prime}( | \omega | r^{\prime}) - d_{n}H_{n - 1 +{\frac{D}{2}}}^{( 1 ) \prime}( | \omega | r^{\prime}) ={\frac{( 2 n + D - 2 ) \Gamma \left({\frac{D - 2}{2}}\right)}{4 ( \pi r^{\prime})^{\frac{D}{2}}| \omega |}}.
Z={{V_{p+1}\Gamma(0)}\over{2\pi\alpha}}{{\left({{\mbox{T}}\over{2\pi}}\right)}^{(p+1)/2}}\int d\tau{{\mbox{det}(\psi,B)}\over{\sqrt{\mbox{det}\hat{(B,B)}}}}{{\sqrt{\mbox{Det}'{\hat{P}^\dagger}\hat{P}}}\over{\mbox{Vol}(\mbox{CKV})}}{{\left({{{\mbox{Det}_N}'\hat{\Delta}}\over{\int{d^2}\xi\sqrt{\hat{g}}}}\right)}^{-(p+2)/2}}
\mathbf{x}^{AA^{\prime}}\rightarrow\mathbf{x}_{1}^{AA^{\prime}}=-\mu_{1}^{AA^{\prime}}\mathbf{m}-\mu_{1}^{AA^{\prime}}\overline{\mathbf{m}}+i\nu_{1}^{AA^{\prime}}\mathbf{m}\overline{\mathbf{m}}
\frac{2\omega_0+\omega_1}{3}u_{j-\frac{1}{2}}^{-}+ \frac{\omega_0+2(\omega_1+\omega_2)}{3}u_{j+\frac{1}{2}}^{-}+\frac{1}{3}\omega_2u_{j+\frac{3}{2}}^{-}= \frac{1}{6}\omega_0\bar{u}_{j-1}+\frac{5(\omega_0+\omega_1)+\omega_2}{6}\bar{u}_j+\frac{\omega_1+5\omega_2}{6}\bar{u}_{j+1}.
\begin{array}{rcl}e^{\frac{4}{3}\hat{\phi}}= - \hat{\hat{g}}_{yy}\, , & \hat{g}_{\hat{\mu}\hat{\nu}}= e^{\frac{2}{3}\hat{\phi}}\left[\hat{\hat{g}}_{\hat{\mu}\hat{\nu}}-\hat{\hat{g}}_{\hat{\mu}y}\hat{\hat{g}}_{\hat{\nu}y}/\hat{\hat{g}}_{yy}\right]\, ,\end{array}
X^+_{\mu,\nu}= A^+_{K+2m}\cdots A^+_{K+2}A^+_{K+1}B^+_{\nu+n-1}\cdots B^+_{\nu+1}B^+_{\nu}, X^-_{\mu,\nu}= A^-_{K-2m+1}\cdots A^-_{K-1}A^-_K B^-_{\nu-n+1}\cdots B^-_{\nu-1}B^-_{\nu},
\sum_{i=1}^k [\tilde g(|x|^2 + \tau^2 + 2 |x|\tau |\langle \hat x, \xi_i \rangle|) + \tilde g(|x|^2 + \tau^2 - 2 |x|\tau |\langle \hat x, \xi_i \rangle|) - 2\tilde g(|x|^2)] \leq \Big{(}\tilde g(|x|^2 + \tau^2 + 2 |x|\tau) + \tilde g(|x|^2 + \tau^2 - 2 |x|\tau) - 2 \tilde g(|x|^2) \Big{)}+ 2(k - 1) \Big{(}\tilde g(|x|^2 + \tau^2) - \tilde g(|x|^2) \Big{)},
H_{q-2\ell}\left(\overline{T}_K(t)\right)H_{2\ell}\left(\mathcal{T}^{\prime}_K(t)\right) =H_{q-2\ell}(B(h_K(t,\cdot)))H_{2\ell}(B(h^\prime_K(t,\cdot))) &=I^B_q\left(h^{\otimes q-2\ell}_K(t,\cdot)\otimes h^{\prime\,\otimes 2\ell}_K(t,\cdot)\right)
J_{n}( k_{1}, \cdots , k_{n}) = \int_{0}^{1}d \tau_{1}\cdots \int_{0}^{1}d \tau_{n}\exp \left[ \frac{i}{2}\sum_{a < b}^{n}( k_{a}\times k_{b}) ( 2 \tau_{a b}- \epsilon ( \tau_{a b}) ) \right] .
\begin{aligned}Q_a(X,Y) = -6\det(a)D(X, Y, a) + 9D(X,a,a)D(Y,a,a), \Phi_a(X,Y) = 4\det(a)^3 (X \times a) \times (Y \times a) + \frac 12(\det(a)^2Q_a(X,Y)-Q_a(X,a)Q_a(Y,a))a. \end{aligned}
\begin{array}{r c l}{\displaystyle \left[ a_{q}, a_{q}^{\dagger}\right]}{=}{\left[ N + 1 + b \gamma \right]_{q}- \left[ N + b \gamma \right]_{q}~ ,}{\displaystyle \left[ N , a_{q}\right]}{=}{- a_{q}~ , ~ ~ ~ \displaystyle \left[ N , a_{q}^{\dagger}\right] = a_{q}^{\dagger}~ .}\end{array}
<{\cal{H}}> ={\frac{1}{8}}G^{- 1}( x x ; \chi ) +{\frac{1}{2}}\lim_{x \rightarrow y}\nabla_{x}\nabla_{y}G ( x , y ; \chi ) -{\frac{\alpha_{0}}{\beta^{2}}}e^{-{\frac{\beta^{2}}{2}}G ( x , x ; \chi )}
\frac{d}{dt}\|\Lambda^{m}u\|_{2}^{2}+\|\Lambda^{m+\alpha}u\|_{2}^{2}\leq C\|\Lambda^{m+\alpha}u\|_{2}\|\Lambda^{m-\alpha+1}(u\otimes u)\|_{2}\leq \|\Lambda^{m+\alpha}u\|_{2}\|\Lambda^{m-\alpha+1}u\|_{2}\|u\|_{\infty}.
\begin{array}{ll}N_5= \int_{\mathbb R}(\zeta_k+\varphi_k)[(2 c_1a_k-\omega b_k) \xi_{n_k}+ (c_2b_k-\omega a_k) \nu_{n_k}] &+ ((\zeta_k \xi_{n_k})' + (\varphi_k \xi_{n_k})' )(2 c_3a'_k-\omega c_4 b'_k)-\omega c_4 a'_k ((\zeta_k \nu_{n_k})' +(\varphi_k \nu_{n_k})'),\end{array}
(\Delta \otimes \mathrm{id})(R) = R_{13}R_{23}, (\mathrm{id}\otimes \Delta)(R) = R_{13}R_{12}, (S \otimes \mathrm{id})(R) = \big(\mathrm{id}\otimes S^{-1}\big)(R) = R^{-1}, (S \otimes S)(R) = R, R_{12}R_{13}R_{23}= R_{23}R_{13}R_{12}
\gamma^{8}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \gamma^{a}= \left( \begin{array}{c c}{0}&{\lambda^{i}}\\{- \lambda^{i}}&{0}\\ \end{array}\right) , i = 1 , . . . . , 7 ,
|\mathcal{I}_{i,n}^{(1)}||\mathcal{I}_{j,n}^{(2)}|=(|\mathcal{I}_{i,n}^{(1)}\cap\mathcal{I}_{j,n}^{(2)}|+|\mathcal{I}_{i,n}^{(1)}\setminus \mathcal{I}_{j,n}^{(2)}|)(|\mathcal{I}_{i,n}^{(1)}\cap\mathcal{I}_{j,n}^{(2)}|+|\mathcal{I}_{j,n}^{(2)}\setminus \mathcal{I}_{i,n}^{(1)}|).
T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta S_{matter}}{\delta g^{\mu\nu}}=\frac{-2}{\sqrt{-g}}\frac{\partial(\sqrt{-g}L_{matter})}{\partial g^{\mu\nu}}=-2\frac{\partial L_{matter}}{\partial g^{\mu\nu}}+g_{\mu\nu}L_{matter},
\begin{array}{rcl}f(x)\!\! \!=\! \!\! 8x^2\!-\!1+h_1(x)\cos4 x+h_2(x)\cos2 x+h_3(x)\sin2 x begin{align*}1.0 ex]\!\! \!=\! \!\! 8x^2\!-\!1\!+\!\cos4 x\!+\!8x^2\cos2 x\!+\!\left(4\left(\frac{8}{45}\!-\!b_1(x)\right)x^5\!-\!4 x\right)\sin2 x>0,\end{array}
\tilde{W}: = \frac q{1 + q}\, \tilde{K}^{- 1}+ \frac 1{1 + q}\, \tilde{K}+ q^{- \frac 1 2}( q \! - \! 1 )^{2}\, \tilde{K}^{- 1}\, \tilde{L}^{-}\, \tilde{L}^{+}.
-\Omega \, \dot{z}_n = -\Omega\, d\psi(y_n)[\dot{y}_n] = -(d\psi(y_n)^*)^{-1}[\Omega \, \dot{y}_n) ] = (d\psi(y_n)^*)^{-1}[2c_{\mathbb{H}}(C) dH_C(y_n) + \eta_n] = 2c_{\mathbb{H}}(C) (d\psi(y_n)^*)^{-1}[dH_C(y_n)] + \theta_n,
{}[i][j] = \frac{\alpha^2[i][j]h_{ud}[j]P_{s}[i]h_{su}[i]}{(\alpha^2[i][j]h_{ud}[j]+1)\sigma^2}= \frac{P_{u}[j]h_{ud}[j]P_{s}[i]h_{su}[i]}{(P_{u}[j]h_{ud}[j] + P_{s}[i]h_{su}[i])\sigma^2 + \sigma^4}.
d_{1}a^{\ast}d_{2}\widetilde{\psi}(w) =d_{1}\overline{N}'(a^{\ast})(d_{2}\widetilde{\psi}(w)) &=d_{1}c_{k}^{-1}\pi_{e_{k}a}'i \hat{\psi}_{i}\Pi_{2}(d_{2}w)+d_{1}c_{k}^{-1}\pi_{e_{k}a}j'\sigma_{2}'\overline{\psi}_{2}(\Pi_{3}(d_{2}w)))
\frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{2\pi i \frac{jr}{m}}}{s+2\sin^2\left(\pi\frac{(j+\alpha)}{m}\right)}= e^{-2\pi i \alpha \ell /m}\cdot \frac{U_{m-\ell-1}(s+1)+e^{2\pi i \alpha}U_{\ell-1}(s+1)}{T_{m}(s+1)-\cos2\pi \alpha}.
\|\mathbf{X}\|_{p-var}:=\sup_{\mathcal{D}\subset[0,1]}\left(\sum_{t_i\in\mathcal{D}}d_{CC}(\mathbf{X}_{t_i},\mathbf{X}_{t_{i+1}})^p\right)^{\frac{1}{p}}=\sup_{\mathcal{D}\subset[0,1]}\left(\sum_{t_i\in\mathcal{D}}\|\mathbf{X}_{t_i,t_{i+1}}\|_{CC}^p\right)^{\frac{1}{p}}.
\widetilde{p}_2(n)u_{n+2}+\widetilde{p}_1(n)u_{n+1}+\widetilde{p}_0(n)u_{n}= 0, \widetilde{p}_2(n) = (5n + 6)(n + 2)(60n + 43), \widetilde{p}_1(n) = 300 (216000n^3 + 759600n^2 + 836940n + 290603) \widetilde{p}_0(n) = 2^{12}\cdot 3^6 \cdot 5^2 (5n + 4)(5n + 3)(60n + 103).
\int_{\Omega}\langle TT_{\omega}x,TT_{\omega}x\rangle_{\mathcal{A}}d\mu(\omega)=\int_{\Omega}\langle T_{\omega}Tx,T_{\omega}Tx\rangle_{\mathcal{A}}d\mu(\omega) \geq A\langle K^{\ast}Tx,K^{\ast}Tx\rangle_{\mathcal{A}}&=A\langle TK^{\ast}x,TK^{\ast}x\rangle_{\mathcal{A}}&\geq A\|T^{\dagger}\|^{-2}\langle K^{\ast}x,K^{\ast}x\rangle_{\mathcal{A}}
\|\partial^{(m_1,m_2)}\widehat{d_j}\|^2_{L^2(\mathcal{W}^+_J)}\lesssim \|g\|^2_{\beta,\infty}\|\omega\|^2_{\beta,2}\begin{cases}0 \quad ,\, m_1\neq0, 2^{-2jm_2s\alpha}2^{js(1-\alpha)}\ell_J^{-2\beta-1},\, m_1=0. \end{cases}
\mathcal{P}_n(e^{i\theta}a+e^{-i\theta}b) = (e^{i\theta}a+e^{-i\theta}b)(e^{i(n-1)\theta}a^{n-1}+e^{-i(n-1)\theta}b^{n-1}) &\quad\quad\quad-ab(e^{i(n-2)\theta}a^{n-2}+e^{-i(n-2)\theta}b^{n-2}) &= e^{in\theta}a^n + e^{-in\theta}b^n.
(\alpha_0 - \beta_0) \cdot \alpha_1 = -\frac{1}{2^3}(a_1 - a_{-1}) + \frac{7}{2^2 \cdot 3}(a_2 + a_{-2}) + \frac{2}{3}(a_3 + a_{-3}) + \frac{1}{2^3}v_{(1,2)}- v_{(2,3)}+ \frac{8}{3}(a_2 + a_{-2}) \cdot v_{(1,3)}
\beta_1 =\frac{(1-|a|^4)(1+|a|^2)^2}{1-|a|^6}(\rho^2-\overline{a}^3\rho)=\frac{\overline{a}^4}{a^2}(1+|a|^2); \beta_2 =\frac{(1-|a|^4)(1+|a|^2)^2}{1-|a|^6}(-2\rho+\overline{a}^3+\overline{a}^3|\rho|^2)=\frac{\overline{a}^2}{a}(1+|a|^2)(2+|a|^2); \beta_3 =\frac{(1-|a|^4)(1+|a|^2)^2}{1-|a|^6}(1-\overline{a}^3\overline{\rho})=(1+|a|^2)^2.
\tilde{\alpha}(v)=uvu^{-1}=v, \tilde{\beta}(v)=vvv^{-1}=v, a*v=uau^{-1}vv^{-1}=uau^{-1}=\tilde{\alpha}(a), v*a=uvu^{-1}av^{-1}=vav^{-1}=\tilde{\beta}(a),
[\eta]_{m}=\eta\left(\eta^2+2m+2\right)^{-\frac{1}{2}}&=\frac{\eta}{\sqrt{2m+2}}\left(1+\frac{\eta^2}{2m+2}\right)^{-\frac{1}{2}}&=\frac{\eta}{\sqrt{2m+2}}\left(1+r(\eta,m)\right),
g(\gamma)=\begin{cases}\,\,0, \gamma \leq -\beta^{\star}\eta^{\star}, \,\,\frac{\gamma}{\eta^{\star}}+\beta^{\star}, -\beta^{\star}\eta^{\star}\leq \gamma \leq \eta^{\star}-\beta^{\star}\eta^{\star}, \,\,1, \gamma \geq \eta^{\star}-\beta^{\star}\eta^{\star}\end{cases}
d(T^3x_\ell,T^3x_\nu) =|x_{8\ell}-x_{8\nu}|=\Bigl|\frac1{8\ell}-\frac1{8\nu}\Bigr| = \frac12\Bigl|\frac1{4\ell}-\frac1{4\nu}\Bigr| = \frac12 |x_{4\ell}-x_{4\nu}| = \frac12 d(T^2x_\ell,T^2x_\nu).
\sum_{\nu=0}^{\infty}\frac{A(\nu)^2}{D(a, b)^2}t^{\nu}= \frac{4}{1-t}+ \frac{a^2}{1-a^2t}+ \frac{a^{-2}}{1-a^{-2}t}+ \frac{b^2}{1-b^2t}+ \frac{b^{-2}}{1-b^{-2}t}& - \frac{2ab}{1-abt}- \frac{2(ab)^{-1}}{1-(ab)^{-1}t}- \frac{2ab^{-1}}{1-ab^{-1}t}- \frac{2a^{-1}b}{1-a^{-1}bt}.
m^{-1/2}E[J^m_{s,t}] = -E\left[\int_s^t (\nabla_z\chi)(r,q_r^m,z_r^m) \cdot (-\nabla_q V(r,q_r^m)-\partial_r\psi(r,q_r^m)+\tilde F(r,q_r^m))dr\right] &-E\left[\int_s^t(\nabla_q\chi)(r,q_r^m,z_r^m)\cdot z_r^m dr\right]+O(m^{1/2}),
F(t)=Z_{G}(t+1,t^{2}+1,t^{3}+1,t^{4}+1)=\frac{(t+1)^{6}+9(t+1)^{2}(t^{2}+1)^{2}+8(t^{3}+1)^{2}+6(t^{2}+1)(t^{4}+1)}{24}
\sum_{i+j=\ell}(-1)^j e_i(z) G_j(z) \:=\: e_\ell(z) + \sum_{j=1}^{\ell}\sum_{a=1}^{k}(-1)^{j+a-1}e_{\ell-j}(z) s_{(j,1^{a-1})}(z) \:=\: e_\ell(z) + \sum_{a=1}^k (-1)^{a-1}\left[\sum_{j=1}^{\ell}(-1)^{j}e_{\ell-j}(z) s_{(j,1^{a-1})}(z) \right]\ .
h_1=t^{1/2}(\log t)^{-1/2}\left(1+(\log t)^{-1}\right),h_2=t^{-1/2}(\log t)^{-1/2}h_3=t^{1/2}(\log t)^{1/2},h_4=t^{-1/2}(\log t)^{1/2}\left(1+(\log t)^{-1}\right)
\overset{\circ\circ}S_k(X^{\frak C\boxplus\tau})= S_k(X^{\frak C\boxplus\tau},\widehat{\mathcal U}^{\frak C\boxplus\tau})\cap (\mathcal R^{\frak C})^{-1}(\overset{\circ}S_k(X,\widehat{\mathcal U})).
\left( \mathcal{R}^{\left( e \right)}\mathcal{T}^{\left( e \right)}\right) \mathcal{V}^{\left( e \right)}= \mathcal{V}^{\left( e \right)}, \quad \mathrm{v e r s u s}\quad \mathcal{R}^{\left( e \right)}\left( \mathcal{T}^{\left( e \right)}\mathcal{V}^{\left( e \right)}\right) = 0 .
\frac{a'(t)}{a(t)}+\frac{\alpha'(t)}{\alpha(t)}+\frac{2n}{t}\geq V^*\left(\frac{G(t)}{ta(t)}\right) + V\left(\frac{\Gamma(t)}{t\alpha(t)}\right) \geq \frac{G(t)}{ta(t)}\cdot\frac{\Gamma(t)}{t\alpha(t)}= \frac{1}{t^2a(t)\alpha(t)}\sum_{i=1}^{n}a_i(t)\alpha_i(t).
\phi_{\alpha_1}(t) = \sqrt{\frac{\pi}{4H_0}}e^{- \frac{3}{2}H_0 t}H_\nu^{(1)}(z),\phi_{\alpha_2}(t) = \sqrt{\frac{\pi}{4H_0}}e^{- \frac{3}{2}H_0 t}H_\nu^{(2)}(z),
([C_{1},C_{2},\cdots,C_{s}]\cdot A)^{\bot H}=([C_{1}^{q},C_{2}^{q},\cdots,C_{s}^{q}]\cdot A^{(q)})^{\bot E}&=[(C_{1}^{q})^{\bot E},(C_{2}^{q})^{\bot E},\cdots,(C_{s}^{q})^{\bot E}]\cdot [(A^{(q)})^{-1}]^{t}&=[C_{1}^{\bot H},C_{2}^{\bot H},\cdots,C_{s}^{\bot H}]\cdot [(A^{(q)})^{-1}]^{t}&\subseteq[C_{1},C_{2},\cdots,C_{s}]\cdot [(A^{(q)})^{-1}]^{t}.
\sum_{k=1}^{n}\frac{y^k}{m^{n-k}}\prod_{h=0}^{k-1}\left(1-\frac{h}{m}\right)a_{n,k}=y\left[\sum_{k=1}^{n-1}\frac{y^k}{m^{n-k-1}}\prod_{h=0}^{k-1}\left(1-\frac{h}{m}\right)a_{n-1,k}+\frac{1-y}{m}\sum_{k=1}^{n-1}\frac{y^{k-1}}{m^{n-k-1}}k\prod_{h=0}^{k-1}\left(1-\frac{h}{m}\right)a_{n-1,k}\right] &
p_{G}= ( \nu - 1 ) m_{\nu}^{\nu - 1}\! \left( \frac{\ddot{a}}{n^{2}a}- \frac{\dot{n}\dot{a}}{n^{3}a}\right) + \frac{1}{2}( \nu - 1 ) ( \nu - 2 ) m_{\nu}^{\nu - 1}\! \left( \frac{\dot{a}^{2}}{n^{2}a^{2}}+ \frac{k}{a^{2}}\right) .
d s^{2}= \left({\frac{1 - | Z |^{2}}{1 - | Z_{\infty}|^{2}}}\right)^{- 1}\, d t^{2}- \left({\frac{1 - | Z |^{2}}{1 - | Z_{\infty}|^{2}}}\right) \, d \vec{x}^{2}\ .
Y_{m+k+1-n,n-1}=\Delta^2Z_{m+k+1-n,n}-\Delta^1Z_{m+k+2-n,n-1}+\xi_1u^iC_{i(2n-m-k-1)(1)}({\partial\over\partial u^*_1})^{2n-m-k-1}Q_1({\partial\over\partial u^*_2})^1Q_2,
G_{u,v}L_{\phi,\psi}f(x) = \int_a^b\Big(\big(-\partial_y^2+V(y)\big)G_{u,v}(x,y)\Big)f(y)\d y +\lim_{y\to a}\big(G_{u,v}(x,y)f'(y)-\partial_yG_{u,v}(x,y)f(y)\big) -\lim_{y\to b}\big(G_{u,v}(x,y)f'(y)-\partial_yG_{u,v}(x,y)f(y)\big) =f(x)+v(x)W(u,f;a)-u(x)W(v,f;b)= f(x).
{\mathcal A}^{(k)}:=\frac{1}{2}\,\sum_{i=1}^k \frac{\partial^2}{\partial z_i^2}+\theta\,\sum_{i=1}^k \, \sum_{\substack{1\le j\le k j\neq i}}\frac{1}{z_i-z_j}\,\frac{\partial}{\partial z_i},
[ (\alpha_1, \alpha_2), (\alpha_1, \alpha_2)]_c = ( [\alpha_1, \alpha_1], [\alpha_1, \alpha_2] + [\alpha_2, \alpha_1], [\alpha_2, \alpha_2]) = ([\alpha_1, \alpha_1], 2[\alpha_1, \alpha_2], [\alpha_2, \alpha_2]).
I = \left| \frac{\partial \chi}{\partial c_{m}}\right| \int{\cal D}\{c_{m}\}{\cal D}\{c \}{\cal D}\{\bar{c}\}{\cal D}\{a \}\exp \left\{- \sum_{m \neq 0}c_{m}^{2}E_{m}- \pi c_{0}^{2}M - c \bar{c}M \right\}
\dot{x}{}^{a}\rightarrow \dot{f}{}^{- 1}\dot{x}{}^{a}, \acute{x}^{a}\rightarrow \acute{\varphi}{}^{- 1}\acute{x}{}^{a}- ( \acute{\varphi}\dot{f}){}^{- 1}\acute{f}\dot{x}{}^{a}.
\int x^k\, d\nu_{m}^a = \frac{1}{d_{l+1}\cdots d_m}\sum_{j=1}^{d_{l+1}\cdots d_m}\int\,x^k\,d\lambda_j &=\frac{1}{d_{l+1}\cdots d_m}\sum_{j=1}^{d_{l+1}\cdots d_m}\frac{s_k(F_l(z)-\beta^j)}{d_1\cdots d_l}&=\frac{1}{d_{l+1}\cdots d_m}\sum_{j=1}^{d_{l+1}\cdots d_m}\frac{s_k(F_l(z))}{d_1\cdots d_l}=\frac{s_k(F_l(z))}{d_1\cdots d_l},
\hat B_{\ell}(\varepsilon,s ) = \left \{\Big | \mu(s) -\frac{1}{t_{\ell+1}- t_{\ell}}\sum_{m= t_\ell}^{t_{\ell+1}-1}\mathbf{1}_{\{Y(m) =s \}}\Big | < \frac{\varepsilon}{2}\right \}.
\bar{\cal X}_{, i}^{\mu}\bar{\cal X}_{, j}^{\nu}{\cal G}_{\mu \nu}= \bar{g}_{i j}, \; \; \bar{\cal X}_{, i}^{\mu}\bar{\eta}^{\nu}{\cal G}_{\mu \nu}= 0 , \; \; \bar{\eta}^{\mu}\bar{\eta}^{\nu}{\cal G}_{\mu \nu}= \varepsilon
P_{l}^{m}(x)=(-1)^{m}\cdot2^{l}\cdot(1-x^{2})^{m/2}\cdot\sum_{k=m}^{l}\frac{k!}{(k-m)!}\cdot x^{k-m}\cdot(\begin{matrix}l\\k\end{matrix})(\begin{matrix}\frac{l+k-1}{2}\\l\end{matrix})
\Biggl [ -{\frac{g^{2}}{( 2 \pi )^{2}}}{\frac{d^{2}}{d z^{2}}}+{\frac{g^{2}}{2}}\bigl [ C ( z ) + D_{1 / 2}( z ) \bigr ] \biggr ] \bar{\zeta}( z ) = \epsilon_{1}\bar{\zeta}( z ) .
(x_2 \otimes g) \epsilon_{0,1}^* \smile (x_1 \otimes h) \epsilon_{1,0}^* = \chi_{g,1}^1 q^{1-1}(-1)^1(x_1 x_2 \otimes gh) \epsilon_{1,1}^* &=-\chi_{g,1}(x_1 x_2 \otimes gh) \epsilon_{1,1}^* &=-(x_1 x_2 \otimes gh) \epsilon_{1,1}^*
\cos\!\left(\frac{2\pi n}q\right) =1-\frac{q}{q-1}\chi_0(n) + \frac{\sqrt{q}}{q-1}\sum_{\substack{\chi\pmod*{q}\chi(-1)=1 \chi\neq\chi_0}}\overline{\epsilon_\chi}\chi(n), \sin\!\left(\frac{2\pi n}q\right) =\frac{\sqrt{q}}{q-1}\sum_{\substack{\chi\pmod*{q}\chi(-1)=-1}}\overline{\epsilon_\chi}\chi(n),
G_{l + \nu}( r^{\prime \prime}, r^{\prime}; E ) = \sum_{n = 0}^{\infty}\prod_{\alpha = 1}^{n}\left[ \int_{0}^{\infty}d r_{\alpha}V ( r_{\alpha}) \right] \prod_{\beta = 0}^{n}\left[ G_{l + \nu}^{( 0 )}( r_{\beta + 1}, r_{\beta}; E ) \right] .
\Delta_{\rho,j}f(x^j_k) = \sum_{i: \rho(x^j_k,x^j_i)\leq 2^{-j+2}}\left(f(x^j_i)-f(x^j_k)\right) &= \Bigl(\sum_{i: \rho(x^j_k,x^j_i)\leq 2^{-j+2}}f(x^j_i)\Bigr)-3 f(x^j_k).
| S^{-1}\int_0^S h(u_{z}^{N,M}(\tau),y_{z}^{N,M}(\tau),z)d\tau - \int_{U\times Y}h(u,y,z)\mu^{N,M}(du,dy|z))|\leq \phi_h(S)\ \ \forall z\in Z, \ \ \ \ \ \ \ \ \ \lim_{S\rightarrow\infty}\phi_h(S)= 0.
\nu_{i}\left(\mathcal{F}_{i}(x)\right)=\begin{cases}1 \mathcal{F}_{i}(x)\geq y_{i}^{0}, n_{i}(\mathcal{F}_i(x)) y_{i}^{0}\geq \mathcal{F}_{i}(x)\geq y_{i}^{1}, 0 \mathcal{F}_{i}(x)\leq y_{i}^{1},\end{cases}
P_{\sigma}^{G G}= \alpha^{2}x \sigma \left( \frac{2 ( K^{\prime})^{2}}{x^{2}}e^{2 \gamma \psi}+ ( H^{\prime})^{2}+ ( \psi^{\prime})^{2}\right) \ , \ \P_{H}^{G G}= 2 \sigma H K^{2}\ .
\sum\limits_{j\neq i}^{n}\frac{2}{z_{i}-z_{j}}+\frac{\frac{1}{\gamma}}{z_{i}}+\frac{\frac{1}{2}\big(\delta-\frac{1}{\gamma}\big)}{z_{i}+1}+\frac{\frac{1}{2}\big(\delta-\frac{1}{\gamma}\big)}{z_{i}-1}=0.
\left| \int_{\mathbb S^1}B_xD^m JD^mB\, dx \right|\leq \ \|B_x\|_{L^\infty}\|D^mB\|_{L^2}\|D^mJ\|_{L^2}\left| \int_{\mathbb S^1}BD^m JD^mB_x\, dx \right|\leq \ \|B\|_{L^\infty}\|D^mB_x\|_{L^2}\|D^mJ\|_{L^2}\left| \int_{\mathbb S^1}JD^m B_xD^mB\, dx \right|\leq \ \|J_x\|_{L^\infty}\|D^mB\|_{L^2}^2.
\langle \bar{\psi}\psi \rangle \Big|_{m^2d^2={\cal O}(0.1)}\; \approx \; \frac{1}{4 \pi^2}\; m \; \rho \; \; c_0 \Bigl( m^2 \, d^2 \Bigr)\Big|_{m^2d^2={\cal O}(0.1)}\; \approx \; \; 0.058 \; \frac{1}{4 \pi^2}\; \rho \, / \, d \; ,
\frac{\,\mathrm{d}P_{s}}{\,\mathrm{d}s}(M^{F}(\tau )):=\lim_{h\downarrow 0}\frac{P_{s}(M^{F}(\tau ))-P_{s-h}(M^{F}(\tau ))}{h}~\mbox{in $L^2(\Omega)$}.
\frac{\partial}{\partial r}\Re{f_{\tau, \theta}(r + iy)}=\frac{4\tau}{1 - \tau^{2}}\cos^{2}\theta + \frac{1 - \tau^{2}}{\tau}\frac{r^{2}- r(\tau + \frac{1}{\tau}) + 1 - y^{2}}{\big((r - \frac{1}{\tau})^{2}+ y^{2}\big) \big((r - \tau)^{2}+ y^{2}\big)}.
\widetilde Y_{\tau}^{t,x}=\int_\tau^Te^{-(s-\tau){A}}G B(s,X^{t,x}_s)\,ds+\int_\tau^Te^{-(s-\tau){A}}\widetilde Z_s^{t,x}\, B(s,X^{t,x}_s)\,ds- \int_\tau^Te^{-(s-\tau){A}}Z^{t,x}_{s}\;d\widetilde W_s,
\int_{|z|< 1}\log(|z|)d\mathrm{arg}(z)d\mathrm{arg}(z-1)=-\frac{1}{2 i}\sum_{n\geq 0}\int_{|z|< 1}\log(|z|)d\arg(z)z^ndz+\frac{1}{2 i}\sum_{n\geq 0}\int_{|z|< 1}\log(|z|)d\arg(z)\overline z^nd\overline z,
B=\frac{1}{4}[\begin{matrix}\frac{1}{a}&\frac{1}{b}&\frac{1}{b}&\frac{1}{a}\\ \frac{1}{b}&-\frac{1}{c}&\frac{1}{c}&-\frac{1}{b}\\ \frac{1}{b}&\frac{1}{c}&-\frac{1}{c}&-\frac{1}{b}\\ \frac{1}{a}&-\frac{1}{b}&-\frac{1}{b}&\frac{1}{a}\end{matrix}],
\overline{H}\left(\bigsqcup_{a \in \mathcal{A}}^{P_A}G_a^{\wedge 3}\right) = 3 \overline{H}\left(\bigsqcup_{a \in \mathcal{A}}^{P_A}G_a\right) = 3\sum_{a \in \mathcal{A}}P_A(a) \overline{H}(G_a) = 3;
\Pi_{\delta}( x ) = m_{*}^{2}( x ) -{\frac{g^{2}}{2}}\int{\frac{d^{4}x^{\prime}d^{4}p}{( 2 \pi )^{4}}}\;{\frac{d^{3}k}{( 2 \pi )^{3}E_{k}}}\;{\frac{e^{- i p ( x^{\prime}- x )}}{p^{2}- m_{*}^{2}+ i p_{0}0^{+}}}\; f_{0}( x^{\prime},{\bf k}) \; ,
F_{1}( q^{2}) = Z + \int \frac{{\mathrm d}^{2}{\vec{k}}_{\perp}{\mathrm d}x}{1 6 \pi^{3}}\Big [ \psi_{+ \frac{1}{2}}^{\uparrow \ *}( x ,{\vec{k}^{\prime}}_{\perp}) \psi_{+ \frac{1}{2}}^{\uparrow}( x ,{\vec{k}}_{\perp}) + \psi_{- \frac{1}{2}}^{\uparrow \ *}( x ,{\vec{k}^{\prime}}_{\perp}) \psi_{- \frac{1}{2}}^{\uparrow}( x ,{\vec{k}}_{\perp}) \Big ] \ ,