harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
w={}\frac{1}{2(a^2+c^2)}\bigg( 1 - a^2 \left( x+ \frac{b}{a\sqrt{D}}\right)^2{}- b\left( x + \frac{b}{a\sqrt{D}}\right) \left( y- \frac{2a}{\sqrt{D}}\right) - c^2 \left( y -\frac{2a}{\sqrt{D}}\right)^2 \bigg).
\tilde H(z,w,\xi, (\Phi_{ij}^{(\beta)})_{|\beta| \leq k-1}):=\eta_0(z,w)\tilde h_0(\xi, (\Phi_{ij}^{(\beta)})_{|\beta| \leq k-1})+ \cdots +\eta_{s}(z,w)\tilde h_{s}(\xi, (\Phi_{ij}^{(\beta)})_{|\beta| \leq k-1})=0.
f =2\pi\tau_2^2\left(\frac{2t^3}{3}\right)-\frac{\tau_2^2\chi}{2(2\pi)^2}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\frac{1}{|m\tau+n|^3}&=4\pi \tau_2^2\left(\frac{t^3}{3}-\frac{\chi}{8(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}\right)\, &>4\pi\tau_2^2R_1(t,\tau).
\Sigma_1=(-n^2-4l^2+n+4ln)\big(\Phi_b^{\sigma}\big)^2+\frac{1}{b^{2n}}\sum_{\substack{j=1}}^{n}\sum_{\lambda,N=1}^{b^n}\varphi_{b,\varepsilon_j(\lambda,N,\Sigma)}^{\sigma}\left(\frac{N}{b^j}\right)\varphi_{b,\varepsilon_j(\lambda,N,\Sigma^{\ast})}^{\overline{\sigma}}\left(\frac{N}{b^j}\right)
{\mathbb S}_{k_-}(u_-,s) ={\mathbb S}_{k_-}^{\alpha_{k^*}}(u_+(0)) + \int_0^1 u_+^*\eta_{k_-}+ \int_{2/3}^{s}u_-^*\eta_{k_-}&={\mathbb S}_{k_-}(u_+,1) +{\mathbb S}_{k_-}^{\alpha_{k^*}}(u_-(s)) -{\mathbb S}_{k_-}^{\alpha_{k^*}}(\beta^n) & \leq{\mathbb S}_{k_+}(u_+,1).
r\circ\,s\,\circ\,S_0\,\circ\,\overline{\Omega}=r\circ S_1\circ R^{NC}_{1}\circ \Omega_Y =r\circ\,R^{PC}_{1}\circ T_1 \circ \Omega_Y =R^{PC}_{2}\circ p'\circ T_1 \circ \Omega_Y \stackrel{R^{PC}_{2}\circ p'(\eta)}{\leadsto}0.
\int \dfrac{x}{(- x^2 (1-q)^2;q^2)_\infty}\left(\frac{q^{-1}-q^{-\nu}[\nu]_q^2}{x}-\frac{cq^{-\nu}[\nu]^2}{x^2}+ c+qx \right) J_\nu^{(2)}( x |q^2) d_qx = & \frac{x}{q(- x^2 (1-q)^2;q^2)_\infty}\left( J_\nu^{(2)}( x |q^2)-(c+x) D_{q^{-1}}J_\nu^{(2)}( x |q^2) \right),
V_{\theta}={\frac{1}{( 4 \pi )^{2}}}\left[{\frac{{\cal I}_{K}}{z_{<}^{4}}}+{\frac{{\cal I}_{I}}{z_{>}^{4}}}+{\frac{{\cal V}( \tau )}{z_{>}^{4}}}\right] +{\frac{\beta_{4}( \beta \theta + 1 )}{( 4 \pi )^{2}}}\left[{\frac{1}{z_{+}^{4}}}\ln \left({\frac{z_{+}}{z_{0}}}\right) +{\frac{1}{z_{-}^{4}}}\ln \left({\frac{z_{-}}{z_{0}}}\right) \right]
\begin{aligned}\mathcal{B}^{(2)}=-\mathcal{C}^{-1}\left(\mathcal{B}^{(2)}\right)^T\mathcal{C}=\frac{1}{8}\begin{pmatrix}0 8 0 0 0 0 0 0 8 8 0 0 -u_3 -u_2 0 0 8 0 0 0 0 0 8 0 u_{4,\zeta}0 0 -u_2 0 8 0 u_{4,\zeta}0 u_3 0 0 \end{pmatrix}\ ,\end{aligned}
I^{(1)}(a) \; = \; \frac{2\pi}{T}D_{ia}(T/2\pi) \;+\; \mathcal{O}\Big(T^{-1/2+\varepsilon}\big( 1+ |a|^{1/6}\big)\Big). &\; = \; \zeta(1+ia)+\zeta(1-ia)\Big(\frac{T}{2\pi}\Big)^{-ia}\frac{1}{1-ia}\;+\; \mathcal{O}\big(T^{-1/3}(1+|a|^{1/6})\big),
\frac{1}{|\Omega |}\sum_{z_{1},\ldots ,z_{N}\in \Omega}\sum_{x\in \bigcap_{i=1}^{N}\Omega_{z_{i}p_{i}}}\sum_{\theta \in \Omega_{g}(x|z_{1}p_{1},\ldots ,z_{N}p_{N})}\prod_{i=1}^{N}\frac{\overline{\phi_{i}}\left( c_{i}^{z_{i}},x^{z_{i}}\right)}{|\Omega_{p_{i}}|}\prod_{\xi \in \mathcal{O}(x|\theta )}\mathcal{D}_{\xi}(z_{1}p_{1},\ldots ,z_{N}p_{N})
\exp \left( \sum_{g = 0}^{\infty}x^{2 g - 2}\mathcal{F}_{g}\right) = e^{- \frac{D ( 0 , 0 , 0 )}{2}\zeta ( 1 )}\prod_{( \ell , n , \gamma , j ) > 0}( 1 - p^{\ell}q^{n}\zeta^{\gamma}y^{j})^{D ( \ell n , \gamma , j )}.
\mathbf{sinh}\,\mathbf{z}=(e^{\sinh (\ln r)\cos \theta},\cosh (\ln r)\sin \theta ), \mathbf{cos}\,\mathbf{z}=(e^{\cosh \theta \cos (\ln r)},-\sinh \theta \sin (\ln r)), \mathbf{sin}\,\mathbf{z}=(e^{\cosh \theta \sin (\ln r)},\sinh \theta \cos (\ln r)).
T_{F, \widetilde{G},N}^L(f_1,\dots,f_4) \approx \frac{1}{N^2}\sum\limits_{\substack{\mathbf{n}\in \mathbb{Z}^4 \Vert L \mathbf{n}\Vert_\infty \leqslant 10 - \delta (\begin{smallmatrix}5 1 \end{smallmatrix})L\mathbf{n}= 0}}\Big(\prod\limits_{j=1}^4 f_j(n_j)\Big) F(\mathbf{n}).
c_* \gamma_f = \sum_{x \in E_N}\xi_f(x) \{c(g_x \rho),c(g_x \rho^2)\}& = \sum_{x \in E_N}\xi_f(x) \{g_{x^c}\rho^2,g_{x^c}\rho\}& = - \sum_{x \in E_N}\xi_f(x^c) \{g_x \rho,g_x \rho^2\}.
Lu_n = \sum_{2 \leq j+\alpha \leq n}\!\! a_{j,\alpha}(t,x) \sum_{|\vec{k}|+|\vec{m}|=n}u_{k_1}\cdots u_{k_j}\frac{\partial u_{m_1}}{\partial x}\cdots \frac{\partial u_{m_{\alpha}}}{\partial x}
\alpha_{i}= \frac{g_{i}^{2}}{4 \pi}~ \left( i = 1 , 2 , 3 , ~ ~ g_{1}= \sqrt{\frac{5}{3}}g^{\prime}\right) , ~ ~ \alpha_{T}= \frac{h_{T}^{2}}{4 \pi}, ~ ~ \alpha_{\lambda}= \frac{\lambda}{4 \pi},
|D\gamma|+|D^2\gamma|\leq C(n,\Omega) \quad\Omega_{\mu_{1}}, \sum_{1\leq i\leq n}\gamma^iD_j\gamma^i=0, \sum_{1\leq i\leq n}\gamma^iD_i\gamma^j= 0, \,|\gamma|=1 \quad \quad\Omega_{\mu_{1}}.
\begin{cases}w^1_{10}v^2_{01}-w^1_{01}v^2_{10}-w^2_{10}v^1_{01}+w^2_{01}v^1_{10}=0\,, w^2_{10}v^2_{01}-w^2_{01}v^2_{10}+w^1_{10}v^1_{10}-w^1_{01}v^1_{10}=0\,,\end{cases}
\sigma_{1}= \begin{pmatrix}0 1 1 0\end{pmatrix}, \sigma_{2}= \begin{pmatrix}0 -i i 0\end{pmatrix}, \sigma_{3}= \begin{pmatrix}1 0 0 -1\end{pmatrix}, \sigma_{4}= \begin{pmatrix}1 0 0 1\end{pmatrix}.
\ell_{1}^{\mu}( \tau ) = \bar{\ell}_{0}^{\mu}+{\frac{1}{2}}\delta^{\mu}+ V^{\mu}\tau , \, \ell_{2}^{\mu}( \tau ) = \bar{\ell}_{0}^{\mu}-{\frac{1}{2}}\delta^{\mu}+ V^{\mu}\tau ,
I_2 \leq C_1 r_n\sqrt{2}\frac{n}{N}\exp\left(-C_2 \frac{n}{N}\right) = C_1\sqrt{2}\left(r_n \sqrt{\frac{n}{N}}\right) \left(\sqrt{\frac{n}{N}}\exp\left(-C_2 \frac{n}{N}\right)\right)
{\rm d}^{\,2}f(\bar{x}\,|\,0)(d)\, =\, \max_{\lambda}\left\{\,\lambda\, d_{\,1}^{\,2}- Q_{11}d_{\,1}^{\,2}\, \mid \,\lambda = Q_{22}\in (0,1)\,\right\}= (Q_{22}- Q_{11})\,d_1^2.
\|\sin\Theta(\hat V, V)\|_F^2 = \sum_{i=1}^r \sin^2(\cos^{-1}(\sigma_r)) = \sum_{i=1}^r (1 - \sigma_r^2) = r - \sum_{i=1}^r \sigma_i^2 = r- \|\hat V^{\intercal}V\|_F^2.
A = \partial^2_{F}W(0,\mbox{Id})(\nabla \bar u-\mbox{Id})+ \partial_{\phi}\partial_F W(0,\mbox{Id})\bar \phi- \partial_FW(\bar \phi,\nabla \bar u), & B = \partial_\phi W(\bar \phi,\nabla \bar u)-\partial^2_{\phi}W(0,\mbox{Id})\bar \phi + \partial_{\phi}\partial_F W(0,\mbox{Id})(\nabla \bar u - \mbox{Id}).
c(\iota_{D}^{*}W^{-}_{i}(\nu^{j}))=\frac{(1-x)^{1+m_{j,1}-k-g_{Y}}}{(1-2x)^{1+m_{j,2}-2k-g_{Y}}}e^{-\frac{\theta}{1-x}+\frac{4\theta}{1-2x}}.
G^{({\cal Q})}(x_{1}, x_{2}, x_{3}, x_{4}) = \langle{\cal Q}^{i_{1}j_{1}}(x_1){\cal Q}^{i_{2}j_{2}}(x_2){\cal Q}^{i_{3}j_{3}}(x_3){\cal Q}^{i_{4}j_{4}}(x_4)\rangle \;
\Bbbk\star\emptyset=\emptyset\star\Bbbk =\Bbbk (\Bbbk,2)\star\Bbbk'=\Bbbk\star(\Bbbk',2) =(\Bbbk\star\Bbbk',2) (\Bbbk,3)\star(\Bbbk',3) =(\Bbbk\star(\Bbbk',3),3)+((\Bbbk,3)\star\Bbbk',3)+(\Bbbk\star\Bbbk',2,2,2).
{\bar{A}_1^{T}(d) \over (4\pi)^{d/2}}={\Gamma\left(\frac d2-1\right) \over 12 \gamma (d-3)!}\left[d(d-7)+(d-1)\left(\gamma^2 -1\right) +12\left(1-\gamma\right)\right]+\delta_{d,2}~~~,
\|\sin \Theta(U, \hat{U})\| \leq \frac{\alpha z_{21}+ \beta z_{12}}{\alpha^2 - \beta^2 - z_{21}^2\wedge z_{12}^2}\wedge 1, \|\sin \Theta(U, \hat{U})\|_F \leq \frac{\alpha\|Z_{21}\|_F + \beta\|Z_{12}\|_F}{\alpha^2 - \beta^2 - z_{21}^2\wedge z_{12}^2}\wedge \sqrt{r}.
B_{n,k}(0,\dots,\, q!c_q, (q+1)!c_{q+1}, 0,\dots) =\sum_{\kappa\le k,\, \nu\le n}\binom{n}{\nu}B_{\nu,\kappa}(0,\dots, q!c_q, 0,\dots) B_{n-\nu,k-\kappa}(0,\dots, (q+1)!c_{q+1}, 0,\dots),
u_n(\theta) = \sup \left\{\sum_{i=1}^n \frac{1}{x_i}: (x_i)_{i=1}^n \in U_n(\theta) \right\}& = \sup \left\{\sum_{i=1}^n \frac{1}{x_i}: (x_i)_{i=1}^n \in U_n(\theta) x_1 < x_1^* \right\}= \sup \left\{\sum_{i=1}^n \frac{1}{x_i}: (x_i)_{i=1}^n \in U_n^{(1)}(\theta) \right\}.
\phi_m=N_m e^{{5 \over 4}k \rho}\left[J_{3/2}\left({2m \over k}\right)Y_{5/2}\left({2m \over k}e^{{k \over 2}\rho}\right)-Y_{3/2}\left({2m \over k}\right)J_{5/2}\left({2m \over k}e^{{k \over 2}\rho}\right)\right]
P^{(i)}_{|\frac{p}{N_{0}}\rightarrow \infty}(\gamma_{})=\frac{m!(m-i+1)^{1-i}}{(i-1)!(m-i)!(n-i+1)!}&\times \left(\frac{\gamma_{}\kappa^{2}_{R}m}{(1-\gamma_{}\kappa^{2}_{T})}\right)^{n-i+1}+o\left(\left(\frac{p}{N_{0}}\right)^{-(n-i+1)}\right).
{\cal H}_{( m )}-{\cal E}_{( m )}= \int_{\Sigma}\sqrt{\sigma}d^{2}z Q_{( m )}^{\mu \nu}( t ) l_{\mu}p_{\nu}+ \int_{C_{t}}\sqrt{\gamma}d^{2}y Q_{( m )}^{\mu \nu}( t ) u_{\mu}n_{\nu},
\Omega_{5,\lambda}:= \left\{\sup_{t_\lambda \le s \le t}\big| X(s) \big| \le C_8 \right\}\cap \left\{\sup_{\theta_3 (t_\lambda, t) \le s \le t}\big| Z \big( s, \theta_3 (t_\lambda, t) \big) - 1 \big|^2 \le \frac{\ 1 \}{4}\right\}.
k_{i}= k[ 1-\sigma_{2}+ 2^{-i}(\sigma_{2}-\sigma_{1}) ] \,, i=0\,,1\,,2\,,\dots t_{i}= \frac{t}{2}[ 1-\sigma_{2}+ 2^{-i}(\sigma_{2}-\sigma_{1}) ] \,, i=0\,,1\,,2\,,\dots
h'_1 =\frac{1}{4(h_2-h_1)}\left(\alpha^3-\alpha^1\right), h'_2 =\frac{1}{4(h_2-h_1)}\left(\alpha^3-\alpha^1\right)+\frac{1}{4(h_3-h_2)}\left(\alpha^4-\alpha^2\right), h'_3 =\frac{1}{4(h_3-h_2)}\left(\alpha^4-\alpha^2\right).
l^{(q,r)}_a(x) = \overline{Z}^{(q,r)}_a (x)-\frac{\psi^\prime (0+)}{q}Z^{(q,r)}_a(x) +\frac{\psi^\prime(0+)}{q}+ r \frac{\psi^\prime(0+)}{q}\overline{W}^{(q + r)}(x).
\mathcal{I}\left( ab\right) =\mathcal{I}\left( a\right) +1_{n,2}\left(\left( a\otimes a\right) \mathcal{I}\left( b\right) \right) +1_{n,2}\left(\left( a-1\right) \otimes \left( a\left( b-1\right) \right) \right) ,\forall a,b\in \mathcal{G}_{n}
\Phi ( r ) = C_{1}e^{- \frac{i \omega \ell^{2}}{2 r_{H}}\ln ( \frac{r - r_{H}}{r + r_{H}})}+ C_{2}e^{\frac{i \omega \ell^{2}}{2 r_{H}}\ln ( \frac{r - r_{H}}{r + r_{H}})}.
M = \frac{q_{0}^{- 1}}{6 H}\left( \begin{matrix}{{{( \frac{{\partial^{2}}V}{\partial{T_{1}}^{2}})}_{0}}}&{{{( \frac{{\partial^{2}}V}{\partial{T_{1}}\partial{T_{2}}})}_{0}}}\\{{{( \frac{{\partial^{2}}V}{\partial{T_{1}}\partial{T_{2}}})}_{0}}}&{{{( \frac{{\partial^{2}}V}{\partial{T_{2}}^{2}})}_{0}}}\\ \end{matrix}\right)
\ll \frac{W^{k+1}B^{k+1}\mathfrak{S}_{WB}(\mathcal{L})^2 (\log{R})^{k-1}}{\phi(WB)^{k+1}}\prod_{p\nmid WB}\Bigl(1+\frac{\omega(p)-1}{p+O(k)}\Bigr)\Bigl(1-\frac{1}{p}\Bigr)^{k-1}&\qquad\times \idotsint\limits_{t_1,\dots,t_k\ge 0, t_m=0}F_2(t_1,\dots,t_k)^2\prod_{i\ne m}dt_i.
\lefteqn{\|{\mathcal M}b_j\|_{L^p([-1,1]^n)}}&\lesssim\left\|{\mathcal M}f \cdot \chi_{{\mathcal O}_j}\right\|_{L^p([-1,1]^n)}+\left(\sum_{l=1}^\infty\left(2^{-\frac{n\theta}{p}}\left\|2^j \chi_{{\mathcal O}_j}\right\|_{L^{p}([-2^l,2^l]^n)}\right)^{\frac{1}{\kappa}}\right)^{\kappa}.
h_1(x) = h_3(x) = \begin{pmatrix}x_2 -x_1 + x_2(1-x_1^2)- 1 \frac{x_1^2+ x_2^2}{2}\end{pmatrix}, h_2(x) = \begin{pmatrix}x_2 -x_1 + x_2(1-x_1^2)+ 1 \frac{x_1^2+ x_2^2}{2}, \end{pmatrix},
\cos\beta_{1}\cos\beta_{2}\cos\beta_{3}=\frac{1}{4}\left\vert\ 1+X(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})+i[\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}]\ \right\vert \Theta(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}) =2\arg(1+X(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})+i[\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}])
\mu_{2n}\left( z\right) =2{\displaystyle\int\limits_{0}^{z}}x^{2n}e^{-x^{2}}dx={\displaystyle\int\limits_{0}^{z^{2}}}s^{n-\frac{1}{2}}e^{-s}ds=\widehat{\gamma}\left( n+\frac{1}{2},z^{2}\right),
\mathcal{A}_j \mathfrak{n}(\varphi(s_i)) w = \mathfrak{n}(\varphi(s_i)) \mathcal{A}_j w + [\mathcal{A}_j, \mathfrak{n}(\varphi(s_i))] w = \begin{cases}(\delta_j+1) w j\neq i (\delta_i+2) w j=i. \end{cases}
1-L(\rho+w,{\rm sym}^m f)M_x(\rho+w,{\rm sym}^m f) & = L(\rho+w,{\rm sym}^m f)G_f(\rho+w)\sum_{\substack{n> x (n,P(z))=1}}\frac{\mu(n)\lambda_{{\rm sym}^m f}(n)}{n^{-\rho-w}}\ll (kN)^{\varepsilon}.
w_m(\lambda, r,r') := I_{|m+\alpha|}(\sqrt{|\lambda|}\, r)\, K_{|m+\alpha|}(\sqrt{|\lambda|}\, r') - \frac{\big(r/r'\big)^{|m+\alpha|}}{2\, |m+\alpha|}\, + \, \frac{\delta_{m,k(\alpha)}\ |\lambda|^{\mu}\pi\, 4^{-\mu}\, (r r')^{\mu}}{2 \sin(\mu \pi)\, \mu^2(\alpha)\, \Gamma^2(\mu )},
<W_{1}W_{2}W_{3}>= \frac{<[W_{1},[W_{2},W_{3}]]>}{(p_{1}-2m_{1})(p_{3}+2m_{1}+2m_{2})}+\frac{<[W_{2},[W_{1},W_{3}]]>}{(p_{2}-2m_{2})(p_{3}+2m_{1}+2m_{2})}.
\mathcal M(a_1\dots a_n;a) \cong (\mathcal M\wr\widetilde{\mathbb E}_{\mathcal G})(a_1\dots a_n,a)_{\mu_n}&\xrightarrow[\cong]{H}\mathcal C(H(a_1\dots a_n),H(a))_{\mu_n}\ .
L_n^{\alpha}(x) = \sum\limits_{j=0}^n\frac{(\alpha +1)_n\cdot (-n)_j}{(\alpha+1)_j \cdot n!}\cdot \frac{x^j}{j!}= \sum\limits_{j=0}^n\frac{(-1)^j}{j!}\cdot{n+\alpha \choose n - j}\cdot x^j
\Psi_{j}(k,t,y,s) = U^{\theta}(k,t) - u_{\theta}(y,s) - \frac{(-k\Delta x - y)^{2}}{2 \eta}- \frac{(t - s)^{2}}{2 \eta}- bt \quad- \sqrt{\theta}(T + 1 - t)^{-1}+ g_{\eta,\nu}(-\nu y) - \alpha (k\Delta x)^{2}.
\begin{aligned}\pi^{Sp(n)}_{d,e}(d,e,0,\ldots,0,0), begin{align*}.3ex] \dim \pi^{Sp(n)}_{d,e}= \frac{(d+e+2n-1)(d-e+1)\prod_{j=1}^{2n-3}(d+j+1)(e+j)}{(2n -1)(2n-2)\cdot [(2n-3)!]^2}. \end{aligned}
\begin{aligned}\mathbb{E}\bigg[\Big(\int_{E_\varepsilon}\delta f(s,v(s))+f_z(s)p^0(s)\delta\sigma(s,v(s))+p^0(s)\delta b(s,v(s))+p^1(s)\widehat{\delta b(s,v(s))}ds\Big)^4\bigg]\leq \varepsilon^{2}\rho(\varepsilon).\end{aligned}
-\frac{1}{3}\frac{1}{2^{9}}\frac{1}{\pi^4}\big{[}\Omega_5\big{]}^{\frac{1}{2}}cot{\theta_4^0}\sum_{N=1}^{\infty}\frac{C_N^{\frac{3}{2}}(cos{\theta_4^0})}{N(N + 3)}C_N^{\frac{3}{2}}(cos{\theta_4})\big{[}\frac{L^2 - 2(N + 1)(N + 2)}{2a^{2}}\big{]}\frac{1}{\epsilon}+ F. T.
\frac{|\|z\|^\gamma-\|w\|^\gamma|}{\|\zeta\|^\gamma}\|\tilde{k}_0(\zeta,z)\| \leq \|z-w\| \sum_{\ell=0}^{\gamma-1}\frac{1}{\|\zeta\|^{\gamma-\ell}\|\zeta-z\|^{2n-(\gamma-\ell)}}\lesssim \|z-w\| \left(\frac{1}{\|\zeta\|^{2n}}+ \frac{1}{\|\zeta-z\|^{2n}}\right).
Z_{\lambda}=\Theta_{\lambda}+\frac{1}{2}P_{\sigma\nu}\omega^{\sigma}\wedge\omega^{\nu}+Q_{\sigma,\nu}^{j}\omega^{\sigma}\wedge\omega_{j}^{\nu}+\frac{1}{2}R_{\sigma,\nu}^{i,j}\omega_{i}^{\sigma}\wedge\omega_{j}^{\nu},
R_{ab}(\xi )=\left ( \begin{array}{cccc}1 0 0 0 0 \frac{\xi^{-1}-\xi}{q^{-1}\xi^{-1}-q\xi}\frac{q^{-1}-q}{q^{-1}\xi^{-1}-q\xi}0 0 \frac{q^{-1}-q}{q^{-1}\xi^{-1}-q\xi}\frac{\xi^{-1}-\xi}{q^{-1}\xi^{-1}-q\xi}0 0 0 0 1 \end{array}\right)\, ,
E_{in}(z) = \frac{\sqrt{3}}{2\pi}n^{2\beta}\left(\frac{f(z)}{z}\right)^{\frac{2\beta}{3}}\left(\mathbb I - B_n^{(3)}(z) \right) \left(\mathbb I - B_n^{(2)}(z) \right)\left( \mathbb I - B_n^{(1)}(z) \right)E_n(z)
\int_0^t \int \Big(u^\eta\cdot w_t - (v\cdot \nabla) u^\eta \cdot w +\frac{A\gamma}{\gamma-1}(\phi^\eta)^{\frac{2\gamma-2}{\delta-1}}w \Big)xs = -\int u_0 \cdot w(0,x)+\int_0^t \int \Big(((\phi^\eta)^2+\eta^2) Lu^\eta\cdot w-\nabla (\phi^\eta)^2 \cdot Q(v) \cdot w\Big) xs.
\alpha_1= \delta_n^k-\alpha_k, \{\alpha_j= \delta_n^k+\alpha_{j-1}- \alpha_k\}_{j=2}^k \{\alpha_j=\alpha_{j+(i-1)}-\alpha_{i-1}\}_{j=1}^{k+1-i}, \ \ \delta_{n}^k=\alpha_{k+2-i}+\alpha_{i-1}, \ \{\delta_n^k+\alpha_{j}- \alpha_{i-1}= \alpha_{j+k+2-i}\}_{j=1}^{i-2}\ \k\geq i \geq 2
P\bigg( \bigg| A^3 - \big(N(\frac{\epsilon}{4},\frac{\delta}{4})\big)^{-1}\sum_{i=1}^{N(\frac{\epsilon}{4},\frac{\delta}{4})}A^0_i - Z^{k+1}_t(\gamma) \bigg| \geq \epsilon \bigg) \leq \delta\ \ \ \textrm{as desired.}
\tilde{u}_{\nu}=u_{\nu}\left[ \lambda \delta +\bar{\lambda}\left( 1-\beta_{0}x^{-1}\right) u_{0}-\bar{\lambda}\sum\nolimits_{\nu =0}^{d-2}\gamma_{1}^{d-1-\nu}\left( x^{-1}u_{\nu +1}\right) \right]^{-1}
b_{\scriptscriptstyle{L - 1 , L + 1}}/ b_{\scriptscriptstyle{L , L + 1}}= \frac{( i k a - 1 ) \, a \, \psi_{\scriptscriptstyle{L , L + 1}}^{>}( k , a )}{( \gamma_{\scriptscriptstyle{L + 1 , L + 1}}+ \frac{1}{2}) \, \psi_{\scriptscriptstyle{L + 1 , L + 1}}^{>}( k , a )}\, ( 1 + \mathrm{O}( k a )^{2}) \, ,
\Phi_{p}|_{L_{+}}=\sum_{\sigma \in \mathfrak{S}_{k}}\prod_{1\le i<j \le k}\left(1+\frac{(\alpha+\beta p_{\sigma(j)})(\gamma+\delta p_{\sigma(i)})}{p_{\sigma(j)}-p_{\sigma(i)}}\right)\prod_{i=1}^{k}p_{\sigma(i)}^{-\epsilon_{i}}.
w_{2p}'(t) \,\le\, -\frac{a}{p}\min_{\beta = 2,4}\biggl\{\Bigl( \frac{w_{2p}(t)}{\overline w_p(t)}\Bigr)^{\beta p}\biggr\}w_{2p}(t) + pb\alpha^2 w_{2p}(t)~, t > 0~,
A_m\geq n^{-1}\sigma^2 C_1\sum_{j=i_0}^{m-1}j^{\beta/r}\left(C_1 m^{\beta/r}-C_2 j^{\beta/r}\right)\geq K_1 n^{-1}m^{\beta/r}\sum_{j=i_0}^{m-1}j^{\beta/r}.
\mathfrak{M}(-n\,|\tau,\,\lambda_1,\,\lambda_2) = \prod\limits_{j=0}^{n-1}\frac{\Gamma(1+\lambda_1+\frac{(j+1)}{\tau}) \,\Gamma(1+\lambda_2+\frac{(j+1)}{\tau})\Gamma(1-\frac{1}{\tau})}{\Gamma(1+\lambda_1+\lambda_2+\frac{(j+1)}{\tau})\,\Gamma(1+\frac{j}{\tau})}.
\pi_\alpha^{0}* \tilde{H}^{(1)}* \pi_\alpha^{0}= \pi_\alpha^{0}* H^{(1)}* \pi_\alpha^{0}+ i \hbar \pi_\alpha^{0}\underbrace{\Delta\Gamma(X_{\lambda_\alpha})}_{\mbox{``Berry phase'' term}}\pi_\alpha^{0}
\rho_a\rho_x =\left(\sum_{k=0}^d \rho_{a,k}\tau^k\right)(x+[1]_x\tau+\tau^2) &=\sum_{k=0}^d\left(x^{2^k}\rho_{a,k}\tau^k+[1]_x^{2^k}\rho_{a,k}\tau^{k+1}+\rho_{a,k}\tau^{k+2}\right).
\left< W ( C ) \right>_{\mathrm{w e a k - f i e l d}}= \left< W ( C ) \right>_{a_{\mu}}\int{\cal D}h_{\mu \nu}\exp \left[ - \int d^{4}x \left( \frac{\Lambda^{2}}{2 4 \zeta}H_{\mu \nu \lambda}^{2}+ \frac 1 4 h_{\mu \nu}^{2}\right) + \frac{i g}{4}\int_{\Sigma_{\mathrm{m i n .}}}^{}d \sigma_{\mu \nu}h_{\mu \nu}\right] ,
f(X,Y) = \frac{R(X^\ell,Y^\ell)}{{\Big ( \big(\prod\limits_{j=0}^{\ell-1}{\delta_j}\big)X^\ell - (-1)^{\ell-1}\big(\prod\limits_{j=0}^{\ell-1}{\gamma_j}\big) Y^\ell \Big )}^t}\cdot
{\cal D}( g ) = \left( \begin{array}{c c c}{{z_{1}^{\prime}}}&{{- \bar{z}_{2}^{\prime}}}&{{- \chi}}\\{{z_{2}^{\prime}}}&{{\bar{z}_{1}^{\prime}}}&{{- \bar{\chi}}}\\{{\theta^{\prime}}}&{{- \bar{\theta}^{\prime}}}&{{\lambda}}\\ \end{array}\right) \, , \quad \quad \sum_{i}| z_{i}^{\prime}|^{2}+ \bar{\theta}^{\prime}\theta^{\prime}= 1
P(\varphi_{\alpha,\beta}^{\lambda_{\alpha}})({\bf y}) =g_{\alpha,\beta}^{\lambda_{\alpha}}\,, g_{\alpha,\beta}^{\lambda_{\alpha}}= -\sum_{\gamma\le\beta, |\gamma|\ge1}\binom{\beta}{\gamma}\big[(\partial_{\bf y}^{\gamma}V_{\alpha}^{\lambda_{\alpha}})({\bf y})\big](\partial_{\bf y}^{\beta-\gamma}\psi_{\alpha}^{\lambda_{\alpha}})({\bf y})\,.
f_1(w) =(2\pi)^{-1}\int_{\mathbb{R}}e^{-ihw}\frac{1}{2}e^{-|h|H}dh= \frac{1}{2}(2\pi)^{-1}\int_{\mathbb{R}}e^{-ihw}e^{-|h|H}dh =\frac{1}{2\pi}\frac{H}{H^2+w^2},
\gamma_{k+1}^{{\scriptscriptstyle (\mu)}}=\frac{\left(\gamma_{k}^{{\scriptscriptstyle (\mu)}}-\gamma_{k}\right)}{\mu\left(\gamma_{k}^{{\scriptscriptstyle (\mu)}}-\gamma_{k}\right)+\delta_{k+1}},\qquad\gamma_{0}^{{\scriptscriptstyle (\mu)}}=\frac{1}{\mu},
( \Delta w )_{\mathrm{m a g n}}= \frac{e^{2}{\cal B}^{2}}{1 2 \pi}\ \sum_{n = 1}^{\infty}\exp \left\{- n \frac{\pi m^{2}}{e{\cal E}}\right\}+ O \left( \frac{{\cal B}^{4}}{{\cal E}^{4}}\right) .
(X_1\otimes_{h A_2}...\otimes_{hA_{n-1}}X_{n-1})^{\natural_{A_2,...,A_{n-1}normal}}\otimes_{eh A_n'}X_n^\natural =X_1^\natural\otimes_{eh A_2'}...\otimes_{eh A_n'}X_n^\natural &\subset(X_1\otimes_{h A_2}...\otimes_{hA_n}X_n)^{\natural_{A_2,...,A_n normal}},
I_1=\int_0^{+\infty}dt_1\,dt_2\,\exp\Bigl[-\frac{t_1^2}{2\alpha}-\frac{t_2^2}{2\alpha}-\frac{t_1t_2}{2\alpha}\Bigr]\sin(t_1\frac{\alpha_1}{\alpha})\sin(t_2\frac{\alpha_1}{\alpha}).
I_{n,k,d}(z_1,\dots,z_n;\epsilon) =\frac{(4\pi)^{\hat{d}/2}\Gamma(\hat{d}/2)}{(4\pi)^{d/2}\Gamma(d/2)}\int\frac{d^{\hat{d}}p}{(2\pi)^{\hat{d}}}\left(p^2\right)^r\prod_{i=1}^n\frac{1}{\Delta_i},
\psi(x,S',Q') - \psi(x,S,Q) = \psi(0,S',Q') - \psi(0,S,Q) + \int_0^x \frac{\partial \psi(y,S',Q')}{\partial y}- \frac{\partial \psi(y,S,Q)}{\partial y}dy & \le \psi(0,S',Q') - \psi(0,S,Q) + \int_0^{x'}\frac{\partial \psi(y,S',Q')}{\partial y}- \frac{\partial \psi(y,S,Q)}{\partial y}dy & = \psi(x',S',Q') - \psi(x',S,Q) \: .
\omega_{j}= t r ( \gamma^{- 1}d \gamma ) e^{2 \pi \sqrt{- 1}\frac{\sqrt{3}}{3}a q ( j ) \mathbf{\sigma}_{3}}( \gamma^{- 1}d \gamma ) e^{- 2 \pi \sqrt{- 1}\frac{\sqrt{3}}{3}a q ( j ) \mathbf{\sigma}_{3}},
E_{L_{1}}= (\lambda_{1, k}^{j - 1})_{j \in [r], k \in [0, s - 1]}, F_{L_1}= (c^{1}_{1, b(1, k)})_{k \in [0, s - 1]},E_{V}= (\lambda_{i, b_{i}}^{j - 1})_{j \in [r], i \in [2, n]}, F_{V}= (\mu^{(b)}_{i, 1})_{i \in [2, n]}.
T \Vert f'_h - f' \Vert_1 = \int_0^T |f'(t+h) - f'(t)| \mathrm{d}t = \int_0^T |g_1'(t) - g_2'(t)| \mathrm{d}t &\leq \int_0^T g_1'(t) \mathrm{d}t + \int_0^T g_2'(t) \mathrm{d}t &= g_1(T) - g_1(0) + g_2(T) - g_2(0) &= V_0^T(g) - V_0^0(g) + V_0^T(g) - g(T) - V_0^0(g) + g(0) &= 2 V_0^T(g) - g(T) + g(0) &= 2 V_0^T(g),
\overset{4}{R}(X,Y,AZ,AW) = \overset{4}{R}(AX,AY,Z,W), \overset{4}{R}(X,AY,AZ,W) = \overset{4}{R}(AX,Y,Z,AW), \overset{4}{R}(AX,AY,AZ,AW) = \overset{4}{R}(X,Y,Z,W).
\left[2\atop 1\right]_q+\left[2\atop 1\right]_q\left[2\atop 1\right]_q+\left[2\atop 1\right]_q+\left[3\atop 1\right]_q+\left[3\atop 1\right]_q\left[3\atop 1\right]_q = \left[2\atop 1\right]_q+\left[2\atop 1\right]_q\left[2\atop 1\right]_q+\left[3\atop 2\right]_q\left[2\atop 1\right]_q+\left[3\atop 1\right]_q+\left[5\atop 1\right]_q.
\rho_{1M}^{(N)}(\mathbf{v}_{1})=\frac{n_{o}}{\pi^{3/2}\left(2T_{o}/m\right)^{3/2}}\exp \left\{-\frac{m\left( \mathbf{v}_{1}-\mathbf{V}_{o}\right)^{2}}{2T_{o}}\right\},
{J}_1(\tau) := f \cos(\tau \mathcal{A}^{1/2}) f^{-1}- f_0 \cos(\tau (\mathcal{A}^0)^{1/2}) f_0^{-1},{J}_2(\tau) := f{\mathcal{A}}^{-1/2}\sin(\tau{\mathcal{A}}^{1/2}) f^* - f_0 ({\mathcal{A}}^0)^{-1/2}\sin(\tau ({\mathcal{A}}^0)^{1/2}) f_0.
\sum^{d}_{i=2}E_0A^*E_iE^*_0(\theta_i-\theta_1) = E_0E^*_0(\vartheta_1-\vartheta_d), \sum^{d}_{i=2}E^*_0AE^*_iE_0(\theta^*_1-\theta^*_i) = E^*_0E_0(\vartheta_1-\vartheta_d).
\mathfrak{P}_{\Lambda}\ni\sigma\longmapsto\Psi_{\#}\left( \sigma\right):=\prod_{\mathsf{s}\in\sigma}\left( \frac{1-p}{p}\right)^{\left\vert\mathsf{s}\right\vert}q^{\left\Vert \mathsf{s}\right\Vert_{\#}}\in\mathbb{R}^{+}\;,\quad\#=
\int d^{2}x \left\{2 \chi_{\ell}^{\dagger}\left( \partial_{z}\partial_{\bar{z}}+ \partial_{\bar{z}}\partial_{z}\right) \phi_{\ell}- 2 \phi_{\ell}^{\dagger}\left( \partial_{z}\partial_{\bar{z}}+ \partial_{\bar{z}}\partial_{z}\right) \chi_{\ell}\right\}= - 4 \sin \delta_{\ell}\, .
\begin{cases}g_i(x^*) \le 0, h_j(x^*) = 0, \frac{d{L(x, \lambda, \nu)}}{dx}|_{x = x^*, \lambda = \lambda^*, \nu = \nu^*}= 0, \lambda_i^*g_i(x^*) = 0, \lambda_i^* \ge 0.\end{cases}
\frac{1}{N}\sum_{n=0}^{N-1}T^{[\gamma n+\ell]}f(x) =\frac{1}{N}\sum_{n=0}^{N-1}\exp\Big(2\pi i \frac{m[n\gamma +\ell]}{\gamma}\Big)f(x) &=\exp\Big(2\pi i \frac{m\ell}{\gamma}\Big)f(x)\cdot\frac{1}{N}\sum_{n=0}^{N-1}\exp\Big(-2\pi i\frac{m\{n\gamma +\ell\}}{\gamma}\Big).
\tfrac d{dt}W_t = \frac{p^{P_t}\delta e^{\delta t}\eta(t)-p^{P_t}e^{\delta t}\eta'(t)}{\eta(t)^2}= W_t\cdot\frac{\delta\eta(t)-\vartheta p^{P_t}e^{\delta t}}{\eta(t)}= W_t(\delta-\vartheta W_t) = w(W_t).
+\left[ \frac{A_{1}B_{1}}{B_{2}}-\left( \frac{B_{1}}{B_{2}}\right)^{\prime}\right] ff_{c_{2}}-\frac{B_{1}}{B_{2}}f^{\prime}f_{c_{2}}-\frac{B_{1}}{B_{2}}ff_{c_{2}}^{\prime}=B_{1}.
R_2 =\sum_{\beta \in F^{\times}}\sum_{d=1}^{6}\sum_{X \in M_{m_{1},m_{2}}^{d,\beta}}\Theta_{\theta}(h)\overline{\psi_A({m_1}^{-1}X)}=\sum_{\beta \in F^{\times}}\sum_{d=1}^{6}(-1)^{6-1}(1-q) \cdots (1-q^{d-1})(\# M_{m_{1},m_{2}}^{d,\beta})\overline{\psi_0(\beta)}
\epsilon([m_t]) = \epsilon([g_t^{-1}m g_t]) &= \epsilon([g_t^{-1}g_t])- \epsilon([g_t^{-1}])\epsilon([g_t]) [\textrm{Skein relation}] &= \epsilon([e]) - \epsilon([g_t^{-1}])\epsilon([g_t]) = 0,
|I^{\star}| = \mathcal{O}\left( a^{10-2h}\; d^{-6}\; \underset{i}{\max}\left\Vert \overset{1}{\mathbb{P}}\left(\tilde{E}_{i}^{T}\right) \right\Vert^{2}+ a^{8 - 2h}\, d^{-8}\underset{i}{\max}\left\Vert \overset{3}{\mathbb{P}}\left(\tilde{E}_{i}^{T}\right) \right\Vert^{2}\right).
([p,v]+_1[pk_1,u])+_2([pk_2,v']+_1[pk_2k'_1,u'])=[p,(v+_1k_1u)+_2(k_2v'+_1k_2k'_1u')] = [p,(v+_2k_2v')+_1(k_1u+_2k_2k'_1u')]=([p,v]+_2[pk_2,v'])+_1([pk_1,u]+_2[pk_2k'_1,u']).
\Tilde{\Phi}_1(\alpha_0,\eta)=\dfrac{1}{L_m^2}\bigg(\bigg(\tilde{N_1}+\frac{N_{1M}\tilde{L_1}}{L_{1m}}\bigg)\bigg[\frac{\eta^{3-\nu}}{3-\nu}-\alpha_0^2\dfrac{\eta^{1-\nu}}{1-\nu}+\frac{2\alpha_0^2}{(3-\nu)(1-\nu)}\bigg]+\tilde{L_1}\frac{\eta^{1-\nu}}{1-\nu}\bigg),
L^{2}({\cal A}_{\gamma}\times{\cal F}_{\gamma}) = \bigoplus_{\rho_{e}\in \mathrm{R e p}( G ) , \rho_{v}\in S}\quad \bigotimes_{v \in V}\bigl ( \bigotimes_{t ( e ) = v}\rho_{e}^{*}\otimes \bigotimes_{s ( e ) = v}\rho_{e}\otimes \rho_{v}\bigr )
{\cal V}_{N;g}\, =\,{\cal C}_{g}^{open}\! < \Omega| \int [{\mbox d}m]^{g}_{N}\exp \left\{\frac{1}{2}\sum_{i=1}^{N}\sqrt{2 \alpha'}p^{(i)}\cdot\left.\! \left[ \sqrt{2 \alpha'}p^{(i)}+{\alpha}_{1}^{(i)}\partial_{z}\right] \log V'_{i}(z)\right|_{z=0}\right\}
\Phi = U_{1}^{- 1}({\bf r}) \left( \begin{matrix}{{t_{4}-{\frac{1}{2 y_{R}}}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{t_{2}-{\frac{1}{2 y_{L}}}+{\frac{1}{2 y_{R}}}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{t_{2}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{t_{1}+{\frac{1}{2 y_{L}}}}}\\ \end{matrix}\right) U_{1}({\bf r})
\bigg(\int_D \lvert u(x)\rvert^q\delta_{\partial D}^{(q/p)(n-s p+\beta)-n}(x)\,dx\bigg)^{p/q}\le C\int_D \int_{B(x,\tau \delta_{\partial D}(x))}\frac{\lvert u(x)-u(y)\rvert^p}{\lvert x-y\rvert^{n+s p}}\,dy\,\delta_{\partial D}^\beta(x)\,dx\,,